diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_postprocess.srt
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+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_postprocess.srt
@@ -0,0 +1,3708 @@
+1
+00:00:04,910 --> 00:00:08,190
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:08,190 --> 00:00:12,270
+والصلاة والسلام على سيد المرسلين سيدنا محمد على
+
+3
+00:00:12,270 --> 00:00:18,470
+آله وصحبه أجمعين هذه هي المحاضرة رقم 22 في مساق
+
+4
+00:00:18,470 --> 00:00:23,930
+تحليل حقيقة نيل لطلاب وطالبات الجامعة الإسلاميةقسم
+
+5
+00:00:23,930 --> 00:00:29,530
+الرياضيات في كلية العلوم عنوان المحاضرة اليوم
+
+6
+00:00:29,530 --> 00:00:32,850
+هنكمل chapter تمانية هيكون في عندي اللي هو
+
+7
+00:00:32,850 --> 00:00:37,130
+applications على اللي هي تمانية اللي هي واحد و
+
+8
+00:00:37,130 --> 00:00:41,630
+تمانية اتنين المحاضرة اليوم اللي هي هنحكي عن the
+
+9
+00:00:41,630 --> 00:00:46,730
+exponential and the logarithmic functions هنحكي عن
+
+10
+00:00:46,730 --> 00:00:51,830
+اللي هو دالة ال E to the X ودالة ال Lin أو دالة ال
+
+11
+00:00:51,830 --> 00:00:57,660
+logالان هنثبت اللي هو من خلال .. في البداية هنحكي
+
+12
+00:00:57,660 --> 00:01:03,280
+عن اللي هو ال exponential function أو هنثبت اللي
+
+13
+00:01:03,280 --> 00:01:05,740
+هو وجود ال exponential function
+
+14
+00:01:09,490 --> 00:01:13,630
+احنا استخدمناها قبل هيك مجرد أمثلة بعيدا عن اللي
+
+15
+00:01:13,630 --> 00:01:18,730
+هو انه اللي هي مفترضنا انه معلومات موجودة مسبقا
+
+16
+00:01:18,730 --> 00:01:22,170
+ولا تناقض اللي هو انه نثبتها اليوم لإنه اثباتها
+
+17
+00:01:22,170 --> 00:01:25,410
+اليوم لا يعتمد على اللي حكيناه سابقا بما يخص
+
+18
+00:01:25,410 --> 00:01:29,350
+بأمثلة اللي ذكرت فيها ال exponential الان ال
+
+19
+00:01:29,350 --> 00:01:34,310
+exponential function بدنا نثبت وجودها ايش اللي
+
+20
+00:01:34,310 --> 00:01:38,820
+بنقوله نشوف عبر ال theorem 8 3 1 theoremبقول there
+
+21
+00:01:38,820 --> 00:01:44,060
+exists a function U دلّة E من R ل R إذا في عندنا
+
+22
+00:01:44,060 --> 00:01:49,680
+دلّة اسمها E من R ل R such that الـ E prime of X
+
+23
+00:01:49,680 --> 00:01:55,280
+لها بساول EX لكل X element in Rإتنين E of Zero
+
+24
+00:01:55,280 --> 00:01:58,820
+بساوي إيش واحد يعني الآن النظرية دي بتقولي إنه
+
+25
+00:01:58,820 --> 00:02:02,920
+يوجد عندنا دالة domainها كل ال R و rangeها بروح
+
+26
+00:02:02,920 --> 00:02:10,220
+بصوب في ال R هذه الدالة تحقق شرطين اللي هي E prime
+
+27
+00:02:10,220 --> 00:02:14,500
+of X بساوي E of X و E of Zero بساوي إيش واحدالان
+
+28
+00:02:14,500 --> 00:02:19,920
+نجي اللي هو بدنا نثبت وجود هذه الدالة ايه التي
+
+29
+00:02:19,920 --> 00:02:25,380
+تحقق اللي هو الخواص اللي عندنا المذكورة فيه اللي
+
+30
+00:02:25,380 --> 00:02:29,960
+هي نظريا الان خلينا عشان نروح باتجاه اثبات وجود
+
+31
+00:02:29,960 --> 00:02:34,080
+هذه الدالة خلينا ناخد we inductively define a
+
+32
+00:02:34,080 --> 00:02:37,900
+sequence of continuous functions as follows بدي
+
+33
+00:02:37,900 --> 00:02:43,590
+الآن اعرف اللي هو sequence of functionsالأولى
+
+34
+00:02:43,590 --> 00:02:48,790
+اسمها E1 of X بتساوي واحد زائد X طبعا هذه الدالة
+
+35
+00:02:48,790 --> 00:02:52,790
+موجودة E1 of X بتساوي واحد زائد X هي دالة خطية
+
+36
+00:02:52,790 --> 00:02:58,590
+الآن بدي أعرف ال E2 ال E2 of X بساوي ال
+
+37
+00:02:58,590 --> 00:03:04,170
+integration واحد زائد ال integration من سفر لعند X
+
+38
+00:03:04,170 --> 00:03:13,140
+اللي هو E1 of X D أو E1 of T DTإذاً الـ E2 بدي
+
+39
+00:03:13,140 --> 00:03:17,960
+أجيبها من مين؟ من الـ E1 طبعاً بتيجي الـ E والـ E2
+
+40
+00:03:17,960 --> 00:03:23,500
+اللي هي عبارة عن واحد زائد هذه واحد زائد T اللي هو
+
+41
+00:03:23,500 --> 00:03:27,730
+تفاضلها زي ما انتوا عارفين تكملها اللي هي Tزائد
+
+42
+00:03:27,730 --> 00:03:32,910
+اللي هي T تربيع على اتنين من صفر لعند X ويسوا واحد
+
+43
+00:03:32,910 --> 00:03:38,490
+زائد X زائد X على X تربيع على اتنين هذه الدالة من
+
+44
+00:03:38,490 --> 00:03:45,020
+هي E2 of X E3 of Xبتعرفها بنفس الأسلوب بالساوي 1
+
+45
+00:03:45,020 --> 00:03:49,840
+زائد الـ integration من 0 ل X لدالة اللي وجدتها
+
+46
+00:03:49,840 --> 00:03:56,140
+قبلها اللي هي E2 of T DT بضل أسائي ساير في التعريف
+
+47
+00:03:56,140 --> 00:04:03,760
+بقول in general بتعرف ال E N زائد 1 of Xبساوة الـ
+
+48
+00:04:03,760 --> 00:04:11,420
+1 زي ال integration من 0 ل X EN of T DT إذن الآن
+
+49
+00:04:11,420 --> 00:04:16,020
+عرفت اللي هي ال E1 of X بساوة 1 زي X ومنها عرفت
+
+50
+00:04:16,020 --> 00:04:21,840
+اللي هي E2 وE2 عرفت منها E3 وE3 عرفت منها E4 وهكذا
+
+51
+00:04:21,840 --> 00:04:26,120
+ال EN زي ال 1 of X هتساوة 1 زي ال integration من 0
+
+52
+00:04:26,120 --> 00:04:31,440
+ل X لل EN اللي جاب الها ديof D D T لكل N element
+
+53
+00:04:31,440 --> 00:04:36,720
+in N و لكل X element in M in R نيجي الآن نطلع على
+
+54
+00:04:36,720 --> 00:04:40,920
+الملاحظات اللي بدنا نحكيها عشان نستخدمها الان 1
+
+55
+00:04:40,920 --> 00:04:46,840
+زائد X دالة متصلة مش متصلة أصلا بس هي أصلا قابلة
+
+56
+00:04:46,840 --> 00:04:51,900
+للتفاضل أيضا اللي هي differentiable الان بناء عليه
+
+57
+00:04:51,900 --> 00:05:00,140
+مدام ال E1 is continuous هتكون اللي هي E2E2 هي
+
+58
+00:05:00,140 --> 00:05:03,940
+الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 الـ E1
+
+59
+00:05:03,940 --> 00:05:04,180
+هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 الـ
+
+60
+00:05:04,180 --> 00:05:07,840
+E1 هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1
+
+61
+00:05:07,840 --> 00:05:11,080
+الـ E1 هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ
+
+62
+00:05:11,080 --> 00:05:12,820
+E1 الـ E1 هي الانتجراش للـ E1 الـ E1 هي الانتجراش
+
+63
+00:05:12,820 --> 00:05:14,980
+للـ E1 الـ E1 هي الانتجراش للـ E1 الـ E1 هي
+
+64
+00:05:14,980 --> 00:05:18,980
+الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 الـ E1
+
+65
+00:05:18,980 --> 00:05:22,820
+هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 ال
+
+66
+00:05:25,750 --> 00:05:27,270
+الآن F.E.L
+
+67
+00:05:31,050 --> 00:05:34,950
+in on R then it is integrable over any bounded
+
+68
+00:05:34,950 --> 00:05:39,590
+interval زي ما قلنا مدام هذا continuous اللي هي E
+
+69
+00:05:39,590 --> 00:05:45,210
+اللي هي 2 continuous ومن هنا هتطلع E 3 continuous
+
+70
+00:05:45,210 --> 00:05:49,530
+وE 4 continuous إذا صارت هذي دايما continuous وده
+
+71
+00:05:49,530 --> 00:05:53,170
+continuous وده integration exist و by fundamental
+
+72
+00:05:53,170 --> 00:05:57,250
+theorem هذا ال integration كله على بعض can be
+
+73
+00:05:57,250 --> 00:06:01,160
+differentiatedوهتكون الهو ايش is differentiable
+
+74
+00:06:01,160 --> 00:06:05,520
+ومنه so EN زائد واحد is well defined by the above
+
+75
+00:06:05,520 --> 00:06:09,300
+formula moreover زي ما قلت it is from fundamental
+
+76
+00:06:09,300 --> 00:06:12,400
+theorem of calculus ال second form اللي هي سبعة
+
+77
+00:06:12,400 --> 00:06:15,920
+تلاتة خمسة هيكون عند ال EN زائد واحد is
+
+78
+00:06:15,920 --> 00:06:19,280
+differentiable ومش هيك وتفاضل هذه زي ما احنا
+
+79
+00:06:19,280 --> 00:06:24,540
+عارفين بساوي بنشيل ال integration طبعا التفاضل بلغ
+
+80
+00:06:24,540 --> 00:06:29,220
+ال integrationبصير en of x فبصير عندي ال en زائد
+
+81
+00:06:29,220 --> 00:06:34,240
+واحد prime of x موجودة ويساوي en x for all n
+
+82
+00:06:34,240 --> 00:06:37,720
+element in N إذن الآن عملنا sequence ال sequence
+
+83
+00:06:37,720 --> 00:06:40,580
+هذه طلعت sequence of differentiable functions و ال
+
+84
+00:06:40,580 --> 00:06:43,860
+derivative لل en زائد واحد prime هي بترجع لمين
+
+85
+00:06:43,860 --> 00:06:50,320
+بتيجي اللي هي ال en of xطيب الان هذا كله و ده
+
+86
+00:06:50,320 --> 00:06:53,900
+خليني اسميها تلاتة و خليني نحضر حالنا نصل للي بدنا
+
+87
+00:06:53,900 --> 00:06:59,500
+ياجولكم وين هنصل في الآخر هوصلكم انه اللي هو ال
+
+88
+00:06:59,500 --> 00:07:05,400
+limit لهذه ال sequence او لهذه ال sequence هي ال E
+
+89
+00:07:05,400 --> 00:07:12,300
+of X اللي انا بثبت وجودهاوستكون اللي هي بتحقق
+
+90
+00:07:12,300 --> 00:07:16,440
+الشروط اللي حكيناها خلّينا نشوف ما لاستعجلش نشوف
+
+91
+00:07:16,440 --> 00:07:19,900
+إيش اللي بدنا نصلله إذا اللي عملناه كوّننا
+
+92
+00:07:19,900 --> 00:07:23,380
+sequence sequence زي ما قلنا اللي هي ال sequence
+
+93
+00:07:23,380 --> 00:07:28,160
+بدأت عشان يبقى لكم الذاكرين E1 of X بساوي واحد
+
+94
+00:07:28,160 --> 00:07:34,760
+زائد X E and زائد واحد of X بساوي الانتجرأت واحد
+
+95
+00:07:35,450 --> 00:07:39,790
+بسوء واحدة زي الـ integration من صفر لعدد X E N of
+
+96
+00:07:39,790 --> 00:07:47,450
+T DT خلّين هذولة أمامنا طبعاً N element in N عندي
+
+97
+00:07:47,450 --> 00:07:54,630
+و X أي element وين in Rطيب إذا الان سار عندي اللي
+
+98
+00:07:54,630 --> 00:07:58,430
+هو ال .. ده اللي هذي is differential لان بقوللي
+
+99
+00:07:58,430 --> 00:08:03,230
+انه ابن الدعي ويمكن لو حد شاف قبل بشوية ما وصلنا ل
+
+100
+00:08:03,230 --> 00:08:08,450
+E2 E2 كانت عبارة عن واحد زي دكستر بيه على اتنين
+
+101
+00:08:08,450 --> 00:08:13,070
+اللي هو اللي هي اتنين factorialلو كملنا هيكون E N
+
+102
+00:08:13,070 --> 00:08:16,910
+of X بتساوي 1 زياد X على 1 factorial X تربيه على 2
+
+103
+00:08:16,910 --> 00:08:21,170
+factorial زياد X وصف N على 100 على N factorial لكل
+
+104
+00:08:21,170 --> 00:08:25,750
+X element in R هذه الـ N صحيحة اللي هي لكل N
+
+105
+00:08:25,750 --> 00:08:31,470
+element in R اللي هو طبعا اثباتها سهل نثبتها by
+
+106
+00:08:31,470 --> 00:08:36,010
+induction خلينا نشوف كيفنا نثبت أربعة by induction
+
+107
+00:08:58,380 --> 00:09:01,020
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+108
+00:09:01,020 --> 00:09:04,640
+أربعة
+
+109
+00:09:04,640 --> 00:09:06,540
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+110
+00:09:06,540 --> 00:09:06,720
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+111
+00:09:06,720 --> 00:09:06,900
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+112
+00:09:06,900 --> 00:09:08,610
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أfour is
+
+113
+00:09:08,610 --> 00:09:16,450
+a true four and بتساوي كتابة يعني صار عند الالالـ
+
+114
+00:09:16,450 --> 00:09:23,550
+E K of X بتساوي 1 زائد X على 1 factorial زائد X
+
+115
+00:09:23,550 --> 00:09:29,170
+تربيع على 2 factorial زائد X أُس K على K factorial
+
+116
+00:09:29,170 --> 00:09:33,890
+هذا لما نفرض إن اللي هي .. اللي هي الأربعة is true
+
+117
+00:09:33,890 --> 00:09:39,150
+for N بتساوي K بدنا نثبت إن E K زائد 1 هتطلع اللي
+
+118
+00:09:39,150 --> 00:09:43,870
+هي هذهزائد x أس k على k زائد واحد فيكتوريا اللي
+
+119
+00:09:43,870 --> 00:09:48,790
+على ال net بتطلع أربعة صحيحة for k زائد واحد ده
+
+120
+00:09:48,790 --> 00:09:56,070
+نحسب خلّينا نحسب الآن E k زائد واحد of x حسب اللي
+
+121
+00:09:56,070 --> 00:10:01,370
+احنا مفترضينه أو معرفين ال sequence على أساسه E k
+
+122
+00:10:01,370 --> 00:10:04,830
+زائد واحد of x أيش بتساوي واحد زائد ال integration
+
+123
+00:10:04,830 --> 00:10:14,410
+من صفر ل X E k of T DTمظبوط؟ طيب الان بدي اعوض عن
+
+124
+00:10:14,410 --> 00:10:20,030
+E K of T احنا مفترضينها صحيحة ل K اذا E K of X
+
+125
+00:10:20,030 --> 00:10:24,230
+هيها اذا باجي بعوض بيصير Y ساوي واحد زائد ال
+
+126
+00:10:24,230 --> 00:10:29,250
+integration من صفر ل X E K of T اللي هي واحد زائد
+
+127
+00:10:29,250 --> 00:10:33,170
+T زائد T تربيع طبعا واحد فيكتوريال اللي هي واحد
+
+128
+00:10:33,170 --> 00:10:40,080
+على اتنين فيكتوريال زائد لما أصل X أو T أس Kعلى K
+
+129
+00:10:40,080 --> 00:10:46,520
+فكتوريال الكل إشماله DT لأن اللي هي أكيد وضحت
+
+130
+00:10:46,520 --> 00:10:49,660
+الصورة بدل الفاضل نطلع قيمة التفاصيل الكامل و نطلع
+
+131
+00:10:49,660 --> 00:10:54,360
+قيمة التكامل و يساوي واحد هيوزايد ما هو جاعد زاد
+
+132
+00:10:54,360 --> 00:11:00,520
+هذا ال integration اللي هو عبارة عن Tزائد T تربيع
+
+133
+00:11:00,520 --> 00:11:06,140
+على 2 في 1 يعني 2 factorial زائد T تكعيب على 3 في
+
+134
+00:11:06,140 --> 00:11:10,920
+2 factorial يعني 3 factorial زائد لما أصل لآخر 1 T
+
+135
+00:11:10,920 --> 00:11:15,640
+أُس K زائد 1 على K زائد 1 في K factorial هي K زائد
+
+136
+00:11:15,640 --> 00:11:22,230
+1 لكل factorial هذا الكلام من 0 لمين؟لعند X واضحة
+
+137
+00:11:22,230 --> 00:11:26,550
+الصورة ويساوي عبارة عن لما اعوض من 0 ل X بيصير 1
+
+138
+00:11:26,550 --> 00:11:32,090
+زائد X زائد X تربيع على 2 factorial لما أصل لأخر
+
+139
+00:11:32,090 --> 00:11:38,210
+واحد X K زائد 1 على K زائد 1فاكتوريال إذا فعلاً
+
+140
+00:11:38,210 --> 00:11:42,110
+طلعت عندي a k زائد واحد بساوي هذا المقدار يعني
+
+141
+00:11:42,110 --> 00:11:46,450
+بمعنى آخر أربعة طلعت ال true for n بتساوي k زائد
+
+142
+00:11:46,450 --> 00:11:51,830
+واحد إذا من كل هذا ال induction بنكون أثبتنا أن e
+
+143
+00:11:51,830 --> 00:11:56,430
+n of x بساوي هذا الكلام لكل x element in R لكل n
+
+144
+00:11:56,430 --> 00:12:00,630
+element in Nإذاً هذه صورة اللي هي الـEN of X صورة
+
+145
+00:12:00,630 --> 00:12:05,150
+غير الصورة اللي عرفناها فوق استنتجناها منها الآن
+
+146
+00:12:05,150 --> 00:12:11,770
+خلّينا نروح باتجاه إثبات إنه ال limit للـEN هذه أو
+
+147
+00:12:11,770 --> 00:12:17,910
+الـEK أو الـEN أو اللي هي is uniformly convergent
+
+148
+00:12:18,680 --> 00:12:23,900
+to some function هذه الـ some function هي اللي
+
+149
+00:12:23,900 --> 00:12:29,120
+بتدعي انها هتكون ال exponential طيب اللي بنسميها
+
+150
+00:12:29,120 --> 00:12:32,840
+exponential بعد شوية اللي هي بتحقق شرطين اللي
+
+151
+00:12:32,840 --> 00:12:35,400
+حكينا عنهم لسه احنا بنعرف ال exponential احنا
+
+152
+00:12:35,400 --> 00:12:38,340
+بنعرف الدالة هذه لكن لأن معلوماتنا عارفين من اول
+
+153
+00:12:38,340 --> 00:12:43,000
+ال exponentialالان احنا الدالة هذه اللى هنثبت
+
+154
+00:12:43,000 --> 00:12:46,540
+وجودها اليوم هي اللى بعد شوية بعد ما نثبت ال
+
+155
+00:12:46,540 --> 00:12:50,880
+uniqueness لها بنسميها ال exponential زي ما هنشوف
+
+156
+00:12:50,880 --> 00:12:56,620
+بعد شوية الان اذا ال sequence اللى عندى هذه
+
+157
+00:12:56,620 --> 00:13:05,300
+هيوضعها الان قلنا E Nof X هذه المعادلة التانية هي
+
+158
+00:13:05,300 --> 00:13:11,560
+واحد زائد X زائد X تربيع على اتنين factorial زائد
+
+159
+00:13:11,560 --> 00:13:15,600
+X أُس N على N factorial وهذا الشكل طبعا أنتوا مش
+
+160
+00:13:15,600 --> 00:13:20,240
+غريب عليكم بتعرفوه الآن خليني أخد الان let A أكبر
+
+161
+00:13:20,240 --> 00:13:24,120
+من سفر بيه given افترض ان A اني اللي هو real
+
+162
+00:13:24,120 --> 00:13:28,660
+number أكبر من مين من سفر باخده arbitrarily لكن
+
+163
+00:13:28,660 --> 00:13:33,470
+خليني نحكي عن A محددةالان if absolute value of X
+
+164
+00:13:33,470 --> 00:13:37,910
+أصغر أو تساوي A يعني بتحكي الحديث هان على ال Xات
+
+165
+00:13:37,910 --> 00:13:43,590
+اللي في ال R اللي من عند ناقص A لعند مين ال A يعني
+
+166
+00:13:43,590 --> 00:13:48,610
+ال absolute value لهن هذولة أصغر أو تساوي ال A
+
+167
+00:13:48,610 --> 00:13:52,390
+يعني في الفترة المغلقة اللي أمامي اللي بين ناقص ال
+
+168
+00:13:52,390 --> 00:13:58,870
+A و ال A بتتحدث شوف الآن احسبلي Em of X ناقص En of
+
+169
+00:13:58,870 --> 00:14:03,750
+Xو يساوي و بدنا نفترض لكم ان الان ايه شمالها اكبر
+
+170
+00:14:03,750 --> 00:14:07,910
+من الان و الان اكبر من اتنين ايه اتنين ايه مكتوب
+
+171
+00:14:07,910 --> 00:14:10,850
+اتنين ايه لغرض الحسابات اللي جيت منشوفها بنفع
+
+172
+00:14:10,850 --> 00:14:13,270
+تلاتة ايه، بنفع اربعة ايه، بنفع خمسة ايه، بنفع ستة
+
+173
+00:14:13,270 --> 00:14:16,550
+ايه، بنفع كله بنفع هذا الكلام على اساس انك ماينفعش
+
+174
+00:14:16,550 --> 00:14:19,550
+اقول نص ايه او تلت ايه او ربع ايه لأنها مش هتقد
+
+175
+00:14:19,550 --> 00:14:23,910
+الغرض اللي بدي انالأن مين اخترت أنا اخترت اللي هو
+
+176
+00:14:23,910 --> 00:14:29,650
+الأمات والأنات اللي أكبر منين من اتنين في الـA
+
+177
+00:14:29,650 --> 00:14:34,390
+اللي هي قيبة مين الـA اللي هي نص الفترة اللي عندي
+
+178
+00:14:34,390 --> 00:14:39,210
+أو طول نص الفترة طيب، احسبلي ايه M of X ناقص Y of
+
+179
+00:14:39,210 --> 00:14:44,360
+X؟E M of X اللي هي عبارة عن واحد زائد X زائد X
+
+180
+00:14:44,360 --> 00:14:49,360
+تربيع و هجابل في الطريق من ال X M لأنه M أكبر X
+
+181
+00:14:49,360 --> 00:14:52,720
+تربيع على M فكتوريال زائد X M زائد واحد على M زائد
+
+182
+00:14:52,720 --> 00:14:57,280
+واحد فكتوريال لما أصل الأخر واحدة من X أُس M على M
+
+183
+00:14:57,280 --> 00:15:03,400
+فكتوريال يعني الآن هذي هيكون زي هيك لما أصل طبعا X
+
+184
+00:15:03,400 --> 00:15:10,500
+M زائد واحد على Mزائد واحد factorial زائد لمّا أصل
+
+185
+00:15:10,500 --> 00:15:15,760
+لعند X أُس M على M factorial هذه مين هي هذه عبارة
+
+186
+00:15:15,760 --> 00:15:24,220
+عن ال EM يعني ال EM هتساوي اللي هي E of XE N of X
+
+187
+00:15:24,220 --> 00:15:30,020
+زائد المتبقي هذا الان حاصل طرح الاتنتين هيكون
+
+188
+00:15:30,020 --> 00:15:33,480
+عبارة عن لإن المفترض ال M أكبر من L زي ما قلنا
+
+189
+00:15:33,480 --> 00:15:37,700
+حاصل طرح اللي هيكون اللي هو المتبقي هذا X N زائد
+
+190
+00:15:37,700 --> 00:15:40,840
+واحد على N زائد واحد فيكتوريال لما أصل ل X M على M
+
+191
+00:15:40,840 --> 00:15:46,240
+إيش فيكتوريال ماشي الحال إذا الأن أوصلنا لحاصل طرح
+
+192
+00:15:46,240 --> 00:15:51,340
+دولة بساوي هذا المقدر نيجي نكمل الأن عندي
+
+193
+00:15:53,560 --> 00:15:58,460
+عند ال absolute value لل X أصغر من 100 من A
+
+194
+00:16:10,590 --> 00:16:17,130
+زائد absolute value زائد xn زائد اتنين على n زائد
+
+195
+00:16:17,130 --> 00:16:22,710
+اتنين factorial زائد لما أصل لأخر واحد x أسم م على
+
+196
+00:16:22,710 --> 00:16:28,150
+m factorial ماشي الحال طيب الآن ان ان كل x من
+
+197
+00:16:28,150 --> 00:16:31,270
+هدولة ال absolute value هي أصغر يساوي منين ايه إذا
+
+198
+00:16:31,270 --> 00:16:36,650
+صار هذا المقدار أصغر أو يساوي هذا المقدار اللي هنا
+
+199
+00:16:36,650 --> 00:16:45,550
+أصغر أو يساوي Mزائد واحد على N آسف A أُس N زائد
+
+200
+00:16:45,550 --> 00:16:49,970
+واحد لأن ال absolute value X أصغر تساوي A إذا X N
+
+201
+00:16:49,970 --> 00:16:54,510
+زائد واحد أصغر أو ساوي ال A أُس N زائد واحد على N
+
+202
+00:16:54,510 --> 00:17:00,870
+زائد واحد factorial زائد التاني اللي هو A N زائد
+
+203
+00:17:00,870 --> 00:17:07,130
+اتنينعلى n زائد 2 factorial زائد لما أصل لآخر واحد
+
+204
+00:17:07,130 --> 00:17:14,870
+اللي هو عبارة عن a أس m على m factorial ماشي الحال
+
+205
+00:17:14,870 --> 00:17:21,080
+طيبالان خليني اخد من هدولة ال a n زائد واحد على ال
+
+206
+00:17:21,080 --> 00:17:24,480
+n زائد واحد factorial عامل مشترك قبل ما اصلح هذه
+
+207
+00:17:24,480 --> 00:17:28,520
+الخطوة لسه اه خليني اخد عامل مشترك بيصير عندي a n
+
+208
+00:17:28,520 --> 00:17:33,180
+زائد واحد على n زائد واحد factorial فاش اللي بيضل
+
+209
+00:17:33,180 --> 00:17:39,420
+هنا فيه اللي بيضل هنا واحد زائد هنا بيضل a على n
+
+210
+00:17:39,420 --> 00:17:46,280
+زائد اتنين اكيد زائد اللي بعدها a تربيع على nزائد
+
+211
+00:17:46,280 --> 00:17:52,160
+اتنين فان زائد تلاتة زائد لما أصل لآخر واحد مين
+
+212
+00:17:52,160 --> 00:17:57,080
+آخر واحد بشيل منه ان زائد واحد بيصير a أس ام ناقص
+
+213
+00:17:57,080 --> 00:18:03,000
+ان ناقص واحد على اللي بيضل من ان زائد اتنين مضروب
+
+214
+00:18:03,000 --> 00:18:09,250
+لعند ان ام ناقص ان ناقص واحد هذول الأنماشي الحال
+
+215
+00:18:09,250 --> 00:18:13,470
+هذا أخد تمين يا جماعة ماعليش دخل الكلام مع بعضه بس
+
+216
+00:18:13,470 --> 00:18:18,130
+أكيد أنتوا مستوعبين إيش بقول طلعت ال a n زائد واحد
+
+217
+00:18:18,130 --> 00:18:21,090
+على n زائد واحد في كتوريا العام المشترك طلع عندي
+
+218
+00:18:21,090 --> 00:18:26,550
+هذا المقدار الان عندي أكيد ال n زائد اتنين أكبر من
+
+219
+00:18:26,550 --> 00:18:32,050
+مين من ال end فمقلوبه أصغرأه فبصير عندى اللى هو
+
+220
+00:18:32,050 --> 00:18:38,050
+عندى اللى هى هذا المقدار a n على l زائد 2 أصغر أو
+
+221
+00:18:38,050 --> 00:18:42,170
+شاوي a على n و اللى بعده a تربيع أصغر أو شاوي ا ا
+
+222
+00:18:42,170 --> 00:18:44,370
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+223
+00:18:44,370 --> 00:18:44,410
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+224
+00:18:44,410 --> 00:18:45,090
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+225
+00:18:45,090 --> 00:18:46,010
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+226
+00:18:46,010 --> 00:18:47,730
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+227
+00:18:47,730 --> 00:18:48,010
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+228
+00:18:51,320 --> 00:18:55,380
+أكبر من أنه وهذه أكبر من أنه فمقلوبهم بيصير هذا
+
+229
+00:18:55,380 --> 00:19:00,980
+المقدار أصغر اللي هو أو يساوي المقدار اللي بعده
+
+230
+00:19:00,980 --> 00:19:07,730
+بيصير عندى هذا المقدارأصغر أو يساوي هذا زي ما هو و
+
+231
+00:19:07,730 --> 00:19:13,670
+هذه واحد هذه a على n لأنها هتكون أكبر a تربيع على
+
+232
+00:19:13,670 --> 00:19:19,170
+n تربيع ان تربيع هتكبر لما أصل ل a m minus n نقص
+
+233
+00:19:19,170 --> 00:19:25,420
+واحد على m minus n نقص واحد لأن هنا دولة عددهمن
+
+234
+00:19:25,420 --> 00:19:30,200
+نقص واحد من الانز هيكون ان طبعا استبدالت كل واحدة
+
+235
+00:19:30,200 --> 00:19:34,400
+بان فهيطلع بالشكل هذا وبتظل ال inequality صحيحة
+
+236
+00:19:34,400 --> 00:19:38,540
+خلصنا من هذا طيب دعونا نشوف كيف أضبح لكم هذا دعونا
+
+237
+00:19:38,540 --> 00:19:48,770
+نخلص منه نجي الآن اللي هو نرجعانكمل الحسبة هذا
+
+238
+00:19:48,770 --> 00:19:57,670
+الان المقدار عندى اللى هو المقدار هذا هيه اتطلعوا
+
+239
+00:19:57,670 --> 00:20:06,710
+عندى ال a ال a على n ايش انا ماخد ال n ماخد ال n
+
+240
+00:20:06,710 --> 00:20:12,930
+اكبر من مين من اتنين a يعني ال a اصغر من n على مين
+
+241
+00:20:13,680 --> 00:20:21,900
+على اتنين مظبوط ال A أصغر من N على اتنين واضحة جسم
+
+242
+00:20:21,900 --> 00:20:25,660
+ال N على اتنين و N على اتنين صارت N على اتنين أكبر
+
+243
+00:20:25,660 --> 00:20:29,940
+من مين من ايه بدي استخدمها الآن هذه ال N بصير أصغر
+
+244
+00:20:29,940 --> 00:20:34,020
+أو يساوي هذا المقدار بصير أصغر أو يساوي هذا بحكي
+
+245
+00:20:34,020 --> 00:20:39,590
+عن هذا بصير أصغر أو يساوي اللي هو Aأُس ان زائد
+
+246
+00:20:39,590 --> 00:20:45,410
+واحد على ان زائد واحد الكل فكتوريال فيه اللي هو
+
+247
+00:20:45,410 --> 00:20:56,830
+واحد زائد ا على ان زائد ا تربية على ان تربية زائد
+
+248
+00:20:56,830 --> 00:21:06,530
+ا اللي هو ا تكعيب على ان تكعيبزائد مش بتصل لهذا
+
+249
+00:21:06,530 --> 00:21:11,670
+زائد و بتظل ماشيها يشمالها إلى ما لا نهاية يعني
+
+250
+00:21:11,670 --> 00:21:17,110
+بتضيف عليها a على n أس ام ناقص n a على n أس ام
+
+251
+00:21:17,110 --> 00:21:20,290
+ناقص n زائد واحد وهكذا تظلها إلى ما لا نهاية يعني
+
+252
+00:21:20,290 --> 00:21:23,850
+بتحولها لمين ل infinite series طيب احنا قلنا ال a
+
+253
+00:21:23,850 --> 00:21:28,670
+أصغر من n على 2يعني ال a n على n أصغر من مين؟ من
+
+254
+00:21:28,670 --> 00:21:32,930
+نص عرفته الأن إيش بدأ أسأل صار عندي هذا المقدار
+
+255
+00:21:32,930 --> 00:21:39,310
+أصغر أو يساوي a أس n زائد واحد على n زائد واحد لكل
+
+256
+00:21:39,310 --> 00:21:45,690
+factorial مضروب في مين؟ اللي هو واحد زائد نص زائد
+
+257
+00:21:45,690 --> 00:21:50,670
+نص تربيع أه لأن a على n أصغر من مين يا جماعة؟ من
+
+258
+00:21:50,670 --> 00:21:57,850
+نص زائدتربيع زاد نص تتعيد زاد الآخرين لأن هذه صارت
+
+259
+00:21:57,850 --> 00:22:02,470
+عبارة عن geometric series مجموحها هذه بتساوي مجموح
+
+260
+00:22:02,470 --> 00:22:07,050
+هذه عبارة عن واحد على واحد ناقصه اللي هو نص وهي
+
+261
+00:22:07,050 --> 00:22:12,730
+ساوي جداش اتنينماشي إذا هذا المقدار اللي هو صار
+
+262
+00:22:12,730 --> 00:22:16,510
+عبارة عن اتنين في هذا المقدار اللي هو وصلنا له
+
+263
+00:22:16,510 --> 00:22:20,650
+أصغر أو يساوي a n زائد واحد على n زائد واحد
+
+264
+00:22:20,650 --> 00:22:23,890
+فيكتوريا مضروب في مين؟ في اتنين يعني حسابات هذه
+
+265
+00:22:23,890 --> 00:22:28,770
+طيب إذا صار عندي الآن هاي اللي بدي أصله و اللي
+
+266
+00:22:28,770 --> 00:22:31,630
+كاتبه باختصار هو لإنه بيعتبر ان الباقية حساباتي
+
+267
+00:22:31,630 --> 00:22:36,950
+بتعرفوا تحسبوها em-n of x of x أصغر أو يساوي هذا
+
+268
+00:22:36,950 --> 00:22:43,450
+المقدار على مين؟ على اتنينالان شوف ما يليه الان as
+
+269
+00:22:43,450 --> 00:22:49,590
+n goes to infinity هذا المقدار هذا المقدار as n
+
+270
+00:22:49,590 --> 00:22:54,230
+goes to infinity ال limit بتروح للصفر لأن a n زائد
+
+271
+00:22:54,230 --> 00:22:57,110
+واحد على n زائد واحد كل factorial as n goes to
+
+272
+00:22:57,110 --> 00:23:01,190
+infinity ايش بده يروح للصفر يعني بمعنى اخر مدام
+
+273
+00:23:01,190 --> 00:23:04,430
+بده يروح للصفر طبعا وذا كبرت الان بتكبر مين برضه
+
+274
+00:23:04,430 --> 00:23:07,850
+مباشرة ام لأن ام اكبر منها اذا four
+
+275
+00:23:10,880 --> 00:23:19,800
+for large ن و أم we get
+
+276
+00:23:23,490 --> 00:23:31,050
+E M of X ناقص E N of X اللي هي أصغر أو يساوي من
+
+277
+00:23:31,050 --> 00:23:38,070
+هذه هذه أصلا for large N as N goes to infinity هذا
+
+278
+00:23:38,070 --> 00:23:43,470
+المقدار بيصير أصغر من أي ي في الدنيا اللي هي بيصير
+
+279
+00:23:43,470 --> 00:23:49,330
+أصغر من مين من ي أو أصغر أو يساوي ي as N و M
+
+280
+00:23:49,330 --> 00:23:54,630
+become very very largeماشي الحال لإنه ال limit لل
+
+281
+00:23:54,630 --> 00:23:59,110
+a n على n فكتوره ليش بيساوي بيساوي سفر إذا صار
+
+282
+00:23:59,110 --> 00:24:05,950
+عندي الآن اللي هي E m of X E n of X أصغر وشوى Y
+
+283
+00:24:05,950 --> 00:24:15,250
+لكل X وين موجودة في الفترة A ماقص A أو A ماشي
+
+284
+00:24:15,250 --> 00:24:23,160
+الحال الآن بناء عليهالان لكل X بغض النظر على ال X
+
+285
+00:24:23,160 --> 00:24:28,000
+اللي هو بيكون عندي أنه I'm very large وبيعطيني هذا
+
+286
+00:24:28,000 --> 00:24:32,560
+أصغر وسوء من إبسلون وهذه اللي سميناها Cauchy's
+
+287
+00:24:32,560 --> 00:24:37,470
+criterion for uniformاللي هو convergence في ال ..
+
+288
+00:24:37,470 --> 00:24:43,590
+اللي هو section 81 بناء عليها بطلع عندى E and اللي
+
+289
+00:24:43,590 --> 00:24:47,090
+هو converges uniformly to some function مش
+
+290
+00:24:47,090 --> 00:24:51,410
+عارفينها اللي هي .. اللي هي converges uniformly on
+
+291
+00:24:51,410 --> 00:24:59,770
+mean سالب a أو a باشي الحالة الالالآن عندي it
+
+292
+00:24:59,770 --> 00:25:02,610
+follows the sequence a and a converge uniformly on
+
+293
+00:25:02,610 --> 00:25:07,210
+the interval ناقص a و a where a أكبر من 0 لكن a
+
+294
+00:25:07,210 --> 00:25:11,670
+أكبر من 0 was a شمالها arbitrarily يعني اللي وصلنا
+
+295
+00:25:11,670 --> 00:25:18,110
+له ما ياليا يا جماعة اللي وصلنا له أنهعندي اللي هي
+
+296
+00:25:18,110 --> 00:25:23,210
+الـ sequence هذه لو جيت أخدت for every a element
+
+297
+00:25:23,210 --> 00:25:27,130
+in R و a أكبر من سفر لو وصلنا له اللي كانت a
+
+298
+00:25:27,130 --> 00:25:32,130
+arbitrarily وجدنا أنه هيكون عند الـ EL converges
+
+299
+00:25:32,130 --> 00:25:39,550
+uniformly on ماقص a و مين و a ماشي الحال طيب بناء
+
+300
+00:25:39,550 --> 00:25:48,200
+عليه لو أنت جيت أخدت خد الآن letX element in R، أي
+
+301
+00:25:48,200 --> 00:25:53,540
+X element in R، مدام X في R، إذا أكيد الـ X عشر،
+
+302
+00:25:53,540 --> 00:25:59,250
+عشرين، تلاتين، مليون، ممكن ما تكون،بقدر ألاقي A
+
+303
+00:25:59,250 --> 00:26:05,890
+بحيث أن X تنتمي إلى الفترة من ناقص A لعند مين لعند
+
+304
+00:26:05,890 --> 00:26:11,590
+إيه يعني مثلا ال X لجيتها 100 باخد ال A 110 بصير
+
+305
+00:26:11,590 --> 00:26:17,850
+ال X بين اللي هي ناقص 110 و100 إيش ما تيجي X بلاقي
+
+306
+00:26:17,850 --> 00:26:24,470
+لها Aإذا there exists a such that x element in a
+
+307
+00:26:24,470 --> 00:26:31,410
+minus a أكبر من صفر طبعا minus a ومين أو a أكيد طب
+
+308
+00:26:31,410 --> 00:26:34,390
+ما هو احنا قد أثبتنا انه لكل a element in R
+
+309
+00:26:38,870 --> 00:26:44,730
+Converge uniformly للـ A اللي لاجناها هذه On ناقص
+
+310
+00:26:44,730 --> 00:26:49,690
+A و A لهذه الـ A يعني بمعنى معناه مادام Converge
+
+311
+00:26:49,690 --> 00:26:55,030
+uniformly لإذا E N of Xاللي هو converges point
+
+312
+00:26:55,030 --> 00:26:58,250
+-wise to some function عند مين؟ عند الـ X اللي
+
+313
+00:26:58,250 --> 00:27:03,670
+أخدتها هانا هذه إذا الأن for any X element in R
+
+314
+00:27:03,670 --> 00:27:11,450
+for any X element in R هنلاقي E N of X convergesما
+
+315
+00:27:11,450 --> 00:27:15,250
+دام ال E N of X is convergent for every X element
+
+316
+00:27:15,250 --> 00:27:22,230
+in R إذا صارت عندي limit ال E N of X as N goes to
+
+317
+00:27:22,230 --> 00:27:27,830
+infinity exists for every X element in R ما دام
+
+318
+00:27:27,830 --> 00:27:34,210
+exist حسب اللي قلناها إذا الآن شرعت لتعريف الدالة
+
+319
+00:27:34,210 --> 00:27:40,600
+التالية الآن أنا بقول إنه احنا صار عندنااللي هو
+
+320
+00:27:40,600 --> 00:27:47,760
+مشروع ان نقول we define E من R لR by limit E N of
+
+321
+00:27:47,760 --> 00:27:53,940
+X اللي هي موجودة سميتها إيش اسمها E of X for X
+
+322
+00:27:53,940 --> 00:28:00,760
+element of Rوجود اللي هو دالة اسمها E أنا سميتها E
+
+323
+00:28:00,760 --> 00:28:06,400
+من R لعند R بحيث ان E of X هي limit ال E أنه VIX
+
+324
+00:28:06,400 --> 00:28:11,480
+اللي أثبتت وجوده بدي أثبتلك الأن أن E of X هذه هي
+
+325
+00:28:11,480 --> 00:28:16,120
+الدالة المبتغة اللي بتحقق الشروط اللي طلبها في
+
+326
+00:28:16,120 --> 00:28:21,740
+النظرية نشوف كيف طيب صلوا على النبي عليه الصلاة
+
+327
+00:28:21,740 --> 00:28:22,160
+والسلام
+
+328
+00:28:35,970 --> 00:28:39,830
+الان عندى ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+329
+00:28:39,830 --> 00:28:39,910
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+330
+00:28:39,910 --> 00:28:39,970
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+331
+00:28:39,970 --> 00:28:40,290
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+332
+00:28:40,290 --> 00:28:42,410
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+333
+00:28:42,410 --> 00:28:42,810
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+334
+00:28:42,810 --> 00:28:44,810
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+335
+00:28:44,810 --> 00:28:49,770
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+336
+00:28:49,770 --> 00:28:51,450
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+337
+00:28:51,450 --> 00:28:51,610
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+338
+00:28:51,610 --> 00:28:52,610
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+339
+00:28:52,610 --> 00:28:55,710
+.. ال ..
+
+340
+00:29:00,330 --> 00:29:06,370
+converge such that x element ناقص a أو a كلها هذه
+
+341
+00:29:06,370 --> 00:29:12,330
+الـen converged uniformly الان لمن؟ للـe معايا
+
+342
+00:29:12,330 --> 00:29:19,190
+converged uniformly للـe for every أو on ناقص a أو
+
+343
+00:29:19,190 --> 00:29:23,010
+a لأنه سمناها إيش اللي هي ال limit بتروحلها اسمها
+
+344
+00:29:23,010 --> 00:29:27,670
+إيه الآن en اتفجنا إنه أثبتناها إنها إيش معالها
+
+345
+00:29:28,810 --> 00:29:34,050
+continuous مادام continuous و en اللي هو طبعا
+
+346
+00:29:34,050 --> 00:29:36,330
+continuous مش بس على ناقص a و a continuous وين
+
+347
+00:29:36,330 --> 00:29:39,690
+مكان احنا الآن هنحكي عن ناقص a و a continuous صارت
+
+348
+00:29:39,690 --> 00:29:42,910
+ال en continuous sequence converts uniformly to e
+
+349
+00:29:42,910 --> 00:29:48,110
+على ناقص a و a إذا صارت ال e هذه حسب نظريتنا هتكون
+
+350
+00:29:48,110 --> 00:29:53,560
+continuous نظرية الأولىفي سيكشن 8 اللي هو 2 هتكون
+
+351
+00:29:53,560 --> 00:29:57,800
+الـ E is continuous على ناقص A و A قولنا if E N is
+
+352
+00:29:57,800 --> 00:30:00,200
+a sequence of continuous functions that converge
+
+353
+00:30:00,200 --> 00:30:03,200
+uniformly to some function then this function or
+
+354
+00:30:03,200 --> 00:30:06,880
+some function is continuous on ناقص A و Eإذا صارت
+
+355
+00:30:06,880 --> 00:30:10,200
+ال E continuous على نقص A و A يعني ال E continuous
+
+356
+00:30:10,200 --> 00:30:14,520
+عند من؟ عند ال X و since X was arbitrary in R then
+
+357
+00:30:14,520 --> 00:30:21,180
+E هتكون is continuous على كل ال R إذا صارت عندي E
+
+358
+00:30:21,180 --> 00:30:26,560
+is continuous at any X element in R الآن لو جينا
+
+359
+00:30:26,560 --> 00:30:31,600
+حسبنا ال E N of 0 و E 1 of 0 و E 1 of 0 أشو
+
+360
+00:30:31,600 --> 00:30:39,410
+بالساوية؟ هي واحدE اللي هو N زائد واحد of Zero
+
+361
+00:30:39,410 --> 00:30:45,630
+اللي هو بساوي من سفر لسفر هيطلع سفر وهذا واحد يعني
+
+362
+00:30:45,630 --> 00:30:52,470
+دايما ال E N of Zero هتطلع إيش؟ Zero مدام E N of
+
+363
+00:30:52,470 --> 00:31:00,670
+Zero Zero وبما أنه E is continuous إذن limit ال Eن
+
+364
+00:31:00,670 --> 00:31:05,930
+of zero as n goes to infinity بسوء اللي هو E of
+
+365
+00:31:05,930 --> 00:31:10,990
+zero أو E of X أسف E of X as n goes to infinity
+
+366
+00:31:10,990 --> 00:31:18,450
+بسوء E of X وبما أنه اللي هواللي هي ال ال ال ال ال
+
+367
+00:31:18,450 --> 00:31:20,590
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+368
+00:31:20,590 --> 00:31:21,370
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+369
+00:31:21,370 --> 00:31:22,190
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+370
+00:31:22,190 --> 00:31:25,670
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+371
+00:31:25,670 --> 00:31:30,470
+ال ال ال ال ال ال ال ال ال ال
+
+372
+00:31:30,470 --> 00:31:32,430
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+373
+00:31:32,430 --> 00:31:32,790
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+374
+00:31:32,790 --> 00:31:32,950
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+375
+00:31:32,950 --> 00:31:35,550
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+376
+00:31:35,550 --> 00:31:39,680
+ال الأنا كنت بخلطها أنا بقول E n of 0 بساوي 1
+
+377
+00:31:39,680 --> 00:31:43,540
+limit E n of X بساوي E of X وال E is continuous
+
+378
+00:31:43,540 --> 00:31:48,900
+إذا الحيكون ال E of 0 هذه هتساوي إيش؟ هيساوي 1
+
+379
+00:31:48,900 --> 00:31:53,800
+لأنه limit E n of 0 point twice بساوي E of 0 اللي
+
+380
+00:31:53,800 --> 00:31:58,580
+هي 0الان e n of zero بتساوي واحد for all and infim
+
+381
+00:31:58,580 --> 00:32:04,940
+therefore e of zero بتساوي واحد لإن اللي هي e
+
+382
+00:32:04,940 --> 00:32:11,240
+نفسها is continuous إذا حصلنا على اللي هو الجزء
+
+383
+00:32:11,240 --> 00:32:16,080
+الأول أذكركم في الظاهرية على اللي هو المطلوب أو
+
+384
+00:32:16,080 --> 00:32:20,770
+الخاصية الأولىالخاصية الأولى هى الخاصية الأولى هى
+
+385
+00:32:20,770 --> 00:32:25,430
+عندى أثبتنا .. التانى أسف أثبتنا E اللى هى E of
+
+386
+00:32:25,430 --> 00:32:29,370
+Zero بالساوية واحدة بدى أثبتلك الأن E prime of X
+
+387
+00:32:29,370 --> 00:32:34,750
+هتساوي E of X عشان أثبتها بدنا نذكركم في نظرية
+
+388
+00:32:34,750 --> 00:32:45,750
+أخدناها المرة الماضية اللى هى بتقول ما يليه أذكركم
+
+389
+00:32:45,750 --> 00:32:51,350
+يا جماعةبالنظرية اللي بستخدمها كانت ان عندي لو في
+
+390
+00:32:51,350 --> 00:32:56,330
+عندي fn sequence of functions و هذه fn ال
+
+391
+00:32:56,330 --> 00:33:00,050
+differentiable لأن fn برايم بتروح ل g uniformly
+
+392
+00:33:00,050 --> 00:33:08,330
+ماشي on some interval on some j وكان عندي fn of x
+
+393
+00:33:08,330 --> 00:33:11,050
+not converge
+
+394
+00:33:12,940 --> 00:33:21,240
+converts to f of x not مثلا on j اه for x not
+
+395
+00:33:21,240 --> 00:33:31,300
+element in j then بقول لي اللي هو ال f هذه f prime
+
+396
+00:33:31,300 --> 00:33:39,230
+existsو FN converts uniformly to this F او FN
+
+397
+00:33:39,230 --> 00:33:46,950
+converts uniformly to this F and F' هي مين هي الـ
+
+398
+00:33:46,950 --> 00:33:50,370
+G هذا حكيناه المرة الماضية طيب، بدأ استخدمها الآن
+
+399
+00:33:50,370 --> 00:33:54,510
+شوف عنده، ان شوف عنده الـ N أثبتنا او الـ N
+
+400
+00:33:54,510 --> 00:33:57,110
+interval لاقصية و ايه؟ we have the uniform
+
+401
+00:33:57,110 --> 00:34:02,350
+convergence of the sequence EN اللي هو هتكون EN
+
+402
+00:34:04,720 --> 00:34:12,400
+converge uniformly ماشي الحالة of you of me of
+
+403
+00:34:12,400 --> 00:34:15,900
+تلاتة اللي هو تلاتة اللي حكيناها قبل بشوية خليني
+
+404
+00:34:15,900 --> 00:34:21,540
+بس نذكركم فيها اللي هي تلاتة هي عندي اللي هي تلاتة
+
+405
+00:34:21,840 --> 00:34:25,980
+عندي الان conversion formally to some function E
+
+406
+00:34:25,980 --> 00:34:30,140
+سمناها الان بناء عليه بدلا من الان زائد واحد هي
+
+407
+00:34:30,140 --> 00:34:34,820
+سافسة الان إذا صارت الان زائد واحد prime برضه
+
+408
+00:34:34,820 --> 00:34:38,140
+conversion هادي هادي أصلا conversion formally to
+
+409
+00:34:38,140 --> 00:34:46,280
+مين؟ to ال E اللي عندي الان وضحت الصورة الان صارت
+
+410
+00:34:46,280 --> 00:34:53,410
+الصورة واضحة الان اللي حصلنا انهانتبه عليا، on any
+
+411
+00:34:53,410 --> 00:34:56,010
+interval ناقص a و a, we have the inferred
+
+412
+00:34:56,010 --> 00:35:01,170
+convergence of E n وفي ضوء اللي هو تلاتة, we also
+
+413
+00:35:01,170 --> 00:35:04,290
+have the inferred convergence E n prime of the
+
+414
+00:35:04,290 --> 00:35:08,350
+derivatives الآن من ال theorem 8.2.3 اللي حكيتها
+
+415
+00:35:08,350 --> 00:35:13,700
+هنا، مدام F n prime converges ل Gماشي الحال و ال
+
+416
+00:35:13,700 --> 00:35:17,160
+.. او ال Fn X0 بتروح ل F of X0 for X0 اللي بتنجح
+
+417
+00:35:17,160 --> 00:35:21,880
+اللي هو اكتر من النقطة و اصلا على اساس عرفنا ان Fn
+
+418
+00:35:21,880 --> 00:35:27,420
+of X بالساوي اللي هو مين E of X بتروح لمين لل E of
+
+419
+00:35:27,420 --> 00:35:31,780
+X اذا اكيد هذه متحققة كمان بالنسبة لمين لل E N
+
+420
+00:35:31,780 --> 00:35:36,760
+صارت هذه متحققة و هذه متحققة من التحقق هذههذه يا
+
+421
+00:35:36,760 --> 00:35:42,140
+عزيزي التحقق هذه إذا إذا هيكون عنده ال F N بتروح
+
+422
+00:35:42,140 --> 00:35:46,020
+لل F اللي هي ال E N بتروح لل E صنادر أثبتناها بس
+
+423
+00:35:46,020 --> 00:35:51,720
+إيش المهم أنه ال F prime لهذه اللي بتروحلها هذه
+
+424
+00:35:51,720 --> 00:35:57,080
+اللي هي ال E prime هي مين اللي هي اللي بتروحلها ال
+
+425
+00:35:57,080 --> 00:36:02,340
+E N ال F ال F N prime عند ال E N زائد واحد prime
+
+426
+00:36:02,340 --> 00:36:03,260
+بتروح لل E
+
+427
+00:36:06,460 --> 00:36:13,560
+إذا من نظرية إيان ستذهب إلى الـ E برضه وليس كذلك
+
+428
+00:36:13,560 --> 00:36:20,250
+الـ E' لهذههي هذه يعني بمعنى آخر E N زي دول برايم
+
+429
+00:36:20,250 --> 00:36:25,830
+برايم limitها ال E برايم زي ما هي مين ال E يعني
+
+430
+00:36:25,830 --> 00:36:29,950
+مدام الان limitها هي ال E و هي نفسها limitها من
+
+431
+00:36:29,950 --> 00:36:33,370
+النظرية اللي بنستنتجها E برايم إذا صارت ال E هي
+
+432
+00:36:33,370 --> 00:36:38,610
+مين ال E برايم أو بطريقة أخرى limit ال E N برايم
+
+433
+00:36:38,610 --> 00:36:43,440
+حسب النظرية بتساوي ال E برايمومن جهة أخرى limit en
+
+434
+00:36:43,440 --> 00:36:50,500
+براي هو نفسه limit en نفسهاو limit en اللي هي طبعا
+
+435
+00:36:50,500 --> 00:36:53,280
+انا مسميها n و n نقص واحد احنا مسميينها n و n زاد
+
+436
+00:36:53,280 --> 00:36:57,740
+واحد طبيعي en برايم هي limit من limit en نقص واحد
+
+437
+00:36:57,740 --> 00:37:02,620
+لحالها لأن هذه بتساوي هذه و هذه اصلا limit احنا
+
+438
+00:37:02,620 --> 00:37:07,140
+اثبتنا ايش بساوي ال E اذا صارت عند ال E برايم هي
+
+439
+00:37:07,140 --> 00:37:10,720
+ايش بتساوي ال E يعني هذه ال E برايم استنتاجا من
+
+440
+00:37:10,720 --> 00:37:16,260
+النظريةو هذه اللي هو من اللي أثبتنا في البداية و
+
+441
+00:37:16,260 --> 00:37:19,800
+أيضا نقدر نستخدم .. لسه نتجه من النظرية لكن أحنا
+
+442
+00:37:19,800 --> 00:37:25,540
+أثبتنا فيها قبل ما نستعمل المظلةطيب هذا الكلام
+
+443
+00:37:25,540 --> 00:37:30,680
+صحيح وين لكل x وين موجودة في ناقص a و a لكن a احنا
+
+444
+00:37:30,680 --> 00:37:35,600
+أخدناها أجمالها arbitrary يعني الآن بيصير عندي
+
+445
+00:37:35,600 --> 00:37:40,740
+اللي هو لكل x بلقي لواحدة زي هيك بتحقق الكلام هذا
+
+446
+00:37:40,740 --> 00:37:44,560
+الآن فبصير عندي a prime of x بسوية of x لكل x
+
+447
+00:37:44,560 --> 00:37:50,720
+element in R وهيك بيكون احنا أثبتنا وجود اللي هو
+
+448
+00:37:50,720 --> 00:37:58,390
+ال E of Xأثبتنا وجود دالة أثبتنا
+
+449
+00:37:58,390 --> 00:38:06,100
+وجود دالة سمناها E of Xهذه الدالة بتحقق شرطين اللي
+
+450
+00:38:06,100 --> 00:38:11,900
+هو E prime of X بساوي E of X وبتحقق الشرط التاني E
+
+451
+00:38:11,900 --> 00:38:20,680
+of 0 بساوي 1 السؤال الآن هل في غيرها؟ هل في غيرها
+
+452
+00:38:20,680 --> 00:38:25,860
+ولا لأ؟ طبعا لحظة الحظ أنه لا يوجد دالة غير هذه
+
+453
+00:38:25,860 --> 00:38:31,140
+الدالة اللي بتحقق هذا الكلامبس قبل ما نثبت الـ
+
+454
+00:38:31,140 --> 00:38:35,740
+uniqueness خلّينا ناخد بعض النتائج بشكل سريع و
+
+455
+00:38:35,740 --> 00:38:39,900
+نتائج سهلة و اللي هي ان شاء الله متاخدش وقتها
+
+456
+00:38:39,900 --> 00:38:43,040
+شوفوا يا جماعة صلوا على النبي عليه الصلاة والسلام
+
+457
+00:38:43,040 --> 00:38:48,080
+الـ function E has a derivative of every order and
+
+458
+00:38:48,080 --> 00:38:51,260
+E N of X بيساوي E of X for all N element in N و X
+
+459
+00:38:51,260 --> 00:38:54,480
+element in Rيعني الآن لو فضلناها كمان مرة و مرتين
+
+460
+00:38:54,480 --> 00:38:56,980
+و تلاتة و أربعة هتطلع أيش نفس الدالة طبعا هذا
+
+461
+00:38:56,980 --> 00:39:00,360
+الكلام سهل و by induction زي ما انتوا عارفين أول
+
+462
+00:39:00,360 --> 00:39:03,120
+حاجة ماثبتها فوران بالساوية واحد فوران بالساوية
+
+463
+00:39:03,120 --> 00:39:06,480
+واحد ماحنا أثبتنا E prime of X أيش بالساوية E of X
+
+464
+00:39:06,480 --> 00:39:11,070
+إذا هذه is true فوران بالساوية واحدطيب لان افترض
+
+465
+00:39:11,070 --> 00:39:15,510
+انها هذه صحيحة في القرآن بيساوي K بيصير E K of X
+
+466
+00:39:15,510 --> 00:39:20,010
+بيساوي E of X لما ان اتبتها ل K زائد واحد طيب ل K
+
+467
+00:39:20,010 --> 00:39:23,870
+زائد واحد خد A K زائد واحد of X ايش هذه؟ هذه اللي
+
+468
+00:39:23,870 --> 00:39:30,840
+هي A K of X الكل اشمالها إبراهيمالان a k of x
+
+469
+00:39:30,840 --> 00:39:33,540
+فرضناها ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+470
+00:39:33,540 --> 00:39:33,940
+بالساوية ايه بالساوية ايه بالساوية ايه بالساوية
+
+471
+00:39:33,940 --> 00:39:34,880
+ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+472
+00:39:34,880 --> 00:39:35,180
+ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+473
+00:39:35,180 --> 00:39:37,200
+بالساوية ايه بالساوية ايه بالساوية ايه بالساوية
+
+474
+00:39:37,200 --> 00:39:43,180
+ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+475
+00:39:43,180 --> 00:39:54,500
+بالساوية ايه بالساوية ايه بالسا
+
+476
+00:39:55,640 --> 00:40:01,620
+وضلك فاضل إلى ما لا نهاية حيظل تطلع نفس الدالة نجي
+
+477
+00:40:01,620 --> 00:40:07,340
+الآن لـ Corollary أو الخاصية اللي بعدها لهذه
+
+478
+00:40:07,340 --> 00:40:15,040
+الدالة هذه الدالة Fx أكبر من 0 بقولك دايماً حيكون
+
+479
+00:40:15,040 --> 00:40:21,280
+الواحد زائد X strictly أصغر من مين من E of X من E
+
+480
+00:40:21,280 --> 00:40:32,910
+of X ان انعندي اللي هي أربعة أربعة عشان تشوف أيش
+
+481
+00:40:32,910 --> 00:40:40,550
+أربعة الأربعة اللي هي ال E prime of X أظن هيك كنا
+
+482
+00:40:40,550 --> 00:40:51,590
+حاكين ال E الناس في ال E of X عندي
+
+483
+00:40:51,590 --> 00:40:53,390
+E
+
+484
+00:40:55,380 --> 00:41:03,750
+انا of Xأقبى أو أصغر strictly من E N زائد واحد of
+
+485
+00:41:03,750 --> 00:41:09,850
+X لكل X أكبر من مين من صفر ليه؟ لأن هذه هيزيد
+
+486
+00:41:09,850 --> 00:41:12,630
+عليها term اللي هو X أسوان زائد واحد على N زائد
+
+487
+00:41:12,630 --> 00:41:15,630
+واحد فكتوريا وهذا ال term الاكساتي اللي أكبر من
+
+488
+00:41:15,630 --> 00:41:19,810
+صفر أشماله اللي هو مودة إذا صارت ال sequence اللي
+
+489
+00:41:19,810 --> 00:41:25,010
+عندي يا جماعة هذه عبارة عن strictly increasing
+
+490
+00:41:25,680 --> 00:41:29,120
+sequence strictly increasing sequence بناءً عليه
+
+491
+00:41:29,120 --> 00:41:36,240
+هيكون عند E1 of X أصغر strictly من E2 of X وهذا
+
+492
+00:41:36,240 --> 00:41:42,020
+أصغر strictly من E N of X لكل N أكبر أو يساوي
+
+493
+00:41:42,020 --> 00:41:51,440
+تلاتة وال X أشمالها أكبر من سفر ماشي الحال طيب E1
+
+494
+00:41:51,440 --> 00:42:02,170
+of X مين هي؟ واحد زاد Xوهذا أصغر من E2 of X وهذا
+
+495
+00:42:02,170 --> 00:42:08,170
+أصغر من E N of X إذا نخد ال limit لكل الجهات as N
+
+496
+00:42:08,170 --> 00:42:10,670
+goes to infinity وهذا independent of N وهذا
+
+497
+00:42:10,670 --> 00:42:15,590
+independent of N إذا سيصبح أصغر أو يساوي limit E N
+
+498
+00:42:15,590 --> 00:42:20,430
+of X as N goes to infinityو limit E n of X as n
+
+499
+00:42:20,430 --> 00:42:25,350
+goes to infinity هو عبارة عن مين قلنا عنه E of X
+
+500
+00:42:25,350 --> 00:42:31,350
+إذا صارت 1 زائد X strictly أصغر من E of X أو E of
+
+501
+00:42:31,350 --> 00:42:39,850
+X أكبر strictly من 1 زائد Xطيب نيجي اللي هو نثبت
+
+502
+00:42:39,850 --> 00:42:45,310
+الـ uniqueness للدالة اللي أثبتنا وجودها الآن إذا
+
+503
+00:42:45,310 --> 00:42:51,290
+أثبتنا وجود دالة سميناها E of X هذه الدالة تحقق
+
+504
+00:42:51,290 --> 00:43:01,390
+الشرطين اللي قلناهن اللي هو E of E primeof X بساوي
+
+505
+00:43:01,390 --> 00:43:07,430
+E of X لكل X element in R الشرط الأول والشرط
+
+506
+00:43:07,430 --> 00:43:14,350
+التاني اللي هو E of Zero بتساوي إياش واحد الآن هذه
+
+507
+00:43:14,350 --> 00:43:17,390
+الشرط ده اللي أكبرنا وجودها لنتبت إنها إياش وحيدة
+
+508
+00:43:17,700 --> 00:43:22,860
+الأن الـ function E من R لـ R that satisfies I and
+
+509
+00:43:22,860 --> 00:43:29,200
+I I هكذا of theorem 8.3.1 is unique إذا ما لها هذه
+
+510
+00:43:29,200 --> 00:43:32,940
+الدالة وحيدة ونشوف كيف نثبت وحيدة طبعا انتوا
+
+511
+00:43:32,940 --> 00:43:36,140
+عارفين ال .. الاستراتيجية تثبت انها وحيدة ونفترض
+
+512
+00:43:36,140 --> 00:43:39,340
+ان في دالتين وفي الآخر يا بنسأل لتناقض أو بنسأل
+
+513
+00:43:39,340 --> 00:43:43,260
+لإن الدالتين إشمالهم متساويتين خلينا نشوف
+
+514
+00:43:47,010 --> 00:43:52,110
+لأن let E1 and E2 be two functions on R ماشي الحال
+
+515
+00:43:52,110 --> 00:43:59,470
+E1 وE2 عبارة عن دالتين من R لR تحقيقان البرو بارتز
+
+516
+00:43:59,470 --> 00:44:03,010
+I and I I of T والمتمنى هو تلاتة واحد اللي كتبه ان
+
+517
+00:44:03,010 --> 00:44:08,700
+انا على الجنب هناكالان و خلّينا نسمي E1-E2 يتساوي
+
+518
+00:44:08,700 --> 00:44:14,840
+F رايحكم باتجاهة F يتساوي 0 إذا أثبتنا F يتساوي 0
+
+519
+00:44:14,840 --> 00:44:21,630
+إذا سيصبح E1 يتساوي E2 يكون خلصناطيب then فضلي هذه
+
+520
+00:44:21,630 --> 00:44:24,510
+.. فضلي هذه لأن هذا قبل التفاضل و هذا قبل التفاضل
+
+521
+00:44:24,510 --> 00:44:27,350
+إذا F prime of X بيساوي E1 prime of X ناقص E2
+
+522
+00:44:27,350 --> 00:44:32,230
+prime of X E1 prime of X إيش هتساوي اللي هو نفسها
+
+523
+00:44:32,230 --> 00:44:35,730
+لأن مفترضين احنا و E2 prime of X برضه هتساوي نفسها
+
+524
+00:44:35,730 --> 00:44:39,770
+إذا E1 ناقص E2 يعني مين بيساوي F يعني صارت F prime
+
+525
+00:44:39,770 --> 00:44:44,610
+تبعتنا مين هي بيساوي F of X لكل X element المين
+
+526
+00:44:44,610 --> 00:44:50,090
+إنا احسب اللي دي إيش يا جماعة احسب لولي F في صفرأف
+
+527
+00:44:50,090 --> 00:44:53,350
+اف صفر بيساوي ا واحد اف صفر ناقص اتنين اف صفر ا
+
+528
+00:44:53,350 --> 00:44:56,710
+واحد اف صفر واحد و اتنين اف صفر برضه اش بتساوي
+
+529
+00:44:56,710 --> 00:44:59,510
+واحد لإن مفترضين ا واحد و اتنين بتحفق اللي هي
+
+530
+00:44:59,510 --> 00:45:02,970
+الشطن الأمامنا إذا بيساوي واحد ناقص واحد و اي ساوي
+
+531
+00:45:02,970 --> 00:45:04,350
+اش صفر
+
+532
+00:45:07,320 --> 00:45:13,380
+by induction f double prime هتساوي f نفسها f
+
+533
+00:45:13,380 --> 00:45:18,060
+triple بتساوي f نفسها fn of x بيساوي f of x وعملت
+
+534
+00:45:18,060 --> 00:45:21,920
+قبل بشوية واحدة زيها إذا by induction fn of x
+
+535
+00:45:21,920 --> 00:45:28,280
+بيساوي f of x لكل x element in R إذا حققت الآن f
+
+536
+00:45:28,280 --> 00:45:34,500
+of 0 بيساوي 0 و fn of x بيساوي مين؟ f of x شوف
+
+537
+00:45:34,500 --> 00:45:42,870
+الآنأذا صار عندي F of Zero بساوي Zero أو F
+
+538
+00:46:10,090 --> 00:46:17,830
+بساوة 0طب ايش علاقتنا ابوها الجيت بتشوف عشان بدنا
+
+539
+00:46:17,830 --> 00:46:20,690
+نطبق اللي هو taylor's theorem تبعت ال derivative
+
+540
+00:46:20,690 --> 00:46:25,930
+نص اللي اللي بدنا يعني برهان حلو let x element in
+
+541
+00:46:25,930 --> 00:46:29,550
+R be arbitrary أخدنا إذا x element in R arbitrary
+
+542
+00:46:29,550 --> 00:46:33,810
+x اللي يعني let I x be the closed interval with
+
+543
+00:46:33,810 --> 00:46:40,760
+end point 0x يعني I xهناخدها بتساوي 0 و X أو اللي
+
+544
+00:46:40,760 --> 00:46:47,080
+هي X و 0 على .. على اللي هو اللي هي حسب X اللي هي
+
+545
+00:46:47,080 --> 00:46:51,980
+سالبة أو مجبة إذا أخدنا ال I X عبارة عن اللي هي ال
+
+546
+00:46:51,980 --> 00:46:56,600
+closed interval اللي within point 0 X الآن F is
+
+547
+00:46:56,600 --> 00:47:01,020
+continuous on I Xعلى الـ closed interval مدام
+
+548
+00:47:01,020 --> 00:47:03,320
+continuous على closed interval and function that
+
+549
+00:47:03,320 --> 00:47:06,460
+is continuous على closed interval then it is
+
+550
+00:47:06,460 --> 00:47:09,200
+bounded on this interval يعني بمعنى أخر there
+
+551
+00:47:09,200 --> 00:47:12,880
+exist k بحيث أن absolute value of f of t أصغر أو
+
+552
+00:47:12,880 --> 00:47:17,960
+يساوي k for all t element in I X يعني f is bounded
+
+553
+00:47:17,960 --> 00:47:23,580
+on this intervalالان بدنا نطبق if we apply
+
+554
+00:47:23,580 --> 00:47:27,540
+taylor's theorem 6 4 1 to F بدنا نطبقها على
+
+555
+00:47:27,540 --> 00:47:31,800
+النقطتين النقطة الأولى اللى هى ال X X و X naught 0
+
+556
+00:47:31,800 --> 00:47:35,640
+يعني بنطبقها على مين اللى هى X X اللى هى طبعا
+
+557
+00:47:35,640 --> 00:47:41,370
+انتوا متذكرين F of Xأف of X تيلر سيوريم there
+
+558
+00:47:41,370 --> 00:47:46,370
+exists C in اللي هي ال interval مثلا X not و X
+
+559
+00:47:46,370 --> 00:47:52,690
+such that F of X بيساوي اللي هو F of X not زائد F
+
+560
+00:47:52,690 --> 00:47:58,490
+prime of X not على واحد factorial في X minus X not
+
+561
+00:47:58,490 --> 00:48:05,910
+زائد زائد Fn of x0 على n factorial في x minus x0
+
+562
+00:48:05,910 --> 00:48:11,610
+الكلوس n زائد ال remainder اللي هو fn زائد واحد of
+
+563
+00:48:11,610 --> 00:48:15,730
+اللي هي ال c اللي لاجيناها there exists c اللي هي
+
+564
+00:48:15,730 --> 00:48:19,170
+of c طبعا ال c هنا هتعتمد على ال n اللي احنا
+
+565
+00:48:19,170 --> 00:48:24,850
+اختارناها واشتغلنا عليها هو مسميها cn على اللي هو
+
+566
+00:48:24,850 --> 00:48:28,670
+n زائد واحد factorial في x
+
+567
+00:48:33,180 --> 00:48:37,920
+الـ N هو
+
+568
+00:48:37,920 --> 00:48:42,580
+ما خدها لعند مين بدل ما هو remainder N زائد واحد
+
+569
+00:48:42,580 --> 00:48:45,880
+نسميه remainder N نفس الشيء مافيش مشكلة احنا
+
+570
+00:48:45,880 --> 00:48:51,880
+كملناها يعني بدل ما اشتغل على N زي ما اشتغل على N
+
+571
+00:48:51,880 --> 00:48:54,460
+زائد واحد اشتغل على ال N هو اما هي ال Taylor's
+
+572
+00:48:54,460 --> 00:48:59,460
+theorem طيب المهم انتبهوا عن ديلنبن Taylor's
+
+573
+00:48:59,460 --> 00:49:03,580
+theorem على ال interval Ix اللي X0 بيساوي 0
+
+574
+00:49:03,580 --> 00:49:08,040
+وخلّينا نستخدم Fk of 0 اللي كتبتها هناك اللي هو
+
+575
+00:49:08,040 --> 00:49:12,020
+هتساوي Fk of 0 هتساوي 0 دايما اللي هو و لكل k
+
+576
+00:49:12,020 --> 00:49:15,740
+element n it follows thatfor each n unlimited on
+
+577
+00:49:15,740 --> 00:49:20,000
+there exist a point cn unlimited on x هذه اللي هي
+
+578
+00:49:20,000 --> 00:49:23,920
+ال cn بتعتمد على مين على n يعني الأن لو أخدت ببدل
+
+579
+00:49:23,920 --> 00:49:28,860
+اللي هنا على اللي هي on بتساوي مثلا اتنين أخد on
+
+580
+00:49:28,860 --> 00:49:31,660
+بتساوي تلتة أخد بتساوي on بتساوي أربعة ده هتختلف
+
+581
+00:49:31,660 --> 00:49:35,740
+من ال cn اللي بنلاقيها such that f of x بيساوي f
+
+582
+00:49:35,740 --> 00:49:39,420
+of zero زيد f prime of zero على واحد factorial في
+
+583
+00:49:39,420 --> 00:49:44,390
+xطبعاً نقص صفر X هتظلها زائد FN نقص واحد لأن نقص
+
+584
+00:49:44,390 --> 00:49:47,630
+واحد factorial في X minus X note اللي هي صفر طبعاً
+
+585
+00:49:47,630 --> 00:49:52,830
+هتصير X نقص واحد زائد FN of CN على N factorial X
+
+586
+00:49:52,830 --> 00:49:56,530
+أثن يعني هنا ناخد ال remainder اللي هو RN مش RN
+
+587
+00:49:56,530 --> 00:50:01,410
+زائد واحد زي ما نعمله طبعاً ماتفرجشبتزيده و بتشتغل
+
+588
+00:50:01,410 --> 00:50:04,910
+عليه نفس الاشي الان بس ال cn اللي بتختلف من n زائد
+
+589
+00:50:04,910 --> 00:50:07,650
+واحد بصير cn زائد واحد مثلا لأنه حاجة تانية ممكن
+
+590
+00:50:07,650 --> 00:50:11,290
+تيجي غير ال cn اللي لاجيناها حسب النظرية و يساوي
+
+591
+00:50:11,290 --> 00:50:16,350
+لأن كل هدول التاريج زي شمال هي الصفار لإنه fk of
+
+592
+00:50:16,350 --> 00:50:21,290
+00 إذا هذا سفر وهذا سفر طبعا هذا عند مين محسب عند
+
+593
+00:50:21,290 --> 00:50:25,800
+السفر وهذا عند السفر إلى آخرهإذا كل هذول هيكون
+
+594
+00:50:25,800 --> 00:50:30,700
+الصفار هيظل عندى بس القيمة هذه ايش القيمة هذه اللي
+
+595
+00:50:30,700 --> 00:50:35,320
+هي fn of cn طبيعي هتكون f of min of cn لأنه احنا
+
+596
+00:50:35,320 --> 00:50:41,070
+عرفنا انه ال derivative f primeأف دابل برايم أف
+
+597
+00:50:41,070 --> 00:50:45,710
+تريبل كل نياش بيساوي الأف إذا حيكون أف of CN على N
+
+598
+00:50:45,710 --> 00:50:50,710
+فيكتوريال في X أُس N إذا وصلنا له الآن أنه بعد ما
+
+599
+00:50:50,710 --> 00:50:57,170
+طبقنا Taylor's theorem طلع عندي الآن F of X هذه
+
+600
+00:50:57,170 --> 00:51:04,710
+اللي بنصبه إلى أن نثبتها بتساوي صفر طلعت عندي F of
+
+601
+00:51:04,710 --> 00:51:10,230
+X بيساوي F of CN على N فيكتوريال في X أُس Nالان خد
+
+602
+00:51:10,230 --> 00:51:14,510
+لل absolute value لها absolute value لل F of X
+
+603
+00:51:14,510 --> 00:51:20,090
+هيصير اصغر او يساوي ال absolute value او بيساوي
+
+604
+00:51:20,090 --> 00:51:26,870
+بالظبط absolute value F of C N فال absolute value
+
+605
+00:51:27,400 --> 00:51:33,440
+للـ X أُس N على N فكتوريال، مظبوط؟ لكن F of C N
+
+606
+00:51:33,440 --> 00:51:37,200
+هذه is .. الـ F is bounded على الفترة اللي لجيناها
+
+607
+00:51:37,200 --> 00:51:40,180
+اللي هي الفترة اليمين اللي قلت عنها اللي هي الـ I
+
+608
+00:51:40,180 --> 00:51:44,380
+X مدان bounded قلنا هذه أصغر أو يساوي الـ K إذا
+
+609
+00:51:44,380 --> 00:51:47,380
+هذه بيصير أصغر أو يساوي الـ K في ال absolute value
+
+610
+00:51:47,380 --> 00:51:52,750
+X أس N على N فكتوريال نيجي الآنهذه as n goes to
+
+611
+00:51:52,750 --> 00:51:56,770
+infinity هذه independent of n هذه أكبره يساوي سفر
+
+612
+00:51:56,770 --> 00:51:59,430
+و أصغره يساوي هذه as n goes to infinity هذه بتروح
+
+613
+00:51:59,430 --> 00:52:05,590
+للسفر إذا هذه بتروح أشمالها بدها تصير f of x تساوي
+
+614
+00:52:05,590 --> 00:52:11,230
+السفر ومنه ال E1 بتساوي ال E2 لأن f بتساوي E1 ناقص
+
+615
+00:52:11,230 --> 00:52:17,350
+Eإتنين وهو المطلوب وهذا الكلام صحيح لكل X element
+
+616
+00:52:17,350 --> 00:52:25,050
+in R لأن X اللي أخدناها arbitrary element in R هيك
+
+617
+00:52:25,050 --> 00:52:31,990
+بنكون أثبتنا وجود ال E واحد ال E التي تحقق الشرطين
+
+618
+00:52:31,990 --> 00:52:38,590
+اللي عندنا وفي نفس الوقت أثبتنا أن هذه الدالة اللي
+
+619
+00:52:38,590 --> 00:52:46,680
+بتحقق الشرطينهي دالة وحيدةمدام الدالة واحدة إذن
+
+620
+00:52:46,680 --> 00:52:53,280
+الآن يعني شرّعنا إنه نعطيها اسم إنه نقدر نعرف
+
+621
+00:52:53,280 --> 00:52:59,200
+الدالة اللي بتحقق هذول الشرطين إنها اسمها كذا the
+
+622
+00:52:59,200 --> 00:53:02,660
+unique definition ثمانية تلاتة خمسة the unique
+
+623
+00:53:02,660 --> 00:53:06,140
+function E من R لR such that E prime of X في سوء E
+
+624
+00:53:06,140 --> 00:53:10,120
+of X for all X elements in R وتحقق ال E زيرو
+
+625
+00:53:10,120 --> 00:53:15,190
+بالسوء واحد موجودةووحيدة إذا بحق اللي أقول is
+
+626
+00:53:15,190 --> 00:53:18,450
+called the exponential function وهي الـ
+
+627
+00:53:18,450 --> 00:53:22,350
+exponential اللي أنتوا مبسوطين عليها و بتستخدموها
+
+628
+00:53:22,350 --> 00:53:27,530
+أثبتنا الآن من خلال اللي هو اللي هي الترتيب اللي
+
+629
+00:53:27,530 --> 00:53:31,690
+رتبناه في المادة differentiability integrability و
+
+630
+00:53:31,690 --> 00:53:36,350
+بعدين اللي هو sequences of functions وصلنا إلى
+
+631
+00:53:36,350 --> 00:53:41,190
+اللي هو ال exponential هذه function existsالتي
+
+632
+00:53:41,190 --> 00:53:43,850
+سمناها الـ Exponential Function
+
+633
+00:53:48,710 --> 00:53:53,850
+بدنا نسمي الـ E of واحد قيمة الدالة القيمة الدالة
+
+634
+00:53:53,850 --> 00:53:58,650
+الـ E هذه عند الواحد بدنا نسميها E E و هذا اللي
+
+635
+00:53:58,650 --> 00:54:04,410
+بنسميه Ehlers number و احنا يعني بعد هيك هيصير
+
+636
+00:54:04,410 --> 00:54:09,410
+عندي اللي هو الرمز E of X ان هو ال exponential X
+
+637
+00:54:09,410 --> 00:54:15,710
+او أسهلنا في الاستخدام ال E of X ايش هتساويالـ E X
+
+638
+00:54:15,710 --> 00:54:21,530
+هذا الـ E واحد قيمة الدالة عند مين عند اللي هو
+
+639
+00:54:21,530 --> 00:54:26,750
+الرقم واحد سمناها E ماشي الحالة الان E of X
+
+640
+00:54:26,750 --> 00:54:32,610
+بالساوي E to the X هذه عبارة عن اللي هي الدالة E
+
+641
+00:54:32,610 --> 00:54:37,550
+of X بالساوي X notation لها لكن بعد شوية هنلاقي ال
+
+642
+00:54:37,550 --> 00:54:42,310
+notation consistent of اللي هو مع مين مع اللي هو
+
+643
+00:54:42,310 --> 00:54:47,480
+ال exponentيعني هيصير الـ E to the exponent X هو
+
+644
+00:54:47,480 --> 00:54:51,960
+عبارة عن بالظبط اللي هو قيمة الـ E of X لأن الـ E
+
+645
+00:54:51,960 --> 00:54:54,620
+of X بيساوي E to the X و ال E of واحد بيساوي E
+
+646
+00:54:54,620 --> 00:54:59,440
+واحد بيصير عندي E لما أرفحها للقوة X يطلع قيمتها
+
+647
+00:54:59,440 --> 00:55:05,840
+هي عبارة عن اللي هي قيمة دل E of X حسب اللي احنا
+
+648
+00:55:05,840 --> 00:55:11,550
+معرفينه هنايعني من الإحسابات ومن تعريفنا للدالة
+
+649
+00:55:11,550 --> 00:55:23,070
+هيكون فيه consistent it is consistent في الحالتين
+
+650
+00:55:23,070 --> 00:55:28,990
+طيبنجي الآن لما الكلام هذا ال number E can be
+
+651
+00:55:28,990 --> 00:55:32,570
+obtained as a limit and thereby approximated in
+
+652
+00:55:32,570 --> 00:55:36,410
+several different ways طبيعا احنا مادام عنده اللي
+
+653
+00:55:36,410 --> 00:55:42,990
+هو عبارة عن limit لل E N of Xاللي هي بتطلع عندى
+
+654
+00:55:42,990 --> 00:55:47,450
+limit عند ال .. ال .. ال .. ال end of واحد لأنها
+
+655
+00:55:47,450 --> 00:55:52,130
+continuous بصير عندى مدام limit إذا بصير أقدر أقرب
+
+656
+00:55:52,130 --> 00:55:56,490
+هذه القيمة اللي هي و نحصل على قيمة تقريبية بال E و
+
+657
+00:55:56,490 --> 00:56:00,010
+هذا مش شغلنا شغل تبعين اللي هو numerical analysis
+
+658
+00:56:00,010 --> 00:56:05,530
+أو الناس اللي بدأت .. اللي هي تشتغل في التقريب أو
+
+659
+00:56:05,530 --> 00:56:09,270
+مش شغل نهاني يعني بصديإلى أن الـ use of notation E
+
+660
+00:56:09,270 --> 00:56:15,050
+و X of E X زي ما قلنا كله تمام تمام consistent إلى
+
+661
+00:56:15,050 --> 00:56:19,910
+أن يجي اللي هو لبعض الخواص الأخرى لهذه اللي هي
+
+662
+00:56:19,910 --> 00:56:24,130
+الدالة إلى أن الـ exponential function satisfies
+
+663
+00:56:24,130 --> 00:56:26,150
+the following properties
+
+664
+00:56:32,620 --> 00:56:35,960
+أول حاجة أن الـ E of X لا تساوي سفر لكل X element
+
+665
+00:56:35,960 --> 00:56:40,140
+in R هي أول حاجة أن هذه الدالة دايما لا تساوي سفر
+
+666
+00:56:40,140 --> 00:56:45,980
+طبعا شغالة على R E of X زي Y بسوء E of X في E of Y
+
+667
+00:56:45,980 --> 00:56:50,880
+for all X element .. Y element in R الآن E of R
+
+668
+00:56:50,880 --> 00:56:56,080
+هنا اللي هي E of R قيمة الـ function هذه هتطلع لنا
+
+669
+00:56:56,080 --> 00:57:00,260
+بالظبطهي عبارة عن الـ E اللي قلنا عنها الرقم اللي
+
+670
+00:57:00,260 --> 00:57:06,000
+قبل و شوية لما نرفع للقوة R فبصير الان consistency
+
+671
+00:57:06,000 --> 00:57:11,440
+between the definition of the exponential and the
+
+672
+00:57:11,440 --> 00:57:15,700
+exponent of E to the power R بس هذي الـ R إيش اللي
+
+673
+00:57:15,700 --> 00:57:21,880
+من الين؟ NQ لان ماشي الحالة ده نشوف نبرهن اللي هو
+
+674
+00:57:21,880 --> 00:57:27,390
+الأولى by contradictionبدي افترض انه في عندي E of
+
+675
+00:57:27,390 --> 00:57:31,930
+α for some α element in R والـ E of α ايش تساوي
+
+676
+00:57:31,930 --> 00:57:35,870
+تساوي سفر ونوصل لcontradiction لان suppose that
+
+677
+00:57:35,870 --> 00:57:39,550
+there exists α element in R such that E of α ايش
+
+678
+00:57:39,550 --> 00:57:44,750
+بساوي بساوي سفرالآن and let G α be the closed
+
+679
+00:57:44,750 --> 00:57:49,710
+interval with endpoints mean α و 0 يعني الـ G of α
+
+680
+00:57:49,710 --> 00:57:54,230
+زي اللي فوق يا بساوي اللي هو 0 ألفة أو ألف و زيره
+
+681
+00:57:54,230 --> 00:57:59,440
+حسب قيمة الألف موجبة أو سالبة طيبالان زي ما قلت
+
+682
+00:57:59,440 --> 00:58:02,480
+قبل بشوية بما ان E is continuous on a closed
+
+683
+00:58:02,480 --> 00:58:07,700
+interval Iα او Jα اللي هو مسميها then there exist
+
+684
+00:58:07,700 --> 00:58:10,280
+case such that اللي هو ال absolute value E of D
+
+685
+00:58:10,280 --> 00:58:16,030
+أصفر سوى K لكل T element in Gالأن مشابه للي قبل دي
+
+686
+00:58:16,030 --> 00:58:19,930
+ربالكم الان بدنا نستخدم mean taylor theorem اللي
+
+687
+00:58:19,930 --> 00:58:24,890
+هو على اللي هي ال function اللي عندي هذه اللي هو
+
+688
+00:58:24,890 --> 00:58:29,310
+بس في مين الآن في ال end points ألف ومين ألف وصفر
+
+689
+00:58:29,310 --> 00:58:32,610
+اذا there exist cn element in dn such that مسرع
+
+690
+00:58:32,610 --> 00:58:35,950
+لإنه قاعد بعيد نفس البرهان اللي قبله شوية such
+
+691
+00:58:35,950 --> 00:58:42,200
+that اللي هو E of صفربطبق عند e of 0 بساوي e of
+
+692
+00:58:42,200 --> 00:58:44,340
+alpha زائد e prime of alpha على واحد factorial في
+
+693
+00:58:44,340 --> 00:58:47,280
+ناقص alpha اللي هي zero ناقص alpha زائد e n ناقص
+
+694
+00:58:47,280 --> 00:58:50,780
+واحد على ناقص واحد factorial alpha ناقص alpha أُس
+
+695
+00:58:50,780 --> 00:58:54,160
+n ناقص واحد لما أصل لعند ال remainder e n of alpha
+
+696
+00:58:54,160 --> 00:59:00,060
+على n factorial في ناقص alpha أُس n الآن اخلق هذه
+
+697
+00:59:00,060 --> 00:59:05,770
+مش e alpha هذه الـCnهذه الـ C أنا مخربط بصيحة الله
+
+698
+00:59:05,770 --> 00:59:11,430
+ماشي الحالة الان E of Zero طبيعي ايش يساوي واحد
+
+699
+00:59:11,430 --> 00:59:14,510
+ماحنا عارفين ان خلاص عرضناها الدالة هذه E of Zero
+
+700
+00:59:14,510 --> 00:59:17,630
+بتساوي واحد اتساوي واحد يساوي E of Zero و يساوي
+
+701
+00:59:20,060 --> 00:59:23,400
+الأن E of Alpha فرضناها إيش بالساوية سفر هيك
+
+702
+00:59:23,400 --> 00:59:28,280
+مفترضينها E prime of Alpha اللي هي نفس E of Alpha
+
+703
+00:59:28,280 --> 00:59:32,880
+إذا سفر برضه وهذا نفس الشيء إذا كلنا دول أصفر مع
+
+704
+00:59:32,880 --> 00:59:36,100
+دمين الأخيرة إذا هتساوي E of Cn على N factorial
+
+705
+00:59:36,100 --> 00:59:42,340
+ناقص Alpha أسعى وزي ما قلنا إنه احنا الآن هذا
+
+706
+00:59:42,340 --> 00:59:48,390
+المقدار عندي صار .. نوضح لكم هذهنشوف كيف نصل لـ
+
+707
+00:59:48,390 --> 00:59:53,770
+contradiction هال gate صار عند مايا ليه يا جماعة
+
+708
+00:59:53,770 --> 00:59:57,190
+صارت عند الواحد هذه اللي بيساوي E to the zero
+
+709
+00:59:57,190 --> 01:00:03,630
+بيساوي E to the zero اللي هي أصغر أو يساوي ال
+
+710
+01:00:03,630 --> 01:00:07,510
+absolute value لهذه بيساويها بعد إنه أصغر أو يساوي
+
+711
+01:00:07,510 --> 01:00:14,420
+بيساوي ال absolute value E of CL على N factorialفي
+
+712
+01:00:14,420 --> 01:00:22,320
+absolute value ناقص Alpha أُس N و هذه زي ما قلنا E
+
+713
+01:00:22,320 --> 01:00:25,880
+of C N إيش ما اللي قلنا قبل بشوية هذه أصغر أوي و
+
+714
+01:00:25,880 --> 01:00:32,960
+ساوي K إذا K على N factorial في ناقص Alpha أُس N
+
+715
+01:00:32,960 --> 01:00:38,710
+ماشي الآن هذا المقدارas n goes to infinity بيروح
+
+716
+01:00:38,710 --> 01:00:42,910
+لـ0 هذا independent of n و هذا independent of n
+
+717
+01:00:42,910 --> 01:00:48,650
+صار عندى الان هذا الان هي الواحد أصغر أو يساوي
+
+718
+01:00:48,650 --> 01:00:53,390
+اللي هو هذا المقدار خد ال limit as n goes to
+
+719
+01:00:53,390 --> 01:00:57,430
+infinity بيصير الواحد أصغر أو يساوي سفر و هذا إيش
+
+720
+01:00:57,430 --> 01:01:03,250
+ماله contradictionإذا صار في عندي الفرضية الأولى
+
+721
+01:01:03,250 --> 01:01:07,770
+خاطئة إذا there is no such Alpha اللي هو بتكون
+
+722
+01:01:07,770 --> 01:01:10,530
+عندها ال K of Alpha بتساوي سفر إذا ال K of Alpha
+
+723
+01:01:10,530 --> 01:01:16,090
+بتساوي أكبر لا تساوي سفر دائما إذا صار عندي اللي
+
+724
+01:01:16,090 --> 01:01:22,150
+هو قيمة اللي هي E to the X لا تساوي سفر أبدا نيجي
+
+725
+01:01:22,150 --> 01:01:29,110
+الآن إن بدنا نثبت أنهنثبت الخاصية اللي بعدها اللي
+
+726
+01:01:29,110 --> 01:01:38,430
+هي عبارة عن E of X زائد Y بساوة E of X في E of Y
+
+727
+01:01:38,430 --> 01:01:42,670
+لكل X و Y element in R هذه الخاصية اللي بدنا
+
+728
+01:01:42,670 --> 01:01:48,410
+نثبتها شوف طريقته حلوة في الإثبات بيقول ليه نفترض
+
+729
+01:01:48,410 --> 01:01:53,030
+Y fixed arbitrary لكن ليهاش fixed بنحكي عن Y محددة
+
+730
+01:01:53,030 --> 01:01:59,760
+arbitrary لكن نحكي عن Y محددةالان مادام Y اللي هي
+
+731
+01:01:59,760 --> 01:02:05,240
+ال E of Y أكيد يشملها لا تساوي سفر اتفجنا عليها
+
+732
+01:02:05,240 --> 01:02:11,520
+الان عرفلي الان function G من R لR عرفها كيف؟G of
+
+733
+01:02:11,520 --> 01:02:15,720
+X المتغير X الان Y اللي هي عبارة عن اشي Fix ثابت
+
+734
+01:02:15,720 --> 01:02:19,080
+بحكي عنه لكن كان arbitrary اللي هي G of X بيساوي E
+
+735
+01:02:19,080 --> 01:02:24,420
+of X زائد Y على E of Y for X element in R أخدنا
+
+736
+01:02:24,420 --> 01:02:30,060
+هذه الدالة الآن شوف هذه الدالة اعمله G prime لها G
+
+737
+01:02:30,060 --> 01:02:33,920
+prime of X بالفضل بالنسبالي XY ثابتة اللي هي بيصير
+
+738
+01:02:33,920 --> 01:02:38,280
+هذه E prime of X زائد Y على E of Y عدد ما لاش
+
+739
+01:02:38,280 --> 01:02:43,080
+علاقة فيه Y ساويالـ E' هي نفس مين؟ الـ E of X زي Y
+
+740
+01:02:43,080 --> 01:02:48,720
+على E of Y ماشي؟ طيب، الآن صار هذا الـ E X زي Y
+
+741
+01:02:48,720 --> 01:02:53,280
+على E of Y هو مين؟ رجع G إذا رجعت إن الـ G' مش
+
+742
+01:02:53,280 --> 01:03:01,770
+بيساوي الـ G الآن andأحسب لي g of 0 بساوية e of 0
+
+743
+01:03:01,770 --> 01:03:06,070
+زائد y على e of y يعني بساوية واحدة إذا الدالة
+
+744
+01:03:06,070 --> 01:03:11,050
+اللي عرفناها هذه g of x طلعت لي g prime لها نفسها
+
+745
+01:03:11,050 --> 01:03:15,310
+وطلعت لي g of 0 واحدة وعلمة قبل قليل عمال بيقول
+
+746
+01:03:15,310 --> 01:03:18,610
+مافيش لغير واحدة في الدنيا بتهبها الخاصيتين اللي
+
+747
+01:03:18,610 --> 01:03:23,190
+هي مين ال E of X إذا هذه غصب عننا لازم تطلع مين ال
+
+748
+01:03:23,190 --> 01:03:27,080
+E of X because of the uniqueness of Eإذا صحيح أنا
+
+749
+01:03:27,080 --> 01:03:31,000
+عندي E of X بساوي E X Z Y على E E of Y يعني E X Z
+
+750
+01:03:31,000 --> 01:03:37,940
+Y بساوي E of Y في ال E of X و هو المطلوب طيب نيجي
+
+751
+01:03:37,940 --> 01:03:46,800
+الآن نثبت اللي بعدها اللي هي بدنا نثبت E of R
+
+752
+01:03:46,800 --> 01:03:54,870
+بساوي E to the RR المنتنين LQ اللي هو rational
+
+753
+01:03:54,870 --> 01:04:00,870
+number طيب جماعة فيكم تصل على النبي عليه الصلاة
+
+754
+01:04:00,870 --> 01:04:09,370
+والسلام لنشوف كيف نيجي
+
+755
+01:04:09,370 --> 01:04:14,690
+البرهن اللي هو الجزء الثالث أو هذا جزء من النظرية
+
+756
+01:04:14,690 --> 01:04:18,830
+أول
+
+757
+01:04:18,830 --> 01:04:24,560
+حاجةهذه صحيحة لكل أنهي المتنانة كيف؟ by induction
+
+758
+01:04:24,560 --> 01:04:53,330
+E of X بساوي E of Xأكيد أكيد أكيد أكيد أكيد
+
+759
+01:04:53,330 --> 01:04:56,270
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+760
+01:04:56,270 --> 01:04:56,330
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+761
+01:04:56,330 --> 01:04:57,110
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+762
+01:04:57,110 --> 01:05:00,170
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+763
+01:05:00,170 --> 01:05:02,740
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكE
+
+764
+01:05:02,740 --> 01:05:08,660
+of K في X زائد X ويساوي من الخاصية اللي احنا لسه
+
+765
+01:05:08,660 --> 01:05:12,880
+ماخلصنهاش هذه بساوي E للأول زايد في E للتاني اللي
+
+766
+01:05:12,880 --> 01:05:17,960
+هي E للأول اللي هو KX في E للتاني اللي هي EX
+
+767
+01:05:17,960 --> 01:05:22,780
+ويساوي EKX اللي هو E of X أُس K لأنه فرضينا أنا
+
+768
+01:05:22,780 --> 01:05:31,260
+صحيحة اللي هي K بيصير عندي E of X أُس K في E of Xو
+
+769
+01:05:31,260 --> 01:05:39,800
+يساوي E of X الكل أُس K زائد واحد إذا صارت
+
+770
+01:05:39,800 --> 01:05:49,580
+هذه is true for all N element in N طيب .. الآن شوف
+
+771
+01:05:49,580 --> 01:05:53,920
+ما يليه نلاحظ ما يليه
+
+772
+01:06:02,460 --> 01:06:06,080
+Lit x بتساوي ايش؟ واحدة على ان ناخد الـ x ايش
+
+773
+01:06:06,080 --> 01:06:10,540
+بتساوي واحدة على ان؟ بتصير هذا E of واحد لأن ان في
+
+774
+01:06:10,540 --> 01:06:13,840
+واحدة على ان واحد ال E of واحد رمزناها من رمز ايش
+
+775
+01:06:13,840 --> 01:06:17,940
+يا جماعة؟ E وهذا اللي بدأ اصليه هذا رمزناها من رمز
+
+776
+01:06:17,940 --> 01:06:22,420
+E وال E عبارة عن رقم ال E of واحد هي قيمة الدالة
+
+777
+01:06:22,420 --> 01:06:28,100
+اللي تبعتنا عند اللي هي الواحد سمينها Eالان E of N
+
+778
+01:06:28,100 --> 01:06:33,380
+في 1 على N تساوي اللي هي عبارة عن ال X هذه اللي هي
+
+779
+01:06:33,380 --> 01:06:40,720
+عبارة عن 1 على N E of 1 على N الكل اسمه أُس N يعني
+
+780
+01:06:40,720 --> 01:06:48,380
+الآن عندي هذه ال X E N X بساوي E X أُس N X مين 1
+
+781
+01:06:48,380 --> 01:06:53,210
+على N E of 1 على N أُس Nإذا صار عندى الآن هي هذا
+
+782
+01:06:53,210 --> 01:06:59,010
+القيمة بساوية هذا إذا بمعنى آخر E of واحدة الان
+
+783
+01:06:59,010 --> 01:07:05,830
+بساوية E العدد أس واحد على N هذا الان E of واحدة
+
+784
+01:07:05,830 --> 01:07:14,050
+الان بساوية E أس واحد على N الآن من جهة أخرى لو
+
+785
+01:07:14,050 --> 01:07:23,700
+جيت E أس minus Mهتساوي واحد على E أُس M خد
+
+786
+01:07:23,700 --> 01:07:26,760
+ال
+
+787
+01:07:26,760 --> 01:07:33,460
+E of M ناقص M اللي هي بيساوي E of Zero اللي هي
+
+788
+01:07:33,460 --> 01:07:41,020
+بيساوي واحد صح وإيه ساوي E of Mفي E of ناقص M حسب
+
+789
+01:07:41,020 --> 01:07:45,900
+الخاصية اللي هي زائد ناقص M هذه X وهذه ال Y E
+
+790
+01:07:45,900 --> 01:07:49,760
+الأولى في E التانية الان صار عندى هذا المقدار
+
+791
+01:07:49,760 --> 01:07:57,600
+بساوي واحد يعني إذا E of ناقص M بساوي بنجل هدهان
+
+792
+01:07:57,600 --> 01:08:03,260
+واحد على E of M إذا صارت عندى E of ناقص M بساوي
+
+793
+01:08:03,260 --> 01:08:09,360
+واحد على E of M وال E of M ال E of Mمن فوق E of M
+
+794
+01:08:09,360 --> 01:08:14,280
+هذا X بواحد يعني بيساوي E of واحد أُس M يعني عبارة
+
+795
+01:08:14,280 --> 01:08:19,500
+عن E أُس M وهي فوق إيش واحد الآن هذه اللي هو عدد
+
+796
+01:08:19,500 --> 01:08:23,800
+عادي اللي هو E مرفوع للأُس M اللي هو نفسه E to the
+
+797
+01:08:23,800 --> 01:08:27,360
+minus M لأن هذه معلومة قديمة لذا انصار عندي الآن E
+
+798
+01:08:27,360 --> 01:08:34,820
+of minus M بيساوي E أُس ماقص M وال E أُس واحد على
+
+799
+01:08:34,820 --> 01:08:43,460
+N بيساوي Eأُس واحد على N نجي
+
+800
+01:08:43,460 --> 01:08:46,780
+الآن احنا غايتنا مين؟ E of R الـ R إيش اللي نقدر
+
+801
+01:08:46,780 --> 01:08:50,640
+نكتبها؟ الـ R في الـ Q إذا الـ R بتنكتب على صورة M
+
+802
+01:08:50,640 --> 01:08:54,780
+على N إذا نخد R في اللي نمتني Q إذا there exists M
+
+803
+01:08:54,780 --> 01:08:58,820
+و N اي واحدة في زد واحدة في N صحيح ده؟ R بتساوي M
+
+804
+01:08:58,820 --> 01:09:03,320
+على N بتثبتلك إن E of M على N بساوي E to the M على
+
+805
+01:09:03,320 --> 01:09:12,580
+N شوف كيفE of M على N ايش هيساوي E أُس واحدة ل N
+
+806
+01:09:12,580 --> 01:09:21,400
+الكل A شماله أُس M أُس M E of واحدة ل N أُس M مين
+
+807
+01:09:21,400 --> 01:09:28,870
+اللي شرعلي اللي هوعندي اللي هي E-M و هي سالبة طلعت
+
+808
+01:09:28,870 --> 01:09:33,870
+E أص مينس M و اللي شرعلي E و واحدة ل N و N مودبة
+
+809
+01:09:33,870 --> 01:09:41,130
+هي E أص واحد على N فبصير عندي الأن E أص M على N أص
+
+810
+01:09:41,130 --> 01:09:45,970
+M على N بساوي E أص واحد على N هذه اللي جوا الكل
+
+811
+01:09:45,970 --> 01:09:53,650
+أسمين الكل أص M ماشي الحال هو يساوي E
+
+812
+01:09:58,010 --> 01:10:03,730
+هذه شرعات لإي و واحدة الان كل اس ام وهذا عدد عادي
+
+813
+01:10:03,730 --> 01:10:08,730
+الان وهذا عدد عادية بيصير اي أس ام عالميا على ان و
+
+814
+01:10:08,730 --> 01:10:14,850
+هذه بيكون عندي اي أس اي او ام على ان بساوي اللي هو
+
+815
+01:10:14,850 --> 01:10:18,870
+عبارة عن اي أس ام على ان و ام على ان was our
+
+816
+01:10:18,870 --> 01:10:21,710
+arbitral rational number اذا صار عندي اي أس ار
+
+817
+01:10:21,710 --> 01:10:24,210
+بساوي اي أس ار
+
+818
+01:10:29,400 --> 01:10:35,160
+نجي الآن للنظرية اللي هي تمانية تلاتة سبعة اللي هي
+
+819
+01:10:35,160 --> 01:10:39,980
+النظرية الأخيرة في اللي هي الحديث عن اللي هو ال
+
+820
+01:10:39,980 --> 01:10:45,260
+exponential function و بعدها طبعا بنحكي عن اللي هو
+
+821
+01:10:45,260 --> 01:10:48,060
+ال logarithmic function بس خلينا الآن نحكي عالميا
+
+822
+01:10:48,060 --> 01:10:54,340
+انكمل نظريتنا على ال exponential function بنشوف
+
+823
+01:10:55,540 --> 01:11:00,240
+الأن بقولي الـ exponential function E is strictly
+
+824
+01:11:00,240 --> 01:11:04,840
+increasing on R يعني الـ derivative إلها أكبر من
+
+825
+01:11:04,840 --> 01:11:11,460
+سفر ماشي؟ and this range اللي هو Y المتنرسج ده Y
+
+826
+01:11:11,460 --> 01:11:16,720
+أكبر من سفر يعني الـ E هتكون بالظبط ده اللي من R
+
+827
+01:11:16,720 --> 01:11:24,080
+بتصب إلى R positive إلى R positive يعني عبارة عن
+
+828
+01:11:24,080 --> 01:11:31,370
+Zeroومالة نهاية هي التي هي range دالة range دالة
+
+829
+01:11:31,370 --> 01:11:34,610
+هذه هي من سفر إلى مالة نهاية لأن كتابة على سورة
+
+830
+01:11:34,610 --> 01:11:40,210
+function is on to طيب مش هيك limit E of X لما X
+
+831
+01:11:40,210 --> 01:11:43,470
+تروح لسالب مالة نهاية بتساوي سفر and limit E of X
+
+832
+01:11:43,470 --> 01:11:47,110
+لما X تروح لمالة نهاية ايش بتساوي؟ بتساوي مالة
+
+833
+01:11:47,110 --> 01:11:55,670
+نهايةيعني الدالة هذه في حالة ال ال E of X اللي
+
+834
+01:11:55,670 --> 01:11:59,070
+احنا سميناها ال X exponential لل X هذه الدالة هذه
+
+835
+01:11:59,070 --> 01:12:03,130
+الدالة زي ما انتوا عارفين لما X تروح إلى سالب ماله
+
+836
+01:12:03,130 --> 01:12:06,830
+نهاية هذا هيروح للسفر ال E of X هتروح للسفر و لما
+
+837
+01:12:06,830 --> 01:12:10,890
+ال X تروح لماله نهاية ال E of X هتروح إيش إلى ماله
+
+838
+01:12:10,890 --> 01:12:14,810
+نهاية نشوف أول شي بالنسبة لمن؟ لل range
+
+839
+01:12:19,570 --> 01:12:24,250
+الان we know that E of 0 بيساوي Eاش واحد اكبر من
+
+840
+01:12:24,250 --> 01:12:29,770
+Eاش من سفر اكيد،ظبط ولا لا؟ and E of X ده بيساوي
+
+841
+01:12:29,770 --> 01:12:36,390
+سفر for X element in R صار عندى الان هاي دالة E of
+
+842
+01:12:36,390 --> 01:12:38,650
+0 إلها واحد،ماشي؟
+
+843
+01:12:40,360 --> 01:12:46,000
+أو E of X مش سفر يعني إيه شمالها لا تقطع محور
+
+844
+01:12:46,000 --> 01:12:51,910
+السينات إطلاقااللي E of X نفسها و 0 من سفرها الأن
+
+845
+01:12:51,910 --> 01:12:57,110
+أنا بقول مستحيل تكون في إيها قيم سالبة يعني مستحيل
+
+846
+01:12:57,110 --> 01:13:02,030
+نلاقي E of X naught اللي هي أصغر من 0 ليش؟ لأنه لو
+
+847
+01:13:02,030 --> 01:13:05,990
+لجينا E of X naught سالبة لو هالجيت عندنا X naught
+
+848
+01:13:05,990 --> 01:13:09,910
+و هيها قيمتها E of X naught اللي هو سالبة يعني
+
+849
+01:13:09,910 --> 01:13:16,460
+هتكون تحتطب ما هي الدالة متصلة مدام متصلة إذا غصب
+
+850
+01:13:16,460 --> 01:13:20,940
+عنها هتيجي تقطع اللي هو ما هتيجي منها لها هتقطع
+
+851
+01:13:20,940 --> 01:13:24,760
+محور السينات إذا هتصير سفر وهذا Contradiction من
+
+852
+01:13:24,760 --> 01:13:28,380
+أين الكلام هذا؟ By Bolzano Intermediate Value
+
+853
+01:13:28,380 --> 01:13:32,760
+Theorem بما أنه احنا في عندنا X0 فرضناها ال E of
+
+854
+01:13:32,760 --> 01:13:39,260
+X0 أصغر من سفر وفي عندي اللي هي E of Zero بساوي
+
+855
+01:13:39,260 --> 01:13:40,980
+واحد اللي هي
+
+856
+01:13:44,440 --> 01:13:48,620
+أه لو فرضنا ان في اكس نوت اكس نوت تكون اللي هو
+
+857
+01:13:48,620 --> 01:13:56,300
+سالبة اكس نوت اصغر من سفر فعندي E of X naught اصغر
+
+858
+01:13:56,300 --> 01:14:05,740
+من سفر وعندي اللي هو في اللي هو E of X بساوي
+
+859
+01:14:05,740 --> 01:14:14,470
+واحد اه وهادي اكبر من سفرأه وفرضنا وجود هذه وفرضنا
+
+860
+01:14:14,470 --> 01:14:18,810
+وجود هذه اذا by intermediate value theorem غصبن
+
+861
+01:14:18,810 --> 01:14:23,650
+عنها there exists c element الفترة اللي احنا بنحكي
+
+862
+01:14:23,650 --> 01:14:28,510
+عنها R such that f of هذه الـ c اللي هي E of C
+
+863
+01:14:28,510 --> 01:14:33,330
+بتساول كمية اللي بينين اللي هي اسمها سفر وهذا
+
+864
+01:14:33,330 --> 01:14:41,860
+contradictionبالتالي بيحقق البلزان انها مستمرة على
+
+865
+01:14:41,860 --> 01:14:47,060
+اي انتقال مغلق على كل أرض لانها بيحقق البلزان انها
+
+866
+01:14:47,060 --> 01:14:47,260
+مستمرة على اي انتقال مغلق على كل أرض لانها بيحقق
+
+867
+01:14:47,260 --> 01:14:47,280
+البلزان انها مستمرة على اي انتقال مغلق على كل أرض
+
+868
+01:14:47,280 --> 01:14:47,700
+لانها بيحقق البلزان انها مستمرة على اي انتقال مغلق
+
+869
+01:14:47,700 --> 01:14:48,880
+على كل أرض لانها بيحقق البلزان انها مستمرة على اي
+
+870
+01:14:48,880 --> 01:14:49,100
+انتقال مغلق على كل أرض لانها بيحقق البلزان انها
+
+871
+01:14:49,100 --> 01:14:51,280
+مستمرة على اي انتقال مغلق على كل أرض لانها بيحقق
+
+872
+01:14:51,280 --> 01:14:54,390
+البلزان انها مستمرة على اي انتقمن سفر طيب مدام ال
+
+873
+01:14:54,390 --> 01:14:57,010
+E of X أكبر من سفر و احنا بنعرف ان ال E prime of X
+
+874
+01:14:57,010 --> 01:15:00,690
+بيتساوي E of X اذا صارت ال E prime أكبر من مين من
+
+875
+01:15:00,690 --> 01:15:04,890
+سفر كل X element ال R ومن ثم هذا معناته ان E is
+
+876
+01:15:04,890 --> 01:15:10,450
+strictly increasing on R اذا اللي هو المطلوب الأول
+
+877
+01:15:10,450 --> 01:15:15,810
+أثبتناه ان E is strictly increasing on R طيب
+
+878
+01:15:24,230 --> 01:15:30,250
+عندي اتنين strictly increasing ال function مظبوط
+
+879
+01:15:30,250 --> 01:15:41,510
+يعني اتنين اصغر من مين من اياش من ال E ليش لأن
+
+880
+01:15:41,510 --> 01:15:53,730
+عندي الواحد E واحد او E of Xقلنا أصغر strictly ولا
+
+881
+01:15:53,730 --> 01:15:57,890
+أكبر strictly من واحد زاد X صح؟ هذه الكورولا
+
+882
+01:15:57,890 --> 01:16:02,810
+remained تمانية تلاتة تلاتة اثبتناها إذا صار عند E
+
+883
+01:16:02,810 --> 01:16:08,050
+of واحدأكبر strictly من واحد زائد وا أكبر strictly
+
+884
+01:16:08,050 --> 01:16:13,970
+من واحد زائد واحد وال E of واحد مين هو E أس واحد
+
+885
+01:16:13,970 --> 01:16:17,390
+مش هيك اتفاجنا يعني هذا إياش اتنين يعني ال E أكبر
+
+886
+01:16:17,390 --> 01:16:22,450
+strictly من مين من الاتنين اتفاجنا طيب اتفاجنا إذا
+
+887
+01:16:22,450 --> 01:16:26,410
+صار عند ال E of اتنين أكبر strictly ال E أكبر من
+
+888
+01:16:26,410 --> 01:16:29,950
+مين من اتنين خلينا نبدأ عليها هذه ال E أكبر من
+
+889
+01:16:29,950 --> 01:16:41,000
+إياش من اتنينالان اكيد عنده ال E of R اللي بيسوي E
+
+890
+01:16:41,000 --> 01:16:48,440
+R او E N خليني اقول E N اكبر من اتنين هتكون اصغر
+
+891
+01:16:48,440 --> 01:16:53,700
+اللي هي اكبر من اتنين أُس N الان as N goes to
+
+892
+01:16:53,700 --> 01:16:57,680
+infinity اذا
+
+893
+01:16:57,680 --> 01:17:04,230
+اكيد هذه هتروح لمين لإنفينيتيوالـ E نفسها اللي هي
+
+894
+01:17:04,230 --> 01:17:08,750
+is strictly increasing is strictly increasing إذا
+
+895
+01:17:08,750 --> 01:17:12,370
+ال E of X تبعتها برضه هتروح لمين ل Infinity لما X
+
+896
+01:17:12,370 --> 01:17:18,650
+تروح إلى وين إلى مالة نهاية ليش لأن لكل L في X
+
+897
+01:17:18,650 --> 01:17:24,670
+أكبر منهابحيث انه اللي هو تصير E of X اللي هي
+
+898
+01:17:24,670 --> 01:17:30,110
+بيساوي E to the X أكبر من E to the N لكل N في X
+
+899
+01:17:30,110 --> 01:17:33,850
+أكبر منها بتصير E of X أكبر منها لأنها strictly
+
+900
+01:17:33,850 --> 01:17:37,710
+increasing فلان لما تروح هذه إلى مالة نهاية الكبار
+
+901
+01:17:37,710 --> 01:17:41,170
+تبعتها وهذه طبعا أكبر دايما موجودة هذا هتروح إلى
+
+902
+01:17:41,170 --> 01:17:45,540
+أين برضه إلى مالة نهاية يعني limit ال E of Xلما X
+
+903
+01:17:45,540 --> 01:17:48,980
+تروح إلى مالة نهاية بتساوي مالة نهاية وقلكم اللي
+
+904
+01:17:48,980 --> 01:17:54,040
+هو لو تجربته اللي هو تثبته أنه بما أنه limit E of
+
+905
+01:17:54,040 --> 01:17:58,840
+N بتروح إلى مالة نهاية by اللي هي limit definition
+
+906
+01:17:58,840 --> 01:18:03,020
+إذا limit E of X لما X تروح إلى مالة نهاية بتساوي
+
+907
+01:18:03,020 --> 01:18:08,080
+مالة نهاية الآن similarly for mean for اللي هو
+
+908
+01:18:08,080 --> 01:18:11,120
+limit
+
+909
+01:18:12,620 --> 01:18:16,560
+الـ E of X لما X تروح لسالب ملا نهاية هذه الان
+
+910
+01:18:16,560 --> 01:18:22,840
+بيصير اتنين أو minus N أكبر من E أو minus N ماشي
+
+911
+01:18:22,840 --> 01:18:28,000
+الحال الان ال limit لهذه as N goes to infinity
+
+912
+01:18:28,000 --> 01:18:31,940
+أكبر أو ساوي ال limit لهذه as N goes to infinity
+
+913
+01:18:31,940 --> 01:18:33,420
+الان
+
+914
+01:18:35,300 --> 01:18:40,100
+هذه اش مالها بيساوي سفر وهذه اوتوماتيك هيساوي ايش؟
+
+915
+01:18:40,100 --> 01:18:44,840
+هيساوي سفر الان as N goes to infinity يعني ال E to
+
+916
+01:18:44,840 --> 01:18:49,720
+the minus N بتروح للسفر من وين؟من اليمين، مظبوط أو
+
+917
+01:18:49,720 --> 01:18:53,060
+تروح للـ Zero، مظبوط لأن at E minus N تروح لما
+
+918
+01:18:53,060 --> 01:18:57,440
+لنهاية As N goes to infinity E to the minus N تروح
+
+919
+01:18:57,440 --> 01:19:03,460
+لإنفينيتي لأن as X بتروح إلى سالب ما لنهاية ال E
+
+920
+01:19:03,460 --> 01:19:07,560
+to the minus X برضه هتروح إلى وين؟ إلى السفر يعني
+
+921
+01:19:07,560 --> 01:19:11,220
+ال E to the X نفسها لما ناخد ال limit لما X تروح
+
+922
+01:19:11,220 --> 01:19:14,580
+بدل ما لنهاية تروح إلى مين؟ سالب ما لنهاية برضه
+
+923
+01:19:14,580 --> 01:19:20,200
+هتساوي إيش؟ السفرإذا صار عندى limit E ضو X لما X
+
+924
+01:19:20,200 --> 01:19:23,560
+تروح السلمة لها يابسة أو سفر و بعتمد على اللى هو
+
+925
+01:19:23,560 --> 01:19:26,780
+الجهتين على ال strictly increasing تبع ال E و
+
+926
+01:19:26,780 --> 01:19:30,580
+التفاصيل اللى هى ال definition بتظهر عندكم أنا
+
+927
+01:19:30,580 --> 01:19:34,220
+مافصلتاش لإنه هو إيه هذا القائم
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..1a92f380e3d2541bd6eb536cd8e39a74f37bf59e
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_raw.json
@@ -0,0 +1 @@
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اللي هو انه اللي هي مفترضنا انه معلومات موجودة مسبقا ولا تناقض اللي هو انه نثبتها اليوم لإنه اثباتها اليوم لا يعتمد على اللي حكيناه سابقا بما يخص بأمثلة اللي ذكرت فيها ال exponential الان ال exponential function بدنا نثبت وجودها ايش اللي بنقوله نشوف عبر ال theorem 8 3 1 theorem", "tokens": [995, 5016, 8315, 44713, 9778, 40448, 8315, 11296, 12174, 36150, 39896, 4117, 3714, 7435, 2288, 3215, 5551, 2304, 12984, 37977, 45030, 25708, 995, 18871, 13672, 1829, 31439, 16472, 3224, 13672, 1829, 39896, 3714, 5172, 2655, 43042, 8315, 16472, 3224, 20449, 1211, 20498, 9307, 3714, 29245, 23328, 3660, 3714, 35457, 4587, 995, 49429, 6055, 8315, 4587, 11242, 13672, 1829, 31439, 16472, 3224, 8717, 12984, 3555, 2655, 11296, 45595, 20498, 5296, 28814, 1863, 3224, 1975, 12984, 3555, 9307, 11296, 45595, 20498, 20193, 7251, 34268, 2304, 3215, 15844, 13672, 1829, 11331, 4117, 1829, 8315, 3224, 8608, 16758, 4587, 995, 4724, 15042, 7251, 9778, 9381, 4724, 10721, 2304, 12984, 37977, 13672, 1829, 29910, 37983, 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{"start": 84.17, "end": 84.77, "word": " سابقا", "probability": 0.9915771484375}, {"start": 84.77, "end": 85.07, "word": " بما", "probability": 0.963134765625}, {"start": 85.07, "end": 85.41, "word": " يخص", "probability": 0.9890950520833334}, {"start": 85.41, "end": 86.03, "word": " بأمثلة", "probability": 0.9357421875}, {"start": 86.03, "end": 86.27, "word": " اللي", "probability": 0.978271484375}, {"start": 86.27, "end": 86.71, "word": " ذكرت", "probability": 0.9295247395833334}, {"start": 86.71, "end": 87.07, "word": " فيها", "probability": 0.9912109375}, {"start": 87.07, "end": 87.33, "word": " ال", "probability": 0.96142578125}, {"start": 87.33, "end": 87.75, "word": " exponential", "probability": 0.36181640625}, {"start": 87.75, "end": 89.05, "word": " الان", "probability": 0.738525390625}, {"start": 89.05, "end": 89.35, "word": " ال", "probability": 0.98779296875}, {"start": 89.35, "end": 90.03, "word": " exponential", "probability": 0.93310546875}, {"start": 90.03, "end": 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"probability": 0.5517578125}, {"start": 96.81, "end": 97.25, "word": " 1", "probability": 0.9638671875}, {"start": 97.25, "end": 97.69, "word": " theorem", "probability": 0.63818359375}], "temperature": 1.0}, {"id": 5, "seek": 11326, "start": 98.28, "end": 113.26, "text": "بقول there exists a function U دلّة E من R ل R إذا في عندنا دلّة اسمها E من R ل R such that الـ E prime of X لها بساول EX لكل X element in R", "tokens": [3555, 39648, 456, 8198, 257, 2445, 624, 11778, 1211, 11703, 3660, 462, 9154, 497, 5296, 497, 11933, 15730, 8978, 43242, 8315, 11778, 1211, 11703, 3660, 24525, 2304, 11296, 462, 9154, 497, 5296, 497, 1270, 300, 2423, 39184, 462, 5835, 295, 1783, 5296, 11296, 4724, 3794, 995, 12610, 16385, 5296, 28820, 1783, 4478, 294, 497], "avg_logprob": -0.38238636797124687, "compression_ratio": 1.2466666666666666, "no_speech_prob": 4.172325134277344e-07, "words": [{"start": 98.28, "end": 98.64, "word": "بقول", "probability": 0.71044921875}, {"start": 98.64, "end": 98.82, "word": " 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104.06, "word": " عندنا", "probability": 0.950439453125}, {"start": 104.06, "end": 104.46, "word": " دلّة", "probability": 0.885986328125}, {"start": 104.46, "end": 104.96, "word": " اسمها", "probability": 0.86181640625}, {"start": 104.96, "end": 105.22, "word": " E", "probability": 0.9248046875}, {"start": 105.22, "end": 105.58, "word": " من", "probability": 0.98388671875}, {"start": 105.58, "end": 105.96, "word": " R", "probability": 0.98486328125}, {"start": 105.96, "end": 106.08, "word": " ل", "probability": 0.978515625}, {"start": 106.08, "end": 106.28, "word": " R", "probability": 0.91943359375}, {"start": 106.28, "end": 106.58, "word": " such", "probability": 0.57861328125}, {"start": 106.58, "end": 106.96, "word": " that", "probability": 0.9296875}, {"start": 106.96, "end": 108.0, "word": " الـ", "probability": 0.3525390625}, {"start": 108.0, "end": 108.8, "word": " E", "probability": 0.84716796875}, {"start": 108.8, "end": 109.2, "word": " prime", "probability": 0.449951171875}, {"start": 109.2, "end": 109.4, "word": " of", "probability": 0.93359375}, {"start": 109.4, "end": 109.68, "word": " X", "probability": 0.923828125}, {"start": 109.68, "end": 110.12, "word": " لها", "probability": 0.725341796875}, {"start": 110.12, "end": 110.74, "word": " بساول", "probability": 0.6851806640625}, {"start": 110.74, "end": 111.46, "word": " EX", "probability": 0.54541015625}, {"start": 111.46, "end": 112.08, "word": " لكل", "probability": 0.937255859375}, {"start": 112.08, "end": 112.48, "word": " X", "probability": 0.97021484375}, {"start": 112.48, "end": 112.8, "word": " element", "probability": 0.6630859375}, {"start": 112.8, "end": 113.02, "word": " in", "probability": 0.469970703125}, {"start": 113.02, "end": 113.26, "word": " R", "probability": 0.99853515625}], "temperature": 1.0}, {"id": 6, "seek": 13352, "start": 113.9, "end": 133.52, "text": "إتنين E of Zero بساوي إيش واحد يعني الآن النظرية دي بتقولي إنه يوجد عندنا دالة domainها كل ال R و rangeها بروح بصوب في ال R هذه الدالة تحقق شرطين اللي هي E prime of X بساوي E of X و E of Zero بساوي إيش واحد", "tokens": [28814, 2655, 1863, 9957, 462, 295, 17182, 4724, 3794, 995, 45865, 11933, 1829, 8592, 36764, 24401, 37495, 22653, 6024, 48506, 28239, 19913, 2288, 10632, 11778, 1829, 39894, 39648, 1829, 36145, 3224, 7251, 29245, 3215, 43242, 8315, 11778, 6027, 3660, 9274, 11296, 28242, 2423, 497, 4032, 3613, 11296, 4724, 32887, 5016, 4724, 9381, 37746, 8978, 2423, 497, 29538, 32748, 6027, 3660, 6055, 5016, 4587, 4587, 13412, 2288, 9566, 9957, 13672, 1829, 39896, 462, 5835, 295, 1783, 4724, 3794, 995, 45865, 462, 295, 1783, 4032, 462, 295, 17182, 4724, 3794, 995, 45865, 11933, 1829, 8592, 36764, 24401], "avg_logprob": -0.24527995303894082, "compression_ratio": 1.6331658291457287, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 113.9, "end": 114.5, "word": "إتنين", "probability": 0.593170166015625}, {"start": 114.5, "end": 114.74, "word": " E", "probability": 0.5458984375}, 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"end": 119.72, "word": " عندنا", "probability": 0.905029296875}, {"start": 119.72, "end": 120.24, "word": " دالة", "probability": 0.98681640625}, {"start": 120.24, "end": 120.96, "word": " domainها", "probability": 0.5577392578125}, {"start": 120.96, "end": 121.28, "word": " كل", "probability": 0.98486328125}, {"start": 121.28, "end": 121.48, "word": " ال", "probability": 0.94580078125}, {"start": 121.48, "end": 121.76, "word": " R", "probability": 0.72119140625}, {"start": 121.76, "end": 121.96, "word": " و", "probability": 0.90673828125}, {"start": 121.96, "end": 122.52, "word": " rangeها", "probability": 0.787109375}, {"start": 122.52, "end": 122.92, "word": " بروح", "probability": 0.94921875}, {"start": 122.92, "end": 123.36, "word": " بصوب", "probability": 0.7373860677083334}, {"start": 123.36, "end": 123.48, "word": " في", "probability": 0.91845703125}, {"start": 123.48, "end": 123.62, "word": " ال", "probability": 0.9716796875}, {"start": 123.62, "end": 123.92, "word": " R", "probability": 0.93798828125}, {"start": 123.92, "end": 124.7, "word": " هذه", "probability": 0.34130859375}, {"start": 124.7, "end": 126.14, "word": " الدالة", "probability": 0.9814453125}, {"start": 126.14, "end": 128.7, "word": " تحقق", "probability": 0.89794921875}, {"start": 128.7, "end": 129.38, "word": " شرطين", "probability": 0.9434814453125}, {"start": 129.38, "end": 129.56, "word": " اللي", "probability": 0.7213134765625}, {"start": 129.56, "end": 129.72, "word": " هي", "probability": 0.74462890625}, {"start": 129.72, "end": 129.96, "word": " E", "probability": 0.59423828125}, {"start": 129.96, "end": 130.22, "word": " prime", "probability": 0.5693359375}, {"start": 130.22, "end": 130.36, "word": " of", "probability": 0.9443359375}, {"start": 130.36, "end": 130.56, "word": " X", "probability": 0.96337890625}, {"start": 130.56, "end": 130.84, "word": " بساوي", "probability": 0.938232421875}, {"start": 130.84, "end": 131.02, "word": " E", "probability": 0.986328125}, {"start": 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continuous functions as follows بدي الآن اعرف اللي هو sequence of functions", "tokens": [6027, 7649, 8717, 7435, 1829, 13672, 1829, 31439, 47525, 8315, 8717, 12984, 3555, 2655, 49610, 23328, 29538, 32748, 6027, 3660, 1975, 1829, 3224, 38392, 6055, 5016, 4587, 4587, 13672, 1829, 31439, 33962, 2407, 33546, 13672, 1829, 43242, 8315, 9673, 8848, 4117, 13063, 3660, 8978, 3224, 13672, 1829, 39896, 8717, 19913, 16572, 995, 2423, 7649, 16490, 1211, 9957, 995, 6225, 8592, 7649, 8717, 32887, 5016, 4724, 9307, 7435, 40294, 1975, 12984, 3555, 9307, 49610, 23328, 29538, 32748, 6027, 3660, 16490, 1211, 9957, 995, 8717, 47283, 3215, 321, 31612, 3413, 6964, 257, 8310, 295, 10957, 6828, 382, 10002, 47525, 1829, 6024, 48506, 1975, 3615, 28480, 13672, 1829, 31439, 8310, 295, 6828], "avg_logprob": -0.19133523363958704, "compression_ratio": 1.8675213675213675, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 133.88, "end": 134.5, "word": "الان", "probability": 0.6571044921875}, {"start": 134.5, 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112, "seek": 251430, "start": 2485.68, "end": 2514.3, "text": "sequence strictly increasing sequence بناءً عليه هيكون عند E1 of X أصغر strictly من E2 of X وهذا أصغر strictly من E N of X لكل N أكبر أو يساوي تلاتة وال X أشمالها أكبر من سفر ماشي الحال طيب E1 of X مين هي؟ واحد زاد X", "tokens": [11834, 655, 20792, 5662, 8310, 44945, 16606, 14111, 47356, 39896, 30544, 43242, 462, 16, 295, 1783, 5551, 9381, 17082, 2288, 20792, 9154, 462, 17, 295, 1783, 37037, 15730, 5551, 9381, 17082, 2288, 20792, 9154, 462, 426, 295, 1783, 5296, 28820, 426, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 6055, 1211, 9307, 3660, 16070, 1783, 5551, 8592, 2304, 6027, 11296, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 3714, 33599, 1829, 21542, 6027, 23032, 1829, 3555, 462, 16, 295, 1783, 3714, 9957, 39896, 22807, 36764, 24401, 30767, 18513, 1783], "avg_logprob": -0.22349016719989562, "compression_ratio": 1.5445544554455446, "no_speech_prob": 1.728534698486328e-05, "words": [{"start": 2485.68, "end": 2486.4, 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2538.43, "text": "وهذا أصغر من E2 of X وهذا أصغر من E N of X إذا نخد ال limit لكل الجهات as N goes to infinity وهذا independent of N وهذا independent of N إذا سيصبح أصغر أو يساوي limit E N of X as N goes to infinity", "tokens": [2407, 3224, 15730, 5551, 9381, 17082, 2288, 9154, 462, 17, 295, 1783, 37037, 15730, 5551, 9381, 17082, 2288, 9154, 462, 426, 295, 1783, 11933, 15730, 8717, 9778, 3215, 2423, 4948, 5296, 28820, 25724, 3224, 9307, 382, 426, 1709, 281, 13202, 37037, 15730, 6695, 295, 426, 37037, 15730, 6695, 295, 426, 11933, 15730, 8608, 1829, 9381, 49628, 5551, 9381, 17082, 2288, 34051, 7251, 3794, 995, 45865, 4948, 462, 426, 295, 1783, 382, 426, 1709, 281, 13202], "avg_logprob": -0.2925575661816095, "compression_ratio": 1.6948051948051948, "no_speech_prob": 1.1324882507324219e-06, "words": [{"start": 2515.03, "end": 2515.51, "word": "وهذا", "probability": 0.77880859375}, {"start": 2515.51, "end": 2516.31, "word": " أصغر", "probability": 0.942626953125}, {"start": 2516.31, "end": 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0.82794189453125}, {"start": 2848.2, "end": 2848.64, "word": " النقطة", "probability": 0.91845703125}, {"start": 2848.64, "end": 2849.0, "word": " الأولى", "probability": 0.8946940104166666}, {"start": 2849.0, "end": 2849.12, "word": " اللى", "probability": 0.4920654296875}, {"start": 2849.12, "end": 2849.28, "word": " هى", "probability": 0.8310546875}, {"start": 2849.28, "end": 2849.44, "word": " ال", "probability": 0.54150390625}, {"start": 2849.44, "end": 2849.7, "word": " X", "probability": 0.55029296875}, {"start": 2849.7, "end": 2850.28, "word": " X", "probability": 0.8515625}, {"start": 2850.28, "end": 2850.76, "word": " و", "probability": 0.69677734375}, {"start": 2850.76, "end": 2851.06, "word": " X", "probability": 0.734375}, {"start": 2851.06, "end": 2851.4, "word": " naught", "probability": 0.1597900390625}, {"start": 2851.4, "end": 2851.8, "word": " 0", "probability": 0.3076171875}, {"start": 2851.8, "end": 2852.22, "word": " يعني", "probability": 0.75390625}, {"start": 2852.22, "end": 2852.76, "word": " بنطبقها", "probability": 0.85517578125}, {"start": 2852.76, "end": 2852.94, "word": " على", "probability": 0.6630859375}, {"start": 2852.94, "end": 2853.3, "word": " مين", "probability": 0.888671875}, {"start": 2853.3, "end": 2853.9, "word": " اللى", "probability": 0.87646484375}, {"start": 2853.9, "end": 2854.26, "word": " هى", "probability": 0.96630859375}, {"start": 2854.26, "end": 2854.58, "word": " X", "probability": 0.90185546875}, {"start": 2854.58, "end": 2854.82, "word": " X", "probability": 0.650390625}, {"start": 2854.82, "end": 2855.04, "word": " اللى", "probability": 0.890625}, {"start": 2855.04, "end": 2855.3, "word": " هى", "probability": 0.93310546875}, {"start": 2855.3, "end": 2855.64, "word": " طبعا", "probability": 0.8924560546875}, {"start": 2855.64, "end": 2855.78, "word": " انتوا", "probability": 0.5472005208333334}, {"start": 2855.78, "end": 2856.48, "word": " متذكرين", "probability": 0.9085693359375}, {"start": 2856.48, "end": 2857.26, "word": " F", "probability": 0.36328125}, {"start": 2857.26, "end": 2857.48, "word": " of", "probability": 0.7353515625}, {"start": 2857.48, "end": 2857.92, "word": " X", "probability": 0.99609375}], "temperature": 1.0}, {"id": 129, "seek": 288061, "start": 2858.89, "end": 2880.61, "text": "أف of X تيلر سيوريم there exists C in اللي هي ال interval مثلا X not و X such that F of X بيساوي اللي هو F of X not زائد F prime of X not على واحد factorial في X minus X not زائد زائد F", "tokens": [10721, 5172, 295, 1783, 6055, 26895, 2288, 8608, 1829, 13063, 32640, 456, 8198, 383, 294, 13672, 1829, 39896, 2423, 15035, 50113, 15040, 1783, 406, 4032, 1783, 1270, 300, 479, 295, 1783, 4724, 1829, 3794, 995, 45865, 13672, 1829, 31439, 479, 295, 1783, 406, 30767, 16373, 3215, 479, 5835, 295, 1783, 406, 15844, 36764, 24401, 36916, 8978, 1783, 3175, 1783, 406, 30767, 16373, 3215, 30767, 16373, 3215, 479], "avg_logprob": -0.41613050331087675, "compression_ratio": 1.463855421686747, 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2863.89, "word": " ال", "probability": 0.92529296875}, {"start": 2863.89, "end": 2864.29, "word": " interval", "probability": 0.509765625}, {"start": 2864.29, "end": 2864.73, "word": " مثلا", "probability": 0.931884765625}, {"start": 2864.73, "end": 2865.01, "word": " X", "probability": 0.84326171875}, {"start": 2865.01, "end": 2865.37, "word": " not", "probability": 0.12457275390625}, {"start": 2865.37, "end": 2865.99, "word": " و", "probability": 0.77734375}, {"start": 2865.99, "end": 2866.37, "word": " X", "probability": 0.76708984375}, {"start": 2866.37, "end": 2867.19, "word": " such", "probability": 0.806640625}, {"start": 2867.19, "end": 2867.57, "word": " that", "probability": 0.9541015625}, {"start": 2867.57, "end": 2867.85, "word": " F", "probability": 0.52001953125}, {"start": 2867.85, "end": 2868.03, "word": " of", "probability": 0.8701171875}, {"start": 2868.03, "end": 2868.29, "word": " X", "probability": 0.9912109375}, {"start": 2868.29, "end": 2868.91, "word": " بيساوي", "probability": 0.7962890625}, {"start": 2868.91, "end": 2869.45, "word": " اللي", "probability": 0.92822265625}, {"start": 2869.45, "end": 2869.73, "word": " هو", "probability": 0.9765625}, {"start": 2869.73, "end": 2870.21, "word": " F", "probability": 0.857421875}, {"start": 2870.21, "end": 2870.93, "word": " of", "probability": 0.94677734375}, {"start": 2870.93, "end": 2871.19, "word": " X", "probability": 0.9892578125}, {"start": 2871.19, "end": 2871.63, "word": " not", "probability": 0.91796875}, {"start": 2871.63, "end": 2872.45, "word": " زائد", "probability": 0.7755533854166666}, {"start": 2872.45, "end": 2872.69, "word": " F", "probability": 0.95263671875}, {"start": 2872.69, "end": 2873.23, "word": " prime", "probability": 0.68505859375}, {"start": 2873.23, "end": 2873.53, "word": " of", "probability": 0.97705078125}, {"start": 2873.53, "end": 2873.77, "word": " X", "probability": 0.99072265625}, {"start": 2873.77, "end": 2874.27, "word": " not", "probability": 0.9541015625}, {"start": 2874.27, "end": 2874.93, "word": " على", "probability": 0.70361328125}, {"start": 2874.93, "end": 2875.89, "word": " واحد", "probability": 0.812255859375}, {"start": 2875.89, "end": 2876.45, "word": " factorial", "probability": 0.236328125}, {"start": 2876.45, "end": 2877.13, "word": " في", "probability": 0.94140625}, {"start": 2877.13, "end": 2877.41, "word": " X", "probability": 0.95361328125}, {"start": 2877.41, "end": 2877.75, "word": " minus", "probability": 0.9765625}, {"start": 2877.75, "end": 2878.05, "word": " X", "probability": 0.9951171875}, {"start": 2878.05, "end": 2878.49, "word": " not", "probability": 0.958984375}, {"start": 2878.49, "end": 2879.41, "word": " زائد", "probability": 0.9661458333333334}, {"start": 2879.41, "end": 2880.21, "word": " زائد", "probability": 0.96142578125}, {"start": 2880.21, "end": 2880.61, "word": " F", "probability": 0.9814453125}], "temperature": 1.0}, {"id": 130, "seek": 290867, "start": 2880.93, "end": 2908.67, "text": "n of x0 على n factorial في x minus x0 الكلوس n زائد ال remainder اللي هو fn زائد واحد of اللي هي ال c اللي لاجيناها there exists c اللي هي of c طبعا ال c هنا هتعتمد على ال n اللي احنا اختارناها واشتغلنا عليها هو مسميها cn على اللي هو n زائد واحد factorial في x", "tokens": [77, 295, 2031, 15, 15844, 297, 36916, 8978, 2031, 3175, 2031, 15, 2423, 28820, 2407, 3794, 297, 30767, 16373, 3215, 2423, 29837, 13672, 1829, 31439, 283, 77, 30767, 16373, 3215, 36764, 24401, 295, 13672, 1829, 39896, 2423, 269, 13672, 1829, 20193, 7435, 1829, 8315, 11296, 456, 8198, 269, 13672, 1829, 39896, 295, 269, 23032, 3555, 3615, 995, 2423, 269, 34105, 8032, 2655, 34268, 2304, 3215, 15844, 2423, 297, 13672, 1829, 1975, 5016, 8315, 1975, 46456, 9640, 8315, 11296, 4032, 33599, 2655, 17082, 1211, 8315, 25894, 11296, 31439, 3714, 38251, 1829, 11296, 269, 77, 15844, 13672, 1829, 31439, 297, 30767, 16373, 3215, 36764, 24401, 36916, 8978, 2031], "avg_logprob": -0.23817172479406695, "compression_ratio": 1.8942307692307692, "no_speech_prob": 3.403425216674805e-05, "words": [{"start": 2880.93, "end": 2881.43, "word": "n", "probability": 0.1082763671875}, {"start": 2881.43, "end": 2882.13, "word": " of", "probability": 0.634765625}, {"start": 2882.13, "end": 2882.91, "word": " x0", "probability": 0.47906494140625}, {"start": 2882.91, "end": 2883.51, "word": " على", "probability": 0.391845703125}, {"start": 2883.51, "end": 2883.75, "word": " n", "probability": 0.841796875}, {"start": 2883.75, "end": 2884.35, "word": " factorial", "probability": 0.251708984375}, {"start": 2884.35, "end": 2884.67, "word": " في", "probability": 0.74609375}, {"start": 2884.67, "end": 2884.91, "word": " x", "probability": 0.91015625}, {"start": 2884.91, "end": 2885.27, "word": " minus", "probability": 0.8447265625}, {"start": 2885.27, "end": 2885.91, "word": " x0", "probability": 0.960205078125}, {"start": 2885.91, "end": 2886.73, "word": " الكلوس", "probability": 0.57720947265625}, {"start": 2886.73, "end": 2887.01, "word": " n", "probability": 0.6630859375}, {"start": 2887.01, "end": 2887.77, "word": " زائد", "probability": 0.8720703125}, {"start": 2887.77, "end": 2887.93, "word": " ال", "probability": 0.83642578125}, {"start": 2887.93, "end": 2888.55, "word": " remainder", "probability": 0.31689453125}, {"start": 2888.55, "end": 2889.25, "word": " اللي", "probability": 0.74853515625}, {"start": 2889.25, "end": 2889.39, "word": " هو", "probability": 0.97509765625}, {"start": 2889.39, "end": 2889.89, "word": " fn", "probability": 0.642333984375}, {"start": 2889.89, "end": 2890.37, "word": " زائد", "probability": 0.9453125}, {"start": 2890.37, "end": 2890.89, "word": " واحد", "probability": 0.841796875}, {"start": 2890.89, "end": 2891.61, "word": " of", "probability": 0.9697265625}, {"start": 2891.61, "end": 2891.95, "word": " اللي", "probability": 0.889404296875}, {"start": 2891.95, "end": 2892.09, "word": " هي", "probability": 0.6875}, {"start": 2892.09, "end": 2892.31, "word": " ال", "probability": 0.658203125}, {"start": 2892.31, "end": 2892.43, "word": " c", "probability": 0.32373046875}, {"start": 2892.43, "end": 2892.61, "word": " اللي", "probability": 0.982666015625}, {"start": 2892.61, "end": 2893.29, "word": " لاجيناها", "probability": 0.77626953125}, {"start": 2893.29, "end": 2893.83, "word": " there", "probability": 0.68017578125}, {"start": 2893.83, "end": 2894.15, "word": " exists", "probability": 0.7734375}, {"start": 2894.15, "end": 2894.67, "word": " c", "probability": 0.85986328125}, {"start": 2894.67, "end": 2895.49, "word": " اللي", "probability": 0.96142578125}, {"start": 2895.49, "end": 2895.73, "word": " هي", "probability": 0.87353515625}, {"start": 2895.73, "end": 2896.01, "word": " of", "probability": 0.98046875}, {"start": 2896.01, "end": 2896.49, "word": " c", "probability": 0.89501953125}, {"start": 2896.49, "end": 2897.23, "word": " طبعا", "probability": 0.9158935546875}, {"start": 2897.23, "end": 2897.31, "word": " ال", "probability": 0.95263671875}, {"start": 2897.31, "end": 2897.43, "word": " c", "probability": 0.76513671875}, {"start": 2897.43, "end": 2897.61, "word": " هنا", "probability": 0.421875}, {"start": 2897.61, "end": 2898.13, "word": " هتعتمد", "probability": 0.781787109375}, {"start": 2898.13, "end": 2898.27, "word": " على", "probability": 0.8349609375}, {"start": 2898.27, "end": 2898.39, "word": " ال", "probability": 0.95703125}, {"start": 2898.39, "end": 2898.53, "word": " n", "probability": 0.91259765625}, {"start": 2898.53, "end": 2898.77, "word": " اللي", "probability": 0.98583984375}, {"start": 2898.77, "end": 2899.17, "word": " احنا", "probability": 0.9267578125}, {"start": 2899.17, "end": 2900.11, "word": " اختارناها", "probability": 0.878515625}, {"start": 2900.11, "end": 2900.61, "word": " واشتغلنا", "probability": 0.9521484375}, {"start": 2900.61, "end": 2901.07, "word": " عليها", "probability": 0.993896484375}, {"start": 2901.07, "end": 2901.55, "word": " هو", "probability": 0.62060546875}, {"start": 2901.55, "end": 2901.97, "word": " مسميها", "probability": 0.871826171875}, {"start": 2901.97, "end": 2902.45, "word": " cn", "probability": 0.808349609375}, {"start": 2902.45, "end": 2904.23, "word": " على", "probability": 0.7822265625}, {"start": 2904.23, "end": 2904.59, "word": " اللي", "probability": 0.989013671875}, {"start": 2904.59, "end": 2904.85, "word": " هو", "probability": 0.982421875}, {"start": 2904.85, "end": 2905.55, "word": " n", "probability": 0.857421875}, {"start": 2905.55, "end": 2906.25, "word": " زائد", "probability": 0.9645182291666666}, {"start": 2906.25, "end": 2906.69, "word": " واحد", "probability": 0.9765625}, {"start": 2906.69, "end": 2907.27, "word": " factorial", "probability": 0.68798828125}, {"start": 2907.27, "end": 2908.21, "word": " في", "probability": 0.9833984375}, {"start": 2908.21, "end": 2908.67, "word": " x", "probability": 0.96533203125}], "temperature": 1.0}, {"id": 131, "seek": 293692, "start": 2913.18, "end": 2936.92, "text": "الـ N هو ما خدها لعند مين بدل ما هو remainder N زائد واحد نسميه remainder N نفس الشيء مافيش مشكلة احنا كملناها يعني بدل ما اشتغل على N زي ما اشتغل على N زائد واحد اشتغل على ال N هو اما هي ال Taylor's theorem طيب المهم انتبهوا عن دي", "tokens": [6027, 39184, 426, 31439, 19446, 16490, 3215, 11296, 5296, 3615, 41260, 3714, 9957, 47525, 1211, 19446, 31439, 29837, 426, 30767, 16373, 3215, 36764, 24401, 8717, 38251, 1829, 3224, 29837, 426, 8717, 36178, 25124, 1829, 38207, 19446, 41185, 8592, 37893, 28820, 3660, 1975, 5016, 8315, 9122, 42213, 8315, 11296, 37495, 22653, 47525, 1211, 19446, 1975, 8592, 2655, 17082, 1211, 15844, 426, 30767, 1829, 19446, 1975, 8592, 2655, 17082, 1211, 15844, 426, 30767, 16373, 3215, 36764, 24401, 1975, 8592, 2655, 17082, 1211, 15844, 2423, 426, 31439, 1975, 15042, 39896, 2423, 12060, 311, 20904, 23032, 1829, 3555, 9673, 16095, 16472, 2655, 3555, 3224, 14407, 18871, 11778, 1829], "avg_logprob": -0.2776785827818371, "compression_ratio": 1.776190476190476, "no_speech_prob": 1.430511474609375e-06, "words": [{"start": 2913.1800000000003, "end": 2913.82, "word": "الـ", "probability": 0.17578125}, {"start": 2913.82, "end": 2913.98, "word": " N", "probability": 0.346435546875}, {"start": 2913.98, "end": 2917.92, "word": " هو", "probability": 0.392578125}, {"start": 2917.92, "end": 2918.1, "word": " ما", "probability": 0.7236328125}, {"start": 2918.1, "end": 2918.58, "word": " خدها", "probability": 0.7674153645833334}, {"start": 2918.58, "end": 2919.26, "word": " لعند", "probability": 0.87646484375}, {"start": 2919.26, "end": 2919.62, "word": " مين", "probability": 0.934814453125}, {"start": 2919.62, "end": 2920.38, "word": " بدل", "probability": 0.94580078125}, {"start": 2920.38, "end": 2920.46, "word": " ما", "probability": 0.9306640625}, {"start": 2920.46, "end": 2920.82, "word": " هو", "probability": 0.94970703125}, {"start": 2920.82, "end": 2921.46, "word": " remainder", "probability": 0.42724609375}, {"start": 2921.46, "end": 2922.0, "word": " N", "probability": 0.461181640625}, {"start": 2922.0, "end": 2922.32, "word": " زائد", "probability": 0.5600179036458334}, {"start": 2922.32, "end": 2922.58, "word": " واحد", "probability": 0.89794921875}, {"start": 2922.58, "end": 2923.06, "word": " نسميه", "probability": 0.8895263671875}, {"start": 2923.06, "end": 2923.38, "word": " remainder", "probability": 0.92529296875}, {"start": 2923.38, "end": 2923.7, "word": " N", "probability": 0.81201171875}, {"start": 2923.7, "end": 2924.22, "word": " نفس", "probability": 0.921142578125}, {"start": 2924.22, "end": 2924.68, "word": " الشيء", "probability": 0.767333984375}, {"start": 2924.68, "end": 2924.98, "word": " مافيش", "probability": 0.5638020833333334}, {"start": 2924.98, "end": 2925.46, "word": " مشكلة", "probability": 0.98095703125}, {"start": 2925.46, "end": 2925.88, "word": " احنا", "probability": 0.6276041666666666}, {"start": 2925.88, "end": 2926.94, "word": " كملناها", "probability": 0.8466796875}, {"start": 2926.94, "end": 2927.82, "word": " يعني", "probability": 0.857421875}, {"start": 2927.82, "end": 2928.6, "word": " بدل", "probability": 0.994384765625}, {"start": 2928.6, "end": 2928.68, "word": " ما", "probability": 0.76220703125}, {"start": 2928.68, "end": 2929.56, "word": " اشتغل", "probability": 0.91357421875}, {"start": 2929.56, "end": 2929.74, "word": " على", "probability": 0.8232421875}, {"start": 2929.74, "end": 2930.02, "word": " N", "probability": 0.91357421875}, {"start": 2930.02, "end": 2930.22, "word": " زي", "probability": 0.6436767578125}, {"start": 2930.22, "end": 2930.32, "word": " ما", "probability": 0.8916015625}, {"start": 2930.32, "end": 2930.76, "word": " اشتغل", "probability": 0.96328125}, {"start": 2930.76, "end": 2931.76, "word": " على", "probability": 0.64404296875}, {"start": 2931.76, "end": 2931.88, "word": " N", "probability": 0.75830078125}, {"start": 2931.88, "end": 2932.12, "word": " زائد", "probability": 0.8180338541666666}, {"start": 2932.12, "end": 2932.24, 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{"start": 3443.0, "end": 3443.48, "word": " by", "probability": 0.623046875}, {"start": 3443.48, "end": 3444.08, "word": " contradiction", "probability": 0.88818359375}], "temperature": 1.0}, {"id": 155, "seek": 346121, "start": 3444.81, "end": 3461.21, "text": "بدي افترض انه في عندي E of α for some α element in R والـ E of α ايش تساوي تساوي سفر ونوصل لcontradiction لان suppose that there exists α element in R such that E of α ايش بساوي بساوي سفر", "tokens": [3555, 16254, 1975, 5172, 2655, 43042, 16472, 3224, 8978, 43242, 1829, 462, 295, 5691, 337, 512, 5691, 4478, 294, 497, 16070, 39184, 462, 295, 5691, 1975, 1829, 8592, 6055, 3794, 995, 45865, 6055, 3794, 995, 45865, 8608, 5172, 2288, 4032, 1863, 2407, 36520, 5296, 9000, 6206, 4105, 5296, 7649, 7297, 300, 456, 8198, 5691, 4478, 294, 497, 1270, 300, 462, 295, 5691, 1975, 1829, 8592, 4724, 3794, 995, 45865, 4724, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.3331980457553616, "compression_ratio": 1.5582822085889572, 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"probability": 0.9130859375}, {"start": 3449.91, "end": 3450.13, "word": " in", "probability": 0.93603515625}, {"start": 3450.13, "end": 3450.39, "word": " R", "probability": 0.990234375}, {"start": 3450.39, "end": 3450.75, "word": " والـ", "probability": 0.4222412109375}, {"start": 3450.75, "end": 3450.85, "word": " E", "probability": 0.73583984375}, {"start": 3450.85, "end": 3450.97, "word": " of", "probability": 0.90087890625}, {"start": 3450.97, "end": 3451.19, "word": " α", "probability": 0.7587890625}, {"start": 3451.19, "end": 3451.39, "word": " ايش", "probability": 0.67724609375}, {"start": 3451.39, "end": 3451.93, "word": " تساوي", "probability": 0.94677734375}, {"start": 3451.93, "end": 3452.61, "word": " تساوي", "probability": 0.83349609375}, {"start": 3452.61, "end": 3452.87, "word": " سفر", "probability": 0.7195638020833334}, {"start": 3452.87, "end": 3453.21, "word": " ونوصل", "probability": 0.8011474609375}, {"start": 3453.21, "end": 3454.07, "word": " لcontradiction", "probability": 0.78802490234375}, {"start": 3454.07, "end": 3454.73, "word": " لان", "probability": 0.4833984375}, {"start": 3454.73, "end": 3455.15, "word": " suppose", "probability": 0.84423828125}, {"start": 3455.15, "end": 3455.87, "word": " that", "probability": 0.9228515625}, {"start": 3455.87, "end": 3456.21, "word": " there", "probability": 0.912109375}, {"start": 3456.21, "end": 3456.73, "word": " exists", "probability": 0.48583984375}, {"start": 3456.73, "end": 3457.07, "word": " α", "probability": 0.5263671875}, {"start": 3457.07, "end": 3457.47, "word": " element", "probability": 0.9423828125}, {"start": 3457.47, "end": 3457.77, "word": " in", "probability": 0.95166015625}, {"start": 3457.77, "end": 3458.03, "word": " R", "probability": 0.98974609375}, {"start": 3458.03, "end": 3458.37, "word": " such", "probability": 0.94921875}, {"start": 3458.37, "end": 3458.77, "word": " that", "probability": 0.9521484375}, {"start": 3458.77, "end": 3458.95, "word": " E", "probability": 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3464.75, "end": 3465.37, "word": " interval", "probability": 0.94677734375}, {"start": 3465.37, "end": 3465.65, "word": " with", "probability": 0.85205078125}, {"start": 3465.65, "end": 3466.33, "word": " endpoints", "probability": 0.70751953125}, {"start": 3466.33, "end": 3466.79, "word": " mean", "probability": 0.42626953125}, {"start": 3466.79, "end": 3467.33, "word": " α", "probability": 0.5439453125}, {"start": 3467.33, "end": 3467.99, "word": " و", "probability": 0.53173828125}, {"start": 3467.99, "end": 3468.33, "word": " 0", "probability": 0.4638671875}, {"start": 3468.33, "end": 3469.05, "word": " يعني", "probability": 0.77978515625}, {"start": 3469.05, "end": 3469.23, "word": " الـ", "probability": 0.6767578125}, {"start": 3469.23, "end": 3469.35, "word": " G", "probability": 0.64501953125}, {"start": 3469.35, "end": 3469.49, "word": " of", "probability": 0.10418701171875}, {"start": 3469.49, "end": 3469.71, "word": " α", "probability": 0.56689453125}, {"start": 3469.71, 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"word": " زيره", "probability": 0.5272623697916666}, {"start": 3474.23, "end": 3474.55, "word": " حسب", "probability": 0.970458984375}, {"start": 3474.55, "end": 3474.95, "word": " قيمة", "probability": 0.994140625}, {"start": 3474.95, "end": 3475.23, "word": " الألف", "probability": 0.815185546875}, {"start": 3475.23, "end": 3475.75, "word": " موجبة", "probability": 0.8274739583333334}, {"start": 3475.75, "end": 3476.03, "word": " أو", "probability": 0.96826171875}, {"start": 3476.03, "end": 3476.45, "word": " سالبة", "probability": 0.97021484375}, {"start": 3476.45, "end": 3477.41, "word": " طيب", "probability": 0.9137369791666666}], "temperature": 1.0}, {"id": 157, "seek": 349244, "start": 3478.44, "end": 3492.44, "text": "الان زي ما قلت قبل بشوية بما ان E is continuous on a closed interval Iα او Jα اللي هو مسميها then there exist case such that اللي هو ال absolute value E of D أصفر سوى K لكل T element in G", "tokens": [6027, 7649, 30767, 1829, 19446, 12174, 1211, 2655, 12174, 36150, 4724, 8592, 2407, 10632, 4724, 15042, 16472, 462, 307, 10957, 322, 257, 5395, 15035, 286, 1529, 1975, 2407, 508, 1529, 13672, 1829, 31439, 3714, 38251, 1829, 11296, 550, 456, 2514, 1389, 1270, 300, 13672, 1829, 31439, 2423, 8236, 2158, 462, 295, 413, 5551, 9381, 5172, 2288, 8608, 2407, 7578, 591, 5296, 28820, 314, 4478, 294, 460], "avg_logprob": -0.3003731414453307, "compression_ratio": 1.3172043010752688, "no_speech_prob": 0.0, "words": [{"start": 3478.44, "end": 3478.92, "word": "الان", "probability": 0.722900390625}, {"start": 3478.92, "end": 3479.14, "word": " زي", "probability": 0.6126708984375}, {"start": 3479.14, "end": 3479.2, "word": " ما", "probability": 0.93408203125}, {"start": 3479.2, "end": 3479.44, "word": " قلت", "probability": 0.8175455729166666}, {"start": 3479.44, "end": 3479.68, "word": " قبل", "probability": 0.956787109375}, {"start": 3479.68, "end": 3480.16, "word": " بشوية", "probability": 0.8553466796875}, {"start": 3480.16, "end": 3480.52, "word": " 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بيصير اي أس ام عالميا على ان و هذه بيكون عندي اي أس اي او ام على ان بساوي اللي هو عبارة عن اي أس ام على ان و ام على ان was our arbitral rational number اذا صار عندي اي أس ار بساوي اي أس ار", "tokens": [3224, 24192, 13412, 2288, 3615, 9307, 5296, 28814, 1829, 4032, 36764, 24401, 3660, 2423, 7649, 28242, 24525, 1975, 2304, 37037, 15730, 6225, 3215, 3215, 6225, 995, 16254, 2423, 7649, 37037, 15730, 6225, 3215, 3215, 6225, 18513, 10632, 4724, 1829, 9381, 13546, 1975, 1829, 5551, 3794, 1975, 2304, 6225, 45340, 25528, 15844, 16472, 4032, 29538, 4724, 1829, 30544, 43242, 1829, 1975, 1829, 5551, 3794, 1975, 1829, 1975, 2407, 1975, 2304, 15844, 16472, 4724, 3794, 995, 45865, 13672, 1829, 31439, 6225, 3555, 9640, 3660, 18871, 1975, 1829, 5551, 3794, 1975, 2304, 15844, 16472, 4032, 1975, 2304, 15844, 16472, 390, 527, 14931, 2155, 15090, 1230, 1975, 15730, 20328, 9640, 43242, 1829, 1975, 1829, 5551, 3794, 1975, 2288, 4724, 3794, 995, 45865, 1975, 1829, 5551, 3794, 1975, 2288], "avg_logprob": 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" لأن", "probability": 0.578857421875}, {"start": 4293.85, "end": 4294.21, "word": " كتابة", "probability": 0.6968994140625}, {"start": 4294.21, "end": 4294.33, "word": " على", "probability": 0.80859375}, {"start": 4294.33, "end": 4294.61, "word": " سورة", "probability": 0.6950276692708334}, {"start": 4294.61, "end": 4294.95, "word": " function", "probability": 0.69287109375}, {"start": 4294.95, "end": 4295.29, "word": " is", "probability": 0.826171875}, {"start": 4295.29, "end": 4295.53, "word": " on", "probability": 0.65087890625}, {"start": 4295.53, "end": 4295.69, "word": " to", "probability": 0.28125}, {"start": 4295.69, "end": 4297.63, "word": " طيب", "probability": 0.857421875}, {"start": 4297.63, "end": 4298.19, "word": " مش", "probability": 0.63623046875}, {"start": 4298.19, "end": 4298.57, "word": " هيك", "probability": 0.965087890625}, {"start": 4298.57, "end": 4299.31, "word": " limit", "probability": 0.9033203125}, {"start": 4299.31, "end": 4299.53, "word": " E", 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"probability": 0.93212890625}, {"start": 4303.01, "end": 4303.21, "word": " E", "probability": 0.958984375}, {"start": 4303.21, "end": 4303.31, "word": " of", "probability": 0.96728515625}, {"start": 4303.31, "end": 4303.47, "word": " X", "probability": 0.994140625}, {"start": 4303.47, "end": 4303.69, "word": " لما", "probability": 0.977783203125}, {"start": 4303.69, "end": 4303.85, "word": " X", "probability": 0.98876953125}, {"start": 4303.85, "end": 4304.13, "word": " تروح", "probability": 0.9918619791666666}, {"start": 4304.13, "end": 4304.49, "word": " لمالة", "probability": 0.84033203125}, {"start": 4304.49, "end": 4304.75, "word": " نهاية", "probability": 0.8494466145833334}, {"start": 4304.75, "end": 4304.87, "word": " ايش", "probability": 0.7002766927083334}, {"start": 4304.87, "end": 4305.95, "word": " بتساوي؟", "probability": 0.838330078125}, {"start": 4305.95, "end": 4306.61, "word": " بتساوي", "probability": 0.94580078125}, {"start": 4306.61, "end": 4307.11, "word": " مالة", "probability": 0.9803059895833334}, {"start": 4307.11, "end": 4307.51, "word": " نهاية", "probability": 0.96875}], "temperature": 1.0}, {"id": 190, "seek": 433481, "start": 4308.37, "end": 4334.81, "text": "يعني الدالة هذه في حالة ال ال E of X اللي احنا سميناها ال X exponential لل X هذه الدالة هذه الدالة زي ما انتوا عارفين لما X تروح إلى سالب ماله نهاية هذا هيروح للسفر ال E of X هتروح للسفر و لما ال X تروح لماله نهاية ال E of X هتروح إيش إلى ماله نهاية نشوف أول شي بالنسبة لمن؟ لل range", "tokens": [40228, 22653, 32748, 6027, 3660, 29538, 8978, 11331, 6027, 3660, 2423, 2423, 462, 295, 1783, 13672, 1829, 1975, 5016, 8315, 8608, 2304, 1829, 8315, 11296, 2423, 1783, 21510, 24976, 1783, 29538, 32748, 6027, 3660, 29538, 32748, 6027, 3660, 30767, 1829, 19446, 16472, 2655, 14407, 6225, 9640, 5172, 9957, 5296, 15042, 1783, 6055, 32887, 5016, 30731, 8608, 6027, 3555, 3714, 6027, 3224, 8717, 11296, 10632, 23758, 39896, 32887, 5016, 24976, 3794, 5172, 2288, 2423, 462, 295, 1783, 8032, 2655, 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4350.57, "end": 4350.89, "word": " element", "probability": 0.90380859375}, {"start": 4350.89, "end": 4351.17, "word": " in", "probability": 0.958984375}, {"start": 4351.17, "end": 4351.41, "word": " R", "probability": 0.99853515625}, {"start": 4351.41, "end": 4352.19, "word": " صار", "probability": 0.961669921875}, {"start": 4352.19, "end": 4352.57, "word": " عندى", "probability": 0.763427734375}, {"start": 4352.57, "end": 4353.05, "word": " الان", "probability": 0.6820068359375}, {"start": 4353.05, "end": 4354.13, "word": " هاي", "probability": 0.5015869140625}, {"start": 4354.13, "end": 4354.75, "word": " دالة", "probability": 0.9690755208333334}, {"start": 4354.75, "end": 4356.17, "word": " E", "probability": 0.96044921875}, {"start": 4356.17, "end": 4356.39, "word": " of", "probability": 0.9736328125}, {"start": 4356.39, "end": 4356.71, "word": " 0", "probability": 0.716796875}, {"start": 4356.71, "end": 4356.95, "word": " إلها", "probability": 0.69189453125}, {"start": 4356.95, "end": 4358.65, "word": " واحد،ماشي؟", "probability": 0.7939453125}], "temperature": 1.0}, {"id": 192, "seek": 436797, "start": 4360.36, "end": 4367.98, "text": "أو E of X مش سفر يعني إيه شمالها لا تقطع محور السينات إطلاقا", "tokens": [10721, 2407, 462, 295, 1783, 37893, 8608, 5172, 2288, 37495, 22653, 11933, 1829, 3224, 13412, 2304, 6027, 11296, 20193, 6055, 47432, 3615, 3714, 5016, 13063, 21136, 1829, 8315, 2655, 11933, 9566, 15040, 4587, 995], "avg_logprob": -0.22377232313156128, "compression_ratio": 1.10752688172043, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 4360.36, "end": 4360.72, "word": "أو", "probability": 0.6383056640625}, {"start": 4360.72, "end": 4360.92, "word": " E", "probability": 0.88037109375}, {"start": 4360.92, "end": 4361.1, "word": " of", "probability": 0.525390625}, {"start": 4361.1, "end": 4361.52, "word": " X", "probability": 0.9931640625}, {"start": 4361.52, "end": 4362.12, "word": " مش", "probability": 0.79296875}, {"start": 4362.12, "end": 4362.84, "word": " سفر", "probability": 0.8170572916666666}, {"start": 4362.84, "end": 4363.74, "word": " يعني", "probability": 0.888427734375}, {"start": 4363.74, "end": 4363.92, "word": " إيه", "probability": 0.7306315104166666}, {"start": 4363.92, "end": 4364.5, "word": " شمالها", "probability": 0.8397216796875}, {"start": 4364.5, "end": 4364.98, "word": " لا", "probability": 0.3349609375}, {"start": 4364.98, "end": 4365.58, "word": " تقطع", "probability": 0.99267578125}, {"start": 4365.58, "end": 4366.0, "word": " محور", "probability": 0.9842122395833334}, {"start": 4366.0, "end": 4366.84, "word": " السينات", "probability": 0.8851318359375}, {"start": 4366.84, "end": 4367.98, "word": " إطلاقا", "probability": 0.87880859375}], "temperature": 1.0}, {"id": 193, "seek": 439071, "start": 4368.77, "end": 4390.71, "text": "اللي E of X نفسها و 0 من سفرها الأن أنا بقول مستحيل تكون في إيها قيم سالبة يعني مستحيل نلاقي E of X naught اللي هي أصغر من 0 ليش؟ لأنه لو لجينا E of X naught سالبة لو هالجيت عندنا X naught و هيها قيمتها E of X naught اللي هو سالبة يعني هتكون تحت", "tokens": [6027, 20292, 462, 295, 1783, 8717, 36178, 11296, 4032, 1958, 9154, 8608, 5172, 2288, 11296, 16247, 1863, 41850, 4724, 39648, 3714, 14851, 5016, 26895, 6055, 30544, 8978, 11933, 1829, 11296, 12174, 32640, 8608, 6027, 49401, 37495, 22653, 3714, 14851, 5016, 26895, 8717, 15040, 38436, 462, 295, 1783, 13138, 13672, 1829, 39896, 5551, 9381, 17082, 2288, 9154, 1958, 32239, 8592, 22807, 5296, 33456, 3224, 45164, 5296, 7435, 1829, 8315, 462, 295, 1783, 13138, 8608, 6027, 49401, 45164, 8032, 6027, 7435, 36081, 43242, 8315, 1783, 13138, 4032, 39896, 11296, 12174, 32640, 2655, 11296, 462, 295, 1783, 13138, 13672, 1829, 31439, 8608, 6027, 49401, 37495, 22653, 8032, 2655, 30544, 6055, 33753], "avg_logprob": -0.23795870848751943, "compression_ratio": 1.9356435643564356, "no_speech_prob": 5.125999450683594e-06, "words": [{"start": 4368.77, "end": 4369.07, "word": "اللي", "probability": 0.682373046875}, 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4388.05, "word": " X", "probability": 0.998046875}, {"start": 4388.05, "end": 4388.51, "word": " naught", "probability": 0.96142578125}, {"start": 4388.51, "end": 4389.15, "word": " اللي", "probability": 0.992431640625}, {"start": 4389.15, "end": 4389.33, "word": " هو", "probability": 0.615234375}, {"start": 4389.33, "end": 4389.71, "word": " سالبة", "probability": 0.9777018229166666}, {"start": 4389.71, "end": 4389.91, "word": " يعني", "probability": 0.952392578125}, {"start": 4389.91, "end": 4390.27, "word": " هتكون", "probability": 0.8204752604166666}, {"start": 4390.27, "end": 4390.71, "word": " تحت", "probability": 0.894287109375}], "temperature": 1.0}, {"id": 194, "seek": 442097, "start": 4391.6, "end": 4420.98, "text": "طب ما هي الدالة متصلة مدام متصلة إذا غصب عنها هتيجي تقطع اللي هو ما هتيجي منها لها هتقطع محور السينات إذا هتصير سفر وهذا Contradiction من أين الكلام هذا؟ By Bolzano Intermediate Value Theorem بما أنه احنا في عندنا X0 فرضناها ال E of X0 أصغر من سفر وفي عندي اللي هي E of Zero بساوي واحد اللي هي", "tokens": [9566, 3555, 19446, 39896, 32748, 6027, 3660, 44650, 36520, 3660, 3714, 3215, 10943, 44650, 36520, 3660, 11933, 15730, 32771, 9381, 3555, 18871, 11296, 8032, 31371, 7435, 1829, 6055, 47432, 3615, 13672, 1829, 31439, 19446, 8032, 31371, 7435, 1829, 9154, 11296, 5296, 11296, 8032, 2655, 47432, 3615, 3714, 5016, 13063, 21136, 1829, 8315, 2655, 11933, 15730, 8032, 2655, 9381, 13546, 8608, 5172, 2288, 37037, 15730, 4839, 6206, 4105, 9154, 5551, 9957, 2423, 28820, 10943, 23758, 22807, 3146, 14331, 89, 3730, 5751, 3130, 473, 39352, 440, 37956, 4724, 15042, 14739, 3224, 1975, 5016, 8315, 8978, 43242, 8315, 1783, 15, 6156, 43042, 8315, 11296, 2423, 462, 295, 1783, 15, 5551, 9381, 17082, 2288, 9154, 8608, 5172, 2288, 4032, 41185, 18871, 16254, 13672, 1829, 39896, 462, 295, 17182, 4724, 3794, 995, 45865, 36764, 24401, 13672, 1829, 39896], "avg_logprob": -0.23810633716743385, "compression_ratio": 1.7169117647058822, "no_speech_prob": 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4709.34, "end": 4709.58, "word": " limit", "probability": 0.97216796875}, {"start": 4709.58, "end": 4710.14, "word": " لهذه", "probability": 0.990234375}, {"start": 4710.14, "end": 4710.94, "word": " as", "probability": 0.93408203125}, {"start": 4710.94, "end": 4711.1, "word": " N", "probability": 0.9814453125}, {"start": 4711.1, "end": 4711.34, "word": " goes", "probability": 0.95556640625}, {"start": 4711.34, "end": 4711.48, "word": " to", "probability": 0.97021484375}, {"start": 4711.48, "end": 4711.94, "word": " infinity", "probability": 0.875}, {"start": 4711.94, "end": 4713.42, "word": " الان", "probability": 0.725830078125}], "temperature": 1.0}, {"id": 207, "seek": 472738, "start": 4715.3, "end": 4727.38, "text": "هذه اش مالها بيساوي سفر وهذه اوتوماتيك هيساوي ايش؟ هيساوي سفر الان as N goes to infinity يعني ال E to the minus N بتروح للسفر من وين؟", "tokens": [3224, 24192, 1975, 8592, 3714, 6027, 11296, 4724, 1829, 3794, 995, 45865, 8608, 5172, 2288, 37037, 24192, 1975, 35473, 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0.7228800455729166}, {"start": 4718.9, "end": 4719.4, "word": " هيساوي", "probability": 0.82427978515625}, {"start": 4719.4, "end": 4720.1, "word": " ايش؟", "probability": 0.545013427734375}, {"start": 4720.1, "end": 4720.62, "word": " هيساوي", "probability": 0.9117431640625}, {"start": 4720.62, "end": 4721.1, "word": " سفر", "probability": 0.9763997395833334}, {"start": 4721.1, "end": 4722.32, "word": " الان", "probability": 0.7042236328125}, {"start": 4722.32, "end": 4722.76, "word": " as", "probability": 0.6025390625}, {"start": 4722.76, "end": 4723.0, "word": " N", "probability": 0.348876953125}, {"start": 4723.0, "end": 4723.36, "word": " goes", "probability": 0.884765625}, {"start": 4723.36, "end": 4723.56, "word": " to", "probability": 0.958984375}, {"start": 4723.56, "end": 4724.02, "word": " infinity", "probability": 0.8466796875}, {"start": 4724.02, "end": 4724.34, "word": " يعني", "probability": 0.897216796875}, {"start": 4724.34, "end": 4724.52, "word": " ال", 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0.9881184895833334}, {"start": 4750.02, "end": 4750.16, "word": " ال", "probability": 0.9521484375}, {"start": 4750.16, "end": 4750.42, "word": " limit", "probability": 0.93017578125}, {"start": 4750.42, "end": 4750.7, "word": " لما", "probability": 0.892822265625}, {"start": 4750.7, "end": 4750.92, "word": " X", "probability": 0.97314453125}, {"start": 4750.92, "end": 4751.22, "word": " تروح", "probability": 0.9864908854166666}, {"start": 4751.22, "end": 4751.56, "word": " بدل", "probability": 0.9150390625}, {"start": 4751.56, "end": 4751.68, "word": " ما", "probability": 0.953125}, {"start": 4751.68, "end": 4752.1, "word": " لنهاية", "probability": 0.921630859375}, {"start": 4752.1, "end": 4752.4, "word": " تروح", "probability": 0.947265625}, {"start": 4752.4, "end": 4752.54, "word": " إلى", "probability": 0.97265625}, {"start": 4752.54, "end": 4753.34, "word": " مين؟", "probability": 0.8841145833333334}, {"start": 4753.34, "end": 4753.72, "word": " سالب", "probability": 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a01d5dd629de031a23b14cbea0e12552fb084a37
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2PcqkFMSAOE_raw.srt
@@ -0,0 +1,3764 @@
+1
+00:00:04,910 --> 00:00:08,190
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:08,190 --> 00:00:12,270
+والصلاة والسلام على سيد المرسلين سيدنا محمد على
+
+3
+00:00:12,270 --> 00:00:18,470
+آله وصحبه أجمعين هذه هي المحاضرة رقم 22 في مساق
+
+4
+00:00:18,470 --> 00:00:23,930
+تحليل حقيقة نيل لطلاب وطالبات الجامعة الإسلاميةقسم
+
+5
+00:00:23,930 --> 00:00:29,530
+الرياضيات في كلية العلوم عنوان المحاضرة اليوم
+
+6
+00:00:29,530 --> 00:00:32,850
+هنكمل chapter تمانية هيكون في عندي اللي هو
+
+7
+00:00:32,850 --> 00:00:37,130
+applications على اللي هي تمانية اللي هي واحد و
+
+8
+00:00:37,130 --> 00:00:41,630
+تمانية اتنين المحاضرة اليوم اللي هي هنحكي عن the
+
+9
+00:00:41,630 --> 00:00:46,730
+exponential and the logarithmic functions هنحكي عن
+
+10
+00:00:46,730 --> 00:00:51,830
+اللي هو دالة ال E to the X ودالة ال Lin أو دالة ال
+
+11
+00:00:51,830 --> 00:00:57,660
+logالان هنثبت اللي هو من خلال .. في البداية هنحكي
+
+12
+00:00:57,660 --> 00:01:03,280
+عن اللي هو ال exponential function أو هنثبت اللي
+
+13
+00:01:03,280 --> 00:01:05,740
+هو وجود ال exponential function
+
+14
+00:01:09,490 --> 00:01:13,630
+احنا استخدمناها قبل هيك مجرد أمثلة بعيدا عن اللي
+
+15
+00:01:13,630 --> 00:01:18,730
+هو انه اللي هي مفترضنا انه معلومات موجودة مسبقا
+
+16
+00:01:18,730 --> 00:01:22,170
+ولا تناقض اللي هو انه نثبتها اليوم لإنه اثباتها
+
+17
+00:01:22,170 --> 00:01:25,410
+اليوم لا يعتمد على اللي حكيناه سابقا بما يخص
+
+18
+00:01:25,410 --> 00:01:29,350
+بأمثلة اللي ذكرت فيها ال exponential الان ال
+
+19
+00:01:29,350 --> 00:01:34,310
+exponential function بدنا نثبت وجودها ايش اللي
+
+20
+00:01:34,310 --> 00:01:38,820
+بنقوله نشوف عبر ال theorem 8 3 1 theoremبقول there
+
+21
+00:01:38,820 --> 00:01:44,060
+exists a function U دلّة E من R ل R إذا في عندنا
+
+22
+00:01:44,060 --> 00:01:49,680
+دلّة اسمها E من R ل R such that الـ E prime of X
+
+23
+00:01:49,680 --> 00:01:55,280
+لها بساول EX لكل X element in Rإتنين E of Zero
+
+24
+00:01:55,280 --> 00:01:58,820
+بساوي إيش واحد يعني الآن النظرية دي بتقولي إنه
+
+25
+00:01:58,820 --> 00:02:02,920
+يوجد عندنا دالة domainها كل ال R و rangeها بروح
+
+26
+00:02:02,920 --> 00:02:10,220
+بصوب في ال R هذه الدالة تحقق شرطين اللي هي E prime
+
+27
+00:02:10,220 --> 00:02:14,500
+of X بساوي E of X و E of Zero بساوي إيش واحدالان
+
+28
+00:02:14,500 --> 00:02:19,920
+نجي اللي هو بدنا نثبت وجود هذه الدالة ايه التي
+
+29
+00:02:19,920 --> 00:02:25,380
+تحقق اللي هو الخواص اللي عندنا المذكورة فيه اللي
+
+30
+00:02:25,380 --> 00:02:29,960
+هي نظريا الان خلينا عشان نروح باتجاه اثبات وجود
+
+31
+00:02:29,960 --> 00:02:34,080
+هذه الدالة خلينا ناخد we inductively define a
+
+32
+00:02:34,080 --> 00:02:37,900
+sequence of continuous functions as follows بدي
+
+33
+00:02:37,900 --> 00:02:43,590
+الآن اعرف اللي هو sequence of functionsالأولى
+
+34
+00:02:43,590 --> 00:02:48,790
+اسمها E1 of X بتساوي واحد زائد X طبعا هذه الدالة
+
+35
+00:02:48,790 --> 00:02:52,790
+موجودة E1 of X بتساوي واحد زائد X هي دالة خطية
+
+36
+00:02:52,790 --> 00:02:58,590
+الآن بدي أعرف ال E2 ال E2 of X بساوي ال
+
+37
+00:02:58,590 --> 00:03:04,170
+integration واحد زائد ال integration من سفر لعند X
+
+38
+00:03:04,170 --> 00:03:13,140
+اللي هو E1 of X D أو E1 of T DTإذاً الـ E2 بدي
+
+39
+00:03:13,140 --> 00:03:17,960
+أجيبها من مين؟ من الـ E1 طبعاً بتيجي الـ E والـ E2
+
+40
+00:03:17,960 --> 00:03:23,500
+اللي هي عبارة عن واحد زائد هذه واحد زائد T اللي هو
+
+41
+00:03:23,500 --> 00:03:27,730
+تفاضلها زي ما انتوا عارفين تكملها اللي هي Tزائد
+
+42
+00:03:27,730 --> 00:03:32,910
+اللي هي T تربيع على اتنين من صفر لعند X ويسوا واحد
+
+43
+00:03:32,910 --> 00:03:38,490
+زائد X زائد X على X تربيع على اتنين هذه الدالة من
+
+44
+00:03:38,490 --> 00:03:45,020
+هي E2 of X E3 of Xبتعرفها بنفس الأسلوب بالساوي 1
+
+45
+00:03:45,020 --> 00:03:49,840
+زائد الـ integration من 0 ل X لدالة اللي وجدتها
+
+46
+00:03:49,840 --> 00:03:56,140
+قبلها اللي هي E2 of T DT بضل أسائي ساير في التعريف
+
+47
+00:03:56,140 --> 00:04:03,760
+بقول in general بتعرف ال E N زائد 1 of Xبساوة الـ
+
+48
+00:04:03,760 --> 00:04:11,420
+1 زي ال integration من 0 ل X EN of T DT إذن الآن
+
+49
+00:04:11,420 --> 00:04:16,020
+عرفت اللي هي ال E1 of X بساوة 1 زي X ومنها عرفت
+
+50
+00:04:16,020 --> 00:04:21,840
+اللي هي E2 وE2 عرفت منها E3 وE3 عرفت منها E4 وهكذا
+
+51
+00:04:21,840 --> 00:04:26,120
+ال EN زي ال 1 of X هتساوة 1 زي ال integration من 0
+
+52
+00:04:26,120 --> 00:04:31,440
+ل X لل EN اللي جاب الها ديof D D T لكل N element
+
+53
+00:04:31,440 --> 00:04:36,720
+in N و لكل X element in M in R نيجي الآن نطلع على
+
+54
+00:04:36,720 --> 00:04:40,920
+الملاحظات اللي بدنا نحكيها عشان نستخدمها الان 1
+
+55
+00:04:40,920 --> 00:04:46,840
+زائد X دالة متصلة مش متصلة أصلا بس هي أصلا قابلة
+
+56
+00:04:46,840 --> 00:04:51,900
+للتفاضل أيضا اللي هي differentiable الان بناء عليه
+
+57
+00:04:51,900 --> 00:05:00,140
+مدام ال E1 is continuous هتكون اللي هي E2E2 هي
+
+58
+00:05:00,140 --> 00:05:03,940
+الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 الـ E1
+
+59
+00:05:03,940 --> 00:05:04,180
+هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 الـ
+
+60
+00:05:04,180 --> 00:05:07,840
+E1 هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1
+
+61
+00:05:07,840 --> 00:05:11,080
+الـ E1 هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ
+
+62
+00:05:11,080 --> 00:05:12,820
+E1 الـ E1 هي الانتجراش للـ E1 الـ E1 هي الانتجراش
+
+63
+00:05:12,820 --> 00:05:14,980
+للـ E1 الـ E1 هي الانتجراش للـ E1 الـ E1 هي
+
+64
+00:05:14,980 --> 00:05:18,980
+الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 الـ E1
+
+65
+00:05:18,980 --> 00:05:22,820
+هي الانتجراش للـ E1 الـ E1 هي الانتجراش للـ E1 ال
+
+66
+00:05:25,750 --> 00:05:27,270
+الآن F.E.L
+
+67
+00:05:31,050 --> 00:05:34,950
+in on R then it is integrable over any bounded
+
+68
+00:05:34,950 --> 00:05:39,590
+interval زي ما قلنا مدام هذا continuous اللي هي E
+
+69
+00:05:39,590 --> 00:05:45,210
+اللي هي 2 continuous ومن هنا هتطلع E 3 continuous
+
+70
+00:05:45,210 --> 00:05:49,530
+وE 4 continuous إذا صارت هذي دايما continuous وده
+
+71
+00:05:49,530 --> 00:05:53,170
+continuous وده integration exist و by fundamental
+
+72
+00:05:53,170 --> 00:05:57,250
+theorem هذا ال integration كله على بعض can be
+
+73
+00:05:57,250 --> 00:06:01,160
+differentiatedوهتكون الهو ايش is differentiable
+
+74
+00:06:01,160 --> 00:06:05,520
+ومنه so EN زائد واحد is well defined by the above
+
+75
+00:06:05,520 --> 00:06:09,300
+formula moreover زي ما قلت it is from fundamental
+
+76
+00:06:09,300 --> 00:06:12,400
+theorem of calculus ال second form اللي هي سبعة
+
+77
+00:06:12,400 --> 00:06:15,920
+تلاتة خمسة هيكون عند ال EN زائد واحد is
+
+78
+00:06:15,920 --> 00:06:19,280
+differentiable ومش هيك وتفاضل هذه زي ما احنا
+
+79
+00:06:19,280 --> 00:06:24,540
+عارفين بساوي بنشيل ال integration طبعا التفاضل بلغ
+
+80
+00:06:24,540 --> 00:06:29,220
+ال integrationبصير en of x فبصير عندي ال en زائد
+
+81
+00:06:29,220 --> 00:06:34,240
+واحد prime of x موجودة ويساوي en x for all n
+
+82
+00:06:34,240 --> 00:06:37,720
+element in N إذن الآن عملنا sequence ال sequence
+
+83
+00:06:37,720 --> 00:06:40,580
+هذه طلعت sequence of differentiable functions و ال
+
+84
+00:06:40,580 --> 00:06:43,860
+derivative لل en زائد واحد prime هي بترجع لمين
+
+85
+00:06:43,860 --> 00:06:50,320
+بتيجي اللي هي ال en of xطيب الان هذا كله و ده
+
+86
+00:06:50,320 --> 00:06:53,900
+خليني اسميها تلاتة و خليني نحضر حالنا نصل للي بدنا
+
+87
+00:06:53,900 --> 00:06:59,500
+ياجولكم وين هنصل في الآخر هوصلكم انه اللي هو ال
+
+88
+00:06:59,500 --> 00:07:05,400
+limit لهذه ال sequence او لهذه ال sequence هي ال E
+
+89
+00:07:05,400 --> 00:07:12,300
+of X اللي انا بثبت وجودهاوستكون اللي هي بتحقق
+
+90
+00:07:12,300 --> 00:07:16,440
+الشروط اللي حكيناها خلّينا نشوف ما لاستعجلش نشوف
+
+91
+00:07:16,440 --> 00:07:19,900
+إيش اللي بدنا نصلله إذا اللي عملناه كوّننا
+
+92
+00:07:19,900 --> 00:07:23,380
+sequence sequence زي ما قلنا اللي هي ال sequence
+
+93
+00:07:23,380 --> 00:07:28,160
+بدأت عشان يبقى لكم الذاكرين E1 of X بساوي واحد
+
+94
+00:07:28,160 --> 00:07:34,760
+زائد X E and زائد واحد of X بساوي الانتجرأت واحد
+
+95
+00:07:35,450 --> 00:07:39,790
+بسوء واحدة زي الـ integration من صفر لعدد X E N of
+
+96
+00:07:39,790 --> 00:07:47,450
+T DT خلّين هذولة أمامنا طبعاً N element in N عندي
+
+97
+00:07:47,450 --> 00:07:54,630
+و X أي element وين in Rطيب إذا الان سار عندي اللي
+
+98
+00:07:54,630 --> 00:07:58,430
+هو ال .. ده اللي هذي is differential لان بقوللي
+
+99
+00:07:58,430 --> 00:08:03,230
+انه ابن الدعي ويمكن لو حد شاف قبل بشوية ما وصلنا ل
+
+100
+00:08:03,230 --> 00:08:08,450
+E2 E2 كانت عبارة عن واحد زي دكستر بيه على اتنين
+
+101
+00:08:08,450 --> 00:08:13,070
+اللي هو اللي هي اتنين factorialلو كملنا هيكون E N
+
+102
+00:08:13,070 --> 00:08:16,910
+of X بتساوي 1 زياد X على 1 factorial X تربيه على 2
+
+103
+00:08:16,910 --> 00:08:21,170
+factorial زياد X وصف N على 100 على N factorial لكل
+
+104
+00:08:21,170 --> 00:08:25,750
+X element in R هذه الـ N صحيحة اللي هي لكل N
+
+105
+00:08:25,750 --> 00:08:31,470
+element in R اللي هو طبعا اثباتها سهل نثبتها by
+
+106
+00:08:31,470 --> 00:08:36,010
+induction خلينا نشوف كيفنا نثبت أربعة by induction
+
+107
+00:08:58,380 --> 00:09:01,020
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+108
+00:09:01,020 --> 00:09:04,640
+أربعة
+
+109
+00:09:04,640 --> 00:09:06,540
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+110
+00:09:06,540 --> 00:09:06,720
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+111
+00:09:06,720 --> 00:09:06,900
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+112
+00:09:06,900 --> 00:09:06,900
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+113
+00:09:06,900 --> 00:09:06,900
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+114
+00:09:06,900 --> 00:09:06,900
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+115
+00:09:06,900 --> 00:09:06,900
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أربعة
+
+116
+00:09:06,900 --> 00:09:08,610
+أربعة أربعة أربعة أربعة أربعة أربعة أربعة أfour is
+
+117
+00:09:08,610 --> 00:09:16,450
+a true four and بتساوي كتابة يعني صار عند الالالـ
+
+118
+00:09:16,450 --> 00:09:23,550
+E K of X بتساوي 1 زائد X على 1 factorial زائد X
+
+119
+00:09:23,550 --> 00:09:29,170
+تربيع على 2 factorial زائد X أُس K على K factorial
+
+120
+00:09:29,170 --> 00:09:33,890
+هذا لما نفرض إن اللي هي .. اللي هي الأربعة is true
+
+121
+00:09:33,890 --> 00:09:39,150
+for N بتساوي K بدنا نثبت إن E K زائد 1 هتطلع اللي
+
+122
+00:09:39,150 --> 00:09:43,870
+هي هذهزائد x أس k على k زائد واحد فيكتوريا اللي
+
+123
+00:09:43,870 --> 00:09:48,790
+على ال net بتطلع أربعة صحيحة for k زائد واحد ده
+
+124
+00:09:48,790 --> 00:09:56,070
+نحسب خلّينا نحسب الآن E k زائد واحد of x حسب اللي
+
+125
+00:09:56,070 --> 00:10:01,370
+احنا مفترضينه أو معرفين ال sequence على أساسه E k
+
+126
+00:10:01,370 --> 00:10:04,830
+زائد واحد of x أيش بتساوي واحد زائد ال integration
+
+127
+00:10:04,830 --> 00:10:14,410
+من صفر ل X E k of T DTمظبوط؟ طيب الان بدي اعوض عن
+
+128
+00:10:14,410 --> 00:10:20,030
+E K of T احنا مفترضينها صحيحة ل K اذا E K of X
+
+129
+00:10:20,030 --> 00:10:24,230
+هيها اذا باجي بعوض بيصير Y ساوي واحد زائد ال
+
+130
+00:10:24,230 --> 00:10:29,250
+integration من صفر ل X E K of T اللي هي واحد زائد
+
+131
+00:10:29,250 --> 00:10:33,170
+T زائد T تربيع طبعا واحد فيكتوريال اللي هي واحد
+
+132
+00:10:33,170 --> 00:10:40,080
+على اتنين فيكتوريال زائد لما أصل X أو T أس Kعلى K
+
+133
+00:10:40,080 --> 00:10:46,520
+فكتوريال الكل إشماله DT لأن اللي هي أكيد وضحت
+
+134
+00:10:46,520 --> 00:10:49,660
+الصورة بدل الفاضل نطلع قيمة التفاصيل الكامل و نطلع
+
+135
+00:10:49,660 --> 00:10:54,360
+قيمة التكامل و يساوي واحد هيوزايد ما هو جاعد زاد
+
+136
+00:10:54,360 --> 00:11:00,520
+هذا ال integration اللي هو عبارة عن Tزائد T تربيع
+
+137
+00:11:00,520 --> 00:11:06,140
+على 2 في 1 يعني 2 factorial زائد T تكعيب على 3 في
+
+138
+00:11:06,140 --> 00:11:10,920
+2 factorial يعني 3 factorial زائد لما أصل لآخر 1 T
+
+139
+00:11:10,920 --> 00:11:15,640
+أُس K زائد 1 على K زائد 1 في K factorial هي K زائد
+
+140
+00:11:15,640 --> 00:11:22,230
+1 لكل factorial هذا الكلام من 0 لمين؟لعند X واضحة
+
+141
+00:11:22,230 --> 00:11:26,550
+الصورة ويساوي عبارة عن لما اعوض من 0 ل X بيصير 1
+
+142
+00:11:26,550 --> 00:11:32,090
+زائد X زائد X تربيع على 2 factorial لما أصل لأخر
+
+143
+00:11:32,090 --> 00:11:38,210
+واحد X K زائد 1 على K زائد 1فاكتوريال إذا فعلاً
+
+144
+00:11:38,210 --> 00:11:42,110
+طلعت عندي a k زائد واحد بساوي هذا المقدار يعني
+
+145
+00:11:42,110 --> 00:11:46,450
+بمعنى آخر أربعة طلعت ال true for n بتساوي k زائد
+
+146
+00:11:46,450 --> 00:11:51,830
+واحد إذا من كل هذا ال induction بنكون أثبتنا أن e
+
+147
+00:11:51,830 --> 00:11:56,430
+n of x بساوي هذا الكلام لكل x element in R لكل n
+
+148
+00:11:56,430 --> 00:12:00,630
+element in Nإذاً هذه صورة اللي هي الـEN of X صورة
+
+149
+00:12:00,630 --> 00:12:05,150
+غير الصورة اللي عرفناها فوق استنتجناها منها الآن
+
+150
+00:12:05,150 --> 00:12:11,770
+خلّينا نروح باتجاه إثبات إنه ال limit للـEN هذه أو
+
+151
+00:12:11,770 --> 00:12:17,910
+الـEK أو الـEN أو اللي هي is uniformly convergent
+
+152
+00:12:18,680 --> 00:12:23,900
+to some function هذه الـ some function هي اللي
+
+153
+00:12:23,900 --> 00:12:29,120
+بتدعي انها هتكون ال exponential طيب اللي بنسميها
+
+154
+00:12:29,120 --> 00:12:32,840
+exponential بعد شوية اللي هي بتحقق شرطين اللي
+
+155
+00:12:32,840 --> 00:12:35,400
+حكينا عنهم لسه احنا بنعرف ال exponential احنا
+
+156
+00:12:35,400 --> 00:12:38,340
+بنعرف الدالة هذه لكن لأن معلوماتنا عارفين من اول
+
+157
+00:12:38,340 --> 00:12:43,000
+ال exponentialالان احنا الدالة هذه اللى هنثبت
+
+158
+00:12:43,000 --> 00:12:46,540
+وجودها اليوم هي اللى بعد شوية بعد ما نثبت ال
+
+159
+00:12:46,540 --> 00:12:50,880
+uniqueness لها بنسميها ال exponential زي ما هنشوف
+
+160
+00:12:50,880 --> 00:12:56,620
+بعد شوية الان اذا ال sequence اللى عندى هذه
+
+161
+00:12:56,620 --> 00:13:05,300
+هيوضعها الان قلنا E Nof X هذه المعادلة التانية هي
+
+162
+00:13:05,300 --> 00:13:11,560
+واحد زائد X زائد X تربيع على اتنين factorial زائد
+
+163
+00:13:11,560 --> 00:13:15,600
+X أُس N على N factorial وهذا الشكل طبعا أنتوا مش
+
+164
+00:13:15,600 --> 00:13:20,240
+غريب عليكم بتعرفوه الآن خليني أخد الان let A أكبر
+
+165
+00:13:20,240 --> 00:13:24,120
+من سفر بيه given افترض ان A اني اللي هو real
+
+166
+00:13:24,120 --> 00:13:28,660
+number أكبر من مين من سفر باخده arbitrarily لكن
+
+167
+00:13:28,660 --> 00:13:33,470
+خليني نحكي عن A محددةالان if absolute value of X
+
+168
+00:13:33,470 --> 00:13:37,910
+أصغر أو تساوي A يعني بتحكي الحديث هان على ال Xات
+
+169
+00:13:37,910 --> 00:13:43,590
+اللي في ال R اللي من عند ناقص A لعند مين ال A يعني
+
+170
+00:13:43,590 --> 00:13:48,610
+ال absolute value لهن هذولة أصغر أو تساوي ال A
+
+171
+00:13:48,610 --> 00:13:52,390
+يعني في الفترة المغلقة اللي أمامي اللي بين ناقص ال
+
+172
+00:13:52,390 --> 00:13:58,870
+A و ال A بتتحدث شوف الآن احسبلي Em of X ناقص En of
+
+173
+00:13:58,870 --> 00:14:03,750
+Xو يساوي و بدنا نفترض لكم ان الان ايه شمالها اكبر
+
+174
+00:14:03,750 --> 00:14:07,910
+من الان و الان اكبر من اتنين ايه اتنين ايه مكتوب
+
+175
+00:14:07,910 --> 00:14:10,850
+اتنين ايه لغرض الحسابات اللي جيت منشوفها بنفع
+
+176
+00:14:10,850 --> 00:14:13,270
+تلاتة ايه، بنفع اربعة ايه، بنفع خمسة ايه، بنفع ستة
+
+177
+00:14:13,270 --> 00:14:16,550
+ايه، بنفع كله بنفع هذا الكلام على اساس انك ماينفعش
+
+178
+00:14:16,550 --> 00:14:19,550
+اقول نص ايه او تلت ايه او ربع ايه لأنها مش هتقد
+
+179
+00:14:19,550 --> 00:14:23,910
+الغرض اللي بدي انالأن مين اخترت أنا اخترت اللي هو
+
+180
+00:14:23,910 --> 00:14:29,650
+الأمات والأنات اللي أكبر منين من اتنين في الـA
+
+181
+00:14:29,650 --> 00:14:34,390
+اللي هي قيبة مين الـA اللي هي نص الفترة اللي عندي
+
+182
+00:14:34,390 --> 00:14:39,210
+أو طول نص الفترة طيب، احسبلي ايه M of X ناقص Y of
+
+183
+00:14:39,210 --> 00:14:44,360
+X؟E M of X اللي هي عبارة عن واحد زائد X زائد X
+
+184
+00:14:44,360 --> 00:14:49,360
+تربيع و هجابل في الطريق من ال X M لأنه M أكبر X
+
+185
+00:14:49,360 --> 00:14:52,720
+تربيع على M فكتوريال زائد X M زائد واحد على M زائد
+
+186
+00:14:52,720 --> 00:14:57,280
+واحد فكتوريال لما أصل الأخر واحدة من X أُس M على M
+
+187
+00:14:57,280 --> 00:15:03,400
+فكتوريال يعني الآن هذي هيكون زي هيك لما أصل طبعا X
+
+188
+00:15:03,400 --> 00:15:10,500
+M زائد واحد على Mزائد واحد factorial زائد لمّا أصل
+
+189
+00:15:10,500 --> 00:15:15,760
+لعند X أُس M على M factorial هذه مين هي هذه عبارة
+
+190
+00:15:15,760 --> 00:15:24,220
+عن ال EM يعني ال EM هتساوي اللي هي E of XE N of X
+
+191
+00:15:24,220 --> 00:15:30,020
+زائد المتبقي هذا الان حاصل طرح الاتنتين هيكون
+
+192
+00:15:30,020 --> 00:15:33,480
+عبارة عن لإن المفترض ال M أكبر من L زي ما قلنا
+
+193
+00:15:33,480 --> 00:15:37,700
+حاصل طرح اللي هيكون اللي هو المتبقي هذا X N زائد
+
+194
+00:15:37,700 --> 00:15:40,840
+واحد على N زائد واحد فيكتوريال لما أصل ل X M على M
+
+195
+00:15:40,840 --> 00:15:46,240
+إيش فيكتوريال ماشي الحال إذا الأن أوصلنا لحاصل طرح
+
+196
+00:15:46,240 --> 00:15:51,340
+دولة بساوي هذا المقدر نيجي نكمل الأن عندي
+
+197
+00:15:53,560 --> 00:15:58,460
+عند ال absolute value لل X أصغر من 100 من A
+
+198
+00:16:10,590 --> 00:16:17,130
+زائد absolute value زائد xn زائد اتنين على n زائد
+
+199
+00:16:17,130 --> 00:16:22,710
+اتنين factorial زائد لما أصل لأخر واحد x أسم م على
+
+200
+00:16:22,710 --> 00:16:28,150
+m factorial ماشي الحال طيب الآن ان ان كل x من
+
+201
+00:16:28,150 --> 00:16:31,270
+هدولة ال absolute value هي أصغر يساوي منين ايه إذا
+
+202
+00:16:31,270 --> 00:16:36,650
+صار هذا المقدار أصغر أو يساوي هذا المقدار اللي هنا
+
+203
+00:16:36,650 --> 00:16:45,550
+أصغر أو يساوي Mزائد واحد على N آسف A أُس N زائد
+
+204
+00:16:45,550 --> 00:16:49,970
+واحد لأن ال absolute value X أصغر تساوي A إذا X N
+
+205
+00:16:49,970 --> 00:16:54,510
+زائد واحد أصغر أو ساوي ال A أُس N زائد واحد على N
+
+206
+00:16:54,510 --> 00:17:00,870
+زائد واحد factorial زائد التاني اللي هو A N زائد
+
+207
+00:17:00,870 --> 00:17:07,130
+اتنينعلى n زائد 2 factorial زائد لما أصل لآخر واحد
+
+208
+00:17:07,130 --> 00:17:14,870
+اللي هو عبارة عن a أس m على m factorial ماشي الحال
+
+209
+00:17:14,870 --> 00:17:21,080
+طيبالان خليني اخد من هدولة ال a n زائد واحد على ال
+
+210
+00:17:21,080 --> 00:17:24,480
+n زائد واحد factorial عامل مشترك قبل ما اصلح هذه
+
+211
+00:17:24,480 --> 00:17:28,520
+الخطوة لسه اه خليني اخد عامل مشترك بيصير عندي a n
+
+212
+00:17:28,520 --> 00:17:33,180
+زائد واحد على n زائد واحد factorial فاش اللي بيضل
+
+213
+00:17:33,180 --> 00:17:39,420
+هنا فيه اللي بيضل هنا واحد زائد هنا بيضل a على n
+
+214
+00:17:39,420 --> 00:17:46,280
+زائد اتنين اكيد زائد اللي بعدها a تربيع على nزائد
+
+215
+00:17:46,280 --> 00:17:52,160
+اتنين فان زائد تلاتة زائد لما أصل لآخر واحد مين
+
+216
+00:17:52,160 --> 00:17:57,080
+آخر واحد بشيل منه ان زائد واحد بيصير a أس ام ناقص
+
+217
+00:17:57,080 --> 00:18:03,000
+ان ناقص واحد على اللي بيضل من ان زائد اتنين مضروب
+
+218
+00:18:03,000 --> 00:18:09,250
+لعند ان ام ناقص ان ناقص واحد هذول الأنماشي الحال
+
+219
+00:18:09,250 --> 00:18:13,470
+هذا أخد تمين يا جماعة ماعليش دخل الكلام مع بعضه بس
+
+220
+00:18:13,470 --> 00:18:18,130
+أكيد أنتوا مستوعبين إيش بقول طلعت ال a n زائد واحد
+
+221
+00:18:18,130 --> 00:18:21,090
+على n زائد واحد في كتوريا العام المشترك طلع عندي
+
+222
+00:18:21,090 --> 00:18:26,550
+هذا المقدار الان عندي أكيد ال n زائد اتنين أكبر من
+
+223
+00:18:26,550 --> 00:18:32,050
+مين من ال end فمقلوبه أصغرأه فبصير عندى اللى هو
+
+224
+00:18:32,050 --> 00:18:38,050
+عندى اللى هى هذا المقدار a n على l زائد 2 أصغر أو
+
+225
+00:18:38,050 --> 00:18:42,170
+شاوي a على n و اللى بعده a تربيع أصغر أو شاوي ا ا
+
+226
+00:18:42,170 --> 00:18:44,370
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+227
+00:18:44,370 --> 00:18:44,410
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+228
+00:18:44,410 --> 00:18:45,090
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+229
+00:18:45,090 --> 00:18:46,010
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+230
+00:18:46,010 --> 00:18:47,730
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+231
+00:18:47,730 --> 00:18:47,730
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+232
+00:18:47,730 --> 00:18:48,010
+ا ا ا ا ا ا ا ا ا ا ا ا ا ا
+
+233
+00:18:51,320 --> 00:18:55,380
+أكبر من أنه وهذه أكبر من أنه فمقلوبهم بيصير هذا
+
+234
+00:18:55,380 --> 00:19:00,980
+المقدار أصغر اللي هو أو يساوي المقدار اللي بعده
+
+235
+00:19:00,980 --> 00:19:07,730
+بيصير عندى هذا المقدارأصغر أو يساوي هذا زي ما هو و
+
+236
+00:19:07,730 --> 00:19:13,670
+هذه واحد هذه a على n لأنها هتكون أكبر a تربيع على
+
+237
+00:19:13,670 --> 00:19:19,170
+n تربيع ان تربيع هتكبر لما أصل ل a m minus n نقص
+
+238
+00:19:19,170 --> 00:19:25,420
+واحد على m minus n نقص واحد لأن هنا دولة عددهمن
+
+239
+00:19:25,420 --> 00:19:30,200
+نقص واحد من الانز هيكون ان طبعا استبدالت كل واحدة
+
+240
+00:19:30,200 --> 00:19:34,400
+بان فهيطلع بالشكل هذا وبتظل ال inequality صحيحة
+
+241
+00:19:34,400 --> 00:19:38,540
+خلصنا من هذا طيب دعونا نشوف كيف أضبح لكم هذا دعونا
+
+242
+00:19:38,540 --> 00:19:48,770
+نخلص منه نجي الآن اللي هو نرجعانكمل الحسبة هذا
+
+243
+00:19:48,770 --> 00:19:57,670
+الان المقدار عندى اللى هو المقدار هذا هيه اتطلعوا
+
+244
+00:19:57,670 --> 00:20:06,710
+عندى ال a ال a على n ايش انا ماخد ال n ماخد ال n
+
+245
+00:20:06,710 --> 00:20:12,930
+اكبر من مين من اتنين a يعني ال a اصغر من n على مين
+
+246
+00:20:13,680 --> 00:20:21,900
+على اتنين مظبوط ال A أصغر من N على اتنين واضحة جسم
+
+247
+00:20:21,900 --> 00:20:25,660
+ال N على اتنين و N على اتنين صارت N على اتنين أكبر
+
+248
+00:20:25,660 --> 00:20:29,940
+من مين من ايه بدي استخدمها الآن هذه ال N بصير أصغر
+
+249
+00:20:29,940 --> 00:20:34,020
+أو يساوي هذا المقدار بصير أصغر أو يساوي هذا بحكي
+
+250
+00:20:34,020 --> 00:20:39,590
+عن هذا بصير أصغر أو يساوي اللي هو Aأُس ان زائد
+
+251
+00:20:39,590 --> 00:20:45,410
+واحد على ان زائد واحد الكل فكتوريال فيه اللي هو
+
+252
+00:20:45,410 --> 00:20:56,830
+واحد زائد ا على ان زائد ا تربية على ان تربية زائد
+
+253
+00:20:56,830 --> 00:21:06,530
+ا اللي هو ا تكعيب على ان تكعيبزائد مش بتصل لهذا
+
+254
+00:21:06,530 --> 00:21:11,670
+زائد و بتظل ماشيها يشمالها إلى ما لا نهاية يعني
+
+255
+00:21:11,670 --> 00:21:17,110
+بتضيف عليها a على n أس ام ناقص n a على n أس ام
+
+256
+00:21:17,110 --> 00:21:20,290
+ناقص n زائد واحد وهكذا تظلها إلى ما لا نهاية يعني
+
+257
+00:21:20,290 --> 00:21:23,850
+بتحولها لمين ل infinite series طيب احنا قلنا ال a
+
+258
+00:21:23,850 --> 00:21:28,670
+أصغر من n على 2يعني ال a n على n أصغر من مين؟ من
+
+259
+00:21:28,670 --> 00:21:32,930
+نص عرفته الأن إيش بدأ أسأل صار عندي هذا المقدار
+
+260
+00:21:32,930 --> 00:21:39,310
+أصغر أو يساوي a أس n زائد واحد على n زائد واحد لكل
+
+261
+00:21:39,310 --> 00:21:45,690
+factorial مضروب في مين؟ اللي هو واحد زائد نص زائد
+
+262
+00:21:45,690 --> 00:21:50,670
+نص تربيع أه لأن a على n أصغر من مين يا جماعة؟ من
+
+263
+00:21:50,670 --> 00:21:57,850
+نص زائدتربيع زاد نص تتعيد زاد الآخرين لأن هذه صارت
+
+264
+00:21:57,850 --> 00:22:02,470
+عبارة عن geometric series مجموحها هذه بتساوي مجموح
+
+265
+00:22:02,470 --> 00:22:07,050
+هذه عبارة عن واحد على واحد ناقصه اللي هو نص وهي
+
+266
+00:22:07,050 --> 00:22:12,730
+ساوي جداش اتنينماشي إذا هذا المقدار اللي هو صار
+
+267
+00:22:12,730 --> 00:22:16,510
+عبارة عن اتنين في هذا المقدار اللي هو وصلنا له
+
+268
+00:22:16,510 --> 00:22:20,650
+أصغر أو يساوي a n زائد واحد على n زائد واحد
+
+269
+00:22:20,650 --> 00:22:23,890
+فيكتوريا مضروب في مين؟ في اتنين يعني حسابات هذه
+
+270
+00:22:23,890 --> 00:22:28,770
+طيب إذا صار عندي الآن هاي اللي بدي أصله و اللي
+
+271
+00:22:28,770 --> 00:22:31,630
+كاتبه باختصار هو لإنه بيعتبر ان الباقية حساباتي
+
+272
+00:22:31,630 --> 00:22:36,950
+بتعرفوا تحسبوها em-n of x of x أصغر أو يساوي هذا
+
+273
+00:22:36,950 --> 00:22:43,450
+المقدار على مين؟ على اتنينالان شوف ما يليه الان as
+
+274
+00:22:43,450 --> 00:22:49,590
+n goes to infinity هذا المقدار هذا المقدار as n
+
+275
+00:22:49,590 --> 00:22:54,230
+goes to infinity ال limit بتروح للصفر لأن a n زائد
+
+276
+00:22:54,230 --> 00:22:57,110
+واحد على n زائد واحد كل factorial as n goes to
+
+277
+00:22:57,110 --> 00:23:01,190
+infinity ايش بده يروح للصفر يعني بمعنى اخر مدام
+
+278
+00:23:01,190 --> 00:23:04,430
+بده يروح للصفر طبعا وذا كبرت الان بتكبر مين برضه
+
+279
+00:23:04,430 --> 00:23:07,850
+مباشرة ام لأن ام اكبر منها اذا four
+
+280
+00:23:10,880 --> 00:23:19,800
+for large ن و أم we get
+
+281
+00:23:23,490 --> 00:23:31,050
+E M of X ناقص E N of X اللي هي أصغر أو يساوي من
+
+282
+00:23:31,050 --> 00:23:38,070
+هذه هذه أصلا for large N as N goes to infinity هذا
+
+283
+00:23:38,070 --> 00:23:43,470
+المقدار بيصير أصغر من أي ي في الدنيا اللي هي بيصير
+
+284
+00:23:43,470 --> 00:23:49,330
+أصغر من مين من ي أو أصغر أو يساوي ي as N و M
+
+285
+00:23:49,330 --> 00:23:54,630
+become very very largeماشي الحال لإنه ال limit لل
+
+286
+00:23:54,630 --> 00:23:59,110
+a n على n فكتوره ليش بيساوي بيساوي سفر إذا صار
+
+287
+00:23:59,110 --> 00:24:05,950
+عندي الآن اللي هي E m of X E n of X أصغر وشوى Y
+
+288
+00:24:05,950 --> 00:24:15,250
+لكل X وين موجودة في الفترة A ماقص A أو A ماشي
+
+289
+00:24:15,250 --> 00:24:23,160
+الحال الآن بناء عليهالان لكل X بغض النظر على ال X
+
+290
+00:24:23,160 --> 00:24:28,000
+اللي هو بيكون عندي أنه I'm very large وبيعطيني هذا
+
+291
+00:24:28,000 --> 00:24:32,560
+أصغر وسوء من إبسلون وهذه اللي سميناها Cauchy's
+
+292
+00:24:32,560 --> 00:24:37,470
+criterion for uniformاللي هو convergence في ال ..
+
+293
+00:24:37,470 --> 00:24:43,590
+اللي هو section 81 بناء عليها بطلع عندى E and اللي
+
+294
+00:24:43,590 --> 00:24:47,090
+هو converges uniformly to some function مش
+
+295
+00:24:47,090 --> 00:24:51,410
+عارفينها اللي هي .. اللي هي converges uniformly on
+
+296
+00:24:51,410 --> 00:24:59,770
+mean سالب a أو a باشي الحالة الالالآن عندي it
+
+297
+00:24:59,770 --> 00:25:02,610
+follows the sequence a and a converge uniformly on
+
+298
+00:25:02,610 --> 00:25:07,210
+the interval ناقص a و a where a أكبر من 0 لكن a
+
+299
+00:25:07,210 --> 00:25:11,670
+أكبر من 0 was a شمالها arbitrarily يعني اللي وصلنا
+
+300
+00:25:11,670 --> 00:25:18,110
+له ما ياليا يا جماعة اللي وصلنا له أنهعندي اللي هي
+
+301
+00:25:18,110 --> 00:25:23,210
+الـ sequence هذه لو جيت أخدت for every a element
+
+302
+00:25:23,210 --> 00:25:27,130
+in R و a أكبر من سفر لو وصلنا له اللي كانت a
+
+303
+00:25:27,130 --> 00:25:32,130
+arbitrarily وجدنا أنه هيكون عند الـ EL converges
+
+304
+00:25:32,130 --> 00:25:39,550
+uniformly on ماقص a و مين و a ماشي الحال طيب بناء
+
+305
+00:25:39,550 --> 00:25:48,200
+عليه لو أنت جيت أخدت خد الآن letX element in R، أي
+
+306
+00:25:48,200 --> 00:25:53,540
+X element in R، مدام X في R، إذا أكيد الـ X عشر،
+
+307
+00:25:53,540 --> 00:25:59,250
+عشرين، تلاتين، مليون، ممكن ما تكون،بقدر ألاقي A
+
+308
+00:25:59,250 --> 00:26:05,890
+بحيث أن X تنتمي إلى الفترة من ناقص A لعند مين لعند
+
+309
+00:26:05,890 --> 00:26:11,590
+إيه يعني مثلا ال X لجيتها 100 باخد ال A 110 بصير
+
+310
+00:26:11,590 --> 00:26:17,850
+ال X بين اللي هي ناقص 110 و100 إيش ما تيجي X بلاقي
+
+311
+00:26:17,850 --> 00:26:24,470
+لها Aإذا there exists a such that x element in a
+
+312
+00:26:24,470 --> 00:26:31,410
+minus a أكبر من صفر طبعا minus a ومين أو a أكيد طب
+
+313
+00:26:31,410 --> 00:26:34,390
+ما هو احنا قد أثبتنا انه لكل a element in R
+
+314
+00:26:38,870 --> 00:26:44,730
+Converge uniformly للـ A اللي لاجناها هذه On ناقص
+
+315
+00:26:44,730 --> 00:26:49,690
+A و A لهذه الـ A يعني بمعنى معناه مادام Converge
+
+316
+00:26:49,690 --> 00:26:55,030
+uniformly لإذا E N of Xاللي هو converges point
+
+317
+00:26:55,030 --> 00:26:58,250
+-wise to some function عند مين؟ عند الـ X اللي
+
+318
+00:26:58,250 --> 00:27:03,670
+أخدتها هانا هذه إذا الأن for any X element in R
+
+319
+00:27:03,670 --> 00:27:11,450
+for any X element in R هنلاقي E N of X convergesما
+
+320
+00:27:11,450 --> 00:27:15,250
+دام ال E N of X is convergent for every X element
+
+321
+00:27:15,250 --> 00:27:22,230
+in R إذا صارت عندي limit ال E N of X as N goes to
+
+322
+00:27:22,230 --> 00:27:27,830
+infinity exists for every X element in R ما دام
+
+323
+00:27:27,830 --> 00:27:34,210
+exist حسب اللي قلناها إذا الآن شرعت لتعريف الدالة
+
+324
+00:27:34,210 --> 00:27:40,600
+التالية الآن أنا بقول إنه احنا صار عندنااللي هو
+
+325
+00:27:40,600 --> 00:27:47,760
+مشروع ان نقول we define E من R لR by limit E N of
+
+326
+00:27:47,760 --> 00:27:53,940
+X اللي هي موجودة سميتها إيش اسمها E of X for X
+
+327
+00:27:53,940 --> 00:28:00,760
+element of Rوجود اللي هو دالة اسمها E أنا سميتها E
+
+328
+00:28:00,760 --> 00:28:06,400
+من R لعند R بحيث ان E of X هي limit ال E أنه VIX
+
+329
+00:28:06,400 --> 00:28:11,480
+اللي أثبتت وجوده بدي أثبتلك الأن أن E of X هذه هي
+
+330
+00:28:11,480 --> 00:28:16,120
+الدالة المبتغة اللي بتحقق الشروط اللي طلبها في
+
+331
+00:28:16,120 --> 00:28:21,740
+النظرية نشوف كيف طيب صلوا على النبي عليه الصلاة
+
+332
+00:28:21,740 --> 00:28:22,160
+والسلام
+
+333
+00:28:35,970 --> 00:28:39,830
+الان عندى ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+334
+00:28:39,830 --> 00:28:39,910
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+335
+00:28:39,910 --> 00:28:39,970
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+336
+00:28:39,970 --> 00:28:40,290
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+337
+00:28:40,290 --> 00:28:42,410
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+338
+00:28:42,410 --> 00:28:42,810
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+339
+00:28:42,810 --> 00:28:44,810
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+340
+00:28:44,810 --> 00:28:44,810
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+341
+00:28:44,810 --> 00:28:49,770
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+342
+00:28:49,770 --> 00:28:51,450
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+343
+00:28:51,450 --> 00:28:51,610
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+344
+00:28:51,610 --> 00:28:51,610
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+345
+00:28:51,610 --> 00:28:52,610
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+346
+00:28:52,610 --> 00:28:55,710
+.. ال ..
+
+347
+00:29:00,330 --> 00:29:06,370
+converge such that x element ناقص a أو a كلها هذه
+
+348
+00:29:06,370 --> 00:29:12,330
+الـen converged uniformly الان لمن؟ للـe معايا
+
+349
+00:29:12,330 --> 00:29:19,190
+converged uniformly للـe for every أو on ناقص a أو
+
+350
+00:29:19,190 --> 00:29:23,010
+a لأنه سمناها إيش اللي هي ال limit بتروحلها اسمها
+
+351
+00:29:23,010 --> 00:29:27,670
+إيه الآن en اتفجنا إنه أثبتناها إنها إيش معالها
+
+352
+00:29:28,810 --> 00:29:34,050
+continuous مادام continuous و en اللي هو طبعا
+
+353
+00:29:34,050 --> 00:29:36,330
+continuous مش بس على ناقص a و a continuous وين
+
+354
+00:29:36,330 --> 00:29:39,690
+مكان احنا الآن هنحكي عن ناقص a و a continuous صارت
+
+355
+00:29:39,690 --> 00:29:42,910
+ال en continuous sequence converts uniformly to e
+
+356
+00:29:42,910 --> 00:29:48,110
+على ناقص a و a إذا صارت ال e هذه حسب نظريتنا هتكون
+
+357
+00:29:48,110 --> 00:29:53,560
+continuous نظرية الأولىفي سيكشن 8 اللي هو 2 هتكون
+
+358
+00:29:53,560 --> 00:29:57,800
+الـ E is continuous على ناقص A و A قولنا if E N is
+
+359
+00:29:57,800 --> 00:30:00,200
+a sequence of continuous functions that converge
+
+360
+00:30:00,200 --> 00:30:03,200
+uniformly to some function then this function or
+
+361
+00:30:03,200 --> 00:30:06,880
+some function is continuous on ناقص A و Eإذا صارت
+
+362
+00:30:06,880 --> 00:30:10,200
+ال E continuous على نقص A و A يعني ال E continuous
+
+363
+00:30:10,200 --> 00:30:14,520
+عند من؟ عند ال X و since X was arbitrary in R then
+
+364
+00:30:14,520 --> 00:30:21,180
+E هتكون is continuous على كل ال R إذا صارت عندي E
+
+365
+00:30:21,180 --> 00:30:26,560
+is continuous at any X element in R الآن لو جينا
+
+366
+00:30:26,560 --> 00:30:31,600
+حسبنا ال E N of 0 و E 1 of 0 و E 1 of 0 أشو
+
+367
+00:30:31,600 --> 00:30:39,410
+بالساوية؟ هي واحدE اللي هو N زائد واحد of Zero
+
+368
+00:30:39,410 --> 00:30:45,630
+اللي هو بساوي من سفر لسفر هيطلع سفر وهذا واحد يعني
+
+369
+00:30:45,630 --> 00:30:52,470
+دايما ال E N of Zero هتطلع إيش؟ Zero مدام E N of
+
+370
+00:30:52,470 --> 00:31:00,670
+Zero Zero وبما أنه E is continuous إذن limit ال Eن
+
+371
+00:31:00,670 --> 00:31:05,930
+of zero as n goes to infinity بسوء اللي هو E of
+
+372
+00:31:05,930 --> 00:31:10,990
+zero أو E of X أسف E of X as n goes to infinity
+
+373
+00:31:10,990 --> 00:31:18,450
+بسوء E of X وبما أنه اللي هواللي هي ال ال ال ال ال
+
+374
+00:31:18,450 --> 00:31:20,590
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+375
+00:31:20,590 --> 00:31:21,370
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+376
+00:31:21,370 --> 00:31:22,190
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+377
+00:31:22,190 --> 00:31:25,670
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+378
+00:31:25,670 --> 00:31:30,470
+ال ال ال ال ال ال ال ال ال ال
+
+379
+00:31:30,470 --> 00:31:32,430
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+380
+00:31:32,430 --> 00:31:32,790
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+381
+00:31:32,790 --> 00:31:32,790
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+382
+00:31:32,790 --> 00:31:32,950
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+383
+00:31:32,950 --> 00:31:32,950
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+384
+00:31:32,950 --> 00:31:32,950
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+385
+00:31:32,950 --> 00:31:32,950
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+386
+00:31:32,950 --> 00:31:35,550
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+387
+00:31:35,550 --> 00:31:39,680
+ال الأنا كنت بخلطها أنا بقول E n of 0 بساوي 1
+
+388
+00:31:39,680 --> 00:31:43,540
+limit E n of X بساوي E of X وال E is continuous
+
+389
+00:31:43,540 --> 00:31:48,900
+إذا الحيكون ال E of 0 هذه هتساوي إيش؟ هيساوي 1
+
+390
+00:31:48,900 --> 00:31:53,800
+لأنه limit E n of 0 point twice بساوي E of 0 اللي
+
+391
+00:31:53,800 --> 00:31:58,580
+هي 0الان e n of zero بتساوي واحد for all and infim
+
+392
+00:31:58,580 --> 00:32:04,940
+therefore e of zero بتساوي واحد لإن اللي هي e
+
+393
+00:32:04,940 --> 00:32:11,240
+نفسها is continuous إذا حصلنا على اللي هو الجزء
+
+394
+00:32:11,240 --> 00:32:16,080
+الأول أذكركم في الظاهرية على اللي هو المطلوب أو
+
+395
+00:32:16,080 --> 00:32:20,770
+الخاصية الأولىالخاصية الأولى هى الخاصية الأولى هى
+
+396
+00:32:20,770 --> 00:32:25,430
+عندى أثبتنا .. التانى أسف أثبتنا E اللى هى E of
+
+397
+00:32:25,430 --> 00:32:29,370
+Zero بالساوية واحدة بدى أثبتلك الأن E prime of X
+
+398
+00:32:29,370 --> 00:32:34,750
+هتساوي E of X عشان أثبتها بدنا نذكركم في نظرية
+
+399
+00:32:34,750 --> 00:32:45,750
+أخدناها المرة الماضية اللى هى بتقول ما يليه أذكركم
+
+400
+00:32:45,750 --> 00:32:51,350
+يا جماعةبالنظرية اللي بستخدمها كانت ان عندي لو في
+
+401
+00:32:51,350 --> 00:32:56,330
+عندي fn sequence of functions و هذه fn ال
+
+402
+00:32:56,330 --> 00:33:00,050
+differentiable لأن fn برايم بتروح ل g uniformly
+
+403
+00:33:00,050 --> 00:33:08,330
+ماشي on some interval on some j وكان عندي fn of x
+
+404
+00:33:08,330 --> 00:33:11,050
+not converge
+
+405
+00:33:12,940 --> 00:33:21,240
+converts to f of x not مثلا on j اه for x not
+
+406
+00:33:21,240 --> 00:33:31,300
+element in j then بقول لي اللي هو ال f هذه f prime
+
+407
+00:33:31,300 --> 00:33:39,230
+existsو FN converts uniformly to this F او FN
+
+408
+00:33:39,230 --> 00:33:46,950
+converts uniformly to this F and F' هي مين هي الـ
+
+409
+00:33:46,950 --> 00:33:50,370
+G هذا حكيناه المرة الماضية طيب، بدأ استخدمها الآن
+
+410
+00:33:50,370 --> 00:33:54,510
+شوف عنده، ان شوف عنده الـ N أثبتنا او الـ N
+
+411
+00:33:54,510 --> 00:33:57,110
+interval لاقصية و ايه؟ we have the uniform
+
+412
+00:33:57,110 --> 00:34:02,350
+convergence of the sequence EN اللي هو هتكون EN
+
+413
+00:34:04,720 --> 00:34:12,400
+converge uniformly ماشي الحالة of you of me of
+
+414
+00:34:12,400 --> 00:34:15,900
+تلاتة اللي هو تلاتة اللي حكيناها قبل بشوية خليني
+
+415
+00:34:15,900 --> 00:34:21,540
+بس نذكركم فيها اللي هي تلاتة هي عندي اللي هي تلاتة
+
+416
+00:34:21,840 --> 00:34:25,980
+عندي الان conversion formally to some function E
+
+417
+00:34:25,980 --> 00:34:30,140
+سمناها الان بناء عليه بدلا من الان زائد واحد هي
+
+418
+00:34:30,140 --> 00:34:34,820
+سافسة الان إذا صارت الان زائد واحد prime برضه
+
+419
+00:34:34,820 --> 00:34:38,140
+conversion هادي هادي أصلا conversion formally to
+
+420
+00:34:38,140 --> 00:34:46,280
+مين؟ to ال E اللي عندي الان وضحت الصورة الان صارت
+
+421
+00:34:46,280 --> 00:34:53,410
+الصورة واضحة الان اللي حصلنا انهانتبه عليا، on any
+
+422
+00:34:53,410 --> 00:34:56,010
+interval ناقص a و a, we have the inferred
+
+423
+00:34:56,010 --> 00:35:01,170
+convergence of E n وفي ضوء اللي هو تلاتة, we also
+
+424
+00:35:01,170 --> 00:35:04,290
+have the inferred convergence E n prime of the
+
+425
+00:35:04,290 --> 00:35:08,350
+derivatives الآن من ال theorem 8.2.3 اللي حكيتها
+
+426
+00:35:08,350 --> 00:35:13,700
+هنا، مدام F n prime converges ل Gماشي الحال و ال
+
+427
+00:35:13,700 --> 00:35:17,160
+.. او ال Fn X0 بتروح ل F of X0 for X0 اللي بتنجح
+
+428
+00:35:17,160 --> 00:35:21,880
+اللي هو اكتر من النقطة و اصلا على اساس عرفنا ان Fn
+
+429
+00:35:21,880 --> 00:35:27,420
+of X بالساوي اللي هو مين E of X بتروح لمين لل E of
+
+430
+00:35:27,420 --> 00:35:31,780
+X اذا اكيد هذه متحققة كمان بالنسبة لمين لل E N
+
+431
+00:35:31,780 --> 00:35:36,760
+صارت هذه متحققة و هذه متحققة من التحقق هذههذه يا
+
+432
+00:35:36,760 --> 00:35:42,140
+عزيزي التحقق هذه إذا إذا هيكون عنده ال F N بتروح
+
+433
+00:35:42,140 --> 00:35:46,020
+لل F اللي هي ال E N بتروح لل E صنادر أثبتناها بس
+
+434
+00:35:46,020 --> 00:35:51,720
+إيش المهم أنه ال F prime لهذه اللي بتروحلها هذه
+
+435
+00:35:51,720 --> 00:35:57,080
+اللي هي ال E prime هي مين اللي هي اللي بتروحلها ال
+
+436
+00:35:57,080 --> 00:36:02,340
+E N ال F ال F N prime عند ال E N زائد واحد prime
+
+437
+00:36:02,340 --> 00:36:03,260
+بتروح لل E
+
+438
+00:36:06,460 --> 00:36:13,560
+إذا من نظرية إيان ستذهب إلى الـ E برضه وليس كذلك
+
+439
+00:36:13,560 --> 00:36:20,250
+الـ E' لهذههي هذه يعني بمعنى آخر E N زي دول برايم
+
+440
+00:36:20,250 --> 00:36:25,830
+برايم limitها ال E برايم زي ما هي مين ال E يعني
+
+441
+00:36:25,830 --> 00:36:29,950
+مدام الان limitها هي ال E و هي نفسها limitها من
+
+442
+00:36:29,950 --> 00:36:33,370
+النظرية اللي بنستنتجها E برايم إذا صارت ال E هي
+
+443
+00:36:33,370 --> 00:36:38,610
+مين ال E برايم أو بطريقة أخرى limit ال E N برايم
+
+444
+00:36:38,610 --> 00:36:43,440
+حسب النظرية بتساوي ال E برايمومن جهة أخرى limit en
+
+445
+00:36:43,440 --> 00:36:50,500
+براي هو نفسه limit en نفسهاو limit en اللي هي طبعا
+
+446
+00:36:50,500 --> 00:36:53,280
+انا مسميها n و n نقص واحد احنا مسميينها n و n زاد
+
+447
+00:36:53,280 --> 00:36:57,740
+واحد طبيعي en برايم هي limit من limit en نقص واحد
+
+448
+00:36:57,740 --> 00:37:02,620
+لحالها لأن هذه بتساوي هذه و هذه اصلا limit احنا
+
+449
+00:37:02,620 --> 00:37:07,140
+اثبتنا ايش بساوي ال E اذا صارت عند ال E برايم هي
+
+450
+00:37:07,140 --> 00:37:10,720
+ايش بتساوي ال E يعني هذه ال E برايم استنتاجا من
+
+451
+00:37:10,720 --> 00:37:16,260
+النظريةو هذه اللي هو من اللي أثبتنا في البداية و
+
+452
+00:37:16,260 --> 00:37:19,800
+أيضا نقدر نستخدم .. لسه نتجه من النظرية لكن أحنا
+
+453
+00:37:19,800 --> 00:37:25,540
+أثبتنا فيها قبل ما نستعمل المظلةطيب هذا الكلام
+
+454
+00:37:25,540 --> 00:37:30,680
+صحيح وين لكل x وين موجودة في ناقص a و a لكن a احنا
+
+455
+00:37:30,680 --> 00:37:35,600
+أخدناها أجمالها arbitrary يعني الآن بيصير عندي
+
+456
+00:37:35,600 --> 00:37:40,740
+اللي هو لكل x بلقي لواحدة زي هيك بتحقق الكلام هذا
+
+457
+00:37:40,740 --> 00:37:44,560
+الآن فبصير عندي a prime of x بسوية of x لكل x
+
+458
+00:37:44,560 --> 00:37:50,720
+element in R وهيك بيكون احنا أثبتنا وجود اللي هو
+
+459
+00:37:50,720 --> 00:37:58,390
+ال E of Xأثبتنا وجود دالة أثبتنا
+
+460
+00:37:58,390 --> 00:38:06,100
+وجود دالة سمناها E of Xهذه الدالة بتحقق شرطين اللي
+
+461
+00:38:06,100 --> 00:38:11,900
+هو E prime of X بساوي E of X وبتحقق الشرط التاني E
+
+462
+00:38:11,900 --> 00:38:20,680
+of 0 بساوي 1 السؤال الآن هل في غيرها؟ هل في غيرها
+
+463
+00:38:20,680 --> 00:38:25,860
+ولا لأ؟ طبعا لحظة الحظ أنه لا يوجد دالة غير هذه
+
+464
+00:38:25,860 --> 00:38:31,140
+الدالة اللي بتحقق هذا الكلامبس قبل ما نثبت الـ
+
+465
+00:38:31,140 --> 00:38:35,740
+uniqueness خلّينا ناخد بعض النتائج بشكل سريع و
+
+466
+00:38:35,740 --> 00:38:39,900
+نتائج سهلة و اللي هي ان شاء الله متاخدش وقتها
+
+467
+00:38:39,900 --> 00:38:43,040
+شوفوا يا جماعة صلوا على النبي عليه الصلاة والسلام
+
+468
+00:38:43,040 --> 00:38:48,080
+الـ function E has a derivative of every order and
+
+469
+00:38:48,080 --> 00:38:51,260
+E N of X بيساوي E of X for all N element in N و X
+
+470
+00:38:51,260 --> 00:38:54,480
+element in Rيعني الآن لو فضلناها كمان مرة و مرتين
+
+471
+00:38:54,480 --> 00:38:56,980
+و تلاتة و أربعة هتطلع أيش نفس الدالة طبعا هذا
+
+472
+00:38:56,980 --> 00:39:00,360
+الكلام سهل و by induction زي ما انتوا عارفين أول
+
+473
+00:39:00,360 --> 00:39:03,120
+حاجة ماثبتها فوران بالساوية واحد فوران بالساوية
+
+474
+00:39:03,120 --> 00:39:06,480
+واحد ماحنا أثبتنا E prime of X أيش بالساوية E of X
+
+475
+00:39:06,480 --> 00:39:11,070
+إذا هذه is true فوران بالساوية واحدطيب لان افترض
+
+476
+00:39:11,070 --> 00:39:15,510
+انها هذه صحيحة في القرآن بيساوي K بيصير E K of X
+
+477
+00:39:15,510 --> 00:39:20,010
+بيساوي E of X لما ان اتبتها ل K زائد واحد طيب ل K
+
+478
+00:39:20,010 --> 00:39:23,870
+زائد واحد خد A K زائد واحد of X ايش هذه؟ هذه اللي
+
+479
+00:39:23,870 --> 00:39:30,840
+هي A K of X الكل اشمالها إبراهيمالان a k of x
+
+480
+00:39:30,840 --> 00:39:33,540
+فرضناها ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+481
+00:39:33,540 --> 00:39:33,940
+بالساوية ايه بالساوية ايه بالساوية ايه بالساوية
+
+482
+00:39:33,940 --> 00:39:34,880
+ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+483
+00:39:34,880 --> 00:39:34,880
+بالساوية ايه بالساوية ايه بالساوية ايه بالساوية
+
+484
+00:39:34,880 --> 00:39:35,180
+ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+485
+00:39:35,180 --> 00:39:37,200
+بالساوية ايه بالساوية ايه بالساوية ايه بالساوية
+
+486
+00:39:37,200 --> 00:39:43,180
+ايه بالساوية ايه بالساوية ايه بالساوية ايه
+
+487
+00:39:43,180 --> 00:39:54,500
+بالساوية ايه بالساوية ايه بالسا
+
+488
+00:39:55,640 --> 00:40:01,620
+وضلك فاضل إلى ما لا نهاية حيظل تطلع نفس الدالة نجي
+
+489
+00:40:01,620 --> 00:40:07,340
+الآن لـ Corollary أو الخاصية اللي بعدها لهذه
+
+490
+00:40:07,340 --> 00:40:15,040
+الدالة هذه الدالة Fx أكبر من 0 بقولك دايماً حيكون
+
+491
+00:40:15,040 --> 00:40:21,280
+الواحد زائد X strictly أصغر من مين من E of X من E
+
+492
+00:40:21,280 --> 00:40:32,910
+of X ان انعندي اللي هي أربعة أربعة عشان تشوف أيش
+
+493
+00:40:32,910 --> 00:40:40,550
+أربعة الأربعة اللي هي ال E prime of X أظن هيك كنا
+
+494
+00:40:40,550 --> 00:40:51,590
+حاكين ال E الناس في ال E of X عندي
+
+495
+00:40:51,590 --> 00:40:53,390
+E
+
+496
+00:40:55,380 --> 00:41:03,750
+انا of Xأقبى أو أصغر strictly من E N زائد واحد of
+
+497
+00:41:03,750 --> 00:41:09,850
+X لكل X أكبر من مين من صفر ليه؟ لأن هذه هيزيد
+
+498
+00:41:09,850 --> 00:41:12,630
+عليها term اللي هو X أسوان زائد واحد على N زائد
+
+499
+00:41:12,630 --> 00:41:15,630
+واحد فكتوريا وهذا ال term الاكساتي اللي أكبر من
+
+500
+00:41:15,630 --> 00:41:19,810
+صفر أشماله اللي هو مودة إذا صارت ال sequence اللي
+
+501
+00:41:19,810 --> 00:41:25,010
+عندي يا جماعة هذه عبارة عن strictly increasing
+
+502
+00:41:25,680 --> 00:41:29,120
+sequence strictly increasing sequence بناءً عليه
+
+503
+00:41:29,120 --> 00:41:36,240
+هيكون عند E1 of X أصغر strictly من E2 of X وهذا
+
+504
+00:41:36,240 --> 00:41:42,020
+أصغر strictly من E N of X لكل N أكبر أو يساوي
+
+505
+00:41:42,020 --> 00:41:51,440
+تلاتة وال X أشمالها أكبر من سفر ماشي الحال طيب E1
+
+506
+00:41:51,440 --> 00:42:02,170
+of X مين هي؟ واحد زاد Xوهذا أصغر من E2 of X وهذا
+
+507
+00:42:02,170 --> 00:42:08,170
+أصغر من E N of X إذا نخد ال limit لكل الجهات as N
+
+508
+00:42:08,170 --> 00:42:10,670
+goes to infinity وهذا independent of N وهذا
+
+509
+00:42:10,670 --> 00:42:15,590
+independent of N إذا سيصبح أصغر أو يساوي limit E N
+
+510
+00:42:15,590 --> 00:42:20,430
+of X as N goes to infinityو limit E n of X as n
+
+511
+00:42:20,430 --> 00:42:25,350
+goes to infinity هو عبارة عن مين قلنا عنه E of X
+
+512
+00:42:25,350 --> 00:42:31,350
+إذا صارت 1 زائد X strictly أصغر من E of X أو E of
+
+513
+00:42:31,350 --> 00:42:39,850
+X أكبر strictly من 1 زائد Xطيب نيجي اللي هو نثبت
+
+514
+00:42:39,850 --> 00:42:45,310
+الـ uniqueness للدالة اللي أثبتنا وجودها الآن إذا
+
+515
+00:42:45,310 --> 00:42:51,290
+أثبتنا وجود دالة سميناها E of X هذه الدالة تحقق
+
+516
+00:42:51,290 --> 00:43:01,390
+الشرطين اللي قلناهن اللي هو E of E primeof X بساوي
+
+517
+00:43:01,390 --> 00:43:07,430
+E of X لكل X element in R الشرط الأول والشرط
+
+518
+00:43:07,430 --> 00:43:14,350
+التاني اللي هو E of Zero بتساوي إياش واحد الآن هذه
+
+519
+00:43:14,350 --> 00:43:17,390
+الشرط ده اللي أكبرنا وجودها لنتبت إنها إياش وحيدة
+
+520
+00:43:17,700 --> 00:43:22,860
+الأن الـ function E من R لـ R that satisfies I and
+
+521
+00:43:22,860 --> 00:43:29,200
+I I هكذا of theorem 8.3.1 is unique إذا ما لها هذه
+
+522
+00:43:29,200 --> 00:43:32,940
+الدالة وحيدة ونشوف كيف نثبت وحيدة طبعا انتوا
+
+523
+00:43:32,940 --> 00:43:36,140
+عارفين ال .. الاستراتيجية تثبت انها وحيدة ونفترض
+
+524
+00:43:36,140 --> 00:43:39,340
+ان في دالتين وفي الآخر يا بنسأل لتناقض أو بنسأل
+
+525
+00:43:39,340 --> 00:43:43,260
+لإن الدالتين إشمالهم متساويتين خلينا نشوف
+
+526
+00:43:47,010 --> 00:43:52,110
+لأن let E1 and E2 be two functions on R ماشي الحال
+
+527
+00:43:52,110 --> 00:43:59,470
+E1 وE2 عبارة عن دالتين من R لR تحقيقان البرو بارتز
+
+528
+00:43:59,470 --> 00:44:03,010
+I and I I of T والمتمنى هو تلاتة واحد اللي كتبه ان
+
+529
+00:44:03,010 --> 00:44:08,700
+انا على الجنب هناكالان و خلّينا نسمي E1-E2 يتساوي
+
+530
+00:44:08,700 --> 00:44:14,840
+F رايحكم باتجاهة F يتساوي 0 إذا أثبتنا F يتساوي 0
+
+531
+00:44:14,840 --> 00:44:21,630
+إذا سيصبح E1 يتساوي E2 يكون خلصناطيب then فضلي هذه
+
+532
+00:44:21,630 --> 00:44:24,510
+.. فضلي هذه لأن هذا قبل التفاضل و هذا قبل التفاضل
+
+533
+00:44:24,510 --> 00:44:27,350
+إذا F prime of X بيساوي E1 prime of X ناقص E2
+
+534
+00:44:27,350 --> 00:44:32,230
+prime of X E1 prime of X إيش هتساوي اللي هو نفسها
+
+535
+00:44:32,230 --> 00:44:35,730
+لأن مفترضين احنا و E2 prime of X برضه هتساوي نفسها
+
+536
+00:44:35,730 --> 00:44:39,770
+إذا E1 ناقص E2 يعني مين بيساوي F يعني صارت F prime
+
+537
+00:44:39,770 --> 00:44:44,610
+تبعتنا مين هي بيساوي F of X لكل X element المين
+
+538
+00:44:44,610 --> 00:44:50,090
+إنا احسب اللي دي إيش يا جماعة احسب لولي F في صفرأف
+
+539
+00:44:50,090 --> 00:44:53,350
+اف صفر بيساوي ا واحد اف صفر ناقص اتنين اف صفر ا
+
+540
+00:44:53,350 --> 00:44:56,710
+واحد اف صفر واحد و اتنين اف صفر برضه اش بتساوي
+
+541
+00:44:56,710 --> 00:44:59,510
+واحد لإن مفترضين ا واحد و اتنين بتحفق اللي هي
+
+542
+00:44:59,510 --> 00:45:02,970
+الشطن الأمامنا إذا بيساوي واحد ناقص واحد و اي ساوي
+
+543
+00:45:02,970 --> 00:45:04,350
+اش صفر
+
+544
+00:45:07,320 --> 00:45:13,380
+by induction f double prime هتساوي f نفسها f
+
+545
+00:45:13,380 --> 00:45:18,060
+triple بتساوي f نفسها fn of x بيساوي f of x وعملت
+
+546
+00:45:18,060 --> 00:45:21,920
+قبل بشوية واحدة زيها إذا by induction fn of x
+
+547
+00:45:21,920 --> 00:45:28,280
+بيساوي f of x لكل x element in R إذا حققت الآن f
+
+548
+00:45:28,280 --> 00:45:34,500
+of 0 بيساوي 0 و fn of x بيساوي مين؟ f of x شوف
+
+549
+00:45:34,500 --> 00:45:42,870
+الآنأذا صار عندي F of Zero بساوي Zero أو F
+
+550
+00:46:10,090 --> 00:46:17,830
+بساوة 0طب ايش علاقتنا ابوها الجيت بتشوف عشان بدنا
+
+551
+00:46:17,830 --> 00:46:20,690
+نطبق اللي هو taylor's theorem تبعت ال derivative
+
+552
+00:46:20,690 --> 00:46:25,930
+نص اللي اللي بدنا يعني برهان حلو let x element in
+
+553
+00:46:25,930 --> 00:46:29,550
+R be arbitrary أخدنا إذا x element in R arbitrary
+
+554
+00:46:29,550 --> 00:46:33,810
+x اللي يعني let I x be the closed interval with
+
+555
+00:46:33,810 --> 00:46:40,760
+end point 0x يعني I xهناخدها بتساوي 0 و X أو اللي
+
+556
+00:46:40,760 --> 00:46:47,080
+هي X و 0 على .. على اللي هو اللي هي حسب X اللي هي
+
+557
+00:46:47,080 --> 00:46:51,980
+سالبة أو مجبة إذا أخدنا ال I X عبارة عن اللي هي ال
+
+558
+00:46:51,980 --> 00:46:56,600
+closed interval اللي within point 0 X الآن F is
+
+559
+00:46:56,600 --> 00:47:01,020
+continuous on I Xعلى الـ closed interval مدام
+
+560
+00:47:01,020 --> 00:47:03,320
+continuous على closed interval and function that
+
+561
+00:47:03,320 --> 00:47:06,460
+is continuous على closed interval then it is
+
+562
+00:47:06,460 --> 00:47:09,200
+bounded on this interval يعني بمعنى أخر there
+
+563
+00:47:09,200 --> 00:47:12,880
+exist k بحيث أن absolute value of f of t أصغر أو
+
+564
+00:47:12,880 --> 00:47:17,960
+يساوي k for all t element in I X يعني f is bounded
+
+565
+00:47:17,960 --> 00:47:23,580
+on this intervalالان بدنا نطبق if we apply
+
+566
+00:47:23,580 --> 00:47:27,540
+taylor's theorem 6 4 1 to F بدنا نطبقها على
+
+567
+00:47:27,540 --> 00:47:31,800
+النقطتين النقطة الأولى اللى هى ال X X و X naught 0
+
+568
+00:47:31,800 --> 00:47:35,640
+يعني بنطبقها على مين اللى هى X X اللى هى طبعا
+
+569
+00:47:35,640 --> 00:47:41,370
+انتوا متذكرين F of Xأف of X تيلر سيوريم there
+
+570
+00:47:41,370 --> 00:47:46,370
+exists C in اللي هي ال interval مثلا X not و X
+
+571
+00:47:46,370 --> 00:47:52,690
+such that F of X بيساوي اللي هو F of X not زائد F
+
+572
+00:47:52,690 --> 00:47:58,490
+prime of X not على واحد factorial في X minus X not
+
+573
+00:47:58,490 --> 00:48:05,910
+زائد زائد Fn of x0 على n factorial في x minus x0
+
+574
+00:48:05,910 --> 00:48:11,610
+الكلوس n زائد ال remainder اللي هو fn زائد واحد of
+
+575
+00:48:11,610 --> 00:48:15,730
+اللي هي ال c اللي لاجيناها there exists c اللي هي
+
+576
+00:48:15,730 --> 00:48:19,170
+of c طبعا ال c هنا هتعتمد على ال n اللي احنا
+
+577
+00:48:19,170 --> 00:48:24,850
+اختارناها واشتغلنا عليها هو مسميها cn على اللي هو
+
+578
+00:48:24,850 --> 00:48:28,670
+n زائد واحد factorial في x
+
+579
+00:48:33,180 --> 00:48:37,920
+الـ N هو
+
+580
+00:48:37,920 --> 00:48:42,580
+ما خدها لعند مين بدل ما هو remainder N زائد واحد
+
+581
+00:48:42,580 --> 00:48:45,880
+نسميه remainder N نفس الشيء مافيش مشكلة احنا
+
+582
+00:48:45,880 --> 00:48:51,880
+كملناها يعني بدل ما اشتغل على N زي ما اشتغل على N
+
+583
+00:48:51,880 --> 00:48:54,460
+زائد واحد اشتغل على ال N هو اما هي ال Taylor's
+
+584
+00:48:54,460 --> 00:48:59,460
+theorem طيب المهم انتبهوا عن ديلنبن Taylor's
+
+585
+00:48:59,460 --> 00:49:03,580
+theorem على ال interval Ix اللي X0 بيساوي 0
+
+586
+00:49:03,580 --> 00:49:08,040
+وخلّينا نستخدم Fk of 0 اللي كتبتها هناك اللي هو
+
+587
+00:49:08,040 --> 00:49:12,020
+هتساوي Fk of 0 هتساوي 0 دايما اللي هو و لكل k
+
+588
+00:49:12,020 --> 00:49:15,740
+element n it follows thatfor each n unlimited on
+
+589
+00:49:15,740 --> 00:49:20,000
+there exist a point cn unlimited on x هذه اللي هي
+
+590
+00:49:20,000 --> 00:49:23,920
+ال cn بتعتمد على مين على n يعني الأن لو أخدت ببدل
+
+591
+00:49:23,920 --> 00:49:28,860
+اللي هنا على اللي هي on بتساوي مثلا اتنين أخد on
+
+592
+00:49:28,860 --> 00:49:31,660
+بتساوي تلتة أخد بتساوي on بتساوي أربعة ده هتختلف
+
+593
+00:49:31,660 --> 00:49:35,740
+من ال cn اللي بنلاقيها such that f of x بيساوي f
+
+594
+00:49:35,740 --> 00:49:39,420
+of zero زيد f prime of zero على واحد factorial في
+
+595
+00:49:39,420 --> 00:49:44,390
+xطبعاً نقص صفر X هتظلها زائد FN نقص واحد لأن نقص
+
+596
+00:49:44,390 --> 00:49:47,630
+واحد factorial في X minus X note اللي هي صفر طبعاً
+
+597
+00:49:47,630 --> 00:49:52,830
+هتصير X نقص واحد زائد FN of CN على N factorial X
+
+598
+00:49:52,830 --> 00:49:56,530
+أثن يعني هنا ناخد ال remainder اللي هو RN مش RN
+
+599
+00:49:56,530 --> 00:50:01,410
+زائد واحد زي ما نعمله طبعاً ماتفرجشبتزيده و بتشتغل
+
+600
+00:50:01,410 --> 00:50:04,910
+عليه نفس الاشي الان بس ال cn اللي بتختلف من n زائد
+
+601
+00:50:04,910 --> 00:50:07,650
+واحد بصير cn زائد واحد مثلا لأنه حاجة تانية ممكن
+
+602
+00:50:07,650 --> 00:50:11,290
+تيجي غير ال cn اللي لاجيناها حسب النظرية و يساوي
+
+603
+00:50:11,290 --> 00:50:16,350
+لأن كل هدول التاريج زي شمال هي الصفار لإنه fk of
+
+604
+00:50:16,350 --> 00:50:21,290
+00 إذا هذا سفر وهذا سفر طبعا هذا عند مين محسب عند
+
+605
+00:50:21,290 --> 00:50:25,800
+السفر وهذا عند السفر إلى آخرهإذا كل هذول هيكون
+
+606
+00:50:25,800 --> 00:50:30,700
+الصفار هيظل عندى بس القيمة هذه ايش القيمة هذه اللي
+
+607
+00:50:30,700 --> 00:50:35,320
+هي fn of cn طبيعي هتكون f of min of cn لأنه احنا
+
+608
+00:50:35,320 --> 00:50:41,070
+عرفنا انه ال derivative f primeأف دابل برايم أف
+
+609
+00:50:41,070 --> 00:50:45,710
+تريبل كل نياش بيساوي الأف إذا حيكون أف of CN على N
+
+610
+00:50:45,710 --> 00:50:50,710
+فيكتوريال في X أُس N إذا وصلنا له الآن أنه بعد ما
+
+611
+00:50:50,710 --> 00:50:57,170
+طبقنا Taylor's theorem طلع عندي الآن F of X هذه
+
+612
+00:50:57,170 --> 00:51:04,710
+اللي بنصبه إلى أن نثبتها بتساوي صفر طلعت عندي F of
+
+613
+00:51:04,710 --> 00:51:10,230
+X بيساوي F of CN على N فيكتوريال في X أُس Nالان خد
+
+614
+00:51:10,230 --> 00:51:14,510
+لل absolute value لها absolute value لل F of X
+
+615
+00:51:14,510 --> 00:51:20,090
+هيصير اصغر او يساوي ال absolute value او بيساوي
+
+616
+00:51:20,090 --> 00:51:26,870
+بالظبط absolute value F of C N فال absolute value
+
+617
+00:51:27,400 --> 00:51:33,440
+للـ X أُس N على N فكتوريال، مظبوط؟ لكن F of C N
+
+618
+00:51:33,440 --> 00:51:37,200
+هذه is .. الـ F is bounded على الفترة اللي لجيناها
+
+619
+00:51:37,200 --> 00:51:40,180
+اللي هي الفترة اليمين اللي قلت عنها اللي هي الـ I
+
+620
+00:51:40,180 --> 00:51:44,380
+X مدان bounded قلنا هذه أصغر أو يساوي الـ K إذا
+
+621
+00:51:44,380 --> 00:51:47,380
+هذه بيصير أصغر أو يساوي الـ K في ال absolute value
+
+622
+00:51:47,380 --> 00:51:52,750
+X أس N على N فكتوريال نيجي الآنهذه as n goes to
+
+623
+00:51:52,750 --> 00:51:56,770
+infinity هذه independent of n هذه أكبره يساوي سفر
+
+624
+00:51:56,770 --> 00:51:59,430
+و أصغره يساوي هذه as n goes to infinity هذه بتروح
+
+625
+00:51:59,430 --> 00:52:05,590
+للسفر إذا هذه بتروح أشمالها بدها تصير f of x تساوي
+
+626
+00:52:05,590 --> 00:52:11,230
+السفر ومنه ال E1 بتساوي ال E2 لأن f بتساوي E1 ناقص
+
+627
+00:52:11,230 --> 00:52:17,350
+Eإتنين وهو المطلوب وهذا الكلام صحيح لكل X element
+
+628
+00:52:17,350 --> 00:52:25,050
+in R لأن X اللي أخدناها arbitrary element in R هيك
+
+629
+00:52:25,050 --> 00:52:31,990
+بنكون أثبتنا وجود ال E واحد ال E التي تحقق الشرطين
+
+630
+00:52:31,990 --> 00:52:38,590
+اللي عندنا وفي نفس الوقت أثبتنا أن هذه الدالة اللي
+
+631
+00:52:38,590 --> 00:52:46,680
+بتحقق الشرطينهي دالة وحيدةمدام الدالة واحدة إذن
+
+632
+00:52:46,680 --> 00:52:53,280
+الآن يعني شرّعنا إنه نعطيها اسم إنه نقدر نعرف
+
+633
+00:52:53,280 --> 00:52:59,200
+الدالة اللي بتحقق هذول الشرطين إنها اسمها كذا the
+
+634
+00:52:59,200 --> 00:53:02,660
+unique definition ثمانية تلاتة خمسة the unique
+
+635
+00:53:02,660 --> 00:53:06,140
+function E من R لR such that E prime of X في سوء E
+
+636
+00:53:06,140 --> 00:53:10,120
+of X for all X elements in R وتحقق ال E زيرو
+
+637
+00:53:10,120 --> 00:53:15,190
+بالسوء واحد موجودةووحيدة إذا بحق اللي أقول is
+
+638
+00:53:15,190 --> 00:53:18,450
+called the exponential function وهي الـ
+
+639
+00:53:18,450 --> 00:53:22,350
+exponential اللي أنتوا مبسوطين عليها و بتستخدموها
+
+640
+00:53:22,350 --> 00:53:27,530
+أثبتنا الآن من خلال اللي هو اللي هي الترتيب اللي
+
+641
+00:53:27,530 --> 00:53:31,690
+رتبناه في المادة differentiability integrability و
+
+642
+00:53:31,690 --> 00:53:36,350
+بعدين اللي هو sequences of functions وصلنا إلى
+
+643
+00:53:36,350 --> 00:53:41,190
+اللي هو ال exponential هذه function existsالتي
+
+644
+00:53:41,190 --> 00:53:43,850
+سمناها الـ Exponential Function
+
+645
+00:53:48,710 --> 00:53:53,850
+بدنا نسمي الـ E of واحد قيمة الدالة القيمة الدالة
+
+646
+00:53:53,850 --> 00:53:58,650
+الـ E هذه عند الواحد بدنا نسميها E E و هذا اللي
+
+647
+00:53:58,650 --> 00:54:04,410
+بنسميه Ehlers number و احنا يعني بعد هيك هيصير
+
+648
+00:54:04,410 --> 00:54:09,410
+عندي اللي هو الرمز E of X ان هو ال exponential X
+
+649
+00:54:09,410 --> 00:54:15,710
+او أسهلنا في الاستخدام ال E of X ايش هتساويالـ E X
+
+650
+00:54:15,710 --> 00:54:21,530
+هذا الـ E واحد قيمة الدالة عند مين عند اللي هو
+
+651
+00:54:21,530 --> 00:54:26,750
+الرقم واحد سمناها E ماشي الحالة الان E of X
+
+652
+00:54:26,750 --> 00:54:32,610
+بالساوي E to the X هذه عبارة عن اللي هي الدالة E
+
+653
+00:54:32,610 --> 00:54:37,550
+of X بالساوي X notation لها لكن بعد شوية هنلاقي ال
+
+654
+00:54:37,550 --> 00:54:42,310
+notation consistent of اللي هو مع مين مع اللي هو
+
+655
+00:54:42,310 --> 00:54:47,480
+ال exponentيعني هيصير الـ E to the exponent X هو
+
+656
+00:54:47,480 --> 00:54:51,960
+عبارة عن بالظبط اللي هو قيمة الـ E of X لأن الـ E
+
+657
+00:54:51,960 --> 00:54:54,620
+of X بيساوي E to the X و ال E of واحد بيساوي E
+
+658
+00:54:54,620 --> 00:54:59,440
+واحد بيصير عندي E لما أرفحها للقوة X يطلع قيمتها
+
+659
+00:54:59,440 --> 00:55:05,840
+هي عبارة عن اللي هي قيمة دل E of X حسب اللي احنا
+
+660
+00:55:05,840 --> 00:55:11,550
+معرفينه هنايعني من الإحسابات ومن تعريفنا للدالة
+
+661
+00:55:11,550 --> 00:55:23,070
+هيكون فيه consistent it is consistent في الحالتين
+
+662
+00:55:23,070 --> 00:55:28,990
+طيبنجي الآن لما الكلام هذا ال number E can be
+
+663
+00:55:28,990 --> 00:55:32,570
+obtained as a limit and thereby approximated in
+
+664
+00:55:32,570 --> 00:55:36,410
+several different ways طبيعا احنا مادام عنده اللي
+
+665
+00:55:36,410 --> 00:55:42,990
+هو عبارة عن limit لل E N of Xاللي هي بتطلع عندى
+
+666
+00:55:42,990 --> 00:55:47,450
+limit عند ال .. ال .. ال .. ال end of واحد لأنها
+
+667
+00:55:47,450 --> 00:55:52,130
+continuous بصير عندى مدام limit إذا بصير أقدر أقرب
+
+668
+00:55:52,130 --> 00:55:56,490
+هذه القيمة اللي هي و نحصل على قيمة تقريبية بال E و
+
+669
+00:55:56,490 --> 00:56:00,010
+هذا مش شغلنا شغل تبعين اللي هو numerical analysis
+
+670
+00:56:00,010 --> 00:56:05,530
+أو الناس اللي بدأت .. اللي هي تشتغل في التقريب أو
+
+671
+00:56:05,530 --> 00:56:09,270
+مش شغل نهاني يعني بصديإلى أن الـ use of notation E
+
+672
+00:56:09,270 --> 00:56:15,050
+و X of E X زي ما قلنا كله تمام تمام consistent إلى
+
+673
+00:56:15,050 --> 00:56:19,910
+أن يجي اللي هو لبعض الخواص الأخرى لهذه اللي هي
+
+674
+00:56:19,910 --> 00:56:24,130
+الدالة إلى أن الـ exponential function satisfies
+
+675
+00:56:24,130 --> 00:56:26,150
+the following properties
+
+676
+00:56:32,620 --> 00:56:35,960
+أول حاجة أن الـ E of X لا تساوي سفر لكل X element
+
+677
+00:56:35,960 --> 00:56:40,140
+in R هي أول حاجة أن هذه الدالة دايما لا تساوي سفر
+
+678
+00:56:40,140 --> 00:56:45,980
+طبعا شغالة على R E of X زي Y بسوء E of X في E of Y
+
+679
+00:56:45,980 --> 00:56:50,880
+for all X element .. Y element in R الآن E of R
+
+680
+00:56:50,880 --> 00:56:56,080
+هنا اللي هي E of R قيمة الـ function هذه هتطلع لنا
+
+681
+00:56:56,080 --> 00:57:00,260
+بالظبطهي عبارة عن الـ E اللي قلنا عنها الرقم اللي
+
+682
+00:57:00,260 --> 00:57:06,000
+قبل و شوية لما نرفع للقوة R فبصير الان consistency
+
+683
+00:57:06,000 --> 00:57:11,440
+between the definition of the exponential and the
+
+684
+00:57:11,440 --> 00:57:15,700
+exponent of E to the power R بس هذي الـ R إيش اللي
+
+685
+00:57:15,700 --> 00:57:21,880
+من الين؟ NQ لان ماشي الحالة ده نشوف نبرهن اللي هو
+
+686
+00:57:21,880 --> 00:57:27,390
+الأولى by contradictionبدي افترض انه في عندي E of
+
+687
+00:57:27,390 --> 00:57:31,930
+α for some α element in R والـ E of α ايش تساوي
+
+688
+00:57:31,930 --> 00:57:35,870
+تساوي سفر ونوصل لcontradiction لان suppose that
+
+689
+00:57:35,870 --> 00:57:39,550
+there exists α element in R such that E of α ايش
+
+690
+00:57:39,550 --> 00:57:44,750
+بساوي بساوي سفرالآن and let G α be the closed
+
+691
+00:57:44,750 --> 00:57:49,710
+interval with endpoints mean α و 0 يعني الـ G of α
+
+692
+00:57:49,710 --> 00:57:54,230
+زي اللي فوق يا بساوي اللي هو 0 ألفة أو ألف و زيره
+
+693
+00:57:54,230 --> 00:57:59,440
+حسب قيمة الألف موجبة أو سالبة طيبالان زي ما قلت
+
+694
+00:57:59,440 --> 00:58:02,480
+قبل بشوية بما ان E is continuous on a closed
+
+695
+00:58:02,480 --> 00:58:07,700
+interval Iα او Jα اللي هو مسميها then there exist
+
+696
+00:58:07,700 --> 00:58:10,280
+case such that اللي هو ال absolute value E of D
+
+697
+00:58:10,280 --> 00:58:16,030
+أصفر سوى K لكل T element in Gالأن مشابه للي قبل دي
+
+698
+00:58:16,030 --> 00:58:19,930
+ربالكم الان بدنا نستخدم mean taylor theorem اللي
+
+699
+00:58:19,930 --> 00:58:24,890
+هو على اللي هي ال function اللي عندي هذه اللي هو
+
+700
+00:58:24,890 --> 00:58:29,310
+بس في مين الآن في ال end points ألف ومين ألف وصفر
+
+701
+00:58:29,310 --> 00:58:32,610
+اذا there exist cn element in dn such that مسرع
+
+702
+00:58:32,610 --> 00:58:35,950
+لإنه قاعد بعيد نفس البرهان اللي قبله شوية such
+
+703
+00:58:35,950 --> 00:58:42,200
+that اللي هو E of صفربطبق عند e of 0 بساوي e of
+
+704
+00:58:42,200 --> 00:58:44,340
+alpha زائد e prime of alpha على واحد factorial في
+
+705
+00:58:44,340 --> 00:58:47,280
+ناقص alpha اللي هي zero ناقص alpha زائد e n ناقص
+
+706
+00:58:47,280 --> 00:58:50,780
+واحد على ناقص واحد factorial alpha ناقص alpha أُس
+
+707
+00:58:50,780 --> 00:58:54,160
+n ناقص واحد لما أصل لعند ال remainder e n of alpha
+
+708
+00:58:54,160 --> 00:59:00,060
+على n factorial في ناقص alpha أُس n الآن اخلق هذه
+
+709
+00:59:00,060 --> 00:59:05,770
+مش e alpha هذه الـCnهذه الـ C أنا مخربط بصيحة الله
+
+710
+00:59:05,770 --> 00:59:11,430
+ماشي الحالة الان E of Zero طبيعي ايش يساوي واحد
+
+711
+00:59:11,430 --> 00:59:14,510
+ماحنا عارفين ان خلاص عرضناها الدالة هذه E of Zero
+
+712
+00:59:14,510 --> 00:59:17,630
+بتساوي واحد اتساوي واحد يساوي E of Zero و يساوي
+
+713
+00:59:20,060 --> 00:59:23,400
+الأن E of Alpha فرضناها إيش بالساوية سفر هيك
+
+714
+00:59:23,400 --> 00:59:28,280
+مفترضينها E prime of Alpha اللي هي نفس E of Alpha
+
+715
+00:59:28,280 --> 00:59:32,880
+إذا سفر برضه وهذا نفس الشيء إذا كلنا دول أصفر مع
+
+716
+00:59:32,880 --> 00:59:36,100
+دمين الأخيرة إذا هتساوي E of Cn على N factorial
+
+717
+00:59:36,100 --> 00:59:42,340
+ناقص Alpha أسعى وزي ما قلنا إنه احنا الآن هذا
+
+718
+00:59:42,340 --> 00:59:48,390
+المقدار عندي صار .. نوضح لكم هذهنشوف كيف نصل لـ
+
+719
+00:59:48,390 --> 00:59:53,770
+contradiction هال gate صار عند مايا ليه يا جماعة
+
+720
+00:59:53,770 --> 00:59:57,190
+صارت عند الواحد هذه اللي بيساوي E to the zero
+
+721
+00:59:57,190 --> 01:00:03,630
+بيساوي E to the zero اللي هي أصغر أو يساوي ال
+
+722
+01:00:03,630 --> 01:00:07,510
+absolute value لهذه بيساويها بعد إنه أصغر أو يساوي
+
+723
+01:00:07,510 --> 01:00:14,420
+بيساوي ال absolute value E of CL على N factorialفي
+
+724
+01:00:14,420 --> 01:00:22,320
+absolute value ناقص Alpha أُس N و هذه زي ما قلنا E
+
+725
+01:00:22,320 --> 01:00:25,880
+of C N إيش ما اللي قلنا قبل بشوية هذه أصغر أوي و
+
+726
+01:00:25,880 --> 01:00:32,960
+ساوي K إذا K على N factorial في ناقص Alpha أُس N
+
+727
+01:00:32,960 --> 01:00:38,710
+ماشي الآن هذا المقدارas n goes to infinity بيروح
+
+728
+01:00:38,710 --> 01:00:42,910
+لـ0 هذا independent of n و هذا independent of n
+
+729
+01:00:42,910 --> 01:00:48,650
+صار عندى الان هذا الان هي الواحد أصغر أو يساوي
+
+730
+01:00:48,650 --> 01:00:53,390
+اللي هو هذا المقدار خد ال limit as n goes to
+
+731
+01:00:53,390 --> 01:00:57,430
+infinity بيصير الواحد أصغر أو يساوي سفر و هذا إيش
+
+732
+01:00:57,430 --> 01:01:03,250
+ماله contradictionإذا صار في عندي الفرضية الأولى
+
+733
+01:01:03,250 --> 01:01:07,770
+خاطئة إذا there is no such Alpha اللي هو بتكون
+
+734
+01:01:07,770 --> 01:01:10,530
+عندها ال K of Alpha بتساوي سفر إذا ال K of Alpha
+
+735
+01:01:10,530 --> 01:01:16,090
+بتساوي أكبر لا تساوي سفر دائما إذا صار عندي اللي
+
+736
+01:01:16,090 --> 01:01:22,150
+هو قيمة اللي هي E to the X لا تساوي سفر أبدا نيجي
+
+737
+01:01:22,150 --> 01:01:29,110
+الآن إن بدنا نثبت أنهنثبت الخاصية اللي بعدها اللي
+
+738
+01:01:29,110 --> 01:01:38,430
+هي عبارة عن E of X زائد Y بساوة E of X في E of Y
+
+739
+01:01:38,430 --> 01:01:42,670
+لكل X و Y element in R هذه الخاصية اللي بدنا
+
+740
+01:01:42,670 --> 01:01:48,410
+نثبتها شوف طريقته حلوة في الإثبات بيقول ليه نفترض
+
+741
+01:01:48,410 --> 01:01:53,030
+Y fixed arbitrary لكن ليهاش fixed بنحكي عن Y محددة
+
+742
+01:01:53,030 --> 01:01:59,760
+arbitrary لكن نحكي عن Y محددةالان مادام Y اللي هي
+
+743
+01:01:59,760 --> 01:02:05,240
+ال E of Y أكيد يشملها لا تساوي سفر اتفجنا عليها
+
+744
+01:02:05,240 --> 01:02:11,520
+الان عرفلي الان function G من R لR عرفها كيف؟G of
+
+745
+01:02:11,520 --> 01:02:15,720
+X المتغير X الان Y اللي هي عبارة عن اشي Fix ثابت
+
+746
+01:02:15,720 --> 01:02:19,080
+بحكي عنه لكن كان arbitrary اللي هي G of X بيساوي E
+
+747
+01:02:19,080 --> 01:02:24,420
+of X زائد Y على E of Y for X element in R أخدنا
+
+748
+01:02:24,420 --> 01:02:30,060
+هذه الدالة الآن شوف هذه الدالة اعمله G prime لها G
+
+749
+01:02:30,060 --> 01:02:33,920
+prime of X بالفضل بالنسبالي XY ثابتة اللي هي بيصير
+
+750
+01:02:33,920 --> 01:02:38,280
+هذه E prime of X زائد Y على E of Y عدد ما لاش
+
+751
+01:02:38,280 --> 01:02:43,080
+علاقة فيه Y ساويالـ E' هي نفس مين؟ الـ E of X زي Y
+
+752
+01:02:43,080 --> 01:02:48,720
+على E of Y ماشي؟ طيب، الآن صار هذا الـ E X زي Y
+
+753
+01:02:48,720 --> 01:02:53,280
+على E of Y هو مين؟ رجع G إذا رجعت إن الـ G' مش
+
+754
+01:02:53,280 --> 01:03:01,770
+بيساوي الـ G الآن andأحسب لي g of 0 بساوية e of 0
+
+755
+01:03:01,770 --> 01:03:06,070
+زائد y على e of y يعني بساوية واحدة إذا الدالة
+
+756
+01:03:06,070 --> 01:03:11,050
+اللي عرفناها هذه g of x طلعت لي g prime لها نفسها
+
+757
+01:03:11,050 --> 01:03:15,310
+وطلعت لي g of 0 واحدة وعلمة قبل قليل عمال بيقول
+
+758
+01:03:15,310 --> 01:03:18,610
+مافيش لغير واحدة في الدنيا بتهبها الخاصيتين اللي
+
+759
+01:03:18,610 --> 01:03:23,190
+هي مين ال E of X إذا هذه غصب عننا لازم تطلع مين ال
+
+760
+01:03:23,190 --> 01:03:27,080
+E of X because of the uniqueness of Eإذا صحيح أنا
+
+761
+01:03:27,080 --> 01:03:31,000
+عندي E of X بساوي E X Z Y على E E of Y يعني E X Z
+
+762
+01:03:31,000 --> 01:03:37,940
+Y بساوي E of Y في ال E of X و هو المطلوب طيب نيجي
+
+763
+01:03:37,940 --> 01:03:46,800
+الآن نثبت اللي بعدها اللي هي بدنا نثبت E of R
+
+764
+01:03:46,800 --> 01:03:54,870
+بساوي E to the RR المنتنين LQ اللي هو rational
+
+765
+01:03:54,870 --> 01:04:00,870
+number طيب جماعة فيكم تصل على النبي عليه الصلاة
+
+766
+01:04:00,870 --> 01:04:09,370
+والسلام لنشوف كيف نيجي
+
+767
+01:04:09,370 --> 01:04:14,690
+البرهن اللي هو الجزء الثالث أو هذا جزء من النظرية
+
+768
+01:04:14,690 --> 01:04:18,830
+أول
+
+769
+01:04:18,830 --> 01:04:24,560
+حاجةهذه صحيحة لكل أنهي المتنانة كيف؟ by induction
+
+770
+01:04:24,560 --> 01:04:53,330
+E of X بساوي E of Xأكيد أكيد أكيد أكيد أكيد
+
+771
+01:04:53,330 --> 01:04:56,270
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+772
+01:04:56,270 --> 01:04:56,330
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+773
+01:04:56,330 --> 01:04:57,110
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+774
+01:04:57,110 --> 01:05:00,170
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+775
+01:05:00,170 --> 01:05:00,170
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+776
+01:05:00,170 --> 01:05:00,170
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد
+
+777
+01:05:00,170 --> 01:05:02,740
+أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكيد أكE
+
+778
+01:05:02,740 --> 01:05:08,660
+of K في X زائد X ويساوي من الخاصية اللي احنا لسه
+
+779
+01:05:08,660 --> 01:05:12,880
+ماخلصنهاش هذه بساوي E للأول زايد في E للتاني اللي
+
+780
+01:05:12,880 --> 01:05:17,960
+هي E للأول اللي هو KX في E للتاني اللي هي EX
+
+781
+01:05:17,960 --> 01:05:22,780
+ويساوي EKX اللي هو E of X أُس K لأنه فرضينا أنا
+
+782
+01:05:22,780 --> 01:05:31,260
+صحيحة اللي هي K بيصير عندي E of X أُس K في E of Xو
+
+783
+01:05:31,260 --> 01:05:39,800
+يساوي E of X الكل أُس K زائد واحد إذا صارت
+
+784
+01:05:39,800 --> 01:05:49,580
+هذه is true for all N element in N طيب .. الآن شوف
+
+785
+01:05:49,580 --> 01:05:53,920
+ما يليه نلاحظ ما يليه
+
+786
+01:06:02,460 --> 01:06:06,080
+Lit x بتساوي ايش؟ واحدة على ان ناخد الـ x ايش
+
+787
+01:06:06,080 --> 01:06:10,540
+بتساوي واحدة على ان؟ بتصير هذا E of واحد لأن ان في
+
+788
+01:06:10,540 --> 01:06:13,840
+واحدة على ان واحد ال E of واحد رمزناها من رمز ايش
+
+789
+01:06:13,840 --> 01:06:17,940
+يا جماعة؟ E وهذا اللي بدأ اصليه هذا رمزناها من رمز
+
+790
+01:06:17,940 --> 01:06:22,420
+E وال E عبارة عن رقم ال E of واحد هي قيمة الدالة
+
+791
+01:06:22,420 --> 01:06:28,100
+اللي تبعتنا عند اللي هي الواحد سمينها Eالان E of N
+
+792
+01:06:28,100 --> 01:06:33,380
+في 1 على N تساوي اللي هي عبارة عن ال X هذه اللي هي
+
+793
+01:06:33,380 --> 01:06:40,720
+عبارة عن 1 على N E of 1 على N الكل اسمه أُس N يعني
+
+794
+01:06:40,720 --> 01:06:48,380
+الآن عندي هذه ال X E N X بساوي E X أُس N X مين 1
+
+795
+01:06:48,380 --> 01:06:53,210
+على N E of 1 على N أُس Nإذا صار عندى الآن هي هذا
+
+796
+01:06:53,210 --> 01:06:59,010
+القيمة بساوية هذا إذا بمعنى آخر E of واحدة الان
+
+797
+01:06:59,010 --> 01:07:05,830
+بساوية E العدد أس واحد على N هذا الان E of واحدة
+
+798
+01:07:05,830 --> 01:07:14,050
+الان بساوية E أس واحد على N الآن من جهة أخرى لو
+
+799
+01:07:14,050 --> 01:07:23,700
+جيت E أس minus Mهتساوي واحد على E أُس M خد
+
+800
+01:07:23,700 --> 01:07:26,760
+ال
+
+801
+01:07:26,760 --> 01:07:33,460
+E of M ناقص M اللي هي بيساوي E of Zero اللي هي
+
+802
+01:07:33,460 --> 01:07:41,020
+بيساوي واحد صح وإيه ساوي E of Mفي E of ناقص M حسب
+
+803
+01:07:41,020 --> 01:07:45,900
+الخاصية اللي هي زائد ناقص M هذه X وهذه ال Y E
+
+804
+01:07:45,900 --> 01:07:49,760
+الأولى في E التانية الان صار عندى هذا المقدار
+
+805
+01:07:49,760 --> 01:07:57,600
+بساوي واحد يعني إذا E of ناقص M بساوي بنجل هدهان
+
+806
+01:07:57,600 --> 01:08:03,260
+واحد على E of M إذا صارت عندى E of ناقص M بساوي
+
+807
+01:08:03,260 --> 01:08:09,360
+واحد على E of M وال E of M ال E of Mمن فوق E of M
+
+808
+01:08:09,360 --> 01:08:14,280
+هذا X بواحد يعني بيساوي E of واحد أُس M يعني عبارة
+
+809
+01:08:14,280 --> 01:08:19,500
+عن E أُس M وهي فوق إيش واحد الآن هذه اللي هو عدد
+
+810
+01:08:19,500 --> 01:08:23,800
+عادي اللي هو E مرفوع للأُس M اللي هو نفسه E to the
+
+811
+01:08:23,800 --> 01:08:27,360
+minus M لأن هذه معلومة قديمة لذا انصار عندي الآن E
+
+812
+01:08:27,360 --> 01:08:34,820
+of minus M بيساوي E أُس ماقص M وال E أُس واحد على
+
+813
+01:08:34,820 --> 01:08:43,460
+N بيساوي Eأُس واحد على N نجي
+
+814
+01:08:43,460 --> 01:08:46,780
+الآن احنا غايتنا مين؟ E of R الـ R إيش اللي نقدر
+
+815
+01:08:46,780 --> 01:08:50,640
+نكتبها؟ الـ R في الـ Q إذا الـ R بتنكتب على صورة M
+
+816
+01:08:50,640 --> 01:08:54,780
+على N إذا نخد R في اللي نمتني Q إذا there exists M
+
+817
+01:08:54,780 --> 01:08:58,820
+و N اي واحدة في زد واحدة في N صحيح ده؟ R بتساوي M
+
+818
+01:08:58,820 --> 01:09:03,320
+على N بتثبتلك إن E of M على N بساوي E to the M على
+
+819
+01:09:03,320 --> 01:09:12,580
+N شوف كيفE of M على N ايش هيساوي E أُس واحدة ل N
+
+820
+01:09:12,580 --> 01:09:21,400
+الكل A شماله أُس M أُس M E of واحدة ل N أُس M مين
+
+821
+01:09:21,400 --> 01:09:28,870
+اللي شرعلي اللي هوعندي اللي هي E-M و هي سالبة طلعت
+
+822
+01:09:28,870 --> 01:09:33,870
+E أص مينس M و اللي شرعلي E و واحدة ل N و N مودبة
+
+823
+01:09:33,870 --> 01:09:41,130
+هي E أص واحد على N فبصير عندي الأن E أص M على N أص
+
+824
+01:09:41,130 --> 01:09:45,970
+M على N بساوي E أص واحد على N هذه اللي جوا الكل
+
+825
+01:09:45,970 --> 01:09:53,650
+أسمين الكل أص M ماشي الحال هو يساوي E
+
+826
+01:09:58,010 --> 01:10:03,730
+هذه شرعات لإي و واحدة الان كل اس ام وهذا عدد عادي
+
+827
+01:10:03,730 --> 01:10:08,730
+الان وهذا عدد عادية بيصير اي أس ام عالميا على ان و
+
+828
+01:10:08,730 --> 01:10:14,850
+هذه بيكون عندي اي أس اي او ام على ان بساوي اللي هو
+
+829
+01:10:14,850 --> 01:10:18,870
+عبارة عن اي أس ام على ان و ام على ان was our
+
+830
+01:10:18,870 --> 01:10:21,710
+arbitral rational number اذا صار عندي اي أس ار
+
+831
+01:10:21,710 --> 01:10:24,210
+بساوي اي أس ار
+
+832
+01:10:29,400 --> 01:10:35,160
+نجي الآن للنظرية اللي هي تمانية تلاتة سبعة اللي هي
+
+833
+01:10:35,160 --> 01:10:39,980
+النظرية الأخيرة في اللي هي الحديث عن اللي هو ال
+
+834
+01:10:39,980 --> 01:10:45,260
+exponential function و بعدها طبعا بنحكي عن اللي هو
+
+835
+01:10:45,260 --> 01:10:48,060
+ال logarithmic function بس خلينا الآن نحكي عالميا
+
+836
+01:10:48,060 --> 01:10:54,340
+انكمل نظريتنا على ال exponential function بنشوف
+
+837
+01:10:55,540 --> 01:11:00,240
+الأن بقولي الـ exponential function E is strictly
+
+838
+01:11:00,240 --> 01:11:04,840
+increasing on R يعني الـ derivative إلها أكبر من
+
+839
+01:11:04,840 --> 01:11:11,460
+سفر ماشي؟ and this range اللي هو Y المتنرسج ده Y
+
+840
+01:11:11,460 --> 01:11:16,720
+أكبر من سفر يعني الـ E هتكون بالظبط ده اللي من R
+
+841
+01:11:16,720 --> 01:11:24,080
+بتصب إلى R positive إلى R positive يعني عبارة عن
+
+842
+01:11:24,080 --> 01:11:31,370
+Zeroومالة نهاية هي التي هي range دالة range دالة
+
+843
+01:11:31,370 --> 01:11:34,610
+هذه هي من سفر إلى مالة نهاية لأن كتابة على سورة
+
+844
+01:11:34,610 --> 01:11:40,210
+function is on to طيب مش هيك limit E of X لما X
+
+845
+01:11:40,210 --> 01:11:43,470
+تروح لسالب مالة نهاية بتساوي سفر and limit E of X
+
+846
+01:11:43,470 --> 01:11:47,110
+لما X تروح لمالة نهاية ايش بتساوي؟ بتساوي مالة
+
+847
+01:11:47,110 --> 01:11:55,670
+نهايةيعني الدالة هذه في حالة ال ال E of X اللي
+
+848
+01:11:55,670 --> 01:11:59,070
+احنا سميناها ال X exponential لل X هذه الدالة هذه
+
+849
+01:11:59,070 --> 01:12:03,130
+الدالة زي ما انتوا عارفين لما X تروح إلى سالب ماله
+
+850
+01:12:03,130 --> 01:12:06,830
+نهاية هذا هيروح للسفر ال E of X هتروح للسفر و لما
+
+851
+01:12:06,830 --> 01:12:10,890
+ال X تروح لماله نهاية ال E of X هتروح إيش إلى ماله
+
+852
+01:12:10,890 --> 01:12:14,810
+نهاية نشوف أول شي بالنسبة لمن؟ لل range
+
+853
+01:12:19,570 --> 01:12:24,250
+الان we know that E of 0 بيساوي Eاش واحد اكبر من
+
+854
+01:12:24,250 --> 01:12:29,770
+Eاش من سفر اكيد،ظبط ولا لا؟ and E of X ده بيساوي
+
+855
+01:12:29,770 --> 01:12:36,390
+سفر for X element in R صار عندى الان هاي دالة E of
+
+856
+01:12:36,390 --> 01:12:38,650
+0 إلها واحد،ماشي؟
+
+857
+01:12:40,360 --> 01:12:46,000
+أو E of X مش سفر يعني إيه شمالها لا تقطع محور
+
+858
+01:12:46,000 --> 01:12:51,910
+السينات إطلاقااللي E of X نفسها و 0 من سفرها الأن
+
+859
+01:12:51,910 --> 01:12:57,110
+أنا بقول مستحيل تكون في إيها قيم سالبة يعني مستحيل
+
+860
+01:12:57,110 --> 01:13:02,030
+نلاقي E of X naught اللي هي أصغر من 0 ليش؟ لأنه لو
+
+861
+01:13:02,030 --> 01:13:05,990
+لجينا E of X naught سالبة لو هالجيت عندنا X naught
+
+862
+01:13:05,990 --> 01:13:09,910
+و هيها قيمتها E of X naught اللي هو سالبة يعني
+
+863
+01:13:09,910 --> 01:13:16,460
+هتكون تحتطب ما هي الدالة متصلة مدام متصلة إذا غصب
+
+864
+01:13:16,460 --> 01:13:20,940
+عنها هتيجي تقطع اللي هو ما هتيجي منها لها هتقطع
+
+865
+01:13:20,940 --> 01:13:24,760
+محور السينات إذا هتصير سفر وهذا Contradiction من
+
+866
+01:13:24,760 --> 01:13:28,380
+أين الكلام هذا؟ By Bolzano Intermediate Value
+
+867
+01:13:28,380 --> 01:13:32,760
+Theorem بما أنه احنا في عندنا X0 فرضناها ال E of
+
+868
+01:13:32,760 --> 01:13:39,260
+X0 أصغر من سفر وفي عندي اللي هي E of Zero بساوي
+
+869
+01:13:39,260 --> 01:13:40,980
+واحد اللي هي
+
+870
+01:13:44,440 --> 01:13:48,620
+أه لو فرضنا ان في اكس نوت اكس نوت تكون اللي هو
+
+871
+01:13:48,620 --> 01:13:56,300
+سالبة اكس نوت اصغر من سفر فعندي E of X naught اصغر
+
+872
+01:13:56,300 --> 01:14:05,740
+من سفر وعندي اللي هو في اللي هو E of X بساوي
+
+873
+01:14:05,740 --> 01:14:14,470
+واحد اه وهادي اكبر من سفرأه وفرضنا وجود هذه وفرضنا
+
+874
+01:14:14,470 --> 01:14:18,810
+وجود هذه اذا by intermediate value theorem غصبن
+
+875
+01:14:18,810 --> 01:14:23,650
+عنها there exists c element الفترة اللي احنا بنحكي
+
+876
+01:14:23,650 --> 01:14:28,510
+عنها R such that f of هذه الـ c اللي هي E of C
+
+877
+01:14:28,510 --> 01:14:33,330
+بتساول كمية اللي بينين اللي هي اسمها سفر وهذا
+
+878
+01:14:33,330 --> 01:14:41,860
+contradictionبالتالي بيحقق البلزان انها مستمرة على
+
+879
+01:14:41,860 --> 01:14:47,060
+اي انتقال مغلق على كل أرض لانها بيحقق البلزان انها
+
+880
+01:14:47,060 --> 01:14:47,260
+مستمرة على اي انتقال مغلق على كل أرض لانها بيحقق
+
+881
+01:14:47,260 --> 01:14:47,280
+البلزان انها مستمرة على اي انتقال مغلق على كل أرض
+
+882
+01:14:47,280 --> 01:14:47,700
+لانها بيحقق البلزان انها مستمرة على اي انتقال مغلق
+
+883
+01:14:47,700 --> 01:14:48,880
+على كل أرض لانها بيحقق البلزان انها مستمرة على اي
+
+884
+01:14:48,880 --> 01:14:49,100
+انتقال مغلق على كل أرض لانها بيحقق البلزان انها
+
+885
+01:14:49,100 --> 01:14:51,280
+مستمرة على اي انتقال مغلق على كل أرض لانها بيحقق
+
+886
+01:14:51,280 --> 01:14:54,390
+البلزان انها مستمرة على اي انتقمن سفر طيب مدام ال
+
+887
+01:14:54,390 --> 01:14:57,010
+E of X أكبر من سفر و احنا بنعرف ان ال E prime of X
+
+888
+01:14:57,010 --> 01:15:00,690
+بيتساوي E of X اذا صارت ال E prime أكبر من مين من
+
+889
+01:15:00,690 --> 01:15:04,890
+سفر كل X element ال R ومن ثم هذا معناته ان E is
+
+890
+01:15:04,890 --> 01:15:10,450
+strictly increasing on R اذا اللي هو المطلوب الأول
+
+891
+01:15:10,450 --> 01:15:15,810
+أثبتناه ان E is strictly increasing on R طيب
+
+892
+01:15:24,230 --> 01:15:30,250
+عندي اتنين strictly increasing ال function مظبوط
+
+893
+01:15:30,250 --> 01:15:41,510
+يعني اتنين اصغر من مين من اياش من ال E ليش لأن
+
+894
+01:15:41,510 --> 01:15:53,730
+عندي الواحد E واحد او E of Xقلنا أصغر strictly ولا
+
+895
+01:15:53,730 --> 01:15:57,890
+أكبر strictly من واحد زاد X صح؟ هذه الكورولا
+
+896
+01:15:57,890 --> 01:16:02,810
+remained تمانية تلاتة تلاتة اثبتناها إذا صار عند E
+
+897
+01:16:02,810 --> 01:16:08,050
+of واحدأكبر strictly من واحد زائد وا أكبر strictly
+
+898
+01:16:08,050 --> 01:16:13,970
+من واحد زائد واحد وال E of واحد مين هو E أس واحد
+
+899
+01:16:13,970 --> 01:16:17,390
+مش هيك اتفاجنا يعني هذا إياش اتنين يعني ال E أكبر
+
+900
+01:16:17,390 --> 01:16:22,450
+strictly من مين من الاتنين اتفاجنا طيب اتفاجنا إذا
+
+901
+01:16:22,450 --> 01:16:26,410
+صار عند ال E of اتنين أكبر strictly ال E أكبر من
+
+902
+01:16:26,410 --> 01:16:29,950
+مين من اتنين خلينا نبدأ عليها هذه ال E أكبر من
+
+903
+01:16:29,950 --> 01:16:41,000
+إياش من اتنينالان اكيد عنده ال E of R اللي بيسوي E
+
+904
+01:16:41,000 --> 01:16:48,440
+R او E N خليني اقول E N اكبر من اتنين هتكون اصغر
+
+905
+01:16:48,440 --> 01:16:53,700
+اللي هي اكبر من اتنين أُس N الان as N goes to
+
+906
+01:16:53,700 --> 01:16:57,680
+infinity اذا
+
+907
+01:16:57,680 --> 01:17:04,230
+اكيد هذه هتروح لمين لإنفينيتيوالـ E نفسها اللي هي
+
+908
+01:17:04,230 --> 01:17:08,750
+is strictly increasing is strictly increasing إذا
+
+909
+01:17:08,750 --> 01:17:12,370
+ال E of X تبعتها برضه هتروح لمين ل Infinity لما X
+
+910
+01:17:12,370 --> 01:17:18,650
+تروح إلى وين إلى مالة نهاية ليش لأن لكل L في X
+
+911
+01:17:18,650 --> 01:17:24,670
+أكبر منهابحيث انه اللي هو تصير E of X اللي هي
+
+912
+01:17:24,670 --> 01:17:30,110
+بيساوي E to the X أكبر من E to the N لكل N في X
+
+913
+01:17:30,110 --> 01:17:33,850
+أكبر منها بتصير E of X أكبر منها لأنها strictly
+
+914
+01:17:33,850 --> 01:17:37,710
+increasing فلان لما تروح هذه إلى مالة نهاية الكبار
+
+915
+01:17:37,710 --> 01:17:41,170
+تبعتها وهذه طبعا أكبر دايما موجودة هذا هتروح إلى
+
+916
+01:17:41,170 --> 01:17:45,540
+أين برضه إلى مالة نهاية يعني limit ال E of Xلما X
+
+917
+01:17:45,540 --> 01:17:48,980
+تروح إلى مالة نهاية بتساوي مالة نهاية وقلكم اللي
+
+918
+01:17:48,980 --> 01:17:54,040
+هو لو تجربته اللي هو تثبته أنه بما أنه limit E of
+
+919
+01:17:54,040 --> 01:17:58,840
+N بتروح إلى مالة نهاية by اللي هي limit definition
+
+920
+01:17:58,840 --> 01:18:03,020
+إذا limit E of X لما X تروح إلى مالة نهاية بتساوي
+
+921
+01:18:03,020 --> 01:18:08,080
+مالة نهاية الآن similarly for mean for اللي هو
+
+922
+01:18:08,080 --> 01:18:11,120
+limit
+
+923
+01:18:12,620 --> 01:18:16,560
+الـ E of X لما X تروح لسالب ملا نهاية هذه الان
+
+924
+01:18:16,560 --> 01:18:22,840
+بيصير اتنين أو minus N أكبر من E أو minus N ماشي
+
+925
+01:18:22,840 --> 01:18:28,000
+الحال الان ال limit لهذه as N goes to infinity
+
+926
+01:18:28,000 --> 01:18:31,940
+أكبر أو ساوي ال limit لهذه as N goes to infinity
+
+927
+01:18:31,940 --> 01:18:33,420
+الان
+
+928
+01:18:35,300 --> 01:18:40,100
+هذه اش مالها بيساوي سفر وهذه اوتوماتيك هيساوي ايش؟
+
+929
+01:18:40,100 --> 01:18:44,840
+هيساوي سفر الان as N goes to infinity يعني ال E to
+
+930
+01:18:44,840 --> 01:18:49,720
+the minus N بتروح للسفر من وين؟من اليمين، مظبوط أو
+
+931
+01:18:49,720 --> 01:18:53,060
+تروح للـ Zero، مظبوط لأن at E minus N تروح لما
+
+932
+01:18:53,060 --> 01:18:57,440
+لنهاية As N goes to infinity E to the minus N تروح
+
+933
+01:18:57,440 --> 01:19:03,460
+لإنفينيتي لأن as X بتروح إلى سالب ما لنهاية ال E
+
+934
+01:19:03,460 --> 01:19:07,560
+to the minus X برضه هتروح إلى وين؟ إلى السفر يعني
+
+935
+01:19:07,560 --> 01:19:11,220
+ال E to the X نفسها لما ناخد ال limit لما X تروح
+
+936
+01:19:11,220 --> 01:19:14,580
+بدل ما لنهاية تروح إلى مين؟ سالب ما لنهاية برضه
+
+937
+01:19:14,580 --> 01:19:20,200
+هتساوي إيش؟ السفرإذا صار عندى limit E ضو X لما X
+
+938
+01:19:20,200 --> 01:19:23,560
+تروح السلمة لها يابسة أو سفر و بعتمد على اللى هو
+
+939
+01:19:23,560 --> 01:19:26,780
+الجهتين على ال strictly increasing تبع ال E و
+
+940
+01:19:26,780 --> 01:19:30,580
+التفاصيل اللى هى ال definition بتظهر عندكم أنا
+
+941
+01:19:30,580 --> 01:19:34,220
+مافصلتاش لإنه هو إيه هذا القائم
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..29f7bdb318a0ec760f77982c59dffc5fcb30ecd2
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw.srt
@@ -0,0 +1,1376 @@
+1
+00:00:04,940 --> 00:00:07,660
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:07,660 --> 00:00:10,500
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه 
+
+3
+00:00:10,500 --> 00:00:17,340
+أجمعين هذه هي المحاضرة رقم 23 في مساق تحليل حقيقي
+
+4
+00:00:17,340 --> 00:00:22,200
+لنيل طلاب وطالبات الجامعة الإسلامية قسم الرياضيات
+
+5
+00:00:22,200 --> 00:00:27,900
+كلية العلوم، المحاضرة اللي هي اليوم هي عبارة عن
+
+6
+00:00:27,900 --> 00:00:33,180
+تكملة لـ section 8.3 الجزء الثاني من .. من
+
+7
+00:00:33,180 --> 00:00:36,580
+.. من هذا .. اللي هو الـ chapter الجزء الأول من هذا
+
+8
+00:00:36,580 --> 00:00:39,320
+الـ section، الجزء الأول تحدثنا عن الـ exponential 
+
+9
+00:00:39,320 --> 00:00:44,670
+function وكيف أثبتنا وجودها وأخذنا خواصها، الآن بدنا
+
+10
+00:00:44,670 --> 00:00:47,910
+نحكي عن الجزء الثاني من اللي هو الـ section اللي هو
+
+11
+00:00:47,910 --> 00:00:51,050
+الـ logarithmic function، الـ logarithmic function
+
+12
+00:00:51,050 --> 00:00:55,290
+اللي هو نشوف كيف بدنا نثبت وجودها وكيف اللي هو
+
+13
+00:00:55,290 --> 00:01:00,410
+نأخذ خواصها بنفس البناء اللي أو نبنِي البناء اللي
+
+14
+00:01:00,410 --> 00:01:05,820
+بنيناه المحاضرة الماضية، الآن لما حكينا عن الـ
+
+15
+00:01:05,820 --> 00:01:10,020
+exponential function E، جئنا أن الـ exponential E
+
+16
+00:01:10,020 --> 00:01:12,780
+is strictly increasing differentiable function
+
+17
+00:01:12,780 --> 00:01:18,160
+with domain R and range اللي هو Y أكبر من 0، يعني
+
+18
+00:01:18,160 --> 00:01:22,480
+لما حكينا عن الـ E، حكينا عن الـ E من R اللي هي الـ
+
+19
+00:01:22,480 --> 00:01:26,600
+exponential لعند الفترة 0 و ∞، هذه الـ
+
+20
+00:01:26,600 --> 00:01:31,560
+function هي rangeها وهي domainها وكانت strictly
+
+21
+00:01:31,560 --> 00:01:35,700
+increasing، Strictly increasing معناته اللي هو عبارة
+
+22
+00:01:35,700 --> 00:01:40,020
+عن 1-1، يعني بمعنى آخر في لها الـ function هذه on
+
+23
+00:01:40,020 --> 00:01:43,940
+to two وكانت differentiable، الآن الـ function اللي هي
+
+24
+00:01:43,940 --> 00:01:46,840
+الـ exponential طبعًا ما أنتم عارفين كيف رسمتها، لو
+
+25
+00:01:46,840 --> 00:01:50,460
+جئنا جربنا نرسمها هنلاقي الرسمة اللي هو هذه اللي
+
+26
+00:01:50,460 --> 00:01:56,180
+هي عبارة عن رسمة الـ exponential، الآن أنا بدي أجيء
+
+27
+00:01:56,180 --> 00:02:00,300
+اللي هو من خلال اللي هو الـ function الـ exponential
+
+28
+00:02:00,300 --> 00:02:05,980
+أعرف الـ inverse لها وأسميه اللي هو logarithmic
+
+29
+00:02:05,980 --> 00:02:10,860
+function أو بدي أسميه الـ logarithm الطبيعي اللي هي
+
+30
+00:02:10,860 --> 00:02:16,020
+الـ ln function، مشروع الكلام اه لأن ايه عبارة عن
+
+31
+00:02:16,020 --> 00:02:19,240
+function one to one و one to one، إذا صار الـ
+
+32
+00:02:19,240 --> 00:02:23,580
+inverse لها موجود لأنها strictly increasing، إذا
+
+33
+00:02:23,580 --> 00:02:29,560
+صار الـ L من عند zero و ∞ لعند الـ R، هادي
+
+34
+00:02:29,560 --> 00:02:34,100
+اللي هي الـ function الجديدة هي اللي بدي أسميها الـ
+
+35
+00:02:34,100 --> 00:02:38,320
+logarithmic function وهي رسمتها اللي أمامنا اللي
+
+36
+00:02:38,320 --> 00:02:42,460
+هي الـ inverse لهذه الدالة اللي بدي أعرفها الآن
+
+37
+00:02:42,460 --> 00:02:47,000
+وتعريفها الآن صار شرعي بناء على وجود الـ exponential
+
+38
+00:02:47,000 --> 00:02:50,930
+اللي بدي أعرفه اللي هو الـ inverse سبقتها، العمل
+
+39
+00:02:50,930 --> 00:02:57,030
+المعرفي للـ E هو
+
+40
+00:02:57,030 --> 00:03:02,850
+الـ Logarithm أو الـ Natural Logarithm اللي هي It
+
+41
+00:03:02,850 --> 00:03:07,870
+will be denoted by L or by ln، الأكثر شيوعًا طبعًا
+
+42
+00:03:07,870 --> 00:03:11,810
+اللي هو مين الـ ln؟ لأن بما أن الـ E و L انعكاس
+
+43
+00:03:11,810 --> 00:03:17,110
+لبعض، إذا أكيد الـ E composite L composite E of X
+
+44
+00:03:17,110 --> 00:03:22,920
+هيساوي الـ X، لكل الـ x واللي موجودة في الـ R لأن
+
+45
+00:03:22,920 --> 00:03:26,540
+الـ E بتشتغل على كل الإكسات اللي في الـ R لأن
+
+46
+00:03:26,540 --> 00:03:30,740
+بينما E composite L of Y، E composite L of Y، الـ L
+
+47
+00:03:30,740 --> 00:03:34,660
+بتشتغل .. بتشتغل مين على مين؟ بس على الـ positive، E
+
+48
+00:03:34,660 --> 00:03:38,240
+composite L of Y بيساوي لكل Y element in R و Y
+
+49
+00:03:38,240 --> 00:03:44,900
+أشملها أكبر من 0، الآن connotations .. connotations
+
+50
+00:03:44,900 --> 00:03:49,860
+بناء عليه الـ N، الـ E of X لأن الـ N هي الـ L والـ
+
+51
+00:03:49,860 --> 00:03:53,880
+E هي الـ E، وعندي الـ E to the N اللي هو بسوء الـ Y
+
+52
+00:03:53,880 --> 00:03:57,780
+وهو بسوء الـ X بناء على أن الواحدة inverse للتانية
+
+53
+00:04:01,010 --> 00:04:04,750
+أو كل واحدة inverse للأخرى، الـ logarithm is a
+
+54
+00:04:04,750 --> 00:04:08,630
+strictly increasing function L with domain اللي هو
+
+55
+00:04:08,630 --> 00:04:12,150
+مين اللي هو الـ domain اللي عندي اللي هو تعرفنا هيك
+
+56
+00:04:12,150 --> 00:04:16,210
+أصلًا، الآن الـ derivative of L is given by L prime
+
+57
+00:04:16,210 --> 00:04:19,750
+of X ايش بتساوي؟ واحدة على X، for X أكبر من صفر، الآن
+
+58
+00:04:19,750 --> 00:04:23,430
+الـ logarithm satisfy the functional equation، تحقق
+
+59
+00:04:23,430 --> 00:04:27,010
+المعادلة الدالية التالية اللي هي L of X في Y بيساوي
+
+60
+00:04:27,010 --> 00:04:31,000
+L of X زائد L of Y، for X أكبر من صفر، Y أكبر من
+
+61
+00:04:31,000 --> 00:04:34,560
+صفر، Y أكبر
+
+62
+00:04:34,560 --> 00:04:38,260
+من صفر
+
+63
+00:04:38,260 --> 00:04:40,560
+Y أكبر من صفر، Y أكبر من صفر، Y أكبر من صفر، Y أكبر
+
+64
+00:04:40,560 --> 00:04:40,580
+من صفر، Y أكبر من صفر، Y أكبر من صفر، Y أكبر من
+
+65
+00:04:40,580 --> 00:04:40,640
+أكبر من صفر، Y أكبر من صفر، Y أكبر من صفر، Y أكبر من
+
+66
+00:04:40,640 --> 00:04:40,700
+صفر، Y أكبر من صفر، Y أكبر من صفر، Y أكبر من
+
+67
+00:04:40,700 --> 00:04:41,020
+صفر، Y أكبر من صفر، Y أكبر من صفر، Y أكبر من
+
+68
+00:04:41,020 --> 00:04:47,140
+أكبر من صفر، Y أكبر من صفر، Y أكبر من صفر، Y أكبر، L of
+
+69
+00:04:47,140 --> 00:04:51,420
+X<sup>R</sup> بيساوي R لـ L of X، لأن X أكبر من 0 و R المتر كيوها
+
+70
+00:04:51,420 --> 00:04:54,840
+كلهم، اللي هي خواص أنتم بتعرفوها قبل هيك، بس الآن
+
+71
+00:04:54,840 --> 00:04:58,440
+بدنا نبرهنها ونثبت صحتها، limit L of X لما X تروح
+
+72
+00:04:58,440 --> 00:05:01,740
+لـ 0 من اليمين بيساوي -∞، and limit L of X
+
+73
+00:05:01,740 --> 00:05:07,340
+لما X تروح لـ ∞ بتساوي ∞، خلينا
+
+74
+00:05:07,340 --> 00:05:14,840
+احنا نشوف نبرهن اللي هي اللي مطلوب، الآن الـ L is
+
+75
+00:05:14,840 --> 00:05:17,560
+strictly increasing with domain X element alone
+
+76
+00:05:17,560 --> 00:05:20,880
+and range R follows from the fact that E is
+
+77
+00:05:20,880 --> 00:05:24,840
+strictly increasing with domain R and range اللي
+
+78
+00:05:24,840 --> 00:05:33,320
+هو اللي عندي، الآن عندي الـ L is strictly increasing
+
+79
+00:05:33,320 --> 00:05:37,560
+بناء على مين؟ على الـ E نفسها strictly increasing،
+
+80
+00:05:37,560 --> 00:05:48,560
+الآن E Composite L، E Composite L of Y ايش بتساوي؟ Y
+
+81
+00:05:48,560 --> 00:05:55,320
+لكل Y و Y الموجودة، لكل Y element in صفر و ∞،
+
+82
+00:05:55,320 --> 00:06:00,780
+مظبوط؟ الآن فاضل للجهتين، الآن طبعًا احنا بنعرف
+
+83
+00:06:00,780 --> 00:06:06,360
+أنه من الأصل مدامة الـ E is اللي هو differentiable،
+
+84
+00:06:06,360 --> 00:06:10,480
+أكيد اللي هي الـ inverse إلها is differentiable by
+
+85
+00:06:10,480 --> 00:06:14,680
+theorem 6.9، كده مش عارف ايش في اللي هو chapter 6
+
+86
+00:06:14,680 --> 00:06:18,200
+قدامة الـ function اللي هي is differentiable، الـ
+
+87
+00:06:18,200 --> 00:06:20,880
+inverse إلها برضه is differentiable في حالة وجودها،
+
+88
+00:06:20,880 --> 00:06:27,480
+الآن E فاضل للجهتين بيصير عندي E prime of L of Y
+
+89
+00:06:27,480 --> 00:06:36,600
+في L prime of Y، بيساوي ايش؟ واحد، ماشي الحال، الآن واضح
+
+90
+00:06:36,600 --> 00:06:41,900
+أن هذا حاصل الضرب صار أكبر من مين؟ strictly من 0،
+
+91
+00:06:41,900 --> 00:06:47,920
+وبما أن الـ E is strictly increasing، أثبتنا E' of
+
+92
+00:06:47,920 --> 00:06:53,420
+L of Y is strictly أكبر من 0، إذا بيظلها L' of Y is
+
+93
+00:06:53,420 --> 00:06:57,180
+strictly أكبر من 0 لكل Y هنا، إذا صارت الـ L is
+
+94
+00:06:57,180 --> 00:07:02,740
+strictly increasing، الآن طبعًا الـ domain مدام أن هذه
+
+95
+00:07:02,740 --> 00:07:07,000
+الـ inverse لـ الـ E، الـ domain اللي هو الـ inverse هو
+
+96
+00:07:07,000 --> 00:07:11,380
+range الـ function الأصلية وبيصير sub wave في
+
+97
+00:07:11,380 --> 00:07:18,460
+الفرع، إذا الآن احنا أثبتنا أن الـ L is strictly
+
+98
+00:07:18,460 --> 00:07:23,530
+increasing، الآن و rangeها اللي هو صار domain اللي
+
+99
+00:07:23,530 --> 00:07:28,310
+هو أو range اللي هو الـ .. هذي اللي صار domainها
+
+100
+00:07:28,310 --> 00:07:32,990
+domain الـ L وهذي صارت اللي هو range الـ L زي ما
+
+101
+00:07:32,990 --> 00:07:37,330
+قلنا قبل بشوية أو عندي، الآن بدنا نثبت اللي هو
+
+102
+00:07:37,330 --> 00:07:42,070
+الجزء الثاني من النظرية، خليني أكتب هنا عشان نتذكر
+
+103
+00:07:42,070 --> 00:07:48,430
+ايش اللي بدنا نثبته؟ الآن بدنا نثبت، اين أثبتنا
+
+104
+00:07:48,430 --> 00:07:54,230
+الأولى، اللي في النص بتقلّي prime of X، اللي نكتبهن
+
+105
+00:07:54,230 --> 00:07:59,330
+اللي بدنا نثبتهن عشان نتذكرهن
+
+106
+00:07:59,330 --> 00:08:12,240
+L prime of X بيساوي واحد على X،
+
+107
+00:08:12,240 --> 00:08:19,680
+اثنين اللي هو L of XY بيساوي L X زائد L Y، طبعًا الـ Y
+
+108
+00:08:19,680 --> 00:08:25,780
+هناك، L of 1 بيساوي صفر، L of E بيساوي 1، كلهم
+
+109
+00:08:25,780 --> 00:08:34,440
+بسيطات، L prime L of X to the R بيساوي R L of X، و Limit
+
+110
+00:08:34,440 --> 00:08:39,720
+L of X لما X تروح إلى صفر من اليمين بيساوي -∞، لما
+
+111
+00:08:39,720 --> 00:08:45,140
+لـ ∞، و Limit لـ L of X لما X تروح إلى ∞
+
+112
+00:08:45,140 --> 00:08:48,520
+بيساوي ∞، خليني أشوف أن دول على السريع فانوا
+
+113
+00:08:48,520 --> 00:08:53,640
+كلها شغلات، يعني بأعتقد أنه سهل أنك تثبتها
+
+114
+00:08:55,730 --> 00:09:02,210
+عندي، لأن زي ما عملت قبل بشوية اللي هو لما فضلت هذه
+
+115
+00:09:02,210 --> 00:09:07,290
+تفاضله E composite L of X لما عملتها قبل بشوية
+
+116
+00:09:07,290 --> 00:09:14,250
+اللي هي كانت عندي هين أعمل E composite L of X الكل
+
+117
+00:09:14,250 --> 00:09:19,370
+اللي هي بيساوي الـ X فاضل هذا يصير E prime
+
+118
+00:09:26,060 --> 00:09:29,100
+بالنسبة لـ x، أقل الـ prime of x،
+
+119
+00:09:34,780 --> 00:09:40,160
+اللي هو واحد على الـ E prime of L of X، إذا الـ E
+
+120
+00:09:40,160 --> 00:09:44,300
+اللي قلنا prime of X بيساوي واحد على E prime composite
+
+121
+00:09:44,300 --> 00:09:48,340
+L of X، والـ E prime هي نفس الـ E زي ما قلنا إذا
+
+122
+00:09:48,340 --> 00:09:51,140
+بيصير واحد على E composite L of X، إلى الـ E
+
+123
+00:09:51,140 --> 00:09:54,640
+composite L of X زي ما قلنا ايش بتساوي؟ بيساوي X،
+
+124
+00:09:54,640 --> 00:09:57,660
+فبيساوي واحد على X، فاللي قلنا prime بيساوي واحد على X،
+
+125
+00:09:57,660 --> 00:10:02,910
+لكل X في الموجودة في الفترة صفر و ∞، نيجي الآن نشوف
+
+126
+00:10:02,910 --> 00:10:06,710
+اللي هي اللي بعدها، الخاصية اللي بعدها خلينا نثبت
+
+127
+00:10:06,710 --> 00:10:12,690
+اللي هو L of X في Y بيساوي L of X زائد مين؟ زائد L of
+
+128
+00:10:12,690 --> 00:10:17,270
+Y، برضه الإثبات سهل وانتبهوا معايا وسهل عندي الآن،
+
+129
+00:10:24,240 --> 00:10:27,240
+F X أكبر من صفر، Y أكبر من صفر، F X أكبر من صفر، Y
+
+130
+00:10:27,240 --> 00:10:27,740
+أكبر من صفر، F X أكبر من صفر، Y أكبر من صفر، F X أكبر
+
+131
+00:10:27,740 --> 00:10:27,900
+من صفر، F X أكبر من صفر، F X أكبر من صفر، F X أكبر من
+
+132
+00:10:27,900 --> 00:10:28,100
+صفر، F X أكبر من صفر، F X أكبر من صفر، F X أكبر من
+
+133
+00:10:28,100 --> 00:10:28,700
+صفر، F X أكبر من صفر، F X أكبر من صفر، F X أكبر من
+
+134
+00:10:28,700 --> 00:10:28,920
+صفر، F X أكبر من صفر، F X أكبر من صفر، F X أكبر من
+
+135
+00:10:28,920 --> 00:10:32,100
+صفر، F X أكبر من صفر، F X أكبر من صفر، F X أكبر من
+
+136
+00:10:32,100 --> 00:10:43,400
+صفر، F X أكبر من صفر، F X أكبر من صفر، F X
+
+137
+00:10:43,400 --> 00:10:51,850
+أكبر من صفر، F X، لأن الـ E والـ L انعكاس بعض، الآن من
+
+138
+00:10:51,850 --> 00:10:55,130
+الخاصية تبع الـ exponential بدنا نصل لمن؟ للـ
+
+139
+00:10:55,130 --> 00:11:00,970
+logarithmic، إذا أضرب لـ X في Y بيطلع عند X في Y
+
+140
+00:11:00,970 --> 00:11:05,190
+بتساوي E of U في E of V، E of U في E of V ايش
+
+141
+00:11:05,190 --> 00:11:10,010
+بتساوي؟ E of U زائد V، أثبتناها، إذا من هذا الكلام
+
+142
+00:11:10,970 --> 00:11:15,270
+خذ الـ L للجهتين لأنه اللي هي الـ inverse لبعض،
+
+143
+00:11:15,270 --> 00:11:20,450
+بيصير عندي L of X في Y بيساوي L of E of U زائد V،
+
+144
+00:11:20,450 --> 00:11:24,410
+اللي هي ايش بتساوي؟ U زائد V، U اللي هي عبارة عن L
+
+145
+00:11:24,410 --> 00:11:30,750
+of X، و V عبارة عن L of Y، إذا أثبتت L of X زائد X
+
+146
+00:11:30,750 --> 00:11:39,370
+في Y بيساوي L of X زائد L of Y، الآن عندي اللي هي E
+
+147
+00:11:39,370 --> 00:11:47,050
+of صفر بيساوي 1، خذ لي الـ L للجهتين بيصير L of
+
+148
+00:11:47,050 --> 00:11:53,010
+E of صفر بيساوي L of 1، الـ L of E of صفر هذيك
+
+149
+00:11:53,010 --> 00:11:59,450
+inverse للتانية بيساوي صفر، نفس الشيء، الـ L of E of
+
+150
+00:11:59,450 --> 00:12:07,270
+1 بيساوي L of E، مظبوط؟ الـ L of E of 1 بيساوي 1،
+
+151
+00:12:07,270 --> 00:12:12,230
+بيصير L of E بيساوي 1، بيصير أثبتنا L of E بيساوي 1،
+
+152
+00:12:12,230 --> 00:12:19,730
+و L of 1 بيساوي 0، وهذا الكلام كلام سهل، طيب بيصير
+
+153
+00:12:19,730 --> 00:12:23,710
+علاقة أنه سهل اللي بنحكيه، الآن
+
+154
+00:12:27,210 --> 00:12:32,730
+نأتي نثبت اللي هو اللي هي L of X to the R بيساوي
+
+155
+00:12:32,730 --> 00:12:37,010
+صفر بيساوي R في L of X، هذه برضه بنثبتها By
+
+156
+00:12:37,010 --> 00:12:41,910
+Mathematical Induction، عملناها قبل هيكة باللي هو
+
+157
+00:12:41,910 --> 00:12:47,330
+الـ section اللي قبله أو اللي هو الـ exponential،
+
+158
+00:12:47,330 --> 00:12:52,830
+على السريع نشوف التفاصيل لأنه التفاصيل معادة
+
+159
+00:13:07,050 --> 00:13:11,430
+التفاصيل حتلاقيها معادة، فخليني بسرعة نمر عليها،
+
+160
+00:13:11,430 --> 00:13:17,730
+عندي we show by induction L of X بيساوي L L of X زي ما
+
+161
+00:13:17,730 --> 00:13:21,850
+قلنا، عشان نثبتها هذه، أثبتنا اللي قبل بشوية L of
+
+162
+00:13:21,850 --> 00:13:27,190
+X، عارف مش الرزوم أصلًا أفصل ولا لأ، لكن بدي أفصل لو 
+
+163
+00:13:27,190 --> 00:13:32,790
+أنتوا عندي لو اتفقنا أن نفصل ولا لأ، L of X في Y
+
+164
+00:13:32,790 --> 00:13:39,580
+بيساوي L of X في L of Y، of x was n بيساوي n في L of x،
+
+165
+00:13:39,580 --> 00:13:44,060
+طبعًا for n بتساوي واحد اللي هي a trivial، نفترض
+
+166
+00:13:44,060 --> 00:13:48,180
+أنها صحيحة لـ L لـ n بتساوي k، بيصير L of x was k
+
+167
+00:13:48,180 --> 00:13:53,080
+بيساوي k L of x، الآن بدنا نحسب لـ L of x was k زائد
+
+168
+00:13:53,080 --> 00:13:59,480
+واحد اللي هي بيساوي L of x was k في x، هذه الـ L لها
+
+169
+00:13:59,480 --> 00:14:04,760
+حسب اللي هي الخاصية هذه بسوء L الأولى X plus K في
+
+170
+00:14:04,760 --> 00:14:09,860
+L التانية، L of X اللي هو مفترض إنها صحيحة على K ده
+
+17
+
+201
+00:16:44,910 --> 00:16:49,490
+عندي لأي rational number صار عندي L of X  R بيساوي R
+
+202
+00:16:49,490 --> 00:16:55,670
+L of X لكل R اللي بنتمي لـ NQ نيجي الآن لـ اللي هي
+
+203
+00:16:55,670 --> 00:16:57,230
+الجزء الأخير من النظرية
+
+204
+00:17:08,180 --> 00:17:11,120
+الكلام مشابه للي حكيناها قبل بشوية في الإثبات اللي
+
+205
+00:17:11,120 --> 00:17:15,260
+هو ال limit تبع ال exponential عند 2 أصغر من E
+
+206
+00:17:15,260 --> 00:17:19,380
+وقلنا ليش الآن ال E n هنا بيصير 2 أصغر من E أصغر 
+
+207
+00:17:19,380 --> 00:17:19,920
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+208
+00:17:19,920 --> 00:17:21,140
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+209
+00:17:21,140 --> 00:17:22,840
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+210
+00:17:22,840 --> 00:17:23,240
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+211
+00:17:23,240 --> 00:17:26,700
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+212
+00:17:26,700 --> 00:17:30,620
+من E أصغ
+
+213
+00:17:33,960 --> 00:17:39,580
+لكن اللي هو L of E N بيساوي N and اللي هي L of E 
+
+214
+00:17:39,580 --> 00:17:44,020
+minus N بيساوي نقص N خليني في الذاكرة هذول إذا for
+
+215
+00:17:44,020 --> 00:17:47,480
+every N element in R there exists X element in R
+
+216
+00:17:47,480 --> 00:17:52,670
+بحيث أن X أكبر من مين من N لكل N element in N في X
+
+217
+00:17:52,670 --> 00:17:56,350
+element in R أكيد X أكبر من مين من E N لأنه أخذت
+
+218
+00:17:56,350 --> 00:18:00,590
+N بين إيديا حسبت ال E N طلع عندي رقم أخذت ال X أكبر
+
+219
+00:18:00,590 --> 00:18:03,310
+منها كيد بالله طيب لأنه unbounded real numbers
+
+220
+00:18:03,310 --> 00:18:06,710
+then L في هذه أكبر أو من L في هذه لأنه ال L
+
+221
+00:18:06,710 --> 00:18:10,810
+strictly increasing إذا صار L في X أكبر من مين من
+
+222
+00:18:10,810 --> 00:18:16,130
+ال NA ألف E أن يعني أكبر من الآن لأن limit هذه as
+
+223
+00:18:16,130 --> 00:18:21,210
+x goes to infinity اللي هو بيكون أكبر أو يساوي اللي
+
+224
+00:18:21,210 --> 00:18:23,950
+هي limit هذه as n goes to infinity ويساوي infinity
+
+225
+00:18:23,950 --> 00:18:27,550
+لأن لكل لما الآن تروح لما لا نهاية أكيد ال X بتروح
+
+226
+00:18:27,550 --> 00:18:31,270
+لمين إلى ما لا نهاية فصار عندي هذه بتروح إلى ما
+
+227
+00:18:31,270 --> 00:18:34,970
+لا نهاية الآن صار عندي limit L of X لما X تروح لما
+
+228
+00:18:34,970 --> 00:18:41,040
+لا نهاية بيساوي ما لا نهاية similarly الآن لكل نقص any
+
+229
+00:18:41,040 --> 00:18:43,100
+element in z positive بلاقي x element in r
+
+230
+00:18:43,100 --> 00:18:45,740
+positive بحيث أن x أكبر من صفر و أصغر من صفر و
+
+231
+00:18:45,740 --> 00:18:46,340
+أصغر من صفر و أصغر من صفر و أصغر من صفر و أصغر من
+
+232
+00:18:46,340 --> 00:18:47,840
+صفر و أصغر من صفر و أصغر من صفر و أصغر من صفر و
+
+233
+00:18:47,840 --> 00:18:51,500
+أصغر من صفر و أصغر من صفر و أصغر من صفر و أصغر من
+
+234
+00:18:51,500 --> 00:18:57,960
+صفر و أصغر من صفر و أصغر من صفر و أصغر من صفر و
+
+235
+00:18:57,960 --> 00:19:02,520
+أصغر من صفر و أصغر من صفر و أصغر من صفر و أصغر من
+
+236
+00:19:02,520 --> 00:19:09,620
+صفر و أصغر من صفر و أصغر من صفر و أطيب then L of X
+
+237
+00:19:09,620 --> 00:19:13,660
+هيكون أصغر أو يساوي ال E to the minus N يعني L of X
+
+238
+00:19:13,660 --> 00:19:17,800
+اللي هي أصغر أو يساوي ناقص N إذا as N goes to
+
+239
+00:19:17,800 --> 00:19:22,700
+infinity as N goes to infinity اللي هو ال E to the
+
+240
+00:19:22,700 --> 00:19:26,540
+minus N بيروح للـ 0 من اليمين إذا ال X بتروح للـ 0 من
+
+241
+00:19:26,540 --> 00:19:30,820
+اليمين إذا عندي ال X بتروح للـ 0 من اليمين اللي هو
+
+242
+00:19:30,820 --> 00:19:34,580
+أصغر لما الـ limit L of X أصغر من limit E to the
+
+243
+00:19:34,580 --> 00:19:37,900
+minus N لما هذا يروح لـ 0 من اليمين أو بمعنى آخر
+
+244
+00:19:37,900 --> 00:19:41,680
+لما الـ N تروح لما لا نهاية و هذا بيروح لمين؟ لـ
+
+245
+00:19:41,680 --> 00:19:44,760
+Infinity إذا limit L of X لما X تروح لما لا نهاية
+
+246
+00:19:44,760 --> 00:19:50,920
+بيساوي سالب Infinity و هو المطلوب hence limit L of
+
+247
+00:19:50,920 --> 00:19:54,460
+X لما X تروح لـ 0 من اليمين بيساوي سالب Infinity
+
+248
+00:19:54,460 --> 00:19:58,500
+طيب
+
+249
+00:20:12,260 --> 00:20:16,320
+الآن سارعنا الآن نقدر أن اللي هو نحكي عن ال bar
+
+250
+00:20:16,320 --> 00:20:20,080
+functions بدنا نعرف ال bar functions اللي هي بناء
+
+251
+00:20:20,080 --> 00:20:25,060
+على اللي حكيناه و اللي هي موضوع ال bar functions
+
+252
+00:20:25,060 --> 00:20:28,880
+كل ما فيه تقريبًا يعني بنعتبره exercises احنا لكن
+
+253
+00:20:28,880 --> 00:20:32,520
+خلينا نعرف التعريفات و النظريات بتكون اللي هي
+
+254
+00:20:32,520 --> 00:20:35,900
+معاكم exercises بسيطة بناء على التعريف اللي
+
+255
+00:20:35,900 --> 00:20:40,720
+بنعرفها اللي هناخد α يليمنتان R و X أكبر من 0 The
+
+256
+00:20:40,720 --> 00:20:43,320
+number X to the Alpha is defined to be .. الآن بدي
+
+257
+00:20:43,320 --> 00:20:46,940
+أعرف حاجة اسمها X to the Alpha X to the Alpha بدي
+
+258
+00:20:46,940 --> 00:20:49,900
+أعرفها .. إيش بدي أعرفها؟ بإيش أنا معرف عندي من
+
+259
+00:20:49,900 --> 00:20:54,440
+الأصل ال exponential معرفة .. خلصنا منها و ال len
+
+260
+00:20:54,440 --> 00:20:59,000
+معرفة .. إذا E to the Alpha في Ln ال X هذه المقدار
+
+261
+00:20:59,000 --> 00:21:03,820
+لهذا معرف و هذا معرف بدي أسمي X to the Min to the
+
+262
+00:21:03,820 --> 00:21:07,400
+Alpha اللي هو في الواقع عبارة عن Min E to the
+
+263
+00:21:07,400 --> 00:21:12,040
+Alpha L of X L of X معرفة و ال E معرفة إذا كل هذه
+
+264
+00:21:12,040 --> 00:21:16,000
+معرفة بتسميها X to the main to the alpha الآن صارت
+
+265
+00:21:16,000 --> 00:21:19,540
+عندي يعني قيمة ال X under this function اللي
+
+266
+00:21:19,540 --> 00:21:23,120
+عرفتها لجديدة يعني إذا بتسميها دي ال function ال 
+
+267
+00:21:23,120 --> 00:21:28,240
+R of X إيش عرفتها أنا بتساوي X to the alpha؟ يعني
+
+268
+00:21:28,240 --> 00:21:31,680
+كل ال X بتصير يشمل X to the Alpha و X أكبر من 0
+
+269
+00:21:31,680 --> 00:21:36,780
+هذه ال X to the Alpha هي اللي بدي أسميها ال power
+
+270
+00:21:36,780 --> 00:21:42,000
+function بدي أسميها power function with exponent
+
+271
+00:21:42,000 --> 00:21:47,540
+mean Alpha و ال X هي أشمالها المتغيرة اللي أكبر من
+
+272
+00:21:47,540 --> 00:21:54,340
+0 شوف الآن نشوف بعض الخواص اللي هو هذه اللي هي
+
+273
+00:21:54,340 --> 00:21:56,460
+الدالة طيب
+
+274
+00:22:04,730 --> 00:22:08,690
+الآن if x أكبر من 0 and alpha بساوي m على n where
+
+275
+00:22:08,690 --> 00:22:12,770
+m element in z و n element in n then we define x
+
+276
+00:22:12,770 --> 00:22:17,790
+to the alpha بيساوي x to the m أسواحد على n in
+
+277
+00:22:17,790 --> 00:22:23,110
+section mean خمسة ستة هتعرفناها زمان أنه في حالة
+
+278
+00:22:23,110 --> 00:22:26,570
+بس اللي هي ال rational number عرفنا x to the m على
+
+279
+00:22:26,570 --> 00:22:30,630
+n بيساوي x to the m لكل أسواحد على n ماشي الحال
+
+280
+00:22:30,630 --> 00:22:34,510
+.. الآن بدنا نشوف هذا التعريف مطابق لتعريفنا اليوم
+
+281
+00:22:34,510 --> 00:22:41,670
+ولا لأ hence we have لن ال X to the Alpha لن ال X
+
+282
+00:22:41,670 --> 00:22:45,370
+to the Alpha لن ال X to the Alpha بيساوي Alpha لن
+
+283
+00:22:45,370 --> 00:22:51,540
+ال X عرفناها هذه طيب where X to the Alpha بيساوي E
+
+284
+00:22:51,540 --> 00:22:56,260
+to the Lin X to the Min to the Alpha اللي هو بيساوي
+
+285
+00:22:56,260 --> 00:23:01,020
+E to the Alpha في Min في Lin ال X كلام كله سهل ال
+
+286
+00:23:01,020 --> 00:23:05,740
+X to the Alpha هو اللي عرفناها اللي عبارة عن E
+
+287
+00:23:05,740 --> 00:23:10,570
+بتصير to the Lin X to the Alpha لأن استبدلت الـ x
+
+288
+00:23:10,570 --> 00:23:15,090
+to the alpha بقيمتها اللي هي عبارة عن اللي هي
+
+289
+00:23:15,090 --> 00:23:18,630
+alpha ln x اللي هي بيساوي e to the ln x to the
+
+290
+00:23:18,630 --> 00:23:24,510
+mean to the alpha إذا سواء احنا بالتعريف اللي هو
+
+291
+00:23:24,510 --> 00:23:28,190
+احنا هذا بال exponent أو بال ال function اللي
+
+292
+00:23:28,190 --> 00:23:33,110
+عرفناها بالشكل هذا هيطلع عندي اللي هو القيمتين نفس
+
+293
+00:23:33,110 --> 00:23:34,790
+القيمة طيب
+
+294
+00:23:37,160 --> 00:23:42,300
+نيجي الآن لبعض الخواص اللي هي تبعت ال exponential
+
+295
+00:23:42,300 --> 00:23:47,180
+ال power function و الخواص هنتركه لكم إياه لأنها
+
+296
+00:23:47,180 --> 00:23:54,600
+مباشرة على التعريف تبعنا مباشرة
+
+297
+00:23:54,600 --> 00:24:01,100
+على اللي هي التعريف اللي عندنا و هيكون في عندي
+
+298
+00:24:01,100 --> 00:24:05,760
+الآن النظرية الأولى اللي هي 8 3 11 لو كانت Alpha
+
+299
+00:24:05,760 --> 00:24:11,340
+element in R و X و Y اللي هو تنتمي للفترة Zero و
+
+300
+00:24:11,340 --> 00:24:16,500
+ثمانية Zero و ما لا نهاية آسف then معلش عشان ده طلعت
+
+301
+00:24:16,500 --> 00:24:20,450
+الكهرباء قعدت نقلف الكهرباء أنا If α element in R و X
+
+302
+00:24:20,450 --> 00:24:26,350
+Y belongs to 0 α then 1 to the α بيساوي 1 و X to the
+
+303
+00:24:26,350 --> 00:24:30,270
+α أكبر من 0 و X Y to the α بيساوي X to the α و Y to
+
+304
+00:24:30,270 --> 00:24:35,590
+the α و X على Y to the α بيساوي X to the α على Y to
+
+305
+00:24:35,590 --> 00:24:39,370
+the α هذه اللي هي النظرية طبعًا اللي هي مباشرة على
+
+306
+00:24:39,370 --> 00:24:45,100
+تعريفنا اللي هو X to the α بيساوي Eof α لن X يعني
+
+307
+00:24:45,100 --> 00:24:49,800
+بدك تيجي تستخدم تعريفك اللي هو اللي عرفناه وعليه
+
+308
+00:24:49,800 --> 00:24:53,480
+اللي هو بتبدأ تشتغل و تبني اللي هو اللي هي
+
+309
+00:24:53,480 --> 00:24:57,680
+القوانين اللي بنحكي عنها اللي هو تعريفنا اللي هو X
+
+310
+00:24:58,570 --> 00:25:05,210
+to the alpha بتساوي E of alpha len اللي هي L of X
+
+311
+00:25:05,210 --> 00:25:10,650
+أو حسب ال notation تبعتنا E to the alpha len ال X
+
+312
+00:25:10,650 --> 00:25:15,710
+هذا الآن التعريف اللي عليه بدك اللي هو تبدأ اللي
+
+313
+00:25:15,710 --> 00:25:23,290
+هو تشتغل على اللي هي النظرية و تبرهنها اللي عندنا
+
+314
+00:25:23,290 --> 00:25:27,630
+نظرية الأولى اللي ذكرناها قبل بشوية اللي هي هذه
+
+315
+00:25:27,630 --> 00:25:32,090
+النظرية على التعريف مباشرة و نظرية ثانية أيضًا برضه
+
+316
+00:25:32,090 --> 00:25:35,670
+من الخواص إذا كانت Alpha و Beta element ر و X في
+
+317
+00:25:35,670 --> 00:25:40,640
+الفترة Zero ولا نهاية إذا X تو دا ألفة زيادة بيتا
+
+318
+00:25:40,640 --> 00:25:44,040
+برضه نفس الأشياء طبعًا هتلاقي اللي هو أنت لما تيجي
+
+319
+00:25:44,040 --> 00:25:48,320
+تفرد هذه هتصير تستخدم خواص المعرفة اللي هي
+
+320
+00:25:48,320 --> 00:25:52,360
+بواسطتها معرفة هتستخدم خواص ال X exponential و الن
+
+321
+00:25:52,360 --> 00:25:55,420
+اللي قبله بشوية هتلاقي حالك بتصل X تو دا ألفة
+
+322
+00:25:55,420 --> 00:25:58,310
+زيادة بيتا بيساوي X تو دا ألفة في X تو دا بيتا و نفس
+
+323
+00:25:58,310 --> 00:26:05,170
+الشيء x²α²β بيساوي x α beta و بيساوي x²β²α و هتيجي
+
+324
+00:26:05,170 --> 00:26:08,830
+.. اللي هي كلها قوانين احنا بنعرفها x²-α بيساوي 1
+
+325
+00:26:08,830 --> 00:26:12,270
+على x²α و نفس الشيء إذا كانت alpha أصغر من beta
+
+326
+00:26:12,270 --> 00:26:17,770
+هيكون x²α أصغر من x²β لما أن x تكون أكبر من 1 و
+
+327
+00:26:17,770 --> 00:26:22,130
+هذه كلها بتكون x resources معاكم اللي هي مباشرة
+
+328
+00:26:22,130 --> 00:26:31,830
+على هذه التعريف النظرية اللي عندنا على السريع let
+
+329
+00:26:31,830 --> 00:26:35,010
+alpha element in R then the function x بتروح للـ
+
+330
+00:26:35,010 --> 00:26:37,670
+x alpha من 0 و 1 to R is continuous and
+
+331
+00:26:37,670 --> 00:26:41,210
+differentiable and اللي هو ال derivative لل x to
+
+332
+00:26:41,210 --> 00:26:43,630
+the alpha بيساوي alpha to the x to the alpha minus 1
+
+333
+00:26:43,630 --> 00:26:47,650
+for x element in 0 و 1 طبيعي أصلاً هي composition
+
+334
+00:26:47,650 --> 00:26:54,490
+of two هي عندي function هذه اللي هي continuous هذا
+
+335
+00:26:54,490 --> 00:26:57,650
+كلها على بعض ال E كمان continuous ده اللي هتطلع
+
+336
+00:26:57,650 --> 00:26:59,850
+هذا continuous و هذا continuous و نفس الشيء ال
+
+337
+00:26:59,850 --> 00:27:02,870
+differentiability إذا أكيد اللي هي ال function
+
+338
+00:27:02,870 --> 00:27:05,890
+اللي عندنا X to the X to the Alpha حسب تعريفنا is
+
+339
+00:27:05,890 --> 00:27:09,410
+continuous and differentiable و لو بدك تسمي اللي
+
+340
+00:27:09,410 --> 00:27:13,710
+هو هذه ال derivative و بدك تبدأ تفاضل دي اتفاضل DX
+
+341
+00:27:13,710 --> 00:27:17,050
+Alpha يعني بدك تتفاضل هذه كيف تتفاضل هذه ال
+
+342
+00:27:17,050 --> 00:27:20,860
+exponential اللي هي E to the Alpha لن ال X في
+
+343
+00:27:20,860 --> 00:27:25,580
+التفاضل اللي هو اللي جوا اللي هو Alpha في واحد على
+
+344
+00:27:25,580 --> 00:27:30,120
+X ماشي الحال اللي هي بمعنى آخر بصير عندي اللي هو
+
+345
+00:27:30,120 --> 00:27:35,260
+عبارة عن E to the Alpha لن ال X اللي هي عبارة عن
+
+346
+00:27:35,260 --> 00:27:38,480
+ال X to the Alpha نفسها في التفاضل هذه اللي هي
+
+347
+00:27:38,480 --> 00:27:43,080
+Alpha على X بيساوي Alpha أس X اللي هي هذه بتطلع
+
+348
+00:27:43,080 --> 00:27:46,680
+ناقص واحد Alpha ناقص واحد for X element in zero
+
+349
+00:27:46,680 --> 00:27:53,250
+و ما لا نهاية الآن بعض الملاحظات الأخرى اللي بيقولك
+
+350
+00:27:53,250 --> 00:28:01,010
+إياها على هذه الدالة بيقول بيقول لك اللي هو عندي إذا
+
+351
+00:28:01,010 --> 00:28:07,610
+كانت Alpha أكبر من 0 فبيصير
+
+352
+00:28:07,610 --> 00:28:11,970
+عندي اللي هي ال function من X and X alpha is
+
+353
+00:28:11,970 --> 00:28:15,930
+strictly increasing على فترة 0 و ما لا نهاية طبيعي
+
+354
+00:28:15,930 --> 00:28:19,890
+لما Alpha أكبر من 0 هيصير عندي الآن المقدار هذا
+
+355
+00:28:19,890 --> 00:28:24,120
+بيظل موجب هذه ألف أكبر من صفر بيكون بيبقى أكبر من
+
+356
+00:28:24,120 --> 00:28:27,180
+صفر إذا صارت عندي اللي هي ال derivative أكبر من
+
+357
+00:28:27,180 --> 00:28:31,160
+صفر إذا صارت عند الدالة strictly increasing لو كانت
+
+358
+00:28:31,160 --> 00:28:34,520
+ألف أصغر من صفر هتصير اللي هي العكس strictly
+
+359
+00:28:34,520 --> 00:28:38,420
+decreasing لإنه هتكون هذه سالبة و هذه مدلّة موجبة
+
+360
+00:28:38,420 --> 00:28:42,180
+بتظل هذه كلها موجبة إذا صارت strictly decreasing
+
+361
+00:28:42,180 --> 00:28:45,360
+عند ألف بتساوي صفر بيكون احنا ال derivative للواحد
+
+362
+00:28:45,360 --> 00:28:48,880
+اللي هو بيكون عبارة عن constant function اللي هو
+
+363
+00:28:48,880 --> 00:28:51,300
+في حالة ألف بتساوي صفر
+
+364
+00:28:53,910 --> 00:29:02,970
+الآن نيجي إلى اللي هي هيك بنكون اللي هو وصلنا
+
+365
+00:29:02,970 --> 00:29:09,190
+لآخر تعريف بده يعرف اللي هو ال log function للأساس
+
+366
+00:29:09,190 --> 00:29:13,090
+a احنا اللي عرفناه ال len اللي هو للأساس e بمعنى
+
+367
+00:29:13,090 --> 00:29:16,530
+آخر كيف بده يعرفها الآن احنا لسه ما عرفش الأساسات
+
+368
+00:29:16,530 --> 00:29:19,930
+هذا كصمنا ال len و ال exponential الآن بدنا نعرف
+
+369
+00:29:19,930 --> 00:29:25,920
+اللي هو نسمي ال log  اللوغاريثم للأساس A نفترض أن A
+
+370
+00:29:25,920 --> 00:29:28,860
+أكبر من 0 و A لا تساوي 1 it is sometimes useful to
+
+371
+00:29:28,860 --> 00:29:34,820
+define the function log للأساس A كما يلي الآن log A
+
+372
+00:29:34,820 --> 00:29:39,560
+of X كده اللي بيساوي لن ال X على لن ال A حيث ال A
+
+373
+00:29:39,560 --> 00:29:43,940
+عدد ثابت ماشي الحال هذه الآن صارت اللي هي ال log
+
+374
+00:29:
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..965fc060c36529ff49bca3920d56a7cdece6db4e
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw_postprocess.srt
@@ -0,0 +1,1560 @@
+1
+00:00:04,940 --> 00:00:07,660
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:07,660 --> 00:00:10,500
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه
+
+3
+00:00:10,500 --> 00:00:17,340
+أجمعين هذه هي المحاضرة رقم 23 في مساق تحليل حقيقة
+
+4
+00:00:17,340 --> 00:00:22,200
+نيل طلاب وطالبات الجامعة الإسلامية قسم الرياضيات
+
+5
+00:00:22,200 --> 00:00:27,900
+كلية العلومالمحاضرة اللي هي اليوم هي عبارة عن
+
+6
+00:00:27,900 --> 00:00:33,180
+تكملة ل section تمانية تلاتة الجزء الثاني من .. من
+
+7
+00:00:33,180 --> 00:00:36,580
+.. من هذا .. اللي هو ال chapter الجزء الأول من هذا
+
+8
+00:00:36,580 --> 00:00:39,320
+ال section الجزء الأول اتحدثنا عن ال exponential
+
+9
+00:00:39,320 --> 00:00:44,670
+function وكيف أثبتنا وجودهاواخدنا خواصها الآن بدنا
+
+10
+00:00:44,670 --> 00:00:47,910
+نحكي عن الجزء التاني من اللي هو ال section اللي هو
+
+11
+00:00:47,910 --> 00:00:51,050
+ال logarithmic function ال logarithmic function
+
+12
+00:00:51,050 --> 00:00:55,290
+اللي هو نشوف كيف بدنا نثبت وجودها وكيف اللي هو
+
+13
+00:00:55,290 --> 00:01:00,410
+ناخد خواصها بنفس البناء اللي أو نبنع البناء اللي
+
+14
+00:01:00,410 --> 00:01:05,820
+بنناه المحاضرة الماضيةالان لما حكينا عن ال
+
+15
+00:01:05,820 --> 00:01:10,020
+exponential function E لجينا ان ال exponential E
+
+16
+00:01:10,020 --> 00:01:12,780
+is strictly increasing differentiable function
+
+17
+00:01:12,780 --> 00:01:18,160
+with domain R and range اللي هو Y أكبر من 0 يعني
+
+18
+00:01:18,160 --> 00:01:22,480
+لما حكينا عن ال E حكينا عن ال E من R اللي هي ال
+
+19
+00:01:22,480 --> 00:01:26,600
+exponential لعند الفترة 0 و مالا نهاية هذه ال
+
+20
+00:01:26,600 --> 00:01:31,560
+function هي rangeها و هي domainها و كانت strictly
+
+21
+00:01:31,560 --> 00:01:35,700
+increasingStrictly increasing معناته اللي هو عبارة
+
+22
+00:01:35,700 --> 00:01:40,020
+عن 120 يعني بمعنى أخر في إلها ال function هذه on
+
+23
+00:01:40,020 --> 00:01:43,940
+two وكانت differentiable الآن ال function اللي هي
+
+24
+00:01:43,940 --> 00:01:46,840
+ال exponential طبعا ما أنتوا عارفين كيف رسمتها لو
+
+25
+00:01:46,840 --> 00:01:50,460
+جينا جربنا نرسمها هنلاقي الرسمة اللي هو هذه اللي
+
+26
+00:01:50,460 --> 00:01:56,180
+هي عبارة عن رسمة ال exponentialالان انا بدى اجى
+
+27
+00:01:56,180 --> 00:02:00,300
+اللى هو من خلال اللى هو ال function ال exponential
+
+28
+00:02:00,300 --> 00:02:05,980
+اعرف ال inverse لها واسميه اللى هو logarithmic
+
+29
+00:02:05,980 --> 00:02:10,860
+function او بدى اسميه ال logarithm الطبيعى اللى هى
+
+30
+00:02:10,860 --> 00:02:16,020
+ال len functionمشروع الكلام اه لإن ايه عبارة عن
+
+31
+00:02:16,020 --> 00:02:19,240
+function one to one و one to Hana إذا صار ال
+
+32
+00:02:19,240 --> 00:02:23,580
+inverse لها موجود لإنها strictly increasing إذا
+
+33
+00:02:23,580 --> 00:02:29,560
+صار ال L من عند zero و مالة نهاية لعند ال R هادي
+
+34
+00:02:29,560 --> 00:02:34,100
+اللي هيالـ function الجديدة هي اللي بدي أسميها الـ
+
+35
+00:02:34,100 --> 00:02:38,320
+logarithmic function وهي رسمتها اللي أمامنا اللي
+
+36
+00:02:38,320 --> 00:02:42,460
+هي ال inverse لهذه الدالة اللي بدي أعرفها الآن
+
+37
+00:02:42,460 --> 00:02:47,000
+وتعريف الآن صار شرعي بناء على وجود ال exponential
+
+38
+00:02:47,000 --> 00:02:50,930
+اللي بدي أعرفه اللي هو ال inverse سبعتهاالعمل
+
+39
+00:02:50,930 --> 00:02:57,030
+المعرفي للـ E هو
+
+40
+00:02:57,030 --> 00:03:02,850
+الـ Logarithm أو الـ Nature Logarithm اللي هي It
+
+41
+00:03:02,850 --> 00:03:07,870
+will be denoted by L or by Lin الأكتر شيوعا طبعا
+
+42
+00:03:07,870 --> 00:03:11,810
+اللي هو مين الـ Lin لأن بما أن الـ E و L انفرس
+
+43
+00:03:11,810 --> 00:03:17,110
+لبعض إذا أكيد الـ E composite L composite E of X
+
+44
+00:03:17,110 --> 00:03:22,920
+هيساوي الـ Xلكل الـ x و اللي موجودة في الـ R لأن
+
+45
+00:03:22,920 --> 00:03:26,540
+الـ E بتشتغل على كل الإكسات اللي في الـ R لأن
+
+46
+00:03:26,540 --> 00:03:30,740
+بينما E composite L of Y E composite L of Y الـ L
+
+47
+00:03:30,740 --> 00:03:34,660
+بتشتغل .. بتشتغل مين على مين بس على الـ positive E
+
+48
+00:03:34,660 --> 00:03:38,240
+composite L of Y بيساوي لكل Y element in R و Y
+
+49
+00:03:38,240 --> 00:03:44,900
+أشملها أكبر من 0الان connotations .. connotations
+
+50
+00:03:44,900 --> 00:03:49,860
+بناء عليه الـ N الـ E of X لأن الـ N هي الـ L والـ
+
+51
+00:03:49,860 --> 00:03:53,880
+E هي الـ E وعندي الـ E to the N اللي هو بسوء الـ Y
+
+52
+00:03:53,880 --> 00:03:57,780
+وهو بسوء الـ X بناء على أن الواحدة inverse للتانية
+
+53
+00:04:01,010 --> 00:04:04,750
+أو كل واحدة inverse للاخرى الـ logarithm is a
+
+54
+00:04:04,750 --> 00:04:08,630
+strictly increasing function L with domain اللي هو
+
+55
+00:04:08,630 --> 00:04:12,150
+مين اللي هو ال domain اللي عندي اللي هو تعرفنا هيك
+
+56
+00:04:12,150 --> 00:04:16,210
+أصلا الان ال derivative of L is given by L prime
+
+57
+00:04:16,210 --> 00:04:19,750
+of X ايش بتساوي واحدة ل X for X أكبر من سفر الان
+
+58
+00:04:19,750 --> 00:04:23,430
+ال logarithm satisfy the functional equation تحقق
+
+59
+00:04:23,430 --> 00:04:27,010
+المعادلة الدالية التالية اللي هي L of X في Y بساوي
+
+60
+00:04:27,010 --> 00:04:31,000
+L of X زائد LL of Y for X أكبر من سفر Y أكبر من
+
+61
+00:04:31,000 --> 00:04:34,560
+سفر Y أكبر
+
+62
+00:04:34,560 --> 00:04:38,260
+من سفر
+
+63
+00:04:38,260 --> 00:04:40,560
+Y أكبر من سفر Y أكبر من سفر Y أكبر من سفر Y أكبر
+
+64
+00:04:40,560 --> 00:04:40,580
+من سفر Y أكبر من سفر Y أكبر من سفر Y أكبر من سفر Y
+
+65
+00:04:40,580 --> 00:04:40,640
+أكبر من سفر Y أكبر من سفر Y أكبر من سفر Y أكبر من
+
+66
+00:04:40,640 --> 00:04:40,700
+سفر Y أكبر من سفر Y أكبر من سفر Y أكبر من سفر Y
+
+67
+00:04:40,700 --> 00:04:41,020
+سفر Y أكبر من سفر Y أكبر من سفر Y أكبر من سفر Y
+
+68
+00:04:41,020 --> 00:04:47,140
+أكبر من سفر Y أكبر من سفر Y أكبر من سفر Y أكبرL of
+
+69
+00:04:47,140 --> 00:04:51,420
+XR بيساوي R لL of X لإن X أكبر من 0 وR المتر كيوه
+
+70
+00:04:51,420 --> 00:04:54,840
+كلهن اللي هي خواص انتوا بتعرفوهم قبل هيك بس الآن
+
+71
+00:04:54,840 --> 00:04:58,440
+بدنا نبرهنهم و نثبت صحتهم limit L of X لما X تروح
+
+72
+00:04:58,440 --> 00:05:01,740
+لـ0 من اليمين بساوي سالب infinity and limit L of X
+
+73
+00:05:01,740 --> 00:05:07,340
+لما X تروح لمال النهاية بتساوي مال النهاية خلينا
+
+74
+00:05:07,340 --> 00:05:14,840
+احنا نشوف نبرهن اللي هي اللي مطلوبالأن الـ L is
+
+75
+00:05:14,840 --> 00:05:17,560
+strictly increasing with domain X element alone
+
+76
+00:05:17,560 --> 00:05:20,880
+and range R follows from the fact that E is
+
+77
+00:05:20,880 --> 00:05:24,840
+strictly increasing with domain R and range اللي
+
+78
+00:05:24,840 --> 00:05:33,320
+هو اللي عندي الان عندي ال L is strictly increasing
+
+79
+00:05:33,320 --> 00:05:37,560
+بناء على مين على ال E نفسها strictly increasing
+
+80
+00:05:37,560 --> 00:05:48,560
+الان EComposite L E Composite L of Y ايش بتساوي؟ Y
+
+81
+00:05:48,560 --> 00:05:55,320
+لكل Y وY الموجودة لكل Y element in سفر وملا نهاية،
+
+82
+00:05:55,320 --> 00:06:00,780
+مظبوط؟ الان فاضلول الجهتين الان طبعا احنا بنعرف
+
+83
+00:06:00,780 --> 00:06:06,360
+انه من الاصل مدامة ال E isاللي هو differentiable
+
+84
+00:06:06,360 --> 00:06:10,480
+أكيد اللي هي ال inverse إلها is differentiable by
+
+85
+00:06:10,480 --> 00:06:14,680
+theorem 6 9 كده مش عارف إيش في اللي هو chapter 6
+
+86
+00:06:14,680 --> 00:06:18,200
+قدامة ال function اللي هي is differentiable ال
+
+87
+00:06:18,200 --> 00:06:20,880
+inverse إلها برضه is differentiable في حالة وجودها
+
+88
+00:06:20,880 --> 00:06:27,480
+الآن E فاضل الجهتين بيصير عندي E prime of L of Y
+
+89
+00:06:27,480 --> 00:06:36,600
+فL prime of Y بساوي إيش واحد ماشي الحال الآنواضح
+
+90
+00:06:36,600 --> 00:06:41,900
+أن هذا حاصل الضرب صار أكبر من مين strictly من 0
+
+91
+00:06:41,900 --> 00:06:47,920
+وبما أن الـ E is strictly increasing أثبتنا E' of
+
+92
+00:06:47,920 --> 00:06:53,420
+L of Y is strictly أكبر من 0 إذا بيظلها L' of Y is
+
+93
+00:06:53,420 --> 00:06:57,180
+strictly أكبر من 0 لكل Y هنا إذا صارت الـ L is
+
+94
+00:06:57,180 --> 00:07:02,740
+strictly increasingالان طبعا ال domain مدام ان هذه
+
+95
+00:07:02,740 --> 00:07:07,000
+ال inverse ل ال E ال domain اللي هو ال inverse هو
+
+96
+00:07:07,000 --> 00:07:11,380
+range ال function الأصلية و بيصير sub wave في
+
+97
+00:07:11,380 --> 00:07:18,460
+الفئر اذا الان احنا اثبتنا ان ال is strictly
+
+98
+00:07:18,460 --> 00:07:23,530
+increasingالان و rangeها اللي هو صار domain اللي
+
+99
+00:07:23,530 --> 00:07:28,310
+هو او range اللي هو ال .. هذي اللي صار domainها
+
+100
+00:07:28,310 --> 00:07:32,990
+domain ال L و هذي صارت اللي هو range ال L زي ما
+
+101
+00:07:32,990 --> 00:07:37,330
+قلنا قبل بشوية او عندي الان بدنا نثبت اللي هو
+
+102
+00:07:37,330 --> 00:07:42,070
+الجزء الثاني من النظرية خليني نكتب هنا عشان نتذكر
+
+103
+00:07:42,070 --> 00:07:48,430
+ايش اللي بدنا نثبته الان بدنا نثبت اين أثبتنا
+
+104
+00:07:48,430 --> 00:07:54,230
+الأولىاللي فى النص بتقلّى prime of X اللى نكتبهن
+
+105
+00:07:54,230 --> 00:07:59,330
+اللى بدنا نثبتهن عشان نتذكرهن
+
+106
+00:07:59,330 --> 00:08:12,240
+قلّى prime I او VIII قلّى primeof X بسوة واحدة ل X
+
+107
+00:08:12,240 --> 00:08:19,680
+اتنين اللي هو L of XY بسوة L X زائد L Y طبعا ال Y
+
+108
+00:08:19,680 --> 00:08:25,780
+هناك ال L of واحد بسوة سفر L of E بسوة واحد كلهم
+
+109
+00:08:25,780 --> 00:08:34,440
+بسيطاتL prime L of X to the R سوى R L of X و Limit
+
+110
+00:08:34,440 --> 00:08:39,720
+L of X لما X تروح إلى Zero من اليمين سوى سالب لما
+
+111
+00:08:39,720 --> 00:08:45,140
+لنهاية و Limit لL of X لما X تروح إلى ما لنهاية
+
+112
+00:08:45,140 --> 00:08:48,520
+سوى ما لنهاية خلّيني أشوف أن دول على السريع فانوا
+
+113
+00:08:48,520 --> 00:08:53,640
+كلها شغلات يعني بأعتقد أنه سهل أنك نثبتها
+
+114
+00:08:55,730 --> 00:09:02,210
+عندي لأن زي ما عملت قبل بشوية اللي هو لما فضلت هذه
+
+115
+00:09:02,210 --> 00:09:07,290
+تفاضله E composite L of X لما عملتها قبل بشوية
+
+116
+00:09:07,290 --> 00:09:14,250
+اللي هي كانت عندي هين أعمل E composite L of X الكل
+
+117
+00:09:14,250 --> 00:09:19,370
+اللي هي بساوي ال X فاضل هذا يصير E prime
+
+118
+00:09:26,060 --> 00:09:29,100
+بنسبة لـ x أقل الـ prime of x
+
+119
+00:09:34,780 --> 00:09:40,160
+اللي هو واحد على الـ E prime of L of X إذا الـ E
+
+120
+00:09:40,160 --> 00:09:44,300
+القلي برايم of X بيساوي واحد على E prime composite
+
+121
+00:09:44,300 --> 00:09:48,340
+L of X والـ E prime هي نفس الـ E زي ما قلنا إذا
+
+122
+00:09:48,340 --> 00:09:51,140
+بيصير واحد على E composite L of X إلى ال E
+
+123
+00:09:51,140 --> 00:09:54,640
+composite L of X زي ما قلنا إيش بتساوي بساوي X
+
+124
+00:09:54,640 --> 00:09:57,660
+فبساوي واحد على X فالقلي برايم بيساوي واحد على X
+
+125
+00:09:57,660 --> 00:10:02,910
+لكل X في الموجودة في الفترة Zeroو 1 نيجي الآن نشوف
+
+126
+00:10:02,910 --> 00:10:06,710
+اللي هي اللي بعدها الخاصية اللي بعدها خلّينا نثبت
+
+127
+00:10:06,710 --> 00:10:12,690
+اللي هو L of X في Y بسوا L of X زائد مين زائد L of
+
+128
+00:10:12,690 --> 00:10:17,270
+Y برضه الإثبات سهل وانتبهوا معايا وسهلس عندى الآن
+
+129
+00:10:24,240 --> 00:10:27,240
+F X أكبر من صفر Y أكبر من صفر F X أكبر من صفر Y
+
+130
+00:10:27,240 --> 00:10:27,740
+أكبر من صفر F X أكبر من صفر Y أكبر من صفر F X أكبر
+
+131
+00:10:27,740 --> 00:10:27,900
+من صفر F X أكبر من صفر F X أكبر من صفر F X أكبر من
+
+132
+00:10:27,900 --> 00:10:28,100
+صفر F X أكبر من صفر F X أكبر من صفر F X أكبر من
+
+133
+00:10:28,100 --> 00:10:28,700
+صفر F X أكبر من صفر F X أكبر من صفر F X أكبر من
+
+134
+00:10:28,700 --> 00:10:28,920
+صفر F X أكبر من صفر F X أكبر من صفر F X أكبر من
+
+135
+00:10:28,920 --> 00:10:32,100
+صفر F X أكبر من صفر F X أكبر من صفر F X أكبر من
+
+136
+00:10:32,100 --> 00:10:43,400
+صفر F X أكبر من صفر F X أكبر من صفر F X
+
+137
+00:10:43,400 --> 00:10:51,850
+أكبر من صفر F Xلأن الـ E والـ L انفرسة بعض الان من
+
+138
+00:10:51,850 --> 00:10:55,130
+الخاصية تبع الـ exponential بدنا نصل لمن؟ لل
+
+139
+00:10:55,130 --> 00:11:00,970
+logarithmic إذا أضرب ل X في Y بيطلع عند X في Y
+
+140
+00:11:00,970 --> 00:11:05,190
+بتساوي E of U في E of V E of U في E of V إيش
+
+141
+00:11:05,190 --> 00:11:10,010
+بتساوي؟ E of U زائد V أثبتناها إذا من هذا الكلام
+
+142
+00:11:10,970 --> 00:11:15,270
+خُد الـ L للجهتين لإنه اللي هي ال inverse لبعض
+
+143
+00:11:15,270 --> 00:11:20,450
+بيصير عندي L of X في Y بساوي L of E of U زائد V
+
+144
+00:11:20,450 --> 00:11:24,410
+اللي هي إيش بتساوي U زائد V U اللي هي عبارة عن L
+
+145
+00:11:24,410 --> 00:11:30,750
+of X و V عبارة عن L of Y إذا أثبتت L of X زائد X
+
+146
+00:11:30,750 --> 00:11:39,370
+في Y بساوي L of X زائد L of Yالان عندى اللى هى E
+
+147
+00:11:39,370 --> 00:11:47,050
+of Zero بيساوي واحد خدلى ال L الجهتين بيصير ZL of
+
+148
+00:11:47,050 --> 00:11:53,010
+E of Zero بيساوي L of واحد ال L of E of Zero هدك
+
+149
+00:11:53,010 --> 00:11:59,450
+inverse التانية بيساوي Zero نفس الاشيالـ L of E of
+
+150
+00:11:59,450 --> 00:12:07,270
+1 بيساوي L of E، مظبوط؟ الـ L of E of 1 بيساوي 1،
+
+151
+00:12:07,270 --> 00:12:12,230
+بيصير L of E بيساوي 1 بيصير أثبتنا L of E بيساوي 1
+
+152
+00:12:12,230 --> 00:12:19,730
+و L of 1 بيساوي 0 وهذا الكلام كلام سهل، طيب بيصير
+
+153
+00:12:19,730 --> 00:12:23,710
+علاقة أنه سهل اللي بنحكيه، الآن
+
+154
+00:12:27,210 --> 00:12:32,730
+نأتي نثبت اللي هو اللي هي L of X to the R بساوة
+
+155
+00:12:32,730 --> 00:12:37,010
+Zero بساوة R في L of X هذه برضه بقاش By
+
+156
+00:12:37,010 --> 00:12:41,910
+Mathematical Induction عملناها قبل هيكة باللي هو
+
+157
+00:12:41,910 --> 00:12:47,330
+ال section اللي جابله أو اللي هو ال exponential
+
+158
+00:12:47,330 --> 00:12:52,830
+على السريع نشوف التفاصيل لإنه التفاصيل معادة
+
+159
+00:13:07,050 --> 00:13:11,430
+التفاصيل هتلاجوها معادة فخليني بسرعة نمر عليها
+
+160
+00:13:11,430 --> 00:13:17,730
+عندي we show by induction L of X بسوا L L of X زي
+
+161
+00:13:17,730 --> 00:13:21,850
+ما قلنا عشان لثبتها هذه أثبتنا اللي قبل بشوية L of
+
+162
+00:13:21,850 --> 00:13:27,190
+X عارفش الرزوم أصلا أفصل ولا لأ لكن بدي أفصل لو
+
+163
+00:13:27,190 --> 00:13:32,790
+أنتوا عندى لو اتفجنا أن نفصل ولا لأ L of X في Y
+
+164
+00:13:32,790 --> 00:13:39,580
+بسوا L of X في L of Yof x was n بساوي n في L of x
+
+165
+00:13:39,580 --> 00:13:44,060
+طبعا for n بتساوي واحد اللي هي a trivial نفترض
+
+166
+00:13:44,060 --> 00:13:48,180
+أنها صحيحة ل L ل n بتساوي k بيصير L of x was k
+
+167
+00:13:48,180 --> 00:13:53,080
+بساوي k L of x الآن بدنا نحسب ل L of x was k زائد
+
+168
+00:13:53,080 --> 00:13:59,480
+واحد اللي هي بساوي L of x was k في xهذه الـ L لها
+
+169
+00:13:59,480 --> 00:14:04,760
+حسب اللي هي الخاصية هذه بسوء L الأولى X plus K في
+
+170
+00:14:04,760 --> 00:14:09,860
+L التاني L of X اللي هو مفترض إنها صحيحة على K ده
+
+171
+00:14:09,860 --> 00:14:19,600
+بسوء K في L of X آسف زائد هذه بسوء K L of Xلأنها
+
+172
+00:14:19,600 --> 00:14:26,400
+صحيحة لـ K زائد L of X ويساوي K زائد واحد في L of
+
+173
+00:14:26,400 --> 00:14:31,520
+X إذا صارت هذه صحيحة اللي هي L ل K زائد واحد إذا
+
+174
+00:14:31,520 --> 00:14:36,760
+صارت صحيحة لكل من لكل N element in N حسب اللي هو
+
+175
+00:14:36,760 --> 00:14:43,080
+ال induction اللي بنحكي فيه إذا الأن أثبتنا أن L
+
+176
+00:14:43,080 --> 00:14:49,710
+of X هو N لL of X لكل اللي هي عنديالان by VI اللي
+
+177
+00:14:49,710 --> 00:14:53,530
+هو زي .. مشابه للي حكيناها قبل بشوية بالظبط في حد
+
+178
+00:14:53,530 --> 00:14:58,890
+ال exponential بس خليني مش وشكلة بتعيده الان شوف L
+
+179
+00:14:58,890 --> 00:15:03,680
+of XM minus M أيش بتساوي؟ L of واحداللي هي لأن هذا
+
+180
+00:15:03,680 --> 00:15:05,860
+X هو السفر اللي هي L of واحد L of واحد مش قولنا
+
+181
+00:15:05,860 --> 00:15:11,040
+عنها سفر هو يسوى L of XM في XM minus واحد ال
+
+182
+00:15:11,040 --> 00:15:16,300
+logarithmic بطلح انجمع L of XM زائد L of X minus M
+
+183
+00:15:16,300 --> 00:15:21,920
+لأن هذه أثبتناها عبارة عن M L of X زائدL of X
+
+184
+00:15:21,920 --> 00:15:27,500
+minus M صار عندي الان M في L of X زائد L of X
+
+185
+00:15:27,500 --> 00:15:32,040
+minus M بسوا سفر انجلي هذا على الجهة الثانية بطلع
+
+186
+00:15:32,040 --> 00:15:35,760
+L of X minus M اللي قعدت لحالها بسوا ناقص M في L
+
+187
+00:15:35,760 --> 00:15:41,720
+of X اذا صار عندي الان لكل M سواء موجبة او سالبة
+
+188
+00:15:41,720 --> 00:15:48,940
+بطلع عندي اللي هي L of X قص Mبساوة M في L of X
+
+189
+00:15:48,940 --> 00:15:53,860
+سواء كانت وجبة أو سالبة نيجي الآن منها بدنا ناخد
+
+190
+00:15:53,860 --> 00:15:57,360
+لمين لأ اللي هي ال R لان therefore for any M
+
+191
+00:15:57,360 --> 00:16:02,800
+element in Z و N element in Nعندي احسبلي الان L of
+
+192
+00:16:02,800 --> 00:16:07,020
+X أس M على N بس ضربليها في N بعد إذنك Y ساوي اللي
+
+193
+00:16:07,020 --> 00:16:12,460
+هي L of X أس M على N لكل ما له أس N لإنه صحيحة هذه
+
+194
+00:16:12,460 --> 00:16:18,080
+لل N اللي هي في N واتفجنا عليها الان هذه بتساوي
+
+195
+00:16:18,080 --> 00:16:21,520
+هذه واضحة لإن هذه هي ال X تبعتنا وهذه ال N بتطلع
+
+196
+00:16:21,520 --> 00:16:27,410
+براالان هذه الان مع الان بيصير L of X plus M L of
+
+197
+00:16:27,410 --> 00:16:30,870
+X plus M قبل بشوية بقى قولنا عنها بيساوي M L of X
+
+198
+00:16:30,870 --> 00:16:34,270
+سواء كانت M positive أو اللي هو negative صار هذه
+
+199
+00:16:34,270 --> 00:16:40,930
+بتساوي هذه إذا انجلي الآنالان هذه ان بيصير عندي L
+
+200
+00:16:40,930 --> 00:16:44,910
+of X of M على N بيسوي M على N في L of X إذا صار
+
+201
+00:16:44,910 --> 00:16:49,490
+عندي لأي rational number صار عندي L of X R بيسوي R
+
+202
+00:16:49,490 --> 00:16:55,670
+L of X لكل R اللي بنتميين NQ نيجي الآن لأ اللي هي
+
+203
+00:16:55,670 --> 00:16:57,230
+الجزء الأخير من النظرية
+
+204
+00:17:08,180 --> 00:17:11,120
+الكلام مشابه للي حكيناها قبل بشوية في الإثبات اللي
+
+205
+00:17:11,120 --> 00:17:15,260
+هو ال limit تبع ال exponential عند 2 أصغر من E
+
+206
+00:17:15,260 --> 00:17:19,380
+وقلنا ليش الان ال E n هنا بيصير 2 أصغر من E أصغر
+
+207
+00:17:19,380 --> 00:17:19,920
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+208
+00:17:19,920 --> 00:17:21,140
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+209
+00:17:21,140 --> 00:17:22,840
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+210
+00:17:22,840 --> 00:17:23,240
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+211
+00:17:23,240 --> 00:17:26,700
+من E أصغر من E أصغر من E أصغر من E أصغر من E أصغر
+
+212
+00:17:26,700 --> 00:17:30,620
+من E أصغ
+
+213
+00:17:33,960 --> 00:17:39,580
+لكن اللي هو L of E N بسوء N and اللي هي L of E
+
+214
+00:17:39,580 --> 00:17:44,020
+minus N بسوء نقص N خليني في الذاكر هذول اذا for
+
+215
+00:17:44,020 --> 00:17:47,480
+every N element in R there exists X element in R
+
+216
+00:17:47,480 --> 00:17:52,670
+بحيث ان X أكبر من مين من Nلكل N element in N في X
+
+217
+00:17:52,670 --> 00:17:56,350
+element in R أكيد X أكبر من مين من E N لأنه أخسرت
+
+218
+00:17:56,350 --> 00:18:00,590
+N بين إيديا حسبت ال E N طلع عند رقم أخدت ال X أكبر
+
+219
+00:18:00,590 --> 00:18:03,310
+منها كيد بالله طيب لأنه unbounded real numbers
+
+220
+00:18:03,310 --> 00:18:06,710
+then L في هذه أكبر أو من L في هذه لأنه ال L
+
+221
+00:18:06,710 --> 00:18:10,810
+strictly increasing إذا صار L في X أكبر من مين من
+
+222
+00:18:10,810 --> 00:18:16,130
+ال NA ألوف E أن يعني أكبر من الان لأن limit هذه as
+
+223
+00:18:16,130 --> 00:18:21,210
+x goes to infinity اللي هو بيكون أكبر أساوي اللي
+
+224
+00:18:21,210 --> 00:18:23,950
+هي limit هذه as n goes to infinity ويساوى infinity
+
+225
+00:18:23,950 --> 00:18:27,550
+لأن لكل لما الان تروح لما لنهاية أكيد ال X بتروح
+
+226
+00:18:27,550 --> 00:18:31,270
+لمين إلى ما لنهاية فصار عندي هذه بتروح إلى ما
+
+227
+00:18:31,270 --> 00:18:34,970
+لنهاية الآن صار عندي limit L of X لما X تروح لما
+
+228
+00:18:34,970 --> 00:18:41,040
+لنهاية بيساوي ما لنهاية similarlyالان لكل نقص any
+
+229
+00:18:41,040 --> 00:18:43,100
+element in z positive بلاقي x element in r
+
+230
+00:18:43,100 --> 00:18:45,740
+positive بحيث ان x أكبر من صفر و أصغر من صفر و
+
+231
+00:18:45,740 --> 00:18:46,340
+أصغر من صفر و أصغر من صفر و أصغر من صفر و أصغر من
+
+232
+00:18:46,340 --> 00:18:47,840
+صفر و أصغر من صفر و أصغر من صفر و أصغر من صفر و
+
+233
+00:18:47,840 --> 00:18:51,500
+أصغر من صفر و أصغر من صفر و أصغر من صفر و أصغر من
+
+234
+00:18:51,500 --> 00:18:57,960
+صفر و أصغر من صفر و أصغر من صفر و أصغر من صفر و
+
+235
+00:18:57,960 --> 00:19:02,520
+أصغر من صفر و أصغر من صفر و أصغر من صفر و أصغر من
+
+236
+00:19:02,520 --> 00:19:09,620
+صفر و أصغر من صفر و أصغر من صفر و أطيب then L of X
+
+237
+00:19:09,620 --> 00:19:13,660
+هيكون أصغر يساوي ال E to the minus N يعني L of X
+
+238
+00:19:13,660 --> 00:19:17,800
+اللي هي أصغر يساوي ناقص N إذا as N goes to
+
+239
+00:19:17,800 --> 00:19:22,700
+infinity as N goes to infinity اللي هو ال E to the
+
+240
+00:19:22,700 --> 00:19:26,540
+minus N بيروح لل0 من اليمين إذا ال X بتروح لل0 من
+
+241
+00:19:26,540 --> 00:19:30,820
+اليمين إذا عندي ال X بتروح لل 0 من اليميناللي هو
+
+242
+00:19:30,820 --> 00:19:34,580
+أصغر لما الـ limit L of X أصغر من limit E to the
+
+243
+00:19:34,580 --> 00:19:37,900
+minus N لما هذا يروح لـ 0 من اليمين أو بمعنى أخر
+
+244
+00:19:37,900 --> 00:19:41,680
+لما الـ N تروح لمهة لنهاية و هذا بيروح لمين؟ لـ
+
+245
+00:19:41,680 --> 00:19:44,760
+Infinity إذا limit L of X لما X تروح لمهة لنهاية
+
+246
+00:19:44,760 --> 00:19:50,920
+بساوي سالب Infinity و هو المطلوب hence limit L of
+
+247
+00:19:50,920 --> 00:19:54,460
+X لما X تروح لـ 0 من اليمين بساوي سالب Infinity
+
+248
+00:19:54,460 --> 00:19:58,500
+طيب
+
+249
+00:20:12,260 --> 00:20:16,320
+الان سارعنا الان نقدر ان اللي هو نحكي عن ال bar
+
+250
+00:20:16,320 --> 00:20:20,080
+functions بدنا نعرف ال bar functions اللي هي بناء
+
+251
+00:20:20,080 --> 00:20:25,060
+على اللي حكيناه و اللي هي موضوع ال bar functions
+
+252
+00:20:25,060 --> 00:20:28,880
+كل ما فيه تقريبا يعني بنعتبره exercises احنا لكن
+
+253
+00:20:28,880 --> 00:20:32,520
+خلينا نعرف التعريفات و النظريات بتكون اللي هي
+
+254
+00:20:32,520 --> 00:20:35,900
+معاكم exercises بسيطة بناء على التعريف اللي
+
+255
+00:20:35,900 --> 00:20:40,720
+بنعرفها اللي هناخد α يليمنتان R و X أكبر من 0The
+
+256
+00:20:40,720 --> 00:20:43,320
+number X to the Alpha is defined to be .. الآن بدي
+
+257
+00:20:43,320 --> 00:20:46,940
+أعرف حاجة اسمها X to the Alpha X to the Alpha بدي
+
+258
+00:20:46,940 --> 00:20:49,900
+أعرفها .. إيش بدي أعرفها؟ بإيشي أنا معرف عندي من
+
+259
+00:20:49,900 --> 00:20:54,440
+الأصل ال exponential معرفة .. خلصنا منها و ال len
+
+260
+00:20:54,440 --> 00:20:59,000
+معرفة .. إذا E to the Alpha في لن ال X هذه المقدار
+
+261
+00:20:59,000 --> 00:21:03,820
+لهذا معرف و هذا معرف بدي أسمي X to the Min to the
+
+262
+00:21:03,820 --> 00:21:07,400
+Alpha اللي هو في الواقع عبارة عن Min E to the
+
+263
+00:21:07,400 --> 00:21:12,040
+AlphaL of X L of X معرفة والـ E معرفة إذا كل هذه
+
+264
+00:21:12,040 --> 00:21:16,000
+معرفة بتسميها X to the main to the alpha الآن صارت
+
+265
+00:21:16,000 --> 00:21:19,540
+عندي يعني قيمة الـ X under this function اللي
+
+266
+00:21:19,540 --> 00:21:23,120
+عرفتها لجديدة يعني إذا بتسميها دي ال function الـ
+
+267
+00:21:23,120 --> 00:21:28,240
+R of X إيش عرفتها أنا بتساوي X to the alpha؟يعني
+
+268
+00:21:28,240 --> 00:21:31,680
+كل ال X بتصير يشمل X to the Alpha و X أكبر من 0
+
+269
+00:21:31,680 --> 00:21:36,780
+هذه ال X to the Alpha هي اللي بدي أسميها ال power
+
+270
+00:21:36,780 --> 00:21:42,000
+function بدي أسميها power function with exponent
+
+271
+00:21:42,000 --> 00:21:47,540
+mean Alpha و ال X هي أشمالها المتغيرة اللي أكبر من
+
+272
+00:21:47,540 --> 00:21:54,340
+0 شوف الآن نشوف بعض الخواص اللي هو هذه اللي هي
+
+273
+00:21:54,340 --> 00:21:56,460
+الدالة طيب
+
+274
+00:22:04,730 --> 00:22:08,690
+الان if x أكبر من 0 and alpha بساوة m على n where
+
+275
+00:22:08,690 --> 00:22:12,770
+m element in z و n element in n then we define x
+
+276
+00:22:12,770 --> 00:22:17,790
+to the alpha بساوة x to the m أسواحد على n in
+
+277
+00:22:17,790 --> 00:22:23,110
+section mean خمسة ستة هتعرفناها زمان انه في حالة
+
+278
+00:22:23,110 --> 00:22:26,570
+بس اللي هي ال rational number عرفنا x to the m على
+
+279
+00:22:26,570 --> 00:22:30,630
+n بساوة x to the m لكل أسواحد على nماشي الحال
+
+280
+00:22:30,630 --> 00:22:34,510
+..الان بدنا نشوف هذا التعريف مطابق لتعريفنا اليوم
+
+281
+00:22:34,510 --> 00:22:41,670
+ولا لأ hence we have لن ال X to the Alpha لن ال X
+
+282
+00:22:41,670 --> 00:22:45,370
+to the Alpha لن ال X to the Alpha بساوي Alpha لن
+
+283
+00:22:45,370 --> 00:22:51,540
+ال Xعرفناها هذه طيب where X to the Alpha بيسوا E
+
+284
+00:22:51,540 --> 00:22:56,260
+to the Lin X to the Min to the Alpha اللي هو بيسوا
+
+285
+00:22:56,260 --> 00:23:01,020
+E to the Alpha في Min في Lin ال X كلام كله سهل ال
+
+286
+00:23:01,020 --> 00:23:05,740
+X to the Alpha هو اللي عرفناها اللي عبارة عن E
+
+287
+00:23:05,740 --> 00:23:10,570
+بتصير to the Lin X to the Alphaلأنه استبدلت الـ x
+
+288
+00:23:10,570 --> 00:23:15,090
+to the alpha بقيمتها اللي هي عبارة عن اللي هي
+
+289
+00:23:15,090 --> 00:23:18,630
+alpha ln x اللي هي بالساوية e to the ln x to the
+
+290
+00:23:18,630 --> 00:23:24,510
+mean to the alpha إذا سواء احنا بالتعريف اللي هو
+
+291
+00:23:24,510 --> 00:23:28,190
+احنا هذا بال exponent أو بال ال function اللي
+
+292
+00:23:28,190 --> 00:23:33,110
+عرفناها بالشكل هذا هيطلع عندي اللي هو القيمتين نفس
+
+293
+00:23:33,110 --> 00:23:34,790
+القيمة طيب
+
+294
+00:23:37,160 --> 00:23:42,300
+نجي الآن لبعض الخواص اللي هي تبعت ال exponential
+
+295
+00:23:42,300 --> 00:23:47,180
+ال power function و الخواص هنتركه لكم إياه لأنها
+
+296
+00:23:47,180 --> 00:23:54,600
+مباشرة على التعريف تبعنا مباشرة
+
+297
+00:23:54,600 --> 00:24:01,100
+على اللي هي التعريف اللي عندنا و هيكون في عندي
+
+298
+00:24:01,100 --> 00:24:05,760
+الآن النظرية الأولىاللي هي 8 3 11 لو كانت Alpha
+
+299
+00:24:05,760 --> 00:24:11,340
+element in R و X و Y اللي هو تنتمي للفترة Zero و
+
+300
+00:24:11,340 --> 00:24:16,500
+تمانية Zero و ما لنهاية آسف then معلش عشان ده طلعت
+
+301
+00:24:16,500 --> 00:24:20,450
+الكهربا قعد نقلف الكهربا أناIf α element in R و X
+
+302
+00:24:20,450 --> 00:24:26,350
+Y belongs to 0 α then 1 to the α بسوء 1 و X to the
+
+303
+00:24:26,350 --> 00:24:30,270
+α أكبر من 0 و X Y to the α بسوء X to the α و Y to
+
+304
+00:24:30,270 --> 00:24:35,590
+the α و X على Y to the α بسوء X to the α على Y to
+
+305
+00:24:35,590 --> 00:24:39,370
+the α هذه اللي هي النظرية طبعا اللي هي مباشرة على
+
+306
+00:24:39,370 --> 00:24:45,100
+تعريفنا اللي هو X to the α بسوء Eof α لن X يعني
+
+307
+00:24:45,100 --> 00:24:49,800
+بدك تيجي تستخدم تعريفك اللي هو اللي عرفناه وعليه
+
+308
+00:24:49,800 --> 00:24:53,480
+اللي هو بتبدأ تشتغل و تبني اللي هو اللي هي
+
+309
+00:24:53,480 --> 00:24:57,680
+القوانين اللي بنحكي عنها اللي هو تعريفنا اللي هو X
+
+310
+00:24:58,570 --> 00:25:05,210
+to the alpha بتساوي E of alpha len اللي هي L of X
+
+311
+00:25:05,210 --> 00:25:10,650
+أو حسب ال notation تبعتنا E to the alpha len ال X
+
+312
+00:25:10,650 --> 00:25:15,710
+هذا الآن التعريف اللي عليه بدك اللي هو تبدأ اللي
+
+313
+00:25:15,710 --> 00:25:23,290
+هو تشتغل على اللي هي النظريةوتبرهنها اللي عندنا
+
+314
+00:25:23,290 --> 00:25:27,630
+نظرية الأولى اللي ذكرناها قبل بشوية اللي هي هذه
+
+315
+00:25:27,630 --> 00:25:32,090
+النظرية على التعريف مباشرة ونظرية تانية أيضا برضه
+
+316
+00:25:32,090 --> 00:25:35,670
+من الخواص إذا كانت Alpha و Beta element ر و X في
+
+317
+00:25:35,670 --> 00:25:40,640
+الفترة Zero ولا نهاية إذا Xتو دا ألفة زيادة بيتا
+
+318
+00:25:40,640 --> 00:25:44,040
+برضه نفس الاشياء طبعا هتلاقي اللي هو انت لما تيجي
+
+319
+00:25:44,040 --> 00:25:48,320
+تفرد هذه هتصير تستخدم خواص المعرفة اللي هي
+
+320
+00:25:48,320 --> 00:25:52,360
+بواسطتها معرفة هتستخدم خواص ال X exponential والن
+
+321
+00:25:52,360 --> 00:25:55,420
+اللي قبله بشوية هتلاقي حالك بتصل X تو دا ألفة
+
+322
+00:25:55,420 --> 00:25:58,310
+زيادة بيتا بسوء X تو دا ألفة في X تو دا بيتاو نفس
+
+323
+00:25:58,310 --> 00:26:05,170
+الشيء x²α²β بيساوي x α beta و يساوي x²β²α و هتيجي
+
+324
+00:26:05,170 --> 00:26:08,830
+.. اللي هي كلها قوانين احنا بنعرفها x²-α بيساوي 1
+
+325
+00:26:08,830 --> 00:26:12,270
+على x²α و نفس الشيء إذا كانت alpha أصغر من beta
+
+326
+00:26:12,270 --> 00:26:17,770
+هيكون x²α أصغر من x²β لما ان x تكون أكبر من 1 و
+
+327
+00:26:17,770 --> 00:26:22,130
+هذه كلها بتكون x resources معاكم اللي هي مباشرة
+
+328
+00:26:22,130 --> 00:26:31,830
+على هذه التعريفالنظرية اللي عندنا على السريع let
+
+329
+00:26:31,830 --> 00:26:35,010
+alpha element in R then the function x بالتروح للـ
+
+330
+00:26:35,010 --> 00:26:37,670
+x alpha من 0 و 1 to R is continuous and
+
+331
+00:26:37,670 --> 00:26:41,210
+differentiable and اللي هو ال derivative لل x to
+
+332
+00:26:41,210 --> 00:26:43,630
+the alpha بسوء alpha to the x to the alpha minus 1
+
+333
+00:26:43,630 --> 00:26:47,650
+for x element in 0 و 1 طبيعي أصلا هي composition
+
+334
+00:26:47,650 --> 00:26:54,490
+of two هي عندى function هذه اللي هي continuousهذا
+
+335
+00:26:54,490 --> 00:26:57,650
+كلها على بعض الـ E كمان continuous ده اللي هتطلع
+
+336
+00:26:57,650 --> 00:26:59,850
+هذا continuous و هذا continuous و نفس الاشي ال
+
+337
+00:26:59,850 --> 00:27:02,870
+differentiability إذا أكيد اللي هي ال function
+
+338
+00:27:02,870 --> 00:27:05,890
+اللي عندنا X to the X to the Alpha حسب تعريفنا is
+
+339
+00:27:05,890 --> 00:27:09,410
+continuous and differentiable و لو بدك تسمي اللي
+
+340
+00:27:09,410 --> 00:27:13,710
+هو هذه ال derivative و بدك تبدأ تفاضل دي اتفاضل DX
+
+341
+00:27:13,710 --> 00:27:17,050
+Alpha يعني بدك تتفاضل هذه كيف تتفاضل هذه ال
+
+342
+00:27:17,050 --> 00:27:20,860
+exponential اللي هي E to the Alpha لإن ال Xفى
+
+343
+00:27:20,860 --> 00:27:25,580
+التفاضل اللى هو اللى جوا اللى هو Alpha فى واحد على
+
+344
+00:27:25,580 --> 00:27:30,120
+X ماشي الحال اللى هى بمعنى آخر بصير عندى اللى هو
+
+345
+00:27:30,120 --> 00:27:35,260
+عبارة عن E to the Alpha لن ال X اللى هى عبارة عن
+
+346
+00:27:35,260 --> 00:27:38,480
+ال X to the Alpha نفسها فى التفاضل هذه اللى هى
+
+347
+00:27:38,480 --> 00:27:43,080
+Alpha على X بيساوى Alpha أس X اللى هى هذه بتطلع
+
+348
+00:27:43,080 --> 00:27:46,680
+ناقص واحد Alpha مانس واحد for X element in zero
+
+349
+00:27:46,680 --> 00:27:53,250
+وملا نهايةالان بعض الملاحظات الأخرى اللي بيقولك
+
+350
+00:27:53,250 --> 00:28:01,010
+إياها على هذه الدالة بيقولبقول لك اللي هو عندي إذا
+
+351
+00:28:01,010 --> 00:28:07,610
+كانت Alpha أكبر من 0 فبصير
+
+352
+00:28:07,610 --> 00:28:11,970
+عندي اللي هي ال function من X and X alpha is
+
+353
+00:28:11,970 --> 00:28:15,930
+strictly increasing على فترة 0 و ملا نهاية طبيعي
+
+354
+00:28:15,930 --> 00:28:19,890
+لما Alpha أكبر من 0 هيصير عندي الان المقدر هذا
+
+355
+00:28:19,890 --> 00:28:24,120
+بظله موجبو هذه ألف أكبر من سفر بيكون بيبدو أكبر من
+
+356
+00:28:24,120 --> 00:28:27,180
+سفر إذا صارت عندي اللي هي ال derivative أكبر من
+
+357
+00:28:27,180 --> 00:28:31,160
+سفر إذا صارت عند الدلة strictly increasing لو كانت
+
+358
+00:28:31,160 --> 00:28:34,520
+ألف أصغر من سفر هتصير اللي هي العكس strictly
+
+359
+00:28:34,520 --> 00:28:38,420
+decreasing لإنه هتكون هذه سالبة وهذه مدلة موجبة
+
+360
+00:28:38,420 --> 00:28:42,180
+بتظل هذه كلها موجبة إذا صارت strictly decreasing
+
+361
+00:28:42,180 --> 00:28:45,360
+عند ألف بتساوي سفر بيكون احنا ال derivative للواحد
+
+362
+00:28:45,360 --> 00:28:48,880
+اللي هو بيكون عبارة عن constant function اللي هو
+
+363
+00:28:48,880 --> 00:28:51,300
+في حالة الألف بتساوي سفر
+
+364
+00:28:53,910 --> 00:29:02,970
+الان نيجي الى اللي هي هيك بنكون اللي هو وصلنا
+
+365
+00:29:02,970 --> 00:29:09,190
+لآخر تعريف بده يعرف اللي هو ال log function للأساس
+
+366
+00:29:09,190 --> 00:29:13,090
+a احنا اللي عرفناه ال len اللي هو للأساس e بمعنى
+
+367
+00:29:13,090 --> 00:29:16,530
+آخر كيف بده اعرفه الآن احنا لسه ماعرفش الأساسات
+
+368
+00:29:16,530 --> 00:29:19,930
+هدا كصمنا ال len و ال exponential الان بدنا نعرف
+
+369
+00:29:19,930 --> 00:29:25,920
+اللي هو نسمي ال logاللغاريثم للأساس A نفترض أن A
+
+370
+00:29:25,920 --> 00:29:28,860
+أكبر من 0 و A لات سوى 1 it is sometimes useful to
+
+371
+00:29:28,860 --> 00:29:34,820
+define the function log للأساس A كمالي الان log A
+
+372
+00:29:34,820 --> 00:29:39,560
+of X كده اللي بيساوي لن ال X على لن ال A حيث ال A
+
+373
+00:29:39,560 --> 00:29:43,940
+عدد ثابت ماشي الحال هذه الآن صارت اللي هي ال log
+
+374
+00:29:43,940 --> 00:29:49,540
+العامة هي نفس ال exponential بس مضروبة في ثابتالأن
+
+375
+00:29:49,540 --> 00:29:52,920
+إذا الـ exponential الأصلية عليها هو معرف الان
+
+376
+00:29:52,920 --> 00:29:59,140
+بقولك إنه اللي هي هذه بنسميها log أو ال logarithm
+
+377
+00:29:59,140 --> 00:30:04,620
+للأساس A لو كان الأساس E هذا بصير لن ال E واحد
+
+378
+00:30:04,620 --> 00:30:09,320
+بنصير نرجع لن ال X اللي هي الدالة الأصلية إذا لو
+
+379
+00:30:09,320 --> 00:30:14,000
+كانت ال A هي ال E بنرجع للدالة الأصلية اللي هي زي
+
+380
+00:30:14,000 --> 00:30:17,060
+ما قلنا is called the logarithm of X to the base A
+
+381
+00:30:19,560 --> 00:30:23,400
+Yields دا الـ logarithm العادي الان اللي مشهور
+
+382
+00:30:23,400 --> 00:30:28,020
+عندنا للحسابات اللي هو للأساس عشرة اللي هو بنسمي
+
+383
+00:30:28,020 --> 00:30:32,220
+اللي هو log to the base عشرة أو اللي بنسمي common
+
+384
+00:30:32,220 --> 00:30:36,720
+logarithm اللي هو اللي بنستخدمه عادة في الحسابات و
+
+385
+00:30:36,720 --> 00:30:41,620
+هيك بكون عندنا احنا انهينا اللي هو ال section اللي
+
+386
+00:30:41,620 --> 00:30:46,180
+هو تمانية تلاتة و بكون خلصنا اللي هيالجزء الثاني
+
+387
+00:30:46,180 --> 00:30:52,240
+من المحاضرة اللي هو ما يتعلق بال .. اللي هو ال
+
+388
+00:30:52,240 --> 00:30:54,660
+logarithmic function وال power function وال
+
+389
+00:30:54,660 --> 00:31:00,040
+logarithmic للأساس اللي هو زي ما قلنا ايه و إلى
+
+390
+00:31:00,040 --> 00:31:00,640
+لقاء
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..3e864527b80816413881b5d0081ce94efca24b1d
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/2__gojdcGsw_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2322, "start": 4.94, "end": 23.22, "text": "بسم الله الرحمن الرحيم الحمد لله رب العالمين والصلاة والسلام على سيدنا محمد وعلى آله وصحبه أجمعين هذه هي المحاضرة رقم 23 في مساق تحليل حقيقة نيل طلاب وطالبات الجامعة الإسلامية قسم الرياضيات كلية العلوم", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21542, 2304, 3215, 24976, 3224, 12602, 3555, 18863, 45340, 9957, 16070, 9381, 15040, 3660, 16070, 3794, 37440, 15844, 8608, 25708, 8315, 3714, 35571, 3215, 4032, 3615, 23942, 19753, 43761, 4032, 9381, 5016, 3555, 3224, 5551, 7435, 2304, 3615, 9957, 29538, 39896, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 6673, 8978, 47524, 995, 4587, 6055, 5016, 20292, 1211, 11331, 38436, 28671, 8717, 26895, 23032, 15040, 3555, 4032, 9566, 6027, 3555, 9307, 25724, 10943, 27884, 33688, 3794, 37440, 10632, 12174, 38251, 2423, 16572, 46958, 1829, 9307, 28242, 10632, 18863, 1211, 20498], "avg_logprob": -0.09043560350182081, "compression_ratio": 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اللى هو من خلال اللى هو ال function ال exponential اعرف ال inverse لها واسميه اللى هو logarithmic function او بدى اسميه ال logarithm الطبيعى اللى هى ال len function", "tokens": [6027, 7649, 1975, 8315, 47525, 7578, 1975, 7435, 7578, 13672, 7578, 31439, 9154, 16490, 1211, 6027, 13672, 7578, 31439, 2423, 2445, 2423, 21510, 1975, 3615, 28480, 2423, 17340, 5296, 11296, 4032, 32277, 2304, 1829, 3224, 13672, 7578, 31439, 41473, 355, 13195, 2445, 1975, 2407, 47525, 7578, 24525, 2304, 1829, 3224, 2423, 41473, 32674, 41950, 21292, 3615, 7578, 13672, 7578, 8032, 7578, 2423, 40116, 2445], "avg_logprob": -0.2514423076923077, "compression_ratio": 1.7483443708609272, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 112.82, "end": 113.72, "word": "الان", "probability": 0.814208984375}, {"start": 113.72, "end": 114.8, "word": " انا", "probability": 0.75341796875}, {"start": 114.8, "end": 115.66, "word": " بدى", "probability": 0.462646484375}, {"start": 115.66, "end": 116.18, "word": " 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"start": 342.14, "end": 364.44, "text": "Composite L E Composite L of Y ايش بتساوي؟ Y لكل Y وY الموجودة لكل Y element in سفر وملا نهاية، مظبوط؟ الان فاضلول الجهتين الان طبعا احنا بنعرف انه من الاصل مدامة ال E is", "tokens": [34, 8586, 329, 642, 441, 462, 6620, 329, 642, 441, 295, 398, 1975, 1829, 8592, 39894, 3794, 995, 45865, 22807, 398, 5296, 28820, 398, 4032, 56, 9673, 29245, 23328, 3660, 5296, 28820, 398, 4478, 294, 8608, 5172, 2288, 4032, 2304, 15040, 8717, 11296, 10632, 12399, 3714, 19913, 3555, 2407, 9566, 22807, 2423, 7649, 6156, 46958, 1211, 12610, 25724, 3224, 2655, 9957, 2423, 7649, 23032, 3555, 3615, 995, 1975, 5016, 8315, 44945, 3615, 28480, 16472, 3224, 9154, 2423, 33546, 1211, 3714, 3215, 10943, 3660, 2423, 462, 307], "avg_logprob": -0.3234554556594498, "compression_ratio": 1.4640883977900552, "no_speech_prob": 4.76837158203125e-07, "words": [{"start": 342.14, "end": 342.84, "word": "Composite", "probability": 0.7455902099609375}, {"start": 342.84, "end": 343.18, "word": 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"probability": 0.74365234375}, {"start": 706.23, "end": 707.05, "word": " of", "probability": 0.81982421875}, {"start": 707.05, "end": 707.33, "word": " E", "probability": 0.9892578125}, {"start": 707.33, "end": 707.57, "word": " of", "probability": 0.953125}, {"start": 707.57, "end": 708.03, "word": " Zero", "probability": 0.9130859375}, {"start": 708.03, "end": 709.25, "word": " بيساوي", "probability": 0.95849609375}, {"start": 709.25, "end": 709.57, "word": " L", "probability": 0.96826171875}, {"start": 709.57, "end": 709.95, "word": " of", "probability": 0.94091796875}, {"start": 709.95, "end": 710.43, "word": " واحد", "probability": 0.989990234375}, {"start": 710.43, "end": 711.51, "word": " ال", "probability": 0.880859375}, {"start": 711.51, "end": 711.77, "word": " L", "probability": 0.92236328125}, {"start": 711.77, "end": 711.95, "word": " of", "probability": 0.93017578125}, {"start": 711.95, "end": 712.13, "word": " E", "probability": 0.99658203125}, {"start": 712.13, "end": 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برضه بقاش By Mathematical Induction عملناها قبل هيكة باللي هو ال section اللي جابله أو اللي هو ال exponential على السريع نشوف التفاصيل لإنه التفاصيل معادة", "tokens": [1863, 10721, 31371, 8717, 12984, 3555, 2655, 13672, 1829, 31439, 13672, 1829, 39896, 441, 295, 1783, 281, 264, 497, 4724, 3794, 995, 2407, 3660, 17182, 4724, 3794, 995, 2407, 3660, 497, 8978, 441, 295, 1783, 29538, 4724, 43042, 3224, 4724, 4587, 33599, 3146, 15776, 8615, 804, 2333, 27549, 6225, 42213, 8315, 11296, 12174, 36150, 39896, 4117, 3660, 4724, 6027, 20292, 31439, 2423, 3541, 13672, 1829, 10874, 16758, 43761, 34051, 13672, 1829, 31439, 2423, 21510, 15844, 21136, 16572, 3615, 8717, 8592, 38688, 16712, 5172, 33546, 26895, 5296, 28814, 1863, 3224, 16712, 5172, 33546, 26895, 20449, 18513, 3660], "avg_logprob": -0.21633375797075094, "compression_ratio": 1.6, "no_speech_prob": 4.76837158203125e-07, "words": [{"start": 747.21, "end": 747.67, "word": "نأتي", "probability": 0.5808919270833334}, {"start": 747.67, "end": 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{"start": 758.53, "end": 759.51, "word": " عملناها", "probability": 0.9417724609375}, {"start": 759.51, "end": 760.19, "word": " قبل", "probability": 0.910888671875}, {"start": 760.19, "end": 760.83, "word": " هيكة", "probability": 0.6533203125}, {"start": 760.83, "end": 761.51, "word": " باللي", "probability": 0.5071207682291666}, {"start": 761.51, "end": 761.91, "word": " هو", "probability": 0.98388671875}, {"start": 761.91, "end": 762.69, "word": " ال", "probability": 0.8076171875}, {"start": 762.69, "end": 763.33, "word": " section", "probability": 0.52294921875}, {"start": 763.33, "end": 763.59, "word": " اللي", "probability": 0.97802734375}, {"start": 763.59, "end": 764.09, "word": " جابله", "probability": 0.8982747395833334}, {"start": 764.09, "end": 765.55, "word": " أو", "probability": 0.69189453125}, {"start": 765.55, "end": 765.79, "word": " اللي", "probability": 0.9736328125}, {"start": 765.79, "end": 766.03, "word": " هو", "probability": 0.9951171875}, {"start": 766.03, 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ولا لأ لكن بدي أفصل لو أنتوا عندى لو اتفجنا أن نفصل ولا لأ L of X في Y بسوا L of X في L of Y", "tokens": [6027, 2655, 5172, 33546, 26895, 8032, 2655, 15040, 7435, 2407, 11296, 20449, 18513, 3660, 6156, 9778, 1211, 9957, 1829, 4724, 3794, 2288, 27884, 8717, 29973, 25894, 11296, 18871, 16254, 321, 855, 538, 33371, 441, 295, 1783, 4724, 3794, 14407, 441, 441, 295, 1783, 30767, 1829, 19446, 12174, 1211, 8315, 6225, 8592, 7649, 5296, 12984, 3555, 2655, 11296, 29538, 5551, 12984, 3555, 2655, 8315, 13672, 1829, 12174, 36150, 4724, 8592, 2407, 10632, 441, 295, 1783, 6225, 9640, 5172, 8592, 34892, 11622, 20498, 5551, 9381, 15040, 5551, 5172, 36520, 49429, 5296, 10721, 44381, 4724, 16254, 5551, 5172, 36520, 45164, 14739, 2655, 14407, 43242, 7578, 45164, 1975, 2655, 5172, 7435, 8315, 14739, 8717, 5172, 36520, 49429, 5296, 10721, 441, 295, 1783, 8978, 398, 4724, 3794, 14407, 441, 295, 1783, 8978, 441, 295, 398], "avg_logprob": -0.22292461104065409, "compression_ratio": 1.7593360995850622, 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X", "probability": 0.93017578125}, {"start": 1290.18, "end": 1290.42, "word": " أكبر", "probability": 0.9454752604166666}, {"start": 1290.42, "end": 1290.6, "word": " من", "probability": 0.99462890625}, {"start": 1290.6, "end": 1291.68, "word": " 0", "probability": 0.5107421875}, {"start": 1291.68, "end": 1292.62, "word": " هذه", "probability": 0.317138671875}, {"start": 1292.62, "end": 1292.92, "word": " ال", "probability": 0.7421875}, {"start": 1292.92, "end": 1293.2, "word": " X", "probability": 0.9599609375}, {"start": 1293.2, "end": 1293.42, "word": " to", "probability": 0.955078125}, {"start": 1293.42, "end": 1293.56, "word": " the", "probability": 0.923828125}, {"start": 1293.56, "end": 1293.94, "word": " Alpha", "probability": 0.9296875}, {"start": 1293.94, "end": 1294.6, "word": " هي", "probability": 0.95654296875}, {"start": 1294.6, "end": 1294.78, "word": " اللي", "probability": 0.937744140625}, {"start": 1294.78, "end": 1294.96, "word": " بدي", "probability": 0.4967041015625}, {"start": 1294.96, "end": 1295.72, "word": " أسميها", "probability": 0.8746337890625}, {"start": 1295.72, "end": 1296.4, "word": " ال", "probability": 0.794921875}, {"start": 1296.4, "end": 1296.78, "word": " power", "probability": 0.59033203125}, {"start": 1296.78, "end": 1297.62, "word": " function", "probability": 0.94873046875}, {"start": 1297.62, "end": 1297.94, "word": " بدي", "probability": 0.7880859375}, {"start": 1297.94, "end": 1298.56, "word": " أسميها", "probability": 0.98486328125}, {"start": 1298.56, "end": 1299.58, "word": " power", "probability": 0.49072265625}, {"start": 1299.58, "end": 1300.32, "word": " function", "probability": 0.95703125}, {"start": 1300.32, "end": 1301.1, "word": " with", "probability": 0.9140625}, {"start": 1301.1, "end": 1302.0, "word": " exponent", "probability": 0.93115234375}, {"start": 1302.0, "end": 1302.74, "word": " mean", "probability": 0.39013671875}, {"start": 1302.74, "end": 1303.34, "word": " Alpha", "probability": 0.8076171875}, {"start": 1303.34, "end": 1303.86, "word": " و", "probability": 0.62939453125}, {"start": 1303.86, "end": 1304.0, "word": " ال", "probability": 0.6044921875}, {"start": 1304.0, "end": 1304.26, "word": " X", "probability": 0.9912109375}, {"start": 1304.26, "end": 1304.5, "word": " هي", "probability": 0.9609375}, {"start": 1304.5, "end": 1305.18, "word": " أشمالها", "probability": 0.817578125}, {"start": 1305.18, "end": 1306.2, "word": " المتغيرة", "probability": 0.9569091796875}, {"start": 1306.2, "end": 1306.52, "word": " اللي", "probability": 0.953857421875}, {"start": 1306.52, "end": 1307.0, "word": " أكبر", "probability": 0.9806315104166666}, {"start": 1307.0, "end": 1307.54, "word": " من", "probability": 0.998046875}, {"start": 1307.54, "end": 1307.84, "word": " 0", "probability": 0.97412109375}, {"start": 1307.84, "end": 1308.68, "word": " شوف", "probability": 0.87841796875}, {"start": 1308.68, "end": 1309.12, "word": " الآن", "probability": 0.725341796875}, {"start": 1309.12, "end": 1311.2, "word": " نشوف", "probability": 0.7921549479166666}, {"start": 1311.2, "end": 1311.56, "word": " بعض", "probability": 0.9814453125}, {"start": 1311.56, "end": 1312.2, "word": " الخواص", "probability": 0.6770833333333334}, {"start": 1312.2, "end": 1312.54, "word": " اللي", "probability": 0.9638671875}, {"start": 1312.54, "end": 1312.76, "word": " هو", "probability": 0.75}, {"start": 1312.76, "end": 1313.18, "word": " هذه", "probability": 0.7490234375}, {"start": 1313.18, "end": 1313.88, "word": " اللي", "probability": 0.96875}, {"start": 1313.88, "end": 1314.34, "word": " هي", "probability": 0.974609375}, {"start": 1314.34, "end": 1316.1, "word": " الدالة", "probability": 0.9205729166666666}, {"start": 1316.1, "end": 1316.46, "word": " طيب", "probability": 0.9596354166666666}], "temperature": 1.0}, {"id": 56, "seek": 134947, "start": 1324.73, "end": 1349.47, "text": "الان if x أكبر من 0 and alpha بساوة m على n where m element in z و n element in n then we define x to the alpha بساوة x to the m أسواحد على n in section mean خمسة ستة هتعرفناها زمان انه في حالة بس اللي هي ال rational number عرفنا x to the m على n بساوة x to the m لكل أسواحد على n", "tokens": [6027, 7649, 498, 2031, 5551, 4117, 26890, 9154, 1958, 293, 8961, 4724, 3794, 995, 2407, 3660, 275, 15844, 297, 689, 275, 4478, 294, 710, 4032, 297, 4478, 294, 297, 550, 321, 6964, 2031, 281, 264, 8961, 4724, 3794, 995, 2407, 3660, 2031, 281, 264, 275, 5551, 3794, 14407, 5016, 3215, 15844, 297, 294, 3541, 914, 16490, 2304, 3794, 3660, 8608, 2655, 3660, 8032, 2655, 3615, 28480, 8315, 11296, 30767, 2304, 7649, 16472, 3224, 8978, 11331, 6027, 3660, 4724, 3794, 13672, 1829, 39896, 2423, 15090, 1230, 6225, 28480, 8315, 2031, 281, 264, 275, 15844, 297, 4724, 3794, 995, 2407, 3660, 2031, 281, 264, 275, 5296, 28820, 5551, 3794, 14407, 5016, 3215, 15844, 297], "avg_logprob": -0.22289823641819237, "compression_ratio": 1.7616822429906542, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 1324.73, "end": 1325.25, "word": "الان", "probability": 0.5794677734375}, {"start": 1325.25, "end": 1325.77, "word": " if", "probability": 0.273193359375}, {"start": 1325.77, "end": 1326.09, "word": " x", "probability": 0.51806640625}, {"start": 1326.09, "end": 1326.45, "word": " أكبر", "probability": 0.86083984375}, {"start": 1326.45, "end": 1326.65, "word": " من", "probability": 0.9931640625}, {"start": 1326.65, "end": 1326.91, "word": " 0", "probability": 0.304931640625}, {"start": 1326.91, "end": 1327.25, "word": " and", "probability": 0.68017578125}, {"start": 1327.25, "end": 1327.47, "word": " alpha", "probability": 0.31787109375}, {"start": 1327.47, "end": 1327.93, "word": " بساوة", "probability": 0.653466796875}, {"start": 1327.93, "end": 1328.11, "word": " m", "probability": 0.47314453125}, {"start": 1328.11, "end": 1328.27, "word": " على", "probability": 0.69677734375}, {"start": 1328.27, "end": 1328.47, "word": " n", "probability": 0.921875}, {"start": 1328.47, "end": 1328.69, "word": " where", "probability": 0.6435546875}, {"start": 1328.69, "end": 1328.91, "word": " m", "probability": 0.89697265625}, {"start": 1328.91, "end": 1329.29, "word": " element", "probability": 0.77099609375}, {"start": 1329.29, "end": 1329.49, "word": " in", "probability": 0.685546875}, {"start": 1329.49, "end": 1329.63, "word": " z", "probability": 0.9033203125}, {"start": 1329.63, "end": 1329.81, "word": " و", "probability": 0.95751953125}, {"start": 1329.81, "end": 1329.95, "word": " n", "probability": 0.51318359375}, {"start": 1329.95, "end": 1330.23, "word": " element", "probability": 0.95458984375}, {"start": 1330.23, "end": 1330.47, "word": " in", "probability": 0.96044921875}, {"start": 1330.47, "end": 1330.69, "word": " n", "probability": 0.7236328125}, {"start": 1330.69, "end": 1331.33, "word": " then", "probability": 0.58935546875}, {"start": 1331.33, "end": 1331.63, "word": " we", "probability": 0.94287109375}, {"start": 1331.63, "end": 1332.33, "word": " define", "probability": 0.958984375}, {"start": 1332.33, "end": 1332.77, "word": " x", "probability": 0.94384765625}, {"start": 1332.77, "end": 1332.97, "word": " to", "probability": 0.70166015625}, {"start": 1332.97, "end": 1333.09, "word": " the", "probability": 0.91650390625}, {"start": 1333.09, "end": 1333.49, "word": " alpha", "probability": 0.82373046875}, {"start": 1333.49, "end": 1334.55, "word": " بساوة", "probability": 0.869580078125}, {"start": 1334.55, "end": 1335.27, "word": " x", "probability": 0.92822265625}, {"start": 1335.27, "end": 1335.67, "word": " to", "probability": 0.9404296875}, {"start": 1335.67, "end": 1335.83, "word": " the", "probability": 0.93017578125}, {"start": 1335.83, "end": 1336.05, "word": " m", "probability": 0.95361328125}, {"start": 1336.05, "end": 1336.77, "word": " أسواحد", "probability": 0.6794921875}, {"start": 1336.77, "end": 1336.91, "word": " على", "probability": 0.92333984375}, {"start": 1336.91, "end": 1337.17, "word": " n", "probability": 0.95361328125}, {"start": 1337.17, "end": 1337.79, "word": " in", "probability": 0.58447265625}, {"start": 1337.79, "end": 1338.15, "word": " section", "probability": 0.8583984375}, {"start": 1338.15, "end": 1338.65, "word": " mean", "probability": 0.583984375}, {"start": 1338.65, "end": 1339.17, "word": " خمسة", "probability": 0.8216552734375}, {"start": 1339.17, "end": 1339.69, "word": " ستة", "probability": 0.9557291666666666}, {"start": 1339.69, "end": 1340.83, "word": " هتعرفناها", "probability": 0.8129475911458334}, {"start": 1340.83, "end": 1341.47, "word": " زمان", "probability": 0.9879557291666666}, {"start": 1341.47, "end": 1342.53, "word": " انه", "probability": 0.6505126953125}, {"start": 1342.53, "end": 1342.67, "word": " في", "probability": 0.830078125}, {"start": 1342.67, "end": 1343.11, "word": " حالة", "probability": 0.9923502604166666}, {"start": 1343.11, "end": 1343.43, "word": " بس", "probability": 0.595458984375}, {"start": 1343.43, "end": 1343.61, "word": " اللي", "probability": 0.940673828125}, {"start": 1343.61, "end": 1343.81, "word": " هي", "probability": 0.85693359375}, {"start": 1343.81, "end": 1343.91, "word": " ال", "probability": 0.51220703125}, {"start": 1343.91, "end": 1344.17, "word": " rational", "probability": 0.7802734375}, {"start": 1344.17, "end": 1344.65, "word": " number", "probability": 0.98291015625}, {"start": 1344.65, "end": 1345.25, "word": " عرفنا", "probability": 0.9578450520833334}, {"start": 1345.25, "end": 1345.81, "word": " x", "probability": 0.904296875}, {"start": 1345.81, "end": 1345.99, "word": " to", "probability": 0.94384765625}, {"start": 1345.99, "end": 1346.13, "word": " the", "probability": 0.9384765625}, {"start": 1346.13, "end": 1346.35, "word": " m", "probability": 0.98388671875}, {"start": 1346.35, "end": 1346.57, "word": " على", "probability": 0.896484375}, {"start": 1346.57, "end": 1346.89, "word": " n", "probability": 0.97607421875}, {"start": 1346.89, "end": 1347.49, "word": " بساوة", "probability": 0.98046875}, {"start": 1347.49, "end": 1347.73, "word": " x", "probability": 0.990234375}, {"start": 1347.73, "end": 1347.91, "word": " to", "probability": 0.978515625}, {"start": 1347.91, "end": 1348.07, "word": " the", "probability": 0.92431640625}, {"start": 1348.07, "end": 1348.21, "word": " m", "probability": 0.982421875}, {"start": 1348.21, "end": 1348.55, "word": " لكل", "probability": 0.9072265625}, {"start": 1348.55, "end": 1349.09, "word": " أسواحد", "probability": 0.91455078125}, {"start": 1349.09, "end": 1349.21, "word": " على", "probability": 0.9599609375}, {"start": 1349.21, "end": 1349.47, "word": " n", "probability": 0.97412109375}], "temperature": 1.0}, {"id": 57, "seek": 136579, "start": 1349.91, "end": 1365.79, "text": "ماشي الحال ..الان بدنا نشوف هذا التعريف مطابق لتعريفنا اليوم ولا لأ hence we have لن ال X to the Alpha لن ال X to the Alpha لن ال X to the Alpha بساوي Alpha لن ال X", "tokens": [2304, 33599, 1829, 21542, 6027, 4386, 6027, 7649, 47525, 8315, 8717, 8592, 38688, 23758, 16712, 3615, 16572, 5172, 3714, 9566, 16758, 4587, 5296, 2655, 3615, 16572, 5172, 8315, 45595, 20498, 49429, 5296, 10721, 16678, 321, 362, 5296, 1863, 2423, 1783, 281, 264, 20588, 5296, 1863, 2423, 1783, 281, 264, 20588, 5296, 1863, 2423, 1783, 281, 264, 20588, 4724, 3794, 995, 45865, 20588, 5296, 1863, 2423, 1783], "avg_logprob": -0.24113806459441114, "compression_ratio": 1.6830985915492958, "no_speech_prob": 2.753734588623047e-05, "words": [{"start": 1349.91, "end": 1350.27, "word": "ماشي", "probability": 0.721435546875}, {"start": 1350.27, "end": 1350.63, "word": " الحال", "probability": 0.971923828125}, {"start": 1350.63, "end": 1351.75, "word": " ..الان", "probability": 0.4761555989583333}, {"start": 1351.75, "end": 1352.05, "word": " بدنا", "probability": 0.91943359375}, {"start": 1352.05, "end": 1352.31, "word": " نشوف", "probability": 0.9934895833333334}, {"start": 1352.31, "end": 1352.55, "word": " هذا", "probability": 0.69873046875}, {"start": 1352.55, "end": 1353.11, "word": " التعريف", "probability": 0.994384765625}, {"start": 1353.11, "end": 1353.65, "word": " مطابق", "probability": 0.8740234375}, {"start": 1353.65, "end": 1354.19, "word": " لتعريفنا", "probability": 0.9681803385416666}, {"start": 1354.19, "end": 1354.51, "word": " اليوم", "probability": 0.966796875}, {"start": 1354.51, "end": 1354.67, "word": " ولا", "probability": 0.79736328125}, {"start": 1354.67, "end": 1355.11, "word": " لأ", "probability": 0.7279052734375}, {"start": 1355.11, "end": 1355.81, "word": " hence", "probability": 0.325927734375}, {"start": 1355.81, "end": 1356.09, "word": " we", "probability": 0.92724609375}, {"start": 1356.09, "end": 1358.37, "word": " have", "probability": 0.97021484375}, {"start": 1358.37, "end": 1359.15, "word": " لن", "probability": 0.53448486328125}, {"start": 1359.15, "end": 1359.33, "word": " ال", "probability": 0.8095703125}, {"start": 1359.33, "end": 1359.55, "word": " X", "probability": 0.62841796875}, {"start": 1359.55, "end": 1359.77, "word": " to", "probability": 0.7978515625}, {"start": 1359.77, "end": 1359.89, "word": " the", "probability": 0.91748046875}, {"start": 1359.89, "end": 1360.29, "word": " Alpha", "probability": 0.22216796875}, {"start": 1360.29, "end": 1361.39, "word": " لن", "probability": 0.800537109375}, {"start": 1361.39, "end": 1361.51, "word": " ال", "probability": 0.72509765625}, {"start": 1361.51, "end": 1361.67, "word": " X", "probability": 0.99462890625}, {"start": 1361.67, "end": 1361.77, "word": " to", "probability": 0.55712890625}, {"start": 1361.77, "end": 1361.91, "word": " the", "probability": 0.90673828125}, {"start": 1361.91, "end": 1362.25, "word": " Alpha", "probability": 0.9443359375}, {"start": 1362.25, "end": 1363.11, "word": " لن", "probability": 0.821044921875}, {"start": 1363.11, "end": 1363.17, "word": " ال", "probability": 0.90087890625}, {"start": 1363.17, "end": 1363.29, "word": " X", "probability": 0.99755859375}, {"start": 1363.29, "end": 1363.41, "word": " to", "probability": 0.7626953125}, {"start": 1363.41, "end": 1363.51, "word": " the", "probability": 0.9130859375}, {"start": 1363.51, "end": 1363.75, "word": " Alpha", "probability": 0.9697265625}, {"start": 1363.75, "end": 1364.61, "word": " بساوي", "probability": 0.734375}, {"start": 1364.61, "end": 1364.97, "word": " Alpha", "probability": 0.7861328125}, {"start": 1364.97, "end": 1365.37, "word": " لن", "probability": 0.992431640625}, {"start": 1365.37, "end": 1365.51, "word": " ال", "probability": 0.87158203125}, {"start": 1365.51, "end": 1365.79, "word": " X", "probability": 0.99462890625}], "temperature": 1.0}, {"id": 58, "seek": 138840, "start": 1366.78, "end": 1388.4, "text": "عرفناها هذه طيب where X to the Alpha بيسوا E to the Lin X to the Min to the Alpha اللي هو بيسوا E to the Alpha في Min في Lin ال X كلام كله سهل ال X to the Alpha هو اللي عرفناها اللي عبارة عن E بتصير to the Lin X to the Alpha", "tokens": [3615, 28480, 8315, 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"probability": 0.982177734375}, {"start": 1760.17, "end": 1760.43, "word": " هو", "probability": 0.990234375}, {"start": 1760.43, "end": 1761.45, "word": " نسمي", "probability": 0.8556315104166666}, {"start": 1761.45, "end": 1761.75, "word": " ال", "probability": 0.93603515625}, {"start": 1761.75, "end": 1762.21, "word": " log", "probability": 0.97412109375}], "temperature": 1.0}, {"id": 75, "seek": 178846, "start": 1762.62, "end": 1788.46, "text": "اللغاريثم للأساس A نفترض أن A أكبر من 0 و A لات سوى 1 it is sometimes useful to define the function log للأساس A كمالي الان log A of X كده اللي بيساوي لن ال X على لن ال A حيث ال A عدد ثابت ماشي الحال هذه الآن صارت اللي هي ال log العامة هي نفس ال exponential بس مضروبة في ثابت", "tokens": [6027, 1211, 17082, 9640, 1829, 12984, 2304, 24976, 10721, 3794, 32277, 316, 8717, 5172, 2655, 43042, 14739, 316, 5551, 4117, 26890, 9154, 1958, 4032, 316, 5296, 9307, 8608, 2407, 7578, 502, 309, 307, 2171, 4420, 281, 6964, 264, 2445, 3565, 24976, 10721, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg.srt
new file mode 100644
index 0000000000000000000000000000000000000000..199327320de180568e11042da59d293a6fe81f15
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg.srt
@@ -0,0 +1,3236 @@
+1
+00:00:04,940 --> 00:00:11,820
+بسم الله الرحمن الرحيم اليوم المحاضرة رقم 13 مساق 
+
+2
+00:00:11,820 --> 00:00:17,100
+تحليل حقيقي 2 لطلاب وطالبات الجامعة الإسلامية كلية
+
+3
+00:00:17,100 --> 00:00:21,800
+العلوم قسم رياضيات إن شاء الله ستكون المحاضرة على
+
+4
+00:00:21,800 --> 00:00:28,620
+جزئين، الجزء الأول هنستمر في الحديث عن سبعة اثنين
+
+5
+00:00:28,620 --> 00:00:34,240
+وهو الحديث أو سبعة ثلاثة وهو الحديث سبعة اثنين في
+
+6
+00:00:34,240 --> 00:00:40,130
+البداية وهو الحديث عن الخواص التكامل الريماني، حكينا
+
+7
+00:00:40,130 --> 00:00:45,310
+عن اللي هو مجموع دالتين قابلتين للتكامل اللي هو 
+
+8
+00:00:45,310 --> 00:00:48,510
+قابل
+
+9
+00:00:48,510 --> 00:00:52,510
+التكامل و حاصل ضرب ثابت في دالة قابلة للتكامل برضه
+
+10
+00:00:52,510 --> 00:00:58,630
+قابل للتكامل، واليوم هنكمل الأمر، الحديث عن
+
+11
+00:00:58,630 --> 00:01:04,550
+composition of two integrable functions هل قابل
+
+12
+00:01:04,550 --> 00:01:08,270
+التكامل أم لا، وإذا قابل التكامل بدنا نبرهن، وإذا مش
+
+13
+00:01:08,270 --> 00:01:14,750
+قابل بدنا نجيب counter example، الآن الجزء الثاني 
+
+14
+00:01:14,750 --> 00:01:18,110
+هيكون اللي هو برضه تطبيق اللي هو composition
+
+15
+00:01:18,110 --> 00:01:25,410
+theorem اللي هي هنشوف بعد شوية ايش المتطلب أن يكون
+
+16
+00:01:25,410 --> 00:01:30,190
+composition of two functions is integrable، أيضًا 
+
+17
+00:01:30,190 --> 00:01:35,230
+هنتحدث عن اللي هو نوظفها في إثبات اللي هو أن 
+
+18
+00:01:35,230 --> 00:01:38,950
+الدالة الأسية والـ absolute value of the function 
+
+19
+00:01:38,950 --> 00:01:43,530
+F،  و أيضًا اللي هو مقلوب الدالة 1 على F في حالة F
+
+20
+00:01:43,530 --> 00:01:50,090
+لا تساوي 0 على ال domain المُعطى أنها تكون قابلة 
+
+21
+00:01:50,090 --> 00:01:54,830
+للتكامل، و أيضًا اللي هو هيكون في عندنا اللي هو تطبيق
+
+22
+00:01:54,830 --> 00:01:59,640
+آخر اللي هو حاصل ضرب دالتين كيف يكون integrable أو
+
+23
+00:01:59,640 --> 00:02:03,080
+كيف تكون integrable في حالة كلا الدالتين integrable
+
+24
+00:02:03,080 --> 00:02:06,200
+نبدأ الآن في اللي هو النظرية الـ composition
+
+25
+00:02:06,200 --> 00:02:09,600
+theorem، وطولوا روحكم علينا شوية على اللي هو 
+
+26
+00:02:09,600 --> 00:02:14,930
+البرهان، والبرهان شوية بده تركيز، والبرهان طويل
+
+27
+00:02:14,930 --> 00:02:20,110
+شوية، خلّينا إن نعمل الآن focusing على نص النظرية، و
+
+28
+00:02:20,110 --> 00:02:25,410
+بعدين بنبدأ نحكي عن البرهان، نعمل outline للبرهان، و
+
+29
+00:02:25,410 --> 00:02:27,610
+من ثم ندخل لتفاصيل البرهان.
+
+30
+00:02:30,230 --> 00:02:35,470
+بنأخذ I عبارة عن closed bounded interval A و B، و J
+
+31
+00:02:35,470 --> 00:02:38,550
+عبارة عن closed bounded interval سميناها C و D، 
+
+32
+00:02:38,550 --> 00:02:44,830
+ونفترض أن F من I لعند R، يعني F عبارة عن دالة من عند 
+
+33
+00:02:44,830 --> 00:02:49,350
+الـ A والـ B لعند R، أنها تكون integrable on I and 
+
+34
+00:02:49,350 --> 00:02:52,050
+Phi من J لعند R،
+
+35
+00:02:55,070 --> 00:02:58,390
+النظرية بتستلزم أن نقول continuous لأن الـ Integra
+
+36
+00:02:58,390 --> 00:03:03,270
+بالحالها مش هتعطي اللي هو النتيجة زي ما هنشوف قدام
+
+37
+00:03:03,270 --> 00:03:07,610
+في اللي هو counter example، الآن فرضنا أن Phi من J 
+
+38
+00:03:07,610 --> 00:03:12,890
+لعند R is continuous، وبدنا نفترض أن F of I .. F of
+
+39
+00:03:12,890 --> 00:03:17,450
+I جزئية من مين؟ من J عشان نعرف .. نعرف الـ
+
+40
+00:03:17,450 --> 00:03:21,670
+composition بين الـ two functions، الآن في ضوء هذه 
+
+41
+00:03:21,670 --> 00:03:26,050
+المعطيات، إن الـ function F is integrable و الـ
+
+42
+00:03:26,050 --> 00:03:30,410
+function Phi is continuous لازم يطلع عندي الآن Phi 
+
+43
+00:03:30,410 --> 00:03:37,450
+composite F is integrable on I، mean on I، إذن F من I
+
+44
+00:03:37,450 --> 00:03:43,130
+لـ R integrable، Phi من J لعند R continuous، الآن Phi
+
+45
+00:03:43,130 --> 00:03:46,170
+composite F is continuous، composite integrable، 
+
+46
+00:03:46,170 --> 00:03:53,500
+هيعطيني integrable function on I، الآن بدنا نثبت، الآن 
+
+47
+00:03:53,500 --> 00:04:00,600
+فاي composite F من I لعند R إنّها is integrable.
+
+48
+00:04:00,600 --> 00:04:04,900
+الآن كيف بدي أثبتها؟ بدي أثبتها .. بدي ألاقي
+
+49
+00:04:04,900 --> 00:04:09,040
+partition ..  ده أقول لكل إبسيلون أكبر من 0 بدي ألاقي
+
+50
+00:04:09,040 --> 00:04:16,060
+partition P element in P of I such that L of أو U
+
+51
+00:04:16,060 --> 00:04:24,200
+of P و F هذا اللي جبتّه ما عرف أسأل، آسف أنا فعلاً 
+
+52
+00:04:24,200 --> 00:04:30,690
+composed of G and F، فاي كومبوزيت F ناقص L of P و فاي
+
+53
+00:04:30,690 --> 00:04:35,730
+كومبوزيت F، أن يكون هذا المقدار أصغر من إبسيلون، إذا 
+
+54
+00:04:35,730 --> 00:04:40,590
+وصلت لهذه النتيجة معناته أنه أنا أثبتت أنه اللي هي 
+
+55
+00:04:40,590 --> 00:04:46,490
+الـ فاي كومبوزيت F is integrable بناء على اللي هو الـ
+
+56
+00:04:46,490 --> 00:04:48,870
+criterion of integrability اللي حكينا عنها
+
+57
+00:04:48,870 --> 00:04:53,700
+المحاضرة قبل الماضية، الآن هذا الهدف اللي بدي أوصله،
+
+58
+00:04:53,700 --> 00:04:58,020
+هيك بدي أوصل، هيك، الآن عشان أوصل هيك بدي اللي هو 
+
+59
+00:04:58,020 --> 00:05:02,900
+أستخدم اللي هو المعطيات اللي موجودة عندي الآن،
+
+60
+00:05:02,900 --> 00:05:08,380
+هستخدم أمرين، هستخدم أكيد الـ continuity للـ Phi، وهي
+
+61
+00:05:08,380 --> 00:05:14,320
+continuous على closed bounded interval، إذا حسب 
+
+62
+00:05:14,320 --> 00:05:18,080
+نظرية في تحليل واحد هتكون Phi is uniformly 
+
+63
+00:05:18,080 --> 00:05:23,880
+continuous، وهذا هستغله في الوصول إلى هدفي، الآن هذه
+
+64
+00:05:23,880 --> 00:05:27,960
+المعلومة بعد شوية هنخزنها ونحطها في .. في مكان ما 
+
+65
+00:05:27,960 --> 00:05:33,420
+لحين نستخدمها مع اللي هو أن F is integrable، مزام F 
+
+66
+00:05:33,420 --> 00:05:37,240
+is integrable، إذن نبلج Partition I بحيث أن الـ
+
+67
+00:05:37,240 --> 00:05:40,500
+U للـ P والـ F ناقص الـ L للـ P والـ F أصغر من مين؟ من
+
+68
+00:05:40,500 --> 00:05:44,220
+some .. من الـ Epsilon، Epsilon تخدمني، هبدأ أسميها
+
+69
+00:05:44,220 --> 00:05:47,490
+Epsilon، هبدأ أسميها Delta، أنا حأور، المهم أكيد مدام 
+
+70
+00:05:47,490 --> 00:05:51,610
+الـ Phi عندي integrability للـ F هتحقق أنه لكل اللي
+
+71
+00:05:51,610 --> 00:05:56,010
+هو Epsilon there exists partition P، وحنلاقي أنه الـ
+
+72
+00:05:56,010 --> 00:05:59,510
+partition هذا اللي .. اللي نفع للـ F هو اللي هينفع
+
+73
+00:05:59,510 --> 00:06:04,300
+للـ Phi composite F، هنجمج المعلومتين التنتين مع بعض
+
+74
+00:06:04,300 --> 00:06:06,680
+اللي هي الـ .. الـ .. الـ .. الـ .. الـ uniform
+
+75
+00:06:06,680 --> 00:06:10,260
+continuity للـ Phi مع الـ integrability للـ F للوصول
+
+76
+00:06:10,260 --> 00:06:14,920
+إلى نتيجتنا وهي هذه النتيجة، هذه خلينا نقول الـ
+
+77
+00:06:14,920 --> 00:06:19,660
+outline للبرهان، نبدأ الآن في تفاصيل البرهان وطوّل 
+
+78
+00:06:19,660 --> 00:06:25,400
+روحكم عليها في اللي هو تفاصيل البرهان للوصول 
+
+79
+00:06:25,400 --> 00:06:31,780
+للنتيجة اللي حكيتها اللي كتبت على اللوح، بدي أقول 
+
+80
+00:06:31,780 --> 00:06:39,240
+الآن أول حاجة given epsilon أكبر من صفر، أنا أخذت
+
+81
+00:06:39,240 --> 00:06:44,600
+أي epsilon أكبر من صفر، بدي أوصل لـ U P Phi 
+
+82
+00:06:44,600 --> 00:06:50,610
+composite F ناقص الـ L P Phi composite F أصغر من إبسيلون،
+
+83
+00:06:50,610 --> 00:06:55,150
+for some اللي هو partition P، إذا وصلت لهيك بكون خلصت
+
+84
+00:06:55,150 --> 00:06:58,770
+اللي هو نظريتي، طيب سلموا على النبي عليه الصلاة 
+
+85
+00:06:58,770 --> 00:07:03,850
+والسلام، لأن Phi عندي، Phi عندي continuous على
+
+86
+00:07:03,850 --> 00:07:08,950
+مين؟ على اللي هي الـ J، الـ J عبارة عن closed bounded
+
+87
+00:07:08,950 --> 00:07:12,650
+interval، مدام continuous عليها، إذن uniformly
+
+88
+00:07:12,650 --> 00:07:16,690
+continuous، ماشي الحال، إذا مدام uniformly 
+
+89
+00:07:16,690 --> 00:07:23,170
+continuous إذا أكيد .. أكيد حتكون اللي هو، مدام ..
+
+90
+00:07:23,170 --> 00:07:26,750
+حتى في الـ continuity أكيد حتكون ايش مالها؟ bounded،
+
+91
+00:07:26,750 --> 00:07:34,710
+إذن بقدر اللي هو أقول K بتساوي الـ supremum للـ Phi 
+
+92
+00:07:34,710 --> 00:07:41,540
+of T such that T element في الـ C و D التي هي الـ J،
+
+93
+00:07:41,540 --> 00:07:45,900
+هذه الآن بقدر أحكي عن حاجة اسمها supremum، اه طبعًا
+
+94
+00:07:45,900 --> 00:07:51,720
+مش هي كلكم كمان، والـ K هذه هتكون attains for
+
+95
+00:07:51,720 --> 00:07:55,500
+some T بين C و D، ليه؟ لأنه Phi is continuous on a
+
+96
+00:07:55,500 --> 00:07:58,340
+closed bounded interval then it attains its 
+
+97
+00:07:58,340 --> 00:08:00,920
+absolute maximum and absolute minimum on this
+
+98
+00:08:00,920 --> 00:08:05,120
+interval، إذن أكيد في عندي K بتساوي الـ supremum، لو
+
+99
+00:08:05,120 --> 00:08:09,200
+سلوك يدفع له 5 T، T Element in C و D، سميليها دي K، ليش
+
+100
+00:08:09,200 --> 00:08:13,280
+هتجي هتتَعرف ليش هتستخدمها في الوصول إلى هدفي، إذا 
+
+101
+00:08:13,280 --> 00:08:16,960
+الآن اللي عملته لحد الآن أخذت epsilon arbitrarily،
+
+102
+00:08:16,960 --> 00:08:21,860
+أخذت اللي هو الـ supremum لهذا المقدار وسميته K، هل 
+
+103
+00:08:21,860 --> 00:08:24,440
+الـ supremum موجود؟ اه الـ supremum موجود و maximum
+
+104
+00:08:24,440 --> 00:08:27,480
+كمان لأن الـ Phi is continuous on a closed
+
+105
+00:08:27,480 --> 00:08:31,840
+bounded interval C و D، طيب
+
+106
+00:08:34,510 --> 00:08:39,830
+الآن بدي آخذ بعيد إذنكم حاجة اسميها let epsilon
+
+107
+00:08:39,830 --> 00:08:45,490
+برايم بتساوي اللي هو epsilon على b ناقص a زائد
+
+108
+00:08:45,490 --> 00:08:49,670
+اثنين K، الـ K هذه اللي فوق، زائد اثنين K، والـ b والـ
+
+109
+00:08:49,670 --> 00:08:54,010
+a اللي هي طول الفترة اللي هي I اللي أنا عمال بشتغل
+
+110
+00:08:54,010 --> 00:09:01,100
+عليها، معرف عليها F، ليش هيك؟ بغرض الحسابات بعد شوية 
+
+111
+00:09:01,100 --> 00:09:06,060
+هتشوفوا ليش، ولو أصلاً احنا في النهاية الـ Epsilon 
+
+112
+00:09:06,060 --> 00:09:10,140
+الـ prime هذه ما كتبناش بالشكل هذا، وطلع عندي اللي
+
+113
+00:09:10,140 --> 00:09:19,940
+هو الـ U of P أو Phi composite F ناقص الـ L Phi
+
+114
+00:09:19,940 --> 00:09:26,070
+composite F يكون أصغر من إبسيلون مضروبة في something
+
+115
+00:09:26,070 --> 00:09:32,270
+أي شيء something ثابت، برضه هتقدي الغرض لأنه اللي هي
+
+116
+00:09:32,270 --> 00:09:35,310
+زي ما قلنا المرة الماضية، مدام صحيحة لكل epsilon
+
+117
+00:09:35,310 --> 00:09:38,150
+في الدنيا إذا صحيح، واحد على n خذ الـ limit
+
+118
+00:09:38,150 --> 00:09:41,530
+للجهتين as n goes to infinity بيصير اللي هو هذا
+
+119
+00:09:41,530 --> 00:09:45,670
+اللي هو يؤدي الغرض أو الـ epsilon اللي هنا بتكون
+
+120
+00:09:45,670 --> 00:09:50,730
+اللي هو بيصير يعني تؤدي غرض أي أصغر من epsilon في
+
+121
+00:09:50,730 --> 00:09:53,510
+الدنيا لأن الـ epsilon مدام انضربت ضرب في ثابت
+
+122
+00:09:53,510 --> 00:09:57,930
+بقدر الـ epsilon أصغرها جد ما بدي، وتؤدي الغرض.
+
+123
+00:09:58,610 --> 00:10:05,280
+المفروض فاهمين، طيب نيجي الآن نرجع نقول أنه أخذت
+
+124
+00:10:05,280 --> 00:10:08,980
+إبسيلون برايم بتساوي إبسيلون على P minus A زائد 2K على
+
+125
+00:10:08,980 --> 00:10:12,680
+أساس أنه اللي هي تطلع عندنا الحسابات في الآخر
+
+126
+00:10:12,680 --> 00:10:16,480
+مرتبة وخالصة اللي هي أصغر من إبسيلون، طبعًا هو
+
+127
+00:10:16,480 --> 00:10:20,720
+الأشرف أن إبسيلون برايم بالشكل هذا أصلاً هو برهن
+
+128
+00:10:20,720 --> 00:10:23,680
+النظرية أو برهنا النظرية وفي الآخر طلع أن إبسيلون 
+
+129
+00:10:23,680 --> 00:10:28,440
+مضروبة في رقم، جيت اللي هو رتبت حالي بحيث أنه حسبت 
+
+130
+00:10:28,440 --> 00:10:32,320
+أنه عشان أطلع إبسيلون لحالها خذ إبسيلون برايم بالشكل 
+
+131
+00:10:32,320 --> 00:10:39,070
+هذا، طيب نشوف .. نشوف الآن عندي Phi زي ما وعدناكم 
+
+132
+00:10:39,070 --> 00:10:49,690
+Phi is continuous on اللي هو C و D، مدام Phi 
+
+133
+00:10:49,690 --> 00:10:53,010
+continuous on C و D يا جماعة إذا زي ما قلنا قبل
+
+134
+00:10:53,010 --> 00:11:01,660
+بشوية، إذا Phi is uniformly continuous on C و D، ايش
+
+135
+00:11:01,660 --> 00:11:07,680
+معنى uniformly continuous يعني اللي هو for every 
+
+136
+00:11:07,680 --> 00:11:12,100
+إبسيلون في الدنيا، for every إبسيلون .. خلّيني أخذ
+
+137
+00:11:12,100 --> 00:11:15,020
+إبسيلون الـ prime، for every إبسيلون الـ prime أكبر
+
+138
+00:11:15,020 --> 00:11:21,680
+من صفر there exists delta prime such that
+
+139
+00:11:21,680 --> 00:11:29,560
+uniformly continuous، For every S و T element in C و D،
+
+140
+00:11:29,560 --> 00:11:38,690
+تحقق S minus T أصغر من Delta prime يؤدي إلى Phi of
+
+141
+00:11:38,690 --> 00:11:43,690
+S ناقص Phi of T أصغر من مين؟ من Epsilon prime اللي
+
+142
+00:11:43,690 --> 00:11:49,450
+أخدتها عندي لأي Epsilon في الدنيا، هذا الكلام 
+
+143
+00:11:49,450 --> 00:11:52,210
+بيتحقق من دون أن أ هم، Epsilon prime اللي حكيت عنها
+
+144
+00:11:52,210 --> 00:11:58,090
+فوق، طيب إذا الآن اللي استخدمته بما أن Phi
+
+145
+00:11:58,090 --> 00:12:02,010
+continuous على C و D، إذا Phi is uniformly 
+
+146
+00:12:02,010 --> 00:12:08,210
+continuous on مين؟ on C و D، إذن الآن حسب التعريف
+
+147
+00:12:08,210 --> 00:12:11,530
+الـ uniformly continuous لأي إبسيلون في الدنيا، من 
+
+148
+00:12:11,530 --> 00:12:14,530
+ضمنهم الـ إبسيلون الـ prime الأكبر من 0، there exists
+
+149
+00:12:14,530 --> 00:12:17,430
+delta prime خاصة بالإبسيلون الـ prime بحيث أنه لما 
+
+150
+00:12:17,430 --> 00:12:20,770
+S و T في الـ C و الـ D ويكون الـ S minus T أصغر
+
+151
+00:12:20,770 --> 00:12:24,950
+من delta prime يعطيني أن Phi S ناقص Phi T أصغر من مين؟ من
+
+152
+00:12:24,950 --> 00:12:30,990
+إبسيلون الـ prime، الآن أنا في Delta معينة بدي
+
+153
+00:12:30,990 --> 00:12:35,830
+ألاقيها أربطها بهذه واتحققلي هذا الكلام، شوفوا كيف 
+
+154
+00:12:35,830 --> 00:12:44,030
+اصبروا عليها، الآن if عندي delta prime هذه أصغر من
+
+155
+00:12:44,030 --> 00:12:49,430
+epsilon prime، then there exists Delta بتساوي Delta 
+
+156
+00:12:49,430 --> 00:12:53,130
+prime، بتاخذ Delta، ايش بتساوي؟ أسميها بتسمي Delta 
+
+157
+00:12:53,130 --> 00:12:56,370
+prime، مين يا جماعة؟ Delta، وهاد الـ Delta اللي
+
+158
+00:12:56,370 --> 00:12:58,790
+سميتها اللي هي Delta prime، اللي سميتها Delta، هاد
+
+159
+00:12:58,790 --> 00:13:03,670
+الـ Delta هتحققها ليش؟ لأنها نفسها، يعني بمعنى آخر
+
+160
+00:13:03,670 --> 00:13:08,150
+إذا كانت الـ S minus T أصغر من Delta prime بيعطيني
+
+161
+00:13:08,150 --> 00:13:10,830
+automatic أصغر من Delta prime اللي سميتها Delta،
+
+162
+00:13:10,830 --> 00:13:15,810
+بيعطيني Phi of S ناقص Phi of T أصغر من مين؟ من
+
+163
+00:13:15,810 --> 00:13:23,530
+Epsilon prime، طيب تشوفوا الآن if delta prime أكبر 
+
+164
+00:13:23,530 --> 00:13:26,430
+أو يساوي epsilon prime يا أي حالتين ما فيش غير هيك،
+
+165
+00:13:26,430 --> 00:13:28,830
+يا delta prime أصغر من epsilon يا delta prime أكبر
+
+166
+00:13:28,830 --> 00:13:32,090
+ساوي epsilon prime، الآن if delta .. أهل جاي تفهموا
+
+167
+00:13:32,090 --> 00:13:34,530
+ليش عملت هيك، if delta prime أكبر ساوي epsilon 
+
+168
+00:13:34,530 --> 00:13:41,100
+prime then اللي هو there exists Delta أصغر من 
+
+169
+00:13:41,100 --> 00:13:45,540
+Epsilon prime، بلاجي ولا بلاجي؟ Epsilon prime أكبر
+
+170
+00:13:45,540 --> 00:13:50,200
+من 0 أكيد between، أنا إبسيلون prime أكبر من 0، إذا 
+
+171
+00:13:50,200 --> 00:13:52,620
+أنا أكيد بلاجي بين Epsilon prime، أكبر من 0، بلاجي
+
+172
+00:13:52,620 --> 00:13:57,340
+Delta، بلاجي عدد نهائي من الأعداد اللي هو Delta أكبر
+
+173
+00:13:57,340 --> 00:14:01,400
+من صفر وأصغر من إبسيلون prime، إذن بلاجي Delta أصغر 
+
+174
+0
+
+201
+00:16:13,950 --> 00:16:22,170
+برايم such that لكل S وT element in C وD إذا حققت
+
+202
+00:16:22,170 --> 00:16:27,610
+الخاصية S minus T أصغر من Delta بيعطيني على طول 5S
+
+203
+00:16:27,610 --> 00:16:34,130
+ناقص 5T أصغر من مين من إبسلون برايم إذن هي
+
+204
+00:16:34,130 --> 00:16:41,710
+معلومة أخرى بدي أخذنها لأنني هحتاجها هي كمان معلومة
+
+205
+00:16:41,710 --> 00:16:46,270
+الآن اسمحولي أمسح هذا ال hand عشان أخزن معلومتي
+
+206
+00:16:46,270 --> 00:16:49,770
+اللي وصلت إليها مع المعلومات اللي موجودة عندي فوق
+
+207
+00:16:49,770 --> 00:16:57,090
+طيب الآن شطبنا اللي عندنا hand خلصنا منه ووصلنا
+
+208
+00:16:57,090 --> 00:17:03,030
+إلى المعلومة التالية اللي بدأ أخزنها الآن مع اللي
+
+209
+00:17:03,030 --> 00:17:10,620
+مخزن فوق لأن there exists Delta أكبر من 0 وأصغر من
+
+210
+00:17:10,620 --> 00:17:18,140
+Y' Such that for every S وT element in C وD إذا 
+
+211
+00:17:18,140 --> 00:17:25,340
+حقق S minus T أصغر من Delta بيعطيني اللي هو 5S
+
+212
+00:17:25,340 --> 00:17:34,140
+minus 5T أصغر من Y' وهذا سمونيها 1 سمونيها 2
+
+213
+00:17:34,140 --> 00:17:38,630
+سمونيها Star اللي بدكم إياها ماشي الحال هذا الآن
+
+214
+00:17:38,630 --> 00:17:44,170
+وصلت إليه و أنا بدي أستخدمه بعد شوية
+
+215
+00:17:49,890 --> 00:17:54,390
+اللي اللي بيحب يتابع على التلخيص
+
+216
+00:17:54,390 --> 00:17:58,130
+هاي اللي وصلت إليها الآن هايها there exists delta و
+
+217
+00:17:58,130 --> 00:18:01,390
+الـ delta أصغر من epsilon prime if S و T element
+
+218
+00:18:01,390 --> 00:18:04,470
+in J and S minus T أصغر من Delta then Phi of S
+
+219
+00:18:04,470 --> 00:18:08,030
+ناقص Phi of T أصغر من إبسلون برايم هذه اللي وصلنا
+
+220
+00:18:08,030 --> 00:18:15,670
+إليها اللي قدرنا أن نصل إليها عشان بعد شوية بتستخدمها 
+
+221
+00:18:15,670 --> 00:18:20,950
+انتبه عليها الآن الآن استغلقنا معلومة الـ if I is
+
+222
+00:18:20,950 --> 00:18:24,490
+continuous وحصلنا على معلومة مهمة جداً هي هذه
+
+223
+00:18:24,490 --> 00:18:28,630
+المعلومة الآن بدي أستخدم المعلومة الموازية لها أن
+
+224
+00:18:28,630 --> 00:18:37,770
+F is integrable الآن عندي F is integrable on I إذا
+
+225
+00:18:37,770 --> 00:18:43,010
+حسب then حسب اللي هو ال .. ال .. ال integrability
+
+226
+00:18:43,010 --> 00:18:48,570
+criterion اللي حكينا عنها there exists partition B
+
+227
+00:18:48,570 --> 00:18:56,690
+element in B of I بجزء مين I such that اللي هو U
+
+228
+00:18:56,690 --> 00:19:05,920
+of B وF معقس L بيوقف أصغر من أي إبسلون في الدنيا
+
+229
+00:19:05,920 --> 00:19:08,360
+الإبسلون اللي في الدنيا الإبسلون اللي بدها
+
+230
+00:19:08,360 --> 00:19:12,040
+تستخدمها اللي حتة في الدنيا اللي هي مين هي Delta
+
+231
+00:19:12,040 --> 00:19:17,000
+تربيع أصغر من مين من Delta تربيع بقدر أه بقدر طبعا
+
+232
+00:19:17,000 --> 00:19:19,960
+مش احنا بنقول لكل إبسلون أكبر من صفر مادام F is
+
+233
+00:19:19,960 --> 00:19:23,040
+integrable لذا لكل إبسلون أكبر من صفر there exist
+
+234
+00:19:23,040 --> 00:19:27,590
+ال partition B بحيث أنه هذا أصغر من إبسلون الآن
+
+235
+00:19:27,590 --> 00:19:30,330
+أبسط من اللي بحكي عنها دلتة تربيع دلفة التربيع
+
+236
+00:19:30,330 --> 00:19:35,650
+أكبر من صفر إذا for delta تربيع there exists B
+
+237
+00:19:35,650 --> 00:19:40,070
+element in B of I such that U ناقص L أصغر من مين
+
+238
+00:19:40,070 --> 00:19:44,650
+من Delta تربيع هذا ال partition B بيجزئلي مين؟
+
+239
+00:19:44,650 --> 00:19:51,670
+بيجزئلي I بعد أذنكم سموليها بيه X0 و X1 لعند مين؟
+
+240
+00:19:51,670 --> 00:19:55,230
+لعند X سموليها ال partition بس عشان أتعامل معاها
+
+241
+00:19:55,660 --> 00:19:57,540
+اللي هو partition للـ I partition للـ I معناه
+
+242
+00:19:57,540 --> 00:20:00,880
+بجزّه I جزّه أو ل X note X واحد X أنت جداش أعدادها
+
+243
+00:20:00,880 --> 00:20:07,080
+مش عارف حسب اللي هو اللي جناه بيه طيب شوف الآن بدي
+
+244
+00:20:07,080 --> 00:20:13,520
+أعمل اللي هو شغلة بحيث عن الآن أقدر أستخدمها اللي
+
+245
+00:20:13,520 --> 00:20:18,360
+هي أصل للي بدي من خلالها الآن هجيتوا تشوفوا ليش
+
+246
+00:20:18,360 --> 00:20:23,400
+جزعين خدوا A الآن يا جماعة صارت دلتة بين إيديا
+
+247
+00:20:23,400 --> 00:20:28,520
+لجهتها دلتة طيب الآن خدوا ايه 6 هي عبارة عن كل ال
+
+248
+00:20:28,520 --> 00:20:34,160
+indices K اللي هان هذولة و 0,1,2,3 كده هذه جزء
+
+249
+00:20:34,160 --> 00:20:38,520
+اثنين ال I هي عند X note لعند X end هذه الفترة
+
+250
+00:20:38,520 --> 00:20:44,000
+اللي هي من A لعند B يا جماعة لعند B ماشي الحال طيب
+
+251
+00:20:44,000 --> 00:20:49,380
+خدوله A هي عبارة عن كل ال K بحيث ان M K ناقص M K
+
+252
+00:20:49,380 --> 00:20:54,900
+small تكون أصغر من Delta وخدول ال B بيساوي كل ال K
+
+253
+00:20:54,900 --> 00:20:59,740
+such that M K ناقص M K أكبر يساوي Delta إيش هذول
+
+254
+00:20:59,740 --> 00:21:04,700
+عاملا؟ هذول بس ال indices عند من هنا صفر و واحد و
+
+255
+00:21:04,700 --> 00:21:08,480
+اثنين عند من؟ عند ال K إذا اتجت جزءات ال indices
+
+256
+00:21:08,480 --> 00:21:15,500
+هذول إلى جزء ايه؟ الجزء اللي هو عندي اللي بخاصية
+
+257
+00:21:15,500 --> 00:21:19,980
+اللي هو ال mk ناقص mk أصغر من دلتا بده حطه فهذا ال
+
+258
+00:21:19,980 --> 00:21:24,280
+set إذا هذه عبارة عن إيش مجموعة جزئية من ال .. من
+
+259
+00:21:24,280 --> 00:21:27,680
+ال .. من ال .. من ال .. من ال .. من ال K من صفر
+
+260
+00:21:27,680 --> 00:21:36,490
+لعند مين لعند واحد لعند أنا .. لعند أنا الآن هذه P
+
+261
+00:21:36,490 --> 00:21:40,870
+هي عبارة عن المتبقى منهن، مين المتبقى اللي الـ M K
+
+262
+00:21:40,870 --> 00:21:44,170
+نقص M K أصغر أكبر أو شوية دلتا، يعني بمعنى آخر
+
+263
+00:21:44,170 --> 00:21:49,430
+أداة التجزئة لهذه اللي هي الخاصية أنه M K نقص M K
+
+264
+00:21:49,430 --> 00:21:52,010
+أصغر من مين مين دلتا، أنتو عارفين إيش الـ M K
+
+265
+00:21:52,010 --> 00:21:55,730
+Capital و M K Small؟ أكيد، الـ M K Capital هي
+
+266
+00:21:55,730 --> 00:21:59,710
+عبارة عن الـ supremum لل F of X such that X
+
+267
+00:21:59,710 --> 00:22:04,740
+element in X K minus 1 لعند الـ X K و الـ mk small
+
+268
+00:22:04,740 --> 00:22:09,560
+بيساوي الـ infimum لل F of X such that X element
+
+269
+00:22:09,560 --> 00:22:16,640
+in XK minus واحد والـ XK إذن يا جماعة اللي جزألي
+
+270
+00:22:16,640 --> 00:22:21,900
+الـ A و الـ B هو خاصيته إن الـ mk capital يعني
+
+271
+00:22:21,900 --> 00:22:25,340
+أعلى قيمة هي .. هي .. هي .. هي عندي أنا XK minus
+
+272
+00:22:25,340 --> 00:22:29,860
+واحد وهي XK فرضنا أن الدالة هي الدالة زي هيك مثلا
+
+273
+00:22:29,860 --> 00:22:35,250
+في المنطقة هذه الآن هي أعلى قيمة وهي أقل قيمة
+
+274
+00:22:35,250 --> 00:22:41,070
+الحاصل طرح أعلى قيمة و أقل قيمة في كل فترة .. كل
+
+275
+00:22:41,070 --> 00:22:44,770
+sub interval بادي بقول هل هذا أكبر من .. أصغر من
+
+276
+00:22:44,770 --> 00:22:49,270
+delta ولا أكبر يساوي delta اللي خاصيتهم الفرق
+
+277
+00:22:49,270 --> 00:22:53,930
+بينهم أصغر من delta بحط هنا الاندسيز هذا كهن واللي
+
+278
+00:22:53,930 --> 00:22:59,930
+أكبر بحطه نهن و بحط تجزيتها ليش هذه؟ هذه طريقة
+
+279
+00:22:59,930 --> 00:23:07,130
+للوصول إلى اللي بديها و هتشوفه الآن طيب نيجي الآن
+
+280
+00:23:07,130 --> 00:23:20,690
+لللي هو نشوف الـ K F K Element A مدام K Element A
+
+281
+00:23:22,110 --> 00:23:26,470
+إذا تتحقق الخاصية هذه يعني Mk نقص Mk أصغر من مين؟
+
+282
+00:23:26,470 --> 00:23:32,430
+من Delta خدولي الآن أي X و Y في الفترة مين اللي هي
+
+283
+00:23:32,430 --> 00:23:39,330
+XK minus واحد لعند XK شوفوا إيش اللي بديها ال .. ال
+
+284
+00:23:39,330 --> 00:23:43,130
+.. ال .. ال .. ال X و ال Y هنا هي ال X و ال Y وهى
+
+285
+00:23:43,130 --> 00:23:46,430
+رسمة الدلة في المنطقة هذه ال X و ال Y في داخل هدول
+
+286
+00:23:46,430 --> 00:23:53,930
+طيب الآن ال X و ال Y هنا إذا أكيد ال F of X ال F
+
+287
+00:23:53,930 --> 00:24:00,090
+of X ناقص F of Y نقطتين هنا نقطتين من هنا لهنا صور
+
+288
+00:24:00,090 --> 00:24:03,750
+هنا صور هنا يعني ممكن صورة واحدة هنا و صورة
+
+289
+00:24:03,750 --> 00:24:08,440
+التانية هنا يعني ممكن صورة التالتة هنا وصورة الأولى
+
+290
+00:24:08,440 --> 00:24:15,500
+هنا يعني بمعنى آخر لو ال X هنا و ال Y هنا هي صورة
+
+291
+00:24:15,500 --> 00:24:21,400
+ال X وهي صورة مين ال Y لو جيتوا .. أخدتوا .. هي
+
+292
+00:24:21,400 --> 00:24:26,760
+صورة ال Y لو أخدتوا الفرق بين هذه و بين هذه الفرق
+
+293
+00:24:26,760 --> 00:24:31,020
+بين ما هو قيمة الدالة F of X هنا و قيمة الدالة F
+
+294
+00:24:31,020 --> 00:24:36,280
+of Y هنا أكيد .. أكيد .. أكيد الفرق بينهم هيكون
+
+295
+00:24:36,280 --> 00:24:42,320
+أصغر أو يساوي M K ناقص من M K لسبب بسيط أصلا لأن
+
+296
+00:24:42,320 --> 00:24:49,660
+أصلا F of X هذه أصغر أو يساوي ما هو M K أكيد و F
+
+297
+00:24:49,660 --> 00:24:54,400
+of Y أصغر أو يساويها ماشي الحال و نفس ال .. يعني ..
+
+298
+00:24:54,400 --> 00:24:57,080
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+299
+00:24:57,080 --> 00:24:57,100
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+300
+00:24:57,100 --> 00:24:57,360
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+301
+00:24:57,360 --> 00:24:57,720
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+302
+00:24:57,720 --> 00:24:58,100
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+303
+00:24:58,100 --> 00:24:59,480
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+304
+00:24:59,480 --> 00:25:00,040
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+305
+00:25:00,040 --> 00:25:07,580
+و .. و .. و .. و .. و
+
+306
+00:25:07,580 --> 00:25:08,160
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+307
+00:25:08,160 --> 00:25:11,200
+.. و .. و .. و .. 
+
+308
+00:25:19,040 --> 00:25:24,000
+فبيظل قيمة f of x ناقص f of y أصغر من مين Mk ناقص
+
+309
+00:25:24,000 --> 00:25:28,460
+Mk اللي مش واضح له من خلال الرسمة اللي عندي هنا
+
+310
+00:25:31,440 --> 00:25:34,520
+يعني الآن خلّيني أرى أضحى لحسن تكون مش واضحة للبعض
+
+311
+00:25:34,520 --> 00:25:41,360
+هاي الرسمة هاي XK minus واحد وهي XK خلّيني أكبر
+
+312
+00:25:41,360 --> 00:25:46,180
+الرسمة عشان تكون أوضح وهي XK لأن نفترضها أن هاي
+
+313
+00:25:46,180 --> 00:25:53,520
+رسمتنا هيك وطلعت زي هيك ماشي وهي عندي أعلى نقطة
+
+314
+00:25:53,520 --> 00:25:58,710
+خلّيني أزغر هذه شوية وهي أصغر النقطة هذه اللي
+
+315
+00:25:58,710 --> 00:26:03,970
+بتتمثل لـ M K وهذه بتتمثل لـ M K Small هاي المسافة
+
+316
+00:26:03,970 --> 00:26:09,030
+بينهم الفرق بينهم الآن الثانية لو جيت أخدت أي نقطة
+
+317
+00:26:09,030 --> 00:26:17,190
+هنا X و أي نقطة Y هنا هاي X وهي Y المسافة بينها دي
+
+318
+00:26:17,190 --> 00:26:21,310
+و بينها دي قيمتها دي بيصير F of X أنا بصير F of X
+
+319
+00:26:22,380 --> 00:26:26,700
+وهنا بيصير F of Y الفرق بين F of X و F of Y أكيد
+
+320
+00:26:26,700 --> 00:26:31,140
+أصغر من الفرق هذا وكل النقاط اللي بين هذه و هذه
+
+321
+00:26:31,140 --> 00:26:37,340
+هيكون يا إما زي الفرق هذا أو أصغر منه إذا أكيد
+
+322
+00:26:37,340 --> 00:26:45,820
+عندي صار المفروض وضح الأمر هيكون عندي اللي هو هذا
+
+323
+00:26:45,820 --> 00:26:50,910
+المقدار أصغر من شو هذا طب ليه ليش هذا؟ هذا من مين للـ
+
+324
+00:26:50,910 --> 00:26:53,990
+K اللي في الـ A و الـ K اللي في الـ A شخصيتها Mk
+
+325
+00:26:53,990 --> 00:26:57,610
+نقصها Mk أصغر من مين؟ إذا صارت أصغر من الـ Delta
+
+326
+00:26:57,610 --> 00:27:04,470
+إذا صارت F of X و F of Y أصغر من مين؟ من Delta طيب
+
+327
+00:27:04,470 --> 00:27:08,430
+مادام F of X إيش علاقات هنا F of X و F of Y ما هي
+
+328
+00:27:08,430 --> 00:27:13,950
+أصلا احنا مفترضين من الأول أن F of I جزئية من الـ
+
+329
+00:27:13,950 --> 00:27:18,430
+J اللي هي عبارة عن C و D مظبوط ولا لأ إذا حيصير
+
+330
+00:27:18,430 --> 00:27:23,600
+عندي F of X و F of Y موجودات في الـ C و D F of X و
+
+331
+00:27:23,600 --> 00:27:27,000
+F of Y موجودات في الـ C و الـ D و بتحقق المسافة
+
+332
+00:27:27,000 --> 00:27:32,200
+أصغر من Delta بينهم إذا حسب واحد اللي احنا أثبتنا
+
+333
+00:27:32,200 --> 00:27:35,960
+أي نقطتين في الـ S و الـ D المسافة بينهم أصغر من
+
+334
+00:27:35,960 --> 00:27:39,600
+Delta لازم صورة ال .. صورة Phi of S و Phi of T
+
+335
+00:27:39,600 --> 00:27:42,540
+المسافة بينهم أصغر من مين؟ من Epsilon و Prime إذا
+
+336
+00:27:42,540 --> 00:27:47,640
+هذا Automatic هيعطيني Phi of النقطة الأولى اللي هي
+
+337
+00:27:47,640 --> 00:27:56,290
+F of X ناقص Phi أفضل النقطة الثانية يكون أصغر من
+
+338
+00:27:56,290 --> 00:28:06,890
+مين من إبسلون برايم إذا صار عندي الآن لكل اللي عندي
+
+339
+00:28:06,890 --> 00:28:13,740
+ال K اللي في ال A و ال X و Y في هذه المنطقة F of X
+
+340
+00:28:13,740 --> 00:28:17,920
+minus F of Y طلع ليه أصغر من Delta حتى ملي الـ 5 F
+
+341
+00:28:17,920 --> 00:28:25,000
+of X ناقص 5 F of Y أصغر من مين؟ من Y برايم ماشي
+
+342
+00:28:25,000 --> 00:28:29,820
+الحال هذا الكلام لكل X و Y موجودات في الـ XK minus
+
+343
+00:28:29,820 --> 00:28:33,460
+واحد و الـ XK لخص الآن اللي وصلنا له لأنه بدي
+
+344
+00:28:33,460 --> 00:28:36,920
+أستخدمه خليني ألخص خليني أشيل
+
+345
+00:28:41,000 --> 00:28:45,820
+خلّيني بس ساعدك أكتب لكم في مكان هنا اتحملوني أنه
+
+346
+00:28:45,820 --> 00:28:54,640
+عشان اللوح شوية A بيساوي كل ال K مقصد MK أصغر من
+
+347
+00:28:54,640 --> 00:29:05,600
+دلتا وعند B بيساوي كل الـ K بحيث أن M K نقص M K أكبر
+
+348
+00:29:05,600 --> 00:29:10,060
+أو يساوي دلتة خلّيني أقولها في الذاكرة الآن خلّيني
+
+349
+00:29:10,060 --> 00:29:16,140
+أمسح حاجة و ألخص معلومتي إيش معلومتي بتقول دي لـ K
+
+350
+00:29:16,140 --> 00:29:26,610
+اللي في الـ A ماشي الحال لكل X و Y في الفترة X K-1
+
+351
+00:29:26,610 --> 00:29:32,750
+لعندي الـ xk بيطلع عندي الـ Phi f of x يعني Phi
+
+352
+00:29:32,750 --> 00:29:40,990
+composite f of x نقص Phi composite f of y أشماله
+
+353
+00:29:40,990 --> 00:29:48,600
+أصغر من إبسلون برايم واضح؟ إذن اللي وصلنا له لـ K
+
+354
+00:29:48,600 --> 00:29:53,700
+اللي في الـ A عندي لـ K اللي في الـ A هذا الـ
+
+355
+00:29:53,700 --> 00:29:58,380
+absolute value أظهر من مين؟ من إبسلون برايم لكل
+
+356
+00:29:58,380 --> 00:30:05,580
+مين لكل XY في ال XK minus XK minus 1 طيب نكمل لكم
+
+357
+00:30:05,580 --> 00:30:10,860
+عليها هذا خلصنا منه صار عندي الآن شوفوا
+
+358
+00:30:19,430 --> 00:30:24,070
+الشغل عميدي يا جماعة على الفترة هذه طيب شوف عندي
+
+359
+00:30:24,070 --> 00:30:31,590
+Phi composite F of X ناقص Phi composite F of Y
+
+360
+00:30:31,590 --> 00:30:37,770
+أكبر من Y' و أكبر من مين من ناقص Y برايم بدأ أخد
+
+361
+00:30:37,770 --> 00:30:43,970
+هذه الجهة خليني أخد الجهة من الجهتين و أبدأ أشتغل
+
+362
+00:30:43,970 --> 00:30:51,300
+عليها واضحة هذه هذه أصغر من Y' أكبر من ناقص Y بديش
+
+363
+00:30:51,300 --> 00:30:56,980
+هذا بتشغل هذه الآن لان هذا الكلام صحيح لمن؟ لكل X
+
+364
+00:30:56,980 --> 00:31:02,740
+و Y في ال sub interval هذه اللي هي لكل مين ال case؟
+
+365
+00:31:02,740 --> 00:31:06,600
+ال case اللي من A بس طيب شوف إذا صار عندي Phi
+
+366
+00:31:06,600 --> 00:31:14,580
+composite F of X أصغر من Y' بزايد Phi composite F
+
+367
+00:31:14,580 --> 00:31:21,720
+of Y ماشي يا جماعة .. ماشي هذا الكلام لكل x و y في
+
+368
+00:31:21,720 --> 00:31:26,980
+.. اللي هو
+
+401
+00:34:24,860 --> 00:34:32,200
+الناقص Phi composite F of Y such that Y element in
+
+402
+00:34:32,200 --> 00:34:39,360
+YK as if XK minus 1 عند XK هذا الـ Supremum له
+
+403
+00:34:39,360 --> 00:34:48,670
+أظهر أو يساوي Y' ناقص M K tilde طيب هات ايش بتساوي
+
+404
+00:34:48,670 --> 00:34:56,670
+طلعون الناقص برا بيساوي ناقص الـ Infimum معايا لأ
+
+405
+00:34:56,670 --> 00:34:59,790
+لأنه مدام ناقص واحد طلع إذا بيقلب الـ Supremum ل
+
+406
+00:34:59,790 --> 00:35:05,530
+Infimum Phi composite F of Y such that Y element
+
+407
+00:35:05,530 --> 00:35:09,610
+in XK minus واحد لعند ال XK عرفتوا ايش اللي بده
+
+408
+00:35:09,610 --> 00:35:16,060
+أقوله هاد مين هي يا شباب هذه عبارة عن الـ M K tilde
+
+409
+00:35:16,060 --> 00:35:19,680
+اللي هي الـ Infimum للـ Phi Composite F كتبت
+
+410
+00:35:19,680 --> 00:35:22,800
+التلدة عشان ترمز لمين؟ للـ Phi Composite F وفي عندي
+
+411
+00:35:22,800 --> 00:35:28,460
+ناقص قبلها إذا صارت عندي ناقص الـ M K tilde أصغر
+
+412
+00:35:28,460 --> 00:35:33,400
+يساوي Epsilon ناقص الـ M K tilde بدي أجيب هذه هنا
+
+413
+00:35:33,400 --> 00:35:42,400
+بيصير عندي إذا M K tilde أصغر ناقص Epsilon ناقص M K tilde
+
+414
+00:35:42,400 --> 00:35:51,080
+smaller أصغر أو يساوي مين؟ Epsilon إذا اللي وصلت له ما
+
+415
+00:35:51,080 --> 00:35:59,440
+يلي وهذه بده اصلها أنه لكل الـ a طلع عندي M K tilde
+
+416
+00:35:59,440 --> 00:36:05,300
+ناقص M K prime أصغر اش هو ايش؟ Epsilon خلينا نسجلها
+
+417
+00:36:05,300 --> 00:36:11,830
+عشان نبني عليها صار عندي الآن يا جماعة طولوا روحكم 
+
+418
+00:36:11,830 --> 00:36:18,670
+عليا إن شاء الله مش هم طوالين نخلص M K tilde ناقص M
+
+419
+00:36:18,670 --> 00:36:22,890
+K tilde smaller أصغر يمشي هو Epsilon هذا لكل K وين
+
+420
+00:36:22,890 --> 00:36:30,330
+موجودة في الـ A لكل K في الـ A طيب هذا اللي احنا ايش
+
+421
+00:36:30,330 --> 00:36:38,150
+ماله وصلنا له صار عندي الان الـ M K tilde ناقص M K
+
+422
+00:36:38,150 --> 00:36:52,190
+tilde أصغر من الـ Epsilon لكل K فيه اللي هي N A الان
+
+423
+00:36:52,190 --> 00:36:58,330
+خد الـ summation الـ summation للـ M K tilde ناقص M K
+
+424
+00:36:58,330 --> 00:37:05,490
+tilde smaller K element in A اللي هو هذا اللي هو أصغر
+
+425
+00:37:05,490 --> 00:37:09,310
+أو يساوي في .. أصغر أو يساوي .. الآن قدوا .. قدوا
+
+426
+00:37:09,310 --> 00:37:11,470
+.. قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+427
+00:37:11,470 --> 00:37:12,010
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+428
+00:37:12,010 --> 00:37:13,570
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+429
+00:37:13,570 --> 00:37:15,170
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+430
+00:37:15,170 --> 00:37:17,810
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+431
+00:37:17,810 --> 00:37:31,850
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا 
+
+432
+00:37:31,850 --> 00:37:39,180
+.. K element in A العداد هذا بعيد بعدد الـ K's اللي
+
+433
+00:37:39,180 --> 00:37:43,180
+في الـ A يعني Epsilon مضروب في عدد الـ K's اللي
+
+434
+00:37:43,180 --> 00:37:48,060
+موجودة في مين؟ في A هذا أكيد أكيد هذا أصغر أو
+
+435
+00:37:48,060 --> 00:37:55,320
+يساوي الـ summation للإبسلون على كل الـ K من عند
+
+436
+00:37:55,320 --> 00:38:00,980
+صفر لعند مين؟ لعند اللي هو H N
+
+437
+00:38:03,340 --> 00:38:09,620
+بس أنا الان بدي أجي أودي الـ summation و أضربه في
+
+438
+00:38:09,620 --> 00:38:13,460
+مين؟ في طول الفترة لأنه هي لازم لطول الفترة بعد
+
+439
+00:38:13,460 --> 00:38:20,660
+اسمهم هذا أضربه ليه؟ في XK بدي أضربه بس أوسع عقله
+
+440
+00:38:20,660 --> 00:38:23,700
+معلش بدي أضربه يعني هذا الـ summation اللي حصلت
+
+441
+00:38:23,700 --> 00:38:29,320
+عليه بدي أخد الـ summation MK tilde نقص mk tilde
+
+442
+00:38:29,320 --> 00:38:34,540
+و أضربه يا شباب في xk minus xk minus واحد k element
+
+443
+00:38:34,540 --> 00:38:39,000
+in a هيصير أصغر أو يساوي الـ summation للإيبسلون في
+
+444
+00:38:39,000 --> 00:38:48,620
+xk minus xk minus واحد على الان k element in a أكيد
+
+445
+00:38:48,620 --> 00:38:52,120
+أصغر زي ما قلنا من الـ summation اللي فوق epsilon
+
+446
+00:38:52,120 --> 00:38:59,980
+في xk minus xk minus 1 k من 0 لعند n ليش؟ هذا ال
+
+447
+00:38:59,980 --> 00:39:02,420
+summation بس على العناصر اللي في الـ A هذا ال
+
+448
+00:39:02,420 --> 00:39:05,260
+summation على كله على اللي في الـ A وعلى اللي في ال
+
+449
+00:39:05,260 --> 00:39:08,760
+B على كل بقى العناصر اللي هي الـ partition اللي
+
+450
+00:39:08,760 --> 00:39:15,420
+عنده من 0 ل 1 2 عند مين؟ لعند الـ N طيب الـ N هذا
+
+451
+00:39:15,420 --> 00:39:20,200
+المقدر هي اللي بدي إياها الآن هذا bridge كان أو جسر
+
+452
+00:39:20,200 --> 00:39:23,860
+للوصول للي بدي إياها وحتى ممكن نهان مباشرة لهذه هذه
+
+453
+00:39:23,860 --> 00:39:27,760
+أصغر أو يساوي هذا هذا مش هو اللي بتطلع عليه بتطلع
+
+454
+00:39:27,760 --> 00:39:32,640
+عليه هذا وبتطلع عليه هذا هذا خليه لأنه أصغر يساوي
+
+455
+00:39:32,640 --> 00:39:36,460
+مين؟ هذا المقدار هذا المقدار ايش بيساوي؟ Epsilon في
+
+456
+00:39:36,460 --> 00:39:43,790
+مين؟ x note ناقص x واحد زائد x واحد ناقص x اتنين زائد
+
+457
+00:39:43,790 --> 00:39:47,810
+x اتنين ناقص x ثلاثة لما أصل لآخر واحد يضل اللي
+
+458
+00:39:47,810 --> 00:39:53,330
+عبارة عن y في x n ناقص x note عملناها كتير ويساوي
+
+459
+00:39:53,330 --> 00:39:58,710
+y prime ويساوي y prime في مين؟ في b minus a لأن ال
+
+460
+00:39:58,710 --> 00:40:03,950
+xn هي عبارة عن b و ال x note هي مين؟ كانت هي ال a
+
+461
+00:40:03,950 --> 00:40:10,110
+إذا اللي وصلت له يا جماعة الان وصلت إلى ما يلي إنه
+
+462
+00:40:10,110 --> 00:40:19,170
+الـ summation لل mk tilde ناقص mk tilde smaller لكي
+
+463
+00:40:19,170 --> 00:40:25,790
+element ايه في xk minus xk minus واحد أصغر أو
+
+464
+00:40:25,790 --> 00:40:34,550
+يساوي epsilon prime في b minus ايش؟ minus a واضح
+
+465
+00:40:34,550 --> 00:40:42,680
+الانهذه خزينها لأنه بدي إياها بعد شوية اتحملونا
+
+466
+00:40:42,680 --> 00:40:46,880
+البرهان طويل شوية هي وصلنا إليها summation ازرع
+
+467
+00:40:46,880 --> 00:40:50,980
+شوية Epsilon رابع في b minus a لأن بدنا نيجي لمين؟
+
+468
+00:40:50,980 --> 00:40:59,140
+يا جماعة بدنا نيجي للي هو انتبهوا عليها بدنا
+
+469
+00:40:59,140 --> 00:41:03,280
+نيجي للـ M للـ K اللي موجودة في مين؟ هذا الـ K في
+
+470
+00:41:03,280 --> 00:41:06,820
+الـ A ضال عند مين؟ احنا جزأتها إلى جزئين اللي هي
+
+471
+00:41:06,820 --> 00:41:11,340
+الـ indices اللي عنده K اللي هي في الـ A و K اللي
+
+472
+00:41:11,340 --> 00:41:18,060
+هي في مين؟ في الـ B خد الآن K element in B شوف
+
+473
+00:41:18,060 --> 00:41:26,980
+ايش؟ الـ MK tilde إلهانقص mk tilde إله أيش بتساوي؟
+
+474
+00:41:26,980 --> 00:41:32,380
+بتساوي حسب التعريف Supremum للـ Phi Composite F of
+
+475
+00:41:32,380 --> 00:41:38,240
+X للـ K اللي وين؟ في الـ B اللي هي mk tilde نقص mk
+
+476
+00:41:38,240 --> 00:41:42,540
+tilde اللي هي smaller Supremum للـ Phi F of X اللي
+
+477
+00:41:42,540 --> 00:41:47,980
+هي الأولى هذه Such that X element in XK minus واحد
+
+478
+00:41:47,980 --> 00:41:54,860
+و XK اللي هي زائد أو ناقص اللي هي مين؟ هذه الـ
+
+479
+00:41:54,860 --> 00:42:01,200
+Infimum لـ Phi composite F of X such that X
+
+480
+00:42:01,200 --> 00:42:10,220
+والموجودة في الـ X K minus واحد والـ X K مظبوط ولا
+
+481
+00:42:10,220 --> 00:42:18,690
+لأ احنا قلنا الـ Supremum لكل الـ Phi of T T على كل
+
+482
+00:42:18,690 --> 00:42:24,130
+الـ C والـ D بيساوي ايش اسمه؟ بيساوي K هذه عبارة
+
+483
+00:42:24,130 --> 00:42:29,790
+عن مين؟ شايفين مابديش أعيدك بقى أكتب مرتين خليني
+
+484
+00:42:29,790 --> 00:42:33,890
+أكتبها على شورة بتصير Phi of F of X يعني بتصير
+
+485
+00:42:33,890 --> 00:42:42,250
+عبارة عن مين؟ Phi of F of X معايا؟
+
+486
+00:42:43,040 --> 00:42:48,280
+وهاد ايش اسمها؟ بدي أدخل السالب جوا يا جماعة أو
+
+487
+00:42:48,280 --> 00:42:56,440
+قبل ما أدخله هد بيصير Phi Phi of F of X بعد
+
+488
+00:42:56,440 --> 00:43:01,560
+اذنكم بدي أدخل السالب جوا الـ Infimum هتصير ايش ما
+
+489
+00:43:01,560 --> 00:43:06,240
+لها؟ Supremum إذا بيصير هذا زائد وهد بتصير
+
+490
+00:43:06,240 --> 00:43:14,100
+Supremum والسالب بدخل جوا طيب .. عندي الـ Phi F of 
+
+491
+00:43:14,100 --> 00:43:19,800
+X والـ Phi F of X للـ X اللي موجودة هنا الـ F of X
+
+492
+00:43:19,800 --> 00:43:25,460
+هدولة من مين جايت؟ من الفترة C و D لأن زي ما قلنا F
+
+493
+00:43:25,460 --> 00:43:30,680
+of A و F of I subset من الجهة اللي هي C و D لذا كل
+
+494
+00:43:30,680 --> 00:43:35,040
+الـ element هنا موجود هنا يعني بمعنى آخر الـ ال
+
+495
+00:43:35,040 --> 00:43:38,720
+Supremum من اللي هنا اللي هي الـ K عندي الـ absolute
+
+496
+00:43:38,720 --> 00:43:45,400
+value للـ Phi of T أكيد أكبر أو يساوي الـ Phi سالب
+
+497
+00:43:45,400 --> 00:43:51,080
+Phi of T و أصغر أو يساوي الـ Phi of T أو بالسوية
+
+498
+00:43:51,080 --> 00:43:55,280
+حتى أكيد
+
+499
+00:43:55,280 --> 00:43:57,900
+خليني أخدها على خطوتين بتسويها دي لأن ال
+
+500
+00:43:57,900 --> 00:44:06,260
+absolute value Phi of T أكبر يساوي مين؟ سالب ال Phi
+
+501
+00:44:06,260 --> 00:44:11,300
+of T فبيصير عندي الـ Supremum اللي عندي اللي هالة
+
+502
+00:44:11,300 --> 00:44:18,410
+هذا Supremum لهذه أكيد أصغر أو يساوي اللي هو الـ
+
+503
+00:44:18,410 --> 00:44:22,170
+Supremum هذا اللي هو K لأن الـ Supremum هذا عالمين
+
+504
+00:44:22,170 --> 00:44:26,370
+على كل الفترة C و D هنا اللي هي عالمين على X
+
+505
+00:44:26,370 --> 00:44:30,890
+element in XK مانس واحد و XK لأن هذا Similarly
+
+506
+00:44:30,890 --> 00:44:35,910
+أكيد الـ Supremum للـ Absolute Value هيكون أكبر أو
+
+507
+00:44:35,910 --> 00:44:39,070
+يساوي الـ Supremum لهذه من جهتين إنه Absolute
+
+508
+00:44:39,070 --> 00:44:42,790
+Value وفي نفس الوقت اللي هو المجموع هذه جزئية من
+
+509
+00:44:42,790 --> 00:44:48,380
+اللي فوق إذن ده برضه بيكون أصغر أو يساوي K إذا صار
+
+510
+00:44:48,380 --> 00:44:53,800
+automatic اللي هو هذه زائد هذه أصغر أو يساوي مين؟
+
+511
+00:44:53,800 --> 00:45:01,440
+اتنين K من أي حاجة أصغر أو يساوي اتنين K الان هذه
+
+512
+00:45:01,440 --> 00:45:06,160
+الـ Supremum زائد الـ Supremum هذه بس لا ياش لسانك
+
+513
+00:45:06,160 --> 00:45:11,900
+يا شباب وهذه K أصغر يساوي K زائد K عشان التلخيص
+
+514
+00:45:11,900 --> 00:45:19,460
+بيصير اشمله اتنين K واضح إذا نخزن التانية اللي هي K
+
+515
+00:45:19,460 --> 00:45:24,480
+element in B بعد إذنكم بيصير عندي الـ M K نقص بلاش
+
+516
+00:45:24,480 --> 00:45:30,200
+التفاصيل هذه خلصنا منها بيصير عندي الـ M K نقص M K
+
+517
+00:45:30,200 --> 00:45:37,180
+smaller أصغر من tilde tilde طبعا أصغر من مين؟ من اتنين K
+
+518
+00:45:37,180 --> 00:45:43,280
+طبعا الـ K هنا index الـ K هذه Capital هذه الـ K مين؟
+
+519
+00:45:43,280 --> 00:45:48,590
+الـ K اللي فوق هذه هذا الـ K هو الـ index للـ B وهذا
+
+520
+00:45:48,590 --> 00:45:50,690
+الـ K هو الـ index للـ B يعني الـ K هذه ليست إعاده
+
+521
+00:45:50,690 --> 00:45:57,370
+هذه K هي الـ Supremum اللي فوق طيب نجمح
+
+522
+00:45:57,370 --> 00:46:03,190
+معلوماتنا ونبدأ نجمحها الآن ونكون حصلنا على اللي
+
+523
+00:46:03,190 --> 00:46:08,410
+بده 2K اه لب تلخيص 2K يا شباب لما تشوفوها لب
+
+524
+00:46:08,410 --> 00:46:13,770
+تلخيص 2K الان أوجدت الـ summation على مين؟ على ايه؟
+
+525
+00:46:13,770 --> 00:46:16,710
+بدي أوجد الـ summation على الـ P باقي الـ indices لأن
+
+526
+00:46:16,710 --> 00:46:21,650
+هالجات بدي أقولكم ليش بالضيقة الان إذا صار عندي ال
+
+527
+00:46:21,650 --> 00:46:29,930
+summation لل Mk ناقص Mk tilde tilde في Xk minus Xk
+
+528
+00:46:29,930 --> 00:46:40,680
+minus واحد K element in B الآن مظبوط أصغر أو يساوي
+
+529
+00:46:40,680 --> 00:46:45,120
+اللي هو 2K برضه على الـ summation هذا الـ summation
+
+530
+00:46:45,120 --> 00:46:53,040
+اللي هو 2K Capital Capital هذه مالهاش علاقة بالـ K
+
+531
+00:46:53,040 --> 00:47:00,840
+الـ element in B هنا في XK minus XK minus 1 أخدت ال
+
+532
+00:47:00,840 --> 00:47:03,700
+summation على الجهتين على كل الـ K الـ element in B
+
+533
+00:47:06,360 --> 00:47:20,220
+ماشي طيب وين أكتب طيب اتحملوني هلجيت عندي خليني
+
+534
+00:47:20,220 --> 00:47:25,300
+أكتب هنا أشرح أنا شوية بهمش خليني أشرح على ال ..
+
+535
+00:47:25,300 --> 00:47:33,740
+كل الـ B كل الـ B احنا من تعريف كل الـ B بدي
+
+536
+00:47:33,740 --> 00:47:39,160
+يكون الـ mk ناقص mk أكبر شو مين؟ Delta صح؟
+
+537
+00:47:39,160 --> 00:47:41,980
+بيصير عندي الآن Delta يعني أصغر شو mk ناقص m
+
+538
+00:47:41,980 --> 00:47:47,000
+k الـ small ايه بيسمعوا على الـ Delta هنا؟ هو أنا
+
+539
+00:47:47,000 --> 00:47:50,920
+على الـ Delta بيصير هذا ايه شبال؟ هو واحد معايا؟
+
+540
+00:47:50,920 --> 00:48:02,180
+فبيصير عندي الآن الـ summation على الواحد XK-XK- XK
+
+541
+00:48:02,180 --> 00:48:07,650
+- XK- XK- XK- XK- XK- XK- XK أصغر أو يساوي mk نقص
+
+542
+00:48:07,650 --> 00:48:13,130
+mk على Delta ماشي خد الـ summation للجهتين ال
+
+543
+00:48:13,130 --> 00:48:19,030
+summation للجهتين xk minus xk minus واحد وانا ال
+
+544
+00:48:19,030 --> 00:48:24,310
+summation xk minus xk minus واحد k element in b
+
+545
+00:48:24,310 --> 00:48:28,850
+لأن هذا صح للـ b بس لـ k element in b مظبوط ممكن
+
+546
+00:48:28,850 --> 00:48:33,880
+حصلنا على هذا هي اللي موصول وهذا واحد على Delta
+
+547
+00:48:33,880 --> 00:48:38,720
+طلعها برا في الـ summation اللي عندي هذا ماشي وصلنا
+
+548
+00:48:38,720 --> 00:48:45,620
+ل هيك إذا الآن صار عندي هذا أصغر أو يساوي واحد على
+
+549
+00:48:45,620 --> 00:48:56,360
+Delta في مين؟ في الان هذا اللي هو ال UPUF طب أنا دي
+
+550
+00:48:56,360 --> 00:49:00,500
+MK و MK اه مش MK tilde هذه اه
+
+551
+00:49:03,350 --> 00:49:07,230
+ليش؟ لأنه .. اشكروا إن أنا كتبت هنا Mk و Mk tilde
+
+552
+00:49:07,230 --> 00:49:15,130
+لأن هذه الـ A الـ B الـ B الـ K اللي فيها مصنفة
+
+553
+00:49:15,130 --> 00:49:19,530
+على أساس Mk اسمه نقص Mk أكبر زي Delta فبيصير عندي
+
+554
+00:49:19,530 --> 00:49:25,800
+الآن الـ Mk نقص Mk اللي هو أكبر أو يساوي Delta مش
+
+555
+00:49:25,800 --> 00:49:31,520
+Delta هذه هذه لمين؟ اللي مصنف عليها الـ B اللي هي
+
+556
+00:49:31,520 --> 00:49:35,860
+بالنسبة للـ F هذه مش لـ Alpha Composite F إذا هذا
+
+557
+00:49:35,860 --> 00:49:38,560
+أكبر أو يساوي Delta على Delta بيصير أكبر أو يساوي
+
+558
+00:49:38,560 --> 00:49:41,740
+واحد على الـ summation اللي عملناها قبل بشوية أصغر
+
+559
+00:49:41,740 --> 00:49:44,720
+أو يساوي واحد على Delta في الـ summation هذا الآن
+
+560
+00:49:44,720 --> 00:49:55,810
+.. الآن هذا مين هو؟ هو عبارة عن الـ UP و F مش كله
+
+561
+00:49:55,810 --> 00:50:01,210
+حتى جزء منه لأن الـ U P و F ايش ماله الـ M كده تبعته
+
+562
+00:50:01,210 --> 00:50:08,310
+الـ U P و F ايش هو يا شباب؟ هو عبارة عن الـ summation
+
+563
+00:50:08,310 --> 00:50:14,090
+K من عند واحد لعند N أو من Zero لعند N من واحد
+
+564
+00:50:14,090 --> 00:50:25,770
+لعند N مظبوط لمين؟ لـ الـ M ك في xk minus xk minus
+
+565
+00:50:25,770 --> 00:50:30,070
+واحد هذا الـ summation على مين؟ على كل الـ case من
+
+566
+00:50:30,070 --> 00:50:35,290
+واحد لعند n بينما هذا الـ summation لمين؟ بس للجزء
+
+567
+00:50:35,290 --> 00:50:39,130
+اللي هو في مين؟ في الـ B فأكيد الـ summation على هذه
+
+568
+00
+
+601
+00:53:28,540 --> 00:53:32,980
+الـ summation اللي وصلت إليه الـ summation للـ xk
+
+602
+00:53:32,980 --> 00:53:41,600
+ناقص xk minus 1 k element in B أصغر من إبسلون
+
+603
+00:53:41,600 --> 00:53:46,920
+إبراهيم الآن
+
+604
+00:53:46,920 --> 00:53:53,760
+بس إحنا أثبتنا إن الـ summation هذا أيه أصغر أو
+
+605
+00:53:53,760 --> 00:54:00,430
+يساوي 2k اثنين كيف ال summation هذا مظبوط بدي أعوض
+
+606
+00:54:00,430 --> 00:54:06,850
+الآن أشيل هذا بيصير عندي هاي شايفين يا شباب هاي
+
+607
+00:54:06,850 --> 00:54:14,070
+عند ال summation هذا أثبته أنه أصغر أو يساوي اللي
+
+608
+00:54:14,070 --> 00:54:24,580
+هو epsilon prime واضح ومن أربعة بيصير عندي أعوض ال
+
+609
+00:54:24,580 --> 00:54:28,980
+summation أصغر من 2 كيف ال summation اللي أثبتناه
+
+610
+00:54:28,980 --> 00:54:34,840
+بحيته بس قبل شوية ال summation أصغر من 2 كيف ال
+
+611
+00:54:34,840 --> 00:54:40,140
+summation بشيل هذا اللي هو هذا وأقول أصغر من مين
+
+612
+00:54:40,140 --> 00:54:43,760
+من epsilon prime بيصير هذا المقدار اللي هو هذا ال
+
+613
+00:54:43,760 --> 00:54:45,680
+summation هيه
+
+614
+00:54:47,550 --> 00:54:54,290
+هيو هذا هيو أصغر من من 2k في ash في ال epsilon
+
+615
+00:54:54,290 --> 00:55:00,110
+prime إذا طلع عندي الجزء الثاني summation خلصنا من
+
+616
+00:55:00,110 --> 00:55:07,610
+هذا صار عندي الجزء الثاني اللي أثبته summation لل
+
+617
+00:55:07,610 --> 00:55:19,240
+mk tilde ناقص mk tilde في xk-xk-1 k element in B أصغر
+
+618
+00:55:19,240 --> 00:55:26,980
+من مين طلع من epsilon prime في 2k إبسلون برايم في
+
+619
+00:55:26,980 --> 00:55:36,650
+2k الآن خلصنا احسبولي إذا احنا لقينا ال partition B
+
+620
+00:55:36,650 --> 00:55:44,330
+لكل option أكبر من 0 لقينا B بحيث أنه U B of I
+
+621
+00:55:44,330 --> 00:55:54,210
+composite F ناقص ال B of I composite F هذا أشماله
+
+622
+00:55:54,210 --> 00:55:59,310
+بيساوى اللي هو الـ summation اللي هيضع على كل الـ K
+
+623
+00:55:59,310 --> 00:56:02,390
+والـ summation اللي هيضع على كل الـ K والـ K
+
+624
+00:56:02,390 --> 00:56:05,910
+جزّأناه لجزئين إيش في الـ A وإيش في الـ B إذا
+
+625
+00:56:05,910 --> 00:56:14,110
+بيساوي اللي هو الـ summation ل الـ A الـ MK ناقص
+
+626
+00:56:14,110 --> 00:56:23,140
+MK tilde tilde في xk-xk-1 k من واحد ل عند n صح ولا لأ
+
+627
+00:56:23,140 --> 00:56:28,220
+آه طبعًا هو التعريف هذا اللي هو بيساوي الآن الـ K
+
+628
+00:56:28,220 --> 00:56:32,460
+اللي عندك جزّأته لجزئين واحد في الـ A واحد في مين؟
+
+629
+00:56:32,460 --> 00:56:37,080
+في الـ B، إذاً هذا الـ summation الكل بيساوي ال
+
+630
+00:56:37,080 --> 00:56:41,340
+summation على K في الـ A زائد ال summation نفسه
+
+631
+00:56:41,340 --> 00:56:45,720
+على K وين ما له؟ في الـ B ال summation لمين؟ لهذه
+
+632
+00:56:45,720 --> 00:56:53,280
+اللي هي MK tilde ناقص MK small في XK minus XK
+
+633
+00:56:53,280 --> 00:57:01,450
+minus واحد زائد MK tilde ناقص mk small في xk minus
+
+634
+00:57:01,450 --> 00:57:08,990
+xk minus واحد آه طيب يعني الآن هذا ال U وهذا ال L
+
+635
+00:57:08,990 --> 00:57:12,030
+صار بيساوي هذا الجزء وهذا الجزء وهذا السبب أصلاً
+
+636
+00:57:12,030 --> 00:57:17,590
+اللي خلّاه أجزاء هذا طلع لي الآن مين هو طلع لي أصغر
+
+637
+00:57:17,590 --> 00:57:22,970
+من epsilon prime أصغر من epsilon prime في b minus
+
+638
+00:57:22,970 --> 00:57:32,330
+a والثاني طلع أصغر من E' في 2K E' في 2K يعني هدول
+
+639
+00:57:32,330 --> 00:57:40,140
+الاثنين مع بعض إيش بيساويون؟ بيساويون E' فيه 2k بيزاق
+
+640
+00:57:40,140 --> 00:57:56,220
+بيزاق بيزاق بيزاق بيزاق بيزاق بيزاق
+
+641
+00:57:56,310 --> 00:58:04,950
+على b minus a زي اثنين k في اثنين k زي b minus a
+
+642
+00:58:04,950 --> 00:58:10,030
+وهذه بتروح مع حد بيساوي إيش إبسلون إذا اللي وصلت له
+
+643
+00:58:11,120 --> 00:58:17,600
+لكل إبسلون أكبر من صفر لكل
+
+644
+00:58:17,600 --> 00:58:22,480
+إبسلون أكبر من صفر لكل إبسلون أكبر من صفر لكل
+
+645
+00:58:22,480 --> 00:58:25,660
+إبسلون أكبر من صفر لكل إبسلون أكبر من صفر لكل
+
+646
+00:58:25,660 --> 00:58:27,320
+إبسلون أكبر من صفر لكل إبسلون أكبر من صفر لكل
+
+647
+00:58:27,320 --> 00:58:28,360
+إبسلون أكبر من صفر لكل إبسلون أكبر من صفر لكل
+
+648
+00:58:28,360 --> 00:58:32,200
+إبسلون أكبر من صفر لكل إبسلون أكبر من صفر لكل
+
+649
+00:58:32,200 --> 00:58:34,180
+إبسلون أكبر من صفر لكل إبسلون أكبر من صفر لكل
+
+650
+00:58:34,180 --> 00:58:36,880
+إبسلون أكبر من صفر لكل إبسلون أكبر من صفر لكل
+
+651
+00:58:36,880 --> 00:58:47,340
+إبسلون أكبر من صفر طيب الآن نجي بدنا اللي هو نأخذ
+
+652
+00:58:47,340 --> 00:58:55,660
+اللي هو تطبيقات على هذه النظرية ونشوف
+
+653
+00:58:55,660 --> 00:59:02,600
+كيف بدنا نطبق هذه النظرية للوصول إلى نتائج أخرى
+
+654
+00:59:02,600 --> 00:59:09,280
+تتعلق بخواص التكامل أو بخواص تكامل ريمان لأن
+
+655
+00:59:09,280 --> 00:59:15,300
+عرفنا وجاوبنا على اللي هو المجموع وعلى حسب ضرب K
+
+656
+00:59:15,300 --> 00:59:20,100
+في الـ .. في الـ F ولقينا على ال composition وتحت أي
+
+657
+00:59:20,100 --> 00:59:24,360
+ضارب كان F I composite F is integrable الآن نكمل
+
+658
+00:59:24,360 --> 00:59:30,120
+اللي هو نأخذ اللي هو ال corollary اللي عندي اللي
+
+659
+00:59:30,120 --> 00:59:39,150
+هي بتقول لي اللي هي من ثلاث فروع لو كانت I عبارة عن
+
+660
+00:59:39,150 --> 00:59:43,990
+الـ closed interval A وB وF من I لعند R is
+
+661
+00:59:43,990 --> 00:59:47,370
+integrable on I then the absolute value of a
+
+662
+00:59:47,370 --> 00:59:50,630
+function F obtained by the absolute value is
+
+663
+00:59:50,630 --> 00:59:54,410
+integrable on I والـ absolute value of integration
+
+664
+00:59:54,410 --> 00:59:56,470
+أصغر أو يساوي الـ integration للـ absolute value للـ
+
+665
+00:59:56,470 --> 01:00:02,810
+F اللي هي أصغر أو يساوي K K في ال B minus A حيث ال K
+
+666
+01:00:02,810 --> 01:00:06,750
+هذه من أين جاية؟ هي ال bound لل F of X لأن F is
+
+667
+01:00:06,750 --> 01:00:12,260
+integrable إذا أكيد bounded فن element in F يعني
+
+668
+01:00:12,260 --> 01:00:15,280
+اللي بتقول لي هذا باختصار ده كانت F is integrable ال
+
+669
+01:00:15,280 --> 01:00:18,100
+absolute value لل F إيش ما لها؟ integrable وال
+
+670
+01:00:18,100 --> 01:00:20,160
+absolute value لل integration أصغر أو سوى ال
+
+671
+01:00:20,160 --> 01:00:22,860
+integration لل absolute value أصغر أو سوى K اللي هي
+
+672
+01:00:22,860 --> 01:00:26,580
+ال maximum لل أو خلّيني أقول ال bound لل F of X
+
+673
+01:00:26,580 --> 01:00:31,080
+absolute value ل F of X في B minus A الجزء الثاني
+
+674
+01:00:31,080 --> 01:00:35,460
+اللي هو هتطلع لي برضه الـ Fn is integrable لأن Fn
+
+675
+01:00:35,460 --> 01:00:41,560
+هو integer
+
+676
+01:00:42,100 --> 01:00:46,780
+الآن إذا كان الشغل الثاني إذا كان فيه Delta بحيث
+
+677
+01:00:46,780 --> 01:00:50,400
+أن F of X أكبر أو يساوي Delta يعني F of X أكبر أو
+
+678
+01:00:50,400 --> 01:00:53,900
+يساوي Delta أكبر أو يساوي Delta ليش؟ على أساس إنه
+
+679
+01:00:53,900 --> 01:00:58,140
+يضمن مقلوبه هيكون bounded بيصير 1 على F of X أصغر
+
+680
+01:00:58,140 --> 01:01:00,680
+أو يساوي 1 على Delta يعني بمعنى أكثر F is bounded
+
+681
+01:01:00,680 --> 01:01:04,720
+إذن بحقله يحكي عن ال Integrability ل 1 على F ويقول
+
+682
+01:01:04,720 --> 01:01:07,780
+لي تحت الظرف هذا لو كانت F of X أكبر أو يساوي Delta
+
+683
+01:01:08,470 --> 01:01:12,210
+و Delta أكبر من 0 هيكون 1 على الـ F إيش ما له؟ is
+
+684
+01:01:12,210 --> 01:01:15,830
+integrable on I خلّيني أترجح واحدة واحدة اللي
+
+685
+01:01:15,830 --> 01:01:24,690
+عنها ال gate F عندي من I لعند ب R is integrable إذا
+
+686
+01:01:24,690 --> 01:01:29,350
+there exists K اللي هو أكبر من 0 such that
+
+687
+01:01:29,350 --> 01:01:33,330
+absolute value of F of X أصغر أو يساوي إيش؟ K مدام
+
+688
+01:01:33,330 --> 01:01:36,430
+أن تقرأ بالـ F إذا أكيد is bounded إذا ال absolute
+
+689
+01:01:36,430 --> 01:01:43,030
+value of X أصغر أو يساوي ال K طيب الآن يا جماعة
+
+690
+01:01:43,030 --> 01:01:47,230
+عندي هاد ال corollary أصلاً اللي جايبله بدي أظبط لي
+
+691
+01:01:47,230 --> 01:01:54,070
+two functions بدي أعرف الآن صارت عند ال F of I ال
+
+692
+01:01:54,070 --> 01:02:00,230
+F of I صارت ال F of I أكيد subset بين ناقص K ومين؟
+
+693
+01:02:00,230 --> 01:02:01,210
+أو K
+
+694
+01:02:07,180 --> 01:02:19,960
+طيب يعني الآن لو جيت عرفت Phi من اللي هو عنده ناقص
+
+695
+01:02:19,960 --> 01:02:27,990
+K لعند ال K لعند ال R عرفتها على أساس فاي ما هو
+
+696
+01:02:27,990 --> 01:02:30,950
+absolute value إذا مضحكت تجيب ال absolute value
+
+697
+01:02:30,950 --> 01:02:38,250
+فاي of T بيساوي absolute value لمين؟ لل T ماشي الحال
+
+698
+01:02:38,250 --> 01:02:42,430
+هي الدالة عندي أكيد ال absolute value إيش ما لها؟
+
+699
+01:02:42,430 --> 01:02:47,370
+is continuous مدام ال absolute value الفاي is
+
+700
+01:02:47,370 --> 01:02:52,870
+continuous و ال F معطيله إياه integrable أيضا by
+
+701
+01:02:52,870 --> 01:03:00,670
+the above theorem فاي composite F is integrable فاي
+
+702
+01:03:00,670 --> 01:03:07,610
+composite F of T بيساوي فاي of F of T اللي هي
+
+703
+01:03:07,610 --> 01:03:12,330
+بتساوي إيش؟ واحدة absolute value فاي of T بتساوي
+
+704
+01:03:12,330 --> 01:03:17,490
+absolute value لمين؟ للـ F of T إذا صار عندي
+
+705
+01:03:17,490 --> 01:03:23,460
+absolute value للـ F is integrable إذاً مدامة Phi
+
+706
+01:03:23,460 --> 01:03:27,460
+is continuous و F integrable لأن حسب النظرية Phi
+
+707
+01:03:27,460 --> 01:03:30,640
+composite F is integrable و Phi composite F هي
+
+708
+01:03:30,640 --> 01:03:33,400
+طلعت من الـ absolute value للـ F طبعًا Phi
+
+709
+01:03:33,400 --> 01:03:36,880
+composite F من وين هتشتغل؟ من عند اللي هي ال
+
+710
+01:03:36,880 --> 01:03:42,920
+interval I لعند مين؟ لعند R لأن Phi اللي هو F of T
+
+711
+01:03:42,920 --> 01:03:49,980
+هيجي يقعد في الفترة هذه و Phi هترسل اللي بيجي لوين؟
+
+712
+01:03:49,980 --> 01:03:55,250
+لعند I ماضح ها طيب الآن عندي بدي أثبت أن ال
+
+713
+01:03:55,250 --> 01:03:58,590
+integration ال absolute value ال integration أصغر
+
+714
+01:03:58,590 --> 01:04:01,350
+أو يساوي ال integration ل absolute value هذه صارت
+
+715
+01:04:01,350 --> 01:04:07,670
+بعضها معلومات صادقة اللي هو عندي ال F أكبر أو يساوي
+
+716
+01:04:07,670 --> 01:04:11,610
+اللي هي absolute value لل F بالسالب وأصغر أو يساوي
+
+717
+01:04:11,610 --> 01:04:14,790
+ال absolute value ل 100 لل F صارت الآن احنا أثبتنا
+
+718
+01:04:14,790 --> 01:04:17,990
+أن ال absolute value is integrable إذا الآن بما أن
+
+719
+01:04:17,990 --> 01:04:21,270
+F بين هذولة الدالتين حسب نظريها أخذناها المرة
+
+720
+01:04:21,270 --> 01:04:24,910
+الماضية أو remark إذا بيصير ال integration لل
+
+721
+01:04:24,910 --> 01:04:28,590
+absolute value لل F من A ل B أصغر أو يساوي ال
+
+722
+01:04:28,590 --> 01:04:32,330
+integration لل F من A ل B أصغر أو يساوي ال
+
+723
+01:04:32,330 --> 01:04:38,190
+integration لل F من A ل B وهذا إيش معناه؟ هذا
+
+724
+01:04:38,190 --> 01:04:43,610
+معناه إذا ال absolute value لل integration لل F
+
+725
+01:04:43,610 --> 01:04:49,150
+من A ل B أصغر أو يساوي اللي هو ال integration لأ لأبسل
+
+726
+01:04:49,150 --> 01:04:57,690
+و لأ لأف من A ل B ماشي الحال و طبيعي عندي هذه
+
+727
+01:04:57,690 --> 01:05:02,270
+أثبتناها بهذه و طبيعي مدام عندنا لسه ما
+
+728
+01:05:07,890 --> 01:05:12,470
+أو ملاحظة سابقة لبس الـ integration من a ل b لبس ال
+
+729
+01:05:12,470 --> 01:05:18,070
+value of f of x أصغر أو يساوي اللي هو من k في طول
+
+730
+01:05:18,070 --> 01:05:27,050
+الفترة ب - a مع هذه together بنحصل على اللي هو
+
+731
+01:05:27,050 --> 01:05:30,750
+المطلوب ال absolute value ال integration أصغر أو يساوي
+
+732
+01:05:30,750 --> 01:05:34,690
+ال integration لل absolute value أصغر أو يساوي K في ال
+
+733
+01:05:34,690 --> 01:05:39,510
+B minus A هذه اللي هو الجزء الأول من ال corollary
+
+734
+01:05:39,510 --> 01:05:44,310
+خلينا نشوف الجزء الثاني من ال corollary برضه اللي
+
+735
+01:05:44,310 --> 01:05:51,270
+هو برهانه سهل و برهانه اللي هو مشابه للمنطق اللي
+
+736
+01:05:51,270 --> 01:05:57,070
+حكيناها عندنا قبل بشوية طيب صلى الله علينا يا عزيزي
+
+737
+01:05:57,070 --> 01:06:01,410
+عليه الصلاة والسلام الآن يا شباب عندي اللي هو
+
+738
+01:06:01,410 --> 01:06:05,110
+الجزء الثاني بدنا نثبت أن الـ F أس n is integrable
+
+739
+01:06:05,110 --> 01:06:09,110
+في حالة الـ F integrable لأن أكيد شكلنا بدنا نيجي
+
+740
+01:06:09,110 --> 01:06:16,950
+عند الـ F من I لعند R is integrable الآن عندي اللي
+
+741
+01:06:16,950 --> 01:06:21,470
+هو .. بدي أسأل أن أثبت أنها Integrable فلو جيت أنا
+
+742
+01:06:21,470 --> 01:06:30,170
+عرفت F F أو اللي هو Phi function من اللي هو domain
+
+743
+01:06:30,170 --> 01:06:35,150
+اللي هو F of I من اللي هو range ال F range ال F
+
+744
+01:06:35,150 --> 01:06:45,640
+range ال F لعند ال R بحيث إن أقول Phi of T بيساوي T
+
+745
+01:06:45,640 --> 01:06:51,120
+أس n five of T إيش بيساوي؟ بيساوي T أس n أو
+
+746
+01:06:51,120 --> 01:06:54,180
+إذا كان بدك اللي هو بدك إيش تكون ال range ل F خلاص
+
+747
+01:06:54,180 --> 01:06:57,460
+من عارفها من وين؟ من اللي من اللي قبل بشوية من ناقص
+
+748
+01:06:57,460 --> 01:07:01,980
+K لعند K عارفين ليه؟ لأن ال absolute value لل F of
+
+749
+01:07:01,980 --> 01:07:06,840
+X أصغر مش هو ال K إذا أكيد لما ترمي ال F of X
+
+750
+01:07:06,840 --> 01:07:11,410
+هتكون ترمي في الفترة من ناقص K لعند K أو subset
+
+751
+01:07:11,410 --> 01:07:15,970
+بيصير الـ F of I subset من ناقص K و K إذاً صارت
+
+752
+01:07:15,970 --> 01:07:22,510
+الـ Phi composite F Phi composite F is defined من
+
+753
+01:07:22,510 --> 01:07:31,140
+I لعند ال R وعندي Phi composite F of T بيساوي Phi
+
+754
+01:07:31,140 --> 01:07:37,200
+of F of T ويساوي F of T اللي هو Phi of T بيساوي T
+
+755
+01:07:37,200 --> 01:07:41,560
+أس N بيصير Phi of F of T اللي هي عبارة عن F of T
+
+756
+01:07:41,560 --> 01:07:48,540
+يا جماعة أس N الآن كل الأمور طيبة ومنيحة ليش؟
+
+757
+01:07:48,540 --> 01:07:53,260
+ووصلنا كمان عندي F-integrable و F-continuous
+
+758
+01:07:53,260 --> 01:07:57,420
+هتتدللوا الـ C إيش ما له؟ اللي هو الـ T أس n is
+
+759
+01:07:57,420 --> 01:08:03,540
+continuous اللي جزء من ال polynomial بيصير عندي FI
+
+760
+01:08:03,540 --> 01:08:06,660
+is continuous و F-integrable لذلك حسب النظرية اللي
+
+761
+01:08:06,660 --> 01:08:10,680
+قبل بشوية إيش هيطلع عندي؟ FI composite F is
+
+762
+01:08:10,680 --> 01:08:16,440
+integrable بمعنى أن F N is integrable فكرة معادة
+
+763
+01:08:16,440 --> 01:08:22,570
+أصلًا أنه سنة يعني اللي هو الأمور واضحة الآن ضال
+
+764
+01:08:22,570 --> 01:08:30,190
+عند مين؟ اللي هو الـ .. الـ .. الـ .. الـ .. نثبت الجزء
+
+765
+01:08:30,190 --> 01:08:36,010
+الثالث من ال corollary اللي هي بدنا نثبت أنه لو
+
+766
+01:08:36,010 --> 01:08:43,490
+كانت ال F of X أكبر أو يساوي واحد على دلتا اللي هي
+
+767
+01:08:43,490 --> 01:08:47,530
+for every x element in I وعندي there exists دلتا
+
+768
+01:08:47,530 --> 01:08:52,390
+كمان وصفر ساشرات إذا كان هذا متحقق إذن هتطلع عندي
+
+769
+01:08:52,390 --> 01:08:55,970
+اللي هي الواحد على ال F بيدثبت أن الواحد على ال F
+
+770
+01:08:55,970 --> 01:08:58,950
+إيش؟ is integrable بدنا نعمل .. نجيب continuous
+
+771
+01:08:58,950 --> 01:09:02,890
+function أكيد كل جمجال نجيب مقلوبة دالة يعني في حوض
+
+772
+01:09:02,890 --> 01:09:09,800
+five من ناقص K لعند K لعند R وخد فاي عارفين ما هو
+
+773
+01:09:09,800 --> 01:09:13,000
+ال scale كده سبب اللي قلت قبل شوية فايف دي إيش
+
+774
+01
+
+801
+01:12:01,040 --> 01:12:04,940
+بشوية و Corollary برضه إشمالها G تربيع برضه
+
+802
+01:12:04,940 --> 01:12:10,060
+Integrable صح ولا لا يا جماعة؟ صح طيب F تربيع
+
+803
+01:12:10,060 --> 01:12:14,260
+Integrable و G تربيع Integrable و برضه من نظرية
+
+804
+01:12:14,260 --> 01:12:18,580
+سابقة مدام F و G Integrable إذا F زائد G برضه إيش 
+
+805
+01:12:18,580 --> 01:12:24,980
+Integrable؟ لأ F زائد G تربيع Integrable كمان لأن F
+
+806
+01:12:24,980 --> 01:12:27,800
+و G Integrable أدت لـ F زائد G Integrable و F زائد
+
+807
+01:12:27,800 --> 01:12:31,640
+G Integrable أدت من القرول اللي قبل شوية أنه تربيعها
+
+808
+01:12:31,640 --> 01:12:40,520
+يكون إيش Integrable؟ طيب خلصنا إذا إذا F تربيع زائد
+
+809
+01:12:40,520 --> 01:12:49,780
+G تربيع is integrable صح؟ مظبوط و ناقص F تربيع
+
+810
+01:12:49,780 --> 01:12:54,220
+و ناقص G تربيع برضه integrable مظبوط؟ برضه لأنه اللي
+
+811
+01:12:54,220 --> 01:13:00,060
+هو عندي اللي هو ثابت في الـ integrable و ثابت في
+
+812
+01:13:00,060 --> 01:13:03,560
+الـ integrable integrable ومجموع انصار الـ integrable إذا
+
+813
+01:13:03,560 --> 01:13:09,020
+انصار هذا integrable زائد F زائد G الكل تربيع هذه
+
+814
+01:13:09,020 --> 01:13:12,980
+integrable وهذه integrable وهذه integrable مجموحين
+
+815
+01:13:12,980 --> 01:13:17,520
+هذا إيش بيساويهذا؟ هذا integrable و هذا
+
+816
+01:13:17,520 --> 01:13:19,860
+integrable و هذا integrable إذا المجموع integrable
+
+817
+01:13:19,860 --> 01:13:23,220
+إذا هذا كله على بعضه integrable طب هذا مين هو؟ هذا
+
+818
+01:13:23,220 --> 01:13:29,160
+عبارة عن F تربيع زائد G تربيع ناقص اللي هو إيش؟ زائد
+
+819
+01:13:29,160 --> 01:13:35,760
+2FG بيصير عبارة عن 2F مضروب في G صارت 2FG integrable
+
+820
+01:13:35,760 --> 01:13:40,840
+طب لو جينا قولنا خلّي نص هنا يعني ضربنا ثابت في
+
+821
+01:13:40,840 --> 01:13:43,960
+integrable إذا هيطلع اللي هو كله integrable إذا FG
+
+822
+01:13:43,960 --> 01:13:48,520
+إيش مالها؟ is integrable صار عندي الآن FG
+
+823
+01:13:48,520 --> 01:13:54,040
+integrable تابعًا لأن F تربيع و G تربيع و F زائد G 
+
+824
+01:13:54,040 --> 01:13:59,860
+الكل تربيع حاصل جمعهم وضرب النص في هنا ثابت وبيطلع 
+
+825
+01:13:59,860 --> 01:14:04,720
+عبارة عن integrable function نيجي لآخر اللي هو
+
+826
+01:14:04,720 --> 01:14:11,850
+النقطة في الـ في الـ في الـ section 7-2 اللي 
+
+827
+01:14:11,850 --> 01:14:14,790
+هو السؤال اللي سألناه في الأول قولنا لو كانت F is
+
+828
+01:14:14,790 --> 01:14:19,570
+integrable function و Phi is integrable هل Phi
+
+829
+01:14:19,570 --> 01:14:24,310
+composite F is integrable قولنا أكيد هو لو في
+
+830
+01:14:24,310 --> 01:14:28,430
+البداية قولنا أن Phi composite F need not to be
+
+831
+01:14:28,430 --> 01:14:32,280
+integrable إذا the composition of two integrable
+
+832
+01:14:32,280 --> 01:14:35,720
+functions need not to be integrable يعني هذا إعلان
+
+833
+01:14:35,720 --> 01:14:38,340
+the composition of two integrable functions need
+
+834
+01:14:38,340 --> 01:14:42,420
+not to be integrable but if phi the first one is
+
+835
+01:14:42,420 --> 01:14:46,340
+continuous then phi composite of f is integrable
+
+836
+01:14:46,340 --> 01:14:50,620
+كما شفنا في المثال اللي هو النظرية الأولى طيب نيجي
+
+837
+01:14:50,620 --> 01:14:53,000
+الآن للمثال الأخير بيقولنا يا جماعة the
+
+838
+01:14:53,000 --> 01:14:55,760
+composition of integrable functions need not to be
+
+839
+01:14:55,760 --> 01:14:59,540
+integrable في عندي سؤالين أصلا معكم homework
+
+840
+01:14:59,540 --> 01:15:03,860
+السؤال الأول بيقول لنا لو كانت f of x بتساوي واحد
+
+841
+01:15:03,860 --> 01:15:07,540
+ده كانت x بتساوي صفر و zero ده كانت x is
+
+842
+01:15:07,540 --> 01:15:11,760
+irrational و كانت f of x بتساوي واحد لما x بيساوي
+
+843
+01:15:11,760 --> 01:15:16,160
+m على n حيث الـ m و الـ n عبارة عن integers و العامل
+
+844
+01:15:16,160 --> 01:15:19,740
+المشترك الأعلى بينهم إيش؟ إشمالها بادر بيساوي واحد
+
+845
+01:15:19,740 --> 01:15:23,540
+يعني شيلنا كل العامل المشترك اللي بينهم وكتبنا x
+
+846
+01:15:23,540 --> 01:15:28,900
+بتساوي m على n طيب، هذه عبارة عن دالة معرفة من الـ 0
+
+847
+01:15:28,900 --> 01:15:32,460
+والـ 1 لعند الـ R يعني هذي X is rational well في
+
+848
+01:15:32,460 --> 01:15:36,320
+الفترة 0 و 1 و X بيبقى بيساوي M على N يعني rational
+
+849
+01:15:36,320 --> 01:15:41,260
+في الفترة 0 و 1 وفرضنا للـ X بتساوي صفر قيمتها إيش
+
+850
+01:15:41,260 --> 01:15:48,070
+بتساوي؟ بتساوي 1 هذي الآن by exercise 7.1.11 مطلوب
+
+851
+01:15:48,070 --> 01:15:51,790
+منك أنك تثبت أن F is إشمالها is integrable on I
+
+852
+01:15:51,790 --> 01:15:55,070
+ويوم ما تعرفوش تحلوه إن شاء الله بنحلوه وبنصوره
+
+853
+01:15:55,070 --> 01:16:00,030
+بإذن الله الآن السؤال الثاني اللي هعتمد عليه برضه 
+
+854
+01:16:00,030 --> 01:16:03,170
+أنه الـ function الثاني اللي هي G من I لعند R هذه 
+
+855
+01:16:03,170 --> 01:16:07,790
+سهلة أصلا لإثباتها be defined by G of X step
+
+856
+01:16:07,790 --> 01:16:12,090
+function بسيطة يعني أو خلّيني أقول فيها jump و بس 
+
+857
+01:16:12,090 --> 01:16:17,030
+jump على نقطة بس جيوب x بيساوي 0 إذا كانت x بيساوي
+
+858
+01:16:17,030 --> 01:16:21,410
+0 و بيساوي 1 إذا كانت x في الفترة من 0 لعند 1 اللي 
+
+859
+01:16:21,410 --> 01:16:25,410
+هو الـ closed عند الواحد يعني الآن هذا الـ function 
+
+860
+01:16:25,410 --> 01:16:29,110
+بيقول لي برضه في الـ exercise مطلوب برضه في exercise
+
+861
+01:16:29,110 --> 01:16:33,090
+717 برضه اللي هو homework معكم في الـ exercise
+
+862
+01:16:33,090 --> 01:16:36,270
+بيقول لك اثبت أن g إيش معناها؟ is integrable function
+
+863
+01:16:36,920 --> 01:16:40,960
+إذا الـ F والـ G عبارة عن two integrable functions
+
+864
+01:16:40,960 --> 01:16:46,780
+two integrable functions هاي الـ F من I لعند R والـ
+
+865
+01:16:46,780 --> 01:16:55,200
+G اللي هو من عند I لعند R لاحظ أن الـ G فيه لغة 
+
+866
+01:16:55,200 --> 01:17:00,600
+تانية zero لمين؟ يا واحد عندي هذه فئتها الـ 1 و 0 و
+
+867
+01:17:00,600 --> 01:17:06,800
+في عندنا قيم أخرى اللي هو كثيرة طيب، الآن عندي .. 
+
+868
+01:17:06,800 --> 01:17:09,660
+لو جيت حسبت الـ G composed of F مش غريبة عليكم 
+
+869
+01:17:09,660 --> 01:17:14,480
+الدالة هذه G composed of F of X خلينا نحسبها مع 
+
+870
+01:17:14,480 --> 01:17:20,860
+بعض وبنقول خلصنا اللي هو S section أو قولنا أن
+
+871
+01:17:20,860 --> 01:17:24,320
+الـ G composed of F هذه اللي أنتو بتعرفوها أن ها
+
+872
+01:17:24,320 --> 01:17:36,430
+is not integrable G composite F of اللي هو عندي
+
+873
+01:17:36,430 --> 01:17:46,900
+الـ G الـ F of X هيها لاخد G of اللي هو Zero بيساوي 
+
+874
+01:17:46,900 --> 01:17:55,920
+g of f of 0 بيساوي g of f of 0 بيساوي
+
+875
+01:17:55,920 --> 01:17:57,880
+g of f of 0 بيساوي g of f of 0 بيساوي g of f of 0
+
+876
+01:17:57,880 --> 01:18:06,220
+بيساوي g of f of 0 بيساوي g of f of 0 بيساوي g of f
+
+877
+01:18:06,220 --> 01:18:07,060
+of 0 بيساوي g of f of 0 بيساوي g of f of 0 بيساوي g
+
+878
+01:18:07,060 --> 01:18:07,140
+of f of 0 بيساوي g of f of 0 بيساوي g of f of 0
+
+879
+01:18:07,140 --> 01:18:08,100
+بيساوي g of f of 0 بيساوي g of f of 0 بيساوي g of f
+
+880
+01:18:08,100 --> 01:18:09,880
+of 0 بيساوي g of f of 0 بيساوي g of f of 0 بيساوي g
+
+881
+01:18:09,880 --> 01:18:14,160
+of f of 0 بيساوي g of f of 0 بيساوي g of f of 0
+
+882
+01:18:14,160 --> 01:18:17,110
+of 0 بيساوي g of f of X اللي هو irrational يعني 
+
+883
+01:18:17,110 --> 01:18:22,230
+irrational G of irrational اللي هو اللي هو الموجود 
+
+884
+01:18:22,230 --> 01:18:25,910
+في الفترة Zero و واحد G of irrational إيش هيساوي
+
+885
+01:18:25,910 --> 01:18:31,290
+حسب التعريف .. التعريف هنا بيساوي G of F of
+
+886
+01:18:31,290 --> 01:18:38,750
+irrational irrational هاي بيساوي اللي هو G of F of 
+
+887
+01:18:38,750 --> 01:18:44,240
+irrational إيش بيساوي صفر G of 0 إيش بيساوي هنا G 
+
+888
+01:18:44,240 --> 01:18:52,420
+of 0 بيساوي Zero معرفليها بيساوي إيش؟ صفر الثالثة
+
+889
+01:18:54,350 --> 01:18:58,630
+Composite F of مين ضال؟ Of اللي هي irrational
+
+890
+01:18:58,630 --> 01:19:02,530
+البجيات غير الـ Zero هذا أصلا هو جسمها irrational
+
+891
+01:19:02,530 --> 01:19:08,410
+و rational جهتين واحدة واحدة واحدة Zero لحالها و
+
+892
+01:19:08,410 --> 01:19:11,650
+واحدة كل الـ rational اللي بين Zero و واحد ما عدا
+
+893
+01:19:11,650 --> 01:19:18,870
+الـ Zero اللي هو G of X اللي هي عبارة عن M على N
+
+894
+01:19:19,590 --> 01:19:23,630
+اللي هي rational في الواقع رياشونال كلهم معدن اللي 
+
+895
+01:19:23,630 --> 01:19:28,790
+فوق اللي فوق حسبناها طلعت واحد ويساوي G of F of M
+
+896
+01:19:28,790 --> 01:19:34,350
+على N ويساوي G of F of M على N أيش بتساوي؟ واحد على
+
+897
+01:19:34,350 --> 01:19:38,910
+N ويساوي G of واحد على N أيش بتحسبها هذه؟ هذا أصلا
+
+898
+01:19:38,910 --> 01:19:42,070
+الدالة دائما بتساوي واحد معدن عند سفر حتى سفر عشان
+
+899
+01:19:42,070 --> 01:19:46,750
+عملنا المشكلة G of واحد على N أيش بتساوي؟ واحد هو
+
+900
+01:19:46,750 --> 01:19:53,580
+يساوي واحد من هذا كله صار عندي g composite f of x
+
+901
+01:19:53,580 --> 01:20:06,710
+بتساوي 0 if x إشمالها is irrational ويساوي 1 1 إذا
+
+902
+01:20:06,710 --> 01:20:11,790
+كانت X is هاي الـ rational كله ما عدا السفر وهي
+
+903
+01:20:11,790 --> 01:20:16,670
+السفر برضه طلع واحد if X is rational وهذه دالة
+
+904
+01:20:16,670 --> 01:20:20,610
+تبعتنا اللي اعتمدناها المرة الماضية أن ها is not 
+
+905
+01:20:20,610 --> 01:20:24,590
+integrable function لحظوا أنتو بس خليني نقول شوف
+
+906
+01:20:24,590 --> 01:20:32,660
+هالغرابة مش غرابة لأ هو مهم بالكلام أنه اللي خلف
+
+907
+01:20:32,660 --> 01:20:37,380
+الموضوع لو كانت G continuous على كل الـ domain على
+
+908
+01:20:37,380 --> 01:20:40,820
+طول الـ G continuous ده هتطلع أن Integrable بالغصب عنها
+
+909
+01:20:40,820 --> 01:20:44,560
+من النظرية اللي قبل شوية لكن اللي خلف الموضوع شغلة
+
+910
+01:20:44,560 --> 01:20:49,440
+واحدة شايفين هالدالة هاد الـ G of X الـ G of X G of
+
+911
+01:20:49,440 --> 01:20:55,000
+X بتساوي اللي هو صفر عند الصفر ومن عند الـ zero
+
+912
+01:20:55,000 --> 01:21:01,130
+لعند الواحد هنا الدالة قيمتها إيش بتساوي؟ واحد، هاي
+
+913
+01:21:01,130 --> 01:21:04,270
+قيمتها واحد، بين Zero والواحد قيمتها واحد، هذا
+
+914
+01:21:04,270 --> 01:21:09,540
+الـ G of X اللي عندي يعني هذا كل حال عالي العالي ما 
+
+915
+01:21:09,540 --> 01:21:14,120
+عدا عند من عند الصفر فيه jump point هذه النقطة
+
+916
+01:21:14,120 --> 01:21:20,200
+الوحيدة اللي فيها discontinuity هي اللي .. اللي 
+
+917
+01:21:20,200 --> 01:21:25,340
+أنا بنشرتلي أنها تصير decomposed F is continuous و
+
+918
+01:21:25,340 --> 01:21:29,200
+هذا عشان نعرف جداش الرياضيات أو جداش التحليل دقيق
+
+919
+01:21:29,780 --> 01:21:35,960
+دقيق أن أن احنا نقطة واحدة .. نقطة واحدة اللي كانت 
+
+920
+01:21:35,960 --> 01:21:40,360
+عندها point of discontinuity جالت لي أن الـ 
+
+921
+01:21:40,360 --> 01:21:44,680
+decomposite F need not to be integrable وهي المثال
+
+922
+01:21:44,680 --> 01:21:50,720
+أمامكم و .. و هيك بيكون احنا خلصنا الـ section 
+
+923
+01:21:50,720 --> 01:21:56,730
+الثاني من اللي هو chapter 7 وهاي الـ homework عندكم 
+
+924
+01:21:56,730 --> 01:22:02,630
+مطلوبة 1,2,4,7,10,17,18,19 وإن شاء الله المرة 
+
+925
+01:22:02,630 --> 01:22:08,950
+القادمة بنكمل وبنشرح 7.3 اللي هو الـ fundamental
+
+926
+01:22:08,950 --> 01:22:10,550
+theorem of calculus 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..81f586541a8319406e778228554b452ad626c747
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg_postprocess.srt
@@ -0,0 +1,3704 @@
+1
+00:00:04,940 --> 00:00:11,820
+بسم الله الرحمن الرحيم اليوم المحاضرة رقم 13 مساق
+
+2
+00:00:11,820 --> 00:00:17,100
+تحليل حقيقه 2 لطلاب طالبات الجامعة الإسلامية كلية
+
+3
+00:00:17,100 --> 00:00:21,800
+العلوم قسم رياضيات ان شاء الله ستكون المحاضرة على
+
+4
+00:00:21,800 --> 00:00:28,620
+جزئينالجزء الأول هنستمر في الحديث عن سبعة اتنين
+
+5
+00:00:28,620 --> 00:00:34,240
+وهو الحديث أو سبعة تلاتة وهو الحديث سبعة اتنين في
+
+6
+00:00:34,240 --> 00:00:40,130
+البداية وهو الحديث عن الخواص التكامل الريمانحكينا
+
+7
+00:00:40,130 --> 00:00:45,310
+عن اللي هو مجموع دالتين قابلات التكامل اللي هو
+
+8
+00:00:45,310 --> 00:00:48,510
+قابل
+
+9
+00:00:48,510 --> 00:00:52,510
+التكامل وعاصل ضرب ثابت في دالة قابل التكامل برضه
+
+10
+00:00:52,510 --> 00:00:58,630
+قابل التكاملواليوم هنكمل الأمر الحديث عن
+
+11
+00:00:58,630 --> 00:01:04,550
+composition of two integrable functions هل قابل
+
+12
+00:01:04,550 --> 00:01:08,270
+التكامل أم لا وإذا قابل التكامل بدنا نبرهن وإذا مش
+
+13
+00:01:08,270 --> 00:01:14,750
+قابل بدنا نجيه counter exampleالان الجزء الثاني
+
+14
+00:01:14,750 --> 00:01:18,110
+هيكون اللي هو برضه تطبيق اللي هو composition
+
+15
+00:01:18,110 --> 00:01:25,410
+theorem اللي هي هنشوف بعد شوية ايش المتطلب ان يكون
+
+16
+00:01:25,410 --> 00:01:30,190
+composition of two functions is integrableأيضًا
+
+17
+00:01:30,190 --> 00:01:35,230
+هنتحدث عن اللي هو نوظفها في إثبات اللي هو أن
+
+18
+00:01:35,230 --> 00:01:38,950
+الدالة الأسية والـ absolute value of the function
+
+19
+00:01:38,950 --> 00:01:43,530
+F و أيضًا اللي هو مقلوب الدالة 1 على F في حالة F
+
+20
+00:01:43,530 --> 00:01:50,090
+لا تساوي 0 على ال domain المُعطَع إنها تكون قابلة
+
+21
+00:01:50,090 --> 00:01:54,830
+للتكامل و أيضًا اللي هو هيكون في عنا اللي هو تطبيق
+
+22
+00:01:54,830 --> 00:01:59,640
+أخر اللي هوحاصل ضرب دالتين كيف يكون انتجرابل أو
+
+23
+00:01:59,640 --> 00:02:03,080
+كيف تكون انتجرابل في حالة كلا الدالتين انتجرابل
+
+24
+00:02:03,080 --> 00:02:06,200
+نبدأ الآن في اللي هو النظرية الـ composition
+
+25
+00:02:06,200 --> 00:02:09,600
+theorem واطولوا روحكم علينا شوية على اللي هو
+
+26
+00:02:09,600 --> 00:02:14,930
+البرهان و البرهان شوية بده تركيز و البرهانطويل
+
+27
+00:02:14,930 --> 00:02:20,110
+شوية خلّينا ان نعمل الان focusing على نص النظرية و
+
+28
+00:02:20,110 --> 00:02:25,410
+بعدين بنبدأ نحكي عن البرهان نعمل outline للبرهان و
+
+29
+00:02:25,410 --> 00:02:27,610
+من ثم ندخل لتفاصيل البرهان
+
+30
+00:02:30,230 --> 00:02:35,470
+بنأخد I عبارة عن close bounded interval A وB وJ
+
+31
+00:02:35,470 --> 00:02:38,550
+عبارة عن close bounded interval سميناها C وD
+
+32
+00:02:38,550 --> 00:02:44,830
+ونفترض أن F من I لعند R يعني F عبارة عن دلة من عند
+
+33
+00:02:44,830 --> 00:02:49,350
+ال A و ال B لعند R أنها تكون integrable on I and
+
+34
+00:02:49,350 --> 00:02:52,050
+Phi من J لعند R
+
+35
+00:02:55,070 --> 00:02:58,390
+النظرية بتستلزم أن نقول continuous لأن ال Integra
+
+36
+00:02:58,390 --> 00:03:03,270
+بالحالها مش هتعطي اللي هو النتيجة زي ما هنشوف قدام
+
+37
+00:03:03,270 --> 00:03:07,610
+في اللي هو counter example الآن فرضنا أن فايل من J
+
+38
+00:03:07,610 --> 00:03:12,890
+لعند R is continuous وبدنا نفترض أن F of I .. F of
+
+39
+00:03:12,890 --> 00:03:17,450
+I جزئية من مين؟ من J عشان نعرف .. نعرف ال
+
+40
+00:03:17,450 --> 00:03:21,670
+composition بين ال two functions الآن في ضوء هذه
+
+41
+00:03:21,670 --> 00:03:26,050
+المعطياتإن الـ function F is integrable و الـ
+
+42
+00:03:26,050 --> 00:03:30,410
+function Phi is continuous لازم يطلع عندي الآن Phi
+
+43
+00:03:30,410 --> 00:03:37,450
+composite F is integrable on mean on I إذن F من I
+
+44
+00:03:37,450 --> 00:03:43,130
+لR انتجرابل Phi من J لعند R continuous الآن Phi
+
+45
+00:03:43,130 --> 00:03:46,170
+composite J continuous composite integrable
+
+46
+00:03:46,170 --> 00:03:53,500
+هيعطيني انتجرابل function on I الآن بدنا نثبتالأن
+
+47
+00:03:53,500 --> 00:04:00,600
+فاي composite F من I لعند R انها is integrable
+
+48
+00:04:00,600 --> 00:04:04,900
+الان كيف بدى أثبتها؟ بدى أثبتها .. بدى ألاقي
+
+49
+00:04:04,900 --> 00:04:09,040
+partition .. ده أقول لكل يبسن أكبر من 0 بدلاقي
+
+50
+00:04:09,040 --> 00:04:16,060
+partition B element in B of I such that L of أو U
+
+51
+00:04:16,060 --> 00:04:24,200
+ofB و F هذا اللي لجيته ماعرف أسأل أسف انا فعلا
+
+52
+00:04:24,200 --> 00:04:30,690
+composed of G and Fفاي كومبوزيت F ناقص L of B وفاي
+
+53
+00:04:30,690 --> 00:04:35,730
+كومبوزيت F أن يكون هذا المقدار أصغر من إبسلون إذا
+
+54
+00:04:35,730 --> 00:04:40,590
+وصلت لهذه النتيجة معناته أنه أنا أثبتت أنه اللي هي
+
+55
+00:04:40,590 --> 00:04:46,490
+الفاي كومبوزيت F is integrable بناء على اللي هو ال
+
+56
+00:04:46,490 --> 00:04:48,870
+criterion of integrability اللي حكينا عنها
+
+57
+00:04:48,870 --> 00:04:53,700
+المحاضرة قبل الباضيةالان هذا الهدف اللي بده أصله
+
+58
+00:04:53,700 --> 00:04:58,020
+هيك بده أصل هيك الان عشان أصل هيك بدي اللي هو
+
+59
+00:04:58,020 --> 00:05:02,900
+أستخدم اللي هو المعطيات اللي موجودة عندى الان
+
+60
+00:05:02,900 --> 00:05:08,380
+هستخدم أمرين هستخدم اكيد ال continuity للـ Phiوهي
+
+61
+00:05:08,380 --> 00:05:14,320
+continuous على closed bounded interval إذا حسب
+
+62
+00:05:14,320 --> 00:05:18,080
+نظرية في تحليل واحد هتكون Phi is uniformly
+
+63
+00:05:18,080 --> 00:05:23,880
+continuous وهذا هستغلها في الوصول إلى هدفيالان هذه
+
+64
+00:05:23,880 --> 00:05:27,960
+المعلومة بعد شوية هنخزنها ونحطها في .. في مكان ما
+
+65
+00:05:27,960 --> 00:05:33,420
+لحين نستخدمها مع اللي هو ان F is integrable مزام F
+
+66
+00:05:33,420 --> 00:05:37,240
+is integrable اذا نبلج Partitional I بحيث انه الـ
+
+67
+00:05:37,240 --> 00:05:40,500
+U للـ B والـ F نقص الـ B والـ F أصغر من مين؟ من
+
+68
+00:05:40,500 --> 00:05:44,220
+some .. من الـ Epsilon Epsilon تخدمني، هبدأ أسميها
+
+69
+00:05:44,220 --> 00:05:47,490
+Epsilon، هبدأ أسميها Delta، أنا حورالمهم أكيد مدام
+
+70
+00:05:47,490 --> 00:05:51,610
+الفيه عندي integrability للـ F هتحقق أنه لكل اللي
+
+71
+00:05:51,610 --> 00:05:56,010
+هو Epsilon there exists partition B وحنلاقي أنه ال
+
+72
+00:05:56,010 --> 00:05:59,510
+partition هذا اللي .. اللي نفع للـ F هو اللي هينفع
+
+73
+00:05:59,510 --> 00:06:04,300
+للـ Phi composite Fهنجمج المعلومتين التنتين مع بعض
+
+74
+00:06:04,300 --> 00:06:06,680
+اللي هي ال .. ال .. ال .. ال .. الـ uniform
+
+75
+00:06:06,680 --> 00:06:10,260
+continuity للفعل مع الintegrability للأف للوصول
+
+76
+00:06:10,260 --> 00:06:14,920
+إلى نتيجتنا وهي هذه النتيجة هذه خلينا نقول ال
+
+77
+00:06:14,920 --> 00:06:19,660
+outline للبرهان نبدأ الآن في تفاصيل البرهان وطول
+
+78
+00:06:19,660 --> 00:06:25,400
+روحكم عليها في اللي هو تفاصيل البرهانللوصول
+
+79
+00:06:25,400 --> 00:06:31,780
+للنتيجة اللى حكيتها اللى كتبت على اللوح بدي اقول
+
+80
+00:06:31,780 --> 00:06:39,240
+الان اول حاجة given epsilon أكبر من سفر انا اخدت
+
+81
+00:06:39,240 --> 00:06:44,600
+اي epsilon أكبر من سفر بدي اصل ال U بي فاي
+
+82
+00:06:44,600 --> 00:06:50,610
+composite F ناقص ال بيفاي كومبوزيت F أزر من إبسلون
+
+83
+00:06:50,610 --> 00:06:55,150
+for some اللي هو بارتي شنبي إذا وصلت لهك بكون خلصت
+
+84
+00:06:55,150 --> 00:06:58,770
+اللي هو نظريتي طيب سلموا على النبي عليه الصلاة
+
+85
+00:06:58,770 --> 00:07:03,850
+والسلام اللي لأن فاي عندى فاي عندى continuous على
+
+86
+00:07:03,850 --> 00:07:08,950
+مين على اللي هي ال J ال J عبارة عن closed bounded
+
+87
+00:07:08,950 --> 00:07:12,650
+intervalمدام continuous عليها إذا uniformly
+
+88
+00:07:12,650 --> 00:07:16,690
+continuous ماشي الحال إذا مدام uniformly
+
+89
+00:07:16,690 --> 00:07:23,170
+continuous إذا أكيد .. أكيد حتكون اللي هو مدام ..
+
+90
+00:07:23,170 --> 00:07:26,750
+حتى في ال continuity أكيد حتكون إيش ما لها bounded
+
+91
+00:07:26,750 --> 00:07:34,710
+إذا بقدر اللي هو أقول K بتساوي ال supremum لل Phi
+
+92
+00:07:34,710 --> 00:07:41,540
+of T such that T elementالـ C و D التي هي الـ J
+
+93
+00:07:41,540 --> 00:07:45,900
+هذه الآن بقدر احكي عن حاجة اسمها Supremum اه طبعا
+
+94
+00:07:45,900 --> 00:07:51,720
+مش هي ككلكم كمان و الـ K هذه هتكون attains for
+
+95
+00:07:51,720 --> 00:07:55,500
+some T بين C و D ليه؟ لأنه فايز continuous on a
+
+96
+00:07:55,500 --> 00:07:58,340
+closed bounded interval then it attains its
+
+97
+00:07:58,340 --> 00:08:00,920
+absolute maximum and absolute minimum on this
+
+98
+00:08:00,920 --> 00:08:05,120
+interval إذن أكيد في عنديK بيتساوى الـ Supremum لو
+
+99
+00:08:05,120 --> 00:08:09,200
+سلوك يدفع له 5T T Element in C وD سميليها دي K ليش
+
+100
+00:08:09,200 --> 00:08:13,280
+هتجهت بتعرف ليش هتستخدمها في الوصول إلى هدفي إذا
+
+101
+00:08:13,280 --> 00:08:16,960
+الأن اللي عملته لحد الآن أخدت epsilon arbitrarily
+
+102
+00:08:16,960 --> 00:08:21,860
+أخدت اللي هو ال supremum لهذا المقدار وسميته K هل
+
+103
+00:08:21,860 --> 00:08:24,440
+ال supremum موجود؟ أه ال supremum موجود و maximum
+
+104
+00:08:24,440 --> 00:08:27,480
+كمان لإن ال Phi is continuously on a closed
+
+105
+00:08:27,480 --> 00:08:31,840
+bounded interval COD طيب
+
+106
+00:08:34,510 --> 00:08:39,830
+الان بدي اخد بعيد اذنكم حاجة اسميها let epsilon
+
+107
+00:08:39,830 --> 00:08:45,490
+برايم بالساوي اللي هو epsilon على b ناقص a زائد
+
+108
+00:08:45,490 --> 00:08:49,670
+اتنين k ال k هذه اللي فوق زائد اتنين k وال b و ال
+
+109
+00:08:49,670 --> 00:08:54,010
+a اللي هي طول الفترة اللي هي I اللي انا عمال بشتغل
+
+110
+00:08:54,010 --> 00:09:01,100
+عليها معرف عليها Fليش هيك؟ بغرض الحسابات بعد شوية
+
+111
+00:09:01,100 --> 00:09:06,060
+هتشوفوا ليش و لو أصلا احنا في النهاية الـ Epsilon
+
+112
+00:09:06,060 --> 00:09:10,140
+الـ Prime هذه ما كتبناش بالشكل هذا و طلع عندي اللي
+
+113
+00:09:10,140 --> 00:09:19,940
+هو الـ U of B أو Phi Composite F ناقص الـ B Phi
+
+114
+00:09:19,940 --> 00:09:26,070
+Composite Fيكون أصغر من إبسلون مضروبة في something
+
+115
+00:09:26,070 --> 00:09:32,270
+أي إشي something ثابتبرضه هتقدي الغرض لإنه اللي هي
+
+116
+00:09:32,270 --> 00:09:35,310
+زي ما قلنا المرة الماضية المدام صحيحة لكل epsilon
+
+117
+00:09:35,310 --> 00:09:38,150
+في الدنيا إذا صحيح الواحد على n خد ال limit
+
+118
+00:09:38,150 --> 00:09:41,530
+للجهتين as n goes to infinity بيصير اللي هو هذا
+
+119
+00:09:41,530 --> 00:09:45,670
+اللي هو يؤد الغرض أو ال epsilon اللي هنا بتكون
+
+120
+00:09:45,670 --> 00:09:50,730
+اللي هو بيصير يعني تؤدي غرض أي أصغر من epsilon في
+
+121
+00:09:50,730 --> 00:09:53,510
+الدنيا لإن ال epsilon مدام اتمضروبة ضرب في التابط
+
+122
+00:09:53,510 --> 00:09:57,930
+بقدر ال epsilon أزغرها جد ما بدي و تؤدي الغرض
+
+123
+00:09:58,610 --> 00:10:05,280
+المفروض فاهمين طيب نيجي الآننرجع نقول أنه أخدت
+
+124
+00:10:05,280 --> 00:10:08,980
+إبسلون مراي بيساوي إبسلون على P minus A زاد 2K على
+
+125
+00:10:08,980 --> 00:10:12,680
+أساس أنه اللي هي تطلع عندنا الحسابات في الآخر
+
+126
+00:10:12,680 --> 00:10:16,480
+مرتبة وخالصة اللي هي أصغر من إبسلون طبعا هو
+
+127
+00:10:16,480 --> 00:10:20,720
+الاشعرف أن إبسلون مراي بالشكل هذا أصلا هو برهن
+
+128
+00:10:20,720 --> 00:10:23,680
+النظرية أو برهننا النظرية وفي الآخر طلع أن إبسلون
+
+129
+00:10:23,680 --> 00:10:28,440
+مضروبة في رقم جيت اللي هو رتبت حالي بحيث أنه حسبت
+
+130
+00:10:28,440 --> 00:10:32,320
+أنه عشان أطلح إبسلون لحالها خد إبسلون مبران بالشكل
+
+131
+00:10:32,320 --> 00:10:39,070
+هذاطيب نشوف .. نشوف الآن عندي فاي زي ما وعدناكم
+
+132
+00:10:39,070 --> 00:10:49,690
+فاي is continuous on اللي هو cod مدام فاي
+
+133
+00:10:49,690 --> 00:10:53,010
+continuous on cod يا جماعة إذا زي ما قلنا قبل
+
+134
+00:10:53,010 --> 00:11:01,660
+بشوية إذا5 is uniformly continuous on COD أيش
+
+135
+00:11:01,660 --> 00:11:07,680
+معناة uniformly continuous يعني اللي هو for every
+
+136
+00:11:07,680 --> 00:11:12,100
+إبسلون في الدنيا for every إبسلون .. خلّيني أخد
+
+137
+00:11:12,100 --> 00:11:15,020
+إبسلون الـ prime for every إبسلون الـ prime أكبر
+
+138
+00:11:15,020 --> 00:11:21,680
+من سفر there exists delta prime such that
+
+139
+00:11:21,680 --> 00:11:29,560
+uniformly continuousFor every S وT element in C وD
+
+140
+00:11:29,560 --> 00:11:38,690
+تحقق S minus T أصغر من Delta Prime يؤدي إلى Phiof
+
+141
+00:11:38,690 --> 00:11:43,690
+S ناقص Phi of T أصغر من مين من Epsilon برايم اللي
+
+142
+00:11:43,690 --> 00:11:49,450
+أخدتها عندي لأي Epsilon في الدنيا هذا الكلام
+
+143
+00:11:49,450 --> 00:11:52,210
+بيتحقق من دون أن هم Epsilon برايم اللي حكيت عنها
+
+144
+00:11:52,210 --> 00:11:58,090
+فوق طيب إذا الأن اللي استخدمته بما أن Phi
+
+145
+00:11:58,090 --> 00:12:02,010
+continuous على C وD إذا Phi is uniformly
+
+146
+00:12:02,010 --> 00:12:08,210
+continuous on مين؟ on C أو Dإذن الآن حسب التعريف
+
+147
+00:12:08,210 --> 00:12:11,530
+الـ Uniformly Continuous لأي إبسلون في الدنيا من
+
+148
+00:12:11,530 --> 00:12:14,530
+ضمنهن الإبسلون الـ prime الأكبر من 0 there exists
+
+149
+00:12:14,530 --> 00:12:17,430
+delta prime خاصة بالإبسلون الـ prime بحيث أنه لما
+
+150
+00:12:17,430 --> 00:12:20,770
+S و T في الـ C و الـ D و يكون الـ S minus T أصغر
+
+151
+00:12:20,770 --> 00:12:24,950
+من delta prime يعطيني أن 5S ناقص 5T أصغر من مين من
+
+152
+00:12:24,950 --> 00:12:30,990
+إبسلون الـ primeالان انا في delta معينة بدي
+
+153
+00:12:30,990 --> 00:12:35,830
+ألاقيها أربطها بهذه واتحققلي هذا الكلام شوفوا كيف
+
+154
+00:12:35,830 --> 00:12:44,030
+اصبروا عليها الان if عندي delta prime هذه أصغر من
+
+155
+00:12:44,030 --> 00:12:49,430
+epsilon prime thenThere exists Delta بتساوي Delta
+
+156
+00:12:49,430 --> 00:12:53,130
+Prime بتاخد Delta إيش بتساوي أسميه بتسمي Delta
+
+157
+00:12:53,130 --> 00:12:56,370
+Prime مين يا جماعة؟ Delta و هاد ال Delta اللي
+
+158
+00:12:56,370 --> 00:12:58,790
+سميتها اللي هي Delta Prime اللي سميتها Delta هاد
+
+159
+00:12:58,790 --> 00:13:03,670
+ال Delta هتحققها ليش؟ لأنها نفسها يعني بمعنى آخر
+
+160
+00:13:03,670 --> 00:13:08,150
+إذا كانت ال S minus T أصغر من Delta Prime بيعطيني
+
+161
+00:13:08,150 --> 00:13:10,830
+Automatic أصغر من Delta Prime اللي سميتها Delta
+
+162
+00:13:10,830 --> 00:13:15,810
+بيعطيني Phi of S ناقص Phi of T أصغر من مين؟ من
+
+163
+00:13:15,810 --> 00:13:23,530
+Epsilon Primeطيب تشوفوا الان if delta prime أكبر
+
+164
+00:13:23,530 --> 00:13:26,430
+أو يساوي epsilon prime يا أي حالتين مافيش غير هيك
+
+165
+00:13:26,430 --> 00:13:28,830
+يا delta prime أصغر من epsilon يا delta prime أكبر
+
+166
+00:13:28,830 --> 00:13:32,090
+ساوي epsilon prime الان if delta .. أهل جاي تفهموا
+
+167
+00:13:32,090 --> 00:13:34,530
+ليش عملت هيك if delta prime أكبر ساوي epsilon
+
+168
+00:13:34,530 --> 00:13:41,100
+prime thenاللي هو there exists Delta أصغر من
+
+169
+00:13:41,100 --> 00:13:45,540
+Epsilon برايم بلاجي ولا بلاجيش Epsilon برايم أكبر
+
+170
+00:13:45,540 --> 00:13:50,200
+من 0 أكيد between انا ابسلون برايم أكبر من 0 إذا
+
+171
+00:13:50,200 --> 00:13:52,620
+أنا أكيد بلاجي بين Epsilon برايم أكبر من 0 بلاجي
+
+172
+00:13:52,620 --> 00:13:57,340
+Deltaبلاجي عدد نهائي من الأعداد اللي هو Delta أكبر
+
+173
+00:13:57,340 --> 00:14:01,400
+من سفر وأصغر من إبسل و براين إذاً بلاجي Delta أصغر
+
+174
+00:14:01,400 --> 00:14:04,820
+من إبسل و براين بلاجي أه بلاجي لإن بين الـ two
+
+175
+00:14:04,820 --> 00:14:09,820
+real numbers اللي هو بين السفر وبين أي positive
+
+176
+00:14:09,820 --> 00:14:14,360
+real number في infinite number of numbers بينهم
+
+177
+00:14:14,360 --> 00:14:17,600
+سميت واحد قلت there exists Delta أصغر من إبسل و
+
+178
+00:14:17,600 --> 00:14:21,860
+براين such that .. طيب إيش بدك فيها هذه؟ such that
+
+179
+00:14:21,860 --> 00:14:27,400
+.. أحنا الـ gate تشوفواIf S minus T أصغر من Delta
+
+180
+00:14:27,400 --> 00:14:34,000
+فهذه الـ Delta If S minus T أصغر من Delta إذا أكيد
+
+181
+00:14:34,000 --> 00:14:37,860
+هذه الـ Delta أصغر من مين؟ المختار هي أصغر من Y'
+
+182
+00:14:38,320 --> 00:14:42,480
+وY' أصغر من مين؟ أصغر أو يسوى Delta Prime من هنا
+
+183
+00:14:43,910 --> 00:14:50,290
+الان if S-C أصغر من Delta إذا .. إذا حيث الـ S-C
+
+184
+00:14:50,290 --> 00:14:54,910
+هتكون أصغر من Y' واللي بدورها SY' أصغر بساوي Delta
+
+185
+00:14:54,910 --> 00:14:59,090
+.. أصغر بساوي Delta Prime إذا صارت عندي S-C أصغر
+
+186
+00:14:59,090 --> 00:15:03,010
+من Delta قطعًا S-C أصغر من مين؟ من Delta Prime
+
+187
+00:15:03,010 --> 00:15:07,450
+وبناءً على اللي كتبته بالأحمر هذا كل إشي S-C أصغر
+
+188
+00:15:07,450 --> 00:15:15,270
+من Delta Prime إشي بيعطيني5s-5t أصغر من إبسلون
+
+189
+00:15:15,270 --> 00:15:20,550
+برايم إذا يا جماعة سواء دلتا برايم أصغر من إبسلون
+
+190
+00:15:20,550 --> 00:15:25,710
+برايم أو دلتا برايم أصغر من إبسلون إبسلون برايم
+
+191
+00:15:25,710 --> 00:15:33,020
+بقدر ألاقي دلتاتتحقق فيها الخاصية لما S minus T
+
+192
+00:15:33,020 --> 00:15:36,900
+أصغر من Delta لما S minus T أصغر من Delta يعطيني
+
+193
+00:15:36,900 --> 00:15:39,900
+هذا المقدار أصغر من Epsilon Prime هذا المقدار أصغر
+
+194
+00:15:39,900 --> 00:15:44,580
+من Epsilon Prime إذا أنا في النهاية there exist
+
+195
+00:15:44,580 --> 00:15:50,690
+لوصلت له there exist Deltaأكبر من 0 ونفس الوجد
+
+196
+00:15:50,690 --> 00:15:55,170
+أشمالها Delta أصغر من يبسلون برايم لأنه في الحالة
+
+197
+00:15:55,170 --> 00:15:59,650
+ده أصغر من يبسلون برايم وفي الحالة الأولى برضه الـ
+
+198
+00:15:59,650 --> 00:16:03,630
+Delta أصغر من يبسلون برايم لأنه اختار الدلتة هي
+
+199
+00:16:03,630 --> 00:16:08,490
+Delta إبرايم إذن في كل الحالات هي المربط الفلسفي
+
+200
+00:16:08,490 --> 00:16:13,950
+الآن there exists Delta أكبر من 0 وأصغر من يبسلون
+
+201
+00:16:13,950 --> 00:16:22,170
+برايم such thatلكل S وT element in C وD إذا حققت
+
+202
+00:16:22,170 --> 00:16:27,610
+الخاصية S minus T أصغر من Delta بيعطيني على طول 5S
+
+203
+00:16:27,610 --> 00:16:34,130
+ناقص 5T أصغر من مين من إبسلون إبراهيم إذن هي
+
+204
+00:16:34,130 --> 00:16:41,710
+معلومة أخرى بدي أخذنها لأنني هحتاجهاهي كمان معلومة
+
+205
+00:16:41,710 --> 00:16:46,270
+الان اسمحولي أمسح هذا ال hand عشان أخزن معلومتي
+
+206
+00:16:46,270 --> 00:16:49,770
+اللي وصلت إلها مع المعلومات اللي موجودة عندى فوق
+
+207
+00:16:49,770 --> 00:16:57,090
+طيب الان شطبنا اللي عندنا hand خلصنا منه ووصلنا
+
+208
+00:16:57,090 --> 00:17:03,030
+إلى المعلومة التالية اللي بدأ أخزنها الآن مع اللي
+
+209
+00:17:03,030 --> 00:17:10,620
+مخزن فوق لأن there existDelta أكبر من 0 وأصغر من
+
+210
+00:17:10,620 --> 00:17:18,140
+Y' Such that for every S وT element in C وD إذا
+
+211
+00:17:18,140 --> 00:17:25,340
+حقق S minus T أصغر من Delta بيعطيني اللي هو 5S
+
+212
+00:17:25,340 --> 00:17:34,140
+minus 5T أصغر من Y' وهذا سمونيها 1 سمونيها 2
+
+213
+00:17:34,140 --> 00:17:38,630
+سمونيها Star اللي بدكم إياها ماشي الحالهذا الان
+
+214
+00:17:38,630 --> 00:17:44,170
+وصلت له وانا بدي استخدمه بعد شوية
+
+215
+00:17:49,890 --> 00:17:54,390
+اللي اللي بحب يتابع على التلخيص
+
+216
+00:17:54,390 --> 00:17:58,130
+هاي اللي وصلت إليه الآن هايها there exists delta و
+
+217
+00:17:58,130 --> 00:18:01,390
+الـ delta أصغر من epsilon prime if S و T element
+
+218
+00:18:01,390 --> 00:18:04,470
+in J and S minus T أصغر من Delta then Phi of S
+
+219
+00:18:04,470 --> 00:18:08,030
+ناقص Phi of T أصغر من إبسلون برايم هذه اللي وصلنا
+
+220
+00:18:08,030 --> 00:18:15,670
+إليها اللي قدرنا أن نصلها عشان بعد شوية بتستخدمها
+
+221
+00:18:15,670 --> 00:18:20,950
+انتبه عليها الآنالآن استغلقنا معلومة الـ if I is
+
+222
+00:18:20,950 --> 00:18:24,490
+continuous وحصلنا على معلومة مهمة جداً هي هذه
+
+223
+00:18:24,490 --> 00:18:28,630
+المعلومة الآن بدي أستخدم المعلومة الموازية لها أن
+
+224
+00:18:28,630 --> 00:18:37,770
+F is integrable الان عندي F is integrable on I إذا
+
+225
+00:18:37,770 --> 00:18:43,010
+حسب then حسب اللي هو ال .. ال .. ال integrability
+
+226
+00:18:43,010 --> 00:18:48,570
+criterionاللي حكينا عنها there exists partition B
+
+227
+00:18:48,570 --> 00:18:56,690
+element in B of I بجزء مين I such that اللي هو U
+
+228
+00:18:56,690 --> 00:19:05,920
+of B وFمعقس L بيوقف أصغر من أي إبسلون في الدنيا
+
+229
+00:19:05,920 --> 00:19:08,360
+الإبسلون اللي في الدنيا الإبسلون اللي بدها
+
+230
+00:19:08,360 --> 00:19:12,040
+استخدمها اللي حتة في الدنيا اللي هي مين هي Delta
+
+231
+00:19:12,040 --> 00:19:17,000
+تربيع أصغر من مين من Delta تربيع بقدر أه بقدر طبعا
+
+232
+00:19:17,000 --> 00:19:19,960
+مش احنا بنقول لكل إبسلون أكبر من سفر مادام F is
+
+233
+00:19:19,960 --> 00:19:23,040
+integrable لذا لكل إبسلون أكبر من سفر there exist
+
+234
+00:19:23,040 --> 00:19:27,590
+ال partition B بحيث أنه هذا أصغر من إبسلونالأن
+
+235
+00:19:27,590 --> 00:19:30,330
+أبسط من اللي بحكي عنها دلتة تربيع دلفة التربيع
+
+236
+00:19:30,330 --> 00:19:35,650
+أكبر من صفر إذا for delta تربيع there exists B
+
+237
+00:19:35,650 --> 00:19:40,070
+element in B of I such that U ناقص L أصغر من مين
+
+238
+00:19:40,070 --> 00:19:44,650
+من Delta تربيع هذا ال partition B بيجزئلي مين؟
+
+239
+00:19:44,650 --> 00:19:51,670
+بيجزئلي I بعد أذنكم سموليها بيه X0 و X1 لعند مين؟
+
+240
+00:19:51,670 --> 00:19:55,230
+لعند X سموليها ال partition بس عشان أتعامل معاها
+
+241
+00:19:55,660 --> 00:19:57,540
+اللي هو partition للـ I partition للـ I معناه
+
+242
+00:19:57,540 --> 00:20:00,880
+بجزّه I جزّه او ل X note X واحد X انت جداش أعدادها
+
+243
+00:20:00,880 --> 00:20:07,080
+مش عارف حسب اللي هو اللي جناه بيه طيب شوف الآن بدي
+
+244
+00:20:07,080 --> 00:20:13,520
+أعمل اللي هو شغلة بحيث عن الان أقدر أستخدمها اللي
+
+245
+00:20:13,520 --> 00:20:18,360
+هي أصل للي بدي من خلالها الآن هجيتوا تشوفوا ليش
+
+246
+00:20:18,360 --> 00:20:23,400
+جزعين خدوا Aالان يا جماعة صارت دلتة بين إيديا
+
+247
+00:20:23,400 --> 00:20:28,520
+لجهتها دلتة طيب الان خدوا ايه 6 هي عبارة عن كل ال
+
+248
+00:20:28,520 --> 00:20:34,160
+indices K اللي هان هذولة و 0,1,2,3 كده هذه جزء
+
+249
+00:20:34,160 --> 00:20:38,520
+اثنين ال I هي عند X note لعند X end هذه الفترة
+
+250
+00:20:38,520 --> 00:20:44,000
+اللي هي من A لعند B يا جماعة لعند Bماشي الحال طيب
+
+251
+00:20:44,000 --> 00:20:49,380
+خدوله A هي عبارة عن كل ال K بحيث ان M K ناقص M K
+
+252
+00:20:49,380 --> 00:20:54,900
+small تكون أصغر من Delta وخدول ال B بيساوي كل ال K
+
+253
+00:20:54,900 --> 00:20:59,740
+such that M K ناقص M K أكبر يساوي Delta إيش هذول
+
+254
+00:20:59,740 --> 00:21:04,700
+عاملا؟ هذول بس ال indices عند من هنا صفر و واحد و
+
+255
+00:21:04,700 --> 00:21:08,480
+اتنين عند من؟ عند ال K إذا اتجت جزءات ال indices
+
+256
+00:21:08,480 --> 00:21:15,500
+هذول إلى جزء ايه؟الجزء اللى هو عندى اللى بخاصية
+
+257
+00:21:15,500 --> 00:21:19,980
+اللى هو ال mk ناقص mk أصغر من دلتا بده حطه فهذا ال
+
+258
+00:21:19,980 --> 00:21:24,280
+set إذا هذه عبارة عن إيش مجموعة جزئية من ال .. من
+
+259
+00:21:24,280 --> 00:21:27,680
+ال .. من ال .. من ال .. من ال .. من ال ك من سفر
+
+260
+00:21:27,680 --> 00:21:36,490
+لعند مين لعند واحد لعند أنا .. لعند أناالأن هذه P
+
+261
+00:21:36,490 --> 00:21:40,870
+هي عبارة عن المتبقى منهن، مين المتبقى اللي الـ M K
+
+262
+00:21:40,870 --> 00:21:44,170
+نقص M K أصغر أكبر أو شويه دلتا، يعني بمعنى أخر
+
+263
+00:21:44,170 --> 00:21:49,430
+أداة التجزئة لهذه اللي هي الخاصية أنه M K نقص M K
+
+264
+00:21:49,430 --> 00:21:52,010
+أصغر مين مين دلتا، أنتوا عارفين إيش الـ M K
+
+265
+00:21:52,010 --> 00:21:55,730
+Capital و M K Small؟ أكيد، الـ M K Capital هي
+
+266
+00:21:55,730 --> 00:21:59,710
+عبارة عن الـ supremum لل F of X such that X
+
+267
+00:21:59,710 --> 00:22:04,740
+element in X K minus 1 لعند الـ X Kو الـ mk small
+
+268
+00:22:04,740 --> 00:22:09,560
+بيساوي الـ infimum لل F of X such that X element
+
+269
+00:22:09,560 --> 00:22:16,640
+in XK minus واحد والـ XK إذن يا جماعة اللي جزألي
+
+270
+00:22:16,640 --> 00:22:21,900
+الـ A و الـ B هو خاصيته إن الـ mk capital يعني
+
+271
+00:22:21,900 --> 00:22:25,340
+أعلى قيمة هي .. هي .. هي .. هي عندي أنا XK minus
+
+272
+00:22:25,340 --> 00:22:29,860
+واحد وهي XK فرضنا أن الدالة هي الدالة زي هيك مثلا
+
+273
+00:22:29,860 --> 00:22:35,250
+في المنطقة هذهالان هي أعلى قيمة وهي أقل قيمة
+
+274
+00:22:35,250 --> 00:22:41,070
+الحاصل طرح أعلى قيمة و أقل قيمة في كل فترة .. كل
+
+275
+00:22:41,070 --> 00:22:44,770
+sub interval بادي بقول هل هذا أكبر من .. أصغر من
+
+276
+00:22:44,770 --> 00:22:49,270
+delta ولا أكبر يساوي delta اللي خاصيتهم الفرق
+
+277
+00:22:49,270 --> 00:22:53,930
+بينهم أصغر من delta بحط هناالاندسيز هذا كهن واللي
+
+278
+00:22:53,930 --> 00:22:59,930
+أكبر بحطه نهن و بحط تجزيتها ليش هذه؟ هذه طريقة
+
+279
+00:22:59,930 --> 00:23:07,130
+للوصول إلى اللي بديها و هتشوفه الآن طيب نيجي الآن
+
+280
+00:23:07,130 --> 00:23:20,690
+لاللي هو نشوف الـ K F K Element A مدام K Element A
+
+281
+00:23:22,110 --> 00:23:26,470
+إذا تتحقق الخاصية هذه يعني Mk نقص Mk أصغر من مين؟
+
+282
+00:23:26,470 --> 00:23:32,430
+من Delta خدولي الآن أي X و Y في الفترة مين اللي هي
+
+283
+00:23:32,430 --> 00:23:39,330
+XK minus واحد لعند XK شوفوا إيش اللي بديهاال .. ال
+
+284
+00:23:39,330 --> 00:23:43,130
+.. ال .. ال .. ال X و ال Y هنا هى ال X و ال Y وهى
+
+285
+00:23:43,130 --> 00:23:46,430
+رسمة الدلة فى المنطقة هذه ال X و ال Y فى داخل هدول
+
+286
+00:23:46,430 --> 00:23:53,930
+طيب الان ال X و ال Y هنا إذا أكيد ال F of X ال F
+
+287
+00:23:53,930 --> 00:24:00,090
+of X ناقص F of Y نقطتين هنا نقطتين من هنا لهنا صور
+
+288
+00:24:00,090 --> 00:24:03,750
+هنا صور هنا يعني ممكن صورة واحدة هنا و صورة
+
+289
+00:24:03,750 --> 00:24:08,440
+التانية هنايعني ممكن صورة التالتة هنا وصورة الأولى
+
+290
+00:24:08,440 --> 00:24:15,500
+هنا يعني بمعنى آخر لو ال X هنا و ال Y هنا هي صورة
+
+291
+00:24:15,500 --> 00:24:21,400
+ال X وهي صورة مين ال Y لو جيتوا .. أخدتوا .. هي
+
+292
+00:24:21,400 --> 00:24:26,760
+صورة ال Y لو أخدتوا الفرق بين هذه و بين هذهالفرق
+
+293
+00:24:26,760 --> 00:24:31,020
+بين ما هو قيمة الدالة F of X هنا و قيمة الدالة F
+
+294
+00:24:31,020 --> 00:24:36,280
+of Y هنا أكيد .. أكيد .. أكيد الفرق بينهم هيكون
+
+295
+00:24:36,280 --> 00:24:42,320
+أصغر أو يساوي M K ناقص من M K لسبب بسيط أصلا لأن
+
+296
+00:24:42,320 --> 00:24:49,660
+أصلا F of X هذه أصغر أو يساوي ما هو M K أكيد و F
+
+297
+00:24:49,660 --> 00:24:54,400
+of Y أصغر أو يساويها ماشي الحالو نفس ال .. يعني ..
+
+298
+00:24:54,400 --> 00:24:57,080
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+299
+00:24:57,080 --> 00:24:57,100
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+300
+00:24:57,100 --> 00:24:57,360
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+301
+00:24:57,360 --> 00:24:57,720
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+302
+00:24:57,720 --> 00:24:58,100
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+303
+00:24:58,100 --> 00:24:59,480
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+304
+00:24:59,480 --> 00:25:00,040
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+305
+00:25:00,040 --> 00:25:07,580
+و .. و .. و .. و .. و
+
+306
+00:25:07,580 --> 00:25:08,160
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+307
+00:25:08,160 --> 00:25:11,200
+.. و .. و .. و ..
+
+308
+00:25:19,040 --> 00:25:24,000
+فبيظل قيمة f of x ناقص f of y أصغر شاومين مك ناقص
+
+309
+00:25:24,000 --> 00:25:28,460
+مك اللي مش واضح له من خلال الرسمة اللي عندي هنا
+
+310
+00:25:31,440 --> 00:25:34,520
+يعني الآن خلّيني أرى أضحى لحسن تكون مش واضحة للبعض
+
+311
+00:25:34,520 --> 00:25:41,360
+هاي الرسمة هاي XK minus واحد وهي XK خلّيني أكبر
+
+312
+00:25:41,360 --> 00:25:46,180
+الرسمة عشان تكون أوضح وهي XK لأن نفترضها أن هاي
+
+313
+00:25:46,180 --> 00:25:53,520
+رسمتنا هيك وطلعت زي هيك ماشي وهي عندي أعلى نقطة
+
+314
+00:25:53,520 --> 00:25:58,710
+خلّيني أزغر هذه شويةوهي أصغر النقطة هذي اللي
+
+315
+00:25:58,710 --> 00:26:03,970
+بتتمثل لـ M K وهذه بتتمثل لـ M K Small هاي المسافة
+
+316
+00:26:03,970 --> 00:26:09,030
+بينهم الفرق بينهم الآن الثانية لو جيت أخدت أي نقطة
+
+317
+00:26:09,030 --> 00:26:17,190
+هنا X و أي نقطة Y هنا هاي X وهي Y المسافة بينها دي
+
+318
+00:26:17,190 --> 00:26:21,310
+و بينها دي قيمتها دي بيصير F of X أنا بصير F of X
+
+319
+00:26:22,380 --> 00:26:26,700
+وهنا بيصير F of Y الفرق بين F of X و F of Y أكيد
+
+320
+00:26:26,700 --> 00:26:31,140
+أصغر من الفرق هذا وكل النقاط اللي بين هذه و هذه
+
+321
+00:26:31,140 --> 00:26:37,340
+هيكون يا إما زي الفرق هذا أو أصغر منه إذا أكيد
+
+322
+00:26:37,340 --> 00:26:45,820
+عندي صار المفروض وضح الأمر هيكون عندي اللي هو هذا
+
+323
+00:26:45,820 --> 00:26:50,910
+المقدار أصغر شهو هذا طب ليه ليش هذا؟هذا من مين للـ
+
+324
+00:26:50,910 --> 00:26:53,990
+K اللي في الـ A والـ K اللي في الـ A شخصيتها Mk
+
+325
+00:26:53,990 --> 00:26:57,610
+نقصها Mk أصغر من مين؟ إذا صارت أصغر من الـ Delta
+
+326
+00:26:57,610 --> 00:27:04,470
+إذا صارت F of X و F of Y أصغر من مين؟من Delta طيب
+
+327
+00:27:04,470 --> 00:27:08,430
+مادام F of X ايش علاقات هنا F of X و F of Y ما هي
+
+328
+00:27:08,430 --> 00:27:13,950
+اصلا احنا مفترضين من الاول ان F of I جزئية من الـ
+
+329
+00:27:13,950 --> 00:27:18,430
+J اللي هي عبارة عن C و D مظبوط ولا لأ اذا حيصير
+
+330
+00:27:18,430 --> 00:27:23,600
+عندي F of X و F of Y موجودات في الـ C و DF of X و
+
+331
+00:27:23,600 --> 00:27:27,000
+F of Y موجودات في الـ C و الـ D و بتحقق المسافة
+
+332
+00:27:27,000 --> 00:27:32,200
+أصغر من Delta بينهم إذا حسب واحد اللي احنا أثبتنا
+
+333
+00:27:32,200 --> 00:27:35,960
+أي نقطتين في الـ S و الـ D المسافة بينهم أصغر من
+
+334
+00:27:35,960 --> 00:27:39,600
+Delta لازم صورة ال .. صورة Phi of S و Phi of T
+
+335
+00:27:39,600 --> 00:27:42,540
+المسافة بينهم أصغر من مين؟ من Epsilon و Prime إذا
+
+336
+00:27:42,540 --> 00:27:47,640
+هذا Automatic هيعطيني Phi of النقطة الأولى اللي هي
+
+337
+00:27:47,640 --> 00:27:56,290
+F of X ناقصفاي أفضل النقطة الثانية يكون أصغر من
+
+338
+00:27:56,290 --> 00:28:06,890
+مين من إبسلون براي إذا صار عندي الآن لكل اللي عندي
+
+339
+00:28:06,890 --> 00:28:13,740
+ال K اللي في ال A وال X و Y في هذه المنطقةF of X
+
+340
+00:28:13,740 --> 00:28:17,920
+minus F of Y طلع ليه أصغر من Delta حتى ملي الـ 5 F
+
+341
+00:28:17,920 --> 00:28:25,000
+of X ناقص 5 F of Y أصغر من مين؟ من Y إبراهيم ماشي
+
+342
+00:28:25,000 --> 00:28:29,820
+الحال هذا الكلام لكل X و Y موجودات في الـ XK minus
+
+343
+00:28:29,820 --> 00:28:33,460
+واحد والـ XK لخص الآن اللي وصلنا له لأنه بدي
+
+344
+00:28:33,460 --> 00:28:36,920
+أستخدمه خليني ألخص خليني أشيل
+
+345
+00:28:41,000 --> 00:28:45,820
+خلّيني بس ساعدك اكتبلكم في مكان هنا اتحملوني انه
+
+346
+00:28:45,820 --> 00:28:54,640
+عشان اللوح شوية A بساوي كل ال K مقصد MK اصغر من
+
+347
+00:28:54,640 --> 00:29:05,600
+دلتا وعند B بساوي كلالـ K بحيث أن M K نقص M K أكبر
+
+348
+00:29:05,600 --> 00:29:10,060
+أو يساوي دلتة خلّيني أقولها في الذاكرة الآن خلّيني
+
+349
+00:29:10,060 --> 00:29:16,140
+أمسح حاجة و ألخص معلومتي إيش معلومتي بتقول دي لـ K
+
+350
+00:29:16,140 --> 00:29:26,610
+اللي في الـ A ماشي الحال لكل X و Y في الفترة X K-1
+
+351
+00:29:26,610 --> 00:29:32,750
+لعندي الـ xk بيطلع عندى الـ Phi f of x يعني Phi
+
+352
+00:29:32,750 --> 00:29:40,990
+composite f of x نقص Phi composite f of y أشماله
+
+353
+00:29:40,990 --> 00:29:48,600
+أصغر من إبسلون براي واضح؟إذن اللي وصلناله لـ K
+
+354
+00:29:48,600 --> 00:29:53,700
+اللي في الـ A عندي لـ K اللي في الـ A هذا الـ
+
+355
+00:29:53,700 --> 00:29:58,380
+absolute value أظهر من مين؟ من إبسلون إبراهيم لكل
+
+356
+00:29:58,380 --> 00:30:05,580
+مين لكل XY في ال XK minus XK minus 1 طيب نكمل لكم
+
+357
+00:30:05,580 --> 00:30:10,860
+عليها هذا خلصنا منه صار عندي الآن شوفوا
+
+358
+00:30:19,430 --> 00:30:24,070
+الشغل عميدي يا جماعة على الفترة هذه طيب شوف عندي
+
+359
+00:30:24,070 --> 00:30:31,590
+Phi composite F of X ناقص Phi composite F of Y
+
+360
+00:30:31,590 --> 00:30:37,770
+أكبر من Y و أكبر من مين من ناقص Y برايم بدأ أخد
+
+361
+00:30:37,770 --> 00:30:43,970
+هذه الجهة خليني أخد الجهة من الجهتين و أبدأ أشتغل
+
+362
+00:30:43,970 --> 00:30:51,300
+عليها واضحة هذههذه أصغر من Y' أكبر من ناقص Y بديش
+
+363
+00:30:51,300 --> 00:30:56,980
+هذا بتشغل هذه الان لان هذا الكلام صحيح لمن؟ لكل X
+
+364
+00:30:56,980 --> 00:31:02,740
+وY في ال sub interval هذه اللي هي لكل مين ال case؟
+
+365
+00:31:02,740 --> 00:31:06,600
+ال case اللي من A بس طيب شوف إذا صار عندي Phi
+
+366
+00:31:06,600 --> 00:31:14,580
+composite F of X أصغر من Y' بزايد Phi composite F
+
+367
+00:31:14,580 --> 00:31:21,720
+of Yماشي يا جماعة .. ماشي هذا الكلام لكل x و y في
+
+368
+00:31:21,720 --> 00:31:26,980
+.. اللي هو xk minus .. من xk minus 1 عند xk اللي
+
+369
+00:31:26,980 --> 00:31:31,600
+هنا .. هذا لكل x و y ثبتلي y .. ثبتلي y .. خلينا
+
+370
+00:31:31,600 --> 00:31:34,380
+نحكي عن y محدد .. arbitrary y لكن خلينا .. ثبتها
+
+371
+00:31:34,380 --> 00:31:40,180
+.. خلينا نحكي عن arbitrary fixed yبظل y' زي الـ
+
+372
+00:31:40,180 --> 00:31:47,140
+phi composite f of y is true أكبر من هذه لكل x
+
+373
+00:31:47,140 --> 00:31:52,500
+element in xk minus 1 لعند مين؟ لعند الـ xk أكيد
+
+374
+00:31:52,500 --> 00:32:00,060
+أكيد هذا أكبر من هذا لكل x well موجودة في xk minus
+
+375
+00:32:00,060 --> 00:32:04,540
+1 لعند مين؟ لعند الـ xk إذا صار هذا عبارة عن إيش
+
+376
+00:32:04,540 --> 00:32:08,880
+يا جماعة عبارة عن upper bound لهذا ال setمدام
+
+377
+00:32:08,880 --> 00:32:13,900
+upper bound إلها إذا هيكون أكبر أو يساوي ال least
+
+378
+00:32:13,900 --> 00:32:19,340
+upper bound يعني بمعنى آخر ال supremum لفاي
+
+379
+00:32:19,340 --> 00:32:25,280
+composite F of X such that X element in XK minus 1
+
+380
+00:32:25,280 --> 00:32:31,240
+لعند ال XK اللي هو أصغر أو يساوي Y prime زي فاي
+
+381
+00:32:31,240 --> 00:32:38,520
+composite F of mean of Y for any fixed Yطيب، هذا
+
+382
+00:32:38,520 --> 00:32:45,140
+مين هو؟ هذا عبارة عن الـ M K وسميها Delta أساس أو
+
+383
+00:32:45,140 --> 00:32:48,080
+تلدا أساس اللي هي خاصة بمين الآن في الـ Phi
+
+384
+00:32:48,080 --> 00:32:53,100
+Composite F عشان نميزها بالـ M K اسمه بالـ M K
+
+385
+00:32:53,100 --> 00:32:59,400
+اللي خاصة بالـ F اللي عندي، ماشي الحال، طيبصار
+
+386
+00:32:59,400 --> 00:33:02,800
+عندى MKتل ده بساوي الـ Supremum لهذا أصغر يساوي
+
+387
+00:33:02,800 --> 00:33:07,680
+هذا هذا صار عدد صار العدد هذا أصغر من يبسلون برايم
+
+388
+00:33:07,680 --> 00:33:11,640
+في كومبوزيت F of Y for any fixed Y يعني صحيح على
+
+389
+00:33:11,640 --> 00:33:16,320
+كل Y وين موجودة في الفترة هذه إذا صار عندى الآن
+
+390
+00:33:16,320 --> 00:33:21,680
+جيبولي هذه على الجهة هذه وهذه على الجهة هذه بيصير
+
+391
+00:33:21,680 --> 00:33:28,050
+عندى اللي هو ناقص في كومبوزيت F of Yأصغر أو ساوي
+
+392
+00:33:28,050 --> 00:33:34,770
+أبسلون و براين ناقص مين مك تلدة وإذا بدك ممكن نجيب
+
+393
+00:33:34,770 --> 00:33:40,050
+اللي هو الابسلون و البراين على الجهة الثانية مش
+
+394
+00:33:40,050 --> 00:33:50,310
+مشكلة صار عندي الآن هذا يا جماعة صار
+
+395
+00:33:50,310 --> 00:33:58,380
+عندي هذا اللي هو عبارة عن upper boundلهذا المقدار
+
+396
+00:33:58,380 --> 00:34:07,020
+أكيد ولا لا؟ عارفين ليش؟ لأن هذا الأن أكبر أو
+
+397
+00:34:07,020 --> 00:34:11,000
+يساوي هذا المقدار لكل Y لأن كل المخدين Y
+
+398
+00:34:11,000 --> 00:34:14,600
+arbitrarily fixed لكن arbitrarily، إذا صحيح على
+
+399
+00:34:14,600 --> 00:34:19,640
+كله، إذا بيصير عندي الآن اللي هو هذا أكبر أو يساوي
+
+400
+00:34:19,640 --> 00:34:24,260
+ال supremum لهذا، يعني بمعنى أخر ال supremum، لا
+
+401
+00:34:24,860 --> 00:34:32,200
+الناقص Phi composite F of Y such that Y element in
+
+402
+00:34:32,200 --> 00:34:39,360
+YK as if XK minus 1 عند XK هذا الـ Supremum له
+
+403
+00:34:39,360 --> 00:34:48,670
+أظهر أو يسوى Y' ناقص M K تلدةطيب هات ايش بتساوي
+
+404
+00:34:48,670 --> 00:34:56,670
+طلعون الناقص برا بيساوي ناقص الانف مام معايا لأ
+
+405
+00:34:56,670 --> 00:34:59,790
+لأنه مدام ناقص واحد طلع اذا بيقلب ال super mum ل
+
+406
+00:34:59,790 --> 00:35:05,530
+انف مام فاي composite F of Y such that Y element
+
+407
+00:35:05,530 --> 00:35:09,610
+in XK minus واحد لعند ال XK عرفتوا ايش اللي بده
+
+408
+00:35:09,610 --> 00:35:16,060
+اقوله هاد مين هي يا شبابهذه عبارة عن الـ M K تلدة
+
+409
+00:35:16,060 --> 00:35:19,680
+اللي هي الـ infimum للـ Phi Composite F كتبت
+
+410
+00:35:19,680 --> 00:35:22,800
+التلدة عشان ترمز لمين للـ Phi Composite F وفي عندي
+
+411
+00:35:22,800 --> 00:35:28,460
+ناقص قبلها إذا صارت عندي ناقص الـ M K تلدة أصغر
+
+412
+00:35:28,460 --> 00:35:33,400
+يساوي إبسلون ناقص الـ M K تلدة بدي أجيب هذه هنا
+
+413
+00:35:33,400 --> 00:35:42,400
+بيصير عندي إذا M K تلدةأصغر نقص اف ناقص مك تلد ا
+
+414
+00:35:42,400 --> 00:35:51,080
+small اصغر او يساوي مين ابسلون اذا اللي وصلتله ما
+
+415
+00:35:51,080 --> 00:35:59,440
+يلي وهذه بده اصلها انه لكلف ال a طلع عندي مك تلد
+
+416
+00:35:59,440 --> 00:36:05,300
+ناقص مك prime اصغر اش هو ايش ابسلون خلينا نسجلها
+
+417
+00:36:05,300 --> 00:36:11,830
+عشان نبني عليهاصار عندى الآن يا جماعة طولوا روحكم
+
+418
+00:36:11,830 --> 00:36:18,670
+عليا ان شاء الله مش هم طوالين نخلص M K تلدى ناقص M
+
+419
+00:36:18,670 --> 00:36:22,890
+K تلدى small أصغر يمشي هو إبسلون هذا لكل K وين
+
+420
+00:36:22,890 --> 00:36:30,330
+موجودة في ال A لكل K في ال A طيب هذا اللى احنا إيش
+
+421
+00:36:30,330 --> 00:36:38,150
+ماله أوصلنا إلهصار عندى الان الـ M K تلدى ناقص M K
+
+422
+00:36:38,150 --> 00:36:52,190
+تلدى أصغر من الـ E' لكل K فيه اللى هى N A الان
+
+423
+00:36:52,190 --> 00:36:58,330
+خد ال summation ال summation لل M K تلدى ناقص M K
+
+424
+00:36:58,330 --> 00:37:05,490
+تلدى small K element in A اللى هوهذا اللي هو أصغر
+
+425
+00:37:05,490 --> 00:37:09,310
+أو يساوي في .. أصغر أو يساوي .. الآن قدوا .. قدوا
+
+426
+00:37:09,310 --> 00:37:11,470
+.. قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+427
+00:37:11,470 --> 00:37:12,010
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+428
+00:37:12,010 --> 00:37:13,570
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+429
+00:37:13,570 --> 00:37:15,170
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+430
+00:37:15,170 --> 00:37:17,810
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+431
+00:37:17,810 --> 00:37:31,850
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا
+
+432
+00:37:31,850 --> 00:37:39,180
+..K element in A العداد هذا بعيد بعدد الـ K's اللي
+
+433
+00:37:39,180 --> 00:37:43,180
+في الـ A يعني إبسلون مضروب في عدد الـ K's اللي
+
+434
+00:37:43,180 --> 00:37:48,060
+موجودة في مين في إيه هذا أكيد أكيد هذا أصغر أو
+
+435
+00:37:48,060 --> 00:37:55,320
+يساوي ال summation للإبسلون على كل الـ K من عند
+
+436
+00:37:55,320 --> 00:38:00,980
+صفر لعند مين لعند اللي هو H N
+
+437
+00:38:03,340 --> 00:38:09,620
+بس انا الان بدي اجي اودي ال summation و اضربه فى
+
+438
+00:38:09,620 --> 00:38:13,460
+مين فى طول الفترة لإنه هي لازم لطول الفترة بعد
+
+439
+00:38:13,460 --> 00:38:20,660
+اسمهم هذا اضربه ليه فى XK بدي اضربه بس اوسع عقله
+
+440
+00:38:20,660 --> 00:38:23,700
+معلش بدي اضربه يعني هذا ال summation اللى حصلت
+
+441
+00:38:23,700 --> 00:38:29,320
+عليه بدي اخد ال summation MK تلدهنقص mk تلدى
+
+442
+00:38:29,320 --> 00:38:34,540
+وادربه يا شباب في xk minus xk minus واحد k element
+
+443
+00:38:34,540 --> 00:38:39,000
+in a هيصير اصغر او يساوي ال summation للإيبسلون في
+
+444
+00:38:39,000 --> 00:38:48,620
+xk minus xk minus واحد على الان k element in aأكيد
+
+445
+00:38:48,620 --> 00:38:52,120
+أصغر زي ما قلنا من ال summation اللي فوق epsilon
+
+446
+00:38:52,120 --> 00:38:59,980
+في xk minus xk minus 1k من 0 لعند n ليش؟ هذا ال
+
+447
+00:38:59,980 --> 00:39:02,420
+summation بس على العراصر اللي في ال A هذا ال
+
+448
+00:39:02,420 --> 00:39:05,260
+summation على كله على اللي في ال A وعلى اللي في ال
+
+449
+00:39:05,260 --> 00:39:08,760
+B على كل بقى العراصر اللي هي ال partition اللي
+
+450
+00:39:08,760 --> 00:39:15,420
+عنده من 0 ل1 اتنين عند مين لعند ال N طيب ال N هذا
+
+451
+00:39:15,420 --> 00:39:20,200
+المقدر هي اللي بديهالأن هذا bridge كان أو جسر
+
+452
+00:39:20,200 --> 00:39:23,860
+للوصول للي بديها و حتى ممكن نهان مباشرة لهذه هذه
+
+453
+00:39:23,860 --> 00:39:27,760
+أصغر أو يساوي هذا هذا مش هو اللي بتطلع عليه بتطلع
+
+454
+00:39:27,760 --> 00:39:32,640
+عليه هذا و بتطلع عليه هذا هذا خلّيه لأن أصغر يساوي
+
+455
+00:39:32,640 --> 00:39:36,460
+مين هذا المقدار هذا المقدار ايش بساوي epsilon في
+
+456
+00:39:36,460 --> 00:39:43,790
+مينx note ناقص x واحد زاد x واحد ناقص x اتنين زاد
+
+457
+00:39:43,790 --> 00:39:47,810
+x اتنين ناقص x ثلاثة لما أصل لآخر واحد يضل اللي
+
+458
+00:39:47,810 --> 00:39:53,330
+عبارة عن y في x n ناقص x note عملناها كتير ويساوي
+
+459
+00:39:53,330 --> 00:39:58,710
+y prime ويساوي y prime في مين في b minus a لأن ال
+
+460
+00:39:58,710 --> 00:40:03,950
+xn هي عبارة عن b و ال x note هي مين كانت هي ال a
+
+461
+00:40:03,950 --> 00:40:10,110
+إذا اللي وصلتله يا جماعةالان وصلت الى ما يلي انه
+
+462
+00:40:10,110 --> 00:40:19,170
+ال summation لل mk تلدى ناقص mk تلدى small لكي
+
+463
+00:40:19,170 --> 00:40:25,790
+element ايه في xk minus xk minus واحد اصغر او
+
+464
+00:40:25,790 --> 00:40:34,550
+يساوي epsilon prime في b minus ايش minus a واضح
+
+465
+00:40:34,550 --> 00:40:42,680
+الانهذه خزينها لإنه بدّيها بعد شوية اتحملونا
+
+466
+00:40:42,680 --> 00:40:46,880
+البرهان طويل شوية هي وصلنا إليها summation أزرع
+
+467
+00:40:46,880 --> 00:40:50,980
+شويه أبسلون رابع في b minus a لإن بدنا نيجي لمين
+
+468
+00:40:50,980 --> 00:40:59,140
+يا جماعة بدنا نيجي لاللي هوانتبهوا عليها، بدنا
+
+469
+00:40:59,140 --> 00:41:03,280
+نيجي للـ M للـ K اللي موجودة في مين؟ هذا الـ K في
+
+470
+00:41:03,280 --> 00:41:06,820
+الـ A، ضال عند مين؟ احنا جزأتها إلى جزئين اللي هي
+
+471
+00:41:06,820 --> 00:41:11,340
+الـ indices اللي عنده، K اللي هي في الـ A و K اللي
+
+472
+00:41:11,340 --> 00:41:18,060
+هي في مين؟ في الـ B خد الآن K element in B، شوف
+
+473
+00:41:18,060 --> 00:41:26,980
+إيش؟ الـ MK تلد إلهانقص mk تلد إلها أيش بتساوي؟
+
+474
+00:41:26,980 --> 00:41:32,380
+بتساوي حسب التعريف Supremum للـ Phi Composite F of
+
+475
+00:41:32,380 --> 00:41:38,240
+X للـ K اللي وين في الـ B اللي هي mk تلدها نقص mk
+
+476
+00:41:38,240 --> 00:41:42,540
+تلدها اللي هي small Supremum للـ Phi F of X اللي
+
+477
+00:41:42,540 --> 00:41:47,980
+هي الأولى هذه Such that X element in XK minus واحد
+
+478
+00:41:47,980 --> 00:41:54,860
+و XK اللي هي زائدأو ناقص اللي هي مين هذه الـ
+
+479
+00:41:54,860 --> 00:42:01,200
+infimum لأ فاي composite F of X such that X
+
+480
+00:42:01,200 --> 00:42:10,220
+والموجودة في ال X K minus واحد وال X K مظبوط ولا
+
+481
+00:42:10,220 --> 00:42:18,690
+لأ احنا قلناالـ Supremum لكل الـ Phi of T T على كل
+
+482
+00:42:18,690 --> 00:42:24,130
+الـ C والـ D بيساوي ايش اسمه؟ بيساوي K هذه عبارة
+
+483
+00:42:24,130 --> 00:42:29,790
+عن مين؟ شايفين، مابديش أعيدك، بقى أكتب مرتين خليني
+
+484
+00:42:29,790 --> 00:42:33,890
+أكتبها على شورة بتصير Phi of F of X يعني بتصير
+
+485
+00:42:33,890 --> 00:42:42,250
+عبارة عن مين؟ Phi of F of X معايا؟
+
+486
+00:42:43,040 --> 00:42:48,280
+وهاد ايش اسمها؟ بدي أدخل السالب جوا يا جماعة أو
+
+487
+00:42:48,280 --> 00:42:56,440
+قبل ما أدخله، هد بيصير فاي فاي of F of X بعد
+
+488
+00:42:56,440 --> 00:43:01,560
+أذنكم، بدي أدخل السالب جوا ال infimum هتصير إيش ما
+
+489
+00:43:01,560 --> 00:43:06,240
+لها؟ supremum إذا بيصير هذا زائد وهد بتصير
+
+490
+00:43:06,240 --> 00:43:14,100
+supremum والسالب بدخل جواطيب .. عندي الـ Phi F of
+
+491
+00:43:14,100 --> 00:43:19,800
+Xوالـ Phi F of X للـ X اللي موجودة هنا الـ F of X
+
+492
+00:43:19,800 --> 00:43:25,460
+هدولة من مين جايت من الفترة C و D لأن زي ما قلنا F
+
+493
+00:43:25,460 --> 00:43:30,680
+of A و F of I subset من الجهة اللي هي C و D لذا كل
+
+494
+00:43:30,680 --> 00:43:35,040
+ال element هنا موجود هنا يعني بمعنى أخر ال .. ال
+
+495
+00:43:35,040 --> 00:43:38,720
+supreme من اللي هنا اللي هي ال K عندي ال absolute
+
+496
+00:43:38,720 --> 00:43:45,400
+value للـ Phi of T أكيد أكبر أو يساويالـ Phi سالب
+
+497
+00:43:45,400 --> 00:43:51,080
+Phi of T و أصغر أو يساوي الـ Phi of T أو بالسوية
+
+498
+00:43:51,080 --> 00:43:55,280
+حتى أكيد
+
+499
+00:43:55,280 --> 00:43:57,900
+خلّيني أخدها على خطواتين بتسويها دي لأن ال
+
+500
+00:43:57,900 --> 00:44:06,260
+absolute value Phi of T أكبر سوى مين؟ سالب ال Phi
+
+501
+00:44:06,260 --> 00:44:11,300
+of T فبصير عندي ال supremum اللي عندي اللي هالة
+
+502
+00:44:11,300 --> 00:44:18,410
+هذاSupermom لهذه اكيد اصغر او يساوياللي هو الـ
+
+503
+00:44:18,410 --> 00:44:22,170
+Supremum هذا اللي هو K لأن الـ Supremum هذا عالمين
+
+504
+00:44:22,170 --> 00:44:26,370
+على كل الفترة C وD هنا اللي هي عالمين على X
+
+505
+00:44:26,370 --> 00:44:30,890
+element in XK مانس واحد و XK لأن هذا Similarly
+
+506
+00:44:30,890 --> 00:44:35,910
+أكيد الـ Supremum للـ Absolute Value هيكون أكبر أو
+
+507
+00:44:35,910 --> 00:44:39,070
+يساوي الـ Supremum لهذه من جهتين أنه Absolute
+
+508
+00:44:39,070 --> 00:44:42,790
+Value وفي نفس الوقت اللي هو المجموع هذه جزئية من
+
+509
+00:44:42,790 --> 00:44:48,380
+اللي فوق إذن ده برضه بيكون أصغر أو يساويK اذا صار
+
+510
+00:44:48,380 --> 00:44:53,800
+automatic اللي هو هذه زائد هذه اصغر او يساوي مين
+
+511
+00:44:53,800 --> 00:45:01,440
+اتنين K من اي حاجة اصغر او يساوي اتنين K الان هذه
+
+512
+00:45:01,440 --> 00:45:06,160
+ال supremum زائد ال supremum هذه بس لا ياش لسانك
+
+513
+00:45:06,160 --> 00:45:11,900
+يا شباب و هذه K اصغر يساوي K زائد K عشان التلخيص
+
+514
+00:45:11,900 --> 00:45:19,460
+بيصير اشمله اتنين Kواضح إذا نخزن التانية اللي هي K
+
+515
+00:45:19,460 --> 00:45:24,480
+element in B بعد إذنكم بيصير عندي ال M K نقص بلاش
+
+516
+00:45:24,480 --> 00:45:30,200
+التفاصيل هذه خلصنا منها بيصير عندي ال M K نقص M K
+
+517
+00:45:30,200 --> 00:45:37,180
+small أصغر من تلد تلد طبعا أصغر من مين؟ من اتنين K
+
+518
+00:45:37,180 --> 00:45:43,280
+طبعا ال K هنا index ال K هذه capital هذه ال K مين؟
+
+519
+00:45:43,280 --> 00:45:48,590
+ال K اللي فوق هذههذا الـ K هو الاندكس للـ B وهذا
+
+520
+00:45:48,590 --> 00:45:50,690
+الـ K هو الاندكس للـ B يعني الـ K هذه ليست إيادة
+
+521
+00:45:50,690 --> 00:45:57,370
+هذه K هي الـ Supremum اللي فوق طيب نجمح
+
+522
+00:45:57,370 --> 00:46:03,190
+معلوماتنا ونبدأ نجمحها الآن ونكون حصلنا على اللي
+
+523
+00:46:03,190 --> 00:46:08,410
+بدهد اتنين Kأه لب تلخيص 2K يا شباب لما تشوفوها لب
+
+524
+00:46:08,410 --> 00:46:13,770
+تلخيص 2K الان اوجدت ال summation على مين؟ على ايه؟
+
+525
+00:46:13,770 --> 00:46:16,710
+بدي اوجد ال summation على ال P باقي ال indices لإن
+
+526
+00:46:16,710 --> 00:46:21,650
+هالجات بدي أقولكم ليش بالضيقة الان اذا صار عندي ال
+
+527
+00:46:21,650 --> 00:46:29,930
+summation لل Mk ناقص Mk تلدة تلدة في Xk minus Xk
+
+528
+00:46:29,930 --> 00:46:40,680
+minus واحد K element in B الآنمظبوط أصغر أو يساوي
+
+529
+00:46:40,680 --> 00:46:45,120
+اللي هو 2K برضه على ال summation هذا ال summation
+
+530
+00:46:45,120 --> 00:46:53,040
+اللي هو 2K كابيتال كابيتال هذه مالهاش علاقة بال K
+
+531
+00:46:53,040 --> 00:47:00,840
+ال element in B هنا في XK minus XK minus 1 أخدت ال
+
+532
+00:47:00,840 --> 00:47:03,700
+summation على الجهتين على كل ال K ال element in B
+
+533
+00:47:06,360 --> 00:47:20,220
+ماشي طيب وين أكتب طيب اتحملوني هلجيت عندي خليني
+
+534
+00:47:20,220 --> 00:47:25,300
+أكتب هنا أشرح أنا شوية بهمش خليني أشرح على ال ..
+
+535
+00:47:25,300 --> 00:47:33,740
+الكلف ال B الكلف ال B احنا من تعريف الكلف ال Bبدي
+
+536
+00:47:33,740 --> 00:47:39,160
+يكون الام كي ناقص ام كي أكبر شو مين؟ Delta صح؟
+
+537
+00:47:39,160 --> 00:47:41,980
+بيصير عندى الآن Delta يعني أصغر شو ام كي ناقص ام
+
+538
+00:47:41,980 --> 00:47:47,000
+كي اله small ايه بيسمعوا على ال Delta هنا؟ هو انا
+
+539
+00:47:47,000 --> 00:47:50,920
+على ال Delta بيصير هذا إيه شبال، هو واحد، معايا؟
+
+540
+00:47:50,920 --> 00:48:02,180
+فبيصير عندى الآن ال summation على الواحدXK-XK- XK
+
+541
+00:48:02,180 --> 00:48:07,650
+- XK- XK- XK- XK- XK- XK- XKأصغر أو يساوي mk نقص
+
+542
+00:48:07,650 --> 00:48:13,130
+mk على delta ماشي خد ال summation للجهتين ال
+
+543
+00:48:13,130 --> 00:48:19,030
+summation للجهتين xk minus xk minus واحد وانا ال
+
+544
+00:48:19,030 --> 00:48:24,310
+summation xk minus xk minus واحد k element in b
+
+545
+00:48:24,310 --> 00:48:28,850
+لأن هذا صح لل b بس لكي element in b مظبوط ممكن
+
+546
+00:48:28,850 --> 00:48:33,880
+حصلنا على هذا هيهاللي موصول وهذا واحد على دلتا
+
+547
+00:48:33,880 --> 00:48:38,720
+طلعها برا في ال summation اللي عندي هذا ماشي وصلنا
+
+548
+00:48:38,720 --> 00:48:45,620
+ل هيك إذا الآن صار عندي هذا أصغر أوي ساوي واحد على
+
+549
+00:48:45,620 --> 00:48:56,360
+دلتا في مين؟ في الان هذا اللي هو ال UPUF طب أنا دي
+
+550
+00:48:56,360 --> 00:49:00,500
+MK وMK اه مش MK تلد هذه اه
+
+551
+00:49:03,350 --> 00:49:07,230
+ليش؟ لأنه .. اشكروا ان انا كتبين هنا Mk و Mk تلتة
+
+552
+00:49:07,230 --> 00:49:15,130
+لأن هذه الـ A الـ B الـ B الـ K اللي فيها مصنفة
+
+553
+00:49:15,130 --> 00:49:19,530
+على أساس Mk اسمه نقص Mk أكبر زي دلتة فبصير عندى
+
+554
+00:49:19,530 --> 00:49:25,800
+الآنالـ Mk نقص Mk اللي هو أكبر أو يساوي Delta مش
+
+555
+00:49:25,800 --> 00:49:31,520
+Delta هذه هذه لمين؟ اللي مصنف عليها الـ B اللي هي
+
+556
+00:49:31,520 --> 00:49:35,860
+بالنسبة للـ F هذه مش لـ Alpha Composite F إذا هذا
+
+557
+00:49:35,860 --> 00:49:38,560
+أكبر أو يساوي Delta على Delta بيصير أكبر أو يساوي
+
+558
+00:49:38,560 --> 00:49:41,740
+واحد على ال summation اللي عملناها قبل بشوية أصغر
+
+559
+00:49:41,740 --> 00:49:44,720
+أو يساوي واحد على Delta في ال summation هذا الآن
+
+560
+00:49:44,720 --> 00:49:55,810
+.. الآن هذا مين هو؟ هو عبارة عن الـ UP و F مش كله
+
+561
+00:49:55,810 --> 00:50:01,210
+حتى جزء منه لأن ال U P و F ايش ماله ال M كده تبعته
+
+562
+00:50:01,210 --> 00:50:08,310
+ال U P و F ايش هو يا شباب هو عبارة عن ال summation
+
+563
+00:50:08,310 --> 00:50:14,090
+كي من عند واحد لعند N او من Zero لعند N من واحد
+
+564
+00:50:14,090 --> 00:50:25,770
+لعند N مظبوط لمين لأ ال Mك في xk minus xk minus
+
+565
+00:50:25,770 --> 00:50:30,070
+واحد هذا ال summation على مين؟ على كل ال case من
+
+566
+00:50:30,070 --> 00:50:35,290
+واحد لعند n بينما هذا ال summation لمين؟ بس للجزء
+
+567
+00:50:35,290 --> 00:50:39,130
+اللي هو في مين؟ في ال B فأكيد ال summation على هذه
+
+568
+00:50:39,130 --> 00:50:44,410
+أظهر أوي ساوي ال summation على هذه باشي لأن ال
+
+569
+00:50:44,410 --> 00:50:48,430
+summation على هذه ناقص هذه برضه بظهر أوي ساوي هذا
+
+570
+00:50:48,430 --> 00:50:55,190
+ناقص هذاليش؟ لأن الـ M K ناقص M نعمل كمية على جهة
+
+571
+00:50:55,190 --> 00:50:59,290
+نعملها
+
+572
+00:50:59,290 --> 00:51:03,690
+.. نعملها
+
+573
+00:51:03,690 --> 00:51:04,770
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+574
+00:51:04,770 --> 00:51:09,950
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+575
+00:51:09,950 --> 00:51:10,210
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+576
+00:51:10,210 --> 00:51:10,270
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+577
+00:51:10,270 --> 00:51:10,410
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+578
+00:51:10,410 --> 00:51:18,420
+.. نعملها .. نعملها .. نعملها ..أيش هيساوي الـ
+
+579
+00:51:18,420 --> 00:51:24,940
+summation للـ mk ناقص mk small في xk minus xk
+
+580
+00:51:24,940 --> 00:51:30,880
+minus واحد هذا k من عند واحد لعند n مظبوط ولا لأ؟
+
+581
+00:51:30,880 --> 00:51:35,360
+الان أكيد أكيد هذا لأن هذا موجبة وهذا موجبة لأن
+
+582
+00:51:35,360 --> 00:51:42,060
+هذا أكيد أكبر أو يساوي ال summation لل mk ناقص mk
+
+583
+00:51:43,610 --> 00:51:51,390
+xk-xk-1 k element in B لأن هذوله جزء من هذوله يعني
+
+584
+00:51:51,390 --> 00:51:56,150
+نقصت من ال summation هذا بعض ال terms اللي أكبر أو
+
+585
+00:51:56,150 --> 00:51:57,810
+يساوي 0 إذا نقص ال summation
+
+586
+00:52:12,960 --> 00:52:19,320
+طيب المفروض ان هذا واضح خلّينا الان نتطلع عليه
+
+587
+00:52:19,320 --> 00:52:25,060
+عندى اذا ال summation لل XK minus XK minus واحد K
+
+588
+00:52:25,060 --> 00:52:30,860
+element بيطلعلي اصغر او يساوي واحد على دلتا في ال
+
+589
+00:52:30,860 --> 00:52:39,040
+U ناقص L بس هذا انا مخزنه لسه عندى ال U ناقص Lاللي
+
+590
+00:52:39,040 --> 00:52:40,940
+هو هذا كله على الـ partition اللي بنشتغل عليه اللي
+
+591
+00:52:40,940 --> 00:52:45,860
+لاجيناه في الأول الـU-L هذه أصغر من مين طلعت أصلا
+
+592
+00:52:45,860 --> 00:52:48,880
+من دلتة تربيع لأن هذا اعتمدنا عليه على
+
+593
+00:52:48,880 --> 00:52:53,660
+الـintegrability للـF الأول إذا لازمتني هنا اللي
+
+594
+00:52:53,660 --> 00:52:57,800
+هو هذا أصغر من دلتة تربيع في واحد على دلتة بيصير
+
+595
+00:52:57,800 --> 00:53:04,160
+أصغر من مين؟ من دلتة وهذا كلهفى ضوء الـ Delta اللى
+
+596
+00:53:04,160 --> 00:53:08,060
+أنا بدأت فيها أصغر من مين أو أثبتتها أصغر من
+
+597
+00:53:08,060 --> 00:53:11,860
+Epsilon إبراهيم اللى وجهتها أصغر من Epsilon
+
+598
+00:53:11,860 --> 00:53:16,340
+إبراهيم إذا صار عندى ال summation هذا أصغر من Ash
+
+599
+00:53:16,340 --> 00:53:22,280
+من Epsilon إبراهيم لمين؟ لل K اللى Ash في B إذا
+
+600
+00:53:22,280 --> 00:53:25,340
+اللى وصلت إليه الآن ال summation
+
+601
+00:53:28,540 --> 00:53:32,980
+الـ summation اللي وصلت إليه الـ summation للـ xk
+
+602
+00:53:32,980 --> 00:53:41,600
+ناقص xk minus 1 k element in B أصغر من إبسلون
+
+603
+00:53:41,600 --> 00:53:46,920
+إبراهيم الآن
+
+604
+00:53:46,920 --> 00:53:53,760
+بس إحنا أثبتنا إن الـ summation هذا أيه أصغر أو
+
+605
+00:53:53,760 --> 00:54:00,430
+يساوي 2kاتنين كيف ال summation هذا مظبوط بدي اعوض
+
+606
+00:54:00,430 --> 00:54:06,850
+الان اشيل هذا بيصير عندي هاي شايفين يا شباب هاي
+
+607
+00:54:06,850 --> 00:54:14,070
+عند ال summation هذا اثبته انه اصغر او يساوي اللي
+
+608
+00:54:14,070 --> 00:54:24,580
+هو epsilon prime واضح ومن اربعةبصير عندى بعوض ال
+
+609
+00:54:24,580 --> 00:54:28,980
+summation أصغر من 2 كيف ال summation اللى أثبتناه
+
+610
+00:54:28,980 --> 00:54:34,840
+بحيته بس قبل شوية ال summation أصغر من 2 كيف ال
+
+611
+00:54:34,840 --> 00:54:40,140
+summation بشيل هذا اللى هو هذا و أقول أصغر من مين
+
+612
+00:54:40,140 --> 00:54:43,760
+من epsilon prime بصير هذا المقدار اللى هو هذا ال
+
+613
+00:54:43,760 --> 00:54:45,680
+summation هيه
+
+614
+00:54:47,550 --> 00:54:54,290
+هيو هذا هيو أصغر من من 2k في ash في ال epsilon
+
+615
+00:54:54,290 --> 00:55:00,110
+prime إذا طلع عندي الجزء الثاني summation خلصنا من
+
+616
+00:55:00,110 --> 00:55:07,610
+هذا صار عندي الجزء الثاني اللي أثبته summation لل
+
+617
+00:55:07,610 --> 00:55:19,240
+mk تلدى ناقص mk تلدىفي xk-xk-1 k element in B أصغر
+
+618
+00:55:19,240 --> 00:55:26,980
+من مين طلع من epsilon prime في 2k ابسلون برايم في
+
+619
+00:55:26,980 --> 00:55:36,650
+2k الآن خلصنااحسبولي إذا احنا لقينا ال partition B
+
+620
+00:55:36,650 --> 00:55:44,330
+لكل option أكبر من 0 لقينا B بحيث أنه U B of I
+
+621
+00:55:44,330 --> 00:55:54,210
+composite F ناقص ال B of I composite F هذا أشماله
+
+622
+00:55:54,210 --> 00:55:59,310
+بساوياللي هو الـ summation اللي هيضع على كل الـ K
+
+623
+00:55:59,310 --> 00:56:02,390
+و الـ summation اللي هيضع على كل الـ K و الـ K
+
+624
+00:56:02,390 --> 00:56:05,910
+جزّانها لجزين إيش في الـ A و إيش في الـ B إذا
+
+625
+00:56:05,910 --> 00:56:14,110
+بيساوي اللي هو الـ summation ل الـ A الـ MK ناقص
+
+626
+00:56:14,110 --> 00:56:23,140
+MK تلدة تلدةفي xk-xk-1 k من واحد ل عند n صح ولا لأ
+
+627
+00:56:23,140 --> 00:56:28,220
+اه طبعا هو التعريف هذا اللي هو بيثاويالان الـ K
+
+628
+00:56:28,220 --> 00:56:32,460
+اللي عندك جزّالت لجزين واحد في الـ A واحد في مين؟
+
+629
+00:56:32,460 --> 00:56:37,080
+في الـ B، إذاً هذا الـ summation الكل بيساوي ال
+
+630
+00:56:37,080 --> 00:56:41,340
+summation على K في الـ A زائد ال summation نفسه
+
+631
+00:56:41,340 --> 00:56:45,720
+على K وين ماله؟ في الـ B ال summation لمين؟ لهذه
+
+632
+00:56:45,720 --> 00:56:53,280
+اللي هي MK تلد ناقص MK تلد small في XK minus XK
+
+633
+00:56:53,280 --> 00:57:01,450
+minus واحد زائد MKتلدة ناقص mk small في xk minus
+
+634
+00:57:01,450 --> 00:57:08,990
+xk minus واحد اه طيب يعني الان هذا ال U و هذا ال L
+
+635
+00:57:08,990 --> 00:57:12,030
+صار بساوي هذا الجزء و هذا الجزء و هذا السبب أصلا
+
+636
+00:57:12,030 --> 00:57:17,590
+اللي خلانه أجزء هذا طلع لي الان مين هو طلع لي أصغر
+
+637
+00:57:17,590 --> 00:57:22,970
+من epsilon prime أصغر من epsilon prime في b minus
+
+638
+00:57:22,970 --> 00:57:32,330
+aوالتاني طلع أصغر من E' في 2K E' في 2K يعني هدول
+
+639
+00:57:32,330 --> 00:57:40,140
+الاتنين مع بعض إيش بيساون؟ بيساون E' فيه2k بيزاق
+
+640
+00:57:40,140 --> 00:57:56,220
+بيزاق بيزاق بيزاق بيزاق بيزاق بيزاق
+
+641
+00:57:56,310 --> 00:58:04,950
+على b minus a زي اتنين k في اتنين k زي b minus a
+
+642
+00:58:04,950 --> 00:58:10,030
+وهذه بتروح مع حد بيساوي ايش ابسلون اذا اللي وصلتله
+
+643
+00:58:11,120 --> 00:58:17,600
+لكل إبسلون أكبر من سفر لكل
+
+644
+00:58:17,600 --> 00:58:22,480
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+645
+00:58:22,480 --> 00:58:25,660
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+646
+00:58:25,660 --> 00:58:27,320
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+647
+00:58:27,320 --> 00:58:28,360
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+648
+00:58:28,360 --> 00:58:32,200
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+649
+00:58:32,200 --> 00:58:34,180
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+650
+00:58:34,180 --> 00:58:36,880
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+651
+00:58:36,880 --> 00:58:47,340
+إبسلون أكبر من سفرطيب الآن نيجي بدنا اللي هو ناخد
+
+652
+00:58:47,340 --> 00:58:55,660
+اللي هو تطبيقات على هذه النظرية ونشوف
+
+653
+00:58:55,660 --> 00:59:02,600
+كيف بدنا نطبق هذه النظرية للوصول إلى نتاج أخرى
+
+654
+00:59:02,600 --> 00:59:09,280
+تتعلق بخواص التكامل أو بخواص تكامل لريمان لأن
+
+655
+00:59:09,280 --> 00:59:15,300
+عرفناو جوابنا على اللي هو المجموع وعلى حسب ضرب K
+
+656
+00:59:15,300 --> 00:59:20,100
+في ال .. في ال F ولقنا على ال composition و تحت اي
+
+657
+00:59:20,100 --> 00:59:24,360
+ضارف كان ف I composite F is integrable الان نكمل
+
+658
+00:59:24,360 --> 00:59:30,120
+اللي هو ناخد اللي هو ال corollary اللي عندي اللي
+
+659
+00:59:30,120 --> 00:59:39,150
+هي بتقولي اللي هي من تلت فروع لو كانتI عبارة عن
+
+660
+00:59:39,150 --> 00:59:43,990
+الـ closed interval A و B و F من I لعند R is
+
+661
+00:59:43,990 --> 00:59:47,370
+integrable on I then the absolute value of a
+
+662
+00:59:47,370 --> 00:59:50,630
+function F obtained by the absolute value is
+
+663
+00:59:50,630 --> 00:59:54,410
+integrable on Iوالـ absolute value of integration
+
+664
+00:59:54,410 --> 00:59:56,470
+أصغر يساوي الـ integration للـ absolute value للـ
+
+665
+00:59:56,470 --> 01:00:02,810
+F اللي هي أصغر يساوي K K في ال B minus A حيث ال K
+
+666
+01:00:02,810 --> 01:00:06,750
+هذه من أين جاية هي ال bound لل F of X لأن F is
+
+667
+01:00:06,750 --> 01:00:12,260
+integrable إذا أكيد boundedفن element in F يعني
+
+668
+01:00:12,260 --> 01:00:15,280
+اللي بتقولي هذا باختصار ده كانت F is integrable ال
+
+669
+01:00:15,280 --> 01:00:18,100
+absolute value لل F إيش ما لها integrable وال
+
+670
+01:00:18,100 --> 01:00:20,160
+absolute value لل integration أصغر سوى ال
+
+671
+01:00:20,160 --> 01:00:22,860
+integration لل absolute value أصغر سوى K اللي هي
+
+672
+01:00:22,860 --> 01:00:26,580
+ال maximum لل أو خلّيني أقول ال bound لل F of X
+
+673
+01:00:26,580 --> 01:00:31,080
+absolute value ل F of X في B minus Aالجزء الثاني
+
+674
+01:00:31,080 --> 01:00:35,460
+اللي هو هتطلع لي برضه الـ Fn is integrable لأن Fn
+
+675
+01:00:35,460 --> 01:00:41,560
+هو إنتجر
+
+676
+01:00:42,100 --> 01:00:46,780
+الآن إذا كان الشغل الثاني إذا كان فيه Delta بحيث
+
+677
+01:00:46,780 --> 01:00:50,400
+أن F of X أكبر يساوي Delta يعني F of X أكبر أو
+
+678
+01:00:50,400 --> 01:00:53,900
+يساوي Delta أكبر يساوي Delta ليش؟ على أساس إنه
+
+679
+01:00:53,900 --> 01:00:58,140
+يضمن مقلوب هيكون bounded بيصير 1 على F of X أصغر
+
+680
+01:00:58,140 --> 01:01:00,680
+أو يساوي 1 على Delta يعني بمعنى أكتر F is bounded
+
+681
+01:01:00,680 --> 01:01:04,720
+إذن بحقله يحكي عن ال Integrability ل1 على F ويقول
+
+682
+01:01:04,720 --> 01:01:07,780
+لي تحت الظرف هذا لو كانت F of X أكبر يساوي Delta
+
+683
+01:01:08,470 --> 01:01:12,210
+وDelta أكبر من 0 هيكون 1 على الـ F إيه اشمالها is
+
+684
+01:01:12,210 --> 01:01:15,830
+integrable on I خلّيني أتبرجح واحدة واحدة اللي
+
+685
+01:01:15,830 --> 01:01:24,690
+عنها ال gate F عندي من I لعين بR is integrable إذا
+
+686
+01:01:24,690 --> 01:01:29,350
+there exists K اللي هو أكبر من 0 such that
+
+687
+01:01:29,350 --> 01:01:33,330
+absolute value of F of Xأصغر أو يساوي إياش K مدام
+
+688
+01:01:33,330 --> 01:01:36,430
+ان تقرأ بالـ F إذا أكيد is bounded إذا ال absolute
+
+689
+01:01:36,430 --> 01:01:43,030
+value of X أصغر أو يساوي ال K طيب الآن يا جماعة
+
+690
+01:01:43,030 --> 01:01:47,230
+عندي هاد ال corollary أصلا اللي جابلها بدي أظبطلي
+
+691
+01:01:47,230 --> 01:01:54,070
+two functions بدي أعرف الآن صارت عند ال F of I ال
+
+692
+01:01:54,070 --> 01:02:00,230
+F of I صارت ال F of I أكيد subset بين نقص K ومين
+
+693
+01:02:00,230 --> 01:02:01,210
+أو K
+
+694
+01:02:07,180 --> 01:02:19,960
+طيب يعني الآن لو جيت عرفت Phi من اللي هو عنده ناقص
+
+695
+01:02:19,960 --> 01:02:27,990
+K لعند ال K لعند ال R عرفتها على أساسفاي ما هو
+
+696
+01:02:27,990 --> 01:02:30,950
+absolute value إذا مضحكت تجيب ال absolute value
+
+697
+01:02:30,950 --> 01:02:38,250
+فاي of T بساوة absolute value لمن؟ لل T ماشي الحال
+
+698
+01:02:38,250 --> 01:02:42,430
+هي الدلال عندى أكيد ال absolute value إيش ما لها
+
+699
+01:02:42,430 --> 01:02:47,370
+is continuous مدام ال absolute value الفاي is
+
+700
+01:02:47,370 --> 01:02:52,870
+continuous و ال F معطيله إيها integrable أيضا by
+
+701
+01:02:52,870 --> 01:03:00,670
+the above theoremفاي كومبوزيت F is integrable فاي
+
+702
+01:03:00,670 --> 01:03:07,610
+كومبوزيت F of T بيساوي فاي of F of T اللي هي
+
+703
+01:03:07,610 --> 01:03:12,330
+بتساوي ايش؟ واحدة absolute value فاي of T بتساوي
+
+704
+01:03:12,330 --> 01:03:17,490
+absolute value لمين؟ للـ F of T إذا شار عندي
+
+705
+01:03:17,490 --> 01:03:23,460
+absolute value للـ F is integrableإذاً مدامة Phi
+
+706
+01:03:23,460 --> 01:03:27,460
+is continuous و F integrable لأن حسب النظرية Phi
+
+707
+01:03:27,460 --> 01:03:30,640
+composite F is integrable و Phi composite F هي
+
+708
+01:03:30,640 --> 01:03:33,400
+طلعت من الـ absolute value للـ F طبعا Phi
+
+709
+01:03:33,400 --> 01:03:36,880
+composite F من وين هتشتغل من عند اللي هي ال
+
+710
+01:03:36,880 --> 01:03:42,920
+interval I لعند مين لعند R لأن Phi اللي هو F of T
+
+711
+01:03:42,920 --> 01:03:49,980
+هيجي يقعد في الفترة هذه و Phi هترسل اللي بيجي لوين
+
+712
+01:03:49,980 --> 01:03:55,250
+لعند Iماضح ها طيب الان عندي بدي أثبت أن ال
+
+713
+01:03:55,250 --> 01:03:58,590
+integration ال absolute value ال integration أصغر
+
+714
+01:03:58,590 --> 01:04:01,350
+ساوي ال integration ل absolute value هذه صارت
+
+715
+01:04:01,350 --> 01:04:07,670
+بعضها معلومات صادقة اللي هو عندي ال F أكبر أو ساوي
+
+716
+01:04:07,670 --> 01:04:11,610
+اللي هي absolute value لل F بالسالب و أصغر أو ساوي
+
+717
+01:04:11,610 --> 01:04:14,790
+ال absolute value ل 100 لل F صارت الان احنا أتبعت
+
+718
+01:04:14,790 --> 01:04:17,990
+أن ال absolute value is integrableإذا الان بما أن
+
+719
+01:04:17,990 --> 01:04:21,270
+F بين هذولة الدالتين حسب نظريها أخدناها المرة
+
+720
+01:04:21,270 --> 01:04:24,910
+الماضية أو remark إذا بيصير ال integration لل
+
+721
+01:04:24,910 --> 01:04:28,590
+absolute value لل F من A ل B أصغر أساوي ال
+
+722
+01:04:28,590 --> 01:04:32,330
+integration لل F من A ل B أصغر أساوي ال
+
+723
+01:04:32,330 --> 01:04:38,190
+integration لل F من A ل B وهذا إيش معناته؟ هذا
+
+724
+01:04:38,190 --> 01:04:43,610
+معناته إذا ال absolute value لل integration لل F
+
+725
+01:04:43,610 --> 01:04:49,150
+من A ل B أصغر أساوياللي هو ال integration لأ لأبسل
+
+726
+01:04:49,150 --> 01:04:57,690
+و لأ لأف من A لB ماشي الحالة و طبيعي عندي هذه
+
+727
+01:04:57,690 --> 01:05:02,270
+أثبتناها بهذه و طبيعي مدام عنا لسه ما
+
+728
+01:05:07,890 --> 01:05:12,470
+أو ملاحظة سابقة لبس الـ integration من a لb لبس ال
+
+729
+01:05:12,470 --> 01:05:18,070
+value of f of x أصغر أو شاوي اللي هو من k في طول
+
+730
+01:05:18,070 --> 01:05:27,050
+الفترة بـ-aمع هذه together بنحصل على اللي هو
+
+731
+01:05:27,050 --> 01:05:30,750
+المطلوب ال absolute value ال integration أصغر سوا
+
+732
+01:05:30,750 --> 01:05:34,690
+ال integration لل absolute value أصغر سوا K في ال
+
+733
+01:05:34,690 --> 01:05:39,510
+B minus A هذه اللي هو الجزء الأول من ال corollary
+
+734
+01:05:39,510 --> 01:05:44,310
+خلينا نشوف الجزء الثاني من ال corollary برضه اللي
+
+735
+01:05:44,310 --> 01:05:51,270
+هو برهانه سهل و برهانه اللي هو مشابهللمنطق اللي
+
+736
+01:05:51,270 --> 01:05:57,070
+حكيناها عندنا قبل بشوية طيب صلى الله عننا يا عزيزي
+
+737
+01:05:57,070 --> 01:06:01,410
+عليه الصلاة والسلام الآن يا شباب عندي اللي هو
+
+738
+01:06:01,410 --> 01:06:05,110
+الجزء الثاني بدنا نثبت أن الـ F أسئن is integrable
+
+739
+01:06:05,110 --> 01:06:09,110
+في حالة الـ F integrable لأن أكيد شكلنا بدنا نيجي
+
+740
+01:06:09,110 --> 01:06:16,950
+عند الـ F من I لعند R is integrableالان عندي اللي
+
+741
+01:06:16,950 --> 01:06:21,470
+هو .. بدي أسأل أن أثبت انها Integrable فلو جيت أنا
+
+742
+01:06:21,470 --> 01:06:30,170
+عرفت F F أو اللي هو Phi function من اللي هو domain
+
+743
+01:06:30,170 --> 01:06:35,150
+اللي هو F of I من اللي هو range ال F range ال F
+
+744
+01:06:35,150 --> 01:06:45,640
+range ال F لعند ال R بحيث ان اقول Phi of Tبساوي T
+
+745
+01:06:45,640 --> 01:06:51,120
+أُس أنت five of T أيش بيساوي بيساوي T أُس أنت أو
+
+746
+01:06:51,120 --> 01:06:54,180
+إذا كان بدك اللي هو بدكاش تكون ال range ل F خلاص
+
+747
+01:06:54,180 --> 01:06:57,460
+من عارفها من وين من اللي من اللي قبل بشوية من ناقص
+
+748
+01:06:57,460 --> 01:07:01,980
+K لعند K عارفين ليه؟ لأن ال absolute value لل F of
+
+749
+01:07:01,980 --> 01:07:06,840
+X أصغر مش هو ال K إذا أكيد لما ترمي ال F of X
+
+750
+01:07:06,840 --> 01:07:11,410
+هتكون ترمي في الفترة من ناقص K لعندكأو subset
+
+751
+01:07:11,410 --> 01:07:15,970
+بيصير الـ F of I subset من ناقص K و K إذاً صارت
+
+752
+01:07:15,970 --> 01:07:22,510
+الـ Phi composite F Phi composite F is defined من
+
+753
+01:07:22,510 --> 01:07:31,140
+I لعند ال R وعندي Phi composite F ofT بيساوي Phi
+
+754
+01:07:31,140 --> 01:07:37,200
+of F of T ويساوي F of T اللي هو Phi of T بيساوي T
+
+755
+01:07:37,200 --> 01:07:41,560
+أُس N بيصير Phi of F of T اللي هي عبارة عن F of T
+
+756
+01:07:41,560 --> 01:07:48,540
+يا جماعة أُس N الآن كل الأمور طيبة ومنيحة ليش
+
+757
+01:07:48,540 --> 01:07:53,260
+ووصلنا كمانعندي F-integrable و F-continuous
+
+758
+01:07:53,260 --> 01:07:57,420
+هتتدللوا الـ C أشمالها اللي هو الـ T أسان is
+
+759
+01:07:57,420 --> 01:08:03,540
+continuous اللي جزء من البلونوميال بيصير عندي FI
+
+760
+01:08:03,540 --> 01:08:06,660
+is continuous و F-integrable لذلك حسب النظرية اللي
+
+761
+01:08:06,660 --> 01:08:10,680
+قبل و شوية إيش هيطلع عندي FI composite F is
+
+762
+01:08:10,680 --> 01:08:16,440
+integrable بمعنى أن ف FN is integrable فكرة معادة
+
+763
+01:08:16,440 --> 01:08:22,570
+أصلالأنه سنة يعني اللي هو الأمور واضحة الآن ضال
+
+764
+01:08:22,570 --> 01:08:30,190
+عند مين اللي هو ال .. ال .. ال .. ال .. نثبت الجزء
+
+765
+01:08:30,190 --> 01:08:36,010
+الثالث من ال corollary اللي هي بدنا نثبت أنه لو
+
+766
+01:08:36,010 --> 01:08:43,490
+كانت ال F of X أكبر أو شويه واحد على دلتااللي هي
+
+767
+01:08:43,490 --> 01:08:47,530
+for every x element in I وعندي there exists دلتا
+
+768
+01:08:47,530 --> 01:08:52,390
+كمان وصفر ساشرات إذا كان هذا متحقق إذن هتطلع عندى
+
+769
+01:08:52,390 --> 01:08:55,970
+اللي هي الواحد على ال F بيدثبت أن الواحد على ال F
+
+770
+01:08:55,970 --> 01:08:58,950
+إيش is integrable بدنا نعمل .. نجيب continuous
+
+771
+01:08:58,950 --> 01:09:02,890
+function أكيد كل جمجال نجيب مقلوبة دلة يعني في حوض
+
+772
+01:09:02,890 --> 01:09:09,800
+five من ناقص K لعند Kلعند R وخد فاي عارفين ما هو
+
+773
+01:09:09,800 --> 01:09:13,000
+ال scale كده سبب اللي قلت قبل شوية فايف دي ايش
+
+774
+01:09:13,000 --> 01:09:17,600
+متساوي اكيد كلكم حيقول واحد على تي ماشي الحال واحد
+
+775
+01:09:17,600 --> 01:09:26,180
+على تي بس اه لأ عدم ديش اقدر اعمل hand افعل عشان
+
+776
+01:09:26,180 --> 01:09:31,250
+مانبدعش في السفرة يا شباب خدوها من عند Deltaلعند
+
+777
+01:09:31,250 --> 01:09:36,470
+مين؟ لعند K ليش .. ليش .. ليش اللي هو عكيفك؟ أه F
+
+778
+01:09:36,470 --> 01:09:40,670
+of X اللي هو absolute value أصغر أو تسوى K يعني F
+
+779
+01:09:40,670 --> 01:09:46,160
+of X أصغر أو تسوى K وأكبر أو تسوى نقص Kو أثناء هذه
+
+780
+01:09:46,160 --> 01:10:12,580
+أكيد ماعطينيها أكبر شو مين دلتا
+
+781
+01:10:14,710 --> 01:10:21,230
+اللي هو .. إيه الكلام هذا؟ مشروع هو بيصير عنده ..
+
+782
+01:10:21,230 --> 01:10:24,170
+اللي هو Phi of T بسواحدة على T مافيش أي مشكلة،
+
+783
+01:10:24,170 --> 01:10:27,470
+مافيش مشاكل أسفار، مافيش مشاكل كده، إذا صارت ده
+
+784
+01:10:27,470 --> 01:10:31,490
+اللي إيش مالها؟ continuous continuous هفهم إن فهمت
+
+785
+01:10:31,490 --> 01:10:34,810
+القصة Phi composite of F continuous و integrable
+
+786
+01:10:34,810 --> 01:10:38,310
+إذا كلها على بعضها integrable يعني Phi of T بتصير
+
+787
+01:10:38,310 --> 01:10:44,750
+Phiof F of T يعني بتساوي واحد على F of T يعني صارت
+
+788
+01:10:44,750 --> 01:10:51,650
+الدالة واحد على F is integrable وهك
+
+789
+01:10:51,650 --> 01:10:55,670
+مكون احنا خلصنا اللي هو ال corollary ضال عندى الآن
+
+790
+01:10:55,670 --> 01:11:01,570
+نجاوب على السؤال الثاني هل حاصل ضرب ضرب دالتين
+
+791
+01:11:01,570 --> 01:11:06,650
+integrable is integrable بنقول اه integrable يعني
+
+792
+01:11:06,650 --> 01:11:11,380
+بمعنى اخر بقول ليهالنظرية اللي هي اللي بعدها 7,2,7
+
+793
+01:11:11,380 --> 01:11:18,020
+بتقول ما يلي بتقول لو كانت F و G integrable يعني
+
+794
+01:11:18,020 --> 01:11:25,480
+لو كان عندي خليني اكتب بالأصمر أفضل F و G من I
+
+795
+01:11:25,480 --> 01:11:33,700
+لعند R كانت integrable functions بيعطيني هذا ان FG
+
+796
+01:11:33,700 --> 01:11:42,500
+من I لعند R برضه ايش ماله؟Integrable Function الـ
+
+797
+01:11:42,500 --> 01:11:46,740
+L صار إن حصيلة من المعلومات بتسهل علي الوصول
+
+798
+01:11:46,740 --> 01:11:53,820
+للنتيجة و Proof و Proof بما أن F is Integrable إذن
+
+799
+01:11:53,820 --> 01:11:57,700
+فال Corollary اللي قبل بشوية أكيد F تربيع اللي هي
+
+800
+01:11:57,700 --> 01:12:01,040
+Integrable و G Integrable من النظرية اللي قبل
+
+801
+01:12:01,040 --> 01:12:04,940
+بشوية و Corollary برضه إشمالها G تربيع برضه
+
+802
+01:12:04,940 --> 01:12:10,060
+Integrable صح ولا لا يا جماعة صحطيب F تربيع
+
+803
+01:12:10,060 --> 01:12:14,260
+Integrable و G تربيع Integrable و برضه من نظرية
+
+804
+01:12:14,260 --> 01:12:18,580
+سابقة مدام F و G Integrable إذا F زائد G برضه إيش
+
+805
+01:12:18,580 --> 01:12:24,980
+Integrable لأ F زائد G تربيع Integrable كمان لأن F
+
+806
+01:12:24,980 --> 01:12:27,800
+و G Integrable أدت ل F زائد G Integrable و F زائد
+
+807
+01:12:27,800 --> 01:12:31,640
+G Integrable أدت من القرول اللي قبل شوية أنه تربيح
+
+808
+01:12:31,640 --> 01:12:40,520
+يكون إيش Integrable طيب خلصنا إذا إذاF تربيع زائد
+
+809
+01:12:40,520 --> 01:12:49,780
+D تربيع is integrable صح؟ مظبوط وناقص F تربيع
+
+810
+01:12:49,780 --> 01:12:54,220
+وناقص D تربيع برضه integrable مظبوط برضهلأنه اللي
+
+811
+01:12:54,220 --> 01:13:00,060
+هو عندي اللي هو ثابت في هذه integrable و ثابت في
+
+812
+01:13:00,060 --> 01:13:03,560
+ال integrable integrable ومجموح انصار انتجرابل إذا
+
+813
+01:13:03,560 --> 01:13:09,020
+انصار هذا integrable زائد F زائد G لكل تربيع هذه
+
+814
+01:13:09,020 --> 01:13:12,980
+integrable وهذه integrable وهذه integrable مجموحين
+
+815
+01:13:12,980 --> 01:13:17,520
+هذا إيش بيساويهذا .. هذا integrable و هذا
+
+816
+01:13:17,520 --> 01:13:19,860
+integrable و هذا integrable إذا المجموع integrable
+
+817
+01:13:19,860 --> 01:13:23,220
+إذا هذا كله على بعض integrable طب هذا مين هو؟ هذا
+
+818
+01:13:23,220 --> 01:13:29,160
+عبارة عن F تربيع زاد G تربيع ناقص اللي هو إيش زاد
+
+819
+01:13:29,160 --> 01:13:35,760
+2FG بيصير عبارة عن 2F main G صارت 2FG integrable
+
+820
+01:13:35,760 --> 01:13:40,840
+طب لو جينا قولنا خل نص هنا يعني ضربنا ثابت في
+
+821
+01:13:40,840 --> 01:13:43,960
+integrable إذا هيطلع اللي هو كله integrable إذا FG
+
+822
+01:13:43,960 --> 01:13:48,520
+إيش ما لهاis integrable صار عندي الان fg
+
+823
+01:13:48,520 --> 01:13:54,040
+integrable تابعا لأن f تربيع و g تربيع و f زائد g
+
+824
+01:13:54,040 --> 01:13:59,860
+كل تربيع حاصل جمحين و ضرب النصفي هنا ثابت و بيطلع
+
+825
+01:13:59,860 --> 01:14:04,720
+عبارة عن integrable function نيجي لآخر اللي هو
+
+826
+01:14:04,720 --> 01:14:11,850
+نقطة في الفي ال .. في ال .. في ال section 7-2 اللي
+
+827
+01:14:11,850 --> 01:14:14,790
+هو السؤال اللي سألناه في الأول قولنا لو كانت F is
+
+828
+01:14:14,790 --> 01:14:19,570
+integrable function و Phi is integrable هل Phi
+
+829
+01:14:19,570 --> 01:14:24,310
+composite F is integrable قولنا أكيد هو لو .. في
+
+830
+01:14:24,310 --> 01:14:28,430
+البداية قولنا أن Phi composite F need not to be
+
+831
+01:14:28,430 --> 01:14:32,280
+integrableإذا the composition of two integrable
+
+832
+01:14:32,280 --> 01:14:35,720
+functions need not to be integrable يعني هذا إعلان
+
+833
+01:14:35,720 --> 01:14:38,340
+the composition of two integrable functions need
+
+834
+01:14:38,340 --> 01:14:42,420
+not to be integrable but if phi the first one is
+
+835
+01:14:42,420 --> 01:14:46,340
+continuous then phi composite of f is integrable
+
+836
+01:14:46,340 --> 01:14:50,620
+كما شفنا في العقل اللي هو النظرية الأولى طيب نيجي
+
+837
+01:14:50,620 --> 01:14:53,000
+الآن المثال الأخير بيقولنا يا جماعة the
+
+838
+01:14:53,000 --> 01:14:55,760
+composition of integrable functions need not to be
+
+839
+01:14:55,760 --> 01:14:59,540
+integrableفي عندى سؤالين اصلا معاكم homework
+
+840
+01:14:59,540 --> 01:15:03,860
+السؤال الاول بيقوللنا لو كانت f of x بتساوي واحد
+
+841
+01:15:03,860 --> 01:15:07,540
+ده كانت x بتساوي سفر و zero ده كانت x is
+
+842
+01:15:07,540 --> 01:15:11,760
+irrational و كانت f of x بتساوي واحد لما x بيساوي
+
+843
+01:15:11,760 --> 01:15:16,160
+m على n حيث ال m و ال n عبارة عن integers و العامل
+
+844
+01:15:16,160 --> 01:15:19,740
+المشترك الاعلى بينهم ايه اشمله بادر بيساوي واحد
+
+845
+01:15:19,740 --> 01:15:23,540
+يعني شولنا كل العامل المشترك اللى بينهم و كتبنا x
+
+846
+01:15:23,540 --> 01:15:28,900
+بتساوي m على nطيب، هذه عبارة عن دالة معرفة من الـ0
+
+847
+01:15:28,900 --> 01:15:32,460
+والـ1 لعند الـR يعني هذي X is rational well في
+
+848
+01:15:32,460 --> 01:15:36,320
+الفترة 0 و1 و X بيبقى بيساوي M على N يعني rational
+
+849
+01:15:36,320 --> 01:15:41,260
+في الفترة 0 و1 وفرضنا لل X بتساوي صفر قيمته إيش
+
+850
+01:15:41,260 --> 01:15:48,070
+بتساوي؟ بتساوي 1 هذي الآن by exercise 7.1.11مطلوب
+
+851
+01:15:48,070 --> 01:15:51,790
+منك أنك تثبت أن F is يشملها is integrable on I
+
+852
+01:15:51,790 --> 01:15:55,070
+ويوم ما تعرفوش تحلوه ان شاء الله بنحلوه و بنصوره
+
+853
+01:15:55,070 --> 01:16:00,030
+بإذن الله الآن السؤال التاني اللي هعتمد عليه برضه
+
+854
+01:16:00,030 --> 01:16:03,170
+أنه ال function التاني اللي هي G من I لعند R هذه
+
+855
+01:16:03,170 --> 01:16:07,790
+سهلة أصلا لإثباتها be defined by G of X step
+
+856
+01:16:07,790 --> 01:16:12,090
+function بسيطة يعنيأو خلّيني أقول فيها jump و بس
+
+857
+01:16:12,090 --> 01:16:17,030
+jump على نقطة بس جيوب x بيساوي 0 إذا كانت x بيساوي
+
+858
+01:16:17,030 --> 01:16:21,410
+0 و بيساوي 1 إذا كانت x في الفترة من 0 لعند 1 اللي
+
+859
+01:16:21,410 --> 01:16:25,410
+هو a closed عند الواحد يعني الآن هذا ال function
+
+860
+01:16:25,410 --> 01:16:29,110
+بيقوللي برضه في ال exercise مطلوب برضه في exercise
+
+861
+01:16:29,110 --> 01:16:33,090
+717 برضه اللي هو homework معكم في ال exercise
+
+862
+01:16:33,090 --> 01:16:36,270
+بيقولك اثبت ان g ايش معناها is integrable function
+
+863
+01:16:36,920 --> 01:16:40,960
+إذا الـ F والـ G عبارة عن two integrable functions
+
+864
+01:16:40,960 --> 01:16:46,780
+two integrable functions هاي ال F من I لعند R والـ
+
+865
+01:16:46,780 --> 01:16:55,200
+G اللي هو من عند I لعند R لاحظ ان الـ G فيه لغة
+
+866
+01:16:55,200 --> 01:17:00,600
+تانية zero لمين يا واحدعندي هذه فئلها الـ 1 و 0 و
+
+867
+01:17:00,600 --> 01:17:06,800
+في عندنا قيم أخرى اللي هو كثيرة طيب، الآن عندي ..
+
+868
+01:17:06,800 --> 01:17:09,660
+لو جيت حسب الـ G composed of F مش غريبة عليكم
+
+869
+01:17:09,660 --> 01:17:14,480
+الدالة هذه G composed of F of X خلينا نحسبها مع
+
+870
+01:17:14,480 --> 01:17:20,860
+بعض و بنقول خلصنا اللي هو S section أو قولنا ان
+
+871
+01:17:20,860 --> 01:17:24,320
+الـ G composed of F هذه اللي انتوا بتعرفوها انها
+
+872
+01:17:24,320 --> 01:17:36,430
+is not integrableجي كومبوزيت F of اللي هو عندي
+
+873
+01:17:36,430 --> 01:17:46,900
+الجي الـ F of X هيها لاخد جي of اللي هو Zeroبساوي
+
+874
+01:17:46,900 --> 01:17:55,920
+g of f of 0 بساوي g of f of 0 بساوي
+
+875
+01:17:55,920 --> 01:17:57,880
+g of f of 0 بساوي g of f of 0 بساوي g of f of 0
+
+876
+01:17:57,880 --> 01:18:06,220
+بساوي g of f of 0 بساوي g of f of 0 بساوي g of f
+
+877
+01:18:06,220 --> 01:18:07,060
+of 0 بساوي g of f of 0 بساوي g of f of 0 بساوي g
+
+878
+01:18:07,060 --> 01:18:07,140
+of f of 0 بساوي g of f of 0 بساوي g of f of 0
+
+879
+01:18:07,140 --> 01:18:08,100
+بساوي g of f of 0 بساوي g of f of 0 بساوي g of f
+
+880
+01:18:08,100 --> 01:18:09,880
+of 0 بساوي g of f of 0 بساوي g of f of 0 بساوي g
+
+881
+01:18:09,880 --> 01:18:14,160
+of f of 0 بساوي g of f of 0 بساوي g of f of 0
+
+882
+01:18:14,160 --> 01:18:17,110
+of 0 بساوي g of f ofX اللي هو irrational يعني
+
+883
+01:18:17,110 --> 01:18:22,230
+irrational G of irrational اللي هو اللي هو الموجود
+
+884
+01:18:22,230 --> 01:18:25,910
+في الفترة Zero و واحد G of irrational إيش هيساوي
+
+885
+01:18:25,910 --> 01:18:31,290
+حسب التعريف .. التعريف هنا بيساوي G of F of
+
+886
+01:18:31,290 --> 01:18:38,750
+irrational irrational هاي بيساوي اللي هو G of F of
+
+887
+01:18:38,750 --> 01:18:44,240
+irrational إيش بيساوي سفرو G of 0 ايش بيسوي هنا G
+
+888
+01:18:44,240 --> 01:18:52,420
+of 0 بيسوي Zero معرفليها بيسوي ايش صفر ا الثالثة
+
+889
+01:18:54,350 --> 01:18:58,630
+Composite F of مين ضال؟ Of اللي هي irrational
+
+890
+01:18:58,630 --> 01:19:02,530
+البجيات غير الـ Zero هذا أصلا هو جسمها irrational
+
+891
+01:19:02,530 --> 01:19:08,410
+و rational جهتين واحدة واحدة واحدة Zero لحالها و
+
+892
+01:19:08,410 --> 01:19:11,650
+واحدة كل ال rational اللي بين Zero و واحد ما عدا
+
+893
+01:19:11,650 --> 01:19:18,870
+ال Zero اللي هو G of X اللي هي عبارة عن M على N
+
+894
+01:19:19,590 --> 01:19:23,630
+اللي هي rational في الواقع رياشونال كلهم معدن اللي
+
+895
+01:19:23,630 --> 01:19:28,790
+فوق اللي فوق حسبناها طلعت واحد ويساوي G of F of M
+
+896
+01:19:28,790 --> 01:19:34,350
+على N ويساوي G of F of M على N أيش بتساوي واحد على
+
+897
+01:19:34,350 --> 01:19:38,910
+N ويساوي G of واحد على N أيش بتحسبها هذه هذا اصلا
+
+898
+01:19:38,910 --> 01:19:42,070
+الدالة دايما بتساوي واحد معدن عند سفر حتى سفر عشان
+
+899
+01:19:42,070 --> 01:19:46,750
+تعملنا المشكلة G of واحد على N أيش بتساوي واحد هو
+
+900
+01:19:46,750 --> 01:19:53,580
+يساوي واحدمن هذا كله صار عندي g composite f of x
+
+901
+01:19:53,580 --> 01:20:06,710
+بتساوي 0 if x أشمالها is irrational ويساوي 1 1إذا
+
+902
+01:20:06,710 --> 01:20:11,790
+كانت X is هاي ال rational كله ما عدا السفر وهي
+
+903
+01:20:11,790 --> 01:20:16,670
+السفر برضه طلع واحد if X is rational وهدد دالة
+
+904
+01:20:16,670 --> 01:20:20,610
+تبعتنا اللي اعتبتناها المرة الماضية انها is not
+
+905
+01:20:20,610 --> 01:20:24,590
+integrable function لحظوا انتوا بس خليني نقول شوف
+
+906
+01:20:24,590 --> 01:20:32,660
+هالغرابة مش غرابة لأ هو مهم بالكلامإنه اللي خلف
+
+907
+01:20:32,660 --> 01:20:37,380
+الموضوع لو كانت G continuous على كل ال domain على
+
+908
+01:20:37,380 --> 01:20:40,820
+طول الـ G continuous ده هتطلع ان انتجر بالغصب عنها
+
+909
+01:20:40,820 --> 01:20:44,560
+من النظرية اللي قبل شوية لكن اللي خلف الموضوع شغلة
+
+910
+01:20:44,560 --> 01:20:49,440
+واحدة شايفين هالدالة هاد ال G of X الـ G of X G of
+
+911
+01:20:49,440 --> 01:20:55,000
+X بتساوي اللي هو سفر عند السفر ومن عند ال zero
+
+912
+01:20:55,000 --> 01:21:01,130
+لعند الواحد هناالدالة قيمتها إيش بتساوي؟ واحد، هاي
+
+913
+01:21:01,130 --> 01:21:04,270
+قيمتها واحد، بين Zero والواحد قيمتها واحد، هذا
+
+914
+01:21:04,270 --> 01:21:09,540
+الـG of X اللي عندييعني هذا كل حال عالي العالي ما
+
+915
+01:21:09,540 --> 01:21:14,120
+عدا عند من عند السفر فيه jump point هذه النقطة
+
+916
+01:21:14,120 --> 01:21:20,200
+الوحيدة اللي فيها discontinuity هي اللي .. اللي
+
+917
+01:21:20,200 --> 01:21:25,340
+انا بنشرتلي انها تصير decomposed F is continuous و
+
+918
+01:21:25,340 --> 01:21:29,200
+هذا عشان نعرف جداش الرياضيات او جداش التحليل دقيق
+
+919
+01:21:29,780 --> 01:21:35,960
+دقيق ان ان احنا نقطة واحدة .. نقطة واحدة اللي كانت
+
+920
+01:21:35,960 --> 01:21:40,360
+عندها point of discontinuity جالتلي ان الـ
+
+921
+01:21:40,360 --> 01:21:44,680
+decomposite F need not to be integrable وهي المثال
+
+922
+01:21:44,680 --> 01:21:50,720
+أمامكم و .. و هيك بيكون احنا خلصنا ال section
+
+923
+01:21:50,720 --> 01:21:56,730
+التاني من اللي هو chapter 7و هاي ال homework عندكم
+
+924
+01:21:56,730 --> 01:22:02,630
+مطلوبة 1,2,4,7,10,17,18,19 و ان شاء الله المرة
+
+925
+01:22:02,630 --> 01:22:08,950
+القادمة بنكمل و بنشرح 7.3 اللي هو ال fundamental
+
+926
+01:22:08,950 --> 01:22:10,550
+theorem of calculus
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..c96612faa649a2441b934d1c672b6e792202dcaa
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/64kJ1lF4GJg_raw.srt
@@ -0,0 +1,3732 @@
+1
+00:00:04,940 --> 00:00:11,820
+بسم الله الرحمن الرحيم اليوم المحاضرة رقم 13 مساق
+
+2
+00:00:11,820 --> 00:00:17,100
+تحليل حقيقه 2 لطلاب طالبات الجامعة الإسلامية كلية
+
+3
+00:00:17,100 --> 00:00:21,800
+العلوم قسم رياضيات ان شاء الله ستكون المحاضرة على
+
+4
+00:00:21,800 --> 00:00:28,620
+جزئينالجزء الأول هنستمر في الحديث عن سبعة اتنين
+
+5
+00:00:28,620 --> 00:00:34,240
+وهو الحديث أو سبعة تلاتة وهو الحديث سبعة اتنين في
+
+6
+00:00:34,240 --> 00:00:40,130
+البداية وهو الحديث عن الخواص التكامل الريمانحكينا
+
+7
+00:00:40,130 --> 00:00:45,310
+عن اللي هو مجموع دالتين قابلات التكامل اللي هو
+
+8
+00:00:45,310 --> 00:00:48,510
+قابل
+
+9
+00:00:48,510 --> 00:00:52,510
+التكامل وعاصل ضرب ثابت في دالة قابل التكامل برضه
+
+10
+00:00:52,510 --> 00:00:58,630
+قابل التكاملواليوم هنكمل الأمر الحديث عن
+
+11
+00:00:58,630 --> 00:01:04,550
+composition of two integrable functions هل قابل
+
+12
+00:01:04,550 --> 00:01:08,270
+التكامل أم لا وإذا قابل التكامل بدنا نبرهن وإذا مش
+
+13
+00:01:08,270 --> 00:01:14,750
+قابل بدنا نجيه counter exampleالان الجزء الثاني
+
+14
+00:01:14,750 --> 00:01:18,110
+هيكون اللي هو برضه تطبيق اللي هو composition
+
+15
+00:01:18,110 --> 00:01:25,410
+theorem اللي هي هنشوف بعد شوية ايش المتطلب ان يكون
+
+16
+00:01:25,410 --> 00:01:30,190
+composition of two functions is integrableأيضًا
+
+17
+00:01:30,190 --> 00:01:35,230
+هنتحدث عن اللي هو نوظفها في إثبات اللي هو أن
+
+18
+00:01:35,230 --> 00:01:38,950
+الدالة الأسية والـ absolute value of the function
+
+19
+00:01:38,950 --> 00:01:43,530
+F و أيضًا اللي هو مقلوب الدالة 1 على F في حالة F
+
+20
+00:01:43,530 --> 00:01:50,090
+لا تساوي 0 على ال domain المُعطَع إنها تكون قابلة
+
+21
+00:01:50,090 --> 00:01:54,830
+للتكامل و أيضًا اللي هو هيكون في عنا اللي هو تطبيق
+
+22
+00:01:54,830 --> 00:01:59,640
+أخر اللي هوحاصل ضرب دالتين كيف يكون انتجرابل أو
+
+23
+00:01:59,640 --> 00:02:03,080
+كيف تكون انتجرابل في حالة كلا الدالتين انتجرابل
+
+24
+00:02:03,080 --> 00:02:06,200
+نبدأ الآن في اللي هو النظرية الـ composition
+
+25
+00:02:06,200 --> 00:02:09,600
+theorem واطولوا روحكم علينا شوية على اللي هو
+
+26
+00:02:09,600 --> 00:02:14,930
+البرهان و البرهان شوية بده تركيز و البرهانطويل
+
+27
+00:02:14,930 --> 00:02:20,110
+شوية خلّينا ان نعمل الان focusing على نص النظرية و
+
+28
+00:02:20,110 --> 00:02:25,410
+بعدين بنبدأ نحكي عن البرهان نعمل outline للبرهان و
+
+29
+00:02:25,410 --> 00:02:27,610
+من ثم ندخل لتفاصيل البرهان
+
+30
+00:02:30,230 --> 00:02:35,470
+بنأخد I عبارة عن close bounded interval A وB وJ
+
+31
+00:02:35,470 --> 00:02:38,550
+عبارة عن close bounded interval سميناها C وD
+
+32
+00:02:38,550 --> 00:02:44,830
+ونفترض أن F من I لعند R يعني F عبارة عن دلة من عند
+
+33
+00:02:44,830 --> 00:02:49,350
+ال A و ال B لعند R أنها تكون integrable on I and
+
+34
+00:02:49,350 --> 00:02:52,050
+Phi من J لعند R
+
+35
+00:02:55,070 --> 00:02:58,390
+النظرية بتستلزم أن نقول continuous لأن ال Integra
+
+36
+00:02:58,390 --> 00:03:03,270
+بالحالها مش هتعطي اللي هو النتيجة زي ما هنشوف قدام
+
+37
+00:03:03,270 --> 00:03:07,610
+في اللي هو counter example الآن فرضنا أن فايل من J
+
+38
+00:03:07,610 --> 00:03:12,890
+لعند R is continuous وبدنا نفترض أن F of I .. F of
+
+39
+00:03:12,890 --> 00:03:17,450
+I جزئية من مين؟ من J عشان نعرف .. نعرف ال
+
+40
+00:03:17,450 --> 00:03:21,670
+composition بين ال two functions الآن في ضوء هذه
+
+41
+00:03:21,670 --> 00:03:26,050
+المعطياتإن الـ function F is integrable و الـ
+
+42
+00:03:26,050 --> 00:03:30,410
+function Phi is continuous لازم يطلع عندي الآن Phi
+
+43
+00:03:30,410 --> 00:03:37,450
+composite F is integrable on mean on I إذن F من I
+
+44
+00:03:37,450 --> 00:03:43,130
+لR انتجرابل Phi من J لعند R continuous الآن Phi
+
+45
+00:03:43,130 --> 00:03:46,170
+composite J continuous composite integrable
+
+46
+00:03:46,170 --> 00:03:53,500
+هيعطيني انتجرابل function on I الآن بدنا نثبتالأن
+
+47
+00:03:53,500 --> 00:04:00,600
+فاي composite F من I لعند R انها is integrable
+
+48
+00:04:00,600 --> 00:04:04,900
+الان كيف بدى أثبتها؟ بدى أثبتها .. بدى ألاقي
+
+49
+00:04:04,900 --> 00:04:09,040
+partition .. ده أقول لكل يبسن أكبر من 0 بدلاقي
+
+50
+00:04:09,040 --> 00:04:16,060
+partition B element in B of I such that L of أو U
+
+51
+00:04:16,060 --> 00:04:24,200
+ofB و F هذا اللي لجيته ماعرف أسأل أسف انا فعلا
+
+52
+00:04:24,200 --> 00:04:30,690
+composed of G and Fفاي كومبوزيت F ناقص L of B وفاي
+
+53
+00:04:30,690 --> 00:04:35,730
+كومبوزيت F أن يكون هذا المقدار أصغر من إبسلون إذا
+
+54
+00:04:35,730 --> 00:04:40,590
+وصلت لهذه النتيجة معناته أنه أنا أثبتت أنه اللي هي
+
+55
+00:04:40,590 --> 00:04:46,490
+الفاي كومبوزيت F is integrable بناء على اللي هو ال
+
+56
+00:04:46,490 --> 00:04:48,870
+criterion of integrability اللي حكينا عنها
+
+57
+00:04:48,870 --> 00:04:53,700
+المحاضرة قبل الباضيةالان هذا الهدف اللي بده أصله
+
+58
+00:04:53,700 --> 00:04:58,020
+هيك بده أصل هيك الان عشان أصل هيك بدي اللي هو
+
+59
+00:04:58,020 --> 00:05:02,900
+أستخدم اللي هو المعطيات اللي موجودة عندى الان
+
+60
+00:05:02,900 --> 00:05:08,380
+هستخدم أمرين هستخدم اكيد ال continuity للـ Phiوهي
+
+61
+00:05:08,380 --> 00:05:14,320
+continuous على closed bounded interval إذا حسب
+
+62
+00:05:14,320 --> 00:05:18,080
+نظرية في تحليل واحد هتكون Phi is uniformly
+
+63
+00:05:18,080 --> 00:05:23,880
+continuous وهذا هستغلها في الوصول إلى هدفيالان هذه
+
+64
+00:05:23,880 --> 00:05:27,960
+المعلومة بعد شوية هنخزنها ونحطها في .. في مكان ما
+
+65
+00:05:27,960 --> 00:05:33,420
+لحين نستخدمها مع اللي هو ان F is integrable مزام F
+
+66
+00:05:33,420 --> 00:05:37,240
+is integrable اذا نبلج Partitional I بحيث انه الـ
+
+67
+00:05:37,240 --> 00:05:40,500
+U للـ B والـ F نقص الـ B والـ F أصغر من مين؟ من
+
+68
+00:05:40,500 --> 00:05:44,220
+some .. من الـ Epsilon Epsilon تخدمني، هبدأ أسميها
+
+69
+00:05:44,220 --> 00:05:47,490
+Epsilon، هبدأ أسميها Delta، أنا حورالمهم أكيد مدام
+
+70
+00:05:47,490 --> 00:05:51,610
+الفيه عندي integrability للـ F هتحقق أنه لكل اللي
+
+71
+00:05:51,610 --> 00:05:56,010
+هو Epsilon there exists partition B وحنلاقي أنه ال
+
+72
+00:05:56,010 --> 00:05:59,510
+partition هذا اللي .. اللي نفع للـ F هو اللي هينفع
+
+73
+00:05:59,510 --> 00:06:04,300
+للـ Phi composite Fهنجمج المعلومتين التنتين مع بعض
+
+74
+00:06:04,300 --> 00:06:06,680
+اللي هي ال .. ال .. ال .. ال .. الـ uniform
+
+75
+00:06:06,680 --> 00:06:10,260
+continuity للفعل مع الintegrability للأف للوصول
+
+76
+00:06:10,260 --> 00:06:14,920
+إلى نتيجتنا وهي هذه النتيجة هذه خلينا نقول ال
+
+77
+00:06:14,920 --> 00:06:19,660
+outline للبرهان نبدأ الآن في تفاصيل البرهان وطول
+
+78
+00:06:19,660 --> 00:06:25,400
+روحكم عليها في اللي هو تفاصيل البرهانللوصول
+
+79
+00:06:25,400 --> 00:06:31,780
+للنتيجة اللى حكيتها اللى كتبت على اللوح بدي اقول
+
+80
+00:06:31,780 --> 00:06:39,240
+الان اول حاجة given epsilon أكبر من سفر انا اخدت
+
+81
+00:06:39,240 --> 00:06:44,600
+اي epsilon أكبر من سفر بدي اصل ال U بي فاي
+
+82
+00:06:44,600 --> 00:06:50,610
+composite F ناقص ال بيفاي كومبوزيت F أزر من إبسلون
+
+83
+00:06:50,610 --> 00:06:55,150
+for some اللي هو بارتي شنبي إذا وصلت لهك بكون خلصت
+
+84
+00:06:55,150 --> 00:06:58,770
+اللي هو نظريتي طيب سلموا على النبي عليه الصلاة
+
+85
+00:06:58,770 --> 00:07:03,850
+والسلام اللي لأن فاي عندى فاي عندى continuous على
+
+86
+00:07:03,850 --> 00:07:08,950
+مين على اللي هي ال J ال J عبارة عن closed bounded
+
+87
+00:07:08,950 --> 00:07:12,650
+intervalمدام continuous عليها إذا uniformly
+
+88
+00:07:12,650 --> 00:07:16,690
+continuous ماشي الحال إذا مدام uniformly
+
+89
+00:07:16,690 --> 00:07:23,170
+continuous إذا أكيد .. أكيد حتكون اللي هو مدام ..
+
+90
+00:07:23,170 --> 00:07:26,750
+حتى في ال continuity أكيد حتكون إيش ما لها bounded
+
+91
+00:07:26,750 --> 00:07:34,710
+إذا بقدر اللي هو أقول K بتساوي ال supremum لل Phi
+
+92
+00:07:34,710 --> 00:07:41,540
+of T such that T elementالـ C و D التي هي الـ J
+
+93
+00:07:41,540 --> 00:07:45,900
+هذه الآن بقدر احكي عن حاجة اسمها Supremum اه طبعا
+
+94
+00:07:45,900 --> 00:07:51,720
+مش هي ككلكم كمان و الـ K هذه هتكون attains for
+
+95
+00:07:51,720 --> 00:07:55,500
+some T بين C و D ليه؟ لأنه فايز continuous on a
+
+96
+00:07:55,500 --> 00:07:58,340
+closed bounded interval then it attains its
+
+97
+00:07:58,340 --> 00:08:00,920
+absolute maximum and absolute minimum on this
+
+98
+00:08:00,920 --> 00:08:05,120
+interval إذن أكيد في عنديK بيتساوى الـ Supremum لو
+
+99
+00:08:05,120 --> 00:08:09,200
+سلوك يدفع له 5T T Element in C وD سميليها دي K ليش
+
+100
+00:08:09,200 --> 00:08:13,280
+هتجهت بتعرف ليش هتستخدمها في الوصول إلى هدفي إذا
+
+101
+00:08:13,280 --> 00:08:16,960
+الأن اللي عملته لحد الآن أخدت epsilon arbitrarily
+
+102
+00:08:16,960 --> 00:08:21,860
+أخدت اللي هو ال supremum لهذا المقدار وسميته K هل
+
+103
+00:08:21,860 --> 00:08:24,440
+ال supremum موجود؟ أه ال supremum موجود و maximum
+
+104
+00:08:24,440 --> 00:08:27,480
+كمان لإن ال Phi is continuously on a closed
+
+105
+00:08:27,480 --> 00:08:31,840
+bounded interval COD طيب
+
+106
+00:08:34,510 --> 00:08:39,830
+الان بدي اخد بعيد اذنكم حاجة اسميها let epsilon
+
+107
+00:08:39,830 --> 00:08:45,490
+برايم بالساوي اللي هو epsilon على b ناقص a زائد
+
+108
+00:08:45,490 --> 00:08:49,670
+اتنين k ال k هذه اللي فوق زائد اتنين k وال b و ال
+
+109
+00:08:49,670 --> 00:08:54,010
+a اللي هي طول الفترة اللي هي I اللي انا عمال بشتغل
+
+110
+00:08:54,010 --> 00:09:01,100
+عليها معرف عليها Fليش هيك؟ بغرض الحسابات بعد شوية
+
+111
+00:09:01,100 --> 00:09:06,060
+هتشوفوا ليش و لو أصلا احنا في النهاية الـ Epsilon
+
+112
+00:09:06,060 --> 00:09:10,140
+الـ Prime هذه ما كتبناش بالشكل هذا و طلع عندي اللي
+
+113
+00:09:10,140 --> 00:09:19,940
+هو الـ U of B أو Phi Composite F ناقص الـ B Phi
+
+114
+00:09:19,940 --> 00:09:26,070
+Composite Fيكون أصغر من إبسلون مضروبة في something
+
+115
+00:09:26,070 --> 00:09:32,270
+أي إشي something ثابتبرضه هتقدي الغرض لإنه اللي هي
+
+116
+00:09:32,270 --> 00:09:35,310
+زي ما قلنا المرة الماضية المدام صحيحة لكل epsilon
+
+117
+00:09:35,310 --> 00:09:38,150
+في الدنيا إذا صحيح الواحد على n خد ال limit
+
+118
+00:09:38,150 --> 00:09:41,530
+للجهتين as n goes to infinity بيصير اللي هو هذا
+
+119
+00:09:41,530 --> 00:09:45,670
+اللي هو يؤد الغرض أو ال epsilon اللي هنا بتكون
+
+120
+00:09:45,670 --> 00:09:50,730
+اللي هو بيصير يعني تؤدي غرض أي أصغر من epsilon في
+
+121
+00:09:50,730 --> 00:09:53,510
+الدنيا لإن ال epsilon مدام اتمضروبة ضرب في التابط
+
+122
+00:09:53,510 --> 00:09:57,930
+بقدر ال epsilon أزغرها جد ما بدي و تؤدي الغرض
+
+123
+00:09:58,610 --> 00:10:05,280
+المفروض فاهمين طيب نيجي الآننرجع نقول أنه أخدت
+
+124
+00:10:05,280 --> 00:10:08,980
+إبسلون مراي بيساوي إبسلون على P minus A زاد 2K على
+
+125
+00:10:08,980 --> 00:10:12,680
+أساس أنه اللي هي تطلع عندنا الحسابات في الآخر
+
+126
+00:10:12,680 --> 00:10:16,480
+مرتبة وخالصة اللي هي أصغر من إبسلون طبعا هو
+
+127
+00:10:16,480 --> 00:10:20,720
+الاشعرف أن إبسلون مراي بالشكل هذا أصلا هو برهن
+
+128
+00:10:20,720 --> 00:10:23,680
+النظرية أو برهننا النظرية وفي الآخر طلع أن إبسلون
+
+129
+00:10:23,680 --> 00:10:28,440
+مضروبة في رقم جيت اللي هو رتبت حالي بحيث أنه حسبت
+
+130
+00:10:28,440 --> 00:10:32,320
+أنه عشان أطلح إبسلون لحالها خد إبسلون مبران بالشكل
+
+131
+00:10:32,320 --> 00:10:39,070
+هذاطيب نشوف .. نشوف الآن عندي فاي زي ما وعدناكم
+
+132
+00:10:39,070 --> 00:10:49,690
+فاي is continuous on اللي هو cod مدام فاي
+
+133
+00:10:49,690 --> 00:10:53,010
+continuous on cod يا جماعة إذا زي ما قلنا قبل
+
+134
+00:10:53,010 --> 00:11:01,660
+بشوية إذا5 is uniformly continuous on COD أيش
+
+135
+00:11:01,660 --> 00:11:07,680
+معناة uniformly continuous يعني اللي هو for every
+
+136
+00:11:07,680 --> 00:11:12,100
+إبسلون في الدنيا for every إبسلون .. خلّيني أخد
+
+137
+00:11:12,100 --> 00:11:15,020
+إبسلون الـ prime for every إبسلون الـ prime أكبر
+
+138
+00:11:15,020 --> 00:11:21,680
+من سفر there exists delta prime such that
+
+139
+00:11:21,680 --> 00:11:29,560
+uniformly continuousFor every S وT element in C وD
+
+140
+00:11:29,560 --> 00:11:38,690
+تحقق S minus T أصغر من Delta Prime يؤدي إلى Phiof
+
+141
+00:11:38,690 --> 00:11:43,690
+S ناقص Phi of T أصغر من مين من Epsilon برايم اللي
+
+142
+00:11:43,690 --> 00:11:49,450
+أخدتها عندي لأي Epsilon في الدنيا هذا الكلام
+
+143
+00:11:49,450 --> 00:11:52,210
+بيتحقق من دون أن هم Epsilon برايم اللي حكيت عنها
+
+144
+00:11:52,210 --> 00:11:58,090
+فوق طيب إذا الأن اللي استخدمته بما أن Phi
+
+145
+00:11:58,090 --> 00:12:02,010
+continuous على C وD إذا Phi is uniformly
+
+146
+00:12:02,010 --> 00:12:08,210
+continuous on مين؟ on C أو Dإذن الآن حسب التعريف
+
+147
+00:12:08,210 --> 00:12:11,530
+الـ Uniformly Continuous لأي إبسلون في الدنيا من
+
+148
+00:12:11,530 --> 00:12:14,530
+ضمنهن الإبسلون الـ prime الأكبر من 0 there exists
+
+149
+00:12:14,530 --> 00:12:17,430
+delta prime خاصة بالإبسلون الـ prime بحيث أنه لما
+
+150
+00:12:17,430 --> 00:12:20,770
+S و T في الـ C و الـ D و يكون الـ S minus T أصغر
+
+151
+00:12:20,770 --> 00:12:24,950
+من delta prime يعطيني أن 5S ناقص 5T أصغر من مين من
+
+152
+00:12:24,950 --> 00:12:30,990
+إبسلون الـ primeالان انا في delta معينة بدي
+
+153
+00:12:30,990 --> 00:12:35,830
+ألاقيها أربطها بهذه واتحققلي هذا الكلام شوفوا كيف
+
+154
+00:12:35,830 --> 00:12:44,030
+اصبروا عليها الان if عندي delta prime هذه أصغر من
+
+155
+00:12:44,030 --> 00:12:49,430
+epsilon prime thenThere exists Delta بتساوي Delta
+
+156
+00:12:49,430 --> 00:12:53,130
+Prime بتاخد Delta إيش بتساوي أسميه بتسمي Delta
+
+157
+00:12:53,130 --> 00:12:56,370
+Prime مين يا جماعة؟ Delta و هاد ال Delta اللي
+
+158
+00:12:56,370 --> 00:12:58,790
+سميتها اللي هي Delta Prime اللي سميتها Delta هاد
+
+159
+00:12:58,790 --> 00:13:03,670
+ال Delta هتحققها ليش؟ لأنها نفسها يعني بمعنى آخر
+
+160
+00:13:03,670 --> 00:13:08,150
+إذا كانت ال S minus T أصغر من Delta Prime بيعطيني
+
+161
+00:13:08,150 --> 00:13:10,830
+Automatic أصغر من Delta Prime اللي سميتها Delta
+
+162
+00:13:10,830 --> 00:13:15,810
+بيعطيني Phi of S ناقص Phi of T أصغر من مين؟ من
+
+163
+00:13:15,810 --> 00:13:23,530
+Epsilon Primeطيب تشوفوا الان if delta prime أكبر
+
+164
+00:13:23,530 --> 00:13:26,430
+أو يساوي epsilon prime يا أي حالتين مافيش غير هيك
+
+165
+00:13:26,430 --> 00:13:28,830
+يا delta prime أصغر من epsilon يا delta prime أكبر
+
+166
+00:13:28,830 --> 00:13:32,090
+ساوي epsilon prime الان if delta .. أهل جاي تفهموا
+
+167
+00:13:32,090 --> 00:13:34,530
+ليش عملت هيك if delta prime أكبر ساوي epsilon
+
+168
+00:13:34,530 --> 00:13:41,100
+prime thenاللي هو there exists Delta أصغر من
+
+169
+00:13:41,100 --> 00:13:45,540
+Epsilon برايم بلاجي ولا بلاجيش Epsilon برايم أكبر
+
+170
+00:13:45,540 --> 00:13:50,200
+من 0 أكيد between انا ابسلون برايم أكبر من 0 إذا
+
+171
+00:13:50,200 --> 00:13:52,620
+أنا أكيد بلاجي بين Epsilon برايم أكبر من 0 بلاجي
+
+172
+00:13:52,620 --> 00:13:57,340
+Deltaبلاجي عدد نهائي من الأعداد اللي هو Delta أكبر
+
+173
+00:13:57,340 --> 00:14:01,400
+من سفر وأصغر من إبسل و براين إذاً بلاجي Delta أصغر
+
+174
+00:14:01,400 --> 00:14:04,820
+من إبسل و براين بلاجي أه بلاجي لإن بين الـ two
+
+175
+00:14:04,820 --> 00:14:09,820
+real numbers اللي هو بين السفر وبين أي positive
+
+176
+00:14:09,820 --> 00:14:14,360
+real number في infinite number of numbers بينهم
+
+177
+00:14:14,360 --> 00:14:17,600
+سميت واحد قلت there exists Delta أصغر من إبسل و
+
+178
+00:14:17,600 --> 00:14:21,860
+براين such that .. طيب إيش بدك فيها هذه؟ such that
+
+179
+00:14:21,860 --> 00:14:27,400
+.. أحنا الـ gate تشوفواIf S minus T أصغر من Delta
+
+180
+00:14:27,400 --> 00:14:34,000
+فهذه الـ Delta If S minus T أصغر من Delta إذا أكيد
+
+181
+00:14:34,000 --> 00:14:37,860
+هذه الـ Delta أصغر من مين؟ المختار هي أصغر من Y'
+
+182
+00:14:38,320 --> 00:14:42,480
+وY' أصغر من مين؟ أصغر أو يسوى Delta Prime من هنا
+
+183
+00:14:43,910 --> 00:14:50,290
+الان if S-C أصغر من Delta إذا .. إذا حيث الـ S-C
+
+184
+00:14:50,290 --> 00:14:54,910
+هتكون أصغر من Y' واللي بدورها SY' أصغر بساوي Delta
+
+185
+00:14:54,910 --> 00:14:59,090
+.. أصغر بساوي Delta Prime إذا صارت عندي S-C أصغر
+
+186
+00:14:59,090 --> 00:15:03,010
+من Delta قطعًا S-C أصغر من مين؟ من Delta Prime
+
+187
+00:15:03,010 --> 00:15:07,450
+وبناءً على اللي كتبته بالأحمر هذا كل إشي S-C أصغر
+
+188
+00:15:07,450 --> 00:15:15,270
+من Delta Prime إشي بيعطيني5s-5t أصغر من إبسلون
+
+189
+00:15:15,270 --> 00:15:20,550
+برايم إذا يا جماعة سواء دلتا برايم أصغر من إبسلون
+
+190
+00:15:20,550 --> 00:15:25,710
+برايم أو دلتا برايم أصغر من إبسلون إبسلون برايم
+
+191
+00:15:25,710 --> 00:15:33,020
+بقدر ألاقي دلتاتتحقق فيها الخاصية لما S minus T
+
+192
+00:15:33,020 --> 00:15:36,900
+أصغر من Delta لما S minus T أصغر من Delta يعطيني
+
+193
+00:15:36,900 --> 00:15:39,900
+هذا المقدار أصغر من Epsilon Prime هذا المقدار أصغر
+
+194
+00:15:39,900 --> 00:15:44,580
+من Epsilon Prime إذا أنا في النهاية there exist
+
+195
+00:15:44,580 --> 00:15:50,690
+لوصلت له there exist Deltaأكبر من 0 ونفس الوجد
+
+196
+00:15:50,690 --> 00:15:55,170
+أشمالها Delta أصغر من يبسلون برايم لأنه في الحالة
+
+197
+00:15:55,170 --> 00:15:59,650
+ده أصغر من يبسلون برايم وفي الحالة الأولى برضه الـ
+
+198
+00:15:59,650 --> 00:16:03,630
+Delta أصغر من يبسلون برايم لأنه اختار الدلتة هي
+
+199
+00:16:03,630 --> 00:16:08,490
+Delta إبرايم إذن في كل الحالات هي المربط الفلسفي
+
+200
+00:16:08,490 --> 00:16:13,950
+الآن there exists Delta أكبر من 0 وأصغر من يبسلون
+
+201
+00:16:13,950 --> 00:16:22,170
+برايم such thatلكل S وT element in C وD إذا حققت
+
+202
+00:16:22,170 --> 00:16:27,610
+الخاصية S minus T أصغر من Delta بيعطيني على طول 5S
+
+203
+00:16:27,610 --> 00:16:34,130
+ناقص 5T أصغر من مين من إبسلون إبراهيم إذن هي
+
+204
+00:16:34,130 --> 00:16:41,710
+معلومة أخرى بدي أخذنها لأنني هحتاجهاهي كمان معلومة
+
+205
+00:16:41,710 --> 00:16:46,270
+الان اسمحولي أمسح هذا ال hand عشان أخزن معلومتي
+
+206
+00:16:46,270 --> 00:16:49,770
+اللي وصلت إلها مع المعلومات اللي موجودة عندى فوق
+
+207
+00:16:49,770 --> 00:16:57,090
+طيب الان شطبنا اللي عندنا hand خلصنا منه ووصلنا
+
+208
+00:16:57,090 --> 00:17:03,030
+إلى المعلومة التالية اللي بدأ أخزنها الآن مع اللي
+
+209
+00:17:03,030 --> 00:17:10,620
+مخزن فوق لأن there existDelta أكبر من 0 وأصغر من
+
+210
+00:17:10,620 --> 00:17:18,140
+Y' Such that for every S وT element in C وD إذا
+
+211
+00:17:18,140 --> 00:17:25,340
+حقق S minus T أصغر من Delta بيعطيني اللي هو 5S
+
+212
+00:17:25,340 --> 00:17:34,140
+minus 5T أصغر من Y' وهذا سمونيها 1 سمونيها 2
+
+213
+00:17:34,140 --> 00:17:38,630
+سمونيها Star اللي بدكم إياها ماشي الحالهذا الان
+
+214
+00:17:38,630 --> 00:17:44,170
+وصلت له وانا بدي استخدمه بعد شوية
+
+215
+00:17:49,890 --> 00:17:54,390
+اللي اللي بحب يتابع على التلخيص
+
+216
+00:17:54,390 --> 00:17:58,130
+هاي اللي وصلت إليه الآن هايها there exists delta و
+
+217
+00:17:58,130 --> 00:18:01,390
+الـ delta أصغر من epsilon prime if S و T element
+
+218
+00:18:01,390 --> 00:18:04,470
+in J and S minus T أصغر من Delta then Phi of S
+
+219
+00:18:04,470 --> 00:18:08,030
+ناقص Phi of T أصغر من إبسلون برايم هذه اللي وصلنا
+
+220
+00:18:08,030 --> 00:18:15,670
+إليها اللي قدرنا أن نصلها عشان بعد شوية بتستخدمها
+
+221
+00:18:15,670 --> 00:18:20,950
+انتبه عليها الآنالآن استغلقنا معلومة الـ if I is
+
+222
+00:18:20,950 --> 00:18:24,490
+continuous وحصلنا على معلومة مهمة جداً هي هذه
+
+223
+00:18:24,490 --> 00:18:28,630
+المعلومة الآن بدي أستخدم المعلومة الموازية لها أن
+
+224
+00:18:28,630 --> 00:18:37,770
+F is integrable الان عندي F is integrable on I إذا
+
+225
+00:18:37,770 --> 00:18:43,010
+حسب then حسب اللي هو ال .. ال .. ال integrability
+
+226
+00:18:43,010 --> 00:18:48,570
+criterionاللي حكينا عنها there exists partition B
+
+227
+00:18:48,570 --> 00:18:56,690
+element in B of I بجزء مين I such that اللي هو U
+
+228
+00:18:56,690 --> 00:19:05,920
+of B وFمعقس L بيوقف أصغر من أي إبسلون في الدنيا
+
+229
+00:19:05,920 --> 00:19:08,360
+الإبسلون اللي في الدنيا الإبسلون اللي بدها
+
+230
+00:19:08,360 --> 00:19:12,040
+استخدمها اللي حتة في الدنيا اللي هي مين هي Delta
+
+231
+00:19:12,040 --> 00:19:17,000
+تربيع أصغر من مين من Delta تربيع بقدر أه بقدر طبعا
+
+232
+00:19:17,000 --> 00:19:19,960
+مش احنا بنقول لكل إبسلون أكبر من سفر مادام F is
+
+233
+00:19:19,960 --> 00:19:23,040
+integrable لذا لكل إبسلون أكبر من سفر there exist
+
+234
+00:19:23,040 --> 00:19:27,590
+ال partition B بحيث أنه هذا أصغر من إبسلونالأن
+
+235
+00:19:27,590 --> 00:19:30,330
+أبسط من اللي بحكي عنها دلتة تربيع دلفة التربيع
+
+236
+00:19:30,330 --> 00:19:35,650
+أكبر من صفر إذا for delta تربيع there exists B
+
+237
+00:19:35,650 --> 00:19:40,070
+element in B of I such that U ناقص L أصغر من مين
+
+238
+00:19:40,070 --> 00:19:44,650
+من Delta تربيع هذا ال partition B بيجزئلي مين؟
+
+239
+00:19:44,650 --> 00:19:51,670
+بيجزئلي I بعد أذنكم سموليها بيه X0 و X1 لعند مين؟
+
+240
+00:19:51,670 --> 00:19:55,230
+لعند X سموليها ال partition بس عشان أتعامل معاها
+
+241
+00:19:55,660 --> 00:19:57,540
+اللي هو partition للـ I partition للـ I معناه
+
+242
+00:19:57,540 --> 00:20:00,880
+بجزّه I جزّه او ل X note X واحد X انت جداش أعدادها
+
+243
+00:20:00,880 --> 00:20:07,080
+مش عارف حسب اللي هو اللي جناه بيه طيب شوف الآن بدي
+
+244
+00:20:07,080 --> 00:20:13,520
+أعمل اللي هو شغلة بحيث عن الان أقدر أستخدمها اللي
+
+245
+00:20:13,520 --> 00:20:18,360
+هي أصل للي بدي من خلالها الآن هجيتوا تشوفوا ليش
+
+246
+00:20:18,360 --> 00:20:23,400
+جزعين خدوا Aالان يا جماعة صارت دلتة بين إيديا
+
+247
+00:20:23,400 --> 00:20:28,520
+لجهتها دلتة طيب الان خدوا ايه 6 هي عبارة عن كل ال
+
+248
+00:20:28,520 --> 00:20:34,160
+indices K اللي هان هذولة و 0,1,2,3 كده هذه جزء
+
+249
+00:20:34,160 --> 00:20:38,520
+اثنين ال I هي عند X note لعند X end هذه الفترة
+
+250
+00:20:38,520 --> 00:20:44,000
+اللي هي من A لعند B يا جماعة لعند Bماشي الحال طيب
+
+251
+00:20:44,000 --> 00:20:49,380
+خدوله A هي عبارة عن كل ال K بحيث ان M K ناقص M K
+
+252
+00:20:49,380 --> 00:20:54,900
+small تكون أصغر من Delta وخدول ال B بيساوي كل ال K
+
+253
+00:20:54,900 --> 00:20:59,740
+such that M K ناقص M K أكبر يساوي Delta إيش هذول
+
+254
+00:20:59,740 --> 00:21:04,700
+عاملا؟ هذول بس ال indices عند من هنا صفر و واحد و
+
+255
+00:21:04,700 --> 00:21:08,480
+اتنين عند من؟ عند ال K إذا اتجت جزءات ال indices
+
+256
+00:21:08,480 --> 00:21:15,500
+هذول إلى جزء ايه؟الجزء اللى هو عندى اللى بخاصية
+
+257
+00:21:15,500 --> 00:21:19,980
+اللى هو ال mk ناقص mk أصغر من دلتا بده حطه فهذا ال
+
+258
+00:21:19,980 --> 00:21:24,280
+set إذا هذه عبارة عن إيش مجموعة جزئية من ال .. من
+
+259
+00:21:24,280 --> 00:21:27,680
+ال .. من ال .. من ال .. من ال .. من ال ك من سفر
+
+260
+00:21:27,680 --> 00:21:36,490
+لعند مين لعند واحد لعند أنا .. لعند أناالأن هذه P
+
+261
+00:21:36,490 --> 00:21:40,870
+هي عبارة عن المتبقى منهن، مين المتبقى اللي الـ M K
+
+262
+00:21:40,870 --> 00:21:44,170
+نقص M K أصغر أكبر أو شويه دلتا، يعني بمعنى أخر
+
+263
+00:21:44,170 --> 00:21:49,430
+أداة التجزئة لهذه اللي هي الخاصية أنه M K نقص M K
+
+264
+00:21:49,430 --> 00:21:52,010
+أصغر مين مين دلتا، أنتوا عارفين إيش الـ M K
+
+265
+00:21:52,010 --> 00:21:55,730
+Capital و M K Small؟ أكيد، الـ M K Capital هي
+
+266
+00:21:55,730 --> 00:21:59,710
+عبارة عن الـ supremum لل F of X such that X
+
+267
+00:21:59,710 --> 00:22:04,740
+element in X K minus 1 لعند الـ X Kو الـ mk small
+
+268
+00:22:04,740 --> 00:22:09,560
+بيساوي الـ infimum لل F of X such that X element
+
+269
+00:22:09,560 --> 00:22:16,640
+in XK minus واحد والـ XK إذن يا جماعة اللي جزألي
+
+270
+00:22:16,640 --> 00:22:21,900
+الـ A و الـ B هو خاصيته إن الـ mk capital يعني
+
+271
+00:22:21,900 --> 00:22:25,340
+أعلى قيمة هي .. هي .. هي .. هي عندي أنا XK minus
+
+272
+00:22:25,340 --> 00:22:29,860
+واحد وهي XK فرضنا أن الدالة هي الدالة زي هيك مثلا
+
+273
+00:22:29,860 --> 00:22:35,250
+في المنطقة هذهالان هي أعلى قيمة وهي أقل قيمة
+
+274
+00:22:35,250 --> 00:22:41,070
+الحاصل طرح أعلى قيمة و أقل قيمة في كل فترة .. كل
+
+275
+00:22:41,070 --> 00:22:44,770
+sub interval بادي بقول هل هذا أكبر من .. أصغر من
+
+276
+00:22:44,770 --> 00:22:49,270
+delta ولا أكبر يساوي delta اللي خاصيتهم الفرق
+
+277
+00:22:49,270 --> 00:22:53,930
+بينهم أصغر من delta بحط هناالاندسيز هذا كهن واللي
+
+278
+00:22:53,930 --> 00:22:59,930
+أكبر بحطه نهن و بحط تجزيتها ليش هذه؟ هذه طريقة
+
+279
+00:22:59,930 --> 00:23:07,130
+للوصول إلى اللي بديها و هتشوفه الآن طيب نيجي الآن
+
+280
+00:23:07,130 --> 00:23:20,690
+لاللي هو نشوف الـ K F K Element A مدام K Element A
+
+281
+00:23:22,110 --> 00:23:26,470
+إذا تتحقق الخاصية هذه يعني Mk نقص Mk أصغر من مين؟
+
+282
+00:23:26,470 --> 00:23:32,430
+من Delta خدولي الآن أي X و Y في الفترة مين اللي هي
+
+283
+00:23:32,430 --> 00:23:39,330
+XK minus واحد لعند XK شوفوا إيش اللي بديهاال .. ال
+
+284
+00:23:39,330 --> 00:23:43,130
+.. ال .. ال .. ال X و ال Y هنا هى ال X و ال Y وهى
+
+285
+00:23:43,130 --> 00:23:46,430
+رسمة الدلة فى المنطقة هذه ال X و ال Y فى داخل هدول
+
+286
+00:23:46,430 --> 00:23:53,930
+طيب الان ال X و ال Y هنا إذا أكيد ال F of X ال F
+
+287
+00:23:53,930 --> 00:24:00,090
+of X ناقص F of Y نقطتين هنا نقطتين من هنا لهنا صور
+
+288
+00:24:00,090 --> 00:24:03,750
+هنا صور هنا يعني ممكن صورة واحدة هنا و صورة
+
+289
+00:24:03,750 --> 00:24:08,440
+التانية هنايعني ممكن صورة التالتة هنا وصورة الأولى
+
+290
+00:24:08,440 --> 00:24:15,500
+هنا يعني بمعنى آخر لو ال X هنا و ال Y هنا هي صورة
+
+291
+00:24:15,500 --> 00:24:21,400
+ال X وهي صورة مين ال Y لو جيتوا .. أخدتوا .. هي
+
+292
+00:24:21,400 --> 00:24:26,760
+صورة ال Y لو أخدتوا الفرق بين هذه و بين هذهالفرق
+
+293
+00:24:26,760 --> 00:24:31,020
+بين ما هو قيمة الدالة F of X هنا و قيمة الدالة F
+
+294
+00:24:31,020 --> 00:24:36,280
+of Y هنا أكيد .. أكيد .. أكيد الفرق بينهم هيكون
+
+295
+00:24:36,280 --> 00:24:42,320
+أصغر أو يساوي M K ناقص من M K لسبب بسيط أصلا لأن
+
+296
+00:24:42,320 --> 00:24:49,660
+أصلا F of X هذه أصغر أو يساوي ما هو M K أكيد و F
+
+297
+00:24:49,660 --> 00:24:54,400
+of Y أصغر أو يساويها ماشي الحالو نفس ال .. يعني ..
+
+298
+00:24:54,400 --> 00:24:57,080
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+299
+00:24:57,080 --> 00:24:57,080
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+300
+00:24:57,080 --> 00:24:57,100
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+301
+00:24:57,100 --> 00:24:57,360
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+302
+00:24:57,360 --> 00:24:57,720
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+303
+00:24:57,720 --> 00:24:58,100
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+304
+00:24:58,100 --> 00:24:58,100
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+305
+00:24:58,100 --> 00:24:59,480
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+306
+00:24:59,480 --> 00:25:00,040
+و .. و .. و .. و .. و .. و .. و .. و .. و .. و ..
+
+307
+00:25:00,040 --> 00:25:07,580
+و .. و .. و .. و .. و
+
+308
+00:25:07,580 --> 00:25:08,160
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+309
+00:25:08,160 --> 00:25:11,200
+.. و .. و .. و ..
+
+310
+00:25:19,040 --> 00:25:24,000
+فبيظل قيمة f of x ناقص f of y أصغر شاومين مك ناقص
+
+311
+00:25:24,000 --> 00:25:28,460
+مك اللي مش واضح له من خلال الرسمة اللي عندي هنا
+
+312
+00:25:31,440 --> 00:25:34,520
+يعني الآن خلّيني أرى أضحى لحسن تكون مش واضحة للبعض
+
+313
+00:25:34,520 --> 00:25:41,360
+هاي الرسمة هاي XK minus واحد وهي XK خلّيني أكبر
+
+314
+00:25:41,360 --> 00:25:46,180
+الرسمة عشان تكون أوضح وهي XK لأن نفترضها أن هاي
+
+315
+00:25:46,180 --> 00:25:53,520
+رسمتنا هيك وطلعت زي هيك ماشي وهي عندي أعلى نقطة
+
+316
+00:25:53,520 --> 00:25:58,710
+خلّيني أزغر هذه شويةوهي أصغر النقطة هذي اللي
+
+317
+00:25:58,710 --> 00:26:03,970
+بتتمثل لـ M K وهذه بتتمثل لـ M K Small هاي المسافة
+
+318
+00:26:03,970 --> 00:26:09,030
+بينهم الفرق بينهم الآن الثانية لو جيت أخدت أي نقطة
+
+319
+00:26:09,030 --> 00:26:17,190
+هنا X و أي نقطة Y هنا هاي X وهي Y المسافة بينها دي
+
+320
+00:26:17,190 --> 00:26:21,310
+و بينها دي قيمتها دي بيصير F of X أنا بصير F of X
+
+321
+00:26:22,380 --> 00:26:26,700
+وهنا بيصير F of Y الفرق بين F of X و F of Y أكيد
+
+322
+00:26:26,700 --> 00:26:31,140
+أصغر من الفرق هذا وكل النقاط اللي بين هذه و هذه
+
+323
+00:26:31,140 --> 00:26:37,340
+هيكون يا إما زي الفرق هذا أو أصغر منه إذا أكيد
+
+324
+00:26:37,340 --> 00:26:45,820
+عندي صار المفروض وضح الأمر هيكون عندي اللي هو هذا
+
+325
+00:26:45,820 --> 00:26:50,910
+المقدار أصغر شهو هذا طب ليه ليش هذا؟هذا من مين للـ
+
+326
+00:26:50,910 --> 00:26:53,990
+K اللي في الـ A والـ K اللي في الـ A شخصيتها Mk
+
+327
+00:26:53,990 --> 00:26:57,610
+نقصها Mk أصغر من مين؟ إذا صارت أصغر من الـ Delta
+
+328
+00:26:57,610 --> 00:27:04,470
+إذا صارت F of X و F of Y أصغر من مين؟من Delta طيب
+
+329
+00:27:04,470 --> 00:27:08,430
+مادام F of X ايش علاقات هنا F of X و F of Y ما هي
+
+330
+00:27:08,430 --> 00:27:13,950
+اصلا احنا مفترضين من الاول ان F of I جزئية من الـ
+
+331
+00:27:13,950 --> 00:27:18,430
+J اللي هي عبارة عن C و D مظبوط ولا لأ اذا حيصير
+
+332
+00:27:18,430 --> 00:27:23,600
+عندي F of X و F of Y موجودات في الـ C و DF of X و
+
+333
+00:27:23,600 --> 00:27:27,000
+F of Y موجودات في الـ C و الـ D و بتحقق المسافة
+
+334
+00:27:27,000 --> 00:27:32,200
+أصغر من Delta بينهم إذا حسب واحد اللي احنا أثبتنا
+
+335
+00:27:32,200 --> 00:27:35,960
+أي نقطتين في الـ S و الـ D المسافة بينهم أصغر من
+
+336
+00:27:35,960 --> 00:27:39,600
+Delta لازم صورة ال .. صورة Phi of S و Phi of T
+
+337
+00:27:39,600 --> 00:27:42,540
+المسافة بينهم أصغر من مين؟ من Epsilon و Prime إذا
+
+338
+00:27:42,540 --> 00:27:47,640
+هذا Automatic هيعطيني Phi of النقطة الأولى اللي هي
+
+339
+00:27:47,640 --> 00:27:56,290
+F of X ناقصفاي أفضل النقطة الثانية يكون أصغر من
+
+340
+00:27:56,290 --> 00:28:06,890
+مين من إبسلون براي إذا صار عندي الآن لكل اللي عندي
+
+341
+00:28:06,890 --> 00:28:13,740
+ال K اللي في ال A وال X و Y في هذه المنطقةF of X
+
+342
+00:28:13,740 --> 00:28:17,920
+minus F of Y طلع ليه أصغر من Delta حتى ملي الـ 5 F
+
+343
+00:28:17,920 --> 00:28:25,000
+of X ناقص 5 F of Y أصغر من مين؟ من Y إبراهيم ماشي
+
+344
+00:28:25,000 --> 00:28:29,820
+الحال هذا الكلام لكل X و Y موجودات في الـ XK minus
+
+345
+00:28:29,820 --> 00:28:33,460
+واحد والـ XK لخص الآن اللي وصلنا له لأنه بدي
+
+346
+00:28:33,460 --> 00:28:36,920
+أستخدمه خليني ألخص خليني أشيل
+
+347
+00:28:41,000 --> 00:28:45,820
+خلّيني بس ساعدك اكتبلكم في مكان هنا اتحملوني انه
+
+348
+00:28:45,820 --> 00:28:54,640
+عشان اللوح شوية A بساوي كل ال K مقصد MK اصغر من
+
+349
+00:28:54,640 --> 00:29:05,600
+دلتا وعند B بساوي كلالـ K بحيث أن M K نقص M K أكبر
+
+350
+00:29:05,600 --> 00:29:10,060
+أو يساوي دلتة خلّيني أقولها في الذاكرة الآن خلّيني
+
+351
+00:29:10,060 --> 00:29:16,140
+أمسح حاجة و ألخص معلومتي إيش معلومتي بتقول دي لـ K
+
+352
+00:29:16,140 --> 00:29:26,610
+اللي في الـ A ماشي الحال لكل X و Y في الفترة X K-1
+
+353
+00:29:26,610 --> 00:29:32,750
+لعندي الـ xk بيطلع عندى الـ Phi f of x يعني Phi
+
+354
+00:29:32,750 --> 00:29:40,990
+composite f of x نقص Phi composite f of y أشماله
+
+355
+00:29:40,990 --> 00:29:48,600
+أصغر من إبسلون براي واضح؟إذن اللي وصلناله لـ K
+
+356
+00:29:48,600 --> 00:29:53,700
+اللي في الـ A عندي لـ K اللي في الـ A هذا الـ
+
+357
+00:29:53,700 --> 00:29:58,380
+absolute value أظهر من مين؟ من إبسلون إبراهيم لكل
+
+358
+00:29:58,380 --> 00:30:05,580
+مين لكل XY في ال XK minus XK minus 1 طيب نكمل لكم
+
+359
+00:30:05,580 --> 00:30:10,860
+عليها هذا خلصنا منه صار عندي الآن شوفوا
+
+360
+00:30:19,430 --> 00:30:24,070
+الشغل عميدي يا جماعة على الفترة هذه طيب شوف عندي
+
+361
+00:30:24,070 --> 00:30:31,590
+Phi composite F of X ناقص Phi composite F of Y
+
+362
+00:30:31,590 --> 00:30:37,770
+أكبر من Y و أكبر من مين من ناقص Y برايم بدأ أخد
+
+363
+00:30:37,770 --> 00:30:43,970
+هذه الجهة خليني أخد الجهة من الجهتين و أبدأ أشتغل
+
+364
+00:30:43,970 --> 00:30:51,300
+عليها واضحة هذههذه أصغر من Y' أكبر من ناقص Y بديش
+
+365
+00:30:51,300 --> 00:30:56,980
+هذا بتشغل هذه الان لان هذا الكلام صحيح لمن؟ لكل X
+
+366
+00:30:56,980 --> 00:31:02,740
+وY في ال sub interval هذه اللي هي لكل مين ال case؟
+
+367
+00:31:02,740 --> 00:31:06,600
+ال case اللي من A بس طيب شوف إذا صار عندي Phi
+
+368
+00:31:06,600 --> 00:31:14,580
+composite F of X أصغر من Y' بزايد Phi composite F
+
+369
+00:31:14,580 --> 00:31:21,720
+of Yماشي يا جماعة .. ماشي هذا الكلام لكل x و y في
+
+370
+00:31:21,720 --> 00:31:26,980
+.. اللي هو xk minus .. من xk minus 1 عند xk اللي
+
+371
+00:31:26,980 --> 00:31:31,600
+هنا .. هذا لكل x و y ثبتلي y .. ثبتلي y .. خلينا
+
+372
+00:31:31,600 --> 00:31:34,380
+نحكي عن y محدد .. arbitrary y لكن خلينا .. ثبتها
+
+373
+00:31:34,380 --> 00:31:40,180
+.. خلينا نحكي عن arbitrary fixed yبظل y' زي الـ
+
+374
+00:31:40,180 --> 00:31:47,140
+phi composite f of y is true أكبر من هذه لكل x
+
+375
+00:31:47,140 --> 00:31:52,500
+element in xk minus 1 لعند مين؟ لعند الـ xk أكيد
+
+376
+00:31:52,500 --> 00:32:00,060
+أكيد هذا أكبر من هذا لكل x well موجودة في xk minus
+
+377
+00:32:00,060 --> 00:32:04,540
+1 لعند مين؟ لعند الـ xk إذا صار هذا عبارة عن إيش
+
+378
+00:32:04,540 --> 00:32:08,880
+يا جماعة عبارة عن upper bound لهذا ال setمدام
+
+379
+00:32:08,880 --> 00:32:13,900
+upper bound إلها إذا هيكون أكبر أو يساوي ال least
+
+380
+00:32:13,900 --> 00:32:19,340
+upper bound يعني بمعنى آخر ال supremum لفاي
+
+381
+00:32:19,340 --> 00:32:25,280
+composite F of X such that X element in XK minus 1
+
+382
+00:32:25,280 --> 00:32:31,240
+لعند ال XK اللي هو أصغر أو يساوي Y prime زي فاي
+
+383
+00:32:31,240 --> 00:32:38,520
+composite F of mean of Y for any fixed Yطيب، هذا
+
+384
+00:32:38,520 --> 00:32:45,140
+مين هو؟ هذا عبارة عن الـ M K وسميها Delta أساس أو
+
+385
+00:32:45,140 --> 00:32:48,080
+تلدا أساس اللي هي خاصة بمين الآن في الـ Phi
+
+386
+00:32:48,080 --> 00:32:53,100
+Composite F عشان نميزها بالـ M K اسمه بالـ M K
+
+387
+00:32:53,100 --> 00:32:59,400
+اللي خاصة بالـ F اللي عندي، ماشي الحال، طيبصار
+
+388
+00:32:59,400 --> 00:33:02,800
+عندى MKتل ده بساوي الـ Supremum لهذا أصغر يساوي
+
+389
+00:33:02,800 --> 00:33:07,680
+هذا هذا صار عدد صار العدد هذا أصغر من يبسلون برايم
+
+390
+00:33:07,680 --> 00:33:11,640
+في كومبوزيت F of Y for any fixed Y يعني صحيح على
+
+391
+00:33:11,640 --> 00:33:16,320
+كل Y وين موجودة في الفترة هذه إذا صار عندى الآن
+
+392
+00:33:16,320 --> 00:33:21,680
+جيبولي هذه على الجهة هذه وهذه على الجهة هذه بيصير
+
+393
+00:33:21,680 --> 00:33:28,050
+عندى اللي هو ناقص في كومبوزيت F of Yأصغر أو ساوي
+
+394
+00:33:28,050 --> 00:33:34,770
+أبسلون و براين ناقص مين مك تلدة وإذا بدك ممكن نجيب
+
+395
+00:33:34,770 --> 00:33:40,050
+اللي هو الابسلون و البراين على الجهة الثانية مش
+
+396
+00:33:40,050 --> 00:33:50,310
+مشكلة صار عندي الآن هذا يا جماعة صار
+
+397
+00:33:50,310 --> 00:33:58,380
+عندي هذا اللي هو عبارة عن upper boundلهذا المقدار
+
+398
+00:33:58,380 --> 00:34:07,020
+أكيد ولا لا؟ عارفين ليش؟ لأن هذا الأن أكبر أو
+
+399
+00:34:07,020 --> 00:34:11,000
+يساوي هذا المقدار لكل Y لأن كل المخدين Y
+
+400
+00:34:11,000 --> 00:34:14,600
+arbitrarily fixed لكن arbitrarily، إذا صحيح على
+
+401
+00:34:14,600 --> 00:34:19,640
+كله، إذا بيصير عندي الآن اللي هو هذا أكبر أو يساوي
+
+402
+00:34:19,640 --> 00:34:24,260
+ال supremum لهذا، يعني بمعنى أخر ال supremum، لا
+
+403
+00:34:24,860 --> 00:34:32,200
+الناقص Phi composite F of Y such that Y element in
+
+404
+00:34:32,200 --> 00:34:39,360
+YK as if XK minus 1 عند XK هذا الـ Supremum له
+
+405
+00:34:39,360 --> 00:34:48,670
+أظهر أو يسوى Y' ناقص M K تلدةطيب هات ايش بتساوي
+
+406
+00:34:48,670 --> 00:34:56,670
+طلعون الناقص برا بيساوي ناقص الانف مام معايا لأ
+
+407
+00:34:56,670 --> 00:34:59,790
+لأنه مدام ناقص واحد طلع اذا بيقلب ال super mum ل
+
+408
+00:34:59,790 --> 00:35:05,530
+انف مام فاي composite F of Y such that Y element
+
+409
+00:35:05,530 --> 00:35:09,610
+in XK minus واحد لعند ال XK عرفتوا ايش اللي بده
+
+410
+00:35:09,610 --> 00:35:16,060
+اقوله هاد مين هي يا شبابهذه عبارة عن الـ M K تلدة
+
+411
+00:35:16,060 --> 00:35:19,680
+اللي هي الـ infimum للـ Phi Composite F كتبت
+
+412
+00:35:19,680 --> 00:35:22,800
+التلدة عشان ترمز لمين للـ Phi Composite F وفي عندي
+
+413
+00:35:22,800 --> 00:35:28,460
+ناقص قبلها إذا صارت عندي ناقص الـ M K تلدة أصغر
+
+414
+00:35:28,460 --> 00:35:33,400
+يساوي إبسلون ناقص الـ M K تلدة بدي أجيب هذه هنا
+
+415
+00:35:33,400 --> 00:35:42,400
+بيصير عندي إذا M K تلدةأصغر نقص اف ناقص مك تلد ا
+
+416
+00:35:42,400 --> 00:35:51,080
+small اصغر او يساوي مين ابسلون اذا اللي وصلتله ما
+
+417
+00:35:51,080 --> 00:35:59,440
+يلي وهذه بده اصلها انه لكلف ال a طلع عندي مك تلد
+
+418
+00:35:59,440 --> 00:36:05,300
+ناقص مك prime اصغر اش هو ايش ابسلون خلينا نسجلها
+
+419
+00:36:05,300 --> 00:36:11,830
+عشان نبني عليهاصار عندى الآن يا جماعة طولوا روحكم
+
+420
+00:36:11,830 --> 00:36:18,670
+عليا ان شاء الله مش هم طوالين نخلص M K تلدى ناقص M
+
+421
+00:36:18,670 --> 00:36:22,890
+K تلدى small أصغر يمشي هو إبسلون هذا لكل K وين
+
+422
+00:36:22,890 --> 00:36:30,330
+موجودة في ال A لكل K في ال A طيب هذا اللى احنا إيش
+
+423
+00:36:30,330 --> 00:36:38,150
+ماله أوصلنا إلهصار عندى الان الـ M K تلدى ناقص M K
+
+424
+00:36:38,150 --> 00:36:52,190
+تلدى أصغر من الـ E' لكل K فيه اللى هى N A الان
+
+425
+00:36:52,190 --> 00:36:58,330
+خد ال summation ال summation لل M K تلدى ناقص M K
+
+426
+00:36:58,330 --> 00:37:05,490
+تلدى small K element in A اللى هوهذا اللي هو أصغر
+
+427
+00:37:05,490 --> 00:37:09,310
+أو يساوي في .. أصغر أو يساوي .. الآن قدوا .. قدوا
+
+428
+00:37:09,310 --> 00:37:11,470
+.. قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+429
+00:37:11,470 --> 00:37:11,470
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+430
+00:37:11,470 --> 00:37:12,010
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+431
+00:37:12,010 --> 00:37:13,570
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+432
+00:37:13,570 --> 00:37:15,170
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+433
+00:37:15,170 --> 00:37:15,170
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+434
+00:37:15,170 --> 00:37:17,810
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا .. قدوا ..
+
+435
+00:37:17,810 --> 00:37:31,850
+قدوا .. قدوا .. قدوا .. قدوا .. قدوا
+
+436
+00:37:31,850 --> 00:37:39,180
+..K element in A العداد هذا بعيد بعدد الـ K's اللي
+
+437
+00:37:39,180 --> 00:37:43,180
+في الـ A يعني إبسلون مضروب في عدد الـ K's اللي
+
+438
+00:37:43,180 --> 00:37:48,060
+موجودة في مين في إيه هذا أكيد أكيد هذا أصغر أو
+
+439
+00:37:48,060 --> 00:37:55,320
+يساوي ال summation للإبسلون على كل الـ K من عند
+
+440
+00:37:55,320 --> 00:38:00,980
+صفر لعند مين لعند اللي هو H N
+
+441
+00:38:03,340 --> 00:38:09,620
+بس انا الان بدي اجي اودي ال summation و اضربه فى
+
+442
+00:38:09,620 --> 00:38:13,460
+مين فى طول الفترة لإنه هي لازم لطول الفترة بعد
+
+443
+00:38:13,460 --> 00:38:20,660
+اسمهم هذا اضربه ليه فى XK بدي اضربه بس اوسع عقله
+
+444
+00:38:20,660 --> 00:38:23,700
+معلش بدي اضربه يعني هذا ال summation اللى حصلت
+
+445
+00:38:23,700 --> 00:38:29,320
+عليه بدي اخد ال summation MK تلدهنقص mk تلدى
+
+446
+00:38:29,320 --> 00:38:34,540
+وادربه يا شباب في xk minus xk minus واحد k element
+
+447
+00:38:34,540 --> 00:38:39,000
+in a هيصير اصغر او يساوي ال summation للإيبسلون في
+
+448
+00:38:39,000 --> 00:38:48,620
+xk minus xk minus واحد على الان k element in aأكيد
+
+449
+00:38:48,620 --> 00:38:52,120
+أصغر زي ما قلنا من ال summation اللي فوق epsilon
+
+450
+00:38:52,120 --> 00:38:59,980
+في xk minus xk minus 1k من 0 لعند n ليش؟ هذا ال
+
+451
+00:38:59,980 --> 00:39:02,420
+summation بس على العراصر اللي في ال A هذا ال
+
+452
+00:39:02,420 --> 00:39:05,260
+summation على كله على اللي في ال A وعلى اللي في ال
+
+453
+00:39:05,260 --> 00:39:08,760
+B على كل بقى العراصر اللي هي ال partition اللي
+
+454
+00:39:08,760 --> 00:39:15,420
+عنده من 0 ل1 اتنين عند مين لعند ال N طيب ال N هذا
+
+455
+00:39:15,420 --> 00:39:20,200
+المقدر هي اللي بديهالأن هذا bridge كان أو جسر
+
+456
+00:39:20,200 --> 00:39:23,860
+للوصول للي بديها و حتى ممكن نهان مباشرة لهذه هذه
+
+457
+00:39:23,860 --> 00:39:27,760
+أصغر أو يساوي هذا هذا مش هو اللي بتطلع عليه بتطلع
+
+458
+00:39:27,760 --> 00:39:32,640
+عليه هذا و بتطلع عليه هذا هذا خلّيه لأن أصغر يساوي
+
+459
+00:39:32,640 --> 00:39:36,460
+مين هذا المقدار هذا المقدار ايش بساوي epsilon في
+
+460
+00:39:36,460 --> 00:39:43,790
+مينx note ناقص x واحد زاد x واحد ناقص x اتنين زاد
+
+461
+00:39:43,790 --> 00:39:47,810
+x اتنين ناقص x ثلاثة لما أصل لآخر واحد يضل اللي
+
+462
+00:39:47,810 --> 00:39:53,330
+عبارة عن y في x n ناقص x note عملناها كتير ويساوي
+
+463
+00:39:53,330 --> 00:39:58,710
+y prime ويساوي y prime في مين في b minus a لأن ال
+
+464
+00:39:58,710 --> 00:40:03,950
+xn هي عبارة عن b و ال x note هي مين كانت هي ال a
+
+465
+00:40:03,950 --> 00:40:10,110
+إذا اللي وصلتله يا جماعةالان وصلت الى ما يلي انه
+
+466
+00:40:10,110 --> 00:40:19,170
+ال summation لل mk تلدى ناقص mk تلدى small لكي
+
+467
+00:40:19,170 --> 00:40:25,790
+element ايه في xk minus xk minus واحد اصغر او
+
+468
+00:40:25,790 --> 00:40:34,550
+يساوي epsilon prime في b minus ايش minus a واضح
+
+469
+00:40:34,550 --> 00:40:42,680
+الانهذه خزينها لإنه بدّيها بعد شوية اتحملونا
+
+470
+00:40:42,680 --> 00:40:46,880
+البرهان طويل شوية هي وصلنا إليها summation أزرع
+
+471
+00:40:46,880 --> 00:40:50,980
+شويه أبسلون رابع في b minus a لإن بدنا نيجي لمين
+
+472
+00:40:50,980 --> 00:40:59,140
+يا جماعة بدنا نيجي لاللي هوانتبهوا عليها، بدنا
+
+473
+00:40:59,140 --> 00:41:03,280
+نيجي للـ M للـ K اللي موجودة في مين؟ هذا الـ K في
+
+474
+00:41:03,280 --> 00:41:06,820
+الـ A، ضال عند مين؟ احنا جزأتها إلى جزئين اللي هي
+
+475
+00:41:06,820 --> 00:41:11,340
+الـ indices اللي عنده، K اللي هي في الـ A و K اللي
+
+476
+00:41:11,340 --> 00:41:18,060
+هي في مين؟ في الـ B خد الآن K element in B، شوف
+
+477
+00:41:18,060 --> 00:41:26,980
+إيش؟ الـ MK تلد إلهانقص mk تلد إلها أيش بتساوي؟
+
+478
+00:41:26,980 --> 00:41:32,380
+بتساوي حسب التعريف Supremum للـ Phi Composite F of
+
+479
+00:41:32,380 --> 00:41:38,240
+X للـ K اللي وين في الـ B اللي هي mk تلدها نقص mk
+
+480
+00:41:38,240 --> 00:41:42,540
+تلدها اللي هي small Supremum للـ Phi F of X اللي
+
+481
+00:41:42,540 --> 00:41:47,980
+هي الأولى هذه Such that X element in XK minus واحد
+
+482
+00:41:47,980 --> 00:41:54,860
+و XK اللي هي زائدأو ناقص اللي هي مين هذه الـ
+
+483
+00:41:54,860 --> 00:42:01,200
+infimum لأ فاي composite F of X such that X
+
+484
+00:42:01,200 --> 00:42:10,220
+والموجودة في ال X K minus واحد وال X K مظبوط ولا
+
+485
+00:42:10,220 --> 00:42:18,690
+لأ احنا قلناالـ Supremum لكل الـ Phi of T T على كل
+
+486
+00:42:18,690 --> 00:42:24,130
+الـ C والـ D بيساوي ايش اسمه؟ بيساوي K هذه عبارة
+
+487
+00:42:24,130 --> 00:42:29,790
+عن مين؟ شايفين، مابديش أعيدك، بقى أكتب مرتين خليني
+
+488
+00:42:29,790 --> 00:42:33,890
+أكتبها على شورة بتصير Phi of F of X يعني بتصير
+
+489
+00:42:33,890 --> 00:42:42,250
+عبارة عن مين؟ Phi of F of X معايا؟
+
+490
+00:42:43,040 --> 00:42:48,280
+وهاد ايش اسمها؟ بدي أدخل السالب جوا يا جماعة أو
+
+491
+00:42:48,280 --> 00:42:56,440
+قبل ما أدخله، هد بيصير فاي فاي of F of X بعد
+
+492
+00:42:56,440 --> 00:43:01,560
+أذنكم، بدي أدخل السالب جوا ال infimum هتصير إيش ما
+
+493
+00:43:01,560 --> 00:43:06,240
+لها؟ supremum إذا بيصير هذا زائد وهد بتصير
+
+494
+00:43:06,240 --> 00:43:14,100
+supremum والسالب بدخل جواطيب .. عندي الـ Phi F of
+
+495
+00:43:14,100 --> 00:43:19,800
+Xوالـ Phi F of X للـ X اللي موجودة هنا الـ F of X
+
+496
+00:43:19,800 --> 00:43:25,460
+هدولة من مين جايت من الفترة C و D لأن زي ما قلنا F
+
+497
+00:43:25,460 --> 00:43:30,680
+of A و F of I subset من الجهة اللي هي C و D لذا كل
+
+498
+00:43:30,680 --> 00:43:35,040
+ال element هنا موجود هنا يعني بمعنى أخر ال .. ال
+
+499
+00:43:35,040 --> 00:43:38,720
+supreme من اللي هنا اللي هي ال K عندي ال absolute
+
+500
+00:43:38,720 --> 00:43:45,400
+value للـ Phi of T أكيد أكبر أو يساويالـ Phi سالب
+
+501
+00:43:45,400 --> 00:43:51,080
+Phi of T و أصغر أو يساوي الـ Phi of T أو بالسوية
+
+502
+00:43:51,080 --> 00:43:55,280
+حتى أكيد
+
+503
+00:43:55,280 --> 00:43:57,900
+خلّيني أخدها على خطواتين بتسويها دي لأن ال
+
+504
+00:43:57,900 --> 00:44:06,260
+absolute value Phi of T أكبر سوى مين؟ سالب ال Phi
+
+505
+00:44:06,260 --> 00:44:11,300
+of T فبصير عندي ال supremum اللي عندي اللي هالة
+
+506
+00:44:11,300 --> 00:44:18,410
+هذاSupermom لهذه اكيد اصغر او يساوياللي هو الـ
+
+507
+00:44:18,410 --> 00:44:22,170
+Supremum هذا اللي هو K لأن الـ Supremum هذا عالمين
+
+508
+00:44:22,170 --> 00:44:26,370
+على كل الفترة C وD هنا اللي هي عالمين على X
+
+509
+00:44:26,370 --> 00:44:30,890
+element in XK مانس واحد و XK لأن هذا Similarly
+
+510
+00:44:30,890 --> 00:44:35,910
+أكيد الـ Supremum للـ Absolute Value هيكون أكبر أو
+
+511
+00:44:35,910 --> 00:44:39,070
+يساوي الـ Supremum لهذه من جهتين أنه Absolute
+
+512
+00:44:39,070 --> 00:44:42,790
+Value وفي نفس الوقت اللي هو المجموع هذه جزئية من
+
+513
+00:44:42,790 --> 00:44:48,380
+اللي فوق إذن ده برضه بيكون أصغر أو يساويK اذا صار
+
+514
+00:44:48,380 --> 00:44:53,800
+automatic اللي هو هذه زائد هذه اصغر او يساوي مين
+
+515
+00:44:53,800 --> 00:45:01,440
+اتنين K من اي حاجة اصغر او يساوي اتنين K الان هذه
+
+516
+00:45:01,440 --> 00:45:06,160
+ال supremum زائد ال supremum هذه بس لا ياش لسانك
+
+517
+00:45:06,160 --> 00:45:11,900
+يا شباب و هذه K اصغر يساوي K زائد K عشان التلخيص
+
+518
+00:45:11,900 --> 00:45:19,460
+بيصير اشمله اتنين Kواضح إذا نخزن التانية اللي هي K
+
+519
+00:45:19,460 --> 00:45:24,480
+element in B بعد إذنكم بيصير عندي ال M K نقص بلاش
+
+520
+00:45:24,480 --> 00:45:30,200
+التفاصيل هذه خلصنا منها بيصير عندي ال M K نقص M K
+
+521
+00:45:30,200 --> 00:45:37,180
+small أصغر من تلد تلد طبعا أصغر من مين؟ من اتنين K
+
+522
+00:45:37,180 --> 00:45:43,280
+طبعا ال K هنا index ال K هذه capital هذه ال K مين؟
+
+523
+00:45:43,280 --> 00:45:48,590
+ال K اللي فوق هذههذا الـ K هو الاندكس للـ B وهذا
+
+524
+00:45:48,590 --> 00:45:50,690
+الـ K هو الاندكس للـ B يعني الـ K هذه ليست إيادة
+
+525
+00:45:50,690 --> 00:45:57,370
+هذه K هي الـ Supremum اللي فوق طيب نجمح
+
+526
+00:45:57,370 --> 00:46:03,190
+معلوماتنا ونبدأ نجمحها الآن ونكون حصلنا على اللي
+
+527
+00:46:03,190 --> 00:46:08,410
+بدهد اتنين Kأه لب تلخيص 2K يا شباب لما تشوفوها لب
+
+528
+00:46:08,410 --> 00:46:13,770
+تلخيص 2K الان اوجدت ال summation على مين؟ على ايه؟
+
+529
+00:46:13,770 --> 00:46:16,710
+بدي اوجد ال summation على ال P باقي ال indices لإن
+
+530
+00:46:16,710 --> 00:46:21,650
+هالجات بدي أقولكم ليش بالضيقة الان اذا صار عندي ال
+
+531
+00:46:21,650 --> 00:46:29,930
+summation لل Mk ناقص Mk تلدة تلدة في Xk minus Xk
+
+532
+00:46:29,930 --> 00:46:40,680
+minus واحد K element in B الآنمظبوط أصغر أو يساوي
+
+533
+00:46:40,680 --> 00:46:45,120
+اللي هو 2K برضه على ال summation هذا ال summation
+
+534
+00:46:45,120 --> 00:46:53,040
+اللي هو 2K كابيتال كابيتال هذه مالهاش علاقة بال K
+
+535
+00:46:53,040 --> 00:47:00,840
+ال element in B هنا في XK minus XK minus 1 أخدت ال
+
+536
+00:47:00,840 --> 00:47:03,700
+summation على الجهتين على كل ال K ال element in B
+
+537
+00:47:06,360 --> 00:47:20,220
+ماشي طيب وين أكتب طيب اتحملوني هلجيت عندي خليني
+
+538
+00:47:20,220 --> 00:47:25,300
+أكتب هنا أشرح أنا شوية بهمش خليني أشرح على ال ..
+
+539
+00:47:25,300 --> 00:47:33,740
+الكلف ال B الكلف ال B احنا من تعريف الكلف ال Bبدي
+
+540
+00:47:33,740 --> 00:47:39,160
+يكون الام كي ناقص ام كي أكبر شو مين؟ Delta صح؟
+
+541
+00:47:39,160 --> 00:47:41,980
+بيصير عندى الآن Delta يعني أصغر شو ام كي ناقص ام
+
+542
+00:47:41,980 --> 00:47:47,000
+كي اله small ايه بيسمعوا على ال Delta هنا؟ هو انا
+
+543
+00:47:47,000 --> 00:47:50,920
+على ال Delta بيصير هذا إيه شبال، هو واحد، معايا؟
+
+544
+00:47:50,920 --> 00:48:02,180
+فبيصير عندى الآن ال summation على الواحدXK-XK- XK
+
+545
+00:48:02,180 --> 00:48:07,650
+- XK- XK- XK- XK- XK- XK- XKأصغر أو يساوي mk نقص
+
+546
+00:48:07,650 --> 00:48:13,130
+mk على delta ماشي خد ال summation للجهتين ال
+
+547
+00:48:13,130 --> 00:48:19,030
+summation للجهتين xk minus xk minus واحد وانا ال
+
+548
+00:48:19,030 --> 00:48:24,310
+summation xk minus xk minus واحد k element in b
+
+549
+00:48:24,310 --> 00:48:28,850
+لأن هذا صح لل b بس لكي element in b مظبوط ممكن
+
+550
+00:48:28,850 --> 00:48:33,880
+حصلنا على هذا هيهاللي موصول وهذا واحد على دلتا
+
+551
+00:48:33,880 --> 00:48:38,720
+طلعها برا في ال summation اللي عندي هذا ماشي وصلنا
+
+552
+00:48:38,720 --> 00:48:45,620
+ل هيك إذا الآن صار عندي هذا أصغر أوي ساوي واحد على
+
+553
+00:48:45,620 --> 00:48:56,360
+دلتا في مين؟ في الان هذا اللي هو ال UPUF طب أنا دي
+
+554
+00:48:56,360 --> 00:49:00,500
+MK وMK اه مش MK تلد هذه اه
+
+555
+00:49:03,350 --> 00:49:07,230
+ليش؟ لأنه .. اشكروا ان انا كتبين هنا Mk و Mk تلتة
+
+556
+00:49:07,230 --> 00:49:15,130
+لأن هذه الـ A الـ B الـ B الـ K اللي فيها مصنفة
+
+557
+00:49:15,130 --> 00:49:19,530
+على أساس Mk اسمه نقص Mk أكبر زي دلتة فبصير عندى
+
+558
+00:49:19,530 --> 00:49:25,800
+الآنالـ Mk نقص Mk اللي هو أكبر أو يساوي Delta مش
+
+559
+00:49:25,800 --> 00:49:31,520
+Delta هذه هذه لمين؟ اللي مصنف عليها الـ B اللي هي
+
+560
+00:49:31,520 --> 00:49:35,860
+بالنسبة للـ F هذه مش لـ Alpha Composite F إذا هذا
+
+561
+00:49:35,860 --> 00:49:38,560
+أكبر أو يساوي Delta على Delta بيصير أكبر أو يساوي
+
+562
+00:49:38,560 --> 00:49:41,740
+واحد على ال summation اللي عملناها قبل بشوية أصغر
+
+563
+00:49:41,740 --> 00:49:44,720
+أو يساوي واحد على Delta في ال summation هذا الآن
+
+564
+00:49:44,720 --> 00:49:55,810
+.. الآن هذا مين هو؟ هو عبارة عن الـ UP و F مش كله
+
+565
+00:49:55,810 --> 00:50:01,210
+حتى جزء منه لأن ال U P و F ايش ماله ال M كده تبعته
+
+566
+00:50:01,210 --> 00:50:08,310
+ال U P و F ايش هو يا شباب هو عبارة عن ال summation
+
+567
+00:50:08,310 --> 00:50:14,090
+كي من عند واحد لعند N او من Zero لعند N من واحد
+
+568
+00:50:14,090 --> 00:50:25,770
+لعند N مظبوط لمين لأ ال Mك في xk minus xk minus
+
+569
+00:50:25,770 --> 00:50:30,070
+واحد هذا ال summation على مين؟ على كل ال case من
+
+570
+00:50:30,070 --> 00:50:35,290
+واحد لعند n بينما هذا ال summation لمين؟ بس للجزء
+
+571
+00:50:35,290 --> 00:50:39,130
+اللي هو في مين؟ في ال B فأكيد ال summation على هذه
+
+572
+00:50:39,130 --> 00:50:44,410
+أظهر أوي ساوي ال summation على هذه باشي لأن ال
+
+573
+00:50:44,410 --> 00:50:48,430
+summation على هذه ناقص هذه برضه بظهر أوي ساوي هذا
+
+574
+00:50:48,430 --> 00:50:55,190
+ناقص هذاليش؟ لأن الـ M K ناقص M نعمل كمية على جهة
+
+575
+00:50:55,190 --> 00:50:59,290
+نعملها
+
+576
+00:50:59,290 --> 00:51:03,690
+.. نعملها
+
+577
+00:51:03,690 --> 00:51:04,770
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+578
+00:51:04,770 --> 00:51:09,950
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+579
+00:51:09,950 --> 00:51:10,210
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+580
+00:51:10,210 --> 00:51:10,270
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+581
+00:51:10,270 --> 00:51:10,270
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+582
+00:51:10,270 --> 00:51:10,270
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+583
+00:51:10,270 --> 00:51:10,410
+.. نعملها .. نعملها .. نعملها .. نعملها .. نعملها
+
+584
+00:51:10,410 --> 00:51:18,420
+.. نعملها .. نعملها .. نعملها ..أيش هيساوي الـ
+
+585
+00:51:18,420 --> 00:51:24,940
+summation للـ mk ناقص mk small في xk minus xk
+
+586
+00:51:24,940 --> 00:51:30,880
+minus واحد هذا k من عند واحد لعند n مظبوط ولا لأ؟
+
+587
+00:51:30,880 --> 00:51:35,360
+الان أكيد أكيد هذا لأن هذا موجبة وهذا موجبة لأن
+
+588
+00:51:35,360 --> 00:51:42,060
+هذا أكيد أكبر أو يساوي ال summation لل mk ناقص mk
+
+589
+00:51:43,610 --> 00:51:51,390
+xk-xk-1 k element in B لأن هذوله جزء من هذوله يعني
+
+590
+00:51:51,390 --> 00:51:56,150
+نقصت من ال summation هذا بعض ال terms اللي أكبر أو
+
+591
+00:51:56,150 --> 00:51:57,810
+يساوي 0 إذا نقص ال summation
+
+592
+00:52:12,960 --> 00:52:19,320
+طيب المفروض ان هذا واضح خلّينا الان نتطلع عليه
+
+593
+00:52:19,320 --> 00:52:25,060
+عندى اذا ال summation لل XK minus XK minus واحد K
+
+594
+00:52:25,060 --> 00:52:30,860
+element بيطلعلي اصغر او يساوي واحد على دلتا في ال
+
+595
+00:52:30,860 --> 00:52:39,040
+U ناقص L بس هذا انا مخزنه لسه عندى ال U ناقص Lاللي
+
+596
+00:52:39,040 --> 00:52:40,940
+هو هذا كله على الـ partition اللي بنشتغل عليه اللي
+
+597
+00:52:40,940 --> 00:52:45,860
+لاجيناه في الأول الـU-L هذه أصغر من مين طلعت أصلا
+
+598
+00:52:45,860 --> 00:52:48,880
+من دلتة تربيع لأن هذا اعتمدنا عليه على
+
+599
+00:52:48,880 --> 00:52:53,660
+الـintegrability للـF الأول إذا لازمتني هنا اللي
+
+600
+00:52:53,660 --> 00:52:57,800
+هو هذا أصغر من دلتة تربيع في واحد على دلتة بيصير
+
+601
+00:52:57,800 --> 00:53:04,160
+أصغر من مين؟ من دلتة وهذا كلهفى ضوء الـ Delta اللى
+
+602
+00:53:04,160 --> 00:53:08,060
+أنا بدأت فيها أصغر من مين أو أثبتتها أصغر من
+
+603
+00:53:08,060 --> 00:53:11,860
+Epsilon إبراهيم اللى وجهتها أصغر من Epsilon
+
+604
+00:53:11,860 --> 00:53:16,340
+إبراهيم إذا صار عندى ال summation هذا أصغر من Ash
+
+605
+00:53:16,340 --> 00:53:22,280
+من Epsilon إبراهيم لمين؟ لل K اللى Ash في B إذا
+
+606
+00:53:22,280 --> 00:53:25,340
+اللى وصلت إليه الآن ال summation
+
+607
+00:53:28,540 --> 00:53:32,980
+الـ summation اللي وصلت إليه الـ summation للـ xk
+
+608
+00:53:32,980 --> 00:53:41,600
+ناقص xk minus 1 k element in B أصغر من إبسلون
+
+609
+00:53:41,600 --> 00:53:46,920
+إبراهيم الآن
+
+610
+00:53:46,920 --> 00:53:53,760
+بس إحنا أثبتنا إن الـ summation هذا أيه أصغر أو
+
+611
+00:53:53,760 --> 00:54:00,430
+يساوي 2kاتنين كيف ال summation هذا مظبوط بدي اعوض
+
+612
+00:54:00,430 --> 00:54:06,850
+الان اشيل هذا بيصير عندي هاي شايفين يا شباب هاي
+
+613
+00:54:06,850 --> 00:54:14,070
+عند ال summation هذا اثبته انه اصغر او يساوي اللي
+
+614
+00:54:14,070 --> 00:54:24,580
+هو epsilon prime واضح ومن اربعةبصير عندى بعوض ال
+
+615
+00:54:24,580 --> 00:54:28,980
+summation أصغر من 2 كيف ال summation اللى أثبتناه
+
+616
+00:54:28,980 --> 00:54:34,840
+بحيته بس قبل شوية ال summation أصغر من 2 كيف ال
+
+617
+00:54:34,840 --> 00:54:40,140
+summation بشيل هذا اللى هو هذا و أقول أصغر من مين
+
+618
+00:54:40,140 --> 00:54:43,760
+من epsilon prime بصير هذا المقدار اللى هو هذا ال
+
+619
+00:54:43,760 --> 00:54:45,680
+summation هيه
+
+620
+00:54:47,550 --> 00:54:54,290
+هيو هذا هيو أصغر من من 2k في ash في ال epsilon
+
+621
+00:54:54,290 --> 00:55:00,110
+prime إذا طلع عندي الجزء الثاني summation خلصنا من
+
+622
+00:55:00,110 --> 00:55:07,610
+هذا صار عندي الجزء الثاني اللي أثبته summation لل
+
+623
+00:55:07,610 --> 00:55:19,240
+mk تلدى ناقص mk تلدىفي xk-xk-1 k element in B أصغر
+
+624
+00:55:19,240 --> 00:55:26,980
+من مين طلع من epsilon prime في 2k ابسلون برايم في
+
+625
+00:55:26,980 --> 00:55:36,650
+2k الآن خلصنااحسبولي إذا احنا لقينا ال partition B
+
+626
+00:55:36,650 --> 00:55:44,330
+لكل option أكبر من 0 لقينا B بحيث أنه U B of I
+
+627
+00:55:44,330 --> 00:55:54,210
+composite F ناقص ال B of I composite F هذا أشماله
+
+628
+00:55:54,210 --> 00:55:59,310
+بساوياللي هو الـ summation اللي هيضع على كل الـ K
+
+629
+00:55:59,310 --> 00:56:02,390
+و الـ summation اللي هيضع على كل الـ K و الـ K
+
+630
+00:56:02,390 --> 00:56:05,910
+جزّانها لجزين إيش في الـ A و إيش في الـ B إذا
+
+631
+00:56:05,910 --> 00:56:14,110
+بيساوي اللي هو الـ summation ل الـ A الـ MK ناقص
+
+632
+00:56:14,110 --> 00:56:23,140
+MK تلدة تلدةفي xk-xk-1 k من واحد ل عند n صح ولا لأ
+
+633
+00:56:23,140 --> 00:56:28,220
+اه طبعا هو التعريف هذا اللي هو بيثاويالان الـ K
+
+634
+00:56:28,220 --> 00:56:32,460
+اللي عندك جزّالت لجزين واحد في الـ A واحد في مين؟
+
+635
+00:56:32,460 --> 00:56:37,080
+في الـ B، إذاً هذا الـ summation الكل بيساوي ال
+
+636
+00:56:37,080 --> 00:56:41,340
+summation على K في الـ A زائد ال summation نفسه
+
+637
+00:56:41,340 --> 00:56:45,720
+على K وين ماله؟ في الـ B ال summation لمين؟ لهذه
+
+638
+00:56:45,720 --> 00:56:53,280
+اللي هي MK تلد ناقص MK تلد small في XK minus XK
+
+639
+00:56:53,280 --> 00:57:01,450
+minus واحد زائد MKتلدة ناقص mk small في xk minus
+
+640
+00:57:01,450 --> 00:57:08,990
+xk minus واحد اه طيب يعني الان هذا ال U و هذا ال L
+
+641
+00:57:08,990 --> 00:57:12,030
+صار بساوي هذا الجزء و هذا الجزء و هذا السبب أصلا
+
+642
+00:57:12,030 --> 00:57:17,590
+اللي خلانه أجزء هذا طلع لي الان مين هو طلع لي أصغر
+
+643
+00:57:17,590 --> 00:57:22,970
+من epsilon prime أصغر من epsilon prime في b minus
+
+644
+00:57:22,970 --> 00:57:32,330
+aوالتاني طلع أصغر من E' في 2K E' في 2K يعني هدول
+
+645
+00:57:32,330 --> 00:57:40,140
+الاتنين مع بعض إيش بيساون؟ بيساون E' فيه2k بيزاق
+
+646
+00:57:40,140 --> 00:57:56,220
+بيزاق بيزاق بيزاق بيزاق بيزاق بيزاق
+
+647
+00:57:56,310 --> 00:58:04,950
+على b minus a زي اتنين k في اتنين k زي b minus a
+
+648
+00:58:04,950 --> 00:58:10,030
+وهذه بتروح مع حد بيساوي ايش ابسلون اذا اللي وصلتله
+
+649
+00:58:11,120 --> 00:58:17,600
+لكل إبسلون أكبر من سفر لكل
+
+650
+00:58:17,600 --> 00:58:22,480
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+651
+00:58:22,480 --> 00:58:25,660
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+652
+00:58:25,660 --> 00:58:27,320
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+653
+00:58:27,320 --> 00:58:28,360
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+654
+00:58:28,360 --> 00:58:32,200
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+655
+00:58:32,200 --> 00:58:34,180
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+656
+00:58:34,180 --> 00:58:36,880
+إبسلون أكبر من سفر لكل إبسلون أكبر من سفر لكل
+
+657
+00:58:36,880 --> 00:58:47,340
+إبسلون أكبر من سفرطيب الآن نيجي بدنا اللي هو ناخد
+
+658
+00:58:47,340 --> 00:58:55,660
+اللي هو تطبيقات على هذه النظرية ونشوف
+
+659
+00:58:55,660 --> 00:59:02,600
+كيف بدنا نطبق هذه النظرية للوصول إلى نتاج أخرى
+
+660
+00:59:02,600 --> 00:59:09,280
+تتعلق بخواص التكامل أو بخواص تكامل لريمان لأن
+
+661
+00:59:09,280 --> 00:59:15,300
+عرفناو جوابنا على اللي هو المجموع وعلى حسب ضرب K
+
+662
+00:59:15,300 --> 00:59:20,100
+في ال .. في ال F ولقنا على ال composition و تحت اي
+
+663
+00:59:20,100 --> 00:59:24,360
+ضارف كان ف I composite F is integrable الان نكمل
+
+664
+00:59:24,360 --> 00:59:30,120
+اللي هو ناخد اللي هو ال corollary اللي عندي اللي
+
+665
+00:59:30,120 --> 00:59:39,150
+هي بتقولي اللي هي من تلت فروع لو كانتI عبارة عن
+
+666
+00:59:39,150 --> 00:59:43,990
+الـ closed interval A و B و F من I لعند R is
+
+667
+00:59:43,990 --> 00:59:47,370
+integrable on I then the absolute value of a
+
+668
+00:59:47,370 --> 00:59:50,630
+function F obtained by the absolute value is
+
+669
+00:59:50,630 --> 00:59:54,410
+integrable on Iوالـ absolute value of integration
+
+670
+00:59:54,410 --> 00:59:56,470
+أصغر يساوي الـ integration للـ absolute value للـ
+
+671
+00:59:56,470 --> 01:00:02,810
+F اللي هي أصغر يساوي K K في ال B minus A حيث ال K
+
+672
+01:00:02,810 --> 01:00:06,750
+هذه من أين جاية هي ال bound لل F of X لأن F is
+
+673
+01:00:06,750 --> 01:00:12,260
+integrable إذا أكيد boundedفن element in F يعني
+
+674
+01:00:12,260 --> 01:00:15,280
+اللي بتقولي هذا باختصار ده كانت F is integrable ال
+
+675
+01:00:15,280 --> 01:00:18,100
+absolute value لل F إيش ما لها integrable وال
+
+676
+01:00:18,100 --> 01:00:20,160
+absolute value لل integration أصغر سوى ال
+
+677
+01:00:20,160 --> 01:00:22,860
+integration لل absolute value أصغر سوى K اللي هي
+
+678
+01:00:22,860 --> 01:00:26,580
+ال maximum لل أو خلّيني أقول ال bound لل F of X
+
+679
+01:00:26,580 --> 01:00:31,080
+absolute value ل F of X في B minus Aالجزء الثاني
+
+680
+01:00:31,080 --> 01:00:35,460
+اللي هو هتطلع لي برضه الـ Fn is integrable لأن Fn
+
+681
+01:00:35,460 --> 01:00:41,560
+هو إنتجر
+
+682
+01:00:42,100 --> 01:00:46,780
+الآن إذا كان الشغل الثاني إذا كان فيه Delta بحيث
+
+683
+01:00:46,780 --> 01:00:50,400
+أن F of X أكبر يساوي Delta يعني F of X أكبر أو
+
+684
+01:00:50,400 --> 01:00:53,900
+يساوي Delta أكبر يساوي Delta ليش؟ على أساس إنه
+
+685
+01:00:53,900 --> 01:00:58,140
+يضمن مقلوب هيكون bounded بيصير 1 على F of X أصغر
+
+686
+01:00:58,140 --> 01:01:00,680
+أو يساوي 1 على Delta يعني بمعنى أكتر F is bounded
+
+687
+01:01:00,680 --> 01:01:04,720
+إذن بحقله يحكي عن ال Integrability ل1 على F ويقول
+
+688
+01:01:04,720 --> 01:01:07,780
+لي تحت الظرف هذا لو كانت F of X أكبر يساوي Delta
+
+689
+01:01:08,470 --> 01:01:12,210
+وDelta أكبر من 0 هيكون 1 على الـ F إيه اشمالها is
+
+690
+01:01:12,210 --> 01:01:15,830
+integrable on I خلّيني أتبرجح واحدة واحدة اللي
+
+691
+01:01:15,830 --> 01:01:24,690
+عنها ال gate F عندي من I لعين بR is integrable إذا
+
+692
+01:01:24,690 --> 01:01:29,350
+there exists K اللي هو أكبر من 0 such that
+
+693
+01:01:29,350 --> 01:01:33,330
+absolute value of F of Xأصغر أو يساوي إياش K مدام
+
+694
+01:01:33,330 --> 01:01:36,430
+ان تقرأ بالـ F إذا أكيد is bounded إذا ال absolute
+
+695
+01:01:36,430 --> 01:01:43,030
+value of X أصغر أو يساوي ال K طيب الآن يا جماعة
+
+696
+01:01:43,030 --> 01:01:47,230
+عندي هاد ال corollary أصلا اللي جابلها بدي أظبطلي
+
+697
+01:01:47,230 --> 01:01:54,070
+two functions بدي أعرف الآن صارت عند ال F of I ال
+
+698
+01:01:54,070 --> 01:02:00,230
+F of I صارت ال F of I أكيد subset بين نقص K ومين
+
+699
+01:02:00,230 --> 01:02:01,210
+أو K
+
+700
+01:02:07,180 --> 01:02:19,960
+طيب يعني الآن لو جيت عرفت Phi من اللي هو عنده ناقص
+
+701
+01:02:19,960 --> 01:02:27,990
+K لعند ال K لعند ال R عرفتها على أساسفاي ما هو
+
+702
+01:02:27,990 --> 01:02:30,950
+absolute value إذا مضحكت تجيب ال absolute value
+
+703
+01:02:30,950 --> 01:02:38,250
+فاي of T بساوة absolute value لمن؟ لل T ماشي الحال
+
+704
+01:02:38,250 --> 01:02:42,430
+هي الدلال عندى أكيد ال absolute value إيش ما لها
+
+705
+01:02:42,430 --> 01:02:47,370
+is continuous مدام ال absolute value الفاي is
+
+706
+01:02:47,370 --> 01:02:52,870
+continuous و ال F معطيله إيها integrable أيضا by
+
+707
+01:02:52,870 --> 01:03:00,670
+the above theoremفاي كومبوزيت F is integrable فاي
+
+708
+01:03:00,670 --> 01:03:07,610
+كومبوزيت F of T بيساوي فاي of F of T اللي هي
+
+709
+01:03:07,610 --> 01:03:12,330
+بتساوي ايش؟ واحدة absolute value فاي of T بتساوي
+
+710
+01:03:12,330 --> 01:03:17,490
+absolute value لمين؟ للـ F of T إذا شار عندي
+
+711
+01:03:17,490 --> 01:03:23,460
+absolute value للـ F is integrableإذاً مدامة Phi
+
+712
+01:03:23,460 --> 01:03:27,460
+is continuous و F integrable لأن حسب النظرية Phi
+
+713
+01:03:27,460 --> 01:03:30,640
+composite F is integrable و Phi composite F هي
+
+714
+01:03:30,640 --> 01:03:33,400
+طلعت من الـ absolute value للـ F طبعا Phi
+
+715
+01:03:33,400 --> 01:03:36,880
+composite F من وين هتشتغل من عند اللي هي ال
+
+716
+01:03:36,880 --> 01:03:42,920
+interval I لعند مين لعند R لأن Phi اللي هو F of T
+
+717
+01:03:42,920 --> 01:03:49,980
+هيجي يقعد في الفترة هذه و Phi هترسل اللي بيجي لوين
+
+718
+01:03:49,980 --> 01:03:55,250
+لعند Iماضح ها طيب الان عندي بدي أثبت أن ال
+
+719
+01:03:55,250 --> 01:03:58,590
+integration ال absolute value ال integration أصغر
+
+720
+01:03:58,590 --> 01:04:01,350
+ساوي ال integration ل absolute value هذه صارت
+
+721
+01:04:01,350 --> 01:04:07,670
+بعضها معلومات صادقة اللي هو عندي ال F أكبر أو ساوي
+
+722
+01:04:07,670 --> 01:04:11,610
+اللي هي absolute value لل F بالسالب و أصغر أو ساوي
+
+723
+01:04:11,610 --> 01:04:14,790
+ال absolute value ل 100 لل F صارت الان احنا أتبعت
+
+724
+01:04:14,790 --> 01:04:17,990
+أن ال absolute value is integrableإذا الان بما أن
+
+725
+01:04:17,990 --> 01:04:21,270
+F بين هذولة الدالتين حسب نظريها أخدناها المرة
+
+726
+01:04:21,270 --> 01:04:24,910
+الماضية أو remark إذا بيصير ال integration لل
+
+727
+01:04:24,910 --> 01:04:28,590
+absolute value لل F من A ل B أصغر أساوي ال
+
+728
+01:04:28,590 --> 01:04:32,330
+integration لل F من A ل B أصغر أساوي ال
+
+729
+01:04:32,330 --> 01:04:38,190
+integration لل F من A ل B وهذا إيش معناته؟ هذا
+
+730
+01:04:38,190 --> 01:04:43,610
+معناته إذا ال absolute value لل integration لل F
+
+731
+01:04:43,610 --> 01:04:49,150
+من A ل B أصغر أساوياللي هو ال integration لأ لأبسل
+
+732
+01:04:49,150 --> 01:04:57,690
+و لأ لأف من A لB ماشي الحالة و طبيعي عندي هذه
+
+733
+01:04:57,690 --> 01:05:02,270
+أثبتناها بهذه و طبيعي مدام عنا لسه ما
+
+734
+01:05:07,890 --> 01:05:12,470
+أو ملاحظة سابقة لبس الـ integration من a لb لبس ال
+
+735
+01:05:12,470 --> 01:05:18,070
+value of f of x أصغر أو شاوي اللي هو من k في طول
+
+736
+01:05:18,070 --> 01:05:27,050
+الفترة بـ-aمع هذه together بنحصل على اللي هو
+
+737
+01:05:27,050 --> 01:05:30,750
+المطلوب ال absolute value ال integration أصغر سوا
+
+738
+01:05:30,750 --> 01:05:34,690
+ال integration لل absolute value أصغر سوا K في ال
+
+739
+01:05:34,690 --> 01:05:39,510
+B minus A هذه اللي هو الجزء الأول من ال corollary
+
+740
+01:05:39,510 --> 01:05:44,310
+خلينا نشوف الجزء الثاني من ال corollary برضه اللي
+
+741
+01:05:44,310 --> 01:05:51,270
+هو برهانه سهل و برهانه اللي هو مشابهللمنطق اللي
+
+742
+01:05:51,270 --> 01:05:57,070
+حكيناها عندنا قبل بشوية طيب صلى الله عننا يا عزيزي
+
+743
+01:05:57,070 --> 01:06:01,410
+عليه الصلاة والسلام الآن يا شباب عندي اللي هو
+
+744
+01:06:01,410 --> 01:06:05,110
+الجزء الثاني بدنا نثبت أن الـ F أسئن is integrable
+
+745
+01:06:05,110 --> 01:06:09,110
+في حالة الـ F integrable لأن أكيد شكلنا بدنا نيجي
+
+746
+01:06:09,110 --> 01:06:16,950
+عند الـ F من I لعند R is integrableالان عندي اللي
+
+747
+01:06:16,950 --> 01:06:21,470
+هو .. بدي أسأل أن أثبت انها Integrable فلو جيت أنا
+
+748
+01:06:21,470 --> 01:06:30,170
+عرفت F F أو اللي هو Phi function من اللي هو domain
+
+749
+01:06:30,170 --> 01:06:35,150
+اللي هو F of I من اللي هو range ال F range ال F
+
+750
+01:06:35,150 --> 01:06:45,640
+range ال F لعند ال R بحيث ان اقول Phi of Tبساوي T
+
+751
+01:06:45,640 --> 01:06:51,120
+أُس أنت five of T أيش بيساوي بيساوي T أُس أنت أو
+
+752
+01:06:51,120 --> 01:06:54,180
+إذا كان بدك اللي هو بدكاش تكون ال range ل F خلاص
+
+753
+01:06:54,180 --> 01:06:57,460
+من عارفها من وين من اللي من اللي قبل بشوية من ناقص
+
+754
+01:06:57,460 --> 01:07:01,980
+K لعند K عارفين ليه؟ لأن ال absolute value لل F of
+
+755
+01:07:01,980 --> 01:07:06,840
+X أصغر مش هو ال K إذا أكيد لما ترمي ال F of X
+
+756
+01:07:06,840 --> 01:07:11,410
+هتكون ترمي في الفترة من ناقص K لعندكأو subset
+
+757
+01:07:11,410 --> 01:07:15,970
+بيصير الـ F of I subset من ناقص K و K إذاً صارت
+
+758
+01:07:15,970 --> 01:07:22,510
+الـ Phi composite F Phi composite F is defined من
+
+759
+01:07:22,510 --> 01:07:31,140
+I لعند ال R وعندي Phi composite F ofT بيساوي Phi
+
+760
+01:07:31,140 --> 01:07:37,200
+of F of T ويساوي F of T اللي هو Phi of T بيساوي T
+
+761
+01:07:37,200 --> 01:07:41,560
+أُس N بيصير Phi of F of T اللي هي عبارة عن F of T
+
+762
+01:07:41,560 --> 01:07:48,540
+يا جماعة أُس N الآن كل الأمور طيبة ومنيحة ليش
+
+763
+01:07:48,540 --> 01:07:53,260
+ووصلنا كمانعندي F-integrable و F-continuous
+
+764
+01:07:53,260 --> 01:07:57,420
+هتتدللوا الـ C أشمالها اللي هو الـ T أسان is
+
+765
+01:07:57,420 --> 01:08:03,540
+continuous اللي جزء من البلونوميال بيصير عندي FI
+
+766
+01:08:03,540 --> 01:08:06,660
+is continuous و F-integrable لذلك حسب النظرية اللي
+
+767
+01:08:06,660 --> 01:08:10,680
+قبل و شوية إيش هيطلع عندي FI composite F is
+
+768
+01:08:10,680 --> 01:08:16,440
+integrable بمعنى أن ف FN is integrable فكرة معادة
+
+769
+01:08:16,440 --> 01:08:22,570
+أصلالأنه سنة يعني اللي هو الأمور واضحة الآن ضال
+
+770
+01:08:22,570 --> 01:08:30,190
+عند مين اللي هو ال .. ال .. ال .. ال .. نثبت الجزء
+
+771
+01:08:30,190 --> 01:08:36,010
+الثالث من ال corollary اللي هي بدنا نثبت أنه لو
+
+772
+01:08:36,010 --> 01:08:43,490
+كانت ال F of X أكبر أو شويه واحد على دلتااللي هي
+
+773
+01:08:43,490 --> 01:08:47,530
+for every x element in I وعندي there exists دلتا
+
+774
+01:08:47,530 --> 01:08:52,390
+كمان وصفر ساشرات إذا كان هذا متحقق إذن هتطلع عندى
+
+775
+01:08:52,390 --> 01:08:55,970
+اللي هي الواحد على ال F بيدثبت أن الواحد على ال F
+
+776
+01:08:55,970 --> 01:08:58,950
+إيش is integrable بدنا نعمل .. نجيب continuous
+
+777
+01:08:58,950 --> 01:09:02,890
+function أكيد كل جمجال نجيب مقلوبة دلة يعني في حوض
+
+778
+01:09:02,890 --> 01:09:09,800
+five من ناقص K لعند Kلعند R وخد فاي عارفين ما هو
+
+779
+01:09:09,800 --> 01:09:13,000
+ال scale كده سبب اللي قلت قبل شوية فايف دي ايش
+
+780
+01:09:13,000 --> 01:09:17,600
+متساوي اكيد كلكم حيقول واحد على تي ماشي الحال واحد
+
+781
+01:09:17,600 --> 01:09:26,180
+على تي بس اه لأ عدم ديش اقدر اعمل hand افعل عشان
+
+782
+01:09:26,180 --> 01:09:31,250
+مانبدعش في السفرة يا شباب خدوها من عند Deltaلعند
+
+783
+01:09:31,250 --> 01:09:36,470
+مين؟ لعند K ليش .. ليش .. ليش اللي هو عكيفك؟ أه F
+
+784
+01:09:36,470 --> 01:09:40,670
+of X اللي هو absolute value أصغر أو تسوى K يعني F
+
+785
+01:09:40,670 --> 01:09:46,160
+of X أصغر أو تسوى K وأكبر أو تسوى نقص Kو أثناء هذه
+
+786
+01:09:46,160 --> 01:10:12,580
+أكيد ماعطينيها أكبر شو مين دلتا
+
+787
+01:10:14,710 --> 01:10:21,230
+اللي هو .. إيه الكلام هذا؟ مشروع هو بيصير عنده ..
+
+788
+01:10:21,230 --> 01:10:24,170
+اللي هو Phi of T بسواحدة على T مافيش أي مشكلة،
+
+789
+01:10:24,170 --> 01:10:27,470
+مافيش مشاكل أسفار، مافيش مشاكل كده، إذا صارت ده
+
+790
+01:10:27,470 --> 01:10:31,490
+اللي إيش مالها؟ continuous continuous هفهم إن فهمت
+
+791
+01:10:31,490 --> 01:10:34,810
+القصة Phi composite of F continuous و integrable
+
+792
+01:10:34,810 --> 01:10:38,310
+إذا كلها على بعضها integrable يعني Phi of T بتصير
+
+793
+01:10:38,310 --> 01:10:44,750
+Phiof F of T يعني بتساوي واحد على F of T يعني صارت
+
+794
+01:10:44,750 --> 01:10:51,650
+الدالة واحد على F is integrable وهك
+
+795
+01:10:51,650 --> 01:10:55,670
+مكون احنا خلصنا اللي هو ال corollary ضال عندى الآن
+
+796
+01:10:55,670 --> 01:11:01,570
+نجاوب على السؤال الثاني هل حاصل ضرب ضرب دالتين
+
+797
+01:11:01,570 --> 01:11:06,650
+integrable is integrable بنقول اه integrable يعني
+
+798
+01:11:06,650 --> 01:11:11,380
+بمعنى اخر بقول ليهالنظرية اللي هي اللي بعدها 7,2,7
+
+799
+01:11:11,380 --> 01:11:18,020
+بتقول ما يلي بتقول لو كانت F و G integrable يعني
+
+800
+01:11:18,020 --> 01:11:25,480
+لو كان عندي خليني اكتب بالأصمر أفضل F و G من I
+
+801
+01:11:25,480 --> 01:11:33,700
+لعند R كانت integrable functions بيعطيني هذا ان FG
+
+802
+01:11:33,700 --> 01:11:42,500
+من I لعند R برضه ايش ماله؟Integrable Function الـ
+
+803
+01:11:42,500 --> 01:11:46,740
+L صار إن حصيلة من المعلومات بتسهل علي الوصول
+
+804
+01:11:46,740 --> 01:11:53,820
+للنتيجة و Proof و Proof بما أن F is Integrable إذن
+
+805
+01:11:53,820 --> 01:11:57,700
+فال Corollary اللي قبل بشوية أكيد F تربيع اللي هي
+
+806
+01:11:57,700 --> 01:12:01,040
+Integrable و G Integrable من النظرية اللي قبل
+
+807
+01:12:01,040 --> 01:12:04,940
+بشوية و Corollary برضه إشمالها G تربيع برضه
+
+808
+01:12:04,940 --> 01:12:10,060
+Integrable صح ولا لا يا جماعة صحطيب F تربيع
+
+809
+01:12:10,060 --> 01:12:14,260
+Integrable و G تربيع Integrable و برضه من نظرية
+
+810
+01:12:14,260 --> 01:12:18,580
+سابقة مدام F و G Integrable إذا F زائد G برضه إيش
+
+811
+01:12:18,580 --> 01:12:24,980
+Integrable لأ F زائد G تربيع Integrable كمان لأن F
+
+812
+01:12:24,980 --> 01:12:27,800
+و G Integrable أدت ل F زائد G Integrable و F زائد
+
+813
+01:12:27,800 --> 01:12:31,640
+G Integrable أدت من القرول اللي قبل شوية أنه تربيح
+
+814
+01:12:31,640 --> 01:12:40,520
+يكون إيش Integrable طيب خلصنا إذا إذاF تربيع زائد
+
+815
+01:12:40,520 --> 01:12:49,780
+D تربيع is integrable صح؟ مظبوط وناقص F تربيع
+
+816
+01:12:49,780 --> 01:12:54,220
+وناقص D تربيع برضه integrable مظبوط برضهلأنه اللي
+
+817
+01:12:54,220 --> 01:13:00,060
+هو عندي اللي هو ثابت في هذه integrable و ثابت في
+
+818
+01:13:00,060 --> 01:13:03,560
+ال integrable integrable ومجموح انصار انتجرابل إذا
+
+819
+01:13:03,560 --> 01:13:09,020
+انصار هذا integrable زائد F زائد G لكل تربيع هذه
+
+820
+01:13:09,020 --> 01:13:12,980
+integrable وهذه integrable وهذه integrable مجموحين
+
+821
+01:13:12,980 --> 01:13:17,520
+هذا إيش بيساويهذا .. هذا integrable و هذا
+
+822
+01:13:17,520 --> 01:13:19,860
+integrable و هذا integrable إذا المجموع integrable
+
+823
+01:13:19,860 --> 01:13:23,220
+إذا هذا كله على بعض integrable طب هذا مين هو؟ هذا
+
+824
+01:13:23,220 --> 01:13:29,160
+عبارة عن F تربيع زاد G تربيع ناقص اللي هو إيش زاد
+
+825
+01:13:29,160 --> 01:13:35,760
+2FG بيصير عبارة عن 2F main G صارت 2FG integrable
+
+826
+01:13:35,760 --> 01:13:40,840
+طب لو جينا قولنا خل نص هنا يعني ضربنا ثابت في
+
+827
+01:13:40,840 --> 01:13:43,960
+integrable إذا هيطلع اللي هو كله integrable إذا FG
+
+828
+01:13:43,960 --> 01:13:48,520
+إيش ما لهاis integrable صار عندي الان fg
+
+829
+01:13:48,520 --> 01:13:54,040
+integrable تابعا لأن f تربيع و g تربيع و f زائد g
+
+830
+01:13:54,040 --> 01:13:59,860
+كل تربيع حاصل جمحين و ضرب النصفي هنا ثابت و بيطلع
+
+831
+01:13:59,860 --> 01:14:04,720
+عبارة عن integrable function نيجي لآخر اللي هو
+
+832
+01:14:04,720 --> 01:14:11,850
+نقطة في الفي ال .. في ال .. في ال section 7-2 اللي
+
+833
+01:14:11,850 --> 01:14:14,790
+هو السؤال اللي سألناه في الأول قولنا لو كانت F is
+
+834
+01:14:14,790 --> 01:14:19,570
+integrable function و Phi is integrable هل Phi
+
+835
+01:14:19,570 --> 01:14:24,310
+composite F is integrable قولنا أكيد هو لو .. في
+
+836
+01:14:24,310 --> 01:14:28,430
+البداية قولنا أن Phi composite F need not to be
+
+837
+01:14:28,430 --> 01:14:32,280
+integrableإذا the composition of two integrable
+
+838
+01:14:32,280 --> 01:14:35,720
+functions need not to be integrable يعني هذا إعلان
+
+839
+01:14:35,720 --> 01:14:38,340
+the composition of two integrable functions need
+
+840
+01:14:38,340 --> 01:14:42,420
+not to be integrable but if phi the first one is
+
+841
+01:14:42,420 --> 01:14:46,340
+continuous then phi composite of f is integrable
+
+842
+01:14:46,340 --> 01:14:50,620
+كما شفنا في العقل اللي هو النظرية الأولى طيب نيجي
+
+843
+01:14:50,620 --> 01:14:53,000
+الآن المثال الأخير بيقولنا يا جماعة the
+
+844
+01:14:53,000 --> 01:14:55,760
+composition of integrable functions need not to be
+
+845
+01:14:55,760 --> 01:14:59,540
+integrableفي عندى سؤالين اصلا معاكم homework
+
+846
+01:14:59,540 --> 01:15:03,860
+السؤال الاول بيقوللنا لو كانت f of x بتساوي واحد
+
+847
+01:15:03,860 --> 01:15:07,540
+ده كانت x بتساوي سفر و zero ده كانت x is
+
+848
+01:15:07,540 --> 01:15:11,760
+irrational و كانت f of x بتساوي واحد لما x بيساوي
+
+849
+01:15:11,760 --> 01:15:16,160
+m على n حيث ال m و ال n عبارة عن integers و العامل
+
+850
+01:15:16,160 --> 01:15:19,740
+المشترك الاعلى بينهم ايه اشمله بادر بيساوي واحد
+
+851
+01:15:19,740 --> 01:15:23,540
+يعني شولنا كل العامل المشترك اللى بينهم و كتبنا x
+
+852
+01:15:23,540 --> 01:15:28,900
+بتساوي m على nطيب، هذه عبارة عن دالة معرفة من الـ0
+
+853
+01:15:28,900 --> 01:15:32,460
+والـ1 لعند الـR يعني هذي X is rational well في
+
+854
+01:15:32,460 --> 01:15:36,320
+الفترة 0 و1 و X بيبقى بيساوي M على N يعني rational
+
+855
+01:15:36,320 --> 01:15:41,260
+في الفترة 0 و1 وفرضنا لل X بتساوي صفر قيمته إيش
+
+856
+01:15:41,260 --> 01:15:48,070
+بتساوي؟ بتساوي 1 هذي الآن by exercise 7.1.11مطلوب
+
+857
+01:15:48,070 --> 01:15:51,790
+منك أنك تثبت أن F is يشملها is integrable on I
+
+858
+01:15:51,790 --> 01:15:55,070
+ويوم ما تعرفوش تحلوه ان شاء الله بنحلوه و بنصوره
+
+859
+01:15:55,070 --> 01:16:00,030
+بإذن الله الآن السؤال التاني اللي هعتمد عليه برضه
+
+860
+01:16:00,030 --> 01:16:03,170
+أنه ال function التاني اللي هي G من I لعند R هذه
+
+861
+01:16:03,170 --> 01:16:07,790
+سهلة أصلا لإثباتها be defined by G of X step
+
+862
+01:16:07,790 --> 01:16:12,090
+function بسيطة يعنيأو خلّيني أقول فيها jump و بس
+
+863
+01:16:12,090 --> 01:16:17,030
+jump على نقطة بس جيوب x بيساوي 0 إذا كانت x بيساوي
+
+864
+01:16:17,030 --> 01:16:21,410
+0 و بيساوي 1 إذا كانت x في الفترة من 0 لعند 1 اللي
+
+865
+01:16:21,410 --> 01:16:25,410
+هو a closed عند الواحد يعني الآن هذا ال function
+
+866
+01:16:25,410 --> 01:16:29,110
+بيقوللي برضه في ال exercise مطلوب برضه في exercise
+
+867
+01:16:29,110 --> 01:16:33,090
+717 برضه اللي هو homework معكم في ال exercise
+
+868
+01:16:33,090 --> 01:16:36,270
+بيقولك اثبت ان g ايش معناها is integrable function
+
+869
+01:16:36,920 --> 01:16:40,960
+إذا الـ F والـ G عبارة عن two integrable functions
+
+870
+01:16:40,960 --> 01:16:46,780
+two integrable functions هاي ال F من I لعند R والـ
+
+871
+01:16:46,780 --> 01:16:55,200
+G اللي هو من عند I لعند R لاحظ ان الـ G فيه لغة
+
+872
+01:16:55,200 --> 01:17:00,600
+تانية zero لمين يا واحدعندي هذه فئلها الـ 1 و 0 و
+
+873
+01:17:00,600 --> 01:17:06,800
+في عندنا قيم أخرى اللي هو كثيرة طيب، الآن عندي ..
+
+874
+01:17:06,800 --> 01:17:09,660
+لو جيت حسب الـ G composed of F مش غريبة عليكم
+
+875
+01:17:09,660 --> 01:17:14,480
+الدالة هذه G composed of F of X خلينا نحسبها مع
+
+876
+01:17:14,480 --> 01:17:20,860
+بعض و بنقول خلصنا اللي هو S section أو قولنا ان
+
+877
+01:17:20,860 --> 01:17:24,320
+الـ G composed of F هذه اللي انتوا بتعرفوها انها
+
+878
+01:17:24,320 --> 01:17:36,430
+is not integrableجي كومبوزيت F of اللي هو عندي
+
+879
+01:17:36,430 --> 01:17:46,900
+الجي الـ F of X هيها لاخد جي of اللي هو Zeroبساوي
+
+880
+01:17:46,900 --> 01:17:55,920
+g of f of 0 بساوي g of f of 0 بساوي
+
+881
+01:17:55,920 --> 01:17:57,880
+g of f of 0 بساوي g of f of 0 بساوي g of f of 0
+
+882
+01:17:57,880 --> 01:18:06,220
+بساوي g of f of 0 بساوي g of f of 0 بساوي g of f
+
+883
+01:18:06,220 --> 01:18:07,060
+of 0 بساوي g of f of 0 بساوي g of f of 0 بساوي g
+
+884
+01:18:07,060 --> 01:18:07,140
+of f of 0 بساوي g of f of 0 بساوي g of f of 0
+
+885
+01:18:07,140 --> 01:18:08,100
+بساوي g of f of 0 بساوي g of f of 0 بساوي g of f
+
+886
+01:18:08,100 --> 01:18:09,880
+of 0 بساوي g of f of 0 بساوي g of f of 0 بساوي g
+
+887
+01:18:09,880 --> 01:18:14,160
+of f of 0 بساوي g of f of 0 بساوي g of f of 0
+
+888
+01:18:14,160 --> 01:18:14,160
+بساوي g of f of 0 بساوي g of f of 0 بساوي g of f
+
+889
+01:18:14,160 --> 01:18:17,110
+of 0 بساوي g of f ofX اللي هو irrational يعني
+
+890
+01:18:17,110 --> 01:18:22,230
+irrational G of irrational اللي هو اللي هو الموجود
+
+891
+01:18:22,230 --> 01:18:25,910
+في الفترة Zero و واحد G of irrational إيش هيساوي
+
+892
+01:18:25,910 --> 01:18:31,290
+حسب التعريف .. التعريف هنا بيساوي G of F of
+
+893
+01:18:31,290 --> 01:18:38,750
+irrational irrational هاي بيساوي اللي هو G of F of
+
+894
+01:18:38,750 --> 01:18:44,240
+irrational إيش بيساوي سفرو G of 0 ايش بيسوي هنا G
+
+895
+01:18:44,240 --> 01:18:52,420
+of 0 بيسوي Zero معرفليها بيسوي ايش صفر ا الثالثة
+
+896
+01:18:54,350 --> 01:18:58,630
+Composite F of مين ضال؟ Of اللي هي irrational
+
+897
+01:18:58,630 --> 01:19:02,530
+البجيات غير الـ Zero هذا أصلا هو جسمها irrational
+
+898
+01:19:02,530 --> 01:19:08,410
+و rational جهتين واحدة واحدة واحدة Zero لحالها و
+
+899
+01:19:08,410 --> 01:19:11,650
+واحدة كل ال rational اللي بين Zero و واحد ما عدا
+
+900
+01:19:11,650 --> 01:19:18,870
+ال Zero اللي هو G of X اللي هي عبارة عن M على N
+
+901
+01:19:19,590 --> 01:19:23,630
+اللي هي rational في الواقع رياشونال كلهم معدن اللي
+
+902
+01:19:23,630 --> 01:19:28,790
+فوق اللي فوق حسبناها طلعت واحد ويساوي G of F of M
+
+903
+01:19:28,790 --> 01:19:34,350
+على N ويساوي G of F of M على N أيش بتساوي واحد على
+
+904
+01:19:34,350 --> 01:19:38,910
+N ويساوي G of واحد على N أيش بتحسبها هذه هذا اصلا
+
+905
+01:19:38,910 --> 01:19:42,070
+الدالة دايما بتساوي واحد معدن عند سفر حتى سفر عشان
+
+906
+01:19:42,070 --> 01:19:46,750
+تعملنا المشكلة G of واحد على N أيش بتساوي واحد هو
+
+907
+01:19:46,750 --> 01:19:53,580
+يساوي واحدمن هذا كله صار عندي g composite f of x
+
+908
+01:19:53,580 --> 01:20:06,710
+بتساوي 0 if x أشمالها is irrational ويساوي 1 1إذا
+
+909
+01:20:06,710 --> 01:20:11,790
+كانت X is هاي ال rational كله ما عدا السفر وهي
+
+910
+01:20:11,790 --> 01:20:16,670
+السفر برضه طلع واحد if X is rational وهدد دالة
+
+911
+01:20:16,670 --> 01:20:20,610
+تبعتنا اللي اعتبتناها المرة الماضية انها is not
+
+912
+01:20:20,610 --> 01:20:24,590
+integrable function لحظوا انتوا بس خليني نقول شوف
+
+913
+01:20:24,590 --> 01:20:32,660
+هالغرابة مش غرابة لأ هو مهم بالكلامإنه اللي خلف
+
+914
+01:20:32,660 --> 01:20:37,380
+الموضوع لو كانت G continuous على كل ال domain على
+
+915
+01:20:37,380 --> 01:20:40,820
+طول الـ G continuous ده هتطلع ان انتجر بالغصب عنها
+
+916
+01:20:40,820 --> 01:20:44,560
+من النظرية اللي قبل شوية لكن اللي خلف الموضوع شغلة
+
+917
+01:20:44,560 --> 01:20:49,440
+واحدة شايفين هالدالة هاد ال G of X الـ G of X G of
+
+918
+01:20:49,440 --> 01:20:55,000
+X بتساوي اللي هو سفر عند السفر ومن عند ال zero
+
+919
+01:20:55,000 --> 01:21:01,130
+لعند الواحد هناالدالة قيمتها إيش بتساوي؟ واحد، هاي
+
+920
+01:21:01,130 --> 01:21:04,270
+قيمتها واحد، بين Zero والواحد قيمتها واحد، هذا
+
+921
+01:21:04,270 --> 01:21:09,540
+الـG of X اللي عندييعني هذا كل حال عالي العالي ما
+
+922
+01:21:09,540 --> 01:21:14,120
+عدا عند من عند السفر فيه jump point هذه النقطة
+
+923
+01:21:14,120 --> 01:21:20,200
+الوحيدة اللي فيها discontinuity هي اللي .. اللي
+
+924
+01:21:20,200 --> 01:21:25,340
+انا بنشرتلي انها تصير decomposed F is continuous و
+
+925
+01:21:25,340 --> 01:21:29,200
+هذا عشان نعرف جداش الرياضيات او جداش التحليل دقيق
+
+926
+01:21:29,780 --> 01:21:35,960
+دقيق ان ان احنا نقطة واحدة .. نقطة واحدة اللي كانت
+
+927
+01:21:35,960 --> 01:21:40,360
+عندها point of discontinuity جالتلي ان الـ
+
+928
+01:21:40,360 --> 01:21:44,680
+decomposite F need not to be integrable وهي المثال
+
+929
+01:21:44,680 --> 01:21:50,720
+أمامكم و .. و هيك بيكون احنا خلصنا ال section
+
+930
+01:21:50,720 --> 01:21:56,730
+التاني من اللي هو chapter 7و هاي ال homework عندكم
+
+931
+01:21:56,730 --> 01:22:02,630
+مطلوبة 1,2,4,7,10,17,18,19 و ان شاء الله المرة
+
+932
+01:22:02,630 --> 01:22:08,950
+القادمة بنكمل و بنشرح 7.3 اللي هو ال fundamental
+
+933
+01:22:08,950 --> 01:22:10,550
+theorem of calculus
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/7kAGhF5ugUM.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/7kAGhF5ugUM.srt
new file mode 100644
index 0000000000000000000000000000000000000000..055892e654d4ac2f5a7d033f2cf54a699e5ad42e
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/7kAGhF5ugUM.srt
@@ -0,0 +1,2222 @@
+1
+00:00:10,580 --> 00:00:15,760
+إيش يا أبو حسين؟ بسم الله الرحمن الرحيم الحمد لله
+
+2
+00:00:15,760 --> 00:00:18,980
+رب العالمين والصلاة والسلام على سيد المرسلين سيدنا 
+
+3
+00:00:18,980 --> 00:00:24,540
+محمد وعلى آله وصحبه أجمعين هذه هي المحاضرة رقم 17
+
+4
+00:00:24,540 --> 00:00:29,900
+في مساق تحليل حقيقي 2 لطلاب وطالبات الجامعة
+
+5
+00:00:29,900 --> 00:00:37,170
+الإسلامية كلية العلوم قسم الرياضيات بدأنا اليوم
+
+6
+00:00:37,170 --> 00:00:41,490
+استكمال للي بدأنا المرة الماضية حكينا المرة
+
+7
+00:00:41,490 --> 00:00:48,070
+الماضية عن الـ integration as a limit قلنا احنا
+
+8
+00:00:48,070 --> 00:00:51,970
+قبل كنا معرفين ال integration أنه موجود على أساس
+
+9
+00:00:51,970 --> 00:00:56,950
+أن ال upper integral U of F يساوي ال lower
+
+10
+00:00:56,950 --> 00:01:01,710
+integral L of F قلنا المرة الماضية أن احنا بدنا
+
+11
+00:01:01,710 --> 00:01:07,300
+نروح باتجاه أنه نصل في النهاية أن هذا الكلام اللي
+
+12
+00:01:07,300 --> 00:01:11,440
+هو الـ Integrability يكافئ أن نصل أن ال limit
+
+13
+00:01:11,440 --> 00:01:14,760
+طبعا وفي حالة وجوده هنسميه بساوة ال integration من
+
+14
+00:01:14,760 --> 00:01:19,940
+a ل b of x dx قلنا المرة الماضية أن هنصل في 
+
+15
+00:01:19,940 --> 00:01:25,600
+النهاية في نهاية ال section 7-4 أن هذا اللي هو ال
+
+16
+00:01:25,600 --> 00:01:28,840
+condition أو هذا ال definition يكافئ أن نقول
+
+17
+00:01:28,840 --> 00:01:35,640
+limit اللي هو SB و F as normal B أو ال mesh تبع الـ B
+
+18
+00:01:35,640 --> 00:01:41,100
+بيروح لـ 0 و في هذه الحالة إذا كان هذا exist بيصير
+
+19
+00:01:41,100 --> 00:01:45,380
+.. هيساوي ال integration من A ل B F of X DX يعني
+
+20
+00:01:45,380 --> 00:01:49,500
+بمعنى آخر .. بمعنى آخر أن اللي هو هذا طبعا الـ
+
+21
+00:01:49,500 --> 00:01:55,460
+Riemann Sum أن الـ limit للRiemann sum as الـ
+
+22
+00:01:55,460 --> 00:02:00,020
+partition B بيروح للـ 0 لو كان exist حيصير اللي هي
+
+23
+00:02:00,020 --> 00:02:04,220
+F is integrable والـ integration لها هو عبارة عن
+
+24
+00:02:04,220 --> 00:02:09,520
+قيمة هذا الـ limitولو كان ال integration exist
+
+25
+00:02:09,520 --> 00:02:14,040
+طبعا بالمفهوم اللي هو هذا هيكون عنده ال limit هذا
+
+26
+00:02:14,040 --> 00:02:18,300
+exist وهيساوي قيمة مين ال integration هذا اللي 
+
+27
+00:02:18,300 --> 00:02:22,260
+حنصل إليه في النهاية اللي حكيناه المرة الماضية
+
+28
+00:02:22,260 --> 00:02:27,380
+بشكل سريع اللي هو عرفنا إيش هو معنى اللي هو ال
+
+29
+00:02:27,380 --> 00:02:32,080
+Riemann sum اللي أمامنا هذا بعد هيك اللي هو إيجينا
+
+30
+00:02:32,080 --> 00:02:36,600
+أخدنا اللي هي الخاصية الأولى قلنا دائما الRiemann
+
+31
+00:02:36,600 --> 00:02:41,140
+sum بين اللي هو ال lower sum و بين ال upper sum و
+
+32
+00:02:41,140 --> 00:02:46,080
+بعدين اللي هو إيجينا قلنا اللي هو النظرية ال .. ال
+
+33
+00:02:46,080 --> 00:02:47,220
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+34
+00:02:47,220 --> 00:02:47,280
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+35
+00:02:47,280 --> 00:02:48,000
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+36
+00:02:48,000 --> 00:02:48,120
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+37
+00:02:48,120 --> 00:02:49,580
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+38
+00:02:49,580 --> 00:02:49,700
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+39
+00:02:49,700 --> 00:02:50,220
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+40
+00:02:50,220 --> 00:02:59,260
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال ..إذا
+
+41
+00:02:59,260 --> 00:03:03,200
+كانت F Integrable إذا لكل Y أكبر من 0 نجد نلاقي
+
+42
+00:03:03,200 --> 00:03:07,220
+partition بي إبسلون و by such that إذا كانت بي أي
+
+43
+00:03:07,220 --> 00:03:10,600
+اللي هو refinement partition refinement للبي
+
+44
+00:03:10,600 --> 00:03:14,260
+إبسلون يعني يحتوي البي إبسلون هيكون وكان عند ال
+
+45
+00:03:14,260 --> 00:03:18,540
+SPUF عبارة عن اللي هو الRiemann sum بيكون الـ absolute
+
+46
+00:03:18,540 --> 00:03:21,300
+value ال integration بين S بي و F مقصد ال
+
+47
+00:03:21,300 --> 00:03:24,840
+integration إيش ماله أصغر من مين من إبسلون يعني لو
+
+48
+00:03:24,840 --> 00:03:28,460
+F is integrable هنلاقي بي إبسلون بحيث كل ال B
+
+49
+00:03:28,460 --> 00:03:31,540
+بتحتوي بي إبسلون هيكون ال absolute value بين
+
+50
+00:03:31,540 --> 00:03:36,400
+الRiemann sum هذا أصغر من مين من اللي هي إيش إبسلون
+
+51
+00:03:36,820 --> 00:03:41,900
+طيب بعد هي كتب عن البرهان هذه وصار في عندي الآن
+
+52
+00:03:41,900 --> 00:03:46,240
+اللي هو النظرية الثانية أيضا برهناها والنظرية
+
+53
+00:03:46,240 --> 00:03:52,110
+الثانية كانت تقول أن لو كانت F is bounded الآن ..
+
+54
+00:03:52,110 --> 00:03:57,250
+وقدرنا .. وبدنا الآن نفترض أن لجينا اللي هو
+
+55
+00:03:57,250 --> 00:04:01,830
+number a يحقق ماله إيش هذا number a يحقق أن لكل
+
+56
+00:04:01,830 --> 00:04:04,890
+epsilon أكبر من 0 ونلاقي بي ابسلون بحيث أن أي بي
+
+57
+00:04:04,890 --> 00:04:11,040
+يحتوي بي ابسلون و S اللي هي أي Riemann sum و then اللي
+
+58
+00:04:11,040 --> 00:04:16,060
+هي s,b وf ناقص a أصغر من إبسلون إذا كان هذا الكلام
+
+59
+00:04:16,060 --> 00:04:19,480
+كله على بعض اتحقق لما نلاقي a بحيث يقول كل إبسلون
+
+60
+00:04:19,480 --> 00:04:24,180
+أكبر من سفر اللي هو نلاقي بي إبسلون لو كان بي
+
+61
+00:04:24,180 --> 00:04:27,740
+إبسلون بحيث أن بي بتحتوي بي إبسلون وهذا ال Riemann
+
+62
+00:04:27,740 --> 00:04:32,030
+sum هيعطيني هذا هذا كله على بعض ال conditions زي ما
+
+63
+00:04:32,030 --> 00:04:35,410
+قلنا المرة الماضية بيعطينا أن F إيش مالها
+
+64
+00:04:35,410 --> 00:04:39,270
+integrable وهذا الـ A اللي حقق هذه الشروط هنا هو
+
+65
+00:04:39,270 --> 00:04:44,050
+قيمة ال integration هذا اللي حكيناه المرة الماضية
+
+66
+00:04:44,050 --> 00:04:51,120
+وبعدين اللي هو احنا إيجينا وعرفنا شو معنى ال mesh أو
+
+67
+00:04:51,120 --> 00:04:54,800
+ال norm إذا بذكركم ال mesh أو ال norm هو عبارة عن ال
+
+68
+00:04:54,800 --> 00:05:00,140
+supremum لاللي هي أطوال ال subintervals تبعات ال
+
+69
+00:05:00,140 --> 00:05:03,060
+partition قلنا ال norm للبيبس أو ال supremum لهذه
+
+70
+00:05:03,060 --> 00:05:07,280
+اللي هي الأطوال تبعات ال subintervals اللي ناتجة
+
+71
+00:05:07,280 --> 00:05:11,550
+من ال partition B هذا اسمنا ال mesh أو ال norm أخدنا
+
+72
+00:05:11,550 --> 00:05:17,230
+بعض اللي هي الخواص وصُجفنا عند مين؟ عند Darboux's
+
+73
+00:05:17,230 --> 00:05:20,690
+theorem اليوم بدنا نبدأ نشرح في مين في Darboux's
+
+74
+00:05:20,690 --> 00:05:24,230
+theorem نشوف إيش Darboux's theorem بتقول الآن بقول
+
+75
+00:05:24,230 --> 00:05:28,990
+لي let i بتساوي a و b اللي هي bounded interval و F
+
+76
+00:05:28,990 --> 00:05:32,910
+من I لعنب R Integrable on I in the sense of
+
+77
+00:05:32,910 --> 00:05:38,050
+definition mean 716 يعني بقول لي نفترض أن F is an
+
+78
+00:05:38,050 --> 00:05:42,830
+integrable function حسب تعريف mean 716 يعني ال
+
+79
+00:05:42,830 --> 00:05:46,470
+upper integral بساوة ال lower integral بساوة ال
+
+80
+00:05:46,470 --> 00:05:50,350
+integration هذا اللي هو ال definition 716 الآن
+
+81
+00:05:50,350 --> 00:05:56,710
+بقول لي then for every epsilon أكبر من 0 then for
+
+82
+00:05:56,710 --> 00:06:02,450
+every epsilon أكبر من صفر there exists delta such
+
+83
+00:06:02,450 --> 00:06:07,150
+that الآن لو كان P is any partition of I وال
+
+84
+00:06:07,150 --> 00:06:11,710
+normal P أصغر من Delta وكان عند ال S P و F اللي
+
+85
+00:06:11,710 --> 00:06:15,810
+هو is any corresponding Riemann sum then ال
+
+86
+00:06:15,810 --> 00:06:19,450
+integration بين ال S P .. البعد بين ال integration
+
+87
+00:06:19,450 --> 00:06:23,670
+بين ال S P وال integration أصغر من مين أصغر من
+
+88
+00:06:23,670 --> 00:06:28,990
+Epsilon بس
+
+89
+00:06:28,990 --> 00:06:29,910
+بقى في دقيقة يا أبو حسن
+
+90
+00:06:59,460 --> 00:07:03,900
+أيوة أنا في محاضرة .. أنا في محاضرة وبسجل .. أيوة
+
+91
+00:07:03,900 --> 00:07:08,400
+أه في
+
+92
+00:07:08,400 --> 00:07:16,360
+كلية العلوم؟ من قالك في كلية العلوم؟
+
+93
+00:07:16,360 --> 00:07:18,500
+لأ يا عم أنا روحت على دراسات العلوم اللي قالولي
+
+94
+00:07:18,500 --> 00:07:27,580
+طلعن أه .. بعرفش .. بعرفش كلام هو صحيح ولا لأ لأ
+
+95
+00:07:27,580 --> 00:07:33,080
+لأبقى طيب يا راجل أنا في المحاضرة ماشي مع السلام
+
+96
+00:07:33,080 --> 00:07:40,800
+.. ماشي بقى كله مروح مع السلام ليش؟
+
+97
+00:07:40,800 --> 00:07:44,040
+مع السلام .. بس شوف كيف مع السلام
+
+98
+00:08:32,780 --> 00:08:36,900
+اعطيني بس تليفون هذا في العلوم 1600 أنا عارف الله
+
+99
+00:08:36,900 --> 00:08:41,780
+ده
+
+100
+00:08:41,780 --> 00:08:45,660
+الدس
+
+101
+00:08:45,660 --> 00:08:50,040
+اه هو كيد روح إذا أبو بلال الموجود روح
+
+102
+00:09:15,660 --> 00:09:18,500
+عليكم السلام وعليكم يا أبو ساما كيف حالك؟ كيف صحتك؟
+
+103
+00:09:18,500 --> 00:09:21,980
+الله بعرفيك الحمد لله الله عزيزي عنك هل في اللي
+
+104
+00:09:21,980 --> 00:09:28,260
+عندكم رسالة تابعة السناق؟ وتابعة ياسر؟ طب .. طب
+
+105
+00:09:28,260 --> 00:09:33,820
+.. مين عندك؟ قاعد بتسأل عليه لو .. لو أبو أكرم
+
+106
+00:09:33,820 --> 00:09:36,880
+أخذن بس وحطن في غرفتي لأن أنا قاعد بصور والله
+
+107
+00:09:36,880 --> 00:09:42,060
+يبقى اه .. اه .. اه .. الله يبارك فيك يا عزيزي أنا
+
+108
+00:09:42,060 --> 00:09:50,370
+بجيبها المفتاح هو أه أه ماشي
+
+109
+00:09:50,370 --> 00:09:54,770
+لو حتى أنا يعني أنا موجود بس بنتأخر يعني التصوير
+
+110
+00:09:54,770 --> 00:09:59,950
+ممكن أظل التلاتة ماشي ماشي الله أبقى أرفع حاجة
+
+111
+00:09:59,950 --> 00:10:03,290
+لأن بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+112
+00:10:03,290 --> 00:10:03,490
+بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+113
+00:10:03,490 --> 00:10:06,830
+بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+114
+00:10:06,830 --> 00:10:08,250
+ب
+
+115
+00:10:27,230 --> 00:10:30,330
+إذن Darboux's theorem زي ما قلنا إن لو كانت F
+
+116
+00:10:30,330 --> 00:10:34,600
+integrable function بقول إذا لكل إبسلون أكبر من 0
+
+117
+00:10:34,600 --> 00:10:38,780
+بقدر ألاقي Delta أكبر من 0 بحيث أن لو كان بي أي
+
+118
+00:10:38,780 --> 00:10:41,620
+partition بحيث أن ال .. اللي هو ال .. ال .. ال
+
+119
+00:10:41,620 --> 00:10:44,980
+norm للـ B أصغر من Delta و S, B و F أي
+
+120
+00:10:44,980 --> 00:10:49,420
+corresponding Riemann sum بالنسبة للـ B هيعطيني أن
+
+121
+00:10:49,420 --> 00:10:52,860
+ال absolute value بين اللي هو أو البعد بين S, B و
+
+122
+00:10:52,860 --> 00:10:57,780
+F ناقص ال integration أصغر من إبسلون يعني وكأنه
+
+123
+00:10:57,780 --> 00:11:02,190
+بقول من تحت بطنانه ال integration ال integration
+
+124
+00:11:02,190 --> 00:11:08,710
+لل F لو F is integrable بيعطينا أن بمفهوم اللي هو
+
+125
+00:11:08,710 --> 00:11:18,140
+سبعة واحد ستة بكون عنده limit ال SB وF as normal B
+
+126
+00:11:18,140 --> 00:11:24,320
+بيروح للسفر بساوة بالظبط integration من A لB F of X
+
+127
+00:11:24,320 --> 00:11:29,260
+DX لأن إيش معناته هذا؟ معناته هذا اللي هو حسب الـ
+
+128
+00:11:29,260 --> 00:11:32,160
+Epsilon Delta Definition for every Epsilon أكبر من
+
+129
+00:11:32,160 --> 00:11:36,740
+سفر there exists Delta أكبر من سفر such that if
+
+130
+00:11:36,740 --> 00:11:43,170
+normal B أصغر، ناقص السفر أصغر من دلتا then هيعطيني
+
+131
+00:11:43,170 --> 00:11:46,810
+الـ absolute value للـ S, P و F ناقص ال
+
+132
+00:11:46,810 --> 00:11:52,650
+integration من A ل B F of X DX كله أصغر من مين من
+
+133
+00:11:52,650 --> 00:11:56,790
+إبسلون إذا اللي بدنا نتبته بدنا لكل إبسلون نلاقي
+
+134
+00:11:56,790 --> 00:12:01,870
+Delta بحيث أن لما أي partition نورمه أصغر من Delta
+
+135
+00:12:01,870 --> 00:12:05,510
+يعطيني الفرق بين الـ Riemann Sum و بين الـ
+
+136
+00:12:05,510 --> 00:12:08,910
+Integration يكون أصغر من مين؟ من Epsilon هذه اللي
+
+137
+00:12:08,910 --> 00:12:11,990
+هي اللي بتقولنا Darboux's Theorem أن لو F is
+
+138
+00:12:11,990 --> 00:12:15,610
+integrable إذا اللي هو الـ limit لاللي هو الـ
+
+139
+00:12:15,610 --> 00:12:18,730
+Riemann Sum هو حيساوي هذا ال integration اللي
+
+140
+00:12:18,730 --> 00:12:23,330
+فرضنا احنا إيش مالها أنه موجود خلينا نشوف كيف بدنا
+
+141
+00:12:23,330 --> 00:12:29,380
+نبرهن النظرية وركزوا معايا وتبعوا إيش اللي بدنا
+
+142
+00:12:29,380 --> 00:12:35,200
+نحكيه عشان نصل للي هو المطلوب طيب صلى الله عليه
+
+143
+00:12:35,200 --> 00:12:41,370
+الصلاة والسلام أول إشي بما أن F is integrable الـ
+
+144
+00:12:41,370 --> 00:12:42,290
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+145
+00:12:42,290 --> 00:12:42,330
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+146
+00:12:42,330 --> 00:12:43,190
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+147
+00:12:43,190 --> 00:12:43,510
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+148
+00:12:43,510 --> 00:12:44,470
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+149
+00:12:44,470 --> 00:12:50,750
+ال .. ال .. ال .. ال .. ال ..
+
+150
+00:12:50,750 --> 00:12:52,070
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+151
+00:12:52,070 --> 00:12:53,590
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+152
+00:12:53,590 --> 00:12:53,750
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+153
+00:12:53,750 --> 00:12:54,490
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+154
+00:12:54,490 --> 00:12:54,910
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+155
+00:12:54,910 --> 00:13:02,270
+.. ال .. ال .. ال .. إذن الآن افترضت epsilon
+
+156
+00:13:02,270 --> 00:13:06,870
+arbitrary بدي أدور على الـ delta اللي لما ال
+
+157
+00:13:06,870 --> 00:13:10,630
+normal بيه أصغر من delta يعطيني الفرق بين هذه و
+
+158
+00:13:10,630 --> 00:13:14,710
+هذه أصغر من مين من epsilon لذا هيك رايح أنا لأن
+
+159
+00:13:14,710 --> 00:13:19,400
+إذن بما أن افنتي جربت لكل epsilon أكبر من 0 بقدر
+
+160
+00:13:19,400 --> 00:13:23,980
+ألاقي partition الآن سميته بي إبسلون اللي هو x0 و
+
+161
+00:13:23,980 --> 00:13:29,400
+x1 لعندي xn such that ال U ال upper sum لل بي
+
+162
+00:13:29,400 --> 00:13:32,340
+إبسلون و F ناقص ال lower sum لل بي إبسلون و F أصغر
+
+163
+00:13:32,340 --> 00:13:35,780
+من مين من إبسلون سميتهاش أنا إبسلون على تلاتة
+
+164
+00:13:35,780 --> 00:13:38,700
+للحسابات طبعا أصغر من إبسلون في الدنيا من ضمنها
+
+165
+00:13:38,700 --> 00:13:45,040
+الإبسلون على تلاتة هذا حسب الRiemann criterion أو ال
+
+166
+00:13:45,040 --> 00:13:51,190
+integrable criterion طيب moreover لأن لو كانت عندي
+
+167
+00:13:51,190 --> 00:13:55,650
+ال B أي B partition بحتوي ال بي إبسلون أكيد هيكون
+
+168
+00:13:55,650 --> 00:13:59,550
+اللي هو ال .. ال .. ال .. هذا بيصير refinement
+
+169
+00:13:59,550 --> 00:14:04,410
+مدام refinement ال refinement التحسين ال lower
+
+170
+00:14:04,410 --> 00:14:09,670
+بعله و ال upper بقاشه بينزل يعني بمعنى أخر هيكون
+
+171
+00:14:09,670 --> 00:14:13,030
+ال lower لل refine .. لل refinement أو التحسين
+
+172
+00:14:13,030 --> 00:14:17,240
+أكبر أو يساوي مين ال lower للأصل ال بي إبسلون وأكيد
+
+173
+00:14:17,240 --> 00:14:20,840
+الـ lower لل B و ال F أكيد أصغر أو يساوي ال lower لل B
+
+174
+00:14:20,840 --> 00:14:24,780
+و ال F in general و ال upper لل B و F قلنا مدام
+
+175
+00:14:24,780 --> 00:14:29,600
+تحسين إذا أكيد بنزل بصغر عن مين عن اللي صار له
+
+176
+00:14:29,600 --> 00:14:34,960
+تحسين اللي هو مين U, B, Y و F إذا صار عندي الآن ..
+
+177
+00:14:34,960 --> 00:14:45,230
+هي عندي ال L .. ال L, B, Y و F وهذا L, B و
+
+201
+00:16:34,740 --> 00:16:37,220
+integrable لإنها مدام bounded أو integrable
+
+202
+00:16:37,220 --> 00:16:41,460
+السبريمم لها على كل الفترة موجود، خُد الـ Supremum
+
+203
+00:16:41,460 --> 00:16:45,200
+لقيمة الدالة F of X للـ X element in the interval
+
+204
+00:16:45,200 --> 00:16:49,940
+سمّيها إيش بتساوي؟ سمّيها بتساوي M Capital، ماشي
+
+205
+00:16:49,940 --> 00:16:54,360
+الحال، الآن بادعي الآن إن دلتا اللي هتظبط معنا و
+
+206
+00:16:54,360 --> 00:16:57,320
+تجيبلنا اللي هو القيمة اللي بدنا إياها اللي هي
+
+207
+00:16:57,320 --> 00:17:00,180
+أصغر من إبسلون، اللي هي دلتا بتساوي إبسلون على
+
+208
+00:17:00,180 --> 00:17:05,140
+اثنا عشر N في مين في M، N مين هذه، N اللي ظهرت في ال
+
+209
+00:17:05,140 --> 00:17:09,750
+partition هنا اللي هو عدد عناصر الـ partition اللي
+
+210
+00:17:09,750 --> 00:17:14,490
+جزّأ لي الفترة هي N زائد 1، هذا مين الـ partition اللي
+
+211
+00:17:14,490 --> 00:17:19,790
+لجيته في الريمان من خلال الـ integrability للأف، إذاً 
+
+212
+00:17:19,790 --> 00:17:24,370
+where الـ N زائد واحد is the number of parts in B
+
+213
+00:17:24,370 --> 00:17:29,190
+إبسلون يعني الـ Delta بتساوي إبسلون على 12 في N
+
+214
+00:17:29,190 --> 00:17:33,370
+الـ N ارتبطت بالـ partition اللي لجته والـ M هي 
+
+215
+00:17:33,370 --> 00:17:39,250
+عبارة عن الـ supremum لكل الدالة على كل الفترة
+
+216
+00:17:39,250 --> 00:17:41,410
+المذكورة، نيجي الآن
+
+217
+00:17:46,350 --> 00:17:53,750
+Now let Q بيساوي Y not، Y واحد، و Y اثنين لعند Y M
+
+218
+00:17:53,750 --> 00:17:59,890
+مش N، Y M ماشي بـ any partition of I with ال
+
+219
+00:17:59,890 --> 00:18:03,550
+absolute value للـ Q أو الـ norm للـ Q أصغر من Delta
+
+220
+00:18:04,420 --> 00:18:08,500
+إيش يا جماعة قاعد أنا أحاول أصل للي بديها؟ إيش 
+
+221
+00:18:08,500 --> 00:18:12,080
+غرضي؟ إن لكل epsilon أكبر من 0 بدي ألاقي delta
+
+222
+00:18:12,080 --> 00:18:16,600
+بحيث أنه أي partition أصغر من delta norm يعطيني
+
+223
+00:18:16,600 --> 00:18:21,320
+الفرق بين ريمان صمّه وال integration أصغر من epsilon
+
+224
+00:18:21,320 --> 00:18:27,660
+وكأنه الآن بدأت أحط أيدي على الجرح وأصل أقول let Q
+
+225
+00:18:27,660 --> 00:18:34,780
+بيساوي Y0، Y1، YM بأي partition Normal أصغر من مين؟
+
+226
+00:18:34,780 --> 00:18:40,260
+من Delta، متوقع أنا أنه في النهاية عشان أخلص
+
+227
+00:18:40,260 --> 00:18:47,200
+النظرية، أثبت لك أنه الـ absolute value لـ اللي هو الـ
+
+228
+00:18:47,200 --> 00:18:55,230
+S الـ UF ناقص الـ integration من قلبي f of x dx يطلع
+
+229
+00:18:55,230 --> 00:19:00,290
+لي أصغر من مين؟ من إبسلون، إذا أثبتت هذا بناء على
+
+230
+00:19:00,290 --> 00:19:05,830
+اختياري للـ Delta اللي عندي وبناء على كل الـ cues
+
+231
+00:19:05,830 --> 00:19:10,490
+اللي نورهم أصغر من Delta يعطيني هذا ناقص هذا أصغر
+
+232
+00:19:10,490 --> 00:19:14,210
+من إبسلون وبقول خلصت وبرهنت اللي هي نظريتي
+
+233
+00:19:14,210 --> 00:19:19,350
+Darboux's theorem، خلينا نشوف كيف بدنا نروح لهذه
+
+234
+00:19:19,350 --> 00:19:24,810
+اللي هي نثبتها إنها أصغر من إبسلون، تعال شوية الآن
+
+235
+00:19:24,810 --> 00:19:31,630
+لأن let Q*) بساوي Q اتحادين بـ إبسلون هذه بـ إبسلون
+
+236
+00:19:31,630 --> 00:19:37,210
+مين اللي لجيتها فوق اللي هي عبارة عن بـ إبسلون
+
+237
+00:19:38,430 --> 00:19:43,870
+بـ إبسلون هي الفترة من a لعند b بـ إبسلون اللي هي أش
+
+238
+00:19:43,870 --> 00:19:50,850
+جزّأت للفترة هذه، هذي x note، هذي x1، هذي x2، هذي x3
+
+239
+00:19:50,850 --> 00:19:56,430
+لما أصل xn ناقص واحد، لما أصل هنا عند مين؟ xn اللي
+
+240
+00:19:56,430 --> 00:20:01,390
+هي مين الـ b؟ هذي مين جزّأها جزء الفترة هذه واحدة
+
+241
+00:20:01,390 --> 00:20:08,190
+اسمها بـ إبسلون، نيجي الآن اتجزّأت الفترة بمين؟ بالـ Q
+
+242
+00:20:08,190 --> 00:20:17,270
+الـ Q مين هي؟ بدأت بـ Y note طبعا هي ايه Y1 و Y2 و
+
+243
+00:20:17,270 --> 00:20:24,950
+Y3 لعند مؤصل لعند Y M ناقص واحد، وهنا اللي هي YM
+
+244
+00:20:24,950 --> 00:20:29,730
+اللي هي طبعاً مين هي؟ اللي هي الـ B هذه الـ Q جزّقتها
+
+245
+00:20:29,730 --> 00:20:33,430
+بشكلها اللي أمامي، مع العلم إن الـ Q هي تبعتنا
+
+246
+00:20:33,430 --> 00:20:38,950
+لنورمها أصغر من مين؟ من Delta، الآن بيقول لي خد Q*)
+
+247
+00:20:38,950 --> 00:20:46,150
+إيش بتساوي؟ الـ Q اتحاد من الـ B إبسلون، صارت الآن
+
+248
+00:20:46,150 --> 00:20:54,880
+الـ Q هذه بساوي بـ بي ستار اتحاد لـ Q، هذه مين هي الـ 
+
+249
+00:20:54,880 --> 00:21:01,100
+Q star؟ طبعاً النقطة الأولى X not هي نفسها Y not
+
+250
+00:21:01,100 --> 00:21:06,840
+والـ Xn هي مين الـ Yn؟ اللي يعني ممكن يكون
+
+251
+00:21:06,840 --> 00:21:11,380
+المختلفات اللي هي من X واحد لعند Xn ناقص واحد يكون
+
+252
+00:21:11,380 --> 00:21:16,890
+المختلفات عن هدول الاتحاد، أقصى حدودهم يعني إذا بدهم
+
+253
+00:21:16,890 --> 00:21:21,430
+يعملوا لي sub intervals زيادات، هيكون أقصى حدودهم
+
+254
+00:21:21,430 --> 00:21:27,290
+اللي لنا اللي هي X1 و X2 و X3 و X-1، ما تنطبقش ولا
+
+255
+00:21:27,290 --> 00:21:33,810
+على أي Y1 ولا Y2 ولا Y3 ولا Y أم minus 1، متى ما
+
+256
+00:21:33,810 --> 00:21:38,410
+انطبقتش معناته أنه هيكونوا وين؟ موجودات في داخل الـ
+
+257
+00:21:38,410 --> 00:21:42,630
+intervals، معناته كل واحدة عملت لها إلا عن فترة تاني
+
+258
+00:21:42,630 --> 00:21:46,550
+جديدة، اللي هو في الفترة الثانية الإجداد، يعني لو إجت
+
+259
+00:21:46,550 --> 00:21:51,390
+هنا الـ X1، هنا مثلاً بيكون هذه عملت فترة تانية لو
+
+260
+00:21:51,390 --> 00:21:58,930
+إجينا سمينا هذه المنطقة اللي هو M1 star، اللي هو M1
+
+261
+00:21:58,930 --> 00:22:02,470
+مين اللي هو الـ maximum أو الـ supreme على مين؟ على
+
+262
+00:22:02,470 --> 00:22:09,510
+الـ partition الجديد Q star، وهذه سميناها هتكون M2*)
+
+263
+00:22:09,510 --> 00:22:15,670
+هيصير الـ M1*) بتختلف لأن هذه عبارة عن الـ
+
+264
+00:22:15,670 --> 00:22:20,490
+Supremum على المنطقة هذه بينما بالنسبة للـ Q لو
+
+265
+00:22:20,490 --> 00:22:27,030
+إجينا لـ Maximum تبعها أو Supremum تبعها هنسميه M1
+
+266
+00:22:27,030 --> 00:22:32,970
+هذا الـ M1 قد يختلف عن الـ M1 Star وعن مين؟ الـ M2
+
+267
+00:22:32,970 --> 00:22:37,610
+Star لأن الـ M1 على هذه كلها بينما الـ M1 Star على
+
+268
+00:22:37,610 --> 00:22:41,070
+هذه والـ M2 Star على هذه اللي هو الـ Supremum
+
+269
+00:22:41,070 --> 00:22:44,670
+للدالة على الجزء من هذا أو الجزء من هذا قد يختلف
+
+270
+00:22:44,670 --> 00:22:49,860
+عن الـ Supremum عن الجزء هذا يعني الآن الـ الـ الـ 
+
+271
+00:22:49,860 --> 00:22:55,060
+الـ الـ subintervals جديدة هتولد لي M1 maximum
+
+272
+00:22:55,060 --> 00:23:00,700
+لعناصر للـ function على اللي هي الكلي بختلف عن مين
+
+273
+00:23:00,700 --> 00:23:07,240
+عن الجزء قد يختلف، الـ N لكن اللي .. اللي .. لو كانت
+
+274
+00:23:07,240 --> 00:23:12,420
+اللي هي عدد هنا دولة اللي هي الـ N أكبر من الـ N
+
+275
+00:23:12,420 --> 00:23:19,420
+هتتوزع .. هتتوزع اللي هو الـ X1 والـ X2 لإذ
+
+276
+00:23:19,420 --> 00:23:23,280
+اختلفن كمان واتوزعن مختلفات هيتوزعن وين؟ في
+
+277
+00:23:23,280 --> 00:23:28,870
+المناطق اللي هيجبل ما نخلص الـ YM يعني هتلاقيها كل
+
+278
+00:23:28,870 --> 00:23:33,970
+واحدة مكان لأن الـ M أكبر من الـ N اللي بتضال بدون
+
+279
+00:23:33,970 --> 00:23:39,870
+ما تدخلها واحدة من الـ X1 أو X2 أو X3، هتسير في ال
+
+280
+00:23:39,870 --> 00:23:43,810
+partition الجديد Q هي نفسها في الأصل في الـ
+
+281
+00:23:43,810 --> 00:23:50,140
+partition main الـ Q، يعني هيكون الـ M هنا star والـ
+
+282
+00:23:50,140 --> 00:23:56,440
+M نفسها هيكون زي بعض يعني هيكون التنتين بنفس
+
+283
+00:23:56,440 --> 00:24:02,600
+القيمة عشان هيك لو إجينا طرحنا اللي هو .. اللي هي
+
+284
+00:24:02,600 --> 00:24:09,860
+قيمة الـ lower sum، الـ upper sum على الـ Q star مع الـ
+
+285
+00:24:09,860 --> 00:24:16,340
+upper sum على الـ Q هيصير عندي اللي هنا يظهر اللي
+
+286
+00:24:16,340 --> 00:24:22,810
+هو M من العادية أو M من الـ Star ناقص الـ M من الـ
+
+287
+00:24:22,810 --> 00:24:27,350
+Star لأنه بيكون قد يكون المختلفات لكن اللي هان مش
+
+288
+00:24:27,350 --> 00:24:31,050
+هيظهر لأنه هيصير الـ M ناقص M من الـ Star صفر، إذن
+
+289
+00:24:31,050 --> 00:24:36,110
+هيصير عندي ظهور هذولة في أكام فترة عن أكثر هاي
+
+290
+00:24:36,110 --> 00:24:41,710
+واحدة تنتين، بضل أمشي لـ N ناقص واحد لإن عددنا دولة
+
+291
+00:24:41,710 --> 00:24:45,330
+من X1 عند XN ناقص واحدة كم واحدة؟ N ناقص واحد يعني
+
+292
+00:24:45,330 --> 00:24:49,130
+على الأكثر هيكون عدد الـ sub intervals اللي إجداد
+
+293
+00:24:49,130 --> 00:24:54,390
+عاملًا اللي هو 2 في N ناقص واحد اللي اتجسمت تنتين
+
+294
+00:24:54,390 --> 00:25:00,590
+تنتين الآن فبصير عندي الـ summation هذا لو إجينا لو
+
+295
+00:25:00,590 --> 00:25:08,230
+أخدنا لك الـ U الـ U star معاللي الـ F ناقص الـ U
+
+296
+00:25:08,230 --> 00:25:14,910
+الـ QF هيكون عبارة عن إيش يا جماعة؟ اللي هنا هيتصفر
+
+297
+00:25:14,910 --> 00:25:19,070
+حيساوي الصميه، كم واحدة من هذولة هيكون على الأكثر
+
+298
+00:25:19,070 --> 00:25:23,790
+الشكل هذا واحدة اسمها مثلاً MK، MJ زي ما هو مسميها
+
+299
+00:25:23,790 --> 00:25:28,870
+والتاني M star K، هذولة على الأكثر، على الأكثر، على
+
+300
+00:25:28,870 --> 00:25:34,900
+الأكثر هيكون عددهم جداش، عددهم عبارة عن اثنين في N
+
+301
+00:25:34,900 --> 00:25:40,240
+ناقص واحد مضروبات في طول الفترة هذه، الفترة الصغيرة
+
+302
+00:25:40,240 --> 00:25:43,680
+اللي هي .. اللي هي الفترة تبعت .. اللي هي مين
+
+303
+00:25:43,680 --> 00:25:48,720
+الـ Q*) اللي هي .. واسميها مثلاً ZK ناقص ZK ماينص
+
+304
+00:25:48,720 --> 00:25:53,700
+واحد يعني عددهم هذولة اللي بظهر الـ U ناقص الـ U
+
+305
+00:25:53,700 --> 00:25:59,270
+بظهر عبارة عن كم واحدة من هذولة؟ summation لهم عبارة عن
+
+306
+00:25:59,270 --> 00:26:04,390
+الـ two في N ناقص واحد من الـ terms، نجي الآن إيش
+
+307
+00:26:04,390 --> 00:26:12,830
+اللي كاتبوا؟ شوف قلنا let Q star بيساوي Q اتحاد من
+
+308
+00:26:12,830 --> 00:26:17,930
+الـ B إبسلون، الـ Q star صار refinement لمين؟ للـ B
+
+309
+00:26:17,930 --> 00:26:23,620
+إبسلون، إذا الـ Q*) حسب اللي حكيته هناك has at most
+
+310
+00:26:23,620 --> 00:26:28,540
+N ناقص واحد more than points than main Q، على
+
+311
+00:26:28,540 --> 00:26:32,660
+الأكثر هيكون عدد اللي في الـ Q*) العناقص اللي في
+
+312
+00:26:32,660 --> 00:26:38,040
+Q*) عند Canon اللي في الـ Q اللي في ال .. أمسح
+
+313
+00:26:38,040 --> 00:26:45,900
+اللي فوق بس زي ما قلت اللي في الـ By اللي في الـ
+
+314
+00:26:45,900 --> 00:26:54,870
+By هي عبارة عن X0، X1، Xm-1 لعند Xm، اللي في الـ Q هي
+
+315
+00:26:54,870 --> 00:27:04,190
+عبارة عن Y0، Y1، Ym-1 لعند Ym، اللي في الـ Q*) إيش
+
+316
+00:27:04,190 --> 00:27:10,910
+هيصير إنه أكيد هذي و هذي هم نفسهم، فأكيد لو بدأنا بـ
+
+317
+00:27:10,910 --> 00:27:17,650
+Z0 وانتهينا بأخر واحدة Z مين؟ ZB مثلاً هيكون اللي
+
+318
+00:27:17,650 --> 00:27:23,390
+هو على الأكثر، على الأكثر، على الأكثر الـ Q*) بتزيد عن
+
+319
+00:27:23,390 --> 00:27:30,170
+الـ P بجداش بالنقاط هذولة اللي لو كانت مش مكررات من
+
+320
+00:27:30,170 --> 00:27:35,430
+هنا خالص اللي هي بيصير عندي في داخل هذه عدد عناصر
+
+321
+00:27:35,430 --> 00:27:41,890
+الـ Q*) أكثر من عدد عناصر الـ Q بـ على الأكثر ناقص واحد
+
+322
+00:27:42,200 --> 00:27:50,720
+وكل هذا تبع اللي هو النقاط اتكرر معها دولة أو ما
+
+323
+00:27:50,720 --> 00:27:57,000
+اتكررنش ما اتكررنش بالمرة، بيكون بالظبط عددهم اللي
+
+324
+00:27:57,000 --> 00:28:03,000
+هو الـ YM زائد الـ N ناقص واحد لو ما اتكررنش ولا
+
+325
+00:28:03,000 --> 00:28:07,700
+واحدة، يعني بمعنى آخر الـ Q start زيد عن الـ Q بـ N
+
+326
+00:28:07,700 --> 00:28:10,740
+ناقص واحد من النقاط
+
+327
+00:28:12,960 --> 00:28:17,560
+طيب زي ما قال إذن So Q star تحتوي بي إبسلون and Q
+
+328
+00:28:17,560 --> 00:28:21,780
+star has at most N نقص واحد more points than Q
+
+329
+00:28:21,780 --> 00:28:26,220
+namely those points among اللي هي النقاط اللي من
+
+330
+00:28:26,220 --> 00:28:30,080
+مين؟ اللي بيكون الزيادات من الـ X واحد لعندي الـ X N
+
+331
+00:28:30,080 --> 00:28:35,200
+الـ X N نقص واحد that is belongs to B star but not
+
+332
+00:28:35,200 --> 00:28:40,560
+to مين؟ to Q، اللي هدول اللي بتكون الزيادات طيب نطلع
+
+333
+00:28:40,560 --> 00:28:48,130
+لفوق، الآن بما أن الـ Q صار تحتوى لـ Q إذا أكيد الـ
+
+334
+00:28:48,130 --> 00:28:53,850
+UQ ناقص الـ UQ صار أكبر يساوي 0، طبعاً عارفين ليش؟ لأن
+
+335
+00:28:53,850 --> 00:28:58,890
+الـ refinement دائماً للـ upper sum بيجلّه، ماشي عشان
+
+336
+00:28:58,890 --> 00:29:02,670
+يصير refinement عشان يبطل يطلع فوق ينزل إشي بسيط
+
+337
+00:29:02,670 --> 00:29:07,730
+فهذا في الـ refinement معناته بيصير الـ UQ أصغر من
+
+338
+00:29:07,730 --> 00:29:11,790
+مين؟ أكبر من مين؟ من الـ UQ، UQ star، لأنه في
+
+339
+00:29:11,790 --> 00:29:15,630
+الـ refinement الأعلى بيجلّه، مدام بيجلّه إذا هذا UQ
+
+340
+00:29:15,630 --> 00:29:20,610
+ناقص UQ star أكبر أو يساوي صفر، الآن زي ما سمينا if
+
+341
+00:29:20,610 --> 00:29:25,170
+we write UQ star بيساوي Z node، Z واحد، عند مين؟ عند
+
+342
+00:29:25,170 --> 00:29:34,610
+ZB then، لو إجينا بدنا نحسب الآن UQ F ناقص UQ star
+
+343
+00:29:34,610 --> 00:29:42,230
+و F، خليني أوضح لك 
+
+344
+00:29:42,230 --> 00:29:53,010
+خليني أكتبها ما عليش، عندي U Q و
+
+345
+00:29:53,010 --> 00:29:59,290
+F ناقص U Q Star و F
+
+346
+00:30:02,130 --> 00:30:07,370
+الـ summation الأولى العبارة عن الـ summation له
+
+347
+00:30:07,370 --> 00:30:19,310
+اللي هي M J في 100 في الـ Q اللي هو Y J minus Y J
+
+348
+00:30:19,310 --> 00:30:25,130
+minus 1 J من عند 1 لعند الـ M ناقص الـ summation
+
+349
+00:30:25,130 --> 00:30:37,430
+للتانية اللي هي MK Star في مين؟ في Z K-Z K-1، K من
+
+350
+00:30:37,430 --> 00:30:42,750
+عند 1 لعند مين؟ لعند الـ B اللي فرضناها، إيش اسمها
+
+351
+00:30:42,750 --> 00:30:49,190
+اللي هي الـ B زي ما قلت قبل بشوية، لو كان هذولة
+
+352
+00:30:49,190 --> 00:30:53,870
+اللي صاروا زيادات هنا، اللي صاروا زيادات هنا الـ X
+
+353
+00:30:53,870 --> 00:30:59,410
+نقطة من X واحد لعند X أناقس واحد، لو توزع بين هذولة
+
+354
+00:31:00,990 --> 00:31:04,390
+يعني بمعنى آخر كل هدف في حالة الـ M أكبر من N
+
+355
+00:31:04,390 --> 00:31:10,430
+اتوزع بينهن هيعمل اللي هو الـ intervals زي ما قلنا
+
+356
+00:31:10,430 --> 00:31:15,850
+2 في N ناقص 1 في هذه الـ intervals هيكون اللي هو
+
+357
+00:31:15,850 --> 00:31:22,460
+الـ Mj و الـ Mk قد يكون المختلفات، لكن في الباقي
+
+358
+00:31:22,460 --> 00:31:28,400
+اللي أصلاً ما دخلهنش إش جديد وضلت الـ Q زي الـ Q*)
+
+359
+00:31:28,400 --> 00:31:35,680
+فبتكون M تبعت الـ Q و M*) تبعت الـ Q*) زي بعض، فبصير
+
+360
+00:31:35,680 --> 00:31:40,360
+هذه ناقص هذه بساوي 0، اللي هي الـ Mj والمك بساوي 0
+
+361
+00:31:40,360 --> 00:31:47,070
+يعني بمعنى آخر الـ terms اللي هتكون على صورة MJ ناقص
+
+362
+00:31:47,070 --> 00:31:52,110
+MK star على الأكثر، على الأكثر اللي ممكن يكون المختلفات
+
+363
+00:31:52,110 --> 00:31:57,550
+عبارة عن جداش جماعة؟ اثنين في N ناقص واحد، يعني هذول
+
+364
+00:31:57,550 --> 00:32:01,950
+عدد هنا على الأكثر طبعاً في ZK minus ZK minus واحد
+
+365
+00:32:01,950 --> 00:32:04,770
+تبعي
+
+401
+00:34:37,460 --> 00:34:41,680
+الدولة على الأكثر اللي هو بالشكل هذا إذا الآن لو
+
+402
+00:34:41,680 --> 00:34:51,060
+أجينا بدي أحسبلك اللي هو الـ M J نقص M K Star هذه
+
+403
+00:34:51,060 --> 00:35:00,000
+هيكون أصغر أو يساوي مج زائد مك
+
+404
+00:35:01,360 --> 00:35:05,180
+وكل واحدة طبعاً هذا supremum على فترتها وهذا
+
+405
+00:35:05,180 --> 00:35:09,240
+supremum للـ F على فترتها أكيد أصغر أو يساوي الـ M
+
+406
+00:35:09,240 --> 00:35:13,160
+زائد الـ M لأن هذا الـ M عبارة عن الـ supremum على كل
+
+407
+00:35:13,160 --> 00:35:16,360
+الـ interval A وB هذا على جزءه هذا على كل الـ
+
+408
+00:35:16,360 --> 00:35:19,240
+interval A وB هذا على جزءه يعني في النهاية هذا
+
+409
+00:35:19,240 --> 00:35:25,460
+أصغر أو يساوي قداش 2 في M يعني هيصير عندي الآن الـ
+
+410
+00:35:25,460 --> 00:35:32,980
+absolute value للـ M J ناقص M K Starفي ZK-ZK-1
+
+411
+00:35:32,980 --> 00:35:39,970
+هيكون أصغر أو يساوي 2M في مين؟ في اللي هي طول الـ
+
+412
+00:35:39,970 --> 00:35:43,550
+interval هذه طول الـ interval هذه طب هذه الـ
+
+413
+00:35:43,550 --> 00:35:50,170
+interval zk minus zk minus 1 أكيد أصغر أو يساوي
+
+414
+00:35:50,170 --> 00:35:56,410
+مين؟ norm لـ Q star مين norm لـ Q star؟ اللي هو الـ
+
+415
+00:35:56,410 --> 00:36:01,510
+supremum على مين؟ على أطوال الـ subintervals الـ
+
+416
+00:36:01,510 --> 00:36:05,750
+supremum أكيد أكبر أو يساوي واحدة منهم طول واحدة
+
+417
+00:36:05,750 --> 00:36:13,190
+منهم كلهم وهذه دائماً لما نزيد العدد قلنا لما نكون
+
+418
+00:36:13,190 --> 00:36:19,630
+Q يحتوي Q*) automatic norm لـ Q*) زي ما قلنا المرة
+
+419
+00:36:19,630 --> 00:36:24,710
+الماضية هيكون أصغر أو يساوي .. آسف لما يكون Q
+
+420
+00:36:24,710 --> 00:36:28,370
+substantive لـ Q*) يكون norm لـ Q*) عكسه أصغر أو
+
+421
+00:36:28,370 --> 00:36:34,420
+يساوي norm من لـ Q إذا الـ N هذا أصغر أو يساوي هذا و
+
+422
+00:36:34,420 --> 00:36:38,940
+هذا نفسه أصغر أو يساوي مين؟ الـ Q و الـ Normal Q
+
+423
+00:36:38,940 --> 00:36:42,720
+أصغر من مين؟ من دلتا ما احنا على أساسه اللي هي
+
+424
+00:36:42,720 --> 00:36:48,120
+اشتغلنا إذا صار عندي هذا Normal أصغر من دلتا يعني
+
+425
+00:36:48,120 --> 00:36:53,380
+نرجع لهذا هذا المقدار كل واحد لحاله أصغر من اثنين
+
+426
+00:36:53,380 --> 00:36:58,350
+أم في مين بيصير؟ في دلتا الآن عدد هذول اللي ممكن
+
+427
+00:36:58,350 --> 00:37:07,870
+يظهر في UQ ناقص UQ star عبارة عن 2M ناقص 1 يعني في
+
+428
+00:37:07,870 --> 00:37:14,330
+النهاية هذا أصغر أو يساوي اللي هو عدد الـ 2M ناقص
+
+429
+00:37:14,330 --> 00:37:22,940
+1 مضروب في مين؟ في اللي هي 2M 2M في Delta لأن هذا
+
+430
+00:37:22,940 --> 00:37:28,390
+نفسه أصغر يساوي 2M و هذا نفسه أصغر يساوي Delta الآن
+
+431
+00:37:28,390 --> 00:37:33,030
+هدولة بتكرر على الأكثر اثنين فئة ناقص واحد هذا ..
+
+432
+00:37:33,030 --> 00:37:36,750
+هذا بتكرر في هذا المقدار على الأكثر في هذا و
+
+433
+00:37:36,750 --> 00:37:40,370
+الباقية هيتصفر زي ما قلنا إذا صار هيكون هذا أصغر
+
+434
+00:37:40,370 --> 00:37:44,750
+أو يساوي عدد تكرار من اثنين فئة ناقص واحد في اثنين
+
+435
+00:37:44,750 --> 00:37:51,010
+M في مين في Delta فإذا صار عندي الآن الـ UQF ناقص
+
+436
+00:37:51,010 --> 00:37:55,050
+الـ UQ star و F أصغر أو يساوي اثنين فئة ناقص واحد في
+
+437
+00:37:55,050 --> 00:38:01,350
+اثنين M في مين في delta مضروب اللي هو هذا عبارة عن
+
+438
+00:38:01,350 --> 00:38:13,090
+قداش يا جماعة نيجي نحسبها عشان نصل للي بدنا إياه صار
+
+439
+00:38:13,090 --> 00:38:26,230
+عندي اللي هو الـ U Q F U star و F Q F ناقص U
+
+440
+00:38:32,930 --> 00:38:38,230
+UQF-UQF UQF
+
+441
+00:38:38,230 --> 00:38:44,130
+-UQF UQF
+
+442
+00:38:44,130 --> 00:38:45,750
+-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF
+
+443
+00:38:45,750 --> 00:38:45,850
+-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF
+
+444
+00:38:45,850 --> 00:38:54,050
+-UQF UQF-UQF UQF-UQF UQF في اللي هو M في Delta اللي
+
+445
+00:38:54,050 --> 00:38:59,870
+هو أكيد هذا أصغر من أربعة في N في M في Delta مظهرة
+
+446
+00:38:59,870 --> 00:39:03,790
+ولا لأ؟ لأن الـ N أكبر من مين؟ من الـ N نقص واحد
+
+447
+00:39:03,790 --> 00:39:06,370
+الـ Delta إيش اخترناها؟ ما هي الحسابات من الأول
+
+448
+00:39:06,370 --> 00:39:12,010
+حاسبينها احنا اخترنا الـ Delta فوق عبارة عن قداش
+
+449
+00:39:12,010 --> 00:39:15,710
+لأن احنا كنا عارفين حالنا أصلاً بنصل لهذه المرحلة
+
+450
+00:39:15,710 --> 00:39:19,730
+فبدنا نخلي هذه إبسيلون على ثلاثة ليش إبسيلون على ثلاثة؟ بعد
+
+451
+00:39:19,730 --> 00:39:26,470
+شوية تشوف اللي هي عبارة عن إبسيلون هذه إبسيلون على 12N في مين
+
+452
+00:39:26,470 --> 00:39:31,170
+في N هذه الآن لما أشيل هذه و أضرب هذه فيها بيصير
+
+453
+00:39:31,170 --> 00:39:34,770
+أربعة في واحد والآن مع الآن و الـ M مع الـ M وهذي
+
+454
+00:39:34,770 --> 00:39:41,750
+بيصير قداش ثلاثة إذا صار عندي الآن U Q و F ناقص U
+
+455
+00:39:41,750 --> 00:39:48,000
+Q Star و F أصغر من إبسيلون على ثلاثة يعني بمعنى آخر
+
+456
+00:39:48,000 --> 00:39:55,380
+صار عندي الـ U Q و F أصغر من إبسيلون من على ثلاثة زائد U
+
+457
+00:39:55,380 --> 00:40:01,420
+Q star من و F ماشي الحال إذا هذه الـ inequality
+
+458
+00:40:01,420 --> 00:40:06,160
+الثانية اللي وصلناها اللي هتوصلني في النهاية للي
+
+459
+00:40:06,160 --> 00:40:11,490
+بدنا إياه ونشوف كيف هتوصلني طيب يا جماعة صلوا على النبي
+
+460
+00:40:11,490 --> 00:40:17,670
+عليه الصلاة والسلام تنسوش أنه احنا في الأول حصلنا
+
+461
+00:40:17,670 --> 00:40:21,730
+على U بي و أف ماينص أل بي و أف أصغر من مين من
+
+462
+00:40:21,730 --> 00:40:24,930
+إبسيلون على ثلاثة وحصلنا على U بي و إبسلون هذا أصغر
+
+463
+00:40:24,930 --> 00:40:27,910
+من إبسيلون على ثلاثة هنحتاجها بعد شوية خليني أكتبها
+
+464
+00:40:27,910 --> 00:40:35,630
+الثانية عند حصلنا قبل هيك على الـ U بي و أف ماينص أل 
+
+465
+00:40:35,630 --> 00:40:42,970
+بي و أف أصغر من إبسيلون على ثلاثة وطبعاً أكيد اللي هي
+
+466
+00:40:42,970 --> 00:40:46,870
+أصغر من u بي إبسلون و اف عشان لا نرجع الهاشمون
+
+467
+00:40:46,870 --> 00:40:51,970
+اقصال بي إبسلون و اف كله أصغر من إبسلون على ثلاثة
+
+468
+00:40:51,970 --> 00:41:00,930
+هاي كمان هدول الاثنتين حصلناها ماشي طيب الآن
+
+469
+00:41:00,930 --> 00:41:09,000
+اللي عملنا هذا اللي هو مع الـ upper sum بنعمل نفسه
+
+470
+00:41:09,000 --> 00:41:14,680
+مع مين مع اللي هو lower sum هنلاقي اللي هو LQ يعني
+
+471
+00:41:14,680 --> 00:41:19,060
+LQ star و F ناقص LQ F هنلاقيه هنا على الأكثر
+
+472
+00:41:19,060 --> 00:41:23,020
+باختلاف أنه هو الاثنين فئة ناقص واحد من التلزمه و و و
+
+473
+00:41:23,020 --> 00:41:26,640
+الأخر و بنكمل نفس الشيء بطل عند LQ star و F ناقص
+
+474
+00:41:26,640 --> 00:41:30,600
+إبشر على ثلاثة أصغر من مين من LQ F إذا similarly
+
+475
+00:41:30,600 --> 00:41:35,120
+for هذه اللي عملناها هذه ما حد بنعمله مع اللي هو
+
+476
+00:41:35,730 --> 00:41:47,930
+الـ L L of Q star و F ماقص اللي هو إبسلون على ثلاثة
+
+477
+00:41:47,930 --> 00:41:55,890
+أصغر من L Q O mean O F ماشي الحال، طيب and exactly
+
+478
+00:41:55,890 --> 00:42:00,310
+زي ما بنقول similar argument symbolize that اللي
+
+479
+00:42:00,310 --> 00:42:07,870
+كتبناها لنبدأ نجمع معلوماتنا عشان نصل لإنه الـ
+
+480
+00:42:07,870 --> 00:42:11,530
+Riemann Sum نقص الـ integration يكون إيه شماله
+
+481
+00:42:11,530 --> 00:42:15,890
+اللي هو أصغر من إبسيلون، انتبهوا كيف عشان يمكن هيك
+
+482
+00:42:15,890 --> 00:42:23,210
+تكون أسهل لنا نرسم خط الأعداد وهنا هخلينا اللي هو
+
+483
+00:42:23,210 --> 00:42:43,920
+نحط الـ U أكي و FU Q F هي أصغر من
+
+484
+00:42:43,920 --> 00:42:55,420
+إبسيلون على ثلاثة زي U U أستار U F عندي
+
+485
+00:42:55,420 --> 00:42:57,860
+U Q F
+
+486
+00:43:00,380 --> 00:43:07,380
+بس خليني عشان أحط الـ L قبل لـ L قبل هه عشان مفيش
+
+487
+00:43:07,380 --> 00:43:14,800
+وسح أنا جماعة بعد أذنكم خليني بس هكتب الأول أشنب
+
+488
+00:43:14,800 --> 00:43:27,870
+ده بمين بـ L L Q و F الـ Refinement طبعاً لـ L, Q,
+
+489
+00:43:27,970 --> 00:43:36,450
+Star و F قبلها زي ما أنتم عارفين L, Q و F خليني
+
+490
+00:43:36,450 --> 00:43:41,090
+أعمل هذه قبلها عشان تظبط أحسن L, Q, Star و F ناقص
+
+491
+00:43:41,090 --> 00:43:50,640
+بإبسيلون على ثلاثة وهذه LUQF معايا هي هذه بتكون قبل هذه
+
+492
+00:43:50,640 --> 00:43:59,300
+طيب الـ QF أكيد بعدها مين UQF صح ولا لأ أكيد طيب
+
+493
+00:43:59,300 --> 00:44:07,420
+بعد الـ UQF إيش بتيجي بتيجي هنا إبسيلون على ثلاثة زائد
+
+494
+00:44:07,420 --> 00:44:15,340
+UQ star و F مظبوط مظبوط أكيد؟ طيب إذا أنا أجيت لخصت
+
+495
+00:44:15,340 --> 00:44:21,120
+المعلومة هذه و لخصت المعلومة هذه اللي هي أصغر من
+
+496
+00:44:21,120 --> 00:44:28,460
+إبسيلون على ثلاثة UQF UQF ولا الـ PUF هذه المسافة
+
+497
+00:44:28,460 --> 00:44:32,910
+بينهم إيه أشمل يا جماعة؟ أصغر من إبسيلون على ثلاثة
+
+498
+00:44:32,910 --> 00:44:37,130
+واضح أه؟ ومن هنا قال لك U square of F نقص هذه وهذه
+
+499
+00:44:37,130 --> 00:44:43,050
+إبسيلون وهذه بالترتيب اللي احنا حصلنا عليه طيب الآن
+
+500
+00:44:43,050 --> 00:44:48,230
+طبيعي اللي هو الـ remansum و الـ integration هيكون
+
+501
+00:44:48,230 --> 00:44:52,030
+بين الـ lower و بين الـ main اللي هو إيش الأبر إذن
+
+502
+00:44:52,030 --> 00:44:57,110
+أكيد أكيد أكيد الـ remansum تبعنا هي بدينا نصل للي
+
+503
+00:44:57,110 --> 00:45:04,400
+وجدنا إياها الـ CQF واللي هو الـ integration من a إلى
+
+504
+00:45:04,400 --> 00:45:08,720
+b إلى الـ F هيكون بين الـ lower و بين الـ upper الـ
+
+505
+00:45:08,720 --> 00:45:11,720
+integration موجود لأن مراس الدور جللنا إذا هذوله
+
+506
+00:45:11,720 --> 00:45:17,800
+صار اللي هو بين هذه و بين هذه إذا صار هذه و هذه
+
+507
+00:45:17,800 --> 00:45:26,120
+بين هذه الاثنتين وهذه طبعاً الـ U Q F و اللي قاله Q F
+
+508
+00:45:26,120 --> 00:45:31,780
+واضح .. واضح أنه اللي هو وين في داخل الفترة اللي
+
+509
+00:45:31,780 --> 00:45:36,660
+هي L Q Star و F ناقص إبسيلون على ثلاثة و U Q Star و F
+
+510
+00:45:36,660 --> 00:45:42,500
+إبسيلون على ثلاثة يعني بين .. هذه طبعاً .. بين هذه و
+
+511
+00:45:42,500 --> 00:45:47,200
+بين مين و بين الـ interval هذه يعني subset من مين
+
+512
+00:45:47,200 --> 00:45:53,460
+جايل من I إبسيلون اللي هي I Epsilon بتساوي اللي هو
+
+513
+00:45:53,460 --> 00:45:59,440
+الفترة هذه من هنا لعند الـ hand هو اللي مسمينها
+
+514
+00:45:59,440 --> 00:46:05,740
+احنا شماله I Epsilon بين L Q Star و F نقص
+
+515
+00:46:05,740 --> 00:46:11,860
+Epsilon على ثلاثة و U Q Star F و Epsilon على ثلاثة
+
+516
+00:46:11,860 --> 00:46:20,060
+الآن U Q Star و F ناقص قالك U Star و F ما مسحتاش
+
+517
+00:46:20,060 --> 00:46:28,200
+لسه أنا U Q Star F و قالك U Star F أصغر من مين من
+
+518
+00:46:28,200 --> 00:46:36,520
+إبسيلون على ثلاثة لأ اللي هي U
+
+519
+00:46:36,520 --> 00:46:42,670
+Q Star F وقال الـ Q star F حصلنا عليها قبل هيك أنها
+
+520
+00:46:42,670 --> 00:46:46,650
+أصغر من مين من إبسيلون على ثلاثة طبعاً من وين بنحصلها؟
+
+521
+00:46:46,650 --> 00:46:49,650
+ممكن نحصلها مبقى من هنا ده كما حصلناش أن الـ Q
+
+522
+00:46:49,650 --> 00:46:54,430
+star و F و الـ By بتحتوي الـ By فهيكون الـ U اللي
+
+523
+00:46:54,430 --> 00:47:00,090
+حصلنا أولاً بالنسبة لمن؟ للي هي الـ By حصلنا قبل
+
+524
+00:47:00,090 --> 00:47:06,400
+هيك أن الـ U بإبسيلون و الـ L بإبسيلون الفرق بينهم إين
+
+525
+00:47:06,400 --> 00:47:10,840
+إبسيلون على ثلاثة و الـ Q star refinement إذا أكيد
+
+526
+00:47:10,840 --> 00:47:14,920
+المسافة بين الـ other طبعاً في الـ refinement المسافة
+
+527
+00:47:14,920 --> 00:47:20,440
+بين الـ other و الـ other هتصغر لكل refinement الـ U
+
+528
+00:47:20,440 --> 00:47:28,340
+بما أن هذه refinement إذا الـ U لأ الـ Q star و F
+
+529
+00:47:28,340 --> 00:47:34,960
+ناقص L Q star و F أصغر أو يساوي لأنها صارت تحسين
+
+530
+00:47:34,960 --> 00:47:38,400
+عليها مازالت صارت تحسين يعني بتجلل مسافة بين الـ
+
+531
+00:47:38,400 --> 00:47:40,780
+Upper و الـ Lower عشان نروح في الآخر للـ integration
+
+532
+00:47:40,780 --> 00:47:51,520
+ناقص أصغر أو يساوي لـ L B epsilon و F U ناقص L B
+
+533
+00:47:51,520 --> 00:47:56,980
+epsilon و F اللي هي هذه أصغر من 100 من إبسلون وينها
+
+534
+00:47:56,980 --> 00:48:03,400
+أصغر من إبسلون على ثلاثة أصغر من إبسلون على ثلاثة إذا
+
+535
+00:48:03,400 --> 00:48:10,320
+فعلاً عرفنا من وين إجت هذه أصغر من مين من إبسلون على
+
+536
+00:48:10,320 --> 00:48:18,460
+ثلاثة إذا صار عندي الـ length لهذه الـ length لهذه
+
+537
+00:48:18,460 --> 00:48:23,240
+سهل نوجده بعليش خليني أكتب فوق عشان لو جبت لكم الـ
+
+538
+00:48:23,240 --> 00:48:29,780
+length إلها عشان أقول لكم إنه هذه و هذه هيكون اللي
+
+539
+00:48:29,780 --> 00:48:38,900
+هو بالبعد المطلوب احسب لي الآن هذه ناقص هذه بيصير
+
+540
+00:48:38,900 --> 00:48:48,940
+عبارة عن طول الـ interval mean Iy بساوي الـ UQ star
+
+541
+00:48:48,940 --> 00:48:55,780
+و F زائد إبسيلون على ثلاثة ناقص تبعين معيار الـ Q
+
+542
+00:48:55,780 --> 00:49:01,720
+star و F ناقص ناقص بيصير زائد إبسيلون على ثلاثة و
+
+543
+00:49:01,720 --> 00:49:04,900
+يساوي اللي هو إبسيلون على ثلاثة و إبسيلون على ثلاثة
+
+544
+00:49:04,900 --> 00:49:12,180
+اثنين إبسيلون على ثلاثة زائد U Q star و F ناقص الـ Q
+
+545
+00:49:12,180 --> 00:49:18,800
+star و F اللي هم هذولة هذه اللي كتبتها لسه ماحتنش
+
+546
+00:49:18,800 --> 00:49:23,400
+أصغر من مين من إبسيلون على ثلاثة فبتصير هذه طولها
+
+547
+00:49:23,400 --> 00:49:26,700
+أصغر من اثنين إبسيلون على ثلاثة زائد إبسيلون على
+
+548
+00:49:26,700 --> 00:49:31,620
+ثلاثة ويساوي إبسيلون إذا صار عندي طول الـ interval
+
+549
+00:49:31,620 --> 00:49:40,520
+اللي أسي Qf و الـ integration A وB موجودة في كل
+
+550
+00:49:40,520 --> 00:49:46,660
+الفترة هذه اللي هي المسافة بينهم عبارة عن أصغر من
+
+551
+00:49:46,660 --> 00:49:53,460
+مين من Epsilon إذا صار عندي الآن الـ integration
+
+552
+00:49:53,460 --> 00:50:05,480
+أين أكتب الـ integration إذا صار عندي إذا صار عندي
+
+553
+00:50:05,480 --> 00:50:12,380
+المسافة بين هذه و هذه SQF نقص الـ integration من A B
+
+554
+00:50:12,380 --> 00:50:16,900
+ل F أصغر من مين يا جماعة من إبسيلون من مين هذا تحت
+
+555
+00:50:16,900 --> 00:50:23,360
+شرط مين إنه normally Q هذا أصغر من A H من دلتا
+
+556
+00:50:23,360 --> 00:50:29,200
+أصغر من دلتا بقول إيه احنا خلصنا اللي بدنا إياه
+
+557
+00:50:29,200 --> 00:50:32,280
+نوقف
+
+558
+00:50:32,280 --> 00:50:33,020
+الجيجا بحثها
+
+559
+00:51:39,830 --> 00:51:44,830
+أه يا أبو حسن بقول هيك احنا خلصنا برهان نظرية أول
+
+560
+00:51:44,830 --> 00:51:48,490
+حد يقول لي أنه اللي هو المسافة بين U وQ قال الـ Q
+
+561
+00:51:48,490 --> 00:51:50,950
+أصغر من إبسيلون على ثلاثة لأ احنا ماقلناش هيك
+
+562
+00:51:50,950 --> 00:51:53,950
+المسافة بين الأصغر من إبسيلون على ثلاثة اللي هو الـ
+
+563
+00:51:53,950 --> 00:51:58,190
+P إبسيلون الـ U P إبسيلون و الـ L P إبسيلون وأي
+
+564
+00:51:58,190 --> 00:52:01,010
+refinement له بس الـ Q هنا مش refinement هذا بس
+
+565
+00:52:01,010 --> 00:52:04,830
+اللي بنعرف أنه norm لـ Q أصغر من مين من دلتا أصلاً
+
+566
+00:52:04,830 --> 00:52:08,9
+
+601
+00:54:59,420 --> 00:55:04,470
+عبارة عن اللي هو بن..اللي هي دمج بين اللي هي
+
+602
+00:55:04,470 --> 00:55:08,190
+نظريات السابقة المرة الماضية نشوف كيف اللي هو
+
+603
+00:55:08,190 --> 00:55:14,100
+البرهن برهان بسيط لإنه بعتمد عليه بشكل سريع لـ y
+
+604
+00:55:14,100 --> 00:55:17,440
+أكبر من 0 و delta أكبر من 0 as in the theorem اللي
+
+605
+00:55:17,440 --> 00:55:20,180
+هو اللي في الـ theorem اللي فوق المقصود فيها النص
+
+606
+00:55:20,180 --> 00:55:25,820
+هذه قدت بي إبسلون  partition of I with نفترض أنه
+
+607
+00:55:25,820 --> 00:55:30,080
+norm بي إبسلون أصغر من دلتا هذا المعطى ماشي
+
+608
+00:55:30,080 --> 00:55:36,240
+الحال نفترض أنه إبسلون أكبر من 0 arbitrarily و
+
+609
+00:55:36,240 --> 00:55:41,190
+delta أكبر من 0 أو نفترض أن بي إبسلون أي partition
+
+610
+00:55:41,190 --> 00:55:45,310
+بحيث أنه نورمه إيه شماله أصغر من دلتا لكن لو جينا 
+
+611
+00:55:45,310 --> 00:55:48,930
+أخدنا بيه أي partition آخر و بيه محتوى بيه إبسلون 
+
+612
+00:55:48,930 --> 00:55:53,770
+هيكون نورم ال partition بيه هذا اللي هو ال
+
+613
+00:55:53,770 --> 00:55:57,470
+refinement أكيد أصغر أو يساوي نورم مين الأصلي اللي
+
+614
+00:55:57,470 --> 00:56:01,030
+هو أصغر من دلتا صار عنده الآن لكل إبسلون أكبر من
+
+615
+00:56:01,030 --> 00:56:06,470
+صفر there exists دلتا بحيث أنه نورم البي إبسلون 
+
+616
+00:56:06,470 --> 00:56:11,740
+أصغر من دلتا ولأي partition بي refinement لبيبسلون
+
+617
+00:56:11,740 --> 00:56:17,260
+صار norm أصغر من delta إذا من اللي هو ال.. بواسط
+
+618
+00:56:17,260 --> 00:56:21,740
+اللي هو النظرية السابقة حيكون عند ال S minus البي
+
+619
+00:56:21,740 --> 00:56:27,580
+أصغر من ابسلون يؤدي إلى هذا بيبساو ال integration
+
+620
+00:56:27,580 --> 00:56:32,020
+لإن إيش كانت نظرية سبعة أربعة تلاتة بتشترط تشترط
+
+621
+00:56:32,020 --> 00:56:33,840
+إنه نلاقي لكل ابسلون
+
+622
+00:56:36,480 --> 00:56:42,920
+أي بارتشن بي ابسلون بحيث أن أي بي بتحتول بي ابسلون
+
+623
+00:56:43,910 --> 00:56:48,530
+يؤدي إلى هذا أصغر من إبسلون كله على بعض بيؤدي F
+
+624
+00:56:48,530 --> 00:56:53,790
+أشماله is integrable إذا بيواصل 7 4 3 مدام حصلنا
+
+625
+00:56:53,790 --> 00:56:57,910
+على ال S بي و F نقص ال V أصغر من إبسلون بيكون هذا
+
+626
+00:56:57,910 --> 00:57:01,190
+اللي حصلنا عليه اللي هو P بساوة ال integration لل
+
+627
+00:57:01,190 --> 00:57:06,630
+F على الفترة من A لB يعني بمعنى أخر احنا اللي
+
+628
+00:57:06,630 --> 00:57:13,530
+حصلنا الآن من النظريتين أن F is integrable F is
+
+629
+00:57:13,530 --> 00:57:19,050
+integrable if and only if limit الـ S,B وF من
+
+630
+00:57:19,050 --> 00:57:23,890
+normal B بيروح للسفر exist وبساوي قيمة ال
+
+631
+00:57:23,890 --> 00:57:28,550
+integration يعني الآن ال integration أوصلنا له عبر
+
+632
+00:57:28,550 --> 00:57:34,330
+طريقين إما عبر طريق اللي هو limit للرمان sumيكون
+
+633
+00:57:34,330 --> 00:57:38,690
+exist أو عبر الطريق الآخر لأن ال upper.. ال upper
+
+634
+00:57:38,690 --> 00:57:42,930
+integral والlower integral يكون جد بعض ومتساويين
+
+635
+00:57:42,930 --> 00:57:52,690
+وحيكون في هذه الحالة هم ال integration وقف
+
+636
+00:57:52,690 --> 00:57:59,570
+هنا يا أبو حسن كم ساعة.. كم
+
+637
+00:57:59,570 --> 00:58:00,810
+عندك الوقت؟
+
+638
+00:58:03,290 --> 00:58:10,650
+نخلصها هيك و نبدأ تانية؟ لأن الجزء الثاني هذا اللي
+
+639
+00:58:10,650 --> 00:58:17,070
+هو الimproper integral بس بطولش بياخد نص ساعة انت 
+
+640
+00:58:17,070 --> 00:58:22,090
+قاعد معايا للجد إيش؟ أكيد؟ مديش أضغط عليك يا أبو
+
+641
+00:58:22,090 --> 00:58:30,390
+حسن طيب لأ خلينا نخلص هذه ما هو درس 17 ها؟ هلجيت
+
+642
+00:58:30,390 --> 00:58:31,270
+18؟
+
+643
+00:58:41,590 --> 00:58:48,290
+هي الان بتبدأ في المحاضرة 18 خلصنا 17 كيف؟
+
+644
+00:58:54,810 --> 00:58:59,850
+هيك مكون احنا خلصنا المحاضر رقم 17 اللي هو ال
+
+645
+00:58:59,850 --> 00:59:03,610
+integration as a limit بعد شوية هنكمل ال section
+
+646
+00:59:03,610 --> 00:59:08,490
+هذا مظبوط بس هنكمله بموضوع جديد اللي هو عبارة عن
+
+647
+00:59:08,490 --> 00:59:12,510
+الimproper integral وإلى لقاء آخر والسلام عليكم
+
+648
+00:59:12,510 --> 00:59:13,670
+ورحمة الله وبركاته
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/7kAGhF5ugUM_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/7kAGhF5ugUM_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8adbb2bf14facdd870dbb8e38ff1dbe111ee4abb
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/7kAGhF5ugUM_raw.srt
@@ -0,0 +1,2628 @@
+1
+00:00:10,580 --> 00:00:15,760
+إيش يا أبو حسين؟ بسم الله الرحمن الرحيم الحمد لله
+
+2
+00:00:15,760 --> 00:00:18,980
+رب العالمين والصلاة والسلام على سيد المرسلين سيدنا
+
+3
+00:00:18,980 --> 00:00:24,540
+محمد وعلى آله وصحبه أجمعين هذه هي المحاضرة رقم 17
+
+4
+00:00:24,540 --> 00:00:29,900
+في مساق تحليل حقيقة 2 لطلاب وطالبات الجامعة
+
+5
+00:00:29,900 --> 00:00:37,170
+الإسلامية كلية العلود قسم الرياضيةبدأنا اليوم
+
+6
+00:00:37,170 --> 00:00:41,490
+استكمال للي بدأنا المرة الماضية حكينا المرة
+
+7
+00:00:41,490 --> 00:00:48,070
+الماضية عن الـ integration as a limit قلنا احنا
+
+8
+00:00:48,070 --> 00:00:51,970
+قبل كنا معرفين ال integration انه موجود على أساس
+
+9
+00:00:51,970 --> 00:00:56,950
+ان ال upper integral U of F يساوي ال lower
+
+10
+00:00:56,950 --> 00:01:01,710
+integral L of F قلنا المرة الماضية ان احنا بدنا
+
+11
+00:01:01,710 --> 00:01:07,300
+نروح باتجاهإنه نصل في النهاية إنه هذا الكلام اللي
+
+12
+00:01:07,300 --> 00:01:11,440
+هو الـ Integrability يكافئ إن نصل إنه ال limit
+
+13
+00:01:11,440 --> 00:01:14,760
+طبعا وفي حالة وجوده هنسميه بساوة ال integration من
+
+14
+00:01:14,760 --> 00:01:19,940
+a ل b of x dx قلنا المرة الماضية إنه هنصل في
+
+15
+00:01:19,940 --> 00:01:25,600
+النهاية في نهاية ال section 7-4 إنه هذا اللي هو ال
+
+16
+00:01:25,600 --> 00:01:28,840
+condition أو هذا ال definition يكافئ إنه نقول
+
+17
+00:01:28,840 --> 00:01:35,640
+limit اللي هو SB و F as normal B أو المش تبع الـ B
+
+18
+00:01:35,640 --> 00:01:41,100
+بيروح لـ 0 و في هذه الحالة إذا كان هذا exist بيصير
+
+19
+00:01:41,100 --> 00:01:45,380
+.. هيساوي ال integration من A ل B F of X DX يعني
+
+20
+00:01:45,380 --> 00:01:49,500
+بمعنى آخر .. بمعنى آخر أنه اللي هو هذا طبعا الـ
+
+21
+00:01:49,500 --> 00:01:55,460
+Riemann Sum أنه الـ limitلرمان sum as الـ
+
+22
+00:01:55,460 --> 00:02:00,020
+partition B بيروح للـ 0 لو كان exist حسير اللي هي
+
+23
+00:02:00,020 --> 00:02:04,220
+F is integrable والـ integration لها هو عبارة عن
+
+24
+00:02:04,220 --> 00:02:09,520
+قيمة هذا الـ limitولو كان ال integration exist
+
+25
+00:02:09,520 --> 00:02:14,040
+طبعا بالمفهوم اللي هو هذا هيكون عنده ال limit هذا
+
+26
+00:02:14,040 --> 00:02:18,300
+exist و هيساوي قيمة مين ال integration هذا اللي
+
+27
+00:02:18,300 --> 00:02:22,260
+حنصل إليه في النهاية اللي حكيناه المرة الماضية
+
+28
+00:02:22,260 --> 00:02:27,380
+بشكل سريع اللي هو عرفنا إيش هو معنات اللي هو ال
+
+29
+00:02:27,380 --> 00:02:32,080
+remand صم اللي أمامنا هذابعد هيكة اللي هو إيجينا
+
+30
+00:02:32,080 --> 00:02:36,600
+أخدنا اللي هي الخاصية الأولى قلنا دايما الريمان
+
+31
+00:02:36,600 --> 00:02:41,140
+sum بين اللي هو ال lower sum و بين ال upper sum و
+
+32
+00:02:41,140 --> 00:02:46,080
+بعدين اللي هو إيجينا قلنا اللي هو النظرية ال .. ال
+
+33
+00:02:46,080 --> 00:02:47,220
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+34
+00:02:47,220 --> 00:02:47,280
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+35
+00:02:47,280 --> 00:02:47,280
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+36
+00:02:47,280 --> 00:02:48,000
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+37
+00:02:48,000 --> 00:02:48,120
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+38
+00:02:48,120 --> 00:02:49,580
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+39
+00:02:49,580 --> 00:02:49,700
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+40
+00:02:49,700 --> 00:02:50,220
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+41
+00:02:50,220 --> 00:02:59,260
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال ..إذا
+
+42
+00:02:59,260 --> 00:03:03,200
+كانت F Integrable إذا لكل Y أكبر من 0 نجدر نلاقي
+
+43
+00:03:03,200 --> 00:03:07,220
+partition بي إبسلون و by such that إذا كانت بي أي
+
+44
+00:03:07,220 --> 00:03:10,600
+اللي هو refinement partition refinement للبي
+
+45
+00:03:10,600 --> 00:03:14,260
+إبسلون يعني يحتوي البي إبسلون هيكون و كان عند ال
+
+46
+00:03:14,260 --> 00:03:18,540
+SPUF عبارة عن اللي هو الريمان sumبكون الـ absolute
+
+47
+00:03:18,540 --> 00:03:21,300
+value ال integration بين S بي و F مقصد ال
+
+48
+00:03:21,300 --> 00:03:24,840
+integration أش ماله أصغر من مين من إبسلون يعني لو
+
+49
+00:03:24,840 --> 00:03:28,460
+F is integrable هنلاقي بي إبسلون بحيث كل ال B
+
+50
+00:03:28,460 --> 00:03:31,540
+بتحتوي بي إبسلون هيكون ال absolute value بين
+
+51
+00:03:31,540 --> 00:03:36,400
+الرمان صموا هذا أصغر من مين من اللي هي إيش إبسلون
+
+52
+00:03:36,820 --> 00:03:41,900
+طيب بعد هي كتب عن البرهناها هذه وصار في عندي الآن
+
+53
+00:03:41,900 --> 00:03:46,240
+اللي هو النظرية الثانية أيضا برهناها و النظرية
+
+54
+00:03:46,240 --> 00:03:52,110
+الثانية كانت تقول أنه لو كانت F is boundedالان ..
+
+55
+00:03:52,110 --> 00:03:57,250
+وقدرنا .. وبدنا الان نفترض انه لجينا اللي هو
+
+56
+00:03:57,250 --> 00:04:01,830
+number a يحقق مالي ايش هذا number a يحقق انه لكل
+
+57
+00:04:01,830 --> 00:04:04,890
+epsilon أكبر من 0 ونلاقي بي ابسلون بحيث انه أي بي
+
+58
+00:04:04,890 --> 00:04:11,040
+يحتوي بي ابسلون و S اللي هي أي رمان sumوthen اللي
+
+59
+00:04:11,040 --> 00:04:16,060
+هي s,b وf ناقص a أصغر من إبسلون إذا كان هذا الكلام
+
+60
+00:04:16,060 --> 00:04:19,480
+كله على بعض اتحقق لما نلاقي a بحيث يقول كل إبسلون
+
+61
+00:04:19,480 --> 00:04:24,180
+أكبر من سفر اللي هو نلاقي بي إبسلون لو كان بي
+
+62
+00:04:24,180 --> 00:04:27,740
+إبسلون بحيث أن بي بتحتوي بي إبسلون وهذا ال remand
+
+63
+00:04:27,740 --> 00:04:32,030
+sum هيعطيني هذههذا كله على بعض ال conditions زي ما
+
+64
+00:04:32,030 --> 00:04:35,410
+قلنا المرة الماضية بيعطينا ان F اي شمالها
+
+65
+00:04:35,410 --> 00:04:39,270
+integrable وهذا الـ A اللي حقق هذه الشروط هنا هو
+
+66
+00:04:39,270 --> 00:04:44,050
+قيمة ال integration هذا اللي حكيناه المرة الماضية
+
+67
+00:04:44,050 --> 00:04:51,120
+و بعدين اللي هوأحنا إجينا وعرفنا شو معنات ال مش أو
+
+68
+00:04:51,120 --> 00:04:54,800
+ال norm إذا بذكركم ال مش أو ال norm هو عبارة عن ال
+
+69
+00:04:54,800 --> 00:05:00,140
+supremum لاللي هي أطوال ال subintervals تبعات ال
+
+70
+00:05:00,140 --> 00:05:03,060
+partition قلنا ال norm للبيبس أو ال supremum لهذه
+
+71
+00:05:03,060 --> 00:05:07,280
+اللي هي الأطوال تبعات ال subintervals اللي ناتجة
+
+72
+00:05:07,280 --> 00:05:11,550
+من ال partition Bهذا اسمنا المش أو ال norm أخدنا
+
+73
+00:05:11,550 --> 00:05:17,230
+بعض اللي هي الخواص وصُجفنا عند مين؟ عند Darabox's
+
+74
+00:05:17,230 --> 00:05:20,690
+theorem اليوم بدنا نبدأ نشرح في مين في Darabox's
+
+75
+00:05:20,690 --> 00:05:24,230
+theorem نشوف أيش Darabox's theorem بتقول الآن بقول
+
+76
+00:05:24,230 --> 00:05:28,990
+لي let i بتساوي a و bاللي هي bounded interval و F
+
+77
+00:05:28,990 --> 00:05:32,910
+من I لعنب R Integrable on I in the sense of
+
+78
+00:05:32,910 --> 00:05:38,050
+definition mean 716 يعني بقول لي نفترض أن F is an
+
+79
+00:05:38,050 --> 00:05:42,830
+integrable function حسب تعريف mean 716 يعني ال
+
+80
+00:05:42,830 --> 00:05:46,470
+upper integral بسوء ال lower integral بسوء ال
+
+81
+00:05:46,470 --> 00:05:50,350
+integration هذا اللي هو ال definition 716 الآن
+
+82
+00:05:50,350 --> 00:05:56,710
+بقول لي then for every epsilon أكبر من 0then for
+
+83
+00:05:56,710 --> 00:06:02,450
+every epsilon أكبر من صفر there exists delta such
+
+84
+00:06:02,450 --> 00:06:07,150
+thatالان لو كان P is any partition of I و ال
+
+85
+00:06:07,150 --> 00:06:11,710
+normal P أصغر من Delta و كان عند ال S P و F اللي
+
+86
+00:06:11,710 --> 00:06:15,810
+هو is any corresponding Riemann sum then ال
+
+87
+00:06:15,810 --> 00:06:19,450
+integration بين ال S P .. البعد بين ال integration
+
+88
+00:06:19,450 --> 00:06:23,670
+بين ال S P و ال integration أصغر من مين أصغر من
+
+89
+00:06:23,670 --> 00:06:28,990
+Epsilon بس
+
+90
+00:06:28,990 --> 00:06:29,910
+بقى في دقيقة يا أبو حسن
+
+91
+00:06:59,460 --> 00:07:03,900
+أيوة أنا في محاضرة .. أنا في محاضرة وبسجل .. أيوة
+
+92
+00:07:03,900 --> 00:07:08,400
+أه في
+
+93
+00:07:08,400 --> 00:07:16,360
+كلية العلوم؟ من قالك في كلية العلوم؟
+
+94
+00:07:16,360 --> 00:07:18,500
+لأ يا عم أنا روحت على دراسات العلوم اللي قالولي
+
+95
+00:07:18,500 --> 00:07:27,580
+طلعن أه .. بعرفش .. بعرفش كلام هو صحيح ولا لأ لأ
+
+96
+00:07:27,580 --> 00:07:33,080
+لأبقى طيب يا راجل أنا في المحاضرة ماشي مع السلام
+
+97
+00:07:33,080 --> 00:07:40,800
+.. ماشي بقى كله مروح مع السلام ليش؟
+
+98
+00:07:40,800 --> 00:07:44,040
+مع السلام .. بس شوف كيف مع السلام
+
+99
+00:08:32,780 --> 00:08:36,900
+اعطيني بس تليفون هذا في العلوم 1600 انا عارف الله
+
+100
+00:08:36,900 --> 00:08:41,780
+ده
+
+101
+00:08:41,780 --> 00:08:45,660
+الدس
+
+102
+00:08:45,660 --> 00:08:50,040
+اه هو كيد روح اذا ابو بلال الموجود روح
+
+103
+00:09:15,660 --> 00:09:18,500
+عليكم السلام عليك يا أبو ساما كيف حالك؟ كيف صحتك؟
+
+104
+00:09:18,500 --> 00:09:21,980
+الله بعرفيك الحمد لله الله عزيزي عنك هل في إلي
+
+105
+00:09:21,980 --> 00:09:28,260
+عندكم رسالة تابعة السناق؟ و تابعة ياسر؟ طب .. طب
+
+106
+00:09:28,260 --> 00:09:33,820
+.. مين عندك؟ قاعد بتسأل عليه لو .. لو أبو أكرم
+
+107
+00:09:33,820 --> 00:09:36,880
+أخدن بس و حطن في غرفتي لأنه أنا قاعد بصور والله
+
+108
+00:09:36,880 --> 00:09:42,060
+يبقى اه .. اه .. اه .. الله يبارك فيا يا عزيزي انا
+
+109
+00:09:42,060 --> 00:09:50,370
+بجيبها المفتاح هوأه أه ماشي
+
+110
+00:09:50,370 --> 00:09:54,770
+لو حتى أنا يعني أنا موجود بس بنتأخر يعني التصوير
+
+111
+00:09:54,770 --> 00:09:59,950
+ممكن أظل التلاتة ماشي ماشي الله أبقى أرفع حاجة
+
+112
+00:09:59,950 --> 00:10:03,290
+لإنه بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+113
+00:10:03,290 --> 00:10:03,490
+بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+114
+00:10:03,490 --> 00:10:03,490
+بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+115
+00:10:03,490 --> 00:10:03,490
+بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+116
+00:10:03,490 --> 00:10:06,830
+بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس بس
+
+117
+00:10:06,830 --> 00:10:08,250
+ب
+
+118
+00:10:27,230 --> 00:10:30,330
+أدن درابكس الـ theorem زي ما قلنا إن لو كانت F
+
+119
+00:10:30,330 --> 00:10:34,600
+انتجرة بالـ functionبقول إذا لكل إبسلون أكبر من 0
+
+120
+00:10:34,600 --> 00:10:38,780
+بقدر ألاقي Delta أكبر من 0 بحيث أنه لو كان بي أي
+
+121
+00:10:38,780 --> 00:10:41,620
+partition بحيث أنه ال .. اللي هو ال .. ال .. ال
+
+122
+00:10:41,620 --> 00:10:44,980
+norm للـ B أصغر من Delta و S, B و F أي
+
+123
+00:10:44,980 --> 00:10:49,420
+corresponding remand sum بالنسبة للـ B هيعطيني أنه
+
+124
+00:10:49,420 --> 00:10:52,860
+ال absolute value بين اللي هو أو البعد بين S, B و
+
+125
+00:10:52,860 --> 00:10:57,780
+F ناقص ال integration أصغر من إبسلون يعني و كأنه
+
+126
+00:10:57,780 --> 00:11:02,190
+بقول من تحت بطنانه ال integration ال integration
+
+127
+00:11:02,190 --> 00:11:08,710
+لل F لو F is integrable بيعطينا انه بمفهوم اللي هو
+
+128
+00:11:08,710 --> 00:11:18,140
+سبعة واحد ستة بكون عنده limit ال SB وF as normal B
+
+129
+00:11:18,140 --> 00:11:24,320
+بروح للسفر بسوا بالظبط integration من A لB F of X
+
+130
+00:11:24,320 --> 00:11:29,260
+DX لأن ايش معناته هذا؟ معناته هذا اللي هو حسب الـ
+
+131
+00:11:29,260 --> 00:11:32,160
+Epsilon Delta Definition for every Epsilon أكبر من
+
+132
+00:11:32,160 --> 00:11:36,740
+سفر there exists Delta أكبر من سفر such that if
+
+133
+00:11:36,740 --> 00:11:43,170
+normal B أصغر، ناقص السفر أصغر من دلتاthen هيعطيني
+
+134
+00:11:43,170 --> 00:11:46,810
+الـ absolute value للـ S, P و F ناقص ال
+
+135
+00:11:46,810 --> 00:11:52,650
+integration من A ل B F of X DX كله أصغر من مين من
+
+136
+00:11:52,650 --> 00:11:56,790
+إبسلون إذا اللي بدنا نتبته بدنا لكل إبسلون نلاقي
+
+137
+00:11:56,790 --> 00:12:01,870
+Deltaبحيث أنه لما أي partition نورمه أصغر من Delta
+
+138
+00:12:01,870 --> 00:12:05,510
+يعطيني الفرق بين الـ Riemann Sum و بين الـ
+
+139
+00:12:05,510 --> 00:12:08,910
+Integration يكون أصغر من مين؟ من Epsilon هذه اللي
+
+140
+00:12:08,910 --> 00:12:11,990
+هي اللي بتقولنا درابوكس Theorem أنه لو F is
+
+141
+00:12:11,990 --> 00:12:15,610
+integrable إذا اللي هو الـ limit لاللي هو الـ
+
+142
+00:12:15,610 --> 00:12:18,730
+Riemann Sum هو حيساوي هذا ال integration اللي
+
+143
+00:12:18,730 --> 00:12:23,330
+فرضنا إحنا إشماله أنه موجود خلينا نشوف كيف بدنا
+
+144
+00:12:23,330 --> 00:12:29,380
+انبرهن النظرية و ركزوا معاياوتبعوا إيش اللي بدنا
+
+145
+00:12:29,380 --> 00:12:35,200
+نحكيه عشان نصل للي هو المطلوب طيب صلى الله عليه
+
+146
+00:12:35,200 --> 00:12:41,370
+الصلاة والسلام أول إشي بما أن F is integrableالـ
+
+147
+00:12:41,370 --> 00:12:42,290
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+148
+00:12:42,290 --> 00:12:42,330
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+149
+00:12:42,330 --> 00:12:43,190
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+150
+00:12:43,190 --> 00:12:43,510
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+151
+00:12:43,510 --> 00:12:44,470
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+152
+00:12:44,470 --> 00:12:50,750
+ال .. ال .. ال .. ال .. ال ..
+
+153
+00:12:50,750 --> 00:12:52,070
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+154
+00:12:52,070 --> 00:12:53,590
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+155
+00:12:53,590 --> 00:12:53,750
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+156
+00:12:53,750 --> 00:12:54,490
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+157
+00:12:54,490 --> 00:12:54,910
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+158
+00:12:54,910 --> 00:12:54,910
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+159
+00:12:54,910 --> 00:12:54,910
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+160
+00:12:54,910 --> 00:13:02,270
+.. ال .. ال .. ال .. الإذن الآن أخفرضت epsilon
+
+161
+00:13:02,270 --> 00:13:06,870
+arbitrary بدي أدور على الـ delta اللي لما ال
+
+162
+00:13:06,870 --> 00:13:10,630
+normal بيه أصغر من delta يعطيني الفرق بين هذه و
+
+163
+00:13:10,630 --> 00:13:14,710
+هذه أصغر من مين من epsilon لذا هيك رايح أنا لأن
+
+164
+00:13:14,710 --> 00:13:19,400
+إذن بما أن أفنتي جربت لكل epsilon أكبر من 0بقدر
+
+165
+00:13:19,400 --> 00:13:23,980
+ألاقي partition الآن سميته بي إبسلون اللي هو x0 و
+
+166
+00:13:23,980 --> 00:13:29,400
+x1 لعندي xn such that ال U ال upper sum لل بي
+
+167
+00:13:29,400 --> 00:13:32,340
+إبسلون و F ناقص ال lower sum لل بي إبسلون و F أصغر
+
+168
+00:13:32,340 --> 00:13:35,780
+من مين من إبسلون سميتهاش أنا إبسلون على تلاتة
+
+169
+00:13:35,780 --> 00:13:38,700
+للحسابات طبعا أصغر من إبسلون في الدنيا من ضمنها
+
+170
+00:13:38,700 --> 00:13:45,040
+الإبسلون على تلاتة هذا حسب الريمان criterion أو ال
+
+171
+00:13:45,040 --> 00:13:51,190
+integrable criterionطيب moreover لأن لو كانت عندي
+
+172
+00:13:51,190 --> 00:13:55,650
+ال B أي B partition بحتوى ال بي إبسلون أكيد هيكون
+
+173
+00:13:55,650 --> 00:13:59,550
+اللي هو ال .. ال .. ال .. هذا بيصير refinement
+
+174
+00:13:59,550 --> 00:14:04,410
+مدام refinement ال refinement التحسين ال lower
+
+175
+00:14:04,410 --> 00:14:09,670
+بعله و ال upper بقاشه بينزل يعني بمعنى أخر هيكون
+
+176
+00:14:09,670 --> 00:14:13,030
+ال lower لل refine .. لل refinement أو التحسين
+
+177
+00:14:13,030 --> 00:14:17,240
+أكبر أوي سوى مين ال lower للأصل ال بي إبسلونوأكيد
+
+178
+00:14:17,240 --> 00:14:20,840
+الـ lower لل B و ال F أكيد أصغر أو سوء القبر لل B
+
+179
+00:14:20,840 --> 00:14:24,780
+و ال F in general و القبر لل B و F قلنا مدامه
+
+180
+00:14:24,780 --> 00:14:29,600
+تحسين إذا أكيد بنزل بصغر عن مين عن اللي صار له
+
+181
+00:14:29,600 --> 00:14:34,960
+تحسين اللي هو مين U, B, Y و F إذا صار عندي الآن ..
+
+182
+00:14:34,960 --> 00:14:45,230
+هي عندي ال L .. ال L, B, Y و F وهذا L, B و Fوهذا
+
+183
+00:14:45,230 --> 00:14:51,650
+الـ U, B و F وهذا الـ U, B, E, F واضح إن المسافة
+
+184
+00:14:51,650 --> 00:14:58,430
+بين الـ U, B, E, F و الـ L, B, E, F أكيد أكبر أو
+
+185
+00:14:58,430 --> 00:15:02,550
+تساوي المسافة بين الـ U و الـ L للـ B و الـ F يعني
+
+186
+00:15:02,550 --> 00:15:05,710
+هذه المسافة واضح إنها أكبر أو تساوي هذه المسافة
+
+187
+00:15:05,710 --> 00:15:12,000
+يعني بمعنى آخر هيكون الـ U, B, Fنقص lbf أصغر أو
+
+188
+00:15:12,000 --> 00:15:15,240
+يساوي لـ U بي إبسلون نقص ل L بي إبسلون و F هذا
+
+189
+00:15:15,240 --> 00:15:21,260
+لمين؟ عدي ال .. ال .. ال بي إبسلون من فوق حصلنا
+
+190
+00:15:21,260 --> 00:15:24,940
+عنها أصغر من إبسلون ع تلاتة إذا صار عند ال U بي إف
+
+191
+00:15:24,940 --> 00:15:29,500
+و ال L بي إف المسافة بينهم برضه أصغر من مين؟ من
+
+192
+00:15:29,500 --> 00:15:35,360
+إبسلون ع تلاتة إذا الآن صار عند أيب لجأت الـ
+
+193
+00:15:35,360 --> 00:15:39,840
+بإبسلون و أي بي بيحتوى الـ بإبسلون لجئنا المسافة
+
+194
+00:15:39,840 --> 00:15:47,880
+بين الـ U بي و F ناقص الـ L بي و F أصغر من مين من
+
+195
+00:15:47,880 --> 00:15:53,710
+إبسلون على 3ماشي الحال هذا اللي حصلناه في الأول
+
+196
+00:15:53,710 --> 00:15:57,170
+الآن احنا زي ما قلنا ادورتنا ع مين يا جماعة
+
+197
+00:15:57,170 --> 00:16:01,210
+ادورتنا على الـ Delta اللي لما ال Normal بيه أصغر
+
+198
+00:16:01,210 --> 00:16:04,330
+من Delta يعطينا المسافة بين الريمان صموا ال
+
+199
+00:16:04,330 --> 00:16:08,670
+integration أصغر من إيش من أبسلون الآن أنا بتدعي
+
+200
+00:16:08,670 --> 00:16:15,060
+أو الكاتب بتدعي أنههناخد Delta بقيمة معينة و هتشوف
+
+201
+00:16:15,060 --> 00:16:17,960
+إنها فعلاً هذه الـ Delta بتحقق طبعاً الـ Delta هذه
+
+202
+00:16:17,960 --> 00:16:22,160
+نتيجة إحسابات اللي هو بيبقى مشة على برهان كامل و
+
+203
+00:16:22,160 --> 00:16:26,400
+في الآخر لجأ الإحساب اللي بيخلي اللي هو قيمة هنا
+
+204
+00:16:26,400 --> 00:16:30,540
+Epsilon الدلتة اللي بياخدها الآن طيب، قبل ما نقول
+
+205
+00:16:30,540 --> 00:16:34,560
+إيش الـ Delta بدنا نعطي الرمز الـ F is bounded,
+
+206
+00:16:34,740 --> 00:16:37,220
+integrable لإنها مدام bounded أو integrable
+
+207
+00:16:37,220 --> 00:16:41,460
+السبريمم إلها على كل الفترة موجودخُد الـ Supremum
+
+208
+00:16:41,460 --> 00:16:45,200
+لقيمة الدالة F of X للـ X element in the interval
+
+209
+00:16:45,200 --> 00:16:49,940
+I سميها إيش بتساوي؟ سميها بتساوي M Capital، ماشي
+
+210
+00:16:49,940 --> 00:16:54,360
+الحال، الان بادعي الان إن دلتا اللي هتظبط معنى و
+
+211
+00:16:54,360 --> 00:16:57,320
+تجيبلنا اللي هو القيمة اللي بدنا إياها اللي هي
+
+212
+00:16:57,320 --> 00:17:00,180
+أصغر من إبسلون، اللي هي دلتا بتساوي إبسلون على
+
+213
+00:17:00,180 --> 00:17:05,140
+اتناشر N في مين في M N مين هذه، N اللي ظهرت في ال
+
+214
+00:17:05,140 --> 00:17:09,750
+partition هنا اللي هو عدد عناصر ال partitionاللي
+
+215
+00:17:09,750 --> 00:17:14,490
+جزأني الفترة هي N زائد 1 هذا مين ال partition اللي
+
+216
+00:17:14,490 --> 00:17:19,790
+لجيته في الريمان من خلال الintegrability للأفإذاً
+
+217
+00:17:19,790 --> 00:17:24,370
+where الـ N زائد واحد is the number of parts in B
+
+218
+00:17:24,370 --> 00:17:29,190
+إبسلون يعني الـ Delta بتساوي إبسلون على 12 في N
+
+219
+00:17:29,190 --> 00:17:33,370
+الـ N ارتبطت بال partition اللي لجيته والـ M هي
+
+220
+00:17:33,370 --> 00:17:39,250
+عبارة عن ال supremum لكل الدالة على كل الفترة
+
+221
+00:17:39,250 --> 00:17:41,410
+المذكورة نيجي الآن
+
+222
+00:17:46,350 --> 00:17:53,750
+Now let Q بيساوي Y not Y واحد و Y اتنين لعند Y M
+
+223
+00:17:53,750 --> 00:17:59,890
+مش N، Y M ماشي بـany partition of I with ال
+
+224
+00:17:59,890 --> 00:18:03,550
+absolute value للـQ أو ال norm للـQ أصغر من Delta
+
+225
+00:18:04,420 --> 00:18:08,500
+أيش يا جماعة قاعد أنا أحاول أصل للي بديها؟ أيش
+
+226
+00:18:08,500 --> 00:18:12,080
+غرضي؟ إن لكل epsilon أكبر من 0 بدي ألاقي delta
+
+227
+00:18:12,080 --> 00:18:16,600
+بحيث أنه أي partition أصغر من delta norm يعطيني
+
+228
+00:18:16,600 --> 00:18:21,320
+الفرق بين رمان صمه وال integration أصغر من epsilon
+
+229
+00:18:21,320 --> 00:18:27,660
+وكأنه الآن بدأت أحط أيدي على الجرح وأصل أقول let Q
+
+230
+00:18:27,660 --> 00:18:34,780
+بيساوي Y0 Y1 YM بأي partitionNormal أصغر من مين؟
+
+231
+00:18:34,780 --> 00:18:40,260
+من Delta متوقع أنا أنه في النهاية عشان أخلص
+
+232
+00:18:40,260 --> 00:18:47,200
+النظرية أثبتلك أنه ال absolute value لأ اللي هو ال
+
+233
+00:18:47,200 --> 00:18:55,230
+S ال UFناقص ال integration من قلبي f of x dx يطلع
+
+234
+00:18:55,230 --> 00:19:00,290
+لي أصغر من مين من إبسلون إذا أثبتت هذا بناء على
+
+235
+00:19:00,290 --> 00:19:05,830
+إختياري للـ Delta اللي عندي وبناء على كل ال cues
+
+236
+00:19:05,830 --> 00:19:10,490
+اللي نورهم أصغر من Delta يعطيني هذا ناقص هذا أصغر
+
+237
+00:19:10,490 --> 00:19:14,210
+من إبسلون وبقول خلصت وبرهنت اللي هي نظريتي
+
+238
+00:19:14,210 --> 00:19:19,350
+درابوكسيز theoremخلّينا نشوف كيف بدنا نروح لهذه
+
+239
+00:19:19,350 --> 00:19:24,810
+اللي هي نثبتها إنها أصغر من إبسلون تعالي شوية الآن
+
+240
+00:19:24,810 --> 00:19:31,630
+لأن let Q*) بسوى Q اتحادين بإبسلون هذه بي إبسلون
+
+241
+00:19:31,630 --> 00:19:37,210
+مين اللي لجيتها فوق اللي هي عبارة عن بي إبسلون
+
+242
+00:19:38,430 --> 00:19:43,870
+بإبسلون هي الفترة من a لعند b بإبسلون اللي هي أش
+
+243
+00:19:43,870 --> 00:19:50,850
+جزأة للفترة هذه هذي x note هذي x1 هذي x2 هذي x3
+
+244
+00:19:50,850 --> 00:19:56,430
+لما أصل xn ناقص واحد لما أصل هنا عند مين xn اللي
+
+245
+00:19:56,430 --> 00:20:01,390
+هي مين ال b هذي مين جزأها جزء الفترة هذه واحدة
+
+246
+00:20:01,390 --> 00:20:08,190
+اسمها بإبسلوننيجي الآن اتجزأت الفترة بمين بالـ Q
+
+247
+00:20:08,190 --> 00:20:17,270
+الـ Q مين هي بدأت ب Y note طبعا هي ايه Y1 و Y2 و
+
+248
+00:20:17,270 --> 00:20:24,950
+Y3 لعند مؤصل لعند Y M ناقص واحدوهنا اللي هي YM
+
+249
+00:20:24,950 --> 00:20:29,730
+اللي هي طبعاً مين هي اللي هي الـB هذه الـQ جزقتها
+
+250
+00:20:29,730 --> 00:20:33,430
+بشكلها اللي أمامي مع العلم إن الـQ هي تبعتنا
+
+251
+00:20:33,430 --> 00:20:38,950
+لنورمها أصغر من مين من Delta الآن بيقوللي خد Q*)
+
+252
+00:20:38,950 --> 00:20:46,150
+إيش بتساوي؟ الـQ اتحاد من الـBإبسلون صارت الآن
+
+253
+00:20:46,150 --> 00:20:54,880
+الـQ هذهبساوة بي بي ستار اتحاد لـ Q هذه مين هي الـ
+
+254
+00:20:54,880 --> 00:21:01,100
+Q star طبعا النقطة الأولى X not هي نفسها Y not
+
+255
+00:21:01,100 --> 00:21:06,840
+والـ Xn هي مين الـ Yn اللي يعني ممكن يكون
+
+256
+00:21:06,840 --> 00:21:11,380
+المختلفات اللي هي من X واحد لعند Xn نقص واحد يكون
+
+257
+00:21:11,380 --> 00:21:16,890
+المختلفات عن هدول الاتحادأقصى حدودهم يعني إذا بدهم
+
+258
+00:21:16,890 --> 00:21:21,430
+يعملنّي sub intervals زيادات هيكون أقصى حدودهم
+
+259
+00:21:21,430 --> 00:21:27,290
+اللي لنا اللي هي X1 و X2 و X3 و X-1 ماتنطبقش ولا
+
+260
+00:21:27,290 --> 00:21:33,810
+على أي Y1 ولا Y2 ولا Y3 ولا Y أم minus 1 متى ما
+
+261
+00:21:33,810 --> 00:21:38,410
+انطبقلنش معناته أنه هيكونين وين موجودات في داخل ال
+
+262
+00:21:38,410 --> 00:21:42,630
+intervals معناته كل واحدة عملتلها إلا عن فترة تاني
+
+263
+00:21:42,630 --> 00:21:46,550
+جداداللي هو في الفترة تانية الإجداد يعني لو إجت
+
+264
+00:21:46,550 --> 00:21:51,390
+هنا الـ X1 هنا مثلا بيكون هذه عملة فترة تانية لو
+
+265
+00:21:51,390 --> 00:21:58,930
+إجينا سمينا هذه المنطقة اللي هو M1 star اللي هو M1
+
+266
+00:21:58,930 --> 00:22:02,470
+مين اللي هو ال maximum أو ال supreme على مين على
+
+267
+00:22:02,470 --> 00:22:09,510
+ال partition الجديد Q starوهذه سمناها هتكون M2*)
+
+268
+00:22:09,510 --> 00:22:15,670
+هيصير الـ M1*) بتختلف لأن هذه عبارة عن الـ
+
+269
+00:22:15,670 --> 00:22:20,490
+Supremum على المنطقة هذه بينما بالنسبة للـ Q لو
+
+270
+00:22:20,490 --> 00:22:27,030
+أجينا لـ Maximum تبعها أو Supremum تبعها هنسميه M1
+
+271
+00:22:27,030 --> 00:22:32,970
+هذا الـ M1قد يختلف عن الـ M1 Star وعن مين الـ M2
+
+272
+00:22:32,970 --> 00:22:37,610
+Star لأن الـ M1 على هذه كلها بينما الـ M1 Star على
+
+273
+00:22:37,610 --> 00:22:41,070
+هذه والـ M2 Star على هذه اللي هو الـ Supremum
+
+274
+00:22:41,070 --> 00:22:44,670
+للدالة على الجزء من هذا أو الجزء من هذا قد يختلف
+
+275
+00:22:44,670 --> 00:22:49,860
+عن الـ Supremum عن الجزء هذا يعني الآنالـ الـ الـ
+
+276
+00:22:49,860 --> 00:22:55,060
+الـ الـ subintervals جديدة هتولدلي M1 maximum
+
+277
+00:22:55,060 --> 00:23:00,700
+لعناصر للـ function على اللي هي الكلي بختلف عن مين
+
+278
+00:23:00,700 --> 00:23:07,240
+عن الجزء قد يختلفالـ N لكن اللي .. اللي .. لو كانت
+
+279
+00:23:07,240 --> 00:23:12,420
+اللي هي عدد هنا دولة اللي هي الـ N أكبر من الـ N
+
+280
+00:23:12,420 --> 00:23:19,420
+هتتوزع .. هتتوزع اللي هو الـ X1 والـ X2 لإذا
+
+281
+00:23:19,420 --> 00:23:23,280
+اختلفن كمان واتوزعن مختلفات هيتوزعن وين؟ في
+
+282
+00:23:23,280 --> 00:23:28,870
+المناطق اللي هيجبل ما نخلص الـ YM يعني هتلاقيها كل
+
+283
+00:23:28,870 --> 00:23:33,970
+واحدة مكان لأن الـ M أكبر من الـ N اللي بتضال بدون
+
+284
+00:23:33,970 --> 00:23:39,870
+ما تدخلها واحدة من الـ X1 أو X2 أو X3 هتسير في ال
+
+285
+00:23:39,870 --> 00:23:43,810
+partition الجديد Q هي نفسها في الاصل في ال
+
+286
+00:23:43,810 --> 00:23:50,140
+partition main الـ Q يعني هيكون الـ M هناstar و ال
+
+287
+00:23:50,140 --> 00:23:56,440
+M نفسها هيكون زي بعر يعني هيكون التنتين بنفس
+
+288
+00:23:56,440 --> 00:24:02,600
+القيمة عشان هيك لو أجينا طرحنا اللي هو .. اللي هي
+
+289
+00:24:02,600 --> 00:24:09,860
+قيمة ال lower sum ال upper sum على ال Q star مع ال
+
+290
+00:24:09,860 --> 00:24:16,340
+upper sum على ال Q هيصير عندي اللي هنا يظهر اللي
+
+291
+00:24:16,340 --> 00:24:22,810
+هو Mمن العادية أو M من الـ Star ناقص الـ M من الـ
+
+292
+00:24:22,810 --> 00:24:27,350
+Star لأنه بيكون قد يكون المختلفات لكن اللي هان مش
+
+293
+00:24:27,350 --> 00:24:31,050
+هيظهر لأنه هيصير الـ M ناقص M من الـ Star سفر إذن
+
+294
+00:24:31,050 --> 00:24:36,110
+هيصير عندي ظهور هذولة في أكام فترة عن أكثر هاي
+
+295
+00:24:36,110 --> 00:24:41,710
+واحدة تنتينبضل أمشي ل N ناقص واحد لإن عددنا دولة
+
+296
+00:24:41,710 --> 00:24:45,330
+من X1 عند XN ناقص واحدة كم واحدة N ناقص واحد يعني
+
+297
+00:24:45,330 --> 00:24:49,130
+على الأكثر هيكون عدد ال sub intervals اللي اجداد
+
+298
+00:24:49,130 --> 00:24:54,390
+عملاً اللي هو 2 في N ناقص واحد اللي اتجسمت تنتين
+
+299
+00:24:54,390 --> 00:25:00,590
+تنتين الآن فبصير عندي ال summation هذا لو جينا لو
+
+300
+00:25:00,590 --> 00:25:08,230
+أخدنا لك ال U ال U star معاللي الـ F ناقص الـ U
+
+301
+00:25:08,230 --> 00:25:14,910
+الـ QF هيكون عبارة عن إيش يا جماعة اللي هنا هيتصفر
+
+302
+00:25:14,910 --> 00:25:19,070
+حيساوي الصميه أكم واحدة من هذولة هيكون على الأكثر
+
+303
+00:25:19,070 --> 00:25:23,790
+الشكل هذا واحدة اسمها مثلا MK MJ زي ما هو مسميها
+
+304
+00:25:23,790 --> 00:25:28,870
+والتاني M star K هذولة على الأكثر على الأكثر على
+
+305
+00:25:28,870 --> 00:25:34,900
+الأكثر هيكون عددهم جداشعددهم عبارة عن اتنين فئة
+
+306
+00:25:34,900 --> 00:25:40,240
+ناقص واحد مضروبات في طول الفترة هذه الفترة الصغيرة
+
+307
+00:25:40,240 --> 00:25:43,680
+اللي هي .. اللي هي الفترة تبعت .. اللي هي مين
+
+308
+00:25:43,680 --> 00:25:48,720
+الـQ*) اللي هي .. واسميها مثلا ZK ناقص ZK مايس
+
+309
+00:25:48,720 --> 00:25:53,700
+واحد يعني عددهم هذولة اللي بظهر الـU ناقص الـU
+
+310
+00:25:53,700 --> 00:25:59,270
+بظهر عبارة عن أكم واحدة من هذولة سماشن لهمعبارة عن
+
+311
+00:25:59,270 --> 00:26:04,390
+الـ two فئة ناقص واحد من الـ terms نجي الآن إيش
+
+312
+00:26:04,390 --> 00:26:12,830
+اللي كاتبوا شوف قلنا لت Q star بيساوي Q اتحاد من
+
+313
+00:26:12,830 --> 00:26:17,930
+الـ B إبسلون الـ Q star صار refinement لمين؟ للـ B
+
+314
+00:26:17,930 --> 00:26:23,620
+إبسلونإذا الـ Q*) حسب اللي حكيته هناك has at most
+
+315
+00:26:23,620 --> 00:26:28,540
+N ناقص واحد more than points than main Q على
+
+316
+00:26:28,540 --> 00:26:32,660
+الأكتر هيكون عدد اللي في الـ Q*) العناقص اللي في
+
+317
+00:26:32,660 --> 00:26:38,040
+Q*) عند Canon اللي في الـ Q اللي في ال .. أمسح
+
+318
+00:26:38,040 --> 00:26:45,900
+اللي فوق بس زي ما قلت اللي في الـ By اللي في الـ
+
+319
+00:26:45,900 --> 00:26:54,870
+By هي عبارة عن X0X1 Xm-1 لعند Xm اللي في ال Q هي
+
+320
+00:26:54,870 --> 00:27:04,190
+عبارة عن Y0 Y1 Ym-1 لعند Ym اللي في ال Q*) إيش
+
+321
+00:27:04,190 --> 00:27:10,910
+هيصير انه أكيد هذي و هذي هم نفسهم فأكيد لو بدأنا ب
+
+322
+00:27:10,910 --> 00:27:17,650
+Z0 و انتهينا بأخر واحدة Z مين ZB مثلا هيكون اللي
+
+323
+00:27:17,650 --> 00:27:23,390
+هوعال أكتر، عال أكتر، عال أكتر الـQ*) بتزيد عن
+
+324
+00:27:23,390 --> 00:27:30,170
+الـP بجداش بالنقاط هذولة اللي لو كانت مش مكررات من
+
+325
+00:27:30,170 --> 00:27:35,430
+هنا خالص اللي هي بيصير عندى في داخل هذه عدد عناصر
+
+326
+00:27:35,430 --> 00:27:41,890
+الـQ*) أكتر من عدد عناصر الـQ بعال أكتر أناقص واحد
+
+327
+00:27:42,200 --> 00:27:50,720
+وكل هذا تبع اللي هو النقاط اتكرر المعها دولة او ما
+
+328
+00:27:50,720 --> 00:27:57,000
+اتكررنش ما اتكررنش بالمرة بيكون بالظبط عددهم اللي
+
+329
+00:27:57,000 --> 00:28:03,000
+هو ال YM زائد ال N ناقص واحد لو ما اتكررنش ولا
+
+330
+00:28:03,000 --> 00:28:07,700
+واحدة يعني بمعنى اخر ال Q start زيد عن ال Q ب N
+
+331
+00:28:07,700 --> 00:28:10,740
+ناقص واحد من النقاط
+
+332
+00:28:12,960 --> 00:28:17,560
+طيب زي ما قال إذن So Q star تحتوي بي إبسلون and Q
+
+333
+00:28:17,560 --> 00:28:21,780
+star has at most N نقص واحد more points than Q
+
+334
+00:28:21,780 --> 00:28:26,220
+namely those points among اللي هي النقاط اللي من
+
+335
+00:28:26,220 --> 00:28:30,080
+مين اللي بيكون الزيادات من ال X واحد لعندي ال X N
+
+336
+00:28:30,080 --> 00:28:35,200
+ال X N نقص واحد that is belongs to B star but not
+
+337
+00:28:35,200 --> 00:28:40,560
+to مين to Q اللي هدول اللي بتكون الزيادات طيب نطلع
+
+338
+00:28:40,560 --> 00:28:48,130
+لفوقالآن بما أن الـ Q صار تحتوى لـ Q إذا أكيد الـ
+
+339
+00:28:48,130 --> 00:28:53,850
+UQ ناقص الـ UQ صار أكبر سوى 0 طبعا عارفين ليش لأن
+
+340
+00:28:53,850 --> 00:28:58,890
+الـ refinement دائما لل upper sum بيجلله ماشي عشان
+
+341
+00:28:58,890 --> 00:29:02,670
+يصير refinement عشان يبطل يطلع فوق ينزل إشي بسيط
+
+342
+00:29:02,670 --> 00:29:07,730
+فهذا في الـ refinement معناته بيصير الـ UQأصغر من
+
+343
+00:29:07,730 --> 00:29:11,790
+مين؟ أكبر من مين؟ من الـUQ UQ star لأنه في
+
+344
+00:29:11,790 --> 00:29:15,630
+الـrefinement الأعلى بجلد مدام بجلد إذا هذا UQ
+
+345
+00:29:15,630 --> 00:29:20,610
+ناقص UQ star أكبر أو يساوي سفر الآن زي ما سمنا if
+
+346
+00:29:20,610 --> 00:29:25,170
+we write UQ star بيساوي Z node Z واحد عند مين؟ عند
+
+347
+00:29:25,170 --> 00:29:34,610
+ZB then لو أجينا بدنا نحسب الآن UQ F ناقص UQ star
+
+348
+00:29:34,610 --> 00:29:42,230
+وFخلّيني اوضّح لك
+
+349
+00:29:42,230 --> 00:29:53,010
+خلّيني اكتبها ماعليش عندي U Q و
+
+350
+00:29:53,010 --> 00:29:59,290
+F ناقص U Q Star و F
+
+351
+00:30:02,130 --> 00:30:07,370
+الـ summation الأولى العبارة عن الـ summation له
+
+352
+00:30:07,370 --> 00:30:19,310
+اللي هي M J في 100 في الـ Q اللي هو Y J minus Y J
+
+353
+00:30:19,310 --> 00:30:25,130
+minus 1 J من عند 1 لعند الـ M ناقص ال summation
+
+354
+00:30:25,130 --> 00:30:37,430
+للتانية اللي هي MK Star في مين؟ في Z K-Z K-1 K من
+
+355
+00:30:37,430 --> 00:30:42,750
+عند 1 لعند مين؟ لعند الـ B اللي فرضناها إيش اسمها
+
+356
+00:30:42,750 --> 00:30:49,190
+اللي هي الـ B زي ما قولت قبل بشويةلو كان هذولة
+
+357
+00:30:49,190 --> 00:30:53,870
+اللي صارن زيادات هنا اللي صارن زيادات هنا ال X
+
+358
+00:30:53,870 --> 00:30:59,410
+نقطة من X واحد لعند X اناقس واحد لو توزع بين هذولة
+
+359
+00:31:00,990 --> 00:31:04,390
+يعني بمعنى أخر كل هدف في حالة الـ M أكبر من N
+
+360
+00:31:04,390 --> 00:31:10,430
+اتوزع بينهين هيعمل اللي هو ال intervals زي ما قلنا
+
+361
+00:31:10,430 --> 00:31:15,850
+2 في N ناقص 1 في هذه ال intervals هيكون اللي هو
+
+362
+00:31:15,850 --> 00:31:22,460
+الـ Mj و الـ Mk قد يكون المختلفاتلكن في الباقي
+
+363
+00:31:22,460 --> 00:31:28,400
+اللي أصلا مادخلهنش إش جديد وضلت الـQ زي الـQ*)
+
+364
+00:31:28,400 --> 00:31:35,680
+فبتكون M تبعت الـQ و M*) تبعت الـQ*) زي بعض فبصير
+
+365
+00:31:35,680 --> 00:31:40,360
+هذه ناقص هذه بساوي 0 اللي هي الـMj والمك بساوي 0
+
+366
+00:31:40,360 --> 00:31:47,070
+يعني بمعنى آخرالـ terms اللي هتكون ع صورة MJ ناقص
+
+367
+00:31:47,070 --> 00:31:52,110
+MK star ع الأكثر ع الأكثر اللي ممكن يكون المختلفات
+
+368
+00:31:52,110 --> 00:31:57,550
+عبارة عن جداش جماعة اتنين في N ناقص واحد يعني هذول
+
+369
+00:31:57,550 --> 00:32:01,950
+عدد هنا ع الأكتر طبعا في ZK minus ZK minus واحد
+
+370
+00:32:01,950 --> 00:32:04,770
+تبعي اللي هو التجزئة الجديدة لإنه هذه التجزئة اللي
+
+371
+00:32:04,770 --> 00:32:08,950
+بتجسمها ل sub intervals يعني ع الأكتر هذول عدد هنا
+
+372
+00:32:08,950 --> 00:32:13,180
+جداش اتنين في N ناقص واحدهذا أنا قلته لما في حالتي
+
+373
+00:32:13,180 --> 00:32:18,610
+الأم أكبر من أنافي حالة الـ M أصغر أو يساوي N
+
+374
+00:32:18,610 --> 00:32:23,890
+بيصير اللي هو في الشيء .. بيصير اللي هو غصبًا عنها
+
+375
+00:32:23,890 --> 00:32:30,270
+مدام الـ N أكبر وبدها تدخل الـ X1 عليهم غصبًا عنها
+
+376
+00:32:30,270 --> 00:32:35,210
+هيكون في الداخل انطبج على مين؟ على إيش من هان يعني
+
+377
+00:32:35,210 --> 00:32:39,630
+مش هيعمل إن 2 في N ناقص 1 هيعمل إن أجل من هيك إلا
+
+378
+00:32:39,630 --> 00:32:42,150
+في حالة اللي هي M بتساوي N بتطلع اللي هو على
+
+379
+00:32:42,150 --> 00:32:47,560
+الأكثر ممكن يطلع 2 في N-1 لأنه ما بيظلش فيه عنده
+
+380
+00:32:47,560 --> 00:32:53,580
+اللي هو مكان بين هذوله يدخلن فبضطر ان اللي هو اكيد
+
+381
+00:32:53,580 --> 00:32:58,140
+هيكون من منين من جواك اللي هي هذه او بكون اللي جوا
+
+382
+00:32:58,140 --> 00:33:03,760
+برضه مش هيزيدن عن اتنين في الـ-1 اذا اللي هي دخلن
+
+383
+00:33:04,020 --> 00:33:09,620
+على أماكن غير مين حتى في حالة ال M ال M أصغر من N
+
+384
+00:33:09,620 --> 00:33:14,000
+لأ هيدخلن هيدخلن يعملن sub intervals هيختلفن برضه
+
+385
+00:33:14,000 --> 00:33:18,640
+نفس الشيء يعني لو دخلت X1 هنا و X2 هنا فبصير برضه
+
+386
+00:33:18,640 --> 00:33:22,550
+يعملن sub intervals في مجموحينلن يتجاوز ان اتنين
+
+387
+00:33:22,550 --> 00:33:29,230
+في ان ناقص واحد لأن عدد ان ناقص واحد من النقاط
+
+388
+00:33:29,230 --> 00:33:33,550
+الجديدة ال X1 X2 عند Xn ناقص واحد ان شاء الله تكون
+
+389
+00:33:33,550 --> 00:33:39,490
+واضحة هذه الصورة يعني مختصرها انههن اللي ممكن ما
+
+390
+00:33:39,490 --> 00:33:47,270
+يحفظلش على اللي هو الـ الـ الـ الـ الـ الـ الـ الـ
+
+391
+00:33:47,270 --> 00:33:48,310
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+392
+00:33:48,310 --> 00:33:48,310
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+393
+00:33:48,310 --> 00:33:48,670
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+394
+00:33:48,670 --> 00:33:48,690
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+395
+00:33:48,690 --> 00:33:48,810
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+396
+00:33:48,810 --> 00:33:49,030
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+397
+00:33:49,030 --> 00:33:49,030
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+398
+00:33:49,030 --> 00:33:52,480
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الاللي
+
+399
+00:33:52,480 --> 00:33:57,120
+هي بس اللي هي N ناقص واحد من النقاط اللي هتصنعني
+
+400
+00:33:57,120 --> 00:34:02,180
+بس اتنين في N ناقص واحد على الأكثر من ال .. اللي
+
+401
+00:34:02,180 --> 00:34:10,840
+هي ال M واحد أو ال Mj تختلف عن ال M كستار طيب،
+
+402
+00:34:10,840 --> 00:34:13,940
+ماشي الحالة ان شاء الله تكون المضحكة هذه لأن هي بس
+
+403
+00:34:13,940 --> 00:34:17,840
+هذه النقطة اللي بدها توضيح في الباقي هيكون بإذن
+
+404
+00:34:17,840 --> 00:34:23,040
+الله سعيد إذا صار زي ما قلنانرجع لهذا then U Q F
+
+405
+00:34:23,040 --> 00:34:27,200
+نقص U Q Star و F can be written as the sum of at
+
+406
+00:34:27,200 --> 00:34:31,920
+most اتنين في M نقص واحد terms of the form M J نقص
+
+407
+00:34:31,920 --> 00:34:37,460
+M K Star في Z K minus Z K minus واحد لأن عدد هذه
+
+408
+00:34:37,460 --> 00:34:41,680
+الدولة على الأكثر اللي هو بالشكل هذا إذا الآن لو
+
+409
+00:34:41,680 --> 00:34:51,060
+أجينا بدي أحسبلك اللي هو الـ M J نقص M K Starهذه
+
+410
+00:34:51,060 --> 00:35:00,000
+هيكون أصغر أو يساوي مج زائد مك
+
+411
+00:35:01,360 --> 00:35:05,180
+وكل واحدة طبعاً هذا supremum على فترتها وهذا
+
+412
+00:35:05,180 --> 00:35:09,240
+supremum لل F على فترتها أكيد أصغر أو يساوي ال M
+
+413
+00:35:09,240 --> 00:35:13,160
+زائد ال M لأن هذا ال M عبارة عن ال supremum على كل
+
+414
+00:35:13,160 --> 00:35:16,360
+ال interval A وB هذا على جزءه هذا على كل ال
+
+415
+00:35:16,360 --> 00:35:19,240
+interval A وB هذا على جزءه يعني في النهاية هذا
+
+416
+00:35:19,240 --> 00:35:25,460
+أصغر أو يساوي قداش 2 في M يعني هيصير عندي الآن ال
+
+417
+00:35:25,460 --> 00:35:32,980
+absolute value لل M J ناقص M K Starفي ZK-ZK-1
+
+418
+00:35:32,980 --> 00:35:39,970
+هيكون أصغر أو ساوى 2Mفي مين؟ في اللي هي طول الـ
+
+419
+00:35:39,970 --> 00:35:43,550
+interval هذه طول الـ interval هذه طب هذه الـ
+
+420
+00:35:43,550 --> 00:35:50,170
+interval zk minus zk minus 1 أكيد أصغر أو يساوي
+
+421
+00:35:50,170 --> 00:35:56,410
+مين؟ norm لـ Q star مين norm لـ Q star؟ اللي هو ال
+
+422
+00:35:56,410 --> 00:36:01,510
+supremum على مين؟ على أطوال ال subintervals ال
+
+423
+00:36:01,510 --> 00:36:05,750
+supremum أكيد أكبر أو يساوي واحدة منهم طول واحدة
+
+424
+00:36:05,750 --> 00:36:13,190
+منهملكلهم وهذه دائماً لما نزيد العدد قلنا لما نكون
+
+425
+00:36:13,190 --> 00:36:19,630
+Q يحتوي Q*) automatic norm لـ Q*) زي ما قلنا المرة
+
+426
+00:36:19,630 --> 00:36:24,710
+الفاتة هيكون أصغر أو يشوي .. آسف لما يكون Q
+
+427
+00:36:24,710 --> 00:36:28,370
+substantive لـ Q*) يكون norm لـ Q*) عكسه أصغر أو
+
+428
+00:36:28,370 --> 00:36:34,420
+يشوي norm من لـ Qإذا الـ N هذا أصغر أو يساوي هذا و
+
+429
+00:36:34,420 --> 00:36:38,940
+هذا نفسه أصغر أو يساوي مين؟ الـ Q و الـ Normal Q
+
+430
+00:36:38,940 --> 00:36:42,720
+أصغر من مين؟ من دلتا ما احنا على أساسه اللي هي
+
+431
+00:36:42,720 --> 00:36:48,120
+اشتغلنا إذا صار عندي هذا Normal أصغر من دلتا يعني
+
+432
+00:36:48,120 --> 00:36:53,380
+نرجع لهذا هذا المقدار كل واحد لحاله أصغر من اتنين
+
+433
+00:36:53,380 --> 00:36:58,350
+أم في مين بيصير؟ في دلتاالأن عدد هذول اللي ممكن
+
+434
+00:36:58,350 --> 00:37:07,870
+يظهر في UQ ناقص UQ star عبارة عن 2M ناقص 1 يعني في
+
+435
+00:37:07,870 --> 00:37:14,330
+النهاية هذا أصغر أو يساوي اللي هو عدد الـ 2M ناقص
+
+436
+00:37:14,330 --> 00:37:22,940
+1 مضروب في مين؟ في اللي هي 2M2M في Delta لأن هذا
+
+437
+00:37:22,940 --> 00:37:28,390
+نفسه أصغر يساوي 2M و هذا نفسه أصغر يساوي Deltaالان
+
+438
+00:37:28,390 --> 00:37:33,030
+هدولة بتكرر على الأكتر اتنين فئة ناقص واحد هذا ..
+
+439
+00:37:33,030 --> 00:37:36,750
+هذا بتكرر في هذا المقدار على الأكتر في هذا و
+
+440
+00:37:36,750 --> 00:37:40,370
+الباقية هيتصفر زي ما قلنا إذا صار هيكون هذا أصغر
+
+441
+00:37:40,370 --> 00:37:44,750
+أو يساوي عدد تكرار من اتنين فئة ناقص واحد في اتنين
+
+442
+00:37:44,750 --> 00:37:51,010
+M في مين في Delta فإذا صار عندي الآن ال UQF ناقص
+
+443
+00:37:51,010 --> 00:37:55,050
+ال UQ star و F أصغر أو ساوي اتنين فئة ناقص واحد في
+
+444
+00:37:55,050 --> 00:38:01,350
+اتنين M في مينفى delta نصبوت اللى هو هذا عبارة عن
+
+445
+00:38:01,350 --> 00:38:13,090
+قداشة جماعة نيجى نحسبها عشان نصل للي بدنايا صار
+
+446
+00:38:13,090 --> 00:38:26,230
+عندى اللى هو ال U Q F U star و F Q F ناقص U
+
+447
+00:38:32,930 --> 00:38:38,230
+UQF-UQF UQF
+
+448
+00:38:38,230 --> 00:38:44,130
+-UQF UQF
+
+449
+00:38:44,130 --> 00:38:45,750
+-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF
+
+450
+00:38:45,750 --> 00:38:45,850
+-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF
+
+451
+00:38:45,850 --> 00:38:45,850
+-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF
+
+452
+00:38:45,850 --> 00:38:45,850
+-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF-UQF UQF
+
+453
+00:38:45,850 --> 00:38:54,050
+-UQF UQF-UQF UQF-UQF UQFفى اللى هو M فى Delta اللى
+
+454
+00:38:54,050 --> 00:38:59,870
+هو أكيد هذا أصغر من أربعة فى N فى M فى Deltaمظهورة
+
+455
+00:38:59,870 --> 00:39:03,790
+ولا لأ؟ لأن الـ N أكبر من مين؟ من الـ N نقص واحد
+
+456
+00:39:03,790 --> 00:39:06,370
+الـ Delta إيش اخترناها؟ ما هي الحسابات من الأول
+
+457
+00:39:06,370 --> 00:39:12,010
+حاسبينها احنا اخترنا الـ Delta فوق عبارة عن جداش
+
+458
+00:39:12,010 --> 00:39:15,710
+لإن احنا كنا عارفين حالنا أصلا بنصل لها المرحلة
+
+459
+00:39:15,710 --> 00:39:19,730
+فبدنا نخلي هذه ي على تلاتة ليش ي على تلاتة؟ بعد
+
+460
+00:39:19,730 --> 00:39:26,470
+شوية تشوف اللي هي عبارة عن ي هذه ي على 12N في مين
+
+461
+00:39:26,470 --> 00:39:31,170
+في Nهذه الآن لما أشيل هذه و أضرب هذه فيها بيصير
+
+462
+00:39:31,170 --> 00:39:34,770
+أربعة في واحد و الان مع الان و الام مع الام و هذي
+
+463
+00:39:34,770 --> 00:39:41,750
+بيصير جداش تلاتة إذا صار عندي الان U Q و F ناقص U
+
+464
+00:39:41,750 --> 00:39:48,000
+Q Star و F أصغر من epsilon ع تلاتةيعني بمعنى آخر
+
+465
+00:39:48,000 --> 00:39:55,380
+صار عند الـ U Q و F أصغر من إبسل من ع تلاتة زائد U
+
+466
+00:39:55,380 --> 00:40:01,420
+Q star من و F ماشي الحال إذا هذه ال in quality
+
+467
+00:40:01,420 --> 00:40:06,160
+التانية اللي وصلناها اللي هتوصلني في النهاية للي
+
+468
+00:40:06,160 --> 00:40:11,490
+بديه ونشوف كيف هتوصلنيطيب يا جماعة صلوا على النبي
+
+469
+00:40:11,490 --> 00:40:17,670
+عليه الصلاة والسلام تنسوش أنه احنا في الأول حصلنا
+
+470
+00:40:17,670 --> 00:40:21,730
+على U بي و أف ماطس أل بي و أف أصغر من مين من
+
+471
+00:40:21,730 --> 00:40:24,930
+إبسلون على تلاتة وحصلنا على U بي و إبسلون هذا أصغر
+
+472
+00:40:24,930 --> 00:40:27,910
+من إبسلون على تلاتة هنحتاجها بعد شوية خليني اكتبها
+
+473
+00:40:27,910 --> 00:40:35,630
+التانية عند حصلنا قبل هيك على ال U بي و أف ماطس أل
+
+474
+00:40:35,630 --> 00:40:42,970
+بي و أف أصغرمن epsilon ع تلاتة وطبعا اكيد اللي هي
+
+475
+00:40:42,970 --> 00:40:46,870
+اصغر من u بي ابسلون و اف عشان لا نرجع الهاشمون
+
+476
+00:40:46,870 --> 00:40:51,970
+اقصال بي ابسلون و اف كله اصغر من ابسلون ع تلاتة
+
+477
+00:40:51,970 --> 00:41:00,930
+هاي كمان هدول التنتين حصلناها ماشي طيب الان
+
+478
+00:41:00,930 --> 00:41:09,000
+اللي عملنا هذا اللي هو مع ال upper sumبنعمل نفسه
+
+479
+00:41:09,000 --> 00:41:14,680
+مع مين مع اللي هو our sum هنلاقي اللي هو LQ يعني
+
+480
+00:41:14,680 --> 00:41:19,060
+LQ star و F ناقص LQ F هنلاقيه هنا على الأكتر
+
+481
+00:41:19,060 --> 00:41:23,020
+باختلاف أنه هو التو فئة ناقص واحد من التلزمه و و و
+
+482
+00:41:23,020 --> 00:41:26,640
+الاخر و بنكمل نفس الاشي بطل عند LQ star و F ناقص
+
+483
+00:41:26,640 --> 00:41:30,600
+إبشر على تلاتة أصغر من مين من LQ F إذا similarly
+
+484
+00:41:30,600 --> 00:41:35,120
+for هذه اللي عملناها هذه ما حد بنعمله مع اللي هو
+
+485
+00:41:35,730 --> 00:41:47,930
+الـ L L of Q star و F ماقص اللي هو إبسلون ع تلاتة
+
+486
+00:41:47,930 --> 00:41:55,890
+أصغر من L Q O mean O F ماشي الحال، طيب and exactly
+
+487
+00:41:55,890 --> 00:42:00,310
+زي ما بنقول similar argument symbolize that اللي
+
+488
+00:42:00,310 --> 00:42:07,870
+كتبناهلنبدأ نجمع معلوماتنا عشان نصل لإنه الـ
+
+489
+00:42:07,870 --> 00:42:11,530
+Riemann Sum نقص الـ integration يكون إيه شماله
+
+490
+00:42:11,530 --> 00:42:15,890
+اللي هو أصغر من إبسلون، انتبهوا كيف عشان يمكن هيك
+
+491
+00:42:15,890 --> 00:42:23,210
+تكون أسهل لنا نرسم خط الأعداد وهنا هخلينا اللي هو
+
+492
+00:42:23,210 --> 00:42:43,920
+نحط الـ U أكي و FU Q F هي أصغر من
+
+493
+00:42:43,920 --> 00:42:55,420
+أبسلون على تلاتة زي U U أستار U F عندي
+
+494
+00:42:55,420 --> 00:42:57,860
+U Q F
+
+495
+00:43:00,380 --> 00:43:07,380
+بس خليني عشان أحط ال L قبل ل L قبل هه عشان مفيش
+
+496
+00:43:07,380 --> 00:43:14,800
+وسح أنا جماعة بعد أذنكم خليني بس هكتب الأول أشنب
+
+497
+00:43:14,800 --> 00:43:27,870
+ده بمين ب L L Q و Fالـ Refinement طبعاً لـ L, Q,
+
+498
+00:43:27,970 --> 00:43:36,450
+Star و F قبلها زي ما انتوا عارفين L, Q و F خليني
+
+499
+00:43:36,450 --> 00:43:41,090
+أعمل هذه قبلها عشان تظبط أحسن L, Q, Star و F ناقص
+
+500
+00:43:41,090 --> 00:43:50,640
+بصناعة تلاتة وهذه LUQF معايا هى هذه بتكون قبل هذه
+
+501
+00:43:50,640 --> 00:43:59,300
+طيب ال QF أكيد بعدها مين UQF صح ولا لأ أكيد طيب
+
+502
+00:43:59,300 --> 00:44:07,420
+بعد ال UQF إيش بتيجي بتيجي هنا إبسلون ع تلاتة زائد
+
+503
+00:44:07,420 --> 00:44:15,340
+UQ star و F مظبوطمظبوط اكيد؟ طيب اذا انا اجيت لخصت
+
+504
+00:44:15,340 --> 00:44:21,120
+المعلومة هذه و لخصت المعلومة هذه اللي هي أصغر من
+
+505
+00:44:21,120 --> 00:44:28,460
+epsilon ع تلاتة UQF UQF ولا ال PUF هذه المسافة
+
+506
+00:44:28,460 --> 00:44:32,910
+بينهم ايه اشمل يا جماعة؟أصغر من إبسلون ع تلاتة
+
+507
+00:44:32,910 --> 00:44:37,130
+واضح أه؟ ومن هنا قال لك U square of F نقص هذه وهذه
+
+508
+00:44:37,130 --> 00:44:43,050
+إبسلون وهذه بالترتيب اللي احنا حصلنا عليه طيب الان
+
+509
+00:44:43,050 --> 00:44:48,230
+طبيعي اللي هو ال remansum و ال integration هيكون
+
+510
+00:44:48,230 --> 00:44:52,030
+بين ال lower و بين main اللي هو إياش الأبر إذن
+
+511
+00:44:52,030 --> 00:44:57,110
+أكيد أكيد أكيد ال remansum تبعنا هي بدينا نصل للي
+
+512
+00:44:57,110 --> 00:45:04,400
+جدنا إياها ال CQFواللي هو ال integration من a إلى
+
+513
+00:45:04,400 --> 00:45:08,720
+b إلى ال F هيكون بين ال lower و بين ال upper ال
+
+514
+00:45:08,720 --> 00:45:11,720
+integration موجود لأن مراس الدور جللنا إذا هذوله
+
+515
+00:45:11,720 --> 00:45:17,800
+صار اللي هو بين هذه و بين هذه إذا صار هذه و هذه
+
+516
+00:45:17,800 --> 00:45:26,120
+بين هذه الدنتينوهذه طبعا ال U Q F و اللي قاله Q F
+
+517
+00:45:26,120 --> 00:45:31,780
+واضح .. واضح انه اللي هو وين في داخل الفترة اللي
+
+518
+00:45:31,780 --> 00:45:36,660
+هي L Q Star و F ناقص أبسل على ثلاثة و U Q Star و F
+
+519
+00:45:36,660 --> 00:45:42,500
+أبسل على تلاتة يعني بين .. هذه طبعا .. بين هذه و
+
+520
+00:45:42,500 --> 00:45:47,200
+بين مين و بين ال interval هذه يعني subset من مين
+
+521
+00:45:47,200 --> 00:45:53,460
+جايل من I أبسلوناللي هي I Epsilon بتساوي اللي هو
+
+522
+00:45:53,460 --> 00:45:59,440
+الفترة هذه من هنا لعند الـ hand هو اللي مسمينها
+
+523
+00:45:59,440 --> 00:46:05,740
+احنا اشمالها I Epsilon بين L Q Star و F نقص
+
+524
+00:46:05,740 --> 00:46:11,860
+Epsilon ع تلاتة و U Q Star F و Epsilon ع تلاتة
+
+525
+00:46:11,860 --> 00:46:20,060
+الآنU Q Star و F ناقص قالك U Star و F ما مسحتاش
+
+526
+00:46:20,060 --> 00:46:28,200
+لسه أنا U Q Star F و قالك U Star F أصغر من مين من
+
+527
+00:46:28,200 --> 00:46:36,520
+إبسلون ع تلاتة لأ اللي هي U
+
+528
+00:46:36,520 --> 00:46:42,670
+Q Star Fوقال الـ Q star F حصلنا عليها قبل هيك أنها
+
+529
+00:46:42,670 --> 00:46:46,650
+أصغر من مين من إبسل ع تلاتة طبعا من وين بنحصلها؟
+
+530
+00:46:46,650 --> 00:46:49,650
+ممكن نحصلها مبقى من هنا ده كما حصلناش أن الـ Q
+
+531
+00:46:49,650 --> 00:46:54,430
+star و F و الـ By بتحتوي الـ By فهيكون الـ U اللي
+
+532
+00:46:54,430 --> 00:47:00,090
+حصلنا أولا بالنسبة لمن؟ لاللي هي الـ By حصلنا قبل
+
+533
+00:47:00,090 --> 00:47:06,400
+هيك أن الـ Uبإبسلون و ال ال بإبسلون الفرق بين همين
+
+534
+00:47:06,400 --> 00:47:10,840
+إبسلون على تلاتة و ال Q star refinement إذا أكيد
+
+535
+00:47:10,840 --> 00:47:14,920
+المسافة بين ال other طبعا في ال refinement المسافة
+
+536
+00:47:14,920 --> 00:47:20,440
+بين ال other و ال other هتصغر لكل refinement ال U
+
+537
+00:47:20,440 --> 00:47:28,340
+بما أن هذه refinement إذا ال Uلأ الـ Q star و F
+
+538
+00:47:28,340 --> 00:47:34,960
+ناقص L Q star و F أصغر أو يساوي لأنها صارت تحسين
+
+539
+00:47:34,960 --> 00:47:38,400
+عليها مازالت صارت تحسين يعني بتجلل مسافة بين ال
+
+540
+00:47:38,400 --> 00:47:40,780
+Upper و الLower عشان نروح في الآخر لل integration
+
+541
+00:47:40,780 --> 00:47:51,520
+ناقص أصغر أو يساوي لL B epsilon و F U ناقص L B
+
+542
+00:47:51,520 --> 00:47:56,980
+epsilon و F اللي هي هذه أصغر من 100من إبسلون وينها
+
+543
+00:47:56,980 --> 00:48:03,400
+أصغر من إبسلون ع تلاتة أصغر من إبسلون ع تلاتة إذا
+
+544
+00:48:03,400 --> 00:48:10,320
+فعلا عرفنا من وين إجت هذه أصغر من مين من إبسلون ع
+
+545
+00:48:10,320 --> 00:48:18,460
+تلاتة إذا صار عندي ال length لهذهالـ length لهذه
+
+546
+00:48:18,460 --> 00:48:23,240
+سهل نوجده بعليش خلّيني أكتب فوق عشان لو جبتلكم ال
+
+547
+00:48:23,240 --> 00:48:29,780
+length إلها عشان أقولكم إنه هذه و هذه هيكون اللي
+
+548
+00:48:29,780 --> 00:48:38,900
+هو بالبعد المطلوب تحسبلي الآن هذه ناقص هذه بصير
+
+549
+00:48:38,900 --> 00:48:48,940
+عبارة عن طول ال interval mean Iy بساوي ال UQ star
+
+550
+00:48:48,940 --> 00:48:55,780
+و F زائد إبسلون على تلاتة ناقص تبعين معيار ال Q
+
+551
+00:48:55,780 --> 00:49:01,720
+star و F ناقص ناقص بيصير زائد إبسلون على تلاتة و
+
+552
+00:49:01,720 --> 00:49:04,900
+يساوي اللي هو إبسلون على تلاتة و إبسلون على تلاتة
+
+553
+00:49:04,900 --> 00:49:12,180
+اتنين إبسلون على تلاتة زائد U Q star و F ناقص ال Q
+
+554
+00:49:12,180 --> 00:49:18,800
+star و F اللي هم هذولةهذه اللي كتبتها لسه ماحتنش
+
+555
+00:49:18,800 --> 00:49:23,400
+أصغر من مين من يبسلون على تلاتة فبتصير هذه طولها
+
+556
+00:49:23,400 --> 00:49:26,700
+أصغر من اتنين يبسلون على تلاتة زائد يبسلون على
+
+557
+00:49:26,700 --> 00:49:31,620
+تلاتة ويساوي يبسلون إذا صار عندي طول الـ interval
+
+558
+00:49:31,620 --> 00:49:40,520
+اللي أسي Qf والـ integration A وBموجودة في كل
+
+559
+00:49:40,520 --> 00:49:46,660
+الفترة هذه اللي هي المسافة بينهم عبارة عن أصغر من
+
+560
+00:49:46,660 --> 00:49:53,460
+مين من Epsilon إذا صار عندي الآن ال integration
+
+561
+00:49:53,460 --> 00:50:05,480
+أين أكتب ال integration إذا صار عندي إذا صار عندي
+
+562
+00:50:05,480 --> 00:50:12,380
+المسافة بين هذه و هذهSQF نقص ال integration من A B
+
+563
+00:50:12,380 --> 00:50:16,900
+ل F أصغر من مين يا جماعة من يبسط من مين هذا تحت
+
+564
+00:50:16,900 --> 00:50:23,360
+شرط مين إنه normally Q هذا أصغر من A H من دلتا
+
+565
+00:50:23,360 --> 00:50:29,200
+أصغر من دلتا بقول إيه أحنا خلصنا اللي بدنا إياه
+
+566
+00:50:29,200 --> 00:50:32,280
+نوقف
+
+567
+00:50:32,280 --> 00:50:33,020
+الجيجا بحثها
+
+568
+00:51:39,830 --> 00:51:44,830
+أه يا أبو حسن بقول هيك أحنا خلصنا برهان نظرية أول
+
+569
+00:51:44,830 --> 00:51:48,490
+حد يقول لي أنه اللي هو المسافة بين U وQ قال الـ Q
+
+570
+00:51:48,490 --> 00:51:50,950
+أصغر من يبسلون على تلاتة لأ احنا ماقلناش هيك
+
+571
+00:51:50,950 --> 00:51:53,950
+البسافة بين الأصغر من يبسلون على تلاتة اللي هو الـ
+
+572
+00:51:53,950 --> 00:51:58,190
+P يبسلون الـ U P يبسلون والـ L P يبسلون وأي
+
+573
+00:51:58,190 --> 00:52:01,010
+refinement له بس الـ Q هنا مش refinement هذا بس
+
+574
+00:52:01,010 --> 00:52:04,830
+اللي بنعرف أنه norm لـ Q أصغر من مين من دلتا أصلا
+
+575
+00:52:04,830 --> 00:52:08,960
+احنا بدنا نصل لإنهالمسافة بين هذولة اللي هو أصغر
+
+576
+00:52:08,960 --> 00:52:13,300
+من إبسلون وعلى طول منها بنستنتج هذه فكيف وصلنا
+
+577
+00:52:13,300 --> 00:52:16,160
+بإننا قلنا اللي هو ال integration هذه في هذه
+
+578
+00:52:16,160 --> 00:52:19,780
+الفترة والفترة هذه جزء من هذه وهذه قلنا هذه طولها
+
+579
+00:52:19,780 --> 00:52:22,960
+على الأكثر بساوي إبسلون عشان هيك صارت المسافة بين
+
+580
+00:52:22,960 --> 00:52:28,000
+هذه وهذا أصغر من إبسلون وهو المطلوب طيب نيجي الآن
+
+581
+00:52:28,000 --> 00:52:34,060
+ده اللي هي النظرية اللي بعدها theorem
+
+582
+00:52:35,670 --> 00:52:45,070
+لت I بتساوي A وB و لت F من I لعند R بيكون فنكشف
+
+583
+00:52:46,090 --> 00:52:49,330
+suppose there exists a number B such that لكل Y
+
+584
+00:52:49,330 --> 00:52:54,110
+أكبر من 0 if B is any partition of I with normal B
+
+585
+00:52:54,110 --> 00:52:58,030
+أصغر من Delta and S of Bi is any corresponding
+
+586
+00:52:58,030 --> 00:53:01,950
+Riemann sum اللي هو then لهذا أصغر من إبسلون then
+
+587
+00:53:01,950 --> 00:53:06,310
+F is integrable on I in the sense of definition
+
+588
+00:53:06,310 --> 00:53:12,590
+716 يعني وكأننا جاعدين بنقول conversely إيش بنقول؟
+
+589
+00:53:12,590 --> 00:53:13,610
+بنقول
+
+590
+00:53:16,710 --> 00:53:24,890
+إنه لو كان عندي اللي هو limit لو كان عندي limit
+
+591
+00:53:24,890 --> 00:53:32,330
+اللي هو الـ S أو B و F as normal B بروح للسفر
+
+592
+00:53:32,330 --> 00:53:38,190
+exist و بسوء some number B إذن هذا الـ B بسوء ال
+
+593
+00:53:38,190 --> 00:53:43,410
+integration من A لB F of X DX يعني بقوللي لو كان
+
+594
+00:53:43,410 --> 00:53:50,850
+هذا متحققلو كان هذا متحقق بيعطيني إنه هذا هو هيطلع
+
+595
+00:53:50,850 --> 00:53:55,110
+اللي هي هتطلع F-integrable وهيكون بيه هذا اللي هو
+
+596
+00:53:55,110 --> 00:53:59,030
+ال limit هذا هو قيمة مين ال integration ليش؟ لأن
+
+597
+00:53:59,030 --> 00:54:02,490
+هذا اصلا شو معناته؟هذا معناته for every epsilon
+
+598
+00:54:02,490 --> 00:54:07,490
+أكبر من صفر there exist اللي هو delta أكبر من صفر
+
+599
+00:54:07,490 --> 00:54:14,370
+such that b أصغر من delta بيؤدي إلى S B و F ناقص
+
+600
+00:54:14,370 --> 00:54:17,630
+ال B اللي مفترضين أصغر من مين ال epsilon هذا هو
+
+601
+00:54:17,630 --> 00:54:23,290
+معناته هذا يعني أن اللي هو المعطىطبعا f bounded من
+
+602
+00:54:23,290 --> 00:54:30,590
+رأس الدور لجهنا number b بحيث ان لكل epsilon او fb
+
+603
+00:54:30,590 --> 00:54:33,270
+is any partition بحيث ان normal b أعلى من delta
+
+604
+00:54:34,000 --> 00:54:39,600
+وكانت اللي هي المسافة بين الـ B أصغر من Epsilon
+
+605
+00:54:39,600 --> 00:54:44,440
+هذا .. ده متحقق اللي هو هذا كله عبارة هو عبارة عن
+
+606
+00:54:44,440 --> 00:54:47,940
+limit S بي و F as and non-invalid اللي هو سفر
+
+607
+00:54:47,940 --> 00:54:52,160
+بساول B إذا .. إذا اتحقق هذا بتكون اللي هي F
+
+608
+00:54:52,160 --> 00:54:55,440
+integrable وقيمة الـ B هذه اللي هي قيمة ال limit
+
+609
+00:54:55,440 --> 00:54:59,420
+بساول integration هذه اللي هي عبارة عن .. النظرية
+
+610
+00:54:59,420 --> 00:55:04,470
+عبارة عن اللي هو بن ..اللي هي دمج بين اللي هي
+
+611
+00:55:04,470 --> 00:55:08,190
+نظريات السابقة المرة الماضية نشوف كيف اللي هو
+
+612
+00:55:08,190 --> 00:55:14,100
+البرهن برهن بسيط لإنه بعتمد عليه بشكل سريعلت y
+
+613
+00:55:14,100 --> 00:55:17,440
+أكبر من 0 و delta أكبر من 0 as in the theorem اللي
+
+614
+00:55:17,440 --> 00:55:20,180
+هو اللي في الـ theorem اللي فوق المقصود فيها النص
+
+615
+00:55:20,180 --> 00:55:25,820
+هذه قدت بي إبسلون بي partition of I with نفترض أنه
+
+616
+00:55:25,820 --> 00:55:30,080
+norm بي إبسلون أصغر من 200 من دلتا هذا المعطى ماشي
+
+617
+00:55:30,080 --> 00:55:36,240
+الحال نفترض أنه إبسلون أكبر من 0 arbitrarily و
+
+618
+00:55:36,240 --> 00:55:41,190
+delta أكبر من 0أو نفترض أن بي إبسلون أي partition
+
+619
+00:55:41,190 --> 00:55:45,310
+بحيث أنه نورمه إيه شماله أصغر من دلتا لكن لو جينا
+
+620
+00:55:45,310 --> 00:55:48,930
+أخدنا بيه أي partition آخر و بيه محتوى بيه إبسلون
+
+621
+00:55:48,930 --> 00:55:53,770
+هيكون نورم ال partition بيه هذا اللي هو ال
+
+622
+00:55:53,770 --> 00:55:57,470
+refinement أكيد أصغر أو سوى نورم مين الأصلي اللي
+
+623
+00:55:57,470 --> 00:56:01,030
+هو أصغر من دلتا صار عنده الآن لكل إبسلون أكبر من
+
+624
+00:56:01,030 --> 00:56:06,470
+سفر there exists دلتا بحيث أنه نورم البي إبسلون
+
+625
+00:56:06,470 --> 00:56:11,740
+أصغر من دلتاولأي partition بي refinement لبيبسلون
+
+626
+00:56:11,740 --> 00:56:17,260
+صار norm أصغر من delta إذا من اللي هو ال .. بواسط
+
+627
+00:56:17,260 --> 00:56:21,740
+اللي هو النظرية السابقة حيكون عند ال S minus البي
+
+628
+00:56:21,740 --> 00:56:27,580
+أصغر من ابسلون يؤدي إلى هذا بيبساو ال integration
+
+629
+00:56:27,580 --> 00:56:32,020
+لإن إيش كانت نظرية سبعة أربعة تلاتة بتشترط تشترط
+
+630
+00:56:32,020 --> 00:56:33,840
+إنه نلاقي لكل ابسلون
+
+631
+00:56:36,480 --> 00:56:42,920
+أي بارتشن بي ابسلون بحيث أن أي بي بتحتول بي ابسلون
+
+632
+00:56:43,910 --> 00:56:48,530
+يؤدي إلى هذا أصغر من إبسلون كله على بعض بيؤدي F
+
+633
+00:56:48,530 --> 00:56:53,790
+أشمالها is integrable إذا بيواصل 7 4 3 مدام حصلنا
+
+634
+00:56:53,790 --> 00:56:57,910
+على ال S بي و F نقص ال V أصغر من إبسلون بيكون هذا
+
+635
+00:56:57,910 --> 00:57:01,190
+اللي حصلنا عليه اللي هو P بساوة ال integration لل
+
+636
+00:57:01,190 --> 00:57:06,630
+F على الفترة من A لB يعني بمعنى أخر احنا اللي
+
+637
+00:57:06,630 --> 00:57:13,530
+حصلنا الآن من النظريتين أن F is integrableF is
+
+638
+00:57:13,530 --> 00:57:19,050
+integrable if and only if limit الـ S,B وF من
+
+639
+00:57:19,050 --> 00:57:23,890
+normal B بيروح للسفر exist وبساوي قيمة ال
+
+640
+00:57:23,890 --> 00:57:28,550
+integration يعني الآن ال integration أوصلنا له عبر
+
+641
+00:57:28,550 --> 00:57:34,330
+طريقين إما عبر طريق اللي هو limit للرمان sumيكون
+
+642
+00:57:34,330 --> 00:57:38,690
+exist أو عبر الطريق الآخر لأن ال upper .. ال upper
+
+643
+00:57:38,690 --> 00:57:42,930
+integral والlower integral يكون جد بعض ومتساويين
+
+644
+00:57:42,930 --> 00:57:52,690
+وحيكون في هذه الحالة هم ال integration وقف
+
+645
+00:57:52,690 --> 00:57:59,570
+هنا يا أبو حسن كم ساعة .. كم
+
+646
+00:57:59,570 --> 00:58:00,810
+عندك الوقت؟
+
+647
+00:58:03,290 --> 00:58:10,650
+نخلصها هيك و نبدأ تانية؟ لأن الجزء الثاني هذا اللي
+
+648
+00:58:10,650 --> 00:58:17,070
+هو الimprover integral بس بطولش بياخد نص ساعة انت
+
+649
+00:58:17,070 --> 00:58:22,090
+قاعد معايا للجد إيش؟ أكيد؟ مديش أضغط عليك يا أبو
+
+650
+00:58:22,090 --> 00:58:30,390
+حسن طيب لأ خلينا نخلص هذه ما هو درس 17 ها؟ هلجيت
+
+651
+00:58:30,390 --> 00:58:31,270
+18؟
+
+652
+00:58:41,590 --> 00:58:48,290
+هي الان بتبدأ في المحاضرة 18 خلصنا 17 كيف؟
+
+653
+00:58:54,810 --> 00:58:59,850
+هك مكون احنا خلصنا المحاضر رقم 17 اللي هو ال
+
+654
+00:58:59,850 --> 00:59:03,610
+integration as a limit بعد شوية هنكمل ال section
+
+655
+00:59:03,610 --> 00:59:08,490
+هذا مظبوط بس هنكمله بموضوع جديد اللي هو عبارة عن
+
+656
+00:59:08,490 --> 00:59:12,510
+الimprover integral وإلى لقاء آخر والسلام عليكم
+
+657
+00:59:12,510 --> 00:59:13,670
+ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/A0riuruZYaI.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/A0riuruZYaI.srt
new file mode 100644
index 0000000000000000000000000000000000000000..020a1aad2b8ea95b833a22958d4063df923ef659
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/A0riuruZYaI.srt
@@ -0,0 +1,1765 @@
+1
+00:00:05,740 --> 00:00:08,920
+بسم الله الرحمن الرحيم الجزء الثاني من المحاضرة
+
+2
+00:00:08,920 --> 00:00:14,780
+التاسعة هيكون اللي هو عبارة عن discussion أو مناقشة
+
+3
+00:00:14,780 --> 00:00:22,060
+لـ Section 6.2 و 6.3 اللي هو مناقشة لـ Main Value 
+
+4
+00:00:22,060 --> 00:00:25,220
+Theorem and its Applications ومناقشة أيضًا لـ
+
+5
+00:00:25,220 --> 00:00:30,560
+L'Hopital's Rule نيجي الآن لـ 6.2 الأسئلة المطلوبة
+
+6
+00:00:30,560 --> 00:00:35,720
+هي كما يلي نبدأ في سؤال 6.2  ادخلنا على الكتاب 
+
+7
+00:00:35,720 --> 00:00:39,160
+خلينا نشوف الـ .. الـ .. الـ .. المثال .. السؤال من
+
+8
+00:00:39,160 --> 00:00:45,790
+الكتاب نبدأ الآن بسؤال 5 السؤال الخامس هو كما يلي
+
+9
+00:00:45,790 --> 00:00:49,550
+let a أكبر من صفر و b أكبر من صفر و a strictly
+
+10
+00:00:49,550 --> 00:00:55,570
+أكبر من b طبعًا كل العلاقة strictly وبنفترض أن n
+
+11
+00:00:55,570 --> 00:01:00,290
+أكبر أو يساوي 2 prove that a أس 1 على n ناقص b
+
+12
+00:01:00,290 --> 00:01:05,290
+أس 1 على n أصغر من a - b أس 1 على n الآن
+
+13
+00:01:05,290 --> 00:01:10,490
+لو جينا لاحظنا على اللي هو المطلوب عند 5 بيقول لي
+
+14
+00:01:10,490 --> 00:01:16,710
+أن a أكبر من b أكبر من 0 أو n أكبر أو يساوي 2
+
+15
+00:01:16,710 --> 00:01:23,070
+بيقول لي prove that أن a أس 1 على n ناقص b أس 1
+
+16
+00:01:23,070 --> 00:01:30,710
+على n أصغر من a - b الكل أس 1 على n نيجي 
+
+17
+00:01:30,710 --> 00:01:36,890
+للبرهان لو جينا لاحظنا إنه عندي الـ .. نيجي للسؤال
+
+18
+00:01:36,890 --> 00:01:40,770
+بس كيف نفكر في السؤال هو ما أعطيني hint في الكتاب
+
+19
+00:01:40,770 --> 00:01:45,190
+لكن خلينا نشوف كيف .. كيف حصل على الـ hint لو ادينا
+
+20
+00:01:45,190 --> 00:01:50,490
+جسمنا الجهتين هذا مش من ضمن الحل طبعًا عندي a على b
+
+21
+00:01:51,290 --> 00:01:55,270
+الكل أس 1 على n ناقص جسمة على اللي هو b أس 
+
+22
+00:01:55,270 --> 00:01:59,510
+1 على n للجهتين طبعًا والـ b طبعًا موجبة ففيش شيء
+
+23
+00:01:59,510 --> 00:02:05,290
+بتغير بيصير a على b - 1 الكل أس 1 على n
+
+24
+00:02:05,290 --> 00:02:10,710
+لو نجلنا هذه a على b أس 1 على n ناقص الـ a على 
+
+25
+00:02:10,710 --> 00:02:16,150
+b - 1 أس 1 على n أصغر من مين؟ من 1 الآن
+
+26
+00:02:16,700 --> 00:02:21,940
+عندي هذا الآن كله على بعضه هو نفسه الـ F ممكن
+
+27
+00:02:21,940 --> 00:02:27,680
+نستقل الدالة من خلاله إنه ناخد الـ F of x لتـ F of x
+
+28
+00:02:27,680 --> 00:02:35,690
+و هتوصلنا بساوي x أس 1 على n ناقص x - 1
+
+29
+00:02:35,690 --> 00:02:42,810
+أس 1 على n وطبعًا هو الآن معطيني في السؤال a 
+
+30
+00:02:42,810 --> 00:02:47,790
+أكبر من b أكبر من 0 لو طلعنا نلاقي الدالة هذه
+
+31
+00:02:47,790 --> 00:02:56,610
+ولاحظنا أوجدنا الـ f prime لها f prime of x خلينا
+
+32
+00:02:56,610 --> 00:03:02,130
+ناخد الـ x عنده الـ x أكبر أو يساوي 1 و هنشوف
+
+33
+00:03:02,130 --> 00:03:05,730
+اللي هو ليش عنده الـ x أكبر أو يساوي 1 برضه
+
+34
+00:03:05,730 --> 00:03:09,070
+بتظبط في حالتنا لإن اللي بنينا على أساسها اللي هي
+
+35
+00:03:09,070 --> 00:03:13,290
+الـ a على الـ b نفسها أكبر strictly من مين؟ من 1
+
+36
+00:03:13,290 --> 00:03:17,150
+فالأمور متناسقة مع بعض و لو بدنا نطبق حللها جي جي
+
+37
+00:03:17,150 --> 00:03:20,670
+اللي هو تطبيق معقول اللي هنقف الـ prime of x بساوي
+
+38
+00:03:20,670 --> 00:03:26,200
+1 على n في x أس 1 على n - 1 ناقص اللي
+
+39
+00:03:26,200 --> 00:03:30,720
+هو 1 على n في x - 1 أس 1 على n -
+
+40
+00:03:30,720 --> 00:03:36,520
+1 ويساوي 1 على n في x أس 1 على n - 1
+
+41
+00:03:36,520 --> 00:03:44,840
+ناقص اللي هو x - 1 اللي هو على 1 على n 
+
+42
+00:03:44,840 --> 00:03:50,460
+ناقص 1 الآن لو طلعنا للي عندي هذا
+
+43
+00:03:53,450 --> 00:03:56,930
+لو طلعنا للمقدار اللي عندي الـ x أكبر أو يساوي إيش؟
+
+44
+00:03:56,930 --> 00:04:02,450
+1 يعني الآن الـ x أكبر يساوي 1 لذا لما عندي
+
+45
+00:04:02,450 --> 00:04:06,950
+الأس اللي هان أس إيش؟ ماله بالسالب أو صفر على سواء
+
+46
+00:04:06,950 --> 00:04:10,130
+الظروف اللي هو بالسالب معناته اللي هو بده يصير
+
+47
+00:04:10,130 --> 00:04:16,070
+1 على x الـ 1 على x عبارة عن كسر، مظبوط؟ الآن
+
+48
+00:04:16,070 --> 00:04:20,430
+بيصير عندي المقدار اللي عندي العلاقة بين هذا و هذا
+
+49
+00:04:20,940 --> 00:04:26,360
+x أكيد أكبر من x - 1 صح ولا لأ؟ لكن لأن
+
+50
+00:04:26,360 --> 00:04:31,240
+مقلوبها هيصير إيش ماله؟ هيصير أصغر فهيصير المقدار
+
+51
+00:04:31,240 --> 00:04:35,440
+هذا أصغر من هذا المقدار ماشي؟ فبيصير عندي المقدار
+
+52
+00:04:35,440 --> 00:04:40,180
+هذا كله على بعضه أصغر من مين؟ من صفر فالآن صارت عندي
+
+53
+00:04:40,180 --> 00:04:45,720
+f' أصغر strictly من مين؟ من صفر إذا صار عندي إذا f
+
+54
+00:04:45,720 --> 00:04:47,160
+is strictly
+
+55
+00:04:50,230 --> 00:04:54,310
+decreasing بدي استخدم الخاصية هذه مدام strictly
+
+56
+00:04:54,310 --> 00:04:59,390
+decreasing و أنا عندي a أكبر من b هسه عندي a على
+
+57
+00:04:59,390 --> 00:05:06,110
+b أكبر من 1 والـ b طبعًا لا تساوي صفر إذا بما أن
+
+58
+00:05:06,110 --> 00:05:11,930
+F is strictly decreasing إذا F of a على b أكبر من
+
+59
+00:05:11,930 --> 00:05:18,440
+F of 1 أف of a على b دلتنا بيجيب العوض فوق بيصير
+
+60
+00:05:18,440 --> 00:05:26,380
+عندي اللي هو .. آسف أصغر عندي أف of a على b إيش
+
+61
+00:05:26,380 --> 00:05:30,920
+هتساويك؟ قولوا معايا اللي هو a على b أس 1 على n
+
+62
+00:05:30,920 --> 00:05:40,360
+ناقص a على b - 1 الكل أس 1 على n هذا إيش
+
+63
+00:05:40,360 --> 00:05:45,310
+ماله؟ أصغر من مين؟ من F of 1 F of 1 حسب لي F
+
+64
+00:05:45,310 --> 00:05:50,290
+of 1 هذه 1 وهذه صفر فبيصير عبارة عن أصغر من
+
+65
+00:05:50,290 --> 00:05:53,350
+1 طبعًا هذه اللي هو بنعمل عملية عكسية للي
+
+66
+00:05:53,350 --> 00:05:57,530
+عملناها فوق فبيصير عندي اضرب الجهتين في b أس 1
+
+67
+00:05:57,530 --> 00:06:04,260
+على n فبيصير a أس 1 على n ناقص a - b أس 1 على 
+
+68
+00:06:04,260 --> 00:06:10,940
+n أصغر من مين؟ من b أس 1 على n ضربت كله في مين؟
+
+69
+00:06:10,940 --> 00:06:14,980
+في b أس 1 على n إن جلّي الآن بيصير عندي a أس
+
+70
+00:06:14,980 --> 00:06:20,320
+1 على n ناقص b أس 1 على n أصغر من a - b
+
+71
+00:06:20,320 --> 00:06:26,530
+الكل أس 1 على n وهو المطلوب وهذا اللي هو بده إياه
+
+72
+00:06:26,530 --> 00:06:32,150
+يلا إيه بعده؟ خلينا نيجي للسؤال اللي المطلوب الآخر
+
+73
+00:06:32,150 --> 00:06:38,570
+اللي هو use the mean value theorem سؤال 6 use the
+
+74
+00:06:38,570 --> 00:06:42,830
+mean value theorem to prove that sin x - sin y
+
+75
+00:06:42,830 --> 00:06:47,070
+أصغر أو يساوي x - y for all x, y in R هذا
+
+76
+00:06:47,070 --> 00:06:50,610
+السؤال حلنا زيه بالضبط اللي هو mean اللي هو الـ
+
+77
+00:06:50,610 --> 00:06:52,070
+cosine، مظبوط؟
+
+78
+00:06:57,300 --> 00:07:03,320
+الآن ما أعرفش فيه داعي نحله ولا إن هو نفسي أو مصور
+
+79
+00:07:03,320 --> 00:07:06,740
+عملناه
+
+80
+00:07:06,740 --> 00:07:12,000
+ولا لا؟ اللي هي الـ cosine عملناها بنفس الأسلوب ومش
+
+81
+00:07:12,000 --> 00:07:14,340
+هيختلف اللي هو الحل
+
+82
+00:07:22,460 --> 00:07:27,840
+أحله ولا خلصت؟ الآن عندي اللي هو 7 use the mean
+
+83
+00:07:27,840 --> 00:07:32,580
+value theorem to prove that x - 1 على x أصغر من x
+
+84
+00:07:32,580 --> 00:07:39,160
+أصغر من x - 1 for x أكبر من 1 اللي هو عند ..
+
+85
+00:07:39,160 --> 00:07:45,440
+احنا حلينا واحد زاد x هذه الآن اللي هي عبارة عن
+
+86
+00:07:45,440 --> 00:07:51,040
+مين؟ عن اللي هو ln x الدالة اللي هي F of x بساوي ln 
+
+87
+00:07:51,040 --> 00:07:56,440
+x وبنستخدم الـ mean value theorem وبنفس الأسلوب فيه
+
+88
+00:07:56,440 --> 00:08:00,540
+داعي نحله؟ خلينا نحله، خلينا نحله عشان بنصور
+
+89
+00:08:00,540 --> 00:08:12,360
+الآن سؤال 7 الآن عندي بدأ أثبت أن ln x أصغر من
+
+90
+00:08:12,360 --> 00:08:20,000
+x - 1 وأكبر من x - 1 عالميًا على x
+
+91
+00:08:20,000 --> 00:08:26,400
+solution العالمين value theorem العالمين value
+
+92
+00:08:26,400 --> 00:08:29,960
+theorem نحله العالمين value theorem لأنه لسه ما
+
+93
+00:08:29,960 --> 00:08:35,980
+خدناش اللي هو Taylor's theorem مش
+
+94
+00:08:35,980 --> 00:08:39,340
+فاهم
+
+95
+00:08:39,340 --> 00:08:42,110
+عليك بينفع الحل باستخدام Taylor and x not
+
+96
+00:08:42,110 --> 00:08:44,690
+بيساويها يعني الـ mainly طيب آه احنا .. احنا عشان
+
+97
+00:08:44,690 --> 00:08:48,850
+لسه ما خدناش Taylor's theorem بدنا نحلها على مين؟
+
+98
+00:08:48,850 --> 00:08:51,970
+على الـ mean value theorem ليش؟ لأنه احنا لسه
+
+99
+00:08:51,970 --> 00:09:00,070
+ما خدناش Taylor's theorem طيب الآن let F of x
+
+100
+00:09:00,070 --> 00:09:07,330
+بتساوي ln الـ x؟ وعندي الـ x أكبر من مين؟ أكبر أو
+
+101
+00:09:07,330 --> 00:09:11,800
+تساوي الواحد، ولا لأ؟ عندي الـ x أكبر من 100 من 0
+
+102
+00:09:11,800 --> 00:09:14,940
+for
+
+103
+00:09:14,940 --> 00:09:23,100
+x أكبر من 0 مش عايز أني ماشي الحين نيجي اللي هو أن
+
+104
+00:09:23,100 --> 00:09:26,540
+نستخدم الـ mean value theorem continuous و closed و
+
+105
+00:09:26,540 --> 00:09:31,300
+differentiable وكل الأمور هذه أشملها متحققة إذا
+
+106
+00:09:31,300 --> 00:09:38,150
+there exist c element in a و b c عندي اللي هو
+
+107
+00:09:38,150 --> 00:09:42,290
+element معايا؟
+
+108
+00:09:42,290 --> 00:09:52,610
+طيب لأن let F of x بيساوي ln x و x أكبر من 0 وعندي
+
+109
+00:09:52,610 --> 00:09:57,430
+المطلوب في الـ inequality اللي هي x أكبر من 1 يعني
+
+110
+00:09:57,430 --> 00:10:02,870
+x سنتمي إلى الـ 1 وما لا نهاية معايا؟ إذا there
+
+111
+00:10:02,870 --> 00:10:11,590
+exists c لذن بدي أطبق الآن we apply mean value
+
+112
+00:10:11,590 --> 00:10:18,570
+theorem on وين؟ قول معايا on 1 و x there exists
+
+113
+00:10:18,570 --> 00:10:25,090
+c element 1 و x such that معايا such that اللي
+
+114
+00:10:25,090 --> 00:10:36,890
+هو f prime of c يساوي f of x ناقص f 1 على x ناقص
+
+115
+00:10:36,890 --> 00:10:48,110
+1 يا أخي ناقص 1 آه فالآن عندي f of x جدّيش؟ ويساوي ln 
+
+116
+00:10:48,110 --> 00:10:56,450
+الـ x ناقص ln الـ 1 جدّيش؟ صفر على x - 1 وهذا
+
+117
+00:10:56,450 --> 00:11:02,050
+مين؟ هو f prime of c عبارة عن ln الـ 1 على c مظبوط؟
+
+118
+00:11:02,870 --> 00:11:06,570
+إذا صار عندي الآن بدأ أجيب لإن الـ x أصغر من مين؟ من
+
+119
+00:11:06,570 --> 00:11:12,270
+x - 1 صار عندي إذا ln الـ x بساوي 1 على
+
+120
+00:11:12,270 --> 00:11:18,390
+c في x - 1 واللي عند c أشملها باخدها أنا
+
+121
+00:11:18,390 --> 00:11:23,050
+أكبر من 1 مدام أكبر من 1 إذا 1 على c اللي
+
+122
+00:11:23,050 --> 00:11:28,390
+هو أصغر من 1 مظبوط؟ إذا هذا أكيد هذا أصغر من x
+
+123
+00:11:28,390 --> 00:11:35,440
+- 1 مظبوط؟ لأن الـ 1 على الـ c أشملها عبارة
+
+124
+00:11:35,440 --> 00:11:43,990
+عن كسر الآن من جهة أخرى عندي اللي هو ln الـ x بساوي
+
+125
+00:11:43,990 --> 00:11:49,410
+1 على c في x - 1 لكن الـ x أنا c إيش
+
+126
+00:11:49,410 --> 00:11:54,190
+معناها بين الـ 1 والـ x يعني c أصغر من مين؟ c أصغر
+
+127
+00:11:54,190 --> 00:11:59,430
+من x يعني الـ 1 على x اللي هو أصغر من مين الآن؟ c
+
+128
+00:11:59,430 --> 00:12:05,890
+أصغر من x إذا 1 على c أكبر من 1 على x فبيصير
+
+129
+00:12:05,890 --> 00:12:13,190
+عند هذا أكبر من 1 على x في x - 1 يعني بمعنى آخر
+
+130
+00:12:13,190 --> 00:12:18,230
+صار عند ln الـ x اللي هو أكبر من x - 1 على الـ x
+
+131
+00:12:18,230 --> 00:12:21,550
+هي عند الـ inequality الثانية و هي عند الـ
+
+132
+00:12:21,550 --> 00:12:26,690
+inequality الأولى من التنتين إذا ln الـ x أشملها
+
+133
+00:12:26,690 --> 00:12:32,790
+أصغر من x - 1 مظبوط؟ اللي بعمله و أكبر من x
+
+134
+00:12:32,790 --> 00:12:41,190
+- 1 عالميًا على x وهو المطلوب إيش السؤال اللي
+
+135
+00:12:41,190 --> 00:12:49,550
+بعده؟ الآن سؤال 8 Let F من a لعند b
+
+136
+00:13:11,390 --> 00:13:17,630
+سؤال 8 let
+
+137
+00:13:17,630 --> 00:13:30,470
+f من a و b لعند r أشمل بـ continuous أو
+
+138
+00:13:30,470 --> 00:13:37,190
+differentiable on
+
+139
+00:13:37,190 --> 00:13:45,370
+mean on open interval a و b show
+
+140
+00:13:45,370 --> 00:13:51,730
+that if limit f prime of x عندي limit f prime of x
+
+141
+00:13:51,730 --> 00:14:01,630
+لما x تروح للـ a بساوي a capital then اللي هي f
+
+142
+00:14:01,630 --> 00:14:11,990
+prime of a f prime of a exists and equals a
+
+143
+00:14:11,990 --> 00:14:21,750
+solution أو proof ما أعطيني 
+
+144
+00:14:21,750 --> 00:14:28,650
+f is differentiable على اللي هي الفترة من a و b أو
+
+145
+00:14:28,650 --> 00:14:33,350
+continuous طبعًا open أو continuous على close من a
+
+146
+00:14:34,120 --> 00:14:39,860
+و b عندي limit f prime of x معطيني إياها لما x تروح إلى
+
+147
+00:14:39,860 --> 00:14:50,180
+الـ a إيش؟ بساوي بساوي اللي هو a capital اطلع
+
+148
+00:14:50,180 --> 00:14:58,560
+لفوق الآن عندي الآن الـ f prime of a تعريفها اللي هي
+
+149
+00:14:58,560 --> 00:15:04,600
+limit f of x ناقص f of a على x - a لما x تروح
+
+150
+00:15:04,600 --> 00:15:09,940
+للـ a ماشي الحال هذا اللي هو إذا كان الـ limit had
+
+151
+00:15:09,940 --> 00:15:13,500
+exist إذا أثبتنا إن الـ limit had exist بتكون الـ f
+
+152
+00:15:13,500 --> 00:15:18,440
+prime of a أشملها اللي هي exist ماشي الحال الآن
+
+153
+00:15:18,440 --> 00:15:21,200
+هو ما أعطيني limit f prime of x لما x تروح لـ a
+
+154
+00:15:21,200 --> 00:15:28,290
+أشملها هي الـ exist واضح آه؟ لأن بدي أطبق الـ mean
+
+155
+00:15:28,290 --> 00:15:33,290
+value theorem لأن في البداية على أي x وين؟ في
+
+156
+00:15:33,290 --> 00:15:39,170
+الفترة a و b في الفترة a و b لو أخذنا x في الـ a و b
+
+157
+00:15:39,170 --> 00:15:44,380
+بالـ mean value theorem بين على الـ a والـ x على
+
+158
+00:15:44,380 --> 00:15:47,920
+الفترة الـ a والـ x there exists cx ما لها
+
+159
+00:15:47,920 --> 00:15:52,340
+between x and a such that f of x ناقص f of a بساوي
+
+160
+00:15:52,340 --> 00:15:57,240
+f prime c of x في x - a هذه اللي هي تطبيق الـ 
+
+161
+00:15:57,240 --> 00:16:02,860
+mean value theorem على الفترة a و b and so و منه
+
+162
+00:16:02,860 --> 00:16:07,700
+اللي هي بنقول f prime c of x بساوي اللي هو f of x
+
+163
+00:16:07,700 --> 00:16:15,920
+ناقص f of a على x - a الآن عندي 
+
+164
+00:16:15,920 --> 00:16:22,520
+.. خلينا نيجي ناخد هذه المنطقة آه عشان لسه بدأت
+
+165
+00:16:22,520 --> 00:16:26,600
+بين f prime of a أشملها موجودة بساوي limit الـ f
+
+166
+00:16:26,600 --> 00:16:31,180
+prime cx لما x تروح لمين؟ للـ a، ماشي الحال، هذه
+
+167
+00:16:31,180 --> 00:16:36,610
+الآن لو exist بتكون f prime of a أشملها exist الآن
+
+168
+00:16:36,610 --> 00:16:40,510
+لاحظ احنا طبقنا اللي هي الـ mean value theorem على
+
+169
+00:16:40,510 --> 00:16:46,910
+مين؟ على الفترة من a لعند مين؟ لعند x لجينا الـ cx
+
+170
+00:16:46,910 --> 00:16:53,290
+وين موجودة؟ بين الـ a والـ x الآن لما cx تروح للـ a
+
+171
+00:16:53,290 --> 00:17:00,050
+أكيد الـ x هتروح لمين؟ للـ a ماشي الحال طيب الآن
+
+172
+00:17:00,050 --> 00:17:05,700
+بيصير عندي هو ما أعطيني أصلاً limit f prime of x لما x
+
+173
+
+201
+00:19:28,610 --> 00:19:31,610
+function f من R لـ R with absolute minimum at x
+
+202
+00:19:31,610 --> 00:19:35,210
+بالساوية صفر but such that f prime takes on both
+
+203
+00:19:35,210 --> 00:19:39,110
+positive and negative values on both sides of الـلي
+
+204
+00:19:39,110 --> 00:19:45,000
+هي x بتساوي  عياش بساوية صفر ماشي الحال إذا هذا الآن
+
+205
+00:19:45,000 --> 00:19:49,920
+هذا الحديث هو سؤالنا اللى عندنا اللي بدنا نحكي فيه
+
+206
+00:19:49,920 --> 00:19:52,880
+اللي هو exercise قداش تسعة ارجع لي على exercise
+
+207
+00:19:52,880 --> 00:19:56,560
+تسعة إذا الـ exercise تسعة هو عبارة عن إيش بيقول ليه
+
+208
+00:19:56,560 --> 00:20:00,280
+أن الـ converses of this theorem need not to be true
+
+209
+00:20:00,280 --> 00:20:04,930
+in general بالظبط إيش بيقول؟ بيقول لـ F من R لـ R بي
+
+210
+00:20:04,930 --> 00:20:08,910
+define by F of X بيساوي 2 X plus 4 زائد X plus 4
+
+211
+00:20:08,910 --> 00:20:12,770
+Sine 1 علي X For X لا تساوي صفر عند F of 0 إيش 
+
+212
+00:20:12,770 --> 00:20:16,450
+بيساوي Zero إذا أنا معرفت دالة الـ F بهذه الطريقة
+
+213
+00:20:16,450 --> 00:20:20,450
+بهذه لما X لا تساوي صفر وعند X بيساوي صفر عرفتها F 
+
+214
+00:20:20,450 --> 00:20:25,120
+of 0 بيساوي إيش؟ Zero بقول لشهدات أول إشي F has an
+
+215
+00:20:25,120 --> 00:20:30,440
+absolute minimum when عند الـ 0 but that its
+
+216
+00:20:30,440 --> 00:20:34,820
+derivative has both positive and negative values
+
+217
+00:20:34,820 --> 00:20:40,280
+everywhere اللى هو إيش في neighborhood حوالين مين؟
+
+218
+00:20:40,280 --> 00:20:49,380
+حوالين الصفر واضح طيب، نشوف الآن، عملية فيه
+
+219
+00:20:49,380 --> 00:20:54,280
+absolute minimum مش صعبة، اللى هي بس خلينا نتطلع
+
+220
+00:20:54,280 --> 00:20:58,140
+على الحسابات، لأن الحسابات بتاخد وجه، فخلينا نتطلع
+
+221
+00:20:58,140 --> 00:21:02,900
+على اللى عندنا محسوبة وخلاص لأن لأي x اللى تنتمي لـ R
+
+222
+00:21:02,900 --> 00:21:07,360
+أكيد الـ x أس 4 إيه شمالها أكبر أو يساوي صفر والـ
+
+223
+00:21:07,360 --> 00:21:11,900
+sign الواحد على x اكبير أكيد أكبر أو يساوي مين؟ ناقص
+
+224
+00:21:11,900 --> 00:21:16,300
+واحد اضرب الجهتين في x أس 4 فبيصير x أس 4 في
+
+225
+00:21:16,300 --> 00:21:20,680
+هذا أكبر أو يساوي ناقص إيش؟ x أس 4 أضيف للجهتين
+
+226
+00:21:20,680 --> 00:21:26,110
+اتنين x أس 4 فبيصير 2x أس 4 زائد هذا أكبر أو
+
+227
+00:21:26,110 --> 00:21:30,470
+يساوي اللى ضفت 2x أس 4 زائد اللى هو ياش ناقص اللى
+
+228
+00:21:30,470 --> 00:21:34,930
+هي x أس 4 اللى هو بيطلع قداش x أس 4 اللى هو أكيد
+
+229
+00:21:34,930 --> 00:21:39,270
+أكبر أو يساوي 0 صارت عند قيمة الـ function f of x
+
+230
+00:21:39,270 --> 00:21:44,770
+اللى احنا بنحكي عنها دايما أكبر أو يساوي 0 اللى هو
+
+231
+00:21:44,770 --> 00:21:49,650
+مين بيساوي اللى عرفناه F 0 إذا صار في عندي F has
+
+232
+00:21:49,650 --> 00:21:58,450
+absolute minimum at mean at zero لكن عندي لطبيعة
+
+233
+00:21:58,450 --> 00:22:02,590
+الـ sine و طبيعة الـ cosine لو جيت الآن أخدت أي
+
+234
+00:22:02,590 --> 00:22:06,990
+neighborhood حوالين ناقص delta و delta بدي أثبت لك
+
+235
+00:22:06,990 --> 00:22:12,410
+أن F prime مرة ممكن تسوي لي أصغر من صفر و مرة تكون
+
+236
+00:22:12,410 --> 00:22:21,190
+شمالها أكبر من صفر واضح؟ إذا تعالى شوف عندي خذ لأي
+
+237
+00:22:21,190 --> 00:22:25,370
+neighborhood حوالين اللى هو الـ Zero خده من ناقص
+
+238
+00:22:25,370 --> 00:22:29,850
+Delta و Delta لأي Delta في الدنيا أو لأي أمثلون في
+
+239
+00:22:29,850 --> 00:22:34,450
+الدنيا عندي هي الـ neighborhood اللى بحكي فيه
+
+240
+00:22:40,490 --> 00:22:45,390
+النقطة الداخلية اللى احنا مستهدفين فيها اللى هي الصفر
+
+241
+00:22:45,390 --> 00:22:51,500
+خذ أي neighbor حواليه ناقص دلتا أو دلتا تقدر ألاقي
+
+242
+00:22:51,500 --> 00:22:58,220
+n أكبر أو يساوي اتنين very large اللي
+
+243
+00:22:58,220 --> 00:23:04,540
+بيكون n شمالها very close to zero ماشي يعني مهما
+
+244
+00:23:04,540 --> 00:23:09,300
+زغرت لي الـ delta براجيلك n كبيرة كفاية أنها تضلها
+
+245
+00:23:09,300 --> 00:23:18,320
+في هذا الجوار و تحقق ما يلي إيش أخذتها؟ أخذت النقطة
+
+246
+00:23:18,320 --> 00:23:22,460
+واحد على اتنين and by طبعا الآن هذا بيزغرها جد ما
+
+247
+00:23:22,460 --> 00:23:28,120
+بده بتكبير الآن ونفس الاشي 2 على 4n زائد واحد في 
+
+248
+00:23:28,120 --> 00:23:30,780
+π باقي طبعا ليش أخذت هيك؟ عشان واحدة تخلي لي الـ sign
+
+249
+00:23:30,780 --> 00:23:34,620
+صفر وواحدة تخلي لي الـ cosine إيه عشان صفر واضح و
+
+250
+00:23:34,620 --> 00:23:37,580
+في نفس الوجه بتخلي لي الـ sign واحد والـ cosine واحد
+
+251
+00:23:37,580 --> 00:23:44,380
+بشكل معاكس ده نشوف إشي اللي بقوله الآن هذه الآن و
+
+252
+00:23:44,380 --> 00:23:47,380
+هذه الآن اخترت الآن اللى تخليني إياها موجودة في
+
+253
+00:23:47,380 --> 00:23:51,060
+ناقص delta و delta هذولا النقطتين في وين؟ في الجوار
+
+254
+00:23:51,060 --> 00:23:56,880
+اللى اعطيتني إياه أي جوار بتعطيني يا بدأ جيلك الآن
+
+255
+00:23:56,880 --> 00:24:01,300
+المناسبة إليه احسب لي الآن F prime F prime of X
+
+256
+00:24:01,300 --> 00:24:04,040
+بتعرف .. نعرف نحسبها خلينا نحسب F prime of X على
+
+257
+00:24:04,040 --> 00:24:08,580
+جهة  لإنه مش هحسبها عشان تكون قدامكم F prime of X
+
+258
+00:24:10,740 --> 00:24:18,560
+F prime of x إيش بتساوي؟ 8 x تكعيب زائد 4
+
+259
+00:24:18,560 --> 00:24:28,300
+x تكعيب sign 1 على x ناقص اللى هو ليش
+
+260
+00:24:28,300 --> 00:24:34,800
+زائد ناقص x أصبح 4 بيصير x تربيع sign 1 على
+
+261
+00:24:34,800 --> 00:24:41,650
+x صحيح؟ هذه اليومين F prime of x بدّي الآن نعوض على
+
+262
+00:24:41,650 --> 00:24:47,310
+F عند النقطة أول إشي النقطة الأولى عبارة عن 1
+
+263
+00:24:47,310 --> 00:24:52,450
+على اتنين and by عوضنا عنها في 8 x تكعيب هي
+
+264
+00:24:52,450 --> 00:24:57,210
+8 في x تكعيب زائد 4 هي x تكعيب وهي sin
+
+265
+00:24:57,210 --> 00:25:01,820
+1 على x بيصير sin 2 and by عشان هيك الاختيار
+
+266
+00:25:01,820 --> 00:25:05,020
+زائد 1 على 2 and by أس 4 اللي هي ..
+
+267
+00:25:05,020 --> 00:25:11,580
+اللى هي .. بحها دي بيصير أس 2 اه وهاي الناقص
+
+268
+00:25:11,580 --> 00:25:16,900
+فاهمين؟ x تربيع ها دي محادي اللى هي بيصير x تربيع
+
+269
+00:25:16,900 --> 00:25:23,800
+بالسالب واضح هه في cosine من اللى هو 2 nبي الآن
+
+270
+00:25:23,800 --> 00:25:27,580
+هذا المقدار اللي في النص كله إيش حاله؟ هيصير صفر
+
+271
+00:25:27,580 --> 00:25:32,160
+لأن sin 2 nبي صفر هذا المقدار عبارة عن اللي هو
+
+272
+00:25:32,160 --> 00:25:34,860
+8 بتروح مع 8 اللي هم يصير 1 على nبي
+
+273
+00:25:34,860 --> 00:25:39,880
+كل تكعيب هذا الـ cosine إيش بيساوي؟ 1 إذا بيظل من
+
+274
+00:25:39,880 --> 00:25:47,480
+عنده اللي هو هذا سالب تربيع إيه الآن اخذ 1 على
+
+275
+00:25:47,480 --> 00:25:52,120
+nبي عامل المشترك تربيع بيظل عندي اللى هنا عبارة عن
+
+276
+00:25:52,120 --> 00:25:59,560
+اللى هي 4 nبي لأنه ماخذ هنا اللى هي 1 على 2 nبي
+
+277
+00:25:59,560 --> 00:26:03,720
+لكل تربيع اللي هو عامل المشترك بيظل 4 على nبي ناقص
+
+278
+00:26:03,720 --> 00:26:08,840
+إيه شمالها 1 واضح الـ 4 على nبي للـ n أكبر أو يساوي
+
+279
+00:26:08,840 --> 00:26:14,660
+2 أن هذا المقدار بيصير يصغر لدرجة أنه أصغر من
+
+280
+00:26:14,660 --> 00:26:18,320
+الـ 1 للـ n أكبر يساوي 2 إذا صار هذا المقدار عبارة
+
+281
+00:26:18,320 --> 00:26:22,460
+عن سالب في موجبة بيطلع أصغر من 0 إذا F prime طلعت
+
+282
+00:26:22,460 --> 00:26:29,390
+أصغر من 0 في نفس الوقت F prime لـ 2 على كذا على 4n
+
+283
+00:26:29,390 --> 00:26:34,130
+زائد 1 في π اضربها احسبها بيصير 8 وعوض وعوض
+
+284
+00:26:34,130 --> 00:26:38,150
+وعوض الآن مش الـ sign اللي هتت cancel هتت cancel
+
+285
+00:26:38,150 --> 00:26:43,110
+مين؟ الـ cosine فهيطلع عندي هذا المقدار هيه وهيه هذا
+
+286
+00:26:43,110 --> 00:26:48,390
+المقدار التنين المقدارين موجبات، إذا هيكون أكبر من
+
+287
+00:26:48,390 --> 00:26:53,510
+مين؟ من صفر، إذا في كل المنطقة، في كل المنطقة،
+
+288
+00:26:53,510 --> 00:26:58,050
+هتلاقي اللي هو جانب بعض، جانب بعض، مرة أكبر من
+
+289
+00:26:58,050 --> 00:27:00,530
+صفر، مرة أصغر من صفر، مرة أكبر من صفر، مرة أصغر من
+
+290
+00:27:00,530 --> 00:27:05,070
+صفر، ففيش عندك اللى هو تغير إشارتها من موجب إلى
+
+291
+00:27:05,070 --> 00:27:09,750
+سالب، لأ، هم جانب بعض، هذولة تنفج يا واحدة تنساش في
+
+292
+00:27:09,750 --> 00:27:14,570
+جهة واحدة من الـ neighborhood مش على الجهتين هو الـ
+
+293
+00:27:14,570 --> 00:27:18,270
+test اللى بنعرفه أشمله بيقول لك لو اتغير من قبل الـ ..
+
+294
+00:27:18,270 --> 00:27:21,690
+قبل .. في الـ neighborhood قبل النقطة الـ X note من
+
+295
+00:27:21,690 --> 00:27:26,990
+موجب إلى سالب بيصير maximum من سالب إلى موجب بيصير
+
+296
+00:27:26,990 --> 00:27:34,400
+موجب بيصير minimum الآن احنا لجينا relative minimum
+
+297
+00:27:34,400 --> 00:27:38,800
+أو absolute minimum لكن في النقاط من ناقص ذلت
+
+298
+00:27:38,800 --> 00:27:44,180
+العين دلت هاين اختارت لك هنا وهنا جان بعض وجرب
+
+299
+00:27:44,180 --> 00:27:48,650
+كمان بلا جيلك كمان اللي هو بحيث انها تكون موجبة و
+
+300
+00:27:48,650 --> 00:27:53,050
+سالبة وموجبة وسالبة يعني فش derivative هنا تكون
+
+301
+00:27:53,050 --> 00:27:57,690
+موجبة كلها وهنا سالبة كلها عشان تحكم max .. تقول
+
+302
+00:27:57,690 --> 00:28:01,190
+اللي هو .. اللى هي الـ .. و .. و .. و فقط اللى هي
+
+303
+00:28:01,190 --> 00:28:04,750
+الـ converse in and the converse need not to be
+
+304
+00:28:04,750 --> 00:28:09,990
+true in general واضح؟ في الجوارين
+
+305
+00:28:09,990 --> 00:28:14,830
+.. هذا .. هذا من هنا .. من هنا لعين دلها فش إشارة
+
+306
+00:28:14,830 --> 00:28:19,040
+واحدة ومن هنا لعين دلها فش إشارة واحدة إذا the
+
+307
+00:28:19,040 --> 00:28:22,320
+converse need not to be true in general واضح و
+
+308
+00:28:22,320 --> 00:28:30,400
+الله واضح وأعيد واضح يا محمد اه طيب اللى بعده
+
+309
+00:28:30,400 --> 00:28:34,120
+برضه في الـ .. في الـ .. في الـ .. في المثال ارجع لي
+
+310
+00:28:34,120 --> 00:28:38,700
+للكتاب عشان أقول لك هذا برضه بخدمين السؤال قبلها
+
+311
+00:28:38,700 --> 00:28:42,160
+انزل اطلع قبل النظرية قبل النظرية قبل النظرية في
+
+312
+00:28:42,160 --> 00:28:45,520
+عندي مثال هيو أو لا .. أيوة الـ remark الـ remark
+
+313
+00:28:45,520 --> 00:28:52,760
+أيوة اطلع الـ remark هذه الآن إذا بتتذكروا سؤال عشر
+
+314
+00:28:52,760 --> 00:28:55,260
+اللي بناحكيها الجد حكيها إن احنا قولنا it is
+
+315
+00:28:55,260 --> 00:28:59,420
+reasonable to define a function to be increasing
+
+316
+00:28:59,420 --> 00:29:04,170
+at a point إذا كانت هناك مقارنة للمقارنة في
+
+317
+00:29:04,170 --> 00:29:13,750
+المقارنة اللى حوالها، إذا
+
+318
+00:29:13,750 --> 00:29:18,810
+كانت هناك مقارنة في المقارنة اللى حوالها، إذا كانت
+
+319
+00:29:18,810 --> 00:29:19,510
+هناك مقارنة في المقارنة اللي حوالها،
+
+320
+00:29:22,640 --> 00:29:27,460
+من الممكن أن يكون الـ derivative صحيح بشكل صحيح في
+
+321
+00:29:27,460 --> 00:29:35,840
+نقطة فالعمل يزيد في هذه النقطة ولكن هذا الوضع غير
+
+322
+00:29:35,840 --> 00:29:40,960
+صحيح يعني هذا الكلام إيش ماله؟ ليس شرط أنه صحيح أنه
+
+323
+00:29:40,960 --> 00:29:44,500
+عندي اللى هنلاقي اللى هو الـ derivative صحيح بشكل
+
+324
+00:29:44,500 --> 00:29:49,220
+صحيح لكن مافيش .. ماقدرش نقول عنها إيش مالها؟ is
+
+325
+00:29:49,220 --> 00:29:53,460
+increasing at this point هذا في الـ Interval صحيح
+
+326
+00:29:53,460 --> 00:29:57,020
+لكن في الـ Point إيش ماله؟ need not to be true in
+
+327
+00:29:57,020 --> 00:30:01,860
+general وهي مثال G of X هي اللي هو بحيث أنه G'
+
+328
+00:30:02,240 --> 00:30:07,940
+للـ 0 بيساوي 1 لكن الـ G is not increasing in any
+
+329
+00:30:07,940 --> 00:30:14,260
+neighbourhood حوالين مين؟ الصفر، هذا مثال على G' of
+
+330
+00:30:14,260 --> 00:30:20,200
+0 بيساوي 1 اللى هو strictly أكبر من 0 but اللى هو في
+
+331
+00:30:20,200 --> 00:30:24,940
+هذا الـ .. عند هذا النقطة اللي هو في الجوار تبعها
+
+332
+00:30:24,940 --> 00:30:30,260
+لا يمكن انها تكون اللى هي increasing في أي جوار
+
+333
+00:30:30,260 --> 00:30:35,610
+حواليها هذا مثلنا يلا خلينا نحل السؤال الآن هاي
+
+334
+00:30:35,610 --> 00:30:40,090
+سؤالنا let g من R لـ R be defined by g of X بيساوي X
+
+335
+00:30:40,090 --> 00:30:44,410
+زي 2 X ثانوية صين واحدة لـ X for X تتساوي صفر and g
+
+336
+00:30:44,410 --> 00:30:47,550
+of 0 بيساوي 0 show that g prime of 0 إيش بيساوي
+
+337
+00:30:47,550 --> 00:30:53,030
+واحد but in every neighborhood of zero, the
+
+338
+00:30:53,030 --> 00:30:57,430
+derivative g prime takes on both positive and
+
+339
+00:30:57,430 --> 00:31:01,710
+negative values. Thus, g is not monotonic in any
+
+340
+00:31:01,710 --> 00:31:05,830
+neighborhood. هنلاقي في المنطقة هذه وين ما كان،
+
+341
+00:31:05,830 --> 00:31:13,210
+هنلاقي الـ g prime اللى هي أكبر من صفر وأصغر من صفر
+
+342
+00:31:13,210 --> 00:31:18,410
+يعني مش هنلاقي في أي neighborhood أن الـ G' أكبر
+
+343
+00:31:18,410 --> 00:31:25,410
+من صفر لحالها أو الـ G' للـ X هي أكبر من حالها أو
+
+344
+00:31:25,410 --> 00:31:33,910
+أصغر من حالها في كل .. من صفر في كل الجوار G' مش
+
+345
+00:31:33,910 --> 00:31:38,930
+هتكون أصغر من صفر في كل الجوار ولا أكبر من صفر في
+
+346
+00:31:38,930 --> 00:31:46,630
+كل الجوار هنلاقي لها متذبذبة في الإشارة كيف؟
+
+347
+00:31:48,510 --> 00:31:54,930
+الحل شبهه صحيح الحل شبهه بس هذا الآن موظف لمين؟ لا
+
+348
+00:31:54,930 --> 00:31:58,990
+اللى هي اللى هو counter example على اللى هي الـ
+
+349
+00:31:58,990 --> 00:32:02,630
+remark اللى عندنا وده counter example على اللي هي
+
+350
+00:32:02,630 --> 00:32:08,010
+النظرية هذك بوضح إن الـ theorem need not to be true
+
+351
+00:32:08,010 --> 00:32:12,170
+in general وهنا اللي هو بوضح أنه اللي هو الـ
+
+352
+00:32:12,170 --> 00:32:18,000
+strictly increasing عند نقطة لأ .. لأ .. لأ .. لأ
+
+353
+00:32:18,000 --> 00:32:18,380
+.. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ ..
+
+354
+00:32:18,380 --> 00:32:18,540
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+355
+00:32:18,540 --> 00:32:19,120
+.. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ ..
+
+356
+00:32:19,120 --> 00:32:19,800
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+357
+00:32:19,800 --> 00:32:26,540
+.. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+358
+00:32:26,540 --> 00:32:29,660
+..
+
+359
+00:32:29,660 --> 00:32:31,300
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+360
+00:32:31,300 --> 00:32:32,100
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+361
+00:32:32,100 --> 00:32:39,020
+.. لأ .. لأ
+
+362
+00:32:39,020 --> 00:32:46,020
+.. لأ إيش بعمل قاعد؟ بدي أحاول أثبت إنه g prime of 0
+
+363
+00:32:46,020 --> 00:32:51,600
+اللى هو بيساوي واحد هاي التعريف هاي عنده y ساوي
+
+364
+00:32:51,600 --> 00:32:55,180
+جسمنا عليه بيصير 1 زائد 2 limit x sin 1
+
+365
+00:32:55,180 --> 00:33:00,380
+على x اللى هو هذا إيه شماله؟ بيساوي صفر لأن الـ
+
+366
+00:33:00,380 --> 00:33:03,720
+absolute value للـ X sin 1 على X أكبر بيساوي صفر
+
+367
+00:33:03,720 --> 00:33:07,740
+وأصغر أو يساوي الـ absolute value للـ X اللى هو
+
+368
+00:33:07,740 --> 00:33:11,020
+by Sandwich theorem اللى هو الـ limit هذا إيش بيساوي؟
+
+369
+00:33:11,020 --> 00:33:15,040
+صفر إذا الـ limit
+
+401
+00:35:47,840 --> 00:35:52,340
+differentiable على R من جهة أخرى لاحظ إن النص
+
+402
+00:35:52,340 --> 00:35:55,160
+بين الـ 0 والـ 1، مين هو الـ 0 والـ 1؟ الـ 0 والـ
+
+403
+00:35:55,160 --> 00:35:58,060
+1 هي قيم الـ function هذه اللي بحكي عنها، يعني الـ
+
+404
+00:35:58,060 --> 00:36:03,840
+0 والـ 1 هتكون الـ 0 عبارة عن أشوف X1 و الـ A1
+
+405
+00:36:03,840 --> 00:36:09,390
+أشوف X2، مين X1 و X2؟ اخترت الـ X1 أصغر من 0؟و X2
+
+406
+00:36:09,390 --> 00:36:13,690
+أكبر من 0 صار عند نقطتين X1 و X2 واحد أصغر من 0
+
+407
+00:36:13,690 --> 00:36:18,030
+واحد أكبر من 0 يعني التمتين عملين لفترة إذا صار
+
+408
+00:36:18,030 --> 00:36:24,870
+عند نص في الفترة بين 0 و 1 اللي هو عبارة عن بين H
+
+409
+00:36:24,870 --> 00:36:31,520
+of X1 و H of X2 لكن H of X1 و H of X2 من هم؟F' of
+
+410
+00:36:31,520 --> 00:36:36,900
+X1 و F' of X2 إذا صارت اللي هي Darboux theorem
+
+411
+00:36:36,900 --> 00:36:41,260
+محققة F is differentiable و نص تنتمي للفترة بيه
+
+412
+00:36:41,260 --> 00:36:47,340
+اللي هي F' of X1 و F' of X2 of X2 إذا حسب اللي هي
+
+413
+00:36:47,340 --> 00:36:51,340
+Darboux theorem أي حاجة بينهم لازم يكون لها أصل
+
+414
+00:36:51,340 --> 00:36:57,190
+إذا there exists Cبين الـ x1 و الـ x2 بحيث أنه
+
+415
+00:36:57,190 --> 00:37:02,450
+اللي هو f prime للـ c هذه اللي اللي جاتها بين x1 و
+
+416
+00:37:02,450 --> 00:37:08,770
+x2 هي مين النص يعني زي بمسح كل القيم اللي بين f of
+
+417
+00:37:08,770 --> 00:37:13,080
+x1 و f of x2 مش هيك تدربت في تستيرون بتقولإذا صارت
+
+418
+00:37:13,080 --> 00:37:17,460
+عيدي في c بين هذه و هذه بحيث أن f prime of c بساوة
+
+419
+00:37:17,460 --> 00:37:21,480
+نص يعني h of c بساوة نص طب هذا مستحيل لأن اصلا h
+
+420
+00:37:21,480 --> 00:37:25,100
+of c .. هذا مستحيل ليش
+
+421
+00:37:25,100 --> 00:37:28,760
+مستحيل؟ لأن h أصلا ماتاخدش رقم تاني، يا صفر يا
+
+422
+00:37:28,760 --> 00:37:32,020
+واحد إذا contradiction، مدام contradiction إذا
+
+423
+00:37:32,020 --> 00:37:37,940
+there is no function f من R لR بحيث أن f prime of
+
+424
+00:37:37,940 --> 00:37:41,080
+x بساوة f of x for every x
+
+425
+00:37:45,230 --> 00:38:01,830
+في ضايق سؤالين خلنا نمر عليهم هذا
+
+426
+00:38:01,830 --> 00:38:06,130
+حلنا زيها اللي هو let I be an interval and let F
+
+427
+00:38:06,130 --> 00:38:11,110
+من I لR be differentiable on I مفترضين أن F عبارة
+
+428
+00:38:11,110 --> 00:38:16,000
+عن اللي هو differentiable function على an interval
+
+429
+00:38:16,000 --> 00:38:19,820
+I وقول لي show that إذا كانت F prime is positive
+
+430
+00:38:19,820 --> 00:38:25,000
+on I لو F prime أكبر من 0 على I هتكون ال F أشمالها
+
+431
+00:38:25,000 --> 00:38:29,800
+strictly increasing on I طبعا اللي هو على السريع
+
+432
+00:38:29,800 --> 00:38:34,940
+لنفترض F prime أكبر من 0 لكل X element on I ماشي
+
+433
+00:38:34,940 --> 00:38:41,240
+هو نفسه؟ خلاص اللي بده السؤال موجود في الشريح اللي
+
+434
+00:38:41,240 --> 00:38:44,300
+هو ال main value theorem اتبكّل الله خليني أقوله
+
+435
+00:38:44,300 --> 00:38:50,490
+عنهالأن let I be an interval and let F من I لعند R
+
+436
+00:38:50,490 --> 00:38:55,050
+سواء الـ14 بيه differentiable on I show that if
+
+437
+00:38:55,050 --> 00:38:59,210
+the derivative F' is never zero on I يعني إذا كانت
+
+438
+00:38:59,210 --> 00:39:04,230
+F' أكبر من صفر اللي هي تسوى صفر on I then either
+
+439
+00:39:04,230 --> 00:39:08,430
+F' أكبر من صفر for all X limited on I أو F' أصغر
+
+440
+00:39:08,430 --> 00:39:13,480
+من صفر لكل X جمالهايعني بيقولّي إذا كانت ال F
+
+441
+00:39:13,480 --> 00:39:17,760
+differentiable على ال interval I وكانت ال F'
+
+442
+00:39:18,160 --> 00:39:23,240
+بتسويش 0 في هذه الحالة ال F' يا هتكون كلها موجبة
+
+443
+00:39:23,240 --> 00:39:28,940
+على ال I يا كلها سالبة عالميا على ال I مدامت مغيرتش
+
+444
+00:39:28,940 --> 00:39:32,760
+شرطها بالمرة يعني بمعنى أخر يعني مش هتغير شرطها
+
+445
+00:39:32,760 --> 00:39:38,260
+بالمرة مدامت ال F' مش 0 على الفترة إذا إشارتها
+
+446
+00:39:38,260 --> 00:39:43,650
+واحدةيا F أكبر من 0، يا F أكبر من .. أصغر من 0،
+
+447
+00:39:43,650 --> 00:39:47,070
+هدا في ضوء أن F is differentiable، بدنا نفترض
+
+448
+00:39:47,070 --> 00:39:51,170
+العكس ونصل لcontradiction، suppose on the contrary
+
+449
+00:39:51,170 --> 00:39:56,290
+نفترض أن F prime of X أكبر من 0 لنقات .. لبعض
+
+450
+00:39:56,290 --> 00:40:00,390
+النقات في I، و F prime of X أكبر من .. أصغر من 0
+
+451
+00:40:00,390 --> 00:40:04,660
+لبعض النقات في I، يعني مخلوطةوبدّا نصلّمين ل
+
+452
+00:40:04,660 --> 00:40:08,640
+contradiction طيب then there exist a و b element
+
+453
+00:40:08,640 --> 00:40:12,220
+in I such that أف برايم of a أشماله أكبر من صفر و
+
+454
+00:40:12,220 --> 00:40:15,520
+أف برايم بي أصغر من صفر لإن أنا مفترض الآن افترضت
+
+455
+00:40:15,520 --> 00:40:18,200
+أنه في نقاط اللي هي أكبر من صفر عندها ال
+
+456
+00:40:18,200 --> 00:40:20,940
+derivative وفي نقاط أصغر من صفر عندها ال
+
+457
+00:40:20,940 --> 00:40:25,040
+derivative ماشي الحال إذا بنلاقي a و b بالحقق هذه
+
+458
+00:40:25,490 --> 00:40:28,710
+الان اكيد مدام هذا أكبر من صفر وهذا أصغر من صفر،
+
+459
+00:40:28,710 --> 00:40:32,910
+إذا الصفر بين الـF'A والـF'B لأنها واحدة موجبة
+
+460
+00:40:32,910 --> 00:40:36,850
+بواحدة مين، مدام الصفر بينهم، إذا حسب Darboux's
+
+461
+00:40:36,850 --> 00:40:40,990
+theorem، هي بتمسح كل المنطقة اللي بين الصور، لازم
+
+462
+00:40:40,990 --> 00:40:45,570
+كلها يكون اللي هي أصولإذن by Darboux's theorem
+
+463
+00:40:45,570 --> 00:40:49,430
+there exists c element in I such that f prime of c
+
+464
+00:40:49,430 --> 00:40:52,970
+بيساوي صفر وهذا اللي هو بناقض الفرضية اللي احنا
+
+465
+00:40:52,970 --> 00:40:57,570
+فرضناها أن f prime لا تساوي صفر على كل ال I إذن f
+
+466
+00:40:57,570 --> 00:41:01,730
+prime لا تساوي صفر على كل ال I معناته و
+
+467
+00:41:01,730 --> 00:41:05,670
+differentiable طبعا معناته يا إما كلهن ال f prime
+
+468
+00:41:05,670 --> 00:41:11,950
+موجبات يا إما كلهن إيه شمالهن سالباتيعني يا
+
+469
+00:41:11,950 --> 00:41:15,710
+increasing strictly increasing يا strictly يا
+
+470
+00:41:15,710 --> 00:41:23,230
+شمالها decreasing مافيش أي تغيير للرسم والشعر طيب
+
+471
+00:41:23,230 --> 00:41:28,820
+نيجي لآخر السؤاللت I Be An Interval Prove That If
+
+472
+00:41:28,820 --> 00:41:32,980
+F Is Differentiable On I And The Derivative Of F'
+
+473
+00:41:33,300 --> 00:41:37,820
+Is Bounded On I، Then F Satisfies Lipschitz
+
+474
+00:41:37,820 --> 00:41:43,920
+Condition On I، ماشي الحل، بتقول لي، بتقوللي
+
+475
+00:41:43,920 --> 00:41:48,460
+السؤال ما يعني، Lipschitz Condition
+
+476
+00:41:53,580 --> 00:41:58,020
+خلّينا نشوف إيش اللي بنقوله إذا كانت F
+
+477
+00:41:58,020 --> 00:42:10,320
+differentiable on I و F' bounded on I اثبت إن F
+
+478
+00:42:10,320 --> 00:42:15,680
+satisfies Lipschitz conditions ماشي؟ إيش Lipschitz
+
+479
+00:42:15,680 --> 00:42:23,450
+condition؟ إنه there exists K أكبر من 0 such that F
+
+480
+00:42:23,450 --> 00:42:30,130
+of X ناقص F of Y أصغر أو ساوي لبس اليود فيها ل X
+
+481
+00:42:30,130 --> 00:42:34,330
+minus Y في مين؟ في K اللي كنا نقول عنها على طول
+
+482
+00:42:34,330 --> 00:42:40,630
+هذي بتعطي إيش ما لها uniformly continuous لكل X و
+
+483
+00:42:40,630 --> 00:42:44,190
+Y لكل X و Y في الفترة اللي بنحكي عنها اللي بشت
+
+484
+00:42:44,190 --> 00:42:48,790
+غل condition إيش ما لها متحققةإذا الـ Absolute هذا
+
+485
+00:42:48,790 --> 00:42:54,670
+معناه اللي بش تسكندشه نيجي نحققها هذه نيجي نحقق
+
+486
+00:42:54,670 --> 00:42:55,710
+اللي هو
+
+487
+00:42:58,230 --> 00:43:02,930
+الـ Lipschitz condition لت x و y element in I بيه
+
+488
+00:43:02,930 --> 00:43:06,130
+such that x strictly أصغر من y
+
+489
+00:43:27,180 --> 00:43:30,960
+إذا منها ناخد ال absolute value للجهتين بتطلع عندي
+
+490
+00:43:30,960 --> 00:43:36,260
+اللي أماني واضح لكن F' is bounded مدان bounded F'
+
+491
+00:43:36,740 --> 00:43:41,940
+إذا there exist K بحيث أن F' of C أصغر سوى K لكل C
+
+492
+00:43:41,940 --> 00:43:47,550
+إيه شمالها من ضمن ال C اللي فوقأه لأن أف برايم is
+
+493
+00:43:47,550 --> 00:43:51,870
+bounded على كل ال I و هذه ال K بتنفع لكل مين لكل
+
+494
+00:43:51,870 --> 00:43:54,570
+ال C's اللي في ال I إذا صارت أف برايم في C أصغر أو
+
+495
+00:43:54,570 --> 00:43:58,270
+شويه K بنعود فوق بيصير أف of X ناقص أف of Y أصغر
+
+496
+00:43:58,270 --> 00:44:02,150
+أو شويه K في X minus Y و ال X و ال Y كانت نقش
+
+497
+00:44:02,150 --> 00:44:06,870
+مالهين arbitrary إذا صار هذه أصغر أو شويه K في X
+
+498
+00:44:06,870 --> 00:44:11,790
+minus Y ودخلت المتساوية لأن في حالة ال X بتساوي Y
+
+499
+00:44:11,790 --> 00:44:21,600
+اللي هو it is trivialtherefore I satisfy اللي هو
+
+500
+00:44:21,600 --> 00:44:25,820
+Lipschitz condition و الله يعطيكوا العافية 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/A0riuruZYaI_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/A0riuruZYaI_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..b748194a393e0c1e1c2316d7c4b9a9991c329f6f
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/A0riuruZYaI_raw.srt
@@ -0,0 +1,2004 @@
+1
+00:00:05,740 --> 00:00:08,920
+بسم الله الرحمن الرحيم الجزء الثاني من المحاضرة
+
+2
+00:00:08,920 --> 00:00:14,780
+التاسعة هيكون اللي هو عبارة عن discussionأو مناقشة
+
+3
+00:00:14,780 --> 00:00:22,060
+لـ Section 6.2 و6.3 اللي هو مناقشة لـ Main Value
+
+4
+00:00:22,060 --> 00:00:25,220
+Theorem and its Applications ومناقشة أيضًا لـ
+
+5
+00:00:25,220 --> 00:00:30,560
+Lobitals Rules نيجي الآن لـ 6.2 الأسئلة المطلوبة
+
+6
+00:00:30,560 --> 00:00:35,720
+هي كما يلي نبدأ في سؤال 6.2 ادخلنا على الكتاب
+
+7
+00:00:35,720 --> 00:00:39,160
+خلينا نشوف ال .. ال .. ال .. المثال .. السؤال من
+
+8
+00:00:39,160 --> 00:00:45,790
+الكتاب نبدأ الآن بسؤال 5السؤال الخامسة هو كما يلي
+
+9
+00:00:45,790 --> 00:00:49,550
+let a أكبر من سفر و b أكبر من سفر و a strictly
+
+10
+00:00:49,550 --> 00:00:55,570
+أكبر من b طبعا كل العلاقة strictly وبنفترض أن n
+
+11
+00:00:55,570 --> 00:01:00,290
+أكبر أساوي اتنين prove that a أس واحدة لأ ناقص b
+
+12
+00:01:00,290 --> 00:01:05,290
+أس واحدة لأ ن أصغر من a minus b أس واحد عليا الان
+
+13
+00:01:05,290 --> 00:01:10,490
+لو جينا لاحظنا على اللي هو المطلوب عند خمسةبقولي
+
+14
+00:01:10,490 --> 00:01:16,710
+أن a أكبر من b أكبر من 0 أو n أكبر أو يساوي 2
+
+15
+00:01:16,710 --> 00:01:23,070
+بقولي prove that أن a أس واحد على n ناقص b أس واحد
+
+16
+00:01:23,070 --> 00:01:30,710
+على n أصغر من a ناقص b الكل أس واحد على n نيجي
+
+17
+00:01:30,710 --> 00:01:36,890
+للبرهان لو جينا لاحظناإنه عندي ال .. نيجي للسؤال
+
+18
+00:01:36,890 --> 00:01:40,770
+بس كيف نفكر في السؤال هو ماعطيني hint في الكتاب
+
+19
+00:01:40,770 --> 00:01:45,190
+لكن خلينا نشوف كيف .. كيف حصل على ال hint لو ادينا
+
+20
+00:01:45,190 --> 00:01:50,490
+جسمنا الجهتين هذا مش من ضمن الحل طبعا عندي a على b
+
+21
+00:01:51,290 --> 00:01:55,270
+الكل أس واحدة لأن ناقص جسمة على اللي هو بي أس
+
+22
+00:01:55,270 --> 00:01:59,510
+واحدة لأن الجهتين طبعا والبي طبعا موجبة ففيش إشي
+
+23
+00:01:59,510 --> 00:02:05,290
+بتغير بيصير a على بي ناقص واحد الكل أس واحدة لأن
+
+24
+00:02:05,290 --> 00:02:10,710
+لو نجلنا هذه a على بي أس واحدة لأن ناقص ال a على
+
+25
+00:02:10,710 --> 00:02:16,150
+بي ناقص واحد أس واحدة لأن أصغر من مين من واحد الان
+
+26
+00:02:16,700 --> 00:02:21,940
+عندي هذا الأن كله على بعض هو نفسه اللهان فممكن
+
+27
+00:02:21,940 --> 00:02:27,680
+نستقل الدالة من خلاله أنه ناخد ال F of X لت F of X
+
+28
+00:02:27,680 --> 00:02:35,690
+و هتوصلنا بساوي X أسواحة ده الأن ناقصx ناقص واحد
+
+29
+00:02:35,690 --> 00:02:42,810
+واس واحد علان وطبعا هو الان معطيني في السؤال a
+
+30
+00:02:42,810 --> 00:02:47,790
+أكبر من b أكبر من 0 لو طلعنا نلاقي الدالة هذه
+
+31
+00:02:47,790 --> 00:02:56,610
+ولاحظنا أوجدنا ال f prime لها f prime of xخلّينا
+
+32
+00:02:56,610 --> 00:03:02,130
+ناخد ال X عنده ال X أكبر أو يساوي واحد و هنشوف
+
+33
+00:03:02,130 --> 00:03:05,730
+اللي هو ليش عنده ال X أكبر أو يساوي واحد برضه
+
+34
+00:03:05,730 --> 00:03:09,070
+بتظبط في حالتنا لإن اللي بنينا على أساسها اللي هي
+
+35
+00:03:09,070 --> 00:03:13,290
+ال A على B نفسها أكبر strictly من مين من واحد
+
+36
+00:03:13,290 --> 00:03:17,150
+فالأمور متناسقة مع بعض و لو بدنا نطبق حللها جي جي
+
+37
+00:03:17,150 --> 00:03:20,670
+اللي هو تطبيق معقول اللي هنقف ال prime of X بساوي
+
+38
+00:03:20,670 --> 00:03:26,200
+واحد على N في X أسوا واحد على N ناقص واحد ناقصاللي
+
+39
+00:03:26,200 --> 00:03:30,720
+هو واحدة لان في x ناقص واحد واس واحدة لان ناقص
+
+40
+00:03:30,720 --> 00:03:36,520
+واحد ويسوى واحدة لان في x اس واحدة لان ناقص واحد
+
+41
+00:03:36,520 --> 00:03:44,840
+ناقص اللي هو x ناقص واحد اللي هو على واحدة لان
+
+42
+00:03:44,840 --> 00:03:50,460
+ناقص واحد الآن لو طلعنا للي عندي هذا
+
+43
+00:03:53,450 --> 00:03:56,930
+لو طلعنا للمقدار اللي عندي ال X أكبر أو يساوي إيش
+
+44
+00:03:56,930 --> 00:04:02,450
+واحد يعني الآن ال X أكبر يساوي واحد لذا لما عندي
+
+45
+00:04:02,450 --> 00:04:06,950
+الأس اللي هان أس إيش ماله بالسالب أو صفر على سوء
+
+46
+00:04:06,950 --> 00:04:10,130
+الظروف اللي هو بالسالب معناته اللي هو بده يصير
+
+47
+00:04:10,130 --> 00:04:16,070
+واحد على X الواحد على X عبارة عن كسر، مظبوط؟ الآن
+
+48
+00:04:16,070 --> 00:04:20,430
+بيصير عندي المقدار اللي عندي العلاقة بين هذا و هذا
+
+49
+00:04:20,940 --> 00:04:26,360
+x أكيد أكبر من x نقص واحد صح ولا لأ لكن لأن
+
+50
+00:04:26,360 --> 00:04:31,240
+مقلوبها هيصير ايش ماله هيصير أصغر فهيصير المقدار
+
+51
+00:04:31,240 --> 00:04:35,440
+هذا أصغر من هذا المقدار ماشي فبصير عندي المقدار
+
+52
+00:04:35,440 --> 00:04:40,180
+هذا كله على بعض أصغر من مين من سفر فالآن صارت عندي
+
+53
+00:04:40,180 --> 00:04:45,720
+f' أصغر strictly من مين من سفر إذا صار عندي إذا f
+
+54
+00:04:45,720 --> 00:04:47,160
+is strictly
+
+55
+00:04:50,230 --> 00:04:54,310
+Decreasing بدى استخدم الخاصية هذه مدام Strictly
+
+56
+00:04:54,310 --> 00:04:59,390
+Decreasing وانا عندى a أكبر من b هسينا عندى a على
+
+57
+00:04:59,390 --> 00:05:06,110
+b أكبر من واحد و ال b طبعا لا تسوى سفر اذا بما ان
+
+58
+00:05:06,110 --> 00:05:11,930
+F is strictly decreasing اذا F of a على b أكبر من
+
+59
+00:05:11,930 --> 00:05:18,440
+F of واحدأف of a على b دلتنا بيجيب العوض فوق بيصير
+
+60
+00:05:18,440 --> 00:05:26,380
+عندي اللي هو .. آسف أصغر عندي أف of a على b إيش
+
+61
+00:05:26,380 --> 00:05:30,920
+هتساويك؟ قولوا معايا اللي هو a على b أس واحد على n
+
+62
+00:05:30,920 --> 00:05:40,360
+ناقص a على b ناقص واحد وكل أس واحد على n هذا إيش
+
+63
+00:05:40,360 --> 00:05:45,310
+ماله؟ أصغر من مين؟من F of واحد F of واحد حسب لي F
+
+64
+00:05:45,310 --> 00:05:50,290
+of واحد هذه واحد وهذه سفر فبصير عبارة عن اصغر من
+
+65
+00:05:50,290 --> 00:05:53,350
+واحد طبعا هذه اللي هو بنعمل عملية عكسية للي
+
+66
+00:05:53,350 --> 00:05:57,530
+عملناها فوق فبصير عندى اضرب الجهتين في B أس واحدة
+
+67
+00:05:57,530 --> 00:06:04,260
+لان فبصير A أس واحدة لان ناقصA ناقص B أس واحد على
+
+68
+00:06:04,260 --> 00:06:10,940
+N أصغر من مين من B أس واحد على N ضربت كله في مين؟
+
+69
+00:06:10,940 --> 00:06:14,980
+في B أس واحد على N إن جلّي الأن بصير عندي A أس
+
+70
+00:06:14,980 --> 00:06:20,320
+واحد على N ناقص B أس واحد على N أصغر من A minus B
+
+71
+00:06:20,320 --> 00:06:26,530
+الكل أس واحد على Nوهو المطلوب وهذا اللي هو بدهيا
+
+72
+00:06:26,530 --> 00:06:32,150
+يلا إيه بعده؟ خلينا نيجي للسؤال اللي المطلوب الآخر
+
+73
+00:06:32,150 --> 00:06:38,570
+اللي هو use the mean value theorem سؤال 6 use the
+
+74
+00:06:38,570 --> 00:06:42,830
+mean value theorem to prove that sin x minus sin y
+
+75
+00:06:42,830 --> 00:06:47,070
+أصغر أو يسوى x minus y for all x, y in R هذا
+
+76
+00:06:47,070 --> 00:06:50,610
+السؤال حلنا زيه بالظبط اللي هو mean اللي هو ال
+
+77
+00:06:50,610 --> 00:06:52,070
+cosine، مظبوط؟
+
+78
+00:06:57,300 --> 00:07:03,320
+الان ماعرفش فيه داعي نحله ولا ان هو نفسي او مصور
+
+79
+00:07:03,320 --> 00:07:06,740
+عملناه
+
+80
+00:07:06,740 --> 00:07:12,000
+ولا لا اللي هي ال cosine عملناها بنفس الأسلوب ومش
+
+81
+00:07:12,000 --> 00:07:14,340
+هيختلف اللي هو الحل
+
+82
+00:07:22,460 --> 00:07:27,840
+أحلو ولا خلصت؟ الان عندي اللي هو سبعة use the mean
+
+83
+00:07:27,840 --> 00:07:32,580
+value theorem to prove that X-1 على X أصغر من X
+
+84
+00:07:32,580 --> 00:07:39,160
+أصغر من X واحد for X أكبر من واحد اللي هو عند ..
+
+85
+00:07:39,160 --> 00:07:45,440
+احنا حلينا واحد زاد X هذه الآن اللي هي عبارة عن
+
+86
+00:07:45,440 --> 00:07:51,040
+مين؟عن اللي هو ln LX الدلة اللي هي F of X بساوي ln
+
+87
+00:07:51,040 --> 00:07:56,440
+LX وبنستخدم ال mean value theorem وبنفس الأسلوب في
+
+88
+00:07:56,440 --> 00:08:00,540
+داعي نحله؟ خلّينا نحله، خلّينا نحله عشان بنصور
+
+89
+00:08:00,540 --> 00:08:12,360
+الآن سؤال سبعة الآن عندي بدأ أثبت أن ln LX أصغر من
+
+90
+00:08:12,360 --> 00:08:20,000
+X ناقص واحدو أكبر من x minus واحد عالميا على x
+
+91
+00:08:20,000 --> 00:08:26,400
+solution العالمين value theorem عالمين value
+
+92
+00:08:26,400 --> 00:08:29,960
+theorem نحله عالمين value theorem لأنه لسه ما
+
+93
+00:08:29,960 --> 00:08:35,980
+خدناش اللي هو Taylor's theorem مش
+
+94
+00:08:35,980 --> 00:08:39,340
+فاهم
+
+95
+00:08:39,340 --> 00:08:42,110
+عليكبينفع الحلقة باستخدام Taylor and X not
+
+96
+00:08:42,110 --> 00:08:44,690
+بيساومها يعني ال mainly طيب اه احنا .. احنا عشان
+
+97
+00:08:44,690 --> 00:08:48,850
+لسه ماخدناش Taylor's theorem بدنا نحلها على مين؟
+
+98
+00:08:48,850 --> 00:08:51,970
+على ال mean value theorem ليش؟ لأنه احنا لسه
+
+99
+00:08:51,970 --> 00:09:00,070
+ماخدناش Taylor's theorem طيب الان let F of X
+
+100
+00:09:00,070 --> 00:09:07,330
+بتساوي لين ال X؟ وعندي ال X أكبر من مين؟ أكبر أو
+
+101
+00:09:07,330 --> 00:09:11,800
+تساوي الواحد، ولا لأ؟عندي الـ x أكبر من 100 من 0
+
+102
+00:09:11,800 --> 00:09:14,940
+for
+
+103
+00:09:14,940 --> 00:09:23,100
+x أكبر من 0 مش عايز أني ماشي الحين نيجي اللي هو أن
+
+104
+00:09:23,100 --> 00:09:26,540
+نستخدم ال main value theorem continuous و claw و
+
+105
+00:09:26,540 --> 00:09:31,300
+differentiable وكل الأمور هذه أشملها متحققة إذا
+
+106
+00:09:31,300 --> 00:09:38,150
+there exist c element in a و bC عندي اللي هو
+
+107
+00:09:38,150 --> 00:09:42,290
+element معايا؟
+
+108
+00:09:42,290 --> 00:09:52,610
+طيب لأن let F of X بيساوي لن X و X أكبر من 0 وعندي
+
+109
+00:09:52,610 --> 00:09:57,430
+المطلوب في ال inequality اللي هي X أكبر من 1 يعني
+
+110
+00:09:57,430 --> 00:10:02,870
+X سنتمي إلى الواحد وما لا نهاية معايا؟ إذا there
+
+111
+00:10:02,870 --> 00:10:11,590
+exists Cلذن بدى اطبق الان we apply mean value
+
+112
+00:10:11,590 --> 00:10:18,570
+theorem on وين قول معايا on واحد و X there exists
+
+113
+00:10:18,570 --> 00:10:25,090
+C element واحد و X such that معايا such that اللي
+
+114
+00:10:25,090 --> 00:10:36,890
+هو F prime of C سوى F of X نقص F واحدعلى x ناقص
+
+115
+00:10:36,890 --> 00:10:48,110
+اياش ناقص واحد اه فالان عندي f of x جديش ويساوي لن
+
+116
+00:10:48,110 --> 00:10:56,450
+ال x ناقص لن الواحد جديش سفر على x ناقص واحد وهذا
+
+117
+00:10:56,450 --> 00:11:02,050
+مين هو f prime of c عبارة عن لن الواحد على c مظبوط
+
+118
+00:11:02,870 --> 00:11:06,570
+إذا صار عندى الآن بدأ أجيب لإن ال X أصغر من مين من
+
+119
+00:11:06,570 --> 00:11:12,270
+X ماينس واحد صار عندى إذا لإن ال X بساوي واحد على
+
+120
+00:11:12,270 --> 00:11:18,390
+C في X ماينس واحد واللي عند C أشمالة باخدها أنا
+
+121
+00:11:18,390 --> 00:11:23,050
+أكبر من واحد مدام أكبر من واحد إذا واحد على C اللي
+
+122
+00:11:23,050 --> 00:11:28,390
+هو أصغر من واحد مظبوط إذا هذا أكيد هذا أصغر من X
+
+123
+00:11:28,390 --> 00:11:35,440
+ماينس واحدمظبوط؟ لأن الواحد على الـC أشمالها عبارة
+
+124
+00:11:35,440 --> 00:11:43,990
+عن كسر الآن من جهة أخرى عندي اللي هو لن ال Xبساوي
+
+125
+00:11:43,990 --> 00:11:49,410
+واحد على C في X ناقص واحد لكن الـ X أنا C إيش
+
+126
+00:11:49,410 --> 00:11:54,190
+معناها بين الواحد والإكس يعني C أصغر من مين C أصغر
+
+127
+00:11:54,190 --> 00:11:59,430
+من X يعني الواحد على X اللي هو أصغر من مين الآن C
+
+128
+00:11:59,430 --> 00:12:05,890
+أصغر من X إذا واحد على C أكبر من واحد على X فبصير
+
+129
+00:12:05,890 --> 00:12:13,190
+عند هذا أكبرمن 1 على x في x minus 1 يعني بمعنى آخر
+
+130
+00:12:13,190 --> 00:12:18,230
+صار عند لن ال x اللي هو أكبر من x minus 1 على ال x
+
+131
+00:12:18,230 --> 00:12:21,550
+هي عند ال inequality التانية و هي عند ال
+
+132
+00:12:21,550 --> 00:12:26,690
+inequality الأولى من التنتين إذا لن ال x أشمله
+
+133
+00:12:26,690 --> 00:12:32,790
+أصغر من x minus 1 مظبوط اللي بعمله و أكبر من x
+
+134
+00:12:32,790 --> 00:12:41,190
+minus 1 عالميا على xوهو المطلوب إيش السؤال اللي
+
+135
+00:12:41,190 --> 00:12:49,550
+بعده؟ الآن سؤال تمانية Let F من A لعند B
+
+136
+00:13:11,390 --> 00:13:17,630
+سؤال تمانية let
+
+137
+00:13:17,630 --> 00:13:30,470
+f من a و b لعند r اشمل بي continuous او
+
+138
+00:13:30,470 --> 00:13:37,190
+differentiable on
+
+139
+00:13:37,190 --> 00:13:45,370
+meanon open interval a و b show
+
+140
+00:13:45,370 --> 00:13:51,730
+that if limit f prime of x عندي limit f prime of x
+
+141
+00:13:51,730 --> 00:14:01,630
+لما x تروح لل a بساوي a capital then اللي هي f
+
+142
+00:14:01,630 --> 00:14:11,990
+prime of aF prime of A exists and equals A
+
+143
+00:14:11,990 --> 00:14:21,750
+solution أو proof ماعطيني
+
+144
+00:14:21,750 --> 00:14:28,650
+F is differentiable على اللي هي الفترة من A وB أو
+
+145
+00:14:28,650 --> 00:14:33,350
+continuous طبعاً open أو continuous على close من A
+
+146
+00:14:34,120 --> 00:14:39,860
+وب عندي limit f prime of x معطينيها لما x تروح إلى
+
+147
+00:14:39,860 --> 00:14:50,180
+ال a ايش بساوي بساوي اللي هو a capital اطلع
+
+148
+00:14:50,180 --> 00:14:58,560
+لفوق الان عندي الان ال f prime of a تعريفهااللي هي
+
+149
+00:14:58,560 --> 00:15:04,600
+limit F of X ناقص F of A على X minus A لما X تروح
+
+150
+00:15:04,600 --> 00:15:09,940
+للـA ماشي الحال هذا اللي هو إذا كان ال limit had
+
+151
+00:15:09,940 --> 00:15:13,500
+exist إذا أثبتنا إن ال limit had exist بتكون ال F
+
+152
+00:15:13,500 --> 00:15:18,440
+prime of A أشمالها اللي هي exist ماشي الحال الآن
+
+153
+00:15:18,440 --> 00:15:21,200
+هو ماعطيني limit F prime of X لما X تروح لـA
+
+154
+00:15:21,200 --> 00:15:28,290
+أشمالها هي ال existواضح اه؟ لان بدي اطبق الـ Mean
+
+155
+00:15:28,290 --> 00:15:33,290
+Value Theorem لان في البداية على أي X وين في
+
+156
+00:15:33,290 --> 00:15:39,170
+الفترة A وB في الفترة A وB لو أخدنا X في ال A وB
+
+157
+00:15:39,170 --> 00:15:44,380
+بالـ Mean Value Theoremبين على الـ A و الـ X على
+
+158
+00:15:44,380 --> 00:15:47,920
+الفترة الـ A و الـ X there exists CX ما لها
+
+159
+00:15:47,920 --> 00:15:52,340
+between X and A such that F of X ناقص F of A بساوي
+
+160
+00:15:52,340 --> 00:15:57,240
+F prime C of X في X minus A هذه اللي هي تطبيق الـ
+
+161
+00:15:57,240 --> 00:16:02,860
+Mean Value Theorem على الفترة A و B and so و منه
+
+162
+00:16:02,860 --> 00:16:07,700
+اللي هي بنقول F prime C of X بساوي اللي هو F of X
+
+163
+00:16:07,700 --> 00:16:15,920
+ناقص F of A على X minus Aالان عندى
+
+164
+00:16:15,920 --> 00:16:22,520
+.. خلّينا نيجى ناخد هذه المنطقة اه عشان لسه بدات
+
+165
+00:16:22,520 --> 00:16:26,600
+بين F prime of A أشمالها موجودة بساوي limit ال F
+
+166
+00:16:26,600 --> 00:16:31,180
+prime CX لما X تروح لمين؟ لل A، ماشي الحال، هذه
+
+167
+00:16:31,180 --> 00:16:36,610
+الآن لو exist بتكون F prime of A أشمالها existالان
+
+168
+00:16:36,610 --> 00:16:40,510
+لاحظ احنا طبقنا اللى هى ال mean value theorem على
+
+169
+00:16:40,510 --> 00:16:46,910
+مين على الفترة من a لعند مين لعند x لجينا ال cx
+
+170
+00:16:46,910 --> 00:16:53,290
+وين موجودة بين ال a و ال x الان لما cx تروح لل a
+
+171
+00:16:53,290 --> 00:17:00,050
+اكيد ال x هتروح لمين لل a ماشي الحال طيب الان
+
+172
+00:17:00,050 --> 00:17:05,700
+بيصير عندىهو ماعطيني أصلا limit f prime of x لما x
+
+173
+00:17:05,700 --> 00:17:10,600
+تروح للإيه exist ماشي فبصير عندي لأن limit f prime
+
+174
+00:17:11,520 --> 00:17:17,760
+of CX لما ال X تروح لل A هي نفسها as X goes to A
+
+175
+00:17:17,760 --> 00:17:21,700
+CX وين هتروح لما X تروح لل A اتوماتيك CX هتروح لل
+
+176
+00:17:21,700 --> 00:17:26,480
+A فبصير عندى لقى limit F prime of CX لما X تروح لل
+
+177
+00:17:26,480 --> 00:17:29,880
+A هو نفس limit F prime of CX لما A CX تروح لل A
+
+178
+00:17:29,880 --> 00:17:34,580
+وهذا هو ماعطينى ايش اسمه ان existence هو ايه، اذا
+
+179
+00:17:34,580 --> 00:17:39,500
+صار هذا ال limitexist يعني هذا اللي هو اللي مساوي
+
+180
+00:17:39,500 --> 00:17:42,620
+لل limit هذا اللي بنين عليه إذا حيكون ال F prime
+
+181
+00:17:42,620 --> 00:17:46,920
+of A exist و برضه حيساوي مين؟ حيساوي إيه؟ اطلع على
+
+182
+00:17:46,920 --> 00:17:53,100
+لدك اللي بالك نخلص نعم لا اطلع على اللي بعده أيوة
+
+183
+00:17:53,100 --> 00:17:55,420
+السؤال اللي بعده
+
+184
+00:18:06,430 --> 00:18:13,030
+كبر هذا السؤال اللي سألتني عنه يا محمد
+
+185
+00:18:13,030 --> 00:18:18,450
+قبل هيك اطلعلي .. خليه بس اطلعلي على الكتاب على
+
+186
+00:18:18,450 --> 00:18:28,770
+628 النظرية 628 انزل انزل 628 اطلعلي عليها النظرية
+
+187
+00:18:28,770 --> 00:18:36,630
+عشان نقولك ايش هو السؤال علىلأنه مهم نعرف عن إيش،
+
+188
+00:18:36,630 --> 00:18:40,410
+الآن إذا بتتذكروا أخدنا اللي هو ال first
+
+189
+00:18:40,410 --> 00:18:45,390
+derivative test for extrema بتقول إذا كان لجينا
+
+190
+00:18:45,390 --> 00:18:47,870
+neighborhood Hannah subset من I such that F double
+
+191
+00:18:47,870 --> 00:18:51,950
+prime أكبر ساعة و سفر و X ال .. ال .. ال .. لو F
+
+192
+00:18:51,950 --> 00:18:54,350
+double prime أكبر ساعة و سفر مرة ع اليمين و مرة ع
+
+193
+00:18:54,350 --> 00:18:59,350
+اليسار إذا F has إيش مالها relative إيش مالها
+
+194
+00:18:59,350 --> 00:19:05,280
+maximum اللي هو اللي بتغير شرطهامن موجب إلى سالب
+
+195
+00:19:05,280 --> 00:19:09,260
+فبتكون عندي relative maximum، الآن هل العكس صحيح؟
+
+196
+00:19:09,260 --> 00:19:12,440
+يعني لو كان في عندنا relative maximum، هل شرط إنها
+
+197
+00:19:12,440 --> 00:19:17,950
+تغير إشارتها في اللي دا؟أطلع لفوق شوية عشان أورجيك
+
+198
+00:19:17,950 --> 00:19:21,170
+السؤال وين كان موجود هان remark the converse of
+
+199
+00:19:21,170 --> 00:19:25,410
+the first derivative test is not true مهم الكلام
+
+200
+00:19:25,410 --> 00:19:28,610
+هذا for example there exists a differentiable
+
+201
+00:19:28,610 --> 00:19:31,610
+function f من R لR with absolute minimum at x
+
+202
+00:19:31,610 --> 00:19:35,210
+بالساوية سفر but such that f prime takes on both
+
+203
+00:19:35,210 --> 00:19:39,110
+positive and negative values on both sides of اللي
+
+204
+00:19:39,110 --> 00:19:45,000
+هي x بتساوي عياش بساوية سفرماشي الحال إذا هذا الآن
+
+205
+00:19:45,000 --> 00:19:49,920
+هذا الحديث هو سؤالنا اللي عندنا اللي بدنا نحكي فيه
+
+206
+00:19:49,920 --> 00:19:52,880
+اللي هو exercise قداش تسعة ارجعلي على exercise
+
+207
+00:19:52,880 --> 00:19:56,560
+تسعة إذا ال exercise تسعة هو عبارة عن إيش يقول ليه
+
+208
+00:19:56,560 --> 00:20:00,280
+ان ال converts of this theorem need not to be true
+
+209
+00:20:00,280 --> 00:20:04,930
+in generalبالظبط ايش بيقول؟ بيقول لت F من R ل R بي
+
+210
+00:20:04,930 --> 00:20:08,910
+define by F of X بيساوة 2 X plus 4 زاد X plus 4
+
+211
+00:20:08,910 --> 00:20:12,770
+Sine 1 علي X For X لا تساوي سفر عند F of 0 ايش
+
+212
+00:20:12,770 --> 00:20:16,450
+بيساوي Zero إذا أنا معرفت دالة ال F بهذه الطريقة
+
+213
+00:20:16,450 --> 00:20:20,450
+بهذه لما X لا تساوي سفر وعند X بيساوي سفر عرفها F
+
+214
+00:20:20,450 --> 00:20:25,120
+of 0 بيساوي أيش؟ Zeroبقول لشهدات أول إشي F has an
+
+215
+00:20:25,120 --> 00:20:30,440
+absolute minimum when عند الـ 0 but that its
+
+216
+00:20:30,440 --> 00:20:34,820
+derivative has both positive and negative values
+
+217
+00:20:34,820 --> 00:20:40,280
+everywhere اللي هو إيش في neighborhood حوالين مين؟
+
+218
+00:20:40,280 --> 00:20:49,380
+حوالين الصفر واضحطيب، نشوف الآن، عملية فيه
+
+219
+00:20:49,380 --> 00:20:54,280
+absolute minimum مش صعبة، اللي هي بس خلينا نتطلع
+
+220
+00:20:54,280 --> 00:20:58,140
+على الحسابات، لإن الحسابات بتاخد وجه، فخلينا نتطلع
+
+221
+00:20:58,140 --> 00:21:02,900
+على اللي عندنا محسوبة وخالصلأن لأي x الينتن ار
+
+222
+00:21:02,900 --> 00:21:07,360
+اكيد ال x الاربعة ايه شماله اكبر ويساوي سفر وال
+
+223
+00:21:07,360 --> 00:21:11,900
+sign الواحد على x اكبير اكيد اكبر ويساوي مين ناقص
+
+224
+00:21:11,900 --> 00:21:16,300
+واحد اضرب الجهتين في x الاربعة فبصير x الاربعة في
+
+225
+00:21:16,300 --> 00:21:20,680
+هذا اكبر ويساوي ناقص ايش x الاربعة اضيف للجهتين
+
+226
+00:21:20,680 --> 00:21:26,110
+اتنين x الاربعةفبصير 2x أس 4 زائد هذا أكبر أو
+
+227
+00:21:26,110 --> 00:21:30,470
+يساوي اللي ضفت 2x أس 4 زائد اللي هو ياش ناقص اللي
+
+228
+00:21:30,470 --> 00:21:34,930
+هي x أس 4 اللي هو بيطلع جداش x أس 4 اللي هو أكيد
+
+229
+00:21:34,930 --> 00:21:39,270
+أكبر أو يساوي 0 صارت عند قيمة ال function f of x
+
+230
+00:21:39,270 --> 00:21:44,770
+اللي احنا بنحكي عنها دايما أكبر أو يساوي 0 اللي هو
+
+231
+00:21:44,770 --> 00:21:49,650
+مين بيساوي اللي عرفناه F0 إذا صار في عندي F has
+
+232
+00:21:49,650 --> 00:21:58,450
+absoluteminimum at mean at zero لكن عندي لطبيعة
+
+233
+00:21:58,450 --> 00:22:02,590
+الـ sine و طبيعة ال cosine لو جيت الآن أخدت أي
+
+234
+00:22:02,590 --> 00:22:06,990
+neighborhood حوالين ناقص delta و delta بدي أثبتلك
+
+235
+00:22:06,990 --> 00:22:12,410
+أن F prime مرة ممكن تسويلي أصغر من سفر و مرة تكون
+
+236
+00:22:12,410 --> 00:22:21,190
+أشمالها أكبر من سفر واضح؟ إذا تعالى شوفعندي خد لأي
+
+237
+00:22:21,190 --> 00:22:25,370
+neighborhood حوالين اللي هو الـ Zero خده من ناقص
+
+238
+00:22:25,370 --> 00:22:29,850
+Delta وDelta لأي Delta في الدنيا أو لأي أمثلون في
+
+239
+00:22:29,850 --> 00:22:34,450
+الدنيا عندي هي ال neighborhood اللي بحكي فيه
+
+240
+00:22:40,490 --> 00:22:45,390
+النقطة الداخلية اللي احنا مستهد فينا اللي هي الصفر
+
+241
+00:22:45,390 --> 00:22:51,500
+خد أي neighbor حواليه ناقص دلتا او ديلتابقدر ألادي
+
+242
+00:22:51,500 --> 00:22:58,220
+n اكبر او يساوي اتنين very large اللي
+
+243
+00:22:58,220 --> 00:23:04,540
+بيكون ان اشمالهن very close to zero ماشي يعني مهما
+
+244
+00:23:04,540 --> 00:23:09,300
+زغرتلي ال delta براجيلك n كبيرة كفاية انها تضلها
+
+245
+00:23:09,300 --> 00:23:18,320
+في هذا الجوار وتحقق ما يليه ايش أخدتها اخدت النقطة
+
+246
+00:23:18,320 --> 00:23:22,460
+واحد على اتنين and byطبعا الان هذا بزغرها جد ما
+
+247
+00:23:22,460 --> 00:23:28,120
+بده بتكبير الان و نفس الاشي 2 على 4 ان زاد واحد في
+
+248
+00:23:28,120 --> 00:23:30,780
+باقي طبعا ليش أخدت هيك عشان واحدة تخليلي ال sign
+
+249
+00:23:30,780 --> 00:23:34,620
+صفر و واحدة تخليلي ال cosine إيه عشان صفر واضح و
+
+250
+00:23:34,620 --> 00:23:37,580
+في نفس الوجه بتخليلي ال sign واحد و ال cosine واحد
+
+251
+00:23:37,580 --> 00:23:44,380
+بشكل معاكس ده فنشوف أشي اللي بقولهالان هذه الان و
+
+252
+00:23:44,380 --> 00:23:47,380
+هذه الان اخترت الان اللي تخليني إياها موجودة في
+
+253
+00:23:47,380 --> 00:23:51,060
+نقص delta و delta هذولا النقطين في وين في الجوار
+
+254
+00:23:51,060 --> 00:23:56,880
+اللي اعطيتني إياه أي جوار بتعطيني يا بدأ جيلك الان
+
+255
+00:23:56,880 --> 00:24:01,300
+المناسبة إليه احسبلي الان F prime F prime of X
+
+256
+00:24:01,300 --> 00:24:04,040
+بتعرف .. نعرف نحسبها خلينا نحسب F prime of X على
+
+257
+00:24:04,040 --> 00:24:08,580
+جهها لإنه مش هحسبها عشان تكون قدامكم F prime of X
+
+258
+00:24:10,740 --> 00:24:18,560
+أف برايم of x ايش بتساوي تمانية x تكعيب زائد أربعة
+
+259
+00:24:18,560 --> 00:24:28,300
+x تكعيب sign واحد على x ناقص اللي هو ليش
+
+260
+00:24:28,300 --> 00:24:34,800
+زائد ناقص x أصبح أربعة بيصير x تربيع sign واحد على
+
+261
+00:24:34,800 --> 00:24:41,650
+x صحيح؟هذه اليومين f prime of x بدّي الآن نعوض على
+
+262
+00:24:41,650 --> 00:24:47,310
+f عند النقطة أول إشي النقطة الأولى عبارة عن واحد
+
+263
+00:24:47,310 --> 00:24:52,450
+على اتنين and by عوضنا عنها في تمانية x سكعيب هي
+
+264
+00:24:52,450 --> 00:24:57,210
+تمانية في x سكعيب زاد أربعة هي x سكعيب وهي sin
+
+265
+00:24:57,210 --> 00:25:01,820
+واحد على x بيصير sin اتنين and byعشان هيك الاختيار
+
+266
+00:25:01,820 --> 00:25:05,020
+زائد واحد على اتنين and by أُس أربع اللي هي ..
+
+267
+00:25:05,020 --> 00:25:11,580
+اللي هي .. بحها دي بيصير أُس اتنين أه و هاي الناقص
+
+268
+00:25:11,580 --> 00:25:16,900
+فاهمين؟ X تربيع ها دي محادي اللي هي بيصير X تربيع
+
+269
+00:25:16,900 --> 00:25:23,800
+بالسالبواضح هه في cosine من اللي هو 2 unbi الان
+
+270
+00:25:23,800 --> 00:25:27,580
+هذا المقدار اللي في النص كله ايش حباله هيصير سفر
+
+271
+00:25:27,580 --> 00:25:32,160
+لأنه sin 2 unbi سفر هذا المقدار عبارة عن اللي هو
+
+272
+00:25:32,160 --> 00:25:34,860
+تمانية بتروح مع تمانية اللي هم يصير واحد على unbi
+
+273
+00:25:34,860 --> 00:25:39,880
+كل تكعيب هذا ال cosine ايش بيساوي واحد اذا بيظل من
+
+274
+00:25:39,880 --> 00:25:47,480
+عنده اللي هو هذا سالب تربيع ايه الانواخد 1 على
+
+275
+00:25:47,480 --> 00:25:52,120
+unbi عام المشترك تربيع بيظل عندى اللى هنا عبارة عن
+
+276
+00:25:52,120 --> 00:25:59,560
+اللى هى 4 unbi لأنه ماخد هنا اللى هى 1 على 2 unbi
+
+277
+00:25:59,560 --> 00:26:03,720
+لكل تربيع اللى هو عام المشترك بيظل 4 على unbi ناقص
+
+278
+00:26:03,720 --> 00:26:08,840
+ايه اش واحدواضح الاربع على n by لل n أكبر أو يساوى
+
+279
+00:26:08,840 --> 00:26:14,660
+2 أن هذا المقدار بصير يصغر لدرجة أنه أصغر من
+
+280
+00:26:14,660 --> 00:26:18,320
+الواحد لل n أكبر يساوى 2 إذا صار هذا المقدار عبارة
+
+281
+00:26:18,320 --> 00:26:22,460
+عن سالب في موجة بيطلع أصغر من 0 إذا F براين طلعت
+
+282
+00:26:22,460 --> 00:26:29,390
+أصغر من 0في نفس الوقت أف برايم ل 2 على كذا على 4n
+
+283
+00:26:29,390 --> 00:26:34,130
+زائد 1 في πاي أضربها أحسبها بيصير تمانية وعوض وعوض
+
+284
+00:26:34,130 --> 00:26:38,150
+وعوض الان مش ال sign اللي هتت cancel هتت cancel
+
+285
+00:26:38,150 --> 00:26:43,110
+مين ال cosine فهيطلع عندى هذا المقدار هيه وهيه هذا
+
+286
+00:26:43,110 --> 00:26:48,390
+المقدارالتنين المقدارين موجبات، إذا هيكون أكبر من
+
+287
+00:26:48,390 --> 00:26:53,510
+مين؟ من سفر، إذا في كل المنطقة، في كل المنطقة،
+
+288
+00:26:53,510 --> 00:26:58,050
+هتلاقي اللي هو جانب بعض، جانب بعض، مرة أكبر من
+
+289
+00:26:58,050 --> 00:27:00,530
+سفر، مرة أصغر من سفر، مرة أكبر من سفر، مرة أصغر من
+
+290
+00:27:00,530 --> 00:27:05,070
+سفر، ففيش عندك اللي هو تغير إشارتها من موجب إلى
+
+291
+00:27:05,070 --> 00:27:09,750
+سالب، لأ، هم جانب بعض، هذولة تنفج يا واحدة تنساشفي
+
+292
+00:27:09,750 --> 00:27:14,570
+جهة واحدة من الـ neighborhood مش على الجهتين هو ال
+
+293
+00:27:14,570 --> 00:27:18,270
+test اللي بنعرفه أشمل بيقولك لو اتغير من قبل ال ..
+
+294
+00:27:18,270 --> 00:27:21,690
+قبل .. في ال neighborhood قبل النقطة ال X note من
+
+295
+00:27:21,690 --> 00:27:26,990
+موجب إلى سالب بيصير maximum من سالب إلى موجب بيصير
+
+296
+00:27:26,990 --> 00:27:34,400
+موجب بيصير minimum الآنأحنا لجينا relative minimum
+
+297
+00:27:34,400 --> 00:27:38,800
+أو absolute minimum لكن في النقاط من ناقص ذلت
+
+298
+00:27:38,800 --> 00:27:44,180
+العين دلت هاين اختارتلك هنا وهنا جان بعض و جرب
+
+299
+00:27:44,180 --> 00:27:48,650
+كمان بلا جيلك كمان اللي هو بحيث انها تكون موجبةو
+
+300
+00:27:48,650 --> 00:27:53,050
+سالبة و موجبة و سالبة يعني فش derivative هنا تكون
+
+301
+00:27:53,050 --> 00:27:57,690
+موجبة كلها و هنا سالبة كلها عشان تحكم max .. تقول
+
+302
+00:27:57,690 --> 00:28:01,190
+اللي هو .. اللي هي ال .. و .. و .. و فقط اللي هي
+
+303
+00:28:01,190 --> 00:28:04,750
+ال converse in and the converse need not to be
+
+304
+00:28:04,750 --> 00:28:09,990
+true in general واضح؟ في الجوارين
+
+305
+00:28:09,990 --> 00:28:14,830
+.. هذا .. هذا من هنا .. من هنا لعين دلها فش إشارة
+
+306
+00:28:14,830 --> 00:28:19,040
+واحدة و من هنا لعين دلها فش إشارة واحدةإذا the
+
+307
+00:28:19,040 --> 00:28:22,320
+converse need not to be true in general واضح و
+
+308
+00:28:22,320 --> 00:28:30,400
+الله واضح و أعيد واضح يا محمد اه طيب اللي بعده
+
+309
+00:28:30,400 --> 00:28:34,120
+برضه في ال .. في ال .. في ال .. في المثال ارجعلي
+
+310
+00:28:34,120 --> 00:28:38,700
+للكتاب عشان أقولك هذا برضه بخدمين السؤال قبلها
+
+311
+00:28:38,700 --> 00:28:42,160
+انزل اطلع قبل النظرية قبل النظرية قبل النظرية في
+
+312
+00:28:42,160 --> 00:28:45,520
+عندي مثال هيو او لا .. ايو ال remark ال remark
+
+313
+00:28:45,520 --> 00:28:52,760
+أيوة اطلع ال remark هذهالان اذا بتتذكروا سؤال عشر
+
+314
+00:28:52,760 --> 00:28:55,260
+اللي بناحكيها الجد حكيها ان احنا قولنا it is
+
+315
+00:28:55,260 --> 00:28:59,420
+reasonable to define a function to be increasing
+
+316
+00:28:59,420 --> 00:29:04,170
+at a pointإذا كانت هناك مقارنة للمقارنة في
+
+317
+00:29:04,170 --> 00:29:13,750
+المقارنة اللي حوالها، إذا
+
+318
+00:29:13,750 --> 00:29:18,810
+كانت هناك مقارنة في المقارنة اللي حوالها، إذا كانت
+
+319
+00:29:18,810 --> 00:29:19,510
+هناك مقارنة في المقارنة اللي حوالها،
+
+320
+00:29:22,640 --> 00:29:27,460
+من الممكن أن يكون الـ derivative صحيح بشكل صحيح في
+
+321
+00:29:27,460 --> 00:29:35,840
+نقطة فالعمل يزيد في هذه النقطة ولكن هذا الوضع غير
+
+322
+00:29:35,840 --> 00:29:40,960
+صحيح يعني هذا الكلام إيش ماله ليس شرط انه صحيح انه
+
+323
+00:29:40,960 --> 00:29:44,500
+عندي اللي هنلاقي اللي هو الـ derivative صحيح بشكل
+
+324
+00:29:44,500 --> 00:29:49,220
+صحيح لكن مافيش .. ماقدرش نقول عنها إيش مالها is
+
+325
+00:29:49,220 --> 00:29:53,460
+increasing at this pointهذا في الـ Interval صحيح
+
+326
+00:29:53,460 --> 00:29:57,020
+لكن في الـ Point إيش ماله need not to be true in
+
+327
+00:29:57,020 --> 00:30:01,860
+general وهي مثال G of X هيه اللي هو بحيث انه G'
+
+328
+00:30:02,240 --> 00:30:07,940
+للـ 0 بساوي 1 لكن الـ G is not increasing in any
+
+329
+00:30:07,940 --> 00:30:14,260
+neighbourhood حوالين مين الصفر، هذا مثال على G' of
+
+330
+00:30:14,260 --> 00:30:20,200
+0 بساوي 1 اللي هو strictly أكبر من 0 butاللي هو في
+
+331
+00:30:20,200 --> 00:30:24,940
+هذا ال .. عند هذا النقطة اللي هو في الجوار تبعها
+
+332
+00:30:24,940 --> 00:30:30,260
+لا يمكن انها تكون اللي هي increasing في أي جوار
+
+333
+00:30:30,260 --> 00:30:35,610
+حواليهاهذا مثلنا يلّا خلّيني نحل السؤال الان هاي
+
+334
+00:30:35,610 --> 00:30:40,090
+سؤالنا let g من R لR be defined by g of X بيساوي X
+
+335
+00:30:40,090 --> 00:30:44,410
+زي 2 X ثانوية صين واحدة ل X for X تتساوي سفر and g
+
+336
+00:30:44,410 --> 00:30:47,550
+of 0 بيساوي 0 show that g prime of 0 ايش بيساوي
+
+337
+00:30:47,550 --> 00:30:53,030
+واحد but in every neighborhoodof zero, the
+
+338
+00:30:53,030 --> 00:30:57,430
+derivative g prime takes on both positive and
+
+339
+00:30:57,430 --> 00:31:01,710
+negative values. Thus, g is not monotonic in any
+
+340
+00:31:01,710 --> 00:31:05,830
+neighborhood. هنلاقي في المنطقة هذه، وين ما كان،
+
+341
+00:31:05,830 --> 00:31:13,210
+هنلاقي الـ g prime اللي هيأكبر من سفر وأصغر من سفر
+
+342
+00:31:13,210 --> 00:31:18,410
+يعني مش هنلاقي في أي neighborhood أن الـ G' أكبر
+
+343
+00:31:18,410 --> 00:31:25,410
+من سفر لحالها أو الـ G' للـ X هي أكبر من حالها أو
+
+344
+00:31:25,410 --> 00:31:33,910
+أصغر من حالها في كل .. من سفر في كل الجوارG' مش
+
+345
+00:31:33,910 --> 00:31:38,930
+هتكون أصغر من سفر في كل الجوار ولا أكبر من سفر في
+
+346
+00:31:38,930 --> 00:31:46,630
+كل الجوار هنلاقي إلها متذبذبة في الإشارة كيف؟
+
+347
+00:31:48,510 --> 00:31:54,930
+الحل شبهه صحيح الحل شبهه بس هذا الان موظف لمين لا
+
+348
+00:31:54,930 --> 00:31:58,990
+اللي هي اللي هو counter example على اللي هي ال
+
+349
+00:31:58,990 --> 00:32:02,630
+remark اللي عندنا وده counter example على اللي هي
+
+350
+00:32:02,630 --> 00:32:08,010
+النظرية هذك بوضح ان ال theorem need not to be true
+
+351
+00:32:08,010 --> 00:32:12,170
+in general وهنا اللي هو بوضح انه اللي هو ال
+
+352
+00:32:12,170 --> 00:32:18,000
+strictly increasing عند نقطةلأ .. لأ .. لأ .. لأ
+
+353
+00:32:18,000 --> 00:32:18,380
+.. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ ..
+
+354
+00:32:18,380 --> 00:32:18,540
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+355
+00:32:18,540 --> 00:32:19,120
+.. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ ..
+
+356
+00:32:19,120 --> 00:32:19,800
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+357
+00:32:19,800 --> 00:32:26,540
+.. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+358
+00:32:26,540 --> 00:32:29,660
+..
+
+359
+00:32:29,660 --> 00:32:31,300
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+360
+00:32:31,300 --> 00:32:31,300
+.. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ ..
+
+361
+00:32:31,300 --> 00:32:32,100
+لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ .. لأ
+
+362
+00:32:32,100 --> 00:32:39,020
+.. لأ .. لأ
+
+363
+00:32:39,020 --> 00:32:46,020
+.. لأأيش بعمل قاعد بدي أحاول أثبت انه g prime of 0
+
+364
+00:32:46,020 --> 00:32:51,600
+اللي هو بيساوي واحد هاي التعريف هاي عنده y ساوي
+
+365
+00:32:51,600 --> 00:32:55,180
+جسمنا عليه بيصير واحد زاد اتنين limit x sin واحد
+
+366
+00:32:55,180 --> 00:33:00,380
+على x اللي هو هذا ايه شماله؟بساوة صفر لأن الـ
+
+367
+00:33:00,380 --> 00:33:03,720
+absolute value للـ X صين واحد على X أكبر بساوة صفر
+
+368
+00:33:03,720 --> 00:33:07,740
+و أصغر أو يساوي الـ absolute value للـ X اللي هو
+
+369
+00:33:07,740 --> 00:33:11,020
+by Sandwich theorem اللي هو ال limit هذا إيش بساوة
+
+370
+00:33:11,020 --> 00:33:15,040
+صفر إذا ال limit على بعض و كله إيش بساوة واحد إذا
+
+371
+00:33:15,040 --> 00:33:17,740
+D prime of zero بساوة واحد اطلع لفوق
+
+372
+00:33:20,460 --> 00:33:26,260
+عندي for x تتساوى 0 الـ g prime سهل أن انا اجدها
+
+373
+00:33:26,260 --> 00:33:30,300
+عبارة عن واحد زائد تفاضل قاعد اربع x sin واحد على
+
+374
+00:33:30,300 --> 00:33:34,080
+x ناقص اتنين cosine واحد على x لان بالظبط as above
+
+375
+00:33:34,080 --> 00:33:37,520
+for any neighborhood زي اللي حكينا قبل شوية for
+
+376
+00:33:37,520 --> 00:33:43,340
+any neighborhood ناقص دلتا ودلتا حول الصفر بقدر
+
+377
+00:33:43,340 --> 00:33:49,200
+الاقي اد اكبر سواء واحد بحيث انههذا وهذا يكون في
+
+378
+00:33:49,200 --> 00:33:54,160
+الجوار لكن جي برايم عند الأولى أصغر من سفر وجي
+
+379
+00:33:54,160 --> 00:33:58,480
+برايم عند الثانية أشمالها أكبر من سفر بحسابات
+
+380
+00:33:58,480 --> 00:34:01,580
+مشابهة أي سؤال؟
+
+381
+00:34:05,050 --> 00:34:09,570
+طيب نيجي لسؤال 12 سؤال 12 بيقول إذا كانت h of x
+
+382
+00:34:09,570 --> 00:34:13,590
+بيساوي 0 إذا كانت x أصغر من 0 و 1 إذا كانت x أكبر
+
+383
+00:34:13,590 --> 00:34:16,650
+بيساوي 0 prove that there does not exist a
+
+384
+00:34:16,650 --> 00:34:21,310
+function f من R لR such that f prime of x أشمالها
+
+385
+00:34:21,310 --> 00:34:26,190
+بيساوي h of x هذا حلنا زيه برضه اللي هو درابوكس
+
+386
+00:34:26,190 --> 00:34:29,270
+theorem using ايش درابوكس theorem ايش بنقول
+
+387
+00:34:29,270 --> 00:34:37,360
+suppose notمظبوط صبّق الان هي عند H of X هيهابقول
+
+388
+00:34:37,360 --> 00:34:42,920
+لأثبت أنه فيش ولا function F لو فضلناها بتطلع مين؟
+
+389
+00:34:42,920 --> 00:34:46,540
+H of X بدنا نفترض العكس، نفترض أنه في function
+
+390
+00:34:46,540 --> 00:34:51,260
+اسمها F بحيث أن F' إيش بتساوي H صارت H نفسها
+
+391
+00:34:51,260 --> 00:34:54,400
+differentiable، ماشي آسف، F إيش ما لا؟
+
+392
+00:34:54,400 --> 00:34:57,880
+differentiable، مظبوط؟ مدام F differentiable، إذا
+
+393
+00:34:57,880 --> 00:35:03,120
+by مين؟ By Daraboux's theorem اللي هو there existو
+
+394
+00:35:03,120 --> 00:35:07,640
+طبعا و احنا عارفين النص بين مين؟ بين الـ 0 و الـ 1
+
+395
+00:35:07,640 --> 00:35:12,880
+اذا by Daraboux's theorem there exist اللي هو ..
+
+396
+00:35:12,880 --> 00:35:19,840
+اللي هي C بين الـ X1 و X2 بحيث انه G of C بسوء أفه
+
+397
+00:35:19,840 --> 00:35:24,940
+prime of C و يسوى نص which is impossible أسرعت
+
+398
+00:35:24,940 --> 00:35:32,940
+عليكم، مظبوط؟ الآن يا جماعةبدي افترض ان العكس صحيح
+
+399
+00:35:32,940 --> 00:35:37,880
+يعني بدي افترض ان F من R ل R بحيث ان F prime of X
+
+400
+00:35:37,880 --> 00:35:42,740
+ايش بتساوي؟ X لكل X element in R ماشي الحال اتفجن
+
+401
+00:35:42,740 --> 00:35:47,840
+الهن اذا F نفسها بناء على هذا الحديث F is
+
+402
+00:35:47,840 --> 00:35:52,340
+differentiable على Rمن جهة أخرى أخرى لاحظ إن النص
+
+403
+00:35:52,340 --> 00:35:55,160
+بين الـ 0 والـ 1، مين هو الـ 0 والـ 1؟ الـ 0 والـ
+
+404
+00:35:55,160 --> 00:35:58,060
+1 هي قيم الـ function هذه اللي بحكي عنها، يعني الـ
+
+405
+00:35:58,060 --> 00:36:03,840
+0 والـ 1 هتكون الـ 0 عبارة عن أشوف X1 و الـ A1
+
+406
+00:36:03,840 --> 00:36:09,390
+أشوف X2، مين X1 و X2؟ اخترت الـ X1 أصغر من 0؟و X2
+
+407
+00:36:09,390 --> 00:36:13,690
+أكبر من 0 صار عند نقطين X1 و X2 واحد أصغر من 0
+
+408
+00:36:13,690 --> 00:36:18,030
+واحد أكبر من 0 يعني التمتين عملين لفترة إذا صار
+
+409
+00:36:18,030 --> 00:36:24,870
+عند نص في الفترة بين 0 و 1 اللي هو عبارة عن بين H
+
+410
+00:36:24,870 --> 00:36:31,520
+of X1 و H of X2 لكن H of X1 و H of X2 من هم؟F' of
+
+411
+00:36:31,520 --> 00:36:36,900
+X1 و F' of X2 إذا صارت اللي هي درابوكس theorem
+
+412
+00:36:36,900 --> 00:36:41,260
+محققة F is differentiable و نص تنتمي للفترة بيه
+
+413
+00:36:41,260 --> 00:36:47,340
+اللي هي F' of X1 و F' of X2 of X2 إذا حسب اللي هي
+
+414
+00:36:47,340 --> 00:36:51,340
+درابوكس theorem أي حاجة بينهم لازم يكون لها أصل
+
+415
+00:36:51,340 --> 00:36:57,190
+إذا there exists Cبين الـ x1 و الـ x2 بحيث أنه
+
+416
+00:36:57,190 --> 00:37:02,450
+اللي هو f prime للـ c هذه اللي اللي جاتها بين x1 و
+
+417
+00:37:02,450 --> 00:37:08,770
+x2 هي مين النص يعني زي بمسح كل القيم اللي بين f of
+
+418
+00:37:08,770 --> 00:37:13,080
+x1 و f of x2 مش هيك تدربت في تستيرون بتقولإذا صارت
+
+419
+00:37:13,080 --> 00:37:17,460
+عيدي في c بين هذه و هذه بحيث أن f prime of c بساوة
+
+420
+00:37:17,460 --> 00:37:21,480
+نص يعني g of c بساوة نص طب هذا مستحيل لأن اصلا g
+
+421
+00:37:21,480 --> 00:37:25,100
+.. h طبعا هذا مش g .. h of c .. هذا مستحيل ليش
+
+422
+00:37:25,100 --> 00:37:28,760
+مستحيل؟ لأن h أصلا ماتاخدش رقم تاني، يا سفر يا
+
+423
+00:37:28,760 --> 00:37:32,020
+واحد إذا contradiction، مدام contradiction إذا
+
+424
+00:37:32,020 --> 00:37:37,940
+there is no function f من R لR بحيث أن f prime of
+
+425
+00:37:37,940 --> 00:37:41,080
+x بساوة f of x for every x
+
+426
+00:37:45,230 --> 00:38:01,830
+في ضايق سؤالين خلنا نمر عليهم هذا
+
+427
+00:38:01,830 --> 00:38:06,130
+حلنا زيها اللي هو let I be an interval and let F
+
+428
+00:38:06,130 --> 00:38:11,110
+من I لR be differentiable on I مفترضين أن F عبارة
+
+429
+00:38:11,110 --> 00:38:16,000
+عن اللي هو differentiable function علىan interval
+
+430
+00:38:16,000 --> 00:38:19,820
+I وقول لي show that إذا كانت F prime is positive
+
+431
+00:38:19,820 --> 00:38:25,000
+on I لو F prime أكبر من 0 على I هتكون ال F أشمالها
+
+432
+00:38:25,000 --> 00:38:29,800
+strictly increasing on I طبعا اللي هو على السريع
+
+433
+00:38:29,800 --> 00:38:34,940
+لنفترض F prime أكبر من 0 لكل X element on I ماشي
+
+434
+00:38:34,940 --> 00:38:41,240
+هو نفسه؟ خلاص اللي بده السؤال موجود في الشريح اللي
+
+435
+00:38:41,240 --> 00:38:44,300
+هو ال main value theorem اتبكّل الله خليني أقوله
+
+436
+00:38:44,300 --> 00:38:50,490
+عنهالأن let I be an interval and let F من I لعند R
+
+437
+00:38:50,490 --> 00:38:55,050
+سواء الـ14 بيه differentiable on I show that if
+
+438
+00:38:55,050 --> 00:38:59,210
+the derivative F' is never zero on I يعني إذا كانت
+
+439
+00:38:59,210 --> 00:39:04,230
+F' أكبر من سفر اللي هي تسوى سفر on I then either
+
+440
+00:39:04,230 --> 00:39:08,430
+F' أكبر من سفر for all X limited on I أو F' أصغر
+
+441
+00:39:08,430 --> 00:39:13,480
+من سفر لكل XA جمالهايعني بيقولّي إذا كانت ال F
+
+442
+00:39:13,480 --> 00:39:17,760
+differentiable على ال interval I وكانت ال F'
+
+443
+00:39:18,160 --> 00:39:23,240
+بتسويش 0 في هذه الحالة ال F' يا هتكون كلها موجة
+
+444
+00:39:23,240 --> 00:39:28,940
+على ال I يا كلها سالب عالميا على ال I مدامت مغيرتش
+
+445
+00:39:28,940 --> 00:39:32,760
+شرطها بالمرة يعني بمعنى أخر يعني مش هتغير شرطها
+
+446
+00:39:32,760 --> 00:39:38,260
+بالمرة مدامت ال F' مش 0 على الفترة إذا إشارتها
+
+447
+00:39:38,260 --> 00:39:43,650
+واحدةيا F أكبر من 0، يا F أكبر من .. أصغر من 0،
+
+448
+00:39:43,650 --> 00:39:47,070
+هدا في ضوء أن F is differentiable، بدنا نفترض
+
+449
+00:39:47,070 --> 00:39:51,170
+العكس ونصل لcontradiction، suppose on the contrary
+
+450
+00:39:51,170 --> 00:39:56,290
+نفترض أن F prime of X أكبر من 0 لنقات .. لبعض
+
+451
+00:39:56,290 --> 00:40:00,390
+النقات في I، و F prime of X أكبر من .. أصغر من 0
+
+452
+00:40:00,390 --> 00:40:04,660
+لبعض النقات في I، يعني مخلوطةوبدّا نصلّمين ل
+
+453
+00:40:04,660 --> 00:40:08,640
+contradiction طيب then there exist a و b element
+
+454
+00:40:08,640 --> 00:40:12,220
+in I such that أف برايم of a أشماله أكبر من سفر و
+
+455
+00:40:12,220 --> 00:40:15,520
+أف برايم بي أصغر من سفر لإن أنا مفترض الآن افترضت
+
+456
+00:40:15,520 --> 00:40:18,200
+أنه في نقاط اللي هي أكبر من سفر عندها ال
+
+457
+00:40:18,200 --> 00:40:20,940
+derivative وفي نقاط أصغر من سفر عندها ال
+
+458
+00:40:20,940 --> 00:40:25,040
+derivative ماشي الحال إذا بنلاقي a و b بالحقق هذه
+
+459
+00:40:25,490 --> 00:40:28,710
+الان اكيد مدام هذا أكبر من صفر وهذا أصغر من صفر،
+
+460
+00:40:28,710 --> 00:40:32,910
+إذا الصفر بين الـF'A والـF'B لأنها واحدة موجة
+
+461
+00:40:32,910 --> 00:40:36,850
+بواحدة مين، مدام الصفر بينهم، إذا حسب Darabowski's
+
+462
+00:40:36,850 --> 00:40:40,990
+theorem، هي بتمسح كل المنطقة اللي بين الصور، لازم
+
+463
+00:40:40,990 --> 00:40:45,570
+كلها يكون اللي هي أصولإذن by Daraboux's theorem
+
+464
+00:40:45,570 --> 00:40:49,430
+there exists c element in I such that f prime of c
+
+465
+00:40:49,430 --> 00:40:52,970
+بيساوي سفر وهذا اللي هو بناقض الفرضية اللي احنا
+
+466
+00:40:52,970 --> 00:40:57,570
+فرضناها أن f prime لا تساوي سفر على كل ال I إذن f
+
+467
+00:40:57,570 --> 00:41:01,730
+prime لا تساوي سفر على كل ال I معناته و
+
+468
+00:41:01,730 --> 00:41:05,670
+differentiable طبعا معناته يا إما كلهن ال f prime
+
+469
+00:41:05,670 --> 00:41:11,950
+وجبات يا إما كلهن إيه شمالهن سالباتيعني يا
+
+470
+00:41:11,950 --> 00:41:15,710
+increasing strictly increasing يا strictly يا
+
+471
+00:41:15,710 --> 00:41:23,230
+شمالها decreasing مافيش أي تغيير للرسم والشعر طيب
+
+472
+00:41:23,230 --> 00:41:28,820
+نيجي لآخر السؤاللت I Be An Interval Prove That If
+
+473
+00:41:28,820 --> 00:41:32,980
+F Is Differentiable On I And The Derivative Of F'
+
+474
+00:41:33,300 --> 00:41:37,820
+Is Bounded On I، Then F Satisfies Lipschitz
+
+475
+00:41:37,820 --> 00:41:43,920
+Condition On I، ماشي الحل، بتقول لي، بتقوللي
+
+476
+00:41:43,920 --> 00:41:48,460
+السؤال ما يعني، Lipschitz Condition
+
+477
+00:41:53,580 --> 00:41:58,020
+خلّينا نشوف إيش اللي بنقوله إذا كانت F
+
+478
+00:41:58,020 --> 00:42:10,320
+differentiable on I و F' bounded on I اثبت إن F
+
+479
+00:42:10,320 --> 00:42:15,680
+satisfies Lipschitz conditions ماشي؟ إيش Lipschitz
+
+480
+00:42:15,680 --> 00:42:23,450
+condition؟ إنه there exists K أكبر من 0 such thatF
+
+481
+00:42:23,450 --> 00:42:30,130
+of X ناقص F of Y أصغر أو ساوي لبس اليود فيها ل X
+
+482
+00:42:30,130 --> 00:42:34,330
+minus Y في مين؟ في K اللي كنا نقول عنها على طول
+
+483
+00:42:34,330 --> 00:42:40,630
+هذي بتعطي إيش ما لها uniformly continuous لكل X و
+
+484
+00:42:40,630 --> 00:42:44,190
+Y لكل X و Y في الفترة اللي بنحكي عنها اللي بشت
+
+485
+00:42:44,190 --> 00:42:48,790
+condition إيش ما لها متحققةإذا الـ Absolute هذا
+
+486
+00:42:48,790 --> 00:42:54,670
+معناه اللي بش تسكندشه نيجي نحققها هذه نيجي نحقق
+
+487
+00:42:54,670 --> 00:42:55,710
+اللي هو
+
+488
+00:42:58,230 --> 00:43:02,930
+الـ Lipschitz condition لت x و y element in I بيه
+
+489
+00:43:02,930 --> 00:43:06,130
+such that x strictly أصغر من 100 من y
+
+490
+00:43:27,180 --> 00:43:30,960
+إذا منها ناخد ال absolute value للجهتين بتطلع عندي
+
+491
+00:43:30,960 --> 00:43:36,260
+اللي أماني واضح لكن F' is bounded مدان bounded F'
+
+492
+00:43:36,740 --> 00:43:41,940
+إذا there exist K بحيث أن F' of C أصغر سوى K لكل C
+
+493
+00:43:41,940 --> 00:43:47,550
+إيه شمالها من ضمن ال C اللي فوقأه لأن أف برايم is
+
+494
+00:43:47,550 --> 00:43:51,870
+bounded على كل ال I و هذه ال K بتنفع لكل مين لكل
+
+495
+00:43:51,870 --> 00:43:54,570
+ال C's اللي في ال I إذا صارت أف برايم في C أصغر أو
+
+496
+00:43:54,570 --> 00:43:58,270
+شويه K بنعود فوق بيصير أف of X ناقص أف of Y أصغر
+
+497
+00:43:58,270 --> 00:44:02,150
+أو شويه K في X minus Y و ال X و ال Y كانت نقش
+
+498
+00:44:02,150 --> 00:44:06,870
+مالهين arbitrary إذا صار هذه أصغر أو شويه K في X
+
+499
+00:44:06,870 --> 00:44:11,790
+minus Y ودخلت المتساوية لأن في حالة ال X بتساوي Y
+
+500
+00:44:11,790 --> 00:44:21,600
+اللي هو it is trivialtherefore I satisfy اللي هو
+
+501
+00:44:21,600 --> 00:44:25,820
+Lipschitz condition و الله يعطيكوا العافية
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ.srt
new file mode 100644
index 0000000000000000000000000000000000000000..abc902f555fe3c44c80e1744c2e9345062ade1cb
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ.srt
@@ -0,0 +1,1364 @@
+1
+00:00:04,910 --> 00:00:09,510
+بسم الله الرحمن الرحيم هذه المحاضرة الخامسة في
+
+2
+00:00:09,510 --> 00:00:15,630
+مساق تحليل حقيقي 2 لطلبة كلية العلوم تخصص 
+
+3
+00:00:15,630 --> 00:00:22,110
+رياضيات في الجامعة الإسلامية بغزة المحاضرة
+
+4
+00:00:22,110 --> 00:00:26,550
+اليوم هي جزئين الجزء الأول اللي هو عبارة عن
+
+5
+00:00:26,550 --> 00:00:31,350
+discussion لـ 6.1 اللي هو مناقشة لموضوع اللي هو
+
+6
+00:00:31,350 --> 00:00:37,410
+derivative أو الاشتقاق. الجزء الثاني هنكمل اللي هو
+
+7
+00:00:37,410 --> 00:00:43,210
+الحديث عن 6.2 اللي هو الـ Mean Value Theorem أو
+
+8
+00:00:43,210 --> 00:00:47,730
+نظرية القيمة المتوسطة. هناخذ بعض التطبيقات لنبدأ
+
+9
+00:00:47,730 --> 00:00:51,750
+الآن اللي هي بالأسئلة اللي احنا طلبناها منكم 
+
+10
+00:00:51,750 --> 00:00:57,230
+تحلوها كواجب المرة الماضية أو التي قبلها وكانت
+
+11
+00:00:57,230 --> 00:01:01,210
+الأسئلة هي عبارة عن السؤال الرابع والسؤال السابع 
+
+12
+00:01:02,000 --> 00:01:07,420
+والسؤال التاسع والسؤال الثالث عشر. بالنسبة للسؤال
+
+13
+00:01:07,420 --> 00:01:13,600
+الثالث اللي هو عبارة عن اللي هي برهان نظرية 6.1.3 A
+
+14
+00:01:13,600 --> 00:01:19,200
+وB وهذه البراهين براهين سهلة اللي كانت الـ F 
+
+15
+00:01:19,200 --> 00:01:22,960
+differentiable. الـ F differentiable والـ Alpha
+
+16
+00:01:22,960 --> 00:01:26,560
+عبارة عن constant بيعطينا الـ Alpha F برضه is
+
+17
+00:01:26,560 --> 00:01:30,200
+differentiable. ولو كانت الـ F والـ G 
+
+18
+00:01:30,200 --> 00:01:32,700
+differentiable بيعطينا الـ F زائد G is
+
+19
+00:01:32,700 --> 00:01:35,660
+differentiable. واحنا برهننا حالة الضرب وحالة 
+
+20
+00:01:35,660 --> 00:01:41,500
+القسمة، وهدول حالات تعتبر سهلة مباشرة على التعريف
+
+21
+00:01:41,500 --> 00:01:45,900
+لذلك هنبدأ إن شاء الله في الحديث أو في حل الأسئلة
+
+22
+00:01:45,900 --> 00:01:52,540
+على السؤال الرابع اللي هو بقول لي عندي معطيني F
+
+23
+00:01:52,540 --> 00:02:01,760
+من R لـ R بي defined by F of X بيساوي اللي هو X
+
+24
+00:02:01,760 --> 00:02:06,420
+تربيع إذا كانت X rational وبساوي صفر إذا كانت X 
+
+25
+00:02:06,420 --> 00:02:09,700
+irrational. اثبت إن الـ F الـ differentiable بالقدر X
+
+26
+00:02:09,700 --> 00:02:15,550
+بتساوي صفر أو جد اللي هي F prime of 0. لاحظ أن الـ 
+
+27
+00:02:15,550 --> 00:02:16,610
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ 
+
+28
+00:02:16,610 --> 00:02:17,090
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+29
+00:02:17,090 --> 00:02:17,910
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+30
+00:02:17,910 --> 00:02:20,450
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+31
+00:02:20,450 --> 00:02:27,990
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+32
+00:02:27,990 --> 00:02:28,350
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+33
+00:02:28,350 --> 00:02:29,630
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+34
+00:02:29,630 --> 00:02:29,690
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+35
+00:02:29,690 --> 00:02:41,470
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ 
+
+36
+00:02:41,470 --> 00:02:52,150
+Find this value. الآن عشان نوجد 
+
+37
+00:02:52,150 --> 00:02:56,730
+اللي هي نثبت أن F prime of 0 موجودة. خلّيني ألاحظ
+
+38
+00:02:56,730 --> 00:03:00,950
+مالي عند الـ function اللي هي X تربيع لما X 
+
+39
+00:03:00,950 --> 00:03:07,150
+rational وصفر لما X is irrational. الآن بدنا
+
+40
+00:03:07,150 --> 00:03:11,770
+نتوقع أول شيء اللي هو لانه هتلزمني جاي اللي بعدين
+
+41
+00:03:11,770 --> 00:03:15,890
+find this value. بدنا نتوقع ايش قيمة اللي هو الـ
+
+42
+00:03:15,890 --> 00:03:20,950
+derivative عند الصفر. لاحظ أنه اللي هو .. اللي ..
+
+43
+00:03:20,950 --> 00:03:25,630
+اللي لو بدنا نقول إنه الـ derivative ممكن تكون صفر
+
+44
+00:03:25,630 --> 00:03:30,510
+كون إن f of x صفر أو لو بدها تكون والـ
+
+45
+00:03:30,510 --> 00:03:36,560
+derivative هنا لو قولنا f of 2x بدو يكون اللي هو لو
+
+46
+00:03:36,560 --> 00:03:40,820
+بدو يكون اللي عند الـ zero اللي هي F prime بدو في
+
+47
+00:03:40,820 --> 00:03:47,220
+النهاية تروح لـ اللي هيكون برضه قريبة من إنه نقول
+
+48
+00:03:47,220 --> 00:03:52,580
+أو نأكد إنها صفر عشان هي كالظن الغالب إن F prime
+
+49
+00:03:52,580 --> 00:03:58,660
+هتكون ايش صفر. هذه مجرد تفكيرات. الآن بدي أثبت لك إنه
+
+50
+00:03:58,660 --> 00:04:03,080
+فعلًا هي الـ derivative بتساوي 0. كيف بدي أثبتها؟ 
+
+51
+00:04:03,080 --> 00:04:11,280
+بدي أثبت لك إنه الـ limit لـ F of X ناقص F of 0 على 
+
+52
+00:04:11,280 --> 00:04:17,600
+X minus 0 لما X تروح لـ 0 بتساوي 0. بدي أثبت لك
+
+53
+00:04:17,600 --> 00:04:24,240
+هيها. الآن واضح إن X تقول إلى الصفر. X تقول إلى الصفر 
+
+54
+00:04:24,240 --> 00:04:29,200
+هتمر بالـ rational والـ rational. عشان هيك صعب إن أنا
+
+55
+00:04:29,200 --> 00:04:32,600
+أتحدث عن اللي هو إيجاد الـ derivative مباشرة من
+
+56
+00:04:32,600 --> 00:04:36,130
+الآن أو من الـ two branches اللي عندي. لا بقدر أخد من
+
+57
+00:04:36,130 --> 00:04:40,030
+اليمين ولا أخد من اليسار لإنه عندي من اليمين أو من 
+
+58
+00:04:40,030 --> 00:04:44,130
+اليسار هيكون عندي اللي هو قابلت اللي هي الأعداد الـ
+
+59
+00:04:44,130 --> 00:04:48,110
+rational للـ rational. فعشانك أسلم شيء إنه نستخدم
+
+60
+00:04:48,110 --> 00:04:52,650
+التعريف في إثبات هذا. يعني الآن بدي أثبت هذا الكلام
+
+61
+00:04:53,270 --> 00:04:58,090
+كيف بدي أثبته؟ بدي أثبت ما يعني بدي أصل لكل إبسلون
+
+62
+00:04:58,090 --> 00:05:03,190
+أكبر من 0 بدلاج دلتا أكبر من 0 بدلاجية حاجيلك دلتا
+
+63
+00:05:03,190 --> 00:05:08,350
+بحيث إنه لما يكون الـ absolute value لـ F of X ناقص
+
+64
+00:05:08,350 --> 00:05:14,630
+F of Zero على X minus Zero يكون ناقص صفر طبعًا
+
+65
+00:05:14,630 --> 00:05:19,790
+صفر هذي أصغر من إبسلون. هذا متى؟ whenever
+
+66
+00:05:22,670 --> 00:05:28,470
+x-0 أكبر من 0 وأصغر من مين؟ من دلتا. هذا اللي بده 
+
+67
+00:05:28,470 --> 00:05:33,150
+يُثبتُه أو هذا اللي بده أثبته. خلّينا نشوف إذا خلّينا
+
+68
+00:05:33,150 --> 00:05:36,530
+نشوف كيف بدنا نوجده عشان أثبت إن الـ limit هذا
+
+69
+00:05:36,530 --> 00:05:43,110
+بيساوي 0. لاحظ القيمة اللي عندي أول شيء الـ absolute 
+
+70
+00:05:43,110 --> 00:05:50,530
+value of f of x ناقص f of 0 على x-0 ايش هتساوي؟
+
+71
+00:05:51,590 --> 00:05:57,390
+هتساوي اللي هو الـ absolute value of f of X عندي يا X
+
+72
+00:05:57,390 --> 00:06:01,070
+تربيع يمين يا صفر. خلّينها زي ما هي أول شيء f of X
+
+73
+00:06:01,070 --> 00:06:06,010
+ناقص f of Zero اللي هو جداش بيساوي صفر لإنه f 
+
+74
+00:06:06,010 --> 00:06:10,130
+of Zero بيساوي صفر لإنه Zero rational على إذن هذا 
+
+75
+00:06:10,130 --> 00:06:16,230
+صفر على X ناقص صفر اللي هو X. absolute value هذا 
+
+76
+00:06:16,230 --> 00:06:22,790
+القيمة الآن لاحظ بتساوي يا إما اللي هو x تربيع على 
+
+77
+00:06:22,790 --> 00:06:30,750
+x absolute value في حالة x is rational أو بتساوي 
+
+78
+00:06:30,750 --> 00:06:37,450
+اللي هو صفر في حالة x أشمالها is irrational لأن
+
+79
+00:06:37,450 --> 00:06:42,030
+مقيمة f of x يا x تربيع يا صفر حسب اللي هو كتبناه
+
+80
+00:06:42,030 --> 00:06:46,790
+حاليًا. الآن هذا بالظبط هو عبارة عن الـ absolute
+
+81
+00:06:46,790 --> 00:06:57,590
+value للـ X if X is rational. Zero if X is
+
+82
+00:06:57,590 --> 00:07:03,950
+irrational. الآن الصورة وضحت. خلّينا نسمي هذا اللي 
+
+83
+00:07:03,950 --> 00:07:09,900
+هو واحد. الآن حضرت عشان أصل للنهاية اللي هاني كاتبها
+
+84
+00:07:09,900 --> 00:07:13,800
+هنا وأشوف ايش الـ delta اللي بتطلع عندي. الآن بدعي
+
+85
+00:07:13,800 --> 00:07:17,380
+مالي for every epsilon أكبر من صفر أنا بقول there
+
+86
+00:07:17,380 --> 00:07:21,800
+exists delta هتساوي من الـ epsilon. هتجد تشوفوا ليش 
+
+87
+00:07:21,800 --> 00:07:26,980
+there exists delta بساوي epsilon such that if x
+
+88
+00:07:26,980 --> 00:07:35,210
+minus 0 أكبر من صفر أصغر من delta. then هذا معناته
+
+89
+00:07:35,210 --> 00:07:37,870
+ايه أن ابسلوت فيها الـ X أكبر من صفر وأصغر من
+
+90
+00:07:37,870 --> 00:07:41,870
+مين؟ من دلتا. إذا اخترت دلتا ايش بتساوي؟ وهي اللي
+
+91
+00:07:41,870 --> 00:07:47,850
+هتخلص من الموضوع. then اللي هو F of X نقص F of Zero
+
+92
+00:07:47,850 --> 00:07:50,970
+على 
+
+93
+00:07:50,970 --> 00:07:57,790
+X minus Zero هو طبعًا نقص الصفر اللي قلنا عنها الـ
+
+94
+00:07:57,790 --> 00:08:01,520
+derivative المتوقع عليا. هذه بالظبط اللي فوق هذا هو
+
+95
+00:08:01,520 --> 00:08:05,920
+اللي فوق طلع ايش عندي؟ هذا بيساوي هذا المقدار إذا
+
+96
+00:08:05,920 --> 00:08:09,760
+كانت X rational وصفر إذا كانت X irrational. يعني
+
+97
+00:08:09,760 --> 00:08:15,400
+بمعنى آخر بيساوي absolute value لـ X if X is 
+
+98
+00:08:15,400 --> 00:08:23,530
+rational وصفر if X is irrational. in both cases اللي
+
+99
+00:08:23,530 --> 00:08:27,590
+هي إذا كان بيساوي absolute value للـ X هيكون أصغر
+
+100
+00:08:27,590 --> 00:08:31,250
+من Delta اللي أنا اخترتها أشمالها Epsilon هيكون
+
+101
+00:08:31,250 --> 00:08:34,570
+أصغر من Epsilon. وأيضًا هذه متحققة لإن الـ Epsilon
+
+102
+00:08:34,570 --> 00:08:38,850
+دائمًا أشمالها أكبر من 0. إذا اللي حصلته إنه لكل 
+
+103
+00:08:38,850 --> 00:08:42,490
+Epsilon أكبر من 0 اللي جيت Delta لما يكون هذا أصغر
+
+104
+00:08:42,490 --> 00:08:47,130
+من Delta بيعطيني هذا أصغر من Epsilon وهذا يعني 
+
+105
+00:08:47,130 --> 00:08:54,370
+hence limit هذا المقدار اللي هو f of x ناقص f of 0
+
+106
+00:08:54,370 --> 00:09:01,690
+على x minus 0 لما x تروح للصفر بساوية اللي هو
+
+107
+00:09:01,690 --> 00:09:12,870
+الصفر وهذا هو تعريف من الـ f prime عند 0. that is f 
+
+108
+00:09:12,870 --> 00:09:18,740
+prime at 0 بساوية 0. وفي نفس الواجب طبعًا أثبتنا الـ
+
+109
+00:09:18,740 --> 00:09:25,960
+existence للـ F prime عند الـ zero. أي سؤال؟ طيب
+
+110
+00:09:25,960 --> 00:09:30,840
+نيجي الآن نشوف السؤال الثاني خلينا السؤال الرابع 
+
+111
+00:09:30,840 --> 00:09:44,220
+نيجي لسؤال سبعة الآن
+
+112
+00:09:44,220 --> 00:09:48,100
+سؤال سبعة ايش اللي بيقوله؟ سؤال سبعة اللي بيقوله سؤال
+
+113
+00:09:48,100 --> 00:09:52,420
+سبعة مالي عندي
+
+114
+00:09:52,420 --> 00:09:55,740
+suppose
+
+115
+00:09:55,740 --> 00:09:59,320
+that F من R لـ R is differentiable at C يعني نفترض 
+
+116
+00:09:59,320 --> 00:10:04,520
+أنه F قبل الاشتقاق عند C. ونفترض ان f of c قيمة الـ
+
+117
+00:10:04,520 --> 00:10:07,920
+function عند c بساوية 0. لأن بقول لشهود الـ absolute
+
+118
+00:10:07,920 --> 00:10:10,960
+value لل f of x اللي هي نسميها d of x is 
+
+119
+00:10:10,960 --> 00:10:14,080
+differentiable at c if and only if f شمالها f
+
+120
+00:10:14,080 --> 00:10:21,740
+prime of c بتساوي 0. إذا ناخد لك f من R لعند R و
+
+121
+00:10:21,740 --> 00:10:30,810
+جايلك أن f prime عند c exist معطيكي إياها أو ومعطيك
+
+122
+00:10:30,810 --> 00:10:37,390
+اللي هو f of c بتساوي صفر وبقول لي prove that أنه 
+
+123
+00:10:37,390 --> 00:10:42,370
+g of x بيساوي الـ absolute value لل f of x is
+
+124
+00:10:42,370 --> 00:10:49,810
+differentiable at c if and only if f prime عند الـ
+
+125
+00:10:49,810 --> 00:10:57,880
+c ايش بتساوي؟ بساوي صفر. مظبوط؟ طيب شوف الآن خلينا
+
+126
+00:10:57,880 --> 00:11:03,380
+افترض أول شيء إن f prime عند c ايش بتساوي؟ صفر. نقول
+
+127
+00:11:03,380 --> 00:11:09,320
+suppose proof suppose
+
+128
+00:11:09,320 --> 00:11:15,580
+suppose 
+
+129
+00:11:15,580 --> 00:11:23,120
+that f prime at c بتساوي صفر. ايش هذا بيعني؟ إن then
+
+130
+00:11:23,120 --> 00:11:25,940
+limit
+
+131
+00:11:27,220 --> 00:11:36,680
+F of X ناقص F of C على X minus C لما X تروح للـ C 
+
+132
+00:11:36,680 --> 00:11:40,240
+اللي هو بيساوي صفر لأن هذا تعريف مين؟ F prime 
+
+133
+00:11:40,240 --> 00:11:46,000
+بيساوي صفر وF of C ايش معطيني هو؟ بيساوي صفر لأن 
+
+134
+00:11:46,000 --> 00:11:53,260
+بيساوي limit F of X على X minus C لما X تروح لمين؟ 
+
+135
+00:11:53,260 --> 00:11:58,730
+للـ C. مدامة الـ limit هذه موجودة إذا الـ limit من
+
+136
+00:11:58,730 --> 00:12:02,270
+اليمين والـ limit من اليسار ايش برضه مالها؟ موجودة
+
+137
+00:12:02,270 --> 00:12:06,710
+ماشي الحال. أنا الآن غرضي إن أو اثبت إن g of x بسبب
+
+138
+00:12:06,710 --> 00:12:09,190
+absolute value of f of x ايش مالها؟ is 
+
+139
+00:12:09,190 --> 00:12:14,110
+differentiable at c. ماشي الآن ايش اللي بدي أثبته؟ 
+
+140
+00:12:14,110 --> 00:12:22,370
+بمعنى آخر بدي أثبت لك إنه limit الـ G of X ناقص G of
+
+141
+00:12:22,370 --> 00:12:29,430
+C على X minus C لما X تروح للـ C exist بايش؟ إذا
+
+142
+00:12:29,430 --> 00:12:31,930
+أثبتوا معناته أثبتوا إن الـ G is differentiable
+
+143
+00:12:31,930 --> 00:12:36,970
+at C أو بمعنى آخر بالدثبت limit لـ absolute value
+
+144
+00:12:36,970 --> 00:12:42,660
+للـ F of X مدلة g of x ناقص الـ absolute value of f
+
+145
+00:12:42,660 --> 00:12:48,960
+of c على x minus c لما x بتروح للـ c exist. بدي أشوف 
+
+146
+00:12:48,960 --> 00:12:53,240
+هدا لسه exist ولا لأ. يعني بدي أثبت اللي هو limit
+
+147
+00:12:53,240 --> 00:12:58,900
+absolute value of f of x على x minus c as x بتروح 
+
+148
+00:12:58,900 --> 00:13:07,520
+للـ c أشمالها exist. ماشي الحال. طيب الآن واضح
+
+149
+00:13:08,360 --> 00:13:13,000
+عندي من اللي فوق الـ limit لل f of x على x minus c
+
+150
+00:13:13,000 --> 00:13:19,160
+as x بتروح للـ c إنه ايش بيساوي؟ بيساوي صفر. مظبوط؟
+
+151
+00:13:19,160 --> 00:13:27,160
+إذا هيكون عندي limit اللي هو limit absolute value
+
+152
+00:13:27,160 --> 00:13:33,400
+للـ limit. خليني أكتبها بس في طريقة أخرى الصفر هذا 
+
+153
+00:13:33,400 --> 00:13:43,200
+اللي بدأتبته اه إنه exist عندي الـ absolute value لـ
+
+154
+00:13:43,200 --> 00:13:52,020
+limit f of x على x minus c لما x تروح للـ c اللي هو
+
+155
+00:13:52,020 --> 00:13:58,020
+من اليمين ومن اليسار exist. مظبوط؟ واضحة ويساوي
+
+156
+00:13:58,020 --> 00:14:05,160
+اللي هو limit absolute value لل f of x على x minus 
+
+157
+00:14:05,160 --> 00:14:12,420
+c as absolute value as X بتروح لمين؟ للـ C. بايش الحال؟ 
+
+158
+00:14:12,420 --> 00:14:16,820
+الآن من اليمين ومن اليسار كله هيكون موجود
+
+159
+00:14:16,820 --> 00:14:20,480
+بناء على هذا إنه موجود. خلّيني أخد من اليمين وهنا
+
+160
+00:14:20,480 --> 00:14:25,860
+من اليمين فبصير هذا عبارة عن limit absolute value 
+
+161
+00:14:25,860 --> 00:14:32,140
+لـ F of X على X minus C لما X بتروح لـ C من وين؟ من
+
+162
+00:14:32,140 --> 00:14:38,330
+اليمين. يعني هذا المقدار صار موجود وايه بيساوي؟ بيساوي 
+
+163
+00:14:38,330 --> 00:14:42,350
+صفر هذا خليه لإنه هذا اللي بدنا نصلّه. لأن limit 
+
+164
+00:14:42,350 --> 00:14:47,310
+صار عندي. معنى آخر limit absolute value of f of x
+
+165
+00:14:47,310 --> 00:14:53,850
+على x minus c لما x تروح للـ c من اليمين بيساوي صفر.
+
+166
+00:14:53,850 --> 00:14:58,360
+exist لاحظ قاعد رايح لأثبت أن هذا exist. ده خد الآن 
+
+167
+00:14:58,360 --> 00:15:03,520
+من وين؟ من اليسار خد لأن احسب similarly عندي صفر
+
+168
+00:15:03,520 --> 00:15:08,840
+بيساوي absolute value of limit f of x على x minus c 
+
+169
+00:15:08,840 --> 00:15:15,100
+لما x تروح لـ c من وين؟ من اليسار. ويساوي عبارة عن 
+
+170
+00:15:15,100 --> 00:15:19,960
+limit absolute value of f of x على absolute value
+
+171
+00:15:19,960 --> 00:15:25
+
+201
+00:18:34,960 --> 00:18:39,660
+بشوية بنفس الأسلوب على x minus c لما x تروح ل c من
+
+202
+00:18:39,660 --> 00:18:47,020
+وين؟ من اليمين بساوي limit للـ absolute value لـ f
+
+203
+00:18:47,020 --> 00:18:53,780
+of x على x minus c لما x تروح ل c من اليمين لأن الـ
+
+204
+00:18:53,780 --> 00:19:01,840
+x أكبر من الـ c  واضح؟ الآن من جهة أخرى احسبلي الـ
+
+205
+00:19:01,840 --> 00:19:08,060
+absolute value للـ limit للـ F of X طبعا هذا إيش
+
+206
+00:19:08,060 --> 00:19:15,620
+هيساوي؟ بسوى G prime of C، مظبوط؟ بسوى G prime of
+
+207
+00:19:15,620 --> 00:19:23,290
+C، موجود، limit F of X على x minus c لما x تروح ل c 
+
+208
+00:19:23,290 --> 00:19:28,570
+من وين؟ من اليسار بساوي زي ما عملنا قبل شوية سالب
+
+209
+00:19:28,570 --> 00:19:33,530
+limit absolute value of f of x على x minus c لما x
+
+210
+00:19:33,530 --> 00:19:41,050
+تروح ل c من وين؟ من اليسار الآن واضح بما أن هذه
+
+211
+00:19:41,050 --> 00:19:51,470
+exist إذا هذا المقدار و هذا المقدار زي بعض، الآن هذا
+
+212
+00:19:51,470 --> 00:19:56,590
+بيساوي ناقص هذا أو بمعنى آخر نشيل الناقص من هنا و
+
+213
+00:19:56,590 --> 00:20:03,390
+نضربه هنا صار عندي المقدارين هدول بما أنه متساويين
+
+214
+00:20:03,390 --> 00:20:10,310
+لأن الاتنين إيش بيساوين؟ الـ G prime of C مظبوط إذا
+
+215
+00:20:10,310 --> 00:20:13,390
+صار عندي هذا المقدار بيساوي هذا المقدار إذا صار
+
+216
+00:20:13,390 --> 00:20:20,940
+عندي الـ limit F of X على X minus C لما X تروح للـ C
+
+217
+00:20:20,940 --> 00:20:29,560
+من اليمين absolute value بساوي ناقص limit F of X
+
+218
+00:20:29,560 --> 00:20:34,600
+على X minus C لما X تروح للـ C من وين؟ من اليسار
+
+219
+00:20:34,600 --> 00:20:41,500
+واضحة؟ لكن أصلا عندي هذا
+
+220
+00:20:43,370 --> 00:20:49,010
+بساوي اللي هو الاتنين بساوي نفس القيمة ماشي الحال
+
+221
+00:20:49,010 --> 00:20:55,850
+إذا لازم عندي من هنا جي برايم و جي برايم اللي هو
+
+222
+00:20:55,850 --> 00:21:00,210
+صار بنساوي نفس القيمة إذا هيطلع الـ جي برايم إيش
+
+223
+00:21:00,210 --> 00:21:05,530
+ماله بساوي صفر، إذا صار المقدار هذا كله إيش بده يساوي؟
+
+224
+00:21:05,530 --> 00:21:08,650
+صفر وهذا بيساوي صفر يعني الـ limit من اليمين و
+
+225
+00:21:08,650 --> 00:21:12,070
+limit من اليسار متساويين إذا صارت limit F of X على
+
+226
+00:21:12,070 --> 00:21:23,310
+X minus C لما X تروح للـ C بدها تساوي صفر أي سؤال؟
+
+227
+00:21:23,310 --> 00:21:31,130
+زي ما حكينا، الآن إحنا قلنا إنه لو كانت D بدينا D
+
+228
+00:21:31,130 --> 00:21:34,600
+بيساوي F of X، الـ differential بالـ C جبنا g prime
+
+229
+00:21:34,600 --> 00:21:37,500
+و كتبناها بالصورة اللي أمامنا بعدين أخدنا الـ
+
+230
+00:21:37,500 --> 00:21:40,120
+absolute value لـ limit f of x علي x minus c لم x
+
+231
+00:21:40,120 --> 00:21:45,200
+تروح ل c positive طلعت عندي بساوي g prime of c و
+
+232
+00:21:45,200 --> 00:21:48,680
+أخدنا اللي هي ناقص هذه طلعت عندي برضه g prime
+
+233
+00:21:48,680 --> 00:21:53,920
+الاتنين الـ C إذا صار هذا بساوي هذا المقدار و احنا
+
+234
+00:21:53,920 --> 00:21:57,640
+بنعرف في الأصل ان f prime of c exist يعني
+
+235
+00:21:57,640 --> 00:22:01,900
+differentiable يعني الـ limit هذا اللي هو موجودو من
+
+236
+00:22:01,900 --> 00:22:05,100
+اليمين و من اليسار زي بعض يعني يعني هذا بده يساوي
+
+237
+00:22:05,100 --> 00:22:07,500
+هذا من اليمين و هذا من اليسار يعني في الواقع هذا
+
+238
+00:22:07,500 --> 00:22:10,960
+اللي جوا هو نفسه اللي جوا بساوي limit of f of X يعني
+
+239
+00:22:10,960 --> 00:22:15,780
+بمعنى آخر absolute value لـ limit of f of X على X
+
+240
+00:22:15,780 --> 00:22:22,100
+minus C لما X تروح للـ C هو نفسه سالب limit للـ f of
+
+241
+00:22:22,100 --> 00:22:27,400
+of X على X minus C لما X تروح للـ C من وين؟ من
+
+242
+00:22:27,400 --> 00:22:31,540
+اليسار هو نفسه للـ C من وين تبت هذ؟ لأن إحنا بنقول
+
+243
+00:22:31,540 --> 00:22:34,560
+f prime of c exist يعني الـ limit هذا موجود و من
+
+244
+00:22:34,560 --> 00:22:37,360
+اليمين و من اليسار زي بعض، إذا صار هذا المقدار نفس
+
+245
+00:22:37,360 --> 00:22:41,200
+هذا المقدار زي ما قلنا، إذا صار عند هذا المقدار بساوي
+
+246
+00:22:41,200 --> 00:22:46,800
+صفر لأن الاتنين بسينا بساوي صفر إذا صار عندي limit F
+
+247
+00:22:46,800 --> 00:22:50,880
+of X على X minus C اللي هو الـ absolute value as X
+
+248
+00:22:50,880 --> 00:22:56,100
+بتروح للـ C بساوي صفر ومن ثم اللي جوا بساوي صفر هو
+
+249
+00:22:56,100 --> 00:22:58,740
+مين؟ هو اللي جوا هذا اللي كنا بدنا نصله من الأول
+
+250
+00:22:58,740 --> 00:23:04,260
+اللي هو F prime of C بتساوي صفر وهو المطلوب هذا
+
+251
+00:23:04,260 --> 00:23:10,300
+توضيح بشكل كامل ل اللي صار في اللي هو الاتجاه
+
+252
+00:23:10,300 --> 00:23:16,200
+الثاني، نيجينا سؤال بعد ما خلصنا السؤال سبعة، نجي
+
+253
+00:23:16,200 --> 00:23:23,420
+لسؤال تسعة، تسعة إيش اللي بقوله؟ تسعة نشوف إيش سؤال
+
+254
+00:23:23,420 --> 00:23:31,360
+تسعة بقول ونحل سؤال تسعة، سؤال تسعة بقول ليه
+
+255
+00:23:31,360 --> 00:23:39,110
+باختصار انه لو كان عنده الـ function of عبارة عن من
+
+256
+00:23:39,110 --> 00:23:43,130
+R لـ R even function طبعا عارفين إيش الـ even اللي
+
+257
+00:23:43,130 --> 00:23:49,130
+هو F ناقص X بيساوي F X لكل X موجودة في الـ R and
+
+258
+00:23:49,130 --> 00:23:54,310
+has a derivative at every point then F prime is an
+
+259
+00:23:54,310 --> 00:23:58,190
+odd function يعني بيقول لي لو كانت باختصار يعني لو
+
+260
+00:23:58,190 --> 00:24:01,730
+كانت الـ F even و الـ derivative موجودة بتكون الـ
+
+261
+00:24:01,730 --> 00:24:07,010
+derivative odd و لو كانت الـ derivative odd بتكون
+
+262
+00:24:07,010 --> 00:24:11,350
+الـ function الـ G prime إيه شمالها is even، انحل
+
+263
+00:24:11,350 --> 00:24:16,970
+واحدة من هنا والتانية similarly زيها، الآن لنفترض F
+
+264
+00:24:16,970 --> 00:24:29,890
+من عند R لعند R be an odd differentiable function
+
+265
+00:24:29,890 --> 00:24:34,970
+ماشي الحال، show that
+
+266
+00:24:36,040 --> 00:24:44,360
+F' is even، a proof، بدنا نثبت أنه لو كانت الـ F
+
+267
+00:24:44,360 --> 00:24:49,940
+اللي هي odd function بدو يكون عنده و
+
+268
+00:24:49,940 --> 00:24:52,320
+differentiable بدو يكون عنده derivative إلها إيه
+
+269
+00:24:52,320 --> 00:25:03,000
+إيش even، لأن let C element in R be arbitrary and
+
+270
+00:25:03,000 --> 00:25:04,180
+fixed
+
+271
+00:25:06,310 --> 00:25:15,930
+NR ناخد الـ R ناخد الـ C أي اللي هو real number in
+
+272
+00:25:15,930 --> 00:25:23,360
+R لكن نحكي عن أي شيء محدد الآن F prime of C بدأ
+
+273
+00:25:23,360 --> 00:25:29,060
+أثبتلك أنه هو بساوي F prime of ناقص C يعني F prime
+
+274
+00:25:29,060 --> 00:25:34,920
+is even، إذا خد F prime ناقص C وابدأ حسب وواصلك في
+
+275
+00:25:34,920 --> 00:25:39,140
+النهاية بساوي F prime of C إذا F prime is even، بساوي
+
+276
+00:25:39,140 --> 00:25:49,380
+limit اللي هي F of X ناقص F of minus C على X minus
+
+277
+00:25:50,570 --> 00:25:57,070
+اللي هو minus C لما X تروح لمين؟ لـ minus C مظبوط
+
+278
+00:25:57,070 --> 00:26:05,550
+طيب و يساوي limit الـ F إيش معطينا إياها؟ عبارة عن odd
+
+279
+00:26:05,550 --> 00:26:12,250
+إيش يعني odd؟ يعني F of ناقص X بيساوي ناقص F of X
+
+280
+00:26:12,250 --> 00:26:16,590
+مظبوط، هذه اللي هي إيش ما لها؟ odd function اللي هو
+
+281
+00:26:16,590 --> 00:26:28,550
+بيساوي limit F of الآن اللي هي ناقص f of ناقص x
+
+282
+00:26:28,550 --> 00:26:34,150
+واضح أه؟
+
+283
+00:26:34,150 --> 00:26:37,150
+f of ناقص x بيساوي ناقص f of x يعني f of x بيساوي
+
+284
+00:26:37,150 --> 00:26:41,230
+ناقص f of ناقص x، فالجد تعرفوا ليش عملت هيك؟ لأن
+
+285
+00:26:41,230 --> 00:26:48,570
+ناقص f of ناقص c الآن f is odd مظبوط بيصير زائد f
+
+286
+00:26:48,570 --> 00:26:49,330
+of c
+
+287
+00:26:52,220 --> 00:26:58,060
+على خد الآن ناقص من هنا عامل مشترك بيصير عبارة عن
+
+288
+00:26:58,060 --> 00:27:08,220
+ناقص X الآن ناقص الـ C لما X تروح لمين؟ لسالب الـ C
+
+289
+00:27:08,220 --> 00:27:14,040
+الـ X بتروح لسالب الـ C إذا و فقط إذا سالب الـ X
+
+290
+00:27:14,040 --> 00:27:22,160
+بتروح لمين؟ إلى الـ C الآن خد لي Y بساوي سالب الـ X
+
+291
+00:27:22,160 --> 00:27:29,240
+واستبدل عشان نوضح لك إياه بساوي limit الآن خد الآن
+
+292
+00:27:29,240 --> 00:27:38,320
+عندي F حكما كان نقص اكسميا الـ Y limit ناقص F of Y
+
+293
+00:27:38,320 --> 00:27:49,390
+زائد F of C على اللي هو Y بتروح إلى الـ C وهنا y
+
+294
+00:27:49,390 --> 00:27:57,430
+ناقص الـ c وهنا في عندي إيش برضه ناقص برا لأن خد من
+
+295
+00:27:57,430 --> 00:28:01,670
+هنا ناقص عام المشترك أو ضيعه مع الناقص اللي هنا
+
+296
+00:28:01,670 --> 00:28:11,160
+ما واضح أه؟ بيصير عندي y ساوي limit f of y نقص f of c
+
+297
+00:28:11,160 --> 00:28:16,800
+على y minus c لما y تروح للـ c وهذا عبارة عن f
+
+298
+00:28:16,800 --> 00:28:22,440
+prime لمين؟ للـ c، بدأنا بـ f prime نقص c وانتهينا بـ f
+
+299
+00:28:22,440 --> 00:28:31,520
+prime للـ c لذا therefore f prime is even whenever
+
+300
+00:28:31,520 --> 00:28:37,160
+f is odd and f is differentiable
+
+301
+00:28:40,710 --> 00:28:56,810
+السؤال الأخير السؤال 13، السؤال
+
+302
+00:28:56,810 --> 00:29:03,990
+13 هو كما يلي، إيش
+
+303
+00:29:03,990 --> 00:29:09,200
+اللي بقول السؤال 13، سؤال 13 بيقول إذا كانت F من R
+
+304
+00:29:09,200 --> 00:29:12,380
+لـ R is differentiable at C element in R show that
+
+305
+00:29:12,380 --> 00:29:16,840
+F prime of C بساوي limit N في F of C زائد 1 على N ناقص
+
+306
+00:29:16,840 --> 00:29:20,620
+F of C as N goes to infinity، يعني لو كانت F
+
+307
+00:29:20,620 --> 00:29:24,900
+differentiable عند الـ C element in R بنقدر نكتب
+
+308
+00:29:24,900 --> 00:29:27,920
+الـ derivative اللي هي F prime of C على صورة limit
+
+309
+00:29:27,920 --> 00:29:32,320
+N F of C زائد 1 على N ناقص F of C as N goes to
+
+310
+00:29:32,320 --> 00:29:38,120
+infinity، لكن بيقول لي by example show that the
+
+311
+00:29:38,120 --> 00:29:44,000
+existence of this limit، this limit need not اللي
+
+312
+00:29:44,000 --> 00:29:52,300
+هو imply the existence of the derivative، نيجي
+
+313
+00:29:52,300 --> 00:30:03,960
+الآن للجزء الأول عندي F من R لـ R و F prime of C
+
+314
+00:30:03,960 --> 00:30:12,120
+exist، لأنه يقول لي prove that F prime of C can be
+
+315
+00:30:12,120 --> 00:30:20,760
+written as limit on F of C زائد واحدة على N ناقص F
+
+316
+00:30:20,760 --> 00:30:26,180
+of C as N goes to infinity، هذا الجزء الأول، الجزء
+
+317
+00:30:26,180 --> 00:30:32,400
+الثاني حنجي الآن نقول prove للجزء الأول قبل ما نقول
+
+318
+00:30:32,400 --> 00:30:37,700
+الـ proof نذكركم بس بنظرية سابقة في الـ real واحد
+
+319
+00:30:37,700 --> 00:30:43,840
+إنه لو عندي limit f of x لما x تروح لل c لو كانت
+
+320
+00:30:43,840 --> 00:30:52,780
+بتساوي L بكون عندي أي sequence xₙ بتروح للـ C لازم
+
+321
+00:30:52,780 --> 00:30:59,020
+يتحقق لها limit f of xₙ as n goes to infinity
+
+322
+00:30:59,020 --> 00:31:05,530
+بساوي برضه الـ كنا نتحدث بالحديث، باستبداء الحديث
+
+323
+00:31:05,530 --> 00:31:08,670
+عن الـ limit العادية للـ function S X بتروح للـ C
+
+324
+00:31:08,670 --> 00:31:13,470
+إلى limit لمين؟ للـ sequence أو limit للـ sequences
+
+325
+00:31:13,470 --> 00:31:17,290
+الآن بنستخلص هذه اللي هو المعلومة في إثبات اللي
+
+326
+00:31:17,290 --> 00:31:22,630
+بدنا إياه، عند الآن since F prime of C exist بما أنه
+
+327
+00:31:22,630 --> 00:31:29,250
+الـ derivative عند C موجودة إذا أكيد عندي صار f
+
+328
+00:31:29,250 --> 00:31:37,010
+prime of c بساوي limit f of x أو اللي هي الـ x عند c
+
+329
+00:31:38,080 --> 00:31:46,140
+زائد h ناقص f of c على h as h goes to mean to zero
+
+330
+00:31:46,140 --> 00:31:49,860
+اللي هو التعريف التعريف الـ derivative أو الشكل
+
+331
+00:31:49,860 --> 00:31:52,980
+الآخر للتعريف الـ derivative f of c زائد الـ
+
+332
+00:31:52,980 --> 00:31:55,560
+increment ناقص f of c على الـ increment as الـ
+
+333
+00:31:55,560 --> 00:32:01,600
+increment goes to mean to zero، ماشي الحل، الـ L بما
+
+334
+00:32:01,600 --> 00:32:07,480
+أنه اللي هو الـ 1 على N sequence بتروح للصفر وهذا
+
+335
+00:32:07,480 --> 00:32:11,440
+الـ limit exist لأن حسب النظرية اللي حكيتها قبل
+
+336
+00:32:11,440 --> 00:32:21,100
+بشوية بكون عندي لأن butكذا then F prime of C can
+
+337
+00:32:21,100 --> 00:32:26,580
+be اللي هو إعادة اللي هي rewritten as a limit of a
+
+338
+00:32:26,580 --> 00:32:36,800
+sequence F of اللي هو limit limit F of C زائد but
+
+339
+00:32:36,800 --> 00:32:40,000
+الـ H اللي هي تروح للصفر صارت mean الـ sequence
+
+340
+00:32:40,000 --> 00:32:46,380
+واحدة على N تروح للصفر، واحدة على N ناقص F of C على
+
+341
+00:32:46,380 --> 00:32:52,640
+واحدة على N as N goes to infinity، مدامت الـ sequence
+
+342
+00:32:52,640 --> 00:32:57,100
+واحدة على N بتروح للصفر صارت الـ F of واحدة على N اللي
+
+343
+00:32:57,100 --> 00:33:00,900
+هي عبارة عن F of C زائد واحدة على N لإن الـ C عبارة
+
+344
+00:33:00,900 --> 00:33:07,740
+عن إياش فالت واضح أه؟ الآن هذا بيساوي اللي هو
+
+345
+00:33:07,740 --> 00:33:14,600
+limit الآن as n goes to infinity اللي هو أكيد اللي
+
+346
+00:33:14,600 --> 00:33:21,920
+هو بيصير عندي f n في الجوس f of c زائد واحدة على n
+
+347
+00:33:21,920 --> 00:33:28,260
+ناقص f of c اللي هو as n goes to infinity هو هذا
+
+348
+00:33:28,260 --> 00:33:35,150
+صار f prime of c وهو المطلوب، الآن conversely the
+
+349
+00:33:35,150 --> 00:33:37,890
+converse need not to be true in general، هيك بيقول
+
+350
+00:33:37,890 --> 00:33:45,390
+ليه؟ يعني بيقول لي if if بيقول لي أو اللي بيقول if f
+
+351
+00:33:45,390 --> 00:33:53,670
+برا اللي هو limit f of c زاد واحدة على n ناقص f of c
+
+352
+00:33:53,670 --> 00:34:01,610
+الكل مضروب في n as n goes to infinity exist if كده
+
+353
+00:34:02,500 --> 00:34:12,640
+then f prime at c need not be exist أصلا مش انه
+
+354
+00:34:12,640 --> 00:34:16,180
+يكون بيساوي هذا أو لا need not to be اشمله exist
+
+355
+00:34:16,180 --> 00:34:19,640
+لأن لو كان exist على طول بيساوي أما need not to be
+
+356
+00:34:19,640 --> 00:34:28,200
+exist وما أخد مثال، جالي consider consider consider
+
+357
+00:34:28,200 --> 00:34:37,000
+f of x بساوي الـ absolute value للـ X وخد عند الـ C إيش
+
+358
+00:34:37,000 --> 00:34:46,440
+بتساوي صفر واضح F prime of 0 does not exist لأنها
+
+359
+00:34:46,440 --> 00:34:48,780
+عبارة عن corner point، إنتوا عارفين إحنا اللي هو
+
+360
+00:34:48,780 --> 00:34:52,480
+الـ F prime عند الـ zero للـ absolute value does not
+
+361
+00:34:52,480 --> 00:34:59,740
+exist لكن هذه متحققة، ليش؟ but limit
+
+362
+00:35:01,280 --> 00:35:10,900
+N في F of 0 زائد 1 على N نقص F of 0 as N goes to
+
+363
+00:35:10,900 --> 00:35:18,840
+infinity بساوي limit N F of 0 زائد 1 على N يعني F
+
+364
+00:35:18,840 --> 00:35:23,560
+of 1 على N، F of X بساوي absolute value X ف 1 على N
+
+365
+00:35:23,560 --> 00:35:28,960
+مظبوط أه as N goes to infinity طبعا الـ N بتروح
+
+366
+00:35:29,220 --> 00:35:35,100
+يصير عبارة عن as 1 إذا فعلا جبنا مثال أن الـ
+
+367
+00:35:35,100 --> 00:35:39,720
+limit هذه تكون exist و بساوي واحد but الـ F prime عند
+
+368
+00:35:39,720 --> 00:35:42,640
+هذا النقطة C اللي هي 00 في هذه الحالة does not
+
+369
+00:35:42,640 --> 00:35:47,320
+exist، بيكون هيك إحنا انتهينا من الجزء الأول من
+
+370
+00:35:47,320 --> 00:35:54,870
+المحاضرة الخامسة اللي هو discussion لأو مناقشة لـ
+
+3
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0fe1e77efbcbf108e3fabf9108b97f46e2f98add
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_postprocess.srt
@@ -0,0 +1,1500 @@
+1
+00:00:04,910 --> 00:00:09,510
+بسم الله الرحمن الرحيم هذه المحاضرة الخامسة في
+
+2
+00:00:09,510 --> 00:00:15,630
+مساق تحليل حقيقه اتنين لطلبة كلية العلوم تخصص
+
+3
+00:00:15,630 --> 00:00:22,110
+رياضيات في الجامعة الإسلامية بغزا المحاضرة
+
+4
+00:00:22,110 --> 00:00:26,550
+اليوم هي جزئين الجزء الاول اللي هو عبارة عن
+
+5
+00:00:26,550 --> 00:00:31,350
+discussion لست واحد اللي هو مناقشة لموضوع اللي هو
+
+6
+00:00:31,350 --> 00:00:37,410
+derivative او الاشتقاءالجزء الثاني هنكمل اللي هو
+
+7
+00:00:37,410 --> 00:00:43,210
+الحديث عن 6.2 اللي هو الـ Mean Value Theorem أو
+
+8
+00:00:43,210 --> 00:00:47,730
+نظرية القيمة المتوسطة هناخذ بعض التطبيقات لنبدأ
+
+9
+00:00:47,730 --> 00:00:51,750
+الآن اللي هي بالأسئلة اللي احنا طلبناها منكم
+
+10
+00:00:51,750 --> 00:00:57,230
+تحلوها كواجب المرة الماضية أو التي قبلها وكانت
+
+11
+00:00:57,230 --> 00:01:01,210
+الأسئلة هي عبارة عن السؤال الرابع والسؤال السابع
+
+12
+00:01:02,000 --> 00:01:07,420
+والسؤال التاسع و السؤال تلتاشر بالنسبة للسؤال
+
+13
+00:01:07,420 --> 00:01:13,600
+الثالث اللي هو عبارة عن اللي هي برهان نظرية 613A
+
+14
+00:01:13,600 --> 00:01:19,200
+وB وهذه البرهين براهين سهلة اللي كانت ال F
+
+15
+00:01:19,200 --> 00:01:22,960
+differentiableالـ F differentiable والـ Alpha
+
+16
+00:01:22,960 --> 00:01:26,560
+عبارة عن constant بيعطينا الـ Alpha F برضه is
+
+17
+00:01:26,560 --> 00:01:30,200
+differentiable و لو كانت الـ F والـ G
+
+18
+00:01:30,200 --> 00:01:32,700
+differentiable بيعطينا الـ F زائد G is
+
+19
+00:01:32,700 --> 00:01:35,660
+differentiable و احنا برهننا حالة الضرب و حالة
+
+20
+00:01:35,660 --> 00:01:41,500
+القسمة و هدول حالات تعتبر سهلة مباشرة على التعريف
+
+21
+00:01:41,500 --> 00:01:45,900
+لذلك هنبدأ ان شاء الله في الحديث او في حل الأسئلة
+
+22
+00:01:45,900 --> 00:01:52,540
+على السؤال الرابع اللي هو ببول ماليعندي معطيني F
+
+23
+00:01:52,540 --> 00:02:01,760
+من R لـ R بي defined by F of X بساوي اللي هو X
+
+24
+00:02:01,760 --> 00:02:06,420
+تربيع إذا كانت X rational وبساوي سفر إذا كانت X
+
+25
+00:02:06,420 --> 00:02:09,700
+irrational اثبت إن الـ F الـ differential بالقدر X
+
+26
+00:02:09,700 --> 00:02:15,550
+بتساوي سفر أو جد اللي هي F prime of 0لاحظ أن الـ
+
+27
+00:02:15,550 --> 00:02:16,610
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+28
+00:02:16,610 --> 00:02:17,090
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+29
+00:02:17,090 --> 00:02:17,910
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+30
+00:02:17,910 --> 00:02:20,450
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+31
+00:02:20,450 --> 00:02:27,990
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+32
+00:02:27,990 --> 00:02:28,350
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+33
+00:02:28,350 --> 00:02:29,630
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+34
+00:02:29,630 --> 00:02:29,690
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+35
+00:02:29,690 --> 00:02:41,470
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+36
+00:02:41,470 --> 00:02:52,150
+الـ الـ الـFind this value a proof الآن عشان نوجد
+
+37
+00:02:52,150 --> 00:02:56,730
+اللي هي نثبت أن F prime of 0 موجودة خلّيني ألاحظ
+
+38
+00:02:56,730 --> 00:03:00,950
+مالي عند ال function اللي هي X تربيع لما X
+
+39
+00:03:00,950 --> 00:03:07,150
+rational و Zero لما Xاش is irrationalالان بدنا
+
+40
+00:03:07,150 --> 00:03:11,770
+نتوقع اول اشي اللي هو لانه هتلزمني جاي اللي بعدين
+
+41
+00:03:11,770 --> 00:03:15,890
+find this value بدنا نتوقع ايش قيمة اللي هو ال
+
+42
+00:03:15,890 --> 00:03:20,950
+derivative عند الصفر لاحظ انه اللي هو .. اللي ..
+
+43
+00:03:20,950 --> 00:03:25,630
+اللي لو بدنا نقول انه ال derivative ممكن تكون zero
+
+44
+00:03:25,630 --> 00:03:30,510
+كون ان off of x صفر او لو بدها تكون و ال
+
+45
+00:03:30,510 --> 00:03:36,560
+derivative هنا لو قولنا رفل 2xبدو يكون اللي هو لو
+
+46
+00:03:36,560 --> 00:03:40,820
+بدو يكون اللي عند الـ zero اللي هي F prime بدو في
+
+47
+00:03:40,820 --> 00:03:47,220
+النهاية تروح ل اللي هيكون برضه قريبة من انه نقول
+
+48
+00:03:47,220 --> 00:03:52,580
+او نأكد انها صفر عشان هي كالظن الغالب ان F prime
+
+49
+00:03:52,580 --> 00:03:58,660
+هتكون ايش صفر هذه مجرد تفكيرات الانبدي أثبت لك إنه
+
+50
+00:03:58,660 --> 00:04:03,080
+فعلًا هي الـ derivative بتساوي 0 كيف بدي أثبتها؟
+
+51
+00:04:03,080 --> 00:04:11,280
+بدي أثبت لك إنه الـ limit لـ F of X نقص F of 0 على
+
+52
+00:04:11,280 --> 00:04:17,600
+X minus 0 لما X تروح لـ 0 بتساوي 0 بدي أثبت لك
+
+53
+00:04:17,600 --> 00:04:24,240
+هيهاالان واضح ان X تقول الى الصفر X تقول الى الصفر
+
+54
+00:04:24,240 --> 00:04:29,200
+هتمر بال rational وال rational عشان هيك صعب ان انا
+
+55
+00:04:29,200 --> 00:04:32,600
+اتحدث عن اللي هو ايجاد ال derivative مباشرة من
+
+56
+00:04:32,600 --> 00:04:36,130
+الان او من ال two branches اللي عندىلا بقدر أخد من
+
+57
+00:04:36,130 --> 00:04:40,030
+اليمين ولا أخد من اليسار لإنه عندي من اليمين أو من
+
+58
+00:04:40,030 --> 00:04:44,130
+اليسار هيكون عندي اللي هو قابلت اللي هي الأعداد ال
+
+59
+00:04:44,130 --> 00:04:48,110
+rational لل rational فعشانك أسلم إشي إنه نستخدم
+
+60
+00:04:48,110 --> 00:04:52,650
+التعريف في إثبات هذا يعني الآن بدي أثبت هذا الكلام
+
+61
+00:04:53,270 --> 00:04:58,090
+كيف بدي أثبته؟ بدي أثبت ما يعني بدي أصل لكل إبسلون
+
+62
+00:04:58,090 --> 00:05:03,190
+أكبر من 0 بدلاج دلتا أكبر من 0 بدلاجية حاجيلك دلتا
+
+63
+00:05:03,190 --> 00:05:08,350
+بحيث أنه لما يكون ال absolute value ل F of X ناقص
+
+64
+00:05:08,350 --> 00:05:14,630
+F of Zero على X minus Zero يكون ناقص Zero طبعا
+
+65
+00:05:14,630 --> 00:05:19,790
+Zero هذي أصغر من إبسلون هذا متى؟ whenever
+
+66
+00:05:22,670 --> 00:05:28,470
+x-0 أكبر من 0 وأصغر من مين من دلتا هذا اللي بده
+
+67
+00:05:28,470 --> 00:05:33,150
+أصله أو هذا اللي بده أثبته خلّينا نشوف إذا خلّينا
+
+68
+00:05:33,150 --> 00:05:36,530
+نشوف كيف بدنا نوجده عشان أثبت إن ال limit هذا
+
+69
+00:05:36,530 --> 00:05:43,110
+بساوي 0 لاحظ القيمة اللي عندي أول إشي ال absolute
+
+70
+00:05:43,110 --> 00:05:50,530
+value of f of x ناقص f of 0 على x-0 إيش هتساوي؟
+
+71
+00:05:51,590 --> 00:05:57,390
+هتساوي اللي هو ال absolute value of of X عندى يا X
+
+72
+00:05:57,390 --> 00:06:01,070
+تربيع يمين يا سفر خلّينها زي ما هي أول إشي of of X
+
+73
+00:06:01,070 --> 00:06:06,010
+ناقص of of Zero اللي هو جداش بيساوي Zero لإنه of
+
+74
+00:06:06,010 --> 00:06:10,130
+of Zero بيساوي Zero لإنه Zero rational على إذن هذا
+
+75
+00:06:10,130 --> 00:06:16,230
+سفر على X ناقص سفر اللي هو X absolute value هذا
+
+76
+00:06:16,230 --> 00:06:22,790
+القيمة الآن لاحظ بتساوييا إما اللي هو x تربيع على
+
+77
+00:06:22,790 --> 00:06:30,750
+x absolute value في حالة x is rational أو بتساوي
+
+78
+00:06:30,750 --> 00:06:37,450
+اللي هو سفر في حالة x أشمالها is irrational لأن
+
+79
+00:06:37,450 --> 00:06:42,030
+مقيمة f of x يا x تربيع يا سفر حسب اللي هو كتبناه
+
+80
+00:06:42,030 --> 00:06:46,790
+حاليًاالآن هذا بالظبط هو عبارة عن الـ absolute
+
+81
+00:06:46,790 --> 00:06:57,590
+value لل X if X is rational Zero if X is
+
+82
+00:06:57,590 --> 00:07:03,950
+irrational الآن الصورة وضحت خلّينا نسمي هذا اللي
+
+83
+00:07:03,950 --> 00:07:09,900
+هو واحدالان حضرت عشان اصل للنهاية اللى هاني كاتبها
+
+84
+00:07:09,900 --> 00:07:13,800
+هنا واشوف ايش ال delta اللى بتطلع عندى الان بتدعي
+
+85
+00:07:13,800 --> 00:07:17,380
+مالي for every epsilon أكبر من سفر انا بقول there
+
+86
+00:07:17,380 --> 00:07:21,800
+exists delta هتساوي من ال epsilon هتجد تشوفوا ليش
+
+87
+00:07:21,800 --> 00:07:26,980
+there exists delta بساوي epsilon such that if x
+
+88
+00:07:26,980 --> 00:07:35,210
+minus 0 أكبر من سفره أصغر من delta thenهذا معناته
+
+89
+00:07:35,210 --> 00:07:37,870
+إيش أن أبسل يوت فيها ال X أكبر من صفر وأصغر من
+
+90
+00:07:37,870 --> 00:07:41,870
+مين؟ من دلتا إذا اخترت دلتا إيش بتساوي و هي اللي
+
+91
+00:07:41,870 --> 00:07:47,850
+هتخلص من الموضوع then اللي هو F of X نقص F of Zero
+
+92
+00:07:47,850 --> 00:07:50,970
+على
+
+93
+00:07:50,970 --> 00:07:57,790
+X minus Zero هو طبعا نقص الصفر اللي قلنا عنها ال
+
+94
+00:07:57,790 --> 00:08:01,520
+derivative المتوقع عليا هذه بالظبطاللي فوق هذا هو
+
+95
+00:08:01,520 --> 00:08:05,920
+اللي فوق طلع إيش عندي هذا بساوي هذا المقدار إذا
+
+96
+00:08:05,920 --> 00:08:09,760
+كانت X rational و 0 إذا كانت X irrational يعني
+
+97
+00:08:09,760 --> 00:08:15,400
+بمعنى أخر بساوي absolute value ل X if X is
+
+98
+00:08:15,400 --> 00:08:23,530
+rational و 0 if X is irrational in both casesاللي
+
+99
+00:08:23,530 --> 00:08:27,590
+هي إذا كان بساوي absolute value للـ X هيكون أصغر
+
+100
+00:08:27,590 --> 00:08:31,250
+من Delta اللي انا اخترتها أشملها Epsilon هيكون
+
+101
+00:08:31,250 --> 00:08:34,570
+أصغر من Epsilon و أيضا هذه متحققة لإن الـ Epsilon
+
+102
+00:08:34,570 --> 00:08:38,850
+دائما أشملها أكبر من 0 إذا اللي حصلته أنه لكل
+
+103
+00:08:38,850 --> 00:08:42,490
+Epsilon أكبر من 0 اللي جيت Delta لما يكون هذا أصغر
+
+104
+00:08:42,490 --> 00:08:47,130
+من Delta بيعطيني هذا أصغر من Epsilon وهذا يعني
+
+105
+00:08:47,130 --> 00:08:54,370
+hence limitهذا المقدار اللي هو f of x ناقص f of 0
+
+106
+00:08:54,370 --> 00:09:01,690
+على x minus 0 لما x تروح للصفر بساوية اللي هو
+
+107
+00:09:01,690 --> 00:09:12,870
+الصفر وهذا هو تعريف من ال f prime عند 0 that is f
+
+108
+00:09:12,870 --> 00:09:18,740
+prime at 0 بساوية 0 وفي نفس الواجب طبعا أثبتناالـ
+
+109
+00:09:18,740 --> 00:09:25,960
+existence للـ F prime عند الـ zero أي سؤال طيب
+
+110
+00:09:25,960 --> 00:09:30,840
+نيجي الآن نشوف السؤال الثاني خلّال السؤال الأربعة
+
+111
+00:09:30,840 --> 00:09:44,220
+نيجي لسؤال سبعة الان
+
+112
+00:09:44,220 --> 00:09:48,100
+سؤال سبعة اش اللي بقوله سؤال سبعةاللي بيقوله سؤال
+
+113
+00:09:48,100 --> 00:09:52,420
+سبعة مالي عندي
+
+114
+00:09:52,420 --> 00:09:55,740
+suppose
+
+115
+00:09:55,740 --> 00:09:59,320
+that F من R لR is differentiable at C يعني نفترض
+
+116
+00:09:59,320 --> 00:10:04,520
+أنه F قبل الاشتقاق عند Cونفترض ان f of c قيمة الـ
+
+117
+00:10:04,520 --> 00:10:07,920
+function عند c بساوة 0 لان بقول لشهدات ال absolute
+
+118
+00:10:07,920 --> 00:10:10,960
+value لل f of x اللي هي نسميها d of x is
+
+119
+00:10:10,960 --> 00:10:14,080
+differentiable at c if and only if a شمالها f
+
+120
+00:10:14,080 --> 00:10:21,740
+prime of c بتساوة 0 إذا ناخد لك f من R لعند R و
+
+121
+00:10:21,740 --> 00:10:30,810
+جايلك أن f prime عند c exist معطيكي إياهاأو ومعطيك
+
+122
+00:10:30,810 --> 00:10:37,390
+اللي هو f of c بتساوي سفر وبقول لي prove that أنه
+
+123
+00:10:37,390 --> 00:10:42,370
+g of x بساوي ال absolute value لل f of x is
+
+124
+00:10:42,370 --> 00:10:49,810
+differentiable at c if and only if f prime عند ال
+
+125
+00:10:49,810 --> 00:10:57,880
+c ايش بتساوي بساوي سفر، مظبوط؟ طيب شوفالان خلينا
+
+126
+00:10:57,880 --> 00:11:03,380
+افترض اول اشي ان f prime عند c ايش بتساوي سفر نقول
+
+127
+00:11:03,380 --> 00:11:09,320
+suppose proof suppose
+
+128
+00:11:09,320 --> 00:11:15,580
+suppose
+
+129
+00:11:15,580 --> 00:11:23,120
+that f prime at c بتساوي سفر ايش هذا بيعني ان then
+
+130
+00:11:23,120 --> 00:11:25,940
+limit
+
+131
+00:11:27,220 --> 00:11:36,680
+F of X ناقص F of C على X minus C لما X تروح للـ C
+
+132
+00:11:36,680 --> 00:11:40,240
+اللي هو بيساوي سفر لأن هذا تعريف مين F براين
+
+133
+00:11:40,240 --> 00:11:46,000
+بيساوي سفر و F of C إيش معطنية هو بيساوي سفر لأن
+
+134
+00:11:46,000 --> 00:11:53,260
+بيساوي limit F of X على X minus C لما X تروح لمين
+
+135
+00:11:53,260 --> 00:11:58,730
+للـ C مدامة ال limit هذه موجودةإذا ال limit من
+
+136
+00:11:58,730 --> 00:12:02,270
+اليمين وال limit من اليسار أيش برضه مالها موجودة
+
+137
+00:12:02,270 --> 00:12:06,710
+ماشي الحال انا الآن غرضي ان او اثبت ان g of x بسبب
+
+138
+00:12:06,710 --> 00:12:09,190
+absolute value of f of x أيش مالها is
+
+139
+00:12:09,190 --> 00:12:14,110
+differentiable at c ماشي الان ايش اللي بدي اثبته
+
+140
+00:12:14,110 --> 00:12:22,370
+بمعنى اخر بدي اثبتلك انه limitالـ G of X ناقص G of
+
+141
+00:12:22,370 --> 00:12:29,430
+C على X minus C لما X تروح للـ C exist باشي إذا
+
+142
+00:12:29,430 --> 00:12:31,930
+أثبتوا معناته أثبتتوا إن الـ G is differentiable
+
+143
+00:12:31,930 --> 00:12:36,970
+at C أو بمعنى آخر بالدثبت limit لـ absolute value
+
+144
+00:12:36,970 --> 00:12:42,660
+لل F of Xمدلة g of x ناقص ال absolute value of f
+
+145
+00:12:42,660 --> 00:12:48,960
+of c على x minus c لما x بتروح لل c exist بدي أشوف
+
+146
+00:12:48,960 --> 00:12:53,240
+هدا لسه exist ولا لأ يعني بدي أثبت اللي هو limit
+
+147
+00:12:53,240 --> 00:12:58,900
+absolute value of f of x على x minus c as x بتروح
+
+148
+00:12:58,900 --> 00:13:07,520
+لل c أشماله exist ماشي الحال طيب الآن واضح
+
+149
+00:13:08,360 --> 00:13:13,000
+عندي من اللي فوق ال limit لل f of x على x minus c
+
+150
+00:13:13,000 --> 00:13:19,160
+as x بتروح لل c انه ايش بيساوي بيساوي صفر، مظبوط؟
+
+151
+00:13:19,160 --> 00:13:27,160
+اذا هيكون عندي limit اللي هو limit absolute value
+
+152
+00:13:27,160 --> 00:13:33,400
+لل limit خليني اكتبها بس في طريقة أخرى الصفر هذا
+
+153
+00:13:33,400 --> 00:13:43,200
+اللي بدأتبته اه انه existعندي ال absolute value ل
+
+154
+00:13:43,200 --> 00:13:52,020
+limit f of x على x minus c لما x تروح لل c اللي هو
+
+155
+00:13:52,020 --> 00:13:58,020
+من اليمين و من اليسار exist، مظبوط؟ واضحة و يساوي
+
+156
+00:13:58,020 --> 00:14:05,160
+اللي هو limit absolute value لل f of x على x minus
+
+157
+00:14:05,160 --> 00:14:12,420
+c as absolute valueas X بتروح لمين؟ للـ C باشي
+
+158
+00:14:12,420 --> 00:14:16,820
+الحال الآن من اليمين ومن اليسار كله هيكون موجود
+
+159
+00:14:16,820 --> 00:14:20,480
+بناء على هذا أنه موجود خلّيني أخد من اليمين وهنا
+
+160
+00:14:20,480 --> 00:14:25,860
+من اليمين فبصير هذا عبارة عن limit absolute value
+
+161
+00:14:25,860 --> 00:14:32,140
+ل F of X على X minus C لما X بتروح ل C من وين؟ من
+
+162
+00:14:32,140 --> 00:14:38,330
+اليمين يعني هذا المقدارصار موجود وإيش بساوي بساوي
+
+163
+00:14:38,330 --> 00:14:42,350
+سفر هذا خلّيه لإنه هذا اللي بدنا نصلّه لأن limit
+
+164
+00:14:42,350 --> 00:14:47,310
+صار عندي معنى آخر limit absolute value of f of x
+
+165
+00:14:47,310 --> 00:14:53,850
+على x minus c لما x تروح لل c من اليمين بساوي سفر
+
+166
+00:14:53,850 --> 00:14:58,360
+existلاحظا قاعد رايح لأثبت أن هذا exist ده خد الآن
+
+167
+00:14:58,360 --> 00:15:03,520
+من وين؟ من اليسار خد لأن احسب similarly عندي سفر
+
+168
+00:15:03,520 --> 00:15:08,840
+بساوي absolute value of limit f of x على x minus c
+
+169
+00:15:08,840 --> 00:15:15,100
+لما x تروح ل c من وين؟ من اليسار ويساوي عبارة عن
+
+170
+00:15:15,100 --> 00:15:19,960
+limit absolute value of f of x على absolute value
+
+171
+00:15:19,960 --> 00:15:25,900
+of x minus c لما x تروح ل c من وين؟ من اليسارهذه
+
+172
+00:15:25,900 --> 00:15:34,580
+نفسها بساوي limit أو بساوي سالب limit f of x
+
+173
+00:15:34,580 --> 00:15:42,270
+absolute value على x minus c لما x تروحللـ C من
+
+174
+00:15:42,270 --> 00:15:47,030
+اليسار، ليش؟ لأن X أصغر من C، إذا X minus C سالبة
+
+175
+00:15:47,030 --> 00:15:50,230
+إذا الـ absolute value سالب إليها واخدت السالب برا
+
+176
+00:15:50,230 --> 00:15:54,990
+هذا الآن المخضر بيساوي سفر، إذا هذا لحاله برضه إيش
+
+177
+00:15:54,990 --> 00:16:01,270
+ماله؟ سفر، إذا limit absolute value للـ F of X على
+
+178
+00:16:01,270 --> 00:16:04,870
+X minus C، لما X تروح للـ C من اليسار، برضه إيش
+
+179
+00:16:04,870 --> 00:16:10,450
+بيساوي؟ بيساوي سفر، لاحظ إن ال limit من اليمينوالـ
+
+180
+00:16:10,450 --> 00:16:15,890
+limit من اليسار موجود وبساوي 0 متساويين يعني الآن
+
+181
+00:16:15,890 --> 00:16:23,130
+ال limit هذا صار ايش بساوي بساوي 0 إذا الان هذا
+
+182
+00:16:23,130 --> 00:16:29,870
+الآن بنقوله كله تحت هذا وبقول hence اللي هو g
+
+183
+00:16:29,870 --> 00:16:38,310
+prime of c بساوي limit of g of x نقص g of c على x
+
+184
+00:16:38,310 --> 00:16:46,870
+minus cas x → c بساوي حسب اللي عندى من هنا و من
+
+185
+00:16:46,870 --> 00:16:55,310
+هنا و من هنا هيساوي سفر الان conversely بتأفترض
+
+186
+00:16:55,310 --> 00:16:58,090
+طبعا ان ال conversely هيكون الخطوات كثير مشابهه
+
+187
+00:16:58,090 --> 00:17:03,350
+لللي هنا يعني كثير اللي استخدمته هنا هستخدمه في
+
+188
+00:17:03,350 --> 00:17:03,930
+اللي بعدها
+
+189
+00:17:12,670 --> 00:17:17,790
+Conversely suppose that
+
+190
+00:17:17,790 --> 00:17:25,890
+g of x سواء absolute value او f of x اللي
+
+191
+00:17:25,890 --> 00:17:33,550
+هو is differentiable at c بدا وجدلك الان اثبتلك ان
+
+192
+00:17:33,550 --> 00:17:39,650
+f prime of c ايش ما لها بتساوي سفر يعني بدا اثبت
+
+193
+00:17:39,650 --> 00:17:46,880
+limitf of x ناقص f of c اللي هي سفر على x minus c
+
+194
+00:17:46,880 --> 00:17:51,160
+لما x تروح لل c ايش بيساوي بساوي سفر بكون خلصت
+
+195
+00:17:51,160 --> 00:17:58,500
+الان issue مشابه عند الان limit
+
+196
+00:17:58,500 --> 00:18:05,180
+هذا exist عند ال c اذا عندي صار g prime of c exist
+
+197
+00:18:05,180 --> 00:18:10,430
+ويساوي حسب الحديث اللي هنا limitabsolute value of
+
+198
+00:18:10,430 --> 00:18:18,570
+f of x ناقص ال absolute value of f of c صفر على x
+
+199
+00:18:18,570 --> 00:18:26,310
+minus c لما x تروح ل c اشماله exist معايا؟ طيب شوف
+
+200
+00:18:26,310 --> 00:18:34,960
+الآن عندى اذا احسبلي limitf of x اللي حسبناها قبل
+
+201
+00:18:34,960 --> 00:18:39,660
+بشوية بنفس الأسلوب على x minus c لما x تروح ل c من
+
+202
+00:18:39,660 --> 00:18:47,020
+وين؟ من اليمين بساوي limit لل absolute value ل f
+
+203
+00:18:47,020 --> 00:18:53,780
+of x على x minus c لما x تروح ل c من اليمين لأن ال
+
+204
+00:18:53,780 --> 00:19:01,840
+x أكبر من ال c واضحالآن من جهة أخرى احسبلي الـ
+
+205
+00:19:01,840 --> 00:19:08,060
+absolute value لل limit لل F of X طبعا هذا ايش
+
+206
+00:19:08,060 --> 00:19:15,620
+هيساوي بسوى G prime of C، مظبوط؟ بسوى G prime of
+
+207
+00:19:15,620 --> 00:19:23,290
+C، موجود، limit F of Xعلى x minus c لما x تروح ل c
+
+208
+00:19:23,290 --> 00:19:28,570
+من وين؟ من اليسار بساوي زي ما عملنا قبل شوية سالب
+
+209
+00:19:28,570 --> 00:19:33,530
+limit absolute value of f of x على x minus c لما x
+
+210
+00:19:33,530 --> 00:19:41,050
+تروح ل c من وين؟ من اليسار الآن واضح بما أن هذه
+
+211
+00:19:41,050 --> 00:19:51,470
+exist إذا هذا المقدار و هذا المقدار زي بعضالان هذا
+
+212
+00:19:51,470 --> 00:19:56,590
+بيساوي ناقص هذا او بمعنى اخر نشيل الناقص من هنا و
+
+213
+00:19:56,590 --> 00:20:03,390
+نضربه هنا صار عندى المقدرين هدولة بما انه متساويين
+
+214
+00:20:03,390 --> 00:20:10,310
+لإن الاتنين ايش بيساوين ال G prime of C مظبوط اذا
+
+215
+00:20:10,310 --> 00:20:13,390
+صار عندى هذا المقدار بيساوي هذا المقدار اذا صار
+
+216
+00:20:13,390 --> 00:20:20,940
+عندى ال limitF of X على X minus C لما X تروح للـ C
+
+217
+00:20:20,940 --> 00:20:29,560
+من اليمين absolute value بساوي ناقص limit F of X
+
+218
+00:20:29,560 --> 00:20:34,600
+على X minus C لما X تروح للـ C من وين؟ من اليسار
+
+219
+00:20:34,600 --> 00:20:41,500
+واضحة؟ لكن أصلا عندى هذا
+
+220
+00:20:43,370 --> 00:20:49,010
+بساوي اللي هو التنين بساوي نفس القيمة ماشي الحال
+
+221
+00:20:49,010 --> 00:20:55,850
+إذا لازم عندي من هنا جي برايم و جي برايم اللي هو
+
+222
+00:20:55,850 --> 00:21:00,210
+صار بنساوي نفس القيمة إذا هيطلع الجي برايم إيش
+
+223
+00:21:00,210 --> 00:21:05,530
+ماله بساوي سفرإذا صار المقدار هذا كله إش بده يساوي
+
+224
+00:21:05,530 --> 00:21:08,650
+صفر وهذا بيساوي صفر يعني ال limit من اليمين و
+
+225
+00:21:08,650 --> 00:21:12,070
+limit من اليسار متساويين إذا صارت limit F of X على
+
+226
+00:21:12,070 --> 00:21:23,310
+X minus C لما X تروح للـC بدها تساوي صفر أي سؤال؟
+
+227
+00:21:23,310 --> 00:21:31,130
+زي ما حكينا، الآن إحنا قلنا إنه لو كانت D بدينا D
+
+228
+00:21:31,130 --> 00:21:34,600
+بيساوي F of X ال differential بالأد Cجيبنا g prime
+
+229
+00:21:34,600 --> 00:21:37,500
+و كتبناها بالصورة اللي أمامنا بعدين أخدنا ال
+
+230
+00:21:37,500 --> 00:21:40,120
+absolute value لlimit f of x علي x minus c لم x
+
+231
+00:21:40,120 --> 00:21:45,200
+تروح ل c positive طلعت عندي بساوي g prime of c و
+
+232
+00:21:45,200 --> 00:21:48,680
+أخدنا اللي هي ناقص هذه طلعت عندي برضه g prime
+
+233
+00:21:48,680 --> 00:21:53,920
+الاتنين ال c إذا صار هذا بساوي هذا المقدار و احنا
+
+234
+00:21:53,920 --> 00:21:57,640
+بنعرف في الأصل ان f prime of c exist يعني
+
+235
+00:21:57,640 --> 00:22:01,900
+differentiable يعني ال limit هذا اللي هو موجودو من
+
+236
+00:22:01,900 --> 00:22:05,100
+اليمين و من اليسار زي بعض يعني يعني هذا بده يساوي
+
+237
+00:22:05,100 --> 00:22:07,500
+هذا من اليمين و هذا من اليسار يعني في الواقع هذا
+
+238
+00:22:07,500 --> 00:22:10,960
+اللي جوا هو نفسه اللي جوا بساوي limit of of X يعني
+
+239
+00:22:10,960 --> 00:22:15,780
+بمعنى أخر absolute value ل limit of of X على X
+
+240
+00:22:15,780 --> 00:22:22,100
+minus C لما X تروح لل C هو نفسه سالب limit لل of
+
+241
+00:22:22,100 --> 00:22:27,400
+of X على X minus C لما X تروح لل C من وين من
+
+242
+00:22:27,400 --> 00:22:31,540
+اليسار هو نفسه لل C من وين تبت هذالأنه احنا بنقول
+
+243
+00:22:31,540 --> 00:22:34,560
+f prime of c exist يعني ال limit هذا موجود و من
+
+244
+00:22:34,560 --> 00:22:37,360
+اليمين و من اليسار زي بعر إذا صار هذا المقدر نفس
+
+245
+00:22:37,360 --> 00:22:41,200
+هذا المقدر زي ما قلنا إذا صار عند هذا المقدر بساوة
+
+246
+00:22:41,200 --> 00:22:46,800
+سفر لإن التنين بسينا بساوة سفر إذا صار عنده limitF
+
+247
+00:22:46,800 --> 00:22:50,880
+of X على X minus C اللي هو الـ absolute value as X
+
+248
+00:22:50,880 --> 00:22:56,100
+بتروح للـ C بساوة صفر ومن ثم اللي جوا بساوة صفر هو
+
+249
+00:22:56,100 --> 00:22:58,740
+مين هو اللي جوا هذا اللي كنا بدنا نصله من الأول
+
+250
+00:22:58,740 --> 00:23:04,260
+اللي هو F prime of C بتساوة صفر وهو المطلوب هذا
+
+251
+00:23:04,260 --> 00:23:10,300
+توضيح بشكل كامل ل اللي صار في اللي هو الاتجاه
+
+252
+00:23:10,300 --> 00:23:16,200
+الثاني نيجينا سؤال بعد ما خلصنا السؤال سبعةنجي
+
+253
+00:23:16,200 --> 00:23:23,420
+لسؤال تسعة تسعة ايش اللي بقوله تسعة نشوف ايش سؤال
+
+254
+00:23:23,420 --> 00:23:31,360
+تسعة بقول ونحل سؤال تسعة سؤال تسعة بقول ليه
+
+255
+00:23:31,360 --> 00:23:39,110
+باختصار انه لو كان عنده ال function ofعبارة عن من
+
+256
+00:23:39,110 --> 00:23:43,130
+R لـ R even function طبعا عارفين ايش ال even اللي
+
+257
+00:23:43,130 --> 00:23:49,130
+هو F ناقص X بيساوي F X لكل X موجودة في الـ R and
+
+258
+00:23:49,130 --> 00:23:54,310
+has a derivative at every point then F prime is an
+
+259
+00:23:54,310 --> 00:23:58,190
+odd function يعني بيقول لي لو كانت باختصار يعني لو
+
+260
+00:23:58,190 --> 00:24:01,730
+كانت ال F even و ال derivative موجودة بتكون ال
+
+261
+00:24:01,730 --> 00:24:07,010
+derivative odd و لو كانت ال derivative oddبتكون
+
+262
+00:24:07,010 --> 00:24:11,350
+الـ function الـ G prime إيه شمالها is even انحل
+
+263
+00:24:11,350 --> 00:24:16,970
+واحدة من هنا والتانية similarly زيها الآن لنفترض F
+
+264
+00:24:16,970 --> 00:24:29,890
+من عند R لعند R be an odd differentiable function
+
+265
+00:24:29,890 --> 00:24:34,970
+ماشي الحال show that
+
+266
+00:24:36,040 --> 00:24:44,360
+F' is even a proof بدنا نثبت أنه لو كانت الـ F
+
+267
+00:24:44,360 --> 00:24:49,940
+اللي هي odd function بدو يكون عنده و
+
+268
+00:24:49,940 --> 00:24:52,320
+differentiable بدو يكون عنده derivative إلها إيه
+
+269
+00:24:52,320 --> 00:25:03,000
+إيش even لأن let C element in R be arbitrary and
+
+270
+00:25:03,000 --> 00:25:04,180
+fixed
+
+271
+00:25:06,310 --> 00:25:15,930
+NR ناخد الـ R ناخد الـ C أي اللي هو real number in
+
+272
+00:25:15,930 --> 00:25:23,360
+R لكن نحكي عن اي شي محدد الان F prime of Cبدأ
+
+273
+00:25:23,360 --> 00:25:29,060
+أثبتلك أنه هو بسوء F prime of ناقص C يعني F prime
+
+274
+00:25:29,060 --> 00:25:34,920
+is even إذا خد F prime ناقص C وابدأ حسب وواصلك في
+
+275
+00:25:34,920 --> 00:25:39,140
+النهاية بسوء F prime of C إذا F prime is even بسوء
+
+276
+00:25:39,140 --> 00:25:49,380
+limit اللي هي F of X ناقص F of minus C على X minus
+
+277
+00:25:50,570 --> 00:25:57,070
+اللي هو minus C لما X تروح لمين لـ minus C مظبوط
+
+278
+00:25:57,070 --> 00:26:05,550
+طيب و يساوي limit ال F إيش معتنيها عبارة عن odd
+
+279
+00:26:05,550 --> 00:26:12,250
+إيش يعني odd يعني F of ناقص X بيساوي ناقص F of X
+
+280
+00:26:12,250 --> 00:26:16,590
+مظبوط هذه اللي هي إيش ما لها odd function اللي هو
+
+281
+00:26:16,590 --> 00:26:28,550
+بيساوي limit F ofالان اللي هي ناقص f of ناقص x
+
+282
+00:26:28,550 --> 00:26:34,150
+واضح أه؟
+
+283
+00:26:34,150 --> 00:26:37,150
+f of ناقص x بيساوي ناقص f of x يعني f of x بيساوي
+
+284
+00:26:37,150 --> 00:26:41,230
+ناقص f of ناقص x فالجد تعرفوا ليش بعملت هيك لأن
+
+285
+00:26:41,230 --> 00:26:48,570
+ناقص f of ناقص c الان f is odd مظبوط بيصير زائد f
+
+286
+00:26:48,570 --> 00:26:49,330
+of c
+
+287
+00:26:52,220 --> 00:26:58,060
+على خد الان ناقص من هنا عامل مشترك بيصير عبارة عن
+
+288
+00:26:58,060 --> 00:27:08,220
+ناقص X الان ناقص ال C لما X تروح لمين لسالب ال C
+
+289
+00:27:08,220 --> 00:27:14,040
+ال X بتروح لسالب ال C إذا و فقط إذا سالب ال X
+
+290
+00:27:14,040 --> 00:27:22,160
+بتروح لمين إلى ال C الان خد لي Yبساوي سالب ال X
+
+291
+00:27:22,160 --> 00:27:29,240
+واستبدل عشان نضح لك إياه بساوي limit الان خد الان
+
+292
+00:27:29,240 --> 00:27:38,320
+عندي F حكما كان نقص اكسميا ال Y limit ناقص F of Y
+
+293
+00:27:38,320 --> 00:27:49,390
+زائد F of C على اللي هو Y بتروح إلى ال Cوهنا y
+
+294
+00:27:49,390 --> 00:27:57,430
+ناقص ال c وهنا في عندي ايش برضه ناقص برا لان خد من
+
+295
+00:27:57,430 --> 00:28:01,670
+هنا ناقص عام المشترك او ضيعه مع الناقص اللي هنا
+
+296
+00:28:01,670 --> 00:28:11,160
+ماضح اه بيصير عندي y ساوي limit f of yنقص f of c
+
+297
+00:28:11,160 --> 00:28:16,800
+على y minus c لما y تروح لل c وهذا عبارة عن f
+
+298
+00:28:16,800 --> 00:28:22,440
+prime لمين لل c بدأنا ب f prime نقص c وانتهينا ب f
+
+299
+00:28:22,440 --> 00:28:31,520
+prime لل c لذا therefore f prime is even whenever
+
+300
+00:28:31,520 --> 00:28:37,160
+f is odd and f is differentiable
+
+301
+00:28:40,710 --> 00:28:56,810
+السؤال الأخير السؤال 13 السؤال
+
+302
+00:28:56,810 --> 00:29:03,990
+13 هو كما يلي اش
+
+303
+00:29:03,990 --> 00:29:09,200
+اللي بقول السؤال 13سؤال 13 بيقول إذا كانت F من R
+
+304
+00:29:09,200 --> 00:29:12,380
+لـ R is differentiable at C element in R show that
+
+305
+00:29:12,380 --> 00:29:16,840
+F prime of C سوى limit N في F of C زائد 1 لأن ناقص
+
+306
+00:29:16,840 --> 00:29:20,620
+F of C as N goes to infinity يعني لو كانت F
+
+307
+00:29:20,620 --> 00:29:24,900
+differentiable عند الـ C element in R بنقدر نكتب
+
+308
+00:29:24,900 --> 00:29:27,920
+ال derivative اللي هي F prime of C على سورة limit
+
+309
+00:29:27,920 --> 00:29:32,320
+N F of C زائد 1 لأن ناقص F of C as N goes to
+
+310
+00:29:32,320 --> 00:29:38,120
+infinityلكن بيقول لي by example show that the
+
+311
+00:29:38,120 --> 00:29:44,000
+existence of this limit this limit need not اللي
+
+312
+00:29:44,000 --> 00:29:52,300
+هو imply the existence of the derivative نيجي
+
+313
+00:29:52,300 --> 00:30:03,960
+الآن للجزء الأول عندي F من R لRو F prime of C
+
+314
+00:30:03,960 --> 00:30:12,120
+exist لانه يقول لي prove that F prime of C can be
+
+315
+00:30:12,120 --> 00:30:20,760
+written as limit on F of C زائد واحدة لان ناقص F
+
+316
+00:30:20,760 --> 00:30:26,180
+of C as N goes to infinity هذا الجزء الأول الجزء
+
+317
+00:30:26,180 --> 00:30:32,400
+الثاني حنجي الان نقول prove للجزء الأولقبل ما نقول
+
+318
+00:30:32,400 --> 00:30:37,700
+الـ proof نذكركم بس بنظرية سابقة في الـ real واحد
+
+319
+00:30:37,700 --> 00:30:43,840
+إنه لو عندي limit f of x لما x تروح لل c لو كانت
+
+320
+00:30:43,840 --> 00:30:52,780
+بتساوي Lبكون عندى أي sequence xₙ بتروح للـ C لازم
+
+321
+00:30:52,780 --> 00:30:59,020
+يتحقق لها limit f of xₙ as n goes to infinity
+
+322
+00:30:59,020 --> 00:31:05,530
+بساوي برضه الـكنا نتحدث بالحديث، باستبداء الحديث
+
+323
+00:31:05,530 --> 00:31:08,670
+عن الـ limit العادية للـ function S X بتروح للـ C
+
+324
+00:31:08,670 --> 00:31:13,470
+إلى limit لمين للـ sequence أو limit للـ sequences
+
+325
+00:31:13,470 --> 00:31:17,290
+الآن بنستخل هذه اللي هو المعلومة في إثبات اللي
+
+326
+00:31:17,290 --> 00:31:22,630
+بدنايا عند الآن since F prime of C exist بما أنه
+
+327
+00:31:22,630 --> 00:31:29,250
+ال derivative عند C موجودةإذا أكيد عندي صار f
+
+328
+00:31:29,250 --> 00:31:37,010
+prime of c بسهولمة f of x أو اللي هي الـ x عند c
+
+329
+00:31:38,080 --> 00:31:46,140
+زائد h ناقص f of c على h as h goes to mean to zero
+
+330
+00:31:46,140 --> 00:31:49,860
+اللي هو التعريف التعريف ال derivative أو الشكل
+
+331
+00:31:49,860 --> 00:31:52,980
+الآخر للتعريف ال derivative f of c زائد ال
+
+332
+00:31:52,980 --> 00:31:55,560
+increment ناقص f of c على ال increment as ال
+
+333
+00:31:55,560 --> 00:32:01,600
+increment goes to mean to zero ماشي الحل ال L بما
+
+334
+00:32:01,600 --> 00:32:07,480
+أنه اللي هو ال 1 على N sequence بتروح للسفروهذا
+
+335
+00:32:07,480 --> 00:32:11,440
+الـ limit exist لأن حسب النظرية اللي حكيتها قبل
+
+336
+00:32:11,440 --> 00:32:21,100
+بشوية بكون عندي لأن but كذا then F prime of C can
+
+337
+00:32:21,100 --> 00:32:26,580
+be اللي هو إعادة اللي هي rewritten as a limit of a
+
+338
+00:32:26,580 --> 00:32:36,800
+sequence F of اللي هو limit limitF of C زائد but
+
+339
+00:32:36,800 --> 00:32:40,000
+الـ H اللي هي تروح للصفر صارت mean ال sequence
+
+340
+00:32:40,000 --> 00:32:46,380
+واحدة ل N تروح للصفر واحدة ل N ناقص F of C على
+
+341
+00:32:46,380 --> 00:32:52,640
+واحدة ل N as N goes to infinity مدامت ال sequence
+
+342
+00:32:52,640 --> 00:32:57,100
+واحدة ل N بتروح للصفر صارت ال F of واحدة ل N اللي
+
+343
+00:32:57,100 --> 00:33:00,900
+هي عبارة عن F of C زائد واحدة ل N لإن الـ C عبارة
+
+344
+00:33:00,900 --> 00:33:07,740
+عن إياش فالتةواضح أه؟ الان هذا بيساوي اللي هو
+
+345
+00:33:07,740 --> 00:33:14,600
+limit الان as n goes to infinity اللي هو أكيد اللي
+
+346
+00:33:14,600 --> 00:33:21,920
+هو بيصير عندي f n في الجوس f of c زائد واحدة ل n
+
+347
+00:33:21,920 --> 00:33:28,260
+ناقص f of c اللي هو as n goes to infinity هو هذا
+
+348
+00:33:28,260 --> 00:33:35,150
+صار f prime of c وهو المطلوبالان conversely the
+
+349
+00:33:35,150 --> 00:33:37,890
+converse need not to be true in general هيك بيقول
+
+350
+00:33:37,890 --> 00:33:45,390
+ليه يعني بيقوللي if if بيقوللي او اللي بيقول if f
+
+351
+00:33:45,390 --> 00:33:53,670
+برا اللي هو limit f of c زاد واحدة ل n ناقص f of c
+
+352
+00:33:53,670 --> 00:34:01,610
+الكل مضروف في n as n goes to infinity exist if كده
+
+353
+00:34:02,500 --> 00:34:12,640
+then f prime at c need not be exist اصلا مش انه
+
+354
+00:34:12,640 --> 00:34:16,180
+يكون بيسوي هذا او لا need not to be اشمله exist
+
+355
+00:34:16,180 --> 00:34:19,640
+لان لو كان exist على طول بيسوي اما need not to be
+
+356
+00:34:19,640 --> 00:34:28,200
+exist ومااخد مثال جالي consider consider consider
+
+357
+00:34:28,200 --> 00:34:37,000
+f of xبساوي ال absolute value لل X وخد عند ال C أش
+
+358
+00:34:37,000 --> 00:34:46,440
+بتساوي سفر واضح F prime of 0 does not exist لأنها
+
+359
+00:34:46,440 --> 00:34:48,780
+عبارة عن corner point، انتوا عارفين احنا اللي هو
+
+360
+00:34:48,780 --> 00:34:52,480
+ال F prime عند ال zero لل absolute value does not
+
+361
+00:34:52,480 --> 00:34:59,740
+exist لكن هذه متحققة، ليش؟ but limit
+
+362
+00:35:01,280 --> 00:35:10,900
+N في F of 0 زائد 1 على N نقص F of 0 as N goes to
+
+363
+00:35:10,900 --> 00:35:18,840
+infinity بساوي limit N F of 0 زائد 1 على N يعني F
+
+364
+00:35:18,840 --> 00:35:23,560
+of 1 على N F of X بساوي absolute value X ف1 على N
+
+365
+00:35:23,560 --> 00:35:28,960
+مظبوط أه as N goes to infinity طبعا الـ N بتروح
+
+366
+00:35:29,220 --> 00:35:35,100
+يصير عبارة عن ash واحد إذا فعلا جبنا مثال أن ال
+
+367
+00:35:35,100 --> 00:35:39,720
+limit هذه تكون exist و سوى واحد but ال F prime عند
+
+368
+00:35:39,720 --> 00:35:42,640
+هذا النقطة C اللي هي 00 في هذه الحالة does not
+
+369
+00:35:42,640 --> 00:35:47,320
+exist بيكون هيك احنا انتهينا من الجزء الأول من
+
+370
+00:35:47,320 --> 00:35:54,870
+المحاضرة الخامسةاللي هو discussion لأو مناقشة ل
+
+371
+00:35:54,870 --> 00:35:59,910
+section 6-1 اللي هو the derivative والآن سنكمل
+
+372
+00:35:59,910 --> 00:36:05,690
+الحديث في الجزء الثاني من المحاضرة اللي هو عن اللي
+
+373
+00:36:05,690 --> 00:36:09,250
+هو the mean value theorem أو اللي هي نكمل
+
+374
+00:36:09,250 --> 00:36:11,910
+applications على mean value theorem وناخد اللي هو
+
+375
+00:36:11,910 --> 00:36:12,030
+ال
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..aa8a496058b0c26fa52efb39cc735f8adb72d0f8
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_raw.json
@@ -0,0 +1 @@
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870.32, "word": " بتروح", "probability": 0.8341471354166666}, {"start": 870.32, "end": 870.44, "word": " ل", "probability": 0.8330078125}, {"start": 870.44, "end": 870.56, "word": " C", "probability": 0.435791015625}, {"start": 870.56, "end": 870.76, "word": " من", "probability": 0.9365234375}, {"start": 870.76, "end": 871.98, "word": " وين؟", "probability": 0.8779296875}, {"start": 871.98, "end": 872.14, "word": " من", "probability": 0.98828125}, {"start": 872.14, "end": 872.64, "word": " اليمين", "probability": 0.8865559895833334}, {"start": 872.64, "end": 873.36, "word": " يعني", "probability": 0.903076171875}, {"start": 873.36, "end": 874.1, "word": " هذا", "probability": 0.9482421875}, {"start": 874.1, "end": 874.96, "word": " المقدار", "probability": 0.9228515625}], "temperature": 1.0}, {"id": 35, "seek": 89419, "start": 875.57, "end": 894.19, "text": "صار موجود وإيش بساوي بساوي سفر هذا خلّيه لإنه هذا اللي بدنا نصلّه لأن limit صار عندي معنى آخر limit absolute value of f of x على x minus c لما x تروح لل c من اليمين بساوي سفر exist", "tokens": [9381, 9640, 3714, 29245, 23328, 4032, 28814, 1829, 8592, 4724, 3794, 995, 45865, 4724, 3794, 995, 45865, 8608, 5172, 2288, 23758, 16490, 1211, 11703, 1829, 3224, 5296, 28814, 1863, 3224, 23758, 13672, 1829, 47525, 8315, 8717, 36520, 11703, 3224, 5296, 33456, 4948, 20328, 9640, 18871, 16254, 20449, 1863, 7578, 19753, 34740, 4948, 8236, 2158, 295, 283, 295, 2031, 15844, 2031, 3175, 269, 5296, 15042, 2031, 6055, 32887, 5016, 24976, 269, 9154, 2423, 32640, 9957, 4724, 3794, 995, 45865, 8608, 5172, 2288, 2514], "avg_logprob": -0.22025602014668016, "compression_ratio": 1.5274725274725274, "no_speech_prob": 1.6689300537109375e-06, "words": [{"start": 875.57, "end": 875.95, "word": "صار", "probability": 0.63916015625}, {"start": 875.95, "end": 876.37, "word": " موجود", "probability": 0.9705403645833334}, {"start": 876.37, "end": 876.61, "word": " وإيش", "probability": 0.729736328125}, {"start": 876.61, "end": 877.17, "word": " بساوي", "probability": 0.838134765625}, {"start": 877.17, "end": 878.33, "word": " بساوي", "probability": 0.794189453125}, {"start": 878.33, "end": 878.75, "word": " سفر", "probability": 0.7451171875}, {"start": 878.75, "end": 879.53, "word": " هذا", "probability": 0.638671875}, {"start": 879.53, "end": 880.01, "word": " خلّيه", "probability": 0.83818359375}, {"start": 880.01, "end": 880.19, "word": " لإنه", "probability": 0.7928466796875}, {"start": 880.19, "end": 880.33, "word": " هذا", "probability": 0.8857421875}, {"start": 880.33, "end": 880.45, "word": " اللي", "probability": 0.96875}, {"start": 880.45, "end": 880.63, "word": " بدنا", "probability": 0.88134765625}, {"start": 880.63, "end": 881.19, "word": " نصلّه", "probability": 0.8170166015625}, {"start": 881.19, "end": 881.77, "word": " لأن", "probability": 0.774658203125}, {"start": 881.77, "end": 882.35, "word": " limit", "probability": 0.9052734375}, {"start": 882.35, "end": 882.81, "word": " صار", "probability": 0.9736328125}, {"start": 882.81, "end": 883.09, "word": " عندي", "probability": 0.79931640625}, {"start": 883.09, "end": 883.47, "word": " معنى", "probability": 0.96337890625}, {"start": 883.47, "end": 883.89, "word": " آخر", "probability": 0.9248046875}, {"start": 883.89, "end": 884.85, "word": " limit", "probability": 0.94287109375}, {"start": 884.85, "end": 885.89, "word": " absolute", "probability": 0.63623046875}, {"start": 885.89, "end": 886.35, "word": " value", "probability": 0.9873046875}, {"start": 886.35, "end": 886.55, "word": " of", "probability": 0.394775390625}, {"start": 886.55, "end": 886.77, "word": " f", "probability": 0.65869140625}, {"start": 886.77, "end": 886.93, "word": " of", "probability": 0.44384765625}, {"start": 886.93, "end": 887.31, "word": " x", "probability": 0.68017578125}, {"start": 887.31, "end": 888.33, "word": " على", "probability": 0.68310546875}, {"start": 888.33, "end": 888.73, "word": " x", "probability": 0.83935546875}, {"start": 888.73, "end": 889.17, "word": " minus", "probability": 0.69091796875}, {"start": 889.17, "end": 889.47, "word": " c", "probability": 0.83056640625}, {"start": 889.47, "end": 889.79, "word": " لما", "probability": 0.891845703125}, {"start": 889.79, "end": 890.07, "word": " x", "probability": 0.8466796875}, {"start": 890.07, "end": 890.45, "word": " تروح", "probability": 0.994140625}, {"start": 890.45, "end": 890.71, "word": " لل", "probability": 0.68359375}, {"start": 890.71, "end": 891.07, "word": " c", "probability": 0.358642578125}, {"start": 891.07, "end": 892.07, "word": " من", "probability": 0.9921875}, {"start": 892.07, "end": 892.85, "word": " اليمين", "probability": 0.8310546875}, {"start": 892.85, "end": 893.47, "word": " بساوي", "probability": 0.92236328125}, {"start": 893.47, "end": 893.85, "word": " سفر", "probability": 0.9275716145833334}, {"start": 893.85, "end": 894.19, "word": " exist", "probability": 0.6865234375}], "temperature": 1.0}, {"id": 36, "seek": 92472, "start": 895.12, "end": 924.72, "text": "لاحظا قاعد رايح لأثبت أن هذا exist ده خد الآن من وين؟ من اليسار خد لأن احسب similarly عندي سفر بساوي absolute value of limit f of x على x minus c لما x تروح ل c من وين؟ من اليسار ويساوي عبارة عن limit absolute value of f of x على absolute value of x minus c لما x تروح ل c من وين؟ من اليسار", "tokens": [15040, 5016, 19913, 995, 12174, 995, 22488, 12602, 47302, 5016, 5296, 10721, 12984, 3555, 2655, 14739, 23758, 2514, 11778, 3224, 16490, 3215, 6024, 48506, 9154, 4032, 9957, 22807, 9154, 45595, 3794, 9640, 16490, 3215, 5296, 33456, 1975, 5016, 35457, 14138, 18871, 16254, 8608, 5172, 2288, 4724, 3794, 995, 45865, 8236, 2158, 295, 4948, 283, 295, 2031, 15844, 2031, 3175, 269, 5296, 15042, 2031, 6055, 32887, 5016, 5296, 269, 9154, 4032, 9957, 22807, 9154, 45595, 3794, 9640, 4032, 1829, 3794, 995, 45865, 6225, 3555, 9640, 3660, 18871, 4948, 8236, 2158, 295, 283, 295, 2031, 15844, 8236, 2158, 295, 2031, 3175, 269, 5296, 15042, 2031, 6055, 32887, 5016, 5296, 269, 9154, 4032, 9957, 22807, 9154, 45595, 3794, 9640], "avg_logprob": -0.22729700905644995, "compression_ratio": 2.053921568627451, "no_speech_prob": 6.735324859619141e-06, "words": [{"start": 895.12, "end": 895.72, "word": "لاحظا", "probability": 0.7103271484375}, {"start": 895.72, "end": 895.98, "word": " قاعد", "probability": 0.633056640625}, {"start": 895.98, "end": 896.32, "word": " رايح", "probability": 0.7725423177083334}, {"start": 896.32, "end": 897.18, "word": " لأثبت", "probability": 0.821630859375}, {"start": 897.18, "end": 897.28, "word": " أن", "probability": 0.6357421875}, {"start": 897.28, "end": 897.48, "word": " هذا", "probability": 0.76904296875}, {"start": 897.48, "end": 897.8, "word": " exist", "probability": 0.422119140625}, {"start": 897.8, "end": 898.02, "word": " ده", "probability": 0.542236328125}, {"start": 898.02, "end": 898.16, "word": " خد", "probability": 0.810546875}, {"start": 898.16, "end": 898.36, "word": " الآن", "probability": 0.53778076171875}, {"start": 898.36, "end": 898.5, "word": " من", "probability": 0.94580078125}, {"start": 898.5, "end": 898.88, "word": " وين؟", "probability": 0.7643229166666666}, {"start": 898.88, "end": 898.96, "word": " من", "probability": 0.97265625}, {"start": 898.96, "end": 899.42, "word": " اليسار", "probability": 0.9921875}, {"start": 899.42, "end": 900.3, "word": " خد", "probability": 0.906494140625}, {"start": 900.3, "end": 900.56, "word": " لأن", "probability": 0.763427734375}, {"start": 900.56, "end": 900.98, "word": " احسب", "probability": 0.8865559895833334}, {"start": 900.98, "end": 901.78, "word": " similarly", "probability": 0.5146484375}, {"start": 901.78, "end": 902.7, "word": " عندي", "probability": 0.5999755859375}, {"start": 902.7, "end": 903.52, "word": " سفر", "probability": 0.6846923828125}, {"start": 903.52, "end": 904.06, "word": " بساوي", "probability": 0.8065185546875}, {"start": 904.06, "end": 904.62, "word": " absolute", "probability": 0.55029296875}, {"start": 904.62, "end": 905.1, "word": " value", "probability": 0.986328125}, {"start": 905.1, "end": 905.22, "word": " of", "probability": 0.32763671875}, {"start": 905.22, "end": 905.72, "word": " limit", "probability": 0.78515625}, {"start": 905.72, "end": 906.6, "word": " f", "probability": 0.2342529296875}, {"start": 906.6, "end": 906.8, "word": " of", "probability": 0.36962890625}, {"start": 906.8, "end": 907.22, "word": " x", "probability": 0.66162109375}, {"start": 907.22, "end": 907.96, "word": " على", "probability": 0.5625}, {"start": 907.96, "end": 908.22, "word": " x", "probability": 0.82177734375}, {"start": 908.22, "end": 908.54, "word": " minus", "probability": 0.75634765625}, {"start": 908.54, "end": 908.84, "word": " c", "probability": 0.8994140625}, {"start": 908.84, "end": 909.12, "word": " لما", "probability": 0.8759765625}, {"start": 909.12, "end": 909.36, "word": " x", "probability": 0.861328125}, {"start": 909.36, "end": 909.66, "word": " تروح", "probability": 0.9881184895833334}, {"start": 909.66, "end": 909.8, "word": " ل", "probability": 0.62646484375}, {"start": 909.8, "end": 909.96, "word": " c", "probability": 0.5986328125}, {"start": 909.96, "end": 910.18, "word": " من", "probability": 0.98486328125}, {"start": 910.18, "end": 911.48, "word": " وين؟", "probability": 0.9677734375}, {"start": 911.48, "end": 911.7, "word": " من", "probability": 0.9794921875}, {"start": 911.7, "end": 912.76, "word": " اليسار", "probability": 0.9978841145833334}, {"start": 912.76, "end": 913.72, "word": " ويساوي", "probability": 0.8728515625}, {"start": 913.72, "end": 914.84, "word": " عبارة", "probability": 0.9930419921875}, {"start": 914.84, "end": 915.1, "word": " عن", "probability": 0.9951171875}, {"start": 915.1, "end": 915.9, "word": " limit", "probability": 0.98486328125}, {"start": 915.9, "end": 917.02, "word": " absolute", "probability": 0.951171875}, {"start": 917.02, "end": 917.46, "word": " value", "probability": 0.98876953125}, {"start": 917.46, "end": 917.62, "word": " of", "probability": 0.64501953125}, {"start": 917.62, "end": 917.82, "word": " f", "probability": 0.939453125}, {"start": 917.82, "end": 918.02, "word": " of", "probability": 0.91455078125}, {"start": 918.02, "end": 918.44, "word": " x", "probability": 0.9912109375}, {"start": 918.44, "end": 919.16, "word": " على", "probability": 0.7431640625}, {"start": 919.16, "end": 919.6, "word": " absolute", "probability": 0.94580078125}, {"start": 919.6, "end": 919.96, "word": " value", "probability": 0.98583984375}, {"start": 919.96, "end": 920.04, "word": " of", "probability": 0.875}, {"start": 920.04, "end": 920.2, "word": " x", "probability": 0.9892578125}, {"start": 920.2, "end": 920.44, "word": " minus", "probability": 0.986328125}, {"start": 920.44, "end": 920.94, "word": " c", "probability": 0.98681640625}, {"start": 920.94, "end": 921.7, "word": " لما", "probability": 0.97802734375}, {"start": 921.7, "end": 921.96, "word": " x", "probability": 0.96923828125}, {"start": 921.96, "end": 922.28, "word": " تروح", "probability": 0.9952799479166666}, {"start": 922.28, "end": 922.42, "word": " ل", "probability": 0.8642578125}, {"start": 922.42, "end": 922.6, "word": " c", "probability": 0.9384765625}, {"start": 922.6, "end": 922.8, "word": " من", "probability": 0.99267578125}, {"start": 922.8, "end": 923.62, "word": " وين؟", "probability": 0.98974609375}, {"start": 923.62, "end": 924.18, "word": " من", "probability": 0.9912109375}, {"start": 924.18, "end": 924.72, "word": " اليسار", "probability": 0.9983723958333334}], "temperature": 1.0}, {"id": 37, "seek": 94102, "start": 925.52, "end": 941.02, "text": "هذه نفسها بساوي limit أو بساوي سالب limit f of x absolute value على x minus c لما x تروح", "tokens": [3224, 24192, 8717, 36178, 11296, 4724, 3794, 995, 45865, 4948, 34051, 4724, 3794, 995, 45865, 8608, 6027, 3555, 4948, 283, 295, 2031, 8236, 2158, 15844, 2031, 3175, 269, 5296, 15042, 2031, 6055, 32887, 5016], "avg_logprob": -0.1620535739830562, "compression_ratio": 1.1401869158878504, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 925.52, "end": 925.9, "word": "هذه", "probability": 0.818115234375}, {"start": 925.9, "end": 926.64, "word": " نفسها", "probability": 0.9835611979166666}, {"start": 926.64, "end": 927.44, "word": " بساوي", "probability": 0.760498046875}, {"start": 927.44, "end": 928.56, "word": " limit", "probability": 0.83740234375}, {"start": 928.56, "end": 929.92, "word": " أو", "probability": 0.6162109375}, {"start": 929.92, "end": 930.42, "word": " بساوي", "probability": 0.957275390625}, {"start": 930.42, "end": 931.38, "word": " سالب", "probability": 0.9080403645833334}, {"start": 931.38, "end": 932.8, "word": " limit", "probability": 0.9296875}, {"start": 932.8, "end": 934.02, "word": " f", "probability": 0.23681640625}, {"start": 934.02, "end": 934.28, "word": " of", "probability": 0.55859375}, {"start": 934.28, "end": 934.58, "word": " x", "probability": 0.794921875}, {"start": 934.58, "end": 934.96, "word": " absolute", "probability": 0.88427734375}, {"start": 934.96, "end": 935.64, "word": " value", "probability": 0.99267578125}, {"start": 935.64, "end": 936.64, "word": " على", "probability": 0.67578125}, {"start": 936.64, "end": 937.58, "word": " x", "probability": 0.83642578125}, {"start": 937.58, "end": 938.1, "word": " minus", "probability": 0.7470703125}, {"start": 938.1, "end": 938.6, "word": " c", "probability": 0.8095703125}, {"start": 938.6, "end": 940.08, "word": " لما", "probability": 0.830810546875}, {"start": 940.08, "end": 940.4, "word": " x", "probability": 0.84765625}, {"start": 940.4, "end": 941.02, "word": " تروح", "probability": 0.9894205729166666}], "temperature": 1.0}, {"id": 38, "seek": 96921, "start": 941.69, "end": 969.21, "text": "للـ C من اليسار، ليش؟ لأن X أصغر من C، إذا X minus C سالبة إذا الـ absolute value سالب إليها واخدت السالب برا هذا الآن المخضر بيساوي سفر، إذا هذا لحاله برضه إيش ماله؟ سفر، إذا limit absolute 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20193, 5016, 19913, 36145, 2423, 4948, 9154, 45595, 2304, 9957], "avg_logprob": -0.2059563770390197, "compression_ratio": 2.059071729957806, "no_speech_prob": 1.9669532775878906e-06, "words": [{"start": 941.69, "end": 942.03, "word": "للـ", "probability": 0.6399739583333334}, {"start": 942.03, "end": 942.15, "word": " C", "probability": 0.5087890625}, {"start": 942.15, "end": 942.27, "word": " من", "probability": 0.95751953125}, {"start": 942.27, "end": 942.77, "word": " اليسار،", "probability": 0.82489013671875}, {"start": 942.77, "end": 943.73, "word": " ليش؟", "probability": 0.755126953125}, {"start": 943.73, "end": 944.07, "word": " لأن", "probability": 0.8916015625}, {"start": 944.07, "end": 944.59, "word": " X", "probability": 0.89013671875}, {"start": 944.59, "end": 944.99, "word": " أصغر", "probability": 0.9705810546875}, {"start": 944.99, "end": 945.17, "word": " من", "probability": 0.9931640625}, {"start": 945.17, "end": 945.61, "word": " C،", "probability": 0.669921875}, 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955.79, "word": " ماله؟", "probability": 0.82879638671875}, {"start": 955.79, "end": 956.23, "word": " سفر،", "probability": 0.9503173828125}, {"start": 956.23, "end": 956.53, "word": " إذا", "probability": 0.938720703125}, {"start": 956.53, "end": 957.87, "word": " limit", "probability": 0.91552734375}, {"start": 957.87, "end": 958.91, "word": " absolute", "probability": 0.92236328125}, {"start": 958.91, "end": 959.39, "word": " value", "probability": 0.9892578125}, {"start": 959.39, "end": 959.61, "word": " للـ", "probability": 0.5020751953125}, {"start": 959.61, "end": 959.77, "word": " F", "probability": 0.61474609375}, {"start": 959.77, "end": 959.93, "word": " of", "probability": 0.93115234375}, {"start": 959.93, "end": 960.37, "word": " X", "probability": 0.99658203125}, {"start": 960.37, "end": 961.27, "word": " على", "probability": 0.73779296875}, {"start": 961.27, "end": 961.63, "word": " X", "probability": 0.986328125}, {"start": 961.63, "end": 961.95, "word": " minus", "probability": 0.9794921875}, {"start": 961.95, "end": 962.37, "word": " C،", "probability": 0.752197265625}, {"start": 962.37, "end": 962.53, "word": " لما", "probability": 0.955322265625}, {"start": 962.53, "end": 962.77, "word": " X", "probability": 0.9677734375}, {"start": 962.77, "end": 963.05, "word": " تروح", "probability": 0.9830729166666666}, {"start": 963.05, "end": 963.29, "word": " للـ", "probability": 0.748291015625}, {"start": 963.29, "end": 963.39, "word": " C", "probability": 0.82958984375}, {"start": 963.39, "end": 963.53, "word": " من", "probability": 0.9892578125}, {"start": 963.53, "end": 964.39, "word": " اليسار،", "probability": 0.8795166015625}, {"start": 964.39, "end": 964.67, "word": " برضه", "probability": 0.9876302083333334}, {"start": 964.67, "end": 964.87, "word": " إيش", "probability": 0.9547526041666666}, {"start": 964.87, "end": 965.73, "word": " بيساوي؟", "probability": 0.9657389322916666}, {"start": 965.73, "end": 966.13, "word": " بيساوي", 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996.91, "word": " of", "probability": 0.95263671875}, {"start": 996.91, "end": 997.31, "word": " c", "probability": 0.93017578125}, {"start": 997.31, "end": 998.03, "word": " على", "probability": 0.7041015625}, {"start": 998.03, "end": 998.31, "word": " x", "probability": 0.95849609375}, {"start": 998.31, "end": 998.61, "word": " minus", "probability": 0.91259765625}, {"start": 998.61, "end": 999.03, "word": " c", "probability": 0.9267578125}], "temperature": 1.0}, {"id": 40, "seek": 102393, "start": 999.39, "end": 1023.93, "text": "as x → c بساوي حسب اللي عندى من هنا و من هنا و من هنا هيساوي سفر الان conversely بتأفترض طبعا ان ال conversely هيكون الخطوات كثير مشابهه لللي هنا يعني كثير اللي استخدمته هنا هستخدمه في اللي بعدها", "tokens": [296, 2031, 41600, 269, 4724, 3794, 995, 45865, 11331, 35457, 13672, 1829, 43242, 7578, 9154, 34105, 4032, 9154, 34105, 4032, 9154, 34105, 39896, 3794, 995, 45865, 8608, 5172, 2288, 2423, 7649, 2615, 736, 39894, 10721, 5172, 2655, 43042, 23032, 3555, 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1003.31, "end": 1004.53, "word": " عندى", "probability": 0.594482421875}, {"start": 1004.53, "end": 1006.19, "word": " من", "probability": 0.7734375}, {"start": 1006.19, "end": 1006.45, "word": " هنا", "probability": 0.5595703125}, {"start": 1006.45, "end": 1006.69, "word": " و", "probability": 0.826171875}, {"start": 1006.69, "end": 1006.87, "word": " من", "probability": 0.732421875}, {"start": 1006.87, "end": 1007.15, "word": " هنا", "probability": 0.98974609375}, {"start": 1007.15, "end": 1007.33, "word": " و", "probability": 0.95458984375}, {"start": 1007.33, "end": 1007.47, "word": " من", "probability": 0.96875}, {"start": 1007.47, "end": 1007.79, "word": " هنا", "probability": 0.9912109375}, {"start": 1007.79, "end": 1008.99, "word": " هيساوي", "probability": 0.6663818359375}, {"start": 1008.99, "end": 1009.89, "word": " سفر", "probability": 0.85546875}, {"start": 1009.89, "end": 1011.45, "word": " الان", "probability": 0.65771484375}, {"start": 1011.45, "end": 1012.27, "word": " conversely", "probability": 0.82080078125}, {"start": 1012.27, "end": 1015.31, "word": " بتأفترض", "probability": 0.46146240234375}, {"start": 1015.31, "end": 1015.67, "word": " طبعا", "probability": 0.8831787109375}, {"start": 1015.67, "end": 1015.79, "word": " ان", "probability": 0.3798828125}, {"start": 1015.79, "end": 1015.97, "word": " ال", "probability": 0.74267578125}, {"start": 1015.97, "end": 1016.33, "word": " conversely", "probability": 0.5279541015625}, {"start": 1016.33, "end": 1016.73, "word": " هيكون", "probability": 0.859130859375}, {"start": 1016.73, "end": 1017.19, "word": " الخطوات", "probability": 0.9783935546875}, {"start": 1017.19, "end": 1017.53, "word": " كثير", "probability": 0.7532552083333334}, {"start": 1017.53, "end": 1018.09, "word": " مشابهه", "probability": 0.83551025390625}, {"start": 1018.09, "end": 1018.41, "word": " لللي", "probability": 0.7587890625}, {"start": 1018.41, "end": 1018.63, "word": " هنا", "probability": 0.2490234375}, {"start": 1018.63, "end": 1019.77, "word": " يعني", "probability": 0.881591796875}, {"start": 1019.77, "end": 1020.09, "word": " كثير", "probability": 0.88427734375}, {"start": 1020.09, "end": 1020.17, "word": " اللي", "probability": 0.84326171875}, {"start": 1020.17, "end": 1020.71, "word": " استخدمته", "probability": 0.9327392578125}, {"start": 1020.71, "end": 1020.87, "word": " هنا", "probability": 0.9052734375}, {"start": 1020.87, "end": 1021.81, "word": " هستخدمه", "probability": 0.89150390625}, {"start": 1021.81, "end": 1023.35, "word": " في", "probability": 0.75341796875}, {"start": 1023.35, "end": 1023.47, "word": " اللي", "probability": 0.963623046875}, {"start": 1023.47, "end": 1023.93, "word": " بعدها", "probability": 0.70556640625}], "temperature": 1.0}, {"id": 41, "seek": 105991, "start": 1032.67, "end": 1059.91, "text": "Conversely suppose that g of x سواء absolute value او f of x اللي هو is differentiable at c بدا وجدلك الان اثبتلك ان f prime of c ايش ما لها بتساوي سفر يعني بدا اثبت limit", "tokens": [9838, 4308, 356, 7297, 300, 290, 295, 2031, 8608, 2407, 16606, 8236, 2158, 1975, 2407, 283, 295, 2031, 13672, 1829, 31439, 307, 819, 9364, 412, 269, 4724, 28259, 49610, 3215, 23275, 2423, 7649, 1975, 12984, 3555, 2655, 23275, 16472, 283, 5835, 295, 269, 1975, 1829, 8592, 19446, 5296, 11296, 39894, 3794, 995, 45865, 8608, 5172, 2288, 37495, 22653, 4724, 28259, 1975, 12984, 3555, 2655, 4948], "avg_logprob": -0.41477272456342523, "compression_ratio": 1.3275862068965518, "no_speech_prob": 0.0, "words": [{"start": 1032.6699999999998, "end": 1033.55, "word": "Conversely", "probability": 0.5113525390625}, {"start": 1033.55, "end": 1034.43, "word": " suppose", "probability": 0.2186279296875}, {"start": 1034.43, "end": 1037.79, "word": " that", "probability": 0.88525390625}, {"start": 1037.79, "end": 1039.37, "word": " g", "probability": 0.45751953125}, {"start": 1039.37, "end": 1039.59, "word": " of", "probability": 0.405517578125}, {"start": 1039.59, "end": 1040.05, "word": " x", "probability": 0.869140625}, {"start": 1040.05, "end": 1040.97, "word": " سواء", "probability": 0.4903157552083333}, {"start": 1040.97, "end": 1041.29, "word": " absolute", "probability": 0.213623046875}, {"start": 1041.29, "end": 1041.75, "word": " value", "probability": 0.97705078125}, {"start": 1041.75, "end": 1041.89, "word": " او", "probability": 0.71484375}, {"start": 1041.89, "end": 1042.07, "word": " f", "probability": 0.414306640625}, {"start": 1042.07, "end": 1042.23, "word": " of", "probability": 0.85693359375}, {"start": 1042.23, "end": 1042.71, "word": " x", "probability": 0.90478515625}, {"start": 1042.71, "end": 1045.89, "word": " اللي", "probability": 0.5924072265625}, {"start": 1045.89, "end": 1046.13, "word": " هو", "probability": 0.9931640625}, {"start": 1046.13, "end": 1046.63, "word": " is", "probability": 0.74609375}, {"start": 1046.63, "end": 1048.85, "word": " differentiable", "probability": 0.962646484375}, {"start": 1048.85, "end": 1050.19, "word": " 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"start": 1446.41, "end": 1474.97, "text": "بتكون الـ function الـ G prime إيه شمالها is even انحل واحدة من هنا والتانية similarly زيها الآن لنفترض F من عند R لعند R be an odd differentiable function ماشي الحال show that", "tokens": [3555, 2655, 30544, 2423, 39184, 2445, 2423, 39184, 460, 5835, 11933, 1829, 3224, 13412, 2304, 6027, 11296, 307, 754, 16472, 5016, 1211, 36764, 24401, 3660, 9154, 34105, 16070, 2655, 7649, 10632, 1034, 2202, 356, 30767, 1829, 11296, 6024, 48506, 5296, 1863, 5172, 2655, 43042, 479, 9154, 43242, 497, 5296, 3615, 41260, 497, 312, 364, 7401, 819, 9364, 2445, 3714, 33599, 1829, 21542, 6027, 855, 300], "avg_logprob": -0.2904829473206491, "compression_ratio": 1.358695652173913, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 1446.41, "end": 1447.01, "word": "بتكون", "probability": 0.67919921875}, {"start": 1447.01, "end": 1447.19, "word": " الـ", "probability": 0.821533203125}, {"start": 1447.19, "end": 1447.51, "word": " function", "probability": 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limit f of xₙ as n goes to infinity بساوي برضه الـ", "tokens": [3555, 30544, 43242, 7578, 36632, 8310, 2031, 37781, 247, 39894, 32887, 5016, 24976, 39184, 383, 5296, 31377, 2304, 7251, 2655, 5016, 4587, 4587, 5296, 11296, 4948, 283, 295, 2031, 37781, 247, 382, 297, 1709, 281, 13202, 4724, 3794, 995, 45865, 4724, 43042, 3224, 2423, 39184], "avg_logprob": -0.26460598344388214, "compression_ratio": 1.1044776119402986, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 1845.7, "end": 1846.14, "word": "بكون", "probability": 0.7431640625}, {"start": 1846.14, "end": 1846.94, "word": " عندى", "probability": 0.75390625}, {"start": 1846.94, "end": 1847.32, "word": " أي", "probability": 0.443603515625}, {"start": 1847.32, "end": 1848.24, "word": " sequence", "probability": 0.7919921875}, {"start": 1848.24, "end": 1849.42, "word": " xₙ", "probability": 0.4777933756510417}, {"start": 1849.42, "end": 1850.62, "word": " بتروح", "probability": 0.9111328125}, {"start": 1850.62, "end": 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"word": " to", "probability": 0.953125}, {"start": 1858.34, "end": 1859.02, "word": " infinity", "probability": 0.8505859375}, {"start": 1859.02, "end": 1860.04, "word": " بساوي", "probability": 0.7733154296875}, {"start": 1860.04, "end": 1860.42, "word": " برضه", "probability": 0.9654947916666666}, {"start": 1860.42, "end": 1861.9, "word": " الـ", "probability": 0.41357421875}], "temperature": 1.0}, {"id": 73, "seek": 188539, "start": 1861.93, "end": 1885.39, "text": "كنا نتحدث بالحديث، باستبداء الحديث عن الـ limit العادية للـ function S X بتروح للـ C إلى limit لمين للـ sequence أو limit للـ sequences الآن بنستخل هذه اللي هو المعلومة في إثبات اللي بدنايا عند الآن since F prime of C exist بما أنه ال derivative عند C موجودة", "tokens": [4117, 8315, 8717, 2655, 24401, 12984, 20666, 5016, 16254, 12984, 12399, 4724, 995, 14851, 44510, 16606, 21542, 16254, 12984, 18871, 2423, 39184, 4948, 18863, 18513, 10632, 24976, 39184, 2445, 318, 1783, 39894, 32887, 5016, 24976, 39184, 383, 30731, 4948, 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1880.99, "end": 1881.79, "word": " exist", "probability": 0.6103515625}, {"start": 1881.79, "end": 1882.23, "word": " بما", "probability": 0.908935546875}, {"start": 1882.23, "end": 1882.63, "word": " أنه", "probability": 0.610595703125}, {"start": 1882.63, "end": 1883.03, "word": " ال", "probability": 0.97265625}, {"start": 1883.03, "end": 1883.51, "word": " derivative", "probability": 0.63916015625}, {"start": 1883.51, "end": 1884.01, "word": " عند", "probability": 0.986328125}, {"start": 1884.01, "end": 1884.41, "word": " C", "probability": 0.93310546875}, {"start": 1884.41, "end": 1885.39, "word": " موجودة", "probability": 0.992431640625}], "temperature": 1.0}, {"id": 74, "seek": 189701, "start": 1886.03, "end": 1897.01, "text": "إذا أكيد عندي صار f prime of c بسهولمة f of x أو اللي هي الـ x عند c", "tokens": [28814, 15730, 5551, 4117, 25708, 18871, 16254, 20328, 9640, 283, 5835, 295, 269, 4724, 3794, 3224, 12610, 46007, 283, 295, 2031, 34051, 13672, 1829, 39896, 2423, 39184, 2031, 43242, 269], "avg_logprob": -0.4697580779752424, "compression_ratio": 1.0957446808510638, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1886.03, "end": 1886.45, "word": "إذا", "probability": 0.44818115234375}, {"start": 1886.45, "end": 1887.09, "word": " أكيد", "probability": 0.7860514322916666}, {"start": 1887.09, "end": 1887.77, "word": " عندي", "probability": 0.6884765625}, {"start": 1887.77, "end": 1888.97, "word": " صار", "probability": 0.6317138671875}, {"start": 1888.97, "end": 1889.25, "word": " f", "probability": 0.3076171875}, {"start": 1889.25, "end": 1889.89, "word": " prime", "probability": 0.379150390625}, {"start": 1889.89, "end": 1890.11, "word": " of", "probability": 0.908203125}, {"start": 1890.11, "end": 1890.53, "word": " c", "probability": 0.69580078125}, {"start": 1890.53, "end": 1891.97, "word": " بسهولمة", "probability": 0.6477783203125}, {"start": 1891.97, "end": 1893.21, "word": " f", "probability": 0.85595703125}, {"start": 1893.21, "end": 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mean to zero ماشي الحل ال L بما أنه اللي هو ال 1 على N sequence بتروح للسفر", "tokens": [11622, 16373, 3215, 276, 8717, 995, 4587, 9381, 283, 295, 269, 15844, 276, 382, 276, 1709, 281, 914, 281, 4018, 13672, 1829, 31439, 16712, 3615, 16572, 5172, 16712, 3615, 16572, 5172, 2423, 13760, 34051, 25124, 28820, 6024, 95, 34740, 24976, 2655, 3615, 16572, 5172, 2423, 13760, 283, 295, 269, 30767, 16373, 3215, 2423, 26200, 8717, 995, 4587, 9381, 283, 295, 269, 15844, 2423, 26200, 382, 2423, 26200, 1709, 281, 914, 281, 4018, 3714, 33599, 1829, 21542, 1211, 2423, 441, 4724, 15042, 14739, 3224, 13672, 1829, 31439, 2423, 502, 15844, 426, 8310, 39894, 32887, 5016, 24976, 3794, 5172, 2288], "avg_logprob": -0.1991792938323936, "compression_ratio": 1.9447236180904524, "no_speech_prob": 1.2516975402832031e-06, "words": [{"start": 1898.08, "end": 1898.78, "word": "زائد", "probability": 0.6513671875}, {"start": 1898.78, "end": 1899.4, "word": " h", "probability": 0.31591796875}, {"start": 1899.4, "end": 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{"start": 1920.08, "end": 1920.4, "word": " L", "probability": 0.57421875}, {"start": 1920.4, "end": 1921.6, "word": " بما", "probability": 0.94482421875}, {"start": 1921.6, "end": 1922.16, "word": " أنه", "probability": 0.656494140625}, {"start": 1922.16, "end": 1922.98, "word": " اللي", "probability": 0.978271484375}, {"start": 1922.98, "end": 1923.24, "word": " هو", "probability": 0.99267578125}, {"start": 1923.24, "end": 1923.46, "word": " ال", "probability": 0.9765625}, {"start": 1923.46, "end": 1923.76, "word": " 1", "probability": 0.47607421875}, {"start": 1923.76, "end": 1924.08, "word": " على", "probability": 0.8564453125}, {"start": 1924.08, "end": 1924.36, "word": " N", "probability": 0.5458984375}, {"start": 1924.36, "end": 1925.24, "word": " sequence", "probability": 0.916015625}, {"start": 1925.24, "end": 1925.68, "word": " بتروح", "probability": 0.9581705729166666}, {"start": 1925.68, "end": 1926.38, "word": " للسفر", "probability": 0.796142578125}], "temperature": 1.0}, {"id": 76, "seek": 195258, "start": 1927.04, "end": 1952.58, "text": "وهذا الـ limit exist لأن حسب النظرية اللي حكيتها قبل بشوية بكون عندي لأن but كذا then F prime of C can be اللي هو إعادة اللي هي rewritten as a limit of a sequence F of اللي هو limit limit", "tokens": [2407, 3224, 15730, 2423, 39184, 4948, 2514, 5296, 33456, 11331, 35457, 28239, 19913, 2288, 10632, 13672, 1829, 11331, 4117, 36081, 11296, 12174, 36150, 4724, 8592, 2407, 10632, 4724, 30544, 18871, 16254, 5296, 33456, 457, 9122, 15730, 550, 479, 5835, 295, 383, 393, 312, 13672, 1829, 31439, 11933, 3615, 18513, 3660, 13672, 1829, 39896, 319, 26859, 382, 257, 4948, 295, 257, 8310, 479, 295, 13672, 1829, 31439, 4948, 4948], "avg_logprob": -0.27513587129288825, "compression_ratio": 1.4719101123595506, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 1927.04, "end": 1927.48, "word": "وهذا", "probability": 0.7101236979166666}, {"start": 1927.48, "end": 1927.6, "word": " الـ", "probability": 0.42626953125}, 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0.9724934895833334}, {"start": 2031.51, "end": 2031.63, "word": " ل", "probability": 0.974609375}, {"start": 2031.63, "end": 2031.95, "word": " n", "probability": 0.4501953125}, {"start": 2031.95, "end": 2032.85, "word": " ناقص", "probability": 0.9140625}, {"start": 2032.85, "end": 2033.09, "word": " f", "probability": 0.89892578125}, {"start": 2033.09, "end": 2033.29, "word": " of", "probability": 0.94140625}, {"start": 2033.29, "end": 2033.67, "word": " c", "probability": 0.98193359375}, {"start": 2033.67, "end": 2034.69, "word": " الكل", "probability": 0.6812744140625}, {"start": 2034.69, "end": 2035.17, "word": " مضروف", "probability": 0.88818359375}, {"start": 2035.17, "end": 2035.35, "word": " في", "probability": 0.90478515625}, {"start": 2035.35, "end": 2035.65, "word": " n", "probability": 0.8427734375}, {"start": 2035.65, "end": 2036.33, "word": " as", "probability": 0.88427734375}, {"start": 2036.33, "end": 2036.51, "word": " n", "probability": 0.939453125}, {"start": 2036.51, "end": 2036.73, "word": " goes", "probability": 0.91650390625}, {"start": 2036.73, "end": 2036.87, "word": " to", "probability": 0.91162109375}, {"start": 2036.87, "end": 2037.35, "word": " infinity", "probability": 0.892578125}, {"start": 2037.35, "end": 2038.69, "word": " exist", "probability": 0.495361328125}, {"start": 2038.69, "end": 2041.13, "word": " if", "probability": 0.7333984375}, {"start": 2041.13, "end": 2041.61, "word": " كده", "probability": 0.95849609375}], "temperature": 1.0}, {"id": 80, "seek": 206992, "start": 2042.5, "end": 2069.92, "text": "then f prime at c need not be exist اصلا مش انه يكون بيسوي هذا او لا need not to be اشمله exist لان لو كان exist على طول بيسوي اما need not to be exist ومااخد مثال جالي consider consider consider f of x", "tokens": [19096, 283, 5835, 412, 269, 643, 406, 312, 2514, 1975, 9381, 15040, 37893, 16472, 3224, 7251, 30544, 4724, 1829, 3794, 45865, 23758, 1975, 2407, 20193, 643, 406, 281, 312, 1975, 8592, 2304, 43761, 2514, 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"probability": 0.8857421875}], "temperature": 1.0}, {"id": 81, "seek": 209973, "start": 2070.66, "end": 2099.74, "text": "بساوي ال absolute value لل X وخد عند ال C أش بتساوي سفر واضح F prime of 0 does not exist لأنها عبارة عن corner point، انتوا عارفين احنا اللي هو ال F prime عند ال zero لل absolute value does not exist لكن هذه متحققة، ليش؟ but limit", "tokens": [3555, 3794, 995, 45865, 2423, 8236, 2158, 24976, 1783, 4032, 9778, 3215, 43242, 2423, 383, 5551, 8592, 39894, 3794, 995, 45865, 8608, 5172, 2288, 4032, 46958, 5016, 479, 5835, 295, 1958, 775, 406, 2514, 5296, 33456, 11296, 6225, 3555, 9640, 3660, 18871, 4538, 935, 12399, 16472, 2655, 14407, 6225, 9640, 5172, 9957, 1975, 5016, 8315, 13672, 1829, 31439, 2423, 479, 5835, 43242, 2423, 4018, 24976, 8236, 2158, 775, 406, 2514, 44381, 29538, 44650, 5016, 4587, 28671, 12399, 32239, 8592, 22807, 457, 4948], "avg_logprob": -0.2682605292423662, "compression_ratio": 1.4703196347031964, "no_speech_prob": 2.682209014892578e-06, "words": 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2090.64, "word": " zero", "probability": 0.438232421875}, {"start": 2090.64, "end": 2090.86, "word": " لل", "probability": 0.65673828125}, {"start": 2090.86, "end": 2091.64, "word": " absolute", "probability": 0.89501953125}, {"start": 2091.64, "end": 2092.06, "word": " value", "probability": 0.990234375}, {"start": 2092.06, "end": 2092.26, "word": " does", "probability": 0.5595703125}, {"start": 2092.26, "end": 2092.48, "word": " not", "probability": 0.96875}, {"start": 2092.48, "end": 2092.96, "word": " exist", "probability": 0.9736328125}, {"start": 2092.96, "end": 2094.62, "word": " لكن", "probability": 0.93603515625}, {"start": 2094.62, "end": 2095.16, "word": " هذه", "probability": 0.837890625}, {"start": 2095.16, "end": 2096.66, "word": " متحققة،", "probability": 0.866015625}, {"start": 2096.66, "end": 2097.34, "word": " ليش؟", "probability": 0.8447265625}, {"start": 2097.34, "end": 2097.76, "word": " but", "probability": 0.62939453125}, {"start": 2097.76, "end": 2099.74, "word": " limit", "probability": 0.78955078125}], "temperature": 1.0}, {"id": 82, "seek": 212896, "start": 2101.28, "end": 2128.96, "text": "N في F of 0 زائد 1 على N نقص F of 0 as N goes to infinity بساوي limit N F of 0 زائد 1 على N يعني F of 1 على N F of X بساوي absolute value X ف1 على N مظبوط أه as N goes to infinity طبعا الـ N بتروح", "tokens": [45, 8978, 479, 295, 1958, 30767, 16373, 3215, 502, 15844, 426, 8717, 4587, 9381, 479, 295, 1958, 382, 426, 1709, 281, 13202, 4724, 3794, 995, 45865, 4948, 426, 479, 295, 1958, 30767, 16373, 3215, 502, 15844, 426, 37495, 22653, 479, 295, 502, 15844, 426, 479, 295, 1783, 4724, 3794, 995, 45865, 8236, 2158, 1783, 6156, 16, 15844, 426, 3714, 19913, 3555, 2407, 9566, 5551, 3224, 382, 426, 1709, 281, 13202, 23032, 3555, 3615, 995, 2423, 39184, 426, 39894, 32887, 5016], "avg_logprob": -0.1947337904094178, "compression_ratio": 1.59375, "no_speech_prob": 1.2516975402832031e-06, "words": [{"start": 2101.28, "end": 2101.92, "word": "N", "probability": 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{"start": 2120.54, "end": 2120.8, "word": " absolute", "probability": 0.81103515625}, {"start": 2120.8, "end": 2121.12, "word": " value", "probability": 0.99169921875}, {"start": 2121.12, "end": 2121.38, "word": " X", "probability": 0.87744140625}, {"start": 2121.38, "end": 2122.1, "word": " ف1", "probability": 0.5560302734375}, {"start": 2122.1, "end": 2123.2, "word": " على", "probability": 0.890625}, {"start": 2123.2, "end": 2123.56, "word": " N", "probability": 0.982421875}, {"start": 2123.56, "end": 2124.34, "word": " مظبوط", "probability": 0.90498046875}, {"start": 2124.34, "end": 2124.88, "word": " أه", "probability": 0.621337890625}, {"start": 2124.88, "end": 2125.56, "word": " as", "probability": 0.46826171875}, {"start": 2125.56, "end": 2125.76, "word": " N", "probability": 0.98095703125}, {"start": 2125.76, "end": 2126.0, "word": " goes", "probability": 0.95263671875}, {"start": 2126.0, "end": 2126.18, "word": " to", "probability": 0.96484375}, {"start": 2126.18, "end": 2126.62, "word": " infinity", "probability": 0.89697265625}, {"start": 2126.62, "end": 2127.96, "word": " طبعا", "probability": 0.9835205078125}, {"start": 2127.96, "end": 2128.18, "word": " الـ", "probability": 0.7005615234375}, {"start": 2128.18, "end": 2128.3, "word": " N", "probability": 0.8740234375}, {"start": 2128.3, "end": 2128.96, "word": " بتروح", "probability": 0.9820963541666666}], "temperature": 1.0}, {"id": 83, "seek": 214900, "start": 2129.22, "end": 2149.0, "text": "يصير عبارة عن ash واحد إذا فعلا جبنا مثال أن ال limit هذه تكون exist و سوى واحد but ال F prime عند هذا النقطة C اللي هي 00 في هذه الحالة does not exist بيكون هيك احنا انتهينا من الجزء الأول من المحاضرة الخامسة", "tokens": [1829, 9381, 13546, 6225, 3555, 9640, 3660, 18871, 12588, 36764, 24401, 11933, 15730, 6156, 3615, 15040, 10874, 3555, 8315, 50113, 6027, 14739, 2423, 4948, 29538, 6055, 30544, 2514, 4032, 8608, 2407, 7578, 36764, 24401, 457, 2423, 479, 5835, 43242, 23758, 28239, 47432, 3660, 383, 13672, 1829, 39896, 7143, 8978, 29538, 21542, 6027, 3660, 775, 406, 2514, 4724, 1829, 30544, 39896, 4117, 1975, 5016, 8315, 16472, 47395, 1829, 8315, 9154, 25724, 11622, 38207, 16247, 12610, 9154, 9673, 5016, 46958, 25720, 33962, 10943, 3794, 3660], "avg_logprob": -0.2736235009062858, "compression_ratio": 1.5896226415094339, "no_speech_prob": 2.4259090423583984e-05, "words": [{"start": 2129.22, "end": 2129.7, "word": "يصير", "probability": 0.8001302083333334}, {"start": 2129.7, "end": 2130.6, "word": " عبارة", "probability": 0.986083984375}, {"start": 2130.6, "end": 2130.86, "word": " عن", "probability": 0.96044921875}, {"start": 2130.86, "end": 2131.4, "word": " ash", "probability": 0.177734375}, {"start": 2131.4, "end": 2132.62, "word": " واحد", "probability": 0.573486328125}, {"start": 2132.62, "end": 2133.48, "word": " إذا", "probability": 0.5986328125}, {"start": 2133.48, "end": 2133.9, "word": " فعلا", "probability": 0.9117838541666666}, {"start": 2133.9, "end": 2134.22, "word": " جبنا", "probability": 0.8313802083333334}, {"start": 2134.22, "end": 2134.74, "word": " مثال", "probability": 0.989990234375}, {"start": 2134.74, "end": 2134.94, "word": " أن", "probability": 0.40966796875}, {"start": 2134.94, "end": 2135.1, "word": " ال", "probability": 0.58203125}, {"start": 2135.1, "end": 2135.36, "word": " limit", "probability": 0.7470703125}, {"start": 2135.36, "end": 2135.6, "word": " هذه", "probability": 0.40478515625}, {"start": 2135.6, "end": 2135.9, "word": " تكون", "probability": 0.875244140625}, {"start": 2135.9, "end": 2136.28, "word": " exist", "probability": 0.75341796875}, {"start": 2136.28, "end": 2136.48, "word": " و", "probability": 0.81982421875}, {"start": 2136.48, "end": 2136.72, "word": " سوى", "probability": 0.464111328125}, {"start": 2136.72, "end": 2137.22, "word": " واحد", "probability": 0.9765625}, {"start": 2137.22, "end": 2138.02, "word": " but", "probability": 0.6240234375}, {"start": 2138.02, "end": 2138.2, "word": " ال", "probability": 0.77587890625}, {"start": 2138.2, "end": 2138.42, "word": " F", "probability": 0.5244140625}, {"start": 2138.42, "end": 2138.98, "word": " prime", "probability": 0.5380859375}, {"start": 2138.98, "end": 2139.72, "word": " عند", "probability": 0.97998046875}, {"start": 2139.72, "end": 2139.96, "word": " هذا", "probability": 0.408935546875}, {"start": 2139.96, "end": 2140.38, "word": " النقطة", "probability": 0.9733072916666666}, {"start": 2140.38, "end": 2140.5, "word": " C", "probability": 0.62255859375}, {"start": 2140.5, "end": 2140.62, "word": " اللي", "probability": 0.8740234375}, {"start": 2140.62, "end": 2140.78, "word": " هي", "probability": 0.82666015625}, {"start": 2140.78, "end": 2141.2, "word": " 00", "probability": 0.188232421875}, {"start": 2141.2, "end": 2141.5, "word": " في", "probability": 0.76220703125}, {"start": 2141.5, "end": 2141.66, "word": " هذه", "probability": 0.77001953125}, {"start": 2141.66, "end": 2142.04, "word": " الحالة", "probability": 0.9759114583333334}, {"start": 2142.04, "end": 2142.26, "word": " does", "probability": 0.63916015625}, {"start": 2142.26, "end": 2142.64, "word": " not", "probability": 0.96826171875}, {"start": 2142.64, "end": 2143.58, "word": " exist", "probability": 0.97265625}, {"start": 2143.58, "end": 2144.28, "word": " بيكون", "probability": 0.583251953125}, {"start": 2144.28, "end": 2144.52, "word": " هيك", "probability": 0.82275390625}, {"start": 2144.52, "end": 2144.74, "word": " احنا", "probability": 0.771728515625}, {"start": 2144.74, "end": 2145.5, "word": " انتهينا", "probability": 0.972412109375}, {"start": 2145.5, "end": 2146.06, "word": " من", "probability": 0.9921875}, {"start": 2146.06, "end": 2146.48, "word": " الجزء", "probability": 0.96435546875}, {"start": 2146.48, "end": 2147.02, "word": " الأول", "probability": 0.92822265625}, {"start": 2147.02, "end": 2147.32, "word": " من", "probability": 0.9443359375}, {"start": 2147.32, "end": 2148.12, "word": " المحاضرة", "probability": 0.9747314453125}, {"start": 2148.12, "end": 2149.0, "word": " الخامسة", "probability": 0.990966796875}], "temperature": 1.0}, {"id": 84, "seek": 217203, "start": 2150.43, "end": 2172.03, "text": "اللي هو discussion لأو مناقشة ل section 6-1 اللي هو the derivative والآن سنكمل الحديث في الجزء الثاني من المحاضرة اللي هو عن اللي هو the mean value theorem أو اللي هي نكمل applications على mean value theorem وناخد اللي هو ال", "tokens": [6027, 20292, 31439, 5017, 5296, 10721, 2407, 3714, 8315, 4587, 8592, 3660, 5296, 3541, 1386, 12, 16, 13672, 1829, 31439, 264, 13760, 16070, 148, 48506, 8608, 1863, 24793, 1211, 21542, 16254, 12984, 8978, 25724, 11622, 38207, 6024, 104, 7649, 1829, 9154, 9673, 5016, 46958, 25720, 13672, 1829, 31439, 18871, 13672, 1829, 31439, 264, 914, 2158, 20904, 34051, 13672, 1829, 39896, 8717, 24793, 1211, 5821, 15844, 914, 2158, 20904, 4032, 1863, 47283, 3215, 13672, 1829, 31439, 2423], "avg_logprob": -0.21266233534007878, "compression_ratio": 1.6683673469387754, "no_speech_prob": 1.6689300537109375e-06, "words": [{"start": 2150.4300000000003, "end": 2151.15, "word": "اللي", "probability": 0.753173828125}, {"start": 2151.15, "end": 2151.51, "word": " هو", "probability": 0.9287109375}, {"start": 2151.51, "end": 2152.81, "word": " discussion", "probability": 0.88818359375}, {"start": 2152.81, "end": 2153.89, "word": " لأو", "probability": 0.6074625651041666}, {"start": 2153.89, "end": 2154.65, "word": " مناقشة", "probability": 0.948046875}, {"start": 2154.65, "end": 2154.87, "word": " ل", "probability": 0.88916015625}, {"start": 2154.87, "end": 2155.27, "word": " section", "probability": 0.271240234375}, {"start": 2155.27, "end": 2155.59, "word": " 6", "probability": 0.587890625}, {"start": 2155.59, "end": 2155.85, "word": "-1", "probability": 0.6712646484375}, {"start": 2155.85, "end": 2156.03, "word": " اللي", "probability": 0.73388671875}, {"start": 2156.03, "end": 2156.17, "word": " هو", "probability": 0.9755859375}, {"start": 2156.17, "end": 2156.29, "word": " the", "probability": 0.42041015625}, {"start": 2156.29, "end": 2157.47, "word": " derivative", "probability": 0.853515625}, {"start": 2157.47, "end": 2158.57, "word": " والآن", "probability": 0.7032877604166666}, {"start": 2158.57, "end": 2159.91, "word": " سنكمل", "probability": 0.985595703125}, {"start": 2159.91, "end": 2160.83, "word": " الحديث", "probability": 0.9915364583333334}, {"start": 2160.83, "end": 2160.99, "word": " في", "probability": 0.91357421875}, {"start": 2160.99, "end": 2161.29, "word": " الجزء", "probability": 0.9462890625}, {"start": 2161.29, "end": 2161.65, "word": " الثاني", "probability": 0.9615478515625}, {"start": 2161.65, "end": 2161.81, "word": " من", "probability": 0.970703125}, {"start": 2161.81, "end": 2162.55, "word": " المحاضرة", "probability": 0.9903564453125}, {"start": 2162.55, "end": 2163.81, "word": " اللي", "probability": 0.9267578125}, {"start": 2163.81, "end": 2164.15, "word": " هو", "probability": 0.9775390625}, {"start": 2164.15, "end": 2164.75, "word": " عن", "probability": 0.96630859375}, {"start": 2164.75, "end": 2165.69, "word": " اللي", "probability": 0.7099609375}, {"start": 2165.69, "end": 2166.01, "word": " هو", "probability": 0.9833984375}, {"start": 2166.01, "end": 2166.35, "word": " the", "probability": 0.74462890625}, {"start": 2166.35, "end": 2166.59, "word": " mean", "probability": 0.73193359375}, {"start": 2166.59, "end": 2166.91, "word": " value", "probability": 0.9765625}, {"start": 2166.91, "end": 2167.41, "word": " theorem", "probability": 0.87939453125}, {"start": 2167.41, "end": 2167.79, "word": " أو", "probability": 0.73193359375}, {"start": 2167.79, "end": 2168.41, "word": " اللي", "probability": 0.943603515625}, {"start": 2168.41, "end": 2168.67, "word": " هي", "probability": 0.92578125}, {"start": 2168.67, "end": 2169.25, "word": " نكمل", "probability": 0.9290364583333334}, {"start": 2169.25, "end": 2169.95, "word": " applications", "probability": 0.58984375}, {"start": 2169.95, "end": 2170.31, "word": " على", "probability": 0.7041015625}, {"start": 2170.31, "end": 2170.49, "word": " mean", "probability": 0.685546875}, {"start": 2170.49, "end": 2170.81, "word": " value", "probability": 0.98876953125}, {"start": 2170.81, "end": 2171.17, "word": " theorem", "probability": 0.8916015625}, {"start": 2171.17, "end": 2171.65, "word": " وناخد", "probability": 0.9600830078125}, {"start": 2171.65, "end": 2171.75, "word": " اللي", "probability": 0.975341796875}, {"start": 2171.75, "end": 2171.91, "word": " هو", "probability": 0.98095703125}, {"start": 2171.91, "end": 2172.03, "word": " ال", "probability": 0.311767578125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2173.5270625, "duration_after_vad": 2117.446874999994}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0fe1e77efbcbf108e3fabf9108b97f46e2f98add
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Apc0JvRJbDQ_raw.srt
@@ -0,0 +1,1500 @@
+1
+00:00:04,910 --> 00:00:09,510
+بسم الله الرحمن الرحيم هذه المحاضرة الخامسة في
+
+2
+00:00:09,510 --> 00:00:15,630
+مساق تحليل حقيقه اتنين لطلبة كلية العلوم تخصص
+
+3
+00:00:15,630 --> 00:00:22,110
+رياضيات في الجامعة الإسلامية بغزا المحاضرة
+
+4
+00:00:22,110 --> 00:00:26,550
+اليوم هي جزئين الجزء الاول اللي هو عبارة عن
+
+5
+00:00:26,550 --> 00:00:31,350
+discussion لست واحد اللي هو مناقشة لموضوع اللي هو
+
+6
+00:00:31,350 --> 00:00:37,410
+derivative او الاشتقاءالجزء الثاني هنكمل اللي هو
+
+7
+00:00:37,410 --> 00:00:43,210
+الحديث عن 6.2 اللي هو الـ Mean Value Theorem أو
+
+8
+00:00:43,210 --> 00:00:47,730
+نظرية القيمة المتوسطة هناخذ بعض التطبيقات لنبدأ
+
+9
+00:00:47,730 --> 00:00:51,750
+الآن اللي هي بالأسئلة اللي احنا طلبناها منكم
+
+10
+00:00:51,750 --> 00:00:57,230
+تحلوها كواجب المرة الماضية أو التي قبلها وكانت
+
+11
+00:00:57,230 --> 00:01:01,210
+الأسئلة هي عبارة عن السؤال الرابع والسؤال السابع
+
+12
+00:01:02,000 --> 00:01:07,420
+والسؤال التاسع و السؤال تلتاشر بالنسبة للسؤال
+
+13
+00:01:07,420 --> 00:01:13,600
+الثالث اللي هو عبارة عن اللي هي برهان نظرية 613A
+
+14
+00:01:13,600 --> 00:01:19,200
+وB وهذه البرهين براهين سهلة اللي كانت ال F
+
+15
+00:01:19,200 --> 00:01:22,960
+differentiableالـ F differentiable والـ Alpha
+
+16
+00:01:22,960 --> 00:01:26,560
+عبارة عن constant بيعطينا الـ Alpha F برضه is
+
+17
+00:01:26,560 --> 00:01:30,200
+differentiable و لو كانت الـ F والـ G
+
+18
+00:01:30,200 --> 00:01:32,700
+differentiable بيعطينا الـ F زائد G is
+
+19
+00:01:32,700 --> 00:01:35,660
+differentiable و احنا برهننا حالة الضرب و حالة
+
+20
+00:01:35,660 --> 00:01:41,500
+القسمة و هدول حالات تعتبر سهلة مباشرة على التعريف
+
+21
+00:01:41,500 --> 00:01:45,900
+لذلك هنبدأ ان شاء الله في الحديث او في حل الأسئلة
+
+22
+00:01:45,900 --> 00:01:52,540
+على السؤال الرابع اللي هو ببول ماليعندي معطيني F
+
+23
+00:01:52,540 --> 00:02:01,760
+من R لـ R بي defined by F of X بساوي اللي هو X
+
+24
+00:02:01,760 --> 00:02:06,420
+تربيع إذا كانت X rational وبساوي سفر إذا كانت X
+
+25
+00:02:06,420 --> 00:02:09,700
+irrational اثبت إن الـ F الـ differential بالقدر X
+
+26
+00:02:09,700 --> 00:02:15,550
+بتساوي سفر أو جد اللي هي F prime of 0لاحظ أن الـ
+
+27
+00:02:15,550 --> 00:02:16,610
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+28
+00:02:16,610 --> 00:02:17,090
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+29
+00:02:17,090 --> 00:02:17,910
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+30
+00:02:17,910 --> 00:02:20,450
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+31
+00:02:20,450 --> 00:02:27,990
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+32
+00:02:27,990 --> 00:02:28,350
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+33
+00:02:28,350 --> 00:02:29,630
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+34
+00:02:29,630 --> 00:02:29,690
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+35
+00:02:29,690 --> 00:02:41,470
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+36
+00:02:41,470 --> 00:02:52,150
+الـ الـ الـFind this value a proof الآن عشان نوجد
+
+37
+00:02:52,150 --> 00:02:56,730
+اللي هي نثبت أن F prime of 0 موجودة خلّيني ألاحظ
+
+38
+00:02:56,730 --> 00:03:00,950
+مالي عند ال function اللي هي X تربيع لما X
+
+39
+00:03:00,950 --> 00:03:07,150
+rational و Zero لما Xاش is irrationalالان بدنا
+
+40
+00:03:07,150 --> 00:03:11,770
+نتوقع اول اشي اللي هو لانه هتلزمني جاي اللي بعدين
+
+41
+00:03:11,770 --> 00:03:15,890
+find this value بدنا نتوقع ايش قيمة اللي هو ال
+
+42
+00:03:15,890 --> 00:03:20,950
+derivative عند الصفر لاحظ انه اللي هو .. اللي ..
+
+43
+00:03:20,950 --> 00:03:25,630
+اللي لو بدنا نقول انه ال derivative ممكن تكون zero
+
+44
+00:03:25,630 --> 00:03:30,510
+كون ان off of x صفر او لو بدها تكون و ال
+
+45
+00:03:30,510 --> 00:03:36,560
+derivative هنا لو قولنا رفل 2xبدو يكون اللي هو لو
+
+46
+00:03:36,560 --> 00:03:40,820
+بدو يكون اللي عند الـ zero اللي هي F prime بدو في
+
+47
+00:03:40,820 --> 00:03:47,220
+النهاية تروح ل اللي هيكون برضه قريبة من انه نقول
+
+48
+00:03:47,220 --> 00:03:52,580
+او نأكد انها صفر عشان هي كالظن الغالب ان F prime
+
+49
+00:03:52,580 --> 00:03:58,660
+هتكون ايش صفر هذه مجرد تفكيرات الانبدي أثبت لك إنه
+
+50
+00:03:58,660 --> 00:04:03,080
+فعلًا هي الـ derivative بتساوي 0 كيف بدي أثبتها؟
+
+51
+00:04:03,080 --> 00:04:11,280
+بدي أثبت لك إنه الـ limit لـ F of X نقص F of 0 على
+
+52
+00:04:11,280 --> 00:04:17,600
+X minus 0 لما X تروح لـ 0 بتساوي 0 بدي أثبت لك
+
+53
+00:04:17,600 --> 00:04:24,240
+هيهاالان واضح ان X تقول الى الصفر X تقول الى الصفر
+
+54
+00:04:24,240 --> 00:04:29,200
+هتمر بال rational وال rational عشان هيك صعب ان انا
+
+55
+00:04:29,200 --> 00:04:32,600
+اتحدث عن اللي هو ايجاد ال derivative مباشرة من
+
+56
+00:04:32,600 --> 00:04:36,130
+الان او من ال two branches اللي عندىلا بقدر أخد من
+
+57
+00:04:36,130 --> 00:04:40,030
+اليمين ولا أخد من اليسار لإنه عندي من اليمين أو من
+
+58
+00:04:40,030 --> 00:04:44,130
+اليسار هيكون عندي اللي هو قابلت اللي هي الأعداد ال
+
+59
+00:04:44,130 --> 00:04:48,110
+rational لل rational فعشانك أسلم إشي إنه نستخدم
+
+60
+00:04:48,110 --> 00:04:52,650
+التعريف في إثبات هذا يعني الآن بدي أثبت هذا الكلام
+
+61
+00:04:53,270 --> 00:04:58,090
+كيف بدي أثبته؟ بدي أثبت ما يعني بدي أصل لكل إبسلون
+
+62
+00:04:58,090 --> 00:05:03,190
+أكبر من 0 بدلاج دلتا أكبر من 0 بدلاجية حاجيلك دلتا
+
+63
+00:05:03,190 --> 00:05:08,350
+بحيث أنه لما يكون ال absolute value ل F of X ناقص
+
+64
+00:05:08,350 --> 00:05:14,630
+F of Zero على X minus Zero يكون ناقص Zero طبعا
+
+65
+00:05:14,630 --> 00:05:19,790
+Zero هذي أصغر من إبسلون هذا متى؟ whenever
+
+66
+00:05:22,670 --> 00:05:28,470
+x-0 أكبر من 0 وأصغر من مين من دلتا هذا اللي بده
+
+67
+00:05:28,470 --> 00:05:33,150
+أصله أو هذا اللي بده أثبته خلّينا نشوف إذا خلّينا
+
+68
+00:05:33,150 --> 00:05:36,530
+نشوف كيف بدنا نوجده عشان أثبت إن ال limit هذا
+
+69
+00:05:36,530 --> 00:05:43,110
+بساوي 0 لاحظ القيمة اللي عندي أول إشي ال absolute
+
+70
+00:05:43,110 --> 00:05:50,530
+value of f of x ناقص f of 0 على x-0 إيش هتساوي؟
+
+71
+00:05:51,590 --> 00:05:57,390
+هتساوي اللي هو ال absolute value of of X عندى يا X
+
+72
+00:05:57,390 --> 00:06:01,070
+تربيع يمين يا سفر خلّينها زي ما هي أول إشي of of X
+
+73
+00:06:01,070 --> 00:06:06,010
+ناقص of of Zero اللي هو جداش بيساوي Zero لإنه of
+
+74
+00:06:06,010 --> 00:06:10,130
+of Zero بيساوي Zero لإنه Zero rational على إذن هذا
+
+75
+00:06:10,130 --> 00:06:16,230
+سفر على X ناقص سفر اللي هو X absolute value هذا
+
+76
+00:06:16,230 --> 00:06:22,790
+القيمة الآن لاحظ بتساوييا إما اللي هو x تربيع على
+
+77
+00:06:22,790 --> 00:06:30,750
+x absolute value في حالة x is rational أو بتساوي
+
+78
+00:06:30,750 --> 00:06:37,450
+اللي هو سفر في حالة x أشمالها is irrational لأن
+
+79
+00:06:37,450 --> 00:06:42,030
+مقيمة f of x يا x تربيع يا سفر حسب اللي هو كتبناه
+
+80
+00:06:42,030 --> 00:06:46,790
+حاليًاالآن هذا بالظبط هو عبارة عن الـ absolute
+
+81
+00:06:46,790 --> 00:06:57,590
+value لل X if X is rational Zero if X is
+
+82
+00:06:57,590 --> 00:07:03,950
+irrational الآن الصورة وضحت خلّينا نسمي هذا اللي
+
+83
+00:07:03,950 --> 00:07:09,900
+هو واحدالان حضرت عشان اصل للنهاية اللى هاني كاتبها
+
+84
+00:07:09,900 --> 00:07:13,800
+هنا واشوف ايش ال delta اللى بتطلع عندى الان بتدعي
+
+85
+00:07:13,800 --> 00:07:17,380
+مالي for every epsilon أكبر من سفر انا بقول there
+
+86
+00:07:17,380 --> 00:07:21,800
+exists delta هتساوي من ال epsilon هتجد تشوفوا ليش
+
+87
+00:07:21,800 --> 00:07:26,980
+there exists delta بساوي epsilon such that if x
+
+88
+00:07:26,980 --> 00:07:35,210
+minus 0 أكبر من سفره أصغر من delta thenهذا معناته
+
+89
+00:07:35,210 --> 00:07:37,870
+إيش أن أبسل يوت فيها ال X أكبر من صفر وأصغر من
+
+90
+00:07:37,870 --> 00:07:41,870
+مين؟ من دلتا إذا اخترت دلتا إيش بتساوي و هي اللي
+
+91
+00:07:41,870 --> 00:07:47,850
+هتخلص من الموضوع then اللي هو F of X نقص F of Zero
+
+92
+00:07:47,850 --> 00:07:50,970
+على
+
+93
+00:07:50,970 --> 00:07:57,790
+X minus Zero هو طبعا نقص الصفر اللي قلنا عنها ال
+
+94
+00:07:57,790 --> 00:08:01,520
+derivative المتوقع عليا هذه بالظبطاللي فوق هذا هو
+
+95
+00:08:01,520 --> 00:08:05,920
+اللي فوق طلع إيش عندي هذا بساوي هذا المقدار إذا
+
+96
+00:08:05,920 --> 00:08:09,760
+كانت X rational و 0 إذا كانت X irrational يعني
+
+97
+00:08:09,760 --> 00:08:15,400
+بمعنى أخر بساوي absolute value ل X if X is
+
+98
+00:08:15,400 --> 00:08:23,530
+rational و 0 if X is irrational in both casesاللي
+
+99
+00:08:23,530 --> 00:08:27,590
+هي إذا كان بساوي absolute value للـ X هيكون أصغر
+
+100
+00:08:27,590 --> 00:08:31,250
+من Delta اللي انا اخترتها أشملها Epsilon هيكون
+
+101
+00:08:31,250 --> 00:08:34,570
+أصغر من Epsilon و أيضا هذه متحققة لإن الـ Epsilon
+
+102
+00:08:34,570 --> 00:08:38,850
+دائما أشملها أكبر من 0 إذا اللي حصلته أنه لكل
+
+103
+00:08:38,850 --> 00:08:42,490
+Epsilon أكبر من 0 اللي جيت Delta لما يكون هذا أصغر
+
+104
+00:08:42,490 --> 00:08:47,130
+من Delta بيعطيني هذا أصغر من Epsilon وهذا يعني
+
+105
+00:08:47,130 --> 00:08:54,370
+hence limitهذا المقدار اللي هو f of x ناقص f of 0
+
+106
+00:08:54,370 --> 00:09:01,690
+على x minus 0 لما x تروح للصفر بساوية اللي هو
+
+107
+00:09:01,690 --> 00:09:12,870
+الصفر وهذا هو تعريف من ال f prime عند 0 that is f
+
+108
+00:09:12,870 --> 00:09:18,740
+prime at 0 بساوية 0 وفي نفس الواجب طبعا أثبتناالـ
+
+109
+00:09:18,740 --> 00:09:25,960
+existence للـ F prime عند الـ zero أي سؤال طيب
+
+110
+00:09:25,960 --> 00:09:30,840
+نيجي الآن نشوف السؤال الثاني خلّال السؤال الأربعة
+
+111
+00:09:30,840 --> 00:09:44,220
+نيجي لسؤال سبعة الان
+
+112
+00:09:44,220 --> 00:09:48,100
+سؤال سبعة اش اللي بقوله سؤال سبعةاللي بيقوله سؤال
+
+113
+00:09:48,100 --> 00:09:52,420
+سبعة مالي عندي
+
+114
+00:09:52,420 --> 00:09:55,740
+suppose
+
+115
+00:09:55,740 --> 00:09:59,320
+that F من R لR is differentiable at C يعني نفترض
+
+116
+00:09:59,320 --> 00:10:04,520
+أنه F قبل الاشتقاق عند Cونفترض ان f of c قيمة الـ
+
+117
+00:10:04,520 --> 00:10:07,920
+function عند c بساوة 0 لان بقول لشهدات ال absolute
+
+118
+00:10:07,920 --> 00:10:10,960
+value لل f of x اللي هي نسميها d of x is
+
+119
+00:10:10,960 --> 00:10:14,080
+differentiable at c if and only if a شمالها f
+
+120
+00:10:14,080 --> 00:10:21,740
+prime of c بتساوة 0 إذا ناخد لك f من R لعند R و
+
+121
+00:10:21,740 --> 00:10:30,810
+جايلك أن f prime عند c exist معطيكي إياهاأو ومعطيك
+
+122
+00:10:30,810 --> 00:10:37,390
+اللي هو f of c بتساوي سفر وبقول لي prove that أنه
+
+123
+00:10:37,390 --> 00:10:42,370
+g of x بساوي ال absolute value لل f of x is
+
+124
+00:10:42,370 --> 00:10:49,810
+differentiable at c if and only if f prime عند ال
+
+125
+00:10:49,810 --> 00:10:57,880
+c ايش بتساوي بساوي سفر، مظبوط؟ طيب شوفالان خلينا
+
+126
+00:10:57,880 --> 00:11:03,380
+افترض اول اشي ان f prime عند c ايش بتساوي سفر نقول
+
+127
+00:11:03,380 --> 00:11:09,320
+suppose proof suppose
+
+128
+00:11:09,320 --> 00:11:15,580
+suppose
+
+129
+00:11:15,580 --> 00:11:23,120
+that f prime at c بتساوي سفر ايش هذا بيعني ان then
+
+130
+00:11:23,120 --> 00:11:25,940
+limit
+
+131
+00:11:27,220 --> 00:11:36,680
+F of X ناقص F of C على X minus C لما X تروح للـ C
+
+132
+00:11:36,680 --> 00:11:40,240
+اللي هو بيساوي سفر لأن هذا تعريف مين F براين
+
+133
+00:11:40,240 --> 00:11:46,000
+بيساوي سفر و F of C إيش معطنية هو بيساوي سفر لأن
+
+134
+00:11:46,000 --> 00:11:53,260
+بيساوي limit F of X على X minus C لما X تروح لمين
+
+135
+00:11:53,260 --> 00:11:58,730
+للـ C مدامة ال limit هذه موجودةإذا ال limit من
+
+136
+00:11:58,730 --> 00:12:02,270
+اليمين وال limit من اليسار أيش برضه مالها موجودة
+
+137
+00:12:02,270 --> 00:12:06,710
+ماشي الحال انا الآن غرضي ان او اثبت ان g of x بسبب
+
+138
+00:12:06,710 --> 00:12:09,190
+absolute value of f of x أيش مالها is
+
+139
+00:12:09,190 --> 00:12:14,110
+differentiable at c ماشي الان ايش اللي بدي اثبته
+
+140
+00:12:14,110 --> 00:12:22,370
+بمعنى اخر بدي اثبتلك انه limitالـ G of X ناقص G of
+
+141
+00:12:22,370 --> 00:12:29,430
+C على X minus C لما X تروح للـ C exist باشي إذا
+
+142
+00:12:29,430 --> 00:12:31,930
+أثبتوا معناته أثبتتوا إن الـ G is differentiable
+
+143
+00:12:31,930 --> 00:12:36,970
+at C أو بمعنى آخر بالدثبت limit لـ absolute value
+
+144
+00:12:36,970 --> 00:12:42,660
+لل F of Xمدلة g of x ناقص ال absolute value of f
+
+145
+00:12:42,660 --> 00:12:48,960
+of c على x minus c لما x بتروح لل c exist بدي أشوف
+
+146
+00:12:48,960 --> 00:12:53,240
+هدا لسه exist ولا لأ يعني بدي أثبت اللي هو limit
+
+147
+00:12:53,240 --> 00:12:58,900
+absolute value of f of x على x minus c as x بتروح
+
+148
+00:12:58,900 --> 00:13:07,520
+لل c أشماله exist ماشي الحال طيب الآن واضح
+
+149
+00:13:08,360 --> 00:13:13,000
+عندي من اللي فوق ال limit لل f of x على x minus c
+
+150
+00:13:13,000 --> 00:13:19,160
+as x بتروح لل c انه ايش بيساوي بيساوي صفر، مظبوط؟
+
+151
+00:13:19,160 --> 00:13:27,160
+اذا هيكون عندي limit اللي هو limit absolute value
+
+152
+00:13:27,160 --> 00:13:33,400
+لل limit خليني اكتبها بس في طريقة أخرى الصفر هذا
+
+153
+00:13:33,400 --> 00:13:43,200
+اللي بدأتبته اه انه existعندي ال absolute value ل
+
+154
+00:13:43,200 --> 00:13:52,020
+limit f of x على x minus c لما x تروح لل c اللي هو
+
+155
+00:13:52,020 --> 00:13:58,020
+من اليمين و من اليسار exist، مظبوط؟ واضحة و يساوي
+
+156
+00:13:58,020 --> 00:14:05,160
+اللي هو limit absolute value لل f of x على x minus
+
+157
+00:14:05,160 --> 00:14:12,420
+c as absolute valueas X بتروح لمين؟ للـ C باشي
+
+158
+00:14:12,420 --> 00:14:16,820
+الحال الآن من اليمين ومن اليسار كله هيكون موجود
+
+159
+00:14:16,820 --> 00:14:20,480
+بناء على هذا أنه موجود خلّيني أخد من اليمين وهنا
+
+160
+00:14:20,480 --> 00:14:25,860
+من اليمين فبصير هذا عبارة عن limit absolute value
+
+161
+00:14:25,860 --> 00:14:32,140
+ل F of X على X minus C لما X بتروح ل C من وين؟ من
+
+162
+00:14:32,140 --> 00:14:38,330
+اليمين يعني هذا المقدارصار موجود وإيش بساوي بساوي
+
+163
+00:14:38,330 --> 00:14:42,350
+سفر هذا خلّيه لإنه هذا اللي بدنا نصلّه لأن limit
+
+164
+00:14:42,350 --> 00:14:47,310
+صار عندي معنى آخر limit absolute value of f of x
+
+165
+00:14:47,310 --> 00:14:53,850
+على x minus c لما x تروح لل c من اليمين بساوي سفر
+
+166
+00:14:53,850 --> 00:14:58,360
+existلاحظا قاعد رايح لأثبت أن هذا exist ده خد الآن
+
+167
+00:14:58,360 --> 00:15:03,520
+من وين؟ من اليسار خد لأن احسب similarly عندي سفر
+
+168
+00:15:03,520 --> 00:15:08,840
+بساوي absolute value of limit f of x على x minus c
+
+169
+00:15:08,840 --> 00:15:15,100
+لما x تروح ل c من وين؟ من اليسار ويساوي عبارة عن
+
+170
+00:15:15,100 --> 00:15:19,960
+limit absolute value of f of x على absolute value
+
+171
+00:15:19,960 --> 00:15:25,900
+of x minus c لما x تروح ل c من وين؟ من اليسارهذه
+
+172
+00:15:25,900 --> 00:15:34,580
+نفسها بساوي limit أو بساوي سالب limit f of x
+
+173
+00:15:34,580 --> 00:15:42,270
+absolute value على x minus c لما x تروحللـ C من
+
+174
+00:15:42,270 --> 00:15:47,030
+اليسار، ليش؟ لأن X أصغر من C، إذا X minus C سالبة
+
+175
+00:15:47,030 --> 00:15:50,230
+إذا الـ absolute value سالب إليها واخدت السالب برا
+
+176
+00:15:50,230 --> 00:15:54,990
+هذا الآن المخضر بيساوي سفر، إذا هذا لحاله برضه إيش
+
+177
+00:15:54,990 --> 00:16:01,270
+ماله؟ سفر، إذا limit absolute value للـ F of X على
+
+178
+00:16:01,270 --> 00:16:04,870
+X minus C، لما X تروح للـ C من اليسار، برضه إيش
+
+179
+00:16:04,870 --> 00:16:10,450
+بيساوي؟ بيساوي سفر، لاحظ إن ال limit من اليمينوالـ
+
+180
+00:16:10,450 --> 00:16:15,890
+limit من اليسار موجود وبساوي 0 متساويين يعني الآن
+
+181
+00:16:15,890 --> 00:16:23,130
+ال limit هذا صار ايش بساوي بساوي 0 إذا الان هذا
+
+182
+00:16:23,130 --> 00:16:29,870
+الآن بنقوله كله تحت هذا وبقول hence اللي هو g
+
+183
+00:16:29,870 --> 00:16:38,310
+prime of c بساوي limit of g of x نقص g of c على x
+
+184
+00:16:38,310 --> 00:16:46,870
+minus cas x → c بساوي حسب اللي عندى من هنا و من
+
+185
+00:16:46,870 --> 00:16:55,310
+هنا و من هنا هيساوي سفر الان conversely بتأفترض
+
+186
+00:16:55,310 --> 00:16:58,090
+طبعا ان ال conversely هيكون الخطوات كثير مشابهه
+
+187
+00:16:58,090 --> 00:17:03,350
+لللي هنا يعني كثير اللي استخدمته هنا هستخدمه في
+
+188
+00:17:03,350 --> 00:17:03,930
+اللي بعدها
+
+189
+00:17:12,670 --> 00:17:17,790
+Conversely suppose that
+
+190
+00:17:17,790 --> 00:17:25,890
+g of x سواء absolute value او f of x اللي
+
+191
+00:17:25,890 --> 00:17:33,550
+هو is differentiable at c بدا وجدلك الان اثبتلك ان
+
+192
+00:17:33,550 --> 00:17:39,650
+f prime of c ايش ما لها بتساوي سفر يعني بدا اثبت
+
+193
+00:17:39,650 --> 00:17:46,880
+limitf of x ناقص f of c اللي هي سفر على x minus c
+
+194
+00:17:46,880 --> 00:17:51,160
+لما x تروح لل c ايش بيساوي بساوي سفر بكون خلصت
+
+195
+00:17:51,160 --> 00:17:58,500
+الان issue مشابه عند الان limit
+
+196
+00:17:58,500 --> 00:18:05,180
+هذا exist عند ال c اذا عندي صار g prime of c exist
+
+197
+00:18:05,180 --> 00:18:10,430
+ويساوي حسب الحديث اللي هنا limitabsolute value of
+
+198
+00:18:10,430 --> 00:18:18,570
+f of x ناقص ال absolute value of f of c صفر على x
+
+199
+00:18:18,570 --> 00:18:26,310
+minus c لما x تروح ل c اشماله exist معايا؟ طيب شوف
+
+200
+00:18:26,310 --> 00:18:34,960
+الآن عندى اذا احسبلي limitf of x اللي حسبناها قبل
+
+201
+00:18:34,960 --> 00:18:39,660
+بشوية بنفس الأسلوب على x minus c لما x تروح ل c من
+
+202
+00:18:39,660 --> 00:18:47,020
+وين؟ من اليمين بساوي limit لل absolute value ل f
+
+203
+00:18:47,020 --> 00:18:53,780
+of x على x minus c لما x تروح ل c من اليمين لأن ال
+
+204
+00:18:53,780 --> 00:19:01,840
+x أكبر من ال c واضحالآن من جهة أخرى احسبلي الـ
+
+205
+00:19:01,840 --> 00:19:08,060
+absolute value لل limit لل F of X طبعا هذا ايش
+
+206
+00:19:08,060 --> 00:19:15,620
+هيساوي بسوى G prime of C، مظبوط؟ بسوى G prime of
+
+207
+00:19:15,620 --> 00:19:23,290
+C، موجود، limit F of Xعلى x minus c لما x تروح ل c
+
+208
+00:19:23,290 --> 00:19:28,570
+من وين؟ من اليسار بساوي زي ما عملنا قبل شوية سالب
+
+209
+00:19:28,570 --> 00:19:33,530
+limit absolute value of f of x على x minus c لما x
+
+210
+00:19:33,530 --> 00:19:41,050
+تروح ل c من وين؟ من اليسار الآن واضح بما أن هذه
+
+211
+00:19:41,050 --> 00:19:51,470
+exist إذا هذا المقدار و هذا المقدار زي بعضالان هذا
+
+212
+00:19:51,470 --> 00:19:56,590
+بيساوي ناقص هذا او بمعنى اخر نشيل الناقص من هنا و
+
+213
+00:19:56,590 --> 00:20:03,390
+نضربه هنا صار عندى المقدرين هدولة بما انه متساويين
+
+214
+00:20:03,390 --> 00:20:10,310
+لإن الاتنين ايش بيساوين ال G prime of C مظبوط اذا
+
+215
+00:20:10,310 --> 00:20:13,390
+صار عندى هذا المقدار بيساوي هذا المقدار اذا صار
+
+216
+00:20:13,390 --> 00:20:20,940
+عندى ال limitF of X على X minus C لما X تروح للـ C
+
+217
+00:20:20,940 --> 00:20:29,560
+من اليمين absolute value بساوي ناقص limit F of X
+
+218
+00:20:29,560 --> 00:20:34,600
+على X minus C لما X تروح للـ C من وين؟ من اليسار
+
+219
+00:20:34,600 --> 00:20:41,500
+واضحة؟ لكن أصلا عندى هذا
+
+220
+00:20:43,370 --> 00:20:49,010
+بساوي اللي هو التنين بساوي نفس القيمة ماشي الحال
+
+221
+00:20:49,010 --> 00:20:55,850
+إذا لازم عندي من هنا جي برايم و جي برايم اللي هو
+
+222
+00:20:55,850 --> 00:21:00,210
+صار بنساوي نفس القيمة إذا هيطلع الجي برايم إيش
+
+223
+00:21:00,210 --> 00:21:05,530
+ماله بساوي سفرإذا صار المقدار هذا كله إش بده يساوي
+
+224
+00:21:05,530 --> 00:21:08,650
+صفر وهذا بيساوي صفر يعني ال limit من اليمين و
+
+225
+00:21:08,650 --> 00:21:12,070
+limit من اليسار متساويين إذا صارت limit F of X على
+
+226
+00:21:12,070 --> 00:21:23,310
+X minus C لما X تروح للـC بدها تساوي صفر أي سؤال؟
+
+227
+00:21:23,310 --> 00:21:31,130
+زي ما حكينا، الآن إحنا قلنا إنه لو كانت D بدينا D
+
+228
+00:21:31,130 --> 00:21:34,600
+بيساوي F of X ال differential بالأد Cجيبنا g prime
+
+229
+00:21:34,600 --> 00:21:37,500
+و كتبناها بالصورة اللي أمامنا بعدين أخدنا ال
+
+230
+00:21:37,500 --> 00:21:40,120
+absolute value لlimit f of x علي x minus c لم x
+
+231
+00:21:40,120 --> 00:21:45,200
+تروح ل c positive طلعت عندي بساوي g prime of c و
+
+232
+00:21:45,200 --> 00:21:48,680
+أخدنا اللي هي ناقص هذه طلعت عندي برضه g prime
+
+233
+00:21:48,680 --> 00:21:53,920
+الاتنين ال c إذا صار هذا بساوي هذا المقدار و احنا
+
+234
+00:21:53,920 --> 00:21:57,640
+بنعرف في الأصل ان f prime of c exist يعني
+
+235
+00:21:57,640 --> 00:22:01,900
+differentiable يعني ال limit هذا اللي هو موجودو من
+
+236
+00:22:01,900 --> 00:22:05,100
+اليمين و من اليسار زي بعض يعني يعني هذا بده يساوي
+
+237
+00:22:05,100 --> 00:22:07,500
+هذا من اليمين و هذا من اليسار يعني في الواقع هذا
+
+238
+00:22:07,500 --> 00:22:10,960
+اللي جوا هو نفسه اللي جوا بساوي limit of of X يعني
+
+239
+00:22:10,960 --> 00:22:15,780
+بمعنى أخر absolute value ل limit of of X على X
+
+240
+00:22:15,780 --> 00:22:22,100
+minus C لما X تروح لل C هو نفسه سالب limit لل of
+
+241
+00:22:22,100 --> 00:22:27,400
+of X على X minus C لما X تروح لل C من وين من
+
+242
+00:22:27,400 --> 00:22:31,540
+اليسار هو نفسه لل C من وين تبت هذالأنه احنا بنقول
+
+243
+00:22:31,540 --> 00:22:34,560
+f prime of c exist يعني ال limit هذا موجود و من
+
+244
+00:22:34,560 --> 00:22:37,360
+اليمين و من اليسار زي بعر إذا صار هذا المقدر نفس
+
+245
+00:22:37,360 --> 00:22:41,200
+هذا المقدر زي ما قلنا إذا صار عند هذا المقدر بساوة
+
+246
+00:22:41,200 --> 00:22:46,800
+سفر لإن التنين بسينا بساوة سفر إذا صار عنده limitF
+
+247
+00:22:46,800 --> 00:22:50,880
+of X على X minus C اللي هو الـ absolute value as X
+
+248
+00:22:50,880 --> 00:22:56,100
+بتروح للـ C بساوة صفر ومن ثم اللي جوا بساوة صفر هو
+
+249
+00:22:56,100 --> 00:22:58,740
+مين هو اللي جوا هذا اللي كنا بدنا نصله من الأول
+
+250
+00:22:58,740 --> 00:23:04,260
+اللي هو F prime of C بتساوة صفر وهو المطلوب هذا
+
+251
+00:23:04,260 --> 00:23:10,300
+توضيح بشكل كامل ل اللي صار في اللي هو الاتجاه
+
+252
+00:23:10,300 --> 00:23:16,200
+الثاني نيجينا سؤال بعد ما خلصنا السؤال سبعةنجي
+
+253
+00:23:16,200 --> 00:23:23,420
+لسؤال تسعة تسعة ايش اللي بقوله تسعة نشوف ايش سؤال
+
+254
+00:23:23,420 --> 00:23:31,360
+تسعة بقول ونحل سؤال تسعة سؤال تسعة بقول ليه
+
+255
+00:23:31,360 --> 00:23:39,110
+باختصار انه لو كان عنده ال function ofعبارة عن من
+
+256
+00:23:39,110 --> 00:23:43,130
+R لـ R even function طبعا عارفين ايش ال even اللي
+
+257
+00:23:43,130 --> 00:23:49,130
+هو F ناقص X بيساوي F X لكل X موجودة في الـ R and
+
+258
+00:23:49,130 --> 00:23:54,310
+has a derivative at every point then F prime is an
+
+259
+00:23:54,310 --> 00:23:58,190
+odd function يعني بيقول لي لو كانت باختصار يعني لو
+
+260
+00:23:58,190 --> 00:24:01,730
+كانت ال F even و ال derivative موجودة بتكون ال
+
+261
+00:24:01,730 --> 00:24:07,010
+derivative odd و لو كانت ال derivative oddبتكون
+
+262
+00:24:07,010 --> 00:24:11,350
+الـ function الـ G prime إيه شمالها is even انحل
+
+263
+00:24:11,350 --> 00:24:16,970
+واحدة من هنا والتانية similarly زيها الآن لنفترض F
+
+264
+00:24:16,970 --> 00:24:29,890
+من عند R لعند R be an odd differentiable function
+
+265
+00:24:29,890 --> 00:24:34,970
+ماشي الحال show that
+
+266
+00:24:36,040 --> 00:24:44,360
+F' is even a proof بدنا نثبت أنه لو كانت الـ F
+
+267
+00:24:44,360 --> 00:24:49,940
+اللي هي odd function بدو يكون عنده و
+
+268
+00:24:49,940 --> 00:24:52,320
+differentiable بدو يكون عنده derivative إلها إيه
+
+269
+00:24:52,320 --> 00:25:03,000
+إيش even لأن let C element in R be arbitrary and
+
+270
+00:25:03,000 --> 00:25:04,180
+fixed
+
+271
+00:25:06,310 --> 00:25:15,930
+NR ناخد الـ R ناخد الـ C أي اللي هو real number in
+
+272
+00:25:15,930 --> 00:25:23,360
+R لكن نحكي عن اي شي محدد الان F prime of Cبدأ
+
+273
+00:25:23,360 --> 00:25:29,060
+أثبتلك أنه هو بسوء F prime of ناقص C يعني F prime
+
+274
+00:25:29,060 --> 00:25:34,920
+is even إذا خد F prime ناقص C وابدأ حسب وواصلك في
+
+275
+00:25:34,920 --> 00:25:39,140
+النهاية بسوء F prime of C إذا F prime is even بسوء
+
+276
+00:25:39,140 --> 00:25:49,380
+limit اللي هي F of X ناقص F of minus C على X minus
+
+277
+00:25:50,570 --> 00:25:57,070
+اللي هو minus C لما X تروح لمين لـ minus C مظبوط
+
+278
+00:25:57,070 --> 00:26:05,550
+طيب و يساوي limit ال F إيش معتنيها عبارة عن odd
+
+279
+00:26:05,550 --> 00:26:12,250
+إيش يعني odd يعني F of ناقص X بيساوي ناقص F of X
+
+280
+00:26:12,250 --> 00:26:16,590
+مظبوط هذه اللي هي إيش ما لها odd function اللي هو
+
+281
+00:26:16,590 --> 00:26:28,550
+بيساوي limit F ofالان اللي هي ناقص f of ناقص x
+
+282
+00:26:28,550 --> 00:26:34,150
+واضح أه؟
+
+283
+00:26:34,150 --> 00:26:37,150
+f of ناقص x بيساوي ناقص f of x يعني f of x بيساوي
+
+284
+00:26:37,150 --> 00:26:41,230
+ناقص f of ناقص x فالجد تعرفوا ليش بعملت هيك لأن
+
+285
+00:26:41,230 --> 00:26:48,570
+ناقص f of ناقص c الان f is odd مظبوط بيصير زائد f
+
+286
+00:26:48,570 --> 00:26:49,330
+of c
+
+287
+00:26:52,220 --> 00:26:58,060
+على خد الان ناقص من هنا عامل مشترك بيصير عبارة عن
+
+288
+00:26:58,060 --> 00:27:08,220
+ناقص X الان ناقص ال C لما X تروح لمين لسالب ال C
+
+289
+00:27:08,220 --> 00:27:14,040
+ال X بتروح لسالب ال C إذا و فقط إذا سالب ال X
+
+290
+00:27:14,040 --> 00:27:22,160
+بتروح لمين إلى ال C الان خد لي Yبساوي سالب ال X
+
+291
+00:27:22,160 --> 00:27:29,240
+واستبدل عشان نضح لك إياه بساوي limit الان خد الان
+
+292
+00:27:29,240 --> 00:27:38,320
+عندي F حكما كان نقص اكسميا ال Y limit ناقص F of Y
+
+293
+00:27:38,320 --> 00:27:49,390
+زائد F of C على اللي هو Y بتروح إلى ال Cوهنا y
+
+294
+00:27:49,390 --> 00:27:57,430
+ناقص ال c وهنا في عندي ايش برضه ناقص برا لان خد من
+
+295
+00:27:57,430 --> 00:28:01,670
+هنا ناقص عام المشترك او ضيعه مع الناقص اللي هنا
+
+296
+00:28:01,670 --> 00:28:11,160
+ماضح اه بيصير عندي y ساوي limit f of yنقص f of c
+
+297
+00:28:11,160 --> 00:28:16,800
+على y minus c لما y تروح لل c وهذا عبارة عن f
+
+298
+00:28:16,800 --> 00:28:22,440
+prime لمين لل c بدأنا ب f prime نقص c وانتهينا ب f
+
+299
+00:28:22,440 --> 00:28:31,520
+prime لل c لذا therefore f prime is even whenever
+
+300
+00:28:31,520 --> 00:28:37,160
+f is odd and f is differentiable
+
+301
+00:28:40,710 --> 00:28:56,810
+السؤال الأخير السؤال 13 السؤال
+
+302
+00:28:56,810 --> 00:29:03,990
+13 هو كما يلي اش
+
+303
+00:29:03,990 --> 00:29:09,200
+اللي بقول السؤال 13سؤال 13 بيقول إذا كانت F من R
+
+304
+00:29:09,200 --> 00:29:12,380
+لـ R is differentiable at C element in R show that
+
+305
+00:29:12,380 --> 00:29:16,840
+F prime of C سوى limit N في F of C زائد 1 لأن ناقص
+
+306
+00:29:16,840 --> 00:29:20,620
+F of C as N goes to infinity يعني لو كانت F
+
+307
+00:29:20,620 --> 00:29:24,900
+differentiable عند الـ C element in R بنقدر نكتب
+
+308
+00:29:24,900 --> 00:29:27,920
+ال derivative اللي هي F prime of C على سورة limit
+
+309
+00:29:27,920 --> 00:29:32,320
+N F of C زائد 1 لأن ناقص F of C as N goes to
+
+310
+00:29:32,320 --> 00:29:38,120
+infinityلكن بيقول لي by example show that the
+
+311
+00:29:38,120 --> 00:29:44,000
+existence of this limit this limit need not اللي
+
+312
+00:29:44,000 --> 00:29:52,300
+هو imply the existence of the derivative نيجي
+
+313
+00:29:52,300 --> 00:30:03,960
+الآن للجزء الأول عندي F من R لRو F prime of C
+
+314
+00:30:03,960 --> 00:30:12,120
+exist لانه يقول لي prove that F prime of C can be
+
+315
+00:30:12,120 --> 00:30:20,760
+written as limit on F of C زائد واحدة لان ناقص F
+
+316
+00:30:20,760 --> 00:30:26,180
+of C as N goes to infinity هذا الجزء الأول الجزء
+
+317
+00:30:26,180 --> 00:30:32,400
+الثاني حنجي الان نقول prove للجزء الأولقبل ما نقول
+
+318
+00:30:32,400 --> 00:30:37,700
+الـ proof نذكركم بس بنظرية سابقة في الـ real واحد
+
+319
+00:30:37,700 --> 00:30:43,840
+إنه لو عندي limit f of x لما x تروح لل c لو كانت
+
+320
+00:30:43,840 --> 00:30:52,780
+بتساوي Lبكون عندى أي sequence xₙ بتروح للـ C لازم
+
+321
+00:30:52,780 --> 00:30:59,020
+يتحقق لها limit f of xₙ as n goes to infinity
+
+322
+00:30:59,020 --> 00:31:05,530
+بساوي برضه الـكنا نتحدث بالحديث، باستبداء الحديث
+
+323
+00:31:05,530 --> 00:31:08,670
+عن الـ limit العادية للـ function S X بتروح للـ C
+
+324
+00:31:08,670 --> 00:31:13,470
+إلى limit لمين للـ sequence أو limit للـ sequences
+
+325
+00:31:13,470 --> 00:31:17,290
+الآن بنستخل هذه اللي هو المعلومة في إثبات اللي
+
+326
+00:31:17,290 --> 00:31:22,630
+بدنايا عند الآن since F prime of C exist بما أنه
+
+327
+00:31:22,630 --> 00:31:29,250
+ال derivative عند C موجودةإذا أكيد عندي صار f
+
+328
+00:31:29,250 --> 00:31:37,010
+prime of c بسهولمة f of x أو اللي هي الـ x عند c
+
+329
+00:31:38,080 --> 00:31:46,140
+زائد h ناقص f of c على h as h goes to mean to zero
+
+330
+00:31:46,140 --> 00:31:49,860
+اللي هو التعريف التعريف ال derivative أو الشكل
+
+331
+00:31:49,860 --> 00:31:52,980
+الآخر للتعريف ال derivative f of c زائد ال
+
+332
+00:31:52,980 --> 00:31:55,560
+increment ناقص f of c على ال increment as ال
+
+333
+00:31:55,560 --> 00:32:01,600
+increment goes to mean to zero ماشي الحل ال L بما
+
+334
+00:32:01,600 --> 00:32:07,480
+أنه اللي هو ال 1 على N sequence بتروح للسفروهذا
+
+335
+00:32:07,480 --> 00:32:11,440
+الـ limit exist لأن حسب النظرية اللي حكيتها قبل
+
+336
+00:32:11,440 --> 00:32:21,100
+بشوية بكون عندي لأن but كذا then F prime of C can
+
+337
+00:32:21,100 --> 00:32:26,580
+be اللي هو إعادة اللي هي rewritten as a limit of a
+
+338
+00:32:26,580 --> 00:32:36,800
+sequence F of اللي هو limit limitF of C زائد but
+
+339
+00:32:36,800 --> 00:32:40,000
+الـ H اللي هي تروح للصفر صارت mean ال sequence
+
+340
+00:32:40,000 --> 00:32:46,380
+واحدة ل N تروح للصفر واحدة ل N ناقص F of C على
+
+341
+00:32:46,380 --> 00:32:52,640
+واحدة ل N as N goes to infinity مدامت ال sequence
+
+342
+00:32:52,640 --> 00:32:57,100
+واحدة ل N بتروح للصفر صارت ال F of واحدة ل N اللي
+
+343
+00:32:57,100 --> 00:33:00,900
+هي عبارة عن F of C زائد واحدة ل N لإن الـ C عبارة
+
+344
+00:33:00,900 --> 00:33:07,740
+عن إياش فالتةواضح أه؟ الان هذا بيساوي اللي هو
+
+345
+00:33:07,740 --> 00:33:14,600
+limit الان as n goes to infinity اللي هو أكيد اللي
+
+346
+00:33:14,600 --> 00:33:21,920
+هو بيصير عندي f n في الجوس f of c زائد واحدة ل n
+
+347
+00:33:21,920 --> 00:33:28,260
+ناقص f of c اللي هو as n goes to infinity هو هذا
+
+348
+00:33:28,260 --> 00:33:35,150
+صار f prime of c وهو المطلوبالان conversely the
+
+349
+00:33:35,150 --> 00:33:37,890
+converse need not to be true in general هيك بيقول
+
+350
+00:33:37,890 --> 00:33:45,390
+ليه يعني بيقوللي if if بيقوللي او اللي بيقول if f
+
+351
+00:33:45,390 --> 00:33:53,670
+برا اللي هو limit f of c زاد واحدة ل n ناقص f of c
+
+352
+00:33:53,670 --> 00:34:01,610
+الكل مضروف في n as n goes to infinity exist if كده
+
+353
+00:34:02,500 --> 00:34:12,640
+then f prime at c need not be exist اصلا مش انه
+
+354
+00:34:12,640 --> 00:34:16,180
+يكون بيسوي هذا او لا need not to be اشمله exist
+
+355
+00:34:16,180 --> 00:34:19,640
+لان لو كان exist على طول بيسوي اما need not to be
+
+356
+00:34:19,640 --> 00:34:28,200
+exist ومااخد مثال جالي consider consider consider
+
+357
+00:34:28,200 --> 00:34:37,000
+f of xبساوي ال absolute value لل X وخد عند ال C أش
+
+358
+00:34:37,000 --> 00:34:46,440
+بتساوي سفر واضح F prime of 0 does not exist لأنها
+
+359
+00:34:46,440 --> 00:34:48,780
+عبارة عن corner point، انتوا عارفين احنا اللي هو
+
+360
+00:34:48,780 --> 00:34:52,480
+ال F prime عند ال zero لل absolute value does not
+
+361
+00:34:52,480 --> 00:34:59,740
+exist لكن هذه متحققة، ليش؟ but limit
+
+362
+00:35:01,280 --> 00:35:10,900
+N في F of 0 زائد 1 على N نقص F of 0 as N goes to
+
+363
+00:35:10,900 --> 00:35:18,840
+infinity بساوي limit N F of 0 زائد 1 على N يعني F
+
+364
+00:35:18,840 --> 00:35:23,560
+of 1 على N F of X بساوي absolute value X ف1 على N
+
+365
+00:35:23,560 --> 00:35:28,960
+مظبوط أه as N goes to infinity طبعا الـ N بتروح
+
+366
+00:35:29,220 --> 00:35:35,100
+يصير عبارة عن ash واحد إذا فعلا جبنا مثال أن ال
+
+367
+00:35:35,100 --> 00:35:39,720
+limit هذه تكون exist و سوى واحد but ال F prime عند
+
+368
+00:35:39,720 --> 00:35:42,640
+هذا النقطة C اللي هي 00 في هذه الحالة does not
+
+369
+00:35:42,640 --> 00:35:47,320
+exist بيكون هيك احنا انتهينا من الجزء الأول من
+
+370
+00:35:47,320 --> 00:35:54,870
+المحاضرة الخامسةاللي هو discussion لأو مناقشة ل
+
+371
+00:35:54,870 --> 00:35:59,910
+section 6-1 اللي هو the derivative والآن سنكمل
+
+372
+00:35:59,910 --> 00:36:05,690
+الحديث في الجزء الثاني من المحاضرة اللي هو عن اللي
+
+373
+00:36:05,690 --> 00:36:09,250
+هو the mean value theorem أو اللي هي نكمل
+
+374
+00:36:09,250 --> 00:36:11,910
+applications على mean value theorem وناخد اللي هو
+
+375
+00:36:11,910 --> 00:36:12,030
+ال
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY.srt
new file mode 100644
index 0000000000000000000000000000000000000000..2f6385a402083484440427ad8b5675229aebca11
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY.srt
@@ -0,0 +1,3106 @@
+1
+00:00:04,720 --> 00:00:09,780
+بسم الله الرحمن الرحيم هذه هي المحاضرة رقم 11 في
+
+2
+00:00:09,780 --> 00:00:16,040
+مساق تحليل حقيقي 2 لطلاب وطالبات الجامعة الإسلامية
+
+3
+00:00:16,040 --> 00:00:24,760
+كلية العلوم وهي المحاضرة الأولى بعد إعلان الطوارئ
+
+4
+00:00:24,760 --> 00:00:32,580
+بخصوص أو بمواجهة فيروس كورونا المنتشر  اتحدتنا 
+
+5
+00:00:32,580 --> 00:00:37,960
+المرة الماضية بدأنا في اللي هو chapter 7 اللي كان
+
+6
+00:00:37,960 --> 00:00:41,280
+الحديث عن ال Riemann integral أو تكامل ريمان
+
+7
+00:00:41,280 --> 00:00:45,860
+بدأنا في ال section الأول اللي هو تحت عنوان Riemann
+
+8
+00:00:45,860 --> 00:00:50,740
+integrability عرفنا شغلتين حاجة اسمها ال upper sum
+
+9
+00:00:50,740 --> 00:00:55,960
+و حاجة اسمها ال lower sum و قلنا اللي هو ال lower
+
+10
+00:00:55,960 --> 00:01:00,290
+sum هو عبارة عن الـ summation للـ mk الـ mk هذه
+
+11
+00:01:00,290 --> 00:01:06,410
+تمثل في xk minus xk minus 1 حيث mk كانت تمثل أو m
+
+12
+00:01:06,410 --> 00:01:10,910
+small k كانت تمثل عبارة عن الـ infimum للدالة على
+
+13
+00:01:10,910 --> 00:01:15,520
+الفترة اللي هي المذكورة الآن ال .. ال .. ال other
+
+14
+00:01:15,520 --> 00:01:19,380
+sum هو عبارة عن ال summation لنفس ال sum العلوي
+
+15
+00:01:19,380 --> 00:01:24,520
+ولكن بدلا منها اللي هي M K capital اللي كانت تمثل
+
+16
+00:01:24,520 --> 00:01:28,280
+ال supremum لـ ال F of X وال X على اللي هي في
+
+17
+00:01:28,280 --> 00:01:34,110
+الفترة المذكورة اللي عندي الآن أخدنا أول لمّة المرة
+
+18
+00:01:34,110 --> 00:01:38,190
+الماضية وقلنا إذا كانت F من I لـ R bounded و B any
+
+19
+00:01:38,190 --> 00:01:43,230
+partition of I بدي يكون ال lower اللي هو sum لأي
+
+20
+00:01:43,230 --> 00:01:47,810
+partition B و function F أصغر أو يساوي ال upper
+
+21
+00:01:47,810 --> 00:01:52,470
+sum لنفس ال partition و لنفس اللي هي ال function F
+
+22
+00:01:52,470 --> 00:01:58,710
+بعد هي طبعا خطينا خطوة أخرى و جينا عرفنا اللي هو شو
+
+23
+00:01:58,710 --> 00:02:03,110
+معناته انها تكون اللي هو ال partition Q refinement
+
+24
+00:02:03,110 --> 00:02:08,610
+لل partition B قلنا Q اللي هو تحسين لـ B إذا كانت B
+
+25
+00:02:08,610 --> 00:02:13,940
+بي عبارة عن مجموعة جزئية من Q و بناء عليه اللي هو
+
+26
+00:02:13,940 --> 00:02:19,440
+قلنا ان أي اللي هو sub interval xk-1xk من ال
+
+27
+00:02:19,440 --> 00:02:23,520
+partition B يمكن كتابتها على صورة union of sub
+
+28
+00:02:23,520 --> 00:02:27,680
+intervals من اللي هو التحسين اللي هو ∈Q
+
+29
+00:02:31,060 --> 00:02:36,380
+الآن جينا اللي هو بناء على هذا التعريف جينا قولنا
+
+30
+00:02:36,380 --> 00:02:40,480
+لو كانت F is من I لعند R is bounded و B is any
+
+31
+00:02:40,480 --> 00:02:45,780
+partition of I و Q refinement لل B مدام اللي هو Q
+
+32
+00:02:45,780 --> 00:02:50,420
+-refinement إذا ال lower sum هيعلى و ال upper sum
+
+33
+00:02:50,420 --> 00:02:54,820
+هينزل على أساس انه اللي هو في النهاية يلتقي ال
+
+34
+00:02:54,820 --> 00:02:58,740
+upper مع ال lower و نصل لاللي هو ال integrability
+
+35
+00:02:58,740 --> 00:03:02,800
+أو معنى ال integrability كما سنرى لاحقا على الأقل
+
+36
+00:03:02,800 --> 00:03:06,560
+في اللي هو يكون واضح من خلال الرسم في اللي هي
+
+37
+00:03:06,560 --> 00:03:13,890
+الدوال الموجبة كما ذكرنا سابقا اللي بحكيه إنه لو
+
+38
+00:03:13,890 --> 00:03:17,330
+كانت عندي اللي هو F من I لـ R bounded و B partition
+
+39
+00:03:17,330 --> 00:03:22,750
+و Q و refinement للـ B هيكون عندي lower sum لل
+
+40
+00:03:22,750 --> 00:03:28,570
+partition B أصغر أو يساوي lower sum للتحسين عماله
+
+41
+00:03:28,570 --> 00:03:32,850
+التحسين بكبر لما بده يصل لفعلا المساحة تحت المنحنة
+
+42
+00:03:32,850 --> 00:03:39,470
+في حالة الدوال الموجبة القبر صم لـ Q و F سيبدأ يصغر
+
+43
+00:03:39,470 --> 00:03:42,850
+ويكون أصغر من الساوي اللي هو القبر صم لـ P و F
+
+44
+00:03:42,850 --> 00:03:47,310
+التحسين يعني سيصغره بمعنى آخر سيبدأ يلتقوا إلى
+
+45
+00:03:47,310 --> 00:03:51,770
+أسفل لما نصل إلى اللي هو مساواة في حالة الـ
+
+46
+00:03:51,770 --> 00:03:55,730
+Integrability لما نكون عندنا أخدنا الـ Supremum
+
+47
+00:03:55,730 --> 00:04:01,340
+لكل L و F الـ L و الـ infimum لكل الـ U بيصير بنسمي
+
+48
+00:04:01,340 --> 00:04:04,800
+بعد شوية حاجة اسمها ال lower integral و ال upper
+
+49
+00:04:04,800 --> 00:04:08,980
+integral و برهننا هذه النظرية و بعدين جينا لللمة و
+
+50
+00:04:08,980 --> 00:04:13,520
+بعدين جينا لللمة أخرى اللي هو لو كانت F من I لـ R
+
+51
+00:04:13,520 --> 00:04:17,600
+bounded و B1 و B2 اي partitions الآن لأي partitions
+
+52
+00:04:17,600 --> 00:04:22,480
+هيكون ال lower دايما بغض النظر عن ال partition اللي
+
+53
+00:04:22,480 --> 00:04:26,120
+هو هيكون أصغر أو يساوي الأبر بغض النظر عن ال
+
+54
+00:04:26,120 --> 00:04:28,660
+partition B2 يعني مش لنفس ال partition زي ما قلنا
+
+55
+00:04:28,660 --> 00:04:33,180
+في اللمبة 7 1 1 لأ لأي two partitions دايما
+
+56
+00:04:33,180 --> 00:04:37,170
+ال lower ما هو هيكون تحت أسفل المنحنى و ال upper
+
+57
+00:04:37,170 --> 00:04:41,350
+هيكون أعلى المنحنى بغض النظر عن ال partitions اللي
+
+58
+00:04:41,350 --> 00:04:45,690
+عندي طبعا التمثيل هذا في حال اللي هو ال F is a
+
+59
+00:04:45,690 --> 00:04:49,130
+positive function على ال interval المذكورة الآن
+
+60
+00:04:49,130 --> 00:04:52,730
+بعد هيك اجينا وعرفنا شو معناه ال lower integral
+
+61
+00:04:52,730 --> 00:04:55,510
+و شو معناه ال upper integral قلنا ال lower
+
+62
+00:04:55,510 --> 00:05:00,050
+integral كما هو متوقع سميناه ال F هو عبارة عن ال
+
+63
+00:05:00,050 --> 00:05:05,660
+supremum لل lowers و ال .. و ال .. و ال .. و ال
+
+64
+00:05:05,660 --> 00:05:10,440
+upper هو عبارة عن الـ infimum لل uppers حتى لو
+
+65
+00:05:10,440 --> 00:05:14,300
+التقت الـ L of F مع الـ U of F اللي هو من أعلى مع
+
+66
+00:05:14,300 --> 00:05:17,940
+الأسفل هيكونوا التقوا بالظبط عند .. من مساحة تحت
+
+67
+00:05:17,940 --> 00:05:20,960
+المنحنى في حالة الدالة الموجبة و هذه .. في هذه
+
+68
+00:05:20,960 --> 00:05:24,300
+الحالة بنسمي إذا كانت ال upper تساوي ال lower
+
+69
+00:05:24,300 --> 00:05:27,920
+بنسمي ال function على هذه الفترة is integral و هذا
+
+70
+00:05:27,920 --> 00:05:31,900
+الكلام كله تحدثنا فيه عشان هيك أنا مسرع شوية هو
+
+71
+00:05:31,900 --> 00:05:37,160
+حكينا إنه اللي هو دايما ال lower sum لل F أخدنا
+
+72
+00:05:37,160 --> 00:05:40,760
+نظرية قلنا ال lower integral أسف ال lower
+
+73
+00:05:40,760 --> 00:05:45,100
+integral دايما أصغر يساوي مين ال upper integral
+
+74
+00:05:45,100 --> 00:05:46,860
+إذن الآن النظرية
+
+75
+00:05:49,600 --> 00:05:52,760
+الإعلان المهم اللي هو لو كانت F من I لعند R
+
+76
+00:05:52,760 --> 00:05:56,000
+bounded function على closed bounded interval A و B
+
+77
+00:05:56,000 --> 00:05:59,720
+بدي يكون ال lower integral L of F أصغر أو ساوي ال
+
+78
+00:05:59,720 --> 00:06:04,930
+upper integral U of F بصورة عامة هذه هي الـ
+
+79
+00:06:04,930 --> 00:06:08,050
+definition اللي ذكرته قبل و شوية نقول عن الـ
+
+80
+00:06:08,050 --> 00:06:15,150
+function F على bounded sub interval A و B أو
+
+81
+00:06:15,150 --> 00:06:18,650
+closed bounded interval A و B و كانت الـ F عبارة
+
+82
+00:06:18,650 --> 00:06:22,080
+عن bounded function بنعرف أن الـ F is Riemann
+
+83
+00:06:22,080 --> 00:06:26,840
+integrable on I إذا كانت ال lower of F بساوي ال
+
+84
+00:06:26,840 --> 00:06:30,080
+upper of F معناته صارت اللي هي ال F is Riemann
+
+85
+00:06:30,080 --> 00:06:34,460
+integrable if and only if ال lower sum يساوي ال
+
+86
+00:06:34,460 --> 00:06:39,240
+upper sum هذا كله ذكرناه المرة الماضية و أيضا
+
+87
+00:06:39,240 --> 00:06:42,520
+عرفنا .. قلنا في هذه الحلقة أن كل integration من A
+
+88
+00:06:42,520 --> 00:06:47,890
+لـ B هو ال lower أو ال upper المتساويين و عرفنا أيضا
+
+89
+00:06:47,890 --> 00:06:50,610
+تعريف آخر قلنا ال integration من a لـ b بساوي ناقص
+
+90
+00:06:50,610 --> 00:06:53,830
+ال integration من b لـ a و عرفنا ال integration من a
+
+91
+00:06:53,830 --> 00:06:59,410
+لـ a بساوي صفر هذا كله حكينا المرة الماضية و مش هيك
+
+92
+00:06:59,410 --> 00:07:03,820
+كمان و أخدنا المثال اللي هو أثبتنا إنه اللي هو g of
+
+93
+00:07:03,820 --> 00:07:07,880
+x بيساوي x is integrable on i استنادًا على إنه
+
+94
+00:07:07,880 --> 00:07:11,020
+أوجدنا ال lower sum ال lower integral و ال upper
+
+95
+00:07:11,020 --> 00:07:13,300
+integral أثبتنا إن ال lower integral و ال upper
+
+96
+00:07:13,300 --> 00:07:16,300
+integral are equal و من ثم أثبتنا إنه ال
+
+97
+00:07:16,300 --> 00:07:19,940
+integration exist لل function x على الفترة 0 و 1
+
+98
+00:07:19,940 --> 00:07:26,810
+و أوجدنا قيمة ال integration في حينه وصلنا إلى اللي
+
+99
+00:07:26,810 --> 00:07:33,290
+هو مثالنا التالي أنه لو كانت F من I لعند .. F من I
+
+100
+00:07:33,290 --> 00:07:40,090
+اللي هي 01 لعند ال .. ال R be defined by أخدنا
+
+101
+00:07:40,090 --> 00:07:45,990
+الدالة كما يلي اللي هو قولنا أن F of X F of X
+
+102
+00:07:45,990 --> 00:07:53,040
+بساوي واحد إذا كانت x rational number element in Q
+
+103
+00:07:53,040 --> 00:08:00,100
+و بساوي 0 إذا كانت x element in ℝ\Q أو element in Q
+
+104
+00:08:00,100 --> 00:08:04,660
+complement اللي هي ال rational numbers الآن بدنا
+
+105
+00:08:04,660 --> 00:08:11,200
+نثبت بقول show that this function F طبعا أنا method
+
+106
+00:08:11,200 --> 00:08:17,340
+F على اللي هي الـ Q تقاطع طبعا الـ 0 و 1 اللي هي
+
+107
+00:08:17,340 --> 00:08:19,920
+الـ interval اللي بدأ عليها التقاطع الـ 0 و 1
+
+108
+00:08:19,920 --> 00:08:25,160
+بمعنى إن دالة F صارت من I اللي هي عبارة عن 0 و 1
+
+109
+00:08:26,040 --> 00:08:29,380
+اللي عند R واضح ان الدالة هذه is a bounded
+
+110
+00:08:29,380 --> 00:08:33,560
+function الآن بدأ أثبت لكم ان هذا الدالة is not
+
+111
+00:08:33,560 --> 00:08:38,120
+integrable on this interval is not integrable on
+
+112
+00:08:38,120 --> 00:08:44,100
+this interval الآن علشان أصل اللي هو اللي هي ان
+
+113
+00:08:44,100 --> 00:08:48,180
+هذا الدالة غير قابلة تكامل بالنسبة لتكامل بالنسبة
+
+114
+00:08:48,180 --> 00:08:55,650
+لتكامل ريمان بدي اخد الان B أخذوا أي partition X0
+
+115
+00:08:55,650 --> 00:09:02,550
+X1 لعند Xn هذا any partition لإيه ال interval اللي
+
+116
+00:09:02,550 --> 00:09:06,790
+هي الفترة مين Zero و واحد يعني بمعنى أتيت للفترة
+
+117
+00:09:06,790 --> 00:09:14,030
+Zero و واحد و جزقتها X0 X1 لعند مؤصل لعند مين لعند
+
+118
+00:09:14,030 --> 00:09:18,610
+Xn اللي هي إيش بتساوي بتساوي واحد الآن هذا ال
+
+119
+00:09:18,610 --> 00:09:22,130
+partition اخدته arbitrarily اللي هو partition
+
+120
+00:09:22,130 --> 00:09:31,790
+لفترة L عندي الآن بدي احسب ال L of B و F لهذا ال
+
+121
+00:09:31,790 --> 00:09:34,990
+partition أيش بتساوي حسب اللي عرفناها سابقا بتساوي
+
+122
+00:09:34,990 --> 00:09:40,610
+ال summation لل M K في X K minus X K minus واحد K
+
+123
+00:09:40,610 --> 00:09:46,330
+من عند واحد لعند مين لعند اللي هي N و يساوي الآن ال
+
+124
+00:09:46,330 --> 00:09:52,450
+M K عرفناها ال M K هي عبارة عن ال infimum لقيمة ال
+
+125
+00:09:52,450 --> 00:09:56,470
+function F of X حيث X تنتمي إلى الفترة X K minus
+
+126
+00:09:56,470 --> 00:10:02,210
+واحد لعند X K طبعا اللي هي F of X معرفة على
+
+127
+00:10:02,210 --> 00:10:05,130
+أساسيها يا إما واحد يا إما صفر حسب إنها تكون
+
+128
+00:10:05,130 --> 00:10:08,030
+rational أو إيش ال irrational يعني ال function F
+
+129
+00:10:08,030 --> 00:10:11,910
+أصلا اللي هي قيمتين بس إذا الآن ال infimum لل F of
+
+130
+00:10:11,910 --> 00:10:16,630
+X عندها يا هيكون واحد يا هيكون صفر ليش؟ لأن أصلا
+
+131
+00:10:16,630 --> 00:10:23,340
+الفترة هذه فيها أي فترة subinterval xk-1xk فيها
+
+132
+00:10:23,340 --> 00:10:27,620
+rational و irrational إذا قيمة ال F of x في الفترة
+
+133
+00:10:27,620 --> 00:10:31,260
+هتلاقي عند قيم واحد هتلاقي أكيد عند قيم أش بتساوي
+
+134
+00:10:31,260 --> 00:10:35,000
+بساوي صفر إذا ال infimum في هذه الحالة هو عبارة عن
+
+135
+00:10:35,000 --> 00:10:42,320
+إيش يساوي صفر إذا ال summation لـ 0 في xk-xk-1 كامل
+
+136
+00:10:42,320 --> 00:10:46,120
+عند واحد عند أنه طبيعي هذا بديهي إيش هيساوي بساوي
+
+137
+00:10:46,120 --> 00:10:52,410
+صفر إذن الآن L of F بي و F ساوي صفر فور أي partition
+
+138
+00:10:52,410 --> 00:10:59,370
+بيه إذا ال L of F اللي هي عبارة عن الأو الـ
+
+139
+00:10:59,370 --> 00:11:06,070
+Supremum الـ Supremum لكل الـ L of B و F Such that
+
+140
+00:11:06,070 --> 00:11:09,890
+B element in the set of all partitions اللي هو B
+
+141
+00:11:09,890 --> 00:11:14,090
+of I هيكون ال .. ال Supremum اللي هين صفر لأن كل
+
+142
+00:11:14,090 --> 00:11:18,350
+اللي هين أصلا إيش بتنطلع .. تنطلع بساوي صفر إذا
+
+143
+00:11:18,350 --> 00:11:23,070
+هذا إيش هيساوي يا شباب؟ هو يساوي Zero إذا طلع عندي
+
+144
+00:11:23,070 --> 00:11:28,150
+ال lower sum بساوي 0 الآن بدي أحسبلكم مين أحسبلكم
+
+145
+00:11:28,150 --> 00:11:31,390
+ال upper sum سامحوني أكتب هنا بس عساس اللي يبقى
+
+146
+00:11:31,390 --> 00:11:39,670
+كله مكتوب عندي نوجد ال upper sum ال upper sum اللي
+
+147
+00:11:39,670 --> 00:11:45,510
+هي ال U of P U of F بساوي summation للـ M K capital في XK
+
+148
+00:11:45,510 --> 00:11:51,740
+minus XK minus واحد K من عند واحد لعند ال N الآن
+
+149
+00:11:51,740 --> 00:11:55,900
+هذا بيساوي ال Mk زي ما قلنا قبل هيك ال Mk بدل ما
+
+150
+00:11:55,900 --> 00:11:59,400
+هي ال M في مميزة استعريفها ال Mk بتساوي ال
+
+151
+00:11:59,400 --> 00:12:03,780
+supremum لهذه ال F و زي ما قلنا ال F هذه في داخلها
+
+152
+00:12:03,780 --> 00:12:08,400
+يا واحد يا زيرو نظرا لإن اللي هو أي sub interval
+
+153
+00:12:08,400 --> 00:12:11,820
+هيكون فيها rational و irrational و تبعا إلها هيكون
+
+154
+00:12:11,820 --> 00:12:16,060
+قيمة ال function في داخلها واحد أو صفر و احنا بنبحث
+
+155
+00:12:16,060 --> 00:12:19,280
+عن ال supremum إذا هيكون ال supremum في كل الأحوال
+
+156
+00:12:19,280 --> 00:12:25,600
+ال Mk بتساوي واحدمضروبة في xk-xk-1 k من عند 1 لعند
+
+157
+00:12:25,600 --> 00:12:30,700
+n نفردها هذه و يساوي اللي هنبصير k من عند 1 يعني x1
+
+158
+00:12:30,700 --> 00:12:39,570
+-x0 زائد x2-x1 زائد إلى آخر لما أصل لعند xn-1 ناقص xn
+
+159
+00:12:39,570 --> 00:12:44,550
+ناقص واحد طبعا واضح انه عندي ال x واحد هت cancel مع
+
+160
+00:12:44,550 --> 00:12:48,270
+ناقص x واحد و ال x اتنين مع ناقص x اتنين لما نصل
+
+161
+00:12:48,270 --> 00:12:52,450
+للاخر هيكون في عندي ات cancel الجميع بس ضال عندي
+
+162
+00:12:52,450 --> 00:12:58,170
+ال xn و ال x note و هد بتساوي xn ناقص x note و
+
+163
+00:12:58,170 --> 00:13:01,750
+يساوي ال xn طبعا إيش هي عبارة عن واحد و ال x note
+
+164
+00:13:01,750 --> 00:13:07,190
+إيش هي شبه صفر و يساوي واحد ناقص صفر و يساوي واحد
+
+165
+00:13:07,570 --> 00:13:12,930
+إذا طلع عندي الـ Upper Sum لأي Bar تشهم بيه، هيتلع
+
+166
+00:13:12,930 --> 00:13:18,510
+إيش بساوة؟ بساوة واحدة إذا الآن لما بدي أخد ال U
+
+167
+00:13:18,510 --> 00:13:23,510
+of F اللي هو Upper Integral هيساوة عبارة عن ال
+
+168
+00:13:23,510 --> 00:13:30,220
+Infimum لمين؟ للـ U, B و F such that B element in
+
+169
+00:13:30,220 --> 00:13:34,580
+the set of all partitions B of I و الـ U, B و F
+
+170
+00:13:34,580 --> 00:13:38,840
+قيمته ثابتة for أي partition بيساوي واحد إذا الـ
+
+171
+00:13:38,840 --> 00:13:43,300
+infimum لكل اللي هنا عبارة عن برضه إيش بيساوي واحد
+
+172
+00:13:43,300 --> 00:13:47,410
+صار عندي الآن lower integral و ال upper integral
+
+173
+00:13:47,410 --> 00:13:51,110
+have different values واحد بيساوي صفر واحد بيساوي
+
+174
+00:13:51,110 --> 00:13:56,890
+واحد و بناء عليه بتكون عنده اللي هو ال F is not
+
+175
+00:13:56,890 --> 00:14:03,150
+Riemann integrable أي سؤال؟ طيب ماشي الحاجة
+
+176
+00:14:03,150 --> 00:14:10,930
+الآنصار عندي أخدنا مثلين المثال الأول اللي هو
+
+177
+
+201
+00:16:16,410 --> 00:16:20,150
+اللي هو طريقة للحكم على ال integrability  أخرى غير 
+
+202
+00:16:20,150 --> 00:16:26,430
+التعريف اللي هو بتقول F is integrable
+
+203
+00:16:28,870 --> 00:16:32,890
+if and only if طبعاً هذه لمين الـ F؟ F عبارة عن زي
+
+204
+00:16:32,890 --> 00:16:36,190
+ما أنتم عارفين bounded function لإنه كل شغلنا
+
+205
+00:16:36,190 --> 00:16:40,010
+أصلاً على اللي هو remaining integrability أنه نفترض
+
+206
+00:16:40,010 --> 00:16:43,210
+أنه الـ F bounded عشان اللي هو تكون ال supremum و
+
+207
+00:16:43,210 --> 00:16:46,330
+ال infimum اللي مبني عليها التعريف تكون مضمون إنها 
+
+208
+00:16:46,330 --> 00:16:49,870
+موجودة عشان هيك بنحكي أن F is a bounded function طيب
+
+209
+00:16:49,870 --> 00:16:56,140
+إذا الـ F is integrable if and only if اللي هي لكل
+
+210
+00:16:56,140 --> 00:17:00,620
+إبسيلون أكبر من 0 there exists a partition P إبسيلون
+
+211
+00:17:00,620 --> 00:17:04,240
+هذا ال P اللي هو ال partition يعتمد عالميا على
+
+212
+00:17:04,240 --> 00:17:07,820
+إبسيلون لكل إبسيلون بالله دي partition P إبسيلون لمين
+
+213
+00:17:07,820 --> 00:17:11,840
+ال partition طبعاً لل interval اللي عندنا اللي هي I
+
+214
+00:17:11,840 --> 00:17:17,040
+there exists P إبسيلون a partition of I such that
+
+215
+00:17:17,040 --> 00:17:25,210
+الـ U  P إبسيلون  والـ F ناقص الـ L P إبسيلون  و
+
+216
+00:17:25,210 --> 00:17:31,050
+F يكون أصغر من مين من إبسيلون الآن إذا كان لجينا
+
+217
+00:17:31,050 --> 00:17:34,190
+لكل إبسيلون لجينا ب P إبسيلون بحيث هذا يتحقق معناته F
+
+218
+00:17:34,190 --> 00:17:37,590
+is integrable and conversely if F is integrable
+
+219
+00:17:37,590 --> 00:17:42,550
+أكيد لكل إبسيلون هلجي ب P إبسيلون بحيث أن هذا يتحقق
+
+220
+00:17:42,550 --> 00:17:48,170
+خلونا نيجي الآن نبرهن و نشوف كيف بدنا نبرهن
+
+221
+00:17:48,170 --> 00:17:53,880
+نظريتنا الآن بدنا نفترض ان F أي شمالها is
+
+222
+00:17:53,880 --> 00:18:00,240
+integrable ونصل منها للي عيدي اللي هي ال partition
+
+223
+00:18:00,240 --> 00:18:07,040
+اللي مذكور مدام F is integrable إذا كانت ..الآن
+
+224
+00:18:07,040 --> 00:18:16,100
+بنقول suppose that F is integrable مدام integrable
+
+225
+00:18:16,100 --> 00:18:22,090
+يا شباب أكيد عنده اللي هو الـ U of F بيساوي L of
+
+226
+00:18:22,090 --> 00:18:29,490
+إيش؟ Of F، مظبوط؟ أكيد الـ L of .. الـ U of F
+
+227
+00:18:29,490 --> 00:18:33,410
+بساوي الـ L of F إيش اللي بيدلّ عليه؟ بيدلّ عليه لأي
+
+228
+00:18:33,410 --> 00:18:35,810
+إبسيلون أكبر من صفر بدلّ عليه بإبسيلون، شوفوا كيف
+
+229
+00:18:35,810 --> 00:18:41,090
+بنلجيه، الآن نفترض إن إبسيلون let إبسيلون أكبر من
+
+230
+00:18:41,090 --> 00:18:47,810
+صفر be given، ماشي الحال عندي الـ U of F هو إيش يا 
+
+231
+00:18:47,810 --> 00:18:54,910
+شباب؟ هو عبارة عن الـ infimum للـ L of P و F such
+
+232
+00:18:54,910 --> 00:19:01,170
+that P element in P of I، مظبوط؟ إذا الـ U of F
+
+233
+00:19:01,170 --> 00:19:05,830
+عبارة عن infimum يعني هو عبارة عن greatest lower
+
+234
+00:19:05,830 --> 00:19:11,630
+bound لو هذا الـ greatest lower bound ضفنا إليه أي
+
+235
+00:19:11,630 --> 00:19:17,230
+كمية هيبطل lower bound لأن هو أصلاً إيش اسمه
+
+236
+00:19:17,230 --> 00:19:23,510
+greatest lower bound إذا لو الـU of F ضفت له Y على 
+
+237
+00:19:23,510 --> 00:19:28,610
+2 مثلاً طبعاً هذا المقدار هيبطل lower bound إيش
+
+238
+00:19:28,610 --> 00:19:33,690
+معناته بطل lower bound؟ يعني بمعنى آخر هيكون في
+
+239
+00:19:33,690 --> 00:19:44,080
+عندي شيء أصغر منه هيكون عندي أصغر من ال U of F أو ب
+
+240
+00:19:44,080 --> 00:19:49,740
+واحد مثلاً و F for some P واحد إذا لما نشيل
+
+241
+00:19:49,740 --> 00:19:54,600
+من ال infimum إبسيلون
+
+242
+00:19:54,600 --> 00:19:57,740
+على اثنين هيبطل هذا lower bound يعني بمعنى آخر
+
+243
+00:19:57,740 --> 00:20:03,280
+هلاقي اللي هو lower bound
+
+244
+00:20:07,660 --> 00:20:12,140
+عند الـ Y أكبر من 0 خلّيني أبدأ لـ Y أكبر من 0 ب
+
+245
+00:20:12,140 --> 00:20:17,940
+given إذا عندي الآن بدي أثبت لك بدي أجيب لك 
+
+246
+00:20:17,940 --> 00:20:21,440
+partition ب P إبسيلون بحيث أنه هذا ناقص هذا يكون أصغر
+
+247
+00:20:21,440 --> 00:20:26,070
+من مين من إبسيلون شوف كيف بدي أعمل الآن أنا عندي ال
+
+248
+00:20:26,070 --> 00:20:30,710
+U of F إيش بيساوي يا جماعة ال U of F بيساوي ال
+
+249
+00:20:30,710 --> 00:20:39,590
+infimum لل L P of F ب element I طيب ال U of F آسف يا
+
+250
+00:20:39,590 --> 00:20:43,010
+جماعة ال U of F بيساوي ال infimum لمين لل U P of F
+
+251
+00:20:43,720 --> 00:20:49,780
+ماشي الحال الآن إيش معناه إنه هذا infimum معناته
+
+252
+00:20:49,780 --> 00:20:55,100
+هذا هو عبارة عن ال greatest lower bound مدام ال
+
+253
+00:20:55,100 --> 00:20:59,450
+greatest lower bound إذا ال lower bound هذا أو الـ
+
+254
+00:20:59,450 --> 00:21:03,330
+Greatest Lower Bound لو ضفت له أي رقم إبسيلون على
+
+255
+00:21:03,330 --> 00:21:06,370
+اثنين مثلاً بالذنب إبسيلون على اثنين ال game
+
+256
+00:21:06,370 --> 00:21:09,670
+بتعرفوا ليش إبسيلون على اثنين يعني لو ضفت له أي رقم
+
+257
+00:21:09,670 --> 00:21:14,730
+بيبطل Lower Bound إيش معناته بيبطل Lower Bound
+
+258
+00:21:14,730 --> 00:21:21,830
+يعني هيصير هذا أكبر من ال U P1 of F for some P
+
+259
+00:21:21,830 --> 00:21:27,510
+واحدة لأنه بطل إيش ماله هذا بطل lower bound بطل يكون
+
+260
+00:21:27,510 --> 00:21:32,670
+أصغر من الكل من هان يعني هلاجي واحد من هان هو مش
+
+261
+00:21:32,670 --> 00:21:36,830
+أصغر منه أو بمعنى آخر U, P, 1 و F أصغر من اللي هو
+
+262
+00:21:36,830 --> 00:21:44,850
+هذا المقدار طيب similarly ال L of F هي عبارة عن ال
+
+263
+00:21:44,850 --> 00:21:52,050
+supremum لل L, P و F such that P elemented P of I
+
+264
+00:21:53,040 --> 00:21:57,780
+بنفس الطريقة يا جماعة ال L of F هي عبارة عن إيش ال
+
+265
+00:21:57,780 --> 00:22:04,040
+least upper bound يعني هذا لو least upper bound
+
+266
+00:22:04,040 --> 00:22:07,200
+Upper bound لو كان أصغر و أحد لو هذا least upper
+
+267
+00:22:07,200 --> 00:22:11,000
+bound راحت منه عدد ولو صغير جداً وليكن إبسيلون على
+
+268
+00:22:11,000 --> 00:22:16,800
+اثنين هيبطل هذا عبارة عن upper bound يعني هلاقي
+
+269
+00:22:16,800 --> 00:22:24,280
+واحد من اللي هان اللي هو L of P2 وF مثلاً أكبر منه
+
+270
+00:22:24,280 --> 00:22:30,560
+لأنه هيبطل هذا إيش ماله upper bound لأنه هو ال least
+
+271
+00:22:30,560 --> 00:22:35,780
+لما طلعت منه بطل من ال upper bounds يعني لجيت واحد
+
+272
+00:22:35,780 --> 00:22:43,120
+من هذول أكبر منه إذا صار في عندي اللي هو لجيت ب
+
+273
+00:22:43,120 --> 00:22:53,960
+واحد بحقق الأولى و P اثنين بحقق الثانية لأن خد الآن
+
+274
+00:22:53,960 --> 00:22:58,840
+خد لي P إبسيلون هذا اللي بدي إياه هذا يحقق اللي اللي
+
+275
+00:22:58,840 --> 00:23:03,600
+بدي إياه خد P إبسيلون إيش بيساوي ال P واحد اللي لجيتة
+
+276
+00:23:03,600 --> 00:23:12,050
+هنا اتحاد الـ P2 اللي لجيتة هان اتحاد مين؟ P2 صار
+
+277
+00:23:12,050 --> 00:23:18,510
+عندي .. صار عندي الآن مع بعض خليني أمسح فأقوم،
+
+278
+00:23:18,510 --> 00:23:22,170
+مهمش؟
+
+279
+00:23:22,170 --> 00:23:31,670
+صار
+
+280
+00:23:31,670 --> 00:23:41,110
+عندي ما يليه؟ صار عندي الآن L of F ناقص
+
+281
+00:23:41,110 --> 00:23:53,490
+Y على 2 أصغر من L of P2 F اللي هو أكيد أصغر أو
+
+282
+00:23:53,490 --> 00:23:59,750
+يساوي L of P of F قُقَف، ليش؟ لأن الـ P  يا جماعة عبارة
+
+283
+00:23:59,750 --> 00:24:04,570
+عن refinement للـ P2 وال lower لما يصير فيه تحسين
+
+284
+00:24:04,570 --> 00:24:08,510
+بكبر، بروح نحو اتجاه نيل اللي هو المنحنة التالية
+
+285
+00:24:08,510 --> 00:24:14,010
+مساحة تحت المنحنة الكلية لأن أيضاً لو جيت قلت الـU
+
+286
+00:24:14,010 --> 00:24:24,400
+of F زائد إبسيلون على اثنين هتلاقيها أكبر من ال U P
+
+287
+00:24:24,400 --> 00:24:30,410
+واحد و F وأكيد عندي الـ U of F زائد إبسيلون على اثنين
+
+288
+00:24:30,410 --> 00:24:34,490
+أكبر من الـ U P1 of F هيكون هذا أكبر أو يساوي الـ
+
+289
+00:24:34,490 --> 00:24:39,610
+U P إبسيلون of F لأن الـ P إبسيلون refinement لمين
+
+290
+00:24:39,610 --> 00:24:44,790
+برضه؟ للـ P1 بزام refinement إذن اللي هو التحسين
+
+291
+00:24:44,790 --> 00:24:50,270
+بيزغر القبر وبروح ناحية اللي هو المنحنة إذن الآن
+
+292
+00:24:50,270 --> 00:24:55,480
+من الـ tip 2 هذولة وجدنا طبعاً احنا تنسوش إن احنا
+
+293
+00:24:55,480 --> 00:24:59,100
+مفترضين من رأس الدول إن U of F is integrable يعني
+
+294
+00:24:59,100 --> 00:25:05,340
+مفترضين إن ال L of F إيش بتساوي U of F تنسواش هذه
+
+295
+00:25:05,340 --> 00:25:12,540
+ليه جيت نطرح مع بعض دول بيصير عندي حصل على L of F
+
+296
+00:25:12,540 --> 00:25:22,970
+ناقص Y على 2 اللي هو أصغر من L of P, إبسيلون و F
+
+297
+00:25:22,970 --> 00:25:30,110
+والـ L و الـ U of F زائد إبسيلون على 2 أكبر من U, 
+
+298
+00:25:30,190 --> 00:25:35,290
+P, إبسيلون و F أنا إيش غرضي؟ غرضي أثبت إن U, P,
+
+299
+00:25:35,350 --> 00:25:39,130
+إبسيلون و F ناقص L, P, إبسيلون و F أصغر من إبسيلون
+
+300
+00:25:39,130 --> 00:25:42,890
+يلا اترحل من بعض، إذا بيصير عندي، بتحصل البدكية
+
+301
+00:25:42,890 --> 00:25:48,570
+بيصير عندي الآن، بطرح حياة دي، بقول U, P, إبسيلون و
+
+302
+00:25:48,570 --> 00:25:53,240
+F ناقص لأنه لما نضرب هذا في ناقص هتنعكس يعني هتصير
+
+303
+00:25:53,240 --> 00:25:57,100
+هذه جماعة ناقص وهذه زائد وهذه هتنعكس هيك وهيصير
+
+304
+00:25:57,100 --> 00:26:05,600
+الاش ناقص بيصير عندي U P و F ناقص ال P و F هيصير
+
+305
+00:26:05,600 --> 00:26:11,820
+أصغر من مين من ناقص L of F ناخد هذا قبل زي ما احنا
+
+306
+00:26:11,820 --> 00:26:20,690
+مرتبينها U of F زائد Y على 2 ناقص L of F زي إبسيلون
+
+307
+00:26:20,690 --> 00:26:24,790
+على 2 وطبعاً إحنا جايلين إن F is integrable يعني
+
+308
+00:26:24,790 --> 00:26:28,770
+الـ U of F بيساوي L of F إذا هذي بتروح مع هذي بظل
+
+309
+00:26:28,770 --> 00:26:33,610
+إيش ماله؟ بظل إبسيلون إذا إحنا لكل إبسيلون أكبر من
+
+310
+00:26:33,610 --> 00:26:36,950
+صفر لجينا ب P إبسيلون هي في الواقع P إبسيلون اللي
+
+311
+00:26:36,950 --> 00:26:39,970
+لجيناه ب P واحد اتحاد P اتنين حيث P واحد اللي
+
+312
+00:26:39,970 --> 00:26:44,120
+لجيناه هان وال P اتنين اللي لجيناه هان such that U
+
+313
+00:26:44,120 --> 00:26:49,120
+P Y of F ناقص L P Y of F أصغر من اللي هو إبسيلون
+
+314
+00:26:49,120 --> 00:26:57,480
+وهو المطلوب أي سؤال؟ طيب، ماشي يا شباب، الآن خلصنا
+
+315
+00:26:57,480 --> 00:27:04,420
+الجزء الأول من النظرية أثبتنا اللي بدنا إياه إنه اللي
+
+316
+00:27:04,420 --> 00:27:08,860
+هو هذه العلاقة صحيحة لأننا نفترض إنه suppose that
+
+317
+00:27:08,860 --> 00:27:12,380
+star holds اللي هي star ليه هذه نفترض إن لكل
+
+318
+00:27:12,380 --> 00:27:15,380
+epsilon أكبر من 0 there exists P of epsilon such
+
+319
+00:27:15,380 --> 00:27:18,680
+that U P Epsilon و F ناقصها دي أصغر من إبسيلون و
+
+320
+00:27:18,680 --> 00:27:24,280
+بدنا نصل من خلالها لإيش لأن ال F is integrable
+
+321
+00:27:24,280 --> 00:27:25,760
+فنشوف
+
+322
+00:27:40,480 --> 00:27:47,440
+البرهان بسيط لو طلعنا عليه مباشرة على اللوح الآن
+
+323
+00:27:47,440 --> 00:27:52,140
+بدنا نفترض أن هذه تتحقق اللي هو نفترض أنه لكل Y
+
+324
+00:27:52,140 --> 00:27:57,020
+أكبر من 0 يوجد P إبسيلون بحيث أن هذا اللي هي تتحقق
+
+325
+00:27:57,020 --> 00:28:01,620
+عشان أصل بدي أوصلكم في النهاية أن L of F هي إيش
+
+326
+00:28:01,620 --> 00:28:05,810
+بتساوي U of F شوف كيف بدي أوصلها اللي هبدأ أقول لكم اللي
+
+327
+00:28:05,810 --> 00:28:08,650
+هو نفترض إنه زي ما قلنا إنه star holds اللي حكينا
+
+328
+00:28:08,650 --> 00:28:12,990
+عنها لأن for any partition P هيكون ال L P of F
+
+329
+00:28:12,990 --> 00:28:17,970
+أصغر أو يساوي L of F و ال U of F أصغر أو يساوي مين ال U
+
+330
+00:28:17,970 --> 00:28:25,970
+P of F واضح؟ إذا أصبح عيدي الآن L يمكن أوضح لكم على
+
+331
+00:28:25,970 --> 00:28:35,790
+اللوح L of P of F ده ال P أي partition أصغر مظبوط
+
+332
+00:28:35,790 --> 00:28:40,910
+أو يساوي ال L of F لـ F لأنه ال L of F يا جماعة
+
+333
+00:28:40,910 --> 00:28:47,330
+هو ال supremum اللي هنا و ال U P of F
+
+334
+00:28:47,330 --> 00:28:53,130
+أكبر أو يساوي ال U of F لأنه ال U of F يا جماعة
+
+335
+00:28:53,130 --> 00:28:57,950
+هو عبارة عن مين عبارة عن ال infimum أترحلوا الجهة
+
+336
+00:28:57,950 --> 00:29:01,410
+التانية هذا طبعاً لكل مين لكل ال partitions اللي في
+
+337
+00:29:01,410 --> 00:29:06,750
+الدنيا من ضمنهم ال P إبسيلون اللي احنا معطينا إياه
+
+338
+00:29:06,750 --> 00:29:12,250
+في ال .. اللي هو نص النظرية إذا بسيري عندي لأن لو
+
+339
+00:29:12,250 --> 00:29:19,990
+أجيب طرحة ال U F ناقص L F ال U F ناقص L F هيصير إيش
+
+340
+00:29:19,990 --> 00:29:23,850
+ماله يا جماعة؟ يعني طريقة رحلة من هذه بيصير أصغر
+
+341
+00:29:23,850 --> 00:29:28,510
+أو يساوي لإن هذه بتضربها في ناقص و هذه ناقص و
+
+342
+00:29:28,510 --> 00:29:32,750
+بتنقلب زي ما عملنا قبل و شوية بيصير أصغر أو يساوي
+
+343
+00:29:32,750 --> 00:29:43,640
+U P of F ناقص ال P of F هذا الكلام صحيح لإيش لكل
+
+344
+00:29:43,640 --> 00:29:48,200
+partition في الدنيا من ضمنها المين ال partition
+
+345
+00:29:48,200 --> 00:29:52,580
+المواطع لنا يعني حيصير عند هذا ينطبق برضه على ال
+
+346
+00:29:52,580 --> 00:29:57,500
+P إبسيلون إذا صار هذا أصغر يسوي P إبسيلون ناقص ال
+
+347
+00:29:57,500 --> 00:30:01,740
+P إبسيلون of F طيب هم يعطيين إن هذا المقدار إيش
+
+348
+00:30:01,740 --> 00:30:06,260
+ماله أصغر من إبسيلون لأي إبسيلون في الدنيا هو أنا
+
+349
+00:30:06,260 --> 00:30:10,580
+بعرف إن هذا المقدار نفسه أكبر أو يساوي إيش صفر صار
+
+350
+00:30:10,580 --> 00:30:17,720
+عندي الآن ال U of F ناقص ال L of F دائماً أصغر من
+
+351
+00:30:17,720 --> 00:30:23,420
+إبسيلون و أكبر أو يساوي صفر لكل إبسيلون أكبر من صفر
+
+352
+00:30:23,420 --> 00:30:28,260
+إذا على طول من نظرية في تحليل واحد هيكون هذا اللي
+
+353
+00:30:28,260 --> 00:30:36,250
+عندي إذا U of F ناقص L of F بيساوي صفر إذا U of F
+
+354
+00:30:36,250 --> 00:30:44,850
+بيساوي L of F وهذا يعني F is a Riemann Integral هو
+
+355
+00:30:44,850 --> 00:30:52,110
+المطلوب بيكون إحنا هيك أثبتنا اللي هو integrable
+
+356
+00:30:52,110 --> 00:30:58,350
+criterion أو اللي هو طريقة لتحديد اللي هو ال
+
+357
+00:30:58,350 --> 00:31:02,270
+function is integrable أو لا غير اللي هو طريقة
+
+358
+00:31:02,270 --> 00:31:07,110
+التعريف الآن في عند كورولاري بعدها كورولاري
+
+359
+00:31:07,110 --> 00:31:11,530
+الكورولاري
+
+360
+00:31:11,530 --> 00:31:16,230
+هي تقول كما يلي الآن بدنا نترجم الحديث بدل ما كان
+
+361
+00:31:16,230 --> 00:31:22,710
+بإبسيلون نحكي عن مين؟ عن اللي هو sequence of
+
+362
+00:31:22,710 --> 00:31:29,870
+partitions طبعاً هو هذا معهود التحويل في نظريات
+
+363
+00:31:29,870 --> 00:31:34,590
+مشابهة في حتى في كورسات أخرى خلّينا نشوف عندنا
+
+364
+00:31:34,590 --> 00:31:38,790
+اللي هو النظرية إيش ب .. أو الكورلري إيش بتقول
+
+365
+00:31:38,790 --> 00:31:43,350
+بتقول let I بيساوي A و B and let F من I ل R be a
+
+366
+00:31:43,350 --> 00:31:48,600
+bounded function لأن لو فرضنا ب أن أن element none
+
+367
+00:31:48,600 --> 00:31:52,300
+is a sequence of partitions of I بحيث أن ال limit
+
+368
+00:31:52
+
+401
+00:34:22,710 --> 00:34:29,110
+بسبب إن الـ UBK أو الـ UBK زاد واحد أو الآخر ناقص
+
+402
+00:34:29,110 --> 00:34:32,590
+القللها أصغر من 100 من إبسلون إذا الـ criterion
+
+403
+00:34:32,590 --> 00:34:36,550
+اللي في الـ corollary حققت إذا صارت عندي هذه إذا F
+
+404
+00:34:36,550 --> 00:34:41,830
+is integrable يعني مش بي إبسلون واحد اللي جينا له
+
+405
+00:34:41,830 --> 00:34:46,550
+من عند بك وطالع كل الـ partitions هذه اللي هي بك و
+
+406
+00:34:46,550 --> 00:34:49,950
+بك زائد واحد و بك زائد اثنين بتعمل عمل الـ P
+
+407
+00:34:49,950 --> 00:34:53,510
+إبسلون اللي في وين في النظرية إذا الـ F أشمالها
+
+408
+00:34:53,510 --> 00:34:58,750
+صارت الـ F عبارة عن Integrable من النظرية السابقة
+
+409
+00:34:58,750 --> 00:35:05,200
+الآن الدور دلّ أن نثبت مين أن الـ limit للأل اللي هو
+
+410
+00:35:05,200 --> 00:35:09,260
+هذا المقدار هو limit لهذا المقدار بساوي إيش اللي
+
+411
+00:35:09,260 --> 00:35:12,540
+جوا طبعا هو لو كانت يا جماعة limit هذا ناقص هذا صفر
+
+412
+00:35:12,540 --> 00:35:15,920
+مش معناته الـ limit الأول و الـ limit الثاني exist  هاي
+
+413
+00:35:15,920 --> 00:35:20,610
+مثلا هذه لو كانت هذه Un تربيع وهذه Un  ... آسف
+
+414
+00:35:20,610 --> 00:35:24,610
+Un تربيع و Un تربيع أو Un و Un limit Un ناقص Un على
+
+415
+00:35:24,610 --> 00:35:27,850
+طول صفر لكن لا الـ limit الأولى عدد ولا الـ limit الثاني 
+
+416
+00:35:27,850 --> 00:35:33,270
+عدد اثنين اثنين بروحين إلى ما لا نهاية فالآن لكن في
+
+417
+00:35:33,270 --> 00:35:37,390
+هذه الحالة نظرا للمعطيات اللي موجودة و اللي هي 
+
+418
+00:35:37,390 --> 00:35:40,930
+طبيعي اللي هو اللي بنحكي فيه NuF is bounded  و ببو
+
+419
+00:35:40,930 --> 00:35:45,550
+و الآخر هو استخدام السابق هيطلع عندي فعلا الـ
+
+420
+00:35:45,550 --> 00:35:50,840
+limit للأل بيساوي limit للأبر بيساوي قيمة الـ
+
+421
+00:35:50,840 --> 00:36:08,200
+integration ماشي اطلعوا يا جماعة عندي الآن خليني
+
+422
+00:36:08,200 --> 00:36:15,240
+أطلع .. نحط البرهان أمامنا نبدأ الآن خلصت اللي هو F
+
+423
+00:36:15,240 --> 00:36:21,500
+is integrable بدي أستخدم زي الـ corollary بشوية بالظبط اللي
+
+424
+00:36:21,500 --> 00:36:28,570
+هو تعريف الـ L of F و U of F و إيش تعريف هذا ما
+
+425
+00:36:28,570 --> 00:36:32,250
+بديش أعيد اللي حكيته قبل بشوية الآن بما أن الـ L of 
+
+426
+00:36:32,250 --> 00:36:42,370
+F عبارة عن اللي هي supremum للـ L B of F إذا لكل
+
+427
+00:36:42,370 --> 00:36:46,610
+ي ساوي واحدة لأن there exists B N partition of I
+
+428
+00:36:46,610 --> 00:36:50,390
+such that L of F ناقص واحدة لأن أصغر من مين من L B
+
+429
+00:36:50,390 --> 00:36:54,070
+N of F زي ما قلت قبل شوية، لو الـ supremum طرحنا
+
+430
+00:36:54,070 --> 00:36:57,270
+اللي هي الـ least upper bound، طرحنا منه أي عدد، 1
+
+431
+00:36:57,270 --> 00:37:01,330
+على N، إذا هلاجي، هيبطل إيش ماله upper bound، إيش
+
+432
+00:37:01,330 --> 00:37:04,190
+معناه يبطل upper bound؟ هيلاجي واحد من المجموعة
+
+433
+00:37:04,190 --> 00:37:08,230
+أكبر منه، وهذا فعلا اللي لاجينا BN، بحيث أن الـ BN
+
+434
+00:37:08,230 --> 00:37:13,910
+و F أكبر من الـ F ناقص واحدة لأن الآن من هذا .. من
+
+435
+00:37:13,910 --> 00:37:16,670
+.. من .. من ال .. ال .. ال .. ناخد هذا على الطرف
+
+436
+00:37:16,670 --> 00:37:19,970
+الثاني على الطرف هنا و ناخد هذا على الطرف هذا بصير
+
+437
+00:37:19,970 --> 00:37:23,890
+عندي L of F ناقص Lb of F أصغر من واحدة الآن و أنا
+
+438
+00:37:23,890 --> 00:37:28,670
+بعرف أن هذا دايما أكبر يساوي هذا لأن هذا الـ
+
+439
+00:37:28,670 --> 00:37:32,410
+supremum منهم إذا أنا هيكون أكبر يساوي صفر الآن
+
+440
+00:37:32,410 --> 00:37:38,440
+خذوا الـ limit للجهتين as n goes to infinity بيصير
+
+441
+00:37:38,440 --> 00:37:43,160
+عندي هذا اللي هي limit
+
+442
+00:37:43,160 --> 00:37:48,060
+لـ L بيقل نوف أف حيث الـ لـ L أوف أف لإنه حيثير الـ
+
+443
+00:37:48,060 --> 00:37:53,720
+limit هذا أيش بيساوي بيساوي صفر واضحالان يا جماعة
+
+444
+00:37:53,720 --> 00:37:58,400
+عندي الـ L of F ناقص الـ P L of F أصغر من واحدة لأنه
+
+445
+00:37:58,400 --> 00:38:03,660
+أكبر يساوي صفر لأن معله الآن هذا لكل إبسلون اللي هو
+
+446
+00:38:03,660 --> 00:38:07,300
+واحدة لأن لجينا partition يعني للإبسلون بيساوي
+
+447
+00:38:07,300 --> 00:38:09,360
+واحد لجينا بيه واحد للإبسلون بيساوي اثنين بيه
+
+448
+00:38:09,360 --> 00:38:12,160
+اثنين للإبسلون تلاتة بيه تلاتة إذا صار عندي
+
+449
+00:38:12,160 --> 00:38:16,580
+sequence of اللي هي إيش partitions الآن طلع عندي
+
+450
+00:38:16,580 --> 00:38:20,460
+دائما دائما دائما L of F ناقص الـ PN of F أصغر من
+
+451
+00:38:20,460 --> 00:38:25,440
+واحد على N لكل N الآن as N goes to infinity هذا الـ
+
+452
+00:38:25,440 --> 00:38:30,620
+limit هيصير صفر وهذا صفر إذا هيصير limit هذا as N
+
+453
+00:38:30,620 --> 00:38:33,200
+goes to infinity بيساوي صفر لكن الـ L of F أصلا
+
+454
+00:38:33,200 --> 00:38:37,860
+independent of N إذا هيصير limit PN of F as N goes
+
+455
+00:38:37,860 --> 00:38:43,720
+to infinity بيساوي L of F لأن المواضحات لوش يا شباب
+
+456
+00:38:43,720 --> 00:38:52,360
+limit L of F ناقص L B N و F هيساوي 0 as N goes to
+
+457
+00:38:52,360 --> 00:38:56,920
+infinity و هذا المقدار عبارة عن مقدار ثابت
+
+458
+00:38:56,920 --> 00:39:04,740
+independent of N إذا هيصير عندي limit L B N و F
+
+459
+00:39:04,740 --> 00:39:10,540
+بتساوي limit L of F ناقص
+
+460
+00:39:17,660 --> 00:39:27,170
+L of P L و F زائد L of F مظبوط؟ طيب هذا الآن
+
+461
+00:39:27,170 --> 00:39:33,590
+المقدار معروف أنه بيساوي 0 و هذا ثابت إذا بقدر أوزع
+
+462
+00:39:33,590 --> 00:39:37,210
+الـ limit على الجهتيين و أنا مرتاح إذا بيساوي limit
+
+463
+00:39:37,210 --> 00:39:42,950
+الأول اللي هو 0 زائد limit الثاني نفسه لأنه ثابت
+
+464
+00:39:42,950 --> 00:39:46,450
+إذا صار عندي limit L P N of F as N goes to
+
+465
+00:39:46,450 --> 00:39:54,710
+infinity بيساوي L of F كما هو حكينا عنه حقامن جهة
+
+466
+00:39:54,710 --> 00:39:58,650
+أخرى limit الـ U ناقص limit الـ L المعطيلة هي بيساوي
+
+467
+00:39:58,650 --> 00:40:03,490
+0 إذا صار عندي سهل أن أوجد mean برضه limit الـ U
+
+468
+00:40:03,490 --> 00:40:08,990
+اللي هو limit الـ U إيش هيساوي ناقص اللي هو .. الـ
+
+469
+00:40:08,990 --> 00:40:12,670
+.. ال .. هيساوي limit الـ L of F يساوي الـ U of F
+
+470
+00:40:12,670 --> 00:40:17,270
+يساوي الـ integration أكثر توضيحا and بيهاجدى
+
+471
+00:40:17,270 --> 00:40:22,420
+أعتقد أنه واضح لكن خلّينا نوضحه بشكل أكبر عشان
+
+472
+00:40:22,420 --> 00:40:34,100
+ما يضلش مشي عندي limit U P N of F إيش هيساوي هيساوي
+
+473
+00:40:34,100 --> 00:40:44,560
+اللي هو limit U
+
+474
+00:40:44,560 --> 00:40:49,740
+P N of F
+
+475
+00:40:52,490 --> 00:40:58,090
+limit U P N و F ناقص
+
+476
+00:40:58,090 --> 00:41:07,810
+L of P N of F زائد L P N of F واضحة يا شباب أه؟
+
+477
+00:41:07,810 --> 00:41:12,690
+الآن هذا مضمون أنه موجود و صفر وهذا مضمون و
+
+478
+00:41:12,690 --> 00:41:18,350
+أثبتناه إيش بيساوي L of F إذا إيش صار بيساوي؟
+
+479
+00:41:18,350 --> 00:41:21,810
+بيساوي اللي هو هذا صفر إذا صار بيساوي L of F
+
+480
+00:41:21,810 --> 00:41:25,510
+إذا صار هذا برضه بيساوي L of F لكن أنا مثبت قبل
+
+481
+00:41:25,510 --> 00:41:29,610
+بشوية أن الـ F is integrable يعني الـ U في F إيش
+
+482
+00:41:29,610 --> 00:41:34,550
+هتساوي؟ اللي هو L of F وهذه أثبتناها إيش بتساوي؟
+
+483
+00:41:34,550 --> 00:41:39,350
+limit U P N و F وهذه نفسها أثبتناها قبل بشوية إيش
+
+484
+00:41:39,350 --> 00:41:46,070
+بتساوي؟ limit L P N و F وهو المطلوب طبعا مدام
+
+485
+00:41:46,070 --> 00:41:49,370
+integration هذه وهذه هي عبارة عن الـ integration
+
+486
+00:41:49,370 --> 00:41:55,410
+على الفترة اللي بنحكي عنها الانقلا بيه لقيمة الـ F
+
+487
+00:41:55,410 --> 00:42:01,710
+لأن صار عندي كل القيام هذه متساوية وصار عندي إيجاد
+
+488
+00:42:01,710 --> 00:42:07,610
+الـ limit للـ U P N و F أو limit للـ L P N و F يكفي
+
+489
+00:42:07,610 --> 00:42:11,330
+أنه نوجد فيه قيمة الـ integration بعد ما أثبتناه أو
+
+490
+00:42:11,330 --> 00:42:16,260
+تحت الظروف اللي هي في الـ corollary الأخرى لنأتي بدنا
+
+491
+00:42:16,260 --> 00:42:22,660
+نبرهن اللي هو المثال اللي برهناه المرة الماضية
+
+492
+00:42:22,660 --> 00:42:27,660
+بالتعريف بدنا نبرهنه بواسطة اللي هي الـ corollary
+
+493
+00:42:27,660 --> 00:42:33,060
+اللي عندنا بدنا نبرهن اللي هو اللي هي نثبت أنه
+
+494
+00:42:36,170 --> 00:42:42,070
+نثبت أن F of X بساوة X يا شباب عبارة عن Integrable
+
+495
+00:42:42,070 --> 00:42:45,770
+أو اللي سمناها G of X بساوة X is Integrable على
+
+496
+00:42:45,770 --> 00:42:48,830
+الفترة Zero واحد المرة الماضية أثبتناها كيف
+
+497
+00:42:48,830 --> 00:42:54,230
+أثبتناها زي ما أنتم متذكرين بواسط التعريف مظبوط؟
+
+498
+00:42:54,230 --> 00:42:58,610
+طيب جيبنا اللي هو الـ upper sum و الـ lower sum و
+
+499
+00:42:58,610 --> 00:43:02,930
+بعدين جيبنا الـ upper integral و الـ lower integral
+
+500
+00:43:02,930 --> 00:43:04,750
+و أثبتنا ان الـ upper integral بساوي الـ lower
+
+501
+00:43:04,750 --> 00:43:08,760
+integral و خلصناه الآن بدنا نثبتها بطريقتنا اللي هي
+
+502
+00:43:08,760 --> 00:43:12,540
+على الـ corollary اللي جابل بشوية عند g of x بساوة x على
+
+503
+00:43:12,540 --> 00:43:16,600
+الفترة 0 بواحد هي المطلوب اثباتها show that g is
+
+504
+00:43:16,600 --> 00:43:20,940
+integrable على هذه الفترة الآن بديش أعيد اللي
+
+505
+00:43:20,940 --> 00:43:23,850
+حكيته المرة الماضية اللي حكيناه المرة الماضية من
+
+506
+00:43:23,850 --> 00:43:26,910
+الـ example اللي أثبتنا فيه انها Integra بالواسطة
+
+507
+00:43:26,910 --> 00:43:31,470
+اللي هي التعريف أخدنا P N اللي هو صفر واحد على N
+
+508
+00:43:31,470 --> 00:43:34,910
+واثنين على N وان ناقص واحد على N لعند الواحد أخدنا
+
+509
+00:43:34,910 --> 00:43:39,290
+اللي هو عبارة عن Any Partition اللي هو بالطريقة
+
+510
+00:43:39,290 --> 00:43:43,710
+اللي أمامي يعني حسب N بيصير بختلف الـ Partition أما
+
+511
+00:43:43,710 --> 00:43:46,730
+إيش فكرة الـ Partition زي ما قلنا المرة الفائتة من
+
+512
+00:43:46,730 --> 00:43:51,350
+0 لعند 1 جزأناها إلى أجزاء متساوية إلى N من
+
+513
+00:43:51,350 --> 00:43:55,350
+الأجزاء المتساوية صار طول كل sub integral عبارة عن
+
+514
+00:43:55,350 --> 00:44:00,850
+إيش عبارة عن واحد على N وأوجدنا في حينه اللي هو
+
+515
+00:44:00,850 --> 00:44:08,470
+الـ U P N O G ولاجناها بتساوي إيش ده بتتذكروا نص في
+
+516
+00:44:08,470 --> 00:44:13,710
+واحد زائد واحد على N وأوجدنا برضه الـ P N O G
+
+517
+00:44:13,710 --> 00:44:18,910
+ولاجناها عبارة عن نص في واحد ناقص واحد على N كما
+
+518
+00:44:18,910 --> 00:44:26,960
+أذكر ماشي الحال فعلا طيب الآن صار عندي الـ PN في
+
+519
+00:44:26,960 --> 00:44:29,780
+الواقع عبارة عن sequence of partitions
+
+520
+00:44:35,190 --> 00:44:39,110
+هذه صارت sequence of partitions بواحد بتعود عن أنا
+
+521
+00:44:39,110 --> 00:44:42,750
+بواحد باتنين كده بتلاتة كده بأربعة كده وفي كل
+
+522
+00:44:42,750 --> 00:44:47,270
+الأحوال الـ U, B, N و G انجر لهيها و الـ L, B, N
+
+523
+00:44:47,270 --> 00:44:51,490
+و G بساوي إيش المقدار الأمامي الآن عشان أثبت إنها
+
+524
+00:44:51,490 --> 00:44:56,590
+integrable يكفي من النظرية الـ corollary إن قاعد أقول
+
+525
+00:44:56,590 --> 00:45:03,290
+طب خلينا نشوف limit الـ U, B, N و G ناقص الـ L, B, 
+
+526
+00:45:03,450 --> 00:45:09,840
+N و G as n goes to infinity بيساوي إيه يا عشان بيساوي
+
+527
+00:45:09,840 --> 00:45:16,360
+صفر نشوفها صح ولا لأ طبعا أكيد مع حسبة بسيطة
+
+528
+00:45:16,360 --> 00:45:24,580
+احسبليها بيساوي limit نص في واحد زائد واحدة على ن زائد
+
+529
+00:45:24,580 --> 00:45:28,600
+نص في واحد ناقص واحدة على N as n goes to infinity
+
+530
+00:45:28,600 --> 00:45:32,260
+هذه صفر وهذه صفر وهذه نص وهذه نص وهذه نص وهذه نص
+
+531
+00:45:32,260 --> 00:45:32,340
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+532
+00:45:32,340 --> 00:45:32,920
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+533
+00:45:32,920 --> 00:45:38,340
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+534
+00:45:38,340 --> 00:45:45,580
+وهذه اللي هو y بساوي نص ناقص النص y بساوي صفر واضح أهي
+
+535
+00:45:45,580 --> 00:45:50,120
+عندي اللي هو هاد صفر و هاد صفر و نص ناقص النص y
+
+536
+00:45:50,120 --> 00:45:54,920
+بساوي صفر مادام صفر إذا إيش أشمالها إذا the function
+
+537
+00:45:54,920 --> 00:46:02,720
+d of x بساوة x is integrable on zero واحد هذا by
+
+538
+00:46:02,720 --> 00:46:06,560
+mean by الـ corollary اللي قبل بشوية
+
+539
+00:46:15,600 --> 00:46:19,420
+الآن بقيمة الـ integration إيش هيكون قيمة الـ limit
+
+540
+00:46:19,420 --> 00:46:24,280
+limit u, b, n, g هيقدر الغرض و limit l of n of g
+
+541
+00:46:24,280 --> 00:46:29,320
+هيقدر الغرض ليش لأن limit هذا اصلا هيطلع لك limit
+
+542
+00:46:29,320 --> 00:46:35,710
+الـ u, b, n و g هو عبارة عن قيمة integration من A لـ
+
+543
+00:46:35,710 --> 00:46:42,370
+B حسب النظرية الـ Corollary و بيساوي limit 2.5 في 1
+
+544
+00:46:42,370 --> 00:46:48,600
+زائد 1 ل N و بيساوي هذا تروح ل 0 و بيساوي نص و أيضا
+
+545
+00:46:48,600 --> 00:46:54,080
+لو جربت حسبت limit ال b, n و g طبعا هيطلع نفس
+
+546
+00:46:54,080 --> 00:46:58,200
+الجواب و إلا إن كان هناك مشكلة لدينا limit نص في
+
+547
+00:46:58,200 --> 00:47:02,340
+واحد ناقص واحد على n و بيساوي برضه جدياش نص إذن
+
+548
+00:47:02,340 --> 00:47:06,700
+قيمة الـ integration بساوة نص إذن هذه طريقة أخرى
+
+549
+00:47:06,700 --> 00:47:11,780
+لحساب اللي هو أو لإثبات أن g of x بساوة x is
+
+550
+00:47:11,780 --> 00:47:17,560
+integrable الآن بدنا ندخل على أمر آخر الأمر هو في
+
+551
+00:47:17,560 --> 00:47:24,500
+الواقع يا شباب أنه بدنا نشوف إيش في من عائلات
+
+552
+00:47:24,500 --> 00:47:28,400
+الدوال عائلات الدوال إنها تكون Integrable الآن
+
+553
+00:47:28,400 --> 00:47:31,720
+بدنا نيجي نحوش الدوال الـ Integrable إحنا عرفنا بس
+
+554
+00:47:31,720 --> 00:47:36,500
+ده لأ D of X سو X is Integrable و هنلاقي زيها لكن
+
+555
+00:47:36,500 --> 00:47:40,960
+الآن بدنا نيجي نحكي عن دوال اللي هو عائلات من
+
+556
+00:47:40,960 --> 00:47:45,480
+الدوال الأول عائلة من العائلات المهمة اللي هي الـ
+
+557
+00:47:45,480 --> 00:47:48,960
+monotone functions يعني الدوال اللي بتكون يا
+
+558
+00:47:48,960 --> 00:47:52,360
+increasing على كل الفترة يا decreasing على كل
+
+559
+00:47:52,360 --> 00:47:59,320
+الفترة بقولكم هذا على اللي هي F من F الـ function
+
+560
+00:47:59,320 --> 00:48:04,000
+لو كانت bounded لو كانت monotone على اللي هي
+
+561
+00:48:04,000 --> 00:48:09,660
+closed bounded interval I على طول integrable إذن
+
+562
+00:48:09,660 --> 00:48:16,700
+عيلة كبيرة عيلة الدوال اللي بتكون يا increasing يا
+
+563
+00:48:16,700 --> 00:48:22,380
+decreasing على كل الفترة a و b هذه مضمون انها تكون
+
+564
+00:48:22,380 --> 00:48:26,980
+الدوال إيش أشمالها عبارة عن integrable functions
+
+565
+00:48:26,980 --> 00:48:35,090
+إذا أول إعلان الآن اللي هو any monotone function on
+
+566
+00:48:35,090 --> 00:48:40,850
+a closed bounded interval is integrable وهذا اللي
+
+567
+00:48:40,850 --> 00:48:47,090
+هو عنواننا integrability of monotone functions لـ I
+
+568
+00:48:47,090 --> 00:48:51,650
+بتساوي A و B و لـ F من I لـ R بيكون مونوتون فانكشن
+
+569
+00:48:51,650 --> 00:48:57,370
+on I ثم F أشمالها is integrable on I نفترض أن F
+
+570
+00:48:57,370 --> 00
+
+601
+00:51:39,910 --> 00:51:45,510
+increasing إذن هيكون الـ Mk هو F of XK رسم عنها
+
+602
+00:51:45,990 --> 00:51:51,210
+بساوة F of XK طيب الـ MK Small أكيد هيقولنا كلكم
+
+603
+00:51:51,210 --> 00:51:54,010
+هتقوله بتساوة الـ M في معمع الفترة مدام الـ M في
+
+604
+00:51:54,010 --> 00:51:57,370
+معمع الفترة إذا أول واحدة فيهم إذا هو F of XK
+
+605
+00:51:57,370 --> 00:52:04,530
+ناقص واحد، مظبوط شباب؟ طيب، مظبوط هذه اللي هي
+
+606
+00:52:04,530 --> 00:52:10,750
+مالها الدالة اللي بتكون monotone عندي،
+
+607
+00:52:10,750 --> 00:52:21,840
+احسبلي الآن الـ U P N و F ناقص الـ L P N و F إيش هيساوي؟
+
+608
+00:52:21,840 --> 00:52:26,840
+جهزة الأمور بساوي الـ summation لمين؟ للـ MK
+
+609
+00:52:26,840 --> 00:52:36,190
+Capital في XK ناقص XK ناقص واحدك من عند 1 لعند n
+
+610
+00:52:36,190 --> 00:52:44,370
+ناقص summation mk في xk ناقص xk ناقص 1 k من عند 1
+
+611
+00:52:44,370 --> 00:52:49,440
+لعند n ويساوي الـ Mk وجدناها، اللي هي عبارة عن مين؟
+
+612
+00:52:49,440 --> 00:52:55,420
+F of Xk ويساوي الـ summation للـ F of Xk في مين
+
+613
+00:52:55,420 --> 00:53:00,200
+مضروبة؟ في هذه، هذه كده طولها ثابت، ما احنا هيك
+
+614
+00:53:00,200 --> 00:53:02,540
+على هذا الأساس إذا اخترنا الـ sequence of
+
+615
+00:53:02,540 --> 00:53:05,320
+partitions اللي عندنا، اللي هو طوالث في
+
+616
+00:53:05,320 --> 00:53:09,580
+subintervals ثابتة، كل واحد اسمه يشمله B ناقص A
+
+617
+00:53:09,580 --> 00:53:14,250
+على N، هذا K من واحد إن عندنا ناقص خلّيني أضعه في
+
+618
+00:53:14,250 --> 00:53:18,550
+summation 1 ناقص نفس القصة اللي هي mk Small إيش
+
+619
+00:53:18,550 --> 00:53:23,710
+هي يا جماعة اتفاقنا عبارة عن f of xk ناقص 1 في
+
+620
+00:53:23,710 --> 00:53:27,610
+هذه خلّيني أخدها عامل مشترك بعد إذنكم لإن هو
+
+621
+00:53:27,610 --> 00:53:31,070
+موجودة هنا وموجودة هنا خلّيني أطلّحها برا واضحة
+
+622
+00:53:31,070 --> 00:53:36,070
+أشيل هذه برا بيصير اللي هو مضروبة في b ناقص a على
+
+623
+00:53:36,910 --> 00:53:42,350
+وأصلا هذه ثابتة بالنسبة للـ summation ليش؟ لأن الـ
+
+624
+00:53:42,350 --> 00:53:46,110
+summation العداد K من واحد عندنا أن هذه N ثابتة
+
+625
+00:53:46,110 --> 00:53:52,870
+بالنسبة للـ K لذلك بتساوي B ناقص A على N مضروبة في
+
+626
+00:53:52,870 --> 00:53:59,280
+مين في الـ summation للـ f of xk ناقص f of xk
+
+627
+00:53:59,280 --> 00:54:03,840
+ناقص واحد k من عند واحد لعند مين يا جماعة لعند n
+
+628
+00:54:03,840 --> 00:54:07,920
+بعدين this is and could بدي أفرطها هذه بيصير عندي y
+
+629
+00:54:07,920 --> 00:54:14,880
+ساوي هذا اللي هو مين كله اللي جاعت بحسبه الـ u,b,n
+
+630
+00:54:14,880 --> 00:54:19,320
+و F ناقص الـ L,b,n و F هيسوي هذا المقدار اللي هو b
+
+631
+00:54:19,320 --> 00:54:25,970
+ناقص a على المضروب فيه الآن الـ summation عبارة عن
+
+632
+00:54:25,970 --> 00:54:32,630
+k ب 1 بصير f of x1 ناقص f of x0 اللي بعدها
+
+633
+00:54:32,630 --> 00:54:40,170
+k ب 2 زائد f of x2 ناقص f of x1 اللي بعدها زائد
+
+634
+00:54:40,170 --> 00:54:46,130
+f of x3 ناقص f of x2 لما نقضى اللي ماشي لأخر
+
+635
+00:54:46,130 --> 00:54:53,140
+واحد بكون عندي f of xn ناقص f of xn ناقص واحد
+
+636
+00:54:57,360 --> 00:55:04,840
+F of X1 بيطير مع سالب F of X1 و F of X2 بيطير مع
+
+637
+00:55:04,840 --> 00:55:09,840
+سالب F of X2 و هكذا بظل ماشي لما كله يروح مع كله
+
+638
+00:55:09,840 --> 00:55:16,640
+ما عدا بظل عندي اللي هو أول قيمة اللي هي F of X
+
+639
+00:55:16,640 --> 00:55:22,660
+note بالسالب مع F of Xn الأخيرة بالموجب بيصير عندي
+
+640
+00:55:22,660 --> 00:55:34,740
+y ساوي B ناقص A على N في F of Xn ناقص F of X0 طبعا
+
+641
+00:55:34,740 --> 00:55:39,420
+Xn آخر واحدة اللي هي B و X0 أول واحدة اللي هي A
+
+642
+00:55:39,420 --> 00:55:46,580
+إذن هيساوي هذا عبارة عن B ناقص A في F of B ناقص
+
+643
+00:55:46,580 --> 00:55:53,150
+F of A الكل هذا مجسوم على مين يا جماعة؟ على N هذا
+
+644
+00:55:53,150 --> 00:55:58,090
+عبارة عن ثابت وهذا عبارة عن ثابت وهذا الان هي اللي
+
+645
+00:55:58,090 --> 00:56:03,670
+بدلالتها كُتِب الـ partitions إذا صار عندي الآن بعد
+
+646
+00:56:03,670 --> 00:56:07,230
+كل اللي حكيته هنا أخدت الـ Pn بالشكل اللي أمامي
+
+647
+00:56:07,230 --> 00:56:12,190
+عبارة عن sequence of partitions ووصلنا إلى ما
+
+648
+00:56:12,190 --> 00:56:17,890
+لاقيا ليه يا جماعة اللي هو وصلنا إن الـ U Pn و
+
+649
+00:56:17,890 --> 00:56:26,260
+F ناقص الـ L Pn و F أصغر أو يساوي طبعا أكيد أكبر
+
+650
+00:56:26,260 --> 00:56:29,600
+يساوي صفر لأن هذا دايما أكبر يساوي هذا أصغر يساوي
+
+651
+00:56:29,600 --> 00:56:36,140
+B ناقص A في F of B أكيد عرفت أشهد يساوي ناقص F of A
+
+652
+00:56:36,140 --> 00:56:41,720
+على اللي هو N الـ N خذ الـ limit للجهتين as N goes
+
+653
+00:56:41,720 --> 00:56:45,160
+to infinity as N goes to infinity هذا goes to zero
+
+654
+00:56:45,160 --> 00:56:51,730
+وهذا أصلا صفر فبيصير عندي limit إذا .. إذا limit الـ
+
+655
+00:56:51,730 --> 00:56:59,290
+U Pn و F ناقص الـ L Pn و F as N goes to infinity
+
+656
+00:56:59,290 --> 00:57:04,210
+بساوي صفر إذا صار عندي sequence of partitions تحقق
+
+657
+00:57:04,210 --> 00:57:07,730
+لهذا إذا حسب الـ Corollary اللي حكيتها قبل بشوية
+
+658
+00:57:07,730 --> 00:57:17,130
+إذا F is integrable وهو المطلوب إذا صار عندي أي
+
+659
+00:57:17,130 --> 00:57:21,850
+increasing function is integrable similarly for
+
+660
+00:57:21,850 --> 00:57:26,170
+decreasing لماذا similarly لأنه تصبح الدالة بدل ما
+
+661
+00:57:26,170 --> 00:57:30,950
+هي طالع هيك تصبح أشمالها نازلة نزول الدالة فبيصير
+
+662
+00:57:30,950 --> 00:57:35,430
+عندك اللي هو الـ maximum هي الأولى أو الـ supremum
+
+663
+00:57:35,430 --> 00:57:41,920
+هي F بيصير الـ Mk بسوء F of Xk ناقص واحد والـ MK
+
+664
+00:57:41,920 --> 00:57:46,760
+بسوء F of Xk وبيتكملوا البرهان بنفس الطريقة، هيطلع
+
+665
+00:57:46,760 --> 00:57:52,040
+عندكم البرهان automatic وبشكل سهل وبشكل سلس،
+
+666
+00:57:52,040 --> 00:57:58,500
+similar أي سؤال؟ إذن الإعلان اللي أعلنناه قبل
+
+667
+00:57:58,500 --> 00:58:04,800
+بشوية أنه any monotone function يعني any increasing
+
+668
+00:58:04,800 --> 00:58:08,780
+function on a closed bounded interval must be
+
+669
+00:58:08,780 --> 00:58:15,740
+integrable and any decreasing function on a closed
+
+670
+00:58:15,740 --> 00:58:20,420
+bounded interval برضه must be integrable إذا صار
+
+671
+00:58:20,420 --> 00:58:26,060
+في عنا عائلة كاملة من الدوال القابلة للتكامل بواسط
+
+672
+00:58:26,060 --> 00:58:32,160
+التكامل Riemann الآن بدنا ننتقل إلى عائلة أخرى
+
+673
+00:58:32,160 --> 00:58:38,300
+وعائلة لا تقل أهمية عن هذه العائلة وعائلة يعني
+
+674
+00:58:38,300 --> 00:58:45,560
+محبوبة عنها اللي هي اللي
+
+675
+00:58:45,560 --> 00:58:53,960
+هي عائلة الدوال المتصلة اللي هي integrable of
+
+676
+00:58:53,960 --> 00:58:56,140
+continuous functions
+
+677
+00:59:00,850 --> 00:59:06,130
+اللي هي integrability of continuous functions
+
+678
+00:59:06,130 --> 00:59:12,210
+النظرية بتقول ما يلي لت أ .. طبعا عندي الـ function
+
+679
+00:59:12,210 --> 00:59:16,410
+على closed bounded interval لت F من I لـ R be
+
+680
+00:59:16,410 --> 00:59:20,750
+continuous on I then F is integrable on I إذا الآن
+
+681
+00:59:20,750 --> 00:59:24,230
+any continuous function on a closed bounded
+
+682
+00:59:24,230 --> 00:59:29,600
+interval must be integrable كمان مرة any continuous
+
+683
+00:59:29,600 --> 00:59:33,840
+function on a closed bounded interval must be
+
+684
+00:59:33,840 --> 00:59:38,920
+integrable طبعا حيزنا في البرهان شغلة اللي هي
+
+685
+00:59:38,920 --> 00:59:45,080
+أخدناها سابقا إنّه any continuous function on a
+
+686
+00:59:45,080 --> 00:59:49,840
+closed bounded interval must attain its maximum
+
+687
+00:59:49,840 --> 00:59:57,800
+and minimum on this interval بمعنى
+
+688
+00:59:57,800 --> 01:00:01,970
+آخر هنلاقيلو كانت F is continuous على اللي هو الـ
+
+689
+01:00:01,970 --> 01:00:05,990
+A و الـ B هنلاقي نقطة في داخل الفترة A و B بحيث
+
+690
+01:00:05,990 --> 01:00:09,570
+أنها تكون الـ F عندها نقطة maximum و هنلاقي نقطة
+
+691
+01:00:09,570 --> 01:00:12,750
+أخرى في داخل هذه الفترة هنلاقي اللي هي الـ
+
+692
+01:00:12,750 --> 01:00:15,930
+function عندها إشمالها is minimum طبعا absolute و
+
+693
+01:00:15,930 --> 01:00:24,180
+absolute نيجي نشوف .. نيجي للبرهان الآن .. نظرية
+
+694
+01:00:24,180 --> 01:00:28,540
+أخرى أيضا .. برضه خلينا نقولها أن لو كانت F is
+
+695
+01:00:28,540 --> 01:00:32,580
+continuous على closed bounded interval then F is
+
+696
+01:00:32,580 --> 01:00:43,820
+uniformly continuous الآن F .. عند F من A و B لعند
+
+697
+01:00:43,820 --> 01:00:51,090
+R is continuous on A و B continuous عالمين على
+
+698
+01:00:51,090 --> 01:00:54,450
+closed bounded interval في عندنا نظرية اللي بتقول
+
+699
+01:00:54,450 --> 01:00:58,270
+any continuous function في الواحد on a closed
+
+700
+01:00:58,270 --> 01:01:02,230
+bounded interval must be uniformly continuous إذا
+
+701
+01:01:02,230 --> 01:01:10,030
+then F is uniformly continuous
+
+702
+01:01:10,730 --> 01:01:15,290
+on A و B إيش يعني uniformly continuous يعني لكل
+
+703
+01:01:15,290 --> 01:01:18,810
+إبسيلون أكبر من صفر there exist دلتا أكبر من صفر
+
+704
+01:01:18,810 --> 01:01:22,990
+دلتا بس بتعتمد عالميا على إبسيلون بتنفع لكل الـ X و
+
+705
+01:01:22,990 --> 01:01:30,810
+الـ Y such that if X ناقص Y أو U ناقص V زي ما هو
+
+706
+01:01:30,810 --> 01:01:40,200
+مسميها U ناقص V أصغر من دلتا then F of U ناقص F of
+
+707
+01:01:40,200 --> 01:01:49,800
+V أصغر من إبسيلون لكل U V element in A أو B إذن لكل y
+
+708
+01:01:49,800 --> 01:01:52,900
+أكبر من 0 there exists دلتا أشمة لا بتعتمد
+
+709
+01:01:52,900 --> 01:01:56,280
+على مين بس على إبسيلون في حالة الـ continuity
+
+710
+01:01:56,280 --> 01:02:01,560
+العادية بنقول احنا limit F of x as x بتروح للـ a
+
+711
+01:02:01,560 --> 01:02:07,260
+بساوي إيش F of a بنقول الآن إنّه limit F of x بساوي
+
+712
+01:02:07,260 --> 01:02:13,450
+F of a as x بتروح للـ a لكل إبسيلون أكبر من 0 هصب مين
+
+713
+01:02:13,450 --> 01:02:18,070
+بالـ a هذه there exist دلتا و لكل a element in A
+
+714
+01:02:18,070 --> 01:02:23,270
+there exist Delta such that اللي لما يكون x ناقص
+
+715
+01:02:23,270 --> 01:02:27,190
+a أصغر من دلتا بيعطينا F of x ناقص F of a أصغر من
+
+716
+01:02:27,190 --> 01:02:32,550
+إبسيلون يعني بيكون الـ دلتا هنا تعتمد على الإبسيلون
+
+717
+01:02:32,550 --> 01:02:37,550
+وتعتمد على الـ a اللي عندها الـ continuity لكن في الـ
+
+718
+01:02:37,550 --> 01:02:41,890
+Uniform Continuous الـ Delta اللي اللي جيت هنا
+
+719
+01:02:41,890 --> 01:02:47,510
+بتنفع لكل اللي هو الـ .. الـ .. الـ .. الـ .. الـ .. الـ
+
+720
+01:02:47,510 --> 01:02:47,890
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ
+
+721
+01:02:47,890 --> 01:02:47,910
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ
+
+722
+01:02:47,910 --> 01:02:48,290
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ
+
+723
+01:02:48,290 --> 01:02:48,530
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ
+
+724
+01:02:48,530 --> 01:02:50,270
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ
+
+725
+01:02:50,270 --> 01:02:50,330
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ..
+
+726
+01:02:50,330 --> 01:02:50,410
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ..
+
+727
+01:02:50,410 --> 01:02:54,250
+.. الـ .. الـ .. الـ .. الـ .. الـ ..
+
+728
+01:02:54,250 --> 01:02:59,970
+الـ .. الـ F is continuous، إذن ينفرم الـ
+
+729
+01:02:59,970 --> 01:03:02,650
+continuous، إيش معناه؟ كل إبسيلون أكبر من صفر،
+
+730
+01:03:02,650 --> 01:03:05,490
+there exists دلتا، بحيث إن U ناقص V أصغر من دلتا
+
+731
+01:03:05,490 --> 01:03:08,690
+يعطيني F of U ناقص F of V أصغر من إياش، من إبسيلون،
+
+732
+01:03:08,690 --> 01:03:11,870
+وخلي هذه إيش مالها في الذاكرة، أنا عشان الحسابات
+
+733
+01:03:11,870 --> 01:03:15,890
+بدي أخليها إبسيلون على مين، على B ناقص A، طول
+
+734
+01:03:15,890 --> 01:03:19,070
+الفترة اللي أنا بشتغل عليها، هتشوفوا ليش كتبت هيك،
+
+735
+01:03:19,070 --> 01:03:22,720
+بس لا الحسابات بقدر أه بقدر لإن هو أصغر .. أه بقدر
+
+736
+01:03:22,720 --> 01:03:25,140
+خليه أصغر من الـ إبسيلون في الدنيا من ضمن إن كل
+
+737
+01:03:25,140 --> 01:03:27,520
+الـ إبسيلون عن الـ B ناقص A لكل الـ إبسيلونات
+
+738
+01:03:27,520 --> 01:03:34,040
+اللي في الدنيا طيب لأن شوف وين بده أروح، بده أروح
+
+739
+01:03:34,040 --> 01:03:39,250
+لـ Integrability لـ الـ function F الآن خذ n الآن
+
+740
+01:03:39,250 --> 01:03:42,190
+بدنا نعمل partitions بدنا نجيب sequence of
+
+741
+01:03:42,190 --> 01:03:45,890
+partitions Bn الـ sequence of partitions هذه هي
+
+742
+01:03:45,890 --> 01:03:50,010
+اللي هتخدمني هتخدمني متى؟ بعد شوية بتشوفوا ليش
+
+743
+01:03:50,010 --> 01:03:54,110
+هتخدمني لكل الأنات اللي وين ما لإن اللي أكبر من الـ
+
+744
+01:03:54,110 --> 01:03:59,370
+B ناقص A على مين؟ على الـ دلتا اللي لجيتها إذا
+
+745
+01:03:59,370 --> 01:04:04,130
+اللي صارت الـ دلتا بين إيديا إذا بقولك الآن choose
+
+746
+01:04:05,140 --> 01:04:12,320
+N element in N such that مالها N أكبر من B ناقص
+
+747
+01:04:12,320 --> 01:04:17,620
+A على مين على دلتا برضه B ناقص A عشان الحسبان N
+
+748
+01:04:17,620 --> 01:04:21,240
+أكبر من B ناقص A على مين على دلتا يعني الآن أنا
+
+749
+01:04:21,240 --> 01:04:26,920
+بحكي بحكي عن الأنات اللي بكون أكبر من B ناقص A
+
+750
+01:04:26,920 --> 01:04:31,460
+على مين على الـ دلتا اللي لجيتها فوق عندي في اللي هو
+
+751
+01:04:31,460 --> 01:04:37,080
+هذا واحد ماشي الحال الآن بتبدأ أكون partitions من
+
+752
+01:04:37,080 --> 01:04:42,480
+مين؟ من الأمنات هذه الآن بتاخد بيئا اني partitions
+
+753
+01:04:42,480 --> 01:04:47,220
+اني partitions برضه أشماله with equal اللي هي إيش
+
+754
+01:04:47,220 --> 01:04:54,240
+sub intervals يعني X نوت و X واحد لعند Xn يعني
+
+755
+01:04:54,240 --> 01:04:58,540
+طول كل فترة منهم برضه طول .. بتاخد اللي هي كلهم
+
+756
+01:04:58,540 --> 01:05:03,590
+أشمالين من عند A لعند B يكون إن كل الـ sub intervals
+
+757
+01:05:03,590 --> 01:05:08,530
+جت بعض يعني بمعنى آخر هيكون الـ Xk ناقص Xk ناقص
+
+758
+01:05:08,530 --> 01:05:12,970
+واحد بيساوي B ناقص A على N طول كل فترة إيش بيساوي
+
+759
+01:05:12,970 --> 01:05:16,950
+B ناقص A على عدد الفترات اللي هي N فبيكون B ناقص
+
+760
+01:05:16,950 --> 01:05:21,250
+A عالميا على N أنا حرّن إنهم ألاقي في النهاية
+
+761
+01:05:21,250 --> 01:05:25,630
+sequence of partitions Bn limit الـ U Bn و F ناقص
+
+762
+01:05:25,630 --> 01:05:29,290
+الـ L Bn و F بساوي 0 بيكون F is integrable خلصنا
+
+763
+01:05:29,290 --> 01:05:32,950
+أنا قاعد اخترت اللي هي sequence of partitions بناء
+
+764
+01:05:32,950 --> 01:05:36,390
+على الـ دلتا اللي لجيتها في الـ uniformity و
+
+765
+01:05:36,390 --> 01:05:39,830
+الأنات اللي أكبر منها و حطيت الـ partition اللي هو
+
+766
+01:05:39,830 --> 01:05:44,710
+بشكل اللي هو تكون الـ subintervals كلها لها نفس
+
+767
+01:05:44,710 --> 01:05:46,230
+الطول طيب
+
+768
+01:05:48,470 --> 01:05:56,170
+الآن زي ما قلت قبل بشوية negation على الفترة xk و
+
+769
+01:05:56,170 --> 01:06:01,710
+ناقص 1 و xk هذا الـ sub interval الـ function is
+
+770
+01:06:01,710 --> 0
+
+801
+01:08:19,950 --> 01:08:26,220
+وين موجودة في هذه الآن في مين؟ في xk minus xk minus
+
+802
+01:08:26,220 --> 01:08:31,520
+واحد xk minus xk minus واحد كامل عند واحد اللي
+
+803
+01:08:31,520 --> 01:08:37,020
+عندها انهدمين هي الـ U الآن زيها مين الـ L ناقص
+
+804
+01:08:37,020 --> 01:08:42,600
+summation الآن مين عند الـ MK small الـ F of VK الـ
+
+805
+01:08:42,600 --> 01:08:48,920
+VK اللي لاجيناها برضه هنا اللي هي F of VK مضروبة في
+
+806
+01:08:48,920 --> 01:08:54,380
+طول الفترة xk-xk-1 كامل عند واحد لعند مين يا جماعة
+
+807
+01:08:54,380 --> 01:09:00,000
+لعندنا خذولي الآن xk-xk-1 عامل مشترك وطبعا أنا
+
+808
+01:09:00,000 --> 01:09:03,220
+مفترضها اللي هو وماخدها إلا طول الـ intervals أو
+
+809
+01:09:03,220 --> 01:09:07,080
+sub intervals متساوي يعني طول هذه وطول هذه هو
+
+810
+01:09:07,080 --> 01:09:10,600
+عبارة عن b-a على n زي النظرية السابقة ويساوي الـ
+
+811
+01:09:10,600 --> 01:09:17,850
+summation للـ F of uk ناقص F of vk الكل مضروب في مين
+
+812
+01:09:17,850 --> 01:09:26,030
+يا جماعة في B-A على N N أو K من عند واحد لعند إيش
+
+813
+01:09:26,030 --> 01:09:36,830
+لعند N أي سؤال؟ طيب شوفوا الآن اسمحولي بس هنا أشتغل 
+
+814
+01:09:36,830 --> 01:09:39,730
+شوية طيب
+
+815
+01:09:42,370 --> 01:09:47,490
+طولوا روحكم عندنا خلصنا يعني جربنا إذا صار عندي الـ
+
+816
+01:09:47,490 --> 01:09:51,870
+U, P, N و F عشان تعرفوا أين رايح أنا هي الـ U, P, N
+
+817
+01:09:51,870 --> 01:09:55,550
+و F ناقص الـ P, N و F أكبر يساوي الصفر اللي جيته
+
+818
+01:09:55,550 --> 01:10:00,610
+أصغر يساوي المقدار اللي أمامي الآن خلوني أطلع حد
+
+819
+01:10:00,610 --> 01:10:06,400
+برا بعد اسمكم هي B- a على n لأنها عبارة عن ثابت في
+
+820
+01:10:06,400 --> 01:10:13,020
+الـ summation f of u,k ناقص f of v,k كامنة عند واحد
+
+821
+01:10:13,020 --> 01:10:20,340
+لأنك لاحظوا ما يلي يا جماعة عند الـ u,k و الـ v,k
+
+822
+01:10:20,340 --> 01:10:27,480
+وين موجودة في الـ x,k minus واحد و الـ x,k اه مظبوط
+
+823
+01:10:27,480 --> 01:10:32,540
+يعني الآن اللي هي طول
+
+824
+01:10:34,610 --> 01:10:41,150
+طول الفترة طول الفترة xk على جهة دي بس يا شباب
+
+825
+01:10:41,150 --> 01:10:46,850
+ناقص xk minus واحد طول الفترة لإن هاد صارت هي xk
+
+826
+01:10:46,850 --> 01:10:53,480
+minus واحد وهي xk ماشي جوات هنا الـ mean الـ UK و الـ 
+
+827
+01:10:53,480 --> 01:10:56,860
+VK في مكان الماد، التنتين المهم جوا التنتين يعني
+
+828
+01:10:56,860 --> 01:10:59,840
+المسافة بين اللي برا هدول، حتى لو كان التنتين
+
+829
+01:10:59,840 --> 01:11:04,120
+زيهن، بكون أصغر أو يساوي المسافة بين التنتين هدول
+
+830
+01:11:04,120 --> 01:11:11,560
+أصغر أو يساوي المسافة هذه، اللي هو UK ناقص VK،
+
+831
+01:11:11,560 --> 01:11:17,010
+مظبوط؟ آسف، العكس، أكبر شكرا سامحونا أكبر أيّاش أو
+
+832
+01:11:17,010 --> 01:11:21,310
+يساوي صارت المسافة بين الجهتين هدولة أكيد أصغر
+
+833
+01:11:21,310 --> 01:11:24,970
+أصغر من المسافة الكلية اللي هنا بينهم طيب المسافة
+
+834
+01:11:24,970 --> 01:11:28,890
+بين هذه وهذه احنا ماخدينها أصلا طول الـ interval
+
+835
+01:11:28,890 --> 01:11:29,670
+إيش بتساوي
+
+836
+01:11:33,030 --> 01:11:39,710
+على N إذا المسافة هذه أصلا عبارة عن B-A عالمين على
+
+837
+01:11:39,710 --> 01:11:43,890
+N سامحوني أني بنكتبها هنا إذا صار عندي اللي هو الـ
+
+838
+01:11:43,890 --> 01:11:50,230
+U K ناقص الـ V K أصغر أو يساوي B-A عالمين على N إذا
+
+839
+01:11:50,230 --> 01:11:58,530
+يا شباب لاحظولي هنا U K ناقص V K صارت أصغر أو
+
+840
+01:11:58,530 --> 01:12:06,440
+يساوي B-A عالمين على n واضحة؟ طيب نشوف إيش معناه هذا
+
+841
+01:12:06,440 --> 01:12:08,700
+الكلام وإيش اللي بتقوله؟ ليش بتقولنا هذا الكلام
+
+842
+01:12:08,700 --> 01:12:13,620
+بقوله عشان كلام مهم هذا هو شوفوا فهميته أنا مختار
+
+843
+01:12:13,620 --> 01:12:19,260
+الـ n اكبر من مين؟ من b minus a على delta يعني بمعنى
+
+844
+01:12:19,260 --> 01:12:24,280
+آخر إيش معنات هذا؟ يعني b minus a على n أصغر من مين
+
+845
+01:12:24,280 --> 01:12:29,640
+من دلتا يعني هذا المقدار أصغر من إيش يا جماعة؟ من
+
+846
+01:12:29,640 --> 01:12:35,160
+دلتا صارت الـ U K ناقص الـ V K أصغر من مين؟ من دلتا
+
+847
+01:12:35,160 --> 01:12:40,800
+واحنا بنقول أن المقاطع اللي بتحقق فيها الـ U ناقص V
+
+848
+01:12:40,800 --> 01:12:44,560
+أصغر من دلتا بكون عندي F of U ناقص F of V أصغر من
+
+849
+01:12:44,560 --> 01:12:50,240
+مين يا جماعة؟ من Epsilon على B minus A إذا صارت
+
+850
+01:12:50,240 --> 01:12:55,100
+عندي المقاطين هذولة الـ U K ناقص V K اللي أصغر من
+
+851
+01:12:55,100 --> 01:13:04,770
+دلتا إذا أكيد من واحد هيكون عندي F of U K ناقص F of
+
+852
+01:13:04,770 --> 01:13:12,670
+V K إيش هيكون؟ هيكون عبارة عن أصغر من Y على B
+
+853
+01:13:12,670 --> 01:13:20,410
+minus A إذا الآن بيحجلني أني أقول أن هذا المقدار
+
+854
+01:13:20,410 --> 01:13:24,400
+اللي عندي اللي هو طبعا هذا الكبير وهذا الصغير طبعا
+
+855
+01:13:24,400 --> 01:13:27,800
+على absolute value نفسه بيحجل إن أقول هذا أصغر أو
+
+856
+01:13:27,800 --> 01:13:34,680
+ساوي B minus A على N مضروب في مين؟ في الـ summation
+
+857
+01:13:34,680 --> 01:13:42,480
+للـ Epsilon على B minus A كامن عند واحد لعند مين
+
+858
+01:13:42,480 --> 01:13:50,100
+لعند N واضح
+
+859
+01:13:52,720 --> 01:13:58,440
+إذا صار عندي الآن اللي هو اللي أثبته أنه U P N و F
+
+860
+01:13:58,440 --> 01:14:04,220
+ناقص سؤال P N و F أصغر أو يساوي P minus A على N في
+
+861
+01:14:04,220 --> 01:14:10,300
+.. خلّيني أجف و أقولها .. أه .. انتظر .. أعملني ..
+
+862
+01:14:13,410 --> 01:14:18,250
+إذا اللي وصلنا له يا جماعة اللي هو أن الـ U P N و F
+
+863
+01:14:18,250 --> 01:14:21,650
+ناقص الـ P N و F أكبر يساوي سفر و في نفس الوقت أصغر
+
+864
+01:14:21,650 --> 01:14:24,430
+يساوي P Minus A على N في الـ summation للـ Epsilon ع
+
+865
+01:14:24,430 --> 01:14:28,630
+P Minus A اللي هو كامن عند واحد ل N يعني هذه قاعدة
+
+866
+01:14:28,630 --> 01:14:34,050
+عمالها كل مرة بتعد Epsilon ع P Minus A كم مرة
+
+867
+01:14:34,050 --> 01:14:37,670
+بتعد N من المرات إذا حيصير عندي هذا عبارة عن P
+
+868
+01:14:37,670 --> 01:14:43,170
+Minus A على N و اللي نعد هنا N من القيمة اللي هذه
+
+869
+01:14:43,170 --> 01:14:47,630
+يعني N في epsilon على B minus A الآن B minus A مع
+
+870
+01:14:47,630 --> 01:14:51,970
+الـ B minus A و الـ N مع الـ N فبصير أوصلنا احنا إلى
+
+871
+01:14:51,970 --> 01:14:55,950
+ما يلي انتبهوا للنتيجة النهائية اللي بتوصلي لي
+
+872
+01:14:55,950 --> 01:15:04,310
+اللي هو المطلوب النتيجة اللي وصلت لها انه إذا U P N
+
+873
+01:15:04,310 --> 01:15:12,430
+و F نقص L P N و F أكبر أو يساوي سفر وأصغر من مين
+
+874
+01:15:12,430 --> 01:15:19,190
+من إبسلون لكل إبسلون أكبر من سفر إذا اللي جوا غصب
+
+875
+01:15:19,190 --> 01:15:24,310
+عنه إذا لازم يكون اللي هو لو أخدت الـ limit هيظل
+
+876
+01:15:24,310 --> 01:15:30,220
+limit الـ U بن لأن هذا أصلا صحيح علميا للأنات
+
+877
+01:15:30,220 --> 01:15:34,200
+الكبيرة فلما أخد الـ limit أزنه جوزه infinity بضل
+
+878
+01:15:34,200 --> 01:15:38,280
+في الـ safe side يعني بضل في اللي بتحقق فيه هذا إذا
+
+879
+01:15:38,280 --> 01:15:45,630
+الـ limit U, B, N و F ناقص L, B, N و F هيكون أكبر أو
+
+880
+01:15:45,630 --> 01:15:50,490
+يساوي سفر و أصغر أو يساوي إبسلون و هذا الكلام as
+
+881
+01:15:50,490 --> 01:15:53,530
+and goes to infinity و هذا الكلام صحيح برضه لمين
+
+882
+01:15:53,530 --> 01:15:58,790
+لكل إبسلون أكبر من سفر إذا غصب عنها هيكون اللي جوا
+
+883
+01:15:58,790 --> 01:16:08,780
+limit UPN و F-Lبان و اف لازم يساوي إيش سفر از انقص
+
+884
+01:16:08,780 --> 01:16:14,780
+انفينيتي وهذا by corollary اللي قبل بشوية هيعني ان
+
+885
+01:16:14,780 --> 01:16:23,310
+اف is integrable Hence, F is integrable وهكذا
+
+886
+01:16:23,310 --> 01:16:27,610
+أثبتنا العائلة الثانية من الدوال الـ continuous
+
+887
+01:16:27,610 --> 01:16:31,170
+function على closed bounded interval is a
+
+888
+01:16:31,170 --> 01:16:34,210
+continuous function is an integrable function
+
+889
+01:16:34,210 --> 01:16:40,790
+وهكذا خلصنا الـ section الأول و الـ homework هيا
+
+890
+01:16:40,790 --> 01:16:46,750
+اللي موجودة في التلخيص و المرة القادمة إن شاء الله
+
+891
+01:16:47,630 --> 01:16:57,970
+إن شاء الله بكون احنا بنبدأ في اللي هي الـ section
+
+892
+01:16:57,970 --> 01:17:03,510
+اللي بعده اللي هو properties of the Riemann
+
+893
+01:17:03,510 --> 01:17:04,470
+Integral 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..615cab0444a228171429cbca4aa8e393f68d4efc
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_postprocess.srt
@@ -0,0 +1,3572 @@
+1
+00:00:04,720 --> 00:00:09,780
+بسم الله الرحمن الرحيم هذه هي المحاضرة رقم 11 في
+
+2
+00:00:09,780 --> 00:00:16,040
+مساق تحليل حقيقي 2 لطلاب وطالبات الجامعة الإسلامية
+
+3
+00:00:16,040 --> 00:00:24,760
+كلية العلوم وهي المحاضرة الأولى بعد إعلان الطوارئ
+
+4
+00:00:24,760 --> 00:00:32,580
+بخصوص أو بمواجهة فيروس كورونا المنتشر اتحدتنا
+
+5
+00:00:32,580 --> 00:00:37,960
+المرة الماضيةبدأنا في اللي هو chapter 7 اللي كان
+
+6
+00:00:37,960 --> 00:00:41,280
+الحديث عن ال riman integral أو تكامل ال riman
+
+7
+00:00:41,280 --> 00:00:45,860
+بدأنا في ال section الأول اللي هو تحت عنوان riman
+
+8
+00:00:45,860 --> 00:00:50,740
+integrability عرفنا شغلتين حاجة اسمها ال upper sum
+
+9
+00:00:50,740 --> 00:00:55,960
+و حاجة اسمها ال lower sum و قلنا اللي هو ال lower
+
+10
+00:00:55,960 --> 00:01:00,290
+sumهو عبارة عن الـ summation للـ mk الـ mk هذه
+
+11
+00:01:00,290 --> 00:01:06,410
+تمثل في xk minus xk minus 1 حيث mk كانت تمثل أو m
+
+12
+00:01:06,410 --> 00:01:10,910
+small k كانت تمثل عبارة عن الـ infimum للدالة على
+
+13
+00:01:10,910 --> 00:01:15,520
+الفترة اللي هي المذكورةالان ال .. ال .. ال other
+
+14
+00:01:15,520 --> 00:01:19,380
+sum هو عبارة عن ال summation لنفس ال sum العلوي
+
+15
+00:01:19,380 --> 00:01:24,520
+ولكن بدلا منها اللي هي M K capital اللي كانت تمثل
+
+16
+00:01:24,520 --> 00:01:28,280
+ال supremum ل ال F of X وال X على اللي هي في
+
+17
+00:01:28,280 --> 00:01:34,110
+الفترة المذكورة اللي عنديالآن أخدنا أول لمّة المرة
+
+18
+00:01:34,110 --> 00:01:38,190
+الماضية وقلنا إذا كانت F من I لR bounded و B any
+
+19
+00:01:38,190 --> 00:01:43,230
+partition of I بدي يكون ال lower اللي هو sum لأي
+
+20
+00:01:43,230 --> 00:01:47,810
+partition B و function F أصغر أو يساوي ال upper
+
+21
+00:01:47,810 --> 00:01:52,470
+sum لنفس ال partition و لنفس اللي هي ال function F
+
+22
+00:01:52,470 --> 00:01:58,710
+بعد هي طبعا خطينا خطوة أخرىو جينا عرفنا اللي هو شو
+
+23
+00:01:58,710 --> 00:02:03,110
+معناته انها تكون اللي هو ال partition Q refinement
+
+24
+00:02:03,110 --> 00:02:08,610
+لل partition B قلنا Q اللي هو تحسين ل B إذا كانت B
+
+25
+00:02:08,610 --> 00:02:13,940
+بي عبارة عن مجموعة جزئية من Qو بناء عليه اللي هو
+
+26
+00:02:13,940 --> 00:02:19,440
+قلنا ان اي اللي هو sub interval xk-1xk من ال
+
+27
+00:02:19,440 --> 00:02:23,520
+partition B يمكن كتابتها على صورة union of sub
+
+28
+00:02:23,520 --> 00:02:27,680
+intervals من اللي هو التحسين اللي هو EQ
+
+29
+00:02:31,060 --> 00:02:36,380
+الان جينا اللي هو بناء على هذا التعريف جينا قولنا
+
+30
+00:02:36,380 --> 00:02:40,480
+لو كانت F is من I لعند R is bounded و B is any
+
+31
+00:02:40,480 --> 00:02:45,780
+partition of I و Q refinement لل Bمدام اللي هو Q
+
+32
+00:02:45,780 --> 00:02:50,420
+-refinement إذا ال lower sum هيعلى و ال upper sum
+
+33
+00:02:50,420 --> 00:02:54,820
+هينزل على أساس انه اللي هو في النهاية يلتقي ال
+
+34
+00:02:54,820 --> 00:02:58,740
+upper مع ال lower و نصل لاللي هو ال integrability
+
+35
+00:02:58,740 --> 00:03:02,800
+أو معنى ال integrability كما سنرى لاحقا على الأقل
+
+36
+00:03:02,800 --> 00:03:06,560
+في اللي هو يكون واضح من خلال الرسم في اللي هي
+
+37
+00:03:06,560 --> 00:03:13,890
+الدوال الموجبة كما ذكرنا سابقااللي بحكيه إنه لو
+
+38
+00:03:13,890 --> 00:03:17,330
+كانت عندي اللي هو F من I ل R bounded وB partition
+
+39
+00:03:17,330 --> 00:03:22,750
+وQ وrefinement للـ B هيكون عندي lower sum لل
+
+40
+00:03:22,750 --> 00:03:28,570
+partition B أصغر أو يساوي lower sum للتحسين عماله
+
+41
+00:03:28,570 --> 00:03:32,850
+التحسين بكبر لما بده يصل لفعلا المساحة تحت المنحنة
+
+42
+00:03:32,850 --> 00:03:39,470
+في حالة الدوال الموجبةالقبر صم لـ Q و F سيبدأ يصغر
+
+43
+00:03:39,470 --> 00:03:42,850
+ويكون أصغر من الساوي اللي هو القبر صم لـ P و F
+
+44
+00:03:42,850 --> 00:03:47,310
+التحسين يعني سيصغره بمعنى آخر سيبدأ يلتقوا إلى
+
+45
+00:03:47,310 --> 00:03:51,770
+أسفل لما نصل إلى اللي هو مساواة في حالة الـ
+
+46
+00:03:51,770 --> 00:03:55,730
+Integrability لما نكون عندنا أخدنا الـ Supremum
+
+47
+00:03:55,730 --> 00:04:01,340
+لكل L و Fالـ L والـ infimum لكل الـ U بيصير بنسمي
+
+48
+00:04:01,340 --> 00:04:04,800
+بعد شوية حاجة اسمها ال lower integral وال upper
+
+49
+00:04:04,800 --> 00:04:08,980
+integral وبرهننا هذه النظرية و بعدين جينا لللمة و
+
+50
+00:04:08,980 --> 00:04:13,520
+بعدين جينا لللمة أخرىاللي هو لو كانت F من I ل R
+
+51
+00:04:13,520 --> 00:04:17,600
+bounded وB1 وB2 اي partitions الآن لأي partitions
+
+52
+00:04:17,600 --> 00:04:22,480
+هيكون الlower دايما بغض النظر عن ال partition اللي
+
+53
+00:04:22,480 --> 00:04:26,120
+هو هيكون أصغر أو يساوي الأبر بغض النظر عن ال
+
+54
+00:04:26,120 --> 00:04:28,660
+partition B2 يعني مش لنفس ال partition زي ما قلنا
+
+55
+00:04:28,660 --> 00:04:33,180
+في اللمبة 7 1 1 لأ لأي two partitions دايما
+
+56
+00:04:33,180 --> 00:04:37,170
+الlower ما هو هيكون تحتأسفل المنحنى والـ Upper
+
+57
+00:04:37,170 --> 00:04:41,350
+هيكون أعلى المنحنى بغض النظر عن ال partitions اللي
+
+58
+00:04:41,350 --> 00:04:45,690
+عندى طبعا التمثيل هذا في حال اللي هو ال F is a
+
+59
+00:04:45,690 --> 00:04:49,130
+positive function على ال interval المذكورة الآن
+
+60
+00:04:49,130 --> 00:04:52,730
+بعد هيكة اجينا وعرفنا شو معناه ال lower integral
+
+61
+00:04:52,730 --> 00:04:55,510
+وشو معناه ال upper integral قولنا ال lower
+
+62
+00:04:55,510 --> 00:05:00,050
+integral كما هو متوقع سميناه الف F هو عبارة عن ال
+
+63
+00:05:00,050 --> 00:05:05,660
+supremum لل lowersوال .. و ال .. و ال .. و ال
+
+64
+00:05:05,660 --> 00:05:10,440
+upper هو عبارة عن الـ infimum لل uppers حتى لو
+
+65
+00:05:10,440 --> 00:05:14,300
+التقت الالو أف مع الالو أف اللي هو من أعلى مع
+
+66
+00:05:14,300 --> 00:05:17,940
+الأسفل هيكونوا التقوا بالظبط عند .. من مساحة تحت
+
+67
+00:05:17,940 --> 00:05:20,960
+المنحنى في حالة الدالة الموجبة وهذه .. في هذه
+
+68
+00:05:20,960 --> 00:05:24,300
+الحالة بنسمي إذا كانت ال upper تساوي ال lower
+
+69
+00:05:24,300 --> 00:05:27,920
+بنسمي ال function على هذه الفترة is integral و هذا
+
+70
+00:05:27,920 --> 00:05:31,900
+الكلام كله تحدثنا فيه عشان هيك أنا مسرع شويةهو
+
+71
+00:05:31,900 --> 00:05:37,160
+حكينا إنه اللي هو دايما ال lower sum لل F أخدنا
+
+72
+00:05:37,160 --> 00:05:40,760
+نظرية قولنا ال lower integral أسف ال lower
+
+73
+00:05:40,760 --> 00:05:45,100
+integral دايما أصغر يساوي مين ال upper integral
+
+74
+00:05:45,100 --> 00:05:46,860
+إذن الآن النظرية
+
+75
+00:05:49,600 --> 00:05:52,760
+الإعلان المهم اللي هو لو كانت F من I لعند R
+
+76
+00:05:52,760 --> 00:05:56,000
+bounded function على closed bounded interval A وB
+
+77
+00:05:56,000 --> 00:05:59,720
+بدي يكون ال lower integral L of F أصغر أو ساوي ال
+
+78
+00:05:59,720 --> 00:06:04,930
+upper integral U of F بصورة عامةهذه هي الـ
+
+79
+00:06:04,930 --> 00:06:08,050
+definition اللي ذكرته قبل و شوية نقول عن الـ
+
+80
+00:06:08,050 --> 00:06:15,150
+function F على bounded sub interval A و B أو
+
+81
+00:06:15,150 --> 00:06:18,650
+closed bounded interval A و B و كانت الـ F عبارة
+
+82
+00:06:18,650 --> 00:06:22,080
+عن bounded functionبنعرف أن الـ F is remain
+
+83
+00:06:22,080 --> 00:06:26,840
+integrable on I إذا كانت ال lower of F بساوي ال
+
+84
+00:06:26,840 --> 00:06:30,080
+upper of F معناته صارت اللي هي ال F is remain
+
+85
+00:06:30,080 --> 00:06:34,460
+integrable if and only if ال lower sum يساوي ال
+
+86
+00:06:34,460 --> 00:06:39,240
+upper sum هذا كله ذكرناه المرة الماضية و أيضا
+
+87
+00:06:39,240 --> 00:06:42,520
+عرفنا .. قلنا في هذا الحلقة أن كل integration من A
+
+88
+00:06:42,520 --> 00:06:47,890
+ل B هو ال lower أو ال upper المتساويينوعرفنا ايضا
+
+89
+00:06:47,890 --> 00:06:50,610
+تعريف اخر قولنا ال integration من a ل b بساوي ناقص
+
+90
+00:06:50,610 --> 00:06:53,830
+ال integration من b ل a وعرفنا ال integration من a
+
+91
+00:06:53,830 --> 00:06:59,410
+ل a بساوي صفر هذا كله حكينا المرة الماضية ومش هيك
+
+92
+00:06:59,410 --> 00:07:03,820
+كمان واخدنا المثالاللي هو أثبتنا إنه اللي هو g of
+
+93
+00:07:03,820 --> 00:07:07,880
+x بيساوي x is integrable on i استنادًا على إنه
+
+94
+00:07:07,880 --> 00:07:11,020
+أوجدنا ال lower sum ال lower integral و ال upper
+
+95
+00:07:11,020 --> 00:07:13,300
+integral أثبتنا إن ال lower integral و ال upper
+
+96
+00:07:13,300 --> 00:07:16,300
+integral are equal و من ثم أثبتنا إنه ال
+
+97
+00:07:16,300 --> 00:07:19,940
+integration exist لل function x على الفترة 0 و 1
+
+98
+00:07:19,940 --> 00:07:26,810
+وأوجدنا قيمة ال integration في حينهوصلنا إلى اللي
+
+99
+00:07:26,810 --> 00:07:33,290
+هو مثالنا التالي أنه لو كانت F من I لعند .. F من I
+
+100
+00:07:33,290 --> 00:07:40,090
+اللي هي 01 لعند ال .. ال R be defined by أخدنا
+
+101
+00:07:40,090 --> 00:07:45,990
+الدالة كما يلي اللي هو قولنا أن F of X F of X
+
+102
+00:07:45,990 --> 00:07:53,040
+بساوي واحدإذا كانت x rational number element in Q
+
+103
+00:07:53,040 --> 00:08:00,100
+وبساوي 0 إذا كانت x element in IQ أو element in Q
+
+104
+00:08:00,100 --> 00:08:04,660
+complement اللي هي ال rational numbers الآن بدنا
+
+105
+00:08:04,660 --> 00:08:11,200
+نثبت بقول show that this function Fطبعا أنا method
+
+106
+00:08:11,200 --> 00:08:17,340
+F على اللي هي الـ Q تقاطع طبعا الـ 0 و 1 اللي هي
+
+107
+00:08:17,340 --> 00:08:19,920
+الـ interval اللي بدأ عليها التقاطع الـ 0 و 1
+
+108
+00:08:19,920 --> 00:08:25,160
+بمعنى إن دالت F صارت من I اللي هي عبارة عن 0 و 1
+
+109
+00:08:26,040 --> 00:08:29,380
+اللي عند R واضح ان الدالة هذه is a bounded
+
+110
+00:08:29,380 --> 00:08:33,560
+function الان بدأ أثبت لكم ان هذا الدالة is not
+
+111
+00:08:33,560 --> 00:08:38,120
+integrable on this interval is not integrable on
+
+112
+00:08:38,120 --> 00:08:44,100
+this interval الان علشان أصل اللي هو اللي هي ان
+
+113
+00:08:44,100 --> 00:08:48,180
+هذا الدالة غير قابلة تكامل بالنسبة لتكامل بالنسبة
+
+114
+00:08:48,180 --> 00:08:55,650
+لتكامل الريمان بدي اخد الان Bأخدوا أي partition X0
+
+115
+00:08:55,650 --> 00:09:02,550
+X1 لعند Xn هذا any partition لإيه ال interval اللي
+
+116
+00:09:02,550 --> 00:09:06,790
+هي الفترة مين Zero و واحد يعني بمعنى أتيت للفترة
+
+117
+00:09:06,790 --> 00:09:14,030
+Zero و واحد و جزقتها X0 X1 لعند مؤصل لعند مين لعند
+
+118
+00:09:14,030 --> 00:09:18,610
+Xn اللي هي إيش بتساوي بتساوي واحدالان هذا ال
+
+119
+00:09:18,610 --> 00:09:22,130
+partition اخدته arbitrarily اللى هو partition
+
+120
+00:09:22,130 --> 00:09:31,790
+لفترة L عندى الان بدى احسب ال Lof B و F لهذا ال
+
+121
+00:09:31,790 --> 00:09:34,990
+partition أيش بتساوي حسب اللي عرفناها سابقا بتساوي
+
+122
+00:09:34,990 --> 00:09:40,610
+ال summation لل M K في X K minus X K minus واحد K
+
+123
+00:09:40,610 --> 00:09:46,330
+من عند واحد لعند مين لعند اللي هي N ويساوي الآن ال
+
+124
+00:09:46,330 --> 00:09:52,450
+M K عرفناها ال M K هي عبارة عن ال infimum لقيمة ال
+
+125
+00:09:52,450 --> 00:09:56,470
+function F of X حيث X تنتمي إلى الفترة X K minus
+
+126
+00:09:56,470 --> 00:10:02,210
+واحد لعند X Kطبعاً اللي هي F of X معرفة على
+
+127
+00:10:02,210 --> 00:10:05,130
+أساسيها يا إما واحد يا إما سفر حسب إنها تكون
+
+128
+00:10:05,130 --> 00:10:08,030
+rational أو إيش ال rational يعني ال function F
+
+129
+00:10:08,030 --> 00:10:11,910
+أصلا اللي هي قيمتين بس إذا الأن ال infimum لل F of
+
+130
+00:10:11,910 --> 00:10:16,630
+X عندها يا هيكون واحد يا هيكون سفر ليش؟ لأن أصلا
+
+131
+00:10:16,630 --> 00:10:23,340
+الفترة هذهفيها أي فترة subinterval xk-1xk فيها
+
+132
+00:10:23,340 --> 00:10:27,620
+rational وirrational إذا قيمة ال up of x في الفترة
+
+133
+00:10:27,620 --> 00:10:31,260
+هتلاقي عند قيم واحد هتلاقي أكيد عند قيم أش بتساوي
+
+134
+00:10:31,260 --> 00:10:35,000
+بساوي سفر إذا ال infimum في هذه الحالة هو عبارة عن
+
+135
+00:10:35,000 --> 00:10:42,320
+إيش يساوي سفر إذا ال summation ل 0 في xk-xk-1 كامل
+
+136
+00:10:42,320 --> 00:10:46,120
+عند واحد عند أنه طبيعي هذا بديهي إيش هيساوي بساوي
+
+137
+00:10:46,120 --> 00:10:52,410
+سفرأذن الان L of F بي و F ساوة سفر فور أي بورتيشن
+
+138
+00:10:52,410 --> 00:10:59,370
+بيه إذا ال L of F اللي هي عبارة عن الأو الـ
+
+139
+00:10:59,370 --> 00:11:06,070
+Supremum الـ Supremum لكل الـ L of B و F Such that
+
+140
+00:11:06,070 --> 00:11:09,890
+B element in the set of all partitions اللي هو B
+
+141
+00:11:09,890 --> 00:11:14,090
+of I هيكون ال .. ال Supremum اللي هين صفر لإن كل
+
+142
+00:11:14,090 --> 00:11:18,350
+اللي هين أصلا إشق بتنطلع .. تنطلع بساوي صفر إذا
+
+143
+00:11:18,350 --> 00:11:23,070
+هذا إيش هيساوي يا شباب؟ هو يساوي Zeroإذا طلع عندى
+
+144
+00:11:23,070 --> 00:11:28,150
+ال lower sum بساوة 0 الان بدي أحسبلكم مين أحسبلكم
+
+145
+00:11:28,150 --> 00:11:31,390
+ال upper sum سامحوني أكتب هنا بس عساس اللي يبقى
+
+146
+00:11:31,390 --> 00:11:39,670
+كله مكتوب عندى نوجد ال upper sum ال upper sum اللي
+
+147
+00:11:39,670 --> 00:11:45,510
+هي ال UPUF بساوة summation للان K capital في XK
+
+148
+00:11:45,510 --> 00:11:51,740
+minus XK minus واحد K من عند واحد لعند الانالان
+
+149
+00:11:51,740 --> 00:11:55,900
+هذا بيساوي ال Mk زي ما قلنا قبل هيك ال Mk بدل ما
+
+150
+00:11:55,900 --> 00:11:59,400
+هي ال M في مميزة استعريفها ال Mk بتساوي ال
+
+151
+00:11:59,400 --> 00:12:03,780
+supremum لهذه ال 6 وزي ما قلنا ال 6 هذه في داخلها
+
+152
+00:12:03,780 --> 00:12:08,400
+يا واحد يا زيرو نظرا لإن اللي هو أي sub interval
+
+153
+00:12:08,400 --> 00:12:11,820
+هيكون فيها rational و irrational وتبعا إلها هيكون
+
+154
+00:12:11,820 --> 00:12:16,060
+قيمة ال function في داخلها واحد أو سفر واحنا بنبحث
+
+155
+00:12:16,060 --> 00:12:19,280
+عن ال supremum إذا هيكون ال supremum في كل الأحوال
+
+156
+00:12:19,280 --> 00:12:25,600
+ال Mk بتساوي واحدمضروبة في xk-xk-1 k من عند 1 لعند
+
+157
+00:12:25,600 --> 00:12:30,700
+n نفردها هذه ويساوي اللي هنبصير k من عند 1 يعني x1
+
+158
+00:12:30,700 --> 00:12:39,570
+-x0 زاد x2-x1 زاد إلى أخير لما أصل لعند xn-1نقص xn
+
+159
+00:12:39,570 --> 00:12:44,550
+نقص واحد طبعا واضح انه عندي ال x واحد هت cancel مع
+
+160
+00:12:44,550 --> 00:12:48,270
+ناقص x واحد و ال x اتنين مع ناقص x اتنين لما نصل
+
+161
+00:12:48,270 --> 00:12:52,450
+للاخر هيكون في عندي ات cancel الجميع بس ضال عندي
+
+162
+00:12:52,450 --> 00:12:58,170
+ال xn و ال x note و هد بتساوي xn ناقص x note و
+
+163
+00:12:58,170 --> 00:13:01,750
+يساوي ال xn طبعا ايش هي عبارة عن واحد و ال x note
+
+164
+00:13:01,750 --> 00:13:07,190
+ايش هي شبه سفر و يساوي واحد ناقص سفر و يساوي واحد
+
+165
+00:13:07,570 --> 00:13:12,930
+إذا طلع عندي الـ Upper Sum لأي Bar تشهم بيه، هيتلع
+
+166
+00:13:12,930 --> 00:13:18,510
+ايش بساوة؟ بساوة واحدة إذا الان لما بدي أخد ال U
+
+167
+00:13:18,510 --> 00:13:23,510
+of F اللي هو Upper Integral هيساوة عبارة عن ال
+
+168
+00:13:23,510 --> 00:13:30,220
+Infimumلمين؟ للـ U, B وF such that B element in
+
+169
+00:13:30,220 --> 00:13:34,580
+the set of all partitions B of I والـ U, B وF
+
+170
+00:13:34,580 --> 00:13:38,840
+قيمته ثابتة for أي partition بيساوي واحد إذا الـ
+
+171
+00:13:38,840 --> 00:13:43,300
+infimum لكل اللي هنا عبارة عن برضه إيش بيساوي واحد
+
+172
+00:13:43,300 --> 00:13:47,410
+صار عندي الآنlower integral و ال upper integral
+
+173
+00:13:47,410 --> 00:13:51,110
+have different values واحد بيساوي صفر واحد بيساوي
+
+174
+00:13:51,110 --> 00:13:56,890
+واحد وبناء عليه بتكون عنده اللي هو ال F is not
+
+175
+00:13:56,890 --> 00:14:03,150
+Riemann integrable اي سؤال؟ طيب ماشي الحاجة
+
+176
+00:14:03,150 --> 00:14:10,930
+الانصار عندى اخدنا مثلين المثال الأولاللي هو
+
+177
+00:14:10,930 --> 00:14:16,770
+أثبتنا إن off of x بيساوي x is integrable على
+
+178
+00:14:16,770 --> 00:14:22,010
+الفترة 0 و1 و أثبتناها بواسطة التعريف وأيضا أثبتنا
+
+179
+00:14:22,010 --> 00:14:26,190
+مثال آخر لbounded function أيضا وكانت is not
+
+180
+00:14:26,190 --> 00:14:31,250
+remain integrable اللي هي off of x بيساوي 1 إذا
+
+181
+00:14:31,250 --> 00:14:36,090
+كانت x rational ويساوي 0 إذا كانت x irrational هذا
+
+182
+00:14:36,090 --> 00:14:43,040
+اللي هو المثال الثانيالان نيجي لاللي هو criterion
+
+183
+00:14:43,040 --> 00:14:49,280
+مهمة اللي احنا بنسميها اللي هي عبارة عن remain
+
+184
+00:14:49,280 --> 00:14:54,800
+criterion for
+
+185
+00:14:54,800 --> 00:15:01,490
+integrabilityأحنا طبعا اتحدثنا عن الـ Remain
+
+186
+00:15:01,490 --> 00:15:05,330
+Integrability كيف نثبت أنه Remain Integrable عن
+
+187
+00:15:05,330 --> 00:15:09,470
+طريق التعريف طبعا الآن مش دايما بدنا نثبت عن طريق
+
+188
+00:15:09,470 --> 00:15:14,350
+التعريف إذا بدنا اللي هو طرق أخرى نحاول اللي هو
+
+189
+00:15:14,350 --> 00:15:21,780
+نوسع اللي هيإمكانياتنا في الحكم على الدالة إنها
+
+190
+00:15:21,780 --> 00:15:26,860
+integrable أو مش integrable وهذه الإمكانية الأخرى
+
+191
+00:15:26,860 --> 00:15:31,400
+غير التعريف هي اللي بنسميها اللي هو الريمان
+
+192
+00:15:31,400 --> 00:15:36,500
+integrability criterion أو criterion for
+
+193
+00:15:36,500 --> 00:15:41,340
+integrability نشوف أيش بيقول النظرية
+
+194
+00:15:43,700 --> 00:15:48,060
+لت I بساوة A وB و لت F من I لـ R بيـ bounded نفترض
+
+195
+00:15:48,060 --> 00:15:51,360
+أن F عبارة عن إيه اشمال يا جماعة؟ bounded function
+
+196
+00:15:51,360 --> 00:15:57,500
+then F is integrable on I if and only if for each
+
+197
+00:15:57,500 --> 00:16:00,340
+epsilon أكبر من 0 there exists a partition B
+
+198
+00:16:00,340 --> 00:16:04,660
+epsilon of I such that U B epsilon نقص الـ B
+
+199
+00:16:04,660 --> 00:16:10,730
+epsilon إيه إشماله أصغر من اللي هو إبسلونإذن اللي
+
+200
+00:16:10,730 --> 00:16:16,410
+هو واضح إنه عندي فصار فيه test لل integrability أو
+
+201
+00:16:16,410 --> 00:16:20,150
+اللي هو طريقة للحكم على ال integrability أخرى غير
+
+202
+00:16:20,150 --> 00:16:26,430
+التعريف اللي هو بتقول F is integrable
+
+203
+00:16:28,870 --> 00:16:32,890
+if and only if طبعاً هذه لمين الـ F؟ F عبارة عن زي
+
+204
+00:16:32,890 --> 00:16:36,190
+ما انتوا عارفين bounded function لإنه كل شغلنا
+
+205
+00:16:36,190 --> 00:16:40,010
+أصلا على اللي هو remaining integrability أنه نفترض
+
+206
+00:16:40,010 --> 00:16:43,210
+أنه الـ F bounded عشان اللي هو تكون ال supremum و
+
+207
+00:16:43,210 --> 00:16:46,330
+ال infimum اللي مبني عليها التعريف تكون مضمون إنها
+
+208
+00:16:46,330 --> 00:16:49,870
+موجودة عشان هيك بنحكي أن F is bounded function طيب
+
+209
+00:16:49,870 --> 00:16:56,140
+إذا الـ F is integrable if and only ifاللي هي لكل
+
+210
+00:16:56,140 --> 00:17:00,620
+إبسلون أكبر من 0 there exists a partition P إبسلون
+
+211
+00:17:00,620 --> 00:17:04,240
+هذا ال P اللي هو ال partition يعتمد عالميا على
+
+212
+00:17:04,240 --> 00:17:07,820
+إبسلون لكل إبسلون بالله دي partition P إبسلون لمين
+
+213
+00:17:07,820 --> 00:17:11,840
+ال partition طبعا لل interval اللي عندنا اللي هي R
+
+214
+00:17:11,840 --> 00:17:17,040
+there exists P إبسلون a partition of I such that
+
+215
+00:17:17,040 --> 00:17:25,210
+ال P ال U ال P إبسلونوالـ F ناقص الـ L بي إبسلون و
+
+216
+00:17:25,210 --> 00:17:31,050
+F يكون أصغر من مين من إبسلون الآن إذا كان لجينا
+
+217
+00:17:31,050 --> 00:17:34,190
+لكل إبسلون لجينا بي إبسلون بحيث هذا يتحقق معناته F
+
+218
+00:17:34,190 --> 00:17:37,590
+is integrable and conversely if F is integrable
+
+219
+00:17:37,590 --> 00:17:42,550
+أكيد لكل إبسلون هلجي بي إبسلون بحيث أن هذا يتحقق
+
+220
+00:17:42,550 --> 00:17:48,170
+خلونا نيجي الآن نبرهم و نشوف كيف بدنا نبرهم
+
+221
+00:17:48,170 --> 00:17:53,880
+نظريتناالان بدنا نفترض ان F اي شمالها is
+
+222
+00:17:53,880 --> 00:18:00,240
+integrable ونصل منها للي عيدي اللي هي ال partition
+
+223
+00:18:00,240 --> 00:18:07,040
+اللي مذكور مدام F is integrable اذا كانت ..الان
+
+224
+00:18:07,040 --> 00:18:16,100
+بنقول suppose that F is integrable مدام integrable
+
+225
+00:18:16,100 --> 00:18:22,090
+يا شبابأكيد عنده اللي هو الـ U of F بساوي L of
+
+226
+00:18:22,090 --> 00:18:29,490
+إيش؟ Of F، مظبوط؟ أكيد الـ L of .. الـ U of F
+
+227
+00:18:29,490 --> 00:18:33,410
+بساوي الـ L of F إيش اللي بيدثبته؟ بيدثبته لأي
+
+228
+00:18:33,410 --> 00:18:35,810
+إبسلون أكبر من سفر بدلاجي بإبسلون، شوفوا كيف
+
+229
+00:18:35,810 --> 00:18:41,090
+بنلاجيه، الآن نفترض إن إبسلون let إبسلون أكبر من
+
+230
+00:18:41,090 --> 00:18:47,810
+سفر be given، ماشي الحالةعندي الـ U of F هو إيش يا
+
+231
+00:18:47,810 --> 00:18:54,910
+شباب؟ هو عبارة عن الـ infimum للـ L of B و F such
+
+232
+00:18:54,910 --> 00:19:01,170
+that B element in B of I، مظبوط؟ إذا الـ U of F
+
+233
+00:19:01,170 --> 00:19:05,830
+عبارة عن infimum يعني هو عبارة عن greatest lower
+
+234
+00:19:05,830 --> 00:19:11,630
+boundلو هذا الـ greatest lower bound ضفنا إليه أي
+
+235
+00:19:11,630 --> 00:19:17,230
+كمية هيبط اللاور باوند لأن هو أصلا إيش اسمه
+
+236
+00:19:17,230 --> 00:19:23,510
+greatest lower bound إذا لو الـU of F ضفتله Y على
+
+237
+00:19:23,510 --> 00:19:28,610
+2 مثلا طبعا هذا المقدار هيبط اللاور باوند إيش
+
+238
+00:19:28,610 --> 00:19:33,690
+معناته بط اللاور باوند؟ يعني بمعنى آخر هيكون في
+
+239
+00:19:33,690 --> 00:19:44,080
+عند إشي أصغر منههيكون عندى أصغر من ال U of F أو بي
+
+240
+00:19:44,080 --> 00:19:49,740
+واحد مثلا و F for some mean بي واحد إذا لما نشيل
+
+241
+00:19:49,740 --> 00:19:54,600
+من ال infimum إبسلون
+
+242
+00:19:54,600 --> 00:19:57,740
+على اتنين هيبطل هذا lower bound يعني بمعنى أخر
+
+243
+00:19:57,740 --> 00:20:03,280
+هلاقي اللي هو lower bound
+
+244
+00:20:07,660 --> 00:20:12,140
+عند الـ Y أكبر من 0 خلّيني أبدأ لت Y أكبر من 0 بي
+
+245
+00:20:12,140 --> 00:20:17,940
+given إذا عندي الآن بدي أثبت لك بدي أجيب لك
+
+246
+00:20:17,940 --> 00:20:21,440
+partition بيبسلون بحيث أنه هذا ناقص هذا يكون أصغر
+
+247
+00:20:21,440 --> 00:20:26,070
+من مين من ابسلون شوف كيف بدي أعملالان انا عندي ال
+
+248
+00:20:26,070 --> 00:20:30,710
+U of F ايش بيساوي يا جماعة ال U of F بيساوي ال
+
+249
+00:20:30,710 --> 00:20:39,590
+infimum لل L بيقف ب element I طيب ال U of F اسف يا
+
+250
+00:20:39,590 --> 00:20:43,010
+جماعة ال U of F بيساوي ال infimum لمين لل U بيقف
+
+251
+00:20:43,720 --> 00:20:49,780
+ماشي الحال الان ايش معناه انه هذا infimum معناته
+
+252
+00:20:49,780 --> 00:20:55,100
+هذا هو عبارة عن ال greatest lower bound مدام ال
+
+253
+00:20:55,100 --> 00:20:59,450
+greatest lower bound اذا ال lower bound هذاأو الـ
+
+254
+00:20:59,450 --> 00:21:03,330
+Greatest Lower Bound لو ضفتله أي رقم يابسون على
+
+255
+00:21:03,330 --> 00:21:06,370
+اتنين مثلا بالذنب يابسون على اتنين الـ game
+
+256
+00:21:06,370 --> 00:21:09,670
+بتعرفوا ليش يابسون على اتنين يعني لو ضفتله أي رقم
+
+257
+00:21:09,670 --> 00:21:14,730
+بيبطل Lower Bound إيش معناته بيبطل Lower Bound
+
+258
+00:21:14,730 --> 00:21:21,830
+يعني هيصير هذا أكبر من ال UB 1 of F for some B
+
+259
+00:21:21,830 --> 00:21:27,510
+وحدهالأنه بطل أش ماله هذا بطل lower bound بطل يكون
+
+260
+00:21:27,510 --> 00:21:32,670
+أصغر من الكل من هان يعني هلاجي واحد من هان هو مش
+
+261
+00:21:32,670 --> 00:21:36,830
+أصغر منه أو بمعنى أخر U, P, 1 و F أصغر من اللي هو
+
+262
+00:21:36,830 --> 00:21:44,850
+هذا المقدر طيب similarly ال L of F هي عبارة عن ال
+
+263
+00:21:44,850 --> 00:21:52,050
+supremum لل L, P و F such that P elemented P of I
+
+264
+00:21:53,040 --> 00:21:57,780
+بنفس الطريقة يا جماعة ال L of F هي عبارة عن إيش ال
+
+265
+00:21:57,780 --> 00:22:04,040
+least upper bound يعني هذا لو least upper bound
+
+266
+00:22:04,040 --> 00:22:07,200
+Upper bound لو كان أصغر و أحد لو هذا least upper
+
+267
+00:22:07,200 --> 00:22:11,000
+bound راحت منه عدد ولو صغير جدا و ليكن يبسلون على
+
+268
+00:22:11,000 --> 00:22:16,800
+اتنينهيبطل هذا عبارة عن upper bound يعني هلاقي
+
+269
+00:22:16,800 --> 00:22:24,280
+واحد من اللي هان اللي هو L of B2 وF مثلا أكبر منه
+
+270
+00:22:24,280 --> 00:22:30,560
+لأنه هيبطل هذا أشماله upper bound لأنه هو ال least
+
+271
+00:22:30,560 --> 00:22:35,780
+لما طلعت منه بطل من ال upper bounds يعني لجيت واحد
+
+272
+00:22:35,780 --> 00:22:43,120
+من هذول أكبر منه إذا صار في عندي اللي هولجيت بي
+
+273
+00:22:43,120 --> 00:22:53,960
+واحد بحقق الأولى و بيتنين بحقق التانية لأن خد الآن
+
+274
+00:22:53,960 --> 00:22:58,840
+خدلي بي إبسلون هذا اللي بديها هذا الحقق اللي اللي
+
+275
+00:22:58,840 --> 00:23:03,600
+بديها خد بي إبسلون إيش بساوي ال بي واحد اللي لجيته
+
+276
+00:23:03,600 --> 00:23:12,050
+هنااتحاد الـ B2 اللي لجيته هان اتحاد مين؟ B2 صار
+
+277
+00:23:12,050 --> 00:23:18,510
+عندى .. صار عندى الآن مع بعض خليني أمسح فأقوم،
+
+278
+00:23:18,510 --> 00:23:22,170
+مهمش؟
+
+279
+00:23:22,170 --> 00:23:31,670
+صار
+
+280
+00:23:31,670 --> 00:23:41,110
+عندى ما يليه؟ صار عندى الآنL of F نقص
+
+281
+00:23:41,110 --> 00:23:53,490
+Y على 2 أصغر من L of B2 Fاللي هو أكيد أصغر أو
+
+282
+00:23:53,490 --> 00:23:59,750
+يساوي L of By قُقَف، ليش؟ لأن الـBy يا جماعة عبارة
+
+283
+00:23:59,750 --> 00:24:04,570
+عن refinement للـB2 والlower لما يصير فيه تحسين
+
+284
+00:24:04,570 --> 00:24:08,510
+بكبر، بروح نحو اتجاه نيل اللي هو المنحنة التالية
+
+285
+00:24:08,510 --> 00:24:14,010
+مساحة تحت المنحنة الكلية لأن أيضا لو جيت قولة الـU
+
+286
+00:24:14,010 --> 00:24:24,400
+of Fزاد إبسلون على اتنين هتلاقيها أكبر من ال U بي
+
+287
+00:24:24,400 --> 00:24:30,410
+واحد و Fوأكيد عندي الـ U of F زائد إبسلون عدنان
+
+288
+00:24:30,410 --> 00:24:34,490
+أكبر من الـ U B1 of F هيكون هذا أكبر أو يساوي الـ
+
+289
+00:24:34,490 --> 00:24:39,610
+U B epsilon of F لأن الـ B epsilon refinement لمين
+
+290
+00:24:39,610 --> 00:24:44,790
+برضه؟ للـ B1 بزام refinement إذن اللي هو التحسين
+
+291
+00:24:44,790 --> 00:24:50,270
+بيزغر القبر وبروح ناحية اللي هو المنحنة إذن الآن
+
+292
+00:24:50,270 --> 00:24:55,480
+من الـ tip 2 هذولةواجدنا طبعا احنا تنسوش ان احنا
+
+293
+00:24:55,480 --> 00:24:59,100
+مفترضين من رأس الدول ان U of F is integrable يعني
+
+294
+00:24:59,100 --> 00:25:05,340
+مفترضين ان ال L of F ايش بتساوي U of F تنسواش هذه
+
+295
+00:25:05,340 --> 00:25:12,540
+ليه جيت تنتر مع بعض دول بصير عندى الحصل على L of F
+
+296
+00:25:12,540 --> 00:25:22,970
+ناقص Y على 2 اللي هو اصغر من Lof B, Epsilon و F
+
+297
+00:25:22,970 --> 00:25:30,110
+والـ L و الـ U of F زائد Epsilon على 2 أكبر من U,
+
+298
+00:25:30,190 --> 00:25:35,290
+B, Epsilon و F أنا إيش غرضي؟ غرضي أثبت إن U, B,
+
+299
+00:25:35,350 --> 00:25:39,130
+Epsilon و F ناقص L, B, Epsilon و F أصغر من Epsilon
+
+300
+00:25:39,130 --> 00:25:42,890
+يلّا اترحل من بعض، إذا بصير عندي، بتحصل البدكية
+
+301
+00:25:42,890 --> 00:25:48,570
+بصير عندي الآن، بطرح حياة دي، بقول U, B, Epsilon و
+
+302
+00:25:48,570 --> 00:25:53,240
+Fناقص لأنه لما نضرب هذا في ناقص هتنعكس يعني هتصير
+
+303
+00:25:53,240 --> 00:25:57,100
+هذه جماعة ناقص وهذه زائد وهذه هتنعكس هيك وهيصير
+
+304
+00:25:57,100 --> 00:26:05,600
+الاش ناقص بيصير عندي U بي و F ناقص ال بي و F هيصير
+
+305
+00:26:05,600 --> 00:26:11,820
+أصغر من مين من ناقص L of F ناخد هذا قبل زي ما احنا
+
+306
+00:26:11,820 --> 00:26:20,690
+مرتبينها U of F زائد ي على 2 ناقصL of F زي إبسلون
+
+307
+00:26:20,690 --> 00:26:24,790
+على 2 وطبعا إحنا جايلين إن F is integrable يعني
+
+308
+00:26:24,790 --> 00:26:28,770
+الـ U of F بسوء L of F إذا هذي بتروح مع هذي بظل
+
+309
+00:26:28,770 --> 00:26:33,610
+إيه شماله؟ بظل إبسلون إذا إحنا لكل إبسلون أكبر من
+
+310
+00:26:33,610 --> 00:26:36,950
+سفر لجينا بي إبسلون هي في الواقع بي إبسلون اللي
+
+311
+00:26:36,950 --> 00:26:39,970
+لجيناها بي واحد اتحاد بي اتنين حيث بي واحد اللي
+
+312
+00:26:39,970 --> 00:26:44,120
+لجيناه هان وال بي اتنين اللي لجيناه هانsuch that U
+
+313
+00:26:44,120 --> 00:26:49,120
+P Y of F نقص L P Y of F أصغر من اللي هو Epsilon
+
+314
+00:26:49,120 --> 00:26:57,480
+وهو المطلوب أي سؤال؟ طيب، ماشي يا شباب، الآن خلصنا
+
+315
+00:26:57,480 --> 00:27:04,420
+الجزء الأول من النظريةأثبتنا اللي بدنا يا انه اللي
+
+316
+00:27:04,420 --> 00:27:08,860
+هو هذه العلاقة صحيحة لأننا نفترض انه suppose that
+
+317
+00:27:08,860 --> 00:27:12,380
+star holds اللي هي star ليه هذه نفترض ان لكل
+
+318
+00:27:12,380 --> 00:27:15,380
+epsilon أكبر من 0 there exists B of epsilon such
+
+319
+00:27:15,380 --> 00:27:18,680
+that U B Epsilon و F نقصها دي أصغر من Epsilon و
+
+320
+00:27:18,680 --> 00:27:24,280
+بدنا نصل من خلالها لإيش لأن ال F is integrable
+
+321
+00:27:24,280 --> 00:27:25,760
+فنشوف
+
+322
+00:27:40,480 --> 00:27:47,440
+البرهام بسيط لو طلعنا عليه مباشرة على اللوح الآن
+
+323
+00:27:47,440 --> 00:27:52,140
+بدنا نفترض أن هذه تتحقق اللي هو نفترض أنه لكل ي
+
+324
+00:27:52,140 --> 00:27:57,020
+أكبر من 0 يوجد بي إبسلون بحيث أن هذا اللي هي تتحقق
+
+325
+00:27:57,020 --> 00:28:01,620
+علشان أصل بدي أصلكم في النهاية أن L of F هي إيش
+
+326
+00:28:01,620 --> 00:28:05,810
+بتساوي U of F شوف كيف ده صلهااللي هبدأ أقوللك اللي
+
+327
+00:28:05,810 --> 00:28:08,650
+هو نفترض انه زي ما قلنا انه star holds اللي حكينا
+
+328
+00:28:08,650 --> 00:28:12,990
+عنها لأن for any partition B هيكون ال L B of F
+
+329
+00:28:12,990 --> 00:28:17,970
+أصغر أساوي L of F و ال U of F أصغر أساوي مين ال U
+
+330
+00:28:17,970 --> 00:28:25,970
+B of F واضح؟ إذا أصار عيدي الآن L يمكن أوضحلكم على
+
+331
+00:28:25,970 --> 00:28:35,790
+اللوح L of B of F ده ال B أي partition أصغرمظبوط
+
+332
+00:28:35,790 --> 00:28:40,910
+أو يساوي ال L of A H of F لانه ال L of F يا جماعة
+
+333
+00:28:40,910 --> 00:28:47,330
+هو ال supremum اللي هنا و ال U بيقف
+
+334
+00:28:47,330 --> 00:28:53,130
+أكبر أو يساوي ال U of F لانه ال U of F H يا جماعة
+
+335
+00:28:53,130 --> 00:28:57,950
+هو عبارة عن مين عبارة عن ال infimumأترحولي الجهة
+
+336
+00:28:57,950 --> 00:29:01,410
+تانية هذا طبعا لكل مين لكل ال partitions اللي في
+
+337
+00:29:01,410 --> 00:29:06,750
+الدنيا من ضمنهم ال P Epsilon اللي احنا ماعطينايا
+
+338
+00:29:06,750 --> 00:29:12,250
+في ال .. اللي هو نص النظرية اذا بسيري عندى لأن لو
+
+339
+00:29:12,250 --> 00:29:19,990
+اجيب طرحة ال U F ناقص L F ال U F ناقص L Fهيصير ايش
+
+340
+00:29:19,990 --> 00:29:23,850
+ماله يا جماعة؟ يعني طريقة رحلة من هذه بيصير أصغر
+
+341
+00:29:23,850 --> 00:29:28,510
+أو يساوي لإن هذه بتضربها في ناقص و هذه ناقص و
+
+342
+00:29:28,510 --> 00:29:32,750
+بتنقلب زي ما عملنا قبل و شوية بيصير أصغر أو يساوي
+
+343
+00:29:32,750 --> 00:29:43,640
+U P of Fنقص ال B of F هذا الكلام صحيح لإيش لكل
+
+344
+00:29:43,640 --> 00:29:48,200
+partition في الدنيا من ضمنها المين ال partition
+
+345
+00:29:48,200 --> 00:29:52,580
+المواطع لنا يعني حيصير عند هذا ينطبق برضه على ال
+
+346
+00:29:52,580 --> 00:29:57,500
+بي إبسلون إذا صار هذا أصغر يسوي بي إبسلون نقص ال
+
+347
+00:29:57,500 --> 00:30:01,740
+بي إبسلون of F طيب هم يعطيين إن هذا المقدار إيش
+
+348
+00:30:01,740 --> 00:30:06,260
+ماله أصغر من إبسلون لأي إبسلون في الدنياهو أنا
+
+349
+00:30:06,260 --> 00:30:10,580
+بعرف أن هذا المقدار نفسه أكبر أو يساوي إيش سفر صار
+
+350
+00:30:10,580 --> 00:30:17,720
+عندى الآن ال U of F ناقص ال L of F دايما أصغر من
+
+351
+00:30:17,720 --> 00:30:23,420
+إبسلون و أكبر أو يساوي سفر لكل إبسلون أكبر من سفر
+
+352
+00:30:23,420 --> 00:30:28,260
+إذا على طول من نظرية في تحليل واحد هيكون هذا اللى
+
+353
+00:30:28,260 --> 00:30:36,250
+عندى إذا U of Fنقص L of F بيساوي سفر إذا U of F
+
+354
+00:30:36,250 --> 00:30:44,850
+بيساوي L of F وهذا يعني F is a Riemann Integral هو
+
+355
+00:30:44,850 --> 00:30:52,110
+المطلوب بيكون احنا هيك أثبتنا اللي هوIntegrable
+
+356
+00:30:52,110 --> 00:30:58,350
+criterion أو اللي هو طريقة لتحديد اللي هو ال
+
+357
+00:30:58,350 --> 00:31:02,270
+function is integrable أو لا غير اللي هو طريقة
+
+358
+00:31:02,270 --> 00:31:07,110
+التعريف الآن في عند كورولاري بعدها كورولاري
+
+359
+00:31:07,110 --> 00:31:11,530
+الكورولاري
+
+360
+00:31:11,530 --> 00:31:16,230
+هي تقول كما يليه الآن بدنا نترجم الحديث بدل ما كان
+
+361
+00:31:16,230 --> 00:31:22,710
+بإبسلون نحكي عن مين؟ عن اللي هوsequence of
+
+362
+00:31:22,710 --> 00:31:29,870
+partitions طبعاً هو هذا معهود التحويل في نظريات
+
+363
+00:31:29,870 --> 00:31:34,590
+مشابهة في حتى في كورسات أخرى خلّينا نشوف عندنا
+
+364
+00:31:34,590 --> 00:31:38,790
+اللي هو النظرية ايش ب .. او الكورلري ايش بتقول
+
+365
+00:31:38,790 --> 00:31:43,350
+بتقول let I بساوة A و B and let F من I لR be a
+
+366
+00:31:43,350 --> 00:31:48,600
+bounded functionلأن لو فرضنا بي أن أن element none
+
+367
+00:31:48,600 --> 00:31:52,300
+is a sequence of partitions of I بحيث أن ال limit
+
+368
+00:31:52,300 --> 00:31:55,920
+هذا بيساوي سفر then f is integrable and ال limit
+
+369
+00:31:55,920 --> 00:31:58,040
+لل integration بيساوي ال integration بيساوي ال
+
+370
+00:31:58,040 --> 00:32:04,480
+limit ال أسف ال limit لل lower p and f بيساوي ال
+
+371
+00:32:04,480 --> 00:32:07,020
+limit لل upper p and f اللي هو بيساوي قيمة ال
+
+372
+00:32:07,020 --> 00:32:13,000
+integration حتى ال converse جماعة اللي هواللي قبل
+
+373
+00:32:13,000 --> 00:32:16,820
+بشوية اللي كان .. اللي هي كانت F أندولي F لأن لو
+
+374
+00:32:16,820 --> 00:32:21,520
+كانت F is integrable أكيد هلاقي sequence من
+
+375
+00:32:21,520 --> 00:32:25,380
+partitions بحيث أنه ال limit اللي حاصل الطرح بساوي
+
+376
+00:32:25,380 --> 00:32:29,760
+سفر اللي هو البرهان مشابه لإي اللي حكيناه فيه اللي
+
+377
+00:32:29,760 --> 00:32:34,500
+هو برهان إيجاد الـ B epsilon ولكن هنا بنجداللي هو
+
+378
+00:32:34,500 --> 00:32:37,380
+الـ Epsilon بساوية واحدة لان فبنلاقي اللي هو ال
+
+379
+00:32:37,380 --> 00:32:42,380
+sequence هذه في ال corollary .. في من مشابه .. شيء
+
+380
+00:32:42,380 --> 00:32:45,900
+مشابه في برهان النظرية الأولى اللي قبل بشوية و
+
+381
+00:32:45,900 --> 00:32:49,360
+ياريت تجربوها عندكم خلينا ناخد اللي .. اللي موجود
+
+382
+00:32:49,360 --> 00:32:54,160
+حاليا اللي هو الاتجاه هذا ان لو لجينا sequence of
+
+383
+00:32:54,160 --> 00:32:59,430
+partitionsوكان ال limit لل U P N و F نقص ال P N و
+
+384
+00:32:59,430 --> 00:33:02,870
+F بساوة سفر إذا هتكون F is integrable و هتكون ال
+
+385
+00:33:02,870 --> 00:33:07,990
+limit للأولى بساوة limit للثانية بساوة قيمة ال
+
+386
+00:33:07,990 --> 00:33:08,970
+integration
+
+387
+00:33:12,900 --> 00:33:17,500
+عند ما أعطيني limit هذا إيش بساوي؟ سفر خلّينا ندخل
+
+388
+00:33:17,500 --> 00:33:20,580
+على التعريف مباشرة تعريف ال limit بتعرفوا تعريف ال
+
+389
+00:33:20,580 --> 00:33:23,140
+limit يا شباب؟ اللي هو لكل إبسلون أكبر من سفر
+
+390
+00:33:23,140 --> 00:33:26,160
+there exist k such that لكل أن أكبر سوى k بيصير
+
+391
+00:33:26,160 --> 00:33:33,200
+هذا ناقص هذا أصغر من مين؟ من اللي هو إبسلون وهذا
+
+392
+00:33:33,200 --> 00:33:35,980
+على طول يعطينا as integral خلّيش تشوف أيش اللي
+
+393
+00:33:35,980 --> 00:33:43,000
+بيقوله لأن since عند ما أعطيني limitU P N و F ناقص
+
+394
+00:33:43,000 --> 00:33:51,240
+ال P N و F as N goes to infinity بساوة سفر، مظبوط؟
+
+395
+00:33:51,240 --> 00:33:56,220
+هي كل ما أعطينيها نجي للتعريف، إذا تعريف ال
+
+396
+00:33:56,220 --> 00:33:58,880
+sequence عادية for every epsilon أكبر من سفر there
+
+397
+00:33:58,880 --> 00:34:02,680
+exists K element in N such that for every N أكبر
+
+398
+00:34:02,680 --> 00:34:13,090
+سوا K اللي هو عندي ال U P N و F ناقصبن و F أصغر من
+
+399
+00:34:13,090 --> 00:34:19,010
+إبسلون إذا مش لجينا بارتشن واحد لجينا بارتشن بك و
+
+400
+00:34:19,010 --> 00:34:22,710
+بك زاد واحد و بك زاد اتنين و بك زاد تلاتة كلهم
+
+401
+00:34:22,710 --> 00:34:29,110
+بسبب إن ال UBK أو ال UBK زاد واحد أو الاخره ناقص
+
+402
+00:34:29,110 --> 00:34:32,590
+القللها أصغر من 100 من إبسلون إذا ال criterion
+
+403
+00:34:32,590 --> 00:34:36,550
+اللي في الكورولاريت حققت إذا صارت عندي هذه إذا F
+
+404
+00:34:36,550 --> 00:34:41,830
+is integrableيعني مش بي إبسلون واحد اللي جينا لأ
+
+405
+00:34:41,830 --> 00:34:46,550
+من عند بك وطالع كل ال partitions هذه اللي هي بك و
+
+406
+00:34:46,550 --> 00:34:49,950
+بك زائد واحد و بك زائد اتنين بتعمل عمل ال بي
+
+407
+00:34:49,950 --> 00:34:53,510
+إبسلون اللي في وين في النظرية إذا ال F أشمالها
+
+408
+00:34:53,510 --> 00:34:58,750
+صارت ال F عبارة عن Integrable من النظرية السابقة
+
+409
+00:34:58,750 --> 00:35:05,200
+الآن الدور دل أن نثبت مين أن ال limitلأ اللي هو
+
+410
+00:35:05,200 --> 00:35:09,260
+هذا المقدار هو limit لهذا المقدار بساوي إيش اللي
+
+411
+00:35:09,260 --> 00:35:12,540
+جوا طبعا هو لو كانت يا جماعة limit هذا نقص هذا سفر
+
+412
+00:35:12,540 --> 00:35:15,920
+مش معناته ال limit الأول و limit الثاني exist هاي
+
+413
+00:35:15,920 --> 00:35:20,610
+مثلا هذه لو كانت هذه un تربيع وهذه unUnterm .. آسف
+
+414
+00:35:20,610 --> 00:35:24,610
+Unتربيع و Unتربيع أو Un و Un limit Un نقص Un على
+
+415
+00:35:24,610 --> 00:35:27,850
+طول سفر لكن لا limit الأولى عدد ولا limit الثاني
+
+416
+00:35:27,850 --> 00:35:33,270
+عدد اتنين اتين بروحين إلى مالا نهاية فالان لكن في
+
+417
+00:35:33,270 --> 00:35:37,390
+هذه الحالة نظرا للمعطيات اللي موجودة و اللي هي
+
+418
+00:35:37,390 --> 00:35:40,930
+طبيعي اللي هو اللي بنحكي فيه NuF is bounded و ببو
+
+419
+00:35:40,930 --> 00:35:45,550
+و الاخره هو استخدام السابق هيطلع عندي فعلا ال
+
+420
+00:35:45,550 --> 00:35:50,840
+limitللأل بيساوي limit للأبر بيساوي قيمة ال
+
+421
+00:35:50,840 --> 00:36:08,200
+integration ماشي اطلعوا يا جماعة عندي الآن خليني
+
+422
+00:36:08,200 --> 00:36:15,240
+أطلع .. نحط البرهان أمامنابدي الآن خلصت اللي هو F
+
+423
+00:36:15,240 --> 00:36:21,500
+is integrable بدي أستخدم زي جابل بشوية بالظبط اللي
+
+424
+00:36:21,500 --> 00:36:28,570
+هو تعريف ال L of F و U of F و إيش تعريفهاذا ما
+
+425
+00:36:28,570 --> 00:36:32,250
+بديش اعيد اللي حكيته قبل بشوية الان بما ان ال L of
+
+426
+00:36:32,250 --> 00:36:42,370
+F عبارة عن اللي هي supremum لل L B of F اذا لكل
+
+427
+00:36:42,370 --> 00:36:46,610
+ي ساوي واحدة لان there exists B N partition of I
+
+428
+00:36:46,610 --> 00:36:50,390
+such that L of F نقص واحدة لان اصغر من مين من L B
+
+429
+00:36:50,390 --> 00:36:54,070
+N of Fزي ما قلت قبل شوية، لو الـ Supremum طرحنا
+
+430
+00:36:54,070 --> 00:36:57,270
+اللي هي الـ Least Upper Bound، طرحنا منه أي عدد، 1
+
+431
+00:36:57,270 --> 00:37:01,330
+على N، إذا هلاجي، هيبطل إيش ماله Upper Bound، إيش
+
+432
+00:37:01,330 --> 00:37:04,190
+معناه يبطل Upper Bound؟ هيلاجي واحد من المجموعة
+
+433
+00:37:04,190 --> 00:37:08,230
+أكبر منه، وهذا فعلا اللي لاجينا BN، بحيث أن الـ BN
+
+434
+00:37:08,230 --> 00:37:13,910
+و F أكبر من الـ F ناقص واحدة لأنالان من هذا .. من
+
+435
+00:37:13,910 --> 00:37:16,670
+.. من .. من ال .. ال .. ال .. ناخد هذا على الطرف
+
+436
+00:37:16,670 --> 00:37:19,970
+الثاني على الطرف هنا و ناخد هذا على الطرف هذا بصير
+
+437
+00:37:19,970 --> 00:37:23,890
+عندي L of F نقص Lb of F أصغر من واحدة الآن و أنا
+
+438
+00:37:23,890 --> 00:37:28,670
+بعرف أن هذا دايما أكبر يساوي هذا لأن هذا ال
+
+439
+00:37:28,670 --> 00:37:32,410
+supremum منهم إذا انا هيكون أكبر يساوي سفر الان
+
+440
+00:37:32,410 --> 00:37:38,440
+خدوا ال limit للجهتينas n goes to infinity بيصير
+
+441
+00:37:38,440 --> 00:37:43,160
+عندي هذا اللي هي limit
+
+442
+00:37:43,160 --> 00:37:48,060
+لL بيقل نوف أف حيثالوي لL أوف أف لإنه حيثير ال
+
+443
+00:37:48,060 --> 00:37:53,720
+limit هذا أيش بيساوي بيساوي سفر واضحالان يا جماعة
+
+444
+00:37:53,720 --> 00:37:58,400
+عندي ال L of F ناقص ال P L of F أصغر من واحدة لأنه
+
+445
+00:37:58,400 --> 00:38:03,660
+أكبر سوى سفر لان معله الان هذا لكل إبسلون اللي هو
+
+446
+00:38:03,660 --> 00:38:07,300
+واحدة لان لجينا partition يعني للإبسلون بيساوي
+
+447
+00:38:07,300 --> 00:38:09,360
+واحد لجينا بيه واحد للإبسلون بيساوي اتنين بيه
+
+448
+00:38:09,360 --> 00:38:12,160
+اتنين للإبسلون تلاتة بيه تلاتة إذا صار عندي
+
+449
+00:38:12,160 --> 00:38:16,580
+sequence of اللي هي إياش partitionsالأن طلع عندي
+
+450
+00:38:16,580 --> 00:38:20,460
+دائما دائما دائما L of F نقص ال PN of F أصغر من
+
+451
+00:38:20,460 --> 00:38:25,440
+واحد على N لكل N الأن as N goes to infinity هذا ال
+
+452
+00:38:25,440 --> 00:38:30,620
+limit هيصير سفر وهذا سفر إذا هيصير limit هذا as N
+
+453
+00:38:30,620 --> 00:38:33,200
+goes to infinity بساوي سفر لكن ال L of F أصلا
+
+454
+00:38:33,200 --> 00:38:37,860
+independent of N إذا هيصير limit PN of F as N goes
+
+455
+00:38:37,860 --> 00:38:43,720
+to infinity بساوي L of F لأن المواضحات لوش يا شباب
+
+456
+00:38:43,720 --> 00:38:52,360
+limitL of F ناقص L B N و F هيساوي 0 as N goes to
+
+457
+00:38:52,360 --> 00:38:56,920
+infinity و هذا المقدار عبارة عن مقدار ثابت
+
+458
+00:38:56,920 --> 00:39:04,740
+independent of N إذا هيصير عندي limit L B N و F
+
+459
+00:39:04,740 --> 00:39:10,540
+بتساوي limit L of F ناقص
+
+460
+00:39:17,660 --> 00:39:27,170
+L of P L و F زائد L of Fمظبوط؟ طيب هذا الأن
+
+461
+00:39:27,170 --> 00:39:33,590
+المقدار معروف أنه بساوي 0 و هذا ثابت إذا بقدر أوزع
+
+462
+00:39:33,590 --> 00:39:37,210
+ال limit على الجهتيين و أنا مرتاح إذا بساوي limit
+
+463
+00:39:37,210 --> 00:39:42,950
+الأول اللي هو 0 زائد limit الثاني نفسه لأنه ثابت
+
+464
+00:39:42,950 --> 00:39:46,450
+إذا صار عندي limit L P N of F as N goes to
+
+465
+00:39:46,450 --> 00:39:54,710
+infinity بساوي L of F كما هو حكينا عنه حقامن جهة
+
+466
+00:39:54,710 --> 00:39:58,650
+أخرى limit الـU ناقص limit الـL المعطيلة هي بيساوي
+
+467
+00:39:58,650 --> 00:40:03,490
+0 إذا صار عندى سهل أن أوجد mean برضه limit الـU
+
+468
+00:40:03,490 --> 00:40:08,990
+اللي هو limit الـU إيش هيساوي ناقص اللي هو .. ال
+
+469
+00:40:08,990 --> 00:40:12,670
+.. ال .. هيساوي limit الـL of F يساوي الـU of F
+
+470
+00:40:12,670 --> 00:40:17,270
+يساوي ال integration أكثر توضيحاً and بيهاجدى
+
+471
+00:40:17,270 --> 00:40:22,420
+أعتقد أنه واضح لكن خلّينا نوضحه بشكل أكبرعشان
+
+472
+00:40:22,420 --> 00:40:34,100
+مايضلش مشي عندى limit U P N of F ايش هيساوي هيساوي
+
+473
+00:40:34,100 --> 00:40:44,560
+اللي هو limit U
+
+474
+00:40:44,560 --> 00:40:49,740
+P N of F
+
+475
+00:40:52,490 --> 00:40:58,090
+limit U P N و F ناقص
+
+476
+00:40:58,090 --> 00:41:07,810
+L of P N of F زائد L P N of F واضحة يا شباب أه؟
+
+477
+00:41:07,810 --> 00:41:12,690
+الآن هذا مضمون أنه موجود و سفر وهذا مضمون و
+
+478
+00:41:12,690 --> 00:41:18,350
+أثبتناه إيش بيساوي L of Fإذا إيش صار بيساوي؟
+
+479
+00:41:18,350 --> 00:41:21,810
+بيساوي اللي هو هذا صفر إذا صار بيساوي أقل و فأف
+
+480
+00:41:21,810 --> 00:41:25,510
+إذا صار هذا برضه بيساوي أقل و فأف لكن أنا مثبت قبل
+
+481
+00:41:25,510 --> 00:41:29,610
+بشوية أن ال F is integrable يعني ال U في F إيش
+
+482
+00:41:29,610 --> 00:41:34,550
+هتساوي؟ اللي أقل و فأف وهذه أثبتناها إيش بتساوي؟
+
+483
+00:41:34,550 --> 00:41:39,350
+limit U P N و F وهذه نفسها أثبتناها قبل بشوية إيش
+
+484
+00:41:39,350 --> 00:41:46,070
+بتساوي؟ limit L P N و Fوهو المطلوب طبعاً مدام
+
+485
+00:41:46,070 --> 00:41:49,370
+integration هذه وهذه هي عبارة عن ال integration
+
+486
+00:41:49,370 --> 00:41:55,410
+على الفترة اللي بنحكي عنها الإنقلا بيه لقيمة ال F
+
+487
+00:41:55,410 --> 00:42:01,710
+لأن صار عندي كل القيام هذه متساوية وصار عندي إيجاد
+
+488
+00:42:01,710 --> 00:42:07,610
+ال limit لل U P N و F أو limit لل L P N و F يكفي
+
+489
+00:42:07,610 --> 00:42:11,330
+أنه نوجد فيه قيمة ال integration بعد ما أثبتناه أو
+
+490
+00:42:11,330 --> 00:42:16,260
+تحت الظروف اللي هي في الكورو الأخرىلأ نيجي بدنا
+
+491
+00:42:16,260 --> 00:42:22,660
+نبرهن اللي هو المثال اللي برهناه المرة الماضية
+
+492
+00:42:22,660 --> 00:42:27,660
+بالتعريف بدنا نبرهنه بواسطة اللي هي ال corollary
+
+493
+00:42:27,660 --> 00:42:33,060
+اللي عندنا بدنا نبرهن اللي هو اللي هي نثبت أنه
+
+494
+00:42:36,170 --> 00:42:42,070
+نثبت أن F of X بساوة X يا شباب عبارة عن Integrable
+
+495
+00:42:42,070 --> 00:42:45,770
+أو اللي سمناها G of X بساوة X is Integrable على
+
+496
+00:42:45,770 --> 00:42:48,830
+الفترة Zero واحد المرة الماضية أثبتناها كيف
+
+497
+00:42:48,830 --> 00:42:54,230
+أثبتناها زي ما أنتم متذكرين بواسط التعريفمظبوط؟
+
+498
+00:42:54,230 --> 00:42:58,610
+طيب جيبنا اللي هو ال upper sum و ال lower sum و
+
+499
+00:42:58,610 --> 00:43:02,930
+بعدين جيبنا ال upper integral و ال lower integral
+
+500
+00:43:02,930 --> 00:43:04,750
+و أثبتنا ان ال upper integral بساوي ال lower
+
+501
+00:43:04,750 --> 00:43:08,760
+integral و خلصناهالآن بدنا نثبتها بطريقتنا اللي هي
+
+502
+00:43:08,760 --> 00:43:12,540
+على الكورولر اللي جابل بشوية عند g of x ساوة x على
+
+503
+00:43:12,540 --> 00:43:16,600
+الفترة 0 بواحد هي المطلوب اثباتها show that g is
+
+504
+00:43:16,600 --> 00:43:20,940
+integrable على هذه الفترة الآن بديش اعيد اللي
+
+505
+00:43:20,940 --> 00:43:23,850
+حكيته المرة الماضية اللي حكيناه المرة الماضيةمن
+
+506
+00:43:23,850 --> 00:43:26,910
+الـ Example اللي أثبتنا فيه انها Integra بالبواسطة
+
+507
+00:43:26,910 --> 00:43:31,470
+اللي هي التعريف أخدنا P N اللي هو Zero واحد على N
+
+508
+00:43:31,470 --> 00:43:34,910
+واثنين على N وان نقص واحد على N لعند الواحد أخدنا
+
+509
+00:43:34,910 --> 00:43:39,290
+اللي هو عبارة عن Any Partition اللي هو بالطريقة
+
+510
+00:43:39,290 --> 00:43:43,710
+اللي أمامي يعني حسب N بيصير بختلف ال Partition أما
+
+511
+00:43:43,710 --> 00:43:46,730
+إيش فكرة ال Partition زي ما قلنا المرة الفائتة من
+
+512
+00:43:46,730 --> 00:43:51,350
+0 لعند 1 جزأناها إلى أجزاء متساوية إلى N من
+
+513
+00:43:51,350 --> 00:43:55,350
+الأجزاء المتساويةصار طول كل sub integral عبارة عن
+
+514
+00:43:55,350 --> 00:44:00,850
+إيهاش عبارة عن واحد على N وأوجدنا في حينه اللي هو
+
+515
+00:44:00,850 --> 00:44:08,470
+ال U P N O G ولاجناها بتساوي إيه ده بتتذكروا نص في
+
+516
+00:44:08,470 --> 00:44:13,710
+واحد زائد واحد على N وأوجدنا برضه ال P N O G
+
+517
+00:44:13,710 --> 00:44:18,910
+ولاجناها عبارة عن نص في واحد ناقص واحد على N كما
+
+518
+00:44:18,910 --> 00:44:26,960
+أذكر ماشي الحال فعلا طيبالان صار عندى الـ PN في
+
+519
+00:44:26,960 --> 00:44:29,780
+الواقع عبارة عن sequence of partitions
+
+520
+00:44:35,190 --> 00:44:39,110
+هذه صارت sequence of partitions بواحد بتعود عن انا
+
+521
+00:44:39,110 --> 00:44:42,750
+بواحد باتنين كده بتلاتة كده باربعة كده وفي كل
+
+522
+00:44:42,750 --> 00:44:47,270
+الأحوال الـ U, B, N و G انجروا لهيها و الـ L, B, N
+
+523
+00:44:47,270 --> 00:44:51,490
+و G بساوي إيش المقدار الأمامي الآن عشان أثبت إنها
+
+524
+00:44:51,490 --> 00:44:56,590
+integrable يكفي من النظرية الكورولاري إن قاعد أقول
+
+525
+00:44:56,590 --> 00:45:03,290
+طب خلينا نشوف limit الـ U, B, N و G ناقص الـ L, B,
+
+526
+00:45:03,450 --> 00:45:09,840
+N و Gas n goes to infinity بساوة إيه يا عشان بساوة
+
+527
+00:45:09,840 --> 00:45:16,360
+سفر نشوفها صح ولا لأ طبعا أكيد مع حسبة بسيطة
+
+528
+00:45:16,360 --> 00:45:24,580
+احسبليها بساوة limit نصفي واحد زياد واحدة لان زياد
+
+529
+00:45:24,580 --> 00:45:28,600
+نص في واحد ناقص واحدة لان as n goes to infinity
+
+530
+00:45:28,600 --> 00:45:32,260
+هذه سفر وهذه سفر وهذه نص وهذه نص وهذه نص وهذه نص
+
+531
+00:45:32,260 --> 00:45:32,340
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+532
+00:45:32,340 --> 00:45:32,920
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+533
+00:45:32,920 --> 00:45:38,340
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+534
+00:45:38,340 --> 00:45:45,580
+وهذهاللي هو y ساوي نص نقص النص y ساوي سفر واضح أهي
+
+535
+00:45:45,580 --> 00:45:50,120
+عندي اللي هو هاد سفر و هاد سفر و نص نقص النص y
+
+536
+00:45:50,120 --> 00:45:54,920
+ساوي سفر مادام سفر إذا إيه شمالها إذا the function
+
+537
+00:45:54,920 --> 00:46:02,720
+d of x ساوي x is integrable on zero واحد هذا by
+
+538
+00:46:02,720 --> 00:46:06,560
+mean by ال corollary اللي قبل بشوية
+
+539
+00:46:15,600 --> 00:46:19,420
+الان بقيمة ال integration ايش هيكون قيمة ال limit
+
+540
+00:46:19,420 --> 00:46:24,280
+limit u, b, n, g هيقدر الغرض و limit l of n of g
+
+541
+00:46:24,280 --> 00:46:29,320
+هيقدر الغرض ليش لأن limit هذا اصلا هيطلع لك limit
+
+542
+00:46:29,320 --> 00:46:35,710
+ال u, b, n و gهو عبارة عن قيمة integration من A ل
+
+543
+00:46:35,710 --> 00:46:42,370
+B حسب النظرية الـ Corollary و يساوي limit 2.5 في 1
+
+544
+00:46:42,370 --> 00:46:48,600
+زائد 1 ل N و يساوي هذا تروح ل 0و يساوى نص و أيضا
+
+545
+00:46:48,600 --> 00:46:54,080
+لو جربت حسب تاب limit ال b, n و g طبعا هيطلع نفس
+
+546
+00:46:54,080 --> 00:46:58,200
+الجواب و إلا إن كان هناك مشكلة لدينا limit نص في
+
+547
+00:46:58,200 --> 00:47:02,340
+واحد ناقص واحد على n و يساوى برضه جدياش نص إذن
+
+548
+00:47:02,340 --> 00:47:06,700
+قيمة ال integration بساوة نص إذن هذه طريقة أخرى
+
+549
+00:47:06,700 --> 00:47:11,780
+لحساب اللي هو أو لإثبات أن g of x بساوة x is
+
+550
+00:47:11,780 --> 00:47:17,560
+integrableالآن بدنا ندخل على أمر آخر الأمر هو في
+
+551
+00:47:17,560 --> 00:47:24,500
+الواقع يا شباب أنه بدنا نشوف إيش في من عائلات
+
+552
+00:47:24,500 --> 00:47:28,400
+الدوال عائلات الدوال إنها تكون Integrable الآن
+
+553
+00:47:28,400 --> 00:47:31,720
+بدنا نيجي نحوش الدوال الـ Integrable إحنا عرفنا بس
+
+554
+00:47:31,720 --> 00:47:36,500
+ده لأ D of X سو X is Integrable و هنلاقي زيها لكن
+
+555
+00:47:36,500 --> 00:47:40,960
+الآن بدنا نيجي نحكي عن دوال اللي هو عائلات من
+
+556
+00:47:40,960 --> 00:47:45,480
+الدوالأول عائلة من العائلات المهمة اللي هي الـ
+
+557
+00:47:45,480 --> 00:47:48,960
+monotone functions يعني الدوال اللي بتكون يا
+
+558
+00:47:48,960 --> 00:47:52,360
+increasing على كل الفترة يا decreasing على كل
+
+559
+00:47:52,360 --> 00:47:59,320
+الفترة بقولكم هذا على اللي هي F من F ال function
+
+560
+00:47:59,320 --> 00:48:04,000
+لو كانت bounded لو كانت monotone على اللي هي
+
+561
+00:48:04,000 --> 00:48:09,660
+closed bounded interval I على طول integrable إذن
+
+562
+00:48:09,660 --> 00:48:16,700
+عيلة كبيرةعيلة الدوال اللي بتكون يا increasing يا
+
+563
+00:48:16,700 --> 00:48:22,380
+decreasing على كل الفترة a و b هذه مضمون انها تكون
+
+564
+00:48:22,380 --> 00:48:26,980
+الدوال ايه شمالها عبارة عن integrable functions
+
+565
+00:48:26,980 --> 00:48:35,090
+اذا اول اعلان الان اللي هو any monotone functionon
+
+566
+00:48:35,090 --> 00:48:40,850
+a closed bounded interval is integrable وهذا اللي
+
+567
+00:48:40,850 --> 00:48:47,090
+هو عنواننا integrability of monotone functionsلت I
+
+568
+00:48:47,090 --> 00:48:51,650
+بتساوي A و B و لت F من I لR بيكون مونوتون فانكشن
+
+569
+00:48:51,650 --> 00:48:57,370
+on I ثم F أشمالها is integrable on I نفترض أن F
+
+570
+00:48:57,370 --> 00:49:03,110
+انتقل هو مونوتور نصل إليها integrable نفترض أن F
+
+571
+00:49:03,110 --> 00:49:09,440
+مثلا increasingونصل إنها integrable وsimilarly
+
+572
+00:49:09,440 --> 00:49:14,560
+وفعلا similarly لو كانت f is decreasing هتكون برضه
+
+573
+00:49:14,560 --> 00:49:21,500
+is integrable خلّينا مع بعض شباب نفترض suppose
+
+574
+00:49:21,500 --> 00:49:31,240
+that f is increasing يعني الدالة على الفترة هي a
+
+575
+00:49:31,240 --> 00:49:37,760
+وb مثلاوالدالة هتكون أشمالها تزايدية يا هيك يا هيك
+
+576
+00:49:37,760 --> 00:49:41,620
+طبعا حسب مش مشكلة بتفرج أشمالها ماشي الحال اللي هي
+
+577
+00:49:41,620 --> 00:49:47,060
+الدالة أشمالها is increasing is increasing بتدخل
+
+578
+00:49:47,060 --> 00:49:53,500
+الآن بي أن اللي هو عبارة عن any partition اللي هو
+
+579
+00:49:55,970 --> 00:50:00,970
+بس بدي أجزه زي منهج اللي عملته معه اللي هو f of x
+
+580
+00:50:00,970 --> 00:50:05,370
+بساوة x اللي قبله شوية بدي أجزه إلى أجزاء متساوية
+
+581
+00:50:05,370 --> 00:50:11,130
+يعني بدي أجز الفقرة a و b إلى أجزاء متساوية الأولى
+
+582
+00:50:11,130 --> 00:50:15,730
+بدي أسميها x note اللي بعدها x1 اللي بعدها x2 لما
+
+583
+00:50:15,730 --> 00:50:21,780
+أصل لآخر 11 أسميها xnونكون طول كل واحدة متساوية
+
+584
+00:50:21,780 --> 00:50:26,340
+للثانية إذا الان إذا بدي أجزئها إلى N إلى N من ال
+
+585
+00:50:26,340 --> 00:50:30,240
+sub intervals بيصير طول كل فترة عبارة عن B minus A
+
+586
+00:50:30,240 --> 00:50:35,080
+على مين؟ على N طول الفترة على عدد الفترات اللي
+
+587
+00:50:35,080 --> 00:50:39,400
+بتديهاأنا بدّي الفترات إيش عددها؟ N فبيصير عند بي
+
+588
+00:50:39,400 --> 00:50:43,320
+ماينس إيه على مين على N هذا طول الفترة يعني الفترة
+
+589
+00:50:43,320 --> 00:50:47,740
+النموذجية لأي K X K minus X K minus واحد هيتمكن
+
+590
+00:50:47,740 --> 00:50:52,560
+طولها عبارة عن بي ماينس إيه على N for every K كيه
+
+591
+00:50:52,560 --> 00:50:56,000
+طبعا إيه اشمالها؟ يا ما بيساوي Zero يا واحد، لعند
+
+592
+00:50:56,000 --> 00:51:01,100
+ماصل، لعند مين؟ لعند N أو K بيساوي واحد لعند N
+
+593
+00:51:04,810 --> 00:51:10,230
+xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+594
+00:51:10,230 --> 00:51:14,230
+-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+595
+00:51:14,230 --> 00:51:14,590
+-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+596
+00:51:14,590 --> 00:51:17,070
+-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+597
+00:51:17,070 --> 00:51:25,770
+-xn-n-xn-n-xn-n-xn-n-xn-nلأ لو جينا هذا increasing
+
+598
+00:51:25,770 --> 00:51:30,170
+مدام increasing إذن الـ Mk اللي هي عبارة عن ال
+
+599
+00:51:30,170 --> 00:51:35,510
+supremum اللي هنا supremum لهذه Mk ال supremum
+
+600
+00:51:35,510 --> 00:51:39,910
+اللي هنا هيكون على أخر واحدة لأن أجملها ده لا is
+
+601
+00:51:39,910 --> 00:51:45,510
+increasing إذن هيكون ال Mk هو F of XK رسم عنها
+
+602
+00:51:45,990 --> 00:51:51,210
+بساوة F of X K طيب ال M K Small أكيد هيقولنا كلكم
+
+603
+00:51:51,210 --> 00:51:54,010
+هتقوله بتساوة ال M في معمع الفترة مدام ال M في
+
+604
+00:51:54,010 --> 00:51:57,370
+معمع الفترة إذا أول واحدة فيهم إذا هو F of X K
+
+605
+00:51:57,370 --> 00:52:04,530
+minus واحد، مظبوط شباب؟ طيب، مظبوط هذه اللي هي
+
+606
+00:52:04,530 --> 00:52:10,750
+مالها الدالة اللي بتكون monotone عندي،
+
+607
+00:52:10,750 --> 00:52:21,840
+احسبلي الآن ال U P N و Fنقص الـ BNOF إيش هيساوي؟
+
+608
+00:52:21,840 --> 00:52:26,840
+جهزة الأمور بساوي الـ summation لمين؟ للـ MK
+
+609
+00:52:26,840 --> 00:52:36,190
+Capital في XK نقص XK minus واحدك من عند 1 لعند n
+
+610
+00:52:36,190 --> 00:52:44,370
+نقص summation mk في xk minus xk minus 1 k من عند 1
+
+611
+00:52:44,370 --> 00:52:49,440
+لعند n ويساويالـ Mk وجدناها، اللي هي عبارة عن مين؟
+
+612
+00:52:49,440 --> 00:52:55,420
+F of Xk ويساوي الـ summation للـ F of Xk في مين
+
+613
+00:52:55,420 --> 00:53:00,200
+مضروبة؟ في هذه، هذه كده طولها ثابت، ما احنا هيك
+
+614
+00:53:00,200 --> 00:53:02,540
+على هذا الأساس اذا اخترنا الـ sequence of
+
+615
+00:53:02,540 --> 00:53:05,320
+partitions اللي عندنا، اللي هو طوالث في
+
+616
+00:53:05,320 --> 00:53:09,580
+subintervals ثابتة، كل واحد اسمه يشمله B minus A
+
+617
+00:53:09,580 --> 00:53:14,250
+على N، هذا K من واحدإن عندنا ناقص خلّيني أضعه في
+
+618
+00:53:14,250 --> 00:53:18,550
+summation 1 ناقص نفس القصة اللي هي m k small إيش
+
+619
+00:53:18,550 --> 00:53:23,710
+هي يا جماعة اتفاقنا عبارة عن f of x k minus 1 في
+
+620
+00:53:23,710 --> 00:53:27,610
+هذه خلّيني أخدها عامل مشترك بعد إذنكم لإن هو
+
+621
+00:53:27,610 --> 00:53:31,070
+موجودة هنا وموجودة هنا خلّيني أطلّحها برا واضحة
+
+622
+00:53:31,070 --> 00:53:36,070
+أشيل هذه برا بيصير اللي هو مضروبة في b minus a على
+
+623
+00:53:36,910 --> 00:53:42,350
+وأصلا هذه ثابتة بالنسبة لل summation ليش؟ لأن ال
+
+624
+00:53:42,350 --> 00:53:46,110
+summation العداد K من واحد عندنا أن هذه N ثابتة
+
+625
+00:53:46,110 --> 00:53:52,870
+بالنسبة لل K لذلك بتساوي B minus A على N مضروبة في
+
+626
+00:53:52,870 --> 00:53:59,280
+مين في ال summationللـ f of x k minus f of x k
+
+627
+00:53:59,280 --> 00:54:03,840
+minus واحد k من عند واحد لعند مين يا جماعة لعند ان
+
+628
+00:54:03,840 --> 00:54:07,920
+بعدين the is and could بدي افرطها هذه بيصير عندي y
+
+629
+00:54:07,920 --> 00:54:14,880
+ساوي هذا اللي هو مين كله اللي جاعت بحسبه ال u,b,n
+
+630
+00:54:14,880 --> 00:54:19,320
+وf ناقص ال b,n وf هيسوي هذا المقدار اللي هو b
+
+631
+00:54:19,320 --> 00:54:25,970
+minus a على المضروب فيهالان ال summation عبارة عن
+
+632
+00:54:25,970 --> 00:54:32,630
+k ب 1 بصير f of x 1 ناقص f of x naught اللي بعدها
+
+633
+00:54:32,630 --> 00:54:40,170
+k ب 2 زائد f of x 2 ناقص f of x 1 اللي بعدها زائد
+
+634
+00:54:40,170 --> 00:54:46,130
+f of x 3 ناقص f of x 2 لما نقضى اللي ماشي لأخر
+
+635
+00:54:46,130 --> 00:54:53,140
+واحد بكون عندى f of x nناقص f of x n ناقص واحد
+
+636
+00:54:57,360 --> 00:55:04,840
+F of X1 بيطير مع سالب F of X1 و F of X2 بيطير مع
+
+637
+00:55:04,840 --> 00:55:09,840
+سالب F of X2 و هكذا بظل ماشي لما كله يروح مع كله
+
+638
+00:55:09,840 --> 00:55:16,640
+ما عدا بظل عندي اللي هو أول قيمة اللي هي F of X
+
+639
+00:55:16,640 --> 00:55:22,660
+note بالسالب مع F of Xn الأخيرة بالموجب بيصير عندي
+
+640
+00:55:22,660 --> 00:55:34,740
+Y ساوي B minus A على Nفي F of Xn ناقص F of X0 طبعا
+
+641
+00:55:34,740 --> 00:55:39,420
+Xn آخر واحدة اللي هي B و X0 أول واحدة اللي هي A
+
+642
+00:55:39,420 --> 00:55:46,580
+إذن هيساوي هذا عبارة عن B minus A في F of B minus
+
+643
+00:55:46,580 --> 00:55:53,150
+F of A الكل هذا مجسوم على مين يا جماعة؟ على Nهذا
+
+644
+00:55:53,150 --> 00:55:58,090
+عبارة عن ثابت وهذا عبارة عن ثابت وهذا الان هي اللي
+
+645
+00:55:58,090 --> 00:56:03,670
+بدلالتها كُتِب ال partitions إذا صار عندى الآن بعد
+
+646
+00:56:03,670 --> 00:56:07,230
+كل اللي حكيته هان أخدت ال بي أن بالشكل اللي أمامي
+
+647
+00:56:07,230 --> 00:56:12,190
+عبارة عن sequence of partitions ووصلنا إلى ما
+
+648
+00:56:12,190 --> 00:56:17,890
+لاقيا ليه يا جماعة اللي هو وصلنا ان ال you بي أن و
+
+649
+00:56:17,890 --> 00:56:26,260
+أف ناقص ال بي أن و أفأصغر أو يساوي طبعا أكيد أكبر
+
+650
+00:56:26,260 --> 00:56:29,600
+يساوي صفر لأن هذا دايما أكبر يساوي هذا أصغر يساوي
+
+651
+00:56:29,600 --> 00:56:36,140
+B minus A ف F of B أكيد عرفت أشهد أساوي نقص F of A
+
+652
+00:56:36,140 --> 00:56:41,720
+على اللي هو N الـ N خد ال limit للجهتين as N goes
+
+653
+00:56:41,720 --> 00:56:45,160
+to infinity as N goes to infinity هذا goes to zero
+
+654
+00:56:45,160 --> 00:56:51,730
+وهذا أصلا صفر فبصير عندي limit إذا .. إذا limitالـ
+
+655
+00:56:51,730 --> 00:56:59,290
+U P N و F ناقص الـ L P N و F as N goes to infinity
+
+656
+00:56:59,290 --> 00:57:04,210
+بساوي سفر إذا صار عندي sequence of partitions تحقق
+
+657
+00:57:04,210 --> 00:57:07,730
+لهذا إذا حسب الـ Corollary اللي حكيتها قبل بشوية
+
+658
+00:57:07,730 --> 00:57:17,130
+إذا F is integrable وهو المطلوب إذا صار عندي أي
+
+659
+00:57:17,130 --> 00:57:21,850
+increasing function is integrablesimilarly for
+
+660
+00:57:21,850 --> 00:57:26,170
+decreasing لماذا similarly لأنه تصبح الدالة بدل ما
+
+661
+00:57:26,170 --> 00:57:30,950
+هي طالع هيك تصبح أشمالها نازلة نزول الدالة فبصير
+
+662
+00:57:30,950 --> 00:57:35,430
+عندك اللي هو ال maximum هي الأولى أو ال supremum
+
+663
+00:57:35,430 --> 00:57:41,920
+هي أفبصير الـ MK بسوء F of XK minus واحد والـ MK
+
+664
+00:57:41,920 --> 00:57:46,760
+بسوء F of XK وبيتكملوا البرهان بنفس الطريقة، هيطلع
+
+665
+00:57:46,760 --> 00:57:52,040
+عندكم البرهان automatic وبشكل سهل وبشكل سلس،
+
+666
+00:57:52,040 --> 00:57:58,500
+similar أي سؤال؟ إذن الإعلان اللي أعلنناه قبل
+
+667
+00:57:58,500 --> 00:58:04,800
+بشوية أنه any monotone functionيعني any increasing
+
+668
+00:58:04,800 --> 00:58:08,780
+function on a closed bounded interval must be
+
+669
+00:58:08,780 --> 00:58:15,740
+integrable and any decreasing function on a closed
+
+670
+00:58:15,740 --> 00:58:20,420
+bounded interval برضه must be integrable إذا صار
+
+671
+00:58:20,420 --> 00:58:26,060
+في عنا عائلة كاملة من الدوال القابلة للتكامل بواسط
+
+672
+00:58:26,060 --> 00:58:32,160
+التكامل remandالآن بدنا ننتقل إلى عائلة أخرى
+
+673
+00:58:32,160 --> 00:58:38,300
+وعائلة لا تقل أهمية عن هذه العائلة وعائلة يعني
+
+674
+00:58:38,300 --> 00:58:45,560
+محبوبة عنها اللي هي اللي
+
+675
+00:58:45,560 --> 00:58:53,960
+هي عائلة الدوال المتصلة اللي هي integrable of
+
+676
+00:58:53,960 --> 00:58:56,140
+continuous functions
+
+677
+00:59:00,850 --> 00:59:06,130
+اللي هي integrability of continuous functions
+
+678
+00:59:06,130 --> 00:59:12,210
+النظرية بتقول ما يلي لت أ .. طبعا عندي ال function
+
+679
+00:59:12,210 --> 00:59:16,410
+على closed bounded interval لت F من I لR be
+
+680
+00:59:16,410 --> 00:59:20,750
+continuous on I then F is integrable on I إذا الآن
+
+681
+00:59:20,750 --> 00:59:24,230
+any continuous function on a closed bounded
+
+682
+00:59:24,230 --> 00:59:29,600
+interval must be integrableكمان مرة any continuous
+
+683
+00:59:29,600 --> 00:59:33,840
+function on a closed bounded interval must be
+
+684
+00:59:33,840 --> 00:59:38,920
+integrable طبعا حيزنا في البرهان شغلة اللي هي
+
+685
+00:59:38,920 --> 00:59:45,080
+أخدناها سابقا انهAny continuous function on a
+
+686
+00:59:45,080 --> 00:59:49,840
+closed bounded interval must attain its maximum
+
+687
+00:59:49,840 --> 00:59:57,800
+and minimum on this interval بمعنى
+
+688
+00:59:57,800 --> 01:00:01,970
+آخر هنلاقيلو كانت F is continuous على اللي هو الـ
+
+689
+01:00:01,970 --> 01:00:05,990
+A و الـ B هنلاقي نقطة في داخل الفترة A و B بحيث
+
+690
+01:00:05,990 --> 01:00:09,570
+انها تكون الـ F عندها نقطة maximum و هنلاقي نقطة
+
+691
+01:00:09,570 --> 01:00:12,750
+أخرى في داخل هذه الفترة هنلاقي اللي هي الـ
+
+692
+01:00:12,750 --> 01:00:15,930
+function عندها إشمالها is minimum طبعا absolute و
+
+693
+01:00:15,930 --> 01:00:24,180
+absolute نيجي نشوف .. نيجي للبرهانالان .. نظرية
+
+694
+01:00:24,180 --> 01:00:28,540
+أخرى أيضا .. برضه خلينا نقولها أن لو كانت F is
+
+695
+01:00:28,540 --> 01:00:32,580
+continuous على closed bounded interval then F is
+
+696
+01:00:32,580 --> 01:00:43,820
+uniformly continuous الان F .. عند F من A و B لعند
+
+697
+01:00:43,820 --> 01:00:51,090
+R is continuous on A و Bcontinuous عالمين على
+
+698
+01:00:51,090 --> 01:00:54,450
+closed bounded interval في عندنا نظرية اللي بتقول
+
+699
+01:00:54,450 --> 01:00:58,270
+any continuous function في الواحد on a closed
+
+700
+01:00:58,270 --> 01:01:02,230
+bounded interval must be uniformly continuous إذا
+
+701
+01:01:02,230 --> 01:01:10,030
+then f is uniformly continuous
+
+702
+01:01:10,730 --> 01:01:15,290
+on a و b ايش يعني uniformly continuous يعني لكل
+
+703
+01:01:15,290 --> 01:01:18,810
+ابسلون اكبر من سفر there exist دلتا اكبر من سفر
+
+704
+01:01:18,810 --> 01:01:22,990
+دلتا بس بتعتمد عالميا على ابسلون بتنفع لكل ال X و
+
+705
+01:01:22,990 --> 01:01:30,810
+ال Y such that if X minus Y او U ناقص V زي ما هو
+
+706
+01:01:30,810 --> 01:01:40,200
+مسميها U minus Vأصغر من دلتا then F of U ناقص F of
+
+707
+01:01:40,200 --> 01:01:49,800
+V أصغر من إبسلون لكل UV element in A أو Bإذن لكل y
+
+708
+01:01:49,800 --> 01:01:52,900
+أكبر من 0 there exists Delta دلتا أشمة لا بتعتمد
+
+709
+01:01:52,900 --> 01:01:56,280
+على مين بس على إبسلون في حالة ال continuity
+
+710
+01:01:56,280 --> 01:02:01,560
+العادية بنقول إحنا limit f of x as x بتروح لل a
+
+711
+01:02:01,560 --> 01:02:07,260
+بساوي ايش f of a بنقول الآن إنه limit f of x بساوي
+
+712
+01:02:07,260 --> 01:02:13,450
+f of a as x بتروح لل a لكل إبسلون أكبر من 0هصب مين
+
+713
+01:02:13,450 --> 01:02:18,070
+بالـ a هذه there exist دي و لكل a element in a
+
+714
+01:02:18,070 --> 01:02:23,270
+there exist delta such that اللي لما يكون x minus
+
+715
+01:02:23,270 --> 01:02:27,190
+a أصغر من delta بيعطينا f of x نقص f of a أصغر من
+
+716
+01:02:27,190 --> 01:02:32,550
+إبسلون يعني بيكون الـ delta هنا تعتمد على الإبسلون
+
+717
+01:02:32,550 --> 01:02:37,550
+وتعتمد على ال a اللي عندها ال continuityلكن في الـ
+
+718
+01:02:37,550 --> 01:02:41,890
+Uniform Continuous الـ Delta اللي اللي جيت هنا
+
+719
+01:02:41,890 --> 01:02:47,510
+بتنفع لكل اللي هو ال .. ال .. ال .. ال .. ال .. ال
+
+720
+01:02:47,510 --> 01:02:47,890
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+721
+01:02:47,890 --> 01:02:47,910
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+722
+01:02:47,910 --> 01:02:48,290
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+723
+01:02:48,290 --> 01:02:48,530
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+724
+01:02:48,530 --> 01:02:50,270
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+725
+01:02:50,270 --> 01:02:50,330
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+726
+01:02:50,330 --> 01:02:50,410
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+727
+01:02:50,410 --> 01:02:54,250
+.. ال .. ال .. ال .. ال .. ال ..
+
+728
+01:02:54,250 --> 01:02:59,970
+ال .. ال ..F is continuous، إذن يُنفرم الـ
+
+729
+01:02:59,970 --> 01:03:02,650
+continuous، إيش معناته؟ كل إبسلون أكبر من صفر،
+
+730
+01:03:02,650 --> 01:03:05,490
+there exists دلتا، بحيث إن U minus V أصغر من دلتا
+
+731
+01:03:05,490 --> 01:03:08,690
+يعطيني F of U ناقص F of V أصغر من إياش، من إبسلون،
+
+732
+01:03:08,690 --> 01:03:11,870
+وخلّي هذه إيش مالها في الذاكرة، أنا عشان الحسابات
+
+733
+01:03:11,870 --> 01:03:15,890
+بدي أخليها إبسلون على مين، على B minus A، طول
+
+734
+01:03:15,890 --> 01:03:19,070
+الفترة اللي أنا بشتغل عليها، هتشوفوا ليش كتبت هيك،
+
+735
+01:03:19,070 --> 01:03:22,720
+بس لا الحساباتبقدر أه بقدر لإن هو أصغر .. أه بقدر
+
+736
+01:03:22,720 --> 01:03:25,140
+خليه أصغر من الـ Epsilon في الدنيا من ضمن إن كل
+
+737
+01:03:25,140 --> 01:03:27,520
+الـ Epsilon عن الـ B minus A لكل الـ Epsilonات
+
+738
+01:03:27,520 --> 01:03:34,040
+اللي في الدنيا طيب لأن شوف وين بده أروح، بده أروح
+
+739
+01:03:34,040 --> 01:03:39,250
+لـ Integrability لـ الـ function Fالأن خُد n الان
+
+740
+01:03:39,250 --> 01:03:42,190
+بدنا نعمل partitions بدنا نجيب sequence of
+
+741
+01:03:42,190 --> 01:03:45,890
+partitions Bn ال sequence of partitions هذه هي
+
+742
+01:03:45,890 --> 01:03:50,010
+اللي هتخدمني هتخدمني متى؟ بعد شوية بتشوفوا ليش
+
+743
+01:03:50,010 --> 01:03:54,110
+هتخدمني لكل الأنات اللي وين ما لإن اللي أكبر من ال
+
+744
+01:03:54,110 --> 01:03:59,370
+B minus A على مين؟ على ال delta اللي لجيتها إذا
+
+745
+01:03:59,370 --> 01:04:04,130
+اللي صارت ال delta بين إيديا إذا بقولك الآن choose
+
+746
+01:04:05,140 --> 01:04:12,320
+N element in N such that ما لها N أكبر من B minus
+
+747
+01:04:12,320 --> 01:04:17,620
+A على مين على دلتا برضه B minus A عشان الحسبان N
+
+748
+01:04:17,620 --> 01:04:21,240
+أكبر من B minus A على مين على دلتا يعني الآن أنا
+
+749
+01:04:21,240 --> 01:04:26,920
+بحكي بحكي عن الأنات اللي بكون أكبر من B minus A
+
+750
+01:04:26,920 --> 01:04:31,460
+على مين على الدلتا اللي لجيتها فوق عندي في اللي هو
+
+751
+01:04:31,460 --> 01:04:37,080
+هذا واحدماشي الحالة الان بتبدأ اكوّن partitions من
+
+752
+01:04:37,080 --> 01:04:42,480
+مين؟ من الأمنات هذه الان بتاخد بيئا اني partitions
+
+753
+01:04:42,480 --> 01:04:47,220
+اني partitions برضه أشماله with equal اللي هي إيش
+
+754
+01:04:47,220 --> 01:04:54,240
+sub intervals يعني X نوت و X واحد لعند X ان يعني
+
+755
+01:04:54,240 --> 01:04:58,540
+طول كل فترة منهم برضه طول .. بتاخد اللي هي كلهم
+
+756
+01:04:58,540 --> 01:05:03,590
+أشمالينمن عند A لعند B يكون ان كل ال sub intervals
+
+757
+01:05:03,590 --> 01:05:08,530
+جت بعض يعني بمعنى أخر هيكون ال XK minus XK minus
+
+758
+01:05:08,530 --> 01:05:12,970
+واحد بيساوي B minus A على N طول كل فترة ايش بيساوي
+
+759
+01:05:12,970 --> 01:05:16,950
+B minus A على عدد الفترات اللي هي N فبكون B minus
+
+760
+01:05:16,950 --> 01:05:21,250
+A عالميا على Nأنا حرّن انهم ألاقي في النهاية
+
+761
+01:05:21,250 --> 01:05:25,630
+sequence of partitions BN limit الـ U B N و F نقص
+
+762
+01:05:25,630 --> 01:05:29,290
+الـ L B N و F بساوي 0 بيكون F is integrable خلصنا
+
+763
+01:05:29,290 --> 01:05:32,950
+أنا قاعد اخترت اللي هي sequence of partitions بناء
+
+764
+01:05:32,950 --> 01:05:36,390
+على الـ Delta اللي لجيتها في ال uniformity و
+
+765
+01:05:36,390 --> 01:05:39,830
+الأنات اللي أكبر منها و حطيت ال partition اللي هو
+
+766
+01:05:39,830 --> 01:05:44,710
+بشكل اللي هو تكون ال subintervals كلها لها نفس
+
+767
+01:05:44,710 --> 01:05:46,230
+الطول طيب
+
+768
+01:05:48,470 --> 01:05:56,170
+الآن زي ما قلت قبل بشوية negation على الفترة xk و
+
+769
+01:05:56,170 --> 01:06:01,710
+minus 1 و xk هذا ال sub interval ال function is
+
+770
+01:06:01,710 --> 01:06:03,550
+continuous عليها لأنها continuous على كل ال
+
+771
+01:06:03,550 --> 01:06:06,970
+interval a و b مدام continuous عليها إذا it
+
+772
+01:06:06,970 --> 01:06:10,310
+attains its maximum and its minimum on this
+
+773
+01:06:10,310 --> 01:06:16,100
+intervalأكيد؟ لقد حد كده قبل مشوية إذن بما أن الـ
+
+774
+01:06:16,100 --> 01:06:18,480
+F is continuously not closed bound in الـ interval
+
+775
+01:06:18,480 --> 01:06:26,800
+هذه إذن there exist سميها U K و V K element in X K
+
+776
+01:06:26,800 --> 01:06:33,720
+minus واحد و X K such that F of U K هي ال maximum
+
+777
+01:06:33,720 --> 01:06:41,190
+على كل هذه ال maximum معناته هي ال supremum of Kأو
+
+778
+01:06:41,190 --> 01:06:47,110
+أيضًا عند الـ VK F of VK هي الـ Minimum على هذه
+
+779
+01:06:47,110 --> 01:06:52,310
+مضمون موجود قلناها هي طبعًا مدامة Minimum على كل
+
+780
+01:06:52,310 --> 01:06:56,150
+الفترة إذًا هي الـ Infimum اللي ببحث عنها اليمين
+
+781
+01:06:56,150 --> 01:07:00,850
+MK على هذه الفترة وها المفتاح أصلا في استخدام الـ
+
+782
+01:07:00,850 --> 01:07:06,120
+Continuity إن ضمن لوجود نقطةعندها ال maximum في
+
+783
+01:07:06,120 --> 01:07:10,540
+هذه المنطقة وضمن اللي وجود نقطة هنا ضمن اللي وجود
+
+784
+01:07:10,540 --> 01:07:13,680
+ال minimum عندها و ال minimum و ال maximum مدام
+
+785
+01:07:13,680 --> 01:07:18,100
+عند نقاط محددة في الفترة هي هتتلاقى و تنقل هي
+
+786
+01:07:18,100 --> 01:07:21,780
+عبارة عن ال supremum و ال infimum على ال sub
+
+787
+01:07:21,780 --> 01:07:29,160
+interval أي سؤال طيب الآن ضال علينا إيش نسوي إن
+
+788
+01:07:29,160 --> 01:07:35,610
+نحسب الحسابات اللي هو نبدأ في حساباتنااللي هو نحسب
+
+789
+01:07:35,610 --> 01:07:44,230
+الـ U و نحسب مين يا جماعة أو نحسب اللي هو ال ال ال
+
+790
+01:07:44,230 --> 01:07:44,930
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+791
+01:07:44,930 --> 01:07:44,950
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+792
+01:07:44,950 --> 01:07:45,050
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+793
+01:07:45,050 --> 01:07:45,070
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+794
+01:07:45,070 --> 01:07:45,130
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+795
+01:07:45,130 --> 01:07:53,950
+ال ال الت حسبولي
+
+796
+01:07:54,430 --> 01:07:59,790
+نقص L, P, N و F طبعا أكيد هذا أكبر يساوي سفر حفظنا
+
+797
+01:07:59,790 --> 01:08:04,050
+عن غيره هذا أنه أكبر أو يساوي سفر معروف ضل بدي
+
+798
+01:08:04,050 --> 01:08:08,850
+أحسبهم مع بعض ال summation عبارة عن بساوي
+
+799
+01:08:08,850 --> 01:08:13,610
+summation اللي هو ال M K صارت من ال M K capital
+
+800
+01:08:13,610 --> 01:08:19,950
+اللي هي F of U K صح ولا لأ يا جماعة أه صح و ال U K
+
+801
+01:08:19,950 --> 01:08:26,220
+وين موجودة في هذهالان في مين؟ في xk minus xk minus
+
+802
+01:08:26,220 --> 01:08:31,520
+واحد xk minus xk minus واحد كامل عند واحد اللي
+
+803
+01:08:31,520 --> 01:08:37,020
+عندها انهدمين هي الـ U الان زيها مين ال L ناقص
+
+804
+01:08:37,020 --> 01:08:42,600
+summation الان مين عند ال MK small ال F of VK ال
+
+805
+01:08:42,600 --> 01:08:48,920
+VK اللي لاجيناها برضه هان اللي هي F of VKمضروبة في
+
+806
+01:08:48,920 --> 01:08:54,380
+طول الفترة xk-xk-1 كامل عند واحد لعند مين يا جماعة
+
+807
+01:08:54,380 --> 01:09:00,000
+لعندنا خدولي الآن xk-xk-1 عامل مشترك وطبعا أنا
+
+808
+01:09:00,000 --> 01:09:03,220
+مفترضها اللي هو وماخدها إلا طول ال intervals أو
+
+809
+01:09:03,220 --> 01:09:07,080
+sub intervals متساوي يعني طول هذه وطول هذه هو
+
+810
+01:09:07,080 --> 01:09:10,600
+عبارة عن b-a على n زي النظرية السابقة ويساوي ال
+
+811
+01:09:10,600 --> 01:09:17,850
+summation لل F of uk ناقص F of vkالكل مضروب في مين
+
+812
+01:09:17,850 --> 01:09:26,030
+يا جماعة في B-A على N N أو K من عند واحد لعند إيش
+
+813
+01:09:26,030 --> 01:09:36,830
+لعند N أي سؤال طيب شوفوا الآن اسمحولي بس هنا أشتغل
+
+814
+01:09:36,830 --> 01:09:39,730
+شوية طيب
+
+815
+01:09:42,370 --> 01:09:47,490
+طولوا روحكم عندنا خلصنا يعني جربنا اذا صار عندي ال
+
+816
+01:09:47,490 --> 01:09:51,870
+U, P, N و F عشان تعرفوا أين رايح أنا هي ال U, P, N
+
+817
+01:09:51,870 --> 01:09:55,550
+و F ناقص ال P, N و F أكبر يساوي الصفر اللي جيته
+
+818
+01:09:55,550 --> 01:10:00,610
+أصغر يساوي المقدار اللي أمامي الان خلوني أطلع حد
+
+819
+01:10:00,610 --> 01:10:06,400
+برا بعد اسمكم هي B- a على n لأنها عبارة عن ثابت في
+
+820
+01:10:06,400 --> 01:10:13,020
+ال summation f of u,k ناقص f of v,k كامنة عند واحد
+
+821
+01:10:13,020 --> 01:10:20,340
+لأنك لاحظوا ما يلي يا جماعة عند ال u,k و ال v,k
+
+822
+01:10:20,340 --> 01:10:27,480
+وين موجودة في ال x,k minus واحد و ال x,k اه مظبوط
+
+823
+01:10:27,480 --> 01:10:32,540
+يعني الآن اللي هي طول
+
+824
+01:10:34,610 --> 01:10:41,150
+طول الفترة طول الفترة xk على جهة دي بس يا شباب
+
+825
+01:10:41,150 --> 01:10:46,850
+ناقص xk minus واحد طول الفترة لإن هاد صارت هي xk
+
+826
+01:10:46,850 --> 01:10:53,480
+minus واحد وهي xk ماشي جوات هنا ال mean ال UKو الـ
+
+827
+01:10:53,480 --> 01:10:56,860
+VK في مكان الماد، التنتين المهم جوا التنتين يعني
+
+828
+01:10:56,860 --> 01:10:59,840
+المسافة بين اللي برا هدول، حتى لو كان التنتين
+
+829
+01:10:59,840 --> 01:11:04,120
+زيهن، بكون أصغر أو يساوي المسافة بين التنتين هدول
+
+830
+01:11:04,120 --> 01:11:11,560
+أصغر أو يساوي المسافة هذه، اللي هو UK ناقص VK،
+
+831
+01:11:11,560 --> 01:11:17,010
+مظبوط؟ آسف، العكس، أكبر شكراسامحونا أكبر اياش أو
+
+832
+01:11:17,010 --> 01:11:21,310
+يساوي صارت المسافة بين الجهتين هدولة أكيد أصغر
+
+833
+01:11:21,310 --> 01:11:24,970
+أشهر من المسافة الكلية اللي هنا بينهم طيب المسافة
+
+834
+01:11:24,970 --> 01:11:28,890
+بين هذه وهذه احنا ماخدينها أصلا طول ال interval
+
+835
+01:11:28,890 --> 01:11:29,670
+ايش بتساوي
+
+836
+01:11:33,030 --> 01:11:39,710
+على N إذا المسافة هذه أصلا عبارة عن B-A عالمين على
+
+837
+01:11:39,710 --> 01:11:43,890
+N سامحوني أني بنكتبها هنا إذا صار عندي اللي هو ال
+
+838
+01:11:43,890 --> 01:11:50,230
+U K ناقص ال V K أصغر أو يساوي B-A عالمين على N إذا
+
+839
+01:11:50,230 --> 01:11:58,530
+يا شباب راحظولي هنا U K ناقص V K صارت أصغر أو
+
+840
+01:11:58,530 --> 01:12:06,440
+يساوي B-A عالمينعلى n واضحة طيب نشوف ايش معناه هذا
+
+841
+01:12:06,440 --> 01:12:08,700
+الكلام واش اللي بتقوله ليش بتقولنا هذا الكلام
+
+842
+01:12:08,700 --> 01:12:13,620
+بقوله عشان كلام مهم هذا هو شوفوا فهميته انا مختار
+
+843
+01:12:13,620 --> 01:12:19,260
+ال n اكبر من مين من b minus a على delta يعني بمعنى
+
+844
+01:12:19,260 --> 01:12:24,280
+اخر ايش معنات هذا يعني b minus a على n اصغر من مين
+
+845
+01:12:24,280 --> 01:12:29,640
+من دلتا يعني هذا المقدارأصغر من إيش يا جماعة؟ من
+
+846
+01:12:29,640 --> 01:12:35,160
+دلتا صارت ال U K ناقص ال V K أصغر من مين؟ من دلتا
+
+847
+01:12:35,160 --> 01:12:40,800
+واحنا بنقول أن المقاطع اللي بتحقق فيها ال U ناقص V
+
+848
+01:12:40,800 --> 01:12:44,560
+أصغر من دلتا بكون عندي F of U ناقص F of V أصغر من
+
+849
+01:12:44,560 --> 01:12:50,240
+مين يا جماعة؟ من Epsilon على B minus A إذا صارت
+
+850
+01:12:50,240 --> 01:12:55,100
+عندي المقاطين هذولة ال U K ناقص V K اللي أصغر من
+
+851
+01:12:55,100 --> 01:13:04,770
+دلتاإذا أكيد من واحد هيكون عندي F of U K ناقص F of
+
+852
+01:13:04,770 --> 01:13:12,670
+V K إيش هيكون؟ هيكون عبارة عن أصغر من Y على B
+
+853
+01:13:12,670 --> 01:13:20,410
+minus A إذا الآن بحجلني أني أقول أن هذا المقدار
+
+854
+01:13:20,410 --> 01:13:24,400
+اللي عندياللي هو طبعا هذا الكبير وهذا الصغير طبعا
+
+855
+01:13:24,400 --> 01:13:27,800
+على absolute value نفسه بحجل إن أقول هذا أصغر أو
+
+856
+01:13:27,800 --> 01:13:34,680
+ساوي B minus A على N مضروب في مين في ال summation
+
+857
+01:13:34,680 --> 01:13:42,480
+لل Epsilon على B minus A كامن عند واحد لعند مين
+
+858
+01:13:42,480 --> 01:13:50,100
+لعند N واضح
+
+859
+01:13:52,720 --> 01:13:58,440
+إذا صار عندى الآن اللى هو اللى أثبته أنه U P N و F
+
+860
+01:13:58,440 --> 01:14:04,220
+ناطق سؤال P N و F أصغر أو ساوى P minus A على N في
+
+861
+01:14:04,220 --> 01:14:10,300
+.. خلّيني أجف و أقولها .. أه .. انتظر .. أعملنى ..
+
+862
+01:14:13,410 --> 01:14:18,250
+إذا اللي وصلنا له يا جماعة اللي هو أن ال U P N و F
+
+863
+01:14:18,250 --> 01:14:21,650
+نقص ال P N و F أكبر يساوي سفر و في نفس الوقت أصغر
+
+864
+01:14:21,650 --> 01:14:24,430
+يساوي P Minus A على N في ال summation لل Epsilon ع
+
+865
+01:14:24,430 --> 01:14:28,630
+P Minus A اللي هو كامن عند واحد ل N يعني هذه قاعدة
+
+866
+01:14:28,630 --> 01:14:34,050
+عمالها كل مرة بتعد Epsilon ع P Minus A أكم مرة
+
+867
+01:14:34,050 --> 01:14:37,670
+بتعد N من المرات إذا حيصير عندي هذا عبارة عن P
+
+868
+01:14:37,670 --> 01:14:43,170
+Minus A على N و اللي نعد هناN من القمية اللي هذه
+
+869
+01:14:43,170 --> 01:14:47,630
+يعني N في epsilon على B minus A الان B minus A مع
+
+870
+01:14:47,630 --> 01:14:51,970
+ال B minus A و ال N مع ال N فبصير أوصلنا احنا إلى
+
+871
+01:14:51,970 --> 01:14:55,950
+ما يلي انتبهوا للنتيجة النهائية اللي بتوصلي لي
+
+872
+01:14:55,950 --> 01:15:04,310
+اللي هو المطلوب النتيجة اللي وصلت لها انه اذاU P N
+
+873
+01:15:04,310 --> 01:15:12,430
+و F نقص L P N و F أكبر أو يساوي سفر وأصغر من مين
+
+874
+01:15:12,430 --> 01:15:19,190
+من إبسلون لكل إبسلون أكبر من سفر إذا اللي جوا غصب
+
+875
+01:15:19,190 --> 01:15:24,310
+عنه إذا لازم يكون اللي هو لو أخدت ال limit هيظل
+
+876
+01:15:24,310 --> 01:15:30,220
+limit ال Uبن لأن هذا أصلا صحيح علميا للأنات
+
+877
+01:15:30,220 --> 01:15:34,200
+الكبيرة فلما أخد ال limit أزنه جوزه infinity بضل
+
+878
+01:15:34,200 --> 01:15:38,280
+في ال safe side يعني بضل في اللي بتحقق فيه هذا إذا
+
+879
+01:15:38,280 --> 01:15:45,630
+ال limit U, B, N و F ماقص L, B, N و Fهيكون أكبر أو
+
+880
+01:15:45,630 --> 01:15:50,490
+يساوي سفر و أصغر أو يساوي إبسلون و هذا الكلام as
+
+881
+01:15:50,490 --> 01:15:53,530
+and goes to infinity و هذا الكلام صحيح برضه لمين
+
+882
+01:15:53,530 --> 01:15:58,790
+لكل إبسلون أكبر من سفر إذا غصب عنها هيكون اللي جوا
+
+883
+01:15:58,790 --> 01:16:08,780
+limit UPN و F-Lبان و اف لازم يساوي ايش سفر از انقص
+
+884
+01:16:08,780 --> 01:16:14,780
+انفينيتي وهذا by corollary اللي قبل بشوية هيعني ان
+
+885
+01:16:14,780 --> 01:16:23,310
+اف is integrableHence, F is integrable وهكذا
+
+886
+01:16:23,310 --> 01:16:27,610
+أثبتنا العائلة الثانية من الدوال الـ continuous
+
+887
+01:16:27,610 --> 01:16:31,170
+function على closed bounded interval is a
+
+888
+01:16:31,170 --> 01:16:34,210
+continuous function is an integrable function
+
+889
+01:16:34,210 --> 01:16:40,790
+وهكذا خلصنا الـ section الأول و الـ homework هيا
+
+890
+01:16:40,790 --> 01:16:46,750
+اللي موجودة في التلخيص و المرة القادمة ان شاء الله
+
+891
+01:16:47,630 --> 01:16:57,970
+إن شاء الله بكون احنا بنبدأ في اللي هي ال section
+
+892
+01:16:57,970 --> 01:17:03,510
+اللي بعده اللي هو properties of the Riemann
+
+893
+01:17:03,510 --> 01:17:04,470
+Integral
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..f417a739af19a6a40a72d5a3a13b8e122386dbda
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 3361, "start": 4.72, "end": 33.62, "text": "بسم الله الرحمن الرحيم هذه هي المحاضرة رقم 11 في مساق تحليل حقيقي 2 لطلاب وطالبات الجامعة الإسلامية كلية العلوم وهي المحاضرة الأولى بعد إعلان الطوارئ بخصوص أو بمواجهة فيروس كورونا المنتشر اتحدتنا المرة الماضية", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 29538, 39896, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 2975, 8978, 47524, 995, 4587, 6055, 5016, 20292, 1211, 11331, 38436, 38436, 568, 5296, 9566, 15040, 3555, 4032, 9566, 6027, 3555, 9307, 25724, 10943, 27884, 33688, 3794, 37440, 10632, 9122, 20292, 3660, 18863, 1211, 20498, 37037, 1829, 9673, 5016, 46958, 25720, 16247, 12610, 7578, 39182, 11933, 30241, 7649, 41950, 2407, 9640, 19986, 4724, 9778, 9381, 2407, 9381, 34051, 4724, 2304, 2407, 26108, 3224, 3660, 8978, 32887, 3794, 9122, 13063, 2407, 8315, 9673, 29399, 46309, 1975, 2655, 24401, 2655, 8315, 9673, 25720, 9673, 46958, 10632], "avg_logprob": -0.07702517986876294, 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"probability": 0.734375}, {"start": 63.93, "end": 64.33, "word": " كانت", "probability": 0.962158203125}, {"start": 64.33, "end": 65.57, "word": " تمثل", "probability": 0.8660481770833334}, {"start": 65.57, "end": 66.03, "word": " أو", "probability": 0.498779296875}, {"start": 66.03, "end": 66.41, "word": " m", "probability": 0.70654296875}, {"start": 66.41, "end": 66.83, "word": " small", "probability": 0.5380859375}, {"start": 66.83, "end": 67.09, "word": " k", "probability": 0.9560546875}, {"start": 67.09, "end": 67.51, "word": " كانت", "probability": 0.965576171875}, {"start": 67.51, "end": 67.99, "word": " تمثل", "probability": 0.9895833333333334}, {"start": 67.99, "end": 68.39, "word": " عبارة", "probability": 0.9932861328125}, {"start": 68.39, "end": 68.57, "word": " عن", "probability": 0.99609375}, {"start": 68.57, "end": 68.81, "word": " الـ", "probability": 0.6275634765625}, {"start": 68.81, "end": 69.43, "word": " infimum", "probability": 0.9051106770833334}, {"start": 69.43, "end": 70.67, "word": " للدالة", "probability": 0.8614501953125}, {"start": 70.67, "end": 70.91, "word": " على", "probability": 0.89599609375}, {"start": 70.91, "end": 71.55, "word": " الفترة", "probability": 0.9884440104166666}, {"start": 71.55, "end": 71.89, "word": " اللي", "probability": 0.909423828125}, {"start": 71.89, "end": 72.17, "word": " هي", "probability": 0.880859375}, {"start": 72.17, "end": 73.03, "word": " المذكورة", "probability": 0.99375}], "temperature": 1.0}, {"id": 4, "seek": 9076, "start": 73.56, "end": 90.76, "text": "الان ال .. ال .. ال other sum هو عبارة عن ال summation لنفس ال sum العلوي ولكن بدلا منها اللي هي M K capital اللي كانت تمثل ال supremum ل ال F of X وال X على اللي هي في الفترة المذكورة اللي عندي", "tokens": [6027, 7649, 2423, 4386, 2423, 4386, 2423, 661, 2408, 31439, 6225, 3555, 9640, 3660, 18871, 2423, 28811, 5296, 1863, 36178, 2423, 2408, 18863, 1211, 45865, 35525, 19452, 47525, 15040, 9154, 11296, 13672, 1829, 39896, 376, 591, 4238, 13672, 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0.955078125}, {"start": 76.38, "end": 76.7, "word": " عبارة", "probability": 0.9952392578125}, {"start": 76.7, "end": 76.86, "word": " عن", "probability": 0.99609375}, {"start": 76.86, "end": 76.98, "word": " ال", "probability": 0.896484375}, {"start": 76.98, "end": 77.44, "word": " summation", "probability": 0.34521484375}, {"start": 77.44, "end": 77.94, "word": " لنفس", "probability": 0.9505208333333334}, {"start": 77.94, "end": 78.06, "word": " ال", "probability": 0.95361328125}, {"start": 78.06, "end": 78.26, "word": " sum", "probability": 0.383056640625}, {"start": 78.26, "end": 79.38, "word": " العلوي", "probability": 0.9248046875}, {"start": 79.38, "end": 80.26, "word": " ولكن", "probability": 0.871826171875}, {"start": 80.26, "end": 80.62, "word": " بدلا", "probability": 0.962890625}, {"start": 80.62, "end": 81.14, "word": " منها", "probability": 0.981689453125}, {"start": 81.14, "end": 81.42, "word": " اللي", "probability": 0.88916015625}, {"start": 81.42, "end": 81.78, "word": " هي", "probability": 0.81787109375}, {"start": 81.78, "end": 82.58, "word": " M", "probability": 0.4521484375}, {"start": 82.58, "end": 83.04, "word": " K", "probability": 0.51806640625}, {"start": 83.04, "end": 83.56, "word": " capital", "probability": 0.53515625}, {"start": 83.56, "end": 83.8, "word": " اللي", "probability": 0.949951171875}, {"start": 83.8, "end": 84.1, "word": " كانت", "probability": 0.976318359375}, {"start": 84.1, "end": 84.52, "word": " تمثل", "probability": 0.9775390625}, {"start": 84.52, "end": 84.66, "word": " ال", "probability": 0.6904296875}, {"start": 84.66, "end": 85.28, "word": " supremum", "probability": 0.828125}, {"start": 85.28, "end": 85.56, "word": " ل", "probability": 0.85205078125}, {"start": 85.56, "end": 85.82, "word": " ال", "probability": 0.318359375}, {"start": 85.82, "end": 86.0, "word": " F", "probability": 0.70263671875}, {"start": 86.0, "end": 86.16, "word": " of", "probability": 0.78515625}, {"start": 86.16, "end": 86.44, "word": " X", "probability": 0.9716796875}, {"start": 86.44, "end": 86.96, "word": " وال", "probability": 0.60400390625}, {"start": 86.96, "end": 87.3, "word": " X", "probability": 0.55908203125}, {"start": 87.3, "end": 87.64, "word": " على", "probability": 0.75048828125}, {"start": 87.64, "end": 87.98, "word": " اللي", "probability": 0.981689453125}, {"start": 87.98, "end": 88.14, "word": " هي", "probability": 0.83642578125}, {"start": 88.14, "end": 88.28, "word": " في", "probability": 0.72265625}, {"start": 88.28, "end": 88.78, "word": " الفترة", "probability": 0.9890950520833334}, {"start": 88.78, "end": 89.86, "word": " المذكورة", "probability": 0.99111328125}, {"start": 89.86, "end": 90.42, "word": " اللي", "probability": 0.986083984375}, {"start": 90.42, "end": 90.76, "word": " عندي", "probability": 0.7822265625}], "temperature": 1.0}, {"id": 5, "seek": 11595, "start": 91.05, "end": 115.95, "text": "الآن أخدنا أول لمّة المرة الماضية وقلنا إذا كانت F من I لR bounded و B any partition of I بدي يكون ال lower اللي هو sum لأي partition B و function F أصغر أو يساوي ال upper sum لنفس ال partition و لنفس اللي هي ال function F بعد هي طبعا خطينا خطوة أخرى", "tokens": [6027, 148, 48506, 5551, 9778, 3215, 8315, 5551, 12610, 32767, 11703, 3660, 9673, 25720, 9673, 46958, 10632, 4032, 4587, 1211, 8315, 11933, 15730, 25961, 2655, 479, 9154, 286, 5296, 49, 37498, 4032, 363, 604, 24808, 295, 286, 4724, 16254, 7251, 30544, 2423, 3126, 13672, 1829, 31439, 2408, 5296, 10721, 1829, 24808, 363, 4032, 2445, 479, 5551, 9381, 17082, 2288, 34051, 7251, 3794, 995, 45865, 2423, 6597, 2408, 5296, 1863, 36178, 2423, 24808, 4032, 5296, 1863, 36178, 13672, 1829, 39896, 2423, 2445, 479, 39182, 39896, 23032, 3555, 3615, 995, 16490, 9566, 1829, 8315, 16490, 9566, 2407, 3660, 5551, 34740, 7578], "avg_logprob": -0.2101562462747097, "compression_ratio": 1.6457399103139014, "no_speech_prob": 8.344650268554688e-07, "words": [{"start": 91.05, "end": 91.61, "word": "الآن", "probability": 0.6083984375}, {"start": 91.61, "end": 92.15, "word": " أخدنا", "probability": 0.881103515625}, {"start": 92.15, "end": 92.49, "word": " أول", "probability": 0.976318359375}, {"start": 92.49, "end": 93.09, "word": " لمّة", "probability": 0.71630859375}, {"start": 93.09, "end": 94.11, "word": " المرة", "probability": 0.947021484375}, {"start": 94.11, "end": 94.75, "word": " الماضية", "probability": 0.9939778645833334}, {"start": 94.75, "end": 95.27, "word": " وقلنا", "probability": 0.8489990234375}, {"start": 95.27, "end": 95.45, "word": " إذا", "probability": 0.893798828125}, {"start": 95.45, "end": 95.77, "word": " كانت", "probability": 0.8896484375}, {"start": 95.77, "end": 95.93, "word": " F", "probability": 0.477783203125}, {"start": 95.93, "end": 96.07, "word": " من", "probability": 0.95947265625}, {"start": 96.07, "end": 96.25, "word": " I", "probability": 0.97705078125}, {"start": 96.25, "end": 96.57, "word": " لR", "probability": 0.6402587890625}, {"start": 96.57, "end": 97.17, "word": " bounded", "probability": 0.71826171875}, {"start": 97.17, "end": 97.65, "word": " و", "probability": 0.76025390625}, {"start": 97.65, "end": 97.85, "word": " B", "probability": 0.19482421875}, {"start": 97.85, "end": 98.19, "word": " any", "probability": 0.52490234375}, {"start": 98.19, "end": 98.63, "word": " partition", "probability": 0.9208984375}, {"start": 98.63, "end": 98.93, "word": " of", "probability": 0.97119140625}, {"start": 98.93, "end": 99.27, "word": " I", "probability": 0.98876953125}, {"start": 99.27, "end": 99.55, "word": " بدي", "probability": 0.5367431640625}, {"start": 99.55, "end": 99.87, "word": " يكون", "probability": 0.6761474609375}, {"start": 99.87, "end": 100.15, "word": " ال", "probability": 0.93359375}, {"start": 100.15, "end": 100.65, "word": " lower", "probability": 0.6025390625}, {"start": 100.65, "end": 101.31, "word": " اللي", "probability": 0.854248046875}, {"start": 101.31, "end": 101.59, "word": " هو", "probability": 0.98046875}, {"start": 101.59, "end": 102.05, "word": " sum", "probability": 0.81689453125}, {"start": 102.05, "end": 103.23, "word": " لأي", "probability": 0.8489583333333334}, {"start": 103.23, "end": 104.33, "word": " partition", "probability": 0.6484375}, {"start": 104.33, "end": 104.95, "word": " B", "probability": 0.70166015625}, {"start": 104.95, "end": 105.31, "word": " و", "probability": 0.9833984375}, {"start": 105.31, "end": 105.75, "word": " function", "probability": 0.490234375}, {"start": 105.75, "end": 106.31, "word": " F", "probability": 0.97509765625}, {"start": 106.31, "end": 107.07, "word": " أصغر", "probability": 0.9805908203125}, {"start": 107.07, "end": 107.19, "word": " أو", "probability": 0.869140625}, {"start": 107.19, "end": 107.47, "word": " يساوي", "probability": 0.6431884765625}, {"start": 107.47, "end": 107.59, "word": " ال", "probability": 0.8681640625}, {"start": 107.59, "end": 107.81, "word": " upper", "probability": 0.59716796875}, {"start": 107.81, "end": 108.19, "word": " sum", "probability": 0.921875}, {"start": 108.19, "end": 108.69, "word": " لنفس", "probability": 0.9850260416666666}, {"start": 108.69, "end": 108.83, "word": " ال", "probability": 0.9609375}, {"start": 108.83, "end": 109.29, "word": " partition", "probability": 0.87548828125}, {"start": 109.29, "end": 109.99, "word": " و", "probability": 0.83935546875}, {"start": 109.99, "end": 110.53, "word": " لنفس", "probability": 0.9345703125}, {"start": 110.53, "end": 110.79, "word": " اللي", "probability": 0.7308349609375}, {"start": 110.79, "end": 111.19, "word": " هي", "probability": 0.93408203125}, {"start": 111.19, "end": 111.45, "word": " ال", "probability": 0.943359375}, {"start": 111.45, "end": 111.85, "word": " function", "probability": 0.95654296875}, {"start": 111.85, "end": 112.47, "word": " F", "probability": 0.9755859375}, {"start": 112.47, "end": 113.35, "word": " بعد", "probability": 0.9033203125}, {"start": 113.35, "end": 113.57, "word": " هي", "probability": 0.95556640625}, {"start": 113.57, "end": 114.03, "word": " طبعا", "probability": 0.8714599609375}, {"start": 114.03, "end": 115.03, "word": " خطينا", "probability": 0.894287109375}, {"start": 115.03, "end": 115.43, "word": " خطوة", "probability": 0.991455078125}, {"start": 115.43, "end": 115.95, "word": " أخرى", "probability": 0.9890950520833334}], "temperature": 1.0}, {"id": 6, "seek": 13141, "start": 116.45, "end": 131.41, "text": "و جينا عرفنا اللي هو شو معناته انها تكون اللي هو ال partition Q refinement لل partition B قلنا Q اللي هو تحسين ل B إذا كانت B بي عبارة عن مجموعة جزئية من Q", "tokens": [2407, 10874, 1829, 8315, 6225, 28480, 8315, 13672, 1829, 31439, 13412, 2407, 20449, 8315, 47395, 16472, 11296, 6055, 30544, 13672, 1829, 31439, 2423, 24808, 1249, 1895, 30229, 24976, 24808, 363, 12174, 1211, 8315, 1249, 13672, 1829, 31439, 6055, 5016, 3794, 9957, 5296, 363, 11933, 15730, 25961, 2655, 363, 4724, 1829, 6225, 3555, 9640, 3660, 18871, 3714, 7435, 2304, 2407, 27884, 10874, 11622, 19986, 10632, 9154, 1249], "avg_logprob": -0.2333255610359249, "compression_ratio": 1.5125, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 116.45, "end": 116.71, "word": "و", "probability": 0.74755859375}, {"start": 116.71, "end": 117.01, "word": " جينا", "probability": 0.7931315104166666}, {"start": 117.01, "end": 117.79, "word": " عرفنا", "probability": 0.8984375}, {"start": 117.79, "end": 118.03, "word": " اللي", "probability": 0.63671875}, {"start": 118.03, "end": 118.29, "word": " هو", "probability": 0.97314453125}, {"start": 118.29, "end": 118.71, "word": " شو", "probability": 0.6107177734375}, {"start": 118.71, "end": 119.19, "word": " معناته", "probability": 0.8776041666666666}, {"start": 119.19, "end": 119.39, "word": " انها", "probability": 0.696533203125}, {"start": 119.39, "end": 119.93, "word": " تكون", "probability": 0.9794921875}, {"start": 119.93, "end": 120.49, "word": " اللي", "probability": 0.863525390625}, {"start": 120.49, "end": 120.89, "word": " هو", "probability": 0.9638671875}, {"start": 120.89, "end": 121.31, "word": " ال", "probability": 0.6318359375}, {"start": 121.31, "end": 121.89, "word": " partition", "probability": 0.386474609375}, {"start": 121.89, "end": 122.35, "word": " Q", "probability": 0.1663818359375}, {"start": 122.35, "end": 123.11, "word": " refinement", "probability": 0.63232421875}, {"start": 123.11, "end": 123.39, "word": " لل", "probability": 0.720703125}, {"start": 123.39, "end": 123.85, "word": " partition", "probability": 0.70263671875}, {"start": 123.85, "end": 124.21, "word": " B", "probability": 0.8115234375}, {"start": 124.21, "end": 125.07, "word": " قلنا", "probability": 0.7986653645833334}, {"start": 125.07, "end": 125.51, "word": " Q", "probability": 0.373046875}, {"start": 125.51, "end": 125.81, "word": " اللي", "probability": 0.96044921875}, {"start": 125.81, "end": 125.95, "word": " هو", "probability": 0.96240234375}, {"start": 125.95, "end": 126.55, "word": " تحسين", "probability": 0.9896240234375}, {"start": 126.55, "end": 126.73, "word": " ل", "probability": 0.84423828125}, {"start": 126.73, "end": 127.01, "word": " B", "probability": 0.5771484375}, {"start": 127.01, "end": 127.85, "word": " إذا", "probability": 0.8515625}, {"start": 127.85, "end": 128.31, "word": " كانت", "probability": 0.9716796875}, {"start": 128.31, "end": 128.61, "word": " B", "probability": 0.7841796875}, {"start": 128.61, "end": 128.93, "word": " بي", "probability": 0.631591796875}, {"start": 128.93, "end": 129.35, "word": " عبارة", "probability": 0.9488525390625}, {"start": 129.35, "end": 129.49, "word": " عن", "probability": 0.98876953125}, {"start": 129.49, "end": 130.01, "word": " مجموعة", "probability": 0.96728515625}, {"start": 130.01, "end": 130.67, "word": " جزئية", "probability": 0.9501953125}, {"start": 130.67, "end": 131.13, "word": " من", "probability": 0.99267578125}, {"start": 131.13, "end": 131.41, "word": " Q", "probability": 0.806640625}], "temperature": 1.0}, {"id": 7, "seek": 14768, "start": 131.66, "end": 147.68, "text": "و بناء عليه اللي هو قلنا ان اي اللي هو sub interval xk-1xk من ال partition B يمكن كتابتها على صورة union of sub intervals من اللي هو التحسين اللي هو EQ", "tokens": [2407, 44945, 16606, 47356, 13672, 1829, 31439, 12174, 1211, 8315, 16472, 1975, 1829, 13672, 1829, 31439, 1422, 15035, 2031, 74, 12, 16, 87, 74, 9154, 2423, 24808, 363, 7251, 43020, 9122, 2655, 16758, 2655, 11296, 15844, 20328, 13063, 3660, 11671, 295, 1422, 26651, 9154, 13672, 1829, 31439, 16712, 5016, 3794, 9957, 13672, 1829, 31439, 33580], "avg_logprob": -0.23353794323546545, "compression_ratio": 1.457516339869281, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 131.66, "end": 132.0, "word": "و", "probability": 0.54638671875}, {"start": 132.0, "end": 132.4, "word": " بناء", "probability": 0.884521484375}, {"start": 132.4, "end": 132.98, "word": " عليه", "probability": 0.479248046875}, {"start": 132.98, "end": 133.82, "word": " اللي", 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"probability": 0.9345703125}, {"start": 433.14, "end": 433.3, "word": " upper", "probability": 0.90576171875}, {"start": 433.3, "end": 433.78, "word": " integral", "probability": 0.794921875}, {"start": 433.78, "end": 434.04, "word": " are", "probability": 0.8681640625}, {"start": 434.04, "end": 434.46, "word": " equal", "probability": 0.91650390625}, {"start": 434.46, "end": 434.9, "word": " و", "probability": 0.98193359375}, {"start": 434.9, "end": 435.08, "word": " من", "probability": 0.7373046875}, {"start": 435.08, "end": 435.36, "word": " ثم", "probability": 0.892333984375}, {"start": 435.36, "end": 435.92, "word": " أثبتنا", "probability": 0.99619140625}, {"start": 435.92, "end": 436.18, "word": " إنه", "probability": 0.75927734375}, {"start": 436.18, "end": 436.3, "word": " ال", "probability": 0.91796875}, {"start": 436.3, "end": 436.78, "word": " integration", "probability": 0.96728515625}, {"start": 436.78, "end": 437.58, "word": " exist", "probability": 0.73046875}, {"start": 437.58, "end": 438.16, "word": " لل", "probability": 0.6025390625}, {"start": 438.16, "end": 438.54, "word": " function", "probability": 0.794921875}, {"start": 438.54, "end": 438.84, "word": " x", "probability": 0.90966796875}, {"start": 438.84, "end": 438.98, "word": " على", "probability": 0.77490234375}, {"start": 438.98, "end": 439.26, "word": " الفترة", "probability": 0.9475911458333334}, {"start": 439.26, "end": 439.5, "word": " 0", "probability": 0.32275390625}, {"start": 439.5, "end": 439.62, "word": " و", "probability": 0.6083984375}, {"start": 439.62, "end": 439.94, "word": " 1", "probability": 0.88818359375}, {"start": 439.94, "end": 440.9, "word": " وأوجدنا", "probability": 0.8662109375}, {"start": 440.9, "end": 441.3, "word": " قيمة", "probability": 0.9715169270833334}, {"start": 441.3, "end": 441.42, "word": " ال", "probability": 0.98974609375}, {"start": 441.42, "end": 442.02, "word": " integration", "probability": 0.96484375}, {"start": 442.02, "end": 442.6, 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F من I اللي هي 01 لعند ال .. ال R be defined by أخدنا الدالة كما يلي اللي هو قولنا أن F of X F of X بساوي واحد", "tokens": [2407, 36520, 8315, 30731, 13672, 1829, 31439, 50113, 6027, 8315, 16712, 6027, 1829, 14739, 3224, 45164, 25961, 2655, 479, 9154, 286, 5296, 3615, 41260, 4386, 479, 9154, 286, 13672, 1829, 39896, 23185, 5296, 3615, 41260, 2423, 4386, 2423, 497, 312, 7642, 538, 5551, 9778, 3215, 8315, 32748, 6027, 3660, 9122, 15042, 7251, 20292, 13672, 1829, 31439, 12174, 12610, 8315, 14739, 479, 295, 1783, 479, 295, 1783, 4724, 3794, 995, 45865, 36764, 24401], "avg_logprob": -0.20783389757757317, "compression_ratio": 1.6158536585365855, "no_speech_prob": 5.185604095458984e-06, "words": [{"start": 444.9100000000001, "end": 445.69000000000005, "word": "وصلنا", "probability": 0.8396809895833334}, {"start": 445.69000000000005, "end": 446.47, "word": " إلى", "probability": 0.491455078125}, {"start": 446.47, "end": 446.81, "word": " اللي", "probability": 0.766845703125}, {"start": 446.81, 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"word": " numbers", "probability": 0.39794921875}, {"start": 482.52, "end": 483.76, "word": " الآن", "probability": 0.6888427734375}, {"start": 483.76, "end": 484.66, "word": " بدنا", "probability": 0.756103515625}, {"start": 484.66, "end": 485.14, "word": " نثبت", "probability": 0.9949951171875}, {"start": 485.14, "end": 485.46, "word": " بقول", "probability": 0.670654296875}, {"start": 485.46, "end": 485.7, "word": " show", "probability": 0.75634765625}, {"start": 485.7, "end": 486.26, "word": " that", "probability": 0.9375}, {"start": 486.26, "end": 487.06, "word": " this", "probability": 0.94677734375}, {"start": 487.06, "end": 487.72, "word": " function", "probability": 0.96630859375}, {"start": 487.72, "end": 488.76, "word": " F", "probability": 0.5966796875}], "temperature": 1.0}, {"id": 24, "seek": 50516, "start": 490.3, "end": 505.16, "text": "طبعا أنا method F على اللي هي الـ Q تقاطع طبعا الـ 0 و 1 اللي هي الـ interval اللي بدأ عليها التقاطع الـ 0 و 1 بمعنى إن دالت F صارت من I اللي هي عبارة عن 0 و 1", "tokens": [9566, 3555, 3615, 995, 41850, 3170, 479, 15844, 13672, 1829, 39896, 2423, 39184, 1249, 6055, 4587, 41193, 3615, 23032, 3555, 3615, 995, 2423, 39184, 1958, 4032, 502, 13672, 1829, 39896, 2423, 39184, 15035, 13672, 1829, 47525, 10721, 25894, 11296, 16712, 4587, 41193, 3615, 2423, 39184, 1958, 4032, 502, 4724, 2304, 3615, 1863, 7578, 36145, 11778, 6027, 2655, 479, 20328, 9640, 2655, 9154, 286, 13672, 1829, 39896, 6225, 3555, 9640, 3660, 18871, 1958, 4032, 502], "avg_logprob": -0.22541667381922403, "compression_ratio": 1.6601307189542485, "no_speech_prob": 1.5139579772949219e-05, "words": [{"start": 490.3, "end": 490.68, "word": "طبعا", "probability": 0.8153076171875}, {"start": 490.68, "end": 490.88, "word": " أنا", "probability": 0.286865234375}, {"start": 490.88, "end": 491.2, "word": " method", "probability": 0.935546875}, {"start": 491.2, "end": 491.56, "word": " F", "probability": 0.65771484375}, {"start": 491.56, "end": 492.48, "word": " على", "probability": 0.6259765625}, {"start": 492.48, "end": 492.84, "word": " اللي", "probability": 0.76806640625}, {"start": 492.84, "end": 493.04, "word": " هي", "probability": 0.7900390625}, {"start": 493.04, "end": 493.26, "word": " الـ", "probability": 0.852783203125}, {"start": 493.26, "end": 493.62, "word": " Q", "probability": 0.50048828125}, {"start": 493.62, "end": 494.76, "word": " تقاطع", "probability": 0.966552734375}, {"start": 494.76, "end": 495.34, "word": " طبعا", "probability": 0.8717041015625}, {"start": 495.34, "end": 495.68, "word": " الـ", "probability": 0.945068359375}, {"start": 495.68, "end": 495.88, "word": " 0", "probability": 0.55419921875}, {"start": 495.88, "end": 496.1, "word": " و", "probability": 0.84765625}, {"start": 496.1, "end": 496.46, "word": " 1", "probability": 0.61279296875}, {"start": 496.46, "end": 497.24, "word": " اللي", "probability": 0.963134765625}, {"start": 497.24, "end": 497.34, "word": " هي", "probability": 0.857421875}, {"start": 497.34, "end": 497.46, "word": " الـ", "probability": 0.7230224609375}, {"start": 497.46, "end": 497.78, "word": " interval", "probability": 0.748046875}, {"start": 497.78, "end": 497.92, "word": " اللي", "probability": 0.971435546875}, {"start": 497.92, "end": 498.12, "word": " بدأ", "probability": 0.4183349609375}, {"start": 498.12, "end": 498.5, "word": " عليها", "probability": 0.98046875}, {"start": 498.5, "end": 499.18, "word": " التقاطع", "probability": 0.78466796875}, {"start": 499.18, "end": 499.32, "word": " الـ", "probability": 0.896484375}, {"start": 499.32, "end": 499.5, "word": " 0", "probability": 0.869140625}, {"start": 499.5, "end": 499.64, "word": " و", "probability": 0.89208984375}, {"start": 499.64, "end": 499.92, "word": " 1", "probability": 0.333251953125}, {"start": 499.92, "end": 500.86, "word": " بمعنى", "probability": 0.97265625}, {"start": 500.86, "end": 501.02, "word": " إن", "probability": 0.39501953125}, {"start": 501.02, "end": 501.58, "word": " دالت", 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واضح ان الدالة هذه is a bounded function الان بدأ أثبت لكم ان هذا الدالة is not integrable on this interval is not integrable on this interval الان علشان أصل اللي هو اللي هي ان هذا الدالة غير قابلة تكامل بالنسبة لتكامل بالنسبة لتكامل الريمان بدي اخد الان B", "tokens": [6027, 20292, 43242, 497, 4032, 46958, 5016, 16472, 32748, 6027, 3660, 29538, 307, 257, 37498, 2445, 2423, 7649, 47525, 10721, 5551, 12984, 3555, 2655, 5296, 24793, 16472, 23758, 32748, 6027, 3660, 307, 406, 16200, 424, 638, 322, 341, 15035, 307, 406, 16200, 424, 638, 322, 341, 15035, 2423, 7649, 11203, 8592, 7649, 5551, 36520, 13672, 1829, 31439, 13672, 1829, 39896, 16472, 23758, 32748, 6027, 3660, 32771, 13546, 12174, 16758, 37977, 6055, 4117, 10943, 1211, 20666, 1863, 35457, 3660, 5296, 2655, 4117, 10943, 1211, 20666, 1863, 35457, 3660, 5296, 2655, 4117, 10943, 1211, 2423, 16572, 2304, 7649, 4724, 16254, 1975, 9778, 3215, 2423, 7649, 363], "avg_logprob": -0.25803570974440804, "compression_ratio": 1.975609756097561, 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"end": 523.86, "word": " هي", "probability": 0.89013671875}, {"start": 523.86, "end": 524.1, "word": " ان", "probability": 0.8271484375}, {"start": 524.1, "end": 524.38, "word": " هذا", "probability": 0.60546875}, {"start": 524.38, "end": 524.76, "word": " الدالة", "probability": 0.9822591145833334}, {"start": 524.76, "end": 525.02, "word": " غير", "probability": 0.988525390625}, {"start": 525.02, "end": 525.56, "word": " قابلة", "probability": 0.9327799479166666}, {"start": 525.56, "end": 526.3, "word": " تكامل", "probability": 0.79241943359375}, {"start": 526.3, "end": 526.9, "word": " بالنسبة", "probability": 0.980224609375}, {"start": 526.9, "end": 527.54, "word": " لتكامل", "probability": 0.8990234375}, {"start": 527.54, "end": 528.18, "word": " بالنسبة", "probability": 0.9180908203125}, {"start": 528.18, "end": 529.18, "word": " لتكامل", "probability": 0.96396484375}, {"start": 529.18, "end": 529.74, "word": " الريمان", "probability": 0.6346435546875}, {"start": 529.74, "end": 530.78, "word": " بدي", "probability": 0.68212890625}, {"start": 530.78, "end": 531.28, "word": " اخد", "probability": 0.7665201822916666}, {"start": 531.28, "end": 531.7, "word": " الان", "probability": 0.765869140625}, {"start": 531.7, "end": 532.16, "word": " B", "probability": 0.326416015625}], "temperature": 1.0}, {"id": 26, "seek": 55680, "start": 532.83, "end": 556.81, "text": "أخدوا أي partition X0 X1 لعند Xn هذا any partition لإيه ال interval اللي هي الفترة مين Zero و واحد يعني بمعنى أتيت للفترة Zero و واحد و جزقتها X0 X1 لعند مؤصل لعند مين لعند Xn اللي هي إيش بتساوي بتساوي واحد", "tokens": [10721, 9778, 3215, 14407, 36632, 24808, 1783, 15, 1783, 16, 5296, 3615, 41260, 1783, 77, 23758, 604, 24808, 5296, 28814, 1829, 3224, 2423, 15035, 13672, 1829, 39896, 27188, 2655, 25720, 3714, 9957, 17182, 4032, 36764, 24401, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 5551, 31371, 2655, 24976, 5172, 2655, 25720, 17182, 4032, 36764, 24401, 4032, 10874, 11622, 38149, 11296, 1783, 15, 1783, 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539.31, "end": 539.61, "word": " any", "probability": 0.388427734375}, {"start": 539.61, "end": 540.17, "word": " partition", "probability": 0.93798828125}, {"start": 540.17, "end": 541.57, "word": " لإيه", "probability": 0.577880859375}, {"start": 541.57, "end": 541.73, "word": " ال", "probability": 0.59912109375}, {"start": 541.73, "end": 542.27, "word": " interval", "probability": 0.64697265625}, {"start": 542.27, "end": 542.55, "word": " اللي", "probability": 0.6539306640625}, {"start": 542.55, "end": 542.73, "word": " هي", "probability": 0.77099609375}, {"start": 542.73, "end": 543.15, "word": " الفترة", "probability": 0.9724934895833334}, {"start": 543.15, "end": 543.57, "word": " مين", "probability": 0.56915283203125}, {"start": 543.57, "end": 544.35, "word": " Zero", "probability": 0.166748046875}, {"start": 544.35, "end": 544.77, "word": " و", "probability": 0.94580078125}, {"start": 544.77, "end": 545.07, "word": " واحد", "probability": 0.891845703125}, {"start": 545.07, 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L عندى الان بدى احسب ال L", "tokens": [6027, 7649, 23758, 2423, 24808, 1975, 9778, 3215, 47395, 19071, 3289, 13672, 7578, 31439, 24808, 5296, 5172, 2655, 25720, 441, 43242, 7578, 2423, 7649, 47525, 7578, 1975, 5016, 35457, 2423, 441], "avg_logprob": -0.39892579056322575, "compression_ratio": 1.201834862385321, "no_speech_prob": 4.172325134277344e-07, "words": [{"start": 557.35, "end": 557.79, "word": "الان", "probability": 0.759521484375}, {"start": 557.79, "end": 558.51, "word": " هذا", "probability": 0.69677734375}, {"start": 558.51, "end": 558.61, "word": " ال", "probability": 0.78515625}, {"start": 558.61, "end": 558.95, "word": " partition", "probability": 0.357177734375}, {"start": 558.95, "end": 559.55, "word": " اخدته", "probability": 0.7041015625}, {"start": 559.55, "end": 560.31, "word": " arbitrarily", "probability": 0.9189453125}, {"start": 560.31, "end": 561.21, "word": " اللى", "probability": 0.663330078125}, {"start": 561.21, "end": 561.49, "word": " هو", "probability": 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integrable أكيد لكل إبسلون هلجي بي إبسلون بحيث أن هذا يتحقق خلونا نيجي الآن نبرهم و نشوف كيف بدنا نبرهم نظريتنا", "tokens": [2407, 6027, 39184, 479, 8717, 995, 4587, 9381, 2423, 39184, 441, 4724, 1829, 11933, 3555, 3794, 1211, 11536, 4032, 479, 7251, 30544, 5551, 9381, 17082, 2288, 9154, 3714, 9957, 9154, 11933, 3555, 3794, 1211, 11536, 6024, 48506, 11933, 15730, 25961, 5296, 7435, 1829, 8315, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 5296, 7435, 1829, 8315, 4724, 1829, 11933, 3555, 3794, 1211, 11536, 4724, 5016, 1829, 12984, 23758, 7251, 2655, 5016, 4587, 4587, 20449, 8315, 47395, 479, 307, 16200, 424, 638, 293, 2615, 736, 498, 479, 307, 16200, 424, 638, 5551, 4117, 25708, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 8032, 1211, 7435, 1829, 4724, 1829, 11933, 3555, 3794, 1211, 11536, 4724, 5016, 1829, 12984, 14739, 23758, 7251, 2655, 5016, 4587, 4587, 16490, 1211, 2407, 8315, 8717, 1829, 7435, 1829, 6024, 48506, 8717, 26890, 16095, 4032, 8717, 8592, 38688, 9122, 33911, 47525, 8315, 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"start": 1069.54, "end": 1096.8, "text": "الان بدنا نفترض ان F اي شمالها is integrable ونصل منها للي عيدي اللي هي ال partition اللي مذكور مدام F is integrable اذا كانت ..الان بنقول suppose that F is integrable مدام integrable يا شباب", "tokens": [6027, 7649, 47525, 8315, 8717, 5172, 2655, 43042, 16472, 479, 1975, 1829, 13412, 2304, 6027, 11296, 307, 16200, 424, 638, 4032, 1863, 36520, 9154, 11296, 5296, 20292, 6225, 1829, 16254, 13672, 1829, 39896, 2423, 24808, 13672, 1829, 3714, 8848, 4117, 13063, 3714, 3215, 10943, 479, 307, 16200, 424, 638, 1975, 15730, 25961, 2655, 4386, 6027, 7649, 44945, 39648, 7297, 300, 479, 307, 16200, 424, 638, 3714, 3215, 10943, 3572, 712, 35186, 13412, 3555, 16758], "avg_logprob": -0.2714583396911621, "compression_ratio": 1.5862068965517242, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1069.54, "end": 1070.02, "word": "الان", "probability": 0.7315673828125}, {"start": 1070.02, "end": 1070.9, "word": " بدنا", "probability": 0.8486328125}, 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{"start": 1079.1, "end": 1079.46, "word": " هي", "probability": 0.47998046875}, {"start": 1079.46, "end": 1079.86, "word": " ال", "probability": 0.818359375}, {"start": 1079.86, "end": 1080.24, "word": " partition", "probability": 0.5419921875}, {"start": 1080.24, "end": 1080.72, "word": " اللي", "probability": 0.899169921875}, {"start": 1080.72, "end": 1081.18, "word": " مذكور", "probability": 0.85791015625}, {"start": 1081.18, "end": 1082.24, "word": " مدام", "probability": 0.7459309895833334}, {"start": 1082.24, "end": 1082.74, "word": " F", "probability": 0.79833984375}, {"start": 1082.74, "end": 1083.12, "word": " is", "probability": 0.91455078125}, {"start": 1083.12, "end": 1084.2, "word": " integrable", "probability": 0.89501953125}, {"start": 1084.2, "end": 1085.1, "word": " اذا", "probability": 0.6558837890625}, {"start": 1085.1, "end": 1086.34, "word": " كانت", "probability": 0.99365234375}, {"start": 1086.34, "end": 1087.04, "word": " ..الان", "probability": 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إيش؟ Of F، مظبوط؟ أكيد الـ L of .. الـ U of F بساوي الـ L of F إيش اللي بيدثبته؟ بيدثبته لأي إبسلون أكبر من سفر بدلاجي بإبسلون، شوفوا كيف بنلاجيه، الآن نفترض إن إبسلون let إبسلون أكبر من سفر be given، ماشي الحالة", "tokens": [10721, 4117, 25708, 43242, 3224, 13672, 1829, 31439, 2423, 39184, 624, 295, 479, 4724, 3794, 995, 45865, 441, 295, 11933, 1829, 8592, 22807, 2720, 479, 12399, 3714, 19913, 3555, 2407, 9566, 22807, 5551, 4117, 25708, 2423, 39184, 441, 295, 4386, 2423, 39184, 624, 295, 479, 4724, 3794, 995, 45865, 2423, 39184, 441, 295, 479, 11933, 1829, 8592, 13672, 1829, 4724, 25708, 12984, 3555, 47395, 22807, 4724, 25708, 12984, 3555, 47395, 5296, 10721, 1829, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 47525, 1211, 26108, 1829, 4724, 28814, 3555, 3794, 1211, 11536, 12399, 13412, 38688, 14407, 9122, 33911, 44945, 15040, 7435, 1829, 3224, 12399, 6024, 48506, 8717, 5172, 2655, 43042, 36145, 11933, 3555, 3794, 1211, 11536, 718, 11933, 3555, 3794, 1211, 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"probability": 0.8505859375}, {"start": 1108.97, "end": 1109.19, "word": " of", "probability": 0.98046875}, {"start": 1109.19, "end": 1109.49, "word": " F", "probability": 0.9970703125}, {"start": 1109.49, "end": 1110.23, "word": " بساوي", "probability": 0.941650390625}, {"start": 1110.23, "end": 1110.37, "word": " الـ", "probability": 0.6917724609375}, {"start": 1110.37, "end": 1110.47, "word": " L", "probability": 0.935546875}, {"start": 1110.47, "end": 1110.59, "word": " of", "probability": 0.96533203125}, {"start": 1110.59, "end": 1110.89, "word": " F", "probability": 0.99462890625}, {"start": 1110.89, "end": 1111.73, "word": " إيش", "probability": 0.7797037760416666}, {"start": 1111.73, "end": 1111.87, "word": " اللي", "probability": 0.98388671875}, {"start": 1111.87, "end": 1112.73, "word": " بيدثبته؟", "probability": 0.7488606770833334}, {"start": 1112.73, "end": 1113.17, "word": " بيدثبته", "probability": 0.95546875}, {"start": 1113.17, "end": 1113.41, "word": " لأي", "probability": 0.7613932291666666}, {"start": 1113.41, "end": 1113.71, "word": " إبسلون", "probability": 0.6693115234375}, {"start": 1113.71, "end": 1113.93, "word": " أكبر", "probability": 0.9796549479166666}, {"start": 1113.93, "end": 1114.09, "word": " من", "probability": 0.9970703125}, {"start": 1114.09, "end": 1114.35, "word": " سفر", "probability": 0.8115234375}, {"start": 1114.35, "end": 1114.87, "word": " بدلاجي", "probability": 0.535064697265625}, {"start": 1114.87, "end": 1115.41, "word": " بإبسلون،", "probability": 0.9476841517857143}, {"start": 1115.41, "end": 1115.69, "word": " شوفوا", "probability": 0.92138671875}, {"start": 1115.69, "end": 1115.81, "word": " كيف", "probability": 0.734130859375}, {"start": 1115.81, "end": 1116.81, "word": " بنلاجيه،", "probability": 0.6963297526041666}, {"start": 1116.81, "end": 1117.13, "word": " الآن", "probability": 0.775390625}, {"start": 1117.13, "end": 1117.67, "word": " نفترض", "probability": 0.971923828125}, {"start": 1117.67, 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of F هو إيش يا شباب؟ هو عبارة عن الـ infimum للـ L of B و F such that B element in B of I، مظبوط؟ إذا الـ U of F عبارة عن infimum يعني هو عبارة عن greatest lower bound", "tokens": [3615, 1863, 16254, 2423, 39184, 624, 295, 479, 31439, 11933, 1829, 8592, 35186, 13412, 3555, 16758, 22807, 31439, 6225, 3555, 9640, 3660, 18871, 2423, 39184, 1536, 332, 449, 24976, 39184, 441, 295, 363, 4032, 479, 1270, 300, 363, 4478, 294, 363, 295, 286, 12399, 3714, 19913, 3555, 2407, 9566, 22807, 11933, 15730, 2423, 39184, 624, 295, 479, 6225, 3555, 9640, 3660, 18871, 1536, 332, 449, 37495, 22653, 31439, 6225, 3555, 9640, 3660, 18871, 6636, 3126, 5472], "avg_logprob": -0.15858360931470797, "compression_ratio": 1.455621301775148, "no_speech_prob": 0.0, "words": [{"start": 1125.19, "end": 1125.73, "word": "عندي", "probability": 0.767578125}, {"start": 1125.73, "end": 1126.01, "word": " الـ", "probability": 0.83349609375}, {"start": 1126.01, "end": 1126.13, "word": " U", "probability": 0.56787109375}, 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ايش", "probability": 0.6603597005208334}, {"start": 1227.91, "end": 1228.25, "word": " بيساوي", "probability": 0.9197265625}, {"start": 1228.25, "end": 1228.39, "word": " يا", "probability": 0.8310546875}, {"start": 1228.39, "end": 1228.71, "word": " جماعة", "probability": 0.9832356770833334}, {"start": 1228.71, "end": 1228.85, "word": " ال", "probability": 0.89306640625}, {"start": 1228.85, "end": 1228.97, "word": " U", "probability": 0.9033203125}, {"start": 1228.97, "end": 1229.11, "word": " of", "probability": 0.95703125}, {"start": 1229.11, "end": 1229.43, "word": " F", "probability": 0.99462890625}, {"start": 1229.43, "end": 1230.55, "word": " بيساوي", "probability": 0.84296875}, {"start": 1230.55, "end": 1230.71, "word": " ال", "probability": 0.962890625}, {"start": 1230.71, "end": 1231.37, "word": " infimum", "probability": 0.7484537760416666}, {"start": 1231.37, "end": 1231.77, "word": " لل", "probability": 0.485107421875}, {"start": 1231.77, "end": 1232.13, "word": " L", 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1258.54, "text": "ماشي الحال الان ايش معناه انه هذا infimum معناته هذا هو عبارة عن ال greatest lower bound مدام ال greatest lower bound اذا ال lower bound هذا", "tokens": [2304, 33599, 1829, 21542, 6027, 2423, 7649, 1975, 1829, 8592, 20449, 8315, 3224, 16472, 3224, 23758, 1536, 332, 449, 20449, 8315, 47395, 23758, 31439, 6225, 3555, 9640, 3660, 18871, 2423, 6636, 3126, 5472, 3714, 3215, 10943, 2423, 6636, 3126, 5472, 1975, 15730, 2423, 3126, 5472, 23758], "avg_logprob": -0.3124999911227125, "compression_ratio": 1.558139534883721, "no_speech_prob": 9.953975677490234e-06, "words": [{"start": 1243.72, "end": 1244.04, "word": "ماشي", "probability": 0.8440755208333334}, {"start": 1244.04, "end": 1244.4, "word": " الحال", "probability": 0.959716796875}, {"start": 1244.4, "end": 1245.38, "word": " الان", "probability": 0.581298828125}, {"start": 1245.38, "end": 1245.74, "word": " ايش", "probability": 0.7503255208333334}, {"start": 1245.74, "end": 1246.38, "word": " معناه", "probability": 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مثلا بالذنب يابسون على اتنين الـ game بتعرفوا ليش يابسون على اتنين يعني لو ضفتله أي رقم بيبطل Lower Bound إيش معناته بيبطل Lower Bound يعني هيصير هذا أكبر من ال UB 1 of F for some B وحدها", "tokens": [10721, 2407, 2423, 39184, 3769, 377, 25523, 363, 554, 45164, 48812, 5172, 2655, 43761, 36632, 12602, 4587, 2304, 7251, 16758, 3794, 11536, 15844, 1975, 2655, 1863, 9957, 50113, 15040, 20666, 8848, 1863, 3555, 7251, 16758, 3794, 11536, 15844, 1975, 2655, 1863, 9957, 2423, 39184, 1216, 39894, 3615, 28480, 14407, 32239, 8592, 7251, 16758, 3794, 11536, 15844, 1975, 2655, 1863, 9957, 37495, 22653, 45164, 48812, 5172, 2655, 43761, 36632, 12602, 4587, 2304, 4724, 1829, 3555, 9566, 1211, 25523, 363, 554, 11933, 1829, 8592, 20449, 8315, 47395, 4724, 1829, 3555, 9566, 1211, 25523, 363, 554, 37495, 22653, 39896, 9381, 13546, 23758, 5551, 4117, 26890, 9154, 2423, 624, 33, 502, 295, 479, 337, 512, 363, 4032, 24401, 11296], "avg_logprob": -0.23343210590296778, "compression_ratio": 1.875, 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{"start": 1273.83, "end": 1274.19, "word": " Lower", "probability": 0.87109375}, {"start": 1274.19, "end": 1274.73, "word": " Bound", "probability": 0.9638671875}, {"start": 1274.73, "end": 1275.53, "word": " يعني", "probability": 0.75830078125}, {"start": 1275.53, "end": 1276.29, "word": " هيصير", "probability": 0.88525390625}, {"start": 1276.29, "end": 1276.71, "word": " هذا", "probability": 0.9716796875}, {"start": 1276.71, "end": 1277.37, "word": " أكبر", "probability": 0.9539388020833334}, {"start": 1277.37, "end": 1277.99, "word": " من", "probability": 0.994140625}, {"start": 1277.99, "end": 1278.21, "word": " ال", "probability": 0.9365234375}, {"start": 1278.21, "end": 1279.27, "word": " UB", "probability": 0.4979248046875}, {"start": 1279.27, "end": 1280.07, "word": " 1", "probability": 0.298583984375}, {"start": 1280.07, "end": 1280.57, "word": " of", "probability": 0.869140625}, {"start": 1280.57, "end": 1281.05, "word": " F", "probability": 0.9443359375}, {"start": 1281.05, 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انه suppose that star holds اللي هي star ليه هذه نفترض ان لكل epsilon أكبر من 0 there exists B of epsilon such that U B Epsilon و F نقصها دي أصغر من Epsilon و بدنا نصل من خلالها لإيش لأن ال F is integrable فنشوف", "tokens": [10721, 12984, 3555, 2655, 8315, 13672, 1829, 47525, 8315, 35186, 16472, 3224, 13672, 1829, 31439, 29538, 18863, 15040, 28671, 20328, 5016, 1829, 5016, 3660, 5296, 33456, 8315, 8717, 5172, 2655, 43042, 16472, 3224, 7297, 300, 3543, 9190, 13672, 1829, 39896, 3543, 32239, 3224, 29538, 8717, 5172, 2655, 43042, 16472, 5296, 28820, 17889, 5551, 4117, 26890, 9154, 1958, 456, 8198, 363, 295, 17889, 1270, 300, 624, 363, 462, 16592, 4032, 479, 8717, 4587, 9381, 11296, 11778, 1829, 5551, 9381, 17082, 2288, 9154, 462, 16592, 4032, 47525, 8315, 8717, 36520, 9154, 16490, 1211, 6027, 11296, 5296, 28814, 1829, 8592, 5296, 33456, 2423, 479, 307, 16200, 424, 638, 6156, 1863, 8592, 38688], "avg_logprob": -0.2977272862737829, "compression_ratio": 1.6514522821576763, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 1622.18, "end": 1622.88, "word": "أثبتنا", "probability": 0.840576171875}, {"start": 1622.88, "end": 1623.1, "word": " اللي", "probability": 0.627197265625}, {"start": 1623.1, "end": 1623.46, "word": " بدنا", "probability": 0.9169921875}, {"start": 1623.46, "end": 1623.72, "word": " يا", "probability": 0.194091796875}, {"start": 1623.72, "end": 1624.16, "word": " انه", "probability": 0.4786376953125}, {"start": 1624.16, "end": 1624.42, "word": " اللي", "probability": 0.71923828125}, {"start": 1624.42, "end": 1624.6, "word": " هو", "probability": 0.94970703125}, {"start": 1624.6, "end": 1624.8, "word": " هذه", "probability": 0.63671875}, {"start": 1624.8, "end": 1625.16, "word": " العلاقة", "probability": 0.9578450520833334}, {"start": 1625.16, "end": 1626.08, "word": " صحيحة", "probability": 0.901171875}, {"start": 1626.08, "end": 1626.6, "word": " لأننا", "probability": 0.590576171875}, {"start": 1626.6, "end": 1627.1, "word": " نفترض", "probability": 0.9346923828125}, {"start": 1627.1, "end": 1627.62, "word": " انه", "probability": 0.4951171875}, {"start": 1627.62, "end": 1628.4, "word": " suppose", "probability": 0.6474609375}, {"start": 1628.4, "end": 1628.86, "word": " that", "probability": 0.8828125}, {"start": 1628.86, "end": 1629.3, "word": " star", "probability": 0.50830078125}, {"start": 1629.3, "end": 1629.76, "word": " holds", "probability": 0.82373046875}, {"start": 1629.76, "end": 1629.96, "word": " اللي", "probability": 0.95361328125}, {"start": 1629.96, "end": 1630.12, "word": " هي", "probability": 0.8671875}, {"start": 1630.12, "end": 1630.4, "word": " star", "probability": 0.410400390625}, {"start": 1630.4, "end": 1630.72, "word": " ليه", "probability": 0.611328125}, {"start": 1630.72, "end": 1631.02, "word": " هذه", "probability": 0.607421875}, {"start": 1631.02, "end": 1631.9, "word": " نفترض", "probability": 0.9337158203125}, {"start": 1631.9, "end": 1632.04, "word": " ان", "probability": 0.69189453125}, {"start": 1632.04, "end": 1632.38, "word": " لكل", "probability": 0.869384765625}, {"start": 1632.38, "end": 1632.82, "word": " epsilon", "probability": 0.1429443359375}, {"start": 1632.82, "end": 1633.12, "word": " أكبر", "probability": 0.9319661458333334}, {"start": 1633.12, "end": 1633.24, "word": " من", "probability": 0.88818359375}, {"start": 1633.24, "end": 1633.44, "word": " 0", "probability": 0.499267578125}, {"start": 1633.44, "end": 1633.66, "word": " there", "probability": 0.74462890625}, {"start": 1633.66, "end": 1633.96, "word": " exists", "probability": 0.6494140625}, {"start": 1633.96, "end": 1634.16, "word": " B", "probability": 0.363037109375}, {"start": 1634.16, "end": 1634.36, "word": " of", "probability": 0.8115234375}, {"start": 1634.36, "end": 1634.8, "word": " epsilon", "probability": 0.86328125}, {"start": 1634.8, "end": 1635.38, "word": " such", "probability": 0.88818359375}, {"start": 1635.38, "end": 1635.74, "word": " that", "probability": 0.9501953125}, {"start": 1635.74, "end": 1635.92, "word": " U", "probability": 0.669921875}, {"start": 1635.92, "end": 1636.06, "word": " B", "probability": 0.2247314453125}, {"start": 1636.06, "end": 1636.46, "word": " Epsilon", "probability": 0.6431884765625}, {"start": 1636.46, "end": 1636.56, "word": " و", "probability": 0.8095703125}, {"start": 1636.56, "end": 1636.74, "word": " F", "probability": 0.71923828125}, {"start": 1636.74, "end": 1637.38, "word": " نقصها", "probability": 0.7440185546875}, {"start": 1637.38, "end": 1637.44, "word": " دي", "probability": 0.592041015625}, {"start": 1637.44, "end": 1637.76, "word": " أصغر", "probability": 0.9619140625}, {"start": 1637.76, "end": 1637.86, "word": " من", "probability": 0.97998046875}, {"start": 1637.86, "end": 1638.24, "word": " Epsilon", "probability": 0.748291015625}, {"start": 1638.24, "end": 1638.68, "word": " و", "probability": 0.8330078125}, {"start": 1638.68, "end": 1638.94, "word": " بدنا", "probability": 0.8525390625}, {"start": 1638.94, "end": 1639.22, "word": " نصل", "probability": 0.971923828125}, {"start": 1639.22, "end": 1639.42, "word": " من", "probability": 0.99658203125}, {"start": 1639.42, "end": 1639.94, "word": " خلالها", "probability": 0.9847412109375}, {"start": 1639.94, "end": 1640.56, "word": " لإيش", "probability": 0.760009765625}, {"start": 1640.56, "end": 1641.54, "word": " لأن", "probability": 0.613525390625}, {"start": 1641.54, "end": 1641.98, "word": " ال", "probability": 0.77685546875}, {"start": 1641.98, "end": 1642.38, "word": " F", "probability": 0.58935546875}, {"start": 1642.38, "end": 1642.98, "word": " is", "probability": 0.955078125}, {"start": 1642.98, "end": 1644.28, "word": " integrable", "probability": 0.8779296875}, {"start": 1644.28, "end": 1645.76, "word": " فنشوف", "probability": 0.8577880859375}], "temperature": 1.0}, {"id": 73, "seek": 168416, "start": 1660.48, "end": 1684.16, "text": "البرهام بسيط لو طلعنا عليه مباشرة على اللوح الآن بدنا نفترض أن هذه تتحقق اللي هو نفترض أنه لكل ي أكبر من 0 يوجد بي إبسلون بحيث أن هذا اللي هي تتحقق علشان أصل بدي أصلكم في النهاية أن L of F هي إيش بتساوي U of F شوف كيف ده صلها", "tokens": [6027, 26890, 3224, 10943, 4724, 3794, 1829, 9566, 45164, 23032, 1211, 3615, 8315, 47356, 3714, 3555, 33599, 25720, 15844, 13672, 2407, 5016, 6024, 48506, 47525, 8315, 8717, 5172, 2655, 43042, 14739, 29538, 6055, 2655, 5016, 4587, 4587, 13672, 1829, 31439, 8717, 5172, 2655, 43042, 14739, 3224, 5296, 28820, 7251, 5551, 4117, 26890, 9154, 1958, 7251, 29245, 3215, 4724, 1829, 11933, 3555, 3794, 1211, 11536, 4724, 5016, 1829, 12984, 14739, 23758, 13672, 1829, 39896, 6055, 2655, 5016, 4587, 4587, 11203, 8592, 7649, 5551, 36520, 4724, 16254, 5551, 36520, 24793, 8978, 28239, 11296, 10632, 14739, 441, 295, 479, 39896, 11933, 1829, 8592, 39894, 3794, 995, 45865, 624, 295, 479, 13412, 38688, 9122, 33911, 11778, 3224, 20328, 1211, 11296], "avg_logprob": -0.20232371591095233, "compression_ratio": 1.7889908256880733, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 1660.48, "end": 1661.04, "word": "البرهام", "probability": 0.70050048828125}, {"start": 1661.04, "end": 1661.46, "word": " بسيط", "probability": 0.9910888671875}, {"start": 1661.46, "end": 1661.6, "word": " لو", "probability": 0.9033203125}, {"start": 1661.6, "end": 1662.06, "word": " طلعنا", "probability": 0.9676513671875}, {"start": 1662.06, "end": 1662.34, "word": " عليه", "probability": 0.98291015625}, {"start": 1662.34, "end": 1663.1, "word": " مباشرة", "probability": 0.935546875}, {"start": 1663.1, "end": 1664.56, "word": " على", "probability": 0.822265625}, {"start": 1664.56, "end": 1665.08, "word": " اللوح", "probability": 0.9855143229166666}, {"start": 1665.08, "end": 1667.44, "word": " الآن", "probability": 0.5789794921875}, {"start": 1667.44, "end": 1668.0, "word": " بدنا", "probability": 0.720458984375}, {"start": 1668.0, "end": 1668.58, "word": " نفترض", "probability": 0.9874267578125}, {"start": 1668.58, "end": 1668.74, "word": " أن", "probability": 0.5087890625}, {"start": 1668.74, "end": 1669.0, "word": " هذه", "probability": 0.30029296875}, {"start": 1669.0, "end": 1669.8, "word": " تتحقق", "probability": 0.98388671875}, {"start": 1669.8, "end": 1670.4, "word": " اللي", "probability": 0.7177734375}, {"start": 1670.4, "end": 1670.5, "word": " هو", "probability": 0.97265625}, {"start": 1670.5, "end": 1670.96, "word": " نفترض", "probability": 0.985595703125}, {"start": 1670.96, "end": 1671.42, "word": " أنه", "probability": 0.777587890625}, {"start": 1671.42, "end": 1671.88, "word": " لكل", "probability": 0.9716796875}, {"start": 1671.88, "end": 1672.14, "word": " ي", "probability": 0.256103515625}, {"start": 1672.14, "end": 1672.5, "word": " أكبر", "probability": 0.9503580729166666}, {"start": 1672.5, "end": 1672.68, "word": " من", "probability": 0.99560546875}, {"start": 1672.68, "end": 1673.02, "word": " 0", "probability": 0.365478515625}, {"start": 1673.02, "end": 1673.88, "word": " يوجد", "probability": 0.935546875}, {"start": 1673.88, "end": 1674.14, "word": " بي", "probability": 0.928955078125}, {"start": 1674.14, "end": 1674.64, "word": " إبسلون", "probability": 0.86865234375}, {"start": 1674.64, "end": 1675.04, "word": " بحيث", "probability": 0.919921875}, {"start": 1675.04, "end": 1675.2, "word": " أن", "probability": 0.83642578125}, {"start": 1675.2, "end": 1675.58, "word": " هذا", "probability": 0.6630859375}, {"start": 1675.58, "end": 1675.9, "word": " اللي", "probability": 0.927001953125}, {"start": 1675.9, "end": 1676.16, "word": " هي", "probability": 0.85400390625}, {"start": 1676.16, "end": 1677.02, "word": " تتحقق", "probability": 0.9892578125}, {"start": 1677.02, "end": 1678.36, "word": " علشان", "probability": 0.76220703125}, {"start": 1678.36, "end": 1678.74, "word": " أصل", "probability": 0.9267578125}, {"start": 1678.74, "end": 1679.04, "word": " بدي", "probability": 0.4677734375}, {"start": 1679.04, "end": 1679.66, "word": " أصلكم", "probability": 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9381, 17082, 2288, 5551, 3794, 995, 45865, 441, 295, 479, 4032, 2423, 624, 295, 479, 5551, 9381, 17082, 2288, 5551, 3794, 995, 45865, 3714, 9957, 2423, 624, 363, 295, 479, 4032, 46958, 5016, 22807, 11933, 15730, 5551, 9381, 9640, 6225, 1829, 16254, 6024, 48506, 441, 7251, 43020, 5551, 2407, 11242, 5016, 23275, 2304, 15844, 13672, 2407, 5016, 441, 295, 363, 295, 479, 11778, 3224, 2423, 363, 36632, 24808, 5551, 9381, 17082, 2288], "avg_logprob": -0.25614753047950933, "compression_ratio": 1.7167381974248928, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 1684.61, "end": 1684.85, "word": "اللي", "probability": 0.859619140625}, {"start": 1684.85, "end": 1685.11, "word": " هبدأ", "probability": 0.6607869466145834}, {"start": 1685.11, "end": 1685.59, "word": " أقوللك", "probability": 0.71044921875}, {"start": 1685.59, "end": 1685.81, "word": " اللي", "probability": 0.802490234375}, {"start": 1685.81, "end": 1685.97, "word": " هو", "probability": 0.974609375}, {"start": 1685.97, 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1881.43, "end": 1904.23, "text": "sequence of partitions طبعاً هو هذا معهود التحويل في نظريات مشابهة في حتى في كورسات أخرى خلّينا نشوف عندنا اللي هو النظرية ايش ب .. او الكورلري ايش بتقول بتقول let I بساوة A و B and let F من I لR be a bounded function", "tokens": [11834, 655, 295, 644, 2451, 23032, 3555, 3615, 995, 14111, 31439, 23758, 20449, 3224, 23328, 16712, 5016, 2407, 26895, 8978, 8717, 19913, 16572, 9307, 37893, 16758, 3224, 3660, 8978, 11331, 49975, 8978, 9122, 13063, 3794, 9307, 5551, 34740, 7578, 16490, 1211, 11703, 9957, 995, 8717, 8592, 38688, 43242, 8315, 13672, 1829, 31439, 28239, 19913, 2288, 10632, 1975, 1829, 8592, 4724, 4386, 1975, 2407, 33251, 13063, 1211, 16572, 1975, 1829, 8592, 39894, 39648, 39894, 39648, 718, 286, 4724, 3794, 995, 2407, 3660, 316, 4032, 363, 293, 718, 479, 9154, 286, 5296, 49, 312, 257, 37498, 2445], "avg_logprob": -0.24316405799860755, "compression_ratio": 1.4521739130434783, "no_speech_prob": 9.953975677490234e-06, "words": [{"start": 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اللي هو ال sequence هذه في ال corollary .. في من مشابه .. شيء مشابه في برهان النظرية الأولى اللي قبل بشوية و ياريت تجربوها عندكم خلينا ناخد اللي .. اللي موجود حاليا اللي هو الاتجاه هذا ان لو لجينا sequence of partitions", "tokens": [6027, 20292, 31439, 2423, 39184, 462, 16592, 4724, 3794, 995, 2407, 10632, 36764, 24401, 3660, 5296, 7649, 6156, 3555, 1863, 15040, 38436, 13672, 1829, 31439, 2423, 8310, 29538, 8978, 2423, 1181, 1833, 822, 4386, 8978, 9154, 37893, 16758, 3224, 4386, 44049, 38207, 37893, 16758, 3224, 8978, 4724, 2288, 3224, 7649, 28239, 19913, 2288, 10632, 16247, 12610, 7578, 13672, 1829, 12174, 36150, 4724, 8592, 2407, 10632, 4032, 7251, 9640, 36081, 6055, 7435, 25513, 2407, 11296, 43242, 24793, 16490, 20292, 8315, 8717, 47283, 3215, 13672, 1829, 4386, 13672, 1829, 3714, 29245, 23328, 11331, 6027, 25528, 13672, 1829, 31439, 2423, 9307, 7435, 40294, 23758, 16472, 45164, 5296, 7435, 1829, 8315, 8310, 295, 644, 2451], "avg_logprob": -0.27525111261223045, "compression_ratio": 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"end": 1957.66, "word": " sequence", "probability": 0.611328125}, {"start": 1957.66, "end": 1958.24, "word": " هذه", "probability": 0.481201171875}, {"start": 1958.24, "end": 1958.92, "word": " في", "probability": 0.6611328125}, {"start": 1958.92, "end": 1959.14, "word": " ال", "probability": 0.677734375}, {"start": 1959.14, "end": 1959.72, "word": " corollary", "probability": 0.6469930013020834}, {"start": 1959.72, "end": 1959.82, "word": " ..", "probability": 0.4609375}, {"start": 1959.82, "end": 1959.96, "word": " في", "probability": 0.75244140625}, {"start": 1959.96, "end": 1960.36, "word": " من", "probability": 0.69287109375}, {"start": 1960.36, "end": 1961.8, "word": " مشابه", "probability": 0.6602376302083334}, {"start": 1961.8, "end": 1961.8, "word": " ..", "probability": 0.7939453125}, {"start": 1961.8, "end": 1962.38, "word": " شيء", "probability": 0.666015625}, {"start": 1962.38, "end": 1962.9, "word": " مشابه", "probability": 0.98876953125}, {"start": 1962.9, "end": 1963.16, "word": " في", "probability": 0.8359375}, {"start": 1963.16, "end": 1963.5, "word": " برهان", "probability": 0.9024658203125}, {"start": 1963.5, "end": 1963.96, "word": " النظرية", "probability": 0.9599609375}, {"start": 1963.96, "end": 1964.82, "word": " الأولى", "probability": 0.97802734375}, {"start": 1964.82, "end": 1965.02, "word": " اللي", "probability": 0.91259765625}, {"start": 1965.02, "end": 1965.26, "word": " قبل", "probability": 0.971923828125}, {"start": 1965.26, "end": 1965.72, "word": " بشوية", "probability": 0.9617919921875}, {"start": 1965.72, "end": 1965.9, "word": " و", "probability": 0.9609375}, {"start": 1965.9, "end": 1966.22, "word": " ياريت", "probability": 0.71484375}, {"start": 1966.22, "end": 1966.98, "word": " تجربوها", "probability": 0.9638671875}, {"start": 1966.98, "end": 1967.66, "word": " عندكم", "probability": 0.9873046875}, {"start": 1967.66, "end": 1968.34, "word": " خلينا", "probability": 0.781982421875}, {"start": 1968.34, "end": 1968.6, "word": " ناخد", "probability": 0.9523111979166666}, {"start": 1968.6, "end": 1968.8, "word": " اللي", "probability": 0.968994140625}, {"start": 1968.8, "end": 1968.8, "word": " ..", "probability": 0.39990234375}, {"start": 1968.8, "end": 1968.96, "word": " اللي", "probability": 0.994384765625}, {"start": 1968.96, "end": 1969.36, "word": " موجود", "probability": 0.9890950520833334}, {"start": 1969.36, "end": 1969.88, "word": " حاليا", "probability": 0.70849609375}, {"start": 1969.88, "end": 1970.9, "word": " اللي", "probability": 0.6544189453125}, {"start": 1970.9, "end": 1971.12, "word": " هو", "probability": 0.91064453125}, {"start": 1971.12, "end": 1971.8, "word": " الاتجاه", "probability": 0.9072265625}, {"start": 1971.8, "end": 1972.02, "word": " هذا", "probability": 0.95654296875}, {"start": 1972.02, "end": 1972.16, "word": " ان", "probability": 0.27392578125}, {"start": 1972.16, "end": 1972.34, "word": " لو", "probability": 0.703125}, {"start": 1972.34, "end": 1972.88, "word": " لجينا", "probability": 0.69940185546875}, {"start": 1972.88, "end": 1973.54, "word": " sequence", "probability": 0.96875}, {"start": 1973.54, "end": 1974.16, "word": " of", "probability": 0.982421875}, {"start": 1974.16, "end": 1974.78, "word": " partitions", "probability": 0.96435546875}], "temperature": 1.0}, {"id": 86, "seek": 198897, "start": 1975.55, "end": 1988.97, "text": "وكان ال limit لل U P N و F نقص ال P N و F بساوة سفر إذا هتكون F is integrable و هتكون ال limit للأولى بساوة limit للثانية بساوة قيمة ال integration", "tokens": [2407, 41361, 2423, 4948, 24976, 624, 430, 426, 4032, 479, 8717, 4587, 9381, 2423, 430, 426, 4032, 479, 4724, 3794, 995, 2407, 3660, 8608, 5172, 2288, 11933, 15730, 8032, 2655, 30544, 479, 307, 16200, 424, 638, 4032, 8032, 2655, 30544, 2423, 4948, 24976, 10721, 12610, 7578, 4724, 3794, 995, 2407, 3660, 4948, 24976, 12984, 7649, 10632, 4724, 3794, 995, 2407, 3660, 12174, 32640, 3660, 2423, 10980], "avg_logprob": -0.30876865582679636, "compression_ratio": 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P", "probability": 0.9443359375}, {"start": 1979.19, "end": 1979.33, "word": " N", "probability": 0.609375}, {"start": 1979.33, "end": 1979.43, "word": " و", "probability": 0.44287109375}, {"start": 1979.43, "end": 1979.63, "word": " F", "probability": 0.8251953125}, {"start": 1979.63, "end": 1980.03, "word": " بساوة", "probability": 0.653564453125}, {"start": 1980.03, "end": 1980.37, "word": " سفر", "probability": 0.7236735026041666}, {"start": 1980.37, "end": 1981.03, "word": " إذا", "probability": 0.6217041015625}, {"start": 1981.03, "end": 1981.39, "word": " هتكون", "probability": 0.7235921223958334}, {"start": 1981.39, "end": 1981.55, "word": " F", "probability": 0.7333984375}, {"start": 1981.55, "end": 1981.69, "word": " is", "probability": 0.9169921875}, {"start": 1981.69, "end": 1982.25, "word": " integrable", "probability": 0.81298828125}, {"start": 1982.25, "end": 1982.37, "word": " و", "probability": 0.8330078125}, {"start": 1982.37, "end": 1982.73, "word": " هتكون", "probability": 0.7281901041666666}, {"start": 1982.73, "end": 1982.87, "word": " ال", "probability": 0.63134765625}, {"start": 1982.87, "end": 1983.25, "word": " limit", "probability": 0.96630859375}, {"start": 1983.25, "end": 1984.49, "word": " للأولى", "probability": 0.890625}, {"start": 1984.49, "end": 1984.91, "word": " بساوة", "probability": 0.96865234375}, {"start": 1984.91, "end": 1985.17, "word": " limit", "probability": 0.51416015625}, {"start": 1985.17, "end": 1985.89, "word": " للثانية", "probability": 0.96240234375}, {"start": 1985.89, "end": 1986.61, "word": " بساوة", "probability": 0.9810546875}, {"start": 1986.61, "end": 1986.99, "word": " قيمة", "probability": 0.9939778645833334}, {"start": 1986.99, "end": 1987.99, "word": " ال", "probability": 0.7001953125}, {"start": 1987.99, "end": 1988.97, "word": " integration", "probability": 0.94970703125}], "temperature": 1.0}, {"id": 87, "seek": 201892, "start": 1992.9, "end": 2018.92, "text": "عند ما أعطيني limit هذا إيش بساوي؟ سفر خلّينا ندخل على التعريف مباشرة تعريف ال limit بتعرفوا تعريف ال limit يا شباب؟ اللي هو لكل إبسلون أكبر من سفر there exist k such that لكل أن أكبر سوى k بيصير هذا ناقص هذا أصغر من مين؟ من اللي هو إبسلون وهذا على طول يعطينا as integral خلّيش تشوف أيش اللي بيقوله لأن since عند ما أعطيني limit", "tokens": [3615, 41260, 19446, 5551, 3615, 9566, 9957, 1829, 4948, 23758, 11933, 1829, 8592, 4724, 3794, 995, 45865, 22807, 8608, 5172, 2288, 16490, 1211, 11703, 9957, 995, 8717, 3215, 9778, 1211, 15844, 16712, 3615, 16572, 5172, 3714, 3555, 33599, 25720, 37279, 16572, 5172, 2423, 4948, 39894, 3615, 28480, 14407, 37279, 16572, 5172, 2423, 4948, 35186, 13412, 3555, 16758, 22807, 13672, 1829, 31439, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 456, 2514, 350, 1270, 300, 5296, 28820, 14739, 5551, 4117, 26890, 8608, 2407, 7578, 350, 4724, 1829, 9381, 13546, 23758, 8717, 995, 4587, 9381, 23758, 5551, 9381, 17082, 2288, 9154, 3714, 9957, 22807, 9154, 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1.0}, {"id": 109, "seek": 255306, "start": 2533.86, "end": 2553.06, "text": "لأ نيجي بدنا نبرهن اللي هو المثال اللي برهناه المرة الماضية بالتعريف بدنا نبرهنه بواسطة اللي هي ال corollary اللي عندنا بدنا نبرهن اللي هو اللي هي نثبت أنه", "tokens": [1211, 10721, 8717, 1829, 7435, 1829, 47525, 8315, 8717, 26890, 3224, 1863, 13672, 1829, 31439, 9673, 12984, 6027, 13672, 1829, 4724, 2288, 3224, 8315, 3224, 9673, 25720, 9673, 46958, 10632, 20666, 2655, 3615, 16572, 5172, 47525, 8315, 8717, 26890, 3224, 1863, 3224, 4724, 14407, 3794, 9566, 3660, 13672, 1829, 39896, 2423, 1181, 1833, 822, 13672, 1829, 43242, 8315, 47525, 8315, 8717, 26890, 3224, 1863, 13672, 1829, 31439, 13672, 1829, 39896, 8717, 12984, 3555, 2655, 14739, 3224], "avg_logprob": -0.1412337716523703, "compression_ratio": 1.9361702127659575, "no_speech_prob": 1.6093254089355469e-06, "words": [{"start": 2533.86, "end": 2534.2, "word": "لأ", "probability": 0.697998046875}, {"start": 2534.2, "end": 2534.7, "word": " نيجي", "probability": 0.7794189453125}, {"start": 2534.7, "end": 2536.26, "word": " بدنا", "probability": 0.6842041015625}, {"start": 2536.26, "end": 2537.92, "word": " نبرهن", "probability": 0.84368896484375}, {"start": 2537.92, "end": 2539.04, "word": " اللي", "probability": 0.933837890625}, {"start": 2539.04, "end": 2539.4, "word": " هو", "probability": 0.98974609375}, {"start": 2539.4, "end": 2540.54, "word": " المثال", "probability": 0.9602864583333334}, {"start": 2540.54, "end": 2541.08, "word": " اللي", "probability": 0.974853515625}, {"start": 2541.08, "end": 2541.7, "word": " برهناه", "probability": 0.848974609375}, {"start": 2541.7, "end": 2541.98, "word": " المرة", "probability": 0.90771484375}, {"start": 2541.98, "end": 2542.66, "word": " الماضية", "probability": 0.9925130208333334}, {"start": 2542.66, "end": 2543.88, "word": " بالتعريف", "probability": 0.9822265625}, {"start": 2543.88, "end": 2544.14, "word": " بدنا", "probability": 0.855224609375}, {"start": 2544.14, "end": 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"end": 2551.78, "word": " اللي", "probability": 0.833251953125}, {"start": 2551.78, "end": 2552.04, "word": " هي", "probability": 0.8818359375}, {"start": 2552.04, "end": 2552.6, "word": " نثبت", "probability": 0.970458984375}, {"start": 2552.6, "end": 2553.06, "word": " أنه", "probability": 0.7271728515625}], "temperature": 1.0}, {"id": 110, "seek": 257305, "start": 2556.17, "end": 2573.05, "text": "نثبت أن F of X بساوة X يا شباب عبارة عن Integrable أو اللي سمناها G of X بساوة X is Integrable على الفترة Zero واحد المرة الماضية أثبتناها كيف أثبتناها زي ما أنتم متذكرين بواسط التعريف", "tokens": [1863, 12984, 3555, 2655, 14739, 479, 295, 1783, 4724, 3794, 995, 2407, 3660, 1783, 35186, 13412, 3555, 16758, 6225, 3555, 9640, 3660, 18871, 5681, 1146, 424, 638, 34051, 13672, 1829, 8608, 2304, 8315, 11296, 460, 295, 1783, 4724, 3794, 995, 2407, 3660, 1783, 307, 5681, 1146, 424, 638, 15844, 27188, 2655, 25720, 17182, 36764, 24401, 9673, 25720, 9673, 46958, 10632, 5551, 12984, 3555, 2655, 8315, 11296, 9122, 33911, 5551, 12984, 3555, 2655, 8315, 11296, 30767, 1829, 19446, 14739, 39237, 44650, 8848, 37983, 9957, 4724, 14407, 3794, 9566, 16712, 3615, 16572, 5172], "avg_logprob": -0.1830842447993548, "compression_ratio": 1.6536312849162011, "no_speech_prob": 4.827976226806641e-06, "words": [{"start": 2556.1700000000005, "end": 2556.8500000000004, "word": "نثبت", "probability": 0.85491943359375}, {"start": 2556.8500000000004, "end": 2557.53, "word": " أن", "probability": 0.505859375}, {"start": 2557.53, "end": 2557.85, "word": " F", "probability": 0.18896484375}, {"start": 2557.85, "end": 2558.07, "word": " of", "probability": 0.1610107421875}, {"start": 2558.07, "end": 2558.25, "word": " X", "probability": 0.974609375}, {"start": 2558.25, "end": 2558.69, "word": " بساوة", "probability": 0.589111328125}, {"start": 2558.69, "end": 2558.91, "word": " X", "probability": 0.97314453125}, {"start": 2558.91, "end": 2559.07, "word": " يا", "probability": 0.66259765625}, {"start": 2559.07, "end": 2559.63, "word": " شباب", "probability": 0.9915364583333334}, {"start": 2559.63, "end": 2560.65, "word": " عبارة", "probability": 0.9876708984375}, {"start": 2560.65, "end": 2560.99, "word": " عن", "probability": 0.9931640625}, {"start": 2560.99, "end": 2562.07, "word": " Integrable", "probability": 0.7947998046875}, {"start": 2562.07, "end": 2562.31, "word": " أو", "probability": 0.7080078125}, {"start": 2562.31, "end": 2562.51, "word": " اللي", "probability": 0.746826171875}, {"start": 2562.51, "end": 2562.99, "word": " سمناها", "probability": 0.815185546875}, {"start": 2562.99, "end": 2563.15, "word": " G", "probability": 0.63818359375}, {"start": 2563.15, "end": 2563.29, "word": " of", "probability": 0.80126953125}, {"start": 2563.29, "end": 2563.63, "word": " X", "probability": 0.99658203125}, {"start": 2563.63, "end": 2564.51, "word": " بساوة", "probability": 0.98046875}, {"start": 2564.51, "end": 2564.77, "word": " X", "probability": 0.97802734375}, {"start": 2564.77, "end": 2564.95, "word": " is", "probability": 0.68896484375}, {"start": 2564.95, "end": 2565.55, "word": " Integrable", "probability": 0.8748779296875}, {"start": 2565.55, "end": 2565.77, "word": " على", "probability": 0.92138671875}, {"start": 2565.77, "end": 2566.13, "word": " الفترة", "probability": 0.9772135416666666}, {"start": 2566.13, "end": 2566.37, "word": " Zero", "probability": 0.5146484375}, {"start": 2566.37, "end": 2566.67, "word": " واحد", "probability": 0.8330078125}, {"start": 2566.67, "end": 2567.35, "word": " المرة", "probability": 0.89697265625}, {"start": 2567.35, "end": 2567.71, "word": " الماضية", "probability": 0.98095703125}, {"start": 2567.71, "end": 2568.33, "word": " أثبتناها", "probability": 0.8650716145833334}, {"start": 2568.33, "end": 2568.83, "word": " كيف", "probability": 0.919677734375}, {"start": 2568.83, "end": 2570.23, "word": " أثبتناها", "probability": 0.9192708333333334}, {"start": 2570.23, "end": 2570.47, "word": " زي", "probability": 0.937744140625}, {"start": 2570.47, "end": 2570.53, "word": " ما", "probability": 0.9619140625}, {"start": 2570.53, "end": 2570.73, "word": " أنتم", "probability": 0.640380859375}, {"start": 2570.73, "end": 2571.67, "word": " متذكرين", "probability": 0.829345703125}, {"start": 2571.67, "end": 2572.33, "word": " بواسط", "probability": 0.9693603515625}, {"start": 2572.33, "end": 2573.05, "word": " التعريف", "probability": 0.9212646484375}], "temperature": 1.0}, {"id": 111, "seek": 258589, "start": 2573.79, "end": 2585.89, "text": "مظبوط؟ طيب جيبنا اللي هو ال upper sum و ال lower sum و بعدين جيبنا ال upper integral و ال lower integral و أثبتنا ان ال upper integral بساوي ال lower integral و خلصناه", "tokens": [2304, 19913, 3555, 2407, 9566, 22807, 23032, 1829, 3555, 10874, 1829, 3555, 8315, 13672, 1829, 31439, 2423, 6597, 2408, 4032, 2423, 3126, 2408, 4032, 39182, 9957, 10874, 1829, 3555, 8315, 2423, 6597, 11573, 4032, 2423, 3126, 11573, 4032, 5551, 12984, 3555, 2655, 8315, 16472, 2423, 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2578.01, "word": " ال", "probability": 0.46240234375}, {"start": 2578.01, "end": 2578.21, "word": " lower", "probability": 0.83935546875}, {"start": 2578.21, "end": 2578.51, "word": " sum", "probability": 0.978515625}, {"start": 2578.51, "end": 2578.61, "word": " و", "probability": 0.8349609375}, {"start": 2578.61, "end": 2578.89, "word": " بعدين", "probability": 0.93798828125}, {"start": 2578.89, "end": 2579.15, "word": " جيبنا", "probability": 0.990234375}, {"start": 2579.15, "end": 2579.25, "word": " ال", "probability": 0.9228515625}, {"start": 2579.25, "end": 2579.49, "word": " upper", "probability": 0.90087890625}, {"start": 2579.49, "end": 2581.87, "word": " integral", "probability": 0.4345703125}, {"start": 2581.87, "end": 2582.17, "word": " و", "probability": 0.97509765625}, {"start": 2582.17, "end": 2582.25, "word": " ال", "probability": 0.44921875}, {"start": 2582.25, "end": 2582.43, "word": " lower", "probability": 0.90625}, {"start": 2582.43, "end": 2582.93, "word": " integral", "probability": 0.87353515625}, {"start": 2582.93, "end": 2583.09, "word": " و", "probability": 0.93017578125}, {"start": 2583.09, "end": 2583.47, "word": " أثبتنا", "probability": 0.96748046875}, {"start": 2583.47, "end": 2583.55, "word": " ان", "probability": 0.42138671875}, {"start": 2583.55, "end": 2583.61, "word": " ال", "probability": 0.9521484375}, {"start": 2583.61, "end": 2583.77, "word": " upper", "probability": 0.884765625}, {"start": 2583.77, "end": 2584.15, "word": " integral", "probability": 0.86572265625}, {"start": 2584.15, "end": 2584.49, "word": " بساوي", "probability": 0.62628173828125}, {"start": 2584.49, "end": 2584.59, "word": " ال", "probability": 0.88525390625}, {"start": 2584.59, "end": 2584.75, "word": " lower", "probability": 0.93603515625}, {"start": 2584.75, "end": 2585.09, "word": " integral", "probability": 0.83935546875}, {"start": 2585.09, "end": 2585.25, "word": " و", "probability": 0.984375}, {"start": 2585.25, "end": 2585.89, "word": " خلصناه", "probability": 0.93349609375}], "temperature": 1.0}, {"id": 112, "seek": 260342, "start": 2586.36, "end": 2603.42, "text": "الآن بدنا نثبتها بطريقتنا اللي هي على الكورولر اللي جابل بشوية عند g of x ساوة x على الفترة 0 بواحد هي المطلوب اثباتها show that g is integrable على هذه الفترة الآن بديش اعيد اللي حكيته المرة الماضية اللي حكيناه المرة الماضية", "tokens": [6027, 148, 48506, 47525, 8315, 8717, 12984, 3555, 2655, 11296, 4724, 9566, 16572, 38149, 8315, 13672, 1829, 39896, 15844, 33251, 13063, 12610, 2288, 13672, 1829, 10874, 16758, 1211, 4724, 8592, 2407, 10632, 43242, 290, 295, 2031, 8608, 995, 2407, 3660, 2031, 15844, 27188, 2655, 25720, 1958, 4724, 14407, 24401, 39896, 9673, 9566, 1211, 37746, 1975, 12984, 3555, 9307, 11296, 855, 300, 290, 307, 16200, 424, 638, 15844, 29538, 27188, 2655, 25720, 6024, 48506, 4724, 16254, 8592, 1975, 3615, 25708, 13672, 1829, 11331, 4117, 36081, 3224, 9673, 25720, 9673, 46958, 10632, 13672, 1829, 11331, 4117, 1829, 8315, 3224, 9673, 25720, 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{"start": 2589.92, "end": 2590.44, "word": " بشوية", "probability": 0.880859375}, {"start": 2590.44, "end": 2591.38, "word": " عند", "probability": 0.826171875}, {"start": 2591.38, "end": 2591.58, "word": " g", "probability": 0.48486328125}, {"start": 2591.58, "end": 2591.72, "word": " of", "probability": 0.11102294921875}, {"start": 2591.72, "end": 2591.88, "word": " x", "probability": 0.90771484375}, {"start": 2591.88, "end": 2592.22, "word": " ساوة", "probability": 0.438507080078125}, {"start": 2592.22, "end": 2592.4, "word": " x", "probability": 0.95947265625}, {"start": 2592.4, "end": 2592.54, "word": " على", "probability": 0.662109375}, {"start": 2592.54, "end": 2592.82, "word": " الفترة", "probability": 0.9739583333333334}, {"start": 2592.82, "end": 2593.1, "word": " 0", "probability": 0.410888671875}, {"start": 2593.1, "end": 2593.74, "word": " بواحد", "probability": 0.85009765625}, {"start": 2593.74, "end": 2594.2, "word": " هي", "probability": 0.8759765625}, {"start": 2594.2, 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"end": 2600.8, "word": " اعيد", "probability": 0.77392578125}, {"start": 2600.8, "end": 2600.94, "word": " اللي", "probability": 0.949951171875}, {"start": 2600.94, "end": 2601.32, "word": " حكيته", "probability": 0.982177734375}, {"start": 2601.32, "end": 2601.6, "word": " المرة", "probability": 0.947265625}, {"start": 2601.6, "end": 2602.14, "word": " الماضية", "probability": 0.9943033854166666}, {"start": 2602.14, "end": 2602.32, "word": " اللي", "probability": 0.926513671875}, {"start": 2602.32, "end": 2602.68, "word": " حكيناه", "probability": 0.951171875}, {"start": 2602.68, "end": 2602.92, "word": " المرة", "probability": 0.971435546875}, {"start": 2602.92, "end": 2603.42, "word": " الماضية", "probability": 0.9962565104166666}], "temperature": 1.0}, {"id": 113, "seek": 263245, "start": 2603.69, "end": 2632.45, "text": "من الـ Example اللي أثبتنا فيه انها Integra بالبواسطة اللي هي التعريف أخدنا P N اللي هو Zero واحد على N واثنين على N وان نقص واحد على N لعند الواحد أخدنا اللي هو 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"probability": 0.87939453125}, {"start": 2652.01, "end": 2652.71, "word": " P", "probability": 0.95947265625}, {"start": 2652.71, "end": 2653.05, "word": " N", "probability": 0.95166015625}, {"start": 2653.05, "end": 2653.29, "word": " O", "probability": 0.9951171875}, {"start": 2653.29, "end": 2653.71, "word": " G", "probability": 0.9912109375}, {"start": 2653.71, "end": 2654.63, "word": " ولاجناها", "probability": 0.88017578125}, {"start": 2654.63, "end": 2655.01, "word": " عبارة", "probability": 0.9947509765625}, {"start": 2655.01, "end": 2655.23, "word": " عن", "probability": 0.9921875}, {"start": 2655.23, "end": 2655.67, "word": " نص", "probability": 0.993896484375}, {"start": 2655.67, "end": 2656.27, "word": " في", "probability": 0.9619140625}, {"start": 2656.27, "end": 2656.69, "word": " واحد", "probability": 0.990478515625}, {"start": 2656.69, "end": 2657.11, "word": " ناقص", "probability": 0.9686279296875}, {"start": 2657.11, "end": 2657.55, "word": " واحد", "probability": 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"probability": 0.8359375}, {"start": 2824.46, "end": 2824.8, "word": " نص", "probability": 0.993408203125}, {"start": 2824.8, "end": 2825.3, "word": " إذن", "probability": 0.7887369791666666}, {"start": 2825.3, "end": 2825.46, "word": " هذه", "probability": 0.8154296875}, {"start": 2825.46, "end": 2825.8, "word": " طريقة", "probability": 0.9679361979166666}, {"start": 2825.8, "end": 2826.7, "word": " أخرى", "probability": 0.9908854166666666}, {"start": 2826.7, "end": 2827.52, "word": " لحساب", "probability": 0.9609375}, {"start": 2827.52, "end": 2828.0, "word": " اللي", "probability": 0.92919921875}, {"start": 2828.0, "end": 2828.42, "word": " هو", "probability": 0.994140625}, {"start": 2828.42, "end": 2829.4, "word": " أو", "probability": 0.54052734375}, {"start": 2829.4, "end": 2829.92, "word": " لإثبات", "probability": 0.878125}, {"start": 2829.92, "end": 2830.08, "word": " أن", "probability": 0.5888671875}, {"start": 2830.08, "end": 2830.3, "word": " g", "probability": 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functions يعني الدوال اللي بتكون يا increasing على كل الفترة يا decreasing على كل الفترة بقولكم هذا على اللي هي F من F ال function لو كانت bounded لو كانت monotone على اللي هي closed bounded interval I على طول integrable إذن عيلة كبيرة", "tokens": [10721, 12610, 6225, 16373, 37977, 9154, 18863, 16373, 1211, 9307, 9673, 16095, 3660, 13672, 1829, 39896, 2423, 39184, 1108, 310, 546, 6828, 37495, 22653, 32748, 2407, 6027, 13672, 1829, 39894, 30544, 35186, 5662, 15844, 28242, 27188, 2655, 25720, 35186, 23223, 15844, 28242, 27188, 2655, 25720, 4724, 39648, 24793, 23758, 15844, 13672, 1829, 39896, 479, 9154, 479, 2423, 2445, 45164, 25961, 2655, 37498, 45164, 25961, 2655, 1108, 310, 546, 15844, 13672, 1829, 39896, 5395, 37498, 15035, 286, 15844, 23032, 12610, 16200, 424, 638, 11933, 8848, 1863, 6225, 26895, 3660, 9122, 3555, 48923], "avg_logprob": -0.13850203577591025, "compression_ratio": 1.8844444444444444, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2861.64, "end": 2862.1, 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"end": 2874.0, "word": " بقولكم", "probability": 0.9300130208333334}, {"start": 2874.0, "end": 2874.58, "word": " هذا", "probability": 0.97314453125}, {"start": 2874.58, "end": 2875.84, "word": " على", "probability": 0.76318359375}, {"start": 2875.84, "end": 2876.18, "word": " اللي", "probability": 0.9658203125}, {"start": 2876.18, "end": 2876.58, "word": " هي", "probability": 0.91064453125}, {"start": 2876.58, "end": 2877.3, "word": " F", "probability": 0.56201171875}, {"start": 2877.3, "end": 2877.58, "word": " من", "probability": 0.8876953125}, {"start": 2877.58, "end": 2877.9, "word": " F", "probability": 0.2413330078125}, {"start": 2877.9, "end": 2878.8, "word": " ال", "probability": 0.7119140625}, {"start": 2878.8, "end": 2879.32, "word": " function", "probability": 0.685546875}, {"start": 2879.32, "end": 2879.72, "word": " لو", "probability": 0.97119140625}, {"start": 2879.72, "end": 2880.18, "word": " كانت", "probability": 0.9755859375}, {"start": 2880.18, "end": 2880.76, "word": " bounded", "probability": 0.93408203125}, {"start": 2880.76, "end": 2881.56, "word": " لو", "probability": 0.873046875}, {"start": 2881.56, "end": 2882.0, "word": " كانت", "probability": 0.9931640625}, {"start": 2882.0, "end": 2882.56, "word": " monotone", "probability": 0.9444986979166666}, {"start": 2882.56, "end": 2882.96, "word": " على", "probability": 0.845703125}, {"start": 2882.96, "end": 2883.64, "word": " اللي", "probability": 0.97412109375}, {"start": 2883.64, "end": 2884.0, "word": " هي", "probability": 0.9443359375}, {"start": 2884.0, "end": 2885.14, "word": " closed", "probability": 0.62109375}, {"start": 2885.14, "end": 2885.56, "word": " bounded", "probability": 0.8740234375}, {"start": 2885.56, "end": 2886.54, "word": " interval", "probability": 0.96875}, {"start": 2886.54, "end": 2887.1, "word": " I", "probability": 0.9306640625}, {"start": 2887.1, "end": 2888.02, "word": " على", "probability": 0.91015625}, {"start": 2888.02, "end": 2888.46, "word": " طول", "probability": 0.991943359375}, {"start": 2888.46, "end": 2889.3, "word": " integrable", "probability": 0.761962890625}, {"start": 2889.3, "end": 2889.66, "word": " إذن", "probability": 0.4013671875}, {"start": 2889.66, "end": 2889.92, "word": " عيلة", "probability": 0.9171549479166666}, {"start": 2889.92, "end": 2890.3, "word": " كبيرة", "probability": 0.9778645833333334}], "temperature": 1.0}, {"id": 125, "seek": 291332, "start": 2890.84, "end": 2913.32, "text": "عيلة الدوال اللي بتكون يا increasing يا decreasing على كل الفترة a و b هذه مضمون انها تكون الدوال ايه شمالها عبارة عن integrable functions اذا اول اعلان الان اللي هو any monotone function", "tokens": [3615, 26895, 3660, 32748, 2407, 6027, 13672, 1829, 39894, 30544, 35186, 5662, 35186, 23223, 15844, 28242, 27188, 2655, 25720, 257, 4032, 272, 29538, 3714, 11242, 2304, 11536, 16472, 11296, 6055, 30544, 32748, 2407, 6027, 1975, 1829, 3224, 13412, 2304, 6027, 11296, 6225, 3555, 9640, 3660, 18871, 16200, 424, 638, 6828, 1975, 15730, 1975, 12610, 1975, 30241, 7649, 2423, 7649, 13672, 1829, 31439, 604, 1108, 310, 546, 2445], "avg_logprob": -0.21829044621656923, "compression_ratio": 1.5875706214689265, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2890.84, "end": 2891.72, "word": "عيلة", "probability": 0.9122721354166666}, {"start": 2891.72, "end": 2892.8, "word": " الدوال", "probability": 0.9033203125}, {"start": 2892.8, "end": 2893.82, "word": " اللي", "probability": 0.64794921875}, {"start": 2893.82, "end": 2894.44, "word": " بتكون", "probability": 0.958740234375}, {"start": 2894.44, "end": 2894.82, "word": " يا", "probability": 0.529296875}, {"start": 2894.82, "end": 2895.5, "word": " increasing", "probability": 0.91455078125}, {"start": 2895.5, "end": 2896.7, "word": " يا", "probability": 0.7763671875}, {"start": 2896.7, "end": 2897.42, "word": " decreasing", "probability": 0.97705078125}, {"start": 2897.42, "end": 2898.6, "word": " على", "probability": 0.904296875}, {"start": 2898.6, "end": 2898.88, "word": " كل", "probability": 0.984375}, {"start": 2898.88, "end": 2899.5, "word": " الفترة", "probability": 0.9871419270833334}, {"start": 2899.5, "end": 2899.9, "word": " a", "probability": 0.35107421875}, {"start": 2899.9, "end": 2899.98, "word": " و", "probability": 0.90869140625}, {"start": 2899.98, "end": 2900.28, "word": " b", "probability": 0.53857421875}, {"start": 2900.28, "end": 2901.16, "word": " هذه", "probability": 0.71240234375}, {"start": 2901.16, "end": 2901.86, "word": " مضمون", "probability": 0.9796142578125}, {"start": 2901.86, "end": 2902.1, "word": " انها", "probability": 0.707763671875}, {"start": 2902.1, "end": 2902.38, "word": " تكون", "probability": 0.966552734375}, {"start": 2902.38, "end": 2902.78, "word": " الدوال", "probability": 0.91015625}, {"start": 2902.78, "end": 2903.04, "word": " ايه", "probability": 0.6365559895833334}, {"start": 2903.04, "end": 2903.48, "word": " شمالها", "probability": 0.82562255859375}, {"start": 2903.48, "end": 2904.64, "word": " عبارة", "probability": 0.9227294921875}, {"start": 2904.64, "end": 2905.02, "word": " عن", "probability": 0.98876953125}, {"start": 2905.02, "end": 2906.12, "word": " integrable", "probability": 0.7100830078125}, {"start": 2906.12, "end": 2906.98, "word": " functions", "probability": 0.84912109375}, {"start": 2906.98, "end": 2907.64, "word": " اذا", "probability": 0.681884765625}, {"start": 2907.64, "end": 2908.2, "word": " اول", "probability": 0.785400390625}, {"start": 2908.2, "end": 2908.8, "word": " اعلان", "probability": 0.9150390625}, {"start": 2908.8, "end": 2909.38, "word": " الان", "probability": 0.6268310546875}, {"start": 2909.38, "end": 2910.5, "word": " اللي", "probability": 0.949951171875}, {"start": 2910.5, "end": 2910.84, "word": " هو", "probability": 0.94775390625}, {"start": 2910.84, "end": 2912.0, "word": " any", "probability": 0.428955078125}, {"start": 2912.0, "end": 2912.62, "word": " monotone", "probability": 0.88916015625}, {"start": 2912.62, "end": 2913.32, "word": " function", "probability": 0.96337890625}], "temperature": 1.0}, {"id": 126, "seek": 292529, "start": 2914.67, "end": 2925.29, "text": "on a closed bounded interval is integrable وهذا اللي هو عنواننا integrability of monotone functions", "tokens": [266, 257, 5395, 37498, 15035, 307, 16200, 424, 638, 37037, 15730, 13672, 1829, 31439, 18871, 2407, 7649, 8315, 3572, 2310, 295, 1108, 310, 546, 6828], "avg_logprob": -0.3587740258528636, "compression_ratio": 1.1262135922330097, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 2914.67, "end": 2915.09, "word": "on", "probability": 0.02874755859375}, {"start": 2915.09, "end": 2915.73, "word": " a", "probability": 0.97216796875}, {"start": 2915.73, "end": 2916.29, "word": " closed", "probability": 0.75439453125}, {"start": 2916.29, "end": 2916.81, "word": " bounded", "probability": 0.890625}, {"start": 2916.81, "end": 2917.77, "word": " interval", "probability": 0.95751953125}, {"start": 2917.77, "end": 2918.83, "word": " is", "probability": 0.83251953125}, {"start": 2918.83, "end": 2920.13, "word": " integrable", "probability": 0.7224934895833334}, {"start": 2920.13, "end": 2920.71, "word": " وهذا", "probability": 0.6890869140625}, {"start": 2920.71, "end": 2920.85, "word": " اللي", "probability": 0.818603515625}, {"start": 2920.85, "end": 2921.09, "word": " هو", "probability": 0.98046875}, {"start": 2921.09, "end": 2921.83, "word": " عنواننا", "probability": 0.83203125}, {"start": 2921.83, "end": 2923.27, "word": " integrability", "probability": 0.55914306640625}, {"start": 2923.27, "end": 2923.57, "word": " of", "probability": 0.9560546875}, {"start": 2923.57, "end": 2924.49, "word": " monotone", "probability": 0.9488932291666666}, {"start": 2924.49, "end": 2925.29, "word": " functions", "probability": 0.833984375}], "temperature": 1.0}, {"id": 127, "seek": 294413, "start": 2926.29, "end": 2944.13, "text": "لت I بتساوي A و B و لت F من I لR بيكون مونوتون فانكشن on I ثم F أشمالها is integrable on I نفترض أن F انتقل هو مونوتور نصل إليها integrable نفترض أن F مثلا increasing", "tokens": [1211, 2655, 286, 39894, 3794, 995, 45865, 316, 4032, 363, 4032, 5296, 2655, 479, 9154, 286, 5296, 49, 4724, 1829, 30544, 3714, 11536, 35473, 11536, 6156, 7649, 4117, 8592, 1863, 322, 286, 38637, 2304, 479, 5551, 8592, 2304, 6027, 11296, 307, 16200, 424, 638, 322, 286, 8717, 5172, 2655, 43042, 14739, 479, 16472, 2655, 4587, 1211, 31439, 3714, 11536, 35473, 13063, 8717, 36520, 11933, 20292, 11296, 16200, 424, 638, 8717, 5172, 2655, 43042, 14739, 479, 50113, 15040, 5662], "avg_logprob": -0.4005142382428616, "compression_ratio": 1.4850299401197604, "no_speech_prob": 3.0994415283203125e-06, "words": [{"start": 2926.29, "end": 2926.75, "word": "لت", "probability": 0.4788055419921875}, {"start": 2926.75, "end": 2927.09, "word": " I", "probability": 0.62939453125}, {"start": 2927.09, "end": 2927.99, "word": " بتساوي", "probability": 0.669525146484375}, {"start": 2927.99, "end": 2928.17, "word": " A", "probability": 0.66552734375}, {"start": 2928.17, "end": 2928.27, "word": " و", "probability": 0.87744140625}, {"start": 2928.27, "end": 2928.41, "word": " B", "probability": 0.51611328125}, {"start": 2928.41, "end": 2928.67, "word": " و", "probability": 0.45556640625}, {"start": 2928.67, "end": 2928.89, "word": " لت", "probability": 0.791259765625}, {"start": 2928.89, "end": 2929.07, "word": " F", "probability": 0.9453125}, {"start": 2929.07, "end": 2929.21, "word": " من", "probability": 0.96533203125}, {"start": 2929.21, "end": 2929.41, "word": " I", "probability": 0.98583984375}, {"start": 2929.41, "end": 2929.83, "word": " لR", "probability": 0.7646484375}, {"start": 2929.83, "end": 2930.09, "word": " بيكون", "probability": 0.7440592447916666}, {"start": 2930.09, "end": 2930.83, "word": " مونوتون", "probability": 0.487152099609375}, {"start": 2930.83, "end": 2931.65, "word": " فانكشن", "probability": 0.766015625}, {"start": 2931.65, "end": 2932.17, "word": " on", "probability": 0.496337890625}, {"start": 2932.17, "end": 2932.49, "word": " I", "probability": 0.98291015625}, {"start": 2932.49, "end": 2932.85, "word": " ثم", "probability": 0.55859375}, {"start": 2932.85, "end": 2933.15, "word": " F", "probability": 0.414306640625}, {"start": 2933.15, "end": 2933.89, "word": " أشمالها", "probability": 0.758642578125}, {"start": 2933.89, "end": 2934.25, "word": " is", "probability": 0.75537109375}, {"start": 2934.25, "end": 2935.13, "word": " integrable", "probability": 0.83447265625}, {"start": 2935.13, "end": 2935.31, "word": " on", "probability": 0.86962890625}, {"start": 2935.31, "end": 2935.65, "word": " I", "probability": 0.99365234375}, {"start": 2935.65, "end": 2936.65, "word": " نفترض", "probability": 0.9495849609375}, {"start": 2936.65, "end": 2936.87, "word": " أن", "probability": 0.73876953125}, {"start": 2936.87, "end": 2937.37, "word": " F", "probability": 0.82666015625}, {"start": 2937.37, "end": 2939.27, "word": " انتقل", "probability": 0.652679443359375}, {"start": 2939.27, "end": 2939.49, "word": " هو", "probability": 0.68212890625}, {"start": 2939.49, "end": 2940.21, "word": " مونوتور", "probability": 0.7879638671875}, {"start": 2940.21, "end": 2940.73, "word": " نصل", "probability": 0.37451171875}, {"start": 2940.73, "end": 2941.01, "word": " إليها", "probability": 0.6671142578125}, {"start": 2941.01, "end": 2941.65, "word": " integrable", "probability": 0.7091878255208334}, {"start": 2941.65, "end": 2942.73, "word": " نفترض", "probability": 0.9361572265625}, {"start": 2942.73, "end": 2942.89, "word": " أن", "probability": 0.8623046875}, {"start": 2942.89, "end": 2943.11, "word": " F", "probability": 0.9267578125}, {"start": 2943.11, "end": 2943.47, "word": " مثلا", "probability": 0.912353515625}, {"start": 2943.47, "end": 2944.13, "word": " increasing", "probability": 0.8134765625}], "temperature": 1.0}, {"id": 128, "seek": 297182, "start": 2945.06, "end": 2971.82, "text": "ونصل إنها integrable وsimilarly وفعلا similarly لو كانت f is decreasing هتكون برضه is integrable خلّينا مع بعض شباب نفترض suppose that f is increasing يعني الدالة على الفترة هي a وb مثلا", "tokens": [2407, 1863, 36520, 36145, 11296, 16200, 424, 638, 4032, 30937, 2202, 356, 4032, 5172, 3615, 15040, 14138, 45164, 25961, 2655, 283, 307, 23223, 8032, 2655, 30544, 4724, 43042, 3224, 307, 16200, 424, 638, 16490, 1211, 11703, 1829, 8315, 20449, 45030, 11242, 13412, 3555, 16758, 8717, 5172, 2655, 43042, 7297, 300, 283, 307, 5662, 37495, 22653, 32748, 6027, 3660, 15844, 27188, 2655, 25720, 39896, 257, 4032, 65, 50113, 15040], "avg_logprob": -0.25226449102595233, "compression_ratio": 1.4184782608695652, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2945.06, "end": 2945.64, "word": "ونصل", "probability": 0.5909830729166666}, {"start": 2945.64, "end": 2946.34, "word": " إنها", "probability": 0.713134765625}, {"start": 2946.34, "end": 2947.7, "word": " integrable", "probability": 0.690673828125}, {"start": 2947.7, "end": 2949.44, "word": " وsimilarly", "probability": 0.8006591796875}, {"start": 2949.44, "end": 2950.06, "word": " وفعلا", "probability": 0.90087890625}, {"start": 2950.06, "end": 2950.86, "word": " similarly", "probability": 0.4501953125}, {"start": 2950.86, "end": 2951.5, "word": " لو", "probability": 0.8203125}, {"start": 2951.5, "end": 2951.88, "word": " كانت", "probability": 0.952880859375}, {"start": 2951.88, "end": 2952.06, "word": " f", "probability": 0.4140625}, {"start": 2952.06, "end": 2952.28, "word": " is", "probability": 0.857421875}, {"start": 2952.28, "end": 2952.98, "word": " decreasing", "probability": 0.958984375}, {"start": 2952.98, "end": 2953.92, "word": " هتكون", "probability": 0.7679850260416666}, {"start": 2953.92, "end": 2954.56, "word": " برضه", "probability": 0.9794921875}, {"start": 2954.56, "end": 2955.52, "word": " is", "probability": 0.9111328125}, {"start": 2955.52, "end": 2956.3, "word": " integrable", "probability": 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"probability": 0.87744140625}, {"start": 3563.61, "end": 3564.23, "word": " bounded", "probability": 0.947265625}, {"start": 3564.23, "end": 3564.99, "word": " interval", "probability": 0.97119140625}, {"start": 3564.99, "end": 3565.51, "word": " must", "probability": 0.76123046875}, {"start": 3565.51, "end": 3566.09, "word": " be", "probability": 0.9677734375}, {"start": 3566.09, "end": 3566.85, "word": " integrable", "probability": 0.8995768229166666}], "temperature": 1.0}, {"id": 154, "seek": 358256, "start": 3567.58, "end": 3582.56, "text": "كمان مرة any continuous function on a closed bounded interval must be integrable طبعا حيزنا في البرهان شغلة اللي هي أخدناها سابقا انه", "tokens": [24793, 7649, 3714, 25720, 604, 10957, 2445, 322, 257, 5395, 37498, 15035, 1633, 312, 16200, 424, 638, 23032, 3555, 3615, 995, 11331, 1829, 11622, 8315, 8978, 2423, 26890, 3224, 7649, 13412, 17082, 37977, 13672, 1829, 39896, 5551, 9778, 3215, 8315, 11296, 8608, 16758, 4587, 995, 16472, 3224], 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3582.56, "word": " انه", "probability": 0.767333984375}], "temperature": 1.0}, {"id": 155, "seek": 359888, "start": 3582.96, "end": 3598.88, "text": "Any continuous function on a closed bounded interval must attain its maximum and minimum on this interval بمعنى آخر هنلاقي", "tokens": [29647, 10957, 2445, 322, 257, 5395, 37498, 15035, 1633, 23766, 1080, 6674, 293, 7285, 322, 341, 15035, 4724, 2304, 3615, 1863, 7578, 19753, 34740, 8032, 1863, 15040, 38436], "avg_logprob": -0.36072199509061614, "compression_ratio": 1.1239669421487604, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 3582.96, "end": 3583.24, "word": "Any", "probability": 0.07537841796875}, {"start": 3583.24, "end": 3583.8, "word": " continuous", "probability": 0.806640625}, {"start": 3583.8, "end": 3584.54, "word": " function", "probability": 0.96923828125}, {"start": 3584.54, "end": 3584.9, "word": " on", "probability": 0.92919921875}, {"start": 3584.9, "end": 3585.08, "word": " a", "probability": 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.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..", "tokens": [1211, 19452, 8978, 2423, 39184, 1156, 8629, 14674, 12549, 2423, 39184, 18183, 13672, 1829, 13672, 1829, 10874, 36081, 34105, 39894, 1863, 5172, 3615, 5296, 28820, 13672, 1829, 31439, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 2423, 4386, 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إذا غصب عنها هيكون اللي جوا limit UPN و F-L", "tokens": [3224, 1829, 30544, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 8608, 5172, 2288, 4032, 5551, 9381, 17082, 2288, 34051, 7251, 3794, 995, 45865, 11933, 3555, 3794, 1211, 11536, 4032, 23758, 2423, 28820, 10943, 382, 293, 1709, 281, 13202, 4032, 23758, 2423, 28820, 10943, 20328, 5016, 1829, 5016, 4724, 43042, 3224, 32767, 9957, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 11933, 15730, 32771, 9381, 3555, 18871, 11296, 39896, 30544, 13672, 1829, 10874, 14407, 4948, 624, 15466, 4032, 479, 12, 43], "avg_logprob": -0.20791903544556012, "compression_ratio": 1.591160220994475, "no_speech_prob": 3.725290298461914e-05, "words": [{"start": 4543.91, "end": 4544.57, "word": "هيكون", "probability": 0.786376953125}, {"start": 4544.57, "end": 4545.27, "word": " أكبر", "probability": 0.9025065104166666}, {"start": 4545.27, "end": 4545.63, "word": " أو", "probability": 0.8681640625}, {"start": 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4550.65, "end": 4550.81, "word": " goes", "probability": 0.72265625}, {"start": 4550.81, "end": 4550.91, "word": " to", "probability": 0.97021484375}, {"start": 4550.91, "end": 4551.25, "word": " infinity", "probability": 0.76904296875}, {"start": 4551.25, "end": 4551.75, "word": " و", "probability": 0.869140625}, {"start": 4551.75, "end": 4551.93, "word": " هذا", "probability": 0.955078125}, {"start": 4551.93, "end": 4552.29, "word": " الكلام", "probability": 0.96142578125}, {"start": 4552.29, "end": 4552.79, "word": " صحيح", "probability": 0.9954833984375}, {"start": 4552.79, "end": 4553.09, "word": " برضه", "probability": 0.9420572916666666}, {"start": 4553.09, "end": 4553.53, "word": " لمين", "probability": 0.84228515625}, {"start": 4553.53, "end": 4553.91, "word": " لكل", "probability": 0.697021484375}, {"start": 4553.91, "end": 4554.29, "word": " إبسلون", "probability": 0.96044921875}, {"start": 4554.29, "end": 4554.77, "word": " أكبر", "probability": 0.9812825520833334}, {"start": 4554.77, "end": 4555.31, "word": " من", "probability": 0.99365234375}, {"start": 4555.31, "end": 4555.61, "word": " سفر", "probability": 0.9840494791666666}, {"start": 4555.61, "end": 4555.83, "word": " إذا", "probability": 0.618408203125}, {"start": 4555.83, "end": 4556.63, "word": " غصب", "probability": 0.9295247395833334}, {"start": 4556.63, "end": 4557.07, "word": " عنها", "probability": 0.5478515625}, {"start": 4557.07, "end": 4557.71, "word": " هيكون", "probability": 0.6708984375}, {"start": 4557.71, "end": 4558.33, "word": " اللي", "probability": 0.922607421875}, {"start": 4558.33, "end": 4558.79, "word": " جوا", "probability": 0.885986328125}, {"start": 4558.79, "end": 4559.61, "word": " limit", "probability": 0.64453125}, {"start": 4559.61, "end": 4560.95, "word": " UPN", "probability": 0.446044921875}, {"start": 4560.95, "end": 4561.71, "word": " و", "probability": 0.8623046875}, {"start": 4561.71, "end": 4562.13, "word": " F", "probability": 0.7216796875}, {"start": 4562.13, "end": 4563.19, "word": "-L", "probability": 0.53143310546875}], "temperature": 1.0}, {"id": 196, "seek": 457706, "start": 4563.52, "end": 4577.06, "text": "بان و اف لازم يساوي ايش سفر از انقص انفينيتي وهذا by corollary اللي قبل بشوية هيعني ان اف is integrable", "tokens": [3555, 7649, 4032, 1975, 5172, 5296, 31377, 2304, 7251, 3794, 995, 45865, 1975, 1829, 8592, 8608, 5172, 2288, 1975, 11622, 16472, 4587, 9381, 16472, 5172, 9957, 1829, 31371, 37037, 15730, 538, 1181, 1833, 822, 13672, 1829, 12174, 36150, 4724, 8592, 2407, 10632, 39896, 3615, 22653, 16472, 1975, 5172, 307, 16200, 424, 638], "avg_logprob": -0.36969340297411074, "compression_ratio": 1.2635658914728682, "no_speech_prob": 1.1742115020751953e-05, "words": [{"start": 4563.52, "end": 4564.22, "word": "بان", "probability": 0.40869140625}, {"start": 4564.22, "end": 4564.58, "word": " و", "probability": 0.8427734375}, {"start": 4564.58, "end": 4565.04, "word": " اف", "probability": 0.58502197265625}, {"start": 4565.04, "end": 4566.1, "word": " لازم", "probability": 0.8336588541666666}, {"start": 4566.1, "end": 4566.56, "word": " يساوي", "probability": 0.9423828125}, {"start": 4566.56, "end": 4567.16, "word": " ايش", "probability": 0.5628255208333334}, {"start": 4567.16, "end": 4568.0, "word": " سفر", "probability": 0.7197265625}, {"start": 4568.0, "end": 4568.3, "word": " از", "probability": 0.5137939453125}, {"start": 4568.3, "end": 4568.78, "word": " انقص", "probability": 0.7032063802083334}, {"start": 4568.78, "end": 4569.42, "word": " انفينيتي", "probability": 0.6779296875}, {"start": 4569.42, "end": 4570.84, "word": " وهذا", "probability": 0.881591796875}, {"start": 4570.84, "end": 4571.3, "word": " by", "probability": 0.55029296875}, {"start": 4571.3, "end": 4572.04, "word": " corollary", "probability": 0.8015950520833334}, {"start": 4572.04, "end": 4572.28, "word": " اللي", "probability": 0.90478515625}, {"start": 4572.28, "end": 4572.54, "word": " قبل", "probability": 0.906982421875}, {"start": 4572.54, "end": 4573.26, "word": " بشوية", "probability": 0.915283203125}, {"start": 4573.26, "end": 4574.5, "word": " هيعني", "probability": 0.7845052083333334}, {"start": 4574.5, "end": 4574.78, "word": " ان", "probability": 0.93310546875}, {"start": 4574.78, "end": 4575.4, "word": " اف", "probability": 0.6392822265625}, {"start": 4575.4, "end": 4575.86, "word": " is", "probability": 0.8349609375}, {"start": 4575.86, "end": 4577.06, "word": " integrable", "probability": 0.777587890625}], "temperature": 1.0}, {"id": 197, "seek": 460675, "start": 4578.37, "end": 4606.75, "text": "Hence, F is integrable وهكذا أثبتنا العائلة الثانية من الدوال الـ continuous function على closed bounded interval is a continuous function is an integrable function وهكذا خلصنا الـ section الأول و الـ homework هيا اللي موجودة في التلخيص و المرة القادمة ان شاء الله", "tokens": [39, 655, 11, 479, 307, 16200, 424, 638, 4032, 3224, 4117, 15730, 5551, 12984, 3555, 2655, 8315, 18863, 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4583.31, "end": 4584.43, "word": " أثبتنا", "probability": 0.88486328125}, {"start": 4584.43, "end": 4584.99, "word": " العائلة", "probability": 0.9601236979166666}, {"start": 4584.99, "end": 4585.75, "word": " الثانية", "probability": 0.849853515625}, {"start": 4585.75, "end": 4586.25, "word": " من", "probability": 0.94580078125}, {"start": 4586.25, "end": 4586.85, "word": " الدوال", "probability": 0.92724609375}, {"start": 4586.85, "end": 4587.13, "word": " الـ", "probability": 0.2310791015625}, {"start": 4587.13, "end": 4587.61, "word": " continuous", "probability": 0.66845703125}, {"start": 4587.61, "end": 4588.33, "word": " function", "probability": 0.9033203125}, {"start": 4588.33, "end": 4588.83, "word": " على", "probability": 0.724609375}, {"start": 4588.83, "end": 4589.17, "word": " closed", "probability": 0.55224609375}, {"start": 4589.17, "end": 4589.53, "word": " bounded", "probability": 0.87890625}, {"start": 4589.53, "end": 4590.25, "word": " interval", "probability": 0.9560546875}, {"start": 4590.25, "end": 4590.91, "word": " is", "probability": 0.43212890625}, {"start": 4590.91, "end": 4591.17, "word": " a", "probability": 0.78271484375}, {"start": 4591.17, "end": 4591.59, "word": " continuous", "probability": 0.87158203125}, {"start": 4591.59, "end": 4592.37, "word": " function", "probability": 0.95361328125}, {"start": 4592.37, "end": 4592.77, "word": " is", "probability": 0.54833984375}, {"start": 4592.77, "end": 4593.11, "word": " an", "probability": 0.8486328125}, {"start": 4593.11, "end": 4593.65, "word": " integrable", "probability": 0.9156901041666666}, {"start": 4593.65, "end": 4594.21, "word": " function", "probability": 0.962890625}, {"start": 4594.21, "end": 4594.97, "word": " وهكذا", "probability": 0.8243408203125}, {"start": 4594.97, "end": 4597.73, "word": " خلصنا", "probability": 0.8372802734375}, {"start": 4597.73, "end": 4598.27, "word": " الـ", "probability": 0.423828125}, {"start": 4598.27, "end": 4598.99, "word": " section", "probability": 0.371826171875}, {"start": 4598.99, "end": 4599.69, "word": " الأول", "probability": 0.96533203125}, {"start": 4599.69, "end": 4599.91, "word": " و", "probability": 0.384765625}, {"start": 4599.91, "end": 4600.05, "word": " الـ", "probability": 0.32379150390625}, {"start": 4600.05, "end": 4600.45, "word": " homework", "probability": 0.8388671875}, {"start": 4600.45, "end": 4600.79, "word": " هيا", "probability": 0.207550048828125}, {"start": 4600.79, "end": 4601.65, "word": " اللي", "probability": 0.5518798828125}, {"start": 4601.65, "end": 4602.41, "word": " موجودة", "probability": 0.9193115234375}, {"start": 4602.41, "end": 4602.73, "word": " في", "probability": 0.87744140625}, {"start": 4602.73, "end": 4604.03, "word": " التلخيص", "probability": 0.97509765625}, {"start": 4604.03, "end": 4604.75, "word": " و", "probability": 0.6494140625}, {"start": 4604.75, "end": 4605.11, "word": " المرة", "probability": 0.936279296875}, {"start": 4605.11, "end": 4605.71, "word": " القادمة", "probability": 0.9931640625}, {"start": 4605.71, "end": 4606.05, "word": " ان", "probability": 0.474365234375}, {"start": 4606.05, "end": 4606.43, "word": " شاء", "probability": 0.98681640625}, {"start": 4606.43, "end": 4606.75, "word": " الله", "probability": 0.94580078125}], "temperature": 1.0}, {"id": 198, "seek": 462447, "start": 4607.63, "end": 4624.47, "text": "إن شاء الله بكون احنا بنبدأ في اللي هي ال section اللي بعده اللي هو properties of the Riemann Integral", "tokens": [28814, 1863, 13412, 16606, 21984, 4724, 30544, 1975, 5016, 8315, 44945, 44510, 10721, 8978, 13672, 1829, 39896, 2423, 3541, 13672, 1829, 39182, 3224, 13672, 1829, 31439, 7221, 295, 264, 497, 4907, 969, 23894, 304], "avg_logprob": -0.30937499829701015, "compression_ratio": 1.2231404958677685, "no_speech_prob": 0.0, "words": [{"start": 4607.63, "end": 4607.91, "word": "إن", "probability": 0.50543212890625}, {"start": 4607.91, "end": 4608.15, "word": " شاء", "probability": 0.988037109375}, {"start": 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4621.09, "end": 4621.25, "word": " هو", "probability": 0.96923828125}, {"start": 4621.25, "end": 4621.77, "word": " properties", "probability": 0.4912109375}, {"start": 4621.77, "end": 4622.87, "word": " of", "probability": 0.9755859375}, {"start": 4622.87, "end": 4623.05, "word": " the", "probability": 0.87890625}, {"start": 4623.05, "end": 4623.51, "word": " Riemann", "probability": 0.6896565755208334}, {"start": 4623.51, "end": 4624.47, "word": " Integral", "probability": 0.717041015625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 4625.4381875, "duration_after_vad": 4507.979374999997}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_raw.srt
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+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EHWkgcDuyyY_raw.srt
@@ -0,0 +1,3624 @@
+1
+00:00:04,720 --> 00:00:09,780
+بسم الله الرحمن الرحيم هذه هي المحاضرة رقم 11 في
+
+2
+00:00:09,780 --> 00:00:16,040
+مساق تحليل حقيقي 2 لطلاب وطالبات الجامعة الإسلامية
+
+3
+00:00:16,040 --> 00:00:24,760
+كلية العلوم وهي المحاضرة الأولى بعد إعلان الطوارئ
+
+4
+00:00:24,760 --> 00:00:32,580
+بخصوص أو بمواجهة فيروس كورونا المنتشر اتحدتنا
+
+5
+00:00:32,580 --> 00:00:37,960
+المرة الماضيةبدأنا في اللي هو chapter 7 اللي كان
+
+6
+00:00:37,960 --> 00:00:41,280
+الحديث عن ال riman integral أو تكامل ال riman
+
+7
+00:00:41,280 --> 00:00:45,860
+بدأنا في ال section الأول اللي هو تحت عنوان riman
+
+8
+00:00:45,860 --> 00:00:50,740
+integrability عرفنا شغلتين حاجة اسمها ال upper sum
+
+9
+00:00:50,740 --> 00:00:55,960
+و حاجة اسمها ال lower sum و قلنا اللي هو ال lower
+
+10
+00:00:55,960 --> 00:01:00,290
+sumهو عبارة عن الـ summation للـ mk الـ mk هذه
+
+11
+00:01:00,290 --> 00:01:06,410
+تمثل في xk minus xk minus 1 حيث mk كانت تمثل أو m
+
+12
+00:01:06,410 --> 00:01:10,910
+small k كانت تمثل عبارة عن الـ infimum للدالة على
+
+13
+00:01:10,910 --> 00:01:15,520
+الفترة اللي هي المذكورةالان ال .. ال .. ال other
+
+14
+00:01:15,520 --> 00:01:19,380
+sum هو عبارة عن ال summation لنفس ال sum العلوي
+
+15
+00:01:19,380 --> 00:01:24,520
+ولكن بدلا منها اللي هي M K capital اللي كانت تمثل
+
+16
+00:01:24,520 --> 00:01:28,280
+ال supremum ل ال F of X وال X على اللي هي في
+
+17
+00:01:28,280 --> 00:01:34,110
+الفترة المذكورة اللي عنديالآن أخدنا أول لمّة المرة
+
+18
+00:01:34,110 --> 00:01:38,190
+الماضية وقلنا إذا كانت F من I لR bounded و B any
+
+19
+00:01:38,190 --> 00:01:43,230
+partition of I بدي يكون ال lower اللي هو sum لأي
+
+20
+00:01:43,230 --> 00:01:47,810
+partition B و function F أصغر أو يساوي ال upper
+
+21
+00:01:47,810 --> 00:01:52,470
+sum لنفس ال partition و لنفس اللي هي ال function F
+
+22
+00:01:52,470 --> 00:01:58,710
+بعد هي طبعا خطينا خطوة أخرىو جينا عرفنا اللي هو شو
+
+23
+00:01:58,710 --> 00:02:03,110
+معناته انها تكون اللي هو ال partition Q refinement
+
+24
+00:02:03,110 --> 00:02:08,610
+لل partition B قلنا Q اللي هو تحسين ل B إذا كانت B
+
+25
+00:02:08,610 --> 00:02:13,940
+بي عبارة عن مجموعة جزئية من Qو بناء عليه اللي هو
+
+26
+00:02:13,940 --> 00:02:19,440
+قلنا ان اي اللي هو sub interval xk-1xk من ال
+
+27
+00:02:19,440 --> 00:02:23,520
+partition B يمكن كتابتها على صورة union of sub
+
+28
+00:02:23,520 --> 00:02:27,680
+intervals من اللي هو التحسين اللي هو EQ
+
+29
+00:02:31,060 --> 00:02:36,380
+الان جينا اللي هو بناء على هذا التعريف جينا قولنا
+
+30
+00:02:36,380 --> 00:02:40,480
+لو كانت F is من I لعند R is bounded و B is any
+
+31
+00:02:40,480 --> 00:02:45,780
+partition of I و Q refinement لل Bمدام اللي هو Q
+
+32
+00:02:45,780 --> 00:02:50,420
+-refinement إذا ال lower sum هيعلى و ال upper sum
+
+33
+00:02:50,420 --> 00:02:54,820
+هينزل على أساس انه اللي هو في النهاية يلتقي ال
+
+34
+00:02:54,820 --> 00:02:58,740
+upper مع ال lower و نصل لاللي هو ال integrability
+
+35
+00:02:58,740 --> 00:03:02,800
+أو معنى ال integrability كما سنرى لاحقا على الأقل
+
+36
+00:03:02,800 --> 00:03:06,560
+في اللي هو يكون واضح من خلال الرسم في اللي هي
+
+37
+00:03:06,560 --> 00:03:13,890
+الدوال الموجبة كما ذكرنا سابقااللي بحكيه إنه لو
+
+38
+00:03:13,890 --> 00:03:17,330
+كانت عندي اللي هو F من I ل R bounded وB partition
+
+39
+00:03:17,330 --> 00:03:22,750
+وQ وrefinement للـ B هيكون عندي lower sum لل
+
+40
+00:03:22,750 --> 00:03:28,570
+partition B أصغر أو يساوي lower sum للتحسين عماله
+
+41
+00:03:28,570 --> 00:03:32,850
+التحسين بكبر لما بده يصل لفعلا المساحة تحت المنحنة
+
+42
+00:03:32,850 --> 00:03:39,470
+في حالة الدوال الموجبةالقبر صم لـ Q و F سيبدأ يصغر
+
+43
+00:03:39,470 --> 00:03:42,850
+ويكون أصغر من الساوي اللي هو القبر صم لـ P و F
+
+44
+00:03:42,850 --> 00:03:47,310
+التحسين يعني سيصغره بمعنى آخر سيبدأ يلتقوا إلى
+
+45
+00:03:47,310 --> 00:03:51,770
+أسفل لما نصل إلى اللي هو مساواة في حالة الـ
+
+46
+00:03:51,770 --> 00:03:55,730
+Integrability لما نكون عندنا أخدنا الـ Supremum
+
+47
+00:03:55,730 --> 00:04:01,340
+لكل L و Fالـ L والـ infimum لكل الـ U بيصير بنسمي
+
+48
+00:04:01,340 --> 00:04:04,800
+بعد شوية حاجة اسمها ال lower integral وال upper
+
+49
+00:04:04,800 --> 00:04:08,980
+integral وبرهننا هذه النظرية و بعدين جينا لللمة و
+
+50
+00:04:08,980 --> 00:04:13,520
+بعدين جينا لللمة أخرىاللي هو لو كانت F من I ل R
+
+51
+00:04:13,520 --> 00:04:17,600
+bounded وB1 وB2 اي partitions الآن لأي partitions
+
+52
+00:04:17,600 --> 00:04:22,480
+هيكون الlower دايما بغض النظر عن ال partition اللي
+
+53
+00:04:22,480 --> 00:04:26,120
+هو هيكون أصغر أو يساوي الأبر بغض النظر عن ال
+
+54
+00:04:26,120 --> 00:04:28,660
+partition B2 يعني مش لنفس ال partition زي ما قلنا
+
+55
+00:04:28,660 --> 00:04:33,180
+في اللمبة 7 1 1 لأ لأي two partitions دايما
+
+56
+00:04:33,180 --> 00:04:37,170
+الlower ما هو هيكون تحتأسفل المنحنى والـ Upper
+
+57
+00:04:37,170 --> 00:04:41,350
+هيكون أعلى المنحنى بغض النظر عن ال partitions اللي
+
+58
+00:04:41,350 --> 00:04:45,690
+عندى طبعا التمثيل هذا في حال اللي هو ال F is a
+
+59
+00:04:45,690 --> 00:04:49,130
+positive function على ال interval المذكورة الآن
+
+60
+00:04:49,130 --> 00:04:52,730
+بعد هيكة اجينا وعرفنا شو معناه ال lower integral
+
+61
+00:04:52,730 --> 00:04:55,510
+وشو معناه ال upper integral قولنا ال lower
+
+62
+00:04:55,510 --> 00:05:00,050
+integral كما هو متوقع سميناه الف F هو عبارة عن ال
+
+63
+00:05:00,050 --> 00:05:05,660
+supremum لل lowersوال .. و ال .. و ال .. و ال
+
+64
+00:05:05,660 --> 00:05:10,440
+upper هو عبارة عن الـ infimum لل uppers حتى لو
+
+65
+00:05:10,440 --> 00:05:14,300
+التقت الالو أف مع الالو أف اللي هو من أعلى مع
+
+66
+00:05:14,300 --> 00:05:17,940
+الأسفل هيكونوا التقوا بالظبط عند .. من مساحة تحت
+
+67
+00:05:17,940 --> 00:05:20,960
+المنحنى في حالة الدالة الموجبة وهذه .. في هذه
+
+68
+00:05:20,960 --> 00:05:24,300
+الحالة بنسمي إذا كانت ال upper تساوي ال lower
+
+69
+00:05:24,300 --> 00:05:27,920
+بنسمي ال function على هذه الفترة is integral و هذا
+
+70
+00:05:27,920 --> 00:05:31,900
+الكلام كله تحدثنا فيه عشان هيك أنا مسرع شويةهو
+
+71
+00:05:31,900 --> 00:05:37,160
+حكينا إنه اللي هو دايما ال lower sum لل F أخدنا
+
+72
+00:05:37,160 --> 00:05:40,760
+نظرية قولنا ال lower integral أسف ال lower
+
+73
+00:05:40,760 --> 00:05:45,100
+integral دايما أصغر يساوي مين ال upper integral
+
+74
+00:05:45,100 --> 00:05:46,860
+إذن الآن النظرية
+
+75
+00:05:49,600 --> 00:05:52,760
+الإعلان المهم اللي هو لو كانت F من I لعند R
+
+76
+00:05:52,760 --> 00:05:56,000
+bounded function على closed bounded interval A وB
+
+77
+00:05:56,000 --> 00:05:59,720
+بدي يكون ال lower integral L of F أصغر أو ساوي ال
+
+78
+00:05:59,720 --> 00:06:04,930
+upper integral U of F بصورة عامةهذه هي الـ
+
+79
+00:06:04,930 --> 00:06:08,050
+definition اللي ذكرته قبل و شوية نقول عن الـ
+
+80
+00:06:08,050 --> 00:06:15,150
+function F على bounded sub interval A و B أو
+
+81
+00:06:15,150 --> 00:06:18,650
+closed bounded interval A و B و كانت الـ F عبارة
+
+82
+00:06:18,650 --> 00:06:22,080
+عن bounded functionبنعرف أن الـ F is remain
+
+83
+00:06:22,080 --> 00:06:26,840
+integrable on I إذا كانت ال lower of F بساوي ال
+
+84
+00:06:26,840 --> 00:06:30,080
+upper of F معناته صارت اللي هي ال F is remain
+
+85
+00:06:30,080 --> 00:06:34,460
+integrable if and only if ال lower sum يساوي ال
+
+86
+00:06:34,460 --> 00:06:39,240
+upper sum هذا كله ذكرناه المرة الماضية و أيضا
+
+87
+00:06:39,240 --> 00:06:42,520
+عرفنا .. قلنا في هذا الحلقة أن كل integration من A
+
+88
+00:06:42,520 --> 00:06:47,890
+ل B هو ال lower أو ال upper المتساويينوعرفنا ايضا
+
+89
+00:06:47,890 --> 00:06:50,610
+تعريف اخر قولنا ال integration من a ل b بساوي ناقص
+
+90
+00:06:50,610 --> 00:06:53,830
+ال integration من b ل a وعرفنا ال integration من a
+
+91
+00:06:53,830 --> 00:06:59,410
+ل a بساوي صفر هذا كله حكينا المرة الماضية ومش هيك
+
+92
+00:06:59,410 --> 00:07:03,820
+كمان واخدنا المثالاللي هو أثبتنا إنه اللي هو g of
+
+93
+00:07:03,820 --> 00:07:07,880
+x بيساوي x is integrable on i استنادًا على إنه
+
+94
+00:07:07,880 --> 00:07:11,020
+أوجدنا ال lower sum ال lower integral و ال upper
+
+95
+00:07:11,020 --> 00:07:13,300
+integral أثبتنا إن ال lower integral و ال upper
+
+96
+00:07:13,300 --> 00:07:16,300
+integral are equal و من ثم أثبتنا إنه ال
+
+97
+00:07:16,300 --> 00:07:19,940
+integration exist لل function x على الفترة 0 و 1
+
+98
+00:07:19,940 --> 00:07:26,810
+وأوجدنا قيمة ال integration في حينهوصلنا إلى اللي
+
+99
+00:07:26,810 --> 00:07:33,290
+هو مثالنا التالي أنه لو كانت F من I لعند .. F من I
+
+100
+00:07:33,290 --> 00:07:40,090
+اللي هي 01 لعند ال .. ال R be defined by أخدنا
+
+101
+00:07:40,090 --> 00:07:45,990
+الدالة كما يلي اللي هو قولنا أن F of X F of X
+
+102
+00:07:45,990 --> 00:07:53,040
+بساوي واحدإذا كانت x rational number element in Q
+
+103
+00:07:53,040 --> 00:08:00,100
+وبساوي 0 إذا كانت x element in IQ أو element in Q
+
+104
+00:08:00,100 --> 00:08:04,660
+complement اللي هي ال rational numbers الآن بدنا
+
+105
+00:08:04,660 --> 00:08:11,200
+نثبت بقول show that this function Fطبعا أنا method
+
+106
+00:08:11,200 --> 00:08:17,340
+F على اللي هي الـ Q تقاطع طبعا الـ 0 و 1 اللي هي
+
+107
+00:08:17,340 --> 00:08:19,920
+الـ interval اللي بدأ عليها التقاطع الـ 0 و 1
+
+108
+00:08:19,920 --> 00:08:25,160
+بمعنى إن دالت F صارت من I اللي هي عبارة عن 0 و 1
+
+109
+00:08:26,040 --> 00:08:29,380
+اللي عند R واضح ان الدالة هذه is a bounded
+
+110
+00:08:29,380 --> 00:08:33,560
+function الان بدأ أثبت لكم ان هذا الدالة is not
+
+111
+00:08:33,560 --> 00:08:38,120
+integrable on this interval is not integrable on
+
+112
+00:08:38,120 --> 00:08:44,100
+this interval الان علشان أصل اللي هو اللي هي ان
+
+113
+00:08:44,100 --> 00:08:48,180
+هذا الدالة غير قابلة تكامل بالنسبة لتكامل بالنسبة
+
+114
+00:08:48,180 --> 00:08:55,650
+لتكامل الريمان بدي اخد الان Bأخدوا أي partition X0
+
+115
+00:08:55,650 --> 00:09:02,550
+X1 لعند Xn هذا any partition لإيه ال interval اللي
+
+116
+00:09:02,550 --> 00:09:06,790
+هي الفترة مين Zero و واحد يعني بمعنى أتيت للفترة
+
+117
+00:09:06,790 --> 00:09:14,030
+Zero و واحد و جزقتها X0 X1 لعند مؤصل لعند مين لعند
+
+118
+00:09:14,030 --> 00:09:18,610
+Xn اللي هي إيش بتساوي بتساوي واحدالان هذا ال
+
+119
+00:09:18,610 --> 00:09:22,130
+partition اخدته arbitrarily اللى هو partition
+
+120
+00:09:22,130 --> 00:09:31,790
+لفترة L عندى الان بدى احسب ال Lof B و F لهذا ال
+
+121
+00:09:31,790 --> 00:09:34,990
+partition أيش بتساوي حسب اللي عرفناها سابقا بتساوي
+
+122
+00:09:34,990 --> 00:09:40,610
+ال summation لل M K في X K minus X K minus واحد K
+
+123
+00:09:40,610 --> 00:09:46,330
+من عند واحد لعند مين لعند اللي هي N ويساوي الآن ال
+
+124
+00:09:46,330 --> 00:09:52,450
+M K عرفناها ال M K هي عبارة عن ال infimum لقيمة ال
+
+125
+00:09:52,450 --> 00:09:56,470
+function F of X حيث X تنتمي إلى الفترة X K minus
+
+126
+00:09:56,470 --> 00:10:02,210
+واحد لعند X Kطبعاً اللي هي F of X معرفة على
+
+127
+00:10:02,210 --> 00:10:05,130
+أساسيها يا إما واحد يا إما سفر حسب إنها تكون
+
+128
+00:10:05,130 --> 00:10:08,030
+rational أو إيش ال rational يعني ال function F
+
+129
+00:10:08,030 --> 00:10:11,910
+أصلا اللي هي قيمتين بس إذا الأن ال infimum لل F of
+
+130
+00:10:11,910 --> 00:10:16,630
+X عندها يا هيكون واحد يا هيكون سفر ليش؟ لأن أصلا
+
+131
+00:10:16,630 --> 00:10:23,340
+الفترة هذهفيها أي فترة subinterval xk-1xk فيها
+
+132
+00:10:23,340 --> 00:10:27,620
+rational وirrational إذا قيمة ال up of x في الفترة
+
+133
+00:10:27,620 --> 00:10:31,260
+هتلاقي عند قيم واحد هتلاقي أكيد عند قيم أش بتساوي
+
+134
+00:10:31,260 --> 00:10:35,000
+بساوي سفر إذا ال infimum في هذه الحالة هو عبارة عن
+
+135
+00:10:35,000 --> 00:10:42,320
+إيش يساوي سفر إذا ال summation ل 0 في xk-xk-1 كامل
+
+136
+00:10:42,320 --> 00:10:46,120
+عند واحد عند أنه طبيعي هذا بديهي إيش هيساوي بساوي
+
+137
+00:10:46,120 --> 00:10:52,410
+سفرأذن الان L of F بي و F ساوة سفر فور أي بورتيشن
+
+138
+00:10:52,410 --> 00:10:59,370
+بيه إذا ال L of F اللي هي عبارة عن الأو الـ
+
+139
+00:10:59,370 --> 00:11:06,070
+Supremum الـ Supremum لكل الـ L of B و F Such that
+
+140
+00:11:06,070 --> 00:11:09,890
+B element in the set of all partitions اللي هو B
+
+141
+00:11:09,890 --> 00:11:14,090
+of I هيكون ال .. ال Supremum اللي هين صفر لإن كل
+
+142
+00:11:14,090 --> 00:11:18,350
+اللي هين أصلا إشق بتنطلع .. تنطلع بساوي صفر إذا
+
+143
+00:11:18,350 --> 00:11:23,070
+هذا إيش هيساوي يا شباب؟ هو يساوي Zeroإذا طلع عندى
+
+144
+00:11:23,070 --> 00:11:28,150
+ال lower sum بساوة 0 الان بدي أحسبلكم مين أحسبلكم
+
+145
+00:11:28,150 --> 00:11:31,390
+ال upper sum سامحوني أكتب هنا بس عساس اللي يبقى
+
+146
+00:11:31,390 --> 00:11:39,670
+كله مكتوب عندى نوجد ال upper sum ال upper sum اللي
+
+147
+00:11:39,670 --> 00:11:45,510
+هي ال UPUF بساوة summation للان K capital في XK
+
+148
+00:11:45,510 --> 00:11:51,740
+minus XK minus واحد K من عند واحد لعند الانالان
+
+149
+00:11:51,740 --> 00:11:55,900
+هذا بيساوي ال Mk زي ما قلنا قبل هيك ال Mk بدل ما
+
+150
+00:11:55,900 --> 00:11:59,400
+هي ال M في مميزة استعريفها ال Mk بتساوي ال
+
+151
+00:11:59,400 --> 00:12:03,780
+supremum لهذه ال 6 وزي ما قلنا ال 6 هذه في داخلها
+
+152
+00:12:03,780 --> 00:12:08,400
+يا واحد يا زيرو نظرا لإن اللي هو أي sub interval
+
+153
+00:12:08,400 --> 00:12:11,820
+هيكون فيها rational و irrational وتبعا إلها هيكون
+
+154
+00:12:11,820 --> 00:12:16,060
+قيمة ال function في داخلها واحد أو سفر واحنا بنبحث
+
+155
+00:12:16,060 --> 00:12:19,280
+عن ال supremum إذا هيكون ال supremum في كل الأحوال
+
+156
+00:12:19,280 --> 00:12:25,600
+ال Mk بتساوي واحدمضروبة في xk-xk-1 k من عند 1 لعند
+
+157
+00:12:25,600 --> 00:12:30,700
+n نفردها هذه ويساوي اللي هنبصير k من عند 1 يعني x1
+
+158
+00:12:30,700 --> 00:12:39,570
+-x0 زاد x2-x1 زاد إلى أخير لما أصل لعند xn-1نقص xn
+
+159
+00:12:39,570 --> 00:12:44,550
+نقص واحد طبعا واضح انه عندي ال x واحد هت cancel مع
+
+160
+00:12:44,550 --> 00:12:48,270
+ناقص x واحد و ال x اتنين مع ناقص x اتنين لما نصل
+
+161
+00:12:48,270 --> 00:12:52,450
+للاخر هيكون في عندي ات cancel الجميع بس ضال عندي
+
+162
+00:12:52,450 --> 00:12:58,170
+ال xn و ال x note و هد بتساوي xn ناقص x note و
+
+163
+00:12:58,170 --> 00:13:01,750
+يساوي ال xn طبعا ايش هي عبارة عن واحد و ال x note
+
+164
+00:13:01,750 --> 00:13:07,190
+ايش هي شبه سفر و يساوي واحد ناقص سفر و يساوي واحد
+
+165
+00:13:07,570 --> 00:13:12,930
+إذا طلع عندي الـ Upper Sum لأي Bar تشهم بيه، هيتلع
+
+166
+00:13:12,930 --> 00:13:18,510
+ايش بساوة؟ بساوة واحدة إذا الان لما بدي أخد ال U
+
+167
+00:13:18,510 --> 00:13:23,510
+of F اللي هو Upper Integral هيساوة عبارة عن ال
+
+168
+00:13:23,510 --> 00:13:30,220
+Infimumلمين؟ للـ U, B وF such that B element in
+
+169
+00:13:30,220 --> 00:13:34,580
+the set of all partitions B of I والـ U, B وF
+
+170
+00:13:34,580 --> 00:13:38,840
+قيمته ثابتة for أي partition بيساوي واحد إذا الـ
+
+171
+00:13:38,840 --> 00:13:43,300
+infimum لكل اللي هنا عبارة عن برضه إيش بيساوي واحد
+
+172
+00:13:43,300 --> 00:13:47,410
+صار عندي الآنlower integral و ال upper integral
+
+173
+00:13:47,410 --> 00:13:51,110
+have different values واحد بيساوي صفر واحد بيساوي
+
+174
+00:13:51,110 --> 00:13:56,890
+واحد وبناء عليه بتكون عنده اللي هو ال F is not
+
+175
+00:13:56,890 --> 00:14:03,150
+Riemann integrable اي سؤال؟ طيب ماشي الحاجة
+
+176
+00:14:03,150 --> 00:14:10,930
+الانصار عندى اخدنا مثلين المثال الأولاللي هو
+
+177
+00:14:10,930 --> 00:14:16,770
+أثبتنا إن off of x بيساوي x is integrable على
+
+178
+00:14:16,770 --> 00:14:22,010
+الفترة 0 و1 و أثبتناها بواسطة التعريف وأيضا أثبتنا
+
+179
+00:14:22,010 --> 00:14:26,190
+مثال آخر لbounded function أيضا وكانت is not
+
+180
+00:14:26,190 --> 00:14:31,250
+remain integrable اللي هي off of x بيساوي 1 إذا
+
+181
+00:14:31,250 --> 00:14:36,090
+كانت x rational ويساوي 0 إذا كانت x irrational هذا
+
+182
+00:14:36,090 --> 00:14:43,040
+اللي هو المثال الثانيالان نيجي لاللي هو criterion
+
+183
+00:14:43,040 --> 00:14:49,280
+مهمة اللي احنا بنسميها اللي هي عبارة عن remain
+
+184
+00:14:49,280 --> 00:14:54,800
+criterion for
+
+185
+00:14:54,800 --> 00:15:01,490
+integrabilityأحنا طبعا اتحدثنا عن الـ Remain
+
+186
+00:15:01,490 --> 00:15:05,330
+Integrability كيف نثبت أنه Remain Integrable عن
+
+187
+00:15:05,330 --> 00:15:09,470
+طريق التعريف طبعا الآن مش دايما بدنا نثبت عن طريق
+
+188
+00:15:09,470 --> 00:15:14,350
+التعريف إذا بدنا اللي هو طرق أخرى نحاول اللي هو
+
+189
+00:15:14,350 --> 00:15:21,780
+نوسع اللي هيإمكانياتنا في الحكم على الدالة إنها
+
+190
+00:15:21,780 --> 00:15:26,860
+integrable أو مش integrable وهذه الإمكانية الأخرى
+
+191
+00:15:26,860 --> 00:15:31,400
+غير التعريف هي اللي بنسميها اللي هو الريمان
+
+192
+00:15:31,400 --> 00:15:36,500
+integrability criterion أو criterion for
+
+193
+00:15:36,500 --> 00:15:41,340
+integrability نشوف أيش بيقول النظرية
+
+194
+00:15:43,700 --> 00:15:48,060
+لت I بساوة A وB و لت F من I لـ R بيـ bounded نفترض
+
+195
+00:15:48,060 --> 00:15:51,360
+أن F عبارة عن إيه اشمال يا جماعة؟ bounded function
+
+196
+00:15:51,360 --> 00:15:57,500
+then F is integrable on I if and only if for each
+
+197
+00:15:57,500 --> 00:16:00,340
+epsilon أكبر من 0 there exists a partition B
+
+198
+00:16:00,340 --> 00:16:04,660
+epsilon of I such that U B epsilon نقص الـ B
+
+199
+00:16:04,660 --> 00:16:10,730
+epsilon إيه إشماله أصغر من اللي هو إبسلونإذن اللي
+
+200
+00:16:10,730 --> 00:16:16,410
+هو واضح إنه عندي فصار فيه test لل integrability أو
+
+201
+00:16:16,410 --> 00:16:20,150
+اللي هو طريقة للحكم على ال integrability أخرى غير
+
+202
+00:16:20,150 --> 00:16:26,430
+التعريف اللي هو بتقول F is integrable
+
+203
+00:16:28,870 --> 00:16:32,890
+if and only if طبعاً هذه لمين الـ F؟ F عبارة عن زي
+
+204
+00:16:32,890 --> 00:16:36,190
+ما انتوا عارفين bounded function لإنه كل شغلنا
+
+205
+00:16:36,190 --> 00:16:40,010
+أصلا على اللي هو remaining integrability أنه نفترض
+
+206
+00:16:40,010 --> 00:16:43,210
+أنه الـ F bounded عشان اللي هو تكون ال supremum و
+
+207
+00:16:43,210 --> 00:16:46,330
+ال infimum اللي مبني عليها التعريف تكون مضمون إنها
+
+208
+00:16:46,330 --> 00:16:49,870
+موجودة عشان هيك بنحكي أن F is bounded function طيب
+
+209
+00:16:49,870 --> 00:16:56,140
+إذا الـ F is integrable if and only ifاللي هي لكل
+
+210
+00:16:56,140 --> 00:17:00,620
+إبسلون أكبر من 0 there exists a partition P إبسلون
+
+211
+00:17:00,620 --> 00:17:04,240
+هذا ال P اللي هو ال partition يعتمد عالميا على
+
+212
+00:17:04,240 --> 00:17:07,820
+إبسلون لكل إبسلون بالله دي partition P إبسلون لمين
+
+213
+00:17:07,820 --> 00:17:11,840
+ال partition طبعا لل interval اللي عندنا اللي هي R
+
+214
+00:17:11,840 --> 00:17:17,040
+there exists P إبسلون a partition of I such that
+
+215
+00:17:17,040 --> 00:17:25,210
+ال P ال U ال P إبسلونوالـ F ناقص الـ L بي إبسلون و
+
+216
+00:17:25,210 --> 00:17:31,050
+F يكون أصغر من مين من إبسلون الآن إذا كان لجينا
+
+217
+00:17:31,050 --> 00:17:34,190
+لكل إبسلون لجينا بي إبسلون بحيث هذا يتحقق معناته F
+
+218
+00:17:34,190 --> 00:17:37,590
+is integrable and conversely if F is integrable
+
+219
+00:17:37,590 --> 00:17:42,550
+أكيد لكل إبسلون هلجي بي إبسلون بحيث أن هذا يتحقق
+
+220
+00:17:42,550 --> 00:17:48,170
+خلونا نيجي الآن نبرهم و نشوف كيف بدنا نبرهم
+
+221
+00:17:48,170 --> 00:17:53,880
+نظريتناالان بدنا نفترض ان F اي شمالها is
+
+222
+00:17:53,880 --> 00:18:00,240
+integrable ونصل منها للي عيدي اللي هي ال partition
+
+223
+00:18:00,240 --> 00:18:07,040
+اللي مذكور مدام F is integrable اذا كانت ..الان
+
+224
+00:18:07,040 --> 00:18:16,100
+بنقول suppose that F is integrable مدام integrable
+
+225
+00:18:16,100 --> 00:18:22,090
+يا شبابأكيد عنده اللي هو الـ U of F بساوي L of
+
+226
+00:18:22,090 --> 00:18:29,490
+إيش؟ Of F، مظبوط؟ أكيد الـ L of .. الـ U of F
+
+227
+00:18:29,490 --> 00:18:33,410
+بساوي الـ L of F إيش اللي بيدثبته؟ بيدثبته لأي
+
+228
+00:18:33,410 --> 00:18:35,810
+إبسلون أكبر من سفر بدلاجي بإبسلون، شوفوا كيف
+
+229
+00:18:35,810 --> 00:18:41,090
+بنلاجيه، الآن نفترض إن إبسلون let إبسلون أكبر من
+
+230
+00:18:41,090 --> 00:18:47,810
+سفر be given، ماشي الحالةعندي الـ U of F هو إيش يا
+
+231
+00:18:47,810 --> 00:18:54,910
+شباب؟ هو عبارة عن الـ infimum للـ L of B و F such
+
+232
+00:18:54,910 --> 00:19:01,170
+that B element in B of I، مظبوط؟ إذا الـ U of F
+
+233
+00:19:01,170 --> 00:19:05,830
+عبارة عن infimum يعني هو عبارة عن greatest lower
+
+234
+00:19:05,830 --> 00:19:11,630
+boundلو هذا الـ greatest lower bound ضفنا إليه أي
+
+235
+00:19:11,630 --> 00:19:17,230
+كمية هيبط اللاور باوند لأن هو أصلا إيش اسمه
+
+236
+00:19:17,230 --> 00:19:23,510
+greatest lower bound إذا لو الـU of F ضفتله Y على
+
+237
+00:19:23,510 --> 00:19:28,610
+2 مثلا طبعا هذا المقدار هيبط اللاور باوند إيش
+
+238
+00:19:28,610 --> 00:19:33,690
+معناته بط اللاور باوند؟ يعني بمعنى آخر هيكون في
+
+239
+00:19:33,690 --> 00:19:44,080
+عند إشي أصغر منههيكون عندى أصغر من ال U of F أو بي
+
+240
+00:19:44,080 --> 00:19:49,740
+واحد مثلا و F for some mean بي واحد إذا لما نشيل
+
+241
+00:19:49,740 --> 00:19:54,600
+من ال infimum إبسلون
+
+242
+00:19:54,600 --> 00:19:57,740
+على اتنين هيبطل هذا lower bound يعني بمعنى أخر
+
+243
+00:19:57,740 --> 00:20:03,280
+هلاقي اللي هو lower bound
+
+244
+00:20:07,660 --> 00:20:12,140
+عند الـ Y أكبر من 0 خلّيني أبدأ لت Y أكبر من 0 بي
+
+245
+00:20:12,140 --> 00:20:17,940
+given إذا عندي الآن بدي أثبت لك بدي أجيب لك
+
+246
+00:20:17,940 --> 00:20:21,440
+partition بيبسلون بحيث أنه هذا ناقص هذا يكون أصغر
+
+247
+00:20:21,440 --> 00:20:26,070
+من مين من ابسلون شوف كيف بدي أعملالان انا عندي ال
+
+248
+00:20:26,070 --> 00:20:30,710
+U of F ايش بيساوي يا جماعة ال U of F بيساوي ال
+
+249
+00:20:30,710 --> 00:20:39,590
+infimum لل L بيقف ب element I طيب ال U of F اسف يا
+
+250
+00:20:39,590 --> 00:20:43,010
+جماعة ال U of F بيساوي ال infimum لمين لل U بيقف
+
+251
+00:20:43,720 --> 00:20:49,780
+ماشي الحال الان ايش معناه انه هذا infimum معناته
+
+252
+00:20:49,780 --> 00:20:55,100
+هذا هو عبارة عن ال greatest lower bound مدام ال
+
+253
+00:20:55,100 --> 00:20:59,450
+greatest lower bound اذا ال lower bound هذاأو الـ
+
+254
+00:20:59,450 --> 00:21:03,330
+Greatest Lower Bound لو ضفتله أي رقم يابسون على
+
+255
+00:21:03,330 --> 00:21:06,370
+اتنين مثلا بالذنب يابسون على اتنين الـ game
+
+256
+00:21:06,370 --> 00:21:09,670
+بتعرفوا ليش يابسون على اتنين يعني لو ضفتله أي رقم
+
+257
+00:21:09,670 --> 00:21:14,730
+بيبطل Lower Bound إيش معناته بيبطل Lower Bound
+
+258
+00:21:14,730 --> 00:21:21,830
+يعني هيصير هذا أكبر من ال UB 1 of F for some B
+
+259
+00:21:21,830 --> 00:21:27,510
+وحدهالأنه بطل أش ماله هذا بطل lower bound بطل يكون
+
+260
+00:21:27,510 --> 00:21:32,670
+أصغر من الكل من هان يعني هلاجي واحد من هان هو مش
+
+261
+00:21:32,670 --> 00:21:36,830
+أصغر منه أو بمعنى أخر U, P, 1 و F أصغر من اللي هو
+
+262
+00:21:36,830 --> 00:21:44,850
+هذا المقدر طيب similarly ال L of F هي عبارة عن ال
+
+263
+00:21:44,850 --> 00:21:52,050
+supremum لل L, P و F such that P elemented P of I
+
+264
+00:21:53,040 --> 00:21:57,780
+بنفس الطريقة يا جماعة ال L of F هي عبارة عن إيش ال
+
+265
+00:21:57,780 --> 00:22:04,040
+least upper bound يعني هذا لو least upper bound
+
+266
+00:22:04,040 --> 00:22:07,200
+Upper bound لو كان أصغر و أحد لو هذا least upper
+
+267
+00:22:07,200 --> 00:22:11,000
+bound راحت منه عدد ولو صغير جدا و ليكن يبسلون على
+
+268
+00:22:11,000 --> 00:22:16,800
+اتنينهيبطل هذا عبارة عن upper bound يعني هلاقي
+
+269
+00:22:16,800 --> 00:22:24,280
+واحد من اللي هان اللي هو L of B2 وF مثلا أكبر منه
+
+270
+00:22:24,280 --> 00:22:30,560
+لأنه هيبطل هذا أشماله upper bound لأنه هو ال least
+
+271
+00:22:30,560 --> 00:22:35,780
+لما طلعت منه بطل من ال upper bounds يعني لجيت واحد
+
+272
+00:22:35,780 --> 00:22:43,120
+من هذول أكبر منه إذا صار في عندي اللي هولجيت بي
+
+273
+00:22:43,120 --> 00:22:53,960
+واحد بحقق الأولى و بيتنين بحقق التانية لأن خد الآن
+
+274
+00:22:53,960 --> 00:22:58,840
+خدلي بي إبسلون هذا اللي بديها هذا الحقق اللي اللي
+
+275
+00:22:58,840 --> 00:23:03,600
+بديها خد بي إبسلون إيش بساوي ال بي واحد اللي لجيته
+
+276
+00:23:03,600 --> 00:23:12,050
+هنااتحاد الـ B2 اللي لجيته هان اتحاد مين؟ B2 صار
+
+277
+00:23:12,050 --> 00:23:18,510
+عندى .. صار عندى الآن مع بعض خليني أمسح فأقوم،
+
+278
+00:23:18,510 --> 00:23:22,170
+مهمش؟
+
+279
+00:23:22,170 --> 00:23:31,670
+صار
+
+280
+00:23:31,670 --> 00:23:41,110
+عندى ما يليه؟ صار عندى الآنL of F نقص
+
+281
+00:23:41,110 --> 00:23:53,490
+Y على 2 أصغر من L of B2 Fاللي هو أكيد أصغر أو
+
+282
+00:23:53,490 --> 00:23:59,750
+يساوي L of By قُقَف، ليش؟ لأن الـBy يا جماعة عبارة
+
+283
+00:23:59,750 --> 00:24:04,570
+عن refinement للـB2 والlower لما يصير فيه تحسين
+
+284
+00:24:04,570 --> 00:24:08,510
+بكبر، بروح نحو اتجاه نيل اللي هو المنحنة التالية
+
+285
+00:24:08,510 --> 00:24:14,010
+مساحة تحت المنحنة الكلية لأن أيضا لو جيت قولة الـU
+
+286
+00:24:14,010 --> 00:24:24,400
+of Fزاد إبسلون على اتنين هتلاقيها أكبر من ال U بي
+
+287
+00:24:24,400 --> 00:24:30,410
+واحد و Fوأكيد عندي الـ U of F زائد إبسلون عدنان
+
+288
+00:24:30,410 --> 00:24:34,490
+أكبر من الـ U B1 of F هيكون هذا أكبر أو يساوي الـ
+
+289
+00:24:34,490 --> 00:24:39,610
+U B epsilon of F لأن الـ B epsilon refinement لمين
+
+290
+00:24:39,610 --> 00:24:44,790
+برضه؟ للـ B1 بزام refinement إذن اللي هو التحسين
+
+291
+00:24:44,790 --> 00:24:50,270
+بيزغر القبر وبروح ناحية اللي هو المنحنة إذن الآن
+
+292
+00:24:50,270 --> 00:24:55,480
+من الـ tip 2 هذولةواجدنا طبعا احنا تنسوش ان احنا
+
+293
+00:24:55,480 --> 00:24:59,100
+مفترضين من رأس الدول ان U of F is integrable يعني
+
+294
+00:24:59,100 --> 00:25:05,340
+مفترضين ان ال L of F ايش بتساوي U of F تنسواش هذه
+
+295
+00:25:05,340 --> 00:25:12,540
+ليه جيت تنتر مع بعض دول بصير عندى الحصل على L of F
+
+296
+00:25:12,540 --> 00:25:22,970
+ناقص Y على 2 اللي هو اصغر من Lof B, Epsilon و F
+
+297
+00:25:22,970 --> 00:25:30,110
+والـ L و الـ U of F زائد Epsilon على 2 أكبر من U,
+
+298
+00:25:30,190 --> 00:25:35,290
+B, Epsilon و F أنا إيش غرضي؟ غرضي أثبت إن U, B,
+
+299
+00:25:35,350 --> 00:25:39,130
+Epsilon و F ناقص L, B, Epsilon و F أصغر من Epsilon
+
+300
+00:25:39,130 --> 00:25:42,890
+يلّا اترحل من بعض، إذا بصير عندي، بتحصل البدكية
+
+301
+00:25:42,890 --> 00:25:48,570
+بصير عندي الآن، بطرح حياة دي، بقول U, B, Epsilon و
+
+302
+00:25:48,570 --> 00:25:53,240
+Fناقص لأنه لما نضرب هذا في ناقص هتنعكس يعني هتصير
+
+303
+00:25:53,240 --> 00:25:57,100
+هذه جماعة ناقص وهذه زائد وهذه هتنعكس هيك وهيصير
+
+304
+00:25:57,100 --> 00:26:05,600
+الاش ناقص بيصير عندي U بي و F ناقص ال بي و F هيصير
+
+305
+00:26:05,600 --> 00:26:11,820
+أصغر من مين من ناقص L of F ناخد هذا قبل زي ما احنا
+
+306
+00:26:11,820 --> 00:26:20,690
+مرتبينها U of F زائد ي على 2 ناقصL of F زي إبسلون
+
+307
+00:26:20,690 --> 00:26:24,790
+على 2 وطبعا إحنا جايلين إن F is integrable يعني
+
+308
+00:26:24,790 --> 00:26:28,770
+الـ U of F بسوء L of F إذا هذي بتروح مع هذي بظل
+
+309
+00:26:28,770 --> 00:26:33,610
+إيه شماله؟ بظل إبسلون إذا إحنا لكل إبسلون أكبر من
+
+310
+00:26:33,610 --> 00:26:36,950
+سفر لجينا بي إبسلون هي في الواقع بي إبسلون اللي
+
+311
+00:26:36,950 --> 00:26:39,970
+لجيناها بي واحد اتحاد بي اتنين حيث بي واحد اللي
+
+312
+00:26:39,970 --> 00:26:44,120
+لجيناه هان وال بي اتنين اللي لجيناه هانsuch that U
+
+313
+00:26:44,120 --> 00:26:49,120
+P Y of F نقص L P Y of F أصغر من اللي هو Epsilon
+
+314
+00:26:49,120 --> 00:26:57,480
+وهو المطلوب أي سؤال؟ طيب، ماشي يا شباب، الآن خلصنا
+
+315
+00:26:57,480 --> 00:27:04,420
+الجزء الأول من النظريةأثبتنا اللي بدنا يا انه اللي
+
+316
+00:27:04,420 --> 00:27:08,860
+هو هذه العلاقة صحيحة لأننا نفترض انه suppose that
+
+317
+00:27:08,860 --> 00:27:12,380
+star holds اللي هي star ليه هذه نفترض ان لكل
+
+318
+00:27:12,380 --> 00:27:15,380
+epsilon أكبر من 0 there exists B of epsilon such
+
+319
+00:27:15,380 --> 00:27:18,680
+that U B Epsilon و F نقصها دي أصغر من Epsilon و
+
+320
+00:27:18,680 --> 00:27:24,280
+بدنا نصل من خلالها لإيش لأن ال F is integrable
+
+321
+00:27:24,280 --> 00:27:25,760
+فنشوف
+
+322
+00:27:40,480 --> 00:27:47,440
+البرهام بسيط لو طلعنا عليه مباشرة على اللوح الآن
+
+323
+00:27:47,440 --> 00:27:52,140
+بدنا نفترض أن هذه تتحقق اللي هو نفترض أنه لكل ي
+
+324
+00:27:52,140 --> 00:27:57,020
+أكبر من 0 يوجد بي إبسلون بحيث أن هذا اللي هي تتحقق
+
+325
+00:27:57,020 --> 00:28:01,620
+علشان أصل بدي أصلكم في النهاية أن L of F هي إيش
+
+326
+00:28:01,620 --> 00:28:05,810
+بتساوي U of F شوف كيف ده صلهااللي هبدأ أقوللك اللي
+
+327
+00:28:05,810 --> 00:28:08,650
+هو نفترض انه زي ما قلنا انه star holds اللي حكينا
+
+328
+00:28:08,650 --> 00:28:12,990
+عنها لأن for any partition B هيكون ال L B of F
+
+329
+00:28:12,990 --> 00:28:17,970
+أصغر أساوي L of F و ال U of F أصغر أساوي مين ال U
+
+330
+00:28:17,970 --> 00:28:25,970
+B of F واضح؟ إذا أصار عيدي الآن L يمكن أوضحلكم على
+
+331
+00:28:25,970 --> 00:28:35,790
+اللوح L of B of F ده ال B أي partition أصغرمظبوط
+
+332
+00:28:35,790 --> 00:28:40,910
+أو يساوي ال L of A H of F لانه ال L of F يا جماعة
+
+333
+00:28:40,910 --> 00:28:47,330
+هو ال supremum اللي هنا و ال U بيقف
+
+334
+00:28:47,330 --> 00:28:53,130
+أكبر أو يساوي ال U of F لانه ال U of F H يا جماعة
+
+335
+00:28:53,130 --> 00:28:57,950
+هو عبارة عن مين عبارة عن ال infimumأترحولي الجهة
+
+336
+00:28:57,950 --> 00:29:01,410
+تانية هذا طبعا لكل مين لكل ال partitions اللي في
+
+337
+00:29:01,410 --> 00:29:06,750
+الدنيا من ضمنهم ال P Epsilon اللي احنا ماعطينايا
+
+338
+00:29:06,750 --> 00:29:12,250
+في ال .. اللي هو نص النظرية اذا بسيري عندى لأن لو
+
+339
+00:29:12,250 --> 00:29:19,990
+اجيب طرحة ال U F ناقص L F ال U F ناقص L Fهيصير ايش
+
+340
+00:29:19,990 --> 00:29:23,850
+ماله يا جماعة؟ يعني طريقة رحلة من هذه بيصير أصغر
+
+341
+00:29:23,850 --> 00:29:28,510
+أو يساوي لإن هذه بتضربها في ناقص و هذه ناقص و
+
+342
+00:29:28,510 --> 00:29:32,750
+بتنقلب زي ما عملنا قبل و شوية بيصير أصغر أو يساوي
+
+343
+00:29:32,750 --> 00:29:43,640
+U P of Fنقص ال B of F هذا الكلام صحيح لإيش لكل
+
+344
+00:29:43,640 --> 00:29:48,200
+partition في الدنيا من ضمنها المين ال partition
+
+345
+00:29:48,200 --> 00:29:52,580
+المواطع لنا يعني حيصير عند هذا ينطبق برضه على ال
+
+346
+00:29:52,580 --> 00:29:57,500
+بي إبسلون إذا صار هذا أصغر يسوي بي إبسلون نقص ال
+
+347
+00:29:57,500 --> 00:30:01,740
+بي إبسلون of F طيب هم يعطيين إن هذا المقدار إيش
+
+348
+00:30:01,740 --> 00:30:06,260
+ماله أصغر من إبسلون لأي إبسلون في الدنياهو أنا
+
+349
+00:30:06,260 --> 00:30:10,580
+بعرف أن هذا المقدار نفسه أكبر أو يساوي إيش سفر صار
+
+350
+00:30:10,580 --> 00:30:17,720
+عندى الآن ال U of F ناقص ال L of F دايما أصغر من
+
+351
+00:30:17,720 --> 00:30:23,420
+إبسلون و أكبر أو يساوي سفر لكل إبسلون أكبر من سفر
+
+352
+00:30:23,420 --> 00:30:28,260
+إذا على طول من نظرية في تحليل واحد هيكون هذا اللى
+
+353
+00:30:28,260 --> 00:30:36,250
+عندى إذا U of Fنقص L of F بيساوي سفر إذا U of F
+
+354
+00:30:36,250 --> 00:30:44,850
+بيساوي L of F وهذا يعني F is a Riemann Integral هو
+
+355
+00:30:44,850 --> 00:30:52,110
+المطلوب بيكون احنا هيك أثبتنا اللي هوIntegrable
+
+356
+00:30:52,110 --> 00:30:58,350
+criterion أو اللي هو طريقة لتحديد اللي هو ال
+
+357
+00:30:58,350 --> 00:31:02,270
+function is integrable أو لا غير اللي هو طريقة
+
+358
+00:31:02,270 --> 00:31:07,110
+التعريف الآن في عند كورولاري بعدها كورولاري
+
+359
+00:31:07,110 --> 00:31:11,530
+الكورولاري
+
+360
+00:31:11,530 --> 00:31:16,230
+هي تقول كما يليه الآن بدنا نترجم الحديث بدل ما كان
+
+361
+00:31:16,230 --> 00:31:22,710
+بإبسلون نحكي عن مين؟ عن اللي هوsequence of
+
+362
+00:31:22,710 --> 00:31:29,870
+partitions طبعاً هو هذا معهود التحويل في نظريات
+
+363
+00:31:29,870 --> 00:31:34,590
+مشابهة في حتى في كورسات أخرى خلّينا نشوف عندنا
+
+364
+00:31:34,590 --> 00:31:38,790
+اللي هو النظرية ايش ب .. او الكورلري ايش بتقول
+
+365
+00:31:38,790 --> 00:31:43,350
+بتقول let I بساوة A و B and let F من I لR be a
+
+366
+00:31:43,350 --> 00:31:48,600
+bounded functionلأن لو فرضنا بي أن أن element none
+
+367
+00:31:48,600 --> 00:31:52,300
+is a sequence of partitions of I بحيث أن ال limit
+
+368
+00:31:52,300 --> 00:31:55,920
+هذا بيساوي سفر then f is integrable and ال limit
+
+369
+00:31:55,920 --> 00:31:58,040
+لل integration بيساوي ال integration بيساوي ال
+
+370
+00:31:58,040 --> 00:32:04,480
+limit ال أسف ال limit لل lower p and f بيساوي ال
+
+371
+00:32:04,480 --> 00:32:07,020
+limit لل upper p and f اللي هو بيساوي قيمة ال
+
+372
+00:32:07,020 --> 00:32:13,000
+integration حتى ال converse جماعة اللي هواللي قبل
+
+373
+00:32:13,000 --> 00:32:16,820
+بشوية اللي كان .. اللي هي كانت F أندولي F لأن لو
+
+374
+00:32:16,820 --> 00:32:21,520
+كانت F is integrable أكيد هلاقي sequence من
+
+375
+00:32:21,520 --> 00:32:25,380
+partitions بحيث أنه ال limit اللي حاصل الطرح بساوي
+
+376
+00:32:25,380 --> 00:32:29,760
+سفر اللي هو البرهان مشابه لإي اللي حكيناه فيه اللي
+
+377
+00:32:29,760 --> 00:32:34,500
+هو برهان إيجاد الـ B epsilon ولكن هنا بنجداللي هو
+
+378
+00:32:34,500 --> 00:32:37,380
+الـ Epsilon بساوية واحدة لان فبنلاقي اللي هو ال
+
+379
+00:32:37,380 --> 00:32:42,380
+sequence هذه في ال corollary .. في من مشابه .. شيء
+
+380
+00:32:42,380 --> 00:32:45,900
+مشابه في برهان النظرية الأولى اللي قبل بشوية و
+
+381
+00:32:45,900 --> 00:32:49,360
+ياريت تجربوها عندكم خلينا ناخد اللي .. اللي موجود
+
+382
+00:32:49,360 --> 00:32:54,160
+حاليا اللي هو الاتجاه هذا ان لو لجينا sequence of
+
+383
+00:32:54,160 --> 00:32:59,430
+partitionsوكان ال limit لل U P N و F نقص ال P N و
+
+384
+00:32:59,430 --> 00:33:02,870
+F بساوة سفر إذا هتكون F is integrable و هتكون ال
+
+385
+00:33:02,870 --> 00:33:07,990
+limit للأولى بساوة limit للثانية بساوة قيمة ال
+
+386
+00:33:07,990 --> 00:33:08,970
+integration
+
+387
+00:33:12,900 --> 00:33:17,500
+عند ما أعطيني limit هذا إيش بساوي؟ سفر خلّينا ندخل
+
+388
+00:33:17,500 --> 00:33:20,580
+على التعريف مباشرة تعريف ال limit بتعرفوا تعريف ال
+
+389
+00:33:20,580 --> 00:33:23,140
+limit يا شباب؟ اللي هو لكل إبسلون أكبر من سفر
+
+390
+00:33:23,140 --> 00:33:26,160
+there exist k such that لكل أن أكبر سوى k بيصير
+
+391
+00:33:26,160 --> 00:33:33,200
+هذا ناقص هذا أصغر من مين؟ من اللي هو إبسلون وهذا
+
+392
+00:33:33,200 --> 00:33:35,980
+على طول يعطينا as integral خلّيش تشوف أيش اللي
+
+393
+00:33:35,980 --> 00:33:43,000
+بيقوله لأن since عند ما أعطيني limitU P N و F ناقص
+
+394
+00:33:43,000 --> 00:33:51,240
+ال P N و F as N goes to infinity بساوة سفر، مظبوط؟
+
+395
+00:33:51,240 --> 00:33:56,220
+هي كل ما أعطينيها نجي للتعريف، إذا تعريف ال
+
+396
+00:33:56,220 --> 00:33:58,880
+sequence عادية for every epsilon أكبر من سفر there
+
+397
+00:33:58,880 --> 00:34:02,680
+exists K element in N such that for every N أكبر
+
+398
+00:34:02,680 --> 00:34:13,090
+سوا K اللي هو عندي ال U P N و F ناقصبن و F أصغر من
+
+399
+00:34:13,090 --> 00:34:19,010
+إبسلون إذا مش لجينا بارتشن واحد لجينا بارتشن بك و
+
+400
+00:34:19,010 --> 00:34:22,710
+بك زاد واحد و بك زاد اتنين و بك زاد تلاتة كلهم
+
+401
+00:34:22,710 --> 00:34:29,110
+بسبب إن ال UBK أو ال UBK زاد واحد أو الاخره ناقص
+
+402
+00:34:29,110 --> 00:34:32,590
+القللها أصغر من 100 من إبسلون إذا ال criterion
+
+403
+00:34:32,590 --> 00:34:36,550
+اللي في الكورولاريت حققت إذا صارت عندي هذه إذا F
+
+404
+00:34:36,550 --> 00:34:41,830
+is integrableيعني مش بي إبسلون واحد اللي جينا لأ
+
+405
+00:34:41,830 --> 00:34:46,550
+من عند بك وطالع كل ال partitions هذه اللي هي بك و
+
+406
+00:34:46,550 --> 00:34:49,950
+بك زائد واحد و بك زائد اتنين بتعمل عمل ال بي
+
+407
+00:34:49,950 --> 00:34:53,510
+إبسلون اللي في وين في النظرية إذا ال F أشمالها
+
+408
+00:34:53,510 --> 00:34:58,750
+صارت ال F عبارة عن Integrable من النظرية السابقة
+
+409
+00:34:58,750 --> 00:35:05,200
+الآن الدور دل أن نثبت مين أن ال limitلأ اللي هو
+
+410
+00:35:05,200 --> 00:35:09,260
+هذا المقدار هو limit لهذا المقدار بساوي إيش اللي
+
+411
+00:35:09,260 --> 00:35:12,540
+جوا طبعا هو لو كانت يا جماعة limit هذا نقص هذا سفر
+
+412
+00:35:12,540 --> 00:35:15,920
+مش معناته ال limit الأول و limit الثاني exist هاي
+
+413
+00:35:15,920 --> 00:35:20,610
+مثلا هذه لو كانت هذه un تربيع وهذه unUnterm .. آسف
+
+414
+00:35:20,610 --> 00:35:24,610
+Unتربيع و Unتربيع أو Un و Un limit Un نقص Un على
+
+415
+00:35:24,610 --> 00:35:27,850
+طول سفر لكن لا limit الأولى عدد ولا limit الثاني
+
+416
+00:35:27,850 --> 00:35:33,270
+عدد اتنين اتين بروحين إلى مالا نهاية فالان لكن في
+
+417
+00:35:33,270 --> 00:35:37,390
+هذه الحالة نظرا للمعطيات اللي موجودة و اللي هي
+
+418
+00:35:37,390 --> 00:35:40,930
+طبيعي اللي هو اللي بنحكي فيه NuF is bounded و ببو
+
+419
+00:35:40,930 --> 00:35:45,550
+و الاخره هو استخدام السابق هيطلع عندي فعلا ال
+
+420
+00:35:45,550 --> 00:35:50,840
+limitللأل بيساوي limit للأبر بيساوي قيمة ال
+
+421
+00:35:50,840 --> 00:36:08,200
+integration ماشي اطلعوا يا جماعة عندي الآن خليني
+
+422
+00:36:08,200 --> 00:36:15,240
+أطلع .. نحط البرهان أمامنابدي الآن خلصت اللي هو F
+
+423
+00:36:15,240 --> 00:36:21,500
+is integrable بدي أستخدم زي جابل بشوية بالظبط اللي
+
+424
+00:36:21,500 --> 00:36:28,570
+هو تعريف ال L of F و U of F و إيش تعريفهاذا ما
+
+425
+00:36:28,570 --> 00:36:32,250
+بديش اعيد اللي حكيته قبل بشوية الان بما ان ال L of
+
+426
+00:36:32,250 --> 00:36:42,370
+F عبارة عن اللي هي supremum لل L B of F اذا لكل
+
+427
+00:36:42,370 --> 00:36:46,610
+ي ساوي واحدة لان there exists B N partition of I
+
+428
+00:36:46,610 --> 00:36:50,390
+such that L of F نقص واحدة لان اصغر من مين من L B
+
+429
+00:36:50,390 --> 00:36:54,070
+N of Fزي ما قلت قبل شوية، لو الـ Supremum طرحنا
+
+430
+00:36:54,070 --> 00:36:57,270
+اللي هي الـ Least Upper Bound، طرحنا منه أي عدد، 1
+
+431
+00:36:57,270 --> 00:37:01,330
+على N، إذا هلاجي، هيبطل إيش ماله Upper Bound، إيش
+
+432
+00:37:01,330 --> 00:37:04,190
+معناه يبطل Upper Bound؟ هيلاجي واحد من المجموعة
+
+433
+00:37:04,190 --> 00:37:08,230
+أكبر منه، وهذا فعلا اللي لاجينا BN، بحيث أن الـ BN
+
+434
+00:37:08,230 --> 00:37:13,910
+و F أكبر من الـ F ناقص واحدة لأنالان من هذا .. من
+
+435
+00:37:13,910 --> 00:37:16,670
+.. من .. من ال .. ال .. ال .. ناخد هذا على الطرف
+
+436
+00:37:16,670 --> 00:37:19,970
+الثاني على الطرف هنا و ناخد هذا على الطرف هذا بصير
+
+437
+00:37:19,970 --> 00:37:23,890
+عندي L of F نقص Lb of F أصغر من واحدة الآن و أنا
+
+438
+00:37:23,890 --> 00:37:28,670
+بعرف أن هذا دايما أكبر يساوي هذا لأن هذا ال
+
+439
+00:37:28,670 --> 00:37:32,410
+supremum منهم إذا انا هيكون أكبر يساوي سفر الان
+
+440
+00:37:32,410 --> 00:37:38,440
+خدوا ال limit للجهتينas n goes to infinity بيصير
+
+441
+00:37:38,440 --> 00:37:43,160
+عندي هذا اللي هي limit
+
+442
+00:37:43,160 --> 00:37:48,060
+لL بيقل نوف أف حيثالوي لL أوف أف لإنه حيثير ال
+
+443
+00:37:48,060 --> 00:37:53,720
+limit هذا أيش بيساوي بيساوي سفر واضحالان يا جماعة
+
+444
+00:37:53,720 --> 00:37:58,400
+عندي ال L of F ناقص ال P L of F أصغر من واحدة لأنه
+
+445
+00:37:58,400 --> 00:38:03,660
+أكبر سوى سفر لان معله الان هذا لكل إبسلون اللي هو
+
+446
+00:38:03,660 --> 00:38:07,300
+واحدة لان لجينا partition يعني للإبسلون بيساوي
+
+447
+00:38:07,300 --> 00:38:09,360
+واحد لجينا بيه واحد للإبسلون بيساوي اتنين بيه
+
+448
+00:38:09,360 --> 00:38:12,160
+اتنين للإبسلون تلاتة بيه تلاتة إذا صار عندي
+
+449
+00:38:12,160 --> 00:38:16,580
+sequence of اللي هي إياش partitionsالأن طلع عندي
+
+450
+00:38:16,580 --> 00:38:20,460
+دائما دائما دائما L of F نقص ال PN of F أصغر من
+
+451
+00:38:20,460 --> 00:38:25,440
+واحد على N لكل N الأن as N goes to infinity هذا ال
+
+452
+00:38:25,440 --> 00:38:30,620
+limit هيصير سفر وهذا سفر إذا هيصير limit هذا as N
+
+453
+00:38:30,620 --> 00:38:33,200
+goes to infinity بساوي سفر لكن ال L of F أصلا
+
+454
+00:38:33,200 --> 00:38:37,860
+independent of N إذا هيصير limit PN of F as N goes
+
+455
+00:38:37,860 --> 00:38:43,720
+to infinity بساوي L of F لأن المواضحات لوش يا شباب
+
+456
+00:38:43,720 --> 00:38:52,360
+limitL of F ناقص L B N و F هيساوي 0 as N goes to
+
+457
+00:38:52,360 --> 00:38:56,920
+infinity و هذا المقدار عبارة عن مقدار ثابت
+
+458
+00:38:56,920 --> 00:39:04,740
+independent of N إذا هيصير عندي limit L B N و F
+
+459
+00:39:04,740 --> 00:39:10,540
+بتساوي limit L of F ناقص
+
+460
+00:39:17,660 --> 00:39:27,170
+L of P L و F زائد L of Fمظبوط؟ طيب هذا الأن
+
+461
+00:39:27,170 --> 00:39:33,590
+المقدار معروف أنه بساوي 0 و هذا ثابت إذا بقدر أوزع
+
+462
+00:39:33,590 --> 00:39:37,210
+ال limit على الجهتيين و أنا مرتاح إذا بساوي limit
+
+463
+00:39:37,210 --> 00:39:42,950
+الأول اللي هو 0 زائد limit الثاني نفسه لأنه ثابت
+
+464
+00:39:42,950 --> 00:39:46,450
+إذا صار عندي limit L P N of F as N goes to
+
+465
+00:39:46,450 --> 00:39:54,710
+infinity بساوي L of F كما هو حكينا عنه حقامن جهة
+
+466
+00:39:54,710 --> 00:39:58,650
+أخرى limit الـU ناقص limit الـL المعطيلة هي بيساوي
+
+467
+00:39:58,650 --> 00:40:03,490
+0 إذا صار عندى سهل أن أوجد mean برضه limit الـU
+
+468
+00:40:03,490 --> 00:40:08,990
+اللي هو limit الـU إيش هيساوي ناقص اللي هو .. ال
+
+469
+00:40:08,990 --> 00:40:12,670
+.. ال .. هيساوي limit الـL of F يساوي الـU of F
+
+470
+00:40:12,670 --> 00:40:17,270
+يساوي ال integration أكثر توضيحاً and بيهاجدى
+
+471
+00:40:17,270 --> 00:40:22,420
+أعتقد أنه واضح لكن خلّينا نوضحه بشكل أكبرعشان
+
+472
+00:40:22,420 --> 00:40:34,100
+مايضلش مشي عندى limit U P N of F ايش هيساوي هيساوي
+
+473
+00:40:34,100 --> 00:40:44,560
+اللي هو limit U
+
+474
+00:40:44,560 --> 00:40:49,740
+P N of F
+
+475
+00:40:52,490 --> 00:40:58,090
+limit U P N و F ناقص
+
+476
+00:40:58,090 --> 00:41:07,810
+L of P N of F زائد L P N of F واضحة يا شباب أه؟
+
+477
+00:41:07,810 --> 00:41:12,690
+الآن هذا مضمون أنه موجود و سفر وهذا مضمون و
+
+478
+00:41:12,690 --> 00:41:18,350
+أثبتناه إيش بيساوي L of Fإذا إيش صار بيساوي؟
+
+479
+00:41:18,350 --> 00:41:21,810
+بيساوي اللي هو هذا صفر إذا صار بيساوي أقل و فأف
+
+480
+00:41:21,810 --> 00:41:25,510
+إذا صار هذا برضه بيساوي أقل و فأف لكن أنا مثبت قبل
+
+481
+00:41:25,510 --> 00:41:29,610
+بشوية أن ال F is integrable يعني ال U في F إيش
+
+482
+00:41:29,610 --> 00:41:34,550
+هتساوي؟ اللي أقل و فأف وهذه أثبتناها إيش بتساوي؟
+
+483
+00:41:34,550 --> 00:41:39,350
+limit U P N و F وهذه نفسها أثبتناها قبل بشوية إيش
+
+484
+00:41:39,350 --> 00:41:46,070
+بتساوي؟ limit L P N و Fوهو المطلوب طبعاً مدام
+
+485
+00:41:46,070 --> 00:41:49,370
+integration هذه وهذه هي عبارة عن ال integration
+
+486
+00:41:49,370 --> 00:41:55,410
+على الفترة اللي بنحكي عنها الإنقلا بيه لقيمة ال F
+
+487
+00:41:55,410 --> 00:42:01,710
+لأن صار عندي كل القيام هذه متساوية وصار عندي إيجاد
+
+488
+00:42:01,710 --> 00:42:07,610
+ال limit لل U P N و F أو limit لل L P N و F يكفي
+
+489
+00:42:07,610 --> 00:42:11,330
+أنه نوجد فيه قيمة ال integration بعد ما أثبتناه أو
+
+490
+00:42:11,330 --> 00:42:16,260
+تحت الظروف اللي هي في الكورو الأخرىلأ نيجي بدنا
+
+491
+00:42:16,260 --> 00:42:22,660
+نبرهن اللي هو المثال اللي برهناه المرة الماضية
+
+492
+00:42:22,660 --> 00:42:27,660
+بالتعريف بدنا نبرهنه بواسطة اللي هي ال corollary
+
+493
+00:42:27,660 --> 00:42:33,060
+اللي عندنا بدنا نبرهن اللي هو اللي هي نثبت أنه
+
+494
+00:42:36,170 --> 00:42:42,070
+نثبت أن F of X بساوة X يا شباب عبارة عن Integrable
+
+495
+00:42:42,070 --> 00:42:45,770
+أو اللي سمناها G of X بساوة X is Integrable على
+
+496
+00:42:45,770 --> 00:42:48,830
+الفترة Zero واحد المرة الماضية أثبتناها كيف
+
+497
+00:42:48,830 --> 00:42:54,230
+أثبتناها زي ما أنتم متذكرين بواسط التعريفمظبوط؟
+
+498
+00:42:54,230 --> 00:42:58,610
+طيب جيبنا اللي هو ال upper sum و ال lower sum و
+
+499
+00:42:58,610 --> 00:43:02,930
+بعدين جيبنا ال upper integral و ال lower integral
+
+500
+00:43:02,930 --> 00:43:04,750
+و أثبتنا ان ال upper integral بساوي ال lower
+
+501
+00:43:04,750 --> 00:43:08,760
+integral و خلصناهالآن بدنا نثبتها بطريقتنا اللي هي
+
+502
+00:43:08,760 --> 00:43:12,540
+على الكورولر اللي جابل بشوية عند g of x ساوة x على
+
+503
+00:43:12,540 --> 00:43:16,600
+الفترة 0 بواحد هي المطلوب اثباتها show that g is
+
+504
+00:43:16,600 --> 00:43:20,940
+integrable على هذه الفترة الآن بديش اعيد اللي
+
+505
+00:43:20,940 --> 00:43:23,850
+حكيته المرة الماضية اللي حكيناه المرة الماضيةمن
+
+506
+00:43:23,850 --> 00:43:26,910
+الـ Example اللي أثبتنا فيه انها Integra بالبواسطة
+
+507
+00:43:26,910 --> 00:43:31,470
+اللي هي التعريف أخدنا P N اللي هو Zero واحد على N
+
+508
+00:43:31,470 --> 00:43:34,910
+واثنين على N وان نقص واحد على N لعند الواحد أخدنا
+
+509
+00:43:34,910 --> 00:43:39,290
+اللي هو عبارة عن Any Partition اللي هو بالطريقة
+
+510
+00:43:39,290 --> 00:43:43,710
+اللي أمامي يعني حسب N بيصير بختلف ال Partition أما
+
+511
+00:43:43,710 --> 00:43:46,730
+إيش فكرة ال Partition زي ما قلنا المرة الفائتة من
+
+512
+00:43:46,730 --> 00:43:51,350
+0 لعند 1 جزأناها إلى أجزاء متساوية إلى N من
+
+513
+00:43:51,350 --> 00:43:55,350
+الأجزاء المتساويةصار طول كل sub integral عبارة عن
+
+514
+00:43:55,350 --> 00:44:00,850
+إيهاش عبارة عن واحد على N وأوجدنا في حينه اللي هو
+
+515
+00:44:00,850 --> 00:44:08,470
+ال U P N O G ولاجناها بتساوي إيه ده بتتذكروا نص في
+
+516
+00:44:08,470 --> 00:44:13,710
+واحد زائد واحد على N وأوجدنا برضه ال P N O G
+
+517
+00:44:13,710 --> 00:44:18,910
+ولاجناها عبارة عن نص في واحد ناقص واحد على N كما
+
+518
+00:44:18,910 --> 00:44:26,960
+أذكر ماشي الحال فعلا طيبالان صار عندى الـ PN في
+
+519
+00:44:26,960 --> 00:44:29,780
+الواقع عبارة عن sequence of partitions
+
+520
+00:44:35,190 --> 00:44:39,110
+هذه صارت sequence of partitions بواحد بتعود عن انا
+
+521
+00:44:39,110 --> 00:44:42,750
+بواحد باتنين كده بتلاتة كده باربعة كده وفي كل
+
+522
+00:44:42,750 --> 00:44:47,270
+الأحوال الـ U, B, N و G انجروا لهيها و الـ L, B, N
+
+523
+00:44:47,270 --> 00:44:51,490
+و G بساوي إيش المقدار الأمامي الآن عشان أثبت إنها
+
+524
+00:44:51,490 --> 00:44:56,590
+integrable يكفي من النظرية الكورولاري إن قاعد أقول
+
+525
+00:44:56,590 --> 00:45:03,290
+طب خلينا نشوف limit الـ U, B, N و G ناقص الـ L, B,
+
+526
+00:45:03,450 --> 00:45:09,840
+N و Gas n goes to infinity بساوة إيه يا عشان بساوة
+
+527
+00:45:09,840 --> 00:45:16,360
+سفر نشوفها صح ولا لأ طبعا أكيد مع حسبة بسيطة
+
+528
+00:45:16,360 --> 00:45:24,580
+احسبليها بساوة limit نصفي واحد زياد واحدة لان زياد
+
+529
+00:45:24,580 --> 00:45:28,600
+نص في واحد ناقص واحدة لان as n goes to infinity
+
+530
+00:45:28,600 --> 00:45:32,260
+هذه سفر وهذه سفر وهذه نص وهذه نص وهذه نص وهذه نص
+
+531
+00:45:32,260 --> 00:45:32,260
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+532
+00:45:32,260 --> 00:45:32,260
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+533
+00:45:32,260 --> 00:45:32,260
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+534
+00:45:32,260 --> 00:45:32,260
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+535
+00:45:32,260 --> 00:45:32,340
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+536
+00:45:32,340 --> 00:45:32,920
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+537
+00:45:32,920 --> 00:45:38,340
+وهذه نص وهذه نص وهذه نص وهذه نص وهذه نص
+
+538
+00:45:38,340 --> 00:45:45,580
+وهذهاللي هو y ساوي نص نقص النص y ساوي سفر واضح أهي
+
+539
+00:45:45,580 --> 00:45:50,120
+عندي اللي هو هاد سفر و هاد سفر و نص نقص النص y
+
+540
+00:45:50,120 --> 00:45:54,920
+ساوي سفر مادام سفر إذا إيه شمالها إذا the function
+
+541
+00:45:54,920 --> 00:46:02,720
+d of x ساوي x is integrable on zero واحد هذا by
+
+542
+00:46:02,720 --> 00:46:06,560
+mean by ال corollary اللي قبل بشوية
+
+543
+00:46:15,600 --> 00:46:19,420
+الان بقيمة ال integration ايش هيكون قيمة ال limit
+
+544
+00:46:19,420 --> 00:46:24,280
+limit u, b, n, g هيقدر الغرض و limit l of n of g
+
+545
+00:46:24,280 --> 00:46:29,320
+هيقدر الغرض ليش لأن limit هذا اصلا هيطلع لك limit
+
+546
+00:46:29,320 --> 00:46:35,710
+ال u, b, n و gهو عبارة عن قيمة integration من A ل
+
+547
+00:46:35,710 --> 00:46:42,370
+B حسب النظرية الـ Corollary و يساوي limit 2.5 في 1
+
+548
+00:46:42,370 --> 00:46:48,600
+زائد 1 ل N و يساوي هذا تروح ل 0و يساوى نص و أيضا
+
+549
+00:46:48,600 --> 00:46:54,080
+لو جربت حسب تاب limit ال b, n و g طبعا هيطلع نفس
+
+550
+00:46:54,080 --> 00:46:58,200
+الجواب و إلا إن كان هناك مشكلة لدينا limit نص في
+
+551
+00:46:58,200 --> 00:47:02,340
+واحد ناقص واحد على n و يساوى برضه جدياش نص إذن
+
+552
+00:47:02,340 --> 00:47:06,700
+قيمة ال integration بساوة نص إذن هذه طريقة أخرى
+
+553
+00:47:06,700 --> 00:47:11,780
+لحساب اللي هو أو لإثبات أن g of x بساوة x is
+
+554
+00:47:11,780 --> 00:47:17,560
+integrableالآن بدنا ندخل على أمر آخر الأمر هو في
+
+555
+00:47:17,560 --> 00:47:24,500
+الواقع يا شباب أنه بدنا نشوف إيش في من عائلات
+
+556
+00:47:24,500 --> 00:47:28,400
+الدوال عائلات الدوال إنها تكون Integrable الآن
+
+557
+00:47:28,400 --> 00:47:31,720
+بدنا نيجي نحوش الدوال الـ Integrable إحنا عرفنا بس
+
+558
+00:47:31,720 --> 00:47:36,500
+ده لأ D of X سو X is Integrable و هنلاقي زيها لكن
+
+559
+00:47:36,500 --> 00:47:40,960
+الآن بدنا نيجي نحكي عن دوال اللي هو عائلات من
+
+560
+00:47:40,960 --> 00:47:45,480
+الدوالأول عائلة من العائلات المهمة اللي هي الـ
+
+561
+00:47:45,480 --> 00:47:48,960
+monotone functions يعني الدوال اللي بتكون يا
+
+562
+00:47:48,960 --> 00:47:52,360
+increasing على كل الفترة يا decreasing على كل
+
+563
+00:47:52,360 --> 00:47:59,320
+الفترة بقولكم هذا على اللي هي F من F ال function
+
+564
+00:47:59,320 --> 00:48:04,000
+لو كانت bounded لو كانت monotone على اللي هي
+
+565
+00:48:04,000 --> 00:48:09,660
+closed bounded interval I على طول integrable إذن
+
+566
+00:48:09,660 --> 00:48:16,700
+عيلة كبيرةعيلة الدوال اللي بتكون يا increasing يا
+
+567
+00:48:16,700 --> 00:48:22,380
+decreasing على كل الفترة a و b هذه مضمون انها تكون
+
+568
+00:48:22,380 --> 00:48:26,980
+الدوال ايه شمالها عبارة عن integrable functions
+
+569
+00:48:26,980 --> 00:48:35,090
+اذا اول اعلان الان اللي هو any monotone functionon
+
+570
+00:48:35,090 --> 00:48:40,850
+a closed bounded interval is integrable وهذا اللي
+
+571
+00:48:40,850 --> 00:48:47,090
+هو عنواننا integrability of monotone functionsلت I
+
+572
+00:48:47,090 --> 00:48:51,650
+بتساوي A و B و لت F من I لR بيكون مونوتون فانكشن
+
+573
+00:48:51,650 --> 00:48:57,370
+on I ثم F أشمالها is integrable on I نفترض أن F
+
+574
+00:48:57,370 --> 00:49:03,110
+انتقل هو مونوتور نصل إليها integrable نفترض أن F
+
+575
+00:49:03,110 --> 00:49:09,440
+مثلا increasingونصل إنها integrable وsimilarly
+
+576
+00:49:09,440 --> 00:49:14,560
+وفعلا similarly لو كانت f is decreasing هتكون برضه
+
+577
+00:49:14,560 --> 00:49:21,500
+is integrable خلّينا مع بعض شباب نفترض suppose
+
+578
+00:49:21,500 --> 00:49:31,240
+that f is increasing يعني الدالة على الفترة هي a
+
+579
+00:49:31,240 --> 00:49:37,760
+وb مثلاوالدالة هتكون أشمالها تزايدية يا هيك يا هيك
+
+580
+00:49:37,760 --> 00:49:41,620
+طبعا حسب مش مشكلة بتفرج أشمالها ماشي الحال اللي هي
+
+581
+00:49:41,620 --> 00:49:47,060
+الدالة أشمالها is increasing is increasing بتدخل
+
+582
+00:49:47,060 --> 00:49:53,500
+الآن بي أن اللي هو عبارة عن any partition اللي هو
+
+583
+00:49:55,970 --> 00:50:00,970
+بس بدي أجزه زي منهج اللي عملته معه اللي هو f of x
+
+584
+00:50:00,970 --> 00:50:05,370
+بساوة x اللي قبله شوية بدي أجزه إلى أجزاء متساوية
+
+585
+00:50:05,370 --> 00:50:11,130
+يعني بدي أجز الفقرة a و b إلى أجزاء متساوية الأولى
+
+586
+00:50:11,130 --> 00:50:15,730
+بدي أسميها x note اللي بعدها x1 اللي بعدها x2 لما
+
+587
+00:50:15,730 --> 00:50:21,780
+أصل لآخر 11 أسميها xnونكون طول كل واحدة متساوية
+
+588
+00:50:21,780 --> 00:50:26,340
+للثانية إذا الان إذا بدي أجزئها إلى N إلى N من ال
+
+589
+00:50:26,340 --> 00:50:30,240
+sub intervals بيصير طول كل فترة عبارة عن B minus A
+
+590
+00:50:30,240 --> 00:50:35,080
+على مين؟ على N طول الفترة على عدد الفترات اللي
+
+591
+00:50:35,080 --> 00:50:39,400
+بتديهاأنا بدّي الفترات إيش عددها؟ N فبيصير عند بي
+
+592
+00:50:39,400 --> 00:50:43,320
+ماينس إيه على مين على N هذا طول الفترة يعني الفترة
+
+593
+00:50:43,320 --> 00:50:47,740
+النموذجية لأي K X K minus X K minus واحد هيتمكن
+
+594
+00:50:47,740 --> 00:50:52,560
+طولها عبارة عن بي ماينس إيه على N for every K كيه
+
+595
+00:50:52,560 --> 00:50:56,000
+طبعا إيه اشمالها؟ يا ما بيساوي Zero يا واحد، لعند
+
+596
+00:50:56,000 --> 00:51:01,100
+ماصل، لعند مين؟ لعند N أو K بيساوي واحد لعند N
+
+597
+00:51:04,810 --> 00:51:10,230
+xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+598
+00:51:10,230 --> 00:51:14,230
+-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+599
+00:51:14,230 --> 00:51:14,590
+-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+600
+00:51:14,590 --> 00:51:17,070
+-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n-xn-n
+
+601
+00:51:17,070 --> 00:51:25,770
+-xn-n-xn-n-xn-n-xn-n-xn-nلأ لو جينا هذا increasing
+
+602
+00:51:25,770 --> 00:51:30,170
+مدام increasing إذن الـ Mk اللي هي عبارة عن ال
+
+603
+00:51:30,170 --> 00:51:35,510
+supremum اللي هنا supremum لهذه Mk ال supremum
+
+604
+00:51:35,510 --> 00:51:39,910
+اللي هنا هيكون على أخر واحدة لأن أجملها ده لا is
+
+605
+00:51:39,910 --> 00:51:45,510
+increasing إذن هيكون ال Mk هو F of XK رسم عنها
+
+606
+00:51:45,990 --> 00:51:51,210
+بساوة F of X K طيب ال M K Small أكيد هيقولنا كلكم
+
+607
+00:51:51,210 --> 00:51:54,010
+هتقوله بتساوة ال M في معمع الفترة مدام ال M في
+
+608
+00:51:54,010 --> 00:51:57,370
+معمع الفترة إذا أول واحدة فيهم إذا هو F of X K
+
+609
+00:51:57,370 --> 00:52:04,530
+minus واحد، مظبوط شباب؟ طيب، مظبوط هذه اللي هي
+
+610
+00:52:04,530 --> 00:52:10,750
+مالها الدالة اللي بتكون monotone عندي،
+
+611
+00:52:10,750 --> 00:52:21,840
+احسبلي الآن ال U P N و Fنقص الـ BNOF إيش هيساوي؟
+
+612
+00:52:21,840 --> 00:52:26,840
+جهزة الأمور بساوي الـ summation لمين؟ للـ MK
+
+613
+00:52:26,840 --> 00:52:36,190
+Capital في XK نقص XK minus واحدك من عند 1 لعند n
+
+614
+00:52:36,190 --> 00:52:44,370
+نقص summation mk في xk minus xk minus 1 k من عند 1
+
+615
+00:52:44,370 --> 00:52:49,440
+لعند n ويساويالـ Mk وجدناها، اللي هي عبارة عن مين؟
+
+616
+00:52:49,440 --> 00:52:55,420
+F of Xk ويساوي الـ summation للـ F of Xk في مين
+
+617
+00:52:55,420 --> 00:53:00,200
+مضروبة؟ في هذه، هذه كده طولها ثابت، ما احنا هيك
+
+618
+00:53:00,200 --> 00:53:02,540
+على هذا الأساس اذا اخترنا الـ sequence of
+
+619
+00:53:02,540 --> 00:53:05,320
+partitions اللي عندنا، اللي هو طوالث في
+
+620
+00:53:05,320 --> 00:53:09,580
+subintervals ثابتة، كل واحد اسمه يشمله B minus A
+
+621
+00:53:09,580 --> 00:53:14,250
+على N، هذا K من واحدإن عندنا ناقص خلّيني أضعه في
+
+622
+00:53:14,250 --> 00:53:18,550
+summation 1 ناقص نفس القصة اللي هي m k small إيش
+
+623
+00:53:18,550 --> 00:53:23,710
+هي يا جماعة اتفاقنا عبارة عن f of x k minus 1 في
+
+624
+00:53:23,710 --> 00:53:27,610
+هذه خلّيني أخدها عامل مشترك بعد إذنكم لإن هو
+
+625
+00:53:27,610 --> 00:53:31,070
+موجودة هنا وموجودة هنا خلّيني أطلّحها برا واضحة
+
+626
+00:53:31,070 --> 00:53:36,070
+أشيل هذه برا بيصير اللي هو مضروبة في b minus a على
+
+627
+00:53:36,910 --> 00:53:42,350
+وأصلا هذه ثابتة بالنسبة لل summation ليش؟ لأن ال
+
+628
+00:53:42,350 --> 00:53:46,110
+summation العداد K من واحد عندنا أن هذه N ثابتة
+
+629
+00:53:46,110 --> 00:53:52,870
+بالنسبة لل K لذلك بتساوي B minus A على N مضروبة في
+
+630
+00:53:52,870 --> 00:53:59,280
+مين في ال summationللـ f of x k minus f of x k
+
+631
+00:53:59,280 --> 00:54:03,840
+minus واحد k من عند واحد لعند مين يا جماعة لعند ان
+
+632
+00:54:03,840 --> 00:54:07,920
+بعدين the is and could بدي افرطها هذه بيصير عندي y
+
+633
+00:54:07,920 --> 00:54:14,880
+ساوي هذا اللي هو مين كله اللي جاعت بحسبه ال u,b,n
+
+634
+00:54:14,880 --> 00:54:19,320
+وf ناقص ال b,n وf هيسوي هذا المقدار اللي هو b
+
+635
+00:54:19,320 --> 00:54:25,970
+minus a على المضروب فيهالان ال summation عبارة عن
+
+636
+00:54:25,970 --> 00:54:32,630
+k ب 1 بصير f of x 1 ناقص f of x naught اللي بعدها
+
+637
+00:54:32,630 --> 00:54:40,170
+k ب 2 زائد f of x 2 ناقص f of x 1 اللي بعدها زائد
+
+638
+00:54:40,170 --> 00:54:46,130
+f of x 3 ناقص f of x 2 لما نقضى اللي ماشي لأخر
+
+639
+00:54:46,130 --> 00:54:53,140
+واحد بكون عندى f of x nناقص f of x n ناقص واحد
+
+640
+00:54:57,360 --> 00:55:04,840
+F of X1 بيطير مع سالب F of X1 و F of X2 بيطير مع
+
+641
+00:55:04,840 --> 00:55:09,840
+سالب F of X2 و هكذا بظل ماشي لما كله يروح مع كله
+
+642
+00:55:09,840 --> 00:55:16,640
+ما عدا بظل عندي اللي هو أول قيمة اللي هي F of X
+
+643
+00:55:16,640 --> 00:55:22,660
+note بالسالب مع F of Xn الأخيرة بالموجب بيصير عندي
+
+644
+00:55:22,660 --> 00:55:34,740
+Y ساوي B minus A على Nفي F of Xn ناقص F of X0 طبعا
+
+645
+00:55:34,740 --> 00:55:39,420
+Xn آخر واحدة اللي هي B و X0 أول واحدة اللي هي A
+
+646
+00:55:39,420 --> 00:55:46,580
+إذن هيساوي هذا عبارة عن B minus A في F of B minus
+
+647
+00:55:46,580 --> 00:55:53,150
+F of A الكل هذا مجسوم على مين يا جماعة؟ على Nهذا
+
+648
+00:55:53,150 --> 00:55:58,090
+عبارة عن ثابت وهذا عبارة عن ثابت وهذا الان هي اللي
+
+649
+00:55:58,090 --> 00:56:03,670
+بدلالتها كُتِب ال partitions إذا صار عندى الآن بعد
+
+650
+00:56:03,670 --> 00:56:07,230
+كل اللي حكيته هان أخدت ال بي أن بالشكل اللي أمامي
+
+651
+00:56:07,230 --> 00:56:12,190
+عبارة عن sequence of partitions ووصلنا إلى ما
+
+652
+00:56:12,190 --> 00:56:17,890
+لاقيا ليه يا جماعة اللي هو وصلنا ان ال you بي أن و
+
+653
+00:56:17,890 --> 00:56:26,260
+أف ناقص ال بي أن و أفأصغر أو يساوي طبعا أكيد أكبر
+
+654
+00:56:26,260 --> 00:56:29,600
+يساوي صفر لأن هذا دايما أكبر يساوي هذا أصغر يساوي
+
+655
+00:56:29,600 --> 00:56:36,140
+B minus A ف F of B أكيد عرفت أشهد أساوي نقص F of A
+
+656
+00:56:36,140 --> 00:56:41,720
+على اللي هو N الـ N خد ال limit للجهتين as N goes
+
+657
+00:56:41,720 --> 00:56:45,160
+to infinity as N goes to infinity هذا goes to zero
+
+658
+00:56:45,160 --> 00:56:51,730
+وهذا أصلا صفر فبصير عندي limit إذا .. إذا limitالـ
+
+659
+00:56:51,730 --> 00:56:59,290
+U P N و F ناقص الـ L P N و F as N goes to infinity
+
+660
+00:56:59,290 --> 00:57:04,210
+بساوي سفر إذا صار عندي sequence of partitions تحقق
+
+661
+00:57:04,210 --> 00:57:07,730
+لهذا إذا حسب الـ Corollary اللي حكيتها قبل بشوية
+
+662
+00:57:07,730 --> 00:57:17,130
+إذا F is integrable وهو المطلوب إذا صار عندي أي
+
+663
+00:57:17,130 --> 00:57:21,850
+increasing function is integrablesimilarly for
+
+664
+00:57:21,850 --> 00:57:26,170
+decreasing لماذا similarly لأنه تصبح الدالة بدل ما
+
+665
+00:57:26,170 --> 00:57:30,950
+هي طالع هيك تصبح أشمالها نازلة نزول الدالة فبصير
+
+666
+00:57:30,950 --> 00:57:35,430
+عندك اللي هو ال maximum هي الأولى أو ال supremum
+
+667
+00:57:35,430 --> 00:57:41,920
+هي أفبصير الـ MK بسوء F of XK minus واحد والـ MK
+
+668
+00:57:41,920 --> 00:57:46,760
+بسوء F of XK وبيتكملوا البرهان بنفس الطريقة، هيطلع
+
+669
+00:57:46,760 --> 00:57:52,040
+عندكم البرهان automatic وبشكل سهل وبشكل سلس،
+
+670
+00:57:52,040 --> 00:57:58,500
+similar أي سؤال؟ إذن الإعلان اللي أعلنناه قبل
+
+671
+00:57:58,500 --> 00:58:04,800
+بشوية أنه any monotone functionيعني any increasing
+
+672
+00:58:04,800 --> 00:58:08,780
+function on a closed bounded interval must be
+
+673
+00:58:08,780 --> 00:58:15,740
+integrable and any decreasing function on a closed
+
+674
+00:58:15,740 --> 00:58:20,420
+bounded interval برضه must be integrable إذا صار
+
+675
+00:58:20,420 --> 00:58:26,060
+في عنا عائلة كاملة من الدوال القابلة للتكامل بواسط
+
+676
+00:58:26,060 --> 00:58:32,160
+التكامل remandالآن بدنا ننتقل إلى عائلة أخرى
+
+677
+00:58:32,160 --> 00:58:38,300
+وعائلة لا تقل أهمية عن هذه العائلة وعائلة يعني
+
+678
+00:58:38,300 --> 00:58:45,560
+محبوبة عنها اللي هي اللي
+
+679
+00:58:45,560 --> 00:58:53,960
+هي عائلة الدوال المتصلة اللي هي integrable of
+
+680
+00:58:53,960 --> 00:58:56,140
+continuous functions
+
+681
+00:59:00,850 --> 00:59:06,130
+اللي هي integrability of continuous functions
+
+682
+00:59:06,130 --> 00:59:12,210
+النظرية بتقول ما يلي لت أ .. طبعا عندي ال function
+
+683
+00:59:12,210 --> 00:59:16,410
+على closed bounded interval لت F من I لR be
+
+684
+00:59:16,410 --> 00:59:20,750
+continuous on I then F is integrable on I إذا الآن
+
+685
+00:59:20,750 --> 00:59:24,230
+any continuous function on a closed bounded
+
+686
+00:59:24,230 --> 00:59:29,600
+interval must be integrableكمان مرة any continuous
+
+687
+00:59:29,600 --> 00:59:33,840
+function on a closed bounded interval must be
+
+688
+00:59:33,840 --> 00:59:38,920
+integrable طبعا حيزنا في البرهان شغلة اللي هي
+
+689
+00:59:38,920 --> 00:59:45,080
+أخدناها سابقا انهAny continuous function on a
+
+690
+00:59:45,080 --> 00:59:49,840
+closed bounded interval must attain its maximum
+
+691
+00:59:49,840 --> 00:59:57,800
+and minimum on this interval بمعنى
+
+692
+00:59:57,800 --> 01:00:01,970
+آخر هنلاقيلو كانت F is continuous على اللي هو الـ
+
+693
+01:00:01,970 --> 01:00:05,990
+A و الـ B هنلاقي نقطة في داخل الفترة A و B بحيث
+
+694
+01:00:05,990 --> 01:00:09,570
+انها تكون الـ F عندها نقطة maximum و هنلاقي نقطة
+
+695
+01:00:09,570 --> 01:00:12,750
+أخرى في داخل هذه الفترة هنلاقي اللي هي الـ
+
+696
+01:00:12,750 --> 01:00:15,930
+function عندها إشمالها is minimum طبعا absolute و
+
+697
+01:00:15,930 --> 01:00:24,180
+absolute نيجي نشوف .. نيجي للبرهانالان .. نظرية
+
+698
+01:00:24,180 --> 01:00:28,540
+أخرى أيضا .. برضه خلينا نقولها أن لو كانت F is
+
+699
+01:00:28,540 --> 01:00:32,580
+continuous على closed bounded interval then F is
+
+700
+01:00:32,580 --> 01:00:43,820
+uniformly continuous الان F .. عند F من A و B لعند
+
+701
+01:00:43,820 --> 01:00:51,090
+R is continuous on A و Bcontinuous عالمين على
+
+702
+01:00:51,090 --> 01:00:54,450
+closed bounded interval في عندنا نظرية اللي بتقول
+
+703
+01:00:54,450 --> 01:00:58,270
+any continuous function في الواحد on a closed
+
+704
+01:00:58,270 --> 01:01:02,230
+bounded interval must be uniformly continuous إذا
+
+705
+01:01:02,230 --> 01:01:10,030
+then f is uniformly continuous
+
+706
+01:01:10,730 --> 01:01:15,290
+on a و b ايش يعني uniformly continuous يعني لكل
+
+707
+01:01:15,290 --> 01:01:18,810
+ابسلون اكبر من سفر there exist دلتا اكبر من سفر
+
+708
+01:01:18,810 --> 01:01:22,990
+دلتا بس بتعتمد عالميا على ابسلون بتنفع لكل ال X و
+
+709
+01:01:22,990 --> 01:01:30,810
+ال Y such that if X minus Y او U ناقص V زي ما هو
+
+710
+01:01:30,810 --> 01:01:40,200
+مسميها U minus Vأصغر من دلتا then F of U ناقص F of
+
+711
+01:01:40,200 --> 01:01:49,800
+V أصغر من إبسلون لكل UV element in A أو Bإذن لكل y
+
+712
+01:01:49,800 --> 01:01:52,900
+أكبر من 0 there exists Delta دلتا أشمة لا بتعتمد
+
+713
+01:01:52,900 --> 01:01:56,280
+على مين بس على إبسلون في حالة ال continuity
+
+714
+01:01:56,280 --> 01:02:01,560
+العادية بنقول إحنا limit f of x as x بتروح لل a
+
+715
+01:02:01,560 --> 01:02:07,260
+بساوي ايش f of a بنقول الآن إنه limit f of x بساوي
+
+716
+01:02:07,260 --> 01:02:13,450
+f of a as x بتروح لل a لكل إبسلون أكبر من 0هصب مين
+
+717
+01:02:13,450 --> 01:02:18,070
+بالـ a هذه there exist دي و لكل a element in a
+
+718
+01:02:18,070 --> 01:02:23,270
+there exist delta such that اللي لما يكون x minus
+
+719
+01:02:23,270 --> 01:02:27,190
+a أصغر من delta بيعطينا f of x نقص f of a أصغر من
+
+720
+01:02:27,190 --> 01:02:32,550
+إبسلون يعني بيكون الـ delta هنا تعتمد على الإبسلون
+
+721
+01:02:32,550 --> 01:02:37,550
+وتعتمد على ال a اللي عندها ال continuityلكن في الـ
+
+722
+01:02:37,550 --> 01:02:41,890
+Uniform Continuous الـ Delta اللي اللي جيت هنا
+
+723
+01:02:41,890 --> 01:02:47,510
+بتنفع لكل اللي هو ال .. ال .. ال .. ال .. ال .. ال
+
+724
+01:02:47,510 --> 01:02:47,890
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+725
+01:02:47,890 --> 01:02:47,910
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+726
+01:02:47,910 --> 01:02:48,290
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+727
+01:02:48,290 --> 01:02:48,530
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+728
+01:02:48,530 --> 01:02:48,530
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+729
+01:02:48,530 --> 01:02:50,270
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+730
+01:02:50,270 --> 01:02:50,330
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+731
+01:02:50,330 --> 01:02:50,330
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+732
+01:02:50,330 --> 01:02:50,410
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+733
+01:02:50,410 --> 01:02:50,410
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+734
+01:02:50,410 --> 01:02:54,250
+.. ال .. ال .. ال .. ال .. ال ..
+
+735
+01:02:54,250 --> 01:02:59,970
+ال .. ال ..F is continuous، إذن يُنفرم الـ
+
+736
+01:02:59,970 --> 01:03:02,650
+continuous، إيش معناته؟ كل إبسلون أكبر من صفر،
+
+737
+01:03:02,650 --> 01:03:05,490
+there exists دلتا، بحيث إن U minus V أصغر من دلتا
+
+738
+01:03:05,490 --> 01:03:08,690
+يعطيني F of U ناقص F of V أصغر من إياش، من إبسلون،
+
+739
+01:03:08,690 --> 01:03:11,870
+وخلّي هذه إيش مالها في الذاكرة، أنا عشان الحسابات
+
+740
+01:03:11,870 --> 01:03:15,890
+بدي أخليها إبسلون على مين، على B minus A، طول
+
+741
+01:03:15,890 --> 01:03:19,070
+الفترة اللي أنا بشتغل عليها، هتشوفوا ليش كتبت هيك،
+
+742
+01:03:19,070 --> 01:03:22,720
+بس لا الحساباتبقدر أه بقدر لإن هو أصغر .. أه بقدر
+
+743
+01:03:22,720 --> 01:03:25,140
+خليه أصغر من الـ Epsilon في الدنيا من ضمن إن كل
+
+744
+01:03:25,140 --> 01:03:27,520
+الـ Epsilon عن الـ B minus A لكل الـ Epsilonات
+
+745
+01:03:27,520 --> 01:03:34,040
+اللي في الدنيا طيب لأن شوف وين بده أروح، بده أروح
+
+746
+01:03:34,040 --> 01:03:39,250
+لـ Integrability لـ الـ function Fالأن خُد n الان
+
+747
+01:03:39,250 --> 01:03:42,190
+بدنا نعمل partitions بدنا نجيب sequence of
+
+748
+01:03:42,190 --> 01:03:45,890
+partitions Bn ال sequence of partitions هذه هي
+
+749
+01:03:45,890 --> 01:03:50,010
+اللي هتخدمني هتخدمني متى؟ بعد شوية بتشوفوا ليش
+
+750
+01:03:50,010 --> 01:03:54,110
+هتخدمني لكل الأنات اللي وين ما لإن اللي أكبر من ال
+
+751
+01:03:54,110 --> 01:03:59,370
+B minus A على مين؟ على ال delta اللي لجيتها إذا
+
+752
+01:03:59,370 --> 01:04:04,130
+اللي صارت ال delta بين إيديا إذا بقولك الآن choose
+
+753
+01:04:05,140 --> 01:04:12,320
+N element in N such that ما لها N أكبر من B minus
+
+754
+01:04:12,320 --> 01:04:17,620
+A على مين على دلتا برضه B minus A عشان الحسبان N
+
+755
+01:04:17,620 --> 01:04:21,240
+أكبر من B minus A على مين على دلتا يعني الآن أنا
+
+756
+01:04:21,240 --> 01:04:26,920
+بحكي بحكي عن الأنات اللي بكون أكبر من B minus A
+
+757
+01:04:26,920 --> 01:04:31,460
+على مين على الدلتا اللي لجيتها فوق عندي في اللي هو
+
+758
+01:04:31,460 --> 01:04:37,080
+هذا واحدماشي الحالة الان بتبدأ اكوّن partitions من
+
+759
+01:04:37,080 --> 01:04:42,480
+مين؟ من الأمنات هذه الان بتاخد بيئا اني partitions
+
+760
+01:04:42,480 --> 01:04:47,220
+اني partitions برضه أشماله with equal اللي هي إيش
+
+761
+01:04:47,220 --> 01:04:54,240
+sub intervals يعني X نوت و X واحد لعند X ان يعني
+
+762
+01:04:54,240 --> 01:04:58,540
+طول كل فترة منهم برضه طول .. بتاخد اللي هي كلهم
+
+763
+01:04:58,540 --> 01:05:03,590
+أشمالينمن عند A لعند B يكون ان كل ال sub intervals
+
+764
+01:05:03,590 --> 01:05:08,530
+جت بعض يعني بمعنى أخر هيكون ال XK minus XK minus
+
+765
+01:05:08,530 --> 01:05:12,970
+واحد بيساوي B minus A على N طول كل فترة ايش بيساوي
+
+766
+01:05:12,970 --> 01:05:16,950
+B minus A على عدد الفترات اللي هي N فبكون B minus
+
+767
+01:05:16,950 --> 01:05:21,250
+A عالميا على Nأنا حرّن انهم ألاقي في النهاية
+
+768
+01:05:21,250 --> 01:05:25,630
+sequence of partitions BN limit الـ U B N و F نقص
+
+769
+01:05:25,630 --> 01:05:29,290
+الـ L B N و F بساوي 0 بيكون F is integrable خلصنا
+
+770
+01:05:29,290 --> 01:05:32,950
+أنا قاعد اخترت اللي هي sequence of partitions بناء
+
+771
+01:05:32,950 --> 01:05:36,390
+على الـ Delta اللي لجيتها في ال uniformity و
+
+772
+01:05:36,390 --> 01:05:39,830
+الأنات اللي أكبر منها و حطيت ال partition اللي هو
+
+773
+01:05:39,830 --> 01:05:44,710
+بشكل اللي هو تكون ال subintervals كلها لها نفس
+
+774
+01:05:44,710 --> 01:05:46,230
+الطول طيب
+
+775
+01:05:48,470 --> 01:05:56,170
+الآن زي ما قلت قبل بشوية negation على الفترة xk و
+
+776
+01:05:56,170 --> 01:06:01,710
+minus 1 و xk هذا ال sub interval ال function is
+
+777
+01:06:01,710 --> 01:06:03,550
+continuous عليها لأنها continuous على كل ال
+
+778
+01:06:03,550 --> 01:06:06,970
+interval a و b مدام continuous عليها إذا it
+
+779
+01:06:06,970 --> 01:06:10,310
+attains its maximum and its minimum on this
+
+780
+01:06:10,310 --> 01:06:16,100
+intervalأكيد؟ لقد حد كده قبل مشوية إذن بما أن الـ
+
+781
+01:06:16,100 --> 01:06:18,480
+F is continuously not closed bound in الـ interval
+
+782
+01:06:18,480 --> 01:06:26,800
+هذه إذن there exist سميها U K و V K element in X K
+
+783
+01:06:26,800 --> 01:06:33,720
+minus واحد و X K such that F of U K هي ال maximum
+
+784
+01:06:33,720 --> 01:06:41,190
+على كل هذه ال maximum معناته هي ال supremum of Kأو
+
+785
+01:06:41,190 --> 01:06:47,110
+أيضًا عند الـ VK F of VK هي الـ Minimum على هذه
+
+786
+01:06:47,110 --> 01:06:52,310
+مضمون موجود قلناها هي طبعًا مدامة Minimum على كل
+
+787
+01:06:52,310 --> 01:06:56,150
+الفترة إذًا هي الـ Infimum اللي ببحث عنها اليمين
+
+788
+01:06:56,150 --> 01:07:00,850
+MK على هذه الفترة وها المفتاح أصلا في استخدام الـ
+
+789
+01:07:00,850 --> 01:07:06,120
+Continuity إن ضمن لوجود نقطةعندها ال maximum في
+
+790
+01:07:06,120 --> 01:07:10,540
+هذه المنطقة وضمن اللي وجود نقطة هنا ضمن اللي وجود
+
+791
+01:07:10,540 --> 01:07:13,680
+ال minimum عندها و ال minimum و ال maximum مدام
+
+792
+01:07:13,680 --> 01:07:18,100
+عند نقاط محددة في الفترة هي هتتلاقى و تنقل هي
+
+793
+01:07:18,100 --> 01:07:21,780
+عبارة عن ال supremum و ال infimum على ال sub
+
+794
+01:07:21,780 --> 01:07:29,160
+interval أي سؤال طيب الآن ضال علينا إيش نسوي إن
+
+795
+01:07:29,160 --> 01:07:35,610
+نحسب الحسابات اللي هو نبدأ في حساباتنااللي هو نحسب
+
+796
+01:07:35,610 --> 01:07:44,230
+الـ U و نحسب مين يا جماعة أو نحسب اللي هو ال ال ال
+
+797
+01:07:44,230 --> 01:07:44,230
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+798
+01:07:44,230 --> 01:07:44,930
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+799
+01:07:44,930 --> 01:07:44,930
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+800
+01:07:44,930 --> 01:07:44,950
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+801
+01:07:44,950 --> 01:07:45,050
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+802
+01:07:45,050 --> 01:07:45,070
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+803
+01:07:45,070 --> 01:07:45,130
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+804
+01:07:45,130 --> 01:07:45,130
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+805
+01:07:45,130 --> 01:07:45,130
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+806
+01:07:45,130 --> 01:07:45,130
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+807
+01:07:45,130 --> 01:07:45,130
+ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال ال
+
+808
+01:07:45,130 --> 01:07:53,950
+ال ال الت حسبولي
+
+809
+01:07:54,430 --> 01:07:59,790
+نقص L, P, N و F طبعا أكيد هذا أكبر يساوي سفر حفظنا
+
+810
+01:07:59,790 --> 01:08:04,050
+عن غيره هذا أنه أكبر أو يساوي سفر معروف ضل بدي
+
+811
+01:08:04,050 --> 01:08:08,850
+أحسبهم مع بعض ال summation عبارة عن بساوي
+
+812
+01:08:08,850 --> 01:08:13,610
+summation اللي هو ال M K صارت من ال M K capital
+
+813
+01:08:13,610 --> 01:08:19,950
+اللي هي F of U K صح ولا لأ يا جماعة أه صح و ال U K
+
+814
+01:08:19,950 --> 01:08:26,220
+وين موجودة في هذهالان في مين؟ في xk minus xk minus
+
+815
+01:08:26,220 --> 01:08:31,520
+واحد xk minus xk minus واحد كامل عند واحد اللي
+
+816
+01:08:31,520 --> 01:08:37,020
+عندها انهدمين هي الـ U الان زيها مين ال L ناقص
+
+817
+01:08:37,020 --> 01:08:42,600
+summation الان مين عند ال MK small ال F of VK ال
+
+818
+01:08:42,600 --> 01:08:48,920
+VK اللي لاجيناها برضه هان اللي هي F of VKمضروبة في
+
+819
+01:08:48,920 --> 01:08:54,380
+طول الفترة xk-xk-1 كامل عند واحد لعند مين يا جماعة
+
+820
+01:08:54,380 --> 01:09:00,000
+لعندنا خدولي الآن xk-xk-1 عامل مشترك وطبعا أنا
+
+821
+01:09:00,000 --> 01:09:03,220
+مفترضها اللي هو وماخدها إلا طول ال intervals أو
+
+822
+01:09:03,220 --> 01:09:07,080
+sub intervals متساوي يعني طول هذه وطول هذه هو
+
+823
+01:09:07,080 --> 01:09:10,600
+عبارة عن b-a على n زي النظرية السابقة ويساوي ال
+
+824
+01:09:10,600 --> 01:09:17,850
+summation لل F of uk ناقص F of vkالكل مضروب في مين
+
+825
+01:09:17,850 --> 01:09:26,030
+يا جماعة في B-A على N N أو K من عند واحد لعند إيش
+
+826
+01:09:26,030 --> 01:09:36,830
+لعند N أي سؤال طيب شوفوا الآن اسمحولي بس هنا أشتغل
+
+827
+01:09:36,830 --> 01:09:39,730
+شوية طيب
+
+828
+01:09:42,370 --> 01:09:47,490
+طولوا روحكم عندنا خلصنا يعني جربنا اذا صار عندي ال
+
+829
+01:09:47,490 --> 01:09:51,870
+U, P, N و F عشان تعرفوا أين رايح أنا هي ال U, P, N
+
+830
+01:09:51,870 --> 01:09:55,550
+و F ناقص ال P, N و F أكبر يساوي الصفر اللي جيته
+
+831
+01:09:55,550 --> 01:10:00,610
+أصغر يساوي المقدار اللي أمامي الان خلوني أطلع حد
+
+832
+01:10:00,610 --> 01:10:06,400
+برا بعد اسمكم هي B- a على n لأنها عبارة عن ثابت في
+
+833
+01:10:06,400 --> 01:10:13,020
+ال summation f of u,k ناقص f of v,k كامنة عند واحد
+
+834
+01:10:13,020 --> 01:10:20,340
+لأنك لاحظوا ما يلي يا جماعة عند ال u,k و ال v,k
+
+835
+01:10:20,340 --> 01:10:27,480
+وين موجودة في ال x,k minus واحد و ال x,k اه مظبوط
+
+836
+01:10:27,480 --> 01:10:32,540
+يعني الآن اللي هي طول
+
+837
+01:10:34,610 --> 01:10:41,150
+طول الفترة طول الفترة xk على جهة دي بس يا شباب
+
+838
+01:10:41,150 --> 01:10:46,850
+ناقص xk minus واحد طول الفترة لإن هاد صارت هي xk
+
+839
+01:10:46,850 --> 01:10:53,480
+minus واحد وهي xk ماشي جوات هنا ال mean ال UKو الـ
+
+840
+01:10:53,480 --> 01:10:56,860
+VK في مكان الماد، التنتين المهم جوا التنتين يعني
+
+841
+01:10:56,860 --> 01:10:59,840
+المسافة بين اللي برا هدول، حتى لو كان التنتين
+
+842
+01:10:59,840 --> 01:11:04,120
+زيهن، بكون أصغر أو يساوي المسافة بين التنتين هدول
+
+843
+01:11:04,120 --> 01:11:11,560
+أصغر أو يساوي المسافة هذه، اللي هو UK ناقص VK،
+
+844
+01:11:11,560 --> 01:11:17,010
+مظبوط؟ آسف، العكس، أكبر شكراسامحونا أكبر اياش أو
+
+845
+01:11:17,010 --> 01:11:21,310
+يساوي صارت المسافة بين الجهتين هدولة أكيد أصغر
+
+846
+01:11:21,310 --> 01:11:24,970
+أشهر من المسافة الكلية اللي هنا بينهم طيب المسافة
+
+847
+01:11:24,970 --> 01:11:28,890
+بين هذه وهذه احنا ماخدينها أصلا طول ال interval
+
+848
+01:11:28,890 --> 01:11:29,670
+ايش بتساوي
+
+849
+01:11:33,030 --> 01:11:39,710
+على N إذا المسافة هذه أصلا عبارة عن B-A عالمين على
+
+850
+01:11:39,710 --> 01:11:43,890
+N سامحوني أني بنكتبها هنا إذا صار عندي اللي هو ال
+
+851
+01:11:43,890 --> 01:11:50,230
+U K ناقص ال V K أصغر أو يساوي B-A عالمين على N إذا
+
+852
+01:11:50,230 --> 01:11:58,530
+يا شباب راحظولي هنا U K ناقص V K صارت أصغر أو
+
+853
+01:11:58,530 --> 01:12:06,440
+يساوي B-A عالمينعلى n واضحة طيب نشوف ايش معناه هذا
+
+854
+01:12:06,440 --> 01:12:08,700
+الكلام واش اللي بتقوله ليش بتقولنا هذا الكلام
+
+855
+01:12:08,700 --> 01:12:13,620
+بقوله عشان كلام مهم هذا هو شوفوا فهميته انا مختار
+
+856
+01:12:13,620 --> 01:12:19,260
+ال n اكبر من مين من b minus a على delta يعني بمعنى
+
+857
+01:12:19,260 --> 01:12:24,280
+اخر ايش معنات هذا يعني b minus a على n اصغر من مين
+
+858
+01:12:24,280 --> 01:12:29,640
+من دلتا يعني هذا المقدارأصغر من إيش يا جماعة؟ من
+
+859
+01:12:29,640 --> 01:12:35,160
+دلتا صارت ال U K ناقص ال V K أصغر من مين؟ من دلتا
+
+860
+01:12:35,160 --> 01:12:40,800
+واحنا بنقول أن المقاطع اللي بتحقق فيها ال U ناقص V
+
+861
+01:12:40,800 --> 01:12:44,560
+أصغر من دلتا بكون عندي F of U ناقص F of V أصغر من
+
+862
+01:12:44,560 --> 01:12:50,240
+مين يا جماعة؟ من Epsilon على B minus A إذا صارت
+
+863
+01:12:50,240 --> 01:12:55,100
+عندي المقاطين هذولة ال U K ناقص V K اللي أصغر من
+
+864
+01:12:55,100 --> 01:13:04,770
+دلتاإذا أكيد من واحد هيكون عندي F of U K ناقص F of
+
+865
+01:13:04,770 --> 01:13:12,670
+V K إيش هيكون؟ هيكون عبارة عن أصغر من Y على B
+
+866
+01:13:12,670 --> 01:13:20,410
+minus A إذا الآن بحجلني أني أقول أن هذا المقدار
+
+867
+01:13:20,410 --> 01:13:24,400
+اللي عندياللي هو طبعا هذا الكبير وهذا الصغير طبعا
+
+868
+01:13:24,400 --> 01:13:27,800
+على absolute value نفسه بحجل إن أقول هذا أصغر أو
+
+869
+01:13:27,800 --> 01:13:34,680
+ساوي B minus A على N مضروب في مين في ال summation
+
+870
+01:13:34,680 --> 01:13:42,480
+لل Epsilon على B minus A كامن عند واحد لعند مين
+
+871
+01:13:42,480 --> 01:13:50,100
+لعند N واضح
+
+872
+01:13:52,720 --> 01:13:58,440
+إذا صار عندى الآن اللى هو اللى أثبته أنه U P N و F
+
+873
+01:13:58,440 --> 01:14:04,220
+ناطق سؤال P N و F أصغر أو ساوى P minus A على N في
+
+874
+01:14:04,220 --> 01:14:10,300
+.. خلّيني أجف و أقولها .. أه .. انتظر .. أعملنى ..
+
+875
+01:14:13,410 --> 01:14:18,250
+إذا اللي وصلنا له يا جماعة اللي هو أن ال U P N و F
+
+876
+01:14:18,250 --> 01:14:21,650
+نقص ال P N و F أكبر يساوي سفر و في نفس الوقت أصغر
+
+877
+01:14:21,650 --> 01:14:24,430
+يساوي P Minus A على N في ال summation لل Epsilon ع
+
+878
+01:14:24,430 --> 01:14:28,630
+P Minus A اللي هو كامن عند واحد ل N يعني هذه قاعدة
+
+879
+01:14:28,630 --> 01:14:34,050
+عمالها كل مرة بتعد Epsilon ع P Minus A أكم مرة
+
+880
+01:14:34,050 --> 01:14:37,670
+بتعد N من المرات إذا حيصير عندي هذا عبارة عن P
+
+881
+01:14:37,670 --> 01:14:43,170
+Minus A على N و اللي نعد هناN من القمية اللي هذه
+
+882
+01:14:43,170 --> 01:14:47,630
+يعني N في epsilon على B minus A الان B minus A مع
+
+883
+01:14:47,630 --> 01:14:51,970
+ال B minus A و ال N مع ال N فبصير أوصلنا احنا إلى
+
+884
+01:14:51,970 --> 01:14:55,950
+ما يلي انتبهوا للنتيجة النهائية اللي بتوصلي لي
+
+885
+01:14:55,950 --> 01:15:04,310
+اللي هو المطلوب النتيجة اللي وصلت لها انه اذاU P N
+
+886
+01:15:04,310 --> 01:15:12,430
+و F نقص L P N و F أكبر أو يساوي سفر وأصغر من مين
+
+887
+01:15:12,430 --> 01:15:19,190
+من إبسلون لكل إبسلون أكبر من سفر إذا اللي جوا غصب
+
+888
+01:15:19,190 --> 01:15:24,310
+عنه إذا لازم يكون اللي هو لو أخدت ال limit هيظل
+
+889
+01:15:24,310 --> 01:15:30,220
+limit ال Uبن لأن هذا أصلا صحيح علميا للأنات
+
+890
+01:15:30,220 --> 01:15:34,200
+الكبيرة فلما أخد ال limit أزنه جوزه infinity بضل
+
+891
+01:15:34,200 --> 01:15:38,280
+في ال safe side يعني بضل في اللي بتحقق فيه هذا إذا
+
+892
+01:15:38,280 --> 01:15:45,630
+ال limit U, B, N و F ماقص L, B, N و Fهيكون أكبر أو
+
+893
+01:15:45,630 --> 01:15:50,490
+يساوي سفر و أصغر أو يساوي إبسلون و هذا الكلام as
+
+894
+01:15:50,490 --> 01:15:53,530
+and goes to infinity و هذا الكلام صحيح برضه لمين
+
+895
+01:15:53,530 --> 01:15:58,790
+لكل إبسلون أكبر من سفر إذا غصب عنها هيكون اللي جوا
+
+896
+01:15:58,790 --> 01:16:08,780
+limit UPN و F-Lبان و اف لازم يساوي ايش سفر از انقص
+
+897
+01:16:08,780 --> 01:16:14,780
+انفينيتي وهذا by corollary اللي قبل بشوية هيعني ان
+
+898
+01:16:14,780 --> 01:16:23,310
+اف is integrableHence, F is integrable وهكذا
+
+899
+01:16:23,310 --> 01:16:27,610
+أثبتنا العائلة الثانية من الدوال الـ continuous
+
+900
+01:16:27,610 --> 01:16:31,170
+function على closed bounded interval is a
+
+901
+01:16:31,170 --> 01:16:34,210
+continuous function is an integrable function
+
+902
+01:16:34,210 --> 01:16:40,790
+وهكذا خلصنا الـ section الأول و الـ homework هيا
+
+903
+01:16:40,790 --> 01:16:46,750
+اللي موجودة في التلخيص و المرة القادمة ان شاء الله
+
+904
+01:16:47,630 --> 01:16:57,970
+إن شاء الله بكون احنا بنبدأ في اللي هي ال section
+
+905
+01:16:57,970 --> 01:17:03,510
+اللي بعده اللي هو properties of the Riemann
+
+906
+01:17:03,510 --> 01:17:04,470
+Integral
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc.srt
new file mode 100644
index 0000000000000000000000000000000000000000..160d0421caa818775798a38d89721f78c307fae5
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc.srt
@@ -0,0 +1,2921 @@
+1
+00:00:05,090 --> 00:00:08,030
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:08,030 --> 00:00:11,070
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه
+
+3
+00:00:11,070 --> 00:00:18,250
+أجمعين هذه المحاضرة رقم 24 مساق تحليل حقيقة 2 طلاب
+
+4
+00:00:18,250 --> 00:00:22,650
+وطالبات الجامعة الإسلامية قسم رياضيات كلية العلوم
+
+5
+00:00:23,470 --> 00:00:26,630
+اليوم هنكمل إن شاء الله الـ section الأخير في
+
+6
+00:00:26,630 --> 00:00:29,870
+chapter 8 اللي هو 8.4 تحت عنوان the
+
+7
+00:00:29,870 --> 00:00:35,170
+trigonometric functions وهو أيضا اللي هو الـ section
+
+8
+00:00:35,170 --> 00:00:42,210
+أو موضوع تطبيق على اللي هو الـ pointwise and
+
+9
+00:00:42,210 --> 00:00:46,030
+uniform convergence للـ sequence of functions وكيف
+
+10
+00:00:46,030 --> 00:00:50,510
+بدنا نعرف بطريقة مشابهة جدا لمعرفناها المرة
+
+11
+00:00:50,510 --> 00:00:53,330
+الماضية أو اللي قبلها بخصوص الـ exponential
+
+12
+00:00:53,330 --> 00:00:58,710
+function هنعرف اليوم اللي هو بنفس الطريقة كيف نعرف 
+
+13
+00:00:58,710 --> 00:01:03,510
+اللي هو الـ sine والـ cosine as a limit اللي هو
+
+14
+00:01:03,510 --> 00:01:07,390
+of a uniformly convergent sequence of functions
+
+15
+00:01:07,950 --> 00:01:11,230
+العنوان is in the trigonometric functions اللي هو
+
+16
+00:01:11,230 --> 00:01:15,450
+section 8.4 النظرية
+
+17
+00:01:15,450 --> 00:01:21,950
+الأولى اللي عندنا اللي مشابهة لنظرية سابقة اللي هي
+
+18
+00:01:21,950 --> 00:01:25,870
+الـ exponential اللي من خلالها بدنا نصل للي هو
+
+19
+00:01:25,870 --> 00:01:31,370
+تعريف اللي هي الـ cosine والـ sine النظرية بتقول
+
+20
+00:01:31,370 --> 00:01:35,190
+مفادها there exist functions يوجد دوال الأمنان في
+
+21
+00:01:35,190 --> 00:01:40,570
+بتوجد دوال C من R إلى R و S من R إلى R such that
+
+22
+00:01:40,570 --> 00:01:44,690
+اللي هي طبعا مستقبلا هنتسمي الـ C اللي هي الـ cosine
+
+23
+00:01:44,690 --> 00:01:48,910
+ومستقبلا هنتسمي الـ S اللي هي الـ sine اللي احنا
+
+24
+00:01:48,910 --> 00:01:54,090
+بنعرفها الآن بنقول في هذا اللي الثانية واحدة اسمها c
+
+25
+00:01:54,090 --> 00:01:59,950
+واحدة s تحقق ما يلي اللي هي c' of x بيساوي
+
+26
+00:01:59,950 --> 00:02:04,870
+ناقص c of x , s' of x بيساوي ناقص s of x ولو
+
+27
+00:02:04,870 --> 00:02:08,730
+استذكرت أنت الـ sine والـ cosine دي ما سيحدث لاحقا
+
+28
+00:02:08,730 --> 00:02:12,270
+طبعا لو استذكرت الـ cosine لما ان... ان... ان... ان
+
+29
+00:02:12,270 --> 00:02:16,130
+فاضلها مرتين هتصير اللي هي ناقص اللي هي نفسها ولو
+
+30
+00:02:16,130 --> 00:02:19,570
+فاضلت الـ sine برضه فاضلها مرتين هتلاقيها إيش هي
+
+31
+00:02:19,960 --> 00:02:23,840
+بتطلع سالب S الخاصية الثانية اللي هي C of Zero
+
+32
+00:02:23,840 --> 00:02:27,780
+بيساوية واحد ولو استذكرت الكوسين كوسين الـ Zero بيساوية
+
+33
+00:02:27,780 --> 00:02:31,240
+واحد ولو استذكرت الكوسين لما تاخدها الـ derivative
+
+34
+00:02:31,240 --> 00:02:35,160
+هتصير عبارة عن سالب Sin عند Zero هتطلع 0 بيساوية
+
+35
+00:02:35,160 --> 00:02:38,960
+Zero و S of Zero بيساوية Zero و S' of Zero
+
+36
+00:02:38,960 --> 00:02:45,330
+بيساوية واحد يعني بمعنى آخر يوجد دالتين الدالتين
+
+37
+00:02:45,330 --> 00:02:49,770
+واحدة من C إلى R اسمها S واحدة اسمها C من R إلى R
+
+38
+00:02:49,770 --> 00:02:56,690
+تحقق الشرطين التاليين c' بيساوي ناقص C و s' بيساوي
+
+39
+00:02:56,690 --> 00:03:01,330
+ناقص S على كل R و C عند الـ 0 هي 1 و c' عند الـ 0
+
+40
+00:03:01,330 --> 00:03:05,650
+بيساوي 0 و S عند الـ 0 بيساوي 0 و s' عند الـ 0 بيساوي 1
+
+41
+00:03:05,650 --> 00:03:11,250
+وبعد شوية هقول لك لا يوجد في الدنيا دالتين بحقق إن
+
+42
+00:03:11,250 --> 00:03:17,300
+الشروط هذول إلا هي واحدة اسمها C واحدة اسمها S
+
+43
+00:03:17,300 --> 00:03:22,640
+يعني التنتين واحدات وهذا يجعلنا نسميهم التسمية بعد
+
+44
+00:03:22,640 --> 00:03:26,380
+ذلك واحدة اسمها cosine واحدة اسمها sine ومن ثم
+
+45
+00:03:26,380 --> 00:03:29,320
+بنجيب كل الخواص اللي احنا بنعرفها عن الـ sine والـ
+
+46
+00:03:29,320 --> 00:03:34,020
+cosine من هذا البناء إذن الآن أنا ببني ببني ببني
+
+47
+00:03:34,020 --> 00:03:36,980
+وجود الـ sine والـ cosine وبعد شوية ببني اللي هو
+
+48
+00:03:36,980 --> 00:03:40,960
+الـ uniqueness طبقا لهذه الشروط اللي موجودة عندي
+
+49
+00:03:42,380 --> 00:03:45,860
+الآن بكيف بدنا نعمل زي ما عملنا المرة الماضية في
+
+50
+00:03:45,860 --> 00:03:48,340
+ال... في ال... اللي هو الـ Exponential عشانك
+
+51
+00:03:48,340 --> 00:03:52,060
+هتلاقيني شوية مسرع لأن اللي بيحضر اللي هي محاضرة
+
+52
+00:03:52,060 --> 00:03:56,620
+الـ Exponential هيلاقي إن هذا فيه كثير من الحديث في
+
+53
+00:03:56,620 --> 00:04:00,640
+إعادة We define the sequence c<sub>n</sub> as n of continuous
+
+54
+00:04:00,640 --> 00:04:05,240
+functions inductively as بدنا نعرف اللي هي... اللي
+
+55
+00:04:05,240 --> 00:04:09,640
+هو two sequences واحدة نسميها c<sub>n</sub> وواحدة نسميها s<sub>n</sub>
+
+56
+00:04:10,270 --> 00:04:13,890
+كيف بدنا نعرفها؟ زي ما عرفنا الـ exponential بنعرف
+
+57
+00:04:13,890 --> 00:04:19,250
+c<sub>1</sub> of x إيش بتساوي واحد و s<sub>1</sub> of x إيش بدنا نسميها
+
+58
+00:04:19,250 --> 00:04:26,490
+بيساو x الآن s<sub>2</sub> of x , s<sub>2</sub> of x هيساوي الـ integration
+
+59
+00:04:26,490 --> 00:04:31,990
+من صفر لعند x c<sub>2</sub> of t dt طب c<sub>2</sub> من وين أجيبها؟ c<sub>2</sub>
+
+60
+00:04:31,990 --> 00:04:36,050
+بتجيبها من هنا C اللي هي واحد زائد واحد يعني c<sub>2</sub>
+
+61
+00:04:36,050 --> 00:04:41,470
+بيساو 1 ناقص الـ integration من صفر ل x لـ s<sub>1</sub> of
+
+62
+00:04:41,470 --> 00:04:48,870
+t اللي هذا dt فبكون جيبت الـ c<sub>2</sub> وبتجيبت الـ s<sub>2</sub> من
+
+63
+00:04:48,870 --> 00:04:55,530
+الـ c<sub>2</sub> لأن s<sub>3</sub> و c<sub>3</sub> بنفس الطريقة in general اللي أنا
+
+64
+00:04:55,530 --> 00:04:59,170
+عملت sequence of functions اللي هو بدأت اللي هي الـ
+
+65
+00:04:59,170 --> 00:05:05,120
+c<sub>1</sub> بـ 1, s<sub>1</sub> بـ x ومن ثم s<sub>n</sub> بتساوي من صفر ل x
+
+66
+00:05:05,120 --> 00:05:10,760
+integration c<sub>n</sub> of t dt يعني كمالة c<sub>n</sub> of t dt و c
+
+67
+00:05:10,760 --> 00:05:13,460
+<sub>n</sub> زائد 1 of x بتساوي 1 نقص integration من
+
+68
+00:05:13,460 --> 00:05:18,700
+صفر ل x s<sub>n</sub> of t dt الآن هو بيقول اللي هو هذه
+
+69
+00:05:18,700 --> 00:05:21,820
+sequence of continuous functions طب sequence of 
+
+70
+00:05:21,820 --> 00:05:25,840
+continuous functions هذا الكلام بده إثبات طيب، الآن
+
+71
+00:05:25,840 --> 00:05:29,500
+عندي اللي هو by induction زي ما أنتم عارفين الآن
+
+72
+00:05:29,500 --> 00:05:33,600
+عندي الـ c<sub>1</sub> continuous لأن الثابتة s<sub>1</sub> continuous
+
+73
+00:05:33,600 --> 00:05:38,880
+لأن هي شمالها بتساوي x بناء عليه هتطلع عندي اللي
+
+74
+00:05:38,880 --> 00:05:44,920
+هي c<sub>2</sub> continuous ومن ثم c<sub>3</sub> و c<sub>4</sub> الآخرين الآن لو
+
+75
+00:05:44,920 --> 00:05:49,000
+بدنا نثبتها by induction بدنا نفترض إنه هذول
+
+76
+00:05:49,000 --> 00:05:52,560
+الـ s<sub>n</sub> والـ c<sub>n</sub>
+
+77
+00:05:53,920 --> 00:06:00,700
+continuous سنثبت هنا continuous لأن s<sub>1</sub> إيش بتساوي
+
+78
+00:06:00,700 --> 00:06:06,860
+x؟ c<sub>1</sub> إيش بتساوي 1؟ continuous إذا صارت هذه اللي
+
+79
+00:06:06,860 --> 00:06:12,340
+هي الجملة is true for n بتساوي 1 نفترض الآن
+
+80
+00:06:12,340 --> 00:06:19,900
+suppose by induction بتهدي suppose that اللي هو star
+
+81
+00:06:19,900 --> 00:06:25,960
+هذه is true for n إيش بتساوي n بتساوي k معناته
+
+82
+00:06:25,960 --> 00:06:33,820
+صارت الـ s<sub>k</sub> والـ c<sub>k</sub> are continuous بتثبت الآن من
+
+83
+00:06:33,820 --> 00:06:38,520
+الصحيحة لـ k زائد 1 يعني بتثبت اللي هو c<sub>k</sub> زائد
+
+84
+00:06:38,520 --> 00:06:42,700
+1 و s<sub>k</sub> زائد 1 إن هنا شمال هنا continuous طيب
+
+85
+00:06:42,700 --> 00:06:48,000
+الآن شوف c<sub>k</sub> زائد 1 بتساوي حسب اللي هي عندي هنا
+
+86
+00:06:48,000 --> 00:06:51,740
+إيش بتساوي اللي هو عبارة عن c<sub>k</sub> واحد of x بتساوي الـ
+
+87
+00:06:51,740 --> 00:06:58,020
+integration 1 ناقص الـ integration من صفر x s<sub>k</sub>...
+
+88
+00:06:58,020 --> 00:07:03,020
+هذا c<sub>k</sub> زائد 1... هذا k... of dt طيب أنا مفترض أن s<sub>k</sub>
+
+89
+00:07:03,020 --> 00:07:06,600
+من الـ hypothesis induction إنها continuous إذا
+
+90
+00:07:06,600 --> 00:07:11,040
+صارت هذه كلها إيش مالها؟ اللي هي s<sub>k</sub> integrable
+
+91
+00:07:11,040 --> 00:07:15,600
+وصارت هذه كلها على بعض by fundamental theorem of
+
+92
+00:07:15,600 --> 00:07:20,980
+calculus اللي هي الـ derivative إلها موجودة إذا
+
+93
+00:07:20,980 --> 00:07:23,440
+صارت هذه كلها الـ derivative إلها وموجودة بالنسبة
+
+94
+00:07:23,440 --> 00:07:27,680
+لـ x إذا صارت مدام هيك هاذ is differentiable إذا
+
+95
+00:07:27,680 --> 00:07:30,940
+continuous إذا صارت اللي هي c<sub>k</sub> زائد 1
+
+96
+00:07:30,940 --> 00:07:35,200
+continuous اللي ثبت الآن is s<sub>k</sub> زائد 1, s<sub>k</sub> زائد
+
+97
+00:07:35,200 --> 00:07:39,280
+1 of x إيش بتساوي حسب اللي التعريف بتساوي الـ
+
+98
+00:07:39,280 --> 00:07:47,660
+integration من صفر لعند x c<sub>k</sub> of t زائد 1 هذا n
+
+99
+00:07:47,660 --> 00:07:52,620
+هذا n k زائد 1, k زائد 1 dt وأنا مثبت فوق إنه
+
+100
+00:07:52,620 --> 00:07:56,160
+c<sub>k</sub> زائد 1 is continuous إذا صارت هذه كلها على
+
+101
+00:07:56,160 --> 00:07:59,500
+بعض integrable اللي هي c<sub>k</sub> زائد 1 صار هذا الـ
+
+102
+00:07:59,500 --> 00:08:02,000
+integration exist ومش هيك by fundamental theorem
+
+103
+00:08:02,000 --> 00:08:04,720
+of calculus برضه الـ derivative له إيه شمالها
+
+104
+00:08:04,720 --> 00:08:07,700
+موجودة إذا صارت هذه الـ differentiable إذا
+
+105
+00:08:07,700 --> 00:08:11,060
+continuous إذا صارت c<sub>k</sub> زائد 1 و s<sub>k</sub> زائد 1 are
+
+106
+00:08:11,060 --> 00:08:16,160
+continuous إذا صارت الجملة هذه صحيحة لـ k زائد 1
+
+107
+00:08:16,160 --> 00:08:21,000
+إذا صارت صحيحة دائما إذا صارت عندي الآن مفرغ منه
+
+108
+00:08:21,000 --> 00:08:26,360
+الـ c<sub>n</sub> والـ s<sub>n</sub> are continuous functions وبناء عليه
+
+109
+00:08:26,360 --> 00:08:29,860
+مدام continuous functions by fundamental theorem
+
+110
+00:08:29,860 --> 00:08:35,460
+of calculus هتكون هذه الـ s<sub>n</sub> differentiable والـ c<sub>n</sub>
+
+111
+00:08:35,460 --> 00:08:39,800
+زائد 1 differentiable ومش هيك كمان وهيعطيني الـ
+
+112
+00:08:39,800 --> 00:08:43,440
+s<sub>n</sub>' of x حسب الـ fundamental theorem of
+
+113
+00:08:43,440 --> 00:08:46,820
+calculus الـ differentiation بتضايق الـ integration
+
+114
+00:08:46,820 --> 00:08:54,580
+بتضال بيساوي c<sub>n</sub> of x الآن والـ derivative لهذه برضه
+
+115
+00:08:54,580 --> 00:08:59,900
+exist c<sub>n</sub> زائد 1' of x اللي هو هي ساوي اللي
+
+116
+00:08:59,900 --> 00:09:10,170
+هو مين هتطلع ناقص s<sub>n</sub> of x إذا صار عندي الآن اللي هي
+
+117
+00:09:10,170 --> 00:09:15,610
+الـ sequence اللي عندي صارت well defined كلها و
+
+118
+00:09:15,610 --> 00:09:20,270
+continuous كلها ولأ و differentiable كمان الـ s<sub>n</sub>
+
+119
+00:09:20,270 --> 00:09:25,150
+' of x بيساوي c<sub>n</sub> of x و c<sub>n</sub> زائد 1' of
+
+120
+00:09:25,150 --> 00:09:31,730
+x بيساوي ناقص s<sub>n</sub> of x زي ما أنا أثبتها وأوضحت لكم
+
+121
+00:09:31,730 --> 00:09:38,710
+إياها هنا الآن induction arguments برضه induction
+
+122
+00:09:38,710 --> 00:09:42,970
+arguments بقول لي we leave this argument for you
+
+123
+00:09:42,970 --> 00:09:46,890
+خلينا نشوفها مع بعض ماهي اللي بنقصده زي ما عملت
+
+124
+00:09:46,890 --> 00:09:49,870
+بالظبط لو طلعت على الخطوات هتلاقيها مشابه لخطوات
+
+125
+00:09:49,870 --> 00:09:54,970
+تبعات الـ exponential عند الـ s<sub>n</sub> of x زي ما قلنا
+
+126
+00:09:54,970 --> 00:10:02,060
+بيساو الـ integration من 0 ل x c<sub>n</sub> of t dt , c<sub>n</sub> زائد
+
+127
+00:10:02,060 --> 00:10:06,140
+1 of x بيساوي 1 مقص الـ integration من c ل x s
+
+128
+00:10:06,140 --> 00:10:15,360
+<sub>n</sub> of t dt ومعطينا طبعا احنا أخذنا الـ c<sub>1</sub> of x
+
+129
+00:10:15,360 --> 00:10:22,530
+بيساو 1 والـ c و s<sub>1</sub> of x بيساو x هذه اللي هي الـ
+
+130
+00:10:22,530 --> 00:10:25,370
+sequence of functions اللي أثبتناها هذه الـ
+
+131
+00:10:25,370 --> 00:10:27,990
+sequence of functions الـ s<sub>n</sub> والـ c<sub>n</sub> زائد 1
+
+132
+00:10:27,990 --> 00:10:32,230
+التنتين are continuous for every n وهمش هي
+
+133
+00:10:32,230 --> 00:10:34,850
+codifferentiable والـ derivative إلها زي ما قلنا
+
+134
+00:10:34,850 --> 00:10:39,430
+s<sub>n</sub>' of x بيساو c<sub>n</sub> of x والـ c<sub>n</sub> زائد 1'
+
+135
+00:10:39,430 --> 00:10:44,190
+of x بيساو ناقص s<sub>n</sub> of x وخلينا نسجلها هذه لأن
+
+136
+00:10:44,190 --> 00:10:52,370
+هنحتاجها بعد شوية اللي هي s<sub>n</sub>' of x بيساوي
+
+137
+00:10:52,370 --> 00:11:00,310
+c<sub>n</sub> of x and c<sub>n</sub> زائد 1' of x بيساوي ناقص s
+
+138
+00:11:00,310 --> 00:11:09,070
+<sub>n</sub> of x طيب الآن بدنا اللي هو by induction نثبت
+
+139
+00:11:09,070 --> 00:11:12,570
+اللي هو c<sub>n</sub> زائد 1 of x بيساوي اللي هو اللي
+
+140
+00:11:12,570 --> 00:11:18,460
+أمامي هذا طبعا أكيد اللي هو بعضكم قال ما هي by
+
+141
+00:11:18,460 --> 00:11:24,960
+induction نتطلع على c<sub>2</sub> of x , c<sub>1</sub> في القرآن بتساوي
+
+142
+00:11:24,960 --> 00:11:31,720
+واحد نثبتها يعني الآن c<sub>2</sub> of x إيش بتساوي حسب
+
+143
+00:11:31,720 --> 00:11:35,960
+القانون بيساوي 1 ماقص الـ integration من صفر ل x
+
+144
+00:11:35,960 --> 00:11:41,020
+s<sub>1</sub> لأن بـ 1 هنا s<sub>1</sub> اللي هي جدش s<sub>1</sub>
+
+145
+00:11:41,020 --> 00:11:46,060
+الـ x integration اللي هي t dt ويساوي التفاضل اللي
+
+146
+00:11:46,060 --> 00:11:49,820
+هذه بيصير 1 ماقص x تربيع على مين على 2 إذا
+
+147
+00:11:49,820 --> 00:11:54,410
+فعلا اللي هي c اللي هي هذه 2 is true for any
+
+148
+00:11:54,410 --> 00:11:58,870
+إيش بتساوي n بتساوي 1 بدنا نثبت الثانية معها
+
+149
+00:11:58,870 --> 00:12:01,730
+اللي هي true for any بتساوي 1 لأنها دي جملة
+
+150
+00:12:01,730 --> 00:12:05,690
+واحدة بدأ أثبتها إنها true for every any الآن for
+
+151
+00:12:05,690 --> 00:12:10,590
+any بتساوي 1 بيصير عندي s<sub>2</sub> of x بدأ أتأكد
+
+152
+00:12:10,590 --> 00:12:15,990
+إنه اللي هي بتحقق اللي هي ال... اللي... إيه هذه الـ
+
+153
+00:12:15,990 --> 00:12:19,960
+equation s<sub>2</sub> of x بيساوي من وين بده اجيبها من
+
+154
+00:12:19,960 --> 00:12:23,480
+التعريف اللي فوق بيساوي الـ integration من صفر ل x
+
+155
+00:12:23,480 --> 00:12:29,180
+c<sub>2</sub> of x إيش c<sub>2</sub> of x هايه اللي هو 1 ناقص t تربيع
+
+156
+00:12:29,180 --> 00:12:34,800
+على 2 لكل ما له dt ويساوي الكملة هذا بيصير
+
+157
+00:12:34,800 --> 00:12:40,200
+عبارة عن x ناقص x تكعيب على 3 وفيه 2 بيصير
+
+158
+00:12:40,200 --> 00:12:44,980
+x تكعيب على مين على 3! اللي هي جدش
+
+159
+00:12:44,980 --> 00:12:50,380
+اللي هي 6 إذا فعلا فعلا طلع عندي اللي هو c<sup> </sup>
+
+160
+00:12:50,380 --> 00:12:55,400
+<sub>2</sub> هي عبارة عن أنا بـ 1 يعني <sup> </sup>s<sub>2</sub> بيساوي x
+
+161
+00:12:55,400 --> 00:12:59,380
+ناقص x تكعيب على 3! يعني صارت الجملة
+
+162
+00:12:59,380 --> 00:13:02,900
+هذه برضه صحيحة for n بيساوي 1 إذا كلها على بعض
+
+163
+00:13:02,900 --> 00:13:09,650
+هذه صارت صحيحة for n بتساوي جدش الآن بدنا نفترض
+
+164
+00:13:09,650 --> 00:13:13,430
+إنها
+
+165
+00:13:13,430 --> 00:13:19,490
+true for n بتساوي k ونجيب منها مين اللي هو k زائد
+
+166
+00:13:19,490 --> 00:13:25,950
+1 نفترض suppose that this is true for n بتساوي
+
+167
+00:13:25,950 --> 00:13:30,690
+k يعني بمعنى آخر صار عندي c<sub>k</sub> زائد 1 بيساوي هذا
+
+168
+00:13:30,690 --> 00:13:37,260
+وهنا 2k وهنا k وهنا 2k وهنا نفس الشي
+
+169
+00:13:37,260 --> 00:13:41,640
+اللي هو نفترض إنها صحيحة for n بتساوي k صارت عبارة
+
+
+201
+00:16:37,130 --> 00:16:41,890
+صارت هذه CK زائد واحد زائد اثنين of X بتساوي هذا
+
+202
+00:16:41,890 --> 00:16:49,290
+المقدار هذا المقدار صحيح صار بالظبط هو هذا اللي هو
+
+203
+00:16:49,290 --> 00:16:53,830
+المقدار اللي عندي أثبتت و كإلمام يعني الآن أن هذا
+
+204
+00:16:54,470 --> 00:16:59,490
+اللي عندي is true for mean for k زائد لواحد لأنه
+
+205
+00:16:59,490 --> 00:17:03,950
+لما أحط مكان n بيساوي k زائد واحد بيصير هذه k زائد
+
+206
+00:17:03,950 --> 00:17:09,030
+اثنين أشوف اللي طلعته صح ولا لأ بيساوي تباينة واحد
+
+207
+00:17:09,030 --> 00:17:12,990
+ناقص x تربيع على اثنين فيكتوريا لما أثر الاخر واحد
+
+208
+00:17:12,990 --> 00:17:17,790
+اللي هي ناقص واحد قص مين k زائد واحد فيكس قص اثنين
+
+209
+00:17:17,790 --> 00:17:20,970
+في k زائد واحد يعني اثنين k زائد اثنين وها اثنين k
+
+210
+00:17:20,970 --> 00:17:25,760
+زائد اثنين كل فيكتوريا الآن صارت هذه صحيحة for n
+
+211
+00:17:25,760 --> 00:17:30,600
+ايش بتساوي k زائد واحد الثاني بنفس الأسلوب بأثبتها
+
+212
+00:17:30,600 --> 00:17:35,260
+صحيحة for ايش for k زائد واحد يعني بدي احسب مين يا 
+
+213
+00:17:35,260 --> 00:17:42,300
+جماعة بدي احسب Skزايد اثنين ايش حسب اللي فوق
+
+214
+00:17:42,300 --> 00:17:47,180
+بتساوي بتساوي ال integration من صفر ل X ل CK زائد
+
+215
+00:17:47,180 --> 00:17:54,020
+اثنين of T DT و باجي بعوضها هنا و بكملها و بتطلع
+
+216
+00:17:54,020 --> 00:17:57,800
+عندي بالظبط ال formula هذه يعني صح بيصير عندي
+
+217
+00:17:57,800 --> 00:18:01,620
+بتعملها لحالك لأن حسابات نفس الأسلوب بتطلع عندي
+
+218
+00:18:01,620 --> 00:18:06,310
+اللي هي هذه صحيحة for mean برضه for an بتساوي k
+
+219
+00:18:06,310 --> 00:18:11,050
+زائد واحد إذا هذا صار المقدار صحيح دائماً for mean
+
+220
+00:18:11,050 --> 00:18:20,430
+for any k for any n element in N الآن واضح أن
+
+221
+00:18:20,430 --> 00:18:26,310
+الخطوات مشابهة لخطوات ال exponential نجي الآن
+
+222
+00:18:26,310 --> 00:18:28,970
+قصّل N prime
+
+223
+00:18:38,770 --> 00:18:45,020
+نجي الآن وين رايح؟ زي اللي بكونت رايحي الأيام الـ
+
+224
+00:18:45,020 --> 00:18:49,060
+exponential هأثبت لك أن الـ sequence هذه converged
+
+225
+00:18:49,060 --> 00:18:52,220
+uniformly وهذه طبعاً هتصبح converged uniformly
+
+226
+00:18:52,220 --> 00:18:55,000
+automatically فهتصبح اللي هي differentiable لأنه بيصير
+
+227
+00:18:55,000 --> 00:18:57,860
+طبق النظرية اللي هي تبعها الـ differentiability
+
+228
+00:18:57,860 --> 00:19:01,200
+بيصير عندي مادام differentiable اللي هي ال
+
+229
+00:19:01,200 --> 00:19:04,580
+derivative اللي لها exist و هتكون ال derivative
+
+230
+00:19:04,580 --> 00:19:07,960
+متحقق الشروط و بيكون خلاص نعم نشوف إيش بقول طيب
+
+231
+00:19:07,960 --> 00:19:12,530
+الآن بعد ما طلعنا هدول let a أكبر من صفر b given
+
+232
+00:19:12,530 --> 00:19:15,790
+نفس الخطوات تبعت ال exponential then if الabsolute
+
+233
+00:19:15,790 --> 00:19:19,970
+value of x أصغر من أو يساوي a و m أكبر من n أكبر من 2a
+
+234
+00:19:19,970 --> 00:19:25,510
+يعني بتاخد اللي هي الأمات لأكبر من n و أكبر من 2a
+
+235
+00:19:25,510 --> 00:19:29,650
+يعني الآن أنا بشتغل على الـ  يا جماعة الفترة من ناقص
+
+236
+00:19:29,650 --> 00:19:34,570
+a لعند a وأخدت ال a arbitrarily أكبر من صفر لكن
+
+237
+00:19:34,570 --> 00:19:41,850
+fixed الآن we have عندي اللي هو .. عندي اللي هي الـ
+
+238
+00:19:41,850 --> 00:19:50,630
+A الـ A على اللي هي 2N بما أنه N أكبر من 2A أقسم
+
+239
+00:19:50,630 --> 00:19:56,790
+الجهتين على 2N هنا على 2N و هنا على 2N بيصير عند الـ
+
+240
+00:19:56,790 --> 00:20:07,710
+A على N .. A على N أصغر من النصف، صح؟ هي a تروح
+
+241
+00:20:07,710 --> 00:20:11,090
+الاثنين مع الاثنين وهذه n مع ال n بيصير a أعليها أصغر
+
+242
+00:20:11,090 --> 00:20:16,050
+من نصف يعني a على 2n أصغر من 1 على 4 قسمت الاثنين
+
+243
+00:20:16,050 --> 00:20:22,400
+عالمياً على 2 إذا لما تكون ال n أكبر من 2a هتطلع
+
+244
+00:20:22,400 --> 00:20:24,640
+عندها get بتعرف ليش هذه لأن بتلزمنا في الحسابات
+
+245
+00:20:24,640 --> 00:20:28,580
+بعد شوية هتكون اللي هي لما الاثنين a أصغر من n
+
+246
+00:20:28,580 --> 00:20:31,960
+بتكون عند a على اثنين n أصغر من مين من ربع المرة
+
+247
+00:20:31,960 --> 00:20:35,460
+اللي فاتت كان لازمنا a على n أصغر من نصف وكملنا ال
+
+248
+00:20:35,460 --> 00:20:40,650
+series اللي متذكر اللي عملنا في اللي هي الـ
+
+249
+00:20:40,650 --> 00:20:43,330
+Exponential أنا بأحكيش تفاصيله و مفترض أن أنتم
+
+250
+00:20:43,330 --> 00:20:47,570
+فاهمين حسب حكينا اللي هو في الـ Exponential الآن
+
+251
+00:20:47,570 --> 00:20:52,290
+للامات اللي أكبر من M أكبر من 2A بدي احسب الـCM
+
+252
+00:20:52,290 --> 00:20:59,870
+ناقص اثنين الـSN عشان تظلها جدامكم اللي فوق هذه عندي
+
+253
+00:20:59,870 --> 00:21:06,960
+الآن الـCM الـ C M الـ C M شايفينها؟ بتظلها ماشية
+
+254
+00:21:06,960 --> 00:21:10,940
+واحد لو ف .. الآن هذه M بدأ الـ N زاد واحد ايش
+
+255
+00:21:10,940 --> 00:21:14,880
+اسمها؟ M بتظلها ماشية واحد ناقص X أربع على اثنين
+
+256
+00:21:14,880 --> 00:21:18,820
+فيكتوريال X أربع على أربع فيكتوريال و الـ M أكبر من
+
+257
+00:21:18,820 --> 00:21:23,160
+الـ N هتجيه يقبل في طريقها من الـ N الـ N بيصير
+
+258
+00:21:23,160 --> 00:21:27,540
+الـ N طبعاً إيه شمالها؟ الـ N عبارة عن في جبل ال
+
+259
+00:21:27,540 --> 00:21:33,590
+term هذا اللي هو الـ N ناقص واحد اه فبيصير زائد اللي
+
+260
+00:21:33,590 --> 00:21:39,350
+هو ناقص واحد أس ان ناقص واحد في X أس اثنين ام ناقص
+
+261
+00:21:39,350 --> 00:21:44,050
+اثنين على اثنين ام ناقص اثنين الكل factorial و
+
+262
+00:21:44,050 --> 00:21:48,390
+بتكمل هذا و بتبقى مكمل ايه أنت لما تصل لعند X أس
+
+263
+00:21:48,390 --> 00:21:54,670
+اثنين ام ناقص اثنين لأن هذا لل ام مش لل ان زائد
+
+264
+00:21:54,670 --> 00:21:59,730
+واحد على اثنين ام ناقص اثنين الكل شماله factorial
+
+265
+00:22:00,420 --> 00:22:08,320
+لما تطرح ال CM نقص ال CN اللي هو لها بيصير المتبقي
+
+266
+00:22:08,320 --> 00:22:12,600
+هيه زي ما عملنا بالظبط قبل هيك فبيصير ال CM نقص ال
+
+267
+00:22:12,600 --> 00:22:18,080
+CN بتساوي ناقص واحد طبعاً في absolute value عند X2N
+
+268
+00:22:18,080 --> 00:22:21,200
+عشان هيك طيرنا مش فارقة كتير نقص و سالب أخذنا له
+
+269
+00:22:21,200 --> 00:22:25,900
+absolute value مش هتفرج معنا ال X2N على 2N
+
+270
+00:22:25,900 --> 00:22:32,490
+vectorial ناقص X أس 2 M زائد 2 اللي بعيدها على 2 M
+
+271
+00:22:32,490 --> 00:22:35,990
+زائد 2 كله factorial لما أصل لآخر term اللي هو X
+
+272
+00:22:35,990 --> 00:22:41,170
+أس 2 M ناقص 2 على 2 M ناقص 2 كله factorial هذا
+
+273
+00:22:41,170 --> 00:22:46,670
+الآن هذا نفسه أخذنا احنا ال absolute value لل X
+
+274
+00:22:46,670 --> 00:22:50,350
+أصغر من A في هذه الفترة احنا شغالين في الفترة ال
+
+275
+00:22:50,350 --> 00:22:54,430
+absolute value X أصغر من A الآن بيصير عند ال X
+
+276
+00:22:54,430 --> 00:23:05,180
+نفسها أس 2n أصغر من أو يساوي هذه اللي هي a أس 2n على
+
+277
+00:23:05,180 --> 00:23:09,240
+2n الكل factorial الأولى زائد استخدمت triangle
+
+278
+00:23:09,240 --> 00:23:13,440
+inequality هذه زائد هذه زائد هذه زائد هذه الآن
+
+279
+00:23:13,440 --> 00:23:18,240
+زائد اللي بعيدها لما أصل لآخر واحدة x أو اللي جاب
+
+280
+00:23:18,240 --> 00:23:23,420
+اللي خليني أكتبها عشان x اللي هي بيصير a أس اثنين ان
+
+281
+00:23:23,420 --> 00:23:28,060
+زائد اثنين على اثنين ان زائد اثنين لكل factorial
+
+282
+00:23:28,060 --> 00:23:37,020
+زائد لما أصل لآخر term عندي هنا لما أصل لآخر term
+
+283
+00:23:37,020 --> 00:23:43,160
+اللي هو زائد 
+
+284
+00:23:43,160 --> 00:23:57,230
+هذا ال term اللي هو زائد A 2M-2 2M-2 كل A شماله
+
+285
+00:23:57,230 --> 00:24:03,930
+factorial خليني أخذ هذا A 2N 2N كل factorial العام
+
+286
+00:24:03,930 --> 00:24:11,460
+المشترك بيظل عندي 1 زائد a تربيع لأنه بيصير a يساوي 2
+
+287
+00:24:11,460 --> 00:24:17,960
+a تربيع على مين على 2n زائد 2 أكيد ال 1 على 2n
+
+288
+00:24:17,960 --> 00:24:23,280
+زائد 2 أصغر من 1 على 2n لأن 2n أصغر من هذه
+
+289
+00:24:23,280 --> 00:24:27,240
+فمقلوبها بيصير أكبر ماشي و بيظل زي ما عملته المرة
+
+290
+00:24:27,240 --> 00:24:32,700
+الفاتة أسحب منه A أس 2 على N و في الآخر أستبدل
+
+291
+00:24:32,700 --> 00:24:37,680
+اللي هو 2N هذه عن الرقم اللي أكبر منه N فبيصير هي
+
+292
+00:24:37,680 --> 00:24:41,400
+مقلوبها أكبر فبتظل اللي هي ال inequality زي هيك
+
+293
+00:24:41,400 --> 00:24:45,840
+و بكون سحبتي 2 من هذه بيصير 2 ام نقص 2 ام نقص ايش
+
+294
+00:24:45,840 --> 00:24:53,350
+نقص 2 وصلنا لعند ال inequality هذه الآن احنا قلنا a
+
+295
+00:24:53,350 --> 00:24:58,950
+على 2n أصغر من مين من ربعه يعني بيصير هذا المقدار
+
+296
+00:24:58,950 --> 00:25:05,310
+كله أصغر من أو يساوي a أس 2n على 2n لكل factorial
+
+297
+00:25:05,310 --> 00:25:09,550
+مضروب في مين أصغر من أو يساوي لأن الربع أكبر منهم بس
+
+298
+00:25:09,550 --> 00:25:14,930
+تبدأ كل واحد وقت ما كانوا إيهاش ربع واحد زاد واحد
+
+299
+00:25:14,930 --> 00:25:23,510
+على أربعة تربيع زائد لما أصل لآخر واحد a اللي هو
+
+300
+00:25:23,510 --> 00:25:31,370
+عبارة عن واحد على أربعة الكل أس اثنين ام ناقص
+
+301
+00:25:31,370 --> 00:25:36,590
+اثنين ام ناقص ايش اثنين هذه الآن ال ال ال ال
+
+302
+00:25:36,590 --> 00:25:39,710
+finite geometric series أصغر من أو يساوي ال infinite
+
+303
+00:25:39,710 --> 00:25:45,770
+اللي هي a أس اثنين ان على اثنين ان لكل factorial
+
+304
+00:25:45,770 --> 00:25:53,570
+في اللي هو الـ summation ده واحد زائد ربع تربيع
+
+305
+00:25:53,570 --> 00:26:00,310
+زائد ربع تكعيب زائد لما أصل ربع أس أربعة زائد إلى
+
+306
+00:26:00,310 --> 00:26:05,070
+ما لا نهاية هذه الآن أساسها جداش وكأنه أساسها كل
+
+307
+00:26:05,070 --> 00:26:11,750
+مرة تمد واحد على ستة عشر فبتصير هذه المجموحة اللي
+
+308
+00:26:11,750 --> 00:26:16,470
+هي واحد مجموحة بتعرفوها واحد ناقص واحد على ستة عشر
+
+309
+00:26:17,250 --> 00:26:21,750
+بـ 1 على 1 ناقص 1 على 16 ويساوي
+
+310
+00:26:21,750 --> 00:26:27,810
+جداش خمسة عشر أو ستة عشر على خمسة عشر لأن هذه واحد
+
+311
+00:26:27,810 --> 00:26:29,870
+ناقص واحد على ستة عشر تطلع خمسة عشر على ستة عشر
+
+312
+00:26:29,870 --> 00:26:34,830
+مقلوبة بيصير ستة عشر على خمسة عشر المفهوم المقصود هو
+
+313
+00:26:34,830 --> 00:26:38,670
+بأسمحها Geometric Series بظل بقول هذه أصغر من أو ساوي
+
+314
+00:26:38,670 --> 00:26:41,930
+الـ summation إلى ما له نهاية و بأستبدل هذا باللي أكبر
+
+315
+00:26:41,930 --> 00:26:46,070
+منها فبتظل هذه أكبر الآن مجموعها بيصير عبارة عن 16
+
+316
+00:26:46,070 --> 00:26:51,010
+في المقدار هذا اللي دله موخود عام المشترك فبيصير
+
+317
+00:26:51,010 --> 00:26:58,940
+عندي الآن يا جماعة اللي هو ال CM نقص الـ CN زي ما
+
+318
+00:26:58,940 --> 00:27:04,460
+حكينا قبل ذلك بالظبط أصغر من هذا المقدار اللي هذا
+
+319
+00:27:04,460 --> 00:27:09,000
+limit و اين بيروح as N goes to infinity بيروح لـ 0
+
+320
+00:27:09,000 --> 00:27:13,760
+إذا بيصير هذا المقدار زي ما قلنا المرة الماضية CM
+
+321
+00:27:13,760 --> 00:27:20,800
+of X نقص CN of X اللي هو أصغر من أو يساوي Y for very
+
+322
+00:27:20,800 --> 00:27:28,700
+large M and N M and N لأنه لما تكبر M كثير N كثير
+
+323
+00:27:28,700 --> 00:27:33,100
+تروح لنا لنهاية لأنه limit بروح لها صفر as N goes
+
+324
+00:27:33,100 --> 00:27:37,080
+to infinity ال M برضه بتكبر إذا for very large M
+
+325
+00:27:37,080 --> 00:27:40,800
+هيكون هذا أصغر من أو يساوي إبسلون لأي إبسلون في
+
+326
+00:27:40,800 --> 00:27:44,480
+الدنيا لأنه هذا بيقترب للصفر فبيصير صغير صغير صغير
+
+327
+00:27:44,480 --> 00:27:48,720
+صغير لدرجة إنه مضروب في هذا يكون أصغر من إبسلون
+
+328
+00:27:48,720 --> 00:27:53,360
+وهذه اللي قلنا عنها اللي هي الـ Cauchy criterion
+
+329
+00:27:53,360 --> 00:28:00,040
+for uniform continuity هذا الكلام كله لمين؟ صحيح
+
+330
+00:28:00,040 --> 00:28:06,880
+لأي X وين على الفترة اللي هو absolute value X أصغر
+
+331
+00:28:06,880 --> 00:28:11,220
+من أو يساوي A يعني على الفترة المغلقة من ناقص A لعند
+
+332
+00:28:11,220 --> 00:28:19,110
+A وهذا بيعطيني أن الـ is uniformly continuous على
+
+333
+00:28:19,110 --> 00:28:24,150
+الفترة من نفس A لعند A is as if is uniformly is
+
+334
+00:28:24,150 --> 00:28:28,330
+uniformly convergence على الفترة من نفس A ل A يعني
+
+335
+00:28:28,330 --> 00:28:35,790
+بمعنى آخر صارت عندي ال sequence هذه اللي هي CNN
+
+336
+00:28:35,790 --> 00:28:43,150
+converges uniformly to some function عالمياً على
+
+337
+00:28:43,150 --> 00:28:50,730
+الفترة من ناقص a لعند مين لعند a الآن طب ما هو اللي
+
+338
+00:28:50,730 --> 00:28:53,930
+عملناها أنا على الفترة هذه نقدر نعمله على أي شيء
+
+339
+00:28:53,930 --> 00:28:58,310
+ثاني يعني بمعنى آخر لو جيت أخدت x element in R
+
+340
+00:28:58,310 --> 00:29:04,400
+هألاقي number a بحيث أن هذا ال number a هي ال x في
+
+341
+00:29:04,400 --> 00:29:10,180
+ال R بقدر ألاقي a بحيث أن ناقص a و a تكون ال x في
+
+342
+00:29:10,180 --> 00:29:15,340
+الفترة بين ناقص a و a يعني بمعنى آخر هأعمل نفس اللي
+
+343
+00:29:15,340 --> 00:29:18,100
+عملته في الأول و هأحصر على c أن uniformly
+
+344
+00:29:18,100 --> 00:29:23,030
+convergence على هذه الفترة يعني بمعنى آخر الآن صار
+
+345
+00:29:23,030 --> 00:29:29,450
+عندي limit cn of x for any x exist بده أسميها هذه
+
+346
+00:29:29,450 --> 00:29:35,190
+limit c of x وبناء عليه بدي أعرف الآن in
+
+347
+00:29:35,190 --> 00:29:38,710
+particular this means that cn of x converge for
+
+348
+00:29:38,710 --> 00:29:42,710
+each x element in R we define c من R لR by c of x
+
+349
+00:29:42,710 --> 00:29:45,650
+بـ تساوي limit cn of x for x element in R
+
+350
+00:29:53,570 --> 00:29:59,630
+الآن .. بما أنه الآن الـCn مرتبط بشكل مرتبط لـC زي ما
+
+351
+00:29:59,630 --> 00:30:04,690
+قلنا أو الـCn of X كلهم مرتبط حسب اللي هو
+
+352
+00:30:04,690 --> 00:30:08,030
+النظرية في الـ pointwise .. الـ uniform convergence
+
+353
+00:30:08,030 --> 00:30:12,360
+اللي هو the limit .. the uniform .. limit أو الـ
+
+354
+00:30:12,360 --> 00:30:15,220
+uniform convergence of a sequence of continuous
+
+355
+00:30:15,220 --> 00:30:18,480
+functions مصمد كونتنيواس الـ limit تبعتها يعني
+
+356
+00:30:18,480 --> 00:30:21,160
+هتطلع عندي C of X continuous مثلاً C of X
+
+357
+00:30:21,160 --> 00:30:25,000
+continuous إذا عندي .. اللي هو صار عندي الـ
+
+358
+00:30:25,000 --> 00:30:28,160
+function هذه continuous اللي هي الـ C اللي احنا
+
+359
+00:30:28,160 --> 00:30:33,730
+بدنا إياها ومش هيك و limitCn of 0 as n goes to
+
+360
+00:30:33,730 --> 00:30:37,270
+infinity بتساوي limit اللي هي Cn of 0 ايش بتساوي
+
+361
+00:30:37,270 --> 00:30:41,910
+واحد وهي تساوي واحد اللي هي عبارة عن مين الـ C of
+
+362
+00:30:41,910 --> 00:30:47,370
+ايش of 0 لأن احنا متفقين الـ C of X بتساوي limit Cn
+
+363
+00:30:47,370 --> 00:30:51,310
+of X و in particular for X بتساوي صفر بتصير limit
+
+364
+00:30:51,310 --> 00:30:55,430
+Cn of 0 بتساوي C of 0 و Cn of 0 كلها واحد أيضاً
+
+365
+00:30:55,430 --> 00:30:57,730
+limit الواحد اللي هو بتساوي واحد يعني الـ C of 0
+
+366
+00:30:57,730 --> 00:31:05,970
+ايش هتساوي؟ هتساوي واحد خصّلت كمان شغلة أن حصلت
+
+367
+00:31:05,970 --> 00:31:10,070
+عندي أن الـ c of zero بيساوي واحد
+
+368
+00:31:13,750 --> 00:31:19,670
+لأ اللي هو الـ S الـ CN بأعمل شيء مشابه له لمين للـ SN
+
+369
+00:31:19,670 --> 00
+
+401
+00:34:04,340 --> 00:34:08,980
+of X وهذه اللي هي الـ function الثانية بنفس الأسلوب
+
+402
+00:34:08,980 --> 00:34:13,380
+وبنفس اللي هو المنطق وبنفس الأسباب بما أن الـ S N
+
+403
+00:34:13,380 --> 00:34:16,860
+conversion formally to S وS N continuous إذا حد
+
+404
+00:34:16,860 --> 00:34:22,340
+طلع الـ S برضه نفس إشمالها continuous وأيضًا limit
+
+405
+00:34:22,340 --> 00:34:28,540
+of S of 0 هو عبارة عن S of 0 وبما أن S of 0 دائمًا
+
+406
+00:34:28,540 --> 00:34:31,640
+0 إذًا limitها سيكون as N goes to infinity 0 يعني
+
+407
+00:34:31,640 --> 00:34:35,300
+ستظهر لدي S of 0 إيش بساوي 0 إذًا الآن طلعنا
+
+408
+00:34:35,300 --> 00:34:45,840
+كمان شغلة أثبتناها أن S of 0 بساوي 0 و C of 0 بساوي
+
+409
+00:34:45,840 --> 00:34:54,400
+1 وعرفنا هذولة الدالتين أسي و أس من اللي هي R لعند
+
+410
+00:34:54,400 --> 00:34:59,680
+مين لعند R إذن الآن احنا الدالتين اللي أثبتنا لإن
+
+411
+00:34:59,680 --> 00:35:05,430
+وجود هنا الآن اللي بحكي عنهم هذولة الدالتين اللي هي
+
+412
+00:35:05,430 --> 00:35:10,190
+الدالة الأولى سميتها أسوة الثانية C ولاقيت إنه C of 
+
+413
+00:35:10,190 --> 00:35:15,790
+0 بساوي واحد و S of 0 بساوي 0 وظل أثبت هذه و أثبت
+
+414
+00:35:15,790 --> 00:35:22,910
+هذه و أثبت هذولة اللي استبعتنا بتحققهم فخلينا نشوف
+
+415
+00:35:22,910 --> 00:35:29,670
+كيف نثبتها طيب الآن أوصلنا لعند اللي هو المنطقة
+
+416
+00:35:29,670 --> 00:35:39,670
+هذه لأن احنا تفجنا على ما يليها تفجنا إن الـ S N
+
+417
+00:35:39,670 --> 00:35:47,170
+تتعامل بشكل عام لتعمل S والـ C N تتعامل بشكل عام
+
+418
+00:35:47,170 --> 00:35:54,280
+لتعمل C لكن الآن احنا أثبتنا في الأول هي إن ما
+
+419
+00:35:54,280 --> 00:36:00,880
+محتاج الـ S N برايم بسوء الـ C N، مظبوط؟ يعني وكأنه
+
+420
+00:36:00,880 --> 00:36:04,860
+بناء على هذا مزام الـ S N برايم بسوء الـ C N فبيصير
+
+421
+00:36:04,860 --> 00:36:09,520
+الـ S N برايم، هذه، مكانها دي S N برايم بيصير الـ S N
+
+422
+00:36:09,520 --> 00:36:15,340
+برايم converges uniformly to some function mean C
+
+423
+00:36:15,340 --> 00:36:22,060
+الآن بما إن S N برايم converges uniformly to C الآن
+
+424
+00:36:22,060 --> 00:36:28,900
+والـ S N converged to S uniformly برضه حسب نظرية
+
+425
+00:36:28,900 --> 00:36:32,360
+بتطلع عندي بما إن s n prime differentiable هتكون
+
+426
+00:36:32,360 --> 00:36:37,140
+الـ limit ه differentiable و الـ c prime اللي هي الـ
+
+427
+00:36:37,140 --> 00:36:42,340
+s prime هذي هي مين الـ c اللي طلعت هنا يعني s n
+
+428
+00:36:42,340 --> 00:36:45,960
+prime differentiable أو s n differentiable و
+
+429
+00:36:45,960 --> 00:36:48,560
+converges to some function إذا بتتذكر كنا نسميها g
+
+430
+00:36:48,560 --> 00:36:52,520
+وهذي كنا نسميها f فكنا نقول بما إن s n بتروح للـ f
+
+431
+00:36:52,520 --> 00:36:56,740
+و الـ s n prime بتروح للـ g إذا نتيجة النظرية هتكون
+
+432
+00:36:56,740 --> 00:37:00,000
+اللي هي الـ F هي الـ Differentiable و الـ Derivative
+
+433
+00:37:00,000 --> 00:37:04,940
+لها إيش بتطلع D يعني الـ Derivative للـ S' إيش
+
+434
+00:37:04,940 --> 00:37:12,260
+هتطلع عبارة عن مين C' آسف C هتطلع مين اللي هي الـ
+
+435
+00:37:12,260 --> 00:37:21,000
+C إذا أثبتت أنا الآن S' of X بتساوي C of X هي
+
+436
+00:37:21,000 --> 00:37:28,190
+اللي أثبتته هنا الآن من جهة أخرى من جهة أخرى احنا
+
+437
+00:37:28,190 --> 00:37:35,850
+أثبتنا اللي هي قبل هيك إن الـ C N برايم بسوء ناقص S N
+
+438
+00:37:35,850 --> 00:37:42,450
+ناقص واحد الـ C N برايم بسوء ناقص S N of واحد يعني
+
+439
+00:37:42,450 --> 00:37:47,790
+بمعنى آخر الـ C N برايم of X أثبتنا بسوء ناقص S N
+
+440
+00:37:47,790 --> 00:37:54,640
+ناقص واحد of X الآن بما إن الـ S N converged زي الـ S
+
+441
+00:37:54,640 --> 00:38:00,260
+خلصنا هذه اه خلصنا هذه المنطقة خليني اشرح بلغة
+
+442
+00:38:00,260 --> 00:38:05,060
+ثانية عشان أميز بين الكلامين عندي الآن انتبهوا
+
+443
+00:38:05,060 --> 00:38:08,860
+بتعمل نفسي إشي بس بالنسبة لمن الآن بالنسبة عشان
+
+444
+00:38:08,860 --> 00:38:13,780
+أجيب الـ derivative للـ S للـ C prime عندي الآن الـ C N
+
+445
+00:38:13,780 --> 00:38:19,740
+prime of X بسوء نقص S N نقص واحد of X بما أن S N
+
+446
+00:38:19,740 --> 00:38:23,500
+راحت للـ S إذا اللي هي الـ derivative هي اللي
+
+447
+00:38:23,500 --> 00:38:30,280
+بتساويها اللي هي C N زائد واحد prime of X هتروح
+
+448
+00:38:30,280 --> 00:38:36,770
+لمين؟ اللي هي مش هي ناقصها لأن S N ناقص واحد اللي
+
+449
+00:38:36,770 --> 00:38:40,730
+هي بساوي ناقص هذه أنجل ناقصها بعد أذنكم يعني بدي
+
+450
+00:38:40,730 --> 00:38:44,810
+أستبدل الـ S N بمين بقيمتها هذه صارت ناقص الـ C N
+
+451
+00:38:44,810 --> 00:38:50,330
+prime of X أيش بتساوي بتروح للـ S أو بمعنى آخر صار
+
+452
+00:38:50,330 --> 00:38:56,620
+عندي من هنا من هنا اللي بتطلع عندي هنا صار عندي
+
+453
+00:38:56,620 --> 00:39:03,640
+الآن من هنا اللي هو c n زائد واحد prime of x بتروح
+
+454
+00:39:03,640 --> 00:39:08,080
+لناقص طبعًا uniformly بتروح لمين ناقص s لإن ناقصها
+
+455
+00:39:08,080 --> 00:39:13,040
+بتروح للـ s إذا هي بتروح لناقص الـ s وفي نفس الوقت
+
+456
+00:39:13,040 --> 00:39:19,840
+أنا بقول الـ C N نفسها بتروح uniform لمين؟ للـ C بنفس
+
+457
+00:39:19,840 --> 00:39:24,740
+الأسلوب اللي قبل بشوية اللي هي بما إنه هذه اللي هي
+
+458
+00:39:24,740 --> 00:39:27,580
+differentiable sequence of functions و converge
+
+459
+00:39:27,580 --> 00:39:31,260
+uniform to some function إذا هذه هتكون اللي هي
+
+460
+00:39:31,260 --> 00:39:35,600
+عبارة عن اللي هي الـ C الأصلية differentiable و الـ
+
+461
+00:39:35,600 --> 00:39:40,070
+derivative فيها لها مين؟ الناقص S بيصير عندي الـ C'
+
+462
+00:39:40,530 --> 00:39:45,970
+of X بسوء ناقص S of X بكون هذه اللي هي النتيجة
+
+463
+00:39:45,970 --> 00:39:50,330
+الثانية اللي طبقنا عليها طلعت عندي اللي هو هذه
+
+464
+00:39:50,330 --> 00:39:54,050
+الخاصية و هذه الخاصية متحققة يعني الآن صار عندي
+
+465
+00:39:54,050 --> 00:40:02,590
+اللي هو تحقق ما يلي إنه تبعنا هذه الـ C' of X
+
+466
+00:40:02,590 --> 00:40:08,530
+بيساوي ناقص S of X وطلع عندي اللي هي S prime of X
+
+467
+00:40:08,530 --> 00:40:16,290
+بيساوي C of X ماشي الحال بيكون هيك احنا ضال علينا
+
+468
+00:40:16,290 --> 00:40:21,230
+شغل أخرى لحاول نثبتها بيكون أثبتنا اللي هو كل
+
+469
+00:40:21,230 --> 00:40:27,630
+الصفات المطلوبة اللي تتحقق في الـ S وحددت هوية الـ S
+
+470
+00:40:27,630 --> 00:40:28,050
+و الـ C
+
+471
+00:40:38,700 --> 00:40:48,060
+الجزء الثاني سهل فضل الـ C N prime of x
+
+472
+00:41:00,590 --> 00:41:05,470
+بتصير C double prime هيها و هذه فضلها بنفعلها
+
+473
+00:41:05,470 --> 00:41:09,550
+لأنها قبل التفاضل اللي فضلناها وبتصير ناقص اللي هي
+
+474
+00:41:09,550 --> 00:41:15,030
+S prime of x اللي هي S prime of x إيش بتساوي C of
+
+475
+00:41:15,030 --> 00:41:20,350
+x فبيصير ناقص مين؟ C of x والآن S N' of X من أين بدي
+
+476
+00:41:20,350 --> 00:41:23,650
+أجيبها من هنا S N' بيساوي اللي هو هذه الـ
+
+477
+00:41:23,650 --> 00:41:26,050
+derivative اللي هي الـ derivative اللي هي عبارة عن
+
+478
+00:41:26,050 --> 00:41:31,530
+ناقص S of X فبيصير عبارة عن ناقص S of X فبيصير عندي
+
+479
+00:41:31,530 --> 00:41:38,910
+هذا برضه إيش تحقق آخر إشي اللي هي C' of Zero C' of
+
+480
+00:41:38,910 --> 00:41:42,410
+Zero بيساوي ناقص S of Zero و S of Zero بيساوي صفر
+
+481
+00:41:42,410 --> 00:41:47,030
+إذا C' of Zero بيساوي Zero الـ S' of Zero
+
+482
+00:41:52,540 --> 00:41:56,800
+وهكذا أثبتنا وجود دالة
+
+483
+00:41:59,570 --> 00:42:04,970
+اللي هي حققت لي اللي هي الشرط الأول هذا و الدالتين
+
+484
+00:42:04,970 --> 00:42:09,710
+آسف والشرط الثاني اه اللي بعده هذه بتكون أثبتنا
+
+485
+00:42:09,710 --> 00:42:15,290
+وجود الـ C و الـ S لإن بعض النتائج الأخرى على اللي هي
+
+486
+00:42:15,290 --> 00:42:18,830
+الدالتين
+
+487
+00:42:18,830 --> 00:42:19,790
+اللي أوجدناهم
+
+488
+00:42:24,500 --> 00:42:28,960
+نشوف إيش الـ Corollary الأولى بقول لي if C and S
+
+489
+00:42:28,960 --> 00:42:33,680
+are the functions in 3x, 8x, 4x, 1 then C' of X
+
+490
+00:42:33,680 --> 00:42:39,560
+بيساوي نقص S of X أثبتناها هيها and طبعًا في طريقنا
+
+491
+00:42:39,560 --> 00:42:44,020
+في البرهان S' of X بيساوي C of X هيها اللي هي خواص
+
+492
+00:42:44,020 --> 00:42:50,340
+اللي هي أثبتناها طيب الآن بقول لي moreover these
+
+493
+00:42:50,340 --> 00:42:54,630
+functions have derivatives of all order يعني بأي
+
+494
+00:42:54,630 --> 00:42:57,710
+order الـ derivative موجودة طبعًا هذا by induction
+
+495
+00:42:57,710 --> 00:43:01,550
+by induction اللي هو بدك تثبت إنه اللي هو اللي هو
+
+496
+00:43:01,550 --> 00:43:06,670
+C N هتكون موجودة و بيساوي ناقص a of x أو زائد a of x
+
+497
+00:43:06,670 --> 00:43:11,450
+حسب اللي هي جداش درجة الـ n و ممكن تساوي C of x أو
+
+498
+00:43:11,450 --> 00:43:16,930
+اللي هي ناقص C of x حسب درجة الـ n اللي موجودة إذا
+
+499
+00:43:16,930 --> 00:43:19,990
+الآن هذه by induction بنقدر نثبت إن الـ derivative
+
+500
+00:43:19,990 --> 00:43:26,390
+اللي هو موجودة for any order ضر بناء على اللي حكيناه
+
+501
+00:43:26,390 --> 00:43:32,300
+نيجي الآن لـ Corollary اللي بعدها بقول الآن الـ
+
+502
+00:43:32,300 --> 00:43:38,280
+function C and S بتحقق اللي هي الـ Pythagorean
+
+503
+00:43:38,280 --> 00:43:42,280
+Identity اللي كنا نعرفها كـ cosine تربيع زائد sine تربيع
+
+504
+00:43:42,280 --> 00:43:47,660
+إيش بساوي بساوي واحد فكرة البرهان سهلة بسمي هذه
+
+505
+00:43:47,660 --> 00:43:51,520
+كلها بسميها function اسمها F of X بقول سمي هذه
+
+506
+00:43:51,520 --> 00:43:56,080
+اللي هي F of X بساوي هذه فضليها بلا بتفضليها F
+
+507
+00:43:56,080 --> 00:44:01,200
+prime of X بساوي 2 في C of X في تفاضل اللي جوا
+
+508
+00:44:01,200 --> 00:44:06,300
+اللي هو ناقص S of X و الثانية زائد 2 في S of X
+
+509
+00:44:06,300 --> 00:44:11,880
+فتفاضلها هي C of X هذه هي هذه بس بالسالب إذا حصل
+
+510
+00:44:11,880 --> 00:44:15,100
+طرح حلو اسم ساوى صفر إذا صارت عند الـ derivative للـ
+
+511
+00:44:15,100 --> 00:44:18,320
+function هذه إشمالها بساوي صفر يعني بمعنى آخر
+
+512
+00:44:18,320 --> 00:44:22,490
+الدالة هي دالة ثابتة يعني F is a constant function
+
+513
+00:44:22,490 --> 00:44:27,210
+يعني قيمة الـ F عند أي قيمة إيش بتساوي مقدار ثابت
+
+514
+00:44:27,210 --> 00:44:31,310
+إذا تفاجأنا إنها ثابتة أسهل إشي أسهل إشي أوجد لي F of
+
+515
+00:44:31,310 --> 00:44:34,670
+Zero عشان أتعرف إيش ده اللي بساوي بالظبط C of
+
+516
+00:44:34,670 --> 00:44:38,730
+Zero واحد S of Zero صفر إذا صار عندي F of Zero
+
+517
+00:44:38,730 --> 00:44:42,170
+بساوي واحد وهي ثابتة إذا صارت F of X دائمًا
+
+518
+00:44:42,170 --> 00:44:46,270
+بساوي واحد يعني هذا المقدار بساوي واحد دائمًا
+
+519
+00:44:46,270 --> 00:44:48,690
+طيب نيجي الآن
+
+520
+00:44:51,700 --> 00:44:57,820
+للـ theorem اللي بعدها الـ functions C and S
+
+521
+00:44:57,820 --> 00:45:02,820
+satisfy the properties I and II of theorem 8-4-1
+
+522
+00:45:02,820 --> 00:45:07,020
+are unique لما بقول لي برضه هتلاقي الـ uniqueness
+
+523
+00:45:07,020 --> 00:45:13,020
+بشبه الـ uniqueness لمن؟ للي هو الـ .. الـ ..
+
+524
+00:45:13,020 --> 00:45:18,030
+قولوا معايا الـ uniqueness للـ exponential و سمناها
+
+525
+00:45:18,030 --> 00:45:21,170
+اللي هو let E واحد .. نفترض إنه في E واحد و E
+
+526
+00:45:21,170 --> 00:45:24,070
+اتنين و سمها فرق بيساوي D و في الآخر روحنا لفرق
+
+527
+00:45:24,070 --> 00:45:28,390
+بيساوي إيش؟ بيساوي صفر هنا نفس الإشي هنشتغل و برضه
+
+528
+00:45:28,390 --> 00:45:31,590
+هنستخدم اللي هو الـ Taylor's theorem زي ما استخدمنا
+
+529
+00:45:31,590 --> 00:45:34,470
+هناك الـ Taylor's theorem يعني هتلاقيه الـ sketch
+
+530
+00:45:34,470 --> 00:45:37,470
+للبرهان هو نفس الـ sketch الأولاني عشان هيك
+
+531
+00:45:37,470 --> 00:45:43,110
+هتلاقوني سريع فيه بدنا نثبت إن الـ C و الـ S are
+
+532
+00:45:43,110 --> 00:45:47,930
+unique functions طبعًا الطريقة let c1 and c2 be two
+
+533
+00:45:47,930 --> 00:45:52,290
+functions on R that satisfy it satisfies مين الـ
+
+534
+00:45:52,290 --> 00:45:55,850
+conditions اللي احنا بنقول عنهم اللي هي الـ I و II
+
+535
+00:45:55,850 --> 00:46:00,930
+اللي هو بحيث إنه c1 double prime of x بساوي c1 of
+
+536
+00:46:00,930 --> 00:46:01,150
+x
+
+537
+00:46:16,750 --> 00:46:18,150
+2
+
+538
+00:46:18,550 --> 00:46:24,010
+لإن نفترض إن دي بتساوي C1 - C2 وبدنا نصل في النهاية
+
+539
+00:46:24,010 --> 00:46:27,410
+إن دي هذي لازم تطلع إيش بتساوي؟ بتساوي 0 مدام دي
+
+540
+00:46:27,410 --> 00:46:32,710
+بتساوي 0 مدام C1 بتساوي إيش؟ C2 لاحظ الآن D W' of
+
+541
+00:46:32,710 --> 00:46:38,190
+X فاضل هذا مرتين بتصير C1 W' - C2 W'
+
+542
+00:46:39,860 --> 00:46:47,500
+الآن بساوي هتطلع إيش بتساوي ناقص اللي هي مين D of
+
+543
+00:46:47,500 --> 00:46:59,700
+X نعملها اللي هي حسابات D of X بساوي D of X بساوي
+
+544
+00:46:59,700 --> 00:47:08,540
+C1 ناقص C2 D prime of X إيش هيساوي اللي هو عبارة
+
+545
+00:47:08,540 --> 00:47:16,670
+عن S واحد ناقص S واحد ناقص بيصير اللي هو مين الـ S
+
+546
+00:47:16,670 --> 00:47:19,810
+واحد اللي هي بالنسبة لهذه اللي أوجدناها ناقص S
+
+547
+00:47:19,810 --> 00:47:25,350
+اتنين بيصير زائد لإن D double prime بساوي بتفاضل
+
+548
+00:47:25,350 --> 00:47:30,030
+هذا كمان مرة اللي هي بترجع مين نفسها C واحد و هذه
+
+549
+00:47:30,030 --> 00:47:38,390
+ترجع نفسها C2 التي هي ناقص D التي هي ناقص D طيب
+
+550
+00:47:38,390 --> 00:47:43,410
+إذا الـ D لإن مفترضين الـ C1 و C2 إشمالها بتحقق
+
+551
+00:47:43,410 --> 00:47:52,470
+الشروط اللي فوق اللي قلنا عنها Double prime بساوي D
+
+552
+00:47:52,470 --> 00:48:02,390
+of X أو D double prime بساوي D double prime بساوي
+
+553
+00:48:06,030 --> 00:48:11,870
+بساوي هذي W' ناقص هذي W' هذي W' ناقص C1 و هذي W' اللي
+
+554
+00:48:11,870 --> 00:48:15,870
+هي ناقص ناقص C زائد صارت D W' بساوي ناقص D و X
+
+555
+00:48:15,870 --> 00:48:21,890
+هيك اللي هو دعنا نقول أسلم طيب الآن بستعجل لإن
+
+556
+00:48:21,890 --> 00:48:29,470
+الكلام معاد يعني أو الأفكار معادة الآن احط لي D of 0
+
+557
+00:48:29,470 --> 00:48:34,530
+إيش هيساوي؟ Zero لإن D of 0 بيساوي هذي ناقص هذي و
+
+558
+00:48:34,530 --> 00:48:37,630
+هذي عند الـ zero واحد و هذي عند الـ zero واحد
+
+559
+00:48:37,630 --> 00:48:40,010
+الاثنتين عند الـ zero واحد انحصل طرحي إن إيش هيساوي
+
+560
+00:48:40,010 --> 00:48:47,160
+صفر الآن D prime عند الـ zero و D W' عند الـ zero و و
+
+561
+00:48:47,160 --> 00:48:48,760
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+562
+00:48:48,760 --> 00:48:50,140
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+563
+00:48:50,140 --> 00:48:52,320
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+564
+00:48:52,320 --> 00:48:52,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+565
+00:48:52,380 --> 00:48:55,440
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+566
+00:48:55,440 --> 00:48:59,220
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+567
+00:48:59,220 --> 00:
+
+601
+00:51:44,970 --> 00:51:50,030
+سواء الآن بنقول for each n unlimited n عملناها
+
+602
+00:51:50,030 --> 00:51:56,510
+عشان هيك بس ماشي عنه بسرعة عملناها قبل هيك there
+
+603
+00:51:56,510 --> 00:51:59,630
+exist a point Cn unlimited Ix such that هذا
+
+604
+00:51:59,630 --> 00:52:03,890
+remainder بتعملي remainderD ب X بيساوي D of 0 زي
+
+605
+00:52:03,890 --> 00:52:06,850
+D prime of 0 زائد زائد زائد زائد X ان نقص واحد
+
+606
+00:52:06,850 --> 00:52:12,630
+لما أصل لمين لل remainder كل هذول أصفار بسبب مين
+
+607
+00:52:12,630 --> 00:52:16,430
+أن D0 وD prime of 0 وD ان نقص واحد وزير وكلهم إيه
+
+608
+00:52:16,430 --> 00:52:20,090
+شمالهم بيساوي أصفار عند الصفر بيظل هذا المقدار
+
+609
+00:52:20,090 --> 00:52:24,010
+يساوي اللي هو المقدار اللي عندي اللي تحت هذا هذا
+
+610
+00:52:24,010 --> 00:52:27,490
+ال derivative هذا ال derivative أنا بعرفش إيش هي
+
+611
+00:52:27,490 --> 00:52:31,230
+ما أنتم عارفين ال derivative للـ .. للـ .. للـ C
+
+612
+00:52:34,160 --> 00:52:41,940
+أو ال derivative للـ C إذا فضلت مرة واحدة بتطلع
+
+613
+00:52:41,940 --> 00:52:47,120
+الـ S أو سلبها فظلت تلت مرات بترجع ليها S بس
+
+614
+00:52:47,120 --> 00:52:53,180
+بالموجة فظلت خمسة بترجع ناقص S فظلت زوجي بتطلع هي
+
+615
+00:52:53,180 --> 00:52:58,920
+نفسها أو سلبها عشان هي كادل أنا بعرفش طبعا هنا
+
+616
+00:52:58,920 --> 00:53:02,400
+عندك اللي هي بتتوزع على مرة derivative تندين
+
+617
+00:53:02,400 --> 00:53:04,620
+derivative تلاتة derivative أربعة derivative و
+
+618
+00:53:04,620 --> 00:53:08,180
+بترجع الدورة زي ما هي لإنه في الأول بيكون اللي هي
+
+619
+00:53:08,180 --> 00:53:15,520
+تفاضل الـC ناقص S بعدها بتتفاضل بترجع اللي هي ناقص
+
+620
+00:53:15,520 --> 00:53:18,770
+حالها بعدها بترجع اللي هي
+
+621
+00:53:18,770 --> 00:53:36,410
+بببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببب
+
+622
+00:53:36,660 --> 00:53:40,180
+اللي هي الـC ويصير نقص C كمان مرة بترجع اللي هي
+
+623
+00:53:40,180 --> 00:53:45,740
+الـ S وكمان مرة بترجع اللي هي الـ C بعد هيك بتصير
+
+624
+00:53:45,740 --> 00:53:51,920
+اللي هي تكرر حالها يعني بمعنى آخر جربوها أنتم
+
+625
+00:53:51,920 --> 00:53:59,920
+هتلاقوا أن حسب الأُس هنا بتطلع اللي هي عبارة عن
+
+626
+00:53:59,920 --> 00:54:11,180
+هي أو سلبها أو الـS أو سلبها بكل الأحوال بغض النظر
+
+627
+00:54:11,180 --> 00:54:15,600
+اللي هي DN of CN هي حاصل طرح التنتين اللي هي سمنها
+
+628
+00:54:15,600 --> 00:54:23,280
+يا D يا T الـ D تبعت الـ C الفرق طبعا والـ T اللي
+
+629
+00:54:23,280 --> 00:54:29,010
+هي الفرق بين الأسات يعني في الآخر زاد أو ناقص و زاد
+
+630
+00:54:29,010 --> 00:54:33,810
+أو ناقص سواء هذه أو سواء هذه حضرنا من الأصل أن ها
+
+631
+00:54:33,810 --> 00:54:38,870
+bounded و كله أصغر أو يساوي من K أصغر أو يساوي إيش
+
+632
+00:54:38,870 --> 00:54:45,860
+K بناء عليه هتطلع هذه هيك absolute value أصغر أو
+
+633
+00:54:45,860 --> 00:54:51,780
+يساوي هذه أي إن كانت K في X أُس N absolute value
+
+634
+00:54:51,780 --> 00:54:56,880
+على N factorial بنفس الحال إذا صلعت عندي هذا
+
+635
+00:54:56,880 --> 00:55:00,860
+المقدار أصغر أو يساوي هذا as N goes to infinity
+
+636
+00:55:00,860 --> 00:55:05,540
+هذا بروح لمن؟ للصفر وهذا independent of N بيصير
+
+637
+00:55:05,540 --> 00:55:08,100
+أكبر أو يساوي الصفر يعني هذا اللي جوا بده يصير صفر
+
+638
+00:55:08,100 --> 00:55:12,930
+إذا ال D of X هاشي بدها تساوي تساوي صفر وبكون
+
+639
+00:55:12,930 --> 00:55:21,090
+أثبتنا أن الـ C1 والـ C2 are equivalent الآن
+
+640
+00:55:21,090 --> 00:55:27,700
+similar arguments بنفس الأسلوب وخليها إليكم أن
+
+641
+00:55:27,700 --> 00:55:32,120
+اللي هي بنثبت أن S إذا كان S1 وS2 are two
+
+642
+00:55:32,120 --> 00:55:36,320
+functions such that بيحققوا اللي هما الخواص اللي
+
+643
+00:55:36,320 --> 00:55:41,700
+بنقولها عنها في النظرية اللي هي S1 double prime
+
+644
+00:55:41,700 --> 00:55:46,400
+بيساوي نفس S1 ونفس ال S2 وهذا عند Zero بيساوي Zero
+
+645
+00:55:46,400 --> 00:55:50,160
+وهذا ال prime عند Zero بيساوي Zero هيتلعلك غصب
+
+646
+00:55:50,160 --> 00:55:52,240
+عنها في الآخر S1 بيساوي S2
+
+647
+00:55:55,620 --> 00:55:59,960
+بس بدل ما تعملوا على ال D يعني سمو اللي هو T
+
+648
+00:55:59,960 --> 00:56:04,040
+بتساوي S1 نقص S2 وطبقوا الشروط وامشوا نفس اللي
+
+649
+00:56:04,040 --> 00:56:07,580
+امشيناها هتلاقوا حالكم أن لازم تطلع ال S1 بيساوي
+
+650
+00:56:07,580 --> 00:56:11,800
+S2 وبناء عليه صار ال ال two functions ال C والS
+
+651
+00:56:11,800 --> 00:56:17,820
+are unique functions طيب
+
+652
+00:56:17,820 --> 00:56:24,720
+ال answer المؤهلين نذهب بالاتجاه نثبت أن افترض أن
+
+653
+00:56:24,720 --> 00:56:27,880
+الـ Definition of the unique function C من R لR و
+
+654
+00:56:27,880 --> 00:56:32,400
+أسم الـ R لR اللي يتحقق CW prime of X بساوية نقطة
+
+655
+00:56:32,400 --> 00:56:36,650
+C of X هو الأصل الـ differential equation التاني S
+
+656
+00:56:36,650 --> 00:56:40,990
+W برامج و X ناقص بساوة ناقص S X لكل X element in R
+
+657
+00:56:40,990 --> 00:56:43,470
+و ال C of Zero اللي هو ال condition ال condition
+
+658
+00:56:43,470 --> 00:56:46,350
+بساوي واحد و C برامج و Zero بساوة Zero و ال S of
+
+659
+00:56:46,350 --> 00:56:49,710
+Zero بساوة Zero و ال S برامج و Zero بساوة Zero
+
+660
+00:56:49,710 --> 00:56:57,150
+الدالتين اللي بحقق إن الكلام هذا اللي أثبتنا أن
+
+661
+00:56:57,150 --> 00:57:02,740
+unique بنسميهم respectively the cosine function and
+
+662
+00:57:02,740 --> 00:57:07,600
+the sine function اللي أنتم بتعرفوها وهي cosine X
+
+663
+00:57:07,600 --> 00:57:12,620
+اللي بتعرفوها وهي sin X اللي أنتم بتعرفوها نيجي
+
+664
+00:57:12,620 --> 00:57:17,930
+الآن ناخد بعض الخواص خلينا ناخد اللي هو الخاصية
+
+665
+00:57:17,930 --> 00:57:23,430
+اللي هي نظرية 4.6 بتقول لي إذا كانت f من R لR is
+
+666
+00:57:23,430 --> 00:57:27,890
+such that f double prime of x بساوي نقص f of x for
+
+667
+00:57:27,890 --> 00:57:30,370
+x element in R يعني هذه ال differential equation
+
+668
+00:57:30,370 --> 00:57:32,810
+بتقول لي هذه ال differential equation هي حلولها
+
+669
+00:57:32,810 --> 00:57:37,690
+مالي then there exist Alpha و Beta Such that F of X
+
+670
+00:57:37,690 --> 00:57:41,390
+بيساوي Alpha CX زائد Beta S of X بيقول لي هذه ال
+
+671
+00:57:41,390 --> 00:57:45,110
+differential equation حالة أن ها عبارة عن linear او
+
+672
+00:57:45,110 --> 00:57:47,870
+خليني أقول combination او linear combination
+
+673
+00:57:47,870 --> 00:57:55,890
+between F C of X S of X خليني أخد نسمي g of x إيش
+
+674
+00:57:55,890 --> 00:57:59,590
+بتساوي اللي هي عبارة عن اللي هي ال c of x و ال a
+
+675
+00:57:59,590 --> 00:58:03,770
+sub x نسمي هذه f of zero و f prime of zero for x
+
+676
+00:58:03,770 --> 00:58:10,330
+element in R الآن لو حسبت ال gw prime of x احسبها
+
+677
+00:58:10,330 --> 00:58:15,270
+فاضل هذه
+
+678
+00:58:15,270 --> 00:58:18,550
+مرتين هدولة ثوابت طبعا و هدولة اللي بتنزلك
+
+679
+00:58:18,550 --> 00:58:23,970
+هتلاقيهم اللي هو بيساوي ناقص g of x آه .. و لو حسبت
+
+680
+00:58:23,970 --> 00:58:26,730
+الـ g of zero .. g of zero هتلاقيها بيساوي of zero
+
+681
+00:58:26,730 --> 00:58:30,110
+لإن هذا صفر وهذا .. هذا واحد وهذا صفر الآن صار
+
+682
+00:58:30,110 --> 00:58:33,850
+عندي gw prime of x بيساوي نقص g of x و g of zero
+
+683
+00:58:33,850 --> 00:58:40,410
+بساوي f of zero الآن احسب الـ g prime of xصار عندي
+
+684
+00:58:40,410 --> 00:58:43,450
+تلات معلومات معلمتين جي دابل برايم of X بيساوي نقص
+
+685
+00:58:43,450 --> 00:58:45,990
+جي of X و جي of Zero بيساوي أف of Zero خلينا نجيب
+
+686
+00:58:45,990 --> 00:58:50,330
+جي برايم of X جي برايم of X مش بتساوي فاضل اللي هي
+
+687
+00:58:50,330 --> 00:58:55,430
+عبارة عن نقص أس of X و هذا تفضلها اللي هي C of X
+
+688
+00:58:55,430 --> 00:58:58,830
+صار عندي اللي هي جي برايم of X بيساوي هذا الكلام
+
+689
+00:58:58,830 --> 00:59:04,720
+زائد هذا الكلام الآن احسب لي g prime of 0 هيصير
+
+690
+00:59:04,720 --> 00:59:08,420
+عبارة عن هذا طبعا صفر وهذا هتصير واحد فبصير F
+
+691
+00:59:08,420 --> 00:59:13,750
+prime of 0 صار عندي الآن g prime of 0 إيش بيساوي يا
+
+692
+00:59:13,750 --> 00:59:18,090
+جماعة؟ جي برايم اف زيرو خليني أطلع لك إياها لفوق
+
+693
+00:59:18,090 --> 00:59:21,750
+صارت جي برايم اف زيرو بيساوي أف برايم اف زيرو و جي
+
+694
+00:59:21,750 --> 00:59:26,730
+اف زيرو بيساوي أف اف زيرو الآن بدنا نشرح نعيد اللي
+
+695
+00:59:26,730 --> 00:59:30,950
+هو therefore الآن the functions h بتساوي f ناقص g
+
+696
+00:59:30,950 --> 00:59:35,990
+is such that يعني عرف لي function h إيش بيساوي؟ f
+
+697
+00:59:35,990 --> 00:59:43,220
+ناقص مين ناقص g الآن الـ H double prime لها لو جيت
+
+698
+00:59:43,220 --> 00:59:46,840
+حسبت الـ H double prime اللي هي هتلاقيها بتساوي
+
+699
+00:59:46,840 --> 00:59:50,260
+ناقص H of X بتحسبوها لحالكم H double prime of X
+
+700
+00:59:50,260 --> 00:59:55,140
+عشان بتساوي ناقص H of X لكل X element in R لو حسبت
+
+701
+00:59:55,140 --> 00:59:59,670
+اللي هي H of Zero هتساوي اللي هي zero لإنه احنا
+
+702
+00:59:59,670 --> 01:00:03,010
+أثبتنا أن g of zero ساوي f of zero وh prime of
+
+703
+01:00:03,010 --> 01:00:07,110
+zero h prime of zero اللي هي عبارة عن f prime نقص
+
+704
+01:00:07,110 --> 01:00:09,210
+g prime عند ال zero التانية و التساويات إذا إيه مش
+
+705
+01:00:09,210 --> 01:00:15,170
+ساوي صفر الآن it then thus it follows as in the
+
+706
+01:00:15,170 --> 01:00:19,430
+proof of the preceding theorem النظرية الماضية أن
+
+707
+01:00:19,430 --> 01:00:22,170
+h of x إيه مش ساوي صفر يعني بدك تعمل نفس اللي
+
+708
+01:00:22,170 --> 01:00:25,180
+عملناها قبل هيك على ال Taylor's theorem و الاخرينو
+
+709
+01:00:25,180 --> 01:00:29,300
+تصل ال H of X بيساوي 0 لكل X ومنه هتقول F of X
+
+710
+01:00:29,300 --> 01:00:33,320
+بساوي G of X مدام F of X بيساوي G of X إذن هذه
+
+711
+01:00:33,320 --> 01:00:38,580
+اللي هي F of X بتطلع F of X بتساوي اللي هي هذا
+
+712
+01:00:38,580 --> 01:00:42,780
+اسمها Alpha وهذا اسمها Beta وبكون عندي هي اللي هو
+
+713
+01:00:42,780 --> 01:00:46,280
+solution لاللي هي ال differential equation اللي
+
+714
+01:00:46,280 --> 01:00:50,260
+احنا حكينا عنها بديش أعيد نفس الكلام عشان هي كأنا
+
+715
+01:00:50,260 --> 01:00:52,780
+اختصرت لأن الحسابات كلها مشابهة
+
+716
+01:00:58,150 --> 01:01:03,430
+الآن بدنا اللي هي 8 4 7 the function C is even and
+
+717
+01:01:03,430 --> 01:01:07,570
+S is odd in the sense that هو بتكون الصحيحة دي
+
+718
+01:01:07,570 --> 01:01:10,610
+الأصل كلها تكون أو معظمها اللي هي عبارة عن
+
+719
+01:01:10,610 --> 01:01:14,430
+exercises لإنه تطبيقات على النظرية اللي قلتها في
+
+720
+01:01:14,430 --> 01:01:19,630
+الأول ال C ناقص X هيساوي C of X يعني عبارة عن even
+
+721
+01:01:19,630 --> 01:01:23,230
+function S ناقص X بيساوي ناقص S of X لكل X element
+
+722
+01:01:23,230 --> 01:01:28,590
+in R الآن if x,y المتنار then we have the addition
+
+723
+01:01:28,590 --> 01:01:32,230
+formula c of x زي اد y بيسبب c of x في c of y نقص
+
+724
+01:01:32,230 --> 01:01:35,950
+s of x وs of y اللي هي اللي بتعرفوها أنتم sign ال
+
+725
+01:01:35,950 --> 01:01:38,350
+x زي اد y بيسبب sign ال x بكسان ال y زي اد كسان ال
+
+726
+01:01:38,350 --> 01:01:42,090
+y في sign ال x وهكذا هي هذه الخواص اللي احنا
+
+727
+01:01:42,090 --> 01:01:47,570
+عارفينها قبل هيك وبدنا نشوف كيف اللي هي أن نبرهن
+
+728
+01:01:47,570 --> 01:01:53,660
+النظرية نشوف البرهان تبع النظرية الآن سمي لي Phi of
+
+729
+01:01:53,660 --> 01:01:59,370
+X بيساوي C of ناقص X لكل X element in R احسب لي الـ
+
+730
+01:01:59,370 --> 01:02:05,010
+phi w prime of x لو أتيت فضلت هذه مرتين هتلاقيها
+
+731
+01:02:05,010 --> 01:02:09,950
+ناقص phi of x وهذا الكلام تفضله سهل و بتجيبه لحالك
+
+732
+01:02:09,950 --> 01:02:15,050
+الآن احسب لي ال phi of 0 phi of 0 هتساوي إيش؟
+
+733
+01:02:15,050 --> 01:02:19,350
+بتساوي واحد phi prime of 0 هتطلع اللي هي عبارة عن
+
+734
+01:02:19,350 --> 01:02:24,430
+اللي هي ال sign ال sign إيش معناها؟ بتساوي 0 عند
+
+735
+01:02:24,430 --> 01:02:30,730
+ال zero الآن صار عندي الآن الفاى هي مين؟ هي الـ C
+
+736
+01:02:30,730 --> 01:02:35,710
+ليش الفاى هي الـ C؟ لأن حققت الفاى اللي فرضتها
+
+737
+01:02:35,710 --> 01:02:40,790
+بساوي C-X شروط الـ C و الـ C is unique إذا الفاى H
+
+738
+01:02:40,790 --> 01:02:47,200
+بتساوي C إذا صار عندي الآن C-X بيساوي C of X الفكرة
+
+739
+01:02:47,200 --> 01:02:52,060
+واضحة أن أنا جبت سميت الـ C-X هي الفاية واتبتت أن
+
+740
+01:02:52,060 --> 01:02:57,060
+هذه الفاية بتحقق الشرطين اللي احنا حكينا عنهم في
+
+741
+01:02:57,060 --> 01:03:01,300
+الأول الدالة اللي بتخلي الـ C is unique فصارت
+
+742
+01:03:01,300 --> 01:03:07,080
+الفاية إيش بتساوي الـ C الآن بمعنى آخر صارت C of X
+
+743
+01:03:07,080 --> 01:03:10,720
+هي C من ناقص X اللي فرضناها الـ Phi in a similar
+
+744
+01:03:10,720 --> 01:03:16,040
+way برضه بدك تعملاش S of ناقص X اللي هو بتسمي اللي
+
+745
+01:03:16,040 --> 01:03:20,480
+هي ال .. الفا .. بتسميها بـ Psi مثلا Psi بتساوي
+
+746
+01:03:20,480 --> 01:03:25,020
+مين؟ بتساوي اللي هو ناقص هتجيبها سامي بصي بيساوي
+
+747
+01:03:25,020 --> 01:03:29,540
+ناقص S ناقص X و تجيب الشروط اللي هي تبعها ال sign
+
+748
+01:03:29,540 --> 01:03:33,500
+هتلاقيها متطابق متحققة و بما أن ال S is unique أو
+
+749
+01:03:33,500 --> 01:03:37,240
+ال sign is unique إذا اللي هي اللي حققتها اللي
+
+750
+01:03:37,240 --> 01:03:41,500
+كتبتها بتساوي ال S وصارين إيه جهتين متساويات بنفس
+
+751
+01:03:41,500 --> 01:03:47,980
+الأسلوب طيب الآن نيجي نثبت اللي هو مين VI شوفوا
+
+752
+01:03:47,980 --> 01:03:52,850
+صلوا على النبي عليه الصلاة والسلام هنفير نبره ان vi
+
+753
+01:03:52,850 --> 01:03:55,830
+let y المتن عار بيجيبنا let f of x بيساوي c of x
+
+754
+01:03:55,830 --> 01:04:00,070
+زي dash زي y for x المتن عار الآن a calculation
+
+755
+01:04:00,070 --> 01:04:04,230
+shows that يعني فضلي هذا مرتين التفاضل سهل والله
+
+756
+01:04:04,230 --> 01:04:08,010
+يا جماعة عشانك أنا يعني ما بديش نضيع واجتنا في
+
+757
+01:04:08,010 --> 01:04:12,190
+التفاضل أو في الحسابات f w prime of x طبعا فاضل
+
+758
+01:04:12,190 --> 01:04:15,430
+بالنسبة لي x y إيش ما على f x ثابت أسيبك منها لأن
+
+759
+01:04:15,430 --> 01:04:18,590
+لو فضلت مرتين هتلاقي ناقص f of x for x element in
+
+760
+01:04:18,590 --> 01:04:22,550
+R مدام f w prime بيساوي ناقص f of x بالحصارة اللي
+
+761
+01:04:22,550 --> 01:04:26,850
+هي حل المعادلة القفاضلية هذه اللي هي بالنظرية اللي
+
+762
+01:04:26,850 --> 01:04:32,880
+قبل شوية هي 8 4 6 هتكون اللي هي ال f of x عبارة عن
+
+763
+01:04:32,880 --> 01:04:36,300
+linear combination هذه اللي هو α C of X زي Beta S
+
+764
+01:04:36,300 --> 01:04:39,840
+of X و الـ F of X مين هي احنا فرضينا C of XY صارت
+
+765
+01:04:39,840 --> 01:04:43,180
+هذه عبارة عن هذه يعني هذه بتساوي هذه من حال
+
+766
+01:04:43,180 --> 01:
+
+801
+01:08:00,350 --> 01:08:03,530
+خليني أطلعلك عليها على السريع لأن حسابات كلها
+
+802
+01:08:03,530 --> 01:08:07,840
+بتعملها لحالك أكيد بتعرف إذا كان الـ x أكبر أو أقل
+
+803
+01:08:07,840 --> 01:08:12,880
+من 0، فهناك Sx بنقص الـ x والـ x والـ c of x بين
+
+804
+01:08:12,880 --> 01:08:16,780
+الواحد ناقص x أربع عذر تو واحد والـ c of x بقدر
+
+805
+01:08:16,780 --> 01:08:18,820
+يكمل الـ polynomial
+
+806
+01:08:21,810 --> 01:08:25,650
+باللي عملناها اللي هي الـCN of X اللي جابله شوية و
+
+807
+01:08:25,650 --> 01:08:28,430
+الـSN of X لأنه في النهاية limitها ال series
+
+808
+01:08:28,430 --> 01:08:30,790
+الأولى as N goes to infinity ما أنتم عارفين اللي
+
+809
+01:08:30,790 --> 01:08:34,990
+هي نقدر نكتب الـC of X على صورة ال series اللي في
+
+810
+01:08:34,990 --> 01:08:38,130
+الأول و الS of X عبارة عن limit ال series الثانية
+
+811
+01:08:38,130 --> 01:08:43,390
+ال terms إضافة term وطرح term  ووجدنا عند حدود بال
+
+812
+01:08:43,390 --> 01:08:47,090
+term بتعمل ال inequality اللي عندنا اللي هو كل هذه
+
+813
+01:08:47,090 --> 01:08:49,810
+الشغلات اللي أخذناها في ال calculus مافيش داعي
+
+814
+01:08:49,810 --> 01:08:55,650
+للتفصيل فيها الآن الأولى عندي ال .. ال .. احنا قلنا
+
+815
+01:08:55,650 --> 01:08:59,990
+sign تربيع زائد plus sign تربيع بساوي واحد يعني في
+
+816
+01:08:59,990 --> 01:09:02,670
+النهاية ال C of T بين ناقص واحد وواحد ده ممكن
+
+817
+01:09:02,670 --> 01:09:06,710
+هتتجاوزها لأنه بتختل اللي هي ال C .. لو كانت أكبر
+
+818
+01:09:06,710 --> 01:09:10,750
+من واحد معناته بـ C .. C of T تربيع زائد S of T
+
+819
+01:09:10,750 --> 01:09:14,550
+تربيع يتجاوز من واحد لكن مجموعها بساوي واحد إذا هذه
+
+820
+01:09:14,550 --> 01:09:18,630
+بين .. يعني هذه اللي وين جاية من C تربيع زائد S
+
+821
+01:09:18,630 --> 01:09:23,030
+تربيع بساوي واحد طيب اللي هنعمل integration لجهتين
+
+822
+01:09:23,030 --> 01:09:26,990
+من صفر لعين DX هذه اللي هي بتطلع اللي هي main ال S
+
+823
+01:09:26,990 --> 01:09:30,750
+of X وهذه بتطلع ناقص X وهذه بتطلع X فالمعمل ال
+
+824
+01:09:30,750 --> 01:09:35,990
+integration هدول الآن بدي 
+
+825
+01:09:35,990 --> 01:09:43,300
+أجيب الثانية فضلي ال S of Tهذه وهذه فاضليها هذه
+
+826
+01:09:43,300 --> 01:09:48,460
+فاضليها بنقل صفر عند X بتطلع اللي هي بين هذه وهذه
+
+827
+01:09:48,460 --> 01:09:53,660
+هيتفاضلوا لهذه وهذه الآن بدنا اللي هو نجيب الواحد
+
+828
+01:09:53,660 --> 01:09:58,280
+ناقص X تربيع اللي هو هذا قيمته اللي هو عبارة عن
+
+829
+01:09:58,280 --> 01:10:02,600
+C في X بساوي واحد ناقص هذه بنضرب الناقص وبنجمع
+
+830
+01:10:02,600 --> 01:10:08,280
+الجهتين واحد بطلع عندي اللي هي ال inequality اللي
+
+831
+01:10:08,280 --> 01:10:13,190
+عندي هذه زي ما قلنا ضربنا في ناقص وجمعنا واحد طلعت
+
+832
+01:10:13,190 --> 01:10:16,590
+هذه أو عوضنا هذه مكان هذه تعويض عادي وبعدين ضربنا
+
+833
+01:10:16,590 --> 01:10:20,690
+في ناقص فبصير عندي هذه المقدار اللي عندي c of x
+
+834
+01:10:20,690 --> 01:10:24,310
+أكبر سواء هذه وناقص هذه واللي بتستخدم الثانية
+
+835
+01:10:24,310 --> 01:10:27,670
+بتطلع نفس الشيء الموضوع موضوع حسابات بحت عشان هيك
+
+836
+01:10:27,670 --> 01:10:31,510
+بحت عشان هيك مافيش داعي للتكميل وأنتم بتكملوا
+
+837
+01:10:31,510 --> 01:10:35,150
+باقي الحسابات اللي هي المطلوبة في هذا اللي هي
+
+838
+01:10:35,150 --> 01:10:42,700
+النظرية الآن الجزء المتبقي هذا حبيته تطلعوا عليه
+
+839
+01:10:42,700 --> 01:10:46,880
+لكن اللي هو مش مطلوب من ضمن حديثنا وهيك ممكن
+
+840
+01:10:46,880 --> 01:10:52,700
+نخلصنا اللي هو section ثمانية أربعة متسمة سبتر
+
+841
+01:10:52,700 --> 01:10:58,420
+ثمانية هذا اللي هو الجزء الثالث من المادة الجزء
+
+842
+01:10:58,420 --> 01:11:00,160
+الأول كان ال differentiation والتاني ال
+
+843
+01:11:00,160 --> 01:11:01,960
+integration والتالت اللي هو point twice
+
+844
+01:11:01,960 --> 01:11:05,700
+convergence المرة الجاية إن شاء الله اللي هو بنبدأ
+
+845
+01:11:05,700 --> 01:11:09,880
+في الجزء الرابع من المادة اللي هو ال series وإن
+
+846
+01:11:09,880 --> 01:11:16,590
+شاء الله بقدر الإمكان هأحكي الجزء الخامس في .. من
+
+847
+01:11:16,590 --> 01:11:20,510
+المادة اللي هو عبارة عن اللي هو Topology in R أو
+
+848
+01:11:20,510 --> 01:11:23,570
+اللي هو اللي هي العلاقة بين اللي هي Topological
+
+849
+01:11:23,570 --> 01:11:27,410
+Spaces وNormed Spaces وHilbert Spaces إلى آخره
+
+850
+01:11:27,410 --> 01:11:33,530
+ويمكن بتكون هي اتجاوزت خلينا نقول حد ال .. ال .. ال
+
+851
+01:11:33,530 --> 01:11:37,150
+.. المطلوب في المادة لكن على أساس إنه لو يكون
+
+852
+01:11:37,150 --> 01:11:42,210
+التصوير كامل للو .. للوصف اللي أنا اقتراحته أو اللي
+
+853
+01:11:42,210 --> 01:11:47,230
+هو الوصف اللي مقترح في الجسم اللي هو لعين نهاية ال
+
+854
+01:11:47,230 --> 01:11:49,990
+series وطبعا هذا الكلام قلنا احنا فيه له سبب لما
+
+855
+01:11:49,990 --> 01:11:53,450
+بدأنا في الأول إنه ال series بتاخدوها في advanced
+
+856
+01:11:53,450 --> 01:11:56,990
+calculus واحد فعشان هيك يمكن اللي هو
+
+857
+01:12:00,350 --> 01:12:03,810
+هيكون منطقي أنه ما نأخذش ال series ونأخذ بدلها
+
+858
+01:12:03,810 --> 01:12:07,110
+اللي هو ال topology إنار أو اللي هو العلاقة بين ال
+
+859
+01:12:07,110 --> 01:12:10,550
+topological spaces وال metric spaces وإلى لقاء و
+
+860
+01:12:10,550 --> 01:12:12,650
+السلام آخر والسلام عليكم ورحمة الله وبركاته 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..72f3c9eb77d499a0c414bd335aaf4a26e8b110db
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_postprocess.srt
@@ -0,0 +1,3440 @@
+1
+00:00:05,090 --> 00:00:08,030
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:08,030 --> 00:00:11,070
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه
+
+3
+00:00:11,070 --> 00:00:18,250
+أجمعين هذه المحاضرة رقم 24 مساق تحليل حقيقة 2 طلاب
+
+4
+00:00:18,250 --> 00:00:22,650
+وطالبات الجامعة الإسلامية قسم رياضيات كلية العلوم
+
+5
+00:00:23,470 --> 00:00:26,630
+اليوم هنكمل ان شاء الله ال section الأخير في
+
+6
+00:00:26,630 --> 00:00:29,870
+chapter 8 اللي هو 8 أربعة تحت عنوان the
+
+7
+00:00:29,870 --> 00:00:35,170
+trigonometric functions وهو أيضا اللي هو section
+
+8
+00:00:35,170 --> 00:00:42,210
+أو موضوع تطبيق على اللي هوالـ pointwise and
+
+9
+00:00:42,210 --> 00:00:46,030
+uniform convergence للـ sequence of functions وكيف
+
+10
+00:00:46,030 --> 00:00:50,510
+بدنا نعرف بطريقة مشابهة جدا لمعرفناها المرة
+
+11
+00:00:50,510 --> 00:00:53,330
+الماضية أو اللي قبلها بخصوص الـ exponential
+
+12
+00:00:53,330 --> 00:00:58,710
+function هنعرف اليوم اللي هو بنفس الطريقة كيف نعرف
+
+13
+00:00:58,710 --> 00:01:03,510
+اللي هو الـ sine و الـ cosine as a limit اللي هو
+
+14
+00:01:03,510 --> 00:01:07,390
+of a uniformly convergent sequence of functions
+
+15
+00:01:07,950 --> 00:01:11,230
+العنوان is in the trigonometric functions اللي هو
+
+16
+00:01:11,230 --> 00:01:15,450
+section 8 أربعة النظرية
+
+17
+00:01:15,450 --> 00:01:21,950
+الأولى اللى عندنا اللى مشابهة لنظرية سابقة اللى هي
+
+18
+00:01:21,950 --> 00:01:25,870
+ال exponential اللى من خلالها بدنا نصل لاللي هو
+
+19
+00:01:25,870 --> 00:01:31,370
+تعريف اللى هي ال cosine و ال sineالنظرية بتقول
+
+20
+00:01:31,370 --> 00:01:35,190
+مالي there exist functions يوجد دوال الأمنان في
+
+21
+00:01:35,190 --> 00:01:40,570
+بتوجود دوال C من R ل R and S من R ل R such that
+
+22
+00:01:40,570 --> 00:01:44,690
+اللي هي طبعا مستقبلا هنتسمي ال C اللي هي ال cosine
+
+23
+00:01:44,690 --> 00:01:48,910
+ومستقبلا هنتسمي ال S اللي هي ال sine اللي احنا
+
+24
+00:01:48,910 --> 00:01:54,090
+بنعرفهاالان بنقول في ده اللي تاني واحدة اسمها c
+
+25
+00:01:54,090 --> 00:01:59,950
+واحدة s تحقق ما يلي اللي هي cw prime of x بيساوي
+
+26
+00:01:59,950 --> 00:02:04,870
+ناقص c of x s w prime of x بيساوي ناقص s of x ولو
+
+27
+00:02:04,870 --> 00:02:08,730
+استذكرت انت ال sine و ال cosine دي ما سيحدث لاحقا
+
+28
+00:02:08,730 --> 00:02:12,270
+طبعا لو استذكرت ال cosine لما ان .. ان .. ان .. ان
+
+29
+00:02:12,270 --> 00:02:16,130
+فاضلها مرتين هتصير اللي هي ناقص اللي هي نفسها و لو
+
+30
+00:02:16,130 --> 00:02:19,570
+فاضلت ال sine برضه فاضلها مرتين هتلاقيها ايش هي
+
+31
+00:02:19,960 --> 00:02:23,840
+بتطلع سالب S الخاصية التانية اللي هي C of Zero
+
+32
+00:02:23,840 --> 00:02:27,780
+بساوة واحد ولو استذكرت الكوسين كوسين ال Zero بساوة
+
+33
+00:02:27,780 --> 00:02:31,240
+واحد ولو استذكرت الكوسين لما تاخدها ال derivative
+
+34
+00:02:31,240 --> 00:02:35,160
+هتصير عبارة عن سالب Sin عند Zero هتطلع H بساوة
+
+35
+00:02:35,160 --> 00:02:38,960
+Zero and S of Zero بساوة Zero وS prime of Zero
+
+36
+00:02:38,960 --> 00:02:45,330
+بساوة واحد يعني بمعنى أخر يوجد دالتين الآندالتين
+
+37
+00:02:45,330 --> 00:02:49,770
+واحدة من C لـ A اسمها S واحدة اسمها C من R ل R
+
+38
+00:02:49,770 --> 00:02:56,690
+تحقق الشرطين التاليين CW' بسوء ناقص C وSW' بسوء
+
+39
+00:02:56,690 --> 00:03:01,330
+ناقص S على كل R وC عند الـ 0 هي 1 وC' عند الـ 0
+
+40
+00:03:01,330 --> 00:03:05,650
+بسوء 0 وS عند الـ 0 بسوء 0 وS' عند الـ 0 بسوء 1
+
+41
+00:03:05,650 --> 00:03:11,250
+وبعد شوية هقولك لا يوجد في الدنيا دالتين بحقق ان
+
+42
+00:03:11,250 --> 00:03:17,300
+الشروط هذولةإلا هي واحدة اسمها C واحدة اسمها S
+
+43
+00:03:17,300 --> 00:03:22,640
+يعني التنتين واحدات وهذا يجعلنا نسميهم التسمية بعد
+
+44
+00:03:22,640 --> 00:03:26,380
+ذلك واحدة اسمها cosine واحدة اسمها sine ومن ثم
+
+45
+00:03:26,380 --> 00:03:29,320
+بنجيب كل الخواص اللي احنا بنعرفه عن ال sine و ال
+
+46
+00:03:29,320 --> 00:03:34,020
+cosine من هذا البناء إذن الآن أنا ببني ببني ببني
+
+47
+00:03:34,020 --> 00:03:36,980
+وجود ال sine و ال cosine و بعد شوية ببني اللي هو
+
+48
+00:03:36,980 --> 00:03:40,960
+ال uniqueness طبقا لهذه الشروط اللي موجودة عندى
+
+49
+00:03:42,380 --> 00:03:45,860
+الان بكيف بدنا نعمل زي ما عملنا المرة الماضية في
+
+50
+00:03:45,860 --> 00:03:48,340
+ال .. في ال .. اللي هو الـ Exponential عشانك
+
+51
+00:03:48,340 --> 00:03:52,060
+هتلاقوني شوية مسرع لإن اللي بيحضر اللي هي محاضرة
+
+52
+00:03:52,060 --> 00:03:56,620
+الـ Exponential هيلاقي إن هذا في كتير من الحديث في
+
+53
+00:03:56,620 --> 00:04:00,640
+إعادة We define the sequence cn as n of continuous
+
+54
+00:04:00,640 --> 00:04:05,240
+functions inductively as بدنا نعرف اللي هي .. اللي
+
+55
+00:04:05,240 --> 00:04:09,640
+هو two sequences واحدة نسميها cn وواحدة نسميها sn
+
+56
+00:04:10,270 --> 00:04:13,890
+كيف بدنا نعرفها؟ زي ما عرفنا الـ exponential بنعرف
+
+57
+00:04:13,890 --> 00:04:19,250
+C1 of X إيش بتساوي واحد وS1 of X إيش بدنا نسميها
+
+58
+00:04:19,250 --> 00:04:26,490
+بساوي X الآن S2 of X S2 of X هيساوي ال integration
+
+59
+00:04:26,490 --> 00:04:31,990
+من صفر لعند X C2 of T DT طب C2 من وين أجيبها؟ C2
+
+60
+00:04:31,990 --> 00:04:36,050
+بتجيبها من هنا C اللي هي واحد زائد واحد يعني C2
+
+61
+00:04:36,050 --> 00:04:41,470
+بساوي واحد ناقص ال integration من صفر ل Xلـ S1 of
+
+62
+00:04:41,470 --> 00:04:48,870
+T اللي هذا DT فبكون جيبت ال C2 و بتجيبت ال S2 من
+
+63
+00:04:48,870 --> 00:04:55,530
+ال C2 لأن S3 و C3 بنفس الطريقة in general اللي أنا
+
+64
+00:04:55,530 --> 00:04:59,170
+عملت sequence of functions اللي هو بدأت اللي هي ال
+
+65
+00:04:59,170 --> 00:05:05,120
+C1 ب1 S1 ب Xومن ثم S N بتساوي من صفر ل X
+
+66
+00:05:05,120 --> 00:05:10,760
+integration C N of T DT يعني كمالة C N of T DT وC
+
+67
+00:05:10,760 --> 00:05:13,460
+N زائد واحد of X بتساوي واحد نقص integration من
+
+68
+00:05:13,460 --> 00:05:18,700
+صفر ل X S N of T DT الآن هو بيقول اللي هو هذه
+
+69
+00:05:18,700 --> 00:05:21,820
+sequence of continuous functions طب sequence of
+
+70
+00:05:21,820 --> 00:05:25,840
+continuous functions هذا الكلام بده إثباتطيب، الان
+
+71
+00:05:25,840 --> 00:05:29,500
+عندى اللى هو by induction زى ما انتوا عارفين الان
+
+72
+00:05:29,500 --> 00:05:33,600
+عندى الـC1 continuous لان الثابتة S1 continuous
+
+73
+00:05:33,600 --> 00:05:38,880
+لان هى شمالها بتساوي X بناء عليه هتطلع عندى اللى
+
+74
+00:05:38,880 --> 00:05:44,920
+هى C2 continuous ومن ثم C3 وC4 الاخرين الان لو
+
+75
+00:05:44,920 --> 00:05:49,000
+بدنا نثبتها by induction بدنا نفترض انه هذولة
+
+76
+00:05:49,000 --> 00:05:52,560
+الـSn والـCn
+
+77
+00:05:53,920 --> 00:06:00,700
+continuous سنثبت هنا continuous لأن S1 ايش بتساوي
+
+78
+00:06:00,700 --> 00:06:06,860
+X؟ C1 ايش بتساوي 1؟ continuous إذا صارت هذه اللي
+
+79
+00:06:06,860 --> 00:06:12,340
+هي الجملة is true for N بتساوي 1 نفترض الآن
+
+80
+00:06:12,340 --> 00:06:19,900
+supposeby induction بتهدي suppose that الهو star
+
+81
+00:06:19,900 --> 00:06:25,960
+هذي is true for n ايش بتساوي n بتساوي k معناته
+
+82
+00:06:25,960 --> 00:06:33,820
+صارت ال S Kوالـ CK are continuous بتثبت الآن من
+
+83
+00:06:33,820 --> 00:06:38,520
+الصحيحة لـ K زائد واحد يعني بتثبت اللي هو CK زائد
+
+84
+00:06:38,520 --> 00:06:42,700
+واحد و SK زائد واحد ان هنا شمال هنا continuous طيب
+
+85
+00:06:42,700 --> 00:06:48,000
+الآن شوف CK زائد واحد بتساوي حسب اللي هي عندي هان
+
+86
+00:06:48,000 --> 00:06:51,740
+ايش بتساوي اللي هو عبارة عن CK واحد of X بتساوي ال
+
+87
+00:06:51,740 --> 00:06:58,020
+integration واحد نقص ال integration من صفر X Sك ..
+
+88
+00:06:58,020 --> 00:07:03,020
+هذا كز واحد .. هذا ك .. of DT طيب أنا مفترض أن S K
+
+89
+00:07:03,020 --> 00:07:06,600
+من ال hypothesis induction إنها continuous إذا
+
+90
+00:07:06,600 --> 00:07:11,040
+صارت هذه كلها إيش مالها؟ اللي هي S K integrable
+
+91
+00:07:11,040 --> 00:07:15,600
+وصارت هذه كلها على بعض by fundamental theorem of
+
+92
+00:07:15,600 --> 00:07:20,980
+calculus اللي هي ال derivative إلها موجودة إذا
+
+93
+00:07:20,980 --> 00:07:23,440
+صارت هذه كلها ال derivative إلها وموجودة بالنسبة
+
+94
+00:07:23,440 --> 00:07:27,680
+لل Xإذا صارت مدام هيك هاذ is differentiable إذا
+
+95
+00:07:27,680 --> 00:07:30,940
+continuous إذا صارت اللي هي ck زائد واحد
+
+96
+00:07:30,940 --> 00:07:35,200
+continuous اللي ثبت الآن is sk زائد واحد sk زائد
+
+97
+00:07:35,200 --> 00:07:39,280
+واحد of x أيش بتساوي حسب اللي التعريف بتساوي ال
+
+98
+00:07:39,280 --> 00:07:47,660
+integration من صفر العند x ck of tزايد واحد هذا N
+
+99
+00:07:47,660 --> 00:07:52,620
+هذا N K زايد واحد K زايد واحد DT وانا مثبت فوق انه
+
+100
+00:07:52,620 --> 00:07:56,160
+CK زايد واحد is continuous إذا صارت هذه كلها على
+
+101
+00:07:56,160 --> 00:07:59,500
+بعض integrable اللي هي CK زايد واحد صار هذا ال
+
+102
+00:07:59,500 --> 00:08:02,000
+integration exist ومش هيك by fundamental theorem
+
+103
+00:08:02,000 --> 00:08:04,720
+of calculus برضه ال derivative له إيه شمالها
+
+104
+00:08:04,720 --> 00:08:07,700
+موجودة إذا صارت هذه ال differentiable إذا
+
+105
+00:08:07,700 --> 00:08:11,060
+continuous إذا صارت CK زايد واحد وSK زايد واحد are
+
+106
+00:08:11,060 --> 00:08:16,160
+continuous إذا صارت الجملة هذه صحيحةلأ K زائد واحد
+
+107
+00:08:16,160 --> 00:08:21,000
+إذا صارت صحيحة دائما إذا صارت عندي الان مفرغ منه
+
+108
+00:08:21,000 --> 00:08:26,360
+ال CN والSN are continuous functions وبناء عليه
+
+109
+00:08:26,360 --> 00:08:29,860
+مدام continuous functions by fundamental theorem
+
+110
+00:08:29,860 --> 00:08:35,460
+of calculus هتكون هذه ال SN differentiable و ال CN
+
+111
+00:08:35,460 --> 00:08:39,800
+زائد واحد differentiable ومش هيك كمان و هيعطيني ال
+
+112
+00:08:39,800 --> 00:08:43,440
+SN prime of X حسب ال fundamental theorem of
+
+113
+00:08:43,440 --> 00:08:46,820
+calculusالـ differentiation بتضايق ال integration
+
+114
+00:08:46,820 --> 00:08:54,580
+بتضال بساوي cn of x الان و ال derivative لهذه برضه
+
+115
+00:08:54,580 --> 00:08:59,900
+exist cn زائد واحد prime of x اللي هو هي ساوي اللي
+
+116
+00:08:59,900 --> 00:09:10,170
+هو مين هتطلع ناقص snof X إذا صار عندي الآن اللي هي
+
+117
+00:09:10,170 --> 00:09:15,610
+ال sequence اللي عندي صارت well defined كلها و
+
+118
+00:09:15,610 --> 00:09:20,270
+continuous كلها و لأ و differentiable كمان ال SN
+
+119
+00:09:20,270 --> 00:09:25,150
+prime of X بيساوي CN of X و CN زائد واحد prime of
+
+120
+00:09:25,150 --> 00:09:31,730
+X بيساوي ناقص SN of X زي ما أنا أثبتها و أوضحت لكم
+
+121
+00:09:31,730 --> 00:09:38,710
+إياها هناالآن induction arguments برضه induction
+
+122
+00:09:38,710 --> 00:09:42,970
+arguments بقول لي we leave this argument for you
+
+123
+00:09:42,970 --> 00:09:46,890
+خلينا نشوفها مع بعض ماهي اللي بنقصده زي ما عملت
+
+124
+00:09:46,890 --> 00:09:49,870
+بالظبط لو طلعت على الخطوات هتلاقيها مشابه لخطوات
+
+125
+00:09:49,870 --> 00:09:54,970
+تبعات ال exponential عند ال Sn of X زي ما قلنا
+
+126
+00:09:54,970 --> 00:10:02,060
+بسوء ال integration من 0 ل X Cn of T dtC N زائد
+
+127
+00:10:02,060 --> 00:10:06,140
+واحد of X بساوة واحد مقص ال integration من C ل X S
+
+128
+00:10:06,140 --> 00:10:15,360
+N of T DT ومعطينا طبعا احنا اخدنا ال C واحد of X
+
+129
+00:10:15,360 --> 00:10:22,530
+بساوة واحدو الـ c و s1 of x بسوة x هذه اللي هي ال
+
+130
+00:10:22,530 --> 00:10:25,370
+sequence of functions اللي أثبتناها هذه ال
+
+131
+00:10:25,370 --> 00:10:27,990
+sequence of functions ال sn و ال cn زائد واحد
+
+132
+00:10:27,990 --> 00:10:32,230
+التنتين are continuous for every n و همش هي
+
+133
+00:10:32,230 --> 00:10:34,850
+codifferentiable و ال derivative إلها زي ما قلنا
+
+134
+00:10:34,850 --> 00:10:39,430
+sn prime of x بسوة cn of x و ال cn زائد واحد prime
+
+135
+00:10:39,430 --> 00:10:44,190
+of x بسوة ناقص sn of x و خلّينا نسجلها هذه لإنه
+
+136
+00:10:44,190 --> 00:10:52,370
+هنحتاجها بعد شويةاللي هي S N prime of X بساوي
+
+137
+00:10:52,370 --> 00:11:00,310
+C N of X and C N زائد واحد prime of X بساوي ناقص S
+
+138
+00:11:00,310 --> 00:11:09,070
+N of X طيب الآن بدنا اللي هو by induction نثبت
+
+139
+00:11:09,070 --> 00:11:12,570
+اللي هو C N زائد واحد of X بساوي اللي هو اللي
+
+140
+00:11:12,570 --> 00:11:18,460
+أمامي هذا طبعا أكيد اللي هوبعضكم قال ما هي by
+
+141
+00:11:18,460 --> 00:11:24,960
+induction نتطلع على C2 of X C1 في القرآن بتساوي
+
+142
+00:11:24,960 --> 00:11:31,720
+واحد نثبتها يعني الان C2of X ايش بتساوي حسب
+
+143
+00:11:31,720 --> 00:11:35,960
+القانون بساوي واحد ماقص ال integration من صفر ل X
+
+144
+00:11:35,960 --> 00:11:41,020
+أس واحد لأنه بواحد هنا أس واحد اللي هي جدش أس واحد
+
+145
+00:11:41,020 --> 00:11:46,060
+ال X integration اللي هي TDT ويساوي التفاضل اللي
+
+146
+00:11:46,060 --> 00:11:49,820
+هذي بيصير واحد ماقص X تربيع على مين على اتنين إذا
+
+147
+00:11:49,820 --> 00:11:54,410
+فعلا اللي هي Cاللي هي هذه اتنين is true for any
+
+148
+00:11:54,410 --> 00:11:58,870
+ايش بتساوي ان بتساوي واحد بدنا نثبت التانية معها
+
+149
+00:11:58,870 --> 00:12:01,730
+اللي هي true for any بتساوي واحد لإنها دي جملة
+
+150
+00:12:01,730 --> 00:12:05,690
+واحدة بدأ أثبتها انها true for every any الان for
+
+151
+00:12:05,690 --> 00:12:10,590
+any بتساوي واحد بيصير عندي أس اتنين of x بدأ أتأكد
+
+152
+00:12:10,590 --> 00:12:15,990
+انه اللي هي بتحقق اللي هي ال .. اللي .. ايه هذه ال
+
+153
+00:12:15,990 --> 00:12:19,960
+equationS2 of X بيساوي من وين بده اجيبها من
+
+154
+00:12:19,960 --> 00:12:23,480
+التعريف اللي فوق بيساوي ال integration من سفل ل X
+
+155
+00:12:23,480 --> 00:12:29,180
+C2 of X ايش C2 of X هايه اللي هو واحد نقص T تربيع
+
+156
+00:12:29,180 --> 00:12:34,800
+على اتنين لكل ما له DT ويساوي الكملة هذا بيصير
+
+157
+00:12:34,800 --> 00:12:40,200
+عبارة عن Xناقص X تكعيب على تلاتة وفيه اتنين بيصير
+
+158
+00:12:40,200 --> 00:12:44,980
+X تكعيب على مين على تلاتة factorial اللي هي جداش
+
+159
+00:12:44,980 --> 00:12:50,380
+اللي هي ستة إذا فعلا فعلا طلع عندي اللي هو C أس
+
+160
+00:12:50,380 --> 00:12:55,400
+اتنين هي عبارة عن انا بواحد يعني أس اتنين بيساوي X
+
+161
+00:12:55,400 --> 00:12:59,380
+ناقص X تكعيب على تلاتة factorial يعني صارت الجملة
+
+162
+00:12:59,380 --> 00:13:02,900
+هذه برضه صحيحة فور أن بيساوي واحد إذا كلها على بعض
+
+163
+00:13:02,900 --> 00:13:09,650
+هذه صارت صحيحة فور أن بتساوي جداشالان بدنا نفترض
+
+164
+00:13:09,650 --> 00:13:13,430
+انها
+
+165
+00:13:13,430 --> 00:13:19,490
+true for n بتساوي k ونجيب منها مين اللي هو k زائد
+
+166
+00:13:19,490 --> 00:13:25,950
+واحد نفترض suppose that this is true for n بتساوي
+
+167
+00:13:25,950 --> 00:13:30,690
+k يعني بمعنى اخر صار عندى ck زائد واحد بساوي هذا
+
+168
+00:13:30,690 --> 00:13:37,260
+وهنا اتنين k وهنا kوهنا اتنين K وهنا نفس الاشي
+
+169
+00:13:37,260 --> 00:13:41,640
+اللي هو نفترض انها صحيحة for N بتساوي K صارت عبارة
+
+170
+00:13:41,640 --> 00:13:48,460
+عن اتنين K و اتنين K أو اس K ماشي الحال نثبت ان
+
+171
+00:13:48,460 --> 00:13:53,480
+هذه صحيحة for من؟ for K بتساوي .. for N بتساوي
+
+172
+00:13:53,480 --> 00:14:02,020
+كدهش؟ K زائد واحد يعني بدي اديب Cك زائد جداش اتنين
+
+173
+00:14:02,020 --> 00:14:07,480
+لأن في الاصل هي ك زائد واحد في الاصل سن زائد واحد
+
+174
+00:14:07,480 --> 00:14:11,620
+فرضتها صحيحة في القرآن بتساوي ك لأن بتثبتها صحيحة
+
+175
+00:14:11,620 --> 00:14:15,520
+لأن بتساوي ك زائد واحد يعني ك زائد واحد واحد بيصير
+
+176
+00:14:15,520 --> 00:14:22,820
+ك زائد اتنين of Xماذا يعني حساب تعريفها؟ تعريفها 1
+
+177
+00:14:22,820 --> 00:14:29,300
+ناقص الانتجريشن من 0 إلى X هذا K زائد 2 إذا هذا K
+
+178
+00:14:29,300 --> 00:14:34,420
+N زائد 1 وهذا N K زائد 2 ماذا؟ هذا سيصبح SK زائد 1
+
+179
+00:14:34,420 --> 00:14:36,340
+of DT
+
+180
+00:14:38,530 --> 00:14:42,870
+و يساوي واحد ناقص الفكرة انت فاهمت من صفر لمين
+
+181
+00:14:42,870 --> 00:14:47,910
+لإكس مين هكامل هكامل S K زائد واحد انا فرضت ان انا
+
+182
+00:14:47,910 --> 00:14:52,750
+اتروه في القرآن بالساوية K اللي هي عبارة عن T ناقص
+
+183
+00:14:52,750 --> 00:15:00,450
+T تكعيب ع تلاتة factorial لما اصل لآخر واحد زائد
+
+184
+00:15:00,450 --> 00:15:12,390
+ناقص واحد أس K فيهT 2K 1 2 T 1 كل ايه اش مالها
+
+185
+00:15:12,390 --> 00:15:16,110
+factorial الكل دي تي حسابات والله يا جماعة اللي
+
+186
+00:15:16,110 --> 00:15:22,660
+حالكم بتعملوها دي تي و يساويواحد ناقص نفتح جوس هذي
+
+187
+00:15:22,660 --> 00:15:26,700
+T بيصير T تربيع على اتنين factorial هذي ايش بيصير
+
+188
+00:15:26,700 --> 00:15:32,820
+ناقص T أُس أربعة على أربعة فى تلاتة factorial عن
+
+189
+00:15:32,820 --> 00:15:36,980
+أربعة factorial زائد لما أصل الأخر واحد ناقص واحد
+
+190
+00:15:36,980 --> 00:15:43,580
+أُس K زي ما هي لأن إشارة هذي T بيصير اتنين K زائد
+
+191
+00:15:43,580 --> 00:15:49,130
+واحد زائد واحد يعني زائد اتنين علىاللي هو اتنين K
+
+192
+00:15:49,130 --> 00:15:54,490
+زائد اتنين في هذا بتطلع اتنين K زائد اتنين الكل
+
+193
+00:15:54,490 --> 00:15:59,550
+ايه شماله؟ factorial ماشي الحال هذا طبعا كله من
+
+194
+00:15:59,550 --> 00:16:04,630
+صفر ل X إذا بتصير هذا عبارة عن X وهذا X وهذا
+
+195
+00:16:04,630 --> 00:16:09,010
+الأخير برضه ايه؟ X اللي هي هذا عبارة عن ايه؟ هو
+
+196
+00:16:09,010 --> 00:16:16,350
+يساوي1 ناقص x تربيع على 2 factorial زائد x أُس 4
+
+197
+00:16:16,350 --> 00:16:20,310
+على 4 factorial ضربت الناقص جوا لإن أنا لما أصل
+
+198
+00:16:20,310 --> 00:16:24,910
+ناقص لما أصل الأخر واحد زائد ناقص واحد K وناقص أنا
+
+199
+00:16:24,910 --> 00:16:31,050
+بصير K زائد 1 في X أُس 2K
+
+200
+00:16:31,910 --> 00:16:37,130
+زائد اتنين على اتنين K زائد اتنين لكل vector يعني
+
+201
+00:16:37,130 --> 00:16:41,890
+صارت هذه CK زائد واحد زائد اتنين of X بتساوي هذا
+
+202
+00:16:41,890 --> 00:16:49,290
+المقدار هذا المقدار صحيح صار بالظبط هو هذا اللي هو
+
+203
+00:16:49,290 --> 00:16:53,830
+المقدار اللي عندي أثبتت و كإلمي يعني الآن ان هذا
+
+204
+00:16:54,470 --> 00:16:59,490
+اللي عندى is true for mean for k زائد لأ واحد لإنه
+
+205
+00:16:59,490 --> 00:17:03,950
+لما احط مكان n بيساوي k زائد واحد بيصير هذه k زائد
+
+206
+00:17:03,950 --> 00:17:09,030
+اتنين اشوف اللى طلعته صح ولا لأ بيساوي تبحانة واحد
+
+207
+00:17:09,030 --> 00:17:12,990
+ناقص x تربيع على اتنين فيكتوريا لما اثر الاخر واحد
+
+208
+00:17:12,990 --> 00:17:17,790
+اللى هي ناقص واحد قص مين k زائد واحد فكس قص اتنين
+
+209
+00:17:17,790 --> 00:17:20,970
+في k زائد واحد يعني اتنين k زائد اتنين وها اتنين k
+
+210
+00:17:20,970 --> 00:17:25,760
+زائد اتنين كل فيكتورياالان صارت هذه صحيحة for n
+
+211
+00:17:25,760 --> 00:17:30,600
+ايش بتساوي k زائد واحد التاني بنفس الأسلوب بثبتها
+
+212
+00:17:30,600 --> 00:17:35,260
+صحيحة for ايش for k زائد واحد يعني بدي احسب مين يا
+
+213
+00:17:35,260 --> 00:17:42,300
+جماعة بدي احسب Sكزايد اتنين ايش حسب اللي فوق
+
+214
+00:17:42,300 --> 00:17:47,180
+بتساوي بساوي ال integration من سفر ل X ل CK زايد
+
+215
+00:17:47,180 --> 00:17:54,020
+اتنين of T DT و باجي بعوضها هنا و بكملها و بتطلع
+
+216
+00:17:54,020 --> 00:17:57,800
+عندى بالظبط ال formula هذه يعني صح بيصير عندى
+
+217
+00:17:57,800 --> 00:18:01,620
+بتعملها لحالك لان حسابات نفس الأسلوب بتطلع عندى
+
+218
+00:18:01,620 --> 00:18:06,310
+اللي هي هذه صحيحة for mean برضهfor an بتساوي k
+
+219
+00:18:06,310 --> 00:18:11,050
+زائد واحد إذا هذا صار المقدار صحيح دائما for mean
+
+220
+00:18:11,050 --> 00:18:20,430
+for any k for any n element in n الان واضح ان
+
+221
+00:18:20,430 --> 00:18:26,310
+الخطوات مشابهة لخطوات ال exponential نيجي الآن
+
+222
+00:18:26,310 --> 00:18:28,970
+قصّل n prime
+
+223
+00:18:38,770 --> 00:18:45,020
+نجي الآنوين رايح؟ زي اللي بكونت رايحي الأيام الـ
+
+224
+00:18:45,020 --> 00:18:49,060
+exponential هثبتلك أن الـ sequence هذه converged
+
+225
+00:18:49,060 --> 00:18:52,220
+uniformly وهذه طبعا هتصبح converged uniformly
+
+226
+00:18:52,220 --> 00:18:55,000
+automatic فهتصبح اللي هي differentiable لأنه بيصير
+
+227
+00:18:55,000 --> 00:18:57,860
+طبقة النظرية اللي هي تبعها الـ differentiability
+
+228
+00:18:57,860 --> 00:19:01,200
+بيصير عند مادام differentiable اللي هي ال
+
+229
+00:19:01,200 --> 00:19:04,580
+derivative اللي لها exist و هتكون ال derivative
+
+230
+00:19:04,580 --> 00:19:07,960
+متحقق الشروط و بيكون خلاص نعم نشوف أش بقول طيب
+
+231
+00:19:07,960 --> 00:19:12,530
+الآنبعد ما طلعنا هدولة let a أكبر من صفر b given
+
+232
+00:19:12,530 --> 00:19:15,790
+نفس الخطوات تبعت ال exponential then if الabsolute
+
+233
+00:19:15,790 --> 00:19:19,970
+value of x أصغر بسوء a and m أكبر من n أكبر من 2a
+
+234
+00:19:19,970 --> 00:19:25,510
+يعني بتداخد اللي هي الامات لأكبر من n و أكبر من 2a
+
+235
+00:19:25,510 --> 00:19:29,650
+يعني الآن أنا بشتغل عالين يا جماعة الفترة من ناقص
+
+236
+00:19:29,650 --> 00:19:34,570
+a لعند a وأخدت ال a arbitrarily أكبر من صفر لكن
+
+237
+00:19:34,570 --> 00:19:41,850
+fixed الان we haveعندي اللي هو ..عندي اللي هي الـ
+
+238
+00:19:41,850 --> 00:19:50,630
+A الـ A على اللي هي 2N بما أنه N أكبر من 2A اجسم
+
+239
+00:19:50,630 --> 00:19:56,790
+الجهتين على 2N هن على 2N و هن على 2N بيصير عند الـ
+
+240
+00:19:56,790 --> 00:20:07,710
+A على N .. A على N أصغر من النص، صح؟هي a تروح
+
+241
+00:20:07,710 --> 00:20:11,090
+التنين مع التنين وهذه n مع ال n بصير a أعليها أصغر
+
+242
+00:20:11,090 --> 00:20:16,050
+من نص يعني a على 2n أصغر من 1 على 4 جسمت التنين
+
+243
+00:20:16,050 --> 00:20:22,400
+عالميا على 2 إذا لما تكون ال n أكبر من 2aهتطلع
+
+244
+00:20:22,400 --> 00:20:24,640
+عندها get بتعرف ليش هذه لإن بتلزمنا في الحسابات
+
+245
+00:20:24,640 --> 00:20:28,580
+بعد شوية هتكون اللي هي لما الاتنين a أصغر من n
+
+246
+00:20:28,580 --> 00:20:31,960
+بتكون عند a على اتنين n أصغر من مين من ربع المرة
+
+247
+00:20:31,960 --> 00:20:35,460
+اللي فاتت كان لازمنا a على n أصغر من نص وكملنا ال
+
+248
+00:20:35,460 --> 00:20:40,650
+series اللي متذكر اللي عملنا في اللي هيالـ
+
+249
+00:20:40,650 --> 00:20:43,330
+Exponential أنا بحكيش تفاصيله و مفترض ان انتوا
+
+250
+00:20:43,330 --> 00:20:47,570
+فاهمين حسب حكينا اللي هو في الـ Exponential الان
+
+251
+00:20:47,570 --> 00:20:52,290
+للأمات اللي أكبر من M أكبر من 2A بدي احسب الـCM
+
+252
+00:20:52,290 --> 00:20:59,870
+ناقص 2 الـSN عشان تظلها جدامكم اللي فوق هذه عندى
+
+253
+00:20:59,870 --> 00:21:06,960
+الان الـCMالـ C M الـ C M شايفينها؟ بتظلها ماشية
+
+254
+00:21:06,960 --> 00:21:10,940
+واحد لو ف .. الآن هذه M بدأ الـ N زاد واحد ايش
+
+255
+00:21:10,940 --> 00:21:14,880
+اسمها؟ M بتظلها ماشية واحد ناقص X أربع على اتنين
+
+256
+00:21:14,880 --> 00:21:18,820
+فيكتوريال X أربع على أربع فيكتوريال والـ M أكبر من
+
+257
+00:21:18,820 --> 00:21:23,160
+الـ N هتجيه يقبل في طريقها من الـ N الـ N بيصير
+
+258
+00:21:23,160 --> 00:21:27,540
+الـ N طبعا إيه شمالها؟ الـ N عبارة عن في جبل ال
+
+259
+00:21:27,540 --> 00:21:33,590
+term هذا اللي هو الـ N ناقص واحدأه فبصير زائد اللي
+
+260
+00:21:33,590 --> 00:21:39,350
+هو ناقص واحد أس ان ناقص واحد في X أس اتنين ام ناقص
+
+261
+00:21:39,350 --> 00:21:44,050
+اتنين على اتنين ام ناقص اتنين الكل factorial و
+
+262
+00:21:44,050 --> 00:21:48,390
+بتكمل هذا و بتبقى مكمل ايه انت لما اتصل لعند X أس
+
+263
+00:21:48,390 --> 00:21:54,670
+اتنين ام ناقص اتنين لأن هذا لل ام مش لل ان زائد
+
+264
+00:21:54,670 --> 00:21:59,730
+واحد على اتنين ام ناقص اتنين الكل اشماله factorial
+
+265
+00:22:00,420 --> 00:22:08,320
+لما تطرح ال CM نقص ال CN اللي هو لها بيصير المتبقى
+
+266
+00:22:08,320 --> 00:22:12,600
+هيه زي ما عملنا بالظبط قبل هيك فبيصير ال CM نقص ال
+
+267
+00:22:12,600 --> 00:22:18,080
+CN بساوي النقص واحد طبعا في absolute value عند X2N
+
+268
+00:22:18,080 --> 00:22:21,200
+عشان هيك طيرنا مش فارقة كتير نقص و سالب أخدنا له
+
+269
+00:22:21,200 --> 00:22:25,900
+absolute value مش هتفرج معنا ال X2N على 2N
+
+270
+00:22:25,900 --> 00:22:32,490
+vectorialناقص X أس 2 M زائد 2 اللي بعيدها على 2 M
+
+271
+00:22:32,490 --> 00:22:35,990
+زائد 2 كله factorial لما أصل لآخر term اللي هو X
+
+272
+00:22:35,990 --> 00:22:41,170
+أس 2 M ناقص 2 على 2 M ناقص 2 كله factorial هذا
+
+273
+00:22:41,170 --> 00:22:46,670
+الأن هذا نفسه أخدنا احنا ال absolute value لل X
+
+274
+00:22:46,670 --> 00:22:50,350
+أصغر من A في هذه الفترة احنا شغالين في الفترة ال
+
+275
+00:22:50,350 --> 00:22:54,430
+absolute value X أصغر من A الآن بيصير عند ال X
+
+276
+00:22:54,430 --> 00:23:05,180
+نفسهاأُس 2n أصغر أو يساوي هذه اللي هي a أُس 2n على
+
+277
+00:23:05,180 --> 00:23:09,240
+2n الكل factorial الأولى زائد استخدمت triangle
+
+278
+00:23:09,240 --> 00:23:13,440
+inequality هذه زائد هذه زائد هذه زائد هذه الآن
+
+279
+00:23:13,440 --> 00:23:18,240
+زائد اللي بعيدها لما أصل لآخر واحدة x أو اللي جاب
+
+280
+00:23:18,240 --> 00:23:23,420
+اللي خليني اكتبها عشان x اللي هي بصير aأس اتنين ان
+
+281
+00:23:23,420 --> 00:23:28,060
+زائد اتنين على اتنين ان زائد اتنين لكل factorial
+
+282
+00:23:28,060 --> 00:23:37,020
+زائد لما أصل لآخر term عندي هنا لما أصل لآخر term
+
+283
+00:23:37,020 --> 00:23:43,160
+اللي هو زائد
+
+284
+00:23:43,160 --> 00:23:57,230
+هذا ال term اللي هو زائدA 2M-2 2M-2 كل A شماله
+
+285
+00:23:57,230 --> 00:24:03,930
+فكتوريا خلّيني أخد هذا A 2N 2N كل فكتوريا العام
+
+286
+00:24:03,930 --> 00:24:11,460
+المشتركبظل عندى 1 زائد a تربيع لأنه بيصير a يسوى 2
+
+287
+00:24:11,460 --> 00:24:17,960
+a تربيع على مين على 2n زائد 2 أكيد ال 1 على 2n
+
+288
+00:24:17,960 --> 00:24:23,280
+زائد 2 أصغر من 1 على 2n لأنه 2n أصغر من هذه
+
+289
+00:24:23,280 --> 00:24:27,240
+فمقلوبها بيصير أكبرماشي و بضل زي ما عملته المرة
+
+290
+00:24:27,240 --> 00:24:32,700
+الفاتة أسحب منه A أس 2 على N و في الآخر أستبدل
+
+291
+00:24:32,700 --> 00:24:37,680
+اللي هو 2N هذه عن الرقم اللي أكبر منه N فبصير هي
+
+292
+00:24:37,680 --> 00:24:41,400
+مقلوبها أكبر فبتضلها اللي هي ال inequality زي هيك
+
+293
+00:24:41,400 --> 00:24:45,840
+و بكون سحبتي 2 من هذه بيصير 2 ام نقص 2 ام نقص ايش
+
+294
+00:24:45,840 --> 00:24:53,350
+نقص 2 وصلنا لعند ال inequality هذه الآنأحنا قلنا a
+
+295
+00:24:53,350 --> 00:24:58,950
+على 2n أصغر من مين من ربعه يعني بيصير هذا المقدار
+
+296
+00:24:58,950 --> 00:25:05,310
+كله أصغر أو يساوي a أس 2n على 2n لكل factorial
+
+297
+00:25:05,310 --> 00:25:09,550
+مضروب في مين أصغر أو يساوي لأن الربع أكبر منهم بس
+
+298
+00:25:09,550 --> 00:25:14,930
+تبدأ كل واحد وقت ما كانوا إيهاش ربع واحد زاد واحد
+
+299
+00:25:14,930 --> 00:25:23,510
+على أربعة تربيعزائد لما أصل لآخر واحد a اللي هو
+
+300
+00:25:23,510 --> 00:25:31,370
+عبارة عن واحد على أربعة الكل أس اتنين ام ناقص
+
+301
+00:25:31,370 --> 00:25:36,590
+اتنين ام ناقص ايش اتنين هذه الان ال ال ال ال
+
+302
+00:25:36,590 --> 00:25:39,710
+finite geometric series أصغر أو يساوي ال infinite
+
+303
+00:25:39,710 --> 00:25:45,770
+اللي هي a أس اتنين ان على اتنين ان لكل factorial
+
+304
+00:25:45,770 --> 00:25:53,570
+في اللي هوالـ summation ده واحد زائد ربع تربيع
+
+305
+00:25:53,570 --> 00:26:00,310
+زائد ربع تكعيب زائد لمّا أصل ربع أس أربعة زائد إلى
+
+306
+00:26:00,310 --> 00:26:05,070
+ما لا نهاية هذه الأن أساسها جداش وكأنه أساسها كل
+
+307
+00:26:05,070 --> 00:26:11,750
+مرة تمد واحد على ست عشر فبتصير هذه المجموحة اللي
+
+308
+00:26:11,750 --> 00:26:16,470
+هي واحد مجموحة بتعرفوها واحد ناقص واحد على ست عشر
+
+309
+00:26:17,250 --> 00:26:21,750
+Passive واحد على واحد ناقص واحد على ست عشر ويساوي
+
+310
+00:26:21,750 --> 00:26:27,810
+جداش خمست عشر او ست عشر على خمست عشر لأن هذه واحد
+
+311
+00:26:27,810 --> 00:26:29,870
+ناقص واحد على ست عشر تطلع خمست عشر على ست عشر
+
+312
+00:26:29,870 --> 00:26:34,830
+مجلوبة بيصير ست عشر على خمست عشرالمفهوم المقصود هو
+
+313
+00:26:34,830 --> 00:26:38,670
+بسمحها Geometric Series بضل بقول هذه أصغر أو ساوي
+
+314
+00:26:38,670 --> 00:26:41,930
+الصماشن إلى ماله نهاية و بزدبدل هذا باللي أكبر
+
+315
+00:26:41,930 --> 00:26:46,070
+منها فبتظل هذه أكبر الان مجموحها بيصير عبارة عن 16
+
+316
+00:26:46,070 --> 00:26:51,010
+في المقدار هذا اللي دله موخود عام المشترك فبيصير
+
+317
+00:26:51,010 --> 00:26:58,940
+عندى الآن يا جماعة اللي هو ال CMنقص الـCN زي ما
+
+318
+00:26:58,940 --> 00:27:04,460
+حكينا قبل ذلك بالظبط أصغر من هذا المقدار اللي هذا
+
+319
+00:27:04,460 --> 00:27:09,000
+limit و اين بيروح as N goes to infinity بيروح لـ 0
+
+320
+00:27:09,000 --> 00:27:13,760
+إذا بيصير هذا المقدار زي ما قلنا المرة الماضية CM
+
+321
+00:27:13,760 --> 00:27:20,800
+of X نقص CN of X اللي هو أصغر أو يسوء Y for very
+
+322
+00:27:20,800 --> 00:27:28,700
+large M and NM and N لأنه لما تكبر M كثير N كثير
+
+323
+00:27:28,700 --> 00:27:33,100
+تروح لنا لنهاية لأنه limit بروح لها سفر as N goes
+
+324
+00:27:33,100 --> 00:27:37,080
+to infinity ال M برضه بتكبر إذا for very large M
+
+325
+00:27:37,080 --> 00:27:40,800
+هيكون هذا أصغر أو يساوي إبسلون لأي إبسلون في
+
+326
+00:27:40,800 --> 00:27:44,480
+الدنيا لأنه هذا بيقدي للسفر فبيصير صغير صغير صغير
+
+327
+00:27:44,480 --> 00:27:48,720
+صغير لدرجة إنه مضروب في هذا يكون أصغر من إبسلون
+
+328
+00:27:48,720 --> 00:27:53,360
+وهذه اللي قلنا عنهااللي هي الـ Cauchy criterion
+
+329
+00:27:53,360 --> 00:28:00,040
+for uniform continuity هذا الكلام كله لمين؟ صحيح
+
+330
+00:28:00,040 --> 00:28:06,880
+لأي X وين على الفترة اللي هو absolute value X أصغر
+
+331
+00:28:06,880 --> 00:28:11,220
+أو يساوي A يعني على الفترة المولقة من ناقص A لعند
+
+332
+00:28:11,220 --> 00:28:19,110
+A وهذا بيعطيني أن الـis uniformly continuous على
+
+333
+00:28:19,110 --> 00:28:24,150
+الفترة من نفس A لعند A is as if is uniformly is
+
+334
+00:28:24,150 --> 00:28:28,330
+uniformly convergence على الفترة من نفس A ل A يعني
+
+335
+00:28:28,330 --> 00:28:35,790
+بمعنى آخر صارت عندي ال sequence هذه اللي هي CNN
+
+336
+00:28:35,790 --> 00:28:43,150
+converts uniformly to some function عالميا على
+
+337
+00:28:43,150 --> 00:28:50,730
+الفترةمن ناقص a لعند مين لعند a الآن طب ما هو اللي
+
+338
+00:28:50,730 --> 00:28:53,930
+عملناها أنا على الفترة هذه نقدر نعمله على أي شيء
+
+339
+00:28:53,930 --> 00:28:58,310
+ثاني يعني بمعنى أخر لو جيت أخدت x element in R
+
+340
+00:28:58,310 --> 00:29:04,400
+هلاجي number aبحيث أن هذا ال number a هي ال x في
+
+341
+00:29:04,400 --> 00:29:10,180
+ال R بقدر ألاقي a بحيث أن ناقص a و a تكون ال x في
+
+342
+00:29:10,180 --> 00:29:15,340
+الفترة بين ناقص a و a يعني بمعنى آخر هعمل نفس اللي
+
+343
+00:29:15,340 --> 00:29:18,100
+عملته في الأول و هحصر على c أن uniformly
+
+344
+00:29:18,100 --> 00:29:23,030
+convergence على هذه الفترةيعني بمعنى آخر الآن صار
+
+345
+00:29:23,030 --> 00:29:29,450
+عندي limit cn of x for any x exist بده أسميها هذه
+
+346
+00:29:29,450 --> 00:29:35,190
+limit c of x وبناء عليه بدي أعرف الآن in
+
+347
+00:29:35,190 --> 00:29:38,710
+particular this means that cn of x converge for
+
+348
+00:29:38,710 --> 00:29:42,710
+each x element in R we define c من R لR by c of x
+
+349
+00:29:42,710 --> 00:29:45,650
+بسواية limit cn of x for x element in R
+
+350
+00:29:53,570 --> 00:29:59,630
+الان .. بما انه الان الـCn مرتبط بشكل مرتبط لـC زي
+
+351
+00:29:59,630 --> 00:30:04,690
+ما قلنا أو الـCn of X كلهم مرتبط حسب اللي هو
+
+352
+00:30:04,690 --> 00:30:08,030
+النظرية في الـpointwise .. الـuniform convergence
+
+353
+00:30:08,030 --> 00:30:12,360
+اللي هو the limit .. the uniform .. limitأو الـ
+
+354
+00:30:12,360 --> 00:30:15,220
+Form convergence of a sequence of continuous
+
+355
+00:30:15,220 --> 00:30:18,480
+functions مصمد كونتنيواس الـ limit تبعتها يعني
+
+356
+00:30:18,480 --> 00:30:21,160
+هتطلع عندي C of X continuous مثلاً C of X
+
+357
+00:30:21,160 --> 00:30:25,000
+continuous إذا عندي .. اللي هو صار عندي الـ
+
+358
+00:30:25,000 --> 00:30:28,160
+function هذه continuous اللي هي الـ C اللي إحنا
+
+359
+00:30:28,160 --> 00:30:33,730
+بدنا إياها ومش هيك و limitCn of 0 as n goes to
+
+360
+00:30:33,730 --> 00:30:37,270
+infinity بساوة limit اللي هي Cn of 0 ايش بتساوي
+
+361
+00:30:37,270 --> 00:30:41,910
+واحد وهي ساوية واحد اللي هي عبارة عن مين الـC of
+
+362
+00:30:41,910 --> 00:30:47,370
+ايش of 0 لأن احنا متفجين الـC of X بساوية limit Cn
+
+363
+00:30:47,370 --> 00:30:51,310
+of X و in particular for X بتساوية سفر بسيرة limit
+
+364
+00:30:51,310 --> 00:30:55,430
+Cn of 0 بساوية C of 0 وCn of 0 كلها واحد ايضا
+
+365
+00:30:55,430 --> 00:30:57,730
+limit الواحد اللي هو بساوية واحد يعني الـC of 0
+
+366
+00:30:57,730 --> 00:31:05,970
+ايش هتساوي؟ هتساوي واحدخصّلت كمان شغلة انه حصلت
+
+367
+00:31:05,970 --> 00:31:10,070
+عندي ان الـ c of zero بيساوي واحد
+
+368
+00:31:13,750 --> 00:31:19,670
+لأ اللي هو الـS الـCN بعمل اشي مشابه له لمين للـSN
+
+369
+00:31:19,670 --> 00:31:25,250
+عشان نثبت الآن اللي أثبتناه انه صار في عندى ده
+
+370
+00:31:25,250 --> 00:31:31,730
+اللي عرفناها اسمها الـC of X اللي عبارة عن limit
+
+371
+00:31:31,730 --> 00:31:38,970
+CN of X حيث الـC من R إلى R طيب
+
+372
+00:31:41,560 --> 00:31:45,380
+الان ناخد ايضا ال absolute value X أصغر من مين من
+
+373
+00:31:45,380 --> 00:31:49,960
+ايه وال M أكبر او يساوي N و أصغر من مين من اتنين
+
+374
+00:31:49,960 --> 00:32:00,360
+ايه الان نحسب ل SM ناقص SN SM ايه قيمتهاDT من صفر
+
+375
+00:32:00,360 --> 00:32:05,400
+الاندكس SN وهي SM
+
+376
+00:32:05,400 --> 00:32:11,240
+ناقص هذه هي قيمتها إذا صار هذه ناقص هذه هي قيمتها
+
+377
+00:32:11,240 --> 00:32:19,680
+الان هذه سهل إثباتها أنها تتعامل ال absolute value
+
+378
+00:32:19,680 --> 00:32:24,200
+لهذهأصغر أو يساوي الـ absolute value لهذه أصغر أو
+
+379
+00:32:24,200 --> 00:32:27,700
+يساوي ال integration من 0 ل X ل absolute value CM
+
+380
+00:32:27,700 --> 00:32:35,600
+of T نقص CN of T اشماله DT ماشي الحال أو بنكمل
+
+381
+00:32:35,600 --> 00:32:39,580
+اللي هو بنستخدم هذه الخاصية اللي هو بيكون أصغر أو
+
+382
+00:32:39,580 --> 00:32:42,820
+يساوي من الحسابات اللي حسبناها قبل شوية هذا
+
+383
+00:32:42,820 --> 00:32:45,380
+حسبناها أصغر من مين الحسابات اللي قبل شوية أصغر أو
+
+384
+00:32:45,380 --> 00:32:52,510
+ساوي A أس 2N على 2N الكل factorialمضروبة في اللي
+
+385
+00:32:52,510 --> 00:32:58,130
+هو ست عشر على مين على خمسة في ال integration من
+
+386
+00:32:58,130 --> 00:33:02,750
+صفر لل X ل DT هذا ال integration ايش بيساوي بيساوي
+
+387
+00:33:02,750 --> 00:33:08,230
+X ماشي و ال X عندي احنا ماخدينها اصغر او يساوي مين
+
+388
+00:33:08,230 --> 00:33:13,010
+ايه اصغر او يساوي ايه فبصير عندي هذا المقدار بيكون
+
+389
+00:33:13,680 --> 00:33:18,020
+لما نبعد مكانه بيطلع X قيمته أصغر أو يساوي A
+
+390
+00:33:18,020 --> 00:33:22,040
+فبيصير هذا مضروب في مين في ايه اللي هو هاي المقدار
+
+391
+00:33:22,040 --> 00:33:26,680
+وهي 16 على 5 وهي ايش ال A صار هذا أصغر أو يساوي
+
+392
+00:33:26,680 --> 00:33:31,600
+هذا المقدار نفس ما عملنا قبل بشوية الان as N goes
+
+393
+00:33:31,600 --> 00:33:36,690
+to infinityهذه قيمتها بيساوي 0 إذا هذا المقدار for
+
+394
+00:33:36,690 --> 00:33:41,470
+very large M و N هيكون أصغر أوي يساوي Y و بالكوشي
+
+395
+00:33:41,470 --> 00:33:45,350
+criterion بيصير عندي اللي هي ال S N converts
+
+396
+00:33:45,350 --> 00:33:49,470
+uniformly و الآن على الفترة نقص A و A و ال A كانت
+
+397
+00:33:49,470 --> 00:33:53,730
+arbitrary إذا S من R ل R بنقدر نعرفها بحيث إنه
+
+398
+00:33:53,730 --> 00:33:57,880
+limit S N of X اللي صارت موجودة على هذهوبناء على
+
+399
+00:33:57,880 --> 00:34:00,460
+الـ A-arbitrary صارت موجودة على كل الـ R زي ما
+
+400
+00:34:00,460 --> 00:34:04,340
+قلنا كيف هذا الكلام بسمي limit S N of X اليمين S
+
+401
+00:34:04,340 --> 00:34:08,980
+of X وهذه اللي هي ال function الثانية بنفس الأسلوب
+
+402
+00:34:08,980 --> 00:34:13,380
+وبنفس اللي هو المنطق وبنفس الأسباب بما أن ال S N
+
+403
+00:34:13,380 --> 00:34:16,860
+conversion formally to S وS N continuous إذا حد
+
+404
+00:34:16,860 --> 00:34:22,340
+طلع ال S برضه نفس إشمالها continuousوأيضًا limit
+
+405
+00:34:22,340 --> 00:34:28,540
+of S of 0 هو عبارة عن S of 0 وبما أن S of 0 دايماً
+
+406
+00:34:28,540 --> 00:34:31,640
+0 إذًا limitها سيكون as N goes to infinity 0 يعني
+
+407
+00:34:31,640 --> 00:34:35,300
+ستظهر لدي S of 0 إيش بالساوة 0 إذًا الآن طلعنا
+
+408
+00:34:35,300 --> 00:34:45,840
+كمان شغلة أثبتناها أن S of 0بساوة 0 و C of 0 بساوة
+
+409
+00:34:45,840 --> 00:34:54,400
+1 وعرفنا هذولة الدالتين أسي و أس من اللي هي R لعند
+
+410
+00:34:54,400 --> 00:34:59,680
+مين لعند R إذن الآن احنا الدالتين اللي أثبتنا لان
+
+411
+00:34:59,680 --> 00:35:05,430
+وجود هنا الآن اللي بحكي عنهم هذولة الدالتيناللي هي
+
+412
+00:35:05,430 --> 00:35:10,190
+الدالة الأولى سميتها أسوة تانية C ولاقيت انه C of
+
+413
+00:35:10,190 --> 00:35:15,790
+0 بساوة واحد و S of 0 بساوة H0 وظل أثبت هذه و أثبت
+
+414
+00:35:15,790 --> 00:35:22,910
+هذه و أثبت هذولة اللي أستبعتنا بتحققهم فخلينا نشوف
+
+415
+00:35:22,910 --> 00:35:29,670
+كيف نثبتها طيب الآن أوصلنا لعند اللي هو المنطقة
+
+416
+00:35:29,670 --> 00:35:39,670
+هذه لأن احنا اتفجنا على ما يليهاتفجنا ان الـ S N
+
+417
+00:35:39,670 --> 00:35:47,170
+تتعامل بشكل عام لتعمل S و الـ C N تتعامل بشكل عام
+
+418
+00:35:47,170 --> 00:35:54,280
+لتعمل Cلكن الان احنا اثبتنا في الأول هي ان ما
+
+419
+00:35:54,280 --> 00:36:00,880
+محتاش ال S N برايم بسوء ال C N، مظبوط؟ يعني و كأنه
+
+420
+00:36:00,880 --> 00:36:04,860
+بناء على هذا مزام ال S N برايم بسوء ال C N فبصير
+
+421
+00:36:04,860 --> 00:36:09,520
+ال S N برايم، هذه، مكانها دي S N برايم بصير ال S N
+
+422
+00:36:09,520 --> 00:36:15,340
+برايم converges uniformly to some function mean C
+
+423
+00:36:15,340 --> 00:36:22,060
+الان بما ان S N برايم converges uniformly to Cالان
+
+424
+00:36:22,060 --> 00:36:28,900
+و الاسن converged to us uniformly برضهحسب نظرية
+
+425
+00:36:28,900 --> 00:36:32,360
+بتطلع عندى بما ان s n prime differentiable هتكون
+
+426
+00:36:32,360 --> 00:36:37,140
+ال limit ه differentiable و ال c prime اللي هي ال
+
+427
+00:36:37,140 --> 00:36:42,340
+s prime هدى هي مين ال c اللي طلعت هن يعني s n
+
+428
+00:36:42,340 --> 00:36:45,960
+prime differentiable او s n differentiable و
+
+429
+00:36:45,960 --> 00:36:48,560
+converts to some function اذا بتتذكر كنا نسميها g
+
+430
+00:36:48,560 --> 00:36:52,520
+و هد كنا نسميها f فكنا نقول بما ان s n بتروح لل f
+
+431
+00:36:52,520 --> 00:36:56,740
+و ال s n prime بتروح لل g إذا نتيجة النظرية هتكون
+
+432
+00:36:56,740 --> 00:37:00,000
+اللي هيالـ F هي الـ Differentiable والـ Derivative
+
+433
+00:37:00,000 --> 00:37:04,940
+لها إيش بتطلع D يعني الـ Derivative للـ S' إيش
+
+434
+00:37:04,940 --> 00:37:12,260
+هتطلع عبارة عن مين C' أسف C هتطلع مين اللي هي الـ
+
+435
+00:37:12,260 --> 00:37:21,000
+C إذا أثبتت أنا الآن S' of X بتساوي C of X هاي
+
+436
+00:37:21,000 --> 00:37:28,190
+اللي أثبتته هنا الآن من جهة أخرى من جهة أخرىإحنا
+
+437
+00:37:28,190 --> 00:37:35,850
+أثبتنا اللي هي قبل هيك أن الـCN برايم بسوء ناقص SN
+
+438
+00:37:35,850 --> 00:37:42,450
+ناقص واحد الـCN برايم بسوء ناقص SN of واحد يعني
+
+439
+00:37:42,450 --> 00:37:47,790
+بمعنى آخر الـCN برايم of X أثبتنا بسوء ناقص SN
+
+440
+00:37:47,790 --> 00:37:54,640
+ناقص واحد of Xالان بما ان ال S Unconverted زي ال S
+
+441
+00:37:54,640 --> 00:38:00,260
+خلصنا هذه اه خلصنا هذه المنطقة خليني اشرح بلغة
+
+442
+00:38:00,260 --> 00:38:05,060
+تانية عشان اميز بين الكلامين عندي الان انتبهوا
+
+443
+00:38:05,060 --> 00:38:08,860
+بتعمل نفسي اشي بس بالنسبة لمن الان بالنسبة عشان
+
+444
+00:38:08,860 --> 00:38:13,780
+اجيب ال derivative لل S لل C primeعندى الان الـCN
+
+445
+00:38:13,780 --> 00:38:19,740
+prime of X بسوء نقص SN نقص واحد of X بما أن SN
+
+446
+00:38:19,740 --> 00:38:23,500
+راحت للـS إذا اللى هى ال derivative هى اللى
+
+447
+00:38:23,500 --> 00:38:30,280
+بتسويها اللى هى CNN زائد واحد prime of X هتروح
+
+448
+00:38:30,280 --> 00:38:36,770
+لمين؟اللي هي مش هي ناقصها لأن S N ناقص واحد اللي
+
+449
+00:38:36,770 --> 00:38:40,730
+هي بساوي ناقص هذه أنجل ناقصها بعد أذنكم يعني بدي
+
+450
+00:38:40,730 --> 00:38:44,810
+أستبدل الـ S N بمين بقيمتها هذه صارت ناقص الـ C N
+
+451
+00:38:44,810 --> 00:38:50,330
+prime of X أيش بتساوي بتروح للـ S أو بمعنى آخر صار
+
+452
+00:38:50,330 --> 00:38:56,620
+عندي من هنا من هنا اللي بتطلع عندي هناصار عندى
+
+453
+00:38:56,620 --> 00:39:03,640
+الان من هنا اللى هو cn زائد واحد prime of x بتروح
+
+454
+00:39:03,640 --> 00:39:08,080
+لناقص طبعا uniformly بتروح لمين ناقص s لأنه ناقصها
+
+455
+00:39:08,080 --> 00:39:13,040
+بتروح لل s إذا هي بتروح لناقص ال sوفي نفس الوقت
+
+456
+00:39:13,040 --> 00:39:19,840
+أنا بقول الـCN نفسها بتروح uniform لمين؟ للـC بنفس
+
+457
+00:39:19,840 --> 00:39:24,740
+الإسلوب اللي قبل بشوية اللي هي بما أنه هذه اللي هي
+
+458
+00:39:24,740 --> 00:39:27,580
+differentiable sequence of functions و converge
+
+459
+00:39:27,580 --> 00:39:31,260
+uniform to some function إذا هذه هتكون اللي هي
+
+460
+00:39:31,260 --> 00:39:35,600
+عبارة عن اللي هي الـC الأصلية differentiable و ال
+
+461
+00:39:35,600 --> 00:39:40,070
+derivative فيه لها مين؟ الناقص Sفبصير عندي الـ C'
+
+462
+00:39:40,530 --> 00:39:45,970
+of X بسوء ناقص S of X بكون هذه اللي هي النتيجة
+
+463
+00:39:45,970 --> 00:39:50,330
+الثانية اللي طبقنا عليها طلعت عندي اللي هو هذه
+
+464
+00:39:50,330 --> 00:39:54,050
+الخاصية و هذه الخاصية متحققة يعني الآن صار عندي
+
+465
+00:39:54,050 --> 00:40:02,590
+اللي هو تحقق ما يليه أنه تبعتنا هذه الـ C'of X
+
+466
+00:40:02,590 --> 00:40:08,530
+بيساوي ناقص S of X وطلع عندي اللي هي S prime of X
+
+467
+00:40:08,530 --> 00:40:16,290
+بيساوي C of X ماشي الحال بيكون هيك احنا ضال علينا
+
+468
+00:40:16,290 --> 00:40:21,230
+شغل اخرى لحاول نثبتها بيكون اثبتنا اللي هو كل
+
+469
+00:40:21,230 --> 00:40:27,630
+الصفات المطلوبة اللي تتحقق في ال S وحددت هوية ال S
+
+470
+00:40:27,630 --> 00:40:28,050
+و ال C
+
+471
+00:40:38,700 --> 00:40:48,060
+الجزء الثاني سهل فضل الـ cw prime of x
+
+472
+00:41:00,590 --> 00:41:05,470
+بتصير c double prime هيها و هذه فضلها بنفعلها
+
+473
+00:41:05,470 --> 00:41:09,550
+لأنها قبل التفاضل اللي فضلناها وبتصير ناقص اللي هي
+
+474
+00:41:09,550 --> 00:41:15,030
+s prime of x اللي هي s prime of x هيش بتساوي c of
+
+475
+00:41:15,030 --> 00:41:20,350
+x فبصير ناقص مين؟ c of xوالآن S W' of X من أين بدي
+
+476
+00:41:20,350 --> 00:41:23,650
+أجيبها من هنا S W' بيساوي اللي هو هذه ال
+
+477
+00:41:23,650 --> 00:41:26,050
+derivative اللي هي ال derivative اللي هي عبارة عن
+
+478
+00:41:26,050 --> 00:41:31,530
+ناقص S of X فبصير عبارة عن ناقص S of X فبصير عندي
+
+479
+00:41:31,530 --> 00:41:38,910
+هذا برضه ايش تحقق اخر اشي اللي هي C' of Zero C' of
+
+480
+00:41:38,910 --> 00:41:42,410
+Zero بيساوي ناقص S of Zero وS of Zero بيساوي صفر
+
+481
+00:41:42,410 --> 00:41:47,030
+اذا C' of Zero بيساوي Zero ال S' of Zero
+
+482
+00:41:52,540 --> 00:41:56,800
+وهكذا اثبتنا وجود دالة
+
+483
+00:41:59,570 --> 00:42:04,970
+اللي هي حققتلي اللي هي الشرط الأول هذا والدالتين
+
+484
+00:42:04,970 --> 00:42:09,710
+اسف والشرط الثاني اه اللي بعده هذه بتكون أثبتنا
+
+485
+00:42:09,710 --> 00:42:15,290
+وجود الـC والـS لأن بعض النتائج الأخرى على اللي هي
+
+486
+00:42:15,290 --> 00:42:18,830
+الدالتين
+
+487
+00:42:18,830 --> 00:42:19,790
+اللي أوجدناهم
+
+488
+00:42:24,500 --> 00:42:28,960
+نشوف ايش الـ Corollary الأولى بقول لي if C and S
+
+489
+00:42:28,960 --> 00:42:33,680
+are the functions in 3x,8x,4x,1 then C' of X
+
+490
+00:42:33,680 --> 00:42:39,560
+بيساوي نقص S of X أثبتناها هيها and طبعا في طريقنا
+
+491
+00:42:39,560 --> 00:42:44,020
+في البرهان S' of X بيساوي C of X هيها اللي هي خواص
+
+492
+00:42:44,020 --> 00:42:50,340
+اللي هي أثبتناها طيب الآن بقول لي moreover these
+
+493
+00:42:50,340 --> 00:42:54,630
+functions have derivatives of all orderيعني بأي
+
+494
+00:42:54,630 --> 00:42:57,710
+order ال derivative موجودة طبعاً هذا by induction
+
+495
+00:42:57,710 --> 00:43:01,550
+by induction اللي هو بدك تثبت انه اللي هو اللي هو
+
+496
+00:43:01,550 --> 00:43:06,670
+cn هتكون موجودة و بيساوي ناقص a of x او زاد a of x
+
+497
+00:43:06,670 --> 00:43:11,450
+حسب اللي هي جداش درجة ال n وممكن تساوي c of x او
+
+498
+00:43:11,450 --> 00:43:16,930
+اللي هي ناقص c of x حسب درجة ال n اللي موجودةاذا
+
+499
+00:43:16,930 --> 00:43:19,990
+الان هذه by induction بنقدر نثبت ان ال derivative
+
+500
+00:43:19,990 --> 00:43:26,390
+اللي هو موجودة for any or ضر بناء على اللي حكيناه
+
+501
+00:43:26,390 --> 00:43:32,300
+نيجي الان لـ Corollary اللي بعدهابقول الآن الـ
+
+502
+00:43:32,300 --> 00:43:38,280
+function C and S بحقق اللي هي الـ Pythagorean
+
+503
+00:43:38,280 --> 00:43:42,280
+Identity اللي كنا نعرفها كصين تربيع زات صين تربيع
+
+504
+00:43:42,280 --> 00:43:47,660
+إش بساوي بساوي واحد فكرة البروحان سهلة بسمي هذه
+
+505
+00:43:47,660 --> 00:43:51,520
+كلها بسميها function اسمها F of X بقول سمي هذه
+
+506
+00:43:51,520 --> 00:43:56,080
+اللي هي F of X بساوي هذه فضليها بلّا بتفضليها F
+
+507
+00:43:56,080 --> 00:44:01,200
+prime of X بساوي اتنينفى C of X فى تفاضل اللى جوا
+
+508
+00:44:01,200 --> 00:44:06,300
+اللى هو ناقص S of X والتانية زائد اتنين فS of X
+
+509
+00:44:06,300 --> 00:44:11,880
+فتفاضلها هي C of X هذه هي هذه بس بالسالد اذا حصل
+
+510
+00:44:11,880 --> 00:44:15,100
+طرح حلو اسم ساوى سفر اذا صارت عند ال derivative لل
+
+511
+00:44:15,100 --> 00:44:18,320
+function هذه اشمالها بساوى سفر يعني بمعنى اخر
+
+512
+00:44:18,320 --> 00:44:22,490
+الدالة هي دالة ثابتةيعني F is a constant function
+
+513
+00:44:22,490 --> 00:44:27,210
+يعني قيمة الـ F عند أي قيمة إيش بتساوي مقدار ثابت
+
+514
+00:44:27,210 --> 00:44:31,310
+إذا اتفاج أنها ثابتة أسهل إشي أسهل إشي أوجدلي F of
+
+515
+00:44:31,310 --> 00:44:34,670
+Zero عشان أتعرف إيش ده اللي بالساوية بالظبط C of
+
+516
+00:44:34,670 --> 00:44:38,730
+Zero واحد S of Zero سفر إذا صار عندي F of Zero
+
+517
+00:44:38,730 --> 00:44:42,170
+بساوية واحد وهي ثابتة إذا صارت F of X دايما
+
+518
+00:44:42,170 --> 00:44:46,270
+بالساوية واحد يعني هذا المقدار بساوية واحد دائما
+
+519
+00:44:46,270 --> 00:44:48,690
+طيب نيجي الآن
+
+520
+00:44:51,700 --> 00:44:57,820
+لأ الـ theorem اللي بعدها الـ functions C and S
+
+521
+00:44:57,820 --> 00:45:02,820
+satisfy the properties I and II of theorem 8-4-1
+
+522
+00:45:02,820 --> 00:45:07,020
+are unique لما بقولّي برضه هتلاقي ال uniqueness
+
+523
+00:45:07,020 --> 00:45:13,020
+بشبه ال uniqueness لمن؟ لأ اللي هو ال .. ال ..
+
+524
+00:45:13,020 --> 00:45:18,030
+قولوا معايا ال uniqueness للـ exponentialو سمناها
+
+525
+00:45:18,030 --> 00:45:21,170
+اللي هو let E واحد .. نفترض أنه في E واحد و E
+
+526
+00:45:21,170 --> 00:45:24,070
+اتنين و سمها فرق بيساوي D و في الآخر روحنا لفرق
+
+527
+00:45:24,070 --> 00:45:28,390
+بيساوي إيش؟ بيساوي سفر هنا نفس الاشي هنشتغل و برضه
+
+528
+00:45:28,390 --> 00:45:31,590
+هنستخدم اللي هو ال Taylor's theorem زي ما استخدمنا
+
+529
+00:45:31,590 --> 00:45:34,470
+هناك ال Taylor's theorem يعني هتلاقيه ال sketch
+
+530
+00:45:34,470 --> 00:45:37,470
+للبرهان هو نفس ال sketch الأولاني عشان هيك
+
+531
+00:45:37,470 --> 00:45:43,110
+هتلاقوني سريع فيه بدنا نثبت أن ال C و ال S are
+
+532
+00:45:43,110 --> 00:45:47,930
+unique functionsطبعا الطريقة let c1 and c2 be two
+
+533
+00:45:47,930 --> 00:45:52,290
+functions on R that satisfy it satisfies مين ال
+
+534
+00:45:52,290 --> 00:45:55,850
+conditions اللي احنا بنقول عنهم اللي هي ال I و I I
+
+535
+00:45:55,850 --> 00:46:00,930
+اللي هو بحيث انه c1 double prime of x بساوي c1 of
+
+536
+00:46:00,930 --> 00:46:01,150
+x
+
+537
+00:46:16,750 --> 00:46:18,150
+2�
+
+538
+00:46:18,550 --> 00:46:24,010
+لأن نفترض أن دي بتساوي C1-C2 وبدنا نصل في النهاية
+
+539
+00:46:24,010 --> 00:46:27,410
+أن دي هذي لازم تطلع إيش بتساوي؟ بتساوي 0 مدام دي
+
+540
+00:46:27,410 --> 00:46:32,710
+بتساوي 0 مدام C1 بتساوي إيش؟ C2 لاحظ الآن دي W' of
+
+541
+00:46:32,710 --> 00:46:38,190
+X فاضل هذا مرتين تصير C1W'-C2W'
+
+542
+00:46:39,860 --> 00:46:47,500
+الان بساوي هتطلع ايش بتساوي ناقص اللي هي مين D of
+
+543
+00:46:47,500 --> 00:46:59,700
+X نعملها اللي هي حسابات D of X بساوي D of X بساوي
+
+544
+00:46:59,700 --> 00:47:08,540
+C1 ناقص C2 D prime of X ايش هيساوي اللي هو عبارة
+
+545
+00:47:08,540 --> 00:47:16,670
+عنأس واحد ناقص اس واحد ناقص بيصير اللي هو مين الاس
+
+546
+00:47:16,670 --> 00:47:19,810
+واحد اللي هي بالنسبة لهذه اللي أوجدناها ناقص اس
+
+547
+00:47:19,810 --> 00:47:25,350
+اتنين بيصير زائد لان دي double prime بساوي بتفاضل
+
+548
+00:47:25,350 --> 00:47:30,030
+هذا كمان مرة اللي هي بترجع مين نفسها C واحدوهذه
+
+549
+00:47:30,030 --> 00:47:38,390
+ترجع نفسها C2 التي هي ناقص D التي هي ناقص D طيب
+
+550
+00:47:38,390 --> 00:47:43,410
+إذا ال D لأنه مفترضين ال C1 وC2 أشمالها بتحقق
+
+551
+00:47:43,410 --> 00:47:52,470
+الشروط اللي فوق اللي قلنا عنهاDouble prime بساوة D
+
+552
+00:47:52,470 --> 00:48:02,390
+of X أو D double prime بساوة D double prime بساوة
+
+553
+00:48:06,030 --> 00:48:11,870
+بساوي هد W' ناقص هد W' هد W' ناقص C1 و هد W' اللي
+
+554
+00:48:11,870 --> 00:48:15,870
+هي ناقص ناقص C زائدة سارة D W' بساوي ناقص D و X
+
+555
+00:48:15,870 --> 00:48:21,890
+هيك اللي هو دعنا نقول أسلم طيب الآن بستعجل لإن
+
+556
+00:48:21,890 --> 00:48:29,470
+الكلام معاد يعنيأو الأفكار معادة الآن احطلي D of 0
+
+557
+00:48:29,470 --> 00:48:34,530
+ايش هيساوي؟ Zero لأن D of 0 بيساوي هذي ناقص هذي و
+
+558
+00:48:34,530 --> 00:48:37,630
+هذي عند ال zero واحد و هذي عند ال zero واحد
+
+559
+00:48:37,630 --> 00:48:40,010
+التنتين عند ال zero واحد انحصل طرحي ان ايش هيساوي
+
+560
+00:48:40,010 --> 00:48:47,160
+سفر الان D prime عند ال zeroوDW' عند الـ zero و و
+
+561
+00:48:47,160 --> 00:48:48,760
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+562
+00:48:48,760 --> 00:48:50,140
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+563
+00:48:50,140 --> 00:48:52,320
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+564
+00:48:52,320 --> 00:48:52,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+565
+00:48:52,380 --> 00:48:55,440
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+566
+00:48:55,440 --> 00:48:59,220
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+567
+00:48:59,220 --> 00:48:59,240
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+568
+00:48:59,240 --> 00:49:00,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+569
+00:49:00,380 --> 00:49:01,520
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+570
+00:49:01,520 --> 00:49:05,260
+.. و
+
+571
+00:49:05,260 --> 00:49:15,970
+.. و .. و .. و .. و .. و .. وبحكي عن الـ
+
+572
+00:49:15,970 --> 00:49:25,410
+derivative الـ dw prime عبارة عن اللي هي ناقص d of
+
+573
+00:49:25,410 --> 00:49:30,410
+0 عند الـ 0 و الـ d عند الـ 00 إذن هذه الـ 0 هذه
+
+574
+00:49:30,410 --> 00:49:34,330
+لمن؟ لـ k اللي هي إتنين و أربعة و ستة و تمانية
+
+575
+00:49:34,330 --> 00:49:37,870
+بنفس السبب هتطلع سفر الفرديات بيجي من مين؟ من هذه
+
+576
+00:49:38,330 --> 00:49:44,430
+فردية الـ Derivative لإيش؟ لأن D' of 0 هتساوي C1'
+
+577
+00:49:44,890 --> 00:49:51,110
+نقص C2' C1' عند الصفر صفر و C2' عند ال 0 0 إذا هذه
+
+578
+00:49:51,110 --> 00:49:56,730
+دي K عند ال 0 بالساوية 0 لكل K سواء زوجية أو إيش
+
+579
+00:49:56,730 --> 00:50:01,230
+أو فردية إذا جهزنا هذا اللي جهزناه قبل هيك يعمل
+
+580
+00:50:01,230 --> 00:50:05,940
+Exponentialوبتنا نطبّك اللي هو مين الـ Taylor's
+
+581
+00:50:05,940 --> 00:50:09,000
+theorem نطبّك الـ Now let x element in R be
+
+582
+00:50:09,000 --> 00:50:12,760
+arbitrary element in R and let I x be the interval
+
+583
+00:50:12,760 --> 00:50:18,940
+within point 0x طبّجنا عليها إذا since D بتساوي C1
+
+584
+00:50:18,940 --> 00:50:25,220
+ناقص C2 وT بتساوي S1 ناقص S2 اللي هي مفترضين S1
+
+585
+00:50:25,220 --> 00:50:28,400
+وS2 اللي هو two functions such that اللي بيحققوا
+
+586
+00:50:28,400 --> 00:50:34,170
+تبعات الـ sineاللي هي بالساوية S1 ايش هي مسميها
+
+587
+00:50:34,170 --> 00:50:39,950
+انا اللي هي بدل C2 prime او هي C2 prime والـ S2
+
+588
+00:50:39,950 --> 00:50:46,510
+اللي هي C2 A prime العكسة
+
+589
+00:50:46,510 --> 00:50:50,530
+ال derivative بيصيح بالنقص are continuous on mean
+
+590
+00:50:50,530 --> 00:50:55,930
+on Ix عارفين continuous انه احنا مفترضين ان نحقق
+
+591
+00:50:55,930 --> 00:51:00,160
+نفس اللي هو اللي قبلمدام continuous الـ D والـ T
+
+592
+00:51:00,160 --> 00:51:07,440
+continuous على closed bounded interval I X إذا في
+
+593
+00:51:07,440 --> 00:51:12,810
+اللي هو K واحد للأولىوK2 للتانية خدوا ال maximum
+
+594
+00:51:12,810 --> 00:51:18,470
+إليها واسموها K إذا في K للجهتين بحيث ان ال D of T
+
+595
+00:51:18,470 --> 00:51:22,930
+bounded علي هذه أصغر شهو T كي على كل الفترة اللي
+
+596
+00:51:22,930 --> 00:51:27,390
+بنحكي عنها IX وT of T أصغر شهو K for all T
+
+597
+00:51:27,390 --> 00:51:30,290
+elements of IX لأنه زي ما أقول continuous function
+
+598
+00:51:30,290 --> 00:51:33,930
+in a closed bounded interval must be boundedالآن
+
+599
+00:51:33,930 --> 00:51:39,170
+جاهزين نطبق مين؟ الـ Taylor's theorem to d من Ix
+
+600
+00:51:39,170 --> 00:51:44,970
+and use the fact دي اللي أثبتناه دي 0 سواء 0 دي K0
+
+601
+00:51:44,970 --> 00:51:50,030
+سواء 0 الآن بنقول for each n unlimited n عملناها
+
+602
+00:51:50,030 --> 00:51:56,510
+عشان هيك بس ماشي عنه بسرعة عملناها قبل هيك there
+
+603
+00:51:56,510 --> 00:51:59,630
+exist a point Cn unlimited Ix such that هذا
+
+604
+00:51:59,630 --> 00:52:03,890
+remainder بتعملي remainderD ب X بيساوي D of 0 زي
+
+605
+00:52:03,890 --> 00:52:06,850
+دي prime of 0 زائد زائد زائد زائد X ان نقص واحد
+
+606
+00:52:06,850 --> 00:52:12,630
+لما أصل لمين لل remainder كل هذول أصفار بسبب مين
+
+607
+00:52:12,630 --> 00:52:16,430
+انه D0 وD prime of 0 وD ان نقص واحد وزير وكلهم إيه
+
+608
+00:52:16,430 --> 00:52:20,090
+شمالهم بيساوي أصفار عند الصفر بيظل هذا المقدار
+
+609
+00:52:20,090 --> 00:52:24,010
+يساوي اللي هو المقدار اللي عندي اللي تحت هذا هذا
+
+610
+00:52:24,010 --> 00:52:27,490
+ال derivative هذا ال derivative أنا بعرفش إيش هي
+
+611
+00:52:27,490 --> 00:52:31,230
+ما أنتوا عارفين ال derivative لل .. لل .. لل C
+
+612
+00:52:34,160 --> 00:52:41,940
+أو ال derivative للـC إذا فضلت مرة واحدة بتطلع
+
+613
+00:52:41,940 --> 00:52:47,120
+الـS أو سلبها فظلت تلت مرات بترجع ليها S بس
+
+614
+00:52:47,120 --> 00:52:53,180
+بالموجة فظلت خمسة بترجع ناقص Sفظلت زوجي بتطلع هي
+
+615
+00:52:53,180 --> 00:52:58,920
+نفسها أو سلبها عشان هي كادل أنا بعرفش طبعا هنا
+
+616
+00:52:58,920 --> 00:53:02,400
+عندك اللي هي بتتوزع على مرة derivative تندين
+
+617
+00:53:02,400 --> 00:53:04,620
+derivative تلاتة derivative أربعة derivative و
+
+618
+00:53:04,620 --> 00:53:08,180
+بترجع الدورة زي ما هي لإنه في الأول بيكون اللي هي
+
+619
+00:53:08,180 --> 00:53:15,520
+تفاضل الـC ناقص S بعدها بتتفاضل بترجع اللي هي ناقص
+
+620
+00:53:15,520 --> 00:53:18,770
+حالهابعدها بترجع اللي هي
+
+621
+00:53:18,770 --> 00:53:36,410
+ببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببب
+
+622
+00:53:36,660 --> 00:53:40,180
+اللي هي الـC ويصير نقص C كمان مرة بترجع اللي هي
+
+623
+00:53:40,180 --> 00:53:45,740
+الـS وكمان مرة بترجع اللي هي الـC بعد هيك بتصير
+
+624
+00:53:45,740 --> 00:53:51,920
+اللي هي تكرر حالها يعني بمعنى آخر جربوها أنتوا
+
+625
+00:53:51,920 --> 00:53:59,920
+هتلاقوا أنه حسب الأُس هنا بتطلع اللي هي عبارة عن
+
+626
+00:53:59,920 --> 00:54:11,180
+هي أو سلبها أو الـS أو سلبهابكل الأحوال بغض النظر
+
+627
+00:54:11,180 --> 00:54:15,600
+اللي هي DN of CN هي حاصل طرح التنتين اللي هي سمنها
+
+628
+00:54:15,600 --> 00:54:23,280
+يا D يا T الـ D تبعت الـ C الفرق طبعا والـ T اللي
+
+629
+00:54:23,280 --> 00:54:29,010
+هي الفرق بين الأسات يعني في الآخرزاد او ناقص و زاد
+
+630
+00:54:29,010 --> 00:54:33,810
+او ناقص سواء هذه او سواء هذه حضرنا من الاصل انها
+
+631
+00:54:33,810 --> 00:54:38,870
+bounded و كله اصغر او يساوي من K اصغر او يساوي ايش
+
+632
+00:54:38,870 --> 00:54:45,860
+K بناء عليه هتطلع هذه هيكabsolute value أصغر أو
+
+633
+00:54:45,860 --> 00:54:51,780
+يساوي هذه أي إن كانت K في X أُس N absolute value
+
+634
+00:54:51,780 --> 00:54:56,880
+على N factorial باشي الحال إذا صلعت عندي هذا
+
+635
+00:54:56,880 --> 00:55:00,860
+المقدار أصغر أو يساوي هذا as N goes to infinity
+
+636
+00:55:00,860 --> 00:55:05,540
+هذا بروح لمن؟ للسفر وهذا independent of N بيصير
+
+637
+00:55:05,540 --> 00:55:08,100
+أكبر أو يساوي السفر يعني هذا اللي جوا بده يصير سفر
+
+638
+00:55:08,100 --> 00:55:12,930
+إذا ال D of X هاشي بدها تساويبتساوي سفر وبكون
+
+639
+00:55:12,930 --> 00:55:21,090
+أثبتنا ان الـ C1 والـ C2 are equivalent الان
+
+640
+00:55:21,090 --> 00:55:27,700
+similar arguments بنفس الاسلوبوخلّيها إليكم إنه
+
+641
+00:55:27,700 --> 00:55:32,120
+اللي هي بنثبت إنه S إذا كان S1 وS2 are two
+
+642
+00:55:32,120 --> 00:55:36,320
+functions such that بيحققوا اللي هما الخواصة اللي
+
+643
+00:55:36,320 --> 00:55:41,700
+بنقولها عنها في النظرية اللي هي S1 double prime
+
+644
+00:55:41,700 --> 00:55:46,400
+بيساوي نفس S1 ونفس ال S2 وهذا عند Zero بيساوي Zero
+
+645
+00:55:46,400 --> 00:55:50,160
+وهذا ال prime عند Zero بيساوي Zero هيتلعلك غصب
+
+646
+00:55:50,160 --> 00:55:52,240
+عنها في الأخر S1 بيساوي S2
+
+647
+00:55:55,620 --> 00:55:59,960
+بس بدل ما تعملوا على ال D يعني ساموا اللي هو T
+
+648
+00:55:59,960 --> 00:56:04,040
+بتساوي S1 نقص S2 وطبقوا الشروط وامشوا نفس اللي
+
+649
+00:56:04,040 --> 00:56:07,580
+امشيناها هتلاقوا حالكم انه لازم تطلع ال S1 بساوي
+
+650
+00:56:07,580 --> 00:56:11,800
+S2 وبناء عليه صار ال ال two functions ال C والS
+
+651
+00:56:11,800 --> 00:56:17,820
+are unique functions طيب
+
+652
+00:56:17,820 --> 00:56:24,720
+ال answer المؤهليننذهب بالاتجاه نثبت أن افترض أن
+
+653
+00:56:24,720 --> 00:56:27,880
+الـ Definition of the unique function C من R لR و
+
+654
+00:56:27,880 --> 00:56:32,400
+أسم الـ R لR اللي يتحقق CW prime of X بساوية نقطة
+
+655
+00:56:32,400 --> 00:56:36,650
+C of Xهو الاصل الـ differential equation التاني S
+
+656
+00:56:36,650 --> 00:56:40,990
+w برامج و X ناقص بساوة ناقص S X لكل X element in R
+
+657
+00:56:40,990 --> 00:56:43,470
+و ال C of Zero اللي هو ال condition ال condition
+
+658
+00:56:43,470 --> 00:56:46,350
+بساوة واحد و C برامج و Zero بساوة Zero و ال S of
+
+659
+00:56:46,350 --> 00:56:49,710
+Zero بساوة Zero و ال S برامج و Zero بساوة Zero
+
+660
+00:56:49,710 --> 00:56:57,150
+الدالتين اللي بحقق إن الكلام هذا اللي أثبتنا إنه
+
+661
+00:56:57,150 --> 00:57:02,740
+uniqueبنسميهم respectively the cosine function and
+
+662
+00:57:02,740 --> 00:57:07,600
+the sine function اللي انتوا بتعرفوها وهي cosine X
+
+663
+00:57:07,600 --> 00:57:12,620
+اللي بتعرفوها وهي sin X اللي انتوا بتعرفوها نيجي
+
+664
+00:57:12,620 --> 00:57:17,930
+الآنلاخد بعض الخواص خلينا ناخد اللي هو الخاصية
+
+665
+00:57:17,930 --> 00:57:23,430
+اللي هي نظرية 4.6 بتقول لي إذا كانت f من R ل R is
+
+666
+00:57:23,430 --> 00:57:27,890
+such that f double prime of x بساعة نقص f of x for
+
+667
+00:57:27,890 --> 00:57:30,370
+x element in R يعني هذه ال differential equation
+
+668
+00:57:30,370 --> 00:57:32,810
+بتقوللي هذه ال differential equation هي حلولها
+
+669
+00:57:32,810 --> 00:57:37,690
+مالي then there exist Alpha و BetaSuch that F of X
+
+670
+00:57:37,690 --> 00:57:41,390
+بيسوء Alpha CX زائد Beta S of X بيقوللي هذه ال
+
+671
+00:57:41,390 --> 00:57:45,110
+differential equation حالة انها عبارة عن linear او
+
+672
+00:57:45,110 --> 00:57:47,870
+خليني اقول combination او linear combination
+
+673
+00:57:47,870 --> 00:57:55,890
+between F C of X S of Xخلّيني أخد نسمي g of x إيش
+
+674
+00:57:55,890 --> 00:57:59,590
+بتساوي اللي هي عبارة عن اللي هي ال c of x و ال a
+
+675
+00:57:59,590 --> 00:58:03,770
+sub x نسمي هذه f of zero و f prime of zero for x
+
+676
+00:58:03,770 --> 00:58:10,330
+element in R الآن لو حسبت ال gw prime of x احسبها
+
+677
+00:58:10,330 --> 00:58:15,270
+فاضل هذه
+
+678
+00:58:15,270 --> 00:58:18,550
+مرتين هدولة ثوابت طبعا و هدولة اللي بتنزلك
+
+679
+00:58:18,550 --> 00:58:23,970
+هتلاقيهم اللي هو بساوي ناقص g of xأه .. و لو حسبت
+
+680
+00:58:23,970 --> 00:58:26,730
+الـ g of zero .. g of zero هتلاقيها بساوي of zero
+
+681
+00:58:26,730 --> 00:58:30,110
+لإن هذا سفر وهذا .. هذا واحد وهذا سفر الان صار
+
+682
+00:58:30,110 --> 00:58:33,850
+عندى gw prime of x بساوي نقص g of x و g of zero
+
+683
+00:58:33,850 --> 00:58:40,410
+بساوي f of zero الان احسب الـ g prime of xصار عندى
+
+684
+00:58:40,410 --> 00:58:43,450
+تلات معلومات معلمتين جي دابل برايم of X بيساوي نقص
+
+685
+00:58:43,450 --> 00:58:45,990
+جي of X و جي of Zero بيساوي أف of Zero خلينا نجيب
+
+686
+00:58:45,990 --> 00:58:50,330
+جي برايم of X جي برايم of X مش بتساوي فاضل اللى هى
+
+687
+00:58:50,330 --> 00:58:55,430
+عبارة عن نقص أس of X و هدا تفضلها اللى هى C of X
+
+688
+00:58:55,430 --> 00:58:58,830
+صار عندى اللى هى جي برايم of X بيساوي هذا الكلام
+
+689
+00:58:58,830 --> 00:59:04,720
+زائد هذا الكلامالان احسبلي g prime of 0 هيصير
+
+690
+00:59:04,720 --> 00:59:08,420
+عبارة عن هذا طبعا سفر وهذا هتصير واحد فبصير F
+
+691
+00:59:08,420 --> 00:59:13,750
+prime of 0 صار عندى الان g prime of 0أيش بيساوي يا
+
+692
+00:59:13,750 --> 00:59:18,090
+جماعة؟ جي برايم اف زيرو خليني أطلعلك إياها لفوق
+
+693
+00:59:18,090 --> 00:59:21,750
+صارت جي برايم اف زيرو بيساوي أف برايم اف زيرو و جي
+
+694
+00:59:21,750 --> 00:59:26,730
+اف زيرو بيساوي أف اف زيرو الآن بديش ناشر نعيد اللي
+
+695
+00:59:26,730 --> 00:59:30,950
+هو therefore الآن the functions h بتساوي f ناقص g
+
+696
+00:59:30,950 --> 00:59:35,990
+is such that يعني عرفلي function h أيش بيساوي؟ f
+
+697
+00:59:35,990 --> 00:59:43,220
+ناقص مين ناقص gالآن الـ H double prime لها لو جيت
+
+698
+00:59:43,220 --> 00:59:46,840
+حسبت الـ H double prime اللي هي هتلاقيها بتساوي
+
+699
+00:59:46,840 --> 00:59:50,260
+ناقص H of X بتحسبوها لحالكم H double prime of X
+
+700
+00:59:50,260 --> 00:59:55,140
+عشان بتساوي ناقص H of X لكل X element in R لو حسبت
+
+701
+00:59:55,140 --> 00:59:59,670
+اللي هي H of Zeroهتساوي اللي هي zero لإنه احنا
+
+702
+00:59:59,670 --> 01:00:03,010
+أثبتنا ان g of zero ساوي f of zero وh prime of
+
+703
+01:00:03,010 --> 01:00:07,110
+zero h prime of zero اللي هي عبارة عن f prime نقص
+
+704
+01:00:07,110 --> 01:00:09,210
+g prime عند ال zero التانية و التساويات إذا إيه مش
+
+705
+01:00:09,210 --> 01:00:15,170
+ساوي سفر الان it then thus it follows as in the
+
+706
+01:00:15,170 --> 01:00:19,430
+proof of the preceding theorem النظرية الماضية إن
+
+707
+01:00:19,430 --> 01:00:22,170
+h of x إيه مش ساوي سفر يعني بدك تعمل نفس اللي
+
+708
+01:00:22,170 --> 01:00:25,180
+عملناها قبل هيك على ال Taylor's theorem و الاخرينو
+
+709
+01:00:25,180 --> 01:00:29,300
+تصل ال H of X بساوية 0 لكل X ومنه هتقول F of X
+
+710
+01:00:29,300 --> 01:00:33,320
+بساوية G of X مدام F of X بساوية G of X إذن هذه
+
+711
+01:00:33,320 --> 01:00:38,580
+اللي هي F of X بتطلع F of X بتساوي اللي هي هذا
+
+712
+01:00:38,580 --> 01:00:42,780
+اسمها Alpha وهذا اسمها Beta وبكون عندي هي اللي هو
+
+713
+01:00:42,780 --> 01:00:46,280
+solution لاللي هي ال differential equation اللي
+
+714
+01:00:46,280 --> 01:00:50,260
+احنا حكينا عنها بديش أعيد نفس الكلام عشان هي كأنا
+
+715
+01:00:50,260 --> 01:00:52,780
+اختصرت لأن الحسابات كلها مشابهة
+
+716
+01:00:58,150 --> 01:01:03,430
+الان بدنا اللي هي 8 4 7 the function C is even and
+
+717
+01:01:03,430 --> 01:01:07,570
+S is odd in the sense that هو بتكون الصحيحة دي
+
+718
+01:01:07,570 --> 01:01:10,610
+الأصل كلها تكون أو معظمها اللي هي عبارة عن
+
+719
+01:01:10,610 --> 01:01:14,430
+exercises لإنها تطبيقات على النظرية اللي قلتها في
+
+720
+01:01:14,430 --> 01:01:19,630
+الأول ال C ناقص X هيسوء C of X يعني عبارة عن even
+
+721
+01:01:19,630 --> 01:01:23,230
+function S ناقص X بيسوء ناقص S of X لكل X element
+
+722
+01:01:23,230 --> 01:01:28,590
+in Rالآن if x,y المتنار then we have the addition
+
+723
+01:01:28,590 --> 01:01:32,230
+formula c of x زي اد y بيسبب c of x في c of y نقص
+
+724
+01:01:32,230 --> 01:01:35,950
+s of x وs of y اللي هي اللي بتعرفوها انتوا sign ال
+
+725
+01:01:35,950 --> 01:01:38,350
+x زي اد y بيسبب sign ال x بكسان ال y زي اد كسان ال
+
+726
+01:01:38,350 --> 01:01:42,090
+y في sign ال x وهكذا هي هذه الخواص اللي احنا
+
+727
+01:01:42,090 --> 01:01:47,570
+عارفينها قبل هيك وبدنا نشوف كيف اللي هي انبرهن
+
+728
+01:01:47,570 --> 01:01:53,660
+النظرية نشوف البرهان تبع النظريةالآن سمّيلي Phi of
+
+729
+01:01:53,660 --> 01:01:59,370
+X بيساوي C of ماقص X لكل X element in Rاحسبلي الـ
+
+730
+01:01:59,370 --> 01:02:05,010
+phi w prime of x لو اتيت فضلت هذه مرتين هتلاقيها
+
+731
+01:02:05,010 --> 01:02:09,950
+ناقص phi of x وهذا الكلام تفضله سهل و بتجيبه لحالك
+
+732
+01:02:09,950 --> 01:02:15,050
+الآن احسبلي ال phi of 0 phi of 0 هتساوي ايش؟
+
+733
+01:02:15,050 --> 01:02:19,350
+بتساوي واحد phi prime of 0 هتطلع اللي هي عبارة عن
+
+734
+01:02:19,350 --> 01:02:24,430
+اللي هي ال sign ال sign ايش معناها؟ بتساوي 0 عند
+
+735
+01:02:24,430 --> 01:02:30,730
+ال zeroالان صار عندى الان الفاى هي مين؟ هي الـC
+
+736
+01:02:30,730 --> 01:02:35,710
+ليش الفاى هي الـC؟ لأن حققت الفاى اللى فرضتها
+
+737
+01:02:35,710 --> 01:02:40,790
+بساوية C-X شروط الـC و الـC is unique إذا الفاى H
+
+738
+01:02:40,790 --> 01:02:47,200
+بتساوي Cإذا صار عندى الان C-X بيساوي C of X الفكرة
+
+739
+01:02:47,200 --> 01:02:52,060
+واضحة أنه أنا جبت سميت الـC-X هي الفاية واتبتت أن
+
+740
+01:02:52,060 --> 01:02:57,060
+هذه الفاية بتحقق الشرطين اللي احنا حكينا عنهم في
+
+741
+01:02:57,060 --> 01:03:01,300
+الأول الدالة اللي بتخلي الـC is unique فصارت
+
+742
+01:03:01,300 --> 01:03:07,080
+الفاية أيش بتساوي الـCالان بمعنى اخر صارت C of X
+
+743
+01:03:07,080 --> 01:03:10,720
+هي C من ناقص X اللي فرضناها الـ Phi in a similar
+
+744
+01:03:10,720 --> 01:03:16,040
+way برضه بدك تعملاش S of ناقص X اللي هو بتسمي اللي
+
+745
+01:03:16,040 --> 01:03:20,480
+هي ال .. الفا .. بتسميها بـ Psi مثلا Psi بتساوي
+
+746
+01:03:20,480 --> 01:03:25,020
+مين؟بتساوي اللي هو ناقص هتجيبها سامي بصي بيساوي
+
+747
+01:03:25,020 --> 01:03:29,540
+ناقص S ناقص X و تجيب الشروط اللي هي تبعها ال sign
+
+748
+01:03:29,540 --> 01:03:33,500
+هتلاقيها متطابق متحققة و بما ان ال S is unique او
+
+749
+01:03:33,500 --> 01:03:37,240
+ال sign is unique اذا اللي هي اللي حققتها اللي
+
+750
+01:03:37,240 --> 01:03:41,500
+كتبتها بتساوي ال S وصارين ايه جهتين متساويات بنفس
+
+751
+01:03:41,500 --> 01:03:47,980
+الأسلوب طيب الآن نيجي نثبت اللي هو مين VI شوفوا
+
+752
+01:03:47,980 --> 01:03:52,850
+صلى على النبي عليه الصلاة والسلامهنفير نبره ان vi
+
+753
+01:03:52,850 --> 01:03:55,830
+let y المتن عار بيجيبنا let f of x بيساوي c of x
+
+754
+01:03:55,830 --> 01:04:00,070
+زي dash زي y for x المتن عار الان a calculation
+
+755
+01:04:00,070 --> 01:04:04,230
+shows that يعني فضلي هذا مرتين التفاضل سهل والله
+
+756
+01:04:04,230 --> 01:04:08,010
+يا جماعة عشانك انا يعني مابديش نضيع واجتنا في
+
+757
+01:04:08,010 --> 01:04:12,190
+التفاضلأو في الحسابات f w prime of x طبعا فاضل
+
+758
+01:04:12,190 --> 01:04:15,430
+بالنسبالي x y ايش ما على f x ثابت أسيبك منها لأن
+
+759
+01:04:15,430 --> 01:04:18,590
+لو فضلت مرتين هتلاقي ناقص f of x for x element in
+
+760
+01:04:18,590 --> 01:04:22,550
+R مدام f w prime بيساوي ناقص f of x بالحصارة اللي
+
+761
+01:04:22,550 --> 01:04:26,850
+هي حل المعادلة القفاضلية هذه اللي هي بالنظرية اللي
+
+762
+01:04:26,850 --> 01:04:32,880
+قبل شويه هي 8 4 6 هتكون اللي هي ال f of xعبارة عن
+
+763
+01:04:32,880 --> 01:04:36,300
+linear combination هذه اللي هو α C of X زي Beta S
+
+764
+01:04:36,300 --> 01:04:39,840
+of X والـ F of X مين هي احنا فرضينا C of XY صارت
+
+765
+01:04:39,840 --> 01:04:43,180
+هذه عبارة عن هذه يعني هذه بتساوي هذه من حال
+
+766
+01:04:43,180 --> 01:04:47,260
+المعادلة التفاضلية هذه بواسطة النظرية هذه وهذه
+
+767
+01:04:47,260 --> 01:04:51,330
+اصلا انا نسميها هيك صارت هذه بتساوي هذا المقطعالان
+
+768
+01:04:51,330 --> 01:04:55,350
+جيبله F prime F prime of X اللي هي هذه التفضيلة
+
+769
+01:04:55,350 --> 01:04:59,070
+بـC ناقص S of X زائد Y وفاضل هذه بيطلع اللي هو
+
+770
+01:04:59,070 --> 01:05:02,910
+عبارة عن ناقص Alpha S of X زائد Beta C of X ثم
+
+771
+01:05:02,910 --> 01:05:09,710
+فضلت ماشي الان خد X بيساوي 0 X بيساوي 0 لما X
+
+772
+01:05:09,710 --> 01:05:18,250
+بتساوي 0بنحصل الآن عندى .. بصير عندى عوض X بتساوي
+
+773
+01:05:18,250 --> 01:05:25,190
+سفر في اللى هى المعادلة اللى عندى هنا بصير عندى S
+
+774
+01:05:25,190 --> 01:05:31,360
+of Zero و C of Zeroبساومين F of Zero F of Zero
+
+775
+01:05:31,360 --> 01:05:40,100
+اللي هي عبارة عن F of Zero هي C of Y هذه فاهمين
+
+776
+01:05:40,100 --> 01:05:44,560
+عليها؟ خليني أقولها واضحة أحسن الآن بدنا ناخد F
+
+777
+01:05:44,560 --> 01:05:52,600
+مين؟ F عند Zero فاشر؟ بيصير هذا C of Y بساومي
+
+778
+01:05:52,600 --> 01:05:58,190
+Alphaفي c of zero واحد وهذه beta s of zero زيرو
+
+779
+01:05:58,190 --> 01:06:03,890
+إذا صارت عندي ال alpha بتساوي c of y هاي واحدة لأن
+
+780
+01:06:03,890 --> 01:06:09,990
+similarly اللي هو ناقص خد عند ال zero عند ال zero
+
+781
+01:06:09,990 --> 01:06:15,570
+بيصير ناقص s of y بساوي هذي بيصير سفر وهذه بيصير
+
+782
+01:06:15,570 --> 01:06:20,070
+beta بساوي beta الآن بنعوض وين في ال formula
+
+783
+01:06:20,070 --> 01:06:26,370
+الأولىبصير عندى الان ال formula اللى عندى اللى هى
+
+784
+01:06:26,370 --> 01:06:30,010
+بتحط مكان C of Y بسوة Alpha و الفورمولة الأولى
+
+785
+01:06:30,010 --> 01:06:35,770
+اللى احنا عملناها بصير عندى اللى هى C
+
+786
+01:06:38,530 --> 01:06:43,570
+Half X زائد Y بساوة Alpha اللي هي اسمها قلنا طلعت
+
+787
+01:06:43,570 --> 01:06:48,310
+عندنا C of Y بصير C of Y في C of X هيها مظبوطة
+
+788
+01:06:48,310 --> 01:06:53,130
+وهذه ناقص S مكان ال Beta بصير ناقص S of Y في S of
+
+789
+01:06:53,130 --> 01:07:01,740
+X صحيحة إذا بكون إحنا خلصنا اللي بدناياه وهذهبتعوض
+
+790
+01:07:01,740 --> 01:07:08,240
+عن اللي هي Alpha C of Y بC نقص C of Y وهذه بتعوضها
+
+791
+01:07:08,240 --> 01:07:13,440
+هنا بتطلع هذه بتساوي نقص هذه اعمل الحساب الأخير
+
+792
+01:07:13,440 --> 01:07:17,400
+وظروف الدقيقتين بنقص بتطلع عندك هذا المخضر يعني
+
+793
+01:07:17,400 --> 01:07:23,840
+هذه التعويض فيها عن قيمة Alpha و Beta بهذه هنا
+
+794
+01:07:23,840 --> 01:07:30,860
+بتطلع هذه وهذه التعويض هذهعن alpha و beta هنا اللي
+
+795
+01:07:30,860 --> 01:07:35,420
+بتطلع الأولى بكون احنا خلصنا اللي هو اثبات هذه
+
+796
+01:07:35,420 --> 01:07:42,800
+اللي هي النظرية وضال عندي اللي هو النظرية هذه
+
+797
+01:07:42,800 --> 01:07:47,350
+الأخيرةوالباقي اللي هو لو اتطلعتوا على اللي هي
+
+798
+01:07:47,350 --> 01:07:52,450
+باقي النظريات اللي هي بس اتطلع لعند الها بيكون
+
+799
+01:07:52,450 --> 01:07:55,990
+احنا بيكون خلصنا chapter اللي هو المطلوب في
+
+800
+01:07:55,990 --> 01:08:00,350
+chapter في تمانية أربعة هذه اللي هي ال theorem
+
+801
+01:08:00,350 --> 01:08:03,530
+خليني أطلعلك عليها على السريع لإن حسابات كلها
+
+802
+01:08:03,530 --> 01:08:07,840
+بتعملها لحالك أكيد بتعرفإذا كان الـ x أكبر أو أقل
+
+803
+01:08:07,840 --> 01:08:12,880
+من 0، فهناك Sx بنقص الـ x والـ x والـ c of x بين
+
+804
+01:08:12,880 --> 01:08:16,780
+الواحد نقصه x أربع عزر تو واحد والـ c of x بقدر
+
+805
+01:08:16,780 --> 01:08:18,820
+أكمل الـ polynomial
+
+806
+01:08:21,810 --> 01:08:25,650
+باللي عملناها اللي هي الـCN of X اللي جابله شوية و
+
+807
+01:08:25,650 --> 01:08:28,430
+الـSN of X لأنه في النهاية limitها ال series
+
+808
+01:08:28,430 --> 01:08:30,790
+الأولى as N goes to infinity ما أنتوا عارفين اللي
+
+809
+01:08:30,790 --> 01:08:34,990
+هي نقدر نكتب الـC of X على صورة ال series اللي في
+
+810
+01:08:34,990 --> 01:08:38,130
+الأول و الS of X عبارة عن limit ال series التانية
+
+811
+01:08:38,130 --> 01:08:43,390
+ال terms إضافة term و طرح term و وجوف عند حدود بال
+
+812
+01:08:43,390 --> 01:08:47,090
+term بتعمل ال inequality اللي عندنا اللي هو كل هذه
+
+813
+01:08:47,090 --> 01:08:49,810
+شغلات اللي أخدناها في ال calculus مافيش داعي
+
+814
+01:08:49,810 --> 01:08:55,650
+للتفصيل فيهاالان الاولىعندي ال .. ال .. احنا قلنا
+
+815
+01:08:55,650 --> 01:08:59,990
+sign تربيع زائد plus sign تربيع بساوي واحد يعني في
+
+816
+01:08:59,990 --> 01:09:02,670
+النهاية ال C of T بين ماقص واحد و واحد ده ممكن
+
+817
+01:09:02,670 --> 01:09:06,710
+هتتجاوزها لأنه بتختل اللي هي ال C .. لو كانت أكبر
+
+818
+01:09:06,710 --> 01:09:10,750
+من واحد معناته بـ C .. C of T تربيع زائد S of T
+
+819
+01:09:10,750 --> 01:09:14,550
+تربيع يتجاوز من واحد لكن مجموح بسوء واحد إذا هدي
+
+820
+01:09:14,550 --> 01:09:18,630
+بين .. يعني هدي اللي وين جاية من C تربيع زائد S
+
+821
+01:09:18,630 --> 01:09:23,030
+تربيع بساوي واحد طيباللي هنعمل integration لجهتين
+
+822
+01:09:23,030 --> 01:09:26,990
+من صفر لعين DX هذه اللي هي بتطلع اللي هي main ال S
+
+823
+01:09:26,990 --> 01:09:30,750
+of X وهذه بتطلع أنخس X وهذه بتطلع X فالمعمل ال
+
+824
+01:09:30,750 --> 01:09:35,990
+integration هدول الان بدي
+
+825
+01:09:35,990 --> 01:09:43,300
+أجيب التانية فضلي ال S of Tهذه وهذه فاضليها هذه
+
+826
+01:09:43,300 --> 01:09:48,460
+فاضليها بنقل صفر عند X بتطلع اللي هي بين هذه وهذه
+
+827
+01:09:48,460 --> 01:09:53,660
+هيتفاضوا لهذه وهذه الان بدنا اللي هو نجيب الواحد
+
+828
+01:09:53,660 --> 01:09:58,280
+ناقص X السربيع اللي هو هذا قيمته اللي هو عبارة عن
+
+829
+01:09:58,280 --> 01:10:02,600
+C في X بالساوية واحد ناقص هذه بنضرب الناقص و بنجمع
+
+830
+01:10:02,600 --> 01:10:08,280
+الجهتين واحد بطلع عندي اللي هي ال inequality اللي
+
+831
+01:10:08,280 --> 01:10:13,190
+عندي هذهزي ما قلنا ضربنا في ناقص و جمعنا واحد طلعت
+
+832
+01:10:13,190 --> 01:10:16,590
+هذه او عوضنا هذه مكان هذه تعويض عادي و بعدين ضربنا
+
+833
+01:10:16,590 --> 01:10:20,690
+في ناقص فبصير عندى هذه المقدار اللى عندى c of x
+
+834
+01:10:20,690 --> 01:10:24,310
+أكبر سواء هذه و ناقص هذه و اللى بتستخدم الثانية
+
+835
+01:10:24,310 --> 01:10:27,670
+بتطلع نفس الاشي الموضوع موضوع حسابات بحت عشان هيك
+
+836
+01:10:27,670 --> 01:10:31,510
+بحت عشان هيك مافي داعي للتكميل و انتوا بتكملوا
+
+837
+01:10:31,510 --> 01:10:35,150
+باقى الحسابات اللى هي المطلوبة في هذا اللى هي
+
+838
+01:10:35,150 --> 01:10:42,700
+النظرية الان الجزء المتبقى هذاحبيته تطلعوا عليه
+
+839
+01:10:42,700 --> 01:10:46,880
+لكن اللي هو مش مطلوب من ضمن حديثنا و هيك ممكن
+
+840
+01:10:46,880 --> 01:10:52,700
+نخلصنااللي هو section تمانية أربعة متسمة سبتر
+
+841
+01:10:52,700 --> 01:10:58,420
+تمانية هذا اللي هو الجزء الثالث من المادة الجزء
+
+842
+01:10:58,420 --> 01:11:00,160
+الأول كان ال differentiation والتاني ال
+
+843
+01:11:00,160 --> 01:11:01,960
+integration والتالت اللي هو point twice
+
+844
+01:11:01,960 --> 01:11:05,700
+convergence المرة الجاية ان شاء الله اللي هو بنبدأ
+
+845
+01:11:05,700 --> 01:11:09,880
+في الجزء الرابع من المادة اللي هو ال series وان
+
+846
+01:11:09,880 --> 01:11:16,590
+شاء الله بقدر الإمكان هأحكي الجزء الخامسفي .. من
+
+847
+01:11:16,590 --> 01:11:20,510
+المادة اللي هو عبارة عن اللي هو Topology in R أو
+
+848
+01:11:20,510 --> 01:11:23,570
+اللي هو اللي هي العلاقة بين اللي هي Topological
+
+849
+01:11:23,570 --> 01:11:27,410
+Spaces وNormed Spaces وHilbert Spaces إلى آخره
+
+850
+01:11:27,410 --> 01:11:33,530
+ويمكن بتكون هي اتجاوزت خلنا نقول حد ال .. ال .. ال
+
+851
+01:11:33,530 --> 01:11:37,150
+.. المطلوب في المادة لكن على أساس إنه لو يكون
+
+852
+01:11:37,150 --> 01:11:42,210
+التصوير كامل للو .. للوصف اللي أنا اقتراحتهأو اللي
+
+853
+01:11:42,210 --> 01:11:47,230
+هو الوصف اللي مقترح في الجسم اللي هو لعين نهاية ال
+
+854
+01:11:47,230 --> 01:11:49,990
+series و طبعا هذا الكلام قلنا احنا فيه له سبب لما
+
+855
+01:11:49,990 --> 01:11:53,450
+بدأنا في الأول انه ال series بتاخدوها في advanced
+
+856
+01:11:53,450 --> 01:11:56,990
+calculus واحد فعشان هيك يمكن اللي هو
+
+857
+01:12:00,350 --> 01:12:03,810
+هيكون منطقي أنه ما ناخدش ال series و ناخد بدلها
+
+858
+01:12:03,810 --> 01:12:07,110
+اللي هو ال topology انار أو اللي هو العلاقة بين ال
+
+859
+01:12:07,110 --> 01:12:10,550
+topological spaces و ال metric spaces و إلى لقاء و
+
+860
+01:12:10,550 --> 01:12:12,650
+السلام آخر والسلام عليكم و رحمه الله وبركاته
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..262b2c615ea8a581870f09feccc7c3ab91b63376
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_raw.json
@@ -0,0 +1 @@
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0.96533203125}, {"start": 1074.02, "end": 1074.34, "word": " عندى", "probability": 0.621826171875}, {"start": 1074.34, "end": 1075.02, "word": " بالظبط", "probability": 0.96337890625}, {"start": 1075.02, "end": 1075.22, "word": " ال", "probability": 0.63623046875}, {"start": 1075.22, "end": 1075.64, "word": " formula", "probability": 0.912109375}, {"start": 1075.64, "end": 1076.14, "word": " هذه", "probability": 0.6162109375}, {"start": 1076.14, "end": 1076.76, "word": " يعني", "probability": 0.900634765625}, {"start": 1076.76, "end": 1077.08, "word": " صح", "probability": 0.96533203125}, {"start": 1077.08, "end": 1077.48, "word": " بيصير", "probability": 0.84033203125}, {"start": 1077.48, "end": 1077.8, "word": " عندى", "probability": 0.94873046875}, {"start": 1077.8, "end": 1078.22, "word": " بتعملها", "probability": 0.926513671875}, {"start": 1078.22, "end": 1078.66, "word": " لحالك", "probability": 0.9588623046875}, {"start": 1078.66, "end": 1078.88, "word": " لان", "probability": 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"probability": 0.0791015625}, {"start": 1085.65, "end": 1086.13, "word": " بتساوي", "probability": 0.6683349609375}, {"start": 1086.13, "end": 1086.31, "word": " k", "probability": 0.35302734375}, {"start": 1086.31, "end": 1086.69, "word": " زائد", "probability": 0.7581380208333334}, {"start": 1086.69, "end": 1087.01, "word": " واحد", "probability": 0.849609375}, {"start": 1087.01, "end": 1087.25, "word": " إذا", "probability": 0.5628662109375}, {"start": 1087.25, "end": 1087.59, "word": " هذا", "probability": 0.51416015625}, {"start": 1087.59, "end": 1087.89, "word": " صار", "probability": 0.957763671875}, {"start": 1087.89, "end": 1088.51, "word": " المقدار", "probability": 0.94873046875}, {"start": 1088.51, "end": 1089.15, "word": " صحيح", "probability": 0.9942626953125}, {"start": 1089.15, "end": 1089.79, "word": " دائما", "probability": 0.88330078125}, {"start": 1089.79, "end": 1090.53, "word": " for", "probability": 0.8671875}, {"start": 1090.53, "end": 1091.05, "word": " mean", "probability": 0.33935546875}, {"start": 1091.05, "end": 1092.05, "word": " for", "probability": 0.9052734375}, {"start": 1092.05, "end": 1092.67, "word": " any", "probability": 0.904296875}, {"start": 1092.67, "end": 1093.49, "word": " k", "probability": 0.5146484375}, {"start": 1093.49, "end": 1094.75, "word": " for", "probability": 0.90283203125}, {"start": 1094.75, "end": 1095.05, "word": " any", "probability": 0.9111328125}, {"start": 1095.05, "end": 1095.41, "word": " n", "probability": 0.32861328125}, {"start": 1095.41, "end": 1096.17, "word": " element", "probability": 0.8701171875}, {"start": 1096.17, "end": 1096.51, "word": " in", "probability": 0.9326171875}, {"start": 1096.51, "end": 1096.85, "word": " n", "probability": 0.311767578125}, {"start": 1096.85, "end": 1098.07, "word": " الان", "probability": 0.7568359375}, {"start": 1098.07, "end": 1099.45, "word": " واضح", "probability": 0.9611002604166666}, {"start": 1099.45, "end": 1100.43, "word": " ان", "probability": 0.427001953125}, {"start": 1100.43, "end": 1101.23, "word": " الخطوات", "probability": 0.955810546875}, {"start": 1101.23, "end": 1102.01, "word": " مشابهة", "probability": 0.916015625}, {"start": 1102.01, "end": 1102.73, "word": " لخطوات", "probability": 0.9619140625}, {"start": 1102.73, "end": 1102.89, "word": " ال", "probability": 0.69677734375}, {"start": 1102.89, "end": 1103.75, "word": " exponential", "probability": 0.46728515625}, {"start": 1103.75, "end": 1105.85, "word": " نيجي", "probability": 0.84228515625}, {"start": 1105.85, "end": 1106.31, "word": " الآن", "probability": 0.725341796875}, {"start": 1106.31, "end": 1108.29, "word": " قصّل", "probability": 0.40655517578125}, {"start": 1108.29, "end": 1108.51, "word": " n", "probability": 0.34521484375}, {"start": 1108.51, "end": 1108.97, "word": " prime", "probability": 0.365234375}], "temperature": 1.0}, {"id": 47, "seek": 112043, "start": 1118.77, "end": 1120.43, "text": "نجي الآن", "tokens": [1863, 7435, 1829, 6024, 48506], "avg_logprob": -1.0065104564030964, "compression_ratio": 0.625, "no_speech_prob": 3.2782554626464844e-06, "words": [{"start": 1118.77, "end": 1120.17, "word": "نجي", "probability": 0.37615966796875}, {"start": 1120.17, "end": 1120.43, "word": " الآن", "probability": 0.6202392578125}], "temperature": 1.0}, {"id": 48, "seek": 114820, "start": 1122.32, "end": 1148.2, "text": "وين رايح؟ زي اللي بكونت رايحي الأيام الـ exponential هثبتلك أن الـ sequence هذه converged uniformly وهذه طبعا هتصبح converged uniformly automatic فهتصبح اللي هي differentiable لأنه بيصير طبقة النظرية اللي هي تبعها الـ differentiability بيصير عند مادام differentiable اللي هي ال derivative اللي لها exist و هتكون ال derivative متحقق الشروط و بيكون خلاص نعم نشوف أش بقول طيب الآن", "tokens": [2407, 9957, 12602, 47302, 5016, 22807, 30767, 1829, 13672, 1829, 4724, 30544, 2655, 12602, 47302, 5016, 1829, 16247, 1829, 10943, 2423, 39184, 21510, 8032, 12984, 3555, 2655, 23275, 14739, 2423, 39184, 8310, 29538, 9652, 3004, 48806, 37037, 24192, 23032, 3555, 3615, 995, 8032, 2655, 9381, 49628, 9652, 3004, 48806, 12509, 6156, 3224, 2655, 9381, 49628, 13672, 1829, 39896, 819, 9364, 5296, 33456, 3224, 4724, 1829, 9381, 13546, 23032, 3555, 28671, 28239, 19913, 2288, 10632, 13672, 1829, 39896, 6055, 3555, 3615, 11296, 2423, 39184, 27372, 2310, 4724, 1829, 9381, 13546, 43242, 3714, 18513, 10943, 819, 9364, 13672, 1829, 39896, 2423, 13760, 13672, 1829, 5296, 11296, 2514, 4032, 8032, 2655, 30544, 2423, 13760, 44650, 5016, 4587, 4587, 25124, 32887, 9566, 4032, 4724, 1829, 30544, 16490, 1211, 33546, 8717, 25957, 8717, 8592, 38688, 5551, 8592, 4724, 39648, 23032, 1829, 3555, 6024, 48506], "avg_logprob": -0.34107142005647934, "compression_ratio": 1.8729096989966556, "no_speech_prob": 4.112720489501953e-06, "words": [{"start": 1122.32, "end": 1122.94, "word": "وين", "probability": 0.21954345703125}, {"start": 1122.94, "end": 1123.48, "word": " رايح؟", "probability": 0.78619384765625}, {"start": 1123.48, "end": 1123.68, "word": " زي", "probability": 0.72265625}, {"start": 1123.68, "end": 1123.76, "word": " اللي", "probability": 0.80029296875}, {"start": 1123.76, "end": 1124.1, "word": " بكونت", "probability": 0.556884765625}, {"start": 1124.1, "end": 1124.52, "word": " رايحي", "probability": 0.8104248046875}, {"start": 1124.52, "end": 1124.86, "word": " الأيام", "probability": 0.719970703125}, {"start": 1124.86, "end": 1125.02, "word": " الـ", "probability": 0.8681640625}, {"start": 1125.02, "end": 1125.5, "word": " exponential", "probability": 0.6494140625}, {"start": 1125.5, "end": 1126.64, "word": " هثبتلك", "probability": 0.85078125}, {"start": 1126.64, "end": 1126.8, "word": " أن", "probability": 0.46337890625}, {"start": 1126.8, "end": 1127.58, "word": " الـ", "probability": 0.593505859375}, {"start": 1127.58, "end": 1128.26, "word": " sequence", "probability": 0.91064453125}, {"start": 1128.26, "end": 1128.54, "word": " هذه", "probability": 0.609375}, {"start": 1128.54, "end": 1129.06, 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{"start": 1134.38, "end": 1134.72, "word": " لأنه", "probability": 0.8611653645833334}, {"start": 1134.72, "end": 1135.0, "word": " بيصير", "probability": 0.7520751953125}, {"start": 1135.0, "end": 1135.84, "word": " طبقة", "probability": 0.9130859375}, {"start": 1135.84, "end": 1136.36, "word": " النظرية", "probability": 0.9395751953125}, {"start": 1136.36, "end": 1136.54, "word": " اللي", "probability": 0.936279296875}, {"start": 1136.54, "end": 1136.66, "word": " هي", "probability": 0.921875}, {"start": 1136.66, "end": 1137.02, "word": " تبعها", "probability": 0.63995361328125}, {"start": 1137.02, "end": 1137.1, "word": " الـ", "probability": 0.62548828125}, {"start": 1137.1, "end": 1137.86, "word": " differentiability", "probability": 0.671875}, {"start": 1137.86, "end": 1138.76, "word": " بيصير", "probability": 0.8829345703125}, {"start": 1138.76, "end": 1138.92, "word": " عند", "probability": 0.830078125}, {"start": 1138.92, "end": 1139.22, "word": " مادام", "probability": 0.58154296875}, {"start": 1139.22, "end": 1139.88, "word": " differentiable", "probability": 0.81396484375}, {"start": 1139.88, "end": 1140.46, "word": " اللي", "probability": 0.656005859375}, {"start": 1140.46, "end": 1140.78, "word": " هي", "probability": 0.95361328125}, {"start": 1140.78, "end": 1141.2, "word": " ال", "probability": 0.66552734375}, {"start": 1141.2, "end": 1143.1, "word": " derivative", "probability": 0.5712890625}, {"start": 1143.1, "end": 1143.36, "word": " اللي", "probability": 0.843017578125}, {"start": 1143.36, "end": 1143.46, "word": " لها", "probability": 0.59930419921875}, {"start": 1143.46, "end": 1143.76, "word": " exist", "probability": 0.57470703125}, {"start": 1143.76, "end": 1143.9, "word": " و", "probability": 0.68359375}, {"start": 1143.9, "end": 1144.16, "word": " هتكون", "probability": 0.764404296875}, {"start": 1144.16, "end": 1144.28, "word": " ال", "probability": 0.8818359375}, {"start": 1144.28, "end": 1144.58, "word": " derivative", 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117693, "start": 1148.95, "end": 1176.93, "text": "بعد ما طلعنا هدولة let a أكبر من صفر b given نفس الخطوات تبعت ال exponential then if الabsolute value of x أصغر بسوء a and m أكبر من n أكبر من 2a يعني بتداخد اللي هي الامات لأكبر من n و أكبر من 2a يعني الآن أنا بشتغل عالين يا جماعة الفترة من ناقص a لعند a وأخدت ال a arbitrarily أكبر من صفر لكن fixed الان we have", "tokens": [3555, 22488, 19446, 23032, 1211, 3615, 8315, 8032, 3215, 12610, 3660, 718, 257, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 272, 2212, 8717, 36178, 33962, 9566, 2407, 9307, 6055, 3555, 34268, 2423, 21510, 550, 498, 2423, 17243, 401, 1169, 2158, 295, 2031, 5551, 9381, 17082, 2288, 4724, 3794, 2407, 38207, 257, 293, 275, 5551, 4117, 26890, 9154, 297, 5551, 4117, 26890, 9154, 568, 64, 37495, 22653, 39894, 3215, 47283, 3215, 13672, 1829, 39896, 2423, 10943, 9307, 5296, 10721, 4117, 26890, 9154, 297, 4032, 5551, 4117, 26890, 9154, 568, 64, 37495, 22653, 6024, 48506, 41850, 4724, 8592, 2655, 17082, 1211, 6225, 6027, 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الإسلوب اللي قبل بشوية اللي هي بما أنه هذه اللي هي differentiable sequence of functions و converge uniform to some function إذا هذه هتكون اللي هي عبارة عن اللي هي الـC الأصلية differentiable و ال derivative فيه لها مين؟ الناقص S", "tokens": [38688, 1829, 8717, 36178, 2423, 30543, 2655, 41850, 4724, 39648, 2423, 39184, 34, 45, 8717, 36178, 11296, 39894, 32887, 5016, 9452, 5296, 2304, 9957, 22807, 24976, 39184, 34, 44945, 36178, 33688, 3794, 1211, 37746, 13672, 1829, 12174, 36150, 4724, 8592, 2407, 10632, 13672, 1829, 39896, 4724, 15042, 14739, 3224, 29538, 13672, 1829, 39896, 819, 9364, 8310, 295, 6828, 4032, 41881, 9452, 281, 512, 2445, 11933, 15730, 29538, 8032, 2655, 30544, 13672, 1829, 39896, 6225, 3555, 9640, 3660, 18871, 13672, 1829, 39896, 2423, 39184, 34, 16247, 36520, 10632, 819, 9364, 4032, 2423, 13760, 8978, 3224, 5296, 11296, 3714, 9957, 22807, 2423, 8315, 4587, 9381, 318], "avg_logprob": -0.2364583276567005, "compression_ratio": 1.7165354330708662, "no_speech_prob": 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let E واحد .. نفترض أنه في E واحد و E اتنين و سمها فرق بيساوي D و في الآخر روحنا لفرق بيساوي إيش؟ بيساوي سفر هنا نفس الاشي هنشتغل و برضه هنستخدم اللي هو ال Taylor's theorem زي ما استخدمنا هناك ال Taylor's theorem يعني هتلاقيه ال sketch للبرهان هو نفس ال sketch الأولاني عشان هيك هتلاقوني سريع فيه بدنا نثبت أن ال C و ال S are unique functions", "tokens": [2407, 8608, 2304, 8315, 11296, 13672, 1829, 31439, 718, 462, 36764, 24401, 4386, 8717, 5172, 2655, 43042, 14739, 3224, 8978, 462, 36764, 24401, 4032, 462, 1975, 2655, 1863, 9957, 4032, 8608, 2304, 11296, 6156, 2288, 4587, 4724, 1829, 3794, 995, 45865, 413, 4032, 8978, 6024, 95, 34740, 12602, 2407, 5016, 8315, 5296, 5172, 2288, 4587, 4724, 1829, 3794, 995, 45865, 11933, 1829, 8592, 22807, 4724, 1829, 3794, 995, 45865, 8608, 5172, 2288, 34105, 8717, 36178, 2423, 33599, 1829, 8032, 1863, 8592, 2655, 17082, 1211, 4032, 4724, 43042, 3224, 8032, 1863, 14851, 9778, 40448, 13672, 1829, 31439, 2423, 12060, 311, 20904, 30767, 1829, 19446, 44713, 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" theorem", "probability": 0.7470703125}, {"start": 2730.39, "end": 2730.95, "word": " زي", "probability": 0.965576171875}, {"start": 2730.95, "end": 2731.03, "word": " ما", "probability": 0.97509765625}, {"start": 2731.03, "end": 2731.59, "word": " استخدمنا", "probability": 0.965087890625}, {"start": 2731.59, "end": 2731.91, "word": " هناك", "probability": 0.9580078125}, {"start": 2731.91, "end": 2732.01, "word": " ال", "probability": 0.89697265625}, {"start": 2732.01, "end": 2732.43, "word": " Taylor's", "probability": 0.8857421875}, {"start": 2732.43, "end": 2732.77, "word": " theorem", "probability": 0.88671875}, {"start": 2732.77, "end": 2733.37, "word": " يعني", "probability": 0.960693359375}, {"start": 2733.37, "end": 2733.95, "word": " هتلاقيه", "probability": 0.836328125}, {"start": 2733.95, "end": 2734.13, "word": " ال", "probability": 0.86083984375}, {"start": 2734.13, "end": 2734.47, "word": " sketch", "probability": 0.87548828125}, {"start": 2734.47, "end": 2735.09, 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0.9405924479166666}, {"start": 3705.71, "end": 3706.13, "word": " كيف", "probability": 0.95556640625}, {"start": 3706.13, "end": 3706.41, "word": " اللي", "probability": 0.91259765625}, {"start": 3706.41, "end": 3706.73, "word": " هي", "probability": 0.88037109375}, {"start": 3706.73, "end": 3707.57, "word": " انبرهن", "probability": 0.8946533203125}, {"start": 3707.57, "end": 3708.49, "word": " النظرية", "probability": 0.948974609375}, {"start": 3708.49, "end": 3709.53, "word": " نشوف", "probability": 0.798583984375}, {"start": 3709.53, "end": 3710.03, "word": " البرهان", "probability": 0.77813720703125}, {"start": 3710.03, "end": 3710.31, "word": " تبع", "probability": 0.8756510416666666}, {"start": 3710.31, "end": 3710.85, "word": " النظرية", "probability": 0.97509765625}], "temperature": 1.0}, {"id": 159, "seek": 371660, "start": 3712.24, "end": 3716.6, "text": "الآن سمّيلي Phi of X بيساوي C of ماقص X لكل X element in R", "tokens": [6027, 148, 48506, 8608, 2304, 11703, 1829, 20292, 41435, 295, 1783, 4724, 1829, 3794, 995, 45865, 383, 295, 19446, 4587, 9381, 1783, 5296, 28820, 1783, 4478, 294, 497], "avg_logprob": -0.4862607840833993, "compression_ratio": 0.9642857142857143, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 3712.2400000000002, "end": 3712.8, "word": "الآن", "probability": 0.4920247395833333}, {"start": 3712.8, "end": 3713.36, "word": " سمّيلي", "probability": 0.696728515625}, {"start": 3713.36, "end": 3713.52, "word": " Phi", "probability": 0.15576171875}, {"start": 3713.52, "end": 3713.66, "word": " of", "probability": 0.70703125}, {"start": 3713.66, "end": 3713.82, "word": " X", "probability": 0.89453125}, {"start": 3713.82, "end": 3714.18, "word": " بيساوي", "probability": 0.5813232421875}, {"start": 3714.18, "end": 3714.46, "word": " C", "probability": 0.70458984375}, {"start": 3714.46, "end": 3714.62, "word": " of", "probability": 0.57421875}, {"start": 3714.62, "end": 3715.0, "word": " ماقص", "probability": 0.627197265625}, {"start": 3715.0, "end": 3715.28, "word": " X", "probability": 0.96240234375}, {"start": 3715.28, "end": 3715.58, "word": " لكل", "probability": 0.6524658203125}, {"start": 3715.58, "end": 3715.82, "word": " X", "probability": 0.978515625}, {"start": 3715.82, "end": 3716.12, "word": " element", "probability": 0.63916015625}, {"start": 3716.12, "end": 3716.32, "word": " in", "probability": 0.35400390625}, {"start": 3716.32, "end": 3716.6, "word": " R", "probability": 0.998046875}], "temperature": 1.0}, {"id": 160, "seek": 374489, "start": 3718.45, "end": 3744.89, "text": "احسبلي الـ phi w prime of x لو اتيت فضلت هذه مرتين هتلاقيها ناقص phi of x وهذا الكلام تفضله سهل و بتجيبه لحالك الآن احسبلي ال phi of 0 phi of 0 هتساوي ايش؟ بتساوي واحد phi prime of 0 هتطلع اللي هي عبارة عن اللي هي ال sign ال sign ايش معناها؟ بتساوي 0 عند ال zero", "tokens": [39319, 35457, 20292, 2423, 39184, 13107, 261, 5835, 295, 2031, 45164, 1975, 31371, 2655, 6156, 11242, 1211, 2655, 29538, 3714, 43500, 9957, 8032, 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"probability": 0.86474609375}, {"start": 3737.01, "end": 3737.25, "word": " of", "probability": 0.982421875}, {"start": 3737.25, "end": 3737.65, "word": " 0", "probability": 0.8203125}, {"start": 3737.65, "end": 3738.45, "word": " هتطلع", "probability": 0.932421875}, {"start": 3738.45, "end": 3738.61, "word": " اللي", "probability": 0.768310546875}, {"start": 3738.61, "end": 3738.73, "word": " هي", "probability": 0.74560546875}, {"start": 3738.73, "end": 3739.09, "word": " عبارة", "probability": 0.9818115234375}, {"start": 3739.09, "end": 3739.35, "word": " عن", "probability": 0.9951171875}, {"start": 3739.35, "end": 3739.89, "word": " اللي", "probability": 0.785888671875}, {"start": 3739.89, "end": 3740.29, "word": " هي", "probability": 0.908203125}, {"start": 3740.29, "end": 3741.49, "word": " ال", "probability": 0.265380859375}, {"start": 3741.49, "end": 3741.87, "word": " sign", "probability": 0.45166015625}, {"start": 3741.87, "end": 3742.05, "word": " ال", "probability": 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{"start": 3749.25, "end": 3750.03, "word": " مين؟", "probability": 0.6241861979166666}, {"start": 3750.03, "end": 3750.35, "word": " هي", "probability": 0.841796875}, {"start": 3750.35, "end": 3750.73, "word": " الـC", "probability": 0.534912109375}, {"start": 3750.73, "end": 3751.07, "word": " ليش", "probability": 0.6405029296875}, {"start": 3751.07, "end": 3751.37, "word": " الفاى", "probability": 0.9231770833333334}, {"start": 3751.37, "end": 3751.51, "word": " هي", "probability": 0.87109375}, {"start": 3751.51, "end": 3752.37, "word": " الـC؟", "probability": 0.872314453125}, {"start": 3752.37, "end": 3752.67, "word": " لأن", "probability": 0.878662109375}, {"start": 3752.67, "end": 3753.67, "word": " حققت", "probability": 0.8758138020833334}, {"start": 3753.67, "end": 3754.75, "word": " الفاى", "probability": 0.9724934895833334}, {"start": 3754.75, "end": 3755.17, "word": " اللى", "probability": 0.939453125}, {"start": 3755.17, "end": 3755.71, "word": " فرضتها", "probability": 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هي C من ناقص X اللي فرضناها الـ Phi in a similar way برضه بدك تعملاش S of ناقص X اللي هو بتسمي اللي هي ال .. الفا .. بتسميها بـ Psi مثلا Psi بتساوي مين؟", "tokens": [6027, 7649, 4724, 2304, 3615, 1863, 7578, 1975, 34740, 20328, 9640, 2655, 383, 295, 1783, 39896, 383, 9154, 8717, 995, 4587, 9381, 1783, 13672, 1829, 6156, 43042, 8315, 11296, 2423, 39184, 41435, 294, 257, 2531, 636, 4724, 43042, 3224, 47525, 4117, 6055, 25957, 1211, 33599, 318, 295, 8717, 995, 4587, 9381, 1783, 13672, 1829, 31439, 39894, 38251, 1829, 13672, 1829, 39896, 2423, 4386, 27188, 995, 4386, 39894, 38251, 1829, 11296, 4724, 39184, 430, 7691, 50113, 15040, 430, 7691, 39894, 3794, 995, 45865, 3714, 9957, 22807], "avg_logprob": -0.37463662582774493, "compression_ratio": 1.5469613259668509, "no_speech_prob": 8.344650268554688e-07, "words": [{"start": 3783.9, "end": 3784.42, "word": "الان", "probability": 0.662841796875}, {"start": 3784.42, "end": 3785.62, "word": " بمعنى", "probability": 0.7813232421875}, {"start": 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"end": 4309.69, "word": " سبب", "probability": 0.9912109375}, {"start": 4309.69, "end": 4309.99, "word": " لما", "probability": 0.876708984375}, {"start": 4309.99, "end": 4310.39, "word": " بدأنا", "probability": 0.7234700520833334}, {"start": 4310.39, "end": 4310.49, "word": " في", "probability": 0.90087890625}, {"start": 4310.49, "end": 4310.87, "word": " الأول", "probability": 0.932861328125}, {"start": 4310.87, "end": 4311.67, "word": " انه", "probability": 0.640380859375}, {"start": 4311.67, "end": 4311.89, "word": " ال", "probability": 0.978515625}, {"start": 4311.89, "end": 4312.35, "word": " series", "probability": 0.97265625}, {"start": 4312.35, "end": 4313.01, "word": " بتاخدوها", "probability": 0.93701171875}, {"start": 4313.01, "end": 4313.13, "word": " في", "probability": 0.927734375}, {"start": 4313.13, "end": 4313.45, "word": " advanced", "probability": 0.60302734375}, {"start": 4313.45, "end": 4313.83, "word": " calculus", "probability": 0.9921875}, {"start": 4313.83, 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4323.09, "word": " و", "probability": 0.9267578125}, {"start": 4323.09, "end": 4323.41, "word": " ناخد", "probability": 0.9342447916666666}, {"start": 4323.41, "end": 4323.81, "word": " بدلها", "probability": 0.9427083333333334}, {"start": 4323.81, "end": 4323.97, "word": " اللي", "probability": 0.848388671875}, {"start": 4323.97, "end": 4324.19, "word": " هو", "probability": 0.9580078125}, {"start": 4324.19, "end": 4324.35, "word": " ال", "probability": 0.654296875}, {"start": 4324.35, "end": 4324.77, "word": " topology", "probability": 0.7568359375}, {"start": 4324.77, "end": 4325.19, "word": " انار", "probability": 0.2349853515625}, {"start": 4325.19, "end": 4325.47, "word": " أو", "probability": 0.64404296875}, {"start": 4325.47, "end": 4325.69, "word": " اللي", "probability": 0.945068359375}, {"start": 4325.69, "end": 4325.83, "word": " هو", "probability": 0.966796875}, {"start": 4325.83, "end": 4326.29, "word": " العلاقة", "probability": 0.9763997395833334}, {"start": 4326.29, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..90baac16b3028f233d1961917fab7505687be3ae
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EIprZTnXBQc_raw.srt
@@ -0,0 +1,3444 @@
+1
+00:00:05,090 --> 00:00:08,030
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:08,030 --> 00:00:11,070
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه
+
+3
+00:00:11,070 --> 00:00:18,250
+أجمعين هذه المحاضرة رقم 24 مساق تحليل حقيقة 2 طلاب
+
+4
+00:00:18,250 --> 00:00:22,650
+وطالبات الجامعة الإسلامية قسم رياضيات كلية العلوم
+
+5
+00:00:23,470 --> 00:00:26,630
+اليوم هنكمل ان شاء الله ال section الأخير في
+
+6
+00:00:26,630 --> 00:00:29,870
+chapter 8 اللي هو 8 أربعة تحت عنوان the
+
+7
+00:00:29,870 --> 00:00:35,170
+trigonometric functions وهو أيضا اللي هو section
+
+8
+00:00:35,170 --> 00:00:42,210
+أو موضوع تطبيق على اللي هوالـ pointwise and
+
+9
+00:00:42,210 --> 00:00:46,030
+uniform convergence للـ sequence of functions وكيف
+
+10
+00:00:46,030 --> 00:00:50,510
+بدنا نعرف بطريقة مشابهة جدا لمعرفناها المرة
+
+11
+00:00:50,510 --> 00:00:53,330
+الماضية أو اللي قبلها بخصوص الـ exponential
+
+12
+00:00:53,330 --> 00:00:58,710
+function هنعرف اليوم اللي هو بنفس الطريقة كيف نعرف
+
+13
+00:00:58,710 --> 00:01:03,510
+اللي هو الـ sine و الـ cosine as a limit اللي هو
+
+14
+00:01:03,510 --> 00:01:07,390
+of a uniformly convergent sequence of functions
+
+15
+00:01:07,950 --> 00:01:11,230
+العنوان is in the trigonometric functions اللي هو
+
+16
+00:01:11,230 --> 00:01:15,450
+section 8 أربعة النظرية
+
+17
+00:01:15,450 --> 00:01:21,950
+الأولى اللى عندنا اللى مشابهة لنظرية سابقة اللى هي
+
+18
+00:01:21,950 --> 00:01:25,870
+ال exponential اللى من خلالها بدنا نصل لاللي هو
+
+19
+00:01:25,870 --> 00:01:31,370
+تعريف اللى هي ال cosine و ال sineالنظرية بتقول
+
+20
+00:01:31,370 --> 00:01:35,190
+مالي there exist functions يوجد دوال الأمنان في
+
+21
+00:01:35,190 --> 00:01:40,570
+بتوجود دوال C من R ل R and S من R ل R such that
+
+22
+00:01:40,570 --> 00:01:44,690
+اللي هي طبعا مستقبلا هنتسمي ال C اللي هي ال cosine
+
+23
+00:01:44,690 --> 00:01:48,910
+ومستقبلا هنتسمي ال S اللي هي ال sine اللي احنا
+
+24
+00:01:48,910 --> 00:01:54,090
+بنعرفهاالان بنقول في ده اللي تاني واحدة اسمها c
+
+25
+00:01:54,090 --> 00:01:59,950
+واحدة s تحقق ما يلي اللي هي cw prime of x بيساوي
+
+26
+00:01:59,950 --> 00:02:04,870
+ناقص c of x s w prime of x بيساوي ناقص s of x ولو
+
+27
+00:02:04,870 --> 00:02:08,730
+استذكرت انت ال sine و ال cosine دي ما سيحدث لاحقا
+
+28
+00:02:08,730 --> 00:02:12,270
+طبعا لو استذكرت ال cosine لما ان .. ان .. ان .. ان
+
+29
+00:02:12,270 --> 00:02:16,130
+فاضلها مرتين هتصير اللي هي ناقص اللي هي نفسها و لو
+
+30
+00:02:16,130 --> 00:02:19,570
+فاضلت ال sine برضه فاضلها مرتين هتلاقيها ايش هي
+
+31
+00:02:19,960 --> 00:02:23,840
+بتطلع سالب S الخاصية التانية اللي هي C of Zero
+
+32
+00:02:23,840 --> 00:02:27,780
+بساوة واحد ولو استذكرت الكوسين كوسين ال Zero بساوة
+
+33
+00:02:27,780 --> 00:02:31,240
+واحد ولو استذكرت الكوسين لما تاخدها ال derivative
+
+34
+00:02:31,240 --> 00:02:35,160
+هتصير عبارة عن سالب Sin عند Zero هتطلع H بساوة
+
+35
+00:02:35,160 --> 00:02:38,960
+Zero and S of Zero بساوة Zero وS prime of Zero
+
+36
+00:02:38,960 --> 00:02:45,330
+بساوة واحد يعني بمعنى أخر يوجد دالتين الآندالتين
+
+37
+00:02:45,330 --> 00:02:49,770
+واحدة من C لـ A اسمها S واحدة اسمها C من R ل R
+
+38
+00:02:49,770 --> 00:02:56,690
+تحقق الشرطين التاليين CW' بسوء ناقص C وSW' بسوء
+
+39
+00:02:56,690 --> 00:03:01,330
+ناقص S على كل R وC عند الـ 0 هي 1 وC' عند الـ 0
+
+40
+00:03:01,330 --> 00:03:05,650
+بسوء 0 وS عند الـ 0 بسوء 0 وS' عند الـ 0 بسوء 1
+
+41
+00:03:05,650 --> 00:03:11,250
+وبعد شوية هقولك لا يوجد في الدنيا دالتين بحقق ان
+
+42
+00:03:11,250 --> 00:03:17,300
+الشروط هذولةإلا هي واحدة اسمها C واحدة اسمها S
+
+43
+00:03:17,300 --> 00:03:22,640
+يعني التنتين واحدات وهذا يجعلنا نسميهم التسمية بعد
+
+44
+00:03:22,640 --> 00:03:26,380
+ذلك واحدة اسمها cosine واحدة اسمها sine ومن ثم
+
+45
+00:03:26,380 --> 00:03:29,320
+بنجيب كل الخواص اللي احنا بنعرفه عن ال sine و ال
+
+46
+00:03:29,320 --> 00:03:34,020
+cosine من هذا البناء إذن الآن أنا ببني ببني ببني
+
+47
+00:03:34,020 --> 00:03:36,980
+وجود ال sine و ال cosine و بعد شوية ببني اللي هو
+
+48
+00:03:36,980 --> 00:03:40,960
+ال uniqueness طبقا لهذه الشروط اللي موجودة عندى
+
+49
+00:03:42,380 --> 00:03:45,860
+الان بكيف بدنا نعمل زي ما عملنا المرة الماضية في
+
+50
+00:03:45,860 --> 00:03:48,340
+ال .. في ال .. اللي هو الـ Exponential عشانك
+
+51
+00:03:48,340 --> 00:03:52,060
+هتلاقوني شوية مسرع لإن اللي بيحضر اللي هي محاضرة
+
+52
+00:03:52,060 --> 00:03:56,620
+الـ Exponential هيلاقي إن هذا في كتير من الحديث في
+
+53
+00:03:56,620 --> 00:04:00,640
+إعادة We define the sequence cn as n of continuous
+
+54
+00:04:00,640 --> 00:04:05,240
+functions inductively as بدنا نعرف اللي هي .. اللي
+
+55
+00:04:05,240 --> 00:04:09,640
+هو two sequences واحدة نسميها cn وواحدة نسميها sn
+
+56
+00:04:10,270 --> 00:04:13,890
+كيف بدنا نعرفها؟ زي ما عرفنا الـ exponential بنعرف
+
+57
+00:04:13,890 --> 00:04:19,250
+C1 of X إيش بتساوي واحد وS1 of X إيش بدنا نسميها
+
+58
+00:04:19,250 --> 00:04:26,490
+بساوي X الآن S2 of X S2 of X هيساوي ال integration
+
+59
+00:04:26,490 --> 00:04:31,990
+من صفر لعند X C2 of T DT طب C2 من وين أجيبها؟ C2
+
+60
+00:04:31,990 --> 00:04:36,050
+بتجيبها من هنا C اللي هي واحد زائد واحد يعني C2
+
+61
+00:04:36,050 --> 00:04:41,470
+بساوي واحد ناقص ال integration من صفر ل Xلـ S1 of
+
+62
+00:04:41,470 --> 00:04:48,870
+T اللي هذا DT فبكون جيبت ال C2 و بتجيبت ال S2 من
+
+63
+00:04:48,870 --> 00:04:55,530
+ال C2 لأن S3 و C3 بنفس الطريقة in general اللي أنا
+
+64
+00:04:55,530 --> 00:04:59,170
+عملت sequence of functions اللي هو بدأت اللي هي ال
+
+65
+00:04:59,170 --> 00:05:05,120
+C1 ب1 S1 ب Xومن ثم S N بتساوي من صفر ل X
+
+66
+00:05:05,120 --> 00:05:10,760
+integration C N of T DT يعني كمالة C N of T DT وC
+
+67
+00:05:10,760 --> 00:05:13,460
+N زائد واحد of X بتساوي واحد نقص integration من
+
+68
+00:05:13,460 --> 00:05:18,700
+صفر ل X S N of T DT الآن هو بيقول اللي هو هذه
+
+69
+00:05:18,700 --> 00:05:21,820
+sequence of continuous functions طب sequence of
+
+70
+00:05:21,820 --> 00:05:25,840
+continuous functions هذا الكلام بده إثباتطيب، الان
+
+71
+00:05:25,840 --> 00:05:29,500
+عندى اللى هو by induction زى ما انتوا عارفين الان
+
+72
+00:05:29,500 --> 00:05:33,600
+عندى الـC1 continuous لان الثابتة S1 continuous
+
+73
+00:05:33,600 --> 00:05:38,880
+لان هى شمالها بتساوي X بناء عليه هتطلع عندى اللى
+
+74
+00:05:38,880 --> 00:05:44,920
+هى C2 continuous ومن ثم C3 وC4 الاخرين الان لو
+
+75
+00:05:44,920 --> 00:05:49,000
+بدنا نثبتها by induction بدنا نفترض انه هذولة
+
+76
+00:05:49,000 --> 00:05:52,560
+الـSn والـCn
+
+77
+00:05:53,920 --> 00:06:00,700
+continuous سنثبت هنا continuous لأن S1 ايش بتساوي
+
+78
+00:06:00,700 --> 00:06:06,860
+X؟ C1 ايش بتساوي 1؟ continuous إذا صارت هذه اللي
+
+79
+00:06:06,860 --> 00:06:12,340
+هي الجملة is true for N بتساوي 1 نفترض الآن
+
+80
+00:06:12,340 --> 00:06:19,900
+supposeby induction بتهدي suppose that الهو star
+
+81
+00:06:19,900 --> 00:06:25,960
+هذي is true for n ايش بتساوي n بتساوي k معناته
+
+82
+00:06:25,960 --> 00:06:33,820
+صارت ال S Kوالـ CK are continuous بتثبت الآن من
+
+83
+00:06:33,820 --> 00:06:38,520
+الصحيحة لـ K زائد واحد يعني بتثبت اللي هو CK زائد
+
+84
+00:06:38,520 --> 00:06:42,700
+واحد و SK زائد واحد ان هنا شمال هنا continuous طيب
+
+85
+00:06:42,700 --> 00:06:48,000
+الآن شوف CK زائد واحد بتساوي حسب اللي هي عندي هان
+
+86
+00:06:48,000 --> 00:06:51,740
+ايش بتساوي اللي هو عبارة عن CK واحد of X بتساوي ال
+
+87
+00:06:51,740 --> 00:06:58,020
+integration واحد نقص ال integration من صفر X Sك ..
+
+88
+00:06:58,020 --> 00:07:03,020
+هذا كز واحد .. هذا ك .. of DT طيب أنا مفترض أن S K
+
+89
+00:07:03,020 --> 00:07:06,600
+من ال hypothesis induction إنها continuous إذا
+
+90
+00:07:06,600 --> 00:07:11,040
+صارت هذه كلها إيش مالها؟ اللي هي S K integrable
+
+91
+00:07:11,040 --> 00:07:15,600
+وصارت هذه كلها على بعض by fundamental theorem of
+
+92
+00:07:15,600 --> 00:07:20,980
+calculus اللي هي ال derivative إلها موجودة إذا
+
+93
+00:07:20,980 --> 00:07:23,440
+صارت هذه كلها ال derivative إلها وموجودة بالنسبة
+
+94
+00:07:23,440 --> 00:07:27,680
+لل Xإذا صارت مدام هيك هاذ is differentiable إذا
+
+95
+00:07:27,680 --> 00:07:30,940
+continuous إذا صارت اللي هي ck زائد واحد
+
+96
+00:07:30,940 --> 00:07:35,200
+continuous اللي ثبت الآن is sk زائد واحد sk زائد
+
+97
+00:07:35,200 --> 00:07:39,280
+واحد of x أيش بتساوي حسب اللي التعريف بتساوي ال
+
+98
+00:07:39,280 --> 00:07:47,660
+integration من صفر العند x ck of tزايد واحد هذا N
+
+99
+00:07:47,660 --> 00:07:52,620
+هذا N K زايد واحد K زايد واحد DT وانا مثبت فوق انه
+
+100
+00:07:52,620 --> 00:07:56,160
+CK زايد واحد is continuous إذا صارت هذه كلها على
+
+101
+00:07:56,160 --> 00:07:59,500
+بعض integrable اللي هي CK زايد واحد صار هذا ال
+
+102
+00:07:59,500 --> 00:08:02,000
+integration exist ومش هيك by fundamental theorem
+
+103
+00:08:02,000 --> 00:08:04,720
+of calculus برضه ال derivative له إيه شمالها
+
+104
+00:08:04,720 --> 00:08:07,700
+موجودة إذا صارت هذه ال differentiable إذا
+
+105
+00:08:07,700 --> 00:08:11,060
+continuous إذا صارت CK زايد واحد وSK زايد واحد are
+
+106
+00:08:11,060 --> 00:08:16,160
+continuous إذا صارت الجملة هذه صحيحةلأ K زائد واحد
+
+107
+00:08:16,160 --> 00:08:21,000
+إذا صارت صحيحة دائما إذا صارت عندي الان مفرغ منه
+
+108
+00:08:21,000 --> 00:08:26,360
+ال CN والSN are continuous functions وبناء عليه
+
+109
+00:08:26,360 --> 00:08:29,860
+مدام continuous functions by fundamental theorem
+
+110
+00:08:29,860 --> 00:08:35,460
+of calculus هتكون هذه ال SN differentiable و ال CN
+
+111
+00:08:35,460 --> 00:08:39,800
+زائد واحد differentiable ومش هيك كمان و هيعطيني ال
+
+112
+00:08:39,800 --> 00:08:43,440
+SN prime of X حسب ال fundamental theorem of
+
+113
+00:08:43,440 --> 00:08:46,820
+calculusالـ differentiation بتضايق ال integration
+
+114
+00:08:46,820 --> 00:08:54,580
+بتضال بساوي cn of x الان و ال derivative لهذه برضه
+
+115
+00:08:54,580 --> 00:08:59,900
+exist cn زائد واحد prime of x اللي هو هي ساوي اللي
+
+116
+00:08:59,900 --> 00:09:10,170
+هو مين هتطلع ناقص snof X إذا صار عندي الآن اللي هي
+
+117
+00:09:10,170 --> 00:09:15,610
+ال sequence اللي عندي صارت well defined كلها و
+
+118
+00:09:15,610 --> 00:09:20,270
+continuous كلها و لأ و differentiable كمان ال SN
+
+119
+00:09:20,270 --> 00:09:25,150
+prime of X بيساوي CN of X و CN زائد واحد prime of
+
+120
+00:09:25,150 --> 00:09:31,730
+X بيساوي ناقص SN of X زي ما أنا أثبتها و أوضحت لكم
+
+121
+00:09:31,730 --> 00:09:38,710
+إياها هناالآن induction arguments برضه induction
+
+122
+00:09:38,710 --> 00:09:42,970
+arguments بقول لي we leave this argument for you
+
+123
+00:09:42,970 --> 00:09:46,890
+خلينا نشوفها مع بعض ماهي اللي بنقصده زي ما عملت
+
+124
+00:09:46,890 --> 00:09:49,870
+بالظبط لو طلعت على الخطوات هتلاقيها مشابه لخطوات
+
+125
+00:09:49,870 --> 00:09:54,970
+تبعات ال exponential عند ال Sn of X زي ما قلنا
+
+126
+00:09:54,970 --> 00:10:02,060
+بسوء ال integration من 0 ل X Cn of T dtC N زائد
+
+127
+00:10:02,060 --> 00:10:06,140
+واحد of X بساوة واحد مقص ال integration من C ل X S
+
+128
+00:10:06,140 --> 00:10:15,360
+N of T DT ومعطينا طبعا احنا اخدنا ال C واحد of X
+
+129
+00:10:15,360 --> 00:10:22,530
+بساوة واحدو الـ c و s1 of x بسوة x هذه اللي هي ال
+
+130
+00:10:22,530 --> 00:10:25,370
+sequence of functions اللي أثبتناها هذه ال
+
+131
+00:10:25,370 --> 00:10:27,990
+sequence of functions ال sn و ال cn زائد واحد
+
+132
+00:10:27,990 --> 00:10:32,230
+التنتين are continuous for every n و همش هي
+
+133
+00:10:32,230 --> 00:10:34,850
+codifferentiable و ال derivative إلها زي ما قلنا
+
+134
+00:10:34,850 --> 00:10:39,430
+sn prime of x بسوة cn of x و ال cn زائد واحد prime
+
+135
+00:10:39,430 --> 00:10:44,190
+of x بسوة ناقص sn of x و خلّينا نسجلها هذه لإنه
+
+136
+00:10:44,190 --> 00:10:52,370
+هنحتاجها بعد شويةاللي هي S N prime of X بساوي
+
+137
+00:10:52,370 --> 00:11:00,310
+C N of X and C N زائد واحد prime of X بساوي ناقص S
+
+138
+00:11:00,310 --> 00:11:09,070
+N of X طيب الآن بدنا اللي هو by induction نثبت
+
+139
+00:11:09,070 --> 00:11:12,570
+اللي هو C N زائد واحد of X بساوي اللي هو اللي
+
+140
+00:11:12,570 --> 00:11:18,460
+أمامي هذا طبعا أكيد اللي هوبعضكم قال ما هي by
+
+141
+00:11:18,460 --> 00:11:24,960
+induction نتطلع على C2 of X C1 في القرآن بتساوي
+
+142
+00:11:24,960 --> 00:11:31,720
+واحد نثبتها يعني الان C2of X ايش بتساوي حسب
+
+143
+00:11:31,720 --> 00:11:35,960
+القانون بساوي واحد ماقص ال integration من صفر ل X
+
+144
+00:11:35,960 --> 00:11:41,020
+أس واحد لأنه بواحد هنا أس واحد اللي هي جدش أس واحد
+
+145
+00:11:41,020 --> 00:11:46,060
+ال X integration اللي هي TDT ويساوي التفاضل اللي
+
+146
+00:11:46,060 --> 00:11:49,820
+هذي بيصير واحد ماقص X تربيع على مين على اتنين إذا
+
+147
+00:11:49,820 --> 00:11:54,410
+فعلا اللي هي Cاللي هي هذه اتنين is true for any
+
+148
+00:11:54,410 --> 00:11:58,870
+ايش بتساوي ان بتساوي واحد بدنا نثبت التانية معها
+
+149
+00:11:58,870 --> 00:12:01,730
+اللي هي true for any بتساوي واحد لإنها دي جملة
+
+150
+00:12:01,730 --> 00:12:05,690
+واحدة بدأ أثبتها انها true for every any الان for
+
+151
+00:12:05,690 --> 00:12:10,590
+any بتساوي واحد بيصير عندي أس اتنين of x بدأ أتأكد
+
+152
+00:12:10,590 --> 00:12:15,990
+انه اللي هي بتحقق اللي هي ال .. اللي .. ايه هذه ال
+
+153
+00:12:15,990 --> 00:12:19,960
+equationS2 of X بيساوي من وين بده اجيبها من
+
+154
+00:12:19,960 --> 00:12:23,480
+التعريف اللي فوق بيساوي ال integration من سفل ل X
+
+155
+00:12:23,480 --> 00:12:29,180
+C2 of X ايش C2 of X هايه اللي هو واحد نقص T تربيع
+
+156
+00:12:29,180 --> 00:12:34,800
+على اتنين لكل ما له DT ويساوي الكملة هذا بيصير
+
+157
+00:12:34,800 --> 00:12:40,200
+عبارة عن Xناقص X تكعيب على تلاتة وفيه اتنين بيصير
+
+158
+00:12:40,200 --> 00:12:44,980
+X تكعيب على مين على تلاتة factorial اللي هي جداش
+
+159
+00:12:44,980 --> 00:12:50,380
+اللي هي ستة إذا فعلا فعلا طلع عندي اللي هو C أس
+
+160
+00:12:50,380 --> 00:12:55,400
+اتنين هي عبارة عن انا بواحد يعني أس اتنين بيساوي X
+
+161
+00:12:55,400 --> 00:12:59,380
+ناقص X تكعيب على تلاتة factorial يعني صارت الجملة
+
+162
+00:12:59,380 --> 00:13:02,900
+هذه برضه صحيحة فور أن بيساوي واحد إذا كلها على بعض
+
+163
+00:13:02,900 --> 00:13:09,650
+هذه صارت صحيحة فور أن بتساوي جداشالان بدنا نفترض
+
+164
+00:13:09,650 --> 00:13:13,430
+انها
+
+165
+00:13:13,430 --> 00:13:19,490
+true for n بتساوي k ونجيب منها مين اللي هو k زائد
+
+166
+00:13:19,490 --> 00:13:25,950
+واحد نفترض suppose that this is true for n بتساوي
+
+167
+00:13:25,950 --> 00:13:30,690
+k يعني بمعنى اخر صار عندى ck زائد واحد بساوي هذا
+
+168
+00:13:30,690 --> 00:13:37,260
+وهنا اتنين k وهنا kوهنا اتنين K وهنا نفس الاشي
+
+169
+00:13:37,260 --> 00:13:41,640
+اللي هو نفترض انها صحيحة for N بتساوي K صارت عبارة
+
+170
+00:13:41,640 --> 00:13:48,460
+عن اتنين K و اتنين K أو اس K ماشي الحال نثبت ان
+
+171
+00:13:48,460 --> 00:13:53,480
+هذه صحيحة for من؟ for K بتساوي .. for N بتساوي
+
+172
+00:13:53,480 --> 00:14:02,020
+كدهش؟ K زائد واحد يعني بدي اديب Cك زائد جداش اتنين
+
+173
+00:14:02,020 --> 00:14:07,480
+لأن في الاصل هي ك زائد واحد في الاصل سن زائد واحد
+
+174
+00:14:07,480 --> 00:14:11,620
+فرضتها صحيحة في القرآن بتساوي ك لأن بتثبتها صحيحة
+
+175
+00:14:11,620 --> 00:14:15,520
+لأن بتساوي ك زائد واحد يعني ك زائد واحد واحد بيصير
+
+176
+00:14:15,520 --> 00:14:22,820
+ك زائد اتنين of Xماذا يعني حساب تعريفها؟ تعريفها 1
+
+177
+00:14:22,820 --> 00:14:29,300
+ناقص الانتجريشن من 0 إلى X هذا K زائد 2 إذا هذا K
+
+178
+00:14:29,300 --> 00:14:34,420
+N زائد 1 وهذا N K زائد 2 ماذا؟ هذا سيصبح SK زائد 1
+
+179
+00:14:34,420 --> 00:14:36,340
+of DT
+
+180
+00:14:38,530 --> 00:14:42,870
+و يساوي واحد ناقص الفكرة انت فاهمت من صفر لمين
+
+181
+00:14:42,870 --> 00:14:47,910
+لإكس مين هكامل هكامل S K زائد واحد انا فرضت ان انا
+
+182
+00:14:47,910 --> 00:14:52,750
+اتروه في القرآن بالساوية K اللي هي عبارة عن T ناقص
+
+183
+00:14:52,750 --> 00:15:00,450
+T تكعيب ع تلاتة factorial لما اصل لآخر واحد زائد
+
+184
+00:15:00,450 --> 00:15:12,390
+ناقص واحد أس K فيهT 2K 1 2 T 1 كل ايه اش مالها
+
+185
+00:15:12,390 --> 00:15:16,110
+factorial الكل دي تي حسابات والله يا جماعة اللي
+
+186
+00:15:16,110 --> 00:15:22,660
+حالكم بتعملوها دي تي و يساويواحد ناقص نفتح جوس هذي
+
+187
+00:15:22,660 --> 00:15:26,700
+T بيصير T تربيع على اتنين factorial هذي ايش بيصير
+
+188
+00:15:26,700 --> 00:15:32,820
+ناقص T أُس أربعة على أربعة فى تلاتة factorial عن
+
+189
+00:15:32,820 --> 00:15:36,980
+أربعة factorial زائد لما أصل الأخر واحد ناقص واحد
+
+190
+00:15:36,980 --> 00:15:43,580
+أُس K زي ما هي لأن إشارة هذي T بيصير اتنين K زائد
+
+191
+00:15:43,580 --> 00:15:49,130
+واحد زائد واحد يعني زائد اتنين علىاللي هو اتنين K
+
+192
+00:15:49,130 --> 00:15:54,490
+زائد اتنين في هذا بتطلع اتنين K زائد اتنين الكل
+
+193
+00:15:54,490 --> 00:15:59,550
+ايه شماله؟ factorial ماشي الحال هذا طبعا كله من
+
+194
+00:15:59,550 --> 00:16:04,630
+صفر ل X إذا بتصير هذا عبارة عن X وهذا X وهذا
+
+195
+00:16:04,630 --> 00:16:09,010
+الأخير برضه ايه؟ X اللي هي هذا عبارة عن ايه؟ هو
+
+196
+00:16:09,010 --> 00:16:16,350
+يساوي1 ناقص x تربيع على 2 factorial زائد x أُس 4
+
+197
+00:16:16,350 --> 00:16:20,310
+على 4 factorial ضربت الناقص جوا لإن أنا لما أصل
+
+198
+00:16:20,310 --> 00:16:24,910
+ناقص لما أصل الأخر واحد زائد ناقص واحد K وناقص أنا
+
+199
+00:16:24,910 --> 00:16:31,050
+بصير K زائد 1 في X أُس 2K
+
+200
+00:16:31,910 --> 00:16:37,130
+زائد اتنين على اتنين K زائد اتنين لكل vector يعني
+
+201
+00:16:37,130 --> 00:16:41,890
+صارت هذه CK زائد واحد زائد اتنين of X بتساوي هذا
+
+202
+00:16:41,890 --> 00:16:49,290
+المقدار هذا المقدار صحيح صار بالظبط هو هذا اللي هو
+
+203
+00:16:49,290 --> 00:16:53,830
+المقدار اللي عندي أثبتت و كإلمي يعني الآن ان هذا
+
+204
+00:16:54,470 --> 00:16:59,490
+اللي عندى is true for mean for k زائد لأ واحد لإنه
+
+205
+00:16:59,490 --> 00:17:03,950
+لما احط مكان n بيساوي k زائد واحد بيصير هذه k زائد
+
+206
+00:17:03,950 --> 00:17:09,030
+اتنين اشوف اللى طلعته صح ولا لأ بيساوي تبحانة واحد
+
+207
+00:17:09,030 --> 00:17:12,990
+ناقص x تربيع على اتنين فيكتوريا لما اثر الاخر واحد
+
+208
+00:17:12,990 --> 00:17:17,790
+اللى هي ناقص واحد قص مين k زائد واحد فكس قص اتنين
+
+209
+00:17:17,790 --> 00:17:20,970
+في k زائد واحد يعني اتنين k زائد اتنين وها اتنين k
+
+210
+00:17:20,970 --> 00:17:25,760
+زائد اتنين كل فيكتورياالان صارت هذه صحيحة for n
+
+211
+00:17:25,760 --> 00:17:30,600
+ايش بتساوي k زائد واحد التاني بنفس الأسلوب بثبتها
+
+212
+00:17:30,600 --> 00:17:35,260
+صحيحة for ايش for k زائد واحد يعني بدي احسب مين يا
+
+213
+00:17:35,260 --> 00:17:42,300
+جماعة بدي احسب Sكزايد اتنين ايش حسب اللي فوق
+
+214
+00:17:42,300 --> 00:17:47,180
+بتساوي بساوي ال integration من سفر ل X ل CK زايد
+
+215
+00:17:47,180 --> 00:17:54,020
+اتنين of T DT و باجي بعوضها هنا و بكملها و بتطلع
+
+216
+00:17:54,020 --> 00:17:57,800
+عندى بالظبط ال formula هذه يعني صح بيصير عندى
+
+217
+00:17:57,800 --> 00:18:01,620
+بتعملها لحالك لان حسابات نفس الأسلوب بتطلع عندى
+
+218
+00:18:01,620 --> 00:18:06,310
+اللي هي هذه صحيحة for mean برضهfor an بتساوي k
+
+219
+00:18:06,310 --> 00:18:11,050
+زائد واحد إذا هذا صار المقدار صحيح دائما for mean
+
+220
+00:18:11,050 --> 00:18:20,430
+for any k for any n element in n الان واضح ان
+
+221
+00:18:20,430 --> 00:18:26,310
+الخطوات مشابهة لخطوات ال exponential نيجي الآن
+
+222
+00:18:26,310 --> 00:18:28,970
+قصّل n prime
+
+223
+00:18:38,770 --> 00:18:45,020
+نجي الآنوين رايح؟ زي اللي بكونت رايحي الأيام الـ
+
+224
+00:18:45,020 --> 00:18:49,060
+exponential هثبتلك أن الـ sequence هذه converged
+
+225
+00:18:49,060 --> 00:18:52,220
+uniformly وهذه طبعا هتصبح converged uniformly
+
+226
+00:18:52,220 --> 00:18:55,000
+automatic فهتصبح اللي هي differentiable لأنه بيصير
+
+227
+00:18:55,000 --> 00:18:57,860
+طبقة النظرية اللي هي تبعها الـ differentiability
+
+228
+00:18:57,860 --> 00:19:01,200
+بيصير عند مادام differentiable اللي هي ال
+
+229
+00:19:01,200 --> 00:19:04,580
+derivative اللي لها exist و هتكون ال derivative
+
+230
+00:19:04,580 --> 00:19:07,960
+متحقق الشروط و بيكون خلاص نعم نشوف أش بقول طيب
+
+231
+00:19:07,960 --> 00:19:12,530
+الآنبعد ما طلعنا هدولة let a أكبر من صفر b given
+
+232
+00:19:12,530 --> 00:19:15,790
+نفس الخطوات تبعت ال exponential then if الabsolute
+
+233
+00:19:15,790 --> 00:19:19,970
+value of x أصغر بسوء a and m أكبر من n أكبر من 2a
+
+234
+00:19:19,970 --> 00:19:25,510
+يعني بتداخد اللي هي الامات لأكبر من n و أكبر من 2a
+
+235
+00:19:25,510 --> 00:19:29,650
+يعني الآن أنا بشتغل عالين يا جماعة الفترة من ناقص
+
+236
+00:19:29,650 --> 00:19:34,570
+a لعند a وأخدت ال a arbitrarily أكبر من صفر لكن
+
+237
+00:19:34,570 --> 00:19:41,850
+fixed الان we haveعندي اللي هو ..عندي اللي هي الـ
+
+238
+00:19:41,850 --> 00:19:50,630
+A الـ A على اللي هي 2N بما أنه N أكبر من 2A اجسم
+
+239
+00:19:50,630 --> 00:19:56,790
+الجهتين على 2N هن على 2N و هن على 2N بيصير عند الـ
+
+240
+00:19:56,790 --> 00:20:07,710
+A على N .. A على N أصغر من النص، صح؟هي a تروح
+
+241
+00:20:07,710 --> 00:20:11,090
+التنين مع التنين وهذه n مع ال n بصير a أعليها أصغر
+
+242
+00:20:11,090 --> 00:20:16,050
+من نص يعني a على 2n أصغر من 1 على 4 جسمت التنين
+
+243
+00:20:16,050 --> 00:20:22,400
+عالميا على 2 إذا لما تكون ال n أكبر من 2aهتطلع
+
+244
+00:20:22,400 --> 00:20:24,640
+عندها get بتعرف ليش هذه لإن بتلزمنا في الحسابات
+
+245
+00:20:24,640 --> 00:20:28,580
+بعد شوية هتكون اللي هي لما الاتنين a أصغر من n
+
+246
+00:20:28,580 --> 00:20:31,960
+بتكون عند a على اتنين n أصغر من مين من ربع المرة
+
+247
+00:20:31,960 --> 00:20:35,460
+اللي فاتت كان لازمنا a على n أصغر من نص وكملنا ال
+
+248
+00:20:35,460 --> 00:20:40,650
+series اللي متذكر اللي عملنا في اللي هيالـ
+
+249
+00:20:40,650 --> 00:20:43,330
+Exponential أنا بحكيش تفاصيله و مفترض ان انتوا
+
+250
+00:20:43,330 --> 00:20:47,570
+فاهمين حسب حكينا اللي هو في الـ Exponential الان
+
+251
+00:20:47,570 --> 00:20:52,290
+للأمات اللي أكبر من M أكبر من 2A بدي احسب الـCM
+
+252
+00:20:52,290 --> 00:20:59,870
+ناقص 2 الـSN عشان تظلها جدامكم اللي فوق هذه عندى
+
+253
+00:20:59,870 --> 00:21:06,960
+الان الـCMالـ C M الـ C M شايفينها؟ بتظلها ماشية
+
+254
+00:21:06,960 --> 00:21:10,940
+واحد لو ف .. الآن هذه M بدأ الـ N زاد واحد ايش
+
+255
+00:21:10,940 --> 00:21:14,880
+اسمها؟ M بتظلها ماشية واحد ناقص X أربع على اتنين
+
+256
+00:21:14,880 --> 00:21:18,820
+فيكتوريال X أربع على أربع فيكتوريال والـ M أكبر من
+
+257
+00:21:18,820 --> 00:21:23,160
+الـ N هتجيه يقبل في طريقها من الـ N الـ N بيصير
+
+258
+00:21:23,160 --> 00:21:27,540
+الـ N طبعا إيه شمالها؟ الـ N عبارة عن في جبل ال
+
+259
+00:21:27,540 --> 00:21:33,590
+term هذا اللي هو الـ N ناقص واحدأه فبصير زائد اللي
+
+260
+00:21:33,590 --> 00:21:39,350
+هو ناقص واحد أس ان ناقص واحد في X أس اتنين ام ناقص
+
+261
+00:21:39,350 --> 00:21:44,050
+اتنين على اتنين ام ناقص اتنين الكل factorial و
+
+262
+00:21:44,050 --> 00:21:48,390
+بتكمل هذا و بتبقى مكمل ايه انت لما اتصل لعند X أس
+
+263
+00:21:48,390 --> 00:21:54,670
+اتنين ام ناقص اتنين لأن هذا لل ام مش لل ان زائد
+
+264
+00:21:54,670 --> 00:21:59,730
+واحد على اتنين ام ناقص اتنين الكل اشماله factorial
+
+265
+00:22:00,420 --> 00:22:08,320
+لما تطرح ال CM نقص ال CN اللي هو لها بيصير المتبقى
+
+266
+00:22:08,320 --> 00:22:12,600
+هيه زي ما عملنا بالظبط قبل هيك فبيصير ال CM نقص ال
+
+267
+00:22:12,600 --> 00:22:18,080
+CN بساوي النقص واحد طبعا في absolute value عند X2N
+
+268
+00:22:18,080 --> 00:22:21,200
+عشان هيك طيرنا مش فارقة كتير نقص و سالب أخدنا له
+
+269
+00:22:21,200 --> 00:22:25,900
+absolute value مش هتفرج معنا ال X2N على 2N
+
+270
+00:22:25,900 --> 00:22:32,490
+vectorialناقص X أس 2 M زائد 2 اللي بعيدها على 2 M
+
+271
+00:22:32,490 --> 00:22:35,990
+زائد 2 كله factorial لما أصل لآخر term اللي هو X
+
+272
+00:22:35,990 --> 00:22:41,170
+أس 2 M ناقص 2 على 2 M ناقص 2 كله factorial هذا
+
+273
+00:22:41,170 --> 00:22:46,670
+الأن هذا نفسه أخدنا احنا ال absolute value لل X
+
+274
+00:22:46,670 --> 00:22:50,350
+أصغر من A في هذه الفترة احنا شغالين في الفترة ال
+
+275
+00:22:50,350 --> 00:22:54,430
+absolute value X أصغر من A الآن بيصير عند ال X
+
+276
+00:22:54,430 --> 00:23:05,180
+نفسهاأُس 2n أصغر أو يساوي هذه اللي هي a أُس 2n على
+
+277
+00:23:05,180 --> 00:23:09,240
+2n الكل factorial الأولى زائد استخدمت triangle
+
+278
+00:23:09,240 --> 00:23:13,440
+inequality هذه زائد هذه زائد هذه زائد هذه الآن
+
+279
+00:23:13,440 --> 00:23:18,240
+زائد اللي بعيدها لما أصل لآخر واحدة x أو اللي جاب
+
+280
+00:23:18,240 --> 00:23:23,420
+اللي خليني اكتبها عشان x اللي هي بصير aأس اتنين ان
+
+281
+00:23:23,420 --> 00:23:28,060
+زائد اتنين على اتنين ان زائد اتنين لكل factorial
+
+282
+00:23:28,060 --> 00:23:37,020
+زائد لما أصل لآخر term عندي هنا لما أصل لآخر term
+
+283
+00:23:37,020 --> 00:23:43,160
+اللي هو زائد
+
+284
+00:23:43,160 --> 00:23:57,230
+هذا ال term اللي هو زائدA 2M-2 2M-2 كل A شماله
+
+285
+00:23:57,230 --> 00:24:03,930
+فكتوريا خلّيني أخد هذا A 2N 2N كل فكتوريا العام
+
+286
+00:24:03,930 --> 00:24:11,460
+المشتركبظل عندى 1 زائد a تربيع لأنه بيصير a يسوى 2
+
+287
+00:24:11,460 --> 00:24:17,960
+a تربيع على مين على 2n زائد 2 أكيد ال 1 على 2n
+
+288
+00:24:17,960 --> 00:24:23,280
+زائد 2 أصغر من 1 على 2n لأنه 2n أصغر من هذه
+
+289
+00:24:23,280 --> 00:24:27,240
+فمقلوبها بيصير أكبرماشي و بضل زي ما عملته المرة
+
+290
+00:24:27,240 --> 00:24:32,700
+الفاتة أسحب منه A أس 2 على N و في الآخر أستبدل
+
+291
+00:24:32,700 --> 00:24:37,680
+اللي هو 2N هذه عن الرقم اللي أكبر منه N فبصير هي
+
+292
+00:24:37,680 --> 00:24:41,400
+مقلوبها أكبر فبتضلها اللي هي ال inequality زي هيك
+
+293
+00:24:41,400 --> 00:24:45,840
+و بكون سحبتي 2 من هذه بيصير 2 ام نقص 2 ام نقص ايش
+
+294
+00:24:45,840 --> 00:24:53,350
+نقص 2 وصلنا لعند ال inequality هذه الآنأحنا قلنا a
+
+295
+00:24:53,350 --> 00:24:58,950
+على 2n أصغر من مين من ربعه يعني بيصير هذا المقدار
+
+296
+00:24:58,950 --> 00:25:05,310
+كله أصغر أو يساوي a أس 2n على 2n لكل factorial
+
+297
+00:25:05,310 --> 00:25:09,550
+مضروب في مين أصغر أو يساوي لأن الربع أكبر منهم بس
+
+298
+00:25:09,550 --> 00:25:14,930
+تبدأ كل واحد وقت ما كانوا إيهاش ربع واحد زاد واحد
+
+299
+00:25:14,930 --> 00:25:23,510
+على أربعة تربيعزائد لما أصل لآخر واحد a اللي هو
+
+300
+00:25:23,510 --> 00:25:31,370
+عبارة عن واحد على أربعة الكل أس اتنين ام ناقص
+
+301
+00:25:31,370 --> 00:25:36,590
+اتنين ام ناقص ايش اتنين هذه الان ال ال ال ال
+
+302
+00:25:36,590 --> 00:25:39,710
+finite geometric series أصغر أو يساوي ال infinite
+
+303
+00:25:39,710 --> 00:25:45,770
+اللي هي a أس اتنين ان على اتنين ان لكل factorial
+
+304
+00:25:45,770 --> 00:25:53,570
+في اللي هوالـ summation ده واحد زائد ربع تربيع
+
+305
+00:25:53,570 --> 00:26:00,310
+زائد ربع تكعيب زائد لمّا أصل ربع أس أربعة زائد إلى
+
+306
+00:26:00,310 --> 00:26:05,070
+ما لا نهاية هذه الأن أساسها جداش وكأنه أساسها كل
+
+307
+00:26:05,070 --> 00:26:11,750
+مرة تمد واحد على ست عشر فبتصير هذه المجموحة اللي
+
+308
+00:26:11,750 --> 00:26:16,470
+هي واحد مجموحة بتعرفوها واحد ناقص واحد على ست عشر
+
+309
+00:26:17,250 --> 00:26:21,750
+Passive واحد على واحد ناقص واحد على ست عشر ويساوي
+
+310
+00:26:21,750 --> 00:26:27,810
+جداش خمست عشر او ست عشر على خمست عشر لأن هذه واحد
+
+311
+00:26:27,810 --> 00:26:29,870
+ناقص واحد على ست عشر تطلع خمست عشر على ست عشر
+
+312
+00:26:29,870 --> 00:26:34,830
+مجلوبة بيصير ست عشر على خمست عشرالمفهوم المقصود هو
+
+313
+00:26:34,830 --> 00:26:38,670
+بسمحها Geometric Series بضل بقول هذه أصغر أو ساوي
+
+314
+00:26:38,670 --> 00:26:41,930
+الصماشن إلى ماله نهاية و بزدبدل هذا باللي أكبر
+
+315
+00:26:41,930 --> 00:26:46,070
+منها فبتظل هذه أكبر الان مجموحها بيصير عبارة عن 16
+
+316
+00:26:46,070 --> 00:26:51,010
+في المقدار هذا اللي دله موخود عام المشترك فبيصير
+
+317
+00:26:51,010 --> 00:26:58,940
+عندى الآن يا جماعة اللي هو ال CMنقص الـCN زي ما
+
+318
+00:26:58,940 --> 00:27:04,460
+حكينا قبل ذلك بالظبط أصغر من هذا المقدار اللي هذا
+
+319
+00:27:04,460 --> 00:27:09,000
+limit و اين بيروح as N goes to infinity بيروح لـ 0
+
+320
+00:27:09,000 --> 00:27:13,760
+إذا بيصير هذا المقدار زي ما قلنا المرة الماضية CM
+
+321
+00:27:13,760 --> 00:27:20,800
+of X نقص CN of X اللي هو أصغر أو يسوء Y for very
+
+322
+00:27:20,800 --> 00:27:28,700
+large M and NM and N لأنه لما تكبر M كثير N كثير
+
+323
+00:27:28,700 --> 00:27:33,100
+تروح لنا لنهاية لأنه limit بروح لها سفر as N goes
+
+324
+00:27:33,100 --> 00:27:37,080
+to infinity ال M برضه بتكبر إذا for very large M
+
+325
+00:27:37,080 --> 00:27:40,800
+هيكون هذا أصغر أو يساوي إبسلون لأي إبسلون في
+
+326
+00:27:40,800 --> 00:27:44,480
+الدنيا لأنه هذا بيقدي للسفر فبيصير صغير صغير صغير
+
+327
+00:27:44,480 --> 00:27:48,720
+صغير لدرجة إنه مضروب في هذا يكون أصغر من إبسلون
+
+328
+00:27:48,720 --> 00:27:53,360
+وهذه اللي قلنا عنهااللي هي الـ Cauchy criterion
+
+329
+00:27:53,360 --> 00:28:00,040
+for uniform continuity هذا الكلام كله لمين؟ صحيح
+
+330
+00:28:00,040 --> 00:28:06,880
+لأي X وين على الفترة اللي هو absolute value X أصغر
+
+331
+00:28:06,880 --> 00:28:11,220
+أو يساوي A يعني على الفترة المولقة من ناقص A لعند
+
+332
+00:28:11,220 --> 00:28:19,110
+A وهذا بيعطيني أن الـis uniformly continuous على
+
+333
+00:28:19,110 --> 00:28:24,150
+الفترة من نفس A لعند A is as if is uniformly is
+
+334
+00:28:24,150 --> 00:28:28,330
+uniformly convergence على الفترة من نفس A ل A يعني
+
+335
+00:28:28,330 --> 00:28:35,790
+بمعنى آخر صارت عندي ال sequence هذه اللي هي CNN
+
+336
+00:28:35,790 --> 00:28:43,150
+converts uniformly to some function عالميا على
+
+337
+00:28:43,150 --> 00:28:50,730
+الفترةمن ناقص a لعند مين لعند a الآن طب ما هو اللي
+
+338
+00:28:50,730 --> 00:28:53,930
+عملناها أنا على الفترة هذه نقدر نعمله على أي شيء
+
+339
+00:28:53,930 --> 00:28:58,310
+ثاني يعني بمعنى أخر لو جيت أخدت x element in R
+
+340
+00:28:58,310 --> 00:29:04,400
+هلاجي number aبحيث أن هذا ال number a هي ال x في
+
+341
+00:29:04,400 --> 00:29:10,180
+ال R بقدر ألاقي a بحيث أن ناقص a و a تكون ال x في
+
+342
+00:29:10,180 --> 00:29:15,340
+الفترة بين ناقص a و a يعني بمعنى آخر هعمل نفس اللي
+
+343
+00:29:15,340 --> 00:29:18,100
+عملته في الأول و هحصر على c أن uniformly
+
+344
+00:29:18,100 --> 00:29:23,030
+convergence على هذه الفترةيعني بمعنى آخر الآن صار
+
+345
+00:29:23,030 --> 00:29:29,450
+عندي limit cn of x for any x exist بده أسميها هذه
+
+346
+00:29:29,450 --> 00:29:35,190
+limit c of x وبناء عليه بدي أعرف الآن in
+
+347
+00:29:35,190 --> 00:29:38,710
+particular this means that cn of x converge for
+
+348
+00:29:38,710 --> 00:29:42,710
+each x element in R we define c من R لR by c of x
+
+349
+00:29:42,710 --> 00:29:45,650
+بسواية limit cn of x for x element in R
+
+350
+00:29:53,570 --> 00:29:59,630
+الان .. بما انه الان الـCn مرتبط بشكل مرتبط لـC زي
+
+351
+00:29:59,630 --> 00:30:04,690
+ما قلنا أو الـCn of X كلهم مرتبط حسب اللي هو
+
+352
+00:30:04,690 --> 00:30:08,030
+النظرية في الـpointwise .. الـuniform convergence
+
+353
+00:30:08,030 --> 00:30:12,360
+اللي هو the limit .. the uniform .. limitأو الـ
+
+354
+00:30:12,360 --> 00:30:15,220
+Form convergence of a sequence of continuous
+
+355
+00:30:15,220 --> 00:30:18,480
+functions مصمد كونتنيواس الـ limit تبعتها يعني
+
+356
+00:30:18,480 --> 00:30:21,160
+هتطلع عندي C of X continuous مثلاً C of X
+
+357
+00:30:21,160 --> 00:30:25,000
+continuous إذا عندي .. اللي هو صار عندي الـ
+
+358
+00:30:25,000 --> 00:30:28,160
+function هذه continuous اللي هي الـ C اللي إحنا
+
+359
+00:30:28,160 --> 00:30:33,730
+بدنا إياها ومش هيك و limitCn of 0 as n goes to
+
+360
+00:30:33,730 --> 00:30:37,270
+infinity بساوة limit اللي هي Cn of 0 ايش بتساوي
+
+361
+00:30:37,270 --> 00:30:41,910
+واحد وهي ساوية واحد اللي هي عبارة عن مين الـC of
+
+362
+00:30:41,910 --> 00:30:47,370
+ايش of 0 لأن احنا متفجين الـC of X بساوية limit Cn
+
+363
+00:30:47,370 --> 00:30:51,310
+of X و in particular for X بتساوية سفر بسيرة limit
+
+364
+00:30:51,310 --> 00:30:55,430
+Cn of 0 بساوية C of 0 وCn of 0 كلها واحد ايضا
+
+365
+00:30:55,430 --> 00:30:57,730
+limit الواحد اللي هو بساوية واحد يعني الـC of 0
+
+366
+00:30:57,730 --> 00:31:05,970
+ايش هتساوي؟ هتساوي واحدخصّلت كمان شغلة انه حصلت
+
+367
+00:31:05,970 --> 00:31:10,070
+عندي ان الـ c of zero بيساوي واحد
+
+368
+00:31:13,750 --> 00:31:19,670
+لأ اللي هو الـS الـCN بعمل اشي مشابه له لمين للـSN
+
+369
+00:31:19,670 --> 00:31:25,250
+عشان نثبت الآن اللي أثبتناه انه صار في عندى ده
+
+370
+00:31:25,250 --> 00:31:31,730
+اللي عرفناها اسمها الـC of X اللي عبارة عن limit
+
+371
+00:31:31,730 --> 00:31:38,970
+CN of X حيث الـC من R إلى R طيب
+
+372
+00:31:41,560 --> 00:31:45,380
+الان ناخد ايضا ال absolute value X أصغر من مين من
+
+373
+00:31:45,380 --> 00:31:49,960
+ايه وال M أكبر او يساوي N و أصغر من مين من اتنين
+
+374
+00:31:49,960 --> 00:32:00,360
+ايه الان نحسب ل SM ناقص SN SM ايه قيمتهاDT من صفر
+
+375
+00:32:00,360 --> 00:32:05,400
+الاندكس SN وهي SM
+
+376
+00:32:05,400 --> 00:32:11,240
+ناقص هذه هي قيمتها إذا صار هذه ناقص هذه هي قيمتها
+
+377
+00:32:11,240 --> 00:32:19,680
+الان هذه سهل إثباتها أنها تتعامل ال absolute value
+
+378
+00:32:19,680 --> 00:32:24,200
+لهذهأصغر أو يساوي الـ absolute value لهذه أصغر أو
+
+379
+00:32:24,200 --> 00:32:27,700
+يساوي ال integration من 0 ل X ل absolute value CM
+
+380
+00:32:27,700 --> 00:32:35,600
+of T نقص CN of T اشماله DT ماشي الحال أو بنكمل
+
+381
+00:32:35,600 --> 00:32:39,580
+اللي هو بنستخدم هذه الخاصية اللي هو بيكون أصغر أو
+
+382
+00:32:39,580 --> 00:32:42,820
+يساوي من الحسابات اللي حسبناها قبل شوية هذا
+
+383
+00:32:42,820 --> 00:32:45,380
+حسبناها أصغر من مين الحسابات اللي قبل شوية أصغر أو
+
+384
+00:32:45,380 --> 00:32:52,510
+ساوي A أس 2N على 2N الكل factorialمضروبة في اللي
+
+385
+00:32:52,510 --> 00:32:58,130
+هو ست عشر على مين على خمسة في ال integration من
+
+386
+00:32:58,130 --> 00:33:02,750
+صفر لل X ل DT هذا ال integration ايش بيساوي بيساوي
+
+387
+00:33:02,750 --> 00:33:08,230
+X ماشي و ال X عندي احنا ماخدينها اصغر او يساوي مين
+
+388
+00:33:08,230 --> 00:33:13,010
+ايه اصغر او يساوي ايه فبصير عندي هذا المقدار بيكون
+
+389
+00:33:13,680 --> 00:33:18,020
+لما نبعد مكانه بيطلع X قيمته أصغر أو يساوي A
+
+390
+00:33:18,020 --> 00:33:22,040
+فبيصير هذا مضروب في مين في ايه اللي هو هاي المقدار
+
+391
+00:33:22,040 --> 00:33:26,680
+وهي 16 على 5 وهي ايش ال A صار هذا أصغر أو يساوي
+
+392
+00:33:26,680 --> 00:33:31,600
+هذا المقدار نفس ما عملنا قبل بشوية الان as N goes
+
+393
+00:33:31,600 --> 00:33:36,690
+to infinityهذه قيمتها بيساوي 0 إذا هذا المقدار for
+
+394
+00:33:36,690 --> 00:33:41,470
+very large M و N هيكون أصغر أوي يساوي Y و بالكوشي
+
+395
+00:33:41,470 --> 00:33:45,350
+criterion بيصير عندي اللي هي ال S N converts
+
+396
+00:33:45,350 --> 00:33:49,470
+uniformly و الآن على الفترة نقص A و A و ال A كانت
+
+397
+00:33:49,470 --> 00:33:53,730
+arbitrary إذا S من R ل R بنقدر نعرفها بحيث إنه
+
+398
+00:33:53,730 --> 00:33:57,880
+limit S N of X اللي صارت موجودة على هذهوبناء على
+
+399
+00:33:57,880 --> 00:34:00,460
+الـ A-arbitrary صارت موجودة على كل الـ R زي ما
+
+400
+00:34:00,460 --> 00:34:04,340
+قلنا كيف هذا الكلام بسمي limit S N of X اليمين S
+
+401
+00:34:04,340 --> 00:34:08,980
+of X وهذه اللي هي ال function الثانية بنفس الأسلوب
+
+402
+00:34:08,980 --> 00:34:13,380
+وبنفس اللي هو المنطق وبنفس الأسباب بما أن ال S N
+
+403
+00:34:13,380 --> 00:34:16,860
+conversion formally to S وS N continuous إذا حد
+
+404
+00:34:16,860 --> 00:34:22,340
+طلع ال S برضه نفس إشمالها continuousوأيضًا limit
+
+405
+00:34:22,340 --> 00:34:28,540
+of S of 0 هو عبارة عن S of 0 وبما أن S of 0 دايماً
+
+406
+00:34:28,540 --> 00:34:31,640
+0 إذًا limitها سيكون as N goes to infinity 0 يعني
+
+407
+00:34:31,640 --> 00:34:35,300
+ستظهر لدي S of 0 إيش بالساوة 0 إذًا الآن طلعنا
+
+408
+00:34:35,300 --> 00:34:45,840
+كمان شغلة أثبتناها أن S of 0بساوة 0 و C of 0 بساوة
+
+409
+00:34:45,840 --> 00:34:54,400
+1 وعرفنا هذولة الدالتين أسي و أس من اللي هي R لعند
+
+410
+00:34:54,400 --> 00:34:59,680
+مين لعند R إذن الآن احنا الدالتين اللي أثبتنا لان
+
+411
+00:34:59,680 --> 00:35:05,430
+وجود هنا الآن اللي بحكي عنهم هذولة الدالتيناللي هي
+
+412
+00:35:05,430 --> 00:35:10,190
+الدالة الأولى سميتها أسوة تانية C ولاقيت انه C of
+
+413
+00:35:10,190 --> 00:35:15,790
+0 بساوة واحد و S of 0 بساوة H0 وظل أثبت هذه و أثبت
+
+414
+00:35:15,790 --> 00:35:22,910
+هذه و أثبت هذولة اللي أستبعتنا بتحققهم فخلينا نشوف
+
+415
+00:35:22,910 --> 00:35:29,670
+كيف نثبتها طيب الآن أوصلنا لعند اللي هو المنطقة
+
+416
+00:35:29,670 --> 00:35:39,670
+هذه لأن احنا اتفجنا على ما يليهاتفجنا ان الـ S N
+
+417
+00:35:39,670 --> 00:35:47,170
+تتعامل بشكل عام لتعمل S و الـ C N تتعامل بشكل عام
+
+418
+00:35:47,170 --> 00:35:54,280
+لتعمل Cلكن الان احنا اثبتنا في الأول هي ان ما
+
+419
+00:35:54,280 --> 00:36:00,880
+محتاش ال S N برايم بسوء ال C N، مظبوط؟ يعني و كأنه
+
+420
+00:36:00,880 --> 00:36:04,860
+بناء على هذا مزام ال S N برايم بسوء ال C N فبصير
+
+421
+00:36:04,860 --> 00:36:09,520
+ال S N برايم، هذه، مكانها دي S N برايم بصير ال S N
+
+422
+00:36:09,520 --> 00:36:15,340
+برايم converges uniformly to some function mean C
+
+423
+00:36:15,340 --> 00:36:22,060
+الان بما ان S N برايم converges uniformly to Cالان
+
+424
+00:36:22,060 --> 00:36:28,900
+و الاسن converged to us uniformly برضهحسب نظرية
+
+425
+00:36:28,900 --> 00:36:32,360
+بتطلع عندى بما ان s n prime differentiable هتكون
+
+426
+00:36:32,360 --> 00:36:37,140
+ال limit ه differentiable و ال c prime اللي هي ال
+
+427
+00:36:37,140 --> 00:36:42,340
+s prime هدى هي مين ال c اللي طلعت هن يعني s n
+
+428
+00:36:42,340 --> 00:36:45,960
+prime differentiable او s n differentiable و
+
+429
+00:36:45,960 --> 00:36:48,560
+converts to some function اذا بتتذكر كنا نسميها g
+
+430
+00:36:48,560 --> 00:36:52,520
+و هد كنا نسميها f فكنا نقول بما ان s n بتروح لل f
+
+431
+00:36:52,520 --> 00:36:56,740
+و ال s n prime بتروح لل g إذا نتيجة النظرية هتكون
+
+432
+00:36:56,740 --> 00:37:00,000
+اللي هيالـ F هي الـ Differentiable والـ Derivative
+
+433
+00:37:00,000 --> 00:37:04,940
+لها إيش بتطلع D يعني الـ Derivative للـ S' إيش
+
+434
+00:37:04,940 --> 00:37:12,260
+هتطلع عبارة عن مين C' أسف C هتطلع مين اللي هي الـ
+
+435
+00:37:12,260 --> 00:37:21,000
+C إذا أثبتت أنا الآن S' of X بتساوي C of X هاي
+
+436
+00:37:21,000 --> 00:37:28,190
+اللي أثبتته هنا الآن من جهة أخرى من جهة أخرىإحنا
+
+437
+00:37:28,190 --> 00:37:35,850
+أثبتنا اللي هي قبل هيك أن الـCN برايم بسوء ناقص SN
+
+438
+00:37:35,850 --> 00:37:42,450
+ناقص واحد الـCN برايم بسوء ناقص SN of واحد يعني
+
+439
+00:37:42,450 --> 00:37:47,790
+بمعنى آخر الـCN برايم of X أثبتنا بسوء ناقص SN
+
+440
+00:37:47,790 --> 00:37:54,640
+ناقص واحد of Xالان بما ان ال S Unconverted زي ال S
+
+441
+00:37:54,640 --> 00:38:00,260
+خلصنا هذه اه خلصنا هذه المنطقة خليني اشرح بلغة
+
+442
+00:38:00,260 --> 00:38:05,060
+تانية عشان اميز بين الكلامين عندي الان انتبهوا
+
+443
+00:38:05,060 --> 00:38:08,860
+بتعمل نفسي اشي بس بالنسبة لمن الان بالنسبة عشان
+
+444
+00:38:08,860 --> 00:38:13,780
+اجيب ال derivative لل S لل C primeعندى الان الـCN
+
+445
+00:38:13,780 --> 00:38:19,740
+prime of X بسوء نقص SN نقص واحد of X بما أن SN
+
+446
+00:38:19,740 --> 00:38:23,500
+راحت للـS إذا اللى هى ال derivative هى اللى
+
+447
+00:38:23,500 --> 00:38:30,280
+بتسويها اللى هى CNN زائد واحد prime of X هتروح
+
+448
+00:38:30,280 --> 00:38:36,770
+لمين؟اللي هي مش هي ناقصها لأن S N ناقص واحد اللي
+
+449
+00:38:36,770 --> 00:38:40,730
+هي بساوي ناقص هذه أنجل ناقصها بعد أذنكم يعني بدي
+
+450
+00:38:40,730 --> 00:38:44,810
+أستبدل الـ S N بمين بقيمتها هذه صارت ناقص الـ C N
+
+451
+00:38:44,810 --> 00:38:50,330
+prime of X أيش بتساوي بتروح للـ S أو بمعنى آخر صار
+
+452
+00:38:50,330 --> 00:38:56,620
+عندي من هنا من هنا اللي بتطلع عندي هناصار عندى
+
+453
+00:38:56,620 --> 00:39:03,640
+الان من هنا اللى هو cn زائد واحد prime of x بتروح
+
+454
+00:39:03,640 --> 00:39:08,080
+لناقص طبعا uniformly بتروح لمين ناقص s لأنه ناقصها
+
+455
+00:39:08,080 --> 00:39:13,040
+بتروح لل s إذا هي بتروح لناقص ال sوفي نفس الوقت
+
+456
+00:39:13,040 --> 00:39:19,840
+أنا بقول الـCN نفسها بتروح uniform لمين؟ للـC بنفس
+
+457
+00:39:19,840 --> 00:39:24,740
+الإسلوب اللي قبل بشوية اللي هي بما أنه هذه اللي هي
+
+458
+00:39:24,740 --> 00:39:27,580
+differentiable sequence of functions و converge
+
+459
+00:39:27,580 --> 00:39:31,260
+uniform to some function إذا هذه هتكون اللي هي
+
+460
+00:39:31,260 --> 00:39:35,600
+عبارة عن اللي هي الـC الأصلية differentiable و ال
+
+461
+00:39:35,600 --> 00:39:40,070
+derivative فيه لها مين؟ الناقص Sفبصير عندي الـ C'
+
+462
+00:39:40,530 --> 00:39:45,970
+of X بسوء ناقص S of X بكون هذه اللي هي النتيجة
+
+463
+00:39:45,970 --> 00:39:50,330
+الثانية اللي طبقنا عليها طلعت عندي اللي هو هذه
+
+464
+00:39:50,330 --> 00:39:54,050
+الخاصية و هذه الخاصية متحققة يعني الآن صار عندي
+
+465
+00:39:54,050 --> 00:40:02,590
+اللي هو تحقق ما يليه أنه تبعتنا هذه الـ C'of X
+
+466
+00:40:02,590 --> 00:40:08,530
+بيساوي ناقص S of X وطلع عندي اللي هي S prime of X
+
+467
+00:40:08,530 --> 00:40:16,290
+بيساوي C of X ماشي الحال بيكون هيك احنا ضال علينا
+
+468
+00:40:16,290 --> 00:40:21,230
+شغل اخرى لحاول نثبتها بيكون اثبتنا اللي هو كل
+
+469
+00:40:21,230 --> 00:40:27,630
+الصفات المطلوبة اللي تتحقق في ال S وحددت هوية ال S
+
+470
+00:40:27,630 --> 00:40:28,050
+و ال C
+
+471
+00:40:38,700 --> 00:40:48,060
+الجزء الثاني سهل فضل الـ cw prime of x
+
+472
+00:41:00,590 --> 00:41:05,470
+بتصير c double prime هيها و هذه فضلها بنفعلها
+
+473
+00:41:05,470 --> 00:41:09,550
+لأنها قبل التفاضل اللي فضلناها وبتصير ناقص اللي هي
+
+474
+00:41:09,550 --> 00:41:15,030
+s prime of x اللي هي s prime of x هيش بتساوي c of
+
+475
+00:41:15,030 --> 00:41:20,350
+x فبصير ناقص مين؟ c of xوالآن S W' of X من أين بدي
+
+476
+00:41:20,350 --> 00:41:23,650
+أجيبها من هنا S W' بيساوي اللي هو هذه ال
+
+477
+00:41:23,650 --> 00:41:26,050
+derivative اللي هي ال derivative اللي هي عبارة عن
+
+478
+00:41:26,050 --> 00:41:31,530
+ناقص S of X فبصير عبارة عن ناقص S of X فبصير عندي
+
+479
+00:41:31,530 --> 00:41:38,910
+هذا برضه ايش تحقق اخر اشي اللي هي C' of Zero C' of
+
+480
+00:41:38,910 --> 00:41:42,410
+Zero بيساوي ناقص S of Zero وS of Zero بيساوي صفر
+
+481
+00:41:42,410 --> 00:41:47,030
+اذا C' of Zero بيساوي Zero ال S' of Zero
+
+482
+00:41:52,540 --> 00:41:56,800
+وهكذا اثبتنا وجود دالة
+
+483
+00:41:59,570 --> 00:42:04,970
+اللي هي حققتلي اللي هي الشرط الأول هذا والدالتين
+
+484
+00:42:04,970 --> 00:42:09,710
+اسف والشرط الثاني اه اللي بعده هذه بتكون أثبتنا
+
+485
+00:42:09,710 --> 00:42:15,290
+وجود الـC والـS لأن بعض النتائج الأخرى على اللي هي
+
+486
+00:42:15,290 --> 00:42:18,830
+الدالتين
+
+487
+00:42:18,830 --> 00:42:19,790
+اللي أوجدناهم
+
+488
+00:42:24,500 --> 00:42:28,960
+نشوف ايش الـ Corollary الأولى بقول لي if C and S
+
+489
+00:42:28,960 --> 00:42:33,680
+are the functions in 3x,8x,4x,1 then C' of X
+
+490
+00:42:33,680 --> 00:42:39,560
+بيساوي نقص S of X أثبتناها هيها and طبعا في طريقنا
+
+491
+00:42:39,560 --> 00:42:44,020
+في البرهان S' of X بيساوي C of X هيها اللي هي خواص
+
+492
+00:42:44,020 --> 00:42:50,340
+اللي هي أثبتناها طيب الآن بقول لي moreover these
+
+493
+00:42:50,340 --> 00:42:54,630
+functions have derivatives of all orderيعني بأي
+
+494
+00:42:54,630 --> 00:42:57,710
+order ال derivative موجودة طبعاً هذا by induction
+
+495
+00:42:57,710 --> 00:43:01,550
+by induction اللي هو بدك تثبت انه اللي هو اللي هو
+
+496
+00:43:01,550 --> 00:43:06,670
+cn هتكون موجودة و بيساوي ناقص a of x او زاد a of x
+
+497
+00:43:06,670 --> 00:43:11,450
+حسب اللي هي جداش درجة ال n وممكن تساوي c of x او
+
+498
+00:43:11,450 --> 00:43:16,930
+اللي هي ناقص c of x حسب درجة ال n اللي موجودةاذا
+
+499
+00:43:16,930 --> 00:43:19,990
+الان هذه by induction بنقدر نثبت ان ال derivative
+
+500
+00:43:19,990 --> 00:43:26,390
+اللي هو موجودة for any or ضر بناء على اللي حكيناه
+
+501
+00:43:26,390 --> 00:43:32,300
+نيجي الان لـ Corollary اللي بعدهابقول الآن الـ
+
+502
+00:43:32,300 --> 00:43:38,280
+function C and S بحقق اللي هي الـ Pythagorean
+
+503
+00:43:38,280 --> 00:43:42,280
+Identity اللي كنا نعرفها كصين تربيع زات صين تربيع
+
+504
+00:43:42,280 --> 00:43:47,660
+إش بساوي بساوي واحد فكرة البروحان سهلة بسمي هذه
+
+505
+00:43:47,660 --> 00:43:51,520
+كلها بسميها function اسمها F of X بقول سمي هذه
+
+506
+00:43:51,520 --> 00:43:56,080
+اللي هي F of X بساوي هذه فضليها بلّا بتفضليها F
+
+507
+00:43:56,080 --> 00:44:01,200
+prime of X بساوي اتنينفى C of X فى تفاضل اللى جوا
+
+508
+00:44:01,200 --> 00:44:06,300
+اللى هو ناقص S of X والتانية زائد اتنين فS of X
+
+509
+00:44:06,300 --> 00:44:11,880
+فتفاضلها هي C of X هذه هي هذه بس بالسالد اذا حصل
+
+510
+00:44:11,880 --> 00:44:15,100
+طرح حلو اسم ساوى سفر اذا صارت عند ال derivative لل
+
+511
+00:44:15,100 --> 00:44:18,320
+function هذه اشمالها بساوى سفر يعني بمعنى اخر
+
+512
+00:44:18,320 --> 00:44:22,490
+الدالة هي دالة ثابتةيعني F is a constant function
+
+513
+00:44:22,490 --> 00:44:27,210
+يعني قيمة الـ F عند أي قيمة إيش بتساوي مقدار ثابت
+
+514
+00:44:27,210 --> 00:44:31,310
+إذا اتفاج أنها ثابتة أسهل إشي أسهل إشي أوجدلي F of
+
+515
+00:44:31,310 --> 00:44:34,670
+Zero عشان أتعرف إيش ده اللي بالساوية بالظبط C of
+
+516
+00:44:34,670 --> 00:44:38,730
+Zero واحد S of Zero سفر إذا صار عندي F of Zero
+
+517
+00:44:38,730 --> 00:44:42,170
+بساوية واحد وهي ثابتة إذا صارت F of X دايما
+
+518
+00:44:42,170 --> 00:44:46,270
+بالساوية واحد يعني هذا المقدار بساوية واحد دائما
+
+519
+00:44:46,270 --> 00:44:48,690
+طيب نيجي الآن
+
+520
+00:44:51,700 --> 00:44:57,820
+لأ الـ theorem اللي بعدها الـ functions C and S
+
+521
+00:44:57,820 --> 00:45:02,820
+satisfy the properties I and II of theorem 8-4-1
+
+522
+00:45:02,820 --> 00:45:07,020
+are unique لما بقولّي برضه هتلاقي ال uniqueness
+
+523
+00:45:07,020 --> 00:45:13,020
+بشبه ال uniqueness لمن؟ لأ اللي هو ال .. ال ..
+
+524
+00:45:13,020 --> 00:45:18,030
+قولوا معايا ال uniqueness للـ exponentialو سمناها
+
+525
+00:45:18,030 --> 00:45:21,170
+اللي هو let E واحد .. نفترض أنه في E واحد و E
+
+526
+00:45:21,170 --> 00:45:24,070
+اتنين و سمها فرق بيساوي D و في الآخر روحنا لفرق
+
+527
+00:45:24,070 --> 00:45:28,390
+بيساوي إيش؟ بيساوي سفر هنا نفس الاشي هنشتغل و برضه
+
+528
+00:45:28,390 --> 00:45:31,590
+هنستخدم اللي هو ال Taylor's theorem زي ما استخدمنا
+
+529
+00:45:31,590 --> 00:45:34,470
+هناك ال Taylor's theorem يعني هتلاقيه ال sketch
+
+530
+00:45:34,470 --> 00:45:37,470
+للبرهان هو نفس ال sketch الأولاني عشان هيك
+
+531
+00:45:37,470 --> 00:45:43,110
+هتلاقوني سريع فيه بدنا نثبت أن ال C و ال S are
+
+532
+00:45:43,110 --> 00:45:47,930
+unique functionsطبعا الطريقة let c1 and c2 be two
+
+533
+00:45:47,930 --> 00:45:52,290
+functions on R that satisfy it satisfies مين ال
+
+534
+00:45:52,290 --> 00:45:55,850
+conditions اللي احنا بنقول عنهم اللي هي ال I و I I
+
+535
+00:45:55,850 --> 00:46:00,930
+اللي هو بحيث انه c1 double prime of x بساوي c1 of
+
+536
+00:46:00,930 --> 00:46:01,150
+x
+
+537
+00:46:16,750 --> 00:46:18,150
+2�
+
+538
+00:46:18,550 --> 00:46:24,010
+لأن نفترض أن دي بتساوي C1-C2 وبدنا نصل في النهاية
+
+539
+00:46:24,010 --> 00:46:27,410
+أن دي هذي لازم تطلع إيش بتساوي؟ بتساوي 0 مدام دي
+
+540
+00:46:27,410 --> 00:46:32,710
+بتساوي 0 مدام C1 بتساوي إيش؟ C2 لاحظ الآن دي W' of
+
+541
+00:46:32,710 --> 00:46:38,190
+X فاضل هذا مرتين تصير C1W'-C2W'
+
+542
+00:46:39,860 --> 00:46:47,500
+الان بساوي هتطلع ايش بتساوي ناقص اللي هي مين D of
+
+543
+00:46:47,500 --> 00:46:59,700
+X نعملها اللي هي حسابات D of X بساوي D of X بساوي
+
+544
+00:46:59,700 --> 00:47:08,540
+C1 ناقص C2 D prime of X ايش هيساوي اللي هو عبارة
+
+545
+00:47:08,540 --> 00:47:16,670
+عنأس واحد ناقص اس واحد ناقص بيصير اللي هو مين الاس
+
+546
+00:47:16,670 --> 00:47:19,810
+واحد اللي هي بالنسبة لهذه اللي أوجدناها ناقص اس
+
+547
+00:47:19,810 --> 00:47:25,350
+اتنين بيصير زائد لان دي double prime بساوي بتفاضل
+
+548
+00:47:25,350 --> 00:47:30,030
+هذا كمان مرة اللي هي بترجع مين نفسها C واحدوهذه
+
+549
+00:47:30,030 --> 00:47:38,390
+ترجع نفسها C2 التي هي ناقص D التي هي ناقص D طيب
+
+550
+00:47:38,390 --> 00:47:43,410
+إذا ال D لأنه مفترضين ال C1 وC2 أشمالها بتحقق
+
+551
+00:47:43,410 --> 00:47:52,470
+الشروط اللي فوق اللي قلنا عنهاDouble prime بساوة D
+
+552
+00:47:52,470 --> 00:48:02,390
+of X أو D double prime بساوة D double prime بساوة
+
+553
+00:48:06,030 --> 00:48:11,870
+بساوي هد W' ناقص هد W' هد W' ناقص C1 و هد W' اللي
+
+554
+00:48:11,870 --> 00:48:15,870
+هي ناقص ناقص C زائدة سارة D W' بساوي ناقص D و X
+
+555
+00:48:15,870 --> 00:48:21,890
+هيك اللي هو دعنا نقول أسلم طيب الآن بستعجل لإن
+
+556
+00:48:21,890 --> 00:48:29,470
+الكلام معاد يعنيأو الأفكار معادة الآن احطلي D of 0
+
+557
+00:48:29,470 --> 00:48:34,530
+ايش هيساوي؟ Zero لأن D of 0 بيساوي هذي ناقص هذي و
+
+558
+00:48:34,530 --> 00:48:37,630
+هذي عند ال zero واحد و هذي عند ال zero واحد
+
+559
+00:48:37,630 --> 00:48:40,010
+التنتين عند ال zero واحد انحصل طرحي ان ايش هيساوي
+
+560
+00:48:40,010 --> 00:48:47,160
+سفر الان D prime عند ال zeroوDW' عند الـ zero و و
+
+561
+00:48:47,160 --> 00:48:48,760
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+562
+00:48:48,760 --> 00:48:50,140
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+563
+00:48:50,140 --> 00:48:50,140
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+564
+00:48:50,140 --> 00:48:52,320
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+565
+00:48:52,320 --> 00:48:52,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+566
+00:48:52,380 --> 00:48:55,440
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+567
+00:48:55,440 --> 00:48:59,220
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+568
+00:48:59,220 --> 00:48:59,240
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+569
+00:48:59,240 --> 00:49:00,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+570
+00:49:00,380 --> 00:49:01,520
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+571
+00:49:01,520 --> 00:49:05,260
+.. و
+
+572
+00:49:05,260 --> 00:49:15,970
+.. و .. و .. و .. و .. و .. وبحكي عن الـ
+
+573
+00:49:15,970 --> 00:49:25,410
+derivative الـ dw prime عبارة عن اللي هي ناقص d of
+
+574
+00:49:25,410 --> 00:49:30,410
+0 عند الـ 0 و الـ d عند الـ 00 إذن هذه الـ 0 هذه
+
+575
+00:49:30,410 --> 00:49:34,330
+لمن؟ لـ k اللي هي إتنين و أربعة و ستة و تمانية
+
+576
+00:49:34,330 --> 00:49:37,870
+بنفس السبب هتطلع سفر الفرديات بيجي من مين؟ من هذه
+
+577
+00:49:38,330 --> 00:49:44,430
+فردية الـ Derivative لإيش؟ لأن D' of 0 هتساوي C1'
+
+578
+00:49:44,890 --> 00:49:51,110
+نقص C2' C1' عند الصفر صفر و C2' عند ال 0 0 إذا هذه
+
+579
+00:49:51,110 --> 00:49:56,730
+دي K عند ال 0 بالساوية 0 لكل K سواء زوجية أو إيش
+
+580
+00:49:56,730 --> 00:50:01,230
+أو فردية إذا جهزنا هذا اللي جهزناه قبل هيك يعمل
+
+581
+00:50:01,230 --> 00:50:05,940
+Exponentialوبتنا نطبّك اللي هو مين الـ Taylor's
+
+582
+00:50:05,940 --> 00:50:09,000
+theorem نطبّك الـ Now let x element in R be
+
+583
+00:50:09,000 --> 00:50:12,760
+arbitrary element in R and let I x be the interval
+
+584
+00:50:12,760 --> 00:50:18,940
+within point 0x طبّجنا عليها إذا since D بتساوي C1
+
+585
+00:50:18,940 --> 00:50:25,220
+ناقص C2 وT بتساوي S1 ناقص S2 اللي هي مفترضين S1
+
+586
+00:50:25,220 --> 00:50:28,400
+وS2 اللي هو two functions such that اللي بيحققوا
+
+587
+00:50:28,400 --> 00:50:34,170
+تبعات الـ sineاللي هي بالساوية S1 ايش هي مسميها
+
+588
+00:50:34,170 --> 00:50:39,950
+انا اللي هي بدل C2 prime او هي C2 prime والـ S2
+
+589
+00:50:39,950 --> 00:50:46,510
+اللي هي C2 A prime العكسة
+
+590
+00:50:46,510 --> 00:50:50,530
+ال derivative بيصيح بالنقص are continuous on mean
+
+591
+00:50:50,530 --> 00:50:55,930
+on Ix عارفين continuous انه احنا مفترضين ان نحقق
+
+592
+00:50:55,930 --> 00:51:00,160
+نفس اللي هو اللي قبلمدام continuous الـ D والـ T
+
+593
+00:51:00,160 --> 00:51:07,440
+continuous على closed bounded interval I X إذا في
+
+594
+00:51:07,440 --> 00:51:12,810
+اللي هو K واحد للأولىوK2 للتانية خدوا ال maximum
+
+595
+00:51:12,810 --> 00:51:18,470
+إليها واسموها K إذا في K للجهتين بحيث ان ال D of T
+
+596
+00:51:18,470 --> 00:51:22,930
+bounded علي هذه أصغر شهو T كي على كل الفترة اللي
+
+597
+00:51:22,930 --> 00:51:27,390
+بنحكي عنها IX وT of T أصغر شهو K for all T
+
+598
+00:51:27,390 --> 00:51:30,290
+elements of IX لأنه زي ما أقول continuous function
+
+599
+00:51:30,290 --> 00:51:33,930
+in a closed bounded interval must be boundedالآن
+
+600
+00:51:33,930 --> 00:51:39,170
+جاهزين نطبق مين؟ الـ Taylor's theorem to d من Ix
+
+601
+00:51:39,170 --> 00:51:44,970
+and use the fact دي اللي أثبتناه دي 0 سواء 0 دي K0
+
+602
+00:51:44,970 --> 00:51:50,030
+سواء 0 الآن بنقول for each n unlimited n عملناها
+
+603
+00:51:50,030 --> 00:51:56,510
+عشان هيك بس ماشي عنه بسرعة عملناها قبل هيك there
+
+604
+00:51:56,510 --> 00:51:59,630
+exist a point Cn unlimited Ix such that هذا
+
+605
+00:51:59,630 --> 00:52:03,890
+remainder بتعملي remainderD ب X بيساوي D of 0 زي
+
+606
+00:52:03,890 --> 00:52:06,850
+دي prime of 0 زائد زائد زائد زائد X ان نقص واحد
+
+607
+00:52:06,850 --> 00:52:12,630
+لما أصل لمين لل remainder كل هذول أصفار بسبب مين
+
+608
+00:52:12,630 --> 00:52:16,430
+انه D0 وD prime of 0 وD ان نقص واحد وزير وكلهم إيه
+
+609
+00:52:16,430 --> 00:52:20,090
+شمالهم بيساوي أصفار عند الصفر بيظل هذا المقدار
+
+610
+00:52:20,090 --> 00:52:24,010
+يساوي اللي هو المقدار اللي عندي اللي تحت هذا هذا
+
+611
+00:52:24,010 --> 00:52:27,490
+ال derivative هذا ال derivative أنا بعرفش إيش هي
+
+612
+00:52:27,490 --> 00:52:31,230
+ما أنتوا عارفين ال derivative لل .. لل .. لل C
+
+613
+00:52:34,160 --> 00:52:41,940
+أو ال derivative للـC إذا فضلت مرة واحدة بتطلع
+
+614
+00:52:41,940 --> 00:52:47,120
+الـS أو سلبها فظلت تلت مرات بترجع ليها S بس
+
+615
+00:52:47,120 --> 00:52:53,180
+بالموجة فظلت خمسة بترجع ناقص Sفظلت زوجي بتطلع هي
+
+616
+00:52:53,180 --> 00:52:58,920
+نفسها أو سلبها عشان هي كادل أنا بعرفش طبعا هنا
+
+617
+00:52:58,920 --> 00:53:02,400
+عندك اللي هي بتتوزع على مرة derivative تندين
+
+618
+00:53:02,400 --> 00:53:04,620
+derivative تلاتة derivative أربعة derivative و
+
+619
+00:53:04,620 --> 00:53:08,180
+بترجع الدورة زي ما هي لإنه في الأول بيكون اللي هي
+
+620
+00:53:08,180 --> 00:53:15,520
+تفاضل الـC ناقص S بعدها بتتفاضل بترجع اللي هي ناقص
+
+621
+00:53:15,520 --> 00:53:18,770
+حالهابعدها بترجع اللي هي
+
+622
+00:53:18,770 --> 00:53:36,410
+ببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببببب
+
+623
+00:53:36,660 --> 00:53:40,180
+اللي هي الـC ويصير نقص C كمان مرة بترجع اللي هي
+
+624
+00:53:40,180 --> 00:53:45,740
+الـS وكمان مرة بترجع اللي هي الـC بعد هيك بتصير
+
+625
+00:53:45,740 --> 00:53:51,920
+اللي هي تكرر حالها يعني بمعنى آخر جربوها أنتوا
+
+626
+00:53:51,920 --> 00:53:59,920
+هتلاقوا أنه حسب الأُس هنا بتطلع اللي هي عبارة عن
+
+627
+00:53:59,920 --> 00:54:11,180
+هي أو سلبها أو الـS أو سلبهابكل الأحوال بغض النظر
+
+628
+00:54:11,180 --> 00:54:15,600
+اللي هي DN of CN هي حاصل طرح التنتين اللي هي سمنها
+
+629
+00:54:15,600 --> 00:54:23,280
+يا D يا T الـ D تبعت الـ C الفرق طبعا والـ T اللي
+
+630
+00:54:23,280 --> 00:54:29,010
+هي الفرق بين الأسات يعني في الآخرزاد او ناقص و زاد
+
+631
+00:54:29,010 --> 00:54:33,810
+او ناقص سواء هذه او سواء هذه حضرنا من الاصل انها
+
+632
+00:54:33,810 --> 00:54:38,870
+bounded و كله اصغر او يساوي من K اصغر او يساوي ايش
+
+633
+00:54:38,870 --> 00:54:45,860
+K بناء عليه هتطلع هذه هيكabsolute value أصغر أو
+
+634
+00:54:45,860 --> 00:54:51,780
+يساوي هذه أي إن كانت K في X أُس N absolute value
+
+635
+00:54:51,780 --> 00:54:56,880
+على N factorial باشي الحال إذا صلعت عندي هذا
+
+636
+00:54:56,880 --> 00:55:00,860
+المقدار أصغر أو يساوي هذا as N goes to infinity
+
+637
+00:55:00,860 --> 00:55:05,540
+هذا بروح لمن؟ للسفر وهذا independent of N بيصير
+
+638
+00:55:05,540 --> 00:55:08,100
+أكبر أو يساوي السفر يعني هذا اللي جوا بده يصير سفر
+
+639
+00:55:08,100 --> 00:55:12,930
+إذا ال D of X هاشي بدها تساويبتساوي سفر وبكون
+
+640
+00:55:12,930 --> 00:55:21,090
+أثبتنا ان الـ C1 والـ C2 are equivalent الان
+
+641
+00:55:21,090 --> 00:55:27,700
+similar arguments بنفس الاسلوبوخلّيها إليكم إنه
+
+642
+00:55:27,700 --> 00:55:32,120
+اللي هي بنثبت إنه S إذا كان S1 وS2 are two
+
+643
+00:55:32,120 --> 00:55:36,320
+functions such that بيحققوا اللي هما الخواصة اللي
+
+644
+00:55:36,320 --> 00:55:41,700
+بنقولها عنها في النظرية اللي هي S1 double prime
+
+645
+00:55:41,700 --> 00:55:46,400
+بيساوي نفس S1 ونفس ال S2 وهذا عند Zero بيساوي Zero
+
+646
+00:55:46,400 --> 00:55:50,160
+وهذا ال prime عند Zero بيساوي Zero هيتلعلك غصب
+
+647
+00:55:50,160 --> 00:55:52,240
+عنها في الأخر S1 بيساوي S2
+
+648
+00:55:55,620 --> 00:55:59,960
+بس بدل ما تعملوا على ال D يعني ساموا اللي هو T
+
+649
+00:55:59,960 --> 00:56:04,040
+بتساوي S1 نقص S2 وطبقوا الشروط وامشوا نفس اللي
+
+650
+00:56:04,040 --> 00:56:07,580
+امشيناها هتلاقوا حالكم انه لازم تطلع ال S1 بساوي
+
+651
+00:56:07,580 --> 00:56:11,800
+S2 وبناء عليه صار ال ال two functions ال C والS
+
+652
+00:56:11,800 --> 00:56:17,820
+are unique functions طيب
+
+653
+00:56:17,820 --> 00:56:24,720
+ال answer المؤهليننذهب بالاتجاه نثبت أن افترض أن
+
+654
+00:56:24,720 --> 00:56:27,880
+الـ Definition of the unique function C من R لR و
+
+655
+00:56:27,880 --> 00:56:32,400
+أسم الـ R لR اللي يتحقق CW prime of X بساوية نقطة
+
+656
+00:56:32,400 --> 00:56:36,650
+C of Xهو الاصل الـ differential equation التاني S
+
+657
+00:56:36,650 --> 00:56:40,990
+w برامج و X ناقص بساوة ناقص S X لكل X element in R
+
+658
+00:56:40,990 --> 00:56:43,470
+و ال C of Zero اللي هو ال condition ال condition
+
+659
+00:56:43,470 --> 00:56:46,350
+بساوة واحد و C برامج و Zero بساوة Zero و ال S of
+
+660
+00:56:46,350 --> 00:56:49,710
+Zero بساوة Zero و ال S برامج و Zero بساوة Zero
+
+661
+00:56:49,710 --> 00:56:57,150
+الدالتين اللي بحقق إن الكلام هذا اللي أثبتنا إنه
+
+662
+00:56:57,150 --> 00:57:02,740
+uniqueبنسميهم respectively the cosine function and
+
+663
+00:57:02,740 --> 00:57:07,600
+the sine function اللي انتوا بتعرفوها وهي cosine X
+
+664
+00:57:07,600 --> 00:57:12,620
+اللي بتعرفوها وهي sin X اللي انتوا بتعرفوها نيجي
+
+665
+00:57:12,620 --> 00:57:17,930
+الآنلاخد بعض الخواص خلينا ناخد اللي هو الخاصية
+
+666
+00:57:17,930 --> 00:57:23,430
+اللي هي نظرية 4.6 بتقول لي إذا كانت f من R ل R is
+
+667
+00:57:23,430 --> 00:57:27,890
+such that f double prime of x بساعة نقص f of x for
+
+668
+00:57:27,890 --> 00:57:30,370
+x element in R يعني هذه ال differential equation
+
+669
+00:57:30,370 --> 00:57:32,810
+بتقوللي هذه ال differential equation هي حلولها
+
+670
+00:57:32,810 --> 00:57:37,690
+مالي then there exist Alpha و BetaSuch that F of X
+
+671
+00:57:37,690 --> 00:57:41,390
+بيسوء Alpha CX زائد Beta S of X بيقوللي هذه ال
+
+672
+00:57:41,390 --> 00:57:45,110
+differential equation حالة انها عبارة عن linear او
+
+673
+00:57:45,110 --> 00:57:47,870
+خليني اقول combination او linear combination
+
+674
+00:57:47,870 --> 00:57:55,890
+between F C of X S of Xخلّيني أخد نسمي g of x إيش
+
+675
+00:57:55,890 --> 00:57:59,590
+بتساوي اللي هي عبارة عن اللي هي ال c of x و ال a
+
+676
+00:57:59,590 --> 00:58:03,770
+sub x نسمي هذه f of zero و f prime of zero for x
+
+677
+00:58:03,770 --> 00:58:10,330
+element in R الآن لو حسبت ال gw prime of x احسبها
+
+678
+00:58:10,330 --> 00:58:15,270
+فاضل هذه
+
+679
+00:58:15,270 --> 00:58:18,550
+مرتين هدولة ثوابت طبعا و هدولة اللي بتنزلك
+
+680
+00:58:18,550 --> 00:58:23,970
+هتلاقيهم اللي هو بساوي ناقص g of xأه .. و لو حسبت
+
+681
+00:58:23,970 --> 00:58:26,730
+الـ g of zero .. g of zero هتلاقيها بساوي of zero
+
+682
+00:58:26,730 --> 00:58:30,110
+لإن هذا سفر وهذا .. هذا واحد وهذا سفر الان صار
+
+683
+00:58:30,110 --> 00:58:33,850
+عندى gw prime of x بساوي نقص g of x و g of zero
+
+684
+00:58:33,850 --> 00:58:40,410
+بساوي f of zero الان احسب الـ g prime of xصار عندى
+
+685
+00:58:40,410 --> 00:58:43,450
+تلات معلومات معلمتين جي دابل برايم of X بيساوي نقص
+
+686
+00:58:43,450 --> 00:58:45,990
+جي of X و جي of Zero بيساوي أف of Zero خلينا نجيب
+
+687
+00:58:45,990 --> 00:58:50,330
+جي برايم of X جي برايم of X مش بتساوي فاضل اللى هى
+
+688
+00:58:50,330 --> 00:58:55,430
+عبارة عن نقص أس of X و هدا تفضلها اللى هى C of X
+
+689
+00:58:55,430 --> 00:58:58,830
+صار عندى اللى هى جي برايم of X بيساوي هذا الكلام
+
+690
+00:58:58,830 --> 00:59:04,720
+زائد هذا الكلامالان احسبلي g prime of 0 هيصير
+
+691
+00:59:04,720 --> 00:59:08,420
+عبارة عن هذا طبعا سفر وهذا هتصير واحد فبصير F
+
+692
+00:59:08,420 --> 00:59:13,750
+prime of 0 صار عندى الان g prime of 0أيش بيساوي يا
+
+693
+00:59:13,750 --> 00:59:18,090
+جماعة؟ جي برايم اف زيرو خليني أطلعلك إياها لفوق
+
+694
+00:59:18,090 --> 00:59:21,750
+صارت جي برايم اف زيرو بيساوي أف برايم اف زيرو و جي
+
+695
+00:59:21,750 --> 00:59:26,730
+اف زيرو بيساوي أف اف زيرو الآن بديش ناشر نعيد اللي
+
+696
+00:59:26,730 --> 00:59:30,950
+هو therefore الآن the functions h بتساوي f ناقص g
+
+697
+00:59:30,950 --> 00:59:35,990
+is such that يعني عرفلي function h أيش بيساوي؟ f
+
+698
+00:59:35,990 --> 00:59:43,220
+ناقص مين ناقص gالآن الـ H double prime لها لو جيت
+
+699
+00:59:43,220 --> 00:59:46,840
+حسبت الـ H double prime اللي هي هتلاقيها بتساوي
+
+700
+00:59:46,840 --> 00:59:50,260
+ناقص H of X بتحسبوها لحالكم H double prime of X
+
+701
+00:59:50,260 --> 00:59:55,140
+عشان بتساوي ناقص H of X لكل X element in R لو حسبت
+
+702
+00:59:55,140 --> 00:59:59,670
+اللي هي H of Zeroهتساوي اللي هي zero لإنه احنا
+
+703
+00:59:59,670 --> 01:00:03,010
+أثبتنا ان g of zero ساوي f of zero وh prime of
+
+704
+01:00:03,010 --> 01:00:07,110
+zero h prime of zero اللي هي عبارة عن f prime نقص
+
+705
+01:00:07,110 --> 01:00:09,210
+g prime عند ال zero التانية و التساويات إذا إيه مش
+
+706
+01:00:09,210 --> 01:00:15,170
+ساوي سفر الان it then thus it follows as in the
+
+707
+01:00:15,170 --> 01:00:19,430
+proof of the preceding theorem النظرية الماضية إن
+
+708
+01:00:19,430 --> 01:00:22,170
+h of x إيه مش ساوي سفر يعني بدك تعمل نفس اللي
+
+709
+01:00:22,170 --> 01:00:25,180
+عملناها قبل هيك على ال Taylor's theorem و الاخرينو
+
+710
+01:00:25,180 --> 01:00:29,300
+تصل ال H of X بساوية 0 لكل X ومنه هتقول F of X
+
+711
+01:00:29,300 --> 01:00:33,320
+بساوية G of X مدام F of X بساوية G of X إذن هذه
+
+712
+01:00:33,320 --> 01:00:38,580
+اللي هي F of X بتطلع F of X بتساوي اللي هي هذا
+
+713
+01:00:38,580 --> 01:00:42,780
+اسمها Alpha وهذا اسمها Beta وبكون عندي هي اللي هو
+
+714
+01:00:42,780 --> 01:00:46,280
+solution لاللي هي ال differential equation اللي
+
+715
+01:00:46,280 --> 01:00:50,260
+احنا حكينا عنها بديش أعيد نفس الكلام عشان هي كأنا
+
+716
+01:00:50,260 --> 01:00:52,780
+اختصرت لأن الحسابات كلها مشابهة
+
+717
+01:00:58,150 --> 01:01:03,430
+الان بدنا اللي هي 8 4 7 the function C is even and
+
+718
+01:01:03,430 --> 01:01:07,570
+S is odd in the sense that هو بتكون الصحيحة دي
+
+719
+01:01:07,570 --> 01:01:10,610
+الأصل كلها تكون أو معظمها اللي هي عبارة عن
+
+720
+01:01:10,610 --> 01:01:14,430
+exercises لإنها تطبيقات على النظرية اللي قلتها في
+
+721
+01:01:14,430 --> 01:01:19,630
+الأول ال C ناقص X هيسوء C of X يعني عبارة عن even
+
+722
+01:01:19,630 --> 01:01:23,230
+function S ناقص X بيسوء ناقص S of X لكل X element
+
+723
+01:01:23,230 --> 01:01:28,590
+in Rالآن if x,y المتنار then we have the addition
+
+724
+01:01:28,590 --> 01:01:32,230
+formula c of x زي اد y بيسبب c of x في c of y نقص
+
+725
+01:01:32,230 --> 01:01:35,950
+s of x وs of y اللي هي اللي بتعرفوها انتوا sign ال
+
+726
+01:01:35,950 --> 01:01:38,350
+x زي اد y بيسبب sign ال x بكسان ال y زي اد كسان ال
+
+727
+01:01:38,350 --> 01:01:42,090
+y في sign ال x وهكذا هي هذه الخواص اللي احنا
+
+728
+01:01:42,090 --> 01:01:47,570
+عارفينها قبل هيك وبدنا نشوف كيف اللي هي انبرهن
+
+729
+01:01:47,570 --> 01:01:53,660
+النظرية نشوف البرهان تبع النظريةالآن سمّيلي Phi of
+
+730
+01:01:53,660 --> 01:01:59,370
+X بيساوي C of ماقص X لكل X element in Rاحسبلي الـ
+
+731
+01:01:59,370 --> 01:02:05,010
+phi w prime of x لو اتيت فضلت هذه مرتين هتلاقيها
+
+732
+01:02:05,010 --> 01:02:09,950
+ناقص phi of x وهذا الكلام تفضله سهل و بتجيبه لحالك
+
+733
+01:02:09,950 --> 01:02:15,050
+الآن احسبلي ال phi of 0 phi of 0 هتساوي ايش؟
+
+734
+01:02:15,050 --> 01:02:19,350
+بتساوي واحد phi prime of 0 هتطلع اللي هي عبارة عن
+
+735
+01:02:19,350 --> 01:02:24,430
+اللي هي ال sign ال sign ايش معناها؟ بتساوي 0 عند
+
+736
+01:02:24,430 --> 01:02:30,730
+ال zeroالان صار عندى الان الفاى هي مين؟ هي الـC
+
+737
+01:02:30,730 --> 01:02:35,710
+ليش الفاى هي الـC؟ لأن حققت الفاى اللى فرضتها
+
+738
+01:02:35,710 --> 01:02:40,790
+بساوية C-X شروط الـC و الـC is unique إذا الفاى H
+
+739
+01:02:40,790 --> 01:02:47,200
+بتساوي Cإذا صار عندى الان C-X بيساوي C of X الفكرة
+
+740
+01:02:47,200 --> 01:02:52,060
+واضحة أنه أنا جبت سميت الـC-X هي الفاية واتبتت أن
+
+741
+01:02:52,060 --> 01:02:57,060
+هذه الفاية بتحقق الشرطين اللي احنا حكينا عنهم في
+
+742
+01:02:57,060 --> 01:03:01,300
+الأول الدالة اللي بتخلي الـC is unique فصارت
+
+743
+01:03:01,300 --> 01:03:07,080
+الفاية أيش بتساوي الـCالان بمعنى اخر صارت C of X
+
+744
+01:03:07,080 --> 01:03:10,720
+هي C من ناقص X اللي فرضناها الـ Phi in a similar
+
+745
+01:03:10,720 --> 01:03:16,040
+way برضه بدك تعملاش S of ناقص X اللي هو بتسمي اللي
+
+746
+01:03:16,040 --> 01:03:20,480
+هي ال .. الفا .. بتسميها بـ Psi مثلا Psi بتساوي
+
+747
+01:03:20,480 --> 01:03:25,020
+مين؟بتساوي اللي هو ناقص هتجيبها سامي بصي بيساوي
+
+748
+01:03:25,020 --> 01:03:29,540
+ناقص S ناقص X و تجيب الشروط اللي هي تبعها ال sign
+
+749
+01:03:29,540 --> 01:03:33,500
+هتلاقيها متطابق متحققة و بما ان ال S is unique او
+
+750
+01:03:33,500 --> 01:03:37,240
+ال sign is unique اذا اللي هي اللي حققتها اللي
+
+751
+01:03:37,240 --> 01:03:41,500
+كتبتها بتساوي ال S وصارين ايه جهتين متساويات بنفس
+
+752
+01:03:41,500 --> 01:03:47,980
+الأسلوب طيب الآن نيجي نثبت اللي هو مين VI شوفوا
+
+753
+01:03:47,980 --> 01:03:52,850
+صلى على النبي عليه الصلاة والسلامهنفير نبره ان vi
+
+754
+01:03:52,850 --> 01:03:55,830
+let y المتن عار بيجيبنا let f of x بيساوي c of x
+
+755
+01:03:55,830 --> 01:04:00,070
+زي dash زي y for x المتن عار الان a calculation
+
+756
+01:04:00,070 --> 01:04:04,230
+shows that يعني فضلي هذا مرتين التفاضل سهل والله
+
+757
+01:04:04,230 --> 01:04:08,010
+يا جماعة عشانك انا يعني مابديش نضيع واجتنا في
+
+758
+01:04:08,010 --> 01:04:12,190
+التفاضلأو في الحسابات f w prime of x طبعا فاضل
+
+759
+01:04:12,190 --> 01:04:15,430
+بالنسبالي x y ايش ما على f x ثابت أسيبك منها لأن
+
+760
+01:04:15,430 --> 01:04:18,590
+لو فضلت مرتين هتلاقي ناقص f of x for x element in
+
+761
+01:04:18,590 --> 01:04:22,550
+R مدام f w prime بيساوي ناقص f of x بالحصارة اللي
+
+762
+01:04:22,550 --> 01:04:26,850
+هي حل المعادلة القفاضلية هذه اللي هي بالنظرية اللي
+
+763
+01:04:26,850 --> 01:04:32,880
+قبل شويه هي 8 4 6 هتكون اللي هي ال f of xعبارة عن
+
+764
+01:04:32,880 --> 01:04:36,300
+linear combination هذه اللي هو α C of X زي Beta S
+
+765
+01:04:36,300 --> 01:04:39,840
+of X والـ F of X مين هي احنا فرضينا C of XY صارت
+
+766
+01:04:39,840 --> 01:04:43,180
+هذه عبارة عن هذه يعني هذه بتساوي هذه من حال
+
+767
+01:04:43,180 --> 01:04:47,260
+المعادلة التفاضلية هذه بواسطة النظرية هذه وهذه
+
+768
+01:04:47,260 --> 01:04:51,330
+اصلا انا نسميها هيك صارت هذه بتساوي هذا المقطعالان
+
+769
+01:04:51,330 --> 01:04:55,350
+جيبله F prime F prime of X اللي هي هذه التفضيلة
+
+770
+01:04:55,350 --> 01:04:59,070
+بـC ناقص S of X زائد Y وفاضل هذه بيطلع اللي هو
+
+771
+01:04:59,070 --> 01:05:02,910
+عبارة عن ناقص Alpha S of X زائد Beta C of X ثم
+
+772
+01:05:02,910 --> 01:05:09,710
+فضلت ماشي الان خد X بيساوي 0 X بيساوي 0 لما X
+
+773
+01:05:09,710 --> 01:05:18,250
+بتساوي 0بنحصل الآن عندى .. بصير عندى عوض X بتساوي
+
+774
+01:05:18,250 --> 01:05:25,190
+سفر في اللى هى المعادلة اللى عندى هنا بصير عندى S
+
+775
+01:05:25,190 --> 01:05:31,360
+of Zero و C of Zeroبساومين F of Zero F of Zero
+
+776
+01:05:31,360 --> 01:05:40,100
+اللي هي عبارة عن F of Zero هي C of Y هذه فاهمين
+
+777
+01:05:40,100 --> 01:05:44,560
+عليها؟ خليني أقولها واضحة أحسن الآن بدنا ناخد F
+
+778
+01:05:44,560 --> 01:05:52,600
+مين؟ F عند Zero فاشر؟ بيصير هذا C of Y بساومي
+
+779
+01:05:52,600 --> 01:05:58,190
+Alphaفي c of zero واحد وهذه beta s of zero زيرو
+
+780
+01:05:58,190 --> 01:06:03,890
+إذا صارت عندي ال alpha بتساوي c of y هاي واحدة لأن
+
+781
+01:06:03,890 --> 01:06:09,990
+similarly اللي هو ناقص خد عند ال zero عند ال zero
+
+782
+01:06:09,990 --> 01:06:15,570
+بيصير ناقص s of y بساوي هذي بيصير سفر وهذه بيصير
+
+783
+01:06:15,570 --> 01:06:20,070
+beta بساوي beta الآن بنعوض وين في ال formula
+
+784
+01:06:20,070 --> 01:06:26,370
+الأولىبصير عندى الان ال formula اللى عندى اللى هى
+
+785
+01:06:26,370 --> 01:06:30,010
+بتحط مكان C of Y بسوة Alpha و الفورمولة الأولى
+
+786
+01:06:30,010 --> 01:06:35,770
+اللى احنا عملناها بصير عندى اللى هى C
+
+787
+01:06:38,530 --> 01:06:43,570
+Half X زائد Y بساوة Alpha اللي هي اسمها قلنا طلعت
+
+788
+01:06:43,570 --> 01:06:48,310
+عندنا C of Y بصير C of Y في C of X هيها مظبوطة
+
+789
+01:06:48,310 --> 01:06:53,130
+وهذه ناقص S مكان ال Beta بصير ناقص S of Y في S of
+
+790
+01:06:53,130 --> 01:07:01,740
+X صحيحة إذا بكون إحنا خلصنا اللي بدناياه وهذهبتعوض
+
+791
+01:07:01,740 --> 01:07:08,240
+عن اللي هي Alpha C of Y بC نقص C of Y وهذه بتعوضها
+
+792
+01:07:08,240 --> 01:07:13,440
+هنا بتطلع هذه بتساوي نقص هذه اعمل الحساب الأخير
+
+793
+01:07:13,440 --> 01:07:17,400
+وظروف الدقيقتين بنقص بتطلع عندك هذا المخضر يعني
+
+794
+01:07:17,400 --> 01:07:23,840
+هذه التعويض فيها عن قيمة Alpha و Beta بهذه هنا
+
+795
+01:07:23,840 --> 01:07:30,860
+بتطلع هذه وهذه التعويض هذهعن alpha و beta هنا اللي
+
+796
+01:07:30,860 --> 01:07:35,420
+بتطلع الأولى بكون احنا خلصنا اللي هو اثبات هذه
+
+797
+01:07:35,420 --> 01:07:42,800
+اللي هي النظرية وضال عندي اللي هو النظرية هذه
+
+798
+01:07:42,800 --> 01:07:47,350
+الأخيرةوالباقي اللي هو لو اتطلعتوا على اللي هي
+
+799
+01:07:47,350 --> 01:07:52,450
+باقي النظريات اللي هي بس اتطلع لعند الها بيكون
+
+800
+01:07:52,450 --> 01:07:55,990
+احنا بيكون خلصنا chapter اللي هو المطلوب في
+
+801
+01:07:55,990 --> 01:08:00,350
+chapter في تمانية أربعة هذه اللي هي ال theorem
+
+802
+01:08:00,350 --> 01:08:03,530
+خليني أطلعلك عليها على السريع لإن حسابات كلها
+
+803
+01:08:03,530 --> 01:08:07,840
+بتعملها لحالك أكيد بتعرفإذا كان الـ x أكبر أو أقل
+
+804
+01:08:07,840 --> 01:08:12,880
+من 0، فهناك Sx بنقص الـ x والـ x والـ c of x بين
+
+805
+01:08:12,880 --> 01:08:16,780
+الواحد نقصه x أربع عزر تو واحد والـ c of x بقدر
+
+806
+01:08:16,780 --> 01:08:18,820
+أكمل الـ polynomial
+
+807
+01:08:21,810 --> 01:08:25,650
+باللي عملناها اللي هي الـCN of X اللي جابله شوية و
+
+808
+01:08:25,650 --> 01:08:28,430
+الـSN of X لأنه في النهاية limitها ال series
+
+809
+01:08:28,430 --> 01:08:30,790
+الأولى as N goes to infinity ما أنتوا عارفين اللي
+
+810
+01:08:30,790 --> 01:08:34,990
+هي نقدر نكتب الـC of X على صورة ال series اللي في
+
+811
+01:08:34,990 --> 01:08:38,130
+الأول و الS of X عبارة عن limit ال series التانية
+
+812
+01:08:38,130 --> 01:08:43,390
+ال terms إضافة term و طرح term و وجوف عند حدود بال
+
+813
+01:08:43,390 --> 01:08:47,090
+term بتعمل ال inequality اللي عندنا اللي هو كل هذه
+
+814
+01:08:47,090 --> 01:08:49,810
+شغلات اللي أخدناها في ال calculus مافيش داعي
+
+815
+01:08:49,810 --> 01:08:55,650
+للتفصيل فيهاالان الاولىعندي ال .. ال .. احنا قلنا
+
+816
+01:08:55,650 --> 01:08:59,990
+sign تربيع زائد plus sign تربيع بساوي واحد يعني في
+
+817
+01:08:59,990 --> 01:09:02,670
+النهاية ال C of T بين ماقص واحد و واحد ده ممكن
+
+818
+01:09:02,670 --> 01:09:06,710
+هتتجاوزها لأنه بتختل اللي هي ال C .. لو كانت أكبر
+
+819
+01:09:06,710 --> 01:09:10,750
+من واحد معناته بـ C .. C of T تربيع زائد S of T
+
+820
+01:09:10,750 --> 01:09:14,550
+تربيع يتجاوز من واحد لكن مجموح بسوء واحد إذا هدي
+
+821
+01:09:14,550 --> 01:09:18,630
+بين .. يعني هدي اللي وين جاية من C تربيع زائد S
+
+822
+01:09:18,630 --> 01:09:23,030
+تربيع بساوي واحد طيباللي هنعمل integration لجهتين
+
+823
+01:09:23,030 --> 01:09:26,990
+من صفر لعين DX هذه اللي هي بتطلع اللي هي main ال S
+
+824
+01:09:26,990 --> 01:09:30,750
+of X وهذه بتطلع أنخس X وهذه بتطلع X فالمعمل ال
+
+825
+01:09:30,750 --> 01:09:35,990
+integration هدول الان بدي
+
+826
+01:09:35,990 --> 01:09:43,300
+أجيب التانية فضلي ال S of Tهذه وهذه فاضليها هذه
+
+827
+01:09:43,300 --> 01:09:48,460
+فاضليها بنقل صفر عند X بتطلع اللي هي بين هذه وهذه
+
+828
+01:09:48,460 --> 01:09:53,660
+هيتفاضوا لهذه وهذه الان بدنا اللي هو نجيب الواحد
+
+829
+01:09:53,660 --> 01:09:58,280
+ناقص X السربيع اللي هو هذا قيمته اللي هو عبارة عن
+
+830
+01:09:58,280 --> 01:10:02,600
+C في X بالساوية واحد ناقص هذه بنضرب الناقص و بنجمع
+
+831
+01:10:02,600 --> 01:10:08,280
+الجهتين واحد بطلع عندي اللي هي ال inequality اللي
+
+832
+01:10:08,280 --> 01:10:13,190
+عندي هذهزي ما قلنا ضربنا في ناقص و جمعنا واحد طلعت
+
+833
+01:10:13,190 --> 01:10:16,590
+هذه او عوضنا هذه مكان هذه تعويض عادي و بعدين ضربنا
+
+834
+01:10:16,590 --> 01:10:20,690
+في ناقص فبصير عندى هذه المقدار اللى عندى c of x
+
+835
+01:10:20,690 --> 01:10:24,310
+أكبر سواء هذه و ناقص هذه و اللى بتستخدم الثانية
+
+836
+01:10:24,310 --> 01:10:27,670
+بتطلع نفس الاشي الموضوع موضوع حسابات بحت عشان هيك
+
+837
+01:10:27,670 --> 01:10:31,510
+بحت عشان هيك مافي داعي للتكميل و انتوا بتكملوا
+
+838
+01:10:31,510 --> 01:10:35,150
+باقى الحسابات اللى هي المطلوبة في هذا اللى هي
+
+839
+01:10:35,150 --> 01:10:42,700
+النظرية الان الجزء المتبقى هذاحبيته تطلعوا عليه
+
+840
+01:10:42,700 --> 01:10:46,880
+لكن اللي هو مش مطلوب من ضمن حديثنا و هيك ممكن
+
+841
+01:10:46,880 --> 01:10:52,700
+نخلصنااللي هو section تمانية أربعة متسمة سبتر
+
+842
+01:10:52,700 --> 01:10:58,420
+تمانية هذا اللي هو الجزء الثالث من المادة الجزء
+
+843
+01:10:58,420 --> 01:11:00,160
+الأول كان ال differentiation والتاني ال
+
+844
+01:11:00,160 --> 01:11:01,960
+integration والتالت اللي هو point twice
+
+845
+01:11:01,960 --> 01:11:05,700
+convergence المرة الجاية ان شاء الله اللي هو بنبدأ
+
+846
+01:11:05,700 --> 01:11:09,880
+في الجزء الرابع من المادة اللي هو ال series وان
+
+847
+01:11:09,880 --> 01:11:16,590
+شاء الله بقدر الإمكان هأحكي الجزء الخامسفي .. من
+
+848
+01:11:16,590 --> 01:11:20,510
+المادة اللي هو عبارة عن اللي هو Topology in R أو
+
+849
+01:11:20,510 --> 01:11:23,570
+اللي هو اللي هي العلاقة بين اللي هي Topological
+
+850
+01:11:23,570 --> 01:11:27,410
+Spaces وNormed Spaces وHilbert Spaces إلى آخره
+
+851
+01:11:27,410 --> 01:11:33,530
+ويمكن بتكون هي اتجاوزت خلنا نقول حد ال .. ال .. ال
+
+852
+01:11:33,530 --> 01:11:37,150
+.. المطلوب في المادة لكن على أساس إنه لو يكون
+
+853
+01:11:37,150 --> 01:11:42,210
+التصوير كامل للو .. للوصف اللي أنا اقتراحتهأو اللي
+
+854
+01:11:42,210 --> 01:11:47,230
+هو الوصف اللي مقترح في الجسم اللي هو لعين نهاية ال
+
+855
+01:11:47,230 --> 01:11:49,990
+series و طبعا هذا الكلام قلنا احنا فيه له سبب لما
+
+856
+01:11:49,990 --> 01:11:53,450
+بدأنا في الأول انه ال series بتاخدوها في advanced
+
+857
+01:11:53,450 --> 01:11:56,990
+calculus واحد فعشان هيك يمكن اللي هو
+
+858
+01:12:00,350 --> 01:12:03,810
+هيكون منطقي أنه ما ناخدش ال series و ناخد بدلها
+
+859
+01:12:03,810 --> 01:12:07,110
+اللي هو ال topology انار أو اللي هو العلاقة بين ال
+
+860
+01:12:07,110 --> 01:12:10,550
+topological spaces و ال metric spaces و إلى لقاء و
+
+861
+01:12:10,550 --> 01:12:12,650
+السلام آخر والسلام عليكم و رحمه الله وبركاته
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk.srt
new file mode 100644
index 0000000000000000000000000000000000000000..fb0c26b8989fd8fa7088841cae475d631782b919
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk.srt
@@ -0,0 +1,2881 @@
+1
+00:00:04,910 --> 00:00:11,030
+بسم الله الرحمن الرحيم المحاضرة الرابعة بعد حالة
+
+2
+00:00:11,030 --> 00:00:19,410
+الطوارئ في مادة أو مساق تحليل رياضي 2 أو تحليل 
+
+3
+00:00:19,410 --> 00:00:23,970
+حقيقي 2 لطلبة وطالبات الجامعة الإسلامية كلية
+
+4
+00:00:23,970 --> 00:00:30,110
+العلوم قسم الرياضيات وهي المحاضرة رقم 14 في المادة
+
+5
+00:00:30,110 --> 00:00:35,970
+أو في المساق عنوان المحاضرة اليوم اللي هو
+
+6
+00:00:35,970 --> 00:00:43,260
+fundamental theorem of calculus بشقيها شق تفاضل
+
+7
+00:00:43,260 --> 00:00:50,640
+التكامل وشق تكامل التفاضل roughly نبدأ اللي هو في
+
+8
+00:00:50,640 --> 00:00:54,960
+النظرية الأولى أو الجزء الأول من النظرية اللي هي
+
+9
+00:00:54,960 --> 00:00:58,760
+the first form of the fundamental theorem of calculus
+
+10
+00:00:58,760 --> 00:01:04,870
+اللي هي الجزء الأول بتقول ما يلي: let f from a to b
+
+11
+00:01:04,870 --> 00:01:08,430
+to R be an integrable function on a to b يعني
+
+12
+00:01:08,430 --> 00:01:12,070
+فرضنا أن الـ function f عبارة عن دالة قابلة
+
+13
+00:01:12,070 --> 00:01:17,190
+للتكامل على الفترة المغلقة a to b and let F capital
+
+14
+00:01:17,190 --> 00:01:22,190
+from a to b to R تحقق الشروط التالية: تحقق أول شيء
+
+15
+00:01:22,190 --> 00:01:25,110
+أنها تكون الـ F capital هذه continuous على الفترة
+
+16
+00:01:25,110 --> 00:01:30,530
+a to b الشرط الثاني: F prime exists and F prime of x
+
+17
+00:01:30,530 --> 00:01:34,130
+بيساوى الـ f small اللي بدأنا فيها دي الفرض منها
+
+18
+00:01:34,130 --> 00:01:39,880
+integrable لكل x element in a to b بنقول الـ
+
+19
+00:01:39,880 --> 00:01:44,900
+integration من A to B للـ F بيساوى F of B ناقص F of A
+
+20
+00:01:44,900 --> 00:01:50,400
+إذا الآن وكأني بقول نفترض أنه الـ F small
+
+21
+00:01:50,400 --> 00:01:56,100
+integrable ونفترض F continuous على closed bounded
+
+22
+00:01:56,100 --> 00:02:01,160
+interval A to B و F نفسها differentiable على الـ
+
+23
+00:02:01,160 --> 00:02:05,260
+open اللي هو interval A to B بشرط أنه الـ F prime
+
+24
+00:02:05,260 --> 00:02:07,940
+ال derivative لها الـ F capital طبعًا بيساوى الـ F
+
+25
+00:02:07,940 --> 00:02:13,240
+small of X وبناءً على كل المعطيات اللي حكيناها بيكون
+
+26
+00:02:13,240 --> 00:02:17,120
+ال integration من a to b لل f small هو عبارة عن f(b)
+
+27
+00:02:17,120 --> 00:02:21,760
+ناقص f of a أو اختصارًا وسهلة للحفظ ال integration
+
+28
+00:02:21,760 --> 00:02:29,670
+من a to b لل f prime of x dx هي عبارة عن وكأنه
+
+29
+00:02:29,670 --> 00:02:36,250
+التكامل بلغ التفاضل بيصير اللي هي f of b ناقص f of
+
+30
+00:02:36,250 --> 00:02:40,610
+a هذا فضوء mean يعني هاي هادة باختصار هيك فضوء
+
+31
+00:02:40,610 --> 00:02:44,710
+mean أن f prime of x تكون موجودة على الفترة من a to 
+
+32
+00:02:44,710 --> 00:02:51,370
+b وأيضًا بتكون إنها integrable نفسها integrable و
+
+33
+00:02:51,370 --> 00:02:56,110
+تكون ال f نفسها continuous على closed من a to b
+
+34
+00:02:56,390 --> 00:03:00,270
+هذه هي النظرية إذا اللي بدنا نثبته الآن أنه قيمة
+
+35
+00:03:00,270 --> 00:03:04,470
+ال integration هذا بيساوى اللي هو اللي أمامنا اللي
+
+36
+00:03:04,470 --> 00:03:09,590
+هي f of b ناقص من ناقص f of a خلينا نشوف إيش
+
+37
+00:03:09,590 --> 00:03:15,030
+البكاء في البحران يا جماعة الآن اللي واضح أنه أنا
+
+38
+00:03:15,030 --> 00:03:18,530
+بده أثبت أنه ال integration هو عبارة عن f of b
+
+39
+00:03:18,530 --> 00:03:24,990
+ناقص f of a شوف أنا كيف بده أوصلك اللي هو ال
+
+40
+00:03:24,990 --> 00:03:28,670
+integration لمنقلة ال b لل f بيساوى f of b ناقص f
+
+41
+00:03:28,670 --> 00:03:32,010
+of a أنا بعرف أول حاجة هو معطيلي ال f small هذه
+
+42
+00:03:32,010 --> 00:03:35,290
+اللي هي ال f prime اللي هي ال f small سميتها هذه
+
+43
+00:03:35,290 --> 00:03:39,240
+معطينيها integrable إذن بواسطة
+
+44
+00:03:39,240 --> 00:03:44,160
+اللي هي .. إذا بتتذكروا Integrable criterion وهي
+
+45
+00:03:44,160 --> 00:03:46,900
+اللي بنستخدمها كثير احنا لإنها .. يعني خلّيني أقول
+
+46
+00:03:46,900 --> 00:03:51,400
+صيغة مهمة في إثبات النظريات اللي هي بما إنه أف
+
+47
+00:03:51,400 --> 00:03:56,380
+integrable إذن لكل Y أكبر من 0 يوجد there exists a
+
+48
+00:03:56,380 --> 00:04:00,600
+partition P اللي هو X0 X1 عند X ل .. عند ..
+
+49
+00:04:00,600 --> 00:04:03,280
+partition لمين؟ طبعًا للفترة اللي أنتم عارفينها
+
+50
+00:04:03,280 --> 00:04:06,800
+اللي احنا بنشتغل عليها الفترة من A لعند مين؟ لعند B
+
+51
+00:04:06,960 --> 00:04:12,920
+There exists a partition P X0 X1 XN of A to B such
+
+52
+00:04:12,920 --> 00:04:20,280
+that the upper sum P F'- the lower sum P F' أصغر من إبسلون F' من
+
+53
+00:04:20,280 --> 00:04:24,000
+هنا من هي يا جماعة؟ هي ال F لأننا مفترضين ال F
+
+54
+00:04:24,000 --> 00:04:29,940
+small أو ال F' is integrable إذا بواسطة ال integrable
+
+55
+00:04:29,940 --> 00:04:34,760
+criterion حصلنا على واحد اللي عنده
+
+56
+00:04:34,760 --> 00:04:40,880
+إلحاد أحكي عنها الآن أو أستخدمها للوصول لهدفي طيب
+
+57
+00:04:40,880 --> 00:04:44,040
+because طبعًا F prime اللي هي same زي ما قلنا F of
+
+58
+00:04:44,040 --> 00:04:48,820
+X is integrable on A to B الآن في شغلة ثانية احنا
+
+59
+00:04:48,820 --> 00:04:55,880
+بدأنا الآن نستخدم اللي هو أمر آخر خلينا نكتب الآن
+
+60
+00:04:55,880 --> 00:05:00,920
+أول شغلة حصلناها عشان تعرف أين أنا رايح اللي هي
+
+61
+00:05:00,920 --> 00:05:06,550
+قلنا F is integrable F التي يسمى F' Integrable إيش
+
+62
+00:05:06,550 --> 00:05:12,270
+أعطتنا يا جماعة؟ أعطتنا أن هناك Partition P هناك P
+
+63
+00:05:12,270 --> 00:05:15,830
+طبعًا for every epsilon أكبر من صفر هناك Partition P
+
+64
+00:05:15,830 --> 00:05:23,390
+such that U, P و F اللي هي الـ F' طبعًا الـ F يمين
+
+65
+00:05:23,390 --> 00:05:29,530
+الـ F' ناقص الـ L, P و F هي أصغر من Y وهذا سميناها
+
+66
+00:05:29,530 --> 00:05:34,710
+إيش؟ سميناها (1) لأن من جهة أخرى يا جماعة بدي أستغل
+
+67
+00:05:34,710 --> 00:05:37,810
+اللي هو اللي معطيني إياه أن الأفكار بتلقى continuous
+
+68
+00:05:37,810 --> 00:05:41,490
+أنا الآن زي ما قلنا جزأنا اللي هي الفترة A to B
+
+69
+00:05:41,490 --> 00:05:52,230
+إلى اللي هي X0 X1 X2 فترة نموذجية Xk-1 to Xk to
+
+70
+00:05:52,230 --> 00:05:56,010
+مين؟ to Xn اللي هي مين؟ الـ B هذا ال Partition
+
+71
+00:05:56,010 --> 00:05:59,930
+اللي جبتّه أنا الآن أنا بعرف أن ال F capital
+
+72
+00:05:59,930 --> 00:06:03,330
+continuous على ال closed interval كلها من A to B
+
+73
+00:06:03,330 --> 00:06:06,870
+إذا الأكيد continuous على كل sub interval موجودة
+
+74
+00:06:06,870 --> 00:06:12,540
+يعني صارت اللي عندي ال F capital continuous on 
+
+75
+00:06:12,540 --> 00:06:18,560
+the Xk-1 to Xk هذه هي ال sub interval اللي
+
+76
+00:06:18,560 --> 00:06:24,080
+ال function f continuous عليها وإحنا مفترضين أيضًا
+
+77
+00:06:24,080 --> 00:06:29,220
+أن ال f prime exist على ال open interval كلها إذا
+
+78
+00:06:29,220 --> 00:06:33,860
+أكيد ال F is differentiable لأن f' exist على ال
+
+79
+00:06:33,860 --> 00:06:37,360
+open interval من a to b إذا أكيد f is
+
+80
+00:06:37,360 --> 00:06:41,980
+differentiable على الفترة xk minus واحد to مين؟
+
+81
+00:06:41,980 --> 00:06:45,140
+to ال xk يعني يا جماعة شروط ال mean value
+
+82
+00:06:45,140 --> 00:06:50,020
+theorem محققة إذا أكيد there exist there exist بما
+
+83
+00:06:50,020 --> 00:06:54,540
+أنه اللي هي شروط ال mean value محققة إذا there
+
+84
+00:06:54,540 --> 00:06:59,620
+exist بسميها اللي هو bk أو tk there exist tk في
+
+85
+00:06:59,620 --> 00:07:06,440
+الفترة xk minus واحد to ال xk such that اللي هو F
+
+86
+00:07:06,440 --> 00:07:11,520
+هي الدالة اللي بأطبق عليها ال mean value theorem يا
+
+87
+00:07:11,520 --> 00:07:19,320
+جماعة F of xk ناقص F of xk minus واحد اللي هو
+
+88
+00:07:19,320 --> 00:07:24,560
+بتساوي اللي هو xk ناقص xk minus واحد طبعًا احنا
+
+89
+00:07:24,560 --> 00:07:29,280
+بنضربها في اللي هي إيش؟ بتساوي اللي هي ال F prime
+
+90
+00:07:30,370 --> 00:07:35,630
+of the tk هذه اللي هي الـ Mean Value Theorem
+
+91
+00:07:35,630 --> 00:07:39,710
+نتيجة تطبيقها على مين يا جماعة؟ على الفترة اللي هي XK
+
+92
+00:07:39,710 --> 00:07:43,770
+minus واحد to XK طبعًا هذا الكلام لكل K لأنه لكل
+
+93
+00:07:43,770 --> 00:07:48,170
+مين؟ لكل ال sub intervals بتتحقق نفس الشروط بناءً
+
+94
+00:07:48,170 --> 00:07:55,010
+عليه احنا بنعرف قبل هيك خليني أبدأ أشتغلها عند ال
+
+95
+00:07:55,010 --> 00:08:04,450
+mk' إبراهيم اللي هي أكيد أصغر أو يساوى الـ F اللي هي
+
+96
+00:08:04,450 --> 00:08:11,370
+الـ F عندك ال mk' إيش أذكركم فيها؟ إيش هي ال mk'؟ ال
+
+97
+00:08:11,370 --> 00:08:15,190
+mk' هي مكتوبة لنكتبها كمان مرة ال mk' هي مين؟ ال
+
+98
+00:08:15,190 --> 00:08:19,610
+infimum لل F' F of X X element of X K minus واحد X
+
+99
+00:08:19,610 --> 00:08:24,950
+K الـ F' مين هذه؟ الـ F small ماشي الحال طيب الـ F أو
+
+100
+00:08:24,950 --> 00:08:30,590
+ال mk' هي عبارة عن ال infimum لمين؟ للـ F' على ال
+
+101
+00:08:30,590 --> 00:08:37,110
+sub interval هذه إذا أكيد أصغر أو يساوى الـ F of أي
+
+102
+00:08:37,110 --> 00:08:41,630
+مقطوع موجودة في الـ sub-interval والـ tk هذه
+
+103
+00:08:41,630 --> 00:08:46,410
+موجودة في هذا الـ sub-interval إذن أكيد F of tk
+
+104
+00:08:46,410 --> 00:08:50,670
+أصغر من الـ mk' لأن الـ mk' هي عبارة عن الـ
+
+105
+00:08:50,670 --> 00:08:57,290
+infimum لقيمة الدالة F' of tk عالميًا على الفترة
+
+106
+00:08:57,290 --> 00:09:01,440
+اللي هي ال sub-interval اللي بنحكي عنها ككل هذه
+
+107
+00:09:01,440 --> 00:09:06,040
+أكيد أصغر أو يساوى من الـ Mk إبراهيم مين الـ Mk
+
+108
+00:09:06,040 --> 00:09:11,060
+إبراهيم هذه؟ هي عبارة عن الـ Supremum لمين؟ لقيمة
+
+109
+00:09:11,060 --> 00:09:16,990
+اللي هي الدالة هذه على الفترة اللي عندها ماشي الحال
+
+110
+00:09:16,990 --> 00:09:23,110
+إذا هذه دائمًا صحيحة طب لو ضربت جهتي كل الأطراف في
+
+111
+00:09:23,110 --> 00:09:27,650
+طول الفترة اللي هي xk ناقص xk minus واحد طبعًا
+
+112
+00:09:27,650 --> 00:09:31,510
+أكيد هذه xk ناقص xk minus واحد كمية موجبة إذا لما
+
+113
+00:09:31,510 --> 00:09:35,010
+أضربها في الأطراف كلها بتظل ال inequality صحيحة
+
+114
+00:09:35,010 --> 00:09:40,610
+و زي ما هي في xk ناقص xk minus واحد وهنا في xk ناقص
+
+115
+00:09:40,610 --> 00:09:46,280
+xk minus واحد مظبوط هذا الكلام أكيد صح طيب لكن هذا
+
+116
+00:09:46,280 --> 00:09:52,220
+المقدار هو هذا إذا باجي بحط هذه هنا معايا يا جماعة
+
+117
+00:09:52,220 --> 00:09:57,000
+بحط هذه في المنطقة هذه المكان القيمة هذه اللي هي
+
+118
+00:09:57,000 --> 00:10:05,190
+هذه إذا صار عندي الآن F of xk ناقص f of xk minus 1
+
+119
+00:10:05,190 --> 00:10:12,070
+أكبر أو يساوى ال mk' في ال xk ناقص xk minus 1
+
+120
+00:10:12,070 --> 00:10:18,490
+وهذا أصغر أو يساوى ال mk' في ال xk ناقص xk minus
+
+121
+00:10:18,490 --> 00:10:24,290
+1 إذا واضح إن أنا حصلت على هذه وهذه صحيحة لكل مين؟
+
+122
+00:10:24,290 --> 00:10:28,150
+لكل k لإنه أخدت أنا هذه فترة عشوائية يعني لـ k
+
+123
+00:10:28,150 --> 00:10:33,790
+بتساوي اللي هو واحد واثنين to مين؟ to n بناءً
+
+124
+00:10:33,790 --> 00:10:38,250
+عليه أنا وصلت الآن إلى ما يلي: وصلت إلى هذه
+
+125
+00:10:38,250 --> 00:10:43,670
+النتيجة شايفين يا جماعة وصلت إلى هذه النتيجة اللي
+
+126
+00:10:43,670 --> 00:10:48,860
+أنا وضحت لكم إياها كيف اجت الآن وال mk prime زي ما
+
+127
+00:10:48,860 --> 00:10:55,280
+قلنا إيش هي وال mk capital prime هيها أمامكم نيجي
+
+128
+00:10:55,280 --> 00:10:59,720
+الآن إيش اللي بده يصير له؟ شوف ال k إيش بده يصير؟ أنا
+
+129
+00:10:59,720 --> 00:11:03,020
+عمال بقول إنه صحيح على كل K إذا لو أخدت ال
+
+130
+00:11:03,020 --> 00:11:05,900
+summation هنا وأخدت ال summation هنا وأخدت ال
+
+131
+00:11:05,900 --> 00:11:10,440
+summation هنا بظل اللي هي ال inequality صحيحة خذ
+
+132
+00:11:10,440 --> 00:11:16,050
+الآن ال summation على الفترة ال summation على
+
+133
+00:11:16,050 --> 00:11:19,530
+ال inequality اللي حكينا عنها هيها بيصير ال
+
+134
+00:11:19,530 --> 00:11:23,630
+summation للـ M k prime في X k X k minus واحد k من
+
+135
+00:11:23,630 --> 00:11:26,630
+عند واحد to M وهنا هيكون أصغر أو يساوى ال
+
+136
+00:11:26,630 --> 00:11:31,090
+summation لل F of X k ناقص F of X k minus واحد
+
+137
+00:11:31,090 --> 00:11:36,190
+اللي هي احنا هنا حصلناها هنا وهذه ال summation
+
+138
+00:11:36,190 --> 00:11:39,490
+عليها وهذه ال summation عليها هي ال summation هذا
+
+139
+00:11:39,850 --> 00:11:43,210
+طيب يا جماعة أكيد أنتم فاكرين هذا ال summation هو
+
+140
+00:11:43,210 --> 00:11:47,770
+عبارة عن مين يا جماعة؟ هو عبارة عن ال lower sum لل
+
+141
+00:11:47,770 --> 00:11:52,370
+partition P على مين؟ على الدالة اللي بنعمل عليها
+
+142
+00:11:52,370 --> 00:11:56,330
+اللي هي ال F' اللي اشتغلنا عليها اللي أخدنا ال F'
+
+143
+00:11:57,150 --> 00:12:01,270
+الـ mk' هي ال infimum لل F' على ال sub interval
+
+144
+00:12:02,160 --> 00:12:07,220
+والآن هذه إيش يا جماعة؟ هي ال upper sum يعني بمعنى
+
+145
+00:12:07,220 --> 00:12:12,960
+آخر صار عندي الآن المقدار هذا هو المقدار هذا اللي
+
+146
+00:12:12,960 --> 00:12:17,680
+محاط بالأزرق هذا المقدار وبالأحمر صار بين اللي هو
+
+147
+00:12:17,680 --> 00:12:22,120
+ال lower sum وبين ال upper sum لكن هذا المقدار
+
+148
+00:12:22,120 --> 00:12:25,860
+الصمّصم اللي احنا عملناه قبل ذلك كتير لو فرضناه
+
+149
+00:12:25,860 --> 00:12:33,860
+هيكون عبارة عن F of X واحد ناقص F of X Note زائد F
+
+150
+00:12:33,860 --> 00:12:40,360
+of X 2 ناقص F of X 1 زائد F of X 3 ناقص F of X 2
+
+151
+00:12:40,360 --> 00:12:45,460
+زائد لما أصل لآخر شيء F of X N ناقص F of X N ناقص
+
+152
+00:12:45,460 --> 00:12:50,520
+1 حلاجي كلّه بيروح وبيظل عندك كل هذا بيروح وبيظل عندك
+
+153
+00:12:50,520 --> 00:12:54,860
+بس F of X N ناقص F of X Note كل
+
+154
+00:12:54,860 --> 00:13:00,340
+term بـ cancel اللي قبله وبيساوي F of Xn اللي هي
+
+155
+00:13:00,340 --> 00:13:07,420
+مين؟ F of B F of X Note اللي هي مين؟ F of A إذا صار
+
+156
+00:13:07,420 --> 00:13:10,920
+عندي الآن الـ summation هذا اللي هو بيساوي F of B
+
+157
+00:13:10,920 --> 00:13:16,440
+ناقص مين؟ ناقص F of A يعني بمعنى آخر بنشيل هذا ونحط
+
+158
+00:13:16,440 --> 00:13:20,580
+مكانه قيمته اللي هو F of B ناقص F of A بيصير F
+
+159
+00:13:20,580 --> 00:13:25,320
+of B minus F of A بين اللي هو L of P و F' وبين
+
+160
+00:13:25,320 --> 00:13:34,230
+الأخرى U of P و F' طيب اطلع على فوق شوية خليني
+
+161
+00:13:34,230 --> 00:13:39,830
+أوضح لك هذه النقطة اللي هتوصلني للي بديها أنا إيش
+
+162
+00:13:39,830 --> 00:13:44,170
+بدي أثبت؟ أنا بدي أثبت أن ال integration خليني بس
+
+163
+00:13:44,170 --> 00:13:51,710
+أشيل هذا بعد إذنكم أنا
+
+164
+00:13:51,710 --> 00:13:58,260
+بدي أثبت أن ال integration لل F prime of x dx أو
+
+165
+00:13:58,260 --> 00:14:06,320
+الـ F of X dx بيساوى F of B ناقص F of A من A to
+
+166
+00:14:06,320 --> 00:14:12,870
+مين؟ to B، هيا، بدي أسهل شوف الآن كيف أنا في
+
+167
+00:14:12,870 --> 00:14:17,590
+الواقع بتوصل لكم هذا ناقص هذا بيكون أكبر أو يساوى صفر
+
+168
+00:14:17,590 --> 00:14:20,990
+وأصغر من epsilon لأي أصغر يساوى epsilon لكل
+
+169
+00:14:20,990 --> 00:14:24,110
+epsilon في الدنيا ومن ثم هيساوى صفر يعني هتحدث
+
+170
+00:14:24,110 --> 00:14:29,090
+مين؟ المساواة اللي ما فهمش عليها هي الآن نجلة تفصيل
+
+171
+00:14:29,090 --> 00:14:39,060
+اتفقنا احنا إنه احنا حصلنا على اللي هو الـ F of B – 
+
+172
+00:14:39,060 --> 00:14:44,520
+F of A بين الـ U وبين الـ L لكن احنا بنقول الـ F
+
+173
+00:14:44,520 --> 00:14:49,420
+نفسها يا جماعة الـ F' هذه is integrable مزام
+
+174
+00:14:49,420 --> 00:14:55,380
+integrable من A to B إذن الـ U of F والـ L of F
+
+201
+00:16:59,720 --> 00:17:02,000
+آسف اللي هو اضرب مين اللي فوق بالسالب يصير هذا
+
+202
+00:17:02,000 --> 00:17:07,640
+ناقص وهذا كله ناقص وهذا عبارة عن ناقص وهذا أكبر أو 
+
+203
+00:17:07,640 --> 00:17:12,760
+يساوي وهذا أكبر أو يساوي وأجمع لبعض أو بلغة أخرى
+
+204
+00:17:13,150 --> 00:17:16,790
+أقول اللي .. اللي .. اللي .. اللي تحت ناقص اللي فوق
+
+205
+00:17:16,790 --> 00:17:20,130
+بيطلع عندي ال integration للـ F prime من A لعند B
+
+206
+00:17:20,130 --> 00:17:24,730
+ناقص اللي هي F of B minus F of A أصغر أو يساوي الـ U
+
+207
+00:17:24,730 --> 00:17:30,770
+ناقص الـ L  و أكبر أو يساوي الـ L ناقص الـ U واضح أنه 
+
+208
+00:17:30,770 --> 00:17:36,480
+هذه عبارة عن ناقص هذه وهذه كمية موجبة إذا صار عند
+
+209
+00:17:36,480 --> 00:17:41,000
+ال absolute value للقيمة هذه أصغر أو يساوي الـ U
+
+210
+00:17:41,000 --> 00:17:49,300
+ناقص مين الـ L كمان مرة هذه سالب هذه وهذه موجبة إذًا
+
+211
+00:17:49,300 --> 00:17:53,540
+هذه الـ Inquality صارت هي عبارة عن مفكوك الـ
+
+212
+00:17:53,540 --> 00:17:57,440
+absolute value للـ F prime ال integration ناقص
+
+213
+00:17:57,440 --> 00:18:01,840
+المقدار هذا أصغر أو يساوي هذه القيمة وهذه القيمة 
+
+214
+00:18:01,840 --> 00:18:05,700
+لسه ما سحنّاش إحنا إيش قلنا عنها؟ أصغر من من إبسلون 
+
+215
+00:18:05,700 --> 00:18:09,280
+وأنا قاعد بشتغل على ال partition اللي جبتّه أعلى 
+
+216
+00:18:09,280 --> 00:18:15,390
+إذا صار عندي الآن اللي وصلت له عبارة عن اللي هو الـ
+
+217
+00:18:15,390 --> 00:18:20,150
+absolute value هذا أصغر من إبسلون وطبعًا أكبر أو يساوي 
+
+218
+00:18:20,150 --> 00:18:24,790
+صفر والإبسلون اللي إحنا شمالها arbitrary يعني
+
+219
+00:18:24,790 --> 00:18:29,430
+المقدار هذا لأي إبسلون في الدنيا لأي إبسلون في
+
+220
+00:18:29,430 --> 00:18:33,770
+الدنيا هذا أصغر من مين؟ من الإبسلون وهو أكبر أو يساوي صفر
+
+221
+00:18:33,970 --> 00:18:38,510
+إذًا لازم يكون هذا المقدار اللي هو عبارة عن صفر 
+
+222
+00:18:38,510 --> 00:18:42,530
+يعني بمعنى آخر ال integration بيساوي اللي هو F of B
+
+223
+00:18:42,530 --> 00:18:47,050
+ناقص F of A وهيك ونكون حصلنا على المطلوب وأثبتنا
+
+224
+00:18:47,050 --> 00:18:51,470
+النظرية طيب
+
+225
+00:18:51,470 --> 00:18:57,850
+الجماعة الآن هذا اللي هو الجزء الأول الجزء الأول من 
+
+226
+00:18:57,850 --> 00:19:00,890
+اللي هو الـ Fundamental Theorem of Calculus اللي
+
+227
+00:19:00,890 --> 00:19:05,750
+هو عبارة عن تكامل التفاضل اللي هي عملية التكامل
+
+228
+00:19:05,750 --> 00:19:10,470
+بتلغي عملية التفاضل زي ما شفنا اللي هو قبل بشوية
+
+229
+00:19:10,470 --> 00:19:15,130
+وشفنا إيش اللي جابه الـ Fundamental Theorem طبعًا
+
+230
+00:19:15,130 --> 00:19:20,270
+هذا الكلام تحت الشروط المذكورة الآن بقول لي .. يمكن
+
+231
+00:19:20,270 --> 00:19:24,810
+حكيت هيدي حتى بقول لي corollary بقول أنا أحيانًا
+
+232
+00:19:24,810 --> 00:19:29,530
+بحب أريح حالي ونقول خلينا نأخذ .. عشان نستذكر الـ
+
+233
+00:19:29,530 --> 00:19:32,790
+.. ال .. ال .. ال Fundamental Theorem الجزء الأول 
+
+234
+00:19:32,790 --> 00:19:37,950
+بشكل سريع بس شوية عندنا نعمل .. في الشروط أنّه قاف 
+
+235
+00:19:37,950 --> 00:19:41,630
+من a لـ b لعند R satisfy the conditions اللي هو F'
+
+236
+00:19:42,150 --> 00:19:45,790
+exists on A وB، ما بتبدأش أجيب سيرة مين الـ F Small؟
+
+237
+00:19:45,790 --> 00:19:49,250
+ما بتبدأش أشغل مين؟ على الـ F Capital لو فرضنا أنّ الـ
+
+238
+00:19:49,250 --> 00:19:54,010
+F هذه من A وB لعند R أنّها الـ F' اللي هي شمالها 
+
+239
+00:19:54,010 --> 00:19:57,890
+exist على الـ A وB يعني فرض أنّها، بس هنا زاد شوية
+
+240
+00:19:57,890 --> 00:20:01,130
+على الشروط اللي جابله بشوية أنّ F is differentiable 
+
+241
+00:20:01,130 --> 00:20:03,790
+على الـ closed interval كنا فرضينها أنّها F
+
+242
+00:20:03,790 --> 00:20:06,740
+differentiable على الـ Open و فرضين أنّها continuous 
+
+243
+00:20:06,740 --> 00:20:10,760
+على اللي هي الـ Closed مدام الآن إذا فيها دي سنة
+
+244
+00:20:10,760 --> 00:20:16,680
+زيادة من الـ .. من اللي هي الشروط F' exists على الـ
+
+245
+00:20:16,680 --> 00:20:19,820
+A والـ B يعني continuous و differentiable على الـ
+
+246
+00:20:19,820 --> 00:20:24,350
+closed open .. على ال closed interval A و B الآن هي
+
+247
+00:20:24,350 --> 00:20:29,550
+الشرط والشرط الثاني فرض أنّ الـ F' مش موجودة و بس الـ 
+
+248
+00:20:29,550 --> 00:20:33,030
+F' انتجرت على الـ A والـ B إذا انتجقت شروط الـ
+
+249
+00:20:33,030 --> 00:20:36,210
+Fundamental Theorem إذا حسب الـ Fundamental Theorem
+
+250
+00:20:36,210 --> 00:20:39,410
+اللي قبله بشوية بكون عنده ال integration من A لـ B
+
+251
+00:20:39,410 --> 00:20:44,190
+لمين؟ الآن للـ f prime اللي كنا نسمّيه في الجبل f
+
+252
+00:20:44,190 --> 00:20:49,310
+small بتساوي f of b ناقص f of a إذا فعلًا هي 
+
+253
+00:20:49,310 --> 00:20:53,130
+automatic حققت شروط الـ Fundamental Theorem وزيادة
+
+254
+00:20:53,130 --> 00:20:59,850
+سنة إذا أكيد النتيجة بتكون صحيحة يعني تكامل الدالة 
+
+255
+00:20:59,850 --> 00:21:05,210
+المتفاضلة بيساوي أصل الدالة اللي هو أصل الدالة عنده
+
+256
+00:21:05,210 --> 00:21:09,070
+النقطة الأولى ناقص النقطة مين؟ بنحكي النقطة الثانية
+
+257
+00:21:09,070 --> 00:21:13,950
+اللي هو عبارة عن integration F of B ناقص مين؟ F of A
+
+258
+00:21:13,950 --> 00:21:19,210
+طب لو كانت X متغير يعني لو أخذنا أي X بين الفترتين
+
+259
+00:21:19,210 --> 00:21:23,550
+وطبقنا النظرية على الـ A اللي عند الـ X فبيصير الـ 
+
+260
+00:21:23,550 --> 00:21:27,130
+integration من A إلى عند الـ X أف برايم بيساوي أف
+
+261
+00:21:27,130 --> 00:21:32,490
+of X ناقص أف of A الـ X هذه هتصبح متغيرة والـ أف of A 
+
+262
+00:21:32,490 --> 00:21:39,190
+ثابتة وهذا بيذكرنا أنّ إحنا قيمة التكامل لدالة
+
+263
+00:21:39,190 --> 00:21:43,790
+بيساوي اللي هو عبارة عن اللي هي الإجابة اللي هي
+
+264
+00:21:43,790 --> 00:21:47,630
+الدالة اللي بتطلع زائد some constant C اللي هو الـ 
+
+265
+00:21:47,630 --> 00:21:53,660
+constant طبعًا لاللي هي طبيعة الدالة لقداش بيطلعلنا
+
+266
+00:21:53,660 --> 00:21:56,420
+اللي هو الـ initial conditions بيسفرلنا إياه الـ
+
+267
+00:21:56,420 --> 00:21:59,620
+constant أو بيعطينا إياه خمسة أربعة ستة أو بيظلوا 
+
+268
+00:21:59,620 --> 00:22:03,720
+بصورة عامة Constancy طيب شوفوا صلوا على النبي عليه
+
+269
+00:22:03,720 --> 00:22:08,090
+الصلاة والسلام الآن بدنا نيجي للجزء الثاني من الـ 
+
+270
+00:22:08,090 --> 00:22:11,470
+Fundamental Theorem خلصنا جزء اللي هو تكامل
+
+271
+00:22:11,470 --> 00:22:16,870
+التفاضل، الآن بنتوقع أنّ نفاضل التكامل، نفاضل 
+
+272
+00:22:16,870 --> 00:22:20,530
+التكامل، تحت شروط، بنشوف اللي هي الـ Fundamental
+
+273
+00:22:20,530 --> 00:22:24,650
+Theorem of Calculus second form أو اللي هو الجزء
+
+274
+00:22:24,650 --> 00:22:27,850
+الثاني من الـ Fundamental Theorem، إيش اللي بيقوله؟
+
+275
+00:22:27,850 --> 00:22:33,540
+اللي بيقول معايا ليه، نفترض أنّ F small من a و b
+
+276
+00:22:33,540 --> 00:22:38,060
+لعند r اللي هو is integrable in a و b إذا افترضناه 
+
+277
+00:22:38,060 --> 00:22:41,460
+أنّه integrable و نفترض أنّ f of x هو بيساوي الـ
+
+278
+00:22:41,460 --> 00:22:46,160
+integration من a لـ x f for all x element in a و b 
+
+279
+00:22:46,160 --> 00:22:52,200
+بس لإن نيجي للنتيجة then هيطلع هذا عبارة عن 
+
+280
+00:22:52,200 --> 00:22:55,360
+continuous function في الـ .. في الـ .. اللي هي main
+
+281
+00:22:55,360 --> 00:23:02,700
+في اللي هي على الفترة a أو b ومش هي كمان ولو كان 
+
+282
+00:23:02,700 --> 00:23:09,830
+الـ F small continuous عند نقطة C في الفترة من A 
+
+283
+00:23:09,830 --> 00:23:14,050
+لعند B أو من A لعند X هتكون الـ F هذه نفسها 
+
+284
+00:23:14,050 --> 00:23:16,950
+differentiable والـ derivative اللي هي إيش بيساوي؟
+
+285
+00:23:16,950 --> 00:23:22,590
+بيساوي قيمة اللي هو الـ F اللي جوا يعني باختصار يا
+
+286
+00:23:22,590 --> 00:23:29,450
+جماعة بتتكرر سهلة بقولك مفترض F Integrable ماشي 
+
+287
+00:23:29,450 --> 00:23:34,690
+الحال هاي بس الآن بقول لو جيت أخذت من A لـ X on
+
+288
+00:23:34,690 --> 00:23:44,670
+طبعًا A و B لو أخذت الآن A و X F small of T DT هاي
+
+289
+00:23:44,670 --> 00:23:51,510
+ده طبعًا الآن صارت كلها تعتمد على مين؟ على X بقولك
+
+290
+00:23:51,510 --> 00:23:55,370
+هذه هذه اللي طلع اللي هو الـ integration للـ
+
+291
+00:23:55,370 --> 00:24:01,130
+integrable function من A لـ X F of T DT هتطلع لي 
+
+292
+00:24:01,130 --> 00:24:05,210
+continuous function لو سميتها F capital زي ما هو
+
+293
+00:24:05,210 --> 00:24:08,730
+مسمّيها F of X بتكون الـ F عندها شمالها is
+
+294
+00:24:08,730 --> 00:24:16,430
+continuous الآن بقولك لو اجيت وقولت أنّ Fis
+
+295
+00:24:16,430 --> 00:24:21,390
+continuous الـ F اللي جوا هذه at C element in A وB
+
+296
+00:24:21,390 --> 00:24:25,650
+لو كانت F is continuous at C element in A وB
+
+297
+00:24:25,650 --> 00:24:31,730
+هيطلع لك بقولك إذا الـ F capital هذه F prime of C
+
+298
+00:24:31,730 --> 00:24:39,870
+exists ومش exist بس والـ F prime of C هتساوي الـ F
+
+299
+00:24:39,870 --> 00:24:46,030
+small of إيش؟ of C يعني وكأنه بقولك لما نفاضل
+
+300
+00:24:46,030 --> 00:24:50,210
+هذه F prime of C اللي هي من إيه؟ اللي عند C يعني
+
+301
+00:24:50,210 --> 00:24:57,370
+بتقول F prime of C بيساوي تفاضل رفلة by DX عند
+
+302
+00:24:57,370 --> 00:25:02,630
+النقطة C لل integration من A لـ X F of T DT
+
+303
+00:25:27,580 --> 00:25:33,180
+C مثلًا بتساوي F of C اللي هي F small of C هو كون
+
+304
+00:25:33,180 --> 00:25:36,540
+الـ F اللي جوا continuous عند النقطة اللي بنحكي
+
+305
+00:25:36,540 --> 00:25:39,940
+عنها طب لو كانت F اللي جوا هيدي continuous وين
+
+306
+00:25:39,940 --> 00:25:43,620
+ما كان على الفترة هيدي بصير خلاص وأنت مغمض عنّاك
+
+307
+00:25:43,620 --> 00:25:47,980
+بتيجي بتقول تفاضل هيدي بتشيل التكامل وبتحط اللي 
+
+308
+00:25:47,980 --> 00:25:53,540
+جوا بدلالة اللي هي الـ X المتغير واضح المفروض طيب
+
+309
+00:25:53,540 --> 00:25:59,660
+نيجي الآن نثبت اللي هي النظرية نثبت شغلتين أنّ الـ F
+
+310
+00:25:59,660 --> 00:26:02,580
+كابتنال continuous عشان أثبت لك continuous هأثبت لك
+
+311
+00:26:02,580 --> 00:26:05,940
+أكثر من continuous هأثبت لك uniform لـ continuous لأ
+
+312
+00:26:05,940 --> 00:26:09,780
+أقول لك هأثبت لك أكثر من uniform لـ continuous هأثبت لك 
+
+313
+00:26:09,780 --> 00:26:12,920
+أنّ الـ F is Lipschitz function مثلًا من Lipschitz 
+
+314
+00:26:12,920 --> 00:26:15,400
+function إذا على طول uniform لـ continuous ومن ثم
+
+315
+00:26:15,400 --> 00:26:18,460
+على طول إيش مالها continuous إيش Lipschitz 
+
+316
+00:26:18,460 --> 00:26:25,240
+function يعني هأثبت لك أنّ Fof x-f of y absolute
+
+317
+00:26:25,240 --> 00:26:30,720
+value أصغر أو يساوي k في x-y had there exist k بحيث
+
+318
+00:26:30,720 --> 00:26:35,260
+أنّ f of x-f of y أصغر من k في xy لكل x و y element
+
+319
+00:26:35,260 --> 00:26:40,020
+in a و b هأثبت لك f بتحقق هذا الشرط هأثبتها ما ده 
+
+320
+00:26:40,020 --> 00:26:42,540
+هأثبتها ما ده هتتحقق هذا الشرط إذا ما على طول اللي
+
+321
+00:26:42,540 --> 00:26:45,680
+هي هتكون lipschitz يعني بمعنى آخر هتكون is
+
+322
+00:26:45,680 --> 00:26:49,480
+continuous هذا الجزء الأول الجزء الثاني يا شباب و
+
+323
+00:26:49,480 --> 00:26:55,760
+يا بنات لأن if F is continuous at C هنثبت أنّ F is
+
+324
+00:26:55,760 --> 00:26:59,320
+differentiable عند C يعني بدأ أثبت لك F is
+
+325
+00:26:59,320 --> 00:27:03,460
+differentiable عند C يعني إيش F is differentiable
+
+326
+00:27:03,460 --> 00:27:14,660
+يعني بدأ أثبت لك أنّ limit F of X زي داتش مثلًا أو F
+
+327
+00:27:14,660 --> 00:27:21,440
+of C زي داتش ناقص F of C على H إذا H بتروح للصفر
+
+328
+00:27:21,440 --> 00:27:27,380
+بتثبت لك إيه إيش بيساوي F capital طبعًا بيساوي F
+
+329
+00:27:27,380 --> 00:27:33,660
+small of C بدي أثبت لك يعني أو بمعنى آخر هأثبت لك لكلّ
+
+330
+00:27:33,660 --> 00:27:37,000
+Y أكبر من 0 داريكزيز ديلتا بحيث أنّ absolute value 
+
+331
+00:27:37,000 --> 00:27:41,320
+أصغر من Delta L على H أصغر من Delta يؤدي إلى F of 
+
+332
+00:27:41,320 --> 00:27:48,260
+C زائد H ناقص F of C على H ناقص F of C بدي أثبت لك
+
+333
+00:27:48,260 --> 00:27:53,180
+يه أصغر من 200 من Epsilon بكون أثبتت فعلًا أنّ هذه 
+
+334
+00:27:53,180 --> 00:27:56,540
+ال limit بالساوية هذه معناته هذه ال limit طبعًا إيش
+
+335
+00:27:56,540 --> 00:28:01,000
+بتعني أنّ F prime F capital prime of C exist و
+
+336
+00:28:01,000 --> 00:28:06,480
+بتساوي F of C اللي هو اللي هو طالبّه أنّ لما تكون 
+
+337
+00:28:06,480 --> 00:28:09,280
+F continuous عند الـ C بتكون F capital
+
+338
+00:28:09,280 --> 00:28:12,040
+differentiable عند الـ C والـ derivative للـ F 
+
+339
+00:28:12,040 --> 00:28:15,700
+capital بتساوي الـ F اسمه نشوف 
+
+340
+00:28:17,960 --> 00:28:21,380
+نثبت الـ Continuity أو نثبت الـ Lipschitz
+
+341
+00:28:21,380 --> 00:28:27,450
+function يعني الأمر سهل إحنا مفترضين يا جماعة أنّ
+
+342
+00:28:27,450 --> 00:28:31,490
+الـ F small هذي is integrable مزام is integrable
+
+343
+00:28:31,490 --> 00:28:37,650
+إذا is bounded S يعني أنّ لكلّ .. نقدر نلاقي K أكبر 
+
+344
+00:28:37,650 --> 00:28:41,830
+من 0 بحيث أنّ الـ absolute value لـ F of X small أصغر
+
+345
+00:28:41,830 --> 00:28:48,070
+سوى K لكلّ X موجودة الـ A والـ B سبب هذه أنّ F نفسها
+
+346
+00:28:48,070 --> 00:28:52,350
+is integrable طيب، بدأ اعتمد على هذه للوصول أنّها
+
+347
+00:28:52,350 --> 00:28:58,620
+لبشت الآن خذ أي x و y في الـ a والـ b و اسمحولي آخذ x
+
+348
+00:28:58,620 --> 00:29:01,760
+أقلّ من y without loss of generality مش عاد بآخذ y
+
+349
+00:29:01,760 --> 00:29:06,100
+أكبر من .. أصغر من x الآن حالة الـ x بتساوي
+
+350
+00:29:06,100 --> 00:29:09,660
+هتلاقيها automatic بتتحقق للي بديها نشوف إيش اللي
+
+351
+00:29:09,660 --> 00:29:13,790
+بديها الآن خذ x و y في الـ a والـ b ونفترض أنّ x
+
+352
+00:29:13,790 --> 00:29:16,610
+أصغر من y without most of generality زي ما قلنا 
+
+353
+00:29:16,610 --> 00:29:22,890
+الآن احسب لي f of x f of y ناقص مين؟ f of x الكبيرة y 
+
+354
+00:29:22,890 --> 00:29:26,630
+والكبيرة x احسب f of y ناقص f of x إيش هيساوي؟ الـ
+
+355
+00:29:26,630 --> 00:29:29,470
+integration حسب التعريف f of x بتساوي الـ 
+
+356
+00:29:29,470 --> 00:29:33,390
+integration من الـ a لـ x لـ f small الآن f of y هي 
+
+357
+00:29:33,390 --> 00:29:36,930
+عبارة عن ال integration من a لـ y، f small، ناقص، f
+
+358
+00:29:36,930 --> 00:29:40,210
+of x إيش بتساوي؟ ال integration من a لـ x لمين؟ للـ
+
+359
+00:29:40,210 --> 00:29:47,290
+F، إذا هذا ناقص هذا، الآن بدي أحول هذه، أجلبها،
+
+360
+00:29:47,290 --> 00:29:52,670
+بيصير زائد ال integration من x لعند مين؟ لعند الـ 
+
+361
+00:29:52,670 --> 00:29:57,640
+a، للـ F صار عندي الآن ال integration من X لعند الـ A 
+
+362
+00:29:57,640 --> 00:30:00,520
+ومن A لعند الـ Y إذا حيصير هذا عبارة عن الـ
+
+363
+00:30:00,520 --> 00:30:05,940
+integration من X لمين؟ من X لعند الـ Y عارفين هذه
+
+364
+00:30:05,940 --> 00:30:09,960
+الخاصية إذا صار عندي الـ F of X F of Y ناقص F of X
+
+365
+00:30:09,960 --> 00:30:15,760
+بيساوي ال integration من X لـ Y لمين؟ للـ F شوف الآن 
+
+366
+00:30:15,760 --> 00:30:20,100
+إيش بدّه أصل؟ بدّه أصل أنّ absolute value F of Y
+
+367
+00:30:20,100 --> 00:30:24,830
+ناقص F of X أصغر أو يساوي K في الـ absolute value X 
+
+368
+00:30:24,830 --> 00:30:31,450
+ناقص Y يعني صارت لبشت نشوف كيف نصلها وأكيد بعضكم
+
+369
+00:30:31,450 --> 00
+
+401
+00:33:11,100 --> 00:33:17,800
+زائد H ناقص F of C على H لما H تروح للصفر بساوي F
+
+402
+00:33:17,800 --> 00:33:23,480
+small of C أو أثبت لكم الفرق بين هاتين أصغر من
+
+403
+00:33:23,480 --> 00:33:28,880
+إبسلون وهذا زي ما قلنا هو اللي هيمثل F' of C ده
+
+404
+00:33:28,880 --> 00:33:34,160
+نشوف كيف طيب الصلاة على النبي صلاة والسلام عليك
+
+405
+00:33:37,390 --> 00:33:42,130
+خلّينا نجيب عند .. أيوة نيجي الآن للبرهان هتلاقوه
+
+406
+00:33:42,130 --> 00:33:49,730
+برضه سهل وفكرته سهلة شوفوا كيف عندي نفترض الآن
+
+407
+00:33:49,730 --> 00:33:53,550
+suppose that f is continuous at c limiting a وb
+
+408
+00:33:53,550 --> 00:33:58,790
+مدام continuous إذن limit
+
+409
+00:34:01,700 --> 00:34:08,460
+F of X as X بتروح للـ C بساوي F of C مظبوط ولا لأ؟
+
+410
+00:34:08,460 --> 00:34:12,300
+أكيد مظبوط فبقى من جهة أخرى خلينا نكتبها بصورة
+
+411
+00:34:12,300 --> 00:34:18,660
+ثانية X ناقص C بتروح للصفر if and only if اللي هو
+
+412
+00:34:18,660 --> 00:34:23,960
+X بتروح لمين للـ C سمولي هذه X minus C إيش اسمها H
+
+413
+00:34:23,960 --> 00:34:31,410
+بصير هذه اللي فوق مين هي limitF of الـ X minus C
+
+414
+00:34:31,410 --> 00:34:36,490
+بتساوي الـ H يعني الـ X بتساوي H زائد C أو C زائد 
+
+415
+00:34:36,490 --> 00:34:41,190
+Hلأن x بتروح للـ c تكافئ أنه .. اللي هو x minus c
+
+416
+00:34:41,190 --> 00:34:44,610
+تروح للصفر، يعني بتكافئ H تروح لمين للصفر، إيش حيث
+
+417
+00:34:44,610 --> 00:34:49,650
+هو هذا؟ F of C، إذا هذه هي تعبير آخر عن ال
+
+418
+00:34:49,650 --> 00:34:56,250
+continuity لل F لما اللي هو اللي هي ال F عند مين،
+
+419
+00:34:56,250 --> 00:34:59,210
+عند ال C، يعني هذه تعبير آخر، ال continuity لل
+
+420
+00:34:59,210 --> 00:35:05,900
+function لما عند النقطة اللي هي Cهذه الآن شو بدي
+
+421
+00:35:05,900 --> 00:35:09,520
+استخدمها للوصول اللي بديها لأن مادام F is
+
+422
+00:35:09,520 --> 00:35:12,600
+continuous ده هذي متحققة هيها مادام F is
+
+423
+00:35:12,600 --> 00:35:15,480
+continuous عن C ده هذي متحققة فمادام هذي متحققة
+
+424
+00:35:15,480 --> 00:35:18,720
+اذا by epsilon delta definition for every epsilon
+
+425
+00:35:18,720 --> 00:35:22,880
+أكبر من 0 there exists delta أكبر من 0 such that
+
+426
+00:35:22,880 --> 00:35:28,950
+لما يكون ال absolute value لل H أصغر من Deltaو
+
+427
+00:35:28,950 --> 00:35:32,230
+طبعا انا وين بشتغل في المنطقة انها تكون C زائد H
+
+428
+00:35:32,230 --> 00:35:36,370
+وين ما لأ في الفترة تبعت من A لعند مين لعند B يعني
+
+429
+00:35:36,370 --> 00:35:41,230
+اخترت ال H صغيرة كفاية بحيث انه C زائد H اضلي وين
+
+430
+00:35:41,230 --> 00:35:45,090
+جاعد في الفترة من A و B لعنده لإن هاي الفترة A وهي
+
+431
+00:35:45,090 --> 00:35:49,330
+Bوالفترة هذه اللي هي I C عندي مثلا في داخلها اللي
+
+432
+00:35:49,330 --> 00:35:55,690
+هي بتختار H دلتتها صغيرة كفاية أنه ضال C زائد H
+
+433
+00:35:55,690 --> 00:36:01,050
+موجودة في الفترة من A لعند B عشان تصير معرفة لإن
+
+434
+00:36:01,050 --> 00:36:05,330
+دالتي أنا عشان تكون معرفة عند C زائد H لازم تكون C
+
+435
+00:36:05,330 --> 00:36:08,490
+زائد H في داخل المنطقة هذه لإن دالتي معرفة على
+
+436
+00:36:08,490 --> 00:36:11,770
+الفترة من A لعند B عشان هيك أقولنا C زائد H لازم
+
+437
+00:36:11,770 --> 00:36:16,630
+تكون في الفترة زائد Bإذن اختيار الـ Delta يعتمد
+
+438
+00:36:16,630 --> 00:36:20,210
+على ال limit ويعتمد على انه أضمن ال C زي داشر ضال
+
+439
+00:36:20,210 --> 00:36:24,510
+هوين في الفترة A وB إذا تعريف ال continuity بيقول
+
+440
+00:36:25,950 --> 00:36:27,530
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أقوم
+
+441
+00:36:27,530 --> 00:36:28,970
+باستخدام صفر يوجد دلتا لكل y أقوم باستخدام صفر
+
+442
+00:36:28,970 --> 00:36:29,950
+يوجد دلتا لكل y أقوم باستخدام صفر يوجد دلتا لكل y
+
+443
+00:36:29,950 --> 00:36:30,650
+أقوم باستخدام صفر يوجد دلتا لكل y أقوم باستخدام
+
+444
+00:36:30,650 --> 00:36:32,710
+صفر يوجد دلتا لكل y أقوم باستخدام صفر يوجد دلتا
+
+445
+00:36:32,710 --> 00:36:35,050
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أقوم
+
+446
+00:36:35,050 --> 00:36:41,110
+باستخدام صفر يوجد دلتا
+
+447
+00:36:41,110 --> 00:36:49,400
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أطيب، الآن
+
+448
+00:36:49,400 --> 00:36:54,020
+وين أنا بدأ أروح؟ بدأ أثبت لكم إنه هذا ناقص هذا
+
+449
+00:36:54,020 --> 00:36:58,240
+أصغر من epsilon عشان ذلك دعنا نحسب F of C زاد H
+
+450
+00:36:58,240 --> 00:37:03,060
+ناقص F of C على H ناقص مين يا جماعة؟ F of C ويساوي
+
+451
+00:37:03,060 --> 00:37:13,170
+الآن F of C زاد H ناقص F of Cخد الآن اللي هي الـ H
+
+452
+00:37:13,170 --> 00:37:16,850
+هدعم المشترك بالـ C الواحدة للـ H ماشي F of C زي
+
+453
+00:37:16,850 --> 00:37:21,910
+دتش تعريفها من A لعند C زي دتش لل F of X DX قلناها
+
+454
+00:37:21,910 --> 00:37:26,090
+هذه اللي ماعطيناها من رأس الدور ناقص ماشي تعريف F
+
+455
+00:37:26,090 --> 00:37:30,230
+of C Capital of C هو ال integration من A لعند C زي
+
+456
+00:37:30,230 --> 00:37:35,630
+دتش الآن دي مش C زي دتش من A لعند مين لعند الـ C
+
+457
+00:37:35,630 --> 00:37:40,110
+واحدة للـ H من A لعند الـ C اللي هي F of C زي دتش
+
+458
+00:37:40,110 --> 00:37:46,360
+هيهاوهي الواحدة الـ H اللي برا ال F of C هيها من A
+
+459
+00:37:46,360 --> 00:37:51,780
+ل C واحدة الـ H ناقص مين F of C هذا من التعريف
+
+460
+00:37:51,780 --> 00:37:55,460
+مباشرة لإنه احنا عرفنا ال F capital of X هي عبارة
+
+461
+00:37:55,460 --> 00:38:01,820
+عن integration من A لعند X F of T DT لأن لما نكون
+
+462
+00:38:01,820 --> 00:38:06,120
+F of C زي دتش بنحطها دي C زي دتش C بنحطها دي عياش
+
+463
+00:38:06,120 --> 00:38:13,460
+C وهي C زي دتش وهي الـ C وهي ساويةالان هذه من ..
+
+464
+00:38:13,460 --> 00:38:18,780
+من .. من عند A لـ C زي ده اتش وهذه من A لعند B
+
+465
+00:38:18,780 --> 00:38:23,980
+لعند C هدولة مع بعض هاي هدولة التنتين بده احسبها
+
+466
+00:38:23,980 --> 00:38:28,120
+مع بعض حسبنا زيين قبل و شوية هو عبارة عن واحد على
+
+467
+00:38:28,120 --> 00:38:32,400
+اتش خدوا عامل مشترك خدوا واحد على اتش عامل مشترك
+
+468
+00:38:32,400 --> 00:38:37,340
+بين الجهتين تصبح واحد على اتش افتح جوس ال
+
+469
+00:38:37,340 --> 00:38:41,660
+integration من A لعند C زي ده اتشلأن بدل الناقص
+
+470
+00:38:41,660 --> 00:38:46,020
+بيصير H اكتب زائد بدل هذا الناقص بيكتب زائد لأن
+
+471
+00:38:46,020 --> 00:38:52,460
+هتنقلب مين الآن من C لعند ال A نظبط اللي هو لنفسها
+
+472
+00:38:52,460 --> 00:38:58,020
+F F هذه الآن من C لعند ال A ومن A لعند ال C زاد H
+
+473
+00:38:58,020 --> 00:39:02,000
+إذا أكيد هذه كلها على بعض يصير ال integration من C
+
+474
+00:39:02,000 --> 00:39:08,930
+ل C زاد H لل F في واحد على H إذا هذه كلهابس
+
+475
+00:39:08,930 --> 00:39:11,350
+ابدلتها بقيمتها اللي قلنا عنها اللي هو ال
+
+476
+00:39:11,350 --> 00:39:15,810
+integration من C ل C زي H اللي اوجدته بدل هذا كله
+
+477
+00:39:15,810 --> 00:39:23,190
+من واحد ال H ل F of X DX ناقص الان هذا هذا اللي هي
+
+478
+00:39:23,190 --> 00:39:28,090
+F of C شوف كيف بده اعملها على جهة تعالي احسب ال
+
+479
+00:39:28,090 --> 00:39:33,660
+integrationالـ integration من C لـ C زائد H للـ
+
+480
+00:39:33,660 --> 00:39:38,180
+constant واحد بعد أذنكم DX إيش هيساوي ال
+
+481
+00:39:38,180 --> 00:39:44,140
+integration هدا عبارة عن C زائد H ناقص C وهيساوي
+
+482
+00:39:44,140 --> 00:39:48,760
+قداش H يعني بمعنى آخر لو جيتها الجد تعرفوا ليش
+
+483
+00:39:48,760 --> 00:39:52,640
+بعمل هيك واحد على H في ال integration من C لـ C
+
+484
+00:39:52,640 --> 00:39:58,720
+زايد H اللي هو الواحد هيساوي إيش؟ واحد، مظبوط ولا
+
+485
+00:39:58,720 --> 00:40:03,450
+لأ؟أكيد مظبوط ماشي هايمل هادي كسمت الديوا تاني
+
+486
+00:40:03,450 --> 00:40:10,570
+علشان تطلع حد واحد الان صار عندى لو ضربت هادي
+
+487
+00:40:11,920 --> 00:40:18,320
+الجهتين في F of C هي F of C بيساوي ايش؟ F of C صار
+
+488
+00:40:18,320 --> 00:40:21,260
+الان F of C بيساوي هذه بدي أشيل F of C تبعتي و أحط
+
+489
+00:40:21,260 --> 00:40:25,360
+مكانها هذه الصورة ليش حطيتها؟ عشان هذه أقدر أتحمل
+
+490
+00:40:25,360 --> 00:40:30,380
+محا محا دي اللي موجودة من الأصل شوف كيف الان شيلت
+
+491
+00:40:30,380 --> 00:40:33,320
+ال F of C و حطيت قيمتها اللي أوجدناها اللي هي
+
+492
+00:40:33,320 --> 00:40:37,660
+عبارة عن F of C على H في ال integration من C لC زي
+
+493
+00:40:37,660 --> 00:40:42,480
+H لمن؟ للواحدماشي لان صار هذا ال integration و هذا
+
+494
+00:40:42,480 --> 00:40:45,920
+ال integration نفس الاشي واحد ل F of X و واحد ل
+
+495
+00:40:45,920 --> 00:40:50,380
+واحد يعني بقدر اشتغل فيهم بيصير عندى خد ال واحد ال
+
+496
+00:40:50,380 --> 00:40:53,960
+H برا عن المشترك خالص بيظل عندى ال integration من
+
+497
+00:40:53,960 --> 00:40:58,720
+C ل C زاد H هذا مين ايش اسمه F of X و هذا F of C
+
+498
+00:40:58,720 --> 00:41:01,580
+دخلتها جوا و مضربتها بالواحد صارت مين بقدر اعلن و
+
+499
+00:41:01,580 --> 00:41:06,520
+ثابت اللي هي ناقص F of C لكل اش ماله DX هذا الكلام
+
+500
+00:41:06,520 --> 00:41:14,340
+كله من C ل C زاد Hصار عندى الآن الصورة هذه هيها اه
+
+501
+00:41:14,340 --> 00:41:20,840
+و هذه الصورة اصلا انا بدياها الان واحد على H أزرع
+
+502
+00:41:20,840 --> 00:41:23,760
+أو ساوي قولنا absolute value of integration أزرع
+
+503
+00:41:23,760 --> 00:41:28,740
+أو ساوي ال integration لل absolute value DX الان
+
+504
+00:41:28,740 --> 00:41:35,320
+هذا المقدار F of X ناقص F of Cقلنا عنها من راس
+
+505
+00:41:35,320 --> 00:41:40,420
+الدور أن F of X ناقص F of C اللي هو أصغر من
+
+506
+00:41:40,420 --> 00:41:44,420
+Epsilon لأن limit F of X لما X تروح للـC أشهر
+
+507
+00:41:44,420 --> 00:41:50,240
+مساوي F of C لأن X الـN هذه في الفترة من C لـC
+
+508
+00:41:50,240 --> 00:41:56,100
+زائداتش يعني بتسمح لي أن أقول limit F of X as X
+
+509
+00:41:56,100 --> 00:42:01,220
+بتروح للـC بتساوي F of C لما X بتروح للـC اللي هي
+
+510
+00:42:01,220 --> 00:42:05,980
+أدش بتروح لمينللسفر مضل في نفس المنطقة لأن قلنا
+
+511
+00:42:05,980 --> 00:42:10,300
+قبل شوية x minus c بتروح للسفر إذا وفقط إذا اللي
+
+512
+00:42:10,300 --> 00:42:13,880
+هي ال H اللي بتروح للسفر اللي هي عبارة عن x minus
+
+513
+00:42:13,880 --> 00:42:18,260
+c يعني هذا المقدار بال continuity لل F عند ال C
+
+514
+00:42:18,260 --> 00:42:21,980
+برضه بكون أصغر من إبسلون على ال absolute value
+
+515
+00:42:21,980 --> 00:42:25,160
+لمين لل H في ال integration من C لC زاد H
+
+516
+00:42:27,990 --> 00:42:33,050
+واضحة اه؟ اكيد هذا الان ايش قيمته؟ C زي داش ناقص C
+
+517
+00:42:33,050 --> 00:42:36,050
+اللي هي عبارة عن H فبصير Y عبارة عن absolute value
+
+518
+00:42:36,050 --> 00:42:42,430
+of H في mean فيه اللي هو الـH الـH اللي هو الـH
+
+519
+00:42:42,430 --> 00:42:42,470
+اللي هو الـH اللي هو الـH اللي هو الـH اللي هو الـH
+
+520
+00:42:42,470 --> 00:42:42,490
+اللي هو الـH اللي هو الـH اللي هو الـH اللي هو الـH
+
+521
+00:42:42,490 --> 00:42:42,570
+الـH اللي هو الـH اللي هو الـH اللي هو الـH اللي
+
+522
+00:42:42,570 --> 00:42:42,710
+هو الـH اللي هو الـH اللي هو الـH اللي هو الـH
+
+523
+00:42:42,710 --> 00:42:46,490
+الـH اللي هو الـH اللي هو الـH
+
+524
+00:42:46,490 --> 00:42:52,450
+اللي هو الـطيب .. بيصير عندها .. بيصير أكبر من 0
+
+525
+00:42:52,450 --> 00:42:55,790
+أسف هذه بحاجة دي بتطلع أيش بتساوي .. بيساوي بساوي
+
+526
+00:42:55,790 --> 00:42:58,150
+.. بقدر أختار H positive؟ أه بقدر أختار H positive
+
+527
+00:42:58,150 --> 00:43:03,130
+طيب و لو حتى ال H negative بتكون similarly بس
+
+528
+00:43:03,130 --> 00:43:06,950
+بتكون .. كل البرهام بيصير .. بترتب على أنه اللي هي
+
+529
+00:43:06,950 --> 00:43:12,890
+ال .. ال C زائد H أصغر من مين من ال C without loss
+
+530
+00:43:12,890 --> 00:43:17,250
+of generality ال H أكبر من 0بصير عندى لان هذا
+
+531
+00:43:17,250 --> 00:43:24,750
+المقدار أصغر من مين من إبسلون إيش يعني اللى عملنا
+
+532
+00:43:24,750 --> 00:43:28,290
+اللى عملنا لكل إبسلون أكبر من Zero لاجئنا Delta
+
+533
+00:43:28,290 --> 00:43:33,350
+بحيث أنه لما H absolute value أصغر من Deltaيعطيني
+
+534
+00:43:33,350 --> 00:43:39,330
+أن هذا المقدار أصغر من مين؟ من إبسلون يعني صار
+
+535
+00:43:39,330 --> 00:43:43,450
+عندي limit هذا المقدار بسوء f of c و هذا المقدار
+
+536
+00:43:43,450 --> 00:43:46,290
+اللي هو f prime of c يعني اعتبتنا f prime of c
+
+537
+00:43:46,290 --> 00:43:56,410
+بسوء f small of c وهو المطلوب طيب نيجي الآن لاللي
+
+538
+00:43:56,410 --> 00:44:02,210
+هي ال corollaryبرضه الى النظرية اللى جاملة بشوية
+
+539
+00:44:02,210 --> 00:44:10,790
+او خلّيني نقول تلخيصها سريع واستذكرها سريعها اللى
+
+540
+00:44:10,790 --> 00:44:17,830
+هو الأولى كانت تكامل والتفاضل هذه تفاضل التكامل
+
+541
+00:44:17,830 --> 00:44:23,470
+طبعا كل واحدة تحت شروطها المذكورة في نظريتها
+
+542
+00:44:25,860 --> 00:44:30,340
+نجي نشوف يا جماعة ال corollary let f من a و b لعند
+
+543
+00:44:30,340 --> 00:44:34,940
+r بي continuous on a و b and let f of x بي سوى ال
+
+544
+00:44:34,940 --> 00:44:37,720
+integration من ال a ل ال x ل ال f إذا فرض f
+
+545
+00:44:37,720 --> 00:44:40,160
+continuous ال f اسمه ال continuous على كل الفترة
+
+546
+00:44:40,160 --> 00:44:44,980
+ريح حاله طبعا أعطى إشي أكبر من النظرية اللي فاتت
+
+547
+00:44:44,980 --> 00:44:47,640
+يعني فرض ال continuity على ال a و ال b ما زم
+
+548
+00:44:47,640 --> 00:44:49,840
+continuous إذا ما بتزمش تقول integrable لإن و أنا
+
+549
+00:44:49,840 --> 00:44:52,340
+continuous function زي ما قلنا إيش ما لها is
+
+550
+00:44:52,340 --> 00:44:58,190
+integrable إذا مدامة f continuousالان بيصير هذه ال
+
+551
+00:44:58,190 --> 00:45:01,810
+integration اللي هي من a لعند ال x نفسها
+
+552
+00:45:01,810 --> 00:45:06,030
+continuous لأ مش continuous بس is differentiable
+
+553
+00:45:06,030 --> 00:45:09,270
+يعني أجم ال continuous يعني هتكون عندي ال f is
+
+554
+00:45:09,270 --> 00:45:13,670
+differentiable و ال f برامي يمين هي ال f then f is
+
+555
+00:45:13,670 --> 00:45:17,730
+differentiable من a و b عند f برامي بيساوي كده إذن
+
+556
+00:45:17,730 --> 00:45:25,000
+باختصار من a ل x اللي هو fof t dt لو فرضنا هذه
+
+557
+00:45:25,000 --> 00:45:33,060
+continuous على ال a و ال b إذا لكل x element in a
+
+558
+00:45:33,060 --> 00:45:38,080
+و ال b بيصير f of x بالساوي هذا is differentiable
+
+559
+00:45:38,080 --> 00:45:45,500
+و ال f prime of x هتساوي f of اللي هي x يعني بمعنى
+
+560
+00:45:45,500 --> 00:45:52,590
+آخر اللي هوانفضل هذا التكامل و نزيل التفاضل و نحط
+
+561
+00:45:52,590 --> 00:46:00,830
+الـ L جوا يعني بمعنى آخر D by DX للـ integration
+
+562
+00:46:00,830 --> 00:46:08,530
+من A لعند X F of D DT لما تكون F continuous هذا
+
+563
+00:46:08,530 --> 00:46:14,610
+بكون ب cancel هذا أو عملية عكسية لهوبتظللنا F of X
+
+564
+00:46:14,610 --> 00:46:19,070
+يعدى الأحقالها أو F of T والـ T هي المتغير أو الـ
+
+565
+00:46:19,070 --> 00:46:23,370
+X هي المتغير هذه اللي هي الكلرية الآن بدوا يلمّوا
+
+566
+00:46:23,370 --> 00:46:26,610
+النظريتين اللي قبل بشوية في نظرية واحدة يلخصن في
+
+567
+00:46:26,610 --> 00:46:30,910
+نظرية واحدة نشوف كيف بدوا يلخصن في نظرية واحدة
+
+568
+00:46:30,910 --> 00:46:40,630
+يعني بدوا يعيد بس اللي هو صياغة النظريتين صلوا
+
+569
+00:46:40,630 --> 00:46:44,540
+على النبي عليه الصلاة
+
+601
+00:49:07,490 --> 00:49:11,230
+التفاضل .. شيل التكامل .. شيل التفاضل بتصير عبارة
+
+602
+00:49:11,230 --> 00:49:15,270
+عن F اللي فوق اللي كانت B ناقص F اللي تحت اللي هي
+
+603
+00:49:15,270 --> 00:49:19,390
+A ويساوي أف of X ناقص أف of A أف of A ما أعطيني إياها من
+
+604
+00:49:19,390 --> 00:49:24,350
+رأس الدوار صفر إذا إيش هتساوي أف of X إذا هذه اللي
+
+605
+00:49:24,350 --> 00:49:28,290
+بدّينا حساباتها بتساوي هذه طلعت إيش بتساوي أف of X
+
+606
+00:49:28,290 --> 00:49:32,830
+يعني أف of X بتساوي هذا المقدار وهو المطلوب طيب
+
+607
+00:49:32,830 --> 00:49:36,730
+هذا الجزء الأول من النظرية اللي هو نجي للجزء
+
+608
+00:49:36,730 --> 00:49:42,240
+الثاني نفترض الآن إن هذه صحيحة وده تعطيها الآن برضه
+
+609
+00:49:42,240 --> 00:49:44,760
+الشروط كلها متحققة لأن الـ F في اسمه ليش مالها
+
+610
+00:49:44,760 --> 00:49:47,340
+continuous لسه إن أنا ما .. ما .. ما لاحظتوش أنا
+
+611
+00:49:47,340 --> 00:49:52,080
+هذا إيه عندي F continuous مدام F continuous ال
+
+612
+00:49:52,080 --> 00:49:57,720
+integration من أعلى عند ال X هذا كله اللي هو اسمه F
+
+613
+00:49:57,720 --> 00:50:01,320
+of X هيكون differentiable حسب النظرية تمام
+
+614
+00:50:01,320 --> 00:50:04,780
+differentiable حسب هذه النظرية إذا F prime of X
+
+615
+00:50:04,780 --> 00:50:11,100
+إيش هتساوي؟ F of X يعني حققنا مين؟ اللي هي I يعني
+
+616
+00:50:11,100 --> 00:50:20,070
+هذا يؤدي إلى هذا نجي الآن لبعض التعريفات اللي 
+
+617
+00:50:20,070 --> 00:50:26,890
+هنتكررها لكم وأنتم هتكونوا اللي هو اتبعوا الأمثلة
+
+618
+00:50:26,890 --> 00:50:32,130
+من خلال ال homework اللي معكم هنعطي اللي هي تعريف
+
+619
+00:50:32,130 --> 00:50:36,830
+ممكن أنتم حتى مر عليكم في ال calculus ومن ثم اللي
+
+620
+00:50:36,830 --> 00:50:41,710
+هنعطي بعض ال counter examples اللي هي هنرجعكم
+
+621
+00:50:41,710 --> 00:50:47,370
+فيها للي هي الأسئلة اللي في الكتاب تشوفوا نصوصها
+
+622
+00:50:47,370 --> 00:50:53,530
+على الأقل و اللي مطلوب تحلوها إيه تحلوها واللي ما
+
+623
+00:50:53,530 --> 00:50:56,910
+بتحلوه يعرفش يحلها في عندنا اللي هو في ال homework
+
+624
+00:50:56,910 --> 00:51:01,230
+الحلول الموجودة بتدرسوها لحالكم ما درستوها برضه
+
+625
+00:51:01,230 --> 00:51:04,310
+ما فهمتوهاش احنا بنعمل فيها discussion ولو عملناها
+
+626
+00:51:04,310 --> 00:51:07,930
+حتى بشكل صوتي على أساس اللي هو احنا يمكن ما نقدرش
+
+627
+00:51:07,930 --> 00:51:12,110
+نصور بشكل كامل اللي هي ال discussions عبر اللي هو
+
+628
+00:51:12,110 --> 00:51:16,270
+التصوير اللي احنا الحالي بنصوره عبر ال power point
+
+629
+00:51:16,270 --> 00:51:22,710
+من البيت إن شاء الله طيب definition let I بتساوي A
+
+630
+00:51:22,710 --> 00:51:27,630
+و B subset من مين؟ من R إذا كانت F small من I ل R
+
+631
+00:51:30,720 --> 00:51:35,400
+then فرضنا أنه في دالة اسم F من I لعند R then the
+
+632
+00:51:35,400 --> 00:51:42,280
+antiderivative of F يعني وكأنها عكس عملية عكس
+
+633
+00:51:42,280 --> 00:51:45,920
+الدالة اللي بنفضلها roughly antiderivative of F on
+
+634
+00:51:45,920 --> 00:51:50,620
+I is a function F من I لعند R such that F prime of
+
+635
+00:51:50,620 --> 00:51:59,830
+X تساوي F of X يعني عندنا دالة F من عند I لعند R لو
+
+636
+00:51:59,830 --> 00:52:05,310
+جينا لجينا F ثانية من عند I لعند R و لجينا F prime
+
+637
+00:52:05,310 --> 00:52:10,450
+of X للي اللي في الـ I بتساوي mean لـ F small of X
+
+638
+00:52:11,290 --> 00:52:15,490
+في هذه الحالة بنسمي ال F capital هذه عبارة عن
+
+639
+00:52:15,490 --> 00:52:19,850
+antiderivative لل F يعني بمعنى آخر لو فضلناها هذه
+
+640
+00:52:19,850 --> 00:52:25,130
+ال antiderivative هتطلع مين؟ ال F الأصلية يعني
+
+641
+00:52:25,130 --> 00:52:30,890
+بتكون عندي اللي هي ال F capital هي ال
+
+642
+00:52:30,890 --> 00:52:35,210
+antiderivative لل F وال F small هي ال derivative
+
+643
+00:52:35,210 --> 00:52:41,910
+لل F واضح أخذتوها في ال calculus حتى نجي الآن ل
+
+644
+00:52:41,910 --> 00:52:45,650
+اللي هو برضه مفهوم أخذتوها في الكاركولاس if f من I
+
+645
+00:52:45,650 --> 00:52:49,750
+لعند R small is integrable لو فرضنا هذه integrable
+
+646
+00:52:49,750 --> 00:52:53,950
+و جينا عرفنا
+
+647
+00:52:53,950 --> 00:52:56,910
+دالة من A لعند X F of X DX قبل شوية قولنا هذا
+
+648
+00:52:56,910 --> 00:53:00,290
+أكيد معرفة مثلًا في integrable ومش معرفة ال
+
+649
+00:53:00,290 --> 00:53:04,110
+function F of X هتطلع continuous هذه اللي هي ال
+
+650
+00:53:04,110 --> 00:53:09,930
+function أو هذا هو اللي بنسميه التكامل المحدد هذه
+
+651
+00:53:09,930 --> 00:53:15,110
+بنسميها الـ Indefinite Integral of F يعني هذا
+
+652
+00:53:15,110 --> 00:53:17,790
+بنسميه Indefinite Integral of F أو هذه الدالة
+
+653
+00:53:17,790 --> 00:53:23,030
+بنسميها اللي هي The Indefinite Integral of F on I
+
+654
+00:53:23,030 --> 00:53:28,230
+برضه هذا برضه شغلات اللي هو أخذتوها سابقًا في اللي
+
+655
+00:53:28,230 --> 00:53:33,780
+هو ال calculus نيجي لبعض الملاحظات بس على اللي هو
+
+656
+00:53:33,780 --> 00:53:42,820
+اللي بتعلق بهذه اللي هي المفاهيم نشوفها وهي طبعًا
+
+657
+00:53:42,820 --> 00:53:48,760
+هذه المفاهيم في عليها counter examples التمييز
+
+658
+00:53:48,760 --> 00:53:56,400
+بينها نشوف المحبة هذه يا شباب طيب أصلي على النبي
+
+659
+00:53:56,400 --> 00:54:03,640
+عليه الصلاة والسلام الآن عندي سؤال سبعة تلاتة اتنين
+
+660
+00:54:03,640 --> 00:54:06,400
+يعني في سبعة تلاتة سؤال اتنين أو سبعة تلاتة سؤال
+
+661
+00:54:06,400 --> 00:54:10,660
+خمسة طبعًا كله في الطبع الثاني هذا اللي هو عندي an
+
+662
+00:54:10,660 --> 00:54:13,540
+integrable function may not have an antiderivative
+
+663
+00:54:13,540 --> 00:54:17,220
+يعني هنلاقي في السؤال هذا integrable function
+
+664
+00:54:17,220 --> 00:54:24,080
+وما لهاش .. وما لهاش antiderivative يعني هنلاقي
+
+665
+00:54:24,080 --> 00:54:30,660
+function F إنّها تكون integrable لكن ما نلاقيش
+
+666
+00:54:30,660 --> 00:54:34,540
+function F اللي ال derivative تبعها بتساوي
+
+667
+00:54:34,540 --> 00:54:39,860
+بتساوي F هذا اللي هو بتشوفوه في الـ 7 3 2 و 7 3 5
+
+668
+00:54:39,860 --> 00:54:43,340
+الأمثلة أو ال counter examples اللي وجودها هنا طيب
+
+669
+00:54:43,340 --> 00:54:47,710
+الملاحظة الثانية الـ function may have an
+
+670
+00:54:47,710 --> 00:54:52,830
+antiderivative يعني عندي F و بتحقق F وفي عندي F
+
+671
+00:54:52,830 --> 00:54:57,310
+كابيتال كمان و F برايم إيش بتساوي بسواء F but اللي
+
+672
+00:54:57,310 --> 00:55:02,210
+هي ال integration لل F إيش ماله does not exist لكن
+
+673
+00:55:02,210 --> 00:55:07,370
+but not integrable وهذا بتجاوب عليه برضه سبعة
+
+674
+00:55:07,370 --> 00:55:12,580
+تلاتة خمسة يعني في function F في لها F' بساوية F
+
+675
+00:55:12,580 --> 00:55:17,640
+لكن ال integration للـ F إيش ماله does not exist
+
+676
+00:55:17,640 --> 00:55:24,900
+تلاتة عندي الملاحظة الثالثة a continuous function
+
+677
+00:55:24,900 --> 00:55:27,800
+always have antiderivative طبعًا ال continuous
+
+678
+00:55:27,800 --> 00:55:35,290
+function أصلاً من القوة بمكان إنّها تخلي عندها يكون
+
+679
+00:55:35,290 --> 00:55:39,490
+لها الـ mean antiderivative الآن a continuous
+
+680
+00:55:39,490 --> 00:55:44,190
+function always have antiderivative وهذا اللي هو
+
+681
+00:55:44,190 --> 00:55:51,910
+مباشرة من ال corollary 7 3 4 اللي كانت F
+
+682
+00:55:51,910 --> 00:55:56,770
+continuous مدام F continuous إذا ال integration من
+
+683
+00:55:56,770 --> 00:56:07,120
+ال X من A لعند ال XA of T DT تساوي F of X اللي هي
+
+684
+00:56:07,120 --> 00:56:10,900
+is differentiable ومش differentiable كمان و F
+
+685
+00:56:10,900 --> 00:56:14,860
+prime of X بتساوي F small of X يعني مدام أف
+
+686
+00:56:14,860 --> 00:56:17,880
+continuous إذا صار لها antiderivative صارت ال F
+
+687
+00:56:17,880 --> 00:56:20,580
+اللي هي ال derivative لها تساوي F of X إذا صارت
+
+688
+00:56:20,580 --> 00:56:24,620
+إيش antiderivative هذه مين هذا اللي هو عبارة عن الـ
+
+689
+00:56:24,620 --> 00:56:28,980
+Corollary 7 3 4 مدام إن F continuous إذا هذه
+
+690
+00:56:28,980 --> 00:56:33,360
+الدالة اللي سميتها F of X is differentiable ومش هي
+
+691
+00:56:33,360 --> 00:56:35,540
+كمان هذا الجزء الثاني من ال fundamental theorem of
+
+692
+00:56:35,540 --> 00:56:39,200
+calculus و ال F prime of X إيش بيساوي F of X يعني
+
+693
+00:56:39,200 --> 00:56:44,280
+ال F capital is an antiderivative of the F small
+
+694
+00:56:44,280 --> 00:56:50,760
+اللي هو النقطة الرابعة the indefinite integral may
+
+695
+00:56:50,760 --> 00:56:57,520
+not be an antiderivative of mean of F الآن الـ
+
+696
+00:56:57,520 --> 00:57:02,400
+indefinite integral اللي
+
+697
+00:57:02,400 --> 00:57:12,000
+هو من A لعند X F of T DT لو سميناه F of X هذا هذا
+
+698
+00:57:12,000 --> 00:57:15,480
+the indefinite integral هذا ممكن يكون هذا يعني إيش
+
+699
+00:57:15,480 --> 00:57:18,480
+بمعنى آخر؟ يعني بدنا نفترض إن ال integration من A
+
+700
+00:57:18,480 --> 00:57:22,580
+ل X F of T DT موجود طبعًا اللي بقول عنه موجود بس
+
+701
+00:57:22,580 --> 00:57:25,040
+تكون F of T is integrable ما حاجناش على control U
+
+702
+00:57:25,040 --> 00:57:29,000
+of T ال F of T is integrable إذا ال A لعندي ال X F
+
+703
+00:57:29,000 --> 00:57:35,100
+of T DT لو ممكن يكون هذا موجود و بنسمي F of X
+
+704
+00:57:35,100 --> 00:57:39,830
+بقول لي the indefinite integral هذا may not have
+
+705
+00:57:39,830 --> 00:57:44,490
+antiderivative of F يعني ممكن ما يكونش هذا بالرغم
+
+706
+00:57:44,490 --> 00:57:48,610
+إنّه هذا موجود ما يكونش ال F capital اللي هو هذا
+
+707
+00:57:48,610 --> 00:57:53,530
+ما يكونش antiderivative لمين لل F small يعني رغم
+
+708
+00:57:53,530 --> 00:57:58,670
+إنّه هذا موجود ليس شرطًا إنّه يكون يتفاضل ولما
+
+709
+00:57:58,670 --> 00:58:03,290
+يتفاضل يطلع اللي جوا يعني مش شرط إنّها تكون ال F
+
+710
+00:58:03,890 --> 00:58:07,170
+اللي هي antiderivative اللي هي مين هو الـ
+
+711
+00:58:07,170 --> 00:58:10,110
+indefinite integral هيك بتقول المعادلة طب إيش
+
+712
+00:58:10,110 --> 00:58:14,150
+دليلكم دليلنا هاي في counter examples هاي سبعة
+
+713
+00:58:14,150 --> 00:58:19,390
+تلاتة أربعة وهذا ناتج ناجم ناجم عدم تحقق هذه أو
+
+714
+00:58:19,390 --> 00:58:23,910
+عدم تحقق ال antiderivative إنّه it may fail to be
+
+715
+00:58:23,910 --> 00:58:28,050
+differentiable at points at the interval ممكن أصلاً
+
+716
+00:58:28,050 --> 00:58:35,440
+إنّ ال F prime ما تكونش موجودة بالمرة عند نقطة اللي هو
+
+717
+00:58:35,440 --> 00:58:39,620
+في داخل الـ interval من a إلى x تمام هذه is not
+
+718
+00:58:39,620 --> 00:58:42,720
+differentiable كيف بدنا نلاقي f prime بتساوي f
+
+719
+00:58:42,720 --> 00:58:48,640
+small إذا قد ينشأ عدم وجود الـ antiderivative منه
+
+720
+00:58:48,640 --> 00:58:52,640
+إنّه هذه ما تكونش differentiable عند نقطة بين ال a
+
+721
+00:58:52,640 --> 00:58:59,090
+وال x هاي أول اللي هو سبب ممكن يؤدي إلى عدم تحقق
+
+722
+00:58:59,090 --> 00:59:03,150
+إنّه يكون الـ Indefinite Integral ليس Anti
+
+723
+00:59:03,150 --> 00:59:09,650
+-derivative للدالة اللي إحنا بنكملها Or الـ
+
+724
+00:59:09,650 --> 00:59:12,850
+Derivative of Indefinite Integral ممكن يكون موجود
+
+725
+00:59:13,450 --> 00:59:18,310
+لكن يختلف عن مين but different from the value of F
+
+726
+00:59:18,310 --> 00:59:22,410
+at any point of the interval وهذا بيجيبوا سبعة
+
+727
+00:59:22,410 --> 00:59:26,990
+تلاتة تمانية يعني بقول لك ممكن تكون يا محلاها و تجيب
+
+728
+00:59:26,990 --> 00:59:32,990
+اللي هو ال F prime of X لكن ما تطلع قيمتها تساوي F
+
+729
+00:59:32,990 --> 00:59:38,780
+of X ففي هذه الحالة بيكون الـ Indefinite Integral
+
+730
+00:59:38,780 --> 00:59:46,320
+ليس اللي هو إيه ماله له Antiderivative عارفين
+
+731
+00:59:46,320 --> 00:59:55,080
+لحاله و هاي في مثال طبعًا نأتي
+
+732
+00:59:55,080 --> 01:00:01,300
+الآن و نبدأ حديث عن الهو evaluation of integrals
+
+733
+01:00:01,300 --> 01:00:07,780
+evaluation of integrals هناخد اللي هو اللي هي اللي
+
+734
+01:00:07,780 --> 01:00:12,660
+كنا نقول عنها اللي هو integration by parts اللي
+
+735
+01:00:12,660 --> 01:00:17,380
+كنا نقول عنه integration by parts نشوف كيف اللي هو
+
+736
+01:00:17,380 --> 01:00:21,360
+البرهن الأول واحدة يعني هناخد اللي هو النظرية
+
+737
+01:00:21,360 --> 01:00:28,430
+الأولى اللي هي في من ضمن اللي هيكيف يمكن أن نجد
+
+738
+01:00:28,430 --> 01:00:33,770
+طرق لإيجاد التكامل؟ evaluation of integrals أول
+
+739
+01:00:33,770 --> 01:00:39,730
+شيء نحكي عنه هو integral by parts التكامل بالتجزئة
+
+740
+01:00:41,640 --> 01:00:48,040
+نضع الأسس النظرية للتكامل بالتجزئة أو نشرع التكامل
+
+741
+01:00:48,040 --> 01:00:50,920
+بالتجزئة من خلال النظرية اللي أمامنا، إيش بتقول
+
+742
+01:00:50,920 --> 01:00:54,720
+لها؟ إذا كانت f و g من a و b، f small و g small
+
+743
+01:00:54,720 --> 01:00:58,520
+طبعًا، لعند R، R integrable on a و b، نفترض إن
+
+744
+01:00:58,520 --> 01:01:02,040
+التنتين إما اللي هين integrable وبدنا نفترض إنّه
+
+745
+01:01:02,040 --> 01:01:06,510
+لهين antiderivatives الآن بدأ افترض إن F لها
+
+746
+01:01:06,510 --> 01:01:09,430
+antiderivative اسمها F capital و G لها
+
+747
+01:01:09,430 --> 01:01:13,910
+antiderivative اسمها G capital on A و B إذا الآن
+
+748
+01:01:13,910 --> 01:01:22,050
+نظريتنا ترتكز في معطياتها على إنّ ال F small ل F و
+
+749
+01:01:22,050 --> 01:01:24,210
+G integrable هي واحد
+
+750
+01:01:36,840 --> 01:01:45,420
+ثم عرض علينا الآن نعمل integration ثم بنيجي بقول
+
+751
+01:01:45,420 --> 01:01:52,610
+له الـ Integration لحاصل ضرب دالتين F of X في G of
+
+752
+01:01:52,610 --> 01:01:58,890
+X DX من A لعين B لأن طيب هذه Integrable عارفين طب F
+
+753
+01:01:58,890 --> 01:02:01,090
+capital Integrable اه Integrable لأنها
+
+754
+01:02:01,090 --> 01:02:03,930
+differentiable لأنها differentiable يعني
+
+755
+01:02:03,930 --> 01:02:07,370
+continuous أكثر من continuous يعني يعني أكيد إيش
+
+756
+01:02:07,370 --> 01:02:11,250
+ما لها Integrable إذا معقول كلامه من A لعند B F of
+
+757
+01:02:11,250 --> 01:02:15,590
+X G of X DX إيش بتساوي؟ بقول لي عبارة عن F capital
+
+758
+01:02:15,590 --> 01:02:21,750
+of B G capital of B ناقص F capital of A G capital
+
+759
+01:02:21,750 --> 01:02:31,040
+of A هذا ناقص الـ integration F small of X G
+
+760
+01:02:31,040 --> 01:02:37,520
+capital of X dx من A لعندي هذه اللي هي ال
+
+761
+01:02:37,520 --> 01:02:41,240
+integration by parts اللي عامة اللي بتدخله أو
+
+762
+01:02:41,240 --> 01:02:45,960
+اللي هي اللي بتقول عنه النظرية إذا if f و g من a
+
+763
+01:02:45,960 --> 01:02:49,260
+لعن بي are integrable on a و b and have
+
+764
+01:02:49,260 --> 01:02:52,140
+antiderivatives f capital and g capital on a و b
+
+765
+01:02:52,140 --> 01:02:56,580
+then ال integration من a لعن بي f of x g x dx
+
+766
+01:02:56,580 --> 01:03:00,080
+تساوي f of b g of b ناقص f of a g of a طبعًا
+
+767
+01:03:00,080 --> 01:03:06,360
+لل capitals ناقص ال integration من a ل b f small g
+
+768
+01:03:06,360 --> 01:03:12,200
+capital dx ماشي الحال بدنا الآن اللي هو البرهان
+
+769
+01:03:12,200 --> 01:03:17,750
+النظري البرهان أسهل يا جماعة هو برهان يعتمد على
+
+770
+01:03:17,750 --> 01:03:22,970
+تفاضل اللي هو حاصل ضرب دالتين ويمكن احنا كنا أصلاً
+
+771
+01:03:22,970 --> 01:03:27,110
+في أثناء أول ما بنبدأ في ال integration by parts
+
+772
+01:03:27,110 --> 01:03:31,870
+سنة عارفينها للطلاب في ال calculus في الواقع في كل
+
+773
+01:03:31,870 --> 01:03:36,490
+سؤال في الأول يعني وكأنّنا بنبرهن النظرية وبعد
+
+774
+01:03:36,490 --> 01:03:40,170
+ذلك بنسير اللي هو حافظين الطريقة ونعملها بشكل
+
+775
+01:03:40,170 --> 01:03:48,220
+سريع طيب نشوف كيف سمي لي اللي هو F capital of X G
+
+776
+01:03:48,220 --> 01:03:52,060
+of capital of X إيش بتساوي H of X طبعًا F is
+
+777
+01:03:52,060 --> 01:03:53,840
+differentiable و G is differentiable زي ما كنا
+
+778
+01:03:53,840 --> 01:
+
+801
+01:05:40,450 --> 01:05:44,830
+بعض إيش ماله صار Integrable إذا صار عندي الآن اللي 
+
+802
+01:05:44,830 --> 01:05:49,550
+هي كل الشروط متحققة إذا من سبعة تلاتة مدامة الـ H
+
+803
+01:05:50,390 --> 01:05:53,570
+الـ H وينها الـ H؟ هي الـ Antiderivative لهذه
+
+804
+01:05:53,570 --> 01:05:57,470
+وهذه Integrable فبصير عندي الآن ال integration من
+
+805
+01:05:57,470 --> 01:06:01,770
+هنا بـ F في G زائد F capital في G DX بتساوي H of B
+
+806
+01:06:01,770 --> 01:06:11,650
+ناقص مين؟ ناقص H of A طيب صار عندي الآن الأمور وضحت
+
+807
+01:06:11,650 --> 01:06:20,490
+وطلعت النتيجة H of B اتطلع عليها وين H of B؟ هيها H 
+
+808
+01:06:20,490 --> 01:06:26,610
+of B بيساوي F capital of B في G
+
+809
+01:06:26,610 --> 01:06:30,930
+of B هيها ماشي F of B في G of B عادلة لو أنتو
+
+810
+01:06:30,930 --> 01:06:36,070
+موجودين لأ على طول قلتولي هيها الآن H of A بيساوي
+
+811
+01:06:36,070 --> 01:06:41,980
+F of A في G of A هيها G of A ماشي إذا إيدينا هذا
+
+812
+01:06:41,980 --> 01:06:46,060
+المقدار عوضنا عن H of B هيها وعوضنا عن H دي هيها
+
+813
+01:06:46,060 --> 01:06:49,960
+هيها وإيش جينا وهذه فصلناها لإنه Integrable إذا
+
+814
+01:06:49,960 --> 01:06:52,940
+فصلناها بنجيب واحدة على الجهة دي وواحدة على الجهة
+
+815
+01:06:52,940 --> 01:06:57,940
+دي وهو المطلوب أعملكم إياها طيب ال integration من
+
+816
+01:06:57,940 --> 01:07:01,780
+A لـ B integration
+
+817
+01:07:01,780 --> 01:07:10,680
+من A لـ B FG capital زائد F capital في G بساوي
+
+818
+01:07:11,790 --> 01:07:24,730
+H of B هي H of B H of B سوى F of B G of B نقص H of
+
+819
+01:07:24,730 --> 01:07:31,770
+A هي F of A G of A من الجهة الثانية بيساوي الـ
+
+820
+01:07:31,770 --> 01:07:35,890
+integration F في G capital زائد ال integration F 
+
+821
+01:07:35,890 --> 01:07:41,070
+capital في G small من A لعند B من A لعند B لأن إن 
+
+822
+01:07:41,070 --> 01:07:44,490
+قلنا هذا على الجهة هذه بيصير عند ال integration
+
+823
+01:07:44,490 --> 01:07:51,350
+المطلوب هذا أكيد فهمتم الأصل من a لعن بيه بساوي الـ
+
+824
+01:07:51,350 --> 01:07:55,570
+integration f في g capital زي f capital في g small
+
+825
+01:07:55,570 --> 01:08:06,170
+لأ خلصنا منه بساوي f of b في f of a في g of a في b
+
+826
+01:08:06,170 --> 01:08:16,030
+ناقص f of a في g of a كل هذا هذا المقدار integration
+
+827
+01:08:16,030 --> 01:08:27,230
+of capital في G من A لـ B وهو المطلوب و هيك احنا
+
+828
+01:08:28,050 --> 01:08:32,490
+بنكون اليوم إن شاء الله وصلنا لعند الـ first
+
+829
+01:08:32,490 --> 01:08:36,250
+substitution theorem المرة القادمة إن شاء الله
+
+830
+01:08:36,250 --> 01:08:45,010
+بنكمل المحاضرة أو بنكمل المادة و بنكمل هذا الـ
+
+831
+01:08:45,010 --> 01:08:49,330
+section في محاضرة قادمة إن شاء الله وإلى لقائنا
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..7a4bdd2a02193f6bf8050f260ff1ab767dceaee2
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk_postprocess.srt
@@ -0,0 +1,3324 @@
+1
+00:00:04,910 --> 00:00:11,030
+بسم الله الرحمن الرحيم المحاضرة الرابعة بعد حالة
+
+2
+00:00:11,030 --> 00:00:19,410
+الطوارئفي مادة او مساق تحليل لرياضي 2 او تحليل
+
+3
+00:00:19,410 --> 00:00:23,970
+حقيقي 2 لطلبة و طالبات الجامعة الإسلامية كلية
+
+4
+00:00:23,970 --> 00:00:30,110
+العلوم قسم الرياضيات وهي المحاضرة رقم 14 في المادة
+
+5
+00:00:30,110 --> 00:00:35,970
+او في المساق عنوان المحاضرة اليوم اللي هو
+
+6
+00:00:35,970 --> 00:00:43,260
+fundamental theorem ofof calculus بشقيها شق تفاضل
+
+7
+00:00:43,260 --> 00:00:50,640
+التكامل وشق تكامل التفاضل roughly نبدأ اللي هو في
+
+8
+00:00:50,640 --> 00:00:54,960
+النظرية الأولة أو الجزء الأول من النظرية اللي هي
+
+9
+00:00:54,960 --> 00:00:58,760
+ال first form ال fundamental theorem of calculus
+
+10
+00:00:58,760 --> 00:01:04,870
+اللي هي الجزء الأول بتقول ما يليlet f from a و b
+
+11
+00:01:04,870 --> 00:01:08,430
+لعند R be an integrable function on a و b يعني
+
+12
+00:01:08,430 --> 00:01:12,070
+فرضنا أن الـ function f عبارة عن دالة قابلة
+
+13
+00:01:12,070 --> 00:01:17,190
+للتكامل على الفترة المغلقة a و b and let f capital
+
+14
+00:01:17,190 --> 00:01:22,190
+من a و b لعند R تحقق الشروط التاليةتحقق أول شيء
+
+15
+00:01:22,190 --> 00:01:25,110
+أنها تكون الاف كابتنال هادي continuous على الفترة
+
+16
+00:01:25,110 --> 00:01:30,530
+a و b الشرط الثاني f prime exists and f prime of x
+
+17
+00:01:30,530 --> 00:01:34,130
+بساوي الاف small اللي بدأنا فيها دي الفرض منها
+
+18
+00:01:34,130 --> 00:01:39,880
+integrable لكل x element in a و bبنقول الـ
+
+19
+00:01:39,880 --> 00:01:44,900
+integration من A لB للـ F بساوي F of B ناقص F of A
+
+20
+00:01:44,900 --> 00:01:50,400
+إذا الآن و كإني بقول نفترض أنه الـ F small
+
+21
+00:01:50,400 --> 00:01:56,100
+integrable و نفترض F continuous على closed bounded
+
+22
+00:01:56,100 --> 00:02:01,160
+interval A و B و F نفسها differentiable على الـ
+
+23
+00:02:01,160 --> 00:02:05,260
+open اللي هو interval A و B بشرط أنه الـ F prime
+
+24
+00:02:05,260 --> 00:02:07,940
+ال derivative لها الـ F capital طبعا بساوي الـ F
+
+25
+00:02:07,940 --> 00:02:13,240
+small of Xوبناء على كل المعطيات اللي حكيناها بيكون
+
+26
+00:02:13,240 --> 00:02:17,120
+ال integration من a لb لل f small هو عبارة عن fb
+
+27
+00:02:17,120 --> 00:02:21,760
+ناقص f of a او اختصارا واسهلك للحفظ ال integration
+
+28
+00:02:21,760 --> 00:02:29,670
+من a لb لل f prime of x dx هي عبارة عنوكأنه
+
+29
+00:02:29,670 --> 00:02:36,250
+التكامل بلغ التفاضل بيصير اللي هي f of b ناقص f of
+
+30
+00:02:36,250 --> 00:02:40,610
+a هذا فضوء mean يعني هاي هادة باختصار هيك فضوء
+
+31
+00:02:40,610 --> 00:02:44,710
+mean أن f prime of x تكون موجودة على الفترة من a و
+
+32
+00:02:44,710 --> 00:02:51,370
+b و أيضا بتكون انهاش integrable نفسها integrable و
+
+33
+00:02:51,370 --> 00:02:56,110
+تكون ال f نفسها continuous على close من a ل عند b
+
+34
+00:02:56,390 --> 00:03:00,270
+هذه هي النظرية إذا اللي بدنا نثبته الآن أنه قيمة
+
+35
+00:03:00,270 --> 00:03:04,470
+ال integration هذا بساوي اللي هو اللي أمامنا اللي
+
+36
+00:03:04,470 --> 00:03:09,590
+هي f of b ناقص من ناقص f of a خلينا نشوف إيش
+
+37
+00:03:09,590 --> 00:03:15,030
+البكاء في البحران يا جماعةالان اللي واضح انه انا
+
+38
+00:03:15,030 --> 00:03:18,530
+بده اثبت انه ال integration هو عبارة عن f of b
+
+39
+00:03:18,530 --> 00:03:24,990
+ناقص f of a شوف انا كيف بده اوصلك اللي هو ال
+
+40
+00:03:24,990 --> 00:03:28,670
+integration لمنقلة ال b لل f بساوي f of b ناقص f
+
+41
+00:03:28,670 --> 00:03:32,010
+of a انا بعرف اول حاجة هو معطيلي ال f small هذه
+
+42
+00:03:32,010 --> 00:03:35,290
+اللي هي ال f prime اللي هي ال f small سميتها هذه
+
+43
+00:03:35,290 --> 00:03:39,240
+معطينيها integrallyMadame Integrable إذن بواسطة
+
+44
+00:03:39,240 --> 00:03:44,160
+اللي هي .. إذا بتتذكروا Integrable criterion وهي
+
+45
+00:03:44,160 --> 00:03:46,900
+اللي بنستخدمها كثير احنا لإنها .. يعني خلّيني أقول
+
+46
+00:03:46,900 --> 00:03:51,400
+صيغة مهمة في إثبات النظريات اللي هي بما إنه أف
+
+47
+00:03:51,400 --> 00:03:56,380
+Integrable إذن لكل Y أكبر من 0 يوجد there exists a
+
+48
+00:03:56,380 --> 00:04:00,600
+partition B اللي هو X0 X1 عند X ل .. عند ..
+
+49
+00:04:00,600 --> 00:04:03,280
+partition لمين طبعاً للفترة اللي أنتوا عارفينها
+
+50
+00:04:03,280 --> 00:04:06,800
+اللي احنا بنشتغل عليها الفترة من A لعند مين لعند B
+
+51
+00:04:06,960 --> 00:04:12,920
+There exists a partition B X0 X1 XN of A و B such
+
+52
+00:04:12,920 --> 00:04:20,280
+that ال other sum B F'-LB F' أصغر من إبسلون F' من
+
+53
+00:04:20,280 --> 00:04:24,000
+هنا من هي يا جماعة؟ هي ال F لأننا مفترضين ال F
+
+54
+00:04:24,000 --> 00:04:29,940
+small أو ال F' is integrable إذا بواسطة الرماء ال
+
+55
+00:04:29,940 --> 00:04:34,760
+integrable criterion حصلنا على واحد اللي عنده
+
+56
+00:04:34,760 --> 00:04:40,880
+إلحاةأحكي عنها الآن أو أستخدمها للوصول لهدفي طيب
+
+57
+00:04:40,880 --> 00:04:44,040
+because طبعا F prime اللي هي meme زي ما قلنا F of
+
+58
+00:04:44,040 --> 00:04:48,820
+X is integrable on A أو B الآن في شغلة تانية احنا
+
+59
+00:04:48,820 --> 00:04:55,880
+بدأ الآن أستخدم اللي هو أمر آخر خلينا نكتب الآن
+
+60
+00:04:55,880 --> 00:05:00,920
+أول شغلة حصلناها عشان تعرف أين أنا رايح اللي هي
+
+61
+00:05:00,920 --> 00:05:06,550
+قلنا F is integrableF التي بيسمى F' Integrable أيش
+
+62
+00:05:06,550 --> 00:05:12,270
+أعطتنا يا جماعة؟ أعطتنا أن هناك بارتيشن B هناك B
+
+63
+00:05:12,270 --> 00:05:15,830
+طبعا for every epsilon أكبر من سفر هناك بارتيشن B
+
+64
+00:05:15,830 --> 00:05:23,390
+such that U, B و F اللي هي الـ F' طبعا الـ F يمين
+
+65
+00:05:23,390 --> 00:05:29,530
+الـ F' ناقص الـ B و F هي أصغر من Y وهذا سمناها
+
+66
+00:05:29,530 --> 00:05:34,710
+إيش؟ سمناها 1لأن من جهة أخرى يا جماعة بدي أستغل
+
+67
+00:05:34,710 --> 00:05:37,810
+اللي هو اللي معطينيه أن الأفكار بتلقوا continuous
+
+68
+00:05:37,810 --> 00:05:41,490
+أنا الآن زي ما قلنا جزاقنا اللي هي الفترة A و B
+
+69
+00:05:41,490 --> 00:05:52,230
+إلىاللي هي X0 X1 X2 فترة نموذجية Xk-1 لعند Xk لعند
+
+70
+00:05:52,230 --> 00:05:56,010
+مين لعند Xn اللي هي مين الـ B هذا ال partition
+
+71
+00:05:56,010 --> 00:05:59,930
+اللي لجيته أنا الآن أنا بعرف أن ال F capital
+
+72
+00:05:59,930 --> 00:06:03,330
+continuous على ال closed interval كلها من A لعند B
+
+73
+00:06:03,330 --> 00:06:06,870
+إذا الأكيد continuous على كل sub interval موجودة
+
+74
+00:06:06,870 --> 00:06:12,540
+يعني صارت اللي عندي ال F capital continuouson a
+
+75
+00:06:12,540 --> 00:06:18,560
+التي هي xk-1 وعند xk هذه هي ال sub interval اللي
+
+76
+00:06:18,560 --> 00:06:24,080
+ال function f continuous عليها وإحنا مفترضين أيضا
+
+77
+00:06:24,080 --> 00:06:29,220
+أن ال f prime exist على ال open interval كلها إذا
+
+78
+00:06:29,220 --> 00:06:33,860
+أكيد ال fis differentiable لأن f' exist على ال
+
+79
+00:06:33,860 --> 00:06:37,360
+open interval من a لعند b إذا أكيد f is
+
+80
+00:06:37,360 --> 00:06:41,980
+differentiable على الفترة xk minus واحد لعند مين
+
+81
+00:06:41,980 --> 00:06:45,140
+لعند ال xk يعني يا جماعة شروط ال mean value
+
+82
+00:06:45,140 --> 00:06:50,020
+theorem محققة إذا أكيد there existthere exist بما
+
+83
+00:06:50,020 --> 00:06:54,540
+انه اللي هي شروط ال mean value محققة اذا there
+
+84
+00:06:54,540 --> 00:06:59,620
+exist بسميها اللي هو bk او tk there exist tk في
+
+85
+00:06:59,620 --> 00:07:06,440
+الفترة xk minus واحد عند ال xk such that اللي هو F
+
+86
+00:07:06,440 --> 00:07:11,520
+هي الدلل اللي بطبق عليها ال mean value theorem يا
+
+87
+00:07:11,520 --> 00:07:19,320
+جماعة F of xk ناقص F of xk minus واحداللي هو
+
+88
+00:07:19,320 --> 00:07:24,560
+بتساوي اللي هو xk ناقص xk مايلوس واحد طبعا احنا
+
+89
+00:07:24,560 --> 00:07:29,280
+بنكسب عليها اللي هي ايش بتساوي اللي هي ال F prime
+
+90
+00:07:30,370 --> 00:07:35,630
+of الهي main TK هذه اللي هي الـ Main Value Theorem
+
+91
+00:07:35,630 --> 00:07:39,710
+نتجتها مطبق على مين يا جماعة على الفترة اللي هي XK
+
+92
+00:07:39,710 --> 00:07:43,770
+minus واحد لعند XK طبعا هذا الكلام لكل K لأنه لكل
+
+93
+00:07:43,770 --> 00:07:48,170
+مين لكل ال sub intervals بتتحقق نفس الشروط بناء
+
+94
+00:07:48,170 --> 00:07:55,010
+عليه احنا بنعرف قبل هيك خليني أبدأ أشتغلها عند ال
+
+95
+00:07:55,010 --> 00:08:04,450
+MK إبراهيماللي هي أكيد أصغر أو يساوي الـ Fاللي هي
+
+96
+00:08:04,450 --> 00:08:11,370
+ال F عندك ال mk' ايش نذاكركم فيها ايش هي ال mk' ال
+
+97
+00:08:11,370 --> 00:08:15,190
+mk' هي مكتوبة لنكتبها كمان مرة ال mk' هي مين ال
+
+98
+00:08:15,190 --> 00:08:19,610
+infimum لل F' F of X X element of X K minus واحد X
+
+99
+00:08:19,610 --> 00:08:24,950
+K ال F' مين هذه ال F small ماشي الحال طيب ال F او
+
+100
+00:08:24,950 --> 00:08:30,590
+ال mk' هي عبارة عن ال infimum لمين لل F' على ال
+
+101
+00:08:30,590 --> 00:08:37,110
+sub interval هذه اذا اكيدأصغر أو يساوي الـ F of أي
+
+102
+00:08:37,110 --> 00:08:41,630
+مقطوين موجودة في الـ sub-interval والـ TK هدوين
+
+103
+00:08:41,630 --> 00:08:46,410
+موجودة في هذا الـ sub-interval إذن أكيد F of TK
+
+104
+00:08:46,410 --> 00:08:50,670
+أصغر من الـ MK' لأن الـ MK' هي عبارة عن الـ
+
+105
+00:08:50,670 --> 00:08:57,290
+infimum لقيمة الدالة F' of TK عالمياً على الفترة
+
+106
+00:08:57,290 --> 00:09:01,440
+اللي هي ال sub-interval اللي بنحكي عنها ككلوهذه
+
+107
+00:09:01,440 --> 00:09:06,040
+أكيد أصغر أو يساوي من الـ MK Ibrahim مين الـ MK
+
+108
+00:09:06,040 --> 00:09:11,060
+Ibrahim هذه؟ هي عبارة عن الـ Supremum لمين؟ لقيمة
+
+109
+00:09:11,060 --> 00:09:16,990
+اللي هي الدالة هذه على الفترة اللي عندهماشي الحال
+
+110
+00:09:16,990 --> 00:09:23,110
+إذا هذه دائما صحيحة طب لو ضربت جهتي كل الأطراف في
+
+111
+00:09:23,110 --> 00:09:27,650
+طول الفترة اللي هي xk minus xk minus واحد طبعا
+
+112
+00:09:27,650 --> 00:09:31,510
+أكيد هذه xk ناقص xk minus واحد كمية موجبة إذا لما
+
+113
+00:09:31,510 --> 00:09:35,010
+أضربها في الأطراف كلها بتظلها ال inequality صحيحة
+
+114
+00:09:35,010 --> 00:09:40,610
+وزي ما هي في xk ناقص xk minus واحد وهنا في xk ناقص
+
+115
+00:09:40,610 --> 00:09:46,280
+xk minus واحد مظبوط هذا الكلام أكيد صحطيب لكن هذا
+
+116
+00:09:46,280 --> 00:09:52,220
+المقدار هو هذا إذا باجي بحط هذه هان معايا جماعة
+
+117
+00:09:52,220 --> 00:09:57,000
+بحط هذه في المنطقة هذه المكان القيمة هذه اللي هي
+
+118
+00:09:57,000 --> 00:10:05,190
+هذه إذا صار عندي الأن Fof xk ناقص f of xk minus 1
+
+119
+00:10:05,190 --> 00:10:12,070
+أكبر أو يساوي ال Mk' في ال xk minus xk minus 1
+
+120
+00:10:12,070 --> 00:10:18,490
+وهذا أصغر أو يساوي ال Mk' في ال xk minus xk minus
+
+121
+00:10:18,490 --> 00:10:24,290
+1إذا واضح إن أنا حصلت على هذه وهذه صحيحة لكل مين
+
+122
+00:10:24,290 --> 00:10:28,150
+لكل كيه لإنه أخدت أنا هذه فترة عشوائية يعني لكي
+
+123
+00:10:28,150 --> 00:10:33,790
+بتساوي اللي هو واحد واتنين لعند مين لعند الان بناء
+
+124
+00:10:33,790 --> 00:10:38,250
+عليه أنا وصلت الآن إلى ما يليه وصلت إلى هذه
+
+125
+00:10:38,250 --> 00:10:43,670
+النتيجة شايفين يا جماعة وصلت إلى هذه النتيجة اللي
+
+126
+00:10:43,670 --> 00:10:48,860
+أنا وضحتلكم إياها كيف إجتالان و ال mk prime زي ما
+
+127
+00:10:48,860 --> 00:10:55,280
+قلنا ايش هي و ال mk capital prime هيها امامكم نيجي
+
+128
+00:10:55,280 --> 00:10:59,720
+الان ايش اللي بده حصله شوف ال K ايش بده حصل انا
+
+129
+00:10:59,720 --> 00:11:03,020
+عمال بقول انه صحيح على كل K اذا لو أخدت ال
+
+130
+00:11:03,020 --> 00:11:05,900
+summation هنا و أخدت ال summation هنا و أخدت ال
+
+131
+00:11:05,900 --> 00:11:10,440
+summation هنا بظل اللي هي ال inequality صحيحة خد
+
+132
+00:11:10,440 --> 00:11:16,050
+الآن ال summationعلى الفترة الـ summation على
+
+133
+00:11:16,050 --> 00:11:19,530
+الinquality اللي حكينا عنها هيها بيصير ال
+
+134
+00:11:19,530 --> 00:11:23,630
+summation لل M K برايم في X K X K minus واحد K من
+
+135
+00:11:23,630 --> 00:11:26,630
+عند واحد لعند M وهنا هيكون أصغر أو ساوى ال
+
+136
+00:11:26,630 --> 00:11:31,090
+summation لل F of X K ناقص F of X K minus واحد
+
+137
+00:11:31,090 --> 00:11:36,190
+اللي هي احنا هنا حصلناها هان وهذه ال summation
+
+138
+00:11:36,190 --> 00:11:39,490
+عليها وهذه ال summation عليها هي ال summation هذا
+
+139
+00:11:39,850 --> 00:11:43,210
+طيب يا جماعة أكيد أنتم اذكرين هذا ال summation هو
+
+140
+00:11:43,210 --> 00:11:47,770
+عبارة عن مين يا جماعة هو عبارة عن ال lower sum لل
+
+141
+00:11:47,770 --> 00:11:52,370
+partition بي على مين على الدالة اللي بنعمل عليها
+
+142
+00:11:52,370 --> 00:11:56,330
+اللي هي ال F' اللي اشتغلنا عليها اللي أخدنا ال F'
+
+143
+00:11:57,150 --> 00:12:01,270
+ال MK' هي ال infimum لل F' على ال sub interval
+
+144
+00:12:02,160 --> 00:12:07,220
+والان هذه يشبه يا جماعة هي ال upper sum يعني بمعنى
+
+145
+00:12:07,220 --> 00:12:12,960
+آخر صار عندى الآن المقدار هذا هي المقدار هذا اللى
+
+146
+00:12:12,960 --> 00:12:17,680
+محاط بالأزرق هذا المقدار و بالأحمر صار بين اللى هو
+
+147
+00:12:17,680 --> 00:12:22,120
+ال lower sum و بين ال upper sum لكن هذا المقدار
+
+148
+00:12:22,120 --> 00:12:25,860
+الصمشي اللى احنا عملناه قبل ذلك كتير لو فرضناه
+
+149
+00:12:25,860 --> 00:12:33,860
+هيكون عبارة عن F of X واحدناقص F of X Note زائد F
+
+150
+00:12:33,860 --> 00:12:40,360
+of X 2 ناقص F of X 1 زائد F of X 3 ناقص F of X 2
+
+151
+00:12:40,360 --> 00:12:45,460
+زائد لما أصل لآخر إشي F of X N ناقص F of X N ناقص
+
+152
+00:12:45,460 --> 00:12:50,520
+1 حلاجي كله بروح و بظل عندك كل هذا بروح و بظل عندك
+
+153
+00:12:50,520 --> 00:12:54,860
+بس F of X ناقص F of X N ناقص مين F of X Note كل
+
+154
+00:12:54,860 --> 00:13:00,340
+term ب cancel اللي قبله و يساويF of Xn اللي هي
+
+155
+00:13:00,340 --> 00:13:07,420
+مين؟ F of B F of Xnot اللي هي مين؟ F of A إذا صار
+
+156
+00:13:07,420 --> 00:13:10,920
+عندي الآن الـSummation هذا اللي هو بيسوي F of B
+
+157
+00:13:10,920 --> 00:13:16,440
+نقص مين؟ نقص F of A يعني بمعنى آخر بنشيل هذا و
+
+158
+00:13:16,440 --> 00:13:20,580
+بنحط مكانه قيمته اللي هو F of B نقص F of A بيصير F
+
+159
+00:13:20,580 --> 00:13:25,320
+of B minus F of A بين اللي هو L of B و F' و بين
+
+160
+00:13:25,320 --> 00:13:34,230
+الأبقى الـU of B و F'طيب اطلع على فوق شوية خليني
+
+161
+00:13:34,230 --> 00:13:39,830
+اوضحلك هذه النقطة اللي هتوصلي للي بديها انا ايش
+
+162
+00:13:39,830 --> 00:13:44,170
+بدي اثبت انا بدي اثبت ان ال integration خليني بس
+
+163
+00:13:44,170 --> 00:13:51,710
+اشيل هذا بعد اذنكم انا
+
+164
+00:13:51,710 --> 00:13:58,260
+بدي اثبت ان ال integration لل F prime of x dxأو
+
+165
+00:13:58,260 --> 00:14:06,320
+الـ F of X DX بساوي F of B ناقص F of A من A لعند
+
+166
+00:14:06,320 --> 00:14:12,870
+مين؟ لعند B، هيا، بدي أسهلشوف الان كيف انا في
+
+167
+00:14:12,870 --> 00:14:17,590
+الواقع بتوصلكم هذا نقص هذا يكون اكبر او يساوي سفر
+
+168
+00:14:17,590 --> 00:14:20,990
+و اصغر من epsilon لأي اصغر يساوي epsilon لكل
+
+169
+00:14:20,990 --> 00:14:24,110
+epsilon في الدنيا و من ثم هيساوي سفر يعني هتحدث
+
+170
+00:14:24,110 --> 00:14:29,090
+مين المساواة اللي مافهمش عليها هي الان نجلة تفصيل
+
+171
+00:14:29,090 --> 00:14:39,060
+اتفجنا احنا انه احنا حصلنا على اللي هوالـ F of D –
+
+172
+00:14:39,060 --> 00:14:44,520
+F of A بين الـ U و بين الـ L لكن احنا بنقول الـ F
+
+173
+00:14:44,520 --> 00:14:49,420
+نفسها يا جماعة الـ F' هذه is integrable مزام
+
+174
+00:14:49,420 --> 00:14:55,380
+Integrable من A لعند Bأذن الـ U of F و الـ L of F
+
+175
+00:14:55,380 --> 00:14:58,560
+اللي هو الـ Upper Integral و الـ Lower Integral
+
+176
+00:14:58,560 --> 00:15:01,960
+اللي هو هيساوي قيمة الـ Integration عارفينوا هذا
+
+177
+00:15:01,960 --> 00:15:05,760
+الكلام احنا بنتعرف شو معناته Integrable الـ U of F
+
+178
+00:15:05,760 --> 00:15:11,640
+هي عبارة عن مين؟ عن الـ infimum لمين؟ للكبار اللي
+
+179
+00:15:11,640 --> 00:15:19,640
+هي U L أو U P و Fوهذا عبارة عن الـ Supremum لنية
+
+180
+00:15:19,640 --> 00:15:28,200
+للـ L, B و F يعني بيصير عندي اللي هو هذا عبارة عن
+
+181
+00:15:28,200 --> 00:15:32,420
+ال U of F من جهة وهذا عبارة عن ال L of F من جهة
+
+182
+00:15:32,420 --> 00:15:35,900
+أخرى هذا ال integration نقلة بيه ال F' ليش؟ لأن ال
+
+183
+00:15:35,900 --> 00:15:40,630
+F' is integrable زي ما قلناهطيب، إيش علاقة الـ L و
+
+184
+00:15:40,630 --> 00:15:44,130
+F بهذه؟ L و F عبارة عن الـ Supremum لكل هذول، إذا
+
+185
+00:15:44,130 --> 00:15:47,450
+أكيد الـ L و F أكبر أو يساوي هذه، إذا صار ال
+
+186
+00:15:47,450 --> 00:15:51,610
+integration هذا اللي عندى أكبر أو يساوي الـ L و F،
+
+187
+00:15:51,610 --> 00:15:56,070
+خلصنا من هذه، لأن من جهة ثانية ال integration لل F
+
+188
+00:15:56,070 --> 00:15:58,510
+براين هو الـ U of F، لأن الـ F is integrable زي ما
+
+189
+00:15:58,510 --> 00:16:04,770
+قلنا، و الـ U of F مين هو؟هو عبارة عن الـ infimum
+
+190
+00:16:04,770 --> 00:16:08,810
+لها دول إذا الـ U of F أصغر أو يشويها مزام أصغر أو
+
+191
+00:16:08,810 --> 00:16:12,210
+يشويها يعني صار ال integration أصغر أو يشويها يعني
+
+192
+00:16:12,210 --> 00:16:18,050
+بمعنى آخر احنا وصلنا ان ال integration تبعناالـ
+
+193
+00:16:18,050 --> 00:16:23,450
+integration اللي هو من A لعند A بي لل F برايم بين
+
+194
+00:16:23,450 --> 00:16:29,530
+ال L و بين مين و بين ال U ماشي الحالة الآن و هذا
+
+195
+00:16:29,530 --> 00:16:35,390
+برضه بين ال L و بين مين ال U اضربلي الآن اللي تحت
+
+196
+00:16:35,390 --> 00:16:41,820
+اللي هو بسالببتنعكس ال inequality وبعدين اجمعل
+
+197
+00:16:41,820 --> 00:16:46,320
+الجهتين ماشي الحال هذي بيصير سالب وهذي بيصير أكبر
+
+198
+00:16:46,320 --> 00:16:51,220
+أو يساوي سالب وهذي بيصير أكبر أو يساوي سالب اجمعل
+
+199
+00:16:51,220 --> 00:16:56,980
+الجهتين بيصير عند ال integrationأو العكس اضرب اللي
+
+200
+00:16:56,980 --> 00:16:59,720
+فوق بالسالب عساس ان اعمل زي ما عامل هو هان اضرب
+
+201
+00:16:59,720 --> 00:17:02,000
+اسف اللي هو اضرب مين اللي فوق بالسالب يصير هذا
+
+202
+00:17:02,000 --> 00:17:07,640
+ناقص وهذا كله ناقص وهذا عبارة عن ناقص وهذا اكبر او
+
+203
+00:17:07,640 --> 00:17:12,760
+يساوي وهذا اكبر او يساوي واجمح لبعض او بلغة اخرى
+
+204
+00:17:13,150 --> 00:17:16,790
+قول اللي .. اللي .. اللي .. اللي تحت ناقص اللي فوق
+
+205
+00:17:16,790 --> 00:17:20,130
+بطلع عندى ال integration لل F prime من A ل عند B
+
+206
+00:17:20,130 --> 00:17:24,730
+ناقص اللي هي F of B minus F of A أصغر يساوي ال U
+
+207
+00:17:24,730 --> 00:17:30,770
+ناقص ال L و أكبر أوي يساوي ال L ناقص ال U واضح أنه
+
+208
+00:17:30,770 --> 00:17:36,480
+هذهعبارة عن ناقص هذه وهذه كمية موجبة إذا صار عند
+
+209
+00:17:36,480 --> 00:17:41,000
+ال absolute value للقيمة هذه أصغر أو يساوي ال U
+
+210
+00:17:41,000 --> 00:17:49,300
+ناقص مين ال L كمان مرة هذه سالب هذهوهذه موجبة إذاً
+
+211
+00:17:49,300 --> 00:17:53,540
+هذه الـ Inquality صارت هي عبارة عن مفكوك الـ
+
+212
+00:17:53,540 --> 00:17:57,440
+absolute value للـ F prime ال integration ناقص
+
+213
+00:17:57,440 --> 00:18:01,840
+المقدار هذا أصغر أو يساوي هذا القيمة وهذا القيمة
+
+214
+00:18:01,840 --> 00:18:05,700
+لسه ما سحنهاش احنا إيش قلنا عنها أصغر من من إبسلان
+
+215
+00:18:05,700 --> 00:18:09,280
+وانا قاعد باشتغل على ال partition اللي لجيته أعلى
+
+216
+00:18:09,280 --> 00:18:15,390
+إذا صار عندى الآن اللي وصلتلهعبارة عن اللي هو ال
+
+217
+00:18:15,390 --> 00:18:20,150
+absolute value هذا أصغر من إبسلون وطبعا أكبر أوي
+
+218
+00:18:20,150 --> 00:18:24,790
+ساوي سفر والإبسلون واز إيه شمالها arbitrary يعني
+
+219
+00:18:24,790 --> 00:18:29,430
+المقدار هذا لأي إبسلون في الدنيا لأي إبسلون في
+
+220
+00:18:29,430 --> 00:18:33,770
+الدنيا هذا أصغر من مين من الإبسلون هو أكبر أوي سفر
+
+221
+00:18:33,970 --> 00:18:38,510
+إذاً لازم يكون هذا المقدار اللي هو عبارة عن سفر
+
+222
+00:18:38,510 --> 00:18:42,530
+يعني بمعنى آخر ال integration بساوي اللي هو F of B
+
+223
+00:18:42,530 --> 00:18:47,050
+نقص F of A و هيك و نكون حصلنا على المطلوب و أثبتنا
+
+224
+00:18:47,050 --> 00:18:51,470
+النظرية طيب
+
+225
+00:18:51,470 --> 00:18:57,850
+الجماعة الان هذا اللي هو الجزء الأولالجزء الأول من
+
+226
+00:18:57,850 --> 00:19:00,890
+اللي هو الـ Fundamental Theorem of Calculus اللي
+
+227
+00:19:00,890 --> 00:19:05,750
+هو عبارة عن تكامل التفاضل اللي هي عملية التكامل
+
+228
+00:19:05,750 --> 00:19:10,470
+هتلغى عملية التفاضل زي ما شفنا اللي هو قبل بشوية
+
+229
+00:19:10,470 --> 00:19:15,130
+وشفنا إيه اللي جالته الـ Fundamental Theorem طبعا
+
+230
+00:19:15,130 --> 00:19:20,270
+هذا الكلام تحت الشروط المذكورةالآن بقول لي .. يمكن
+
+231
+00:19:20,270 --> 00:19:24,810
+حكيت هادى حتى بقول لي corollary بقول انا أحيانا
+
+232
+00:19:24,810 --> 00:19:29,530
+بحب أريح حالي و نقول خلينا ناخد .. عشان نستذكر ال
+
+233
+00:19:29,530 --> 00:19:32,790
+.. ال .. ال .. ال fundamental theorem الجزء الأول
+
+234
+00:19:32,790 --> 00:19:37,950
+بشكل سريع بس شويه عندنا نعمل .. في الشروط ان هو قف
+
+235
+00:19:37,950 --> 00:19:41,630
+من a ل b لعند R satisfy the conditionsاللي هو F'
+
+236
+00:19:42,150 --> 00:19:45,790
+exists on A وB، مابتدش أجيب سيرة مين الـ F Small؟
+
+237
+00:19:45,790 --> 00:19:49,250
+مابتدش أشغل مين؟ على الـ F Capital لو فرضنا إن الـ
+
+238
+00:19:49,250 --> 00:19:54,010
+F هذه من A وB لعند R إنها الـ F' اللي هي الشمالها
+
+239
+00:19:54,010 --> 00:19:57,890
+exist على الـ A وB يعني فرض إنها، بس هنا زاد شوية
+
+240
+00:19:57,890 --> 00:20:01,130
+على الشروط اللي جابل بشوية إن F is differentiable
+
+241
+00:20:01,130 --> 00:20:03,790
+على الـ closed interval كنا فرضينها إنها F
+
+242
+00:20:03,790 --> 00:20:06,740
+differentiable على الـ Openوفرضين إنها continuous
+
+243
+00:20:06,740 --> 00:20:10,760
+على اللي هي الـ Closed مدام الآن إذا فيها دي سنة
+
+244
+00:20:10,760 --> 00:20:16,680
+زيادة من ال .. من اللي هي الشروط F' exists على الـ
+
+245
+00:20:16,680 --> 00:20:19,820
+A و الـ B يعني continuous و differentiable على الـ
+
+246
+00:20:19,820 --> 00:20:24,350
+closed open .. على ال closed interval A و Bالان هي
+
+247
+00:20:24,350 --> 00:20:29,550
+الشرط والشرط الثاني فرض ان ال F' مش موجودة و بس ال
+
+248
+00:20:29,550 --> 00:20:33,030
+F' انتجر بالعلى ال A و ال B إذا انتجقت شروط ال
+
+249
+00:20:33,030 --> 00:20:36,210
+fundamental theorem إذا حسب ال fundamental theorem
+
+250
+00:20:36,210 --> 00:20:39,410
+اللي قبله شوية بكون عنده ال integration من A لB
+
+251
+00:20:39,410 --> 00:20:44,190
+لمين الآنللـ f prime اللي كنا نسميه في الجبل f
+
+252
+00:20:44,190 --> 00:20:49,310
+small بتساوي f of b ناقص f of a إذا فعلا هي
+
+253
+00:20:49,310 --> 00:20:53,130
+automatic حققت شروط ال fundamental theorem وزيادة
+
+254
+00:20:53,130 --> 00:20:59,850
+سنة إذا أكيد النتيجة بتكون صحيحة يعنيتكامل الدالة
+
+255
+00:20:59,850 --> 00:21:05,210
+المتفاضلة بساوء أصل الدالة اللي هو أصل الدالة عنده
+
+256
+00:21:05,210 --> 00:21:09,070
+النقطة الأولى نقص النقطة مين بنحكي النقطة الثانية
+
+257
+00:21:09,070 --> 00:21:13,950
+اللي هو عبارة عن integration F of B نقص مين F of A
+
+258
+00:21:13,950 --> 00:21:19,210
+طب لو كانت X متغير يعني لو أخذنا أي X بين الفترتين
+
+259
+00:21:19,210 --> 00:21:23,550
+وطبّجنا نظرية على الـA اللي عند الـXفبصير ال
+
+260
+00:21:23,550 --> 00:21:27,130
+integration من إلي إلى عند ال X أف براين بساوي أف
+
+261
+00:21:27,130 --> 00:21:32,490
+of X ناقص أف of A X هذه هتصبح متغيرة والأف of A
+
+262
+00:21:32,490 --> 00:21:39,190
+ثابتة وهذا بيذكرنا انه احنا قيمة التكامل لدالة
+
+263
+00:21:39,190 --> 00:21:43,790
+بساوي اللي هو عبارة عن اللي هي الإجابة اللي هي
+
+264
+00:21:43,790 --> 00:21:47,630
+الدالة اللي بتطلع زائد some constant C اللي هو ال
+
+265
+00:21:47,630 --> 00:21:53,660
+constant طبعا لاللي هيطبيعة الديالي لقداش بيطلعلنا
+
+266
+00:21:53,660 --> 00:21:56,420
+اللي هو ال initial conditions بيسفرلنا إياه ال
+
+267
+00:21:56,420 --> 00:21:59,620
+constant أو بيعطينا إياه خمسة أربعة ستة أو بيظلوا
+
+268
+00:21:59,620 --> 00:22:03,720
+بصورة عامة Constancy طيب شوفوا صلوا على النبي عليه
+
+269
+00:22:03,720 --> 00:22:08,090
+الصلاة والسلامالآن بدنا نيجي للجزء الثاني من الـ
+
+270
+00:22:08,090 --> 00:22:11,470
+Fundamental theorem خلصنا جزء اللي هو تكامل
+
+271
+00:22:11,470 --> 00:22:16,870
+التفاضل، الآن بنتوقع إنه انفاضل التكامل، انفاضل
+
+272
+00:22:16,870 --> 00:22:20,530
+التكامل، تحت شروط، بنشوف اللي هي ال Fundamental
+
+273
+00:22:20,530 --> 00:22:24,650
+theorem of calculus second form أو اللي هو الجزء
+
+274
+00:22:24,650 --> 00:22:27,850
+الثاني من ال Fundamental theorem، إيش اللي بيقوله؟
+
+275
+00:22:27,850 --> 00:22:33,540
+اللي بيقول معايا ليه، نفترض إن F smallمن a و b
+
+276
+00:22:33,540 --> 00:22:38,060
+لعند r اللي هو is integrable in a و b إذا انفرضناه
+
+277
+00:22:38,060 --> 00:22:41,460
+انه integrable ونفرض أن f of x هو بيساوي ال
+
+278
+00:22:41,460 --> 00:22:46,160
+integration من a ل x f for all x element in a و b
+
+279
+00:22:46,160 --> 00:22:52,200
+بس لان نيجي للنتيجة thenهيطلع هذا عبارة عن
+
+280
+00:22:52,200 --> 00:22:55,360
+continuous function في ال .. في ال .. اللي هي main
+
+281
+00:22:55,360 --> 00:23:02,700
+في اللي هي على الفترة a أو b ومش هي كمان ولو كان
+
+282
+00:23:02,700 --> 00:23:09,830
+ال Fsmall continuous عند نقطة C في الفترة من A
+
+283
+00:23:09,830 --> 00:23:14,050
+لعند B أو من A لعند X هتكون ال F هذه نفسها
+
+284
+00:23:14,050 --> 00:23:16,950
+differentiable و ال derivative اللي هي إيش بساوي
+
+285
+00:23:16,950 --> 00:23:22,590
+بساوي قيمة اللي هو ال F اللي جوا يعني باختصار يا
+
+286
+00:23:22,590 --> 00:23:29,450
+جماعة بتتكررها سهل بقولك مفترض FIntegrable ماشي
+
+287
+00:23:29,450 --> 00:23:34,690
+الحال هاي بس الان بقول لو جيت اخدت من A ل X on
+
+288
+00:23:34,690 --> 00:23:44,670
+طبعا A و B لو اخدت الان A و X F small of T DT هاي
+
+289
+00:23:44,670 --> 00:23:51,510
+ده طبعا الان صارت كلها تعتمد على مين على X بقولك
+
+290
+00:23:51,510 --> 00:23:55,370
+هذههذه اللي طلع اللي هو الـ integration للـ
+
+291
+00:23:55,370 --> 00:24:01,130
+integrable function من A ل X F of T DT هتطلع لي
+
+292
+00:24:01,130 --> 00:24:05,210
+continuous function لو سميتها F capital زي ما هو
+
+293
+00:24:05,210 --> 00:24:08,730
+مسميها F of X بتكون الـ F عندها شمالها is
+
+294
+00:24:08,730 --> 00:24:16,430
+continuous الآن بقولك لو اجيت و قلت ان Fis
+
+295
+00:24:16,430 --> 00:24:21,390
+continuous الـ F اللي جوا هذه at C element in A وB
+
+296
+00:24:21,390 --> 00:24:25,650
+لو كانت F is continuous at C element in A وB
+
+297
+00:24:25,650 --> 00:24:31,730
+هيطلعلك بقولك إذا الـ F capital هذه F prime of C
+
+298
+00:24:31,730 --> 00:24:39,870
+existsو مش exist بس و ال F prime of C هتساوي ال F
+
+299
+00:24:39,870 --> 00:24:46,030
+small of إيش؟ of C يعني و كأنه بقولك لما نتفاضل
+
+300
+00:24:46,030 --> 00:24:50,210
+هذه F prime of C اللي هي من إيه اللي عند C يعني
+
+301
+00:24:50,210 --> 00:24:57,370
+تقول F prime of C بساوي تفاضل رفلة by DXعنده
+
+302
+00:24:57,370 --> 00:25:02,630
+النقطة C لل integration من A ل X F of T DT
+
+303
+00:25:27,580 --> 00:25:33,180
+C مثلاً بتساوي F of C اللي هي F small of C هو كون
+
+304
+00:25:33,180 --> 00:25:36,540
+ال F اللي جوا continuous عنده النقطة اللي بنحكي
+
+305
+00:25:36,540 --> 00:25:39,940
+عنها طب لو كانت F اللي جوا هادى continuous وين
+
+306
+00:25:39,940 --> 00:25:43,620
+ماكان على الفترة هادى بصير خلاص وانت مغمض عناك
+
+307
+00:25:43,620 --> 00:25:47,980
+بتيجي بتقول تفاضل هادي بتشيل التكامل و بتحطه اللي
+
+308
+00:25:47,980 --> 00:25:53,540
+جوا بدلالة اللي هي ال X المتغيرواضح المفروض طيب
+
+309
+00:25:53,540 --> 00:25:59,660
+نيجي الآن نثبت اللي هي النظرية نثبت شغلتين أن ال F
+
+310
+00:25:59,660 --> 00:26:02,580
+كابتنال continuous عشان أثبتلك continuous هثبتلك
+
+311
+00:26:02,580 --> 00:26:05,940
+أكثر من continuous هثبتلك uniform ل continuous لأ
+
+312
+00:26:05,940 --> 00:26:09,780
+أقولك هثبتلك أكثر من uniform ل continuous هثبتلك
+
+313
+00:26:09,780 --> 00:26:12,920
+أن ال F is Lipschitz function مثلا من Lipschitz
+
+314
+00:26:12,920 --> 00:26:15,400
+function إذا على طول uniform ل continuous ومن ثم
+
+315
+00:26:15,400 --> 00:26:18,460
+على طول إيش ما لها continuous إيش Lipschitz
+
+316
+00:26:18,460 --> 00:26:25,240
+function يعني هثبتلك أن Fof x-f of y absolute
+
+317
+00:26:25,240 --> 00:26:30,720
+value أصغر أو ساوي k في x-y had there exist k بحيث
+
+318
+00:26:30,720 --> 00:26:35,260
+أن f of x-f of y أصغر من k في xy لكل x و y element
+
+319
+00:26:35,260 --> 00:26:40,020
+in a و b هثبت لك f بتحقق هذا الشرط هثبتها ما ده
+
+320
+00:26:40,020 --> 00:26:42,540
+هثبتها ما ده هتحقق هذا الشرط إذا ما على طول اللي
+
+321
+00:26:42,540 --> 00:26:45,680
+هي هتكون lipschitz يعني بمعنى آخر هتكون is
+
+322
+00:26:45,680 --> 00:26:49,480
+continuous هذا الجزء الأول الجزء الثاني يا شباب و
+
+323
+00:26:49,480 --> 00:26:55,760
+يا بناتلأن if F is continuous at C هنثبت أن F is
+
+324
+00:26:55,760 --> 00:26:59,320
+differentiable عند C يعني بدأ أثبتلك F is
+
+325
+00:26:59,320 --> 00:27:03,460
+differentiable عند C يعني ايش F is differentiable
+
+326
+00:27:03,460 --> 00:27:14,660
+يعني بدأ أثبتلك أن limit F of X زي داتش مثلا او F
+
+327
+00:27:14,660 --> 00:27:21,440
+of C زي داتش ناقص F of Cعلى H اذا H بتروح للصفر
+
+328
+00:27:21,440 --> 00:27:27,380
+بتثبت لك ايه ايش بيساوي F capital طبعا بيساوي F
+
+329
+00:27:27,380 --> 00:27:33,660
+small of Cبدي أثبتلك يعني أو بمعنى آخر هأثبتلك لكل
+
+330
+00:27:33,660 --> 00:27:37,000
+Y أكبر من 0 داريكزيز ديلتا بحيث أنه absolute value
+
+331
+00:27:37,000 --> 00:27:41,320
+أصغر من Delta L على H أصغر من Delta يؤدي إلى F of
+
+332
+00:27:41,320 --> 00:27:48,260
+C زائد H ناقص F of C على H ناقص F of C بدي أثبتلك
+
+333
+00:27:48,260 --> 00:27:53,180
+يه أصغر من 200 من Epsilonبكون اثبتت فعلا انه هذه
+
+334
+00:27:53,180 --> 00:27:56,540
+ال limit بالساوية هذه معناته هذه ال limit طبعا ايش
+
+335
+00:27:56,540 --> 00:28:01,000
+بتعني ان F prime F capital prime of C exist و
+
+336
+00:28:01,000 --> 00:28:06,480
+بتساوية F of C اللي هو اللي هو طالبه انه لما تكون
+
+337
+00:28:06,480 --> 00:28:09,280
+F continuous عند ال C بتكون F capital
+
+338
+00:28:09,280 --> 00:28:12,040
+differentiable عند ال C و ال derivative لل F
+
+339
+00:28:12,040 --> 00:28:15,700
+capital بتساوية ال F اسمه فنشوف
+
+340
+00:28:17,960 --> 00:28:21,380
+Media نثبت الـ Continuity أو نثبت الـ Lipschitz
+
+341
+00:28:21,380 --> 00:28:27,450
+function يعني الأمر سهلأحنا مفترضين يا جماعة أن
+
+342
+00:28:27,450 --> 00:28:31,490
+الـF small هذي is integrable مزام is integrable
+
+343
+00:28:31,490 --> 00:28:37,650
+إذا is bounded S يعني أنه لكل .. نقدر نلاقي K أكبر
+
+344
+00:28:37,650 --> 00:28:41,830
+من 0 بحيث أن ال absolute value ل F of X small أصغر
+
+345
+00:28:41,830 --> 00:28:48,070
+سوى K لكل X موجودة ال A و ال B سبب هذه أن F نفسها
+
+346
+00:28:48,070 --> 00:28:52,350
+is integrable طيب، بدأ اعتمد على هذه للوصول إنها
+
+347
+00:28:52,350 --> 00:28:58,620
+لبشتالان خد اي x و y في ال a و ال b واسمحولي اخد x
+
+348
+00:28:58,620 --> 00:29:01,760
+اقل من y without loss of generality مش عاد باخد y
+
+349
+00:29:01,760 --> 00:29:06,100
+اكبر من .. اصغر من x الان حالة ال x بتساوي
+
+350
+00:29:06,100 --> 00:29:09,660
+هتلاقوها automatic بتتحقق للي بديها نشوف ايش اللي
+
+351
+00:29:09,660 --> 00:29:13,790
+بديهاالان خد x و y في ال a و ال b و نفترض ان x
+
+352
+00:29:13,790 --> 00:29:16,610
+أزرار من y without most of generality زي ما قلنا
+
+353
+00:29:16,610 --> 00:29:22,890
+الان احسبلي f of x f of y نقص مين f of x الكبيرة y
+
+354
+00:29:22,890 --> 00:29:26,630
+و الكبيرة x احسب f of y نقص f of x ايش هيساوي ال
+
+355
+00:29:26,630 --> 00:29:29,470
+integration حسب التعريف f of x بتساوي ال
+
+356
+00:29:29,470 --> 00:29:33,390
+integration من ال x ل ال f smallالأن f of y هي
+
+357
+00:29:33,390 --> 00:29:36,930
+عبارة عن ال integration من a ل y، f small، ناقص، f
+
+358
+00:29:36,930 --> 00:29:40,210
+of x إيش بتساوي؟ ال integration من a ل x ل مين؟ لل
+
+359
+00:29:40,210 --> 00:29:47,290
+F، إذا هذا ناقص هذا، الآن بدي أحول هذه، أجلبها،
+
+360
+00:29:47,290 --> 00:29:52,670
+بيصير زائد ال integration من x لعند مين، لعند ال
+
+361
+00:29:52,670 --> 00:29:57,640
+a، لل Fصار عندى الآن ال integration من X لعند ال A
+
+362
+00:29:57,640 --> 00:30:00,520
+و من A لعند ال Y إذا حيصير هذا عبارة عن ال
+
+363
+00:30:00,520 --> 00:30:05,940
+integration من X لمين من X لعند ال Y عارفين هذه
+
+364
+00:30:05,940 --> 00:30:09,960
+الخاصية إذا صار عند ال F of X F of Y ناقص F of X
+
+365
+00:30:09,960 --> 00:30:15,760
+بسوء ال integration من X ل Y لمين لل F شوف الآن
+
+366
+00:30:15,760 --> 00:30:20,100
+إيش بده أصل؟ بده أصل إنه absolute value F of Y
+
+367
+00:30:20,100 --> 00:30:24,830
+ناقص F of Xأصغر أو سوى K في الـ absolute value X
+
+368
+00:30:24,830 --> 00:30:31,450
+مانس Y يعني صارت لبشت لنشوف كيف نصلها وأكيد بعضكم
+
+369
+00:30:31,450 --> 00:30:36,450
+فهم او اتوقع كيف بتسوي لأن صار ال absolute value
+
+370
+00:30:36,450 --> 00:30:40,250
+لهذه بسوى ال absolute value لهذه هيها لأن ال
+
+371
+00:30:40,250 --> 00:30:43,010
+absolute value ال integration بخاصية احنا عارفينها
+
+372
+00:30:43,010 --> 00:30:46,770
+أصغر أو سوى ال integration لل absolute valueوالـ
+
+373
+00:30:46,770 --> 00:30:50,990
+absolute value للـ F في الفترة X وY أشملها .. هي
+
+374
+00:30:50,990 --> 00:30:53,570
+مش في الفترة X وY بس الـ absolute value للـ F الـ
+
+375
+00:30:53,570 --> 00:30:56,170
+absolute value للـ F أصغر أو سوى K على كل الفترة
+
+376
+00:30:56,170 --> 00:31:01,030
+الكبيرة اللي هي A وB فأكيد برضه هتكون أصغر أو سوى
+
+377
+00:31:01,030 --> 00:31:05,670
+K على الفترة الصغيرة فصار عندي أصغر أو سوى K في
+
+378
+00:31:05,670 --> 00:31:07,890
+مين؟ في ال integration من X لعند Y ال integration
+
+379
+00:31:07,890 --> 00:31:13,080
+من X لعند Y هو إيش بساوي؟ اللي هو Y minus Xالـ
+
+380
+00:31:13,080 --> 00:31:19,400
+integration يعني ال integration لـ DT من X لـ DT
+
+381
+00:31:19,400 --> 00:31:28,660
+ال integration لـ DT من X عند Y ايش بيساوي Y minus
+
+382
+00:31:28,660 --> 00:31:36,240
+X Y minus X فهذا صار Y minus X و اللي جوا أصغر
+
+383
+00:31:36,240 --> 00:31:40,790
+بيساوي Kالان هذا على خطوتين سهلات بتصور ان انتوا
+
+384
+00:31:40,790 --> 00:31:44,370
+فاهمينش بحكي من X لعند Y ال integration لل F فده
+
+385
+00:31:44,370 --> 00:31:47,850
+أصغر أو ساوي اللي هو ال integration من X لعند Y
+
+386
+00:31:47,850 --> 00:31:55,370
+هذه بدلة K لكل DT وهذا بساوي K برة في Y minus X
+
+387
+00:31:55,370 --> 00:31:59,230
+اللي هو أكيد حيصير عند ال absolute value هذه أصغر
+
+388
+00:31:59,230 --> 00:32:03,770
+أو ساوي K في ال absolute value Y minus Xالان هذه
+
+389
+00:32:03,770 --> 00:32:07,570
+ال Xات الأصغر من مين؟ من Y لكن زي ما انتوا عارفين
+
+390
+00:32:07,570 --> 00:32:11,150
+حالة ال X بتساوي Y is trivial لأن لما تكون X
+
+391
+00:32:11,150 --> 00:32:14,450
+بتساوي Y هذا سفر و لما تكون X بتساوي Y هذا سفر إذا
+
+392
+00:32:14,450 --> 00:32:18,610
+ال inequality هذه صحيحة دائما إذا الأن صار عندى
+
+393
+00:32:18,610 --> 00:32:25,590
+هذه ال inequality staris true for all x و y
+
+394
+00:32:25,590 --> 00:32:30,390
+limiting a و b لأنه قلنا ال x أصغر من y ال y أصغر
+
+395
+00:32:30,390 --> 00:32:33,990
+من x أكيد similarly و بنفس الأسلوب أو حتى without
+
+396
+00:32:33,990 --> 00:32:38,170
+loss of generality يعني بدون ما نفقد أي إشي من
+
+397
+00:32:38,170 --> 00:32:44,530
+التعميم بنفترض أن x أصغر من y صار
+
+398
+00:32:44,530 --> 00:32:53,230
+عندي يا جماعةالان F مستمر على A وB لأن F بيصبح
+
+399
+00:32:53,230 --> 00:33:02,750
+عملية Lipschitz اللي هي الآن ضال علي أثبت أن الـ F
+
+400
+00:33:02,750 --> 00:33:11,100
+مستخدم وزي ما قلت بدأ أثبت أن الـ Limitللـ F of C
+
+401
+00:33:11,100 --> 00:33:17,800
+زائد H نقص F of C على H لما H تروح للصفر بساوة F
+
+402
+00:33:17,800 --> 00:33:23,480
+small of C أو أثبت لكم الفرق بين هذولتين أصغر من
+
+403
+00:33:23,480 --> 00:33:28,880
+إبسلون وهذا زي ما قلنا هو اللي هيمثل F' of C ده
+
+404
+00:33:28,880 --> 00:33:34,160
+نشوف كيف طيب الصلاة على النبي صلاة والسلام عليك
+
+405
+00:33:37,390 --> 00:33:42,130
+خلّينا نجف عند .. أيوة نيجي الآن البرهان هتلاقوه
+
+406
+00:33:42,130 --> 00:33:49,730
+برضه سهل وفكرته سهلة شوفوا كيف عندي نفترض الآن
+
+407
+00:33:49,730 --> 00:33:53,550
+suppose that f is continuous at c limiting a وb
+
+408
+00:33:53,550 --> 00:33:58,790
+مدام continuous إذن limit
+
+409
+00:34:01,700 --> 00:34:08,460
+F of X as X بتروح للـ C بساوي F of C مظبوط ولا لأ؟
+
+410
+00:34:08,460 --> 00:34:12,300
+أكيد مظبوط فبقى من جهة أخرى خلينا نكتبها بصورة
+
+411
+00:34:12,300 --> 00:34:18,660
+ثانية X ناقص C بتروح للسفر if and only if اللي هو
+
+412
+00:34:18,660 --> 00:34:23,960
+X بتروح لمين للـ C سمولي هذه X minus C إيش اسمها H
+
+413
+00:34:23,960 --> 00:34:31,410
+بصير هذه اللي فوق مين هي limitF of الـ X minus C
+
+414
+00:34:31,410 --> 00:34:36,490
+بتساوي الـ H يعني الـ X بتساوي H زائد C أو C زائد
+
+415
+00:34:36,490 --> 00:34:41,190
+Hلأن x بتروح للـ c تكافئ أنه .. اللي هو x minus c
+
+416
+00:34:41,190 --> 00:34:44,610
+تروح للصفر، يعني بتكافئ H تروح لمين للصفر، إيش حيث
+
+417
+00:34:44,610 --> 00:34:49,650
+هو هذا؟ F of C، إذا هذه هي تعبير آخر عن ال
+
+418
+00:34:49,650 --> 00:34:56,250
+continuity لل F لما اللي هو اللي هي ال F عند مين،
+
+419
+00:34:56,250 --> 00:34:59,210
+عند ال C، يعني هذه تعبير آخر، ال continuity لل
+
+420
+00:34:59,210 --> 00:35:05,900
+function لما عند النقطة اللي هي Cهذه الآن شو بدي
+
+421
+00:35:05,900 --> 00:35:09,520
+استخدمها للوصول اللي بديها لأن مادام F is
+
+422
+00:35:09,520 --> 00:35:12,600
+continuous ده هذي متحققة هيها مادام F is
+
+423
+00:35:12,600 --> 00:35:15,480
+continuous عن C ده هذي متحققة فمادام هذي متحققة
+
+424
+00:35:15,480 --> 00:35:18,720
+اذا by epsilon delta definition for every epsilon
+
+425
+00:35:18,720 --> 00:35:22,880
+أكبر من 0 there exists delta أكبر من 0 such that
+
+426
+00:35:22,880 --> 00:35:28,950
+لما يكون ال absolute value لل H أصغر من Deltaو
+
+427
+00:35:28,950 --> 00:35:32,230
+طبعا انا وين بشتغل في المنطقة انها تكون C زائد H
+
+428
+00:35:32,230 --> 00:35:36,370
+وين ما لأ في الفترة تبعت من A لعند مين لعند B يعني
+
+429
+00:35:36,370 --> 00:35:41,230
+اخترت ال H صغيرة كفاية بحيث انه C زائد H اضلي وين
+
+430
+00:35:41,230 --> 00:35:45,090
+جاعد في الفترة من A و B لعنده لإن هاي الفترة A وهي
+
+431
+00:35:45,090 --> 00:35:49,330
+Bوالفترة هذه اللي هي I C عندي مثلا في داخلها اللي
+
+432
+00:35:49,330 --> 00:35:55,690
+هي بتختار H دلتتها صغيرة كفاية أنه ضال C زائد H
+
+433
+00:35:55,690 --> 00:36:01,050
+موجودة في الفترة من A لعند B عشان تصير معرفة لإن
+
+434
+00:36:01,050 --> 00:36:05,330
+دالتي أنا عشان تكون معرفة عند C زائد H لازم تكون C
+
+435
+00:36:05,330 --> 00:36:08,490
+زائد H في داخل المنطقة هذه لإن دالتي معرفة على
+
+436
+00:36:08,490 --> 00:36:11,770
+الفترة من A لعند B عشان هيك أقولنا C زائد H لازم
+
+437
+00:36:11,770 --> 00:36:16,630
+تكون في الفترة زائد Bإذن اختيار الـ Delta يعتمد
+
+438
+00:36:16,630 --> 00:36:20,210
+على ال limit ويعتمد على انه أضمن ال C زي داشر ضال
+
+439
+00:36:20,210 --> 00:36:24,510
+هوين في الفترة A وB إذا تعريف ال continuity بيقول
+
+440
+00:36:25,950 --> 00:36:27,530
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أقوم
+
+441
+00:36:27,530 --> 00:36:28,970
+باستخدام صفر يوجد دلتا لكل y أقوم باستخدام صفر
+
+442
+00:36:28,970 --> 00:36:29,950
+يوجد دلتا لكل y أقوم باستخدام صفر يوجد دلتا لكل y
+
+443
+00:36:29,950 --> 00:36:30,650
+أقوم باستخدام صفر يوجد دلتا لكل y أقوم باستخدام
+
+444
+00:36:30,650 --> 00:36:32,710
+صفر يوجد دلتا لكل y أقوم باستخدام صفر يوجد دلتا
+
+445
+00:36:32,710 --> 00:36:35,050
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أقوم
+
+446
+00:36:35,050 --> 00:36:41,110
+باستخدام صفر يوجد دلتا
+
+447
+00:36:41,110 --> 00:36:49,400
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أطيب، الآن
+
+448
+00:36:49,400 --> 00:36:54,020
+وين أنا بدأ أروح؟ بدأ أثبت لكم إنه هذا ناقص هذا
+
+449
+00:36:54,020 --> 00:36:58,240
+أصغر من epsilon عشان ذلك دعنا نحسب F of C زاد H
+
+450
+00:36:58,240 --> 00:37:03,060
+ناقص F of C على H ناقص مين يا جماعة؟ F of C ويساوي
+
+451
+00:37:03,060 --> 00:37:13,170
+الآن F of C زاد H ناقص F of Cخد الآن اللي هي الـ H
+
+452
+00:37:13,170 --> 00:37:16,850
+هدعم المشترك بالـ C الواحدة للـ H ماشي F of C زي
+
+453
+00:37:16,850 --> 00:37:21,910
+دتش تعريفها من A لعند C زي دتش لل F of X DX قلناها
+
+454
+00:37:21,910 --> 00:37:26,090
+هذه اللي ماعطيناها من رأس الدور ناقص ماشي تعريف F
+
+455
+00:37:26,090 --> 00:37:30,230
+of C Capital of C هو ال integration من A لعند C زي
+
+456
+00:37:30,230 --> 00:37:35,630
+دتش الآن دي مش C زي دتش من A لعند مين لعند الـ C
+
+457
+00:37:35,630 --> 00:37:40,110
+واحدة للـ H من A لعند الـ C اللي هي F of C زي دتش
+
+458
+00:37:40,110 --> 00:37:46,360
+هيهاوهي الواحدة الـ H اللي برا ال F of C هيها من A
+
+459
+00:37:46,360 --> 00:37:51,780
+ل C واحدة الـ H ناقص مين F of C هذا من التعريف
+
+460
+00:37:51,780 --> 00:37:55,460
+مباشرة لإنه احنا عرفنا ال F capital of X هي عبارة
+
+461
+00:37:55,460 --> 00:38:01,820
+عن integration من A لعند X F of T DT لأن لما نكون
+
+462
+00:38:01,820 --> 00:38:06,120
+F of C زي دتش بنحطها دي C زي دتش C بنحطها دي عياش
+
+463
+00:38:06,120 --> 00:38:13,460
+C وهي C زي دتش وهي الـ C وهي ساويةالان هذه من ..
+
+464
+00:38:13,460 --> 00:38:18,780
+من .. من عند A لـ C زي ده اتش وهذه من A لعند B
+
+465
+00:38:18,780 --> 00:38:23,980
+لعند C هدولة مع بعض هاي هدولة التنتين بده احسبها
+
+466
+00:38:23,980 --> 00:38:28,120
+مع بعض حسبنا زيين قبل و شوية هو عبارة عن واحد على
+
+467
+00:38:28,120 --> 00:38:32,400
+اتش خدوا عامل مشترك خدوا واحد على اتش عامل مشترك
+
+468
+00:38:32,400 --> 00:38:37,340
+بين الجهتين تصبح واحد على اتش افتح جوس ال
+
+469
+00:38:37,340 --> 00:38:41,660
+integration من A لعند C زي ده اتشلأن بدل الناقص
+
+470
+00:38:41,660 --> 00:38:46,020
+بيصير H اكتب زائد بدل هذا الناقص بيكتب زائد لأن
+
+471
+00:38:46,020 --> 00:38:52,460
+هتنقلب مين الآن من C لعند ال A نظبط اللي هو لنفسها
+
+472
+00:38:52,460 --> 00:38:58,020
+F F هذه الآن من C لعند ال A ومن A لعند ال C زاد H
+
+473
+00:38:58,020 --> 00:39:02,000
+إذا أكيد هذه كلها على بعض يصير ال integration من C
+
+474
+00:39:02,000 --> 00:39:08,930
+ل C زاد H لل F في واحد على H إذا هذه كلهابس
+
+475
+00:39:08,930 --> 00:39:11,350
+ابدلتها بقيمتها اللي قلنا عنها اللي هو ال
+
+476
+00:39:11,350 --> 00:39:15,810
+integration من C ل C زي H اللي اوجدته بدل هذا كله
+
+477
+00:39:15,810 --> 00:39:23,190
+من واحد ال H ل F of X DX ناقص الان هذا هذا اللي هي
+
+478
+00:39:23,190 --> 00:39:28,090
+F of C شوف كيف بده اعملها على جهة تعالي احسب ال
+
+479
+00:39:28,090 --> 00:39:33,660
+integrationالـ integration من C لـ C زائد H للـ
+
+480
+00:39:33,660 --> 00:39:38,180
+constant واحد بعد أذنكم DX إيش هيساوي ال
+
+481
+00:39:38,180 --> 00:39:44,140
+integration هدا عبارة عن C زائد H ناقص C وهيساوي
+
+482
+00:39:44,140 --> 00:39:48,760
+قداش H يعني بمعنى آخر لو جيتها الجد تعرفوا ليش
+
+483
+00:39:48,760 --> 00:39:52,640
+بعمل هيك واحد على H في ال integration من C لـ C
+
+484
+00:39:52,640 --> 00:39:58,720
+زايد H اللي هو الواحد هيساوي إيش؟ واحد، مظبوط ولا
+
+485
+00:39:58,720 --> 00:40:03,450
+لأ؟أكيد مظبوط ماشي هايمل هادي كسمت الديوا تاني
+
+486
+00:40:03,450 --> 00:40:10,570
+علشان تطلع حد واحد الان صار عندى لو ضربت هادي
+
+487
+00:40:11,920 --> 00:40:18,320
+الجهتين في F of C هي F of C بيساوي ايش؟ F of C صار
+
+488
+00:40:18,320 --> 00:40:21,260
+الان F of C بيساوي هذه بدي أشيل F of C تبعتي و أحط
+
+489
+00:40:21,260 --> 00:40:25,360
+مكانها هذه الصورة ليش حطيتها؟ عشان هذه أقدر أتحمل
+
+490
+00:40:25,360 --> 00:40:30,380
+محا محا دي اللي موجودة من الأصل شوف كيف الان شيلت
+
+491
+00:40:30,380 --> 00:40:33,320
+ال F of C و حطيت قيمتها اللي أوجدناها اللي هي
+
+492
+00:40:33,320 --> 00:40:37,660
+عبارة عن F of C على H في ال integration من C لC زي
+
+493
+00:40:37,660 --> 00:40:42,480
+H لمن؟ للواحدماشي لان صار هذا ال integration و هذا
+
+494
+00:40:42,480 --> 00:40:45,920
+ال integration نفس الاشي واحد ل F of X و واحد ل
+
+495
+00:40:45,920 --> 00:40:50,380
+واحد يعني بقدر اشتغل فيهم بيصير عندى خد ال واحد ال
+
+496
+00:40:50,380 --> 00:40:53,960
+H برا عن المشترك خالص بيظل عندى ال integration من
+
+497
+00:40:53,960 --> 00:40:58,720
+C ل C زاد H هذا مين ايش اسمه F of X و هذا F of C
+
+498
+00:40:58,720 --> 00:41:01,580
+دخلتها جوا و مضربتها بالواحد صارت مين بقدر اعلن و
+
+499
+00:41:01,580 --> 00:41:06,520
+ثابت اللي هي ناقص F of C لكل اش ماله DX هذا الكلام
+
+500
+00:41:06,520 --> 00:41:14,340
+كله من C ل C زاد Hصار عندى الآن الصورة هذه هيها اه
+
+501
+00:41:14,340 --> 00:41:20,840
+و هذه الصورة اصلا انا بدياها الان واحد على H أزرع
+
+502
+00:41:20,840 --> 00:41:23,760
+أو ساوي قولنا absolute value of integration أزرع
+
+503
+00:41:23,760 --> 00:41:28,740
+أو ساوي ال integration لل absolute value DX الان
+
+504
+00:41:28,740 --> 00:41:35,320
+هذا المقدار F of X ناقص F of Cقلنا عنها من راس
+
+505
+00:41:35,320 --> 00:41:40,420
+الدور أن F of X ناقص F of C اللي هو أصغر من
+
+506
+00:41:40,420 --> 00:41:44,420
+Epsilon لأن limit F of X لما X تروح للـC أشهر
+
+507
+00:41:44,420 --> 00:41:50,240
+مساوي F of C لأن X الـN هذه في الفترة من C لـC
+
+508
+00:41:50,240 --> 00:41:56,100
+زائداتش يعني بتسمح لي أن أقول limit F of X as X
+
+509
+00:41:56,100 --> 00:42:01,220
+بتروح للـC بتساوي F of C لما X بتروح للـC اللي هي
+
+510
+00:42:01,220 --> 00:42:05,980
+أدش بتروح لمينللسفر مضل في نفس المنطقة لأن قلنا
+
+511
+00:42:05,980 --> 00:42:10,300
+قبل شوية x minus c بتروح للسفر إذا وفقط إذا اللي
+
+512
+00:42:10,300 --> 00:42:13,880
+هي ال H اللي بتروح للسفر اللي هي عبارة عن x minus
+
+513
+00:42:13,880 --> 00:42:18,260
+c يعني هذا المقدار بال continuity لل F عند ال C
+
+514
+00:42:18,260 --> 00:42:21,980
+برضه بكون أصغر من إبسلون على ال absolute value
+
+515
+00:42:21,980 --> 00:42:25,160
+لمين لل H في ال integration من C لC زاد H
+
+516
+00:42:27,990 --> 00:42:33,050
+واضحة اه؟ اكيد هذا الان ايش قيمته؟ C زي داش ناقص C
+
+517
+00:42:33,050 --> 00:42:36,050
+اللي هي عبارة عن H فبصير Y عبارة عن absolute value
+
+518
+00:42:36,050 --> 00:42:42,430
+of H في mean فيه اللي هو الـH الـH اللي هو الـH
+
+519
+00:42:42,430 --> 00:42:42,470
+اللي هو الـH اللي هو الـH الـH اللي هو الـH اللي
+
+520
+00:42:42,470 --> 00:42:42,490
+هو الـH اللي هو الـH اللي هو الـH اللي هو الـH
+
+521
+00:42:42,490 --> 00:42:42,570
+الـH اللي هو الـH اللي هو الـH اللي هو الـH اللي
+
+522
+00:42:42,570 --> 00:42:42,710
+هو الـH اللي هو الـH اللي هو الـH اللي هو الـH
+
+523
+00:42:42,710 --> 00:42:46,490
+الـH اللي هو الـH اللي هو الـH
+
+524
+00:42:46,490 --> 00:42:52,450
+اللي هو الـطيب .. بيصير عندها .. بيصير أكبر من 0
+
+525
+00:42:52,450 --> 00:42:55,790
+أسف هذه بحاجة دي بتطلع أيش بتساوي .. بيساوي بساوي
+
+526
+00:42:55,790 --> 00:42:58,150
+.. بقدر أختار H positive؟ أه بقدر أختار H positive
+
+527
+00:42:58,150 --> 00:43:03,130
+طيب و لو حتى ال H negative بتكون similarly بس
+
+528
+00:43:03,130 --> 00:43:06,950
+بتكون .. كل البرهام بيصير .. بترتب على أنه اللي هي
+
+529
+00:43:06,950 --> 00:43:12,890
+ال .. ال C زائد H أصغر من مين من ال C without loss
+
+530
+00:43:12,890 --> 00:43:17,250
+of generality ال H أكبر من 0بصير عندى لان هذا
+
+531
+00:43:17,250 --> 00:43:24,750
+المقدار أصغر من مين من إبسلون إيش يعني اللى عملنا
+
+532
+00:43:24,750 --> 00:43:28,290
+اللى عملنا لكل إبسلون أكبر من Zero لاجئنا Delta
+
+533
+00:43:28,290 --> 00:43:33,350
+بحيث أنه لما H absolute value أصغر من Deltaيعطيني
+
+534
+00:43:33,350 --> 00:43:39,330
+أن هذا المقدار أصغر من مين؟ من إبسلون يعني صار
+
+535
+00:43:39,330 --> 00:43:43,450
+عندي limit هذا المقدار بسوء f of c و هذا المقدار
+
+536
+00:43:43,450 --> 00:43:46,290
+اللي هو f prime of c يعني اعتبتنا f prime of c
+
+537
+00:43:46,290 --> 00:43:56,410
+بسوء f small of c وهو المطلوب طيب نيجي الآن لاللي
+
+538
+00:43:56,410 --> 00:44:02,210
+هي ال corollaryبرضه الى النظرية اللى جاملة بشوية
+
+539
+00:44:02,210 --> 00:44:10,790
+او خلّيني نقول تلخيصها سريع واستذكرها سريعها اللى
+
+540
+00:44:10,790 --> 00:44:17,830
+هو الأولى كانت تكامل والتفاضل هذه تفاضل التكامل
+
+541
+00:44:17,830 --> 00:44:23,470
+طبعا كل واحدة تحت شروطها المذكورة في نظريتها
+
+542
+00:44:25,860 --> 00:44:30,340
+نجي نشوف يا جماعة ال corollary let f من a و b لعند
+
+543
+00:44:30,340 --> 00:44:34,940
+r بي continuous on a و b and let f of x بي سوى ال
+
+544
+00:44:34,940 --> 00:44:37,720
+integration من ال a ل ال x ل ال f إذا فرض f
+
+545
+00:44:37,720 --> 00:44:40,160
+continuous ال f اسمه ال continuous على كل الفترة
+
+546
+00:44:40,160 --> 00:44:44,980
+ريح حاله طبعا أعطى إشي أكبر من النظرية اللي فاتت
+
+547
+00:44:44,980 --> 00:44:47,640
+يعني فرض ال continuity على ال a و ال b ما زم
+
+548
+00:44:47,640 --> 00:44:49,840
+continuous إذا ما بتزمش تقول integrable لإن و أنا
+
+549
+00:44:49,840 --> 00:44:52,340
+continuous function زي ما قلنا إيش ما لها is
+
+550
+00:44:52,340 --> 00:44:58,190
+integrable إذا مدامة f continuousالان بيصير هذه ال
+
+551
+00:44:58,190 --> 00:45:01,810
+integration اللي هي من a لعند ال x نفسها
+
+552
+00:45:01,810 --> 00:45:06,030
+continuous لأ مش continuous بس is differentiable
+
+553
+00:45:06,030 --> 00:45:09,270
+يعني أجم ال continuous يعني هتكون عندي ال f is
+
+554
+00:45:09,270 --> 00:45:13,670
+differentiable و ال f برامي يمين هي ال f then f is
+
+555
+00:45:13,670 --> 00:45:17,730
+differentiable من a و b عند f برامي بيساوي كده إذن
+
+556
+00:45:17,730 --> 00:45:25,000
+باختصار من a ل x اللي هو fof t dt لو فرضنا هذه
+
+557
+00:45:25,000 --> 00:45:33,060
+continuous على ال a و ال b إذا لكل x element in a
+
+558
+00:45:33,060 --> 00:45:38,080
+و ال b بيصير f of x بالساوي هذا is differentiable
+
+559
+00:45:38,080 --> 00:45:45,500
+و ال f prime of x هتساوي f of اللي هي x يعني بمعنى
+
+560
+00:45:45,500 --> 00:45:52,590
+آخر اللي هوانفضل هذا التكامل و نزيل التفاضل و نحط
+
+561
+00:45:52,590 --> 00:46:00,830
+الـ L جوا يعني بمعنى آخر D by DX للـ integration
+
+562
+00:46:00,830 --> 00:46:08,530
+من A لعند X F of D DT لما تكون F continuous هذا
+
+563
+00:46:08,530 --> 00:46:14,610
+بكون ب cancel هذا أو عملية عكسية لهوبتظللنا F of X
+
+564
+00:46:14,610 --> 00:46:19,070
+يعدى الأحقالها أو F of T والـ T هي المتغير أو الـ
+
+565
+00:46:19,070 --> 00:46:23,370
+X هي المتغير هذه اللي هي الكلرية الآن بدوا يلمّوا
+
+566
+00:46:23,370 --> 00:46:26,610
+النظريتين اللي قبل بشوية في نظرية واحدة يلخصن في
+
+567
+00:46:26,610 --> 00:46:30,910
+نظرية واحدة نشوف كيف بدوا يلخصن في نظرية واحدة
+
+568
+00:46:30,910 --> 00:46:40,630
+يعني بدوا يعيد بس اللي هو صياغة النظريتين صلوا
+
+569
+00:46:40,630 --> 00:46:44,540
+على النبي عليه الصلاة والسلامالان fundamental
+
+570
+00:46:44,540 --> 00:46:48,580
+theorem of calculus combined form يعني التنتين
+
+571
+00:46:48,580 --> 00:46:53,790
+الان موجودات في نفس النظرية شوف ايش بقولكlet
+
+572
+00:46:53,790 --> 00:46:57,610
+Fcapital and Fsmall be continuous on a and b فرض
+
+573
+00:46:57,610 --> 00:47:00,710
+أن Fcapital و Fsmall أشمالهم إتن تان continuous
+
+574
+00:47:00,710 --> 00:47:04,670
+على الفترة a و b و فرضك أن F of a بساوة سفر يعني
+
+575
+00:47:04,670 --> 00:47:07,610
+بدي أحط ال initial condition F of a بساوة سفر عسى
+
+576
+00:47:07,610 --> 00:47:11,070
+أن يجمّل الصورة ماحطه عشان سفر بتطلع F of a في
+
+577
+00:47:11,070 --> 00:47:14,290
+الجواب then the following statements are
+
+578
+00:47:14,290 --> 00:47:18,380
+equivalentيعني لما تكون هذا صحيحة هذا صحيحة و لما
+
+579
+00:47:18,380 --> 00:47:22,640
+تكون هذا صحيحة هذا صحيحة نيجي للجزء الأول لو فرضنا
+
+580
+00:47:22,640 --> 00:47:29,160
+أن F prime of X بيساوي F of X إذا حيصير عندي ال F
+
+581
+00:47:29,160 --> 00:47:33,460
+هذه is differential الماشي هيك مفترض أن F prime of
+
+582
+00:47:33,460 --> 00:47:38,620
+X بيساوي F of X بقول إذاإذاً هذه صحيحة، إيش يعني
+
+583
+00:47:38,620 --> 00:47:43,260
+هذه صحيحة؟ يعني ال integration من ال AX لل F of
+
+584
+00:47:43,260 --> 00:47:48,400
+التي هي T DT، هذه DT يا جماعة، بس .. بس نخلف عن
+
+585
+00:47:48,400 --> 00:47:58,200
+هذه، F of T DT، هذا هيساوي مين؟ هيساوي F of X،
+
+586
+00:47:58,200 --> 00:48:03,070
+ناشي؟الان then the following statements are
+
+587
+00:48:03,070 --> 00:48:06,950
+equivalent يعني لما تكون هذه صحيحة بتعطيها دي اه
+
+588
+00:48:06,950 --> 00:48:11,350
+فرضنا ان f prime of x بيساوي f of x اذا حيكون عند
+
+589
+00:48:11,350 --> 00:48:14,290
+ال integration من a ل x f of t dt بيساوي ايش
+
+590
+00:48:14,290 --> 00:48:20,630
+بالظبط؟ f of x كيف ده تبتها؟ الان فرضنا هذا صحيح
+
+591
+00:48:20,630 --> 00:48:24,710
+اذا ال integration من a ل عند ال x f of t اللي هي
+
+592
+00:48:24,710 --> 00:48:33,010
+مين هتصير؟ f prime of tDT هذه قبل بشوية من نظرية
+
+593
+00:48:33,010 --> 00:48:36,470
+الأولى قلنا كامل التفاوض و ليش بتطلع عندك جواب F
+
+594
+00:48:36,470 --> 00:48:42,210
+of X ناقص مين F of A لإن كل الشروط متحققة ماشي
+
+595
+00:48:42,210 --> 00:48:48,110
+الآن هاي كمان مرة يا جماعة فرضنا إن هذا متحقق بدي
+
+596
+00:48:48,110 --> 00:48:51,250
+أحسب هذه أتبتها بالساوية هذه خد ال integration من
+
+597
+00:48:51,250 --> 00:48:55,510
+X F of T DTهو ماعطيني الـ F small هد مين هي الـ F
+
+598
+00:48:55,510 --> 00:48:58,830
+براهن شفت .. شيلت الـ F small إيش حطيت مكانها؟ F
+
+599
+00:48:58,830 --> 00:49:03,510
+براهن الآن هذه قبل بشوية في الـ Corollary فظلنا ..
+
+600
+00:49:03,510 --> 00:49:07,490
+كملنا التفاضل كيف كملنا التفاضل؟ إنه قولنا شيل
+
+601
+00:49:07,490 --> 00:49:11,230
+التفاضل .. شيل التكامل .. شيل التفاضل بتصير عبارة
+
+602
+00:49:11,230 --> 00:49:15,270
+عن F اللي فوق اللي كانت B ناقص F اللي تحت اللي هي
+
+603
+00:49:15,270 --> 00:49:19,390
+A ويساويأف of X نقص أف of A أف of A ماعطينيها من
+
+604
+00:49:19,390 --> 00:49:24,350
+رأس الدوار صفر إذا إيش هتساوي أف of X إذا هذه اللي
+
+605
+00:49:24,350 --> 00:49:28,290
+بدّينا حساباتها بتساوي هذه طلعت إيش بتساوي أف of X
+
+606
+00:49:28,290 --> 00:49:32,830
+يعني أف of X بتساوي هذا المقدار وهو المطلوب طيب
+
+607
+00:49:32,830 --> 00:49:36,730
+هذا الجزء الأول من النظرية اللي هو نيجي للجزء
+
+608
+00:49:36,730 --> 00:49:42,240
+الثاني نفترض الآن إن هذه صحيحة وده تعطيهاالان برضه
+
+609
+00:49:42,240 --> 00:49:44,760
+الشروط كلها متحققة لأن الـ F في اسمه ليش مالها
+
+610
+00:49:44,760 --> 00:49:47,340
+continuous لسه ان انا ما .. ما .. ما ملاحتوش انا
+
+611
+00:49:47,340 --> 00:49:52,080
+هذا ايه عندي F continuous مدام F continuous ال
+
+612
+00:49:52,080 --> 00:49:57,720
+integration من اعلى عند ال X هذا كلهاللي هو اسمه F
+
+613
+00:49:57,720 --> 00:50:01,320
+of X هيكون differentiable حسب النظرية مزام
+
+614
+00:50:01,320 --> 00:50:04,780
+differentiable حسب هذه النظرية إذا F prime of X
+
+615
+00:50:04,780 --> 00:50:11,100
+إيش هتساوي؟ F of X يعني حققنا مين؟ اللي هي I يعني
+
+616
+00:50:11,100 --> 00:50:20,070
+هذا يؤدي إلى هذانجي الآن بعض التعريفات اللي
+
+617
+00:50:20,070 --> 00:50:26,890
+هنتكرها لكم وانتوا هتكونوا اللي هو اتابعوا الأمثلة
+
+618
+00:50:26,890 --> 00:50:32,130
+من خلال ال homework اللي معاكم هنعطي اللي هي تعريف
+
+619
+00:50:32,130 --> 00:50:36,830
+ممكن انتوا حتى مر عليكم في ال calculus ومن ثم اللي
+
+620
+00:50:36,830 --> 00:50:41,710
+هونعطي بعض ال counter examples اللي هي هنرجعكم
+
+621
+00:50:41,710 --> 00:50:47,370
+فيها للي هي الأسئلة اللي في الكتاب تشوفوا نصوصها
+
+622
+00:50:47,370 --> 00:50:53,530
+على الأقل و اللي مطلوب تحلوها ايه تحلوها و اللي ما
+
+623
+00:50:53,530 --> 00:50:56,910
+بتحلوه يعرفش تحلوهافي عندنا اللي هو في ال homework
+
+624
+00:50:56,910 --> 00:51:01,230
+الحلول الموجودة بتدرسوها لحالكم مادرستوها برضه
+
+625
+00:51:01,230 --> 00:51:04,310
+مافهمتوهاش احنا بنعمل فيها discussion ولو عملناها
+
+626
+00:51:04,310 --> 00:51:07,930
+حتى بشكل صوتي على أساس اللي هو احنا يمكن مانقدرش
+
+627
+00:51:07,930 --> 00:51:12,110
+نصور بشكل كامل اللي هي ال discussions عبر اللي هو
+
+628
+00:51:12,110 --> 00:51:16,270
+التصوير اللي احنا الحالي بنصوره عبر ال power point
+
+629
+00:51:16,270 --> 00:51:22,710
+من البيت ان شاء الله طيب definitionlet I بتساوي A
+
+630
+00:51:22,710 --> 00:51:27,630
+وB subset من مين؟ من R إذا كانت F small من I لR
+
+631
+00:51:30,720 --> 00:51:35,400
+then فرضنا أنه في دالة اسم F من I لعند R then the
+
+632
+00:51:35,400 --> 00:51:42,280
+antiderivative of F يعني وكأنها عكس عملية عكس
+
+633
+00:51:42,280 --> 00:51:45,920
+الدالة اللي بنفضلها roughly antiderivative of F on
+
+634
+00:51:45,920 --> 00:51:50,620
+I is a function F من I لعند R such that F prime of
+
+635
+00:51:50,620 --> 00:51:59,830
+X سوى F of X يعني عندنا دالة Fمن عند I لعند R لو
+
+636
+00:51:59,830 --> 00:52:05,310
+جينا لجينا F تانية من عند I لعند R و لجينا F prime
+
+637
+00:52:05,310 --> 00:52:10,450
+of X للي اللي في الـ I بتساوي mean لـ F small of X
+
+638
+00:52:11,290 --> 00:52:15,490
+في هذه الحالة بنسمي ال F capital هذه عبارة عن
+
+639
+00:52:15,490 --> 00:52:19,850
+antiderivative لل F يعني بمعنى أخر لو فضلناها هذه
+
+640
+00:52:19,850 --> 00:52:25,130
+ال antiderivative هتطلع مين؟ ال F الأصلية يعني
+
+641
+00:52:25,130 --> 00:52:30,890
+بتكون عندي اللي هي ال F capital هي ال
+
+642
+00:52:30,890 --> 00:52:35,210
+antiderivative لل F وال F small هي ال derivative
+
+643
+00:52:35,210 --> 00:52:41,910
+لل F واضح أخدته في ال calculus حتى نيجي الآنلأ
+
+644
+00:52:41,910 --> 00:52:45,650
+اللي هو برضه مفهوم اخدته في الكاركولاس if f من I
+
+645
+00:52:45,650 --> 00:52:49,750
+لعند R small is integrable لو فرضنا هذي integrable
+
+646
+00:52:49,750 --> 00:52:53,950
+و جينا عرفنا
+
+647
+00:52:53,950 --> 00:52:56,910
+ادالة من A لعند X F of X DX قبل شوية قولنا هذا
+
+648
+00:52:56,910 --> 00:53:00,290
+أكيد معرفة مثلا في integrable ومش معرفة ال
+
+649
+00:53:00,290 --> 00:53:04,110
+function F of X هتطلع continuous هذه اللي هي ال
+
+650
+00:53:04,110 --> 00:53:09,930
+function أو هذا هو اللي بنسميه التكامل المحدودهذه
+
+651
+00:53:09,930 --> 00:53:15,110
+بنسميها الـ Indefinite Integral of F يعني هذا
+
+652
+00:53:15,110 --> 00:53:17,790
+بنسميه Indefinite Integral of F أو هذه الدلة
+
+653
+00:53:17,790 --> 00:53:23,030
+بنسميها اللي هي The Indefinite Integral of F on I
+
+654
+00:53:23,030 --> 00:53:28,230
+برضه هذا برضه شغلات اللي هو أخدتوها سابقا في اللي
+
+655
+00:53:28,230 --> 00:53:33,780
+هو ال calculus نيجي بعض الملاحظات بسعلى اللي هو
+
+656
+00:53:33,780 --> 00:53:42,820
+اللي بتعلق بهذه اللي هي المفاهيم نشوفها وهي طبعا
+
+657
+00:53:42,820 --> 00:53:48,760
+هذه المفاهيم في عليها counter examples التمييز
+
+658
+00:53:48,760 --> 00:53:56,400
+بينها نشوف المحبة هذه يا شباب طيب أصلي على النبي
+
+659
+00:53:56,400 --> 00:54:03,640
+عليه الصلاة والسلامالان عندي سؤال سبعة تلاتة اتنين
+
+660
+00:54:03,640 --> 00:54:06,400
+يعني في سبعة تلاتة سؤال اتنين او سبعة تلاتة سؤال
+
+661
+00:54:06,400 --> 00:54:10,660
+خمسة طبعا كله في الطبع التاني هذا اللي هو عندي an
+
+662
+00:54:10,660 --> 00:54:13,540
+integrable function may not have an antiderivative
+
+663
+00:54:13,540 --> 00:54:17,220
+يعني هنلاقي في السؤال هذا integrable function
+
+664
+00:54:17,220 --> 00:54:24,080
+ومالهاش .. ومالهاش antiderivative يعني هنلاقي
+
+665
+00:54:24,080 --> 00:54:30,660
+function Fإنها تكون integrable لكن مالناجيش
+
+666
+00:54:30,660 --> 00:54:34,540
+function F اللي ال derivative تبعي تاش بتساوي
+
+667
+00:54:34,540 --> 00:54:39,860
+بتساوي F هذا اللي هو بتشوفوه في الـ 7 3 2 و 7 3 5
+
+668
+00:54:39,860 --> 00:54:43,340
+الأمثلة أو ال counter examples اللي وجودة هنا طيب
+
+669
+00:54:43,340 --> 00:54:47,710
+الملاحظة التانيةالـ function may have an
+
+670
+00:54:47,710 --> 00:54:52,830
+antiderivative يعني عندي F و بتحقق F و في عندي F
+
+671
+00:54:52,830 --> 00:54:57,310
+كابتال كمان و F برايم إيش بتساوي بسوء F but اللي
+
+672
+00:54:57,310 --> 00:55:02,210
+هي ال integration لل F إيش ماله does not exist لكن
+
+673
+00:55:02,210 --> 00:55:07,370
+but not integrable وهذا بتجاوب عليه برضه سبعة
+
+674
+00:55:07,370 --> 00:55:12,580
+تلاتة خمسة يعني في function Fفي إلها F' بساوة F
+
+675
+00:55:12,580 --> 00:55:17,640
+لكن ال integration للـ F إشماله does not exist
+
+676
+00:55:17,640 --> 00:55:24,900
+تلاتة عندي الملاحظة الثالثة a continuous function
+
+677
+00:55:24,900 --> 00:55:27,800
+always have antiderivative طبعا ال continuous
+
+678
+00:55:27,800 --> 00:55:35,290
+function أصلا من القوة بمكان إنهاتخلّي عندها يكون
+
+679
+00:55:35,290 --> 00:55:39,490
+لها الـ mean antiderivative الان a continuous
+
+680
+00:55:39,490 --> 00:55:44,190
+function always have antiderivative وهذا اللي هو
+
+681
+00:55:44,190 --> 00:55:51,910
+مباشرة من ال corollary 7 3 4 اللي كانت F
+
+682
+00:55:51,910 --> 00:55:56,770
+continuous مدام F continuous إذا ال integration من
+
+683
+00:55:56,770 --> 00:56:07,120
+ال X من A لعند ال XA of T DT بساوة F of X اللي هي
+
+684
+00:56:07,120 --> 00:56:10,900
+is differentiable ومش differentiable كمان و F
+
+685
+00:56:10,900 --> 00:56:14,860
+prime of X بتساوي F small of X يعني مدام أف
+
+686
+00:56:14,860 --> 00:56:17,880
+continuous إذا صار لها antiderivative صارت ال F
+
+687
+00:56:17,880 --> 00:56:20,580
+اللي هي ال derivative لها بساوة F of X إذا صارت
+
+688
+00:56:20,580 --> 00:56:24,620
+إيش antiderivativeهد مين هذا اللي هو عبارة عن الـ
+
+689
+00:56:24,620 --> 00:56:28,980
+Corollary 7 3 4 مدام أن F continuous إذا هذه
+
+690
+00:56:28,980 --> 00:56:33,360
+الدالة اللي سميتها F of X is differentiable ومش هي
+
+691
+00:56:33,360 --> 00:56:35,540
+كمان هذا الجزء الثاني من ال fundamental theorem of
+
+692
+00:56:35,540 --> 00:56:39,200
+calculus و ال F prime of X إيش بيساوي F of X يعني
+
+693
+00:56:39,200 --> 00:56:44,280
+ال F capital is an antiderivative of the F small
+
+694
+00:56:44,280 --> 00:56:50,760
+اللي هو النقطة الرابعة the indefinite integralmay
+
+695
+00:56:50,760 --> 00:56:57,520
+not be an antiderivative of mean of F الآن الـ
+
+696
+00:56:57,520 --> 00:57:02,400
+indefinite integral اللي
+
+697
+00:57:02,400 --> 00:57:12,000
+هو من A لعند X F of T DT لو سميناه F of X هذاهذا
+
+698
+00:57:12,000 --> 00:57:15,480
+the indefinite integral هذا ممكن يكون هذا يعني إيش
+
+699
+00:57:15,480 --> 00:57:18,480
+بمعنى آخر؟ يعني بدنا نفترض إن ال integration من A
+
+700
+00:57:18,480 --> 00:57:22,580
+ل X F of T DT موجود طبعا اللي بقول عنه موجود بس
+
+701
+00:57:22,580 --> 00:57:25,040
+تكون F of T is integrable ماحاجيناش على control U
+
+702
+00:57:25,040 --> 00:57:29,000
+of T ال F of T is integrable إذا ال A لعندي ال X F
+
+703
+00:57:29,000 --> 00:57:35,100
+of T DT لو ممكن يكون هذا موجود و بنسمي F of X
+
+704
+00:57:35,100 --> 00:57:39,830
+بقوللي the indefinite integral هذاmay not have
+
+705
+00:57:39,830 --> 00:57:44,490
+antiderivative of F يعني ممكن مايكونش هذا بالرغم
+
+706
+00:57:44,490 --> 00:57:48,610
+انه هذا موجود مايكونش ال F capital اللي هو هذا
+
+707
+00:57:48,610 --> 00:57:53,530
+مايكونش antiderivative لمين لل F small يعني رغم
+
+708
+00:57:53,530 --> 00:57:58,670
+انه هذا موجود ليس شرطا انه يكون يتفاضل ولمّا
+
+709
+00:57:58,670 --> 00:58:03,290
+يتفاضل يطلع اللي جوا يعني مش شرط انها تكون ال F
+
+710
+00:58:03,890 --> 00:58:07,170
+اللي هي antiderivative اللي هي مين هو الـ
+
+711
+00:58:07,170 --> 00:58:10,110
+indefinite integral هيك بتقول المواحدة طب إيش
+
+712
+00:58:10,110 --> 00:58:14,150
+دليلكم دليلنا هاي في counter examples هاي سبعة
+
+713
+00:58:14,150 --> 00:58:19,390
+تلاتة أربعة وهذا ناتب ناجم ناجم عدم تحقق هذه أو
+
+714
+00:58:19,390 --> 00:58:23,910
+عدم تحقق ال antiderivative أنه it may fail to be
+
+715
+00:58:23,910 --> 00:58:28,050
+differentiable at points at the interval ممكن أصلا
+
+716
+00:58:28,050 --> 00:58:35,440
+أن ال F prime تكونش موجودة بالمرةعند نقطة اللي هو
+
+717
+00:58:35,440 --> 00:58:39,620
+في داخل الـ interval من a إلى x مزام هذي is not
+
+718
+00:58:39,620 --> 00:58:42,720
+differentiable كيف بدنا نلاقي f prime بتساوي f
+
+719
+00:58:42,720 --> 00:58:48,640
+small إذا قد ينشأ عدم وجود الـ antiderivative منه
+
+720
+00:58:48,640 --> 00:58:52,640
+إنه هذي ما تكونش differentiable عند نقطة بين ال a
+
+721
+00:58:52,640 --> 00:58:59,090
+و ال x هاي أولاللي هو سبب ممكن يؤدي إلى عدم تحقق
+
+722
+00:58:59,090 --> 00:59:03,150
+إنه يكون الـ Indefinite Integral ليس Anti
+
+723
+00:59:03,150 --> 00:59:09,650
+-derivative للدالة اللي إحنا بنكملها Or الـ
+
+724
+00:59:09,650 --> 00:59:12,850
+Derivative of Indefinite Integral ممكن يكون موجود
+
+725
+00:59:13,450 --> 00:59:18,310
+لكن يختلف عن مين but different from the value of F
+
+726
+00:59:18,310 --> 00:59:22,410
+at any point of the interval وهذا بيجيبوا سبعة
+
+727
+00:59:22,410 --> 00:59:26,990
+تلاتة تمانية يعني بقولك ممكن تكون يا محلاها و تجيب
+
+728
+00:59:26,990 --> 00:59:32,990
+اللي هو ال F prime of X لكن ما تطلع شاشة تساوي F
+
+729
+00:59:32,990 --> 00:59:38,780
+of Xففي هذه الحالة بيكون الـ Indefinite Integral
+
+730
+00:59:38,780 --> 00:59:46,320
+ليس اللي هو إيه شماله له Antiderivative عارفش
+
+731
+00:59:46,320 --> 00:59:55,080
+لحاله و هاي في مثال طبعا نأتي
+
+732
+00:59:55,080 --> 01:00:01,300
+الآن و نبدأ حديثعن الهو evaluation of integrals
+
+733
+01:00:01,300 --> 01:00:07,780
+evaluation of integrals هناخد اللي هو اللي هي اللي
+
+734
+01:00:07,780 --> 01:00:12,660
+كنا نقول عنها اللي هو integration by parts اللي
+
+735
+01:00:12,660 --> 01:00:17,380
+كنا نقول عنه integration by parts نشوف كيف اللي هو
+
+736
+01:00:17,380 --> 01:00:21,360
+البرهن الأول واحدة يعني هناخد اللي هو النظرية
+
+737
+01:00:21,360 --> 01:00:28,430
+الأولى اللي هي في من ضمن اللي هيكيف يمكن أن نجد
+
+738
+01:00:28,430 --> 01:00:33,770
+طرق لإيجاد التكامل؟ evaluation of integrals أول
+
+739
+01:00:33,770 --> 01:00:39,730
+شيء نحكي عنه هو integral by parts التكامل بالتجزيم
+
+740
+01:00:41,640 --> 01:00:48,040
+نضع الأسس النظرية للتكامل بالتجزئة أو نشرع التكامل
+
+741
+01:00:48,040 --> 01:00:50,920
+بالتجزئة من خلال النظرية اللي أمامنا، أيش بتقول
+
+742
+01:00:50,920 --> 01:00:54,720
+ليه؟ إذا كانت f و g من a و b، f small و g small
+
+743
+01:00:54,720 --> 01:00:58,520
+طبعا، لعند R، R integrable on a و b، نفترض إن
+
+744
+01:00:58,520 --> 01:01:02,040
+التنتين إما اللي هين integrable وبدنا نفترض إنه
+
+745
+01:01:02,040 --> 01:01:06,510
+لهين antiderivativesالان بدأ افرض ان F لها
+
+746
+01:01:06,510 --> 01:01:09,430
+antiderivative اسمها F capital و G لها
+
+747
+01:01:09,430 --> 01:01:13,910
+antiderivative اسمها G capital on A و B اذا الان
+
+748
+01:01:13,910 --> 01:01:22,050
+نظريتنا ترتكز في معطياتها على ان ال F small ل F و
+
+749
+01:01:22,050 --> 01:01:24,210
+G integrable هي واحد
+
+750
+01:01:36,840 --> 01:01:45,420
+ثم عرض علينا الآن نعمل integration ثم بنيجي بقول
+
+751
+01:01:45,420 --> 01:01:52,610
+ليهالـ Integration لحاصل ضرب دالتين F of X في G of
+
+752
+01:01:52,610 --> 01:01:58,890
+X DX من A لعين Bلأن طيب هذه Integrable عارفين طب F
+
+753
+01:01:58,890 --> 01:02:01,090
+capital Integrable اه Integrable لإنها
+
+754
+01:02:01,090 --> 01:02:03,930
+differentiable لإنها differentiable يعني
+
+755
+01:02:03,930 --> 01:02:07,370
+continuous أكتر من continuous يعني يعني أكيد إيش
+
+756
+01:02:07,370 --> 01:02:11,250
+ما لها Integrable إذا معقول كلامه من A لعند B F of
+
+757
+01:02:11,250 --> 01:02:15,590
+X G of X DX إيش بتساوي؟ بقولي عبارة عن F capital
+
+758
+01:02:15,590 --> 01:02:21,750
+of B G capital of B نقص F capital of A G capital
+
+759
+01:02:21,750 --> 01:02:31,040
+of A هذاناقص الـ integration F small of X G
+
+760
+01:02:31,040 --> 01:02:37,520
+capital of X dx من A لعندي هذي اللي هي ال
+
+761
+01:02:37,520 --> 01:02:41,240
+integration by parts اللي عامة اللي بتدخلته أو
+
+762
+01:02:41,240 --> 01:02:45,960
+اللي هي اللي بتقول عنه النظريةإذا if f و g من a
+
+763
+01:02:45,960 --> 01:02:49,260
+لعن بي are integrable on a و b and have
+
+764
+01:02:49,260 --> 01:02:52,140
+antiderivatives f capital and g capital on a و b
+
+765
+01:02:52,140 --> 01:02:56,580
+then ال integration من a لعن بي f of x g x dx
+
+766
+01:02:56,580 --> 01:03:00,080
+بيسوّل f of b g of b نقص f of a g of a طبعاً
+
+767
+01:03:00,080 --> 01:03:06,360
+للcapitals نقص ال integration من a لb f small g
+
+768
+01:03:06,360 --> 01:03:12,200
+capital dx ماشي الحال بدنا الآن اللي هو البرهن
+
+769
+01:03:12,200 --> 01:03:17,750
+النظري البرهن أسهل يا جماعةهو برهانة يعتمد على
+
+770
+01:03:17,750 --> 01:03:22,970
+تفاضل اللي هو حاصل ضرب دلتين ويمكن احنا كنا اصلا
+
+771
+01:03:22,970 --> 01:03:27,110
+في اتناء اول ما بنبدأ في ال integration by bars
+
+772
+01:03:27,110 --> 01:03:31,870
+سنة عارفه للطلاب في ال calculus في الواقع في كل
+
+773
+01:03:31,870 --> 01:03:36,490
+سؤال في الأول يعني و كأننا بنبرهن النظرية و بعد
+
+774
+01:03:36,490 --> 01:03:40,170
+ذلك بنسير اللي هو حافظنا الطريقة و نعملها بشكل
+
+775
+01:03:40,170 --> 01:03:48,220
+سريع طيب نشوف كيفسمّيلي اللي هو F capital of X G
+
+776
+01:03:48,220 --> 01:03:52,060
+of capital of X ايش بتساوي H of X طبعاً F is
+
+777
+01:03:52,060 --> 01:03:53,840
+differentiable و G is differentiable زي ما كنا
+
+778
+01:03:53,840 --> 01:03:56,600
+كاتبين هذه لإنه F antiderivative و G
+
+779
+01:03:56,600 --> 01:03:59,860
+antiderivative يعني F prime لها F small و G prime
+
+780
+01:03:59,860 --> 01:04:03,700
+لها G small يعني إذا صارت الـH عبارة عن
+
+781
+01:04:03,700 --> 01:04:07,160
+differentiable و مدام differentiable إذا أكيد الـH
+
+782
+01:04:07,160 --> 01:04:12,340
+شمالها continuous على الفترة A و B ماشي الحال و مش
+
+783
+01:04:12,340 --> 01:04:18,630
+هيك كمانوبنقدر نفضلها شوفوا كيف
+
+784
+01:04:22,920 --> 01:04:26,280
+واضحة كلها .. اه واضحة كلها .. ماشي زي ما قلنا يا
+
+785
+01:04:26,280 --> 01:04:29,440
+جماعة سمينا F capital of X في G capital of X
+
+786
+01:04:29,440 --> 01:04:32,240
+بالساوية H of X F differentiable و G
+
+787
+01:04:32,240 --> 01:04:35,440
+differentiable ثم H differentiable ومن ثم اكيد
+
+788
+01:04:35,440 --> 01:04:39,780
+continuous فضولي H prime of X لو حاصل ضرب دالتين
+
+789
+01:04:39,780 --> 01:04:44,740
+التفاضل الأول في التاني زي اللي هو التاني الأول في
+
+790
+01:04:44,740 --> 01:04:49,840
+التفاضل التاني ويساويتفاضل الـ F' اللي هي F small
+
+791
+01:04:49,840 --> 01:04:54,440
+وهذه تنزل زي ما هي وتفاضل هذه اللي هي G small لذا
+
+792
+01:04:54,440 --> 01:04:59,020
+صارت الـ H' بالساوية هنا يعني وكأنه صار عنده اللي
+
+793
+01:04:59,020 --> 01:05:04,360
+هي الـH هذه بدون الـPrime antiderivative لمن؟ لهذه
+
+794
+01:05:04,360 --> 01:05:09,520
+صارت يعني الـH capital is an antiderivative of FG
+
+795
+01:05:09,520 --> 01:05:17,320
+زي مين زي FGالان كل شيء نيح F وG Integrable و F
+
+796
+01:05:17,320 --> 01:05:21,560
+capital وG continuous اذا اللي هي اكيد F capital
+
+797
+01:05:21,560 --> 01:05:25,520
+وG capital Integrable ومن ثم هيطلع كل هذا اشماله
+
+798
+01:05:25,520 --> 01:05:31,920
+is integrable ماشي كمان مرة F
+
+799
+01:05:33,190 --> 01:05:36,910
+Integrable و G-Integrable و G-Capital و F-Capital
+
+800
+01:05:36,910 --> 01:05:40,450
+continuous إذا صار الـ Integrable إذا هذا كله على
+
+801
+01:05:40,450 --> 01:05:44,830
+بعض إيش ماله صار Integrable إذا صار عندي الآن اللي
+
+802
+01:05:44,830 --> 01:05:49,550
+هي كل الشروط متحققة إذا من سبعة تلاتة مدامة ال H
+
+803
+01:05:50,390 --> 01:05:53,570
+الـ H وينها الـ H ؟ هي الـ Antiderivative لهذه
+
+804
+01:05:53,570 --> 01:05:57,470
+وهذه Integrable فبصير عندى الان ال integration من
+
+805
+01:05:57,470 --> 01:06:01,770
+هنا ب F في G زايد F capital في G DX بتساوي H of B
+
+806
+01:06:01,770 --> 01:06:11,650
+ناقص مين ناقص H of A طيب صار عندى الان الأمور وضحت
+
+807
+01:06:11,650 --> 01:06:20,490
+وطلعت النتيجة H of B اتطلع عليها وين H of B؟ هيهاH
+
+808
+01:06:20,490 --> 01:06:26,610
+of B بيساوي F capital of B في G
+
+809
+01:06:26,610 --> 01:06:30,930
+of B هايها ماشي F of B في G of B عادلة لو انتوا
+
+810
+01:06:30,930 --> 01:06:36,070
+موجودين لأ على طول قلتولي هايها الان H of A بيساوي
+
+811
+01:06:36,070 --> 01:06:41,980
+F of A في G of A هايها G of A ماشي إذا إيديناهذا
+
+812
+01:06:41,980 --> 01:06:46,060
+المقدار أعوضنا عن H of B هيها وعوضنا عن H دي ها
+
+813
+01:06:46,060 --> 01:06:49,960
+ويها وإيش جينا وهذه فصلناها لإنه Integrable إذا
+
+814
+01:06:49,960 --> 01:06:52,940
+فصلناها بنجيب واحدة على الجهة دي وواحدة على الجهة
+
+815
+01:06:52,940 --> 01:06:57,940
+دي وهو المطلوب اعملكم إياها طيب ال integration من
+
+816
+01:06:57,940 --> 01:07:01,780
+A ل B integration
+
+817
+01:07:01,780 --> 01:07:10,680
+من A ل B FG capital زائد F capital في G بساوي
+
+818
+01:07:11,790 --> 01:07:24,730
+H of B هي H of B H of B سوى F of B G of B نقص H of
+
+819
+01:07:24,730 --> 01:07:31,770
+A هي F of AG of A من الجهة الثانية بيسوي ال
+
+820
+01:07:31,770 --> 01:07:35,890
+integration F في G capital زاد ال integration F
+
+821
+01:07:35,890 --> 01:07:41,070
+capital في G small من A لعند B من A لعند B لأن إن
+
+822
+01:07:41,070 --> 01:07:44,490
+قلولي هذا على الجهة هذه بيصير عند ال integration
+
+823
+01:07:44,490 --> 01:07:51,350
+المطلوب هذاأكيد فهمتم الأصل من a لعن بيه بساوي ال
+
+824
+01:07:51,350 --> 01:07:55,570
+integration f في g capital زي f capital في g small
+
+825
+01:07:55,570 --> 01:08:06,170
+لأ خلصنا منه بساوي f of b في f of a في g of a في b
+
+826
+01:08:06,170 --> 01:08:16,030
+نقص f of a في g of a كل هذاهذا المقدار integration
+
+827
+01:08:16,030 --> 01:08:27,230
+of capital في G من A لـ B وهو المطلوب و هيك احنا
+
+828
+01:08:28,050 --> 01:08:32,490
+بنكون اليوم ان شاء الله وصلنا لعند ال first
+
+829
+01:08:32,490 --> 01:08:36,250
+substitution theorem المرة القادمة ان شاء الله
+
+830
+01:08:36,250 --> 01:08:45,010
+بنكمل المحاضرة او بنكمل المادة و بنكمل هذا ال
+
+831
+01:08:45,010 --> 01:08:49,330
+section في محاضرة قادمة ان شاء الله و إلى لقائنا
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..8cf74504e7a08f953b4fc9de0fb1fa38ac2a30c2
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/EvAUsrqrpFk_raw.srt
@@ -0,0 +1,3332 @@
+1
+00:00:04,910 --> 00:00:11,030
+بسم الله الرحمن الرحيم المحاضرة الرابعة بعد حالة
+
+2
+00:00:11,030 --> 00:00:19,410
+الطوارئفي مادة او مساق تحليل لرياضي 2 او تحليل
+
+3
+00:00:19,410 --> 00:00:23,970
+حقيقي 2 لطلبة و طالبات الجامعة الإسلامية كلية
+
+4
+00:00:23,970 --> 00:00:30,110
+العلوم قسم الرياضيات وهي المحاضرة رقم 14 في المادة
+
+5
+00:00:30,110 --> 00:00:35,970
+او في المساق عنوان المحاضرة اليوم اللي هو
+
+6
+00:00:35,970 --> 00:00:43,260
+fundamental theorem ofof calculus بشقيها شق تفاضل
+
+7
+00:00:43,260 --> 00:00:50,640
+التكامل وشق تكامل التفاضل roughly نبدأ اللي هو في
+
+8
+00:00:50,640 --> 00:00:54,960
+النظرية الأولة أو الجزء الأول من النظرية اللي هي
+
+9
+00:00:54,960 --> 00:00:58,760
+ال first form ال fundamental theorem of calculus
+
+10
+00:00:58,760 --> 00:01:04,870
+اللي هي الجزء الأول بتقول ما يليlet f from a و b
+
+11
+00:01:04,870 --> 00:01:08,430
+لعند R be an integrable function on a و b يعني
+
+12
+00:01:08,430 --> 00:01:12,070
+فرضنا أن الـ function f عبارة عن دالة قابلة
+
+13
+00:01:12,070 --> 00:01:17,190
+للتكامل على الفترة المغلقة a و b and let f capital
+
+14
+00:01:17,190 --> 00:01:22,190
+من a و b لعند R تحقق الشروط التاليةتحقق أول شيء
+
+15
+00:01:22,190 --> 00:01:25,110
+أنها تكون الاف كابتنال هادي continuous على الفترة
+
+16
+00:01:25,110 --> 00:01:30,530
+a و b الشرط الثاني f prime exists and f prime of x
+
+17
+00:01:30,530 --> 00:01:34,130
+بساوي الاف small اللي بدأنا فيها دي الفرض منها
+
+18
+00:01:34,130 --> 00:01:39,880
+integrable لكل x element in a و bبنقول الـ
+
+19
+00:01:39,880 --> 00:01:44,900
+integration من A لB للـ F بساوي F of B ناقص F of A
+
+20
+00:01:44,900 --> 00:01:50,400
+إذا الآن و كإني بقول نفترض أنه الـ F small
+
+21
+00:01:50,400 --> 00:01:56,100
+integrable و نفترض F continuous على closed bounded
+
+22
+00:01:56,100 --> 00:02:01,160
+interval A و B و F نفسها differentiable على الـ
+
+23
+00:02:01,160 --> 00:02:05,260
+open اللي هو interval A و B بشرط أنه الـ F prime
+
+24
+00:02:05,260 --> 00:02:07,940
+ال derivative لها الـ F capital طبعا بساوي الـ F
+
+25
+00:02:07,940 --> 00:02:13,240
+small of Xوبناء على كل المعطيات اللي حكيناها بيكون
+
+26
+00:02:13,240 --> 00:02:17,120
+ال integration من a لb لل f small هو عبارة عن fb
+
+27
+00:02:17,120 --> 00:02:21,760
+ناقص f of a او اختصارا واسهلك للحفظ ال integration
+
+28
+00:02:21,760 --> 00:02:29,670
+من a لb لل f prime of x dx هي عبارة عنوكأنه
+
+29
+00:02:29,670 --> 00:02:36,250
+التكامل بلغ التفاضل بيصير اللي هي f of b ناقص f of
+
+30
+00:02:36,250 --> 00:02:40,610
+a هذا فضوء mean يعني هاي هادة باختصار هيك فضوء
+
+31
+00:02:40,610 --> 00:02:44,710
+mean أن f prime of x تكون موجودة على الفترة من a و
+
+32
+00:02:44,710 --> 00:02:51,370
+b و أيضا بتكون انهاش integrable نفسها integrable و
+
+33
+00:02:51,370 --> 00:02:56,110
+تكون ال f نفسها continuous على close من a ل عند b
+
+34
+00:02:56,390 --> 00:03:00,270
+هذه هي النظرية إذا اللي بدنا نثبته الآن أنه قيمة
+
+35
+00:03:00,270 --> 00:03:04,470
+ال integration هذا بساوي اللي هو اللي أمامنا اللي
+
+36
+00:03:04,470 --> 00:03:09,590
+هي f of b ناقص من ناقص f of a خلينا نشوف إيش
+
+37
+00:03:09,590 --> 00:03:15,030
+البكاء في البحران يا جماعةالان اللي واضح انه انا
+
+38
+00:03:15,030 --> 00:03:18,530
+بده اثبت انه ال integration هو عبارة عن f of b
+
+39
+00:03:18,530 --> 00:03:24,990
+ناقص f of a شوف انا كيف بده اوصلك اللي هو ال
+
+40
+00:03:24,990 --> 00:03:28,670
+integration لمنقلة ال b لل f بساوي f of b ناقص f
+
+41
+00:03:28,670 --> 00:03:32,010
+of a انا بعرف اول حاجة هو معطيلي ال f small هذه
+
+42
+00:03:32,010 --> 00:03:35,290
+اللي هي ال f prime اللي هي ال f small سميتها هذه
+
+43
+00:03:35,290 --> 00:03:39,240
+معطينيها integrallyMadame Integrable إذن بواسطة
+
+44
+00:03:39,240 --> 00:03:44,160
+اللي هي .. إذا بتتذكروا Integrable criterion وهي
+
+45
+00:03:44,160 --> 00:03:46,900
+اللي بنستخدمها كثير احنا لإنها .. يعني خلّيني أقول
+
+46
+00:03:46,900 --> 00:03:51,400
+صيغة مهمة في إثبات النظريات اللي هي بما إنه أف
+
+47
+00:03:51,400 --> 00:03:56,380
+Integrable إذن لكل Y أكبر من 0 يوجد there exists a
+
+48
+00:03:56,380 --> 00:04:00,600
+partition B اللي هو X0 X1 عند X ل .. عند ..
+
+49
+00:04:00,600 --> 00:04:03,280
+partition لمين طبعاً للفترة اللي أنتوا عارفينها
+
+50
+00:04:03,280 --> 00:04:06,800
+اللي احنا بنشتغل عليها الفترة من A لعند مين لعند B
+
+51
+00:04:06,960 --> 00:04:12,920
+There exists a partition B X0 X1 XN of A و B such
+
+52
+00:04:12,920 --> 00:04:20,280
+that ال other sum B F'-LB F' أصغر من إبسلون F' من
+
+53
+00:04:20,280 --> 00:04:24,000
+هنا من هي يا جماعة؟ هي ال F لأننا مفترضين ال F
+
+54
+00:04:24,000 --> 00:04:29,940
+small أو ال F' is integrable إذا بواسطة الرماء ال
+
+55
+00:04:29,940 --> 00:04:34,760
+integrable criterion حصلنا على واحد اللي عنده
+
+56
+00:04:34,760 --> 00:04:40,880
+إلحاةأحكي عنها الآن أو أستخدمها للوصول لهدفي طيب
+
+57
+00:04:40,880 --> 00:04:44,040
+because طبعا F prime اللي هي meme زي ما قلنا F of
+
+58
+00:04:44,040 --> 00:04:48,820
+X is integrable on A أو B الآن في شغلة تانية احنا
+
+59
+00:04:48,820 --> 00:04:55,880
+بدأ الآن أستخدم اللي هو أمر آخر خلينا نكتب الآن
+
+60
+00:04:55,880 --> 00:05:00,920
+أول شغلة حصلناها عشان تعرف أين أنا رايح اللي هي
+
+61
+00:05:00,920 --> 00:05:06,550
+قلنا F is integrableF التي بيسمى F' Integrable أيش
+
+62
+00:05:06,550 --> 00:05:12,270
+أعطتنا يا جماعة؟ أعطتنا أن هناك بارتيشن B هناك B
+
+63
+00:05:12,270 --> 00:05:15,830
+طبعا for every epsilon أكبر من سفر هناك بارتيشن B
+
+64
+00:05:15,830 --> 00:05:23,390
+such that U, B و F اللي هي الـ F' طبعا الـ F يمين
+
+65
+00:05:23,390 --> 00:05:29,530
+الـ F' ناقص الـ B و F هي أصغر من Y وهذا سمناها
+
+66
+00:05:29,530 --> 00:05:34,710
+إيش؟ سمناها 1لأن من جهة أخرى يا جماعة بدي أستغل
+
+67
+00:05:34,710 --> 00:05:37,810
+اللي هو اللي معطينيه أن الأفكار بتلقوا continuous
+
+68
+00:05:37,810 --> 00:05:41,490
+أنا الآن زي ما قلنا جزاقنا اللي هي الفترة A و B
+
+69
+00:05:41,490 --> 00:05:52,230
+إلىاللي هي X0 X1 X2 فترة نموذجية Xk-1 لعند Xk لعند
+
+70
+00:05:52,230 --> 00:05:56,010
+مين لعند Xn اللي هي مين الـ B هذا ال partition
+
+71
+00:05:56,010 --> 00:05:59,930
+اللي لجيته أنا الآن أنا بعرف أن ال F capital
+
+72
+00:05:59,930 --> 00:06:03,330
+continuous على ال closed interval كلها من A لعند B
+
+73
+00:06:03,330 --> 00:06:06,870
+إذا الأكيد continuous على كل sub interval موجودة
+
+74
+00:06:06,870 --> 00:06:12,540
+يعني صارت اللي عندي ال F capital continuouson a
+
+75
+00:06:12,540 --> 00:06:18,560
+التي هي xk-1 وعند xk هذه هي ال sub interval اللي
+
+76
+00:06:18,560 --> 00:06:24,080
+ال function f continuous عليها وإحنا مفترضين أيضا
+
+77
+00:06:24,080 --> 00:06:29,220
+أن ال f prime exist على ال open interval كلها إذا
+
+78
+00:06:29,220 --> 00:06:33,860
+أكيد ال fis differentiable لأن f' exist على ال
+
+79
+00:06:33,860 --> 00:06:37,360
+open interval من a لعند b إذا أكيد f is
+
+80
+00:06:37,360 --> 00:06:41,980
+differentiable على الفترة xk minus واحد لعند مين
+
+81
+00:06:41,980 --> 00:06:45,140
+لعند ال xk يعني يا جماعة شروط ال mean value
+
+82
+00:06:45,140 --> 00:06:50,020
+theorem محققة إذا أكيد there existthere exist بما
+
+83
+00:06:50,020 --> 00:06:54,540
+انه اللي هي شروط ال mean value محققة اذا there
+
+84
+00:06:54,540 --> 00:06:59,620
+exist بسميها اللي هو bk او tk there exist tk في
+
+85
+00:06:59,620 --> 00:07:06,440
+الفترة xk minus واحد عند ال xk such that اللي هو F
+
+86
+00:07:06,440 --> 00:07:11,520
+هي الدلل اللي بطبق عليها ال mean value theorem يا
+
+87
+00:07:11,520 --> 00:07:19,320
+جماعة F of xk ناقص F of xk minus واحداللي هو
+
+88
+00:07:19,320 --> 00:07:24,560
+بتساوي اللي هو xk ناقص xk مايلوس واحد طبعا احنا
+
+89
+00:07:24,560 --> 00:07:29,280
+بنكسب عليها اللي هي ايش بتساوي اللي هي ال F prime
+
+90
+00:07:30,370 --> 00:07:35,630
+of الهي main TK هذه اللي هي الـ Main Value Theorem
+
+91
+00:07:35,630 --> 00:07:39,710
+نتجتها مطبق على مين يا جماعة على الفترة اللي هي XK
+
+92
+00:07:39,710 --> 00:07:43,770
+minus واحد لعند XK طبعا هذا الكلام لكل K لأنه لكل
+
+93
+00:07:43,770 --> 00:07:48,170
+مين لكل ال sub intervals بتتحقق نفس الشروط بناء
+
+94
+00:07:48,170 --> 00:07:55,010
+عليه احنا بنعرف قبل هيك خليني أبدأ أشتغلها عند ال
+
+95
+00:07:55,010 --> 00:08:04,450
+MK إبراهيماللي هي أكيد أصغر أو يساوي الـ Fاللي هي
+
+96
+00:08:04,450 --> 00:08:11,370
+ال F عندك ال mk' ايش نذاكركم فيها ايش هي ال mk' ال
+
+97
+00:08:11,370 --> 00:08:15,190
+mk' هي مكتوبة لنكتبها كمان مرة ال mk' هي مين ال
+
+98
+00:08:15,190 --> 00:08:19,610
+infimum لل F' F of X X element of X K minus واحد X
+
+99
+00:08:19,610 --> 00:08:24,950
+K ال F' مين هذه ال F small ماشي الحال طيب ال F او
+
+100
+00:08:24,950 --> 00:08:30,590
+ال mk' هي عبارة عن ال infimum لمين لل F' على ال
+
+101
+00:08:30,590 --> 00:08:37,110
+sub interval هذه اذا اكيدأصغر أو يساوي الـ F of أي
+
+102
+00:08:37,110 --> 00:08:41,630
+مقطوين موجودة في الـ sub-interval والـ TK هدوين
+
+103
+00:08:41,630 --> 00:08:46,410
+موجودة في هذا الـ sub-interval إذن أكيد F of TK
+
+104
+00:08:46,410 --> 00:08:50,670
+أصغر من الـ MK' لأن الـ MK' هي عبارة عن الـ
+
+105
+00:08:50,670 --> 00:08:57,290
+infimum لقيمة الدالة F' of TK عالمياً على الفترة
+
+106
+00:08:57,290 --> 00:09:01,440
+اللي هي ال sub-interval اللي بنحكي عنها ككلوهذه
+
+107
+00:09:01,440 --> 00:09:06,040
+أكيد أصغر أو يساوي من الـ MK Ibrahim مين الـ MK
+
+108
+00:09:06,040 --> 00:09:11,060
+Ibrahim هذه؟ هي عبارة عن الـ Supremum لمين؟ لقيمة
+
+109
+00:09:11,060 --> 00:09:16,990
+اللي هي الدالة هذه على الفترة اللي عندهماشي الحال
+
+110
+00:09:16,990 --> 00:09:23,110
+إذا هذه دائما صحيحة طب لو ضربت جهتي كل الأطراف في
+
+111
+00:09:23,110 --> 00:09:27,650
+طول الفترة اللي هي xk minus xk minus واحد طبعا
+
+112
+00:09:27,650 --> 00:09:31,510
+أكيد هذه xk ناقص xk minus واحد كمية موجبة إذا لما
+
+113
+00:09:31,510 --> 00:09:35,010
+أضربها في الأطراف كلها بتظلها ال inequality صحيحة
+
+114
+00:09:35,010 --> 00:09:40,610
+وزي ما هي في xk ناقص xk minus واحد وهنا في xk ناقص
+
+115
+00:09:40,610 --> 00:09:46,280
+xk minus واحد مظبوط هذا الكلام أكيد صحطيب لكن هذا
+
+116
+00:09:46,280 --> 00:09:52,220
+المقدار هو هذا إذا باجي بحط هذه هان معايا جماعة
+
+117
+00:09:52,220 --> 00:09:57,000
+بحط هذه في المنطقة هذه المكان القيمة هذه اللي هي
+
+118
+00:09:57,000 --> 00:10:05,190
+هذه إذا صار عندي الأن Fof xk ناقص f of xk minus 1
+
+119
+00:10:05,190 --> 00:10:12,070
+أكبر أو يساوي ال Mk' في ال xk minus xk minus 1
+
+120
+00:10:12,070 --> 00:10:18,490
+وهذا أصغر أو يساوي ال Mk' في ال xk minus xk minus
+
+121
+00:10:18,490 --> 00:10:24,290
+1إذا واضح إن أنا حصلت على هذه وهذه صحيحة لكل مين
+
+122
+00:10:24,290 --> 00:10:28,150
+لكل كيه لإنه أخدت أنا هذه فترة عشوائية يعني لكي
+
+123
+00:10:28,150 --> 00:10:33,790
+بتساوي اللي هو واحد واتنين لعند مين لعند الان بناء
+
+124
+00:10:33,790 --> 00:10:38,250
+عليه أنا وصلت الآن إلى ما يليه وصلت إلى هذه
+
+125
+00:10:38,250 --> 00:10:43,670
+النتيجة شايفين يا جماعة وصلت إلى هذه النتيجة اللي
+
+126
+00:10:43,670 --> 00:10:48,860
+أنا وضحتلكم إياها كيف إجتالان و ال mk prime زي ما
+
+127
+00:10:48,860 --> 00:10:55,280
+قلنا ايش هي و ال mk capital prime هيها امامكم نيجي
+
+128
+00:10:55,280 --> 00:10:59,720
+الان ايش اللي بده حصله شوف ال K ايش بده حصل انا
+
+129
+00:10:59,720 --> 00:11:03,020
+عمال بقول انه صحيح على كل K اذا لو أخدت ال
+
+130
+00:11:03,020 --> 00:11:05,900
+summation هنا و أخدت ال summation هنا و أخدت ال
+
+131
+00:11:05,900 --> 00:11:10,440
+summation هنا بظل اللي هي ال inequality صحيحة خد
+
+132
+00:11:10,440 --> 00:11:16,050
+الآن ال summationعلى الفترة الـ summation على
+
+133
+00:11:16,050 --> 00:11:19,530
+الinquality اللي حكينا عنها هيها بيصير ال
+
+134
+00:11:19,530 --> 00:11:23,630
+summation لل M K برايم في X K X K minus واحد K من
+
+135
+00:11:23,630 --> 00:11:26,630
+عند واحد لعند M وهنا هيكون أصغر أو ساوى ال
+
+136
+00:11:26,630 --> 00:11:31,090
+summation لل F of X K ناقص F of X K minus واحد
+
+137
+00:11:31,090 --> 00:11:36,190
+اللي هي احنا هنا حصلناها هان وهذه ال summation
+
+138
+00:11:36,190 --> 00:11:39,490
+عليها وهذه ال summation عليها هي ال summation هذا
+
+139
+00:11:39,850 --> 00:11:43,210
+طيب يا جماعة أكيد أنتم اذكرين هذا ال summation هو
+
+140
+00:11:43,210 --> 00:11:47,770
+عبارة عن مين يا جماعة هو عبارة عن ال lower sum لل
+
+141
+00:11:47,770 --> 00:11:52,370
+partition بي على مين على الدالة اللي بنعمل عليها
+
+142
+00:11:52,370 --> 00:11:56,330
+اللي هي ال F' اللي اشتغلنا عليها اللي أخدنا ال F'
+
+143
+00:11:57,150 --> 00:12:01,270
+ال MK' هي ال infimum لل F' على ال sub interval
+
+144
+00:12:02,160 --> 00:12:07,220
+والان هذه يشبه يا جماعة هي ال upper sum يعني بمعنى
+
+145
+00:12:07,220 --> 00:12:12,960
+آخر صار عندى الآن المقدار هذا هي المقدار هذا اللى
+
+146
+00:12:12,960 --> 00:12:17,680
+محاط بالأزرق هذا المقدار و بالأحمر صار بين اللى هو
+
+147
+00:12:17,680 --> 00:12:22,120
+ال lower sum و بين ال upper sum لكن هذا المقدار
+
+148
+00:12:22,120 --> 00:12:25,860
+الصمشي اللى احنا عملناه قبل ذلك كتير لو فرضناه
+
+149
+00:12:25,860 --> 00:12:33,860
+هيكون عبارة عن F of X واحدناقص F of X Note زائد F
+
+150
+00:12:33,860 --> 00:12:40,360
+of X 2 ناقص F of X 1 زائد F of X 3 ناقص F of X 2
+
+151
+00:12:40,360 --> 00:12:45,460
+زائد لما أصل لآخر إشي F of X N ناقص F of X N ناقص
+
+152
+00:12:45,460 --> 00:12:50,520
+1 حلاجي كله بروح و بظل عندك كل هذا بروح و بظل عندك
+
+153
+00:12:50,520 --> 00:12:54,860
+بس F of X ناقص F of X N ناقص مين F of X Note كل
+
+154
+00:12:54,860 --> 00:13:00,340
+term ب cancel اللي قبله و يساويF of Xn اللي هي
+
+155
+00:13:00,340 --> 00:13:07,420
+مين؟ F of B F of Xnot اللي هي مين؟ F of A إذا صار
+
+156
+00:13:07,420 --> 00:13:10,920
+عندي الآن الـSummation هذا اللي هو بيسوي F of B
+
+157
+00:13:10,920 --> 00:13:16,440
+نقص مين؟ نقص F of A يعني بمعنى آخر بنشيل هذا و
+
+158
+00:13:16,440 --> 00:13:20,580
+بنحط مكانه قيمته اللي هو F of B نقص F of A بيصير F
+
+159
+00:13:20,580 --> 00:13:25,320
+of B minus F of A بين اللي هو L of B و F' و بين
+
+160
+00:13:25,320 --> 00:13:34,230
+الأبقى الـU of B و F'طيب اطلع على فوق شوية خليني
+
+161
+00:13:34,230 --> 00:13:39,830
+اوضحلك هذه النقطة اللي هتوصلي للي بديها انا ايش
+
+162
+00:13:39,830 --> 00:13:44,170
+بدي اثبت انا بدي اثبت ان ال integration خليني بس
+
+163
+00:13:44,170 --> 00:13:51,710
+اشيل هذا بعد اذنكم انا
+
+164
+00:13:51,710 --> 00:13:58,260
+بدي اثبت ان ال integration لل F prime of x dxأو
+
+165
+00:13:58,260 --> 00:14:06,320
+الـ F of X DX بساوي F of B ناقص F of A من A لعند
+
+166
+00:14:06,320 --> 00:14:12,870
+مين؟ لعند B، هيا، بدي أسهلشوف الان كيف انا في
+
+167
+00:14:12,870 --> 00:14:17,590
+الواقع بتوصلكم هذا نقص هذا يكون اكبر او يساوي سفر
+
+168
+00:14:17,590 --> 00:14:20,990
+و اصغر من epsilon لأي اصغر يساوي epsilon لكل
+
+169
+00:14:20,990 --> 00:14:24,110
+epsilon في الدنيا و من ثم هيساوي سفر يعني هتحدث
+
+170
+00:14:24,110 --> 00:14:29,090
+مين المساواة اللي مافهمش عليها هي الان نجلة تفصيل
+
+171
+00:14:29,090 --> 00:14:39,060
+اتفجنا احنا انه احنا حصلنا على اللي هوالـ F of D –
+
+172
+00:14:39,060 --> 00:14:44,520
+F of A بين الـ U و بين الـ L لكن احنا بنقول الـ F
+
+173
+00:14:44,520 --> 00:14:49,420
+نفسها يا جماعة الـ F' هذه is integrable مزام
+
+174
+00:14:49,420 --> 00:14:55,380
+Integrable من A لعند Bأذن الـ U of F و الـ L of F
+
+175
+00:14:55,380 --> 00:14:58,560
+اللي هو الـ Upper Integral و الـ Lower Integral
+
+176
+00:14:58,560 --> 00:15:01,960
+اللي هو هيساوي قيمة الـ Integration عارفينوا هذا
+
+177
+00:15:01,960 --> 00:15:05,760
+الكلام احنا بنتعرف شو معناته Integrable الـ U of F
+
+178
+00:15:05,760 --> 00:15:11,640
+هي عبارة عن مين؟ عن الـ infimum لمين؟ للكبار اللي
+
+179
+00:15:11,640 --> 00:15:19,640
+هي U L أو U P و Fوهذا عبارة عن الـ Supremum لنية
+
+180
+00:15:19,640 --> 00:15:28,200
+للـ L, B و F يعني بيصير عندي اللي هو هذا عبارة عن
+
+181
+00:15:28,200 --> 00:15:32,420
+ال U of F من جهة وهذا عبارة عن ال L of F من جهة
+
+182
+00:15:32,420 --> 00:15:35,900
+أخرى هذا ال integration نقلة بيه ال F' ليش؟ لأن ال
+
+183
+00:15:35,900 --> 00:15:40,630
+F' is integrable زي ما قلناهطيب، إيش علاقة الـ L و
+
+184
+00:15:40,630 --> 00:15:44,130
+F بهذه؟ L و F عبارة عن الـ Supremum لكل هذول، إذا
+
+185
+00:15:44,130 --> 00:15:47,450
+أكيد الـ L و F أكبر أو يساوي هذه، إذا صار ال
+
+186
+00:15:47,450 --> 00:15:51,610
+integration هذا اللي عندى أكبر أو يساوي الـ L و F،
+
+187
+00:15:51,610 --> 00:15:56,070
+خلصنا من هذه، لأن من جهة ثانية ال integration لل F
+
+188
+00:15:56,070 --> 00:15:58,510
+براين هو الـ U of F، لأن الـ F is integrable زي ما
+
+189
+00:15:58,510 --> 00:16:04,770
+قلنا، و الـ U of F مين هو؟هو عبارة عن الـ infimum
+
+190
+00:16:04,770 --> 00:16:08,810
+لها دول إذا الـ U of F أصغر أو يشويها مزام أصغر أو
+
+191
+00:16:08,810 --> 00:16:12,210
+يشويها يعني صار ال integration أصغر أو يشويها يعني
+
+192
+00:16:12,210 --> 00:16:18,050
+بمعنى آخر احنا وصلنا ان ال integration تبعناالـ
+
+193
+00:16:18,050 --> 00:16:23,450
+integration اللي هو من A لعند A بي لل F برايم بين
+
+194
+00:16:23,450 --> 00:16:29,530
+ال L و بين مين و بين ال U ماشي الحالة الآن و هذا
+
+195
+00:16:29,530 --> 00:16:35,390
+برضه بين ال L و بين مين ال U اضربلي الآن اللي تحت
+
+196
+00:16:35,390 --> 00:16:41,820
+اللي هو بسالببتنعكس ال inequality وبعدين اجمعل
+
+197
+00:16:41,820 --> 00:16:46,320
+الجهتين ماشي الحال هذي بيصير سالب وهذي بيصير أكبر
+
+198
+00:16:46,320 --> 00:16:51,220
+أو يساوي سالب وهذي بيصير أكبر أو يساوي سالب اجمعل
+
+199
+00:16:51,220 --> 00:16:56,980
+الجهتين بيصير عند ال integrationأو العكس اضرب اللي
+
+200
+00:16:56,980 --> 00:16:59,720
+فوق بالسالب عساس ان اعمل زي ما عامل هو هان اضرب
+
+201
+00:16:59,720 --> 00:17:02,000
+اسف اللي هو اضرب مين اللي فوق بالسالب يصير هذا
+
+202
+00:17:02,000 --> 00:17:07,640
+ناقص وهذا كله ناقص وهذا عبارة عن ناقص وهذا اكبر او
+
+203
+00:17:07,640 --> 00:17:12,760
+يساوي وهذا اكبر او يساوي واجمح لبعض او بلغة اخرى
+
+204
+00:17:13,150 --> 00:17:16,790
+قول اللي .. اللي .. اللي .. اللي تحت ناقص اللي فوق
+
+205
+00:17:16,790 --> 00:17:20,130
+بطلع عندى ال integration لل F prime من A ل عند B
+
+206
+00:17:20,130 --> 00:17:24,730
+ناقص اللي هي F of B minus F of A أصغر يساوي ال U
+
+207
+00:17:24,730 --> 00:17:30,770
+ناقص ال L و أكبر أوي يساوي ال L ناقص ال U واضح أنه
+
+208
+00:17:30,770 --> 00:17:36,480
+هذهعبارة عن ناقص هذه وهذه كمية موجبة إذا صار عند
+
+209
+00:17:36,480 --> 00:17:41,000
+ال absolute value للقيمة هذه أصغر أو يساوي ال U
+
+210
+00:17:41,000 --> 00:17:49,300
+ناقص مين ال L كمان مرة هذه سالب هذهوهذه موجبة إذاً
+
+211
+00:17:49,300 --> 00:17:53,540
+هذه الـ Inquality صارت هي عبارة عن مفكوك الـ
+
+212
+00:17:53,540 --> 00:17:57,440
+absolute value للـ F prime ال integration ناقص
+
+213
+00:17:57,440 --> 00:18:01,840
+المقدار هذا أصغر أو يساوي هذا القيمة وهذا القيمة
+
+214
+00:18:01,840 --> 00:18:05,700
+لسه ما سحنهاش احنا إيش قلنا عنها أصغر من من إبسلان
+
+215
+00:18:05,700 --> 00:18:09,280
+وانا قاعد باشتغل على ال partition اللي لجيته أعلى
+
+216
+00:18:09,280 --> 00:18:15,390
+إذا صار عندى الآن اللي وصلتلهعبارة عن اللي هو ال
+
+217
+00:18:15,390 --> 00:18:20,150
+absolute value هذا أصغر من إبسلون وطبعا أكبر أوي
+
+218
+00:18:20,150 --> 00:18:24,790
+ساوي سفر والإبسلون واز إيه شمالها arbitrary يعني
+
+219
+00:18:24,790 --> 00:18:29,430
+المقدار هذا لأي إبسلون في الدنيا لأي إبسلون في
+
+220
+00:18:29,430 --> 00:18:33,770
+الدنيا هذا أصغر من مين من الإبسلون هو أكبر أوي سفر
+
+221
+00:18:33,970 --> 00:18:38,510
+إذاً لازم يكون هذا المقدار اللي هو عبارة عن سفر
+
+222
+00:18:38,510 --> 00:18:42,530
+يعني بمعنى آخر ال integration بساوي اللي هو F of B
+
+223
+00:18:42,530 --> 00:18:47,050
+نقص F of A و هيك و نكون حصلنا على المطلوب و أثبتنا
+
+224
+00:18:47,050 --> 00:18:51,470
+النظرية طيب
+
+225
+00:18:51,470 --> 00:18:57,850
+الجماعة الان هذا اللي هو الجزء الأولالجزء الأول من
+
+226
+00:18:57,850 --> 00:19:00,890
+اللي هو الـ Fundamental Theorem of Calculus اللي
+
+227
+00:19:00,890 --> 00:19:05,750
+هو عبارة عن تكامل التفاضل اللي هي عملية التكامل
+
+228
+00:19:05,750 --> 00:19:10,470
+هتلغى عملية التفاضل زي ما شفنا اللي هو قبل بشوية
+
+229
+00:19:10,470 --> 00:19:15,130
+وشفنا إيه اللي جالته الـ Fundamental Theorem طبعا
+
+230
+00:19:15,130 --> 00:19:20,270
+هذا الكلام تحت الشروط المذكورةالآن بقول لي .. يمكن
+
+231
+00:19:20,270 --> 00:19:24,810
+حكيت هادى حتى بقول لي corollary بقول انا أحيانا
+
+232
+00:19:24,810 --> 00:19:29,530
+بحب أريح حالي و نقول خلينا ناخد .. عشان نستذكر ال
+
+233
+00:19:29,530 --> 00:19:32,790
+.. ال .. ال .. ال fundamental theorem الجزء الأول
+
+234
+00:19:32,790 --> 00:19:37,950
+بشكل سريع بس شويه عندنا نعمل .. في الشروط ان هو قف
+
+235
+00:19:37,950 --> 00:19:41,630
+من a ل b لعند R satisfy the conditionsاللي هو F'
+
+236
+00:19:42,150 --> 00:19:45,790
+exists on A وB، مابتدش أجيب سيرة مين الـ F Small؟
+
+237
+00:19:45,790 --> 00:19:49,250
+مابتدش أشغل مين؟ على الـ F Capital لو فرضنا إن الـ
+
+238
+00:19:49,250 --> 00:19:54,010
+F هذه من A وB لعند R إنها الـ F' اللي هي الشمالها
+
+239
+00:19:54,010 --> 00:19:57,890
+exist على الـ A وB يعني فرض إنها، بس هنا زاد شوية
+
+240
+00:19:57,890 --> 00:20:01,130
+على الشروط اللي جابل بشوية إن F is differentiable
+
+241
+00:20:01,130 --> 00:20:03,790
+على الـ closed interval كنا فرضينها إنها F
+
+242
+00:20:03,790 --> 00:20:06,740
+differentiable على الـ Openوفرضين إنها continuous
+
+243
+00:20:06,740 --> 00:20:10,760
+على اللي هي الـ Closed مدام الآن إذا فيها دي سنة
+
+244
+00:20:10,760 --> 00:20:16,680
+زيادة من ال .. من اللي هي الشروط F' exists على الـ
+
+245
+00:20:16,680 --> 00:20:19,820
+A و الـ B يعني continuous و differentiable على الـ
+
+246
+00:20:19,820 --> 00:20:24,350
+closed open .. على ال closed interval A و Bالان هي
+
+247
+00:20:24,350 --> 00:20:29,550
+الشرط والشرط الثاني فرض ان ال F' مش موجودة و بس ال
+
+248
+00:20:29,550 --> 00:20:33,030
+F' انتجر بالعلى ال A و ال B إذا انتجقت شروط ال
+
+249
+00:20:33,030 --> 00:20:36,210
+fundamental theorem إذا حسب ال fundamental theorem
+
+250
+00:20:36,210 --> 00:20:39,410
+اللي قبله شوية بكون عنده ال integration من A لB
+
+251
+00:20:39,410 --> 00:20:44,190
+لمين الآنللـ f prime اللي كنا نسميه في الجبل f
+
+252
+00:20:44,190 --> 00:20:49,310
+small بتساوي f of b ناقص f of a إذا فعلا هي
+
+253
+00:20:49,310 --> 00:20:53,130
+automatic حققت شروط ال fundamental theorem وزيادة
+
+254
+00:20:53,130 --> 00:20:59,850
+سنة إذا أكيد النتيجة بتكون صحيحة يعنيتكامل الدالة
+
+255
+00:20:59,850 --> 00:21:05,210
+المتفاضلة بساوء أصل الدالة اللي هو أصل الدالة عنده
+
+256
+00:21:05,210 --> 00:21:09,070
+النقطة الأولى نقص النقطة مين بنحكي النقطة الثانية
+
+257
+00:21:09,070 --> 00:21:13,950
+اللي هو عبارة عن integration F of B نقص مين F of A
+
+258
+00:21:13,950 --> 00:21:19,210
+طب لو كانت X متغير يعني لو أخذنا أي X بين الفترتين
+
+259
+00:21:19,210 --> 00:21:23,550
+وطبّجنا نظرية على الـA اللي عند الـXفبصير ال
+
+260
+00:21:23,550 --> 00:21:27,130
+integration من إلي إلى عند ال X أف براين بساوي أف
+
+261
+00:21:27,130 --> 00:21:32,490
+of X ناقص أف of A X هذه هتصبح متغيرة والأف of A
+
+262
+00:21:32,490 --> 00:21:39,190
+ثابتة وهذا بيذكرنا انه احنا قيمة التكامل لدالة
+
+263
+00:21:39,190 --> 00:21:43,790
+بساوي اللي هو عبارة عن اللي هي الإجابة اللي هي
+
+264
+00:21:43,790 --> 00:21:47,630
+الدالة اللي بتطلع زائد some constant C اللي هو ال
+
+265
+00:21:47,630 --> 00:21:53,660
+constant طبعا لاللي هيطبيعة الديالي لقداش بيطلعلنا
+
+266
+00:21:53,660 --> 00:21:56,420
+اللي هو ال initial conditions بيسفرلنا إياه ال
+
+267
+00:21:56,420 --> 00:21:59,620
+constant أو بيعطينا إياه خمسة أربعة ستة أو بيظلوا
+
+268
+00:21:59,620 --> 00:22:03,720
+بصورة عامة Constancy طيب شوفوا صلوا على النبي عليه
+
+269
+00:22:03,720 --> 00:22:08,090
+الصلاة والسلامالآن بدنا نيجي للجزء الثاني من الـ
+
+270
+00:22:08,090 --> 00:22:11,470
+Fundamental theorem خلصنا جزء اللي هو تكامل
+
+271
+00:22:11,470 --> 00:22:16,870
+التفاضل، الآن بنتوقع إنه انفاضل التكامل، انفاضل
+
+272
+00:22:16,870 --> 00:22:20,530
+التكامل، تحت شروط، بنشوف اللي هي ال Fundamental
+
+273
+00:22:20,530 --> 00:22:24,650
+theorem of calculus second form أو اللي هو الجزء
+
+274
+00:22:24,650 --> 00:22:27,850
+الثاني من ال Fundamental theorem، إيش اللي بيقوله؟
+
+275
+00:22:27,850 --> 00:22:33,540
+اللي بيقول معايا ليه، نفترض إن F smallمن a و b
+
+276
+00:22:33,540 --> 00:22:38,060
+لعند r اللي هو is integrable in a و b إذا انفرضناه
+
+277
+00:22:38,060 --> 00:22:41,460
+انه integrable ونفرض أن f of x هو بيساوي ال
+
+278
+00:22:41,460 --> 00:22:46,160
+integration من a ل x f for all x element in a و b
+
+279
+00:22:46,160 --> 00:22:52,200
+بس لان نيجي للنتيجة thenهيطلع هذا عبارة عن
+
+280
+00:22:52,200 --> 00:22:55,360
+continuous function في ال .. في ال .. اللي هي main
+
+281
+00:22:55,360 --> 00:23:02,700
+في اللي هي على الفترة a أو b ومش هي كمان ولو كان
+
+282
+00:23:02,700 --> 00:23:09,830
+ال Fsmall continuous عند نقطة C في الفترة من A
+
+283
+00:23:09,830 --> 00:23:14,050
+لعند B أو من A لعند X هتكون ال F هذه نفسها
+
+284
+00:23:14,050 --> 00:23:16,950
+differentiable و ال derivative اللي هي إيش بساوي
+
+285
+00:23:16,950 --> 00:23:22,590
+بساوي قيمة اللي هو ال F اللي جوا يعني باختصار يا
+
+286
+00:23:22,590 --> 00:23:29,450
+جماعة بتتكررها سهل بقولك مفترض FIntegrable ماشي
+
+287
+00:23:29,450 --> 00:23:34,690
+الحال هاي بس الان بقول لو جيت اخدت من A ل X on
+
+288
+00:23:34,690 --> 00:23:44,670
+طبعا A و B لو اخدت الان A و X F small of T DT هاي
+
+289
+00:23:44,670 --> 00:23:51,510
+ده طبعا الان صارت كلها تعتمد على مين على X بقولك
+
+290
+00:23:51,510 --> 00:23:55,370
+هذههذه اللي طلع اللي هو الـ integration للـ
+
+291
+00:23:55,370 --> 00:24:01,130
+integrable function من A ل X F of T DT هتطلع لي
+
+292
+00:24:01,130 --> 00:24:05,210
+continuous function لو سميتها F capital زي ما هو
+
+293
+00:24:05,210 --> 00:24:08,730
+مسميها F of X بتكون الـ F عندها شمالها is
+
+294
+00:24:08,730 --> 00:24:16,430
+continuous الآن بقولك لو اجيت و قلت ان Fis
+
+295
+00:24:16,430 --> 00:24:21,390
+continuous الـ F اللي جوا هذه at C element in A وB
+
+296
+00:24:21,390 --> 00:24:25,650
+لو كانت F is continuous at C element in A وB
+
+297
+00:24:25,650 --> 00:24:31,730
+هيطلعلك بقولك إذا الـ F capital هذه F prime of C
+
+298
+00:24:31,730 --> 00:24:39,870
+existsو مش exist بس و ال F prime of C هتساوي ال F
+
+299
+00:24:39,870 --> 00:24:46,030
+small of إيش؟ of C يعني و كأنه بقولك لما نتفاضل
+
+300
+00:24:46,030 --> 00:24:50,210
+هذه F prime of C اللي هي من إيه اللي عند C يعني
+
+301
+00:24:50,210 --> 00:24:57,370
+تقول F prime of C بساوي تفاضل رفلة by DXعنده
+
+302
+00:24:57,370 --> 00:25:02,630
+النقطة C لل integration من A ل X F of T DT
+
+303
+00:25:27,580 --> 00:25:33,180
+C مثلاً بتساوي F of C اللي هي F small of C هو كون
+
+304
+00:25:33,180 --> 00:25:36,540
+ال F اللي جوا continuous عنده النقطة اللي بنحكي
+
+305
+00:25:36,540 --> 00:25:39,940
+عنها طب لو كانت F اللي جوا هادى continuous وين
+
+306
+00:25:39,940 --> 00:25:43,620
+ماكان على الفترة هادى بصير خلاص وانت مغمض عناك
+
+307
+00:25:43,620 --> 00:25:47,980
+بتيجي بتقول تفاضل هادي بتشيل التكامل و بتحطه اللي
+
+308
+00:25:47,980 --> 00:25:53,540
+جوا بدلالة اللي هي ال X المتغيرواضح المفروض طيب
+
+309
+00:25:53,540 --> 00:25:59,660
+نيجي الآن نثبت اللي هي النظرية نثبت شغلتين أن ال F
+
+310
+00:25:59,660 --> 00:26:02,580
+كابتنال continuous عشان أثبتلك continuous هثبتلك
+
+311
+00:26:02,580 --> 00:26:05,940
+أكثر من continuous هثبتلك uniform ل continuous لأ
+
+312
+00:26:05,940 --> 00:26:09,780
+أقولك هثبتلك أكثر من uniform ل continuous هثبتلك
+
+313
+00:26:09,780 --> 00:26:12,920
+أن ال F is Lipschitz function مثلا من Lipschitz
+
+314
+00:26:12,920 --> 00:26:15,400
+function إذا على طول uniform ل continuous ومن ثم
+
+315
+00:26:15,400 --> 00:26:18,460
+على طول إيش ما لها continuous إيش Lipschitz
+
+316
+00:26:18,460 --> 00:26:25,240
+function يعني هثبتلك أن Fof x-f of y absolute
+
+317
+00:26:25,240 --> 00:26:30,720
+value أصغر أو ساوي k في x-y had there exist k بحيث
+
+318
+00:26:30,720 --> 00:26:35,260
+أن f of x-f of y أصغر من k في xy لكل x و y element
+
+319
+00:26:35,260 --> 00:26:40,020
+in a و b هثبت لك f بتحقق هذا الشرط هثبتها ما ده
+
+320
+00:26:40,020 --> 00:26:42,540
+هثبتها ما ده هتحقق هذا الشرط إذا ما على طول اللي
+
+321
+00:26:42,540 --> 00:26:45,680
+هي هتكون lipschitz يعني بمعنى آخر هتكون is
+
+322
+00:26:45,680 --> 00:26:49,480
+continuous هذا الجزء الأول الجزء الثاني يا شباب و
+
+323
+00:26:49,480 --> 00:26:55,760
+يا بناتلأن if F is continuous at C هنثبت أن F is
+
+324
+00:26:55,760 --> 00:26:59,320
+differentiable عند C يعني بدأ أثبتلك F is
+
+325
+00:26:59,320 --> 00:27:03,460
+differentiable عند C يعني ايش F is differentiable
+
+326
+00:27:03,460 --> 00:27:14,660
+يعني بدأ أثبتلك أن limit F of X زي داتش مثلا او F
+
+327
+00:27:14,660 --> 00:27:21,440
+of C زي داتش ناقص F of Cعلى H اذا H بتروح للصفر
+
+328
+00:27:21,440 --> 00:27:27,380
+بتثبت لك ايه ايش بيساوي F capital طبعا بيساوي F
+
+329
+00:27:27,380 --> 00:27:33,660
+small of Cبدي أثبتلك يعني أو بمعنى آخر هأثبتلك لكل
+
+330
+00:27:33,660 --> 00:27:37,000
+Y أكبر من 0 داريكزيز ديلتا بحيث أنه absolute value
+
+331
+00:27:37,000 --> 00:27:41,320
+أصغر من Delta L على H أصغر من Delta يؤدي إلى F of
+
+332
+00:27:41,320 --> 00:27:48,260
+C زائد H ناقص F of C على H ناقص F of C بدي أثبتلك
+
+333
+00:27:48,260 --> 00:27:53,180
+يه أصغر من 200 من Epsilonبكون اثبتت فعلا انه هذه
+
+334
+00:27:53,180 --> 00:27:56,540
+ال limit بالساوية هذه معناته هذه ال limit طبعا ايش
+
+335
+00:27:56,540 --> 00:28:01,000
+بتعني ان F prime F capital prime of C exist و
+
+336
+00:28:01,000 --> 00:28:06,480
+بتساوية F of C اللي هو اللي هو طالبه انه لما تكون
+
+337
+00:28:06,480 --> 00:28:09,280
+F continuous عند ال C بتكون F capital
+
+338
+00:28:09,280 --> 00:28:12,040
+differentiable عند ال C و ال derivative لل F
+
+339
+00:28:12,040 --> 00:28:15,700
+capital بتساوية ال F اسمه فنشوف
+
+340
+00:28:17,960 --> 00:28:21,380
+Media نثبت الـ Continuity أو نثبت الـ Lipschitz
+
+341
+00:28:21,380 --> 00:28:27,450
+function يعني الأمر سهلأحنا مفترضين يا جماعة أن
+
+342
+00:28:27,450 --> 00:28:31,490
+الـF small هذي is integrable مزام is integrable
+
+343
+00:28:31,490 --> 00:28:37,650
+إذا is bounded S يعني أنه لكل .. نقدر نلاقي K أكبر
+
+344
+00:28:37,650 --> 00:28:41,830
+من 0 بحيث أن ال absolute value ل F of X small أصغر
+
+345
+00:28:41,830 --> 00:28:48,070
+سوى K لكل X موجودة ال A و ال B سبب هذه أن F نفسها
+
+346
+00:28:48,070 --> 00:28:52,350
+is integrable طيب، بدأ اعتمد على هذه للوصول إنها
+
+347
+00:28:52,350 --> 00:28:58,620
+لبشتالان خد اي x و y في ال a و ال b واسمحولي اخد x
+
+348
+00:28:58,620 --> 00:29:01,760
+اقل من y without loss of generality مش عاد باخد y
+
+349
+00:29:01,760 --> 00:29:06,100
+اكبر من .. اصغر من x الان حالة ال x بتساوي
+
+350
+00:29:06,100 --> 00:29:09,660
+هتلاقوها automatic بتتحقق للي بديها نشوف ايش اللي
+
+351
+00:29:09,660 --> 00:29:13,790
+بديهاالان خد x و y في ال a و ال b و نفترض ان x
+
+352
+00:29:13,790 --> 00:29:16,610
+أزرار من y without most of generality زي ما قلنا
+
+353
+00:29:16,610 --> 00:29:22,890
+الان احسبلي f of x f of y نقص مين f of x الكبيرة y
+
+354
+00:29:22,890 --> 00:29:26,630
+و الكبيرة x احسب f of y نقص f of x ايش هيساوي ال
+
+355
+00:29:26,630 --> 00:29:29,470
+integration حسب التعريف f of x بتساوي ال
+
+356
+00:29:29,470 --> 00:29:33,390
+integration من ال x ل ال f smallالأن f of y هي
+
+357
+00:29:33,390 --> 00:29:36,930
+عبارة عن ال integration من a ل y، f small، ناقص، f
+
+358
+00:29:36,930 --> 00:29:40,210
+of x إيش بتساوي؟ ال integration من a ل x ل مين؟ لل
+
+359
+00:29:40,210 --> 00:29:47,290
+F، إذا هذا ناقص هذا، الآن بدي أحول هذه، أجلبها،
+
+360
+00:29:47,290 --> 00:29:52,670
+بيصير زائد ال integration من x لعند مين، لعند ال
+
+361
+00:29:52,670 --> 00:29:57,640
+a، لل Fصار عندى الآن ال integration من X لعند ال A
+
+362
+00:29:57,640 --> 00:30:00,520
+و من A لعند ال Y إذا حيصير هذا عبارة عن ال
+
+363
+00:30:00,520 --> 00:30:05,940
+integration من X لمين من X لعند ال Y عارفين هذه
+
+364
+00:30:05,940 --> 00:30:09,960
+الخاصية إذا صار عند ال F of X F of Y ناقص F of X
+
+365
+00:30:09,960 --> 00:30:15,760
+بسوء ال integration من X ل Y لمين لل F شوف الآن
+
+366
+00:30:15,760 --> 00:30:20,100
+إيش بده أصل؟ بده أصل إنه absolute value F of Y
+
+367
+00:30:20,100 --> 00:30:24,830
+ناقص F of Xأصغر أو سوى K في الـ absolute value X
+
+368
+00:30:24,830 --> 00:30:31,450
+مانس Y يعني صارت لبشت لنشوف كيف نصلها وأكيد بعضكم
+
+369
+00:30:31,450 --> 00:30:36,450
+فهم او اتوقع كيف بتسوي لأن صار ال absolute value
+
+370
+00:30:36,450 --> 00:30:40,250
+لهذه بسوى ال absolute value لهذه هيها لأن ال
+
+371
+00:30:40,250 --> 00:30:43,010
+absolute value ال integration بخاصية احنا عارفينها
+
+372
+00:30:43,010 --> 00:30:46,770
+أصغر أو سوى ال integration لل absolute valueوالـ
+
+373
+00:30:46,770 --> 00:30:50,990
+absolute value للـ F في الفترة X وY أشملها .. هي
+
+374
+00:30:50,990 --> 00:30:53,570
+مش في الفترة X وY بس الـ absolute value للـ F الـ
+
+375
+00:30:53,570 --> 00:30:56,170
+absolute value للـ F أصغر أو سوى K على كل الفترة
+
+376
+00:30:56,170 --> 00:31:01,030
+الكبيرة اللي هي A وB فأكيد برضه هتكون أصغر أو سوى
+
+377
+00:31:01,030 --> 00:31:05,670
+K على الفترة الصغيرة فصار عندي أصغر أو سوى K في
+
+378
+00:31:05,670 --> 00:31:07,890
+مين؟ في ال integration من X لعند Y ال integration
+
+379
+00:31:07,890 --> 00:31:13,080
+من X لعند Y هو إيش بساوي؟ اللي هو Y minus Xالـ
+
+380
+00:31:13,080 --> 00:31:19,400
+integration يعني ال integration لـ DT من X لـ DT
+
+381
+00:31:19,400 --> 00:31:28,660
+ال integration لـ DT من X عند Y ايش بيساوي Y minus
+
+382
+00:31:28,660 --> 00:31:36,240
+X Y minus X فهذا صار Y minus X و اللي جوا أصغر
+
+383
+00:31:36,240 --> 00:31:40,790
+بيساوي Kالان هذا على خطوتين سهلات بتصور ان انتوا
+
+384
+00:31:40,790 --> 00:31:44,370
+فاهمينش بحكي من X لعند Y ال integration لل F فده
+
+385
+00:31:44,370 --> 00:31:47,850
+أصغر أو ساوي اللي هو ال integration من X لعند Y
+
+386
+00:31:47,850 --> 00:31:55,370
+هذه بدلة K لكل DT وهذا بساوي K برة في Y minus X
+
+387
+00:31:55,370 --> 00:31:59,230
+اللي هو أكيد حيصير عند ال absolute value هذه أصغر
+
+388
+00:31:59,230 --> 00:32:03,770
+أو ساوي K في ال absolute value Y minus Xالان هذه
+
+389
+00:32:03,770 --> 00:32:07,570
+ال Xات الأصغر من مين؟ من Y لكن زي ما انتوا عارفين
+
+390
+00:32:07,570 --> 00:32:11,150
+حالة ال X بتساوي Y is trivial لأن لما تكون X
+
+391
+00:32:11,150 --> 00:32:14,450
+بتساوي Y هذا سفر و لما تكون X بتساوي Y هذا سفر إذا
+
+392
+00:32:14,450 --> 00:32:18,610
+ال inequality هذه صحيحة دائما إذا الأن صار عندى
+
+393
+00:32:18,610 --> 00:32:25,590
+هذه ال inequality staris true for all x و y
+
+394
+00:32:25,590 --> 00:32:30,390
+limiting a و b لأنه قلنا ال x أصغر من y ال y أصغر
+
+395
+00:32:30,390 --> 00:32:33,990
+من x أكيد similarly و بنفس الأسلوب أو حتى without
+
+396
+00:32:33,990 --> 00:32:38,170
+loss of generality يعني بدون ما نفقد أي إشي من
+
+397
+00:32:38,170 --> 00:32:44,530
+التعميم بنفترض أن x أصغر من y صار
+
+398
+00:32:44,530 --> 00:32:53,230
+عندي يا جماعةالان F مستمر على A وB لأن F بيصبح
+
+399
+00:32:53,230 --> 00:33:02,750
+عملية Lipschitz اللي هي الآن ضال علي أثبت أن الـ F
+
+400
+00:33:02,750 --> 00:33:11,100
+مستخدم وزي ما قلت بدأ أثبت أن الـ Limitللـ F of C
+
+401
+00:33:11,100 --> 00:33:17,800
+زائد H نقص F of C على H لما H تروح للصفر بساوة F
+
+402
+00:33:17,800 --> 00:33:23,480
+small of C أو أثبت لكم الفرق بين هذولتين أصغر من
+
+403
+00:33:23,480 --> 00:33:28,880
+إبسلون وهذا زي ما قلنا هو اللي هيمثل F' of C ده
+
+404
+00:33:28,880 --> 00:33:34,160
+نشوف كيف طيب الصلاة على النبي صلاة والسلام عليك
+
+405
+00:33:37,390 --> 00:33:42,130
+خلّينا نجف عند .. أيوة نيجي الآن البرهان هتلاقوه
+
+406
+00:33:42,130 --> 00:33:49,730
+برضه سهل وفكرته سهلة شوفوا كيف عندي نفترض الآن
+
+407
+00:33:49,730 --> 00:33:53,550
+suppose that f is continuous at c limiting a وb
+
+408
+00:33:53,550 --> 00:33:58,790
+مدام continuous إذن limit
+
+409
+00:34:01,700 --> 00:34:08,460
+F of X as X بتروح للـ C بساوي F of C مظبوط ولا لأ؟
+
+410
+00:34:08,460 --> 00:34:12,300
+أكيد مظبوط فبقى من جهة أخرى خلينا نكتبها بصورة
+
+411
+00:34:12,300 --> 00:34:18,660
+ثانية X ناقص C بتروح للسفر if and only if اللي هو
+
+412
+00:34:18,660 --> 00:34:23,960
+X بتروح لمين للـ C سمولي هذه X minus C إيش اسمها H
+
+413
+00:34:23,960 --> 00:34:31,410
+بصير هذه اللي فوق مين هي limitF of الـ X minus C
+
+414
+00:34:31,410 --> 00:34:36,490
+بتساوي الـ H يعني الـ X بتساوي H زائد C أو C زائد
+
+415
+00:34:36,490 --> 00:34:41,190
+Hلأن x بتروح للـ c تكافئ أنه .. اللي هو x minus c
+
+416
+00:34:41,190 --> 00:34:44,610
+تروح للصفر، يعني بتكافئ H تروح لمين للصفر، إيش حيث
+
+417
+00:34:44,610 --> 00:34:49,650
+هو هذا؟ F of C، إذا هذه هي تعبير آخر عن ال
+
+418
+00:34:49,650 --> 00:34:56,250
+continuity لل F لما اللي هو اللي هي ال F عند مين،
+
+419
+00:34:56,250 --> 00:34:59,210
+عند ال C، يعني هذه تعبير آخر، ال continuity لل
+
+420
+00:34:59,210 --> 00:35:05,900
+function لما عند النقطة اللي هي Cهذه الآن شو بدي
+
+421
+00:35:05,900 --> 00:35:09,520
+استخدمها للوصول اللي بديها لأن مادام F is
+
+422
+00:35:09,520 --> 00:35:12,600
+continuous ده هذي متحققة هيها مادام F is
+
+423
+00:35:12,600 --> 00:35:15,480
+continuous عن C ده هذي متحققة فمادام هذي متحققة
+
+424
+00:35:15,480 --> 00:35:18,720
+اذا by epsilon delta definition for every epsilon
+
+425
+00:35:18,720 --> 00:35:22,880
+أكبر من 0 there exists delta أكبر من 0 such that
+
+426
+00:35:22,880 --> 00:35:28,950
+لما يكون ال absolute value لل H أصغر من Deltaو
+
+427
+00:35:28,950 --> 00:35:32,230
+طبعا انا وين بشتغل في المنطقة انها تكون C زائد H
+
+428
+00:35:32,230 --> 00:35:36,370
+وين ما لأ في الفترة تبعت من A لعند مين لعند B يعني
+
+429
+00:35:36,370 --> 00:35:41,230
+اخترت ال H صغيرة كفاية بحيث انه C زائد H اضلي وين
+
+430
+00:35:41,230 --> 00:35:45,090
+جاعد في الفترة من A و B لعنده لإن هاي الفترة A وهي
+
+431
+00:35:45,090 --> 00:35:49,330
+Bوالفترة هذه اللي هي I C عندي مثلا في داخلها اللي
+
+432
+00:35:49,330 --> 00:35:55,690
+هي بتختار H دلتتها صغيرة كفاية أنه ضال C زائد H
+
+433
+00:35:55,690 --> 00:36:01,050
+موجودة في الفترة من A لعند B عشان تصير معرفة لإن
+
+434
+00:36:01,050 --> 00:36:05,330
+دالتي أنا عشان تكون معرفة عند C زائد H لازم تكون C
+
+435
+00:36:05,330 --> 00:36:08,490
+زائد H في داخل المنطقة هذه لإن دالتي معرفة على
+
+436
+00:36:08,490 --> 00:36:11,770
+الفترة من A لعند B عشان هيك أقولنا C زائد H لازم
+
+437
+00:36:11,770 --> 00:36:16,630
+تكون في الفترة زائد Bإذن اختيار الـ Delta يعتمد
+
+438
+00:36:16,630 --> 00:36:20,210
+على ال limit ويعتمد على انه أضمن ال C زي داشر ضال
+
+439
+00:36:20,210 --> 00:36:24,510
+هوين في الفترة A وB إذا تعريف ال continuity بيقول
+
+440
+00:36:25,950 --> 00:36:27,530
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أقوم
+
+441
+00:36:27,530 --> 00:36:28,970
+باستخدام صفر يوجد دلتا لكل y أقوم باستخدام صفر
+
+442
+00:36:28,970 --> 00:36:29,950
+يوجد دلتا لكل y أقوم باستخدام صفر يوجد دلتا لكل y
+
+443
+00:36:29,950 --> 00:36:30,650
+أقوم باستخدام صفر يوجد دلتا لكل y أقوم باستخدام
+
+444
+00:36:30,650 --> 00:36:32,710
+صفر يوجد دلتا لكل y أقوم باستخدام صفر يوجد دلتا
+
+445
+00:36:32,710 --> 00:36:35,050
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أقوم
+
+446
+00:36:35,050 --> 00:36:41,110
+باستخدام صفر يوجد دلتا
+
+447
+00:36:41,110 --> 00:36:49,400
+لكل y أقوم باستخدام صفر يوجد دلتا لكل y أطيب، الآن
+
+448
+00:36:49,400 --> 00:36:54,020
+وين أنا بدأ أروح؟ بدأ أثبت لكم إنه هذا ناقص هذا
+
+449
+00:36:54,020 --> 00:36:58,240
+أصغر من epsilon عشان ذلك دعنا نحسب F of C زاد H
+
+450
+00:36:58,240 --> 00:37:03,060
+ناقص F of C على H ناقص مين يا جماعة؟ F of C ويساوي
+
+451
+00:37:03,060 --> 00:37:13,170
+الآن F of C زاد H ناقص F of Cخد الآن اللي هي الـ H
+
+452
+00:37:13,170 --> 00:37:16,850
+هدعم المشترك بالـ C الواحدة للـ H ماشي F of C زي
+
+453
+00:37:16,850 --> 00:37:21,910
+دتش تعريفها من A لعند C زي دتش لل F of X DX قلناها
+
+454
+00:37:21,910 --> 00:37:26,090
+هذه اللي ماعطيناها من رأس الدور ناقص ماشي تعريف F
+
+455
+00:37:26,090 --> 00:37:30,230
+of C Capital of C هو ال integration من A لعند C زي
+
+456
+00:37:30,230 --> 00:37:35,630
+دتش الآن دي مش C زي دتش من A لعند مين لعند الـ C
+
+457
+00:37:35,630 --> 00:37:40,110
+واحدة للـ H من A لعند الـ C اللي هي F of C زي دتش
+
+458
+00:37:40,110 --> 00:37:46,360
+هيهاوهي الواحدة الـ H اللي برا ال F of C هيها من A
+
+459
+00:37:46,360 --> 00:37:51,780
+ل C واحدة الـ H ناقص مين F of C هذا من التعريف
+
+460
+00:37:51,780 --> 00:37:55,460
+مباشرة لإنه احنا عرفنا ال F capital of X هي عبارة
+
+461
+00:37:55,460 --> 00:38:01,820
+عن integration من A لعند X F of T DT لأن لما نكون
+
+462
+00:38:01,820 --> 00:38:06,120
+F of C زي دتش بنحطها دي C زي دتش C بنحطها دي عياش
+
+463
+00:38:06,120 --> 00:38:13,460
+C وهي C زي دتش وهي الـ C وهي ساويةالان هذه من ..
+
+464
+00:38:13,460 --> 00:38:18,780
+من .. من عند A لـ C زي ده اتش وهذه من A لعند B
+
+465
+00:38:18,780 --> 00:38:23,980
+لعند C هدولة مع بعض هاي هدولة التنتين بده احسبها
+
+466
+00:38:23,980 --> 00:38:28,120
+مع بعض حسبنا زيين قبل و شوية هو عبارة عن واحد على
+
+467
+00:38:28,120 --> 00:38:32,400
+اتش خدوا عامل مشترك خدوا واحد على اتش عامل مشترك
+
+468
+00:38:32,400 --> 00:38:37,340
+بين الجهتين تصبح واحد على اتش افتح جوس ال
+
+469
+00:38:37,340 --> 00:38:41,660
+integration من A لعند C زي ده اتشلأن بدل الناقص
+
+470
+00:38:41,660 --> 00:38:46,020
+بيصير H اكتب زائد بدل هذا الناقص بيكتب زائد لأن
+
+471
+00:38:46,020 --> 00:38:52,460
+هتنقلب مين الآن من C لعند ال A نظبط اللي هو لنفسها
+
+472
+00:38:52,460 --> 00:38:58,020
+F F هذه الآن من C لعند ال A ومن A لعند ال C زاد H
+
+473
+00:38:58,020 --> 00:39:02,000
+إذا أكيد هذه كلها على بعض يصير ال integration من C
+
+474
+00:39:02,000 --> 00:39:08,930
+ل C زاد H لل F في واحد على H إذا هذه كلهابس
+
+475
+00:39:08,930 --> 00:39:11,350
+ابدلتها بقيمتها اللي قلنا عنها اللي هو ال
+
+476
+00:39:11,350 --> 00:39:15,810
+integration من C ل C زي H اللي اوجدته بدل هذا كله
+
+477
+00:39:15,810 --> 00:39:23,190
+من واحد ال H ل F of X DX ناقص الان هذا هذا اللي هي
+
+478
+00:39:23,190 --> 00:39:28,090
+F of C شوف كيف بده اعملها على جهة تعالي احسب ال
+
+479
+00:39:28,090 --> 00:39:33,660
+integrationالـ integration من C لـ C زائد H للـ
+
+480
+00:39:33,660 --> 00:39:38,180
+constant واحد بعد أذنكم DX إيش هيساوي ال
+
+481
+00:39:38,180 --> 00:39:44,140
+integration هدا عبارة عن C زائد H ناقص C وهيساوي
+
+482
+00:39:44,140 --> 00:39:48,760
+قداش H يعني بمعنى آخر لو جيتها الجد تعرفوا ليش
+
+483
+00:39:48,760 --> 00:39:52,640
+بعمل هيك واحد على H في ال integration من C لـ C
+
+484
+00:39:52,640 --> 00:39:58,720
+زايد H اللي هو الواحد هيساوي إيش؟ واحد، مظبوط ولا
+
+485
+00:39:58,720 --> 00:40:03,450
+لأ؟أكيد مظبوط ماشي هايمل هادي كسمت الديوا تاني
+
+486
+00:40:03,450 --> 00:40:10,570
+علشان تطلع حد واحد الان صار عندى لو ضربت هادي
+
+487
+00:40:11,920 --> 00:40:18,320
+الجهتين في F of C هي F of C بيساوي ايش؟ F of C صار
+
+488
+00:40:18,320 --> 00:40:21,260
+الان F of C بيساوي هذه بدي أشيل F of C تبعتي و أحط
+
+489
+00:40:21,260 --> 00:40:25,360
+مكانها هذه الصورة ليش حطيتها؟ عشان هذه أقدر أتحمل
+
+490
+00:40:25,360 --> 00:40:30,380
+محا محا دي اللي موجودة من الأصل شوف كيف الان شيلت
+
+491
+00:40:30,380 --> 00:40:33,320
+ال F of C و حطيت قيمتها اللي أوجدناها اللي هي
+
+492
+00:40:33,320 --> 00:40:37,660
+عبارة عن F of C على H في ال integration من C لC زي
+
+493
+00:40:37,660 --> 00:40:42,480
+H لمن؟ للواحدماشي لان صار هذا ال integration و هذا
+
+494
+00:40:42,480 --> 00:40:45,920
+ال integration نفس الاشي واحد ل F of X و واحد ل
+
+495
+00:40:45,920 --> 00:40:50,380
+واحد يعني بقدر اشتغل فيهم بيصير عندى خد ال واحد ال
+
+496
+00:40:50,380 --> 00:40:53,960
+H برا عن المشترك خالص بيظل عندى ال integration من
+
+497
+00:40:53,960 --> 00:40:58,720
+C ل C زاد H هذا مين ايش اسمه F of X و هذا F of C
+
+498
+00:40:58,720 --> 00:41:01,580
+دخلتها جوا و مضربتها بالواحد صارت مين بقدر اعلن و
+
+499
+00:41:01,580 --> 00:41:06,520
+ثابت اللي هي ناقص F of C لكل اش ماله DX هذا الكلام
+
+500
+00:41:06,520 --> 00:41:14,340
+كله من C ل C زاد Hصار عندى الآن الصورة هذه هيها اه
+
+501
+00:41:14,340 --> 00:41:20,840
+و هذه الصورة اصلا انا بدياها الان واحد على H أزرع
+
+502
+00:41:20,840 --> 00:41:23,760
+أو ساوي قولنا absolute value of integration أزرع
+
+503
+00:41:23,760 --> 00:41:28,740
+أو ساوي ال integration لل absolute value DX الان
+
+504
+00:41:28,740 --> 00:41:35,320
+هذا المقدار F of X ناقص F of Cقلنا عنها من راس
+
+505
+00:41:35,320 --> 00:41:40,420
+الدور أن F of X ناقص F of C اللي هو أصغر من
+
+506
+00:41:40,420 --> 00:41:44,420
+Epsilon لأن limit F of X لما X تروح للـC أشهر
+
+507
+00:41:44,420 --> 00:41:50,240
+مساوي F of C لأن X الـN هذه في الفترة من C لـC
+
+508
+00:41:50,240 --> 00:41:56,100
+زائداتش يعني بتسمح لي أن أقول limit F of X as X
+
+509
+00:41:56,100 --> 00:42:01,220
+بتروح للـC بتساوي F of C لما X بتروح للـC اللي هي
+
+510
+00:42:01,220 --> 00:42:05,980
+أدش بتروح لمينللسفر مضل في نفس المنطقة لأن قلنا
+
+511
+00:42:05,980 --> 00:42:10,300
+قبل شوية x minus c بتروح للسفر إذا وفقط إذا اللي
+
+512
+00:42:10,300 --> 00:42:13,880
+هي ال H اللي بتروح للسفر اللي هي عبارة عن x minus
+
+513
+00:42:13,880 --> 00:42:18,260
+c يعني هذا المقدار بال continuity لل F عند ال C
+
+514
+00:42:18,260 --> 00:42:21,980
+برضه بكون أصغر من إبسلون على ال absolute value
+
+515
+00:42:21,980 --> 00:42:25,160
+لمين لل H في ال integration من C لC زاد H
+
+516
+00:42:27,990 --> 00:42:33,050
+واضحة اه؟ اكيد هذا الان ايش قيمته؟ C زي داش ناقص C
+
+517
+00:42:33,050 --> 00:42:36,050
+اللي هي عبارة عن H فبصير Y عبارة عن absolute value
+
+518
+00:42:36,050 --> 00:42:42,430
+of H في mean فيه اللي هو الـH الـH اللي هو الـH
+
+519
+00:42:42,430 --> 00:42:42,470
+اللي هو الـH اللي هو الـH الـH اللي هو الـH اللي
+
+520
+00:42:42,470 --> 00:42:42,490
+هو الـH اللي هو الـH اللي هو الـH اللي هو الـH
+
+521
+00:42:42,490 --> 00:42:42,490
+اللي هو الـH اللي هو الـH اللي هو الـH اللي هو
+
+522
+00:42:42,490 --> 00:42:42,570
+الـH اللي هو الـH اللي هو الـH اللي هو الـH اللي
+
+523
+00:42:42,570 --> 00:42:42,710
+هو الـH اللي هو الـH اللي هو الـH اللي هو الـH
+
+524
+00:42:42,710 --> 00:42:42,710
+اللي هو الـH اللي هو الـH اللي هو الـH اللي هو
+
+525
+00:42:42,710 --> 00:42:46,490
+الـH اللي هو الـH اللي هو الـH
+
+526
+00:42:46,490 --> 00:42:52,450
+اللي هو الـطيب .. بيصير عندها .. بيصير أكبر من 0
+
+527
+00:42:52,450 --> 00:42:55,790
+أسف هذه بحاجة دي بتطلع أيش بتساوي .. بيساوي بساوي
+
+528
+00:42:55,790 --> 00:42:58,150
+.. بقدر أختار H positive؟ أه بقدر أختار H positive
+
+529
+00:42:58,150 --> 00:43:03,130
+طيب و لو حتى ال H negative بتكون similarly بس
+
+530
+00:43:03,130 --> 00:43:06,950
+بتكون .. كل البرهام بيصير .. بترتب على أنه اللي هي
+
+531
+00:43:06,950 --> 00:43:12,890
+ال .. ال C زائد H أصغر من مين من ال C without loss
+
+532
+00:43:12,890 --> 00:43:17,250
+of generality ال H أكبر من 0بصير عندى لان هذا
+
+533
+00:43:17,250 --> 00:43:24,750
+المقدار أصغر من مين من إبسلون إيش يعني اللى عملنا
+
+534
+00:43:24,750 --> 00:43:28,290
+اللى عملنا لكل إبسلون أكبر من Zero لاجئنا Delta
+
+535
+00:43:28,290 --> 00:43:33,350
+بحيث أنه لما H absolute value أصغر من Deltaيعطيني
+
+536
+00:43:33,350 --> 00:43:39,330
+أن هذا المقدار أصغر من مين؟ من إبسلون يعني صار
+
+537
+00:43:39,330 --> 00:43:43,450
+عندي limit هذا المقدار بسوء f of c و هذا المقدار
+
+538
+00:43:43,450 --> 00:43:46,290
+اللي هو f prime of c يعني اعتبتنا f prime of c
+
+539
+00:43:46,290 --> 00:43:56,410
+بسوء f small of c وهو المطلوب طيب نيجي الآن لاللي
+
+540
+00:43:56,410 --> 00:44:02,210
+هي ال corollaryبرضه الى النظرية اللى جاملة بشوية
+
+541
+00:44:02,210 --> 00:44:10,790
+او خلّيني نقول تلخيصها سريع واستذكرها سريعها اللى
+
+542
+00:44:10,790 --> 00:44:17,830
+هو الأولى كانت تكامل والتفاضل هذه تفاضل التكامل
+
+543
+00:44:17,830 --> 00:44:23,470
+طبعا كل واحدة تحت شروطها المذكورة في نظريتها
+
+544
+00:44:25,860 --> 00:44:30,340
+نجي نشوف يا جماعة ال corollary let f من a و b لعند
+
+545
+00:44:30,340 --> 00:44:34,940
+r بي continuous on a و b and let f of x بي سوى ال
+
+546
+00:44:34,940 --> 00:44:37,720
+integration من ال a ل ال x ل ال f إذا فرض f
+
+547
+00:44:37,720 --> 00:44:40,160
+continuous ال f اسمه ال continuous على كل الفترة
+
+548
+00:44:40,160 --> 00:44:44,980
+ريح حاله طبعا أعطى إشي أكبر من النظرية اللي فاتت
+
+549
+00:44:44,980 --> 00:44:47,640
+يعني فرض ال continuity على ال a و ال b ما زم
+
+550
+00:44:47,640 --> 00:44:49,840
+continuous إذا ما بتزمش تقول integrable لإن و أنا
+
+551
+00:44:49,840 --> 00:44:52,340
+continuous function زي ما قلنا إيش ما لها is
+
+552
+00:44:52,340 --> 00:44:58,190
+integrable إذا مدامة f continuousالان بيصير هذه ال
+
+553
+00:44:58,190 --> 00:45:01,810
+integration اللي هي من a لعند ال x نفسها
+
+554
+00:45:01,810 --> 00:45:06,030
+continuous لأ مش continuous بس is differentiable
+
+555
+00:45:06,030 --> 00:45:09,270
+يعني أجم ال continuous يعني هتكون عندي ال f is
+
+556
+00:45:09,270 --> 00:45:13,670
+differentiable و ال f برامي يمين هي ال f then f is
+
+557
+00:45:13,670 --> 00:45:17,730
+differentiable من a و b عند f برامي بيساوي كده إذن
+
+558
+00:45:17,730 --> 00:45:25,000
+باختصار من a ل x اللي هو fof t dt لو فرضنا هذه
+
+559
+00:45:25,000 --> 00:45:33,060
+continuous على ال a و ال b إذا لكل x element in a
+
+560
+00:45:33,060 --> 00:45:38,080
+و ال b بيصير f of x بالساوي هذا is differentiable
+
+561
+00:45:38,080 --> 00:45:45,500
+و ال f prime of x هتساوي f of اللي هي x يعني بمعنى
+
+562
+00:45:45,500 --> 00:45:52,590
+آخر اللي هوانفضل هذا التكامل و نزيل التفاضل و نحط
+
+563
+00:45:52,590 --> 00:46:00,830
+الـ L جوا يعني بمعنى آخر D by DX للـ integration
+
+564
+00:46:00,830 --> 00:46:08,530
+من A لعند X F of D DT لما تكون F continuous هذا
+
+565
+00:46:08,530 --> 00:46:14,610
+بكون ب cancel هذا أو عملية عكسية لهوبتظللنا F of X
+
+566
+00:46:14,610 --> 00:46:19,070
+يعدى الأحقالها أو F of T والـ T هي المتغير أو الـ
+
+567
+00:46:19,070 --> 00:46:23,370
+X هي المتغير هذه اللي هي الكلرية الآن بدوا يلمّوا
+
+568
+00:46:23,370 --> 00:46:26,610
+النظريتين اللي قبل بشوية في نظرية واحدة يلخصن في
+
+569
+00:46:26,610 --> 00:46:30,910
+نظرية واحدة نشوف كيف بدوا يلخصن في نظرية واحدة
+
+570
+00:46:30,910 --> 00:46:40,630
+يعني بدوا يعيد بس اللي هو صياغة النظريتين صلوا
+
+571
+00:46:40,630 --> 00:46:44,540
+على النبي عليه الصلاة والسلامالان fundamental
+
+572
+00:46:44,540 --> 00:46:48,580
+theorem of calculus combined form يعني التنتين
+
+573
+00:46:48,580 --> 00:46:53,790
+الان موجودات في نفس النظرية شوف ايش بقولكlet
+
+574
+00:46:53,790 --> 00:46:57,610
+Fcapital and Fsmall be continuous on a and b فرض
+
+575
+00:46:57,610 --> 00:47:00,710
+أن Fcapital و Fsmall أشمالهم إتن تان continuous
+
+576
+00:47:00,710 --> 00:47:04,670
+على الفترة a و b و فرضك أن F of a بساوة سفر يعني
+
+577
+00:47:04,670 --> 00:47:07,610
+بدي أحط ال initial condition F of a بساوة سفر عسى
+
+578
+00:47:07,610 --> 00:47:11,070
+أن يجمّل الصورة ماحطه عشان سفر بتطلع F of a في
+
+579
+00:47:11,070 --> 00:47:14,290
+الجواب then the following statements are
+
+580
+00:47:14,290 --> 00:47:18,380
+equivalentيعني لما تكون هذا صحيحة هذا صحيحة و لما
+
+581
+00:47:18,380 --> 00:47:22,640
+تكون هذا صحيحة هذا صحيحة نيجي للجزء الأول لو فرضنا
+
+582
+00:47:22,640 --> 00:47:29,160
+أن F prime of X بيساوي F of X إذا حيصير عندي ال F
+
+583
+00:47:29,160 --> 00:47:33,460
+هذه is differential الماشي هيك مفترض أن F prime of
+
+584
+00:47:33,460 --> 00:47:38,620
+X بيساوي F of X بقول إذاإذاً هذه صحيحة، إيش يعني
+
+585
+00:47:38,620 --> 00:47:43,260
+هذه صحيحة؟ يعني ال integration من ال AX لل F of
+
+586
+00:47:43,260 --> 00:47:48,400
+التي هي T DT، هذه DT يا جماعة، بس .. بس نخلف عن
+
+587
+00:47:48,400 --> 00:47:58,200
+هذه، F of T DT، هذا هيساوي مين؟ هيساوي F of X،
+
+588
+00:47:58,200 --> 00:48:03,070
+ناشي؟الان then the following statements are
+
+589
+00:48:03,070 --> 00:48:06,950
+equivalent يعني لما تكون هذه صحيحة بتعطيها دي اه
+
+590
+00:48:06,950 --> 00:48:11,350
+فرضنا ان f prime of x بيساوي f of x اذا حيكون عند
+
+591
+00:48:11,350 --> 00:48:14,290
+ال integration من a ل x f of t dt بيساوي ايش
+
+592
+00:48:14,290 --> 00:48:20,630
+بالظبط؟ f of x كيف ده تبتها؟ الان فرضنا هذا صحيح
+
+593
+00:48:20,630 --> 00:48:24,710
+اذا ال integration من a ل عند ال x f of t اللي هي
+
+594
+00:48:24,710 --> 00:48:33,010
+مين هتصير؟ f prime of tDT هذه قبل بشوية من نظرية
+
+595
+00:48:33,010 --> 00:48:36,470
+الأولى قلنا كامل التفاوض و ليش بتطلع عندك جواب F
+
+596
+00:48:36,470 --> 00:48:42,210
+of X ناقص مين F of A لإن كل الشروط متحققة ماشي
+
+597
+00:48:42,210 --> 00:48:48,110
+الآن هاي كمان مرة يا جماعة فرضنا إن هذا متحقق بدي
+
+598
+00:48:48,110 --> 00:48:51,250
+أحسب هذه أتبتها بالساوية هذه خد ال integration من
+
+599
+00:48:51,250 --> 00:48:55,510
+X F of T DTهو ماعطيني الـ F small هد مين هي الـ F
+
+600
+00:48:55,510 --> 00:48:58,830
+براهن شفت .. شيلت الـ F small إيش حطيت مكانها؟ F
+
+601
+00:48:58,830 --> 00:49:03,510
+براهن الآن هذه قبل بشوية في الـ Corollary فظلنا ..
+
+602
+00:49:03,510 --> 00:49:07,490
+كملنا التفاضل كيف كملنا التفاضل؟ إنه قولنا شيل
+
+603
+00:49:07,490 --> 00:49:11,230
+التفاضل .. شيل التكامل .. شيل التفاضل بتصير عبارة
+
+604
+00:49:11,230 --> 00:49:15,270
+عن F اللي فوق اللي كانت B ناقص F اللي تحت اللي هي
+
+605
+00:49:15,270 --> 00:49:19,390
+A ويساويأف of X نقص أف of A أف of A ماعطينيها من
+
+606
+00:49:19,390 --> 00:49:24,350
+رأس الدوار صفر إذا إيش هتساوي أف of X إذا هذه اللي
+
+607
+00:49:24,350 --> 00:49:28,290
+بدّينا حساباتها بتساوي هذه طلعت إيش بتساوي أف of X
+
+608
+00:49:28,290 --> 00:49:32,830
+يعني أف of X بتساوي هذا المقدار وهو المطلوب طيب
+
+609
+00:49:32,830 --> 00:49:36,730
+هذا الجزء الأول من النظرية اللي هو نيجي للجزء
+
+610
+00:49:36,730 --> 00:49:42,240
+الثاني نفترض الآن إن هذه صحيحة وده تعطيهاالان برضه
+
+611
+00:49:42,240 --> 00:49:44,760
+الشروط كلها متحققة لأن الـ F في اسمه ليش مالها
+
+612
+00:49:44,760 --> 00:49:47,340
+continuous لسه ان انا ما .. ما .. ما ملاحتوش انا
+
+613
+00:49:47,340 --> 00:49:52,080
+هذا ايه عندي F continuous مدام F continuous ال
+
+614
+00:49:52,080 --> 00:49:57,720
+integration من اعلى عند ال X هذا كلهاللي هو اسمه F
+
+615
+00:49:57,720 --> 00:50:01,320
+of X هيكون differentiable حسب النظرية مزام
+
+616
+00:50:01,320 --> 00:50:04,780
+differentiable حسب هذه النظرية إذا F prime of X
+
+617
+00:50:04,780 --> 00:50:11,100
+إيش هتساوي؟ F of X يعني حققنا مين؟ اللي هي I يعني
+
+618
+00:50:11,100 --> 00:50:20,070
+هذا يؤدي إلى هذانجي الآن بعض التعريفات اللي
+
+619
+00:50:20,070 --> 00:50:26,890
+هنتكرها لكم وانتوا هتكونوا اللي هو اتابعوا الأمثلة
+
+620
+00:50:26,890 --> 00:50:32,130
+من خلال ال homework اللي معاكم هنعطي اللي هي تعريف
+
+621
+00:50:32,130 --> 00:50:36,830
+ممكن انتوا حتى مر عليكم في ال calculus ومن ثم اللي
+
+622
+00:50:36,830 --> 00:50:41,710
+هونعطي بعض ال counter examples اللي هي هنرجعكم
+
+623
+00:50:41,710 --> 00:50:47,370
+فيها للي هي الأسئلة اللي في الكتاب تشوفوا نصوصها
+
+624
+00:50:47,370 --> 00:50:53,530
+على الأقل و اللي مطلوب تحلوها ايه تحلوها و اللي ما
+
+625
+00:50:53,530 --> 00:50:56,910
+بتحلوه يعرفش تحلوهافي عندنا اللي هو في ال homework
+
+626
+00:50:56,910 --> 00:51:01,230
+الحلول الموجودة بتدرسوها لحالكم مادرستوها برضه
+
+627
+00:51:01,230 --> 00:51:04,310
+مافهمتوهاش احنا بنعمل فيها discussion ولو عملناها
+
+628
+00:51:04,310 --> 00:51:07,930
+حتى بشكل صوتي على أساس اللي هو احنا يمكن مانقدرش
+
+629
+00:51:07,930 --> 00:51:12,110
+نصور بشكل كامل اللي هي ال discussions عبر اللي هو
+
+630
+00:51:12,110 --> 00:51:16,270
+التصوير اللي احنا الحالي بنصوره عبر ال power point
+
+631
+00:51:16,270 --> 00:51:22,710
+من البيت ان شاء الله طيب definitionlet I بتساوي A
+
+632
+00:51:22,710 --> 00:51:27,630
+وB subset من مين؟ من R إذا كانت F small من I لR
+
+633
+00:51:30,720 --> 00:51:35,400
+then فرضنا أنه في دالة اسم F من I لعند R then the
+
+634
+00:51:35,400 --> 00:51:42,280
+antiderivative of F يعني وكأنها عكس عملية عكس
+
+635
+00:51:42,280 --> 00:51:45,920
+الدالة اللي بنفضلها roughly antiderivative of F on
+
+636
+00:51:45,920 --> 00:51:50,620
+I is a function F من I لعند R such that F prime of
+
+637
+00:51:50,620 --> 00:51:59,830
+X سوى F of X يعني عندنا دالة Fمن عند I لعند R لو
+
+638
+00:51:59,830 --> 00:52:05,310
+جينا لجينا F تانية من عند I لعند R و لجينا F prime
+
+639
+00:52:05,310 --> 00:52:10,450
+of X للي اللي في الـ I بتساوي mean لـ F small of X
+
+640
+00:52:11,290 --> 00:52:15,490
+في هذه الحالة بنسمي ال F capital هذه عبارة عن
+
+641
+00:52:15,490 --> 00:52:19,850
+antiderivative لل F يعني بمعنى أخر لو فضلناها هذه
+
+642
+00:52:19,850 --> 00:52:25,130
+ال antiderivative هتطلع مين؟ ال F الأصلية يعني
+
+643
+00:52:25,130 --> 00:52:30,890
+بتكون عندي اللي هي ال F capital هي ال
+
+644
+00:52:30,890 --> 00:52:35,210
+antiderivative لل F وال F small هي ال derivative
+
+645
+00:52:35,210 --> 00:52:41,910
+لل F واضح أخدته في ال calculus حتى نيجي الآنلأ
+
+646
+00:52:41,910 --> 00:52:45,650
+اللي هو برضه مفهوم اخدته في الكاركولاس if f من I
+
+647
+00:52:45,650 --> 00:52:49,750
+لعند R small is integrable لو فرضنا هذي integrable
+
+648
+00:52:49,750 --> 00:52:53,950
+و جينا عرفنا
+
+649
+00:52:53,950 --> 00:52:56,910
+ادالة من A لعند X F of X DX قبل شوية قولنا هذا
+
+650
+00:52:56,910 --> 00:53:00,290
+أكيد معرفة مثلا في integrable ومش معرفة ال
+
+651
+00:53:00,290 --> 00:53:04,110
+function F of X هتطلع continuous هذه اللي هي ال
+
+652
+00:53:04,110 --> 00:53:09,930
+function أو هذا هو اللي بنسميه التكامل المحدودهذه
+
+653
+00:53:09,930 --> 00:53:15,110
+بنسميها الـ Indefinite Integral of F يعني هذا
+
+654
+00:53:15,110 --> 00:53:17,790
+بنسميه Indefinite Integral of F أو هذه الدلة
+
+655
+00:53:17,790 --> 00:53:23,030
+بنسميها اللي هي The Indefinite Integral of F on I
+
+656
+00:53:23,030 --> 00:53:28,230
+برضه هذا برضه شغلات اللي هو أخدتوها سابقا في اللي
+
+657
+00:53:28,230 --> 00:53:33,780
+هو ال calculus نيجي بعض الملاحظات بسعلى اللي هو
+
+658
+00:53:33,780 --> 00:53:42,820
+اللي بتعلق بهذه اللي هي المفاهيم نشوفها وهي طبعا
+
+659
+00:53:42,820 --> 00:53:48,760
+هذه المفاهيم في عليها counter examples التمييز
+
+660
+00:53:48,760 --> 00:53:56,400
+بينها نشوف المحبة هذه يا شباب طيب أصلي على النبي
+
+661
+00:53:56,400 --> 00:54:03,640
+عليه الصلاة والسلامالان عندي سؤال سبعة تلاتة اتنين
+
+662
+00:54:03,640 --> 00:54:06,400
+يعني في سبعة تلاتة سؤال اتنين او سبعة تلاتة سؤال
+
+663
+00:54:06,400 --> 00:54:10,660
+خمسة طبعا كله في الطبع التاني هذا اللي هو عندي an
+
+664
+00:54:10,660 --> 00:54:13,540
+integrable function may not have an antiderivative
+
+665
+00:54:13,540 --> 00:54:17,220
+يعني هنلاقي في السؤال هذا integrable function
+
+666
+00:54:17,220 --> 00:54:24,080
+ومالهاش .. ومالهاش antiderivative يعني هنلاقي
+
+667
+00:54:24,080 --> 00:54:30,660
+function Fإنها تكون integrable لكن مالناجيش
+
+668
+00:54:30,660 --> 00:54:34,540
+function F اللي ال derivative تبعي تاش بتساوي
+
+669
+00:54:34,540 --> 00:54:39,860
+بتساوي F هذا اللي هو بتشوفوه في الـ 7 3 2 و 7 3 5
+
+670
+00:54:39,860 --> 00:54:43,340
+الأمثلة أو ال counter examples اللي وجودة هنا طيب
+
+671
+00:54:43,340 --> 00:54:47,710
+الملاحظة التانيةالـ function may have an
+
+672
+00:54:47,710 --> 00:54:52,830
+antiderivative يعني عندي F و بتحقق F و في عندي F
+
+673
+00:54:52,830 --> 00:54:57,310
+كابتال كمان و F برايم إيش بتساوي بسوء F but اللي
+
+674
+00:54:57,310 --> 00:55:02,210
+هي ال integration لل F إيش ماله does not exist لكن
+
+675
+00:55:02,210 --> 00:55:07,370
+but not integrable وهذا بتجاوب عليه برضه سبعة
+
+676
+00:55:07,370 --> 00:55:12,580
+تلاتة خمسة يعني في function Fفي إلها F' بساوة F
+
+677
+00:55:12,580 --> 00:55:17,640
+لكن ال integration للـ F إشماله does not exist
+
+678
+00:55:17,640 --> 00:55:24,900
+تلاتة عندي الملاحظة الثالثة a continuous function
+
+679
+00:55:24,900 --> 00:55:27,800
+always have antiderivative طبعا ال continuous
+
+680
+00:55:27,800 --> 00:55:35,290
+function أصلا من القوة بمكان إنهاتخلّي عندها يكون
+
+681
+00:55:35,290 --> 00:55:39,490
+لها الـ mean antiderivative الان a continuous
+
+682
+00:55:39,490 --> 00:55:44,190
+function always have antiderivative وهذا اللي هو
+
+683
+00:55:44,190 --> 00:55:51,910
+مباشرة من ال corollary 7 3 4 اللي كانت F
+
+684
+00:55:51,910 --> 00:55:56,770
+continuous مدام F continuous إذا ال integration من
+
+685
+00:55:56,770 --> 00:56:07,120
+ال X من A لعند ال XA of T DT بساوة F of X اللي هي
+
+686
+00:56:07,120 --> 00:56:10,900
+is differentiable ومش differentiable كمان و F
+
+687
+00:56:10,900 --> 00:56:14,860
+prime of X بتساوي F small of X يعني مدام أف
+
+688
+00:56:14,860 --> 00:56:17,880
+continuous إذا صار لها antiderivative صارت ال F
+
+689
+00:56:17,880 --> 00:56:20,580
+اللي هي ال derivative لها بساوة F of X إذا صارت
+
+690
+00:56:20,580 --> 00:56:24,620
+إيش antiderivativeهد مين هذا اللي هو عبارة عن الـ
+
+691
+00:56:24,620 --> 00:56:28,980
+Corollary 7 3 4 مدام أن F continuous إذا هذه
+
+692
+00:56:28,980 --> 00:56:33,360
+الدالة اللي سميتها F of X is differentiable ومش هي
+
+693
+00:56:33,360 --> 00:56:35,540
+كمان هذا الجزء الثاني من ال fundamental theorem of
+
+694
+00:56:35,540 --> 00:56:39,200
+calculus و ال F prime of X إيش بيساوي F of X يعني
+
+695
+00:56:39,200 --> 00:56:44,280
+ال F capital is an antiderivative of the F small
+
+696
+00:56:44,280 --> 00:56:50,760
+اللي هو النقطة الرابعة the indefinite integralmay
+
+697
+00:56:50,760 --> 00:56:57,520
+not be an antiderivative of mean of F الآن الـ
+
+698
+00:56:57,520 --> 00:57:02,400
+indefinite integral اللي
+
+699
+00:57:02,400 --> 00:57:12,000
+هو من A لعند X F of T DT لو سميناه F of X هذاهذا
+
+700
+00:57:12,000 --> 00:57:15,480
+the indefinite integral هذا ممكن يكون هذا يعني إيش
+
+701
+00:57:15,480 --> 00:57:18,480
+بمعنى آخر؟ يعني بدنا نفترض إن ال integration من A
+
+702
+00:57:18,480 --> 00:57:22,580
+ل X F of T DT موجود طبعا اللي بقول عنه موجود بس
+
+703
+00:57:22,580 --> 00:57:25,040
+تكون F of T is integrable ماحاجيناش على control U
+
+704
+00:57:25,040 --> 00:57:29,000
+of T ال F of T is integrable إذا ال A لعندي ال X F
+
+705
+00:57:29,000 --> 00:57:35,100
+of T DT لو ممكن يكون هذا موجود و بنسمي F of X
+
+706
+00:57:35,100 --> 00:57:39,830
+بقوللي the indefinite integral هذاmay not have
+
+707
+00:57:39,830 --> 00:57:44,490
+antiderivative of F يعني ممكن مايكونش هذا بالرغم
+
+708
+00:57:44,490 --> 00:57:48,610
+انه هذا موجود مايكونش ال F capital اللي هو هذا
+
+709
+00:57:48,610 --> 00:57:53,530
+مايكونش antiderivative لمين لل F small يعني رغم
+
+710
+00:57:53,530 --> 00:57:58,670
+انه هذا موجود ليس شرطا انه يكون يتفاضل ولمّا
+
+711
+00:57:58,670 --> 00:58:03,290
+يتفاضل يطلع اللي جوا يعني مش شرط انها تكون ال F
+
+712
+00:58:03,890 --> 00:58:07,170
+اللي هي antiderivative اللي هي مين هو الـ
+
+713
+00:58:07,170 --> 00:58:10,110
+indefinite integral هيك بتقول المواحدة طب إيش
+
+714
+00:58:10,110 --> 00:58:14,150
+دليلكم دليلنا هاي في counter examples هاي سبعة
+
+715
+00:58:14,150 --> 00:58:19,390
+تلاتة أربعة وهذا ناتب ناجم ناجم عدم تحقق هذه أو
+
+716
+00:58:19,390 --> 00:58:23,910
+عدم تحقق ال antiderivative أنه it may fail to be
+
+717
+00:58:23,910 --> 00:58:28,050
+differentiable at points at the interval ممكن أصلا
+
+718
+00:58:28,050 --> 00:58:35,440
+أن ال F prime تكونش موجودة بالمرةعند نقطة اللي هو
+
+719
+00:58:35,440 --> 00:58:39,620
+في داخل الـ interval من a إلى x مزام هذي is not
+
+720
+00:58:39,620 --> 00:58:42,720
+differentiable كيف بدنا نلاقي f prime بتساوي f
+
+721
+00:58:42,720 --> 00:58:48,640
+small إذا قد ينشأ عدم وجود الـ antiderivative منه
+
+722
+00:58:48,640 --> 00:58:52,640
+إنه هذي ما تكونش differentiable عند نقطة بين ال a
+
+723
+00:58:52,640 --> 00:58:59,090
+و ال x هاي أولاللي هو سبب ممكن يؤدي إلى عدم تحقق
+
+724
+00:58:59,090 --> 00:59:03,150
+إنه يكون الـ Indefinite Integral ليس Anti
+
+725
+00:59:03,150 --> 00:59:09,650
+-derivative للدالة اللي إحنا بنكملها Or الـ
+
+726
+00:59:09,650 --> 00:59:12,850
+Derivative of Indefinite Integral ممكن يكون موجود
+
+727
+00:59:13,450 --> 00:59:18,310
+لكن يختلف عن مين but different from the value of F
+
+728
+00:59:18,310 --> 00:59:22,410
+at any point of the interval وهذا بيجيبوا سبعة
+
+729
+00:59:22,410 --> 00:59:26,990
+تلاتة تمانية يعني بقولك ممكن تكون يا محلاها و تجيب
+
+730
+00:59:26,990 --> 00:59:32,990
+اللي هو ال F prime of X لكن ما تطلع شاشة تساوي F
+
+731
+00:59:32,990 --> 00:59:38,780
+of Xففي هذه الحالة بيكون الـ Indefinite Integral
+
+732
+00:59:38,780 --> 00:59:46,320
+ليس اللي هو إيه شماله له Antiderivative عارفش
+
+733
+00:59:46,320 --> 00:59:55,080
+لحاله و هاي في مثال طبعا نأتي
+
+734
+00:59:55,080 --> 01:00:01,300
+الآن و نبدأ حديثعن الهو evaluation of integrals
+
+735
+01:00:01,300 --> 01:00:07,780
+evaluation of integrals هناخد اللي هو اللي هي اللي
+
+736
+01:00:07,780 --> 01:00:12,660
+كنا نقول عنها اللي هو integration by parts اللي
+
+737
+01:00:12,660 --> 01:00:17,380
+كنا نقول عنه integration by parts نشوف كيف اللي هو
+
+738
+01:00:17,380 --> 01:00:21,360
+البرهن الأول واحدة يعني هناخد اللي هو النظرية
+
+739
+01:00:21,360 --> 01:00:28,430
+الأولى اللي هي في من ضمن اللي هيكيف يمكن أن نجد
+
+740
+01:00:28,430 --> 01:00:33,770
+طرق لإيجاد التكامل؟ evaluation of integrals أول
+
+741
+01:00:33,770 --> 01:00:39,730
+شيء نحكي عنه هو integral by parts التكامل بالتجزيم
+
+742
+01:00:41,640 --> 01:00:48,040
+نضع الأسس النظرية للتكامل بالتجزئة أو نشرع التكامل
+
+743
+01:00:48,040 --> 01:00:50,920
+بالتجزئة من خلال النظرية اللي أمامنا، أيش بتقول
+
+744
+01:00:50,920 --> 01:00:54,720
+ليه؟ إذا كانت f و g من a و b، f small و g small
+
+745
+01:00:54,720 --> 01:00:58,520
+طبعا، لعند R، R integrable on a و b، نفترض إن
+
+746
+01:00:58,520 --> 01:01:02,040
+التنتين إما اللي هين integrable وبدنا نفترض إنه
+
+747
+01:01:02,040 --> 01:01:06,510
+لهين antiderivativesالان بدأ افرض ان F لها
+
+748
+01:01:06,510 --> 01:01:09,430
+antiderivative اسمها F capital و G لها
+
+749
+01:01:09,430 --> 01:01:13,910
+antiderivative اسمها G capital on A و B اذا الان
+
+750
+01:01:13,910 --> 01:01:22,050
+نظريتنا ترتكز في معطياتها على ان ال F small ل F و
+
+751
+01:01:22,050 --> 01:01:24,210
+G integrable هي واحد
+
+752
+01:01:36,840 --> 01:01:45,420
+ثم عرض علينا الآن نعمل integration ثم بنيجي بقول
+
+753
+01:01:45,420 --> 01:01:52,610
+ليهالـ Integration لحاصل ضرب دالتين F of X في G of
+
+754
+01:01:52,610 --> 01:01:58,890
+X DX من A لعين Bلأن طيب هذه Integrable عارفين طب F
+
+755
+01:01:58,890 --> 01:02:01,090
+capital Integrable اه Integrable لإنها
+
+756
+01:02:01,090 --> 01:02:03,930
+differentiable لإنها differentiable يعني
+
+757
+01:02:03,930 --> 01:02:07,370
+continuous أكتر من continuous يعني يعني أكيد إيش
+
+758
+01:02:07,370 --> 01:02:11,250
+ما لها Integrable إذا معقول كلامه من A لعند B F of
+
+759
+01:02:11,250 --> 01:02:15,590
+X G of X DX إيش بتساوي؟ بقولي عبارة عن F capital
+
+760
+01:02:15,590 --> 01:02:21,750
+of B G capital of B نقص F capital of A G capital
+
+761
+01:02:21,750 --> 01:02:31,040
+of A هذاناقص الـ integration F small of X G
+
+762
+01:02:31,040 --> 01:02:37,520
+capital of X dx من A لعندي هذي اللي هي ال
+
+763
+01:02:37,520 --> 01:02:41,240
+integration by parts اللي عامة اللي بتدخلته أو
+
+764
+01:02:41,240 --> 01:02:45,960
+اللي هي اللي بتقول عنه النظريةإذا if f و g من a
+
+765
+01:02:45,960 --> 01:02:49,260
+لعن بي are integrable on a و b and have
+
+766
+01:02:49,260 --> 01:02:52,140
+antiderivatives f capital and g capital on a و b
+
+767
+01:02:52,140 --> 01:02:56,580
+then ال integration من a لعن بي f of x g x dx
+
+768
+01:02:56,580 --> 01:03:00,080
+بيسوّل f of b g of b نقص f of a g of a طبعاً
+
+769
+01:03:00,080 --> 01:03:06,360
+للcapitals نقص ال integration من a لb f small g
+
+770
+01:03:06,360 --> 01:03:12,200
+capital dx ماشي الحال بدنا الآن اللي هو البرهن
+
+771
+01:03:12,200 --> 01:03:17,750
+النظري البرهن أسهل يا جماعةهو برهانة يعتمد على
+
+772
+01:03:17,750 --> 01:03:22,970
+تفاضل اللي هو حاصل ضرب دلتين ويمكن احنا كنا اصلا
+
+773
+01:03:22,970 --> 01:03:27,110
+في اتناء اول ما بنبدأ في ال integration by bars
+
+774
+01:03:27,110 --> 01:03:31,870
+سنة عارفه للطلاب في ال calculus في الواقع في كل
+
+775
+01:03:31,870 --> 01:03:36,490
+سؤال في الأول يعني و كأننا بنبرهن النظرية و بعد
+
+776
+01:03:36,490 --> 01:03:40,170
+ذلك بنسير اللي هو حافظنا الطريقة و نعملها بشكل
+
+777
+01:03:40,170 --> 01:03:48,220
+سريع طيب نشوف كيفسمّيلي اللي هو F capital of X G
+
+778
+01:03:48,220 --> 01:03:52,060
+of capital of X ايش بتساوي H of X طبعاً F is
+
+779
+01:03:52,060 --> 01:03:53,840
+differentiable و G is differentiable زي ما كنا
+
+780
+01:03:53,840 --> 01:03:56,600
+كاتبين هذه لإنه F antiderivative و G
+
+781
+01:03:56,600 --> 01:03:59,860
+antiderivative يعني F prime لها F small و G prime
+
+782
+01:03:59,860 --> 01:04:03,700
+لها G small يعني إذا صارت الـH عبارة عن
+
+783
+01:04:03,700 --> 01:04:07,160
+differentiable و مدام differentiable إذا أكيد الـH
+
+784
+01:04:07,160 --> 01:04:12,340
+شمالها continuous على الفترة A و B ماشي الحال و مش
+
+785
+01:04:12,340 --> 01:04:18,630
+هيك كمانوبنقدر نفضلها شوفوا كيف
+
+786
+01:04:22,920 --> 01:04:26,280
+واضحة كلها .. اه واضحة كلها .. ماشي زي ما قلنا يا
+
+787
+01:04:26,280 --> 01:04:29,440
+جماعة سمينا F capital of X في G capital of X
+
+788
+01:04:29,440 --> 01:04:32,240
+بالساوية H of X F differentiable و G
+
+789
+01:04:32,240 --> 01:04:35,440
+differentiable ثم H differentiable ومن ثم اكيد
+
+790
+01:04:35,440 --> 01:04:39,780
+continuous فضولي H prime of X لو حاصل ضرب دالتين
+
+791
+01:04:39,780 --> 01:04:44,740
+التفاضل الأول في التاني زي اللي هو التاني الأول في
+
+792
+01:04:44,740 --> 01:04:49,840
+التفاضل التاني ويساويتفاضل الـ F' اللي هي F small
+
+793
+01:04:49,840 --> 01:04:54,440
+وهذه تنزل زي ما هي وتفاضل هذه اللي هي G small لذا
+
+794
+01:04:54,440 --> 01:04:59,020
+صارت الـ H' بالساوية هنا يعني وكأنه صار عنده اللي
+
+795
+01:04:59,020 --> 01:05:04,360
+هي الـH هذه بدون الـPrime antiderivative لمن؟ لهذه
+
+796
+01:05:04,360 --> 01:05:09,520
+صارت يعني الـH capital is an antiderivative of FG
+
+797
+01:05:09,520 --> 01:05:17,320
+زي مين زي FGالان كل شيء نيح F وG Integrable و F
+
+798
+01:05:17,320 --> 01:05:21,560
+capital وG continuous اذا اللي هي اكيد F capital
+
+799
+01:05:21,560 --> 01:05:25,520
+وG capital Integrable ومن ثم هيطلع كل هذا اشماله
+
+800
+01:05:25,520 --> 01:05:31,920
+is integrable ماشي كمان مرة F
+
+801
+01:05:33,190 --> 01:05:36,910
+Integrable و G-Integrable و G-Capital و F-Capital
+
+802
+01:05:36,910 --> 01:05:40,450
+continuous إذا صار الـ Integrable إذا هذا كله على
+
+803
+01:05:40,450 --> 01:05:44,830
+بعض إيش ماله صار Integrable إذا صار عندي الآن اللي
+
+804
+01:05:44,830 --> 01:05:49,550
+هي كل الشروط متحققة إذا من سبعة تلاتة مدامة ال H
+
+805
+01:05:50,390 --> 01:05:53,570
+الـ H وينها الـ H ؟ هي الـ Antiderivative لهذه
+
+806
+01:05:53,570 --> 01:05:57,470
+وهذه Integrable فبصير عندى الان ال integration من
+
+807
+01:05:57,470 --> 01:06:01,770
+هنا ب F في G زايد F capital في G DX بتساوي H of B
+
+808
+01:06:01,770 --> 01:06:11,650
+ناقص مين ناقص H of A طيب صار عندى الان الأمور وضحت
+
+809
+01:06:11,650 --> 01:06:20,490
+وطلعت النتيجة H of B اتطلع عليها وين H of B؟ هيهاH
+
+810
+01:06:20,490 --> 01:06:26,610
+of B بيساوي F capital of B في G
+
+811
+01:06:26,610 --> 01:06:30,930
+of B هايها ماشي F of B في G of B عادلة لو انتوا
+
+812
+01:06:30,930 --> 01:06:36,070
+موجودين لأ على طول قلتولي هايها الان H of A بيساوي
+
+813
+01:06:36,070 --> 01:06:41,980
+F of A في G of A هايها G of A ماشي إذا إيديناهذا
+
+814
+01:06:41,980 --> 01:06:46,060
+المقدار أعوضنا عن H of B هيها وعوضنا عن H دي ها
+
+815
+01:06:46,060 --> 01:06:49,960
+ويها وإيش جينا وهذه فصلناها لإنه Integrable إذا
+
+816
+01:06:49,960 --> 01:06:52,940
+فصلناها بنجيب واحدة على الجهة دي وواحدة على الجهة
+
+817
+01:06:52,940 --> 01:06:57,940
+دي وهو المطلوب اعملكم إياها طيب ال integration من
+
+818
+01:06:57,940 --> 01:07:01,780
+A ل B integration
+
+819
+01:07:01,780 --> 01:07:10,680
+من A ل B FG capital زائد F capital في G بساوي
+
+820
+01:07:11,790 --> 01:07:24,730
+H of B هي H of B H of B سوى F of B G of B نقص H of
+
+821
+01:07:24,730 --> 01:07:31,770
+A هي F of AG of A من الجهة الثانية بيسوي ال
+
+822
+01:07:31,770 --> 01:07:35,890
+integration F في G capital زاد ال integration F
+
+823
+01:07:35,890 --> 01:07:41,070
+capital في G small من A لعند B من A لعند B لأن إن
+
+824
+01:07:41,070 --> 01:07:44,490
+قلولي هذا على الجهة هذه بيصير عند ال integration
+
+825
+01:07:44,490 --> 01:07:51,350
+المطلوب هذاأكيد فهمتم الأصل من a لعن بيه بساوي ال
+
+826
+01:07:51,350 --> 01:07:55,570
+integration f في g capital زي f capital في g small
+
+827
+01:07:55,570 --> 01:08:06,170
+لأ خلصنا منه بساوي f of b في f of a في g of a في b
+
+828
+01:08:06,170 --> 01:08:16,030
+نقص f of a في g of a كل هذاهذا المقدار integration
+
+829
+01:08:16,030 --> 01:08:27,230
+of capital في G من A لـ B وهو المطلوب و هيك احنا
+
+830
+01:08:28,050 --> 01:08:32,490
+بنكون اليوم ان شاء الله وصلنا لعند ال first
+
+831
+01:08:32,490 --> 01:08:36,250
+substitution theorem المرة القادمة ان شاء الله
+
+832
+01:08:36,250 --> 01:08:45,010
+بنكمل المحاضرة او بنكمل المادة و بنكمل هذا ال
+
+833
+01:08:45,010 --> 01:08:49,330
+section في محاضرة قادمة ان شاء الله و إلى لقائنا
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8.srt
new file mode 100644
index 0000000000000000000000000000000000000000..2affacb2dfa0e87271e6799b5f2771a2a5bd1920
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8.srt
@@ -0,0 +1,1997 @@
+1
+00:00:05,040 --> 00:00:12,800
+بسم الله الرحمن الرحيم المحاضرة السابعة مساق تحليل
+
+2
+00:00:12,800 --> 00:00:18,640
+الرياضيات 2 اللي طلبت كلية العلوم بالجامعة الإسلامية
+
+3
+00:00:18,640 --> 00:00:23,800
+قسم الرياضيات اليوم 
+
+4
+00:00:23,800 --> 00:00:29,720
+هيكون عندنا استكمال ل .. اللي هي قواعد أو قاعدة
+
+5
+00:00:29,720 --> 00:00:36,330
+L'Hôpital تحدثنا المرة الماضية عن الحالة اللي بتحدث
+
+6
+00:00:36,330 --> 00:00:41,170
+صفر على صفر وكيف نُعالجها وتم اللي هو برهان 
+
+7
+00:00:41,170 --> 00:00:47,910
+النظريات المعنية في ذلك، الآن بدنا نتحدث على الحالة
+
+8
+00:00:47,910 --> 00:00:57,110
+اللي بتحدث عندنا أيضًا 0 على 0 ولكن الـ limit  الـ X
+
+9
+00:00:57,110 --> 00:01:00,830
+بتروح إلى ما لا نهاية أو سالب ما لا نهاية، وقلنا عنها
+
+10
+00:01:00,830 --> 00:01:05,790
+صحيحة برضه، وبعد ذلك هنتحدث عن الحالة infinity على
+
+11
+00:01:05,790 --> 00:01:10,510
+infinity، نيجي الآن لنظريتنا اللي وصلنا عندها 
+
+12
+00:01:10,510 --> 00:01:11,990
+Suppose that
+
+13
+00:01:16,220 --> 00:01:20,260
+F and G are continuous and differentiable على 
+
+14
+00:01:20,260 --> 00:01:24,600
+الفترة من A إلى ما لا نهاية، ونفترض أنه limit للـ F
+
+15
+00:01:24,600 --> 00:01:27,540
+of X لما X تروح إلى ما لا نهاية هي نفس limit G of
+
+16
+00:01:27,540 --> 00:01:31,200
+X لما X تروح إلى ما لا نهاية تساوي إيهاش صفر يعني 
+
+17
+00:01:31,200 --> 00:01:36,080
+هذه هتخلق لي حالة اللي هو صفر على صفر، ومن ثم نشوف 
+
+18
+00:01:36,080 --> 00:01:40,230
+كيف نعالجها، العنوان G of X ذاته ساوة صفر نفترض وG
+
+19
+00:01:40,230 --> 00:01:44,830
+prime of X ذاته ساوة صفر لكل X أكبر من مين؟ من اللي
+
+20
+00:01:44,830 --> 00:01:48,290
+هو المقصود هنا بيه، اللي هي أيش؟ A يعني هذه اللي هي 
+
+21
+00:01:48,290 --> 00:01:52,970
+مين اللي هي المقصود أنه A، يا ما نخلي هذه A يا ما
+
+22
+00:01:52,970 --> 00:01:58,330
+هذه A then limit F of X على G of X لما X تروح لما 
+
+23
+00:01:58,330 --> 00:02:01,930
+النهاية هتكون بالظبط أيش؟ بتساوي limit F prime على
+
+24
+00:02:01,930 --> 00:02:06,050
+G prime of X as X goes to infinity provided that
+
+25
+00:02:06,050 --> 00:02:09,590
+هذا الـ limit بتساوي قيمة معينة أو تساوي infinity
+
+26
+00:02:09,590 --> 00:02:15,730
+أو سالب infinity، الآن احنا في الواقع اللي هنسوي 
+
+27
+00:02:16,910 --> 00:02:21,910
+وكأنه يعني عملية تحويل اللي نخلي بدل X تروح إلى 
+
+28
+00:02:21,910 --> 00:02:26,950
+ما لا نهاية نخلي اللي هو متغير وروح لصفر يعني بدنا
+
+29
+00:02:26,950 --> 00:02:31,490
+نعيد تعريف الدالة بحيث أنه يروح هذا المتغير يعني
+
+30
+00:02:31,490 --> 00:02:34,610
+وكأنه نقول 1 على T ويروح على .. اه 1 على X
+
+31
+00:02:34,610 --> 00:02:37,950
+يروح لصفر لما X تروح إلى ما لا نهاية، ومن ثم نطبق
+
+32
+00:02:37,950 --> 00:02:41,030
+النظريات اللي أخدناها المرة الماضية، دعونا نشوف أيش
+
+33
+00:02:41,030 --> 00:02:47,050
+اللي بنحكيه الآن عند .. الـ الـ function F و G، F و
+
+34
+00:02:47,050 --> 00:02:54,430
+G معرفات من A و ما لا نهاية لعند اللي هو R الآن
+
+35
+00:02:54,430 --> 00:02:57,970
+عندي اللي هو بده أعرف اللي هو عشان أقدر أستخدم
+
+36
+00:02:57,970 --> 00:03:01,610
+نظرية المرة اللي فاتت، المرة الماضية بده أعرف
+
+37
+00:03:01,610 --> 00:03:08,030
+دالتين F capital و G capital من اللي هو عندي من
+
+38
+00:03:08,030 --> 00:03:15,560
+Zero لعند اللي هو 1 على A لعند اللي هو صفر، واضحة
+
+39
+00:03:15,560 --> 00:03:28,640
+الصورة تقريبا أكيد هنقول define F أو G as F of اللي
+
+40
+00:03:28,640 --> 00:03:36,960
+نسميها اللي هو T بتساوي اللي هو G أو F of 1 على T 
+
+41
+00:03:36,960 --> 00:03:42,460
+لما T اللي هي لا تساوي صفر وطبعا T أنا معرفها
+
+42
+00:03:42,460 --> 00:03:47,140
+اللي هي في الفترة من Zero لعند 1 على A وده
+
+43
+00:03:47,140 --> 00:03:51,560
+بيدعرف عند Zero بقول صفر إذا كان T أيش؟ بدها تساوي
+
+44
+00:03:51,560 --> 00:03:59,760
+بتساوي صفر الآن الـ الـ G of T بتساوي نفس الأسلوب 
+
+45
+00:03:59,760 --> 00:04:06,240
+G of 1 على T لما T لا تساوي 0 وصفر إذا كانت T
+
+46
+00:04:06,240 --> 00:04:13,350
+أشمالها بتساوي 0، الآن واضح العلاقة بين الدالتين اللي
+
+47
+00:04:13,350 --> 00:04:19,510
+هو F و F اللي هو على أساس أني أحول الـ limit 
+
+48
+00:04:19,510 --> 00:04:25,650
+المعطى عندي limit F of X لما X تروح إلى ما لا نهاية
+
+49
+00:04:25,650 --> 00:04:30,450
+تصير عندي الـ limit بالنسبة لهذه اللي هي الـ 1 على X 
+
+50
+00:04:30,450 --> 00:04:35,320
+بتاعتنا هتروح لمين؟ في هذه الحالة للصفر، وبنستخدم ومن
+
+51
+00:04:35,320 --> 00:04:40,600
+ثم بنكون حصلنا على المتغير لإن أنا T بروح للصفر as 
+
+52
+00:04:40,600 --> 00:04:44,740
+X بتروح إلينا لنهاية ومن ثم بنستخدم النظريات اللي 
+
+53
+00:04:44,740 --> 00:04:49,940
+أثبتنا صحتها في حالة T بتروح إلى A من اليمين لإن
+
+54
+00:04:49,940 --> 00:04:53,060
+احنا هنروح ل Zero من اليمين هذه اللي هي الخطة اللي 
+
+55
+00:04:53,060 --> 00:04:58,480
+هنمشي عليها، بنشوف أول شيء واضح أن الـ F والـ G الـ F
+
+56
+00:04:59,090 --> 00:05:03,930
+والـ G are differentiable على مين؟ على الفترة من
+
+57
+00:05:03,930 --> 00:05:10,550
+صفر لعند 1 عليه، هاي 1، 2 ومدام 
+
+58
+00:05:10,550 --> 00:05:13,810
+differentiable ده ن continuous عليها ومش هيك كمان
+
+59
+00:05:13,810 --> 00:05:19,230
+لو جيت أخدت أنت هذه بتحسبها لحالك limit F of T as
+
+60
+00:05:19,230 --> 00:05:27,640
+T بتروح للصفر اللي هي بتساوي limit F of 1 على T as 
+
+61
+00:05:27,640 --> 00:05:32,340
+T بتروح للصفر وهتلاقيها أيش؟ هتساوي، هتساوي صفر
+
+62
+00:05:32,340 --> 00:05:36,760
+فصارت اللي هي الـ F الـ capital is continuous على
+
+63
+00:05:36,760 --> 00:05:43,490
+الفترة من 0 لعند 1 على A similarly limit G of T لما
+
+64
+00:05:43,490 --> 00:05:47,650
+T تروح للـ Zero طبعا كل الـ Zero من وين؟ طبيعي من
+
+65
+00:05:47,650 --> 00:05:50,950
+اليمين لأنه احنا عالمنا اللي بنشتغل عليه اللي هي 
+
+66
+00:05:50,950 --> 00:05:53,730
+من عند Zero لعند 1 على أي، من المنطقة المودة بقى
+
+67
+00:05:53,730 --> 00:05:58,750
+إذن هروح للـ Zero من جهات اليمين هيساوي نفس الشيء
+
+68
+00:05:58,750 --> 00:06:03,290
+limit F of 1 على T لما T تروح للـ Zero من 
+
+69
+00:06:03,290 --> 00:06:07,590
+اليمين ويساوي اللي جايها برضه أشمالها بتساوي صفر 
+
+70
+00:06:08,590 --> 00:06:12,970
+الآن صارت الآن الـ F والـ G أشمالين are continuous
+
+71
+00:06:12,970 --> 00:06:17,390
+على الفترة اللي عندنا و differentiable على الفترة 
+
+72
+00:06:17,390 --> 00:06:20,510
+الـ open و .. و .. و .. و .. و .. و .. و .. و .. و
+
+73
+00:06:20,510 --> 00:06:20,590
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+74
+00:06:20,590 --> 00:06:20,790
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+75
+00:06:20,790 --> 00:06:21,070
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+76
+00:06:21,070 --> 00:06:21,290
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+77
+00:06:21,290 --> 00:06:23,330
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+78
+00:06:23,330 --> 00:06:23,350
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+79
+00:06:23,350 --> 00:06:23,650
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+80
+00:06:23,650 --> 00:06:29,210
+.. و .. و .. و .. و .. و .. 
+
+81
+00:06:37,480 --> 00:06:43,680
+بنجي بنحصل على ما يلي، خليني بس اسمحولي أمسح هذه
+
+82
+00:06:43,680 --> 00:06:49,760
+الـ data في الذاكرة عندي، بدي أحسب أنا غاية أن أوجد مين؟
+
+83
+00:06:49,760 --> 00:06:53,380
+يا جماعة limit اللي هو F of X على G of X لما X
+
+84
+00:06:53,380 --> 00:07:00,780
+تروح إلى وين؟ إلى ما لا نهاية لأن limit F of X على G
+
+85
+00:07:00,780 --> 00:07:07,720
+of X لما X تروح إلى ما لا نهاية بتساوي عبارة عن الآن
+
+86
+00:07:07,720 --> 00:07:12,180
+الـ X بتروح إلى ما لا نهاية F and والـ F الواحد على X
+
+87
+00:07:12,180 --> 00:07:20,660
+وين بتروح؟ اللي هو الصفر بتساوي limit F of 1 على
+
+88
+00:07:20,660 --> 00:07:26,040
+T خلينا نسميها على G of 1 على T لما T تروح إلى
+
+89
+00:07:26,040 --> 00:07:32,420
+مين؟ إلى الصفر من اليمين، أو إذا كان عندك اضطراب 
+
+90
+00:07:32,420 --> 00:07:38,640
+Confusion شيء والتكن X ما في مشكلة X لما X تروح لـ 0 
+
+91
+00:07:38,640 --> 00:07:43,660
+من اليمين اللي هي بتصير عند أسف الواحد على X بتروح
+
+92
+00:07:43,660 --> 00:07:47,860
+لـ 0 من اليمين if and only if الـ X بتروح لمين؟ إلى
+
+93
+00:07:47,860 --> 00:07:51,520
+ما لا نهاية، ومن ثم بخطوة ثانية بترجعها للي كتبته 
+
+94
+00:07:51,520 --> 00:07:57,980
+قبل بشوية limit اللي هو عبارة عن F of T على اللي هو
+
+95
+00:07:57,980 --> 00:08:04,420
+G of T لما T تروح لـ 0 من وين؟ من اليمين، باشي
+
+96
+00:08:04,420 --> 00:08:08,120
+الحال، أنا الآن بدي أفضل في الواقع بالنسبة لمن؟
+
+97
+00:08:08,120 --> 00:08:14,040
+بالنسبة له اللي هو الـ X عندي اللي هو وديت واحدة
+
+98
+00:08:14,040 --> 00:08:19,880
+الـ X من اليمين وصارت عندي اللي هي مكان لأ لأ يا
+
+99
+00:08:19,880 --> 00:08:26,200
+جماعة هذا نظام هي بتضلها آسف اه؟ X X وهنا بتصير
+
+100
+00:08:26,200 --> 00:08:30,900
+أننا مين؟ 1 على T وهنا 1 على T، واضح ليش؟ 
+
+101
+00:08:30,900 --> 00:08:36,540
+استبدلنا هنا احنا الواحد على X كله بـ T فصارت هذه
+
+102
+00:08:36,540 --> 00:08:41,320
+اللي هي 1 على T وهذه 1 على T، الآن صارت
+
+103
+00:08:41,320 --> 00:08:44,780
+النظرية السابقة مطبقة لأن الـ T بتروح لمين؟ لـ Zero
+
+104
+00:08:44,780 --> 00:08:49,780
+من لمين؟ لأن هدف الواقع مين؟ هي اللي قلنا عنها شيء
+
+105
+00:08:49,780 --> 00:08:57,850
+هذه بتساوي limit F of T على G of T لما T تروح إلى
+
+106
+00:08:57,850 --> 00:09:01,310
+Zero من وين؟ من اليمين، وهذا متطبق لأن هذا Zero
+
+107
+00:09:01,310 --> 00:09:05,490
+وهذا Zero زي ما عملنا قبل بشوية إذا صار أيش؟ بيساوي 
+
+108
+00:09:05,490 --> 00:09:10,990
+تفاضل الأول على تفاضل الثاني في 100 بالنسبة لعن 
+
+109
+00:09:10,990 --> 00:09:17,220
+نظرية السابقة، إذا بيصير limit التي هي التفاضل هذه 
+
+110
+00:09:17,220 --> 00:09:21,060
+من هنا F prime of T أيش بيساوي يا شباب؟ تفاضل
+
+111
+00:09:21,060 --> 00:09:24,960
+هذه، عن T بتروح للـ 0، إذن هي عبارة عن ناقص 1 على 
+
+112
+00:09:24,960 --> 00:09:34,000
+T تربيع في F prime of 1 على T، تفاضل F of T، F of T 
+
+113
+00:09:34,000 --> 00:09:38,930
+هيها يعني T لا تساوي صفر تفاضلها F prime بساوي ناقص
+
+114
+00:09:38,930 --> 00:09:42,710
+اللي هو F prime الواحد على T فتفاضل اللي جوا اللي
+
+115
+00:09:42,710 --> 00:09:47,450
+هو عبارة عن أيش؟ ناقص 1 على T تربيع، واضح على 
+
+116
+00:09:47,450 --> 00:09:53,430
+similarly ناقص 1 على T تربيع في G prime of 1 
+
+117
+00:09:53,430 --> 00:10:00,200
+على T لما الـ T تروح لمين؟ إلى الـ 0 من اليمين، النقص 1
+
+118
+00:10:00,200 --> 00:10:03,160
+دلت الـ T تربيع، الـ ناقص 1 دلت T تربيع بروحن مع بعض
+
+119
+00:10:03,160 --> 00:10:09,780
+نرجع لأصلنا بيصير limit F prime بده استبدل الآن الـ 
+
+120
+00:10:09,780 --> 00:10:17,800
+1 على T بالـ X على G prime برضه نفس الشيء على X والـ
+
+121
+00:10:17,800 --> 00:10:22,540
+T تستبدل بمين؟ اللي هو عبارة عن 1 على X بده تروح للـ
+
+122
+00:10:22,540 --> 00:10:29,270
+0 من اليمين يعني الـ X بده تروح لمين؟ إلى ما لا نهاية، بيصير
+
+123
+00:10:29,270 --> 00:10:35,510
+عندي اللي هو limit F of X على G of X بساوي limit F
+
+124
+00:10:35,510 --> 00:10:40,350
+prime of X على G prime Y على X وهو المطلوب، نيجي 
+
+125
+00:10:40,350 --> 00:10:44,690
+للنظرية بعدها، النظرية اللي بدها شغل خلينا نشوف 
+
+126
+00:10:44,690 --> 00:10:53,110
+النظرية، اطلع لفوق ونركز نشوف أيش هو النظرية بتحكي 
+
+127
+00:10:53,110 --> 00:10:57,270
+ومن ثم نذهب إلى برهان النظرية
+
+128
+00:11:04,010 --> 00:11:11,750
+عشان مساحة اللوحة شوية خلينا نتبع على التلخيص ومن
+
+129
+00:11:11,750 --> 00:11:19,110
+ثم بنبرهن Theorem 6.3.6 اللي هي باختصار هي حالة ما لا 
+
+130
+00:11:19,110 --> 00:11:22,570
+نهاية على ما لا نهاية، نفترض أن الـ F والـ G are
+
+131
+00:11:22,570 --> 00:11:26,470
+differentiable على الفترة من A و B يعني احنا شغلنا 
+
+132
+00:11:26,470 --> 00:11:34,230
+الآن على الفترة أي كانت هذه الفترة من عند A لعند
+
+133
+00:11:34,230 --> 00:11:41,250
+ما لا نهاية لعند B لأننا مفترضين أن الـ F والـ G اللي هو
+
+134
+00:11:41,250 --> 00:11:46,250
+قابل للاشتقاق على الفترة A و B و limit F of X لما
+
+135
+00:11:46,250 --> 00:11:48,890
+X تروح إلى ما لا نهاية .. إلى Zero من اليمين ..
+
+136
+00:11:48,890 --> 00:11:53,230
+إلى A من اليمين أيش بيساوي؟ Infinity و limit G of X
+
+137
+00:11:53,230 --> 00:11:56,090
+لما X تروح إلى الـ A من اليمين برضه أيش مالها؟
+
+138
+00:11:56,090 --> 00:12:01,370
+Infinity يعني بمعنى آخر صار لما عرضت علينا limit F 
+
+139
+00:12:01,370 --> 00:12:06,980
+of X على G of X لما X راحت إلى الـ A من اليمين
+
+140
+00:12:06,980 --> 00:12:10,660
+وجدنا limit اللي فوق ما لا نهاية و limit اللي تحت
+
+141
+00:12:10,660 --> 00:12:14,260
+ما لا نهاية يعني حصلنا على الكمية الغير معينة
+
+142
+00:12:14,260 --> 00:12:18,220
+Infinity على Infinity إذا هذه النظرية ستعالج هذه 
+
+143
+00:12:18,220 --> 00:12:23,060
+الحالة، طبعا الآن اللي هي بطريقة مشابهة للسابقة
+
+144
+00:12:23,060 --> 00:12:29,080
+والآن احنا طبعا هنقوم ببرهان هذه النظرية، نكمل النص 
+
+145
+00:12:29,820 --> 00:12:35,900
+الآن فرضنا أيضًا أنه اللي هو الـ G of X لا تساوي
+
+146
+00:12:35,900 --> 00:12:40,160
+صفر والـ G prime of X لا تساوي صفر لكل X وإما لها
+
+147
+00:12:40,160 --> 00:12:44,920
+في الفترة A و B يعني مفترض أن الـ G of X لا تساوي
+
+148
+00:12:44,920 --> 00:12:52,860
+صفر على كل الفترة A و B نأتي إلى الجزئية الأولى من
+
+149
+00:12:52,860 --> 00:12:57,380
+النظرية if limit F' على G' بساوي L element in R
+
+150
+00:12:57,380 --> 00:13:02,040
+إذا ف limit F على G برضه أيضا أيش؟ هتساوي L اللي هي 
+
+151
+00:13:02,040 --> 00:13:05,280
+نظريات اللي قبل بشوية بنفس .. بنفس .. خليني أقول 
+
+152
+00:13:05,280 --> 00:13:10,160
+بنفس المنطق ونفس النصوص بس مبدل 0 على 0 بـ ما لا
+
+153
+00:13:10,160 --> 00:13:13,560
+نهاية فلان limit F prime على G prime صارت ما لا
+
+154
+00:13:13,560 --> 00:13:17,700
+نهاية أو سالب ما لا نهاية، برضه النظرية بتظبط، سير 
+
+155
+00:13:17,700 --> 00:13:23,310
+limit اللي عندنا هذه هو بالظبط limit الـ F على G as 
+
+156
+00:13:23,310 --> 00:13:26,170
+X وتروحنا الـ A من اليمين بساوي Infinity وسالب
+
+157
+00:13:26,170 --> 00:13:30,550
+Infinity يعني في النهاية اللي ببحث عنه هذا هو في 
+
+158
+00:13:30,550 --> 00:13:34,650
+الحالة هذه أو في الحالة اللي فوق بساوي limit مين؟
+
+159
+00:13:34,650 --> 00:13:37,750
+الـ F prime على G prime اللي هي الطريقة المحدودة
+
+160
+00:13:37,750 --> 00:13:40,330
+بنفضل اللي فوق وبنفضل اللي تحت وبنوجد limit هين 
+
+161
+00:13:40,330 --> 00:13:44,670
+بيكونين هين limit اللي هي الأصل اللي احنا بدنا إياه
+
+162
+00:13:45,410 --> 00:13:49,590
+الآن نيجي اللي هو in Berlin اللي هو A و B طبعا
+
+163
+00:13:49,590 --> 00:13:54,410
+بنفس الأسلوب الآن 
+
+164
+00:13:54,410 --> 00:14:02,790
+قبل ما نبدأ احنا لو اجينا وقلنا limit F of X لما
+
+165
+00:14:02,790 --> 00:14:06,610
+X تروح لأي شيء مثلا A من اليمين بساوي L
+
+166
+00:14:11,700 --> 00:14:14,480
+وقلنا اللي هو for every Epsilon تعريفها for every
+
+167
+00:14:14,480 --> 00:14:16,660
+Epsilon أكبر من صفر there exists a Delta أكبر من
+
+168
+00:14:16,660 --> 00:14:21,300
+صفر such that if X element in A طبعا من اليمين 
+
+169
+00:14:21,300 --> 
+
+201
+00:17:04,230 --> 00:17:11,420
+بحيث أنه لكل x في الفترة من a إلى a زائد دلتا بيصير
+
+202
+00:17:11,420 --> 00:17:16,780
+عندي لكل x المتنية زائد دلتا بتطلع عندي then f
+
+203
+00:17:16,780 --> 00:17:23,420
+prime of x على g prime of x ناقص الـ L أصغر من مين
+
+204
+00:17:23,420 --> 00:17:32,060
+من الـ epsilon إذا الآن من هذه الـ limit حصلت على أنه
+
+205
+00:17:32,060 --> 00:17:36,540
+لأي إبسلون بين الـ 0 و نصف بقدر ألاقي Delta بحيث
+
+206
+00:17:36,540 --> 00:17:40,540
+لكل الإكسات في المنطقة هذه هذه الـ Inquality إيش
+
+207
+00:17:40,540 --> 00:17:44,880
+ما لها تتحقق الآن بس خليني أكملها دي شوية عشان
+
+208
+00:17:44,880 --> 00:17:52,040
+بستخدمها لشغلات أخرى عند A زائد Delta هذا بس مجرد
+
+209
+00:17:52,040 --> 00:17:56,340
+تعريف الـ Limit F' على الـ G' لما X تروح للـ A من
+
+210
+00:17:56,340 --> 00:17:59,120
+اليمين الآن
+
+211
+00:18:01,110 --> 00:18:04,910
+بدي أختار .. بدي أسهل على حالي بدل من كل مرة أقول
+
+212
+00:18:04,910 --> 00:18:08,130
+a زائد الـ delta وبعد شوية ألاقي delta prime وأخذ
+
+213
+00:18:08,130 --> 00:18:12,010
+الـ minimum بينهم الأخرى أجدفم عليه عشان اللي بحكيه
+
+214
+00:18:12,010 --> 00:18:17,910
+بدي أخذ اللي هو أقول choose c1 element in a و a زائد
+
+215
+00:18:17,910 --> 00:18:24,630
+الـ delta الآن اختارلي c1 من الـ a لعند مين لعند الـ
+
+216
+00:18:24,630 --> 00:18:30,340
+a زائد الـ delta سمّيه c1 يعني مجرد اختيار أنا اخترته 
+
+217
+00:18:30,340 --> 00:18:38,260
+الآن choose .. choose C1 element in A و A زائد
+
+218
+00:18:38,260 --> 00:18:44,140
+دلتا يعني C1 وين موجود؟ موجود في المنطقة التي
+
+219
+00:18:44,140 --> 00:18:50,140
+تتحقق فيها F' على G' ناقص أصغر من ميل من إبسل عشان
+
+220
+00:18:50,140 --> 00:18:56,080
+تستخدمها بعد شوية واضح طيب هذه من جهة الآن من الجهة
+
+221
+00:18:56,080 --> 00:19:03,100
+الثانية عند limit f of x لما x تروح إلى مالة نهاية
+
+222
+00:19:03,100 --> 00:19:09,880
+آسف للـ a من اليمين بسوء أيش هو معطيني هي النظرية
+
+223
+00:19:09,880 --> 00:19:13,540
+هي limit f of x لما x تروح إلى الـ a من اليمين بسوء
+
+224
+00:19:13,540 --> 00:19:15,760
+limit g of x لما x تروح إلى الـ a من اليمين بسوء
+
+225
+00:19:15,760 --> 00:19:23,170
+أيّش ماله مالة نهاية من تعريفها هذا أيّش يعني لكل K
+
+226
+00:19:23,170 --> 00:19:29,270
+اللي هو element in R there exists Delta A' أكبر من
+
+227
+00:19:29,270 --> 00:19:36,590
+0 such that اللي هو F of X أكبر من اللي هو الـ K
+
+228
+00:19:36,590 --> 00:19:42,210
+لكل X و N موجودة في الـ A و A زائد مين Delta
+
+229
+00:19:42,210 --> 00:19:49,300
+Prime، مظبوط ولا لا؟ ماشي الحال الآن لكل x element
+
+230
+00:19:49,300 --> 00:19:52,200
+in A و A زائد الـ delta prime هذه الـ F of X أكبر من
+
+231
+00:19:52,200 --> 00:20:02,420
+مين؟ من K الآن لو كانت الـ A و الـ A .. الآن عندي
+
+232
+00:20:02,420 --> 00:20:07,540
+لكل subset من هذه .. لكل subset .. لو كان عندي A و
+
+233
+00:20:07,540 --> 00:20:10,940
+C2 مثلا subset من الـ A و الـ A زائد الـ delta prime
+
+234
+00:20:12,470 --> 00:20:17,070
+برضه الـ F of X أكبر من كده تحققها تحقق إيش لكل X
+
+235
+00:20:17,070 --> 00:20:24,610
+وين موجودة في الفترة من A لعند مين لعند C2 الآن لو
+
+236
+00:20:24,610 --> 00:20:30,410
+كانت الـ Delta اللي جبتها برا ولا جوا بقدر أختار
+
+237
+00:20:30,410 --> 00:20:36,990
+جوا اللي هو C2 بحيث أنه لو الـ C2 هذي هو اللي
+
+238
+00:20:36,990 --> 00:20:42,450
+اخترتها جوات من الـ A والـ A زائد دلتا بتظل الـ F
+
+239
+00:20:42,450 --> 00:20:47,470
+of X A شمالها أكبر من مين؟ من الـ K، واضح؟ إذا من
+
+240
+00:20:47,470 --> 00:20:52,330
+هذه بدي أستفيد شغلتين بدي أطبّق التعريف هذا لـ K
+
+241
+00:20:52,330 --> 00:20:56,730
+محددة مين الـ K اللي بدي أطبّقها؟ اللي عبارة عن F
+
+242
+00:20:56,730 --> 00:21:02,560
+عند مين؟ عند C1 واضح؟ إذا الآن بما أنه limit f of X
+
+243
+00:21:02,560 --> 00:21:05,680
+ثم X تروح إلى الـA باليمين تساوي Infinity إذا for
+
+244
+00:21:05,680 --> 00:21:13,720
+K .. إذا for f of C1 there exists Delta Prime أي
+
+245
+00:21:13,720 --> 00:21:19,840
+Delta Prime معينة بحيث أن f of X أكبر من 100 من f
+
+246
+00:21:19,840 --> 00:21:27,060
+of C1 لكل X موجودة بين الـA والـA زائد Delta Prime
+
+247
+00:21:27,680 --> 00:21:31,860
+أنا بدي أختار في مين لكل X الموجودة بين الـ A و
+
+248
+00:21:31,860 --> 00:21:37,940
+الـ C2 حيث الـ C2 موجودة في الفترة هذه و بديها من
+
+249
+00:21:37,940 --> 00:21:44,120
+C1 ولجاي إذا عشان هيك بقدر أقول then we can
+
+250
+00:21:44,120 --> 00:21:50,280
+choose C2 بين A و C1 الـ C2 بقدر اختارها فعلاً
+
+251
+00:21:50,280 --> 00:21:55,640
+بختارها بس طبعاً وين تشتري تشريع اللي اختيارها أنها
+
+252
+00:21:55,640 --> 00:22:01,420
+تكون بين الفترة A و A زائد Delta Prime و بديهاها
+
+253
+00:22:01,420 --> 00:22:06,900
+تيجي في داخل الـ A و الـ C1 إذا then we can choose
+
+254
+00:22:06,900 --> 00:22:13,200
+C2 element A of C1 بحيث أن الـ F of X أكبر من مين
+
+255
+00:22:13,200 --> 00:22:22,810
+من F of C1 لكل X وينفي الفترة بين A و C 2 طيب
+
+256
+00:22:22,810 --> 00:22:28,050
+هذه معناته أن الـ F of X هذه لكل الـ Xات اللي هان الـ
+
+257
+00:22:28,050 --> 00:22:34,290
+F of X أكيد أيّ شمالها لها تساوي أيش F of C واحد هل
+
+258
+00:22:34,290 --> 00:22:38,930
+جد تعرف ليش بديها هذه عشان لزوم تعريف شغل معينة
+
+259
+00:22:38,930 --> 00:22:44,770
+بنفعش تكون الـ F of X أيش بتساوي F of C واحد الآن
+
+260
+00:22:44,770 --> 00:22:55,550
+similarly Similarly اللي هو بما أنه limit G of X
+
+261
+00:22:55,550 --> 00:23:00,690
+تساوي مالة نهاية as X بتروح للـ A من اليمين إذا
+
+262
+00:23:00,690 --> 00:23:13,160
+بقدر ألاقي اللي هو C1 برايم بحيث أنه G of X أكبر أو
+
+263
+00:23:13,160 --> 00:23:17,240
+لا تساوي طبعاً أكبر الـ thrifty يعني لا تساوي G of
+
+264
+00:23:17,240 --> 00:23:27,860
+C1 لكل X موجودة في الـ A و C1 إيش إبراهيم اه G C1
+
+265
+00:23:27,860 --> 00:23:34,680
+إبراهيم اه G C1 إبراهيم الآن
+
+266
+00:23:34,680 --> 00:23:40,760
+الـ .. الـ .. الـ .. الباخد الـ Minimum عشان طبعاً هذا
+
+267
+00:23:40,760 --> 00:23:46,120
+اللي لجهة N C1' ممكن اللي هي لكل X في الـ A و C1'
+
+268
+00:23:46,620 --> 00:23:52,840
+و لكل X element in A و C2' بط مانش .. مش عارف مين
+
+269
+00:23:52,840 --> 00:23:58,670
+الأكبر من هذول فبأخذ الـ Minimum من الجهتين و
+
+270
+00:23:58,670 --> 00:24:03,090
+بسمّيها C1 مثلاً أو C2 الـ minimum من التنتين C2
+
+271
+00:24:03,090 --> 00:24:09,550
+فبتصير عندي لأن لكل X في الـ A و الـ minimum بينهن
+
+272
+00:24:09,550 --> 00:24:14,630
+هدى بتظبط و هدى بتظبط يعني الـ G of X لا تساوي F of
+
+273
+00:24:14,630 --> 00:24:20,670
+C1 و الـ F of X لا تساوي مين؟ F of C1 برايم و منه
+
+274
+00:24:20,670 --> 00:24:25,870
+بعرف اللي بديه أو بعمل زي ما هو عامل في الكتاب إيش
+
+275
+00:24:25,870 --> 00:24:29,280
+اللي بقوله؟ نشوف اللي بعدها إذا الآن اللي اتفقنا
+
+276
+00:24:29,280 --> 00:24:38,540
+عليه أنه في الفترة من A لعند A زائد Delta هذه
+
+277
+00:24:38,540 --> 00:24:45,480
+متحققة هاي واحد واللي اتفقنا عليه أن F of X لا
+
+278
+00:24:45,480 --> 00:24:50,140
+تساوي F of C واحد في الفترة من A لعند C 2
+
+279
+00:24:50,140 --> 00:24:55,580
+شمالها متحققة يعني بمعنى آخر الـ Inquality هذه
+
+280
+00:24:55,580 --> 00:25:03,800
+و هذه التنتين محققات من أين؟ من A لعند C الاتنين
+
+281
+00:25:03,800 --> 00:25:05,360
+واضح؟
+
+282
+00:25:11,650 --> 00:25:17,590
+الآن وصلنا لمرحلة أنه نقدر نعرف الدالة اللي بتديها
+
+283
+00:25:17,590 --> 00:25:23,890
+اللي هي F of X بتساوي واحد ناقص F of C واحد على F
+
+284
+00:25:23,890 --> 00:25:30,470
+of X و واحد ناقص G of C واحد على G of X لكل X وين
+
+285
+00:25:30,470 --> 00:25:37,330
+موجودة بين A و C 2 و F of C واحد لا تساوي F of X
+
+286
+00:25:37,330 --> 00:25:43,340
+مظبوط؟ هيا أعملها اه لأن وين بدي أعرف هذه الدالة F
+
+287
+00:25:43,340 --> 00:25:50,480
+من A لعند مين اللي هي بتتحقق عليها خاصية F of X ده
+
+288
+00:25:50,480 --> 00:25:55,080
+تساوى مين F of C واحد و بتتحقق عليها مين اللي هي
+
+289
+00:25:55,080 --> 00:26:00,480
+الخاصية اللي عندي هذه لأن كله هيلزمني هذا إذا الآن
+
+290
+00:26:05,340 --> 00:26:09,980
+اتفقنا عليها دي اه عشان بدي أمسح أنه عندي f prime
+
+291
+00:26:09,980 --> 00:26:13,800
+على الجي برايم ناقصها الأصغر من إبسلون لكل الـ Xات
+
+292
+00:26:13,800 --> 00:26:18,700
+من A لعند A زائد ديلتا لكل الـ Xات اللي هانا
+
+293
+00:26:18,700 --> 00:26:22,040
+واتفقنا أن F of X لا تساوي F of C one برضه في
+
+294
+00:26:22,040 --> 00:26:27,350
+الفرق في المنطقة من إيش من A لـ C2 اتفقنا عليه الآن
+
+295
+00:26:27,350 --> 00:26:33,510
+نيجي نعرف الدالة اللي هتوصلني لهدفي خد الآن f of x
+
+296
+00:26:33,510 --> 00:26:42,670
+بتساوي اللي هو واحد ناقص f of c واحد على f of x
+
+297
+00:26:42,670 --> 00:26:52,450
+على واحد ناقص g of c واحد على g of x الآن g of c
+
+298
+00:26:52,450 --> 00:26:55,810
+واحد مستحيل تساوي g of x من أيّ أسباب اللي حكيناها
+
+299
+00:26:55,810 --> 00:27:01,710
+وليس سبب آخر أيضاً اللي هو ليش لأنه عندي الـ g prime
+
+300
+00:27:01,710 --> 00:27:07,570
+الـ g prime of x ده يساوي صفر إذا حسب Rolle's
+
+301
+00:27:07,570 --> 00:27:13,570
+theorem مستحيل الـ g of c واحد اللي هو تساوي g of
+
+302
+00:27:13,570 --> 00:27:20,530
+x ليش عندي g continuous
+
+303
+00:27:21,550 --> 00:27:29,350
+on a .. طبعاً الـ x واللي موجودة جوها منها الآن من x
+
+304
+00:27:29,350 --> 00:27:37,110
+لعند c واحد صح؟ و g is differentiable on x لعند
+
+305
+00:27:37,110 --> 00:27:44,090
+اللي هي c واحد open واضحة؟
+
+306
+00:27:44,090 --> 00:27:48,670
+لأنه أثر هذا continuous و differentiable من a لعند
+
+307
+00:27:48,670 --> 00:27:52,550
+b العلمي إذا في الجزئية هذه إذا أكيد اللي هم تتحقق
+
+308
+00:27:52,550 --> 00:28:01,270
+هذه الآن لو زي ما بنقول أن جي برا لو لو عندي بدي
+
+309
+00:28:01,270 --> 00:28:09,250
+يكون g of x بسوء g of c واحد معناته حسب role
+
+310
+00:28:09,250 --> 00:28:16,360
+theorem هيعطيني there exist cx element in x و c1
+
+311
+00:28:16,360 --> 00:28:22,800
+such that g prime of cx هي ساوي صفر وهذا مستحيل
+
+312
+00:28:22,800 --> 00:28:27,320
+لإن هو معطيني الـ g prime of x لا تساوي صفر لكل x
+
+313
+00:28:27,320 --> 00:28:34,630
+element من a و b معناته معناه الحديث أنه الـ G of C
+
+314
+00:28:34,630 --> 00:28:41,510
+واحد والـ G of X مستحيل يكون متساويات لكل الإكسات
+
+315
+00:28:41,510 --> 00:28:47,230
+اللي في الفترة من A لعند مين لعند C 2 واضح إذا
+
+316
+00:28:47,230 --> 00:28:50,730
+المقام لا يساوي صفر إذا اللي هو هذي is well
+
+317
+00:28:50,730 --> 00:29:00,950
+defined على مين على الفترة A مظبوط و C2 خلّيها في
+
+318
+00:29:00,950 --> 00:29:05,070
+الذاكرة طيب اطلع لفوق و حسب لل limit الآن هذا
+
+319
+00:29:05,070 --> 00:29:12,070
+عرفناها حفظناها بنضطر أن أمسح الآن احسب لل limit
+
+320
+00:29:12,070 --> 00:29:20,230
+للي فوق limit F of X لما X تروح لي وين لي A من
+
+321
+00:29:20,230 --> 00:29:23,390
+اليمين تساوي واحد
+
+322
+00:29:24,530 --> 00:29:32,650
+ناقص limit F of C1 عدد على F of X لما X تروح للـ A
+
+323
+00:29:32,650 --> 00:29:36,950
+من اليمين على تجرأت لأنه عارف أن ال limit موجودة
+
+324
+00:29:36,950 --> 00:29:42,290
+على التوزيع تجرأت عليه لأنه عارف limit G of C1
+
+325
+00:29:42,290 --> 00:29:49,530
+على G of X لما X تروح لوين للـ A من اليمين واضحة
+
+326
+00:29:49,530 --> 00:29:56,130
+جدّاً هذه ال limit صفر لأن الـ F of X وين بتروح لمالة
+
+327
+00:29:56,130 --> 00:30:00,010
+نهاية هي بداية الموضوع مش بداية الموضوع كان عندنا
+
+328
+00:30:00,010 --> 00:30:05,010
+أن ال limit F of X على G of X لما X تروح إلى A من
+
+329
+00:30:05,010 --> 00:30:08,670
+اليمين اللي هو سببّت لكل القصة أن هذه ال limit
+
+330
+00:30:08,670 --> 00:30:11,230
+مالة نهاية وهذه مالة نهاية صارت مالة نهاية مالة
+
+331
+00:30:11,230 --> 00:30:15,710
+نهاية وهي اللي خلّتني اروح بهذا الاتجاه إذا هذه
+
+332
+00:30:16,610 --> 00:30:19,750
+بتروح إلى ملأ نهاية إذا هذه بتروح إلى صفر وهذه
+
+333
+00:30:19,750 --> 00:30:22,330
+الملأ نهاية بتروح إلى صفر إذا هذا كله على بعض إيش
+
+334
+00:30:22,330 --> 00:30:30,170
+بيساوي؟ بيساوي واحد واضح طيب الآن for every
+
+335
+00:30:30,170 --> 00:30:34,570
+epsilon أكبر من صفر اللي هي اللي فوق خلاص بدنا
+
+336
+00:30:34,570 --> 00:30:39,270
+فيها الآن بيبقى إن ال limit بيساوي واحد إذا there
+
+337
+00:30:39,270 --> 00:30:44,890
+exists Delta أكبر من صفر أيّ إن كانت الـ Delta أكبر
+
+338
+00:30:44,890 --> 00:30:53,790
+من صفر Such that F of X ناقص واحد أصغر من مين من
+
+339
+00:30:53,790 --> 00:30:59,190
+الـ Epsilon epsilon اللي بدنا فيها من الأول هذا لو
+
+340
+00:30:59,190 --> 00:31:04,830
+يعني لكل الـ X اللي في الفترة من a إلى a زائد الـ delta
+
+341
+00:31:04,830 --> 00:31:07,910
+هذه الـ delta الجديدة مش صارت تكون الأولى فأنا
+
+342
+00:31:07,910 --> 00:31:13,330
+عشان أريح حالي بدي اخذ اللي هو أصغر delta بلاقيها
+
+343
+00:31:13,330 --> 00:31:17,910
+و أقول حطها كمان جواها هادي حر أنا مدام بتنفع لي
+
+344
+00:31:17,910 --> 00:31:21,690
+كبيرة أيضاً كده بتنفع لمن؟ للصغيرة فاهمين أنا إيش
+
+345
+00:31:21,690 --> 00:31:28,270
+بقول؟ هذه اللي هي نسمّيها C3 معايا؟ فبيصير عندي الآن
+
+346
+00:31:28,270 --> 00:31:39,200
+F of X لكل X element in A لعند مين؟ C3 أي
+
+347
+00:31:39,200 --> 00:31:46,120
+سؤال؟ F of X ناقص واحد أصغر من Y أكبر من سالب Y
+
+348
+00:31:46,120 --> 00:31:49,960
+تلزمني هذه المنطقة اللي اللي جاي لأن F of X هذه
+
+349
+00:31:49,960 --> 00:31:56,720
+يعني F of X يعطينا F of X أكبر من واحد ناقص Y
+
+350
+00:31:59,390 --> 00:32:03,370
+واضحة؟ لأن هذه جلبها .. اللي هذه طبعاً واحد ناقص
+
+351
+00:32:03,370 --> 00:32:07,850
+إبسلون اللي هي بيصير عنده الـ .. الـ .. الـ إبسلون
+
+352
+00:32:07,850 --> 00:32:12,650
+أصغر من مين؟ من نصف الواحد ناقص إبسلون أكبر من مين؟
+
+353
+00:32:12,650 --> 00:32:20,670
+اللي هي بيصير أكبر من نصف واضحة؟ اللي هي بيصير واحد
+
+354
+00:32:20,670 --> 00:32:28,400
+على f of x اللي هو أصغر من مين؟ هذا لمين؟ لكل
+
+355
+00:32:28,400 --> 00:32:32,760
+الاكسات اللي من A لعند مين C3 وطبيعي الـ
+
+356
+00:32:32,760 --> 00:32:36,880
+inequalities كلها اللي قبل تتحقق هنا لأنها بتتحقق
+
+357
+00:32:36,880 --> 00:32:41,360
+من هنا لهنا بعضها وبعضها بتتحقق من هنا لهنا فأكيد
+
+358
+00:32:41,360 --> 00:32:47,040
+كلها هتتحقق للإكسات اللي مين في الـ A و C3 أصلاً أنا
+
+359
+00:32:47,040 --> 00:32:52,620
+رايح باتجاه مين؟ رايح باتجاه أثبت لكم أنه limit of
+
+360
+00:32:53,680 --> 00:32:59,060
+of x على g of x لما x تروح للـ a من اليمين تساوي
+
+361
+00:32:59,060 --> 00:33:04,880
+limit f prime of x على g prime of x as x تروح للـ a
+
+362
+00:33:04,880 --> 00:33:10,080
+من اليمين اللي هو أيش نسمّيه احنا قال إذا أنا بمهتم
+
+363
+00:33:10,080 --> 00:33:14,920
+بمين الاكسات اللي جنب الـ a لأن أنا رايح للـ a من وين
+
+364
+00:33:14,920 --> 00:33:18,800
+من اليمين فأنا بمهتم بالجوار القريب جداً من الـ a لأن
+
+365
+00:33:18,800 --> 00:33:21,960
+أنا رايح له أصلاً من هنا فالإكسات اللي فيها هي اللي
+
+366
+00:33:21,960 --> 00:33:27,760
+بتلزمني عشان أصل للي بديها خلّي هذه في الذاكرة
+
+367
+00:33:27,760 --> 00:33:34,560
+فصار عندي الآن صار
+
+368
+00:33:34,560 -->
+
+401
+00:36:58,760 --> 00:37:01,840
+ما هي أصلا continuous و differentiable على الفترة
+
+402
+00:37:01,840 --> 00:37:06,500
+دي كلها من ضمنها مين الفترة A وC ثلاثة ومن ضمنها
+
+403
+00:37:06,500 --> 00:37:11,040
+الفترة A وC واحد اللي أنا بدي أطبّق عليها بالـ end
+
+404
+00:37:11,040 --> 00:37:15,900
+point مين الـ end point C واحد هيها و الـ end point
+
+405
+00:37:15,900 --> 00:37:21,890
+الـ X اللي موجودة بين A و B مين؟ C3 واضح واضح اللي
+
+406
+00:37:21,890 --> 00:37:26,350
+بدي .. إذا الآن الآن there exists x i الآن بالظبط
+
+407
+00:37:26,350 --> 00:37:30,630
+بدي أطبّق الـ mean value theorem ع الفترة اللي هي
+
+408
+00:37:30,630 --> 00:37:37,790
+X و C1 اللي الـ F و الـ G differentiable عليها من
+
+409
+00:37:37,790 --> 00:37:40,530
+وين الـ X هذه؟ الـ X اللي فوق هذه اللي بشتغل فيها
+
+410
+00:37:40,530 --> 00:37:45,890
+اللي هي من A لعن مين؟ لعن C3 إذن there exists xi
+
+411
+00:37:45,890 --> 00:37:50,750
+element in x و c واحد such that f prime of xi على
+
+412
+00:37:50,750 --> 00:37:58,770
+g prime of xi في إيش بتساوي؟ f of c واحد أو f of x
+
+413
+00:37:58,770 --> 00:38:05,390
+ناقص f of c واحد فاهمين طبعا على g of x ناقص g of
+
+414
+00:38:05,390 --> 00:38:10,990
+c واحد بس ضربت في ناقص فوق و ناقص صحيح تتعدّى زي ما
+
+415
+00:38:10,990 --> 00:38:20,080
+هي لأن هذه هي هذه إذا صارت عندي F of X على G of X
+
+416
+00:38:20,080 --> 00:38:26,000
+بسوء F prime على G prime في واحد على مين؟ على F of
+
+417
+00:38:26,000 --> 00:38:32,060
+X وهو اللي بده يوصلنا للي بدنا هي معلش أمسح اللي
+
+418
+00:38:32,060 --> 00:38:35,280
+هان بيصير عندي 
+
+419
+00:38:38,180 --> 00:38:45,160
+بيصير عندي الـ F of X على مين؟ F of X على G of X
+
+420
+00:38:45,160 --> 00:38:48,540
+بتساوي
+
+421
+00:38:48,540 --> 00:38:54,120
+اللي هو هدى شلناه و حطينا مكانها مين؟ بتساوي F prime
+
+422
+00:38:54,120 --> 00:39:02,140
+of X I على G prime of X I في مين؟ في واحد على ايه
+
+423
+00:39:02,140 --> 00:39:03,500
+إيش؟ على F
+
+424
+00:39:09,880 --> 00:39:18,360
+واضح أه؟ الآن نمحى هنا هدولة الـ Xات اللي هنا هما
+
+425
+00:39:18,360 --> 00:39:23,380
+الـ Xات اللي وين موجودة؟ بين A وC3 ماشي الحال بدي
+
+426
+00:39:23,380 --> 00:39:29,120
+الآن غايتي أني أوجد F of X على G of X ناقص مين؟
+
+427
+00:39:29,120 --> 00:39:34,600
+ناقص L واضحة؟ إذا نلخص كل اللي اخذناه for every
+
+428
+00:39:34,600 --> 00:39:38,800
+epsilon element in zero نص there exists delta أكبر
+
+429
+00:39:38,800 --> 00:39:48,980
+من صفر such that if اللي هو x أكبر من a أصغر من c3
+
+430
+00:39:48,980 --> 00:39:53,260
+أصغر من c2 أصغر من c1 أصغر من a زي الـ delta اللي
+
+431
+00:39:53,260 --> 00:40:02,070
+لجيناها فاهمين عليها؟ وكون عندي اللي هو قيمة التعويض f
+
+432
+00:40:02,070 --> 00:40:08,110
+of x على g of x ناقص الـ .. بدي أقبل لك يميها هذا
+
+433
+00:40:08,110 --> 00:40:12,510
+اللي هو أصغر من epsilon في something مش مشكلة و
+
+434
+00:40:12,510 --> 00:40:15,810
+epsilon is arbitrary أيضا بيصير limit زي ما بدنا ياها
+
+435
+00:40:15,810 --> 00:40:24,170
+واضح هذا؟ الآن بالظبط بيساوي اللي هو التعويض اللي هو
+
+436
+00:40:24,170 --> 00:40:28,590
+f of x اللي هي عبارة عن واحد
+
+437
+00:40:32,680 --> 00:40:41,680
+أف برايم of x i على جي برايم of x i في واحد على أف
+
+438
+00:40:41,680 --> 00:40:48,180
+of x ناقص اللي
+
+439
+00:40:48,180 --> 00:40:54,460
+واضح؟ لحتى الآن ناخد هذه عامل مشترك برة يصبح واحد
+
+440
+00:40:54,460 --> 00:41:02,140
+على أف of x في أف برايم of x iعلى g prime of xi
+
+441
+00:41:02,140 --> 00:41:13,560
+ناقص L في F of X صح؟ أخدت
+
+442
+00:41:13,560 --> 00:41:17,300
+الواحد على F of X عامل مشترك positive أه positive
+
+443
+00:41:17,300 --> 00:41:21,080
+اللي هو F of X هي أكبر من 2 طلعت اللي هو إيش ماله
+
+444
+00:41:21,080 --> 00:41:27,260
+هذا؟ هنا ده بدي أضيف term و أطرح term عشان أحصل على
+
+445
+00:41:27,260 --> 00:41:31,780
+هذه اللي بدياها أصلا اللي هي الجد تفهموه مش مقصودة
+
+446
+00:41:31,780 --> 00:41:38,900
+1 على f of x في f prime of xi على g prime of xi
+
+447
+00:41:38,900 --> 00:41:50,130
+ناقص L زائد اللي هو L ناقص L F of X إيش سويت؟ استخدمت
+
+448
+00:41:50,130 --> 00:41:56,170
+الـ triangle inequality بإضافة L وطرحة L هذا كله
+
+449
+00:41:56,170 --> 00:42:04,130
+مضروب 100 في 1 على F of X هذا الآن الـ F of X أكبر
+
+450
+00:42:04,130 --> 00:42:12,690
+من 2 حصلناها مظبوط بيصير أصغر من نص واضحة
+
+451
+00:42:13,760 --> 00:42:18,920
+الآن F' على G' لـ XI الـ XI وين لاجيناها؟ في
+
+452
+00:42:18,920 --> 00:42:24,940
+الفترة بين A وC1 ينطبق عليها اللي هو من رأس الدور
+
+453
+00:42:24,940 --> 00:42:34,580
+اللي هي F' على G' of X ناقص
+
+454
+00:42:34,580 --> 00:42:41,200
+L أصغر من الـ Y من أول ما بدينا يعني إنه هذا أصغر
+
+455
+00:42:41,200 --> 00:42:44,460
+من إبسلون لكل الـ Xات اللي موجودة من A لعند مين؟
+
+456
+00:42:44,460 --> 00:42:48,080
+لعند الـA زاد دلتها من ضمنهم مين؟ لك فيبراين كلام
+
+457
+00:42:48,080 --> 00:42:55,940
+دقيق هذه في إبسلون زاد الآن هذا الـ بطلحها برة عن
+
+458
+00:42:55,940 --> 00:43:01,760
+المشترك الـ اللي هي في مين؟ في واحد ناقص F of X
+
+459
+00:43:01,760 --> 00:43:10,240
+واحد ناقص F of X الآن وهذا أصغر من نص في إبسلون
+
+460
+00:43:10,240 --> 00:43:15,440
+زائد absolute value لـ L واحد ناقص F of X هذه الـ
+
+461
+00:43:15,440 --> 00:43:20,260
+Xات اللي موجودة بين A وC تلاتة وهذه اللي هي قبل
+
+462
+00:43:20,260 --> 00:43:26,150
+بشوية أصغر من مين؟ من اللي هو إبسلون أصغر من إبسلون
+
+463
+00:43:26,150 --> 00:43:30,030
+واحد ناقص F of X أصغر من مين؟ من إبسلون مش هثبتنا
+
+464
+00:43:30,030 --> 00:43:33,910
+limit F of X بيساوي واحد قبل بشوية اللي هو واحد
+
+465
+00:43:33,910 --> 00:43:37,050
+ناقص F of X اللي هي أصغر من إيش؟ من إبسلون في
+
+466
+00:43:37,050 --> 00:43:42,850
+إبسلون صار عندي الآن هذا المقدار اللي هو بيساوي
+
+467
+00:43:42,850 --> 00:43:47,030
+epsilon على اتنين في واحد زائد absolute value
+
+468
+00:43:47,030 --> 00:43:52,810
+لـ L لأن الـ epsilon مضروبة فيه إذا as لأن هذا الـ
+
+469
+00:43:52,810 --> 00:43:55,430
+epsilon arbitrary وهذا الكلام صحيح for every
+
+470
+00:43:55,430 --> 00:43:59,710
+epsilon epsilon بين zero و نص إذا هذا المقدار بقدر
+
+471
+00:43:59,710 --> 00:44:04,350
+أصغره جد ما بدي بتصغير epsilon لانه مضروب فيه ضرب
+
+472
+00:44:04,350 --> 00:44:08,630
+إذا صار عندي لكل epsilon أكبر من zero و نصلـ G الـ
+
+473
+00:44:08,630 --> 00:44:12,650
+Delta بحيث أنه الإكسات اللي هنا بيقدّي لي أن هذا
+
+474
+00:44:12,650 --> 00:44:20,810
+المقدار أصغر من أي قيمة بديّها إذا هذا مفهوم limit
+
+475
+00:44:20,810 --> 00:44:31,530
+F of X على G of X as X بتروح لمين؟ للـ A من اليمين
+
+476
+00:44:31,530 --> 00:44:38,920
+بتساوي الـ L وهو المطلوب النظرية اللي حكينا عنها
+
+477
+00:44:38,920 --> 00:44:45,060
+برضه صحيحة في حالة اللي هو the same .. under the
+
+478
+00:44:45,060 --> 00:44:50,160
+same conditions for the calculation of limits as x
+
+479
+00:44:50,160 --> 00:44:55,540
+goes to infinity or x as goes mean to سالبinfinity
+
+480
+00:44:55,540 --> 00:44:59,140
+بس برهان يعني بدون modification عن اللي موجود
+
+481
+00:44:59,140 --> 00:45:03,140
+بتصلي اللي بدكيها نيجي ناخد أمثلة الأمثلة هذه طبعا
+
+482
+00:45:03,140 --> 00:45:09,500
+أمثلة سهلة ومثلة calculus نمر عليها اللي هو ونشوف
+
+483
+00:45:09,500 --> 00:45:16,480
+كيف نطبق نظريتنا أو نظرياتنا كيف نوظفها الحساب هذه
+
+484
+00:45:16,480 --> 00:45:23,260
+النهايات عندي شوفوا صلى الله عليه وسلم
+
+485
+00:45:24,800 --> 00:45:30,720
+Find Limit log sin x على log x as x بتروح للـ 0 من
+
+486
+00:45:30,720 --> 00:45:35,480
+وين؟ من اليمين واحنا عارفين .. مشتغلين على الفترة
+
+487
+00:45:35,480 --> 00:45:40,460
+0 or by يعني فترة بجوار مين؟ الصفر من اليمين و
+
+488
+00:45:40,460 --> 00:45:43,540
+بدنا نروح للـ 0 من اليمين هذا طبعا لو جينا عوضنا
+
+489
+00:45:43,540 --> 00:45:48,080
+هتطلع اللي هو sin 0 0 و أنا 0 فبيصير ماله؟ نهاية على
+
+490
+00:45:48,080 --> 00:45:53,580
+مين؟ على ماله نهاية ك limits الآن صارت عندي ماله
+
+491
+00:45:53,580 --> 00:45:57,880
+نهاية على ماله نهاية إذا إيش بتنسوه؟ إيش نسوي؟
+
+492
+00:45:57,880 --> 00:46:06,400
+راحين لـ 0 من اليمين بيصير عنده اللي هو بالفاضل
+
+493
+00:46:06,400 --> 00:46:11,400
+هذه و بالفاضل هذه بيصير تفاضلها هذه واضحة و هذه
+
+494
+00:46:11,400 --> 00:46:14,060
+تفاضلها هيك ما فيش داعي لنا ندخل في التفاصيل لأن كل
+
+495
+00:46:14,060 --> 00:46:20,320
+بعرفها الآن بنبسط الأمر بيصير X Cos X على Sine X
+
+496
+00:46:20,930 --> 00:46:26,550
+الآن لو جيت اتطلعت لهذه الـ limit و لهذه الـ limit
+
+497
+00:46:26,550 --> 00:46:29,530
+هذه موجودة و هذه موجودة هذه موجودة لأنه تعويض
+
+498
+00:46:29,530 --> 00:46:33,410
+مباشر واحد هذه موجودة لإنه بيصير صفر على صفر بتعملها
+
+499
+00:46:33,410 --> 00:46:36,570
+global rule بيصير واحد على cosine بتطلع واحد برضه
+
+500
+00:46:36,570 --> 00:46:40,590
+إذا قدرت اوزع لإنه عارف الليمتين الموجودة فبيصير
+
+501
+00:46:40,590 --> 00:46:46,220
+عبارة عن واحد في واحد بيساوي إيش؟ واحد الآن خد على
+
+502
+00:46:46,220 --> 00:46:49,900
+الفترة من Zero لعند مين؟ لعند باي على اتنين limit
+
+503
+00:46:49,900 --> 00:46:53,300
+واحد على X ناقص واحد على sign الـ X ثم X تروح لـ Zero
+
+504
+00:46:53,300 --> 00:46:57,140
+من وين؟ من اليمين وهذه طبعا شغلات اللي هو درجنا
+
+505
+00:46:57,140 --> 00:47:01,120
+عليها في الـ calculus بنوحد المقامات وبصير مكتوبة
+
+506
+00:47:01,120 --> 00:47:05,030
+على صورة sign الـ X ناقص X على X في sign الـ X ماشي
+
+507
+00:47:05,030 --> 00:47:10,650
+الحل برضه 0 على 0 بنفضلها بيصير cos X ناقص 1 على
+
+508
+00:47:10,650 --> 00:47:15,470
+اللي هو sin X زائد X فـ cos X لو جينا عوضنا هتطلع
+
+509
+00:47:15,470 --> 00:47:21,250
+برضه 0 على 0 بنشتق كمان مرة بتطلع ناقص sin X على
+
+510
+00:47:21,250 --> 00:47:26,150
+هذا المقدار هذا المقدار هنا بتصفر وهنا بيصير 2
+
+511
+00:47:26,150 --> 00:47:32,680
+وهنا 0 0 على 2 مش بيساوي بيساوي 0 نأتي لهذا المثال
+
+512
+00:47:32,680 --> 00:47:36,440
+معهود الـ let I بتساوي n 1 إلى ما لنهاية الـ
+
+513
+00:47:36,440 --> 00:47:39,760
+interval اللي عليها بدنا نشتغل و بدنا ناخد الـ
+
+514
+00:47:39,760 --> 00:47:43,600
+limit لما X تروح إلى وين؟ إلى ما لنهاية المقدار 1
+
+515
+00:47:43,600 --> 00:47:48,960
+زائد 1 على X و الكل أُس X هذا المقدار الآن بدي
+
+516
+00:47:48,960 --> 00:47:53,460
+أستخدم اللي هو أحوله لـ exponential to the len ما
+
+517
+00:47:53,460 --> 00:47:55,660
+احنا عارفين الـ exponential و الـ len اللي هو n
+
+518
+00:47:55,660 --> 00:47:59,860
+versus لبعض فبيصير عندي E to the X لن الـ 1 زائد 1
+
+519
+00:47:59,860 --> 00:48:05,120
+على X هذا المقدار بدي أشغل على هذاك خطوة أولى بقول
+
+520
+00:48:05,120 --> 00:48:09,260
+limit الـ X لما 1 زائد 1 على X لما X تروح إلى
+
+521
+00:48:09,260 --> 00:48:14,140
+مال نهاية بيصير هذا المقدار limit لما 1 زائد X
+
+522
+00:48:14,140 --> 00:48:18,160
+على 1 على X ليش عملت هيك؟ لأن لما X تروح إلى مال
+
+523
+00:48:18,160 --> 00:48:21,360
+نهاية هذا صفر و لما X تروح إلى مال نهاية هذا صفر و
+
+524
+00:48:21,360 --> 00:48:25,340
+هذا لما 1 فبيصير صفر على صفر بقدر أستخدم الـ loop
+
+525
+00:48:25,340 --> 00:48:30,730
+تال ذرول اللي أنا برهنتها بقي فاضل اللي فوق و فاضل
+
+526
+00:48:30,730 --> 00:48:33,850
+اللي تحت .. فضلنا اللي فوق هي و فضلنا اللي تحت
+
+527
+00:48:33,850 --> 00:48:37,590
+هي .. بروح هذا المقدار مع هذا المقدار .. بظل
+
+528
+00:48:37,590 --> 00:48:42,800
+limit 1 زي 1 على X ناقص 1 لما X تروح لـ مال نهاية ده
+
+529
+00:48:42,800 --> 00:48:47,940
+بيروح لصفر بيطلع جداش 1 لأن 1 بنرد بنعوّض وينها لأن
+
+530
+00:48:47,940 --> 00:48:50,940
+الـ exponential is a continuous function لذا نصحي
+
+531
+00:48:50,940 --> 00:48:54,180
+عن عند limit الـ 1 زي 1 على X لما X تروح لـ مال نهاية
+
+532
+00:48:54,180 --> 00:48:57,920
+to the X وعبارة عن E to the limit لأن الـ
+
+533
+00:48:57,920 --> 00:49:01,760
+exponential عبارة عن continuous function فبيصير E
+
+534
+00:49:01,760 --> 00:49:08,300
+to the 1 ويساوي E هذا اللي هو بيكون هيك احنا أنهينا
+
+535
+00:49:08,300 --> 00:49:16,240
+اللي هي لوبيتالز رولز أو قواعد لوبيتال أو صرنا
+
+536
+00:49:16,240 --> 00:49:20,360
+الآن ندخل
+
+537
+00:49:20,360 --> 00:49:28,120
+على اللي هو الـ chapter الـ section الأخير اللي هو
+
+538
+00:49:28,120 --> 00:49:32,120
+الـ section عبارة عن ستة أربعة اللي هو تايلورز
+
+539
+00:49:32,120 --> 00:49:38,540
+theorem أو تايلورز اللي هي .. هنحكي عن تايلورز
+
+540
+00:49:38,540 --> 00:49:43,640
+polynomial أو تايلورز أيضا اللي هو approximation
+
+541
+00:49:43,640 --> 00:49:49,820
+كيف اللي هو .. إن مرة جاية إن شاء الله كيف بنقرب
+
+542
+00:49:49,820 --> 00:49:54,840
+بعض الدوال اللي هو المحترمة اللي بتكون
+
+543
+00:49:54,840 --> 00:49:58,220
+differentiable first derivative و second
+
+544
+00:49:58,220 --> 00:50:00,400
+derivative و third derivative لأ عند الـ derivative
+
+545
+00:50:00,400 --> 00:50:05,700
+اللي بدنا .. اللي .. اللي هي ما قبل اشتقاق عندها
+
+546
+00:50:05,700 --> 00:50:11,760
+بنقربها بـ polynomial وطبعا أنتو عارفين جداش الـ
+
+547
+00:50:11,760 --> 00:50:17,380
+polynomial من الدوال السهلة سهلة التعامل سواء في
+
+548
+00:50:17,380 --> 00:50:21,500
+تفاضل أو في تكامل أو حتى لو بدنا نحلها ونجيب
+
+549
+00:50:21,500 --> 00:50:26,340
+جذورها ونجيب كذا في شغل عليها كثير فالناس ترغب في
+
+550
+00:50:26,340 --> 00:50:30,540
+إنها تجيب بعض الدوال اللي بتكون أحيانا معقدة و
+
+551
+00:50:30,540 --> 00:50:35,380
+صعبة التكامل أو خلنا نقول بتغلب شوية في التفاضل أو
+
+552
+00:50:35,380 --> 00:50:40,040
+في وجود الجذور أو كده ونحولها إلى اللي هي
+
+553
+00:50:40,040 --> 00:50:44,020
+polynomial طبعا في مقدار خطأ الخطأ هن .. إن شاء
+
+554
+00:50:44,020 --> 00:50:47,520
+الله الـ Taylor's theorem بتقولنا كيف إن هو نوجد
+
+555
+00:50:47,520 --> 00:50:51,320
+هذا الخطأ أو إيش هو الخطأ وايضا Taylor's theorem
+
+556
+00:50:51,320 --> 00:50:55,780
+زي ما أنتو عارفين هنوضح كيف هي عبارة عن تعميم للـ
+
+557
+00:50:55,780 --> 00:51:00,340
+main value theorem هي تعميم للـ main value theorem
+
+558
+00:51:00,340 --> 00:51:05,140
+في حالة اللي هي n بتساوي صفر بتطلع هي بالظبط الـ
+
+559
+00:51:05,140 --> 00:51:09,100
+main value theorem كيف؟
+
+560
+00:51:10,860 --> 00:51:15,100
+الـ Main Value Theorem عبارة الـ Taylor's Theorem
+
+561
+00:51:15,100 --> 00:51:19,900
+عبارة عن تعميم للـ Main Value Theorem خد الآن
+
+562
+00:51:19,900 --> 00:51:25,020
+بتساوي صفر بتطلع لك الـ Main Value Theorem بيشحمها
+
+563
+00:51:25,020 --> 00:51:27,740
+ولحمها وإلى لقاء
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1942f319d791b966afabcd0932d9e6c7034a51af
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_postprocess.srt
@@ -0,0 +1,2252 @@
+1
+00:00:05,040 --> 00:00:12,800
+بسم الله الرحمن الرحيم المحاضرة السابعة مساق تحليل
+
+2
+00:00:12,800 --> 00:00:18,640
+حقيقي 2 اللي طلبت كلية العلوم بالجامعة الإسلامية
+
+3
+00:00:18,640 --> 00:00:23,800
+قسم الرياضيات اليوم
+
+4
+00:00:23,800 --> 00:00:29,720
+هيكون عندنا استكمال ل .. اللي هي قواعد أو قاعدة
+
+5
+00:00:29,720 --> 00:00:36,330
+Lobitalاتحدثنا المرة الماضية عن الحالة اللى بتحدث
+
+6
+00:00:36,330 --> 00:00:41,170
+سفر على سفر وكيف انعلجها وتم اللى هو برهان
+
+7
+00:00:41,170 --> 00:00:47,910
+النظريات المعنية في ذلكالآن بدنا نتحدث على الحالة
+
+8
+00:00:47,910 --> 00:00:57,110
+اللي بتحدث عندنا أيضًا 0 على 0 ولكن ال limit ال X
+
+9
+00:00:57,110 --> 00:01:00,830
+بتروح إلى مالة نهاية أو سالب مالة نهاية وقلنا عنها
+
+10
+00:01:00,830 --> 00:01:05,790
+صحيحة برضه وبعد ذلك هنتحدث عن الحالة infinity على
+
+11
+00:01:05,790 --> 00:01:10,510
+infinity نيجي الآن لنظريتنا اللي وصلنا عندها
+
+12
+00:01:10,510 --> 00:01:11,990
+suppose that
+
+13
+00:01:16,220 --> 00:01:20,260
+F and D are continuous and differentiable على
+
+14
+00:01:20,260 --> 00:01:24,600
+الفترة من A إلى ما لا نهاية ونفترض أنه limit لل F
+
+15
+00:01:24,600 --> 00:01:27,540
+of X لما X تروح إلى ما لا نهاية هي نفس limit D of
+
+16
+00:01:27,540 --> 00:01:31,200
+X لما X تروح إلى ما لا نهاية تساوي إيهاش صفر يعني
+
+17
+00:01:31,200 --> 00:01:36,080
+هذه هتخلقلي حالة اللي هو صفر على صفر ومن ثم نشوف
+
+18
+00:01:36,080 --> 00:01:40,230
+كيف انعليجهاالعنوان D of X ذاته ساوة سفر نفترض وD
+
+19
+00:01:40,230 --> 00:01:44,830
+prime of X ذاته ساوة سفر لكل X أكبر من مين من اللي
+
+20
+00:01:44,830 --> 00:01:48,290
+هو المقصود هنا بيه اللي هي أيش A يعني هذه اللي هي
+
+21
+00:01:48,290 --> 00:01:52,970
+مين اللي هي المقصود أنه A، يا ما نخلي هذه A، يا ما
+
+22
+00:01:52,970 --> 00:01:58,330
+هذه Athen limit f of x على g of x لما x تروح لما
+
+23
+00:01:58,330 --> 00:02:01,930
+النهاية هتكون بالظبط ايش بتساوي limit f prime على
+
+24
+00:02:01,930 --> 00:02:06,050
+g prime of x as x goes to infinity provided that
+
+25
+00:02:06,050 --> 00:02:09,590
+هذا ال limit بتساوي قيمة معينة أو تساوي infinity
+
+26
+00:02:09,590 --> 00:02:15,730
+او سالم infinity الان احنا في الواقع اللي هنسوي
+
+27
+00:02:16,910 --> 00:02:21,910
+وكأنه يعني عملية تحويل اللي نخلّي بدل X تروح إلى
+
+28
+00:02:21,910 --> 00:02:26,950
+ملأ نهاية نخلّي اللي هو متغير وروح لسفر يعني بدنا
+
+29
+00:02:26,950 --> 00:02:31,490
+نعيد تعريف الدالة بحيث أنه يروح هذا المتغير يعني
+
+30
+00:02:31,490 --> 00:02:34,610
+وكأنه نقول واحد على T ويروح على .. اه واحد على X
+
+31
+00:02:34,610 --> 00:02:37,950
+يروح لسفر لما X تروح إلى ملأ نهاية ومن ثم نطبق
+
+32
+00:02:37,950 --> 00:02:41,030
+النظريات اللي أخدناها المرة الماضية دعونا نشوف إيش
+
+33
+00:02:41,030 --> 00:02:47,050
+اللي بنحكيه الآن عند ..الـ الـ function F و G F و
+
+34
+00:02:47,050 --> 00:02:54,430
+G معرفات من A و ماله نهاية لعند اللي هو R الآن
+
+35
+00:02:54,430 --> 00:02:57,970
+عندي اللي هو بده أعرف اللي هو عشان أقدر أستخدم
+
+36
+00:02:57,970 --> 00:03:01,610
+نظرية المرة اللي فاتت المرة الماضية بده أعرف
+
+37
+00:03:01,610 --> 00:03:08,030
+دالتين F capital و G capital من اللي هو عندي من
+
+38
+00:03:08,030 --> 00:03:15,560
+Zero لعند اللي هو واحد على Aلعند الهو asr واضحة
+
+39
+00:03:15,560 --> 00:03:28,640
+الصورة تقريبا اكيد هنقول define f o g as f of اللي
+
+40
+00:03:28,640 --> 00:03:36,960
+نسميها اللي هو T بتساوي اللي هو Gأو F of 1 على T
+
+41
+00:03:36,960 --> 00:03:42,460
+لما T اللي هي لا تساوي سفر و طبعا T أنا معرفها
+
+42
+00:03:42,460 --> 00:03:47,140
+اللي هي في الفترة من Zero لعند واحد على A و ده
+
+43
+00:03:47,140 --> 00:03:51,560
+بيدعرف عند Zero بقول Zero إذا كان T إيش بدها تساوي
+
+44
+00:03:51,560 --> 00:03:59,760
+بتساوي سفر الآن ال .. ال G of Tبتساوي نفس الأسلوب
+
+45
+00:03:59,760 --> 00:04:06,240
+G of 1 على T لما T لا تساوي 0 و 0 إذا كانت T
+
+46
+00:04:06,240 --> 00:04:13,350
+أشمالها بتساوي 0الآن واضح العلاقة بين الدلتين اللي
+
+47
+00:04:13,350 --> 00:04:19,510
+هو F و F اللي هو على أساس أني أحول ال limit
+
+48
+00:04:19,510 --> 00:04:25,650
+المعطاع عندي limit F of X لما X تروح إلى ملا نهاية
+
+49
+00:04:25,650 --> 00:04:30,450
+تصير عندي ال limit بالنسبة لهذه اللي هي ال 1 على X
+
+50
+00:04:30,450 --> 00:04:35,320
+تبعتنا هتروح لمين في هذا الحالة للزفروبستخدم ومن
+
+51
+00:04:35,320 --> 00:04:40,600
+ثم بنكون حصلنا على المتغير لإن أنا T بروح للصفر as
+
+52
+00:04:40,600 --> 00:04:44,740
+X بتروح إلينا لنهاية ومن ثم بنستخدم النظريات اللي
+
+53
+00:04:44,740 --> 00:04:49,940
+أثبتنا صحتها في حالة T بتروح إلى A من اليمين لإن
+
+54
+00:04:49,940 --> 00:04:53,060
+احنا هنروح ل Zero من اليمين هذه اللي هي الخطة اللي
+
+55
+00:04:53,060 --> 00:04:58,480
+هنمشي عليها بنشوف أول إشي واضح أن ال F و ال G ال F
+
+56
+00:04:59,090 --> 00:05:03,930
+والـ G are differentiable على مين على الفترة من
+
+57
+00:05:03,930 --> 00:05:10,550
+سفر لعند واحد عليه هاي واحد اتنين ومدام
+
+58
+00:05:10,550 --> 00:05:13,810
+differentiable ده ن continuous عليها ومش هيك كمان
+
+59
+00:05:13,810 --> 00:05:19,230
+لو جيت أخدت انت هذه بتحسبها لحالك limit F of T as
+
+60
+00:05:19,230 --> 00:05:27,640
+T بتروح للسفر اللي هي بتساوي limitf of 1 على t as
+
+61
+00:05:27,640 --> 00:05:32,340
+t بتروح للصفر و هتلاقيها إيش هتساوي، هتساوي صفر
+
+62
+00:05:32,340 --> 00:05:36,760
+فصارت اللي هي ال F ال capital is continuous على
+
+63
+00:05:36,760 --> 00:05:43,490
+الفترة من 0 لعند 1 على a similarly limit g of tلما
+
+64
+00:05:43,490 --> 00:05:47,650
+T تروح للـ Zero طبعا كل الـ Zero من وين طبيعي من
+
+65
+00:05:47,650 --> 00:05:50,950
+اليمين لأنه أحنا عالمنا اللي بنشتغل عليه اللي هي
+
+66
+00:05:50,950 --> 00:05:53,730
+من عند Zero لعند واحد على أي من المنطقة المودة بقى
+
+67
+00:05:53,730 --> 00:05:58,750
+إذن هروح للـ Zero من جهات اليمين هيساوي نفس الأشي
+
+68
+00:05:58,750 --> 00:06:03,290
+limit F of واحد على T لما T تروح للـ Zero من
+
+69
+00:06:03,290 --> 00:06:07,590
+اليمين ويساوي اللي جيها برضه أشمالها بتساوي صفر
+
+70
+00:06:08,590 --> 00:06:12,970
+الان صارت الان الاف و الجي اشمالين are continuous
+
+71
+00:06:12,970 --> 00:06:17,390
+على الفترة اللي عندنا و differentiable على الفترة
+
+72
+00:06:17,390 --> 00:06:20,510
+ال open و .. و .. و .. و .. و .. و .. و .. و .. و
+
+73
+00:06:20,510 --> 00:06:20,590
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+74
+00:06:20,590 --> 00:06:20,790
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+75
+00:06:20,790 --> 00:06:21,070
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+76
+00:06:21,070 --> 00:06:21,290
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+77
+00:06:21,290 --> 00:06:23,330
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+78
+00:06:23,330 --> 00:06:23,350
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+79
+00:06:23,350 --> 00:06:23,650
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+80
+00:06:23,650 --> 00:06:29,210
+.. و .. و .. و .. و .. و ..
+
+81
+00:06:37,480 --> 00:06:43,680
+بنجي بنحصل على ما يلي خليني بس اسمحولي أمسح هذه
+
+82
+00:06:43,680 --> 00:06:49,760
+دلت في الذاكرة عندي بدي أحسب أنا غاية أن أوجد مين
+
+83
+00:06:49,760 --> 00:06:53,380
+يا جماعة limit اللي هو f of x على g of x لما x
+
+84
+00:06:53,380 --> 00:07:00,780
+تروح إلى وين إلى ما لنهاية لأن limit f of x على g
+
+85
+00:07:00,780 --> 00:07:07,720
+of x لما x تروح إلى ما لنهاية بتساويعبارة عن الان
+
+86
+00:07:07,720 --> 00:07:12,180
+ال X بتروح إلى ما لنهاية F and وال F الواحد على X
+
+87
+00:07:12,180 --> 00:07:20,660
+وين بتروح اللي هو السفر بتساوي limit F of واحد على
+
+88
+00:07:20,660 --> 00:07:26,040
+T خلينا نسميها على G of واحد على T لما T تروح إلى
+
+89
+00:07:26,040 --> 00:07:32,420
+مين إلى السفر من اليمينأو إذا كان عندك اتصار
+
+90
+00:07:32,420 --> 00:07:38,640
+confusion إشي والتكن X مافي مشكلة X لما X تروح ل 0
+
+91
+00:07:38,640 --> 00:07:43,660
+من اليمين اللي هي بتسير عند أسف الواحد على X بتروح
+
+92
+00:07:43,660 --> 00:07:47,860
+ل 0 من اليمين if and only if ال X بتروح لمين إلى
+
+93
+00:07:47,860 --> 00:07:51,520
+ما لنهاية ومن ثم بفخطوة تانية بترجحها للي كتبته
+
+94
+00:07:51,520 --> 00:07:57,980
+قبل بشوية limit اللي هو عبارة عن F of Tعلى اللي هو
+
+95
+00:07:57,980 --> 00:08:04,420
+G of T لما T تروح لـ 0 من وين؟ من اليمين باشي
+
+96
+00:08:04,420 --> 00:08:08,120
+الحال أنا الأن بدي أفضل في الواقع بالنسبة لمن؟
+
+97
+00:08:08,120 --> 00:08:14,040
+بالنسبة له اللي هو الـ X عندي اللي هو وديت واحدة
+
+98
+00:08:14,040 --> 00:08:19,880
+الـ X من اليمين وصارت عندي اللي هي مكان لأ لأ يا
+
+99
+00:08:19,880 --> 00:08:26,200
+جماعة هاد نزام هي بتضلها آسف أه؟X X وهنا بتصير
+
+100
+00:08:26,200 --> 00:08:30,900
+أننا مين واحد على T وهنا واحد على T واضح ليش
+
+101
+00:08:30,900 --> 00:08:36,540
+استبدلنا هنا احنا الواحد على X كله ب T فصارت هذه
+
+102
+00:08:36,540 --> 00:08:41,320
+اللي هي واحد على T وهذه واحد على T الآن صارت
+
+103
+00:08:41,320 --> 00:08:44,780
+النظرية السابقة مطبقة لأن ال T بتروح لمين ل Zero
+
+104
+00:08:44,780 --> 00:08:49,780
+من لمين لأن هدف الواقع مين هي اللي قلنا عنها أشي
+
+105
+00:08:49,780 --> 00:08:57,850
+هذه بتساوي limit ofof T على G of T لما T تروح إلى
+
+106
+00:08:57,850 --> 00:09:01,310
+Zero من وين؟ من اليمين وهذا متطبق لأن هذا Zero
+
+107
+00:09:01,310 --> 00:09:05,490
+وهذا Zero زي ما عملنا قبل بشوية إذا صار إيش بيساوي
+
+108
+00:09:05,490 --> 00:09:10,990
+تفاضل الأول على تفاضل التاني في 100 بالنسبة لعن
+
+109
+00:09:10,990 --> 00:09:17,220
+نظرية السابقة إذا بيصير limitالتي هي التفاضل هادي
+
+110
+00:09:17,220 --> 00:09:21,060
+من هنا F prime of T أيش بيساوي يا شباب؟ تفاضل
+
+111
+00:09:21,060 --> 00:09:24,960
+هادي، عن T بتروح للـ0، إذن هي عبارة عن ناقص 1 على
+
+112
+00:09:24,960 --> 00:09:34,000
+T تربيع في F prime of 1 على T تفاضل F of T، F of T
+
+113
+00:09:34,000 --> 00:09:38,930
+هيهايعني T لا تساوي سفر تفاضلها F prime بساوي ناقص
+
+114
+00:09:38,930 --> 00:09:42,710
+اللي هو F prime الواحد على T فتفاضل اللي جوا اللي
+
+115
+00:09:42,710 --> 00:09:47,450
+هو عبارة عن إيش ناقص واحد على T تربيع واضح على
+
+116
+00:09:47,450 --> 00:09:53,430
+similarly ناقص واحد على T تربيع في G prime of واحد
+
+117
+00:09:53,430 --> 00:10:00,200
+على T لما الـT تروح لمين إلى الـ0 من اليمينالنقص 1
+
+118
+00:10:00,200 --> 00:10:03,160
+دلت ال T تربيع ال نقص 1 دلت T تربيع بروحن مع بعض
+
+119
+00:10:03,160 --> 00:10:09,780
+نرجع لأصلنا بيصير limit F prime بده استبدل الآن ال
+
+120
+00:10:09,780 --> 00:10:17,800
+1 على T بال X على G prime برضه نفس الشي على X و ال
+
+121
+00:10:17,800 --> 00:10:22,540
+T تستبدل بمين اللي هو عبارة عن 1 على X بده تروح لل
+
+122
+00:10:22,540 --> 00:10:29,270
+0 من اليمين يعني ال X بده تروح لمين إلى مالكبصير
+
+123
+00:10:29,270 --> 00:10:35,510
+عندى اللى هو limit f of x على g of x بساوى limit f
+
+124
+00:10:35,510 --> 00:10:40,350
+prime of x على g prime y على x وهو المطلوب نيجى
+
+125
+00:10:40,350 --> 00:10:44,690
+للنظرية البعدها النظرية اللى بدها شغل خلينا نشوف
+
+126
+00:10:44,690 --> 00:10:53,110
+النظرية اطلع لفوق ونركز نشوف ايش هو النظرية بتحكي
+
+127
+00:10:53,110 --> 00:10:57,270
+ومن ثم نذهب الى برهان النظرية
+
+128
+00:11:04,010 --> 00:11:11,750
+عشان مساحة اللوحة شوية خلّينا نتبع على التلخيص ومن
+
+129
+00:11:11,750 --> 00:11:19,110
+ثم بنبرهن theorem 636 اللي هي باختصار هي حالة مالة
+
+130
+00:11:19,110 --> 00:11:22,570
+نهاية على مالة نهاية نفترض أن الـF و الـG are
+
+131
+00:11:22,570 --> 00:11:26,470
+differentiable على الفترة من A و B يعني إحنا شغلنا
+
+132
+00:11:26,470 --> 00:11:34,230
+الآن على الفترة أي إن كانت هذه الفترة من عند Aلعند
+
+133
+00:11:34,230 --> 00:11:41,250
+مي ان لعند بي لأننا مفترضين أن ال F و ال G اللي هو
+
+134
+00:11:41,250 --> 00:11:46,250
+قابل للاشتقاق على الفترة A و B و limit F of X لما
+
+135
+00:11:46,250 --> 00:11:48,890
+X تروح إلى مالة نهائية .. إلى Zero من اليمين ..
+
+136
+00:11:48,890 --> 00:11:53,230
+إلى A من اليمين إيش بيساوي Infinity و limit G of X
+
+137
+00:11:53,230 --> 00:11:56,090
+لما X تروح إلى ال A من اليمين برضه إيش مالها
+
+138
+00:11:56,090 --> 00:12:01,370
+Infinity يعني بمعنى آخر صار لما عرضت علينا limit F
+
+139
+00:12:01,370 --> 00:12:06,980
+of Xعلى G of X لما X راحت إلى الـ A من اليمين
+
+140
+00:12:06,980 --> 00:12:10,660
+وجدنا limit اللي فوق ملا نهاية و limit اللي تحت
+
+141
+00:12:10,660 --> 00:12:14,260
+ملا نهاية يعني حصلنا على الكمية الغير معينة
+
+142
+00:12:14,260 --> 00:12:18,220
+Infinity على Infinity إذا هذه النظرية ستعالج هذه
+
+143
+00:12:18,220 --> 00:12:23,060
+الحالة طبعا الان اللي هي بطريقة مشابهة للسابقة
+
+144
+00:12:23,060 --> 00:12:29,080
+والان احنا طبعا هنقوم ببرهان هذه النظرية نكمل النص
+
+145
+00:12:29,820 --> 00:12:35,900
+الآن فرضنا أيضًا أنه اللي هو الـ g of x لا تساوي
+
+146
+00:12:35,900 --> 00:12:40,160
+سفر والـ g prime of x لا تساوي سفر لكل x و إما لها
+
+147
+00:12:40,160 --> 00:12:44,920
+في الفترة a و b يعني مفترض أن الـ g of x لا تساوي
+
+148
+00:12:44,920 --> 00:12:52,860
+سفر على كل الفترة a و bنأتي إلى الجزئية الأولى من
+
+149
+00:12:52,860 --> 00:12:57,380
+النظرية if limit f' على g' بساوي L element in R
+
+150
+00:12:57,380 --> 00:13:02,040
+إذا فlimit f على g برضه أيضا أيش هتساوي L اللي هي
+
+151
+00:13:02,040 --> 00:13:05,280
+نظريات اللي قبل بشوية بنفس .. بنفس .. خلّيني أقول
+
+152
+00:13:05,280 --> 00:13:10,160
+بنفس المنطق ونفس النصوص بس مبدل 0 على 0 عمالة
+
+153
+00:13:10,160 --> 00:13:13,560
+نهايةفلان limit f prime على g prime صارت مالة
+
+154
+00:13:13,560 --> 00:13:17,700
+نهاية أو سالب مالة نهاية، برضه النظرية بتظبط، سير
+
+155
+00:13:17,700 --> 00:13:23,310
+limit اللي عندنا هذاهو بالظبط limit الـ F على G as
+
+156
+00:13:23,310 --> 00:13:26,170
+X وتروحنا الـ M من اليمين بساوي Infinity وسالب
+
+157
+00:13:26,170 --> 00:13:30,550
+Infinity يعني في النهاية اللي ببحث عنه هذا هو في
+
+158
+00:13:30,550 --> 00:13:34,650
+الحالة هذه أو في الحالة اللي فوق بساوي limit مين
+
+159
+00:13:34,650 --> 00:13:37,750
+الـ F prime على G prime اللي هي الطريقة المحودة
+
+160
+00:13:37,750 --> 00:13:40,330
+بنفضل اللي فوق و بنفضل اللي تحت و بنوجد limit هين
+
+161
+00:13:40,330 --> 00:13:44,670
+بيكونين هين limit اللي هي الأصل اللي احنا بدنايا
+
+162
+00:13:45,410 --> 00:13:49,590
+الان نيجي اللي هو in Berlin اللي هو a و b طبعا
+
+163
+00:13:49,590 --> 00:13:54,410
+بنفس الأسلوب الان
+
+164
+00:13:54,410 --> 00:14:02,790
+قبل ما نبدأ احنا لو اجينا و قولنا limit f of x لما
+
+165
+00:14:02,790 --> 00:14:06,610
+x تروح لأي اشي مثلا a من اليمين بساوة L
+
+166
+00:14:11,700 --> 00:14:14,480
+وقلنا اللي هو for every epsilon تعريفها for every
+
+167
+00:14:14,480 --> 00:14:16,660
+epsilon أكبر من سفر there exists a delta أكبر من
+
+168
+00:14:16,660 --> 00:14:21,300
+سفر such that if x element in A طبعا من اليمين
+
+169
+00:14:21,300 --> 00:14:27,780
+معناته A و A زائد دلتا then اللي هو هيكون عندي f
+
+170
+00:14:27,780 --> 00:14:35,230
+of x ناقص الأصغر من مين من إبسمنالان لو كانت عندي
+
+171
+00:14:35,230 --> 00:14:39,330
+ال epsilon أثبتت انا هذا الكلام ل epsilon ل
+
+172
+00:14:39,330 --> 00:14:45,410
+epsilon لو أثبتت لكل epsilon element in مثلا in
+
+173
+00:14:45,410 --> 00:14:51,970
+zero و نصأو نزيره ربع أو نزيره تلت لو أثبتت أنه
+
+174
+00:14:51,970 --> 00:14:54,950
+لكل إبسلون لجهة دلتة بحيث أن هذا الكلام أصغر من
+
+175
+00:14:54,950 --> 00:15:01,650
+مين من إبسلون برضه بيكون يجزئ عن ال limit ليش؟ لأن
+
+176
+00:15:01,650 --> 00:15:04,950
+المقدار اللي بيكون أصغر من إبسلون اللي هي أصغر من
+
+177
+00:15:04,950 --> 00:15:09,830
+النص أكيد بنفع لمين الدلتة بنفع للمقدار اللي ال
+
+178
+00:15:09,830 --> 00:15:15,450
+إبسلون أكبر من مين من النص لأنه أصلا اللي هي مشكلة
+
+179
+00:15:15,450 --> 00:15:19,620
+أو خلينا نقول ال limit في حد ذاتهاإنه لما الـ A
+
+180
+00:15:19,620 --> 00:15:24,880
+تروحالـ X تروح إلى الـ A من اليمين، بدنا نجمع أن
+
+181
+00:15:24,880 --> 00:15:27,720
+الـ F of X تقترب من مين من الأعلى، يعني التعجيز و
+
+182
+00:15:27,720 --> 00:15:31,100
+كأنه بعجزني أنه يقول لي إنك اتلاقى الـ Delta في
+
+183
+00:15:31,100 --> 00:15:34,600
+حالة الـ Epsilon أكاش اللي بيكون صغيرة فالـ Close
+
+184
+00:15:34,600 --> 00:15:38,960
+اللي هو تمين to Zero، لإنه أصلا المفهوم اللي
+
+185
+00:15:38,960 --> 00:15:43,160
+بتقيله هذا المسافة تضيق، تضيق، تضيق، بحيث أن F of
+
+186
+00:15:43,160 --> 00:15:46,760
+X تقول إلى الـ مين اللي بتعبر عنها؟ الـ Epsilon،
+
+187
+00:15:46,760 --> 00:15:51,190
+لأن بعطيك Epsilon صغيرة جدا جدابدك تلاقي لي Delta
+
+188
+00:15:51,190 --> 00:15:54,870
+أنا الآن الـ Epsilon بين Zero و نص لقيتلك الـ
+
+189
+00:15:54,870 --> 00:15:59,070
+Delta اللي طلعت هنا معناته اللي نفع للـ Epsilon
+
+190
+00:15:59,070 --> 00:16:03,510
+الصغيرة أكيد بيكون أصغر من مين من الأكبيرة إذا هذا
+
+191
+00:16:03,510 --> 00:16:09,250
+يجزء بس هذا قبل ما نبدأ لأنه هسأخدم شغلة اللي هو
+
+192
+00:16:09,250 --> 00:16:14,930
+في هذا الاتجاه نيجي الآن عندي أول إشي المعطى
+
+193
+00:16:14,930 --> 00:16:23,550
+ماتيني LimitF prime of X على G prime of X as X
+
+194
+00:16:23,550 --> 00:16:29,190
+بتروح للـ A من اليمين أيش بساوي؟ بساوي أقل تعريف
+
+195
+00:16:29,190 --> 00:16:32,850
+الـ Limit for every Y أكبر من 0 اللي هتاخد for
+
+196
+00:16:32,850 --> 00:16:39,190
+every Y element in 0 ونص اللي هي لزوم حسابات، بعد
+
+197
+00:16:39,190 --> 00:16:44,650
+شوية هنشوفهاالآن لكل y تنتمي إلى 0.5 وهذا مشروع
+
+198
+00:16:44,650 --> 00:16:49,270
+حسب ما حكيت قبل بشوية there exist delta أكبر من 0
+
+199
+00:16:49,270 --> 00:16:57,650
+such that اللي هو if x element in a و a زيادة دلتا
+
+200
+00:16:57,650 --> 00:17:04,230
+لإن رايح لل a من وين من اليمين لجيت اللي هو دلتا
+
+201
+00:17:04,230 --> 00:17:11,420
+بحيث أنه لكل x في الفترة من a لعيد a زائد دلتابصير
+
+202
+00:17:11,420 --> 00:17:16,780
+عندى لكل x المتنية زائد دلتا بتطلع عندى then f
+
+203
+00:17:16,780 --> 00:17:23,420
+prime of x على g prime of x ناقص ال L أصغر من مين
+
+204
+00:17:23,420 --> 00:17:32,060
+من ال epsilon إذا الآن من هذه ال limitحصلت على أنه
+
+205
+00:17:32,060 --> 00:17:36,540
+لأي إبسلون بين الـ 0 و نص بقدر ألاقي Delta بحيث
+
+206
+00:17:36,540 --> 00:17:40,540
+لكل الإكسات في المنطقة هذه هذه الـ Inquality إيش
+
+207
+00:17:40,540 --> 00:17:44,880
+ما لها تتحقق الآن بس خلّيني أكملها دي شويه عشان
+
+208
+00:17:44,880 --> 00:17:52,040
+بستخدمها لشغلات أخرى عند A زائد Delta هذا بس مجرد
+
+209
+00:17:52,040 --> 00:17:56,340
+تعريف الـ Limit F' على الـ G' لما X تروح للـ A من
+
+210
+00:17:56,340 --> 00:17:59,120
+اليمين الآن
+
+211
+00:18:01,110 --> 00:18:04,910
+بدي أختار .. بدي أسهل على حالي بدل من كل مرة أقول
+
+212
+00:18:04,910 --> 00:18:08,130
+a زي الـ delta و بعد شوية ألاقي delta prime و أخد
+
+213
+00:18:08,130 --> 00:18:12,010
+ال minimum بينهم الاخرى أجدفم عليه عشان اللي بحكيه
+
+214
+00:18:12,010 --> 00:18:17,910
+بدي أخد اللي هو أقول choose c1 element in a و a زي
+
+215
+00:18:17,910 --> 00:18:24,630
+الـ delta الآن اختارلي c1 من ال a لعند مين لعند ال
+
+216
+00:18:24,630 --> 00:18:30,340
+a زي ال delta سميه c1 يعنيمجرد اختيار انا اخترته
+
+217
+00:18:30,340 --> 00:18:38,260
+الان choose .. choose C1 element in A و A زائد
+
+218
+00:18:38,260 --> 00:18:44,140
+دلتا يعني C1 وين موجود؟ موجود في المنطقة التي
+
+219
+00:18:44,140 --> 00:18:50,140
+تتحقق فيها F' ع G' نقص أصغر من ميل من إبسل عشان
+
+220
+00:18:50,140 --> 00:18:56,080
+تستخدمها بعد شويةواضح طيب هذه من جهة الأن من الجهة
+
+221
+00:18:56,080 --> 00:19:03,100
+الثانية عند limit f of x لما x تروح إلى مالة نهاية
+
+222
+00:19:03,100 --> 00:19:09,880
+أسف لل a من اليمين بسوء أيش هو معطينية هي النظرية
+
+223
+00:19:09,880 --> 00:19:13,540
+هي limit f of x لما x تروح إلى ال a من اليمين بسوء
+
+224
+00:19:13,540 --> 00:19:15,760
+limit g of x لما x تروح إلى ال a من اليمين بسوء
+
+225
+00:19:15,760 --> 00:19:23,170
+أيش ماله مالة نهاية من تعريفها هذا إيش يعنيلكل K
+
+226
+00:19:23,170 --> 00:19:29,270
+اللي هو element in R there exists Delta A' أكبر من
+
+227
+00:19:29,270 --> 00:19:36,590
+0 such that اللي هو F of X أكبر من اللي هو الـ K
+
+228
+00:19:36,590 --> 00:19:42,210
+لكل X و N موجودة في الـ A و A زائد مين Delta
+
+229
+00:19:42,210 --> 00:19:49,300
+Prime، مظبوط ولا لا؟ ماشي الحلالآن لكل x element
+
+230
+00:19:49,300 --> 00:19:52,200
+in A و A زي الـ delta prime هذه ال F of X أكبر من
+
+231
+00:19:52,200 --> 00:20:02,420
+مين؟ من K الآن لو كانت ال A و ال A .. الآن عندي
+
+232
+00:20:02,420 --> 00:20:07,540
+لكل subset من هذه .. لكل subset .. لو كان عندي A و
+
+233
+00:20:07,540 --> 00:20:10,940
+C2 مثلا subset من ال A و ال A زي ال delta prime
+
+234
+00:20:12,470 --> 00:20:17,070
+برضه ال F of X أكبر من كده تحققها تحقق إيش لكل X
+
+235
+00:20:17,070 --> 00:20:24,610
+وين موجودة في الفترة من A لعند مين لعند C2 الآن لو
+
+236
+00:20:24,610 --> 00:20:30,410
+كانت ال Delta اللي لجيتها برا ولا جوا بقدر أختار
+
+237
+00:20:30,410 --> 00:20:36,990
+جوا اللي هو C2بحيث أنه لو الـ C2 هذي هو اللي
+
+238
+00:20:36,990 --> 00:20:42,450
+اخترتها جوات من الـ A والـ A زائد دلتا بتظل الـ F
+
+239
+00:20:42,450 --> 00:20:47,470
+of X A شمالها أكبر من مين؟ من الـ K، واضح؟ إذا من
+
+240
+00:20:47,470 --> 00:20:52,330
+هذه بدي أستفيد شغلتين بدي أطبق التعريف هذا لـ K
+
+241
+00:20:52,330 --> 00:20:56,730
+محددة مين الـ K اللي بدي أطبقها؟ اللي عبارة عن F
+
+242
+00:20:56,730 --> 00:21:02,560
+عند مين؟ عند C1واضح؟ إذا الأن بما أنه limit f of X
+
+243
+00:21:02,560 --> 00:21:05,680
+ثم X تروح إلى الـA باليمين يستوى Infinity إذا for
+
+244
+00:21:05,680 --> 00:21:13,720
+K .. إذا for f of C1 there exists Delta Prime أي
+
+245
+00:21:13,720 --> 00:21:19,840
+Delta Prime معينة بحيث أن f of X أكبر من 100 من f
+
+246
+00:21:19,840 --> 00:21:27,060
+of C1 لكل X موجودة بين الـA والـA زائد Delta Prime
+
+247
+00:21:27,680 --> 00:21:31,860
+أنا بدي أختار في مين لكل X الموجودة بين الـ A و
+
+248
+00:21:31,860 --> 00:21:37,940
+الـ C2 حيث الـ C2 موجودة في الفترة هذه و بديها من
+
+249
+00:21:37,940 --> 00:21:44,120
+C1 و لجاي اذا عشان هيك بقدر اقول then we can
+
+250
+00:21:44,120 --> 00:21:50,280
+choose C2 بين A و C1الـ C2 بقدر اختارها فعلا
+
+251
+00:21:50,280 --> 00:21:55,640
+بختارها بس طبعا وين تشتري تشريع اللي اختيارها انها
+
+252
+00:21:55,640 --> 00:22:01,420
+تكون بين الفترة A و A زائد Delta Prime و بدياها
+
+253
+00:22:01,420 --> 00:22:06,900
+تيجي في داخل الـ A و الـ C1 اذا then we can choose
+
+254
+00:22:06,900 --> 00:22:13,200
+C2 element A of C1 بحيث ان ال F of X أكبر من مين
+
+255
+00:22:13,200 --> 00:22:22,810
+من F of C1 لكل X وينفي الفترة بين A وC اتنين طيب
+
+256
+00:22:22,810 --> 00:22:28,050
+هذه معناته ان ال F of X هذه لكل ال Xات اللي هان ال
+
+257
+00:22:28,050 --> 00:22:34,290
+F of X أكيد اي شمالها لها تساوي ايش F of C واحد هل
+
+258
+00:22:34,290 --> 00:22:38,930
+جد تعرف ليش بديها هذه عشان لزوم تعريف شغل معينة
+
+259
+00:22:38,930 --> 00:22:44,770
+بنفعش تكون ال F of X ايش بتساوي F of C واحد الآن
+
+260
+00:22:44,770 --> 00:22:55,550
+similarlySimilarly اللي هو بما انه limit G of X
+
+261
+00:22:55,550 --> 00:23:00,690
+بساوي مالة نهاية as X بتروح للـ A من اليمين إذا
+
+262
+00:23:00,690 --> 00:23:13,160
+بقدر ألاقي اللي هو C1 برايم بحيث انه G of X أكبرأو
+
+263
+00:23:13,160 --> 00:23:17,240
+لا تساوي طبعا أكبر الـ thrifty يعني لا تساوي G of
+
+264
+00:23:17,240 --> 00:23:27,860
+C1 لكل X موجودة في الـ A و C1 إيش إبراهيم اه G C1
+
+265
+00:23:27,860 --> 00:23:34,680
+إبراهيم اه G C1 إبراهيم الآن
+
+266
+00:23:34,680 --> 00:23:40,760
+ال .. ال .. ال .. الباخدالـ Minimum عشان طبعا هذا
+
+267
+00:23:40,760 --> 00:23:46,120
+اللي لجهة N C1' ممكن اللي هي لكل X في الـ A و C1'
+
+268
+00:23:46,620 --> 00:23:52,840
+و لكل X element in A و C2' بط مانش .. مش عارف مين
+
+269
+00:23:52,840 --> 00:23:58,670
+الأكبر من هذول فباخد الـ Minimum من الجهتينو
+
+270
+00:23:58,670 --> 00:24:03,090
+بسميها C1 مثلا او C2 ال minimum من التنتين C2
+
+271
+00:24:03,090 --> 00:24:09,550
+فبتصير عندى لان لكل X في ال A و ال minimum بينهن
+
+272
+00:24:09,550 --> 00:24:14,630
+هدى بتظبط و هدى بتظبط يعنى ال G of X لا تساوي F of
+
+273
+00:24:14,630 --> 00:24:20,670
+C1 و ال F of X لا تساوي مين؟ F of C1 إبرايم و منه
+
+274
+00:24:20,670 --> 00:24:25,870
+بعرف اللى بديه او بعمل زى ما هو عامل في الكتاب إيش
+
+275
+00:24:25,870 --> 00:24:29,280
+اللى بقوله؟ نشوف اللى بعدهاإذا الأن اللي اتفجنا
+
+276
+00:24:29,280 --> 00:24:38,540
+عليه أنه في الفترة من A لعند A زائد Delta هذه
+
+277
+00:24:38,540 --> 00:24:45,480
+متحققة هاي واحد واللي اتفجنا عليه أن F of X لا
+
+278
+00:24:45,480 --> 00:24:50,140
+تساوي F of C واحد في الفترة من A لعند C تنين
+
+279
+00:24:50,140 --> 00:24:55,580
+أشمالها متحققةيعني بمعنى آخر الـ Inquality هذه
+
+280
+00:24:55,580 --> 00:25:03,800
+وهذه التنتين محققات من أين؟ من A لعين C الاتنين
+
+281
+00:25:03,800 --> 00:25:05,360
+واضح؟
+
+282
+00:25:11,650 --> 00:25:17,590
+الان اصلنا لمرحلة انه نقدر نعرف الدالة اللي بتديها
+
+283
+00:25:17,590 --> 00:25:23,890
+اللي هي F of X بتساوي واحد ناقص F of C واحد على F
+
+284
+00:25:23,890 --> 00:25:30,470
+of X وواحد ناقص G of C واحد على G of X لكل X وين
+
+285
+00:25:30,470 --> 00:25:37,330
+موجودة بين A وC تنين و F of C واحد لا تساوي F of X
+
+286
+00:25:37,330 --> 00:25:43,340
+مظبوط؟ هيا اعملهاأه لإن وين بدي أعرف هذه الدالة F
+
+287
+00:25:43,340 --> 00:25:50,480
+من A لعند مين اللي هي بتتحقق عليها خاصية F of X ده
+
+288
+00:25:50,480 --> 00:25:55,080
+تسوى مين F of C واحد و بتتحقق عليها مين اللي هي
+
+289
+00:25:55,080 --> 00:26:00,480
+الخاصية اللي عندي هذه لإن كله هيلزمني هذا إذا الأن
+
+290
+00:26:05,340 --> 00:26:09,980
+اتفجنا عليها دي اه عشان بدي امسح انه عندي f prime
+
+291
+00:26:09,980 --> 00:26:13,800
+ع الجي برايم نقصها الأصغر من يابسولون لكل ال Xات
+
+292
+00:26:13,800 --> 00:26:18,700
+من A لعند A زائد ديلتا لكل ال Xات اللي هانا
+
+293
+00:26:18,700 --> 00:26:22,040
+واتفجنا ان F of X لا تساوي F of C one برضه في
+
+294
+00:26:22,040 --> 00:26:27,350
+الفرق في المنطقة من ايش من A لC2 اتفجنا عليهالان
+
+295
+00:26:27,350 --> 00:26:33,510
+نيجي نعرف الدالة اللي هتوصلني لهدفي خد الان f of x
+
+296
+00:26:33,510 --> 00:26:42,670
+بتساوي اللي هو واحد ناقص f of c واحد على f of x
+
+297
+00:26:42,670 --> 00:26:52,450
+على واحد ناقص g of c واحد على g of xالآن g of c
+
+298
+00:26:52,450 --> 00:26:55,810
+واحد مستحيل تساوي g of x من أي أسباب اللي حكيناها
+
+299
+00:26:55,810 --> 00:27:01,710
+وليس سبب آخر أيضا اللي هو ليش أنه عندي الـ g prime
+
+300
+00:27:01,710 --> 00:27:07,570
+الـ g prime of x ده يساوي سفر إذا حسب Rolle's
+
+301
+00:27:07,570 --> 00:27:13,570
+theorem مستحيل الـ g of c واحد اللي هو تساوي g of
+
+302
+00:27:13,570 --> 00:27:20,530
+x ليش عندي g continuous
+
+303
+00:27:21,550 --> 00:27:29,350
+on a .. طبعا الـ x والي موجودة جوها منها الان من x
+
+304
+00:27:29,350 --> 00:27:37,110
+لعند c واحد صح؟ و g is differentiable on x لعند
+
+305
+00:27:37,110 --> 00:27:44,090
+اللي هي c واحد open واضحة؟
+
+306
+00:27:44,090 --> 00:27:48,670
+لأنه أثر هذا continuous و differentiable من a لعند
+
+307
+00:27:48,670 --> 00:27:52,550
+bالعلمي إذا في الجزية هذه إذا أكيد اللي هم تتحقق
+
+308
+00:27:52,550 --> 00:28:01,270
+هذه الآن لو زي ما بنقول ان جي برا لو لو عندي بدي
+
+309
+00:28:01,270 --> 00:28:09,250
+يكون g of x بسوء g of c واحد معناته حسب role
+
+310
+00:28:09,250 --> 00:28:16,360
+theorem هيعطيني there existcx element in x و c1
+
+311
+00:28:16,360 --> 00:28:22,800
+such that g prime of cx هي ساوى سفر وهذا مستحيل
+
+312
+00:28:22,800 --> 00:28:27,320
+لإن هو معطيني الـ g prime of x لاتساوى سفر لكل x
+
+313
+00:28:27,320 --> 00:28:34,630
+element من a وb معناته معناه الحديثإنه الـ G of C
+
+314
+00:28:34,630 --> 00:28:41,510
+واحد والـ G of X مستحيل يكون متساويات لكل الإكسات
+
+315
+00:28:41,510 --> 00:28:47,230
+اللي في الفترة من A لعند مين لعند C اتنين واضح إذا
+
+316
+00:28:47,230 --> 00:28:50,730
+المقام لا يساوي سفر إذا اللي هو هذي is well
+
+317
+00:28:50,730 --> 00:29:00,950
+defined على مين على الفترة Aمظبوط و C2 خلّيها في
+
+318
+00:29:00,950 --> 00:29:05,070
+الذاكرة طيب اطلع لفوق و حسب لل limit الآن هذا
+
+319
+00:29:05,070 --> 00:29:12,070
+عرفناها احفظناها بنضطر ان امسح الان احسب لل limit
+
+320
+00:29:12,070 --> 00:29:20,230
+للي فوق limit F of X لما X تروح الي وين الي ايه من
+
+321
+00:29:20,230 --> 00:29:23,390
+اليمين بساوى واحد
+
+322
+00:29:24,530 --> 00:29:32,650
+نقص limit F of C1 عدد على F of X لما X تروح للـ A
+
+323
+00:29:32,650 --> 00:29:36,950
+من اليمين على اتجرأت لأنه عارف ان ال limit موجودة
+
+324
+00:29:36,950 --> 00:29:42,290
+على التوزيع اتجرأت عليه لأنه عارف limit G of C1
+
+325
+00:29:42,290 --> 00:29:49,530
+على G of X لما X تروح لوين للـ A من اليمين واضحة
+
+326
+00:29:49,530 --> 00:29:56,130
+جدش هذه ال limit0 لأن ال F of X وين بتروح الى مالة
+
+327
+00:29:56,130 --> 00:30:00,010
+نهاية هي بداية الموضوع مش بداية الموضوع كان عندنا
+
+328
+00:30:00,010 --> 00:30:05,010
+ان ال limit F of X على G of X لما X تروح الى A من
+
+329
+00:30:05,010 --> 00:30:08,670
+اليامين اللي هو سببت لكل القصة ان هذه ال limit
+
+330
+00:30:08,670 --> 00:30:11,230
+مالة نهاية وهذه مالة نهاية صارت مالة نهاية مالة
+
+331
+00:30:11,230 --> 00:30:15,710
+نهاية وهي اللي خلتني اروح بهذا الاتجاه اذا هذه
+
+332
+00:30:16,610 --> 00:30:19,750
+بتروح إلى ملأ نهاية إذا هذه بتروح إلى سفر وهذه
+
+333
+00:30:19,750 --> 00:30:22,330
+الملأ نهاية بتروح إلى سفر إذا هذا كله على بعض إيش
+
+334
+00:30:22,330 --> 00:30:30,170
+بيساوي؟ بيساوي واحد واضح طيب الآن for every
+
+335
+00:30:30,170 --> 00:30:34,570
+epsilon أكبر من سفر اللي هي اللي فوق خلاص بدنا
+
+336
+00:30:34,570 --> 00:30:39,270
+فيها الآن بيبقى إن ال limit بيساوي واحد إذا there
+
+337
+00:30:39,270 --> 00:30:44,890
+existsDelta أكبر من سفر أي إن كانت الـ Delta أكبر
+
+338
+00:30:44,890 --> 00:30:53,790
+من سفر Such that F of X ناقص واحد أصغر من مين من
+
+339
+00:30:53,790 --> 00:30:59,190
+الـ Epsilon Epsilon اللي بدنا فيها من الأول هذا لو
+
+340
+00:30:59,190 --> 00:31:04,830
+يعني لكل الـ Xاللي في الفترة من a ل a زي الـ delta
+
+341
+00:31:04,830 --> 00:31:07,910
+هذه الـ delta الجديدة مش صارت تكون الأولى فأنا
+
+342
+00:31:07,910 --> 00:31:13,330
+عشان أريح حالي بدي أخد اللي هو أصغر delta بلاجيها
+
+343
+00:31:13,330 --> 00:31:17,910
+و أقول حطها كمان جواها هادي حر أنا مدام بتنفع لالي
+
+344
+00:31:17,910 --> 00:31:21,690
+كبيرة ايضا كده بتنفع لمن؟ للصغيرة فاهمين أنا ايش
+
+345
+00:31:21,690 --> 00:31:28,270
+بقول؟ هذه اللي هي نسميها C3 معايا؟ فبصير عندى الآن
+
+346
+00:31:28,270 --> 00:31:39,200
+F of Xلكل X element in A لعند مين؟ C3 أي
+
+347
+00:31:39,200 --> 00:31:46,120
+سؤال؟ F of X ناقص واحد أصغر من Y أكبر من سالب Y
+
+348
+00:31:46,120 --> 00:31:49,960
+تلزمني هذه المنطقة اللي اللي جاي لأن F of X هذه
+
+349
+00:31:49,960 --> 00:31:56,720
+يعني F of X يعطينا F of X أكبر من واحد ناقص Y
+
+350
+00:31:59,390 --> 00:32:03,370
+باضحة؟ لأن هذه جلبها .. اللي هذه طبعا واحد ناقص
+
+351
+00:32:03,370 --> 00:32:07,850
+إبسلون اللي هي بيصير عنده ال .. ال .. ال إبسلون
+
+352
+00:32:07,850 --> 00:32:12,650
+أصغر من مين؟ من نص الواحد ناقص إبسلون أكبر من مين؟
+
+353
+00:32:12,650 --> 00:32:20,670
+اللي هي بيصير أكبر من نص باضحة؟ اللي هي بيصير واحد
+
+354
+00:32:20,670 --> 00:32:28,400
+على f of x اللي هو أصغر من مين؟هذا لمين؟ لكل
+
+355
+00:32:28,400 --> 00:32:32,760
+الاكسات اللي من A لعند مين C3 وطبيعي ال
+
+356
+00:32:32,760 --> 00:32:36,880
+inequalities كلها اللي قبل تتحقق هنا لإنها بتتحقق
+
+357
+00:32:36,880 --> 00:32:41,360
+من هنا لهنا بعضها وبعضها بتتحقق من هنا لهنا فأكيد
+
+358
+00:32:41,360 --> 00:32:47,040
+كلها هتتحقق للإكسات اللي مين في ال A وC3 أصلا أنا
+
+359
+00:32:47,040 --> 00:32:52,620
+رايح باتجاه مين؟ رايح باتجاه أثبت لكم أنه limit of
+
+360
+00:32:53,680 --> 00:32:59,060
+of x على g of x لما x تروح لل a من اليمين بساوي
+
+361
+00:32:59,060 --> 00:33:04,880
+limit f prime of x على g prime of x as x تروح لل a
+
+362
+00:33:04,880 --> 00:33:10,080
+من اليمين اللي هو ايش نسميه احنا قال اذا انا بهمن
+
+363
+00:33:10,080 --> 00:33:14,920
+مين الاكسات اللي جنب ال a لإن انا رايح لل a من وين
+
+364
+00:33:14,920 --> 00:33:18,800
+من اليمين فانا بهمن الجوار القريب جدا من ال a لإن
+
+365
+00:33:18,800 --> 00:33:21,960
+انا رايحله اصلا من هنافالإكسات اللي فيها هي اللي
+
+366
+00:33:21,960 --> 00:33:27,760
+بتلزمني عشان أصل للي بديها خلّي هذه في الذاكرة
+
+367
+00:33:27,760 --> 00:33:34,560
+فصار عندي الآن صار
+
+368
+00:33:34,560 --> 00:33:38,800
+عندي هذا كله حكيناه وخلصنا منه الان نيجي بدنا نصل
+
+369
+00:33:38,800 --> 00:33:44,620
+للي بدنايا بتتذكروا المرة الماضية أنا أصلا جيبت
+
+370
+00:33:44,620 --> 00:33:56,350
+هذهجبت هذه اتطلع فيها و قارنليها مع ال F of X على
+
+371
+00:33:56,350 --> 00:34:01,310
+ال G of X اللي انا بديهاها ال F of X على ال G of X
+
+372
+00:34:01,310 --> 00:34:08,170
+لو جيت قارنتها F of X على ال G of X ايش هتلاقيها؟
+
+373
+00:34:08,170 --> 00:34:14,810
+لو جيت ضربت هذه في F of X وهذه في G of X
+
+374
+00:34:17,460 --> 00:34:22,500
+بنرجع ل F of X ناقص F of C واحد و G of X ناقص G of
+
+375
+00:34:22,500 --> 00:34:26,680
+C واحد طب ايش دخلنا فيه هذه F of X ناقص F of C
+
+376
+00:34:26,680 --> 00:34:30,500
+واحد و G of X ناقص G of C واحد هذه اللي بالكوشي
+
+377
+00:34:30,500 --> 00:34:33,920
+mean value theorem اللي هتجيب لل F prime و G prime
+
+378
+00:34:33,920 --> 00:34:38,340
+اللي انا اصلا موجودات limited فبحصل ع اللي بديا
+
+379
+00:34:40,230 --> 00:34:44,470
+بالظبط الشي اللي بقوله انت يعني ليش هو فكر في هذه
+
+380
+00:34:44,470 --> 00:34:50,470
+بالسبب اللي حكيته F of X على D of X بالساوي F of X
+
+381
+00:34:50,470 --> 00:34:59,890
+على D of X في F of X على D of X على F of X مظبوط؟
+
+382
+00:34:59,890 --> 00:35:09,170
+ماعملش شيء الآن اسحبلي هذه خليهاوحط ليها هذه 1 على
+
+383
+00:35:09,170 --> 00:35:15,210
+f of x ماهي الحالة هذه مين هي في الواقع هي اللي
+
+384
+00:35:15,210 --> 00:35:20,410
+فوق اللي هي عبارة عن اضرب f of x هذه في هذه بصير
+
+385
+00:35:20,410 --> 00:35:32,250
+واحد ويساوي واحد اللي هي ناقص f of c صح؟ f of x
+
+386
+00:35:32,250 --> 00:35:42,660
+ناقص f of cعلى g of x نقص g of c هذا مين هي اللي
+
+387
+00:35:42,660 --> 00:35:46,700
+هي f of x على g of x في f of x عوضت هذه و حطيتها و
+
+388
+00:35:46,700 --> 00:35:51,760
+طلع عندي هذا مضروب في كله في مين في واحدة ال f of
+
+389
+00:35:51,760 --> 00:36:01,080
+x الآن عندي خليني أطبق اللي هو مين الـ Cauchy mean
+
+390
+00:36:01,080 --> 00:36:06,390
+value theorem على الفترة من a لمين ل c تلاتةماشي
+
+391
+00:36:06,390 --> 00:36:11,130
+اللي هو there exist طبعا كله متحقق اللي هو ال F و
+
+392
+00:36:11,130 --> 00:36:15,230
+ال G continuous و differentiable على ال A و ال C 3
+
+393
+00:36:15,230 --> 00:36:21,430
+اذا there exist اللي هو ايه اللي بدكي اسميها اللي
+
+394
+00:36:21,430 --> 00:36:25,690
+هي مثلا there exist gamma او زي ما هو مسميها في
+
+395
+00:36:25,690 --> 00:36:32,230
+الكتاب there exist exi element in A و C تلاتة such
+
+396
+00:36:32,230 --> 00:36:41,500
+thatf prime of xi على g prime of xi بساوي بس ان
+
+397
+00:36:41,500 --> 00:36:44,780
+انا عشان بتطبقها على مين على الفترة f of x لعند
+
+398
+00:36:44,780 --> 00:36:48,800
+مين عشان تطلع عندي f of x of مين و f of c واحد
+
+399
+00:36:48,800 --> 00:36:53,120
+معايا للي اكسات اللي وين اللي موجودة من a لعند c
+
+400
+00:36:53,120 --> 00:36:58,060
+تلاتة بعد اذنكم بتطبقها الكوشي mean value theorem
+
+401
+00:36:58,760 --> 00:37:01,840
+ما هي اصلا continuous و differentiable على الفترة
+
+402
+00:37:01,840 --> 00:37:06,500
+دي كلها من ضمنها مين الفترة A وC ثلاثة ومن ضمنها
+
+403
+00:37:06,500 --> 00:37:11,040
+الفترة A وC واحد اللي انا بده اطبق عليها بال end
+
+404
+00:37:11,040 --> 00:37:15,900
+point مين ال end point C واحد هيها و ال end point
+
+405
+00:37:15,900 --> 00:37:21,890
+ال X اللي موجودة بين A و B مينو C3 واضح واضح اللي
+
+406
+00:37:21,890 --> 00:37:26,350
+بدي .. اذا الان الان there exists x i الان بالظبط
+
+407
+00:37:26,350 --> 00:37:30,630
+بدي اطبق الميكوشي main value term ع الفترة اللي هي
+
+408
+00:37:30,630 --> 00:37:37,790
+X و C1 اللي ال F و ال G differentiable عليها من
+
+409
+00:37:37,790 --> 00:37:40,530
+وين ال X هذه ال X اللي فوق هذه اللي بشتغل فيها
+
+410
+00:37:40,530 --> 00:37:45,890
+اللي هي من A لعن مين لعن C3إذن there exists xi
+
+411
+00:37:45,890 --> 00:37:50,750
+element in x و c واحد such that f prime of xi على
+
+412
+00:37:50,750 --> 00:37:58,770
+g prime of xi في إيش بتساوي f of c واحد أو f of x
+
+413
+00:37:58,770 --> 00:38:05,390
+ناقص f of c واحد فاهمين طبعا على g of x ناقص g of
+
+414
+00:38:05,390 --> 00:38:10,990
+c واحد بس ضربت في ناقص فوق و ناقص صحيح تتعدى زي ما
+
+415
+00:38:10,990 --> 00:38:20,080
+هي لأن هذههي هذه إذا صارت عندي F of X على G of X
+
+416
+00:38:20,080 --> 00:38:26,000
+بسوء F prime على G prime في واحد على مين على F of
+
+417
+00:38:26,000 --> 00:38:32,060
+X وهو اللي بده يوصلنا للي بدنا هي ماعليش أمسح اللي
+
+418
+00:38:32,060 --> 00:38:35,280
+هان بيصير عندى
+
+419
+00:38:38,180 --> 00:38:45,160
+بصير عندى ال F of X على مين F of X على G of X
+
+420
+00:38:45,160 --> 00:38:48,540
+بساوى
+
+421
+00:38:48,540 --> 00:38:54,120
+اللى هو هدى شلناه و حطينا مكانها مين بساوى F prime
+
+422
+00:38:54,120 --> 00:39:02,140
+of X I على G prime of X I فى مين فى واحد على ايه
+
+423
+00:39:02,140 --> 00:39:03,500
+ايش على F
+
+424
+00:39:09,880 --> 00:39:18,360
+واضح أه؟ الآن نمحى هنا هدولة ال Xات اللي هنا هما
+
+425
+00:39:18,360 --> 00:39:23,380
+ال Xات اللي وين موجودة بين A وC3 ماشي الحال بدي
+
+426
+00:39:23,380 --> 00:39:29,120
+الآن غايتي أني أوجد F of X على G of X ناقص مين؟
+
+427
+00:39:29,120 --> 00:39:34,600
+ناقص Lواضحة إذا نلخص كل اللي خدناه for every
+
+428
+00:39:34,600 --> 00:39:38,800
+epsilon element in zero نص there exists delta أكبر
+
+429
+00:39:38,800 --> 00:39:48,980
+من سفر such that if اللي هو x أكبر من a أصغر من c3
+
+430
+00:39:48,980 --> 00:39:53,260
+أصغر من c2 أصغر من c1 أصغر من a زي ال delta اللي
+
+431
+00:39:53,260 --> 00:40:02,070
+لجناها فاهمين عليها وكون عندي اللي هو قيلةالعوّض f
+
+432
+00:40:02,070 --> 00:40:08,110
+of x على g of x ناقص ال .. بدي أقبل لك يميها هذا
+
+433
+00:40:08,110 --> 00:40:12,510
+اللي هو أصغر من epsilon في something مش مشكلة و
+
+434
+00:40:12,510 --> 00:40:15,810
+epsilon is arbitral أيضا بيصير limit زي ما بدنايا
+
+435
+00:40:15,810 --> 00:40:24,170
+واضح هذا الآن بالظبط بيساوي اللي هو العوض اللي هو
+
+436
+00:40:24,170 --> 00:40:28,590
+f of x اللي هي عبارة عن واحد
+
+437
+00:40:32,680 --> 00:40:41,680
+أف برايم of x i على جي برايم of x i في واحد على أف
+
+438
+00:40:41,680 --> 00:40:48,180
+of x ناقص اللى
+
+439
+00:40:48,180 --> 00:40:54,460
+واضح لحتى الان ناخد هذه عامل مشترك برا يصبح واحد
+
+440
+00:40:54,460 --> 00:41:02,140
+على أف of x في أف برايم of x iعلى g prime of xi
+
+441
+00:41:02,140 --> 00:41:13,560
+ناقص L في F of X صح؟ أخدت
+
+442
+00:41:13,560 --> 00:41:17,300
+الواحد على F of X عام المشترك positive أه positive
+
+443
+00:41:17,300 --> 00:41:21,080
+اللي هو F of X هي أكبر من 2 طلعت اللي هو إيش ماله
+
+444
+00:41:21,080 --> 00:41:27,260
+هذا؟هنا ده بدي أضيف term و أطرح term عشان أحصل على
+
+445
+00:41:27,260 --> 00:41:31,780
+هذه اللي بدياها أصلا اللي هي الجد تفهموه مش مقصودة
+
+446
+00:41:31,780 --> 00:41:38,900
+1 على f of x في f prime of xi على g prime of xi
+
+447
+00:41:38,900 --> 00:41:50,130
+ناقص L زائد اللي هوL نقص L F of X ايش سويت استخدمت
+
+448
+00:41:50,130 --> 00:41:56,170
+ال triangle inequality باضافة L وطرحة L هذا كله
+
+449
+00:41:56,170 --> 00:42:04,130
+مضروف 100 في 1 على F of X هذا الآن ال F of X أكبر
+
+450
+00:42:04,130 --> 00:42:12,690
+من 2 حصلناها مظبوط بصير أصغر من نص واضحة
+
+451
+00:42:13,760 --> 00:42:18,920
+الآن F' على G' لـ XI الـ XI وين لاجيناها؟ في
+
+452
+00:42:18,920 --> 00:42:24,940
+الفترة بين A وC1 ينطبق عليها اللي هو من رأس الدور
+
+453
+00:42:24,940 --> 00:42:34,580
+اللي هي F' على G' of X ناقص
+
+454
+00:42:34,580 --> 00:42:41,200
+L أصغر من الـ Y من أول ما بدينايعني إنه هذا أصغر
+
+455
+00:42:41,200 --> 00:42:44,460
+من إبسلون لكل الإكسات اللي موجودة من A لعند مين؟
+
+456
+00:42:44,460 --> 00:42:48,080
+لعند الـA زاد دلتها من ضمنهم مين؟ لك فيبراين كلام
+
+457
+00:42:48,080 --> 00:42:55,940
+دقيق هذه في إبسلون زاد الآن هذا ال بطلحها برا عن
+
+458
+00:42:55,940 --> 00:43:01,760
+المشترك ال اللي هي في مين؟ في واحد ناقص F of X
+
+459
+00:43:01,760 --> 00:43:10,240
+واحد ناقص F of Xالان وهذا أصغر من نص في إبسلون
+
+460
+00:43:10,240 --> 00:43:15,440
+زائد absolute value لأل واحد ماقص F of X هذه ال
+
+461
+00:43:15,440 --> 00:43:20,260
+Xات اللي موجودة بين A وC تلاتة وهذه اللي هي قبل
+
+462
+00:43:20,260 --> 00:43:26,150
+بشوية أصغر من مين من اللي هو إبسلونأصغر من إبسلون
+
+463
+00:43:26,150 --> 00:43:30,030
+واحد ناقص F of X أصغر من مين؟ من إبسلون مش هثبتنا
+
+464
+00:43:30,030 --> 00:43:33,910
+limit F of X بيساوي واحد قبل بشوية اللي هو واحد
+
+465
+00:43:33,910 --> 00:43:37,050
+ناقص F of X اللي هي أصغر من إياش من إبسلون في
+
+466
+00:43:37,050 --> 00:43:42,850
+إبسلونصار عندي الان هذا المقدار اللي هو بساوي
+
+467
+00:43:42,850 --> 00:43:47,030
+epsilon على اتنين في واحد زايد absolute value
+
+468
+00:43:47,030 --> 00:43:52,810
+للالف لان ال epsilon مضروبة فيه اذا as لان هذا ال
+
+469
+00:43:52,810 --> 00:43:55,430
+epsilon arbitrary وهذا الكلام صحيح for every
+
+470
+00:43:55,430 --> 00:43:59,710
+epsilon epsilon بين zero و نص اذا هذا المقدار بقدر
+
+471
+00:43:59,710 --> 00:44:04,350
+ازغره جد ما بدي بتصغير epsilon لانه مضروب فيه ضرب
+
+472
+00:44:04,350 --> 00:44:08,630
+اذا صار عندي لكل epsilon أكبر من zero و نصلـ G الـ
+
+473
+00:44:08,630 --> 00:44:12,650
+Delta بحيث أنه الإكسات اللي هنا بيقدّي لي أن هذا
+
+474
+00:44:12,650 --> 00:44:20,810
+المقدار أصغر من أي قيمة بديّها إذا هذا مفهوم limit
+
+475
+00:44:20,810 --> 00:44:31,530
+F of X على G of X as X بتروح لنين؟ للـ A من اليمين
+
+476
+00:44:31,530 --> 00:44:38,920
+بساوي الـ L وهو المطلوبالثيورم اللي حكينا عنها
+
+477
+00:44:38,920 --> 00:44:45,060
+برضه صحيحة في حالة اللي هو the same .. under the
+
+478
+00:44:45,060 --> 00:44:50,160
+same conditions for the calculation of limits as x
+
+479
+00:44:50,160 --> 00:44:55,540
+goes to infinity or x as goes mean to سالبinfinity
+
+480
+00:44:55,540 --> 00:44:59,140
+بس بورهان يعني بدون modification عن اللي موجود
+
+481
+00:44:59,140 --> 00:45:03,140
+بتصلى اللي بدكيها نيجي ناخد أمثلة الأمثلة هذه طبعا
+
+482
+00:45:03,140 --> 00:45:09,500
+أمثلة سهلة ومثلة calculus نمر عليها اللي هو ونشوف
+
+483
+00:45:09,500 --> 00:45:16,480
+كيف نطبق نظريتنا أو نظرياتنا كيف نوظفها الحساب هذه
+
+484
+00:45:16,480 --> 00:45:23,260
+النهايات عندى شوفوا صلى النبي عليه الصلاة والسلام
+
+485
+00:45:24,800 --> 00:45:30,720
+Find Limitlog sin x على log x as x بتروح للـ 0 من
+
+486
+00:45:30,720 --> 00:45:35,480
+وين؟ من اليمين و احنا عارفين .. مشتغلين على الفترة
+
+487
+00:45:35,480 --> 00:45:40,460
+0 or by يعني فترة بجوار مين؟ الصفر من اليمين و
+
+488
+00:45:40,460 --> 00:45:43,540
+بدنا نروح للـ 0 من اليمين هذا طبعا لو جينا عوضنا
+
+489
+00:45:43,540 --> 00:45:48,080
+هتطلع اللي هو sin 0 0 و أنا 0 فبصير مالة نهاية على
+
+490
+00:45:48,080 --> 00:45:53,580
+مين؟ على مالة نهاية ك limits الآن صارت عندي مالة
+
+491
+00:45:53,580 --> 00:45:57,880
+نهاية على مالة نهاية إذا إيش بتنسوه؟إيش نسوي؟
+
+492
+00:45:57,880 --> 00:46:06,400
+راحين لـ 0 من اليمين بيصير عنده اللي هو بالفاضل
+
+493
+00:46:06,400 --> 00:46:11,400
+هذه و بالفاضل هذه بيصير تفاضلها هذه واضحة و هذه
+
+494
+00:46:11,400 --> 00:46:14,060
+تفاضلها هيك مافيش داعي لنا ندخل في التفاصيل لإن كل
+
+495
+00:46:14,060 --> 00:46:20,320
+بعرفها الآن بنبسط الأمر بيصير X Cos X على Sine X
+
+496
+00:46:20,930 --> 00:46:26,550
+الان لو جيت اتطلعت لهذه ال limit و لهذه ال limit
+
+497
+00:46:26,550 --> 00:46:29,530
+هذه موجودة و هذه موجودة هذه موجودة لإنه تعويض
+
+498
+00:46:29,530 --> 00:46:33,410
+مباشر واحد هذه موجودة لإنه بيصير سفر ع سفر بتعملها
+
+499
+00:46:33,410 --> 00:46:36,570
+global rule بيصير واحد على cosine بتطلع واحد برضه
+
+500
+00:46:36,570 --> 00:46:40,590
+إذا اقدرت اوزع لإنه عارف الليمتين الموجودة فبيصير
+
+501
+00:46:40,590 --> 00:46:46,220
+عبارة عن واحد في واحد بيساوي ايه؟ واحدالان خد على
+
+502
+00:46:46,220 --> 00:46:49,900
+الفترة من Zero لعند مين لعند باي على اتنين limit
+
+503
+00:46:49,900 --> 00:46:53,300
+واحد على X نقص واحد على sign ال X ثم X تروح ل Zero
+
+504
+00:46:53,300 --> 00:46:57,140
+من وين؟ من اليمين وهذه طبعا شغلات اللي هو درجنا
+
+505
+00:46:57,140 --> 00:47:01,120
+عليها في ال calculus بنوحد المقامات وبصير مكتوبة
+
+506
+00:47:01,120 --> 00:47:05,030
+على صورة sign ال X نقص X على X في sign ال Xماشي
+
+507
+00:47:05,030 --> 00:47:10,650
+الحل برضه 0 على 0 بنفضلها بيصير cos X ناقص 1 على
+
+508
+00:47:10,650 --> 00:47:15,470
+اللي هو sin X زاد X فcos X لو جينا عوضنا هتطلع
+
+509
+00:47:15,470 --> 00:47:21,250
+برضه 0 على 0 بنشتق كمان مرة بتطلع ناقص sin X على
+
+510
+00:47:21,250 --> 00:47:26,150
+هذا المقدار هذا المقدار هنا بتصفر وهنا بيصير 2
+
+511
+00:47:26,150 --> 00:47:32,680
+وهنا 0 0 على 2 مش بيساوي بيساوي 0نأتي لهذا مثال
+
+512
+00:47:32,680 --> 00:47:36,440
+معهود الـ let I بيساوين 1 إلى ما لنهاية الـ
+
+513
+00:47:36,440 --> 00:47:39,760
+interval اللي عليها بدنا نشتغل و بدنا ناخد ال
+
+514
+00:47:39,760 --> 00:47:43,600
+limit لما X تروح إلى وين إلى ما لنهاية المقدار 1
+
+515
+00:47:43,600 --> 00:47:48,960
+زائد 1 على X و الكل أُس Xهذا المقدار الان بدي
+
+516
+00:47:48,960 --> 00:47:53,460
+استخدم اللي هو يحوله ل exponential to the len ما
+
+517
+00:47:53,460 --> 00:47:55,660
+احنا عارفين ال exponential و ال len اللي هو n
+
+518
+00:47:55,660 --> 00:47:59,860
+versus لبعض فبصير عندي E to the X لن ال 1 زائد 1
+
+519
+00:47:59,860 --> 00:48:05,120
+على X هذا المقدار بدي اشغل على هذاكخطوة أولى بقول
+
+520
+00:48:05,120 --> 00:48:09,260
+limit ال X لما واحد زائد واحد على X لما X تروح إلى
+
+521
+00:48:09,260 --> 00:48:14,140
+مال نهاية بيصير هذا المقدار limit لما واحد زائد X
+
+522
+00:48:14,140 --> 00:48:18,160
+على واحد على X ليش عملت هيك لأن لما X تروح إلى مال
+
+523
+00:48:18,160 --> 00:48:21,360
+نهاية هذا سفر و لما X تروح إلى مال نهاية هذا سفر و
+
+524
+00:48:21,360 --> 00:48:25,340
+هذا لما واحد فبصير سفر على سفر بقدر استخدم ال loop
+
+525
+00:48:25,340 --> 00:48:30,730
+تالذرول اللي انا برهنتهابقى فاضل اللي فوق و فاضل
+
+526
+00:48:30,730 --> 00:48:33,850
+اللي تحت .. فضلنا اللي فوق هيه و فضلنا اللي تحت
+
+527
+00:48:33,850 --> 00:48:37,590
+هيه .. بروح هذا المقدار مع هذا المقدار .. بظل
+
+528
+00:48:37,590 --> 00:48:42,800
+limit1 زي 1 على X مقص 1 لما X تروح لمدة نهاية ده
+
+529
+00:48:42,800 --> 00:48:47,940
+بروح لصفر بطلع جداش 1 لأن 1 بنرد بنعوض وينها لأن
+
+530
+00:48:47,940 --> 00:48:50,940
+ال exponential is a continuous function لذا نصحي
+
+531
+00:48:50,940 --> 00:48:54,180
+عن عند limit ال 1 زي 1 على X لما X تروح لمدة نهاية
+
+532
+00:48:54,180 --> 00:48:57,920
+to the X وعبارة عن E to the limit لأن ال
+
+533
+00:48:57,920 --> 00:49:01,760
+exponential عبارة عن continuous function فبصير E
+
+534
+00:49:01,760 --> 00:49:08,300
+to the 1 ويساوي Eهذا اللي هو بيكون هيك احنا انهينا
+
+535
+00:49:08,300 --> 00:49:16,240
+اللي هي lobitals rules أو قواعد lobital أو صرنا
+
+536
+00:49:16,240 --> 00:49:20,360
+الآن ندخل
+
+537
+00:49:20,360 --> 00:49:28,120
+على اللي هو ال chapter ال section الأخير اللي هو
+
+538
+00:49:28,120 --> 00:49:32,120
+ال section عبارة عن ستة أربعة اللي هو taylor's
+
+539
+00:49:32,120 --> 00:49:38,540
+theoremأو tailors اللي هي .. هنحكي عن tailors
+
+540
+00:49:38,540 --> 00:49:43,640
+polynomial أو tailors أيضا اللي هو approximation
+
+541
+00:49:43,640 --> 00:49:49,820
+كيف اللي هو .. ان مرة جاية ان شاء الله كيف بنقرب
+
+542
+00:49:49,820 --> 00:49:54,840
+بعض الدوالاللي هو المحترمة اللي بتكون
+
+543
+00:49:54,840 --> 00:49:58,220
+differentiable first derivative و second
+
+544
+00:49:58,220 --> 00:50:00,400
+derivative و third derivative لأ عند ال derivative
+
+545
+00:50:00,400 --> 00:50:05,700
+اللي بدنا .. اللي .. اللي هي ما قبل اشتقاق عندها
+
+546
+00:50:05,700 --> 00:50:11,760
+بنقربها ب polynomial و طبعا انتوا عارفين جدش ال
+
+547
+00:50:11,760 --> 00:50:17,380
+polynomial من الدوال السهلةسهلة التعامل سواء في
+
+548
+00:50:17,380 --> 00:50:21,500
+تفاضل أو في تكامل أو حتى لو بدنا نحلها و نجيب
+
+549
+00:50:21,500 --> 00:50:26,340
+جذورها و نجيب كذا في شغل عليها كثير فالناس ترغب في
+
+550
+00:50:26,340 --> 00:50:30,540
+إنها تجيب بعض الدوال اللي بتكون أحيانا معقدة و
+
+551
+00:50:30,540 --> 00:50:35,380
+صعبة التكامل أو خلنا نقول بتغلب شوية في التفاضل أو
+
+552
+00:50:35,380 --> 00:50:40,040
+في وجود الجذور أو كده و نحولها إلىاللي هي
+
+553
+00:50:40,040 --> 00:50:44,020
+polynomial صبعا في مقدار خطأ الخطأ هن .. ان شاء
+
+554
+00:50:44,020 --> 00:50:47,520
+الله ال Taylor's theorem بتقولنا كيف ان هو نوجد
+
+555
+00:50:47,520 --> 00:50:51,320
+هذا الخطأ او ايش هو الخطأ و ايضا Taylor's theorem
+
+556
+00:50:51,320 --> 00:50:55,780
+زي ما انتوا عارفين هنوضح كيف هي عبارة عن تعميم لل
+
+557
+00:50:55,780 --> 00:51:00,340
+main value theorem هي تعميم لل main value theorem
+
+558
+00:51:00,340 --> 00:51:05,140
+في حالة اللي هي n بتساوي سفر بتطلع هي بالظبط ال
+
+559
+00:51:05,140 --> 00:51:09,100
+main value theorem كيف؟
+
+560
+00:51:10,860 --> 00:51:15,100
+الـ Main Value Theorem عبارة الـ Taylor's Theorem
+
+561
+00:51:15,100 --> 00:51:19,900
+عبارة عن تعميم للـ Main Value Theorem خد الآن
+
+562
+00:51:19,900 --> 00:51:25,020
+بتساوي سفر بتطلعلك الـ Main Value Theorem بيشحمها
+
+563
+00:51:25,020 --> 00:51:27,740
+و لحمها و إلى لقاء
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..c8cbd6234b3619056cef81985abe54a63c4d87a8
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 3026, "start": 5.04, "end": 30.26, "text": "بسم الله الرحمن الرحيم المحاضرة السابعة مساق تحليل حقيقي 2 اللي طلبت كلية العلوم بالجامعة الإسلامية قسم الرياضيات اليوم هيكون عندنا استكمال ل .. اللي هي قواعد أو قاعدة Lobital", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 9673, 5016, 46958, 25720, 21136, 16758, 27884, 47524, 995, 4587, 6055, 5016, 20292, 1211, 11331, 38436, 38436, 568, 13672, 1829, 23032, 46152, 2655, 28242, 10632, 18863, 1211, 20498, 20666, 7435, 10943, 27884, 33688, 3794, 37440, 10632, 12174, 38251, 2423, 16572, 46958, 1829, 9307, 45595, 20498, 39896, 30544, 43242, 8315, 44713, 24793, 6027, 5296, 4386, 13672, 1829, 39896, 12174, 14407, 22488, 34051, 12174, 995, 22488, 3660, 30719, 1686], "avg_logprob": -0.1062297070181215, "compression_ratio": 1.7087912087912087, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 5.04, "end": 5.44, "word": "بسم", "probability": 0.861328125}, {"start": 5.44, "end": 5.64, "word": " الله", "probability": 0.95947265625}, {"start": 5.64, "end": 5.98, "word": " الرحمن", "probability": 0.96044921875}, {"start": 5.98, "end": 6.58, "word": " الرحيم", "probability": 0.99365234375}, {"start": 6.58, "end": 8.78, "word": " المحاضرة", "probability": 0.97265625}, {"start": 8.78, "end": 9.82, "word": " السابعة", "probability": 0.9630533854166666}, {"start": 9.82, "end": 11.4, "word": " مساق", "probability": 0.9026692708333334}, {"start": 11.4, "end": 12.8, "word": " تحليل", "probability": 0.981201171875}, {"start": 12.8, "end": 13.3, "word": " حقيقي", "probability": 0.9744466145833334}, {"start": 13.3, "end": 13.74, "word": " 2", "probability": 0.47314453125}, {"start": 13.74, "end": 14.9, "word": " اللي", "probability": 0.81201171875}, {"start": 14.9, "end": 15.48, "word": " طلبت", "probability": 0.9822591145833334}, {"start": 15.48, "end": 16.0, "word": " كلية", "probability": 0.831298828125}, {"start": 16.0, "end": 16.6, "word": " العلوم", "probability": 0.9519856770833334}, {"start": 16.6, "end": 17.88, "word": " بالجامعة", "probability": 0.94580078125}, {"start": 17.88, "end": 18.64, "word": " الإسلامية", "probability": 0.9681396484375}, {"start": 18.64, "end": 19.72, "word": " قسم", "probability": 0.977294921875}, {"start": 19.72, "end": 20.76, "word": " الرياضيات", "probability": 0.93603515625}, {"start": 20.76, "end": 23.8, "word": " اليوم", "probability": 0.934326171875}, {"start": 23.8, "end": 24.24, "word": " هيكون", "probability": 0.721435546875}, {"start": 24.24, "end": 24.58, "word": " عندنا", "probability": 0.92919921875}, {"start": 24.58, "end": 25.62, "word": " استكمال", "probability": 0.9671223958333334}, {"start": 25.62, "end": 26.28, "word": " ل", "probability": 0.7568359375}, {"start": 26.28, "end": 26.66, "word": " ..", "probability": 0.299072265625}, {"start": 26.66, "end": 26.78, "word": " اللي", "probability": 0.940185546875}, {"start": 26.78, "end": 27.08, "word": " هي", "probability": 0.939453125}, {"start": 27.08, "end": 28.26, "word": " قواعد", "probability": 0.984375}, {"start": 28.26, "end": 28.9, "word": " أو", "probability": 0.81396484375}, {"start": 28.9, "end": 29.72, "word": " قاعدة", "probability": 0.9591064453125}, {"start": 29.72, "end": 30.26, "word": " Lobital", "probability": 0.5982666015625}], "temperature": 1.0}, {"id": 2, "seek": 4477, "start": 31.01, "end": 44.77, "text": "اتحدثنا المرة الماضية عن الحالة اللى بتحدث سفر على سفر وكيف انعلجها وتم اللى هو برهان النظريات المعنية في ذلك", "tokens": [9307, 24401, 12984, 8315, 9673, 25720, 9673, 46958, 10632, 18871, 21542, 6027, 3660, 13672, 7578, 39894, 24401, 12984, 8608, 5172, 2288, 15844, 8608, 5172, 2288, 4032, 4117, 33911, 16472, 30241, 7435, 11296, 4032, 39237, 13672, 7578, 31439, 4724, 2288, 3224, 7649, 28239, 19913, 16572, 9307, 9673, 3615, 1863, 10632, 8978, 29910, 23275], "avg_logprob": -0.1188826617204918, "compression_ratio": 1.6583333333333334, "no_speech_prob": 3.635883331298828e-06, "words": [{"start": 31.01, "end": 32.43, "word": "اتحدثنا", "probability": 0.841064453125}, {"start": 32.43, "end": 32.79, "word": " المرة", "probability": 0.967041015625}, {"start": 32.79, "end": 33.53, "word": " الماضية", "probability": 0.9939778645833334}, {"start": 33.53, "end": 34.09, "word": " عن", "probability": 0.98583984375}, {"start": 34.09, "end": 35.25, "word": " الحالة", "probability": 0.9908854166666666}, {"start": 35.25, "end": 35.77, "word": " اللى", "probability": 0.7734375}, {"start": 35.77, "end": 36.33, "word": " بتحدث", "probability": 0.9674479166666666}, {"start": 36.33, "end": 36.69, "word": " سفر", "probability": 0.8738606770833334}, {"start": 36.69, "end": 36.93, "word": " على", "probability": 0.8291015625}, {"start": 36.93, "end": 37.41, "word": " سفر", "probability": 0.9982096354166666}, {"start": 37.41, "end": 38.11, "word": " وكيف", "probability": 0.9127604166666666}, {"start": 38.11, "end": 38.89, "word": " انعلجها", "probability": 0.722900390625}, {"start": 38.89, "end": 39.91, "word": " وتم", "probability": 0.611083984375}, {"start": 39.91, "end": 40.19, "word": " اللى", "probability": 0.93505859375}, {"start": 40.19, "end": 40.55, "word": " هو", "probability": 0.83837890625}, {"start": 40.55, "end": 41.17, "word": " برهان", "probability": 0.951416015625}, {"start": 41.17, "end": 42.37, "word": " النظريات", "probability": 0.9373779296875}, {"start": 42.37, "end": 44.07, "word": " المعنية", "probability": 0.9775390625}, {"start": 44.07, "end": 44.31, "word": " في", "probability": 0.71533203125}, {"start": 44.31, "end": 44.77, "word": " ذلك", "probability": 0.996826171875}], "temperature": 1.0}, {"id": 3, "seek": 7198, "start": 45.29, "end": 71.99, "text": "الآن بدنا نتحدث على الحالة اللي بتحدث عندنا أيضًا 0 على 0 ولكن ال limit ال X بتروح إلى مالة نهاية أو سالب مالة نهاية وقلنا عنها صحيحة برضه وبعد ذلك هنتحدث عن الحالة infinity على infinity نيجي الآن لنظريتنا اللي وصلنا عندها suppose that", "tokens": [6027, 148, 48506, 47525, 8315, 8717, 2655, 24401, 12984, 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"end": 46.93, "word": " على", "probability": 0.53955078125}, {"start": 46.93, "end": 47.91, "word": " الحالة", "probability": 0.98828125}, {"start": 47.91, "end": 48.55, "word": " اللي", "probability": 0.88623046875}, {"start": 48.55, "end": 49.93, "word": " بتحدث", "probability": 0.97607421875}, {"start": 49.93, "end": 51.31, "word": " عندنا", "probability": 0.86767578125}, {"start": 51.31, "end": 53.63, "word": " أيضًا", "probability": 0.7894287109375}, {"start": 53.63, "end": 53.99, "word": " 0", "probability": 0.54296875}, {"start": 53.99, "end": 54.25, "word": " على", "probability": 0.83642578125}, {"start": 54.25, "end": 54.55, "word": " 0", "probability": 0.98291015625}, {"start": 54.55, "end": 55.37, "word": " ولكن", "probability": 0.78125}, {"start": 55.37, "end": 55.73, "word": " ال", "probability": 0.650390625}, {"start": 55.73, "end": 56.71, "word": " limit", "probability": 0.72900390625}, {"start": 56.71, "end": 56.91, "word": " ال", "probability": 0.71044921875}, {"start": 56.91, "end": 57.11, "word": " X", "probability": 0.6455078125}, {"start": 57.11, "end": 57.49, "word": " بتروح", "probability": 0.9676106770833334}, {"start": 57.49, "end": 57.65, "word": " إلى", "probability": 0.8828125}, {"start": 57.65, "end": 57.95, "word": " مالة", "probability": 0.674560546875}, {"start": 57.95, "end": 58.47, "word": " نهاية", "probability": 0.9298502604166666}, {"start": 58.47, "end": 59.43, "word": " أو", "probability": 0.91064453125}, {"start": 59.43, "end": 59.79, "word": " سالب", "probability": 0.7861328125}, {"start": 59.79, "end": 60.03, "word": " مالة", "probability": 0.9423828125}, {"start": 60.03, "end": 60.25, "word": " نهاية", "probability": 0.9378255208333334}, {"start": 60.25, "end": 60.59, "word": " وقلنا", "probability": 0.73150634765625}, {"start": 60.59, "end": 60.83, "word": " عنها", "probability": 0.994384765625}, {"start": 60.83, "end": 61.35, "word": " صحيحة", "probability": 0.97578125}, {"start": 61.35, "end": 61.77, "word": " برضه", "probability": 0.9866536458333334}, {"start": 61.77, "end": 62.65, "word": " وبعد", "probability": 0.7315673828125}, {"start": 62.65, "end": 63.01, "word": " ذلك", "probability": 0.9931640625}, {"start": 63.01, "end": 63.69, "word": " هنتحدث", "probability": 0.9808349609375}, {"start": 63.69, "end": 63.87, "word": " عن", "probability": 0.99462890625}, {"start": 63.87, "end": 64.31, "word": " الحالة", "probability": 0.99169921875}, {"start": 64.31, "end": 64.85, "word": " infinity", "probability": 0.4892578125}, {"start": 64.85, "end": 65.79, "word": " على", "probability": 0.88916015625}, {"start": 65.79, "end": 66.77, "word": " infinity", "probability": 0.677734375}, {"start": 66.77, "end": 67.49, "word": " نيجي", "probability": 0.850341796875}, {"start": 67.49, "end": 67.73, "word": " الآن", "probability": 0.63720703125}, {"start": 67.73, "end": 68.55, "word": " لنظريتنا", "probability": 0.9010416666666666}, {"start": 68.55, "end": 69.21, "word": " اللي", "probability": 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"end": 870.9, "word": " مين", "probability": 0.63427734375}, {"start": 870.9, "end": 871.3, "word": " من", "probability": 0.93017578125}, {"start": 871.3, "end": 871.68, "word": " إبسمن", "probability": 0.71258544921875}], "temperature": 1.0}, {"id": 37, "seek": 88657, "start": 872.39, "end": 886.57, "text": "الان لو كانت عندي ال epsilon أثبتت انا هذا الكلام ل epsilon ل epsilon لو أثبتت لكل epsilon element in مثلا in zero و نص", "tokens": [6027, 7649, 45164, 25961, 2655, 18871, 16254, 2423, 17889, 5551, 12984, 3555, 2655, 2655, 1975, 8315, 23758, 2423, 28820, 10943, 5296, 17889, 5296, 17889, 45164, 5551, 12984, 3555, 2655, 2655, 5296, 28820, 17889, 4478, 294, 50113, 15040, 294, 4018, 4032, 8717, 9381], "avg_logprob": -0.29851018550784086, "compression_ratio": 1.4615384615384615, "no_speech_prob": 7.152557373046875e-07, "words": [{"start": 872.39, "end": 873.09, "word": "الان", "probability": 0.76318359375}, {"start": 873.09, "end": 874.31, "word": " لو", "probability": 0.6044921875}, 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{"start": 1073.88, "end": 1074.02, "word": " G'", "probability": 0.79736328125}, {"start": 1074.36, "end": 1074.92, "word": " لما", "probability": 0.909423828125}, {"start": 1074.92, "end": 1075.14, "word": " X", "probability": 0.65234375}, {"start": 1075.14, "end": 1075.44, "word": " تروح", "probability": 0.9876302083333334}, {"start": 1075.44, "end": 1075.7, "word": " للـ", "probability": 0.797607421875}, {"start": 1075.7, "end": 1075.94, "word": " A", "probability": 0.85546875}, {"start": 1075.94, "end": 1076.34, "word": " من", "probability": 0.98388671875}, {"start": 1076.34, "end": 1076.86, "word": " اليمين", "probability": 0.7799479166666666}, {"start": 1076.86, "end": 1079.12, "word": " الآن", "probability": 0.758056640625}], "temperature": 1.0}, {"id": 46, "seek": 110727, "start": 1081.11, "end": 1107.27, "text": "بدي أختار .. بدي أسهل على حالي بدل من كل مرة أقول a زي الـ delta و بعد شوية ألاقي delta prime و أخد ال minimum بينهم الاخرى أجدفم عليه عشان اللي بحكيه بدي أخد اللي هو أقول choose c1 element in a و a زي الـ delta الآن اختارلي c1 من ال a لعند مين لعند ال a زي ال delta سميه c1 يعني", "tokens": [3555, 16254, 5551, 46456, 9640, 4386, 4724, 16254, 5551, 3794, 3224, 1211, 15844, 11331, 6027, 1829, 47525, 1211, 9154, 28242, 3714, 25720, 5551, 39648, 257, 30767, 1829, 2423, 39184, 8289, 4032, 39182, 13412, 2407, 10632, 5551, 15040, 38436, 8289, 5835, 4032, 5551, 9778, 3215, 2423, 7285, 49374, 16095, 2423, 47283, 2288, 7578, 5551, 7435, 3215, 5172, 2304, 47356, 6225, 8592, 7649, 13672, 1829, 4724, 5016, 4117, 1829, 3224, 4724, 16254, 5551, 9778, 3215, 13672, 1829, 31439, 5551, 39648, 2826, 269, 16, 4478, 294, 257, 4032, 257, 30767, 1829, 2423, 39184, 8289, 6024, 48506, 1975, 46456, 9640, 20292, 269, 16, 9154, 2423, 257, 5296, 3615, 41260, 3714, 9957, 5296, 3615, 41260, 2423, 257, 30767, 1829, 2423, 8289, 8608, 2304, 1829, 3224, 269, 16, 37495, 22653], "avg_logprob": -0.2472499933242798, "compression_ratio": 1.8008298755186722, "no_speech_prob": 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"probability": 0.818359375}, {"start": 1084.91, "end": 1085.11, "word": " a", "probability": 0.487548828125}, {"start": 1085.11, "end": 1085.27, "word": " زي", "probability": 0.31890869140625}, {"start": 1085.27, "end": 1085.51, "word": " الـ", "probability": 0.760009765625}, {"start": 1085.51, "end": 1085.73, "word": " delta", "probability": 0.40869140625}, {"start": 1085.73, "end": 1086.15, "word": " و", "probability": 0.6611328125}, {"start": 1086.15, "end": 1086.39, "word": " بعد", "probability": 0.8193359375}, {"start": 1086.39, "end": 1086.73, "word": " شوية", "probability": 0.9117838541666666}, {"start": 1086.73, "end": 1087.09, "word": " ألاقي", "probability": 0.7970377604166666}, {"start": 1087.09, "end": 1087.39, "word": " delta", "probability": 0.6279296875}, {"start": 1087.39, "end": 1087.81, "word": " prime", "probability": 0.74853515625}, {"start": 1087.81, "end": 1087.91, "word": " و", "probability": 0.62939453125}, {"start": 1087.91, "end": 1088.13, "word": " أخد", "probability": 0.83984375}, {"start": 1088.13, "end": 1088.25, "word": " ال", "probability": 0.931640625}, {"start": 1088.25, "end": 1088.49, "word": " minimum", "probability": 0.796875}, {"start": 1088.49, "end": 1088.95, "word": " بينهم", "probability": 0.577392578125}, {"start": 1088.95, "end": 1089.59, "word": " الاخرى", "probability": 0.65484619140625}, {"start": 1089.59, "end": 1090.91, "word": " أجدفم", "probability": 0.86455078125}, {"start": 1090.91, "end": 1091.15, "word": " عليه", "probability": 0.62841796875}, {"start": 1091.15, "end": 1091.37, "word": " عشان", "probability": 0.7362467447916666}, {"start": 1091.37, "end": 1091.47, "word": " اللي", "probability": 0.9609375}, {"start": 1091.47, "end": 1092.01, "word": " بحكيه", "probability": 0.98525390625}, {"start": 1092.01, "end": 1092.35, "word": " بدي", "probability": 0.759033203125}, {"start": 1092.35, "end": 1092.79, "word": " أخد", "probability": 0.9267578125}, {"start": 1092.79, "end": 1092.95, "word": " اللي", "probability": 0.97900390625}, {"start": 1092.95, "end": 1093.29, "word": " هو", "probability": 0.9912109375}, {"start": 1093.29, "end": 1094.43, "word": " أقول", "probability": 0.7431640625}, {"start": 1094.43, "end": 1095.13, "word": " choose", "probability": 0.505859375}, {"start": 1095.13, "end": 1095.75, "word": " c1", "probability": 0.729248046875}, {"start": 1095.75, "end": 1096.23, "word": " element", "probability": 0.92919921875}, {"start": 1096.23, "end": 1096.47, "word": " in", "probability": 0.861328125}, {"start": 1096.47, "end": 1096.79, "word": " a", "probability": 0.51025390625}, {"start": 1096.79, "end": 1097.43, "word": " و", "probability": 0.88671875}, {"start": 1097.43, "end": 1097.67, "word": " a", "probability": 0.92333984375}, {"start": 1097.67, "end": 1097.91, "word": " زي", "probability": 0.96923828125}, {"start": 1097.91, "end": 1098.17, "word": " الـ", "probability": 0.874267578125}, {"start": 1098.17, "end": 1098.49, "word": " delta", "probability": 0.7861328125}, {"start": 1098.49, "end": 1099.67, "word": " الآن", "probability": 0.6270751953125}, {"start": 1099.67, "end": 1100.73, "word": " اختارلي", "probability": 0.8856201171875}, {"start": 1100.73, "end": 1102.13, "word": " c1", "probability": 0.941650390625}, {"start": 1102.13, "end": 1102.37, "word": " من", "probability": 0.9814453125}, {"start": 1102.37, "end": 1102.59, "word": " ال", "probability": 0.9423828125}, {"start": 1102.59, "end": 1102.77, "word": " a", "probability": 0.638671875}, {"start": 1102.77, "end": 1103.13, "word": " لعند", "probability": 0.8974609375}, {"start": 1103.13, "end": 1103.61, "word": " مين", "probability": 0.911376953125}, {"start": 1103.61, "end": 1104.49, "word": " لعند", "probability": 0.7928873697916666}, {"start": 1104.49, "end": 1104.63, "word": " ال", "probability": 0.91796875}, {"start": 1104.63, "end": 1104.79, "word": " a", "probability": 0.95556640625}, {"start": 1104.79, "end": 1104.91, "word": " زي", "probability": 0.97314453125}, {"start": 1104.91, "end": 1105.01, "word": " ال", "probability": 0.75048828125}, {"start": 1105.01, "end": 1105.45, "word": " delta", "probability": 0.70166015625}, {"start": 1105.45, "end": 1106.49, "word": " سميه", "probability": 0.917236328125}, {"start": 1106.49, "end": 1106.97, "word": " c1", "probability": 0.9599609375}, {"start": 1106.97, "end": 1107.27, "word": " يعني", "probability": 0.6060791015625}], "temperature": 1.0}, {"id": 47, "seek": 113188, "start": 1108.18, "end": 1131.88, "text": "مجرد اختيار انا اخترته الان choose .. choose C1 element in A و A زائد دلتا يعني C1 وين موجود؟ موجود في المنطقة التي تتحقق فيها F' ع G' نقص أصغر من ميل من إبسل عشان تستخدمها بعد شوية", "tokens": [2304, 7435, 2288, 3215, 1975, 9778, 31371, 9640, 1975, 8315, 1975, 46456, 43500, 3224, 2423, 7649, 2826, 4386, 2826, 383, 16, 4478, 294, 316, 4032, 316, 30767, 16373, 3215, 11778, 1211, 2655, 995, 37495, 22653, 383, 16, 4032, 9957, 3714, 29245, 23328, 22807, 3714, 29245, 23328, 8978, 9673, 1863, 9566, 28671, 38392, 6055, 2655, 5016, 4587, 4587, 8978, 11296, 479, 6, 6225, 460, 6, 8717, 4587, 9381, 5551, 9381, 17082, 2288, 9154, 3714, 26895, 9154, 11933, 3555, 3794, 1211, 6225, 8592, 7649, 6055, 14851, 9778, 40448, 11296, 39182, 13412, 2407, 10632], "avg_logprob": -0.2443953778432763, "compression_ratio": 1.455, "no_speech_prob": 1.1920928955078125e-06, "words": [{"start": 1108.18, "end": 1108.72, "word": "مجرد", "probability": 0.92041015625}, {"start": 1108.72, "end": 1109.26, "word": " اختيار", "probability": 0.8455810546875}, {"start": 1109.26, "end": 1109.6, "word": " انا", "probability": 0.6036376953125}, {"start": 1109.6, "end": 1110.34, "word": " اخترته", "probability": 0.93115234375}, {"start": 1110.34, "end": 1111.34, "word": " الان", "probability": 0.706298828125}, {"start": 1111.34, "end": 1112.1, "word": " choose", "probability": 0.30712890625}, {"start": 1112.1, "end": 1112.26, "word": " ..", "probability": 0.14306640625}, {"start": 1112.26, "end": 1114.08, "word": " choose", "probability": 0.60498046875}, {"start": 1114.08, "end": 1115.7, "word": " C1", "probability": 0.5960693359375}, {"start": 1115.7, "end": 1116.22, "word": " element", "probability": 0.72607421875}, {"start": 1116.22, "end": 1116.5, "word": " in", "probability": 0.8525390625}, {"start": 1116.5, "end": 1116.9, "word": " A", "probability": 0.876953125}, {"start": 1116.9, "end": 1117.8, "word": " و", "probability": 0.73388671875}, {"start": 1117.8, "end": 1117.96, "word": " A", "probability": 0.50830078125}, {"start": 1117.96, "end": 1118.26, "word": " زائد", "probability": 0.5773111979166666}, {"start": 1118.26, "end": 1118.76, "word": " دلتا", "probability": 0.76934814453125}, {"start": 1118.76, "end": 1119.42, "word": " يعني", "probability": 0.86279296875}, {"start": 1119.42, "end": 1120.1, "word": " C1", "probability": 0.905029296875}, {"start": 1120.1, "end": 1120.36, "word": " وين", "probability": 0.6915283203125}, {"start": 1120.36, "end": 1121.34, "word": " موجود؟", "probability": 0.8697509765625}, {"start": 1121.34, "end": 1122.1, "word": " موجود", "probability": 0.97412109375}, {"start": 1122.1, "end": 1122.32, "word": " في", "probability": 0.8935546875}, {"start": 1122.32, "end": 1123.1, "word": " المنطقة", "probability": 0.986083984375}, {"start": 1123.1, "end": 1124.14, "word": " التي", "probability": 0.791015625}, {"start": 1124.14, "end": 1125.12, "word": " تتحقق", "probability": 0.9943359375}, {"start": 1125.12, "end": 1125.52, "word": " فيها", "probability": 0.982421875}, {"start": 1125.52, "end": 1125.82, "word": " F'", "probability": 0.88427734375}, {"start": 1126.24, "end": 1126.42, "word": " ع", "probability": 0.52783203125}, {"start": 1126.42, "end": 1126.66, "word": " G'", "probability": 0.8330078125}, {"start": 1127.0, "end": 1127.4, "word": " نقص", "probability": 0.711669921875}, {"start": 1127.4, "end": 1128.04, "word": " أصغر", "probability": 0.849365234375}, {"start": 1128.04, "end": 1128.24, "word": " من", "probability": 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{"start": 1300.6, "end": 1301.04, "word": " هيك", "probability": 0.6336669921875}, {"start": 1301.04, "end": 1301.76, "word": " بقدر", "probability": 0.9210611979166666}, {"start": 1301.76, "end": 1302.36, "word": " اقول", "probability": 0.81787109375}, {"start": 1302.36, "end": 1303.44, "word": " then", "probability": 0.53369140625}, {"start": 1303.44, "end": 1303.76, "word": " we", "probability": 0.94384765625}, {"start": 1303.76, "end": 1304.12, "word": " can", "probability": 0.92578125}, {"start": 1304.12, "end": 1304.48, "word": " choose", "probability": 0.9111328125}, {"start": 1304.48, "end": 1305.38, "word": " C2", "probability": 0.97216796875}, {"start": 1305.38, "end": 1306.02, "word": " بين", "probability": 0.7412109375}, {"start": 1306.02, "end": 1306.44, "word": " A", "probability": 0.94140625}, {"start": 1306.44, "end": 1306.56, "word": " و", "probability": 0.982421875}, {"start": 1306.56, "end": 1307.34, "word": " C1", "probability": 0.90380859375}], "temperature": 1.0}, {"id": 55, "seek": 133732, "start": 1308.06, "end": 1337.32, "text": "الـ C2 بقدر اختارها فعلا بختارها بس طبعا وين تشتري تشريع اللي اختيارها انها تكون بين الفترة A و A زائد Delta Prime و بدياها تيجي في داخل الـ A و الـ C1 اذا then we can choose C2 element A of C1 بحيث ان ال F of X أكبر من مين من F of C1 لكل X وين", "tokens": [6027, 39184, 383, 17, 4724, 28543, 2288, 1975, 46456, 9640, 11296, 6156, 3615, 15040, 4724, 46456, 9640, 11296, 4724, 3794, 23032, 3555, 3615, 995, 4032, 9957, 6055, 8592, 2655, 16572, 6055, 8592, 16572, 3615, 13672, 1829, 1975, 9778, 31371, 9640, 11296, 16472, 11296, 6055, 30544, 49374, 27188, 2655, 25720, 316, 4032, 316, 30767, 16373, 3215, 18183, 9655, 4032, 4724, 16254, 995, 11296, 6055, 1829, 7435, 1829, 8978, 11778, 47283, 1211, 2423, 39184, 316, 4032, 2423, 39184, 383, 16, 1975, 15730, 550, 321, 393, 2826, 383, 17, 4478, 316, 295, 383, 16, 4724, 5016, 1829, 12984, 16472, 2423, 479, 295, 1783, 5551, 4117, 26890, 9154, 3714, 9957, 9154, 479, 295, 383, 16, 5296, 28820, 1783, 4032, 9957], "avg_logprob": -0.21461003713118723, "compression_ratio": 1.5648535564853556, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1308.06, "end": 1308.34, "word": "الـ", "probability": 0.79296875}, {"start": 1308.34, "end": 1308.72, "word": " C2", "probability": 0.691162109375}, {"start": 1308.72, "end": 1309.14, "word": " بقدر", "probability": 0.9261067708333334}, {"start": 1309.14, "end": 1309.74, "word": " اختارها", "probability": 0.899658203125}, {"start": 1309.74, "end": 1310.28, "word": " فعلا", "probability": 0.8035481770833334}, {"start": 1310.28, "end": 1311.3, "word": " بختارها", "probability": 0.9100341796875}, {"start": 1311.3, "end": 1311.92, "word": " بس", "probability": 0.95361328125}, {"start": 1311.92, "end": 1312.34, "word": " طبعا", "probability": 0.9705810546875}, {"start": 1312.34, "end": 1313.22, "word": " وين", "probability": 0.80322265625}, {"start": 1313.22, "end": 1313.6, "word": " تشتري", "probability": 0.7828369140625}, {"start": 1313.6, "end": 1314.1, "word": " تشريع", "probability": 0.757232666015625}, {"start": 1314.1, "end": 1314.24, "word": " اللي", "probability": 0.6785888671875}, {"start": 1314.24, "end": 1314.96, "word": " اختيارها", "probability": 0.8642578125}, {"start": 1314.96, "end": 1315.64, "word": " انها", "probability": 0.824462890625}, {"start": 1315.64, "end": 1316.02, "word": " تكون", "probability": 0.98486328125}, {"start": 1316.02, "end": 1316.64, "word": " بين", "probability": 0.9697265625}, {"start": 1316.64, "end": 1317.14, "word": " الفترة", "probability": 0.9851888020833334}, {"start": 1317.14, "end": 1317.36, "word": " A", "probability": 0.60693359375}, {"start": 1317.36, "end": 1317.5, "word": " و", "probability": 0.8388671875}, {"start": 1317.5, "end": 1317.84, "word": " A", "probability": 0.66259765625}, {"start": 1317.84, "end": 1318.84, "word": " زائد", "probability": 0.7435709635416666}, {"start": 1318.84, "end": 1319.1, "word": " Delta", 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0.7236328125}, {"start": 1325.06, "end": 1325.96, "word": " then", "probability": 0.380126953125}, {"start": 1325.96, "end": 1326.16, "word": " we", "probability": 0.93310546875}, {"start": 1326.16, "end": 1326.38, "word": " can", "probability": 0.96240234375}, {"start": 1326.38, "end": 1326.9, "word": " choose", "probability": 0.92578125}, {"start": 1326.9, "end": 1327.74, "word": " C2", "probability": 0.9365234375}, {"start": 1327.74, "end": 1328.22, "word": " element", "probability": 0.5458984375}, {"start": 1328.22, "end": 1328.62, "word": " A", "probability": 0.251220703125}, {"start": 1328.62, "end": 1328.9, "word": " of", "probability": 0.69384765625}, {"start": 1328.9, "end": 1329.66, "word": " C1", "probability": 0.9853515625}, {"start": 1329.66, "end": 1330.92, "word": " بحيث", "probability": 0.9599609375}, {"start": 1330.92, "end": 1331.18, "word": " ان", "probability": 0.828125}, {"start": 1331.18, "end": 1331.7, "word": " ال", "probability": 0.431396484375}, {"start": 1331.7, "end": 1331.88, "word": " F", "probability": 0.55224609375}, {"start": 1331.88, "end": 1332.06, "word": " of", "probability": 0.78515625}, {"start": 1332.06, "end": 1332.32, "word": " X", "probability": 0.9892578125}, {"start": 1332.32, "end": 1332.68, "word": " أكبر", "probability": 0.8212890625}, {"start": 1332.68, "end": 1332.9, "word": " من", "probability": 0.9775390625}, {"start": 1332.9, "end": 1333.2, "word": " مين", "probability": 0.7216796875}, {"start": 1333.2, "end": 1333.82, "word": " من", "probability": 0.802734375}, {"start": 1333.82, "end": 1334.22, "word": " F", "probability": 0.9677734375}, {"start": 1334.22, "end": 1334.56, "word": " of", "probability": 0.96240234375}, {"start": 1334.56, "end": 1335.26, "word": " C1", "probability": 0.991455078125}, {"start": 1335.26, "end": 1336.52, "word": " لكل", "probability": 0.94873046875}, {"start": 1336.52, "end": 1336.84, "word": " X", "probability": 0.951171875}, {"start": 1336.84, "end": 1337.32, "word": " وين", "probability": 0.91796875}], "temperature": 1.0}, {"id": 56, "seek": 136637, "start": 1338.49, "end": 1366.37, "text": "في الفترة بين A وC اتنين طيب هذه معناته ان ال F of X هذه لكل ال Xات اللي هان ال F of X أكيد اي شمالها لها تساوي ايش F of C واحد هل جد تعرف ليش بديها هذه عشان لزوم تعريف شغل معينة بنفعش تكون ال F of X ايش بتساوي F of C واحد الآن similarly", "tokens": [41185, 27188, 2655, 25720, 49374, 316, 4032, 34, 1975, 2655, 1863, 9957, 23032, 1829, 3555, 29538, 20449, 8315, 47395, 16472, 2423, 479, 295, 1783, 29538, 5296, 28820, 2423, 1783, 9307, 13672, 1829, 8032, 7649, 2423, 479, 295, 1783, 5551, 4117, 25708, 1975, 1829, 13412, 2304, 6027, 11296, 5296, 11296, 6055, 3794, 995, 45865, 1975, 1829, 8592, 479, 295, 383, 36764, 24401, 8032, 1211, 10874, 3215, 37279, 28480, 32239, 8592, 4724, 16254, 11296, 29538, 6225, 8592, 7649, 5296, 11622, 20498, 37279, 16572, 5172, 13412, 17082, 1211, 20449, 9957, 3660, 4724, 1863, 5172, 3615, 8592, 6055, 30544, 2423, 479, 295, 1783, 1975, 1829, 8592, 39894, 3794, 995, 45865, 479, 295, 383, 36764, 24401, 6024, 48506, 14138], "avg_logprob": -0.25951088407765266, "compression_ratio": 1.7777777777777777, "no_speech_prob": 4.947185516357422e-06, "words": [{"start": 1338.49, "end": 1338.67, "word": "في", "probability": 0.6259765625}, {"start": 1338.67, "end": 1339.27, "word": " الفترة", "probability": 0.9783528645833334}, {"start": 1339.27, "end": 1339.67, "word": " بين", "probability": 0.86962890625}, {"start": 1339.67, "end": 1340.09, "word": " A", "probability": 0.31591796875}, {"start": 1340.09, "end": 1340.85, "word": " وC", "probability": 0.78369140625}, {"start": 1340.85, "end": 1341.43, "word": " اتنين", "probability": 0.685882568359375}, {"start": 1341.43, "end": 1342.81, "word": " طيب", "probability": 0.8268229166666666}, {"start": 1342.81, "end": 1343.43, "word": " هذه", "probability": 0.387939453125}, {"start": 1343.43, "end": 1343.93, "word": " معناته", "probability": 0.9361979166666666}, {"start": 1343.93, "end": 1344.09, "word": " ان", "probability": 0.6806640625}, {"start": 1344.09, "end": 1344.23, "word": " ال", "probability": 0.759765625}, {"start": 1344.23, "end": 1344.39, "word": " F", "probability": 0.6142578125}, {"start": 1344.39, "end": 1344.59, "word": " of", "probability": 0.63671875}, {"start": 1344.59, "end": 1344.83, "word": " X", "probability": 0.9736328125}, {"start": 1344.83, "end": 1345.27, "word": " هذه", "probability": 0.79541015625}, {"start": 1345.27, "end": 1346.85, "word": " لكل", "probability": 0.9482421875}, {"start": 1346.85, "end": 1346.97, "word": " ال", "probability": 0.67333984375}, {"start": 1346.97, "end": 1347.29, "word": " Xات", "probability": 0.49884033203125}, {"start": 1347.29, "end": 1347.51, "word": " اللي", "probability": 0.940673828125}, {"start": 1347.51, "end": 1347.87, "word": " هان", "probability": 0.593994140625}, {"start": 1347.87, "end": 1348.05, "word": " ال", "probability": 0.65087890625}, {"start": 1348.05, "end": 1348.19, "word": " F", 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0.839599609375}, {"start": 1353.63, "end": 1354.29, "word": " هل", "probability": 0.521728515625}, {"start": 1354.29, "end": 1354.43, "word": " جد", "probability": 0.827880859375}, {"start": 1354.43, "end": 1354.71, "word": " تعرف", "probability": 0.835205078125}, {"start": 1354.71, "end": 1354.89, "word": " ليش", "probability": 0.85205078125}, {"start": 1354.89, "end": 1355.31, "word": " بديها", "probability": 0.5580240885416666}, {"start": 1355.31, "end": 1355.65, "word": " هذه", "probability": 0.69384765625}, {"start": 1355.65, "end": 1356.77, "word": " عشان", "probability": 0.8772786458333334}, {"start": 1356.77, "end": 1357.35, "word": " لزوم", "probability": 0.8546549479166666}, {"start": 1357.35, "end": 1357.87, "word": " تعريف", "probability": 0.9920247395833334}, {"start": 1357.87, "end": 1358.17, "word": " شغل", "probability": 0.8546549479166666}, {"start": 1358.17, "end": 1358.93, "word": " معينة", "probability": 0.9744466145833334}, {"start": 1358.93, "end": 1359.77, 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"probability": 0.7197265625}, {"start": 1364.77, "end": 1366.37, "word": " similarly", "probability": 0.47119140625}], "temperature": 1.0}, {"id": 57, "seek": 139171, "start": 1367.43, "end": 1391.71, "text": "Similarly اللي هو بما انه limit G of X بساوي مالة نهاية as X بتروح للـ A من اليمين إذا بقدر ألاقي اللي هو C1 برايم بحيث انه G of X أكبر", "tokens": [39392, 2202, 356, 13672, 1829, 31439, 4724, 15042, 16472, 3224, 4948, 460, 295, 1783, 4724, 3794, 995, 45865, 3714, 6027, 3660, 8717, 11296, 10632, 382, 1783, 39894, 32887, 5016, 24976, 39184, 316, 9154, 2423, 32640, 9957, 11933, 15730, 4724, 28543, 2288, 5551, 15040, 38436, 13672, 1829, 31439, 383, 16, 4724, 23557, 32640, 4724, 5016, 1829, 12984, 16472, 3224, 460, 295, 1783, 5551, 4117, 26890], "avg_logprob": -0.25096153846153846, "compression_ratio": 1.37012987012987, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 1367.43, "end": 1368.71, "word": "Similarly", "probability": 0.8478190104166666}, {"start": 1368.71, 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0.89013671875}, {"start": 3067.56, "end": 3069.1, "word": " كيف؟", "probability": 0.9069010416666666}], "temperature": 1.0}, {"id": 126, "seek": 308774, "start": 3070.86, "end": 3087.74, "text": "الـ Main Value Theorem عبارة الـ Taylor's Theorem عبارة عن تعميم للـ Main Value Theorem خد الآن بتساوي سفر بتطلعلك الـ Main Value Theorem بيشحمها و لحمها و إلى لقاء", "tokens": [6027, 39184, 12383, 39352, 440, 37956, 6225, 3555, 9640, 3660, 2423, 39184, 12060, 311, 440, 37956, 6225, 3555, 9640, 3660, 18871, 37279, 2304, 32640, 24976, 39184, 12383, 39352, 440, 37956, 16490, 3215, 6024, 48506, 39894, 3794, 995, 45865, 8608, 5172, 2288, 39894, 9566, 1211, 3615, 23275, 2423, 39184, 12383, 39352, 440, 37956, 4724, 1829, 8592, 35571, 11296, 4032, 5296, 35571, 11296, 4032, 30731, 5296, 4587, 16606], "avg_logprob": -0.23274253553418972, "compression_ratio": 1.5838926174496644, "no_speech_prob": 0.04443359375, "words": [{"start": 3070.86, "end": 3071.52, "word": "الـ", "probability": 0.58349609375}, {"start": 3071.52, "end": 3071.78, "word": " Main", "probability": 0.254638671875}, {"start": 3071.78, "end": 3072.34, "word": " Value", "probability": 0.8388671875}, {"start": 3072.34, "end": 3072.9, "word": " Theorem", "probability": 0.85791015625}, {"start": 3072.9, "end": 3074.0, "word": " عبارة", "probability": 0.9078369140625}, {"start": 3074.0, "end": 3074.26, "word": " الـ", "probability": 0.6983642578125}, {"start": 3074.26, "end": 3074.78, "word": " Taylor's", "probability": 0.740478515625}, {"start": 3074.78, "end": 3075.1, "word": " Theorem", "probability": 0.6435546875}, {"start": 3075.1, "end": 3075.48, "word": " عبارة", "probability": 0.985595703125}, {"start": 3075.48, "end": 3075.66, "word": " عن", "probability": 0.9921875}, {"start": 3075.66, "end": 3076.4, "word": " تعميم", "probability": 0.94287109375}, {"start": 3076.4, "end": 3077.52, "word": " للـ", "probability": 0.8232421875}, {"start": 3077.52, "end": 3077.78, "word": " Main", "probability": 0.8642578125}, {"start": 3077.78, "end": 3078.28, "word": " Value", "probability": 0.99072265625}, {"start": 3078.28, "end": 3078.74, "word": " Theorem", "probability": 0.904296875}, {"start": 3078.74, "end": 3079.6, "word": " خد", "probability": 0.7626953125}, {"start": 3079.6, "end": 3079.9, "word": " الآن", "probability": 0.63720703125}, {"start": 3079.9, "end": 3080.38, "word": " بتساوي", "probability": 0.8271484375}, {"start": 3080.38, "end": 3080.84, "word": " سفر", "probability": 0.75830078125}, {"start": 3080.84, "end": 3082.24, "word": " بتطلعلك", "probability": 0.7968017578125}, {"start": 3082.24, "end": 3082.5, "word": " الـ", "probability": 0.84326171875}, {"start": 3082.5, "end": 3082.78, "word": " Main", "probability": 0.869140625}, {"start": 3082.78, "end": 3083.32, "word": " Value", "probability": 0.9892578125}, {"start": 3083.32, "end": 3083.96, "word": " Theorem", "probability": 0.89990234375}, {"start": 3083.96, "end": 3085.02, "word": " بيشحمها", "probability": 0.90673828125}, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..24e720eeb67567513db3fd873f3c88af50e167f0
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/GZcIoIj0HL8_raw.srt
@@ -0,0 +1,2256 @@
+1
+00:00:05,040 --> 00:00:12,800
+بسم الله الرحمن الرحيم المحاضرة السابعة مساق تحليل
+
+2
+00:00:12,800 --> 00:00:18,640
+حقيقي 2 اللي طلبت كلية العلوم بالجامعة الإسلامية
+
+3
+00:00:18,640 --> 00:00:23,800
+قسم الرياضيات اليوم
+
+4
+00:00:23,800 --> 00:00:29,720
+هيكون عندنا استكمال ل .. اللي هي قواعد أو قاعدة
+
+5
+00:00:29,720 --> 00:00:36,330
+Lobitalاتحدثنا المرة الماضية عن الحالة اللى بتحدث
+
+6
+00:00:36,330 --> 00:00:41,170
+سفر على سفر وكيف انعلجها وتم اللى هو برهان
+
+7
+00:00:41,170 --> 00:00:47,910
+النظريات المعنية في ذلكالآن بدنا نتحدث على الحالة
+
+8
+00:00:47,910 --> 00:00:57,110
+اللي بتحدث عندنا أيضًا 0 على 0 ولكن ال limit ال X
+
+9
+00:00:57,110 --> 00:01:00,830
+بتروح إلى مالة نهاية أو سالب مالة نهاية وقلنا عنها
+
+10
+00:01:00,830 --> 00:01:05,790
+صحيحة برضه وبعد ذلك هنتحدث عن الحالة infinity على
+
+11
+00:01:05,790 --> 00:01:10,510
+infinity نيجي الآن لنظريتنا اللي وصلنا عندها
+
+12
+00:01:10,510 --> 00:01:11,990
+suppose that
+
+13
+00:01:16,220 --> 00:01:20,260
+F and D are continuous and differentiable على
+
+14
+00:01:20,260 --> 00:01:24,600
+الفترة من A إلى ما لا نهاية ونفترض أنه limit لل F
+
+15
+00:01:24,600 --> 00:01:27,540
+of X لما X تروح إلى ما لا نهاية هي نفس limit D of
+
+16
+00:01:27,540 --> 00:01:31,200
+X لما X تروح إلى ما لا نهاية تساوي إيهاش صفر يعني
+
+17
+00:01:31,200 --> 00:01:36,080
+هذه هتخلقلي حالة اللي هو صفر على صفر ومن ثم نشوف
+
+18
+00:01:36,080 --> 00:01:40,230
+كيف انعليجهاالعنوان D of X ذاته ساوة سفر نفترض وD
+
+19
+00:01:40,230 --> 00:01:44,830
+prime of X ذاته ساوة سفر لكل X أكبر من مين من اللي
+
+20
+00:01:44,830 --> 00:01:48,290
+هو المقصود هنا بيه اللي هي أيش A يعني هذه اللي هي
+
+21
+00:01:48,290 --> 00:01:52,970
+مين اللي هي المقصود أنه A، يا ما نخلي هذه A، يا ما
+
+22
+00:01:52,970 --> 00:01:58,330
+هذه Athen limit f of x على g of x لما x تروح لما
+
+23
+00:01:58,330 --> 00:02:01,930
+النهاية هتكون بالظبط ايش بتساوي limit f prime على
+
+24
+00:02:01,930 --> 00:02:06,050
+g prime of x as x goes to infinity provided that
+
+25
+00:02:06,050 --> 00:02:09,590
+هذا ال limit بتساوي قيمة معينة أو تساوي infinity
+
+26
+00:02:09,590 --> 00:02:15,730
+او سالم infinity الان احنا في الواقع اللي هنسوي
+
+27
+00:02:16,910 --> 00:02:21,910
+وكأنه يعني عملية تحويل اللي نخلّي بدل X تروح إلى
+
+28
+00:02:21,910 --> 00:02:26,950
+ملأ نهاية نخلّي اللي هو متغير وروح لسفر يعني بدنا
+
+29
+00:02:26,950 --> 00:02:31,490
+نعيد تعريف الدالة بحيث أنه يروح هذا المتغير يعني
+
+30
+00:02:31,490 --> 00:02:34,610
+وكأنه نقول واحد على T ويروح على .. اه واحد على X
+
+31
+00:02:34,610 --> 00:02:37,950
+يروح لسفر لما X تروح إلى ملأ نهاية ومن ثم نطبق
+
+32
+00:02:37,950 --> 00:02:41,030
+النظريات اللي أخدناها المرة الماضية دعونا نشوف إيش
+
+33
+00:02:41,030 --> 00:02:47,050
+اللي بنحكيه الآن عند ..الـ الـ function F و G F و
+
+34
+00:02:47,050 --> 00:02:54,430
+G معرفات من A و ماله نهاية لعند اللي هو R الآن
+
+35
+00:02:54,430 --> 00:02:57,970
+عندي اللي هو بده أعرف اللي هو عشان أقدر أستخدم
+
+36
+00:02:57,970 --> 00:03:01,610
+نظرية المرة اللي فاتت المرة الماضية بده أعرف
+
+37
+00:03:01,610 --> 00:03:08,030
+دالتين F capital و G capital من اللي هو عندي من
+
+38
+00:03:08,030 --> 00:03:15,560
+Zero لعند اللي هو واحد على Aلعند الهو asr واضحة
+
+39
+00:03:15,560 --> 00:03:28,640
+الصورة تقريبا اكيد هنقول define f o g as f of اللي
+
+40
+00:03:28,640 --> 00:03:36,960
+نسميها اللي هو T بتساوي اللي هو Gأو F of 1 على T
+
+41
+00:03:36,960 --> 00:03:42,460
+لما T اللي هي لا تساوي سفر و طبعا T أنا معرفها
+
+42
+00:03:42,460 --> 00:03:47,140
+اللي هي في الفترة من Zero لعند واحد على A و ده
+
+43
+00:03:47,140 --> 00:03:51,560
+بيدعرف عند Zero بقول Zero إذا كان T إيش بدها تساوي
+
+44
+00:03:51,560 --> 00:03:59,760
+بتساوي سفر الآن ال .. ال G of Tبتساوي نفس الأسلوب
+
+45
+00:03:59,760 --> 00:04:06,240
+G of 1 على T لما T لا تساوي 0 و 0 إذا كانت T
+
+46
+00:04:06,240 --> 00:04:13,350
+أشمالها بتساوي 0الآن واضح العلاقة بين الدلتين اللي
+
+47
+00:04:13,350 --> 00:04:19,510
+هو F و F اللي هو على أساس أني أحول ال limit
+
+48
+00:04:19,510 --> 00:04:25,650
+المعطاع عندي limit F of X لما X تروح إلى ملا نهاية
+
+49
+00:04:25,650 --> 00:04:30,450
+تصير عندي ال limit بالنسبة لهذه اللي هي ال 1 على X
+
+50
+00:04:30,450 --> 00:04:35,320
+تبعتنا هتروح لمين في هذا الحالة للزفروبستخدم ومن
+
+51
+00:04:35,320 --> 00:04:40,600
+ثم بنكون حصلنا على المتغير لإن أنا T بروح للصفر as
+
+52
+00:04:40,600 --> 00:04:44,740
+X بتروح إلينا لنهاية ومن ثم بنستخدم النظريات اللي
+
+53
+00:04:44,740 --> 00:04:49,940
+أثبتنا صحتها في حالة T بتروح إلى A من اليمين لإن
+
+54
+00:04:49,940 --> 00:04:53,060
+احنا هنروح ل Zero من اليمين هذه اللي هي الخطة اللي
+
+55
+00:04:53,060 --> 00:04:58,480
+هنمشي عليها بنشوف أول إشي واضح أن ال F و ال G ال F
+
+56
+00:04:59,090 --> 00:05:03,930
+والـ G are differentiable على مين على الفترة من
+
+57
+00:05:03,930 --> 00:05:10,550
+سفر لعند واحد عليه هاي واحد اتنين ومدام
+
+58
+00:05:10,550 --> 00:05:13,810
+differentiable ده ن continuous عليها ومش هيك كمان
+
+59
+00:05:13,810 --> 00:05:19,230
+لو جيت أخدت انت هذه بتحسبها لحالك limit F of T as
+
+60
+00:05:19,230 --> 00:05:27,640
+T بتروح للسفر اللي هي بتساوي limitf of 1 على t as
+
+61
+00:05:27,640 --> 00:05:32,340
+t بتروح للصفر و هتلاقيها إيش هتساوي، هتساوي صفر
+
+62
+00:05:32,340 --> 00:05:36,760
+فصارت اللي هي ال F ال capital is continuous على
+
+63
+00:05:36,760 --> 00:05:43,490
+الفترة من 0 لعند 1 على a similarly limit g of tلما
+
+64
+00:05:43,490 --> 00:05:47,650
+T تروح للـ Zero طبعا كل الـ Zero من وين طبيعي من
+
+65
+00:05:47,650 --> 00:05:50,950
+اليمين لأنه أحنا عالمنا اللي بنشتغل عليه اللي هي
+
+66
+00:05:50,950 --> 00:05:53,730
+من عند Zero لعند واحد على أي من المنطقة المودة بقى
+
+67
+00:05:53,730 --> 00:05:58,750
+إذن هروح للـ Zero من جهات اليمين هيساوي نفس الأشي
+
+68
+00:05:58,750 --> 00:06:03,290
+limit F of واحد على T لما T تروح للـ Zero من
+
+69
+00:06:03,290 --> 00:06:07,590
+اليمين ويساوي اللي جيها برضه أشمالها بتساوي صفر
+
+70
+00:06:08,590 --> 00:06:12,970
+الان صارت الان الاف و الجي اشمالين are continuous
+
+71
+00:06:12,970 --> 00:06:17,390
+على الفترة اللي عندنا و differentiable على الفترة
+
+72
+00:06:17,390 --> 00:06:20,510
+ال open و .. و .. و .. و .. و .. و .. و .. و .. و
+
+73
+00:06:20,510 --> 00:06:20,590
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+74
+00:06:20,590 --> 00:06:20,790
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+75
+00:06:20,790 --> 00:06:21,070
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+76
+00:06:21,070 --> 00:06:21,290
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+77
+00:06:21,290 --> 00:06:23,330
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+78
+00:06:23,330 --> 00:06:23,350
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+79
+00:06:23,350 --> 00:06:23,650
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+80
+00:06:23,650 --> 00:06:23,650
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+81
+00:06:23,650 --> 00:06:29,210
+.. و .. و .. و .. و .. و ..
+
+82
+00:06:37,480 --> 00:06:43,680
+بنجي بنحصل على ما يلي خليني بس اسمحولي أمسح هذه
+
+83
+00:06:43,680 --> 00:06:49,760
+دلت في الذاكرة عندي بدي أحسب أنا غاية أن أوجد مين
+
+84
+00:06:49,760 --> 00:06:53,380
+يا جماعة limit اللي هو f of x على g of x لما x
+
+85
+00:06:53,380 --> 00:07:00,780
+تروح إلى وين إلى ما لنهاية لأن limit f of x على g
+
+86
+00:07:00,780 --> 00:07:07,720
+of x لما x تروح إلى ما لنهاية بتساويعبارة عن الان
+
+87
+00:07:07,720 --> 00:07:12,180
+ال X بتروح إلى ما لنهاية F and وال F الواحد على X
+
+88
+00:07:12,180 --> 00:07:20,660
+وين بتروح اللي هو السفر بتساوي limit F of واحد على
+
+89
+00:07:20,660 --> 00:07:26,040
+T خلينا نسميها على G of واحد على T لما T تروح إلى
+
+90
+00:07:26,040 --> 00:07:32,420
+مين إلى السفر من اليمينأو إذا كان عندك اتصار
+
+91
+00:07:32,420 --> 00:07:38,640
+confusion إشي والتكن X مافي مشكلة X لما X تروح ل 0
+
+92
+00:07:38,640 --> 00:07:43,660
+من اليمين اللي هي بتسير عند أسف الواحد على X بتروح
+
+93
+00:07:43,660 --> 00:07:47,860
+ل 0 من اليمين if and only if ال X بتروح لمين إلى
+
+94
+00:07:47,860 --> 00:07:51,520
+ما لنهاية ومن ثم بفخطوة تانية بترجحها للي كتبته
+
+95
+00:07:51,520 --> 00:07:57,980
+قبل بشوية limit اللي هو عبارة عن F of Tعلى اللي هو
+
+96
+00:07:57,980 --> 00:08:04,420
+G of T لما T تروح لـ 0 من وين؟ من اليمين باشي
+
+97
+00:08:04,420 --> 00:08:08,120
+الحال أنا الأن بدي أفضل في الواقع بالنسبة لمن؟
+
+98
+00:08:08,120 --> 00:08:14,040
+بالنسبة له اللي هو الـ X عندي اللي هو وديت واحدة
+
+99
+00:08:14,040 --> 00:08:19,880
+الـ X من اليمين وصارت عندي اللي هي مكان لأ لأ يا
+
+100
+00:08:19,880 --> 00:08:26,200
+جماعة هاد نزام هي بتضلها آسف أه؟X X وهنا بتصير
+
+101
+00:08:26,200 --> 00:08:30,900
+أننا مين واحد على T وهنا واحد على T واضح ليش
+
+102
+00:08:30,900 --> 00:08:36,540
+استبدلنا هنا احنا الواحد على X كله ب T فصارت هذه
+
+103
+00:08:36,540 --> 00:08:41,320
+اللي هي واحد على T وهذه واحد على T الآن صارت
+
+104
+00:08:41,320 --> 00:08:44,780
+النظرية السابقة مطبقة لأن ال T بتروح لمين ل Zero
+
+105
+00:08:44,780 --> 00:08:49,780
+من لمين لأن هدف الواقع مين هي اللي قلنا عنها أشي
+
+106
+00:08:49,780 --> 00:08:57,850
+هذه بتساوي limit ofof T على G of T لما T تروح إلى
+
+107
+00:08:57,850 --> 00:09:01,310
+Zero من وين؟ من اليمين وهذا متطبق لأن هذا Zero
+
+108
+00:09:01,310 --> 00:09:05,490
+وهذا Zero زي ما عملنا قبل بشوية إذا صار إيش بيساوي
+
+109
+00:09:05,490 --> 00:09:10,990
+تفاضل الأول على تفاضل التاني في 100 بالنسبة لعن
+
+110
+00:09:10,990 --> 00:09:17,220
+نظرية السابقة إذا بيصير limitالتي هي التفاضل هادي
+
+111
+00:09:17,220 --> 00:09:21,060
+من هنا F prime of T أيش بيساوي يا شباب؟ تفاضل
+
+112
+00:09:21,060 --> 00:09:24,960
+هادي، عن T بتروح للـ0، إذن هي عبارة عن ناقص 1 على
+
+113
+00:09:24,960 --> 00:09:34,000
+T تربيع في F prime of 1 على T تفاضل F of T، F of T
+
+114
+00:09:34,000 --> 00:09:38,930
+هيهايعني T لا تساوي سفر تفاضلها F prime بساوي ناقص
+
+115
+00:09:38,930 --> 00:09:42,710
+اللي هو F prime الواحد على T فتفاضل اللي جوا اللي
+
+116
+00:09:42,710 --> 00:09:47,450
+هو عبارة عن إيش ناقص واحد على T تربيع واضح على
+
+117
+00:09:47,450 --> 00:09:53,430
+similarly ناقص واحد على T تربيع في G prime of واحد
+
+118
+00:09:53,430 --> 00:10:00,200
+على T لما الـT تروح لمين إلى الـ0 من اليمينالنقص 1
+
+119
+00:10:00,200 --> 00:10:03,160
+دلت ال T تربيع ال نقص 1 دلت T تربيع بروحن مع بعض
+
+120
+00:10:03,160 --> 00:10:09,780
+نرجع لأصلنا بيصير limit F prime بده استبدل الآن ال
+
+121
+00:10:09,780 --> 00:10:17,800
+1 على T بال X على G prime برضه نفس الشي على X و ال
+
+122
+00:10:17,800 --> 00:10:22,540
+T تستبدل بمين اللي هو عبارة عن 1 على X بده تروح لل
+
+123
+00:10:22,540 --> 00:10:29,270
+0 من اليمين يعني ال X بده تروح لمين إلى مالكبصير
+
+124
+00:10:29,270 --> 00:10:35,510
+عندى اللى هو limit f of x على g of x بساوى limit f
+
+125
+00:10:35,510 --> 00:10:40,350
+prime of x على g prime y على x وهو المطلوب نيجى
+
+126
+00:10:40,350 --> 00:10:44,690
+للنظرية البعدها النظرية اللى بدها شغل خلينا نشوف
+
+127
+00:10:44,690 --> 00:10:53,110
+النظرية اطلع لفوق ونركز نشوف ايش هو النظرية بتحكي
+
+128
+00:10:53,110 --> 00:10:57,270
+ومن ثم نذهب الى برهان النظرية
+
+129
+00:11:04,010 --> 00:11:11,750
+عشان مساحة اللوحة شوية خلّينا نتبع على التلخيص ومن
+
+130
+00:11:11,750 --> 00:11:19,110
+ثم بنبرهن theorem 636 اللي هي باختصار هي حالة مالة
+
+131
+00:11:19,110 --> 00:11:22,570
+نهاية على مالة نهاية نفترض أن الـF و الـG are
+
+132
+00:11:22,570 --> 00:11:26,470
+differentiable على الفترة من A و B يعني إحنا شغلنا
+
+133
+00:11:26,470 --> 00:11:34,230
+الآن على الفترة أي إن كانت هذه الفترة من عند Aلعند
+
+134
+00:11:34,230 --> 00:11:41,250
+مي ان لعند بي لأننا مفترضين أن ال F و ال G اللي هو
+
+135
+00:11:41,250 --> 00:11:46,250
+قابل للاشتقاق على الفترة A و B و limit F of X لما
+
+136
+00:11:46,250 --> 00:11:48,890
+X تروح إلى مالة نهائية .. إلى Zero من اليمين ..
+
+137
+00:11:48,890 --> 00:11:53,230
+إلى A من اليمين إيش بيساوي Infinity و limit G of X
+
+138
+00:11:53,230 --> 00:11:56,090
+لما X تروح إلى ال A من اليمين برضه إيش مالها
+
+139
+00:11:56,090 --> 00:12:01,370
+Infinity يعني بمعنى آخر صار لما عرضت علينا limit F
+
+140
+00:12:01,370 --> 00:12:06,980
+of Xعلى G of X لما X راحت إلى الـ A من اليمين
+
+141
+00:12:06,980 --> 00:12:10,660
+وجدنا limit اللي فوق ملا نهاية و limit اللي تحت
+
+142
+00:12:10,660 --> 00:12:14,260
+ملا نهاية يعني حصلنا على الكمية الغير معينة
+
+143
+00:12:14,260 --> 00:12:18,220
+Infinity على Infinity إذا هذه النظرية ستعالج هذه
+
+144
+00:12:18,220 --> 00:12:23,060
+الحالة طبعا الان اللي هي بطريقة مشابهة للسابقة
+
+145
+00:12:23,060 --> 00:12:29,080
+والان احنا طبعا هنقوم ببرهان هذه النظرية نكمل النص
+
+146
+00:12:29,820 --> 00:12:35,900
+الآن فرضنا أيضًا أنه اللي هو الـ g of x لا تساوي
+
+147
+00:12:35,900 --> 00:12:40,160
+سفر والـ g prime of x لا تساوي سفر لكل x و إما لها
+
+148
+00:12:40,160 --> 00:12:44,920
+في الفترة a و b يعني مفترض أن الـ g of x لا تساوي
+
+149
+00:12:44,920 --> 00:12:52,860
+سفر على كل الفترة a و bنأتي إلى الجزئية الأولى من
+
+150
+00:12:52,860 --> 00:12:57,380
+النظرية if limit f' على g' بساوي L element in R
+
+151
+00:12:57,380 --> 00:13:02,040
+إذا فlimit f على g برضه أيضا أيش هتساوي L اللي هي
+
+152
+00:13:02,040 --> 00:13:05,280
+نظريات اللي قبل بشوية بنفس .. بنفس .. خلّيني أقول
+
+153
+00:13:05,280 --> 00:13:10,160
+بنفس المنطق ونفس النصوص بس مبدل 0 على 0 عمالة
+
+154
+00:13:10,160 --> 00:13:13,560
+نهايةفلان limit f prime على g prime صارت مالة
+
+155
+00:13:13,560 --> 00:13:17,700
+نهاية أو سالب مالة نهاية، برضه النظرية بتظبط، سير
+
+156
+00:13:17,700 --> 00:13:23,310
+limit اللي عندنا هذاهو بالظبط limit الـ F على G as
+
+157
+00:13:23,310 --> 00:13:26,170
+X وتروحنا الـ M من اليمين بساوي Infinity وسالب
+
+158
+00:13:26,170 --> 00:13:30,550
+Infinity يعني في النهاية اللي ببحث عنه هذا هو في
+
+159
+00:13:30,550 --> 00:13:34,650
+الحالة هذه أو في الحالة اللي فوق بساوي limit مين
+
+160
+00:13:34,650 --> 00:13:37,750
+الـ F prime على G prime اللي هي الطريقة المحودة
+
+161
+00:13:37,750 --> 00:13:40,330
+بنفضل اللي فوق و بنفضل اللي تحت و بنوجد limit هين
+
+162
+00:13:40,330 --> 00:13:44,670
+بيكونين هين limit اللي هي الأصل اللي احنا بدنايا
+
+163
+00:13:45,410 --> 00:13:49,590
+الان نيجي اللي هو in Berlin اللي هو a و b طبعا
+
+164
+00:13:49,590 --> 00:13:54,410
+بنفس الأسلوب الان
+
+165
+00:13:54,410 --> 00:14:02,790
+قبل ما نبدأ احنا لو اجينا و قولنا limit f of x لما
+
+166
+00:14:02,790 --> 00:14:06,610
+x تروح لأي اشي مثلا a من اليمين بساوة L
+
+167
+00:14:11,700 --> 00:14:14,480
+وقلنا اللي هو for every epsilon تعريفها for every
+
+168
+00:14:14,480 --> 00:14:16,660
+epsilon أكبر من سفر there exists a delta أكبر من
+
+169
+00:14:16,660 --> 00:14:21,300
+سفر such that if x element in A طبعا من اليمين
+
+170
+00:14:21,300 --> 00:14:27,780
+معناته A و A زائد دلتا then اللي هو هيكون عندي f
+
+171
+00:14:27,780 --> 00:14:35,230
+of x ناقص الأصغر من مين من إبسمنالان لو كانت عندي
+
+172
+00:14:35,230 --> 00:14:39,330
+ال epsilon أثبتت انا هذا الكلام ل epsilon ل
+
+173
+00:14:39,330 --> 00:14:45,410
+epsilon لو أثبتت لكل epsilon element in مثلا in
+
+174
+00:14:45,410 --> 00:14:51,970
+zero و نصأو نزيره ربع أو نزيره تلت لو أثبتت أنه
+
+175
+00:14:51,970 --> 00:14:54,950
+لكل إبسلون لجهة دلتة بحيث أن هذا الكلام أصغر من
+
+176
+00:14:54,950 --> 00:15:01,650
+مين من إبسلون برضه بيكون يجزئ عن ال limit ليش؟ لأن
+
+177
+00:15:01,650 --> 00:15:04,950
+المقدار اللي بيكون أصغر من إبسلون اللي هي أصغر من
+
+178
+00:15:04,950 --> 00:15:09,830
+النص أكيد بنفع لمين الدلتة بنفع للمقدار اللي ال
+
+179
+00:15:09,830 --> 00:15:15,450
+إبسلون أكبر من مين من النص لأنه أصلا اللي هي مشكلة
+
+180
+00:15:15,450 --> 00:15:19,620
+أو خلينا نقول ال limit في حد ذاتهاإنه لما الـ A
+
+181
+00:15:19,620 --> 00:15:24,880
+تروحالـ X تروح إلى الـ A من اليمين، بدنا نجمع أن
+
+182
+00:15:24,880 --> 00:15:27,720
+الـ F of X تقترب من مين من الأعلى، يعني التعجيز و
+
+183
+00:15:27,720 --> 00:15:31,100
+كأنه بعجزني أنه يقول لي إنك اتلاقى الـ Delta في
+
+184
+00:15:31,100 --> 00:15:34,600
+حالة الـ Epsilon أكاش اللي بيكون صغيرة فالـ Close
+
+185
+00:15:34,600 --> 00:15:38,960
+اللي هو تمين to Zero، لإنه أصلا المفهوم اللي
+
+186
+00:15:38,960 --> 00:15:43,160
+بتقيله هذا المسافة تضيق، تضيق، تضيق، بحيث أن F of
+
+187
+00:15:43,160 --> 00:15:46,760
+X تقول إلى الـ مين اللي بتعبر عنها؟ الـ Epsilon،
+
+188
+00:15:46,760 --> 00:15:51,190
+لأن بعطيك Epsilon صغيرة جدا جدابدك تلاقي لي Delta
+
+189
+00:15:51,190 --> 00:15:54,870
+أنا الآن الـ Epsilon بين Zero و نص لقيتلك الـ
+
+190
+00:15:54,870 --> 00:15:59,070
+Delta اللي طلعت هنا معناته اللي نفع للـ Epsilon
+
+191
+00:15:59,070 --> 00:16:03,510
+الصغيرة أكيد بيكون أصغر من مين من الأكبيرة إذا هذا
+
+192
+00:16:03,510 --> 00:16:09,250
+يجزء بس هذا قبل ما نبدأ لأنه هسأخدم شغلة اللي هو
+
+193
+00:16:09,250 --> 00:16:14,930
+في هذا الاتجاه نيجي الآن عندي أول إشي المعطى
+
+194
+00:16:14,930 --> 00:16:23,550
+ماتيني LimitF prime of X على G prime of X as X
+
+195
+00:16:23,550 --> 00:16:29,190
+بتروح للـ A من اليمين أيش بساوي؟ بساوي أقل تعريف
+
+196
+00:16:29,190 --> 00:16:32,850
+الـ Limit for every Y أكبر من 0 اللي هتاخد for
+
+197
+00:16:32,850 --> 00:16:39,190
+every Y element in 0 ونص اللي هي لزوم حسابات، بعد
+
+198
+00:16:39,190 --> 00:16:44,650
+شوية هنشوفهاالآن لكل y تنتمي إلى 0.5 وهذا مشروع
+
+199
+00:16:44,650 --> 00:16:49,270
+حسب ما حكيت قبل بشوية there exist delta أكبر من 0
+
+200
+00:16:49,270 --> 00:16:57,650
+such that اللي هو if x element in a و a زيادة دلتا
+
+201
+00:16:57,650 --> 00:17:04,230
+لإن رايح لل a من وين من اليمين لجيت اللي هو دلتا
+
+202
+00:17:04,230 --> 00:17:11,420
+بحيث أنه لكل x في الفترة من a لعيد a زائد دلتابصير
+
+203
+00:17:11,420 --> 00:17:16,780
+عندى لكل x المتنية زائد دلتا بتطلع عندى then f
+
+204
+00:17:16,780 --> 00:17:23,420
+prime of x على g prime of x ناقص ال L أصغر من مين
+
+205
+00:17:23,420 --> 00:17:32,060
+من ال epsilon إذا الآن من هذه ال limitحصلت على أنه
+
+206
+00:17:32,060 --> 00:17:36,540
+لأي إبسلون بين الـ 0 و نص بقدر ألاقي Delta بحيث
+
+207
+00:17:36,540 --> 00:17:40,540
+لكل الإكسات في المنطقة هذه هذه الـ Inquality إيش
+
+208
+00:17:40,540 --> 00:17:44,880
+ما لها تتحقق الآن بس خلّيني أكملها دي شويه عشان
+
+209
+00:17:44,880 --> 00:17:52,040
+بستخدمها لشغلات أخرى عند A زائد Delta هذا بس مجرد
+
+210
+00:17:52,040 --> 00:17:56,340
+تعريف الـ Limit F' على الـ G' لما X تروح للـ A من
+
+211
+00:17:56,340 --> 00:17:59,120
+اليمين الآن
+
+212
+00:18:01,110 --> 00:18:04,910
+بدي أختار .. بدي أسهل على حالي بدل من كل مرة أقول
+
+213
+00:18:04,910 --> 00:18:08,130
+a زي الـ delta و بعد شوية ألاقي delta prime و أخد
+
+214
+00:18:08,130 --> 00:18:12,010
+ال minimum بينهم الاخرى أجدفم عليه عشان اللي بحكيه
+
+215
+00:18:12,010 --> 00:18:17,910
+بدي أخد اللي هو أقول choose c1 element in a و a زي
+
+216
+00:18:17,910 --> 00:18:24,630
+الـ delta الآن اختارلي c1 من ال a لعند مين لعند ال
+
+217
+00:18:24,630 --> 00:18:30,340
+a زي ال delta سميه c1 يعنيمجرد اختيار انا اخترته
+
+218
+00:18:30,340 --> 00:18:38,260
+الان choose .. choose C1 element in A و A زائد
+
+219
+00:18:38,260 --> 00:18:44,140
+دلتا يعني C1 وين موجود؟ موجود في المنطقة التي
+
+220
+00:18:44,140 --> 00:18:50,140
+تتحقق فيها F' ع G' نقص أصغر من ميل من إبسل عشان
+
+221
+00:18:50,140 --> 00:18:56,080
+تستخدمها بعد شويةواضح طيب هذه من جهة الأن من الجهة
+
+222
+00:18:56,080 --> 00:19:03,100
+الثانية عند limit f of x لما x تروح إلى مالة نهاية
+
+223
+00:19:03,100 --> 00:19:09,880
+أسف لل a من اليمين بسوء أيش هو معطينية هي النظرية
+
+224
+00:19:09,880 --> 00:19:13,540
+هي limit f of x لما x تروح إلى ال a من اليمين بسوء
+
+225
+00:19:13,540 --> 00:19:15,760
+limit g of x لما x تروح إلى ال a من اليمين بسوء
+
+226
+00:19:15,760 --> 00:19:23,170
+أيش ماله مالة نهاية من تعريفها هذا إيش يعنيلكل K
+
+227
+00:19:23,170 --> 00:19:29,270
+اللي هو element in R there exists Delta A' أكبر من
+
+228
+00:19:29,270 --> 00:19:36,590
+0 such that اللي هو F of X أكبر من اللي هو الـ K
+
+229
+00:19:36,590 --> 00:19:42,210
+لكل X و N موجودة في الـ A و A زائد مين Delta
+
+230
+00:19:42,210 --> 00:19:49,300
+Prime، مظبوط ولا لا؟ ماشي الحلالآن لكل x element
+
+231
+00:19:49,300 --> 00:19:52,200
+in A و A زي الـ delta prime هذه ال F of X أكبر من
+
+232
+00:19:52,200 --> 00:20:02,420
+مين؟ من K الآن لو كانت ال A و ال A .. الآن عندي
+
+233
+00:20:02,420 --> 00:20:07,540
+لكل subset من هذه .. لكل subset .. لو كان عندي A و
+
+234
+00:20:07,540 --> 00:20:10,940
+C2 مثلا subset من ال A و ال A زي ال delta prime
+
+235
+00:20:12,470 --> 00:20:17,070
+برضه ال F of X أكبر من كده تحققها تحقق إيش لكل X
+
+236
+00:20:17,070 --> 00:20:24,610
+وين موجودة في الفترة من A لعند مين لعند C2 الآن لو
+
+237
+00:20:24,610 --> 00:20:30,410
+كانت ال Delta اللي لجيتها برا ولا جوا بقدر أختار
+
+238
+00:20:30,410 --> 00:20:36,990
+جوا اللي هو C2بحيث أنه لو الـ C2 هذي هو اللي
+
+239
+00:20:36,990 --> 00:20:42,450
+اخترتها جوات من الـ A والـ A زائد دلتا بتظل الـ F
+
+240
+00:20:42,450 --> 00:20:47,470
+of X A شمالها أكبر من مين؟ من الـ K، واضح؟ إذا من
+
+241
+00:20:47,470 --> 00:20:52,330
+هذه بدي أستفيد شغلتين بدي أطبق التعريف هذا لـ K
+
+242
+00:20:52,330 --> 00:20:56,730
+محددة مين الـ K اللي بدي أطبقها؟ اللي عبارة عن F
+
+243
+00:20:56,730 --> 00:21:02,560
+عند مين؟ عند C1واضح؟ إذا الأن بما أنه limit f of X
+
+244
+00:21:02,560 --> 00:21:05,680
+ثم X تروح إلى الـA باليمين يستوى Infinity إذا for
+
+245
+00:21:05,680 --> 00:21:13,720
+K .. إذا for f of C1 there exists Delta Prime أي
+
+246
+00:21:13,720 --> 00:21:19,840
+Delta Prime معينة بحيث أن f of X أكبر من 100 من f
+
+247
+00:21:19,840 --> 00:21:27,060
+of C1 لكل X موجودة بين الـA والـA زائد Delta Prime
+
+248
+00:21:27,680 --> 00:21:31,860
+أنا بدي أختار في مين لكل X الموجودة بين الـ A و
+
+249
+00:21:31,860 --> 00:21:37,940
+الـ C2 حيث الـ C2 موجودة في الفترة هذه و بديها من
+
+250
+00:21:37,940 --> 00:21:44,120
+C1 و لجاي اذا عشان هيك بقدر اقول then we can
+
+251
+00:21:44,120 --> 00:21:50,280
+choose C2 بين A و C1الـ C2 بقدر اختارها فعلا
+
+252
+00:21:50,280 --> 00:21:55,640
+بختارها بس طبعا وين تشتري تشريع اللي اختيارها انها
+
+253
+00:21:55,640 --> 00:22:01,420
+تكون بين الفترة A و A زائد Delta Prime و بدياها
+
+254
+00:22:01,420 --> 00:22:06,900
+تيجي في داخل الـ A و الـ C1 اذا then we can choose
+
+255
+00:22:06,900 --> 00:22:13,200
+C2 element A of C1 بحيث ان ال F of X أكبر من مين
+
+256
+00:22:13,200 --> 00:22:22,810
+من F of C1 لكل X وينفي الفترة بين A وC اتنين طيب
+
+257
+00:22:22,810 --> 00:22:28,050
+هذه معناته ان ال F of X هذه لكل ال Xات اللي هان ال
+
+258
+00:22:28,050 --> 00:22:34,290
+F of X أكيد اي شمالها لها تساوي ايش F of C واحد هل
+
+259
+00:22:34,290 --> 00:22:38,930
+جد تعرف ليش بديها هذه عشان لزوم تعريف شغل معينة
+
+260
+00:22:38,930 --> 00:22:44,770
+بنفعش تكون ال F of X ايش بتساوي F of C واحد الآن
+
+261
+00:22:44,770 --> 00:22:55,550
+similarlySimilarly اللي هو بما انه limit G of X
+
+262
+00:22:55,550 --> 00:23:00,690
+بساوي مالة نهاية as X بتروح للـ A من اليمين إذا
+
+263
+00:23:00,690 --> 00:23:13,160
+بقدر ألاقي اللي هو C1 برايم بحيث انه G of X أكبرأو
+
+264
+00:23:13,160 --> 00:23:17,240
+لا تساوي طبعا أكبر الـ thrifty يعني لا تساوي G of
+
+265
+00:23:17,240 --> 00:23:27,860
+C1 لكل X موجودة في الـ A و C1 إيش إبراهيم اه G C1
+
+266
+00:23:27,860 --> 00:23:34,680
+إبراهيم اه G C1 إبراهيم الآن
+
+267
+00:23:34,680 --> 00:23:40,760
+ال .. ال .. ال .. الباخدالـ Minimum عشان طبعا هذا
+
+268
+00:23:40,760 --> 00:23:46,120
+اللي لجهة N C1' ممكن اللي هي لكل X في الـ A و C1'
+
+269
+00:23:46,620 --> 00:23:52,840
+و لكل X element in A و C2' بط مانش .. مش عارف مين
+
+270
+00:23:52,840 --> 00:23:58,670
+الأكبر من هذول فباخد الـ Minimum من الجهتينو
+
+271
+00:23:58,670 --> 00:24:03,090
+بسميها C1 مثلا او C2 ال minimum من التنتين C2
+
+272
+00:24:03,090 --> 00:24:09,550
+فبتصير عندى لان لكل X في ال A و ال minimum بينهن
+
+273
+00:24:09,550 --> 00:24:14,630
+هدى بتظبط و هدى بتظبط يعنى ال G of X لا تساوي F of
+
+274
+00:24:14,630 --> 00:24:20,670
+C1 و ال F of X لا تساوي مين؟ F of C1 إبرايم و منه
+
+275
+00:24:20,670 --> 00:24:25,870
+بعرف اللى بديه او بعمل زى ما هو عامل في الكتاب إيش
+
+276
+00:24:25,870 --> 00:24:29,280
+اللى بقوله؟ نشوف اللى بعدهاإذا الأن اللي اتفجنا
+
+277
+00:24:29,280 --> 00:24:38,540
+عليه أنه في الفترة من A لعند A زائد Delta هذه
+
+278
+00:24:38,540 --> 00:24:45,480
+متحققة هاي واحد واللي اتفجنا عليه أن F of X لا
+
+279
+00:24:45,480 --> 00:24:50,140
+تساوي F of C واحد في الفترة من A لعند C تنين
+
+280
+00:24:50,140 --> 00:24:55,580
+أشمالها متحققةيعني بمعنى آخر الـ Inquality هذه
+
+281
+00:24:55,580 --> 00:25:03,800
+وهذه التنتين محققات من أين؟ من A لعين C الاتنين
+
+282
+00:25:03,800 --> 00:25:05,360
+واضح؟
+
+283
+00:25:11,650 --> 00:25:17,590
+الان اصلنا لمرحلة انه نقدر نعرف الدالة اللي بتديها
+
+284
+00:25:17,590 --> 00:25:23,890
+اللي هي F of X بتساوي واحد ناقص F of C واحد على F
+
+285
+00:25:23,890 --> 00:25:30,470
+of X وواحد ناقص G of C واحد على G of X لكل X وين
+
+286
+00:25:30,470 --> 00:25:37,330
+موجودة بين A وC تنين و F of C واحد لا تساوي F of X
+
+287
+00:25:37,330 --> 00:25:43,340
+مظبوط؟ هيا اعملهاأه لإن وين بدي أعرف هذه الدالة F
+
+288
+00:25:43,340 --> 00:25:50,480
+من A لعند مين اللي هي بتتحقق عليها خاصية F of X ده
+
+289
+00:25:50,480 --> 00:25:55,080
+تسوى مين F of C واحد و بتتحقق عليها مين اللي هي
+
+290
+00:25:55,080 --> 00:26:00,480
+الخاصية اللي عندي هذه لإن كله هيلزمني هذا إذا الأن
+
+291
+00:26:05,340 --> 00:26:09,980
+اتفجنا عليها دي اه عشان بدي امسح انه عندي f prime
+
+292
+00:26:09,980 --> 00:26:13,800
+ع الجي برايم نقصها الأصغر من يابسولون لكل ال Xات
+
+293
+00:26:13,800 --> 00:26:18,700
+من A لعند A زائد ديلتا لكل ال Xات اللي هانا
+
+294
+00:26:18,700 --> 00:26:22,040
+واتفجنا ان F of X لا تساوي F of C one برضه في
+
+295
+00:26:22,040 --> 00:26:27,350
+الفرق في المنطقة من ايش من A لC2 اتفجنا عليهالان
+
+296
+00:26:27,350 --> 00:26:33,510
+نيجي نعرف الدالة اللي هتوصلني لهدفي خد الان f of x
+
+297
+00:26:33,510 --> 00:26:42,670
+بتساوي اللي هو واحد ناقص f of c واحد على f of x
+
+298
+00:26:42,670 --> 00:26:52,450
+على واحد ناقص g of c واحد على g of xالآن g of c
+
+299
+00:26:52,450 --> 00:26:55,810
+واحد مستحيل تساوي g of x من أي أسباب اللي حكيناها
+
+300
+00:26:55,810 --> 00:27:01,710
+وليس سبب آخر أيضا اللي هو ليش أنه عندي الـ g prime
+
+301
+00:27:01,710 --> 00:27:07,570
+الـ g prime of x ده يساوي سفر إذا حسب Rolle's
+
+302
+00:27:07,570 --> 00:27:13,570
+theorem مستحيل الـ g of c واحد اللي هو تساوي g of
+
+303
+00:27:13,570 --> 00:27:20,530
+x ليش عندي g continuous
+
+304
+00:27:21,550 --> 00:27:29,350
+on a .. طبعا الـ x والي موجودة جوها منها الان من x
+
+305
+00:27:29,350 --> 00:27:37,110
+لعند c واحد صح؟ و g is differentiable on x لعند
+
+306
+00:27:37,110 --> 00:27:44,090
+اللي هي c واحد open واضحة؟
+
+307
+00:27:44,090 --> 00:27:48,670
+لأنه أثر هذا continuous و differentiable من a لعند
+
+308
+00:27:48,670 --> 00:27:52,550
+bالعلمي إذا في الجزية هذه إذا أكيد اللي هم تتحقق
+
+309
+00:27:52,550 --> 00:28:01,270
+هذه الآن لو زي ما بنقول ان جي برا لو لو عندي بدي
+
+310
+00:28:01,270 --> 00:28:09,250
+يكون g of x بسوء g of c واحد معناته حسب role
+
+311
+00:28:09,250 --> 00:28:16,360
+theorem هيعطيني there existcx element in x و c1
+
+312
+00:28:16,360 --> 00:28:22,800
+such that g prime of cx هي ساوى سفر وهذا مستحيل
+
+313
+00:28:22,800 --> 00:28:27,320
+لإن هو معطيني الـ g prime of x لاتساوى سفر لكل x
+
+314
+00:28:27,320 --> 00:28:34,630
+element من a وb معناته معناه الحديثإنه الـ G of C
+
+315
+00:28:34,630 --> 00:28:41,510
+واحد والـ G of X مستحيل يكون متساويات لكل الإكسات
+
+316
+00:28:41,510 --> 00:28:47,230
+اللي في الفترة من A لعند مين لعند C اتنين واضح إذا
+
+317
+00:28:47,230 --> 00:28:50,730
+المقام لا يساوي سفر إذا اللي هو هذي is well
+
+318
+00:28:50,730 --> 00:29:00,950
+defined على مين على الفترة Aمظبوط و C2 خلّيها في
+
+319
+00:29:00,950 --> 00:29:05,070
+الذاكرة طيب اطلع لفوق و حسب لل limit الآن هذا
+
+320
+00:29:05,070 --> 00:29:12,070
+عرفناها احفظناها بنضطر ان امسح الان احسب لل limit
+
+321
+00:29:12,070 --> 00:29:20,230
+للي فوق limit F of X لما X تروح الي وين الي ايه من
+
+322
+00:29:20,230 --> 00:29:23,390
+اليمين بساوى واحد
+
+323
+00:29:24,530 --> 00:29:32,650
+نقص limit F of C1 عدد على F of X لما X تروح للـ A
+
+324
+00:29:32,650 --> 00:29:36,950
+من اليمين على اتجرأت لأنه عارف ان ال limit موجودة
+
+325
+00:29:36,950 --> 00:29:42,290
+على التوزيع اتجرأت عليه لأنه عارف limit G of C1
+
+326
+00:29:42,290 --> 00:29:49,530
+على G of X لما X تروح لوين للـ A من اليمين واضحة
+
+327
+00:29:49,530 --> 00:29:56,130
+جدش هذه ال limit0 لأن ال F of X وين بتروح الى مالة
+
+328
+00:29:56,130 --> 00:30:00,010
+نهاية هي بداية الموضوع مش بداية الموضوع كان عندنا
+
+329
+00:30:00,010 --> 00:30:05,010
+ان ال limit F of X على G of X لما X تروح الى A من
+
+330
+00:30:05,010 --> 00:30:08,670
+اليامين اللي هو سببت لكل القصة ان هذه ال limit
+
+331
+00:30:08,670 --> 00:30:11,230
+مالة نهاية وهذه مالة نهاية صارت مالة نهاية مالة
+
+332
+00:30:11,230 --> 00:30:15,710
+نهاية وهي اللي خلتني اروح بهذا الاتجاه اذا هذه
+
+333
+00:30:16,610 --> 00:30:19,750
+بتروح إلى ملأ نهاية إذا هذه بتروح إلى سفر وهذه
+
+334
+00:30:19,750 --> 00:30:22,330
+الملأ نهاية بتروح إلى سفر إذا هذا كله على بعض إيش
+
+335
+00:30:22,330 --> 00:30:30,170
+بيساوي؟ بيساوي واحد واضح طيب الآن for every
+
+336
+00:30:30,170 --> 00:30:34,570
+epsilon أكبر من سفر اللي هي اللي فوق خلاص بدنا
+
+337
+00:30:34,570 --> 00:30:39,270
+فيها الآن بيبقى إن ال limit بيساوي واحد إذا there
+
+338
+00:30:39,270 --> 00:30:44,890
+existsDelta أكبر من سفر أي إن كانت الـ Delta أكبر
+
+339
+00:30:44,890 --> 00:30:53,790
+من سفر Such that F of X ناقص واحد أصغر من مين من
+
+340
+00:30:53,790 --> 00:30:59,190
+الـ Epsilon Epsilon اللي بدنا فيها من الأول هذا لو
+
+341
+00:30:59,190 --> 00:31:04,830
+يعني لكل الـ Xاللي في الفترة من a ل a زي الـ delta
+
+342
+00:31:04,830 --> 00:31:07,910
+هذه الـ delta الجديدة مش صارت تكون الأولى فأنا
+
+343
+00:31:07,910 --> 00:31:13,330
+عشان أريح حالي بدي أخد اللي هو أصغر delta بلاجيها
+
+344
+00:31:13,330 --> 00:31:17,910
+و أقول حطها كمان جواها هادي حر أنا مدام بتنفع لالي
+
+345
+00:31:17,910 --> 00:31:21,690
+كبيرة ايضا كده بتنفع لمن؟ للصغيرة فاهمين أنا ايش
+
+346
+00:31:21,690 --> 00:31:28,270
+بقول؟ هذه اللي هي نسميها C3 معايا؟ فبصير عندى الآن
+
+347
+00:31:28,270 --> 00:31:39,200
+F of Xلكل X element in A لعند مين؟ C3 أي
+
+348
+00:31:39,200 --> 00:31:46,120
+سؤال؟ F of X ناقص واحد أصغر من Y أكبر من سالب Y
+
+349
+00:31:46,120 --> 00:31:49,960
+تلزمني هذه المنطقة اللي اللي جاي لأن F of X هذه
+
+350
+00:31:49,960 --> 00:31:56,720
+يعني F of X يعطينا F of X أكبر من واحد ناقص Y
+
+351
+00:31:59,390 --> 00:32:03,370
+باضحة؟ لأن هذه جلبها .. اللي هذه طبعا واحد ناقص
+
+352
+00:32:03,370 --> 00:32:07,850
+إبسلون اللي هي بيصير عنده ال .. ال .. ال إبسلون
+
+353
+00:32:07,850 --> 00:32:12,650
+أصغر من مين؟ من نص الواحد ناقص إبسلون أكبر من مين؟
+
+354
+00:32:12,650 --> 00:32:20,670
+اللي هي بيصير أكبر من نص باضحة؟ اللي هي بيصير واحد
+
+355
+00:32:20,670 --> 00:32:28,400
+على f of x اللي هو أصغر من مين؟هذا لمين؟ لكل
+
+356
+00:32:28,400 --> 00:32:32,760
+الاكسات اللي من A لعند مين C3 وطبيعي ال
+
+357
+00:32:32,760 --> 00:32:36,880
+inequalities كلها اللي قبل تتحقق هنا لإنها بتتحقق
+
+358
+00:32:36,880 --> 00:32:41,360
+من هنا لهنا بعضها وبعضها بتتحقق من هنا لهنا فأكيد
+
+359
+00:32:41,360 --> 00:32:47,040
+كلها هتتحقق للإكسات اللي مين في ال A وC3 أصلا أنا
+
+360
+00:32:47,040 --> 00:32:52,620
+رايح باتجاه مين؟ رايح باتجاه أثبت لكم أنه limit of
+
+361
+00:32:53,680 --> 00:32:59,060
+of x على g of x لما x تروح لل a من اليمين بساوي
+
+362
+00:32:59,060 --> 00:33:04,880
+limit f prime of x على g prime of x as x تروح لل a
+
+363
+00:33:04,880 --> 00:33:10,080
+من اليمين اللي هو ايش نسميه احنا قال اذا انا بهمن
+
+364
+00:33:10,080 --> 00:33:14,920
+مين الاكسات اللي جنب ال a لإن انا رايح لل a من وين
+
+365
+00:33:14,920 --> 00:33:18,800
+من اليمين فانا بهمن الجوار القريب جدا من ال a لإن
+
+366
+00:33:18,800 --> 00:33:21,960
+انا رايحله اصلا من هنافالإكسات اللي فيها هي اللي
+
+367
+00:33:21,960 --> 00:33:27,760
+بتلزمني عشان أصل للي بديها خلّي هذه في الذاكرة
+
+368
+00:33:27,760 --> 00:33:34,560
+فصار عندي الآن صار
+
+369
+00:33:34,560 --> 00:33:38,800
+عندي هذا كله حكيناه وخلصنا منه الان نيجي بدنا نصل
+
+370
+00:33:38,800 --> 00:33:44,620
+للي بدنايا بتتذكروا المرة الماضية أنا أصلا جيبت
+
+371
+00:33:44,620 --> 00:33:56,350
+هذهجبت هذه اتطلع فيها و قارنليها مع ال F of X على
+
+372
+00:33:56,350 --> 00:34:01,310
+ال G of X اللي انا بديهاها ال F of X على ال G of X
+
+373
+00:34:01,310 --> 00:34:08,170
+لو جيت قارنتها F of X على ال G of X ايش هتلاقيها؟
+
+374
+00:34:08,170 --> 00:34:14,810
+لو جيت ضربت هذه في F of X وهذه في G of X
+
+375
+00:34:17,460 --> 00:34:22,500
+بنرجع ل F of X ناقص F of C واحد و G of X ناقص G of
+
+376
+00:34:22,500 --> 00:34:26,680
+C واحد طب ايش دخلنا فيه هذه F of X ناقص F of C
+
+377
+00:34:26,680 --> 00:34:30,500
+واحد و G of X ناقص G of C واحد هذه اللي بالكوشي
+
+378
+00:34:30,500 --> 00:34:33,920
+mean value theorem اللي هتجيب لل F prime و G prime
+
+379
+00:34:33,920 --> 00:34:38,340
+اللي انا اصلا موجودات limited فبحصل ع اللي بديا
+
+380
+00:34:40,230 --> 00:34:44,470
+بالظبط الشي اللي بقوله انت يعني ليش هو فكر في هذه
+
+381
+00:34:44,470 --> 00:34:50,470
+بالسبب اللي حكيته F of X على D of X بالساوي F of X
+
+382
+00:34:50,470 --> 00:34:59,890
+على D of X في F of X على D of X على F of X مظبوط؟
+
+383
+00:34:59,890 --> 00:35:09,170
+ماعملش شيء الآن اسحبلي هذه خليهاوحط ليها هذه 1 على
+
+384
+00:35:09,170 --> 00:35:15,210
+f of x ماهي الحالة هذه مين هي في الواقع هي اللي
+
+385
+00:35:15,210 --> 00:35:20,410
+فوق اللي هي عبارة عن اضرب f of x هذه في هذه بصير
+
+386
+00:35:20,410 --> 00:35:32,250
+واحد ويساوي واحد اللي هي ناقص f of c صح؟ f of x
+
+387
+00:35:32,250 --> 00:35:42,660
+ناقص f of cعلى g of x نقص g of c هذا مين هي اللي
+
+388
+00:35:42,660 --> 00:35:46,700
+هي f of x على g of x في f of x عوضت هذه و حطيتها و
+
+389
+00:35:46,700 --> 00:35:51,760
+طلع عندي هذا مضروب في كله في مين في واحدة ال f of
+
+390
+00:35:51,760 --> 00:36:01,080
+x الآن عندي خليني أطبق اللي هو مين الـ Cauchy mean
+
+391
+00:36:01,080 --> 00:36:06,390
+value theorem على الفترة من a لمين ل c تلاتةماشي
+
+392
+00:36:06,390 --> 00:36:11,130
+اللي هو there exist طبعا كله متحقق اللي هو ال F و
+
+393
+00:36:11,130 --> 00:36:15,230
+ال G continuous و differentiable على ال A و ال C 3
+
+394
+00:36:15,230 --> 00:36:21,430
+اذا there exist اللي هو ايه اللي بدكي اسميها اللي
+
+395
+00:36:21,430 --> 00:36:25,690
+هي مثلا there exist gamma او زي ما هو مسميها في
+
+396
+00:36:25,690 --> 00:36:32,230
+الكتاب there exist exi element in A و C تلاتة such
+
+397
+00:36:32,230 --> 00:36:41,500
+thatf prime of xi على g prime of xi بساوي بس ان
+
+398
+00:36:41,500 --> 00:36:44,780
+انا عشان بتطبقها على مين على الفترة f of x لعند
+
+399
+00:36:44,780 --> 00:36:48,800
+مين عشان تطلع عندي f of x of مين و f of c واحد
+
+400
+00:36:48,800 --> 00:36:53,120
+معايا للي اكسات اللي وين اللي موجودة من a لعند c
+
+401
+00:36:53,120 --> 00:36:58,060
+تلاتة بعد اذنكم بتطبقها الكوشي mean value theorem
+
+402
+00:36:58,760 --> 00:37:01,840
+ما هي اصلا continuous و differentiable على الفترة
+
+403
+00:37:01,840 --> 00:37:06,500
+دي كلها من ضمنها مين الفترة A وC ثلاثة ومن ضمنها
+
+404
+00:37:06,500 --> 00:37:11,040
+الفترة A وC واحد اللي انا بده اطبق عليها بال end
+
+405
+00:37:11,040 --> 00:37:15,900
+point مين ال end point C واحد هيها و ال end point
+
+406
+00:37:15,900 --> 00:37:21,890
+ال X اللي موجودة بين A و B مينو C3 واضح واضح اللي
+
+407
+00:37:21,890 --> 00:37:26,350
+بدي .. اذا الان الان there exists x i الان بالظبط
+
+408
+00:37:26,350 --> 00:37:30,630
+بدي اطبق الميكوشي main value term ع الفترة اللي هي
+
+409
+00:37:30,630 --> 00:37:37,790
+X و C1 اللي ال F و ال G differentiable عليها من
+
+410
+00:37:37,790 --> 00:37:40,530
+وين ال X هذه ال X اللي فوق هذه اللي بشتغل فيها
+
+411
+00:37:40,530 --> 00:37:45,890
+اللي هي من A لعن مين لعن C3إذن there exists xi
+
+412
+00:37:45,890 --> 00:37:50,750
+element in x و c واحد such that f prime of xi على
+
+413
+00:37:50,750 --> 00:37:58,770
+g prime of xi في إيش بتساوي f of c واحد أو f of x
+
+414
+00:37:58,770 --> 00:38:05,390
+ناقص f of c واحد فاهمين طبعا على g of x ناقص g of
+
+415
+00:38:05,390 --> 00:38:10,990
+c واحد بس ضربت في ناقص فوق و ناقص صحيح تتعدى زي ما
+
+416
+00:38:10,990 --> 00:38:20,080
+هي لأن هذههي هذه إذا صارت عندي F of X على G of X
+
+417
+00:38:20,080 --> 00:38:26,000
+بسوء F prime على G prime في واحد على مين على F of
+
+418
+00:38:26,000 --> 00:38:32,060
+X وهو اللي بده يوصلنا للي بدنا هي ماعليش أمسح اللي
+
+419
+00:38:32,060 --> 00:38:35,280
+هان بيصير عندى
+
+420
+00:38:38,180 --> 00:38:45,160
+بصير عندى ال F of X على مين F of X على G of X
+
+421
+00:38:45,160 --> 00:38:48,540
+بساوى
+
+422
+00:38:48,540 --> 00:38:54,120
+اللى هو هدى شلناه و حطينا مكانها مين بساوى F prime
+
+423
+00:38:54,120 --> 00:39:02,140
+of X I على G prime of X I فى مين فى واحد على ايه
+
+424
+00:39:02,140 --> 00:39:03,500
+ايش على F
+
+425
+00:39:09,880 --> 00:39:18,360
+واضح أه؟ الآن نمحى هنا هدولة ال Xات اللي هنا هما
+
+426
+00:39:18,360 --> 00:39:23,380
+ال Xات اللي وين موجودة بين A وC3 ماشي الحال بدي
+
+427
+00:39:23,380 --> 00:39:29,120
+الآن غايتي أني أوجد F of X على G of X ناقص مين؟
+
+428
+00:39:29,120 --> 00:39:34,600
+ناقص Lواضحة إذا نلخص كل اللي خدناه for every
+
+429
+00:39:34,600 --> 00:39:38,800
+epsilon element in zero نص there exists delta أكبر
+
+430
+00:39:38,800 --> 00:39:48,980
+من سفر such that if اللي هو x أكبر من a أصغر من c3
+
+431
+00:39:48,980 --> 00:39:53,260
+أصغر من c2 أصغر من c1 أصغر من a زي ال delta اللي
+
+432
+00:39:53,260 --> 00:40:02,070
+لجناها فاهمين عليها وكون عندي اللي هو قيلةالعوّض f
+
+433
+00:40:02,070 --> 00:40:08,110
+of x على g of x ناقص ال .. بدي أقبل لك يميها هذا
+
+434
+00:40:08,110 --> 00:40:12,510
+اللي هو أصغر من epsilon في something مش مشكلة و
+
+435
+00:40:12,510 --> 00:40:15,810
+epsilon is arbitral أيضا بيصير limit زي ما بدنايا
+
+436
+00:40:15,810 --> 00:40:24,170
+واضح هذا الآن بالظبط بيساوي اللي هو العوض اللي هو
+
+437
+00:40:24,170 --> 00:40:28,590
+f of x اللي هي عبارة عن واحد
+
+438
+00:40:32,680 --> 00:40:41,680
+أف برايم of x i على جي برايم of x i في واحد على أف
+
+439
+00:40:41,680 --> 00:40:48,180
+of x ناقص اللى
+
+440
+00:40:48,180 --> 00:40:54,460
+واضح لحتى الان ناخد هذه عامل مشترك برا يصبح واحد
+
+441
+00:40:54,460 --> 00:41:02,140
+على أف of x في أف برايم of x iعلى g prime of xi
+
+442
+00:41:02,140 --> 00:41:13,560
+ناقص L في F of X صح؟ أخدت
+
+443
+00:41:13,560 --> 00:41:17,300
+الواحد على F of X عام المشترك positive أه positive
+
+444
+00:41:17,300 --> 00:41:21,080
+اللي هو F of X هي أكبر من 2 طلعت اللي هو إيش ماله
+
+445
+00:41:21,080 --> 00:41:27,260
+هذا؟هنا ده بدي أضيف term و أطرح term عشان أحصل على
+
+446
+00:41:27,260 --> 00:41:31,780
+هذه اللي بدياها أصلا اللي هي الجد تفهموه مش مقصودة
+
+447
+00:41:31,780 --> 00:41:38,900
+1 على f of x في f prime of xi على g prime of xi
+
+448
+00:41:38,900 --> 00:41:50,130
+ناقص L زائد اللي هوL نقص L F of X ايش سويت استخدمت
+
+449
+00:41:50,130 --> 00:41:56,170
+ال triangle inequality باضافة L وطرحة L هذا كله
+
+450
+00:41:56,170 --> 00:42:04,130
+مضروف 100 في 1 على F of X هذا الآن ال F of X أكبر
+
+451
+00:42:04,130 --> 00:42:12,690
+من 2 حصلناها مظبوط بصير أصغر من نص واضحة
+
+452
+00:42:13,760 --> 00:42:18,920
+الآن F' على G' لـ XI الـ XI وين لاجيناها؟ في
+
+453
+00:42:18,920 --> 00:42:24,940
+الفترة بين A وC1 ينطبق عليها اللي هو من رأس الدور
+
+454
+00:42:24,940 --> 00:42:34,580
+اللي هي F' على G' of X ناقص
+
+455
+00:42:34,580 --> 00:42:41,200
+L أصغر من الـ Y من أول ما بدينايعني إنه هذا أصغر
+
+456
+00:42:41,200 --> 00:42:44,460
+من إبسلون لكل الإكسات اللي موجودة من A لعند مين؟
+
+457
+00:42:44,460 --> 00:42:48,080
+لعند الـA زاد دلتها من ضمنهم مين؟ لك فيبراين كلام
+
+458
+00:42:48,080 --> 00:42:55,940
+دقيق هذه في إبسلون زاد الآن هذا ال بطلحها برا عن
+
+459
+00:42:55,940 --> 00:43:01,760
+المشترك ال اللي هي في مين؟ في واحد ناقص F of X
+
+460
+00:43:01,760 --> 00:43:10,240
+واحد ناقص F of Xالان وهذا أصغر من نص في إبسلون
+
+461
+00:43:10,240 --> 00:43:15,440
+زائد absolute value لأل واحد ماقص F of X هذه ال
+
+462
+00:43:15,440 --> 00:43:20,260
+Xات اللي موجودة بين A وC تلاتة وهذه اللي هي قبل
+
+463
+00:43:20,260 --> 00:43:26,150
+بشوية أصغر من مين من اللي هو إبسلونأصغر من إبسلون
+
+464
+00:43:26,150 --> 00:43:30,030
+واحد ناقص F of X أصغر من مين؟ من إبسلون مش هثبتنا
+
+465
+00:43:30,030 --> 00:43:33,910
+limit F of X بيساوي واحد قبل بشوية اللي هو واحد
+
+466
+00:43:33,910 --> 00:43:37,050
+ناقص F of X اللي هي أصغر من إياش من إبسلون في
+
+467
+00:43:37,050 --> 00:43:42,850
+إبسلونصار عندي الان هذا المقدار اللي هو بساوي
+
+468
+00:43:42,850 --> 00:43:47,030
+epsilon على اتنين في واحد زايد absolute value
+
+469
+00:43:47,030 --> 00:43:52,810
+للالف لان ال epsilon مضروبة فيه اذا as لان هذا ال
+
+470
+00:43:52,810 --> 00:43:55,430
+epsilon arbitrary وهذا الكلام صحيح for every
+
+471
+00:43:55,430 --> 00:43:59,710
+epsilon epsilon بين zero و نص اذا هذا المقدار بقدر
+
+472
+00:43:59,710 --> 00:44:04,350
+ازغره جد ما بدي بتصغير epsilon لانه مضروب فيه ضرب
+
+473
+00:44:04,350 --> 00:44:08,630
+اذا صار عندي لكل epsilon أكبر من zero و نصلـ G الـ
+
+474
+00:44:08,630 --> 00:44:12,650
+Delta بحيث أنه الإكسات اللي هنا بيقدّي لي أن هذا
+
+475
+00:44:12,650 --> 00:44:20,810
+المقدار أصغر من أي قيمة بديّها إذا هذا مفهوم limit
+
+476
+00:44:20,810 --> 00:44:31,530
+F of X على G of X as X بتروح لنين؟ للـ A من اليمين
+
+477
+00:44:31,530 --> 00:44:38,920
+بساوي الـ L وهو المطلوبالثيورم اللي حكينا عنها
+
+478
+00:44:38,920 --> 00:44:45,060
+برضه صحيحة في حالة اللي هو the same .. under the
+
+479
+00:44:45,060 --> 00:44:50,160
+same conditions for the calculation of limits as x
+
+480
+00:44:50,160 --> 00:44:55,540
+goes to infinity or x as goes mean to سالبinfinity
+
+481
+00:44:55,540 --> 00:44:59,140
+بس بورهان يعني بدون modification عن اللي موجود
+
+482
+00:44:59,140 --> 00:45:03,140
+بتصلى اللي بدكيها نيجي ناخد أمثلة الأمثلة هذه طبعا
+
+483
+00:45:03,140 --> 00:45:09,500
+أمثلة سهلة ومثلة calculus نمر عليها اللي هو ونشوف
+
+484
+00:45:09,500 --> 00:45:16,480
+كيف نطبق نظريتنا أو نظرياتنا كيف نوظفها الحساب هذه
+
+485
+00:45:16,480 --> 00:45:23,260
+النهايات عندى شوفوا صلى النبي عليه الصلاة والسلام
+
+486
+00:45:24,800 --> 00:45:30,720
+Find Limitlog sin x على log x as x بتروح للـ 0 من
+
+487
+00:45:30,720 --> 00:45:35,480
+وين؟ من اليمين و احنا عارفين .. مشتغلين على الفترة
+
+488
+00:45:35,480 --> 00:45:40,460
+0 or by يعني فترة بجوار مين؟ الصفر من اليمين و
+
+489
+00:45:40,460 --> 00:45:43,540
+بدنا نروح للـ 0 من اليمين هذا طبعا لو جينا عوضنا
+
+490
+00:45:43,540 --> 00:45:48,080
+هتطلع اللي هو sin 0 0 و أنا 0 فبصير مالة نهاية على
+
+491
+00:45:48,080 --> 00:45:53,580
+مين؟ على مالة نهاية ك limits الآن صارت عندي مالة
+
+492
+00:45:53,580 --> 00:45:57,880
+نهاية على مالة نهاية إذا إيش بتنسوه؟إيش نسوي؟
+
+493
+00:45:57,880 --> 00:46:06,400
+راحين لـ 0 من اليمين بيصير عنده اللي هو بالفاضل
+
+494
+00:46:06,400 --> 00:46:11,400
+هذه و بالفاضل هذه بيصير تفاضلها هذه واضحة و هذه
+
+495
+00:46:11,400 --> 00:46:14,060
+تفاضلها هيك مافيش داعي لنا ندخل في التفاصيل لإن كل
+
+496
+00:46:14,060 --> 00:46:20,320
+بعرفها الآن بنبسط الأمر بيصير X Cos X على Sine X
+
+497
+00:46:20,930 --> 00:46:26,550
+الان لو جيت اتطلعت لهذه ال limit و لهذه ال limit
+
+498
+00:46:26,550 --> 00:46:29,530
+هذه موجودة و هذه موجودة هذه موجودة لإنه تعويض
+
+499
+00:46:29,530 --> 00:46:33,410
+مباشر واحد هذه موجودة لإنه بيصير سفر ع سفر بتعملها
+
+500
+00:46:33,410 --> 00:46:36,570
+global rule بيصير واحد على cosine بتطلع واحد برضه
+
+501
+00:46:36,570 --> 00:46:40,590
+إذا اقدرت اوزع لإنه عارف الليمتين الموجودة فبيصير
+
+502
+00:46:40,590 --> 00:46:46,220
+عبارة عن واحد في واحد بيساوي ايه؟ واحدالان خد على
+
+503
+00:46:46,220 --> 00:46:49,900
+الفترة من Zero لعند مين لعند باي على اتنين limit
+
+504
+00:46:49,900 --> 00:46:53,300
+واحد على X نقص واحد على sign ال X ثم X تروح ل Zero
+
+505
+00:46:53,300 --> 00:46:57,140
+من وين؟ من اليمين وهذه طبعا شغلات اللي هو درجنا
+
+506
+00:46:57,140 --> 00:47:01,120
+عليها في ال calculus بنوحد المقامات وبصير مكتوبة
+
+507
+00:47:01,120 --> 00:47:05,030
+على صورة sign ال X نقص X على X في sign ال Xماشي
+
+508
+00:47:05,030 --> 00:47:10,650
+الحل برضه 0 على 0 بنفضلها بيصير cos X ناقص 1 على
+
+509
+00:47:10,650 --> 00:47:15,470
+اللي هو sin X زاد X فcos X لو جينا عوضنا هتطلع
+
+510
+00:47:15,470 --> 00:47:21,250
+برضه 0 على 0 بنشتق كمان مرة بتطلع ناقص sin X على
+
+511
+00:47:21,250 --> 00:47:26,150
+هذا المقدار هذا المقدار هنا بتصفر وهنا بيصير 2
+
+512
+00:47:26,150 --> 00:47:32,680
+وهنا 0 0 على 2 مش بيساوي بيساوي 0نأتي لهذا مثال
+
+513
+00:47:32,680 --> 00:47:36,440
+معهود الـ let I بيساوين 1 إلى ما لنهاية الـ
+
+514
+00:47:36,440 --> 00:47:39,760
+interval اللي عليها بدنا نشتغل و بدنا ناخد ال
+
+515
+00:47:39,760 --> 00:47:43,600
+limit لما X تروح إلى وين إلى ما لنهاية المقدار 1
+
+516
+00:47:43,600 --> 00:47:48,960
+زائد 1 على X و الكل أُس Xهذا المقدار الان بدي
+
+517
+00:47:48,960 --> 00:47:53,460
+استخدم اللي هو يحوله ل exponential to the len ما
+
+518
+00:47:53,460 --> 00:47:55,660
+احنا عارفين ال exponential و ال len اللي هو n
+
+519
+00:47:55,660 --> 00:47:59,860
+versus لبعض فبصير عندي E to the X لن ال 1 زائد 1
+
+520
+00:47:59,860 --> 00:48:05,120
+على X هذا المقدار بدي اشغل على هذاكخطوة أولى بقول
+
+521
+00:48:05,120 --> 00:48:09,260
+limit ال X لما واحد زائد واحد على X لما X تروح إلى
+
+522
+00:48:09,260 --> 00:48:14,140
+مال نهاية بيصير هذا المقدار limit لما واحد زائد X
+
+523
+00:48:14,140 --> 00:48:18,160
+على واحد على X ليش عملت هيك لأن لما X تروح إلى مال
+
+524
+00:48:18,160 --> 00:48:21,360
+نهاية هذا سفر و لما X تروح إلى مال نهاية هذا سفر و
+
+525
+00:48:21,360 --> 00:48:25,340
+هذا لما واحد فبصير سفر على سفر بقدر استخدم ال loop
+
+526
+00:48:25,340 --> 00:48:30,730
+تالذرول اللي انا برهنتهابقى فاضل اللي فوق و فاضل
+
+527
+00:48:30,730 --> 00:48:33,850
+اللي تحت .. فضلنا اللي فوق هيه و فضلنا اللي تحت
+
+528
+00:48:33,850 --> 00:48:37,590
+هيه .. بروح هذا المقدار مع هذا المقدار .. بظل
+
+529
+00:48:37,590 --> 00:48:42,800
+limit1 زي 1 على X مقص 1 لما X تروح لمدة نهاية ده
+
+530
+00:48:42,800 --> 00:48:47,940
+بروح لصفر بطلع جداش 1 لأن 1 بنرد بنعوض وينها لأن
+
+531
+00:48:47,940 --> 00:48:50,940
+ال exponential is a continuous function لذا نصحي
+
+532
+00:48:50,940 --> 00:48:54,180
+عن عند limit ال 1 زي 1 على X لما X تروح لمدة نهاية
+
+533
+00:48:54,180 --> 00:48:57,920
+to the X وعبارة عن E to the limit لأن ال
+
+534
+00:48:57,920 --> 00:49:01,760
+exponential عبارة عن continuous function فبصير E
+
+535
+00:49:01,760 --> 00:49:08,300
+to the 1 ويساوي Eهذا اللي هو بيكون هيك احنا انهينا
+
+536
+00:49:08,300 --> 00:49:16,240
+اللي هي lobitals rules أو قواعد lobital أو صرنا
+
+537
+00:49:16,240 --> 00:49:20,360
+الآن ندخل
+
+538
+00:49:20,360 --> 00:49:28,120
+على اللي هو ال chapter ال section الأخير اللي هو
+
+539
+00:49:28,120 --> 00:49:32,120
+ال section عبارة عن ستة أربعة اللي هو taylor's
+
+540
+00:49:32,120 --> 00:49:38,540
+theoremأو tailors اللي هي .. هنحكي عن tailors
+
+541
+00:49:38,540 --> 00:49:43,640
+polynomial أو tailors أيضا اللي هو approximation
+
+542
+00:49:43,640 --> 00:49:49,820
+كيف اللي هو .. ان مرة جاية ان شاء الله كيف بنقرب
+
+543
+00:49:49,820 --> 00:49:54,840
+بعض الدوالاللي هو المحترمة اللي بتكون
+
+544
+00:49:54,840 --> 00:49:58,220
+differentiable first derivative و second
+
+545
+00:49:58,220 --> 00:50:00,400
+derivative و third derivative لأ عند ال derivative
+
+546
+00:50:00,400 --> 00:50:05,700
+اللي بدنا .. اللي .. اللي هي ما قبل اشتقاق عندها
+
+547
+00:50:05,700 --> 00:50:11,760
+بنقربها ب polynomial و طبعا انتوا عارفين جدش ال
+
+548
+00:50:11,760 --> 00:50:17,380
+polynomial من الدوال السهلةسهلة التعامل سواء في
+
+549
+00:50:17,380 --> 00:50:21,500
+تفاضل أو في تكامل أو حتى لو بدنا نحلها و نجيب
+
+550
+00:50:21,500 --> 00:50:26,340
+جذورها و نجيب كذا في شغل عليها كثير فالناس ترغب في
+
+551
+00:50:26,340 --> 00:50:30,540
+إنها تجيب بعض الدوال اللي بتكون أحيانا معقدة و
+
+552
+00:50:30,540 --> 00:50:35,380
+صعبة التكامل أو خلنا نقول بتغلب شوية في التفاضل أو
+
+553
+00:50:35,380 --> 00:50:40,040
+في وجود الجذور أو كده و نحولها إلىاللي هي
+
+554
+00:50:40,040 --> 00:50:44,020
+polynomial صبعا في مقدار خطأ الخطأ هن .. ان شاء
+
+555
+00:50:44,020 --> 00:50:47,520
+الله ال Taylor's theorem بتقولنا كيف ان هو نوجد
+
+556
+00:50:47,520 --> 00:50:51,320
+هذا الخطأ او ايش هو الخطأ و ايضا Taylor's theorem
+
+557
+00:50:51,320 --> 00:50:55,780
+زي ما انتوا عارفين هنوضح كيف هي عبارة عن تعميم لل
+
+558
+00:50:55,780 --> 00:51:00,340
+main value theorem هي تعميم لل main value theorem
+
+559
+00:51:00,340 --> 00:51:05,140
+في حالة اللي هي n بتساوي سفر بتطلع هي بالظبط ال
+
+560
+00:51:05,140 --> 00:51:09,100
+main value theorem كيف؟
+
+561
+00:51:10,860 --> 00:51:15,100
+الـ Main Value Theorem عبارة الـ Taylor's Theorem
+
+562
+00:51:15,100 --> 00:51:19,900
+عبارة عن تعميم للـ Main Value Theorem خد الآن
+
+563
+00:51:19,900 --> 00:51:25,020
+بتساوي سفر بتطلعلك الـ Main Value Theorem بيشحمها
+
+564
+00:51:25,020 --> 00:51:27,740
+و لحمها و إلى لقاء
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/POarYOfvYB0_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/POarYOfvYB0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0ba49f1ffa886bad41cc2b88ef525e94baba3aaa
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/POarYOfvYB0_postprocess.srt
@@ -0,0 +1,1572 @@
+1
+00:00:04,890 --> 00:00:08,570
+بسم الله الرحمن الرحيم الان نتحدث عن الجزء الثاني
+
+2
+00:00:08,570 --> 00:00:14,690
+من المحاضرة الخامسة لطلبة قسم رياضيات في كلية
+
+3
+00:00:14,690 --> 00:00:18,850
+العلوم بالجامعة الإسلامية بغزا الجزء الثاني طبعا
+
+4
+00:00:18,850 --> 00:00:22,050
+زي ما وعدناكم في البداية قلنا اللي هو هنكمل فيه
+
+5
+00:00:22,050 --> 00:00:25,970
+اللي هو 6 اتنين اللي هي تطبيقات على ال mean value
+
+6
+00:00:25,970 --> 00:00:28,490
+theorem او applications on mean value theorem
+
+7
+00:00:28,490 --> 00:00:32,390
+هيكون في عندنا two applicationsاللي هو الـ
+
+8
+00:00:32,390 --> 00:00:34,790
+Application الأول على الـ Mean Value Theorem هو
+
+9
+00:00:34,790 --> 00:00:38,290
+عبارة عن اللي هو Bernoulli's Inquality، هنبرهن
+
+10
+00:00:38,290 --> 00:00:42,150
+Bernoulli's Inquality، و بعدين هنتحدث عن اللي هو
+
+11
+00:00:42,150 --> 00:00:47,370
+Daraboux's Theorem، تشوف كيف اللي هو بيصير الأمور،
+
+12
+00:00:47,370 --> 00:00:51,960
+نبدأ الآن في الجزء الأول، اللي هوعبارة عن
+
+13
+00:00:51,960 --> 00:00:55,840
+Bernoulli Inquality Bernoulli's Inquality اللي هي
+
+14
+00:00:55,840 --> 00:00:58,700
+بتقولنا لو كانت عندي Alpha عبارة عن real number
+
+15
+00:00:58,700 --> 00:01:02,500
+أكبر من واحد هيكون عندي الواحد زايد X to the Alpha
+
+16
+00:01:02,500 --> 00:01:06,900
+أكبر يساوي واحد زايد Alpha X لكل X أكبر من مين من
+
+17
+00:01:06,900 --> 00:01:10,780
+ناقص واحد with equality if and only if عند X مين
+
+18
+00:01:10,780 --> 00:01:15,140
+عند X بالساوية سفر عندنا ال equality عند X أيش
+
+19
+00:01:15,140 --> 00:01:20,390
+بتساوي بتساوي سفر شوفزي ما عملنا المرة الماضية
+
+20
+00:01:20,390 --> 00:01:26,070
+اللي هو E to the X اللي هي لما أخدناها وثبتنا إنها
+
+21
+00:01:26,070 --> 00:01:29,550
+أكبر أو يساوي اللي هي الواحد زائد Alpha X
+
+22
+00:01:29,550 --> 00:01:33,990
+والمساوية F and W الساوية اللي هو واحد الان see
+
+23
+00:01:33,990 --> 00:01:42,510
+Bernoulli's inequality Bernoulli's inequality اللي
+
+24
+00:01:42,510 --> 00:01:50,040
+هي كما يعني عندى خد Alpha أكبر من واحدف ألف أكبر
+
+25
+00:01:50,040 --> 00:01:55,740
+من واحد then لو واحد زائد X to the ألف أكبر يساوي
+
+26
+00:01:55,740 --> 00:02:01,460
+واحد زائد Alpha X لكل X أكبر من ناقص واحد والشغل
+
+27
+00:02:01,460 --> 00:02:05,980
+التاني ان ال equality ال equality بتحدث if and
+
+28
+00:02:05,980 --> 00:02:13,300
+only if X بتساوي سفر طبعا الان لو X بتساوي سفر
+
+29
+00:02:13,300 --> 00:02:19,180
+automatic automatic بيصير عندى واحد بساوي واحدA لو
+
+30
+00:02:19,180 --> 00:02:22,920
+X بتساوي صفر Conversely هيطلع من خلال البروحان
+
+31
+00:02:22,920 --> 00:02:27,520
+باقي البروحان تنشوف كيف plan approved صلوا على
+
+32
+00:02:27,520 --> 00:02:32,280
+النبي عليه الصلاة والسلام نسجلها الأولى if X
+
+33
+00:02:32,280 --> 00:02:38,780
+بتساوي صفر it is clearthat واحد زياد اكس تقريبا
+
+34
+00:02:38,780 --> 00:02:43,760
+ألفة اللي هي بتساوي واحد بتساوي واحد زياد ألف اكس
+
+35
+00:02:43,760 --> 00:02:50,320
+هذه ده case اكس ايش بتساوي بتساوي سفر طيب الان بدأ
+
+36
+00:02:50,320 --> 00:02:55,230
+اثبت ال inequality هذههأثبتلك في الواقع اللي هو
+
+37
+00:02:55,230 --> 00:02:58,470
+strict inequality أكبر strictly من واحد زاد Alpha
+
+38
+00:02:58,470 --> 00:03:03,190
+X لكل X أكبر من ناقص واحد ولا تساوي سفر في هذه
+
+39
+00:03:03,190 --> 00:03:06,930
+الحالة بيكون A شمالنا أثبتنا لاتجاه الثاني طيب،
+
+40
+00:03:06,930 --> 00:03:14,900
+بتنشوف كيف، الآن خد X for X أكبر من سفرالان احنا
+
+41
+00:03:14,900 --> 00:03:17,460
+بنحكي عن تطبيقات للـ Mean Value Theorem اذا انا
+
+42
+00:03:17,460 --> 00:03:20,260
+متوقع ان انا اطبقها الـ Mean Value Theorem طبعا
+
+43
+00:03:20,260 --> 00:03:24,480
+لاحظ ان الـ 1 زاد X to the Alpha هذه اللي هي
+
+44
+00:03:24,480 --> 00:03:27,640
+الدالة اللي بدي استخدمها اللي بتطبق عليها الـ Mean
+
+45
+00:03:27,640 --> 00:03:31,060
+Value Theorem وهذه اصلا is differentiable
+
+46
+00:03:31,060 --> 00:03:36,360
+everywhere it is defined قابلة الاشتقاق لكل X أكبر
+
+47
+00:03:36,360 --> 00:03:41,560
+مناقص واحد و لكل Alpha أكبر من 100 من 1ماشي الحالة
+
+48
+00:03:41,560 --> 00:03:50,160
+إذا الأن for x أكبر من سفر we apply the mean value
+
+49
+00:03:50,160 --> 00:03:57,400
+theorem on .. مين هو؟ on f of x بسوا 1 زاد x to
+
+50
+00:03:57,400 --> 00:04:03,340
+the alpha ع ال interval .. on the interval مين هي
+
+51
+00:04:03,340 --> 00:04:07,920
+بنتوقع؟ اللي هي سفرو X زي ما عملنا المرة الماضية
+
+52
+00:04:07,920 --> 00:04:12,380
+لأننا مهتم بمين بالمنطقة 0 و X على أساس أن نحصل
+
+53
+00:04:12,380 --> 00:04:18,260
+على هذه ال inequality زي ما هنشوف كيف واضح أه؟ إيش
+
+54
+00:04:18,260 --> 00:04:20,400
+ال mean value of a theorem بتقول إذا كانت F
+
+55
+00:04:20,400 --> 00:04:23,820
+continuous على هذه و differentiable على ال open و
+
+56
+00:04:23,820 --> 00:04:30,940
+هذا كله متحقق إذا then there exists C element in 0
+
+57
+00:04:30,940 --> 00:04:41,560
+X such that اللي هوF prime of C بساوي اللي هو F of
+
+58
+00:04:41,560 --> 00:04:47,840
+X ناقص F of Zero على X minus Zero اللي مضربة وين
+
+59
+00:04:47,840 --> 00:04:53,820
+وكأنه في X على X minus Zero يعني وكأنه هدهش مضربة
+
+60
+00:04:53,820 --> 00:05:00,260
+في X واضح أه؟الان ال F prime للدالة هذه اللي
+
+61
+00:05:00,260 --> 00:05:03,680
+أخدتها ال F prime إلها أكيد كلكم بعرفى كام إلها
+
+62
+00:05:03,680 --> 00:05:08,800
+على جهة ال F prime بتحسبها عند مين عند C فبصير X
+
+63
+00:05:08,800 --> 00:05:15,780
+في ال F prime إلها Alpha في واحد زايد X أس Alpha
+
+64
+00:05:15,780 --> 00:05:20,440
+ناقص واحد لكن بتحسبها عند مين عند C إذا هذه بتكون
+
+65
+00:05:20,440 --> 00:05:27,340
+C بساوياللي هو f of x قداش اللي هي واحد زائد x to
+
+66
+00:05:27,340 --> 00:05:30,700
+the alpha وهي اللي بديها انا f of zero ايش بتساوي
+
+67
+00:05:30,700 --> 00:05:36,400
+f of zero بتساوي واحد ناقص واحد اللي صار عندي اذا
+
+68
+00:05:36,400 --> 00:05:44,260
+اللي هو واحد زائد x to the alpha بتساوي واحد زائد
+
+69
+00:05:44,260 --> 00:05:51,850
+x في alpha في واحد زائد c أس alpha ناقص واحدمعايا
+
+70
+00:05:51,850 --> 00:06:00,130
+يا شباب هذي سميها واحد واضح
+
+71
+00:06:00,130 --> 00:06:06,770
+اه؟ الان عندى ال alpha أكبر من مين؟ من واحد لأن
+
+72
+00:06:06,770 --> 00:06:12,590
+but alpha ناقص واحد أكبر من مين؟ من سفر لأن ال
+
+73
+00:06:12,590 --> 00:06:19,130
+alpha أكبر من واحدوعندي هذه الـ C أشمالها بين الـ
+
+74
+00:06:19,130 --> 00:06:26,650
+0 والـ X يعني أكبر من 0 and C أكبر من 0 إذا صار
+
+75
+00:06:26,650 --> 00:06:37,420
+عند المقدار هذا 1 زائد C نفسه أكبر من 1ولمّا ترفعه
+
+76
+00:06:37,420 --> 00:06:43,080
+لأس موجب، هيظل أكبر من مين؟ و strictly أكبر من
+
+77
+00:06:43,080 --> 00:06:47,660
+واحد، والمساواة بس عند مين، عند الـC بالساوية سفر،
+
+78
+00:06:47,660 --> 00:06:53,730
+واضح أه؟ طيبالأن صار هذا المقدار أكبر من واحد إذا
+
+79
+00:06:53,730 --> 00:07:00,250
+عندي X موجبة و Alpha موجبة مظبوط ولا لأ لأن X هي
+
+80
+00:07:00,250 --> 00:07:05,790
+أكبر من 0 و Alpha أكبر من 1 إذا أضرب الجهتين في
+
+81
+00:07:05,790 --> 00:07:13,510
+Alpha في X بيصير عندي Alpha في X and so Alpha في X
+
+82
+00:07:13,510 --> 00:07:22,710
+بالضرر أكبر من مين Alpha Xفى الـ 1 زي الـ c plus
+
+83
+00:07:22,710 --> 00:07:27,110
+alpha minus 1 بيبقى أكبر من 100 من alpha في x
+
+84
+00:07:27,110 --> 00:07:30,670
+واضحش اللي سويته ضربت alpha x في الجهتين و ال
+
+85
+00:07:30,670 --> 00:07:33,890
+alpha x موجبة ضلت ال inequality زي ما هي طبعا لا
+
+86
+00:07:33,890 --> 00:07:39,370
+تساوي الصفة بالألف ولا ال X إذا من 1 و من 2 we
+
+87
+00:07:39,370 --> 00:07:44,530
+have done from 1
+
+88
+00:07:44,530 --> 00:07:51,270
+and 2من واحد عند اتنين بيطلع عندى اللى هو واحد
+
+89
+00:07:51,270 --> 00:07:57,130
+زياد اكس طول الف اللى هو بساوى المقدار هذا اللى
+
+90
+00:07:57,130 --> 00:08:02,070
+هذا الجزء منه اثبتنا انه اكبر من مين من الف في اكس
+
+91
+00:08:02,070 --> 00:08:13,800
+لذا صار هذا اكبر من واحد زياد الف اكس اي سؤالإذا
+
+92
+00:08:13,800 --> 00:08:18,120
+في حالة ال X أكبر من 0 وجدنا ال 1 زاد X أو أكبر من
+
+93
+00:08:18,120 --> 00:08:25,580
+1 زاد الف X لأن هنعمل نفس الاشي بس في الحالة مين؟
+
+94
+00:08:25,580 --> 00:08:32,400
+في حالة ال X أصغر من 0 وطبعا أكبر من مين؟ من سالب
+
+95
+00:08:32,400 --> 00:08:37,860
+1 إذا بدي أخد ال X أكبر من 0 و أكبر من نقص 1 وأصغر
+
+96
+00:08:37,860 --> 00:08:42,250
+من مين؟ من 0لاحظ لما تكون X أكبر من ناقص واحد أصغر
+
+97
+00:08:42,250 --> 00:08:46,110
+من صفر برضه نطبق الـ mean value theorem على هذه
+
+98
+00:08:46,110 --> 00:08:53,450
+نفس الدالة الان on the interval mean الان X و صفر
+
+99
+00:08:53,450 --> 00:08:57,850
+لأن X هي مالها الصغيرة على ال interval X و صفر لأن
+
+100
+00:08:57,850 --> 00:09:03,110
+X سالبة طيب إذا برضه there exists C element ال
+
+101
+00:09:03,110 --> 00:09:11,000
+mean in X و Zero is نفس البرهان such that Xف F
+
+102
+00:09:11,000 --> 00:09:15,000
+prime of C بيسوي F of X ناقص F of Zero نفس الحاجة
+
+103
+00:09:15,000 --> 00:09:19,140
+بيصير ال X في Alpha في هذا المقدار و هذا المقدار
+
+104
+00:09:19,140 --> 00:09:23,160
+ماتغيرش ولا شيء بيصير واحد زاد X و ده Alpha بيسوي
+
+105
+00:09:23,160 --> 00:09:26,360
+واحد زاد X في Alpha في واحد زاد C أُس Alpha ناقص
+
+106
+00:09:26,360 --> 00:09:31,560
+واحد الآن بيجي الأسباب هنا ال Alpha ناقص واحد
+
+107
+00:09:31,560 --> 00:09:36,840
+أشمله أكبر من سفر زي ما هو لكن ال C الآن اللي
+
+108
+00:09:36,840 --> 00:09:41,110
+لاجيناها وين موجودةفي المنطقة السالبة ده الـ C إيش
+
+109
+00:09:41,110 --> 00:09:47,830
+هتصير معاينتوش يا بابا هتصير الـ C أصغر من مين من
+
+110
+00:09:47,830 --> 00:09:53,990
+صفر سالبة مدام الـ C سالبة إذا واحد زائد C هتصير
+
+111
+00:09:53,990 --> 00:10:01,310
+أصغر من مين من واحد واحد زائد C أصغر من واحد واضح
+
+112
+00:10:01,310 --> 00:10:09,690
+طيب مدام واحد زائد C أصغر من واحد هيكون عندىand so
+
+113
+00:10:09,690 --> 00:10:19,130
+أضرب الجهتين الآن في اللي هي Alpha Alpha Alpha
+
+114
+00:10:19,130 --> 00:10:22,410
+بتظل زي ما هي لما نضرب في Alpha لأن Alpha إيه
+
+115
+00:10:22,410 --> 00:10:26,370
+معناها موجة ده لو ضربت في Alpha هنا و Alpha هنا
+
+116
+00:10:26,370 --> 00:10:31,600
+بتظل زي ما هيلأن بدنا نضرب في مين؟ في X و X
+
+117
+00:10:31,600 --> 00:10:37,740
+أشمالها سالبة فهتنجلي بالإشارة كمان مرة بيصير and
+
+118
+00:10:37,740 --> 00:10:44,220
+so X في Alpha في واحد زائد C أس Alpha ناقص واحد
+
+119
+00:10:44,220 --> 00:10:51,150
+هتصير أكبر من Alpha في X لأن X سالبةإذا صار عندى
+
+120
+00:10:51,150 --> 00:10:56,590
+هذا المقدار أكبر من هذا لأن من واحد ومن اتنين زى
+
+121
+00:10:56,590 --> 00:11:00,650
+ما عملنا قبل بشوية بتصير المقدار هذا اللى هو أكبر
+
+122
+00:11:00,650 --> 00:11:04,750
+من واحد زايد Alpha X strictly لاحظوا أنه في
+
+123
+00:11:04,750 --> 00:11:11,630
+الجهتين سواء X أكبر من سفر أو أصغر من سفر لجينا أن
+
+124
+00:11:11,630 --> 00:11:16,350
+ال inequality هذه متحقق أو strictly متحقق
+
+125
+00:11:19,030 --> 00:11:25,190
+الان المساوى وين صارت؟ عند السفر فالان عشان ناخد
+
+126
+00:11:25,190 --> 00:11:29,090
+بعين الاعتبار الحالة اللي هي x أكبر من ناقص واحد و
+
+127
+00:11:29,090 --> 00:11:33,430
+أصغر من سفر و حالة x أكبر من سفر و حالة x بتساوي
+
+128
+00:11:33,430 --> 00:11:38,870
+سفر بيصير هذا المقدار أكبر أو يساوي لكل x لكل x
+
+129
+00:11:38,870 --> 00:11:43,760
+ماله أكبر من ناقص واحدلأن الاتجاه الثاني من هذا
+
+130
+00:11:43,760 --> 00:11:48,880
+أثبتناه في الواقع إيش هو إنه لو كان فيه equality X
+
+131
+00:11:48,880 --> 00:11:54,280
+بتساوي 0 أو بمعنى آخر X لا تساوي 0 بيعطي إيه
+
+132
+00:11:54,280 --> 00:12:01,720
+أشماله non equality is not equal وهذا حدث كيف حدث
+
+133
+00:12:01,720 --> 00:12:06,240
+أثبتنا إنه لما X أكبر من نقص واحد وأصغر من 0 أو X
+
+134
+00:12:06,240 --> 00:12:10,720
+أكبر من 0 لجينا إنه هذي strictly أكبر يعني بتساوي
+
+135
+00:12:10,720 --> 00:12:15,260
+hashإذا صار عند الـ equality تحدث if and only if X
+
+136
+00:12:15,260 --> 00:12:22,580
+F بساوة سفر أي سؤال؟
+
+137
+00:12:22,580 --> 00:12:30,240
+طيب نيجي الآن نيجي الآن لل intermediate value of
+
+138
+00:12:30,240 --> 00:12:35,960
+property of derivative خلّيني ألقى في عينتها إذا
+
+139
+00:12:35,960 --> 00:12:42,760
+بتتذكروا في ال intermediate value of propertyاللي
+
+140
+00:12:42,760 --> 00:12:46,660
+هو for a continuous function إذا بتذكروا فيه اللي
+
+141
+00:12:46,660 --> 00:12:53,540
+هو real واحد اللي كانت تقول إذا كانت تقول إنه لو
+
+142
+00:12:53,540 --> 00:13:01,880
+كانت F function continuous على a closed open على
+
+143
+00:13:01,880 --> 00:13:07,020
+closed interval من A لعند B continuous على الفترة
+
+144
+00:13:07,020 --> 00:13:16,060
+هذهو إجيت أخدت أي نقطة K بين ال F of A و بين ال F
+
+145
+00:13:16,060 --> 00:13:22,900
+of B إجيت أخدت أي نقطة .. خد مثلا نقطة K بين F of
+
+146
+00:13:22,900 --> 00:13:31,240
+A .. هي F of A و هي F of B .. هي F of A و هي F of
+
+147
+00:13:31,240 --> 00:13:36,540
+B لو إجيت أخدت أي K بينهم هتلاقي .. هو خلي ناخد K
+
+148
+00:13:36,540 --> 00:13:43,830
+هنا مثلاهتلاقي على الأقل نقطة في الفترة A وBبتكون
+
+149
+00:13:43,830 --> 00:13:49,890
+اللي هو صورتها عبارة عن مين؟ عن الـ K يعني بمعنى
+
+150
+00:13:49,890 --> 00:13:55,970
+أخر there exists C element in A و B such that F of
+
+151
+00:13:55,970 --> 00:14:01,050
+C هي مين الـ K و لا أصلا مش هتلاقيها continuous لو
+
+152
+00:14:01,050 --> 00:14:05,530
+مافشلها فئلة و جينا مدينا خطنا هلجيت و مالجيناش
+
+153
+00:14:05,530 --> 00:14:09,290
+ولا واحدة من هذه النقاط اللي هو تقطع الخط معناته
+
+154
+00:14:09,290 --> 00:14:14,410
+أنه صار فيه jump مش هيكون في continuityالان هنا مش
+
+155
+00:14:14,410 --> 00:14:19,890
+في نقطة هي نقطة هي نقطتين هي تلاتة كلهم صورتين هي
+
+156
+00:14:19,890 --> 00:14:27,610
+C1 هي C2 هي C3 كلهم F of C1 K F of C2 K F of C3 K
+
+157
+00:14:27,610 --> 00:14:33,150
+المهم على الأقل في C element in A وB اللي هي بحيث
+
+158
+00:14:33,150 --> 00:14:36,210
+F of C بالساوة K هذا لو كان اتقفز continuous على
+
+159
+00:14:36,210 --> 00:14:40,650
+closed اللي هي bounded interval اللي عندك الآن
+
+160
+00:14:43,230 --> 00:14:47,090
+الحديث بيقول لي في اشي مشابه في حالة ان ال F' is
+
+161
+00:14:47,090 --> 00:14:53,950
+differentiable الان بيقولك لو كانت K بين ال F' لل
+
+162
+00:14:53,950 --> 00:14:58,970
+A و ال F' لل Bبرضه هنلاقيها بتتحقق اللي هو there
+
+163
+00:14:58,970 --> 00:15:03,810
+exists C بحيث ان F prime of C بتساوي اللي هو الكلب
+
+164
+00:15:03,810 --> 00:15:10,530
+نحكي عنها بالرغم انه مش مفترضين احنا في النظرية
+
+165
+00:15:10,530 --> 00:15:14,190
+اللي هو ال continuity ال continuity لل F prime
+
+166
+00:15:16,020 --> 00:15:19,360
+فضبجنا على الـ F' لكن مش .. لأنهم افترضينا على طول
+
+167
+00:15:19,360 --> 00:15:22,300
+تتطبق الـ Intermediate Value Theorem بس على الدالة
+
+168
+00:15:22,300 --> 00:15:26,980
+الـ F' لكن لأ ما هو مخزون في دالة .. خلال الـ F'
+
+169
+00:15:27,380 --> 00:15:33,140
+من معلومات أهلها أنه في حالة F is differentiable
+
+170
+00:15:33,140 --> 00:15:37,940
+على ال interval Iوالـ F ب .. و لجينا .. وفي عندي
+
+171
+00:15:37,940 --> 00:15:43,120
+أخدت أي K بين الـ F' of A و الـ F' of B هتلاقي C
+
+172
+00:15:43,120 --> 00:15:48,380
+في الـ A و الـ B بحيث انه اللي هي F' of C هي هذه
+
+173
+00:15:48,380 --> 00:15:51,920
+الـ K هنشوفها هذه اللي بنسميها Daraboxes theorem
+
+174
+00:15:51,920 --> 00:15:55,980
+قبل ما نروح للـ Daraboxes theorem خلينا نيجي لللمة
+
+175
+00:15:55,980 --> 00:16:00,680
+اللي جابلة اللي هنستخدمها في إثبات الـ Daraboxes
+
+176
+00:16:00,680 --> 00:16:07,340
+theorem الآنالنظرية هذه أو اللمّة هذه اللي هو
+
+177
+00:16:07,340 --> 00:16:10,880
+خلّينا نستذكرها من خلال الرسم ممكن يكون أسهل لكم
+
+178
+00:16:10,880 --> 00:16:16,680
+من خلال الرسم بتقولّي إنه لو كانت عندنا function F
+
+179
+00:16:16,680 --> 00:16:22,300
+is differentiable على اللي هو interval معينة
+
+180
+00:16:22,300 --> 00:16:29,120
+ولجينا عند C معينة هي F primeعند C موجبة F prime
+
+181
+00:16:29,120 --> 00:16:33,940
+عند C موجبة يعني F prime at some C أكبر من 0 يعني
+
+182
+00:16:33,940 --> 00:16:37,160
+مين المماس إيه اللي شماله اللي هو زاوية حادة عامل
+
+183
+00:16:37,160 --> 00:16:43,660
+زاوية حادة ماشي الان بقولي هتلاقي الان بده يبحث عن
+
+184
+00:16:43,660 --> 00:16:49,260
+نقاط تكون أكبر من مين من F of C الان هتلاقي اللي
+
+185
+00:16:49,260 --> 00:16:56,370
+هو هذا C اللي بتحكي عنها وهنااللي هو في جوار حوالي
+
+186
+00:16:56,370 --> 00:16:59,410
+هذين there exist إذا كان f of rank of c أكبر من
+
+187
+00:16:59,410 --> 00:17:03,630
+سفر there exists delta أكبر من سفر such that f of
+
+188
+00:17:03,630 --> 00:17:11,960
+xهنلاقيها أكبر من F of C لكل X وين متوقع أنا هان
+
+189
+00:17:11,960 --> 00:17:17,300
+لإنها الميل هيه إذا في الوين عشان C ل C زائد Delta
+
+190
+00:17:17,300 --> 00:17:19,620
+مش في الفترة اللي جابلة، في الفترة اللي جابلة
+
+191
+00:17:19,620 --> 00:17:23,960
+هتلاقي اللي هي النقطات أشملها أجل لأنا الآن ببحث
+
+192
+00:17:23,960 --> 00:17:29,420
+.. ببحث لم تبعت هيك بدهااللي هو there exist دلتا
+
+193
+00:17:29,420 --> 00:17:32,120
+أكبر من صفر ساشداد F of X أكبر من F of C لكل X
+
+194
+00:17:32,120 --> 00:17:40,200
+والي موجود من الـ C للـ C زائد دلتا معايا؟ الآن
+
+195
+00:17:40,200 --> 00:17:47,320
+بقولك لو لجينا F prime للـ C عند نقطة ما F prime
+
+196
+00:17:47,320 --> 00:17:54,130
+للـ C هي الـ C التانية لو لجينا F prime للـ Cأصغر
+
+197
+00:17:54,130 --> 00:17:57,210
+من صفر عشان .. عشان ما تحفظهاش حفظ عشان لو انسيتها
+
+198
+00:17:57,210 --> 00:18:00,890
+تستنتجها لحالك لو كان F' للـC أصغر من صفر يعني ميل
+
+199
+00:18:00,890 --> 00:18:06,710
+المماس سلب يعني زاوية منفرجة الآن هتلاقي Delta
+
+200
+00:18:06,710 --> 00:18:13,360
+أكبر من صفر يعني تمثل جواربحيث أن F of X برضه أكبر
+
+201
+00:18:13,360 --> 00:18:19,000
+من F of C لكل X وين موجودة هي وارحة منها لكل X
+
+202
+00:18:19,000 --> 00:18:24,340
+موجودة وين جابلها من C نقص دلتة لعند مين؟ لعند C
+
+203
+00:18:24,340 --> 00:18:29,800
+من C نقص دلتة لعند C هذه بس اللممة للاستذكار، إيش
+
+204
+00:18:29,800 --> 00:18:32,780
+بتقول اللممة؟ تعالى نشوف بالظبط اللي بتقوله اللي
+
+205
+00:18:32,780 --> 00:18:36,220
+حكيته أنا بقول اللممة let I subset من R be an
+
+206
+00:18:36,220 --> 00:18:40,260
+intervaland let F من I ل R و let C element in I
+
+207
+00:18:40,260 --> 00:18:44,700
+and assume that F has a derivative at C نفترض ان
+
+208
+00:18:44,700 --> 00:18:50,120
+ال F الهياشق مشتقة عند النقطة C مقولي إذا F prime
+
+209
+00:18:50,120 --> 00:18:54,200
+أكبر من 0 عند ال C هاي اللي قلته هان إذا نيوجد
+
+210
+00:18:54,200 --> 00:18:58,500
+دلتة أكبر من 0 بحيث أن F of X أكبر من F of C لكل
+
+211
+00:18:58,500 --> 00:19:02,560
+ال X وين موجودة بين ال C و C زائد دلتة اللي هي
+
+212
+00:19:02,560 --> 00:19:03,280
+الجبل هنا
+
+213
+00:19:06,510 --> 00:19:10,590
+الثانية، إذا كانت F' أصغر من 0، حلاقي Delta بحيث
+
+214
+00:19:10,590 --> 00:19:15,430
+أن F of C أصغر واحدة في الجوار، يعني F of C أو في
+
+215
+00:19:15,430 --> 00:19:19,010
+الجزء من الجوار، هذا الجزء اللي لجاي، F of C أصغر
+
+216
+00:19:19,010 --> 00:19:24,370
+من F of X، وين هاي الـ F of X هنا؟ أكبر من F of C،
+
+217
+00:19:24,370 --> 00:19:29,870
+واضح؟ لكل X وين موجودة في الـ C minus Delta والـ C
+
+218
+00:19:31,220 --> 00:19:36,360
+أي سؤال البرهن هو A خلّينا نبرهن بيه نحنا بيه
+
+219
+00:19:36,360 --> 00:19:40,980
+خلّينا نبرهن مع بعض بيه
+
+220
+00:19:40,980 --> 00:19:46,840
+الان
+
+221
+00:19:46,840 --> 00:19:53,820
+بدنا نقول نفترض ان F برايم بيه
+
+222
+00:19:53,820 --> 00:20:02,190
+approved الان ماعطينا F برايم of Cأصغر من سفر،
+
+223
+00:20:02,190 --> 00:20:11,750
+مظبوط؟ لأن بتقول since f prime of c أصغر من سفر
+
+224
+00:20:11,750 --> 00:20:21,070
+then limit f of x ناقص f of c على x minus c لما x
+
+225
+00:20:21,070 --> 00:20:27,540
+تروح لل c إشماله أصغر من سفرماشي، مدام أصغر من سفر
+
+226
+00:20:27,540 --> 00:20:31,080
+عند .. قلنا قبل هيك في نظرية إذا كان اللي هو
+
+227
+00:20:31,080 --> 00:20:35,600
+المقدار هذا أصغر من سفر إذا بقدر ألاقي جوار
+
+228
+00:20:35,600 --> 00:20:41,240
+حواليها بتكون القيم كلها إشمالها أصغر من سفر إذا
+
+229
+00:20:41,240 --> 00:20:48,560
+then there exist V Delta of C اللي هو جوار جوار من
+
+230
+00:20:48,560 --> 00:20:55,260
+عند C minus Delta لعند C زائد Deltaماشي؟ اللي هو
+
+231
+00:20:55,260 --> 00:20:58,980
+احنا أخدناها من هذا المنطقة من C نقص دلتا ل C زائد
+
+232
+00:20:58,980 --> 00:21:06,700
+دلتا such that اللي هو F of X نقص F of C على X
+
+233
+00:21:06,700 --> 00:21:16,220
+minus C أصغر من سفر لكل X وين في V دلتا of C أنا
+
+234
+00:21:16,220 --> 00:21:22,300
+غرضي أني أخلي اللي هو أوجد mean أن F of X أكبر من
+
+235
+00:21:22,300 --> 00:21:25,900
+100من F of C بدا وجد المنطقة اللي بتكون فيها F of
+
+236
+00:21:25,900 --> 00:21:31,880
+X أكبر من مين من F of C لاحظ انه هذا بديها وكأن F
+
+237
+00:21:31,880 --> 00:21:36,480
+of X ناقص F of C بديها أكبر من 0 صح عشان أحصل F of
+
+238
+00:21:36,480 --> 00:21:40,420
+X أكبر من مين من F of C لذن عشان أحصل على هذا
+
+239
+00:21:40,420 --> 00:21:45,000
+المقدار أكبر من 0 وانا بعرف ان هذا كله أصغر من 0
+
+240
+00:21:45,000 --> 00:21:48,880
+بدي المنطقة هذه تكون أصغر من 0 وين هتكون أصغر من
+
+241
+00:21:48,880 --> 00:21:57,510
+0؟ في المنطقة هذه، ماشي الحال؟الان but for every x
+
+242
+00:21:57,510 --> 00:22:05,650
+element in c نقص delta و c ال x minus c أصغر من
+
+243
+00:22:05,650 --> 00:22:15,800
+سفر واضحة؟ إذن الآن من هذه و من هذه henceأف of X
+
+244
+00:22:15,800 --> 00:22:23,300
+ناقص أف of C اللي هي بتساوي X minus C في أف of X
+
+245
+00:22:23,300 --> 00:22:30,300
+ناقص أف of C على X minus C هذا سالب وهذا سالب لذا
+
+246
+00:22:30,300 --> 00:22:35,260
+حصل ضرب بين أشماله هكون أكبر من سفر سالب في سالب
+
+247
+00:22:35,260 --> 00:22:42,720
+موجه مظبوط لهذا الكلام لكل X وين موجودة في الـC
+
+248
+00:22:43,730 --> 00:22:52,150
+-Delta OC وهذا معنى أن F of X أكبر من F of C لنقطة
+
+249
+00:22:52,150 --> 00:22:59,950
+المذكورة لنقات المذكورة واضح؟ أي سؤال؟ نيجي الآن
+
+250
+00:22:59,950 --> 00:23:04,930
+اللي هو انبرهن الـDrabowski's theorem اللي حكينا
+
+251
+00:23:04,930 --> 00:23:08,350
+عنها وقدمنا لها قبل بشوية إيش الـDrabowski's
+
+252
+00:23:08,350 --> 00:23:15,010
+theorem بتقول أنه عندي ال Fهو مستخدم للتحدي في بعض
+
+253
+00:23:15,010 --> 00:23:19,250
+الانترالين
+
+254
+00:23:19,250 --> 00:23:19,950
+A وB
+
+255
+00:23:23,800 --> 00:23:29,540
+اللي هي أي K بين F' و F'
+
+256
+00:23:32,040 --> 00:23:36,580
+of A و F' of B إذا نحن نلاقي الـ C بقيت أن F' of C
+
+257
+00:23:36,580 --> 00:23:42,020
+هي الـ K كمان مرة Drabowski's theorem بتقول if F
+
+258
+00:23:42,020 --> 00:23:47,520
+is differentiable on A و B و if K is any number
+
+259
+00:23:47,520 --> 00:23:53,610
+between F' of A و F' of Bthen there exists at
+
+260
+00:23:53,610 --> 00:23:56,350
+least one point C في الـA والـB بحيث أن الـAlpha
+
+261
+00:23:56,350 --> 00:24:02,650
+Prime of C أيش بتساوي؟ بتساوي K ماشي يا شباب طيب
+
+262
+00:24:02,650 --> 00:24:06,890
+اللي زي ما علقنا قبل بشوية بالرغم إنه إحنا ماحدش
+
+263
+00:24:06,890 --> 00:24:11,810
+جاب سيرة إن الـAlpha Prime هتكون continuousلأ،
+
+264
+00:24:11,810 --> 00:24:16,310
+قلنا ما في الـ F' من مخزون، derivative من مخزون في
+
+265
+00:24:16,310 --> 00:24:21,530
+داخلنا المعلومات أهلها إنها تقدر تصل أنه لأي K بين
+
+266
+00:24:21,530 --> 00:24:27,190
+الـ F A و الـ F' A و الـ F' B بالـ Legacy، بحيث أن
+
+267
+00:24:27,190 --> 00:24:31,170
+F' اللي عند الـ C هي الـ K. إيه المرحلة دلوقتي؟
+
+268
+00:24:31,170 --> 00:24:41,150
+شوف، نفترض أنه Proofنفترض أن G of A أصغر من K أصغر
+
+269
+00:24:41,150 --> 00:24:52,230
+من مين من G أو F prime of A أصغر من F prime of B
+
+270
+00:24:52,230 --> 00:24:56,450
+ماشي الحال واضح
+
+271
+00:24:56,450 --> 00:25:02,430
+هذا فرضناهالآن بدنا نلاقي C بحيث انه F' للـ C هو
+
+272
+00:25:02,430 --> 00:25:06,530
+الـ K الان define هالجيه تعرف ليش اعرفت defining
+
+273
+00:25:06,530 --> 00:25:15,590
+function G of X بساوي K في ال X ناقص F of X عندي
+
+274
+00:25:15,590 --> 00:25:18,790
+اللي هي هذه الدالة هي الدالة اللي هتوصلني للي
+
+275
+00:25:18,790 --> 00:25:23,800
+بديهاالان واضح ان الـ g of x بيساوي kx ناقص f of x
+
+276
+00:25:23,800 --> 00:25:29,660
+انها differentiable اللي هو هذه وين ما كان و هذي
+
+277
+00:25:29,660 --> 00:25:32,080
+differentiable على ال I إذا صار ان التنتين
+
+278
+00:25:32,080 --> 00:25:35,960
+differentiable على مين؟ على ال A و ال B و طبيعي
+
+279
+00:25:35,960 --> 00:25:40,440
+continuous على ال A و ال B إذا بما أن هذا الدالة
+
+280
+00:25:40,440 --> 00:25:44,440
+is continuous على closed bounded interval then it
+
+281
+00:25:44,440 --> 00:25:47,580
+attains its maximum and its minimum on this
+
+282
+00:25:47,580 --> 00:25:51,870
+intervalبتنه نشتغل عليه ال maximum و لو حد بده
+
+283
+00:25:51,870 --> 00:25:58,670
+يشتغل عليه المنه مابنفع برضه الان then g of x
+
+284
+00:25:58,670 --> 00:26:09,810
+attains its maximum on mean on a و bلأن إيش اللي
+
+285
+00:26:09,810 --> 00:26:15,370
+بيدثبتلك هي بيدثبتلك إن الـ G هيكون ال maximum
+
+286
+00:26:15,370 --> 00:26:21,150
+إلها لا عند ال A ولا عند ال B إذا هتكون وين؟ في
+
+287
+00:26:21,150 --> 00:26:25,870
+نقطة داخلية مدام في نقطة داخلية وإحنا عارفين إن
+
+288
+00:26:25,870 --> 00:26:29,490
+الـ G differentiable على كل المنطقة هذه إذا صارت
+
+289
+00:26:29,490 --> 00:26:33,050
+غصب عني إنه لازم مدام عندها maximum و interior
+
+290
+00:26:33,050 --> 00:26:38,390
+point إذا الـ G prime إيش هتساوي؟ هتساوي 0واضح؟
+
+291
+00:26:38,390 --> 00:26:42,590
+مدام جي برايم ما هتساوي سفر هتصير ال F prime of X
+
+292
+00:26:42,590 --> 00:26:46,550
+ال F of some C بتساوي ال K و بيكون خلصنا البرهان
+
+293
+00:26:46,550 --> 00:26:49,950
+تشوفيش اللي بقوله طيب صلوا على النبي عليه الصلاة
+
+294
+00:26:49,950 --> 00:27:05,390
+والسلام الآن عندي ال G prime of X هتساوي K ناقص F
+
+295
+00:27:05,390 --> 00:27:11,520
+prime of مين؟of X صح ولا لأ؟ طيب الآن لو جينا
+
+296
+00:27:11,520 --> 00:27:21,540
+حسبنا الـ G' عند ال A إيش هتلاقيها؟ K ناقص F' عند
+
+297
+00:27:21,540 --> 00:27:28,740
+مين؟ عند ال A صح ولا لأ؟ K ناقص F' عند ال A هتكون
+
+298
+00:27:28,740 --> 00:27:35,260
+أكبر من مين؟ من 0 واضح مدام ال G' على ال A أكبر من
+
+299
+00:27:35,260 --> 00:27:35,520
+0
+
+300
+00:27:39,110 --> 00:27:49,430
+K-F' أكبر من سفر بما أن J' بسوء K-F' أكبر من سفر
+
+301
+00:27:49,430 --> 00:27:53,450
+يعني هذا أكبر من سفر حسب اللي قبل قليل حكينا عنها
+
+302
+00:27:53,450 --> 00:28:06,330
+ثم هناك دلت أكبر من سفر such that G' of X أكبر من
+
+303
+00:28:06,330 --> 00:28:13,400
+G of Aاللي كنا بسمينا C فاهمين عليها الـ G of X
+
+304
+00:28:13,400 --> 00:28:22,800
+أكبر من G of A for every X element inمتذكرين اللي
+
+305
+00:28:22,800 --> 00:28:27,480
+قبل بشوية لما كانت جي برايم لما كانت الجي برايم
+
+306
+00:28:27,480 --> 00:28:32,860
+أكبر من سفر كانت اللي هي وين لجاي X element لغاية
+
+307
+00:28:32,860 --> 00:28:40,920
+أسف X element in C اللي هي A هنا وA زائد دلتا اللي
+
+308
+00:28:40,920 --> 00:28:45,240
+ما اللي قبل بشوية إذا صارت عند جي of X أكبر من 100
+
+309
+00:28:46,010 --> 00:28:50,890
+من الـ G of A واضح لمين؟ لكل الإكسات اللي هي في نص
+
+310
+00:28:50,890 --> 00:28:57,290
+الجوار هذا صارت عندي اللي هي الـ G of A مش maximum
+
+311
+00:28:57,290 --> 00:29:08,650
+لأن في قيم أكبر منها واضح؟ إذا أكيد then G does
+
+312
+00:29:08,650 --> 00:29:18,150
+not have a maximum at Aإذا الفش مش maximum عند
+
+313
+00:29:18,150 --> 00:29:23,990
+مين؟ عند الـ A أي
+
+314
+00:29:23,990 --> 00:29:27,750
+سؤال؟
+
+315
+00:29:27,750 --> 00:29:36,990
+Similarly .. Similarly G' عند ال B معايا يا شباب
+
+316
+00:29:36,990 --> 00:29:44,340
+G' عند ال B بساوي K نقص F' عند ال Bو K ناقص F
+
+317
+00:29:44,340 --> 00:29:49,840
+prime عند الـ B أصغر من مين؟ أصغر من 0 إذا بنفس
+
+318
+00:29:49,840 --> 00:29:54,600
+اللمّة بس على الجهة الثانية then there exists
+
+319
+00:29:54,600 --> 00:30:01,000
+Delta أكبر من 0 such that G of X أكبر من G of B
+
+320
+00:30:01,000 --> 00:30:11,520
+لكل X وان الآن من عند B ناقص Delta لعند الـ B صح؟
+
+321
+00:30:12,880 --> 00:30:19,600
+وهنا end points هذول فمش هتكون برضه مين ال b مش
+
+322
+00:30:19,600 --> 00:30:24,940
+هتكون عندها maximum لأنه منها ولا جوا ال b end
+
+323
+00:30:24,940 --> 00:30:28,920
+point منها ولا جوا ال g of x أكبر من ال g of b إذن
+
+324
+00:30:28,920 --> 00:30:38,160
+مش maximum إذن hence g does not have a maximum at
+
+325
+00:30:38,160 --> 00:30:45,380
+b from هناو from هنا from here and here we have g
+
+326
+00:30:45,380 --> 00:30:48,700
+has
+
+327
+00:30:48,700 --> 00:31:00,180
+the maximum inside a و b، مظبوط؟ يعني بمعنى آخر،
+
+328
+00:31:00,180 --> 00:31:06,100
+مدام احنا أكيد attained its maximum there exists c
+
+329
+00:31:06,100 --> 00:31:13,970
+element in a و b such thatجي هز ا maximum مدام جي
+
+330
+00:31:13,970 --> 00:31:18,770
+هز ا maximum و interior point اذا جي برايم عند ال
+
+331
+00:31:18,770 --> 00:31:23,390
+C ايش بتساوي؟ بتساوي سفر مدام جي برايم عند ال C
+
+332
+00:31:23,390 --> 00:31:28,090
+بتساوي سفر اذا صار عندي hence
+
+333
+00:31:30,200 --> 00:31:33,960
+وهنا اكتب جي برايم عند الـ C ايش بيساوي؟ بيساوي
+
+334
+00:31:33,960 --> 00:31:39,120
+سفر ايش جي برايم بيساوي؟ K ناقص F برايم اذا that
+
+335
+00:31:39,120 --> 00:31:43,020
+is K
+
+336
+00:31:43,020 --> 00:31:50,510
+ان هي بدل جي برايمنقص F' عند C بيساوي سفر أو بمعنى
+
+337
+00:31:50,510 --> 00:31:57,090
+آخر F' عند C أش بيساوي بيساوي K فعلا لقينا C في ال
+
+338
+00:31:57,090 --> 00:32:02,770
+A و ال B بحيث أن F' عند C أش بتساوي بيساوي K كما
+
+339
+00:32:02,770 --> 00:32:13,170
+هو مطلوب هيها أي سؤال الآن
+
+340
+00:32:14,940 --> 00:32:21,600
+عندي المثال الأخير بقول لي use Daraboux's theorem
+
+341
+00:32:21,600 --> 00:32:25,120
+هو في الواقع مثال على Daraboux's theorem بقول لي
+
+342
+00:32:25,120 --> 00:32:32,060
+the function g من
+
+343
+00:32:32,060 --> 00:32:36,080
+ناطس واحد وواحد لعند ال R defined by معرفه g of x
+
+344
+00:32:36,080 --> 00:32:39,320
+أيش بالساوية واحد إذا كانت x أكبر من صفر وأصغر
+
+345
+00:32:39,320 --> 00:32:43,680
+بسوا واحد0 إذا كانت X بيساوي واحد و ناقص واحد إذا
+
+346
+00:32:43,680 --> 00:32:46,180
+كانت X أكبر من ناقص واكبر بيساوي ناقص واحد و
+
+347
+00:32:46,180 --> 00:32:52,360
+بيساوي سفر بقول لي أثبت أن هذه الدالة is not the
+
+348
+00:32:52,360 --> 00:32:57,160
+derivative is not a derivative of any function
+
+349
+00:32:57,160 --> 00:33:01,580
+يعني مافيش function في الدنيا بتكون مشتقتها هي هذه
+
+350
+00:33:01,580 --> 00:33:06,420
+الـG هذه الـG is not a derivative of any function
+
+351
+00:33:07,400 --> 00:33:10,800
+بدا اقول suppose not ونصلى contradiction ماشي
+
+352
+00:33:10,800 --> 00:33:15,440
+الحال اذا الان ال
+
+353
+00:33:15,440 --> 00:33:27,280
+.. النظرية .. ال .. البثال بقول mainly المثال
+
+354
+00:33:27,280 --> 00:33:35,540
+بقول example example show that
+
+355
+00:33:38,010 --> 00:33:47,490
+G من نقص واحد لعند الواحد لعند ال R Defined by G
+
+356
+00:33:47,490 --> 00:33:53,610
+of X بساوي واحد إذا كانت X أكبر من سفر وأصغر يسوء
+
+357
+00:33:53,610 --> 00:33:59,290
+واحد سفر X بتساوي سفر نقص واحد X أكبر يسوء نقص
+
+358
+00:33:59,290 --> 00:34:08,330
+واحد وأصغر من سفر is not a derivativeof any
+
+359
+00:34:08,330 --> 00:34:20,610
+function on ناقص واحد واحد solution او proof اللي
+
+360
+00:34:20,610 --> 00:34:25,930
+قاعدك تقول suppose that
+
+361
+00:34:25,930 --> 00:34:36,550
+there exist f من ناقص واحد واحد لعند ال R suchthat
+
+362
+00:34:36,550 --> 00:34:46,630
+f prime of x بساوي g of x لكل x element ناقص واحد
+
+363
+00:34:46,630 --> 00:34:51,870
+و واحد ماشي الحال ايش هذه الدالة هي في الواقع
+
+364
+00:34:51,870 --> 00:34:59,310
+الدالة هي ال g of x هذه عندي سفر و انا عند الواحد
+
+365
+00:34:59,310 --> 00:35:04,550
+عند الواحد و انا سلب واحد
+
+366
+00:35:07,910 --> 00:35:16,170
+لعند اللي هو سنة واحد هيدّه ده اه لأن افترض انه F
+
+367
+00:35:16,170 --> 00:35:19,130
+بين نقص واحد واحد such that F prime of X بيساوي G
+
+368
+00:35:19,130 --> 00:35:22,650
+of X واضح انه F is different مدهن اذا قلت F prime
+
+369
+00:35:22,650 --> 00:35:25,390
+of X بيساوي G of X لكل X اللي منتقل نقص واحد واحد
+
+370
+00:35:25,390 --> 00:35:30,070
+صارت ال F ايش ما لأ is differentiable مظبوط ولا لأ
+
+371
+00:35:30,070 --> 00:35:39,050
+is differentiable واتنين لأن butالنص ينتبه إلى
+
+372
+00:35:39,050 --> 00:35:43,850
+الفترة ناقص واحد وواحد اللي هي إيش بتساوي اللي هي
+
+373
+00:35:43,850 --> 00:35:50,590
+عبارة عن الفترة هذه عبارة عن اللي هي F prime of
+
+374
+00:35:50,590 --> 00:36:03,650
+ناقص واحد و F prime of واحد باضح
+
+375
+00:36:03,650 --> 00:36:13,290
+أه؟هذه F' هي مين؟ G صح ولا لا؟ فمين؟ وهذه ايه؟ F'
+
+376
+00:36:13,550 --> 00:36:21,810
+G لأن G of واحد واحد، G of ناقص واحد ونص في الفترة
+
+377
+00:36:21,810 --> 00:36:22,170
+هذه
+
+378
+00:36:25,820 --> 00:36:30,060
+أصلا كل قيم الـ F' أو الـ D هي واحد صفر وناقص واحد
+
+379
+00:36:30,060 --> 00:36:34,360
+هذا هيسببلك الخلال مش نص بس تقدر تختار ربع تلت خمس
+
+380
+00:36:34,360 --> 00:36:38,680
+أي حاجة غير الـ Zero و الواحد والسلب واحد ايش ال
+
+381
+00:36:38,680 --> 00:36:49,400
+.. ايش الشد في الأمر but نص element كده and D
+
+382
+00:36:49,400 --> 00:37:00,740
+F is differentiable on ناقص واحد واحدصح؟ then by
+
+383
+00:37:00,740 --> 00:37:09,800
+Darabaux's theorem there exist c element ناقص واحد
+
+384
+00:37:09,800 --> 00:37:19,200
+وواحد such that اللي هو g of c او f prime of c
+
+385
+00:37:19,200 --> 00:37:28,990
+بيساوي ايش نص صح؟Ie بمعنى آخر F prime of C اللي هو
+
+386
+00:37:28,990 --> 00:37:39,590
+G of C بساوة نص Which is impossible Therefore
+
+387
+00:37:39,590 --> 00:37:46,480
+our first assumption is not truethat is there is
+
+388
+00:37:46,480 --> 00:37:50,740
+no F من ناقص واحد لعند واحد .. من واحد ناقص واحد
+
+389
+00:37:50,740 --> 00:37:54,960
+لواحد لعند ال R بحيث أن F' بيساوي G أو بمعنى أخر G
+
+390
+00:37:54,960 --> 00:38:03,680
+is not a derivative of any function وهكذا
+
+391
+00:38:03,680 --> 00:38:07,780
+نكون خلصنا ستة اتنين والمرة اللي جاي ان شاء الله
+
+392
+00:38:07,780 --> 00:38:13,980
+بنكمل في ستة تلاتة ويوم الأحد القادم بنناقش ستة
+
+393
+00:38:13,980 --> 00:38:16,010
+اتنينأسئلة 6
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/POarYOfvYB0_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/POarYOfvYB0_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0ba49f1ffa886bad41cc2b88ef525e94baba3aaa
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/POarYOfvYB0_raw.srt
@@ -0,0 +1,1572 @@
+1
+00:00:04,890 --> 00:00:08,570
+بسم الله الرحمن الرحيم الان نتحدث عن الجزء الثاني
+
+2
+00:00:08,570 --> 00:00:14,690
+من المحاضرة الخامسة لطلبة قسم رياضيات في كلية
+
+3
+00:00:14,690 --> 00:00:18,850
+العلوم بالجامعة الإسلامية بغزا الجزء الثاني طبعا
+
+4
+00:00:18,850 --> 00:00:22,050
+زي ما وعدناكم في البداية قلنا اللي هو هنكمل فيه
+
+5
+00:00:22,050 --> 00:00:25,970
+اللي هو 6 اتنين اللي هي تطبيقات على ال mean value
+
+6
+00:00:25,970 --> 00:00:28,490
+theorem او applications on mean value theorem
+
+7
+00:00:28,490 --> 00:00:32,390
+هيكون في عندنا two applicationsاللي هو الـ
+
+8
+00:00:32,390 --> 00:00:34,790
+Application الأول على الـ Mean Value Theorem هو
+
+9
+00:00:34,790 --> 00:00:38,290
+عبارة عن اللي هو Bernoulli's Inquality، هنبرهن
+
+10
+00:00:38,290 --> 00:00:42,150
+Bernoulli's Inquality، و بعدين هنتحدث عن اللي هو
+
+11
+00:00:42,150 --> 00:00:47,370
+Daraboux's Theorem، تشوف كيف اللي هو بيصير الأمور،
+
+12
+00:00:47,370 --> 00:00:51,960
+نبدأ الآن في الجزء الأول، اللي هوعبارة عن
+
+13
+00:00:51,960 --> 00:00:55,840
+Bernoulli Inquality Bernoulli's Inquality اللي هي
+
+14
+00:00:55,840 --> 00:00:58,700
+بتقولنا لو كانت عندي Alpha عبارة عن real number
+
+15
+00:00:58,700 --> 00:01:02,500
+أكبر من واحد هيكون عندي الواحد زايد X to the Alpha
+
+16
+00:01:02,500 --> 00:01:06,900
+أكبر يساوي واحد زايد Alpha X لكل X أكبر من مين من
+
+17
+00:01:06,900 --> 00:01:10,780
+ناقص واحد with equality if and only if عند X مين
+
+18
+00:01:10,780 --> 00:01:15,140
+عند X بالساوية سفر عندنا ال equality عند X أيش
+
+19
+00:01:15,140 --> 00:01:20,390
+بتساوي بتساوي سفر شوفزي ما عملنا المرة الماضية
+
+20
+00:01:20,390 --> 00:01:26,070
+اللي هو E to the X اللي هي لما أخدناها وثبتنا إنها
+
+21
+00:01:26,070 --> 00:01:29,550
+أكبر أو يساوي اللي هي الواحد زائد Alpha X
+
+22
+00:01:29,550 --> 00:01:33,990
+والمساوية F and W الساوية اللي هو واحد الان see
+
+23
+00:01:33,990 --> 00:01:42,510
+Bernoulli's inequality Bernoulli's inequality اللي
+
+24
+00:01:42,510 --> 00:01:50,040
+هي كما يعني عندى خد Alpha أكبر من واحدف ألف أكبر
+
+25
+00:01:50,040 --> 00:01:55,740
+من واحد then لو واحد زائد X to the ألف أكبر يساوي
+
+26
+00:01:55,740 --> 00:02:01,460
+واحد زائد Alpha X لكل X أكبر من ناقص واحد والشغل
+
+27
+00:02:01,460 --> 00:02:05,980
+التاني ان ال equality ال equality بتحدث if and
+
+28
+00:02:05,980 --> 00:02:13,300
+only if X بتساوي سفر طبعا الان لو X بتساوي سفر
+
+29
+00:02:13,300 --> 00:02:19,180
+automatic automatic بيصير عندى واحد بساوي واحدA لو
+
+30
+00:02:19,180 --> 00:02:22,920
+X بتساوي صفر Conversely هيطلع من خلال البروحان
+
+31
+00:02:22,920 --> 00:02:27,520
+باقي البروحان تنشوف كيف plan approved صلوا على
+
+32
+00:02:27,520 --> 00:02:32,280
+النبي عليه الصلاة والسلام نسجلها الأولى if X
+
+33
+00:02:32,280 --> 00:02:38,780
+بتساوي صفر it is clearthat واحد زياد اكس تقريبا
+
+34
+00:02:38,780 --> 00:02:43,760
+ألفة اللي هي بتساوي واحد بتساوي واحد زياد ألف اكس
+
+35
+00:02:43,760 --> 00:02:50,320
+هذه ده case اكس ايش بتساوي بتساوي سفر طيب الان بدأ
+
+36
+00:02:50,320 --> 00:02:55,230
+اثبت ال inequality هذههأثبتلك في الواقع اللي هو
+
+37
+00:02:55,230 --> 00:02:58,470
+strict inequality أكبر strictly من واحد زاد Alpha
+
+38
+00:02:58,470 --> 00:03:03,190
+X لكل X أكبر من ناقص واحد ولا تساوي سفر في هذه
+
+39
+00:03:03,190 --> 00:03:06,930
+الحالة بيكون A شمالنا أثبتنا لاتجاه الثاني طيب،
+
+40
+00:03:06,930 --> 00:03:14,900
+بتنشوف كيف، الآن خد X for X أكبر من سفرالان احنا
+
+41
+00:03:14,900 --> 00:03:17,460
+بنحكي عن تطبيقات للـ Mean Value Theorem اذا انا
+
+42
+00:03:17,460 --> 00:03:20,260
+متوقع ان انا اطبقها الـ Mean Value Theorem طبعا
+
+43
+00:03:20,260 --> 00:03:24,480
+لاحظ ان الـ 1 زاد X to the Alpha هذه اللي هي
+
+44
+00:03:24,480 --> 00:03:27,640
+الدالة اللي بدي استخدمها اللي بتطبق عليها الـ Mean
+
+45
+00:03:27,640 --> 00:03:31,060
+Value Theorem وهذه اصلا is differentiable
+
+46
+00:03:31,060 --> 00:03:36,360
+everywhere it is defined قابلة الاشتقاق لكل X أكبر
+
+47
+00:03:36,360 --> 00:03:41,560
+مناقص واحد و لكل Alpha أكبر من 100 من 1ماشي الحالة
+
+48
+00:03:41,560 --> 00:03:50,160
+إذا الأن for x أكبر من سفر we apply the mean value
+
+49
+00:03:50,160 --> 00:03:57,400
+theorem on .. مين هو؟ on f of x بسوا 1 زاد x to
+
+50
+00:03:57,400 --> 00:04:03,340
+the alpha ع ال interval .. on the interval مين هي
+
+51
+00:04:03,340 --> 00:04:07,920
+بنتوقع؟ اللي هي سفرو X زي ما عملنا المرة الماضية
+
+52
+00:04:07,920 --> 00:04:12,380
+لأننا مهتم بمين بالمنطقة 0 و X على أساس أن نحصل
+
+53
+00:04:12,380 --> 00:04:18,260
+على هذه ال inequality زي ما هنشوف كيف واضح أه؟ إيش
+
+54
+00:04:18,260 --> 00:04:20,400
+ال mean value of a theorem بتقول إذا كانت F
+
+55
+00:04:20,400 --> 00:04:23,820
+continuous على هذه و differentiable على ال open و
+
+56
+00:04:23,820 --> 00:04:30,940
+هذا كله متحقق إذا then there exists C element in 0
+
+57
+00:04:30,940 --> 00:04:41,560
+X such that اللي هوF prime of C بساوي اللي هو F of
+
+58
+00:04:41,560 --> 00:04:47,840
+X ناقص F of Zero على X minus Zero اللي مضربة وين
+
+59
+00:04:47,840 --> 00:04:53,820
+وكأنه في X على X minus Zero يعني وكأنه هدهش مضربة
+
+60
+00:04:53,820 --> 00:05:00,260
+في X واضح أه؟الان ال F prime للدالة هذه اللي
+
+61
+00:05:00,260 --> 00:05:03,680
+أخدتها ال F prime إلها أكيد كلكم بعرفى كام إلها
+
+62
+00:05:03,680 --> 00:05:08,800
+على جهة ال F prime بتحسبها عند مين عند C فبصير X
+
+63
+00:05:08,800 --> 00:05:15,780
+في ال F prime إلها Alpha في واحد زايد X أس Alpha
+
+64
+00:05:15,780 --> 00:05:20,440
+ناقص واحد لكن بتحسبها عند مين عند C إذا هذه بتكون
+
+65
+00:05:20,440 --> 00:05:27,340
+C بساوياللي هو f of x قداش اللي هي واحد زائد x to
+
+66
+00:05:27,340 --> 00:05:30,700
+the alpha وهي اللي بديها انا f of zero ايش بتساوي
+
+67
+00:05:30,700 --> 00:05:36,400
+f of zero بتساوي واحد ناقص واحد اللي صار عندي اذا
+
+68
+00:05:36,400 --> 00:05:44,260
+اللي هو واحد زائد x to the alpha بتساوي واحد زائد
+
+69
+00:05:44,260 --> 00:05:51,850
+x في alpha في واحد زائد c أس alpha ناقص واحدمعايا
+
+70
+00:05:51,850 --> 00:06:00,130
+يا شباب هذي سميها واحد واضح
+
+71
+00:06:00,130 --> 00:06:06,770
+اه؟ الان عندى ال alpha أكبر من مين؟ من واحد لأن
+
+72
+00:06:06,770 --> 00:06:12,590
+but alpha ناقص واحد أكبر من مين؟ من سفر لأن ال
+
+73
+00:06:12,590 --> 00:06:19,130
+alpha أكبر من واحدوعندي هذه الـ C أشمالها بين الـ
+
+74
+00:06:19,130 --> 00:06:26,650
+0 والـ X يعني أكبر من 0 and C أكبر من 0 إذا صار
+
+75
+00:06:26,650 --> 00:06:37,420
+عند المقدار هذا 1 زائد C نفسه أكبر من 1ولمّا ترفعه
+
+76
+00:06:37,420 --> 00:06:43,080
+لأس موجب، هيظل أكبر من مين؟ و strictly أكبر من
+
+77
+00:06:43,080 --> 00:06:47,660
+واحد، والمساواة بس عند مين، عند الـC بالساوية سفر،
+
+78
+00:06:47,660 --> 00:06:53,730
+واضح أه؟ طيبالأن صار هذا المقدار أكبر من واحد إذا
+
+79
+00:06:53,730 --> 00:07:00,250
+عندي X موجبة و Alpha موجبة مظبوط ولا لأ لأن X هي
+
+80
+00:07:00,250 --> 00:07:05,790
+أكبر من 0 و Alpha أكبر من 1 إذا أضرب الجهتين في
+
+81
+00:07:05,790 --> 00:07:13,510
+Alpha في X بيصير عندي Alpha في X and so Alpha في X
+
+82
+00:07:13,510 --> 00:07:22,710
+بالضرر أكبر من مين Alpha Xفى الـ 1 زي الـ c plus
+
+83
+00:07:22,710 --> 00:07:27,110
+alpha minus 1 بيبقى أكبر من 100 من alpha في x
+
+84
+00:07:27,110 --> 00:07:30,670
+واضحش اللي سويته ضربت alpha x في الجهتين و ال
+
+85
+00:07:30,670 --> 00:07:33,890
+alpha x موجبة ضلت ال inequality زي ما هي طبعا لا
+
+86
+00:07:33,890 --> 00:07:39,370
+تساوي الصفة بالألف ولا ال X إذا من 1 و من 2 we
+
+87
+00:07:39,370 --> 00:07:44,530
+have done from 1
+
+88
+00:07:44,530 --> 00:07:51,270
+and 2من واحد عند اتنين بيطلع عندى اللى هو واحد
+
+89
+00:07:51,270 --> 00:07:57,130
+زياد اكس طول الف اللى هو بساوى المقدار هذا اللى
+
+90
+00:07:57,130 --> 00:08:02,070
+هذا الجزء منه اثبتنا انه اكبر من مين من الف في اكس
+
+91
+00:08:02,070 --> 00:08:13,800
+لذا صار هذا اكبر من واحد زياد الف اكس اي سؤالإذا
+
+92
+00:08:13,800 --> 00:08:18,120
+في حالة ال X أكبر من 0 وجدنا ال 1 زاد X أو أكبر من
+
+93
+00:08:18,120 --> 00:08:25,580
+1 زاد الف X لأن هنعمل نفس الاشي بس في الحالة مين؟
+
+94
+00:08:25,580 --> 00:08:32,400
+في حالة ال X أصغر من 0 وطبعا أكبر من مين؟ من سالب
+
+95
+00:08:32,400 --> 00:08:37,860
+1 إذا بدي أخد ال X أكبر من 0 و أكبر من نقص 1 وأصغر
+
+96
+00:08:37,860 --> 00:08:42,250
+من مين؟ من 0لاحظ لما تكون X أكبر من ناقص واحد أصغر
+
+97
+00:08:42,250 --> 00:08:46,110
+من صفر برضه نطبق الـ mean value theorem على هذه
+
+98
+00:08:46,110 --> 00:08:53,450
+نفس الدالة الان on the interval mean الان X و صفر
+
+99
+00:08:53,450 --> 00:08:57,850
+لأن X هي مالها الصغيرة على ال interval X و صفر لأن
+
+100
+00:08:57,850 --> 00:09:03,110
+X سالبة طيب إذا برضه there exists C element ال
+
+101
+00:09:03,110 --> 00:09:11,000
+mean in X و Zero is نفس البرهان such that Xف F
+
+102
+00:09:11,000 --> 00:09:15,000
+prime of C بيسوي F of X ناقص F of Zero نفس الحاجة
+
+103
+00:09:15,000 --> 00:09:19,140
+بيصير ال X في Alpha في هذا المقدار و هذا المقدار
+
+104
+00:09:19,140 --> 00:09:23,160
+ماتغيرش ولا شيء بيصير واحد زاد X و ده Alpha بيسوي
+
+105
+00:09:23,160 --> 00:09:26,360
+واحد زاد X في Alpha في واحد زاد C أُس Alpha ناقص
+
+106
+00:09:26,360 --> 00:09:31,560
+واحد الآن بيجي الأسباب هنا ال Alpha ناقص واحد
+
+107
+00:09:31,560 --> 00:09:36,840
+أشمله أكبر من سفر زي ما هو لكن ال C الآن اللي
+
+108
+00:09:36,840 --> 00:09:41,110
+لاجيناها وين موجودةفي المنطقة السالبة ده الـ C إيش
+
+109
+00:09:41,110 --> 00:09:47,830
+هتصير معاينتوش يا بابا هتصير الـ C أصغر من مين من
+
+110
+00:09:47,830 --> 00:09:53,990
+صفر سالبة مدام الـ C سالبة إذا واحد زائد C هتصير
+
+111
+00:09:53,990 --> 00:10:01,310
+أصغر من مين من واحد واحد زائد C أصغر من واحد واضح
+
+112
+00:10:01,310 --> 00:10:09,690
+طيب مدام واحد زائد C أصغر من واحد هيكون عندىand so
+
+113
+00:10:09,690 --> 00:10:19,130
+أضرب الجهتين الآن في اللي هي Alpha Alpha Alpha
+
+114
+00:10:19,130 --> 00:10:22,410
+بتظل زي ما هي لما نضرب في Alpha لأن Alpha إيه
+
+115
+00:10:22,410 --> 00:10:26,370
+معناها موجة ده لو ضربت في Alpha هنا و Alpha هنا
+
+116
+00:10:26,370 --> 00:10:31,600
+بتظل زي ما هيلأن بدنا نضرب في مين؟ في X و X
+
+117
+00:10:31,600 --> 00:10:37,740
+أشمالها سالبة فهتنجلي بالإشارة كمان مرة بيصير and
+
+118
+00:10:37,740 --> 00:10:44,220
+so X في Alpha في واحد زائد C أس Alpha ناقص واحد
+
+119
+00:10:44,220 --> 00:10:51,150
+هتصير أكبر من Alpha في X لأن X سالبةإذا صار عندى
+
+120
+00:10:51,150 --> 00:10:56,590
+هذا المقدار أكبر من هذا لأن من واحد ومن اتنين زى
+
+121
+00:10:56,590 --> 00:11:00,650
+ما عملنا قبل بشوية بتصير المقدار هذا اللى هو أكبر
+
+122
+00:11:00,650 --> 00:11:04,750
+من واحد زايد Alpha X strictly لاحظوا أنه في
+
+123
+00:11:04,750 --> 00:11:11,630
+الجهتين سواء X أكبر من سفر أو أصغر من سفر لجينا أن
+
+124
+00:11:11,630 --> 00:11:16,350
+ال inequality هذه متحقق أو strictly متحقق
+
+125
+00:11:19,030 --> 00:11:25,190
+الان المساوى وين صارت؟ عند السفر فالان عشان ناخد
+
+126
+00:11:25,190 --> 00:11:29,090
+بعين الاعتبار الحالة اللي هي x أكبر من ناقص واحد و
+
+127
+00:11:29,090 --> 00:11:33,430
+أصغر من سفر و حالة x أكبر من سفر و حالة x بتساوي
+
+128
+00:11:33,430 --> 00:11:38,870
+سفر بيصير هذا المقدار أكبر أو يساوي لكل x لكل x
+
+129
+00:11:38,870 --> 00:11:43,760
+ماله أكبر من ناقص واحدلأن الاتجاه الثاني من هذا
+
+130
+00:11:43,760 --> 00:11:48,880
+أثبتناه في الواقع إيش هو إنه لو كان فيه equality X
+
+131
+00:11:48,880 --> 00:11:54,280
+بتساوي 0 أو بمعنى آخر X لا تساوي 0 بيعطي إيه
+
+132
+00:11:54,280 --> 00:12:01,720
+أشماله non equality is not equal وهذا حدث كيف حدث
+
+133
+00:12:01,720 --> 00:12:06,240
+أثبتنا إنه لما X أكبر من نقص واحد وأصغر من 0 أو X
+
+134
+00:12:06,240 --> 00:12:10,720
+أكبر من 0 لجينا إنه هذي strictly أكبر يعني بتساوي
+
+135
+00:12:10,720 --> 00:12:15,260
+hashإذا صار عند الـ equality تحدث if and only if X
+
+136
+00:12:15,260 --> 00:12:22,580
+F بساوة سفر أي سؤال؟
+
+137
+00:12:22,580 --> 00:12:30,240
+طيب نيجي الآن نيجي الآن لل intermediate value of
+
+138
+00:12:30,240 --> 00:12:35,960
+property of derivative خلّيني ألقى في عينتها إذا
+
+139
+00:12:35,960 --> 00:12:42,760
+بتتذكروا في ال intermediate value of propertyاللي
+
+140
+00:12:42,760 --> 00:12:46,660
+هو for a continuous function إذا بتذكروا فيه اللي
+
+141
+00:12:46,660 --> 00:12:53,540
+هو real واحد اللي كانت تقول إذا كانت تقول إنه لو
+
+142
+00:12:53,540 --> 00:13:01,880
+كانت F function continuous على a closed open على
+
+143
+00:13:01,880 --> 00:13:07,020
+closed interval من A لعند B continuous على الفترة
+
+144
+00:13:07,020 --> 00:13:16,060
+هذهو إجيت أخدت أي نقطة K بين ال F of A و بين ال F
+
+145
+00:13:16,060 --> 00:13:22,900
+of B إجيت أخدت أي نقطة .. خد مثلا نقطة K بين F of
+
+146
+00:13:22,900 --> 00:13:31,240
+A .. هي F of A و هي F of B .. هي F of A و هي F of
+
+147
+00:13:31,240 --> 00:13:36,540
+B لو إجيت أخدت أي K بينهم هتلاقي .. هو خلي ناخد K
+
+148
+00:13:36,540 --> 00:13:43,830
+هنا مثلاهتلاقي على الأقل نقطة في الفترة A وBبتكون
+
+149
+00:13:43,830 --> 00:13:49,890
+اللي هو صورتها عبارة عن مين؟ عن الـ K يعني بمعنى
+
+150
+00:13:49,890 --> 00:13:55,970
+أخر there exists C element in A و B such that F of
+
+151
+00:13:55,970 --> 00:14:01,050
+C هي مين الـ K و لا أصلا مش هتلاقيها continuous لو
+
+152
+00:14:01,050 --> 00:14:05,530
+مافشلها فئلة و جينا مدينا خطنا هلجيت و مالجيناش
+
+153
+00:14:05,530 --> 00:14:09,290
+ولا واحدة من هذه النقاط اللي هو تقطع الخط معناته
+
+154
+00:14:09,290 --> 00:14:14,410
+أنه صار فيه jump مش هيكون في continuityالان هنا مش
+
+155
+00:14:14,410 --> 00:14:19,890
+في نقطة هي نقطة هي نقطتين هي تلاتة كلهم صورتين هي
+
+156
+00:14:19,890 --> 00:14:27,610
+C1 هي C2 هي C3 كلهم F of C1 K F of C2 K F of C3 K
+
+157
+00:14:27,610 --> 00:14:33,150
+المهم على الأقل في C element in A وB اللي هي بحيث
+
+158
+00:14:33,150 --> 00:14:36,210
+F of C بالساوة K هذا لو كان اتقفز continuous على
+
+159
+00:14:36,210 --> 00:14:40,650
+closed اللي هي bounded interval اللي عندك الآن
+
+160
+00:14:43,230 --> 00:14:47,090
+الحديث بيقول لي في اشي مشابه في حالة ان ال F' is
+
+161
+00:14:47,090 --> 00:14:53,950
+differentiable الان بيقولك لو كانت K بين ال F' لل
+
+162
+00:14:53,950 --> 00:14:58,970
+A و ال F' لل Bبرضه هنلاقيها بتتحقق اللي هو there
+
+163
+00:14:58,970 --> 00:15:03,810
+exists C بحيث ان F prime of C بتساوي اللي هو الكلب
+
+164
+00:15:03,810 --> 00:15:10,530
+نحكي عنها بالرغم انه مش مفترضين احنا في النظرية
+
+165
+00:15:10,530 --> 00:15:14,190
+اللي هو ال continuity ال continuity لل F prime
+
+166
+00:15:16,020 --> 00:15:19,360
+فضبجنا على الـ F' لكن مش .. لأنهم افترضينا على طول
+
+167
+00:15:19,360 --> 00:15:22,300
+تتطبق الـ Intermediate Value Theorem بس على الدالة
+
+168
+00:15:22,300 --> 00:15:26,980
+الـ F' لكن لأ ما هو مخزون في دالة .. خلال الـ F'
+
+169
+00:15:27,380 --> 00:15:33,140
+من معلومات أهلها أنه في حالة F is differentiable
+
+170
+00:15:33,140 --> 00:15:37,940
+على ال interval Iوالـ F ب .. و لجينا .. وفي عندي
+
+171
+00:15:37,940 --> 00:15:43,120
+أخدت أي K بين الـ F' of A و الـ F' of B هتلاقي C
+
+172
+00:15:43,120 --> 00:15:48,380
+في الـ A و الـ B بحيث انه اللي هي F' of C هي هذه
+
+173
+00:15:48,380 --> 00:15:51,920
+الـ K هنشوفها هذه اللي بنسميها Daraboxes theorem
+
+174
+00:15:51,920 --> 00:15:55,980
+قبل ما نروح للـ Daraboxes theorem خلينا نيجي لللمة
+
+175
+00:15:55,980 --> 00:16:00,680
+اللي جابلة اللي هنستخدمها في إثبات الـ Daraboxes
+
+176
+00:16:00,680 --> 00:16:07,340
+theorem الآنالنظرية هذه أو اللمّة هذه اللي هو
+
+177
+00:16:07,340 --> 00:16:10,880
+خلّينا نستذكرها من خلال الرسم ممكن يكون أسهل لكم
+
+178
+00:16:10,880 --> 00:16:16,680
+من خلال الرسم بتقولّي إنه لو كانت عندنا function F
+
+179
+00:16:16,680 --> 00:16:22,300
+is differentiable على اللي هو interval معينة
+
+180
+00:16:22,300 --> 00:16:29,120
+ولجينا عند C معينة هي F primeعند C موجبة F prime
+
+181
+00:16:29,120 --> 00:16:33,940
+عند C موجبة يعني F prime at some C أكبر من 0 يعني
+
+182
+00:16:33,940 --> 00:16:37,160
+مين المماس إيه اللي شماله اللي هو زاوية حادة عامل
+
+183
+00:16:37,160 --> 00:16:43,660
+زاوية حادة ماشي الان بقولي هتلاقي الان بده يبحث عن
+
+184
+00:16:43,660 --> 00:16:49,260
+نقاط تكون أكبر من مين من F of C الان هتلاقي اللي
+
+185
+00:16:49,260 --> 00:16:56,370
+هو هذا C اللي بتحكي عنها وهنااللي هو في جوار حوالي
+
+186
+00:16:56,370 --> 00:16:59,410
+هذين there exist إذا كان f of rank of c أكبر من
+
+187
+00:16:59,410 --> 00:17:03,630
+سفر there exists delta أكبر من سفر such that f of
+
+188
+00:17:03,630 --> 00:17:11,960
+xهنلاقيها أكبر من F of C لكل X وين متوقع أنا هان
+
+189
+00:17:11,960 --> 00:17:17,300
+لإنها الميل هيه إذا في الوين عشان C ل C زائد Delta
+
+190
+00:17:17,300 --> 00:17:19,620
+مش في الفترة اللي جابلة، في الفترة اللي جابلة
+
+191
+00:17:19,620 --> 00:17:23,960
+هتلاقي اللي هي النقطات أشملها أجل لأنا الآن ببحث
+
+192
+00:17:23,960 --> 00:17:29,420
+.. ببحث لم تبعت هيك بدهااللي هو there exist دلتا
+
+193
+00:17:29,420 --> 00:17:32,120
+أكبر من صفر ساشداد F of X أكبر من F of C لكل X
+
+194
+00:17:32,120 --> 00:17:40,200
+والي موجود من الـ C للـ C زائد دلتا معايا؟ الآن
+
+195
+00:17:40,200 --> 00:17:47,320
+بقولك لو لجينا F prime للـ C عند نقطة ما F prime
+
+196
+00:17:47,320 --> 00:17:54,130
+للـ C هي الـ C التانية لو لجينا F prime للـ Cأصغر
+
+197
+00:17:54,130 --> 00:17:57,210
+من صفر عشان .. عشان ما تحفظهاش حفظ عشان لو انسيتها
+
+198
+00:17:57,210 --> 00:18:00,890
+تستنتجها لحالك لو كان F' للـC أصغر من صفر يعني ميل
+
+199
+00:18:00,890 --> 00:18:06,710
+المماس سلب يعني زاوية منفرجة الآن هتلاقي Delta
+
+200
+00:18:06,710 --> 00:18:13,360
+أكبر من صفر يعني تمثل جواربحيث أن F of X برضه أكبر
+
+201
+00:18:13,360 --> 00:18:19,000
+من F of C لكل X وين موجودة هي وارحة منها لكل X
+
+202
+00:18:19,000 --> 00:18:24,340
+موجودة وين جابلها من C نقص دلتة لعند مين؟ لعند C
+
+203
+00:18:24,340 --> 00:18:29,800
+من C نقص دلتة لعند C هذه بس اللممة للاستذكار، إيش
+
+204
+00:18:29,800 --> 00:18:32,780
+بتقول اللممة؟ تعالى نشوف بالظبط اللي بتقوله اللي
+
+205
+00:18:32,780 --> 00:18:36,220
+حكيته أنا بقول اللممة let I subset من R be an
+
+206
+00:18:36,220 --> 00:18:40,260
+intervaland let F من I ل R و let C element in I
+
+207
+00:18:40,260 --> 00:18:44,700
+and assume that F has a derivative at C نفترض ان
+
+208
+00:18:44,700 --> 00:18:50,120
+ال F الهياشق مشتقة عند النقطة C مقولي إذا F prime
+
+209
+00:18:50,120 --> 00:18:54,200
+أكبر من 0 عند ال C هاي اللي قلته هان إذا نيوجد
+
+210
+00:18:54,200 --> 00:18:58,500
+دلتة أكبر من 0 بحيث أن F of X أكبر من F of C لكل
+
+211
+00:18:58,500 --> 00:19:02,560
+ال X وين موجودة بين ال C و C زائد دلتة اللي هي
+
+212
+00:19:02,560 --> 00:19:03,280
+الجبل هنا
+
+213
+00:19:06,510 --> 00:19:10,590
+الثانية، إذا كانت F' أصغر من 0، حلاقي Delta بحيث
+
+214
+00:19:10,590 --> 00:19:15,430
+أن F of C أصغر واحدة في الجوار، يعني F of C أو في
+
+215
+00:19:15,430 --> 00:19:19,010
+الجزء من الجوار، هذا الجزء اللي لجاي، F of C أصغر
+
+216
+00:19:19,010 --> 00:19:24,370
+من F of X، وين هاي الـ F of X هنا؟ أكبر من F of C،
+
+217
+00:19:24,370 --> 00:19:29,870
+واضح؟ لكل X وين موجودة في الـ C minus Delta والـ C
+
+218
+00:19:31,220 --> 00:19:36,360
+أي سؤال البرهن هو A خلّينا نبرهن بيه نحنا بيه
+
+219
+00:19:36,360 --> 00:19:40,980
+خلّينا نبرهن مع بعض بيه
+
+220
+00:19:40,980 --> 00:19:46,840
+الان
+
+221
+00:19:46,840 --> 00:19:53,820
+بدنا نقول نفترض ان F برايم بيه
+
+222
+00:19:53,820 --> 00:20:02,190
+approved الان ماعطينا F برايم of Cأصغر من سفر،
+
+223
+00:20:02,190 --> 00:20:11,750
+مظبوط؟ لأن بتقول since f prime of c أصغر من سفر
+
+224
+00:20:11,750 --> 00:20:21,070
+then limit f of x ناقص f of c على x minus c لما x
+
+225
+00:20:21,070 --> 00:20:27,540
+تروح لل c إشماله أصغر من سفرماشي، مدام أصغر من سفر
+
+226
+00:20:27,540 --> 00:20:31,080
+عند .. قلنا قبل هيك في نظرية إذا كان اللي هو
+
+227
+00:20:31,080 --> 00:20:35,600
+المقدار هذا أصغر من سفر إذا بقدر ألاقي جوار
+
+228
+00:20:35,600 --> 00:20:41,240
+حواليها بتكون القيم كلها إشمالها أصغر من سفر إذا
+
+229
+00:20:41,240 --> 00:20:48,560
+then there exist V Delta of C اللي هو جوار جوار من
+
+230
+00:20:48,560 --> 00:20:55,260
+عند C minus Delta لعند C زائد Deltaماشي؟ اللي هو
+
+231
+00:20:55,260 --> 00:20:58,980
+احنا أخدناها من هذا المنطقة من C نقص دلتا ل C زائد
+
+232
+00:20:58,980 --> 00:21:06,700
+دلتا such that اللي هو F of X نقص F of C على X
+
+233
+00:21:06,700 --> 00:21:16,220
+minus C أصغر من سفر لكل X وين في V دلتا of C أنا
+
+234
+00:21:16,220 --> 00:21:22,300
+غرضي أني أخلي اللي هو أوجد mean أن F of X أكبر من
+
+235
+00:21:22,300 --> 00:21:25,900
+100من F of C بدا وجد المنطقة اللي بتكون فيها F of
+
+236
+00:21:25,900 --> 00:21:31,880
+X أكبر من مين من F of C لاحظ انه هذا بديها وكأن F
+
+237
+00:21:31,880 --> 00:21:36,480
+of X ناقص F of C بديها أكبر من 0 صح عشان أحصل F of
+
+238
+00:21:36,480 --> 00:21:40,420
+X أكبر من مين من F of C لذن عشان أحصل على هذا
+
+239
+00:21:40,420 --> 00:21:45,000
+المقدار أكبر من 0 وانا بعرف ان هذا كله أصغر من 0
+
+240
+00:21:45,000 --> 00:21:48,880
+بدي المنطقة هذه تكون أصغر من 0 وين هتكون أصغر من
+
+241
+00:21:48,880 --> 00:21:57,510
+0؟ في المنطقة هذه، ماشي الحال؟الان but for every x
+
+242
+00:21:57,510 --> 00:22:05,650
+element in c نقص delta و c ال x minus c أصغر من
+
+243
+00:22:05,650 --> 00:22:15,800
+سفر واضحة؟ إذن الآن من هذه و من هذه henceأف of X
+
+244
+00:22:15,800 --> 00:22:23,300
+ناقص أف of C اللي هي بتساوي X minus C في أف of X
+
+245
+00:22:23,300 --> 00:22:30,300
+ناقص أف of C على X minus C هذا سالب وهذا سالب لذا
+
+246
+00:22:30,300 --> 00:22:35,260
+حصل ضرب بين أشماله هكون أكبر من سفر سالب في سالب
+
+247
+00:22:35,260 --> 00:22:42,720
+موجه مظبوط لهذا الكلام لكل X وين موجودة في الـC
+
+248
+00:22:43,730 --> 00:22:52,150
+-Delta OC وهذا معنى أن F of X أكبر من F of C لنقطة
+
+249
+00:22:52,150 --> 00:22:59,950
+المذكورة لنقات المذكورة واضح؟ أي سؤال؟ نيجي الآن
+
+250
+00:22:59,950 --> 00:23:04,930
+اللي هو انبرهن الـDrabowski's theorem اللي حكينا
+
+251
+00:23:04,930 --> 00:23:08,350
+عنها وقدمنا لها قبل بشوية إيش الـDrabowski's
+
+252
+00:23:08,350 --> 00:23:15,010
+theorem بتقول أنه عندي ال Fهو مستخدم للتحدي في بعض
+
+253
+00:23:15,010 --> 00:23:19,250
+الانترالين
+
+254
+00:23:19,250 --> 00:23:19,950
+A وB
+
+255
+00:23:23,800 --> 00:23:29,540
+اللي هي أي K بين F' و F'
+
+256
+00:23:32,040 --> 00:23:36,580
+of A و F' of B إذا نحن نلاقي الـ C بقيت أن F' of C
+
+257
+00:23:36,580 --> 00:23:42,020
+هي الـ K كمان مرة Drabowski's theorem بتقول if F
+
+258
+00:23:42,020 --> 00:23:47,520
+is differentiable on A و B و if K is any number
+
+259
+00:23:47,520 --> 00:23:53,610
+between F' of A و F' of Bthen there exists at
+
+260
+00:23:53,610 --> 00:23:56,350
+least one point C في الـA والـB بحيث أن الـAlpha
+
+261
+00:23:56,350 --> 00:24:02,650
+Prime of C أيش بتساوي؟ بتساوي K ماشي يا شباب طيب
+
+262
+00:24:02,650 --> 00:24:06,890
+اللي زي ما علقنا قبل بشوية بالرغم إنه إحنا ماحدش
+
+263
+00:24:06,890 --> 00:24:11,810
+جاب سيرة إن الـAlpha Prime هتكون continuousلأ،
+
+264
+00:24:11,810 --> 00:24:16,310
+قلنا ما في الـ F' من مخزون، derivative من مخزون في
+
+265
+00:24:16,310 --> 00:24:21,530
+داخلنا المعلومات أهلها إنها تقدر تصل أنه لأي K بين
+
+266
+00:24:21,530 --> 00:24:27,190
+الـ F A و الـ F' A و الـ F' B بالـ Legacy، بحيث أن
+
+267
+00:24:27,190 --> 00:24:31,170
+F' اللي عند الـ C هي الـ K. إيه المرحلة دلوقتي؟
+
+268
+00:24:31,170 --> 00:24:41,150
+شوف، نفترض أنه Proofنفترض أن G of A أصغر من K أصغر
+
+269
+00:24:41,150 --> 00:24:52,230
+من مين من G أو F prime of A أصغر من F prime of B
+
+270
+00:24:52,230 --> 00:24:56,450
+ماشي الحال واضح
+
+271
+00:24:56,450 --> 00:25:02,430
+هذا فرضناهالآن بدنا نلاقي C بحيث انه F' للـ C هو
+
+272
+00:25:02,430 --> 00:25:06,530
+الـ K الان define هالجيه تعرف ليش اعرفت defining
+
+273
+00:25:06,530 --> 00:25:15,590
+function G of X بساوي K في ال X ناقص F of X عندي
+
+274
+00:25:15,590 --> 00:25:18,790
+اللي هي هذه الدالة هي الدالة اللي هتوصلني للي
+
+275
+00:25:18,790 --> 00:25:23,800
+بديهاالان واضح ان الـ g of x بيساوي kx ناقص f of x
+
+276
+00:25:23,800 --> 00:25:29,660
+انها differentiable اللي هو هذه وين ما كان و هذي
+
+277
+00:25:29,660 --> 00:25:32,080
+differentiable على ال I إذا صار ان التنتين
+
+278
+00:25:32,080 --> 00:25:35,960
+differentiable على مين؟ على ال A و ال B و طبيعي
+
+279
+00:25:35,960 --> 00:25:40,440
+continuous على ال A و ال B إذا بما أن هذا الدالة
+
+280
+00:25:40,440 --> 00:25:44,440
+is continuous على closed bounded interval then it
+
+281
+00:25:44,440 --> 00:25:47,580
+attains its maximum and its minimum on this
+
+282
+00:25:47,580 --> 00:25:51,870
+intervalبتنه نشتغل عليه ال maximum و لو حد بده
+
+283
+00:25:51,870 --> 00:25:58,670
+يشتغل عليه المنه مابنفع برضه الان then g of x
+
+284
+00:25:58,670 --> 00:26:09,810
+attains its maximum on mean on a و bلأن إيش اللي
+
+285
+00:26:09,810 --> 00:26:15,370
+بيدثبتلك هي بيدثبتلك إن الـ G هيكون ال maximum
+
+286
+00:26:15,370 --> 00:26:21,150
+إلها لا عند ال A ولا عند ال B إذا هتكون وين؟ في
+
+287
+00:26:21,150 --> 00:26:25,870
+نقطة داخلية مدام في نقطة داخلية وإحنا عارفين إن
+
+288
+00:26:25,870 --> 00:26:29,490
+الـ G differentiable على كل المنطقة هذه إذا صارت
+
+289
+00:26:29,490 --> 00:26:33,050
+غصب عني إنه لازم مدام عندها maximum و interior
+
+290
+00:26:33,050 --> 00:26:38,390
+point إذا الـ G prime إيش هتساوي؟ هتساوي 0واضح؟
+
+291
+00:26:38,390 --> 00:26:42,590
+مدام جي برايم ما هتساوي سفر هتصير ال F prime of X
+
+292
+00:26:42,590 --> 00:26:46,550
+ال F of some C بتساوي ال K و بيكون خلصنا البرهان
+
+293
+00:26:46,550 --> 00:26:49,950
+تشوفيش اللي بقوله طيب صلوا على النبي عليه الصلاة
+
+294
+00:26:49,950 --> 00:27:05,390
+والسلام الآن عندي ال G prime of X هتساوي K ناقص F
+
+295
+00:27:05,390 --> 00:27:11,520
+prime of مين؟of X صح ولا لأ؟ طيب الآن لو جينا
+
+296
+00:27:11,520 --> 00:27:21,540
+حسبنا الـ G' عند ال A إيش هتلاقيها؟ K ناقص F' عند
+
+297
+00:27:21,540 --> 00:27:28,740
+مين؟ عند ال A صح ولا لأ؟ K ناقص F' عند ال A هتكون
+
+298
+00:27:28,740 --> 00:27:35,260
+أكبر من مين؟ من 0 واضح مدام ال G' على ال A أكبر من
+
+299
+00:27:35,260 --> 00:27:35,520
+0
+
+300
+00:27:39,110 --> 00:27:49,430
+K-F' أكبر من سفر بما أن J' بسوء K-F' أكبر من سفر
+
+301
+00:27:49,430 --> 00:27:53,450
+يعني هذا أكبر من سفر حسب اللي قبل قليل حكينا عنها
+
+302
+00:27:53,450 --> 00:28:06,330
+ثم هناك دلت أكبر من سفر such that G' of X أكبر من
+
+303
+00:28:06,330 --> 00:28:13,400
+G of Aاللي كنا بسمينا C فاهمين عليها الـ G of X
+
+304
+00:28:13,400 --> 00:28:22,800
+أكبر من G of A for every X element inمتذكرين اللي
+
+305
+00:28:22,800 --> 00:28:27,480
+قبل بشوية لما كانت جي برايم لما كانت الجي برايم
+
+306
+00:28:27,480 --> 00:28:32,860
+أكبر من سفر كانت اللي هي وين لجاي X element لغاية
+
+307
+00:28:32,860 --> 00:28:40,920
+أسف X element in C اللي هي A هنا وA زائد دلتا اللي
+
+308
+00:28:40,920 --> 00:28:45,240
+ما اللي قبل بشوية إذا صارت عند جي of X أكبر من 100
+
+309
+00:28:46,010 --> 00:28:50,890
+من الـ G of A واضح لمين؟ لكل الإكسات اللي هي في نص
+
+310
+00:28:50,890 --> 00:28:57,290
+الجوار هذا صارت عندي اللي هي الـ G of A مش maximum
+
+311
+00:28:57,290 --> 00:29:08,650
+لأن في قيم أكبر منها واضح؟ إذا أكيد then G does
+
+312
+00:29:08,650 --> 00:29:18,150
+not have a maximum at Aإذا الفش مش maximum عند
+
+313
+00:29:18,150 --> 00:29:23,990
+مين؟ عند الـ A أي
+
+314
+00:29:23,990 --> 00:29:27,750
+سؤال؟
+
+315
+00:29:27,750 --> 00:29:36,990
+Similarly .. Similarly G' عند ال B معايا يا شباب
+
+316
+00:29:36,990 --> 00:29:44,340
+G' عند ال B بساوي K نقص F' عند ال Bو K ناقص F
+
+317
+00:29:44,340 --> 00:29:49,840
+prime عند الـ B أصغر من مين؟ أصغر من 0 إذا بنفس
+
+318
+00:29:49,840 --> 00:29:54,600
+اللمّة بس على الجهة الثانية then there exists
+
+319
+00:29:54,600 --> 00:30:01,000
+Delta أكبر من 0 such that G of X أكبر من G of B
+
+320
+00:30:01,000 --> 00:30:11,520
+لكل X وان الآن من عند B ناقص Delta لعند الـ B صح؟
+
+321
+00:30:12,880 --> 00:30:19,600
+وهنا end points هذول فمش هتكون برضه مين ال b مش
+
+322
+00:30:19,600 --> 00:30:24,940
+هتكون عندها maximum لأنه منها ولا جوا ال b end
+
+323
+00:30:24,940 --> 00:30:28,920
+point منها ولا جوا ال g of x أكبر من ال g of b إذن
+
+324
+00:30:28,920 --> 00:30:38,160
+مش maximum إذن hence g does not have a maximum at
+
+325
+00:30:38,160 --> 00:30:45,380
+b from هناو from هنا from here and here we have g
+
+326
+00:30:45,380 --> 00:30:48,700
+has
+
+327
+00:30:48,700 --> 00:31:00,180
+the maximum inside a و b، مظبوط؟ يعني بمعنى آخر،
+
+328
+00:31:00,180 --> 00:31:06,100
+مدام احنا أكيد attained its maximum there exists c
+
+329
+00:31:06,100 --> 00:31:13,970
+element in a و b such thatجي هز ا maximum مدام جي
+
+330
+00:31:13,970 --> 00:31:18,770
+هز ا maximum و interior point اذا جي برايم عند ال
+
+331
+00:31:18,770 --> 00:31:23,390
+C ايش بتساوي؟ بتساوي سفر مدام جي برايم عند ال C
+
+332
+00:31:23,390 --> 00:31:28,090
+بتساوي سفر اذا صار عندي hence
+
+333
+00:31:30,200 --> 00:31:33,960
+وهنا اكتب جي برايم عند الـ C ايش بيساوي؟ بيساوي
+
+334
+00:31:33,960 --> 00:31:39,120
+سفر ايش جي برايم بيساوي؟ K ناقص F برايم اذا that
+
+335
+00:31:39,120 --> 00:31:43,020
+is K
+
+336
+00:31:43,020 --> 00:31:50,510
+ان هي بدل جي برايمنقص F' عند C بيساوي سفر أو بمعنى
+
+337
+00:31:50,510 --> 00:31:57,090
+آخر F' عند C أش بيساوي بيساوي K فعلا لقينا C في ال
+
+338
+00:31:57,090 --> 00:32:02,770
+A و ال B بحيث أن F' عند C أش بتساوي بيساوي K كما
+
+339
+00:32:02,770 --> 00:32:13,170
+هو مطلوب هيها أي سؤال الآن
+
+340
+00:32:14,940 --> 00:32:21,600
+عندي المثال الأخير بقول لي use Daraboux's theorem
+
+341
+00:32:21,600 --> 00:32:25,120
+هو في الواقع مثال على Daraboux's theorem بقول لي
+
+342
+00:32:25,120 --> 00:32:32,060
+the function g من
+
+343
+00:32:32,060 --> 00:32:36,080
+ناطس واحد وواحد لعند ال R defined by معرفه g of x
+
+344
+00:32:36,080 --> 00:32:39,320
+أيش بالساوية واحد إذا كانت x أكبر من صفر وأصغر
+
+345
+00:32:39,320 --> 00:32:43,680
+بسوا واحد0 إذا كانت X بيساوي واحد و ناقص واحد إذا
+
+346
+00:32:43,680 --> 00:32:46,180
+كانت X أكبر من ناقص واكبر بيساوي ناقص واحد و
+
+347
+00:32:46,180 --> 00:32:52,360
+بيساوي سفر بقول لي أثبت أن هذه الدالة is not the
+
+348
+00:32:52,360 --> 00:32:57,160
+derivative is not a derivative of any function
+
+349
+00:32:57,160 --> 00:33:01,580
+يعني مافيش function في الدنيا بتكون مشتقتها هي هذه
+
+350
+00:33:01,580 --> 00:33:06,420
+الـG هذه الـG is not a derivative of any function
+
+351
+00:33:07,400 --> 00:33:10,800
+بدا اقول suppose not ونصلى contradiction ماشي
+
+352
+00:33:10,800 --> 00:33:15,440
+الحال اذا الان ال
+
+353
+00:33:15,440 --> 00:33:27,280
+.. النظرية .. ال .. البثال بقول mainly المثال
+
+354
+00:33:27,280 --> 00:33:35,540
+بقول example example show that
+
+355
+00:33:38,010 --> 00:33:47,490
+G من نقص واحد لعند الواحد لعند ال R Defined by G
+
+356
+00:33:47,490 --> 00:33:53,610
+of X بساوي واحد إذا كانت X أكبر من سفر وأصغر يسوء
+
+357
+00:33:53,610 --> 00:33:59,290
+واحد سفر X بتساوي سفر نقص واحد X أكبر يسوء نقص
+
+358
+00:33:59,290 --> 00:34:08,330
+واحد وأصغر من سفر is not a derivativeof any
+
+359
+00:34:08,330 --> 00:34:20,610
+function on ناقص واحد واحد solution او proof اللي
+
+360
+00:34:20,610 --> 00:34:25,930
+قاعدك تقول suppose that
+
+361
+00:34:25,930 --> 00:34:36,550
+there exist f من ناقص واحد واحد لعند ال R suchthat
+
+362
+00:34:36,550 --> 00:34:46,630
+f prime of x بساوي g of x لكل x element ناقص واحد
+
+363
+00:34:46,630 --> 00:34:51,870
+و واحد ماشي الحال ايش هذه الدالة هي في الواقع
+
+364
+00:34:51,870 --> 00:34:59,310
+الدالة هي ال g of x هذه عندي سفر و انا عند الواحد
+
+365
+00:34:59,310 --> 00:35:04,550
+عند الواحد و انا سلب واحد
+
+366
+00:35:07,910 --> 00:35:16,170
+لعند اللي هو سنة واحد هيدّه ده اه لأن افترض انه F
+
+367
+00:35:16,170 --> 00:35:19,130
+بين نقص واحد واحد such that F prime of X بيساوي G
+
+368
+00:35:19,130 --> 00:35:22,650
+of X واضح انه F is different مدهن اذا قلت F prime
+
+369
+00:35:22,650 --> 00:35:25,390
+of X بيساوي G of X لكل X اللي منتقل نقص واحد واحد
+
+370
+00:35:25,390 --> 00:35:30,070
+صارت ال F ايش ما لأ is differentiable مظبوط ولا لأ
+
+371
+00:35:30,070 --> 00:35:39,050
+is differentiable واتنين لأن butالنص ينتبه إلى
+
+372
+00:35:39,050 --> 00:35:43,850
+الفترة ناقص واحد وواحد اللي هي إيش بتساوي اللي هي
+
+373
+00:35:43,850 --> 00:35:50,590
+عبارة عن الفترة هذه عبارة عن اللي هي F prime of
+
+374
+00:35:50,590 --> 00:36:03,650
+ناقص واحد و F prime of واحد باضح
+
+375
+00:36:03,650 --> 00:36:13,290
+أه؟هذه F' هي مين؟ G صح ولا لا؟ فمين؟ وهذه ايه؟ F'
+
+376
+00:36:13,550 --> 00:36:21,810
+G لأن G of واحد واحد، G of ناقص واحد ونص في الفترة
+
+377
+00:36:21,810 --> 00:36:22,170
+هذه
+
+378
+00:36:25,820 --> 00:36:30,060
+أصلا كل قيم الـ F' أو الـ D هي واحد صفر وناقص واحد
+
+379
+00:36:30,060 --> 00:36:34,360
+هذا هيسببلك الخلال مش نص بس تقدر تختار ربع تلت خمس
+
+380
+00:36:34,360 --> 00:36:38,680
+أي حاجة غير الـ Zero و الواحد والسلب واحد ايش ال
+
+381
+00:36:38,680 --> 00:36:49,400
+.. ايش الشد في الأمر but نص element كده and D
+
+382
+00:36:49,400 --> 00:37:00,740
+F is differentiable on ناقص واحد واحدصح؟ then by
+
+383
+00:37:00,740 --> 00:37:09,800
+Darabaux's theorem there exist c element ناقص واحد
+
+384
+00:37:09,800 --> 00:37:19,200
+وواحد such that اللي هو g of c او f prime of c
+
+385
+00:37:19,200 --> 00:37:28,990
+بيساوي ايش نص صح؟Ie بمعنى آخر F prime of C اللي هو
+
+386
+00:37:28,990 --> 00:37:39,590
+G of C بساوة نص Which is impossible Therefore
+
+387
+00:37:39,590 --> 00:37:46,480
+our first assumption is not truethat is there is
+
+388
+00:37:46,480 --> 00:37:50,740
+no F من ناقص واحد لعند واحد .. من واحد ناقص واحد
+
+389
+00:37:50,740 --> 00:37:54,960
+لواحد لعند ال R بحيث أن F' بيساوي G أو بمعنى أخر G
+
+390
+00:37:54,960 --> 00:38:03,680
+is not a derivative of any function وهكذا
+
+391
+00:38:03,680 --> 00:38:07,780
+نكون خلصنا ستة اتنين والمرة اللي جاي ان شاء الله
+
+392
+00:38:07,780 --> 00:38:13,980
+بنكمل في ستة تلاتة ويوم الأحد القادم بنناقش ستة
+
+393
+00:38:13,980 --> 00:38:16,010
+اتنينأسئلة 6
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ.srt
new file mode 100644
index 0000000000000000000000000000000000000000..9adc2ef5eb8d60b59e1a8bfd8ccad08d800aad1c
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ.srt
@@ -0,0 +1,3453 @@
+1
+00:00:04,910 --> 00:00:07,650
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:07,650 --> 00:00:10,710
+والصلاة والسلام على سيد المرسلين سيدنا محمد على 
+
+3
+00:00:10,710 --> 00:00:15,510
+آله وصحبه أجمعين هذه هي المحاضرة رقم واحد وعشرين
+
+4
+00:00:15,510 --> 00:00:19,630
+في مساق تحليل حقيقي لطلاب وطالبات الجامعة
+
+5
+00:00:19,630 --> 00:00:26,460
+الإسلامية قسم الرياضيات في كلية العلوم نكمل الحديث
+
+6
+00:00:26,460 --> 00:00:32,660
+عن اللي هو الـ Sequence of Functions خلصنا من
+
+7
+00:00:32,660 --> 00:00:36,180
+سيكشن الـ Pointwise أو الـ Uniform Convergence
+
+8
+00:00:36,180 --> 00:00:40,340
+الآن بدنا نشوف تأثير الـ Pointwise و الـ Uniform
+
+9
+00:00:40,340 --> 00:00:44,700
+Convergence على اللي هو بعض الـ Sequence of
+
+10
+00:00:44,700 --> 00:00:49,040
+Functions إيش بتسوي فيها الآن تحت عنوان 82
+
+11
+00:00:49,040 --> 00:00:53,900
+interchange of limits هنجي نشوف اللي هو الـ
+
+12
+00:00:53,900 --> 00:00:57,160
+sequence of continuous functions لو كانت اللي هي
+
+13
+00:00:57,160 --> 00:01:01,620
+pointwise convergence هل شرط أنه لو لمين بتروح
+
+14
+00:01:01,620 --> 00:01:04,680
+في ال pointwise convergence أن يكون continuous ولا
+
+15
+00:01:04,680 --> 00:01:09,780
+بلزمها دعم شوية نخلي اللي هو بدل ال pointwise
+
+16
+00:01:09,780 --> 00:01:13,220
+convergence نخليه uniform والـ uniform يودي الـ
+
+17
+00:01:13,220 --> 00:01:16,040
+function اللي بتطلع تطلع continuous ولا لأ؟ هنشوف
+
+18
+00:01:16,040 --> 00:01:19,680
+الكلام هذا الشغل الثاني هنشوف أنه لو كانت الـ
+
+19
+00:01:19,680 --> 00:01:23,360
+function is integrable هل الـ sequence of functions
+
+20
+00:01:23,360 --> 00:01:27,640
+is integrable؟ هل الـ limit اللي لما يكون اللي هو
+
+21
+00:01:27,640 --> 00:01:32,080
+الـ Fn بتروح للـ F pointwise أو الـ Fn بتروح للـ F
+
+22
+00:01:32,080 --> 00:01:37,850
+uniform convergence؟ هل هذا واحد منهم بيقدي أنه الـ
+
+23
+00:01:37,850 --> 00:01:39,850
+integration للـ limit بيساوي الـ limit للـ
+
+24
+00:01:39,850 --> 00:01:42,730
+integration يعني الـ limit بتدخل جوا الـ integration
+
+25
+00:01:42,730 --> 00:01:46,990
+ولا لأ؟ هل دخولها يعني بالـ pointwise بحافظ على
+
+26
+00:01:46,990 --> 00:01:50,170
+المساواة ولا لأ بالزمن اللي هو الـ uniform
+
+27
+00:01:50,170 --> 00:01:54,730
+convergence؟ هنشوفه والشغل التالت هل أنه لو كان
+
+28
+00:01:54,730 --> 00:01:59,150
+في عندي sequence of differentiable functions هتطلع
+
+29
+00:01:59,150 --> 00:02:04,040
+أن هو لو كان عندي F converged زي الـ F هيطلع عندي
+
+30
+00:02:04,040 --> 00:02:06,460
+اللي هو الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ..
+
+31
+00:02:06,460 --> 00:02:07,500
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+32
+00:02:07,500 --> 00:02:07,620
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+33
+00:02:07,620 --> 00:02:08,200
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+34
+00:02:08,200 --> 00:02:08,300
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+35
+00:02:08,300 --> 00:02:08,340
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+36
+00:02:08,340 --> 00:02:09,020
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+37
+00:02:09,020 --> 00:02:10,060
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+38
+00:02:10,060 --> 00:02:11,000
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+39
+00:02:11,000 --> 00:02:13,560
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+40
+00:02:13,560 --> 00:02:13,720
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ ال
+
+41
+00:02:13,720 --> 00:02:23,020
+.. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. 
+
+42
+00:02:28,280 --> 00:02:34,340
+المثال الأول اللي أمامنا هاي في عندي example 821 g
+
+43
+00:02:34,340 --> 00:02:38,900
+n of x بساوية x أُس n هذا شفناها يا جماعة على
+
+44
+00:02:38,900 --> 00:02:43,640
+الفترة x 01 is continuous أكيد اللي هي الـ function
+
+45
+00:02:43,640 --> 00:02:47,540
+هذه g 1 of x بساوية x g 2 of x بساوية x تربيع g 3
+
+46
+00:02:47,540 --> 00:02:52,110
+of x بساوية x تكعيب كله عبارة عن دوال متصلة الآن
+
+47
+00:02:52,110 --> 00:02:55,950
+and even continuously differentiable بيقولك أصلا
+
+48
+00:02:55,950 --> 00:03:00,430
+هي طلع مش ده continuous لأ والـ derivative L هتطلع
+
+49
+00:03:00,430 --> 00:03:03,190
+موجودة والـ derivative continuous يعني
+
+50
+00:03:03,190 --> 00:03:06,510
+continuously differentiable بالرغم من أنها القوة
+
+51
+00:03:06,510 --> 00:03:09,650
+اللي موجودة في الـ sequence g of n of x من ناحية
+
+52
+00:03:09,650 --> 00:03:14,220
+أنها continuously differentiable إلا إنه لما جينا
+
+53
+00:03:14,220 --> 00:03:17,880
+ناخد limit GN of X as N goes to infinity طلعت
+
+54
+00:03:17,880 --> 00:03:21,280
+بتساوي G of X اللي هي 0 X في الفترة 01 و1 في
+
+55
+00:03:21,280 --> 00:03:25,180
+الفترة X اللي هو بتساوي 1 يعني اللي طلعتلنا الـ
+
+56
+00:03:25,180 --> 00:03:31,040
+limit إلها طلعتلنا إياها is not continuous يعني
+
+57
+00:03:31,040 --> 00:03:35,080
+بالرغم إن الـ limit موجودة طبعا هذا pointwise
+
+58
+00:03:35,080 --> 00:03:38,520
+convergence اللي هو الـ pointwise convergence هنا
+
+59
+00:03:38,520 --> 00:03:44,660
+طلع لي اللي هو الـ sequence of functions GN limitها
+
+60
+00:03:44,660 --> 00:03:48,820
+مش continuous ماقدرش يطلع اللي هو continuous
+
+61
+00:03:48,820 --> 00:03:52,560
+function هذا الـ pointwise convergence بالرغم إنه
+
+62
+00:03:52,560 --> 00:03:56,740
+GN of X is اللي هو continuously differentiable
+
+63
+00:03:56,740 --> 00:04:03,550
+الآن ناخد مثال آخر إذا إذا كانت GN هي عملية عملية
+
+64
+00:04:03,550 --> 00:04:04,930
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+65
+00:04:04,930 --> 00:04:15,930
+عملية عملية عملية عملية عملية عملية عملية
+
+66
+00:04:15,960 --> 00:04:19,840
+طيب نشوف الـ example اللي بعده الآن الـ example اللي
+
+67
+00:04:19,840 --> 00:04:25,400
+بعده بقول لي اللي هو let FN be هذه let FN من 0
+
+68
+00:04:25,400 --> 00:04:30,460
+لعند 1 من 01 لعند R be defined for N أكبر سواء 2
+
+69
+00:04:30,460 --> 00:04:37,150
+by مراكبين دالة شوف هذه الدالة عندي Fn of X بساوي
+
+70
+00:04:37,150 --> 00:04:42,490
+N تربيع X الـ N هذا دالة المتغير X هذه الـ Fn و
+
+71
+00:04:42,490 --> 00:04:45,030
+الـ N تربيعها دولة اللي هنا اللي بدلنا على اللي هو
+
+72
+00:04:45,030 --> 00:04:49,810
+الـ sequence F1, F2, F3, F4 أما الدالة هي F منين
+
+73
+00:04:49,810 --> 00:04:55,280
+بدلالة X الـ Fn بدلالة X إذاً هذا عدد يعني هذه
+
+74
+00:04:55,280 --> 00:04:59,280
+Linear الدلة N تربيع X في الفترة من Zero لعند مين
+
+75
+00:04:59,280 --> 00:05:03,140
+واحدة على N لو جينا رسمناها اللي هي N تربيع عند
+
+76
+00:05:03,140 --> 00:05:07,620
+طبعا دي خطية بكفي نقطتين نرسم عشان نحط خط بينهم عند
+
+77
+00:05:07,620 --> 00:05:12,900
+X بصفر بتطلع اللي هي الأولى اللي هي N تربيع X صفر
+
+78
+00:05:12,900 --> 00:05:16,540
+الـ X واحدة على N بيصير واحدة على N في N تربيع
+
+79
+00:05:16,540 --> 00:05:19,640
+بتطلع اللي هي واحدة على N هي واحدة على N بتطلع
+
+80
+00:05:19,640 --> 00:05:23,760
+واحدة على N بتطلع قيمة as an آسف n تربيع في واحدة
+
+81
+00:05:23,760 --> 00:05:30,260
+الآن بتطلع n إذا هذه اللي هي الخط هيه للجزء اللي
+
+82
+00:05:30,260 --> 00:05:33,760
+هو الأولاني الجزء الثاني اللي هو بيبدأ من عند
+
+83
+00:05:33,760 --> 00:05:36,520
+واحدة الآن عند مين واحدة الآن عند 2 n خلينا
+
+84
+00:05:36,520 --> 00:05:41,150
+نرسمه هذا برضه إيش ماله ده اللي خطية لإنه إيش ماخده
+
+85
+00:05:41,150 --> 00:05:46,190
+عبارة عن ناقص N تربيع في X ناقص 2 على N هذه ثابت و
+
+86
+00:05:46,190 --> 00:05:50,190
+هذه ثابت بالنسبة لي X الآن X لحقها إذا ده اللي خطية
+
+87
+00:05:50,190 --> 00:05:53,890
+برضه وده الخط خليني ناخد اللي هي X بتساوي 1 على N
+
+88
+00:05:53,890 --> 00:05:58,010
+خذ 1 على N ناقص 2 على N بيطلع ناقص 1 على N في ناقص
+
+89
+00:05:58,010 --> 00:06:01,790
+N تربيع بيطلع 1 على N فعلا 1 على N بيطلع 1 على N و
+
+90
+00:06:01,790 --> 00:06:06,150
+N فبيطلع مين؟ لأن هذا بيطلع 1 الآن في ناقص n
+
+91
+00:06:06,150 --> 00:06:09,850
+تربيع بتطلع اللي هي ناقص 1 الآن في ناقص n
+
+92
+00:06:09,850 --> 00:06:13,990
+تربيع بتطلع اللي هي n n إذا هذه 1 الآن بتطلع
+
+93
+00:06:13,990 --> 00:06:17,590
+أيهاش بالسواء اللي بعيدها النقطة الأخيرة 2 على
+
+94
+00:06:17,590 --> 00:06:20,370
+n 2 على الناقص 2 على n صفر هذه بيصير
+
+95
+00:06:20,370 --> 00:06:24,480
+أيهاش صفر ترجع وين على صفر فبتنزل في الشكل هذا باجي
+
+96
+00:06:24,480 --> 00:06:27,540
+الدالة من عند اللي هو طبعا معرف عالميا على الفترة
+
+97
+00:06:27,540 --> 00:06:31,720
+Zero و1 هي من Zero لعندي 2 على N الآن N
+
+98
+00:06:31,720 --> 00:06:34,340
+بتساوي 1 أو 2 أو 3 أو 4 أو 5 أو
+
+99
+00:06:34,340 --> 00:06:37,580
+6 لبديكة كل دالة من هنا بتختلف عن التانية يعني
+
+100
+00:06:37,580 --> 00:06:43,220
+هذه FN in general F 1 F 2 F 3 بتيجي
+
+101
+00:06:43,220 --> 00:06:46,360
+لهيك أو بتروح لهيك بس بنفس اللي هي الشكل الأمامي
+
+102
+00:06:46,700 --> 00:06:49,900
+إيش عندي في فترة ما 2 على n 1، ما 2 على
+
+103
+00:06:49,900 --> 00:06:53,340
+n 1، الفترة ها دي ماخدين الدالة إيش بتساوي؟
+
+104
+00:06:53,340 --> 00:06:57,440
+بتساوي إيش؟ صفر واضح إن الدالة هي الدالة محترمة
+
+105
+00:06:57,440 --> 00:07:02,880
+دالة is continuous Madame Continuous أكيد شمالها
+
+106
+00:07:02,880 --> 00:07:08,060
+is integrable إذا الدوال هذول F 1 و F 2 و
+
+107
+00:07:08,060 --> 00:07:11,480
+F 3 و F 4 و F 5 و F 6 إلى ما لا
+
+108
+00:07:11,480 --> 00:07:17,380
+نهاية من الـ Fns بتكون عبارة عن .. عبارة عن
+
+109
+00:07:17,380 --> 00:07:21,600
+integrable أو F 2 وطالع Integrable Functions
+
+110
+00:07:21,600 --> 00:07:25,760
+من n وطالع .. من n 1 بيصير في المشكلة طيب
+
+111
+00:07:25,760 --> 00:07:29,520
+Note that Fn of X is continuous لكل n أكبر ساوية
+
+112
+00:07:29,520 --> 00:07:34,180
+2 ناخد الفكرة احنا Hence, Integrable and الـ
+
+113
+00:07:34,180 --> 00:07:38,520
+integration من 0 لأن 1 Fn of X DX إيش بيساوي؟
+
+114
+00:07:38,520 --> 00:07:42,060
+بيساوي اللي هو 1 بتعرفوا تحسبوها هي مساحة ..
+
+115
+00:07:42,060 --> 00:07:48,500
+مساحة المثلت هذا اللي هو 2 على n اللي هو مضروب
+
+116
+00:07:48,500 --> 00:07:54,360
+في مين في n بطلع جداش 2 في نص بطلع 1 عارف أن
+
+117
+00:07:54,360 --> 00:07:56,880
+النص القاعدة في الارتفاع إذا الجداش بطلع عندي اللي
+
+118
+00:07:56,880 --> 00:08:00,100
+هو قيمة الـ integration هذا بساوي 1 لأن هذا
+
+119
+00:08:00,100 --> 00:08:04,060
+المنطقة إيه شمالها صفر إذا اللي هي قيمة الـ
+
+120
+00:08:04,060 --> 00:08:08,480
+integration هذا بساوي 1 يعني الـ function of
+
+121
+00:08:08,480 --> 00:08:12,980
+ands integrable وقيمة اللي هي الـ integration بساوي
+
+122
+00:08:12,980 --> 00:08:18,850
+1 لكل n أكبر أو يساوي 2 n بيساوي 1
+
+123
+00:08:18,850 --> 00:08:21,490
+اعوضها لحالك وشوف إيش اللي بيصير Note that as n
+
+124
+00:08:21,490 --> 00:08:25,490
+goes to infinity اللي
+
+125
+00:08:25,490 --> 00:08:31,310
+هو الفترات اللي عليها عشان نورجيكم إياها طيب هذه
+
+126
+00:08:31,310 --> 00:08:35,230
+الدالة معرفها as n goes to infinity هذه الفترة
+
+127
+00:08:35,230 --> 00:08:39,050
+بتصير عبارة عن اللي هو النقطة zero وهذه الفترة
+
+128
+00:08:39,050 --> 00:08:43,130
+بتصير النقطة zero و 2 على n بتصير اللي هي
+
+129
+00:08:43,130 --> 00:08:48,430
+mean zero لـ عند 1 and so الدالة إيش هتصير limit
+
+130
+00:08:48,430 --> 00:08:54,410
+fn of x بتساوي f of x إيش هتساوي هتساوي zero يعني
+
+131
+00:08:54,410 --> 00:08:59,450
+الآن لما n تروح لما للنهاية هذه كلها إيش مالها
+
+132
+00:08:59,450 --> 00:09:03,870
+بتنطبق عالمين لإن كل ما كبرت n بتفرد هذه بتفرد
+
+133
+00:09:03,870 --> 00:09:07,170
+هذه بتفرد هذه لإن as n goes to infinity هذا الكلام
+
+134
+00:09:07,170 --> 00:09:12,210
+ده اللي هتفرد وتمشي عالمين على اللي هي من صفر
+
+135
+00:09:12,210 --> 00:09:16,250
+لـ عند 1 هتظلها إيش قيمتها قيمتها صفر فاهمين عليش
+
+136
+00:09:16,250 --> 00:09:21,600
+بقول الآن as n goes to infinity هذه الآن بتبدأ اللي
+
+137
+00:09:21,600 --> 00:09:26,000
+هي .. هذه بيصير الآن نص .. هذه المنطقة كلها .. آسف
+
+138
+00:09:26,000 --> 00:09:28,040
+.. هذه المنطقة كلها هذا اللي بيتمد .. مش هدول ..
+
+139
+00:09:28,040 --> 00:09:31,440
+هذه اللي بيتمد اللي هي 2 على n بتجربها n ..
+
+140
+00:09:31,440 --> 00:09:34,220
+بتصغر as n goes to infinity 1 على n بتجربها
+
+141
+00:09:34,220 --> 00:09:37,850
+n لأن as n goes to infinity بيهدفك بروح لمين في دا
+
+142
+00:09:37,850 --> 00:09:42,750
+الصفر يعني بيصير ده لكلها من معرفة كـ zero function
+
+143
+00:09:42,750 --> 00:09:46,370
+يعني limit of n of x as n goes to infinity بيساوي
+
+144
+00:09:46,370 --> 00:09:54,030
+اللي هي صفر لكل x limit الفترة 01 مدام صفر أه إذا
+
+145
+00:09:54,030 --> 00:09:57,530
+صار عنده اللي هو لاحظوا الـ convergence pointwise
+
+146
+00:09:57,530 --> 00:10:01,350
+convergence بعتمد عالميا على x في كل حاجة اللي هو
+
+147
+00:10:01,350 --> 00:10:05,010
+الـ integration من صفر اللي عند 1 f of x dx إيش
+
+148
+00:10:05,010 --> 00:10:09,630
+هيساوي؟ هيساوي صفر لأن قيمة الـ function كله وين
+
+149
+00:10:09,630 --> 00:10:14,510
+هتيجي على اللي هي الصفر فبكون الـ integration اللي
+
+150
+00:10:14,510 --> 00:10:19,770
+بيساوي صفر اللي بدي أنا .. اللي بدي أفله نلاحظ
+
+151
+00:10:20,580 --> 00:10:27,480
+اللي وصلت له أن الـ integration من 0 و1 للـ F of X
+
+152
+00:10:27,480 --> 00:10:33,040
+DX بيساوي 0 أكيد بيساويش 1 اللي هو بيساوي اللي هو
+
+153
+00:10:33,040 --> 00:10:39,510
+مين هذا الـ 1 limit للـ integration من zero and
+
+154
+00:10:39,510 --> 00:10:46,170
+1 F N of X DX as N goes to infinity مظبوط؟ لأن
+
+155
+00:10:46,170 --> 00:10:49,530
+الـ integration هذا طلع دايما إيش حيساوي؟ حيساوي
+
+156
+00:10:49,530 --> 00:10:54,130
+1 بغض النظر عن قيمة الـ N إذا هذا الـ limit للـ
+
+157
+00:10:54,130 --> 00:10:59,270
+integration هاي اللي بتوصله وليس شرط يساوي إيش الـ
+
+158
+00:10:59,270 --> 00:11:03,250
+integration للـ limit رغم إن الـ F طلعت مالها
+
+159
+00:11:03,250 --> 00:11:09,070
+Integrable الـ limit يعني الـ n limit أف ان of X لما
+
+160
+00:11:09,070 --> 00:11:14,810
+n تروح لنهاية اللي عندي إيش بيساوي F of X بالرغم
+
+161
+00:11:14,810 --> 00:11:19,810
+من هيك اللي هو هذا الكلام لكل X وين موجودة لكل X
+
+162
+00:11:19,810 --> 00:11:25,990
+موجودة في الفترة 01 بالرغم من هيك الآن الـ
+
+163
+00:11:25,990 --> 00:11:31,790
+integration limit للـ integration من 0 لـ 1 للـ F of X,
+
+164
+00:11:32,290 --> 00:11:39,570
+N of X, DX ما سواش للـ integration من 0 لـ 1 limit لـ
+
+165
+00:11:39,570 --> 00:11:45,070
+F, N of X يعني إيش اللي بقصده اللي هو ماقدرتش الـ
+
+166
+00:11:45,070 --> 00:11:49,710
+limit تدخل لجوه الـ integration في ضوء الـ pointwise
+
+167
+00:11:49,710 --> 00:11:53,470
+convergence وفي ضوء أن الـ limit نفسها طلعت
+
+168
+00:11:53,470 --> 00:11:58,050
+integrable إذا الآن هنا الـ pointwise convergence
+
+169
+00:11:58,050 --> 00:12:02,570
+قد يعطي أنه اللي هو الـ function تكون اللي هو
+
+170
+00:12:02,570 --> 00:12:07,560
+integrable لكن لا يمكن أنه .. أو .. أيه .. أو ..
+
+171
+00:12:07,560 --> 00:12:11,280
+آسف ليس شرطا أن يكون limit للـ integration بساوي الـ
+
+172
+00:12:11,280 --> 00:12:14,600
+integration للـ limit إذا بدنا دفعة أكبر من الـ
+
+173
+00:12:14,600 --> 00:12:17,820
+pointwise convergence والحالتين الـ continuity و
+
+174
+00:12:17,820 --> 00:12:21,600
+الـ integrability هانفتنتين
+
+201
+00:14:16,630 --> 00:14:20,470
+زي ما أنتم عارفين such that if N أكبر أو يساوي H then
+
+202
+00:14:20,470 --> 00:14:24,610
+F N of X ناقص F of X أصغر من Epsilon على ثلاثة 
+
+203
+00:14:24,610 --> 00:14:27,230
+أصغر من Epsilon في الدنيا من ضمنها مين؟ Epsilon
+
+204
+00:14:27,230 --> 00:14:32,010
+على ثلاثة for all X element in مين؟ in A إذاً الآن
+
+205
+00:14:32,010 --> 00:14:41,540
+أول حاجة لكل Epsilon أكبر من 0 لجينا H such that
+
+206
+00:14:41,540 --> 00:14:46,780
+element of N طبعاً such that for every N أكبر أو
+
+207
+00:14:46,780 --> 00:14:55,100
+يساوي H عندي F N of X نقص F of X أصغر من Epsilon على ثلاثة
+
+208
+00:14:55,100 --> 00:15:04,390
+لكل X element من A هذه أول واحدة لأن let C element
+
+209
+00:15:04,390 --> 00:15:09,870
+in A be arbitrary but fixed ليش؟ بدي أثبت لك أن F
+
+210
+00:15:09,870 --> 00:15:16,050
+continuous on A، F تبعتنا، هي المطلوب كيف بدي
+
+211
+00:15:16,050 --> 00:15:20,150
+أثبتها؟ بدي أخد أي C Fixed لكن Arbitrary إن إيه
+
+212
+00:15:20,150 --> 00:15:23,230
+بما إنه C Arbitrary لو أثبتنا F Continuous and
+
+213
+00:15:23,230 --> 00:15:26,190
+الـC بتكون F Continuous على كل الـMain الـA كيف
+
+214
+00:15:26,190 --> 00:15:31,530
+بدي أثبتها؟ بدي أثبت لك Limit F of X لما X بتروح
+
+215
+00:15:31,530 --> 00:15:36,430
+للـC إيش بتساوي F of C إيش معناه هذا؟ معناه لكل Epsilon
+
+216
+00:15:36,430 --> 00:15:40,910
+أكبر من صفر بدي لاجي K element none such that as
+
+217
+00:15:40,910 --> 00:15:49,500
+تلاقي Delta اللي هو أكبر من صفر such that X minus C
+
+218
+00:15:49,500 --> 00:15:55,280
+أصغر من Delta يعطيني F of X ناقص F of C أصغر من
+
+219
+00:15:55,280 --> 00:16:00,260
+مين؟ من Epsilon هذا اللي بدأ أثبته عشان أصل لمين؟ لل F
+
+220
+00:16:00,260 --> 00:16:04,440
+continuous عند ال C اللي هي كانت arbitrary في ال A
+
+221
+00:16:04,440 --> 00:16:08,620
+إذاً F continuous على كل ال A ده هنشوفه الآن طيب
+
+222
+00:16:10,260 --> 00:16:13,740
+استخدمها معناه الـ for convergence هيه وأخدت C
+
+223
+00:16:13,740 --> 00:16:18,800
+arbitrary point but fixed in A then الآن احسب مالي
+
+224
+00:16:18,800 --> 00:16:24,420
+اف .. طبعاً هذا الكلام بس جاب المبدأ هنا هذا صحيح
+
+225
+00:16:24,420 --> 00:16:31,800
+لكل N أكبر أو يساوي H من ضمنها صحيح للـH يعني
+
+226
+00:16:31,800 --> 00:16:36,260
+هلاجي أن FH of X ناقص F of X أصغر من مين؟ من
+
+227
+00:16:36,260 --> 00:16:41,540
+Epsilon على ثلاثة لكل X element in A برضه صحيح لل F H
+
+228
+00:16:41,540 --> 00:16:48,840
+of C ناقص F of C أصغر من Epsilon على ثلاثة لكل X element A صحيح
+
+229
+00:16:48,840 --> 00:16:52,340
+من ضمن ال X والC وغير و الاخر هي كل العناصر اللي
+
+230
+00:16:52,340 --> 00:16:57,520
+في ال A احسب لي الآن F of X ناقص F of C أصغر أو
+
+231
+00:16:57,520 --> 00:17:03,120
+يساوي F of X ناقص F H of X زائد F H of X ناقص F H
+
+232
+00:17:03,120 --> 00:17:06,980
+of C زائد F H of C ناقص F of C يعني إيش اللي سويته
+
+233
+00:17:06,980 --> 00:17:11,680
+اللي هو ضفت قمتين و طرحتين ضفت ال H of X F H of X
+
+234
+00:17:11,680 --> 00:17:16,320
+وضفت ال F H of C عشان بدي أستخدم اللي فوق عشان أصل
+
+235
+00:17:16,320 --> 00:17:24,660
+للي بديها شوف كيف اللي عندي الـ Fx ناقص Fh of X من
+
+236
+00:17:24,660 --> 00:17:29,320
+اللي فوق أصغر من Epsilon على ثلاثة هذه نزلوها زي ما هي
+
+237
+00:17:29,320 --> 00:17:33,240
+تسيبوكوا منها هذه Fh of C ناقص F of C أصغر من Epsilon
+
+238
+00:17:33,240 --> 00:17:38,760
+برضه من Epsilon على ثلاثة إذاً الآن اللي حصلته لكل Epsilon أكبر
+
+239
+00:17:38,760 --> 00:17:45,620
+من 0 لجيت H أبحث
+
+240
+00:17:46,530 --> 00:17:54,690
+إنه لكل N أكبر أو يساوي H طلع عندي F X ناقص F of C
+
+241
+00:17:54,690 --> 00:18:00,470
+أصغر من Epsilon على ثلاثة وزائد Epsilon على ثلاثة يعني
+
+242
+00:18:00,470 --> 00:18:04,910
+اثنين Epsilon على ثلاثة زائد ال absolute value ل F H
+
+243
+00:18:04,910 --> 00:18:14,170
+of X ناقص F H of C هذه الآن اللي هيسعفني أن هو
+
+244
+00:18:14,170 --> 00:18:18,610
+مفترض لي أن ال FN اللي هي ال sequence كلها
+
+245
+00:18:18,610 --> 00:18:24,010
+continuous عند ال A كلها، إذاً أكيد ال FH اللي هي
+
+246
+00:18:24,010 --> 00:18:27,810
+واحدة من ال sequence is continuous عند مين؟ عند كل
+
+247
+00:18:27,810 --> 00:18:31,930
+العناصر اللي في ال A، من ضمن هنا ال C اللي أخدت
+
+248
+00:18:31,930 --> 00:18:37,490
+fixed but arbitrary و هيك، هتكون أنا بديت أنهي
+
+249
+00:18:37,490 --> 00:18:42,700
+البرهان، شوف كيف هذه حصلناها طيب الآن أكيد هيقول لنا
+
+250
+00:18:42,700 --> 00:18:47,860
+since FH is continuous at C مدام continuous at C
+
+251
+00:18:47,860 --> 00:18:53,800
+إذاً لكل Epsilon أكبر من صفر بقدر ألاقي Delta الـ Delta
+
+252
+00:18:53,800 --> 00:18:57,700
+هذه طبعاً بدي أنوّه لك ويقول لك الـ Delta أكيد هتعتمد
+
+253
+00:18:57,700 --> 00:19:01,920
+على مين؟ على الـ Epsilon وهتعتمد على الـ C لأنه أنا باخد
+
+254
+00:19:01,920 --> 00:19:06,790
+ال continuity لعند مين؟ عند النقطة C وهتعتمد على ال
+
+255
+00:19:06,790 --> 00:19:11,470
+FH لأنه ما أخد ال continuity لمن أنا لل FH إذاً
+
+256
+00:19:11,470 --> 00:19:17,610
+الدلتا هذه هي بتعتمد على ال FH و C لل Epsilon على 3 لأنه
+
+257
+00:19:17,610 --> 00:19:24,650
+أنا بدأ استخدم limit FH of X لما X تروح للـC بسوء
+
+258
+00:19:24,650 --> 00:19:30,590
+FH of C يعني لكل Epsilon أكبر من 0 there exists Delta
+
+259
+00:19:30,590 --> 00:19:35,630
+الـDelta هذه لمين؟ تعتمد على الـC وتعتمد على الـFH
+
+260
+00:19:35,630 --> 00:19:40,470
+وتعتمد على مين؟ على اللي هي الـEpsilon اللي بتاخدها
+
+261
+00:19:40,470 --> 00:19:45,340
+arbitrarily ماشي الحال طيب إذاً التعريف هذا بدي
+
+262
+00:19:45,340 --> 00:19:48,860
+أترجمه لكل إبسلون أكبر من صفر there exist اللي هي
+
+263
+00:19:48,860 --> 00:19:53,300
+Delta أكبر من صفر such that لما X minus C كان أصغر
+
+264
+00:19:53,300 --> 00:19:57,420
+من Delta والـ X في الـ A يعطيني الفرق بين التنتين
+
+265
+00:19:57,420 --> 00:20:00,620
+هدولة أصغر من مين؟ من أي إبسلون في الدنيا من ضمن
+
+266
+00:20:00,620 --> 00:20:04,820
+Epsilon على ثلاثة واضحة الصورة الآن هذه الآن بدي
+
+267
+00:20:04,820 --> 00:20:09,090
+أدمجها مع حد بيصير عندي لكل إبسلون أكبر من صفر
+
+268
+00:20:09,090 --> 00:20:13,330
+there exist Delta أكبر من صفر such that for every
+
+269
+00:20:13,330 --> 00:20:18,610
+X element in A بتحقق X minus C أصغر من Delta اللي
+
+270
+00:20:18,610 --> 00:20:26,230
+هو بيعطيني then F of X ناقص F of C أصغر من 2 Epsilon ع 3
+
+271
+00:20:27,010 --> 00:20:31,930
+ومن هذه أصغر من Epsilon على ثلاثة يعني بمعنى آخر أصغر من
+
+272
+00:20:31,930 --> 00:20:37,270
+Epsilon وهذا بالظبط معناه أنه limit f of x لما x تروح ل
+
+273
+00:20:37,270 --> 00:20:40,570
+c بيساوي f of c وهذا كانت c element in a
+
+274
+00:20:40,570 --> 00:20:46,170
+arbitrarily إذاً f is continuous on a إذاً خلصنا
+
+275
+00:20:46,170 --> 00:20:51,690
+النظرية الأولى اللي هي اللي بتتعلق بال sequence of
+
+276
+00:20:51,690 --> 00:20:52,430
+continuous functions
+
+277
+00:20:58,670 --> 00:21:05,130
+طيب نشوف ال .. ال remark يا جماعة ال
+
+278
+00:21:05,130 --> 00:21:07,430
+remark بيقول لي the limit of a sequence of
+
+279
+00:21:07,430 --> 00:21:10,370
+continuous functions may be continuous but the
+
+280
+00:21:10,370 --> 00:21:13,590
+convergence is pointwise يعني بيقول لك يعني ممكن أنت
+
+281
+00:21:13,590 --> 00:21:17,250
+تلاقي لك اللي هو sequence of functions continuous
+
+282
+00:21:17,250 --> 00:21:20,870
+ويكون ال pointwise convergence ويروح لمين؟ ل
+
+283
+00:21:20,870 --> 00:21:25,660
+continuous functions طبعاً ممكن أكيد و جبلنا دوال ..
+
+284
+00:21:25,660 --> 00:21:30,920
+دوال من هالنوع اللي هو كان عندي اللي هي ال .. إذاً
+
+285
+00:21:30,920 --> 00:21:37,480
+بتتذكروا أظن انتبهنا قبل شوية أن
+
+286
+00:21:37,480 --> 00:21:48,100
+fn of x بساوي x على n هذه continuous طبعاً on R و F
+
+287
+00:21:48,100 --> 00:21:53,840
+of X اللي هي limitها limit F N of X لما N تروح لما
+
+288
+00:21:53,840 --> 00:21:57,940
+النهاية بساوي اللي هو عبارة عن صفر هذه برضه أي
+
+289
+00:21:57,940 --> 00:22:02,000
+شمالها continuous function بالرغم من أن ال F N
+
+290
+00:22:02,000 --> 00:22:06,780
+بتروح لل F pointwise but ال F N does not
+
+291
+00:22:06,780 --> 00:22:10,660
+converge لل F uniformly زي ما أثبتنا يعني ممكن
+
+292
+00:22:10,660 --> 00:22:14,280
+اللي هو ال pointwise convergence يكون limite
+
+293
+00:22:14,280 --> 00:22:16,420
+continuous function لما تكون sequence of
+
+294
+00:22:16,420 --> 00:22:20,810
+continuous functions طبيعي لكن احنا بنقول الضمان
+
+295
+00:22:20,810 --> 00:22:25,010
+أن يطلع continuous أن تكون uniform convergence
+
+296
+00:22:25,010 --> 00:22:30,830
+وجبنا مثال على أن اللي هو اللي هي sequence of
+
+297
+00:22:30,830 --> 00:22:34,130
+continuous functions converge pointwise to some
+
+298
+00:22:34,130 --> 00:22:36,730
+function that is not continuous زي ما شفنا في
+
+299
+00:22:36,730 --> 00:22:41,610
+المثال الأول نيجي الآن نحكي عن موضوع أو الجزء
+
+300
+00:22:41,610 --> 00:22:47,770
+المتعلق بمين؟ الجزء المتعلق بالـ Integrable
+
+301
+00:22:47,770 --> 00:22:54,370
+Functions الآن نشوف إيش العلاج لإنه نضمن الـ Limit
+
+302
+00:22:54,370 --> 00:22:56,730
+للـ Integration بساوي الـ Integration للـ Limit أو
+
+303
+00:22:56,730 --> 00:22:59,850
+اللي هو لما ندخل الـ Limit للـ Integration متى
+
+304
+00:22:59,850 --> 00:23:04,590
+بنقدر ندخله ويظل المحافظة على المساواة Let FN be a
+
+305
+00:23:04,590 --> 00:23:07,750
+sequence of functions that are integrable on A وB
+
+306
+00:23:08,610 --> 00:23:13,110
+and suppose that FN converges uniformly من A و B
+
+307
+00:23:13,110 --> 00:23:19,150
+to F then إذا لما تكون ال convergence uniform then
+
+308
+00:23:19,150 --> 00:23:23,850
+ال F is integrable من A و B هاي واحد اثنين ال
+
+309
+00:23:23,850 --> 00:23:27,210
+integration لل F of X DX من A لعين B بساوي ال limit
+
+310
+00:23:27,210 --> 00:23:29,870
+لل integration يعني بمعنى آخر ال limit لل
+
+311
+00:23:29,870 --> 00:23:33,910
+integration بساوي ال integration لل limit لما نكون
+
+312
+00:23:33,910 --> 00:23:38,490
+هذا ماله uniform convergence خلّينا نشوف اللي هو
+
+313
+00:23:38,490 --> 00:23:43,270
+البرهان الآن خلّينا نسمي الفترة تبعتنا let J
+
+314
+00:23:43,270 --> 00:23:47,950
+بتساوي A وB الآن بما أن FN بتروح لل F uniform اللي
+
+315
+00:23:47,950 --> 00:23:53,540
+في صناعة الكلام هذا الآن تنسوش أن الـ Fn is
+
+316
+00:23:53,540 --> 00:23:56,320
+integrable يعني إيش ما لها؟ bounded يعني ال
+
+317
+00:23:56,320 --> 00:23:58,200
+sequence of bounded functions و زي ما صارت ال
+
+318
+00:23:58,200 --> 00:24:01,980
+sequence of bounded functions إذاً automatic بتتحقق
+
+319
+00:24:01,980 --> 00:24:05,900
+أن اللي هي اللي ما اللي هي تبعت ال norm لل F ناقص
+
+320
+00:24:05,900 --> 00:24:13,130
+Fm اللي هو بتروح للسفر يعني Fn يعني Fn بتروح للـ F
+
+321
+00:24:13,130 --> 00:24:18,850
+uniformly معناه Fn ناقص F اللي هو إيه شماله over
+
+322
+00:24:18,850 --> 00:24:22,650
+some .. over إيه أو اللي مسميها J goes to mean to
+
+323
+00:24:22,650 --> 00:24:26,890
+zero يعني normal Fn ناقص الـ F أصغر من Epsilon يعني
+
+324
+00:24:26,890 --> 00:24:29,190
+forever Epsilon نقوم نسيف it there exists كي لم
+
+325
+00:24:29,190 --> 00:24:31,710
+تنساش that forever أن أكبر من أو يساوي K أف كن ناقص أف
+
+326
+00:24:31,710 --> 00:24:36,150
+أصغر من مين؟ من الـ Epsilon عارفين هالقصة طيب .. طيب نيجي
+
+327
+00:24:36,150 --> 00:24:40,630
+الآن let J يساوي a و b since fn converts uniformly
+
+328
+00:24:40,630 --> 00:24:44,650
+to f و since fn is integrable then fn is bounded
+
+329
+00:24:44,650 --> 00:24:48,830
+then we can use our lemma then for every epsilon
+
+330
+00:24:48,830 --> 00:24:52,510
+there exists k of epsilon such that if n أكبر أو يساوي
+
+331
+00:24:52,510 --> 00:24:58,940
+k of epsilon هذه we have f ناقص fn أصغر من Epsilon
+
+332
+00:24:58,940 --> 00:25:01,540
+على أربعة في B minus A أصغر من Epsilon في الدنيا
+
+333
+00:25:01,540 --> 00:25:04,280
+من ضمنها Epsilon على أربعة في B minus A هذه
+
+334
+00:25:04,280 --> 00:25:08,760
+للحسابات زي العادة ما بنحكي فيها بنحكي عنها وطبعاً
+
+335
+00:25:08,760 --> 00:25:14,400
+في لها تفسير وفسرنا كتير في أوقات
+
+336
+00:25:14,400 --> 00:25:19,580
+سابقة هذه سميها واحد الآن اللي سمينا ال K of
+
+337
+00:25:19,580 --> 00:25:23,520
+Epsilon هذه سمينا هي إيش؟ K للتسهيل بس الآن FK is
+
+338
+00:25:23,520 --> 00:25:28,590
+integrable اه then مدام integrable إذاً there exist
+
+339
+00:25:28,590 --> 00:25:33,730
+a partition بي إبسلون لل FK هذه X not X واحد X and
+
+340
+00:25:33,730 --> 00:25:37,470
+such that ال U بي إبسلون و FK نقص ال إبسلون بي
+
+341
+00:25:37,470 --> 00:25:40,190
+إبسلون و FK أصغر من مين؟ من أي إبسلون في الدنيا
+
+342
+00:25:40,190 --> 00:25:43,370
+نضيب أنّه إبسلون على اثنين عارفين هذي إيش؟ هذي
+
+343
+00:25:43,370 --> 00:25:45,950
+اللي هي ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+344
+00:25:45,950 --> 00:25:46,930
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+345
+00:25:46,930 --> 00:25:46,950
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+346
+00:25:46,950 --> 00:25:47,750
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+347
+00:25:47,750 --> 00:25:49,330
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+348
+00:25:49,330 --> 00:25:49,930
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+349
+00:25:49,930 --> 00:25:53,420
+.. ال .. ال .. ال .. ال .. ال .. الإذا صار عندي
+
+350
+00:25:53,420 --> 00:25:57,640
+الآن من الـ Integrability للـ FK حصلنا ع بارتيشين
+
+351
+00:25:57,640 --> 00:26:01,100
+بي إبسلون بحيث أن ال Upper نقص ال Lower أصغر من
+
+352
+00:26:01,100 --> 00:26:07,800
+الـ Epsilon على 2 لهذا ال بارتيشين الآن عندي فوق F of X
+
+353
+00:26:07,800 --> 00:26:11,520
+نقص FK of X أصغر من Epsilon على 2 على 4 B minus A for
+
+354
+00:26:11,520 --> 00:26:16,410
+every X element in A by 2 by 1 then خلّينا اللي هو
+
+355
+00:26:16,410 --> 00:26:20,850
+نكمل ال .. ال .. ال .. البرهان هي هذه اللي هي
+
+356
+00:26:20,850 --> 00:26:29,290
+خلّينا نكتبها عشان اللي هو نتذكر اللي هي نلم
+
+357
+00:26:29,290 --> 00:26:37,950
+الخطوات تبعتنا اللي عندي FMF ناقص FM أو FN أصغر من
+
+358
+00:26:37,950 --> 00:26:40,910
+إبسلون على أربعة في B minus A هذه B minus A طول
+
+359
+00:26:40,910 --> 00:26:44,970
+الفترة تبعت AJ الـ J الآن الـ partition عندي اللي
+
+360
+00:26:44,970 --> 00:26:52,410
+هو U بإبسلون أو F ناقص FH أو FK ناقص منها ناقص L
+
+361
+00:26:52,410 --> 00:26:59,370
+بإبسلون و FK أصغر من إبسلون على كده؟ على اثنين
+
+362
+00:26:59,370 --> 00:27:07,470
+للحسابات طيب الآن بيقول لي من هذه من هذه أكيد أكبر
+
+363
+00:27:07,470 --> 00:27:12,650
+أو يساوي F ناقص FN of X يعني من هذه بنقدر نحصل هذا
+
+364
+00:27:12,650 --> 00:27:17,850
+صحيح لكل N أكبر أو يساوي K من ضمنها اللي هي N إيش؟
+
+365
+00:27:17,850 --> 00:27:21,890
+بتساوي زي ما احنا متعودين بنستخدم اللي بلزمنا اللي
+
+366
+00:27:21,890 --> 00:27:28,910
+هي FK يعني بيصير هذا أكبر أو يساوي اللي هو f of x
+
+367
+00:27:28,910 --> 00:27:35,250
+ناقص fn of x لكل x element in J لأن من ضمن الأنات
+
+368
+00:27:35,250 --> 00:27:40,010
+هذه اللي هي main الـ K يعني بيصير عنده fn أو زي ما
+
+369
+00:27:40,010 --> 00:27:47,110
+هو مسميها f of x ناقص fk of x absolute value صار
+
+370
+00:27:47,110 --> 00:27:53,430
+أصغر من Epsilon على أربعة بـ بـ بـ بـ بـ بـ بـ بـ
+
+371
+00:27:53,430 --> 00:27:53,490
+بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ
+
+372
+00:27:53,490 --> 00:27:54,170
+بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ بـ
+
+373
+00:27:54,170 --> 00:27:59,490
+بـ بـ بـ بـ ب
+
+401
+00:30:28,770 --> 00:30:31,090
+زي ما أنتو عارفين إيش بتساوي الـ supermom لـ F of X
+
+402
+00:30:31,090 --> 00:30:35,370
+اللي على فترة مين X اللي هي في J ناقص واحد و J
+
+403
+00:30:35,370 --> 00:30:40,670
+أكيد اللي هي الـ ها دي هتكون أصغر من Y على 4 في B
+
+404
+00:30:40,670 --> 00:30:48,220
+ناقص A زائد M J عليها عارفين ليش لأن الـ F of X F
+
+405
+00:30:48,220 --> 00:30:52,440
+of X أصغر أو يساوي هذه القيمة، مظبوط؟ إذا صارت هذه
+
+406
+00:30:52,440 --> 00:30:56,940
+القيمة عبارة عن اللي هو upper bound of the mean
+
+407
+00:30:56,940 --> 00:31:01,580
+للـ F of X، ماشي الحال صارت هذه upper bound، إذا
+
+408
+00:31:01,580 --> 00:31:04,820
+الـ least upper bound، الـ supremum، هيظل أصغر أو 
+
+409
+00:31:04,820 --> 00:31:09,140
+يساوي هذه القيمة، لأن هذه القيمة عبارة عن عدد، صار
+
+410
+00:31:09,140 --> 00:31:13,420
+الآن M J of F أصغر أو يساوي هذا المقدر، وضحت هذه
+
+411
+00:31:13,420 --> 00:31:19,780
+عملناها كتير، إذا صار عندي MJ of F أصغر أو يساوي
+
+412
+00:31:19,780 --> 00:31:28,060
+MJ of FK زائد إبسلون على أربعة في B ناقص A هذا
+
+413
+00:31:28,060 --> 00:31:33,240
+الكلام اللي هو لكل اللي هي من J واحد واتنين عند
+
+414
+00:31:33,240 --> 00:31:36,760
+الـ ناقص واحد الباقي أنتو عارفينه دايما بنعمله اضرب
+
+415
+00:31:36,760 --> 00:31:43,520
+لهنا في XJ ناقص XJ ناقص واحد واضرب هنا في XJ ناقص
+
+416
+00:31:43,520 --> 00:31:49,300
+XJ ناقص واحد وهنا في XJ ناقص XJ ناقص واحد ضربنا
+
+417
+00:31:49,300 --> 00:31:54,200
+اللي هو الترتيب المتباينة في XJ ناقص XJ ناقص
+
+418
+00:31:54,200 --> 00:31:58,780
+واحد وبعدين أخدنا الـ summation J من عند واحد لعند
+
+419
+00:31:58,780 --> 00:32:02,700
+N وأخدنا الـ summation J من عند واحد لعند N و
+
+420
+00:32:02,700 --> 00:32:06,880
+أخدنا الـ summation J من عند واحد لعند N هذا كله
+
+421
+00:32:06,880 --> 00:32:11,020
+على بعضه طبعا كل شغلنا على الـ BY اللي لاقيناها الـ
+
+422
+00:32:11,020 --> 00:32:15,180
+partition اللي لاقيناها فوق اللي هو عندي هذا عبارة
+
+423
+00:32:15,180 --> 00:32:23,430
+عن اللي هي الـ U of اللي هو الـ partition بي إبسلون
+
+424
+00:32:23,430 --> 00:32:28,290
+اللي بدي اشتغل عليه هدول الـ و mean و A اللي هي
+
+425
+00:32:28,290 --> 00:32:34,250
+الـ function F أصغر أو يساوي هذا mean هو الـ U اللي
+
+426
+00:32:34,250 --> 00:32:40,010
+هو الـ بي إبسلون والـ FK وهدول إيش هتكون عبارة عن
+
+427
+00:32:40,010 --> 00:32:44,320
+إيش عشان هيك أخدنا بي مايلوس ايه هتشوفوها الآن هذه
+
+428
+00:32:44,320 --> 00:32:50,400
+Y زائد Y على أربعة في B ناقص A الـ summation هذا
+
+429
+00:32:50,400 --> 00:32:54,160
+للـ sub intervals اللي هو زي ما أنتو عارفين X واحد
+
+430
+00:32:54,160 --> 00:32:57,780
+ناقص X not زائد X اتنين ناقص X واحد لما أصل لـ Xn ناقص
+
+431
+00:32:57,780 --> 00:33:01,600
+Xn ناقص واحد يعني هيضل في الآخر Xn ناقص X not يعني B
+
+432
+00:33:01,600 --> 00:33:06,980
+ناقص A في B ناقص A هذه بتروح مع هذه فبصير عندي
+
+433
+00:33:06,980 --> 00:33:13,610
+الـ U أصغر من الـ U هذا زي إبسلون على أربعة اللي
+
+434
+00:33:13,610 --> 00:33:17,450
+عنده اللي هو الـ U بإبسلون F أصغر يساوي U بإبسلون
+
+435
+00:33:17,450 --> 00:33:21,570
+FK زي إبسلون على أربعة Similarly وفعلا Similarly
+
+436
+00:33:21,570 --> 00:33:25,690
+بعد بدل ما نشتغل على الـ upper اللي فوق بدل ما نشتغل
+
+437
+00:33:25,690 --> 00:33:30,870
+على الـ upper هنا بنشتغل عالميا على الـ lower اللي بيطلع
+
+438
+00:33:30,870 --> 00:33:34,350
+عند الـ الـ ب إبسلون و F هذا بيبقى لكم الـ ب إبسلون
+
+439
+00:33:34,350 --> 00:33:38,030
+و F K ناقص إبسلون على أربع أصغر من الـ ب إبسلون
+
+440
+00:33:38,030 --> 00:33:44,490
+و F إذا اللي وصلنا له ما هي ليه يا جماعة اللي
+
+441
+00:33:44,490 --> 00:33:55,110
+وصلنا له أنه لكل إبسلون أكبر من 0 هناك بي إبسلون
+
+442
+00:33:55,110 --> 00:34:05,870
+مثلًا U بي إبسلون و F أصغر أو يساوي U بي إبسلون و
+
+443
+00:34:05,870 --> 00:34:09,990
+FK زائد إبسلون على أربعة
+
+444
+00:34:13,830 --> 00:34:20,130
+بإبسلون و FK ناقص إبسلون على أربعة أصغره يساوي L
+
+445
+00:34:20,130 --> 00:34:25,830
+بيبسلون و F هذه اللي وصلنا له لحد الآن نيجي الآن
+
+446
+00:34:25,830 --> 00:34:32,230
+نرتبهم مع بعض ونصل للي بدنا إياه شي معهود خدوا الآن
+
+447
+00:34:32,230 --> 00:34:38,530
+U بيبسلون و F ناقص الـ P,Y,F هذه ناقص هذه لما ضروف
+
+448
+00:34:38,530 --> 00:34:43,510
+هذه في ناقص بيصير هذه ناقص هذه معهودة وهذه بيصير
+
+449
+00:34:43,510 --> 00:34:46,690
+ناقص وهذه بيصير زائد وهذه بيصير أكبر أو يساوي
+
+450
+00:34:46,690 --> 00:34:52,570
+بتجمع هذه لهذه يعني بيصير عدى هذه ناقص هذه أصغر من
+
+451
+00:34:52,570 --> 00:34:59,130
+الـ U,P,Y,F هيهازائد هادي ناقص هادي زائد هادي يعني
+
+452
+00:34:59,130 --> 00:35:03,010
+بيصير عنده هادي يو بي إبسلون و أف كيه ناقص البي إبسلون
+
+453
+00:35:03,010 --> 00:35:06,190
+و أف كيه زائد إبسلون على أربعة وإبسلون على أربعة اللي
+
+454
+00:35:06,190 --> 00:35:10,690
+هو إبسلون على مين؟ على اتنين وهذه من الأول لما
+
+455
+00:35:10,690 --> 00:35:15,630
+نستخدمها اللي هو الـ Riemann criterion for
+
+456
+00:35:15,630 --> 00:35:19,640
+integrability للأف كيه اللي هو أوجدتنا الـ Py هو
+
+457
+00:35:19,640 --> 00:35:23,220
+أوجدتنا أن هذه ناقصة أصغر من Y على 2 يعني صار عندي
+
+458
+00:35:23,220 --> 00:35:26,820
+Y على 2 و Y على 2 إبسلون معناته هذه صارت أصغر من
+
+459
+00:35:26,820 --> 00:35:30,080
+إبسلون و بالـ Riemann criterion for integrability
+
+460
+00:35:30,080 --> 00:35:36,560
+بتطلع الـ F is إيش is integrable وهذا اللي هو
+
+461
+00:35:36,560 --> 00:35:41,200
+الإثبات لإن الـ function F is integrable  دلوقتي نثبت
+
+462
+00:35:41,200 --> 00:35:43,500
+اللي هو limit لل integration بساوي integration لل
+
+463
+00:35:43,500 --> 00:35:46,630
+limit وهذا من خلال الخطوات اللي طلعت عندنا حلّجي
+
+464
+00:35:46,630 --> 00:35:55,030
+يطلع بشكل سهل وبشكل سلس و خلينا نشوفه الآن since
+
+465
+00:35:55,030 --> 00:35:58,130
+epsilon was arbitrary then f is integrable on j زي
+
+466
+00:35:58,130 --> 00:36:02,750
+ما قلنا خد ال integration من a ل b f of x dx ناقص الـ
+
+467
+00:36:02,750 --> 00:36:07,950
+integration من a ل b fn of x dx بساوي اللي هو الـ
+
+468
+00:36:07,950 --> 00:36:10,830
+absolute value طبعاً صارت هذه integrable وهذه
+
+469
+00:36:10,830 --> 00:36:14,470
+integrable ماشي الحال إذا ال integration هذا ناقص
+
+470
+00:36:14,470 --> 00:36:17,010
+ال integration هذا بساوي ال integration لهذا ناقص
+
+471
+00:36:17,010 --> 00:36:20,210
+هذا الكل إيش ماله DX من خواص التكامل اللي أخدناها
+
+472
+00:36:20,210 --> 00:36:25,370
+سابقا وهذا نفسه أصغر أو يساوي أصغر يساوي الـ
+
+473
+00:36:25,370 --> 00:36:28,750
+integration من a ل b و أخدناها سابقا absolute value
+
+474
+00:36:28,750 --> 00:36:34,370
+of f of x ناقص fn of x لكل إيش ماله dx الـ absolute
+
+475
+00:36:34,370 --> 00:36:36,490
+value integration أصغر يساوي ال integration للـ
+
+476
+00:36:36,490 --> 00:36:42,340
+absolute value وهذا نفسه اللي هو أصغر أو يساوي norm
+
+477
+00:36:42,340 --> 00:36:47,060
+الـ F ناقص الـ Fn over اللي هي الـ interval J
+
+478
+00:36:47,060 --> 00:36:51,320
+ماشي، الآن بيصير عندي هذا أصغر أو يساوي الـ
+
+479
+00:36:51,320 --> 00:36:56,840
+integration من A لـ B norm الـ F ناقص الـ Fn over J
+
+480
+00:36:56,840 --> 00:37:03,720
+الكل إيش ماله؟ DX الآن هذا يا جماعة ثابت بيطلع
+
+481
+00:37:03,720 --> 00:37:09,520
+بره بيصير عبارة عن يساوي الـ absolute value الـ
+
+482
+00:37:09,520 --> 00:37:13,700
+norm للـ f ناقص الـ fn over j في ال integration من
+
+483
+00:37:13,700 --> 00:37:20,600
+a إلى b dx هذا هيهان وهذا ال integration من a إلى
+
+484
+00:37:20,600 --> 00:37:25,840
+b إلى dx هو عبارة عن b ناقص a صار هذا ناقص هذا
+
+485
+00:37:25,840 --> 00:37:31,200
+أصغر من هذا في b ناقص a الآن as n goes to
+
+486
+00:37:31,200 --> 00:37:34,920
+infinity as n goes to infinity صار هذا أكبر أو
+
+487
+00:37:34,920 --> 00:37:42,930
+يساوي صفر هذا هيوطلع لهذه المنطقة هيها هذه بس خد
+
+488
+00:37:42,930 --> 00:37:46,170
+الـ limit للجهتين as n goes to infinity هذا بيصير
+
+489
+00:37:46,170 --> 00:37:51,590
+الـ limit هنا as n goes to infinity وهنا الـ limit as
+
+490
+00:37:51,590 --> 00:37:54,170
+n goes to infinity طبعا هذا one هيروح لإنه أفن
+
+491
+00:37:54,170 --> 00:37:57,910
+بتروح للأف uniformly هذا بتروح كله لمين للصفر إذا
+
+492
+00:37:57,910 --> 00:38:01,330
+as n goes to infinity الـ limit هذا إيش بيساوي صفر
+
+493
+00:38:01,330 --> 00:38:04,990
+مدام الـ limit هذا صار بيساوي صفر هذا independent
+
+494
+00:38:04,990 --> 00:38:09,170
+of the limit هذا عبارة عن فيش فيه n صار عندي اللي
+
+495
+00:38:09,170 --> 00:38:12,990
+هو مضحكة
+
+496
+00:38:12,990 --> 00:38:19,770
+بتصور الصورة صار عندي الآن limit هذا بيساوي إيش صار
+
+497
+00:38:19,770 --> 00:38:24,770
+بيساوي صفر limit لال absolute value من a لعند b f
+
+498
+00:38:24,770 --> 00:38:32,170
+of x dx ناقص integration fn of x dx من a لعند b
+
+499
+00:38:32,170 --> 00:38:35,710
+limit as angles to infinity limit الـ absolute
+
+500
+00:38:35,710 --> 00:38:38,810
+value صار صفر إذا أكيد الـ limit بدون absolute
+
+501
+00:38:38,810 --> 00:38:42,690
+value بيساوي صفر هذا independent of the limit إذا
+
+502
+00:38:42,690 --> 00:38:48,210
+بيصير بساوي ال integration من a ل b f of x dx بتدخل
+
+503
+00:38:48,210 --> 00:38:54,370
+الـ limit ناقص limit ال integration fn of x dx من a
+
+504
+00:38:54,370 --> 00:38:59,310
+ل عند b ماشي الحال ولا نضحك الصورة هذا صار عليه؟
+
+505
+00:38:59,310 --> 00:39:05,780
+هذا كله على بعضه صار بساوي صفر إذا الـ limit ننجل
+
+506
+00:39:05,780 --> 00:39:08,620
+هذا على الجهة الثانية بيصير limit ال integration
+
+507
+00:39:08,620 --> 00:39:15,080
+من a ل b fn of x dx as n goes to infinity بساوي
+
+508
+00:39:15,080 --> 00:39:20,900
+اللي بيظل هنا من a ل b f of x dx وهذا معناته إنه
+
+509
+00:39:20,900 --> 00:39:24,310
+في حالة الـ uniform convergence الـ FN sequence of
+
+510
+00:39:24,310 --> 00:39:27,030
+integrable functions هتروح لـ integrable function
+
+511
+00:39:27,030 --> 00:39:29,450
+وهيطلع عنده limit لل integration بيساوي الـ
+
+512
+00:39:29,450 --> 00:39:32,150
+integration لـ limit وبهي .. وهيكون كون احنا
+
+513
+00:39:32,150 --> 00:39:37,110
+عالجنا اللي هو النقطة الثانية في اللي هو المحاضرة
+
+514
+00:39:37,110 --> 00:39:42,270
+ضال عندي نقطة أخيرة بس قبلها خليني نذكر هال نظرية
+
+515
+00:39:42,270 --> 00:39:47,470
+اللي هي برهانها خارج نطاق اللي هو الكتاب ونشوف
+
+516
+00:39:47,470 --> 00:39:49,110
+اللي هو
+
+517
+00:39:53,380 --> 00:39:58,540
+إيش بتقول هذه اللي هي النظرية؟ إيش بتقول bounded
+
+518
+00:39:58,540 --> 00:40:02,310
+convergence theorem؟ النظرية بتقول let FN be a
+
+519
+00:40:02,310 --> 00:40:07,150
+sequence of functions that are integrable in A وB
+
+520
+00:40:07,150 --> 00:40:12,530
+مفترضين إنها integrable and suppose that بدي يبعد
+
+521
+00:40:12,530 --> 00:40:14,850
+عن الـ uniform convergence بتقول لو شيلنا الـ
+
+522
+00:40:14,850 --> 00:40:18,550
+uniform convergence ممكن نبدله بإيش إيه؟ ويظل
+
+523
+00:40:18,550 --> 00:40:20,850
+محافظنا على الـ limit لل integration بساوي الـ
+
+524
+00:40:20,850 --> 00:40:24,920
+integration لل limit؟ آه، بنفع، نشوف كيف let FN be a
+
+525
+00:40:24,920 --> 00:40:27,340
+sequence of functions that are integrable in A وB
+
+526
+00:40:27,340 --> 00:40:30,380
+بس بده يدفع ثمن برضه integrable in A وB and
+
+527
+00:40:30,380 --> 00:40:34,400
+suppose that أفأن converts in A وB to an
+
+528
+00:40:34,400 --> 00:40:37,780
+integrable function F يعني أفأن بتروح للـ F على
+
+529
+00:40:37,780 --> 00:40:44,700
+الفترة هذه اللي هو point wise ماجاش uniformly بس
+
+530
+00:40:44,700 --> 00:40:50,770
+الآن بده يفترض شغلة بده يقيد الـ أفأن بشغلة معينة
+
+531
+00:40:50,770 --> 00:40:54,950
+شغلة كبيرة برضه التقييد مش جليل طبعا لكن بيضل إشي
+
+532
+00:40:54,950 --> 00:40:59,830
+يعني منفذ تاني suppose also that there exist B
+
+533
+00:40:59,830 --> 00:41:05,870
+أكبر من صفر بحيث أن FN of X أصغر شهر بيه لكل X
+
+534
+00:41:05,870 --> 00:41:09,750
+element in A و B و لكل N element in B هذا في
+
+535
+00:41:09,750 --> 00:41:14,690
+الواقع اللي هو uniform boundedness ليش؟ لأن هو
+
+536
+00:41:14,690 --> 00:41:21,200
+مفترض أن الـ sequence FN of X أصغر أو يساوي اللي هو
+
+537
+00:41:21,200 --> 00:41:28,140
+B لكل X element in A و B و لكل N element in N هو مش
+
+538
+00:41:28,140 --> 00:41:32,360
+مفترض إن الـ Fn نفسها تكون bounded يعني F1 اللي
+
+539
+00:41:32,360 --> 00:41:35,580
+حالها bounded، F2 bounded، F3 bounded كل إن هيكون
+
+540
+00:41:35,580 --> 00:41:39,940
+bounded صح، بس اللي طالبه أكثر اللي طبعا في حالة
+
+541
+00:41:39,940 --> 00:41:47,640
+الـ F1 bounded، F2 bounded، هيكون عندي Fnof X أصغر
+
+542
+00:41:47,640 --> 00:41:52,480
+أو يساوي BN F4
+
+543
+00:41:53,890 --> 00:41:58,290
+أني اللي هو الآن بنقول عن FN نفسها bounded أنه
+
+544
+00:41:58,290 --> 00:42:02,750
+there exists B لها، B لمن؟ للـ FN لو فرضنا أن FN
+
+545
+00:42:02,750 --> 00:42:05,950
+الحالها bounded، there exists B للـ N سميتها BN
+
+546
+00:42:05,950 --> 00:42:11,850
+شديتها، دي أصغر يساوي BN لكل X element in A و B هذا
+
+547
+00:42:11,850 --> 00:42:16,650
+لو فرضنا أنه كل واحدة على حدة A bounded هو طالب
+
+548
+00:42:16,650 --> 00:42:24,260
+أكثر جاي الـ bound هذا بنفع لكل الـ mean للـ F1 و
+
+549
+00:42:24,260 --> 00:42:30,980
+للـ F2 و للـ F3 و للـ F4 يعني فارض أن FN of X أصغر
+
+550
+00:42:30,980 --> 00:42:35,920
+أو يشوي يعني درج ZEF بأكبر من صفر such that FN of
+
+551
+00:42:35,920 --> 00:42:42,040
+X أصغر شهر و B لكل N element N و لكل X element مين
+
+552
+00:42:42,040 --> 00:42:48,760
+in ال interval اللي هي A أو B طبعا هذا الكلام أكيد
+
+553
+00:42:48,760 --> 00:42:52,940
+بيعطي هذا لكن هذا مش شرط يعطي هذا لأن هذا ال BN للـ
+
+554
+00:42:52,940 --> 00:42:57,540
+F1 فيه B1 الـ F2 بيه F2 F3 بيه F3 F4 بيه F4 هل فيه
+
+555
+00:42:57,540 --> 00:43:01,770
+الـ supremum هدولة ليس شرطًا ما أقدرش نقول أه عشان
+
+556
+00:43:01,770 --> 00:43:06,070
+هيك طلبه كبير طبعا البرنامج خارج عن نطاق الـ course
+
+557
+00:43:06,070 --> 00:43:10,250
+سبعنا الآن تحت هذا الـ condition اللي قلنا عنه
+
+558
+00:43:10,250 --> 00:43:14,190
+condition ما قدرنا نقول وازن then ال integration من
+
+559
+00:43:14,190 --> 00:43:17,450
+a لـ b of x dx بساوي limit لل integration من a b لـ f
+
+560
+00:43:17,450 --> 00:43:20,610
+n of x dx يعني بمعنى آخر limit لل integration بساوي
+
+561
+00:43:20,610 --> 00:43:24,950
+ال integration لل limit الآن نيجي لجزئية الثالث من
+
+562
+00:43:24,950 --> 00:43:29,350
+المحاضرة اللي بتعلق بالـ sequence of
+
+563
+00:43:29,350 --> 00:43:32,550
+differentiable functions خلينا نشوف المثال أول شيء
+
+564
+00:43:32,550 --> 00:43:37,570
+وبعدين نيجي لنظريتنا let fn of x بساوي summation
+
+565
+00:43:37,570 --> 00:43:42,270
+2 to the minus k cosine 3k x k من 1 لعين ده يعني
+
+566
+00:43:42,270 --> 00:43:47,950
+الآن F1 of X بيساوي اللي هو 2 تره ماينس واحد كوساين
+
+567
+00:43:47,950 --> 00:43:52,810
+ثلاثة X F2 of X بيساوي اللي هو 2 تره ماينس واحد
+
+568
+00:43:52,810 --> 00:43:56,930
+كوساين ثلاثة X زائد 2 تره ماينس اثنين كوساين ثلاثة
+
+569
+00:43:56,930 --> 00:44:01,070
+تربيع X وهكذا Fn of X هي عبارة عن الـ summation هذا
+
+570
+00:44:01,070 --> 00:44:04,690
+من واحد لـ N وهذه عبارة عن Sequence of Functions
+
+571
+00:44
+
+601
+00:46:19,940 --> 00:46:20,360
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+602
+00:46:20,360 --> 00:46:21,560
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+603
+00:46:21,560 --> 00:46:21,960
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+604
+00:46:21,960 --> 00:46:23,120
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+605
+00:46:23,120 --> 00:46:24,740
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+606
+00:46:24,740 --> 00:46:28,440
+عامة عامة عامة عامة عامة 
+
+607
+00:46:29,930 --> 00:46:34,590
+هذا أكيد أصغر أو يساوي اللي هو الـ absolute value of
+
+608
+00:46:34,590 --> 00:46:36,130
+summation أصغر أو يساوي الـ summation للـ absolute
+
+609
+00:46:36,130 --> 00:46:39,650
+value والـ cosine أصغر أو يساوي حدث إن صار عبارة عن
+
+610
+00:46:39,650 --> 00:46:41,890
+الـ summation الـ absolute value of summation هذا
+
+611
+00:46:41,890 --> 00:46:44,910
+أصغر أو يساوي two to the minus k كمن عند n زائد
+
+612
+00:46:44,910 --> 00:46:49,430
+واحد إلى مين إلى ما لا نهاية وهذا as n goes to
+
+613
+00:46:49,430 --> 00:46:53,210
+infinity هذا بيروح للـ zero ماشي الحال خليها في
+
+614
+00:46:53,210 --> 00:46:59,350
+الذاكرة الآن صار عندي f of x نقص fn of x أصغر أو
+
+615
+00:46:59,350 --> 00:47:04,450
+يساوي هذا الرقم ماشي مش معتمد على x صار هذا أصغر
+
+616
+00:47:04,450 --> 00:47:08,050
+أو يساوي هذا صار هذا upper bound لهذهمدام upper
+
+617
+00:47:08,050 --> 00:47:10,210
+bound إذا الـ upper bound أكبر أو يساوي الـ least
+
+618
+00:47:10,210 --> 00:47:14,170
+upper bound اللي هو norm للـ F ناقص FN over R أصغر
+
+619
+00:47:14,170 --> 00:47:17,310
+أو يساوي هذا as N goes to infinity هذا بيروح للـ
+
+620
+00:47:17,310 --> 00:47:21,250
+zero إذا الـ F ناقص FN over R بتروح للـ zero وهذا
+
+621
+00:47:21,250 --> 00:47:27,570
+معناته أن FN بتروح للـ F اللي هو on mean on R يعني
+
+622
+00:47:27,570 --> 00:47:31,250
+الـ convergence إيش ما له uniform convergence بالرغم
+
+623
+00:47:31,250 --> 00:47:35,880
+إن الـ uniform convergence حادث والـ Fn sequence of
+
+624
+00:47:35,880 --> 00:47:40,200
+differentiable functions الـ F إيش ما لها طلعتش
+
+625
+00:47:40,200 --> 00:47:44,840
+differentiable إذا بالزمن اللي هو condition أو
+
+626
+00:47:44,840 --> 00:47:49,900
+conditions تضمنلي أنه لما Fn تروح للـ F و Fn
+
+627
+00:47:49,900 --> 00:47:52,580
+sequence of differentiable functions تطلعلي الـ F
+
+628
+00:47:52,580 --> 00:47:56,930
+differentiable خلّينا نشوف نظريتنا اللي هي آخر جزء
+
+629
+00:47:56,930 --> 00:48:00,990
+في الـ section وفي الـ chapter اللي بدنا ناخده منه
+
+630
+00:48:00,990 --> 00:48:07,070
+اللي هي بتجاوبنا على هذا الأمر ونشوف إيش النظرية
+
+631
+00:48:07,070 --> 00:48:12,630
+بتقوله ومن ثم بعد أنفة من نص نبرهن النظرية نشوف
+
+632
+00:48:12,630 --> 00:48:17,400
+let J subset من R be a bounded interval bounded
+
+633
+00:48:17,400 --> 00:48:21,340
+interval open مش open مش فارقة معه a closed مش
+
+634
+00:48:21,340 --> 00:48:24,560
+closed مش فارقة الآن و and let fn be a sequence of
+
+635
+00:48:24,560 --> 00:48:29,360
+functions on j يعني عبقى عندي عندي fn sequence of
+
+636
+00:48:29,360 --> 00:48:33,940
+functions fn sequence of functions على مين؟ على
+
+637
+00:48:33,940 --> 00:48:38,180
+الـ j اللي هي الـ interval اللي عندي لعند ارف واحدة
+
+638
+00:48:38,180 --> 00:48:43,700
+اتنين ارف تلاتة الآخرين وبدنا نفترض اللي هو suppose
+
+639
+00:48:43,700 --> 00:48:49,880
+that there exist x0 element in j بحيث أن fn of x0
+
+640
+00:48:49,880 --> 00:48:56,440
+converges اللي هو to some limit الآن إيش مفترض إن
+
+641
+00:48:56,440 --> 00:49:03,520
+في عندي x0 في الـ J بحيث أن FN of X0 هذه صارت
+
+642
+00:49:03,520 --> 00:49:07,840
+sequence of numbers هذه converges to some number
+
+643
+00:49:07,840 --> 00:49:12,560
+إيش هو مش عارفينه الآن و and that ومعطيك كمان شغلة
+
+644
+00:49:12,560 --> 00:49:18,800
+هذه معطيك إياها اللي هي و and converges عند نقطة محددة
+
+645
+00:49:18,800 --> 00:49:24,320
+اسمها X0 في الـ J و and the sequence FN prime of
+
+646
+00:49:24,320 --> 00:49:29,510
+derivatives exist on J يعني مفترض إن الـ F N'
+
+647
+00:49:29,830 --> 00:49:32,910
+exists يعني F N sequence of differentiable
+
+648
+00:49:32,910 --> 00:49:36,390
+functions مفترض إن F N sequence of differentiable
+
+649
+00:49:36,390 --> 00:49:40,210
+functions ومش هيك وبيقول لك إن هذه الـ F N'
+
+650
+00:49:40,750 --> 00:49:45,230
+converges uniformly on J to some function F يعني
+
+651
+00:49:45,230 --> 00:49:50,950
+بيقول لك F N' exists FN برايم exist لكل N وهذا الـ
+
+652
+00:49:50,950 --> 00:49:55,430
+sequence بيقول لك converges uniformly to some
+
+653
+00:49:55,430 --> 00:50:00,130
+function اسمها مين؟ اسمها J إذن sequence of
+
+654
+00:50:00,130 --> 00:50:05,510
+functions لأن في X0 في الـ J بحيث أن FN of X0
+
+655
+00:50:05,510 --> 00:50:10,830
+نفسها تكون converges FN برايم converges uniformly
+
+656
+00:50:10,830 --> 00:50:17,150
+to some G to some G بكل الأنظمة مع وجود هذه الشروط
+
+657
+00:50:17,150 --> 00:50:23,210
+اللي حكينا عليها then the sequence FN converges
+
+658
+00:50:23,210 --> 00:50:32,330
+uniformly to a function F بعد هذا بعد هذا بنلاقي
+
+659
+00:50:32,330 --> 00:50:37,370
+أن FN غصب عنّا لازم تكون converges uniformly to
+
+660
+00:50:37,370 --> 00:50:44,660
+some function mean F ومش هيك والـ function F هذه
+
+661
+00:50:44,660 --> 00:50:51,260
+تطلع الـ derivative تبعتها هي مين؟ هي الـ G يعني
+
+662
+00:50:51,260 --> 00:51:01,310
+وكأنه أن الـ F N converges uniformly to F اللي هي
+
+663
+00:51:01,310 --> 00:51:07,270
+الـ derivative لها مين؟ إيش الـ G يعني F N برايم
+
+664
+00:51:07,270 --> 00:51:10,510
+بتكون uniformly convergence to some function G
+
+665
+00:51:10,510 --> 00:51:17,890
+هتطلع الـ F N converge uniformly للـ function اللي
+
+666
+00:51:17,890 --> 00:51:24,570
+تفضلها هو مين؟ هو عبارة عن الـ G طيب إذا في عندي
+
+667
+00:51:24,570 --> 00:51:28,210
+شغلتين بدنا نثبت إن FN can form convergence to F
+
+668
+00:51:28,210 --> 00:51:35,690
+والـ F' إيش بتساوي؟ بتساوي مين؟ اللي هي الـ G يعني
+
+669
+00:51:35,690 --> 00:51:42,830
+هتصير إنه الـ FN' اللي هو sequence of
+
+670
+00:51:42,830 --> 00:51:45,770
+differentiable functions بتطلع عندي هذه sequence
+
+671
+00:51:45,770 --> 00:51:49,210
+of differentiable functions converge to function
+
+672
+00:51:49,210 --> 00:51:54,110
+بتكون differentiable ليش؟ لأن F' إيش بتتساوي G؟
+
+673
+00:51:54,110 --> 00:51:57,590
+يعني F is differentiable يعني هذه بتقول إنه لو
+
+674
+00:51:57,590 --> 00:52:00,950
+الشرط ده والشرط ده تتحققوا إذن الـ sequence of
+
+675
+00:52:00,950 --> 00:52:06,290
+differentiable functions FN هتكون converges to ..
+
+676
+00:52:06,290 --> 00:52:09,110
+uniform converges طبعاً to a differentiable
+
+677
+00:52:09,110 --> 00:52:13,310
+function F إذن الشغلتين هدول إحنا بدنا نثبتهم
+
+678
+00:52:13,310 --> 00:52:18,330
+خلينا نشوف كيف نثبت هدول الشغلتين بدنا نثبت إنه
+
+679
+00:52:18,330 --> 00:52:23,460
+FN Converse Uniform للـ F وبدنا نثبت أن F' بيساوي G
+
+680
+00:52:23,460 --> 00:52:29,760
+يعني بدنا نثبت لك في النهاية إنه limit F
+
+681
+00:52:31,240 --> 00:52:37,940
+of X زائد اللي هو F of X زائد Delta X مثلاً أو F of
+
+682
+00:52:37,940 --> 00:52:45,520
+X ناقص F of C على X minus C as X بتروح للـ C بدنا
+
+683
+00:52:45,520 --> 00:52:51,980
+نثبت إنه exist ويساوي G of mean of C يعني بمعنى هذا
+
+684
+00:52:51,980 --> 00:52:56,460
+for arbitrary C element in main NG يعني كأنه في
+
+685
+00:52:56,460 --> 00:53:01,300
+النهاية بدنا نثبت لك إنه هذا exist يعني F prime of
+
+686
+00:53:01,300 --> 00:53:05,840
+C exist ويساوي main G of C أو بمعنى آخر بدنا نثبت
+
+687
+00:53:05,840 --> 00:53:08,320
+لك إنه for every Y colon of Z there exists delta
+
+688
+00:53:08,320 --> 00:53:14,490
+such that X minus C أصغر من Delta يؤدي إلى F of X
+
+689
+00:53:14,490 --> 00:53:20,270
+ناقص F of C على X minus C ناقص G of C يكون أصغر من
+
+690
+00:53:20,270 --> 00:53:24,530
+مين؟ من Y إذا أثبتنا هذا معناته وأثبتنا إن هذا
+
+691
+00:53:24,530 --> 00:53:28,460
+اللي هو limit exist يعني الـ F' exist ويساوي مين؟
+
+692
+00:53:28,460 --> 00:53:31,920
+الـ G of C والـ C كون بتكون آخذها Arbitrary في
+
+693
+00:53:31,920 --> 00:53:36,680
+الـ G فبكون خلصنا أن الـ F' بيساوي الـ G وهذا كله
+
+694
+00:53:36,680 --> 00:53:40,920
+بعد ما نثبت أن الـ F أنه conversion form للمين؟ للـ
+
+695
+00:53:40,920 --> 00:53:48,280
+F و تعالوا لِتفاصيل البرهان طيب نيجي نشوف برهاننا
+
+696
+00:53:48,280 --> 00:53:51,020
+شوفوا طاولوا روحكم علينا
+
+697
+00:53:55,700 --> 00:53:58,860
+عشان نعرف بس نشتغل لبنى حيكة خلّينا نسمي الـ end
+
+698
+00:53:58,860 --> 00:54:02,960
+points جيه واحدة اسمها A واحدة اسمها B يعني let A
+
+699
+00:54:02,960 --> 00:54:06,440
+أزرع من بي بي the end points of J جيه مفتوح مفتوح
+
+700
+00:54:06,440 --> 00:54:08,020
+مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح
+
+701
+00:54:08,020 --> 00:54:08,080
+مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح
+
+702
+00:54:08,080 --> 00:54:08,580
+مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح
+
+703
+00:54:08,580 --> 00:54:13,580
+مفتوح مفتوح مفتوح مفتوح مفتوح خلّيني أخد X بـ
+
+704
+00:54:13,580 --> 00:54:17,700
+Arbitrary Point في الـ J إحنا بنحكي عن أي X وين ما
+
+705
+00:54:17,700 --> 00:54:26,420
+لا في الـ J اللي هي .. if M and N element in N
+
+706
+00:54:26,420 --> 00:54:29,320
+then F minus N is differentiable on the closed
+
+707
+00:54:29,320 --> 00:54:34,550
+interval within points X0 and X أكيد مفترضين إحنا
+
+708
+00:54:34,550 --> 00:54:41,010
+إن الـ F M' exist مظبوط مدام exist إذا F M
+
+709
+00:54:41,010 --> 00:54:44,850
+differentiable و F N differentiable إذا F M نقص F
+
+710
+00:54:44,850 --> 00:54:48,910
+N differentiable على كل مين؟ على كل الـ J اللي أنا
+
+711
+00:54:48,910 --> 00:54:52,950
+بدي اشتغل على فترة محددة أنا أخدت X element in J
+
+712
+00:54:52,950 --> 00:54:57,490
+arbitrary وهو معطيل في النظرية X0 اللي عندها اللي
+
+713
+00:54:57,490 --> 00:55:02,110
+هي Fn of X0 converts بحكي عن X0 هذه المقصودة في
+
+714
+00:55:02,110 --> 00:55:08,170
+النظرية بدي أطبّق عليها عندي بما أن Fm ناقص Fn is
+
+715
+00:55:08,170 --> 00:55:11,730
+differentiable على closed interval within points
+
+716
+00:55:11,730 --> 00:55:18,090
+X0 و X يعني يا X0 و X يا X و X0 حسب موقع كل من X و
+
+717
+00:55:18,090 --> 00:55:23,440
+X0 إذا صارت Fm ناقص Fn عبارة عن الـ differential
+
+718
+00:55:23,440 --> 00:55:26,120
+بالـ function على هذه الـ closed bounded interval
+
+719
+00:55:26,120 --> 00:55:30,320
+إذا أكيد continuous عليها كمان إذا بالـ mean value
+
+720
+00:55:30,320 --> 00:55:33,940
+theorem مطبقة الـ mean value theorem ده طبقها على
+
+721
+00:55:33,940 --> 00:55:39,300
+مين يا جماعة؟ على الـ Fm ناقص Fn يعني هذه دالتنا
+
+722
+00:55:39,300 --> 00:55:43,660
+اللي بدنا نطبق عليها الـ mean value theorem إذا by
+
+723
+00:55:43,660 --> 00:55:47,760
+mean value theorem there exists Y بين الـ X0 والـ
+
+724
+00:55:47,760 --> 00:55:53,380
+X بحيث إن الـ Derivative لهذه بيساوي F M ناقص F N
+
+725
+00:55:53,380 --> 00:55:59,440
+عند اللي هي X ناقص F M ناقص F Note عند الـ X Note
+
+726
+00:55:59,440 --> 00:56:05,640
+بيساوي F M ناقص F N Prime عند مين؟ عند الـ Y by
+
+727
+00:56:05,640 --> 00:56:09,840
+mean value theorem there exists y between x0 and x
+
+728
+00:56:09,840 --> 00:56:13,700
+such that هذا الدالة كلها على بعض عند الـ x fm of x
+
+729
+00:56:13,700 --> 00:56:19,300
+نقص fn of x هذا الدالة ناقص fm of x0 نقص fn of x0
+
+730
+00:56:19,300 --> 00:56:25,060
+هذا الدالة عند النقطة x0 بساوي اللي هي x minus x0 في
+
+731
+00:56:25,060 --> 00:56:30,520
+اللي هو الـ derivative لهذه عند مين؟ عنده النقطة Y
+
+732
+00:56:30,520 --> 00:56:34,700
+اللي هي F M prime of Y نقص F M prime of Y إذا
+
+733
+00:56:34,700 --> 00:56:47,340
+حصلنا على هذه النقطة طيب الآن لو أجيت بس
+
+734
+00:56:47,340 --> 00:56:50,140
+عشان بدي ألاقي مكان لما فيه كلام كثير بدي أكتبه
+
+735
+00:56:50,140 --> 00:56:57,540
+هذا عندي المفروض خلّيني أكتب إن هدول عندي المعطيات
+
+736
+00:56:57,540 --> 00:57:07,950
+اللي هي FN of X0 Converges ومعايا اللي هو FNprime
+
+737
+00:57:07,950 --> 00:57:14,250
+converging formally to the G هذا المطلوب إن FN
+
+738
+00:57:14,250 --> 00:57:20,290
+converging formally to F و F' هي مين؟ الـ G هذه
+
+739
+00:57:20,290 --> 00:57:24,450
+المعطيات وهي المطلوب هذا اللي هو ملخصها خلينا نشوف
+
+740
+00:57:24,450 --> 00:57:31,590
+عشان إنه إحنا نبدأ اللي هو أفسلكم كل خطوة اللي هو
+
+741
+00:57:31,590 --> 00:57:35,270
+ما كانت مش مفسرة في التلخيص نفسلكم إياها بالتفصيل
+
+742
+00:57:36,790 --> 00:57:43,750
+الآن وجدنا هذه يا جماعة ماشي الحال الآن عندي هذا
+
+743
+00:57:43,750 --> 00:57:47,830
+الكلام صحيح لكل x عندي طبعاً x note نحكي عن x note
+
+744
+00:57:47,830 --> 00:57:56,530
+محددة صار عندي الآن fm of x ناقص fn of x هذه صارت
+
+745
+00:57:56,530 --> 00:58:01,600
+absolute value أصغر أو يساوي بدي أنجل هذه على النطق
+
+746
+00:58:01,600 --> 00:58:04,420
+هذه وأقول absolute value هذه بيساوي الـ absolute
+
+747
+00:58:04,420 --> 00:58:07,720
+value هذه وآخذ triangle inequality بيصير أصغر أو
+
+748
+00:58:07,720 --> 00:58:17,260
+يساوي الـ absolute value لهذه Fn of X0 ناقص Fn of
+
+749
+00:58:17,260 --> 00:58:26,650
+X0 زائد الـ absolute value للـ X minus X0 في الـ F M
+
+750
+00:58:26,650 --> 00:58:34,930
+prime of Y ناقص F M F N prime of mean of Y ماشي
+
+751
+00:58:34,930 --> 00:58:39,650
+الحال هذا mean ما له؟ هذا بالـ mean value بالـ
+
+752
+00:58:39,650 --> 00:58:42,550
+triangle inequality بعد ما نجلتها على الجهة
+
+753
+00:58:42,550 --> 00:58:50,310
+الثانية الآن هذا المقدار هذا المقدار اللي هو أكيد
+
+754
+00:58:50,310 --> 00:58:54,790
+هذا بيصير هذا المقدار أصغر يساوي هذا وهذا أصغر
+
+755
+00:58:54,790 --> 00:59:05,090
+يساوي FM of X not FM of X not ناقص FM of X not
+
+756
+00:59:05,090 --> 00:59:12,730
+زائد هذه اللي هي اللي هي موجودة في الفترة من عند A
+
+757
+00:59:12,730 --> 00:59:17,890
+لعند BX و X0 بغض النظر الموقع هنادي أو هنادي في
+
+758
+00:59:17,890 --> 00:59:22,470
+الآخر اللي هي B minus A أكيد أكبر من X minus X0
+
+759
+00:59:22,470 --> 00:59:29,670
+أكيد إذا زائد B minus A فيه هذا هذا أكيد أصغر أو
+
+760
+00:59:29,670 --> 00:59:36,890
+يساوي مين؟ الـ norm لـ F M prime ناقص F N prime over
+
+761
+00:59:36,890 --> 00:59:40,550
+مين؟ الجيل اللي عندي ليش؟ لأن هذا supremum للي فوق
+
+762
+00:59:40,550 --> 00:59:45,630
+صار هذا كله عبارة عن رقم هذا كله عبارة عن رقم هذا
+
+763
+00:59:45,630 --> 00:59:51,650
+الرقم الآن أكبر أو يساوي هذه لكل X وين موجودة في
+
+764
+00:59:51,650 --> 00:59:55,350
+الجيل لأنه أخدت X ما لها من رأس الدور أخدت X is
+
+765
+00:59:55,350 --> 01:00:00,290
+arbitrary point ماشي الحال صار عندي الآن هذا صار
+
+766
+01:00:00,290 --> 01:00:04,650
+upper bound لهذه عدناها مائة مرة إذا صار عند هذا
+
+767
+01:00:04,650 --> 01:00:07,250
+الـ upper bound أكبر أو يساوي الـ least upper bound
+
+768
+01:00:07,250 --> 01:00:10,010
+اللي هو الـ supremum إذا صار عند الـ supremum اللي
+
+769
+01:00:10,010 --> 01:00:15,810
+هو norm الـ FM ناقص FN over J أصغر أو يساوي المقدار
+
+770
+01:00:15,810 --> 01:00:20,570
+هذا وهي اللي هو حصلنا عليه هنا وتفسيره هي اللي
+
+771
+01:00:20,570 --> 01:00:26,130
+فسرت لكم إياه فوق إذا صار عندي FM ناقص FN over J
+
+772
+01:00:26,130 --> 01:00:29,490
+أصغر أو يساوي FN of X node ناقص FN of X node زائد B
+
+773
+01:00:29,490 --> 01:00:36,990
+minus A في الـ norm اللي أمامي الآن لاحظ إن FN of X
+
+774
+01:00:36,990 --> 01:00:41,760
+node converge اللي هو و FN' is uniformly
+
+775
+01:00:41,760
+
+801
+01:03:02,860 --> 01:03:07,640
+الآن بدي أستخدم هذه وأستخدم هذه بدي أستخدم أن الآن
+
+802
+01:03:07,640 --> 01:03:11,020
+و الـ H  بتكون أكبر من H2 والآن والـ H أكبر من مين
+
+803
+01:03:11,020 --> 01:03:15,420
+من H1 إذا for every Epsilon أكبر من الصفر there
+
+804
+01:03:15,420 --> 01:03:18,700
+exist H بتساوي الـ maximum عشان أنا أقدر أستخدم
+
+805
+01:03:18,700 --> 01:03:24,360
+الـ H  الـ maximum بين H1 وH2 such that for every
+
+806
+01:03:24,360 --> 01:03:29,180
+N و M أكبر أو يساوي H أكبر أو يساوي H هيكون عندي
+
+807
+01:03:29,180 --> 01:03:36,260
+norm norm هذه صحيحة لكل M و N في الدنيا الـ FM ناقص
+
+808
+01:03:36,260 --> 01:03:42,800
+FN over J أصغر أو يساوي الأولى زائد الثانية الأولى
+
+809
+01:03:42,800 --> 01:03:47,930
+هذه الأولى أصغر أو يساوي منها  Epsilon على اثنين
+
+810
+01:03:49,210 --> 01:03:52,890
+مظبوط لأن هذه أصغر من Epsilon على 2 لكل الـ N والـ M
+
+811
+01:03:52,890 --> 01:03:56,670
+الـ  أكبر أو يساوي H اللي هي أكيد الـ H أكبر أو يساوي مين؟
+
+812
+01:03:56,670 --> 01:04:00,330
+اللي هي H1 يعني N و M اللي بيكون أكبر أو يساوي H أكيد
+
+813
+01:04:00,330 --> 01:04:04,690
+بيكون الـ H1 اللي أنا بالفعل أستخدمها زائد الثانية
+
+814
+01:04:04,690 --> 01:04:10,090
+برضه الـ F M' ناقص F M' أصغر من مين هنا؟ M' هذا
+
+815
+01:04:10,090 --> 01:04:13,050
+طبعًا، هذي Prime وهذي Prime لأن F M' الـPrime الـ
+
+816
+01:04:13,050 --> 01:04:19,730
+Convergent لمن؟ للـ G Epsilon في 2 في B minus A
+
+817
+01:04:19,730 --> 01:04:24,890
+اللي كان مضروبة في مين في B minus A فبصير هذا
+
+818
+01:04:24,890 --> 01:04:27,250
+Epsilon على اثنين و Epsilon على هذي بيصير روح هذي
+
+819
+01:04:27,250 --> 01:04:30,450
+وهي سوى Epsilon إذا اللي حصلته لكل Epsilon أكبر من
+
+820
+01:04:30,450 --> 01:04:34,830
+zero there exists H such that لكل N أكبر سوى H طلع
+
+821
+01:04:34,830 --> 01:04:39,590
+عندي FN نقص FN أصغر من مين من Epsilon إذا by Cauchy
+
+822
+01:04:39,590 --> 01:04:46,420
+criterion يكون عندي الـ FN converges uniformly to
+
+823
+01:04:46,420 --> 01:04:52,180
+some function mean F مش عارفين لسه عنها حاجة إذا
+
+824
+01:04:52,180 --> 01:04:57,600
+اللي أثبتناه الآن أن F is uniformly convergent to
+
+825
+01:04:57,600 --> 01:05:02,880
+some function F مين هي مش عارفينها لسه أما أثبتنا
+
+826
+01:05:02,880 --> 01:05:07,280
+وجودها أثبتنا أن F converges uniformly لـ F إذا
+
+827
+01:05:07,280 --> 01:05:16,630
+بقدر أجرأ وأقول Limit الـ Fm أو الـ Fn as n goes
+
+828
+01:05:16,630 --> 01:05:21,430
+to infinity تساوي F ما أقدر أتجرأ لإن أثبتت وجودها
+
+829
+01:05:21,430 --> 01:05:27,570
+الآن لاحظوا بس شغلة Fn is differentiable is a
+
+830
+01:05:27,570 --> 01:05:31,360
+continuous مدام continuous وأثبتت FM بتروح
+
+831
+01:05:31,360 --> 01:05:35,080
+converges uniformly لها إذا الـ limit هذه هتطلع برضه
+
+832
+01:05:35,080 --> 01:05:40,260
+إيش ماله is continuous وهذه خلوها في الذاكرة مكون
+
+833
+01:05:40,260 --> 01:05:47,340
+احنا وصلنا أو أثبتنا الجزء الأول من النظرية أن FM
+
+834
+01:05:47,340 --> 01:05:53,820
+converges uniformly to F والـ limit لها سميناها F و
+
+835
+01:05:53,820 --> 01:05:57,720
+هيطلع لنا عبارة عن continuous function شوفوا يا
+
+836
+01:05:57,720 --> 01:05:58,420
+جماعة الآن
+
+837
+01:06:02,600 --> 01:06:09,330
+صار عندي الآن أثبتنا أن Fn تتعامل بشكل مرتبط سمينا
+
+838
+01:06:09,330 --> 01:06:13,050
+Limit Fn بـ F which is continuous because Fn زي
+
+839
+01:06:13,050 --> 01:06:14,670
+ما قلنا is continuous لأنه في الأصل هي
+
+840
+01:06:14,670 --> 01:06:18,610
+differentiable و Fn بتروح للـ F إذن صارت الـ F اللي
+
+841
+01:06:18,610 --> 01:06:23,330
+هي نفسها continuous الآن بدي أستخدم الـ continuity
+
+842
+01:06:23,330 --> 01:06:29,090
+عند الـ F بدي أصل للي بديّه طيب خد C الآن بـ any
+
+843
+01:06:29,090 --> 01:06:35,950
+fixed point in J الآن بدي أروح إلى إثبات أنه الـ
+
+844
+01:06:35,950 --> 01:06:43,810
+Limit للـ F of X ناقص F of C على X minus C as X
+
+845
+01:06:43,810 --> 01:06:46,970
+بتروح للـ C هذه الـ C وينها؟ الـ C أخذناها
+
+846
+01:06:46,970 --> 01:06:51,470
+arbitrary في X in J إذا أثبتت لهذا الـ C اللي هو
+
+847
+01:06:51,470 --> 01:06:56,870
+بيسوي اللي هي G of C معناته إذا أثبتت هذا معناته
+
+848
+01:06:56,870 --> 01:07:02,150
+أثبتت أن F' of C exist وهي ساوي G of C ومعناته
+
+849
+01:07:02,150 --> 01:07:08,310
+أثبتت أن F بتساوي G على كل J لأن C ماخدها
+
+850
+01:07:08,310 --> 01:07:12,610
+arbitrary but fixed يعني إيش بده يثبت في هذا؟ يعني
+
+851
+01:07:12,610 --> 01:07:16,910
+زي ما قلنا في الأول زي ما قلنا في الأول بدنا نثبت
+
+852
+01:07:16,910 --> 01:07:21,970
+لكل الآن لكل Epsilon أكبر من صفر there exists Delta
+
+853
+01:07:21,970 --> 01:07:27,350
+أكبر من صفر such that X minus C أصغر من Delta يؤدي
+
+854
+01:07:27,350 --> 01:07:34,990
+إلى اللي هو F of X ناقص F of C على X minus C ناقص
+
+855
+01:07:34,990 --> 01:07:39,630
+G of C أصغر من Epsilon ده أثبتت بكون خلصت إذا أثبتت
+
+856
+01:07:39,630 --> 01:07:44,780
+هذا بكون خلصت اللي هو الإثبات تنشوف طيب صلوا على
+
+857
+01:07:44,780 --> 01:07:50,120
+النبي عليه الصلاة والسلام نيجي الآن قولنا C N of X
+
+858
+01:07:50,120 --> 01:07:54,020
+point in J F M ناقص F N is differentiable on the
+
+859
+01:07:54,020 --> 01:07:58,020
+closed interval with endpoints C and X الآن F N
+
+860
+01:07:58,020 --> 01:08:00,840
+ناقص F N differentiable على الـ closed interval
+
+861
+01:08:00,840 --> 01:08:06,120
+اللي هي C ومين والـ X الـ XR اللي خدناها اللي هي
+
+862
+01:08:06,120 --> 01:08:10,770
+arbitrary ماشي، الآن بدأ استخدم الـ Mean Value
+
+863
+01:08:10,770 --> 01:08:14,010
+Theorem، then by mean value theorem كمان مرة there
+
+864
+01:08:14,010 --> 01:08:19,250
+exists z between c and x such that fm of x ناقص fn
+
+865
+01:08:19,250 --> 01:08:22,130
+of x النقطة الأولى، ناقص fm of c ناقص fn of c
+
+866
+01:08:22,130 --> 01:08:26,270
+النقطة الثانية بتساوي x minus c مضروبة في fn prime
+
+867
+01:08:26,270 --> 01:08:32,390
+of z ناقص fn prime اللي هو fm prime of z ماشي الحال
+
+868
+01:08:32,390 --> 01:08:35,350
+طبقت الـ Mean Value Theorem على الدالة الـ .. اللي
+
+869
+01:08:35,350 --> 01:08:41,770
+هي Fm ناقص Fn لكل M O N الآن خدوا الحسبة الآن خد X
+
+870
+01:08:41,770 --> 01:08:44,690
+لا تساوي C لإنه .. X لا تساوي C لإن أنا بدأت أدي X
+
+871
+01:08:44,690 --> 01:08:48,430
+عالميا على C يعني X minus C اللي هو أصغر من Delta
+
+872
+01:08:48,430 --> 01:08:52,760
+يعني X بدأت أديها على C الآن اقسموا كل الأطراف على
+
+873
+01:08:52,760 --> 01:09:00,820
+x-c بيصير اللي هي fm ناقص fm of c على x-c ناقص
+
+874
+01:09:00,820 --> 01:09:08,500
+الآن fn of x هي هان ناقص اللي هي هذه اللي هي FN of C
+
+875
+01:09:08,500 --> 01:09:13,300
+هي ناقص وهي ناقص إذا بيطلع صحيح على X minus C
+
+876
+01:09:13,300 --> 01:09:18,400
+بتساوي اللي هو F M prime of Z ناقص FN prime of Z
+
+877
+01:09:18,400 --> 01:09:24,340
+لأنه قسمت على X minus C الآن أكيد هذه .. هذه أصغر
+
+878
+01:09:24,340 --> 01:09:27,460
+أو يساوي Normها لأن Norm هو الـ supremum إلها صار
+
+879
+01:09:27,460 --> 01:09:31,320
+عندي الآن هذا المقدار أصغر أو يساوي هذا المقدار
+
+880
+01:09:31,950 --> 01:09:37,770
+الآن الآن اف ام ابراهيم converges uniformly إذا
+
+881
+01:09:37,770 --> 01:09:40,110
+لكل Epsilon أكبر من صفر there exists delta such
+
+882
+01:09:40,110 --> 01:09:42,390
+that there exists و في Epsilon و such that هذا أصغر
+
+883
+01:09:42,390 --> 01:09:44,990
+من مين some Epsilon أو Epsilon على اثنين أو اللي
+
+884
+01:09:44,990 --> 01:09:52,710
+بدكم إياه الآن إذا نكمل اللي بنقوله شوف Since Fn'
+
+885
+01:09:53,010 --> 01:09:57,510
+converges uniformly on J إذا لكل ي أكبر من 0 there
+
+886
+01:09:57,510 --> 01:10:03,770
+exists H of ي H of ي such that if M و N أكبر سواء
+
+887
+01:10:03,770 --> 01:10:10,230
+H of ي then هذا المقدار اللي هو أصغر من مين من ي
+
+888
+01:10:10,230 --> 01:10:16,590
+يعني صار كل هذا المقدار أصغر من ي لمين؟ لـ الـ M و
+
+889
+01:10:16,590 --> 01:10:25,840
+الـ N لـ أكبر من مين من H of Epsilon لأن صار عندي
+
+890
+01:10:25,840 --> 01:10:31,280
+لأن لكل M و N أكبر سوى H of Epsilon صار المقدار
+
+891
+01:10:31,280 --> 01:10:35,140
+هذا اللي هو أصغر أو سوى هذا اللي هو أصغر من
+
+892
+01:10:35,140 --> 01:10:37,860
+Epsilon خليها في الذاكرة هذه أصغر من مين من
+
+893
+01:10:37,860 --> 01:10:42,740
+Epsilon الآن take the limit of both sides as M goes
+
+894
+01:10:42,740 --> 01:10:46,160
+to infinity بس هان عشان بياخد M رايحة لما لنهاية
+
+895
+01:10:46,160 --> 01:10:50,000
+بدنا نأخذ للـ M أكبر أو يساوي H N عشان لما الـ M
+
+896
+01:10:50,000 --> 01:10:53,340
+تروح لما لنهاية M تبقى مرتاحة زي 100 و M لا تتحركش
+
+897
+01:10:53,340 --> 01:11:01,520
+بدنا إياها تتحرك بيصير عندي as M goes to infinity
+
+898
+01:11:01,520 --> 01:11:08,350
+الـ limit للـ F M of X هيساوي F of X لأنه صارت
+
+899
+01:11:08,350 --> 01:11:14,670
+uniformly الـ Fm أثبتناها بتروح للـ F uniformly
+
+900
+01:11:14,670 --> 01:11:19,830
+إذن Limit Fm of X هي F of X و Limit Fm of C هي F
+
+901
+01:11:19,830 --> 01:11:25,490
+of C على X minus C معايا هذا Fn of X ناقص Fn of C
+
+902
+01:11:25,490 --> 01:11:29,370
+على X minus C زي ما هي لأن أنا إبسلتها زي ما هي
+
+903
+01:11:29,370 --> 01:11:32,890
+مين اللي طيرتها الكبيرة اللي هي M إذا صار هذا
+
+904
+01:11:32,890 --> 01:11:38,130
+المقدار أصغر أو يساوي مين؟ Epsilon في حال مين؟ في
+
+905
+01:11:38,130 --> 01:11:43,930
+حال X لا تساوي C وللآنات اللي أكبر من مين من H of
+
+906
+01:11:43,930 --> 01:11:49,710
+Y أنا تحت هذه الشروط ماشي طيب عمالي قاعد بجرب على
+
+907
+01:11:49,710 --> 01:11:58,400
+اللي هو الهدف تبعي شوف صار عندي الآن هذه الـ
+
+908
+01:11:58,400 --> 01:12:03,740
+Inequality واحد اللي هو متحققها لكن أنا بعرف أنه
+
+909
+01:12:03,740 --> 01:12:07,780
+Limit of F N prime of C أشهر بيساوي G of C لأنه
+
+910
+01:12:07,780 --> 01:12:12,440
+احنا مفترضين من رأس الدور أنه F N prime converges
+
+911
+01:12:12,440 --> 01:12:18,470
+uniformly to مين؟ للـ G .. طيب لأن sense Limit F N
+
+912
+01:12:18,470 --> 01:12:22,330
+prime of C بيساوي G of C إذا there exists N of
+
+913
+01:12:22,330 --> 01:12:25,590
+Epsilon such that for every N أكبر بيساوي N of
+
+914
+01:12:25,590 --> 01:12:29,910
+Epsilon حيكون F N prime of C ناقص G of C اللي هو
+
+915
+01:12:29,910 --> 01:12:38,310
+أصغر من إيش من Epsilon الآن صار عندي هذه متحققة للـ N
+
+916
+01:12:38,310 --> 01:12:43,430
+أكبر سواء H وهذه متحققة للـ N الأكبر سواء N عشان
+
+917
+01:12:43,430 --> 01:12:47,750
+أستخدم التنتين بتاخد الـ maximum بين هذه وهذه و
+
+918
+01:12:47,750 --> 01:12:52,630
+أقول let K بسوء الـ maximum بين H of Epsilon و N
+
+919
+01:12:52,630 --> 01:12:59,130
+Epsilon إذا الآن صار عندي التنتين متحققات لكل N
+
+920
+01:12:59,130 --> 01:13:04,530
+أكبر أو يساوي K ومشروع استخدامه الآن من جهة ثانية
+
+921
+01:13:04,530 --> 01:13:07,750
+بدي أستخدم الـ FK' of C إنها exist
+
+922
+01:13:11,040 --> 01:13:14,540
+إذا صارت عندي هذه زي ما قلنا وهذه هاتين تان بحجله
+
+923
+01:13:14,540 --> 01:13:18,620
+أستخدمهم لكل N أكبر أو ساو كي بدي أستخدم هذه كمان
+
+924
+01:13:18,620 --> 01:13:22,900
+وأربط هذه مع هذه وأخلص اللي بديها since fk prime
+
+925
+01:13:22,900 --> 01:13:26,860
+of c exist مين الـ K هذه الـ K هذه اللي خدتها الـ
+
+926
+01:13:26,860 --> 01:13:31,660
+maximum بين هذه وهذه عشان ينفع أستخدم هذه وهذه
+
+927
+01:13:31,660 --> 01:13:37,760
+وهذه بعد شوية then مدام fk prime of c exist then
+
+928
+01:13:37,760 --> 01:13:42,080
+if x minus c أكبر من صفر أصغر من دلتا دلتا k طبعًا
+
+929
+01:13:42,080 --> 01:13:46,480
+الدلتا هتعتمد على اللي هو الـ Epsilon نسميها دلتا k
+
+930
+01:13:46,480 --> 01:13:49,360
+لأنها اعتمدت على مين؟ على الـ K يعنى then there
+
+931
+01:13:49,360 --> 01:13:53,380
+exists دلتا k بحيث x minus c أصغر من دلتا يعطيني
+
+932
+01:13:53,380 --> 01:13:57,540
+fk of x ناقص fk of c على x minus c اللي هي ناقص fk
+
+933
+01:13:57,540 --> 01:14:04,140
+prime of c أصغر من Epsilon فاهمين هذا؟ مدام fkprime
+
+934
+01:14:04,140 --> 01:14:09,260
+of c بيساوي Limit exist إذا بيساوي Limit f of x ناقص
+
+935
+01:14:09,260 --> 01:14:14,280
+f of c بـ prime على x minus c طبعًا كك كله as x بتروح
+
+936
+01:14:14,280 --> 01:14:17,620
+للـ c صار exist يعني for every Epsilon أكبر من 0
+
+937
+01:14:17,620 --> 01:14:21,740
+exist دلتا بحيث أنه لما تكون x minus c أصغر من
+
+938
+01:14:21,740 --> 01:14:24,360
+دلتا يعطيني هذا ناقص هذا أصغر من Epsilon وهذا هو
+
+939
+01:14:24,360 --> 01:14:28,400
+المكتوب إذن صار عندي هذا موجود وهي اثنين وهي ثلاثة
+
+940
+01:14:28,400 --> 01:14:33,020
+لأن واحد واثنين وثلاثة بدي أصل للي بديّه خلينا
+
+941
+01:14:33,020 --> 01:14:35,720
+نكتب هنا عشان ما نجلبش الصفحة عشان يظلوا كله
+
+942
+01:14:35,720 --> 01:14:43,880
+قدامكم الآن اللي وصلت إليه مالي for every Epsilon
+
+943
+01:14:43,880 --> 01:14:46,660
+أكبر من صفر بقول there exists delta أكبر من صفر
+
+944
+01:14:46,660 --> 01:14:49,740
+such that if
+
+945
+01:14:51,280 --> 01:14:57,660
+x-c أصغر أكبر من صفر وأصغر من دلتا then شوف الآن
+
+946
+01:14:57,660 --> 01:15:02,400
+then إيش اللي هو بدي مين أنا هي هي اللي بديها f of
+
+947
+01:15:02,400 --> 01:15:12,880
+x minus f of c على x minus c ناقص g of c أصغر
+
+948
+01:15:12,880 --> 01:15:18,120
+أو يساوي ماشي الحال أصغر أو يساوي بدي أبدأ الآن
+
+949
+01:15:18,120 --> 01:15:24,240
+اللي هو هي هذه بدي أدخلها دي، عشان؟ أصغر أو ساوي
+
+950
+01:15:24,240 --> 01:15:32,300
+اللي هو F of X ناقص F of C على X minus C ناقص FN
+
+951
+01:15:32,300 --> 01:15:39,840
+of X ناقص FN of C على X minus C زائد اللي دخلتها
+
+952
+01:15:39,840 --> 01:15:51,830
+دي FN of X ناقص FN of C على X minus C ناقص اللي هو
+
+953
+01:15:51,830 --> 01:15:56,850
+إيش
+
+954
+01:15:56,850 --> 01:16:03,170
+الثانية فك
+
+955
+01:16:03,170 --> 01:16:13,890
+prime of c زائد فك
+
+956
+01:16:13,890 --> 01:16:21,820
+prime of c ناقص Fn نقص من اللي ضال G of C بظبط هيك ا
+
+957
+01:16:21,820 --> 01:16:26,460
+أظن تشوف عليها الآن أنا بتلزمنيش كل الـ N بتلزمني
+
+958
+01:16:26,460 --> 01:16:31,480
+بس الـ مين الـ K لأن الآن أنا بحكي على استخدام الـ K
+
+959
+01:16:31,480 --> 01:16:39,280
+K K وهاد أيهاش K طيب الآن نيجي نتطلع لـ الـ K تبعتنا
+
+960
+01:16:39,280 --> 01:16:42,040
+هذه بنحكي عن K محددة الـ K اللي هي maximum بين H
+
+961
+01:16:42,040 --> 01:16:46,320
+وبين N يعني الـ K أكبر أو يساوي H والـ K أكبر أو
+
+962
+01:16:46,320 --> 01:16:51,600
+يساوي مين الـ N يعني بحججلي أستخدم F K of C F K
+
+963
+01:16:51,600 --> 01:16:55,020
+prime of C ناقص G of C أصغر من Epsilon وبحججلي
+
+964
+01:16:55,020 --> 01:16:58,320
+أستخدم هذه ناقص F K of K في K أصغر من مين من
+
+965
+01:16:58,320 --> 01:17:04,540
+Epsilon الآن هذه هذه هي هان الأولى إذا أصغر أو
+
+966
+01:17:04,540 --> 01:17:13,170
+يساوي هذه من واحد Epsilon زائد هذه الثانية من مين
+
+967
+01:17:13,170 --> 01:17:19,250
+من اللي هي ثلاثة هي fk of x نقص fk of c عليك x
+
+968
+01:17:19,250 --> 01:17:23,490
+minus c نقص fk prime أصغر برضه من مين من Epsilon
+
+969
+01:17:23,490 --> 01:17:29,290
+هذه من ثلاثة وهذه من واحد هذه أكيد من مين هستخدمها
+
+970
+01:17:29,290 --> 01:17:35,610
+من اثنين اللي هي N هذه لكل N أكبر سوى N وعندي الـ K
+
+971
+01:17:35,610 --> 01:17:38,190
+أصلًا أكبر أوي سواء لأنها ال
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..81b4124d8d0f8d22c6e0aae9700c2449a1442977
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ_raw.json
@@ -0,0 +1 @@
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0.990234375}, {"start": 720.53, "end": 721.05, "word": " ال", "probability": 0.9443359375}, {"start": 721.05, "end": 721.45, "word": " function", "probability": 0.94287109375}, {"start": 721.45, "end": 722.09, "word": " تكون", "probability": 0.9697265625}, {"start": 722.09, "end": 722.37, "word": " اللي", "probability": 0.968017578125}, {"start": 722.37, "end": 722.57, "word": " هو", "probability": 0.9814453125}, {"start": 722.57, "end": 723.27, "word": " integrable", "probability": 0.75634765625}, {"start": 723.27, "end": 723.99, "word": " لكن", "probability": 0.8662109375}, {"start": 723.99, "end": 724.31, "word": " لا", "probability": 0.81005859375}, {"start": 724.31, "end": 725.13, "word": " يمكن", "probability": 0.89501953125}], "temperature": 1.0}, {"id": 34, "seek": 74506, "start": 725.54, "end": 745.06, "text": "إنه .. أو .. أيه .. أو .. آسف ليس شرطا أن يكون limit لل integration بساوي ال integration لل limit إذا بدنا دفعة أكبر من ال pointwise convergence و الحالتين ال continuity و ال integrability هانفتنتين الدفعة هذه هي ال uniform convergence و هنشوف هذا الكلام", "tokens": [28814, 1863, 3224, 4386, 34051, 4386, 36632, 3224, 4386, 34051, 4386, 19753, 3794, 5172, 32239, 3794, 13412, 2288, 9566, 995, 14739, 7251, 30544, 4948, 24976, 10980, 4724, 3794, 995, 45865, 2423, 10980, 24976, 4948, 11933, 15730, 47525, 8315, 11778, 5172, 27884, 5551, 4117, 26890, 9154, 2423, 935, 3711, 32181, 4032, 21542, 6027, 2655, 9957, 2423, 23807, 4032, 2423, 3572, 2310, 8032, 7649, 5172, 2655, 29399, 9957, 32748, 5172, 27884, 29538, 39896, 2423, 9452, 32181, 4032, 8032, 1863, 8592, 38688, 23758, 2423, 28820, 10943], "avg_logprob": -0.2985491000470661, "compression_ratio": 1.663594470046083, "no_speech_prob": 7.212162017822266e-06, "words": [{"start": 725.54, "end": 726.3, "word": "إنه", "probability": 0.5445149739583334}, {"start": 726.3, "end": 726.72, "word": " ..", "probability": 0.270751953125}, {"start": 726.72, "end": 726.72, "word": " أو", "probability": 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pointwise", "probability": 0.64111328125}, {"start": 735.14, "end": 735.82, "word": " convergence", "probability": 0.9609375}, {"start": 735.82, "end": 736.3, "word": " و", "probability": 0.513671875}, {"start": 736.3, "end": 736.86, "word": " الحالتين", "probability": 0.9169921875}, {"start": 736.86, "end": 736.98, "word": " ال", "probability": 0.93896484375}, {"start": 736.98, "end": 737.58, "word": " continuity", "probability": 0.93798828125}, {"start": 737.58, "end": 737.82, "word": " و", "probability": 0.76513671875}, {"start": 737.82, "end": 737.94, "word": " ال", "probability": 0.89208984375}, {"start": 737.94, "end": 738.64, "word": " integrability", "probability": 0.7010498046875}, {"start": 738.64, "end": 739.8, "word": " هانفتنتين", "probability": 0.6293131510416666}, {"start": 739.8, "end": 740.74, "word": " الدفعة", "probability": 0.9490559895833334}, {"start": 740.74, "end": 741.18, "word": " هذه", "probability": 0.92822265625}, {"start": 741.18, "end": 741.46, "word": " هي", "probability": 0.896484375}, {"start": 741.46, "end": 741.6, "word": " ال", "probability": 0.93115234375}, {"start": 741.6, "end": 742.04, "word": " uniform", "probability": 0.91064453125}, {"start": 742.04, "end": 743.68, "word": " convergence", "probability": 0.9775390625}, {"start": 743.68, "end": 743.88, "word": " و", "probability": 0.791015625}, {"start": 743.88, "end": 744.32, "word": " هنشوف", "probability": 0.9246826171875}, {"start": 744.32, "end": 744.56, "word": " هذا", "probability": 0.98046875}, {"start": 744.56, "end": 745.06, "word": " الكلام", "probability": 0.9609375}], "temperature": 1.0}, {"id": 35, "seek": 76669, "start": 745.25, "end": 766.69, "text": "من خلال النظريات القادمة أول نظرية عندنا هذه النظرية اللي هي هتتعلق بالـ sequence of continuous functions كيف أنه لو كانت اللي هي sequence of continuous functions converts uniformly to some function F then F must be continuous", "tokens": [27842, 16490, 1211, 6027, 28239, 19913, 16572, 9307, 25062, 18513, 46007, 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"probability": 0.800537109375}, {"start": 750.09, "end": 750.45, "word": " هذه", "probability": 0.75048828125}, {"start": 750.45, "end": 751.17, "word": " النظرية", "probability": 0.965576171875}, {"start": 751.17, "end": 751.77, "word": " اللي", "probability": 0.66259765625}, {"start": 751.77, "end": 751.97, "word": " هي", "probability": 0.78466796875}, {"start": 751.97, "end": 752.81, "word": " هتتعلق", "probability": 0.92861328125}, {"start": 752.81, "end": 753.53, "word": " بالـ", "probability": 0.690185546875}, {"start": 753.53, "end": 753.99, "word": " sequence", "probability": 0.50341796875}, {"start": 753.99, "end": 754.31, "word": " of", "probability": 0.974609375}, {"start": 754.31, "end": 755.03, "word": " continuous", "probability": 0.8583984375}, {"start": 755.03, "end": 755.75, "word": " functions", "probability": 0.86328125}, {"start": 755.75, "end": 756.65, "word": " كيف", "probability": 0.95703125}, {"start": 756.65, "end": 757.25, "word": " أنه", "probability": 0.6990966796875}, {"start": 757.25, "end": 757.49, "word": " لو", "probability": 0.98046875}, {"start": 757.49, "end": 757.99, "word": " كانت", "probability": 0.9921875}, {"start": 757.99, "end": 758.15, "word": " اللي", "probability": 0.896728515625}, {"start": 758.15, "end": 758.45, "word": " هي", "probability": 0.95947265625}, {"start": 758.45, "end": 759.31, "word": " sequence", "probability": 0.93701171875}, {"start": 759.31, "end": 759.51, "word": " of", "probability": 0.97412109375}, {"start": 759.51, "end": 760.09, "word": " continuous", "probability": 0.8828125}, {"start": 760.09, "end": 760.79, "word": " functions", "probability": 0.90185546875}, {"start": 760.79, "end": 761.39, "word": " converts", "probability": 0.59912109375}, {"start": 761.39, "end": 762.13, "word": " uniformly", "probability": 0.9609375}, {"start": 762.13, "end": 762.39, "word": " to", "probability": 0.9638671875}, {"start": 762.39, "end": 762.61, "word": " some", "probability": 0.91015625}, {"start": 762.61, "end": 763.19, "word": " function", "probability": 0.96044921875}, {"start": 763.19, "end": 763.57, "word": " F", "probability": 0.50390625}, {"start": 763.57, "end": 764.31, "word": " then", "probability": 0.6611328125}, {"start": 764.31, "end": 764.75, "word": " F", "probability": 0.88037109375}, {"start": 764.75, "end": 765.11, "word": " must", "probability": 0.85400390625}, {"start": 765.11, "end": 765.65, "word": " be", "probability": 0.95751953125}, {"start": 765.65, "end": 766.69, "word": " continuous", "probability": 0.89013671875}], "temperature": 1.0}, {"id": 36, "seek": 77830, "start": 769.5, "end": 778.3, "text": "طيب يا جماعة الان نيجي theorem 8.2.2 بتقول ما يلي الان نتقف ان بيه a sequence of continuous functions", "tokens": [9566, 1829, 3555, 35186, 10874, 15042, 27884, 2423, 7649, 8717, 1829, 7435, 1829, 20904, 1649, 13, 17, 13, 17, 39894, 39648, 19446, 7251, 20292, 2423, 7649, 8717, 2655, 4587, 5172, 16472, 4724, 1829, 3224, 257, 8310, 295, 10957, 6828], "avg_logprob": -0.3326171971857548, "compression_ratio": 1.1544715447154472, "no_speech_prob": 2.2649765014648438e-06, "words": [{"start": 769.5, "end": 770.14, "word": "طيب", "probability": 0.7284342447916666}, {"start": 770.14, "end": 770.22, "word": " يا", "probability": 0.493896484375}, {"start": 770.22, "end": 770.62, "word": " جماعة", "probability": 0.9777018229166666}, {"start": 770.62, "end": 771.64, "word": " الان", "probability": 0.56524658203125}, {"start": 771.64, "end": 771.98, "word": " نيجي", "probability": 0.64031982421875}, {"start": 771.98, "end": 772.28, "word": " theorem", "probability": 0.6611328125}, {"start": 772.28, "end": 772.58, "word": " 8", "probability": 0.552734375}, {"start": 772.58, "end": 772.94, "word": ".2", "probability": 0.608642578125}, {"start": 772.94, "end": 773.28, "word": ".2", "probability": 0.981201171875}, {"start": 773.28, "end": 773.68, "word": " بتقول", "probability": 0.979736328125}, {"start": 773.68, "end": 773.88, "word": " ما", 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على الفترة A subset من R and suppose that اللي هي الـ Fn converges uniformly on A to a function F من A لعند R، بقول إذا ولا يهمك مضمون إن الـ F نفسها تكون إيش ما لها is a continuous function، إذاً", "tokens": [266, 257, 992, 316, 25993, 9154, 497, 12399, 11933, 15730, 14111, 479, 77, 6225, 3555, 9640, 3660, 18871, 44650, 2655, 16758, 27884, 9154, 32748, 2407, 6027, 1829, 9673, 2655, 36520, 3660, 15844, 27188, 2655, 25720, 316, 25993, 9154, 497, 293, 7297, 300, 13672, 1829, 39896, 2423, 39184, 479, 77, 9652, 2880, 48806, 322, 316, 281, 257, 2445, 479, 9154, 316, 5296, 3615, 41260, 497, 12399, 4724, 39648, 11933, 15730, 49429, 7251, 16095, 4117, 3714, 11242, 2304, 11536, 36145, 2423, 39184, 479, 8717, 36178, 11296, 6055, 30544, 11933, 1829, 8592, 19446, 5296, 11296, 307, 257, 10957, 2445, 12399, 11933, 15730, 14111], "avg_logprob": -0.20467202262123033, "compression_ratio": 1.5780590717299579, "no_speech_prob": 2.9206275939941406e-06, "words": [{"start": 779.42, "end": 779.76, "word": 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479], "avg_logprob": -0.38135594634686487, "compression_ratio": 1.4825174825174825, "no_speech_prob": 1.430511474609375e-06, "words": [{"start": 1145.47, "end": 1146.55, "word": "وهتعتمد", "probability": 0.76324462890625}, {"start": 1146.55, "end": 1146.71, "word": " على", "probability": 0.61669921875}, {"start": 1146.71, "end": 1146.79, "word": " ال", "probability": 0.50927734375}, {"start": 1146.79, "end": 1147.25, "word": " FH", "probability": 0.4786376953125}, {"start": 1147.25, "end": 1147.63, "word": " لأنه", "probability": 0.7542317708333334}, {"start": 1147.63, "end": 1147.95, "word": " مااخد", "probability": 0.6890462239583334}, {"start": 1147.95, "end": 1148.05, "word": " ال", "probability": 0.951171875}, {"start": 1148.05, "end": 1148.75, "word": " continuity", "probability": 0.89306640625}, {"start": 1148.75, "end": 1149.23, "word": " لمن", "probability": 0.69091796875}, {"start": 1149.23, "end": 1149.65, "word": " أنا", "probability": 0.42529296875}, {"start": 1149.65, 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مين على اللي هي الـY اللي بتاخدها arbitrarily ماشي الحال طيب", "tokens": [39, 295, 1783, 5296, 15042, 1783, 6055, 32887, 5016, 24976, 39184, 34, 4724, 3794, 2407, 38207, 479, 39, 295, 383, 37495, 22653, 5296, 28820, 398, 5551, 4117, 26890, 9154, 1958, 456, 8198, 18183, 2423, 39184, 40848, 1328, 29538, 32767, 9957, 6055, 34268, 2304, 3215, 15844, 2423, 39184, 34, 34683, 34268, 2304, 3215, 15844, 2423, 39184, 37, 39, 34683, 34268, 2304, 3215, 15844, 3714, 9957, 15844, 13672, 1829, 39896, 2423, 39184, 56, 13672, 1829, 39894, 47283, 3215, 11296, 19071, 3289, 3714, 33599, 1829, 21542, 6027, 23032, 1829, 3555], "avg_logprob": -0.24982242895798248, "compression_ratio": 1.5916230366492146, "no_speech_prob": 0.0003032684326171875, "words": [{"start": 1160.47, "end": 1160.91, "word": "H", "probability": 0.304443359375}, {"start": 1160.91, "end": 1161.51, "word": " of", "probability": 0.2197265625}, {"start": 1161.51, "end": 1161.91, "word": " X", "probability": 0.9736328125}, {"start": 1161.91, 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continuous", "probability": 0.91015625}, {"start": 1336.78, "end": 1337.2, "word": " functions", "probability": 0.6396484375}], "temperature": 1.0}, {"id": 60, "seek": 136289, "start": 1337.43, "end": 1362.89, "text": "طبيعي لكن احنا بنقول الضمان انه يطلع continuous انها تكون uniform convergence وجبنا مثال على انه اللي هو اللي هي sequence of continuous functions converge pointwise to some function that is not continuous زي ما شوفنا في المثال الأول نيجي الآن نحكي عن موضوع او الجزء المتعلق بمين", "tokens": [9566, 21292, 3615, 1829, 44381, 1975, 5016, 8315, 44945, 39648, 6024, 114, 2304, 7649, 16472, 3224, 7251, 9566, 1211, 3615, 10957, 16472, 11296, 6055, 30544, 9452, 32181, 4032, 7435, 3555, 8315, 50113, 6027, 15844, 16472, 3224, 13672, 1829, 31439, 13672, 1829, 39896, 8310, 295, 10957, 6828, 41881, 935, 3711, 281, 512, 2445, 300, 307, 406, 10957, 30767, 1829, 19446, 13412, 38688, 8315, 8978, 9673, 12984, 6027, 16247, 12610, 8717, 1829, 7435, 1829, 6024, 48506, 8717, 5016, 4117, 1829, 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{"start": 1361.27, "end": 1361.61, "word": " الجزء", "probability": 0.98876953125}, {"start": 1361.61, "end": 1362.33, "word": " المتعلق", "probability": 0.9718017578125}, {"start": 1362.33, "end": 1362.89, "word": " بمين", "probability": 0.9596354166666666}], "temperature": 1.0}, {"id": 61, "seek": 138775, "start": 1363.87, "end": 1387.75, "text": "الجزء المتعلق بالـ Integrable Functions الآن نشوف أيش العلاج لإنه نضمن الـ Limit للـ Integration بساوي الـ Integration للـ Limit أو اللي هو لما ندخل الـ Limit للـ Integration متى بنقدر ندخله ويظل المحافظة على المساوية Let FN be a sequence of functions that are integrable on A وB", "tokens": [6027, 7435, 11622, 38207, 9673, 2655, 30241, 4587, 20666, 39184, 5681, 1146, 424, 638, 11166, 3916, 6024, 48506, 8717, 8592, 38688, 36632, 8592, 18863, 1211, 26108, 5296, 28814, 1863, 3224, 8717, 11242, 27842, 2423, 39184, 16406, 270, 24976, 39184, 47713, 4724, 3794, 995, 45865, 2423, 39184, 47713, 24976, 39184, 16406, 270, 34051, 13672, 1829, 31439, 5296, 15042, 8717, 3215, 9778, 1211, 2423, 39184, 16406, 270, 24976, 39184, 47713, 44650, 7578, 44945, 28543, 2288, 8717, 3215, 9778, 43761, 4032, 1829, 19913, 1211, 9673, 5016, 31845, 19913, 3660, 15844, 9673, 3794, 995, 2407, 10632, 961, 479, 45, 312, 257, 8310, 295, 6828, 300, 366, 16200, 424, 638, 322, 316, 4032, 33], "avg_logprob": -0.22656249309127982, "compression_ratio": 1.70995670995671, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 1363.87, "end": 1364.49, "word": "الجزء", "probability": 0.8455810546875}, {"start": 1364.49, "end": 1365.41, "word": " المتعلق", "probability": 0.9761962890625}, {"start": 1365.41, "end": 1366.53, "word": " بالـ", "probability": 0.494384765625}, {"start": 1366.53, "end": 1367.77, "word": " Integrable", "probability": 0.77398681640625}, {"start": 1367.77, "end": 1368.45, "word": " Functions", "probability": 0.6796875}, {"start": 1368.45, "end": 1369.43, "word": " الآن", "probability": 0.5611572265625}, {"start": 1369.43, "end": 1370.83, "word": " نشوف", "probability": 0.7628580729166666}, {"start": 1370.83, "end": 1371.59, "word": " أيش", "probability": 0.6646728515625}, {"start": 1371.59, "end": 1372.25, "word": " العلاج", "probability": 0.9485677083333334}, {"start": 1372.25, "end": 1373.27, "word": " لإنه", "probability": 0.70556640625}, {"start": 1373.27, "end": 1373.97, "word": " نضمن", "probability": 0.9793294270833334}, {"start": 1373.97, "end": 1374.15, "word": " الـ", "probability": 0.616943359375}, {"start": 1374.15, "end": 1374.37, "word": " Limit", "probability": 0.701171875}, {"start": 1374.37, "end": 1374.55, "word": " للـ", "probability": 0.841796875}, {"start": 1374.55, "end": 1375.01, "word": " Integration", "probability": 0.7724609375}, {"start": 1375.01, "end": 1375.35, "word": " بساوي", "probability": 0.5849609375}, {"start": 1375.35, "end": 1375.53, "word": " الـ", "probability": 0.848876953125}, {"start": 1375.53, "end": 1375.95, "word": " Integration", "probability": 0.6982421875}, {"start": 1375.95, "end": 1376.27, "word": " للـ", "probability": 0.786865234375}, {"start": 1376.27, "end": 1376.51, "word": " Limit", "probability": 0.959228515625}, {"start": 1376.51, "end": 1376.73, "word": " أو", "probability": 0.8818359375}, {"start": 1376.73, "end": 1376.89, "word": " اللي", "probability": 0.8818359375}, {"start": 1376.89, "end": 1377.09, "word": " هو", "probability": 0.990234375}, {"start": 1377.09, "end": 1377.65, "word": " لما", "probability": 0.91015625}, {"start": 1377.65, "end": 1378.05, "word": " ندخل", "probability": 0.973388671875}, {"start": 1378.05, "end": 1378.21, "word": " الـ", "probability": 0.887939453125}, {"start": 1378.21, "end": 1378.53, "word": " Limit", "probability": 0.973388671875}, {"start": 1378.53, "end": 1378.97, "word": " للـ", "probability": 0.91064453125}, {"start": 1378.97, "end": 1379.41, "word": " Integration", "probability": 0.869140625}, {"start": 1379.41, "end": 1379.85, "word": " متى", "probability": 0.737060546875}, {"start": 1379.85, "end": 1380.21, "word": " بنقدر", "probability": 0.8751627604166666}, {"start": 1380.21, "end": 1380.63, "word": " ندخله", "probability": 0.990478515625}, {"start": 1380.63, "end": 1380.97, "word": " ويظل", "probability": 0.733154296875}, {"start": 1380.97, "end": 1381.61, "word": " المحافظة", "probability": 0.9453125}, {"start": 1381.61, "end": 1381.75, "word": " على", "probability": 0.947265625}, {"start": 1381.75, "end": 1382.61, "word": " المساوية", "probability": 0.866015625}, {"start": 1382.61, "end": 1383.85, "word": " Let", "probability": 0.282470703125}, {"start": 1383.85, "end": 1384.25, "word": " FN", "probability": 0.69189453125}, {"start": 1384.25, "end": 1384.43, "word": " be", "probability": 0.7421875}, {"start": 1384.43, "end": 1384.59, "word": " a", "probability": 0.970703125}, {"start": 1384.59, "end": 1385.03, "word": " sequence", "probability": 0.9140625}, {"start": 1385.03, "end": 1385.27, "word": " of", "probability": 0.9814453125}, {"start": 1385.27, "end": 1385.77, "word": " functions", "probability": 0.8515625}, {"start": 1385.77, "end": 1386.09, "word": " that", "probability": 0.8974609375}, {"start": 1386.09, "end": 1386.37, "word": " are", "probability": 0.9521484375}, {"start": 1386.37, "end": 1387.07, "word": " integrable", "probability": 0.86865234375}, {"start": 1387.07, "end": 1387.29, "word": " on", "probability": 0.8935546875}, {"start": 1387.29, "end": 1387.47, "word": " A", "probability": 0.90869140625}, {"start": 1387.47, "end": 1387.75, "word": " وB", "probability": 0.7998046875}], "temperature": 1.0}, {"id": 62, "seek": 141255, "start": 1388.61, "end": 1412.55, "text": "and suppose that FN converges uniformly من A و B to F then إذا لما تكون ال convergence uniform then ال F is integrable من A و B هاي واحد اتنين ال integration لل F of X DX من A لعين B بسوء ال limit لل integration يعني بمعنى أخر ال limit لل integration بسوء ال integration لل limit", "tokens": [474, 7297, 300, 479, 45, 9652, 2880, 48806, 9154, 316, 4032, 363, 281, 479, 550, 11933, 15730, 5296, 15042, 6055, 30544, 2423, 32181, 9452, 550, 2423, 479, 307, 16200, 424, 638, 9154, 316, 4032, 363, 8032, 47302, 36764, 24401, 1975, 2655, 1863, 9957, 2423, 10980, 24976, 479, 295, 1783, 48817, 9154, 316, 5296, 3615, 9957, 363, 4724, 3794, 2407, 38207, 2423, 4948, 24976, 10980, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 5551, 34740, 2423, 4948, 24976, 10980, 4724, 3794, 2407, 38207, 2423, 10980, 24976, 4948], "avg_logprob": -0.2796148300863976, "compression_ratio": 1.756218905472637, "no_speech_prob": 4.172325134277344e-07, "words": [{"start": 1388.61, "end": 1388.89, "word": "and", "probability": 0.06512451171875}, {"start": 1388.89, "end": 1389.45, "word": " suppose", "probability": 0.87109375}, {"start": 1389.45, "end": 1390.25, "word": " that", "probability": 0.93359375}, {"start": 1390.25, "end": 1390.99, "word": " FN", "probability": 0.388916015625}, {"start": 1390.99, "end": 1391.87, "word": " converges", "probability": 0.681396484375}, {"start": 1391.87, "end": 1392.41, "word": " uniformly", "probability": 0.9697265625}, {"start": 1392.41, "end": 1392.63, "word": " من", "probability": 0.424560546875}, {"start": 1392.63, "end": 1392.81, "word": " A", "probability": 0.873046875}, {"start": 1392.81, "end": 1392.95, "word": " و", "probability": 0.82470703125}, {"start": 1392.95, "end": 1393.11, "word": " B", "probability": 0.5869140625}, {"start": 1393.11, "end": 1393.33, "word": " to", "probability": 0.6884765625}, {"start": 1393.33, "end": 1393.67, "word": " F", "probability": 0.9912109375}, {"start": 1393.67, "end": 1394.49, "word": " then", "probability": 0.476318359375}, {"start": 1394.49, "end": 1395.39, "word": " إذا", "probability": 0.6038818359375}, {"start": 1395.39, "end": 1395.87, "word": " لما", "probability": 0.890380859375}, {"start": 1395.87, "end": 1396.49, "word": " تكون", "probability": 0.965087890625}, {"start": 1396.49, "end": 1397.09, "word": " ال", "probability": 0.912109375}, {"start": 1397.09, "end": 1398.01, "word": " convergence", "probability": 0.708984375}, {"start": 1398.01, "end": 1398.55, "word": " uniform", "probability": 0.908203125}, {"start": 1398.55, "end": 1399.15, "word": " then", "probability": 0.541015625}, {"start": 1399.15, "end": 1400.07, "word": " ال", "probability": 0.58203125}, {"start": 1400.07, "end": 1400.35, "word": " F", "probability": 0.6982421875}, {"start": 1400.35, "end": 1400.57, "word": " is", "probability": 0.83935546875}, {"start": 1400.57, "end": 1401.23, "word": " integrable", "probability": 0.8733723958333334}, {"start": 1401.23, "end": 1401.39, "word": " من", "probability": 0.61767578125}, {"start": 1401.39, "end": 1401.53, "word": " A", "probability": 0.97265625}, {"start": 1401.53, "end": 1401.65, "word": " و", "probability": 0.98291015625}, {"start": 1401.65, "end": 1401.81, "word": " B", "probability": 0.88232421875}, {"start": 1401.81, "end": 1402.03, "word": " هاي", "probability": 0.5167236328125}, {"start": 1402.03, "end": 1402.55, "word": " واحد", "probability": 0.749755859375}, {"start": 1402.55, "end": 1403.31, "word": " اتنين", "probability": 0.8709716796875}, {"start": 1403.31, "end": 1403.85, "word": " ال", "probability": 0.95263671875}, {"start": 1403.85, "end": 1404.51, "word": " integration", "probability": 0.9755859375}, {"start": 1404.51, "end": 1404.77, "word": " لل", "probability": 0.705078125}, {"start": 1404.77, "end": 1404.97, "word": " F", "probability": 0.87353515625}, {"start": 1404.97, "end": 1405.11, "word": " of", "probability": 0.71435546875}, {"start": 1405.11, "end": 1405.31, "word": " X", "probability": 0.79150390625}, {"start": 1405.31, "end": 1405.65, "word": " DX", "probability": 0.85498046875}, {"start": 1405.65, "end": 1405.87, "word": " من", "probability": 0.9833984375}, {"start": 1405.87, "end": 1406.03, "word": " A", "probability": 0.97119140625}, {"start": 1406.03, "end": 1406.31, "word": " لعين", "probability": 0.628173828125}, {"start": 1406.31, "end": 1406.49, "word": " B", "probability": 0.91796875}, {"start": 1406.49, "end": 1406.87, "word": " بسوء", "probability": 0.6400146484375}, {"start": 1406.87, "end": 1406.99, "word": " ال", "probability": 0.405517578125}, {"start": 1406.99, "end": 1407.21, "word": " limit", "probability": 0.96826171875}, {"start": 1407.21, "end": 1407.39, "word": " لل", "probability": 0.91796875}, {"start": 1407.39, "end": 1408.01, "word": " integration", "probability": 0.96826171875}, {"start": 1408.01, "end": 1408.67, "word": " يعني", "probability": 0.94287109375}, {"start": 1408.67, "end": 1409.01, "word": " بمعنى", "probability": 0.98818359375}, {"start": 1409.01, "end": 1409.27, "word": " أخر", "probability": 0.822509765625}, {"start": 1409.27, "end": 1409.47, "word": " ال", "probability": 0.468017578125}, {"start": 1409.47, "end": 1409.71, "word": " limit", "probability": 0.98828125}, {"start": 1409.71, "end": 1409.87, "word": " لل", "probability": 0.9033203125}, {"start": 1409.87, "end": 1410.47, "word": " integration", "probability": 0.984375}, {"start": 1410.47, "end": 1410.91, "word": " بسوء", "probability": 0.976806640625}, {"start": 1410.91, "end": 1411.03, "word": " ال", "probability": 0.9677734375}, {"start": 1411.03, "end": 1411.63, "word": " integration", "probability": 0.97607421875}, {"start": 1411.63, "end": 1412.01, "word": " لل", "probability": 0.83935546875}, {"start": 1412.01, "end": 1412.55, "word": " limit", "probability": 0.98486328125}], "temperature": 1.0}, {"id": 63, "seek": 142957, "start": 1413.23, "end": 1429.57, "text": "لما نكون هذا ماله uniform convergence خلّينا نشوف اللي هو البرهان الان خلّينا نسمي الفترة تبعتنا let J بتساوي A وB الان بما أن FN بتروح لل F uniform اللي في صناعة الكلام هذا", "tokens": [1211, 15042, 8717, 30544, 23758, 3714, 6027, 3224, 9452, 32181, 16490, 1211, 11703, 9957, 995, 8717, 8592, 38688, 13672, 1829, 31439, 2423, 26890, 3224, 7649, 2423, 7649, 16490, 1211, 11703, 9957, 995, 8717, 38251, 1829, 27188, 2655, 25720, 6055, 3555, 34268, 8315, 718, 508, 39894, 3794, 995, 45865, 316, 4032, 33, 2423, 7649, 4724, 15042, 14739, 479, 45, 39894, 32887, 5016, 24976, 479, 9452, 13672, 1829, 8978, 20328, 8315, 27884, 2423, 28820, 10943, 23758], "avg_logprob": -0.22624999562899273, "compression_ratio": 1.532967032967033, "no_speech_prob": 4.76837158203125e-06, "words": [{"start": 1413.23, "end": 1413.61, "word": "لما", "probability": 0.83740234375}, {"start": 1413.61, "end": 1413.91, "word": " نكون", "probability": 0.87890625}, {"start": 1413.91, "end": 1414.17, "word": " هذا", "probability": 0.92041015625}, {"start": 1414.17, "end": 1414.67, "word": " ماله", "probability": 0.7823079427083334}, {"start": 1414.67, "end": 1415.21, "word": " uniform", "probability": 0.499267578125}, {"start": 1415.21, "end": 1416.31, "word": " convergence", "probability": 0.98388671875}, {"start": 1416.31, "end": 1417.07, "word": " خلّينا", "probability": 0.780859375}, {"start": 1417.07, "end": 1417.63, "word": " نشوف", "probability": 0.9938151041666666}, {"start": 1417.63, "end": 1418.05, "word": " اللي", "probability": 0.91015625}, {"start": 1418.05, "end": 1418.49, "word": " هو", "probability": 0.99462890625}, {"start": 1418.49, "end": 1419.89, "word": " البرهان", "probability": 0.9189453125}, {"start": 1419.89, "end": 1420.63, "word": " الان", "probability": 0.48248291015625}, {"start": 1420.63, "end": 1421.23, "word": " خلّينا", "probability": 0.9125}, {"start": 1421.23, "end": 1421.83, "word": " نسمي", "probability": 0.9710286458333334}, {"start": 1421.83, "end": 1422.29, "word": " الفترة", "probability": 0.9856770833333334}, {"start": 1422.29, "end": 1422.73, "word": " تبعتنا", "probability": 0.867431640625}, {"start": 1422.73, "end": 1422.97, "word": " let", "probability": 0.609375}, {"start": 1422.97, "end": 1423.27, "word": " J", "probability": 0.48779296875}, {"start": 1423.27, "end": 1423.91, "word": " بتساوي", "probability": 0.804443359375}, {"start": 1423.91, "end": 1424.09, "word": " A", "probability": 0.68505859375}, {"start": 1424.09, "end": 1424.41, "word": " وB", "probability": 0.74560546875}, {"start": 1424.41, "end": 1425.35, "word": " الان", "probability": 0.9365234375}, {"start": 1425.35, "end": 1425.69, "word": " بما", "probability": 0.9619140625}, {"start": 1425.69, "end": 1425.87, "word": " أن", "probability": 0.44775390625}, {"start": 1425.87, "end": 1426.29, "word": " FN", "probability": 0.42962646484375}, {"start": 1426.29, "end": 1426.71, "word": " بتروح", "probability": 0.9552408854166666}, {"start": 1426.71, "end": 1426.89, "word": " لل", "probability": 0.72265625}, {"start": 1426.89, "end": 1427.15, "word": " F", "probability": 0.44775390625}, {"start": 1427.15, "end": 1427.69, "word": " uniform", "probability": 0.68359375}, {"start": 1427.69, "end": 1427.95, "word": " اللي", "probability": 0.7451171875}, {"start": 1427.95, "end": 1428.61, "word": " في", "probability": 0.666015625}, {"start": 1428.61, "end": 1428.95, "word": " صناعة", "probability": 0.7434895833333334}, {"start": 1428.95, "end": 1429.33, "word": " الكلام", "probability": 0.9558919270833334}, {"start": 1429.33, "end": 1429.57, "word": " هذا", "probability": 0.95068359375}], "temperature": 1.0}, {"id": 64, "seek": 145024, "start": 1431.6, "end": 1450.24, "text": "الان تنسوش ان الـ Fn is integrable يعني ايش ما لها bounded يعني ال sequence of bounded functions و زي ما صارت ال sequence of bounded functions اذا automatic بتتحقق ان اللي هي اللي ما اللي هي تبعت ال norm لل F ناقص Fm اللي هو بتروح للسفر يعني Fn يعني Fn", "tokens": [6027, 7649, 6055, 1863, 3794, 2407, 8592, 16472, 2423, 39184, 479, 77, 307, 16200, 424, 638, 37495, 22653, 1975, 1829, 8592, 19446, 5296, 11296, 37498, 37495, 22653, 2423, 8310, 295, 37498, 6828, 4032, 30767, 1829, 19446, 20328, 9640, 2655, 2423, 8310, 295, 37498, 6828, 1975, 15730, 12509, 39894, 2655, 5016, 4587, 4587, 16472, 13672, 1829, 39896, 13672, 1829, 19446, 13672, 1829, 39896, 6055, 3555, 34268, 2423, 2026, 24976, 479, 8717, 995, 4587, 9381, 479, 76, 13672, 1829, 31439, 39894, 32887, 5016, 24976, 3794, 5172, 2288, 37495, 22653, 479, 77, 37495, 22653, 479, 77], "avg_logprob": -0.26562501204774736, "compression_ratio": 1.8, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 1431.6000000000001, "end": 1432.16, "word": "الان", "probability": 0.6861572265625}, {"start": 1432.16, "end": 1432.72, "word": " تنسوش", "probability": 0.95390625}, {"start": 1432.72, "end": 1432.92, "word": " ان", "probability": 0.7744140625}, {"start": 1432.92, "end": 1433.04, "word": " الـ", "probability": 0.3724365234375}, {"start": 1433.04, "end": 1433.36, "word": " Fn", "probability": 0.55712890625}, {"start": 1433.36, "end": 1433.54, "word": " is", "probability": 0.54296875}, {"start": 1433.54, "end": 1434.18, "word": " integrable", "probability": 0.8069661458333334}, {"start": 1434.18, "end": 1434.52, "word": " يعني", "probability": 0.852294921875}, {"start": 1434.52, "end": 1434.8, "word": " ايش", "probability": 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2143.5, "end": 2143.76, "word": " limit", "probability": 0.853515625}], "temperature": 1.0}, {"id": 95, "seek": 216629, "start": 2144.03, "end": 2166.29, "text": "وهذا من خلال الخطوات اللي طلعت عندنا حلّجي يطلع بشكل سهل وبشكل سلس وخلّينا نشوفه الان since epsilon was arbitrary then f is integrable on j زي ما قلنا خد ال integration من a لb f of x dx نقص ال integration من a لb fn of x dx بساوي اللي هو", "tokens": [2407, 3224, 15730, 9154, 16490, 1211, 6027, 33962, 9566, 2407, 9307, 13672, 1829, 23032, 1211, 34268, 43242, 8315, 11331, 1211, 11703, 7435, 1829, 7251, 9566, 1211, 3615, 4724, 8592, 28820, 8608, 3224, 1211, 46599, 8592, 28820, 8608, 1211, 3794, 4032, 9778, 1211, 11703, 9957, 995, 8717, 8592, 38688, 3224, 2423, 7649, 1670, 17889, 390, 23211, 550, 283, 307, 16200, 424, 638, 322, 361, 30767, 1829, 19446, 12174, 1211, 8315, 16490, 3215, 2423, 10980, 9154, 257, 5296, 65, 283, 295, 2031, 30017, 8717, 4587, 9381, 2423, 10980, 9154, 257, 5296, 65, 283, 77, 295, 2031, 30017, 4724, 3794, 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0.88525390625}, {"start": 2160.81, "end": 2161.29, "word": " لb", "probability": 0.6551513671875}, {"start": 2161.29, "end": 2161.69, "word": " f", "probability": 0.5888671875}, {"start": 2161.69, "end": 2161.81, "word": " of", "probability": 0.5537109375}, {"start": 2161.81, "end": 2162.01, "word": " x", "probability": 0.9208984375}, {"start": 2162.01, "end": 2162.31, "word": " dx", "probability": 0.383544921875}, {"start": 2162.31, "end": 2162.65, "word": " نقص", "probability": 0.7394205729166666}, {"start": 2162.65, "end": 2162.75, "word": " ال", "probability": 0.56640625}, {"start": 2162.75, "end": 2163.09, "word": " integration", "probability": 0.9658203125}, {"start": 2163.09, "end": 2163.29, "word": " من", "probability": 0.982421875}, {"start": 2163.29, "end": 2163.43, "word": " a", "probability": 0.98095703125}, {"start": 2163.43, "end": 2163.63, "word": " لb", "probability": 0.90771484375}, {"start": 2163.63, "end": 2163.97, "word": " fn", "probability": 0.587158203125}, {"start": 2163.97, "end": 2164.07, "word": " of", "probability": 0.91552734375}, {"start": 2164.07, "end": 2164.27, "word": " x", "probability": 0.9912109375}, {"start": 2164.27, "end": 2164.57, "word": " dx", "probability": 0.90380859375}, {"start": 2164.57, "end": 2165.07, "word": " بساوي", "probability": 0.822021484375}, {"start": 2165.07, "end": 2166.07, "word": " اللي", "probability": 0.92333984375}, {"start": 2166.07, "end": 2166.29, "word": " هو", "probability": 0.98828125}], "temperature": 1.0}, {"id": 96, "seek": 218396, "start": 2167.19, "end": 2183.97, "text": "الـ absolute value طبعاً صارت هذه integrable وهذه integrable ماشي الحال إذا ال integration هذا نقص ال integration هذا بساوي ال integration لهذا نقص هذا الكل إيش ماله DX من خواص التكامل اللي أخدناها سابقاً وهذا نفسه أصغر أو يساوي", "tokens": [6027, 39184, 8236, 2158, 23032, 3555, 3615, 995, 14111, 20328, 9640, 2655, 29538, 16200, 424, 638, 37037, 24192, 16200, 424, 638, 3714, 33599, 1829, 21542, 6027, 11933, 15730, 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هذه", "probability": 0.326904296875}, {"start": 2170.07, "end": 2170.63, "word": " integrable", "probability": 0.7415364583333334}, {"start": 2170.63, "end": 2170.83, "word": " وهذه", "probability": 0.7392578125}, {"start": 2170.83, "end": 2171.51, "word": " integrable", "probability": 0.7631022135416666}, {"start": 2171.51, "end": 2171.89, "word": " ماشي", "probability": 0.704345703125}, {"start": 2171.89, "end": 2172.23, "word": " الحال", "probability": 0.975341796875}, {"start": 2172.23, "end": 2172.81, "word": " إذا", "probability": 0.6181640625}, {"start": 2172.81, "end": 2172.95, "word": " ال", "probability": 0.84814453125}, {"start": 2172.95, "end": 2173.87, "word": " integration", "probability": 0.88916015625}, {"start": 2173.87, "end": 2174.15, "word": " هذا", "probability": 0.515625}, {"start": 2174.15, "end": 2174.47, "word": " نقص", "probability": 0.921875}, {"start": 2174.47, "end": 2174.55, "word": " ال", "probability": 0.8017578125}, {"start": 2174.55, "end": 2174.89, "word": " integration", "probability": 0.9619140625}, {"start": 2174.89, "end": 2175.15, "word": " هذا", "probability": 0.64453125}, {"start": 2175.15, "end": 2175.51, "word": " بساوي", "probability": 0.791259765625}, {"start": 2175.51, "end": 2175.63, "word": " ال", "probability": 0.978515625}, {"start": 2175.63, "end": 2176.15, "word": " integration", "probability": 0.97314453125}, {"start": 2176.15, "end": 2176.63, "word": " لهذا", "probability": 0.923095703125}, {"start": 2176.63, "end": 2177.01, "word": " نقص", "probability": 0.9812825520833334}, {"start": 2177.01, "end": 2177.17, "word": " هذا", "probability": 0.90380859375}, {"start": 2177.17, "end": 2177.47, "word": " الكل", "probability": 0.795654296875}, {"start": 2177.47, "end": 2177.69, "word": " إيش", "probability": 0.5965169270833334}, {"start": 2177.69, "end": 2178.09, "word": " ماله", "probability": 0.8564453125}, {"start": 2178.09, "end": 2178.75, "word": " DX", "probability": 0.2998046875}, {"start": 2178.75, "end": 2178.91, "word": " من", "probability": 0.96826171875}, {"start": 2178.91, "end": 2179.29, "word": " خواص", "probability": 0.83447265625}, {"start": 2179.29, "end": 2179.75, "word": " التكامل", "probability": 0.95263671875}, {"start": 2179.75, "end": 2179.83, "word": " اللي", "probability": 0.98291015625}, {"start": 2179.83, "end": 2180.21, "word": " أخدناها", "probability": 0.916015625}, {"start": 2180.21, "end": 2181.29, "word": " سابقاً", "probability": 0.9349609375}, {"start": 2181.29, "end": 2181.75, "word": " وهذا", "probability": 0.85791015625}, {"start": 2181.75, "end": 2182.53, "word": " نفسه", "probability": 0.9972330729166666}, {"start": 2182.53, "end": 2183.19, "word": " أصغر", "probability": 0.9796142578125}, {"start": 2183.19, "end": 2183.37, "word": " أو", "probability": 0.93798828125}, {"start": 2183.37, "end": 2183.97, "word": " يساوي", "probability": 0.92822265625}], "temperature": 1.0}, {"id": 97, "seek": 219921, "start": 2184.63, "end": 2199.21, "text": "أصغر يساوي ال integration من a لb واخدناها سابقاً absolute value of f of x ماقص fn of x لكل ايش ماله dx ال absolute value integration أصغر يساوي ال integration لل absolute value وهذا نفسه", "tokens": [10721, 9381, 17082, 2288, 7251, 3794, 995, 45865, 2423, 10980, 9154, 257, 5296, 65, 4032, 47283, 3215, 8315, 11296, 8608, 16758, 4587, 995, 14111, 8236, 2158, 295, 283, 295, 2031, 19446, 4587, 9381, 283, 77, 295, 2031, 5296, 28820, 1975, 1829, 8592, 19446, 43761, 30017, 2423, 8236, 2158, 10980, 5551, 9381, 17082, 2288, 7251, 3794, 995, 45865, 2423, 10980, 24976, 8236, 2158, 37037, 15730, 8717, 36178, 3224], "avg_logprob": -0.3249080777168274, "compression_ratio": 1.6688741721854305, "no_speech_prob": 1.0132789611816406e-06, "words": [{"start": 2184.63, "end": 2185.05, "word": "أصغر", "probability": 0.74652099609375}, {"start": 2185.05, "end": 2185.29, "word": " يساوي", "probability": 0.603271484375}, {"start": 2185.29, "end": 2185.37, "word": " ال", "probability": 0.93212890625}, {"start": 2185.37, 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"word": " اللي", "probability": 0.9755859375}, {"start": 2376.05, "end": 2376.33, "word": " هو", "probability": 0.98486328125}, {"start": 2376.33, "end": 2377.11, "word": " المحاضرة", "probability": 0.9833984375}, {"start": 2377.11, "end": 2378.43, "word": " ضال", "probability": 0.58642578125}, {"start": 2378.43, "end": 2378.85, "word": " عندي", "probability": 0.880859375}, {"start": 2378.85, "end": 2379.23, "word": " نقطة", "probability": 0.9933268229166666}, {"start": 2379.23, "end": 2379.65, "word": " أخيرة", "probability": 0.7822265625}, {"start": 2379.65, "end": 2380.01, "word": " بس", "probability": 0.945556640625}, {"start": 2380.01, "end": 2380.41, "word": " قبلها", "probability": 0.9251302083333334}, {"start": 2380.41, "end": 2380.79, "word": " خليني", "probability": 0.7640380859375}, {"start": 2380.79, "end": 2381.23, "word": " نذكر", "probability": 0.7626139322916666}, {"start": 2381.23, "end": 2381.59, "word": " هال", "probability": 0.693115234375}, {"start": 2381.59, "end": 2382.27, "word": " نظرية", "probability": 0.8634033203125}, {"start": 2382.27, "end": 2382.53, "word": " اللي", "probability": 0.967041015625}, {"start": 2382.53, "end": 2382.85, "word": " هي", "probability": 0.72509765625}, {"start": 2382.85, "end": 2383.91, "word": " برهانة", "probability": 0.92431640625}, {"start": 2383.91, "end": 2384.35, "word": " خارج", "probability": 0.9915364583333334}, {"start": 2384.35, "end": 2384.95, "word": " نطاق", "probability": 0.9566650390625}, {"start": 2384.95, "end": 2385.51, "word": " اللي", "probability": 0.983642578125}, {"start": 2385.51, "end": 2385.71, "word": " هو", "probability": 0.99169921875}, {"start": 2385.71, "end": 2386.37, "word": " الكتاب", "probability": 0.9541015625}, {"start": 2386.37, "end": 2386.83, "word": " و", "probability": 0.939453125}, {"start": 2386.83, "end": 2387.47, "word": " نشوف", "probability": 0.9607747395833334}, {"start": 2387.47, "end": 2388.65, "word": " اللي", "probability": 0.974853515625}, {"start": 2388.65, "end": 2389.11, "word": " هو", "probability": 0.99609375}], "temperature": 1.0}, {"id": 106, "seek": 240004, "start": 2393.38, "end": 2400.04, "text": "إيش بتقول هذه اللي هي النظرية؟ إيش بتقول bounded convergence theorem؟", "tokens": [28814, 1829, 8592, 39894, 39648, 29538, 13672, 1829, 39896, 28239, 19913, 2288, 10632, 22807, 11933, 1829, 8592, 39894, 39648, 37498, 32181, 20904, 22807], "avg_logprob": -0.3417968613406022, "compression_ratio": 1.1318681318681318, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2393.38, "end": 2393.78, "word": "إيش", "probability": 0.6481424967447916}, {"start": 2393.78, "end": 2394.18, "word": " بتقول", "probability": 0.98583984375}, {"start": 2394.18, "end": 2394.64, "word": " هذه", "probability": 0.71533203125}, {"start": 2394.64, "end": 2395.06, "word": " اللي", "probability": 0.871337890625}, {"start": 2395.06, "end": 2395.46, "word": " هي", "probability": 0.75927734375}, {"start": 2395.46, "end": 2397.66, "word": " النظرية؟", "probability": 0.836767578125}, {"start": 2397.66, "end": 2397.84, "word": " إيش", "probability": 0.7859700520833334}, {"start": 2397.84, "end": 2398.16, "word": " بتقول", "probability": 0.991455078125}, {"start": 2398.16, "end": 2398.54, "word": " bounded", "probability": 0.450439453125}, {"start": 2398.54, "end": 2399.16, "word": " convergence", "probability": 0.95654296875}, {"start": 2399.16, "end": 2400.04, "word": " theorem؟", "probability": 0.891357421875}], "temperature": 1.0}, {"id": 107, "seek": 242344, "start": 2400.77, "end": 2423.45, "text": "النظرية بتقول let FN be a sequence of functions that are integrable in A وB مفترضين إنها integrable and suppose that بدي يبعد عن ال uniform convergence بتقول لو شيلنا ال uniform convergence ممكن نبدله بإيش إيه؟ ويظل محافظنا على ال limit لل integration بساوي ال integration لل limit؟ آه، بنفع، نشوف كيف", "tokens": [6027, 1863, 19913, 2288, 10632, 39894, 39648, 718, 479, 45, 312, 257, 8310, 295, 6828, 300, 366, 16200, 424, 638, 294, 316, 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"end": 2402.19, "word": " be", "probability": 0.708984375}, {"start": 2402.19, "end": 2402.31, "word": " a", "probability": 0.96044921875}, {"start": 2402.31, "end": 2402.65, "word": " sequence", "probability": 0.9658203125}, {"start": 2402.65, "end": 2402.85, "word": " of", "probability": 0.97216796875}, {"start": 2402.85, "end": 2403.45, "word": " functions", "probability": 0.8681640625}, {"start": 2403.45, "end": 2404.41, "word": " that", "probability": 0.80419921875}, {"start": 2404.41, "end": 2405.61, "word": " are", "probability": 0.94580078125}, {"start": 2405.61, "end": 2406.41, "word": " integrable", "probability": 0.8802083333333334}, {"start": 2406.41, "end": 2406.63, "word": " in", "probability": 0.537109375}, {"start": 2406.63, "end": 2406.83, "word": " A", "probability": 0.89013671875}, {"start": 2406.83, "end": 2407.15, "word": " وB", "probability": 0.705322265625}, {"start": 2407.15, "end": 2408.71, "word": " مفترضين", "probability": 0.928515625}, {"start": 2408.71, "end": 2408.93, "word": " إنها", "probability": 0.7242431640625}, {"start": 2408.93, "end": 2409.71, "word": " integrable", "probability": 0.7195638020833334}, {"start": 2409.71, "end": 2410.51, "word": " and", "probability": 0.7158203125}, {"start": 2410.51, "end": 2411.07, "word": " suppose", "probability": 0.90869140625}, {"start": 2411.07, "end": 2411.49, "word": " that", "probability": 0.8623046875}, {"start": 2411.49, "end": 2412.07, "word": " بدي", "probability": 0.720703125}, {"start": 2412.07, "end": 2412.53, "word": " يبعد", "probability": 0.8194986979166666}, {"start": 2412.53, "end": 2412.77, "word": " عن", "probability": 0.99462890625}, {"start": 2412.77, "end": 2412.91, "word": " ال", "probability": 0.9755859375}, {"start": 2412.91, "end": 2413.19, "word": " uniform", "probability": 0.76220703125}, {"start": 2413.19, "end": 2413.87, "word": " convergence", "probability": 0.94677734375}, {"start": 2413.87, "end": 2414.33, "word": " بتقول", "probability": 0.59197998046875}, {"start": 2414.33, "end": 2414.49, "word": " لو", "probability": 0.95947265625}, {"start": 2414.49, "end": 2414.77, "word": " شيلنا", "probability": 0.9609375}, {"start": 2414.77, "end": 2414.85, "word": " ال", "probability": 0.99169921875}, {"start": 2414.85, "end": 2415.09, "word": " uniform", "probability": 0.86669921875}, {"start": 2415.09, "end": 2415.83, "word": " convergence", "probability": 0.97119140625}, {"start": 2415.83, "end": 2416.71, "word": " ممكن", "probability": 0.975830078125}, {"start": 2416.71, "end": 2417.37, "word": " نبدله", "probability": 0.7626953125}, {"start": 2417.37, "end": 2417.73, "word": " بإيش", "probability": 0.74200439453125}, {"start": 2417.73, "end": 2418.03, "word": " إيه؟", "probability": 0.72760009765625}, {"start": 2418.03, "end": 2418.55, "word": " ويظل", "probability": 0.761474609375}, {"start": 2418.55, "end": 2419.17, "word": " محافظنا", "probability": 0.8232421875}, {"start": 2419.17, "end": 2419.41, "word": " على", "probability": 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some G بكل الأنظم مع وجود هذه الشروط اللي حكينا عليها then the sequence FN converges uniformly to a function F بعد هذا بعد هذا بنلاقي أن FN غصب إن عنا لازم تكون converges uniformly to some function mean F ومش هيك", "tokens": [1353, 512, 460, 4724, 28820, 16247, 1863, 19913, 2304, 20449, 49610, 23328, 29538, 25124, 32887, 9566, 13672, 1829, 11331, 4117, 1829, 8315, 25894, 11296, 550, 264, 8310, 479, 45, 9652, 2880, 48806, 281, 257, 2445, 479, 39182, 23758, 39182, 23758, 44945, 15040, 38436, 14739, 479, 45, 32771, 9381, 3555, 36145, 6225, 8315, 5296, 31377, 2304, 6055, 30544, 9652, 2880, 48806, 281, 512, 2445, 914, 479, 4032, 2304, 8592, 39896, 4117], "avg_logprob": -0.265184846562399, "compression_ratio": 1.4676616915422886, "no_speech_prob": 1.245737075805664e-05, "words": [{"start": 3012.93, "end": 3013.03, "word": "to", "probability": 0.40283203125}, {"start": 3013.03, "end": 3013.29, "word": " some", "probability": 0.87255859375}, {"start": 3013.29, "end": 3013.63, "word": " G", 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"word": " function", "probability": 0.97412109375}, {"start": 3038.03, "end": 3038.49, "word": " mean", "probability": 0.213623046875}, {"start": 3038.49, "end": 3039.51, "word": " F", "probability": 0.884765625}, {"start": 3039.51, "end": 3040.87, "word": " ومش", "probability": 0.9739583333333334}, {"start": 3040.87, "end": 3041.83, "word": " هيك", "probability": 0.952880859375}], "temperature": 1.0}, {"id": 136, "seek": 305948, "start": 3042.54, "end": 3059.48, "text": "والـ function F هذه تطلع الـ derivative تبعتها هي مين؟ هي الـ G يعني وكأنه أن الـ F N converges uniformly to F", "tokens": [2407, 6027, 39184, 2445, 479, 29538, 6055, 9566, 1211, 3615, 2423, 39184, 13760, 6055, 3555, 34268, 11296, 39896, 3714, 9957, 22807, 39896, 2423, 39184, 460, 37495, 22653, 4032, 4117, 33456, 3224, 14739, 2423, 39184, 479, 426, 9652, 2880, 48806, 281, 479], "avg_logprob": -0.2518601261434101, "compression_ratio": 1.1742424242424243, "no_speech_prob": 8.940696716308594e-07, "words": [{"start": 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3049.92, "end": 3050.22, "word": " G", "probability": 0.5791015625}, {"start": 3050.22, "end": 3051.26, "word": " يعني", "probability": 0.906494140625}, {"start": 3051.26, "end": 3052.24, "word": " وكأنه", "probability": 0.858154296875}, {"start": 3052.24, "end": 3053.2, "word": " أن", "probability": 0.45751953125}, {"start": 3053.2, "end": 3054.02, "word": " الـ", "probability": 0.83544921875}, {"start": 3054.02, "end": 3054.32, "word": " F", "probability": 0.830078125}, {"start": 3054.32, "end": 3054.76, "word": " N", "probability": 0.254150390625}, {"start": 3054.76, "end": 3056.42, "word": " converges", "probability": 0.6307373046875}, {"start": 3056.42, "end": 3058.1, "word": " uniformly", "probability": 0.9736328125}, {"start": 3058.1, "end": 3058.94, "word": " to", "probability": 0.93994140625}, {"start": 3058.94, "end": 3059.48, "word": " F", "probability": 0.99072265625}], "temperature": 1.0}, {"id": 137, "seek": 308231, "start": 3060.11, "end": 3082.31, "text": "اللي هي الـ derivative لها مين الاش الـ G يعني F N برايم بتكون uniformly convergence to some function G هتطلع الـ F N converge uniformly لل function اللي تفضلها هو مين هو عبارة عن الـ G", "tokens": [6027, 20292, 39896, 2423, 39184, 13760, 5296, 11296, 3714, 9957, 2423, 33599, 2423, 39184, 460, 37495, 22653, 479, 426, 4724, 23557, 32640, 39894, 30544, 48806, 32181, 281, 512, 2445, 460, 8032, 2655, 9566, 1211, 3615, 2423, 39184, 479, 426, 41881, 48806, 24976, 2445, 13672, 1829, 6055, 5172, 11242, 1211, 11296, 31439, 3714, 9957, 31439, 6225, 3555, 9640, 3660, 18871, 2423, 39184, 460], "avg_logprob": -0.22296626984126985, "compression_ratio": 1.5266272189349113, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 3060.11, "end": 3060.63, "word": "اللي", "probability": 0.864501953125}, {"start": 3060.63, "end": 3061.31, "word": " هي", "probability": 0.94384765625}, {"start": 3061.31, "end": 3061.97, "word": " الـ", "probability": 0.923095703125}, {"start": 3061.97, "end": 3062.79, "word": " 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convergence", "probability": 0.65673828125}, {"start": 3068.97, "end": 3069.29, "word": " to", "probability": 0.68115234375}, {"start": 3069.29, "end": 3069.53, "word": " some", "probability": 0.90673828125}, {"start": 3069.53, "end": 3069.99, "word": " function", "probability": 0.96728515625}, {"start": 3069.99, "end": 3070.51, "word": " G", "probability": 0.857421875}, {"start": 3070.51, "end": 3071.95, "word": " هتطلع", "probability": 0.9419921875}, {"start": 3071.95, "end": 3072.25, "word": " الـ", "probability": 0.83203125}, {"start": 3072.25, "end": 3072.53, "word": " F", "probability": 0.94873046875}, {"start": 3072.53, "end": 3072.99, "word": " N", "probability": 0.88818359375}, {"start": 3072.99, "end": 3074.27, "word": " converge", "probability": 0.40771484375}, {"start": 3074.27, "end": 3075.37, "word": " uniformly", "probability": 0.9560546875}, {"start": 3075.37, "end": 3076.23, "word": " لل", "probability": 0.6533203125}, {"start": 3076.23, "end": 3076.99, "word": " function", "probability": 0.7265625}, {"start": 3076.99, "end": 3077.89, "word": " اللي", "probability": 0.93408203125}, {"start": 3077.89, "end": 3078.97, "word": " تفضلها", "probability": 0.95205078125}, {"start": 3078.97, "end": 3079.27, "word": " هو", "probability": 0.98583984375}, {"start": 3079.27, "end": 3079.93, "word": " مين", "probability": 0.98583984375}, {"start": 3079.93, "end": 3080.63, "word": " هو", "probability": 0.75390625}, {"start": 3080.63, "end": 3081.19, "word": " عبارة", "probability": 0.99658203125}, {"start": 3081.19, "end": 3081.53, "word": " عن", "probability": 0.99755859375}, {"start": 3081.53, "end": 3082.07, "word": " الـ", "probability": 0.919677734375}, {"start": 3082.07, "end": 3082.31, "word": " G", "probability": 0.95556640625}], "temperature": 1.0}, {"id": 138, "seek": 311119, "start": 3082.89, "end": 3111.19, "text": "طيب إذا في عندي شغلتين بدنا نثبت ان FN can form convergence to F والـ F' إيش بتساوي؟ بتساوي مين؟ اللي هي الـ G يعني هتصير إنه الـ FN' اللي هو sequence of differentiable functions بتطلع عندي هذه sequence of differentiable functions converge to function بتكون differentiable", "tokens": [9566, 1829, 3555, 11933, 15730, 8978, 18871, 16254, 13412, 17082, 1211, 2655, 9957, 47525, 8315, 8717, 12984, 3555, 2655, 16472, 479, 45, 393, 1254, 32181, 281, 479, 16070, 39184, 479, 6, 11933, 1829, 8592, 39894, 3794, 995, 45865, 22807, 39894, 3794, 995, 45865, 3714, 9957, 22807, 13672, 1829, 39896, 2423, 39184, 460, 37495, 22653, 8032, 2655, 9381, 13546, 36145, 3224, 2423, 39184, 479, 45, 6, 13672, 1829, 31439, 8310, 295, 819, 9364, 6828, 39894, 9566, 1211, 3615, 18871, 16254, 29538, 8310, 295, 819, 9364, 6828, 41881, 281, 2445, 39894, 30544, 819, 9364], "avg_logprob": -0.29200268304476173, "compression_ratio": 1.7582938388625593, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 3082.89, "end": 3083.63, "word": "طيب", "probability": 0.90576171875}, {"start": 3083.63, "end": 3084.05, "word": " إذا", "probability": 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"end": 3348.48, "text": "إذا صارت Fm ناقص Fn عبارة عن الـ differential بالـ function على هذه الـ closed bounded interval إذا أكيد continuous عليها كمان إذا بالـ mean value theorem مطبقة الـ mean value theorem ده طبقها على مين يا جماعة؟ على الـ Fm ناقص Fn يعني هذه دالتنا اللي بدنا نطبق عليها الـ mean value theorem إذا by mean value theorem there exists Y بين الـ X0 والـ X بحيث", "tokens": [28814, 15730, 20328, 9640, 2655, 479, 76, 8717, 995, 4587, 9381, 479, 77, 6225, 3555, 9640, 3660, 18871, 2423, 39184, 15756, 20666, 39184, 2445, 15844, 29538, 2423, 39184, 5395, 37498, 15035, 11933, 15730, 5551, 4117, 25708, 10957, 25894, 11296, 9122, 2304, 7649, 11933, 15730, 20666, 39184, 914, 2158, 20904, 3714, 9566, 3555, 28671, 2423, 39184, 914, 2158, 20904, 11778, 3224, 23032, 3555, 4587, 11296, 15844, 3714, 9957, 35186, 10874, 15042, 27884, 22807, 15844, 2423, 39184, 479, 76, 8717, 995, 4587, 9381, 479, 77, 37495, 22653, 29538, 11778, 6027, 2655, 8315, 13672, 1829, 47525, 8315, 8717, 9566, 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{"id": 175, "seek": 395842, "start": 3928.94, "end": 3958.42, "text": "مدام continuous و أثبتت FM بتروح converts uniformly لها إذا ال limit هذه هتطلع برضه إيش ماله is continuous و هذه خلوها في الذاكرة مكون احنا أوصلنا أو أثبتنا الجزء الأول من النظرية أن FM converts uniformly to F و ال limit لها سمناها F و هيطلع لنا عبارة عن continuous function شوفوا يا جماعة الآن", "tokens": [2304, 3215, 10943, 10957, 4032, 5551, 12984, 3555, 2655, 2655, 29614, 39894, 32887, 5016, 38874, 48806, 5296, 11296, 11933, 15730, 2423, 4948, 29538, 8032, 2655, 9566, 1211, 3615, 4724, 43042, 3224, 11933, 1829, 8592, 3714, 6027, 3224, 307, 10957, 4032, 29538, 16490, 1211, 2407, 11296, 8978, 32545, 995, 4117, 25720, 3714, 30544, 1975, 5016, 8315, 34051, 36520, 8315, 34051, 5551, 12984, 3555, 2655, 8315, 25724, 11622, 38207, 16247, 12610, 9154, 28239, 19913, 2288, 10632, 14739, 29614, 38874, 48806, 281, 479, 4032, 2423, 4948, 5296, 11296, 8608, 2304, 8315, 11296, 479, 4032, 39896, 9566, 1211, 3615, 5296, 8315, 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"text": "صار عندي الآن", "tokens": [9381, 9640, 18871, 16254, 6024, 48506], "avg_logprob": -0.4040178486279079, "compression_ratio": 0.6857142857142857, "no_speech_prob": 0.0, "words": [{"start": 3962.6, "end": 3963.02, "word": "صار", "probability": 0.828125}, {"start": 3963.02, "end": 3963.3, "word": " عندي", "probability": 0.6588134765625}, {"start": 3963.3, "end": 3963.76, "word": " الآن", "probability": 0.695556640625}], "temperature": 1.0}, {"id": 177, "seek": 399469, "start": 3965.85, "end": 3994.69, "text": "أثبتنا أن Fn تتعامل بشكل مرتبط سمينا limit Fn بسوء F which is continuous because Fn زي ما قلنا is continuous لأنه في الأصل هي differentiable و Fn بتروح لل F إذن صارت ال F اللي هي نفسها continuous الآن بدي أستخدم ال continuity عند ال F بدي أصل اللي بديه طيب خد C الآن ب any fixed point in J الآن بدي أروح إلى إثبات أنه", "tokens": [10721, 12984, 3555, 2655, 8315, 14739, 479, 77, 6055, 2655, 3615, 10943, 1211, 4724, 8592, 28820, 3714, 43500, 3555, 9566, 8608, 2304, 1829, 8315, 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412951, "start": 4111.81, "end": 4129.51, "text": "ماشي الحال طبقت الـ Mean Value Theorem على الدالة ال .. اللي هي Fm ناقص Fn لكل M O N الان خدوا الحسبة الان خد X لا تساوى C لإنه .. X لا تساوى C لإن انا بدأو أدي X عالميا على C يعني X minus C اللي هو أصغر من Delta يعني X بدأو أديها على C", "tokens": [2304, 33599, 1829, 21542, 6027, 23032, 3555, 38149, 2423, 39184, 12302, 39352, 440, 37956, 15844, 32748, 6027, 3660, 2423, 4386, 13672, 1829, 39896, 479, 76, 8717, 995, 4587, 9381, 479, 77, 5296, 28820, 376, 422, 426, 2423, 7649, 16490, 3215, 14407, 21542, 35457, 3660, 2423, 7649, 16490, 3215, 1783, 20193, 6055, 3794, 995, 2407, 7578, 383, 5296, 28814, 1863, 3224, 4386, 1783, 20193, 6055, 3794, 995, 2407, 7578, 383, 5296, 28814, 1863, 1975, 8315, 47525, 10721, 2407, 5551, 16254, 1783, 6225, 45340, 25528, 15844, 383, 37495, 22653, 1783, 3175, 383, 13672, 1829, 31439, 5551, 9381, 17082, 2288, 9154, 18183, 37495, 22653, 1783, 47525, 10721, 2407, 5551, 16254, 11296, 15844, 383], "avg_logprob": -0.24788850599581058, "compression_ratio": 1.688073394495413, "no_speech_prob": 2.6226043701171875e-06, "words": [{"start": 4111.81, "end": 4112.13, "word": "ماشي", "probability": 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"end": 4356.01, "word": " هذه", "probability": 0.69970703125}, {"start": 4356.01, "end": 4356.93, "word": " متحققة", "probability": 0.9581298828125}, {"start": 4356.93, "end": 4358.31, "word": " للان", "probability": 0.4453125}, {"start": 4358.31, "end": 4358.63, "word": " أكبر", "probability": 0.9033203125}, {"start": 4358.63, "end": 4358.97, "word": " سواء", "probability": 0.49462890625}, {"start": 4358.97, "end": 4359.27, "word": " H", "probability": 0.49853515625}, {"start": 4359.27, "end": 4360.05, "word": " و", "probability": 0.54833984375}, {"start": 4360.05, "end": 4360.41, "word": " هذه", "probability": 0.54443359375}, {"start": 4360.41, "end": 4361.25, "word": " متحققة", "probability": 0.9814453125}, {"start": 4361.25, "end": 4362.19, "word": " للان", "probability": 0.90966796875}, {"start": 4362.19, "end": 4362.61, "word": " الأكبر", "probability": 0.9622395833333334}, {"start": 4362.61, "end": 4363.01, "word": " سواء", "probability": 0.9029947916666666}, {"start": 4363.01, "end": 4363.11, "word": " N", "probability": 0.830078125}, {"start": 4363.11, "end": 4363.43, "word": " عشان", "probability": 0.9832356770833334}, {"start": 4363.43, "end": 4364.07, "word": " استخدم", "probability": 0.84765625}, {"start": 4364.07, "end": 4364.87, "word": " التنتين", "probability": 0.8321940104166666}, {"start": 4364.87, "end": 4365.95, "word": " بتاخد", "probability": 0.8052571614583334}, {"start": 4365.95, "end": 4366.09, "word": " ال", "probability": 0.89990234375}, {"start": 4366.09, "end": 4366.53, "word": " maximum", "probability": 0.923828125}, {"start": 4366.53, "end": 4366.91, "word": " بين", "probability": 0.9853515625}, {"start": 4366.91, "end": 4367.21, "word": " هذه", "probability": 0.7568359375}, {"start": 4367.21, "end": 4367.33, "word": " و", "probability": 0.76611328125}, {"start": 4367.33, "end": 4367.59, "word": " هذه", "probability": 0.80029296875}, {"start": 4367.59, "end": 4367.75, "word": " و", "probability": 0.88330078125}, {"start": 4367.75, "end": 4368.15, "word": " اقول", "probability": 0.6724853515625}, {"start": 4368.15, "end": 4368.97, "word": " let", "probability": 0.732421875}, {"start": 4368.97, "end": 4369.57, "word": " K", "probability": 0.796875}, {"start": 4369.57, "end": 4370.63, "word": " بسوء", "probability": 0.582275390625}, {"start": 4370.63, "end": 4370.75, "word": " ال", "probability": 0.8818359375}, {"start": 4370.75, "end": 4371.29, "word": " maximum", "probability": 0.974609375}, {"start": 4371.29, "end": 4371.61, "word": " بين", "probability": 0.89306640625}, {"start": 4371.61, "end": 4371.85, "word": " H", "probability": 0.96875}, {"start": 4371.85, "end": 4372.01, "word": " of", "probability": 0.6962890625}, {"start": 4372.01, "end": 4372.35, "word": " epsilon", "probability": 0.262939453125}, {"start": 4372.35, "end": 4372.51, "word": " و", "probability": 0.81640625}, {"start": 4372.51, "end": 4372.63, "word": " N", "probability": 0.84912109375}, {"start": 4372.63, "end": 4373.11, "word": " epsilon", "probability": 0.9296875}, {"start": 4373.11, "end": 4373.85, "word": " اذا", "probability": 0.665771484375}, {"start": 4373.85, "end": 4374.47, "word": " الان", "probability": 0.794921875}, {"start": 4374.47, "end": 4375.45, "word": " صار", "probability": 0.98583984375}, {"start": 4375.45, "end": 4375.97, "word": " عندى", "probability": 0.945068359375}, {"start": 4375.97, "end": 4376.55, "word": " التنتين", "probability": 0.98779296875}, {"start": 4376.55, "end": 4377.93, "word": " متحققات", "probability": 0.97958984375}, {"start": 4377.93, "end": 4378.79, "word": " لكل", "probability": 0.97705078125}, {"start": 4378.79, "end": 4379.13, "word": " N", "probability": 0.6708984375}, {"start": 4379.13, "end": 4379.59, "word": " أكبر", "probability": 0.9580078125}, {"start": 4379.59, "end": 4379.83, "word": " او", "probability": 0.794677734375}, {"start": 4379.83, "end": 4380.33, "word": " يساوي", "probability": 0.8294677734375}, {"start": 4380.33, "end": 4380.63, "word": " K", "probability": 0.94189453125}, {"start": 4380.63, "end": 4381.37, "word": " ومشروع", "probability": 0.944921875}, {"start": 4381.37, "end": 4382.11, "word": " استخدامه", "probability": 0.95654296875}], "temperature": 1.0}, {"id": 195, "seek": 438775, "start": 4382.67, "end": 4387.75, "text": "الآن من جهة ثانية بدي استخدم الـ FK' of C إنها exist", "tokens": [6027, 148, 48506, 9154, 10874, 3224, 3660, 38637, 7649, 10632, 4724, 16254, 44713, 9778, 40448, 2423, 39184, 479, 42, 6, 295, 383, 36145, 11296, 2514], "avg_logprob": -0.3473557818394441, "compression_ratio": 0.9425287356321839, "no_speech_prob": 0.0, "words": [{"start": 4382.67, "end": 4383.11, "word": "الآن", "probability": 0.5707600911458334}, {"start": 4383.11, "end": 4383.67, "word": " من", "probability": 0.83251953125}, {"start": 4383.67, "end": 4384.01, "word": " جهة", "probability": 0.96630859375}, {"start": 4384.01, "end": 4384.53, "word": " ثانية", "probability": 0.9762369791666666}, {"start": 4384.53, "end": 4385.35, 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0.9130859375}, {"start": 4399.72, "end": 4399.96, "word": " هذه", "probability": 0.85791015625}, {"start": 4399.96, "end": 4400.06, "word": " و", "probability": 0.89453125}, {"start": 4400.06, "end": 4400.66, "word": " أخلص", "probability": 0.969970703125}, {"start": 4400.66, "end": 4400.96, "word": " اللي", "probability": 0.920166015625}, {"start": 4400.96, "end": 4401.48, "word": " بديها", "probability": 0.81640625}, {"start": 4401.48, "end": 4402.14, "word": " since", "probability": 0.58984375}, {"start": 4402.14, "end": 4402.62, "word": " fk", "probability": 0.4166259765625}, {"start": 4402.62, "end": 4402.9, "word": " prime", "probability": 0.64208984375}, {"start": 4402.9, "end": 4403.14, "word": " of", "probability": 0.958984375}, {"start": 4403.14, "end": 4403.4, "word": " c", "probability": 0.74658203125}, {"start": 4403.4, "end": 4403.82, "word": " exist", "probability": 0.52294921875}, {"start": 4403.82, "end": 4404.18, "word": " مين", "probability": 0.712890625}, {"start": 4404.18, "end": 4404.36, "word": " ال", "probability": 0.92529296875}, {"start": 4404.36, "end": 4404.52, "word": " K", "probability": 0.344482421875}, {"start": 4404.52, "end": 4404.96, "word": " هذه", "probability": 0.75146484375}, {"start": 4404.96, "end": 4405.72, "word": " ال", "probability": 0.376953125}, {"start": 4405.72, "end": 4405.94, "word": " K", "probability": 0.81591796875}, {"start": 4405.94, "end": 4406.2, "word": " هذه", "probability": 0.87646484375}, {"start": 4406.2, "end": 4406.36, "word": " اللي", "probability": 0.987060546875}, {"start": 4406.36, "end": 4406.74, "word": " خدتها", "probability": 0.8846435546875}, {"start": 4406.74, "end": 4406.86, "word": " ال", "probability": 0.6005859375}, {"start": 4406.86, "end": 4407.32, "word": " superman", "probability": 0.73583984375}, {"start": 4407.32, "end": 4407.54, "word": " بين", "probability": 0.93310546875}, {"start": 4407.54, "end": 4407.9, "word": " هذه", "probability": 0.85205078125}, {"start": 4407.9, "end": 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444289, "start": 4414.44, "end": 4442.9, "text": "then مدام fk prime of c exist then if x minus c أكبر من صفر أصغر من دلتا دلتا k طبعا الدلتا هتعتمد على اللي هو ال epsilon نسميها دلتا k لإنها اعتمدت على مين؟ على الكل يعنى then there exists دلتا k بحيث x minus c أصغر من دلتا يعطيني fk of x ناقص fk of c على x minus c اللي هي ناقص fk prime of c أصغر من إبسلون فاهمين هذا؟ مدام fk", "tokens": [19096, 3714, 3215, 10943, 283, 74, 5835, 295, 269, 2514, 550, 498, 2031, 3175, 269, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 5551, 9381, 17082, 2288, 9154, 11778, 1211, 2655, 995, 11778, 1211, 2655, 995, 350, 23032, 3555, 3615, 995, 32748, 1211, 2655, 995, 8032, 2655, 34268, 2304, 3215, 15844, 13672, 1829, 31439, 2423, 17889, 8717, 38251, 1829, 11296, 11778, 1211, 2655, 995, 350, 5296, 28814, 1863, 11296, 1975, 34268, 2304, 3215, 2655, 15844, 3714, 9957, 22807, 15844, 2423, 28820, 37495, 1863, 7578, 550, 456, 8198, 11778, 1211, 2655, 995, 350, 4724, 5016, 1829, 12984, 2031, 3175, 269, 5551, 9381, 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يظلوا كله قدامكم الان اللي وصلت اليه مالي for every epsilon أكبر من سفر بقول there exists delta أكبر من سفر such that if", "tokens": [1211, 8848, 1863, 20328, 9640, 43242, 7578, 23758, 3714, 29245, 23328, 37037, 1829, 1975, 2655, 1863, 9957, 37037, 1829, 6055, 1211, 9307, 3660, 5296, 33456, 36764, 24401, 4032, 1975, 2655, 1863, 9957, 4032, 6055, 1211, 9307, 3660, 4724, 16254, 1975, 36520, 13672, 1829, 4724, 16254, 3224, 16490, 1211, 9957, 995, 8717, 4117, 2655, 3555, 34105, 6225, 8592, 7649, 19446, 8717, 7435, 46152, 8592, 31767, 5172, 5016, 3660, 6225, 8592, 7649, 7251, 19913, 1211, 14407, 28242, 3224, 12174, 3215, 10943, 24793, 2423, 7649, 13672, 1829, 4032, 36520, 2655, 45595, 3224, 3714, 6027, 1829, 337, 633, 17889, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 4724, 39648, 456, 8198, 8289, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 1270, 300, 498], "avg_logprob": -0.22311970578917004, "compression_ratio": 1.7076271186440677, "no_speech_prob": 2.562999725341797e-06, "words": 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"probability": 0.97314453125}, {"start": 4702.53, "end": 4702.65, "word": " هو", "probability": 0.978515625}, {"start": 4702.65, "end": 4702.99, "word": " تمانية", "probability": 0.75732421875}, {"start": 4702.99, "end": 4703.33, "word": " اتنين", "probability": 0.9395751953125}, {"start": 4703.33, "end": 4703.71, "word": " وهيك", "probability": 0.8607177734375}, {"start": 4703.71, "end": 4703.95, "word": " بنكون", "probability": 0.913330078125}, {"start": 4703.95, "end": 4704.57, "word": " خلصنا", "probability": 0.99462890625}, {"start": 4704.57, "end": 4705.31, "word": " الحديث", "probability": 0.9866536458333334}, {"start": 4705.31, "end": 4705.43, "word": " اللي", "probability": 0.980224609375}, {"start": 4705.43, "end": 4706.05, "word": " بنتحدثه", "probability": 0.89892578125}, {"start": 4706.05, "end": 4706.13, "word": " في", "probability": 0.958984375}, {"start": 4706.13, "end": 4706.37, "word": " chapter", "probability": 0.473876953125}, {"start": 4706.37, "end": 4706.95, "word": " تمانية", "probability": 0.8582356770833334}, {"start": 4706.95, "end": 4708.21, "word": " وعندنا", "probability": 0.849609375}, {"start": 4708.21, "end": 4708.53, "word": " طبعا", "probability": 0.90869140625}, {"start": 4708.53, "end": 4709.07, "word": " الأسئلة", "probability": 0.971923828125}, {"start": 4709.07, "end": 4709.63, "word": " المطلوبة", "probability": 0.97705078125}, {"start": 4709.63, "end": 4709.91, "word": " هيها", "probability": 0.929931640625}, {"start": 4709.91, "end": 4710.01, "word": " في", "probability": 0.9248046875}, {"start": 4710.01, "end": 4710.37, "word": " نهاية", "probability": 0.88525390625}, {"start": 4710.37, "end": 4710.57, "word": " كل", "probability": 0.91845703125}, {"start": 4710.57, "end": 4710.89, "word": " section", "probability": 0.490478515625}, {"start": 4710.89, "end": 4711.21, "word": " تمانية", "probability": 0.93798828125}, {"start": 4711.21, "end": 4711.39, "word": " واحد", "probability": 0.970703125}, {"start": 4711.39, "end": 4711.77, "word": " وتمانية", "probability": 0.76318359375}, {"start": 4711.77, "end": 4712.17, "word": " اتنين", "probability": 0.9771728515625}, {"start": 4712.17, "end": 4712.93, "word": " وإلى", "probability": 0.9153645833333334}, {"start": 4712.93, "end": 4713.43, "word": " لقاء", "probability": 0.95947265625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 4714.2080625, "duration_after_vad": 4625.11750000001}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ca525981e7f39494eba28daaf4173739677c0df1
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/PVuSVTtYhJQ_raw.srt
@@ -0,0 +1,4028 @@
+1
+00:00:04,910 --> 00:00:07,650
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:07,650 --> 00:00:10,710
+والصلاة والسلام على سيد المسالين سيدنا محمد على
+
+3
+00:00:10,710 --> 00:00:15,510
+آله وصحبه أجمعين هذه هي المحاضرة رقم واحد وعشرين
+
+4
+00:00:15,510 --> 00:00:19,630
+في مساق تحليل حقيقة نين لطلاب وطالبات الجامعة
+
+5
+00:00:19,630 --> 00:00:26,460
+الإسلامية قسم الرياضيات في كلية العلمهنكمل الحديث
+
+6
+00:00:26,460 --> 00:00:32,660
+عن اللي هو الـ Sequence of Functions خلصنا من
+
+7
+00:00:32,660 --> 00:00:36,180
+سيكشن الـ Pointwise أو الـ Uniform Convergence
+
+8
+00:00:36,180 --> 00:00:40,340
+الآن بدنا نشوف تأثير الـ Pointwise و الـ Uniform
+
+9
+00:00:40,340 --> 00:00:44,700
+Convergence على اللي هو بعض الـ Sequence of
+
+10
+00:00:44,700 --> 00:00:49,040
+Functions إيش بتسوي فيهاالان تحت عنوان 82
+
+11
+00:00:49,040 --> 00:00:53,900
+interchange of limits هنجي نشوف اللي هو الـ
+
+12
+00:00:53,900 --> 00:00:57,160
+sequence of continuous functions لو كانت اللي هي
+
+13
+00:00:57,160 --> 00:01:01,620
+pointwise convergence هل صح شرط انه لو لمين بتروح
+
+14
+00:01:01,620 --> 00:01:04,680
+في ال pointwise convergence ان يكون continuous ولا
+
+15
+00:01:04,680 --> 00:01:09,780
+بلزمها دعم شوية نخلي اللي هو بدل ال pointwise
+
+16
+00:01:09,780 --> 00:01:13,220
+convergence نخليه uniform و ال uniform يوديالـ
+
+17
+00:01:13,220 --> 00:01:16,040
+function اللي بتطلع تطلع continuous ولا لأ؟ هنشوف
+
+18
+00:01:16,040 --> 00:01:19,680
+الكلام هذا الشغل الثاني هنشوف أنه لو كانت ال
+
+19
+00:01:19,680 --> 00:01:23,360
+function is integrable هل ال sequence of functions
+
+20
+00:01:23,360 --> 00:01:27,640
+is integrable؟ هل ال limit اللي لما يكون اللي هو
+
+21
+00:01:27,640 --> 00:01:32,080
+ال Fn بتروح لل F point twice أو ال Fn بتروح لل F
+
+22
+00:01:32,080 --> 00:01:37,850
+uniform convergence؟هل هذا واحد منهم بيقدّي انه ال
+
+23
+00:01:37,850 --> 00:01:39,850
+integration لل limit بيساوي ال limit لل
+
+24
+00:01:39,850 --> 00:01:42,730
+integration يعني ال limit بتدخل جوا ال integration
+
+25
+00:01:42,730 --> 00:01:46,990
+ولا لأ؟ هل دخولها يعني بال point wise بحافظ على
+
+26
+00:01:46,990 --> 00:01:50,170
+المساواة ولا لأ بال زمن اللي هو ال uniform
+
+27
+00:01:50,170 --> 00:01:54,730
+convergence؟ هنشوفه والشغل التالثة هل انه لو كان
+
+28
+00:01:54,730 --> 00:01:59,150
+في عندي sequence of differentiable functions هتطلع
+
+29
+00:01:59,150 --> 00:02:04,040
+ان هو لو كان عندي F unconverged زي ال Fهيطلع عندي
+
+30
+00:02:04,040 --> 00:02:06,460
+اللي هو ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+31
+00:02:06,460 --> 00:02:07,500
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+32
+00:02:07,500 --> 00:02:07,620
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+33
+00:02:07,620 --> 00:02:08,200
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+34
+00:02:08,200 --> 00:02:08,300
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+35
+00:02:08,300 --> 00:02:08,340
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+36
+00:02:08,340 --> 00:02:09,020
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+37
+00:02:09,020 --> 00:02:10,060
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+38
+00:02:10,060 --> 00:02:11,000
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+39
+00:02:11,000 --> 00:02:13,560
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+40
+00:02:13,560 --> 00:02:13,560
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+41
+00:02:13,560 --> 00:02:13,720
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+42
+00:02:13,720 --> 00:02:23,020
+.. ال .. ال .. ال .. ال .. ال .. ال ..
+
+43
+00:02:28,280 --> 00:02:34,340
+المثال الأول اللي أمامنا هاي في عندى example 821 g
+
+44
+00:02:34,340 --> 00:02:38,900
+n of x بساوية x أُس n هذا شفناها يا جماعة على
+
+45
+00:02:38,900 --> 00:02:43,640
+الفترة x 01 is continuous أكيد اللي هي ال function
+
+46
+00:02:43,640 --> 00:02:47,540
+هذه g 1 of x بساوية x g 2 of x بساوية x تربيع g 3
+
+47
+00:02:47,540 --> 00:02:52,110
+of x بساوية x تكعيب كله عبارة عن دوال متصلةالان
+
+48
+00:02:52,110 --> 00:02:55,950
+and even continuously differentiable بيقولك أصلا
+
+49
+00:02:55,950 --> 00:03:00,430
+هيطلع مش ده continuous لأ و ال derivative L هتطلع
+
+50
+00:03:00,430 --> 00:03:03,190
+موجودة و ال derivative continuous يعني
+
+51
+00:03:03,190 --> 00:03:06,510
+continuously differentiable بالرغم من أنها القوة
+
+52
+00:03:06,510 --> 00:03:09,650
+اللي موجودة في ال sequence g of n of x من ناحية
+
+53
+00:03:09,650 --> 00:03:14,220
+انها continuously differentiableإلا إنه لما جينا
+
+54
+00:03:14,220 --> 00:03:17,880
+ناخد limit GN of X as N goes to infinity طلعت
+
+55
+00:03:17,880 --> 00:03:21,280
+بتساوي G of X اللي هي 0 X في الفترة 01 و 1 في
+
+56
+00:03:21,280 --> 00:03:25,180
+الفترة X اللي هو بتساوي 1 يعني اللي طلعتلنا ال
+
+57
+00:03:25,180 --> 00:03:31,040
+limit إلها طلعتلنا إياها is not continuousيعني
+
+58
+00:03:31,040 --> 00:03:35,080
+بالرغم إن ال limit موجودة طبعا هذا pointwise
+
+59
+00:03:35,080 --> 00:03:38,520
+convergence اللي هو ال pointwise convergence هنا
+
+60
+00:03:38,520 --> 00:03:44,660
+طلعلي اللي هو ال sequence of functions GN limitها
+
+61
+00:03:44,660 --> 00:03:48,820
+مش continuous ماقدرش يطلع اللي هو continuous
+
+62
+00:03:48,820 --> 00:03:52,560
+function هذا ال pointwise convergence بالرغم إنه
+
+63
+00:03:52,560 --> 00:03:56,740
+GN of X is اللي هو continuously differentiable
+
+64
+00:03:56,740 --> 00:04:03,550
+الآن ناخد مثال آخر إذاإذا كانت GN هي عملية عملية
+
+65
+00:04:03,550 --> 00:04:04,930
+عملية عملية عملية عملية عملية عملية عملية عملية
+
+66
+00:04:04,930 --> 00:04:15,930
+عملية عملية عملية عملية عملية عملية عملية
+
+67
+00:04:15,960 --> 00:04:19,840
+طيب نشوف ال example اللي بعده الان ال example اللي
+
+68
+00:04:19,840 --> 00:04:25,400
+بعده بقول لي اللي هو let FN be هذه let FN من 0
+
+69
+00:04:25,400 --> 00:04:30,460
+لعند 1 من 01 لعند R be defined for N أكبر سواء 2
+
+70
+00:04:30,460 --> 00:04:37,150
+by مراكبين دالة تشوف هذه الدالةعندي Fn of X بساوي
+
+71
+00:04:37,150 --> 00:04:42,490
+N تربيع X الـ N هذا دالة المتغير X هذه الـ Fn و
+
+72
+00:04:42,490 --> 00:04:45,030
+الـ N تربيحة دولة اللي هنا اللي بدلن على اللي هو
+
+73
+00:04:45,030 --> 00:04:49,810
+الـ sequence F1, F2, F3, F4 أما الدالة هي F منين
+
+74
+00:04:49,810 --> 00:04:55,280
+بدلالة X الـ Fn بدلالة Xإذاً هذا عدد يعني هذه
+
+75
+00:04:55,280 --> 00:04:59,280
+Linear الدلة N تربيع X في الفترة من Zero لعند مين
+
+76
+00:04:59,280 --> 00:05:03,140
+واحدة على N لو جينا رسمناها اللي هي N تربيع عند
+
+77
+00:05:03,140 --> 00:05:07,620
+طبعا دي خطية بكفي نقطين نرسم عشان نحط خط بينهم عند
+
+78
+00:05:07,620 --> 00:05:12,900
+X بصفر بتطلع اللي هي الأولى اللي هي N تربيع X صفر
+
+79
+00:05:12,900 --> 00:05:16,540
+ال X واحدة على N بيصير واحدة على N في N تربيع
+
+80
+00:05:16,540 --> 00:05:19,640
+بتطلع اللي هي واحدة على N هي واحدة على N بتطلع
+
+81
+00:05:19,640 --> 00:05:23,760
+واحدة على Nبتطلع قيمة as an أسف an تربيع في واحدة
+
+82
+00:05:23,760 --> 00:05:30,260
+الان بتطلع an إذا هذه اللي هي الخط هيه للجزء اللي
+
+83
+00:05:30,260 --> 00:05:33,760
+هو الأولاني الجزء الثاني اللي هو بيبدأ من عند
+
+84
+00:05:33,760 --> 00:05:36,520
+واحدة الان عند مين واحدة الان عند اتنين ان خلينا
+
+85
+00:05:36,520 --> 00:05:41,150
+نرسمههذا برضه ايش معله ده اللي خطية لإنه ايش ماخده
+
+86
+00:05:41,150 --> 00:05:46,190
+عبارة عن ناقص N تربيع في X ناقص 2 على N هذي ثابت و
+
+87
+00:05:46,190 --> 00:05:50,190
+هذي ثابت بالنسبالي X الان X لحقها إذا ده اللي خطية
+
+88
+00:05:50,190 --> 00:05:53,890
+برضه و ده الخط خليني ناخد اللي هي X بتساوي 1 على N
+
+89
+00:05:53,890 --> 00:05:58,010
+خد 1 على N ناقص 2 على N بيطلع ناقص 1 على N في ناقص
+
+90
+00:05:58,010 --> 00:06:01,790
+N تربيع بيطلع 1 على N فعلا 1 على N بيطلع 1 على N و
+
+91
+00:06:01,790 --> 00:06:06,150
+N فبيطلع مين؟لأن هذا بيطلع واحدة الان في ناقص ان
+
+92
+00:06:06,150 --> 00:06:09,850
+تربيع بتطلع اللي هي ناقص واحدة الان في ناقص ان
+
+93
+00:06:09,850 --> 00:06:13,990
+تربيع بتطلع اللي هي ان ان إذا هذه واحدة الان بتطلع
+
+94
+00:06:13,990 --> 00:06:17,590
+أيهاش بالساوء اللي بعيدها النقطة الأخيرة اتنين على
+
+95
+00:06:17,590 --> 00:06:20,370
+ان اتنين على الناقص اتنين على الان سفر هذي بيصير
+
+96
+00:06:20,370 --> 00:06:24,480
+أيهاش سفر ترجع وين على سفر فبتنزل في الشكل هذاباجي
+
+97
+00:06:24,480 --> 00:06:27,540
+الدالة من عند اللي هو طبعا معرف عالميا على الفترة
+
+98
+00:06:27,540 --> 00:06:31,720
+Zero و واحد هي من Zero لعندي اتنين على N الان N
+
+99
+00:06:31,720 --> 00:06:34,340
+بتساوي واحد او اتنين او تلاتة او اربع او خمس او
+
+100
+00:06:34,340 --> 00:06:37,580
+ستة لبدكية كل دالة من هنا بتختلف عن التانية يعني
+
+101
+00:06:37,580 --> 00:06:43,220
+هذه FN in general F واحد F اتنين F تلاتة بتيجي
+
+102
+00:06:43,220 --> 00:06:46,360
+لهيك او بتروح لهيك بس بنفس اللي هي الشكل الأمامي
+
+103
+00:06:46,700 --> 00:06:49,900
+إيش عندي في فترة ماتنين على ألا واحد، ماتنين على
+
+104
+00:06:49,900 --> 00:06:53,340
+ألا واحد، الفترة ها دي ماخدين الدالة إيش بتساوي؟
+
+105
+00:06:53,340 --> 00:06:57,440
+بتساوي إيش؟ سفر واضح إن الدالة هي الدالة محترمة
+
+106
+00:06:57,440 --> 00:07:02,880
+دالة is continuousMadame Continuous أكيد اشمالها
+
+107
+00:07:02,880 --> 00:07:08,060
+is integrable إذا الدوال هذول أف واحد و أف اتنين و
+
+108
+00:07:08,060 --> 00:07:11,480
+أف تلاتة و أف أربعة و أف خمسة و أف ستة إلى ما لا
+
+109
+00:07:11,480 --> 00:07:17,380
+نهاية من الأفانز بتكون عبارة عن .. عبارة عن
+
+110
+00:07:17,380 --> 00:07:21,600
+integrable أو أف اتنين و طالعIntegrable Functions
+
+111
+00:07:21,600 --> 00:07:25,760
+من أن و طالع .. من أن واحد بيصير في المشكلة طيب
+
+112
+00:07:25,760 --> 00:07:29,520
+Note that Fn of X is continuous لكل أن أكبر ساوية
+
+113
+00:07:29,520 --> 00:07:34,180
+اتنين ناخد الفكرة احنا Hence, Integrable and ال
+
+114
+00:07:34,180 --> 00:07:38,520
+integration من صفر لأن واحد Fn of X DX ايش بيساوي؟
+
+115
+00:07:38,520 --> 00:07:42,060
+بيساوي اللي هو واحد بتعرفوا تحسبوها هي مساحة ..
+
+116
+00:07:42,060 --> 00:07:48,500
+مساحة المتلت هذا اللي هو اتنين على أناللي هو مضروب
+
+117
+00:07:48,500 --> 00:07:54,360
+في مين فان بطلع جداش اتنين في نص بطلع واحد عارف ان
+
+118
+00:07:54,360 --> 00:07:56,880
+النص القاعدة في الارتفاع إذا الجداش بطلع عندى اللي
+
+119
+00:07:56,880 --> 00:08:00,100
+هو قيمة ال integration هذا بساوي واحد لأن هذا
+
+120
+00:08:00,100 --> 00:08:04,060
+المنطقة إيه شمالها سفر إذا اللي هي قيمة ال
+
+121
+00:08:04,060 --> 00:08:08,480
+integration هذا بساوي واحد يعني ال function of
+
+122
+00:08:08,480 --> 00:08:12,980
+ands integrable وقيمة اللي هي ال integration بساوي
+
+123
+00:08:12,980 --> 00:08:18,850
+واحد لكل and أكبر أو يساوي اتنينان بيساوي واحد
+
+124
+00:08:18,850 --> 00:08:21,490
+اعوضها لحالك وشوف ايش اللي بيصير Note that as n
+
+125
+00:08:21,490 --> 00:08:25,490
+goes to infinity اللي
+
+126
+00:08:25,490 --> 00:08:31,310
+هو الفترات اللي عليها عشان نورجيكم إياهاطيب هذه
+
+127
+00:08:31,310 --> 00:08:35,230
+الدالة معرفها as n goes to infinity هذه الفترة
+
+128
+00:08:35,230 --> 00:08:39,050
+بتصير عبارة عن اللي هو النقطة zero و هذه الفترة
+
+129
+00:08:39,050 --> 00:08:43,130
+بتصير النقطة zero و الفترة 2 على n بتصير اللي هي
+
+130
+00:08:43,130 --> 00:08:48,430
+mean zero ل عند واحد and so الدالة إيش هتصير limit
+
+131
+00:08:48,430 --> 00:08:54,410
+fn of x بتساوي f of x إيش هتساوي هتساوي zeroيعني
+
+132
+00:08:54,410 --> 00:08:59,450
+الان لما ان تروح لما لنهاية هذه كلها ايش مالها
+
+133
+00:08:59,450 --> 00:09:03,870
+بتنطبق عالمين لإن كل ما كبرت ان بتفرد هذه بتفرد
+
+134
+00:09:03,870 --> 00:09:07,170
+هذه بتفرد هذه لإن as n goes to infinity هذا الكلام
+
+135
+00:09:07,170 --> 00:09:12,210
+ده اللي هتفرد و تمشي عالمين على اللي هي من سفر
+
+136
+00:09:12,210 --> 00:09:16,250
+لعند واحد هتظلها ايش قيمتها قيمتها سفر فاهمين عليش
+
+137
+00:09:16,250 --> 00:09:21,600
+بقول الان as n goes to infinityهذه الآن بتبدأ اللي
+
+138
+00:09:21,600 --> 00:09:26,000
+هي .. هذي بيصير الان نص .. هذي المنطقة كلها .. آسف
+
+139
+00:09:26,000 --> 00:09:28,040
+.. هذي المنطقة كلها هذا اللي بيتمد .. مش هدوله ..
+
+140
+00:09:28,040 --> 00:09:31,440
+هذي اللي بيتمد اللي هي اتنين على ان بتجربها ان ..
+
+141
+00:09:31,440 --> 00:09:34,220
+بتصغر as and goes to infinity واحدة على ان بتجربها
+
+142
+00:09:34,220 --> 00:09:37,850
+انلأن as n goes to infinity بيهدفك بروح لمين في دا
+
+143
+00:09:37,850 --> 00:09:42,750
+السفر يعني بيصير ده لكلها من معرفة ك zero function
+
+144
+00:09:42,750 --> 00:09:46,370
+يعني limit of n of x as n goes to infinity بيساوي
+
+145
+00:09:46,370 --> 00:09:54,030
+اللي هي سفر لكل x limit الفترة 01 مدام سفرأه إذا
+
+146
+00:09:54,030 --> 00:09:57,530
+صار عنده اللي هو لاحظوا ال convergence point wise
+
+147
+00:09:57,530 --> 00:10:01,350
+convergence بعتمد عالميا على x في كل حاجة اللي هو
+
+148
+00:10:01,350 --> 00:10:05,010
+ال integration من سفر اللي عنده واحد f of x dx إيش
+
+149
+00:10:05,010 --> 00:10:09,630
+هيساوي؟ هيساوي zero لأن قيمة ال function كله وين
+
+150
+00:10:09,630 --> 00:10:14,510
+هتيجي على اللي هي السفر فبكون ال integration اللي
+
+151
+00:10:14,510 --> 00:10:19,770
+بيساوي zero اللي بدي أنا .. اللي بدي أفله نلاحظ
+
+152
+00:10:20,580 --> 00:10:27,480
+اللي وصلتله ان ال integration من 0 و 1 لل F of X
+
+153
+00:10:27,480 --> 00:10:33,040
+DX بيساوي 0 أكيد بيسويش 1 اللي هو بيساوي اللي هو
+
+154
+00:10:33,040 --> 00:10:39,510
+مين هذا الواحد limit لل integrationمن zero and
+
+155
+00:10:39,510 --> 00:10:46,170
+واحد F N of X DX as N goes to infinity مظبوط؟ لأن
+
+156
+00:10:46,170 --> 00:10:49,530
+ال integration هذا طلع دايما إيش حيساوي؟ حيساوي
+
+157
+00:10:49,530 --> 00:10:54,130
+واحد بغض النظر عن قيمة ال N إذا هذا ال limit لل
+
+158
+00:10:54,130 --> 00:10:59,270
+integration هاي اللي بتوصله وليس شرط يساوي إيش ال
+
+159
+00:10:59,270 --> 00:11:03,250
+integration لل limit رغم إن ال F طلعت مالها
+
+160
+00:11:03,250 --> 00:11:09,070
+Integrable ال limit يعني ال N limitأف ان of X لما
+
+161
+00:11:09,070 --> 00:11:14,810
+ان تروح لنهاية اللي عندي ايش بيساوي F of X بالرغم
+
+162
+00:11:14,810 --> 00:11:19,810
+من هيك اللي هو هذا الكلام لكل X وين موجودة لكل X
+
+163
+00:11:19,810 --> 00:11:25,990
+موجودة في الفترة 01 بالرغم من هيكالان الـ
+
+164
+00:11:25,990 --> 00:11:31,790
+integration limit للـ integration من C1 لـ F of X,
+
+165
+00:11:32,290 --> 00:11:39,570
+N of X, DX ما سواش لل integration من 0 ل1 limit لـ
+
+166
+00:11:39,570 --> 00:11:45,070
+F, N of Xيعني ايش اللي بقصده اللي هو ماقدرتش ال
+
+167
+00:11:45,070 --> 00:11:49,710
+limit تدخل لجوه ال integration في ضوء ال pointwise
+
+168
+00:11:49,710 --> 00:11:53,470
+convergence وفي ضوء ان ال limit نفسها طلعت
+
+169
+00:11:53,470 --> 00:11:58,050
+integrable اذا الان هنا ال pointwise convergence
+
+170
+00:11:58,050 --> 00:12:02,570
+قد يعطي انه اللي هو ال function تكون اللي هو
+
+171
+00:12:02,570 --> 00:12:07,560
+integrable لكن لا يمكنإنه .. أو .. أيه .. أو ..
+
+172
+00:12:07,560 --> 00:12:11,280
+آسف ليس شرطا أن يكون limit لل integration بساوي ال
+
+173
+00:12:11,280 --> 00:12:14,600
+integration لل limit إذا بدنا دفعة أكبر من ال
+
+174
+00:12:14,600 --> 00:12:17,820
+pointwise convergence و الحالتين ال continuity و
+
+175
+00:12:17,820 --> 00:12:21,600
+ال integrability هانفتنتين الدفعة هذه هي ال
+
+176
+00:12:21,600 --> 00:12:25,970
+uniform convergence و هنشوف هذا الكلاممن خلال
+
+177
+00:12:25,970 --> 00:12:31,770
+النظريات القادمة أول نظرية عندنا هذه النظرية اللي
+
+178
+00:12:31,770 --> 00:12:35,750
+هي هتتعلق بالـ sequence of continuous functions
+
+179
+00:12:35,750 --> 00:12:40,090
+كيف أنه لو كانت اللي هي sequence of continuous
+
+180
+00:12:40,090 --> 00:12:43,570
+functions converts uniformly to some function F
+
+181
+00:12:43,570 --> 00:12:46,690
+then F must be continuous
+
+182
+00:12:49,500 --> 00:12:54,300
+طيب يا جماعة الان نيجي theorem 8.2.2 بتقول ما يلي
+
+183
+00:12:54,300 --> 00:12:57,520
+الان نتقف ان بيه a sequence of continuous
+
+184
+00:12:57,520 --> 00:13:02,860
+functionson a set A subset من R، إذاً Fn عبارة عن
+
+185
+00:13:02,860 --> 00:13:07,300
+متتابعة من الدوالي المتصلة على الفترة A subset من
+
+186
+00:13:07,300 --> 00:13:12,360
+R and suppose that اللي هي الـ Fn converges
+
+187
+00:13:12,360 --> 00:13:15,640
+uniformly on A to a function F من A لعند R، بقول
+
+188
+00:13:15,640 --> 00:13:21,200
+إذا ولا يهمك مضمون إن الـ F نفسها تكون إيش ما لها
+
+189
+00:13:21,200 --> 00:13:24,400
+is a continuous function، إذاً
+
+190
+00:13:30,480 --> 00:13:36,500
+المقاومة للمعاملات المستمرة هي مستمرة باستخدام
+
+191
+00:13:36,500 --> 00:13:41,880
+المعاملة المستمرة المستمرة المستمرة المستمرة
+
+192
+00:13:41,880 --> 00:13:43,080
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+193
+00:13:43,080 --> 00:13:43,760
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+194
+00:13:43,760 --> 00:13:43,760
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+195
+00:13:43,760 --> 00:13:43,760
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+196
+00:13:43,760 --> 00:13:43,760
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+197
+00:13:43,760 --> 00:13:43,760
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+198
+00:13:43,760 --> 00:13:43,760
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+199
+00:13:43,760 --> 00:13:43,760
+المستمرة المستمرة المستمرة المستمرة المستمرة
+
+200
+00:13:43,760 --> 00:13:50,400
+المستمرة المستمرة المستمرة المستمرة المستأو نصل
+
+201
+00:13:50,400 --> 00:13:54,440
+إلى أن الـ F نفسها continuous عالميا على الـ A
+
+202
+00:13:54,440 --> 00:13:59,280
+element الـ R خلّينا بس نرتب حاجة ده بما أن F
+
+203
+00:13:59,280 --> 00:14:03,180
+unconverts uniform للـ F on A إذا حسب التعريف الـ
+
+204
+00:14:03,180 --> 00:14:05,820
+uniform convergence لكل يسون أكبر من سفر there
+
+205
+00:14:05,820 --> 00:14:10,690
+exists actuallyof epsilon اللي هو such that أو
+
+206
+00:14:10,690 --> 00:14:13,530
+there exists H such that الـH هتتمد على مين؟ على
+
+207
+00:14:13,530 --> 00:14:16,630
+الـEpsilon if N أكبر يسوى H لأنه for convergence
+
+208
+00:14:16,630 --> 00:14:20,470
+زي ما انتوا عارفين such that if N أكبر يسوى H then
+
+209
+00:14:20,470 --> 00:14:24,610
+F N of X ناقص F of X أصغر من Epsilon ع تلاتة حرن
+
+210
+00:14:24,610 --> 00:14:27,230
+أصغر من Epsilon في الدنيا من ضمنها مين؟ Epsilon
+
+211
+00:14:27,230 --> 00:14:32,010
+على تلاتة for all X element in مين؟ in A إذا الأن
+
+212
+00:14:32,010 --> 00:14:41,540
+أول حاجة لكل Epsilonأكبر من 0 لجينا H such that
+
+213
+00:14:41,540 --> 00:14:46,780
+element of N طبعاً such that for every N أكبر أو
+
+214
+00:14:46,780 --> 00:14:55,100
+يساوي H عندي F N of X نقص F of X أصغر من Y ع تلاتة
+
+215
+00:14:55,100 --> 00:15:04,390
+لكل X element من A هذه أول واحدةلأن let C element
+
+216
+00:15:04,390 --> 00:15:09,870
+in A be arbitrary but fixed ليش؟ بدي أثبتلك أن F
+
+217
+00:15:09,870 --> 00:15:16,050
+continuous on A، F تبعتنا، هي المطلوبكيف بدى
+
+218
+00:15:16,050 --> 00:15:20,150
+أثبتها؟ بدى أخد أي C Fixed لكن Arbitrary إن إيه
+
+219
+00:15:20,150 --> 00:15:23,230
+بما إنه C Arbitrary لو أثبتنا F Continuous and
+
+220
+00:15:23,230 --> 00:15:26,190
+الـC بتكون F Continuous على كل الـMain الـA كيف
+
+221
+00:15:26,190 --> 00:15:31,530
+بدى أثبتها؟ بدى أثبتلك Limit F of X لما X بتروح
+
+222
+00:15:31,530 --> 00:15:36,430
+للـC أش بتساوي F of C إيش معناته هذا؟ معناته لكل ي
+
+223
+00:15:36,430 --> 00:15:40,910
+أكبر من صفر بدى لاجي K element none such that as
+
+224
+00:15:40,910 --> 00:15:49,500
+تلاقي Deltaاللي هو أكبر من سفر such that X minus C
+
+225
+00:15:49,500 --> 00:15:55,280
+أصغر من Delta يعطيني F of X ناقص F of C أصغر من
+
+226
+00:15:55,280 --> 00:16:00,260
+مين من Y هذا اللي بدأ أثبته عشان أصل لمين لل F
+
+227
+00:16:00,260 --> 00:16:04,440
+continuous عند ال C اللي هي كانت arbitrary في ال A
+
+228
+00:16:04,440 --> 00:16:08,620
+إذا F continuous على كل ال A ده هنشوفه الآن طيب
+
+229
+00:16:10,260 --> 00:16:13,740
+استخدمها معناه الـ for convergence هيه وأخدت C
+
+230
+00:16:13,740 --> 00:16:18,800
+arbitrary point but fixed in A then الآن احسب مالي
+
+231
+00:16:18,800 --> 00:16:24,420
+اف .. طبعا هذا الكلام بس جاب المبدأ هنا هذا صحيح
+
+232
+00:16:24,420 --> 00:16:31,800
+لكل N أكبر أو يساوي H من ضمنها صحيح للـH يعني
+
+233
+00:16:31,800 --> 00:16:36,260
+هلاجي ان FH of X ناقص F of X أصغر من مين؟ من
+
+234
+00:16:36,260 --> 00:16:41,540
+إبسلون ع تلاتة لكل X element in Aبرضه صحيح لل F H
+
+235
+00:16:41,540 --> 00:16:48,840
+of C ناقص F of C أصغر من F3 لكل X element A صحيح
+
+236
+00:16:48,840 --> 00:16:52,340
+من ضمن ال X والC وغير و الاخر هي كل العناصر اللي
+
+237
+00:16:52,340 --> 00:16:57,520
+في ال Aأحسبلي الآن F of X ناقص F of C أصغر أو
+
+238
+00:16:57,520 --> 00:17:03,120
+أساوي F of X ناقص F H of X زاد F H of X ناقص F H
+
+239
+00:17:03,120 --> 00:17:06,980
+of C زاد F H of C ناقص F of C يعني إيش اللي سويته
+
+240
+00:17:06,980 --> 00:17:11,680
+اللي هو ضفت قمتين و طرحتين ضفت ال H of X F H of X
+
+241
+00:17:11,680 --> 00:17:16,320
+وضفت ال F H of C عشان بدي أستخدم اللي فوق عشان أصل
+
+242
+00:17:16,320 --> 00:17:24,660
+للي بديها شوف كيف اللي عنديالـ Fx ناقص Fh of X من
+
+243
+00:17:24,660 --> 00:17:29,320
+اللي فوق أصغر من ي على تلاتة هذه نزلوها زي ما هي
+
+244
+00:17:29,320 --> 00:17:33,240
+تسيبوكوا منها هذه Fh of C ناقص F of C أصغر من ي
+
+245
+00:17:33,240 --> 00:17:38,760
+برضه من ي على تلاتة إذا الآن اللي حصلته لكل ي أكبر
+
+246
+00:17:38,760 --> 00:17:45,620
+من 0 لجيت H أبحث
+
+247
+00:17:46,530 --> 00:17:54,690
+إنه لكل N أكبر أو سوء H طلع عندي F X ناقص F of C
+
+248
+00:17:54,690 --> 00:18:00,470
+أصغر من Epsilon ع تلاتة وزائد Epsilon ع تلاتة يعني
+
+249
+00:18:00,470 --> 00:18:04,910
+اتنين Epsilon ع تلاتة زائد ال absolute value ل F H
+
+250
+00:18:04,910 --> 00:18:14,170
+of X ناقص F H of C هذه الآناللي هيسعفني ان هو
+
+251
+00:18:14,170 --> 00:18:18,610
+مفترض لي ان ال FN اللي هي ال sequence كلها
+
+252
+00:18:18,610 --> 00:18:24,010
+continuous عند ال A كلها، اذا اكيد ال FH اللي هي
+
+253
+00:18:24,010 --> 00:18:27,810
+واحدة من ال sequence is continuous عند مين، عند كل
+
+254
+00:18:27,810 --> 00:18:31,930
+العناصر اللي في ال A، من ضمن هنا ال C اللي أخدت
+
+255
+00:18:31,930 --> 00:18:37,490
+fixed but arbitrary و هيك، هبتكون أنا بديت أنهي
+
+256
+00:18:37,490 --> 00:18:42,700
+البرهان، شوف كيفهذه حصلناها طيب الأن أكيد هيقولنا
+
+257
+00:18:42,700 --> 00:18:47,860
+since FH is continuous at C مدام continuous at C
+
+258
+00:18:47,860 --> 00:18:53,800
+إذا لكل ي أكبر من سفر بقدر ألاقي Delta الـ Delta
+
+259
+00:18:53,800 --> 00:18:57,700
+هذه طبعا بدي نوّهلك ويقولك الـ Delta أكيد هتعتمد
+
+260
+00:18:57,700 --> 00:19:01,920
+عالميا على الـ Y وهتعتمد على الـ C لأنه أنا باخد
+
+261
+00:19:01,920 --> 00:19:06,790
+ال continuity لعند مين عند النقطة Cوهتعتمد على ال
+
+262
+00:19:06,790 --> 00:19:11,470
+FH لأنه مااخد ال continuity لمن أنا لل FH إذا
+
+263
+00:19:11,470 --> 00:19:17,610
+الدلتا هذه هي بتعتمد على ال FH و C لل Y على 3 لإنه
+
+264
+00:19:17,610 --> 00:19:24,650
+أنا بدأ استخدم limit FH of X لما X تروح للـC بسوء
+
+265
+00:19:24,650 --> 00:19:30,590
+FH of C يعني لكل Y أكبر من 0 there exists Delta
+
+266
+00:19:30,590 --> 00:19:35,630
+الـDelta هذه لمين تعتمد على الـC وتعتمد على الـFH
+
+267
+00:19:35,630 --> 00:19:40,470
+وتعتمد على مين على اللي هي الـY اللي بتاخدها
+
+268
+00:19:40,470 --> 00:19:45,340
+arbitrarily ماشي الحال طيبإذا التعريف هذا بدي
+
+269
+00:19:45,340 --> 00:19:48,860
+أترجمه لكل إبسلون أكبر من صفر there exist اللي هي
+
+270
+00:19:48,860 --> 00:19:53,300
+Delta أكبر من صفر such that لما X minus C كان أصغر
+
+271
+00:19:53,300 --> 00:19:57,420
+من Delta والـ X في الـ A يعطيني الفرق بين التنتين
+
+272
+00:19:57,420 --> 00:20:00,620
+هدولة أصغر من مين من أي إبسلون في الدنيا من ضمن
+
+273
+00:20:00,620 --> 00:20:04,820
+إبسلون ع تلاتة واضحة الصورة الآن هذه الآن بدي
+
+274
+00:20:04,820 --> 00:20:09,090
+أدمجها مع حد بيصير عندي لكل إبسلونأكبر من سفر
+
+275
+00:20:09,090 --> 00:20:13,330
+there exist Delta أكبر من سفر such that for every
+
+276
+00:20:13,330 --> 00:20:18,610
+X element in A بتحقق X minus C أصغر من Delta اللي
+
+277
+00:20:18,610 --> 00:20:26,230
+هو بيعطيني then F of X ناقص F of C أصغر من 2 Y ع 3
+
+278
+00:20:27,010 --> 00:20:31,930
+ومن هذه أصغر من ي على تلاتة يعني بمعنى أخر أصغر من
+
+279
+00:20:31,930 --> 00:20:37,270
+ي وهذا بالظبط معناته أنه limit f of x لما x تروح ل
+
+280
+00:20:37,270 --> 00:20:40,570
+c بيساوي f of c وهذا كانت c element in a
+
+281
+00:20:40,570 --> 00:20:46,170
+arbitrarily إذا f is continuous on a إذا خلصنا
+
+282
+00:20:46,170 --> 00:20:51,690
+النظرية الأولى اللي هي اللي بتتعلق بال sequence of
+
+283
+00:20:51,690 --> 00:20:52,430
+continuous functions
+
+284
+00:20:58,670 --> 00:21:05,130
+طيب نشوف ال .. ال remark يا جماعة ال
+
+285
+00:21:05,130 --> 00:21:07,430
+remark بيقول لي the limit of a sequence of
+
+286
+00:21:07,430 --> 00:21:10,370
+continuous functions may be continuous but the
+
+287
+00:21:10,370 --> 00:21:13,590
+convergence is pointwise يعني بيقولك يعني ممكن أنت
+
+288
+00:21:13,590 --> 00:21:17,250
+تلاقيلك اللي هو sequence of functions continuous
+
+289
+00:21:17,250 --> 00:21:20,870
+ويكون ال pointwise convergence ويروح لمين ل
+
+290
+00:21:20,870 --> 00:21:25,660
+continuous functions طبعا ممكن أكيدو جبلنا دوال ..
+
+291
+00:21:25,660 --> 00:21:30,920
+دوال من هالنوع اللي هو كان عندى اللي هى ال .. اذا
+
+292
+00:21:30,920 --> 00:21:37,480
+بتتذكروا اظن انتبهتنا قبل شوية انه
+
+293
+00:21:37,480 --> 00:21:48,100
+fn of x بساوى x على n هذه continuous طبعا on Rو F
+
+294
+00:21:48,100 --> 00:21:53,840
+of X اللي هي limitها limit F N of X لما N تروح لما
+
+295
+00:21:53,840 --> 00:21:57,940
+نهاية بساوي اللي هو عبارة عن سفر هذه برضه اي
+
+296
+00:21:57,940 --> 00:22:02,000
+شمالها continuous function بالرغم من ان ال F N
+
+297
+00:22:02,000 --> 00:22:06,780
+بتروح لل F point twice but ال F N does not
+
+298
+00:22:06,780 --> 00:22:10,660
+converge لل F uniformly زي ما أثبتنا يعني ممكن
+
+299
+00:22:10,660 --> 00:22:14,280
+اللي هو ال point twice convergence يكون limitه
+
+300
+00:22:14,280 --> 00:22:16,420
+continuous function لما تكون sequence of
+
+301
+00:22:16,420 --> 00:22:20,810
+continuous functionsطبيعي لكن احنا بنقول الضمان
+
+302
+00:22:20,810 --> 00:22:25,010
+انه يطلع continuous انها تكون uniform convergence
+
+303
+00:22:25,010 --> 00:22:30,830
+وجبنا مثال على انه اللي هو اللي هي sequence of
+
+304
+00:22:30,830 --> 00:22:34,130
+continuous functions converge pointwise to some
+
+305
+00:22:34,130 --> 00:22:36,730
+function that is not continuous زي ما شوفنا في
+
+306
+00:22:36,730 --> 00:22:41,610
+المثال الأول نيجي الآن نحكي عن موضوع او الجزء
+
+307
+00:22:41,610 --> 00:22:47,770
+المتعلق بمينالجزء المتعلق بالـ Integrable
+
+308
+00:22:47,770 --> 00:22:54,370
+Functions الآن نشوف أيش العلاج لإنه نضمن الـ Limit
+
+309
+00:22:54,370 --> 00:22:56,730
+للـ Integration بساوي الـ Integration للـ Limit أو
+
+310
+00:22:56,730 --> 00:22:59,850
+اللي هو لما ندخل الـ Limit للـ Integration متى
+
+311
+00:22:59,850 --> 00:23:04,590
+بنقدر ندخله ويظل المحافظة على المساوية Let FN be a
+
+312
+00:23:04,590 --> 00:23:07,750
+sequence of functions that are integrable on A وB
+
+313
+00:23:08,610 --> 00:23:13,110
+and suppose that FN converges uniformly من A و B
+
+314
+00:23:13,110 --> 00:23:19,150
+to F then إذا لما تكون ال convergence uniform then
+
+315
+00:23:19,150 --> 00:23:23,850
+ال F is integrable من A و B هاي واحد اتنين ال
+
+316
+00:23:23,850 --> 00:23:27,210
+integration لل F of X DX من A لعين B بسوء ال limit
+
+317
+00:23:27,210 --> 00:23:29,870
+لل integration يعني بمعنى أخر ال limit لل
+
+318
+00:23:29,870 --> 00:23:33,910
+integration بسوء ال integration لل limitلما نكون
+
+319
+00:23:33,910 --> 00:23:38,490
+هذا ماله uniform convergence خلّينا نشوف اللي هو
+
+320
+00:23:38,490 --> 00:23:43,270
+البرهان الان خلّينا نسمي الفترة تبعتنا let J
+
+321
+00:23:43,270 --> 00:23:47,950
+بتساوي A وB الان بما أن FN بتروح لل F uniform اللي
+
+322
+00:23:47,950 --> 00:23:53,540
+في صناعة الكلام هذاالان تنسوش ان الـ Fn is
+
+323
+00:23:53,540 --> 00:23:56,320
+integrable يعني ايش ما لها bounded يعني ال
+
+324
+00:23:56,320 --> 00:23:58,200
+sequence of bounded functions و زي ما صارت ال
+
+325
+00:23:58,200 --> 00:24:01,980
+sequence of bounded functions اذا automatic بتتحقق
+
+326
+00:24:01,980 --> 00:24:05,900
+ان اللي هي اللي ما اللي هي تبعت ال norm لل F ناقص
+
+327
+00:24:05,900 --> 00:24:13,130
+Fm اللي هو بتروح للسفر يعني Fn يعني Fnبتروح للـ F
+
+328
+00:24:13,130 --> 00:24:18,850
+uniformly معناته Fn ناقص F اللي هو ايه شماله over
+
+329
+00:24:18,850 --> 00:24:22,650
+some .. over ايه او اللي مسميها J goes to mean to
+
+330
+00:24:22,650 --> 00:24:26,890
+zero يعني normal Fn ناقص الـ F أصغر من يبسطه يعني
+
+331
+00:24:26,890 --> 00:24:29,190
+forever يبسطه نقوم نسيف it there exists كي لم
+
+332
+00:24:29,190 --> 00:24:31,710
+تنساش that forever أن أكبر شهو K أفقن ناقص أفقر
+
+333
+00:24:31,710 --> 00:24:36,150
+أصغر من مين من الـ Y عارفين هالقصة طيب .. طيب نيجي
+
+334
+00:24:36,150 --> 00:24:40,630
+الأنلت جيب سواء a و b since fn converts uniformly
+
+335
+00:24:40,630 --> 00:24:44,650
+to f و since fn is integrable then fn is bounded
+
+336
+00:24:44,650 --> 00:24:48,830
+then we can use our lemma then for every epsilon
+
+337
+00:24:48,830 --> 00:24:52,510
+there exists k of epsilon such that if n أكبر سواء
+
+338
+00:24:52,510 --> 00:24:58,940
+k of epsilon هذه we have f ناقص fnأصغر من Epsilon
+
+339
+00:24:58,940 --> 00:25:01,540
+على أربعة في B minus A أصغر من Epsilon في الدنيا
+
+340
+00:25:01,540 --> 00:25:04,280
+من ضمنها Epsilon على أربعة في B minus A هذه
+
+341
+00:25:04,280 --> 00:25:08,760
+للحسابات زي العادة ما بنحكي فيها بحكي عنها و طبعا
+
+342
+00:25:08,760 --> 00:25:14,400
+في لها تفسير و فسرنا كتير في أوقات
+
+343
+00:25:14,400 --> 00:25:19,580
+سابقة هذه سميها واحد الآن اللي سمينا ال K of
+
+344
+00:25:19,580 --> 00:25:23,520
+Epsilon هذه سمينا هي إيش K للتسهيل بس الان FK is
+
+345
+00:25:23,520 --> 00:25:28,590
+integrableأه then مدام integrable اذا there exist
+
+346
+00:25:28,590 --> 00:25:33,730
+a partition بي ابسلون لل FK هذه X not X واحد X and
+
+347
+00:25:33,730 --> 00:25:37,470
+such that ال U بي ابسلون و FK نقص ال ابسلون بي
+
+348
+00:25:37,470 --> 00:25:40,190
+ابسلون و FK أصغر من مين من أي ابسلون في الدنيا
+
+349
+00:25:40,190 --> 00:25:43,370
+نضيب انها ابسلون على اتنين عارفين هذي ايش هذي هذي
+
+350
+00:25:43,370 --> 00:25:45,950
+اللي هي ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+351
+00:25:45,950 --> 00:25:46,930
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+352
+00:25:46,930 --> 00:25:46,950
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+353
+00:25:46,950 --> 00:25:47,750
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+354
+00:25:47,750 --> 00:25:49,330
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+355
+00:25:49,330 --> 00:25:49,930
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+356
+00:25:49,930 --> 00:25:53,420
+.. ال .. ال .. ال .. ال .. ال .. الإذا صار عندى
+
+357
+00:25:53,420 --> 00:25:57,640
+الآن من الـ Integrability للـ FK حصلنا ع بارتيشين
+
+358
+00:25:57,640 --> 00:26:01,100
+بي إبسلون بحيث أن ال Upper نقص ال Lower أصغر من
+
+359
+00:26:01,100 --> 00:26:07,800
+الـ Y على 2 لهذا ال بارتيشين الآن عندى فوق F of X
+
+360
+00:26:07,800 --> 00:26:11,520
+نقص FK of X أصغر من Y على 2 على 4 B minus A for
+
+361
+00:26:11,520 --> 00:26:16,410
+every X element in A by 2 by 1 thenخلّينا اللي هو
+
+362
+00:26:16,410 --> 00:26:20,850
+نكمل ال .. ال .. ال .. البرهان هي هذه اللي هي
+
+363
+00:26:20,850 --> 00:26:29,290
+خلّينا اكتبها عشان اللي هو نتذكر اللي هي نلم
+
+364
+00:26:29,290 --> 00:26:37,950
+الخطوات تبعتنا اللي عندي FMF ناقص FM أو FN أصغر من
+
+365
+00:26:37,950 --> 00:26:40,910
+إبسلون على أربعة في B minus A هذه B minus A طول
+
+366
+00:26:40,910 --> 00:26:44,970
+الفترة تبعت AJ الـ J الآن الـ partition عندي اللي
+
+367
+00:26:44,970 --> 00:26:52,410
+هو U بإبسلون أو F ناقص FH أو FK ناقص منها ناقص L
+
+368
+00:26:52,410 --> 00:26:59,370
+بإبسلون و FK أصغر من إبسلون على كده؟ على اتنين
+
+369
+00:26:59,370 --> 00:27:07,470
+للحسابات طيب الآنبقول لي من هذه من هذه أكيد أكبر
+
+370
+00:27:07,470 --> 00:27:12,650
+أو يساوي F ناقص FN of X يعني من هذه بنقدر نحصل هذا
+
+371
+00:27:12,650 --> 00:27:17,850
+صحيح لكل N أكبر أو يساوي K من ضمنها اللي هي N إيش
+
+372
+00:27:17,850 --> 00:27:21,890
+بتساوي زي ما احنا متعودين بنستخدم اللي بلزمنا اللي
+
+373
+00:27:21,890 --> 00:27:28,910
+هي FK يعني بيصير هذا أكبر أو يساوياللي هو f of x
+
+374
+00:27:28,910 --> 00:27:35,250
+ناقص fn of x لكل x element in J لأن من ضمن الأنات
+
+375
+00:27:35,250 --> 00:27:40,010
+هذه اللي هي main الـ K يعني بيصير عنده fn أو زي ما
+
+376
+00:27:40,010 --> 00:27:47,110
+هو مسميها f of x ناقص fk of x absolute value صار
+
+377
+00:27:47,110 --> 00:27:53,430
+أصغر من ي على أربعةبـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ-
+
+378
+00:27:53,430 --> 00:27:53,490
+بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ
+
+379
+00:27:53,490 --> 00:27:54,170
+- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ-
+
+380
+00:27:54,170 --> 00:27:59,490
+بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ
+
+381
+00:27:59,490 --> 00:27:59,490
+- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ-
+
+382
+00:27:59,490 --> 00:27:59,490
+بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ- بـ
+
+383
+00:27:59,490 --> 00:28:06,630
+- بـ- بـ- بـ- بـفكنا هذه بيصير عندي الان F of X
+
+384
+00:28:06,630 --> 00:28:12,090
+ناقص Y على 4 في B minus A أذر أو يساوي FK of X
+
+385
+00:28:12,090 --> 00:28:16,450
+عارفين طبعا كيف اللي هو أكبر أذر يساوي و أكبر ساوي
+
+386
+00:28:16,450 --> 00:28:21,350
+ناقصها اللي FK نجلت هنا بيصير F of X وجبتها دي هنا
+
+387
+00:28:21,350 --> 00:28:24,310
+بيصير F of X ناقص Y على 4 في B minus A أذر يساوي
+
+388
+00:28:24,310 --> 00:28:30,250
+FK of Xوهذه نفسها fk of x أكيد أصغر أو يساوي ال
+
+389
+00:28:30,250 --> 00:28:34,410
+absolute value لل fk of x زي ما انتوا عارفين وال
+
+390
+00:28:34,410 --> 00:28:36,550
+absolute value لل fk of x أصغر أو يساوي ال
+
+391
+00:28:36,550 --> 00:28:42,230
+supremum لل absolute value له المصورة عامة اللي هو
+
+392
+00:28:42,230 --> 00:28:45,550
+على الفترة xj minus واحد لل xj اللي هو إيش اللي
+
+393
+00:28:45,550 --> 00:28:50,760
+بنسميه اللي هو ال mj اللي هو تبع ال upper sumالان
+
+394
+00:28:50,760 --> 00:28:58,040
+صار عندي ال F of X ال F of X نسجلها هذه F of X أثر
+
+395
+00:28:58,040 --> 00:29:08,440
+أو ساوي Y على 4 في B minus A زائد MJ of FK لكل X
+
+396
+00:29:08,440 --> 00:29:17,970
+لكل إيش؟ اللي هي X X وين في الفترة؟ اللي هو XJأو
+
+397
+00:29:17,970 --> 00:29:23,890
+ناقص واحد و XJ وهذا وين يا جماعة نفسه اللي هو لكل
+
+398
+00:29:23,890 --> 00:29:31,510
+J من واحد و اتنين لعند مين الان طيب لأ
+
+399
+00:29:31,510 --> 00:29:36,450
+هذه حتى لكل X لأن هذه ال X هذه أخدناها arbitrarily
+
+400
+00:29:36,450 --> 00:29:47,720
+في ال J هذه لكل X في ال J نعم خلينا نسجل صح طيبإذا
+
+401
+00:29:47,720 --> 00:29:49,960
+استخدمنا اللي هو معناته ال uniform convergence
+
+402
+00:29:49,960 --> 00:29:54,640
+حصلنا على هذه استخدمنا ال integrability لل FK اللي
+
+403
+00:29:54,640 --> 00:29:57,820
+لاجيناها هذه المرتبطة بال K اللي لاجيناها عشانها
+
+404
+00:29:57,820 --> 00:30:02,340
+اللي هي N أكوا سوا K ومنها استخدمنا اللي هو
+
+405
+00:30:05,130 --> 00:30:10,310
+اللي فوق لل FK بالذات هذه أصغر من هذه وهذه حصلنا
+
+406
+00:30:10,310 --> 00:30:13,730
+على ال inequality هذه عشان بضروح أثبت ال
+
+407
+00:30:13,730 --> 00:30:17,430
+integrability لمين لل F نفسها شوفوا كيفنا نثبت
+
+408
+00:30:17,430 --> 00:30:23,690
+الآن صار عندي الآنF of X أصغر سواء هذا في هذا
+
+409
+00:30:23,690 --> 00:30:28,770
+وعارفين إيش معنات اللي هو M J of F K الان M J of F
+
+410
+00:30:28,770 --> 00:30:31,090
+زي ما انتوا عارفين إيش بتساوي ال supermom ل F of X
+
+411
+00:30:31,090 --> 00:30:35,370
+اللي على فترة مين X اللي هي في J minus واحد وJ
+
+412
+00:30:35,370 --> 00:30:40,670
+أكيد اللي هي ال ها دي هتكون أصغر من Y على 4 في B
+
+413
+00:30:40,670 --> 00:30:48,220
+minus A زائد M J عليها عارفين ليش لأن ال F of XF
+
+414
+00:30:48,220 --> 00:30:52,440
+of X أصغر أو ساوي هذا القيمة، مظبوط؟ إذا صارت هذه
+
+415
+00:30:52,440 --> 00:30:56,940
+القيمة عبارة عن اللي هو upper bound of the mean
+
+416
+00:30:56,940 --> 00:31:01,580
+للـ F of X، ماشي الحال صارت هذه upper bound، إذا
+
+417
+00:31:01,580 --> 00:31:04,820
+ال least upper bound، ال supremum، هيظل أصغر أو
+
+418
+00:31:04,820 --> 00:31:09,140
+ساوي هذه القيمة، لأن هذه القيمة عبارة عن عدد، صار
+
+419
+00:31:09,140 --> 00:31:13,420
+الآن M J of F أصغر أو ساوي هذا المقدر، وضحكة هذه
+
+420
+00:31:13,420 --> 00:31:19,780
+أعملناها كتير، إذا صار عندي MJ of Fأصغر أو يساوي
+
+421
+00:31:19,780 --> 00:31:28,060
+MJ of FK زايد إبسلون على أربعة في B minus A هذا
+
+422
+00:31:28,060 --> 00:31:33,240
+الكلام اللي هو لكل اللي هي من J واحد واتنين عند
+
+423
+00:31:33,240 --> 00:31:36,760
+الناقص واحد الباقي انتوا عارفينه دايما بنعمله اضرب
+
+424
+00:31:36,760 --> 00:31:43,520
+لهنا في XJ ناقص XJ minus واحد واضرب هنا في XJ ناقص
+
+425
+00:31:43,520 --> 00:31:49,300
+XJ minus واحد وهنا في XJ minus XJ minus واحدضربنا
+
+426
+00:31:49,300 --> 00:31:54,200
+اللي هو الترفيه المتباينة في XJ minus XJ minus
+
+427
+00:31:54,200 --> 00:31:58,780
+واحد و بعدين أخدنا ال summation J من عند واحد لعند
+
+428
+00:31:58,780 --> 00:32:02,700
+N و أخدنا ال summation J من عند واحد لعند N و
+
+429
+00:32:02,700 --> 00:32:06,880
+أخدنا ال summation J من عند واحد لعند N هذا كله
+
+430
+00:32:06,880 --> 00:32:11,020
+على بعضه طبعا كل شغلنا على ال BY اللي لاجناها ال
+
+431
+00:32:11,020 --> 00:32:15,180
+partition اللي لاجناها فوق اللي هو عندي هذا عبارة
+
+432
+00:32:15,180 --> 00:32:23,430
+عن اللي هي ال Uof اللي هو الـ partition بي ابسلون
+
+433
+00:32:23,430 --> 00:32:28,290
+اللي بدي اشتغل عليه هدولة الهن و mean و A اللي هي
+
+434
+00:32:28,290 --> 00:32:34,250
+ال function F أصغر أو يساوي هذا mean هو ال U اللي
+
+435
+00:32:34,250 --> 00:32:40,010
+هو ال بي ابسلون و ال FK و هد ايش هتكون عبارة عن
+
+436
+00:32:40,010 --> 00:32:44,320
+ايش عشان هيك أخدنا بي مايلوس ايه هتشوفوهاالآن هذه
+
+437
+00:32:44,320 --> 00:32:50,400
+y زائد y على أربعة في b minus a ال summation هذا
+
+438
+00:32:50,400 --> 00:32:54,160
+لل sub intervals اللي هو زي ما انتوا عارفين x واحد
+
+439
+00:32:54,160 --> 00:32:57,780
+نقص x not زائد x اتنين نقص x واحد لما أصل ل xn نقص
+
+440
+00:32:57,780 --> 00:33:01,600
+xn نقص واحد يعني هيضل في الآخر xn نقص x not يعني b
+
+441
+00:33:01,600 --> 00:33:06,980
+minus a في b minus a هذه بتروح مع هذه فبصير عندي
+
+442
+00:33:06,980 --> 00:33:13,610
+ال uأصغر من الـ U هذا زي إبسلون على أربعة اللي
+
+443
+00:33:13,610 --> 00:33:17,450
+عنده اللي هو الـ U بإبسلون F أصغر يسوي U بإبسلون
+
+444
+00:33:17,450 --> 00:33:21,570
+FK زي إبسلون على أربعة Similarly وفعلا Similarly
+
+445
+00:33:21,570 --> 00:33:25,690
+بعد بدل ما نشتغل على الأبراللي فوق بدل ما نشتغل
+
+446
+00:33:25,690 --> 00:33:30,870
+على الأبر هنا بنشتغل عالميا على اللاور اللي بيطلع
+
+447
+00:33:30,870 --> 00:33:34,350
+عند ال ال ب إبسلون و أف هذا بيبقى لكم ال ب إبسلون
+
+448
+00:33:34,350 --> 00:33:38,030
+و أف كيه ناقص إبسلون على أربع أصغر من ال ب إبسلون
+
+449
+00:33:38,030 --> 00:33:44,490
+و أف إذا اللي وصلنا له ما هي ليه يا جماعة اللي
+
+450
+00:33:44,490 --> 00:33:55,110
+وصلنا له انه لكل إبسلون أكبر من 0هناك بي إبسلون
+
+451
+00:33:55,110 --> 00:34:05,870
+مثلًا U بي إبسلون و F أصغر أو يساوي U بي إبسلون و
+
+452
+00:34:05,870 --> 00:34:09,990
+FK زائد إبسلون على أربعة
+
+453
+00:34:13,830 --> 00:34:20,130
+بإبسلون و FK ناقص إبسلون على أربعة أصغره يساوي L
+
+454
+00:34:20,130 --> 00:34:25,830
+بيبسلون و F هذه اللي وصلنا له لحد الآن نيجي الان
+
+455
+00:34:25,830 --> 00:34:32,230
+نرتبهم مع بعض و نصل للي بدناه اشي معهود خدوا الان
+
+456
+00:34:32,230 --> 00:34:38,530
+U بيبسلون و Fناقص الـ P,Y,F هذه ناقص هذه لما ضروف
+
+457
+00:34:38,530 --> 00:34:43,510
+هذه في ناقص بيصير هذه ناقص هذه معهودة وهذه بيصير
+
+458
+00:34:43,510 --> 00:34:46,690
+ناقص وهذه بيصير زائد وهذه بيصير أكبر أو يساوي
+
+459
+00:34:46,690 --> 00:34:52,570
+بتجمع هذه لهذه يعني بيصير عدى هذه ناقص هذه أصغر من
+
+460
+00:34:52,570 --> 00:34:59,130
+الـ U,P,Y,F هيهازائد هادي ناقص هادي زائد هادي يعني
+
+461
+00:34:59,130 --> 00:35:03,010
+بيصير عنده هادي يو بي إبسل و أف كيه ناقص البي إبسل
+
+462
+00:35:03,010 --> 00:35:06,190
+و أف كيه زائد إبسل على أربعة وإبسل على أربعة اللي
+
+463
+00:35:06,190 --> 00:35:10,690
+هو إبسل على مين؟ على اتنين وهذه من الأول لما
+
+464
+00:35:10,690 --> 00:35:15,630
+نستخدمها اللي هو الـRiemann criterion for
+
+465
+00:35:15,630 --> 00:35:19,640
+integrability للأف كيهاللي هو أوجدتنا الـ Py هو
+
+466
+00:35:19,640 --> 00:35:23,220
+أوجدتنا أن هذه ناقصة أصغر من Y على 2 يعني صار عندي
+
+467
+00:35:23,220 --> 00:35:26,820
+Y على 2 وY على 2 إبسلون معناته هذه صارت أصغر من
+
+468
+00:35:26,820 --> 00:35:30,080
+إبسلون و بالرماني criterion for integrability
+
+469
+00:35:30,080 --> 00:35:36,560
+بتطلع الـ F is إيش is integrable وهذا اللي هو
+
+470
+00:35:36,560 --> 00:35:41,200
+الإثبات لإن ال function F is integrable دل نثبت
+
+471
+00:35:41,200 --> 00:35:43,500
+اللي هو limit لل integration بساوي integration لل
+
+472
+00:35:43,500 --> 00:35:46,630
+limitوهذا من خلال الخطوات اللي طلعت عندنا حلّجي
+
+473
+00:35:46,630 --> 00:35:55,030
+يطلع بشكل سهل وبشكل سلس وخلّينا نشوفه الان since
+
+474
+00:35:55,030 --> 00:35:58,130
+epsilon was arbitrary then f is integrable on j زي
+
+475
+00:35:58,130 --> 00:36:02,750
+ما قلنا خد ال integration من a لb f of x dx نقص ال
+
+476
+00:36:02,750 --> 00:36:07,950
+integration من a لb fn of x dx بساوي اللي هوالـ
+
+477
+00:36:07,950 --> 00:36:10,830
+absolute value طبعاً صارت هذه integrable وهذه
+
+478
+00:36:10,830 --> 00:36:14,470
+integrable ماشي الحال إذا ال integration هذا نقص
+
+479
+00:36:14,470 --> 00:36:17,010
+ال integration هذا بساوي ال integration لهذا نقص
+
+480
+00:36:17,010 --> 00:36:20,210
+هذا الكل إيش ماله DX من خواص التكامل اللي أخدناها
+
+481
+00:36:20,210 --> 00:36:25,370
+سابقاً وهذا نفسه أصغر أو يساويأصغر يساوي ال
+
+482
+00:36:25,370 --> 00:36:28,750
+integration من a لb واخدناها سابقاً absolute value
+
+483
+00:36:28,750 --> 00:36:34,370
+of f of x ماقص fn of x لكل ايش ماله dx ال absolute
+
+484
+00:36:34,370 --> 00:36:36,490
+value integration أصغر يساوي ال integration لل
+
+485
+00:36:36,490 --> 00:36:42,340
+absolute value وهذا نفسهاللي هو أصغر أو يساوي norm
+
+486
+00:36:42,340 --> 00:36:47,060
+الـ F ناقص الـ Fn over اللي هي الـ interval J
+
+487
+00:36:47,060 --> 00:36:51,320
+ماشي، الآن بيصير عندى هذا أصغر أو يساوي ال
+
+488
+00:36:51,320 --> 00:36:56,840
+integration من A ل B norm الـ F ناقص الـ Fn over J
+
+489
+00:36:56,840 --> 00:37:03,720
+الكل إيش ماله؟ DX الآن هذا يا جماعة ثابت بيطلع
+
+490
+00:37:03,720 --> 00:37:09,520
+براته بيصير عبارة عن يساوي ال absolute valueالـ
+
+491
+00:37:09,520 --> 00:37:13,700
+norm للـ f ناقص الـ fn over j في ال integration من
+
+492
+00:37:13,700 --> 00:37:20,600
+a إلى b dx هذا هيهان وهذا ال integration من a إلى
+
+493
+00:37:20,600 --> 00:37:25,840
+b إلى dx هو عبارة عن b minus a صار هذا ناقص هذا
+
+494
+00:37:25,840 --> 00:37:31,200
+أصغر من هذا في b minus a الآن as n goes to
+
+495
+00:37:31,200 --> 00:37:34,920
+infinity as n goes to infinity صار هذا أكبر أو
+
+496
+00:37:34,920 --> 00:37:42,930
+يساوي سفروهذا هيوطلع لهذه المنطقة هيها هذه بس خد
+
+497
+00:37:42,930 --> 00:37:46,170
+ال limit للجهتين as n goes to infinity هذا بيصير
+
+498
+00:37:46,170 --> 00:37:51,590
+ال limit هناas n goes to infinity وهنا ال limit as
+
+499
+00:37:51,590 --> 00:37:54,170
+n goes to infinity طبعا هذا one هيروح لإنه أفن
+
+500
+00:37:54,170 --> 00:37:57,910
+بتروح للأف uniformly هذا بتروح كله لمين للصفر إذا
+
+501
+00:37:57,910 --> 00:38:01,330
+as n goes to infinity ال limit هذا إيش بيساوي صفر
+
+502
+00:38:01,330 --> 00:38:04,990
+مدام ال limit هذا صار بيساوي صفر هذا independent
+
+503
+00:38:04,990 --> 00:38:09,170
+of the limit هذا عبارة عن فيش فيه n صار عندي اللي
+
+504
+00:38:09,170 --> 00:38:12,990
+هو مضحكة
+
+505
+00:38:12,990 --> 00:38:19,770
+بتصور الصورة صار عندي الأن limitهذا بيساوي ايش صار
+
+506
+00:38:19,770 --> 00:38:24,770
+بيساوي سفر limit لال absolute value من a لعند b f
+
+507
+00:38:24,770 --> 00:38:32,170
+of x dx نقص integration fn of x dx من a لعند b
+
+508
+00:38:32,170 --> 00:38:35,710
+limit as angles to infinity limit ال absolute
+
+509
+00:38:35,710 --> 00:38:38,810
+value صار سفر إذا أكيد ال limit بدون absolute
+
+510
+00:38:38,810 --> 00:38:42,690
+value بيساوي سفر هذا independent of the limit إذا
+
+511
+00:38:42,690 --> 00:38:48,210
+بيصير بساوي ال integration من a لb f of x dxبتدخل
+
+512
+00:38:48,210 --> 00:38:54,370
+ال limit ناقص limit ال integration fn of x dx من a
+
+513
+00:38:54,370 --> 00:38:59,310
+ل عند بيه ماشي الحل و لا نضحك الصورة هذا صار عليه؟
+
+514
+00:38:59,310 --> 00:39:05,780
+هذا كله على بعض صار بساوي سفرإذا الـ limit ننجل
+
+515
+00:39:05,780 --> 00:39:08,620
+هذا على الجهة الثانية بيصير limit ال integration
+
+516
+00:39:08,620 --> 00:39:15,080
+من a ل b fn of x dx as n goes to infinity بساوي
+
+517
+00:39:15,080 --> 00:39:20,900
+اللي بيظل هنا من a ل b f of x dx وهذا معناته إنه
+
+518
+00:39:20,900 --> 00:39:24,310
+في حالة ال uniform convergenceالـ FN sequence of
+
+519
+00:39:24,310 --> 00:39:27,030
+integrable functions هتروح لـ integrable function
+
+520
+00:39:27,030 --> 00:39:29,450
+و هيطلع عنده limit لل integration بيساوي ال
+
+521
+00:39:29,450 --> 00:39:32,150
+integration ل limit و بهي .. و هيكون كون احنا
+
+522
+00:39:32,150 --> 00:39:37,110
+علجنا اللي هو النقطة الثانية في اللي هو المحاضرة
+
+523
+00:39:37,110 --> 00:39:42,270
+ضال عندي نقطة أخيرة بس قبلها خليني نذكر هال نظرية
+
+524
+00:39:42,270 --> 00:39:47,470
+اللي هي برهانة خارج نطاق اللي هو الكتاب و نشوف
+
+525
+00:39:47,470 --> 00:39:49,110
+اللي هو
+
+526
+00:39:53,380 --> 00:39:58,540
+إيش بتقول هذه اللي هي النظرية؟ إيش بتقول bounded
+
+527
+00:39:58,540 --> 00:40:02,310
+convergence theorem؟النظرية بتقول let FN be a
+
+528
+00:40:02,310 --> 00:40:07,150
+sequence of functions that are integrable in A وB
+
+529
+00:40:07,150 --> 00:40:12,530
+مفترضين إنها integrable and suppose that بدي يبعد
+
+530
+00:40:12,530 --> 00:40:14,850
+عن ال uniform convergence بتقول لو شيلنا ال
+
+531
+00:40:14,850 --> 00:40:18,550
+uniform convergence ممكن نبدله بإيش إيه؟ ويظل
+
+532
+00:40:18,550 --> 00:40:20,850
+محافظنا على ال limit لل integration بساوي ال
+
+533
+00:40:20,850 --> 00:40:24,920
+integration لل limit؟ آه، بنفع، نشوف كيفلت أفأن بـ
+
+534
+00:40:24,920 --> 00:40:27,340
+sequence of functions that are integrable in A وB
+
+535
+00:40:27,340 --> 00:40:30,380
+بس بده يدفع ثمن برضه integrable in A وB and
+
+536
+00:40:30,380 --> 00:40:34,400
+suppose that أفأن converts in A وB to an
+
+537
+00:40:34,400 --> 00:40:37,780
+integrable function F يعني أفأن بتروح للـ F على
+
+538
+00:40:37,780 --> 00:40:44,700
+الفترة هذه اللي هو point twice ماجالش uniformly بس
+
+539
+00:40:44,700 --> 00:40:50,770
+الآن بده يفترض شغلة بده يقيد الأفأنبشغلة معينة
+
+540
+00:40:50,770 --> 00:40:54,950
+شغلة كبيرة برضه التقييد مش جليل طبعا لكن بيضل اشي
+
+541
+00:40:54,950 --> 00:40:59,830
+يعني منفذ تاني suppose also that there exist B
+
+542
+00:40:59,830 --> 00:41:05,870
+أكبر من سفر بحيث انه FN of X أصغر شهر بيه لكل X
+
+543
+00:41:05,870 --> 00:41:09,750
+element in A و B و لكل N element in B هذا في
+
+544
+00:41:09,750 --> 00:41:14,690
+الواقع اللي هو uniform boundedness ليش؟ لأن هو
+
+545
+00:41:14,690 --> 00:41:21,200
+مفترض ان ال sequence FN of Xأصغر أو يساوي اللي هو
+
+546
+00:41:21,200 --> 00:41:28,140
+B لكل X element in A وB ولكل N element in N هو مش
+
+547
+00:41:28,140 --> 00:41:32,360
+مفترض إن الـ Fn نفسها تكون bounded يعني F1 اللي
+
+548
+00:41:32,360 --> 00:41:35,580
+حالها bounded، F2 bounded، F3 bounded كل إن هيكون
+
+549
+00:41:35,580 --> 00:41:39,940
+bounded صح، بس اللي طالبه أكثر اللي طبعا في حالة
+
+550
+00:41:39,940 --> 00:41:47,640
+الـ F1 bounded، F2 bounded، هيكون عندي Fnof X أصغر
+
+551
+00:41:47,640 --> 00:41:52,480
+أو ساوى BN F4
+
+552
+00:41:53,890 --> 00:41:58,290
+أني اللي هو الآن بنقول عن FN نفسها bounded أنه
+
+553
+00:41:58,290 --> 00:42:02,750
+there exists B لها، B لمن؟ لل FN لو فرضنا أن FN
+
+554
+00:42:02,750 --> 00:42:05,950
+الحالها bounded، there exists B للـN سميتها BN
+
+555
+00:42:05,950 --> 00:42:11,850
+صشدتها، دي أصغر يساوي BN لكل X element in A وB هذا
+
+556
+00:42:11,850 --> 00:42:16,650
+لو فرضنا أنه كل واحدة على حدة A bounded هو طالب
+
+557
+00:42:16,650 --> 00:42:24,260
+أكثر جاي الـ bound هذابنفع لكل الـ mean للـ F1 و
+
+558
+00:42:24,260 --> 00:42:30,980
+للـ F2 و للـ F3 و للـ F4 يعني فارض أن FN of X أصغر
+
+559
+00:42:30,980 --> 00:42:35,920
+أو يشوي يعني درج ZEF بأكبر من صفر such that FN of
+
+560
+00:42:35,920 --> 00:42:42,040
+X أصغر شهر و B لكل N element N و لكل X element مين
+
+561
+00:42:42,040 --> 00:42:48,760
+in ال interval اللي هي A أو B طبعا هذا الكلام أكيد
+
+562
+00:42:48,760 --> 00:42:52,940
+بيعطي هذا لكن هذا مش شرط يعطي هذا لأن هذا ال BN لل
+
+563
+00:42:52,940 --> 00:42:57,540
+F1 فيه B1 ال F2 بيه F2 F3 بيه F3 F4 بيه F4 هل فيه
+
+564
+00:42:57,540 --> 00:43:01,770
+الherosoprimam هدولة ليس شرطاماقدرش نقول أه عشان
+
+565
+00:43:01,770 --> 00:43:06,070
+هيك طلبه كبير طبعا البرنامج خارج عن نطاق ال course
+
+566
+00:43:06,070 --> 00:43:10,250
+سبعنا الان تحت هذا ال condition اللي قلنا عنه
+
+567
+00:43:10,250 --> 00:43:14,190
+condition ماقدرنا نقول وازن then ال integration من
+
+568
+00:43:14,190 --> 00:43:17,450
+a ل b of x dx بسوء limit لل integration من a b ل f
+
+569
+00:43:17,450 --> 00:43:20,610
+n of x dx يعني بمعنى أخر limit لل integration بسوء
+
+570
+00:43:20,610 --> 00:43:24,950
+ال integration لل limitالان نيجي لجزيه الثالث من
+
+571
+00:43:24,950 --> 00:43:29,350
+المحاضرة اللي بتعلق بالـ sequence of
+
+572
+00:43:29,350 --> 00:43:32,550
+differentiable functions خلينا نشوف المثال أول اشي
+
+573
+00:43:32,550 --> 00:43:37,570
+و بعدين نيجي لنظريتنا let fn of x بساوي summation
+
+574
+00:43:37,570 --> 00:43:42,270
+2 to the minus k cosine 3k x k من 1 لعين ده يعني
+
+575
+00:43:42,270 --> 00:43:47,950
+الانF1 of X بيساوي اللي هو 2 تره ماينس واحد كساين
+
+576
+00:43:47,950 --> 00:43:52,810
+تلاتة X F2 of X بيساوي اللي هو 2 تره ماينس واحد
+
+577
+00:43:52,810 --> 00:43:56,930
+كساين تلاتة X زائد 2 تره ماينس اتنين كساين تلاتة
+
+578
+00:43:56,930 --> 00:44:01,070
+تربيع X و هكذاFn of X هي عبارة عن الـSummation هذا
+
+579
+00:44:01,070 --> 00:44:04,690
+من واحد لـ&N وهذه عبارة عن Sequence of Functions
+
+580
+00:44:04,690 --> 00:44:09,070
+واضح أن هذه Sequence of Functions هي عبارة عن
+
+581
+00:44:09,070 --> 00:44:12,570
+Sequence of Differentiable Functions، لماذا؟ لأن
+
+582
+00:44:12,570 --> 00:44:16,450
+كل واحدة منهم عبارة عن Summation، Finite Summation
+
+583
+00:44:16,450 --> 00:44:20,570
+لـCos Function والـCos Function مستمر في كل مكان
+
+584
+00:44:20,570 --> 00:44:25,510
+وDifferentiable في كل مكانإذا صارت عندي الـ F1 و
+
+585
+00:44:25,510 --> 00:44:29,330
+الـ F2 و الـ F3 و الـ Fn عامة R الـ differential
+
+586
+00:44:29,330 --> 00:44:33,930
+of the function on R لكن شوفت على الـ limit احنا
+
+587
+00:44:33,930 --> 00:44:37,890
+خدنا المثال هذا but limit Fn of X as N plus
+
+588
+00:44:37,890 --> 00:44:41,970
+infinity بساوة summation كمن عند واحد إلى ما لا
+
+589
+00:44:41,970 --> 00:44:47,150
+نهاية بتصير ماشي الحالة2 to the minus k cosine 3kx
+
+590
+00:44:47,150 --> 00:44:51,150
+هذه أخدنا مثال على function is continuous
+
+591
+00:44:51,150 --> 00:44:54,590
+everywhere و is not differentiable anywhere و قلنا
+
+592
+00:44:54,590 --> 00:44:59,930
+برهانة خارج نطاق اللي هو اكتبنا فهذه الدالة عبارة
+
+593
+00:44:59,930 --> 00:45:04,790
+عن ال limit هذه اللي هي طلعت هذه الدالة يعني limit
+
+594
+00:45:04,790 --> 00:45:08,590
+ال function هذه دالة هذا ال limit is not
+
+595
+00:45:08,590 --> 00:45:12,710
+differentiable at any x element in R يعني هذا
+
+596
+00:45:12,710 --> 00:45:16,620
+عبارة عن مثالعلى sequence of functions FN
+
+597
+00:45:16,620 --> 00:45:25,100
+differentiable on R but its limit FN هنشوف
+
+598
+00:45:25,100 --> 00:45:32,970
+converts كمان uniformly ل F والـ F andF can
+
+599
+00:45:32,970 --> 00:45:36,950
+converge uniformly to F but الـ F زي ما قلنا is
+
+600
+00:45:36,950 --> 00:45:44,430
+not differentiable at any X element in R وكأنه الـ
+
+601
+00:45:44,430 --> 00:45:50,980
+uniform convergence هنا برضه وصفة ما نفعتشإن نجيب
+
+602
+00:45:50,980 --> 00:45:54,340
+الـ sequence of differentiable functions تطلع
+
+603
+00:45:54,340 --> 00:45:57,220
+differentiable function يعني ال uniform
+
+604
+00:45:57,220 --> 00:46:01,220
+convergence كمان نفسه بده دعمة أو بدنا condition
+
+605
+00:46:01,220 --> 00:46:05,280
+غير هيك يوصلنا للي بدنايا انه اللي هو ال sequence
+
+606
+00:46:05,280 --> 00:46:08,060
+of differentiable functions تقول إلى
+
+607
+00:46:08,060 --> 00:46:13,940
+differentiable function ده نشوف نكمل .. نكملالان
+
+608
+00:46:13,940 --> 00:46:18,300
+هنشوف كيف ينفرج .. ينفرج .. ينفرج ينفرج ينفرج
+
+609
+00:46:18,300 --> 00:46:19,940
+ينفرج ينفرج عامة عامة عامة عامة عامة عامة عامة
+
+610
+00:46:19,940 --> 00:46:20,360
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+611
+00:46:20,360 --> 00:46:21,560
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+612
+00:46:21,560 --> 00:46:21,960
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+613
+00:46:21,960 --> 00:46:23,120
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+614
+00:46:23,120 --> 00:46:24,740
+عامة عامة عامة عامة عامة عامة عامة عامة عامة عامة
+
+615
+00:46:24,740 --> 00:46:28,440
+عامة عامة عامة عامة عامة
+
+616
+00:46:29,930 --> 00:46:34,590
+هذا أكيد أصغر أو ساوي اللي هو ال absolute value of
+
+617
+00:46:34,590 --> 00:46:36,130
+summation أصغر أو ساوي ال summation لل absolute
+
+618
+00:46:36,130 --> 00:46:39,650
+value وال cosine أصغر أو ساوية حدث ان صار عبارة عن
+
+619
+00:46:39,650 --> 00:46:41,890
+ال summation ال absolute value of summation هذا
+
+620
+00:46:41,890 --> 00:46:44,910
+أصغر أو ساوي two to the minus k كمن عند n زائد
+
+621
+00:46:44,910 --> 00:46:49,430
+واحد إلى مين إلى ما لا نهايةوهذا as n goes to
+
+622
+00:46:49,430 --> 00:46:53,210
+infinity هذا بروح لل zero ماشي الحال خليها في
+
+623
+00:46:53,210 --> 00:46:59,350
+الذاكره الآن صار عندي f of x نقص fn of x أصغر أو
+
+624
+00:46:59,350 --> 00:47:04,450
+يساوي هذا الرقم ماشي مش معتمد على x صار هذا أصغر
+
+625
+00:47:04,450 --> 00:47:08,050
+أو ساوي هذا صار هذا upper bound لهذهمدام upper
+
+626
+00:47:08,050 --> 00:47:10,210
+bound إذا ال upper bound أكبر أو ساوي ال least
+
+627
+00:47:10,210 --> 00:47:14,170
+upper bound اللي هو نوم لل F ناقص FN over R أصغر
+
+628
+00:47:14,170 --> 00:47:17,310
+أو ساوي هذا as N goes to infinity هذا بيروح لل
+
+629
+00:47:17,310 --> 00:47:21,250
+zero إذا ال F ناقص FN over R بتروح لل zero وهذا
+
+630
+00:47:21,250 --> 00:47:27,570
+معناته أن FN بتروح لل F اللي هو on mean on R يعني
+
+631
+00:47:27,570 --> 00:47:31,250
+ال convergence إيش ماله uniform convergence بالرغم
+
+632
+00:47:31,250 --> 00:47:35,880
+إن ال uniform convergence حادثوالـ Fn sequence of
+
+633
+00:47:35,880 --> 00:47:40,200
+differentiable functions الـ F إيش ما لها طلعتش
+
+634
+00:47:40,200 --> 00:47:44,840
+differentiable إذا بالزمن اللي هو condition أو
+
+635
+00:47:44,840 --> 00:47:49,900
+conditions تضمنلي أنه لما Fn تروح للـ F و Fn
+
+636
+00:47:49,900 --> 00:47:52,580
+sequence of differentiable functions تطلعلي الـ F
+
+637
+00:47:52,580 --> 00:47:56,930
+differentiable خلّينا نشوف نظريتنا اللي هيأخر جزء
+
+638
+00:47:56,930 --> 00:48:00,990
+في ال section و في ال chapter اللي بدنا ناخده منه
+
+639
+00:48:00,990 --> 00:48:07,070
+اللي هي بتجاوبنا على هذا الأمر و نشوف أيش النظرية
+
+640
+00:48:07,070 --> 00:48:12,630
+بتقوله و من ثم بعد أنفة من نص نبرهن النظرية نشوف
+
+641
+00:48:12,630 --> 00:48:17,400
+let J subset من R be a bounded intervalbounded
+
+642
+00:48:17,400 --> 00:48:21,340
+interval open مش open مش فارقة معاه a closed مش
+
+643
+00:48:21,340 --> 00:48:24,560
+closed مش فارقة الان and let fn be a sequence of
+
+644
+00:48:24,560 --> 00:48:29,360
+functions on j يعني عبقى عندي عندي fn sequence of
+
+645
+00:48:29,360 --> 00:48:33,940
+functions fn sequence of functions على مين؟ على
+
+646
+00:48:33,940 --> 00:48:38,180
+الـ j اللي هي ال interval اللي عندي لعند ارف واحدة
+
+647
+00:48:38,180 --> 00:48:43,700
+اتنين ارف تلاتة الاخرين وبدنا نفترضاللي هو suppose
+
+648
+00:48:43,700 --> 00:48:49,880
+that there exist x0 element in j بحيث أن fn of x0
+
+649
+00:48:49,880 --> 00:48:56,440
+converges اللي هو to some limit الان ايش مفترض ان
+
+650
+00:48:56,440 --> 00:49:03,520
+في عندي x0في الـ J بحيث أن FN of X0 هذه صارت
+
+651
+00:49:03,520 --> 00:49:07,840
+sequence of numbers هذه converges to some number
+
+652
+00:49:07,840 --> 00:49:12,560
+ايش هو مش عارفينه الان and that ومعطيك كمان شغلة
+
+653
+00:49:12,560 --> 00:49:18,800
+هذه معطيكياها اللي هي and converges عند مقطة محددة
+
+654
+00:49:18,800 --> 00:49:24,320
+اسمها X0 في الـ J and the sequence FN prime of
+
+655
+00:49:24,320 --> 00:49:29,510
+derivatives exist on Jيعني مفترض إن الـ F N'
+
+656
+00:49:29,830 --> 00:49:32,910
+exists يعني F N sequence of differentiable
+
+657
+00:49:32,910 --> 00:49:36,390
+functions مفترض إن F N sequence of differentiable
+
+658
+00:49:36,390 --> 00:49:40,210
+functions ومش هيك، وبيقولك إن هذه الـ F N'
+
+659
+00:49:40,750 --> 00:49:45,230
+converts uniformly on J to some function F يعني
+
+660
+00:49:45,230 --> 00:49:50,950
+بيقولك F N' existsFN برايم exist لكل N وهذا الـ
+
+661
+00:49:50,950 --> 00:49:55,430
+sequence بيقول لك converges uniformly to some
+
+662
+00:49:55,430 --> 00:50:00,130
+function اسمها مين؟ اسمها J إذن sequence of
+
+663
+00:50:00,130 --> 00:50:05,510
+functions لأن في X0 في الـ J بحيث أن FN of X0
+
+664
+00:50:05,510 --> 00:50:10,830
+نفسها تكون converges FN برايم converges uniformly
+
+665
+00:50:10,830 --> 00:50:17,150
+to some Gto some G بكل الأنظم مع وجود هذه الشروط
+
+666
+00:50:17,150 --> 00:50:23,210
+اللي حكينا عليها then the sequence FN converges
+
+667
+00:50:23,210 --> 00:50:32,330
+uniformly to a function F بعد هذا بعد هذا بنلاقي
+
+668
+00:50:32,330 --> 00:50:37,370
+أن FN غصب إن عنا لازم تكون converges uniformly to
+
+669
+00:50:37,370 --> 00:50:44,660
+some function mean F ومش هيكوالـ function F هذه
+
+670
+00:50:44,660 --> 00:50:51,260
+تطلع الـ derivative تبعتها هي مين؟ هي الـ G يعني
+
+671
+00:50:51,260 --> 00:51:01,310
+وكأنه أن الـ F N converges uniformly to Fاللي هي
+
+672
+00:51:01,310 --> 00:51:07,270
+الـ derivative لها مين الاش الـ G يعني F N برايم
+
+673
+00:51:07,270 --> 00:51:10,510
+بتكون uniformly convergence to some function G
+
+674
+00:51:10,510 --> 00:51:17,890
+هتطلع الـ F N converge uniformly لل function اللي
+
+675
+00:51:17,890 --> 00:51:24,570
+تفضلها هو مين هو عبارة عن الـ Gطيب إذا في عندي
+
+676
+00:51:24,570 --> 00:51:28,210
+شغلتين بدنا نثبت ان FN can form convergence to F
+
+677
+00:51:28,210 --> 00:51:35,690
+والـ F' إيش بتساوي؟ بتساوي مين؟ اللي هي الـ G يعني
+
+678
+00:51:35,690 --> 00:51:42,830
+هتصير إنه الـ FN' اللي هو sequence of
+
+679
+00:51:42,830 --> 00:51:45,770
+differentiable functions بتطلع عندي هذه sequence
+
+680
+00:51:45,770 --> 00:51:49,210
+of differentiable functions converge to function
+
+681
+00:51:49,210 --> 00:51:54,110
+بتكون differentiableليش؟ لأن F' ايش بتتساوي G؟
+
+682
+00:51:54,110 --> 00:51:57,590
+يعني F is differentiable يعني هذه بتقول إنه لو
+
+683
+00:51:57,590 --> 00:52:00,950
+الشرطة ده و الشرطة ده تتحققوا إذن ال sequence of
+
+684
+00:52:00,950 --> 00:52:06,290
+differentiable functions FN هتكون converges to ..
+
+685
+00:52:06,290 --> 00:52:09,110
+uniform converges طبعا to a differentiable
+
+686
+00:52:09,110 --> 00:52:13,310
+function F إذن الشغلتين هدولة إحنا بدنا نثبتهم
+
+687
+00:52:13,310 --> 00:52:18,330
+خلينا نشوف كيف نثبت هدولة الشغلتين بدنا نثبت إنه
+
+688
+00:52:18,330 --> 00:52:23,460
+FNConverse Uniform للـ F وبدنا نثبت أن F' بيسوا G
+
+689
+00:52:23,460 --> 00:52:29,760
+يعني بدنا نثبت لك في النهاية أنه limit F
+
+690
+00:52:31,240 --> 00:52:37,940
+of X زائد اللي هو F of X زائد Delta X مثلا أو F of
+
+691
+00:52:37,940 --> 00:52:45,520
+X ناقص F of C على X minus C as X بتروح للـ C بدنا
+
+692
+00:52:45,520 --> 00:52:51,980
+نثبت أنه exist ويساوي G of meanof C يعني بمعنى هذا
+
+693
+00:52:51,980 --> 00:52:56,460
+for arbitrary C element in main NG يعني كأنه في
+
+694
+00:52:56,460 --> 00:53:01,300
+النهاية بدنا نثبت لك انه هذا exist يعني F prime of
+
+695
+00:53:01,300 --> 00:53:05,840
+C exist ويساوي main G of C او بمعنى اخر بدنا نثبت
+
+696
+00:53:05,840 --> 00:53:08,320
+لك انه for every Y colon of Z there exists delta
+
+697
+00:53:08,320 --> 00:53:14,490
+such that X minus Cأصغر من Delta يؤدي إلى F of X
+
+698
+00:53:14,490 --> 00:53:20,270
+ناقص F of C على X minus C ناقص G of C يكون أصغر من
+
+699
+00:53:20,270 --> 00:53:24,530
+مين من Y إذا أثبتنا هذا معناته و أثبتنا إن هذا
+
+700
+00:53:24,530 --> 00:53:28,460
+اللي هو limit existيعني الـ F' exist ويساوى مين
+
+701
+00:53:28,460 --> 00:53:31,920
+الـ G of C والـ C كون بتكون أخديها Arbitrary في
+
+702
+00:53:31,920 --> 00:53:36,680
+الـ G فبكون خلصنا أن الـ F' بيساوى الـ G وهذا كله
+
+703
+00:53:36,680 --> 00:53:40,920
+بعد ما نثبت أن الـ F أنه conversion form للمين للـ
+
+704
+00:53:40,920 --> 00:53:48,280
+F و تعالوا لتفاصيل البرهان طيب نيجي نشوف برهاننا
+
+705
+00:53:48,280 --> 00:53:51,020
+شوفوا طاولوا روحكم علينا
+
+706
+00:53:55,700 --> 00:53:58,860
+عشان نعرف بس نشتغل لبنى حيكة خلّينا نسمي ال end
+
+707
+00:53:58,860 --> 00:54:02,960
+points جيه واحدة اسمها A واحدة اسمها B يعني let A
+
+708
+00:54:02,960 --> 00:54:06,440
+أزرع من بي بي the end points of J جيه مفتوح مفتوح
+
+709
+00:54:06,440 --> 00:54:08,020
+مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح
+
+710
+00:54:08,020 --> 00:54:08,080
+مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح
+
+711
+00:54:08,080 --> 00:54:08,580
+مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح مفتوح
+
+712
+00:54:08,580 --> 00:54:13,580
+مفتوح مفتوح مفتوح مفتوح مفتووخلّيني أخد X بـ
+
+713
+00:54:13,580 --> 00:54:17,700
+Arbitrary Point في الـ J أحنا بنحكي عن أي X وين ما
+
+714
+00:54:17,700 --> 00:54:26,420
+لا في الـ J اللي هي .. if M and N element in N
+
+715
+00:54:26,420 --> 00:54:29,320
+then F minus N is differentiable on the closed
+
+716
+00:54:29,320 --> 00:54:34,550
+interval within points X0 and X أكيدمفترضين احنا
+
+717
+00:54:34,550 --> 00:54:41,010
+ان الـ F M' exist مظبوط مدام exist اذا F M
+
+718
+00:54:41,010 --> 00:54:44,850
+differentiable و F N differentiable اذا F M نقص F
+
+719
+00:54:44,850 --> 00:54:48,910
+N differentiable على كل مين؟ على كل الـ J اللي انا
+
+720
+00:54:48,910 --> 00:54:52,950
+بدي اشتغل على فترة محددة انا اخدت X element in J
+
+721
+00:54:52,950 --> 00:54:57,490
+arbitraryوهو معطيل في النظرية X0 اللي عندها اللي
+
+722
+00:54:57,490 --> 00:55:02,110
+هي Fn of X0 converts بحكي عن X0 هذه المقصودة في
+
+723
+00:55:02,110 --> 00:55:08,170
+النظرية بدي أطبّج عليها عندي بما أن Fm ناقص Fn is
+
+724
+00:55:08,170 --> 00:55:11,730
+differentiable على closed interval within points
+
+725
+00:55:11,730 --> 00:55:18,090
+X0 و X يعني يا X0 و X يا X و X0 حسب موقع كل من X و
+
+726
+00:55:18,090 --> 00:55:23,440
+X0إذا صارت Fm ناقص Fn عبارة عن الـ differential
+
+727
+00:55:23,440 --> 00:55:26,120
+بالـ function على هذه الـ closed bounded interval
+
+728
+00:55:26,120 --> 00:55:30,320
+إذا أكيد continuous عليها كمان إذا بالـ mean value
+
+729
+00:55:30,320 --> 00:55:33,940
+theorem مطبقة الـ mean value theorem ده طبقها على
+
+730
+00:55:33,940 --> 00:55:39,300
+مين يا جماعة؟ على الـ Fm ناقص Fn يعني هذه دالتنا
+
+731
+00:55:39,300 --> 00:55:43,660
+اللي بدنا نطبق عليها الـ mean value theorem إذا by
+
+732
+00:55:43,660 --> 00:55:47,760
+mean value theorem there exists Y بين الـ X0 والـ
+
+733
+00:55:47,760 --> 00:55:53,380
+X بحيثإن الـ Derivative لهذه بيساوي F M ناقص F N
+
+734
+00:55:53,380 --> 00:55:59,440
+عند اللي هي X ناقص F M ناقص F Note عند الـ X Note
+
+735
+00:55:59,440 --> 00:56:05,640
+بيساوي F M ناقص F N Prime عند مين؟ عند الـ Yby
+
+736
+00:56:05,640 --> 00:56:09,840
+mean value theorem there exists y between x0 and x
+
+737
+00:56:09,840 --> 00:56:13,700
+such that هذا الدلة كلها على بعض عند ال x fm of x
+
+738
+00:56:13,700 --> 00:56:19,300
+نقص fn of x هذا الدلة ناقص fm of x0 نقص fn of x0
+
+739
+00:56:19,300 --> 00:56:25,060
+هذا الدلة عند النقطة x0 بساوي اللي هي x minus x0فى
+
+740
+00:56:25,060 --> 00:56:30,520
+اللى هو ال derivative لهذه عند مين عنده النقطة Y
+
+741
+00:56:30,520 --> 00:56:34,700
+اللى هى F M prime of Y نقص F M prime of Y إذا
+
+742
+00:56:34,700 --> 00:56:47,340
+حصلنا على هذه النقطة طيب الآن لو أجيت بس
+
+743
+00:56:47,340 --> 00:56:50,140
+عشان بدي ألاقي مكان لما فيه كلام كتير بدي أكتبه
+
+744
+00:56:50,140 --> 00:56:57,540
+هذا عندي المفروضخلّيني أكتب إن هذولة عندي المعطيات
+
+745
+00:56:57,540 --> 00:57:07,950
+اللي هي FN of X0 Converges ومعايا اللي هو FNprime
+
+746
+00:57:07,950 --> 00:57:14,250
+converging formally to the G هذا المطلوب أن FN
+
+747
+00:57:14,250 --> 00:57:20,290
+converging formally to F و F' هي مين الـ G هذه
+
+748
+00:57:20,290 --> 00:57:24,450
+المعطيات وهي المطلوب هذا اللي هو ملخصها خلينا نشوف
+
+749
+00:57:24,450 --> 00:57:31,590
+عشان أنه احنا نبدأ اللي هو أفسلكم كل خطوة اللي هو
+
+750
+00:57:31,590 --> 00:57:35,270
+ما كانت مش مفسرة في التلخيص نفسلكم إياها بالتفصيل
+
+751
+00:57:36,790 --> 00:57:43,750
+الان وجدنا هذه يا جماعة ماشي الحال الان عندي هذا
+
+752
+00:57:43,750 --> 00:57:47,830
+الكلام صحيح لكل x عندي طبعا x note نحكي عن x note
+
+753
+00:57:47,830 --> 00:57:56,530
+محددة صار عندي الان fm of x ناقص fn of x هذه صارت
+
+754
+00:57:56,530 --> 00:58:01,600
+absolute valueأصغر أو يساوي بدي أنجل هذه على النطق
+
+755
+00:58:01,600 --> 00:58:04,420
+هذه وأقول absolute value هذه بساوي ال absolute
+
+756
+00:58:04,420 --> 00:58:07,720
+value هذه وأخد triangle inequality بيصير أصغر أو
+
+757
+00:58:07,720 --> 00:58:17,260
+يساوي ال absolute value لهذه Fn of X0 ناقص Fn of
+
+758
+00:58:17,260 --> 00:58:26,650
+X0 زائد ال absolute value لل X minus X0في الـ F M
+
+759
+00:58:26,650 --> 00:58:34,930
+prime of Y ناقص F M F N prime of mean of Y ماشي
+
+760
+00:58:34,930 --> 00:58:39,650
+الحال هذا mean ماله هذا بال mean value بال
+
+761
+00:58:39,650 --> 00:58:42,550
+triangle inequality بعد ما نجلتها على الجهة
+
+762
+00:58:42,550 --> 00:58:50,310
+الثانية الآن هذا المقدار هذا المقداراللي هو أكيد
+
+763
+00:58:50,310 --> 00:58:54,790
+هذا بيصير هذا المقدار أصغر يساوي هذا وهذا أصغر
+
+764
+00:58:54,790 --> 00:59:05,090
+يساوي FM of X not FM of X not ناقص FM of X not
+
+765
+00:59:05,090 --> 00:59:12,730
+زائد هذه اللي هي اللي هي موجودة في الفترة من عند A
+
+766
+00:59:12,730 --> 00:59:17,890
+لعند BX و X0 بغض النظر الموقع هنادي أو هنادي في
+
+767
+00:59:17,890 --> 00:59:22,470
+الآخر اللي هي B minus A أكيد أكبر من X minus X0
+
+768
+00:59:22,470 --> 00:59:29,670
+أكيد إذا زائد B minus A فيه هذا هذا أكيد أصغر أو
+
+769
+00:59:29,670 --> 00:59:36,890
+يساوي مين ال norm ل F M prime ناقص F N prime over
+
+770
+00:59:36,890 --> 00:59:40,550
+مين الجيل اللي عندي ليش لأن هذا supremum للي فوق
+
+771
+00:59:40,550 --> 00:59:45,630
+صار هذا كله عبارة عن رقمهذا كله عبارة عن رقم هذا
+
+772
+00:59:45,630 --> 00:59:51,650
+الرقم الآن أكبر أو ساوي هذه لكل X وين موجودة في
+
+773
+00:59:51,650 --> 00:59:55,350
+الجيل لأنه أخدت X مالها من رأس الدور أخدت X is
+
+774
+00:59:55,350 --> 01:00:00,290
+arbitrary point ماشي الحال صار عندي الآن هذا صار
+
+775
+01:00:00,290 --> 01:00:04,650
+upper bound لهذه عدناها مائة مرة إذا صار عند هذا
+
+776
+01:00:04,650 --> 01:00:07,250
+ال upper bound أكبر أو ساوي ال least upper bound
+
+777
+01:00:07,250 --> 01:00:10,010
+اللي هو ال supremum إذا صار عند ال supremum اللي
+
+778
+01:00:10,010 --> 01:00:15,810
+هو norm ال FMناقص FN over J أصغر أو ساوي المقدار
+
+779
+01:00:15,810 --> 01:00:20,570
+هذا وهي اللي هو حصلنا عليه هنا وتفسيره هي اللي
+
+780
+01:00:20,570 --> 01:00:26,130
+فسرتلكم إياه فوق إذا صار عندي FM ناقص FN over J
+
+781
+01:00:26,130 --> 01:00:29,490
+أصغر أو ساوي FN of X node ناقص FN of X node زاد B
+
+782
+01:00:29,490 --> 01:00:36,990
+minus A في النور اللي أمامي الآن اللاحظ أن FN of X
+
+783
+01:00:36,990 --> 01:00:41,760
+node convergeاللي هو and FN' is uniformly
+
+784
+01:00:41,760 --> 01:00:46,480
+convergent in J then by the inequality above and
+
+785
+01:00:46,480 --> 01:00:50,520
+theorem 8.11 we have FN is uniformly convergent
+
+786
+01:00:50,520 --> 01:00:55,300
+رماها على طول خلّيني أفسرلكم أيها هذه خلّيني
+
+787
+01:00:55,300 --> 01:01:01,260
+أفسرلكم اللي حكاه هنا وكيف وصل أن FN converges
+
+788
+01:01:01,260 --> 01:01:07,570
+uniformly من مين لل FNنفسرها والتفسير يمكن مرة قبل
+
+789
+01:01:07,570 --> 01:01:17,510
+هيك الان شوفوا معايا بما انه عندي ال FN of X0
+
+790
+01:01:17,510 --> 01:01:25,950
+converges اذا it is Cauchy مظبوط is it is a Cauchy
+
+791
+01:01:25,950 --> 01:01:32,270
+sequence معايا مدام Cauchy sequenceأيضا المعروف
+
+792
+01:01:32,270 --> 01:01:35,790
+ايش معناه الـ Cauchy Sequence ان لكل أبسلون أكبر
+
+793
+01:01:35,790 --> 01:01:43,440
+من سفر there exist اللي هو مثلا نسميها H1اللي هو
+
+794
+01:01:43,440 --> 01:01:49,220
+element in N such that for every N و M أكبر أو
+
+795
+01:01:49,220 --> 01:01:57,500
+يساوي H1 بكون عندي FN of X0 ناقص FN of X0 أصغر من
+
+796
+01:01:57,500 --> 01:02:02,740
+مين؟ من يبسلون على اتنين مثلا يبسلون على اتنين
+
+797
+01:02:02,740 --> 01:02:07,220
+ماشي؟ هذا من وين جبته؟ من أن FN of X0 converges
+
+798
+01:02:07,220 --> 01:02:12,910
+هذه الأولى خليها هذه اسمها واحد الانعندي معطينة
+
+799
+01:02:12,910 --> 01:02:19,030
+برضه اللي هو FN الـ Prime Uniform Convergence
+
+800
+01:02:22,150 --> 01:02:28,730
+Converges uniformly Converges uniformly Converges
+
+801
+01:02:28,730 --> 01:02:32,450
+uniformly Converges
+
+802
+01:02:32,450 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+803
+01:02:34,190 --> 01:02:34,190
+Converges uniformly Converges uniformly Converges
+
+804
+01:02:34,190 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+805
+01:02:34,190 --> 01:02:34,190
+Converges uniformly Converges uniformly Converges
+
+806
+01:02:34,190 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+807
+01:02:34,190 --> 01:02:34,190
+Converges uniformly Converges uniformly Converges
+
+808
+01:02:34,190 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+809
+01:02:34,190 --> 01:02:34,190
+Converges uniformly Converges uniformly Converges
+
+810
+01:02:34,190 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+811
+01:02:34,190 --> 01:02:34,190
+Converges uniformly Converges uniformly Converges
+
+812
+01:02:34,190 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+813
+01:02:34,190 --> 01:02:34,190
+Converges uniformly Converges uniformly Converges
+
+814
+01:02:34,190 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+815
+01:02:34,190 --> 01:02:34,190
+Converges uniformly Converges uniformly Converges
+
+816
+01:02:34,190 --> 01:02:34,190
+uniformly Converges uniformly Converges uniformly
+
+817
+01:02:34,190 --> 01:02:34,190
+Converges usually Converges usually Converges
+
+818
+01:02:34,190 --> 01:02:34,290
+usually Converges usually Converges usually
+
+819
+01:02:34,290 --> 01:02:34,530
+Converges usually Converges usually Converges
+
+820
+01:02:34,530 --> 01:02:38,430
+usually Converges usually Converges usually
+
+821
+01:02:38,430 --> 01:02:44,130
+Converges usually Converges usually Converges
+
+822
+01:02:44,130 --> 01:02:51,420
+usually Converges usually Converges usuallyاللي هو
+
+823
+01:02:51,420 --> 01:03:00,120
+Normal Fn-Fm أصغر من Epsilon على B-A في 2 حرّا أي
+
+824
+01:03:00,120 --> 01:03:02,860
+Epsilon في الدنيا نضب لها Epsilon على B-A في 2
+
+825
+01:03:02,860 --> 01:03:07,640
+الان بدي استخدم هذه واستخدم هذه بدي استخدم ان الان
+
+826
+01:03:07,640 --> 01:03:11,020
+و الام بتكون أكبر من H2 و الان و الام أكبر من مين
+
+827
+01:03:11,020 --> 01:03:15,420
+من H1 إذا for every Epsilonأكبر من الصفر there
+
+828
+01:03:15,420 --> 01:03:18,700
+exist H بتساوي الماكسما عشان انا اقدر استخدم
+
+829
+01:03:18,700 --> 01:03:24,360
+التنتين ال maximum بين H1 وH2 such that for every
+
+830
+01:03:24,360 --> 01:03:29,180
+N و M أكبر أو يساوي H أكبر أو يساوي H هيكون عندى
+
+831
+01:03:29,180 --> 01:03:36,260
+norm norm هذه صحيحة لكل M و N في الدنيا ال FM ناقص
+
+832
+01:03:36,260 --> 01:03:42,800
+FN over J أصغر أو يساوي الأولى زائد التانية الأولى
+
+833
+01:03:42,800 --> 01:03:47,930
+هذهالأولى أصغر أو يساوي منها نائب سن على اتنين
+
+834
+01:03:49,210 --> 01:03:52,890
+مظبوط لأن هذه أصغر من 100 على 2 لكل الـ N و الـ M
+
+835
+01:03:52,890 --> 01:03:56,670
+الأكبر يساوي H اللي هي أكيد الـ H أكبر يساوي مين؟
+
+836
+01:03:56,670 --> 01:04:00,330
+اللي هي H1 يعني N و M اللي بيكون أكبر يساوي H أكيد
+
+837
+01:04:00,330 --> 01:04:04,690
+بيكون الـ H1 اللي أنا بالفعل أستخدمها زائد التانية
+
+838
+01:04:04,690 --> 01:04:10,090
+برضه الـ F M' ناقص F M' أصغر من مين هنا؟ M' هذا
+
+839
+01:04:10,090 --> 01:04:13,050
+طبعا، هذي Prime و هذي Prime لأن F M' الـPrime الـ
+
+840
+01:04:13,050 --> 01:04:19,730
+Convergent لمن؟ لل Gإبسلون في اتنين في B minus A
+
+841
+01:04:19,730 --> 01:04:24,890
+اللي كان مضروبة في مين في B minus A فبصير هذا
+
+842
+01:04:24,890 --> 01:04:27,250
+إبسلون على اتنين و إبسلون على هذي بيصير روح مح هذي
+
+843
+01:04:27,250 --> 01:04:30,450
+و هي سوى إبسلون إذا اللي حصلته لكل إبسلون أكبر من
+
+844
+01:04:30,450 --> 01:04:34,830
+zero there exists H such that لكل N أكبر سوى H طلع
+
+845
+01:04:34,830 --> 01:04:39,590
+عندي FN نقص FN أصغر من مين من إبسلون إذا by Cauchy
+
+846
+01:04:39,590 --> 01:04:46,420
+criterion يكون عند ال FNconverges uniformly to
+
+847
+01:04:46,420 --> 01:04:52,180
+some function mean F مش عارفين لسه عنها حاجة إذا
+
+848
+01:04:52,180 --> 01:04:57,600
+اللي أثبتناه الأن أن F is uniformly convergent to
+
+849
+01:04:57,600 --> 01:05:02,880
+some function F مين هي مش عارفينها لسه أما أثبتنا
+
+850
+01:05:02,880 --> 01:05:07,280
+وجودها أثبتنا أن F converges uniformly ل F إذا
+
+851
+01:05:07,280 --> 01:05:16,630
+بقدر أجرّأ وأقولLimit الـ Fm أو الـ Fn as n goes
+
+852
+01:05:16,630 --> 01:05:21,430
+to infinity تساوي F ماقدر أتجرأ لإن أثبتت وجودها
+
+853
+01:05:21,430 --> 01:05:27,570
+الان لاحظوا بس شغلة Fn is differentiable is a
+
+854
+01:05:27,570 --> 01:05:31,360
+continuousمدام continuous و أثبتت FM بتروح
+
+855
+01:05:31,360 --> 01:05:35,080
+converts uniformly لها إذا ال limit هذه هتطلع برضه
+
+856
+01:05:35,080 --> 01:05:40,260
+إيش ماله is continuous و هذه خلوها في الذاكرة مكون
+
+857
+01:05:40,260 --> 01:05:47,340
+احنا أوصلنا أو أثبتنا الجزء الأول من النظرية أن FM
+
+858
+01:05:47,340 --> 01:05:53,820
+converts uniformly to F و ال limit لها سمناها F و
+
+859
+01:05:53,820 --> 01:05:57,720
+هيطلع لنا عبارة عن continuous function شوفوا يا
+
+860
+01:05:57,720 --> 01:05:58,420
+جماعة الآن
+
+861
+01:06:02,600 --> 01:06:09,330
+صار عندي الآنأثبتنا أن Fn تتعامل بشكل مرتبط سمينا
+
+862
+01:06:09,330 --> 01:06:13,050
+limit Fn بسوء F which is continuous because Fn زي
+
+863
+01:06:13,050 --> 01:06:14,670
+ما قلنا is continuous لأنه في الأصل هي
+
+864
+01:06:14,670 --> 01:06:18,610
+differentiable و Fn بتروح لل F إذن صارت ال F اللي
+
+865
+01:06:18,610 --> 01:06:23,330
+هي نفسها continuous الآن بدي أستخدم ال continuity
+
+866
+01:06:23,330 --> 01:06:29,090
+عند ال F بدي أصل اللي بديه طيب خد C الآن ب any
+
+867
+01:06:29,090 --> 01:06:35,950
+fixed point in J الآن بدي أروح إلى إثبات أنهالـ
+
+868
+01:06:35,950 --> 01:06:43,810
+Limit للـ F of X ناقص F of C على X minus C as X
+
+869
+01:06:43,810 --> 01:06:46,970
+بتروح للـ C هذه الـ C وينها؟ الـ C أخذناها
+
+870
+01:06:46,970 --> 01:06:51,470
+arbitrary في X in J إذا أثبتت لهذا الـ C اللي هو
+
+871
+01:06:51,470 --> 01:06:56,870
+بساوي اللي هي G of Cمعناته إذا أثبتت هذا معناته
+
+872
+01:06:56,870 --> 01:07:02,150
+أثبتت أن F' of C exist وهي ساوي G of C ومعناته
+
+873
+01:07:02,150 --> 01:07:08,310
+أثبتت أن F بتساوي G على كل J لأن C ماخدها
+
+874
+01:07:08,310 --> 01:07:12,610
+arbitrary but fixed يعني إيش بده أثبت في هذا؟يعني
+
+875
+01:07:12,610 --> 01:07:16,910
+زي ما قلنا في الأول زي ما قلنا في الأول بدنا نثبت
+
+876
+01:07:16,910 --> 01:07:21,970
+لكم الآن لكل إبسلون أكبر من سفر there exists Delta
+
+877
+01:07:21,970 --> 01:07:27,350
+أكبر من سفر such that X minus C أصغر من Delta يؤدي
+
+878
+01:07:27,350 --> 01:07:34,990
+إلى اللي هو F of X ناقص F of C على X minus C ناقص
+
+879
+01:07:34,990 --> 01:07:39,630
+G of C أصغر من إبسلون ده أثبتت بكون خلصت إذا أثبتت
+
+880
+01:07:39,630 --> 01:07:44,780
+هذا بكون خلصت اللي هوالإثبات تنشوف طيب صلوا على
+
+881
+01:07:44,780 --> 01:07:50,120
+النبي عليه الصلاة والسلام نيجي الآن قولنا C N of X
+
+882
+01:07:50,120 --> 01:07:54,020
+point in J F M ناقص F N is differentiable on the
+
+883
+01:07:54,020 --> 01:07:58,020
+closed interval with endpoints C and X الان F N
+
+884
+01:07:58,020 --> 01:08:00,840
+ناقص F N differentiable على ال closed interval
+
+885
+01:08:00,840 --> 01:08:06,120
+اللي هي C ومين وال X ال XR اللي خدناها اللي هي
+
+886
+01:08:06,120 --> 01:08:10,770
+arbitraryماشي، الآن بدأ استخدم الـ Mean Value
+
+887
+01:08:10,770 --> 01:08:14,010
+Theorem، then by mean value theorem كمان مرة there
+
+888
+01:08:14,010 --> 01:08:19,250
+exists z between c and x such that fm of x ناقص fn
+
+889
+01:08:19,250 --> 01:08:22,130
+of x النقطة الأولى، ناقص fm of c ناقص fn of c
+
+890
+01:08:22,130 --> 01:08:26,270
+النقطة التانية بساوي x minus c مضروبة في fn prime
+
+891
+01:08:26,270 --> 01:08:32,390
+of z ناقص fn prime اللي هو fm prime of zماشي الحال
+
+892
+01:08:32,390 --> 01:08:35,350
+طبقت الـ Mean Value Theorem على الدالة ال .. اللي
+
+893
+01:08:35,350 --> 01:08:41,770
+هي Fm ناقص Fn لكل M O N الان خدوا الحسبة الان خد X
+
+894
+01:08:41,770 --> 01:08:44,690
+لا تساوى C لإنه .. X لا تساوى C لإن انا بدأو أدي X
+
+895
+01:08:44,690 --> 01:08:48,430
+عالميا على C يعني X minus C اللي هو أصغر من Delta
+
+896
+01:08:48,430 --> 01:08:52,760
+يعني X بدأو أديها على Cالان اجسموا كل الأطراف على
+
+897
+01:08:52,760 --> 01:09:00,820
+x-c بيصير اللي هي fm ناقص fm of c على x-c ناقص
+
+898
+01:09:00,820 --> 01:09:08,500
+الان fn of x هي هاناقص اللي هي هذه اللي هي FN of C
+
+899
+01:09:08,500 --> 01:09:13,300
+هي ناقص وهي ناقص إذا بيطلع صحيح على X minus C
+
+900
+01:09:13,300 --> 01:09:18,400
+بساوي اللي هو F M prime of Z ناقص FN prime of Z
+
+901
+01:09:18,400 --> 01:09:24,340
+لإنه جسمت على X minus C الآن أكيد هذه .. هذه أصغر
+
+902
+01:09:24,340 --> 01:09:27,460
+أو يساوي نورمها لأن نورم هو ال supermom إلها صار
+
+903
+01:09:27,460 --> 01:09:31,320
+عندي الآن هذا المقدار أصغر أو يساوي هذا المقدار
+
+904
+01:09:31,950 --> 01:09:37,770
+الان الان اف ام ابراهيم converged uniformly اذا
+
+905
+01:09:37,770 --> 01:09:40,110
+لكل ابسلون اكبر من سفر there exists delta such
+
+906
+01:09:40,110 --> 01:09:42,390
+there exists such و في ابسلون و such ذات هذا اصغر
+
+907
+01:09:42,390 --> 01:09:44,990
+من مين some ابسلون او ابسلون على اتنين او اللي
+
+908
+01:09:44,990 --> 01:09:52,710
+بدكميا الان اذا نكمل اللي بنقوله شوفSince Fn'
+
+909
+01:09:53,010 --> 01:09:57,510
+converges uniformly on J إذا لكل ي أكبر من 0 there
+
+910
+01:09:57,510 --> 01:10:03,770
+exists H of ي H of ي such that if M و N أكبر سواء
+
+911
+01:10:03,770 --> 01:10:10,230
+H of ي then هذا المقدار اللي هو أصغر من مين من ي
+
+912
+01:10:10,230 --> 01:10:16,590
+يعني صار كل هذا المقدار أصغر من ي لمين؟ لأ ال M و
+
+913
+01:10:16,590 --> 01:10:25,840
+ال N لأأكبر من مين من H of Epsilon لان صار عندي
+
+914
+01:10:25,840 --> 01:10:31,280
+لان لكل M و N أكبر سوى H of Epsilon صار المقدار
+
+915
+01:10:31,280 --> 01:10:35,140
+هذا اللي هو أصغر أو سوى هذا اللي هو أصغر من
+
+916
+01:10:35,140 --> 01:10:37,860
+Epsilon خليها في الذاكرة هذه أصغر من مين من
+
+917
+01:10:37,860 --> 01:10:42,740
+Epsilonالان take the limit of both sides as M goes
+
+918
+01:10:42,740 --> 01:10:46,160
+to infinity بس هان عشان بياخد M رايحة لما لنهاية
+
+919
+01:10:46,160 --> 01:10:50,000
+بدنا ناخد لل M أكبر أو يساوي H N عشان لما ال M
+
+920
+01:10:50,000 --> 01:10:53,340
+تروح لما لنهاية M تبقى مرتاحة زي 100 و M لا تتحركش
+
+921
+01:10:53,340 --> 01:11:01,520
+بدنا إياها تتحرك بيصير عندى as M goes to infinity
+
+922
+01:11:01,520 --> 01:11:08,350
+ال limit لل F M of X هيساوي F of Xلأنه صارت
+
+923
+01:11:08,350 --> 01:11:14,670
+uniformly الـ Fm أثبتناها بتروح للـ F uniformly
+
+924
+01:11:14,670 --> 01:11:19,830
+إذن limit Fm of X هي F of X و limit Fm of C هي F
+
+925
+01:11:19,830 --> 01:11:25,490
+of C على X minus C معايا هذا Fn of X ناقص Fn of C
+
+926
+01:11:25,490 --> 01:11:29,370
+على X minus C زي ما هي لإن أنا إبسلتها زي ما هي
+
+927
+01:11:29,370 --> 01:11:32,890
+مين اللي طيرتها الكبيرة اللي هي M إذا صار هذا
+
+928
+01:11:32,890 --> 01:11:38,130
+المقدار أصغر أو يساوي مين؟ إبسلونفي حال مين؟ في
+
+929
+01:11:38,130 --> 01:11:43,930
+حال X لا تساوي C و للأنات اللي أكبر من مين من H of
+
+930
+01:11:43,930 --> 01:11:49,710
+Y أنا تحت هذه الشروط ماشي طيب عمالي قاعد بجرب على
+
+931
+01:11:49,710 --> 01:11:58,400
+اللي هو الهدف تبعي شوف صار عندي الآنهذه الـ
+
+932
+01:11:58,400 --> 01:12:03,740
+Inquality واحد اللي هو متحققها لكن أنا بعرف أنه
+
+933
+01:12:03,740 --> 01:12:07,780
+limit of F N prime of C أشهر بساوي G of C لأنه
+
+934
+01:12:07,780 --> 01:12:12,440
+احنا مفترضين من رأس الدور أنه F N prime converges
+
+935
+01:12:12,440 --> 01:12:18,470
+uniformly to مين؟للـ G .. طيب لأن sense limit F N
+
+936
+01:12:18,470 --> 01:12:22,330
+prime of C بيسوي G of C إذا there exists N of
+
+937
+01:12:22,330 --> 01:12:25,590
+إبسلون such that for every N أكبر بيسوي N of
+
+938
+01:12:25,590 --> 01:12:29,910
+إبسلون حيكون F N prime of C ناقص G of C اللي هو
+
+939
+01:12:29,910 --> 01:12:38,310
+أصغر من إيش من إبسلونالان صار عندى هذه متحققة للان
+
+940
+01:12:38,310 --> 01:12:43,430
+أكبر سواء H و هذه متحققة للان الأكبر سواء N عشان
+
+941
+01:12:43,430 --> 01:12:47,750
+استخدم التنتين بتاخد ال maximum بين هذه و هذه و
+
+942
+01:12:47,750 --> 01:12:52,630
+اقول let K بسوء ال maximum بين H of epsilon و N
+
+943
+01:12:52,630 --> 01:12:59,130
+epsilon اذا الان صار عندى التنتين متحققات لكل N
+
+944
+01:12:59,130 --> 01:13:04,530
+أكبر او يساوي K ومشروع استخدامهالآن من جهة ثانية
+
+945
+01:13:04,530 --> 01:13:07,750
+بدي استخدم الـ FK' of C إنها exist
+
+946
+01:13:11,040 --> 01:13:14,540
+إذا صارت عندي هذه زي ما قلنا وهذه هاتين تان بحجله
+
+947
+01:13:14,540 --> 01:13:18,620
+استخدمهم لكل ان أكبر أو ساو كي بدي استخدم هذه كمان
+
+948
+01:13:18,620 --> 01:13:22,900
+و أربط هذه مع هذه و أخلص اللي بديها since fk prime
+
+949
+01:13:22,900 --> 01:13:26,860
+of c exist مين ال K هذه ال K هذه اللي خدتها ال
+
+950
+01:13:26,860 --> 01:13:31,660
+superman بين هذه و هذه عشان ينفع استخدم هذه و هذه
+
+951
+01:13:31,660 --> 01:13:37,760
+و هذه بعد شويةthen مدام fk prime of c exist then
+
+952
+01:13:37,760 --> 01:13:42,080
+if x minus c أكبر من صفر أصغر من دلتا دلتا k طبعا
+
+953
+01:13:42,080 --> 01:13:46,480
+الدلتا هتعتمد على اللي هو ال epsilon نسميها دلتا k
+
+954
+01:13:46,480 --> 01:13:49,360
+لإنها اعتمدت على مين؟ على الكل يعنى then there
+
+955
+01:13:49,360 --> 01:13:53,380
+exists دلتا k بحيث x minus c أصغر من دلتا يعطيني
+
+956
+01:13:53,380 --> 01:13:57,540
+fk of x ناقص fk of c على x minus c اللي هي ناقص fk
+
+957
+01:13:57,540 --> 01:14:04,140
+prime of c أصغر من إبسلون فاهمين هذا؟ مدام fkprime
+
+958
+01:14:04,140 --> 01:14:09,260
+of c بيساو limit exist إذا حساو limit f of x ناقص
+
+959
+01:14:09,260 --> 01:14:14,280
+f of c بprime على x minus c طبعا كك كله as x بتروح
+
+960
+01:14:14,280 --> 01:14:17,620
+لل c صار exist يعني forever epsilon أكبر من 0
+
+961
+01:14:17,620 --> 01:14:21,740
+exist دلتا بحيث أنه لما تكون x minus c أصغر من
+
+962
+01:14:21,740 --> 01:14:24,360
+دلتا يعطيني هذا ناقص هذا أصغر من epsilon وهذا هو
+
+963
+01:14:24,360 --> 01:14:28,400
+المكتوبلذن صار عندى هذا موجود وهي اتنين وهي تلاتة
+
+964
+01:14:28,400 --> 01:14:33,020
+لأن واحد و اتنين و تلاتة بدي اصل اللي بديه خلينا
+
+965
+01:14:33,020 --> 01:14:35,720
+نكتب هنا عشان ما نجلبش الصفحة عشان يظلوا كله
+
+966
+01:14:35,720 --> 01:14:43,880
+قدامكم الان اللي وصلت اليه مالي for every epsilon
+
+967
+01:14:43,880 --> 01:14:46,660
+أكبر من سفر بقول there exists delta أكبر من سفر
+
+968
+01:14:46,660 --> 01:14:49,740
+such that if
+
+969
+01:14:51,280 --> 01:14:57,660
+x-c أصغر أكبر من صفر وأصغر من دلتا then شوف الآن
+
+970
+01:14:57,660 --> 01:15:02,400
+then إيش اللي هو بدي مين أنا هي هي اللي بديها f of
+
+971
+01:15:02,400 --> 01:15:12,880
+x minus f of c على x minus c ناقص g of c أصغر
+
+972
+01:15:12,880 --> 01:15:18,120
+أو يساوي ماشي الحال أصغر أو يساوي بدي أبدأ الآن
+
+973
+01:15:18,120 --> 01:15:24,240
+اللي هو هي هذهبدي أدخلها دي، عشان؟ أصغر أو ساوي
+
+974
+01:15:24,240 --> 01:15:32,300
+اللي هو F of X ناقص F of C على X minus C ناقص FN
+
+975
+01:15:32,300 --> 01:15:39,840
+of X ناقص FN of C على X minus C زائد اللي دخلتها
+
+976
+01:15:39,840 --> 01:15:51,830
+دي FN of X ناقص FN of C على X minus C ناقصاللي هو
+
+977
+01:15:51,830 --> 01:15:56,850
+ايش
+
+978
+01:15:56,850 --> 01:16:03,170
+التانية فك
+
+979
+01:16:03,170 --> 01:16:13,890
+prime of c a زائد فك
+
+980
+01:16:13,890 --> 01:16:21,820
+prime of c ناقص فنقص من اللي ضال G of C بظبط هيك ا
+
+981
+01:16:21,820 --> 01:16:26,460
+اظن تشوف عليها الان انا بتلزمنيش كل الانات بتلزمني
+
+982
+01:16:26,460 --> 01:16:31,480
+بس ال مين ال K لأنه الان انا بحكي على استخدام ال K
+
+983
+01:16:31,480 --> 01:16:39,280
+K K و هاد ايهاش K طيب الان نيجي نتطلعالـ K تبعتنا
+
+984
+01:16:39,280 --> 01:16:42,040
+هذه بنحكي عن K محددة الـ K اللي هي superman بين H
+
+985
+01:16:42,040 --> 01:16:46,320
+و بين N يعني الـ K أكبر أو يساوي H و الـ K أكبر أو
+
+986
+01:16:46,320 --> 01:16:51,600
+يساوي مين الـ N يعني بحججلي أستخدم F K of C F K
+
+987
+01:16:51,600 --> 01:16:55,020
+prime of C ناقص G of C أصغر من إبسلون و بحججلي
+
+988
+01:16:55,020 --> 01:16:58,320
+أستخدم هذه ناقص F K of K في K أصغر من مين من
+
+989
+01:16:58,320 --> 01:17:04,540
+إبسلون الآن هذه هذه هي هان الأولى إذا أصغر أو
+
+990
+01:17:04,540 --> 01:17:13,170
+يساوي هذه من واحدepsilon زائد هذه التانية من مين
+
+991
+01:17:13,170 --> 01:17:19,250
+من اللي هي تلاتة هي fk of x نقص fk of c عليك x
+
+992
+01:17:19,250 --> 01:17:23,490
+minus c نقص fk prime أصغر برضه من مين من epsilon
+
+993
+01:17:23,490 --> 01:17:29,290
+هذه من تلاتة وهذه من واحد هذه أكيد من مين هستخدمها
+
+994
+01:17:29,290 --> 01:17:35,610
+من اتنين اللي هي n هذه لكل n أكبر سوى nوعندي الـ K
+
+995
+01:17:35,610 --> 01:17:38,190
+أصلا أكبر أوي سواء لإنها ال maximum بين ال team
+
+996
+01:17:38,190 --> 01:17:42,910
+تان إذا هذه برضه بحق الله أستخدمها أمن اتنين أصغر
+
+997
+01:17:42,910 --> 01:17:47,470
+من إبسلون إذا صار عندي هذا المقدار اللي فوق أصغر
+
+998
+01:17:47,470 --> 01:17:52,290
+من تلاتة إبسلون وإبسلون was arbitrarily إذا صار
+
+999
+01:17:52,290 --> 01:17:57,210
+عندي limit هذا المقدار بساوي G of C إيش معناه
+
+1000
+01:17:57,210 --> 01:18:01,430
+limit هذا المقدار بساوي G of C معناته أنه اللي هو
+
+1001
+01:18:01,430 --> 01:18:08,730
+ال F primeof C هو الـ G of C وزي ما قلنا في الأول
+
+1002
+01:18:08,730 --> 01:18:12,530
+since C was arbitrary in G then صارت اللي هي F
+
+1003
+01:18:12,530 --> 01:18:18,990
+بتساوي G على كل مين على كل الـ G وهيك بنكون خلصنا
+
+1004
+01:18:18,990 --> 01:18:23,330
+النظرية وانهينا ال section اللي هو تمانية اتنين
+
+1005
+01:18:23,330 --> 01:18:26,370
+وهيك بنكون خلصنا الحديث اللي بنتحدثه في chapter
+
+1006
+01:18:26,370 --> 01:18:30,370
+تمانية وعندنا طبعا الأسئلة المطلوبة هيها في نهاية
+
+1007
+01:18:30,370 --> 01:18:33,430
+كل section تمانية واحد وتمانية اتنين وإلى لقاء
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/QjdD_6J2asI.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/QjdD_6J2asI.srt
new file mode 100644
index 0000000000000000000000000000000000000000..9cf515afd4909f7985263dde179fa75f9cf92cc6
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/QjdD_6J2asI.srt
@@ -0,0 +1,1694 @@
+1
+00:00:04,960 --> 00:00:12,580
+بسم الله الرحمن الرحيم هذه المحاضرة رقم 16 في مساق
+
+2
+00:00:12,580 --> 00:00:18,080
+تحليل حقيقي 2 أو تحليل رياضي 2 لطلبة وطالبات كلية
+
+3
+00:00:18,080 --> 00:00:22,820
+العلوم بالجامعة الإسلامية بغزة. كنا سميينها في
+
+4
+00:00:22,820 --> 00:00:29,160
+المحاضرة 15 إنها جزئين، المحاضرة الجزء الأول ذكرناه 
+
+5
+00:00:29,160 --> 00:00:31,820
+في المحاضرة 15 وقلنا الجزء الثاني هنذكره في
+
+6
+00:00:31,820 --> 00:00:35,780
+المحاضرة اللي بعدها كجزء ثاني من 15، بس الآن لطول
+
+7
+00:00:35,780 --> 00:00:40,440
+فترة اللي هي المحاضرة السابقة، هنسمي هذا الجزء الآن 
+
+8
+00:00:40,440 --> 00:00:43,060
+اللي هو كان اسمه الجزء الثاني المحاضرة 15. لأ، بدي 
+
+9
+00:00:43,060 --> 00:00:48,300
+أسميه هو المحاضرة رقم 16، وهو اللي هو بداية سيكشن
+
+10
+00:00:48,300 --> 00:00:52,020
+جديد اللي هو سبعة أربعة، اللي هو the integration as
+
+11
+00:00:52,020 --> 00:00:56,260
+a limit، اللي هو بدنا نصل ال integration عن طريق
+
+12
+00:00:56,260 --> 00:01:01,880
+اللي هو the limit of Riemann sum. بس شغلة سريعة عشان
+
+13
+00:01:01,880 --> 00:01:05,820
+نربط بالسابق باللاحق بالحاضر عشان نفهم إيش اللي
+
+14
+00:01:05,820 --> 00:01:10,780
+بنحكي فيه إحنا. إذا بتتذكرتتذكر، واخدنا على الفترة A
+
+15
+00:01:10,780 --> 00:01:17,200
+وB وعملنا لها تجزئة P بساوي X Note X واحد لعند ال
+
+16
+00:01:17,200 --> 00:01:25,000
+XN وعرفنا اللي هو ال Upper sum اللي هو Upper اللي
+
+17
+00:01:25,000 --> 00:01:31,900
+هي L وP L لو function عندي F فعندي F of X على
+
+18
+00:01:31,900 --> 00:01:39,160
+الفترة من A لعند B، اللي هو ال UpperF of B، وقلنا هو
+
+19
+00:01:39,160 --> 00:01:44,780
+عبارة عن summation لل M K في X K minus X K minus
+
+20
+00:01:44,780 --> 00:01:51,060
+واحد، زي ما أنتم الذاكرين، K من عند واحد لعند N، هذا
+
+21
+00:01:51,060 --> 00:01:54,900
+اللي هو اللي سمناه ال other sum. ال M K طبعا هي ال
+
+22
+00:01:54,900 --> 00:01:57,600
+... زي ما أنتم عارفين، ال supremum على الفترة ... ال
+
+23
+00:01:57,600 --> 00:02:01,040
+supremum لدال ال F، ال supremum لدال ال F على
+
+24
+00:02:01,040 --> 00:02:06,340
+الفترة اللي أمامي. وعندي ال L F of P أو P و F، زي ما
+
+25
+00:02:06,340 --> 00:02:10,300
+إحنا بنكتبها، عبارة عن summation ال M K في X K
+
+26
+00:02:10,300 --> 00:02:14,680
+minus X K minus واحد، K من عند واحد لعندنا. وقلنا لو
+
+27
+00:02:14,680 --> 00:02:19,380
+أتينا ال other sum، أخدناله اللي هو ال infimum ال
+
+28
+00:02:19,380 --> 00:02:24,370
+other sum، وقلنا أخدناله ال infimum، بنسميه U of F
+
+29
+00:02:24,370 --> 00:02:29,070
+اللي هو عبارة عن الـ infimum لـ U of F، و B such
+
+30
+00:02:29,070 --> 00:02:32,370
+that B element in the set of all partition B of I.
+
+31
+00:02:32,370 --> 00:02:36,630
+و أخدنا ال L of F و قلنا هو عبارة عن ال supremum
+
+32
+00:02:36,630 --> 00:02:41,300
+لـ L of F و B، highest B element in the set of all
+
+33
+00:02:41,300 --> 00:02:46,780
+partitions B of I. قلنا إن F هتكون integrable إذا
+
+34
+00:02:46,780 --> 00:02:50,460
+اللي هو ال lower integral بساومين ال upper
+
+35
+00:02:50,460 --> 00:02:55,580
+integral. هذا المدخل اللي دخلنا فيه لإثبات أو
+
+36
+00:02:55,580 --> 00:03:00,220
+لتعريف إن F is integrable. مدخل اللي هو ال upper
+
+37
+00:03:00,220 --> 00:03:03,650
+sum والlower sum أو الـ Upper Integral والـ Lower
+
+38
+00:03:03,650 --> 00:03:07,470
+Integral، يوم ما يتساون بنسميه الـ Integral اللي هو
+
+39
+00:03:07,470 --> 00:03:10,670
+عبارة عن قيمة الـ Integration للـ F على الفترة A
+
+40
+00:03:10,670 --> 00:03:17,640
+وB. الآن، إيش اللي هنسويه اللي هو الجزء الثاني من
+
+41
+00:03:17,640 --> 00:03:21,300
+المحاضرة
+
+42
+00:03:21,300 --> 00:03:24,300
+أو اللي إحنا سمينا المحاضرة 16 اللي هو في ال
+
+43
+00:03:24,300 --> 00:03:29,840
+section 7-4 هندخل لل integration بداخلة أخرى عن
+
+44
+00:03:29,840 --> 00:03:35,280
+طريق حاجة اسمها ال remansum، ومن ثم نقول نثبت في
+
+45
+00:03:35,280 --> 00:03:40,000
+النهاية انه limit ال remansum لما يكون موجود، بكون
+
+46
+00:03:40,000 --> 00:03:45,480
+هو الـ Integration، وهذا مكافئ أو يحدث if and only
+
+47
+00:03:45,480 --> 00:03:50,580
+if إذا حدثت اللي هو معنى التكامل بصيغة ال Upper
+
+48
+00:03:50,580 --> 00:03:54,420
+Sum والLower Sum. يعني الآن .. الآن في عندنا مدخلين
+
+49
+00:03:54,420 --> 00:03:57,660
+للدخول للتكامل، إما عن طريق ال Upper Sum والLower
+
+50
+00:03:57,660 --> 00:04:00,560
+Sum أو ال Upper Integral والLower Integral أو عن
+
+51
+00:04:00,560 --> 00:04:04,200
+طريق اللي هو ال Remain Sum أو اللي هو ال limit
+
+52
+00:04:04,200 --> 00:04:09,000
+للRemain Sum. والتنين لما نكون كل واحد ذاك ال ...
+
+53
+00:04:09,000 --> 00:04:12,740
+الملهمته موجود، و هذا ال upper بساوي ال lower بكون
+
+54
+00:04:12,740 --> 00:04:19,500
+ال R equivalent إيه الآن؟ إيش ال ... ال roman sum؟
+
+55
+00:04:19,500 --> 00:04:24,200
+إحنا برضه التجزية جاءت تجزية هاي P بساوي X note و
+
+56
+00:04:24,200 --> 00:04:29,280
+X واحد لعند ال XN، هاي التجزية، هاي X note و هاي X
+
+57
+00:04:29,280 --> 00:04:32,700
+واحد، وهي الفترة الموضاجية XK minus واحد لعند ال XK.
+
+58
+00:04:33,370 --> 00:04:40,570
+الآن الـ Riemann Sum بنختار في الـ Intervals نقطة
+
+59
+00:04:40,570 --> 00:04:45,830
+عشوائية، لا تقول لي لا أكبر واحدة ولا أصغر واحدة
+
+60
+00:04:45,830 --> 00:04:49,150
+ولا كذا، لأ، نقطة عشوائية وبنسميها مثلا، لو قلت لك
+
+61
+00:04:49,150 --> 00:04:54,970
+الهن، بسميها XIK. الآن XIK هذه صارت element في الـ
+
+62
+00:04:54,970 --> 00:05:02,000
+XK-1 والـ xk. الآن بدي بضرب ال x اللي اللي هي طول
+
+63
+00:05:02,000 --> 00:05:05,140
+الفترة، زي ما ضربتها، زي ما وجدت ال sum اللي هو ال
+
+64
+00:05:05,140 --> 00:05:09,020
+upper sum، هنا بوجد هنا بس بستبدل ال mk و ال mk
+
+65
+00:05:09,020 --> 00:05:13,380
+small و upper بستبدلها بقيمة ال function عند هذه
+
+66
+00:05:13,380 --> 00:05:18,270
+النقطة العشوائية اللي اخترتها، اللي هي بيسميها xik.
+
+67
+00:05:18,270 --> 00:05:24,190
+فبصير xk minus xk minus 1 في ال F عند xik، و باخده
+
+68
+00:05:24,190 --> 00:05:28,290
+ال summation k من عند 1 لعند n، و هذا اللي بيسميه
+
+69
+00:05:28,290 --> 00:05:36,030
+الرمان صم S of B أو مين و F، S of B أو F، و هذا اللي
+
+70
+00:05:36,030 --> 00:05:43,470
+بيسميه الرمان صم. في النهاية، هنصلكم أنه لو كان ال
+
+71
+00:05:43,470 --> 00:05:49,710
+limit للـ S, P و F as الـ N goes to infinity، يعني
+
+72
+00:05:49,710 --> 00:05:53,630
+الـ N اللي هي زغرنا الفترة صارتها الجدّة، و زغرنا
+
+73
+00:05:53,630 --> 00:05:57,090
+صارتها الفترة .. لما تكون N تروح إلى نهاية، معناته
+
+74
+00:05:57,090 --> 00:05:59,590
+أن صارت اللي .. اللي هي، و كأنه الفترات عبارة عن
+
+75
+00:05:59,590 --> 00:06:05,130
+خطوط جنب بعض، هذه الخطوط بتنطبق الـ XK و XK-1 و XIK
+
+76
+00:06:05,130 --> 00:06:10,950
+على بعض، و كأنه بتسير كل خط جنب خط جنب خط إلى قيمة
+
+77
+00:06:10,950 --> 00:06:14,640
+اللي هو مجموع هذه الخطوط زي ما حكينا قبل هيك، اللي
+
+78
+00:06:14,640 --> 00:06:18,820
+بتعمل المساحة تحت المنحنى في حالة الموجب، ومن ثم هي
+
+79
+00:06:18,820 --> 00:06:21,740
+هتعمل لي قيمة ال integration. يعني لما يكون ال
+
+80
+00:06:21,740 --> 00:06:24,420
+limit لـRiemann sum as N goes to infinity exist
+
+81
+00:06:24,420 --> 00:06:29,380
+وسوّت لي رقم، اسمه إيه؟ هذا الرقم هو عبارة عن ال
+
+82
+00:06:29,380 --> 00:06:33,900
+integration إيه لـ BF of X DX على هذه الفترة، هذا
+
+83
+00:06:33,900 --> 00:06:38,400
+اللي في النهاية بدنا نصل إليه أو نوصلكم إنه في
+
+84
+00:06:38,400 --> 00:06:42,680
+حالة وجود هذا ال limit، هيكون هو ال integration
+
+85
+00:06:42,680 --> 00:06:47,040
+اللي بمفهوم ال upper sum والlower sum. يعني
+
+86
+00:06:47,040 --> 00:06:53,160
+الدخلتين بقد إلى نفس القيمة أو إلى نفس النتيجة.
+
+87
+00:06:54,950 --> 00:06:59,130
+يعني في بعض الكتب بتعرفلك أصلا اللي هو ال
+
+88
+00:06:59,130 --> 00:07:04,430
+integration as a limit of remansum. في كتب، و هذا
+
+89
+00:07:04,430 --> 00:07:09,650
+غالبا في ال calculus بيشتغلوه، وفي كتب بتعرفه عن
+
+90
+00:07:09,650 --> 00:07:12,230
+طريق ال other sum و لا other sum. في ال calculus
+
+91
+00:07:12,230 --> 00:07:15,930
+لأنه ملزمش ل Supremum و ل Infimum، لإن الطالب مفهم
+
+92
+00:07:15,930 --> 00:07:20,370
+ال Supremum ومفهم ال Infimum، عنده اللي هو ناضج.
+
+93
+00:07:24,160 --> 00:07:29,980
+بعد هذه المقدمة عن الموضوع، خلّينا ندخل إليه بشكل
+
+94
+00:07:29,980 --> 00:07:34,700
+تفصيلي، The integration as a limit لت I اللي هي ال
+
+95
+00:07:34,700 --> 00:07:36,820
+interval اللي أنا بده اشتغل عليها، و ال function
+
+96
+00:07:36,820 --> 00:07:40,260
+أفنان I لR عبارة عن bounded function، و خدنا ال B
+
+97
+00:07:40,260 --> 00:07:44,440
+هو ال partition اللي أخدناه، و زي ما قلنا، إذا أخدنا
+
+98
+00:07:44,440 --> 00:07:50,470
+XI 1 و XI 2 و XI N عبارة عن arbitrary numbers، اللي
+
+99
+00:07:50,470 --> 00:07:53,330
+هو الـ x i 1 في ال ... في ال ... في ال interval x
+
+100
+00:07:53,330 --> 00:07:57,270
+not x 1، x i 2 في ال interval اللي ... اللي بعدها
+
+101
+00:07:57,270 --> 00:08:00,810
+ايه اخره، يعني ال x i k هي في ال ... الفترة النموذج
+
+102
+00:08:00,810 --> 00:08:06,130
+الذادية x k minus 1 x k، ل k من 1 لعند مين؟ لعند ان.
+
+103
+00:08:06,130 --> 00:08:08,570
+ال sum اللي قولنا عنه اللي هو عبارة عن ال
+
+104
+00:08:08,570 --> 00:08:12,530
+summation لل F of x i k مضروب في x k ناقص x k
+
+105
+00:08:12,530 --> 00:08:17,110
+minus 1. هذا ال summation من 1 لعند ان، هم هو برمزه
+
+106
+00:08:17,110 --> 00:08:21,030
+للرمز SPUF اللي هو بسميه، اللي هو عبارة عن the
+
+107
+00:08:21,030 --> 00:08:25,810
+remand sum، الريمان صم، الريمان صم، الريمان صم.
+
+108
+00:08:25,810 --> 00:08:30,150
+الريمان صم، الريمان صم، الريمان صم، الريمان صم.
+
+109
+00:08:30,150 --> 00:08:31,170
+الريمان صم، الريمان صم، الريمان صم، الريمان صم.
+
+110
+00:08:31,170 --> 00:08:32,670
+الريمان صم، الريمان صم، الريمان صم، الريمان صم.
+
+111
+00:08:32,670 --> 00:08:33,050
+الريمان صم، الريمان صم، الريمان صم، الريمان صم.
+
+112
+00:08:33,050 --> 00:08:33,610
+الريمان صم، الريمان صم، الريمان صم، الريمان صم.
+
+113
+00:08:33,610 --> 00:08:34,330
+الريمان صم، الريمان صم، الريمان صم، الريمان صم.
+
+114
+00:08:34,330 --> 00:08:36,290
+الريمان صم، الريمان صم، الريمان صم، الريمان صم.
+
+115
+00:08:36,290 --> 00:08:43,690
+الريمان صم، الريمان صم، الريمان صم، الريمان صم، الري
+
+116
+00:08:43,710 --> 00:08:48,550
+لأن الـ F of XI K اللي هي النقطة العشوائية هذه
+
+117
+00:08:48,550 --> 00:08:53,190
+أكيد أصغر أو يساوي الـ inf mum على كل النقاط اللي
+
+118
+00:08:53,190 --> 00:08:57,710
+في ال sub interval هذه، وأكبر أو يساوي، وأصغر أو
+
+119
+00:08:57,710 --> 00:09:01,730
+يساوي من ال super mum. يعني F of XI K هذه نقطة
+
+120
+00:09:01,730 --> 00:09:05,490
+عشوائية، أكبر أو يساوي أصغر واحدة أو أصغر اللي هو
+
+121
+00:09:05,490 --> 00:09:10,070
+ال inf mum، وأصغر أو يساوي اللي هي من ال super mum.
+
+122
+00:09:10,610 --> 00:09:13,890
+هذا لكل مين؟ لكل sub interval. يعني هي sub interval
+
+123
+00:09:13,890 --> 00:09:20,810
+xk minus واحد، وهي xk، وهي xik. أكيد اللي هو f of xik
+
+124
+00:09:20,810 --> 00:09:29,210
+هنا أصغر أو يساوي أكبر واحدة، وأصغر أو يساوي أكبر
+
+125
+00:09:29,210 --> 00:09:34,020
+واحدة، وأكبر أو يساوي أصغر واحدة، اللي هي الـ MK.
+
+126
+00:09:34,020 --> 00:09:38,580
+فبكون
+
+127
+00:09:38,580 --> 00:09:43,360
+دايما هي هذه دي، وهذه دي، وهذه دي. خدوا الآن، ضربولي
+
+128
+00:09:43,360 --> 00:09:47,680
+هنا وهنا وهنا، ضربولي في XK-XK-Y، حط طول الفترة.
+
+129
+00:09:48,770 --> 00:09:52,070
+واخدوا الـ summation من عند واحد لعند أن، بيصير هذا
+
+130
+00:09:52,070 --> 00:09:55,670
+بيمثل لي اللي هو الأبر صم، وهذا بيمثل لي الأور صم.
+
+131
+00:09:55,670 --> 00:10:00,070
+وهذا بيمثل لي مين؟ الريمان صم. إذن الريمان صم دائما
+
+132
+00:10:00,070 --> 00:10:05,250
+بين الأبر وبين الأور. لأوله أصلا إحنا ليش بيسميهم
+
+133
+00:10:05,250 --> 00:10:10,210
+أبر و لاور؟ عسى إن كل اللي غيرهم الأبر، كل اللي
+
+134
+00:10:10,210 --> 00:10:14,350
+غيرهم أصغر يسويهم، وكل اللي غيرهم الأور أكبر منهم
+
+135
+00:10:14,350 --> 00:10:21,060
+أو يسويهم. التسمية جزء من المفهوم أو تستقر من
+
+136
+00:10:21,060 --> 00:10:27,020
+المفهوم. يعني بنسميها، مش معقول هي أبر ونسميها لابر
+
+137
+00:10:27,020 --> 00:10:32,520
+أو أبر ونسميها اسم يخالف الأبر. فأكيد اخترنا اسم
+
+138
+00:10:32,520 --> 00:10:37,660
+الأبر عشان هي فعلا أبر. طيب، نجي الى النظرية الأولى.
+
+139
+00:10:37,660 --> 00:10:43,520
+let a و b عبارة عن فترة مغلقة و bounded، و let f من
+
+140
+00:10:43,520 --> 00:10:46,960
+I ل R be integrable on I. نفترض ان اف، إذا احنا لما
+
+141
+00:10:46,960 --> 00:10:49,880
+نحكي integrable، معناته integrable in the sense of
+
+142
+00:10:49,880 --> 00:10:53,340
+definition 716 اللي هو اللي بعتمد على ال U of F
+
+143
+00:10:53,340 --> 00:10:58,080
+بتساوي ال O of F. الآن let F من I ل R بيه Integrable
+
+144
+00:10:58,080 --> 00:11:01,940
+I، زي ما بأكد، in the sense of definition 716، اللي
+
+145
+00:11:01,940 --> 00:11:09,070
+حكيته. إذا لكل ي أكبر من صفر is given، There exists a 
+
+146
+00:11:09,070 --> 00:11:11,470
+partition by بي إبسلون such that if بي is any
+
+147
+00:11:11,470 --> 00:11:15,610
+partition that is a refinement of بي إبسلون، and if
+
+148
+00:11:15,610 --> 00:11:19,930
+أس بي و أف is any remaining sum for أف، then هيكون
+
+149
+00:11:19,930 --> 00:11:24,110
+عنده اللي هو أس بي و أف ناقص ال integration لأف
+
+150
+00:11:24,110 --> 00:11:29,210
+أصغر من مين؟ من إبسلون. بقول باختصار، يعني لو فرضنا
+
+151
+00:11:29,210 --> 00:11:34,490
+أف integrable، اللي هنخد لأي إبسلون في الدنيا بقدر
+
+152
+00:11:34,490 --> 00:11:40,550
+أجيلك partition بي إبسلون بحيث انه اي بي بارتيشن
+
+153
+00:11:40,550 --> 00:11:45,390
+refinement لبيبسلون. refinement معناته البي بحتوى
+
+154
+00:11:45,390 --> 00:11:50,910
+مين؟ البيبسلون، تحسينله. هيكون عندي ال remain sum S
+
+155
+00:11:50,910 --> 00:11:55,150
+بي و F، هذا ناقص integration للف أصغر من مين؟ من
+
+156
+00:11:55,150 --> 00:12:04,090
+ابسلون. يعني الآن، F is integrable يا جماعة، الآن for
+
+157
+00:12:04,090 --> 00:12:08,070
+every epsilon أكبر من 0، there exist بي ابسلون such
+
+158
+00:12:08,070 --> 00:12:12,490
+that for every بي تحتوي بي ابسلون، اللي هو هذا اللي
+
+159
+00:12:12,490 --> 00:12:20,110
+جيته، and for every man some أس للبي والأفهيكون
+
+160
+00:12:20,110 --> 00:12:25,410
+عندي الـ absolute value بين الـ S بي و F ناقص اللي
+
+161
+00:12:25,410 --> 00:12:31,890
+هو ال integration من A ل B لل F أصغر من 100، منافسة
+
+162
+00:12:31,890 --> 00:12:35,790
+يعني جاعت، بقول أخي يعني أصلا يعني .. أني هذا في الآخر
+
+163
+00:12:35,790 --> 00:12:39,910
+في الآخر في الآخر في الآخر هتلاقي الريمان صم اللي
+
+164
+00:12:39,910 --> 00:12:43,890
+هو يكون بينه و بين ال F إيش ما له أصغر من إبسلون.
+
+165
+00:12:43,890 --> 00:12:47,870
+لما تكون ال إيش ال F اللي هي Integral. يعني بلاقي
+
+166
+00:12:47,870 --> 00:12:52,890
+بي إبسلون بضمنك هذا الكلام، بضمنك مش ال بي إبسلون
+
+167
+00:12:52,890 --> 00:12:58,550
+بس، وكل ال refinement. يعني كل ال refinement يعني
+
+168
+00:12:58,550 --> 00:13:05,310
+كل ما نزيد عدد عناصر ال partition يعني كل ما نكبر
+
+169
+00:13:05,310 --> 00:13:09,810
+الآن يعني و كأنه بقودنا لللي بدنا نحكيه قدام انه
+
+170
+00:13:09,810 --> 00:13:15,770
+لما الان تروح لم
+
+201
+00:15:43,990 --> 00:15:50,210
+الـ UBF ناقص الـ BF أصغر من مين؟ من Y صار عندي الـ U
+
+202
+00:15:51,040 --> 00:16:01,320
+بأف ناقص الـ بأف أصغر من إبسلون لكل بي يحتوي من الـ
+
+203
+00:16:01,320 --> 00:16:04,120
+بي إبسلون لكل refinement لمين؟ للـ بي إبسلون اللي
+
+204
+00:16:04,120 --> 00:16:08,280
+لجناه بوسط الـ ريمان الـ criterion خلينا نكمل يا
+
+205
+00:16:08,280 --> 00:16:19,320
+جماعة الآن لكن احنا بنعرف أن أي ريمان صم لـ بي معينة
+
+206
+00:16:19,510 --> 00:16:24,830
+هيكون بين الـ L و بين الـ P أكيد صح ولا لأ؟ طبعًا
+
+207
+00:16:24,830 --> 00:16:28,910
+الـ S بين الـ P و الـ F اللي قلناها قبل شوية بين
+
+208
+00:16:28,910 --> 00:16:34,570
+الـ L P و F و الـ U P و F طيب خد الـ integration الـ
+
+209
+00:16:34,570 --> 00:16:36,970
+integration ما هو الـ F is integrable مزام
+
+210
+00:16:36,970 --> 00:16:41,330
+Integrable إذا هي بتساوي زي ما عملناها كتير بتساوي
+
+211
+00:16:41,330 --> 00:16:45,430
+الـ U في F و بتساوي الـ L في F الـ L في F هي الـ
+
+212
+00:16:45,430 --> 00:16:48,070
+supremum لهذه الأشياء إذا أكيد هذا أكبر يساوي الـ
+
+213
+00:16:48,070 --> 00:16:52,410
+lower اللي هي sum هذا والـ Integration بيساوي الـ U
+
+214
+00:16:52,410 --> 00:16:56,370
+و F و الـ U و F هو عبارة عن الـ M في مهم للـ U و B
+
+215
+00:16:56,370 --> 00:16:59,230
+و F إذا نحن نكون أصغر أو يساوي إذا فعلًا الـ
+
+216
+00:16:59,230 --> 00:17:04,610
+integration بين الـ lower sum و بين الـ upper sum و
+
+217
+00:17:04,610 --> 00:17:09,560
+هذه الفكرة عملناها كتير قبل كده إذا L هو أطرح هذه
+
+218
+00:17:09,560 --> 00:17:15,560
+من هذه تنتهي من بعض بيصير عند الـ S ناقص هذه بين الـ
+
+219
+00:17:15,560 --> 00:17:20,600
+L ناقص الـ U و بين الـ U ناقص الـ L عملناها كتير طرح
+
+220
+00:17:20,600 --> 00:17:24,440
+معادلتين من بعض الآن هي الـ U و الـ U و هي الـ L و هي
+
+221
+00:17:24,440 --> 00:17:26,660
+الـ L و ضرب واحدة في ناقص اللي بدنا نطرح هذه في
+
+222
+00:17:26,660 --> 00:17:31,220
+ناقص تنعكس الإشارات تجمح لبعض و تطلع هذه أصغر أو
+
+223
+00:17:31,220 --> 00:17:36,920
+ساوي هذه و أصغر أو ساوي هذه الآن هذه يعني هذه اللي
+
+224
+00:17:36,920 --> 00:17:41,040
+هو بتساوي سالب هذه يعني صارت الـ absolute value
+
+225
+00:17:41,040 --> 00:17:44,260
+أصغر أو يساوي اللي هو المقدار الموجود في هذا
+
+226
+00:17:44,260 --> 00:17:48,220
+والمقدار الموجود في هذا أصغر من وين؟ هنا لقينا أصغر
+
+227
+00:17:48,220 --> 00:17:51,800
+من إيش؟ من إبسلون يعني صار عند الـ S بي و F ناقص من
+
+228
+00:17:51,800 --> 00:17:58,860
+الـ F أصغر من مين؟ من إبسلون وهو المطلوب نجي للنظرية
+
+229
+00:17:58,860 --> 00:17:59,300
+البايتها
+
+230
+00:18:09,320 --> 00:18:15,260
+النظرية تلاتة أربعة .. سبعة أربعة تلاتة نشوف ..
+
+231
+00:18:15,260 --> 00:18:21,920
+نتبه لنص النظرية و من ثم نيجي إلى اللي هو البرهان
+
+232
+00:18:21,920 --> 00:18:27,380
+إذا من الـ .. من النظرية اللي قبلها شوية لكل إبسلون
+
+233
+00:18:27,380 --> 00:18:32,760
+أكبر من صفر لجينا بي إبسلون such that for every بي
+
+234
+00:18:32,760 --> 00:18:38,220
+يحتوي الـ بي إبسلون أو .. و لكل اللي هو أس بي و أف
+
+235
+00:18:38,980 --> 00:18:43,700
+اللي هو corresponding to this partition بديكون S P
+
+236
+00:18:43,700 --> 00:18:49,000
+و F ناقص الـ integration من هنا لـ P للـ F أصغر من P
+
+237
+00:18:49,000 --> 00:18:53,480
+لـ Y وهذا في حال أن F is integrable هذا عنوان اللي
+
+238
+00:18:53,480 --> 00:18:57,220
+هو اللي هي النظرية اللي قبل بشوية لإنه بتحتاجها
+
+239
+00:18:57,220 --> 00:19:03,130
+بعد شوية طيب شوف الآن إيش النظرية هذي بتقول خليت
+
+240
+00:19:03,130 --> 00:19:05,870
+أفضل about the closed interval A وB اللي هي I اللي
+
+241
+00:19:05,870 --> 00:19:09,110
+عند الـ R بـA bounded function Suppose that there
+
+242
+00:19:09,110 --> 00:19:19,130
+exists a number A يحقق الخاصية التالية يحقق
+
+243
+00:19:19,130 --> 00:19:21,390
+الخاصية التالية
+
+244
+00:19:30,260 --> 00:19:34,660
+نفترض أن يوجد A بحيث أنه لكل إبسلون أكبر من 0
+
+245
+00:19:34,660 --> 00:19:40,140
+بنلاقي partition بي إبسلون و FB يحتوى بي إبسلون
+
+246
+00:19:40,140 --> 00:19:45,100
+and F SB is any remand sum for F corresponding to
+
+247
+00:19:45,100 --> 00:19:52,100
+B تئذن الـ S بي و F ناقص A أصغر من مين؟ من إبسلون
+
+248
+00:19:54,770 --> 00:20:02,070
+لو لجينا .. أنا افترض أن احنا بنلاقي الـ Number A
+
+249
+00:20:02,070 --> 00:20:07,190
+هذا معطى أن Number A، إيش بتميز هذا Number A؟ أن
+
+250
+00:20:07,190 --> 00:20:12,190
+لما يكون عندك لكل ي أكبر من صفر لجينا partition بيه
+
+251
+00:20:12,190 --> 00:20:15,290
+يبسلون بحيث أن لكل refinement بيه تحتوى بيه
+
+252
+00:20:15,290 --> 00:20:19,870
+يبسلون و الـ S, B و F is any remand sum
+
+253
+00:20:22,380 --> 00:20:24,760
+يجب أن يكون الـ a أصغر من إبسلون لأس و الـ b و الـ f
+
+254
+00:20:24,760 --> 00:20:31,980
+ناقص من إبسلون لأس و الـ b و الـ f ناقص من إبسلون لأس
+
+255
+00:20:31,980 --> 00:20:32,420
+و الـ b و الـ f ناقص من إبسلون لأس و الـ b و الـ f ناقص
+
+256
+00:20:32,420 --> 00:20:32,460
+من إبسلون لأس و الـ b و الـ f ناقص من إبسلون لأس و الـ
+
+257
+00:20:32,460 --> 00:20:34,200
+و الـ f ناقص من إبسلون لأس و الـ b و الـ f ناقص من
+
+258
+00:20:34,200 --> 00:20:37,640
+إبسلون لأس و الـ b و الـ f ناقص من إبسلون لأس و الـ b
+
+259
+00:20:37,640 --> 00:20:42,860
+ولكل ي أكبر من 0 بلاقي بي إبسلون بحيث أنه لما الـ
+
+260
+00:20:42,860 --> 00:20:46,780
+بي يحتوي الـ بي إبسلون ويكون اللي هو الـ أس بي و الـ
+
+261
+00:20:46,780 --> 00:20:50,800
+أف ناقص ا أصغر من إبسلون بقول then أف اي شمالها
+
+262
+00:20:50,800 --> 00:20:56,020
+must be integrable in I in the sense طبعًا في 716
+
+263
+00:20:56,020 --> 00:21:00,500
+و الـ a هو من هذا الـ integration يعني بقولنا لو
+
+264
+00:21:00,500 --> 00:21:05,360
+لجينا a بتحقق هذا الكلام هتطلعلكم الـ a يا جماعة
+
+265
+00:21:05,360 --> 00:21:10,070
+هذا هي الـ integration هي قيمة الـ integration الـ
+
+266
+00:21:10,070 --> 00:21:12,630
+integration in the sense of الـ definition اللي
+
+267
+00:21:12,630 --> 00:21:17,730
+حكينا عليه طيب شوف الآن الآن
+
+268
+00:21:17,730 --> 00:21:22,410
+بدي أعطيك شغل عامة وبعدها نخصصها على اللي بدنا إياه
+
+269
+00:21:22,410 --> 00:21:26,370
+الآن give an epsilon أكبر من صفر and be any fixed
+
+270
+00:21:26,370 --> 00:21:31,250
+partition of I ماشي الحال الآن خد أي إبسلون وخد بي
+
+271
+00:21:31,250 --> 00:21:37,020
+أي شمله أي fixed partition thenThere exist S1
+
+272
+00:21:37,020 --> 00:21:41,800
+بيوقف و S2 بيوقف بحيث أن S U بيوقف ناقص S بيوقف
+
+273
+00:21:41,800 --> 00:21:46,800
+أصغر من مين؟ من إبسلون و S2 ناقص الـ أصغر من مين؟ من
+
+274
+00:21:46,800 --> 00:21:57,260
+إبسلون شوف هذه خلينا نشوف كيف عملها هيا أخذنا
+
+275
+00:21:57,260 --> 00:22:02,580
+أي إبسلون في الدنيا و بأي partition الآن بقول لي
+
+276
+00:22:04,500 --> 00:22:11,460
+صار عندي S P و F صار عندي آسف البارتيشن اللي هو P
+
+277
+00:22:11,460 --> 00:22:17,260
+موجود والإبسلون موجود بـانفكس بارتيشن مش بارتيشن
+
+278
+00:22:17,260 --> 00:22:21,320
+معين لأ انفكس بارتيشن الكلام ده بظبط عليه بقول
+
+279
+00:22:21,320 --> 00:22:32,410
+بقدر ألاقي remansum S واحد مسمى P و F بحيث أنه SU
+
+280
+00:22:32,410 --> 00:22:44,070
+B و F ناقص S1 B و F يكون أصغر من مين؟ من Y المنطقي
+
+281
+00:22:44,070 --> 00:22:49,980
+الكلام هذا الأبر هو أعطاني الـ Epsilon جالي اثبت
+
+282
+00:22:49,980 --> 00:22:54,980
+أن بتلاقي S1 Remains sum بحيث أن الفرق بينه وبين
+
+283
+00:22:54,980 --> 00:22:57,340
+الـ A بر للـ B والـ F الـ partition هذا partition معين
+
+284
+00:22:57,340 --> 00:23:01,680
+بحكي الكمبيوتر الـ partition بس fixed أن أخذت أي
+
+285
+00:23:01,680 --> 00:23:07,900
+Epsilon أجدرت ألاقي S1 remains sum بحيث أن هذا ناقص
+
+286
+00:23:07,900 --> 00:23:10,980
+هذا يكون أصغر من إبسلون طبعًا هذا الطبيعي هو أكبر
+
+287
+00:23:10,980 --> 00:23:13,760
+من هذا لكن الفرق بدي أخليه أصغر من إبسلون وأنا
+
+288
+00:23:13,760 --> 00:23:20,440
+بقدر أخليه الآن عندي الـ Mk بتساوي اللي هو عبارة عن
+
+289
+00:23:20,440 --> 00:23:27,580
+الـ supremum للـ F of X such that X element XK ناقص
+
+290
+00:23:27,580 --> 00:23:33,490
+واحد و XK مظبوط؟ الآن هذا اللي هو الـ infimum الـ
+
+291
+00:23:33,490 --> 00:23:37,010
+least upper bound الآن الـ least upper bound لو
+
+292
+00:23:37,010 --> 00:23:43,890
+شيلنا منه نقصنا منه أي عدد صغير كتير كتير وليكن
+
+293
+00:23:43,890 --> 00:23:49,850
+الـ y على مين؟ على b ناقص a ببطل upper bound مدام
+
+294
+00:23:49,850 --> 00:23:55,510
+بطل upper bound إذا بقدر أنا أجي xik element الـ xk
+
+295
+00:23:55,510 --> 00:23:57,230
+ناقص واحد و xk
+
+296
+00:24:00,970 --> 00:24:08,050
+تكون مالها؟ أكبر من مين؟ من اللي هو اللي بطل other
+
+297
+00:24:08,050 --> 00:24:13,600
+bound الآن هذا اللي سويته مع الـ mk بقدر أسويه مع
+
+298
+00:24:13,600 --> 00:24:16,280
+الـ m واحد و الـ m اتنين و الـ m تلاتة إيه منهم كلهم
+
+299
+00:24:16,280 --> 00:24:21,500
+إذا صار الـ summation ك من عند واحد لعند n واضربه
+
+300
+00:24:21,500 --> 00:24:25,940
+كمان في مين؟ xk ناقص xk ناقص واحد مافيش مانع لأن
+
+301
+00:24:25,940 --> 00:24:29,740
+أنا دموجة وأنا دموجة بتظهر زي ما هي xk ناقص xk
+
+302
+00:24:29,740 --> 00:24:32,720
+ناقص واحد واخدة الـ summation هذا ك من عند واحد
+
+303
+00:24:32,720 --> 00:24:38,120
+لعند مين؟ لعند n لجيتها الـ xi k تبعت الـ S واحد هذه
+
+304
+00:24:39,040 --> 00:24:47,580
+الآن هذا remain sum بسميه S1 بيوقف على مين؟ اعتمد
+
+305
+00:24:47,580 --> 00:24:52,620
+على الـ exiled case اللي أنا لجيتها هذا إيش بيساوي؟
+
+306
+00:24:52,620 --> 00:24:59,480
+اللي هو summation Mk في Xk ناقص Xk ناقص واحد
+
+307
+00:24:59,480 --> 00:25:08,840
+ناقص الـ summationXY-XK
+
+308
+00:25:08,840 --> 00:25:10,840
+-1
+
+309
+00:25:17,430 --> 00:25:22,310
+هذا summation سامحوني على ضيق اللوح هذا بيصير الـ
+
+310
+00:25:22,310 --> 00:25:24,350
+summation على الأول زاد الـ summation على الثاني هي
+
+311
+00:25:24,350 --> 00:25:27,630
+الـ summation على الأول كم؟ من عند واحد لعند n ناقص
+
+312
+00:25:27,630 --> 00:25:29,910
+الـ summation على الثاني كم؟ من عند واحد لعند n
+
+313
+00:25:29,910 --> 00:25:34,810
+أخذت الـ y بي و ماينوس أيه عن المشترك برا الـ
+
+314
+00:25:34,810 --> 00:25:37,830
+summation دلت الـ summation لهذا أصغر أو يساوي
+
+315
+00:25:37,830 --> 00:25:44,330
+بيقول لهذا مين هو؟ أو أصغر من رمان صم أس بي و أف و
+
+316
+00:25:44,330 --> 00:25:53,580
+سمينها أس واحدة هذا مين هو؟ أكيد كلكم عارفه U P و F و
+
+317
+00:25:53,580 --> 00:26:00,000
+هذا إيش هو؟ ناقص إيش؟ هذاهذا أكيد y على b ناقص a في
+
+318
+00:26:00,000 --> 00:26:03,680
+هذا المقدار هذا المقدار عبارة عن x naught x واحد
+
+319
+00:26:03,680 --> 00:26:08,440
+ناقص x naught زائد x اتنين ناقص x واحد زائد عرفتها
+
+320
+00:26:08,440 --> 00:26:12,240
+إذا آخر واحد xn ناقص xn ناقص واحد يعني بيظل xn
+
+321
+00:26:12,240 --> 00:26:16,280
+ناقص x naught كله بـcancel بعض xn اللي هي الـ a الـ
+
+322
+00:26:16,280 --> 00:26:20,180
+b و الـ x naught هي الـ a يعني هذا عبارة عن b ناقص
+
+323
+00:26:20,180 --> 00:26:27,450
+a أصلاً من S واحد بيقفيعني صار عندي الـ U بي و F و
+
+324
+00:26:27,450 --> 00:26:35,710
+جيب لي هذا هنا ناقص S1 بي و F إشماله أصغر من إبسلون
+
+325
+00:26:35,710 --> 00:26:43,150
+فعلًا أنا أجدرت S واحد من
+
+326
+00:26:43,150 --> 00:27:00,870
+إبسلون
+
+327
+00:27:00,980 --> 00:27:08,700
+إذا لجيت S1 و S2 بحيث تحقق هذه الخاصية انتبهوا
+
+328
+00:27:08,700 --> 00:27:14,340
+الآن هذا الآن كل اللي حكيته هنا برهان للجزئية اللي
+
+329
+00:27:14,340 --> 00:27:20,770
+كاتبها أنا في البرهان ومش مضحك بشكلها الكامل طيب
+
+330
+00:27:20,770 --> 00:27:23,930
+فيكم أصلي على النبي الله صلي على سيدنا محمد نرجع
+
+331
+00:27:23,930 --> 00:27:28,450
+نقول لكم أن لجينا اللي هو الـ S واحد والـ S اتنين
+
+332
+00:27:28,450 --> 00:27:31,870
+اللي بيحققن هذا طبعًا الـ بي كان إيش ماله؟ any fixed
+
+333
+00:27:31,870 --> 00:27:37,410
+partition الآن معطينا
+
+334
+00:27:37,410 --> 00:27:43,050
+هنا أن هذا معطى هذا معطى ديروا بالكم أن there
+
+335
+00:27:43,050 --> 00:27:46,970
+exists بي ابسلون بحيث أن بي بتحتوي بي ابسلون هذا
+
+336
+00:27:46,970 --> 00:27:52,630
+يتحقق الآن اللي اتحقق على الـ P اللي هام، هيتحقق
+
+337
+00:27:52,630 --> 00:27:57,010
+على الـ P الإبسلون اللي هام ماشي الحال، إذا الآن
+
+338
+00:27:57,010 --> 00:28:01,930
+الـ U P إبسلون و الـ F ناقص الـ L P و F اللي هو
+
+339
+00:28:01,930 --> 00:28:10,460
+بيساو U U P و F ناقص L P و F ماشي إيش هذا؟ أخذت الـ
+
+340
+00:28:10,460 --> 00:28:13,740
+بي إبسلون الـ بي اللي هنا هي الـ بي إبسلون عشان أقول
+
+341
+00:28:13,740 --> 00:28:17,440
+أن أنا اللي حققته على اللي هي الـ بي فده حققته على
+
+342
+00:28:17,440 --> 00:28:20,460
+الـ بي إبسلون أن هذا كان unfixed partition أصغر أو
+
+343
+00:28:20,460 --> 00:28:25,700
+يساوي بالـ triangle inequality UBF ناقص SBF زي S1BF
+
+344
+00:28:25,700 --> 00:28:33,440
+ناقص A زي A ناقص S2BF ماذا فعلت؟ بالـ Triangle
+
+345
+00:28:33,440 --> 00:28:37,380
+Inquality أضفت الـ S واحد وطرحت الـ S واحد وضفت
+
+346
+00:28:37,380 --> 00:28:41,260
+الـ A وطرحت الـ A وضفت الـ S اتنين وطرحت الـ S
+
+347
+00:28:41,260 --> 00:28:44,480
+اتنين وعملت الـ Triangle Inquality وطبقت هذه واحدة
+
+348
+00:28:44,480 --> 00:28:48,260
+كام مرتين، تلاتة، أربعة صار عندي أزرع يساوي هذا، زائد
+
+349
+00:28:48,260 --> 00:28:57,300
+هذا، زائد هذا، زائد هذا، ماشي الحل صار
+
+350
+00:28:57,300 --> 00:28:58,480
+عندي يا جماعة الآن
+
+351
+00:29:12,650 --> 00:29:17,970
+صار عندي اللي هو الآن طبعًا الـ a إيش هي الـ a اللي
+
+352
+00:29:17,970 --> 00:29:20,310
+في النظرية دي بقالكم a is the number in the
+
+353
+00:29:20,310 --> 00:29:24,270
+hypothesis of theorem then by the hypothesis of
+
+354
+00:29:24,270 --> 00:29:31,650
+theorem and star الآن من اللي هو المعطى بيعطينا أن
+
+355
+00:29:31,650 --> 00:29:35,630
+الـ s واحد ناقص اللي a أصغر من إبسلون و الـ s اتنين
+
+356
+00:29:35,630 --> 00:29:39,430
+ناقص اللي a أصغر من إبسلون لأن هذا صحيح اللي فوق
+
+357
+00:29:39,430 --> 00:29:39,850
+لأن
+
+358
+00:29:47,480 --> 00:29:53,200
+أذكركم كمان مرة بالمعطى
+
+359
+00:29:53,200 --> 00:30:00,540
+أن هناك مجموعة بي اس ايبسلون هي اللي بتشتغل في اس
+
+360
+00:30:00,540 --> 00:30:05,040
+بي و أف is any remansum corresponding لأف هيكون
+
+361
+00:30:05,040 --> 00:30:07,860
+الفرق بين هذا وهذا أصغر من ايبسلون من ضمن الـ s
+
+362
+00:30:07,860 --> 00:30:11,740
+واحد والـ s اتنين اللي جاتني قبل شوية إذا صار عندي
+
+363
+00:30:11,740 --> 00:30:20,530
+الكل اللي بشتغلوا مشروع بيصير اللي هو الـ UB ناقص
+
+364
+00:30:20,530 --> 00:30:25,730
+الـ S1 أصغر من إبسلون وهذا أصغر من الإبسلون معطى
+
+365
+00:30:25,730 --> 00:30:29,230
+وهذا أصغر من الإبسلون معطى وهذا وهذا أصغر كل
+
+366
+00:30:29,230 --> 00:30:33,370
+واحد من إبسلون من اللي أتبته هنا قبل بشوية إذا صرت
+
+367
+00:30:33,370 --> 00:30:41,070
+اللي هي كلها أصغر من 4 إبسلون ماشي الحالأبسلون was
+
+368
+00:30:41,070 --> 00:
+
+401
+00:33:10,450 --> 00:33:13,730
+إبسيلون لتبعت النظرية و الـ بي إبسيلون تبعت اللي هو
+
+402
+00:33:13,730 --> 00:33:17,690
+الـ integrability و باخدهن اتحادهن بيكون اللي هو الـ
+
+403
+00:33:17,690 --> 00:33:20,990
+بي إبسيلون اللي لجيتها هنا اللي بدى أستخدم إيه لها
+
+404
+00:33:20,990 --> 00:33:24,770
+الجهتين و أي refinement لها بيه لـ بي إبسيلون و
+
+405
+00:33:24,770 --> 00:33:29,270
+لجديدة بيطلع هذا صح وهذا صح اتنين مع بعض دلوقتي
+
+406
+00:33:29,270 --> 00:33:35,030
+بحجل اللي استخدمهم و أقول إذا الـ integration للـ F
+
+407
+00:33:35,030 --> 00:33:37,830
+من A لـ B نقص A أصغر أو يساوي الـ integration للـ F
+
+408
+00:33:37,830 --> 00:33:41,990
+نقص الـ S زاد الـ S نقص الـ A اللي هو ضيف الـ term و
+
+409
+00:33:41,990 --> 00:33:46,750
+طرحت الـ term هذا هي أصغر من إبسيلون وهذا أصغر من
+
+410
+00:33:46,750 --> 00:33:50,210
+إبسيلون إذا صار هذا أصغر من اتنين إبسيلون صار عندي
+
+411
+00:33:50,210 --> 00:33:53,910
+فيه العدد هذا بين اللي هو اتنين إبسيلون وأكبر أو 
+
+412
+00:33:53,910 --> 00:33:56,870
+يساوي صفر وإبسيلون arbitrary إذا هذا المقدر لازم
+
+413
+00:33:56,870 --> 00:34:00,750
+يساوي صفر إذا الـ integration بساوي مين بساوي ليه
+
+414
+00:34:00,750 --> 00:34:03,510
+وهو المطلوب طيب
+
+415
+00:34:10,110 --> 00:34:17,630
+نجي للي هو الـ definition 7.4.4 الـ
+
+416
+00:34:17,630 --> 00:34:23,270
+definition بسيط بده يعرف حاجة اسمها مش أو norm هو
+
+417
+00:34:23,270 --> 00:34:27,330
+أنا بسميها مش عادة الناس بتسميها norm إيش الـ .. الـ
+
+418
+00:34:27,330 --> 00:34:31,550
+.. الـ مش اللي بده يعرفه؟ بقول لو كان عندك
+
+419
+00:34:31,550 --> 00:34:38,910
+partition P بساوي X0 و X1 لعندي XN الآن كل sub
+
+420
+00:34:38,910 --> 00:34:46,970
+interval لها طول الـ norm للـ P أو مش للـ P هو
+
+421
+00:34:46,970 --> 00:34:51,920
+عبارة عن الـ maximum أو الـ Supremum Maximum لأن الـ
+
+422
+00:34:51,920 --> 00:34:53,860
+Supremum حاسب Maximum لإنه الـ finite دول
+
+423
+00:34:53,860 --> 00:34:59,480
+الـ maximum لـ X1 ناقص X0 و X2 ناقص X1 لأطوال
+
+424
+00:34:59,480 --> 00:35:03,140
+الفترات يعني sub interval لعند الـ XN ناقص XN ناقص
+
+425
+00:35:03,140 --> 00:35:07,880
+1 الـ maximum لهذول أطوال الفترات هو اللي بنسميه
+
+426
+00:35:07,880 --> 00:35:14,000
+مين الـ norm للـ P هو الـ norm للـ P الآن يعني يعني لو
+
+427
+00:35:14,000 --> 00:35:20,670
+كان عندك لو كان عندك هي عند الفترة من صفر لعند
+
+428
+00:35:20,670 --> 00:35:31,410
+ثلاثة مثلا هاي عند P بتساوي صفر نص واحد اتنين 
+
+429
+00:35:31,410 --> 00:35:37,970
+ثلاثة مثلا بكون الـ normal P إيش هيساوي إيش الـ
+
+430
+00:35:37,970 --> 00:35:41,510
+normal P أكيد هتقولوا كلكم ثلاثة ناقص اثنين واحد
+
+431
+00:35:41,510 --> 00:35:44,710
+اثنين ناقص واحد واحد واحد ناقص نص نص واحد إذا إيش
+
+432
+00:35:44,710 --> 00:35:49,110
+هيساوي واحد أكبر أكبر طول sub interval واحد لو جينا
+
+433
+00:35:49,110 --> 00:36:00,350
+أخدنا الـ Q مثلا بساوي اللي هو Zero نص واحد واحد و
+
+434
+00:36:00,350 --> 00:36:08,890
+ثلث اثنين اثنين و نص و ثلاثة وقلنا لكم أوجد الـ Q
+
+435
+00:36:09,620 --> 00:36:13,420
+أكيد كلكم حد يقول الـ Q أشمالها اللي هي بين الواحد و
+
+436
+00:36:13,420 --> 00:36:17,420
+ثلث والاثنين هي أكبر أكبر مسافة اللي هي عبارة عن
+
+437
+00:36:17,420 --> 00:36:23,060
+إيه؟ ثلثين لاحظوا أن الـ P مجموعة جزئية من مين؟ من
+
+438
+00:36:23,060 --> 00:36:33,810
+الـ Q لما .. لما الآن الـ Q كبرت Normها هيكون أصغر
+
+439
+00:36:33,810 --> 00:36:37,590
+أو يساوي Norm من الـ P طبيعي لأنه دخلت نقطة جديدة
+
+440
+00:36:37,590 --> 00:36:43,690
+ممكن تقصر المسافة لكن مش ممكن تزيدها طيب وهذه اللي
+
+441
+00:36:43,690 --> 00:36:47,810
+هي بعض الملاحظات اللي هنا على اللي هي الـ norm أو
+
+442
+00:36:47,810 --> 00:36:55,800
+الـ mesh اللي الآن بعض اللي هو الملاحظات اللي هو
+
+443
+00:36:55,800 --> 00:37:00,780
+different partition of I can have the same مش بقول
+
+444
+00:37:00,780 --> 00:37:03,280
+يعني ممكن different partition للـ .. للـ .. للـ ..
+
+445
+00:37:03,280 --> 00:37:06,140
+للـ interval اللي ده same مش اه بقدر هاد أنا أجيب لك
+
+446
+00:37:06,140 --> 00:37:12,200
+كمان واحد زي هذول خد كمان P' بساوي صفر و نص
+
+447
+00:37:12,200 --> 00:37:20,030
+و واحد و اثنين و اثنين و نص و ثلاثة هذا الـ P'
+
+448
+00:37:20,270 --> 00:37:25,170
+يختلف عن الـ P لكن نورمه التاني الواحد بيساوي واحد
+
+449
+00:37:25,170 --> 00:37:30,530
+يعني الـ different partitions ممكن يكون لها نفس
+
+450
+00:37:30,530 --> 00:37:34,730
+norm و هاي مثال و كثير فيه زي ذلك لأن لو كانت P
+
+451
+00:37:34,730 --> 00:37:39,730
+subset من Q Norm لـ Q أظهر يساوي Norm لمن؟ لـ P و
+
+452
+00:37:39,730 --> 00:37:43,720
+اتبعت السهل هذا خد الـ P subset إذا هيكون عنده اللي
+
+453
+00:37:43,720 --> 00:37:47,080
+هو عدد النقاط اللي في Q أكثر من عدد النقاط اللي في
+
+454
+00:37:47,080 --> 00:37:53,780
+P هكون الـ maximum على الأولى اللي هو أكبر أو يساوي
+
+455
+00:37:53,780 --> 00:37:56,720
+الـ maximum على مين على الثاني لإنه اللي صار فيها
+
+456
+00:37:56,720 --> 00:38:02,070
+زيادات صار فيه إمكانية إنها تقصر الفترات طيب لكن لو
+
+457
+00:38:02,070 --> 00:38:05,330
+كان الـ normal الـ Q أصغر شوي الـ normal الـ P مش شرط
+
+458
+00:38:05,330 --> 00:38:10,110
+أن يكون P subset من مين؟ من الـ Q أو كثير بتلاقي
+
+459
+00:38:10,110 --> 00:38:16,910
+أمثلة زي هي الـ normal
+
+460
+00:38:16,910 --> 00:38:25,160
+P هذا إيش بيساوي؟ Normal P بيساوي واحد الآن عندي ..
+
+461
+00:38:25,160 --> 00:38:32,320
+مش شرط .. يعني هديلك اللي هو Normal Q أصغر أوي
+
+462
+00:38:32,320 --> 00:38:40,040
+يساوي Normal P لكن الـ P اللي هو ليس شرطا أنها تكون
+
+463
+00:38:40,040 --> 00:38:47,500
+subset من مين؟ من إيش؟ من الـ Q Normal P واحد خد
+
+464
+00:38:47,500 --> 00:38:48,060
+الـ Q
+
+465
+00:38:53,000 --> 00:39:03,100
+بساو اللي هو صفر و واحد و اثنين و اثنين و نص و
+
+466
+00:39:03,100 --> 00:39:07,180
+ثلاثة Normal
+
+467
+00:39:07,180 --> 00:39:12,100
+Q بديه ال
+
+468
+00:39:12,100 --> 00:39:19,920
+gate
+
+469
+00:39:19,920 --> 00:39:22,280
+واحد و واحد و نص
+
+470
+00:39:25,990 --> 00:39:34,190
+وهي نص كمان صفر و نص Normal Q بساوة نص مظبوط
+
+471
+00:39:34,190 --> 00:39:45,650
+Normal Q بساوة نص لكن الـ P الـ P خلي هذا ثلث ماشي
+
+472
+00:39:45,650 --> 00:39:51,220
+خلي هذا ثلث صارت الـ P مش subset من مين من الـ Q
+
+473
+00:39:51,220 --> 00:39:54,420
+ممكن تلاقي أفضل كثير هذه الـ P أكيد مش subset من
+
+474
+00:39:54,420 --> 00:40:00,460
+الـ Q Normal P بيبقى واحد صح؟ Normal P واحد و
+
+475
+00:40:00,460 --> 00:40:06,160
+Normal Q إيش بيساوي نص لأن الفرق بينهم أنصاف هذه
+
+476
+00:40:06,160 --> 00:40:12,340
+عندك Normal Q أصغر أو يساوي Normal P لأن هذا نص
+
+477
+00:40:12,340 --> 00:40:14,420
+وهذا واحد but
+
+478
+00:40:17,620 --> 00:40:22,380
+واضح الـ P هي صفر و ثلث و واحد و اثنين و ثلاثة ليس
+
+479
+00:40:22,380 --> 00:40:26,100
+subset من الـ Q صفر و نص و واحد و نص و .. و .. و
+
+480
+00:40:26,100 --> 00:40:27,400
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+481
+00:40:27,400 --> 00:40:28,940
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+482
+00:40:28,940 --> 00:40:29,640
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+483
+00:40:29,640 --> 00:40:30,360
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+484
+00:40:30,360 --> 00:40:30,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+485
+00:40:30,380 --> 00:40:34,960
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+486
+00:40:34,960 --> 00:40:36,340
+.. و .. و .. و .. و .. و .. و .. 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/QjdD_6J2asI_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/QjdD_6J2asI_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..47ba7f5e52432c24d8660d33bdf26e00029c3344
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/QjdD_6J2asI_raw.srt
@@ -0,0 +1,1960 @@
+1
+00:00:04,960 --> 00:00:12,580
+بسم الله الرحمن الرحيم هذه المحاضرة رقم 16 في مساق
+
+2
+00:00:12,580 --> 00:00:18,080
+تحليل حقيقي 2 أو تحليل رياضي 2 لطلبة وطالبات كلية
+
+3
+00:00:18,080 --> 00:00:22,820
+العلوم بالجامعة الإسلامية بغزةكنا سميينها في
+
+4
+00:00:22,820 --> 00:00:29,160
+المحاضرة 15 إنها جزئين المحاضرة الجزء الأول ذكرناه
+
+5
+00:00:29,160 --> 00:00:31,820
+في المحاضرة 15 وقلنا الجزء الثاني هنذكره في
+
+6
+00:00:31,820 --> 00:00:35,780
+المحاضرة اللي بعدها كجزء ثاني من 15 بس الآن لطول
+
+7
+00:00:35,780 --> 00:00:40,440
+فترة اللي هي المحاضرة السابقةهنسمي هذا الجزء الآن
+
+8
+00:00:40,440 --> 00:00:43,060
+اللي هو كان اسمه الجزء الثاني المحاضرة 15 لأ بدي
+
+9
+00:00:43,060 --> 00:00:48,300
+أسميه هو المحاضرة رقم 16 وهو اللي هو بداية سيكشن
+
+10
+00:00:48,300 --> 00:00:52,020
+جديد اللي هو سبعة أربعة اللي هو the integration as
+
+11
+00:00:52,020 --> 00:00:56,260
+a limit اللي هو بدنا نصل ال integration عن طريق
+
+12
+00:00:56,260 --> 00:01:01,880
+اللي هو the limit of Riemann sumبس شغلة سريعة عشان
+
+13
+00:01:01,880 --> 00:01:05,820
+نربط بالسابق باللاحق بالحاضر عشان نفهم إيش اللي
+
+14
+00:01:05,820 --> 00:01:10,780
+بنحكي فيه إحنا إذا بتتظهرتتذكر واخدنا على الفترة A
+
+15
+00:01:10,780 --> 00:01:17,200
+وB وعملنا لها تجزئة P بساوية X Note X واحد لعند ال
+
+16
+00:01:17,200 --> 00:01:25,000
+XN وعرفنا اللي هو ال Upper sum اللي هو Upper اللي
+
+17
+00:01:25,000 --> 00:01:31,900
+هي L وP L لو function عندي F فعندي F of X على
+
+18
+00:01:31,900 --> 00:01:39,160
+الفترة من A لعند B اللي هو ال UpperF of B وقلنا هو
+
+19
+00:01:39,160 --> 00:01:44,780
+عبارة عن summation لل M K في X K minus X K minus
+
+20
+00:01:44,780 --> 00:01:51,060
+واحد زي ما انتم الذاكرين K من عند واحد لعند N هذا
+
+21
+00:01:51,060 --> 00:01:54,900
+اللي هو اللي سمناه ال other sum ال M K طبعا هي ال
+
+22
+00:01:54,900 --> 00:01:57,600
+.. زي ما انتم عارفين ال supremum على الفترة .. ال
+
+23
+00:01:57,600 --> 00:02:01,040
+supremum لدال ال F ال supremum لدال ال F على
+
+24
+00:02:01,040 --> 00:02:06,340
+الفترة اللي أماميوعندي ال L F of P أو P و F زي ما
+
+25
+00:02:06,340 --> 00:02:10,300
+احنا بنكتبها عبارة عن summation ال M K في X K
+
+26
+00:02:10,300 --> 00:02:14,680
+minus X K minus واحد K من عند واحد لعندنا وقلنا لو
+
+27
+00:02:14,680 --> 00:02:19,380
+اتينا ال other sum أخدناله اللي هو ال infimum ال
+
+28
+00:02:19,380 --> 00:02:24,370
+other sum وقلنا أخدناله ال infimumبنسميه U of F
+
+29
+00:02:24,370 --> 00:02:29,070
+اللي هو عبارة عن الـ infimum لـ U of F و B such
+
+30
+00:02:29,070 --> 00:02:32,370
+that B element in the set of all partition B of I
+
+31
+00:02:32,370 --> 00:02:36,630
+و أخدنا ال L of F و قلنا هو عبارة عن ال supremum
+
+32
+00:02:36,630 --> 00:02:41,300
+لـ L of F و Bhighest B element in the set of all
+
+33
+00:02:41,300 --> 00:02:46,780
+partitions B of I قلنا إن F هتكون integrable إذا
+
+34
+00:02:46,780 --> 00:02:50,460
+اللي هو ال lower integral بساومين ال upper
+
+35
+00:02:50,460 --> 00:02:55,580
+integral هذا المدخل اللي دخلنا فيه لإثبات أو
+
+36
+00:02:55,580 --> 00:03:00,220
+لتعريف إن F is integrable مدخل اللي هو ال upper
+
+37
+00:03:00,220 --> 00:03:03,650
+sum والlower sumأو الـ Upper Integral والـ Lower
+
+38
+00:03:03,650 --> 00:03:07,470
+Integral يوم ما يتساون بنسميه الـ Integral اللي هو
+
+39
+00:03:07,470 --> 00:03:10,670
+عبارة عن قيمة الـ Integration للـ F على الفترة A
+
+40
+00:03:10,670 --> 00:03:17,640
+وBالان ايش اللي هنسويه اللي هو الجزء الثاني من
+
+41
+00:03:17,640 --> 00:03:21,300
+المحاضرة
+
+42
+00:03:21,300 --> 00:03:24,300
+او اللي احنا سمينا المحاضرة 16 اللي هو في ال
+
+43
+00:03:24,300 --> 00:03:29,840
+section 7-4 هندخل لل integration بداخلة أخرى عن
+
+44
+00:03:29,840 --> 00:03:35,280
+طريق حاجة اسمها ال remansum ومن ثم نقول نثبت في
+
+45
+00:03:35,280 --> 00:03:40,000
+النهاية انه limit ال remansum لما يكون موجودبكون
+
+46
+00:03:40,000 --> 00:03:45,480
+هو الـ Integration وهذا مكافئ أو يحدث if and only
+
+47
+00:03:45,480 --> 00:03:50,580
+if إذا حدثت اللي هو معنى التكامل بصيغة ال Upper
+
+48
+00:03:50,580 --> 00:03:54,420
+Sum والLower Sum يعني الآن .. الآن في عندنا مدخلين
+
+49
+00:03:54,420 --> 00:03:57,660
+للدخول للتكامل إما عن طريق ال Upper Sum والLower
+
+50
+00:03:57,660 --> 00:04:00,560
+Sum أو ال Upper Integral والLower Integral أو عن
+
+51
+00:04:00,560 --> 00:04:04,200
+طريق اللي هو ال Remain Sum أو اللي هو ال limit
+
+52
+00:04:04,200 --> 00:04:09,000
+للRemain Sumوالتنين لما نكون كل واحد ذاك ال ..
+
+53
+00:04:09,000 --> 00:04:12,740
+الملهمته موجود و هذا ال upper بساوي ال lower بكون
+
+54
+00:04:12,740 --> 00:04:19,500
+ال R equivalent إيه الآن؟ إيش ال .. ال roman sum؟
+
+55
+00:04:19,500 --> 00:04:24,200
+احنا برضه التجزية جاءت تجزية هاي P بساوي X note و
+
+56
+00:04:24,200 --> 00:04:29,280
+X واحد لعند ال XN هاي التجزية هاي X note و هاي X
+
+57
+00:04:29,280 --> 00:04:32,700
+واحد وهي الفترة الموضاجية XK minus واحد لعند ال XK
+
+58
+00:04:33,370 --> 00:04:40,570
+الان الـ Riemann Sum بنختار في الـ Intervals نقطة
+
+59
+00:04:40,570 --> 00:04:45,830
+عشوائية لا تقول لي لا أكبر واحدة ولا أصغر واحدة
+
+60
+00:04:45,830 --> 00:04:49,150
+ولا كذا لأ نقطة عشوائية وبنسميها مثلا لو قلت لك
+
+61
+00:04:49,150 --> 00:04:54,970
+الهن بسميها XIK الان XIK هذه صارت element في الـ
+
+62
+00:04:54,970 --> 00:05:02,000
+XK-1 والـ xk الآن بدي بضرب ال x اللي اللي هي طول
+
+63
+00:05:02,000 --> 00:05:05,140
+الفترة زي ما ضربتها زي ما وجدت ال sum اللي هو ال
+
+64
+00:05:05,140 --> 00:05:09,020
+upper sum هنا بوجد هنا بس بستبدل ال mk و ال mk
+
+65
+00:05:09,020 --> 00:05:13,380
+small و upper بستبدلها بقيمة ال function عند هذه
+
+66
+00:05:13,380 --> 00:05:18,270
+النقطة العشوائية اللي اخترتهااللي هي بيسميها xik
+
+67
+00:05:18,270 --> 00:05:24,190
+فبصير xk minus xk minus 1 في ال F عند xik و باخده
+
+68
+00:05:24,190 --> 00:05:28,290
+ال summation k من عند 1 لعند n و هذا اللي بيسميه
+
+69
+00:05:28,290 --> 00:05:36,030
+الرمان صم S of B أو مين و F S of B أو F و هذا اللي
+
+70
+00:05:36,030 --> 00:05:43,470
+بيسميه الرمان صم في النهاية هنصلكم أنه لو كان ال
+
+71
+00:05:43,470 --> 00:05:49,710
+limitللـ S, P و F as الـ N goes to infinity يعني
+
+72
+00:05:49,710 --> 00:05:53,630
+الـ N اللي هي زغرنا الفترة صارتها الجدّة و زغرنا
+
+73
+00:05:53,630 --> 00:05:57,090
+صارتها الفترة .. لما تكون N تروح إلى نهاية معناته
+
+74
+00:05:57,090 --> 00:05:59,590
+أن صارت اللي .. اللي هي و كأنه الفترات عبارة عن
+
+75
+00:05:59,590 --> 00:06:05,130
+خطوط جنب بعض هذه الخطوط بتنطبق الـ XK و XK-1 و XIK
+
+76
+00:06:05,130 --> 00:06:10,950
+على بعض و كأنه بتسير كل خط جنب خط جنب خط إلى قيمة
+
+77
+00:06:10,950 --> 00:06:14,640
+اللي هو مجموع هذه الخطوط زي ما حكينا قبل هيكاللي
+
+78
+00:06:14,640 --> 00:06:18,820
+بتعمل المساحة تحت المنحنى في حالة الموجب ومن ثم هي
+
+79
+00:06:18,820 --> 00:06:21,740
+هتعمل لي قيمة ال integration يعني لما يكون ال
+
+80
+00:06:21,740 --> 00:06:24,420
+limit لـRiemann sum as N goes to infinity exist
+
+81
+00:06:24,420 --> 00:06:29,380
+وسوّت لي رقم اسمه إيه هذا الرقم هو عبارة عن ال
+
+82
+00:06:29,380 --> 00:06:33,900
+integration إيه لـ BF of X DX على هذه الفترة هذا
+
+83
+00:06:33,900 --> 00:06:38,400
+اللي في النهاية بدنا نصل إليه أو نوصلكم إنه في
+
+84
+00:06:38,400 --> 00:06:42,680
+حالة وجود هذا ال limit هيكون هو ال integration
+
+85
+00:06:42,680 --> 00:06:47,040
+اللي بمفهوم ال upper sum والlower sum يعني
+
+86
+00:06:47,040 --> 00:06:53,160
+الدخلتين بقد إلى نفس القيمة أو إلى نفس النتيجة
+
+87
+00:06:54,950 --> 00:06:59,130
+يعني في بعض الكتب بتعرفلك أصلا اللي هو ال
+
+88
+00:06:59,130 --> 00:07:04,430
+integration as a limit of remansum في كتب و هذا
+
+89
+00:07:04,430 --> 00:07:09,650
+غالبا في ال calculus بيشتغلوه و في كتب بتعرفه عن
+
+90
+00:07:09,650 --> 00:07:12,230
+طريق ال other sum و لا other sum في ال calculus
+
+91
+00:07:12,230 --> 00:07:15,930
+لأنه ملزمش ل Supremum و ل Infimum لإن الطالب مفهم
+
+92
+00:07:15,930 --> 00:07:20,370
+ال Supremum ومفهم ال Infimum عنده اللي هو ناضج
+
+93
+00:07:24,160 --> 00:07:29,980
+بعد هذه المقدمة عن الموضوع خلّينا ندخل إلىه بشكل
+
+94
+00:07:29,980 --> 00:07:34,700
+تفصيلي The integration as a limit لت I اللي هي ال
+
+95
+00:07:34,700 --> 00:07:36,820
+interval اللي انا بده اشتغل عليها و ال function
+
+96
+00:07:36,820 --> 00:07:40,260
+أفنان I لR عبارة عن bounded function و خدنا ال B
+
+97
+00:07:40,260 --> 00:07:44,440
+هو ال partition اللي أخدناه و زي ما قلنا إذا أخدنا
+
+98
+00:07:44,440 --> 00:07:50,470
+XI 1 و XI 2 و XI N عبارة عن arbitrary numbersاللي
+
+99
+00:07:50,470 --> 00:07:53,330
+هو الـ x i 1 في ال .. في ال .. في ال interval x
+
+100
+00:07:53,330 --> 00:07:57,270
+not x 1 x i 2 في ال interval اللي .. اللي بعدها
+
+101
+00:07:57,270 --> 00:08:00,810
+ايه اخره يعني ال x i k هي في ال .. الفترة النموذج
+
+102
+00:08:00,810 --> 00:08:06,130
+الذادية x k minus 1 x k ل k من 1 لعند مين لعند ان
+
+103
+00:08:06,130 --> 00:08:08,570
+ال sum اللي قولنا عنه اللي هو عبارة عن ال
+
+104
+00:08:08,570 --> 00:08:12,530
+summation لل F of x i k مضروب في x k ناقص x k
+
+105
+00:08:12,530 --> 00:08:17,110
+minus 1 هذا ال summation من 1 لعند ان هم هو برمزه
+
+106
+00:08:17,110 --> 00:08:21,030
+للرمز SPUF اللي هو بسميهاللي هو عبارة عن the
+
+107
+00:08:21,030 --> 00:08:25,810
+remand sum الريمان صم الريمان صم الريمان صم
+
+108
+00:08:25,810 --> 00:08:30,150
+الريمان صم الريمان صم الريمان صم الريمان صم
+
+109
+00:08:30,150 --> 00:08:31,170
+الريمان صم الريمان صم الريمان صم الريمان صم
+
+110
+00:08:31,170 --> 00:08:32,670
+الريمان صم الريمان صم الريمان صم الريمان صم
+
+111
+00:08:32,670 --> 00:08:33,050
+الريمان صم الريمان صم الريمان صم الريمان صم
+
+112
+00:08:33,050 --> 00:08:33,610
+الريمان صم الريمان صم الريمان صم الريمان صم
+
+113
+00:08:33,610 --> 00:08:34,330
+الريمان صم الريمان صم الريمان صم الريمان صم
+
+114
+00:08:34,330 --> 00:08:36,290
+الريمان صم الريمان صم الريمان صم الريمان صم
+
+115
+00:08:36,290 --> 00:08:43,690
+الريمان صم الريمان صم الريمان صم الريمان صم الري
+
+116
+00:08:43,710 --> 00:08:48,550
+لأن الـ F of XI K اللي هي النقطة العشوائية هذه
+
+117
+00:08:48,550 --> 00:08:53,190
+أكيد أصغر أو يساوي الـ inf mum على كل النقاط اللي
+
+118
+00:08:53,190 --> 00:08:57,710
+في ال sub interval هذه و أكبر أو يساوي و أصغر أو
+
+119
+00:08:57,710 --> 00:09:01,730
+يساوي من ال super mum يعني F of XI K هذه نقطة
+
+120
+00:09:01,730 --> 00:09:05,490
+عشوائية أكبر أو يساوي أصغر واحدة أو أصغر اللي هو
+
+121
+00:09:05,490 --> 00:09:10,070
+ال inf mum و أصغر أو يساوي اللي هي من ال super mum
+
+122
+00:09:10,610 --> 00:09:13,890
+هذا لكل مين لكل sub interval يعني هي sub interval
+
+123
+00:09:13,890 --> 00:09:20,810
+xk minus واحد وهي xk وهي xik أكيد اللي هو f of xik
+
+124
+00:09:20,810 --> 00:09:29,210
+هنا أصغر أو يساوي أكبر واحدة وأصغر أو يساوي أكبر
+
+125
+00:09:29,210 --> 00:09:34,020
+واحدة وأكبر أو يساوي أصغر واحدةاللي هي الـ MK
+
+126
+00:09:34,020 --> 00:09:38,580
+فبكون
+
+127
+00:09:38,580 --> 00:09:43,360
+دايما هي هذه دي و هذه دي و هذه دي خدوا الآن ضربولي
+
+128
+00:09:43,360 --> 00:09:47,680
+هنا وهنا وهنا ضربولي في XK-XK-Y حط طول الفترة
+
+129
+00:09:48,770 --> 00:09:52,070
+واخدوا الـ summation من عند واحد لعند أن بيصير هذا
+
+130
+00:09:52,070 --> 00:09:55,670
+بيمثل لي اللي هو الأبر صم وهذا بيمثل لي الأور صم
+
+131
+00:09:55,670 --> 00:10:00,070
+وهذا بيمثل لي مين؟ الريمان صم إذن الريمان صم دائما
+
+132
+00:10:00,070 --> 00:10:05,250
+بين الأبر وبين الأور لأوله أصلا نحن ليش بيسميهم
+
+133
+00:10:05,250 --> 00:10:10,210
+أبر و لاور عسى إن كل اللي غيرهم الأبر كل اللي
+
+134
+00:10:10,210 --> 00:10:14,350
+غيرهم أصغر يسويهم وكل اللي غيرهم الأور أكبر منهم
+
+135
+00:10:14,350 --> 00:10:21,060
+أو يسويهم التسمية جزء من المفهومأو تستقر من
+
+136
+00:10:21,060 --> 00:10:27,020
+المفهوم يعني بنسميها مش معقول هي أبر ونسميها لابر
+
+137
+00:10:27,020 --> 00:10:32,520
+أو أبر ونسميها اسم يخالف الأبر فأكيد اخترنا اسم
+
+138
+00:10:32,520 --> 00:10:37,660
+الأبر عشان هي فعلا أبرطيب نجي الى النظرية الأولى
+
+139
+00:10:37,660 --> 00:10:43,520
+let a و b عبارة عن فترة مغلقة و bounded و let f من
+
+140
+00:10:43,520 --> 00:10:46,960
+I ل R be integrable on I نفترض ان اف اذا احنا لما
+
+141
+00:10:46,960 --> 00:10:49,880
+نحكي integrable معناته integrable in the sense of
+
+142
+00:10:49,880 --> 00:10:53,340
+definition 716 اللي هو اللي بعتمد على ال U of F
+
+143
+00:10:53,340 --> 00:10:58,080
+بتساوي ال O of Fالان let F من I ل R بيه Integrable
+
+144
+00:10:58,080 --> 00:11:01,940
+I زي ما بأكد in the sense of definition 716 اللي
+
+145
+00:11:01,940 --> 00:11:09,070
+حكيته إذا لكل ي أكبر من سفر is givenThere exists a
+
+146
+00:11:09,070 --> 00:11:11,470
+partition by بي إبسلون such that if بي is any
+
+147
+00:11:11,470 --> 00:11:15,610
+partition that is a refinement of بي إبسلون and if
+
+148
+00:11:15,610 --> 00:11:19,930
+أس بي و أف is any remaining sum for أف then هيكون
+
+149
+00:11:19,930 --> 00:11:24,110
+عنده اللي هو أس بي و أف ناقص ال integration لأف
+
+150
+00:11:24,110 --> 00:11:29,210
+أصغر من مين من إبسلون بقول باختصار يعني لو فرضنا
+
+151
+00:11:29,210 --> 00:11:34,490
+أف integrable اللي هنخد لأي إبسلون في الدنيا بقدر
+
+152
+00:11:34,490 --> 00:11:40,550
+أجيلك partition بي إبسلونبحيث انه اي بي بارتيشن
+
+153
+00:11:40,550 --> 00:11:45,390
+refinement لبيبسلون refinement معناته البي بحتوى
+
+154
+00:11:45,390 --> 00:11:50,910
+مين البيبسلون تحسينله هيكون عندي ال remain sum S
+
+155
+00:11:50,910 --> 00:11:55,150
+بي و F هذا ناقص integration للف أصغر من مين؟ من
+
+156
+00:11:55,150 --> 00:12:04,090
+ابسلون يعني الآن F is integrable يا جماعةالأن for
+
+157
+00:12:04,090 --> 00:12:08,070
+every epsilon أكبر من 0 there exist بي ابسلون such
+
+158
+00:12:08,070 --> 00:12:12,490
+that for every بي تحتوي بي ابسلون اللي هو هذا اللي
+
+159
+00:12:12,490 --> 00:12:20,110
+جيته and for every man some أس للبي والأفهيكون
+
+160
+00:12:20,110 --> 00:12:25,410
+عندي الـ absolute value بين الـ S بي و F ناقص اللي
+
+161
+00:12:25,410 --> 00:12:31,890
+هو ال integration من A ل B لل F أصغر من 100 منافسة
+
+162
+00:12:31,890 --> 00:12:35,790
+يعني جاعت بقول أخي يعني أصلا يعنيأني هذا في الآخر
+
+163
+00:12:35,790 --> 00:12:39,910
+في الآخر في الآخر في الآخر هتلاقي الريمان صم اللي
+
+164
+00:12:39,910 --> 00:12:43,890
+هو يكون بينه و بين ال F إيش ما له أصغر من إبسلون
+
+165
+00:12:43,890 --> 00:12:47,870
+لما تكون ال إيش ال F اللي هي Integral يعني بلاقي
+
+166
+00:12:47,870 --> 00:12:52,890
+بي إبسلون بضمنك هذا الكلام بضمنك مش ال بي إبسلون
+
+167
+00:12:52,890 --> 00:12:58,550
+بس و كل ال refinement يعني كل ال refinement يعني
+
+168
+00:12:58,550 --> 00:13:05,310
+كل ما نزيد عدد عناصر ال partitionيعني كل ما نكبر
+
+169
+00:13:05,310 --> 00:13:09,810
+الان يعني و كأنه بقودنا لللي بدنا نحكيه قدام انه
+
+170
+00:13:09,810 --> 00:13:15,770
+لما الان تروح لمالة نهاية limit ال remand sum إذا
+
+171
+00:13:15,770 --> 00:13:18,950
+كانت موجودة integrable هيساوي ال integration
+
+172
+00:13:18,950 --> 00:13:23,870
+والعكس هنلاقيه صحيح برضه هيسيه definition طيب في
+
+173
+00:13:23,870 --> 00:13:26,870
+بعض الكتب زي ما قلنا او لو بدنا نسميه definition
+
+174
+00:13:26,870 --> 00:13:29,950
+بنقول equivalent ل definition الأولاني و طبعا
+
+175
+00:13:29,950 --> 00:13:37,040
+هنلاقينا حالة بتتواصلنرجع لنظريتنا نثبتهالأن if F
+
+176
+00:13:37,040 --> 00:13:40,820
+is integrable and Y أكبر من 0 then by Riemann
+
+177
+00:13:40,820 --> 00:13:45,200
+criterion there exists B Epsilon of I such that U
+
+178
+00:13:45,200 --> 00:13:48,980
+ناقص L أصغر من مين من Y هذا أحفظناها زي اسمنا مدام
+
+179
+00:13:48,980 --> 00:13:53,000
+F is integrable and then by Riemann criterion طبعا
+
+180
+00:13:53,000 --> 00:13:55,620
+أكيد هتيجي ع بالكم لأنه مدام قال there exists B
+
+181
+00:13:55,620 --> 00:13:59,120
+Epsilon هكيد أقول أكيد احنا هنستخدم مين اللي بتجيب
+
+182
+00:13:59,120 --> 00:14:01,720
+ال B Epsilon مين اللي كانت تجيب ال B Epsilon اللي
+
+183
+00:14:01,720 --> 00:14:04,870
+هي ال Riemann criterionإذا كان الـ F مستعمل و الـ
+
+184
+00:14:04,870 --> 00:14:07,790
+F أكبر من 0 فبالتالي باستخدام الـ Riemann's
+
+185
+00:14:07,790 --> 00:14:11,850
+criterion يوجد مجموعة بي أبسلون of I مثلًا الـ U
+
+186
+00:14:11,850 --> 00:14:14,950
+بي أبسلون و الـ F نقص الـ بي أبسلون و الـ F أشماله
+
+187
+00:14:14,950 --> 00:14:23,290
+أصغر من مين من أبسلون الآن اخدوا أي refinement بيه
+
+188
+00:14:23,290 --> 00:14:28,590
+بحتوى مين الـ بي أبسلون الـ refinement التحسين
+
+189
+00:14:28,590 --> 00:14:37,240
+اللاور بزيدوال أبر بجل اكيد عارفين ان هذا الكلام
+
+190
+00:14:37,240 --> 00:14:41,220
+اذا صار عند مدام اللي هو ال refinement ال lower
+
+191
+00:14:41,220 --> 00:14:46,940
+بزيد اكيد ال LBUF أصغر بيساوي LBUF اللي هو التحسين
+
+192
+00:14:46,940 --> 00:14:54,760
+و LBUF دايما اصغر بيساوي UBF و ال UBF التحسين بجل
+
+193
+00:14:54,760 --> 00:14:59,800
+عن ال UBEF اذا هذه ال inequality محفوظة عندنا ماشي
+
+194
+00:14:59,800 --> 00:15:08,270
+الحال الآنمن هنا ذولة صارت عندي high ال L بي
+
+195
+00:15:08,270 --> 00:15:15,090
+إبسلون و أف high ال L بي و أف لأنه أكبر high ال U
+
+196
+00:15:15,090 --> 00:15:22,630
+بي و أف high ال U بي إبسلون و أف أكيد أكيد أكيد
+
+197
+00:15:22,630 --> 00:15:27,570
+المسافة بين هذه و هذهأصغر المسافة بين هذه و هذه
+
+198
+00:15:27,570 --> 00:15:33,170
+يعني بمعنى آخر ال UBF ناقص ال LBF المسافة بينهم
+
+199
+00:15:33,170 --> 00:15:39,130
+أصغر أو يساوي ال UBY و ال LBY من هذه و هذه من اللي
+
+200
+00:15:39,130 --> 00:15:43,990
+فوق أشمالها أصغر من Y إذا صارت عندي ال U صار عندي
+
+201
+00:15:43,990 --> 00:15:50,210
+ال UBF ناقص ال BF أصغر من مين؟ من Y صار عندي ال U
+
+202
+00:15:51,040 --> 00:16:01,320
+بأف ناقص ال بأف أصغر من إبسلون لكل بي يحتوي من ال
+
+203
+00:16:01,320 --> 00:16:04,120
+بي إبسلون لكل refinement لمين للبي إبسلون اللي
+
+204
+00:16:04,120 --> 00:16:08,280
+لجناه بوسط الريمان ال criterion خلّينا نكمل يا
+
+205
+00:16:08,280 --> 00:16:19,320
+جماعة الآن لكن احنا بنعرف ان اي ريمان صم لبي معينة
+
+206
+00:16:19,510 --> 00:16:24,830
+هيكون بين الـ L و بين الـ P أكيد صح ولا لأ؟ طبعا
+
+207
+00:16:24,830 --> 00:16:28,910
+الـ S بين الـ P و الـ F اللي قلناها قبل شوية بين
+
+208
+00:16:28,910 --> 00:16:34,570
+الـ L P و F و الـ U P و Fطيب خد ال integration ال
+
+209
+00:16:34,570 --> 00:16:36,970
+integration ما هو ال F is integrable مزام
+
+210
+00:16:36,970 --> 00:16:41,330
+Integrable إذا هي بتساوي زي ما عملناها كتير بساوي
+
+211
+00:16:41,330 --> 00:16:45,430
+ال U في F و بتساوي ال L في F ال L في F هي ال
+
+212
+00:16:45,430 --> 00:16:48,070
+supremum لهذه الأشياء إذا أكيد هذا أكبر يساوي ال
+
+213
+00:16:48,070 --> 00:16:52,410
+lower اللي هي sum هذاوالـ Integration بيساوي الـ U
+
+214
+00:16:52,410 --> 00:16:56,370
+و F و الـ U و F هو عبارة عن الـ M في مهم للـ U و B
+
+215
+00:16:56,370 --> 00:16:59,230
+و F إذا نحن نكون أصغر أو يساوي إذا فعلا ال
+
+216
+00:16:59,230 --> 00:17:04,610
+integration بين ال lower sum و بين ال upper sum و
+
+217
+00:17:04,610 --> 00:17:09,560
+هذه الفكرة عملناها كتير قبل هاتإذا L هو أطرح هذه
+
+218
+00:17:09,560 --> 00:17:15,560
+من هذه تنتهي من بعض بيصير عند ال S ناقص هذه بين ال
+
+219
+00:17:15,560 --> 00:17:20,600
+L ناقص ال U و بين ال U ناقص ال L عملناها كتير طرح
+
+220
+00:17:20,600 --> 00:17:24,440
+معادلتين من بعض الآن هي ال U و ال U و هي ال L و هي
+
+221
+00:17:24,440 --> 00:17:26,660
+ال L و ضرب واحدة في ناقص اللي بدنا نطرح هذه في
+
+222
+00:17:26,660 --> 00:17:31,220
+ناقص تنعكس الأشارات تجمح لبعض و تطلع هذه أصغر أو
+
+223
+00:17:31,220 --> 00:17:36,920
+ساوي هذه و أصغر أو ساوي هذهالأن هذه يعني هذه اللي
+
+224
+00:17:36,920 --> 00:17:41,040
+هو بتساوي سالب هذه يعني صارت ال absolute value
+
+225
+00:17:41,040 --> 00:17:44,260
+أصغر أو يساوي اللي هو المقدار الموجود في هذا
+
+226
+00:17:44,260 --> 00:17:48,220
+والمقدار الموجود في هذا أصغر من وين؟ هنا لقنا أصغر
+
+227
+00:17:48,220 --> 00:17:51,800
+من إياش من إبسلون يعني صار عند ال S بي و F ناقص من
+
+228
+00:17:51,800 --> 00:17:58,860
+ال F أصغر من مين من إبسلون وهو المطلوب نجي للنظرية
+
+229
+00:17:58,860 --> 00:17:59,300
+الباعتها
+
+230
+00:18:09,320 --> 00:18:15,260
+النظرية تلاتة أربعة .. سبعة أربعة تلاتة نشوف ..
+
+231
+00:18:15,260 --> 00:18:21,920
+نتبه لنص النظرية و من ثم نيجي إلى اللي هو البرهان
+
+232
+00:18:21,920 --> 00:18:27,380
+إذا من ال .. من النظرية اللي قبله شوية لكل إبسلون
+
+233
+00:18:27,380 --> 00:18:32,760
+أكبر من سفر لجينا بي إبسلون such that for every بي
+
+234
+00:18:32,760 --> 00:18:38,220
+يحتوي البي إبسلون أو .. و لكل اللي هو أس بي و أف
+
+235
+00:18:38,980 --> 00:18:43,700
+اللي هو corresponding to this partition بديكون S P
+
+236
+00:18:43,700 --> 00:18:49,000
+و F ناقص ال integration من هنا ل P لل F أصغر من P
+
+237
+00:18:49,000 --> 00:18:53,480
+ل Y وهذا في حال أن F is integrable هذا عنوان اللي
+
+238
+00:18:53,480 --> 00:18:57,220
+هو اللي هي النظرية اللي قبل بشوية لإنه بتحتاجها
+
+239
+00:18:57,220 --> 00:19:03,130
+بعد شوية طيب شوف الآن إيش النظرية هذي بتقولخلّيت
+
+240
+00:19:03,130 --> 00:19:05,870
+أفضل about the closed interval A وB اللي هي I اللي
+
+241
+00:19:05,870 --> 00:19:09,110
+عند ال R بـA bounded function Suppose that there
+
+242
+00:19:09,110 --> 00:19:19,130
+exists a number A يحقق الخاصية التالية يحقق
+
+243
+00:19:19,130 --> 00:19:21,390
+الخاصية التالية
+
+244
+00:19:30,260 --> 00:19:34,660
+نفترض أن يوجد A بحيث أنه لايه إبسلون أكبر من 0
+
+245
+00:19:34,660 --> 00:19:40,140
+بنلاقي partition بي إبسلون و FB يحتوى بي إبسلون
+
+246
+00:19:40,140 --> 00:19:45,100
+and F SB is any remand sum for F corresponding to
+
+247
+00:19:45,100 --> 00:19:52,100
+B تئذن ال S بي و F ناقص A أصغر من مين من إبسلون
+
+248
+00:19:54,770 --> 00:20:02,070
+لو لاجينا .. انا افترض ان احنا بنلاقي الـ Number A
+
+249
+00:20:02,070 --> 00:20:07,190
+هذا معطى انه Number A، ايش بتمتح هذا Number A؟ انه
+
+250
+00:20:07,190 --> 00:20:12,190
+لما يكون عندك لكل ي أكبر من سفر لجهة partition بيه
+
+251
+00:20:12,190 --> 00:20:15,290
+يبسلون بحيث انه لكل refinement بيه تحتوى بيه
+
+252
+00:20:15,290 --> 00:20:19,870
+يبسلون و ال S, B و F is any remand sum
+
+253
+00:20:22,380 --> 00:20:24,760
+يجب أن يكون الـ a أصغر من إبسلون لأس و ال b و ال f
+
+254
+00:20:24,760 --> 00:20:31,980
+نقص من إبسلون لأس و ال b و ال f نقص من إبسلون لأس
+
+255
+00:20:31,980 --> 00:20:32,420
+و ال b و ال f نقص من إبسلون لأس و ال b و ال f نقص
+
+256
+00:20:32,420 --> 00:20:32,460
+من إبسلون لأس و ال b و ال f نقص من إبسلون لأس و ال
+
+257
+00:20:32,460 --> 00:20:32,460
+b و ال f نقص من إبسلون لأس و ال b و ال f نقص من
+
+258
+00:20:32,460 --> 00:20:32,460
+إبسلون لأس و ال b و ال f نقص من إبسلون لأس و ال b
+
+259
+00:20:32,460 --> 00:20:34,200
+و ال f نقص من إبسلون لأس و ال b و ال f نقص من
+
+260
+00:20:34,200 --> 00:20:37,640
+إبسلون لأس و ال b و ال f نقص من إبسلون لأس و ال b
+
+261
+00:20:37,640 --> 00:20:42,860
+ولكل ي أكبر من 0 بلاقي بي إبسلون بحيث أنه لما ال
+
+262
+00:20:42,860 --> 00:20:46,780
+بي يحتوي ال بي إبسلون ويكون اللي هو ال أس بي و ال
+
+263
+00:20:46,780 --> 00:20:50,800
+أف ناقص ا أصغر من إبسلون بقول then أف اي شمالها
+
+264
+00:20:50,800 --> 00:20:56,020
+must be integrable in I in the sense طبعاً في 716
+
+265
+00:20:56,020 --> 00:21:00,500
+و ال a هو من هذا ال integration يعني بقولنا لو
+
+266
+00:21:00,500 --> 00:21:05,360
+لجينا a بتحقق هذا الكلام هتطلعلكم ال a يا جماعة
+
+267
+00:21:05,360 --> 00:21:10,070
+هذا هي ال integrationهي قيمة الـ integration الـ
+
+268
+00:21:10,070 --> 00:21:12,630
+integration in the sense of الـ definition اللي
+
+269
+00:21:12,630 --> 00:21:17,730
+حكينا عليه طيب شوف الآن الان
+
+270
+00:21:17,730 --> 00:21:22,410
+بدي أعطيك شغل عامة وبعدها نخصصها على اللي بدناياه
+
+271
+00:21:22,410 --> 00:21:26,370
+الان give an epsilon أكبر من سفر and be any fixed
+
+272
+00:21:26,370 --> 00:21:31,250
+partition of I ماشي الحال الان خد أي إبسلون وخد بي
+
+273
+00:21:31,250 --> 00:21:37,020
+أي شمله أي fixed partition thenThere exist S1
+
+274
+00:21:37,020 --> 00:21:41,800
+بيوقف و S2 بيوقف بحيث أن S U بيوقف نقص S بيوقف
+
+275
+00:21:41,800 --> 00:21:46,800
+أصغر من مين من إبسلون و S2 نقص ال أصغر من مين من
+
+276
+00:21:46,800 --> 00:21:57,260
+إبسلون شوف هذه خلّينا نشوف كيف عملها هيا أخدنا
+
+277
+00:21:57,260 --> 00:22:02,580
+أي إبسلون في الدنيا و بأي partition الآن بقول لي
+
+278
+00:22:04,500 --> 00:22:11,460
+صار عندى S P و F صار عندى اسف البارتيشن اللى هو P
+
+279
+00:22:11,460 --> 00:22:17,260
+موجود والإبسلون موجود بانفكس بارتيشن مش بارتيشن
+
+280
+00:22:17,260 --> 00:22:21,320
+معين لأ انفكس بارتيشن الكلام ده بظبط عليه بقول
+
+281
+00:22:21,320 --> 00:22:32,410
+بقدر ألاقي remansum S واحد مسمى P و F بحيث أنه SU
+
+282
+00:22:32,410 --> 00:22:44,070
+B و F ناقص S1 B و F يكون أصغر من مين من Y المنطقي
+
+283
+00:22:44,070 --> 00:22:49,980
+الكلام هذا الأبرهو اعطاني الـ Epsilon جالي اثبت
+
+284
+00:22:49,980 --> 00:22:54,980
+انه بتلاقي S1 Remains sum بحيث ان الفرق بينه وبين
+
+285
+00:22:54,980 --> 00:22:57,340
+ال A بر لل B والF ال partition هذا partition معين
+
+286
+00:22:57,340 --> 00:23:01,680
+بحكي الاربيوتر ال partition بس fixed انه اخدت اي
+
+287
+00:23:01,680 --> 00:23:07,900
+Epsilonأجدرت ألاقي S1 remains sum بحيث أن هذا نقص
+
+288
+00:23:07,900 --> 00:23:10,980
+هذا يكون أصغر من يبسلون طبعا هذا الطبيعي هو أكبر
+
+289
+00:23:10,980 --> 00:23:13,760
+من هذا لكن الفرق بدي أخليه أصغر من يبسلون وأنا
+
+290
+00:23:13,760 --> 00:23:20,440
+بقدر أخليه الآن عندي ال Mk بساوي اللي هو عبارة عن
+
+291
+00:23:20,440 --> 00:23:27,580
+ال supremum لل F of X such that X element XK minus
+
+292
+00:23:27,580 --> 00:23:33,490
+واحد و XK مظبوط؟الان هذا اللي هو ال infimum ال
+
+293
+00:23:33,490 --> 00:23:37,010
+least upper bound الان ال least upper bound لو
+
+294
+00:23:37,010 --> 00:23:43,890
+شيلنا منه نجفنا منه اي عدد صغير كتير كتير و ليكن
+
+295
+00:23:43,890 --> 00:23:49,850
+ال y على مين على b minus a ببطل upper bound مدام
+
+296
+00:23:49,850 --> 00:23:55,510
+بطل upper bound اذا بقدر اناجي xik element ال xk
+
+297
+00:23:55,510 --> 00:23:57,230
+minus واحد و xk
+
+298
+00:24:00,970 --> 00:24:08,050
+تكون مالها أكبر من مين من اللي هو اللي بطل other
+
+299
+00:24:08,050 --> 00:24:13,600
+boundالان هذا اللي سويته مع الام كيه بقدر اسويه مع
+
+300
+00:24:13,600 --> 00:24:16,280
+الام واحد و الام اتنين و الام تلتة ايه منهم كلهم
+
+301
+00:24:16,280 --> 00:24:21,500
+إذا صار ال summation ك من عند واحد لعند ان و اضربه
+
+302
+00:24:21,500 --> 00:24:25,940
+كمان في مين xk minus xk minus واحد مافيش مانع لإن
+
+303
+00:24:25,940 --> 00:24:29,740
+أنا دموجة و أنا دموجة بتظهر زي ما هي xk ناقص xk
+
+304
+00:24:29,740 --> 00:24:32,720
+minus واحد واخده ال summation هذا ك من عند واحد
+
+305
+00:24:32,720 --> 00:24:38,120
+لعند مين لعند ان لجيتها ال xi k تبعت ال S واحد هذه
+
+306
+00:24:39,040 --> 00:24:47,580
+الان هذا remain sum بسميه S1 بيوف على مين اعتمد
+
+307
+00:24:47,580 --> 00:24:52,620
+على ال exiled case اللي انا لجيتها هذا ايش بيساوي
+
+308
+00:24:52,620 --> 00:24:59,480
+اللي هو summation Mk في Xk minus Xk minus واحد
+
+309
+00:24:59,480 --> 00:25:08,840
+ناقص ال summationXY-XK
+
+310
+00:25:08,840 --> 00:25:10,840
+-1
+
+311
+00:25:17,430 --> 00:25:22,310
+هذا summation سامحوني على الضيق اللوح هذا بيصير ال
+
+312
+00:25:22,310 --> 00:25:24,350
+summation على الأول زاد ال summation على الثانى هي
+
+313
+00:25:24,350 --> 00:25:27,630
+ال summation على الأول كم من عند واحد لعند أن ناقص
+
+314
+00:25:27,630 --> 00:25:29,910
+ال summation على التاني كم من عند واحد لعند أن
+
+315
+00:25:29,910 --> 00:25:34,810
+واخدت ال y بي و ماينوس أيه عن المشترك برا ال
+
+316
+00:25:34,810 --> 00:25:37,830
+summation دلت ال summation لهذا أصغر أو يساوي
+
+317
+00:25:37,830 --> 00:25:44,330
+بيقول لهذا مين هو أو أصغر من رمان صم أس بي و أف و
+
+318
+00:25:44,330 --> 00:25:53,580
+سمنها أس واحدهذا مين هو اكيد كلكم عارفه U P و F و
+
+319
+00:25:53,580 --> 00:26:00,000
+هذا ايش هو ناقص ايش هذاهذا اكيد y على b minus a في
+
+320
+00:26:00,000 --> 00:26:03,680
+هذا المقدار هذا المقدار عبارة عن x naught x واحد
+
+321
+00:26:03,680 --> 00:26:08,440
+ناقص x naught زائد x اتنين ناقص x واحد زائد عرفته
+
+322
+00:26:08,440 --> 00:26:12,240
+اذا اخر واحد xn ناقص xn ناقص واحد يعني بيظل xn
+
+323
+00:26:12,240 --> 00:26:16,280
+ناقص x naught كله ب cancel بعض xn اللي هي ال a ال
+
+324
+00:26:16,280 --> 00:26:20,180
+b و ال x naught هي ال a يعني هذا عبارة عن b minus
+
+325
+00:26:20,180 --> 00:26:27,450
+a اصلا من s واحد بيقفيعني صار عندي الـ U بي و F و
+
+326
+00:26:27,450 --> 00:26:35,710
+جيب لي هذا هنا نقص S1 بي و Fأشماله أصغر من أبسلون
+
+327
+00:26:35,710 --> 00:26:43,150
+فعلا انا اجدت اس واحد من
+
+328
+00:26:43,150 --> 00:27:00,870
+أبسلون
+
+329
+00:27:00,980 --> 00:27:08,700
+إذا لجيت S1 و S2 بحيث تحقق هذه الخاصية انتبهوا
+
+330
+00:27:08,700 --> 00:27:14,340
+الان هذا الان كل اللي حكيته هنا برهان للجزئية اللي
+
+331
+00:27:14,340 --> 00:27:20,770
+كاتبها انا في البرهان و مش مضحك بشكلها الكاملطيب
+
+332
+00:27:20,770 --> 00:27:23,930
+فيكم أصلي على النبي الله مصلي على سيدنا محمد نرجع
+
+333
+00:27:23,930 --> 00:27:28,450
+نقولكم انه لجينا اللي هو الاس واحد والاس اتنين
+
+334
+00:27:28,450 --> 00:27:31,870
+اللي بيحققن هذا طبعا ال بي كان ايش ماله any fixed
+
+335
+00:27:31,870 --> 00:27:37,410
+partition الان معطينا
+
+336
+00:27:37,410 --> 00:27:43,050
+هنا انه هذا معطى هذا معطى ديروا بالكم انه there
+
+337
+00:27:43,050 --> 00:27:46,970
+exists بي ابسلون بحيث ان بي بتحتوي بي ابسلون هذا
+
+338
+00:27:46,970 --> 00:27:52,630
+يتحقق الاناللي اتحقق على الـ P اللي هام، هيتحقق
+
+339
+00:27:52,630 --> 00:27:57,010
+على الـ P الإبسلون اللي هام ماشي الحال، إذا الأن
+
+340
+00:27:57,010 --> 00:28:01,930
+الـ U P إبسلون و الـ F ناقص الـ L P و F اللي هو
+
+341
+00:28:01,930 --> 00:28:10,460
+بساوي U U P و F ناقص L P و Fماشي ايش هذا اخدت ال
+
+342
+00:28:10,460 --> 00:28:13,740
+بي إبسلون ال بي اللي هنا هي ال بي إبسلون عشان اقول
+
+343
+00:28:13,740 --> 00:28:17,440
+انه انا اللي حققته على اللي هي ال بي فده حققته على
+
+344
+00:28:17,440 --> 00:28:20,460
+ال بي إبسلون انه هذا كان unfixed partition اصغر او
+
+345
+00:28:20,460 --> 00:28:25,700
+يساوي بال triangle inequality UBF ناقص SBF زي S1BF
+
+346
+00:28:25,700 --> 00:28:33,440
+ناقص A زي A ناقص S2BFماذا فعلت؟ بالـ Triangle
+
+347
+00:28:33,440 --> 00:28:37,380
+Inquality أضفت الـ S واحد وطرحت الـ S واحد وضفت
+
+348
+00:28:37,380 --> 00:28:41,260
+الـ A وطرحت الـ A وضفت الـ S اتنين وطرحت الـ S
+
+349
+00:28:41,260 --> 00:28:44,480
+اتنين وعملت الـ Triangle Inquality وطبقت هذه واحدة
+
+350
+00:28:44,480 --> 00:28:48,260
+كامتين، تلاتة، أربعة صار عندي أزرع يساوي هذا، زائد
+
+351
+00:28:48,260 --> 00:28:57,300
+هذا، زائد هذا، زائد هذا، ماشي الحل صار
+
+352
+00:28:57,300 --> 00:28:58,480
+عندي يا جماعة الآن
+
+353
+00:29:12,650 --> 00:29:17,970
+صار عندى اللى هو الان طبعا ال a ايش هي ال a اللى
+
+354
+00:29:17,970 --> 00:29:20,310
+في النظرية دى بقالكم a is the number in the
+
+355
+00:29:20,310 --> 00:29:24,270
+hypothesis of theorem then by the hypothesis of
+
+356
+00:29:24,270 --> 00:29:31,650
+theorem and star الان من اللى هو المعطى بيعطينا ان
+
+357
+00:29:31,650 --> 00:29:35,630
+ال s واحد نقص اللى a أصغر من يبسلون و ال s اتنين
+
+358
+00:29:35,630 --> 00:29:39,430
+نقص اللى a أصغر من يبسلون لأن هذا صحيح اللى فوق
+
+359
+00:29:39,430 --> 00:29:39,850
+لان
+
+360
+00:29:47,480 --> 00:29:53,200
+أذكركم كمان مرة بالمعطى
+
+361
+00:29:53,200 --> 00:30:00,540
+ان هناك مجموعة بي اس ايبسلون هي اللي بتشتغل في اس
+
+362
+00:30:00,540 --> 00:30:05,040
+بي و اف is any remansum corresponding لأف هيكون
+
+363
+00:30:05,040 --> 00:30:07,860
+الفرق بين هذا و هذا أصغر من ايبسلون من ضمن الاس
+
+364
+00:30:07,860 --> 00:30:11,740
+واحد والاس اتنين اللي جاتني قبل شوية اذا صار عندي
+
+365
+00:30:11,740 --> 00:30:20,530
+الكل اللي بشتغلوا مشروع بيصير اللي هوالـ UB نقص
+
+366
+00:30:20,530 --> 00:30:25,730
+الـ S1 أصغر من إبسلون و هذا أصغر من الإبسلون مُعطى
+
+367
+00:30:25,730 --> 00:30:29,230
+وهذا أصغر من الإبسلون مُعطى وهذا و هذا أصغر كل
+
+368
+00:30:29,230 --> 00:30:33,370
+واحد من إبسلون من اللي أتبته هنا قبل بشوية إذا صرت
+
+369
+00:30:33,370 --> 00:30:41,070
+اللي هي كلها أصغر من 4 إبسلون ماشي الحالأبسلون was
+
+370
+00:30:41,070 --> 00:30:48,930
+arbitrary then by 4718 F is integrableF is
+
+371
+00:30:48,930 --> 00:30:53,690
+integrable لأنه وجدت بإبسلون لأن U نقص لأخر من
+
+372
+00:30:53,690 --> 00:30:57,430
+أربعة إبسلون بإبسلون وبعض arbitrarily إذا بصير
+
+373
+00:30:57,430 --> 00:31:01,910
+عندي صحيح واحد متضايق من الأربعة إبسلون في اللي
+
+374
+00:31:01,910 --> 00:31:04,430
+فوق خد إبسلون على أربعة إبسلون على أربعة و النظرية
+
+375
+00:31:04,430 --> 00:31:08,730
+الأولى و المعطى برضه بتاخدوا إبسلون على أربعة
+
+376
+00:31:08,730 --> 00:31:19,300
+بطريقة حسابية معاه طيب إذا صار عنديلجينا الان then
+
+377
+00:31:19,300 --> 00:31:22,480
+by theorem 7 1 8 F is integrable and for every
+
+378
+00:31:22,480 --> 00:31:25,360
+epsilon أكبر من 0 there exists B epsilon of I such
+
+379
+00:31:25,360 --> 00:31:28,800
+that FB بتحتول بي ابسلون and B is a partition of I
+
+380
+00:31:28,800 --> 00:31:36,380
+then S of B و F مقصدها أصغر من 100 من إبسلون إيش
+
+381
+00:31:36,380 --> 00:31:39,840
+هذا؟ حنا أثبتنا انه integrable مدام انتجرابل باللي
+
+382
+00:31:39,840 --> 00:31:44,100
+كاتبوا هناواللي كاتبوها هيكون الفرق بين for every
+
+383
+00:31:44,100 --> 00:31:47,320
+ε there exists بي إبسلون و الـ B تحتوى 100 بي
+
+384
+00:31:47,320 --> 00:31:50,040
+إبسلون واحد يقول طب ما هو خايفين الـ بي إبسلون
+
+385
+00:31:50,040 --> 00:31:53,260
+اللي لاجيناها في الأول المعطف في النظرية غير الـ
+
+386
+00:31:53,260 --> 00:31:57,260
+بي إبسلون هذه مش مشكلة الـ بي إبسلون فرطنة تبعة
+
+387
+00:31:57,260 --> 00:32:01,560
+النظرية بي برايم of إبسلونوهذه اللي وجدناها من الـ
+
+388
+00:32:01,560 --> 00:32:05,660
+Integrability للـ P للـ F اللي أثبتناها نسميها بي
+
+389
+00:32:05,660 --> 00:32:09,480
+إبسلون دابل براين خد الـ بي إبسلون اللي بتحكي
+
+390
+00:32:09,480 --> 00:32:14,520
+عليها الآن عشان تنفع للجهتين خدها بساوي بي إبسلون
+
+391
+00:32:14,520 --> 00:32:20,060
+براين اتحاد بي إبسلون دابل براينبصير الآن هذه اللي
+
+392
+00:32:20,060 --> 00:32:26,280
+هي تنفع تنفع للمعطيات الأولى والمعطيات التانية
+
+393
+00:32:26,280 --> 00:32:29,320
+لأنها بيصير refinement للأول و refinement لمن؟
+
+394
+00:32:29,320 --> 00:32:35,980
+للثاني إذا فش فيه مشكلة إذا الآنthen by hypothesis
+
+395
+00:32:35,980 --> 00:32:40,360
+of theorem أخدنا هذا أصغر من أربع إبسلون إذا سبب
+
+396
+00:32:40,360 --> 00:32:43,840
+له هذا أن هو F is integrable مدام F is integrable
+
+397
+00:32:43,840 --> 00:32:48,620
+إذا بالنظرية بنلاقي بي إبسلون بحيث أنه لكل بي
+
+398
+00:32:48,620 --> 00:32:53,000
+بيحتوي بي إبسلون بيكون هذا ناقص هذا أصغر من مين من
+
+399
+00:32:53,000 --> 00:32:58,890
+إبسلون هذا أصبح أصغر من إبسلونبس الآن اثبتنا عند
+
+400
+00:32:58,890 --> 00:33:02,990
+الـ F بي و F ناقص اللي أصغر من إبسلون من النظرية و
+
+401
+00:33:02,990 --> 00:33:06,570
+واحد يقول لي تبعت ال بي هادي بتختلف عن ال بي هادي
+
+402
+00:33:06,570 --> 00:33:10,450
+قلتلك علجتها ال بي هادي و ال بي هادي باخد ال بي
+
+403
+00:33:10,450 --> 00:33:13,730
+إبسلون لتبعت النظرية و ال بي إبسلون تبعت اللي هو
+
+404
+00:33:13,730 --> 00:33:17,690
+ال integrability و باخدهن اتحادهن بيكون اللي هو ال
+
+405
+00:33:17,690 --> 00:33:20,990
+بي إبسلون اللي لجيتها هنااللي بدى أستخدم إيه لها
+
+406
+00:33:20,990 --> 00:33:24,770
+الجهتين و أي refinement لها بيه لـ بي أبسلون و
+
+407
+00:33:24,770 --> 00:33:29,270
+لجديدة بيطلع هذا صح وهذا صح اتتين مع بعض دلوقتي
+
+408
+00:33:29,270 --> 00:33:35,030
+بحجل اللي استخدمهم و أقول إذا ال integration لل F
+
+409
+00:33:35,030 --> 00:33:37,830
+من A ل B نقص A أصغر و أساوي ال integration لل F
+
+410
+00:33:37,830 --> 00:33:41,990
+نقص ال S زاد ال S نقص ال Aاللي هو ضيف ال term و
+
+411
+00:33:41,990 --> 00:33:46,750
+طرحت ال term هذا هي أصغر من إبسلون وهذا أصغر من
+
+412
+00:33:46,750 --> 00:33:50,210
+إبسلون إذا صار هذا أصغر من اتنين إبسلون صار عندي
+
+413
+00:33:50,210 --> 00:33:53,910
+فيه العدد هذا بين اللي هو اتنين إبسلون وأكبر أو
+
+414
+00:33:53,910 --> 00:33:56,870
+ساوي سفر وإبسلون arbitrary إذا هذا المقدر لازم
+
+415
+00:33:56,870 --> 00:34:00,750
+ساوي سفر إذا ال integration بساوي مين بساوي ليه
+
+416
+00:34:00,750 --> 00:34:03,510
+وهو المطلوب طيب
+
+417
+00:34:10,110 --> 00:34:17,630
+نجي لأ اللي هو ال definition سبعة أربعة أربعة ال
+
+418
+00:34:17,630 --> 00:34:23,270
+definition بسيط بده يعرف حاجة اسمها مش أو norm هو
+
+419
+00:34:23,270 --> 00:34:27,330
+انا بسميها مش عادة الناس بتسميها norm ايش ال .. ال
+
+420
+00:34:27,330 --> 00:34:31,550
+.. ال مش اللي بده يعرفه؟ بقول لو كان عندك
+
+421
+00:34:31,550 --> 00:34:38,910
+partition B بساوي X0 و X1 لعندي XNالان كل sub
+
+422
+00:34:38,910 --> 00:34:46,970
+interval لها طول ال norm ل ال B او مش ل ال B هو
+
+423
+00:34:46,970 --> 00:34:51,920
+عبارة عن ال maximumأو الـ Supremum Maximum لأن الـ
+
+424
+00:34:51,920 --> 00:34:53,860
+Supremum حاسب Maximum لإنه الـ finite دول
+
+425
+00:34:53,860 --> 00:34:59,480
+الماكسيمم لا X1 ناقص X0 و X2 ناقص X1 لأطوال
+
+426
+00:34:59,480 --> 00:35:03,140
+الفترات يعني sub interval لعند ال XN ناقص XN ناقص
+
+427
+00:35:03,140 --> 00:35:07,880
+1 الماكسيمم الهن هدول أطوال الفترات هو اللي بنسميه
+
+428
+00:35:07,880 --> 00:35:14,000
+مين النورم للـ B هو النورم للـ B الآن يعني يعني لو
+
+429
+00:35:14,000 --> 00:35:20,670
+كان عندك لو كان عندك هي عند الفترةمن صفر لعين
+
+430
+00:35:20,670 --> 00:35:31,410
+تلاتة مثلا هاي عند بي بتساوي صفر نص واحد اتنين
+
+431
+00:35:31,410 --> 00:35:37,970
+تلاتة مثلا بكون ال normal بي ايش هيساوي ايش ال
+
+432
+00:35:37,970 --> 00:35:41,510
+normal بي اكيد هتقولوا كلكم تلاتة ناقص اتنين واحد
+
+433
+00:35:41,510 --> 00:35:44,710
+اتنين ناقص واحد واحد واحد ناقص نص نص واحد اذا ايش
+
+434
+00:35:44,710 --> 00:35:49,110
+هيساوي واحد اكبر اكبر طول sub interval واحدلو جينا
+
+435
+00:35:49,110 --> 00:36:00,350
+أخدنا الـ Q مثلا بساوي اللي هو Zero نص واحد واحد و
+
+436
+00:36:00,350 --> 00:36:08,890
+تلت اتنين اتنين و نص و تلاتة وقلنا لكم اوجد الـ Q
+
+437
+00:36:09,620 --> 00:36:13,420
+أكيد كلكم حدقول الـ Q أشمالها اللي هي بين الواحد و
+
+438
+00:36:13,420 --> 00:36:17,420
+تلت والتانين هي أكبر أكبر مسافة اللي هي عبارة عن
+
+439
+00:36:17,420 --> 00:36:23,060
+إيه؟ تلتين لاحظوا أن الـ B مجموعة جزئية من مين؟ من
+
+440
+00:36:23,060 --> 00:36:33,810
+الـ Q لما .. لما الآن الـ Q كبرت نورمهاهيكون أصغر
+
+441
+00:36:33,810 --> 00:36:37,590
+أو يساوي norm من الـ B طبيعي لأنه دخلت نقطة جديدة
+
+442
+00:36:37,590 --> 00:36:43,690
+ممكن تزخر المسافة لكن مش ممكن تزيدها طيب وهذه اللي
+
+443
+00:36:43,690 --> 00:36:47,810
+هي بعض الملاحظات اللي هنا على اللي هي ال norm أو
+
+444
+00:36:47,810 --> 00:36:55,800
+ال mesh اللي الآن بعض اللي هوالملاحظات اللي هو
+
+445
+00:36:55,800 --> 00:37:00,780
+different partition of I can have the same مش بقول
+
+446
+00:37:00,780 --> 00:37:03,280
+يعني ممكن different partition لل .. لل .. لل ..
+
+447
+00:37:03,280 --> 00:37:06,140
+للinterval اللي ده same مش اه بقدر هاد انا اجيبلك
+
+448
+00:37:06,140 --> 00:37:12,200
+كمان واحد زي هدول خد كمان بي برايم بساوي سفر و نص
+
+449
+00:37:12,200 --> 00:37:20,030
+و واحد و اتنين و اتنين و نص و تلاتةهذا الـ B'
+
+450
+00:37:20,270 --> 00:37:25,170
+يختلف عن الـ B لكن نورمه التاني الواش بيساوي واحد
+
+451
+00:37:25,170 --> 00:37:30,530
+يعني ال different partitions ممكن يكون لها نفس
+
+452
+00:37:30,530 --> 00:37:34,730
+نورم و هاي مثال و كتير فيه زي ذلك لأن لو كانت B
+
+453
+00:37:34,730 --> 00:37:39,730
+subset من Q نورم لـ Q أظهر يساوي نورم لمن؟ لـ B و
+
+454
+00:37:39,730 --> 00:37:43,720
+اتبعت السهل هذاخد الـ B subset إذا هيكون عنده اللي
+
+455
+00:37:43,720 --> 00:37:47,080
+هو عدد النقاط اللي في Q أكتر من عدد النقاط اللي في
+
+456
+00:37:47,080 --> 00:37:53,780
+B هكون ال maximum على الأولى اللي هو أكبر أو شاوي
+
+457
+00:37:53,780 --> 00:37:56,720
+ال maximum على مين على التاني لإنه اللي صار فيها
+
+458
+00:37:56,720 --> 00:38:02,070
+زيادات صار فيه إمكانية إنها تقصر الفتراتطيب لكن لو
+
+459
+00:38:02,070 --> 00:38:05,330
+كان ال normal ال Q أصغر شويه ال normal ال B مش شرط
+
+460
+00:38:05,330 --> 00:38:10,110
+أن يكون B subset من مين؟ من ال Q أو كتير بتلاقي
+
+461
+00:38:10,110 --> 00:38:16,910
+أمثلة زي هي ال normal
+
+462
+00:38:16,910 --> 00:38:25,160
+B هذا إيش بيساوي؟ normal B بيساوي واحدالان عندي ..
+
+463
+00:38:25,160 --> 00:38:32,320
+مش شرط .. يعني هديلك اللي هو Normal Q أصغر أوي
+
+464
+00:38:32,320 --> 00:38:40,040
+ساوي Normal B لكن الـ B اللي هو ليس شرطا أنها تكون
+
+465
+00:38:40,040 --> 00:38:47,500
+subset من مين؟ من إيش؟ من الـ Q Normal B واحد خد
+
+466
+00:38:47,500 --> 00:38:48,060
+الـ Q
+
+467
+00:38:53,000 --> 00:39:03,100
+بساوي اللي هو سفر و واحد و اتنين و اتنين و نص و
+
+468
+00:39:03,100 --> 00:39:07,180
+تلاتة نورمال
+
+469
+00:39:07,180 --> 00:39:12,100
+كيوه بديه ال
+
+470
+00:39:12,100 --> 00:39:19,920
+gate
+
+471
+00:39:19,920 --> 00:39:22,280
+واحد و واحد و نص
+
+472
+00:39:25,990 --> 00:39:34,190
+وهي نص كمان سفر و نص Normal Q بساوة نص مظبوط
+
+473
+00:39:34,190 --> 00:39:45,650
+Normal Q بساوة نص لكن ال B ال B خلي هذا تلت ماشي
+
+474
+00:39:45,650 --> 00:39:51,220
+خلي هذا تلت صارت ال B مش subset من مينمن الـ Q
+
+475
+00:39:51,220 --> 00:39:54,420
+ممكن تلاقي افضل اكتير هذه الـ B أكيد مش subset من
+
+476
+00:39:54,420 --> 00:40:00,460
+الـ Q Normal B بيبقى واحد صح؟ Normal B واحد و
+
+477
+00:40:00,460 --> 00:40:06,160
+Normal Q ايش بيساوي نص لأن الفرق بينهم انصاص هذه
+
+478
+00:40:06,160 --> 00:40:12,340
+عندك Normal Q أصغر او يساوي Normal B لأن هذا نص
+
+479
+00:40:12,340 --> 00:40:14,420
+وهذا واحد but
+
+480
+00:40:17,620 --> 00:40:22,380
+واضح الـ B هي سفر و تلت و واحد و اتنين و تلاتة ليس
+
+481
+00:40:22,380 --> 00:40:26,100
+subset من الـ Q صفر و نص و واحد و نص و .. و .. و
+
+482
+00:40:26,100 --> 00:40:27,400
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+483
+00:40:27,400 --> 00:40:28,940
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+484
+00:40:28,940 --> 00:40:29,640
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+485
+00:40:29,640 --> 00:40:30,360
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+486
+00:40:30,360 --> 00:40:30,360
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+487
+00:40:30,360 --> 00:40:30,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+488
+00:40:30,380 --> 00:40:30,380
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+489
+00:40:30,380 --> 00:40:34,960
+.. و .. و .. و .. و .. و .. و .. و .. و .. و .. و
+
+490
+00:40:34,960 --> 00:40:36,340
+.. و .. و .. و .. و .. و .. و ..
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg.srt
new file mode 100644
index 0000000000000000000000000000000000000000..3fd6193929a1e2516f7631b41b790a9a74fc1df7
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg.srt
@@ -0,0 +1,2076 @@
+1
+00:00:05,300 --> 00:00:13,060
+بسم الله الرحمن الرحيم المحاضرة السادسة مساق تحليل
+
+2
+00:00:13,060 --> 00:00:18,680
+حقيقة 2 لطلبة قسم رياضيات بكلية العلوم بالجامعة
+
+3
+00:00:18,680 --> 00:00:23,220
+الإسلامية بغزة. الحديث اليوم إن شاء الله هيكون حول
+
+4
+00:00:23,220 --> 00:00:30,200
+L'Hôpital's Rule أو قواعد L'Hôpital's  L'Hôpital's Rule
+
+5
+00:00:30,200 --> 00:00:38,320
+بتمر في calculus في تفاضل وتكامل بتمر من ناحية
+
+6
+00:00:38,320 --> 00:00:45,100
+عملية استخدامها وتوظيفها لحل اللي هي بعض النهايات
+
+7
+00:00:45,100 --> 00:00:50,400
+اللي بيكون ... اللي هو عجزنا عن حلها بطرق عادية
+
+8
+00:00:51,750 --> 00:00:57,550
+هنشوف الآن في الحديث عن الـ L'Hôpital's Rule حول اللي
+
+9
+00:00:57,550 --> 00:01:01,570
+هو كيف اللي هو أن نبرهن هذه اللي هي القواعد كيف نشتق
+
+10
+00:01:01,570 --> 00:01:06,310
+هذه القواعد كيف اللي هو أيضًا بشكل سريع حول اللي
+
+11
+00:01:06,310 --> 00:01:12,490
+هي استخدام هذه القواعد طبعًا 
+
+12
+00:01:12,490 --> 00:01:16,430
+في الأول هنتحدث عن الكميات الغير معينة
+
+13
+00:01:17,020 --> 00:01:23,580
+indeterminate forms اللي هي اللي بتعالجها اللي هي
+
+14
+00:01:23,580 --> 00:01:28,840
+L'Hôpital's Rule عندي على سبيل المثال لو جينا أخدنا
+
+15
+00:01:28,840 --> 00:01:37,090
+الـ limit اللي هو x على x تكعيب على x تربيع مثلًا as x
+
+16
+00:01:37,090 --> 00:01:44,730
+goes to zero limit αx على x لما x تروح للزيرو
+
+17
+00:01:44,730 --> 00:01:50,670
+limit اللي هو x تربيع على x لما x تروح للزيرو
+
+18
+00:01:50,670 --> 00:01:59,900
+limit x على x تكعيب لما x تروح للزيرو limit مثلًا
+
+19
+00:01:59,900 --> 00:02:07,100
+-x على x تكعيب لما x تروح لـ 0 لو طلعنا على
+
+20
+00:02:07,100 --> 00:02:11,440
+اللي هي الـ limits اللي موجودة هنا كلها على صورة
+
+21
+00:02:11,440 --> 00:02:16,260
+اللي هي لو تعويض مباشر هنلاقيها على صورة 0 على 0
+
+22
+00:02:16,960 --> 00:02:21,420
+الآن كمية 0 على 0 بالطرق السابقة كان اللي هو أنه
+
+23
+00:02:21,420 --> 00:02:27,620
+احنا صعب اللي هو نتعامل معها لكن في بعض الأحيان زي
+
+24
+00:02:27,620 --> 00:02:30,520
+الحالة هذه هنلاقي أنه احنا بنعرف نتعامل معها و
+
+25
+00:02:30,520 --> 00:02:36,140
+بنعرف نحكم عليها اللي هلاحظ أنه كلّ رغم أن كلها 0
+
+26
+00:02:36,140 --> 00:02:41,100
+على 0 إلا أنها بتعطي في كل حالة شيء مختلف عن
+
+27
+00:02:41,100 --> 00:02:47,350
+الحالة الثانية الآن هذه مثلًا عبارة عن limit 1 على x
+
+28
+00:02:47,350 --> 00:02:51,030
+لما x تروح للـ 0 طبعًا 1 على x لما x تروح للـ 0 إيش
+
+29
+00:02:51,030 --> 00:02:53,710
+مالها؟ does not exist لأنه من اليمين بتعطي ما لا نهاية
+
+30
+00:02:53,710 --> 00:02:56,790
+ومن اليسار بتعطي سالب ما لا نهاية عشان هيك
+
+31
+00:02:56,790 --> 00:03:02,510
+كده يبقى نقول عنها does not exist لأن limit αx على x 
+
+32
+00:03:02,510 --> 00:03:06,210
+لما x تروح للـ 0 هو يساوي عبارة عن α
+
+33
+00:03:06,960 --> 00:03:09,720
+اللي هو عبارة عن real number لو فرضنا أنه α
+
+34
+00:03:09,720 --> 00:03:13,480
+ماخدينها احنا real number إذا أعطت number عادل
+
+35
+00:03:13,480 --> 00:03:16,320
+أول ما أعطتش اللي هي أنا أعطت does not exist
+
+36
+00:03:16,320 --> 00:03:20,620
+الحالة الثانية اللي هي بتطلع limit x لما x تروح ل
+
+37
+00:03:20,620 --> 00:03:24,330
+0 برضه أعطتنا إيه شمالها real number في الحالة اللي
+
+38
+00:03:24,330 --> 00:03:28,670
+بعدها أعطتنا اللي هو عبارة عن limit 1 على x تربيع
+
+39
+00:03:28,670 --> 00:03:32,790
+لما x تروح للصفر يعني أعطتنا إيش مالها زائد ما لا نهاية
+
+40
+00:03:32,790 --> 00:03:37,070
+في الحالة الثالث الأخيرة هتعطينا اللي هو
+
+41
+00:03:37,070 --> 00:03:42,770
+ناقص limit 1 على x تربيع لما x تروح للزيرو بمعنى
+
+42
+00:03:42,770 --> 00:03:46,700
+آخر سالب ما لا نهاية يعني الـ Indeterminate Form Zero
+
+43
+00:03:46,700 --> 00:03:51,880
+على Zero أعطتنا اللي هو أجوبة أو قيم مختلفة تابعًا
+
+44
+00:03:51,880 --> 00:03:56,280
+لطبيعة كل حالة من الحالات اللي موجودة مرة أعطتنا
+
+45
+00:03:56,280 --> 00:03:59,700
+doesn't exist مرة أعطتنا بـ ما لا نهاية ومرة أعطتنا بـ ما لا
+
+46
+00:03:59,700 --> 00:04:03,640
+نهاية وواحد خمسة ستة ناقص واحد اللي بدناها اللي هو
+
+47
+00:04:03,640 --> 00:04:09,020
+ناقص Infinity و Infinity الآن ال ... ال ... ال ... ال
+
+48
+00:04:09,020 --> 00:04:13,780
+... الـ Indeterminate form هذه اللي الآن يعني بدنا
+
+49
+00:04:13,780 --> 00:04:19,780
+نحاول نعالجها ب ... 
+
+50
+00:04:19,780 --> 00:04:25,400
+نحاول نعالجها بال ... بال ... بالـ L'Hôpital's Rule الـ
+
+51
+00:04:25,400 --> 00:04:28,640
+Indeterminate Form اللي عندنا اللي هو 0 على 0 طبعًا
+
+52
+00:04:28,640 --> 00:04:33,240
+في Indeterminate Form أخرى برضه هتعالجها اللي هي L'
+
+53
+00:04:33,240 --> 00:04:38,620
+Hôpital's Rule أو Rules اللي هي زي Infinity على
+
+54
+00:04:39,400 --> 00:04:43,280
+Infinity أيضًا هذول الشغلتين الأساسيات اللي
+
+55
+00:04:43,280 --> 00:04:47,480
+هتعالجها اللي بطل الزرول مباشرة بنظريات مباشرة
+
+56
+00:04:47,480 --> 00:04:51,240
+عليها أيضًا هتظهر لو ظهرت عندنا مثلًا Infinity ناقص
+
+57
+00:04:51,240 --> 00:04:55,640
+Infinity اللي هو Zero to Infinity Infinity to Zero
+
+58
+00:04:55,640 --> 00:05:01,160
+إلى آخره هذوله حالات أخرى اللي هو بنقدر نحولهم عن
+
+59
+00:05:01,160 --> 00:05:04,800
+طريق الـ ln أو عن طريق الـ exponential أو بطرق
+
+60
+00:05:04,800 --> 00:05:08,820
+معينة لاللي هي الـ formula هذه ومن ثم استخدام اللي
+
+61
+00:05:08,820 --> 00:05:12,240
+هو الـ L'Hôpital's Rule هذه عادة الشغلات اللي كانت
+
+62
+00:05:12,240 --> 00:05:16,460
+تعالجها اليمين اللي هو الـ calculus أو التفاضل اللي
+
+63
+00:05:16,460 --> 00:05:20,500
+أخدناه في سنة أولى أو سنة أولى أو سنة ثانية اطلع
+
+64
+00:05:20,500 --> 00:05:26,260
+لفوق نيجي الآن ناخد النظرية الأولى اللي هي لبتة
+
+65
+00:05:26,260 --> 00:05:30,540
+الزرولي الأولى الـ formula الأولى أو الصورة الأولى
+
+66
+00:05:30,540 --> 00:05:35,660
+نظرية بسيطة و نظرية مرت عليكم و إثباتها أيضًا
+
+67
+00:05:35,660 --> 00:05:41,200
+هتلاحظوا أنه اللي هو بسيط إيش النظرية بتقول؟ بتقول
+
+68
+00:05:41,200 --> 00:05:43,680
+ما يلي عندي
+
+69
+00:05:45,330 --> 00:05:51,250
+Let F be defined على الفترة المغلقة A وB ونفترض أن
+
+70
+00:05:51,250 --> 00:05:55,630
+F of A وG of A مايساوي؟ صفر ونفترض أن G of X لا
+
+71
+00:05:55,630 --> 00:06:00,470
+تساوي صفر في الفترة اللي هي بين A وB ونفترض لو
+
+72
+00:06:00,470 --> 00:06:04,420
+كانت F وG differentiable عند الـ Aو G' عند الـ A
+
+73
+00:06:04,420 --> 00:06:07,880
+لا يساوي صفر مفترضين G' لا يساوي صفر then the
+
+74
+00:06:07,880 --> 00:06:14,240
+limit of F على G at A exist وتساوي F' على G' و
+
+75
+00:06:14,240 --> 00:06:19,580
+إذا كان تحت كل هذا الشروط بيطلع عندي اللي هو ال
+
+76
+00:06:19,580 --> 00:06:24,320
+limit has exist و بالظبط هذا ال limit بيساوي F' على
+
+77
+00:06:24,320 --> 00:06:29,780
+G' of A على G' of A نشوف النظرية و نشوف برهان النظرية
+
+78
+00:06:29,780 --> 00:06:38,960
+theorem عندي ماخد الـ F و الـ G عبارة عن دوال من A
+
+79
+00:06:38,960 --> 00:06:47,280
+و B لعند R مفترض أن الـ F of A بيساوي الـ G of A
+
+80
+00:06:47,280 --> 00:06:53,680
+بيساوي إيش؟ بيساوي صفر و مفترض أن الـ G of X
+
+81
+00:06:58,040 --> 00:07:09,500
+لا تساوي 0 لكل X و M موجودة في الفترة A و B فرضنا
+
+82
+00:07:09,500 --> 00:07:16,260
+كمان F و G differentiable عند الـ A F prime
+
+83
+00:07:16,260 --> 00:07:22,740
+of A و G prime of A exists ستين تين وهذه لا تساوي
+
+84
+00:07:22,740 --> 00:07:29,900
+إيش لا تساوي صفر تحت هذه الظروف كلها بكون عندي limit
+
+85
+00:07:29,900 --> 00:07:38,410
+f of x على g of x as x بتروح لل a طبعًا ال a الفترة
+
+86
+00:07:38,410 --> 00:07:43,390
+اللي عندنا هي فترة من وين من عند a ل عند b إذا
+
+87
+00:07:43,390 --> 00:07:46,270
+أكيد ال x إذا تروح لل a ما فيش مجال لها ال x اللي
+
+88
+00:07:46,270 --> 00:07:49,210
+بتروح لل a اللي من وين من جهة اليمين لأنه هي
+
+89
+00:07:49,210 --> 00:07:52,450
+المنطقة اللي أنا عمال قاعد بشتغل فيها الفترة من a ل
+
+90
+00:07:52,450 --> 00:07:57,870
+b إذا ال x بتروح لل a من وين من اليمين هيساوي اللي
+
+91
+00:07:57,870 --> 00:08:06,930
+هو f prime عند ال a على g prime عند الـ A يعني
+
+92
+00:08:06,930 --> 00:08:09,910
+بمعنى آخر إيش اللي ... إيش ... إيش ... إيش ... كيف هم
+
+93
+00:08:09,910 --> 00:08:13,650
+نطبق هذه النظرية؟ كانت تعرض علينا limit نيجي
+
+94
+00:08:13,650 --> 00:08:19,650
+يقولنا أوجد ال limit لل F of X على G of X لما X
+
+95
+00:08:19,650 --> 00:08:25,090
+تروح لمين؟ لل A من اليمين نيجي الآن ال F of A نعود
+
+96
+00:08:25,090 --> 00:08:29,500
+تعويض مباشر ده طلعت عند 0 على 0 وكانت عندي الشروط
+
+97
+00:08:29,500 --> 00:08:32,520
+هذه مكتملة اللي هي الـF و الـG differentiable و
+
+98
+00:08:32,520 --> 00:08:36,080
+الـF prime و الـG prime موجودات عند الـA على طول
+
+99
+00:08:36,080 --> 00:08:42,420
+نحط هذه إيش بتساوي F prime of A على G prime of A
+
+100
+00:08:42,420 --> 00:08:48,880
+معايا فهذه اللي هي ... اللي هي ... كيفية تطبيق
+
+101
+00:08:48,880 --> 00:08:53,500
+النظرية نيجي لأهلنا برهان النظرية البرهان بسيط
+
+102
+00:08:53,500 --> 00:08:54,680
+عندي
+
+103
+00:08:59,270 --> 00:09:11,330
+خد عندي for x بين a و بين b لو جيت حسبت الهدف
+
+104
+00:09:11,330 --> 00:09:18,970
+الوجود هذه ال f of x على g of x إيش هتساوي؟
+
+105
+00:09:18,970 --> 00:09:28,250
+هتساوي f of x ناقص f of a على g of x ناقص g of a ليش
+
+106
+00:09:28,250 --> 00:09:31,110
+لأن ال f of a و ال d of a ليش ما عطينا إيها بساوي
+
+107
+00:09:31,110 --> 00:09:34,170
+صفر ليش عملت هيك؟ لأ بدأت أعمل أكتر من هي بدأت أعمل
+
+108
+00:09:34,170 --> 00:09:39,110
+أقسم هذا على x minus a وهذا على x ما لها minus a
+
+109
+00:09:40,060 --> 00:09:43,580
+طبيعي ال x لا تساوي ال a الآن أنا باخد ال limit
+
+110
+00:09:43,580 --> 00:09:47,620
+للجهتين وبأجرأ وبأخد موزع لأن أنا ضامن أن ال f
+
+111
+00:09:47,620 --> 00:09:51,260
+prime of A موجودة والـ G prime of A شمالها ومش هيك
+
+112
+00:09:51,260 --> 00:09:54,160
+وكمان الـ G prime of A لا تساوي 0 إذا كل أموري تمام
+
+113
+00:09:54,160 --> 00:09:58,220
+التمام إذا بأخد ال limit للجهتين لما x تروح لل a
+
+114
+00:09:58,220 --> 00:10:04,040
+من اليمين بيساوي ال limit لما x تروح لل a من اليمين
+
+115
+00:10:05,480 --> 00:10:09,560
+والشيء اللي تحت وزعت ليش وزعت؟ ضامن أن ال limit
+
+116
+00:10:09,560 --> 00:10:14,520
+exist وال limit اللي تحت كمان لا تساوي صفر هذه
+
+117
+00:10:14,520 --> 00:10:20,400
+اللي هي عبارة عن مين؟ هذه تعريف F prime of A وهذه
+
+118
+00:10:20,400 --> 00:10:26,660
+تعريف G prime of A بكون أنا حصلت على اللي هو اللي
+
+119
+00:10:26,660 --> 00:10:33,300
+بدي إياه هذه اللي هي النظرية الأولى في اللي هو هذا ال
+
+120
+00:10:33,300 --> 00:10:37,890
+section اللي يعني بتحذير بقول لك أنه أنت يعني انعرضت
+
+121
+00:10:37,890 --> 00:10:46,230
+عليك limit 17x ولا ... قداش؟ مش مشكلة x زيادة 17 أين
+
+122
+00:10:46,230 --> 00:10:51,750
+كانت بنفع x زيادة 17 على 2x زيادة 3 مثلًا لما x تروح
+
+123
+00:10:51,750 --> 00:10:55,500
+لمين للصفر هو مش بيقوم لما نشوف على طول ونروح
+
+124
+00:10:55,500 --> 00:11:00,960
+نفاضل بنفعش أنت يعني فتالت بيصير فاضل الجهتين
+
+125
+00:11:00,960 --> 00:11:06,340
+بيطلع واحد على اثنين لأ هو أنا بقول بتساوي الـ f
+
+126
+00:11:06,340 --> 00:11:10,720
+prime على g prime عند الـ zero لما نكون هدي zero و
+
+127
+00:11:10,720 --> 00:11:16,080
+هدي zero لكن لا هدي zero ولا هدي zero إذا بنفعش
+
+128
+00:11:16,260 --> 00:11:21,280
+تحديد هذا نقول بتساوي اللي هو limit من فعش نقول
+
+129
+00:11:21,280 --> 00:11:24,540
+بتساوي limit واحد على اثنين على اعتبار فضلنا
+
+130
+00:11:24,540 --> 00:11:28,800
+ويساوي نصف وهذا الكلام غير صحيح لأن ال limit زي ما
+
+131
+00:11:28,800 --> 00:11:32,800
+أنتم عارفين لهذا المقدار بالتعويض المباشر هو
+
+132
+00:11:32,800 --> 00:11:42,780
+عبارة عن 17 على 3 هذا كلام سهل ناخذ مثال تطبيقي
+
+133
+00:11:42,780 --> 00:11:48,310
+على النظرية اللي عندنا المثال التطبيقي برضه مثال
+
+134
+00:11:48,310 --> 00:11:56,530
+مباشر عرض علينا الآن example عرض
+
+135
+00:11:56,530 --> 00:12:04,830
+علينا بقول أو جد limit x تربيع زائد x على sin 2x
+
+136
+00:12:04,830 --> 00:12:09,530
+لما x تروح لمين للزيرو بالمناسبة، النظرية اللي قبل
+
+137
+00:12:09,530 --> 00:12:13,070
+بشوية حكينا عنها سواء كانت الـA اللي بتروح لها end
+
+138
+00:12:13,070 --> 00:12:17,150
+point أو نقطة داخلية أو حتى left end point بتظبط
+
+139
+00:12:17,150 --> 00:12:22,150
+عليها النظرية والبرهان similarly ماشي الحال؟ واضح
+
+140
+00:12:22,150 --> 00:12:28,430
+هاه؟ طيب، وإيه السبب؟
+
+141
+00:12:28,430 --> 00:12:34,390
+لأن الاختبار صفر على صفر وهذه differentiable وهذه
+
+142
+00:12:34,390 --> 00:12:38,970
+differentiable كل أمورها ميحة وكويسة ومش هي كمان
+
+143
+00:12:38,970 --> 00:12:43,730
+ولو فضلت هتلاقي اللي هنا لا يساوي صفر إذا على طول
+
+144
+00:12:43,730 --> 00:12:50,410
+بقول 2x عند ال zero بفاضل جاعد وبعوض يعني هذه
+
+145
+00:12:50,410 --> 00:12:56,950
+سميتها وكأنها F وهذه g f of x وهذه g of x بعوض f
+
+146
+00:12:56,950 --> 00:13:02,190
+prime of zero بعوض هنا g prime of zero الآن f
+
+147
+00:13:02,190 --> 00:13:06,490
+prime of zero 2x زائد 1 في صفر بيصير 2 في صفر زائد 1 
+
+148
+00:13:06,490 --> 00:13:11,850
+وتحت اللي هو تفاضلها 2 cos 
+
+149
+00:13:11,850 --> 00:13:16,970
+2x بيصير 2 cos 2x وعوض بصفر بيصير
+
+150
+00:13:16,970 --> 00:13:25,500
+2 cos 2 في صفر وهذا يعني بيساوي اللي هو
+
+151
+00:13:25,500 --> 00:13:30,140
+واحد على اثنين على اعتبار كوصالة Zero بيساوي 1
+
+152
+00:13:30,140 --> 00:13:36,760
+هذه اللي هي تطبيق النظرية اللي عندي نيجي الآن للـ
+
+153
+00:13:36,760 --> 00:13:40,480
+Cauchy Mean Value Theorem الـ Cauchy Mean Value
+
+154
+00:13:40,480 --> 00:13:46,140
+Theorem تعميم للـ Mean Value Theorem اللي احنا
+
+155
+00:13:46,140 --> 00:13:52,050
+عارفينها بدل ما هو على دالة نحكي عن إيش عن دالتين
+
+156
+00:13:52,050 --> 00:14:00,930
+نشوف إيش اللي بيقوله النظرية بتقول ما يلي لأن
+
+157
+00:14:00,930 --> 00:14:11,690
+theorem عند F و G دالتين من A و B مخدهم من A و B
+
+158
+00:14:11,690 --> 00:14:17,390
+لعند R جاي لي نفس شروط الـ mean value theorem
+
+159
+00:14:17,390 --> 00:14:22,150
+العادية بدل ما هي على دالة دالتين جاي لي F و G
+
+160
+00:14:22,150 --> 00:14:35,270
+continuous on A و B and differentiable on (A, B) ماشي
+
+161
+00:14:35,270 --> 00:14:42,230
+الحال ومعطيني أيضًا بقول لي الـ G prime لل X لا تساوي
+
+162
+00:14:42,230 --> 00:14:49,110
+صفر لكل X وين موجودة في الـA و الـB اللي أنا
+
+163
+00:14:49,110 --> 00:14:59,190
+بقول له النتيجة then there exist للنتيجة then then
+
+164
+00:14:59,190 --> 00:15:09,250
+هذا كله معطى if this hold then then
+
+165
+00:15:09,250 --> 00:15:18,160
+there exist C element in A و B such that G أو F
+
+166
+00:15:18,160 --> 00:15:25,820
+prime of C على G prime of C بيساوي F of B ناقص F of
+
+167
+00:15:25,820 --> 00:15:35,360
+A على G of B ناقص G of A الـ proof ده كلام سهل
+
+168
+00:15:35,360 --> 00:15:38,140
+كمان الـ proof نشوف كده
+
+169
+00:15:41,590 --> 00:15:46,050
+عندي يا جماعة أول شيء هو معطيني إيش مالها g prime
+
+170
+00:15:46,050 --> 00:15:51,650
+of x إيش مالها لا تساوي صفر إذا by Rolle's theorem
+
+171
+00:15:51,650 --> 00:15:58,490
+هيكون g of b لا تساوي مين g of a كيف أذكركم أذكركم
+
+172
+00:15:58,490 --> 00:16:03,950
+كيف الآن إيش Rolle's theorem كانت بتقول g من a و b
+
+173
+00:16:03,95
+
+201
+00:18:32,890 --> 00:18:38,260
+النتيجة اللي أنا بديها الآن هي الشكل اللي بديها بدي 
+
+202
+00:18:38,260 --> 00:18:45,680
+آخذ H of X let أو define H of X بساوي اللي هو
+
+203
+00:18:45,680 --> 00:18:52,740
+المقدار هذا اللي بديّه F of B ناقص F of A على G of
+
+204
+00:18:52,740 --> 00:18:57,910
+B ناقص G of A بتخلي المقدار هذا اللي هو يتصفر في
+
+205
+00:18:57,910 --> 00:19:00,650
+حالة ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+206
+00:19:00,650 --> 00:19:00,870
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+207
+00:19:00,870 --> 00:19:00,910
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+208
+00:19:00,910 --> 00:19:00,970
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+209
+00:19:00,970 --> 00:19:01,090
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+210
+00:19:01,090 --> 00:19:01,850
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+211
+00:19:01,850 --> 00:19:01,890
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+212
+00:19:01,890 --> 00:19:02,850
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+213
+00:19:02,850 --> 00:19:07,410
+.. ال .. ال .. ال .. ال .. ال .. ال ..
+
+214
+00:19:07,410 --> 00:19:12,650
+ال ..
+
+215
+00:19:12,650 --> 00:19:16,510
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+216
+00:19:16,510 --> 00:19:24,170
+ال .. ال ..
+
+217
+00:19:25,340 --> 00:19:30,660
+H of A بيطلع صفر لأن بدي اللي هو الجزء الثاني اللي
+
+218
+00:19:30,660 --> 00:19:39,320
+هييجيبلي قيمة مين؟ قيمة الـ F ناقص G، G of B ناقص G of
+
+219
+00:19:39,320 --> 00:19:44,060
+A أو G of X ناقص H of A عشان نروحهم مع بعض ناقص F
+
+220
+00:19:44,060 --> 00:19:50,880
+of X ناقص F of A الآن ليش عملت هيك؟ عشان أحصل H of
+
+221
+00:19:50,880 --> 00:19:53,180
+A، H of A هذه صفر
+
+222
+00:19:55,780 --> 00:20:02,140
+وهدي إيه إيه بيصير صفر، H of B إذا صارت عندي H of A
+
+223
+00:20:02,140 --> 00:20:09,360
+بساوي صفر و H of B حط H of B بيصير هدي بيه هدي
+
+224
+00:20:09,360 --> 00:20:13,780
+بتروح مع هدي بيصير F of B ناقص F of A ناقص F of B 
+
+225
+00:20:13,780 --> 00:20:16,800
+ناقص F of A بيروحوا مع بعض بيصير H برضه بيساوي صفر
+
+226
+00:20:16,800 --> 00:20:22,300
+إذا هدي أكيد برضه بتساوي H of B الآن
+
+227
+00:20:24,650 --> 00:20:28,050
+عندي هذا الـ differentiable و continuous وهذا الـ
+
+228
+00:20:28,050 --> 00:20:31,210
+differentiable و continuous على ما يناسبها من a و
+
+229
+00:20:31,210 --> 00:20:35,750
+b أو على الـ a و b، الـ a و b اللي هو open
+
+230
+00:20:35,750 --> 00:20:38,770
+differentiable وعلى الـ a و b closed continuous
+
+231
+00:20:38,770 --> 00:20:43,350
+تبعاً لها هتطلع إنه ثابت هذا و هدول ثوابت حيث إن
+
+232
+00:20:43,350 --> 00:20:50,330
+عندي هذا كله على بعضه continuous، صار عندي H is
+
+233
+00:20:50,330 --> 00:20:59,130
+continuous on A و B و differentiable on A و B open
+
+234
+00:20:59,130 --> 00:21:03,930
+لأنه أنا من رأس الدورة الـ H of X معرفها اللي هي
+
+235
+00:21:03,930 --> 00:21:10,740
+الـ H for every X element in A و B، H of X بيساوي كده
+
+236
+00:21:10,740 --> 00:21:15,360
+ماعيش يا بابا، صارت الآن أنا حققت كل شروط الـ roles
+
+237
+00:21:15,360 --> 00:21:19,780
+theorem اللي هي الـ H continuous على الـ A و B والـ 
+
+238
+00:21:19,780 --> 00:21:23,360
+differential على الـ A و B open أو H of A بيساوي و
+
+239
+00:21:23,360 --> 00:21:27,260
+H of B بيساوي اللي هي صفر، إذا حسب roles theorem
+
+240
+00:21:27,260 --> 00:21:31,720
+إذا by roles theorem by أو حتى by main value
+
+241
+00:21:31,720 --> 00:21:38,180
+theorem، by roles theorem there exists C element ال
+
+242
+00:21:38,180 --> 00:21:46,290
+A و B such that اتش برايم of c إيش هيساوي؟ صفر، هذا
+
+243
+00:21:46,290 --> 00:21:52,190
+حسب الـ mean value theorem معايا يا جماعة الآن
+
+244
+00:21:52,190 --> 00:21:57,130
+بتفاضل هذه، دلّنا، أوجد تفاضل هذه، تفاضل هذه، معليش
+
+245
+00:21:57,130 --> 00:22:04,760
+خلينا نكتبها لإنه ندخل على العالـ slide بدي أفاضل 
+
+246
+00:22:04,760 --> 00:22:09,740
+هذه، فاضلة H prime of X اللي هي عند الـ C بتساوي
+
+247
+00:22:09,740 --> 00:22:15,580
+صفر، صفر بساوي H prime of C تساوي فاضل هذه وأعوض 
+
+248
+00:22:15,580 --> 00:22:20,420
+عن الـ X بصفر لأن هذا ثابت وهذا ثابت، ده بيصير صفر
+
+249
+00:22:20,420 --> 00:22:23,420
+تفاضل هذا مع هذا لأن مع هذا بيصير G prime of X 
+
+250
+00:22:23,420 --> 00:22:29,220
+مضروب في هذا، إذا صار عندي بساوي F of B ناقص F of A 
+
+251
+00:22:30,080 --> 00:22:36,100
+على g of b ناقص g of a مضروب في مين؟ فاضلة في g
+
+252
+00:22:36,100 --> 00:22:40,920
+prime of x وأنا بدي أحسبها عند مين؟ عند c، إذا g
+
+253
+00:22:40,920 --> 00:22:45,320
+prime of c ناقص تفاضل هذه، هذا ثابت صفر لأن هذا 
+
+254
+00:22:45,320 --> 00:22:50,620
+قداش تفاضلها؟ ناقص f prime عند الـ x وأنا بأخذها عند
+
+255
+00:22:50,620 --> 00:22:56,280
+الـ c اللي هي h of c، فـ h prime of c فبيصير ناقص f 
+
+256
+00:22:56,280 --> 00:23:02,200
+prime of c، الآن بتندجل هذه على جهة هذه، بيصير عندي
+
+257
+00:23:02,200 --> 00:23:10,720
+الآن f prime of c نجلته هان وجسمتها جي برايم of c
+
+258
+00:23:10,720 --> 00:23:15,840
+لأن جي برايم of c لا تساوي صفر، بساوي f of b ناقص f 
+
+259
+00:23:15,840 --> 00:23:25,710
+of a على g of b ناقص g of a وهيك بكون حصلنا على C
+
+260
+00:23:25,710 --> 00:23:31,630
+في الـ A و B بحيث إنه هذه تتحقق وهو المطلوب، هذه اللي
+
+261
+00:23:31,630 --> 00:23:37,190
+بيسميها Cauchy Mean Value Theorem، هذه تعميم لمين؟
+
+262
+00:23:37,190 --> 00:23:41,590
+تعميم اللي هو الـ Mean Value Theorem بس خد اللي هو
+
+263
+00:23:41,590 --> 00:23:47,230
+G of X بتساوي X، G of X بيساوي X و G of X بيساوي X
+
+264
+00:23:47,230 --> 00:23:49,190
+اللي هي تحقق كل الشروط الـ differential
+
+265
+00:23:49,190 --> 00:23:53,470
+بالـ continuous و و الأخره، ماشي الحال بيصير عندي 
+
+266
+00:23:53,470 --> 00:23:58,110
+اللي هي في حالة G of X بتساوي X، معايا يا شباب؟
+
+267
+00:23:58,110 --> 00:24:02,450
+بيصير G of B و G of A وهذه اللي هي G prime اللي 
+
+268
+00:24:02,450 --> 00:24:05,510
+هي واحد فبيصير F prime of C بيساوي F of B ناقص F of 
+
+269
+00:24:05,510 --> 00:24:10,470
+A على B ناقص A، إذا F
+
+270
+00:24:15,650 --> 00:24:21,210
+أمسح البرهان بس الآن 
+
+271
+00:24:21,210 --> 00:24:31,930
+الـ note اللي عندي، الـ note كما يلي، note if
+
+272
+00:24:31,930 --> 00:24:42,820
+g of x بيساوي x then we get from الـ theorem هذه
+
+273
+00:24:42,820 --> 00:24:51,140
+اللي بقول عليها كوشي mean value theorem، we get the
+
+274
+00:24:51,140 --> 00:24:59,500
+mean value theorem، كيف الـ g of x بيساوي x معناه
+
+275
+00:24:59,500 --> 00:25:04,160
+إنّه الـ g of b بيساوي b والـ g of a بيصير a يعني
+
+276
+00:25:04,160 --> 00:25:10,530
+هذا بيصير a, b وهذه بيصير A و G prime of C اللي هي
+
+277
+00:25:10,530 --> 00:25:14,070
+بيصير واحد فبيصير F prime of C بيساوي F of B ناقص F 
+
+278
+00:25:14,070 --> 00:25:20,590
+of A على B ناقص A وهذه هي الـ main value theorem
+
+279
+00:25:21,280 --> 00:25:24,720
+واضح؟ اه، طيب اطلع للي بعده لإنه احنا اللي 
+
+280
+00:25:24,720 --> 00:25:26,820
+بيهمنا هذا الكوشي بين ال value theorem احنا 
+
+281
+00:25:26,820 --> 00:25:31,980
+حكيناها أصلاً عشان خاطر إنه احنا نحكي عن اللي هو
+
+282
+00:25:31,980 --> 00:25:37,960
+L'Hopital's rule، الـ form اللي عنده اللي أمامي دكتور
+
+283
+00:25:37,960 --> 00:25:41,860
+جديد في السؤال، في غيرك؟ مفهوم إنه أنا جسمت على X
+
+284
+00:25:41,860 --> 00:25:46,640
+ناقص A على E ناقص A الطرفين أصلاً أنت بتصير F أو
+
+285
+00:25:46,640 --> 00:25:48,580
+P ناقص F أو A على B ناقص A
+
+286
+00:26:01,110 --> 00:26:06,630
+البرهان مش صعب اللي قلناه، كثير سهل، ما هو الـ C 
+
+287
+00:26:06,630 --> 00:26:09,970
+اللي لاجيناها في الحالة الأولى، هل هي الثانية؟ إذا
+
+288
+00:26:09,970 --> 00:26:13,390
+كنت تتعامل بـ Mean Value Theorem أنت؟ مش بدك تطبّق
+
+289
+00:26:13,390 --> 00:26:16,170
+الـ Value Theory؟ بدك تطبّق الـ Value Theory لأن
+
+290
+00:26:16,170 --> 00:26:21,090
+there exists c1 such that f prime of c1 بيساوي f of 
+
+291
+00:26:21,090 --> 00:26:24,030
+b ناقص f of a على b ناقص a ما فيش فيها مشكلة
+
+292
+00:26:24,030 --> 00:26:27,950
+there exists c2 such that g prime of c2 بيساوي g of 
+
+293
+00:26:27,950 --> 00:26:32,550
+b ناقص g of a على b ناقص a، الـ c1 هذه مش شرط هيكون 
+
+294
+00:26:32,550 --> 00:26:39,630
+هي الـ c2 أو احنا لازم نثبتها هي نفسها اه ف .. و
+
+295
+00:26:39,630 --> 00:26:44,780
+بعدين البرهان سهل ماشي، احنا الواحد يفكر 100% .. 
+
+296
+00:26:44,780 --> 00:26:50,780
+جميل لكن .. يعني .. بس إنه احنا .. الـ C هذه مش
+
+297
+00:26:50,780 --> 00:26:53,700
+ضمنية تساوي الـ C2 وبدنا نثبت أنها تساويها، إذا
+
+298
+00:26:53,700 --> 00:26:59,920
+كانت بتساويها طيب، نيجي لها اللي هي النظرية اللي
+
+299
+00:26:59,920 --> 00:27:05,840
+بعيدها، النظرية اللي بعيدها، أنا مرضيش أمسح اللوح 
+
+300
+00:27:05,840 --> 00:27:08,800
+أساسي، منقررها من الـ theory من الأولى
+
+301
+00:27:11,650 --> 00:27:30,050
+نشوف نطلع إيش اللي بتقوله هذه النظرية، نفس
+
+302
+00:27:30,050 --> 00:27:35,710
+اللي هنا F و G differentiable on A و B، معايا أه؟
+
+303
+00:27:35,710 --> 00:27:42,150
+الآن such that G prime of X ده تساوي صفر، G prime of 
+
+304
+00:27:42,150 --> 00:27:47,970
+X لا تساوي صفر، أنا مش موجودة، For all X elements in 
+
+305
+00:27:47,970 --> 00:27:55,070
+A و B ونفترض إنه limit F of X، limit F of X لما X
+
+306
+00:27:55,070 --> 00:27:59,570
+تروح إلى الـ A من اليمين موجودة وبتساوي إيش؟ صفر
+
+307
+00:27:59,570 --> 00:28:02,490
+و limit G of X لما X تروح إلى الـ A من اليمين
+
+308
+00:28:02,490 --> 00:28:09,580
+بتساوي صفر، إذا صار limit الحاصل القسمة لأن بيساويش f 
+
+309
+00:28:09,580 --> 00:28:14,300
+prime of a على g prime of a بيساويش limit f of x 
+
+310
+00:28:14,300 --> 00:28:19,580
+على g of x لما x تروح لل a باليمين، يعني وكأنه هنا
+
+311
+00:28:19,580 --> 00:28:25,460
+حول الحديث كله من افتراض إنه عند النقطة اللي هي F'
+
+312
+00:28:25,780 --> 00:28:31,240
+و G' موجودة ولا تساوي صفر وحول الحديث من إن التعويض
+
+313
+00:28:31,240 --> 00:28:35,280
+المباشر إن F of A و G of A بتساوي صفر لـ limit لـ ..
+
+314
+00:28:35,280 --> 00:28:39,220
+limit لل function لما X تروح لـ A صفر و limit لل
+
+315
+00:28:39,220 --> 00:28:42,040
+function لما .. اللي .. اللي هو الـ X .. الـ X بتروح
+
+316
+00:28:42,040 --> 00:28:48,920
+لسفر .. للـ A بتساوي صفر، وحول أيضا طلب جي برايم نفسه
+
+317
+00:28:48,920 --> 00:28:54,680
+على كل الفترة اللي تكون لا تساوي صفر مش عند النقطة 
+
+318
+00:28:54,680 --> 00:29:03,060
+بس، وقال لك إنه حتى اللي هو النتيجة هتطلع بـ limit
+
+319
+00:29:03,060 --> 00:29:07,940
+مش هتطلع بـ F prime of A على G prime of A، يعني اللي
+
+320
+00:29:07,940 --> 00:29:14,980
+بقصده إنه الآن طور هذه الحديث هنا إنه لما تعرض
+
+321
+00:29:14,980 --> 00:29:21,580
+علينا limit F of X على G of X لما X تروح للـ A من
+
+322
+00:29:21,580 --> 00:29:24,360
+لمين؟ أو إن كانت حتى لو في الـ interior point الـ X 
+
+323
+00:29:24,360 --> 00:29:31,300
+بتروح للـ A برضه صحيحة، الآن بعدي عندي إذا جيت limit
+
+324
+00:29:31,300 --> 00:29:34,480
+الأولى، اللي أنا بحكي عن limit سيرة مش عن تعويض
+
+325
+00:29:34,480 --> 00:29:38,340
+مباشر زي اللي جابله، الآن limit f of x لما x تروح 
+
+326
+00:29:38,340 --> 00:29:41,780
+ل a من اليمين و limit f of x لما x تروح ل a من 
+
+327
+00:29:41,780 --> 00:29:46,700
+اليمين، إذا هذه موجودة وهذه موجودة وطلع عندي 0
+
+328
+00:29:46,700 --> 00:29:51,060
+على 0، يعني طلع ال limit عبارة عن 0 على 0، هنا بدو 
+
+329
+00:29:51,060 --> 00:29:57,200
+يحدث العلاج، إذا كانت الآن الـ limit اللي طلعت عندي
+
+330
+00:29:57,200 --> 00:30:02,000
+F prime of X على G prime of X لما X تروح للـ A من 
+
+331
+00:30:02,000 --> 00:30:06,780
+اليمين، إذا طلعت عبارة عن قيمة خلاص ارتاح هذه اللي
+
+332
+00:30:06,780 --> 00:30:12,450
+طلعت هي مين؟ فهمت الـ limit، ماشي الحال؟ لو طلعت كمان
+
+333
+00:30:12,450 --> 00:30:18,230
+مرة zero على zero اللي هو وبتحقق كل الشروط اللي 
+
+334
+00:30:18,230 --> 00:30:21,950
+.. اللي في الأول برضه بأعمل كمان مرة بفاضل لما 
+
+335
+00:30:21,950 --> 00:30:26,430
+بتطلع، لكن لو طلعت الـ limit هذه does not exist بسكت 
+
+336
+00:30:26,430 --> 00:30:31,130
+وبجوبي عمش، بدنا ندور على طريقة أخرى واضح؟ الآن
+
+337
+00:30:31,130 --> 00:30:34,990
+لو طلعت هذه infinity أو سالب infinity هذه آسف
+
+338
+00:30:34,990 --> 00:30:38,900
+infinity أو سالب infinity برضه إنّ نظرية صحيحة، اللي
+
+339
+00:30:38,900 --> 00:30:45,290
+هو هدّمين الجزء الثاني من النظرية، if limit f prime 
+
+340
+00:30:45,290 --> 00:30:48,450
+على g prime بساوي L بساوي infinity أو سالب
+
+341
+00:30:48,450 --> 00:30:52,190
+infinity، هتطلع الـ limit على طول للـ F على G اللي
+
+342
+00:30:52,190 --> 00:30:56,350
+ببحث عنها إيش هتساوي؟ برضه الـ infinity أو سالب
+
+343
+00:30:56,350 --> 00:31:00,670
+infinity حسب القيمة هذه، إذا أيّ إن كانت اللي هي الـ
+
+344
+00:31:00,670 --> 00:31:05,190
+limit مادامت exist سواء الـ existence عبارة عن 
+
+345
+00:31:05,190 --> 00:31:09,210
+element in R أو اللي هو عبارة عن ناقص infinity أو 
+
+346
+00:31:09,210 --> 00:31:18,100
+سالب infinity فإنّ النظرية صحيحة، واضح؟ أي سؤال؟ طيب
+
+347
+00:31:18,100 --> 00:31:27,200
+صلّى على النبي عليه الصلاة والسلام، خلينا 
+
+348
+00:31:27,200 --> 00:31:31,360
+نيجي للنظرية ونبرهنها 
+
+349
+00:31:45,860 --> 00:31:50,880
+Theorem كدهش؟ نزلي بس النص يا محمد اللي هي Theorem
+
+350
+00:31:50,880 --> 00:31:59,580
+6 3 3 إيش النظرية بتقول؟ بتقول، ما يعني عندي طبعاً 
+
+351
+00:31:59,580 --> 00:32:04,540
+ماخذ الـ a أصغر من b strictly والـ a ممكن حتى تاخد سالب
+
+352
+00:32:04,540 --> 00:32:07,580
+infinity والـ b تاخد infinity يعني ممكن تكون
+
+353
+00:32:07,580 --> 00:32:11,040
+الفترة من a .. الفترة كلها a ممكن تكون أي فترة sub
+
+354
+00:32:11,040 --> 00:32:16,710
+interval من اللي هو 100 من الـ real numbers، فرضنا f 
+
+355
+00:32:16,710 --> 00:32:26,190
+و g من a و b لعند اللي هو r وإذا كانت a infinity 
+
+356
+00:32:26,190 --> 00:32:30,130
+أو سالب infinity أسف، إذا كانت a سالب infinity أو b
+
+357
+00:32:30,130 --> 00:32:33,610
+infinity بتكون o من زمن بعرفي لأن عشان اللي هو
+
+358
+00:32:33,610 --> 00:32:36,890
+ناخدها من real number لـ real number لذا نفترض f و
+
+359
+00:32:36,890 --> 00:32:46,080
+g من a و b لعند الـ R ونفترض إنّ f و g differentiable
+
+360
+00:32:46,080 --> 00:32:54,620
+on a و b ماشي الحال و differentiable on a و b الآن 
+
+361
+00:32:54,620 --> 00:32:57,900
+مش لازم ي continue وتعندي وكده لإنه أنا هدخل 
+
+362
+00:32:57,900 --> 00:33:02,240
+لجوه شغل هيكون لجوه هل جيت بتشوف واش معنى لجوه 
+
+363
+00:33:02,240 --> 00:33:08,620
+such that g prime of x، g prime of x لا تساوي صفر
+
+364
+00:33:08,620 --> 00:33:17,430
+لكل x وين موجودة في الفترة a و b، الآن بقول لي إذا 
+
+365
+00:33:17,430 --> 00:33:27,170
+كانت limit، limit اللي هي f of x لما x تروح لل a من
+
+366
+00:33:27,170 --> 00:33:32,190
+اليمين بساوي limit g of x لما x تروح لل a من 
+
+367
+00:33:32,190 --> 
+
+401
+00:36:49,960 --> 00:36:54,640
+عليه في ال .. في .. اللي هو التعبير عن ال .. لكل x
+
+402
+00:36:54,640 --> 00:36:58,940
+في الجوار الجوار هذا سهل و .. و .. و .. وعبّرنا 
+
+403
+00:36:58,940 --> 00:36:59,200
+عنه
+
+404
+00:37:02,810 --> 00:37:07,630
+-A هو أكبر من صفر وأصغر من دلتا برضه هيك صح صحيح
+
+405
+00:37:07,630 --> 00:37:11,910
+نفس الشيء طيب
+
+406
+00:37:11,910 --> 00:37:15,050
+الآن
+
+407
+00:37:15,050 --> 00:37:23,590
+هذا الكلام  IA أو بمعنى آخر that is هيكون هذا اللي
+
+408
+00:37:23,590 --> 00:37:31,230
+هو F prime of X ناقص G prime of X أصغر من ناقص L
+
+409
+00:37:31,230 --> 00:37:37,150
+بتغيرها عشان أصغر من Y وأكبر من 200 من سالب Y شيل
+
+410
+00:37:37,150 --> 00:37:43,470
+الـ L هذه بصير اللي هو أصغر من L زائد Y وأكبر من L
+
+411
+00:37:43,470 --> 00:37:49,430
+ناقص Y هذا متحقق لمين؟ لكل X و N موجودة في الفترة
+
+412
+00:37:49,430 --> 00:37:53,550
+من A لعند A زائد Delta واكتب لي هذا سميلي
+
+413
+00:37:56,530 --> 00:38:03,990
+واحد ماشي الحال سميليه واحد الآن عندي اللي هو شروط
+
+414
+00:38:03,990 --> 00:38:06,850
+الـ mean value theorem  الكوشي mean value theorem
+
+415
+00:38:06,850 --> 00:38:10,130
+اللي قبل بشوية الـ F والـ G differentiable على الـ A
+
+416
+00:38:10,130 --> 00:38:16,170
+والـ B اه والـ G prime of X لها تساوي صفر والـ ..
+
+417
+00:38:16,170 --> 00:38:22,490
+أو .. ايش كمان؟ كله متحقق بس خلينا نقول 
+
+418
+00:38:26,140 --> 00:38:31,240
+بدي الآن أوين أطبق الـ mean value theorem بدي
+
+419
+00:38:31,240 --> 00:38:37,720
+أطبقها على اللي هو اللي جوا عندي لداخل الـ mean
+
+420
+00:38:37,720 --> 00:38:44,560
+لداخل الفترة هذه عشان أشتغل وين اللي بتطلع عندي 
+
+421
+00:38:44,560 --> 00:38:48,490
+أضمن تكون هنا عشان القيمة اللي هتطلع عندي اللي
+
+422
+00:38:48,490 --> 00:38:53,170
+بتحققها تكون بتحقق اللي بتطلع عندي وبتحققها دي 
+
+423
+00:38:53,170 --> 00:38:57,730
+عشان هي كصح اللي أعوض مكان بعض ماشي الحال خد الآن
+
+424
+00:38:57,730 --> 00:39:07,030
+لجد تفهموا ايش اللي بقصدهم خد الآن four alpha أكبر 
+
+425
+00:39:07,030 --> 00:39:16,040
+من a وأصغر من beta وأصغر من a زائد delta يعني أنا
+
+426
+00:39:16,040 --> 00:39:20,060
+غرضي إن أنا أشتغل .. إن أنا رايح .. بدي أنا limit
+
+427
+00:39:20,060 --> 00:39:24,580
+أصلا والـ limit بديها لما اروح لمين؟ للـ A يعني بدي
+
+428
+00:39:24,580 --> 00:39:29,760
+في الجوار اللي عيمين الـ A وجاي ناحيتها هذا اللي
+
+429
+00:39:29,760 --> 00:39:31,740
+بهم، اللي ما نيش سغاداش، بيكون التصرف ما فيش ..
+
+430
+00:39:31,740 --> 00:39:36,980
+ما فيش عنده مشكلة الآن هي الـ alpha أخدتها هنا وهي
+
+431
+00:39:36,980 --> 00:39:37,420
+الـ beta
+
+432
+00:39:41,900 --> 00:39:44,540
+أصغر من Alpha أصغر من Beta أصغر من ايه؟ زي الـ
+
+433
+00:39:44,540 --> 00:39:54,060
+Delta؟ By Cauchy Mean Value Theorem there exists
+
+434
+00:39:54,060 --> 00:40:03,220
+سموها هو نسميها U Element in mean in Alpha و Beta
+
+435
+00:40:03,220 --> 00:40:10,110
+والـ Alpha و Beta جزء من هدول، إذا اللي بنطبق 
+
+436
+00:40:10,110 --> 00:40:14,750
+على هذه اللي بنطبق على .. اللي بنطبق على هذه بنطبق
+
+437
+00:40:14,750 --> 00:40:19,790
+على هذه، مظبوط؟ يعني هذه الـ U اللي لجيتها بنطبق
+
+438
+00:40:19,790 --> 00:40:22,210
+عليها الكلام هذا اللي هو F prime of U على D prime
+
+439
+00:40:22,210 --> 00:40:27,510
+of U بين هذه وبين هذه واضح، وهذا الكلام مهم طيب،
+
+440
+00:40:27,510 --> 00:40:34,640
+there exists U such that F prime of U على g prime
+
+441
+00:40:34,640 --> 00:40:39,240
+of u بيساوي ايش يا جماعة؟ بيساوي f of b أو beta
+
+442
+00:40:39,240 --> 00:40:50,260
+ناقص f of alpha على g of beta ناقص g of alpha واضح
+
+443
+00:40:50,260 --> 00:41:00,020
+اه؟ هذا سميلي يمين هو اتنين الآن عند هذه اللي 
+
+444
+00:41:00,020 --> 00:41:05,260
+لجيتها هنا اللي بتحقق هذه هي وين موجودة؟ من ضمن 
+
+445
+00:41:05,260 --> 00:41:12,980
+النقاط اللي بتحقق هذه لكل X هنا وهذه جزء منها إذا
+
+446
+00:41:12,980 --> 00:41:18,100
+F prime of X of U على G prime of U بين الـ L ناقص
+
+447
+00:41:18,100 --> 00:41:22,020
+إبسلون والـ L زائد مين؟ إبسلون وفي نفس الوقت F
+
+448
+00:41:22,020 --> 00:41:25,860
+prime of U هذه اللي لقيتها G prime of U بيساوي هذا 
+
+449
+00:41:25,860 --> 00:41:29,840
+إذا from واحد
+
+450
+00:41:30,390 --> 00:41:40,070
+واتنين we get اللي هو F prime of U على G prime of
+
+451
+00:41:40,070 --> 00:41:47,610
+U بستبدلها وبيصير F of Beta ناقص F of Alpha على G
+
+452
+00:41:47,610 --> 00:41:54,070
+of Beta ناقص G of Alpha بحيث إنه هذا يكون أكبر من 
+
+453
+00:41:54,070 --> 00:42:03,630
+L ناقص Y وأصغر من 100 من L زائد Y لأن هذا صحيح لأي
+
+454
+00:42:03,630 --> 00:42:09,750
+Alpha و Beta بشكلهم اللي موجود Alpha أصغر من Beta
+
+455
+00:42:09,750 --> 00:42:15,570
+و Alpha بين الـ A وبين مين؟ الـ A زائد دلتا إذا
+
+456
+00:42:15,570 --> 00:42:22,150
+الـ alpha هذه حرة في كل المنطقة هذه بنفع يعني الـ alpha
+
+457
+00:42:22,150 --> 00:42:26,530
+هذه لو بدها تروح للـ A حد مش بمنحها الـ alpha تروح لها 
+
+458
+00:42:26,530 --> 00:42:31,060
+لأنه صحيح على كل هذه الـ alpha اللي لجيتها هذه الآن
+
+459
+00:42:31,060 --> 00:42:40,340
+let alpha goes to mean to a من وين؟ من اليمين
+
+460
+00:42:40,340 --> 00:42:45,720
+ماشي؟ مقدر ولن تؤثر لأ على الـ beta ولا عليها زي
+
+461
+00:42:45,720 --> 00:42:50,980
+الدلتية الحرة بتروح لهذه وهذه زي ما هي يعني تصرف
+
+462
+00:42:50,980 --> 00:42:58,540
+alpha يروح لأي a beta بالنسبة لها ثابت ولا تتأثرهذا
+
+463
+00:42:58,540 --> 00:43:03,500
+الكلام مهم إذا صار عندي الآن لما Alpha تروح لأ من
+
+464
+00:43:03,500 --> 00:43:10,700
+اليمين الـ up of Alpha بيصير عندي limit الـ up of 
+
+465
+00:43:10,700 --> 00:43:17,780
+Alpha لما الـ Alpha تروح لأ من اليمين بيساوي هو في 
+
+466
+00:43:17,780 --> 00:43:21,980
+الواقع limitها limitها هذه Alpha اللي كنا نسميها X 
+
+467
+00:43:21,980 --> 00:43:23,980
+مثلا الـ up of X سمى X تروح لأ من اليمين لأ من
+
+468
+00:43:23,980 --> 00:43:29,480
+اليمين الآن الـ Alpha راحت لليمين اليمين limitها
+
+469
+00:43:29,480 --> 00:43:35,020
+أنا ما عطينيها ايش بيساوي؟ بيساوي صفر، ما عطيني
+
+470
+00:43:35,020 --> 00:43:37,080
+limit f of x عندما x تروح لليمين اليمين ايش 
+
+471
+00:43:37,080 --> 00:43:40,540
+بيساوي؟ صفر، أنا سميتها ايش أنا؟ Alpha هذه اللي
+
+472
+00:43:40,540 --> 00:43:46,660
+بتتحرك، إذا هذا الـ limit ايش هيساوي؟ بنفس السبب أو
+
+473
+00:43:46,660 --> 00:43:52,940
+لنفس السبب limitG of Alpha لما Alpha تروح للـ A من
+
+474
+00:43:52,940 --> 00:43:59,680
+اليمين برضه مش هيساوي صفر إذا الآن برجع لهذه برجع
+
+475
+00:43:59,680 --> 00:44:09,060
+لهذه بصير عندي الآن اللي حصلته كله على بعضه لكل ي
+
+476
+00:44:09,060 --> 00:44:17,130
+أكبر من صفر ماشي الحال ل gate Delta بحيث إنه الـ α 
+
+477
+00:44:17,130 --> 00:44:20,050
+و beta بشكل هذا أكبر من ايه وأقل زي الـ delta
+
+478
+00:44:20,050 --> 00:44:24,570
+سيبك من الـ L α خلاص رديتها أنا beta أكبر من ايه وأصغر من ايه؟ زي الـ delta طلع عندي هذا المقدار أصغر
+
+479
+00:44:24,570 --> 00:44:29,170
+من L زي الـ يبسلون وأكبر من L ناقص يبسلون ماشي؟
+
+480
+00:44:29,170 --> 00:44:33,170
+الآن رديت وأخدتله من الـ limit الآن بدي أخد الـ 
+
+481
+00:44:33,170 --> 00:44:37,300
+limit لهذا المقدار كله لما الـ alpha تروح لمين؟ للـ
+
+482
+00:44:37,300 --> 00:44:40,940
+a من اليمين الآن لما أخد الـ limit لهذا زي ما قلت
+
+483
+00:44:40,940 --> 00:44:45,240
+الـ beta مالهاش علاقة بالـ alpha الـ beta ثابتة
+
+484
+00:44:45,240 --> 00:44:48,500
+بالنسبة للـ alpha فالـ alpha تروح للـ a من اليمين زي
+
+485
+00:44:48,500 --> 00:44:50,920
+ما بدها لن تتأثر beta فبيصير limit اللي فوق على 
+
+486
+00:44:50,920 --> 00:44:55,100
+limit اللي اتحد هذا ثابت وهذا ثابت وهذا limit هو
+
+487
+00:44:55,100 --> 00:44:59,520
+اللي هو صفر وهذا limit هو صفر وهذول أعداد إذا
+
+488
+00:44:59,520 --> 00:45:04,300
+أصار عندي الآن عندي take the limit بيصير عندي L
+
+489
+00:45:04,300 --> 00:45:11,220
+ناقص Epsilon أصغر أو يساوي اللي هو F of Beta على G 
+
+490
+00:45:11,220 --> 00:45:18,100
+of Beta أصغر من L زائد Epsilon من وين حصلته هذا؟
+
+491
+00:45:18,100 --> 00:45:23,660
+when I take the limit of this inequality as Alpha
+
+492
+00:45:23,660 --> 00:45:29,160
+goes to A from right واضح؟ هذا حصلت عليه هذا عرفت
+
+493
+00:45:29,160 --> 00:45:37,460
+ثانوية هذا حصلت عليه من وين؟ إن أخدت الـ limit 
+
+494
+00:45:37,460 --> 00:45:40,600
+للجهات التلاتة as Alpha تروح للـ A من اليمين
+
+495
+00:45:40,600 --> 00:45:44,660
+واستخدمت هذه الحقيقة إن هذا صفر وهذا صفر صار هذا
+
+496
+00:45:44,660 --> 00:45:48,780
+المقدار وهذا المقدار بين هذا وهذا إذن اللي حصلت 
+
+497
+00:45:48,780 --> 00:45:53,360
+عليه الآن هو ما يليه لكل epsilon أكبر من سفر لجهة 
+
+498
+00:45:53,360 --> 00:46:02,380
+Delta أكبر من السفر such that هذه beta كانت
+
+499
+00:46:02,380 --> 00:46:07,610
+arbitrary بين a وبين مين؟ اللي هو such that if
+
+500
+00:46:07,610 --> 00:46:14,650
+beta بين ال a والـ a زائد delta we have then ايش
+
+501
+00:46:14,650 --> 00:46:26,150
+اللي حصلنا عليه اللي هو f of beta على f of alpha
+
+502
+00:46:26,150 --> 00:46:31,990
+صارت أصغر أو يساوي L زايد إبسلون وأكبر أو يساوي
+
+503
+00:46:33,100 --> 00:46:38,720
+إبسلون ناقص L أو بمعنى آخر IE F of Beta على F of
+
+504
+00:46:38,720 --> 00:46:47,620
+Alpha ناقص L absolute value أصغر أو يساوي إبسلون 
+
+505
+00:46:47,620 --> 00:46:53,340
+وهذا هذا this means that hence limit 
+
+506
+00:47:03,350 --> 00:47:13,590
+f of beta على f of alpha as اللي هو limit of فدي g
+
+507
+00:47:13,590 --> 00:47:23,970
+مالكم فدي g بساكتين g of beta فدي g of beta limit
+
+508
+00:47:23,970 --> 00:47:28,270
+of beta على g of beta مظبوط؟
+
+509
+00:47:29,140 --> 00:47:33,140
+as اللي هو طبعا الان بيت ايه اشمالها لكل بيتها
+
+510
+00:47:33,140 --> 00:47:37,620
+يعني بيتها وين راحت؟ على الايم اليمين هان لكل بيتها
+
+511
+00:47:37,620 --> 00:47:49,320
+وين؟ في الجوار هذا الان بيساوي الـ اللي هو المطلوب مش 
+
+512
+00:47:49,320 --> 00:47:53,440
+عاجبك بيتك تبقى اكسر أي سؤال
+
+513
+00:47:56,950 --> 00:48:01,030
+بتصور هيك المظهرية واضحة تماما هي تلت خطوات في
+
+514
+00:48:01,030 --> 00:48:05,530
+الواقع تلت خطوات مين هما؟ هو طبعا في الكتاب يعني لو
+
+515
+00:48:05,530 --> 00:48:12,190
+راحظت هتلاقي يعني إنه بدها بس ترتيب الآن لأ بدها 
+
+516
+00:48:12,190 --> 00:48:19,150
+ترتيب طلع عليها الآن عندك هذا الآن استخدمت أنا هذا 
+
+517
+00:48:19,150 --> 00:48:23,370
+في الأول بالعمدان واستخدم هذا استخدمت هذا عشان
+
+518
+00:48:23,370 --> 00:48:27,150
+أقول لك هذه الـ inequality صحيح على كل المنطقة هذه لأن
+
+519
+00:48:27,150 --> 00:48:33,170
+لجي U لأن اللي لجيتها U أنا لجيتها لجيت الـ U و
+
+520
+00:48:33,170 --> 00:48:36,070
+أخدت الفترة هنا عشان أقول لك الـ U اللي لجيتها في
+
+521
+00:48:36,070 --> 00:48:41,570
+داخل الفترة هذه الحديث هذه جابل عن هذه بعمل 
+
+522
+00:48:41,570 --> 00:48:45,750
+confusion عند الطالب لأن اللي لجيتها من قال إنه
+
+523
+00:48:45,750 --> 00:48:49,970
+هنا موجودة ما هي اللي لجيتها هنا؟ وين لجيتها؟ بين A
+
+524
+00:48:49,970 --> 00:48:56,050
+و B بين A و B أو بين Alpha و Beta يعني بده يصير 
+
+525
+00:48:56,050 --> 00:49:05,880
+عندي اللي هي الـ F prime الـ F prime عندي الـ limit 
+
+526
+00:49:05,880 --> 00:49:11,140
+.. الـ limit لل .. خليني أكتبها يا شيخ ليش؟ ها ..
+
+527
+00:49:11,140 --> 00:49:16,120
+هذا بيساوي ال .. اللي هي ايش؟ ما لأ باخد الـ
+
+528
+00:49:16,120 --> 00:49:20,740
+infinity بطلع حد برضه ايش؟ infinity يعني الحالة 
+
+529
+00:49:20,740 --> 00:49:26,250
+التانية اللي هي F بيساوي infinity هيكون الـ limit للـ
+
+530
+00:49:26,250 --> 00:49:33,570
+F على G اشماله بيساوي infinity الآن بده يصير عنده
+
+531
+00:49:33,570 --> 00:49:38,750
+بدل ما أقول limit F prime على G prime بيساوي L بده
+
+532
+00:49:38,750 --> 00:49:43,310
+يصير أشهر بيساوي infinity كيف بنعبر على إنه الرقم 
+
+533
+00:49:43,310 --> 00:49:47,190
+يروح لما لنهاية الـ limit إن ناخد اللي هو صرنا
+
+534
+00:49:47,190 --> 00:49:50,250
+متعارفين for every إبسط إنه كانت عبارة عن ايش؟ صغير
+
+535
+00:49:50,250 --> 00:49:55,190
+for every K Element in R طبعا لو أخدت K positive
+
+536
+00:49:55,190 --> 00:49:58,050
+برضه بنفعله لإذا .. إذا بيكون أكبر من الـ positive 
+
+537
+00:49:58,050 --> 00:50:01,170
+أكيد هيكون أكبر من مين؟ من الـ negative for every K
+
+538
+00:50:01,170 --> 00:50:05,450
+element in R there exists delta such that لكل X في 
+
+539
+00:50:05,450 --> 00:50:12,970
+هذه المنطقة بيطلع اللي هو F prime of X اللي هو هذا
+
+540
+00:50:12,970 --> 00:50:16,630
+.. بقى اللي بمسحه هذا بمسح من التعريف ذاك للتعريف
+
+541
+00:50:16,630 --> 00:50:20,920
+limit F prime على G prime ايش بيساوي؟ ما للهاي بما
+
+542
+00:50:20,920 --> 00:50:23,800
+إن هذا بيساوي ما لنهاي ايضا لأ كل K المتنارة بين X 
+
+543
+00:50:23,800 --> 00:50:28,200
+زي الزلتة such that لما تكون X يعني بين الـ A وبين 
+
+544
+00:50:28,200 --> 00:50:31,060
+Z زي الدلتة يعني روحت في الـ A من اليمين then F
+
+545
+00:50:31,060 --> 00:50:35,800
+برايم على D برايم أكبر من مين؟ من K ماشي الحالة و
+
+546
+00:50:35,800 --> 00:50:39,540
+هذا اللي هو الواحد عندي وهذا كله اللي هو كلام
+
+547
+00:50:39,540 --> 00:50:46,340
+اشماله نفس الشيء متحقق فبيصير عندي بستبدل هذا هذا
+
+548
+00:50:46,340 --> 00:50:50,270
+ما فيش داعي له بيصير التفاصيل هنا وبيصير عندي هذا 
+
+549
+00:50:50,270 --> 00:50:55,110
+لجيتو إذا from واحد عند اتنين we have اللي هو 
+
+550
+00:50:55,110 --> 00:51:01,370
+نعملها مع بعض بستبدل هذا بحطه هان بيصير عندي f 
+
+551
+00:51:01,370 --> 00:51:11,330
+prime f of beta ناقص f of alpha على g of beta ناقص
+
+552
+00:51:11,330 --> 00:51:17,220
+g of alpha أكبر من مين؟ من K ماشي الحال ونفس السبب
+
+553
+00:51:17,220 --> 00:51:21,260
+الأولاني الـ alpha اللي هو limit هذه صفر وlimit
+
+554
+00:51:21,260 --> 00:51:26,260
+هذه صفر المعطيلة هي هو بيصير هذه عبارة عن لما الـ 
+
+555
+00:51:26,260 --> 00:51:28,700
+alpha تروح للـ beta فبيصير عند أف في beta ودي في
+
+556
+00:51:28,700 --> 00:51:33,060
+beta أكبر من مين؟ من K صار عندي الآن لكل K element
+
+557
+00:51:33,060 --> 00:51:38,360
+in R لجهة Delta بحيث إنه لما تكون Beta بين الـ A و
+
+558
+00:51:38,360 --> 00:51:43,660
+A زائد Delta حصلت على هذه أكبر من مين؟ من K وهذا
+
+559
+00:51:43,660 --> 00:51:50,080
+اللي هو ايه اشماله؟ هو تعريف limit F of Beta على G
+
+560
+00:51:50,080 --> 00:51:58,480
+of Beta as Beta روح للـ A من اليمين سوى ملا نهاية 
+
+561
+00:51:58,480 --> 00:52:01,440
+ولو بدنا سالب ملا نهاية بنفس الأسلوب ده بتقول for 
+
+562
+00:52:01,440 --> 00:52:07,660
+every K K سالمة بيصير أصغر ونفس الكلام
+
+563
+00:52:07,660 --> 00:52:13,110
+example examples .. نشوف الـ examples اللي عندنا
+
+564
+00:52:13,110 --> 00:52:20,890
+نيجي للـ examples اللي عندي الأولى يعني limit
+
+565
+00:52:22,440 --> 00:52:28,260
+Sin X على جدر X لما X تروح لـ 0 من وين؟ من اليمين،
+
+566
+00:52:28,260 --> 00:52:33,740
+ليس بسهل، دعونا نشرح عن الوحش وبالكالكولس هذه لأن
+
+567
+00:52:33,740 --> 00:52:36,820
+limit Sin X على جدر X لما X تروح لـ 0 من وين؟ من
+
+568
+00:52:36,820 --> 00:52:40,620
+اليمين، الآن الـ limit اللي فوق لما X تروح لـ 0 من
+
+569
+00:52:40,620 --> 00:52:43,920
+اليمين، صفر واللي تحت صفر، إذا صار عبارة عن 0 على 
+
+570
+00:52:43,920 --> 00:52:4
+
+601
+00:54:28,600 --> 00:54:33,500
+الآن الأخيرة نفس الشيء لأن ال X على X minus واحد
+
+602
+00:54:33,500 --> 00:54:36,560
+برضه نفس الشيء Zero على Zero بيطلع عندي اللي هو
+
+603
+00:54:36,560 --> 00:54:39,240
+بالفاضل هدي بتطلع واحد على X بالفاضل هدي واحد
+
+604
+00:54:39,240 --> 00:54:44,220
+بيصير الآن لما ال X تروح للواحد بيساوي الواحد بطلع
+
+605
+00:54:44,220 --> 00:54:48,920
+لفوق بيكون وصلنا احنا عند مين؟ عند آخر نظرية اللي
+
+606
+00:54:48,920 --> 00:54:56,000
+هي L'Hopital's Rule اللي هي في حالة اللي هي إنّها تطلع 
+
+607
+00:54:56,000 --> 00:54:59,960
+عندي infinity أو ناقص infinity ال limit يعني ال
+
+608
+00:54:59,960 --> 00:55:03,280
+indeterminate form اللي هو infinity على infinity
+
+609
+00:55:03,280 --> 00:55:07,080
+أو ناقص infinity على infinity برضه المرة الجاية إن
+
+610
+00:55:07,080 --> 00:55:07,620
+شاء الله 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..e694fc6c5ca2d3aca7725d06087c34d91f9b3820
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg_postprocess.srt
@@ -0,0 +1,2440 @@
+1
+00:00:05,300 --> 00:00:13,060
+بسم الله الرحمن الرحيم المحاضرة السادسة مساق تحليل
+
+2
+00:00:13,060 --> 00:00:18,680
+حقيقة 2 لطلبة قسم رياضيات بكلية العلوم بالجامعة
+
+3
+00:00:18,680 --> 00:00:23,220
+الإسلامية بغزةالحديث اليوم إن شاء الله هيكون حول
+
+4
+00:00:23,220 --> 00:00:30,200
+ال Lobitals Rules أو قواعد Lobitals Lobitals Rules
+
+5
+00:00:30,200 --> 00:00:38,320
+بتمر في calculus في تفاضل وتكامل بتمر من ناحية
+
+6
+00:00:38,320 --> 00:00:45,100
+عملية استخدامها وتوظيفها لحل اللي هي بعض النهايات
+
+7
+00:00:45,100 --> 00:00:50,400
+اللي بيكون .. اللي هو عجزنا عن حلها بطرق عادية
+
+8
+00:00:51,750 --> 00:00:57,550
+هنشوف الآن في الحديث عن الـ Lobital Zerol حول اللي
+
+9
+00:00:57,550 --> 00:01:01,570
+هو كيف اللي هو انبرهن هذه اللي هي القواعد كيف نشتق
+
+10
+00:01:01,570 --> 00:01:06,310
+هذه القواعد كيف اللي هو أيضًا بشكل سريع حول اللي
+
+11
+00:01:06,310 --> 00:01:12,490
+هي استخدام هذه القواعد طبعًا
+
+12
+00:01:12,490 --> 00:01:16,430
+في الأول هنتحدث عن الكميات الغير معينة
+
+13
+00:01:17,020 --> 00:01:23,580
+indeterminate forms اللي هي اللي بتعالجها اللي هي
+
+14
+00:01:23,580 --> 00:01:28,840
+lobitals rules عندي على سبيل المثال لو جينا أخدنا
+
+15
+00:01:28,840 --> 00:01:37,090
+limitاللي هو x على x تكعيب على x تربيع مثلا as x
+
+16
+00:01:37,090 --> 00:01:44,730
+goes to zero limit alpha x على x لما x تروح للزيرو
+
+17
+00:01:44,730 --> 00:01:50,670
+limit اللي هو x تربيع على x لما x تروح للزيرو
+
+18
+00:01:50,670 --> 00:01:59,900
+limit x على x تكعيب لما x تروح للزيرو limitمثلًا
+
+19
+00:01:59,900 --> 00:02:07,100
+سلب X على X تكعيب لما X تروح لـ 0 لو طلعنا على
+
+20
+00:02:07,100 --> 00:02:11,440
+اللي هي ال limits اللي موجودة هنا كلها على صورة
+
+21
+00:02:11,440 --> 00:02:16,260
+اللي هي لو تعويض مباشر هنلاقيها على صورة 0 على 0
+
+22
+00:02:16,960 --> 00:02:21,420
+الان كمية 0 على 0 بالطرق السابقة كان اللي هو انه
+
+23
+00:02:21,420 --> 00:02:27,620
+احنا صعب اللي هو نتعامل معها لكن في بعض الأحيان زي
+
+24
+00:02:27,620 --> 00:02:30,520
+الحالة هذه حلاقي انه احنا بنعرف نتعامل معها و
+
+25
+00:02:30,520 --> 00:02:36,140
+بنعرف نحكم عليها اللي هلاحظ انه كل الرغم ان كلها 0
+
+26
+00:02:36,140 --> 00:02:41,100
+على 0 الا انها بتعطي في كل حالة اشي مختلف عن
+
+27
+00:02:41,100 --> 00:02:47,350
+الحالة الثانية الان هذه مثلا عبارة عن limit1 على X
+
+28
+00:02:47,350 --> 00:02:51,030
+لما X تروح للـ 0 طبعا 1 على X لما X تروح للـ 0 إيش
+
+29
+00:02:51,030 --> 00:02:53,710
+مالها؟ does not exist لأنه من اليمين بتعطي مالة
+
+30
+00:02:53,710 --> 00:02:56,790
+نهاية ومن اليسار بتعطي سالب مالة نهاية عشان هيك
+
+31
+00:02:56,790 --> 00:03:02,510
+كده يبقى نقول عنها does not exist لأن limit Alpha
+
+32
+00:03:02,510 --> 00:03:06,210
+X على X لما X تروح للـ 0 هو يساوي عبارة عن Alpha
+
+33
+00:03:06,960 --> 00:03:09,720
+اللي هو عبارة عن real number لو فرضنا أنه Alpha
+
+34
+00:03:09,720 --> 00:03:13,480
+ماخدينها احنا real number اذا انا اعطت number عادل
+
+35
+00:03:13,480 --> 00:03:16,320
+اول ما اعطتش اللي هي انا اعطت does not exist
+
+36
+00:03:16,320 --> 00:03:20,620
+الحالة الثانية اللي هي بتطلع limit X لما X تروح ل
+
+37
+00:03:20,620 --> 00:03:24,330
+0 برضه اعطتنا أيه شمالها real numberفي الحالة اللي
+
+38
+00:03:24,330 --> 00:03:28,670
+بعدها أعطتنا اللي هو عبارة عن limit 1 على X تربيع
+
+39
+00:03:28,670 --> 00:03:32,790
+لما X تروح للصفر يعني أعطتنا إيش مالها زائد ملا
+
+40
+00:03:32,790 --> 00:03:37,070
+نهاية في الحالة الثالث الأخيرة هتعطينا اللي هو
+
+41
+00:03:37,070 --> 00:03:42,770
+ناقص limit 1 على X تربيع لما X تروح للزيرو بمعنى
+
+42
+00:03:42,770 --> 00:03:46,700
+أخر سالب ملا نهايةيعني الـ Indeterminate Form Zero
+
+43
+00:03:46,700 --> 00:03:51,880
+على Zero أعطتنا اللي هو أجوبة أو قيم مختلفة تابعًا
+
+44
+00:03:51,880 --> 00:03:56,280
+لطبيعة كل حالة من الحالات اللي موجودة مرة أعطتنا
+
+45
+00:03:56,280 --> 00:03:59,700
+doesn't exist مرة أعطتنا بسوء سفر ومرة أعطتنا بسوء
+
+46
+00:03:59,700 --> 00:04:03,640
+ألف وواحد خمسة ستة ناقص واحد اللي بدناها اللي هو
+
+47
+00:04:03,640 --> 00:04:09,020
+ناقص Infinity و Infinityالان ال .. ال .. ال .. ال
+
+48
+00:04:09,020 --> 00:04:13,780
+.. الـ Indeterminate form هذه اللي الآن يعني بدنا
+
+49
+00:04:13,780 --> 00:04:19,780
+نحاول نعليجها ب ..
+
+50
+00:04:19,780 --> 00:04:25,400
+نحاول نعليجها بال .. بال .. بالـ Lobitals rulesالـ
+
+51
+00:04:25,400 --> 00:04:28,640
+Indeterminate Form اللي عندنا اللي هو 0 على 0 طبعا
+
+52
+00:04:28,640 --> 00:04:33,240
+في Indeterminate Form أخرى برضه هتعالجها اللي هي L
+
+53
+00:04:33,240 --> 00:04:38,620
+'Hôpital's Rule أو Rules اللي هي زي Infinity على
+
+54
+00:04:39,400 --> 00:04:43,280
+Infinity أيضًا هذول الشغلتين الأساسيات اللي
+
+55
+00:04:43,280 --> 00:04:47,480
+هتعالجه اللي بطل الزرول مباشرة بنظريات مباشرة
+
+56
+00:04:47,480 --> 00:04:51,240
+عليها أيضًا هتظهر لو ظهرت عندنا مثلًا Infinity نقص
+
+57
+00:04:51,240 --> 00:04:55,640
+Infinity اللي هو Zero to Infinity Infinity to Zero
+
+58
+00:04:55,640 --> 00:05:01,160
+إلى آخره هذوله حالات أخرىاللي هو بنقدر نحولهم عن
+
+59
+00:05:01,160 --> 00:05:04,800
+طريق الـ ln أو عن طريق ال exponential أو بطرق
+
+60
+00:05:04,800 --> 00:05:08,820
+معينة لاللي هي ال formula هذه ومن ثم استخدام اللي
+
+61
+00:05:08,820 --> 00:05:12,240
+هو ال lobitals rules هذه عادة الشغلات اللي كانت
+
+62
+00:05:12,240 --> 00:05:16,460
+تعالجها اليمين اللي هو ال calculus أو التفاضل اللي
+
+63
+00:05:16,460 --> 00:05:20,500
+أخدناه في سنة أولى أو سنة أولى أو سنة تانية اطلع
+
+64
+00:05:20,500 --> 00:05:26,260
+لفوق نيجي الآن ناخد النظرية الأولىاللي هي لبتة
+
+65
+00:05:26,260 --> 00:05:30,540
+الزرولي الأولى ال formula الأولى أو الصورة الأولى
+
+66
+00:05:30,540 --> 00:05:35,660
+نظرية بسيطة و نظرية مرت عليكم و اثباتها أيضا
+
+67
+00:05:35,660 --> 00:05:41,200
+هتلاحظوا انه اللي هو بسيط ايش النظرية بتقول؟ بتقول
+
+68
+00:05:41,200 --> 00:05:43,680
+مايالي عندي
+
+69
+00:05:45,330 --> 00:05:51,250
+لت F be defined على الفترة المغلقة A وB ونفترض أن
+
+70
+00:05:51,250 --> 00:05:55,630
+F of A وG of A مايساوي؟ سفر ونفترض أن G of X لا
+
+71
+00:05:55,630 --> 00:06:00,470
+تساوي سفر في الفترة اللي هي بين A وB ونفترض لو
+
+72
+00:06:00,470 --> 00:06:04,420
+كانت F وG differentiable عند الـ Aو G' عند الـ A
+
+73
+00:06:04,420 --> 00:06:07,880
+لا يساوي سفر مفترضين G' لا يساوي سفر then the
+
+74
+00:06:07,880 --> 00:06:14,240
+limit of F على G at A exist و تساوي F' على G' و
+
+75
+00:06:14,240 --> 00:06:19,580
+إذا كان تحت كل هذا الشروط بطلع عنده اللي هو ال
+
+76
+00:06:19,580 --> 00:06:24,320
+limit had exist و بالظبط هذا ال limit بساوي F' على
+
+77
+00:06:24,320 --> 00:06:29,780
+F A على G' of A نشوف النظرية و نشوف برهان النظرية
+
+78
+00:06:29,780 --> 00:06:38,960
+theoremعندي ماخد الـ F و الـ G عبارة عن دوال من A
+
+79
+00:06:38,960 --> 00:06:47,280
+و B لعند R مفترض أن الـ F of A بيساوي الـ G of A
+
+80
+00:06:47,280 --> 00:06:53,680
+بيساوي إيش؟ بيساوي سفر و مفترض أن الـ G of X
+
+81
+00:06:58,040 --> 00:07:09,500
+لاتساوى 0 لكل X و M موجودة في الفترة A و B فرضنا
+
+82
+00:07:09,500 --> 00:07:16,260
+كمان F و G differentiable عند الـ A F prime
+
+83
+00:07:16,260 --> 00:07:22,740
+of A و G prime of A exists ستين تين وهذه لاتساوى
+
+84
+00:07:22,740 --> 00:07:29,900
+إيش لاتساوى سفرتحت هذه الظروف كلها بكون عندى limit
+
+85
+00:07:29,900 --> 00:07:38,410
+f of xعلى g of x as x بتروح لل a طبعا ال a الفترة
+
+86
+00:07:38,410 --> 00:07:43,390
+اللي عندنا هي فترة من وين من عند a ل عند b إذا
+
+87
+00:07:43,390 --> 00:07:46,270
+أكيد ال x إذا تروح لل a مافيش مجال لها ال x اللي
+
+88
+00:07:46,270 --> 00:07:49,210
+بتروح لل a اللي من وين من جهة اليمين لأنه هي
+
+89
+00:07:49,210 --> 00:07:52,450
+العالم اللي أنا عمال قاعد بشتغل فيه الفترة من a ل
+
+90
+00:07:52,450 --> 00:07:57,870
+b إذا ال x بتروح لل a من وين من اليمين هيساوي اللي
+
+91
+00:07:57,870 --> 00:08:06,930
+هو f prime عند ال a على g primeعند الـ A يعني
+
+92
+00:08:06,930 --> 00:08:09,910
+بمعنى أخر إيش اللي .. إيش .. إيش .. إيش .. كيف هم
+
+93
+00:08:09,910 --> 00:08:13,650
+نطبّك هذه النظرية؟ كانت تعرض علينا limit نيجي
+
+94
+00:08:13,650 --> 00:08:19,650
+يقولنا أوجد ال limit لل F of X على G of X لما X
+
+95
+00:08:19,650 --> 00:08:25,090
+تروح لمين؟ لل A من الأمين نيجي الآن ال F of A نعود
+
+96
+00:08:25,090 --> 00:08:29,500
+تعويض مباشر ده طلعت عند 0 على 0وكانت عندى الشروط
+
+97
+00:08:29,500 --> 00:08:32,520
+هذه مكتملة اللى هى الـF و الـG differentiable و
+
+98
+00:08:32,520 --> 00:08:36,080
+الـF prime و الـG prime موجودات عند الـA على طول
+
+99
+00:08:36,080 --> 00:08:42,420
+نحط هذه إيش بتساوي F prime of A على G prime of A
+
+100
+00:08:42,420 --> 00:08:48,880
+معايا فهذه اللى هي .. اللى هي .. كيفية تطبيق
+
+101
+00:08:48,880 --> 00:08:53,500
+النظرية نيجي لأهلنا برهان النظرية البرهان بسيط
+
+102
+00:08:53,500 --> 00:08:54,680
+عندى
+
+103
+00:08:59,270 --> 00:09:11,330
+خد عندي four x بين a و بين b لو جيت حسبت الغرض
+
+104
+00:09:11,330 --> 00:09:18,970
+الأوجود هذه ال f of x على g of x إيش هتساوي؟
+
+105
+00:09:18,970 --> 00:09:28,250
+هتساوي f of x ناقص f of aعلى g of x نقص g of a ليش
+
+106
+00:09:28,250 --> 00:09:31,110
+لأن ال f of a و ال d of a ليش ماعطينا إيها بساوة
+
+107
+00:09:31,110 --> 00:09:34,170
+سفر ليش عملت هيك لأ بدت أعمل أكتر من هي بدت أعمل
+
+108
+00:09:34,170 --> 00:09:39,110
+أقسم هذا على x minus a و هذا على x ما لها minus a
+
+109
+00:09:40,060 --> 00:09:43,580
+طبيعي ال X لا تسوى ال A الآن أنا باخد ال limit
+
+110
+00:09:43,580 --> 00:09:47,620
+للجهتين وبتجرأ و باخد موزع لإن أنا ضائع من ال F
+
+111
+00:09:47,620 --> 00:09:51,260
+prime of A موجودة و ال G prime of A شمالها ومش هيك
+
+112
+00:09:51,260 --> 00:09:54,160
+و كمان ال G prime of A لا تسوى 0 إذا كل أموري تمام
+
+113
+00:09:54,160 --> 00:09:58,220
+التمام إذا باخد ال limit للجهتين لما X تروح لل A
+
+114
+00:09:58,220 --> 00:10:04,040
+من اليمين بسوى ال limit لما X تروح لل A من اليمين
+
+115
+00:10:05,480 --> 00:10:09,560
+والاشي اللي تحت وزعت ليش وزعت ضامن أن ال limit
+
+116
+00:10:09,560 --> 00:10:14,520
+exist و ال limit اللي تحت كمان لا تساوي سفر هذه
+
+117
+00:10:14,520 --> 00:10:20,400
+اللي هي عبارة عن مين هذه تعريف F prime of A وهذه
+
+118
+00:10:20,400 --> 00:10:26,660
+تعريف G prime of A بكون أنا حصلت على اللي هو اللي
+
+119
+00:10:26,660 --> 00:10:33,300
+بديها هذه اللي هي النظرية الأولى في اللي هو هذا ال
+
+120
+00:10:33,300 --> 00:10:37,890
+sectionاللي يعني بتحذير بقولك أنه انت يعني انعرضت
+
+121
+00:10:37,890 --> 00:10:46,230
+عليك limit 17x ولا .. قداش؟ مش مشكلة x زياد 17 أين
+
+122
+00:10:46,230 --> 00:10:51,750
+كانت بنفع x زياد 17 على 2x زياد 3 مثلا لما x تروح
+
+123
+00:10:51,750 --> 00:10:55,500
+لمين للزفرهو مش ميقوم لما نشوف على طول و نروح
+
+124
+00:10:55,500 --> 00:11:00,960
+نفاضل بنفعش انت يعني فتالت بيصير فاضل الجهتين
+
+125
+00:11:00,960 --> 00:11:06,340
+بيطلع واحد على اتنين لأ هو انا بقول بتسوي الو أف
+
+126
+00:11:06,340 --> 00:11:10,720
+برايم على جي برايم عند ال zero لما نكون هدي zero و
+
+127
+00:11:10,720 --> 00:11:16,080
+هدي zero لكن لا هدي zero ولا هدي zero إذا بنفعش
+
+128
+00:11:16,260 --> 00:11:21,280
+تحديد هذا نقول بتساوي اللي هو limit من فعش نقول
+
+129
+00:11:21,280 --> 00:11:24,540
+بتساوي limit واحد على اتنين على اعتبار فضلنا
+
+130
+00:11:24,540 --> 00:11:28,800
+ويساوي نص وهذا الكلام غير صحيح لأن ال limit زي ما
+
+131
+00:11:28,800 --> 00:11:32,800
+انتوا عارفين لهذا المقدار بالتعويض المباشر هو
+
+132
+00:11:32,800 --> 00:11:42,780
+عبارة عن 0017 ع 3 هذا كلام سهل ناخد مثال تطبيقي
+
+133
+00:11:42,780 --> 00:11:48,310
+على النظرية اللي عندناالمثال التطبيقي برضه مثال
+
+134
+00:11:48,310 --> 00:11:56,530
+مباشر عرض علينا الآن example عرض
+
+135
+00:11:56,530 --> 00:12:04,830
+علينا بقول او جد limit x تربيع زاد x على sign 2x
+
+136
+00:12:04,830 --> 00:12:09,530
+لما x تروح لمين لزي روبالمناسبة، النظرية اللي قبل
+
+137
+00:12:09,530 --> 00:12:13,070
+بشوية حكينا عنها سواء كانت الـA اللي بتروحلها end
+
+138
+00:12:13,070 --> 00:12:17,150
+point أو نقطة داخلية أو حتى left end point بتظبط
+
+139
+00:12:17,150 --> 00:12:22,150
+عليها النظرية والبرهان similarly ماشي الحال؟ واضح
+
+140
+00:12:22,150 --> 00:12:28,430
+هاه؟ طيب، وإيه السبب؟
+
+141
+00:12:28,430 --> 00:12:34,390
+لأن الإختبار صفر على صفرو هذه differentiable و هذه
+
+142
+00:12:34,390 --> 00:12:38,970
+differentiable كل أمورها ميحة و كويسة و مش هي كمان
+
+143
+00:12:38,970 --> 00:12:43,730
+و لو فضلت هتلاقي اللي هنا لا يساوي سفر اذا على طول
+
+144
+00:12:43,730 --> 00:12:50,410
+بقول 2x عند ال zero بفاضل جاعد و بعوض يعني هذه
+
+145
+00:12:50,410 --> 00:12:56,950
+سميتها و كأنها Fوهذه g f of x وهذه g of x بعوض f
+
+146
+00:12:56,950 --> 00:13:02,190
+prime of zero بعوض هنا g prime of zero الان f
+
+147
+00:13:02,190 --> 00:13:06,490
+prime of zero اتنين x زائد واحد في سفر بيصير اتنين
+
+148
+00:13:06,490 --> 00:13:11,850
+في سفر زائد واحد وتحت اللي هو تفاضلها اتنين cosine
+
+149
+00:13:11,850 --> 00:13:16,970
+اتنين x بيصير اتنين cosine اتنين x وعوض بسفر بيصير
+
+150
+00:13:16,970 --> 00:13:25,500
+اتنين cosine اتنين في zero وهذا يعنيبساوي اللي هو
+
+151
+00:13:25,500 --> 00:13:30,140
+واحد على اتنين على اعتبار كوصالة Zero بساوي واحد
+
+152
+00:13:30,140 --> 00:13:36,760
+هذه اللي هي تطبيق النظرية اللي عندي نيجي الآن للـ
+
+153
+00:13:36,760 --> 00:13:40,480
+Cauchy Mean Value Theorem الـ Cauchy Mean Value
+
+154
+00:13:40,480 --> 00:13:46,140
+Theorem تعميم للـ Mean Value Theorem اللي احنا
+
+155
+00:13:46,140 --> 00:13:52,050
+عارفينهابدل ما هو على دالب نحكي عن إيش عن دالتين
+
+156
+00:13:52,050 --> 00:14:00,930
+نشوف إيش اللي بقوله النظرية بتقول ما يلي لأن
+
+157
+00:14:00,930 --> 00:14:11,690
+theorem عند F و G دالتين من A و B مخدهم من A و B
+
+158
+00:14:11,690 --> 00:14:17,390
+لعند Rجاي لي نفس شروط الـ mean value theorem
+
+159
+00:14:17,390 --> 00:14:22,150
+العادية بدل ما هي على دالها دالتين جاي لي F و G
+
+160
+00:14:22,150 --> 00:14:35,270
+continuous on A و B and differentiable on O B ماشي
+
+161
+00:14:35,270 --> 00:14:42,230
+الحال ومعطيني أيضا بقولي الـ G prime لل X لا تساوي
+
+162
+00:14:42,230 --> 00:14:49,110
+سفرلكل X وين موجودة في الـA و الـB اللي أنا
+
+163
+00:14:49,110 --> 00:14:59,190
+بقولليه النتيجة then there exist لنتيجة then then
+
+164
+00:14:59,190 --> 00:15:09,250
+هذا كله معطى if this hold then then
+
+165
+00:15:09,250 --> 00:15:18,160
+there exist CElement in A و B such that G أو F
+
+166
+00:15:18,160 --> 00:15:25,820
+prime of C على G prime of C بساوي F of B ناقص F of
+
+167
+00:15:25,820 --> 00:15:35,360
+A على G of B ناقص مياً G of A ال proof ده كلام سهل
+
+168
+00:15:35,360 --> 00:15:38,140
+كمان ال proof نشوف كده
+
+169
+00:15:41,590 --> 00:15:46,050
+عندي يا جماعة أول إشي هو معطيني إيش مالها g prime
+
+170
+00:15:46,050 --> 00:15:51,650
+of x إيش مالها لا تساوي سفر إذا by rules theorem
+
+171
+00:15:51,650 --> 00:15:58,490
+هيكون g of b لا تساوي مين g of a كيف أذكركم أذكركم
+
+172
+00:15:58,490 --> 00:16:03,950
+كيف الآن إيش rules theorem كانت بتقول g من a و b
+
+173
+00:16:03,950 --> 00:16:14,130
+لعند r continuous on a و bو differentiable on a و
+
+174
+00:16:14,130 --> 00:16:21,270
+b هذا ما أعطينا إياه هو ماشي الحال بقول لي if g of
+
+175
+00:16:21,270 --> 00:16:28,350
+a بساوي g of b بساوي سفر then هو في الواقع زي ما
+
+176
+00:16:28,350 --> 00:16:31,510
+قلنا أن ال role theorem تنفع لو قلنا g of a بساوي
+
+177
+00:16:31,510 --> 00:16:37,710
+g of b و سكتنا لأنه الشهد في الموضوع أنه المماس
+
+178
+00:16:37,710 --> 00:16:41,920
+يكون معله موازي لمحور الصينةأه و لما تكون الـ G of
+
+179
+00:16:41,920 --> 00:16:45,440
+A بيساوي الـ G of B وسكتنا أي قاطع بينهم هيكون
+
+180
+00:16:45,440 --> 00:16:48,520
+عبارة عن موازي لمحور السينات يعني معناته اللي
+
+181
+00:16:48,520 --> 00:16:52,220
+مماثل اللي باجيه هيكون موازي لهذا يعني موازي لمحور
+
+182
+00:16:52,220 --> 00:16:57,740
+السينات طيب then .. then there exists C element in
+
+183
+00:16:57,740 --> 00:17:03,520
+A و B such that G prime of C بيساوي إيش بيساوي سفر
+
+184
+00:17:03,520 --> 00:17:10,560
+الآن هذا معطى مفرغ منه اللي فوق معطى عنديالان عندى
+
+185
+00:17:10,560 --> 00:17:14,640
+لو كان g of a بسوى g of b بيعطينا انه يوجد صفر بين
+
+186
+00:17:14,640 --> 00:17:18,000
+ال a و ال b بحيث ان g prime of c اشماله لا تسوى
+
+187
+00:17:18,000 --> 00:17:23,260
+صفر لكن هو مفترضلى ان g prime of x لا تسوى صفر لكل
+
+188
+00:17:23,260 --> 00:17:26,540
+x في ال a و ال bيعني الآن الـ Contraposition هو
+
+189
+00:17:26,540 --> 00:17:30,800
+اللي hold hand يعني بمعنى آخر أنه بما أنه G prime
+
+190
+00:17:30,800 --> 00:17:36,160
+of X لا تساوي 0 لكل X element in A و B هيعطينا هذا
+
+191
+00:17:36,160 --> 00:17:40,940
+نفيه B implies Q تكافئ not Q implies not B وهذا
+
+192
+00:17:40,940 --> 00:17:46,240
+لسه عملته أنابما أن g prime of x لا يساوي سفر لكل
+
+193
+00:17:46,240 --> 00:17:51,840
+x طبعا في اللي هو b اذا g of a لا يساوي مين؟ g of
+
+194
+00:17:51,840 --> 00:17:59,900
+b واضح؟ طيب اذا صار عندي الأول حاجة by Rolle's
+
+195
+00:17:59,900 --> 00:18:09,850
+theorem g of a لا يساوي g of b becauseG prime of X
+
+196
+00:18:09,850 --> 00:18:15,670
+لا يساوي سفر لكل X والي موجودة في الـA والـB هذه
+
+197
+00:18:15,670 --> 00:18:22,850
+أول نقطة خلصنا هذه بررناها نيجي الآن زي ما عملنا
+
+198
+00:18:22,850 --> 00:18:26,270
+في إثبات الـMean Value Theorem إذا بتتذكروا بدي
+
+199
+00:18:26,270 --> 00:18:29,630
+أعرف دالة أطبق عليها برضه خلّي رولز الـTheorem
+
+200
+00:18:29,630 --> 00:18:32,890
+مطبقة أو الـMean Value Theorem مطبقة و أحصل عن
+
+201
+00:18:32,890 --> 00:18:38,260
+نتيجة اللي أنا بديهاالأن هي الشكل اللي بديها بدي
+
+202
+00:18:38,260 --> 00:18:45,680
+أخد H of X let أو define H of X بساوة اللي هو
+
+203
+00:18:45,680 --> 00:18:52,740
+المقدار هذا اللي بديه F of B ناقص F of A على G of
+
+204
+00:18:52,740 --> 00:18:57,910
+B minus G of Aبتخلي المقدار هذا اللي هو يتصفر في
+
+205
+00:18:57,910 --> 00:19:00,650
+حالة ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+206
+00:19:00,650 --> 00:19:00,870
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+207
+00:19:00,870 --> 00:19:00,910
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+208
+00:19:00,910 --> 00:19:00,970
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+209
+00:19:00,970 --> 00:19:01,090
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+210
+00:19:01,090 --> 00:19:01,850
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+211
+00:19:01,850 --> 00:19:01,890
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+212
+00:19:01,890 --> 00:19:02,850
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+213
+00:19:02,850 --> 00:19:07,410
+.. ال .. ال .. ال .. ال .. ال .. ال ..
+
+214
+00:19:07,410 --> 00:19:12,650
+ال ..
+
+215
+00:19:12,650 --> 00:19:16,510
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+216
+00:19:16,510 --> 00:19:24,170
+ال .. ال ..
+
+217
+00:19:25,340 --> 00:19:30,660
+H of A بيطلع سفر لإن بدي اللي هو الجزء الثاني اللي
+
+218
+00:19:30,660 --> 00:19:39,320
+هيجيبلي قيمة مين قيمة ال F نقص G G of B ناقص G of
+
+219
+00:19:39,320 --> 00:19:44,060
+A أو G of X ناقص H of A عشان نروحهم مع بعض ناقص F
+
+220
+00:19:44,060 --> 00:19:50,880
+of X ناقص F of A الآن ليش عملت هيك عشان أحصل H of
+
+221
+00:19:50,880 --> 00:19:53,180
+A H of A هذه سفر
+
+222
+00:19:55,780 --> 00:20:02,140
+و هد ايه ايه بيصير سفر H of B إذا صارت عندي H of A
+
+223
+00:20:02,140 --> 00:20:09,360
+بساوي سفر و H of B حط H of B بيصير هدي بيه هدي
+
+224
+00:20:09,360 --> 00:20:13,780
+بتروح مع هدي بيصير F of B ناقص F of A ناقص F of B
+
+225
+00:20:13,780 --> 00:20:16,800
+ناقص F of A بروحن مع بعض بيصير H برضه بيساوي سفر
+
+226
+00:20:16,800 --> 00:20:22,300
+إذا هدي أكيد برضه بتساوي H of B الآن
+
+227
+00:20:24,650 --> 00:20:28,050
+عندي هذا الـ differentiable و continuous و هذا الـ
+
+228
+00:20:28,050 --> 00:20:31,210
+differentiable و continuous على ما يناسبها من a و
+
+229
+00:20:31,210 --> 00:20:35,750
+b أو على ال a و b ال a و b اللي هو open
+
+230
+00:20:35,750 --> 00:20:38,770
+differentiable و على ال a و b closed continuous
+
+231
+00:20:38,770 --> 00:20:43,350
+تبعاً لها هتطلع انه ثابت هذا و هدوله ثوابت حيث ان
+
+232
+00:20:43,350 --> 00:20:50,330
+عندي هذا كله ع بعض continuous صار عنديH is
+
+233
+00:20:50,330 --> 00:20:59,130
+continuous on A وB وdifferentiable on A وB open
+
+234
+00:20:59,130 --> 00:21:03,930
+لأنه أنا من رأس الدورة الـ H of X معرفها اللي هي
+
+235
+00:21:03,930 --> 00:21:10,740
+الـ H for every X element in A وBH of X بيساوي كده
+
+236
+00:21:10,740 --> 00:21:15,360
+ماعيش يا بابا صارت الآن أنا حققت كل شروط ال roles
+
+237
+00:21:15,360 --> 00:21:19,780
+theorem اللي هي ال H continuous على ال A و B و ال
+
+238
+00:21:19,780 --> 00:21:23,360
+differential على ال A و B open أو H of A بيساوي و
+
+239
+00:21:23,360 --> 00:21:27,260
+H of B بيساوي اللي هي سفر إذا حسب roles theorem
+
+240
+00:21:27,260 --> 00:21:31,720
+إذا by roles theorem by أو حتى by main value
+
+241
+00:21:31,720 --> 00:21:38,180
+theorem by roles theorem there exists C element ال
+
+242
+00:21:38,180 --> 00:21:46,290
+A و Bsuch that اتش prime of c ايش هيساوي سفر هذا
+
+243
+00:21:46,290 --> 00:21:52,190
+حسب ال mean value theorem معايا يا جماعة الآن
+
+244
+00:21:52,190 --> 00:21:57,130
+بتفاضل هذه دلنا اوجد تفاضل هذه تفاضل هذه ماعليش
+
+245
+00:21:57,130 --> 00:22:04,760
+خلينا نكتبها لإنه ندخل على العالـ slide بدي أفاضل
+
+246
+00:22:04,760 --> 00:22:09,740
+هذه فاضله H prime of X اللي هي عند الـ C بتساوي
+
+247
+00:22:09,740 --> 00:22:15,580
+سفر سفر بساوي H prime of C تساوي فاضل هذه و أعوض
+
+248
+00:22:15,580 --> 00:22:20,420
+عن الـ X بسفر لأن هذا ثابت و هذا ثابت ده بيصير سفر
+
+249
+00:22:20,420 --> 00:22:23,420
+تفاضل هذا مع هذا لأن مع هذا بيصير G prime of X
+
+250
+00:22:23,420 --> 00:22:29,220
+مضروب في هذا إذا صار عندي بساوي F of B ناقص F of A
+
+251
+00:22:30,080 --> 00:22:36,100
+على g of b ناقص g of a مضروب في مين؟ فضلت في g
+
+252
+00:22:36,100 --> 00:22:40,920
+prime of x وانا بدي أحسبها عند مين؟ عند c، اذا g
+
+253
+00:22:40,920 --> 00:22:45,320
+prime of c ناقص تفاضل هذه، هذا ثابت سفر لأن هذا
+
+254
+00:22:45,320 --> 00:22:50,620
+قداش تفاضلها ناقص f prime عند ال x وانا باخدها عند
+
+255
+00:22:50,620 --> 00:22:56,280
+ال c اللي هي h of c, فh prime of c فبصير ناقص f
+
+256
+00:22:56,280 --> 00:23:02,200
+prime of cالان بتدنجل هذه على جهة هذه بصير عندى
+
+257
+00:23:02,200 --> 00:23:10,720
+الان f prime of c نجلته هان و اجسمها جي برايم of c
+
+258
+00:23:10,720 --> 00:23:15,840
+لأن جي برايم of c لاتساوى سفر بساوة f of b ناقص f
+
+259
+00:23:15,840 --> 00:23:25,710
+of a على g of b ناقص g of a و هيكبكون حصلنا على C
+
+260
+00:23:25,710 --> 00:23:31,630
+في الـA وB بحيث أنه هذه تتحقق وهو المطلوب هذه اللي
+
+261
+00:23:31,630 --> 00:23:37,190
+بيسميها Cauchy Mean Value Theorem هذه تعميم لمين؟
+
+262
+00:23:37,190 --> 00:23:41,590
+تعميم اللي هو الـMean Value Theorem بس خد اللي هو
+
+263
+00:23:41,590 --> 00:23:47,230
+G of X بتساوي XG of X بيساوي X و G of X بيساوي X
+
+264
+00:23:47,230 --> 00:23:49,190
+اللي هي تحقق كل الشروط ال differential
+
+265
+00:23:49,190 --> 00:23:53,470
+بالcontinuous و و الاخره ماشي الحال بيصير عندي
+
+266
+00:23:53,470 --> 00:23:58,110
+اللي هي في حالة G of X بتساوي X معايا يا شباب
+
+267
+00:23:58,110 --> 00:24:02,450
+بيصير G of BB و G of AA و هذه اللي هي G prime اللي
+
+268
+00:24:02,450 --> 00:24:05,510
+هي واحد فبيصير F prime of C بيساوي F of B نقص F of
+
+269
+00:24:05,510 --> 00:24:10,470
+A على B minus A إذا F
+
+270
+00:24:15,650 --> 00:24:21,210
+أمسح البرهان بس الان
+
+271
+00:24:21,210 --> 00:24:31,930
+ال note اللي عندي ال note كما يالي note if
+
+272
+00:24:31,930 --> 00:24:42,820
+g of x بيساوي x then we get fromالـ theorem هذه
+
+273
+00:24:42,820 --> 00:24:51,140
+اللي بقول عليها كوشي mean value theorem we get the
+
+274
+00:24:51,140 --> 00:24:59,500
+mean value theorem كيف الـ g of x بيساوي x معناه
+
+275
+00:24:59,500 --> 00:25:04,160
+انتصار الـ g of b بيساوي b و g of a بيصير a يعني
+
+276
+00:25:04,160 --> 00:25:10,530
+هذا بيصير a, bوهذه بيصير A و G prime of C اللي هي
+
+277
+00:25:10,530 --> 00:25:14,070
+بيصير واحد فبصير F prime of C بيسوي F of B نقص F
+
+278
+00:25:14,070 --> 00:25:20,590
+of A على B minus A وهذه هي ال main value theorem
+
+279
+00:25:21,280 --> 00:25:24,720
+واضح .. اه .. طيب اطلع للي بعده لإن احنا اللي
+
+280
+00:25:24,720 --> 00:25:26,820
+بيهمنا هذا الكوشي بين ال value theorem احنا
+
+281
+00:25:26,820 --> 00:25:31,980
+حكيناها أصلا عشان خاطر انه احنا نحكي عن اللي هو
+
+282
+00:25:31,980 --> 00:25:37,960
+Lobital's rule ال form اللي عنده اللي أمامي دكتور
+
+283
+00:25:37,960 --> 00:25:41,860
+جديد في السؤال في غيرك مفهوم ان انا جسمت على X
+
+284
+00:25:41,860 --> 00:25:46,640
+minus A على E minus A الطرفين اصلا انت بتصير F أو
+
+285
+00:25:46,640 --> 00:25:48,580
+P نقص F أو A على B minus A
+
+286
+00:26:01,110 --> 00:26:06,630
+البرهام مش صعب اللي قلناه، كتير سهل، ما هو الـ C
+
+287
+00:26:06,630 --> 00:26:09,970
+اللي لاجيناها في الحالة الأولى، هل هي التانية؟ إذا
+
+288
+00:26:09,970 --> 00:26:13,390
+كنت تتعامل بجلمين Value Theorem أنت؟مش بدك تطبّق
+
+289
+00:26:13,390 --> 00:26:16,170
+الـ Value Theory بدك تطبّق الـ Value Theory لأن
+
+290
+00:26:16,170 --> 00:26:21,090
+there exists c1 such that f prime of c1 بيسوء f of
+
+291
+00:26:21,090 --> 00:26:24,030
+b minus f of a على b minus a مافيش فيها مشكلة
+
+292
+00:26:24,030 --> 00:26:27,950
+there exists c2 such that g prime of c2 بيسوء g of
+
+293
+00:26:27,950 --> 00:26:32,550
+b نقص g of a على b minus a الـ c1 هذه مش شرط هيكون
+
+294
+00:26:32,550 --> 00:26:39,630
+main الـ c2 او احنا لازم نثبتها هي نفسها اه ف .. و
+
+295
+00:26:39,630 --> 00:26:44,780
+بعدين البرمجان سهلماشي .. نحن الواحد يفكر 100% ..
+
+296
+00:26:44,780 --> 00:26:50,780
+جميل لكن .. يعني .. بس أنه احنا .. الـC هذه مش
+
+297
+00:26:50,780 --> 00:26:53,700
+ضمنية تساوي الـC2 و بدنا نثبت أنها تساويها .. إذا
+
+298
+00:26:53,700 --> 00:26:59,920
+كانت بتساويها طيب .. نيجي لها اللي هي النظرية اللي
+
+299
+00:26:59,920 --> 00:27:05,840
+بعيدها النظرية اللي بعيدها .. أنا مرضيش أمسح اللوح
+
+300
+00:27:05,840 --> 00:27:08,800
+أساسي منقررها من الـtheory من الأولى
+
+301
+00:27:11,650 --> 00:27:30,050
+نشوف نطلع إيش اللي بتقوله هذه النظرية نفس
+
+302
+00:27:30,050 --> 00:27:35,710
+اللي هنا F و G differentiable on A و B معايا أه؟
+
+303
+00:27:35,710 --> 00:27:42,150
+الان such that G prime of X ده تساوي سفرG prime of
+
+304
+00:27:42,150 --> 00:27:47,970
+X لا تساوي سفر أنا مش موجودة For all X elements in
+
+305
+00:27:47,970 --> 00:27:55,070
+A وB ونفترض أنه limit F of X limit F of X لما X
+
+306
+00:27:55,070 --> 00:27:59,570
+تروح إلى الـ A من اليمين موجودة و بتساوي أياش؟ سفر
+
+307
+00:27:59,570 --> 00:28:02,490
+و limit G of X لما X تروح إلى الـ A من اليمين
+
+308
+00:28:02,490 --> 00:28:09,580
+بتساوي سفر اذا صار limit الحاصل القسمةلأن بيسويش f
+
+309
+00:28:09,580 --> 00:28:14,300
+prime of a على g prime of a بيساوياش limit f of x
+
+310
+00:28:14,300 --> 00:28:19,580
+على g of x لما x تروح لل a باليمينيعني وكأنه هنا
+
+311
+00:28:19,580 --> 00:28:25,460
+حول الحديث كله من افتراض أنه عند النقطة اللي هي F'
+
+312
+00:28:25,780 --> 00:28:31,240
+وG' موجودة ولا تساوي سفر وحول الحديث من إن التعويض
+
+313
+00:28:31,240 --> 00:28:35,280
+المباشر أن F of A وG of A بتساوي سفر ل limit لـ ..
+
+314
+00:28:35,280 --> 00:28:39,220
+limit لل function لما X تروح لـ A سفر و limit لل
+
+315
+00:28:39,220 --> 00:28:42,040
+function لما .. اللي .. اللي هو ال X .. ال X بتروح
+
+316
+00:28:42,040 --> 00:28:48,920
+لسفر .. لل A بتساوي سفروحول أيضا طلب جي برايم نفسه
+
+317
+00:28:48,920 --> 00:28:54,680
+على كل الفترة اللي تكون لا تساوي سفر مش عند النقطة
+
+318
+00:28:54,680 --> 00:29:03,060
+بس وقال لك انه حتى اللي هو النتيجة هتطلع ب limit
+
+319
+00:29:03,060 --> 00:29:07,940
+مش هتطلع ب F prime of A على G prime of A يعني اللي
+
+320
+00:29:07,940 --> 00:29:14,980
+بقصده انه الآن طور هذه الحديث هناإنه لما تُعرض
+
+321
+00:29:14,980 --> 00:29:21,580
+علينا limit F of X على G of X لما X تروح للـA من
+
+322
+00:29:21,580 --> 00:29:24,360
+لمين أو إن كانت حتى لو في الـinterior point الـX
+
+323
+00:29:24,360 --> 00:29:31,300
+بتروح للـA برضه صحيحة الآن بعدي عنديإذا لجيت limit
+
+324
+00:29:31,300 --> 00:29:34,480
+الأولى، اللي أنا بحكي عن limit سيرة مش عن تعويض
+
+325
+00:29:34,480 --> 00:29:38,340
+مباشر زي اللي جابله، الآن limit f of x لما x تروح
+
+326
+00:29:38,340 --> 00:29:41,780
+ل a من اليمين و limit f of x لما x تروح ل a من
+
+327
+00:29:41,780 --> 00:29:46,700
+اليمين، إذا هذه موجودة و هذه موجودة و طلع عندي 0
+
+328
+00:29:46,700 --> 00:29:51,060
+على 0، يعني طلع ال limit عبارة عن 0 على 0، هنا بدو
+
+329
+00:29:51,060 --> 00:29:57,200
+يحدث العلاج، إذا كانت الآنالـ limit اللي طلعت عندي
+
+330
+00:29:57,200 --> 00:30:02,000
+F prime of X على G prime of X لما X تروح للـ A من
+
+331
+00:30:02,000 --> 00:30:06,780
+اليمين إذا طلعت عبارة عن قيمة خلاص ارتاح هذه اللي
+
+332
+00:30:06,780 --> 00:30:12,450
+طلعت هي مين فهمت ال limitماشي الحال لو طلعت كمان
+
+333
+00:30:12,450 --> 00:30:18,230
+مرة zero على zero اللي هو و بتحقق كل الشروط اللي
+
+334
+00:30:18,230 --> 00:30:21,950
+.. اللي في الأول برضه بأعمل كمان مرة بفاضل لما
+
+335
+00:30:21,950 --> 00:30:26,430
+بتطلع لكن لو طلعت ال limit هذه does not exist بسكت
+
+336
+00:30:26,430 --> 00:30:31,130
+و بجوبي عمش .. بدنا ندور على طريقة أخرى واضح؟ الآن
+
+337
+00:30:31,130 --> 00:30:34,990
+لو طلعت هذه infinity أو سالب infinity هذه آسف
+
+338
+00:30:34,990 --> 00:30:38,900
+infinity أو سالب infinity برضه أن نظرية صحيحةاللي
+
+339
+00:30:38,900 --> 00:30:45,290
+هو هدمين الجزء الثاني من النظريةif limit f prime
+
+340
+00:30:45,290 --> 00:30:48,450
+على g prime بساوة L بساوة infinity أو سالب
+
+341
+00:30:48,450 --> 00:30:52,190
+infinity هتطلع ال limit على طول لل F على G اللي
+
+342
+00:30:52,190 --> 00:30:56,350
+ببحث عنها إيش هتساوي برضه ال infinity أو سالب
+
+343
+00:30:56,350 --> 00:31:00,670
+infinity حسب القيمة هذه إذا أي إن كانت اللي هي ال
+
+344
+00:31:00,670 --> 00:31:05,190
+limit مادامة exist سواء ال existence عبارة عن
+
+345
+00:31:05,190 --> 00:31:09,210
+element in R أو اللي هو عبارة عن ناقص infinity أو
+
+346
+00:31:09,210 --> 00:31:18,100
+سالب infinity فإن النظرية صحيحةواضح؟ أي سؤال؟ طيب
+
+347
+00:31:18,100 --> 00:31:27,200
+صلى على النبي عليه الصلاة والسلام خلينا
+
+348
+00:31:27,200 --> 00:31:31,360
+نيجي للنظرية ونبرنها
+
+349
+00:31:45,860 --> 00:31:50,880
+Theorem كدهش؟ نزلي بس النص يا محمد اللي هي theorem
+
+350
+00:31:50,880 --> 00:31:59,580
+6 3 3 إيش النظرية بتقول؟ بتقول ما يعني عندي طبعا
+
+351
+00:31:59,580 --> 00:32:04,540
+ماخد ال a أصغر من b strictly و a ممكن حتى تاخد سلب
+
+352
+00:32:04,540 --> 00:32:07,580
+infinity و ال b تاخد infinity يعني ممكن تكون
+
+353
+00:32:07,580 --> 00:32:11,040
+الفترة من a .. الفترة كلها a ممكن تكون أي فترة sub
+
+354
+00:32:11,040 --> 00:32:16,710
+interval من اللي هو 100 من ال real numbersفرضنا f
+
+355
+00:32:16,710 --> 00:32:26,190
+و g من a و b لعند اللي هو r وإذا كانت a infinity
+
+356
+00:32:26,190 --> 00:32:30,130
+أو سلب infinity أسف إذا كانت a سلب infinity أو b
+
+357
+00:32:30,130 --> 00:32:33,610
+infinity بتكون o من زمن بعرفي لأن عشان اللي هو
+
+358
+00:32:33,610 --> 00:32:36,890
+ناخدها من real number ل real number لذا نفترض f و
+
+359
+00:32:36,890 --> 00:32:46,080
+g من a و b لعند ال R ونفترض إن f و gdifferentiable
+
+360
+00:32:46,080 --> 00:32:54,620
+on a و b ماشي الحال و differentiable on a و b الان
+
+361
+00:32:54,620 --> 00:32:57,900
+مش لازم ي continue و تعندى و كده لأنه انا هدخل
+
+362
+00:32:57,900 --> 00:33:02,240
+لجوه شغل هيكون لجوه هل جيت بتشوف واش معنى لجوه
+
+363
+00:33:02,240 --> 00:33:08,620
+such that g prime of x g prime of x لا تسوى سفر
+
+364
+00:33:08,620 --> 00:33:17,430
+لكل x وين موجودة في الفترة a و bالان بقول لي إذا
+
+365
+00:33:17,430 --> 00:33:27,170
+كانت limit limit اللي هي f of x لما x تروح لل a من
+
+366
+00:33:27,170 --> 00:33:32,190
+اليمين بساوي limit g of x لما x تروح لل a من
+
+367
+00:33:32,190 --> 00:33:40,980
+اليمين بساوي سفر هذا كله موضوع الان بدو يوصلبدي
+
+368
+00:33:40,980 --> 00:33:46,200
+أقولك كيف أنا بدي أحصل على نتيجة limit f of x على
+
+369
+00:33:46,200 --> 00:33:51,060
+g of x لما x تروح لل a من اليمين بقولك إذا كان انت
+
+370
+00:33:51,060 --> 00:33:59,180
+if لان ايه if limit f prime of x على g prime of x
+
+371
+00:33:59,180 --> 00:34:06,520
+لما x تروح لل a من اليمين بساوي L thenأتجرأ أقول
+
+372
+00:34:06,520 --> 00:34:12,020
+فاش عندى مشكلة limit f of x اللى ببحث عنها على g
+
+373
+00:34:12,020 --> 00:34:17,360
+of x لما x تروح للإيه من اليمين برضه ايش هيساوي؟
+
+374
+00:34:17,360 --> 00:34:29,740
+هيساوي ال .. من يحياتي هذه هيكون صحيحة طيب خلينا
+
+375
+00:34:29,740 --> 00:34:38,330
+الآن اللى هو البرهن شوف عليهاواضح النظرية شرحناها
+
+376
+00:34:38,330 --> 00:34:45,050
+يعني نص النظرية شرحناها بشكل كامل الان since limit
+
+377
+00:34:45,050 --> 00:34:51,730
+F prime of X على G prime of X لما X تروح ل A من
+
+378
+00:34:51,730 --> 00:35:00,690
+وين؟ من اليمين هيعندي الفترة A وB حتى لو كانت
+
+379
+00:35:00,690 --> 00:35:05,580
+منتده وين ما بدها، هي حاضرة الان عنديX تذهب إلى
+
+380
+00:35:05,580 --> 00:35:09,760
+اليمن اليمين فبدي من الجهة دي طبيعي من اليمين بسوء
+
+381
+00:35:09,760 --> 00:35:15,800
+ال then for every epsilon أكبر من سفر لأي epsilon
+
+382
+00:35:15,800 --> 00:35:20,440
+أكبر من سفر أي epsilon there exists delta أكبر من
+
+383
+00:35:20,440 --> 00:35:24,660
+سفر such that لأن X تذهب إلى الامن وين من اليمين
+
+384
+00:35:24,660 --> 00:35:27,460
+إذن الجوار اللي حوالها هو عبارة عن جوار من A لعند
+
+385
+00:35:27,460 --> 00:35:32,340
+مين ال A زي ال Delta لعند A زي ال Delta صح ولا لأ
+
+386
+00:35:32,340 --> 00:35:36,260
+إذن لكلما دامت ال limit هي دي بساوي L، ده لكل
+
+387
+00:35:36,260 --> 00:35:41,480
+إبسلون بقدر ألاقي Delta بحيث أنه X element في A و
+
+388
+00:35:41,480 --> 00:35:46,400
+A زائد Delta ف X element في A و A زائد Delta، then
+
+389
+00:35:46,400 --> 00:35:52,180
+قطعا .. then قطعا هيكون عندي F prime of X على G
+
+390
+00:35:52,180 --> 00:35:58,030
+prime of X نقص L، ذوكون أصغر من 100 من إبسلونهذا
+
+391
+00:35:58,030 --> 00:36:02,370
+تعريف ال limit لل F prime of X على G prime of X
+
+392
+00:36:02,370 --> 00:36:06,510
+بسهولة لما X تروح لمن؟ لل A من اليمين هو كاتبها
+
+393
+00:36:06,510 --> 00:36:11,110
+هذه A زائد Delta C في الكتاب يعني مسميها C يعني
+
+394
+00:36:11,110 --> 00:36:16,090
+مسمي انه لكل X there exists C بحيث انه لكل X في ال
+
+395
+00:36:16,090 --> 00:36:20,650
+A اللي عند ال A لل C بكون هذا الكلام متحقق أنا
+
+396
+00:36:20,650 --> 00:36:23,510
+حبيت اكتبلك اللي هو التعريف الدارجي اللي انت ده
+
+397
+00:36:23,510 --> 00:36:27,700
+كسله كمان في ال .. في الحلواضح لحد تلاتة
+
+398
+00:36:38,080 --> 00:36:42,520
+ماشي الحال بس هي ك .. هذه .. و هذه ك صح و هذه ك صح
+
+399
+00:36:42,520 --> 00:36:47,040
+و هذه ك صح اللي كاتبها صح بس هذه للطالب أسهله في
+
+400
+00:36:47,040 --> 00:36:49,960
+ال .. في ال .. لإنه .. خلّيني أقول هذا اللي درج
+
+401
+00:36:49,960 --> 00:36:54,640
+عليه في ال .. في .. اللي هو التعبير عن ال .. لكل x
+
+402
+00:36:54,640 --> 00:36:58,940
+في الجوار الجوار هذا سهل و .. و .. و .. وعبّرنا
+
+403
+00:36:58,940 --> 00:36:59,200
+عنه
+
+404
+00:37:02,810 --> 00:37:07,630
+-A هو أكبر من سفر وأصغر من دلتة برضه هيك صح صحيح
+
+405
+00:37:07,630 --> 00:37:11,910
+نفس الاشي طيب
+
+406
+00:37:11,910 --> 00:37:15,050
+الان
+
+407
+00:37:15,050 --> 00:37:23,590
+هذا الكلام IA او بمعنى اخر that is هيكون هذا اللي
+
+408
+00:37:23,590 --> 00:37:31,230
+هو F prime of X ناقص G prime of Xأصغر ناقص L
+
+409
+00:37:31,230 --> 00:37:37,150
+بتغيرها عشان أصغر من Y وأكبر من 200 من سالب Y شيل
+
+410
+00:37:37,150 --> 00:37:43,470
+ال L هذه بصير اللي هو أصغر من L زائد Y وأكبر من L
+
+411
+00:37:43,470 --> 00:37:49,430
+ناقص Y هذا متحقق لمين لكل X و N موجودة في الفترة
+
+412
+00:37:49,430 --> 00:37:53,550
+من A لعند A زائد Delta و اكتب لي هذا سميلي
+
+413
+00:37:56,530 --> 00:38:03,990
+واحد ماشي الحال سميليه واحد الآن عندي اللي هو شروط
+
+414
+00:38:03,990 --> 00:38:06,850
+ال mean value theorem الكوشي mean value theorem
+
+415
+00:38:06,850 --> 00:38:10,130
+اللي قبل بشوية ال F و ال G differentiable على ال A
+
+416
+00:38:10,130 --> 00:38:16,170
+و ال B اه و ال G prime of X لها تساوي سفر و ال ..
+
+417
+00:38:16,170 --> 00:38:22,490
+او .. ايش كمان كله متحقق بس خلينا نقول
+
+418
+00:38:26,140 --> 00:38:31,240
+بدي الآن اوين اطبق الـ mean value theorem بدي
+
+419
+00:38:31,240 --> 00:38:37,720
+اطبقها على اللي هو اللي جوا عندى لداخل الـ mean
+
+420
+00:38:37,720 --> 00:38:44,560
+لداخل الفترة هذه عشان اشتغل وين اللي بتطلع عندى
+
+421
+00:38:44,560 --> 00:38:48,490
+اضمن تكون هناعشان القيمة اللى هتطلع عندى اللى
+
+422
+00:38:48,490 --> 00:38:53,170
+بتحققها تكون بتحقق اللى بتطلع عندى و بتحققها دى
+
+423
+00:38:53,170 --> 00:38:57,730
+عشان هي كصح اللى أعوض مكان بعض ماشي الحال خد الآن
+
+424
+00:38:57,730 --> 00:39:07,030
+لجد تفهموا إيش اللى بقصدهم خد الآن four alpha أكبر
+
+425
+00:39:07,030 --> 00:39:16,040
+من a و أصغر من beta و أصغر من a زائد deltaيعني أنا
+
+426
+00:39:16,040 --> 00:39:20,060
+غرضي ان انا اشتغل .. ان انا رايح .. بدي انا limit
+
+427
+00:39:20,060 --> 00:39:24,580
+اصلا و ال limit بديها لما اروح لمين؟ لل A يعني بدي
+
+428
+00:39:24,580 --> 00:39:29,760
+في الجوار اللي عيمين ال A و جاي ناحيتها هذا اللي
+
+429
+00:39:29,760 --> 00:39:31,740
+بهم، اللي مانيش سغاداش، بيكون التصرف مافيش ..
+
+430
+00:39:31,740 --> 00:39:36,980
+مافيش عنده مشكلة الآن هي ال alpha أخدتها هنا و هي
+
+431
+00:39:36,980 --> 00:39:37,420
+ال beta
+
+432
+00:39:41,900 --> 00:39:44,540
+أصغر من Alpha أصغر من Beta أصغر من ايه زي الـ
+
+433
+00:39:44,540 --> 00:39:54,060
+Delta؟ By Cauchy Mean Value Theorem there exists
+
+434
+00:39:54,060 --> 00:40:03,220
+سموها هو نسميها U Element in mean in Alpha و Beta
+
+435
+00:40:03,220 --> 00:40:10,110
+والـ Alpha و Beta جزء من هذهدُول، إذا اللي بنطبق
+
+436
+00:40:10,110 --> 00:40:14,750
+على هذه اللي بنطبق على .. اللي بنطبق على هذه بنطبق
+
+437
+00:40:14,750 --> 00:40:19,790
+على هذه، مظبوط؟ يعني هذه الـU اللي لجيتها بنطبق
+
+438
+00:40:19,790 --> 00:40:22,210
+عليها الكلام هذا اللي هو F prime of U على D prime
+
+439
+00:40:22,210 --> 00:40:27,510
+of U بين هذه و بين هذه واضح، و هذا الكلام مهم طيب،
+
+440
+00:40:27,510 --> 00:40:34,640
+there exists U such that F prime of Uعلى g prime
+
+441
+00:40:34,640 --> 00:40:39,240
+of u بيساوي ايش يا جماعة بيساوي f of b أو beta
+
+442
+00:40:39,240 --> 00:40:50,260
+ناقص f of alpha على g of beta ناقص g of alpha واضح
+
+443
+00:40:50,260 --> 00:41:00,020
+اه؟ هذا سمولي يمين هو اتنين الآن عند هذه اللي
+
+444
+00:41:00,020 --> 00:41:05,260
+لجيتها هنا اللي بتحقق هذههي وين موجودة من ضمن
+
+445
+00:41:05,260 --> 00:41:12,980
+النقاط اللي بتحقق هذه لكل X هنا وهذه جزء منها إذا
+
+446
+00:41:12,980 --> 00:41:18,100
+F prime of X of U على G prime of U بين الـ L ناقص
+
+447
+00:41:18,100 --> 00:41:22,020
+إبسلون والـ L زائد مين إبسلون وفي نفس الوقت F
+
+448
+00:41:22,020 --> 00:41:25,860
+prime of U هذه اللي لقيتها G prime of U بساوي هذا
+
+449
+00:41:25,860 --> 00:41:29,840
+إذا from واحد
+
+450
+00:41:30,390 --> 00:41:40,070
+واتنين we get اللي هو F prime of U على G prime of
+
+451
+00:41:40,070 --> 00:41:47,610
+U بستبدلها و بيصير F of Beta ناقص F of Alpha على G
+
+452
+00:41:47,610 --> 00:41:54,070
+of Beta ناقص G of Alpha بحيث أنه هذا يكون أكبر من
+
+453
+00:41:54,070 --> 00:42:03,630
+L ناقص Y و أصغر من 100 من L زائد Yلأن هذا صحيح لأي
+
+454
+00:42:03,630 --> 00:42:09,750
+Alpha و Beta بشكلهم اللي موجود Alpha أصغر من Beta
+
+455
+00:42:09,750 --> 00:42:15,570
+و Alpha بين الـA و بين مين الـA زائد دلتا إذا
+
+456
+00:42:15,570 --> 00:42:22,150
+الألفة هذه حرة في كل المنطقة هذه بنفع يعني الألفة
+
+457
+00:42:22,150 --> 00:42:26,530
+هذه لو بدها تروح للـA حد مش بمنحها الألفة تروح لها
+
+458
+00:42:26,530 --> 00:42:31,060
+لأنه صحيح على كل هذه الألفات اللي لجيتها هذهالان
+
+459
+00:42:31,060 --> 00:42:40,340
+let alpha goes to mean to a من وين؟ من اليمين
+
+460
+00:42:40,340 --> 00:42:45,720
+ماشي؟ مقدر و لن تؤثر لأ على ال beta ولا عليها زي
+
+461
+00:42:45,720 --> 00:42:50,980
+الدلتية الحرة بتروح لهذه وهذه زي ما هي يعني تصرف
+
+462
+00:42:50,980 --> 00:42:58,540
+alpha يروح لأي a beta بالنسبالها ثابت ولا تتأثرهذا
+
+463
+00:42:58,540 --> 00:43:03,500
+الكلام مهم إذا صار عندي الآن لما Alpha تروح لأ من
+
+464
+00:43:03,500 --> 00:43:10,700
+اليمين ال up of Alpha بيصير عندي limit ال up of
+
+465
+00:43:10,700 --> 00:43:17,780
+Alpha لما ال Alpha تروح لأ من اليمين بساوي هو في
+
+466
+00:43:17,780 --> 00:43:21,980
+الواقع limitها limitها هذه Alpha اللي كنا نسميها X
+
+467
+00:43:21,980 --> 00:43:23,980
+مثلا ال up of X سمى X تروح لأ من اليمين لأ من
+
+468
+00:43:23,980 --> 00:43:29,480
+اليمينالآن الـ Alpha راحت للإيمن اليمين limitها
+
+469
+00:43:29,480 --> 00:43:35,020
+أنا ماعطينيها إيش بيساوي؟ بيساوي صفر، ماعطيني
+
+470
+00:43:35,020 --> 00:43:37,080
+limit f of x عندما x تروح للإيمن اليمين إيش
+
+471
+00:43:37,080 --> 00:43:40,540
+بيساوي؟ صفر، أنا سميتها إيش أنا؟ Alpha هذه اللي
+
+472
+00:43:40,540 --> 00:43:46,660
+بتتحرك، إذا هذا ال limit إيش هيساوي؟ بنفس السبب أو
+
+473
+00:43:46,660 --> 00:43:52,940
+لنفس السبب limitG of Alpha لما Alpha تروح للـ A من
+
+474
+00:43:52,940 --> 00:43:59,680
+اليمين برضه مش هيساوي سفر إذا الآن برجع لهذه برجع
+
+475
+00:43:59,680 --> 00:44:09,060
+لهذه بصير عندى الآن اللي حصلته كله على بعضه لكل ي
+
+476
+00:44:09,060 --> 00:44:17,130
+أكبر من سفر ماشي الحال ل gate Deltaبحيث أنه الـ α
+
+477
+00:44:17,130 --> 00:44:20,050
+و beta بشكل هذا أكبر من إيه و أقل زي الـ delta
+
+478
+00:44:20,050 --> 00:44:24,570
+سيبك من الـ L α خلاص رديتها أنا beta أكبر من إيه و
+
+479
+00:44:24,570 --> 00:44:29,170
+أصغر من إيه زي الـ delta طلع عندي هذا المقدار أصغر
+
+480
+00:44:29,170 --> 00:44:33,170
+من L زي الـ يبسلون وأكبر من L ناقص يبسلون ماشي؟
+
+481
+00:44:33,170 --> 00:44:37,300
+الآن رديت و أخدتله من ال limitالان بدي أخد ال
+
+482
+00:44:37,300 --> 00:44:40,940
+limit لهذا المقدار كله لما ال alpha تروح لمين؟ لل
+
+483
+00:44:40,940 --> 00:44:45,240
+a من اليمين الان لما أخد ال limit لهذا زي ما قولت
+
+484
+00:44:45,240 --> 00:44:48,500
+ال beta مالهاش علاقة بال alpha ال beta ثابتة
+
+485
+00:44:48,500 --> 00:44:50,920
+بالنسبة لل alpha فال alpha تروح لل a من اليمين زي
+
+486
+00:44:50,920 --> 00:44:55,100
+ما بدها لن تتأثر beta فبصير limit اللي فوق على
+
+487
+00:44:55,100 --> 00:44:59,520
+limit اللي اتحد هذا ثابت و هذا ثابت و هذا limit هو
+
+488
+00:44:59,520 --> 00:45:04,300
+اللي هو صفر و هذا limit هو صفر و هذول أعداد إذا
+
+489
+00:45:04,300 --> 00:45:11,220
+أصار عندي الآنعندي take the limit بيصير عندي L
+
+490
+00:45:11,220 --> 00:45:18,100
+ناقص Epsilon أصغر أو يساوي اللي هو F of Beta على G
+
+491
+00:45:18,100 --> 00:45:23,660
+of Beta أصغر من L زائد Epsilon من وين حصلته هذا؟
+
+492
+00:45:23,660 --> 00:45:29,160
+when I take the limit of this inequality as Alpha
+
+493
+00:45:29,160 --> 00:45:37,460
+goes to A from right واضح؟هذا حصلت عليه هذا عرفت
+
+494
+00:45:37,460 --> 00:45:40,600
+ثانوية هذا حصلت عليه من وين أن أخدت ال limit
+
+495
+00:45:40,600 --> 00:45:44,660
+للجهات التلاتة as Alpha تروح لل A من اليمين
+
+496
+00:45:44,660 --> 00:45:48,780
+واستخدمت هذه الحقيقة أن هذا سفر و هذا سفر صار هذا
+
+497
+00:45:48,780 --> 00:45:53,360
+المقدار و هذا المقدار بين هذا و هذا إذن اللي حصلت
+
+498
+00:45:53,360 --> 00:46:02,380
+عليه الآن هو ما يليه لكل epsilon أكبر من سفر لجهة
+
+499
+00:46:02,380 --> 00:46:07,610
+Deltaأكبر من السفر such that هذه beta كانت
+
+500
+00:46:07,610 --> 00:46:14,650
+arbitrary بين a و بين مين اللي هو such that if
+
+501
+00:46:14,650 --> 00:46:26,150
+beta بين ال a و ال a زائد delta we have then ايش
+
+502
+00:46:26,150 --> 00:46:31,990
+اللي حصلنا عليه اللي هو f of beta على f of alpha
+
+503
+00:46:33,100 --> 00:46:38,720
+صارت أصغر أو يساوي L زايد إبسلون وأكبر أو يساوي
+
+504
+00:46:38,720 --> 00:46:47,620
+إبسلون ناقص L أو بمعنى آخر IE F of Beta على F of
+
+505
+00:46:47,620 --> 00:46:53,340
+Alpha ناقص L absolute value أصغر أو يساوي إبسلون
+
+506
+00:46:53,340 --> 00:47:02,280
+وهذا هذا this means that hence limit
+
+507
+00:47:03,350 --> 00:47:13,590
+f of beta على f of alpha as اللي هو limit of فدي g
+
+508
+00:47:13,590 --> 00:47:23,970
+مالكم فدي g بساكتين g of beta فدي g of beta limit
+
+509
+00:47:23,970 --> 00:47:28,270
+of beta على g of beta مظبوط؟
+
+510
+00:47:29,140 --> 00:47:33,140
+as اللي هو طبعا الان بيت ايه اشمالها لكل بيتها
+
+511
+00:47:33,140 --> 00:47:37,620
+يعني بيتها وين راحت على الايم اليمين هان لكل بيتها
+
+512
+00:47:37,620 --> 00:47:49,320
+وين في الجوار هذا الان بيساوي ال هو المطلوب مش
+
+513
+00:47:49,320 --> 00:47:53,440
+عاجبك بيتك تبقى اكسر اي سؤال
+
+514
+00:47:56,950 --> 00:48:01,030
+بتصور هيك المظهرية واضحة تماما هي تلت خطوات في
+
+515
+00:48:01,030 --> 00:48:05,530
+الواقع تلت خطوات مين هما هو طبعا في الكتاب يعني لو
+
+516
+00:48:05,530 --> 00:48:12,190
+راحظت هتلاقي يعني أنه بدها بس ترتيب الان لأ بدها
+
+517
+00:48:12,190 --> 00:48:19,150
+ترتيب طلع عليها الان عندك هذا الان استخدمت أنا هذا
+
+518
+00:48:19,150 --> 00:48:23,370
+في الأول بالعمدان و استخدم هذا استخدمت هذا عشان
+
+519
+00:48:23,370 --> 00:48:27,150
+أقولك هذه ال inequality صحيح على كل المنطقة هذهلأن
+
+520
+00:48:27,150 --> 00:48:33,170
+لجي U لأن اللي لجيتها U أنا لجيتها لجيت ال U و
+
+521
+00:48:33,170 --> 00:48:36,070
+أخدت الفترة هنا عشان أقول لك ال U اللي لجيتها في
+
+522
+00:48:36,070 --> 00:48:41,570
+داخل الفترة هذه الحديث هذه جابل عن هذه بعمل
+
+523
+00:48:41,570 --> 00:48:45,750
+confusion عند الطالب لأن اللي لجيتها من قال إنه
+
+524
+00:48:45,750 --> 00:48:49,970
+هنا موجودة ما هي اللي لجيتها هنا وين لجيتها بين A
+
+525
+00:48:49,970 --> 00:48:56,050
+و B بين A و B أو بين Alpha و Beta يعني بده يصير
+
+526
+00:48:56,050 --> 00:49:05,880
+عندىاللي هي الـ F prime الـ F prime عندي ال limit
+
+527
+00:49:05,880 --> 00:49:11,140
+.. ال limit لل .. خليني أكتبها يا شيخ ليش؟ ها ..
+
+528
+00:49:11,140 --> 00:49:16,120
+هذا بالساوي ال .. اللي هي إيش ما لأ باخد ال
+
+529
+00:49:16,120 --> 00:49:20,740
+infinity بطلع حد برضه إيش infinity يعني الحالة
+
+530
+00:49:20,740 --> 00:49:26,250
+التانية اللي هي Fبساوة infinity هيكون ال limit لل
+
+531
+00:49:26,250 --> 00:49:33,570
+F على G أشماله بساوة infinity الان بده يصير عنده
+
+532
+00:49:33,570 --> 00:49:38,750
+بدل ما قول limit F prime على G prime بساوة L بده
+
+533
+00:49:38,750 --> 00:49:43,310
+يصير أشهر بساوة infinity كيف بنعبر على إنه الرقم
+
+534
+00:49:43,310 --> 00:49:47,190
+يروح لما لنهاية ال limit أنه ناخد اللي هو صرنا
+
+535
+00:49:47,190 --> 00:49:50,250
+متعارفين for every إبسط إنه كانت عبارة عن إيش صغير
+
+536
+00:49:50,250 --> 00:49:55,190
+for every KElement in R طبعا لو أخدت K positive
+
+537
+00:49:55,190 --> 00:49:58,050
+برضه بنفعله لإذا .. إذا بيكون أكبر من ال positive
+
+538
+00:49:58,050 --> 00:50:01,170
+أكيد هيكون أكبر من مين؟ من ال negative for every K
+
+539
+00:50:01,170 --> 00:50:05,450
+element in R there exists delta such that لكل X في
+
+540
+00:50:05,450 --> 00:50:12,970
+هذه المنطقة بيطلع اللي هو F prime of X اللي هو هذا
+
+541
+00:50:12,970 --> 00:50:16,630
+.. بقى اللي بمسحه هذا بمسح من التعريف ذاك للتعريف
+
+542
+00:50:16,630 --> 00:50:20,920
+limit F prime على G prime إيش بيساوي؟ما للهاي بما
+
+543
+00:50:20,920 --> 00:50:23,800
+أن هذا بيساوي ما لنهاي ايضا لأ كل K المتنارة بين X
+
+544
+00:50:23,800 --> 00:50:28,200
+زي الزلتة such that لما تكون X يعني بين ال A و بين
+
+545
+00:50:28,200 --> 00:50:31,060
+Z زي الدلتة يعني روحت في ال A من اليمين then F
+
+546
+00:50:31,060 --> 00:50:35,800
+برايم على D برايم أكبر من مين من K ماشي الحالة و
+
+547
+00:50:35,800 --> 00:50:39,540
+هذا اللي هو الواحد عندى و هذا كله اللي هو كلام
+
+548
+00:50:39,540 --> 00:50:46,340
+أشمله نفس الاشي متحقق فبصير عندى بستبدل هذا هذا
+
+549
+00:50:46,340 --> 00:50:50,270
+مافيش داعي له بيصير التفاصيل هناو بيصير عندى هذا
+
+550
+00:50:50,270 --> 00:50:55,110
+لجيتو اذا from واحد عند اتنين we have اللى هو
+
+551
+00:50:55,110 --> 00:51:01,370
+نعملها مع بعض بستبدل هذا بحطه هان بيصير عندى f
+
+552
+00:51:01,370 --> 00:51:11,330
+prime f of beta ناقص f of alpha على g of beta ناقص
+
+553
+00:51:11,330 --> 00:51:17,220
+g of alphaأكبر من مين من K ماشي الحال و نفس السبب
+
+554
+00:51:17,220 --> 00:51:21,260
+الأولاني الـ alpha اللي هو limit هذه سفر و limit
+
+555
+00:51:21,260 --> 00:51:26,260
+هذه سفر المعطيلة هي هو بيصير هذه عبارة عن لما ال
+
+556
+00:51:26,260 --> 00:51:28,700
+alpha تروح لل beta فبيصير عند أف في beta و دي في
+
+557
+00:51:28,700 --> 00:51:33,060
+beta أكبر من مين من K صار عندي الآن لكل K element
+
+558
+00:51:33,060 --> 00:51:38,360
+in R لجهة Delta بحيث أنه لما تكون Beta بين ال A و
+
+559
+00:51:38,360 --> 00:51:43,660
+A زائد Deltaحصلت على هذه أكبر من مين؟ من K وهذا
+
+560
+00:51:43,660 --> 00:51:50,080
+اللي هو إيه شماله؟ هو تعريف limit F of Beta على G
+
+561
+00:51:50,080 --> 00:51:58,480
+of Beta as Beta روح لل A من اليمين سوى ملا نهاية
+
+562
+00:51:58,480 --> 00:52:01,440
+ولو بدنا سالب ملا نهاية بنفس الأسلوب ده بتقول for
+
+563
+00:52:01,440 --> 00:52:07,660
+every K K سالمة بيصير أصغر ونفس الكلام
+
+564
+00:52:07,660 --> 00:52:13,110
+exampleexamples .. نشوف ال examples اللي عندنا
+
+565
+00:52:13,110 --> 00:52:20,890
+نيجي لل examples اللي عندى الأولى يعني limit
+
+566
+00:52:22,440 --> 00:52:28,260
+Sin X على جدر X لما X تروح لـ 0 من وين؟ من اليمين،
+
+567
+00:52:28,260 --> 00:52:33,740
+ليس بسهل، دعونا نشرح عن الوحش وبالكالكولس هذه لأن
+
+568
+00:52:33,740 --> 00:52:36,820
+limit Sin X على جدر X لما X تروح لـ 0 من وين؟ من
+
+569
+00:52:36,820 --> 00:52:40,620
+اليمين، الآن الـ limit اللي فوق لما X تروح لـ 0 من
+
+570
+00:52:40,620 --> 00:52:43,920
+اليمين، سفر و اللي تحت سفر، إذا صار عبارة عن 0 على
+
+571
+00:52:43,920 --> 00:52:48,020
+0 وكل أمورها إيش ما لها متحققة ال differential
+
+572
+00:52:48,020 --> 00:52:51,190
+الcontinuous و هو و هو و الاخري و الاخريإذا بنفضل
+
+573
+00:52:51,190 --> 00:52:53,330
+اللي فوق و بنفضل اللي تحت فضلنا اللي فوق و طلعني
+
+574
+00:52:53,330 --> 00:52:58,430
+cos X و اللي تحت 1 على 2 في جدر ال X الآن اللي ..
+
+575
+00:52:58,430 --> 00:53:00,150
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+576
+00:53:00,150 --> 00:53:00,750
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+577
+00:53:00,750 --> 00:53:01,710
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+578
+00:53:01,710 --> 00:53:01,750
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+579
+00:53:01,750 --> 00:53:01,830
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+580
+00:53:01,830 --> 00:53:01,850
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+581
+00:53:01,850 --> 00:53:02,510
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+582
+00:53:02,510 --> 00:53:09,470
+اللي .. اللي ..
+
+583
+00:53:09,470 --> 00:53:10,410
+اللي ..
+
+584
+00:53:15,260 --> 00:53:20,220
+الان اللي بعدها 1-sin x على x تربيه لما x تروح
+
+585
+00:53:20,220 --> 00:53:23,520
+لمين لسه في رجولنا النظرية صحيحة برضه لو كانت اللي
+
+586
+00:53:23,520 --> 00:53:27,900
+.. اللي .. اللي بدنا نروحلها جوا نقطة interior أو
+
+587
+00:53:27,900 --> 00:53:30,700
+على ال end points كلها صحيحة وبنفس الأسلوب البرهان
+
+588
+00:53:30,700 --> 00:53:34,700
+زي ما برهننا عن اليمين بنبرهن في الوسط فبناخد بدل
+
+589
+00:53:34,700 --> 00:53:37,700
+ما هو الجوار من a لعند a زائد delta إذا كانت جوا
+
+590
+00:53:37,700 --> 00:53:40,640
+من a ناقص delta لعند a زائد delta وده كان على
+
+591
+00:53:40,640 --> 00:53:47,360
+الجهة الثانية من a ناقص delta لعند ال aفاهمين
+
+592
+00:53:47,360 --> 00:53:51,060
+عليها هذه اللي هي برضه عبارة عن لو أخدنا limit للي
+
+593
+00:53:51,060 --> 00:53:54,780
+فوق سفر و limit للي تحت سفر إلى أن فضلنا اللي فوق
+
+594
+00:53:54,780 --> 00:53:58,940
+و فضلنا اللي تحت طالع عندي sin x على 2x إلى أن طلع
+
+595
+00:53:58,940 --> 00:54:04,360
+عندي 0 على 0 كمان مرة و متحقق كل أمورها إذا بنشتقل
+
+596
+00:54:04,360 --> 00:54:07,520
+كمان مرة بيصير cosine x على 2 و يساوي نص و هكذا
+
+597
+00:54:07,520 --> 00:54:09,180
+اللي بعدها
+
+598
+00:54:11,950 --> 00:54:16,150
+limit e to the x نقص واحد على x لما x تروح لمين
+
+599
+00:54:16,150 --> 00:54:20,730
+للصفر برضه نفس الاشي هذي بيصير صفر على صفر لlimit
+
+600
+00:54:20,730 --> 00:54:24,750
+الأولى بنشتق أن تطلع عنده واحد
+
+601
+00:54:28,600 --> 00:54:33,500
+الان الأخيرة نفس الشيء لإن ال X على X minus واحد
+
+602
+00:54:33,500 --> 00:54:36,560
+برضه نفس الشيء Zero ع Zero بطلع عندى اللى هو
+
+603
+00:54:36,560 --> 00:54:39,240
+بالفاضل هدى بتطلع واحد على X بالفاضل هدى واحد
+
+604
+00:54:39,240 --> 00:54:44,220
+بيصير الان لما ال X تروح للواحد بساوي الواحد اطلع
+
+605
+00:54:44,220 --> 00:54:48,920
+لفوق بيكون وصلنا احنا عند مين عند اخر نظرية اللى
+
+606
+00:54:48,920 --> 00:54:56,000
+هى Lobitals Ruleاللي هي في حالة اللي هي انها تطلع
+
+607
+00:54:56,000 --> 00:54:59,960
+عندى infinity او ناقص infinity ال limit يعني ال
+
+608
+00:54:59,960 --> 00:55:03,280
+indeterminate form اللي هو infinity على infinity
+
+609
+00:55:03,280 --> 00:55:07,080
+او ناقص infinity على infinity برضه المرة الجاية ان
+
+610
+00:55:07,080 --> 00:55:07,620
+شاء الله
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0246a46ab922bed556b883519bf52736e28310f5
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/SeyE20uTyhg_raw.srt
@@ -0,0 +1,2456 @@
+1
+00:00:05,300 --> 00:00:13,060
+بسم الله الرحمن الرحيم المحاضرة السادسة مساق تحليل
+
+2
+00:00:13,060 --> 00:00:18,680
+حقيقة 2 لطلبة قسم رياضيات بكلية العلوم بالجامعة
+
+3
+00:00:18,680 --> 00:00:23,220
+الإسلامية بغزةالحديث اليوم إن شاء الله هيكون حول
+
+4
+00:00:23,220 --> 00:00:30,200
+ال Lobitals Rules أو قواعد Lobitals Lobitals Rules
+
+5
+00:00:30,200 --> 00:00:38,320
+بتمر في calculus في تفاضل وتكامل بتمر من ناحية
+
+6
+00:00:38,320 --> 00:00:45,100
+عملية استخدامها وتوظيفها لحل اللي هي بعض النهايات
+
+7
+00:00:45,100 --> 00:00:50,400
+اللي بيكون .. اللي هو عجزنا عن حلها بطرق عادية
+
+8
+00:00:51,750 --> 00:00:57,550
+هنشوف الآن في الحديث عن الـ Lobital Zerol حول اللي
+
+9
+00:00:57,550 --> 00:01:01,570
+هو كيف اللي هو انبرهن هذه اللي هي القواعد كيف نشتق
+
+10
+00:01:01,570 --> 00:01:06,310
+هذه القواعد كيف اللي هو أيضًا بشكل سريع حول اللي
+
+11
+00:01:06,310 --> 00:01:12,490
+هي استخدام هذه القواعد طبعًا
+
+12
+00:01:12,490 --> 00:01:16,430
+في الأول هنتحدث عن الكميات الغير معينة
+
+13
+00:01:17,020 --> 00:01:23,580
+indeterminate forms اللي هي اللي بتعالجها اللي هي
+
+14
+00:01:23,580 --> 00:01:28,840
+lobitals rules عندي على سبيل المثال لو جينا أخدنا
+
+15
+00:01:28,840 --> 00:01:37,090
+limitاللي هو x على x تكعيب على x تربيع مثلا as x
+
+16
+00:01:37,090 --> 00:01:44,730
+goes to zero limit alpha x على x لما x تروح للزيرو
+
+17
+00:01:44,730 --> 00:01:50,670
+limit اللي هو x تربيع على x لما x تروح للزيرو
+
+18
+00:01:50,670 --> 00:01:59,900
+limit x على x تكعيب لما x تروح للزيرو limitمثلًا
+
+19
+00:01:59,900 --> 00:02:07,100
+سلب X على X تكعيب لما X تروح لـ 0 لو طلعنا على
+
+20
+00:02:07,100 --> 00:02:11,440
+اللي هي ال limits اللي موجودة هنا كلها على صورة
+
+21
+00:02:11,440 --> 00:02:16,260
+اللي هي لو تعويض مباشر هنلاقيها على صورة 0 على 0
+
+22
+00:02:16,960 --> 00:02:21,420
+الان كمية 0 على 0 بالطرق السابقة كان اللي هو انه
+
+23
+00:02:21,420 --> 00:02:27,620
+احنا صعب اللي هو نتعامل معها لكن في بعض الأحيان زي
+
+24
+00:02:27,620 --> 00:02:30,520
+الحالة هذه حلاقي انه احنا بنعرف نتعامل معها و
+
+25
+00:02:30,520 --> 00:02:36,140
+بنعرف نحكم عليها اللي هلاحظ انه كل الرغم ان كلها 0
+
+26
+00:02:36,140 --> 00:02:41,100
+على 0 الا انها بتعطي في كل حالة اشي مختلف عن
+
+27
+00:02:41,100 --> 00:02:47,350
+الحالة الثانية الان هذه مثلا عبارة عن limit1 على X
+
+28
+00:02:47,350 --> 00:02:51,030
+لما X تروح للـ 0 طبعا 1 على X لما X تروح للـ 0 إيش
+
+29
+00:02:51,030 --> 00:02:53,710
+مالها؟ does not exist لأنه من اليمين بتعطي مالة
+
+30
+00:02:53,710 --> 00:02:56,790
+نهاية ومن اليسار بتعطي سالب مالة نهاية عشان هيك
+
+31
+00:02:56,790 --> 00:03:02,510
+كده يبقى نقول عنها does not exist لأن limit Alpha
+
+32
+00:03:02,510 --> 00:03:06,210
+X على X لما X تروح للـ 0 هو يساوي عبارة عن Alpha
+
+33
+00:03:06,960 --> 00:03:09,720
+اللي هو عبارة عن real number لو فرضنا أنه Alpha
+
+34
+00:03:09,720 --> 00:03:13,480
+ماخدينها احنا real number اذا انا اعطت number عادل
+
+35
+00:03:13,480 --> 00:03:16,320
+اول ما اعطتش اللي هي انا اعطت does not exist
+
+36
+00:03:16,320 --> 00:03:20,620
+الحالة الثانية اللي هي بتطلع limit X لما X تروح ل
+
+37
+00:03:20,620 --> 00:03:24,330
+0 برضه اعطتنا أيه شمالها real numberفي الحالة اللي
+
+38
+00:03:24,330 --> 00:03:28,670
+بعدها أعطتنا اللي هو عبارة عن limit 1 على X تربيع
+
+39
+00:03:28,670 --> 00:03:32,790
+لما X تروح للصفر يعني أعطتنا إيش مالها زائد ملا
+
+40
+00:03:32,790 --> 00:03:37,070
+نهاية في الحالة الثالث الأخيرة هتعطينا اللي هو
+
+41
+00:03:37,070 --> 00:03:42,770
+ناقص limit 1 على X تربيع لما X تروح للزيرو بمعنى
+
+42
+00:03:42,770 --> 00:03:46,700
+أخر سالب ملا نهايةيعني الـ Indeterminate Form Zero
+
+43
+00:03:46,700 --> 00:03:51,880
+على Zero أعطتنا اللي هو أجوبة أو قيم مختلفة تابعًا
+
+44
+00:03:51,880 --> 00:03:56,280
+لطبيعة كل حالة من الحالات اللي موجودة مرة أعطتنا
+
+45
+00:03:56,280 --> 00:03:59,700
+doesn't exist مرة أعطتنا بسوء سفر ومرة أعطتنا بسوء
+
+46
+00:03:59,700 --> 00:04:03,640
+ألف وواحد خمسة ستة ناقص واحد اللي بدناها اللي هو
+
+47
+00:04:03,640 --> 00:04:09,020
+ناقص Infinity و Infinityالان ال .. ال .. ال .. ال
+
+48
+00:04:09,020 --> 00:04:13,780
+.. الـ Indeterminate form هذه اللي الآن يعني بدنا
+
+49
+00:04:13,780 --> 00:04:19,780
+نحاول نعليجها ب ..
+
+50
+00:04:19,780 --> 00:04:25,400
+نحاول نعليجها بال .. بال .. بالـ Lobitals rulesالـ
+
+51
+00:04:25,400 --> 00:04:28,640
+Indeterminate Form اللي عندنا اللي هو 0 على 0 طبعا
+
+52
+00:04:28,640 --> 00:04:33,240
+في Indeterminate Form أخرى برضه هتعالجها اللي هي L
+
+53
+00:04:33,240 --> 00:04:38,620
+'Hôpital's Rule أو Rules اللي هي زي Infinity على
+
+54
+00:04:39,400 --> 00:04:43,280
+Infinity أيضًا هذول الشغلتين الأساسيات اللي
+
+55
+00:04:43,280 --> 00:04:47,480
+هتعالجه اللي بطل الزرول مباشرة بنظريات مباشرة
+
+56
+00:04:47,480 --> 00:04:51,240
+عليها أيضًا هتظهر لو ظهرت عندنا مثلًا Infinity نقص
+
+57
+00:04:51,240 --> 00:04:55,640
+Infinity اللي هو Zero to Infinity Infinity to Zero
+
+58
+00:04:55,640 --> 00:05:01,160
+إلى آخره هذوله حالات أخرىاللي هو بنقدر نحولهم عن
+
+59
+00:05:01,160 --> 00:05:04,800
+طريق الـ ln أو عن طريق ال exponential أو بطرق
+
+60
+00:05:04,800 --> 00:05:08,820
+معينة لاللي هي ال formula هذه ومن ثم استخدام اللي
+
+61
+00:05:08,820 --> 00:05:12,240
+هو ال lobitals rules هذه عادة الشغلات اللي كانت
+
+62
+00:05:12,240 --> 00:05:16,460
+تعالجها اليمين اللي هو ال calculus أو التفاضل اللي
+
+63
+00:05:16,460 --> 00:05:20,500
+أخدناه في سنة أولى أو سنة أولى أو سنة تانية اطلع
+
+64
+00:05:20,500 --> 00:05:26,260
+لفوق نيجي الآن ناخد النظرية الأولىاللي هي لبتة
+
+65
+00:05:26,260 --> 00:05:30,540
+الزرولي الأولى ال formula الأولى أو الصورة الأولى
+
+66
+00:05:30,540 --> 00:05:35,660
+نظرية بسيطة و نظرية مرت عليكم و اثباتها أيضا
+
+67
+00:05:35,660 --> 00:05:41,200
+هتلاحظوا انه اللي هو بسيط ايش النظرية بتقول؟ بتقول
+
+68
+00:05:41,200 --> 00:05:43,680
+مايالي عندي
+
+69
+00:05:45,330 --> 00:05:51,250
+لت F be defined على الفترة المغلقة A وB ونفترض أن
+
+70
+00:05:51,250 --> 00:05:55,630
+F of A وG of A مايساوي؟ سفر ونفترض أن G of X لا
+
+71
+00:05:55,630 --> 00:06:00,470
+تساوي سفر في الفترة اللي هي بين A وB ونفترض لو
+
+72
+00:06:00,470 --> 00:06:04,420
+كانت F وG differentiable عند الـ Aو G' عند الـ A
+
+73
+00:06:04,420 --> 00:06:07,880
+لا يساوي سفر مفترضين G' لا يساوي سفر then the
+
+74
+00:06:07,880 --> 00:06:14,240
+limit of F على G at A exist و تساوي F' على G' و
+
+75
+00:06:14,240 --> 00:06:19,580
+إذا كان تحت كل هذا الشروط بطلع عنده اللي هو ال
+
+76
+00:06:19,580 --> 00:06:24,320
+limit had exist و بالظبط هذا ال limit بساوي F' على
+
+77
+00:06:24,320 --> 00:06:29,780
+F A على G' of A نشوف النظرية و نشوف برهان النظرية
+
+78
+00:06:29,780 --> 00:06:38,960
+theoremعندي ماخد الـ F و الـ G عبارة عن دوال من A
+
+79
+00:06:38,960 --> 00:06:47,280
+و B لعند R مفترض أن الـ F of A بيساوي الـ G of A
+
+80
+00:06:47,280 --> 00:06:53,680
+بيساوي إيش؟ بيساوي سفر و مفترض أن الـ G of X
+
+81
+00:06:58,040 --> 00:07:09,500
+لاتساوى 0 لكل X و M موجودة في الفترة A و B فرضنا
+
+82
+00:07:09,500 --> 00:07:16,260
+كمان F و G differentiable عند الـ A F prime
+
+83
+00:07:16,260 --> 00:07:22,740
+of A و G prime of A exists ستين تين وهذه لاتساوى
+
+84
+00:07:22,740 --> 00:07:29,900
+إيش لاتساوى سفرتحت هذه الظروف كلها بكون عندى limit
+
+85
+00:07:29,900 --> 00:07:38,410
+f of xعلى g of x as x بتروح لل a طبعا ال a الفترة
+
+86
+00:07:38,410 --> 00:07:43,390
+اللي عندنا هي فترة من وين من عند a ل عند b إذا
+
+87
+00:07:43,390 --> 00:07:46,270
+أكيد ال x إذا تروح لل a مافيش مجال لها ال x اللي
+
+88
+00:07:46,270 --> 00:07:49,210
+بتروح لل a اللي من وين من جهة اليمين لأنه هي
+
+89
+00:07:49,210 --> 00:07:52,450
+العالم اللي أنا عمال قاعد بشتغل فيه الفترة من a ل
+
+90
+00:07:52,450 --> 00:07:57,870
+b إذا ال x بتروح لل a من وين من اليمين هيساوي اللي
+
+91
+00:07:57,870 --> 00:08:06,930
+هو f prime عند ال a على g primeعند الـ A يعني
+
+92
+00:08:06,930 --> 00:08:09,910
+بمعنى أخر إيش اللي .. إيش .. إيش .. إيش .. كيف هم
+
+93
+00:08:09,910 --> 00:08:13,650
+نطبّك هذه النظرية؟ كانت تعرض علينا limit نيجي
+
+94
+00:08:13,650 --> 00:08:19,650
+يقولنا أوجد ال limit لل F of X على G of X لما X
+
+95
+00:08:19,650 --> 00:08:25,090
+تروح لمين؟ لل A من الأمين نيجي الآن ال F of A نعود
+
+96
+00:08:25,090 --> 00:08:29,500
+تعويض مباشر ده طلعت عند 0 على 0وكانت عندى الشروط
+
+97
+00:08:29,500 --> 00:08:32,520
+هذه مكتملة اللى هى الـF و الـG differentiable و
+
+98
+00:08:32,520 --> 00:08:36,080
+الـF prime و الـG prime موجودات عند الـA على طول
+
+99
+00:08:36,080 --> 00:08:42,420
+نحط هذه إيش بتساوي F prime of A على G prime of A
+
+100
+00:08:42,420 --> 00:08:48,880
+معايا فهذه اللى هي .. اللى هي .. كيفية تطبيق
+
+101
+00:08:48,880 --> 00:08:53,500
+النظرية نيجي لأهلنا برهان النظرية البرهان بسيط
+
+102
+00:08:53,500 --> 00:08:54,680
+عندى
+
+103
+00:08:59,270 --> 00:09:11,330
+خد عندي four x بين a و بين b لو جيت حسبت الغرض
+
+104
+00:09:11,330 --> 00:09:18,970
+الأوجود هذه ال f of x على g of x إيش هتساوي؟
+
+105
+00:09:18,970 --> 00:09:28,250
+هتساوي f of x ناقص f of aعلى g of x نقص g of a ليش
+
+106
+00:09:28,250 --> 00:09:31,110
+لأن ال f of a و ال d of a ليش ماعطينا إيها بساوة
+
+107
+00:09:31,110 --> 00:09:34,170
+سفر ليش عملت هيك لأ بدت أعمل أكتر من هي بدت أعمل
+
+108
+00:09:34,170 --> 00:09:39,110
+أقسم هذا على x minus a و هذا على x ما لها minus a
+
+109
+00:09:40,060 --> 00:09:43,580
+طبيعي ال X لا تسوى ال A الآن أنا باخد ال limit
+
+110
+00:09:43,580 --> 00:09:47,620
+للجهتين وبتجرأ و باخد موزع لإن أنا ضائع من ال F
+
+111
+00:09:47,620 --> 00:09:51,260
+prime of A موجودة و ال G prime of A شمالها ومش هيك
+
+112
+00:09:51,260 --> 00:09:54,160
+و كمان ال G prime of A لا تسوى 0 إذا كل أموري تمام
+
+113
+00:09:54,160 --> 00:09:58,220
+التمام إذا باخد ال limit للجهتين لما X تروح لل A
+
+114
+00:09:58,220 --> 00:10:04,040
+من اليمين بسوى ال limit لما X تروح لل A من اليمين
+
+115
+00:10:05,480 --> 00:10:09,560
+والاشي اللي تحت وزعت ليش وزعت ضامن أن ال limit
+
+116
+00:10:09,560 --> 00:10:14,520
+exist و ال limit اللي تحت كمان لا تساوي سفر هذه
+
+117
+00:10:14,520 --> 00:10:20,400
+اللي هي عبارة عن مين هذه تعريف F prime of A وهذه
+
+118
+00:10:20,400 --> 00:10:26,660
+تعريف G prime of A بكون أنا حصلت على اللي هو اللي
+
+119
+00:10:26,660 --> 00:10:33,300
+بديها هذه اللي هي النظرية الأولى في اللي هو هذا ال
+
+120
+00:10:33,300 --> 00:10:37,890
+sectionاللي يعني بتحذير بقولك أنه انت يعني انعرضت
+
+121
+00:10:37,890 --> 00:10:46,230
+عليك limit 17x ولا .. قداش؟ مش مشكلة x زياد 17 أين
+
+122
+00:10:46,230 --> 00:10:51,750
+كانت بنفع x زياد 17 على 2x زياد 3 مثلا لما x تروح
+
+123
+00:10:51,750 --> 00:10:55,500
+لمين للزفرهو مش ميقوم لما نشوف على طول و نروح
+
+124
+00:10:55,500 --> 00:11:00,960
+نفاضل بنفعش انت يعني فتالت بيصير فاضل الجهتين
+
+125
+00:11:00,960 --> 00:11:06,340
+بيطلع واحد على اتنين لأ هو انا بقول بتسوي الو أف
+
+126
+00:11:06,340 --> 00:11:10,720
+برايم على جي برايم عند ال zero لما نكون هدي zero و
+
+127
+00:11:10,720 --> 00:11:16,080
+هدي zero لكن لا هدي zero ولا هدي zero إذا بنفعش
+
+128
+00:11:16,260 --> 00:11:21,280
+تحديد هذا نقول بتساوي اللي هو limit من فعش نقول
+
+129
+00:11:21,280 --> 00:11:24,540
+بتساوي limit واحد على اتنين على اعتبار فضلنا
+
+130
+00:11:24,540 --> 00:11:28,800
+ويساوي نص وهذا الكلام غير صحيح لأن ال limit زي ما
+
+131
+00:11:28,800 --> 00:11:32,800
+انتوا عارفين لهذا المقدار بالتعويض المباشر هو
+
+132
+00:11:32,800 --> 00:11:42,780
+عبارة عن 0017 ع 3 هذا كلام سهل ناخد مثال تطبيقي
+
+133
+00:11:42,780 --> 00:11:48,310
+على النظرية اللي عندناالمثال التطبيقي برضه مثال
+
+134
+00:11:48,310 --> 00:11:56,530
+مباشر عرض علينا الآن example عرض
+
+135
+00:11:56,530 --> 00:12:04,830
+علينا بقول او جد limit x تربيع زاد x على sign 2x
+
+136
+00:12:04,830 --> 00:12:09,530
+لما x تروح لمين لزي روبالمناسبة، النظرية اللي قبل
+
+137
+00:12:09,530 --> 00:12:13,070
+بشوية حكينا عنها سواء كانت الـA اللي بتروحلها end
+
+138
+00:12:13,070 --> 00:12:17,150
+point أو نقطة داخلية أو حتى left end point بتظبط
+
+139
+00:12:17,150 --> 00:12:22,150
+عليها النظرية والبرهان similarly ماشي الحال؟ واضح
+
+140
+00:12:22,150 --> 00:12:28,430
+هاه؟ طيب، وإيه السبب؟
+
+141
+00:12:28,430 --> 00:12:34,390
+لأن الإختبار صفر على صفرو هذه differentiable و هذه
+
+142
+00:12:34,390 --> 00:12:38,970
+differentiable كل أمورها ميحة و كويسة و مش هي كمان
+
+143
+00:12:38,970 --> 00:12:43,730
+و لو فضلت هتلاقي اللي هنا لا يساوي سفر اذا على طول
+
+144
+00:12:43,730 --> 00:12:50,410
+بقول 2x عند ال zero بفاضل جاعد و بعوض يعني هذه
+
+145
+00:12:50,410 --> 00:12:56,950
+سميتها و كأنها Fوهذه g f of x وهذه g of x بعوض f
+
+146
+00:12:56,950 --> 00:13:02,190
+prime of zero بعوض هنا g prime of zero الان f
+
+147
+00:13:02,190 --> 00:13:06,490
+prime of zero اتنين x زائد واحد في سفر بيصير اتنين
+
+148
+00:13:06,490 --> 00:13:11,850
+في سفر زائد واحد وتحت اللي هو تفاضلها اتنين cosine
+
+149
+00:13:11,850 --> 00:13:16,970
+اتنين x بيصير اتنين cosine اتنين x وعوض بسفر بيصير
+
+150
+00:13:16,970 --> 00:13:25,500
+اتنين cosine اتنين في zero وهذا يعنيبساوي اللي هو
+
+151
+00:13:25,500 --> 00:13:30,140
+واحد على اتنين على اعتبار كوصالة Zero بساوي واحد
+
+152
+00:13:30,140 --> 00:13:36,760
+هذه اللي هي تطبيق النظرية اللي عندي نيجي الآن للـ
+
+153
+00:13:36,760 --> 00:13:40,480
+Cauchy Mean Value Theorem الـ Cauchy Mean Value
+
+154
+00:13:40,480 --> 00:13:46,140
+Theorem تعميم للـ Mean Value Theorem اللي احنا
+
+155
+00:13:46,140 --> 00:13:52,050
+عارفينهابدل ما هو على دالب نحكي عن إيش عن دالتين
+
+156
+00:13:52,050 --> 00:14:00,930
+نشوف إيش اللي بقوله النظرية بتقول ما يلي لأن
+
+157
+00:14:00,930 --> 00:14:11,690
+theorem عند F و G دالتين من A و B مخدهم من A و B
+
+158
+00:14:11,690 --> 00:14:17,390
+لعند Rجاي لي نفس شروط الـ mean value theorem
+
+159
+00:14:17,390 --> 00:14:22,150
+العادية بدل ما هي على دالها دالتين جاي لي F و G
+
+160
+00:14:22,150 --> 00:14:35,270
+continuous on A و B and differentiable on O B ماشي
+
+161
+00:14:35,270 --> 00:14:42,230
+الحال ومعطيني أيضا بقولي الـ G prime لل X لا تساوي
+
+162
+00:14:42,230 --> 00:14:49,110
+سفرلكل X وين موجودة في الـA و الـB اللي أنا
+
+163
+00:14:49,110 --> 00:14:59,190
+بقولليه النتيجة then there exist لنتيجة then then
+
+164
+00:14:59,190 --> 00:15:09,250
+هذا كله معطى if this hold then then
+
+165
+00:15:09,250 --> 00:15:18,160
+there exist CElement in A و B such that G أو F
+
+166
+00:15:18,160 --> 00:15:25,820
+prime of C على G prime of C بساوي F of B ناقص F of
+
+167
+00:15:25,820 --> 00:15:35,360
+A على G of B ناقص مياً G of A ال proof ده كلام سهل
+
+168
+00:15:35,360 --> 00:15:38,140
+كمان ال proof نشوف كده
+
+169
+00:15:41,590 --> 00:15:46,050
+عندي يا جماعة أول إشي هو معطيني إيش مالها g prime
+
+170
+00:15:46,050 --> 00:15:51,650
+of x إيش مالها لا تساوي سفر إذا by rules theorem
+
+171
+00:15:51,650 --> 00:15:58,490
+هيكون g of b لا تساوي مين g of a كيف أذكركم أذكركم
+
+172
+00:15:58,490 --> 00:16:03,950
+كيف الآن إيش rules theorem كانت بتقول g من a و b
+
+173
+00:16:03,950 --> 00:16:14,130
+لعند r continuous on a و bو differentiable on a و
+
+174
+00:16:14,130 --> 00:16:21,270
+b هذا ما أعطينا إياه هو ماشي الحال بقول لي if g of
+
+175
+00:16:21,270 --> 00:16:28,350
+a بساوي g of b بساوي سفر then هو في الواقع زي ما
+
+176
+00:16:28,350 --> 00:16:31,510
+قلنا أن ال role theorem تنفع لو قلنا g of a بساوي
+
+177
+00:16:31,510 --> 00:16:37,710
+g of b و سكتنا لأنه الشهد في الموضوع أنه المماس
+
+178
+00:16:37,710 --> 00:16:41,920
+يكون معله موازي لمحور الصينةأه و لما تكون الـ G of
+
+179
+00:16:41,920 --> 00:16:45,440
+A بيساوي الـ G of B وسكتنا أي قاطع بينهم هيكون
+
+180
+00:16:45,440 --> 00:16:48,520
+عبارة عن موازي لمحور السينات يعني معناته اللي
+
+181
+00:16:48,520 --> 00:16:52,220
+مماثل اللي باجيه هيكون موازي لهذا يعني موازي لمحور
+
+182
+00:16:52,220 --> 00:16:57,740
+السينات طيب then .. then there exists C element in
+
+183
+00:16:57,740 --> 00:17:03,520
+A و B such that G prime of C بيساوي إيش بيساوي سفر
+
+184
+00:17:03,520 --> 00:17:10,560
+الآن هذا معطى مفرغ منه اللي فوق معطى عنديالان عندى
+
+185
+00:17:10,560 --> 00:17:14,640
+لو كان g of a بسوى g of b بيعطينا انه يوجد صفر بين
+
+186
+00:17:14,640 --> 00:17:18,000
+ال a و ال b بحيث ان g prime of c اشماله لا تسوى
+
+187
+00:17:18,000 --> 00:17:23,260
+صفر لكن هو مفترضلى ان g prime of x لا تسوى صفر لكل
+
+188
+00:17:23,260 --> 00:17:26,540
+x في ال a و ال bيعني الآن الـ Contraposition هو
+
+189
+00:17:26,540 --> 00:17:30,800
+اللي hold hand يعني بمعنى آخر أنه بما أنه G prime
+
+190
+00:17:30,800 --> 00:17:36,160
+of X لا تساوي 0 لكل X element in A و B هيعطينا هذا
+
+191
+00:17:36,160 --> 00:17:40,940
+نفيه B implies Q تكافئ not Q implies not B وهذا
+
+192
+00:17:40,940 --> 00:17:46,240
+لسه عملته أنابما أن g prime of x لا يساوي سفر لكل
+
+193
+00:17:46,240 --> 00:17:51,840
+x طبعا في اللي هو b اذا g of a لا يساوي مين؟ g of
+
+194
+00:17:51,840 --> 00:17:59,900
+b واضح؟ طيب اذا صار عندي الأول حاجة by Rolle's
+
+195
+00:17:59,900 --> 00:18:09,850
+theorem g of a لا يساوي g of b becauseG prime of X
+
+196
+00:18:09,850 --> 00:18:15,670
+لا يساوي سفر لكل X والي موجودة في الـA والـB هذه
+
+197
+00:18:15,670 --> 00:18:22,850
+أول نقطة خلصنا هذه بررناها نيجي الآن زي ما عملنا
+
+198
+00:18:22,850 --> 00:18:26,270
+في إثبات الـMean Value Theorem إذا بتتذكروا بدي
+
+199
+00:18:26,270 --> 00:18:29,630
+أعرف دالة أطبق عليها برضه خلّي رولز الـTheorem
+
+200
+00:18:29,630 --> 00:18:32,890
+مطبقة أو الـMean Value Theorem مطبقة و أحصل عن
+
+201
+00:18:32,890 --> 00:18:38,260
+نتيجة اللي أنا بديهاالأن هي الشكل اللي بديها بدي
+
+202
+00:18:38,260 --> 00:18:45,680
+أخد H of X let أو define H of X بساوة اللي هو
+
+203
+00:18:45,680 --> 00:18:52,740
+المقدار هذا اللي بديه F of B ناقص F of A على G of
+
+204
+00:18:52,740 --> 00:18:57,910
+B minus G of Aبتخلي المقدار هذا اللي هو يتصفر في
+
+205
+00:18:57,910 --> 00:19:00,650
+حالة ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+206
+00:19:00,650 --> 00:19:00,870
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+207
+00:19:00,870 --> 00:19:00,870
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+208
+00:19:00,870 --> 00:19:00,910
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+209
+00:19:00,910 --> 00:19:00,970
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+210
+00:19:00,970 --> 00:19:01,090
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+211
+00:19:01,090 --> 00:19:01,850
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+212
+00:19:01,850 --> 00:19:01,890
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+213
+00:19:01,890 --> 00:19:02,850
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+214
+00:19:02,850 --> 00:19:07,410
+.. ال .. ال .. ال .. ال .. ال .. ال ..
+
+215
+00:19:07,410 --> 00:19:12,650
+ال ..
+
+216
+00:19:12,650 --> 00:19:12,650
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+217
+00:19:12,650 --> 00:19:16,510
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+218
+00:19:16,510 --> 00:19:24,170
+ال .. ال ..
+
+219
+00:19:25,340 --> 00:19:30,660
+H of A بيطلع سفر لإن بدي اللي هو الجزء الثاني اللي
+
+220
+00:19:30,660 --> 00:19:39,320
+هيجيبلي قيمة مين قيمة ال F نقص G G of B ناقص G of
+
+221
+00:19:39,320 --> 00:19:44,060
+A أو G of X ناقص H of A عشان نروحهم مع بعض ناقص F
+
+222
+00:19:44,060 --> 00:19:50,880
+of X ناقص F of A الآن ليش عملت هيك عشان أحصل H of
+
+223
+00:19:50,880 --> 00:19:53,180
+A H of A هذه سفر
+
+224
+00:19:55,780 --> 00:20:02,140
+و هد ايه ايه بيصير سفر H of B إذا صارت عندي H of A
+
+225
+00:20:02,140 --> 00:20:09,360
+بساوي سفر و H of B حط H of B بيصير هدي بيه هدي
+
+226
+00:20:09,360 --> 00:20:13,780
+بتروح مع هدي بيصير F of B ناقص F of A ناقص F of B
+
+227
+00:20:13,780 --> 00:20:16,800
+ناقص F of A بروحن مع بعض بيصير H برضه بيساوي سفر
+
+228
+00:20:16,800 --> 00:20:22,300
+إذا هدي أكيد برضه بتساوي H of B الآن
+
+229
+00:20:24,650 --> 00:20:28,050
+عندي هذا الـ differentiable و continuous و هذا الـ
+
+230
+00:20:28,050 --> 00:20:31,210
+differentiable و continuous على ما يناسبها من a و
+
+231
+00:20:31,210 --> 00:20:35,750
+b أو على ال a و b ال a و b اللي هو open
+
+232
+00:20:35,750 --> 00:20:38,770
+differentiable و على ال a و b closed continuous
+
+233
+00:20:38,770 --> 00:20:43,350
+تبعاً لها هتطلع انه ثابت هذا و هدوله ثوابت حيث ان
+
+234
+00:20:43,350 --> 00:20:50,330
+عندي هذا كله ع بعض continuous صار عنديH is
+
+235
+00:20:50,330 --> 00:20:59,130
+continuous on A وB وdifferentiable on A وB open
+
+236
+00:20:59,130 --> 00:21:03,930
+لأنه أنا من رأس الدورة الـ H of X معرفها اللي هي
+
+237
+00:21:03,930 --> 00:21:10,740
+الـ H for every X element in A وBH of X بيساوي كده
+
+238
+00:21:10,740 --> 00:21:15,360
+ماعيش يا بابا صارت الآن أنا حققت كل شروط ال roles
+
+239
+00:21:15,360 --> 00:21:19,780
+theorem اللي هي ال H continuous على ال A و B و ال
+
+240
+00:21:19,780 --> 00:21:23,360
+differential على ال A و B open أو H of A بيساوي و
+
+241
+00:21:23,360 --> 00:21:27,260
+H of B بيساوي اللي هي سفر إذا حسب roles theorem
+
+242
+00:21:27,260 --> 00:21:31,720
+إذا by roles theorem by أو حتى by main value
+
+243
+00:21:31,720 --> 00:21:38,180
+theorem by roles theorem there exists C element ال
+
+244
+00:21:38,180 --> 00:21:46,290
+A و Bsuch that اتش prime of c ايش هيساوي سفر هذا
+
+245
+00:21:46,290 --> 00:21:52,190
+حسب ال mean value theorem معايا يا جماعة الآن
+
+246
+00:21:52,190 --> 00:21:57,130
+بتفاضل هذه دلنا اوجد تفاضل هذه تفاضل هذه ماعليش
+
+247
+00:21:57,130 --> 00:22:04,760
+خلينا نكتبها لإنه ندخل على العالـ slide بدي أفاضل
+
+248
+00:22:04,760 --> 00:22:09,740
+هذه فاضله H prime of X اللي هي عند الـ C بتساوي
+
+249
+00:22:09,740 --> 00:22:15,580
+سفر سفر بساوي H prime of C تساوي فاضل هذه و أعوض
+
+250
+00:22:15,580 --> 00:22:20,420
+عن الـ X بسفر لأن هذا ثابت و هذا ثابت ده بيصير سفر
+
+251
+00:22:20,420 --> 00:22:23,420
+تفاضل هذا مع هذا لأن مع هذا بيصير G prime of X
+
+252
+00:22:23,420 --> 00:22:29,220
+مضروب في هذا إذا صار عندي بساوي F of B ناقص F of A
+
+253
+00:22:30,080 --> 00:22:36,100
+على g of b ناقص g of a مضروب في مين؟ فضلت في g
+
+254
+00:22:36,100 --> 00:22:40,920
+prime of x وانا بدي أحسبها عند مين؟ عند c، اذا g
+
+255
+00:22:40,920 --> 00:22:45,320
+prime of c ناقص تفاضل هذه، هذا ثابت سفر لأن هذا
+
+256
+00:22:45,320 --> 00:22:50,620
+قداش تفاضلها ناقص f prime عند ال x وانا باخدها عند
+
+257
+00:22:50,620 --> 00:22:56,280
+ال c اللي هي h of c, فh prime of c فبصير ناقص f
+
+258
+00:22:56,280 --> 00:23:02,200
+prime of cالان بتدنجل هذه على جهة هذه بصير عندى
+
+259
+00:23:02,200 --> 00:23:10,720
+الان f prime of c نجلته هان و اجسمها جي برايم of c
+
+260
+00:23:10,720 --> 00:23:15,840
+لأن جي برايم of c لاتساوى سفر بساوة f of b ناقص f
+
+261
+00:23:15,840 --> 00:23:25,710
+of a على g of b ناقص g of a و هيكبكون حصلنا على C
+
+262
+00:23:25,710 --> 00:23:31,630
+في الـA وB بحيث أنه هذه تتحقق وهو المطلوب هذه اللي
+
+263
+00:23:31,630 --> 00:23:37,190
+بيسميها Cauchy Mean Value Theorem هذه تعميم لمين؟
+
+264
+00:23:37,190 --> 00:23:41,590
+تعميم اللي هو الـMean Value Theorem بس خد اللي هو
+
+265
+00:23:41,590 --> 00:23:47,230
+G of X بتساوي XG of X بيساوي X و G of X بيساوي X
+
+266
+00:23:47,230 --> 00:23:49,190
+اللي هي تحقق كل الشروط ال differential
+
+267
+00:23:49,190 --> 00:23:53,470
+بالcontinuous و و الاخره ماشي الحال بيصير عندي
+
+268
+00:23:53,470 --> 00:23:58,110
+اللي هي في حالة G of X بتساوي X معايا يا شباب
+
+269
+00:23:58,110 --> 00:24:02,450
+بيصير G of BB و G of AA و هذه اللي هي G prime اللي
+
+270
+00:24:02,450 --> 00:24:05,510
+هي واحد فبيصير F prime of C بيساوي F of B نقص F of
+
+271
+00:24:05,510 --> 00:24:10,470
+A على B minus A إذا F
+
+272
+00:24:15,650 --> 00:24:21,210
+أمسح البرهان بس الان
+
+273
+00:24:21,210 --> 00:24:31,930
+ال note اللي عندي ال note كما يالي note if
+
+274
+00:24:31,930 --> 00:24:42,820
+g of x بيساوي x then we get fromالـ theorem هذه
+
+275
+00:24:42,820 --> 00:24:51,140
+اللي بقول عليها كوشي mean value theorem we get the
+
+276
+00:24:51,140 --> 00:24:59,500
+mean value theorem كيف الـ g of x بيساوي x معناه
+
+277
+00:24:59,500 --> 00:25:04,160
+انتصار الـ g of b بيساوي b و g of a بيصير a يعني
+
+278
+00:25:04,160 --> 00:25:10,530
+هذا بيصير a, bوهذه بيصير A و G prime of C اللي هي
+
+279
+00:25:10,530 --> 00:25:14,070
+بيصير واحد فبصير F prime of C بيسوي F of B نقص F
+
+280
+00:25:14,070 --> 00:25:20,590
+of A على B minus A وهذه هي ال main value theorem
+
+281
+00:25:21,280 --> 00:25:24,720
+واضح .. اه .. طيب اطلع للي بعده لإن احنا اللي
+
+282
+00:25:24,720 --> 00:25:26,820
+بيهمنا هذا الكوشي بين ال value theorem احنا
+
+283
+00:25:26,820 --> 00:25:31,980
+حكيناها أصلا عشان خاطر انه احنا نحكي عن اللي هو
+
+284
+00:25:31,980 --> 00:25:37,960
+Lobital's rule ال form اللي عنده اللي أمامي دكتور
+
+285
+00:25:37,960 --> 00:25:41,860
+جديد في السؤال في غيرك مفهوم ان انا جسمت على X
+
+286
+00:25:41,860 --> 00:25:46,640
+minus A على E minus A الطرفين اصلا انت بتصير F أو
+
+287
+00:25:46,640 --> 00:25:48,580
+P نقص F أو A على B minus A
+
+288
+00:26:01,110 --> 00:26:06,630
+البرهام مش صعب اللي قلناه، كتير سهل، ما هو الـ C
+
+289
+00:26:06,630 --> 00:26:09,970
+اللي لاجيناها في الحالة الأولى، هل هي التانية؟ إذا
+
+290
+00:26:09,970 --> 00:26:13,390
+كنت تتعامل بجلمين Value Theorem أنت؟مش بدك تطبّق
+
+291
+00:26:13,390 --> 00:26:16,170
+الـ Value Theory بدك تطبّق الـ Value Theory لأن
+
+292
+00:26:16,170 --> 00:26:21,090
+there exists c1 such that f prime of c1 بيسوء f of
+
+293
+00:26:21,090 --> 00:26:24,030
+b minus f of a على b minus a مافيش فيها مشكلة
+
+294
+00:26:24,030 --> 00:26:27,950
+there exists c2 such that g prime of c2 بيسوء g of
+
+295
+00:26:27,950 --> 00:26:32,550
+b نقص g of a على b minus a الـ c1 هذه مش شرط هيكون
+
+296
+00:26:32,550 --> 00:26:39,630
+main الـ c2 او احنا لازم نثبتها هي نفسها اه ف .. و
+
+297
+00:26:39,630 --> 00:26:44,780
+بعدين البرمجان سهلماشي .. نحن الواحد يفكر 100% ..
+
+298
+00:26:44,780 --> 00:26:50,780
+جميل لكن .. يعني .. بس أنه احنا .. الـC هذه مش
+
+299
+00:26:50,780 --> 00:26:53,700
+ضمنية تساوي الـC2 و بدنا نثبت أنها تساويها .. إذا
+
+300
+00:26:53,700 --> 00:26:59,920
+كانت بتساويها طيب .. نيجي لها اللي هي النظرية اللي
+
+301
+00:26:59,920 --> 00:27:05,840
+بعيدها النظرية اللي بعيدها .. أنا مرضيش أمسح اللوح
+
+302
+00:27:05,840 --> 00:27:08,800
+أساسي منقررها من الـtheory من الأولى
+
+303
+00:27:11,650 --> 00:27:30,050
+نشوف نطلع إيش اللي بتقوله هذه النظرية نفس
+
+304
+00:27:30,050 --> 00:27:35,710
+اللي هنا F و G differentiable on A و B معايا أه؟
+
+305
+00:27:35,710 --> 00:27:42,150
+الان such that G prime of X ده تساوي سفرG prime of
+
+306
+00:27:42,150 --> 00:27:47,970
+X لا تساوي سفر أنا مش موجودة For all X elements in
+
+307
+00:27:47,970 --> 00:27:55,070
+A وB ونفترض أنه limit F of X limit F of X لما X
+
+308
+00:27:55,070 --> 00:27:59,570
+تروح إلى الـ A من اليمين موجودة و بتساوي أياش؟ سفر
+
+309
+00:27:59,570 --> 00:28:02,490
+و limit G of X لما X تروح إلى الـ A من اليمين
+
+310
+00:28:02,490 --> 00:28:09,580
+بتساوي سفر اذا صار limit الحاصل القسمةلأن بيسويش f
+
+311
+00:28:09,580 --> 00:28:14,300
+prime of a على g prime of a بيساوياش limit f of x
+
+312
+00:28:14,300 --> 00:28:19,580
+على g of x لما x تروح لل a باليمينيعني وكأنه هنا
+
+313
+00:28:19,580 --> 00:28:25,460
+حول الحديث كله من افتراض أنه عند النقطة اللي هي F'
+
+314
+00:28:25,780 --> 00:28:31,240
+وG' موجودة ولا تساوي سفر وحول الحديث من إن التعويض
+
+315
+00:28:31,240 --> 00:28:35,280
+المباشر أن F of A وG of A بتساوي سفر ل limit لـ ..
+
+316
+00:28:35,280 --> 00:28:39,220
+limit لل function لما X تروح لـ A سفر و limit لل
+
+317
+00:28:39,220 --> 00:28:42,040
+function لما .. اللي .. اللي هو ال X .. ال X بتروح
+
+318
+00:28:42,040 --> 00:28:48,920
+لسفر .. لل A بتساوي سفروحول أيضا طلب جي برايم نفسه
+
+319
+00:28:48,920 --> 00:28:54,680
+على كل الفترة اللي تكون لا تساوي سفر مش عند النقطة
+
+320
+00:28:54,680 --> 00:29:03,060
+بس وقال لك انه حتى اللي هو النتيجة هتطلع ب limit
+
+321
+00:29:03,060 --> 00:29:07,940
+مش هتطلع ب F prime of A على G prime of A يعني اللي
+
+322
+00:29:07,940 --> 00:29:14,980
+بقصده انه الآن طور هذه الحديث هناإنه لما تُعرض
+
+323
+00:29:14,980 --> 00:29:21,580
+علينا limit F of X على G of X لما X تروح للـA من
+
+324
+00:29:21,580 --> 00:29:24,360
+لمين أو إن كانت حتى لو في الـinterior point الـX
+
+325
+00:29:24,360 --> 00:29:31,300
+بتروح للـA برضه صحيحة الآن بعدي عنديإذا لجيت limit
+
+326
+00:29:31,300 --> 00:29:34,480
+الأولى، اللي أنا بحكي عن limit سيرة مش عن تعويض
+
+327
+00:29:34,480 --> 00:29:38,340
+مباشر زي اللي جابله، الآن limit f of x لما x تروح
+
+328
+00:29:38,340 --> 00:29:41,780
+ل a من اليمين و limit f of x لما x تروح ل a من
+
+329
+00:29:41,780 --> 00:29:46,700
+اليمين، إذا هذه موجودة و هذه موجودة و طلع عندي 0
+
+330
+00:29:46,700 --> 00:29:51,060
+على 0، يعني طلع ال limit عبارة عن 0 على 0، هنا بدو
+
+331
+00:29:51,060 --> 00:29:57,200
+يحدث العلاج، إذا كانت الآنالـ limit اللي طلعت عندي
+
+332
+00:29:57,200 --> 00:30:02,000
+F prime of X على G prime of X لما X تروح للـ A من
+
+333
+00:30:02,000 --> 00:30:06,780
+اليمين إذا طلعت عبارة عن قيمة خلاص ارتاح هذه اللي
+
+334
+00:30:06,780 --> 00:30:12,450
+طلعت هي مين فهمت ال limitماشي الحال لو طلعت كمان
+
+335
+00:30:12,450 --> 00:30:18,230
+مرة zero على zero اللي هو و بتحقق كل الشروط اللي
+
+336
+00:30:18,230 --> 00:30:21,950
+.. اللي في الأول برضه بأعمل كمان مرة بفاضل لما
+
+337
+00:30:21,950 --> 00:30:26,430
+بتطلع لكن لو طلعت ال limit هذه does not exist بسكت
+
+338
+00:30:26,430 --> 00:30:31,130
+و بجوبي عمش .. بدنا ندور على طريقة أخرى واضح؟ الآن
+
+339
+00:30:31,130 --> 00:30:34,990
+لو طلعت هذه infinity أو سالب infinity هذه آسف
+
+340
+00:30:34,990 --> 00:30:38,900
+infinity أو سالب infinity برضه أن نظرية صحيحةاللي
+
+341
+00:30:38,900 --> 00:30:45,290
+هو هدمين الجزء الثاني من النظريةif limit f prime
+
+342
+00:30:45,290 --> 00:30:48,450
+على g prime بساوة L بساوة infinity أو سالب
+
+343
+00:30:48,450 --> 00:30:52,190
+infinity هتطلع ال limit على طول لل F على G اللي
+
+344
+00:30:52,190 --> 00:30:56,350
+ببحث عنها إيش هتساوي برضه ال infinity أو سالب
+
+345
+00:30:56,350 --> 00:31:00,670
+infinity حسب القيمة هذه إذا أي إن كانت اللي هي ال
+
+346
+00:31:00,670 --> 00:31:05,190
+limit مادامة exist سواء ال existence عبارة عن
+
+347
+00:31:05,190 --> 00:31:09,210
+element in R أو اللي هو عبارة عن ناقص infinity أو
+
+348
+00:31:09,210 --> 00:31:18,100
+سالب infinity فإن النظرية صحيحةواضح؟ أي سؤال؟ طيب
+
+349
+00:31:18,100 --> 00:31:27,200
+صلى على النبي عليه الصلاة والسلام خلينا
+
+350
+00:31:27,200 --> 00:31:31,360
+نيجي للنظرية ونبرنها
+
+351
+00:31:45,860 --> 00:31:50,880
+Theorem كدهش؟ نزلي بس النص يا محمد اللي هي theorem
+
+352
+00:31:50,880 --> 00:31:59,580
+6 3 3 إيش النظرية بتقول؟ بتقول ما يعني عندي طبعا
+
+353
+00:31:59,580 --> 00:32:04,540
+ماخد ال a أصغر من b strictly و a ممكن حتى تاخد سلب
+
+354
+00:32:04,540 --> 00:32:07,580
+infinity و ال b تاخد infinity يعني ممكن تكون
+
+355
+00:32:07,580 --> 00:32:11,040
+الفترة من a .. الفترة كلها a ممكن تكون أي فترة sub
+
+356
+00:32:11,040 --> 00:32:16,710
+interval من اللي هو 100 من ال real numbersفرضنا f
+
+357
+00:32:16,710 --> 00:32:26,190
+و g من a و b لعند اللي هو r وإذا كانت a infinity
+
+358
+00:32:26,190 --> 00:32:30,130
+أو سلب infinity أسف إذا كانت a سلب infinity أو b
+
+359
+00:32:30,130 --> 00:32:33,610
+infinity بتكون o من زمن بعرفي لأن عشان اللي هو
+
+360
+00:32:33,610 --> 00:32:36,890
+ناخدها من real number ل real number لذا نفترض f و
+
+361
+00:32:36,890 --> 00:32:46,080
+g من a و b لعند ال R ونفترض إن f و gdifferentiable
+
+362
+00:32:46,080 --> 00:32:54,620
+on a و b ماشي الحال و differentiable on a و b الان
+
+363
+00:32:54,620 --> 00:32:57,900
+مش لازم ي continue و تعندى و كده لأنه انا هدخل
+
+364
+00:32:57,900 --> 00:33:02,240
+لجوه شغل هيكون لجوه هل جيت بتشوف واش معنى لجوه
+
+365
+00:33:02,240 --> 00:33:08,620
+such that g prime of x g prime of x لا تسوى سفر
+
+366
+00:33:08,620 --> 00:33:17,430
+لكل x وين موجودة في الفترة a و bالان بقول لي إذا
+
+367
+00:33:17,430 --> 00:33:27,170
+كانت limit limit اللي هي f of x لما x تروح لل a من
+
+368
+00:33:27,170 --> 00:33:32,190
+اليمين بساوي limit g of x لما x تروح لل a من
+
+369
+00:33:32,190 --> 00:33:40,980
+اليمين بساوي سفر هذا كله موضوع الان بدو يوصلبدي
+
+370
+00:33:40,980 --> 00:33:46,200
+أقولك كيف أنا بدي أحصل على نتيجة limit f of x على
+
+371
+00:33:46,200 --> 00:33:51,060
+g of x لما x تروح لل a من اليمين بقولك إذا كان انت
+
+372
+00:33:51,060 --> 00:33:59,180
+if لان ايه if limit f prime of x على g prime of x
+
+373
+00:33:59,180 --> 00:34:06,520
+لما x تروح لل a من اليمين بساوي L thenأتجرأ أقول
+
+374
+00:34:06,520 --> 00:34:12,020
+فاش عندى مشكلة limit f of x اللى ببحث عنها على g
+
+375
+00:34:12,020 --> 00:34:17,360
+of x لما x تروح للإيه من اليمين برضه ايش هيساوي؟
+
+376
+00:34:17,360 --> 00:34:29,740
+هيساوي ال .. من يحياتي هذه هيكون صحيحة طيب خلينا
+
+377
+00:34:29,740 --> 00:34:38,330
+الآن اللى هو البرهن شوف عليهاواضح النظرية شرحناها
+
+378
+00:34:38,330 --> 00:34:45,050
+يعني نص النظرية شرحناها بشكل كامل الان since limit
+
+379
+00:34:45,050 --> 00:34:51,730
+F prime of X على G prime of X لما X تروح ل A من
+
+380
+00:34:51,730 --> 00:35:00,690
+وين؟ من اليمين هيعندي الفترة A وB حتى لو كانت
+
+381
+00:35:00,690 --> 00:35:05,580
+منتده وين ما بدها، هي حاضرة الان عنديX تذهب إلى
+
+382
+00:35:05,580 --> 00:35:09,760
+اليمن اليمين فبدي من الجهة دي طبيعي من اليمين بسوء
+
+383
+00:35:09,760 --> 00:35:15,800
+ال then for every epsilon أكبر من سفر لأي epsilon
+
+384
+00:35:15,800 --> 00:35:20,440
+أكبر من سفر أي epsilon there exists delta أكبر من
+
+385
+00:35:20,440 --> 00:35:24,660
+سفر such that لأن X تذهب إلى الامن وين من اليمين
+
+386
+00:35:24,660 --> 00:35:27,460
+إذن الجوار اللي حوالها هو عبارة عن جوار من A لعند
+
+387
+00:35:27,460 --> 00:35:32,340
+مين ال A زي ال Delta لعند A زي ال Delta صح ولا لأ
+
+388
+00:35:32,340 --> 00:35:36,260
+إذن لكلما دامت ال limit هي دي بساوي L، ده لكل
+
+389
+00:35:36,260 --> 00:35:41,480
+إبسلون بقدر ألاقي Delta بحيث أنه X element في A و
+
+390
+00:35:41,480 --> 00:35:46,400
+A زائد Delta ف X element في A و A زائد Delta، then
+
+391
+00:35:46,400 --> 00:35:52,180
+قطعا .. then قطعا هيكون عندي F prime of X على G
+
+392
+00:35:52,180 --> 00:35:58,030
+prime of X نقص L، ذوكون أصغر من 100 من إبسلونهذا
+
+393
+00:35:58,030 --> 00:36:02,370
+تعريف ال limit لل F prime of X على G prime of X
+
+394
+00:36:02,370 --> 00:36:06,510
+بسهولة لما X تروح لمن؟ لل A من اليمين هو كاتبها
+
+395
+00:36:06,510 --> 00:36:11,110
+هذه A زائد Delta C في الكتاب يعني مسميها C يعني
+
+396
+00:36:11,110 --> 00:36:16,090
+مسمي انه لكل X there exists C بحيث انه لكل X في ال
+
+397
+00:36:16,090 --> 00:36:20,650
+A اللي عند ال A لل C بكون هذا الكلام متحقق أنا
+
+398
+00:36:20,650 --> 00:36:23,510
+حبيت اكتبلك اللي هو التعريف الدارجي اللي انت ده
+
+399
+00:36:23,510 --> 00:36:27,700
+كسله كمان في ال .. في الحلواضح لحد تلاتة
+
+400
+00:36:38,080 --> 00:36:42,520
+ماشي الحال بس هي ك .. هذه .. و هذه ك صح و هذه ك صح
+
+401
+00:36:42,520 --> 00:36:47,040
+و هذه ك صح اللي كاتبها صح بس هذه للطالب أسهله في
+
+402
+00:36:47,040 --> 00:36:49,960
+ال .. في ال .. لإنه .. خلّيني أقول هذا اللي درج
+
+403
+00:36:49,960 --> 00:36:54,640
+عليه في ال .. في .. اللي هو التعبير عن ال .. لكل x
+
+404
+00:36:54,640 --> 00:36:58,940
+في الجوار الجوار هذا سهل و .. و .. و .. وعبّرنا
+
+405
+00:36:58,940 --> 00:36:59,200
+عنه
+
+406
+00:37:02,810 --> 00:37:07,630
+-A هو أكبر من سفر وأصغر من دلتة برضه هيك صح صحيح
+
+407
+00:37:07,630 --> 00:37:11,910
+نفس الاشي طيب
+
+408
+00:37:11,910 --> 00:37:15,050
+الان
+
+409
+00:37:15,050 --> 00:37:23,590
+هذا الكلام IA او بمعنى اخر that is هيكون هذا اللي
+
+410
+00:37:23,590 --> 00:37:31,230
+هو F prime of X ناقص G prime of Xأصغر ناقص L
+
+411
+00:37:31,230 --> 00:37:37,150
+بتغيرها عشان أصغر من Y وأكبر من 200 من سالب Y شيل
+
+412
+00:37:37,150 --> 00:37:43,470
+ال L هذه بصير اللي هو أصغر من L زائد Y وأكبر من L
+
+413
+00:37:43,470 --> 00:37:49,430
+ناقص Y هذا متحقق لمين لكل X و N موجودة في الفترة
+
+414
+00:37:49,430 --> 00:37:53,550
+من A لعند A زائد Delta و اكتب لي هذا سميلي
+
+415
+00:37:56,530 --> 00:38:03,990
+واحد ماشي الحال سميليه واحد الآن عندي اللي هو شروط
+
+416
+00:38:03,990 --> 00:38:06,850
+ال mean value theorem الكوشي mean value theorem
+
+417
+00:38:06,850 --> 00:38:10,130
+اللي قبل بشوية ال F و ال G differentiable على ال A
+
+418
+00:38:10,130 --> 00:38:16,170
+و ال B اه و ال G prime of X لها تساوي سفر و ال ..
+
+419
+00:38:16,170 --> 00:38:22,490
+او .. ايش كمان كله متحقق بس خلينا نقول
+
+420
+00:38:26,140 --> 00:38:31,240
+بدي الآن اوين اطبق الـ mean value theorem بدي
+
+421
+00:38:31,240 --> 00:38:37,720
+اطبقها على اللي هو اللي جوا عندى لداخل الـ mean
+
+422
+00:38:37,720 --> 00:38:44,560
+لداخل الفترة هذه عشان اشتغل وين اللي بتطلع عندى
+
+423
+00:38:44,560 --> 00:38:48,490
+اضمن تكون هناعشان القيمة اللى هتطلع عندى اللى
+
+424
+00:38:48,490 --> 00:38:53,170
+بتحققها تكون بتحقق اللى بتطلع عندى و بتحققها دى
+
+425
+00:38:53,170 --> 00:38:57,730
+عشان هي كصح اللى أعوض مكان بعض ماشي الحال خد الآن
+
+426
+00:38:57,730 --> 00:39:07,030
+لجد تفهموا إيش اللى بقصدهم خد الآن four alpha أكبر
+
+427
+00:39:07,030 --> 00:39:16,040
+من a و أصغر من beta و أصغر من a زائد deltaيعني أنا
+
+428
+00:39:16,040 --> 00:39:20,060
+غرضي ان انا اشتغل .. ان انا رايح .. بدي انا limit
+
+429
+00:39:20,060 --> 00:39:24,580
+اصلا و ال limit بديها لما اروح لمين؟ لل A يعني بدي
+
+430
+00:39:24,580 --> 00:39:29,760
+في الجوار اللي عيمين ال A و جاي ناحيتها هذا اللي
+
+431
+00:39:29,760 --> 00:39:31,740
+بهم، اللي مانيش سغاداش، بيكون التصرف مافيش ..
+
+432
+00:39:31,740 --> 00:39:36,980
+مافيش عنده مشكلة الآن هي ال alpha أخدتها هنا و هي
+
+433
+00:39:36,980 --> 00:39:37,420
+ال beta
+
+434
+00:39:41,900 --> 00:39:44,540
+أصغر من Alpha أصغر من Beta أصغر من ايه زي الـ
+
+435
+00:39:44,540 --> 00:39:54,060
+Delta؟ By Cauchy Mean Value Theorem there exists
+
+436
+00:39:54,060 --> 00:40:03,220
+سموها هو نسميها U Element in mean in Alpha و Beta
+
+437
+00:40:03,220 --> 00:40:10,110
+والـ Alpha و Beta جزء من هذهدُول، إذا اللي بنطبق
+
+438
+00:40:10,110 --> 00:40:14,750
+على هذه اللي بنطبق على .. اللي بنطبق على هذه بنطبق
+
+439
+00:40:14,750 --> 00:40:19,790
+على هذه، مظبوط؟ يعني هذه الـU اللي لجيتها بنطبق
+
+440
+00:40:19,790 --> 00:40:22,210
+عليها الكلام هذا اللي هو F prime of U على D prime
+
+441
+00:40:22,210 --> 00:40:27,510
+of U بين هذه و بين هذه واضح، و هذا الكلام مهم طيب،
+
+442
+00:40:27,510 --> 00:40:34,640
+there exists U such that F prime of Uعلى g prime
+
+443
+00:40:34,640 --> 00:40:39,240
+of u بيساوي ايش يا جماعة بيساوي f of b أو beta
+
+444
+00:40:39,240 --> 00:40:50,260
+ناقص f of alpha على g of beta ناقص g of alpha واضح
+
+445
+00:40:50,260 --> 00:41:00,020
+اه؟ هذا سمولي يمين هو اتنين الآن عند هذه اللي
+
+446
+00:41:00,020 --> 00:41:05,260
+لجيتها هنا اللي بتحقق هذههي وين موجودة من ضمن
+
+447
+00:41:05,260 --> 00:41:12,980
+النقاط اللي بتحقق هذه لكل X هنا وهذه جزء منها إذا
+
+448
+00:41:12,980 --> 00:41:18,100
+F prime of X of U على G prime of U بين الـ L ناقص
+
+449
+00:41:18,100 --> 00:41:22,020
+إبسلون والـ L زائد مين إبسلون وفي نفس الوقت F
+
+450
+00:41:22,020 --> 00:41:25,860
+prime of U هذه اللي لقيتها G prime of U بساوي هذا
+
+451
+00:41:25,860 --> 00:41:29,840
+إذا from واحد
+
+452
+00:41:30,390 --> 00:41:40,070
+واتنين we get اللي هو F prime of U على G prime of
+
+453
+00:41:40,070 --> 00:41:47,610
+U بستبدلها و بيصير F of Beta ناقص F of Alpha على G
+
+454
+00:41:47,610 --> 00:41:54,070
+of Beta ناقص G of Alpha بحيث أنه هذا يكون أكبر من
+
+455
+00:41:54,070 --> 00:42:03,630
+L ناقص Y و أصغر من 100 من L زائد Yلأن هذا صحيح لأي
+
+456
+00:42:03,630 --> 00:42:09,750
+Alpha و Beta بشكلهم اللي موجود Alpha أصغر من Beta
+
+457
+00:42:09,750 --> 00:42:15,570
+و Alpha بين الـA و بين مين الـA زائد دلتا إذا
+
+458
+00:42:15,570 --> 00:42:22,150
+الألفة هذه حرة في كل المنطقة هذه بنفع يعني الألفة
+
+459
+00:42:22,150 --> 00:42:26,530
+هذه لو بدها تروح للـA حد مش بمنحها الألفة تروح لها
+
+460
+00:42:26,530 --> 00:42:31,060
+لأنه صحيح على كل هذه الألفات اللي لجيتها هذهالان
+
+461
+00:42:31,060 --> 00:42:40,340
+let alpha goes to mean to a من وين؟ من اليمين
+
+462
+00:42:40,340 --> 00:42:45,720
+ماشي؟ مقدر و لن تؤثر لأ على ال beta ولا عليها زي
+
+463
+00:42:45,720 --> 00:42:50,980
+الدلتية الحرة بتروح لهذه وهذه زي ما هي يعني تصرف
+
+464
+00:42:50,980 --> 00:42:58,540
+alpha يروح لأي a beta بالنسبالها ثابت ولا تتأثرهذا
+
+465
+00:42:58,540 --> 00:43:03,500
+الكلام مهم إذا صار عندي الآن لما Alpha تروح لأ من
+
+466
+00:43:03,500 --> 00:43:10,700
+اليمين ال up of Alpha بيصير عندي limit ال up of
+
+467
+00:43:10,700 --> 00:43:17,780
+Alpha لما ال Alpha تروح لأ من اليمين بساوي هو في
+
+468
+00:43:17,780 --> 00:43:21,980
+الواقع limitها limitها هذه Alpha اللي كنا نسميها X
+
+469
+00:43:21,980 --> 00:43:23,980
+مثلا ال up of X سمى X تروح لأ من اليمين لأ من
+
+470
+00:43:23,980 --> 00:43:29,480
+اليمينالآن الـ Alpha راحت للإيمن اليمين limitها
+
+471
+00:43:29,480 --> 00:43:35,020
+أنا ماعطينيها إيش بيساوي؟ بيساوي صفر، ماعطيني
+
+472
+00:43:35,020 --> 00:43:37,080
+limit f of x عندما x تروح للإيمن اليمين إيش
+
+473
+00:43:37,080 --> 00:43:40,540
+بيساوي؟ صفر، أنا سميتها إيش أنا؟ Alpha هذه اللي
+
+474
+00:43:40,540 --> 00:43:46,660
+بتتحرك، إذا هذا ال limit إيش هيساوي؟ بنفس السبب أو
+
+475
+00:43:46,660 --> 00:43:52,940
+لنفس السبب limitG of Alpha لما Alpha تروح للـ A من
+
+476
+00:43:52,940 --> 00:43:59,680
+اليمين برضه مش هيساوي سفر إذا الآن برجع لهذه برجع
+
+477
+00:43:59,680 --> 00:44:09,060
+لهذه بصير عندى الآن اللي حصلته كله على بعضه لكل ي
+
+478
+00:44:09,060 --> 00:44:17,130
+أكبر من سفر ماشي الحال ل gate Deltaبحيث أنه الـ α
+
+479
+00:44:17,130 --> 00:44:20,050
+و beta بشكل هذا أكبر من إيه و أقل زي الـ delta
+
+480
+00:44:20,050 --> 00:44:24,570
+سيبك من الـ L α خلاص رديتها أنا beta أكبر من إيه و
+
+481
+00:44:24,570 --> 00:44:29,170
+أصغر من إيه زي الـ delta طلع عندي هذا المقدار أصغر
+
+482
+00:44:29,170 --> 00:44:33,170
+من L زي الـ يبسلون وأكبر من L ناقص يبسلون ماشي؟
+
+483
+00:44:33,170 --> 00:44:37,300
+الآن رديت و أخدتله من ال limitالان بدي أخد ال
+
+484
+00:44:37,300 --> 00:44:40,940
+limit لهذا المقدار كله لما ال alpha تروح لمين؟ لل
+
+485
+00:44:40,940 --> 00:44:45,240
+a من اليمين الان لما أخد ال limit لهذا زي ما قولت
+
+486
+00:44:45,240 --> 00:44:48,500
+ال beta مالهاش علاقة بال alpha ال beta ثابتة
+
+487
+00:44:48,500 --> 00:44:50,920
+بالنسبة لل alpha فال alpha تروح لل a من اليمين زي
+
+488
+00:44:50,920 --> 00:44:55,100
+ما بدها لن تتأثر beta فبصير limit اللي فوق على
+
+489
+00:44:55,100 --> 00:44:59,520
+limit اللي اتحد هذا ثابت و هذا ثابت و هذا limit هو
+
+490
+00:44:59,520 --> 00:45:04,300
+اللي هو صفر و هذا limit هو صفر و هذول أعداد إذا
+
+491
+00:45:04,300 --> 00:45:11,220
+أصار عندي الآنعندي take the limit بيصير عندي L
+
+492
+00:45:11,220 --> 00:45:18,100
+ناقص Epsilon أصغر أو يساوي اللي هو F of Beta على G
+
+493
+00:45:18,100 --> 00:45:23,660
+of Beta أصغر من L زائد Epsilon من وين حصلته هذا؟
+
+494
+00:45:23,660 --> 00:45:29,160
+when I take the limit of this inequality as Alpha
+
+495
+00:45:29,160 --> 00:45:37,460
+goes to A from right واضح؟هذا حصلت عليه هذا عرفت
+
+496
+00:45:37,460 --> 00:45:40,600
+ثانوية هذا حصلت عليه من وين أن أخدت ال limit
+
+497
+00:45:40,600 --> 00:45:44,660
+للجهات التلاتة as Alpha تروح لل A من اليمين
+
+498
+00:45:44,660 --> 00:45:48,780
+واستخدمت هذه الحقيقة أن هذا سفر و هذا سفر صار هذا
+
+499
+00:45:48,780 --> 00:45:53,360
+المقدار و هذا المقدار بين هذا و هذا إذن اللي حصلت
+
+500
+00:45:53,360 --> 00:46:02,380
+عليه الآن هو ما يليه لكل epsilon أكبر من سفر لجهة
+
+501
+00:46:02,380 --> 00:46:07,610
+Deltaأكبر من السفر such that هذه beta كانت
+
+502
+00:46:07,610 --> 00:46:14,650
+arbitrary بين a و بين مين اللي هو such that if
+
+503
+00:46:14,650 --> 00:46:26,150
+beta بين ال a و ال a زائد delta we have then ايش
+
+504
+00:46:26,150 --> 00:46:31,990
+اللي حصلنا عليه اللي هو f of beta على f of alpha
+
+505
+00:46:33,100 --> 00:46:38,720
+صارت أصغر أو يساوي L زايد إبسلون وأكبر أو يساوي
+
+506
+00:46:38,720 --> 00:46:47,620
+إبسلون ناقص L أو بمعنى آخر IE F of Beta على F of
+
+507
+00:46:47,620 --> 00:46:53,340
+Alpha ناقص L absolute value أصغر أو يساوي إبسلون
+
+508
+00:46:53,340 --> 00:47:02,280
+وهذا هذا this means that hence limit
+
+509
+00:47:03,350 --> 00:47:13,590
+f of beta على f of alpha as اللي هو limit of فدي g
+
+510
+00:47:13,590 --> 00:47:23,970
+مالكم فدي g بساكتين g of beta فدي g of beta limit
+
+511
+00:47:23,970 --> 00:47:28,270
+of beta على g of beta مظبوط؟
+
+512
+00:47:29,140 --> 00:47:33,140
+as اللي هو طبعا الان بيت ايه اشمالها لكل بيتها
+
+513
+00:47:33,140 --> 00:47:37,620
+يعني بيتها وين راحت على الايم اليمين هان لكل بيتها
+
+514
+00:47:37,620 --> 00:47:49,320
+وين في الجوار هذا الان بيساوي ال هو المطلوب مش
+
+515
+00:47:49,320 --> 00:47:53,440
+عاجبك بيتك تبقى اكسر اي سؤال
+
+516
+00:47:56,950 --> 00:48:01,030
+بتصور هيك المظهرية واضحة تماما هي تلت خطوات في
+
+517
+00:48:01,030 --> 00:48:05,530
+الواقع تلت خطوات مين هما هو طبعا في الكتاب يعني لو
+
+518
+00:48:05,530 --> 00:48:12,190
+راحظت هتلاقي يعني أنه بدها بس ترتيب الان لأ بدها
+
+519
+00:48:12,190 --> 00:48:19,150
+ترتيب طلع عليها الان عندك هذا الان استخدمت أنا هذا
+
+520
+00:48:19,150 --> 00:48:23,370
+في الأول بالعمدان و استخدم هذا استخدمت هذا عشان
+
+521
+00:48:23,370 --> 00:48:27,150
+أقولك هذه ال inequality صحيح على كل المنطقة هذهلأن
+
+522
+00:48:27,150 --> 00:48:33,170
+لجي U لأن اللي لجيتها U أنا لجيتها لجيت ال U و
+
+523
+00:48:33,170 --> 00:48:36,070
+أخدت الفترة هنا عشان أقول لك ال U اللي لجيتها في
+
+524
+00:48:36,070 --> 00:48:41,570
+داخل الفترة هذه الحديث هذه جابل عن هذه بعمل
+
+525
+00:48:41,570 --> 00:48:45,750
+confusion عند الطالب لأن اللي لجيتها من قال إنه
+
+526
+00:48:45,750 --> 00:48:49,970
+هنا موجودة ما هي اللي لجيتها هنا وين لجيتها بين A
+
+527
+00:48:49,970 --> 00:48:56,050
+و B بين A و B أو بين Alpha و Beta يعني بده يصير
+
+528
+00:48:56,050 --> 00:49:05,880
+عندىاللي هي الـ F prime الـ F prime عندي ال limit
+
+529
+00:49:05,880 --> 00:49:11,140
+.. ال limit لل .. خليني أكتبها يا شيخ ليش؟ ها ..
+
+530
+00:49:11,140 --> 00:49:16,120
+هذا بالساوي ال .. اللي هي إيش ما لأ باخد ال
+
+531
+00:49:16,120 --> 00:49:20,740
+infinity بطلع حد برضه إيش infinity يعني الحالة
+
+532
+00:49:20,740 --> 00:49:26,250
+التانية اللي هي Fبساوة infinity هيكون ال limit لل
+
+533
+00:49:26,250 --> 00:49:33,570
+F على G أشماله بساوة infinity الان بده يصير عنده
+
+534
+00:49:33,570 --> 00:49:38,750
+بدل ما قول limit F prime على G prime بساوة L بده
+
+535
+00:49:38,750 --> 00:49:43,310
+يصير أشهر بساوة infinity كيف بنعبر على إنه الرقم
+
+536
+00:49:43,310 --> 00:49:47,190
+يروح لما لنهاية ال limit أنه ناخد اللي هو صرنا
+
+537
+00:49:47,190 --> 00:49:50,250
+متعارفين for every إبسط إنه كانت عبارة عن إيش صغير
+
+538
+00:49:50,250 --> 00:49:55,190
+for every KElement in R طبعا لو أخدت K positive
+
+539
+00:49:55,190 --> 00:49:58,050
+برضه بنفعله لإذا .. إذا بيكون أكبر من ال positive
+
+540
+00:49:58,050 --> 00:50:01,170
+أكيد هيكون أكبر من مين؟ من ال negative for every K
+
+541
+00:50:01,170 --> 00:50:05,450
+element in R there exists delta such that لكل X في
+
+542
+00:50:05,450 --> 00:50:12,970
+هذه المنطقة بيطلع اللي هو F prime of X اللي هو هذا
+
+543
+00:50:12,970 --> 00:50:16,630
+.. بقى اللي بمسحه هذا بمسح من التعريف ذاك للتعريف
+
+544
+00:50:16,630 --> 00:50:20,920
+limit F prime على G prime إيش بيساوي؟ما للهاي بما
+
+545
+00:50:20,920 --> 00:50:23,800
+أن هذا بيساوي ما لنهاي ايضا لأ كل K المتنارة بين X
+
+546
+00:50:23,800 --> 00:50:28,200
+زي الزلتة such that لما تكون X يعني بين ال A و بين
+
+547
+00:50:28,200 --> 00:50:31,060
+Z زي الدلتة يعني روحت في ال A من اليمين then F
+
+548
+00:50:31,060 --> 00:50:35,800
+برايم على D برايم أكبر من مين من K ماشي الحالة و
+
+549
+00:50:35,800 --> 00:50:39,540
+هذا اللي هو الواحد عندى و هذا كله اللي هو كلام
+
+550
+00:50:39,540 --> 00:50:46,340
+أشمله نفس الاشي متحقق فبصير عندى بستبدل هذا هذا
+
+551
+00:50:46,340 --> 00:50:50,270
+مافيش داعي له بيصير التفاصيل هناو بيصير عندى هذا
+
+552
+00:50:50,270 --> 00:50:55,110
+لجيتو اذا from واحد عند اتنين we have اللى هو
+
+553
+00:50:55,110 --> 00:51:01,370
+نعملها مع بعض بستبدل هذا بحطه هان بيصير عندى f
+
+554
+00:51:01,370 --> 00:51:11,330
+prime f of beta ناقص f of alpha على g of beta ناقص
+
+555
+00:51:11,330 --> 00:51:17,220
+g of alphaأكبر من مين من K ماشي الحال و نفس السبب
+
+556
+00:51:17,220 --> 00:51:21,260
+الأولاني الـ alpha اللي هو limit هذه سفر و limit
+
+557
+00:51:21,260 --> 00:51:26,260
+هذه سفر المعطيلة هي هو بيصير هذه عبارة عن لما ال
+
+558
+00:51:26,260 --> 00:51:28,700
+alpha تروح لل beta فبيصير عند أف في beta و دي في
+
+559
+00:51:28,700 --> 00:51:33,060
+beta أكبر من مين من K صار عندي الآن لكل K element
+
+560
+00:51:33,060 --> 00:51:38,360
+in R لجهة Delta بحيث أنه لما تكون Beta بين ال A و
+
+561
+00:51:38,360 --> 00:51:43,660
+A زائد Deltaحصلت على هذه أكبر من مين؟ من K وهذا
+
+562
+00:51:43,660 --> 00:51:50,080
+اللي هو إيه شماله؟ هو تعريف limit F of Beta على G
+
+563
+00:51:50,080 --> 00:51:58,480
+of Beta as Beta روح لل A من اليمين سوى ملا نهاية
+
+564
+00:51:58,480 --> 00:52:01,440
+ولو بدنا سالب ملا نهاية بنفس الأسلوب ده بتقول for
+
+565
+00:52:01,440 --> 00:52:07,660
+every K K سالمة بيصير أصغر ونفس الكلام
+
+566
+00:52:07,660 --> 00:52:13,110
+exampleexamples .. نشوف ال examples اللي عندنا
+
+567
+00:52:13,110 --> 00:52:20,890
+نيجي لل examples اللي عندى الأولى يعني limit
+
+568
+00:52:22,440 --> 00:52:28,260
+Sin X على جدر X لما X تروح لـ 0 من وين؟ من اليمين،
+
+569
+00:52:28,260 --> 00:52:33,740
+ليس بسهل، دعونا نشرح عن الوحش وبالكالكولس هذه لأن
+
+570
+00:52:33,740 --> 00:52:36,820
+limit Sin X على جدر X لما X تروح لـ 0 من وين؟ من
+
+571
+00:52:36,820 --> 00:52:40,620
+اليمين، الآن الـ limit اللي فوق لما X تروح لـ 0 من
+
+572
+00:52:40,620 --> 00:52:43,920
+اليمين، سفر و اللي تحت سفر، إذا صار عبارة عن 0 على
+
+573
+00:52:43,920 --> 00:52:48,020
+0 وكل أمورها إيش ما لها متحققة ال differential
+
+574
+00:52:48,020 --> 00:52:51,190
+الcontinuous و هو و هو و الاخري و الاخريإذا بنفضل
+
+575
+00:52:51,190 --> 00:52:53,330
+اللي فوق و بنفضل اللي تحت فضلنا اللي فوق و طلعني
+
+576
+00:52:53,330 --> 00:52:58,430
+cos X و اللي تحت 1 على 2 في جدر ال X الآن اللي ..
+
+577
+00:52:58,430 --> 00:53:00,150
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+578
+00:53:00,150 --> 00:53:00,750
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+579
+00:53:00,750 --> 00:53:01,710
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+580
+00:53:01,710 --> 00:53:01,750
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+581
+00:53:01,750 --> 00:53:01,750
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+582
+00:53:01,750 --> 00:53:01,830
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+583
+00:53:01,830 --> 00:53:01,850
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+584
+00:53:01,850 --> 00:53:02,510
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+585
+00:53:02,510 --> 00:53:02,510
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+586
+00:53:02,510 --> 00:53:09,470
+اللي .. اللي ..
+
+587
+00:53:09,470 --> 00:53:10,410
+اللي ..
+
+588
+00:53:15,260 --> 00:53:20,220
+الان اللي بعدها 1-sin x على x تربيه لما x تروح
+
+589
+00:53:20,220 --> 00:53:23,520
+لمين لسه في رجولنا النظرية صحيحة برضه لو كانت اللي
+
+590
+00:53:23,520 --> 00:53:27,900
+.. اللي .. اللي بدنا نروحلها جوا نقطة interior أو
+
+591
+00:53:27,900 --> 00:53:30,700
+على ال end points كلها صحيحة وبنفس الأسلوب البرهان
+
+592
+00:53:30,700 --> 00:53:34,700
+زي ما برهننا عن اليمين بنبرهن في الوسط فبناخد بدل
+
+593
+00:53:34,700 --> 00:53:37,700
+ما هو الجوار من a لعند a زائد delta إذا كانت جوا
+
+594
+00:53:37,700 --> 00:53:40,640
+من a ناقص delta لعند a زائد delta وده كان على
+
+595
+00:53:40,640 --> 00:53:47,360
+الجهة الثانية من a ناقص delta لعند ال aفاهمين
+
+596
+00:53:47,360 --> 00:53:51,060
+عليها هذه اللي هي برضه عبارة عن لو أخدنا limit للي
+
+597
+00:53:51,060 --> 00:53:54,780
+فوق سفر و limit للي تحت سفر إلى أن فضلنا اللي فوق
+
+598
+00:53:54,780 --> 00:53:58,940
+و فضلنا اللي تحت طالع عندي sin x على 2x إلى أن طلع
+
+599
+00:53:58,940 --> 00:54:04,360
+عندي 0 على 0 كمان مرة و متحقق كل أمورها إذا بنشتقل
+
+600
+00:54:04,360 --> 00:54:07,520
+كمان مرة بيصير cosine x على 2 و يساوي نص و هكذا
+
+601
+00:54:07,520 --> 00:54:09,180
+اللي بعدها
+
+602
+00:54:11,950 --> 00:54:16,150
+limit e to the x نقص واحد على x لما x تروح لمين
+
+603
+00:54:16,150 --> 00:54:20,730
+للصفر برضه نفس الاشي هذي بيصير صفر على صفر لlimit
+
+604
+00:54:20,730 --> 00:54:24,750
+الأولى بنشتق أن تطلع عنده واحد
+
+605
+00:54:28,600 --> 00:54:33,500
+الان الأخيرة نفس الشيء لإن ال X على X minus واحد
+
+606
+00:54:33,500 --> 00:54:36,560
+برضه نفس الشيء Zero ع Zero بطلع عندى اللى هو
+
+607
+00:54:36,560 --> 00:54:39,240
+بالفاضل هدى بتطلع واحد على X بالفاضل هدى واحد
+
+608
+00:54:39,240 --> 00:54:44,220
+بيصير الان لما ال X تروح للواحد بساوي الواحد اطلع
+
+609
+00:54:44,220 --> 00:54:48,920
+لفوق بيكون وصلنا احنا عند مين عند اخر نظرية اللى
+
+610
+00:54:48,920 --> 00:54:56,000
+هى Lobitals Ruleاللي هي في حالة اللي هي انها تطلع
+
+611
+00:54:56,000 --> 00:54:59,960
+عندى infinity او ناقص infinity ال limit يعني ال
+
+612
+00:54:59,960 --> 00:55:03,280
+indeterminate form اللي هو infinity على infinity
+
+613
+00:55:03,280 --> 00:55:07,080
+او ناقص infinity على infinity برضه المرة الجاية ان
+
+614
+00:55:07,080 --> 00:55:07,620
+شاء الله
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a76881e09ae9d44f8eb0cff83dad3be105d24616
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0.srt
@@ -0,0 +1,3472 @@
+1
+00:00:05,180 --> 00:00:07,960
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:07,960 --> 00:00:10,980
+والصلاة والسلام على سيد المرسلين سيدنا محمد وعلى 
+
+3
+00:00:10,980 --> 00:00:16,800
+آله وصحبه أجمعين هذه المحاضرة رقم 25 مساق تحليل 
+
+4
+00:00:16,800 --> 00:00:21,880
+حقيقة 2 طلاب وطالبات الجامعة الإسلامية قسم الرياضيات
+
+5
+00:00:21,880 --> 00:00:27,060
+كلية العلوم اليوم هنبدأ إن شاء الله بالـ chapter
+
+6
+00:00:27,060 --> 00:00:32,290
+الرابع في المادة اللي هو الـ chapter 9 أو الجزء 
+
+7
+00:00:32,290 --> 00:00:35,410
+الرابع من المادة اللي هو chapter 9 تحت عنوان
+
+8
+00:00:35,410 --> 00:00:39,670
+infinite series هنحكي اليوم في هذه المحاضرة على
+
+9
+00:00:39,670 --> 00:00:47,900
+اللي هو convergence of infinite series الـ infinite
+
+10
+00:00:47,900 --> 00:00:54,380
+series طبعا هو موضوع معهود للطلاب أو لطلبة العلوم 
+
+11
+00:00:54,380 --> 00:00:58,440
+والهندسة بصورة عامة لأنه تم الحديث عنه في اللي هو
+
+12
+00:00:58,440 --> 00:01:03,340
+calculus باه الآن بنحكي عن اللي هو النظرة
+
+13
+00:01:03,340 --> 00:01:08,300
+التحليلية لـ اللي هو الـ infinite series طبعا كنا
+
+14
+00:01:08,300 --> 00:01:14,180
+حكينا في في chapter ثلاثة في تحليل حقيقي واحد على 
+
+15
+00:01:14,180 --> 00:01:18,540
+اللي هي الـ sequences الآن هذا مبني على اللي هو مين
+
+16
+00:01:18,540 --> 00:01:22,750
+الـ sequences خلينا نشوف التعريف الأول 911
+
+17
+00:01:22,750 --> 00:01:27,130
+definition fx بساوي xn is a sequence in R يعني
+
+18
+00:01:27,130 --> 00:01:32,330
+فرضنا أن xn عبارة عن متتابع في R then the infinite
+
+19
+00:01:32,330 --> 00:01:37,990
+series اللي هي simply the series بنا نحكي عن ..
+
+20
+00:01:37,990 --> 00:01:42,330
+نعرف الآن الـ infinite series أو الـ series بناء على
+
+21
+00:01:42,330 --> 00:01:46,850
+اللي هي الـ sequence x بساوي xn اللي هي the
+
+22
+00:01:46,850 --> 00:01:51,390
+infinite series generated by this sequence x is
+
+23
+00:01:51,390 --> 00:01:54,610
+the sequence S إذن الـ infinite series هي عبارة عن 
+
+24
+00:01:54,610 --> 00:02:00,170
+إيش؟ بتعرفها عبارة عن sequence S بساوي SK بس إيش
+
+25
+00:02:00,170 --> 00:02:05,230
+الـ SK اللي هي اللي بنسميها اللي هي sequence of
+
+26
+00:02:05,230 --> 00:02:11,510
+partial sums يعني الـ S K defined by S1 ايش بتساوي؟
+
+27
+00:02:11,510 --> 00:02:17,270
+X1 S2 ايش بتساوي؟ X1 زائد X2 لاحظوا بقى اجمع
+
+28
+00:02:17,270 --> 00:02:19,430
+العناصر اللي وين موجودة اللي في الـ sequence
+
+29
+00:02:19,430 --> 00:02:24,490
+الأصلية اللي هي بتعمل generation لـ اللي هي ايش؟ الـ
+
+30
+00:02:24,490 --> 00:02:28,710
+sequence of partial sums الـ S K S 3 إيش هتكون X 1
+
+31
+00:02:28,710 --> 00:02:34,290
+زائد X 2 زائد X 3 S K in general هي عبارة عن X 1 زائد 
+
+32
+00:02:34,290 --> 00:02:39,820
+X 2 عند من؟ اللي عند X K أو بمعنى آخر Sk
+
+33
+00:02:39,820 --> 00:02:56,100
+is the sum of the terms of the sequence
+
+
+34
+00:02:56,560 --> 00:03:02,220
+لأن if S converges, we refer to limit S as the sum
+
+35
+00:03:02,220 --> 00:03:06,160
+of infinite series يعني لو كانت الـ SK عندي هذه
+
+36
+00:03:06,160 --> 00:03:10,880
+converges to some limit S، بنقول احنا اللي هو 
+
+37
+00:03:10,880 --> 00:03:16,260
+limit الـ S K اللي هو بيساوي S اللي هو اللي بنسمي S
+
+38
+00:03:16,260 --> 00:03:19,500
+K طبعا goes to infinity أو N اللي بدكيها اللي هي
+
+39
+00:03:19,500 --> 00:03:24,120
+بيساوي الـ summation اللي هي X K كي من عند واحد إلى
+
+40
+00:03:24,120 --> 00:03:29,780
+وين إلى ما لا نهاية سمي هذه S N وهذه S N وهذه S N كل
+
+41
+00:03:29,780 --> 00:03:34,440
+برج للاسهذه اللي بنسميها الـ infinite series اللي
+
+42
+00:03:34,440 --> 00:03:37,300
+هي في حالة الـ convergence أو الـ limit هذه exist
+
+43
+00:03:37,300 --> 00:03:40,520
+تبع الـ sequence of partial sums بنسميها اللي هي 
+
+44
+00:03:40,520 --> 00:03:45,880
+limit exist بنسميها S لأن الـ elements XN are
+
+45
+00:03:45,880 --> 00:03:50,550
+called the terms and S and are called the partial
+
+46
+00:03:50,550 --> 00:03:54,450
+sums of this infinite series يعني هدولة اللي هي الـ
+
+47
+00:03:54,450 --> 00:03:58,390
+.. الـ xk's بنسميها اللي هي الـ terms تبعة الـ
+
+48
+00:03:58,390 --> 00:04:02,170
+infinite series و الـ S and أو الـ Sk بنسميها اللي
+
+49
+00:04:02,170 --> 00:04:06,310
+هي عبارة عن ايش الـ sequence of partial sums أو
+
+50
+00:04:06,310 --> 00:04:10,330
+بنسميهم are partial sums of this infinite series
+
+51
+00:04:10,330 --> 00:04:14,650
+نيجي الآن بما أنه اللي هو في النهاية الـ series
+
+52
+00:04:14,650 --> 00:04:17,960
+اللي عندنا هذه عبارة عن limit في حالة الـ
+
+53
+00:04:17,960 --> 00:04:22,080
+convergence هو عبارة عن limit of a sequence إذا
+
+54
+00:04:22,080 --> 00:04:25,520
+تنطبق عليها قوانين الـ sequences اللي أخدناها في
+
+55
+00:04:25,520 --> 00:04:30,960
+chapter 3 إذا متوقع إنه احنا نستخدم ما تحدثنا فيه
+
+56
+00:04:30,960 --> 00:04:36,420
+في chapter 3 أو نتائجها للحصول على النظريات اللي 
+
+57
+00:04:36,420 --> 00:04:37,520
+احنا بدنا نبرهنها 
+
+58
+00:04:40,050 --> 00:04:44,350
+النظرية الأولى تسعة واحد اثنين theorem a if the
+
+59
+00:04:44,350 --> 00:04:48,610
+series summation xn and summation yn converge يعني
+
+60
+00:04:48,610 --> 00:04:52,490
+السيريز هذه من n من واحد إلى ما لا نهاية لو فرضنا
+
+61
+00:04:52,490 --> 00:04:55,670
+أن هذا الـ sum موجود وهذا الـ sum موجود يعني
+
+62
+00:04:55,670 --> 00:05:00,010
+السيريز هذه اللي هي sequence of partial sums
+
+63
+00:05:00,010 --> 00:05:03,850
+سبعتها الـ limit لها موجودة والـ summation yn
+
+64
+00:05:03,850 --> 00:05:07,310
+converge يعني السيريزسيكوانس الـ partial sums
+
+65
+00:05:07,310 --> 00:05:10,790
+السبعية ده برضه موجودة limit لها then the series
+
+66
+00:05:10,790 --> 00:05:16,190
+summation xn زائد yn برضه converges ومش هيك كمان 
+
+67
+00:05:16,190 --> 00:05:20,510
+وقيمة السيكوه السيريز الجديدة summation xn زائد yn
+
+68
+00:05:20,510 --> 00:05:26,380
+بساوي السيريز الأولى زائد السيريز الثانية الآن اشي
+
+69
+00:05:26,380 --> 00:05:32,100
+مشابه لـ 4 Xn ناقص Yn بيساوي summation Xn ناقص
+
+70
+00:05:32,100 --> 00:05:36,940
+summation Min Yn الآن بي if the series summation
+
+71
+00:05:36,940 --> 00:05:41,560
+Xn is convergent is convergent and C is any real
+
+72
+00:05:41,560 --> 00:05:46,120
+number then the series summation C في Xn برضه
+
+73
+00:05:46,120 --> 00:05:50,960
+هيخليها تظلها convergent و بيطلع عند الـ summation
+
+74
+00:05:50,960 --> 00:05:55,930
+لـ C Xn بيساوي C في الـ summation لـ Xn للـ XL الآن
+
+75
+00:05:55,930 --> 00:06:00,990
+مدام نحكي عن series بده ترجم اللي هو الـ series إلى
+
+76
+00:06:00,990 --> 00:06:04,450
+sequences إلى sequence of partial sums واستخدم
+
+77
+00:06:04,450 --> 00:06:07,730
+اللي هي قوانين الـ limits القديمة يعني الموضوع
+
+78
+00:06:07,730 --> 00:06:13,160
+البرهان أمر سهل إن شاء الله الآن نفترض إنه السميق 
+
+79
+00:06:13,160 --> 00:06:16,580
+.. مدام هذا converts وهذا converts بدي افترض انه 
+
+80
+00:06:16,580 --> 00:06:21,580
+هذه قيمتها بساوي S وهذا قيمتها مثلا بساوي S' يعني
+
+81
+00:06:21,580 --> 00:06:25,300
+جاي let S بتساوي summation xi I من واحد إلى وين 
+
+82
+00:06:25,300 --> 00:06:27,140
+إلى اللي هي
+
+83
+00:06:52,810 --> 00:06:54,210
+كيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكي
+
+84
+00:06:54,500 --> 00:07:01,180
+الـ XI لأن هو جالني إن الـ limit لأن الـ series هذه
+
+85
+00:07:01,180 --> 00:07:05,160
+لما ما حطش اشي معناته من واحد إلى ما لا نهاية لما 
+
+86
+00:07:05,160 --> 00:07:09,090
+ما حطش اشي معناته بقصد الـ finite sum الآن هو جاهز
+
+87
+00:07:09,090 --> 00:07:11,790
+لأن هذا الـ series الـ summation هو موجود وهذا
+
+88
+00:07:11,790 --> 00:07:16,350
+موجود يعني بمعنى آخر الـ limit للـ S اللي هي هذه
+
+89
+00:07:16,350 --> 00:07:20,690
+limit للـ S بساوي الـ series هذه و limit للـ S برايم
+
+90
+00:07:20,690 --> 00:07:24,650
+بساوي الـ series هذه من واحد إلى ما لا نهاية الآن
+
+91
+00:07:24,650 --> 00:07:28,990
+صارت عندي هذا عبارة عن limit of what؟ of sequence
+
+92
+00:07:29,460 --> 00:07:33,960
+هذا عبارة عن limit of what؟ of sequence إذا حسب
+
+93
+00:07:33,960 --> 00:07:39,280
+اللي هو اللي هي النظريات اللي هي في chapter ثلاثة
+
+94
+00:07:39,280 --> 00:07:43,140
+الـ limit إذا كانت الـ limit للـ sequence exist و
+
+95
+00:07:43,140 --> 00:07:45,560
+limit للـ sequence الثانية exist فبكون limit
+
+96
+00:07:45,560 --> 00:07:49,560
+المجموع بتوزع يعني limit S زائد S prime إيش
+
+97
+00:07:49,560 --> 00:07:53,720
+هيساوي؟ limit للـ sequence الأولى زائد limit للـ
+
+98
+00:07:53,720 --> 00:07:56,760
+sequence الثانية هذا من وين جبته؟ هذا جبته من
+
+99
+00:07:56,760 --> 00:08:03,000
+chapter ثلاثة طيب الآن لكن S زي S برايم كـ sequence
+
+100
+00:08:03,000 --> 00:08:06,540
+هذا S زي S برايم مجموعة و sequences يعني الـ
+
+101
+00:08:06,540 --> 00:08:09,300
+sequence of partial sums الأولى زي الـ sequence of
+
+102
+00:08:09,300 --> 00:08:12,880
+partial sums الثانية إيش هتساوي؟ اللي هي عبارة عن
+
+103
+00:08:12,880 --> 00:08:16,760
+الـ sequence اللي الصمشي XI I من عند واحد لعينة K
+
+104
+00:08:16,760 --> 00:08:21,460
+هذا اللي صميتها أنا S K زي مين S K برايم هدولة
+
+105
+00:08:21,460 --> 00:08:25,410
+finite هدولة terms الآن هدولة finite وهدولة finite
+
+106
+00:08:25,410 --> 00:08:31,090
+إذا المجموح من إيش بيساوي summation xi زائد yi من
+
+107
+00:08:31,090 --> 00:08:36,490
+عند واحد لعند مين لعند k هذه هي عبارة عن sk زائد
+
+108
+00:08:36,490 --> 00:08:42,250
+sk برايم صارت عندي sequence جديدة اللي هي s زائد s
+
+109
+00:08:42,250 --> 00:08:47,840
+برايم هذه S زي ده S برايم إيش هتساوي يا جماعة اللي
+
+110
+00:08:47,840 --> 00:08:52,660
+هو الـ sequence اللي عندي اللي الـ .. الـ .. الـ K
+
+111
+00:08:52,660 --> 00:08:56,440
+ثيرم إلها أو اللي هو الـ sequence of partial sums
+
+112
+00:08:56,440 --> 00:09:02,680
+إلها هو عبارة عن XI زي ده YI I من واحد لعند K هذا 
+
+113
+00:09:02,680 --> 00:09:09,500
+الآن صار بساوي هذا طيب الآن بده نبدأ ناخد الـ limit
+
+114
+00:09:09,500 --> 00:09:17,310
+and then الـ summation بناء عليه الـ summation الـ S
+
+115
+00:09:17,310 --> 00:09:22,210
+زي الـ S' اللي هو حسب اللي هنا بيصير limit الـ S
+
+116
+00:09:22,210 --> 00:09:26,710
+زي الـ S' هذه اللي هو limit لهذه الـ sequence إيش 
+
+117
+00:09:26,710 --> 00:09:29,790
+هيصير عند هذا الـ limit لهذه الـ sequence؟ هيعندي
+
+118
+00:09:29,790 --> 00:09:36,150
+هذا بساوي اللي هو هذا المقدار طيب limit S زي الـ
+
+119
+00:09:36,150 --> 00:09:42,010
+S' اللي هو limit هذا المقدار اللي هو limit S زائد S
+
+120
+00:09:42,010 --> 00:09:49,710
+برايم بساوي limit الـ summation XI زائد YI I من عند
+
+121
+00:09:49,710 --> 00:09:54,070
+واحد لعند N أو لعند K زي ما بدي as N goes to
+
+122
+00:09:54,070 --> 00:09:57,390
+infinity هذا notation عبارة عن notation لمين للـ
+
+123
+00:09:57,390 --> 00:10:02,630
+summation لأنه صار هذا exist من أين؟ لأن هذا زائد
+
+124
+00:10:02,630 --> 00:10:06,170
+هذا exist لأن هذا exist بما أن هذا exist نحسس إنه
+
+125
+00:10:06,170 --> 00:10:11,250
+summation xi زائد yi I من عند واحد إلى ما لا نهاية
+
+126
+00:10:11,250 --> 00:10:14,850
+من جهة أخرى هذا أخبرني أيش بيساوي limit S زائد مين
+
+127
+00:10:14,850 --> 00:10:19,630
+S prime limit S اللي هو limit هذه limit هذه إيش
+
+128
+00:10:19,630 --> 00:10:27,710
+هيكون limit S بساوي summation xi I من واحد لمن الآن
+
+129
+00:10:27,710 --> 00:10:33,010
+لأن K راحت إلى وين إلى ما لا نهاية هذه limitها إذا 
+
+130
+00:10:33,010 --> 00:10:36,730
+بتساوي اللي هو I من واحد لعند ما لا نهاية و limit S
+
+131
+00:10:36,730 --> 00:10:41,110
+prime زيها SK بتروح لما لا نهاية أو N بتروح لما لا
+
+132
+00:10:41,110 --> 00:10:44,370
+نهاية بتساوي summation YI I من واحد إلى ما لا 
+
+133
+00:10:44,370 --> 00:10:49,050
+نهاية بناء على أن هذه بتساوي هذه إذا صار المقدار
+
+134
+00:10:49,050 --> 00:10:54,070
+هذا بساوي المقدار هذا زائد المقدار هذا أو بمعنى
+
+135
+00:10:54,070 --> 00:10:57,130
+آخر summation XN زائد YN بساوي summation XN زائد 
+
+136
+00:10:57,130 --> 00:11:00,930
+YN وهو المطلوب
+
+137
+00:11:02,570 --> 00:11:05,930
+Similarly, summation XN ناقص YN بساوي summation XN 
+
+138
+00:11:05,930 --> 00:11:09,410
+ناقص YN بساوي summation XN ناقص YN بساوي summation XN
+
+139
+00:11:09,410 --> 00:11:09,610
+ناقص YN بساوي summation XN ناقص YN بساوي summation XN
+
+140
+00:11:09,610 --> 00:11:09,790
+ناقص YN بساوي summation XN ناقص YN بساوي summation XN
+
+141
+00:11:09,790 --> 00:11:09,830
+ناقص YN بساوي summation XN ناقص YN بساوي summation XN
+
+142
+00:11:09,830 --> 00:11:12,850
+ناقص YN بساوي summation XN ناقص YN بساوي summation XN
+
+143
+00:11:12,850 --> 00:11:15,170
+ناقص YN بساوي summation XN 
+
+144
+00:11:19,400 --> 00:11:25,440
+الـ S ناقص S برايم بساوي limit S ناقص limit S
+
+145
+00:11:25,440 --> 00:11:29,240
+برايم وبنجري نفس اللي جريناها قبل بشوية واللي
+
+146
+00:11:29,240 --> 00:11:34,020
+الثانية لأنه بنحول وكأن حديثنا من series لـ
+
+147
+00:11:34,020 --> 00:11:38,560
+sequence وكل قوانين الـ sequence تم إثباتها سابقا 
+
+148
+00:11:38,560 --> 00:11:43,100
+بـ mean بواسطة اللي هو شعب طر تلاتة أو في تحليل
+
+149
+00:11:43,100 --> 00:11:50,570
+حقيقي واحد الآن الأخيرة الـ summation لـ Cn بيساوي 
+
+150
+00:11:50,570 --> 00:11:56,250
+الـ C في الصممشي للـ Xn برضه اعتمادا على limit الـ C
+
+151
+00:11:56,250 --> 00:12:00,990
+في الـ S بيساوي الـ C في limit الـ S وبنجري نفس اللي 
+
+152
+00:12:00,990 --> 00:12:04,950
+أجرناها قبل بشوية اللي هو تحويلها من .. اللي هو
+
+153
+00:12:04,950 --> 00:12:09,330
+sequence .. من .. من series إلى sequences ومن ثم 
+
+154
+00:12:09,330 --> 00:12:15,690
+استخدام اللي هي قوانين الـ sequences طيب الآن نجي لـ
+
+155
+00:12:15,690 --> 00:12:25,090
+اللي هي اللمة اللي هي اللمة
+
+156
+00:12:25,090 --> 00:12:28,970
+اللي بعدها اللي هي المشهورة
+
+157
+00:12:30,530 --> 00:12:34,010
+وهذه test for divergence في الواقع كنا نقول عنها
+
+158
+00:12:34,010 --> 00:12:39,130
+في calculus باه إيش الـ sequence هذه أو اللي هي 
+
+159
+00:12:39,130 --> 00:12:42,330
+sequence of partial sums أو الـ series بقول إذا
+
+160
+00:12:42,330 --> 00:12:46,490
+كانت الـ series summation xn converges in R إذا كان 
+
+161
+00:12:46,490 --> 00:12:50,650
+اللي هي الـ summation للـ xn converges in R لازم
+
+162
+00:12:50,650 --> 00:12:54,810
+يكون limit اللي هي الـ sequence الأصلية اللي بنكون
+
+163
+00:12:54,810 --> 00:12:58,310
+منها sequence of partial sums لازم limit إيش يساوي
+
+164
+00:12:59,640 --> 00:13:04,000
+يساوي صفر وبناء عليه لو كانت الـ limit لها تساوي 
+
+165
+00:13:04,000 --> 00:13:07,640
+صفر على طول بنحكم على summation للـ Xn إيش ماله
+
+166
+00:13:07,640 --> 00:13:12,580
+diverges وكنا نقول هذا test of divergence مش test
+
+167
+00:13:12,580 --> 00:13:16,360
+of convergence لأن ممكن الـ limit تساوي صفر لكن الـ
+
+168
+00:13:16,360 --> 00:13:19,960
+summation للـ Xn يكون diverges والمثال اللي أنتم
+
+169
+00:13:19,960 --> 00:13:24,160
+مشهور اللي هو summation 1 على n هذا اللي هو limit
+
+170
+00:13:24,680 --> 00:13:29,900
+الواحد على n بيساوي صفر but this series diverges
+
+171
+00:13:29,900 --> 00:13:33,860
+زي ما هنشوف بعد شوية وزي ما احنا اصلا عارفين ولكن
+
+172
+00:13:33,860 --> 00:13:38,780
+احنا هنثبته إثبات الآن كنا اللي هو أخدناها معلومة 
+
+173
+00:13:38,780 --> 00:13:44,890
+في تفاضل وتكامل طيب صلي على النبي عليه الصلاة والسلام الآن 
+
+174
+00:13:44,890 --> 00:13:49,650
+by definition the convergence of summation xn
+
+175
+00:13:49,650 --> 00:13:54,430
+means that limit الـ sk شماله exist إيش الـ sk 
+
+176
+
+201
+00:15:55,140 --> 00:15:59,780
+monotone then the sequence converges if and only if
+
+202
+00:15:59,780 --> 00:16:04,180
+it is bounded هذه اللي هو في حالة مين الـ monotone 
+
+203
+00:16:04,180 --> 00:16:07,620
+functions لكن in general كنا نقول إن الـ sequence
+
+204
+00:16:07,620 --> 00:16:10,360
+is convergent على طول اللي بيعطينا then it is
+
+205
+00:16:10,360 --> 00:16:15,820
+bounded but it may be bounded  بطريقة ما is not 
+
+206
+00:16:15,820 --> 00:16:21,140
+convergent وكنا نقول ناخد مثال ناقص واحد على أن ال
+
+207
+00:16:21,140 --> 00:16:26,200
+sequence هذه أو ناقص واحد أس أن ناقص واحد أس أن ناقص 
+
+208
+00:16:26,200 --> 00:16:29,520
+واحد وثاني هذه الـ sequence bounded لأن قيمتها الـ
+
+209
+00:16:29,520 --> 00:16:32,940
+absolute value لها واحد but اللي هو ال limit اللي
+
+210
+00:16:32,940 --> 00:16:35,500
+لها does not exist لكن في حالة الـ monotone
+
+211
+00:16:35,500 --> 00:16:39,240
+function لأ ال boundedness يكافئ ال convergence
+
+212
+00:16:39,240 --> 00:16:43,880
+لأن هذا بدنا اللي هو الآن نطبق على ال monotone
+
+213
+00:16:43,880 --> 00:16:48,760
+convergence theorem أو نستخدمها في إثبات نظريتنا
+
+214
+00:16:48,760 --> 00:16:52,980
+اللي هي 914 theorem لت XN be a sequence of non
+
+215
+00:16:52,980 --> 00:16:57,500
+-negative real numbers إذا حكينا عن مين عن sequence
+
+216
+00:16:57,500 --> 00:17:01,900
+xn عنصرها إيش ما لها non negative real numbers
+
+217
+00:17:01,900 --> 00:17:08,630
+يعني أكبر أو تساوي صفر ثم الـ summation لـ xn من
+
+218
+00:17:08,630 --> 00:17:12,050
+السيريز تبعتها converges if and only if the
+
+219
+00:17:12,050 --> 00:17:15,750
+sequence S بتساوى Sk of partial sums is bounded
+
+220
+00:17:15,750 --> 00:17:21,110
+يعني الآن السيريز ال summation xn اللي هي هتكون
+
+221
+00:17:21,110 --> 00:17:26,470
+converges إذا و فقط إذا ال Sk إشمالها is bounded in
+
+222
+00:17:26,470 --> 00:17:30,970
+this case ال summation لـ xn بسوى limit لSk بساوى
+
+223
+00:17:30,970 --> 00:17:36,990
+supremum لSk يعني الآن وكأنه بيقوللي إنه limit الـ
+
+224
+00:17:36,990 --> 00:17:45,130
+SK exists if and only if SK is bounded هذه الـ SK
+
+225
+00:17:45,130 --> 00:17:48,410
+هي الـ sequence of partial sums تبع السيريز limit
+
+226
+00:17:48,410 --> 00:17:52,090
+الـ SK هي السيريز نفسها الـ xn من واحد إلى
+
+227
+00:17:52,090 --> 00:17:58,110
+ما لا نهاية الآن باختصار اللي هو هنقوله أنه أصلا هذه
+
+228
+00:17:58,110 --> 00:18:01,810
+الـ Sk sequence of partial sums is an increasing
+
+229
+00:18:01,810 --> 00:18:05,950
+sequence مدام increasing يعني بمعنى آخر monotone
+
+230
+00:18:05,950 --> 00:18:10,090
+أو بالظبط اللي هو خلّيني أقول واحدة من ال monotone
+
+231
+00:18:10,090 --> 00:18:12,530
+هي increasing  increasing هي increasing علنا الآن
+
+232
+00:18:12,530 --> 00:18:17,490
+مدام increasing هيكون عنده الـ Existence limit
+
+233
+00:18:17,490 --> 00:18:21,330
+يكافي الـ Boundedness ويكون خلصنا برهان النظرية
+
+234
+00:18:21,330 --> 00:18:24,310
+حسب الـ monotone convergence theorem ومش هيك وهيكون
+
+235
+00:18:24,310 --> 00:18:30,770
+limit هاش ما لمدام increasing السبريمام لعنصرها
+
+236
+00:18:30,770 --> 00:18:34,410
+هذه الـ monotone convergence theorem خليني نشوف هذا
+
+237
+00:18:34,410 --> 00:18:39,680
+الكلام بالتفصيل اللي حكيته بما أن xn أكبر يساوي 0
+
+238
+00:18:39,680 --> 00:18:43,300
+الـ xn أكبر يساوي 0 then the sequence of partial
+
+239
+00:18:43,300 --> 00:18:48,540
+sums is monotone increasing function ليش؟ لأن S1
+
+240
+00:18:48,540 --> 00:18:55,720
+عبارة عن x1 S1 عبارة عن x1 أكيد أصغر من x1 زاد x2
+
+241
+00:18:55,720 --> 00:19:00,670
+التي هي مين؟ S2 لأن هذا الـ x2 اللي ضفته أكبر أو
+
+242
+00:19:00,670 --> 00:19:06,090
+يساوي 0 وهذا أصغر من S3 اللي هو x1 زي x2 زي x3 in
+
+243
+00:19:06,090 --> 00:19:12,890
+general سيكون الـ Sk أصغر من S k زي 1 أو أصغر أو
+
+244
+00:19:12,890 --> 00:19:16,830
+يساوي حسب اللي هو الإضافة 0 أو كمية موجبة ماشي
+
+245
+00:19:16,830 --> 00:19:21,340
+الحال إذا صارت عندي ال sequence of partial sum اللي
+
+246
+00:19:21,340 --> 00:19:24,560
+هو صارت عبارة عن increasing إذا according to the
+
+247
+00:19:24,560 --> 00:19:27,240
+monotone convergence theorem اللي قلت قبل بشوية
+
+248
+00:19:27,240 --> 00:19:30,480
+اللي هي بالظبط رقمها 3.2 the
+
+249
+00:19:30,480 --> 00:19:37,580
+sequence S of partial sums converges if and only if
+
+250
+00:19:37,580 --> 00:19:41,640
+it is bounded يعني مدام الـ sequence of partial
+
+251
+00:19:41,640 --> 00:19:44,640
+sums as converges يعني الصممش للـ xn يعني السريز
+
+252
+00:19:44,640 --> 00:19:50,000
+هذه اللي هي is convergent if and only if it is
+
+253
+00:19:50,000 --> 00:19:52,260
+bounded اللي هي ال sequence of partial sums طبعاً
+
+254
+00:19:52,670 --> 00:19:57,030
+عند الـ summation لـ xn مش مساوي limit الـsk و الـ
+
+255
+00:19:57,030 --> 00:20:00,430
+sk increasing limit همين حيكون ال sub .. طبعا
+
+256
+00:20:00,430 --> 00:20:02,790
+increasing و صارك bounded مادام bounded then فيه
+
+257
+00:20:02,790 --> 00:20:06,470
+لها bound إذا صار عند ال supremum ل ال x .. ل ال
+
+258
+00:20:06,470 --> 00:20:09,190
+sk هو عبارة عن limit ال sk حسب ال monotone
+
+259
+00:20:09,190 --> 00:20:12,410
+convergence theorem و سوى ال summation ل xn و هو
+
+260
+00:20:12,410 --> 00:20:18,270
+المطلوب طيب هذا لما يكون ال terms سبعات ال
+
+261
+00:20:18,270 --> 00:20:23,510
+sequence يشملها موجبات أو non negative الآن كوشي
+
+262
+00:20:23,510 --> 00:20:29,610
+criterion for اللي هو series احنا زي ما كنا نقول
+
+263
+00:20:29,610 --> 00:20:33,130
+في التحليل الحقيقي واحد احيانا ما بنقدرش نعرف ايش
+
+264
+00:20:33,130 --> 00:20:37,990
+ال limit لكن بنكون عارفين ان ال sequence نفسها
+
+265
+00:20:37,990 --> 00:20:42,190
+converges فبدنا نعبر عن ال convergence لل sequence
+
+266
+00:20:42,190 --> 00:20:45,770
+بطريقة اللي هي ال terms تبعتها مش عن طريقة اللي هي
+
+267
+00:20:45,770 --> 00:20:50,450
+معرفة ال limit فكنا نقول إن الـ xn مثلاً is a
+
+268
+00:20:50,450 --> 00:20:52,850
+Cauchy sequence if and only if for every ε أكبر من
+
+269
+00:20:52,850 --> 00:20:56,190
+0 there exists k element in N such that for every
+
+270
+00:20:56,190 --> 00:21:00,290
+n و m أكبر من k يكون الـ absolute value للـ xn
+
+271
+00:21:00,290 --> 00:21:03,990
+ناقص xm أصغر من ε نحن نجيب اللي هو Cauchy
+
+272
+00:21:03,990 --> 00:21:07,750
+criterion مشابه إليها بس لمين لحالة السيريز ما هي
+
+273
+00:21:07,750 --> 00:21:11,170
+هذه السيريز زي ما قلنا هي عبارة عن في النهاية
+
+274
+00:21:11,170 --> 00:21:17,950
+sequence of partial sums طيب يا جماعة نيجي لكوشي
+
+275
+00:21:17,950 --> 00:21:21,250
+الـ 915 اللي هي quotient criterion for series the
+
+276
+00:21:21,250 --> 00:21:25,370
+series summation xn in R converges if and only if
+
+277
+00:21:25,370 --> 00:21:29,690
+for every ε أكبر من صفر there is a natural number
+
+278
+00:21:29,690 --> 00:21:35,010
+M such that if M و N أكبر أو يساوي M of ε هيكون
+
+279
+00:21:35,010 --> 00:21:40,020
+الأسئلة ماقصة أسئلة M اللي هو بساوى xn زائد 1 الـ M
+
+280
+00:21:40,020 --> 00:21:45,060
+أشملها أكبر أو يساوي N فبصير الـ Sm عبارة عن x1 زي
+
+281
+00:21:45,060 --> 00:21:49,660
+x2 و بياخد في طريقه الـ xn و بعدين بكمل xn زي x1
+
+282
+00:21:49,660 --> 00:21:54,000
+لما أصل عند xm لما نطرح منه الـ Sm اللي هي x1 عند
+
+283
+00:21:54,000 --> 00:21:58,920
+الـ xn بروح كلهم بضال مين اللي هي xn زي x1 و xn زي
+
+284
+00:21:58,920 --> 00:22:03,040
+x2 لما أصل عند مين عند xm هذا حيكون أصغر من مين من
+
+285
+00:22:03,040 --> 00:22:07,560
+ε هذا اللي بنلثبته إنه الـ series summation xn
+
+286
+00:22:07,560 --> 00:22:12,300
+converges if and only if اللي هي الـ Cauchy
+
+287
+00:22:12,300 --> 00:22:15,760
+criterion يعني مين هو لكل ε أكبر من 0 there exists
+
+288
+00:22:15,760 --> 00:22:19,420
+M such that for every M أكبر من M أكبر أو يساوي
+
+289
+00:22:19,420 --> 00:22:22,900
+الـ M اللي راجناها هيكون Absolute value أسمها M
+
+290
+00:22:22,900 --> 00:22:26,980
+ناقص Sn اللي هو بساوية هذا المقدار أصغر من ايش من
+
+291
+00:22:26,980 --> 00:22:31,670
+ε الآن في الواقع احنا قلنا في نظرية سابقة .. نظرية
+
+292
+00:22:31,670 --> 00:22:36,090
+في السابقة في التحليل الحقيقي واحد ان الـ sequence
+
+293
+00:22:36,090 --> 00:22:40,610
+converges if and only if it is Cauchy وهذه اللي
+
+294
+00:22:40,610 --> 00:22:46,950
+هنستخدمها، شوفوا  صلى الله عليه وسلم الصممش للـ xn
+
+295
+00:22:46,950 --> 00:22:51,090
+converges in R if and only if the sequence of
+
+296
+00:22:51,090 --> 00:22:55,990
+partial sums الـ Sn converges in R هذا التعريف الـ
+
+297
+00:22:55,990 --> 00:23:00,150
+sequence Sn converges in R if and only if اللي قلت
+
+298
+00:23:00,150 --> 00:23:04,250
+عنها نظرية 3.5.4 اللي هي the convergent if and only
+
+299
+00:23:04,250 --> 00:23:08,490
+if Sn is a Cauchy sequence ماشي ايش تعريفه Sm
+
+300
+00:23:08,490 --> 00:23:11,850
+Cauchy sequence لكل ε أكبر من صفر there exist M
+
+301
+00:23:11,850 --> 00:23:16,090
+such that if M أكبر من N أكبر أو يساوي الـ M اللي ده
+
+302
+00:23:16,090 --> 00:23:19,850
+جيتها اللي هو هيكون عبارة عن Sm ناقص Sn أصغر
+
+303
+00:23:19,850 --> 00:23:24,330
+من ε هاي اللي عند الهان اللي هي نظرية أصغر من
+
+304
+00:23:24,330 --> 00:23:27,830
+ε اللي هي الـ Cauchy بساوي مين أو بكافي ال
+
+305
+00:23:27,830 --> 00:23:31,330
+convergence هذه حساباتها زي ما قلنا الـ Sm عبارة
+
+306
+00:23:31,330 --> 00:23:39,840
+عن مين x1 زائد لما أصل عند xn زائد xn زائد 1
+
+307
+00:23:39,840 --> 00:23:44,040
+لما أصل عند آخر واحد الـ xm لأن الـ M أكبر من N
+
+308
+00:23:44,040 --> 00:23:47,780
+فهلاجي في طريق اللي هو مين الـ x الـ x1 عند الـ
+
+309
+00:23:47,780 --> 00:23:55,300
+xn ناقص الـ Sn ناقص الـ x1 ناقص الـ x2 ناقص الـ xn هذا
+
+310
+00:23:55,300 --> 00:24:00,750
+الـ absolute value اللي هو هذولة بروح لهنامع هذولة
+
+311
+00:24:00,750 --> 00:24:06,590
+بضل عندي xn زائد واحد زائد xn زائد اتنين زائد xm
+
+312
+00:24:06,590 --> 00:24:10,070
+اللي هي هذه أصغر من مين من ε إذا هذا صار أصغر
+
+313
+00:24:10,070 --> 00:24:13,150
+من ε وهو المطلوب طيب
+
+314
+00:24:17,320 --> 00:24:22,820
+طيب، الآن نعرف حاجة اسمها absolutely convergent
+
+315
+00:24:22,820 --> 00:24:27,380
+absolutely convergent برضه اللي مرت علينا في تفاضل
+
+316
+00:24:27,380 --> 00:24:31,620
+ت بها خلينا نذكركم فيها اللي هو let x بالـ
+
+317
+00:24:31,620 --> 00:24:36,760
+definition 916 let x بساوى xn be a sequence in R we
+
+318
+00:24:36,760 --> 00:24:41,020
+say that the series summation xn is absolutely
+
+319
+00:24:41,020 --> 00:24:42,180
+convergent
+
+320
+00:24:44,650 --> 00:24:52,470
+إذا كانت سلسلة أبسط قيم xn مرتبطة بالـ R
+
+321
+00:24:57,860 --> 00:25:01,700
+if it is convergent but not إيش ما لها absolutely
+
+322
+00:25:01,700 --> 00:25:05,620
+convergent يعني في series في الدنيا أكيد بتكون
+
+323
+00:25:05,620 --> 00:25:11,100
+convergent لكن مش absolutely convergent هذه ال
+
+324
+00:25:11,100 --> 00:25:13,620
+series اللي من هالنوع اللي بتكون convergent لكن مش
+
+325
+00:25:13,620 --> 00:25:16,220
+absolutely convergent بنسميه إيش ما لها
+
+326
+00:25:16,220 --> 00:25:18,820
+conditionally convergent
+
+327
+00:25:20,920 --> 00:25:24,240
+لكن بعد شوية هنقول ان any absolutely convergent
+
+328
+00:25:24,240 --> 00:25:33,240
+هتكون إيش ما لها is convergent طيب الآن نيجي طبعا
+
+329
+00:25:33,240 --> 00:25:37,040
+هذا في R كل الشغل في ال real numbers 9.17 theorem
+
+330
+00:25:37,040 --> 00:25:42,650
+إذا كانت سلسلة مرتبطة بالكامل فهي 
+
+331
+00:25:42,650 --> 00:25:45,530
+مرتبطة بالكامل أيضًا، فهي مرتبطة بالكامل أيضًا،
+
+332
+00:25:45,530 --> 00:25:54,310
+فهي مرتبطة بالكامل أيضًا، فهي مرتبطة
+
+333
+00:25:54,310 --> 00:25:56,430
+بالكامل أيضًا
+
+334
+00:25:57,710 --> 00:26:00,150
+ماذا يعني absolutely يكون convergent؟ يعني معناته
+
+335
+00:26:00,150 --> 00:26:03,270
+الصممش اللي هي absolute value إيش ماله is
+
+336
+00:26:03,270 --> 00:26:07,830
+convergent ماشي then by Cauchy criterion بنطبقها
+
+337
+00:26:07,830 --> 00:26:11,590
+على هذه ال series اللي هو لكل ε أكبر من صفر
+
+338
+00:26:11,590 --> 00:26:14,970
+there exists m of ε element in N such that if
+
+339
+00:26:14,970 --> 00:26:18,190
+m أكبر من n أكبر يساوي m of ε اللي حكيناها قبل
+
+340
+00:26:18,190 --> 00:26:23,100
+شوية then اللي هو اللي عندي الأولى اللي هي n زائد
+
+341
+00:26:23,100 --> 00:26:26,560
+واحد n زائد اتنين لما أصل عند الـ M بس كيف شكل الـ
+
+342
+00:26:26,560 --> 00:26:28,920
+term هنا اللي هو الـ absolute value إذا absolute
+
+343
+00:26:28,920 --> 00:26:33,180
+value لـ Xn زائد واحد زائد .. مين ماخد الـ M أكبر إن
+
+344
+00:26:33,180 --> 00:26:36,720
+اه زائد absolute value Xn زائد اتنين لما أصل عند
+
+345
+00:26:36,720 --> 00:26:40,380
+زائد absolute value ل الـ مين لـ Xm أصغر من ايه يا
+
+346
+00:26:40,380 --> 00:26:47,510
+عياش؟ من ε هذه اللي هي عبارة عن الاسم لهذه ناقص الـ
+
+347
+00:26:47,510 --> 00:26:51,610
+Sm لها الـ Sm فرضا نسميها prime الـ Sm اللي هي
+
+348
+00:26:51,610 --> 00:26:54,610
+عبارة عن absolute value X1 زائد absolute value X2
+
+349
+00:26:54,610 --> 00:26:58,290
+لما أصل عند الـ absolute value Xm وهذه تزيلها منها
+
+350
+00:26:58,290 --> 00:27:02,250
+بيظل الـ absolute value عند Xm n زائد 1 لعند الـ
+
+351
+00:27:02,250 --> 00:27:05,990
+absolute value لـ Xm اللي هي هتكون أصغر من ε حسب الـ
+
+352
+00:27:05,990 --> 00:27:09,330
+Cauchy criterion ها وهذه سهلة بالـ Cauchy criterion
+
+353
+00:27:09,330 --> 00:27:10,010
+مباشرة
+
+354
+00:27:18,700 --> 00:27:23,480
+لو سمنا الـ sequence of partial sums للـ Xn اللي
+
+355
+00:27:23,480 --> 00:27:28,090
+عبارة عن Ssn وهذه سمنها S'm للـ absolute values و
+
+356
+00:27:28,090 --> 00:27:32,430
+Sm prime طيب شوفي لان احسبلي لهذه عشان اثبت هذه ال
+
+357
+00:27:32,430 --> 00:27:37,590
+convergence الآن احسبلي Sm ماقص Sm لمين لم M
+
+358
+00:27:37,590 --> 00:27:42,190
+الأكبر من انه أكبر يساوي مين M of ε بساوي اللي
+
+359
+00:27:42,190 --> 00:27:45,350
+هذولة حسبناها قبل و شوية طلعت xn زائد واحد xn زائد
+
+360
+00:27:45,350 --> 00:27:49,000
+اتنين عند x mean xm هذا الـ Absolute Value بالـ
+
+361
+00:27:49,000 --> 00:27:51,340
+Triangle inequality أصغر من الأولى زاد التانية لما
+
+362
+00:27:51,340 --> 00:27:55,780
+أصل للأخيرة هذا من فوق أصغر من مين؟ من ε إذا
+
+363
+00:27:55,780 --> 00:27:59,220
+لجينا لكل ε أكبر من صفر there exists M such
+
+364
+00:27:59,220 --> 00:28:01,480
+that F M أكبر من M أكبر من M أكبر من M أكبر من M
+
+365
+00:28:01,480 --> 00:28:05,740
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+366
+00:28:05,740 --> 00:28:08,560
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+367
+00:28:08,560 --> 00:28:14,340
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+368
+00:28:17,760 --> 00:28:21,960
+is اللي هي عبارة عن convergence اللي هي طبعا ال
+
+369
+00:28:21,960 --> 00:28:24,960
+sequence of partial sums تبع ال xn then by Cauchy
+
+370
+00:28:24,960 --> 00:28:30,140
+criterion ال summation لل xn is convergent طيب
+
+371
+00:28:30,140 --> 00:28:33,800
+نيجي
+
+372
+00:28:33,800 --> 00:28:42,360
+الآن ناخد example 918 نشوف هذا ال example نشوف كيف
+
+373
+00:28:42,360 --> 00:28:47,970
+بدنا اللي هو انحل أو اللي هو نشرح هذا الـ Example
+
+374
+00:28:47,970 --> 00:28:49,410
+918
+
+375
+00:28:51,320 --> 00:28:55,460
+لت x بساوى an then the geometric series ال
+
+376
+00:28:55,460 --> 00:28:59,820
+summation ل an من واحد اللي مالنهاية هي converges if
+
+377
+00:28:59,820 --> 00:29:04,200
+ال absolute value لل a إشمالها أصغر من واحد إذا ال
+
+378
+00:29:04,200 --> 00:29:07,080
+geometric series هذا هتكون converges معلومة طبعا
+
+379
+00:29:07,080 --> 00:29:10,300
+عارفينها إذا كان ال absolute value لل a أصغر من
+
+380
+00:29
+
+401
+00:30:49,260 --> 00:30:53,640
+زائد a<sub>n</sub> زائد اثنين لما أوصل عند الـ a<sub>m</sub> هذا طبعاً
+
+402
+00:30:53,640 --> 00:30:59,550
+أنا بأخذه في حالة الـ m أكبر من من n وضربت في 1
+
+403
+00:30:59,550 --> 00:31:07,930
+-a لما أضرب وأجمع الـ terms لبعض كله هي cancel كله
+
+404
+00:31:07,930 --> 00:31:12,590
+هيظل بس أول term اللي هو a<sub>m</sub> زائد واحد وآخر term
+
+405
+00:31:12,590 --> 00:31:17,510
+اللي هو a<sub>m</sub> زائد واحد فالآن هذا ضرب هذا هيطلع a<sub>m</sub>
+
+406
+00:31:17,510 --> 00:31:22,180
+زائد واحد ناقص a أو m زائد واحد هذا بيساوي المقدار
+
+407
+00:31:22,180 --> 00:31:27,500
+هذا لكل الـ m's اللي أكبر من N لأن بدي
+
+408
+00:31:27,500 --> 00:31:29,860
+أستخدم شكل بدي أستخدم الـ Cauchy criterion في
+
+409
+00:31:29,860 --> 00:31:34,960
+إثبات الـ convergence احسب لـ S<sub>m</sub> نقص S<sub>n</sub> صرنا
+
+410
+00:31:34,960 --> 00:31:38,080
+حاسبينها وعارفين كيف بتنحسب لـ m الأكبر من N
+
+411
+00:31:38,080 --> 00:31:41,700
+هيطلع عند a<sub>n</sub> زائد واحد لعند a<sub>m</sub> absolute value
+
+412
+00:31:41,700 --> 00:31:48,100
+هذا بيساوي اللي هو حسب اللي هنا عبارة عن الـ
+
+413
+00:31:48,100 --> 00:31:55,100
+absolute value لهذي هي ساوي هذا absolute value على
+
+414
+00:31:55,100 --> 00:32:00,200
+المقدار هذا 1 ناقص a absolute value إذا استبدلت
+
+415
+00:32:00,200 --> 00:32:06,100
+هذا اللي هو هذا على هذا من هذه اللي هي اللي فوق مش
+
+416
+00:32:06,100 --> 00:32:09,920
+الحال إذا صار عند الآن استخدمت اللي فوق هذه أخذت
+
+417
+00:32:09,920 --> 00:32:13,040
+الـ absolute value للجهتين وأخدت اللي هو قيمة هذا
+
+418
+00:32:13,040 --> 00:32:16,400
+صار عبارة عن absolute value للـ a<sub>n</sub> على الـ
+
+419
+00:32:16,400 --> 00:32:21,230
+absolute value اللي هو 1 minus a طيب هذا أصغر أو
+
+420
+00:32:21,230 --> 00:32:24,310
+يساوي بالـ triangle inequality اللي هو الـ absolute
+
+421
+00:32:24,310 --> 00:32:27,810
+value للي فوق زائد الـ absolute value للي جنبه على
+
+422
+00:32:27,810 --> 00:32:33,050
+المقدار هذا نفس اللي هي نفس القيمة ما غيرتهاش لأن
+
+423
+00:32:33,050 --> 00:32:37,610
+الـ absolute value لأصغر من 1 implies that أن الـ
+
+424
+00:32:37,610 --> 00:32:42,090
+limit اللي فوق هذا بيروح لصفر يعني الآن as n goes
+
+425
+00:32:42,090 --> 00:32:46,730
+to infinity as n goes to infinity طبعاً أكيد الـ m
+
+426
+00:32:46,730 --> 00:32:50,330
+هتروح برضه لوين؟ لـ ∞ لأن هي اللي أكبر منها يعني الـ n
+
+427
+00:32:50,330 --> 00:32:53,970
+as n goes to infinity هذا المقدار هيصغر كتير كتير
+
+428
+00:32:53,970 --> 00:32:57,490
+كتير وهذا هيصغر كتير كتير لدرجة أن هذا كله على
+
+429
+00:32:57,490 --> 00:33:03,030
+بعضه as n goes to infinity هيكون اللي هو أصغر من
+
+430
+00:33:03,030 --> 00:33:07,130
+أي رقم في الدنيا هيصير أصغر من ε معناته صار
+
+431
+00:33:07,130 --> 00:33:12,390
+أن دي اللي هو الـ S<sub>m</sub> نقص S<sub>n</sub> أصغر من ε هيكون
+
+432
+00:33:12,390 --> 00:33:16,410
+اللي هو في هذه الحالة أثبتت أنه اللي هو الـ S<sub>n</sub> by
+
+433
+00:33:16,410 --> 00:33:20,970
+Cauchy criterion is convergent اللي هو صار عندي
+
+434
+00:33:20,970 --> 00:33:26,170
+الـ sum إلى الآن convergence طيب المثال اللي هو
+
+435
+00:33:26,170 --> 00:33:30,200
+المثال اللي بعده المثال المشهور اللي هو بـ little
+
+436
+00:33:30,200 --> 00:33:38,500
+bit show that summation للواحد and diverse هذه الـ
+
+437
+00:33:38,500 --> 00:33:42,800
+sequence، هذه الـ series اللي هي series مشهورة
+
+438
+00:33:42,800 --> 00:33:46,380
+اللي كنا نسميها الـ p series الـ p بيساوي 1 في
+
+439
+00:33:46,380 --> 00:33:49,440
+هذه الحالة هذه الـ summation يليها 1 and إيش
+
+440
+00:33:49,440 --> 00:33:54,400
+ماله؟ diverges نشوف to show that summation وحدة لـ N
+
+441
+00:33:54,400 --> 00:33:58,480
+divergent, we construct unbounded subsequence S<sub>k<sub>n</sub></sub>
+
+442
+00:33:58,480 --> 00:34:04,860
+of S<sub>k</sub> قلنا قبل هيك أن إذا كانت الـ sequence
+
+443
+00:34:04,860 --> 00:34:09,800
+converges أكيد هتكون bounded إذا لو كانت الـ
+
+444
+00:34:09,800 --> 00:34:16,060
+sequence unbounded أكيد diverges ماشي الحال طيب I
+
+445
+00:34:16,060 --> 00:34:19,250
+was not convergent بناءً عليه لو احنا جبنا
+
+446
+00:34:19,250 --> 00:34:24,830
+subsequence من الـ sequence تكون unbounded فمن باب
+
+447
+00:34:24,830 --> 00:34:28,530
+الأولى هتصير الـ sequence الأصلية نفسها unbounded
+
+448
+00:34:28,530 --> 00:34:31,450
+مدام أن صارت الـ sequence الأصلية unbounded إذا الـ
+
+449
+00:34:31,450 --> 00:34:36,030
+sequence صارت diverges بناءً عليه عشان أثبت اللي هو
+
+450
+00:34:36,030 --> 00:34:41,030
+الـ sequence of partial sums هذه الـ sequence of
+
+451
+00:34:41,030 --> 00:34:44,430
+partial sums لها اللي هي الـ S<sub>k</sub> مثلاً أو الـ sum
+
+452
+00:34:44,430 --> 00:34:48,530
+عشان اثبتها أنّه diverges الـ sequence of partial
+
+453
+00:34:48,530 --> 00:34:52,370
+sums يعني بمعنى آخر عشان أسمّيها diverges الـ
+
+454
+00:34:52,370 --> 00:34:58,120
+series بدي أجيب unbounded subsequence بدنا نعمل
+
+455
+00:34:58,120 --> 00:35:00,940
+construction unbounded وده اللي هو الـ construction
+
+456
+00:35:00,940 --> 00:35:05,200
+المشهور إذا الآن بدأ أعمل we construct unbounded
+
+457
+00:35:05,200 --> 00:35:08,520
+subsequence as k<sub>n+1</sub> of the sequence of
+
+458
+00:35:08,520 --> 00:35:14,660
+partial sums S<sub>k</sub> of this series طيب شوف الآن كيف
+
+459
+00:35:14,660 --> 00:35:20,780
+بدنا نشتغل اللي هو اللي هي الـ subsequence هذه خدلي
+
+460
+00:35:20,780 --> 00:35:27,830
+الآن بدأ أعمل subsequence as k<sub>n</sub> of mean of this
+
+461
+00:35:27,830 --> 00:35:34,210
+sequence خدلي أول شيء k<sub>1</sub> بيساوي اللي هو عبارة
+
+462
+00:35:34,210 --> 00:35:39,750
+عن 2<sup>1</sup> فبيصير S<sub>k<sub>1</sub></sub> بيساوي اللي هو
+
+463
+00:35:39,750 --> 00:35:45,480
+عبارة عن اللي هي 2 أقصد 1 يعني 2 الـ
+
+464
+00:35:45,480 --> 00:35:50,820
+sequence of partial sums الـ summation 1 لأن n
+
+465
+00:35:50,820 --> 00:35:55,500
+من 1 لعند k ماشي لأن عندي إيش ما أخذتها 2k
+
+466
+00:35:55,500 --> 00:36:01,840
+لأن بيصير عندي اللي هي عبارة عن 1 زائد ½ مظبوط
+
+467
+00:36:01,840 --> 00:36:07,720
+1 زائد ½ أكيد أكبر من من اللي هي أكبر من إيش
+
+468
+00:36:07,720 --> 00:36:14,780
+من الـ 1 طيب خد الآن k<sub>2</sub> بيساوي 2 أقصد 2
+
+469
+00:36:14,780 --> 00:36:22,300
+فبيصير S<sub>k<sub>2</sub></sub> بيساوي اللي هو عبارة عن 1 زائد
+
+470
+00:36:22,300 --> 00:36:29,480
+½ زائد ⅓ زائد ¼ مظبوط اللي هي هذه عبارة عن
+
+471
+00:36:29,480 --> 00:36:38,020
+تساوي S<sub>k<sub>1</sub></sub> أي S<sub>k<sub>1</sub></sub> زائد اللي هو من
+
+472
+00:36:40,830 --> 00:36:47,350
+⅓ زائد إيش زائد ¼ أكيد هتكون هذه خلّيني أكتب
+
+473
+00:36:47,350 --> 00:36:53,410
+4 أكبر وأقول هذه زائد ¼ زائد إيش ¼ لأن
+
+474
+00:36:53,410 --> 00:36:57,430
+الـ ⅓ أكبر من ¼ حطيت بدل هذا فبيصير عند هذا
+
+475
+00:36:57,430 --> 00:37:02,890
+المقدار S<sub>k<sub>2</sub></sub> أكبر من S<sub>k<sub>1</sub></sub> زائد ¼ زائد ¼ يعني
+
+476
+00:37:02,890 --> 00:37:10,820
+بمعنى آخر سيكون أكبر من S<sub>k<sub>1</sub></sub> زائد 1 اللي هو 2
+
+477
+00:37:10,820 --> 00:37:16,820
+في من في الـ ¼ اللي هو بيساوي S<sub>k<sub>1</sub></sub> زائد 2 في
+
+478
+00:37:16,820 --> 00:37:23,440
+1 على 2<sup>2</sup> بإيش اللي هو هذا أكيد أكبر من
+
+479
+00:37:23,440 --> 00:37:27,740
+1 زائد لأن S<sub>k<sub>1</sub></sub> أكبر من من 1 أكبر من
+
+480
+00:37:27,740 --> 00:37:33,900
+1 زائد 2 × 1 على 2<sup>2</sup>
+
+481
+00:37:44,670 --> 00:37:51,130
+خلّينا نحسب S<sub>k<sub>3</sub></sub> ماذا يساوي؟ إذًا k<sub>3</sub> خد
+
+482
+00:37:51,130 --> 00:37:54,070
+k<sub>3</sub> طبعاً واضحة إيش k<sub>3</sub> بدأ أخذها 2 أقصى
+
+483
+00:37:54,070 --> 00:37:57,770
+2 هاخد 2 أقصى 3 اللي هي 8 اللي
+
+484
+00:37:57,770 --> 00:38:04,870
+عبارة عن 1 زائد ½ زائد ⅓ زائد ¼ زائد ⅚
+
+485
+00:38:04,870 --> 00:38:14,050
+زائد ⅙ زائد ⅐ زائد ⅛ ماشي الحال طيب شوف عندي
+
+486
+00:38:14,050 --> 00:38:24,890
+الآن من 1 لعند اللي هو جداش 4 هذا اللي هي S<sub>k<sub>2</sub></sub>
+
+487
+00:38:27,420 --> 00:38:31,960
+الباقية الـ S<sub>k<sub>3</sub></sub> إذاً هذا بيصير عبارة عن S<sub>k</sub>
+
+488
+00:38:31,960 --> 00:38:38,340
+<sub>2</sub> اللي هو يعني أكبر من S<sub>k<sub>2</sub></sub> زائد آخر واحد زي
+
+489
+00:38:38,340 --> 00:38:44,320
+ما أخدت هنا آخر واحد هنا اللي هو ⅛ زائد ⅛ زائد
+
+490
+00:38:44,320 --> 00:38:50,100
+⅛ زائد ⅛ زائد ⅛ 4، 1، 2، 1،
+
+491
+00:38:50,100 --> 00:38:54,880
+2، 3، 4، عيني 4 ماشي الحال هدول الآن
+
+492
+00:38:56,220 --> 00:38:58,780
+من هنا إلى هنا أصغر من هنا إلى هنا عشان هيك بيصير
+
+493
+00:38:58,780 --> 00:39:02,120
+هذا المقدار أكبر من S<sub>k<sub>2</sub></sub> زائد هدوله هدوله إيش
+
+494
+00:39:02,120 --> 00:39:08,400
+بيساووا؟ هو يساوي S<sub>k<sub>2</sub></sub> زائد اللي هو كمّتهم 4
+
+495
+00:39:08,400 --> 00:39:14,940
+أثمان يعني 4 × ⅛ 4
+
+496
+00:39:14,940 --> 00:39:19,880
+× ⅛ هو يساوي الآن 4 × ⅛ يعني عبارة عن
+
+497
+00:39:19,880 --> 00:39:28,720
+½ مظبوط يعني بيساوي اللي هو S<sub>k<sub>2</sub></sub> زائد ½ نشوف من
+
+498
+00:39:28,720 --> 00:39:33,360
+اللي فوق هنا S<sub>k<sub>2</sub></sub> أكبر من 1 زائد 2 على 2
+
+499
+00:39:33,360 --> 00:39:38,800
+إذا حيصير هذا أكبر من 1 زائد 2 على 2
+
+500
+00:39:38,800 --> 00:39:43,500
+<sup>2</sup> 2 على 2<sup>2</sup> هذه 1 ممكن نخليها
+
+501
+00:39:43,500 --> 00:39:50,800
+زي ما هي 1 2 3 4 مش مشكلة ½ اللي
+
+502
+00:39:50,800 --> 00:39:57,500
+هو إيش بيساوي؟ هذا زائد اللي هو ½ هذا S<sub>k<sub>2</sub></sub>
+
+503
+00:39:57,500 --> 00:40:02,720
+أكبر من هذا وهي ½ بيصير زائد ½ وهي ساوي 1
+
+504
+00:40:02,720 --> 00:40:06,560
+زائد 3
+
+505
+00:40:06,560 --> 00:40:13,820
+على من؟ كمّ ½ أنا بس هاد الخلط هنا 2 × ½
+
+506
+00:40:13,820 --> 00:40:21,540
+<sup>2</sup> يعني 2 على 2<sup>2</sup> اللي
+
+507
+00:40:21,540 --> 00:40:22,580
+هي بتساوي
+
+508
+00:40:26,450 --> 00:40:32,410
+و S<sub>k<sub>3</sub></sub> بيساوي 1 زائد 2 على 2<sup>2</sup> زائد ½
+
+509
+00:40:32,410 --> 00:40:38,030
+هذه
+
+510
+00:40:38,030 --> 00:40:46,650
+خلّينا نقول 1 زائد ½ أكبر من بس هيأدي احنا بس
+
+511
+00:40:46,650 --> 00:40:55,560
+اللي هي S<sub>k<sub>1</sub></sub> S<sub>k<sub>1</sub></sub> اللي عبارة عن 1 زائد ½ أكبر من
+
+512
+00:40:55,560 --> 00:41:01,700
+S<sub>k<sub>1</sub></sub> هذه 1 زائد ½ قيمة الـ S خلّيها S<sub>k<sub>1</sub></sub>
+
+513
+00:41:01,700 --> 00:41:04,860
+مش مشكلة S<sub>k<sub>1</sub></sub> لما أنا عوضتها أنا عوضت بـ 1 ده
+
+514
+00:41:04,860 --> 00:41:10,120
+هي 1 زائد ½ هي 1 زائد ½ زائد هذه ماشي
+
+515
+00:41:10,120 --> 00:41:14,220
+الحال ويساوي صار عندها ½ وهنا كمان 2 مع
+
+516
+00:41:14,220 --> 00:41:18,910
+2 بيطلع كمان ½ بيصير 2 على 2 صح بيصير 2
+
+517
+00:41:18,910 --> 00:41:23,330
+على 2 وهذا ½ وكمان ½ هنا الآن لما نعوّض عليها دي
+
+518
+00:41:23,330 --> 00:41:28,830
+برضه بنفس الطريقة اللي صار عندي S<sub>k<sub>2</sub></sub> اللي هو عبارة
+
+519
+00:41:28,830 --> 00:41:36,930
+عن 1 زائد 2 على 2 1 نعود تحت S<sub>k<sub>2</sub></sub> بيصير
+
+520
+00:41:36,930 --> 00:41:42,010
+1 زائد 2 على 2 بيصير عند 1 زائد 2
+
+521
+00:41:42,010 --> 00:41:44,890
+على 2 يعني 1 زائد 3 على من؟ على 2
+
+522
+00:41:44,890 --> 00:41:53,410
+اللي وصلتله يعني الآن الـ S<sub>k<sub>1</sub></sub> أكبر من 1 زائد
+
+523
+00:41:53,410 --> 00:42:00,370
+2 صفر على أو خلّيني أقول بيساوي S<sub>k</sub> بيساوي
+
+524
+00:42:00,370 --> 00:42:08,680
+1 زائد 1 على 2 صفر S<sub>k<sub>3</sub></sub> S<sub>k<sub>2</sub></sub> عبارة عن أكبر
+
+525
+00:42:08,680 --> 00:42:13,440
+من وصلنا آخر شيء أكبر من 1 زائد 2 على 2
+
+526
+00:42:13,440 --> 00:42:21,400
+S<sub>k<sub>3</sub></sub> أكبر يعني هنا 2 هاي 2 S<sub>k<sub>3</sub></sub> اللي هو
+
+527
+00:42:21,400 --> 00:42:28,420
+عبارة عن أكبر من 1 1 زائد 3 على 2
+
+528
+00:42:28,420 --> 00:42:38,580
+لاحظ هنا 3 هنا على 2 هنا في حالة هذه اللي
+
+529
+00:42:38,580 --> 00:42:43,920
+هي 2<sup>2</sup> يعني 2 على 2 وفي حالة
+
+530
+00:42:43,920 --> 00:42:49,980
+الأخيرة أكبر من 1 اللي هو أكبر من 1 زائد ½
+
+531
+00:42:49,980 --> 00:42:54,260
+اللي هو عبارة عن اللي هو 2 أو صفر أو بيساوي
+
+532
+00:42:54,260 --> 00:42:58,940
+أنا 2 أو صفر اللي هو على 2 يعني الآن و
+
+533
+00:42:58,940 --> 00:43:07,250
+كأنه in general in general هنا 3 وهنا k<sub>2</sub> وهنا
+
+534
+00:43:07,250 --> 00:43:13,230
+k<sub>1</sub> in general بيصير عندي S<sub>k<sub>3</sub></sub> أكبر من 1 زائد 3 على
+
+535
+00:43:13,230 --> 00:43:20,550
+2 يعني صار عندي الآن بصورة عامة الـ S<sub>k<sub>n</sub></sub> اللي هو
+
+536
+00:43:20,550 --> 00:43:25,590
+أكبر من 1 زائد n على 2 ومرّينا في الخطوات في
+
+537
+00:43:25,590 --> 00:43:32,870
+أثناءها اللي هي على ما يلي مرّينا على أنّه أكبر من
+
+538
+00:43:32,870 --> 00:43:42,290
+S<sub>k<sub>n-1</sub></sub> زائد 2<sup>n-1</sup> في 1 على 2<sup>n</sup> اللي سميناها S<sub>k<sub>n-1</sub></sub>
+
+539
+00:43:42,290 --> 00:43:52,460
+زائد إيش ½ الآن هذي بدي أثبتها by induction أنا
+
+540
+00:43:52,460 --> 00:43:56,520
+لأن عملتلها لثلاث لكن هذا طبعاً لثلاث بس
+
+541
+00:43:56,520 --> 00:44:02,280
+استقرار لأن in general بدأ أثبتها كما يلي عندي هي
+
+542
+00:44:02,280 --> 00:44:05,860
+S<sub>k<sub>2</sub></sub>
+
+543
+00:44:05,860 --> 00:44:12,700
+اللي هي n - 1 زائد اللي هو 2 أُس n -
+
+544
+00:44:12,700 --> 00:44:16,060
+1 لأن 2 أُس 2 هذه اللي هي n إيش بيساوي
+
+545
+00:44:16,060 --> 00:44:18,800
+3 بيصير 2 أُس 3 ناقص 1 على 2 أُس
+
+546
+00:44:18,800 --> 00:44:22,880
+2 في 1 على 2<sup>3</sup> اللي هي 1 على
+
+547
+00:44:22,880 --> 00:44:27,600
+2<sup>3</sup> اللي هي 1 على 2 أُس n إذا هذه هي
+
+548
+00:44:27,600 --> 00:44:33,920
+هيها هذه اللي هي بتساوي اللي هي S<sub>k</sub> اللي هي n -
+
+549
+00:44:33,920 --> 00:44:39,600
+1 يعني 2 زائد إيش زائد ½ أكبر من من
+
+550
+00:44:39,600 --> 00:44:42,960
+1 زائد 3 على 2 اللي هي 1 زائد n على
+
+551
+00:44:42,960 --> 00:44:47,480
+2 حيث الـ n هنا إيش بتساوي؟ بتساوي 3 إذا هذا
+
+552
+00:44:47,480 --> 00:44:53,130
+الاستقراء الآن بيطلع لي هذه بس هذا مش إثبات بدنا
+
+553
+00:44:53,130 --> 00:44:57,970
+نثبتها for n بتساوي 1 على طول for n بتساوي 1
+
+554
+00:44:57,970 --> 00:45:05,140
+S<sub>k<sub>1</sub></sub> S<sub>k<sub>1</sub></sub> بيساوي اللي هو n بتساوي اللي هي
+
+555
+00:45:05,140 --> 00:45:10,960
+1 S<sub>k<sub>1</sub></sub> بيساوي اللي هو 1 زائد ½ 1 زائد
+
+556
+00:45:10,960 --> 00:45:14,180
+½ طب علشان ما فيش إجابة لك خلّيني أخد أن الـ n إيش
+
+557
+00:45:14,180 --> 00:45:18,120
+بتساوي الـ n بيساوي 2 لأن الـ n = 0 مش معرفة
+
+558
+00:45:18,120 --> 00:45:21,220
+فاهمين عليها إذا أن الـ n بيساوي 2 n بيساوي
+
+559
+00:45:21,220 --> 00:45:25,720
+2 S<sub>k<sub>2</sub></sub> هي S<sub>k<sub>2</sub></sub> عشان ما بديش أعيدها S<sub>k</sub>
+
+560
+00:45:25,720 --> 00:45:34,820
+<sub>2</sub> أكبر من اللي هو اللي هي S<sub>k<sub>1</sub></sub> هيها S<sub>k<sub>2</sub></sub> أكبر من
+
+561
+00:45:34,820 --> 00:45:42,460
+S<sub>k<sub>1</sub></sub> زائد 2<sup>2 - 1</sup> يعني أُس 1 في من في
+
+562
+00:45:42,460 --> 00:45:45,980
+¼ اللي هو 1 على 2<sup>2</sup> إذن هذا صح لأن
+
+563
+00:45:45,980 --> 00:45:52,740
+اللي بعدها اللي هي S<sub>k<sub>1</sub></sub> زائد اللي هي ½ اللي هو هذا
+
+564
+00:45:52,740 --> 00:45:56,200
+ما حد بروح بص له ½ اللي هي أكبر من 1 زائد
+
+565
+00:45:56,200 --> 00:45:59,500
+2 على 2 اللي هي 1 زائد n على 2 اللي
+
+601
+00:49:38,220 --> 00:49:46,730
+تحتاج هذه ما احنا مفترضينها الـSKR اللي هي أكبر
+
+602
+00:49:46,730 --> 00:49:54,160
+من واحد زائد اللي هي R على اثنينهم مفترضين أن هذه 
+
+603
+00:49:54,160 --> 00:49:58,880
+صحيحة إذا الـSKR أكبر من واحد زائد R على اثنين إذا
+
+604
+00:49:58,880 --> 00:50:03,240
+by induction ال hypothesis بيصير هذا أكبر من هذه
+
+605
+00:50:03,240 --> 00:50:07,760
+اللي هو واحد زائد R على اثنين زائد النص الأصلي
+
+606
+00:50:07,760 --> 00:50:12,460
+ويساوي واحد زائد R زائد واحد على من؟ على اثنين
+
+607
+00:50:12,460 --> 00:50:18,330
+إذا صارت عند الـSKR زائد واحد أكبر من 1 زائد R زائد
+
+608
+00:50:18,330 --> 00:50:23,110
+1 على 2 يعني صارت هذه الجملة صحيحة for R زائد 1
+
+609
+00:50:23,110 --> 00:50:27,270
+إذا صحيحة دائماً إذا هيك بيكون أثبتنا by 
+
+610
+00:50:27,270 --> 00:50:32,770
+mathematical induction كل اللي بدنا إياه بخصوص هذا ال
+
+611
+00:50:32,770 --> 00:50:38,060
+subsequence SKR هذه الـ subsequence وضحنا أنها أكبر
+
+612
+00:50:38,060 --> 00:50:41,720
+أو يساوي 1 على اثنين أو اثنين يعني as R goes to
+
+613
+00:50:41,720 --> 00:50:44,540
+infinity هذا المقدار بيروح لما للنهاية يعني هذه ال
+
+614
+00:50:44,540 --> 00:50:51,240
+sequence is unbounded إذا صار عندي اللي هو SKR is
+
+615
+00:50:51,240 --> 00:50:57,440
+unbounded subsequence of the sequence of partial
+
+616
+00:50:57,440 --> 00:51:04,240
+sums SN of summation 1 على N then it must diverge
+
+617
+00:51:04,240 --> 00:51:08,960
+وهو المطلوب بكون هي أثبتنا أن summation 1 على N is
+
+618
+00:51:08,960 --> 00:51:12,820
+a divergent series
+
+619
+00:51:17,210 --> 00:51:22,830
+بنكمل قصة الـ B-Series الـ B-Series المشهورة عندي 
+
+620
+00:51:22,830 --> 00:51:28,750
+summation للواحد N أس بي N من واحد إلى ما لا نهاية
+
+621
+00:51:28,750 --> 00:51:33,050
+هذه الـ Series بدنا نفحص اللي هو متى converge ومتى
+
+622
+00:51:33,050 --> 00:51:37,310
+diverge يعني converge for what بي and diverge
+
+623
+00:51:37,310 --> 00:51:41,850
+for what بي لما جينا عند الـ B بساوي واحد it converge
+
+624
+00:51:41,850 --> 00:51:49,090
+بدنا نشوف الآن for مين؟ for مين؟ for b بين الصفر وبين
+
+625
+00:51:49,090 --> 00:51:52,730
+الواحد show that the b series summation 1 على n b
+
+626
+00:51:52,730 --> 00:51:57,830
+diverges for b اللي أصغر أو يساوي واحد وأكبر من من
+
+627
+00:51:57,830 --> 00:52:01,850
+صفر طبعاً حد يقول لي لو كانت b سالبة لو كانت b
+
+628
+00:52:01,850 --> 00:52:07,690
+سالبة بتطلع أس فوق مثلاً معناها تطارد طيب
+
+629
+00:52:12,030 --> 00:52:17,770
+عندي for n element in n, b element in 0, 1 أن b 
+
+630
+00:52:17,770 --> 00:52:21,330
+أصغر أو يساوي 1 أصغر أو يساوي 1 طبعاً المقصود
+
+631
+00:52:21,330 --> 00:52:24,530
+ب أكبر من .. ب أصغر من سالب واحد هذا اللي بقول
+
+632
+00:52:24,530 --> 00:52:29,370
+عنها اللي بتكبر ال b السالب اللي أصغر من واحد طيب
+
+633
+00:52:29,370 --> 00:52:32,750
+لأن احنا بنحكي عن ب أكبر من صفر أصغر أو يساوي 1
+
+634
+00:52:32,750 --> 00:52:36,490
+شوفوا
+
+635
+00:52:36,490 --> 00:52:42,780
+صلي علينا بيه كلام سهل بنزل واحد ب أصغر أو يساوي
+
+636
+00:52:42,780 --> 00:52:46,680
+من أو أصغر أو يساوي n لأن ال b هنا عبارة عن إيش
+
+637
+00:52:46,680 --> 00:52:49,940
+عن كسر وكأنه بأخذ يدوري هذا هتكون أصغر أو يساوي n
+
+638
+00:52:49,940 --> 00:52:54,100
+سو واحد على n أصغر أو يساوي من واحد على n ب
+
+639
+00:52:54,100 --> 00:52:57,780
+لأنها بتنقل لأن however the sequence of partial
+
+640
+00:52:57,780 --> 00:53:01,080
+sums is unbounded، ال sequence of partial sums
+
+641
+00:53:01,080 --> 00:53:05,200
+لهذه unbounded فما بالك الأكبر منها، إذن أكيد it
+
+642
+00:53:05,200 --> 00:53:09,140
+has a subsequence of un، اللي هي sequence of
+
+643
+00:53:09,140 --> 00:53:12,000
+partial sums which is unbounded، إذن هيكون ال
+
+644
+00:53:12,000 --> 00:53:17,120
+summation واحدة لأن ب divergence هذه السيريزة هي
+
+645
+00:53:17,120 --> 00:53:20,220
+بـ Unbounded subsequence of partial sums أكيد اللي
+
+646
+00:53:20,220 --> 00:53:22,940
+عاجبالنا هذه ال subsequence of partial sums is 
+
+647
+00:53:22,940 --> 00:53:30,700
+unbounded إذا السيريزة هذه is divergent حلّينا
+
+648
+00:53:30,700 --> 00:53:37,160
+جي لمين؟ اللي هي السيريز اللي هي ال B سيريز الثانية 
+
+649
+00:53:37,160 --> 00:53:40,780
+اللي هي for B أكبر من واحد ال power series واحدة 
+
+650
+00:53:40,780 --> 00:53:45,860
+لأن ب أشملها converges هذه الآن في خطواتها كثير 
+
+651
+00:53:45,860 --> 00:53:48,840
+بتشابه اللي هو الخطوات اللي حكينا عليها قبل شوية
+
+652
+00:53:48,840 --> 00:53:52,660
+بس باتجاه ال convergence عشان هيك مش هعيد الخطوات
+
+653
+00:53:52,660 --> 00:53:57,680
+اللي عيدتها في ال B سيريز أو ال واحد سيريز اه اللي
+
+654
+00:53:57,680 --> 00:54:02,560
+هي summation واحد على n الآن for B أكبر من 1 the 
+
+655
+00:54:02,560 --> 00:54:07,480
+power series هذي converges ركزوا الآن معايا first
+
+656
+00:54:07,480 --> 00:54:13,080
+note that Sn أصغر أو يساوي Sm for every M أكبر أو
+
+657
+00:54:13,080 --> 00:54:18,690
+يساوي N يعني بمعنى آخر طيب ما هي أصلاً مبينة 1 على n
+
+658
+00:54:18,690 --> 00:54:23,870
+أس بي دائماً موجبة إذا أكيد مدام موجبة أصغر من أس
+
+659
+00:54:23,870 --> 00:54:28,170
+واحد أصغر من أس ثلاثة أصغر من أس أربعة إذا ال
+
+660
+00:54:28,170 --> 00:54:33,430
+sequence of partial sums أي شمالها is increasing
+
+661
+00:54:33,430 --> 00:54:39,190
+sequence طيب مدام increasing يعني بمعنى أو هتكون
+
+662
+00:54:39,190 --> 00:54:45,840
+أي شمالها monotone Madame Monotone بيكفيني أثبت أنه
+
+663
+00:54:45,840 --> 00:54:52,420
+S K is bounded ماشي بدي أثبت اللي هو ال sequence
+
+664
+00:54:52,420 --> 00:54:58,120
+of partial sums is bounded S K عشان أثبت أنها
+
+665
+00:54:58,120 --> 00:55:02,140
+bounded، شوفوا هذا الشيء الغريب، بس الغرابة بتروح من
+
+666
+00:55:02,140 --> 00:55:08,160
+تصرف ال increasing تبعها الآن SK is bounded عشان
+
+667
+00:55:08,160 --> 00:55:12,460
+أثبتها bounded يكفيني أني أثبت subsequence منها
+
+668
+00:55:12,460 --> 00:55:21,320
+bounded يعني أثبت لك أنه SK1 SK2 SK3 SKN in general
+
+669
+00:55:21,320 --> 00:55:25,860
+هذه ال subsequence يكفي أني أثبت أنها bounded عشان
+
+670
+00:55:25,860 --> 00:55:29,960
+تكون اللي فوق bounded طيب ما بلاقي subsequence
+
+671
+00:55:29,960 --> 00:55:33,580
+ثانية بتكون اللي هي unbounded ما هو لو جيت
+
+672
+00:55:33,580 --> 00:55:36,640
+subsequence ثانية زي ما بتقول أنها unbounded أو لو
+
+673
+00:55:36,640 --> 00:55:40,660
+كانت هذه unbounded في حالة هذه bounded معناته ما
+
+674
+00:55:40,660 --> 00:55:45,640
+هي increasing هذه هتلاقي اللي هو term من هذه بعد
+
+675
+00:55:45,640 --> 00:55:49,080
+الـ .. بعد ال term اللي أنت حكيت عنه من هذه مدام
+
+676
+00:55:49,080 --> 00:55:53,930
+هتلاقيه بعده إذاً معناته بخلي الـ bounded نصف من 
+
+677
+00:55:53,930 --> 00:55:56,970
+في هذه ال sequence إذاً في حالة ال increasing
+
+678
+00:55:56,970 --> 00:56:00,750
+sequence لأنها increasing يكفي أن ألاقي واحدة من 
+
+679
+00:56:00,750 --> 00:56:04,610
+ال subsequences أنها bounded عشان أقول كل ال
+
+680
+00:56:04,610 --> 00:56:08,210
+sequence bounded لأنه لو كانت اللي هو فيه عندك 
+
+681
+00:56:08,210 --> 00:56:14,470
+اللي هو ال sequence أصلاً مش bounded معناته في عنصر
+
+682
+00:56:14,470 --> 00:56:17,550
+من عناصر ال subsequence اللي قولنا عنها bounded
+
+683
+00:56:17,550 --> 00:56:21,370
+تخطت اللي هو اللي بنحكي فيه لأنه دي بتروح لما 
+
+684
+00:56:21,370 --> 00:56:26,890
+للنهاية فعشان هيك غصب عنها مدام لجينا واحدة من ال
+
+685
+00:56:26,890 --> 00:56:31,250
+subsequences لل increasing sequence is bounded
+
+686
+00:56:31,250 --> 00:56:37,810
+هتكون كل ال sequence is bounded طيب، شوفوا عليها،
+
+687
+00:56:37,810 --> 00:56:40,870
+إذا اللي ضال علينا بس نعمل construction ل
+
+688
+00:56:40,870 --> 00:56:46,310
+subsequence skr تكون bounded طبعاً لمن؟ للسيريزة
+
+689
+00:56:46,310 --> 00:56:51,510
+summation 1 على n أس بي اللي ما ياخذوا إشي مشابه للي
+
+690
+00:56:51,510 --> 00:56:55,730
+قبل بشوية عشان هيك مش هعيد ال induction اللي قبل
+
+691
+00:56:55,730 --> 00:57:01,030
+بشوية طريقة ال induction بتعملوها بأسلوب مشابه،
+
+692
+00:57:01,030 --> 00:57:05,190
+إيش بقولت؟ انتبهوا عليه ال K1 بيساوي 2 أس واحد 
+
+693
+00:57:05,190 --> 00:57:09,930
+ناقص واحد، ماشي؟ هناك يقول لنا خذ K1 اثنين، K2
+
+694
+00:57:09,930 --> 00:57:13,950
+اثنين تربيع، لأ هذا يقول لنا خذها KR بيساوي 2 أس R
+
+695
+00:57:13,950 --> 00:57:18,550
+ناقص واحد، ماشي الحال؟ اللي كان غلط حسابات ويساوي
+
+696
+00:57:18,550 --> 00:57:23,790
+واحد، أس K واحد واحد، K2 اثنين أس اثنين ناقص واحد 
+
+697
+00:57:23,790 --> 00:57:29,690
+ويساوي ثلاثة لو جيت حسبة SK2 احسبها لحالك هيصير
+
+698
+00:57:29,690 --> 00:57:35,590
+عبارة عن واحد على واحد على اللي هي زائد لأنه عند P
+
+699
+00:57:35,590 --> 00:57:40,790
+من وين بتبدأ؟ من عند اللي هو عند N من واحد إلى كده
+
+700
+00:57:40,790 --> 00:57:45,430
+بيصير واحد على واحد و P على واحد زائد 1 على 2 أس بي
+
+701
+00:57:45,430 --> 00:57:51,510
+زائد 1 على 3 أس بي هذا أصغر من 1 زائد اللي هو 1 على
+
+702
+00:57:51,510 --> 00:57:55,690
+2 أس بي زائد 1 على 2 أس بي يعني بأخذ الأولى مش 
+
+703
+00:57:55,690 --> 00:57:59,510
+الأخيرة زي ما كنت بأخذ قبل صار عندي 1 زائد 2 على 2
+
+704
+00:57:59,510 --> 00:58:05,200
+أس بي ويساوي 1 زائد 1 على 2 أس بي minus 1 الآن اللي
+
+705
+00:58:05,200 --> 00:58:09,460
+عملته لل S K 2 بدي اعمله لل S K 3 S K 3 اللي هي 2
+
+706
+00:58:09,460 --> 00:58:13,340
+أقص 3 نقص 1 اللي هي S بتطلع 7 اللي هي 8 نقص 1
+
+707
+00:58:13,340 --> 00:58:18,760
+الأولى S K 3 اللي هي عبارة عن S K 2 زائد زي ما
+
+708
+00:58:18,760 --> 00:58:22,040
+عملت قبل شوية ربع أس بي خمس أس بي ست أس بي زائد
+
+709
+00:58:22,040 --> 00:58:25,940
+سبع أس بي الآن بأخذ مش آخر واحدة بأخذ أول واحدة
+
+710
+00:58:25,940 --> 00:58:30,580
+عشان يظل هذا المقدار لهان أكبر من لهان بيصير SK2 زي
+
+711
+00:58:30,580 --> 00:58:37,580
+4 على 4 أُس بي اللي هو عبارة عن 1 زي 1 على 2 أُس
+
+712
+00:58:37,580 --> 00:58:42,580
+بي minus 1 زي 1 على 4 أُس بي minus 1 ماشي الحال
+
+713
+00:58:42,580 --> 00:58:48,320
+الآن سمي لي بقول لي سمي اللي هو 1 على 2 ب minus
+
+714
+00:58:48,320 --> 00:58:53,840
+1 هذه اللي هي سمي لي إياها اسمها إيه ككتلة واحدة عشان
+
+715
+00:58:53,840 --> 00:58:59,680
+ما نضلش نكتبها طيب الآن since ب أكبر من واحد إذا 
+
+716
+00:58:59,680 --> 00:59:04,100
+بدأ ب أكبر من واحد هذه لأن هي مفترضينها صارت ال a
+
+717
+00:59:04,100 --> 00:59:07,520
+اللي عندي هنا بين الصفر وبين الواحد لأن هذا صارت
+
+718
+00:59:07,520 --> 00:59:12,120
+اللي عدد أكبر من اثنين أو أكبر إذا صارت هذه عبارة 
+
+719
+00:59:12,120 --> 00:59:18,140
+عن كسر يعني كسر بين الصفر وبين الواحد الآن by
+
+720
+00:59:18,140 --> 00:59:21,260
+mathematical induction زي اللي عملته قبل شوية we
+
+721
+00:59:21,260 --> 00:59:29,180
+find if اللي هو K R بيساوي 2R-1 هلاقي دائماً S K R
+
+722
+00:59:29,180 --> 00:59:36,400
+أكبر من 0 وأصغر من 1 زائد A زائد A تربيع لما أصل
+
+723
+00:59:36,400 --> 00:59:41,440
+عند A R-1 من ال A ال A ال A اللي هي 1 على 2 أُس B
+
+724
+00:59:41,440 --> 00:59:46,480
+-1 وحالها هذه ال induction بتاعتها سهلة بنفس 
+
+725
+00:59:46,480 --> 00:59:50,500
+الطريقة اللي عملتها أنا قبل بشوية بتفترضها صحيحة ل
+
+726
+00:59:50,500 --> 00:59:57,810
+R بتثبتها صحيحة عند R بساوي 2 هنا أثبتناها لل S K 2
+
+727
+00:59:57,810 --> 01:00:02,570
+هنا أثبتناها فوق وبعدين بتثبتها صحيحة لمن؟ بتفرض
+
+728
+01:00:02,570 --> 01:00:08,150
+صحيتها ل R وبتثبت صحيتها منها ل R زائد 1 بيكون
+
+729
+01:00:08,150 --> 01:00:11,850
+صحيحة دائماً زي ما عملت قبل بشوية بالظبط في الحسبة
+
+730
+01:00:13,320 --> 01:00:17,100
+الآن as r goes to infinity هذه مجهزينها اللي هي a 
+
+731
+01:00:17,100 --> 01:00:21,140
+أصغر من واحد as r goes to infinity هذا المقدار
+
+732
+01:00:21,140 --> 01:00:26,020
+بيروح لواحد على واحد ناقص a ماشي؟ لأن هذه بيصير 
+
+733
+01:00:26,020 --> 01:00:30,360
+اللي هي عبارة عن geometric series مجموعها infinite
+
+734
+01:00:30,360 --> 01:00:35,260
+geometric series بيصير عبارة عن summation a r, r 
+
+735
+01:00:35,260 --> 01:00:42,170
+من صفر من واحد لعند ما لا نهاية مظبوط؟ هو يساوي الـ
+
+736
+01:00:42,170 --> 01:00:47,170
+summation واحد على واحد ناقص اللي هي a حرفينها a 
+
+737
+01:00:47,170 --> 01:00:54,070
+اللي هي واحد على واحد ناقص a r من العين صفر هيها
+
+738
+01:00:54,070 --> 01:00:57,450
+من العين صفر واحد زائد زائد تربيع إلى ما لا نهاية
+
+739
+01:00:57,450 --> 01:00:59,530
+هذه geometric series واحد على واحد ناقص a
+
+740
+01:00:59,530 --> 01:01:06,890
+therefore اللي هي واحد ناقص a هيكون عبارة عن إيش؟
+
+741
+01:01:06,890 --> 01:01:12,970
+bound for the partial sums of bar series ليش؟ لأن
+
+742
+01:01:12,970 --> 01:01:19,670
+صارت ال SKR limit ال SKR as R goes to infinity
+
+743
+01:01:19,670 --> 01:01:25,530
+أكبر أو يساوي أكبر من 0 أكبر أو يساوي 0 وأصغر أو
+
+744
+01:01:25,530 --> 01:01:29,790
+يساوي 100 ال limit اللي طلعت عندي 1 على 1 ناقص 
+
+745
+01:01:29,790 --> 01:01:35,880
+100 ناقص A صارت مدام ال sequence of partial sums 
+
+746
+01:01:35,880 --> 01:01:43,940
+هذه is bounded هيها by واحد على ناقص a اه then by
+
+747
+01:01:43,940 --> 01:01:48,180
+ثانو تسعة واحد اربعة مدامت هذه عبارة عن كل ان
+
+748
+01:01:48,180 --> 01:01:52,780
+عناصر موجبة يكفي عشان نثبت أنها اللي هي converge
+
+749
+01:01:52,780 --> 01:01:58,220
+أنها تكون bounded اللي صارت اللي هي ال series هذه 
+
+750
+01:01:58,220 --> 01:02:03,530
+converges زي ما قلت ماذا قلت؟ هذه صارت ال
+
+751
+01:02:03,530 --> 01:02:09,230
+subsequence of partial sums إيش ما لها؟ صارت عبارة 
+
+752
+01:02:09,230 --> 01:02:13,930
+عن bounded ماشي، مدام bounded إذا أثبتنا إن هناك
+
+753
+01:02:13,930 --> 01:02:17,570
+sub-sequence bounded من an increasing sequence إذا 
+
+754
+01:02:17,570 --> 01:02:19,770
+اتفقنا إن هذا ال increasing sequence سيكون 
+
+755
+01:02:19,770 --> 01:02:23,450
+bounded وهي increasing إذا convergence حسب ال
+
+756
+01:02:23,450 --> 01:02:26,730
+monotone convergence theorem ومن ثم بيكون خلصنا
+
+757
+01:02:26,730 --> 01:02:32,880
+أثبتنا هذا is a convergent series for B أكبر من
+
+758
+01:02:32,880 --> 01:02:33,720
+واحد
+
+759
+01:02:36,760 --> 01:02:41,140
+Show that 1 على N تربيع زائد N من 1 إلى ما لا نهاية 
+
+760
+01:02:41,140 --> 01:02:45,700
+إيش بتساوي؟ بتساوي 1 هذا بتاع مشهور وسؤال مشهور في
+
+761
+01:02:45,700 --> 01:02:50,480
+الـ .. في ال calculus أو سهل إثباته لو جينا طلعنا
+
+762
+01:02:50,480 --> 01:02:53,900
+على اللي قولنا بنعمله partial fraction كنا نقول ال
+
+763
+01:02:53,900 --> 01:02:57,860
+1 على K تربيع زائد K بيساوي 1 على K في K زائد 1
+
+764
+01:02:57,860 --> 01:03:02,440
+ناقص اللي هي عبارة عن 1 على K ناقص 1 على K زائد 1
+
+765
+01:03:03,050 --> 01:03:07,550
+ماشي الحال الآن صارت ال sequence هذا ال series هذه
+
+766
+01:03:07,550 --> 01:03:10,970
+عبارة عن هذا ناقص هذا يعني ال partial sums صارت
+
+767
+01:03:10,970 --> 01:03:16,150
+عبارة عن واحد على N ناقص واحد على N زائد واحد ويساوي
+
+768
+01:03:16,150 --> 01:03:17,150
+كمية
+
+769
+01:03:20,550 --> 01:03:26,150
+كيب واحد بيصير واحد على واحد ناقص نصف اللي بعده
+
+770
+01:03:26,150 --> 01:03:29,830
+بإثنين اللي هو نصف ناقص ثلث اللي بعده ثلث ناقص ربع
+
+771
+01:03:29,830 --> 01:03:33,070
+لما أصل الآخر واحد واحد على n ناقص واحد على 
+
+772
+01:03:33,070 --> 01:03:36,450
+n زائد واحد كلهم بيروحوا مع بعض وضال أول term وآخر
+
+773
+01:03:36,
+
+801
+01:05:56,020 --> 01:06:04,620
+summation لـ Xطيب نأخذ مثال أو أكثر من مثال بقول
+
+802
+01:06:04,620 --> 01:06:07,800
+the harmonic series التي جاءت بشوية هذه إلى
+
+803
+01:06:07,800 --> 01:06:12,940
+summation 1 على n 1 زيادة نصف زيادة ثلث زيادة ربع زيادة
+
+804
+01:06:12,940 --> 01:06:17,560
+1 على n has rearrangements like فيه rearrangements
+
+805
+01:06:17,560 --> 01:06:22,040
+واحد ماذا قال؟ قال: اسمعوا، خلوني أحط لكم التي هي
+
+806
+01:06:22,040 --> 01:06:27,920
+أُجسمها جسمين، الزوجيات و الفرديات، كيف؟ أبدأ أولاً بالزوجيّات
+
+807
+01:06:27,920 --> 01:06:31,540
+نصف و ربع، سدس، ثمن، واحد على اثنين،  إلى أن أصل 
+
+808
+01:06:31,540 --> 01:06:36,020
+ماشي إلى ما لا نهاية، و بينهما أبدأ  بإحضار الفرديات: واحد
+
+809
+01:06:36,020 --> 01:06:38,760
+على واحد، واحدة على ثلاثة، واحدة على خمسة، واحدة على
+
+810
+01:06:38,760 --> 01:06:42,040
+اثنين ناقص واحد، إلى ما لا نهاية، ماشي الحال
+
+811
+01:06:42,040 --> 01:06:48,080
+هذه ال series عنصرها n بالضبط اللي هنا،  بس معمول
+
+812
+01:06:48,080 --> 01:06:51,890
+الذي هنا إعادة الترتيب، واحد ثاني جاء، اسمعوا، يُرتبها
+
+813
+01:06:51,890 --> 01:06:55,450
+بترتيب ثاني،  أي شيء،  بطبيعة الحال، يُرتبها حسب ما يخدمه أو
+
+814
+01:06:55,450 --> 01:06:58,170
+حسب ما يطلع معه في ال... في ال... في الشيء العملي
+
+815
+01:06:58,170 --> 01:07:02,310
+مثلاً الآن أول واحد يقول لك تأخذ الذي هو ال... ال
+
+816
+01:07:02,310 --> 01:07:04,030
+ال... ال... ال... ال... ال... ال... ال... ال... ال... ال
+
+817
+01:07:04,030 --> 01:07:04,050
+ال... ال... ال... ال... ال... ال... ال... ال... ال
+
+818
+01:07:04,050 --> 01:07:04,310
+ال... ال... ال... ال... ال... ال... ال... ال... ال
+
+819
+01:07:04,310 --> 01:07:04,350
+ال... ال... ال... ال... ال... ال... ال... ال... ال
+
+820
+01:07:04,350 --> 01:07:04,430
+ال... ال... ال... ال... ال... ال... ال... ال... ال
+
+821
+01:07:04,430 --> 01:07:05,110
+ال... ال... ال... ال... ال... ال... ال... ال... ال
+
+822
+01:07:05,110 --> 01:07:06,250
+ال... ال... ال... ال... ال... ال... ال... ال... ال
+
+823
+01:07:07,640 --> 01:07:12,480
+الثاني تأخذه اثنين even نصف و ربع آخذ واحد على 
+
+824
+01:07:12,480 --> 01:07:16,560
+المقام even الآن الثالث تأخذه الثلاثة التي بعده
+
+825
+01:07:16,560 --> 01:07:21,740
+ال odd التي هي ثلث خمس سبعة التي بعده الأربعة التي 
+
+826
+01:07:21,740 --> 01:07:27,340
+بعده ال even 6 8 10 12 14 التي بعده ال 5 التي
+
+827
+01:07:27,340 --> 01:07:31,440
+بعده ال odd 9 11 13 15 17 التي
+
+828
+01:07:31,440 --> 01:07:34,300
+بعده ال 6 ال even التي بعده ال 7 ال odd و هكذا و
+
+829
+01:07:34,300 --> 01:07:37,840
+نظل ماشيين إلى ما لا نهاية، هذه برضه rearrangement
+
+830
+01:07:37,840 --> 01:07:43,860
+لمين؟ ل التي هي ال series التي هي summation 1 على 
+
+831
+01:07:53,010 --> 01:07:57,590
+التعريف الرياضي يسمى ال Series يسمى التعريف
+
+832
+01:07:57,590 --> 01:08:00,790
+الرياضي الرياضي الرياضي الرياضي الرياضي الرياضي
+
+833
+01:08:00,790 --> 01:08:06,170
+الرياضي الرياضي الرياضي الرياضييعني هذا ال
+
+834
+01:08:06,170 --> 01:08:10,510
+bijection عبارة عن إعادة ترتيب من السيريز
+
+835
+01:08:10,510 --> 01:08:16,090
+الأصلية إلى السيريز الجديدة لأن هذه ترتيبها I و هذه
+
+836
+01:08:16,090 --> 01:08:19,770
+ترتيبها F of I صارت بس مدام أن ال function one to
+
+837
+01:08:19,770 --> 01:08:24,710
+one و onto معناته أن كله هيستخدم ال terms و كله لا 
+
+838
+01:08:24,710 --> 01:08:28,510
+يستخدم إلا مرة واحدة لأن ال function one to one
+
+839
+01:08:28,510 --> 01:08:30,750
+و onto فصار عند ال Yم
+
+840
+01:08:36,050 --> 01:08:43,730
+Y1 XF1 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+841
+01:08:43,730 --> 01:08:46,350
+Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+842
+01:08:46,350 --> 01:08:52,210
+Y2 XF2 Y2 XF2 Y2 XFما تأثير ال Rearrangements على
+
+843
+01:08:52,210 --> 01:08:56,890
+السيريز الأصلية بقول لك لو كانت السيريز absolutely
+
+844
+01:08:56,890 --> 01:09:00,350
+convergent و هذا الذي سنتطرق إليه،  بس لو كانت السيريز
+
+845
+01:09:00,350 --> 01:09:04,250
+absolutely convergent  رتب كما تريد،  لن يفرق 
+
+846
+01:09:04,250 --> 01:09:08,090
+معنا الذي هو إيه؟ ال convergence يعني let
+
+847
+01:09:08,090 --> 01:09:11,690
+summation xn be an absolutely convergent series in
+
+848
+01:09:11,690 --> 01:09:18,610
+R then any rearrangement of summation xn converge
+
+849
+01:09:18,610 --> 01:09:21,770
+to the same value يعني لو كانت ال series
+
+850
+01:09:21,770 --> 01:09:27,150
+absolutely convergent أي ترتيب لها آخر وفق تعريف
+
+851
+01:09:27,150 --> 01:09:31,390
+الترتيب الذي تحدثنا عنه الذي سيعطي نفس ال
+
+852
+01:09:31,390 --> 01:09:34,510
+limit ستكون converge و لها نفس ال limit  دعونا
+
+853
+01:09:34,510 --> 01:09:38,910
+نرى البرهان و نمر عليه let summation xn converge
+
+854
+01:09:38,910 --> 01:09:44,410
+to x و let yn be a rearrangement of summation xn
+
+855
+01:09:44,410 --> 01:09:46,590
+و let yn be a rearrangement of summation xn و 
+
+856
+01:09:46,590 --> 01:09:46,870
+let yn be a rearrangement of summation xn و let
+
+857
+01:09:46,870 --> 01:09:46,950
+rearrangement of summation xn و let yn be a
+
+858
+01:09:46,950 --> 01:09:47,870
+rearrangement of summation xn و let yn be a
+
+859
+01:09:47,870 --> 01:09:49,130
+rearrangement of summation xn و let yn be a
+
+860
+01:09:49,130 --> 01:09:51,570
+rearrangement of summation xn و let en be a
+
+861
+01:09:51,570 --> 01:09:51,590
+rearrangement of summation xn و let en be a
+
+862
+01:09:51,590 --> 01:09:51,690
+rearrangement of summation xn و let en be a
+
+863
+01:09:51,690 --> 01:09:52,390
+rearrangement of summation xn و let en be a
+
+864
+01:09:52,390 --> 01:09:55,870
+rearrangement of summation xn و let en be a
+
+865
+01:09:55,870 --> 01:09:59,130
+rearrangement of summation xn و let en be a
+
+866
+01:09:59,130 --> 01:10:02,130
+rearrangement of summation xn و let en be 
+
+867
+01:10:07,510 --> 01:10:10,290
+ماذا يعني xn xn xn xn xn xn
+
+868
+01:10:10,290 --> 01:10:10,790
+xn xn xn xn xn xn xn xn
+
+869
+01:10:10,790 --> 01:10:10,850
+xn xn xn xn xn xn xn xn
+
+870
+01:10:10,850 --> 01:10:11,450
+xn xn xn xn xn xn xn xn
+
+871
+01:10:11,450 --> 01:10:12,530
+xn xn xn xn xn xn xn xn
+
+872
+01:10:12,530 --> 01:10:15,790
+xn xn xn xn xn xn xn xn
+
+873
+01:10:15,790 --> 01:10:23,530
+xn xn xn xn xn xn xn xn
+
+874
+01:10:23,530 --> 01:10:30,260
+x هو يعيدها كمان مرة الحديث كمان، بقول إن هذا لا 
+
+875
+01:10:30,260 --> 01:10:34,040
+يُضمن أن ال xn converts إذا لكل y أكبر من 0 there
+
+876
+01:10:34,040 --> 01:10:41,900
+exists n such that fq أكبر من n و التي هي sn x1
+
+877
+01:10:41,900 --> 01:10:50,260
+x2 لعند xn then سيكون عندي ال X K K من عند R
+
+878
+01:10:50,260 --> 01:10:54,540
+مثل الواحد عند Q أصغر من من إبسيلون كانت بطريقة
+
+879
+01:10:54,540 --> 01:10:59,000
+ثانية،  نتحمل،  و مهم،  ليس الآن، الآن ماذا الذي هو
+
+880
+01:10:59,000 --> 01:11:04,760
+summation X N بساوي ماذا؟ بساوي X خلاص، نعتبرنا الذي
+
+881
+01:11:04,760 --> 01:11:08,320
+هو absolute convergence يعني، و نعتبرنا ال terms
+
+882
+01:11:08,320 --> 01:11:12,000
+positive و خلاص و نشتغل عليها summation X N بساوي 
+
+883
+01:11:12,000 --> 01:11:19,230
+ماذا؟ بساوي X الآن for every epsilon أكبر من صفر
+
+884
+01:11:19,230 --> 01:11:25,350
+there exists N such that for every N أو Q سميها
+
+885
+01:11:25,940 --> 01:11:30,460
+أكبر من N أكبر أو يساوي N عندي الذي هو ال summation
+
+886
+01:11:30,460 --> 01:11:35,240
+هذا الذي دعوني أسميه الذي هو ال sequence of
+
+887
+01:11:35,240 --> 01:11:40,580
+partial sums ل SN التي هي SN ناقص X أصغر من ماذا؟
+
+888
+01:11:40,580 --> 01:11:46,400
+من إبسيلون for every N أكبر أو يساوي H N هذا الذي 
+
+889
+01:11:46,400 --> 01:11:49,780
+هو ال limit العادية،  Cauchy criterion لها أنه سيكون
+
+890
+01:11:49,780 --> 01:11:56,890
+عندي ال summation التي هي SN آخذ ال SN و SQ هو 
+
+891
+01:11:56,890 --> 01:12:03,490
+ما يآخذ ال SN ناقص 
+
+892
+01:12:03,490 --> 01:12:09,510
+SP لأكبر Q SQ ناقص SN ليس ليس ممكن أتحدث عن
+
+893
+01:12:09,510 --> 01:12:14,370
+داخل absolute value ناقص SN فينفع SN آه ما هي 
+
+894
+01:12:14,370 --> 01:12:18,480
+صحيحة على كل N أكبر أو سواء منها ماذا؟ N أصغر من 
+
+895
+01:12:18,480 --> 01:12:21,940
+ماذا؟ من إبسيلون، هذا هذا ماذا... ماذا قيمته تساوي
+
+896
+01:12:21,940 --> 01:12:25,700
+summation التي هو حاطها ال absolute value آخذ ال
+
+897
+01:12:25,700 --> 01:12:30,240
+absolute convergence K من عند N زائد واحد إلى ما لا
+
+898
+01:12:30,240 --> 01:12:35,800
+نهاية لعند Q التي هي أصغر من ماذا؟ من إبسيلون،  عارفين
+
+899
+01:12:35,800 --> 01:12:39,640
+هذا آه؟ التي هي SQ نقص SN لأن آخذ ال series
+
+900
+01:12:39,640 --> 01:12:41,620
+ماذا صارت؟ absolute value ليست مشكلة 
+
+901
+01:12:44,600 --> 01:12:48,740
+SN هي ال sequence of partial sums لهذه لل 
+
+902
+01:12:48,740 --> 01:12:52,140
+absolute values ليست لل Xn لل absolute values فصار
+
+903
+01:12:52,140 --> 01:12:57,620
+عندي الآن استخدام الذي هو limit as a limit و as a
+
+904
+01:12:57,620 --> 01:13:02,760
+convergence to X و استخدامها as a Cauchy criterion
+
+905
+01:13:02,760 --> 01:13:06,840
+ماشي؟ قول ما هي هذه بالنسبة لها N و هذه بالنسبة
+
+906
+01:13:06,840 --> 01:13:10,880
+لها N ثانية، لم أستعمل نفس ال N؟ take N
+
+907
+01:13:10,880 --> 01:13:17,610
+maximum بين الجهتين بين علامتين للجهتين طبيعي،  طيب
+
+908
+01:13:17,610 --> 01:13:21,350
+... لا
+
+909
+01:13:21,350 --> 01:13:24,510
+والله نستخدم الأسئلة لمَن؟ لهذه
+
+910
+01:13:27,810 --> 01:13:30,230
+بطبيعي، ما هو مدّعى Absolutely Convergent هذا
+
+911
+01:13:30,230 --> 01:13:33,030
+Convergent مدّعى Convergent هذه مطبقة على من؟
+
+912
+01:13:33,030 --> 01:13:35,930
+على ال Convergence و هذه مطبقة على من؟ على ال 
+
+913
+01:13:35,930 --> 01:13:38,430
+Absolutely Convergent برضه نافع... آه نافع نأخذ
+
+914
+01:13:38,430 --> 01:13:41,570
+الذي هو يطلع له أن واحد هذه و هذا أن اثنين و نأخذه
+
+915
+01:13:41,570 --> 01:13:44,010
+من ال maximum للتان،  تان يصبح لي هذه ال inequality
+
+916
+01:13:44,010 --> 01:13:47,230
+و هذه ال inequality صحيحة...  صحيحتان لمَن؟ لل N التي
+
+917
+01:13:47,230 --> 01:13:50,720
+هي ال maximum الذي عنده،  طيب الآن، إذا أصبح عندي هذه 
+
+918
+01:13:50,720 --> 01:13:54,120
+ال SN بقصد فيها فعلًا التي هي ال sequence of
+
+919
+01:13:54,120 --> 01:14:01,060
+partial sums لسيريز الأصلية Xn و ال SN' هي من؟ ال
+
+920
+01:14:01,060 --> 01:14:05,400
+sequence التي هي تعود إلى ال absolute values هذه طبعًا 
+
+921
+01:14:05,400 --> 01:14:09,700
+حسبناها as q prime نقص as n prime بساوي هذه نقص 
+
+922
+01:14:09,700 --> 01:14:13,060
+هذه من واحد عند q و من واحد عند n التي هي بساوي هذه
+
+923
+01:14:13,060 --> 01:14:16,940
+التي هي هذه حسبناها عارفينها طيب،  ليس هذا الموضوع
+
+924
+01:14:16,940 --> 01:14:23,500
+هذا الأمر سهل، يعني إذا أصبح عندي الآن هي هذه أو هي
+
+925
+01:14:23,500 --> 01:14:32,260
+هذه أو هذه كلها أريد أن أرى كيف أستعملها نحن لماذا نريد 
+
+926
+01:14:32,260 --> 01:14:36,580
+... نريد ال Series الجديدة Summation Yn التي هي 
+
+927
+01:14:36,580 --> 01:14:40,980
+ال Rearrangement انظر هكذا،  لأن Now let M limited M
+
+928
+01:14:40,980 --> 01:14:45,260
+be such that all of the terms X1 لعند ال Xm are
+
+929
+01:14:45,260 --> 01:14:52,680
+contained as a sum and in T M Y1 لعند ال Ym ماذا
+
+930
+01:14:52,680 --> 01:14:55,820
+الذي يقول هذا؟ و هذا الكلام الذي هو الذي نحن
+
+931
+01:14:55,820 --> 01:14:59,760
+نريد أن نتهيأ له و هو الذي يفتح لنا الطريق أمام البرهان
+
+932
+01:15:01,400 --> 01:15:09,360
+جاء لي يحكي لي أن دعونا نختار M في ال N ابحث ال TM
+
+933
+01:15:09,360 --> 01:15:14,740
+هذه التي هي ال Sequence of partial sums لمن؟ لل YN
+
+934
+01:15:14,740 --> 01:15:20,960
+التي هي Y1 زائد Y2 زائد YM بقول لي:  أنا أبدأ أختار
+
+935
+01:15:20,960 --> 01:15:29,370
+ال M هذه على أساس أنه كل ال terms التي جمعتها من ال
+
+936
+01:15:29,370 --> 01:15:34,310
+N التي فوق من X1 لعند XN تكون موجودة من بين هؤلاء 
+
+937
+01:15:34,310 --> 01:15:37,970
+ال terms أستطيع آه أستطيع لأن هؤلاء عشمها hand
+
+938
+01:15:37,970 --> 01:15:44,210
+finite و في النهاية مدام finite إذا سيكون من ضمن 
+
+939
+01:15:44,210 --> 01:15:49,890
+ال Y1 Y2 إلى 100 ألف ألف مليون سميته ال 100 ألف
+
+940
+01:15:49,890 --> 01:15:54,480
+ألف مليون التي هي 100 YN إذًا ال M اخترتها بناءً 
+
+941
+01:15:54,480 --> 01:16:02,060
+على أن أضمن ال X1 و ال X2 لعند ال XN  تين لعند
+
+942
+01:16:02,060 --> 01:16:06,080
+ال XN capital تين هؤلاء من هؤلاء يعني بس إعادة 
+
+943
+01:16:06,080 --> 01:16:10,890
+تسميتهم إذًا الآن الذي... الذي... الذي اخترته أن
+
+944
+01:16:10,890 --> 01:16:15,570
+قلت له let M element in N مهما كانت كبيرة be such
+
+945
+01:16:15,570 --> 01:16:21,510
+that all of the terms X1, X2, XN are contained as
+
+946
+01:16:21,510 --> 01:16:26,610
+a sum and يعني مجموعات من ضمن المجموع هذا Y1 زي Y2
+
+947
+01:16:26,610 --> 01:16:35,330
+عند من؟ YM it follows الآن if M أكبر أو يساوي M
+
+948
+01:16:36,630 --> 01:16:42,970
+then TM ناقص SM ناقص
+
+949
+01:16:42,970 --> 01:16:48,770
+SM ناقص SM ناقص SM ناقص SM ناقص SM ناقص SM
+
+950
+01:16:48,770 --> 01:16:48,910
+SM ناقص SM ناقص SM ناقص SM ناقص SM ناقص SM
+
+951
+01:16:48,910 --> 01:16:48,970
+ناقص SM ناقص SM ناقص SM ناقص SM ناقص SM ناقص
+
+952
+01:16:48,970 --> 01:16:49,030
+SM ناقص SM ناقص SM ناقص SM ناقص SM ناقص SM
+
+953
+01:16:49,030 --> 01:16:49,850
+ناقص SM ناقص SM ناقص SM ناقص SM ناقص SM ناقص
+
+954
+01:16:49,850 --> 01:16:54,450
+SM ناقص SM ناقص S
+
+955
+01:16:56,000 --> 01:16:58,900
+ممكن الذي عندها و ممكن يظل معها عندها نور على نور
+
+956
+01:16:58,900 --> 01:17:04,620
+يعني نريد أن نشيل منه من؟ ال SN ال SN ال N هذه
+
+957
+01:17:04,620 --> 01:17:11,220
+شمالها أكبر من من؟ أكبر أو تساوي ال N طيب إذا هذه
+
+958
+01:17:11,220 --> 01:17:21,340
+ستكون consist of finite sum of terms XK with K
+
+959
+01:17:21,340 --> 01:17:24,600
+أكبر من من؟ من ال N
+
+960
+01:17:29,340 --> 01:17:37,280
+الآن من عند الواحد لعند ال N أنا شيلتهم من ال TM
+
+961
+01:17:37,280 --> 01:17:47,000
+بأشي آه ال SN هذه واحد اثنين لعند التي هي ال N و
+
+962
+01:17:47,000 --> 01:17:53,840
+جلطتها لأن ال N آيه شمالها أكبر أو يساوي M N أكبر
+
+963
+01:17:53,840 --> 01:17:58,980
+أو يساوي M ماشي طيب أصبح عندي ال N ال TM انشال منها
+
+964
+01:17:58,980 --> 01:18:05,180
+ال X1 و ال X2 و ال XN آه إذا الذين بقوا من ال terms
+
+965
+01:18:05,180 --> 01:18:10,940
+هذه أو ماينة التي هنا موجودة ال TM التي موجودة هنا
+
+966
+01:18:10,940 --> 01:18:20,090
+التي فيها أكيد سيكون بعد ال N الذين بقوا ماشي؟ لأنه
+
+967
+01:18:20,090 --> 01:18:24,770
+شيلت منه لأنه عندك S1 لأنه عندك S2 و هذه SN بقى 
+
+968
+01:18:24,770 --> 01:18:31,090
+منها من الذي بعد ال N برضه إذا في النهاية ال terms
+
+969
+01:18:31,090 --> 01:18:36,790
+التي هنا ستكون التي تحتوي على sums of terms ل XK
+
+970
+01:18:36,790 --> 01:18:42,250
+و ال K سيكون أكبر من 100 من ال N لأن الذي ترتيبه
+
+971
+01:18:42,250 --> 01:18:45,770
+من ال N و تنازل كله راح من عملية الطرح هذه
+
+972
+01:18:49,100 --> 01:18:55,660
+إذًا for some q أكبر من n we have Tm ناقص Tn هذا
+
+973
+01:18:55,660 --> 01:19:00,820
+absolute value بساوي ال summation لل XK كي من 
+
+974
+01:19:00,820 --> 01:19:06,420
+أين؟ من n زائد واحد لعند من؟ لعند Q لماذا؟ لأنه أنا 
+
+1001
+01:21:00,940 --> 01:21:03,680
+الأول في شباط التسعة وإلى لقاكم 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..491aa0d4b1fd3ee22065b3d9d5d28184f453bf31
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2840, "start": 5.18, "end": 28.4, "text": "بسم الله الرحمن الرحيم الحمد لله رب العالمين والصلاة والسلام على سيد المرسلين سيدنا محمد وعلى آله وصحبه أجمعين هذه المحاضرة رقم 25 مساق تحليل حقيقة 2 طلاب طالبات الجامعة الإسلامية قسم أرياضيات كلية العلوم اليوم هنبدأ ان شاء الله بال chapter الرابع في المادة", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21542, 2304, 3215, 24976, 3224, 12602, 3555, 18863, 45340, 9957, 16070, 9381, 15040, 3660, 16070, 3794, 37440, 15844, 8608, 25708, 9673, 2288, 3794, 1211, 9957, 8608, 25708, 8315, 3714, 35571, 3215, 4032, 3615, 23942, 19753, 43761, 4032, 9381, 5016, 3555, 3224, 5551, 7435, 2304, 3615, 9957, 29538, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 3552, 47524, 995, 4587, 6055, 5016, 20292, 1211, 11331, 38436, 28671, 568, 23032, 15040, 3555, 23032, 6027, 3555, 9307, 25724, 10943, 27884, 33688, 3794, 37440, 10632, 12174, 38251, 5551, 16572, 46958, 1829, 9307, 28242, 10632, 18863, 1211, 20498, 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series هنحكي اليوم في هذه المحاضرة على اللي هو convergence of infinite series", "tokens": [6027, 20292, 31439, 2423, 7187, 1722, 34051, 25724, 11622, 38207, 34892, 16758, 3615, 9154, 9673, 18513, 3660, 13672, 1829, 31439, 7187, 1722, 6055, 33753, 18871, 2407, 7649, 13785, 2638, 8032, 1863, 5016, 4117, 1829, 45595, 20498, 8978, 29538, 9673, 5016, 46958, 25720, 15844, 13672, 1829, 31439, 32181, 295, 13785, 2638], "avg_logprob": -0.13763786238782547, "compression_ratio": 1.5225806451612902, "no_speech_prob": 1.2755393981933594e-05, "words": [{"start": 28.67, "end": 29.35, "word": "اللي", "probability": 0.667724609375}, {"start": 29.35, "end": 29.85, "word": " هو", "probability": 0.98291015625}, {"start": 29.85, "end": 30.41, "word": " ال", "probability": 0.40234375}, {"start": 30.41, "end": 31.23, "word": " chapter", "probability": 0.84765625}, {"start": 31.23, "end": 31.73, "word": " 9", "probability": 0.80224609375}, {"start": 31.73, "end": 31.99, "word": " أو", "probability": 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"text": "ال infinite series طبعا هو موضوع معهود للطلاب أو لطلبة العلوم والهندسة بصورة عامة لإنه تم الحديث عنه في اللي هو calculus باه الآن بنحكي عن اللي هو النظرة التحليلية ل", "tokens": [6027, 13785, 2638, 23032, 3555, 3615, 995, 31439, 3714, 2407, 11242, 45367, 20449, 3224, 23328, 24976, 9566, 15040, 3555, 34051, 5296, 9566, 46152, 3660, 18863, 1211, 20498, 16070, 3224, 41260, 3794, 3660, 4724, 9381, 13063, 3660, 6225, 10943, 3660, 5296, 28814, 1863, 3224, 46811, 21542, 16254, 12984, 18871, 3224, 8978, 13672, 1829, 31439, 33400, 4724, 40294, 6024, 48506, 44945, 5016, 4117, 1829, 18871, 13672, 1829, 31439, 28239, 19913, 25720, 16712, 5016, 20292, 1211, 10632, 5296], "avg_logprob": -0.16581002916944654, "compression_ratio": 1.553072625698324, "no_speech_prob": 6.198883056640625e-06, "words": [{"start": 44.24, "end": 45.64, "word": "ال", "probability": 0.57470703125}, {"start": 45.64, "end": 47.9, "word": " infinite", "probability": 0.335205078125}, {"start": 47.9, "end": 48.58, "word": 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{"start": 55.88, "end": 56.04, "word": " تم", "probability": 0.98779296875}, {"start": 56.04, "end": 56.54, "word": " الحديث", "probability": 0.9913736979166666}, {"start": 56.54, "end": 57.14, "word": " عنه", "probability": 0.99462890625}, {"start": 57.14, "end": 57.88, "word": " في", "probability": 0.84912109375}, {"start": 57.88, "end": 58.22, "word": " اللي", "probability": 0.6549072265625}, {"start": 58.22, "end": 58.44, "word": " هو", "probability": 0.92919921875}, {"start": 58.44, "end": 58.96, "word": " calculus", "probability": 0.92578125}, {"start": 58.96, "end": 60.1, "word": " باه", "probability": 0.3660888671875}, {"start": 60.1, "end": 60.8, "word": " الآن", "probability": 0.6097412109375}, {"start": 60.8, "end": 61.4, "word": " بنحكي", "probability": 0.9327392578125}, {"start": 61.4, "end": 61.66, "word": " عن", "probability": 0.9873046875}, {"start": 61.66, "end": 62.02, "word": " اللي", "probability": 0.930419921875}, {"start": 62.02, "end": 62.4, "word": " هو", 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{"start": 248.25, "end": 248.85, "word": " of", "probability": 0.9814453125}, {"start": 248.85, "end": 249.25, "word": " this", "probability": 0.9365234375}, {"start": 249.25, "end": 249.73, "word": " infinite", "probability": 0.953125}, {"start": 249.73, "end": 250.33, "word": " series", "probability": 0.9013671875}, {"start": 250.33, "end": 251.11, "word": " نيجي", "probability": 0.8955078125}, {"start": 251.11, "end": 251.41, "word": " الآن", "probability": 0.74072265625}, {"start": 251.41, "end": 251.77, "word": " بما", "probability": 0.869140625}, {"start": 251.77, "end": 252.19, "word": " أنه", "probability": 0.5772705078125}, {"start": 252.19, "end": 252.41, "word": " اللي", "probability": 0.988037109375}, {"start": 252.41, "end": 252.71, "word": " هو", "probability": 0.99609375}, {"start": 252.71, "end": 252.89, "word": " في", "probability": 0.94921875}, {"start": 252.89, "end": 253.67, "word": " النهاية", "probability": 0.9482421875}, {"start": 253.67, "end": 254.27, "word": " ال", "probability": 0.8173828125}, {"start": 254.27, "end": 254.65, "word": " series", "probability": 0.96484375}, {"start": 254.65, "end": 254.95, "word": " اللي", "probability": 0.985595703125}, {"start": 254.95, "end": 255.35, "word": " عندنا", "probability": 0.946533203125}, {"start": 255.35, "end": 255.73, "word": " هذه", "probability": 0.54052734375}], "temperature": 1.0}, {"id": 14, "seek": 27752, "start": 256.16, "end": 277.52, "text": "عبارة عن limit في حالة ال convergence هو عبارة عن limit of a sequence إذا تنطبق عليها قوانين ال sequences اللي أخدناها في chapter 3 إذا متوقع إنه احنا نستخدم ما تحدثنا فيه في chapter 3 أو نتائجها للحصول على النظريات اللي احنا بدنا نبرهنها", "tokens": [3615, 3555, 9640, 3660, 18871, 4948, 8978, 11331, 6027, 3660, 2423, 32181, 31439, 6225, 3555, 9640, 3660, 18871, 4948, 295, 257, 8310, 11933, 15730, 6055, 1863, 9566, 3555, 4587, 25894, 11296, 12174, 2407, 7649, 9957, 2423, 22978, 13672, 1829, 5551, 9778, 3215, 8315, 11296, 8978, 7187, 805, 11933, 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{"start": 293.25, "end": 293.91, "word": " موجود", "probability": 0.9847005208333334}, {"start": 293.91, "end": 294.33, "word": " و", "probability": 0.44970703125}, {"start": 294.33, "end": 294.59, "word": " هذا", "probability": 0.88037109375}, {"start": 294.59, "end": 294.75, "word": " ال", "probability": 0.96533203125}, {"start": 294.75, "end": 294.95, "word": " sum", "probability": 0.9599609375}, {"start": 294.95, "end": 295.39, "word": " موجود", "probability": 0.9905598958333334}, {"start": 295.39, "end": 295.67, "word": " يعني", "probability": 0.916259765625}, {"start": 295.67, "end": 297.63, "word": " السيريز", "probability": 0.818359375}, {"start": 297.63, "end": 298.05, "word": " هذه", "probability": 0.693359375}, {"start": 298.05, "end": 298.69, "word": " اللي", "probability": 0.748291015625}, {"start": 298.69, "end": 298.81, "word": " هي", "probability": 0.90380859375}, {"start": 298.81, "end": 299.17, "word": " sequence", "probability": 0.40625}, {"start": 299.17, "end": 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{"start": 308.73, "end": 309.23, "word": " لها", "probability": 0.741943359375}, {"start": 309.23, "end": 310.11, "word": " then", "probability": 0.390869140625}, {"start": 310.11, "end": 310.39, "word": " the", "probability": 0.494873046875}, {"start": 310.39, "end": 310.79, "word": " series", "probability": 0.8505859375}, {"start": 310.79, "end": 311.45, "word": " summation", "probability": 0.31689453125}, {"start": 311.45, "end": 312.23, "word": " xn", "probability": 0.5301513671875}, {"start": 312.23, "end": 312.71, "word": " زائد", "probability": 0.77197265625}, {"start": 312.71, "end": 313.23, "word": " yn", "probability": 0.9453125}, {"start": 313.23, "end": 314.21, "word": " برضه", "probability": 0.9514973958333334}, {"start": 314.21, "end": 315.07, "word": " converges", "probability": 0.6170654296875}, {"start": 315.07, "end": 315.49, "word": " ومش", "probability": 0.95361328125}, {"start": 315.49, "end": 315.75, "word": " هيك", "probability": 0.91748046875}, {"start": 315.75, "end": 316.19, "word": " كمان", "probability": 0.9894205729166666}, {"start": 316.19, "end": 317.21, "word": " وقيمة", "probability": 0.948486328125}, {"start": 317.21, "end": 318.09, "word": " السيكوه", "probability": 0.6875244140625}, {"start": 318.09, "end": 318.63, "word": " السيريز", "probability": 0.8099365234375}, {"start": 318.63, "end": 319.13, "word": " الجديدة", "probability": 0.8702799479166666}, {"start": 319.13, "end": 319.43, "word": " summation", "probability": 0.81298828125}, {"start": 319.43, "end": 319.87, "word": " xn", "probability": 0.967529296875}, {"start": 319.87, "end": 320.11, "word": " زائد", "probability": 0.8868815104166666}, {"start": 320.11, "end": 320.51, "word": " yn", "probability": 0.97705078125}, {"start": 320.51, "end": 321.35, "word": " بساوي", "probability": 0.82208251953125}, {"start": 321.35, "end": 321.87, "word": " السيريز", "probability": 0.916015625}, {"start": 321.87, "end": 322.47, "word": " الأولى", "probability": 0.9708658854166666}, {"start": 322.47, "end": 322.81, "word": " زائد", "probability": 0.95263671875}, {"start": 322.81, "end": 323.53, "word": " السيريز", "probability": 0.93212890625}, {"start": 323.53, "end": 324.19, "word": " الثانية", "probability": 0.912353515625}], "temperature": 1.0}, {"id": 17, "seek": 35330, "start": 324.66, "end": 353.3, "text": "الان اشي مشابه ل 4 Xn نقص Yn بيسوي summation Xn نقص summation Min Yn الان بي if the series summation Xn is convergent is convergent and C is any real number then the series summation C في Xn برضه هيخليها تظلها convergent و بيطلع عند ال summation ل C Xn بيسوي C في ال summation ل Min", "tokens": [6027, 7649, 1975, 8592, 1829, 37893, 16758, 3224, 5296, 1017, 1783, 77, 8717, 4587, 9381, 398, 77, 4724, 1829, 3794, 45865, 28811, 1783, 77, 8717, 4587, 9381, 28811, 2829, 398, 77, 2423, 7649, 4724, 1829, 498, 264, 2638, 28811, 1783, 77, 307, 9652, 6930, 307, 9652, 6930, 293, 383, 307, 604, 957, 1230, 550, 264, 2638, 28811, 383, 8978, 1783, 77, 4724, 43042, 3224, 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"end": 350.54, "word": " عند", "probability": 0.9296875}, {"start": 350.54, "end": 350.64, "word": " ال", "probability": 0.493408203125}, {"start": 350.64, "end": 350.96, "word": " summation", "probability": 0.92578125}, {"start": 350.96, "end": 351.16, "word": " ل", "probability": 0.591796875}, {"start": 351.16, "end": 351.32, "word": " C", "probability": 0.5732421875}, {"start": 351.32, "end": 351.66, "word": " Xn", "probability": 0.823486328125}, {"start": 351.66, "end": 351.98, "word": " بيسوي", "probability": 0.9027099609375}, {"start": 351.98, "end": 352.18, "word": " C", "probability": 0.943359375}, {"start": 352.18, "end": 352.3, "word": " في", "probability": 0.9287109375}, {"start": 352.3, "end": 352.38, "word": " ال", "probability": 0.35595703125}, {"start": 352.38, "end": 352.7, "word": " summation", "probability": 0.90576171875}, {"start": 352.7, "end": 353.0, "word": " ل", "probability": 0.84326171875}, {"start": 353.0, "end": 353.3, "word": " Min", "probability": 0.87451171875}], "temperature": 1.0}, {"id": 18, "seek": 37031, "start": 353.75, "end": 370.31, "text": "للـ XL الان مدام نحكي عن series بده ترجم اللي هو ال series الى sequences الى sequence of partial sums واستخدم اللي هي قوانين ال limits القديمة يعني الموضوع البرهان أمر سهل ان شاء الله", "tokens": [1211, 1211, 39184, 37210, 2423, 7649, 3714, 3215, 10943, 8717, 5016, 4117, 1829, 18871, 2638, 47525, 3224, 6055, 47341, 2304, 13672, 1829, 31439, 2423, 2638, 2423, 7578, 22978, 2423, 7578, 8310, 295, 14641, 34499, 36764, 14851, 9778, 40448, 13672, 1829, 39896, 12174, 2407, 7649, 9957, 2423, 10406, 25062, 16254, 46007, 37495, 22653, 9673, 2407, 11242, 45367, 2423, 26890, 3224, 7649, 5551, 29973, 8608, 3224, 1211, 16472, 13412, 16606, 21984], "avg_logprob": -0.2195312472326415, "compression_ratio": 1.5243243243243243, "no_speech_prob": 1.6987323760986328e-05, "words": [{"start": 353.75, "end": 354.27, "word": "للـ", "probability": 0.7369791666666666}, {"start": 354.27, "end": 354.65, "word": 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{"start": 736.01, "end": 736.43, "word": " هي", "probability": 0.96435546875}, {"start": 736.43, "end": 737.89, "word": " اللممة", "probability": 0.5899251302083334}], "temperature": 1.0}, {"id": 35, "seek": 74897, "start": 738.65, "end": 748.97, "text": "اللي هي اللمّة اللي بعدها اللي هي المشهورة", "tokens": [6027, 20292, 39896, 13672, 2304, 11703, 3660, 13672, 1829, 39182, 11296, 13672, 1829, 39896, 9673, 8592, 3224, 13063, 3660], "avg_logprob": -0.20527343302965165, "compression_ratio": 1.375, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 738.65, "end": 739.09, "word": "اللي", "probability": 0.816162109375}, {"start": 739.09, "end": 739.71, "word": " هي", "probability": 0.92333984375}, {"start": 739.71, "end": 745.09, "word": " اللمّة", "probability": 0.683349609375}, {"start": 745.09, "end": 745.37, "word": " اللي", "probability": 0.950439453125}, {"start": 745.37, "end": 746.11, "word": " بعدها", "probability": 0.82763671875}, {"start": 746.11, "end": 747.01, "word": " اللي", "probability": 0.8427734375}, {"start": 747.01, "end": 747.25, "word": " هي", "probability": 0.38623046875}, {"start": 747.25, "end": 748.97, "word": " المشهورة", "probability": 0.98916015625}], "temperature": 1.0}, {"id": 36, "seek": 77831, "start": 750.53, "end": 778.31, "text": "وهذه test for divergence في الواقع كنا نقول عنها في calculus باه ايش ال sequence هذه او اللي هي sequence of partial sums او ال series بقول إذا كانت ال series summation xn converges in R إذا كان اللي هي ال summation لل xn converges in R لازم يكون limit اللي هي ال sequence الأصلي اللي بنكون منها sequence of partial sums لازم limit هيش يساوي", "tokens": [2407, 3224, 24192, 1500, 337, 47387, 8978, 2423, 14407, 4587, 3615, 9122, 8315, 8717, 39648, 18871, 11296, 8978, 33400, 4724, 40294, 1975, 1829, 8592, 2423, 8310, 29538, 1975, 2407, 13672, 1829, 39896, 8310, 295, 14641, 34499, 1975, 2407, 2423, 2638, 4724, 39648, 11933, 15730, 25961, 2655, 2423, 2638, 28811, 2031, 77, 9652, 2880, 294, 497, 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summation", "probability": 0.71875}, {"start": 763.63, "end": 764.17, "word": " xn", "probability": 0.4024658203125}, {"start": 764.17, "end": 764.77, "word": " converges", "probability": 0.6673583984375}, {"start": 764.77, "end": 764.95, "word": " in", "probability": 0.853515625}, {"start": 764.95, "end": 765.33, "word": " R", "probability": 0.7802734375}, {"start": 765.33, "end": 766.15, "word": " إذا", "probability": 0.88134765625}, {"start": 766.15, "end": 766.49, "word": " كان", "probability": 0.9912109375}, {"start": 766.49, "end": 766.85, "word": " اللي", "probability": 0.861328125}, {"start": 766.85, "end": 767.25, "word": " هي", "probability": 0.95068359375}, {"start": 767.25, "end": 767.61, "word": " ال", "probability": 0.78125}, {"start": 767.61, "end": 768.05, "word": " summation", "probability": 0.953125}, {"start": 768.05, "end": 768.31, "word": " لل", "probability": 0.5009765625}, {"start": 768.31, "end": 768.65, "word": " xn", "probability": 0.7744140625}, {"start": 768.65, "end": 769.11, "word": " converges", "probability": 0.85400390625}, {"start": 769.11, "end": 769.27, "word": " in", "probability": 0.92724609375}, {"start": 769.27, "end": 769.61, "word": " R", "probability": 0.9853515625}, {"start": 769.61, "end": 770.65, "word": " لازم", "probability": 0.9861653645833334}, {"start": 770.65, "end": 771.03, "word": " يكون", "probability": 0.96923828125}, {"start": 771.03, "end": 771.53, "word": " limit", "probability": 0.67041015625}, {"start": 771.53, "end": 771.91, "word": " اللي", "probability": 0.757568359375}, {"start": 771.91, "end": 772.31, "word": " هي", "probability": 0.97705078125}, {"start": 772.31, "end": 772.85, "word": " ال", "probability": 0.8896484375}, {"start": 772.85, "end": 773.21, "word": " sequence", "probability": 0.98681640625}, {"start": 773.21, "end": 773.69, "word": " الأصلي", "probability": 0.74560546875}, {"start": 773.69, "end": 773.89, "word": " اللي", "probability": 0.858154296875}, {"start": 773.89, "end": 774.81, "word": " بنكون", "probability": 0.85107421875}, {"start": 774.81, "end": 775.17, "word": " منها", "probability": 0.985595703125}, {"start": 775.17, "end": 775.49, "word": " sequence", "probability": 0.80712890625}, {"start": 775.49, "end": 775.69, "word": " of", "probability": 0.7841796875}, {"start": 775.69, "end": 775.91, "word": " partial", "probability": 0.94580078125}, {"start": 775.91, "end": 776.35, "word": " sums", "probability": 0.9296875}, {"start": 776.35, "end": 777.25, "word": " لازم", "probability": 0.9817708333333334}, {"start": 777.25, "end": 777.51, "word": " limit", "probability": 0.489990234375}, {"start": 777.51, "end": 777.83, "word": " هيش", "probability": 0.3597412109375}, {"start": 777.83, "end": 778.31, "word": " يساوي", "probability": 0.9276123046875}], "temperature": 1.0}, {"id": 37, "seek": 80416, "start": 779.64, "end": 804.16, "text": "يساوي سفر وبناء عليه لو كانت ال limit لها تساوي سفر على طول بنحكم على summation لل Xn إيش ماله diverse وكنا نقول هذا test of divergence مش test of convergence لأن ممكن ال limit تساوي سفر لكن ال summation لل Xn يكون diverse والمثال اللي انتوا مشهور اللي هو summation 1 على n هذا اللي هو limit", "tokens": [1829, 3794, 995, 45865, 8608, 5172, 2288, 46599, 1863, 16606, 47356, 45164, 25961, 2655, 2423, 4948, 5296, 11296, 6055, 3794, 995, 45865, 8608, 5172, 2288, 15844, 23032, 12610, 44945, 5016, 24793, 15844, 28811, 24976, 1783, 77, 11933, 1829, 8592, 3714, 6027, 3224, 9521, 4032, 4117, 8315, 8717, 39648, 23758, 1500, 295, 47387, 37893, 1500, 295, 32181, 5296, 33456, 3714, 43020, 2423, 4948, 6055, 3794, 995, 45865, 8608, 5172, 2288, 44381, 2423, 28811, 24976, 1783, 77, 7251, 30544, 9521, 4032, 45340, 12984, 6027, 13672, 1829, 16472, 2655, 14407, 37893, 3224, 13063, 13672, 1829, 31439, 28811, 502, 15844, 297, 23758, 13672, 1829, 31439, 4948], "avg_logprob": -0.28686285481869594, "compression_ratio": 1.9026548672566372, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 779.64, "end": 780.2, "word": "يساوي", "probability": 0.62139892578125}, {"start": 780.2, "end": 780.56, "word": " سفر", "probability": 0.6936848958333334}, {"start": 780.56, "end": 781.46, "word": " وبناء", "probability": 0.74267578125}, {"start": 781.46, "end": 781.86, "word": " عليه", "probability": 0.85107421875}, {"start": 781.86, "end": 782.64, "word": " لو", "probability": 0.81689453125}, {"start": 782.64, "end": 783.08, "word": " كانت", "probability": 0.9560546875}, {"start": 783.08, "end": 783.16, "word": " ال", "probability": 0.49560546875}, {"start": 783.16, "end": 783.38, "word": " limit", "probability": 0.8056640625}, {"start": 783.38, "end": 783.62, "word": " لها", "probability": 0.666259765625}, {"start": 783.62, "end": 784.0, "word": " تساوي", "probability": 0.9605712890625}, {"start": 784.0, "end": 784.44, "word": " سفر", "probability": 0.9231770833333334}, {"start": 784.44, "end": 785.02, "word": " على", "probability": 0.7431640625}, {"start": 785.02, "end": 785.3, "word": " طول", 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"probability": 0.9130859375}, {"start": 790.56, "end": 790.9, "word": " of", "probability": 0.970703125}, {"start": 790.9, "end": 791.68, "word": " divergence", "probability": 0.96044921875}, {"start": 791.68, "end": 792.24, "word": " مش", "probability": 0.802734375}, {"start": 792.24, "end": 792.58, "word": " test", "probability": 0.60107421875}, {"start": 792.58, "end": 792.76, "word": " of", "probability": 0.966796875}, {"start": 792.76, "end": 793.42, "word": " convergence", "probability": 0.96142578125}, {"start": 793.42, "end": 794.34, "word": " لأن", "probability": 0.80859375}, {"start": 794.34, "end": 794.72, "word": " ممكن", "probability": 0.8173828125}, {"start": 794.72, "end": 794.88, "word": " ال", "probability": 0.437744140625}, {"start": 794.88, "end": 795.06, "word": " limit", "probability": 0.96728515625}, {"start": 795.06, "end": 795.5, "word": " تساوي", "probability": 0.98046875}, {"start": 795.5, "end": 795.84, "word": " سفر", "probability": 0.97216796875}, {"start": 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" بشغلة", "probability": 0.7896728515625}], "temperature": 1.0}, {"id": 44, "seek": 96076, "start": 943.36, "end": 960.76, "text": "إنه إحنا في الـ monotone convergence theorem في تحليل حقيقي واحد كنا نقول إنه إذا كانت ال sequence increasing أو decreasing يعني لو كانت monotone then the sequence converge if and only if it is bounded", "tokens": [28814, 1863, 3224, 11933, 5016, 8315, 8978, 2423, 39184, 1108, 310, 546, 32181, 20904, 8978, 6055, 5016, 20292, 1211, 11331, 38436, 38436, 36764, 24401, 9122, 8315, 8717, 39648, 36145, 3224, 11933, 15730, 25961, 2655, 2423, 8310, 5662, 34051, 23223, 37495, 22653, 45164, 25961, 2655, 1108, 310, 546, 550, 264, 8310, 41881, 498, 293, 787, 498, 309, 307, 37498], "avg_logprob": -0.1818061369960591, "compression_ratio": 1.5028901734104045, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 943.36, "end": 943.96, "word": "إنه", "probability": 0.536865234375}, {"start": 943.96, "end": 944.44, "word": " إحنا", "probability": 0.7792154947916666}, {"start": 944.44, "end": 944.82, "word": " في", "probability": 0.96533203125}, {"start": 944.82, "end": 945.24, "word": " الـ", "probability": 0.896728515625}, {"start": 945.24, "end": 945.56, "word": " monotone", "probability": 0.7601725260416666}, {"start": 945.56, "end": 946.16, "word": " convergence", "probability": 0.96337890625}, {"start": 946.16, "end": 946.8, "word": " theorem", "probability": 0.86376953125}, {"start": 946.8, "end": 947.48, "word": " في", "probability": 0.72607421875}, {"start": 947.48, "end": 949.74, "word": " تحليل", "probability": 0.9193115234375}, {"start": 949.74, "end": 950.2, "word": " حقيقي", "probability": 0.9759114583333334}, {"start": 950.2, "end": 950.58, "word": " واحد", "probability": 0.98046875}, {"start": 950.58, "end": 950.8, "word": " كنا", "probability": 0.91064453125}, {"start": 950.8, "end": 951.3, "word": " نقول", "probability": 0.967529296875}, {"start": 951.3, "end": 952.04, "word": " إنه", "probability": 0.736572265625}, {"start": 952.04, "end": 952.14, "word": " إذا", "probability": 0.94970703125}, {"start": 952.14, "end": 952.5, "word": " كانت", "probability": 0.99365234375}, {"start": 952.5, "end": 952.6, "word": " ال", "probability": 0.9111328125}, {"start": 952.6, "end": 952.9, "word": " sequence", "probability": 0.66943359375}, {"start": 952.9, "end": 953.58, "word": " increasing", "probability": 0.98095703125}, {"start": 953.58, "end": 953.84, "word": " أو", "probability": 0.88134765625}, {"start": 953.84, "end": 954.32, "word": " decreasing", "probability": 0.9814453125}, {"start": 954.32, "end": 954.62, "word": " يعني", "probability": 0.80126953125}, {"start": 954.62, "end": 954.78, "word": " لو", "probability": 0.9775390625}, {"start": 954.78, "end": 955.14, "word": " كانت", "probability": 0.9921875}, {"start": 955.14, "end": 955.9, "word": " monotone", "probability": 0.8047688802083334}, {"start": 955.9, "end": 956.62, "word": " then", "probability": 0.6064453125}, {"start": 956.62, "end": 957.44, "word": " the", "probability": 0.66552734375}, {"start": 957.44, "end": 957.92, "word": " sequence", "probability": 0.97705078125}, {"start": 957.92, "end": 958.62, "word": " converge", "probability": 0.497314453125}, {"start": 958.62, "end": 958.88, "word": " if", "probability": 0.931640625}, {"start": 958.88, "end": 959.16, "word": " and", "probability": 0.96044921875}, {"start": 959.16, "end": 959.44, "word": " only", "probability": 0.81787109375}, {"start": 959.44, "end": 959.78, "word": " if", "probability": 0.9677734375}, {"start": 959.78, "end": 960.0, "word": " it", "probability": 0.96630859375}, {"start": 960.0, "end": 960.3, "word": " is", "probability": 0.89697265625}, {"start": 960.3, "end": 960.76, "word": " bounded", "probability": 0.939453125}], "temperature": 1.0}, {"id": 45, "seek": 98522, "start": 962.06, "end": 985.22, "text": "هذه اللي هو في حالة مين الـ monotone functions لكن in general كنا نقول إن الـ sequence is convergent على طول اللي بيعطينا then it is bounded but it may be bounded بط إي شمالها is not convergent وكنا نقول ناخد مثال ناقص واحد على أن ال sequence هذه أو ناقص واحد أس أن ناقص واحد أس أن", "tokens": [3224, 24192, 13672, 1829, 31439, 8978, 11331, 6027, 3660, 3714, 9957, 2423, 39184, 1108, 310, 546, 6828, 44381, 294, 2674, 9122, 8315, 8717, 39648, 36145, 2423, 39184, 8310, 307, 9652, 6930, 15844, 23032, 12610, 13672, 1829, 4724, 40228, 9566, 1829, 8315, 550, 309, 307, 37498, 457, 309, 815, 312, 37498, 4724, 9566, 11933, 1829, 13412, 2304, 6027, 11296, 307, 406, 9652, 6930, 4032, 4117, 8315, 8717, 39648, 8717, 47283, 3215, 50113, 6027, 8717, 995, 4587, 9381, 36764, 24401, 15844, 14739, 2423, 8310, 29538, 34051, 8717, 995, 4587, 9381, 36764, 24401, 5551, 3794, 14739, 8717, 995, 4587, 9381, 36764, 24401, 5551, 3794, 14739], "avg_logprob": -0.27017598013276034, "compression_ratio": 1.7797356828193833, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 962.06, "end": 962.42, "word": "هذه", "probability": 0.65673828125}, {"start": 962.42, "end": 962.52, "word": " اللي", "probability": 0.57135009765625}, {"start": 962.52, "end": 962.7, "word": " هو", "probability": 0.316162109375}, {"start": 962.7, "end": 962.84, "word": " في", "probability": 0.72509765625}, {"start": 962.84, "end": 963.2, "word": " حالة", "probability": 0.9930013020833334}, {"start": 963.2, "end": 963.48, "word": " مين", "probability": 0.454833984375}, {"start": 963.48, "end": 963.72, "word": " الـ", "probability": 0.87548828125}, {"start": 963.72, "end": 964.18, "word": " monotone", "probability": 0.8590494791666666}, {"start": 964.18, "end": 964.88, "word": " functions", "probability": 0.74462890625}, {"start": 964.88, "end": 965.64, "word": " لكن", "probability": 0.6572265625}, {"start": 965.64, "end": 965.9, "word": " in", "probability": 0.45556640625}, {"start": 965.9, "end": 966.34, "word": " general", "probability": 0.93896484375}, {"start": 966.34, "end": 966.68, "word": " كنا", "probability": 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970.2, "end": 970.36, "word": " is", "probability": 0.90234375}, {"start": 970.36, "end": 970.78, "word": " bounded", "probability": 0.91552734375}, {"start": 970.78, "end": 971.54, "word": " but", "probability": 0.7275390625}, {"start": 971.54, "end": 972.2, "word": " it", "probability": 0.95654296875}, {"start": 972.2, "end": 972.46, "word": " may", "probability": 0.8330078125}, {"start": 972.46, "end": 972.82, "word": " be", "probability": 0.96435546875}, {"start": 972.82, "end": 973.46, "word": " bounded", "probability": 0.91259765625}, {"start": 973.46, "end": 973.88, "word": " بط", "probability": 0.48828125}, {"start": 973.88, "end": 974.14, "word": " إي", "probability": 0.6739501953125}, {"start": 974.14, "end": 974.76, "word": " شمالها", "probability": 0.874267578125}, {"start": 974.76, "end": 975.52, "word": " is", "probability": 0.826171875}, {"start": 975.52, "end": 975.82, "word": " not", "probability": 0.95361328125}, {"start": 975.82, "end": 976.68, "word": " convergent", "probability": 0.96923828125}, {"start": 976.68, "end": 977.52, "word": " وكنا", "probability": 0.8274739583333334}, {"start": 977.52, "end": 978.18, "word": " نقول", "probability": 0.71337890625}, {"start": 978.18, "end": 978.62, "word": " ناخد", "probability": 0.9597981770833334}, {"start": 978.62, "end": 979.04, "word": " مثال", "probability": 0.908203125}, {"start": 979.04, "end": 979.5, "word": " ناقص", "probability": 0.83056640625}, {"start": 979.5, "end": 980.08, "word": " واحد", "probability": 0.92822265625}, {"start": 980.08, "end": 980.7, "word": " على", "probability": 0.8896484375}, {"start": 980.7, "end": 980.98, "word": " أن", "probability": 0.154541015625}, {"start": 980.98, "end": 981.14, "word": " ال", "probability": 0.8310546875}, {"start": 981.14, "end": 981.58, "word": " sequence", "probability": 0.814453125}, {"start": 981.58, "end": 982.0, "word": " هذه", "probability": 0.70751953125}, {"start": 982.0, "end": 982.58, "word": " أو", "probability": 0.85009765625}, {"start": 982.58, "end": 982.98, "word": " ناقص", "probability": 0.9827880859375}, {"start": 982.98, "end": 983.3, "word": " واحد", "probability": 0.9794921875}, {"start": 983.3, "end": 983.56, "word": " أس", "probability": 0.642578125}, {"start": 983.56, "end": 983.82, "word": " أن", "probability": 0.7392578125}, {"start": 983.82, "end": 984.42, "word": " ناقص", "probability": 0.9093017578125}, {"start": 984.42, "end": 984.72, "word": " واحد", "probability": 0.98486328125}, {"start": 984.72, "end": 984.98, "word": " أس", "probability": 0.970458984375}, {"start": 984.98, "end": 985.22, "word": " أن", "probability": 0.94482421875}], "temperature": 1.0}, {"id": 46, "seek": 101420, "start": 985.8, "end": 1014.2, "text": "نقص واحد وثاني هذه الـ sequence bounded لأن قيمتها الـ absolute value لها واحد but اللي هو ال limit اللي لها does not exist لكن في حالة الـ monotone function لأ ال boundedness يكافئ ال convergence لأن هذا بدنا اللي هو الآن نطبق على ال monotone convergence theorem أو نستخدمها في إثبات نظريتنا اللي هي 914 theorem لت XN be a sequence of non-negative real numbers", "tokens": [1863, 4587, 9381, 36764, 24401, 4032, 12984, 7649, 1829, 29538, 2423, 39184, 8310, 37498, 5296, 33456, 12174, 32640, 2655, 11296, 2423, 39184, 8236, 2158, 5296, 11296, 36764, 24401, 457, 13672, 1829, 31439, 2423, 4948, 13672, 1829, 5296, 11296, 775, 406, 2514, 44381, 8978, 11331, 6027, 3660, 2423, 39184, 1108, 310, 546, 2445, 5296, 10721, 2423, 37498, 1287, 7251, 4117, 31845, 19986, 2423, 32181, 5296, 33456, 23758, 47525, 8315, 13672, 1829, 31439, 6024, 48506, 8717, 9566, 3555, 4587, 15844, 2423, 1108, 310, 546, 32181, 20904, 34051, 8717, 14851, 9778, 40448, 11296, 8978, 11933, 12984, 3555, 9307, 8717, 19913, 16572, 2655, 8315, 13672, 1829, 39896, 1722, 7271, 20904, 5296, 2655, 1783, 45, 312, 257, 8310, 295, 2107, 12, 28561, 1166, 957, 3547], "avg_logprob": -0.289514449017107, "compression_ratio": 1.679054054054054, "no_speech_prob": 0.0, "words": [{"start": 985.8, "end": 986.2, "word": "نقص", "probability": 0.5895792643229166}, {"start": 986.2, "end": 986.44, "word": " واحد", "probability": 0.9111328125}, {"start": 986.44, "end": 986.74, "word": " وثاني", "probability": 0.5294189453125}, {"start": 986.74, "end": 986.98, "word": " هذه", "probability": 0.283203125}, {"start": 986.98, "end": 987.08, "word": " الـ", "probability": 0.554443359375}, {"start": 987.08, "end": 987.5, "word": " sequence", "probability": 0.54248046875}, {"start": 987.5, "end": 988.0, "word": " bounded", "probability": 0.31591796875}, {"start": 988.0, "end": 988.32, "word": " لأن", "probability": 0.881591796875}, {"start": 988.32, "end": 989.06, "word": " قيمتها", "probability": 0.8603515625}, {"start": 989.06, "end": 989.52, "word": " الـ", "probability": 0.6865234375}, {"start": 989.52, "end": 989.7, "word": " absolute", "probability": 0.84326171875}, {"start": 989.7, "end": 990.14, "word": " value", "probability": 0.982421875}, {"start": 990.14, "end": 990.34, "word": " لها", "probability": 0.830078125}, {"start": 990.34, "end": 990.82, "word": " واحد", "probability": 0.970458984375}, {"start": 990.82, "end": 991.68, "word": " but", "probability": 0.287109375}, {"start": 991.68, "end": 992.02, "word": " اللي", "probability": 0.951904296875}, {"start": 992.02, "end": 992.3, "word": " هو", "probability": 0.94140625}, {"start": 992.3, "end": 992.5, "word": " ال", "probability": 0.6474609375}, {"start": 992.5, "end": 992.76, "word": " limit", "probability": 0.6533203125}, {"start": 992.76, "end": 992.94, "word": " اللي", "probability": 0.719970703125}, {"start": 992.94, "end": 993.04, "word": " لها", "probability": 0.735595703125}, {"start": 993.04, "end": 993.24, "word": " does", "probability": 0.6220703125}, {"start": 993.24, "end": 993.58, "word": " not", "probability": 0.9619140625}, {"start": 993.58, "end": 994.02, "word": " exist", "probability": 0.96484375}, {"start": 994.02, "end": 994.38, "word": " لكن", "probability": 0.84033203125}, {"start": 994.38, "end": 994.6, "word": " في", "probability": 0.97216796875}, {"start": 994.6, "end": 994.96, "word": " حالة", "probability": 0.764404296875}, {"start": 994.96, "end": 995.16, "word": " الـ", "probability": 0.640869140625}, {"start": 995.16, "end": 995.5, "word": " monotone", "probability": 0.9274088541666666}, {"start": 995.5, "end": 996.0, "word": " function", "probability": 0.96630859375}, {"start": 996.0, "end": 996.48, "word": " لأ", "probability": 0.798583984375}, {"start": 996.48, "end": 997.02, "word": " ال", "probability": 0.73828125}, {"start": 997.02, "end": 997.88, "word": " boundedness", "probability": 0.8046875}, {"start": 997.88, "end": 998.54, "word": " يكافئ", "probability": 0.9241943359375}, {"start": 998.54, "end": 998.66, "word": " ال", "probability": 0.884765625}, {"start": 998.66, "end": 999.24, "word": " convergence", "probability": 0.96728515625}, {"start": 999.24, "end": 1000.4, "word": " لأن", "probability": 0.87158203125}, {"start": 1000.4, "end": 1000.76, "word": " هذا", "probability": 0.97216796875}, {"start": 1000.76, "end": 1001.24, "word": " بدنا", "probability": 0.855224609375}, {"start": 1001.24, "end": 1001.44, "word": " اللي", "probability": 0.95751953125}, {"start": 1001.44, "end": 1001.64, "word": " هو", "probability": 0.99365234375}, {"start": 1001.64, "end": 1002.1, "word": " الآن", "probability": 0.758056640625}, {"start": 1002.1, "end": 1003.1, "word": " نطبق", "probability": 0.8916015625}, {"start": 1003.1, "end": 1003.34, "word": " على", "probability": 0.708984375}, {"start": 1003.34, "end": 1003.52, "word": " ال", "probability": 0.955078125}, {"start": 1003.52, "end": 1003.88, "word": " monotone", "probability": 0.8041178385416666}, {"start": 1003.88, "end": 1004.34, "word": " convergence", "probability": 0.962890625}, {"start": 1004.34, "end": 1004.94, "word": " theorem", "probability": 0.92578125}, {"start": 1004.94, "end": 1005.56, "word": " أو", "probability": 0.8349609375}, {"start": 1005.56, "end": 1006.48, "word": " نستخدمها", "probability": 0.99326171875}, {"start": 1006.48, "end": 1007.14, "word": " في", "probability": 0.9677734375}, {"start": 1007.14, "end": 1007.82, "word": " إثبات", "probability": 0.87939453125}, {"start": 1007.82, "end": 1008.76, "word": " نظريتنا", "probability": 0.98271484375}, {"start": 1008.76, "end": 1009.24, "word": " اللي", "probability": 0.58514404296875}, {"start": 1009.24, "end": 1009.36, "word": " هي", "probability": 0.373046875}, {"start": 1009.36, "end": 1009.98, "word": " 914", "probability": 0.4560546875}, {"start": 1009.98, "end": 1010.6, "word": " theorem", "probability": 0.7646484375}, {"start": 1010.6, "end": 1010.98, "word": " لت", "probability": 0.729736328125}, {"start": 1010.98, "end": 1011.48, "word": " XN", "probability": 0.305908203125}, {"start": 1011.48, "end": 1011.7, "word": " be", "probability": 0.46240234375}, {"start": 1011.7, "end": 1011.9, "word": " a", "probability": 0.9677734375}, {"start": 1011.9, "end": 1012.42, "word": " sequence", "probability": 0.9794921875}, {"start": 1012.42, "end": 1012.7, "word": " of", "probability": 0.9755859375}, {"start": 1012.7, "end": 1012.98, "word": " non", "probability": 0.93896484375}, {"start": 1012.98, "end": 1013.44, "word": "-negative", "probability": 0.7276204427083334}, {"start": 1013.44, "end": 1013.72, "word": " real", "probability": 0.95654296875}, {"start": 1013.72, "end": 1014.2, "word": " numbers", "probability": 0.89990234375}], "temperature": 1.0}, {"id": 47, "seek": 102482, "start": 1014.56, "end": 1024.82, "text": "إذا حكينا عن مين عن sequence xn عنصرها إيش ما لها non negative real numbers يعني أكبر أو تساوي سفر", "tokens": [28814, 15730, 11331, 4117, 1829, 8315, 18871, 3714, 9957, 18871, 8310, 2031, 77, 18871, 9381, 2288, 11296, 11933, 1829, 8592, 19446, 5296, 11296, 2107, 3671, 957, 3547, 37495, 22653, 5551, 4117, 26890, 34051, 6055, 3794, 995, 45865, 8608, 5172, 2288], "avg_logprob": -0.3243140185751566, "compression_ratio": 1.1885245901639345, "no_speech_prob": 0.0, "words": [{"start": 1014.56, "end": 1014.86, "word": "إذا", "probability": 0.55902099609375}, {"start": 1014.86, "end": 1015.42, "word": " حكينا", "probability": 0.7396240234375}, {"start": 1015.42, "end": 1015.64, "word": " عن", "probability": 0.943359375}, {"start": 1015.64, "end": 1016.16, "word": " مين", "probability": 0.5765380859375}, {"start": 1016.16, "end": 1016.92, "word": " عن", "probability": 0.5341796875}, {"start": 1016.92, "end": 1017.5, "word": " sequence", "probability": 0.4716796875}, {"start": 1017.5, "end": 1018.32, "word": " xn", "probability": 0.359619140625}, {"start": 1018.32, "end": 1019.02, "word": " عنصرها", "probability": 0.73077392578125}, {"start": 1019.02, "end": 1019.28, "word": " إيش", "probability": 0.7537434895833334}, {"start": 1019.28, "end": 1019.46, "word": " ما", "probability": 0.8232421875}, {"start": 1019.46, "end": 1019.84, "word": " لها", "probability": 0.880859375}, {"start": 1019.84, "end": 1020.6, "word": " non", 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case ال summation لـ xn بسوى limit لSk بسوى سبريمان لSk يعني الآن و كأنه بيقوللي إنه limit", "tokens": [12984, 2304, 2423, 39184, 28811, 5296, 39184, 2031, 77, 9154, 21136, 13546, 1829, 11622, 6055, 3555, 34268, 11296, 38874, 498, 293, 787, 498, 264, 8310, 318, 39894, 3794, 2407, 7578, 7324, 295, 14641, 34499, 307, 37498, 37495, 22653, 6024, 48506, 21136, 13546, 1829, 11622, 2423, 28811, 2031, 77, 13672, 1829, 39896, 8032, 2655, 30544, 38874, 11933, 15730, 4032, 6156, 47432, 11933, 15730, 2423, 7324, 11933, 8592, 2304, 6027, 11296, 307, 37498, 294, 341, 1389, 2423, 28811, 5296, 39184, 2031, 77, 4724, 3794, 2407, 7578, 4948, 5296, 50, 74, 4724, 3794, 2407, 7578, 8608, 3555, 16572, 2304, 7649, 5296, 50, 74, 37495, 22653, 6024, 48506, 4032, 9122, 33456, 3224, 4724, 1829, 39648, 20292, 36145, 3224, 4948], "avg_logprob": -0.3981681001083604, "compression_ratio": 1.7469387755102042, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 1026.49, "end": 1026.99, "word": "ثم", "probability": 0.57550048828125}, {"start": 1026.99, "end": 1027.33, "word": " الـ", "probability": 0.30133056640625}, {"start": 1027.33, "end": 1027.77, "word": " summation", "probability": 0.568359375}, {"start": 1027.77, "end": 1028.07, "word": " لـ", "probability": 0.6090087890625}, {"start": 1028.07, "end": 1028.39, "word": " xn", "probability": 0.31298828125}, {"start": 1028.39, "end": 1028.63, "word": " من", "probability": 0.294189453125}, {"start": 1028.63, "end": 1028.89, "word": " السيريز", "probability": 0.74530029296875}, {"start": 1028.89, "end": 1029.69, "word": " تبعتها", "probability": 0.687408447265625}, {"start": 1029.69, "end": 1030.63, "word": " converts", "probability": 0.346923828125}, {"start": 1030.63, "end": 1031.17, "word": " if", "probability": 0.85595703125}, {"start": 1031.17, "end": 1031.43, "word": " and", "probability": 0.90869140625}, {"start": 1031.43, "end": 1031.69, "word": " only", "probability": 0.818359375}, {"start": 1031.69, "end": 1031.89, "word": " if", "probability": 0.9677734375}, {"start": 1031.89, "end": 1032.05, "word": " the", "probability": 0.2841796875}, {"start": 1032.05, "end": 1032.43, "word": " sequence", "probability": 0.9638671875}, {"start": 1032.43, "end": 1032.65, "word": " S", "probability": 0.6337890625}, {"start": 1032.65, "end": 1033.05, "word": " بتسوى", "probability": 0.4737548828125}, {"start": 1033.05, "end": 1033.45, "word": " Sk", "probability": 0.6220703125}, {"start": 1033.45, "end": 1034.13, "word": " of", "probability": 0.8837890625}, {"start": 1034.13, "end": 1034.51, "word": " partial", "probability": 0.9501953125}, {"start": 1034.51, "end": 1034.93, "word": " sums", "probability": 0.9580078125}, {"start": 1034.93, "end": 1035.23, "word": " is", "probability": 0.939453125}, {"start": 1035.23, "end": 1035.75, "word": " bounded", "probability": 0.939453125}, {"start": 1035.75, "end": 1036.75, "word": " يعني", "probability": 0.89501953125}, {"start": 1036.75, "end": 1037.23, "word": " الآن", "probability": 0.822021484375}, {"start": 1037.23, "end": 1038.27, "word": " السيريز", "probability": 0.869140625}, {"start": 1038.27, "end": 1038.49, "word": " ال", "probability": 0.257568359375}, {"start": 1038.49, "end": 1038.83, "word": " summation", "probability": 0.51953125}, {"start": 1038.83, "end": 1039.63, "word": " xn", "probability": 0.88916015625}, {"start": 1039.63, "end": 1040.21, "word": " اللي", "probability": 0.864501953125}, {"start": 1040.21, "end": 1040.53, "word": " هي", "probability": 0.86083984375}, {"start": 1040.53, "end": 1041.11, "word": " هتكون", "probability": 0.9596354166666666}, {"start": 1041.11, "end": 1041.69, "word": " converts", "probability": 0.6591796875}, {"start": 1041.69, "end": 1041.95, "word": " إذا", "probability": 0.9609375}, {"start": 1041.95, "end": 1042.09, "word": " و", "probability": 0.99267578125}, {"start": 1042.09, "end": 1042.43, "word": " فقط", "probability": 0.772705078125}, {"start": 1042.43, "end": 1042.81, "word": " إذا", "probability": 0.99072265625}, {"start": 1042.81, "end": 1043.27, "word": " ال", "probability": 0.818359375}, {"start": 1043.27, "end": 1043.63, "word": " Sk", "probability": 0.45263671875}, {"start": 1043.63, "end": 1044.43, "word": " إشمالها", "probability": 0.8212890625}, {"start": 1044.43, "end": 1045.07, "word": " is", "probability": 0.91357421875}, {"start": 1045.07, "end": 1045.69, "word": " bounded", "probability": 0.93017578125}, {"start": 1045.69, "end": 1046.47, "word": " in", "probability": 0.787109375}, {"start": 1046.47, "end": 1046.81, "word": " this", "probability": 0.9658203125}, {"start": 1046.81, "end": 1047.43, "word": " case", "probability": 0.9306640625}, {"start": 1047.43, "end": 1047.95, "word": " ال", "probability": 0.42529296875}, {"start": 1047.95, "end": 1048.41, "word": " summation", "probability": 0.9189453125}, {"start": 1048.41, "end": 1048.69, "word": " لـ", "probability": 0.505859375}, {"start": 1048.69, "end": 1048.97, "word": " xn", "probability": 0.855224609375}, {"start": 1048.97, "end": 1049.29, "word": " بسوى", "probability": 0.655029296875}, {"start": 1049.29, "end": 1049.51, "word": " limit", "probability": 0.81103515625}, {"start": 1049.51, "end": 1050.25, "word": " لSk", "probability": 0.700439453125}, {"start": 1050.25, "end": 1050.97, "word": " بسوى", "probability": 0.8037109375}, {"start": 1050.97, "end": 1051.73, "word": " سبريمان", "probability": 0.676171875}, {"start": 1051.73, "end": 1052.89, "word": " لSk", "probability": 0.606689453125}, {"start": 1052.89, "end": 1053.35, "word": " يعني", "probability": 0.969970703125}, {"start": 1053.35, "end": 1053.65, "word": " الآن", "probability": 0.6865234375}, {"start": 1053.65, "end": 1053.81, "word": " و", "probability": 0.4990234375}, {"start": 1053.81, "end": 1054.11, "word": " كأنه", "probability": 0.531494140625}, {"start": 1054.11, "end": 1054.71, "word": " بيقوللي", "probability": 0.71673583984375}, {"start": 1054.71, "end": 1055.11, "word": " إنه", "probability": 0.4486083984375}, {"start": 1055.11, "end": 1055.65, "word": " limit", "probability": 0.83056640625}], "temperature": 1.0}, {"id": 49, "seek": 107299, "start": 1056.63, "end": 1072.99, "text": "الـ SK exists if and only if SK is bounded هذه الـ SK هي الـ sequence of partial sums تبع السيريز limit الـ SK هي السيريز نفسها الـ x and n من واحد إلى مانع نهاية", "tokens": [6027, 39184, 21483, 2514, 82, 498, 293, 787, 498, 21483, 307, 37498, 29538, 2423, 39184, 21483, 39896, 2423, 39184, 8310, 295, 14641, 34499, 6055, 3555, 3615, 21136, 13546, 1829, 11622, 4948, 2423, 39184, 21483, 39896, 21136, 13546, 1829, 11622, 8717, 36178, 11296, 2423, 39184, 2031, 293, 297, 9154, 36764, 24401, 30731, 3714, 7649, 3615, 8717, 11296, 10632], "avg_logprob": -0.4692887895066163, "compression_ratio": 1.4, "no_speech_prob": 5.364418029785156e-07, "words": [{"start": 1056.63, "end": 1056.99, "word": "الـ", "probability": 0.71337890625}, {"start": 1056.99, "end": 1057.37, "word": " SK", "probability": 0.2391357421875}, {"start": 1057.37, "end": 1059.57, "word": " exists", "probability": 0.32232666015625}, {"start": 1059.57, "end": 1059.83, "word": " if", "probability": 0.8818359375}, {"start": 1059.83, "end": 1060.09, "word": " and", "probability": 0.8955078125}, {"start": 1060.09, "end": 1060.35, "word": " only", "probability": 0.92822265625}, {"start": 1060.35, "end": 1060.73, "word": " if", "probability": 0.9794921875}, {"start": 1060.73, "end": 1061.31, "word": " SK", "probability": 0.93798828125}, {"start": 1061.31, "end": 1062.75, "word": " is", "probability": 0.9296875}, {"start": 1062.75, "end": 1063.35, "word": " bounded", "probability": 0.93212890625}, {"start": 1063.35, "end": 1064.37, "word": " هذه", "probability": 0.232421875}, {"start": 1064.37, "end": 1064.81, "word": " الـ", "probability": 0.6343994140625}, {"start": 1064.81, "end": 1065.13, "word": " SK", "probability": 0.365478515625}, {"start": 1065.13, "end": 1065.49, "word": " هي", "probability": 0.6552734375}, {"start": 1065.49, "end": 1065.67, "word": " الـ", "probability": 0.5018310546875}, {"start": 1065.67, "end": 1065.89, "word": " sequence", "probability": 0.75390625}, {"start": 1065.89, "end": 1066.07, "word": " of", "probability": 0.31982421875}, {"start": 1066.07, "end": 1066.31, "word": " partial", "probability": 0.84765625}, {"start": 1066.31, "end": 1066.71, "word": " sums", "probability": 0.9677734375}, {"start": 1066.71, "end": 1067.03, "word": " تبع", "probability": 0.7205403645833334}, {"start": 1067.03, "end": 1067.75, "word": " السيريز", "probability": 0.786865234375}, {"start": 1067.75, "end": 1068.41, "word": " limit", "probability": 0.07867431640625}, {"start": 1068.41, "end": 1068.63, "word": " الـ", "probability": 0.4246826171875}, {"start": 1068.63, "end": 1068.93, "word": " SK", "probability": 0.9658203125}, {"start": 1068.93, "end": 1069.23, "word": " هي", "probability": 0.87060546875}, {"start": 1069.23, "end": 1069.73, "word": " السيريز", "probability": 0.9285888671875}, {"start": 1069.73, "end": 1070.23, "word": " نفسها", "probability": 0.9833984375}, {"start": 1070.23, "end": 1070.45, "word": " الـ", "probability": 0.5147705078125}, {"start": 1070.45, "end": 1070.63, "word": " x", "probability": 0.435546875}, {"start": 1070.63, "end": 1070.91, "word": " and", "probability": 0.076904296875}, {"start": 1070.91, "end": 1071.09, "word": " n", "probability": 0.74755859375}, {"start": 1071.09, "end": 1071.25, "word": " من", "probability": 0.93212890625}, {"start": 1071.25, "end": 1071.79, "word": " واحد", "probability": 0.662109375}, {"start": 1071.79, "end": 1072.09, "word": " إلى", "probability": 0.9091796875}, {"start": 1072.09, "end": 1072.53, "word": " مانع", "probability": 0.5465494791666666}, {"start": 1072.53, "end": 1072.99, "word": " نهاية", "probability": 0.8190104166666666}], "temperature": 1.0}, {"id": 50, "seek": 109369, "start": 1073.57, "end": 1093.69, "text": "الآن باختصار اللي هو هنقوله أنه أصلا هذه ال SQL sequence of partial sums is an increasing sequence مدام increasing يعني بمعنى آخر monotone أو بالظبط اللي هو خلّيني أقول واحدة من ال monotone هي increasing ت increasing هي increasing علنا الآن مدام increasing", "tokens": [6027, 148, 48506, 4724, 47283, 2655, 9381, 9640, 13672, 1829, 31439, 8032, 1863, 39648, 3224, 14739, 3224, 5551, 9381, 15040, 29538, 2423, 19200, 8310, 295, 14641, 34499, 307, 364, 5662, 8310, 3714, 3215, 10943, 5662, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 19753, 34740, 1108, 310, 546, 34051, 20666, 19913, 3555, 9566, 13672, 1829, 31439, 16490, 1211, 11703, 9957, 1829, 5551, 39648, 36764, 24401, 3660, 9154, 2423, 1108, 310, 546, 39896, 5662, 6055, 5662, 39896, 5662, 11203, 8315, 6024, 48506, 3714, 3215, 10943, 5662], "avg_logprob": -0.29577206443337833, "compression_ratio": 1.708133971291866, "no_speech_prob": 0.0, "words": [{"start": 1073.57, "end": 1074.05, "word": "الآن", "probability": 0.5706380208333334}, {"start": 1074.05, "end": 1074.89, "word": " باختصار", "probability": 0.941015625}, {"start": 1074.89, "end": 1075.41, "word": " اللي", "probability": 0.627197265625}, {"start": 1075.41, "end": 1075.55, "word": " هو", "probability": 0.5498046875}, {"start": 1075.55, "end": 1076.87, "word": " هنقوله", "probability": 0.713134765625}, {"start": 1076.87, "end": 1077.17, "word": " أنه", "probability": 0.662353515625}, {"start": 1077.17, "end": 1077.59, "word": " أصلا", "probability": 0.8981119791666666}, {"start": 1077.59, "end": 1078.11, "word": " هذه", "probability": 0.6103515625}, {"start": 1078.11, "end": 1078.73, "word": " ال", "probability": 0.767578125}, {"start": 1078.73, "end": 1079.15, "word": " SQL", "probability": 0.0755615234375}, {"start": 1079.15, "end": 1079.67, "word": " sequence", "probability": 0.5615234375}, {"start": 1079.67, "end": 1079.83, "word": " of", "probability": 0.88427734375}, {"start": 1079.83, "end": 1080.13, "word": " partial", "probability": 0.91845703125}, {"start": 1080.13, "end": 1080.57, "word": " sums", "probability": 0.95703125}, {"start": 1080.57, "end": 1080.91, "word": " is", "probability": 0.78466796875}, {"start": 1080.91, "end": 1081.21, "word": " an", "probability": 0.91650390625}, {"start": 1081.21, "end": 1081.81, "word": " increasing", "probability": 0.93408203125}, {"start": 1081.81, "end": 1082.65, "word": " sequence", "probability": 0.966796875}, {"start": 1082.65, "end": 1083.31, "word": " مدام", "probability": 0.7654622395833334}, {"start": 1083.31, "end": 1083.83, "word": " increasing", "probability": 0.837890625}, {"start": 1083.83, "end": 1084.55, "word": " يعني", "probability": 0.946044921875}, {"start": 1084.55, "end": 1084.91, "word": " بمعنى", "probability": 0.98056640625}, {"start": 1084.91, "end": 1085.25, "word": " آخر", "probability": 0.77783203125}, {"start": 1085.25, "end": 1085.95, "word": " monotone", "probability": 0.698486328125}, {"start": 1085.95, "end": 1086.81, "word": " أو", "probability": 0.853515625}, {"start": 1086.81, "end": 1087.43, 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"word": " increasing", "probability": 0.85498046875}, {"start": 1091.13, "end": 1091.35, "word": " هي", "probability": 0.8798828125}, {"start": 1091.35, "end": 1091.73, "word": " increasing", "probability": 0.990234375}, {"start": 1091.73, "end": 1092.09, "word": " علنا", "probability": 0.738037109375}, {"start": 1092.09, "end": 1092.53, "word": " الآن", "probability": 0.799072265625}, {"start": 1092.53, "end": 1093.13, "word": " مدام", "probability": 0.8893229166666666}, {"start": 1093.13, "end": 1093.69, "word": " increasing", "probability": 0.98681640625}], "temperature": 1.0}, {"id": 51, "seek": 111607, "start": 1094.51, "end": 1116.07, "text": "هيكون عنده الـ Existence limit يكافي الـ Boundedness ويكون خلصنا برهان النظرية حسب الـ Newton convergence theorem ومش هيك وهيكون limit هاش ما لمدام increasing السبريمام لعنصرها هذه الـ Newton convergence theorem خليني نشوف هذا الكلام بالتفصيل اللي حكيته", "tokens": [3224, 1829, 30544, 43242, 3224, 2423, 39184, 2111, 468, 655, 4948, 7251, 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exists k element in N such that for every n و m أكبر من k يكون الـ absolute value للـ xn ناقص xm أصغر من إبسلول نحن نجيب اللي هو Cauchy criterion مشابه إليها بس لمين لحالة السيريز ما هي هذه السيريز زي ما قلنا هي عبارة عن في النهاية sequence of partial sums", "tokens": [5172, 4117, 8315, 8717, 39648, 36145, 2423, 39184, 2031, 77, 50113, 15040, 14111, 307, 257, 7544, 625, 88, 8310, 498, 293, 787, 498, 337, 633, 288, 5551, 4117, 26890, 9154, 1958, 456, 8198, 350, 4478, 294, 426, 1270, 300, 337, 633, 297, 4032, 275, 5551, 4117, 26890, 9154, 350, 7251, 30544, 2423, 39184, 8236, 2158, 24976, 39184, 2031, 77, 8717, 995, 4587, 9381, 2031, 76, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 12610, 8717, 5016, 1863, 8717, 7435, 1829, 3555, 13672, 1829, 31439, 7544, 625, 88, 46691, 37893, 16758, 3224, 11933, 20292, 11296, 4724, 3794, 32767, 9957, 5296, 5016, 6027, 3660, 21136, 13546, 1829, 11622, 19446, 39896, 29538, 21136, 13546, 1829, 11622, 30767, 1829, 19446, 12174, 1211, 8315, 39896, 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1266.97, "end": 1267.51, "word": " السيريز", "probability": 0.75701904296875}, {"start": 1267.51, "end": 1267.61, "word": " ما", "probability": 0.480712890625}, {"start": 1267.61, "end": 1267.75, "word": " هي", "probability": 0.95849609375}, {"start": 1267.75, "end": 1267.95, "word": " هذه", "probability": 0.320556640625}, {"start": 1267.95, "end": 1268.25, "word": " السيريز", "probability": 0.9329833984375}, {"start": 1268.25, "end": 1268.41, "word": " زي", "probability": 0.875244140625}, {"start": 1268.41, "end": 1268.47, "word": " ما", "probability": 0.99169921875}, {"start": 1268.47, "end": 1268.85, "word": " قلنا", "probability": 0.9641927083333334}, {"start": 1268.85, "end": 1269.39, "word": " هي", "probability": 0.74853515625}, {"start": 1269.39, "end": 1269.93, "word": " عبارة", "probability": 0.9970703125}, {"start": 1269.93, "end": 1270.27, "word": " عن", "probability": 0.9970703125}, {"start": 1270.27, "end": 1270.51, "word": " في", "probability": 0.89404296875}, {"start": 1270.51, "end": 1271.17, "word": " النهاية", "probability": 0.9357096354166666}, {"start": 1271.17, "end": 1272.01, "word": " sequence", "probability": 0.90283203125}, {"start": 1272.01, "end": 1272.57, "word": " of", "probability": 0.98046875}, {"start": 1272.57, "end": 1273.23, "word": " partial", "probability": 0.96728515625}, {"start": 1273.23, "end": 1274.05, "word": " sums", "probability": 0.94677734375}], "temperature": 1.0}, {"id": 59, "seek": 129831, "start": 1275.51, "end": 1298.31, "text": "طيب يا جماعة نيجي لكوشيك ال 915 اللي هي quotient criterion for series the series summation xn in R converts if and only if for every ي أكبر من سفر there is a natural number M such that if M و N أكبر أو سوى M of Y هكون الأسئلة ماقصة أسئلة M اللي هو بسوى xn زائد 1", "tokens": [9566, 1829, 3555, 35186, 10874, 15042, 27884, 8717, 1829, 7435, 1829, 5296, 4117, 2407, 8592, 1829, 4117, 2423, 1722, 5211, 13672, 1829, 39896, 9641, 1196, 46691, 337, 2638, 264, 2638, 28811, 2031, 77, 294, 497, 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اللي هو let x بالـ definition 916 let x بسوء xn be a sequence in R we say that the series سماشي xn is absolutely convergent", "tokens": [9566, 1829, 3555, 12399, 2423, 7649, 8717, 3615, 28480, 11331, 26108, 3660, 24525, 2304, 11296, 8236, 356, 9652, 6930, 8236, 356, 9652, 6930, 4724, 43042, 3224, 13672, 1829, 3714, 43500, 25894, 8315, 8978, 6055, 5172, 46958, 1211, 4724, 11296, 16490, 1211, 9957, 995, 8717, 8848, 37983, 24793, 8978, 11296, 13672, 1829, 31439, 718, 2031, 20666, 39184, 7123, 1722, 6866, 718, 2031, 4724, 3794, 2407, 38207, 2031, 77, 312, 257, 8310, 294, 497, 321, 584, 300, 264, 2638, 8608, 2304, 33599, 1829, 2031, 77, 307, 3122, 9652, 6930], "avg_logprob": -0.379261353476481, "compression_ratio": 1.5116279069767442, "no_speech_prob": 8.940696716308594e-07, "words": [{"start": 1457.32, "end": 1457.88, "word": "طيب،", "probability": 0.637176513671875}, {"start": 1457.88, "end": 1458.36, "word": " الان", "probability": 0.6102294921875}, {"start": 1458.36, "end": 1459.88, 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convergent لكن مش absolutely convergent بنسميه إيش ما لها conditionally convergent", "tokens": [351, 309, 307, 9652, 6930, 457, 406, 11933, 1829, 8592, 19446, 5296, 11296, 3122, 9652, 6930, 37495, 22653, 8978, 2638, 8978, 32748, 22653, 995, 5551, 4117, 25708, 39894, 30544, 9652, 6930, 44381, 37893, 3122, 9652, 6930, 29538, 2423, 2638, 13672, 1829, 9154, 8032, 6027, 1863, 45367, 13672, 1829, 39894, 30544, 9652, 6930, 44381, 37893, 3122, 9652, 6930, 44945, 38251, 1829, 3224, 11933, 1829, 8592, 19446, 5296, 11296, 4188, 379, 9652, 6930], "avg_logprob": -0.23480902053415775, "compression_ratio": 1.9883040935672514, "no_speech_prob": 0.0, "words": [{"start": 1497.86, "end": 1498.3, "word": "if", "probability": 0.1131591796875}, {"start": 1498.3, "end": 1498.76, "word": " it", "probability": 0.9619140625}, {"start": 1498.76, "end": 1499.08, "word": " is", "probability": 0.88916015625}, {"start": 1499.08, "end": 1499.78, "word": " convergent", "probability": 0.977783203125}, {"start": 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"word": " is", "probability": 0.90576171875}, {"start": 1526.42, "end": 1527.92, "word": " convergent", "probability": 0.968505859375}, {"start": 1527.92, "end": 1529.46, "word": " طيب", "probability": 0.8806966145833334}, {"start": 1529.46, "end": 1530.84, "word": " الان", "probability": 0.6649169921875}, {"start": 1530.84, "end": 1532.86, "word": " نيجي", "probability": 0.9007568359375}, {"start": 1532.86, "end": 1533.24, "word": " طبعا", "probability": 0.87158203125}, {"start": 1533.24, "end": 1533.38, "word": " هذا", "probability": 0.6513671875}, {"start": 1533.38, "end": 1533.54, "word": " في", "probability": 0.88671875}, {"start": 1533.54, "end": 1533.74, "word": " R", "probability": 0.6884765625}, {"start": 1533.74, "end": 1533.98, "word": " كل", "probability": 0.97900390625}, {"start": 1533.98, "end": 1534.36, "word": " الشغل", "probability": 0.9903971354166666}, {"start": 1534.36, "end": 1535.08, "word": " في", "probability": 0.30224609375}, {"start": 1535.08, "end": 1535.28, 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converts الآن الجزء الثاني b اللي هو بنشوفه لاحقا", "tokens": [1211, 2655, 1975, 4117, 3794, 4724, 3794, 2407, 38207, 257, 297, 550, 264, 33246, 2638, 2423, 28811, 5296, 257, 297, 9154, 36764, 24401, 13672, 1829, 3714, 6027, 1863, 11296, 39896, 38874, 498, 2423, 8236, 2158, 24976, 257, 5551, 8592, 2304, 6027, 11296, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 11933, 15730, 2423, 33246, 2638, 8032, 8848, 995, 8032, 2655, 30544, 38874, 20449, 1211, 20498, 3660, 23032, 3555, 3615, 995, 6225, 9640, 5172, 9957, 11296, 11933, 15730, 25961, 2423, 8236, 2158, 24976, 257, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 6156, 3555, 2655, 30544, 29538, 257, 5551, 8592, 2304, 6027, 11296, 38874, 6024, 48506, 25724, 11622, 38207, 6024, 104, 7649, 1829, 272, 13672, 1829, 31439, 44945, 8592, 38688, 3224, 20193, 5016, 4587, 995], "avg_logprob": -0.29674796844885604, "compression_ratio": 1.8870967741935485, "no_speech_prob": 6.556510925292969e-07, "words": [{"start": 1731.32, "end": 1731.72, "word": 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دي", "probability": 0.890380859375}, {"start": 1987.51, "end": 1987.81, "word": " اللي", "probability": 0.953857421875}, {"start": 1987.81, "end": 1988.17, "word": " هو", "probability": 0.9873046875}], "temperature": 1.0}, {"id": 90, "seek": 200875, "start": 1988.53, "end": 2008.75, "text": "الـ S M نقص S N أصغر من إبسلون هيكون اللي هو في هذه الحالة أثبتت أنه اللي هو الـ S N by Cauchy criterion is convergent اللي هو صار عندي الصممش إلى الآن convergence طيب الديجلة اللي هو المثال اللي بعده المثال المشهور", "tokens": [6027, 39184, 318, 376, 8717, 4587, 9381, 318, 426, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 11536, 39896, 30544, 13672, 1829, 31439, 8978, 29538, 21542, 6027, 3660, 5551, 12984, 3555, 2655, 2655, 14739, 3224, 13672, 1829, 31439, 2423, 39184, 318, 426, 538, 7544, 625, 88, 46691, 307, 9652, 6930, 13672, 1829, 31439, 20328, 9640, 18871, 16254, 31767, 2304, 2304, 8592, 30731, 6024, 48506, 32181, 23032, 1829, 3555, 2423, 16254, 7435, 37977, 13672, 1829, 31439, 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هيكون", "probability": 0.567626953125}, {"start": 1992.39, "end": 1992.71, "word": " اللي", "probability": 0.8544921875}, {"start": 1992.71, "end": 1992.91, "word": " هو", "probability": 0.923828125}, {"start": 1992.91, "end": 1993.05, "word": " في", "probability": 0.935546875}, {"start": 1993.05, "end": 1993.25, "word": " هذه", "probability": 0.71533203125}, {"start": 1993.25, "end": 1993.69, "word": " الحالة", "probability": 0.986328125}, {"start": 1993.69, "end": 1994.27, "word": " أثبتت", "probability": 0.946875}, {"start": 1994.27, "end": 1994.65, "word": " أنه", "probability": 0.617919921875}, {"start": 1994.65, "end": 1995.07, "word": " اللي", "probability": 0.7208251953125}, {"start": 1995.07, "end": 1995.23, "word": " هو", "probability": 0.982421875}, {"start": 1995.23, "end": 1995.45, "word": " الـ", "probability": 0.817626953125}, {"start": 1995.45, "end": 1995.67, "word": " S", "probability": 0.6767578125}, {"start": 1995.67, "end": 1996.07, "word": " N", "probability": 0.759765625}, {"start": 1996.07, "end": 1996.41, "word": " by", "probability": 0.71337890625}, {"start": 1996.41, "end": 1996.85, "word": " Cauchy", "probability": 0.935546875}, {"start": 1996.85, "end": 1997.59, "word": " criterion", "probability": 0.87255859375}, {"start": 1997.59, "end": 1998.13, "word": " is", "probability": 0.90380859375}, {"start": 1998.13, "end": 1999.57, "word": " convergent", "probability": 0.94482421875}, {"start": 1999.57, "end": 2000.27, "word": " اللي", "probability": 0.966064453125}, {"start": 2000.27, "end": 2000.45, "word": " هو", "probability": 0.845703125}, {"start": 2000.45, "end": 2000.77, "word": " صار", "probability": 0.966064453125}, {"start": 2000.77, "end": 2000.97, "word": " عندي", "probability": 0.6231689453125}, {"start": 2000.97, "end": 2001.39, "word": " الصممش", "probability": 0.8369140625}, {"start": 2001.39, "end": 2001.55, "word": " إلى", "probability": 0.499755859375}, {"start": 2001.55, "end": 2001.87, "word": " الآن", "probability": 0.817626953125}, {"start": 2001.87, "end": 2002.85, "word": " convergence", "probability": 0.15869140625}, {"start": 2002.85, "end": 2004.99, "word": " طيب", "probability": 0.8299153645833334}, {"start": 2004.99, "end": 2005.57, "word": " الديجلة", "probability": 0.508056640625}, {"start": 2005.57, "end": 2005.91, "word": " اللي", "probability": 0.975341796875}, {"start": 2005.91, "end": 2006.17, "word": " هو", "probability": 0.9658203125}, {"start": 2006.17, "end": 2006.95, "word": " المثال", "probability": 0.9611002604166666}, {"start": 2006.95, "end": 2007.23, "word": " اللي", "probability": 0.9169921875}, {"start": 2007.23, "end": 2007.55, "word": " بعده", "probability": 0.75146484375}, {"start": 2007.55, "end": 2007.93, "word": " المثال", "probability": 0.9798177083333334}, {"start": 2007.93, "end": 2008.75, "word": " المشهور", "probability": 0.9954833984375}], "temperature": 1.0}, {"id": 91, "seek": 203146, "start": 2009.34, "end": 2031.46, "text": "اللي هو بـ little bit show that summation للواحد and diverse هذه الـ sequence، هذه الـ series اللي هي series مشهورة اللي كنا نسميها الـ B series الـ B بيساوي واحد في هذه الحالة هذه الـ summation يليها واحد and إيش ماله؟ diverse تنشوف", "tokens": [6027, 20292, 31439, 4724, 39184, 707, 857, 855, 300, 2408, 76, 399, 24976, 14407, 24401, 293, 9521, 29538, 2423, 39184, 8310, 12399, 29538, 2423, 39184, 2638, 13672, 1829, 39896, 2638, 37893, 3224, 13063, 3660, 13672, 1829, 9122, 8315, 8717, 38251, 1829, 11296, 2423, 39184, 363, 2638, 2423, 39184, 363, 4724, 1829, 3794, 995, 45865, 36764, 24401, 8978, 29538, 21542, 6027, 3660, 29538, 2423, 39184, 28811, 7251, 20292, 11296, 36764, 24401, 293, 11933, 1829, 8592, 3714, 6027, 3224, 22807, 9521, 6055, 1863, 8592, 38688], "avg_logprob": -0.41294643389327185, "compression_ratio": 1.780104712041885, "no_speech_prob": 1.7285346984863281e-06, "words": [{"start": 2009.34, "end": 2009.7, "word": "اللي", "probability": 0.433837890625}, {"start": 2009.7, "end": 2009.84, "word": " هو", "probability": 0.8037109375}, {"start": 2009.84, "end": 2010.06, "word": " بـ", "probability": 0.165283203125}, {"start": 2010.06, "end": 2010.2, "word": " little", "probability": 0.29443359375}, {"start": 2010.2, "end": 2010.54, "word": " bit", "probability": 0.830078125}, {"start": 2010.54, "end": 2010.82, "word": " show", "probability": 0.85107421875}, {"start": 2010.82, "end": 2011.12, "word": " that", "probability": 0.91455078125}, {"start": 2011.12, "end": 2011.62, "word": " summation", "probability": 0.5111490885416666}, {"start": 2011.62, "end": 2012.2, "word": " للواحد", "probability": 0.884765625}, {"start": 2012.2, "end": 2012.56, "word": " and", "probability": 0.272216796875}, {"start": 2012.56, "end": 2014.14, "word": " diverse", "probability": 0.309326171875}, {"start": 2014.14, "end": 2016.5, "word": " هذه", "probability": 0.154296875}, {"start": 2016.5, "end": 2018.5, "word": " الـ", "probability": 0.530517578125}, {"start": 2018.5, "end": 2019.26, "word": " sequence،", "probability": 0.53814697265625}, {"start": 2019.26, "end": 2019.44, "word": " هذه", "probability": 0.896484375}, {"start": 2019.44, "end": 2019.62, "word": " الـ", "probability": 0.769287109375}, {"start": 2019.62, "end": 2020.12, "word": " series", "probability": 0.89990234375}, {"start": 2020.12, "end": 2020.88, "word": " اللي", "probability": 0.79296875}, {"start": 2020.88, "end": 2021.06, "word": " هي", "probability": 0.8955078125}, {"start": 2021.06, "end": 2021.5, "word": " series", "probability": 0.81005859375}, {"start": 2021.5, "end": 2022.8, "word": " مشهورة", "probability": 0.994384765625}, {"start": 2022.8, "end": 2023.56, "word": " اللي", "probability": 0.94775390625}, {"start": 2023.56, "end": 2023.98, "word": " كنا", "probability": 0.864013671875}, {"start": 2023.98, "end": 2024.42, "word": " نسميها", "probability": 0.9599609375}, {"start": 2024.42, "end": 2024.56, "word": " الـ", "probability": 0.81201171875}, {"start": 2024.56, "end": 2024.68, "word": " B", "probability": 0.5458984375}, {"start": 2024.68, "end": 2025.26, "word": " series", "probability": 0.53271484375}, {"start": 2025.26, "end": 2025.46, "word": " الـ", "probability": 0.7353515625}, {"start": 2025.46, "end": 2025.56, "word": " B", "probability": 0.74951171875}, {"start": 2025.56, "end": 2025.94, "word": " بيساوي", "probability": 0.665234375}, {"start": 2025.94, "end": 2026.24, "word": " واحد", "probability": 0.847412109375}, {"start": 2026.24, "end": 2026.38, "word": " في", "probability": 0.84521484375}, {"start": 2026.38, "end": 2026.58, "word": " هذه", "probability": 0.69921875}, {"start": 2026.58, "end": 2027.1, "word": " الحالة", "probability": 0.822998046875}, {"start": 2027.1, "end": 2027.68, "word": " هذه", "probability": 0.5595703125}, {"start": 2027.68, "end": 2027.84, "word": " الـ", "probability": 0.681396484375}, {"start": 2027.84, "end": 2028.16, "word": " summation", "probability": 0.787109375}, {"start": 2028.16, "end": 2028.5, "word": " يليها", "probability": 0.5756022135416666}, {"start": 2028.5, "end": 2028.84, "word": " واحد", "probability": 0.9765625}, {"start": 2028.84, "end": 2029.16, "word": " and", "probability": 0.6572265625}, {"start": 2029.16, "end": 2029.44, "word": " إيش", "probability": 0.741455078125}, {"start": 2029.44, "end": 2030.14, "word": " ماله؟", "probability": 0.8197021484375}, {"start": 2030.14, "end": 2030.76, "word": " diverse", "probability": 0.64599609375}, {"start": 2030.76, "end": 2031.46, "word": " تنشوف", "probability": 0.69158935546875}], "temperature": 1.0}, {"id": 92, "seek": 205702, "start": 2032.24, "end": 2057.02, "text": "to show that summation وحدة ل N divergent, we construct unbounded subsequence SKN of SK قلنا قبل هيك ان اذا كانت ال sequence converges اكيد هتكون bounded اذا لو كانت ال sequence unbounded اكيد diverges ماشي الحال طيب I was not convergent", "tokens": [1353, 855, 300, 28811, 4032, 24401, 3660, 5296, 426, 18558, 6930, 11, 321, 7690, 517, 18767, 292, 13924, 655, 318, 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"word": " N", "probability": 0.403076171875}, {"start": 2034.4, "end": 2035.18, "word": " divergent,", "probability": 0.808837890625}, {"start": 2035.34, "end": 2035.46, "word": " we", "probability": 0.9296875}, {"start": 2035.46, "end": 2036.18, "word": " construct", "probability": 0.962890625}, {"start": 2036.18, "end": 2037.04, "word": " unbounded", "probability": 0.9274088541666666}, {"start": 2037.04, "end": 2037.86, "word": " subsequence", "probability": 0.984619140625}, {"start": 2037.86, "end": 2038.48, "word": " SKN", "probability": 0.43701171875}, {"start": 2038.48, "end": 2039.44, "word": " of", "probability": 0.90087890625}, {"start": 2039.44, "end": 2039.96, "word": " SK", "probability": 0.86767578125}, {"start": 2039.96, "end": 2041.26, "word": " قلنا", "probability": 0.89892578125}, {"start": 2041.26, "end": 2041.52, "word": " قبل", "probability": 0.917724609375}, {"start": 2041.52, "end": 2042.14, "word": " هيك", "probability": 0.56298828125}, {"start": 2042.14, "end": 2043.2, "word": " ان", "probability": 0.214111328125}, {"start": 2043.2, "end": 2043.98, "word": " اذا", "probability": 0.6734619140625}, {"start": 2043.98, "end": 2044.38, "word": " كانت", "probability": 0.980224609375}, {"start": 2044.38, "end": 2044.52, "word": " ال", "probability": 0.58349609375}, {"start": 2044.52, "end": 2044.86, "word": " sequence", "probability": 0.66064453125}, {"start": 2044.86, "end": 2045.8, "word": " converges", "probability": 0.695068359375}, {"start": 2045.8, "end": 2046.5, "word": " اكيد", "probability": 0.7941080729166666}, {"start": 2046.5, "end": 2046.88, "word": " هتكون", "probability": 0.8888346354166666}, {"start": 2046.88, "end": 2047.42, "word": " bounded", "probability": 0.86279296875}, {"start": 2047.42, "end": 2048.4, "word": " اذا", "probability": 0.7198486328125}, {"start": 2048.4, "end": 2048.86, "word": " لو", "probability": 0.966796875}, {"start": 2048.86, "end": 2049.62, "word": " كانت", "probability": 0.989013671875}, {"start": 2049.62, "end": 2049.8, "word": " ال", "probability": 0.826171875}, {"start": 2049.8, "end": 2050.54, "word": " sequence", "probability": 0.97802734375}, {"start": 2050.54, "end": 2052.04, "word": " unbounded", "probability": 0.93017578125}, {"start": 2052.04, "end": 2052.52, "word": " اكيد", "probability": 0.9567057291666666}, {"start": 2052.52, "end": 2053.54, "word": " diverges", "probability": 0.625}, {"start": 2053.54, "end": 2054.44, "word": " ماشي", "probability": 0.8147786458333334}, {"start": 2054.44, "end": 2054.84, "word": " الحال", "probability": 0.863525390625}, {"start": 2054.84, "end": 2055.42, "word": " طيب", "probability": 0.7406412760416666}, {"start": 2055.42, "end": 2056.06, "word": " I", "probability": 0.458251953125}, {"start": 2056.06, "end": 2056.22, "word": " was", "probability": 0.9306640625}, {"start": 2056.22, "end": 2056.42, "word": " not", "probability": 0.955078125}, {"start": 2056.42, "end": 2057.02, "word": " convergent", "probability": 0.980224609375}], "temperature": 1.0}, {"id": 93, "seek": 207831, "start": 2057.55, "end": 2078.31, "text": "بناءً عليه لو احنا جيبنا subsequence من ال sequence تكون unbounded فمن باب أولى هتصير ال sequence الأصلية نفسها unbounded مدام ان صارت ال sequence الأصلية unbounded إذا ال sequence صارت diverse بناءً عليه عشان أثبت اللي هو ال sequence of partial sums هذه", "tokens": [3555, 1863, 16606, 14111, 47356, 45164, 1975, 5016, 8315, 10874, 1829, 3555, 8315, 13924, 655, 9154, 2423, 8310, 6055, 30544, 517, 18767, 292, 6156, 27842, 4724, 16758, 5551, 12610, 7578, 8032, 2655, 9381, 13546, 2423, 8310, 16247, 36520, 10632, 8717, 36178, 11296, 517, 18767, 292, 3714, 3215, 10943, 16472, 20328, 9640, 2655, 2423, 8310, 16247, 36520, 10632, 517, 18767, 292, 11933, 15730, 2423, 8310, 20328, 9640, 2655, 9521, 44945, 16606, 14111, 47356, 6225, 8592, 7649, 5551, 12984, 3555, 2655, 13672, 1829, 31439, 2423, 8310, 295, 14641, 34499, 29538], "avg_logprob": -0.20312499598170933, "compression_ratio": 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{"start": 3032.61, "end": 3032.77, "word": " ال", "probability": 0.86474609375}, {"start": 3032.77, "end": 3034.17, "word": " subsequence", "probability": 0.69775390625}, {"start": 3034.17, "end": 3035.05, "word": " SKR", "probability": 0.6680908203125}], "temperature": 1.0}, {"id": 136, "seek": 304842, "start": 3035.46, "end": 3048.42, "text": "هذه الـ subsequence وضحى انها اكبر او يساوى 1 على ارضها او اتنين يعني as R goes to infinity هذا المقدار بروح لما لنهاية يعني هذه ال sequence is unbounded اذا صار عندى اللي هو", "tokens": [3224, 24192, 2423, 39184, 13924, 655, 4032, 11242, 5016, 7578, 16472, 11296, 1975, 4117, 26890, 1975, 2407, 7251, 3794, 995, 2407, 7578, 502, 15844, 1975, 43042, 11296, 1975, 2407, 1975, 2655, 1863, 9957, 37495, 22653, 382, 497, 1709, 281, 13202, 23758, 9673, 28543, 9640, 4724, 32887, 5016, 5296, 15042, 5296, 1863, 11296, 10632, 37495, 22653, 29538, 2423, 8310, 307, 517, 18767, 292, 1975, 15730, 20328, 9640, 43242, 7578, 13672, 1829, 31439], "avg_logprob": 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"end": 3323.98, "word": " أني", "probability": 0.669677734375}, {"start": 3323.98, "end": 3324.5, "word": " أثبت", "probability": 0.9876708984375}, {"start": 3324.5, "end": 3324.94, "word": " انها", "probability": 0.762451171875}, {"start": 3324.94, "end": 3325.42, "word": " bounded", "probability": 0.9580078125}, {"start": 3325.42, "end": 3325.86, "word": " عشان", "probability": 0.966796875}, {"start": 3325.86, "end": 3326.2, "word": " تكون", "probability": 0.970947265625}, {"start": 3326.2, "end": 3326.36, "word": " اللي", "probability": 0.6815185546875}, {"start": 3326.36, "end": 3326.64, "word": " فوق", "probability": 0.98583984375}, {"start": 3326.64, "end": 3327.06, "word": " bounded", "probability": 0.93115234375}, {"start": 3327.06, "end": 3328.86, "word": " طب", "probability": 0.817626953125}, {"start": 3328.86, "end": 3328.92, "word": " ما", "probability": 0.66064453125}, {"start": 3328.92, "end": 3329.28, "word": " بلاقي", "probability": 0.8958333333333334}, {"start": 3329.28, "end": 3329.96, "word": " subsequence", "probability": 0.984619140625}, {"start": 3329.96, "end": 3330.3, "word": " تانية", "probability": 0.98193359375}, {"start": 3330.3, "end": 3330.74, "word": " بتكون", "probability": 0.841796875}, {"start": 3330.74, "end": 3330.92, "word": " اللي", "probability": 0.924560546875}, {"start": 3330.92, "end": 3331.12, "word": " هي", "probability": 0.734375}, {"start": 3331.12, "end": 3331.82, "word": " unbounded", "probability": 0.8909505208333334}], "temperature": 1.0}, {"id": 149, "seek": 335006, "start": 3332.42, "end": 3350.06, "text": "ما هو لو لجيت subsequence تانية زي ما بتقول انها unbounded أو لو كانت هذه unbounded في حالة هذه bounded معناته ما هي increasing هذه هتلاقي اللي هو term من هذه بعد ال .. بعد ال term اللي انت حكيت عنه من هذه مادام هتلاقيه بعده", "tokens": [15042, 31439, 45164, 5296, 7435, 36081, 13924, 655, 6055, 7649, 10632, 30767, 1829, 19446, 39894, 39648, 16472, 11296, 517, 18767, 292, 34051, 45164, 25961, 2655, 29538, 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0.92041015625}, {"start": 3818.49, "end": 3818.95, "word": " انزايد", "probability": 0.9598388671875}, {"start": 3818.95, "end": 3819.33, "word": " واحد", "probability": 0.992919921875}, {"start": 3819.33, "end": 3819.97, "word": " لأن", "probability": 0.55859375}, {"start": 3819.97, "end": 3820.21, "word": " هذا", "probability": 0.916015625}, {"start": 3820.21, "end": 3820.47, "word": " صار", "probability": 0.892333984375}, {"start": 3820.47, "end": 3820.65, "word": " اللي", "probability": 0.838623046875}, {"start": 3820.65, "end": 3820.91, "word": " هو", "probability": 0.99072265625}, {"start": 3820.91, "end": 3821.21, "word": " اللي", "probability": 0.7371826171875}, {"start": 3821.21, "end": 3821.79, "word": " هو", "probability": 0.994140625}, {"start": 3821.79, "end": 3822.17, "word": " ال", "probability": 0.93505859375}, {"start": 3822.17, "end": 3823.17, "word": " partial", "probability": 0.84375}, {"start": 3823.17, "end": 3823.69, "word": " sum", "probability": 0.94482421875}, {"start": 3823.69, "end": 3824.19, "word": " او", "probability": 0.4124755859375}, {"start": 3824.19, "end": 3824.29, "word": " ال", "probability": 0.603515625}, {"start": 3824.29, "end": 3824.59, "word": " sequence", "probability": 0.88818359375}, {"start": 3824.59, "end": 3824.77, "word": " of", "probability": 0.767578125}, {"start": 3824.77, "end": 3825.71, "word": " ال", "probability": 0.4365234375}, {"start": 3825.71, "end": 3826.01, "word": " term", "probability": 0.09521484375}, {"start": 3826.01, "end": 3826.53, "word": " of", "probability": 0.64208984375}, {"start": 3826.53, "end": 3826.69, "word": " the", "probability": 0.59423828125}, {"start": 3826.69, "end": 3827.69, "word": " partial", "probability": 0.64501953125}, {"start": 3827.69, "end": 3828.45, "word": " sums", "probability": 0.8701171875}, {"start": 3828.45, "end": 3828.83, "word": " of", "probability": 0.9599609375}, {"start": 3828.83, "end": 3829.17, "word": " this", "probability": 0.904296875}, {"start": 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"probability": 0.76416015625}, {"start": 3833.87, "end": 3834.25, "word": " واحد", "probability": 0.82373046875}, {"start": 3834.25, "end": 3834.63, "word": " نقص", "probability": 0.7239583333333334}, {"start": 3834.63, "end": 3834.83, "word": " limit", "probability": 0.96875}, {"start": 3834.83, "end": 3835.25, "word": " الواحد", "probability": 0.880859375}, {"start": 3835.25, "end": 3835.47, "word": " الآن", "probability": 0.765625}, {"start": 3835.47, "end": 3835.79, "word": " زائد", "probability": 0.76806640625}, {"start": 3835.79, "end": 3836.17, "word": " واحد", "probability": 0.98095703125}, {"start": 3836.17, "end": 3836.67, "word": " هذا", "probability": 0.740234375}, {"start": 3836.67, "end": 3836.95, "word": " بروح", "probability": 0.9051106770833334}, {"start": 3836.95, "end": 3837.51, "word": " للسفر", "probability": 0.73626708984375}, {"start": 3837.51, "end": 3838.09, "word": " ويساوي", "probability": 0.80029296875}, {"start": 3838.09, "end": 3838.51, "word": " واحد", "probability": 0.986328125}, {"start": 3838.51, "end": 3838.99, "word": " لأن", "probability": 0.679443359375}, {"start": 3838.99, "end": 3839.37, "word": " صارت", "probability": 0.9137369791666666}, {"start": 3839.37, "end": 3839.49, "word": " ال", "probability": 0.7998046875}, {"start": 3839.49, "end": 3839.81, "word": " series", "probability": 0.88134765625}, {"start": 3839.81, "end": 3840.01, "word": " اللي", "probability": 0.9228515625}, {"start": 3840.01, "end": 3840.33, "word": " عندي", "probability": 0.79541015625}, {"start": 3840.33, "end": 3840.53, "word": " هذه", "probability": 0.465087890625}, {"start": 3840.53, "end": 3841.11, "word": " الأصلية", "probability": 0.79736328125}, {"start": 3841.11, "end": 3842.05, "word": " converged", "probability": 0.3499755859375}, {"start": 3842.05, "end": 3842.51, "word": " ويساوي", "probability": 0.91142578125}, {"start": 3842.51, "end": 3842.61, "word": " ال", "probability": 0.888671875}, {"start": 3842.61, "end": 3843.13, "word": " convergent", "probability": 0.65087890625}, {"start": 3843.13, "end": 3843.31, "word": " ايش", "probability": 0.7069498697916666}, {"start": 3843.31, "end": 3843.81, "word": " بساوي", "probability": 0.9569091796875}, {"start": 3843.81, "end": 3844.63, "word": " بساوي", "probability": 0.8680419921875}, {"start": 3844.63, "end": 3844.85, "word": " واحد", "probability": 0.98779296875}, {"start": 3844.85, "end": 3845.75, "word": " الآن", "probability": 0.613525390625}, {"start": 3845.75, "end": 3846.29, "word": " اخر", "probability": 0.68798828125}, {"start": 3846.29, "end": 3847.73, "word": " موضوع", "probability": 0.9879150390625}, {"start": 3847.73, "end": 3848.05, "word": " في", "probability": 0.83740234375}, {"start": 3848.05, "end": 3848.95, "word": " اللي", "probability": 0.94482421875}, {"start": 3848.95, "end": 3849.29, "word": " هو", "probability": 0.99072265625}, {"start": 3849.29, "end": 3849.61, "word": " ال", "probability": 0.892578125}, {"start": 3849.61, "end": 3849.91, "word": " section", "probability": 0.89404296875}, {"start": 3849.91, "end": 3850.47, "word": " الأول", "probability": 0.857177734375}], "temperature": 1.0}, {"id": 171, "seek": 386766, "start": 3852.46, "end": 3867.66, "text": "حاجة اسمها rearrangement of series يعني إعادة ترتيب العناصر اللي هو اللي هي ال series أو ال term سبعة ال series ال rearrangement of a series is another series هي في ريز أخر هتكون", "tokens": [5016, 26108, 3660, 24525, 2304, 11296, 39568, 518, 295, 2638, 37495, 22653, 11933, 3615, 18513, 3660, 6055, 2288, 31371, 3555, 18863, 1863, 33546, 2288, 13672, 1829, 31439, 13672, 1829, 39896, 2423, 2638, 34051, 2423, 1433, 8608, 3555, 27884, 2423, 2638, 2423, 39568, 518, 295, 257, 2638, 307, 1071, 2638, 39896, 8978, 12602, 1829, 11622, 5551, 34740, 8032, 2655, 30544], "avg_logprob": -0.2062500019868215, "compression_ratio": 1.5660377358490567, "no_speech_prob": 7.152557373046875e-07, "words": [{"start": 3852.46, "end": 3852.86, "word": "حاجة", "probability": 0.7349853515625}, {"start": 3852.86, "end": 3853.2, "word": " اسمها", "probability": 0.95947265625}, {"start": 3853.2, "end": 3854.22, "word": " rearrangement", "probability": 0.769287109375}, {"start": 3854.22, "end": 3854.42, "word": " of", "probability": 0.97802734375}, {"start": 3854.42, "end": 3854.96, "word": " series", "probability": 0.8984375}, {"start": 3854.96, "end": 3856.08, "word": " يعني", "probability": 0.883056640625}, {"start": 3856.08, "end": 3856.7, "word": " إعادة", "probability": 0.9473876953125}, {"start": 3856.7, "end": 3857.4, "word": " ترتيب", "probability": 0.9580078125}, {"start": 3857.4, "end": 3858.18, "word": " العناصر", "probability": 0.8756103515625}, {"start": 3858.18, "end": 3858.36, "word": " اللي", "probability": 0.88330078125}, {"start": 3858.36, "end": 3858.72, "word": " هو", "probability": 0.73291015625}, {"start": 3858.72, "end": 3859.08, "word": " اللي", "probability": 0.755859375}, {"start": 3859.08, "end": 3859.4, "word": " هي", "probability": 0.94677734375}, {"start": 3859.4, "end": 3859.74, "word": " ال", "probability": 0.8857421875}, {"start": 3859.74, "end": 3860.16, "word": " series", "probability": 0.818359375}, {"start": 3860.16, "end": 3860.76, "word": " أو", "probability": 0.806640625}, {"start": 3860.76, "end": 3860.94, "word": " ال", "probability": 0.38720703125}, {"start": 3860.94, "end": 3861.22, "word": " term", "probability": 0.422607421875}, {"start": 3861.22, "end": 3861.66, "word": " سبعة", "probability": 0.8709309895833334}, {"start": 3861.66, "end": 3861.82, "word": " ال", "probability": 0.8154296875}, {"start": 3861.82, "end": 3862.2, "word": " series", "probability": 0.96826171875}, {"start": 3862.2, "end": 3862.94, "word": " ال", "probability": 0.49169921875}, {"start": 3862.94, "end": 3863.92, "word": " rearrangement", "probability": 0.958251953125}, {"start": 3863.92, "end": 3864.08, "word": " of", "probability": 0.96435546875}, {"start": 3864.08, "end": 3864.28, "word": " a", "probability": 0.8251953125}, {"start": 3864.28, "end": 3864.8, "word": " series", "probability": 0.90673828125}, {"start": 3864.8, "end": 3865.16, "word": " is", "probability": 0.83154296875}, {"start": 3865.16, "end": 3865.56, "word": " another", "probability": 0.93505859375}, {"start": 3865.56, "end": 3866.04, "word": " series", "probability": 0.9052734375}, {"start": 3866.04, "end": 3866.22, "word": " هي", "probability": 0.7998046875}, {"start": 3866.22, "end": 3866.46, "word": " في", "probability": 0.391357421875}, {"start": 3866.46, "end": 3866.78, "word": " ريز", "probability": 0.77685546875}, {"start": 3866.78, "end": 3867.1, "word": " أخر", "probability": 0.935546875}, {"start": 3867.1, "end": 3867.66, "word": " هتكون", "probability": 0.821533203125}], "temperature": 1.0}, {"id": 172, "seek": 388760, "start": 3868.34, "end": 3887.6, "text": "which is obtained from the given one يعني بناخدها من من من الأصلية by using all of the terms exactly one once يعني بإننا نستخدم عناصر ال series مرة واحدة فقط لازم نستخدمها و بنستخدمها بمرة واحدة", "tokens": [13690, 307, 14879, 490, 264, 2212, 472, 37495, 22653, 44945, 47283, 3215, 11296, 9154, 9154, 9154, 16247, 36520, 10632, 538, 1228, 439, 295, 264, 2115, 2293, 472, 1564, 37495, 22653, 4724, 28814, 1863, 8315, 8717, 14851, 9778, 40448, 18871, 33546, 2288, 2423, 2638, 3714, 25720, 36764, 24401, 3660, 6156, 47432, 5296, 31377, 2304, 8717, 14851, 9778, 40448, 11296, 4032, 44945, 14851, 9778, 40448, 11296, 4724, 2304, 25720, 36764, 24401, 3660], "avg_logprob": -0.14018486503144384, "compression_ratio": 1.5494505494505495, "no_speech_prob": 5.7220458984375e-06, "words": [{"start": 3868.34, "end": 3868.82, "word": "which", "probability": 0.399169921875}, {"start": 3868.82, "end": 3869.14, "word": " is", "probability": 0.9482421875}, {"start": 3869.14, "end": 3869.7, "word": " obtained", "probability": 0.90478515625}, {"start": 3869.7, "end": 3870.18, "word": " from", "probability": 0.9296875}, {"start": 3870.18, "end": 3870.5, "word": " the", "probability": 0.8955078125}, {"start": 3870.5, "end": 3870.84, "word": " given", "probability": 0.89306640625}, {"start": 3870.84, "end": 3871.12, "word": " one", "probability": 0.9248046875}, {"start": 3871.12, "end": 3871.42, "word": " يعني", "probability": 0.56976318359375}, {"start": 3871.42, "end": 3871.92, "word": " بناخدها", "probability": 0.9410400390625}, {"start": 3871.92, "end": 3872.1, "word": " من", "probability": 0.9921875}, {"start": 3872.1, "end": 3872.32, "word": " من", "probability": 0.487060546875}, {"start": 3872.32, "end": 3872.54, "word": " من", "probability": 0.448486328125}, {"start": 3872.54, "end": 3873.32, "word": " الأصلية", "probability": 0.7530110677083334}, {"start": 3873.32, "end": 3874.36, "word": " by", "probability": 0.837890625}, {"start": 3874.36, "end": 3875.08, "word": " using", "probability": 0.95751953125}, {"start": 3875.08, "end": 3875.64, "word": " all", "probability": 0.9404296875}, {"start": 3875.64, "end": 3875.9, "word": " of", 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4123.73, "word": " Y2", "probability": 0.917724609375}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9842122395833334}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.91796875}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9845377604166666}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.91845703125}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9860026041666666}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.91748046875}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9864908854166666}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.919921875}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9874674479166666}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.919189453125}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9881184895833334}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.918212890625}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9889322916666666}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.917236328125}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9895833333333334}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.916259765625}, {"start": 4123.73, "end": 4123.73, "word": " XF2", "probability": 0.9895833333333334}, {"start": 4123.73, "end": 4123.73, "word": " Y2", "probability": 0.915283203125}, {"start": 4123.73, "end": 4124.41, "word": " XF2", "probability": 0.9903971354166666}, {"start": 4124.41, "end": 4124.57, "word": " Y2", "probability": 0.9130859375}, {"start": 4124.57, "end": 4125.57, "word": " XF2", "probability": 0.9905598958333334}, {"start": 4125.57, "end": 4125.65, "word": " Y2", "probability": 0.913330078125}, {"start": 4125.65, "end": 4125.65, "word": " XF2", "probability": 0.99072265625}, {"start": 4125.65, "end": 4125.81, "word": " Y2", "probability": 0.9111328125}, {"start": 4125.81, "end": 4125.81, "word": " XF2", "probability": 0.9910481770833334}, {"start": 4125.81, "end": 4126.35, "word": " Y2", "probability": 0.91015625}, {"start": 4126.35, "end": 4126.35, "word": " XF2", "probability": 0.9913736979166666}, {"start": 4126.35, "end": 4126.35, "word": " Y2", "probability": 0.907958984375}, {"start": 4126.35, "end": 4126.35, "word": " XF2", "probability": 0.9910481770833334}, {"start": 4126.35, "end": 4126.35, "word": " Y2", "probability": 0.90869140625}, {"start": 4126.35, "end": 4126.35, "word": " XF2", "probability": 0.9913736979166666}, {"start": 4126.35, "end": 4126.39, "word": " Y2", "probability": 0.904541015625}, {"start": 4126.39, "end": 4127.57, "word": " XF2", "probability": 0.9913736979166666}, {"start": 4127.57, "end": 4128.07, "word": " Y2", "probability": 0.9033203125}, {"start": 4128.07, "end": 4128.65, "word": " XF2", "probability": 0.9913736979166666}, {"start": 4128.65, "end": 4128.65, "word": " Y2", "probability": 0.903564453125}, {"start": 4128.65, "end": 4128.65, "word": " XF", "probability": 0.98828125}], "temperature": 1.0}, {"id": 184, "seek": 415631, "start": 4128.75, "end": 4156.31, "text": "ما تأثير الـ Rearrangements على السيريز الأصلية بقولك لو كانت السيريز absolutely convergent وده اللي هنتطرقله بس لو كانت السيريز absolutely convergent ترتب زي ما بدك مش هتفرج معانا اللي هو إيه ال convergence يعني let summation xn be an absolutely convergent series in R then any rearrangement of summation xn", "tokens": [15042, 6055, 10721, 12984, 13546, 2423, 39184, 1300, 2284, 933, 1117, 15844, 21136, 13546, 1829, 11622, 16247, 36520, 10632, 4724, 39648, 4117, 45164, 25961, 2655, 21136, 13546, 1829, 11622, 3122, 9652, 6930, 4032, 3215, 3224, 13672, 1829, 8032, 29399, 9566, 2288, 4587, 43761, 4724, 3794, 45164, 25961, 2655, 21136, 13546, 1829, 11622, 3122, 9652, 6930, 6055, 43500, 3555, 30767, 1829, 19446, 47525, 4117, 37893, 8032, 2655, 5172, 47341, 20449, 7649, 995, 13672, 1829, 31439, 11933, 1829, 3224, 2423, 32181, 37495, 22653, 718, 28811, 2031, 77, 312, 364, 3122, 9652, 6930, 2638, 294, 497, 550, 604, 39568, 518, 295, 28811, 2031, 77], "avg_logprob": -0.22028186902696012, "compression_ratio": 1.8068669527896997, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 4128.75, "end": 4129.09, "word": "ما", "probability": 0.52685546875}, {"start": 4129.09, "end": 4129.83, "word": " تأثير", "probability": 0.9554443359375}, {"start": 4129.83, "end": 4129.97, "word": " الـ", "probability": 0.865234375}, {"start": 4129.97, "end": 4131.25, "word": " Rearrangements", "probability": 0.76263427734375}, {"start": 4131.25, "end": 4132.21, "word": " على", "probability": 0.7646484375}, {"start": 4132.21, "end": 4133.05, "word": " السيريز", "probability": 0.76800537109375}, {"start": 4133.05, "end": 4133.83, "word": " الأصلية", "probability": 0.8173828125}, {"start": 4133.83, "end": 4134.97, "word": " بقولك", "probability": 0.6038004557291666}, {"start": 4134.97, "end": 4135.15, "word": " لو", "probability": 0.80224609375}, {"start": 4135.15, "end": 4135.69, "word": " كانت", "probability": 0.991943359375}, {"start": 4135.69, "end": 4136.27, "word": " السيريز", "probability": 0.9478759765625}, {"start": 4136.27, "end": 4136.89, "word": " absolutely", "probability": 0.119140625}, {"start": 4136.89, "end": 4137.55, "word": " convergent", "probability": 0.93115234375}, {"start": 4137.55, "end": 4137.79, "word": " وده", "probability": 0.5755208333333334}, {"start": 4137.79, "end": 4137.91, "word": " اللي", "probability": 0.9404296875}, {"start": 4137.91, "end": 4138.65, "word": " هنتطرقله", "probability": 0.865966796875}, {"start": 4138.65, "end": 4139.01, "word": " بس", "probability": 0.932861328125}, {"start": 4139.01, "end": 4139.57, "word": " لو", "probability": 0.87744140625}, {"start": 4139.57, "end": 4139.93, "word": " كانت", "probability": 0.991943359375}, {"start": 4139.93, "end": 4140.35, "word": " السيريز", "probability": 0.9388427734375}, {"start": 4140.35, "end": 4140.87, "word": " absolutely", "probability": 0.38818359375}, {"start": 4140.87, "end": 4141.71, "word": " convergent", "probability": 0.961669921875}, {"start": 4141.71, "end": 4142.75, "word": " ترتب", "probability": 0.8544921875}, {"start": 4142.75, "end": 4143.07, "word": " زي", "probability": 0.98583984375}, {"start": 4143.07, "end": 4143.19, "word": " ما", "probability": 0.9423828125}, {"start": 4143.19, "end": 4143.61, "word": " بدك", "probability": 0.975341796875}, {"start": 4143.61, "end": 4143.77, "word": " مش", "probability": 0.91748046875}, {"start": 4143.77, "end": 4144.25, "word": " هتفرج", "probability": 0.910400390625}, {"start": 4144.25, "end": 4144.93, "word": " معانا", "probability": 0.9685872395833334}, {"start": 4144.93, "end": 4145.45, "word": " اللي", "probability": 0.943115234375}, {"start": 4145.45, "end": 4145.71, "word": " هو", "probability": 0.97119140625}, {"start": 4145.71, "end": 4145.99, "word": " إيه", "probability": 0.7195638020833334}, {"start": 4145.99, "end": 4146.31, "word": " ال", "probability": 0.5654296875}, {"start": 4146.31, "end": 4146.95, "word": " convergence", "probability": 0.9443359375}, {"start": 4146.95, "end": 4147.85, "word": " يعني", "probability": 0.8720703125}, {"start": 4147.85, "end": 4148.09, "word": " let", "probability": 0.513671875}, {"start": 4148.09, "end": 4148.49, "word": " summation", "probability": 0.69140625}, {"start": 4148.49, "end": 4148.95, "word": " xn", "probability": 0.420166015625}, {"start": 4148.95, "end": 4149.11, "word": " be", "probability": 0.73046875}, {"start": 4149.11, "end": 4149.35, "word": " an", "probability": 0.83251953125}, {"start": 4149.35, "end": 4149.85, "word": " absolutely", "probability": 0.77978515625}, {"start": 4149.85, "end": 4150.53, "word": " convergent", "probability": 0.96240234375}, {"start": 4150.53, "end": 4151.11, "word": " series", "probability": 0.87841796875}, {"start": 4151.11, "end": 4151.69, "word": " in", "probability": 0.84375}, {"start": 4151.69, "end": 4151.95, "word": " R", "probability": 0.97509765625}, {"start": 4151.95, "end": 4152.25, "word": " then", "probability": 0.6787109375}, {"start": 4152.25, "end": 4152.67, "word": " any", "probability": 0.84228515625}, {"start": 4152.67, "end": 4154.65, "word": " rearrangement", "probability": 0.79736328125}, {"start": 4154.65, "end": 4154.93, "word": " of", "probability": 0.9404296875}, {"start": 4154.93, "end": 4155.45, "word": " summation", "probability": 0.7646484375}, {"start": 4155.45, "end": 4156.31, "word": " xn", "probability": 0.9775390625}], "temperature": 1.0}, {"id": 185, "seek": 417987, "start": 4157.81, "end": 4179.87, "text": "converge to the same value يعني لو كانت ال series absolutely convergent أي ترتيب إلها آخر وفق تعريف الترتيب اللي حنا حكينا عنه اللي هو هيعطي نفس ال limit هتكون converge ويلها نفس ال limit خلّينا نشوف البرهان ونمر عليه let summation xn converge to x", "tokens": [1671, 331, 432, 281, 264, 912, 2158, 37495, 22653, 45164, 25961, 2655, 2423, 2638, 3122, 9652, 6930, 36632, 6055, 2288, 31371, 3555, 11933, 1211, 11296, 19753, 34740, 4032, 5172, 4587, 37279, 16572, 5172, 16712, 2288, 31371, 3555, 13672, 1829, 11331, 8315, 11331, 4117, 1829, 8315, 18871, 3224, 13672, 1829, 31439, 39896, 3615, 9566, 1829, 8717, 36178, 2423, 4948, 8032, 2655, 30544, 41881, 4032, 26895, 11296, 8717, 36178, 2423, 4948, 16490, 1211, 11703, 9957, 995, 8717, 8592, 38688, 2423, 26890, 3224, 7649, 4032, 1863, 29973, 47356, 718, 28811, 2031, 77, 41881, 281, 2031], "avg_logprob": -0.18985215502400551, "compression_ratio": 1.592920353982301, "no_speech_prob": 1.5497207641601562e-06, "words": [{"start": 4157.8099999999995, "end": 4158.61, "word": "converge", "probability": 0.5975748697916666}, {"start": 4158.61, "end": 4159.07, "word": " to", "probability": 0.93994140625}, {"start": 4159.07, "end": 4159.23, "word": " the", "probability": 0.873046875}, {"start": 4159.23, "end": 4159.77, "word": " same", "probability": 0.8994140625}, {"start": 4159.77, "end": 4160.27, "word": " value", "probability": 0.9697265625}, {"start": 4160.27, "end": 4160.95, "word": " يعني", "probability": 0.7247314453125}, {"start": 4160.95, "end": 4161.09, "word": " لو", "probability": 0.91796875}, {"start": 4161.09, "end": 4161.39, "word": " كانت", "probability": 0.985595703125}, {"start": 4161.39, "end": 4161.51, "word": " ال", "probability": 0.75732421875}, {"start": 4161.51, "end": 4161.77, "word": " series", "probability": 0.763671875}, {"start": 4161.77, "end": 4162.35, "word": " absolutely", "probability": 0.51123046875}, {"start": 4162.35, "end": 4163.17, "word": " convergent", "probability": 0.873779296875}, {"start": 4163.17, "end": 4164.97, "word": " أي", "probability": 0.705078125}, {"start": 4164.97, "end": 4165.75, "word": " ترتيب", "probability": 0.9432373046875}, {"start": 4165.75, "end": 4166.11, "word": " إلها", "probability": 0.6094970703125}, {"start": 4166.11, "end": 4166.51, "word": " آخر", "probability": 0.739990234375}, {"start": 4166.51, "end": 4166.81, "word": " وفق", "probability": 0.9773763020833334}, {"start": 4166.81, "end": 4167.15, "word": " تعريف", "probability": 0.99267578125}, {"start": 4167.15, "end": 4167.61, "word": " الترتيب", "probability": 0.9775390625}, {"start": 4167.61, "end": 4167.71, "word": " اللي", "probability": 0.9580078125}, {"start": 4167.71, "end": 4167.89, "word": " حنا", "probability": 0.5560302734375}, {"start": 4167.89, "end": 4168.21, "word": " حكينا", "probability": 0.9622802734375}, {"start": 4168.21, "end": 4168.67, "word": " عنه", "probability": 0.988525390625}, {"start": 4168.67, "end": 4169.33, "word": " اللي", "probability": 0.93505859375}, {"start": 4169.33, "end": 4169.55, "word": " هو", "probability": 0.9716796875}, {"start": 4169.55, "end": 4170.53, "word": " هيعطي", "probability": 0.94091796875}, {"start": 4170.53, "end": 4171.07, "word": " نفس", "probability": 0.997314453125}, {"start": 4171.07, "end": 4171.39, "word": " ال", "probability": 0.8583984375}, {"start": 4171.39, "end": 4171.67, "word": " limit", "probability": 0.95458984375}, {"start": 4171.67, "end": 4172.07, "word": " هتكون", "probability": 0.7659505208333334}, {"start": 4172.07, "end": 4172.63, "word": " converge", "probability": 0.7548828125}, {"start": 4172.63, "end": 4172.95, "word": " ويلها", "probability": 0.8264973958333334}, {"start": 4172.95, "end": 4173.29, "word": " نفس", "probability": 0.99755859375}, {"start": 4173.29, "end": 4173.51, "word": " ال", "probability": 0.9140625}, {"start": 4173.51, "end": 4173.81, "word": " limit", "probability": 0.98681640625}, {"start": 4173.81, "end": 4174.51, "word": " خلّينا", "probability": 0.7875}, {"start": 4174.51, "end": 4174.81, "word": " نشوف", "probability": 0.9944661458333334}, {"start": 4174.81, "end": 4175.57, "word": " البرهان", "probability": 0.8177490234375}, {"start": 4175.57, "end": 4175.85, "word": " ونمر", "probability": 0.8655598958333334}, {"start": 4175.85, "end": 4176.21, "word": " عليه", "probability": 0.95166015625}, {"start": 4176.21, "end": 4177.13, "word": " let", "probability": 0.802734375}, {"start": 4177.13, "end": 4177.61, "word": " summation", "probability": 0.8076171875}, {"start": 4177.61, "end": 4178.03, "word": " xn", "probability": 0.4605712890625}, {"start": 4178.03, "end": 4178.91, "word": " converge", "probability": 0.50537109375}, {"start": 4178.91, "end": 4179.51, "word": " to", "probability": 0.9677734375}, {"start": 4179.51, "end": 4179.87, "word": " x", "probability": 0.955078125}], "temperature": 1.0}, {"id": 186, "seek": 420213, "start": 4180.61, "end": 4202.13, "text": "and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let yn be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be a rearrangement of summation xn and let en be", "tokens": [474, 718, 17861, 312, 257, 39568, 518, 295, 28811, 2031, 77, 293, 718, 17861, 312, 257, 39568, 518, 295, 28811, 2031, 77, 293, 718, 17861, 312, 257, 39568, 518, 295, 28811, 2031, 77, 293, 718, 17861, 312, 257, 39568, 518, 295, 28811, 2031, 77, 293, 718, 17861, 312, 257, 39568, 518, 295, 28811, 2031, 77, 293, 718, 17861, 312, 257, 39568, 518, 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partial sums لسيريز الأصلية Xn والـ Sn' هي ميان ال sequence اللي هو تبعت ال absolute values", "tokens": [28814, 15730, 20328, 9640, 18871, 16254, 29538, 2423, 39184, 9264, 4724, 4587, 9381, 23328, 8978, 11296, 6156, 30241, 14111, 995, 13672, 1829, 39896, 2423, 8310, 295, 14641, 34499, 5296, 3794, 13546, 1829, 11622, 16247, 36520, 10632, 1783, 77, 16070, 39184, 9264, 6, 39896, 3714, 1829, 7649, 2423, 8310, 13672, 1829, 31439, 6055, 3555, 34268, 2423, 8236, 4190], "avg_logprob": -0.34671335744446724, "compression_ratio": 1.3855421686746987, "no_speech_prob": 0.0, "words": [{"start": 4429.6, "end": 4429.86, "word": "إذا", "probability": 0.6104736328125}, {"start": 4429.86, "end": 4430.16, "word": " صار", "probability": 0.7337646484375}, {"start": 4430.16, "end": 4430.46, "word": " عندي", "probability": 0.656494140625}, {"start": 4430.46, "end": 4430.72, "word": " هذه", "probability": 0.78955078125}, {"start": 4430.72, "end": 4430.84, "word": " الـ", "probability": 0.780517578125}, 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الجديدة Summation Yn اللي هي الـ Rearrangement شوف كده لأن Now let M limited M be such that all of the terms X1 لعند الـ Xm are contained as a sum and in T M Y1 لعند ال Ym إيش اللي بيقولوا هذا؟ وهذا الكلام اللي هو اللي إحنا بدنا نتهيئله و هو اللي بيفتحنا الطريق أمام البرهان", "tokens": [10721, 5016, 8315, 32239, 8592, 47525, 8315, 4386, 47525, 8315, 2423, 39184, 13934, 25724, 16254, 41891, 2746, 2174, 399, 398, 77, 13672, 1829, 39896, 2423, 39184, 1300, 2284, 933, 518, 13412, 38688, 9122, 3215, 3224, 5296, 33456, 823, 718, 376, 5567, 376, 312, 1270, 300, 439, 295, 264, 2115, 1783, 16, 5296, 3615, 41260, 2423, 39184, 1783, 76, 366, 16212, 382, 257, 2408, 293, 294, 314, 376, 398, 16, 5296, 3615, 41260, 2423, 398, 76, 11933, 1829, 8592, 13672, 1829, 4724, 1829, 39648, 14407, 23758, 22807, 37037, 15730, 2423, 28820, 10943, 13672, 1829, 31439, 13672, 1829, 11933, 5016, 8315, 47525, 8315, 8717, 47395, 1829, 19986, 43761, 4032, 31439, 13672, 1829, 4724, 1829, 5172, 2655, 5016, 8315, 41950, 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N capital إذا صار هذا أصغر من إبسلون زي إبسلون أصغر من اتنين إبسلون إذا صار عند الـTM ناقص X أصغر من اتنين إبسلون لكل M أكبر أو يساوي N وهذا يعني أنه", "tokens": [6027, 20292, 31439, 4724, 43042, 3224, 8032, 2655, 30544, 11933, 1829, 3224, 13412, 2304, 6027, 11296, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 11536, 5296, 28814, 1863, 3224, 23758, 5551, 12984, 3555, 2655, 8315, 11296, 20328, 5016, 1829, 5016, 3660, 5296, 28820, 426, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 426, 37037, 24192, 2423, 39184, 45, 39896, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 426, 11933, 15730, 20328, 9640, 43242, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 426, 11933, 15730, 2423, 39184, 45, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 426, 4238, 11933, 15730, 20328, 9640, 23758, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 11536, 30767, 1829, 11933, 3555, 3794, 1211, 11536, 5551, 9381, 17082, 2288, 9154, 1975, 2655, 1863, 9957, 11933, 3555, 3794, 1211, 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{"start": 4831.42, "end": 4831.54, "word": " يساوي", "probability": 0.9652099609375}, {"start": 4831.54, "end": 4831.72, "word": " N", "probability": 0.94189453125}, {"start": 4831.72, "end": 4832.04, "word": " إذا", "probability": 0.70654296875}, {"start": 4832.04, "end": 4832.54, "word": " الـN", "probability": 0.7789713541666666}, {"start": 4832.54, "end": 4832.96, "word": " أكبر", "probability": 0.767578125}, {"start": 4832.96, "end": 4833.06, "word": " أو", "probability": 0.833984375}, {"start": 4833.06, "end": 4833.1, "word": " يساوي", "probability": 0.9871826171875}, {"start": 4833.1, "end": 4833.26, "word": " N", "probability": 0.8662109375}, {"start": 4833.26, "end": 4833.6, "word": " capital", "probability": 0.060791015625}, {"start": 4833.6, "end": 4833.82, "word": " إذا", "probability": 0.861083984375}, {"start": 4833.82, "end": 4834.1, "word": " صار", "probability": 0.949462890625}, {"start": 4834.1, "end": 4834.28, "word": " هذا", "probability": 0.9189453125}, {"start": 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"end": 4837.32, "word": " الـTM", "probability": 0.7078450520833334}, {"start": 4837.32, "end": 4837.8, "word": " ناقص", "probability": 0.872802734375}, {"start": 4837.8, "end": 4838.12, "word": " X", "probability": 0.96728515625}, {"start": 4838.12, "end": 4838.74, "word": " أصغر", "probability": 0.99462890625}, {"start": 4838.74, "end": 4838.86, "word": " من", "probability": 0.9970703125}, {"start": 4838.86, "end": 4839.2, "word": " اتنين", "probability": 0.913330078125}, {"start": 4839.2, "end": 4839.78, "word": " إبسلون", "probability": 0.9935546875}, {"start": 4839.78, "end": 4840.6, "word": " لكل", "probability": 0.9853515625}, {"start": 4840.6, "end": 4840.98, "word": " M", "probability": 0.97607421875}, {"start": 4840.98, "end": 4841.44, "word": " أكبر", "probability": 0.98095703125}, {"start": 4841.44, "end": 4841.62, "word": " أو", "probability": 0.97802734375}, {"start": 4841.62, "end": 4842.3, "word": " يساوي", "probability": 0.9896240234375}, {"start": 4842.3, "end": 4843.36, "word": " N", "probability": 0.9775390625}, {"start": 4843.36, "end": 4844.32, "word": " وهذا", "probability": 0.896484375}, {"start": 4844.32, "end": 4844.76, "word": " يعني", "probability": 0.774658203125}, {"start": 4844.76, "end": 4845.8, "word": " أنه", "probability": 0.7147216796875}], "temperature": 1.0}, {"id": 214, "seek": 486368, "start": 4847.22, "end": 4863.68, "text": "اللي هي ال summation هذي طبعا هي عبارة عن sequence of partial sums لل YM صحيح ال summation لل YM converge و converge to main to X وبهيك بيكون احنا انهينا هنا اللي هو ال section الأول في شبط التسعة وإلى لقاينة", "tokens": [6027, 20292, 39896, 2423, 28811, 8032, 8848, 1829, 23032, 3555, 3615, 995, 39896, 6225, 3555, 9640, 3660, 18871, 8310, 295, 14641, 34499, 24976, 398, 44, 20328, 5016, 1829, 5016, 2423, 28811, 24976, 398, 44, 41881, 4032, 41881, 281, 2135, 281, 1783, 46599, 3224, 1829, 4117, 4724, 1829, 30544, 1975, 5016, 8315, 16472, 3224, 1829, 8315, 34105, 13672, 1829, 31439, 2423, 3541, 16247, 12610, 8978, 13412, 3555, 9566, 16712, 3794, 27884, 4032, 28814, 23942, 5296, 4587, 995, 9957, 3660], "avg_logprob": -0.33030064724668673, "compression_ratio": 1.5625, "no_speech_prob": 1.424551010131836e-05, "words": [{"start": 4847.22, "end": 4847.64, "word": "اللي", "probability": 0.6746826171875}, {"start": 4847.64, "end": 4848.06, "word": " هي", "probability": 0.90625}, {"start": 4848.06, "end": 4848.68, "word": " ال", "probability": 0.833984375}, {"start": 4848.68, "end": 4849.24, "word": " summation", "probability": 0.42041015625}, {"start": 4849.24, "end": 4849.96, "word": " هذي", "probability": 0.4583333333333333}, {"start": 4849.96, "end": 4850.26, "word": " طبعا", "probability": 0.8892822265625}, {"start": 4850.26, "end": 4850.5, "word": " هي", "probability": 0.5576171875}, {"start": 4850.5, "end": 4850.86, "word": " عبارة", "probability": 0.9891357421875}, {"start": 4850.86, "end": 4850.98, "word": " عن", "probability": 0.9951171875}, {"start": 4850.98, "end": 4851.38, "word": " sequence", "probability": 0.6513671875}, {"start": 4851.38, "end": 4851.6, "word": " of", "probability": 0.90673828125}, {"start": 4851.6, "end": 4851.86, "word": " partial", "probability": 0.92724609375}, {"start": 4851.86, "end": 4852.22, "word": " sums", "probability": 0.9580078125}, {"start": 4852.22, "end": 4852.42, "word": " لل", "probability": 0.68701171875}, {"start": 4852.42, "end": 4852.96, "word": " YM", "probability": 0.473876953125}, {"start": 4852.96, "end": 4853.56, "word": " صحيح", "probability": 0.6336669921875}, {"start": 4853.56, "end": 4853.6, "word": " ال", "probability": 0.82666015625}, {"start": 4853.6, "end": 4853.88, "word": " summation", "probability": 0.93359375}, {"start": 4853.88, "end": 4854.1, "word": " لل", "probability": 0.7294921875}, {"start": 4854.1, "end": 4854.58, "word": " YM", "probability": 0.97216796875}, {"start": 4854.58, "end": 4855.48, "word": " converge", "probability": 0.160888671875}, {"start": 4855.48, "end": 4855.68, "word": " و", "probability": 0.75341796875}, {"start": 4855.68, "end": 4856.1, "word": " converge", "probability": 0.740234375}, {"start": 4856.1, "end": 4856.3, "word": " to", "probability": 0.74951171875}, {"start": 4856.3, "end": 4856.62, "word": " main", "probability": 0.2279052734375}, {"start": 4856.62, "end": 4857.06, "word": " to", "probability": 0.5556640625}, {"start": 4857.06, "end": 4857.38, "word": " X", "probability": 0.93994140625}, {"start": 4857.38, "end": 4858.24, "word": " وبهيك", "probability": 0.7889404296875}, {"start": 4858.24, "end": 4858.6, "word": " بيكون", "probability": 0.8082682291666666}, {"start": 4858.6, "end": 4858.88, "word": " احنا", "probability": 0.8968098958333334}, {"start": 4858.88, "end": 4859.54, "word": " انهينا", "probability": 0.80389404296875}, {"start": 4859.54, "end": 4859.88, "word": " هنا", "probability": 0.724609375}, {"start": 4859.88, "end": 4860.42, "word": " اللي", "probability": 0.97119140625}, {"start": 4860.42, "end": 4860.52, "word": " هو", "probability": 0.97705078125}, {"start": 4860.52, "end": 4860.64, "word": " ال", "probability": 0.81298828125}, {"start": 4860.64, "end": 4860.94, "word": " section", "probability": 0.84326171875}, {"start": 4860.94, "end": 4861.38, "word": " الأول", "probability": 0.866455078125}, {"start": 4861.38, "end": 4861.58, "word": " في", "probability": 0.9423828125}, {"start": 4861.58, "end": 4861.8, "word": " شبط", "probability": 0.78515625}, {"start": 4861.8, "end": 4862.32, "word": " التسعة", "probability": 0.7320556640625}, {"start": 4862.32, "end": 4863.08, "word": " وإلى", "probability": 0.7129720052083334}, {"start": 4863.08, "end": 4863.68, "word": " لقاينة", "probability": 0.7857421875}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 4864.0696875, "duration_after_vad": 4763.633125000014}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..33bd48ee9ff23a891530281a92cdd789dd9d71c0
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/UEye6Xfb9c0_raw.srt
@@ -0,0 +1,4076 @@
+1
+00:00:05,180 --> 00:00:07,960
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:07,960 --> 00:00:10,980
+والصلاة والسلام على سيد المرسلين سيدنا محمد وعلى
+
+3
+00:00:10,980 --> 00:00:16,800
+آله وصحبه أجمعين هذه المحاضرة رقم 25 مساق تحليل
+
+4
+00:00:16,800 --> 00:00:21,880
+حقيقة 2 طلاب طالبات الجامعة الإسلامية قسم أرياضيات
+
+5
+00:00:21,880 --> 00:00:27,060
+كلية العلوم اليوم هنبدأ ان شاء الله بال chapter
+
+6
+00:00:27,060 --> 00:00:32,290
+الرابع في المادةاللي هو ال chapter 9 أو الجزء
+
+7
+00:00:32,290 --> 00:00:35,410
+الرابع من المادة اللي هو chapter 9 تحت عنوان
+
+8
+00:00:35,410 --> 00:00:39,670
+infinite series هنحكي اليوم في هذه المحاضرة على
+
+9
+00:00:39,670 --> 00:00:47,900
+اللي هو convergence of infinite seriesال infinite
+
+10
+00:00:47,900 --> 00:00:54,380
+series طبعا هو موضوع معهود للطلاب أو لطلبة العلوم
+
+11
+00:00:54,380 --> 00:00:58,440
+والهندسة بصورة عامة لإنه تم الحديث عنه في اللي هو
+
+12
+00:00:58,440 --> 00:01:03,340
+calculus باه الآن بنحكي عن اللي هو النظرة
+
+13
+00:01:03,340 --> 00:01:08,300
+التحليلية لاللي هو ال infinite series طبعا كنا
+
+14
+00:01:08,300 --> 00:01:14,180
+حكينا في في chapter تلاتة في تحليل حقيقي واحد على
+
+15
+00:01:14,180 --> 00:01:18,540
+اللي هي ال sequences الآن هذا مبني على اللي هو مين
+
+16
+00:01:18,540 --> 00:01:22,750
+ال sequencesخلّينا نشوف التعريف الأول 911
+
+17
+00:01:22,750 --> 00:01:27,130
+definition fx بساوة xn is a sequence in R يعني
+
+18
+00:01:27,130 --> 00:01:32,330
+فرضنا أن xn عبارة عن متتابع في R then the infinite
+
+19
+00:01:32,330 --> 00:01:37,990
+seriesاللي هي simply the series بنا نحكي عن ..
+
+20
+00:01:37,990 --> 00:01:42,330
+نعرف الأن ال infinite series أو ال series بناء على
+
+21
+00:01:42,330 --> 00:01:46,850
+اللي هي ال sequence x بساوي xn اللي هي the
+
+22
+00:01:46,850 --> 00:01:51,390
+infinite series generated by this sequence x is
+
+23
+00:01:51,390 --> 00:01:54,610
+the sequence S إذن ال infinite series هي عبارة عن
+
+24
+00:01:54,610 --> 00:02:00,170
+إيش؟ بتعرفها عبارة عن sequence S بساوي SK بس إيش
+
+25
+00:02:00,170 --> 00:02:05,230
+ال SK اللي هي اللي بنسميها اللي هيsequence of
+
+26
+00:02:05,230 --> 00:02:11,510
+partial sums يعني ال S K defined by S1 ايش بتساوي؟
+
+27
+00:02:11,510 --> 00:02:17,270
+X1 S2 ايش بتساوي؟ X1 زاد X2 لاحظوا بقى اجمع
+
+28
+00:02:17,270 --> 00:02:19,430
+العناصر اللي وين موجودة اللي في ال sequence
+
+29
+00:02:19,430 --> 00:02:24,490
+الأصلية اللي هي بتعمل generation ل اللي هي ايش؟الـ
+
+30
+00:02:24,490 --> 00:02:28,710
+sequence of partial sums الـ S K S 3 إيش هتكون X 1
+
+31
+00:02:28,710 --> 00:02:34,290
+زاد X 2 زاد X 3 S K in general هي عبارة عن X 1 زاد
+
+32
+00:02:34,290 --> 00:02:39,820
+X 2 عند من؟ اللي عند X K أو بمعنى آخرSk
+
+33
+00:02:39,820 --> 00:02:56,100
+-1�
+
+34
+00:02:56,560 --> 00:03:02,220
+لأن if S converges, we refer to limit S as the sum
+
+35
+00:03:02,220 --> 00:03:06,160
+of infinite series يعني لو كانت الـSK عندي هذه
+
+36
+00:03:06,160 --> 00:03:10,880
+converges to some limit S، بنقول احنا اللي هو
+
+37
+00:03:10,880 --> 00:03:16,260
+limitالـ S K اللي هو بيساوي S اللي هو اللي بنسمي S
+
+38
+00:03:16,260 --> 00:03:19,500
+K طبعاً goes to infinity أو N اللي بدكيها اللي هي
+
+39
+00:03:19,500 --> 00:03:24,120
+بيساوي ال summation اللي هي X K كي من عند واحد إلى
+
+40
+00:03:24,120 --> 00:03:29,780
+وين إلى ما لانهية سمي هذه S N وهذه S N وهذه S N كل
+
+41
+00:03:29,780 --> 00:03:34,440
+برج للاسهذه اللي بنسميها الـ infinite series اللي
+
+42
+00:03:34,440 --> 00:03:37,300
+هي في حالة ال convergence أو ال limit هذه exist
+
+43
+00:03:37,300 --> 00:03:40,520
+تبع ال sequence of partial sums بنسميها اللي هي
+
+44
+00:03:40,520 --> 00:03:45,880
+limit exist بنسميها S لأن ال elements XN are
+
+45
+00:03:45,880 --> 00:03:50,550
+called the termsand S and are called the partial
+
+46
+00:03:50,550 --> 00:03:54,450
+sums of this infinite series يعني هدولة اللي هي ال
+
+47
+00:03:54,450 --> 00:03:58,390
+.. ال xk's بنسميها اللي هي ال terms تبعة ال
+
+48
+00:03:58,390 --> 00:04:02,170
+infinite series و ال S and أو ال Sk بنسميها اللي
+
+49
+00:04:02,170 --> 00:04:06,310
+هي عبارة عن أيش ال sequence of partial sums أو
+
+50
+00:04:06,310 --> 00:04:10,330
+بنسميهم are partial sums of this infinite series
+
+51
+00:04:10,330 --> 00:04:14,650
+نيجي الآن بما أنه اللي هو في النهاية ال series
+
+52
+00:04:14,650 --> 00:04:17,960
+اللي عندنا هذهعبارة عن limit في حالة ال
+
+53
+00:04:17,960 --> 00:04:22,080
+convergence هو عبارة عن limit of a sequence إذا
+
+54
+00:04:22,080 --> 00:04:25,520
+تنطبق عليها قوانين ال sequences اللي أخدناها في
+
+55
+00:04:25,520 --> 00:04:30,960
+chapter 3 إذا متوقع إنه احنا نستخدم ما تحدثنا فيه
+
+56
+00:04:30,960 --> 00:04:36,420
+في chapter 3 أو نتائجها للحصول على النظريات اللي
+
+57
+00:04:36,420 --> 00:04:37,520
+احنا بدنا نبرهنها
+
+58
+00:04:40,050 --> 00:04:44,350
+النظرية الأولى تسعة واحد اتنين theorem a if the
+
+59
+00:04:44,350 --> 00:04:48,610
+series summation xn and summation yn converge يعني
+
+60
+00:04:48,610 --> 00:04:52,490
+السيريز هذه من n من واحد إلى ما لا نهاية لو فرضنا
+
+61
+00:04:52,490 --> 00:04:55,670
+أن هذا ال sum موجود و هذا ال sum موجود يعني
+
+62
+00:04:55,670 --> 00:05:00,010
+السيريز هذه اللي هي sequence of partial sums
+
+63
+00:05:00,010 --> 00:05:03,850
+سبعتها ال limit لها موجودة و ال summation yn
+
+64
+00:05:03,850 --> 00:05:07,310
+converge يعني السيريزسيكوانس الـ partial sums
+
+65
+00:05:07,310 --> 00:05:10,790
+السبعية ده برضه موجودة limit لها then the series
+
+66
+00:05:10,790 --> 00:05:16,190
+summation xn زائد yn برضه converges ومش هيك كمان
+
+67
+00:05:16,190 --> 00:05:20,510
+وقيمة السيكوه السيريز الجديدة summation xn زائد yn
+
+68
+00:05:20,510 --> 00:05:26,380
+بساوي السيريز الأولى زائد السيريز الثانيةالان اشي
+
+69
+00:05:26,380 --> 00:05:32,100
+مشابه ل 4 Xn نقص Yn بيسوي summation Xn نقص
+
+70
+00:05:32,100 --> 00:05:36,940
+summation Min Yn الان بي if the series summation
+
+71
+00:05:36,940 --> 00:05:41,560
+Xn is convergent is convergent and C is any real
+
+72
+00:05:41,560 --> 00:05:46,120
+number then the series summation C في Xn برضه
+
+73
+00:05:46,120 --> 00:05:50,960
+هيخليها تظلها convergent و بيطلع عند ال summation
+
+74
+00:05:50,960 --> 00:05:55,930
+ل C Xn بيسوي C في ال summation ل Minللـ XL الان
+
+75
+00:05:55,930 --> 00:06:00,990
+مدام نحكي عن series بده ترجم اللي هو ال series الى
+
+76
+00:06:00,990 --> 00:06:04,450
+sequences الى sequence of partial sums واستخدم
+
+77
+00:06:04,450 --> 00:06:07,730
+اللي هي قوانين ال limits القديمة يعني الموضوع
+
+78
+00:06:07,730 --> 00:06:13,160
+البرهان أمر سهل ان شاء اللهالان نفترض انه السميق
+
+79
+00:06:13,160 --> 00:06:16,580
+.. مدام هذا converts و هذا converts بدي افترض انه
+
+80
+00:06:16,580 --> 00:06:21,580
+هذه قيمتها بساوي S وهذا قيمتها مثلا بساوي S' يعني
+
+81
+00:06:21,580 --> 00:06:25,300
+جاي let S بتساوي summation xi I من واحد إلى وين
+
+82
+00:06:25,300 --> 00:06:27,140
+إلى اللي هي
+
+83
+00:06:52,810 --> 00:06:54,210
+كيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكيكي
+
+84
+00:06:54,500 --> 00:07:01,180
+الـ XI لأن هو جالني ان الـ limit لأن ال series هذه
+
+85
+00:07:01,180 --> 00:07:05,160
+لما ماحطش اشي معناته من واحد إلى مال النهاية لما
+
+86
+00:07:05,160 --> 00:07:09,090
+لاحطش اشي معناته بقصد ال finite sumالان هو جاهز
+
+87
+00:07:09,090 --> 00:07:11,790
+لأن هذا الـ series الـ summation هو موجود وهذا
+
+88
+00:07:11,790 --> 00:07:16,350
+موجود يعني بمعنى آخر ال limit لل S اللي هي هذه
+
+89
+00:07:16,350 --> 00:07:20,690
+limit لل S بساوي ال series هذه و limit لل S برايم
+
+90
+00:07:20,690 --> 00:07:24,650
+بساوي ال series هذه من واحد إلى ما لا نهاية الان
+
+91
+00:07:24,650 --> 00:07:28,990
+صارت عندى هذا عبارة عن limit of what؟ of sequence
+
+92
+00:07:29,460 --> 00:07:33,960
+هذا عبارة عن limit of what؟ of sequence إذا حسب
+
+93
+00:07:33,960 --> 00:07:39,280
+اللي هو اللي هي النظريات اللي هي في chapter تلاتة
+
+94
+00:07:39,280 --> 00:07:43,140
+ال limit إذا كانت ال limit لل sequence exist و
+
+95
+00:07:43,140 --> 00:07:45,560
+limit لل sequence التانية exist فبكون limit
+
+96
+00:07:45,560 --> 00:07:49,560
+المجموع بتوزع يعني limit S زايد S prime إيش
+
+97
+00:07:49,560 --> 00:07:53,720
+هيساوي؟ limit لل sequence الأولى زايد limit لل
+
+98
+00:07:53,720 --> 00:07:56,760
+sequence التانية هذا من وين جبته؟ هذا جبته من
+
+99
+00:07:56,760 --> 00:08:03,000
+chapter تلاتة طيبالان لكن S زي S برايم ك sequence
+
+100
+00:08:03,000 --> 00:08:06,540
+هذا S زي S برايم مجموعة و sequences يعني ال
+
+101
+00:08:06,540 --> 00:08:09,300
+sequence of partial sums الأولى زي ال sequence of
+
+102
+00:08:09,300 --> 00:08:12,880
+partial sums التانية إيش هتساوي؟ اللي هي عبارة عن
+
+103
+00:08:12,880 --> 00:08:16,760
+ال sequence اللي الصمشي XI I من عند واحد لعينة K
+
+104
+00:08:16,760 --> 00:08:21,460
+هذا اللي صميتها أنا S K زي مين S K برايم هدولة
+
+105
+00:08:21,460 --> 00:08:25,410
+finite هدولة termsالان هدولة finite و هدولة finite
+
+106
+00:08:25,410 --> 00:08:31,090
+إذا المجموح من إيش بيساوي summation xi زائد yi من
+
+107
+00:08:31,090 --> 00:08:36,490
+عند واحد لعند مين لعند k هذه هي عبارة عن sk زائد
+
+108
+00:08:36,490 --> 00:08:42,250
+sk برايم صارت عندي sequence جديدة اللي هي s زائد s
+
+109
+00:08:42,250 --> 00:08:47,840
+برايم هذهS زي ده S برايم إيش هتساوي يا جماعة اللي
+
+110
+00:08:47,840 --> 00:08:52,660
+هو ال sequence اللي عندي اللي ال .. ال .. ال K
+
+111
+00:08:52,660 --> 00:08:56,440
+ثيرم إلها أو اللي هو ال sequence of partial sums
+
+112
+00:08:56,440 --> 00:09:02,680
+إلها هو عبارة عن XI زي ده YI I من واحد لعند K هذا
+
+113
+00:09:02,680 --> 00:09:09,500
+الآن صار بساوي هذا طيب الآن بده نبدأ ناخد ال limit
+
+114
+00:09:09,500 --> 00:09:17,310
+and then ال summationبناء عليه الـ summation الـ S
+
+115
+00:09:17,310 --> 00:09:22,210
+زي الـ S' اللي هو حسب اللي هنا بيصير limit الـ S
+
+116
+00:09:22,210 --> 00:09:26,710
+زي الـ S' هذه اللي هو limit لهذه الـ sequence إيش
+
+117
+00:09:26,710 --> 00:09:29,790
+هيصير عند هذا الـ limit لهذه الـ sequence؟ هيعندي
+
+118
+00:09:29,790 --> 00:09:36,150
+هذا بساوي اللي هو هذا المقدار طيب limit S زي الـ
+
+119
+00:09:36,150 --> 00:09:42,010
+S' اللي هو limit هذا المقدار اللي هو limitS زائد S
+
+120
+00:09:42,010 --> 00:09:49,710
+برايم بساوي limit ال summation XI زائد YI I من عند
+
+121
+00:09:49,710 --> 00:09:54,070
+واحد لعند N أو لعند K زي ما بدي as N goes to
+
+122
+00:09:54,070 --> 00:09:57,390
+infinity هذا notation عبارة عن notation لمين لل
+
+123
+00:09:57,390 --> 00:10:02,630
+summation لأنه صار هذا existمن أين؟ لأن هذا زائد
+
+124
+00:10:02,630 --> 00:10:06,170
+هذا exist لأن هذا exist بما أن هذا exist نحسس إنه
+
+125
+00:10:06,170 --> 00:10:11,250
+summation xi زائد iii من عند واحد إلى ما لا نهاية
+
+126
+00:10:11,250 --> 00:10:14,850
+من جهة أخرى هذا أخبرني أيش بيساوي limit S زائد مين
+
+127
+00:10:14,850 --> 00:10:19,630
+S prime limit S اللي هو limit هذه limit هذه أيش
+
+128
+00:10:19,630 --> 00:10:27,710
+هيكون limit Sبساوي summation xi I من واحد لمن الآن
+
+129
+00:10:27,710 --> 00:10:33,010
+لان K راحت الى وين الى ماله نهاية هذه limitها اذا
+
+130
+00:10:33,010 --> 00:10:36,730
+بتساوي اللي هو I من واحد لعند ماله نهاية و limit S
+
+131
+00:10:36,730 --> 00:10:41,110
+prime زيهاSK بتروح لما لا نهاية أو N بتروح لما لا
+
+132
+00:10:41,110 --> 00:10:44,370
+نهاية بتساوي summation YI I من واحد إلى ما لا
+
+133
+00:10:44,370 --> 00:10:49,050
+نهاية بناء على أن هذه بتساوي هذه إذا صار المقدار
+
+134
+00:10:49,050 --> 00:10:54,070
+هذا بساوي المقدار هذا زائد المقدار هذا أو بمعنى
+
+135
+00:10:54,070 --> 00:10:57,130
+آخر summation XN زايد YN بساوي summation XN زايد
+
+136
+00:10:57,130 --> 00:11:00,930
+YN وهو المطلوب
+
+137
+00:11:02,570 --> 00:11:05,930
+Similarly, summation XN نقص YN بسوء summation XN
+
+138
+00:11:05,930 --> 00:11:09,410
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+139
+00:11:09,410 --> 00:11:09,610
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+140
+00:11:09,610 --> 00:11:09,610
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+141
+00:11:09,610 --> 00:11:09,790
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+142
+00:11:09,790 --> 00:11:09,790
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+143
+00:11:09,790 --> 00:11:09,790
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+144
+00:11:09,790 --> 00:11:09,830
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+145
+00:11:09,830 --> 00:11:12,850
+نقص YN بسوء summation XN نقص YN بسوء summation XN
+
+146
+00:11:12,850 --> 00:11:15,170
+نقص YN بسوء summation XN ن
+
+147
+00:11:19,400 --> 00:11:25,440
+الـ S ناقص S برايم بساوي limit S ناقص limit S
+
+148
+00:11:25,440 --> 00:11:29,240
+برايم و بنجري نفس اللي جريناها قبل بشوية و اللي
+
+149
+00:11:29,240 --> 00:11:34,020
+الثانية لأنه بنحول و كأن حديثنا من series ل
+
+150
+00:11:34,020 --> 00:11:38,560
+sequence و كل قوانين ال sequence تم إثباتها سابقا
+
+151
+00:11:38,560 --> 00:11:43,100
+بـ mean بواسطة اللي هو شعب طر تلاتة أو في تحليل
+
+152
+00:11:43,100 --> 00:11:50,570
+حقيقي واحد الان الأخيرة ال summationلـCn بيساوي
+
+153
+00:11:50,570 --> 00:11:56,250
+الـC في الصممشي للـXn برضه اعتمادا على limit الـC
+
+154
+00:11:56,250 --> 00:12:00,990
+في الـS بيساوي الـC في limit الـS وبنجري نفس اللي
+
+155
+00:12:00,990 --> 00:12:04,950
+أجرناها قبل بشوية اللي هو تحويلها من .. اللي هو
+
+156
+00:12:04,950 --> 00:12:09,330
+sequence .. من .. من series إلى sequences ومن ثم
+
+157
+00:12:09,330 --> 00:12:15,690
+استخدام اللي هي قوانين الـsequences طيب الان نجي ل
+
+158
+00:12:15,690 --> 00:12:25,090
+اللي هي اللممةاللي هي اللمّة
+
+159
+00:12:25,090 --> 00:12:28,970
+اللي بعدها اللي هي المشهورة
+
+160
+00:12:30,530 --> 00:12:34,010
+وهذه test for divergence في الواقع كنا نقول عنها
+
+161
+00:12:34,010 --> 00:12:39,130
+في calculus باه ايش ال sequence هذه او اللي هي
+
+162
+00:12:39,130 --> 00:12:42,330
+sequence of partial sums او ال series بقول إذا
+
+163
+00:12:42,330 --> 00:12:46,490
+كانت ال series summation xn converges in R إذا كان
+
+164
+00:12:46,490 --> 00:12:50,650
+اللي هي ال summation لل xn converges in R لازم
+
+165
+00:12:50,650 --> 00:12:54,810
+يكون limit اللي هي ال sequence الأصلي اللي بنكون
+
+166
+00:12:54,810 --> 00:12:58,310
+منها sequence of partial sums لازم limit هيش يساوي
+
+167
+00:12:59,640 --> 00:13:04,000
+يساوي سفر وبناء عليه لو كانت ال limit لها تساوي
+
+168
+00:13:04,000 --> 00:13:07,640
+سفر على طول بنحكم على summation لل Xn إيش ماله
+
+169
+00:13:07,640 --> 00:13:12,580
+diverse وكنا نقول هذا test of divergence مش test
+
+170
+00:13:12,580 --> 00:13:16,360
+of convergence لأن ممكن ال limit تساوي سفر لكن ال
+
+171
+00:13:16,360 --> 00:13:19,960
+summation لل Xn يكون diverse والمثال اللي انتوا
+
+172
+00:13:19,960 --> 00:13:24,160
+مشهور اللي هو summation 1 على n هذا اللي هو limit
+
+173
+00:13:24,680 --> 00:13:29,900
+الواحد على ان بيساوي سفر but this series diverges
+
+174
+00:13:29,900 --> 00:13:33,860
+زي ما هنشوف بعد شوية و زي ما احنا اصلا عارفين ولكن
+
+175
+00:13:33,860 --> 00:13:38,780
+احنا هنثبته اثبات الان كنا اللي هو اخدناها معلومة
+
+176
+00:13:38,780 --> 00:13:44,890
+في تفاضل بام طيبصلى النبي عليه الصلاة والسلام الان
+
+177
+00:13:44,890 --> 00:13:49,650
+by definition the convergence of summation xn
+
+178
+00:13:49,650 --> 00:13:54,430
+means that limit ال sk شماله exist ايش ال sk
+
+179
+00:13:54,430 --> 00:13:59,570
+عارفين ايش ال sk ال sk هو عبارة عن x واحد زاد x
+
+180
+00:13:59,570 --> 00:14:06,190
+اتنين لما اصل عند xk الان بما انه اللي هو limit sk
+
+181
+00:14:06,190 --> 00:14:11,280
+بساوي اللي هو ال summation لل xnلأن نحن مفترضين أن
+
+182
+00:14:11,280 --> 00:14:14,900
+نصمش ال Xn إيش ما له exist مثلا م exist ونقول صمش
+
+183
+00:14:14,900 --> 00:14:20,120
+لل Xn اللي هو بساوي limit ال S K الان من جهة أخرى
+
+184
+00:14:20,120 --> 00:14:24,640
+لو جيت حسبت انت ال S K ناقص S K minus واحد ال S K
+
+185
+00:14:24,640 --> 00:14:28,760
+ناقص S K minus واحد ايش هيساوي؟ هذي من عند واحد
+
+186
+00:14:28,760 --> 00:14:33,020
+لعند K وهذه من عند واحد لعند K minus واحد حصل ترحل
+
+187
+00:14:33,020 --> 00:14:38,230
+هيظلمين ال X Kالان خد ال limit للجهتين S k goes to
+
+188
+00:14:38,230 --> 00:14:42,930
+infinity بصير limit S k as k goes to infinity ناقص
+
+189
+00:14:42,930 --> 00:14:46,450
+limit S k minus واحد لما k تروح لما لا نهاية
+
+190
+00:14:46,450 --> 00:14:52,210
+بيساوي limit X k as k goes to infinity الان اتجرأت
+
+191
+00:14:52,210 --> 00:14:54,590
+ووزعت ال limit لأنه يعرف هذه ال sequence of
+
+192
+00:14:54,590 --> 00:14:58,350
+partial sums اللي بتبدأ من عند واحد وبضالها طالعة
+
+193
+00:14:58,350 --> 00:15:02,310
+limit هل هو summation لل X nexists ما احنا نقوله
+
+194
+00:15:02,310 --> 00:15:06,170
+لأ وهذه اصلا الـ tail تبعتها من عند الواحد وطالع
+
+195
+00:15:06,170 --> 00:15:10,490
+limitها برضه ايش ما ساوي اللي هو صماش ال اكسان لان
+
+196
+00:15:10,490 --> 00:15:13,130
+احنا بنعرف limit ال sequence و ال tail لها زي بعض
+
+197
+00:15:13,130 --> 00:15:16,890
+ما دامت ال limit exist لذن هدولة هذه ناقصت ايش
+
+198
+00:15:16,890 --> 00:15:21,010
+هيطلع سفر لذن صار limit ال اكسان ايش بساوي بساوي
+
+199
+00:15:21,010 --> 00:15:27,480
+سفربكون عندنا الي هو وصلنا ل theorem 914 اللي هي
+
+200
+00:15:27,480 --> 00:15:39,680
+خلينا نشوف ايش هذه ال theorem بتقول ركزوا معانا بس
+
+201
+00:15:39,680 --> 00:15:46,160
+اذكركم بشغلةإنه إحنا في الـ monotone convergence
+
+202
+00:15:46,160 --> 00:15:52,500
+theorem في تحليل حقيقي واحد كنا نقول إنه إذا كانت
+
+203
+00:15:52,500 --> 00:15:55,140
+ال sequence increasing أو decreasing يعني لو كانت
+
+204
+00:15:55,140 --> 00:15:59,780
+monotone then the sequence converge if and only if
+
+205
+00:15:59,780 --> 00:16:04,180
+it is boundedهذه اللي هو في حالة مين الـ monotone
+
+206
+00:16:04,180 --> 00:16:07,620
+functions لكن in general كنا نقول إن الـ sequence
+
+207
+00:16:07,620 --> 00:16:10,360
+is convergent على طول اللي بيعطينا then it is
+
+208
+00:16:10,360 --> 00:16:15,820
+bounded but it may be bounded بط إي شمالها is not
+
+209
+00:16:15,820 --> 00:16:21,140
+convergent وكنا نقول ناخد مثال ناقص واحد على أن ال
+
+210
+00:16:21,140 --> 00:16:26,200
+sequence هذه أو ناقص واحد أس أن ناقص واحد أس أننقص
+
+211
+00:16:26,200 --> 00:16:29,520
+واحد وثاني هذه الـ sequence bounded لأن قيمتها الـ
+
+212
+00:16:29,520 --> 00:16:32,940
+absolute value لها واحد but اللي هو ال limit اللي
+
+213
+00:16:32,940 --> 00:16:35,500
+لها does not exist لكن في حالة الـ monotone
+
+214
+00:16:35,500 --> 00:16:39,240
+function لأ ال boundedness يكافئ ال convergence
+
+215
+00:16:39,240 --> 00:16:43,880
+لأن هذا بدنا اللي هو الآن نطبق على ال monotone
+
+216
+00:16:43,880 --> 00:16:48,760
+convergence theorem أو نستخدمها في إثبات نظريتنا
+
+217
+00:16:48,760 --> 00:16:52,980
+اللي هي 914 theorem لت XN be a sequence of non
+
+218
+00:16:52,980 --> 00:16:57,500
+-negative real numbersإذا حكينا عن مين عن sequence
+
+219
+00:16:57,500 --> 00:17:01,900
+xn عنصرها إيش ما لها non negative real numbers
+
+220
+00:17:01,900 --> 00:17:08,630
+يعني أكبر أو تساوي سفرثم الـ summation لـ xn من
+
+221
+00:17:08,630 --> 00:17:12,050
+السيريز تبعتها converts if and only if the
+
+222
+00:17:12,050 --> 00:17:15,750
+sequence S بتسوى Sk of partial sums is bounded
+
+223
+00:17:15,750 --> 00:17:21,110
+يعني الآن السيريز ال summation xn اللي هي هتكون
+
+224
+00:17:21,110 --> 00:17:26,470
+converts إذا و فقط إذا ال Sk إشمالها is bounded in
+
+225
+00:17:26,470 --> 00:17:30,970
+this case ال summation لـ xn بسوى limit لSk بسوى
+
+226
+00:17:30,970 --> 00:17:36,990
+سبريمان لSk يعني الآن و كأنه بيقوللي إنه limitالـ
+
+227
+00:17:36,990 --> 00:17:45,130
+SK exists if and only if SK is bounded هذه الـ SK
+
+228
+00:17:45,130 --> 00:17:48,410
+هي الـ sequence of partial sums تبع السيريز limit
+
+229
+00:17:48,410 --> 00:17:52,090
+الـ SK هي السيريز نفسها الـ x and n من واحد إلى
+
+230
+00:17:52,090 --> 00:17:58,110
+مانع نهايةالآن باختصار اللي هو هنقوله أنه أصلا هذه
+
+231
+00:17:58,110 --> 00:18:01,810
+ال SQL sequence of partial sums is an increasing
+
+232
+00:18:01,810 --> 00:18:05,950
+sequence مدام increasing يعني بمعنى آخر monotone
+
+233
+00:18:05,950 --> 00:18:10,090
+أو بالظبط اللي هو خلّيني أقول واحدة من ال monotone
+
+234
+00:18:10,090 --> 00:18:12,530
+هي increasing ت increasing هي increasing علنا الآن
+
+235
+00:18:12,530 --> 00:18:17,490
+مدام increasingهيكون عنده الـ Existence limit
+
+236
+00:18:17,490 --> 00:18:21,330
+يكافي الـ Boundedness ويكون خلصنا برهان النظرية
+
+237
+00:18:21,330 --> 00:18:24,310
+حسب الـ Newton convergence theorem ومش هيك وهيكون
+
+238
+00:18:24,310 --> 00:18:30,770
+limit هاش ما لمدام increasing السبريمام لعنصرها
+
+239
+00:18:30,770 --> 00:18:34,410
+هذه الـ Newton convergence theorem خليني نشوف هذا
+
+240
+00:18:34,410 --> 00:18:39,680
+الكلام بالتفصيل اللي حكيتهبما أن xn أكبر يساوي 0
+
+241
+00:18:39,680 --> 00:18:43,300
+الـ xn أكبر يساوي 0 then the sequence of partial
+
+242
+00:18:43,300 --> 00:18:48,540
+sums is monotone increasing function ليش؟ لأن S1
+
+243
+00:18:48,540 --> 00:18:55,720
+عبارة عن x1 S1 عبارة عن x1 أكيد أصغر من x1 زاد x2
+
+244
+00:18:55,720 --> 00:19:00,670
+التي هي مين؟ S2لأن هذا الـ X2 اللي ضفته أكبر أو
+
+245
+00:19:00,670 --> 00:19:06,090
+يساوي 0 وهذا أصغر من S3 اللي هو X1 زي X2 زي X3 in
+
+246
+00:19:06,090 --> 00:19:12,890
+general سيكون ال S K أصغر من S K زي 1 أو أصغر أو
+
+247
+00:19:12,890 --> 00:19:16,830
+يساوي حسب اللي هو الإضافة 0 أو كمية موجبة ماشي
+
+248
+00:19:16,830 --> 00:19:21,340
+الحال إذا صارت عندي ال sequence of partial sumاللي
+
+249
+00:19:21,340 --> 00:19:24,560
+هو صارت عبارة عن increasing إذا according to the
+
+250
+00:19:24,560 --> 00:19:27,240
+monotone convergence theorem اللي قلت قبل بشوية
+
+251
+00:19:27,240 --> 00:19:30,480
+اللي هي بالظبط رقمها تلاتة تلاتة اتنين the
+
+252
+00:19:30,480 --> 00:19:37,580
+sequence S of partial sums converts if and only if
+
+253
+00:19:37,580 --> 00:19:41,640
+it is boundedيعني مدام الـ sequence of partial
+
+254
+00:19:41,640 --> 00:19:44,640
+sums as converges يعني الصممش للـ xn يعني السريز
+
+255
+00:19:44,640 --> 00:19:50,000
+هذه اللي هي is convergent if and only if it is
+
+256
+00:19:50,000 --> 00:19:52,260
+bounded اللي هي ال sequence of partial sums طبعاً
+
+257
+00:19:52,670 --> 00:19:57,030
+عند الـ summation لـ xn مش مساوي limit الـsk و الـ
+
+258
+00:19:57,030 --> 00:20:00,430
+sk increasing limit همين حيكون ال sub .. طبعا
+
+259
+00:20:00,430 --> 00:20:02,790
+increasing و صارك bounded مادام bounded then فيه
+
+260
+00:20:02,790 --> 00:20:06,470
+لها bound إذا صار عند ال supremum ل ال x .. ل ال
+
+261
+00:20:06,470 --> 00:20:09,190
+sk هو عبارة عن limit ال sk حسب ال molten
+
+262
+00:20:09,190 --> 00:20:12,410
+convergence theorem و سوى ال summation ل xn و هو
+
+263
+00:20:12,410 --> 00:20:18,270
+المطلوب طيب هذا لما يكون ال terms سبعات ال
+
+264
+00:20:18,270 --> 00:20:23,510
+sequence يشملها موجبات أو non negativeالانكوشي
+
+265
+00:20:23,510 --> 00:20:29,610
+criterion for اللي هو series احنا زي ما كنا نقول
+
+266
+00:20:29,610 --> 00:20:33,130
+في التحليل الحقيقي واحد احيانا ما بنقدرش نعرف ايش
+
+267
+00:20:33,130 --> 00:20:37,990
+ال limit لكن بنكون عارفين ان ال sequence نفسها
+
+268
+00:20:37,990 --> 00:20:42,190
+converts فبدنا نعبر عن ال convergence لل sequence
+
+269
+00:20:42,190 --> 00:20:45,770
+بطريقة اللي هي ال terms تبعتها مش عن طريقة اللي هي
+
+270
+00:20:45,770 --> 00:20:50,450
+معرفة ال limitفكنا نقول إن الـ xn مثلاً is a
+
+271
+00:20:50,450 --> 00:20:52,850
+Cauchy sequence if and only if for every y أكبر من
+
+272
+00:20:52,850 --> 00:20:56,190
+0 there exists k element in N such that for every
+
+273
+00:20:56,190 --> 00:21:00,290
+n و m أكبر من k يكون الـ absolute value للـ xn
+
+274
+00:21:00,290 --> 00:21:03,990
+ناقص xm أصغر من إبسلول نحن نجيب اللي هو Cauchy
+
+275
+00:21:03,990 --> 00:21:07,750
+criterion مشابه إليها بس لمين لحالة السيريز ما هي
+
+276
+00:21:07,750 --> 00:21:11,170
+هذه السيريز زي ما قلنا هي عبارة عن في النهاية
+
+277
+00:21:11,170 --> 00:21:17,950
+sequence of partial sumsطيب يا جماعة نيجي لكوشيك
+
+278
+00:21:17,950 --> 00:21:21,250
+ال 915 اللي هي quotient criterion for series the
+
+279
+00:21:21,250 --> 00:21:25,370
+series summation xn in R converts if and only if
+
+280
+00:21:25,370 --> 00:21:29,690
+for every ي أكبر من سفر there is a natural number
+
+281
+00:21:29,690 --> 00:21:35,010
+M such that if M و N أكبر أو سوى M of Y هكون
+
+282
+00:21:35,010 --> 00:21:40,020
+الأسئلة ماقصة أسئلة M اللي هو بسوى xn زائد 1الـ M
+
+283
+00:21:40,020 --> 00:21:45,060
+أشملها أكبر أو يساوي N فبصير الـ SM عبارة عن X1 زي
+
+284
+00:21:45,060 --> 00:21:49,660
+X2 و بياخد في طريقه الـ XN و بعدين بكمل XN زي X1
+
+285
+00:21:49,660 --> 00:21:54,000
+لما أصل عند XM لما نطرح منه الـ SM اللي هي X1 عند
+
+286
+00:21:54,000 --> 00:21:58,920
+الـ XN بروح كلهم بضال مين اللي هي XN زي X1 و XN زي
+
+287
+00:21:58,920 --> 00:22:03,040
+X2 لما أصل عند مين عند XM هذا حيكون أصغر من مين من
+
+288
+00:22:03,040 --> 00:22:07,560
+إبسلون هذا اللي بنلثبتهإنه الـ series summation xn
+
+289
+00:22:07,560 --> 00:22:12,300
+converts if and only if اللي هي الـ Cauchy
+
+290
+00:22:12,300 --> 00:22:15,760
+criterion يعني مين هو لكل ي أكبر من 0 there exists
+
+291
+00:22:15,760 --> 00:22:19,420
+M such that for every M أكبر من M أكبر أو يساوي
+
+292
+00:22:19,420 --> 00:22:22,900
+الـ M اللي راجناها هيكون Absolute value أسمها M
+
+293
+00:22:22,900 --> 00:22:26,980
+نقص أس N اللي هو بساوية هذا المقدار أصغر من أيش من
+
+294
+00:22:26,980 --> 00:22:31,670
+يالان في الواقع احنا قلنا في نظرية سابقة .. نظرية
+
+295
+00:22:31,670 --> 00:22:36,090
+في السابقة في التحليل الحقيقي واحد ان الـ sequence
+
+296
+00:22:36,090 --> 00:22:40,610
+converges if and only if it is cauchy وهذه اللي
+
+297
+00:22:40,610 --> 00:22:46,950
+هنستخدمها، شوفواصلى الله عليه وسلم الصممش للـ xn
+
+298
+00:22:46,950 --> 00:22:51,090
+converts in R if and only if the sequence of
+
+299
+00:22:51,090 --> 00:22:55,990
+partial sums الـ Sn converts in R هذا التعريف الـ
+
+300
+00:22:55,990 --> 00:23:00,150
+sequence Sn converts in R if and only if اللي قلت
+
+301
+00:23:00,150 --> 00:23:04,250
+عنها نضية 3.5.4 اللي هي the convergent if and only
+
+302
+00:23:04,250 --> 00:23:08,490
+if Sn is a Cauchy sequenceماشي ايش تعريفه اس ام
+
+303
+00:23:08,490 --> 00:23:11,850
+كوشي سيكوانس لكل يبسلون أكبر من سفر there exist M
+
+304
+00:23:11,850 --> 00:23:16,090
+such that if M أكبر من N أكبر أوي سوء ال M اللي ده
+
+305
+00:23:16,090 --> 00:23:19,850
+جيتها اللي هو هيكون عبارة عن اس ام نقص اس ن أصغر
+
+306
+00:23:19,850 --> 00:23:24,330
+من يبسلون هاي اللي عند الهان اللي هي نظرية أصغر من
+
+307
+00:23:24,330 --> 00:23:27,830
+يبسلون اللي هي الكوشي بساوي مين أو بكافي ال
+
+308
+00:23:27,830 --> 00:23:31,330
+convergence هذه حساباتها زي ما قلنا الاس ام عبارة
+
+309
+00:23:31,330 --> 00:23:39,840
+عن مين X واحدx1 زائد لما أصل عند xn زائد xn زائد 1
+
+310
+00:23:39,840 --> 00:23:44,040
+لما أصل عند آخر واحد ال xm لأن ال M أكبر من N
+
+311
+00:23:44,040 --> 00:23:47,780
+فهلاجي في طريق اللي هو مين ال x ال x واحد عند ال
+
+312
+00:23:47,780 --> 00:23:55,300
+xn ناقص ال sn ناقص ال x1 ناقص ال x2 ناقص ال xn هذا
+
+313
+00:23:55,300 --> 00:24:00,750
+ال absolute value اللي هو هذولة بروح لهنامع هذولة
+
+314
+00:24:00,750 --> 00:24:06,590
+بضل عندى xn زائد واحد زائد xn زائد اتنين زائد xm
+
+315
+00:24:06,590 --> 00:24:10,070
+اللى هى هذه اصغر من مين من ابسلون اذا هذا صار اصغر
+
+316
+00:24:10,070 --> 00:24:13,150
+من ابسلون وهو المطلوب طيب
+
+317
+00:24:17,320 --> 00:24:22,820
+طيب، الان نعرف حاجة اسمها absolutely convergent
+
+318
+00:24:22,820 --> 00:24:27,380
+absolutely convergent برضه اللي مرت علينا في تفاضل
+
+319
+00:24:27,380 --> 00:24:31,620
+بها خلينا نذكركم فيها اللي هو let x بالـ
+
+320
+00:24:31,620 --> 00:24:36,760
+definition 916 let x بسوء xn be a sequence in R we
+
+321
+00:24:36,760 --> 00:24:41,020
+say that the series سماشي xn is absolutely
+
+322
+00:24:41,020 --> 00:24:42,180
+convergent
+
+323
+00:24:44,650 --> 00:24:52,470
+إذا كانت سلسلة أبسط قيم XN مرتبطة بالـ R
+
+324
+00:24:57,860 --> 00:25:01,700
+if it is convergent but not إيش ما لها absolutely
+
+325
+00:25:01,700 --> 00:25:05,620
+convergent يعني في series في الدنيا أكيد بتكون
+
+326
+00:25:05,620 --> 00:25:11,100
+convergent لكن مش absolutely convergent هذه ال
+
+327
+00:25:11,100 --> 00:25:13,620
+series اللي من هالنوع اللي بتكون convergent لكن مش
+
+328
+00:25:13,620 --> 00:25:16,220
+absolutely convergent بنسميه إيش ما لها
+
+329
+00:25:16,220 --> 00:25:18,820
+conditionally convergent
+
+330
+00:25:20,920 --> 00:25:24,240
+لكن بعد شوية هنقول ان any absolutely convergent
+
+331
+00:25:24,240 --> 00:25:33,240
+هتكون ايش ما لها is convergent طيب الان نيجي طبعا
+
+332
+00:25:33,240 --> 00:25:37,040
+هذا في R كل الشغل في ال real numbers تسعة واحد
+
+333
+00:25:37,040 --> 00:25:42,650
+سبعة theoremإذا كانت سلسلة مرتبطة بالكامل فهي
+
+334
+00:25:42,650 --> 00:25:45,530
+مرتبطة بالكامل أيضًا، فهي مرتبطة بالكامل أيضًا،
+
+335
+00:25:45,530 --> 00:25:54,310
+فهي مرتبطة بالكامل أيضًا، فهي مرتبطة
+
+336
+00:25:54,310 --> 00:25:56,430
+بالكامل أيضًا
+
+337
+00:25:57,710 --> 00:26:00,150
+ماذا يعني أبسليوت يكون convergent؟ يعني معناته
+
+338
+00:26:00,150 --> 00:26:03,270
+الصمشي اللي هي أبسليوت value أيش ماله is
+
+339
+00:26:03,270 --> 00:26:07,830
+convergent ماشي then by Cauchy criterion بنطبقها
+
+340
+00:26:07,830 --> 00:26:11,590
+على هذه ال series اللي هو لكل أبسلون أكبر من سفر
+
+341
+00:26:11,590 --> 00:26:14,970
+there exists m of epsilon element and such that if
+
+342
+00:26:14,970 --> 00:26:18,190
+m أكبر من n أكبر يسوى m epsilon اللي حكيناها قبل
+
+343
+00:26:18,190 --> 00:26:23,100
+شويةthen اللي هو اللي عندي الأولى اللي هي n زائد
+
+344
+00:26:23,100 --> 00:26:26,560
+واحد n زائد اتنين لما أصل عند ال M بس كيف شكل ال
+
+345
+00:26:26,560 --> 00:26:28,920
+term هنا اللي هو ال absolute value اذا absolute
+
+346
+00:26:28,920 --> 00:26:33,180
+value ل Xn زائد واحد زائد .. مين ماخد ال M أكبر ان
+
+347
+00:26:33,180 --> 00:26:36,720
+اه زائد absolute value Xn زائد اتنين لما أصل عند
+
+348
+00:26:36,720 --> 00:26:40,380
+زائد absolute value ل ال مين ل Xm أصغر من إيه يا
+
+349
+00:26:40,380 --> 00:26:47,510
+عياش؟ من Y هذه اللي هي عبارة عن الاسم لهذهنقص الـ
+
+350
+00:26:47,510 --> 00:26:51,610
+SM لها الـ SM فرضا نسميها prime الـ SM اللي هي
+
+351
+00:26:51,610 --> 00:26:54,610
+عبارة عن absolute value X1 زائد absolute value X2
+
+352
+00:26:54,610 --> 00:26:58,290
+لما أصل عند ال absolute value Xm وهذه تزيلها منها
+
+353
+00:26:58,290 --> 00:27:02,250
+بيظل ال absolute value عند Xm N زائد 1 لعند ال
+
+354
+00:27:02,250 --> 00:27:05,990
+absolute value ل Xm اللي هي هتكون أصغر من Y حسب ال
+
+355
+00:27:05,990 --> 00:27:09,330
+Cauchy criterion ها وهذه سهلة بال Cauchy criterion
+
+356
+00:27:09,330 --> 00:27:10,010
+مباشرة
+
+357
+00:27:18,700 --> 00:27:23,480
+لو سمنا الـ sequence of partial sums للـ XN اللي
+
+358
+00:27:23,480 --> 00:27:28,090
+عبارة عن SSNوهذه سمنها mean للـ absolute values و
+
+359
+00:27:28,090 --> 00:27:32,430
+SM prime طيب شوفي لان احسبلي لهذه عشان اثبت هذه ال
+
+360
+00:27:32,430 --> 00:27:37,590
+conversion تلان احسبلي SM ماقص SM لمين الامات للام
+
+361
+00:27:37,590 --> 00:27:42,190
+الأكبر من انه أكبر يساوي مين M of Y بساوي اللي
+
+362
+00:27:42,190 --> 00:27:45,350
+هذولة حسبناها قبل و شوية طلعت XN زاد واحد XN زاد
+
+363
+00:27:45,350 --> 00:27:49,000
+اتنين عند X mean XMهذا الـ Absolute Value بالـ
+
+364
+00:27:49,000 --> 00:27:51,340
+Triangling Quality أصغر من الأولى زاد التانية لما
+
+365
+00:27:51,340 --> 00:27:55,780
+أصل للأخيرة هذا من فوق أصغر من مين؟ من إبسلون إذا
+
+366
+00:27:55,780 --> 00:27:59,220
+لجينا لكل إبسلون أكبر من سفر there exists M such
+
+367
+00:27:59,220 --> 00:28:01,480
+that F M أكبر من M أكبر من M أكبر من M أكبر من M
+
+368
+00:28:01,480 --> 00:28:05,740
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+369
+00:28:05,740 --> 00:28:05,740
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+370
+00:28:05,740 --> 00:28:05,740
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+371
+00:28:05,740 --> 00:28:08,560
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+372
+00:28:08,560 --> 00:28:14,340
+أكبر من M أكبر من M أكبر من M أكبر من M أكبر من M
+
+373
+00:28:17,760 --> 00:28:21,960
+is الهى عبارة عن convergence اللى هى طبعا ال
+
+374
+00:28:21,960 --> 00:28:24,960
+sequence of partial sums تبع ال xn then by Cauchy
+
+375
+00:28:24,960 --> 00:28:30,140
+criterion ال summation لل xn is convergent طيب
+
+376
+00:28:30,140 --> 00:28:33,800
+نيجى
+
+377
+00:28:33,800 --> 00:28:42,360
+الان ناخد example 918 نشوف هذا ال example نشوف كيف
+
+378
+00:28:42,360 --> 00:28:47,970
+بدنا اللى هو انحلأو اللي هو نشرح هذا الـ Example
+
+379
+00:28:47,970 --> 00:28:49,410
+918
+
+380
+00:28:51,320 --> 00:28:55,460
+لت اكس بسوء a n then the geometric series ال
+
+381
+00:28:55,460 --> 00:28:59,820
+summation ل a n من واحد اللي مالنها هي converts if
+
+382
+00:28:59,820 --> 00:29:04,200
+ال absolute value لل a أشمالها أصغر من واحد إذا ال
+
+383
+00:29:04,200 --> 00:29:07,080
+geometric series هذا هتكون converts معلومة طبعا
+
+384
+00:29:07,080 --> 00:29:10,300
+عارفينها إذا كان ال absolute value لل a أصغر من
+
+385
+00:29:10,300 --> 00:29:15,680
+واحد فبتكون هذه a أشمالها converts الآن الجزء
+
+386
+00:29:15,680 --> 00:29:27,690
+الثاني b اللي هو بنشوفه لاحقانأتي الآن ونفترض
+
+387
+00:29:27,690 --> 00:29:31,390
+أن الـ summation لـ a أُسن من نقصة واحد إلى ملا
+
+388
+00:29:31,390 --> 00:29:34,930
+نهاية is convergent مادام الـ convergent هذا حسب
+
+389
+00:29:34,930 --> 00:29:40,490
+الحكيمات التي قبل قليل اللي هي على طول limit الـ a
+
+390
+00:29:40,490 --> 00:29:44,480
+n النظرية as n goes to infinity ايش بساوي؟ سفرمدام
+
+391
+00:29:44,480 --> 00:29:47,760
+limit الـ A N بيساوي 0 إذا نكيد الـ absolute value
+
+392
+00:29:47,760 --> 00:29:51,180
+للـ A أصغر من 1، ليش؟ لأن لو كانت ال absolute
+
+393
+00:29:51,180 --> 00:29:54,960
+value للـ A بتساوي 1 مثلا، بيصير ال limit بيساوي
+
+394
+00:29:54,960 --> 00:29:58,520
+1، يعني مش 0، ولو كان هذا ال absolute value للـ A
+
+395
+00:29:58,520 --> 00:30:03,020
+أكبر من 1، بيصير limit الـ A N، بالذات، بيروح إلى
+
+396
+00:30:03,020 --> 00:30:05,560
+.. بيروح إلى وين؟ إلى مالة نهاية أو سلبة مالة
+
+397
+00:30:05,560 --> 00:30:09,000
+نهاية أو دصدات exist حسب اللي هي .. اللي هي الـ A
+
+398
+00:30:09,000 --> 00:30:13,420
+مدبا أو سلبة، بيكون limit A الـ N لو كانت أكبر من
+
+399
+00:30:13,420 --> 00:30:20,340
+1،اللي هي بتكون .. اللي هي diverse الان في النهاية
+
+400
+00:30:20,340 --> 00:30:23,780
+مدامة ال limit لل a n as n goes to infinity بساوة
+
+401
+00:30:23,780 --> 00:30:27,240
+صفر فال absolute value أصبعناها هتكون أصغر من واحد
+
+402
+00:30:27,240 --> 00:30:32,760
+conversely suppose ذات ال absolute value لل a أصغر
+
+403
+00:30:32,760 --> 00:30:39,100
+من واحد بيثبتلك أن ال summation لل a n converts
+
+404
+00:30:39,100 --> 00:30:44,720
+بيثبتلك ال summation لل a n converts فبتنشوفاللاحظ
+
+405
+00:30:44,720 --> 00:30:49,260
+الملاحظة هذه المشهورة عندى لو جيت a n زائد واحد
+
+406
+00:30:49,260 --> 00:30:53,640
+زائد a n زائد اتنين لما اصل عند ال a m هذا طبعا
+
+407
+00:30:53,640 --> 00:30:59,550
+انا باخده في حالة ال m اكبر من مين من nوضربت في 1
+
+408
+00:30:59,550 --> 00:31:07,930
+-a لما أضرب و أجمع ال terms لبعض كله هي cancel كله
+
+409
+00:31:07,930 --> 00:31:12,590
+هيظل بس أول term اللي هو am زائد واحد و آخر term
+
+410
+00:31:12,590 --> 00:31:17,510
+اللي هو am زائد واحد فالان هذا ضرب هذا هيطلع am
+
+411
+00:31:17,510 --> 00:31:22,180
+زائد واحد ناقص a os m زائد واحدهذا بيساوي المقدار
+
+412
+00:31:22,180 --> 00:31:27,500
+هذا لكل الأمات اللي أكبر بالمئة من N لأن بدي
+
+413
+00:31:27,500 --> 00:31:29,860
+أستخدم شكل بدي أستخدم الـ Cauchy criterion في
+
+414
+00:31:29,860 --> 00:31:34,960
+إثبات ال convergence احسب ل S M نقص S N صرنا
+
+415
+00:31:34,960 --> 00:31:38,080
+حاسبينها وعارفين كيف بتنحسب ل الأم الأكبر من N
+
+416
+00:31:38,080 --> 00:31:41,700
+هيطلع عند A N زائد واحد لعند A M absolute value
+
+417
+00:31:41,700 --> 00:31:48,100
+هذا بيساوي اللي هو حسب اللي هنا عبارة عن ال
+
+418
+00:31:48,100 --> 00:31:55,100
+absolute value لهذاهي ساوي هذا absolute value على
+
+419
+00:31:55,100 --> 00:32:00,200
+المقدار هذا واحد ناقص a absolute value إذا استبدلت
+
+420
+00:32:00,200 --> 00:32:06,100
+هذا اللي هو هذا على هذا من هذه اللي هي اللي فوق مش
+
+421
+00:32:06,100 --> 00:32:09,920
+الحال إذا صار عند الآن استخدمت اللي فوق هذه أخدت
+
+422
+00:32:09,920 --> 00:32:13,040
+ال absolute value للجهتين وأخدت اللي هو قيمة هذا
+
+423
+00:32:13,040 --> 00:32:16,400
+صار عبارة عن absolute value للإيهان على ال
+
+424
+00:32:16,400 --> 00:32:21,230
+absolute value اللي هو واحد minus aطيب هذا أصغر أو
+
+425
+00:32:21,230 --> 00:32:24,310
+يساوي بال triangle inequality اللي هو ال absolute
+
+426
+00:32:24,310 --> 00:32:27,810
+value للي فوق زاد ال absolute value للي جنبه على
+
+427
+00:32:27,810 --> 00:32:33,050
+المقدار هذا نفس اللي هي نفس القيمة مابغيرتهاش لأن
+
+428
+00:32:33,050 --> 00:32:37,610
+ال absolute value لأصغر من واحد implies that ان ال
+
+429
+00:32:37,610 --> 00:32:42,090
+limit اللي فوق هذا بيروح لصفر يعني الآن as n goes
+
+430
+00:32:42,090 --> 00:32:46,730
+to infinityas n goes to infinity طبعاً أكيد ال M
+
+431
+00:32:46,730 --> 00:32:50,330
+هتروح برضه لوين لملان هي اللي أكبر منها يعني ال N
+
+432
+00:32:50,330 --> 00:32:53,970
+as n goes to infinity هذا المقدار هيصغر كتير كتير
+
+433
+00:32:53,970 --> 00:32:57,490
+كتير وهذا هيصغر كتير كتير لدرجة أن هذا كله على
+
+434
+00:32:57,490 --> 00:33:03,030
+بعضه as n goes to infinity هيكون اللي هو أصغر من
+
+435
+00:33:03,030 --> 00:33:07,130
+أي رقم في الدنيا هيصير أصغر من أبسلون معناته سارة
+
+436
+00:33:07,130 --> 00:33:12,390
+ان دي اللي هوالـ S M نقص S N أصغر من إبسلون هيكون
+
+437
+00:33:12,390 --> 00:33:16,410
+اللي هو في هذه الحالة أثبتت أنه اللي هو الـ S N by
+
+438
+00:33:16,410 --> 00:33:20,970
+Cauchy criterion is convergent اللي هو صار عندي
+
+439
+00:33:20,970 --> 00:33:26,170
+الصممش إلى الآن convergence طيب الديجلة اللي هو
+
+440
+00:33:26,170 --> 00:33:30,200
+المثال اللي بعده المثال المشهوراللي هو بـ little
+
+441
+00:33:30,200 --> 00:33:38,500
+bit show that summation للواحد and diverse هذه الـ
+
+442
+00:33:38,500 --> 00:33:42,800
+sequence، هذه الـ series اللي هي series مشهورة
+
+443
+00:33:42,800 --> 00:33:46,380
+اللي كنا نسميها الـ B series الـ B بيساوي واحد في
+
+444
+00:33:46,380 --> 00:33:49,440
+هذه الحالة هذه الـ summation يليها واحد and إيش
+
+445
+00:33:49,440 --> 00:33:54,400
+ماله؟ diverse تنشوفto show that summation وحدة ل N
+
+446
+00:33:54,400 --> 00:33:58,480
+divergent, we construct unbounded subsequence SKN
+
+447
+00:33:58,480 --> 00:34:04,860
+of SK قلنا قبل هيك ان اذا كانت ال sequence
+
+448
+00:34:04,860 --> 00:34:09,800
+converges اكيد هتكون bounded اذا لو كانت ال
+
+449
+00:34:09,800 --> 00:34:16,060
+sequence unbounded اكيد diverges ماشي الحال طيب I
+
+450
+00:34:16,060 --> 00:34:19,250
+was not convergentبناءً عليه لو احنا جيبنا
+
+451
+00:34:19,250 --> 00:34:24,830
+subsequence من ال sequence تكون unbounded فمن باب
+
+452
+00:34:24,830 --> 00:34:28,530
+أولى هتصير ال sequence الأصلية نفسها unbounded
+
+453
+00:34:28,530 --> 00:34:31,450
+مدام ان صارت ال sequence الأصلية unbounded إذا ال
+
+454
+00:34:31,450 --> 00:34:36,030
+sequence صارت diverse بناءً عليه عشان أثبت اللي هو
+
+455
+00:34:36,030 --> 00:34:41,030
+ال sequence of partial sums هذهالـ sequence of
+
+456
+00:34:41,030 --> 00:34:44,430
+partial sums لها اللي هي الـ sk مثلا أو ال sun
+
+457
+00:34:44,430 --> 00:34:48,530
+عشان اثبتها انها diverge ال sequence of partial
+
+458
+00:34:48,530 --> 00:34:52,370
+sums يعني بمعنى اخر عشان اسمها الـ diverge ال
+
+459
+00:34:52,370 --> 00:34:58,120
+series بدي اجيبUnbounded subsequence بدنا نعمل
+
+460
+00:34:58,120 --> 00:35:00,940
+construction Unbounded وده اللي هو ال construction
+
+461
+00:35:00,940 --> 00:35:05,200
+المشهور إذا الآن بدأ أعمل we construct Unbounded
+
+462
+00:35:05,200 --> 00:35:08,520
+subsequence as K زائد واحد of the sequence of
+
+463
+00:35:08,520 --> 00:35:14,660
+partial sums SL of this series طيب شوف الآن كيف
+
+464
+00:35:14,660 --> 00:35:20,780
+بدنا نشتغل اللي هو اللي هي ال subsequence هذه خدلي
+
+465
+00:35:20,780 --> 00:35:27,830
+الآن بدأ أعمل subsequence as KRof mean of this
+
+466
+00:35:27,830 --> 00:35:34,210
+sequence خدلي اول اشي K واحد بساوي اللي هو عبارة
+
+467
+00:35:34,210 --> 00:35:39,750
+عن اتنين اس واحد فبصير SK واحد بساوية اللي هو
+
+468
+00:35:39,750 --> 00:35:45,480
+عبارة عن اللي هيتنين اقصد واحد يعني اتنين ال
+
+469
+00:35:45,480 --> 00:35:50,820
+sequence of partial sums ال summation واحدة لان ان
+
+470
+00:35:50,820 --> 00:35:55,500
+من واحد لعند كي ماشي لان عندي اش ماخدها اتنين كي
+
+471
+00:35:55,500 --> 00:36:01,840
+لان بيصير عندي اللي هي عبارة عن واحد زائد نص مظبوط
+
+472
+00:36:01,840 --> 00:36:07,720
+واحد زائد نص اكيد اكبر من مين اللي هي اكبر من اياش
+
+473
+00:36:07,720 --> 00:36:14,780
+من الواحد طيبخد الان كتنين بيساوي اتنين قص اتنين
+
+474
+00:36:14,780 --> 00:36:22,300
+فبيصير أس كتنين بيساوي اللي هو عبارة عن واحد زائد
+
+475
+00:36:22,300 --> 00:36:29,480
+نص زائد تلت زائد ربع تصبوت اللي هي هذه عبارة عن
+
+476
+00:36:29,480 --> 00:36:38,020
+تساوي أس كواحد أي أس كواحد زائد اللي هو مين
+
+477
+00:36:40,830 --> 00:36:47,350
+تلت زائد إياش زائد ربع أكيد هتكون هذه خلّيني أكتب
+
+478
+00:36:47,350 --> 00:36:53,410
+أربعة أكبر و أقول هذه زائد ربع زائد إياش ربع لأن
+
+479
+00:36:53,410 --> 00:36:57,430
+التلت أكبر من ربع حطيت بدل هذا فبصير عند هذا
+
+480
+00:36:57,430 --> 00:37:02,890
+المقدار SK2 أكبر من SK1 زائد ربع زائد ربع يعني
+
+481
+00:37:02,890 --> 00:37:10,820
+بمعنى آخر سيكون أكبر من SK1 زائدواحد اللي هو اتنين
+
+482
+00:37:10,820 --> 00:37:16,820
+في مين في الربع اللي هو بساوي SK واحد زاد اتنين في
+
+483
+00:37:16,820 --> 00:37:23,440
+واحد على اتنين تربية باشي اللي هو هذا اكيد اكبر من
+
+484
+00:37:23,440 --> 00:37:27,740
+واحد زاد لان SK واحد اكبر من مين من واحد اكبر من
+
+485
+00:37:27,740 --> 00:37:33,900
+واحد زاد اتنين اتنين في واحد على اتنين تربية
+
+486
+00:37:44,670 --> 00:37:51,130
+خلّينا نحسب أسكت تلاتةماذا يساوي از كتلاتة خد
+
+487
+00:37:51,130 --> 00:37:54,070
+كتلاتة طبعا واضحة ايش كتلاتة بدأ اخدها اتنين اقصى
+
+488
+00:37:54,070 --> 00:37:57,770
+اتنين هاخد اتنين اقصى تلاتة اللي هي تمانية اللي
+
+489
+00:37:57,770 --> 00:38:04,870
+عبارة عن واحد زائد نص زائد تلت زائد ربع زائد خمس
+
+490
+00:38:04,870 --> 00:38:14,050
+زائد سدس زائد سبع زائد تمد ماشي الحال طيب شوف عندي
+
+491
+00:38:14,050 --> 00:38:24,890
+الان من واحدالعند اللي هو جداش ربع هذا اللي هي SK2
+
+492
+00:38:27,420 --> 00:38:31,960
+الباقية الـ SK ثلاثة إذاً هذا بيصير عبارة عن SK
+
+493
+00:38:31,960 --> 00:38:38,340
+تنين اللي هو يعني أكبر من SK تنين زائد آخر واحد زي
+
+494
+00:38:38,340 --> 00:38:44,320
+ما أخدت هنا آخر واحد هنا اللي هو تمن زائد تمن زائد
+
+495
+00:38:44,320 --> 00:38:50,100
+تمن زائد تمن زائد تمن أربعة، واحد، تنين، واحد،
+
+496
+00:38:50,100 --> 00:38:54,880
+تنين، تلاتة، أربعة، عين أربعة ماشي الحال هدول الآن
+
+497
+00:38:56,220 --> 00:38:58,780
+من هنا إلى هنا أصغر من هنا إلى هنا عشان هيك بيصير
+
+498
+00:38:58,780 --> 00:39:02,120
+هذا المقدار أكبر من SK2 زائد هدوله هدوله إيش
+
+499
+00:39:02,120 --> 00:39:08,400
+بيساول هو يساوي SK2 زائد اللي هو أكمتمهم أربعة
+
+500
+00:39:08,400 --> 00:39:14,940
+أتمان يعني أربعة في تمون أربعة
+
+501
+00:39:14,940 --> 00:39:19,880
+في تمون هو يساوي الآن أربعة في تمون يعني عبارة عن
+
+502
+00:39:19,880 --> 00:39:28,720
+نص مظبوط يعني بيساوي اللي هو SK2زائد نص نشوف من
+
+503
+00:39:28,720 --> 00:39:33,360
+اللي فوق هنا SK2 أكبر من واحد زائد اتنين على اتنين
+
+504
+00:39:33,360 --> 00:39:38,800
+إذا حيصير هذا أكبر من واحد زائد اتنين على اتنين
+
+505
+00:39:38,800 --> 00:39:43,500
+تربيع اتنين على اتنين تربيع هذه واحدة ممكن نخليها
+
+506
+00:39:43,500 --> 00:39:50,800
+زي ما هي واحد اتنين تلاتة أربعة مش مشكلة نص اللي
+
+507
+00:39:50,800 --> 00:39:57,500
+هو إيش بيساوي هذا زائد اللي هو نصهذا اسكي اتنين
+
+508
+00:39:57,500 --> 00:40:02,720
+اكبر من هذا وهي نص بيصير زائد نص وهي ساوي واحد
+
+509
+00:40:02,720 --> 00:40:06,560
+زائد تلاتة
+
+510
+00:40:06,560 --> 00:40:13,820
+على مين اكم نص انا بس هاد الخلبط هنا اتنين في نص
+
+511
+00:40:13,820 --> 00:40:21,540
+تربيع يعني اتنين على اتنين تربيع اللي
+
+512
+00:40:21,540 --> 00:40:22,580
+هي بتساوي
+
+513
+00:40:26,450 --> 00:40:32,410
+وSK3 بساوي واحد زائد اتنين على اتنين تربيع زائد نص
+
+514
+00:40:32,410 --> 00:40:38,030
+هذه
+
+515
+00:40:38,030 --> 00:40:46,650
+خلّينا نقول واحد زائد نص اكبر من بس يهيأدي احنا بس
+
+516
+00:40:46,650 --> 00:40:55,560
+اللي هي SK1 SK1 اللي عبارة عن واحد زائد نصأكبر من
+
+517
+00:40:55,560 --> 00:41:01,700
+S K واحد هذه واحد زائد نص قيمة ال S خليها S K واحد
+
+518
+00:41:01,700 --> 00:41:04,860
+مش مشكلة S K واحد لما أنا عوضتها أنا عوضت بواحد ده
+
+519
+00:41:04,860 --> 00:41:10,120
+هي واحد زائد نص هي واحد زائد نص زائد هذه ماشي
+
+520
+00:41:10,120 --> 00:41:14,220
+الحال و يساوي صار عندها نص و هنا كمان اتنين مع
+
+521
+00:41:14,220 --> 00:41:18,910
+اتنين بطلع كمان نص بصير اتنين على اتنين صحيبصير 2
+
+522
+00:41:18,910 --> 00:41:23,330
+على 2 وهذا نص وكمان نص هنا الآن لما نعوض عليها دي
+
+523
+00:41:23,330 --> 00:41:28,830
+برضه بنفس الطريقة اللي صار عندي SK2 اللي هو عبارة
+
+524
+00:41:28,830 --> 00:41:36,930
+عن واحد زاد اتنين على اتنين واحد نعود تحت SK2 بصير
+
+525
+00:41:36,930 --> 00:41:42,010
+واحد زاد اتنين على اتنين بصير عند واحد زاد اتنين
+
+526
+00:41:42,010 --> 00:41:44,890
+على اتنين يعني واحد زاد تلاتة على من؟ على اتنين
+
+527
+00:41:44,890 --> 00:41:53,410
+اللي وصلتله يعني الآنالاسكي واحد اكبر من واحد زائد
+
+528
+00:41:53,410 --> 00:42:00,370
+اتنين اصفر على او خلّيني اقول بيساوي اسد بيساوي
+
+529
+00:42:00,370 --> 00:42:08,680
+واحد زائد واحد على اتنين اصفرSK3 SK2 عبارة عن أكبر
+
+530
+00:42:08,680 --> 00:42:13,440
+من وصلنا أخر أشي أكبر من واحد زاد اتنين على اتنين
+
+531
+00:42:13,440 --> 00:42:21,400
+SK3 أكبر يعني هان اتنين هاي اتنين SK3 اللي هو
+
+532
+00:42:21,400 --> 00:42:28,420
+عبارة عن أكبر من واحدواحد زائد تلاتة على اتنين
+
+533
+00:42:28,420 --> 00:42:38,580
+لاحظ هنا تلاتة هنا على اتنين هنا في حالة هذه اللي
+
+534
+00:42:38,580 --> 00:42:43,920
+هي اتنين تربيع يعني اتنين على اتنين وفي حالة
+
+535
+00:42:43,920 --> 00:42:49,980
+الأخيرة اكبر من واحداللي هو اكبر من واحد زائد نص
+
+536
+00:42:49,980 --> 00:42:54,260
+اللي هو عبارة عن اللي هو اتنين او السفر او بساوي
+
+537
+00:42:54,260 --> 00:42:58,940
+انا اتنين او السفر اللي هو على اتنين يعني الان و
+
+538
+00:42:58,940 --> 00:43:07,250
+كأنه in general in general هناك ثلاثةوهنا K2 وهنا
+
+539
+00:43:07,250 --> 00:43:13,230
+K1 in general بيصير عندي SK3 أكبر من 1 زائد 3 على
+
+540
+00:43:13,230 --> 00:43:20,550
+2 يعني صار عندي الآن بصورة عامة ال SKR اللي هو
+
+541
+00:43:20,550 --> 00:43:25,590
+أكبر من 1 زائد R على 2 ومرّينا في الخطوات في
+
+542
+00:43:25,590 --> 00:43:32,870
+أثناءها اللي هي علىما يلي مرّينا على أنها أكبر من
+
+543
+00:43:32,870 --> 00:43:42,290
+SKR-1 زاد 2 أُس R-1 في 1 على 2R اللي سمناها SKR-1
+
+544
+00:43:42,290 --> 00:43:52,460
+زاد إياش نص الان هذي بدي أثبتها by inductionانا
+
+545
+00:43:52,460 --> 00:43:56,520
+لان عملتلها لتلاتتين لكن هذا طبعا تلاتتين بس
+
+546
+00:43:56,520 --> 00:44:02,280
+استقرار لان in general بدأ أثبتها كما يالي عندي هي
+
+547
+00:44:02,280 --> 00:44:05,860
+SK اتنين
+
+548
+00:44:05,860 --> 00:44:12,700
+اللي هي R minus واحدزائد اللي هو اتنين اص ار ماينس
+
+549
+00:44:12,700 --> 00:44:16,060
+واحد لان اتنين اص اتنين هذي اللي هى R ايش بيساوي
+
+550
+00:44:16,060 --> 00:44:18,800
+تلاتة بيصير اتنين اص تلاتة ناقص واحد على اتنين اص
+
+551
+00:44:18,800 --> 00:44:22,880
+اتنين فى واحد على اتنين تكعيب اللى هى واحد على
+
+552
+00:44:22,880 --> 00:44:27,600
+اتنين تكعيب اللى هى واحد على اتنين اص ار اذا هذىهي
+
+553
+00:44:27,600 --> 00:44:33,920
+هيها هذه اللي هي بتساوي اللي هي SK اللي هي R minus
+
+554
+00:44:33,920 --> 00:44:39,600
+واحد يعني اتنين زائد ايش زائد نص اكبر من مين من
+
+555
+00:44:39,600 --> 00:44:42,960
+واحد زائد تلاتة على اتنين اللي هي واحد زائد R على
+
+556
+00:44:42,960 --> 00:44:47,480
+اتنين حيث ال R هنا ايش بتساوي بتساوي تلاتة اذا هذا
+
+557
+00:44:47,480 --> 00:44:53,130
+الاستقراء الآنبطلّع لي هذه بس هذا مش اثبات بدنا
+
+558
+00:44:53,130 --> 00:44:57,970
+نثبتها for r بتساوي واحد على طول for r بتساوي واحد
+
+559
+00:44:57,970 --> 00:45:05,140
+از ك واحد از ك واحد بساوي اللي هوR بتساوي اللي هي
+
+560
+00:45:05,140 --> 00:45:10,960
+واحد SK واحد بساوي اللي هو واحد زائد نص واحد زائد
+
+561
+00:45:10,960 --> 00:45:14,180
+نص طب علشان مافيش إجابة لك خلّيني أخد ان ال R إيش
+
+562
+00:45:14,180 --> 00:45:18,120
+بتساوي ال R بيساوي اتنين لإن ال R ر زيرو مش معرفة
+
+563
+00:45:18,120 --> 00:45:21,220
+فاهمين عليها إذا ان ال R بيساوي اتنين R بيساوي
+
+564
+00:45:21,220 --> 00:45:25,720
+اتنين SK اتنين هي SK اتنين عشان مديش أعيدها SK
+
+565
+00:45:25,720 --> 00:45:34,820
+اتنين أكبرمن اللي هو اللي هي SK1 هيها SK2 أكبر من
+
+566
+00:45:34,820 --> 00:45:42,460
+SK1 زائد 2 أُس 2 نقص 1 يعني أُس واحد في مين في
+
+567
+00:45:42,460 --> 00:45:45,980
+رُبع اللي هو واحد على اتنين تربيه إذن هذا صح لأن
+
+568
+00:45:45,980 --> 00:45:52,740
+اللي بعدها اللي هي SK1 زائد اللي هي نصاللي هو هذا
+
+569
+00:45:52,740 --> 00:45:56,200
+ما حد بروح بصيله نص اللي هي أكبر من واحد زائد
+
+570
+00:45:56,200 --> 00:45:59,500
+اتنين على اتنين اللي هي واحد زائد R على اتنين اللي
+
+571
+00:45:59,500 --> 00:46:03,940
+هي R بساوة اتنين إذا هذه true for mean for R
+
+572
+00:46:03,940 --> 00:46:08,160
+بتساوى اتنين لإنه بتبدأ من عند مين من عند R بساوة
+
+573
+00:46:08,160 --> 00:46:15,700
+اتنين الآن بدنا نفترض إنها صحيحة صحيحة for
+
+574
+00:46:17,620 --> 00:46:24,820
+R و نثبت صحيتها في مين لـ R زائد واحد بيكون خلصنا
+
+575
+00:46:24,820 --> 00:46:34,060
+إذا هي اللي بدنا نفترض صحتها لعند مين عند R بتساوي
+
+576
+00:46:34,060 --> 00:46:43,980
+R ماشي و بدنا نثبت صحتها ل SKR زائد واحد SKR زائد
+
+577
+00:46:43,980 --> 00:46:51,130
+واحد إيش بتساويبتساوي واحد طبعا كر زائد واحد زي ما
+
+578
+00:46:51,130 --> 00:46:55,310
+اتفرجنا عبارة عن اتنين اصار زائد واحد بيصير واحد
+
+579
+00:46:55,310 --> 00:47:02,990
+زائد نص زائد تلت زائد لما اصل عند واحد على اتنين
+
+580
+00:47:02,990 --> 00:47:10,210
+اصار ماشي عند اصار هذه اللي هتمثل المين اللي هي
+
+581
+00:47:10,210 --> 00:47:22,240
+skrزائد زائد زائد
+
+582
+00:47:22,240 --> 00:47:33,660
+زائد زائد زائد
+
+583
+00:47:34,490 --> 00:47:40,870
+زائد واحد زائد واحد على اتنين اثار زائد اتنين بضل
+
+584
+00:47:40,870 --> 00:47:46,110
+اضيف واحد لما اصل عند اخر واحدة اللي هي واحد على
+
+585
+00:47:46,110 --> 00:47:51,050
+اتنين اثار زائد واحد هذول المضافات على مين على ال
+
+586
+00:47:51,050 --> 00:47:57,930
+SKR ويساوي SKR زائدالمضافات هي دي .. ده شوف في
+
+587
+00:47:57,930 --> 00:48:01,230
+الجليد ماجدش عددين واحد على اتنين أسقار زائد واحد
+
+588
+00:48:01,230 --> 00:48:05,190
+زائد واحد على اتنين أسقار زائد اتنين زائد .. لما
+
+589
+00:48:05,190 --> 00:48:10,650
+أصل لآخر واحد .. واحد على اتنين أسقار في واحد زائد
+
+590
+00:48:10,650 --> 00:48:16,820
+واحد، مظبوط؟ يعني اتنين أسقار في اتنينهدولة اتنين
+
+591
+00:48:16,820 --> 00:48:19,940
+مضروبات يعني اتنين اصار زائد واحد اللي هى هدى لما
+
+592
+00:48:19,940 --> 00:48:24,680
+اضربها جوا بيصير عبارة عن اتنين اصار زائد اتنين
+
+593
+00:48:24,680 --> 00:48:30,760
+اصار مظبوط طيب يعني وكأن عدد ال terms اللى ضفته
+
+594
+00:48:30,760 --> 00:48:36,020
+هدى هدول عددهن جزء عددهن هدول ايش بيساوي اتنين
+
+595
+00:48:36,020 --> 00:48:44,280
+اصاريعني هيساوي هذا هيكون أكبر من SKR زائد واحد
+
+596
+00:48:44,280 --> 00:48:50,380
+على اتنين اصار زائد واحد أكيد لأن هذا ال term أصغر
+
+597
+00:48:50,380 --> 00:48:54,660
+من هذا ال term زائد واحد على اتنين اصار زائد واحد
+
+598
+00:48:54,660 --> 00:48:57,740
+زي ما عملت قبل زائد لما أصل الاخر واحد واحد على
+
+599
+00:48:57,740 --> 00:49:02,720
+اتنين اصار زائد واحد أكم واحد هدولة عدد هناعدد
+
+600
+00:49:02,720 --> 00:49:06,960
+الاتنين أُس R زي ما عدنا إنها ينهار إذا هذا بيصير
+
+601
+00:49:06,960 --> 00:49:16,980
+عبارة عنأكبر من الـ SKR زائد 2 أُس R في 1 على 2
+
+602
+00:49:16,980 --> 00:49:24,700
+أُس R زائد 1 يعني صارت هذه صحيحة لهنا صحيحة for R
+
+603
+00:49:24,700 --> 00:49:29,580
+بتساوي R زائد 1 لـ R زائد 1 صحيحة لهنا اللي بعدها
+
+604
+00:49:29,580 --> 00:49:35,680
+طبيعي هذه automatic بتطلع SKR زائد نص لأن هذه
+
+605
+00:49:35,680 --> 00:49:38,220
+بيساوي اللي هو بضال بس 2 و اللي فوق بتروح محق اللي
+
+606
+00:49:38,220 --> 00:49:46,730
+تحتالان هذه ما احنا مفترضينها الـSKR اللي هي أكبر
+
+607
+00:49:46,730 --> 00:49:54,160
+من واحد زائد اللي هي R على اتنينهم مفترضين أن هذه
+
+608
+00:49:54,160 --> 00:49:58,880
+صحيحة إذا الـSKR أكبر من واحد زائد R على اتنين إذا
+
+609
+00:49:58,880 --> 00:50:03,240
+by induction ال hypothesis بيصير هذا أكبر مكان هذه
+
+610
+00:50:03,240 --> 00:50:07,760
+اللي هو واحد زائد R على اتنين زائد النص الأصلي
+
+611
+00:50:07,760 --> 00:50:12,460
+ويساوي واحد زائد R زائد واحد على مين؟ على اتنين
+
+612
+00:50:12,460 --> 00:50:18,330
+إذا صارت عند الـSKR زائد واحدأكبر من 1 زائد R زائد
+
+613
+00:50:18,330 --> 00:50:23,110
+1 على 2 يعني صارت هذه الجملة صحيحة for R زائد 1
+
+614
+00:50:23,110 --> 00:50:27,270
+إذا صحيحة دائما إذا هيك بيكون أثبتنا by
+
+615
+00:50:27,270 --> 00:50:32,770
+mathematical induction كل اللي بدنايا بخصوص هذا ال
+
+616
+00:50:32,770 --> 00:50:38,060
+subsequence SKRهذه الـ subsequence وضحى انها اكبر
+
+617
+00:50:38,060 --> 00:50:41,720
+او يساوى 1 على ارضها او اتنين يعني as R goes to
+
+618
+00:50:41,720 --> 00:50:44,540
+infinity هذا المقدار بروح لما لنهاية يعني هذه ال
+
+619
+00:50:44,540 --> 00:50:51,240
+sequence is unbounded اذا صار عندى اللي هوSKR is
+
+620
+00:50:51,240 --> 00:50:57,440
+unbounded subsequence of the sequence of partial
+
+621
+00:50:57,440 --> 00:51:04,240
+sums SN of summation 1 على N then it must diverge
+
+622
+00:51:04,240 --> 00:51:08,960
+وهو المطلوب بكون هي أثبتنا أن summation 1 على N is
+
+623
+00:51:08,960 --> 00:51:12,820
+a divergent series
+
+624
+00:51:17,210 --> 00:51:22,830
+بنكمل قصة الـ B-Series الـ B-Series المشهورة عندي
+
+625
+00:51:22,830 --> 00:51:28,750
+summation للواحد N أس بي N من واحد إلى مالنهاية
+
+626
+00:51:28,750 --> 00:51:33,050
+هذه الـ Series بدنا نفحص اللي هو متى converge و
+
+627
+00:51:33,050 --> 00:51:37,310
+متى diverge يعني converge for what بي and diverge
+
+628
+00:51:37,310 --> 00:51:41,850
+for what بي لجينا عند الـ B بسوا واحد it converge
+
+629
+00:51:41,850 --> 00:51:49,090
+بدنا نشوف الآن for مين بمينfor b بين الصفر و بين
+
+630
+00:51:49,090 --> 00:51:52,730
+الواحد show that the b series summation 1 على n b
+
+631
+00:51:52,730 --> 00:51:57,830
+diverges for b اللي أصغر يساوي واحد و أكبر من مين
+
+632
+00:51:57,830 --> 00:52:01,850
+من صفر طبعا حد يقول لي لو كانت b سالبة لو كانت b
+
+633
+00:52:01,850 --> 00:52:07,690
+سالبة بتطلع أس فوق مثلا معناه تطارد طيب
+
+634
+00:52:12,030 --> 00:52:17,770
+عندي for n element in n, b element in 0, 1 أنقص b
+
+635
+00:52:17,770 --> 00:52:21,330
+أصغر أو أصغر من 1 أصغر أو أصغر من 1 طبعا المقصود
+
+636
+00:52:21,330 --> 00:52:24,530
+بي أكمر من .. بي أصغر من سالب واحد ده اللي بقول
+
+637
+00:52:24,530 --> 00:52:29,370
+عنها اللي بتكبر ال b السالب اللي أصغر من واحد طيب
+
+638
+00:52:29,370 --> 00:52:32,750
+لأن احنا بنحكي عن بي أكبر من صفر أصغر أو أصغر من 1
+
+639
+00:52:32,750 --> 00:52:36,490
+شوفوا
+
+640
+00:52:36,490 --> 00:52:42,780
+صلى عنا بيه كلام سهلبنزيل واحد بي أصغر أو يساوي
+
+641
+00:52:42,780 --> 00:52:46,680
+مين أو أصغر أو يساوي أن لأن ال بي هن عبارة عن إياش
+
+642
+00:52:46,680 --> 00:52:49,940
+عن كسر و كأنه بأخد يدوري ده هتكون أصغر أو يساوي أن
+
+643
+00:52:49,940 --> 00:52:54,100
+سو واحد على أن أصغر أو يساوي مين واحد على أن بي
+
+644
+00:52:54,100 --> 00:52:57,780
+لأنها بتنقلبلأن however the sequence of partial
+
+645
+00:52:57,780 --> 00:53:01,080
+sums is unbounded، الـ sequence of partial sums
+
+646
+00:53:01,080 --> 00:53:05,200
+لهذه unbounded فما بالك الأكبر منها، إذن أكيد it
+
+647
+00:53:05,200 --> 00:53:09,140
+has a subsequence of un، اللي هي sequence of
+
+648
+00:53:09,140 --> 00:53:12,000
+partial sums which is unbounded، إذن هيكون ال
+
+649
+00:53:12,000 --> 00:53:17,120
+summation واحدة لأن بي divergenceهذه السيريزة هي
+
+650
+00:53:17,120 --> 00:53:20,220
+بـ Unbounded subsequence of partial sums اكيد اللي
+
+651
+00:53:20,220 --> 00:53:22,940
+عاجبال هذه الـ subsequence of partial sums is
+
+652
+00:53:22,940 --> 00:53:30,700
+unbounded إذا السيريزة هذه is divergent حلاني
+
+653
+00:53:30,700 --> 00:53:37,160
+جي لمينة اللي هيالسيريز اللي هي ال B سيريز التانية
+
+654
+00:53:37,160 --> 00:53:40,780
+اللي هي for B أكبر من واحد ال power series واحدة
+
+655
+00:53:40,780 --> 00:53:45,860
+لأن بي اشملها converts هذه الآن في خطواتها كتير
+
+656
+00:53:45,860 --> 00:53:48,840
+بتشابه اللي هو الخطوات اللي حكينا عليها قبل شوية
+
+657
+00:53:48,840 --> 00:53:52,660
+بس باتجاه ال convergence عشان هيك مش هعيد الخطوات
+
+658
+00:53:52,660 --> 00:53:57,680
+اللي عيدتها في ال B سيريز او ال واحد سيريز اه اللي
+
+659
+00:53:57,680 --> 00:54:02,560
+هي summation واحد على انالان for B أكبر من 1 the
+
+660
+00:54:02,560 --> 00:54:07,480
+power series هذي converts ركزوا الآن معايا first
+
+661
+00:54:07,480 --> 00:54:13,080
+note that Sn أصغر أو يساوي Sm for every M أكبر أو
+
+662
+00:54:13,080 --> 00:54:18,690
+يساوي Nيعني بمعنى أخر طب ما هي أصلا مبينة 1 على n
+
+663
+00:54:18,690 --> 00:54:23,870
+أص بي دايما موجبة إذا أكيد مدام موجبة أصغر من أس
+
+664
+00:54:23,870 --> 00:54:28,170
+واحد أصغر من أس ثلاثة أصغر من أس أربعة إذا ال
+
+665
+00:54:28,170 --> 00:54:33,430
+sequence of partial sums اي شمالها is increasing
+
+666
+00:54:33,430 --> 00:54:39,190
+sequence طيب مدام increasing يعني بمعنى او هتكون
+
+667
+00:54:39,190 --> 00:54:45,840
+اي شمالها monotoneMadame Monotone بيكفيني أثبت أنه
+
+668
+00:54:45,840 --> 00:54:52,420
+S K is bounded ماشي بدي أثبت اللي هو ال sequence
+
+669
+00:54:52,420 --> 00:54:58,120
+of partial sums is bounded S K عشان أثبت أنها
+
+670
+00:54:58,120 --> 00:55:02,140
+bounded، شوفوا هالاشي الغريب، بس الغرابة بتروح من
+
+671
+00:55:02,140 --> 00:55:08,160
+تصرف ال increasing تبعتهالان SK is bounded عشان
+
+672
+00:55:08,160 --> 00:55:12,460
+أثبتها bounded يكفيني أني أثبت subsequence منها
+
+673
+00:55:12,460 --> 00:55:21,320
+bounded يعني أثبتلك انه SK1 SK2 SK3 SKN in general
+
+674
+00:55:21,320 --> 00:55:25,860
+هذه ال subsequence يكفي أني أثبت انها bounded عشان
+
+675
+00:55:25,860 --> 00:55:29,960
+تكون اللي فوق bounded طب ما بلاقي subsequence
+
+676
+00:55:29,960 --> 00:55:33,580
+تانية بتكون اللي هي unboundedما هو لو لجيت
+
+677
+00:55:33,580 --> 00:55:36,640
+subsequence تانية زي ما بتقول انها unbounded أو لو
+
+678
+00:55:36,640 --> 00:55:40,660
+كانت هذه unbounded في حالة هذه bounded معناته ما
+
+679
+00:55:40,660 --> 00:55:45,640
+هي increasing هذه هتلاقي اللي هو term من هذه بعد
+
+680
+00:55:45,640 --> 00:55:49,080
+ال .. بعد ال term اللي انت حكيت عنه من هذه مادام
+
+681
+00:55:49,080 --> 00:55:53,930
+هتلاقيه بعدهإذاً معناته بخلّي الـ bounded نصف مين
+
+682
+00:55:53,930 --> 00:55:56,970
+في هذه الـ sequence إذاً في حالة ال increasing
+
+683
+00:55:56,970 --> 00:56:00,750
+sequence لإنها increasing يكفي أن ألاقي واحدة منهم
+
+684
+00:56:00,750 --> 00:56:04,610
+ال subsequences انها bounded عشان أقول كل ال
+
+685
+00:56:04,610 --> 00:56:08,210
+sequence bounded لأنه لو كانت اللي هو فيه عندك
+
+686
+00:56:08,210 --> 00:56:14,470
+اللي هو ال sequence أصلا مش bounded معناتهفي عنصر
+
+687
+00:56:14,470 --> 00:56:17,550
+من عناصر الـ subsequence اللي قولنا عنها bounded
+
+688
+00:56:17,550 --> 00:56:21,370
+تخططت اللي هو اللي بنحكي فيه لأنه دي روحي لما
+
+689
+00:56:21,370 --> 00:56:26,890
+لنهاية فعشان هيك غصب عنها مادام لجينا واحدة من ال
+
+690
+00:56:26,890 --> 00:56:31,250
+subsequences لل increasing sequence is bounded
+
+691
+00:56:31,250 --> 00:56:37,810
+هتكون كل ال sequence is boundedطيب، شوفوا عليها،
+
+692
+00:56:37,810 --> 00:56:40,870
+إذا اللي ضال علينا بس نعمل construction ل
+
+693
+00:56:40,870 --> 00:56:46,310
+subsequence skr تكون bounded طبعاً لمن؟ لسريزة
+
+694
+00:56:46,310 --> 00:56:51,510
+summation 1 على n أسبي اللي ماياخدوا إشي مشابه للي
+
+695
+00:56:51,510 --> 00:56:55,730
+قبل بشوية عشان هيك مش هعيد ال induction اللي قبل
+
+696
+00:56:55,730 --> 00:57:01,030
+بشوية طريقة ال induction بتعملوها باسلوب مشابه،
+
+697
+00:57:01,030 --> 00:57:05,190
+إيش بقولت؟ انتبهوا عليهالد K1 بيساوي 2 أس واحد
+
+698
+00:57:05,190 --> 00:57:09,930
+ناقص واحد، ماشي؟ هناك يقول لنا خد K1 اتنين، K2
+
+699
+00:57:09,930 --> 00:57:13,950
+اتنين تربيع، لأ هاد يقول لنا خدها KR بيساوي 2 أس R
+
+700
+00:57:13,950 --> 00:57:18,550
+ناقص واحد، ماشي الحال؟ اللي كان غلط حسابات و يساوي
+
+701
+00:57:18,550 --> 00:57:23,790
+واحد، أس K واحد واحد، K2 اتنين أس اتنين ناقص واحد
+
+702
+00:57:23,790 --> 00:57:29,690
+و يساوي تلاتةلو جيت حسبة SK2 احسبها لحالك هيصير
+
+703
+00:57:29,690 --> 00:57:35,590
+عبارة عن واحد على واحد على اللي هي زائد لأنه عند P
+
+704
+00:57:35,590 --> 00:57:40,790
+من وين بتبدأ من عند اللي هو عند N من واحد إلى كده
+
+705
+00:57:40,790 --> 00:57:45,430
+بيصير واحد على واحد و P على واحدزاد 1 على 2 أس بي
+
+706
+00:57:45,430 --> 00:57:51,510
+زاد 1 على 3 أس بي هذا أصغر من 1 زاد اللي هو 1 على
+
+707
+00:57:51,510 --> 00:57:55,690
+2 أس بي زاد 1 على 2 أس بي يعني باخد الأولة مش
+
+708
+00:57:55,690 --> 00:57:59,510
+الأخيرة زي ما كنت أخد قبل صار عندي 1 زاد 2 على 2
+
+709
+00:57:59,510 --> 00:58:05,200
+أس بي ويساوي 1 زاد 1 على 2 أس بي minus 1الان اللى
+
+710
+00:58:05,200 --> 00:58:09,460
+عملته لل S K 2 بدي اعمله لل S K 3 S K 3 اللى هى 2
+
+711
+00:58:09,460 --> 00:58:13,340
+اقص 3 نقص 1 اللى هى S بتطلع 7 اللى هى 8 نقص 1
+
+712
+00:58:13,340 --> 00:58:18,760
+الأولى S K 3 اللى هى عبارة عن S K 2 زائد زي ما
+
+713
+00:58:18,760 --> 00:58:22,040
+عملت قبل شوية ربع أس بي خمس أس بي ست أس بي زائد
+
+714
+00:58:22,040 --> 00:58:25,940
+سبع أس بي الان باخد مش أخر واحدة باخد أول واحدة
+
+715
+00:58:25,940 --> 00:58:30,580
+عشان يظل هذا المقدار لهان أكبر من لهانبصير SK2 زي
+
+716
+00:58:30,580 --> 00:58:37,580
+4 على 4 أُس بي اللي هو عبارة عن 1 زي 1 على 2 أُس
+
+717
+00:58:37,580 --> 00:58:42,580
+بي minus 1 زي 1 على 4 أُس بي minus 1 ماشي الحال
+
+718
+00:58:42,580 --> 00:58:48,320
+الآن سمّي لي بقول لي سمّي اللي هو 1 على 2 بي minus
+
+719
+00:58:48,320 --> 00:58:53,840
+1 هذه اللي هي سمّي ليها اسمها إيه ككتلة واحد عشان
+
+720
+00:58:53,840 --> 00:58:59,680
+ما نضلش نكتبهاطيب الان since ب أكبر من واحد إذا
+
+721
+00:58:59,680 --> 00:59:04,100
+بدأ ب أكبر من واحد هذه لأن هي مفترضينها صارت ال a
+
+722
+00:59:04,100 --> 00:59:07,520
+اللي عندي هنا بين السفر وبين الواحد لأن ده صارت
+
+723
+00:59:07,520 --> 00:59:12,120
+اللي عدد أكبر من يتنين او أكبر إذا صارت هذه عبارة
+
+724
+00:59:12,120 --> 00:59:18,140
+عن كسر يعني كسر بين السفر وبين الواحد الان by
+
+725
+00:59:18,140 --> 00:59:21,260
+mathematical induction زي اللي عملته قبل شوية we
+
+726
+00:59:21,260 --> 00:59:29,180
+find ifاللي هو K R بيسوا 2R-1 هلاجي دايما S K R
+
+727
+00:59:29,180 --> 00:59:36,400
+أكبر من 0 و أصغر من 1 زائد A زائد A تربيع لما أصل
+
+728
+00:59:36,400 --> 00:59:41,440
+عند A R-1 مين ال A ال A ال A اللي هي 1 على 2 أُس B
+
+729
+00:59:41,440 --> 00:59:46,480
+-1 وحالها هذه ال induction بتاعتها سهلة بنفس
+
+730
+00:59:46,480 --> 00:59:50,500
+الطريقة اللي عملتها أنا قبل بشوية بتفترضها صحيحة ل
+
+731
+00:59:50,500 --> 00:59:57,810
+Rبتثبتها صحيحة عند R بساوة 2 هنا أثبتناها لل S K 2
+
+732
+00:59:57,810 --> 01:00:02,570
+هنا أثبتناها فوق وبعدين بتثبتها صحيحة لمين بتفرض
+
+733
+01:00:02,570 --> 01:00:08,150
+صحيتها ل R و بتثبت صحيتها منها ل R زائد 1 بيكون
+
+734
+01:00:08,150 --> 01:00:11,850
+صحيحة دائما زي ما عملت قبل بشوية بالظبط في الحسابة
+
+735
+01:00:13,320 --> 01:00:17,100
+الان as r goes to infinity هذه مجهزينها اللي هي a
+
+736
+01:00:17,100 --> 01:00:21,140
+أصغر من واحد as r goes to infinity هذا المقدار
+
+737
+01:00:21,140 --> 01:00:26,020
+هيروح لواحد على واحد minus a ماشي لإن هذه بيصير
+
+738
+01:00:26,020 --> 01:00:30,360
+اللي هي عبارة عن geometric series مجموحة infinite
+
+739
+01:00:30,360 --> 01:00:35,260
+geometric series بيصير عبارة عن summation a r, r
+
+740
+01:00:35,260 --> 01:00:42,170
+من صفر من واحد لعند ملا نهايةمظبوط؟ هو يساوي الـ
+
+741
+01:00:42,170 --> 01:00:47,170
+summation واحد على واحد ناقص اللي هي a حرفينها a
+
+742
+01:00:47,170 --> 01:00:54,070
+اللي هي واحد على واحد minus a r من العين سفر هيها
+
+743
+01:00:54,070 --> 01:00:57,450
+من العين سفر واحد زيدي زيدي تربيع إلى ما لانهيها
+
+744
+01:00:57,450 --> 01:00:59,530
+دي geometric series واحد على واحد minus a
+
+745
+01:00:59,530 --> 01:01:06,890
+therefore اللي هي واحد minus aهيكون عبارة عن إيش؟
+
+746
+01:01:06,890 --> 01:01:12,970
+bound for the partial sums of bar series ليش؟ لأن
+
+747
+01:01:12,970 --> 01:01:19,670
+صارت الـSKR limit الـSKR as R goes to infinity
+
+748
+01:01:19,670 --> 01:01:25,530
+أكبر أو يساوي أكبر من 0 أكبر يساوي 0 وأصغر أو
+
+749
+01:01:25,530 --> 01:01:29,790
+يساوي 100 ال limit اللي طلعت عندي 1 على 1 minus
+
+750
+01:01:29,790 --> 01:01:35,880
+100 minus Aصارت مدام الـ sequence of partial sums
+
+751
+01:01:35,880 --> 01:01:43,940
+هذه is bounded هيها by واحد على minus a اه then by
+
+752
+01:01:43,940 --> 01:01:48,180
+ثانو تسعة واحد اربعة مدامت هذه عبارة عن كل ان
+
+753
+01:01:48,180 --> 01:01:52,780
+عناصر موجبة يكفي عشان نثبت انها اللي هي converge
+
+754
+01:01:52,780 --> 01:01:58,220
+انها تكون bounded اللي صارت اللي هي ال series هذه
+
+755
+01:01:58,220 --> 01:02:03,530
+converge زي ما قلتماذا قلت؟ هذه صارت الـ
+
+756
+01:02:03,530 --> 01:02:09,230
+subsequence of partial sums إيش ما لها صارت عبارة
+
+757
+01:02:09,230 --> 01:02:13,930
+عن bounded ماشي، مدام bounded إذا أثبتنا إن هناك
+
+758
+01:02:13,930 --> 01:02:17,570
+sub-sequence bounded من an increasing sequence إذا
+
+759
+01:02:17,570 --> 01:02:19,770
+اتفجنا إن هذا الـ increasing sequence سيكون
+
+760
+01:02:19,770 --> 01:02:23,450
+bounded وهي increasing إذا convergence حسب الـ
+
+761
+01:02:23,450 --> 01:02:26,730
+monotone convergence theorem ومن ثم بيكون خلصنا
+
+762
+01:02:26,730 --> 01:02:32,880
+أثبتنا هذاis a convergent series for B أكبر من
+
+763
+01:02:32,880 --> 01:02:33,720
+واحد
+
+764
+01:02:36,760 --> 01:02:41,140
+Show that 1 على N تربيع زائد N من 1 إلى ملا نهاية
+
+765
+01:02:41,140 --> 01:02:45,700
+ايش بتساوي؟ بتساوي 1 هذا بتاع مشهور وسؤال مشهور في
+
+766
+01:02:45,700 --> 01:02:50,480
+ال .. في ال calculus أو سهل اثباته لو جينا اطلعنا
+
+767
+01:02:50,480 --> 01:02:53,900
+على اللي قولنا بنعمله partial fraction كنا نقول ال
+
+768
+01:02:53,900 --> 01:02:57,860
+1 على K تربيع زائد K بيساوي 1 على K في K زائد 1
+
+769
+01:02:57,860 --> 01:03:02,440
+نار اللي هي عبارة عن 1 على K نار مص 1 على K زائد 1
+
+770
+01:03:03,050 --> 01:03:07,550
+ماشي الحال الآن صارت ال sequence هذا ال series هذه
+
+771
+01:03:07,550 --> 01:03:10,970
+عبارة عن هذا ناقص هذا يعني ال partial sums صارت
+
+772
+01:03:10,970 --> 01:03:16,150
+عبارة عن واحد على N ناقص واحد على N زي واحد ويساوي
+
+773
+01:03:16,150 --> 01:03:17,150
+كمية
+
+774
+01:03:20,550 --> 01:03:26,150
+كيب واحد بيصير واحد على واحد ناقص نص اللي بعده
+
+775
+01:03:26,150 --> 01:03:29,830
+باتنين اللي هو نص ناقص تلت اللي بعده تلت ناقص ربع
+
+776
+01:03:29,830 --> 01:03:33,070
+لما أصل الاخر واحد واحد على ان ناقص واحد على
+
+777
+01:03:33,070 --> 01:03:36,450
+انزايد واحد كلهم بيروحوا مع بعض وضال اول term واخر
+
+778
+01:03:36,450 --> 01:03:40,470
+term الواحد ناقص واحد على انزايد واحد لأن هذا صار
+
+779
+01:03:40,470 --> 01:03:44,770
+اللي هو اللي هو ال partial sum او ال sequence of
+
+780
+01:03:44,770 --> 01:03:50,810
+ال term of the partial sums of this seriesخدوله
+
+781
+01:03:50,810 --> 01:03:54,630
+limit الـ sn as n goes to infinity بساوي واحد نقص
+
+782
+01:03:54,630 --> 01:03:58,090
+limit الواحد الآن زائد واحد هذا بروح للسفر ويساوي
+
+783
+01:03:58,090 --> 01:04:01,110
+واحد لأن صارت ال series اللي عندي هذه الأصلية
+
+784
+01:04:01,110 --> 01:04:04,630
+converged ويساوي ال convergent ايش بساوي بساوي
+
+785
+01:04:04,630 --> 01:04:10,470
+واحد الآن اخر موضوع في اللي هو ال section الأول
+
+786
+01:04:12,460 --> 01:04:16,700
+حاجة اسمها rearrangement of series يعني إعادة
+
+787
+01:04:16,700 --> 01:04:21,220
+ترتيب العناصر اللي هو اللي هي ال series أو ال term
+
+788
+01:04:21,220 --> 01:04:25,160
+سبعة ال series ال rearrangement of a series is
+
+789
+01:04:25,160 --> 01:04:29,140
+another series هي في ريز أخر هتكونwhich is
+
+790
+01:04:29,140 --> 01:04:32,540
+obtained from the given one يعني بناخدها من من من
+
+791
+01:04:32,540 --> 01:04:38,660
+الأصلية by using all of the terms exactly one once
+
+792
+01:04:38,660 --> 01:04:44,020
+يعني بإننا نستخدم عناصر ال series مرة واحدة فقط
+
+793
+01:04:44,020 --> 01:04:49,900
+لازم نستخدمها و بنستخدمها بمرة واحدةbut scrambling
+
+794
+01:04:49,900 --> 01:04:55,720
+the order in which the terms are taken يعني بس إيش
+
+795
+01:04:55,720 --> 01:05:00,820
+بنسوي، بنعيد ترتيب اللي هي أماكن اللي هي S series
+
+796
+01:05:00,820 --> 01:05:06,540
+الآن يعني ال summation XN summation XN هي series
+
+797
+01:05:06,540 --> 01:05:12,950
+قعدنا ترتيبها اللي هي summation YNالأن الـ XL و
+
+798
+01:05:12,950 --> 01:05:18,770
+الـ YN لو جينا حاطناها في مجموعتين هنقول أكيد أكيد
+
+799
+01:05:18,770 --> 01:05:24,890
+الـ X1 لعند الـ XN لعند المالة النهائية هي نفسها
+
+800
+01:05:24,890 --> 01:05:31,940
+Y1 لعند Y2 لعند الـ Yأن إلى ما لا نهاية لكن الـ X1
+
+801
+01:05:31,940 --> 01:05:35,400
+مش شرط الـ Y1 الـ X2 مش شرط الـ Y2 أما هذه كمجموعة
+
+802
+01:05:35,400 --> 01:05:41,900
+هي نفس المجموعة و بنستخدمش ال term الجديد إلا ..
+
+803
+01:05:41,900 --> 01:05:45,840
+بنستخدمش اللي هنا إلا غير مرة واحدة في اللي هي هنا
+
+804
+01:05:45,840 --> 01:05:51,380
+يعني كل اللي هنا موجودة هنا ولكن بكرتيب آخر فبتكون
+
+805
+01:05:51,380 --> 01:05:56,020
+هذه rearrangement of the original series ال
+
+806
+01:05:56,020 --> 01:06:04,620
+summation لـ Xطيب ناخد مثال او اكتر من مثال بقول
+
+807
+01:06:04,620 --> 01:06:07,800
+the harmonic series اللى جابل بشوية هذه الى
+
+808
+01:06:07,800 --> 01:06:12,940
+summation 1 على n 1 زياد نص زياد تلت زياد ربع زياد
+
+809
+01:06:12,940 --> 01:06:17,560
+1 على n has rearrangements like فيه rearrangements
+
+810
+01:06:17,560 --> 01:06:22,040
+واحد ايش قال قال انا اسمعوا خلوني احطلكم اللى هي
+
+811
+01:06:22,040 --> 01:06:27,920
+اجسمها جسمين الزوجيات والفردياتكيف؟ بضأ أولش الزوج
+
+812
+01:06:27,920 --> 01:06:31,540
+نص و ربع، سُدُس، طومن، واحدة على اتنين، ان وضلي
+
+813
+01:06:31,540 --> 01:06:36,020
+ماشي لما لا نهاية و بين هن بضأ احضر الفرديات واحدة
+
+814
+01:06:36,020 --> 01:06:38,760
+على واحد، واحدة على تلت، واحدة على خمسة، واحدة على
+
+815
+01:06:38,760 --> 01:06:42,040
+اتنين، ان ناقص واحد، الى ما لا نهاية، ماشي الحال
+
+816
+01:06:42,040 --> 01:06:48,080
+هذه ال series عنصرها ان بالظبط اللي هنا بس معمول
+
+817
+01:06:48,080 --> 01:06:51,890
+اللي هن إعادة الترتيبواحد تاني جال اسمعه بيرتبها
+
+818
+01:06:51,890 --> 01:06:55,450
+بترتيب تاني اش اللي طبعا بيرتبها حسب ما يخدمه او
+
+819
+01:06:55,450 --> 01:06:58,170
+حسب ما يطلع معاه في ال .. في ال .. في الاشي العملي
+
+820
+01:06:58,170 --> 01:07:02,310
+مثلا الان اول واحد بيقولك بتاخد اللي هو ال .. ال
+
+821
+01:07:02,310 --> 01:07:04,030
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+822
+01:07:04,030 --> 01:07:04,050
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+823
+01:07:04,050 --> 01:07:04,310
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+824
+01:07:04,310 --> 01:07:04,350
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+825
+01:07:04,350 --> 01:07:04,430
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+826
+01:07:04,430 --> 01:07:05,110
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+827
+01:07:05,110 --> 01:07:05,110
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+828
+01:07:05,110 --> 01:07:06,250
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+829
+01:07:07,640 --> 01:07:12,480
+التاني بتاخده اتنين even نص و ربع باخد صوت على
+
+830
+01:07:12,480 --> 01:07:16,560
+المقام even الان التالت بتاخده التلاتة اللي بعده
+
+831
+01:07:16,560 --> 01:07:21,740
+ال odd اللي هو تلت خمس سبع اللي بعده الأربعة اللي
+
+832
+01:07:21,740 --> 01:07:27,340
+بعده ال even6 8 10 1 3 12 اللي بعده الـ 5 اللي
+
+833
+01:07:27,340 --> 01:07:31,440
+بعده الـ odd 9 11 11 11 13 11 15 11 16 17 اللي
+
+834
+01:07:31,440 --> 01:07:34,300
+بعده ال 6 ال even اللي بعده ال 7 ال odd و هكذا و
+
+835
+01:07:34,300 --> 01:07:37,840
+نظل ماشيين إلى مانا نهاية هذه برضه rearrangement
+
+836
+01:07:37,840 --> 01:07:43,860
+لمين لاللي هي ال series اللي هي summation 1 على
+
+837
+01:07:53,010 --> 01:07:57,590
+التعريف الرياضي يسمى الـ Series يسمى التعريف
+
+838
+01:07:57,590 --> 01:08:00,790
+الرياضي الرياضي الرياضي الرياضي الرياضي الرياضي
+
+839
+01:08:00,790 --> 01:08:06,170
+الرياضي الرياضي الرياضي الرياضييعني هذا الـ
+
+840
+01:08:06,170 --> 01:08:10,510
+bijection عبارة عن إعادة الترتيب من السيريز
+
+841
+01:08:10,510 --> 01:08:16,090
+الأصلية إلى السيريز الجديدة لأن هذه ترتيبها I وهذه
+
+842
+01:08:16,090 --> 01:08:19,770
+ترتيبها F of I صارت بس مدام أن الـ function one to
+
+843
+01:08:19,770 --> 01:08:24,710
+one وunto معناته أن كله هيستخدم ال terms وكله لا
+
+844
+01:08:24,710 --> 01:08:28,510
+يستخدم إلا مرة واحدة لأن ال function one to one
+
+845
+01:08:28,510 --> 01:08:30,750
+وunto فصار عند ال Yم
+
+846
+01:08:36,050 --> 01:08:43,730
+Y1 XF1 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+847
+01:08:43,730 --> 01:08:43,730
+Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+848
+01:08:43,730 --> 01:08:43,730
+Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+849
+01:08:43,730 --> 01:08:43,730
+Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+850
+01:08:43,730 --> 01:08:43,730
+Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+851
+01:08:43,730 --> 01:08:46,350
+Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2 Y2 XF2
+
+852
+01:08:46,350 --> 01:08:52,210
+Y2 XF2 Y2 XF2 Y2 XFما تأثير الـ Rearrangements على
+
+853
+01:08:52,210 --> 01:08:56,890
+السيريز الأصلية بقولك لو كانت السيريز absolutely
+
+854
+01:08:56,890 --> 01:09:00,350
+convergent وده اللي هنتطرقله بس لو كانت السيريز
+
+855
+01:09:00,350 --> 01:09:04,250
+absolutely convergent ترتب زي ما بدك مش هتفرج
+
+856
+01:09:04,250 --> 01:09:08,090
+معانا اللي هو إيه ال convergence يعني let
+
+857
+01:09:08,090 --> 01:09:11,690
+summation xn be an absolutely convergent series in
+
+858
+01:09:11,690 --> 01:09:18,610
+R then any rearrangement of summation xnconverge
+
+859
+01:09:18,610 --> 01:09:21,770
+to the same value يعني لو كانت ال series
+
+860
+01:09:21,770 --> 01:09:27,150
+absolutely convergent أي ترتيب إلها آخر وفق تعريف
+
+861
+01:09:27,150 --> 01:09:31,390
+الترتيب اللي حنا حكينا عنه اللي هو هيعطي نفس ال
+
+862
+01:09:31,390 --> 01:09:34,510
+limit هتكون converge ويلها نفس ال limit خلّينا
+
+863
+01:09:34,510 --> 01:09:38,910
+نشوف البرهان ونمر عليه let summation xn converge
+
+864
+01:09:38,910 --> 01:09:44,410
+to xand let yn be a rearrangement of summation xn
+
+865
+01:09:44,410 --> 01:09:46,590
+and let yn be a rearrangement of summation xn and
+
+866
+01:09:46,590 --> 01:09:46,870
+let yn be a rearrangement of summation xn and let
+
+867
+01:09:46,870 --> 01:09:46,870
+yn be a rearrangement of summation xn and let yn
+
+868
+01:09:46,870 --> 01:09:46,870
+be a rearrangement of summation xn and let yn be a
+
+869
+01:09:46,870 --> 01:09:46,870
+rearrangement of summation xn and let yn be a
+
+870
+01:09:46,870 --> 01:09:46,950
+rearrangement of summation xn and let yn be a
+
+871
+01:09:46,950 --> 01:09:47,870
+rearrangement of summation xn and let yn be a
+
+872
+01:09:47,870 --> 01:09:49,130
+rearrangement of summation xn and let yn be a
+
+873
+01:09:49,130 --> 01:09:51,570
+rearrangement of summation xn and let en be a
+
+874
+01:09:51,570 --> 01:09:51,590
+rearrangement of summation xn and let en be a
+
+875
+01:09:51,590 --> 01:09:51,690
+rearrangement of summation xn and let en be a
+
+876
+01:09:51,690 --> 01:09:51,690
+rearrangement of summation xn and let en be a
+
+877
+01:09:51,690 --> 01:09:51,690
+rearrangement of summation xn and let en be a
+
+878
+01:09:51,690 --> 01:09:51,690
+rearrangement of summation xn and let en be a
+
+879
+01:09:51,690 --> 01:09:51,690
+rearrangement of summation xn and let en be a
+
+880
+01:09:51,690 --> 01:09:52,390
+rearrangement of summation xn and let en be a
+
+881
+01:09:52,390 --> 01:09:55,870
+rearrangement of summation xn and let en be a
+
+882
+01:09:55,870 --> 01:09:59,130
+rearrangement of summation xn and let en be a
+
+883
+01:09:59,130 --> 01:10:02,130
+rearrangement of summation xn and let en be
+
+884
+01:10:07,510 --> 01:10:10,290
+ماذا يعني اكسان اكسان اكسان اكسان اكسان اكسان
+
+885
+01:10:10,290 --> 01:10:10,790
+اكسان اكسان اكسان اكسان اكسان اكسان اكسان اكسان
+
+886
+01:10:10,790 --> 01:10:10,850
+اكسان اكسان اكسان اكسان اكسان اكسان اكسان اكسان
+
+887
+01:10:10,850 --> 01:10:11,450
+اكسان اكسان اكسان اكسان اكسان اكسان اكسان اكسان
+
+888
+01:10:11,450 --> 01:10:12,530
+اكسان اكسان اكسان اكسان اكسان اكسان اكسان اكسان
+
+889
+01:10:12,530 --> 01:10:15,790
+اكسان اكسان اكسان اكسان اكسان اكسان اكسان اكسان
+
+890
+01:10:15,790 --> 01:10:23,530
+اكسان اكسان اكسان اكسان اكسان اكسان اكسان اكسان
+
+891
+01:10:23,530 --> 01:10:30,260
+اكسهو هيعيده كمان مرة الحديث كمان بقول إن ده ما
+
+892
+01:10:30,260 --> 01:10:34,040
+تصممش إن الـ xn converts إذا لكل y أكبر من 0 there
+
+893
+01:10:34,040 --> 01:10:41,900
+exists n such that fq أكبر من n and اللي هي sn x1
+
+894
+01:10:41,900 --> 01:10:50,260
+x2 اللي عند xn then هيكون عنديالـ X K K من عند R
+
+895
+01:10:50,260 --> 01:10:54,540
+زي الواحد عند Q أصغر من مين من إبسلون كانت بطريقة
+
+896
+01:10:54,540 --> 01:10:59,000
+تانية نتحمل و مهم مش الان الان إيش اللي هي
+
+897
+01:10:59,000 --> 01:11:04,760
+summation X N بساوة مين بساوة X خلاص نعتبرنا اللي
+
+898
+01:11:04,760 --> 01:11:08,320
+هو absolute convergence يعني و نعتبرنا ال terms
+
+899
+01:11:08,320 --> 01:11:12,000
+positive و خلاص و نشتغل عليها summation X N بساوة
+
+900
+01:11:12,000 --> 01:11:19,230
+مين بساوة Xالان for every epsilon أكبر من سفر
+
+901
+01:11:19,230 --> 01:11:25,350
+there exists N such that for every N أو Q سميها
+
+902
+01:11:25,940 --> 01:11:30,460
+أكبر من N أكبر يساوي N عندي اللي هو ال summation
+
+903
+01:11:30,460 --> 01:11:35,240
+هذا اللي هو خلّيني أسميه اللي هو ال sequence of
+
+904
+01:11:35,240 --> 01:11:40,580
+partial sums ل S N اللي هي S N ناقص X أصغر من مين
+
+905
+01:11:40,580 --> 01:11:46,400
+من إبسلون for every N أكبر أو يساوي H N هذا اللي
+
+906
+01:11:46,400 --> 01:11:49,780
+هو ال limit العادية الكوشي criterion لها أنه هيكون
+
+907
+01:11:49,780 --> 01:11:56,890
+عندي ال summation اللي هي S Nباخد الـ S N و S Q هو
+
+908
+01:11:56,890 --> 01:12:03,490
+ما ياخد الـ S N ناقص
+
+909
+01:12:03,490 --> 01:12:09,510
+S P لأكبر الـ Q S Q ناقص S N مش مش ممكن أتكلم عن
+
+910
+01:12:09,510 --> 01:12:14,370
+داخل absolute value ناقص S N فبنفع S N أه ما هي
+
+911
+01:12:14,370 --> 01:12:18,480
+صحيحة على كل N أكبر سواء من ضمنها مين؟ Nأصغر من
+
+912
+01:12:18,480 --> 01:12:21,940
+مين من إبسلون هذا هذا إيش .. إيش هذا قيمته يساوي
+
+913
+01:12:21,940 --> 01:12:25,700
+summation اللي هو حاطة ال absolute value ماخد ال
+
+914
+01:12:25,700 --> 01:12:30,240
+absolute convergence K من عند N زائد واحد إلى مالة
+
+915
+01:12:30,240 --> 01:12:35,800
+نهائية لعند Q اللي هو أصغر من مين من إبسلون عارفين
+
+916
+01:12:35,800 --> 01:12:39,640
+هذا أه؟ اللي هي أسة Q نقص أسن لأن ماخد ال series
+
+917
+01:12:39,640 --> 01:12:41,620
+إيش صمت absolute value مش مشكلة
+
+918
+01:12:44,600 --> 01:12:48,740
+SN هي الـ sequence of partial sums لهذه للـ
+
+919
+01:12:48,740 --> 01:12:52,140
+absolute values مش للـ XN للـ absolute values فصار
+
+920
+01:12:52,140 --> 01:12:57,620
+عندي الآن استخدام اللي هو limit as a limit و as a
+
+921
+01:12:57,620 --> 01:13:02,760
+convergence to X و استخدامها as a Cauchy criterion
+
+922
+01:13:02,760 --> 01:13:06,840
+ماشي؟ قول ما هي هذه بالنسبة ليها N و هذه بالنسبة
+
+923
+01:13:06,840 --> 01:13:10,880
+ليها N تانية مليش استعملت نفس الـ N؟ take N
+
+924
+01:13:10,880 --> 01:13:17,610
+maximum بين الجهتين بين فعلامين للجهتين طبيعيطيب
+
+925
+01:13:17,610 --> 01:13:21,350
+.. لا
+
+926
+01:13:21,350 --> 01:13:24,510
+والله نستخدم الأسئلة لمين؟ لهذه
+
+927
+01:13:27,810 --> 01:13:30,230
+بطبيعي، ما هو مدانة Absolutely Convergent دا
+
+928
+01:13:30,230 --> 01:13:33,030
+Convergent مدانة Convergent دا مطبقة هذه على مين
+
+929
+01:13:33,030 --> 01:13:35,930
+على الـ Convergence وهذه مطبقة على مين على الـ
+
+930
+01:13:35,930 --> 01:13:38,430
+Absolutely Convergent برضه بنفع .. آه بنفع بناخد
+
+931
+01:13:38,430 --> 01:13:41,570
+اللي هو بطلعله أن واحد هذه و هذا أن اتنين و بناخده
+
+932
+01:13:41,570 --> 01:13:44,010
+من ال maximum للتان تان بيصير لي هذه ال inequality
+
+933
+01:13:44,010 --> 01:13:47,230
+و هذه ال inequality صحيحة .. صحيحات لمين للان اللي
+
+934
+01:13:47,230 --> 01:13:50,720
+هي ال maximum اللي عنده طيب الآنإذا صار عندي هذه
+
+935
+01:13:50,720 --> 01:13:54,120
+الـ Sn بقصود فيها فعلًا اللي هي ال sequence of
+
+936
+01:13:54,120 --> 01:14:01,060
+partial sums لسيريز الأصلية Xn والـ Sn' هي ميان ال
+
+937
+01:14:01,060 --> 01:14:05,400
+sequence اللي هو تبعت ال absolute valuesهذه طبعا
+
+938
+01:14:05,400 --> 01:14:09,700
+حسبناها as q prime نقص as n prime بساوي هذه نقص
+
+939
+01:14:09,700 --> 01:14:13,060
+هذه من واحد عند q ومن واحد عند n اللي هي بساوي هذه
+
+940
+01:14:13,060 --> 01:14:16,940
+اللي هي هذه حسبناها عارفينها طيب مش هذا الموضوع
+
+941
+01:14:16,940 --> 01:14:23,500
+هذا الأمر سهل يعني إذا صار عندي الآن هي هذه أو هي
+
+942
+01:14:23,500 --> 01:14:32,260
+هذه أو هذه كلها بدي أشوف كيف أستعملهاأحنا ليش بدنا
+
+943
+01:14:32,260 --> 01:14:36,580
+.. بدنا الـ Series الجديدة Summation Yn اللي هي
+
+944
+01:14:36,580 --> 01:14:40,980
+الـ Rearrangement شوف كده لأن Now let M limited M
+
+945
+01:14:40,980 --> 01:14:45,260
+be such that all of the terms X1 لعند الـ Xm are
+
+946
+01:14:45,260 --> 01:14:52,680
+contained as a sum and in T M Y1 لعند ال Ym إيش
+
+947
+01:14:52,680 --> 01:14:55,820
+اللي بيقولوا هذا؟ وهذا الكلام اللي هو اللي إحنا
+
+948
+01:14:55,820 --> 01:14:59,760
+بدنا نتهيئله و هو اللي بيفتحنا الطريق أمام البرهان
+
+949
+01:15:01,400 --> 01:15:09,360
+جالي بيحكيلي انه خلّينا نختار M في الـN ابحث الـTM
+
+950
+01:15:09,360 --> 01:15:14,740
+هذه اللي هي الـSequence of partial sums لمين للـYN
+
+951
+01:15:14,740 --> 01:15:20,960
+اللي هي Y1 زائد Y2 زائد YM بقوللي أنا بدأ اختار
+
+952
+01:15:20,960 --> 01:15:29,370
+الـM هذه على أساسإنه كل ال terms اللي لجيتها من ال
+
+953
+01:15:29,370 --> 01:15:34,310
+N اللي فوق من X1 لعند XN تكون موجودة من بين هدول
+
+954
+01:15:34,310 --> 01:15:37,970
+ال terms بقدر أه بقدر لإن هدول عشمها ال hand
+
+955
+01:15:37,970 --> 01:15:44,210
+finite و في النهاية مدام finite إذا هيكون من ضمن
+
+956
+01:15:44,210 --> 01:15:49,890
+ال Y1 Y2 إلى 100 ألف ألف مليون سميته ال 100 ألف
+
+957
+01:15:49,890 --> 01:15:54,480
+ألف مليون اللي هو 100 YNإذاً الأم اخترتها بناءً
+
+958
+01:15:54,480 --> 01:16:02,060
+على إني أضمن الـ X1 والـ X2 لعند الـ XN تين لعند
+
+959
+01:16:02,060 --> 01:16:06,080
+الـ XN كابيتال تين هدولة من هدول يعني بس إعادة
+
+960
+01:16:06,080 --> 01:16:10,890
+تسميتهمإذا الآن اللي .. اللي .. اللي اخترته انه
+
+961
+01:16:10,890 --> 01:16:15,570
+قلتله let M element in N مهما كانت كبيرة be such
+
+962
+01:16:15,570 --> 01:16:21,510
+that all of the terms X1, X2, XN are contained as
+
+963
+01:16:21,510 --> 01:16:26,610
+a sum and يعني مجموعات من ضمن المجموع هذا Y1 زي Y2
+
+964
+01:16:26,610 --> 01:16:35,330
+عند مين؟ YM it follows الآن if M أكبر أو يساوي M
+
+965
+01:16:36,630 --> 01:16:42,970
+then T M ناقص S M ناقص
+
+966
+01:16:42,970 --> 01:16:48,770
+S M ناقص S M ناقص S M ناقص S M ناقص S M ناقص S M
+
+967
+01:16:48,770 --> 01:16:48,770
+ناقص S M ناقص S M ناقص S M ناقص S M ناقص S M ناقص
+
+968
+01:16:48,770 --> 01:16:48,910
+S M ناقص S M ناقص S M ناقص S M ناقص S M ناقص S M
+
+969
+01:16:48,910 --> 01:16:48,970
+ناقص S M ناقص S M ناقص S M ناقص S M ناقص S M ناقص
+
+970
+01:16:48,970 --> 01:16:49,030
+S M ناقص S M ناقص S M ناقص S M ناقص S M ناقص S M
+
+971
+01:16:49,030 --> 01:16:49,850
+ناقص S M ناقص S M ناقص S M ناقص S M ناقص S M ناقص
+
+972
+01:16:49,850 --> 01:16:54,450
+S M ناقص S M ناقص S
+
+973
+01:16:56,000 --> 01:16:58,900
+ممكن اللي عندها و ممكن يظل معاها عندها نور على نور
+
+974
+01:16:58,900 --> 01:17:04,620
+يعني بدنا نشيل منه مين ال S N ال S N ال N هذه
+
+975
+01:17:04,620 --> 01:17:11,220
+أشمالها أكبر من مين أكبر أو تساوي ال N طيب إذا هذه
+
+976
+01:17:11,220 --> 01:17:21,340
+هيكون consist of finite sum of terms X K with K
+
+977
+01:17:21,340 --> 01:17:24,600
+أكبر من مين من ال N
+
+978
+01:17:29,340 --> 01:17:37,280
+الان من عند الواحد لعند الان انا شيلتهم من ال T M
+
+979
+01:17:37,280 --> 01:17:47,000
+باشي اه ال S N هذه واحد اتنين لعند اللي هي ال N و
+
+980
+01:17:47,000 --> 01:17:53,840
+جلطتها لأن ال N ايه شمالها اكبر او يساوي M N اكبر
+
+981
+01:17:53,840 --> 01:17:58,980
+او يساوي Mماشي طيب صار عندي ال N ال TM انشال منها
+
+982
+01:17:58,980 --> 01:18:05,180
+ال X1 وال X2 وال XN اه اذا اللي ضلن من ال terms
+
+983
+01:18:05,180 --> 01:18:10,940
+هذه او ماينة اللي هنا موجودة ال TM اللي موجودة هنا
+
+984
+01:18:10,940 --> 01:18:20,090
+اللي فيها اكيد هيكون بعد ال N اللي ضلنماشي؟ لإنه
+
+985
+01:18:20,090 --> 01:18:24,770
+شيلت منه لإن عندك S1 لإن عندك S2 و هذه S N ضال
+
+986
+01:18:24,770 --> 01:18:31,090
+منها من اللي بعد ال N برضه إذا في النهاية ال terms
+
+987
+01:18:31,090 --> 01:18:36,790
+اللي هنا هيكون اللي بيحتوي على sums of terms ل X K
+
+988
+01:18:36,790 --> 01:18:42,250
+و ال K هيكون أكبر من 100 من ال N لإن اللي ترتيبه
+
+989
+01:18:42,250 --> 01:18:45,770
+من ال N و اتنازل كله راح من عملية الطرح هذه
+
+990
+01:18:49,100 --> 01:18:55,660
+أذا for some q أكبر من n we have Tm ناقص Tn هذا
+
+991
+01:18:55,660 --> 01:19:00,820
+absolute value بساوى ال summation لل X K كي من
+
+992
+01:19:00,820 --> 01:19:06,420
+وين؟ من n زائد واحد لعند مين؟ لعند Q ليش؟لأنه انا
+
+993
+01:19:06,420 --> 01:19:10,320
+شيلت كل العناصر اللي هي انها من عند X1 لعند Xn
+
+994
+01:19:10,320 --> 01:19:14,600
+اللي هي من هنا ماضلش فيها ولا إشي إذا إذا بده يضل
+
+995
+01:19:14,600 --> 01:19:17,940
+مجموح هذا هيكونه تضل المجموع اللي هو من N زائد
+
+996
+01:19:17,940 --> 01:19:21,520
+واحد وطالع لوين مش عارف ماتفرجش معايا ان شاء الله
+
+997
+01:19:21,520 --> 01:19:25,480
+يضلوا لـ 100 ألف ألف ألف مليون بعدها المهم من بعد
+
+998
+01:19:25,480 --> 01:19:33,930
+ال N زائد واحد لعند Q XK أصغر من مين من Y ماشيليش
+
+999
+01:19:33,930 --> 01:19:39,150
+أصغر من إبسلون؟ لأن هذا دايما أصغر من إبسلون لكل
+
+1000
+01:19:39,150 --> 01:19:45,630
+QH أكبر من H من N سواء الـQ هذه أو الـQ الأكبر
+
+1001
+01:19:45,630 --> 01:19:53,010
+منها إيش ضمنك أن هذا Q أكبر من هذا؟ أنا لما اخترت
+
+1002
+01:19:53,010 --> 01:19:56,030
+شيلت كل ال terms، ضالت ال terms اللي بعد ال N
+
+1003
+01:19:56,030 --> 01:19:59,210
+capital لما شيلتها منها طيب، إذن أكيد الـQ هذه
+
+1004
+01:19:59,210 --> 01:20:04,780
+أكبر من ال Nالان ننخص المعلومات اللي هنستخدمها
+
+1005
+01:20:04,780 --> 01:20:08,180
+Therefore if M أكبر أو سوء N, if M أكبر أو سوء N,
+
+1006
+01:20:08,500 --> 01:20:12,440
+we have الـTM ناقص الاكسام أصغر ال triangle
+
+1007
+01:20:12,440 --> 01:20:16,840
+equality TM ناقص SN زي SN ناقص EH ناقص X هذا يقول
+
+1008
+01:20:16,840 --> 01:20:20,560
+إنه أصغر من مين من هنا من إبسلون طيب الSN ناقص X
+
+1009
+01:20:21,200 --> 01:20:24,260
+اللي هو برضه هتكون إيه شمالها أصغر من إبسلون لإنه
+
+1010
+01:20:24,260 --> 01:20:28,180
+هذا أثبتناها صحيحة لكل N أكبر أو يساوي N وهذه الـN
+
+1011
+01:20:28,180 --> 01:20:32,040
+هي أكبر أو يساوي N إذا صار عند أكبر أو يساوي N إذا
+
+1012
+01:20:32,040 --> 01:20:34,580
+الـN أكبر أو يساوي N capital إذا صار هذا أصغر من
+
+1013
+01:20:34,580 --> 01:20:36,960
+إبسلون زي إبسلون أصغر من اتنين إبسلون إذا صار عند
+
+1014
+01:20:36,960 --> 01:20:41,620
+الـTM ناقص X أصغر من اتنين إبسلون لكل M أكبر أو
+
+1015
+01:20:41,620 --> 01:20:50,260
+يساوي N وهذا يعني أنهاللي هي ال summation هذي طبعا
+
+1016
+01:20:50,260 --> 01:20:53,600
+هي عبارة عن sequence of partial sums لل YM صحيح ال
+
+1017
+01:20:53,600 --> 01:20:57,380
+summation لل YM converge و converge to main to X
+
+1018
+01:20:57,380 --> 01:21:00,940
+وبهيك بيكون احنا انهينا هنا اللي هو ال section
+
+1019
+01:21:00,940 --> 01:21:03,680
+الأول في شبط التسعة وإلى لقاينة
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM.srt
new file mode 100644
index 0000000000000000000000000000000000000000..71bb546449958a39d489198783552571c0aa20a3
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM.srt
@@ -0,0 +1,2102 @@
+1
+00:00:05,280 --> 00:00:12,080
+بسم الله الرحمن الرحيم المحاضرة الرابعة مساق تحليل 
+
+2
+00:00:12,080 --> 00:00:18,780
+حقيقة 2 لطلبة الجامعة الإسلامية كلية العلوم قسم 
+
+3
+00:00:18,780 --> 00:00:26,480
+الرياضيات نستكمل الحديث عن نظرية القيمة المتوسطة
+
+4
+00:00:26,480 --> 00:00:33,380
+أو the mean value theorem وما يتعلق بذلك تحدثنا 
+
+5
+00:00:33,380 --> 00:00:40,130
+المرة الماضية عن معنى relative extrema سواء relative
+
+6
+00:00:40,130 --> 00:00:43,290
+maximum أو relative minimum أو بمعنى آخر ال
+
+7
+00:00:43,290 --> 00:00:47,450
+local maximum والـ local minimum
+
+8
+00:00:47,450 --> 00:00:53,910
+وعرفناها المرة الماضية وتحدثنا أيضًا عن اللي هو
+
+9
+00:00:53,910 --> 00:00:58,650
+الـ interior extreme theorem الـ interior extreme
+
+10
+00:00:58,650 --> 00:01:03,430
+theorem اللي هي جاوبتنا على السؤال اللي بنحكي فيه 
+
+11
+00:01:03,770 --> 00:01:09,170
+أين يمكن إيجاد القيم العظمى أو القيم الصغرى 
+
+12
+00:01:09,170 --> 00:01:14,730
+النظرية اللي بتحكي كانت إن لو كانت C عبارة عن 
+
+13
+00:01:14,730 --> 00:01:20,610
+interior point اللي هي الـ interval I اللي عندها
+
+14
+00:01:20,610 --> 00:01:27,220
+اللي هي الـ function F الـ derivative موجودة بقول لو
+
+15
+00:01:27,220 --> 00:01:32,140
+كان عند النقطة هذه في relative extremum سواء
+
+16
+00:01:32,140 --> 00:01:36,740
+minimum أو maximum للـ F عندها نقطة C قطعًا الـ F
+
+17
+00:01:36,740 --> 00:01:42,900
+prime of C هتساوي 0 ومن هنا استقينا أن أين يمكن 
+
+18
+00:01:42,900 --> 00:01:48,060
+إيجاد النقطة العظمى والنقطة الصغرى المحلية وقلنا
+
+19
+00:01:48,060 --> 00:01:52,480
+هي في أحد ثلاث مناطق اللي هو أما F prime بتساوي 0
+
+20
+00:01:53,060 --> 00:01:58,080
+أو F prime does not exist للنقاط الداخلية طبعًا أو
+
+21
+00:01:58,080 --> 00:02:02,120
+اللي هو الـ end points فثلاث مناطق اللي هي بنفحص 
+
+22
+00:02:02,120 --> 00:02:07,360
+فيها اللي هي الـ relative maximum أو الـ relative
+
+23
+00:02:07,360 --> 00:02:11,400
+minimum والـ Corollary اللي بعد هي اللي حكيته عنه
+
+24
+00:02:11,400 --> 00:02:16,400
+قبل بشوية الآن وبعدين تحدثنا.. بدنا نتحدث على
+
+25
+00:02:16,400 --> 00:02:21,180
+الـ Rolle's theorem وقولنا الـ Rolle's theorem اللي هي 
+
+26
+00:02:21,180 --> 00:02:25,840
+كانت بتعالج أنه لو كانت عندي function F من عند A و
+
+27
+00:02:25,840 --> 00:02:31,880
+B لعند R وكانت هذه الـ function is continuous على
+
+28
+00:02:31,880 --> 00:02:37,830
+closed interval اللي هي من a لعند b و
+
+29
+00:02:37,830 --> 00:02:42,030
+differentiable على الـ open interval من a لـ b وأيضًا
+
+30
+00:02:42,030 --> 00:02:47,230
+وضعنا شرط أن f of a بيساوي f of b بيساوي صفر
+
+31
+00:02:47,230 --> 00:02:54,210
+بقولنا الـ f differentiable on a و b والـ F 
+
+32
+00:02:54,210 --> 00:03:00,110
+continuous on A و B كله مع بعض هذا أعطانا أنه there
+
+33
+00:03:00,110 --> 00:03:05,130
+exists C element in A و B بنلاقي النقطة C بين A و B
+
+34
+00:03:05,130 --> 00:03:10,530
+بحيث أن يكون ميل المماس عندها يساوي صفر أو F prime 
+
+35
+00:03:10,530 --> 00:03:16,370
+of C بيساوي صفر أو بمعنى آخر يوجد مماس عند هذه
+
+36
+00:03:16,370 --> 00:03:22,030
+النقطة يوازي محور الـ x بعد ذلك طبعًا تحدثنا عن
+
+37
+00:03:22,030 --> 00:03:27,650
+الـ Mean Value Theorem الـ Mean Value Theorem اللي
+
+38
+00:03:27,650 --> 00:03:31,710
+كانت أيضًا على الرسم خلينا نشوف الـ Mean Value
+
+39
+00:03:31,710 --> 00:03:35,590
+Theorem قالت إن لو كانت الـ function is continuous
+
+40
+00:03:35,590 --> 00:03:40,750
+على closed interval من A لـ B وكانت اللي هي
+
+41
+00:03:40,750 --> 00:03:43,470
+differentiable على الـ open interval A و B
+
+42
+00:03:49,160 --> 00:03:57,260
+الآن الاستنتاج اللي بيكون تعميم للـ Rolle's theorem 
+
+43
+00:03:57,260 --> 00:04:03,350
+إيش هو الاستنتاج؟ إنه بنلاقي نقطة في المنطقة بين A و
+
+44
+00:04:03,350 --> 00:04:10,890
+B بكون المماس عندها موازٍ للقاطع اللي بين مين؟ الـ
+
+45
+00:04:10,890 --> 00:04:14,850
+A والـ B أو بمعنى آخر بكون عند النقطة C اللي 
+
+46
+00:04:14,850 --> 00:04:21,890
+لايجناها اللي هي F prime of C بيساوي F of B ناقص F 
+
+47
+00:04:21,890 --> 00:04:29,310
+of A على B ناقص A وهذا هو الذي يمثل ميل القاطع بين
+
+48
+00:04:29,310 --> 00:04:34,170
+النقطة A أو بين النقطة B أو بين النقطة A و F of A
+
+49
+00:04:34,170 --> 00:04:38,430
+ونقطة B و F of B فبتكون هذه عبارة عن نظرية القيمة 
+
+50
+00:04:38,430 --> 00:04:44,030
+المتوسطة اللي هي تعميم لـ Rolle's theorem بعدها
+
+51
+00:04:44,030 --> 00:04:51,790
+تحدثنا عن اللي هو بعض النتائج قلنا إن لو كانت الـ
+
+52
+00:04:51,790 --> 00:04:56,930
+function F continuous على closed interval I وكانت
+
+53
+00:04:56,930 --> 00:05:00,410
+F differentiable على الـ open interval وكانت اللي
+
+54
+00:05:00,410 --> 00:05:05,090
+هي F prime بتساوي 0 على كل الفترة A و B قلنا هذه
+
+55
+00:05:05,090 --> 00:05:09,430
+الدالة هي الدالة الثابتة هتكون The F is constant
+
+56
+00:05:09,430 --> 00:05:16,400
+على الفترة المغلقة A أو B أيضًا Corollary لهذه
+
+57
+00:05:16,400 --> 00:05:20,840
+النظرية برضه اللي هي معلومات معهودة من الـ calculus
+
+58
+00:05:20,840 --> 00:05:26,340
+اللي هي أنه لو كان في عندنا دالتين F و G وهذولة 
+
+59
+00:05:26,340 --> 00:05:30,600
+الدالتين اللي عندنا اللي هن.. اللي هن نفس المشتقة
+
+60
+00:05:30,600 --> 00:05:36,380
+على الفترة A و B open بكون عندي الفرق بين اللي هي 
+
+61
+00:05:36,380 --> 00:05:39,460
+الـ F والـ G هو عبارة عن ثابت أو بمعنى آخر F
+
+62
+00:05:39,460 --> 00:05:44,880
+بتساوي G زائد C for some constant C على الفترة I
+
+63
+00:05:44,880 --> 00:05:51,160
+بعد هيك اللي دخلنا و عرفنا شو معنى F من I لعند R
+
+64
+00:05:51,160 --> 00:05:55,780
+is increasing على الـ interval قلنا أن F is
+
+65
+00:05:55,780 --> 00:06:03,570
+increasing إذا كانت اللي هي أخذنا أي نقطتين في الـ
+
+66
+00:06:03,570 --> 00:06:09,870
+I X1 و X2 بتحقق إن X1 أصغر من X2 إذا قدنا هذا أن
+
+67
+00:06:09,870 --> 00:06:13,430
+f of X1 أصغر أو يساوي f of X2 من جول عن الـ function 
+
+68
+00:06:13,430 --> 00:06:19,430
+الشمالها is increasing الآن اللي هو similar لتعريف
+
+69
+00:06:19,430 --> 00:06:24,190
+الـ decreasing إن X1 أصغر من X2 تؤدي إلى F of X1
+
+70
+00:06:24,190 --> 00:06:29,230
+أكبر أو يساوي F of X2 بتكون اليد دي اللي شمالها
+
+71
+00:06:29,230 --> 00:06:33,970
+is decreasing على هذه الفترة في المقابل أيضًا في
+
+72
+00:06:33,970 --> 00:06:36,810
+تعريف حاجة اسمها برضه ذكرناه هو strictly
+
+73
+00:06:36,810 --> 00:06:40,230
+increasing أو strictly decreasing strictly 
+
+74
+00:06:40,230 --> 00:06:44,070
+increasing أو strictly decreasing نحكي عن الـ
+
+75
+00:06:44,070 --> 00:06:48,610
+strictly increasing بتكون اللي هي بتحقق إيش؟ لو X1
+
+76
+00:06:48,610 --> 00:06:54,590
+و X2 في الـ I و X1 أصغر من X2 هذا بيؤدي لي إلى f of
+
+77
+00:06:54,590 --> 00:07:00,530
+X1 strictly أصغر من f of X2 مع عدم إمكانية 
+
+78
+00:07:00,530 --> 00:07:06,340
+المساوية الآن similarly for الـ F is اللي هو 
+
+79
+00:07:06,340 --> 00:07:10,820
+strictly decreasing هذا تحدثنا فيه المرة الماضية
+
+80
+00:07:10,820 --> 00:07:14,240
+النظرية 
+
+81
+00:07:14,240 --> 00:07:17,860
+اللي هي let F من I لعند R be differentiable اللي
+
+82
+00:07:17,860 --> 00:07:23,100
+هي أوصلنا عند هذه النظرية خلينا نشوف هذه النظرية و
+
+83
+00:07:23,100 --> 00:07:30,900
+نتفحص إيش هي النظرية بتقول ومن ثم نشوف لماذا اللي هو 
+
+84
+00:07:30,900 --> 00:07:38,960
+هي صحيحة عندي
+
+85
+00:07:38,960 --> 00:07:46,440
+نظرية بتقول لي theorem 6.2.7 اللي بتحكي النظرية
+
+86
+00:07:46,440 --> 00:07:54,580
+بتقول لي لو كان عندي F let F من I لعند R be
+
+87
+00:07:54,580 --> 00:08:05,490
+differentiable on the interval.. the interval I
+
+88
+00:08:05,490 --> 00:08:11,910
+فرضنا إن F is differentiable على الـ interval I الآن
+
+89
+00:08:11,910 --> 00:08:17,690
+اللي بدأنا نقوله عبارة عن test بنستعمل الـ test في 
+
+90
+00:08:17,690 --> 00:08:22,030
+معرفة إن F is increasing أو لأ و F is decreasing 
+
+91
+00:08:22,030 --> 00:08:27,530
+أو لأ بنقول F is increasing 
+
+92
+00:08:29,130 --> 00:08:36,650
+on I إيش الفحص؟ if and only if F prime of X أكبر
+
+93
+00:08:36,650 --> 00:08:44,130
+أو يساوي صفر لكل X موجودة في الـ I والآن F is
+
+94
+00:08:44,130 --> 00:08:52,450
+decreasing on I if and only if F prime of X أصغر
+
+95
+00:08:52,450 --> 00:08:57,750
+أو يساوي صفر لكل X element in I هذه هي النظرية 
+
+96
+00:08:57,750 --> 00:09:04,430
+اللي عندنا وهنبرهن اللي هو A أو B في الكتاب نبرهن 
+
+97
+00:09:04,430 --> 00:09:09,370
+A خلّينا نبرهن مين؟ نبرهن B وهتكون A similarly for
+
+98
+00:09:09,370 --> 00:09:17,550
+B الآن بي.. بدنا نفترض suppose that
+
+99
+00:09:17,550 --> 00:09:30,710
+F is decreasing on I F is decreasing on I بدنا..
+
+100
+00:09:30,710 --> 00:09:36,650
+بدنا نثبت الآن أن الـ F prime of X اللي هي أصغر أو
+
+101
+00:09:36,650 --> 00:09:43,190
+يساوي مين؟ صفر طبعًا F prime of X في النهاية إيش
+
+102
+00:09:43,190 --> 00:09:47,930
+هي؟ عشان نشوف كيف بنفكر أول شيء F prime of X بيساوي
+
+103
+00:09:47,930 --> 00:09:55,080
+limit F of X أو f prime of c مثلًا for arbitrary c f
+
+104
+00:09:55,080 --> 00:10:00,820
+of x ناقص f of c على x ناقص c لما x تروح للـ c إذا 
+
+105
+00:10:00,820 --> 00:10:05,520
+الـ limit هذا أثبتناه أصغر أو يساوي صفر معناته صارت
+
+106
+00:10:05,520 --> 00:10:12,030
+اللي هي الـ f prime أصغر أو يساوي صفر الآن عندي أنا 
+
+107
+00:10:12,030 --> 00:10:16,990
+لو أثبتت هذا المقدار أصغر أو يساوي صفر في المنطقة
+
+108
+00:10:16,990 --> 00:10:20,870
+حول الـ C بقدر أقول إن الـ limit على طول إيش؟ شماله
+
+109
+00:10:20,870 --> 00:10:27,550
+أصغر أو يساوي إيش؟ صفر ده نشوف كيف عندي الآن بدي
+
+110
+00:10:27,550 --> 00:10:32,170
+أثبت إن هي هذه F prime أصغر أو يساوي صفر على كل X
+
+111
+00:10:32,170 --> 00:10:39,450
+element in I بدي أقول let C be an arbitrary
+
+112
+00:10:41,100 --> 00:10:47,160
+and fixed in I يعني بتحكي عنها عن C لكن هي 
+
+113
+00:10:47,160 --> 00:10:54,780
+arbitrary لكن بتحكي عن C معينة الآن عندي يا X أكبر
+
+114
+00:10:54,780 --> 00:11:01,800
+من C يا X شمالها أصغر من C ماشي الحال؟ ليش أخذت 
+
+115
+00:11:01,800 --> 00:11:04,160
+أكبر أو أصغر؟ لأن عمّالي بأخذ limit فالـ X ده
+
+116
+00:11:04,160 --> 00:11:09,060
+بتساوي C في حالة الـ C اللي هو الأمر اللي هيكون 
+
+117
+00:11:09,060 --> 00:11:13,180
+اللي هي عبارة 
+
+118
+00:11:13,180 --> 00:11:17,520
+عن الـ trivial case نشوف كيف، إيش اللي بنحكيه؟
+
+119
+00:11:17,520 --> 00:11:23,440
+الآن X أكبر من C عندي إيش مفترضين إحنا؟ F is
+
+120
+00:11:23,440 --> 00:11:30,920
+decreasing لذا then F of X أصغر أو يساوي F of C
+
+121
+00:11:30,920 --> 00:11:38,100
+مظبوط؟ اه بصير عندي الحسب اللي كتبته قبل بشوية F 
+
+122
+00:11:38,100 --> 00:11:47,680
+of X ناقص F of C اللي هو بيساوي على X ناقص C اللي
+
+123
+00:11:47,680 --> 00:11:52,290
+هو هذا سالب f of x ناقص f of z أصغر أو أكبر من صفر والـ 
+
+124
+00:11:52,290 --> 00:11:57,110
+x ناقص z شماله أكبر من صفر هذا الآن أكبر
+
+125
+00:11:57,110 --> 00:12:02,650
+من سالب أو صفر وهذا موجب إذا هيكون أصغر أو يساوي إيش؟
+
+126
+00:12:02,650 --> 00:12:08,760
+صفر واضح؟ اه Similarly الحالة الثانية من X هذا لكل
+
+127
+00:12:08,760 --> 00:12:12,800
+الـ Xات اللي مين؟ اللي أكبر من C طبعًا والـ Xات أنا من
+
+128
+00:12:12,800 --> 00:12:16,740
+وين بأخذها؟ من I على كل شغلي في داخل من الـ interval
+
+129
+00:12:16,740 --> 00:12:21,320
+I يعني Xات اللي بشتغل فيها داخل I الآن X أصغر من C
+
+130
+00:12:21,320 --> 00:12:24,900
+أكيد Similarly بنفس الطريقة مدام الـ decreasing
+
+131
+00:12:24,900 --> 00:12:31,580
+بيصير F of X أكبر أو يساوي F of C بيصير عندي الآن
+
+132
+00:12:31,580 --> 00:12:40,060
+إذا F of X ناقص F of C وهو موجب وهذه X ناقص C
+
+133
+00:12:40,060 --> 00:12:44,880
+سالب إذا هذولة على X ناقص C برضه هتطلع إيش؟
+
+134
+00:12:44,880 --> 00:12:49,720
+شمالها أصغر أو يساوي صفر إذا صار عندي من الجهتين
+
+135
+00:12:49,720 --> 00:12:55,660
+هنا for any X
+
+136
+00:12:55,660 --> 00:13:03,040
+element in I و X لا تساوي C أنا أصلاً ملزمني حوالين
+
+137
+00:13:03,040 --> 00:13:09,810
+الـ C لكن هذا كله كلام صحيح لـ X ∈ I X ≤ C طالع
+
+138
+00:13:09,810 --> 00:13:17,350
+عندي اللي هو F of X ناقص F of C على X ناقص C أصغر
+
+139
+00:13:17,350 --> 00:13:23,610
+أو يساوي 0 الآن مدام هذا صحيح لكل X إذا الـ limit هذا
+
+140
+00:13:23,610 --> 00:13:27,530
+المقدار لما X تروح للـ C برضه هيظل أصغر أو يساوي 
+
+141
+00:13:27,530 --> 00:13:33,950
+صفر وهذا هو إيش اسمه؟ F prime of C يعني أثبتنا F 
+
+142
+00:13:33,950 --> 00:13:38,530
+prime of C أصغر أو يساوي الصفر وعند C كانت
+
+143
+00:13:38,530 --> 00:13:42,550
+arbitrary and fixed in I إذا صار بناء على هذا
+
+144
+00:13:42,550 --> 00:13:49,050
+الحديث صار F prime of X أصغر أو يساوي صفر لكل X
+
+145
+00:13:49,050 --> 00:13:54,760
+element in I لأنه كانت C is arbitrary هذا اللي هو
+
+146
+00:13:54,760 --> 00:14:00,440
+الجزء الأول من النظرية من اللي هو B إنه بدنا الآن
+
+147
+00:14:00,440 --> 00:14:03,180
+نيجي لـ conversely بدنا نفترض هذا أصحى وأصحى وصفر
+
+148
+00:14:03,180 --> 00:14:09,040
+ونوصل لمين؟ للـ F is decreasing نيجي الآن 
+
+149
+00:14:09,040 --> 00:14:19,240
+conversely ونشوف كيف نبرهن conversely الآن
+
+150
+00:14:19,240 --> 00:14:25,030
+conversely طبعًا ببرهن في AVP برضه أنا conversely
+
+151
+00:14:25,030 --> 00:14:30,810
+suppose that 
+
+152
+00:14:32,170 --> 00:14:39,270
+F prime of X أصغر أو يساوي صفر لكل X موجودة في الآية 
+
+153
+00:14:39,270 --> 00:14:43,790
+ماشي الحال شوفوا صلى الله عليه الصلاة والسلام أنا
+
+154
+00:14:43,790 --> 00:14:47,810
+بدأ أثبت أن F is decreasing يعني بدأ أفترض أن X
+
+155
+00:14:47,810 --> 00:14:53,650
+أصغر أو أخذ أي قيمتين X واحد و X اثنين element 
+
+156
+00:14:53,650 --> 00:14:59,250
+اثنين أخذت أي two elements element x1 و x2 element
+
+157
+00:14:59,250 --> 00:15:06,850
+in I بتحقق such that x1 أصغر من مين؟ من x2 أصغر أو
+
+158
+00:15:06,850 --> 00:15:10,570
+يساوي x.. x1 أصغر أو يساوي x2 بس في حالة الـ x1
+
+159
+00:15:10,570 --> 00:15:14,490
+بيساوي x2 ما هي automatic F of x1 بيساوي F of x2 
+
+160
+00:15:14,490 --> 00:15:19,920
+يعني متحققة إذن الآن x1 أصغر من x2 عملتها strictly 
+
+161
+00:15:19,920 --> 00:15:23,720
+عشان بتعمل منها interval وحالتي الـ x1 بيساوي x2 is 
+
+162
+00:15:23,720 --> 00:15:28,000
+a trivial case تتحقق مدامة x1 أصغر أو يساوي x2 إذن 
+
+163
+00:15:28,000 --> 00:15:32,100
+عشان a function f of x1 بيساوي f of x2 إذا التحقق
+
+164
+00:15:32,100 --> 00:15:39,340
+اللي هي مين؟ الـ decreasing واضحة الآن x1 أصغر من x2 
+
+165
+00:15:39,340 --> 00:15:41,620
+الآن we apply
+
+166
+00:15:45,100 --> 00:15:48,880
+الـ Mean Value Theorem لأن احنا هذه بنبارع corollaries 
+
+167
+00:15:48,880 --> 00:15:52,380
+أو نظريات على الـ Mean Value Theorem فالمتوقع أن
+
+168
+00:15:52,380 --> 00:15:55,240
+نستخدم الـ Mean Value Theorem في إثباتها فهنا 
+
+169
+00:15:55,240 --> 00:15:59,000
+سنستخدم الـ Mean Value Theorem نطبق الـ Mean Value
+
+170
+00:15:59,000 --> 00:16:05,470
+Theorem على الـ interval الموجود بين X1 و X2 طبعًا عندي
+
+171
+00:16:05,470 --> 00:16:10,330
+F هذه X1 و X2 عبارة عن جزئية من مين؟ من الـ I إذا
+
+172
+00:16:10,330 --> 00:16:12,790
+الـ F is differentiable عليها مدام differentiable
+
+173
+00:16:12,790 --> 00:16:16,050
+إذا continuous علي closed و differentiable علي open
+
+174
+00:16:16,050 --> 00:16:19,240
+لأنها differentiable علي كلها إذا شروط الـ Mean
+
+175
+00:16:19,240 --> 00:16:24,580
+Value Theorem مالها؟ متحققة إذا there
+
+201
+00:19:14,600 --> 00:19:23,720
+خليني أمسح النظرية وبرهّنها بيه أنا هو ماشي بدي بس من
+
+202
+00:19:23,720 --> 00:19:30,890
+الآن X بنقول F is strictly increasing إيش معناته
+
+203
+00:19:30,890 --> 00:19:38,470
+إنّه X هنا يكافئ إنّ X1 أصغر من X2 للـ X1 والـ X2 اللي
+
+204
+00:19:38,470 --> 00:19:46,790
+في I such that يؤدي F of X1 strictly أصغر من F of
+
+205
+00:19:46,790 --> 00:19:53,580
+X2 هذا يعني على بعضه إنّ F is strictly increasing
+
+206
+00:19:53,580 --> 00:19:58,440
+similarly for decreasing, for strictly decreasing f
+
+207
+00:19:58,440 --> 00:20:04,360
+is strictly decreasing لما x1 أصغر من x2 تؤدي إلى
+
+208
+00:20:04,360 --> 00:20:10,160
+f of x1 أكبر strictly من مين؟ من f of x2 هذا معناته
+
+209
+00:20:10,160 --> 00:20:16,640
+strictly إيش؟ decreasing الآن السؤال اللي بطرحه 
+
+210
+00:20:16,640 --> 00:20:23,380
+الآن هو بقول لي إنّ أنت حكيت في حالة الـ increasing F
+
+211
+00:20:23,380 --> 00:20:27,980
+is increasing in I if and only if F' أصغر أو يساوي 0
+
+212
+00:20:27,980 --> 00:20:31,960
+يساوي 0 لكل X element in I بقول لي هل هذا الكلام
+
+213
+00:20:31,960 --> 00:20:37,020
+بيضل صحيح في حالة الـ strictly increasing يعني لو
+
+214
+00:20:37,020 --> 00:20:43,480
+كانت بسأل السؤال لو كانت F is strictly
+
+215
+00:20:45,290 --> 00:20:49,790
+increasing أو decreasing اللي بدّك إياها increasing أو
+
+216
+00:20:49,790 --> 00:20:57,630
+decreasing هل هذا بيكون if and only if F prime of
+
+217
+00:20:57,630 --> 00:21:04,270
+X أصغر strictly من 0 لكل X element in I ولا لأ؟
+
+218
+00:21:04,270 --> 00:21:12,010
+الآن لو كان F prime أصغر strictly من 0 لـ X element
+
+219
+00:21:12,010 --> 00:21:17,620
+in I اللي هو الـ conversely هذا اللي هو بيعطينا F's
+
+220
+00:21:17,620 --> 00:21:22,620
+strictly decreasing يعني الاتجاه هذا متوقعين أنّه
+
+221
+00:21:22,620 --> 00:21:27,500
+إحنا إيش؟ شماله يكون صحيح يعني متوقعين أنّه لو F
+
+222
+00:21:27,500 --> 00:21:31,520
+prime أصغر strictly من Zero لكل X element of I
+
+223
+00:21:31,520 --> 00:21:36,260
+يعطينا F's strictly إيش؟  decreasing وزيّه برضه الـ
+
+224
+00:21:36,260 --> 00:21:41,580
+increasing لو كانت F' أكبر strictly من 0 بيعطينا F
+
+225
+00:21:41,580 --> 00:21:46,380
+is strictly increasing هذا متوقع أن يكون صحيح
+
+226
+00:21:46,380 --> 00:21:53,440
+والبرهان هيّه موجود عليه لسه ما شطبّطوش لسه الآن بدأ
+
+227
+00:21:53,440 --> 00:21:56,780
+افترض أنّه بدل ما هي أكبر أصغر أو يساوي بدأ افترضها
+
+228
+00:21:56,780 --> 00:22:02,500
+إيش؟  شمالها Strictly أصغر ماشي الحال الآن بفترض إنّ X1
+
+229
+00:22:02,500 --> 00:22:07,120
+أصغر من X2 و بطبّق كل اللي عملته و لعند الـ hand
+
+230
+00:22:07,120 --> 00:22:14,860
+بشطب هذا بالـ ديشرية و بقف الـ hand و بقول بما أنّه هذا
+
+231
+00:22:14,860 --> 00:22:18,480
+المقدار f prime of c اللي هي f prime of x strictly
+
+232
+00:22:18,480 --> 00:22:22,840
+أصغر من 0 يعني ما في إمكانية أن تساوي 0 وهذا x2
+
+233
+00:22:22,840 --> 00:22:28,320
+ناقص x1 من هذا برضه strictly أكبر من 0 إذا صار هذا
+
+234
+00:22:28,320 --> 00:22:32,520
+عبارة عن strictly أصغر من 0 ولا يوجد إمكانية
+
+235
+00:22:32,520 --> 00:22:36,620
+للمساواة إذا بمعنى آخر صار هذا المقدار strictly
+
+236
+00:22:36,620 --> 00:22:40,680
+أصغر من هذا يعني لما كانت f prime أصغر strictly من
+
+237
+00:22:40,680 --> 00:22:45,770
+0 لكل x element in RI وجدنا إنّ X1 أصغر strictly من
+
+238
+00:22:45,770 --> 00:22:51,010
+X2 تؤدي F of X2 أصغر strictly من F of X1 وهذا يعني
+
+239
+00:22:51,010 --> 00:22:59,690
+strictly decreasing واضح، أه؟ الآن السؤال هل
+
+240
+00:22:59,690 --> 00:23:07,830
+the converse is true؟ هل the converse is true؟
+
+241
+00:23:07,830 --> 00:23:10,650
+الآن يعني
+
+242
+00:23:13,960 --> 00:23:21,020
+الاتجاه هذا صحيح لما يقول الاتجاه الثاني هذا need
+
+243
+00:23:21,020 --> 00:23:26,500
+not to be true in general بدنا نجيب مثال إن تكون F
+
+244
+00:23:26,500 --> 00:23:30,780
+strictly decreasing لكن الـ F براي مش أصغر strictly
+
+245
+00:23:30,780 --> 00:23:37,120
+من مين؟ من 0 على I خد الآن F of X بساوي سالب X تكعيب
+
+246
+00:23:37,120 --> 00:23:42,220
+هذا الدالة، هذا الدالة بنعمل reference مثل F of X
+
+247
+00:23:42,220 --> 00:23:47,120
+ساوى سالب X تكعيب بالشكل هذا، هذا الآن strictly إيش؟
+
+248
+00:23:47,120 --> 00:23:54,800
+شماله decreasing هذه F is strictly decreasing
+
+249
+00:23:55,740 --> 00:24:03,860
+لأنّ لو كانت X1 أصغر من X2 أكيد X1 تكعيب strictly
+
+250
+00:24:03,860 --> 00:24:13,420
+أكبر من X2 إيش؟ شماله تكعيب آسف ناقص مظبوط الآنصار
+
+251
+00:24:13,420 --> 00:24:18,320
+عندي الآن يعني اللي هي الـ F عندي أنا strictly
+
+252
+00:24:18,320 --> 00:24:23,770
+decreasing but طبعًا على كل مين؟ على كل الـ I اللي
+
+253
+00:24:23,770 --> 00:24:27,090
+أخدت منها لها، خد أي I من A لعند B من ناقص خمسة
+
+254
+00:24:27,090 --> 00:24:31,410
+مثلًا لعند خمسة على الفترة اللي بدّك إياها but F
+
+255
+00:24:31,410 --> 00:24:40,250
+prime عند اللي هو نقطة الـ zero إيش بساوي؟ بساوي صفر
+
+256
+00:24:40,250 --> 00:24:45,150
+إذا اللي هو ما أعطانيش الشرط إنّ F prime أصغر من صفر
+
+257
+00:24:45,150 --> 00:24:51,290
+على كل النقاط الـ X اللي وين؟ في الـ I واضح أه الآن
+
+258
+00:24:51,290 --> 00:24:58,210
+معطيني معرف لي
+
+259
+00:24:58,210 --> 00:25:04,150
+اللي هو شو معناته؟ طبعًا إحنا عرفنا الـ increasing و
+
+260
+00:25:04,150 --> 00:25:09,670
+الـ decreasing على الـ interval لأنّها مقارنة بالنقاط
+
+261
+00:25:09,670 --> 00:25:14,510
+فالآن بقول لي طب كيف؟ إيش رأيكم؟ بدنا نحكي عن الـ
+
+262
+00:25:14,510 --> 00:25:18,860
+increasing أو الـ decreasing عند نقطة الـ increasing
+
+263
+00:25:18,860 --> 00:25:21,640
+أو الـ decreasing وعنده ضغطة عمومًا هذه الفلسفة
+
+264
+00:25:21,640 --> 00:25:26,360
+موجودة يعني في أكثر من مجال للتعريف في الرياضيات
+
+265
+00:25:26,360 --> 00:25:31,380
+في المناطق اللي ممكن اللي هو تحتاج أو في المفاهيم
+
+266
+00:25:31,380 --> 00:25:37,020
+اللي ممكن تحتاج لمنطقة لتعريف اللي هو المفهوم
+
+267
+00:25:37,020 --> 00:25:41,400
+المقصود بمنطقة زي الـ interval اللي قبل بشوية يعني
+
+268
+00:25:41,400 --> 00:25:48,200
+مش عند نقطة لأ عند منطقة فبيصير التعريف اللي هو أن
+
+269
+00:25:48,200 --> 00:25:53,920
+يتحقق المفهوم في جوار هذه النقطة اللي مذكورة يعني
+
+270
+00:25:53,920 --> 00:25:59,120
+بنقول إنّ النقطة C تتمتع بهذه الخاصية إذا تمتع
+
+271
+00:25:59,120 --> 00:26:03,940
+الجوار بهذه الخاصية هذه فلسفة التعريف فالآن بدنا
+
+272
+00:26:03,940 --> 00:26:04,260
+نقول
+
+273
+00:26:09,560 --> 00:26:14,260
+F من I لعند R اللي عبارة عن function أيّ إن كانت
+
+274
+00:26:14,260 --> 00:26:18,260
+الـ function هذه واخدنا
+
+275
+00:26:18,260 --> 00:26:28,780
+C element in I بدنا نعرف شو معناته عند النقطة C عند
+
+276
+00:26:28,780 --> 00:26:33,240
+النقطة C شو معناته F is increasing at this point
+
+277
+00:26:33,240 --> 00:26:40,880
+بدنا نشوف شو معناته F is increasing at C element in
+
+278
+00:26:40,880 --> 00:26:48,420
+I معناته إنّه there exist هذا يكافئ there exist V
+
+279
+00:26:48,420 --> 00:26:53,440
+Delta of C يعني U هذا الجوار حوالين من الـ C هذا
+
+280
+00:26:53,440 --> 00:27:03,210
+الجوار C ناقص دلتا و C زائد دلتا ابحث إن تكون F is
+
+281
+00:27:03,210 --> 00:27:08,770
+increasing on this جوار أو on this neighbourhood
+
+282
+00:27:08,770 --> 00:27:14,770
+إذا تحقق الشرط هذا معناته إنّه بنقول عن الـ
+
+283
+00:27:14,770 --> 00:27:18,450
+function F إيش؟ إنها increasing at the point
+
+284
+00:27:18,450 --> 00:27:23,090
+mean C يعني بمعنى آخر إذا تحقق اللي هو there
+
+285
+00:27:23,090 --> 00:27:29,870
+exists V Delta of C such that if X1 و X2 element
+
+286
+00:27:29,870 --> 00:27:42,110
+in V Delta of C with X1 أصغر من X2 أصغر
+
+287
+00:27:42,110 --> 00:27:49,350
+من X2 أصغر
+
+288
+00:27:49,350 --> 00:27:54,840
+من X2 يعني اتحققت الخاصية أو الـ increasing في جوار
+
+289
+00:27:54,840 --> 00:27:58,120
+حول الـ C هذا معناته اللي هو مين؟ اللي هو
+
+290
+00:27:58,120 --> 00:28:01,820
+increasing similarly for strictly increasing ..
+
+291
+00:28:01,820 --> 00:28:07,000
+strictly increasing .. strictly increasing X1 أصغر
+
+292
+00:28:07,000 --> 00:28:11,160
+من X2 يؤدي F of X1 strictly أصغر من F of X2 في هذا
+
+293
+00:28:11,160 --> 00:28:15,480
+الجوار similarly for اللي هو مين؟ decreasing و
+
+294
+00:28:15,480 --> 00:28:20,150
+strictly decreasing بما يتلقى أمهم الـ التلاقم معناته
+
+295
+00:28:20,150 --> 00:28:25,490
+X1 أصغر من X2 بيعطينا F of X1 أكبر أو يساوي F
+
+296
+00:28:25,490 --> 00:28:31,890
+of X2 لـ Decreasing اللي بيكون Strictly أكبر منها
+
+297
+00:28:31,890 --> 00:28:39,410
+طيب .. الآن السؤال .. السؤال اللي بيقول ليه .. طيب
+
+298
+00:28:39,410 --> 00:28:49,320
+.. الآن أنت قلت أنّه لو كانت F prime أكبر من 0 f
+
+299
+00:28:49,320 --> 00:28:57,040
+prime of x أكبر من 0 لكل x element in I اللي
+
+300
+00:28:57,040 --> 00:29:05,200
+بيعطيني اللي هو f is strictly increasing صح ولا
+
+301
+00:29:05,200 --> 00:29:10,560
+لأ؟ قلنا هيك معايا لكن قلنا الـ .. الـ .. الـ
+
+302
+00:29:10,560 --> 00:29:13,600
+converse need not to be true in general جبنا مثال
+
+303
+00:29:15,110 --> 00:29:18,590
+أf of X بساوي X تكعيب للـ increasing أو ناقص X
+
+304
+00:29:18,590 --> 00:29:23,690
+تكعيب للـ decreasing ماشي الحال .. الآن هو بقول طيب
+
+305
+00:29:23,690 --> 00:29:26,330
+لأ أنا بدّي أسأله عن الاتجاه الثاني لأنّه خلاص إحنا
+
+306
+00:29:26,330 --> 00:29:29,990
+شفنا الاتجاه الثاني عند اللي هو الـ interval مش
+
+307
+00:29:29,990 --> 00:29:35,450
+متحقق فالآن بقول لي أنا بدأ أسأل السؤال على اللي هي
+
+308
+00:29:35,450 --> 00:29:39,750
+.. اللي هو هذا السؤال اللي اتحقق في الـ interval هل
+
+309
+00:29:39,750 --> 00:29:44,790
+بتحقق عنده النقطة؟ يعني بمعنى آخر لو كانت عندي a
+
+310
+00:29:44,790 --> 00:29:53,390
+February at c أكبر من 0 هل هذا يؤدي إنّ f is
+
+311
+00:29:53,390 --> 00:30:03,070
+strictly increasing on c أو at c؟ هل هذا يؤدي إنّ
+
+312
+00:30:03,070 --> 00:30:06,090
+لو كانت f' of c أكبر من 0 هل يؤدي إلى إنّ f is
+
+313
+00:30:06,090 --> 00:30:09,910
+strictly increasing at c؟ بقول لك، this need not
+
+314
+00:30:09,910 --> 00:30:14,430
+to be true in general وهذا هو exercise عشرة اللي
+
+315
+00:30:14,430 --> 00:30:18,660
+معاكم homework اللي هو أعطاني function g of x بساوي
+
+316
+00:30:18,660 --> 00:30:23,120
+x زي 2x تربيع sin 1 علي x إذا كانت x بتساوي صفر و
+
+317
+00:30:23,120 --> 00:30:27,780
+0 إذا كانت x بتساوي صفر و بقول لي لو جيت فحصة الـ G
+
+318
+00:30:27,780 --> 00:30:32,960
+prime عند الـ zero هتلاقيها إيش؟ بتساوي 1 ماشي
+
+319
+00:30:32,960 --> 00:30:37,320
+الحال يعني بتساوي واحد يعني strictly أكبر منين؟ من
+
+320
+00:30:37,320 --> 00:30:44,050
+صفر but ولا جوار، there is no neighbourhood around
+
+321
+00:30:44,050 --> 00:30:52,850
+zero in which F is increasing أه فشي دير جي is not
+
+322
+00:30:52,850 --> 00:30:56,850
+increasing in any neighborhood of x بتساوي صفر
+
+323
+00:30:56,850 --> 00:31:02,770
+والتفاصيل إن شاء الله لما في الـ discussion لما
+
+324
+00:31:02,770 --> 00:31:06,550
+نعمل discussion اللي هو الـ section اللي هو 6 2
+
+325
+00:31:06,550 --> 00:31:11,210
+وتجربوا تحلوه في البيت في حال أنتو ما عرفتو تحلوه
+
+326
+00:31:11,210 --> 00:31:15,670
+بنحكي فيه بتفاصيله في الـ discussion إن شاء الله
+
+327
+00:31:22,930 --> 00:31:27,050
+لأ الجوار هتلاقي اللي هو .. اللي هو .. اللي هو
+
+328
+00:31:27,050 --> 00:31:32,830
+مستحيل يعني تلاقي اللي هو جوار تكون X1 أصغر من X2
+
+329
+00:31:32,830 --> 00:31:37,150
+و تضمن إنّ F of X1 أصغر من F of X2 لكل واحدة بتحقق
+
+330
+00:31:37,150 --> 00:31:43,010
+الـ X1 أصغر من X2 يعني هتلاقي نقاط هتلاقي هل جت V
+
+331
+00:31:43,010 --> 00:31:48,510
+Delta of C مثلًا لأ كانت Delta بساوي C ناقص Delta و
+
+332
+00:31:48,510 --> 00:31:53,690
+C زائد Delta ماشي الحال .. الآن هتلاقي إنّه .. إنّه
+
+333
+00:31:53,690 --> 00:31:59,090
+مستحيل يعني إنّه في أيّ .. لو مهما زغرت المسافة
+
+334
+00:31:59,090 --> 00:32:05,110
+بينهم يتلاقي X1 أصغر من X2 تضمن يتلاقي أف of X1
+
+335
+00:32:05,110 --> 00:32:10,050
+أكبر أو يساوي أف of X2 في حالة الـ decreasing أو
+
+336
+00:32:10,050 --> 00:32:13,690
+أصغر أو يساوي في حالة الـ increasing أو اللي بنحكي
+
+337
+00:32:13,690 --> 00:32:16,770
+عنه إيش؟ شماله strictly increasing يعني هتلاقي نقاط
+
+338
+00:32:16,770 --> 00:32:21,030
+جانب بعض وحدة أكبر من التانية و very close .. very
+
+339
+00:32:21,030 --> 00:32:25,870
+close اللي هو .. و تكون .. اللي هو واحدة اللي هو
+
+340
+00:32:25,870 --> 00:32:30,970
+تلاقيها أكبر و التانية تلاقيها نقطش آخر أه .. أه
+
+341
+00:32:30,970 --> 00:32:34,230
+.. أه .. في .. أه .. في .. تلاقيهم يعني الـ .. الـ
+
+342
+00:32:34,230 --> 00:32:37,630
+.. ال .. ال .. طبعًا هو يمكن السبب بالذبذبة لو
+
+343
+00:32:37,630 --> 00:32:43,710
+لاحظت هدي sign واحد على X فالـ X اللي هو بتساوي
+
+344
+00:32:43,710 --> 00:32:48,720
+اتنين .. واحد على اتنين on by أو 2 and by هتلاقي
+
+345
+00:32:48,720 --> 00:32:52,740
+اللي هي إيش؟ بيعملك صفر و إيش؟ بيعملك واحد في حالة
+
+346
+00:32:52,740 --> 00:32:55,920
+الـ F of prime و في حالة الـ F العادية فهيسببلك اللي
+
+347
+00:32:55,920 --> 00:33:01,180
+هو مش هتظلها اللي هي الـ .. اللي هي الـ .. الـ .. الـ
+
+348
+00:33:01,180 --> 00:33:08,300
+.. القيم .. قيم هتلاقيها مستحيل إنّ X1 أصغر من X2 و
+
+349
+00:33:08,300 --> 00:33:12,580
+تظلها على نفس المنوال F of X1 أصغر من F of X2 يعني
+
+350
+00:33:12,580 --> 00:33:18,070
+ممكن تلاقيه لك إنّه جدتين closed لجوه هتلاقي بتحققش
+
+351
+00:33:18,070 --> 00:33:26,150
+هذه يا بتحقق المساواة يا بتحقق الأصغر ولو لجيت لك
+
+352
+00:33:26,150 --> 00:33:30,510
+تنتين بتحقق الأكبر هتلاقي برضه very closed إيش؟ بتحقق
+
+353
+00:33:30,510 --> 00:33:36,810
+الأصغر إن
+
+354
+00:33:36,810 --> 00:33:45,360
+شاء الله لما نحل السؤال بتتضح الأمور تمامًا هذا اللي
+
+355
+00:33:45,360 --> 00:33:54,900
+هو من ضمن الأسئلة اللي معاكم homework سألنا
+
+356
+00:33:54,900 --> 00:34:08,980
+سؤال اللي هو وين يمكن إيجاد الـ relative maximum و
+
+357
+00:34:08,980 --> 00:34:13,430
+relative minimum و جاوبنا عليه في النظرية اللي
+
+358
+00:34:13,430 --> 00:34:17,230
+حكيناها في البداية إنّه توجد عند أحد ثلاث مناطق F'
+
+359
+00:34:17,450 --> 00:34:21,370
+بيساوي صفر، F' لا تساوي صفر لإنّها في حالة الـ
+
+360
+00:34:21,370 --> 00:34:26,410
+interior points أو اللي هو المكان الثالث اللي هو
+
+361
+00:34:26,410 --> 00:34:32,210
+الـ end points الآن عشان نكمل الحديث عن الـ maximum
+
+362
+00:34:32,210 --> 00:34:39,370
+و الـ minimum بدنا نفحص، نفحص هل النقاط هذه مين فيها
+
+363
+00:34:39,370 --> 00:34:43,910
+عندها maximum وفي مين فيها عندها minimum ومين لا
+
+364
+00:34:43,910 --> 00:34:48,530
+عندها maximum ولا عندها minimum وهو الـ test اللي
+
+365
+00:34:48,530 --> 00:34:54,410
+أنّتو اخذتوّه في اللي هو سواء في الثانوية أو حتى
+
+366
+00:34:54,410 --> 00:34:59,330
+في calculus ألف وعشان نساعدكم على إنكم تتذكروا
+
+367
+00:34:59,330 --> 00:35:05,620
+أنا بفضل إنّه نتذكّر من خلال الرسم، الفحص ساعتها
+
+368
+00:35:05,620 --> 00:35:12,780
+بيكون سهل وبيكون أنت بتعرف اللي هو تحدد إنّ هذا
+
+369
+00:35:12,780 --> 00:35:17,020
+النقطة في عندها maximum أو عندها minimum خلينا
+
+370
+00:35:17,020 --> 00:35:22,420
+نستذكر، نستذكر من خلال الرسم النظرية اللي بدنا
+
+371
+00:35:22,420 --> 00:35:26,160
+نحكيها الآن اللي أخذناها في الثانوية أو أخذناها
+
+372
+00:35:26,160 --> 00:35:30,850
+اللي هي تغيير إشارة F برايم اللي هو من موجب إلى
+
+373
+00:35:30,850 --> 00:35
+
+401
+00:38:22,790 --> 00:38:28,850
+نقاط و F اللي هي تحولت إلى F prime من سالب إلى موجب
+
+402
+00:38:28,850 --> 00:38:33,870
+هذه اللي هي معلومة توجيهي أو معلومة 
+
+403
+00:38:33,870 --> 00:38:40,730
+اللي هو calculus الآن احنا بدنا نثبت صحة هذا
+
+404
+00:38:40,730 --> 00:38:41,770
+الحديث
+
+405
+00:38:49,440 --> 00:38:53,520
+نجرها بس من عندنا هنا هي اللي قلته بس نجرها من
+
+406
+00:38:53,520 --> 00:38:57,520
+النظرية let f be continuous on the interval a و b
+
+407
+00:38:57,520 --> 00:39:01,080
+إذا إيش المقصود هنا إن ال function is continuous على
+
+408
+00:39:01,080 --> 00:39:04,780
+فترة a و b و ناخذ c عبارة عن interior point لمين
+
+409
+00:39:05,450 --> 00:39:10,090
+للـ I Assume that F is differentiable إيه على الفترة
+
+410
+00:39:10,090 --> 00:39:13,430
+A وC وC وB يعني مش لازم إنها تكون differentiable
+
+411
+00:39:13,430 --> 00:39:16,170
+عند الـ C أصلاً ممكن تكون مش differentiable و هي
+
+412
+00:39:16,170 --> 00:39:21,290
+نقطة بدنا نبحث عندها بس بالزمن هذه اللي هي من A ل
+
+413
+00:39:21,290 --> 00:39:25,490
+C و من C ل B then الآن
+
+414
+00:39:26,890 --> 00:39:31,430
+إذا كان هناك مقارنة V Delta of C Subset من I بحيث
+
+415
+00:39:31,430 --> 00:39:35,110
+أن F' بتكون أكبر أو يساوي 0 قبلها في الفترة اللي
+
+416
+00:39:35,110 --> 00:39:39,830
+قبل، اللي هو و F' اتحولت إلى أصغر أو يساوي 0، يعني
+
+417
+00:39:39,830 --> 00:39:44,750
+اتحولت الإشارة من موجب إلى سالب في الفترة الأولى
+
+418
+00:39:44,750 --> 00:39:48,210
+للفترة الثانية، هيكون عند relative maximum زي ما
+
+419
+00:39:48,210 --> 00:39:54,260
+حكيناها الآن في الثانية نفس الشيء بس بالعكس if
+
+420
+00:39:54,260 --> 00:39:57,480
+there is a neighborhood اللي هو كذا such that F'
+
+421
+00:39:57,940 --> 00:40:02,160
+أصغر من صفر هيه اللي هو زاوية منفرجة اتحول لF'
+
+422
+00:40:02,540 --> 00:40:06,820
+أكبر من صفر هيه بيعطينا عندي إيش ماله relative
+
+423
+00:40:06,820 --> 00:40:14,040
+minimum طيب ده عندي A و عندي BM مبرهنة خليني نبرهن
+
+424
+00:40:14,040 --> 00:40:18,440
+مين؟ B خليني نشوف مبرهن مع بعض اللي هو B
+
+425
+00:40:29,950 --> 00:40:33,710
+شوفوا صلوا على النبي عليه الصلاة والسلام الآن بيه
+
+426
+00:40:33,710 --> 00:40:39,410
+اللي هو عندي F prime معطيني F prime of X أصغر أو
+
+427
+00:40:39,410 --> 00:40:47,170
+يساوي صفر صح لكل X في الفترة C minus Delta وC
+
+428
+00:40:47,170 --> 00:40:49,710
+ومعطيني أيضاً
+
+429
+00:40:52,140 --> 00:40:57,040
+اللي تحت أنا بحكي عن اللي تحت بيه ال .. البعدها F
+
+430
+00:40:57,040 --> 00:41:02,980
+prime of X أكبر أو يساوي صفر لكل X وين موجودة في
+
+431
+00:41:02,980 --> 00:41:09,740
+الفترة من C لـC زائد إيش؟ لـC زائد دلتا الآن إيش
+
+432
+00:41:09,740 --> 00:41:14,560
+غايتنا نثبت أن F has a relative minimum؟ يعني بدي
+
+433
+00:41:14,560 --> 00:41:21,340
+أثبت أن F of Xأكبر الـ Minimum أصغر أكبر أو يساوي
+
+434
+00:41:21,340 --> 00:41:28,380
+F of C لكل X وين موجودة في V Delta of C إذا أثبتت
+
+435
+00:41:28,380 --> 00:41:32,200
+هذا الكلام معناته صار عندي في Neighborhood حوالي ال
+
+436
+00:41:32,200 --> 00:41:37,140
+F عند ال C أصغر واحدة فيهم يعني في عندها relative
+
+437
+00:41:37,140 --> 00:41:42,560
+minimum اللي هو في المنطقة هذه هقولك إنها بتتحقق
+
+438
+00:41:42,560 --> 00:41:45,700
+هذه وفي المنطقة هذه بتتحقق هذه معناها تقول لكل
+
+439
+00:41:45,700 --> 00:41:49,100
+النقاط اللي هنا لكل النقاط اللي هنا هذه بتتحقق
+
+440
+00:41:49,100 --> 00:41:54,740
+معايا؟ والكلام إن شاء الله سهل الآن طبيعي إحنا
+
+441
+00:41:54,740 --> 00:41:58,920
+قاعدين في جوه ال mean value theorem فأكيد هذا
+
+442
+00:41:58,920 --> 00:42:02,440
+جايبينه في نفس ال section عشان إنه إحنا نطبق ال
+
+443
+00:42:02,440 --> 00:42:07,220
+mean value theorem عشان اللي هو إن نصل لنتائجنا
+
+444
+00:42:07,770 --> 00:42:14,790
+الآن x element in c minus delta of c، مظبوط؟ إذا 
+
+445
+00:42:14,790 --> 00:42:18,590
+صار عندي الآن طبعاً أنا غايتي أن أجيب اللي هو علاقة
+
+446
+00:42:18,590 --> 00:42:23,610
+بين f of c و f of x فالآن ال x في الفترة هذه
+
+447
+00:42:23,610 --> 00:42:28,410
+معناته إنه أنا بدي أطبق ال mean value theorem على
+
+448
+00:42:28,410 --> 00:42:32,790
+ال x وعلى ال c الفترة اللي فيهم يعني بدي أقول لكم
+
+449
+00:42:32,790 --> 00:42:40,940
+consider the interval x و meanOc XOC
+
+450
+00:42:41,980 --> 00:42:47,220
+الـ file الـ data continuous على X وC، ليش؟ لأن
+
+451
+00:42:47,220 --> 00:42:51,120
+continuous على كل ال interval A وB اللي من ضمنها C
+
+452
+00:42:51,120 --> 00:42:55,680
+إذن، هذا أول شرط متحقق لـ mean value theorem
+
+453
+00:42:55,680 --> 00:42:59,900
+وأيضاً F is differentiable على ال open اللي هي
+
+454
+00:42:59,900 --> 00:43:02,940
+مستثنية الـ C بس و اللي بالزمن في ال mean value
+
+455
+00:43:02,940 --> 00:43:08,060
+theorem بس ال open إذن الشرطان المتحققان على هذه
+
+456
+00:43:08,060 --> 00:43:14,470
+الفترة لمين؟ للـ Mean Value Theorem إذا نطبق الـ
+
+457
+00:43:14,470 --> 00:43:18,730
+Mean Value Theorem على هذا الانترفال لكي نحصل على
+
+458
+00:43:18,730 --> 00:43:27,890
+ذلك اللي هو F there exist CX element in X وC such
+
+459
+00:43:27,890 --> 00:43:38,850
+that F of CX فبرايم of CX يساوي F of C ناقصF of X
+
+460
+00:43:38,850 --> 00:43:48,870
+على X C نقص X اللي هو اللي بالذنب هذا أمور جهزة
+
+461
+00:43:48,870 --> 00:43:57,090
+ليش جهزة من هنا إذا F of C نقص F of X يساوي C
+
+462
+00:43:57,090 --> 00:44:04,760
+minus X في F prime of CX الـ CX هي دي وين موجودة في
+
+463
+00:44:04,760 --> 00:44:10,760
+الـ X و C يعني هنا مدام هنا حسب المعطى الـ F prime
+
+464
+00:44:10,760 --> 00:44:15,380
+اللي إيش ماله سالبة إذا هذا أصغر أو يساوي صفر هذا
+
+465
+00:44:15,380 --> 00:44:20,260
+أكبر أو يساوي صفر مع بعض إيش ماله؟ أصغر أو يساوي
+
+466
+00:44:20,260 --> 00:44:24,240
+صفر إذا صار عندي هذا أصغر أو يساوي صفر أو بمعنى
+
+467
+00:44:24,240 --> 00:44:31,560
+آخر F of C أصغر أو يساوي F of X لمين؟ لكل X في هذه
+
+468
+00:44:31,560 --> 00:44:36,440
+الفترة لأن X كانت arbitrary فيها Similarly for
+
+469
+00:44:36,440 --> 00:44:43,700
+اللي هو الجزء الثاني بتاخد الآن X وين موجودة؟ في
+
+470
+00:44:43,700 --> 00:44:48,520
+الـ C لـ C زائد دلتا و برضه اللي عملته فوق حطب
+
+471
+00:44:48,520 --> 00:44:54,400
+جهانة بس عالمياً على الفترة C O X و نفس الحديث
+
+472
+00:44:54,400 --> 00:44:57,580
+الأولاني F is continuous هلأ و differentiable على
+
+473
+00:44:57,580 --> 00:45:01,480
+ال open then there exist اللي هو by mean value
+
+474
+00:45:01,480 --> 00:45:06,780
+theorem there exist C X C prime X element in C O X
+
+475
+00:45:06,780 --> 00:45:20,440
+such that F prime of Cx' يساوي f of x ناقص f of c
+
+476
+00:45:20,440 --> 00:45:27,800
+على x minus c يساوي هذا و زي ما عملنا بالضبط اللي
+
+477
+00:45:27,800 --> 00:45:31,040
+هو بيصير عندي أضرب طرفين في وسطين بيصير هنا x
+
+478
+00:45:31,040 --> 00:45:37,060
+minus c و هذه بتطلع فوق فبيصير عندي هذه x minus c
+
+479
+00:45:37,060 --> 00:45:44,680
+موجبة و هذه f prime cx اللي هو موجبة X minus C
+
+480
+00:45:44,680 --> 00:45:49,780
+موجبة مظبوط و F prime اللي هي أكبر أو يساوي صفر
+
+481
+00:45:49,780 --> 00:45:53,780
+فبيصير هذا المقدار كله على بعض إيش ماله أكبر أو يساوي
+
+482
+00:45:53,780 --> 00:45:58,400
+صفر يعني هذا المقدار أكبر أو يساوي إيش صفر يعني من
+
+483
+00:45:58,400 --> 00:46:05,820
+هنا بيصير عند F of X أكبر أو يساوي F of إيش of C هذا
+
+484
+00:46:05,820 --> 00:46:11,760
+الكلام صحيح برضه لمين لكل X هنا الآن في الجهتين أي
+
+485
+00:46:11,760 --> 00:46:16,040
+إن كان طلعت ال F of C أصغر أو يساوي F of X أصغر أو
+
+486
+00:46:16,040 --> 00:46:21,640
+يساوي F of X لكل X وين موجودة هنا و هنا يعني بمعنى
+
+487
+00:46:21,640 --> 00:46:25,980
+آخر لكل X وين موجودة في ال neighborhood يعني صارت
+
+488
+00:46:25,980 --> 00:46:33,020
+F of X أكبر أو يساوي F of C لكل X في ال V Delta of
+
+489
+00:46:33,020 --> 00:46:41,440
+C وهذا يعني أن F has a local minimum at mean الـ C
+
+490
+00:46:41,440 --> 00:46:44,640
+أي سؤال
+
+491
+00:47:03,950 --> 00:47:07,150
+عندي the converse of the theorem اللي فوقني not to
+
+492
+00:47:07,150 --> 00:47:10,690
+be true in general وهذا اللي هو عبارة عن مين عن
+
+493
+00:47:10,690 --> 00:47:15,090
+exercise 9 برضه exercise 9 من ضمن ال homework اللي
+
+494
+00:47:15,090 --> 00:47:18,090
+معكم اللي هنتحدث فيه إن شاء الله في ال discussion
+
+495
+00:47:18,090 --> 00:47:22,590
+الآن بعض التطبيقات الأخرى على ال mean value
+
+496
+00:47:22,590 --> 00:47:29,600
+theorem وهذا التطبيق بعض التطبيقات اللي هي علمين
+
+497
+00:47:29,600 --> 00:47:34,120
+value theory ممكن شيء اخذنا مشابه له في ال
+
+498
+00:47:34,120 --> 00:47:40,300
+calculus لكن خلينا نشوف هنا يعني إيش اللي بيقوله
+
+499
+00:47:40,300 --> 00:47:47,700
+عندنا التطبيق الأول اللي
+
+500
+00:47:47,700 --> 00:47:54,080
+بيقوله ما هي ليه example example اللي هو
+
+501
+00:48:00,540 --> 00:48:12,120
+show that أكبر
+
+502
+00:48:12,120 --> 00:48:18,080
+أو يساوي واحد زائد X لكل X element in R هذا واحد
+
+503
+00:48:18,080 --> 00:48:24,500
+اثنين اللي هو نفس ال W او A اللي هو E to the X
+
+504
+00:48:25,160 --> 00:48:33,500
+يساوي واحد زائد X F اندولي F X بتساوي صفر و proof
+
+505
+00:48:33,500 --> 00:48:42,480
+خلينا نشوف نبرهن هذا الأمر أو هذا الجزء طيب
+
+506
+00:48:42,480 --> 00:48:45,920
+بناسبة أن E to the X أكبر أو يساوي واحد زائد X لكل
+
+507
+00:48:45,920 --> 00:48:47,880
+X way الموجودة في ال R
+
+508
+00:48:51,200 --> 00:48:55,040
+الآن عندنا طبيعي حالة الـ X بتساوي صفر الـ Z باخد
+
+509
+00:48:55,040 --> 00:48:57,860
+الـ function مين أنا عشان رأس الدولة يعرف عن مين
+
+510
+00:48:57,860 --> 00:49:03,740
+بنحكي let F of X إيش هتساوي طبعاً E to the X طبعاً E
+
+511
+00:49:03,740 --> 00:49:09,120
+to the X معرف هو المكان الآن في حالة ال .. ال ..
+
+512
+00:49:09,120 --> 00:49:12,280
+الأول حاجة لو جينا طلعنا لحالة ال X بتساوي صفر
+
+513
+00:49:12,280 --> 00:49:18,000
+المساواة حادثة ولا لا it was a zero ليه واحد يساوي
+
+514
+00:49:18,000 --> 00:49:21,520
+واحد زائد زيرو اللي هو واحد يعني المساواة حادثة في
+
+515
+00:49:21,520 --> 00:49:25,120
+حالة ال .. ال X بتساوي زيرو الآن خليني أخد ال X
+
+516
+00:49:25,120 --> 00:49:31,970
+إيش ماله لا تساوي أيش؟ صفر الـ X لا تساوي صفر هثبت لك
+
+517
+00:49:31,970 --> 00:49:36,950
+إنه E to the X strictly أكبر من واحد زائد X بكون
+
+518
+00:49:36,950 --> 00:49:42,410
+خلصت هذه و خلصت هذه مع بعض افرض اثنين بنشوف كيف لأن X
+
+519
+00:49:42,410 --> 00:49:46,930
+لا تساوي Zero عندي حالتين X أكبر من صفر و X أصغر
+
+520
+00:49:46,930 --> 00:49:52,130
+من صفر الآن f of x ده اللي هي من الدوال الحلوة اللي
+
+521
+00:49:52,130 --> 00:49:54,230
+differentiable everywhere إذا continuous
+
+522
+00:49:54,230 --> 00:49:56,830
+everywhere إذا ال mean value theorem على أي
+
+523
+00:49:56,830 --> 00:50:01,610
+interval أو sub interval منها من الار متحققة إذا x
+
+524
+00:50:01,610 --> 00:50:04,930
+أكبر من 0 معناته بتتخد أطبق ال mean value theorem
+
+525
+00:50:04,930 --> 00:50:10,910
+من عند ال zero لعند main ال x معايا إذا by mean
+
+526
+00:50:10,910 --> 00:50:16,490
+value theorem there exists element in zero x such
+
+527
+00:50:16,490 --> 00:50:25,720
+thatF prime of C يساوي F of X ناقص F of Zero على X
+
+528
+00:50:25,720 --> 00:50:30,400
+ناقص Zero، مظبوط ولا لأ؟ هذه اللي هي F prime of C
+
+529
+00:50:30,400 --> 00:50:34,120
+طبعاً ال derivative للـE to the X هي نفسها عند الـC
+
+530
+00:50:34,120 --> 00:50:38,560
+في السير E to the C وهذه اللي هي عبارة عن E to the
+
+531
+00:50:38,560 --> 00:50:41,900
+X F of Zero اللي هي E to the Zero اللي هي واحد
+
+532
+00:50:41,900 --> 00:50:46,860
+ناقص واحد على مين على X اضربوا طرفين في وساطين
+
+533
+00:50:46,860 --> 00:50:52,340
+فبيصير E to the X ناقص واحد يساوي X في مين في E
+
+534
+00:50:52,340 --> 00:50:59,020
+أُسي ماشي الحال الآن هذه الآن ال E to the C C
+
+535
+00:50:59,020 --> 00:51:05,100
+موجبة صح ال E to the C هتكون أكبر أو تساوي واحد إن
+
+536
+00:51:05,100 --> 00:51:09,120
+هي أكبر strictly من واحد في الواقع مظبوط E to the
+
+537
+00:51:09,120 --> 00:51:13,960
+C أكبر strictly من مين من الواحد الآن X في ال E to
+
+538
+00:51:13,960 --> 00:51:20,920
+the C هتكون أكبر strictly من مين من اللي هي ال X مع
+
+539
+00:51:20,920 --> 00:51:24,880
+بعض الثنتين بيصير عندي E to the X أكبر من واحد
+
+540
+00:51:24,880 --> 00:51:31,860
+زائد مين زائد ال X الآن في حالة ال X أصغر من صفر
+
+541
+00:51:31,860 --> 00:51:36,600
+في حالة ال X أصغر من صفر بدنا نطبق على مين؟ على
+
+542
+00:51:36,600 --> 00:51:43,380
+الفترة اللي هي X و Zero مظبوط؟ على الفترة X و Zero
+
+543
+00:51:47,630 --> 00:51:53,510
+ب .. كله كله ماشي there exists c element in x و
+
+544
+00:51:53,510 --> 00:52:00,930
+zero such that أقول معايا f prime of c يساوي f of
+
+545
+00:52:00,930 --> 00:52:10,010
+zero ناقص f of x على zero ناقص x هو يساوي واحد
+
+546
+00:52:10,010 --> 00:52:19,700
+ناقص x ناقص e to the x على سالم X هذا إيش يساوي E
+
+547
+00:52:19,700 --> 00:52:25,820
+to the C أضرب الطرفين في وسطين بيصير واحد ناقص E
+
+548
+00:52:25,820 --> 00:52:34,760
+to the X يساوي ناقص X E to the C أضرب اللي هو في
+
+549
+00:52:34,760 --> 00:52:41,840
+ناقص واحد بيصير E to the X ناقص واحد يساوي X في E
+
+550
+00:52:41,840 --> 00:52:50,110
+to the C الآن ال C سالبة مظبوط فبيصير كسر فبيصير عند
+
+551
+00:52:50,110 --> 00:52:56,150
+ال E to the C اللي هو إيش ماله أصغر من واحد فالآن
+
+552
+00:52:56,150 --> 00:53:01,210
+لما نضرب ال X السالبة فيه ال X سالبة هنا فبيصير
+
+553
+00:53:01,210 --> 00:53:08,110
+أكبر من مين من X فبيصير
+
+554
+00:53:08,110 --> 00:53:12,350
+عند E to the X أكبر
+
+555
+00:53:13,170 --> 00:53:21,630
+من واحد زائد مين زائد X الآن صار عندي من الجهتين
+
+556
+00:53:21,630 --> 00:53:26,590
+لما X لا تساوي صفر E to the X strictly أكبر من
+
+557
+00:53:26,590 --> 00:53:32,590
+واحد زائد X إذا X لا تساوي صفر بتعني
+
+558
+00:53:43,580 --> 00:53:50,840
+E to the X أكبر strictly من واحد زائد X هذه واحد
+
+559
+00:53:50,840 --> 00:53:58,260
+الآن اثنين في حالة X بتساوي صفر هذا بيعطينا إن E
+
+560
+00:53:58,260 --> 00:54:03,480
+to the X يساوي واحد زائد X والجهتين هيساوين واحد
+
+561
+00:54:03,480 --> 00:54:09,600
+واحد فE to the Zero واحد الآن الجهتين مع بعض هذول
+
+562
+00:54:11,120 --> 00:54:15,880
+بتدي لمّ الـ x لا تساوي 0 و الـ x بتساوي 0 يعني e
+
+563
+00:54:15,880 --> 00:54:19,740
+to the x بتاخد بعين الاعتبار المساواة هنا بيصير e
+
+564
+00:54:19,740 --> 00:54:24,740
+to the x أكبر أو يساوي 1 زائد x لكل x نين element
+
+565
+00:54:24,740 --> 00:54:29,120
+in R بالضبط في حالة x بتساوي صفر e to the x بتساوي
+
+566
+00:54:29,120 --> 00:54:33,520
+في حالة x لا تساوي صفر e to the x strictly أكبر
+
+567
+00:54:33,520 --> 00:54:40,750
+الآن هذا x بتساوي صفر يؤدي إلى هذا الـ Trivia زي ما
+
+568
+00:54:40,750 --> 00:54:45,510
+قلنا الآن conversely بدنا نثبت هذا هذا أثبتناه في
+
+569
+00:54:45,510 --> 00:54:52,290
+الواقع كيف أثبتناه بدنا نقول بدنا نثبت الآن الاتجاه
+
+570
+00:54:52,290 --> 00:54:57,370
+هذا هذا الاتجاه اللي محيته اتجاه الـ Trivia اللي
+
+571
+00:54:57,370 --> 00:55:02,050
+سهل جداً هذا هذا عشان أثبته بقول Contraposition لو
+
+572
+00:55:02,050 --> 00:55:07,480
+كانت X لا تساوي صفر بيعطينا الـ E to the X لا يساوي
+
+573
+00:55:07,480 --> 00:55:12,080
+واحد زائد X ليش؟ لأنه أثبتته أنه strictly أكبر من
+
+601
+00:58:08,600 --> 00:58:16,550
+x cos c هي sin x إذا صار عند sin x  يظهر يساوي X
+
+602
+00:58:16,550 --> 00:58:22,050
+وأكبر أو يساوي سالب X  هذا الآن لكل X لا تساوي أكبر من
+
+603
+00:58:22,050 --> 00:58:26,490
+صفر but for X بتساوي zero this is trivial إذا صارت 
+
+604
+00:58:26,490 --> 00:58:33,250
+sin ال X بين ناقص X و X لكل X أكبر أو يساوي صفر و
+
+605
+00:58:33,250 --> 00:58:34,970
+المرة القادمة إن شاء الله 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..c7425398db6c6f63dd50c452e38149d4e2c6e661
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 3452, "start": 5.28, "end": 34.52, "text": "بسم الله الرحمن الرحيم المحاضرة الرابعة مساق تحليل حقيقة 2 لطلبة الجامعة الإسلامية كلية العلوم قسم الرياضيات نستكمل الحديث عن نظرية القيمة المتوسطة أو the mean value theorem وما يتعلق بذلك تحدثنا المرة الماضية", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 9673, 5016, 46958, 25720, 34892, 16758, 27884, 47524, 995, 4587, 6055, 5016, 20292, 1211, 11331, 38436, 28671, 568, 5296, 9566, 46152, 3660, 25724, 10943, 27884, 33688, 3794, 37440, 10632, 28242, 10632, 18863, 1211, 20498, 12174, 38251, 2423, 16572, 46958, 1829, 9307, 8717, 14851, 24793, 1211, 21542, 16254, 12984, 18871, 8717, 19913, 2288, 10632, 25062, 32640, 3660, 9673, 2655, 41779, 9566, 3660, 34051, 264, 914, 2158, 20904, 4032, 15042, 7251, 2655, 30241, 4587, 4724, 8848, 23275, 6055, 24401, 12984, 8315, 9673, 25720, 9673, 46958, 10632], "avg_logprob": -0.08701795022538368, "compression_ratio": 1.7632850241545894, "no_speech_prob": 2.682209014892578e-06, "words": [{"start": 5.28, "end": 5.64, "word": "بسم", "probability": 0.887939453125}, {"start": 5.64, "end": 5.86, "word": " الله", "probability": 0.953125}, {"start": 5.86, "end": 6.26, "word": " الرحمن", "probability": 0.9762369791666666}, {"start": 6.26, "end": 6.82, "word": " الرحيم", "probability": 0.9949544270833334}, {"start": 6.82, "end": 8.6, "word": " المحاضرة", "probability": 0.970703125}, {"start": 8.6, "end": 9.52, "word": " الرابعة", "probability": 0.9716796875}, {"start": 9.52, "end": 10.82, "word": " مساق", "probability": 0.92041015625}, {"start": 10.82, "end": 12.08, "word": " تحليل", "probability": 0.9718017578125}, {"start": 12.08, "end": 12.64, "word": " حقيقة", "probability": 0.7757975260416666}, {"start": 12.64, "end": 13.06, "word": " 2", "probability": 0.70947265625}, {"start": 13.06, "end": 15.2, "word": " لطلبة", "probability": 0.950439453125}, {"start": 15.2, "end": 15.64, "word": " الجامعة", "probability": 0.9860026041666666}, {"start": 15.64, "end": 16.48, "word": " الإسلامية", "probability": 0.9822998046875}, {"start": 16.48, "end": 17.62, "word": " كلية", "probability": 0.6817626953125}, {"start": 17.62, "end": 18.22, "word": " العلوم", "probability": 0.9640299479166666}, {"start": 18.22, "end": 18.78, "word": " قسم", "probability": 0.970458984375}, {"start": 18.78, "end": 19.62, "word": " الرياضيات", "probability": 0.94267578125}, {"start": 19.62, "end": 22.5, "word": " نستكمل", "probability": 0.96875}, {"start": 22.5, "end": 23.32, "word": " الحديث", "probability": 0.9934895833333334}, {"start": 23.32, "end": 23.82, "word": " عن", "probability": 0.99560546875}, {"start": 23.82, "end": 25.02, "word": " نظرية", "probability": 0.8294677734375}, {"start": 25.02, "end": 25.46, "word": " القيمة", "probability": 0.9558919270833334}, {"start": 25.46, "end": 26.48, "word": " المتوسطة", "probability": 0.99287109375}, {"start": 26.48, "end": 27.06, "word": " أو", "probability": 0.8203125}, {"start": 27.06, "end": 27.24, "word": " the", "probability": 0.5498046875}, {"start": 27.24, "end": 27.54, "word": " mean", "probability": 0.72021484375}, {"start": 27.54, "end": 27.96, "word": " value", "probability": 0.9482421875}, {"start": 27.96, "end": 28.4, "word": " theorem", "probability": 0.8662109375}, {"start": 28.4, "end": 28.66, "word": " وما", "probability": 0.93310546875}, {"start": 28.66, "end": 29.4, "word": " يتعلق", "probability": 0.991455078125}, {"start": 29.4, "end": 30.08, "word": " بذلك", "probability": 0.9519856770833334}, {"start": 30.08, "end": 33.38, "word": " تحدثنا", "probability": 0.79541015625}, {"start": 33.38, "end": 33.74, "word": " المرة", "probability": 0.95947265625}, {"start": 33.74, "end": 34.52, "word": " الماضية", "probability": 0.9931640625}], "temperature": 1.0}, {"id": 2, "seek": 6343, "start": 35.99, "end": 63.43, "text": "معنى relative extrema سواء relative maximum أو relative minimum أو بمعنى أخر ال absolute ال local maximum و ال local minimum معرفناها المرة الماضية و أتحدثنا أيضًا عن اللي هو ال interior extreme theorem ال interior extreme theorem اللي هي جوابتنا على السؤال اللي بنحكي فيه", "tokens": [2304, 3615, 1863, 7578, 4972, 1279, 265, 1696, 8608, 2407, 16606, 4972, 6674, 34051, 4972, 7285, 34051, 4724, 2304, 3615, 1863, 7578, 5551, 34740, 2423, 8236, 2423, 2654, 6674, 4032, 2423, 2654, 7285, 20449, 28480, 8315, 11296, 9673, 25720, 9673, 46958, 10632, 4032, 5551, 2655, 24401, 12984, 8315, 36632, 11242, 14111, 995, 18871, 13672, 1829, 31439, 2423, 10636, 8084, 20904, 2423, 10636, 8084, 20904, 13672, 1829, 39896, 10874, 14407, 3555, 2655, 8315, 15844, 21136, 33604, 6027, 13672, 1829, 44945, 5016, 4117, 1829, 8978, 3224], "avg_logprob": -0.20790441386839922, "compression_ratio": 1.8349514563106797, "no_speech_prob": 1.430511474609375e-06, "words": [{"start": 35.99000000000001, "end": 36.95, "word": "معنى", "probability": 0.85107421875}, {"start": 36.95, "end": 37.35, "word": " relative", "probability": 0.38818359375}, {"start": 37.35, "end": 38.65, "word": " extrema", "probability": 0.908203125}, {"start": 38.65, "end": 39.75, "word": " سواء", "probability": 0.8805338541666666}, {"start": 39.75, "end": 40.13, "word": " relative", "probability": 0.9287109375}, {"start": 40.13, "end": 40.73, "word": " maximum", "probability": 0.97802734375}, {"start": 40.73, "end": 41.19, "word": " أو", "probability": 0.90283203125}, {"start": 41.19, "end": 41.67, "word": " relative", "probability": 0.8759765625}, {"start": 41.67, "end": 42.09, "word": " minimum", "probability": 0.978515625}, {"start": 42.09, "end": 42.45, "word": " أو", "probability": 0.888671875}, {"start": 42.45, "end": 42.81, "word": " بمعنى", "probability": 0.96484375}, {"start": 42.81, "end": 43.13, "word": " أخر", "probability": 0.752197265625}, {"start": 43.13, "end": 43.29, "word": " ال", "probability": 0.62939453125}, {"start": 43.29, "end": 43.59, "word": " absolute", "probability": 0.1890869140625}, {"start": 43.59, "end": 44.47, "word": " ال", "probability": 0.299072265625}, {"start": 44.47, "end": 45.05, "word": " local", "probability": 0.57373046875}, {"start": 45.05, "end": 45.75, "word": " maximum", "probability": 0.96875}, {"start": 45.75, "end": 46.47, "word": " و", "probability": 0.54248046875}, {"start": 46.47, "end": 46.79, "word": " ال", "probability": 0.93212890625}, {"start": 46.79, "end": 47.15, "word": " local", "probability": 0.85546875}, {"start": 47.15, "end": 47.45, "word": " minimum", "probability": 0.97705078125}, {"start": 47.45, "end": 48.21, "word": " معرفناها", "probability": 0.8594970703125}, {"start": 48.21, "end": 48.53, "word": " المرة", "probability": 0.96484375}, {"start": 48.53, "end": 49.09, "word": " الماضية", "probability": 0.9928385416666666}, {"start": 49.09, "end": 51.17, "word": " و", "probability": 0.28369140625}, {"start": 51.17, "end": 52.09, "word": " أتحدثنا", "probability": 0.865234375}, {"start": 52.09, "end": 52.69, "word": " أيضًا", "probability": 0.8272705078125}, {"start": 52.69, "end": 53.09, "word": " عن", "probability": 0.9892578125}, {"start": 53.09, "end": 53.71, "word": " اللي", "probability": 0.68310546875}, {"start": 53.71, "end": 53.91, "word": " هو", "probability": 0.9794921875}, {"start": 53.91, "end": 54.05, "word": " ال", "probability": 0.8154296875}, {"start": 54.05, "end": 54.49, "word": " interior", "probability": 0.8125}, {"start": 54.49, "end": 56.17, "word": " extreme", "probability": 0.1868896484375}, {"start": 56.17, "end": 57.05, "word": " theorem", "probability": 0.85888671875}, {"start": 57.05, "end": 57.63, "word": " ال", "probability": 0.84765625}, {"start": 57.63, "end": 58.07, "word": " interior", "probability": 0.890625}, {"start": 58.07, "end": 58.65, "word": " extreme", "probability": 0.681640625}, {"start": 58.65, "end": 59.49, "word": " theorem", "probability": 0.896484375}, {"start": 59.49, "end": 59.97, "word": " اللي", "probability": 0.897705078125}, {"start": 59.97, "end": 60.29, "word": " هي", "probability": 0.93115234375}, {"start": 60.29, "end": 61.49, "word": " جوابتنا", "probability": 0.926953125}, {"start": 61.49, "end": 61.75, "word": " على", "probability": 0.91455078125}, {"start": 61.75, "end": 62.33, "word": " السؤال", "probability": 0.9962565104166666}, {"start": 62.33, "end": 62.49, "word": " اللي", "probability": 0.969970703125}, {"start": 62.49, "end": 63.03, "word": " بنحكي", "probability": 0.9583740234375}, {"start": 63.03, "end": 63.43, "word": " فيه", "probability": 0.966552734375}], "temperature": 1.0}, {"id": 3, "seek": 8523, "start": 63.77, "end": 85.23, "text": "أين يمكن إيجاد القيم العظمى أو القيم الصغرى النظرية اللي بتحكي كانت ان لو كانت C عبارة عن interior point اللي هي ال interval I اللي عندها اللي هي ال function F ال derivative موجودة", "tokens": [10721, 9957, 7251, 43020, 11933, 1829, 7435, 18513, 25062, 32640, 18863, 19913, 2304, 7578, 34051, 25062, 32640, 31767, 17082, 2288, 7578, 28239, 19913, 2288, 10632, 13672, 1829, 39894, 5016, 4117, 1829, 25961, 2655, 16472, 45164, 25961, 2655, 383, 6225, 3555, 9640, 3660, 18871, 10636, 935, 13672, 1829, 39896, 2423, 15035, 286, 13672, 1829, 43242, 11296, 13672, 1829, 39896, 2423, 2445, 479, 2423, 13760, 3714, 29245, 23328, 3660], "avg_logprob": -0.1835937491234611, "compression_ratio": 1.6045197740112995, "no_speech_prob": 2.4437904357910156e-06, "words": [{"start": 63.77, "end": 64.19, "word": "أين", "probability": 0.846435546875}, {"start": 64.19, "end": 64.73, "word": " يمكن", "probability": 0.98193359375}, {"start": 64.73, "end": 65.35, "word": " إيجاد", "probability": 0.9375}, {"start": 65.35, "end": 65.73, "word": " القيم", "probability": 0.81689453125}, {"start": 65.73, "end": 66.97, "word": " العظمى", "probability": 0.84930419921875}, {"start": 66.97, "end": 67.59, "word": " أو", "probability": 0.94384765625}, {"start": 67.59, "end": 68.33, "word": " القيم", "probability": 0.94970703125}, {"start": 68.33, "end": 69.17, "word": " الصغرى", "probability": 0.894775390625}, {"start": 69.17, "end": 70.75, "word": " النظرية", "probability": 0.81280517578125}, {"start": 70.75, "end": 71.09, "word": " اللي", "probability": 0.662353515625}, {"start": 71.09, "end": 72.01, "word": " بتحكي", "probability": 0.911376953125}, {"start": 72.01, "end": 72.37, "word": " كانت", "probability": 0.8203125}, {"start": 72.37, "end": 72.55, "word": " ان", "probability": 0.380126953125}, {"start": 72.55, "end": 72.81, "word": " لو", "probability": 0.91162109375}, {"start": 72.81, "end": 73.29, "word": " كانت", "probability": 0.958251953125}, {"start": 73.29, "end": 73.61, "word": " C", "probability": 0.52685546875}, {"start": 73.61, "end": 74.53, "word": " عبارة", "probability": 0.9833984375}, {"start": 74.53, "end": 74.73, "word": " عن", "probability": 0.9912109375}, {"start": 74.73, "end": 75.35, "word": " interior", "probability": 0.6748046875}, {"start": 75.35, "end": 76.29, "word": " point", "probability": 0.98388671875}, {"start": 76.29, "end": 77.55, "word": " اللي", "probability": 0.6368408203125}, {"start": 77.55, "end": 78.05, "word": " هي", "probability": 0.58447265625}, {"start": 78.05, "end": 78.21, "word": " ال", "probability": 0.888671875}, {"start": 78.21, "end": 78.71, "word": " interval", "probability": 0.59423828125}, {"start": 78.71, "end": 79.33, "word": " I", "probability": 0.91845703125}, {"start": 79.33, "end": 79.95, "word": " اللي", "probability": 0.954833984375}, {"start": 79.95, "end": 80.61, "word": " عندها", "probability": 0.991455078125}, {"start": 80.61, "end": 81.41, "word": " اللي", "probability": 0.6514892578125}, {"start": 81.41, "end": 81.59, "word": " هي", "probability": 0.94091796875}, {"start": 81.59, "end": 81.73, "word": " ال", "probability": 0.94921875}, {"start": 81.73, "end": 82.13, "word": " function", "probability": 0.86572265625}, {"start": 82.13, "end": 82.71, "word": " F", "probability": 0.7197265625}, {"start": 82.71, "end": 83.93, "word": " ال", "probability": 0.52099609375}, {"start": 83.93, "end": 84.35, "word": " derivative", "probability": 0.9091796875}, {"start": 84.35, "end": 85.23, "word": " موجودة", "probability": 0.9832763671875}], "temperature": 1.0}, {"id": 4, "seek": 11248, "start": 85.88, "end": 112.48, "text": "بقول لو كان عند النقطة هذه في relative extremum سواء minimum أو maximum لل F عنده نقطة C قطعا ال F prime of C هتساوي 0 ومن هنا استقينا أن أين يمكن إيجاد النقطة العظمى والنقطة الصغيرة المحلية وقلنا هي في أحد ثلاث مناطق اللي هو أما F prime بتساوي 0", "tokens": [3555, 39648, 45164, 25961, 43242, 28239, 47432, 3660, 29538, 8978, 4972, 4040, 449, 8608, 2407, 16606, 7285, 34051, 6674, 24976, 479, 43242, 3224, 8717, 47432, 3660, 383, 12174, 9566, 3615, 995, 2423, 479, 5835, 295, 383, 8032, 2655, 3794, 995, 45865, 1958, 4032, 27842, 34105, 44713, 38436, 8315, 14739, 5551, 9957, 7251, 43020, 11933, 1829, 7435, 18513, 28239, 47432, 3660, 18863, 19913, 2304, 7578, 16070, 1863, 47432, 3660, 31767, 17082, 48923, 9673, 5016, 1211, 10632, 4032, 4587, 1211, 8315, 39896, 8978, 5551, 24401, 38637, 1211, 5718, 104, 9154, 41193, 4587, 13672, 1829, 31439, 5551, 15042, 479, 5835, 39894, 3794, 995, 45865, 1958], "avg_logprob": -0.21601941197821237, "compression_ratio": 1.6375, "no_speech_prob": 6.556510925292969e-07, "words": [{"start": 85.88, "end": 86.96, "word": "بقول", "probability": 0.80078125}, {"start": 86.96, "end": 87.22, "word": " لو", "probability": 0.493408203125}, {"start": 87.22, "end": 87.7, "word": " كان", "probability": 0.9833984375}, {"start": 87.7, "end": 88.44, "word": " عند", "probability": 0.98291015625}, {"start": 88.44, "end": 88.96, "word": " النقطة", "probability": 0.8427734375}, {"start": 88.96, "end": 89.18, "word": " هذه", "probability": 0.71484375}, {"start": 89.18, "end": 89.44, "word": " في", "probability": 0.496826171875}, {"start": 89.44, "end": 89.82, "word": " relative", "probability": 0.62451171875}, {"start": 89.82, "end": 91.16, "word": " extremum", "probability": 0.774658203125}, {"start": 91.16, "end": 92.14, "word": " سواء", "probability": 0.8933919270833334}, {"start": 92.14, "end": 92.58, "word": " minimum", "probability": 0.8935546875}, {"start": 92.58, "end": 93.14, "word": " أو", "probability": 0.71435546875}, {"start": 93.14, "end": 93.86, "word": " maximum", "probability": 0.97509765625}, {"start": 93.86, "end": 94.26, "word": " لل", "probability": 0.85791015625}, {"start": 94.26, "end": 94.6, "word": " F", "probability": 0.46630859375}, {"start": 94.6, "end": 95.02, "word": " عنده", "probability": 0.943603515625}, {"start": 95.02, "end": 95.34, "word": " نقطة", "probability": 0.9488932291666666}, {"start": 95.34, "end": 95.6, "word": " C", "probability": 0.7138671875}, {"start": 95.6, "end": 96.46, "word": " قطعا", "probability": 0.9256591796875}, {"start": 96.46, "end": 96.56, "word": " ال", "probability": 0.83984375}, {"start": 96.56, "end": 96.74, "word": " F", "probability": 0.6708984375}, {"start": 96.74, "end": 97.08, "word": " prime", "probability": 0.44482421875}, {"start": 97.08, "end": 97.24, "word": " of", "probability": 0.83837890625}, {"start": 97.24, "end": 97.42, "word": " C", "probability": 0.90185546875}, {"start": 97.42, "end": 97.96, "word": " هتساوي", "probability": 0.87138671875}, {"start": 97.96, "end": 98.54, "word": " 0", "probability": 0.078857421875}, {"start": 98.54, "end": 99.26, "word": " ومن", "probability": 0.663330078125}, {"start": 99.26, "end": 99.54, "word": " هنا", "probability": 0.99169921875}, {"start": 99.54, "end": 100.38, "word": " استقينا", "probability": 0.725830078125}, {"start": 100.38, "end": 101.0, "word": " أن", "probability": 0.447998046875}, {"start": 101.0, "end": 101.94, "word": " أين", "probability": 0.7454833984375}, {"start": 101.94, "end": 102.9, "word": " يمكن", "probability": 0.98486328125}, {"start": 102.9, "end": 103.56, "word": " إيجاد", "probability": 0.9434814453125}, {"start": 103.56, "end": 104.44, "word": " النقطة", "probability": 0.96875}, {"start": 104.44, "end": 105.04, "word": " العظمى", "probability": 0.885009765625}, {"start": 105.04, "end": 106.12, "word": " والنقطة", "probability": 0.7559814453125}, {"start": 106.12, "end": 106.56, "word": " الصغيرة", "probability": 0.841796875}, {"start": 106.56, "end": 107.32, "word": " المحلية", "probability": 0.981689453125}, {"start": 107.32, "end": 108.06, "word": " وقلنا", "probability": 0.7933349609375}, {"start": 108.06, "end": 108.22, "word": " هي", "probability": 0.6806640625}, {"start": 108.22, "end": 108.4, "word": " في", "probability": 0.93994140625}, {"start": 108.4, "end": 108.6, "word": " أحد", "probability": 0.95166015625}, {"start": 108.6, "end": 108.94, "word": " ثلاث", "probability": 0.8907470703125}, {"start": 108.94, "end": 109.56, "word": " مناطق", "probability": 0.96044921875}, {"start": 109.56, "end": 110.06, "word": " اللي", "probability": 0.85009765625}, {"start": 110.06, "end": 110.3, "word": " هو", "probability": 0.8154296875}, {"start": 110.3, "end": 110.66, "word": " أما", "probability": 0.7388916015625}, {"start": 110.66, "end": 111.1, "word": " F", "probability": 0.91259765625}, {"start": 111.1, "end": 111.64, "word": " prime", "probability": 0.7119140625}, {"start": 111.64, "end": 112.16, "word": " بتساوي", "probability": 0.90966796875}, {"start": 112.16, "end": 112.48, "word": " 0", "probability": 0.8740234375}], "temperature": 1.0}, {"id": 5, "seek": 12800, "start": 113.06, "end": 128.0, "text": "أو F prime does not exist للنقاط الداخلية طبعا أو اللي هو ال end points فتلت مناطق اللي هي بنفحص فيها اللي هي ال relative maximum أو ال relative minimum", "tokens": [10721, 2407, 479, 5835, 775, 406, 2514, 24976, 1863, 4587, 41193, 32748, 47283, 1211, 10632, 23032, 3555, 3615, 995, 34051, 13672, 1829, 31439, 2423, 917, 2793, 6156, 2655, 1211, 2655, 9154, 41193, 4587, 13672, 1829, 39896, 44945, 5172, 5016, 9381, 8978, 11296, 13672, 1829, 39896, 2423, 4972, 6674, 34051, 2423, 4972, 7285], "avg_logprob": -0.19531250674769562, "compression_ratio": 1.4437086092715232, 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الـ Mean Value Theorem اللي كانت أيضا على الرسم خلينا نشوف الـ Mean Value Theorem قالت إن لو كانت الـ function is continuous على closed interval من A لB وكانت اللي هي differentiable على الـ open interval A وB", "tokens": [3555, 22488, 29910, 23275, 23032, 3555, 3615, 995, 6055, 24401, 12984, 8315, 18871, 2423, 39184, 12302, 39352, 440, 37956, 2423, 39184, 12302, 39352, 440, 37956, 13672, 1829, 25961, 2655, 36632, 11242, 995, 15844, 34892, 38251, 16490, 1211, 9957, 995, 8717, 8592, 38688, 2423, 39184, 12302, 39352, 440, 37956, 50239, 2655, 36145, 45164, 25961, 2655, 2423, 39184, 2445, 307, 10957, 15844, 5395, 15035, 9154, 316, 5296, 33, 4032, 41361, 2655, 13672, 1829, 39896, 819, 9364, 15844, 2423, 39184, 1269, 15035, 316, 4032, 33], "avg_logprob": -0.2217620428068092, "compression_ratio": 1.6778846153846154, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 199.61, "end": 200.01, "word": "بعد", "probability": 0.87255859375}, {"start": 200.01, "end": 200.39, "word": " ذلك", 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449.7, "text": "الان similarly for الـ F is اللي هو strictly decreasing هذا اتحدثنا فيه المرة الماضية النظرية اللي هي let F من I لعند R be differentiable اللي هي أوصلنا عند هذه النظرية خلينا نشوف هذه النظرية و نتفحص إيش هي نظرية بتقول و من ثم نشوف لماذا", "tokens": [6027, 7649, 14138, 337, 2423, 39184, 479, 307, 13672, 1829, 31439, 20792, 23223, 23758, 1975, 2655, 24401, 12984, 8315, 8978, 3224, 9673, 25720, 9673, 46958, 10632, 28239, 19913, 2288, 10632, 13672, 1829, 39896, 718, 479, 9154, 286, 5296, 3615, 41260, 497, 312, 819, 9364, 13672, 1829, 39896, 34051, 36520, 8315, 43242, 29538, 28239, 19913, 2288, 10632, 16490, 1211, 9957, 995, 8717, 8592, 38688, 29538, 28239, 19913, 2288, 10632, 4032, 8717, 2655, 5172, 5016, 9381, 11933, 1829, 8592, 39896, 8717, 19913, 2288, 10632, 39894, 39648, 4032, 9154, 38637, 2304, 8717, 8592, 38688, 32767, 45636, 995], "avg_logprob": -0.1837171008712367, "compression_ratio": 1.7031963470319635, "no_speech_prob": 2.384185791015625e-07, "words": 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let F من I لعنب R be differentiable on", "tokens": [6027, 20292, 31439, 39896, 20328, 5016, 1829, 5016, 3660, 18871, 16254, 8717, 19913, 2288, 10632, 39894, 39648, 20292, 20904, 1386, 13, 17, 13, 22, 13672, 1829, 39894, 5016, 4117, 1829, 28239, 19913, 2288, 10632, 39894, 39648, 20292, 45164, 25961, 18871, 16254, 479, 718, 479, 9154, 286, 5296, 3615, 1863, 3555, 497, 312, 819, 9364, 322], "avg_logprob": -0.199358251744083, "compression_ratio": 1.4, "no_speech_prob": 4.303455352783203e-05, "words": [{"start": 450.1, "end": 450.48, "word": "اللي", "probability": 0.799072265625}, {"start": 450.48, "end": 450.9, "word": " هو", "probability": 0.69189453125}, {"start": 450.9, "end": 451.64, "word": " هي", "probability": 0.70361328125}, {"start": 451.64, "end": 453.1, "word": " صحيحة", "probability": 0.98828125}, {"start": 453.1, "end": 458.96, "word": " عندي", "probability": 0.70556640625}, {"start": 458.96, "end": 459.76, "word": " نظرية", "probability": 0.918212890625}, {"start": 459.76, 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538.95, "end": 565.13, "text": "و هنبرهن اللي هو A أو B في الكتاب نبرهن A خلّينا نبرهن مين نبرهن B و هتكون A similarly for B الان بي .. بدنا نفترض suppose that F is decreasing on I", "tokens": [2407, 8032, 1863, 26890, 3224, 1863, 13672, 1829, 31439, 316, 34051, 363, 8978, 33251, 2655, 16758, 8717, 26890, 3224, 1863, 316, 16490, 1211, 11703, 9957, 995, 8717, 26890, 3224, 1863, 3714, 9957, 8717, 26890, 3224, 1863, 363, 4032, 8032, 2655, 30544, 316, 14138, 337, 363, 2423, 7649, 4724, 1829, 4386, 47525, 8315, 8717, 5172, 2655, 43042, 7297, 300, 479, 307, 23223, 322, 286], "avg_logprob": -0.21594238048419356, "compression_ratio": 1.3584905660377358, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 538.95, "end": 539.31, "word": "و", "probability": 0.86181640625}, {"start": 539.31, "end": 540.19, "word": " هنبرهن", "probability": 0.84189453125}, {"start": 540.19, "end": 540.77, "word": " اللي", "probability": 0.8837890625}, {"start": 540.77, "end": 541.27, "word": " هو", 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"word": " that", "probability": 0.84765625}, {"start": 557.55, "end": 559.21, "word": " F", "probability": 0.73876953125}, {"start": 559.21, "end": 560.43, "word": " is", "probability": 0.9482421875}, {"start": 560.43, "end": 561.79, "word": " decreasing", "probability": 0.96435546875}, {"start": 561.79, "end": 564.57, "word": " on", "probability": 0.953125}, {"start": 564.57, "end": 565.13, "word": " I", "probability": 0.96142578125}], "temperature": 1.0}, {"id": 24, "seek": 59025, "start": 567.13, "end": 590.25, "text": "F is decreasing on I بدنا .. بدنا نثبت الآن أن الـ F prime of X اللي هي أصغر أو يساوي مين سفر طبعاً F prime of X في النهاية إيش هي؟ عشان نشوف كيف بنفكر أول إشي F prime of X بساوي limit F of X", "tokens": [37, 307, 23223, 322, 286, 47525, 8315, 4386, 47525, 8315, 8717, 12984, 3555, 2655, 6024, 48506, 14739, 2423, 39184, 479, 5835, 295, 1783, 13672, 1829, 39896, 5551, 9381, 17082, 2288, 34051, 7251, 3794, 995, 45865, 3714, 9957, 8608, 5172, 2288, 23032, 3555, 3615, 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0.919189453125}, {"start": 571.11, "end": 571.49, "word": " نثبت", "probability": 0.99267578125}, {"start": 571.49, "end": 572.03, "word": " الآن", "probability": 0.769775390625}, {"start": 572.03, "end": 573.09, "word": " أن", "probability": 0.39697265625}, {"start": 573.09, "end": 573.71, "word": " الـ", "probability": 0.5751953125}, {"start": 573.71, "end": 573.89, "word": " F", "probability": 0.73974609375}, {"start": 573.89, "end": 574.29, "word": " prime", "probability": 0.421142578125}, {"start": 574.29, "end": 574.53, "word": " of", "probability": 0.95751953125}, {"start": 574.53, "end": 574.95, "word": " X", "probability": 0.951171875}, {"start": 574.95, "end": 575.71, "word": " اللي", "probability": 0.885498046875}, {"start": 575.71, "end": 575.93, "word": " هي", "probability": 0.82275390625}, {"start": 575.93, "end": 576.41, "word": " أصغر", "probability": 0.9874267578125}, {"start": 576.41, "end": 576.65, "word": " أو", "probability": 0.966796875}, {"start": 576.65, "end": 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بدنا الأن نيجي ل conversely بدنا نفترض هذا أصحى و أصحى و صفر و نوصل لمين لل F is decreasing نيجي الأن conversely و نشوف كيف نبرهن conversely الأن conversely", "tokens": [3224, 15730, 13672, 1829, 31439, 25724, 11622, 38207, 16247, 12610, 9154, 28239, 19913, 2288, 10632, 9154, 13672, 1829, 31439, 363, 14739, 3224, 47525, 8315, 16247, 1863, 8717, 1829, 7435, 1829, 5296, 2615, 736, 47525, 8315, 8717, 5172, 2655, 43042, 23758, 5551, 9381, 5016, 7578, 4032, 5551, 9381, 5016, 7578, 4032, 20328, 5172, 2288, 4032, 8717, 2407, 36520, 5296, 2304, 9957, 24976, 479, 307, 23223, 8717, 1829, 7435, 1829, 16247, 1863, 2615, 736, 4032, 8717, 8592, 38688, 9122, 33911, 8717, 26890, 3224, 1863, 2615, 736, 16247, 1863, 2615, 736], "avg_logprob": -0.24947332532218333, "compression_ratio": 1.7351351351351352, "no_speech_prob": 1.4901161193847656e-05, "words": [{"start": 833.32, "end": 834.5, "word": "هذا", "probability": 0.79345703125}, {"start": 834.5, "end": 834.6, "word": " اللي", "probability": 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"probability": 0.9599609375}, {"start": 844.14, "end": 844.72, "word": " لمين", "probability": 0.6819661458333334}, {"start": 844.72, "end": 845.26, "word": " لل", "probability": 0.368408203125}, {"start": 845.26, "end": 845.7, "word": " F", "probability": 0.473876953125}, {"start": 845.7, "end": 846.58, "word": " is", "probability": 0.875}, {"start": 846.58, "end": 847.44, "word": " decreasing", "probability": 0.95947265625}, {"start": 847.44, "end": 848.74, "word": " نيجي", "probability": 0.9647216796875}, {"start": 848.74, "end": 849.04, "word": " الأن", "probability": 0.810546875}, {"start": 849.04, "end": 850.0, "word": " conversely", "probability": 0.91455078125}, {"start": 850.0, "end": 851.06, "word": " و", "probability": 0.98095703125}, {"start": 851.06, "end": 851.48, "word": " نشوف", "probability": 0.9352213541666666}, {"start": 851.48, "end": 851.8, "word": " كيف", "probability": 0.97216796875}, {"start": 851.8, "end": 852.56, "word": " نبرهن", "probability": 0.9239501953125}, {"start": 852.56, "end": 854.64, "word": " conversely", "probability": 0.92724609375}, {"start": 854.64, "end": 859.24, "word": " الأن", "probability": 0.6475830078125}, {"start": 859.24, "end": 860.1, "word": " conversely", "probability": 0.95751953125}], "temperature": 1.0}, {"id": 36, "seek": 87081, "start": 861.01, "end": 870.81, "text": "طبعاً ببرهم في AVP برضه انا conversely suppose that", "tokens": [9566, 3555, 3615, 995, 14111, 4724, 26890, 16095, 8978, 30198, 47, 4724, 43042, 3224, 1975, 8315, 2615, 736, 7297, 300], "avg_logprob": -0.39397321996234713, "compression_ratio": 0.9210526315789473, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 861.01, "end": 861.41, "word": "طبعاً", "probability": 0.72666015625}, {"start": 861.41, "end": 861.79, "word": " ببرهم", "probability": 0.7913411458333334}, {"start": 861.79, "end": 861.99, "word": " في", "probability": 0.8857421875}, {"start": 861.99, "end": 862.57, "word": " AVP", "probability": 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{"start": 882.07, "end": 882.53, "word": " الصلاة", "probability": 0.8375651041666666}, {"start": 882.53, "end": 883.03, "word": " والسلام", "probability": 0.9482421875}, {"start": 883.03, "end": 883.79, "word": " أنا", "probability": 0.74853515625}, {"start": 883.79, "end": 884.07, "word": " بدأ", "probability": 0.598876953125}, {"start": 884.07, "end": 884.31, "word": " أثبت", "probability": 0.868896484375}, {"start": 884.31, "end": 884.43, "word": " أن", "probability": 0.7314453125}, {"start": 884.43, "end": 884.69, "word": " F", "probability": 0.7978515625}, {"start": 884.69, "end": 884.89, "word": " is", "probability": 0.56884765625}, {"start": 884.89, "end": 885.43, "word": " decreasing", "probability": 0.9609375}, {"start": 885.43, "end": 886.47, "word": " يعني", "probability": 0.95703125}, {"start": 886.47, "end": 886.81, "word": " بدأ", "probability": 0.6683349609375}, {"start": 886.81, "end": 887.17, "word": " أفترض", "probability": 0.909912109375}, {"start": 887.17, "end": 887.35, "word": " أن", "probability": 0.779296875}, {"start": 887.35, "end": 887.81, "word": " X", "probability": 0.9814453125}, {"start": 887.81, "end": 888.83, "word": " أصغر", "probability": 0.951171875}, {"start": 888.83, "end": 888.99, "word": " أو", "probability": 0.80615234375}, {"start": 888.99, "end": 889.91, "word": " أخد", "probability": 0.7688802083333334}, {"start": 889.91, "end": 890.19, "word": " أي", "probability": 0.73388671875}, {"start": 890.19, "end": 891.13, "word": " قيمتين", "probability": 0.944580078125}, {"start": 891.13, "end": 891.37, "word": " X", "probability": 0.71044921875}, {"start": 891.37, "end": 891.85, "word": " واحد", "probability": 0.810791015625}, {"start": 891.85, "end": 892.35, "word": " و", "probability": 0.85791015625}, {"start": 892.35, "end": 892.57, "word": " X", "probability": 0.79248046875}, {"start": 892.57, "end": 893.13, "word": " اتنين", "probability": 0.8194580078125}, {"start": 893.13, "end": 893.65, "word": " element", "probability": 0.5068359375}, {"start": 893.65, "end": 894.35, "word": " اتنين", "probability": 0.77813720703125}], "temperature": 1.0}, {"id": 38, "seek": 91553, "start": 895.01, "end": 915.53, "text": "أخدت أي two elements element x1 وx2 element in I تحقق such that x1 أصغر من مين من x2 أصغر أو يساوي x .. x1 أصغر أو يساوي x2 بس في حالة ال x1 بيساوي x2 ما هي automatic F of x1 بيساوي F of x2 يعني متحققة", "tokens": [10721, 9778, 3215, 2655, 36632, 732, 4959, 4478, 2031, 16, 4032, 87, 17, 4478, 294, 286, 6055, 5016, 4587, 4587, 1270, 300, 2031, 16, 5551, 9381, 17082, 2288, 9154, 3714, 9957, 9154, 2031, 17, 5551, 9381, 17082, 2288, 34051, 7251, 3794, 995, 45865, 2031, 4386, 2031, 16, 5551, 9381, 17082, 2288, 34051, 7251, 3794, 995, 45865, 2031, 17, 4724, 3794, 8978, 11331, 6027, 3660, 2423, 2031, 16, 4724, 1829, 3794, 995, 45865, 2031, 17, 19446, 39896, 12509, 479, 295, 2031, 16, 4724, 1829, 3794, 995, 45865, 479, 295, 2031, 17, 37495, 22653, 44650, 5016, 4587, 28671], "avg_logprob": 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0.92822265625}, {"start": 900.83, "end": 901.53, "word": " such", "probability": 0.67236328125}, {"start": 901.53, "end": 901.97, "word": " that", "probability": 0.9501953125}, {"start": 901.97, "end": 902.85, "word": " x1", "probability": 0.92578125}, {"start": 902.85, "end": 904.11, "word": " أصغر", "probability": 0.9556884765625}, {"start": 904.11, "end": 904.37, "word": " من", "probability": 0.98681640625}, {"start": 904.37, "end": 904.87, "word": " مين", "probability": 0.58319091796875}, {"start": 904.87, "end": 905.19, "word": " من", "probability": 0.912109375}, {"start": 905.19, "end": 905.83, "word": " x2", "probability": 0.984619140625}, {"start": 905.83, "end": 906.75, "word": " أصغر", "probability": 0.939208984375}, {"start": 906.75, "end": 906.85, "word": " أو", "probability": 0.94140625}, {"start": 906.85, "end": 907.25, "word": " يساوي", "probability": 0.90673828125}, {"start": 907.25, "end": 907.51, "word": " x", "probability": 0.580078125}, {"start": 907.51, "end": 907.65, "word": " ..", "probability": 0.440673828125}, {"start": 907.65, "end": 908.25, "word": " x1", "probability": 0.6661376953125}, {"start": 908.25, "end": 908.49, "word": " أصغر", "probability": 0.9813232421875}, {"start": 908.49, "end": 908.55, "word": " أو", "probability": 0.62109375}, {"start": 908.55, "end": 908.67, "word": " يساوي", "probability": 0.9862060546875}, {"start": 908.67, "end": 909.09, "word": " x2", "probability": 0.990966796875}, {"start": 909.09, "end": 909.65, "word": " بس", "probability": 0.94775390625}, {"start": 909.65, "end": 909.79, "word": " في", "probability": 0.94189453125}, {"start": 909.79, "end": 910.05, "word": " حالة", "probability": 0.9933268229166666}, {"start": 910.05, "end": 910.15, "word": " ال", "probability": 0.89111328125}, {"start": 910.15, "end": 910.57, "word": " x1", "probability": 0.831787109375}, {"start": 910.57, "end": 910.95, "word": " بيساوي", "probability": 0.837744140625}, {"start": 910.95, "end": 911.45, "word": " x2", "probability": 0.99267578125}, {"start": 911.45, "end": 911.59, "word": " ما", "probability": 0.595703125}, {"start": 911.59, "end": 911.73, "word": " هي", "probability": 0.6845703125}, {"start": 911.73, "end": 912.59, "word": " automatic", "probability": 0.82470703125}, {"start": 912.59, "end": 912.85, "word": " F", "probability": 0.453369140625}, {"start": 912.85, "end": 913.03, "word": " of", "probability": 0.642578125}, {"start": 913.03, "end": 913.41, "word": " x1", "probability": 0.96240234375}, {"start": 913.41, "end": 913.75, "word": " بيساوي", "probability": 0.87041015625}, {"start": 913.75, "end": 913.93, "word": " F", "probability": 0.9765625}, {"start": 913.93, "end": 914.07, "word": " of", "probability": 0.96875}, {"start": 914.07, "end": 914.49, "word": " x2", "probability": 0.993896484375}, {"start": 914.49, "end": 914.71, "word": " يعني", "probability": 0.930419921875}, {"start": 914.71, "end": 915.53, "word": " متحققة", "probability": 0.9498291015625}], "temperature": 1.0}, {"id": 39, "seek": 94162, "start": 915.9, "end": 941.62, "text": "إذن الان x1 أصغر من x2 عملتها strictly عشان بتعمل منها interval وحالتي ال x1 بيساوي x2 is a trivial case تتحقق مدامة x1 أصغر تساوي x2 إذن عشان a function f of x1 بيساوي f of x2 إذا التحقق اللي هي مين ال decreasing واضحة الان x1 أصغر من x2 الآن we apply", "tokens": [28814, 8848, 1863, 2423, 7649, 2031, 16, 5551, 9381, 17082, 2288, 9154, 2031, 17, 6225, 42213, 2655, 11296, 20792, 6225, 8592, 7649, 39894, 25957, 1211, 9154, 11296, 15035, 4032, 5016, 6027, 31371, 2423, 2031, 16, 4724, 1829, 3794, 995, 45865, 2031, 17, 307, 257, 26703, 1389, 6055, 2655, 5016, 4587, 4587, 3714, 3215, 10943, 3660, 2031, 16, 5551, 9381, 17082, 2288, 6055, 3794, 995, 45865, 2031, 17, 11933, 8848, 1863, 6225, 8592, 7649, 257, 2445, 283, 295, 2031, 16, 4724, 1829, 3794, 995, 45865, 283, 295, 2031, 17, 11933, 15730, 16712, 5016, 4587, 4587, 13672, 1829, 39896, 3714, 9957, 2423, 23223, 4032, 46958, 5016, 3660, 2423, 7649, 2031, 16, 5551, 9381, 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0.93154296875}, {"start": 929.62, "end": 929.76, "word": " f", "probability": 0.91064453125}, {"start": 929.76, "end": 929.88, "word": " of", "probability": 0.9345703125}, {"start": 929.88, "end": 931.02, "word": " x2", "probability": 0.9892578125}, {"start": 931.02, "end": 931.26, "word": " إذا", "probability": 0.813720703125}, {"start": 931.26, "end": 932.1, "word": " التحقق", "probability": 0.8665771484375}, {"start": 932.1, "end": 932.58, "word": " اللي", "probability": 0.88818359375}, {"start": 932.58, "end": 932.8, "word": " هي", "probability": 0.7119140625}, {"start": 932.8, "end": 933.16, "word": " مين", "probability": 0.508636474609375}, {"start": 933.16, "end": 933.4, "word": " ال", "probability": 0.748046875}, {"start": 933.4, "end": 933.88, "word": " decreasing", "probability": 0.78466796875}, {"start": 933.88, "end": 935.78, "word": " واضحة", "probability": 0.79302978515625}, {"start": 935.78, "end": 937.2, "word": " الان", "probability": 0.517242431640625}, {"start": 937.2, "end": 938.44, "word": " x1", "probability": 0.899658203125}, {"start": 938.44, "end": 938.68, "word": " أصغر", "probability": 0.9891357421875}, {"start": 938.68, "end": 938.78, "word": " من", "probability": 0.99462890625}, {"start": 938.78, "end": 939.34, "word": " x2", "probability": 0.99658203125}, {"start": 939.34, "end": 940.34, "word": " الآن", "probability": 0.6263427734375}, {"start": 940.34, "end": 941.1, "word": " we", "probability": 0.88427734375}, {"start": 941.1, "end": 941.62, "word": " apply", "probability": 0.9560546875}], "temperature": 1.0}, {"id": 40, "seek": 96420, "start": 945.1, "end": 964.2, "text": "الـ Mean Value Theorem لأن احنا هذه نبارع كورولرز أو نظريات على الـ Mean Value Theorem فالمتوقع أن نستخدم الـ Mean Value Theorem في أثباتها فهنا سنستخدم الـ Mean Value Theorem نطبق الـ Mean Value Theorem على الانترفل الموجود بين X1 و X2", "tokens": [6027, 39184, 12302, 39352, 440, 37956, 5296, 33456, 1975, 5016, 8315, 29538, 8717, 3555, 9640, 3615, 9122, 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"probability": 0.736083984375}, {"start": 946.96, "end": 947.5, "word": " احنا", "probability": 0.748291015625}, {"start": 947.5, "end": 947.76, "word": " هذه", "probability": 0.1513671875}, {"start": 947.76, "end": 948.16, "word": " نبارع", "probability": 0.440673828125}, {"start": 948.16, "end": 948.88, "word": " كورولرز", "probability": 0.550634765625}, {"start": 948.88, "end": 949.04, "word": " أو", "probability": 0.76953125}, {"start": 949.04, "end": 949.72, "word": " نظريات", "probability": 0.982177734375}, {"start": 949.72, "end": 950.32, "word": " على", "probability": 0.79150390625}, {"start": 950.32, "end": 950.5, "word": " الـ", "probability": 0.83251953125}, {"start": 950.5, "end": 950.64, "word": " Mean", "probability": 0.8076171875}, {"start": 950.64, "end": 950.94, "word": " Value", "probability": 0.984375}, {"start": 950.94, "end": 951.24, "word": " Theorem", "probability": 0.916015625}, {"start": 951.24, "end": 952.18, "word": " فالمتوقع", "probability": 0.85966796875}, {"start": 952.18, "end": 952.38, "word": " أن", "probability": 0.44189453125}, {"start": 952.38, "end": 952.96, "word": " نستخدم", "probability": 0.962646484375}, {"start": 952.96, "end": 953.12, "word": " الـ", "probability": 0.863525390625}, {"start": 953.12, "end": 953.24, "word": " Mean", "probability": 0.9013671875}, {"start": 953.24, "end": 953.54, "word": " Value", "probability": 0.986328125}, {"start": 953.54, "end": 953.76, "word": " Theorem", "probability": 0.91455078125}, {"start": 953.76, "end": 953.88, "word": " في", "probability": 0.466796875}, {"start": 953.88, "end": 954.48, "word": " أثباتها", "probability": 0.8984375}, {"start": 954.48, "end": 955.24, "word": " فهنا", "probability": 0.8792317708333334}, {"start": 955.24, "end": 955.98, "word": " سنستخدم", "probability": 0.89033203125}, {"start": 955.98, "end": 956.18, "word": " الـ", "probability": 0.84375}, {"start": 956.18, "end": 956.44, "word": " Mean", "probability": 0.85498046875}, {"start": 956.44, "end": 956.84, "word": " Value", "probability": 0.9853515625}, {"start": 956.84, "end": 957.28, "word": " Theorem", "probability": 0.92333984375}, {"start": 957.28, "end": 958.34, "word": " نطبق", "probability": 0.6715087890625}, {"start": 958.34, "end": 958.34, "word": " الـ", "probability": 0.505523681640625}, {"start": 958.34, "end": 958.66, "word": " Mean", "probability": 0.91748046875}, {"start": 958.66, "end": 959.0, "word": " Value", "probability": 0.98876953125}, {"start": 959.0, "end": 959.64, "word": " Theorem", "probability": 0.910400390625}, {"start": 959.64, "end": 960.4, "word": " على", "probability": 0.6494140625}, {"start": 960.4, "end": 961.28, "word": " الانترفل", "probability": 0.56768798828125}, {"start": 961.28, "end": 961.5, "word": " الموجود", "probability": 0.441162109375}, {"start": 961.5, "end": 961.68, "word": " بين", "probability": 0.57861328125}, {"start": 961.68, "end": 962.88, "word": " X1", "probability": 0.511962890625}, {"start": 962.88, "end": 963.6, "word": " و", "probability": 0.90283203125}, {"start": 963.6, "end": 964.2, "word": " X2", "probability": 0.8427734375}], "temperature": 1.0}, {"id": 41, "seek": 97729, "start": 964.71, "end": 977.29, "text": "طبعاً عندي F هذه X1 و X2 عبارة عن جزئية من مين؟ من الـ I إذا الـ F is differentiable عليها مادام differentiable إذا continuous علي close و differentiable علي open لإنها differentiable علي كلها", "tokens": [9566, 3555, 3615, 995, 14111, 18871, 16254, 479, 29538, 1783, 16, 4032, 1783, 17, 6225, 3555, 9640, 3660, 18871, 10874, 11622, 19986, 10632, 9154, 3714, 9957, 22807, 9154, 2423, 39184, 286, 11933, 15730, 2423, 39184, 479, 307, 819, 9364, 25894, 11296, 3714, 18513, 10943, 819, 9364, 11933, 15730, 10957, 25894, 1998, 4032, 819, 9364, 25894, 1269, 5296, 28814, 1863, 11296, 819, 9364, 25894, 28242, 11296], "avg_logprob": -0.30539771824172046, "compression_ratio": 1.5647058823529412, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 964.71, "end": 965.15, "word": "طبعاً", "probability": 0.796044921875}, {"start": 965.15, "end": 965.47, "word": " عندي", "probability": 0.5469970703125}, {"start": 965.47, "end": 965.83, "word": " F", "probability": 0.64013671875}, {"start": 965.83, "end": 966.75, "word": " هذه", "probability": 0.232177734375}, {"start": 966.75, "end": 967.15, "word": " X1", "probability": 0.8583984375}, {"start": 967.15, "end": 967.27, "word": " و", "probability": 0.94580078125}, {"start": 967.27, "end": 967.69, "word": " X2", "probability": 0.888671875}, {"start": 967.69, "end": 968.19, "word": " عبارة", "probability": 0.99560546875}, {"start": 968.19, "end": 968.39, "word": " عن", "probability": 0.990234375}, {"start": 968.39, "end": 968.81, "word": " جزئية", "probability": 0.9376220703125}, {"start": 968.81, "end": 968.99, "word": " من", "probability": 0.99072265625}, {"start": 968.99, "end": 969.27, "word": " مين؟", "probability": 0.634765625}, {"start": 969.27, "end": 969.39, "word": " من", "probability": 0.9638671875}, {"start": 969.39, "end": 969.65, "word": " الـ", "probability": 0.863525390625}, {"start": 969.65, "end": 969.81, "word": " I", "probability": 0.62353515625}, {"start": 969.81, "end": 970.33, "word": " إذا", "probability": 0.6507568359375}, {"start": 970.33, "end": 970.55, "word": " الـ", "probability": 0.6263427734375}, {"start": 970.55, "end": 970.71, "word": " F", "probability": 0.94140625}, {"start": 970.71, "end": 970.83, "word": " is", "probability": 0.52783203125}, {"start": 970.83, "end": 971.41, "word": " differentiable", "probability": 0.90576171875}, {"start": 971.41, "end": 971.75, "word": " عليها", "probability": 0.983642578125}, {"start": 971.75, "end": 972.05, "word": " مادام", "probability": 0.55908203125}, {"start": 972.05, "end": 972.79, "word": " differentiable", "probability": 0.810546875}, {"start": 972.79, "end": 973.31, "word": " إذا", "probability": 0.751220703125}, {"start": 973.31, "end": 973.97, "word": " continuous", "probability": 0.8564453125}, {"start": 973.97, "end": 974.35, "word": " علي", "probability": 0.494873046875}, {"start": 974.35, "end": 974.87, "word": " close", "probability": 0.28662109375}, {"start": 974.87, "end": 975.01, "word": " و", "probability": 0.98046875}, {"start": 975.01, "end": 975.61, "word": " differentiable", "probability": 0.952880859375}, {"start": 975.61, "end": 975.83, "word": " علي", "probability": 0.69384765625}, {"start": 975.83, "end": 976.05, "word": " open", "probability": 0.76025390625}, {"start": 976.05, "end": 976.27, "word": " لإنها", "probability": 0.68896484375}, {"start": 976.27, "end": 976.67, "word": " differentiable", "probability": 0.88134765625}, {"start": 976.67, "end": 976.83, "word": " علي", "probability": 0.83447265625}, {"start": 976.83, "end": 977.29, "word": " كلها", "probability": 0.8310546875}], "temperature": 1.0}, {"id": 42, "seek": 100608, "start": 977.94, "end": 1006.08, "text": "إذا شروط الـ Mean Value Theorem مالها متحققة إذا there exist then there exist C تنتمي إلى الفترة X1 و X2 يعني موجودة في الـ I اللي هي لإن هذي جزئية من مين من I there exist C such that F prime of C بساوي F of X1 أو F of X2 ناقص", "tokens": [28814, 15730, 13412, 32887, 9566, 2423, 39184, 12302, 39352, 440, 37956, 3714, 6027, 11296, 44650, 5016, 4587, 28671, 11933, 15730, 456, 2514, 550, 456, 2514, 383, 6055, 29399, 2304, 1829, 30731, 27188, 2655, 25720, 1783, 16, 4032, 1783, 17, 37495, 22653, 3714, 29245, 23328, 3660, 8978, 2423, 39184, 286, 13672, 1829, 39896, 5296, 28814, 1863, 8032, 8848, 1829, 10874, 11622, 19986, 10632, 9154, 3714, 9957, 9154, 286, 456, 2514, 383, 1270, 300, 479, 5835, 295, 383, 4724, 3794, 995, 45865, 479, 295, 1783, 16, 34051, 479, 295, 1783, 17, 8717, 995, 4587, 9381], "avg_logprob": -0.25847739773861905, "compression_ratio": 1.444954128440367, "no_speech_prob": 4.76837158203125e-07, "words": [{"start": 977.94, "end": 978.28, "word": "إذا", "probability": 0.6904296875}, {"start": 978.28, "end": 978.98, "word": " شروط", "probability": 0.7991536458333334}, 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"word": " من", "probability": 0.99267578125}, {"start": 1360.68, "end": 1360.9, "word": " 0", "probability": 0.89404296875}, {"start": 1360.9, "end": 1361.22, "word": " لكل", "probability": 0.958251953125}, {"start": 1361.22, "end": 1361.42, "word": " x", "probability": 0.9169921875}, {"start": 1361.42, "end": 1361.68, "word": " element", "probability": 0.62744140625}, {"start": 1361.68, "end": 1361.94, "word": " in", "probability": 0.85107421875}, {"start": 1361.94, "end": 1362.16, "word": " R", "probability": 0.97314453125}], "temperature": 1.0}, {"id": 57, "seek": 139065, "start": 1362.89, "end": 1390.65, "text": "I وجدنا إن X1 أصغر strictly من X2 تؤدي F of X2 أصغر strictly من F of X1 وهذا يعني strictly stick raising واضح، أه؟ الآن السؤال هل the converse is true؟ هل the converse is true؟ الآن يعني", "tokens": [40, 49610, 3215, 8315, 36145, 1783, 16, 5551, 9381, 17082, 2288, 20792, 9154, 1783, 17, 6055, 33604, 16254, 479, 295, 1783, 17, 5551, 9381, 17082, 2288, 20792, 9154, 479, 295, 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تكعيب بالشكل هذا هذا الان strictly ايه شمالها decreasing هذه F is strictly decreasing", "tokens": [3555, 3794, 995, 2407, 8608, 6027, 3555, 1783, 6055, 4117, 3615, 1829, 3555, 23758, 32748, 6027, 3660, 23758, 32748, 6027, 3660, 44945, 25957, 1211, 1895, 260, 655, 50113, 1211, 479, 295, 1783, 8608, 995, 2407, 8608, 6027, 3555, 1783, 6055, 4117, 3615, 1829, 3555, 20666, 8592, 28820, 23758, 23758, 2423, 7649, 20792, 1975, 1829, 3224, 13412, 2304, 6027, 11296, 23223, 29538, 479, 307, 20792, 23223], "avg_logprob": -0.3001893943909443, "compression_ratio": 1.6689655172413793, "no_speech_prob": 6.377696990966797e-06, "words": [{"start": 1415.22, "end": 1415.74, "word": "بساو", "probability": 0.733154296875}, {"start": 1415.74, "end": 1416.28, "word": " سالب", "probability": 0.70751953125}, {"start": 1416.28, "end": 1416.52, "word": " X", "probability": 0.6630859375}, {"start": 1416.52, "end": 1417.12, "word": " تكعيب", "probability": 0.92978515625}, {"start": 1417.12, "end": 1417.96, "word": 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لمنطقة لتعريف", "tokens": [6027, 39184, 5662, 34051, 2423, 39184, 23223, 4032, 43242, 3224, 48812, 17082, 9566, 3660, 6225, 2304, 2407, 15042, 14111, 29538, 27188, 1211, 3794, 5172, 3660, 3714, 29245, 23328, 3660, 37495, 22653, 8978, 5551, 4117, 49115, 9154, 3714, 7435, 6027, 24976, 2655, 3615, 16572, 5172, 8978, 2423, 16572, 46958, 1829, 9307, 8978, 9673, 1863, 41193, 4587, 13672, 1829, 3714, 43020, 13672, 1829, 31439, 6055, 33753, 26108, 34051, 8978, 9673, 5172, 40294, 32640, 13672, 1829, 3714, 43020, 6055, 33753, 26108, 32767, 1863, 9566, 28671, 5296, 2655, 3615, 16572, 5172], "avg_logprob": -0.12260297982191498, "compression_ratio": 1.8333333333333333, "no_speech_prob": 0.0002474784851074219, "words": [{"start": 1518.28, "end": 1518.52, "word": "الـ", "probability": 0.4833984375}, {"start": 1518.52, "end": 1518.86, "word": " increasing", "probability": 0.5703125}, {"start": 1518.86, "end": 1519.1, "word": " أو", "probability": 0.83349609375}, {"start": 1519.1, "end": 1519.22, 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95, "seek": 234478, "start": 2329.44, "end": 2344.78, "text": "نجرها بس من عندنا هنا هي اللي قلته بس نجرها من النظرية let f be continuous on the interval a و b إذا إيش مخصر دين إن ال function is continuous على فترة a و b و ناخد c عبارة عن interior point لمين", "tokens": [1863, 7435, 2288, 11296, 4724, 3794, 9154, 43242, 8315, 34105, 39896, 13672, 1829, 12174, 1211, 47395, 4724, 3794, 8717, 7435, 2288, 11296, 9154, 28239, 19913, 2288, 10632, 718, 283, 312, 10957, 322, 264, 15035, 257, 4032, 272, 11933, 15730, 11933, 1829, 8592, 3714, 9778, 9381, 2288, 11778, 9957, 36145, 2423, 2445, 307, 10957, 15844, 6156, 2655, 25720, 257, 4032, 272, 4032, 8717, 47283, 3215, 269, 6225, 3555, 9640, 3660, 18871, 10636, 935, 32767, 9957], "avg_logprob": -0.3145833277702332, "compression_ratio": 1.5053763440860215, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2329.44, "end": 2330.0, "word": "نجرها", "probability": 0.609130859375}, {"start": 2330.0, "end": 2330.22, "word": " بس", 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3460.35, "word": " Sin", "probability": 0.76953125}, {"start": 3460.35, "end": 3460.91, "word": " X", "probability": 0.9931640625}, {"start": 3460.91, "end": 3462.23, "word": " بيساوي", "probability": 0.80458984375}, {"start": 3462.23, "end": 3462.75, "word": " X", "probability": 0.98193359375}, {"start": 3462.75, "end": 3463.49, "word": " في", "probability": 0.70263671875}, {"start": 3463.49, "end": 3463.95, "word": " Cos", "probability": 0.96142578125}, {"start": 3463.95, "end": 3464.53, "word": " C", "probability": 0.96435546875}, {"start": 3464.53, "end": 3465.87, "word": " لكن", "probability": 0.9365234375}, {"start": 3465.87, "end": 3466.33, "word": " الآن", "probability": 0.6158447265625}, {"start": 3466.33, "end": 3466.57, "word": " أنا", "probability": 0.62451171875}, {"start": 3466.57, "end": 3466.87, "word": " عندي", "probability": 0.79833984375}, {"start": 3466.87, "end": 3467.33, "word": " Cos", "probability": 0.9453125}, {"start": 3467.33, "end": 3467.99, "word": " C", 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0.9443359375}, {"start": 3504.15, "end": 3504.51, "word": " zero", "probability": 0.3857421875}, {"start": 3504.51, "end": 3504.77, "word": " this", "probability": 0.67529296875}, {"start": 3504.77, "end": 3504.93, "word": " is", "probability": 0.958984375}, {"start": 3504.93, "end": 3505.55, "word": " trivial", "probability": 0.607421875}, {"start": 3505.55, "end": 3506.15, "word": " إذا", "probability": 0.7314453125}, {"start": 3506.15, "end": 3506.49, "word": " صار", "probability": 0.88427734375}, {"start": 3506.49, "end": 3506.81, "word": " صين", "probability": 0.6376953125}, {"start": 3506.81, "end": 3506.95, "word": " ال", "probability": 0.54296875}, {"start": 3506.95, "end": 3507.23, "word": " X", "probability": 0.95458984375}, {"start": 3507.23, "end": 3507.53, "word": " بين", "probability": 0.95458984375}, {"start": 3507.53, "end": 3507.87, "word": " نقص", "probability": 0.8719075520833334}, {"start": 3507.87, "end": 3508.13, "word": " X", "probability": 0.98681640625}, {"start": 3508.13, "end": 3508.31, "word": " و", "probability": 0.9501953125}, {"start": 3508.31, "end": 3508.71, "word": " X", "probability": 0.71435546875}, {"start": 3508.71, "end": 3509.67, "word": " لكل", "probability": 0.9677734375}, {"start": 3509.67, "end": 3510.17, "word": " X", "probability": 0.99072265625}, {"start": 3510.17, "end": 3510.89, "word": " أكبر", "probability": 0.974609375}, {"start": 3510.89, "end": 3511.09, "word": " أو", "probability": 0.96826171875}, {"start": 3511.09, "end": 3511.73, "word": " يساوي", "probability": 0.985595703125}, {"start": 3511.73, "end": 3512.87, "word": " صفر", "probability": 0.9715169270833334}, {"start": 3512.87, "end": 3513.25, "word": " و", "probability": 0.5703125}, {"start": 3513.25, "end": 3513.63, "word": " المرة", "probability": 0.7587890625}, {"start": 3513.63, "end": 3514.23, "word": " القادمة", "probability": 0.98974609375}, {"start": 3514.23, "end": 3514.47, "word": " ان", "probability": 0.5361328125}, {"start": 3514.47, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..967f837f77160ac49c7b07ea07fcee4ce25c4efa
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/VMrrs0aJBPM_raw.srt
@@ -0,0 +1,2420 @@
+1
+00:00:05,280 --> 00:00:12,080
+بسم الله الرحمن الرحيم المحاضرة الرابعة مساق تحليل
+
+2
+00:00:12,080 --> 00:00:18,780
+حقيقة 2 لطلبة الجامعة الإسلامية كلية العلوم قسم
+
+3
+00:00:18,780 --> 00:00:26,480
+الرياضيات نستكمل الحديث عن نظرية القيمة المتوسطة
+
+4
+00:00:26,480 --> 00:00:33,380
+أو the mean value theorem وما يتعلق بذلك تحدثنا
+
+5
+00:00:33,380 --> 00:00:40,130
+المرة الماضيةمعنى relative extrema سواء relative
+
+6
+00:00:40,130 --> 00:00:43,290
+maximum أو relative minimum أو بمعنى أخر ال
+
+7
+00:00:43,290 --> 00:00:47,450
+absolute ال local maximum و ال local minimum
+
+8
+00:00:47,450 --> 00:00:53,910
+معرفناها المرة الماضية و أتحدثنا أيضًا عن اللي هو
+
+9
+00:00:53,910 --> 00:00:58,650
+ال interior extreme theorem ال interior extreme
+
+10
+00:00:58,650 --> 00:01:03,430
+theorem اللي هي جوابتنا على السؤال اللي بنحكي فيه
+
+11
+00:01:03,770 --> 00:01:09,170
+أين يمكن إيجاد القيم العظمى أو القيم الصغرى
+
+12
+00:01:09,170 --> 00:01:14,730
+النظرية اللي بتحكي كانت ان لو كانت C عبارة عن
+
+13
+00:01:14,730 --> 00:01:20,610
+interior point اللي هي ال interval I اللي عندها
+
+14
+00:01:20,610 --> 00:01:27,220
+اللي هي ال function F ال derivative موجودةبقول لو
+
+15
+00:01:27,220 --> 00:01:32,140
+كان عند النقطة هذه في relative extremum سواء
+
+16
+00:01:32,140 --> 00:01:36,740
+minimum أو maximum لل F عنده نقطة C قطعا ال F
+
+17
+00:01:36,740 --> 00:01:42,900
+prime of C هتساوي 0 ومن هنا استقينا أن أين يمكن
+
+18
+00:01:42,900 --> 00:01:48,060
+إيجاد النقطة العظمى والنقطة الصغيرة المحلية وقلنا
+
+19
+00:01:48,060 --> 00:01:52,480
+هي في أحد ثلاث مناطق اللي هو أما F prime بتساوي 0
+
+20
+00:01:53,060 --> 00:01:58,080
+أو F prime does not exist للنقاط الداخلية طبعا أو
+
+21
+00:01:58,080 --> 00:02:02,120
+اللي هو ال end points فتلت مناطق اللي هي بنفحص
+
+22
+00:02:02,120 --> 00:02:07,360
+فيها اللي هي ال relative maximum أو ال relative
+
+23
+00:02:07,360 --> 00:02:11,400
+minimumوالـ Corollary اللي بعد هي اللي حكيته عنه
+
+24
+00:02:11,400 --> 00:02:16,400
+قبل بشوية الآن و بعدين اتحدثنا .. بدنا نتحدث على
+
+25
+00:02:16,400 --> 00:02:21,180
+ال roles theorem و قولنا ال roles theorem اللي هي
+
+26
+00:02:21,180 --> 00:02:25,840
+كانت بتعالج أنه لو كانت عندي function F من عند A و
+
+27
+00:02:25,840 --> 00:02:31,880
+B لعند R و كانت هذه ال function is continuous على
+
+28
+00:02:31,880 --> 00:02:37,830
+closed intervalاللي هي من a لعند b و
+
+29
+00:02:37,830 --> 00:02:42,030
+differentiable على ال open interval من a لb و أيضا
+
+30
+00:02:42,030 --> 00:02:47,230
+وضعنا شرط أن f of a بيساوي f of b بيساوي سفر
+
+31
+00:02:47,230 --> 00:02:54,210
+بقولنا ال f differentiable on a و bوالـ F
+
+32
+00:02:54,210 --> 00:03:00,110
+continuous on A وB كله مع بعض هذا أعطانا أنه there
+
+33
+00:03:00,110 --> 00:03:05,130
+exists C element in A وB بنلاقي النقطة C بين A وB
+
+34
+00:03:05,130 --> 00:03:10,530
+بحيث أن يكون ميل المماس عندها يساوي سفر أو F prime
+
+35
+00:03:10,530 --> 00:03:16,370
+of C بساوي سفر أو بمعنى آخر يوجد مماس عند هذه
+
+36
+00:03:16,370 --> 00:03:22,030
+النقطة يوازي محور الـ CNETبعد ذلك طبعا تحدثنا عن
+
+37
+00:03:22,030 --> 00:03:27,650
+الـ Mean Value Theorem الـ Mean Value Theorem اللي
+
+38
+00:03:27,650 --> 00:03:31,710
+كانت أيضا على الرسم خلينا نشوف الـ Mean Value
+
+39
+00:03:31,710 --> 00:03:35,590
+Theorem قالت إن لو كانت الـ function is continuous
+
+40
+00:03:35,590 --> 00:03:40,750
+على closed interval من A لB وكانت اللي هي
+
+41
+00:03:40,750 --> 00:03:43,470
+differentiable على الـ open interval A وB
+
+42
+00:03:49,160 --> 00:03:57,260
+الان الاستنتاج اللى بيكون تعميم للهو Rawls theorem
+
+43
+00:03:57,260 --> 00:04:03,350
+ايش هو الاستنتاج انهبنلاقي نقطة في المنطقة بين A و
+
+44
+00:04:03,350 --> 00:04:10,890
+B بكون المماس عندها موازلة القاطع اللي بين مين الـ
+
+45
+00:04:10,890 --> 00:04:14,850
+A و الـ B أو بمعنى آخر بكون عنده النقطة C اللي
+
+46
+00:04:14,850 --> 00:04:21,890
+لاجناها اللي هي F prime of C بساوي F of B minus F
+
+47
+00:04:21,890 --> 00:04:29,310
+of A على B minus A وهذا هو الذي يمثلميل القاطع بين
+
+48
+00:04:29,310 --> 00:04:34,170
+النقطة A أو بين النقطة B أو بين النقطة A و F of A
+
+49
+00:04:34,170 --> 00:04:38,430
+ونقطة B و F of B فبتكون هذه عبارة عن نظرية القيمة
+
+50
+00:04:38,430 --> 00:04:44,030
+المتوسطة اللي هي تعميم لـ Lowell's theorem بعدها
+
+51
+00:04:44,030 --> 00:04:51,790
+تحدثناعن اللي هو البعض النتائج قلنا إن لو كانت الـ
+
+52
+00:04:51,790 --> 00:04:56,930
+function F continuous على closed interval I وكانت
+
+53
+00:04:56,930 --> 00:05:00,410
+F differentiable على ال open interval وكانت اللي
+
+54
+00:05:00,410 --> 00:05:05,090
+هي F prime بتساوي 0 على كل الفترة A وB قلنا هذه
+
+55
+00:05:05,090 --> 00:05:09,430
+الدالة هي الدالة الثابتة هتكون The F is constant
+
+56
+00:05:09,430 --> 00:05:16,400
+على الفترة المغلقة A O Bأيضًا Corollary لهذه
+
+57
+00:05:16,400 --> 00:05:20,840
+النظرية برضه اللي هي معلومات معهودة من ال calculus
+
+58
+00:05:20,840 --> 00:05:26,340
+اللي هي أنه لو كان في عندنا دالتين F و G و هذولة
+
+59
+00:05:26,340 --> 00:05:30,600
+الدالتين اللي عندنا اللي هن .. اللي هن نفس المشتقة
+
+60
+00:05:30,600 --> 00:05:36,380
+على الفترة A و B Openبكون عندى الفرق بين اللى هى
+
+61
+00:05:36,380 --> 00:05:39,460
+الـ F و الـ G هو عبارة عن ثابت أو بمعنى آخر F
+
+62
+00:05:39,460 --> 00:05:44,880
+بتساوي G زائد C for some constant C على الفترة I
+
+63
+00:05:44,880 --> 00:05:51,160
+بعد هيك اللى دخلنا و أعرفنا شو معنات F من I لعند R
+
+64
+00:05:51,160 --> 00:05:55,780
+is increasing على ال interval قلنا أن F is
+
+65
+00:05:55,780 --> 00:06:03,570
+increasing إذا كانت اللى هىأخدنا أي نقطتين في الـ
+
+66
+00:06:03,570 --> 00:06:09,870
+I X1 و X2 بحقق ان X1 أصغر من X2 إذا قدلنا هذا أن
+
+67
+00:06:09,870 --> 00:06:13,430
+أف X1 أصغر أو أسوأ من أف X2 من جول عن الـ function
+
+68
+00:06:13,430 --> 00:06:19,430
+A شمالها is increasingالان اللي هو similar لتعريف
+
+69
+00:06:19,430 --> 00:06:24,190
+الـ decreasing إن X1 أصغر من X2 تؤدي إلى F of X1
+
+70
+00:06:24,190 --> 00:06:29,230
+أكبر أو يساوي F of X2 بتكون اليد ده اللي أشمالها
+
+71
+00:06:29,230 --> 00:06:33,970
+is decreasing على هذه الفترةفي المقابل أيضًا في
+
+72
+00:06:33,970 --> 00:06:36,810
+تعريف حاجة اسمها برضه ذكرناه هو strictly
+
+73
+00:06:36,810 --> 00:06:40,230
+increasing أو strictly decreasing strictly
+
+74
+00:06:40,230 --> 00:06:44,070
+increasing أو strictly decreasing نحكي عن ال
+
+75
+00:06:44,070 --> 00:06:48,610
+strictly increasing بتكون اللي هي تحقق مالي لو x1
+
+76
+00:06:48,610 --> 00:06:54,590
+و x2 في ال I و x1 أصغر من x2 هذا بيقديلي إلى f of
+
+77
+00:06:54,590 --> 00:07:00,530
+x1 strictly أصغر من f of x2 مع عدم إمكانية
+
+78
+00:07:00,530 --> 00:07:06,340
+المساويةالان similarly for الـ F is اللي هو
+
+79
+00:07:06,340 --> 00:07:10,820
+strictly decreasing هذا اتحدثنا فيه المرة الماضية
+
+80
+00:07:10,820 --> 00:07:14,240
+النظرية
+
+81
+00:07:14,240 --> 00:07:17,860
+اللي هي let F من I لعند R be differentiable اللي
+
+82
+00:07:17,860 --> 00:07:23,100
+هي أوصلنا عند هذه النظرية خلينا نشوف هذه النظرية و
+
+83
+00:07:23,100 --> 00:07:30,900
+نتفحص إيش هي نظرية بتقول و من ثم نشوف لماذااللي هو
+
+84
+00:07:30,900 --> 00:07:38,960
+هي صحيحة عندي
+
+85
+00:07:38,960 --> 00:07:46,440
+نظرية بتقوللي theorem 6.2.7 اللي بتحكي النظرية
+
+86
+00:07:46,440 --> 00:07:54,580
+بتقوللي لو كان عندي F let F من I لعنب R be
+
+87
+00:07:54,580 --> 00:08:05,490
+differentiable onthe interval .. the interval I
+
+88
+00:08:05,490 --> 00:08:11,910
+فرضنا إن F is differentiable على الinterval I الآن
+
+89
+00:08:11,910 --> 00:08:17,690
+اللي بدأ أقوله عبارة عن test بنستعمل ال test في
+
+90
+00:08:17,690 --> 00:08:22,030
+معرفة إن F is increasing أو لأ و F is decreasing
+
+91
+00:08:22,030 --> 00:08:27,530
+أو لأ بنقول F is increasing
+
+92
+00:08:29,130 --> 00:08:36,650
+on I إيش الفحص؟ if and only if F prime of X أكبر
+
+93
+00:08:36,650 --> 00:08:44,130
+أو يساوي سفر لكل X موجودة في الـ I والآن F is
+
+94
+00:08:44,130 --> 00:08:52,450
+decreasing on I if and only if F prime of X أصغر
+
+95
+00:08:52,450 --> 00:08:57,750
+أو يساوي سفر لكل X element in I هذه هي النظرية
+
+96
+00:08:57,750 --> 00:09:04,430
+اللي عندناو هنبرهن اللي هو A أو B في الكتاب نبرهن
+
+97
+00:09:04,430 --> 00:09:09,370
+A خلّينا نبرهن مين نبرهن B و هتكون A similarly for
+
+98
+00:09:09,370 --> 00:09:17,550
+B الان بي .. بدنا نفترض suppose that
+
+99
+00:09:17,550 --> 00:09:30,710
+F is decreasing on IF is decreasing on I بدنا ..
+
+100
+00:09:30,710 --> 00:09:36,650
+بدنا نثبت الآن أن الـ F prime of X اللي هي أصغر أو
+
+101
+00:09:36,650 --> 00:09:43,190
+يساوي مين سفر طبعاً F prime of X في النهاية إيش
+
+102
+00:09:43,190 --> 00:09:47,930
+هي؟ عشان نشوف كيف بنفكر أول إشي F prime of X بساوي
+
+103
+00:09:47,930 --> 00:09:55,080
+limit F of Xأو f prime of c مثلا for arbitrary c f
+
+104
+00:09:55,080 --> 00:10:00,820
+of x نقص f of c على x minus c لما x تروح لل c إذا
+
+105
+00:10:00,820 --> 00:10:05,520
+ال limit هذا أثبتناه أصغر أو يساوي سفر معناته صارت
+
+106
+00:10:05,520 --> 00:10:12,030
+اللي هي ال f prime أصغر أو يساوي سفرالان عندى انا
+
+107
+00:10:12,030 --> 00:10:16,990
+لو اثبتت هذا المقدار اصغر او يساوي سفر في المنطقة
+
+108
+00:10:16,990 --> 00:10:20,870
+حول الـC بقدر اقول ان ال limit على طول ايه اشماله
+
+109
+00:10:20,870 --> 00:10:27,550
+اصغر او يساوي ايه سفر ده نشوف كيف عندى الان بدي
+
+110
+00:10:27,550 --> 00:10:32,170
+اثبت انها هذه F prime اصغر او يساوي سفر على كل X
+
+111
+00:10:32,170 --> 00:10:39,450
+element in I بدي اقول let C be an arbitrary
+
+112
+00:10:41,100 --> 00:10:47,160
+and fixed in I يعني بتحكي عنها عن C لكن هي
+
+113
+00:10:47,160 --> 00:10:54,780
+arbitrary لكن بحكي عن C معينة الآن عندي يا X أكبر
+
+114
+00:10:54,780 --> 00:11:01,800
+من C يا X أشمالها أصغر من Cماشي الحال ليش أخدت
+
+115
+00:11:01,800 --> 00:11:04,160
+أكبر أو أصغر؟ لأن عمّالي بيداخد limit فالـ X ده
+
+116
+00:11:04,160 --> 00:11:09,060
+تساوي C في حالة الـ C اللي هو الأمر اللي هيكون
+
+117
+00:11:09,060 --> 00:11:13,180
+اللي هي عبارة
+
+118
+00:11:13,180 --> 00:11:17,520
+عن الـ trivial case نشوف كيف، إيش اللي بنحكيه؟
+
+119
+00:11:17,520 --> 00:11:23,440
+الآن X أكبر من C عندي إيش مفترضين إحنا؟ F is
+
+120
+00:11:23,440 --> 00:11:30,920
+decreasing لذا then F of Xأصغر أو يساوي F of C
+
+121
+00:11:30,920 --> 00:11:38,100
+مظبوط أه بصير عندي الحسب اللي كتبته قبل و شوية F
+
+122
+00:11:38,100 --> 00:11:47,680
+of X نقص F of C اللي هو بساوي على X minus C اللي
+
+123
+00:11:47,680 --> 00:11:52,290
+هو هذا سالبf of x نقص f of z أصغر أو أكبر من صفر و
+
+124
+00:11:52,290 --> 00:11:57,110
+ال x minus z أشماله أكبر من صفر هذا الأن أكبر
+
+125
+00:11:57,110 --> 00:12:02,650
+أسالب أو صفر وهذا موجب إذا حيكون أصغر أو يساوي أيش
+
+126
+00:12:02,650 --> 00:12:08,760
+صفرواضح اه Similarly الحالة التانية من X هذا لكل
+
+127
+00:12:08,760 --> 00:12:12,800
+ال Xات اللي مين اللي أكبر من C طبعا وال Xات أنا من
+
+128
+00:12:12,800 --> 00:12:16,740
+وين باخدها من I على كل شغلي في داخل من ال interval
+
+129
+00:12:16,740 --> 00:12:21,320
+I يعني Xات اللي بشتغل فيها داخل I الان X أصغر من C
+
+130
+00:12:21,320 --> 00:12:24,900
+أكيد similarly بنفس الطريقة مدام ال decreasing
+
+131
+00:12:24,900 --> 00:12:31,580
+بيصير F of X أكبر أو يساوي F of C بيصير عندي الآن
+
+132
+00:12:31,580 --> 00:12:40,060
+اذا F of Xنقص F of C وهو موجبة وهذه X minus C
+
+133
+00:12:40,060 --> 00:12:44,880
+سالبة إذا هذولة على X minus C برضه هتطلع إيه
+
+134
+00:12:44,880 --> 00:12:49,720
+شمالها أصغر أو يساوي سفر إذا صار عندي من الجهتين
+
+135
+00:12:49,720 --> 00:12:55,660
+هنا for any X
+
+136
+00:12:55,660 --> 00:13:03,040
+element in I و X لا تساوي C أنا أصلا ملزمني حوالين
+
+137
+00:13:03,040 --> 00:13:09,810
+الـ Cلكن هذا كله كلام صحيح لـ X ∈ I X ≤ C طالع
+
+138
+00:13:09,810 --> 00:13:17,350
+عندي اللي هو F of X نعقص F of C على X minus C أصغر
+
+139
+00:13:17,350 --> 00:13:23,610
+و سوى 0 الآن مدام هذا صحيح لكل X إذا ال limitهذا
+
+140
+00:13:23,610 --> 00:13:27,530
+المقدار لما X تروح للـ C برضه هيظل أصغر أو يساوي
+
+141
+00:13:27,530 --> 00:13:33,950
+سفر وهذا هو إيش اسمه F prime of C يعني أثبتنا F
+
+142
+00:13:33,950 --> 00:13:38,530
+prime of C أصغر أو يساوي السفر وعند C كانت
+
+143
+00:13:38,530 --> 00:13:42,550
+arbitrary and fixed in I إذا صار بناء على هذا
+
+144
+00:13:42,550 --> 00:13:49,050
+الحديث صار F prime of X أصغر أو يساوي سفر لكل X
+
+145
+00:13:49,050 --> 00:13:54,760
+element in I لأنه كانت C is arbitraryهذا اللي هو
+
+146
+00:13:54,760 --> 00:14:00,440
+الجزء الأول من النظرية من اللي هو B أنه بدنا الأن
+
+147
+00:14:00,440 --> 00:14:03,180
+نيجي ل conversely بدنا نفترض هذا أصحى و أصحى و صفر
+
+148
+00:14:03,180 --> 00:14:09,040
+و نوصل لمين لل F is decreasing نيجي الأن
+
+149
+00:14:09,040 --> 00:14:19,240
+conversely و نشوف كيف نبرهن conversely الأن
+
+150
+00:14:19,240 --> 00:14:25,030
+converselyطبعاً ببرهم في AVP برضه انا conversely
+
+151
+00:14:25,030 --> 00:14:30,810
+suppose that
+
+152
+00:14:32,170 --> 00:14:39,270
+F prime of X أصغر أو سوى سفر لكل X موجودة في الآية
+
+153
+00:14:39,270 --> 00:14:43,790
+ماشي الحال شوفوا صلى الله عليه الصلاة والسلام أنا
+
+154
+00:14:43,790 --> 00:14:47,810
+بدأ أثبت أن F is decreasing يعني بدأ أفترض أن X
+
+155
+00:14:47,810 --> 00:14:53,650
+أصغر أو أخد أي قيمتين X واحد و X اتنين element
+
+156
+00:14:53,650 --> 00:14:59,250
+اتنينأخدت أي two elements element x1 وx2 element
+
+157
+00:14:59,250 --> 00:15:06,850
+in I تحقق such that x1 أصغر من مين من x2 أصغر أو
+
+158
+00:15:06,850 --> 00:15:10,570
+يساوي x .. x1 أصغر أو يساوي x2 بس في حالة ال x1
+
+159
+00:15:10,570 --> 00:15:14,490
+بيساوي x2 ما هي automatic F of x1 بيساوي F of x2
+
+160
+00:15:14,490 --> 00:15:19,920
+يعني متحققةإذن الان x1 أصغر من x2 عملتها strictly
+
+161
+00:15:19,920 --> 00:15:23,720
+عشان بتعمل منها interval وحالتي ال x1 بيساوي x2 is
+
+162
+00:15:23,720 --> 00:15:28,000
+a trivial case تتحقق مدامة x1 أصغر تساوي x2 إذن
+
+163
+00:15:28,000 --> 00:15:32,100
+عشان a function f of x1 بيساوي f of x2 إذا التحقق
+
+164
+00:15:32,100 --> 00:15:39,340
+اللي هي مين ال decreasing واضحة الان x1 أصغر من x2
+
+165
+00:15:39,340 --> 00:15:41,620
+الآن we apply
+
+166
+00:15:45,100 --> 00:15:48,880
+الـ Mean Value Theorem لأن احنا هذه نبارع كورولرز
+
+167
+00:15:48,880 --> 00:15:52,380
+أو نظريات على الـ Mean Value Theorem فالمتوقع أن
+
+168
+00:15:52,380 --> 00:15:55,240
+نستخدم الـ Mean Value Theorem في أثباتها فهنا
+
+169
+00:15:55,240 --> 00:15:59,000
+سنستخدم الـ Mean Value Theorem نطبق الـ Mean Value
+
+170
+00:15:59,000 --> 00:16:05,470
+Theorem على الانترفل الموجود بين X1 و X2طبعاً عندي
+
+171
+00:16:05,470 --> 00:16:10,330
+F هذه X1 و X2 عبارة عن جزئية من مين؟ من الـ I إذا
+
+172
+00:16:10,330 --> 00:16:12,790
+الـ F is differentiable عليها مادام differentiable
+
+173
+00:16:12,790 --> 00:16:16,050
+إذا continuous علي close و differentiable علي open
+
+174
+00:16:16,050 --> 00:16:19,240
+لإنها differentiable علي كلهاإذا شروط الـ Mean
+
+175
+00:16:19,240 --> 00:16:24,580
+Value Theorem مالها متحققة إذا there exist then
+
+176
+00:16:24,580 --> 00:16:31,060
+there exist C تنتمي إلى الفترة X1 و X2 يعني موجودة
+
+177
+00:16:31,060 --> 00:16:35,220
+في الـ I اللي هي لإن هذي جزئية من مين من I there
+
+178
+00:16:35,220 --> 00:16:43,480
+exist C such that F prime of C بساوي F of X1 أو F
+
+179
+00:16:43,480 --> 00:16:55,670
+of X2 ناقصF of X1 على X2 ناقص X1 واضح أه؟ أضرب
+
+180
+00:16:55,670 --> 00:17:00,510
+طرفين في وسطين أضرب هذه في هذه بيصير عندي F of X2
+
+181
+00:17:00,510 --> 00:17:08,170
+ناقص F of X1 اللي هو بيساوي X2 ناقص X1 في F prime
+
+182
+00:17:08,170 --> 00:17:16,190
+of C الآن عندي X2 ناقص X1 موجبةمظبوط؟ و F prime of
+
+183
+00:17:16,190 --> 00:17:20,290
+C اللي هي سالبة لإنه C واحدة من نقاط ال I و F
+
+184
+00:17:20,290 --> 00:17:24,890
+prime أصغر من سفر فيها كلها إذا صار عندي هذه سالبة
+
+185
+00:17:24,890 --> 00:17:30,150
+و هذه موجبة إذا حاصل ضربهم إيه شماله أصغر أو يساوي
+
+186
+00:17:30,150 --> 00:17:40,640
+سفر إذا صار عندي F of X2 أصغر أو يساوي Fof X1 الان
+
+187
+00:17:40,640 --> 00:17:45,900
+بدنا ب X1 أصغر من X2 وانتهينا ب F of X2 أصغر من F
+
+188
+00:17:45,900 --> 00:17:51,260
+of X1 but .. but .. but .. احنا بنعرف خلينا نكملها
+
+189
+00:17:51,260 --> 00:17:56,300
+كدا من اللي حكيناه في الأول X2 بيساوي X1 implies F
+
+190
+00:17:56,300 --> 00:18:04,100
+of X2 بتساوي F of X1 مظبوط؟ الان من هذه اللي عندي
+
+191
+00:18:04,100 --> 00:18:10,460
+الحقيقة هذهومن اللي أثبتته من هنا لعند هنا التنتين
+
+192
+00:18:10,460 --> 00:18:20,280
+together بنستنتج أنه if x1 أصغر أو يساوي x2 then f
+
+193
+00:18:20,280 --> 00:18:23,660
+of x2
+
+194
+00:18:23,660 --> 00:18:32,800
+أصغر أو يساوي fof x واحد هذا فيضوء أن f prime of x
+
+195
+00:18:32,800 --> 00:18:37,220
+أصغر أو أشهر أو شفر لكل x element in I وهذا يعني
+
+196
+00:18:37,220 --> 00:18:50,740
+أن hence f is decreasing on mean on I واضح طيب
+
+197
+00:18:50,740 --> 00:18:58,280
+الآن بدنا نعرف حاجة اسمهاStrictly increasing
+
+198
+00:18:58,280 --> 00:19:04,300
+Strictly increasing قبل شوية قلتها أنه بنقول أن F
+
+199
+00:19:04,300 --> 00:19:09,140
+is strictly increasing إذا كانت اللي هي strictly
+
+200
+00:19:09,140 --> 00:19:14,600
+increasing بديش أمسح حد لإن أنا اللي بديه بعد شوية
+
+201
+00:19:14,600 --> 00:19:23,720
+خليني أمسح النظرية برهنة بيه أنا هو ماشي بدي بس من
+
+202
+00:19:23,720 --> 00:19:30,890
+الآن X بنقول Fis strictly increasing إيش معناته
+
+203
+00:19:30,890 --> 00:19:38,470
+إنه X هنا يكافئ إن X1 أصغر من X2 لل X1 وال X2 اللي
+
+204
+00:19:38,470 --> 00:19:46,790
+في I such that يؤدي F of X1 strictly أصغر من F of
+
+205
+00:19:46,790 --> 00:19:53,580
+X2 هذا يعني على بعضه إن F is strictlyincreasing
+
+206
+00:19:53,580 --> 00:19:58,440
+similarly for decreasing for strictly decreasing f
+
+207
+00:19:58,440 --> 00:20:04,360
+is strictly decreasing لما x1 أصغر من x2 تؤدي إلى
+
+208
+00:20:04,360 --> 00:20:10,160
+f of x1 أكبر strictly من مين من f of x2 هذا معناته
+
+209
+00:20:10,160 --> 00:20:16,640
+strictly أيش decreasing الآن السؤال اللي بطرحه
+
+210
+00:20:16,640 --> 00:20:23,380
+الآن هو بقولنيإن انت حكيت في حالة الـ increasing F
+
+211
+00:20:23,380 --> 00:20:27,980
+is increasing in I if and only if F' أصغر أوي
+
+212
+00:20:27,980 --> 00:20:31,960
+يساوي 0 لكل X element in I بقول لي هل هذا الكلام
+
+213
+00:20:31,960 --> 00:20:37,020
+بضل صحيح في حالة ال strictly increasing يعني لو
+
+214
+00:20:37,020 --> 00:20:43,480
+كانت بسأل السؤال لو كانت F is strictly
+
+215
+00:20:45,290 --> 00:20:49,790
+increasing او decreasing اللي بدك هي increasing او
+
+216
+00:20:49,790 --> 00:20:57,630
+decreasing هل هذا بيكون if and only if F prime of
+
+217
+00:20:57,630 --> 00:21:04,270
+X أصغر strictly من 0 لكل X element in I ولا لأ؟
+
+218
+00:21:04,270 --> 00:21:12,010
+الآن لو كان F prime أصغر strictly من 0 ل X element
+
+219
+00:21:12,010 --> 00:21:17,620
+in Iاللي هو الـ conversely هذا اللي هو بيعطينا F's
+
+220
+00:21:17,620 --> 00:21:22,620
+strictly decreasing يعني الاتجاه هذا متوقعين أنه
+
+221
+00:21:22,620 --> 00:21:27,500
+إحنا إيه شماله يكون صحيح يعني متوقعين أنه لو F
+
+222
+00:21:27,500 --> 00:21:31,520
+prime أصغر strictly من Zero لكل X element of I
+
+223
+00:21:31,520 --> 00:21:36,260
+يعطينا F's strictly أشماله decreasingوزيه برضه الـ
+
+224
+00:21:36,260 --> 00:21:41,580
+increasing لو كانت F' أكبر strictly من 0 بيعطينا F
+
+225
+00:21:41,580 --> 00:21:46,380
+is strictly increasing هذا متوقع أن يكون صحيح
+
+226
+00:21:46,380 --> 00:21:53,440
+والبرهان هيه موجود علّه لسه ما شطبطوش لسه الآن بدأ
+
+227
+00:21:53,440 --> 00:21:56,780
+افترض أنه بدل ما هي أكبر أصغر أو أسعار بدأ افترضها
+
+228
+00:21:56,780 --> 00:22:02,500
+أشمالهاStrictly أصغر ماشي الحال الان بفترض ان X1
+
+229
+00:22:02,500 --> 00:22:07,120
+أصغر من X2 و بطبق كل اللي عملته و لعند ال hand
+
+230
+00:22:07,120 --> 00:22:14,860
+بشطب هذا بالديشيا و بقف ال hand و بقولبما أنه هذا
+
+231
+00:22:14,860 --> 00:22:18,480
+المقدار f prime of c اللي هي f prime of x strictly
+
+232
+00:22:18,480 --> 00:22:22,840
+أصغر من 0 يعني مافي إمكانية أن تساوي 0 وهذا x2
+
+233
+00:22:22,840 --> 00:22:28,320
+ناقص x1 من هذا برضه strictly أكبر من 0 إذا صار هذا
+
+234
+00:22:28,320 --> 00:22:32,520
+عبارة عن strictly أصغر من 0 ولا يوجد إمكانية
+
+235
+00:22:32,520 --> 00:22:36,620
+للمساوية إذا بمعنى آخر صار هذا المقدار strictly
+
+236
+00:22:36,620 --> 00:22:40,680
+أصغر من هذا يعني لما كانت f prime أصغر strictly من
+
+237
+00:22:40,680 --> 00:22:45,770
+0 لكل x element in RI وجدنا إن X1 أصغر strictly من
+
+238
+00:22:45,770 --> 00:22:51,010
+X2 تؤدي F of X2 أصغر strictly من F of X1 وهذا يعني
+
+239
+00:22:51,010 --> 00:22:59,690
+strictly stick raising واضح، أه؟ الآن السؤال هل
+
+240
+00:22:59,690 --> 00:23:07,830
+the converse is true؟ هل the converse is true؟
+
+241
+00:23:07,830 --> 00:23:10,650
+الآن يعني
+
+242
+00:23:13,960 --> 00:23:21,020
+الاتجاه هذا صحيح لما يقول الاتجاه الثاني هذا need
+
+243
+00:23:21,020 --> 00:23:26,500
+not to be true in general بدنا نجيب مثال ان تكون F
+
+244
+00:23:26,500 --> 00:23:30,780
+strictly decreasing لكن ال F براي مش أصغر strictly
+
+245
+00:23:30,780 --> 00:23:37,120
+من مين؟ من 0 على I خد الآن F of Xبساو سالب X تكعيب
+
+246
+00:23:37,120 --> 00:23:42,220
+هذا الدالة هذا الدالة بنعمل reference مثل F of X
+
+247
+00:23:42,220 --> 00:23:47,120
+ساو سالب X تكعيب بالشكل هذا هذا الان strictly ايه
+
+248
+00:23:47,120 --> 00:23:54,800
+شمالها decreasing هذه F is strictly decreasing
+
+249
+00:23:55,740 --> 00:24:03,860
+لأن لو كانت X1 أصغر من X2 أكيد X1 تكعيب strictly
+
+250
+00:24:03,860 --> 00:24:13,420
+أكبر من X2 إيه شمالها تكعيب آسف ناقص مظبوط الانصار
+
+251
+00:24:13,420 --> 00:24:18,320
+عندي الآن يعني اللي هي ال F عندي أنا strictly
+
+252
+00:24:18,320 --> 00:24:23,770
+decreasingbut طبعاً على كل مين على كل ال I اللي
+
+253
+00:24:23,770 --> 00:24:27,090
+أخدت منها لها خد أي I من A لعند B من ناقص خمسة
+
+254
+00:24:27,090 --> 00:24:31,410
+مثلا لعند خمسة على الفترة اللي بدك إياها but F
+
+255
+00:24:31,410 --> 00:24:40,250
+prime عند اللي هو نقطة ال zero إيش بساوي بساوي سفر
+
+256
+00:24:40,250 --> 00:24:45,150
+إذا اللي هو ماعطنيش الشرط إن F prime أصغر من سفر
+
+257
+00:24:45,150 --> 00:24:51,290
+على كل النقاط ال X اللي وين في ال I واضح أهالان
+
+258
+00:24:51,290 --> 00:24:58,210
+معطيني معرفلي
+
+259
+00:24:58,210 --> 00:25:04,150
+اللي هو شو معناته طبعا احنا عرفنا ال increasing و
+
+260
+00:25:04,150 --> 00:25:09,670
+ال decreasing على ال interval لأنها مقارنة بالنقاط
+
+261
+00:25:09,670 --> 00:25:14,510
+فالان بقولي طب كيف؟ ايش رايكم؟ بدنا نحكي عن ال
+
+262
+00:25:14,510 --> 00:25:18,860
+increasing او ال decreasing عند نقطةالـ increasing
+
+263
+00:25:18,860 --> 00:25:21,640
+أو الـ decreasing و عنده ضغطة عموماً هذه الفلسفة
+
+264
+00:25:21,640 --> 00:25:26,360
+موجودة يعني في أكثر من مجال للتعريف في الرياضيات
+
+265
+00:25:26,360 --> 00:25:31,380
+في المناطق اللي ممكن اللي هو تحتاج أو في المفاهيم
+
+266
+00:25:31,380 --> 00:25:37,020
+اللي ممكن تحتاج لمنطقة لتعريفاللي هو المفهوم
+
+267
+00:25:37,020 --> 00:25:41,400
+المقصود بمنطقة زي الـ interval اللي قبل بشوية يعني
+
+268
+00:25:41,400 --> 00:25:48,200
+مش عند نقطة لأ عند منطقة فبصير التعريف اللي هو أن
+
+269
+00:25:48,200 --> 00:25:53,920
+يتحقق المفهوم في الجوار ده النقطة اللي مذكورة يعني
+
+270
+00:25:53,920 --> 00:25:59,120
+بنقول أنه النقطة C تتمتع بهذه الخاصية إذا تمتع
+
+271
+00:25:59,120 --> 00:26:03,940
+الجوار بهذه الخاصية هذا فلسفة التعريف فالآن بدنا
+
+272
+00:26:03,940 --> 00:26:04,260
+نقول
+
+273
+00:26:09,560 --> 00:26:14,260
+F من I لعند R اللي عبارة عن function أي إن كانت
+
+274
+00:26:14,260 --> 00:26:18,260
+الـ function هذه واخدنا
+
+275
+00:26:18,260 --> 00:26:28,780
+C element in I بدنا نعرف شو معنات عند النقطة C عند
+
+276
+00:26:28,780 --> 00:26:33,240
+النقطة C شو معنات F is increasing at this point
+
+277
+00:26:33,240 --> 00:26:40,880
+بدنا نشوف شو معنات F is increasingat C element in
+
+278
+00:26:40,880 --> 00:26:48,420
+I معناته انه there exist هذا يكافئ there exist V
+
+279
+00:26:48,420 --> 00:26:53,440
+Delta of C يعني U ده الجوار حوالين من الـ C هذا
+
+280
+00:26:53,440 --> 00:27:03,210
+الجوار C ناقص دلتا و C زائد دلتا ابحث ان تكون Fis
+
+281
+00:27:03,210 --> 00:27:08,770
+increasing on this جوار أو on this neighbourhood
+
+282
+00:27:08,770 --> 00:27:14,770
+إذا تحقق الشرط هذا معناته أنه بنقول عن الـ
+
+283
+00:27:14,770 --> 00:27:18,450
+function F أشمالها إنها increasing at the point
+
+284
+00:27:18,450 --> 00:27:23,090
+mean C يعني بمعنى آخر إذا تحقق اللي هو there
+
+285
+00:27:23,090 --> 00:27:29,870
+exists V Delta of C such that if X1 و X2 element
+
+286
+00:27:29,870 --> 00:27:42,110
+in V Delta of C withX1 أصغر من X2 أصغر
+
+287
+00:27:42,110 --> 00:27:49,350
+من X2 أصغر
+
+288
+00:27:49,350 --> 00:27:54,840
+من X2يعني اتحققت الخاصية أو الـ increasing في جوار
+
+289
+00:27:54,840 --> 00:27:58,120
+حول الـ C هذا معناته اللي هو مين؟ اللي هو
+
+290
+00:27:58,120 --> 00:28:01,820
+increasing similarly for strictly increasing ..
+
+291
+00:28:01,820 --> 00:28:07,000
+strictly increasing .. strictly increasing X1 أصغر
+
+292
+00:28:07,000 --> 00:28:11,160
+من X2 يؤد F of X1 strictly أصغر من F of X2 في هذا
+
+293
+00:28:11,160 --> 00:28:15,480
+الجوار similarly for اللي هو مين؟ decreasing و
+
+294
+00:28:15,480 --> 00:28:20,150
+strictly decreasing بما يتلقى أمهمالتلاقم معناته
+
+295
+00:28:20,150 --> 00:28:25,490
+X1 أصغر من X2 بيعطينا F of X1 أصغر أكبر أو يساوي F
+
+296
+00:28:25,490 --> 00:28:31,890
+of X2 لـ Decreasing للي بيكون Strictly أكبر منها
+
+297
+00:28:31,890 --> 00:28:39,410
+طيب .. الآن السؤال .. السؤال اللي بيقول ليه .. طيب
+
+298
+00:28:39,410 --> 00:28:49,320
+.. الآن أنت قلت أنه لو كانت F prime أكبر من 0f
+
+299
+00:28:49,320 --> 00:28:57,040
+prime of x أكبر من 0 لكل x element in I اللي
+
+300
+00:28:57,040 --> 00:29:05,200
+بيعطيني اللي هو f is strictly increasing صح ولا
+
+301
+00:29:05,200 --> 00:29:10,560
+لأ؟ قلنا هيك معايا لكن قلنا ال .. ال .. ال
+
+302
+00:29:10,560 --> 00:29:13,600
+converse need not to be true in general جبنا مثال
+
+303
+00:29:15,110 --> 00:29:18,590
+أف of X بسوء X تكعيب للـ increasing أو ناقص X
+
+304
+00:29:18,590 --> 00:29:23,690
+تكعيب لل decreasingماشي الحال .. الآن هو بقول طيب
+
+305
+00:29:23,690 --> 00:29:26,330
+لأ أنا بديش أسأله عن الاتجاه الثاني لإنه خلاص احنا
+
+306
+00:29:26,330 --> 00:29:29,990
+شفنا الاتجاه الثاني عند اللي هو ال interval مش
+
+307
+00:29:29,990 --> 00:29:35,450
+متحقق فالآن بقوللي أنا بدأ أسأل السؤال على اللي هي
+
+308
+00:29:35,450 --> 00:29:39,750
+.. اللي هو هذا السؤال اللي اتحقق في ال interval هل
+
+309
+00:29:39,750 --> 00:29:44,790
+بتحقق عنده النقطة؟ يعني بمعنى أخر لو كانت عندي a
+
+310
+00:29:44,790 --> 00:29:53,390
+February at c أكبر من 0هل هذا يؤدي أن f is
+
+311
+00:29:53,390 --> 00:30:03,070
+strictly increasing on c أو at c؟ هل هذا يؤدي أن
+
+312
+00:30:03,070 --> 00:30:06,090
+لو كانت f' of c أكبر من 0 هل يؤدي إلى أن f is
+
+313
+00:30:06,090 --> 00:30:09,910
+strictly increasing at c؟ بقول لك، this need not
+
+314
+00:30:09,910 --> 00:30:14,430
+to be true in general وهذا هو exercise عشرة اللي
+
+315
+00:30:14,430 --> 00:30:18,660
+معاكم homework اللي هو أعطاني function g of xبساوي
+
+316
+00:30:18,660 --> 00:30:23,120
+x زي 2x تربيع sin 1 علي x إذا كانت x بتساوي سفر و
+
+317
+00:30:23,120 --> 00:30:27,780
+0 إذا كانت x بتساوي سفر و بقول لي لو جيت فحصة ال G
+
+318
+00:30:27,780 --> 00:30:32,960
+prime عند ال zero هتلاقيها أشمالها بتساوي 1 ماشي
+
+319
+00:30:32,960 --> 00:30:37,320
+الحال يعني بتساوي واحد يعني strictly أكبر منين من
+
+320
+00:30:37,320 --> 00:30:44,050
+سفر butولا جوار، there is no neighbourhood around
+
+321
+00:30:44,050 --> 00:30:52,850
+zero in which F is increasingأه فشي دير جي is not
+
+322
+00:30:52,850 --> 00:30:56,850
+increasing in any neighborhood of x بتساوي سفر
+
+323
+00:30:56,850 --> 00:31:02,770
+والتفاصيل ان شاء الله لما في ال discussion لما
+
+324
+00:31:02,770 --> 00:31:06,550
+نعمل discussion اللي هو ال section اللي هو 6 2
+
+325
+00:31:06,550 --> 00:31:11,210
+وتجربوا تحلوه في البيت في حال انتوا ماعرفتوش تحلوه
+
+326
+00:31:11,210 --> 00:31:15,670
+بنحكي فيه بتفاصيله في ال discussion ان شاء الله
+
+327
+00:31:22,930 --> 00:31:27,050
+لأ الجوار هتلاقي اللي هو .. اللي هو .. اللي هو
+
+328
+00:31:27,050 --> 00:31:32,830
+مستحيل يعني تلاقي اللي هو جوار تكون X1 أصغر من X2
+
+329
+00:31:32,830 --> 00:31:37,150
+و تضمن أن F of X1 أصغر من F of X2 لكل واحدة بتحقق
+
+330
+00:31:37,150 --> 00:31:43,010
+ال X1 أصغر من X2 يعني هتلاقي نقاط هتلاقي هل جت V
+
+331
+00:31:43,010 --> 00:31:48,510
+Delta of C مثلا لأ كانت Delta بساوة C نقص Delta و
+
+332
+00:31:48,510 --> 00:31:53,690
+C زائد Deltaماشي الحال .. الآن هتلاقي أنه .. أنه
+
+333
+00:31:53,690 --> 00:31:59,090
+مستحيل يعني أنه في أي .. لو مهما زغرت المسافة
+
+334
+00:31:59,090 --> 00:32:05,110
+بينهم يتلاقي X1 أصغر من X2 تضمن يتلاقي أف of X1
+
+335
+00:32:05,110 --> 00:32:10,050
+أكبر أو يساوي أف of X2 في حالة ال decreasing أو
+
+336
+00:32:10,050 --> 00:32:13,690
+أصغر أو يساوي في حالة ال increasing أو اللي بنحكي
+
+337
+00:32:13,690 --> 00:32:16,770
+عنه أش ماله strictly increasing يعني هتلاقي نقاط
+
+338
+00:32:16,770 --> 00:32:21,030
+جانب بعضوحدة أكبر من التانية و very close .. very
+
+339
+00:32:21,030 --> 00:32:25,870
+close اللي هو .. و تكون .. اللي هو واحدة اللي هو
+
+340
+00:32:25,870 --> 00:32:30,970
+اتلاقيها أكبر و التانية اتلاقيها نقطش أخر اه .. اه
+
+341
+00:32:30,970 --> 00:32:34,230
+.. اه .. في .. اه .. في .. اتلاقيهم يعني ال .. ال
+
+342
+00:32:34,230 --> 00:32:37,630
+.. ال .. ال .. طبعا هو يمكن السبب بالذبذبحان لو
+
+343
+00:32:37,630 --> 00:32:43,710
+لاحظت هدي sign واحد على X فال X اللي هو بتساوي
+
+344
+00:32:43,710 --> 00:32:48,720
+اتنين .. واحد على اتنين on byأو 2 and by هتلاقي
+
+345
+00:32:48,720 --> 00:32:52,740
+اللي هي إيش بيعملك سفر و إيش بيعملك واحد في حالة
+
+346
+00:32:52,740 --> 00:32:55,920
+ال F of prime و في حالة ال F العادية فهيسببلك اللي
+
+347
+00:32:55,920 --> 00:33:01,180
+هو مش هتظلها اللي هي ال .. اللي هي ال .. ال .. ال
+
+348
+00:33:01,180 --> 00:33:08,300
+.. القيم .. قيم هتلاقيها مستحيل إن X1 أصغر من X2 و
+
+349
+00:33:08,300 --> 00:33:12,580
+تظلها على نفس المنوال F of X1 أصغر من F of X2 يعني
+
+350
+00:33:12,580 --> 00:33:18,070
+ممكن تلاقيه لك إنه جدتينclosed لجوه هتلاقي بتحققش
+
+351
+00:33:18,070 --> 00:33:26,150
+هذه يا بتحقق مسواء يا بتحقق الأصغر ولو لجيتلك
+
+352
+00:33:26,150 --> 00:33:30,510
+تنتين بحقق الأكبر هتلاقي برضه very closed إيش بحقق
+
+353
+00:33:30,510 --> 00:33:36,810
+الأصغر ان
+
+354
+00:33:36,810 --> 00:33:45,360
+شاء الله لما نحل السؤال بتتضح الأمور تماماهذا اللي
+
+355
+00:33:45,360 --> 00:33:54,900
+هو من ضمن الأسئلة اللي معاكم homework سألنا
+
+356
+00:33:54,900 --> 00:34:08,980
+سؤال اللي هو وين يمكن إيجاد ال relative maximum و
+
+357
+00:34:08,980 --> 00:34:13,430
+relative minimumو جاوبنا عليه في النظرية اللي
+
+358
+00:34:13,430 --> 00:34:17,230
+حكيناها في البداية أنه توجد عند أحد ثلاث مناطق F'
+
+359
+00:34:17,450 --> 00:34:21,370
+بيساوي سفر F' لاتساوي سفر لإنها في حالة ال
+
+360
+00:34:21,370 --> 00:34:26,410
+interior points أو اللي هو المكان الثالث اللي هو
+
+361
+00:34:26,410 --> 00:34:32,210
+ال end points الآن عشان نكمل الحديث عن ال maximum
+
+362
+00:34:32,210 --> 00:34:39,370
+و ال minimum بدنا نفحصنفحص هل النقاط هذه مين فيها
+
+363
+00:34:39,370 --> 00:34:43,910
+عندها maximum و في مين فيها عندها minimum و مين لا
+
+364
+00:34:43,910 --> 00:34:48,530
+عندها maximum ولا عندها minimum و هو ال test اللي
+
+365
+00:34:48,530 --> 00:34:54,410
+انتوا أخدتهوا في اللي هو سواء في الثانوية أو حتى
+
+366
+00:34:54,410 --> 00:34:59,330
+في calculus ألف و عشان نساعدكم على انكم تتذكروا
+
+367
+00:34:59,330 --> 00:35:05,620
+انا بفضل انه نتذكروا من خلال الرسم الفحصساعتها
+
+368
+00:35:05,620 --> 00:35:12,780
+بيكون سهل وبكون انت بتعرف اللي هو تحدد ان هذا
+
+369
+00:35:12,780 --> 00:35:17,020
+النقطة في عندها maximum أو عندها minimum خلينا
+
+370
+00:35:17,020 --> 00:35:22,420
+نستذكر نستذكر من خلال الرسم النظرية اللي بدنا
+
+371
+00:35:22,420 --> 00:35:26,160
+نحكيها الآن اللي أخدناها في الثانوية أو أخدناها
+
+372
+00:35:26,160 --> 00:35:30,850
+اللي هي تغيير إشارة أفر براياللي هو من موجب إلى
+
+373
+00:35:30,850 --> 00:35:35,450
+سالب أو من سالب إلى موجب الان نرسم واحدة Maximum
+
+374
+00:35:35,450 --> 00:35:39,210
+وواحدة Minimum هذه النقطة هذه طبعا Maximum بس تذكر
+
+375
+00:35:39,210 --> 00:35:43,190
+في النظرية قاعدة الان هذه النقطة C الجوار اللي
+
+376
+00:35:43,190 --> 00:35:51,710
+حوالها C زائد Delta وC ناقص Delta مظبوط؟ طيب الان
+
+377
+00:35:51,710 --> 00:35:57,430
+عند النقاط المنطقة هذه كلهالاحظ مين المماس إيه
+
+378
+00:35:57,430 --> 00:36:01,710
+شماله بس ما بعمل المماس زاوية حادة إذا مينه إيه
+
+379
+00:36:01,710 --> 00:36:10,970
+شماله موجب يعني F prime of X أكبر أو يساوي سفر لكل
+
+380
+00:36:10,970 --> 00:36:16,950
+X وين موجودة في الجوار V Delta of C من الجهة هذه
+
+381
+00:36:16,950 --> 00:36:23,710
+بالتحديد في المنطقة C minus Delta OC واضح، مفهوم
+
+382
+00:36:24,520 --> 00:36:31,040
+الآن الـ F' هنا أو ميل المماس هنا بيعمل .. أو
+
+383
+00:36:31,040 --> 00:36:36,080
+المماس بيعمل زاوية منفرجة إذا الـ F' عند النقاط
+
+384
+00:36:36,080 --> 00:36:40,480
+هذه اللي هي اللي في المنطقة هذه اللي هي بيكون أصغر
+
+385
+00:36:40,480 --> 00:36:45,660
+أو يساوي سفر لكل X وهي الموجودةC C زائد دلتا وهذا
+
+386
+00:36:45,660 --> 00:36:49,660
+اللي كنا نقول عنه إذا بتتغير إشارة ال derivative
+
+387
+00:36:49,660 --> 00:36:55,940
+حول النقطة C من موجب إلى سالب هذا على بعضه بيعطيني
+
+388
+00:36:55,940 --> 00:37:05,700
+أن F has a local أو relative maximum at meanالـ C
+
+389
+00:37:05,700 --> 00:37:11,420
+Similarly for the other هذه النقطة هان C اللي هي
+
+390
+00:37:11,420 --> 00:37:16,980
+المنمة ماضح عند ميل المناس جابلها سالب لإنه زاوية
+
+391
+00:37:16,980 --> 00:37:22,420
+منفرجة وراح لمين لموجة لإنها زاوية حادة يعني بمعنى
+
+392
+00:37:22,420 --> 00:37:29,680
+آخر F prime of X أصغر أو يساوي 0 لكل X في الـ V أو
+
+393
+00:37:29,680 --> 00:37:36,160
+الجزء من C ما نزلت لعين C فاهمين عليها؟هي C نقص
+
+394
+00:37:36,160 --> 00:37:43,880
+دلتة C وC لعن C زائد دلتة و F prime of X أكبر أو
+
+395
+00:37:43,880 --> 00:37:49,620
+سوى 0 لكل X والي موجودة من C لعن C زائد دلتة
+
+396
+00:37:49,620 --> 00:37:59,200
+هذولتين مع بعض بيعطينا أن F has local minimum أو
+
+397
+00:37:59,200 --> 00:38:05,850
+relative minimum at Cفصار عندى الآن يا جماعة اللى
+
+398
+00:38:05,850 --> 00:38:11,730
+هو ال test التحول شرط الدال ال derivative من موجب
+
+399
+00:38:11,730 --> 00:38:18,410
+إلى سلب local maximum هذه النقاط اللى احنا اخدناه
+
+400
+00:38:18,410 --> 00:38:22,790
+من النظرية السابقة طبعا بنحكي عن ال interior
+
+401
+00:38:22,790 --> 00:38:28,850
+points و Fاللي هي اتحولت F prime من سالب إلى موجب
+
+402
+00:38:28,850 --> 00:38:33,870
+local maximum هذه اللي هي معلومة توجيهي أو معلومة
+
+403
+00:38:33,870 --> 00:38:40,730
+اللي هو calculus ألف الآن احنا بدنا نثبت صحة هذا
+
+404
+00:38:40,730 --> 00:38:41,770
+الحديث
+
+405
+00:38:49,440 --> 00:38:53,520
+نجرها بس من عندنا هنا هي اللي قلته بس نجرها من
+
+406
+00:38:53,520 --> 00:38:57,520
+النظرية let f be continuous on the interval a و b
+
+407
+00:38:57,520 --> 00:39:01,080
+إذا إيش مخصر دين إن ال function is continuous على
+
+408
+00:39:01,080 --> 00:39:04,780
+فترة a و b و ناخد c عبارة عن interior point لمين
+
+409
+00:39:05,450 --> 00:39:10,090
+للـ I Assume that F is differentiable إيه ع الفترة
+
+410
+00:39:10,090 --> 00:39:13,430
+A وC وC وB يعني مش لزمني إنها تكون differentiable
+
+411
+00:39:13,430 --> 00:39:16,170
+عند الـ C أصلا ممكن تكون مش differentiable و هي
+
+412
+00:39:16,170 --> 00:39:21,290
+نقطة بدنا نبحث عندها بس بالزمن هذه اللي هي من A ل
+
+413
+00:39:21,290 --> 00:39:25,490
+C و من C ل B then الآن
+
+414
+00:39:26,890 --> 00:39:31,430
+إذا كان هناك مقارنة V Delta of C Subset من I بحيث
+
+415
+00:39:31,430 --> 00:39:35,110
+أن F' بتكون أكبر أو سوى 0 قبلها في الفترة اللي
+
+416
+00:39:35,110 --> 00:39:39,830
+قبل، اللي هو و F' اتحولت إلى أصغر أو سوى 0، يعني
+
+417
+00:39:39,830 --> 00:39:44,750
+اتحولت الإشارة من موجب إلى سالب في الفترة الأولى
+
+418
+00:39:44,750 --> 00:39:48,210
+للفترة الثانية، هيكون عند relative maximum زي ما
+
+419
+00:39:48,210 --> 00:39:54,260
+حكيناهاالان في الثانية نفس الاشي بس بالعكس if
+
+420
+00:39:54,260 --> 00:39:57,480
+there is a neighborhood اللي هو كذا such that F'
+
+421
+00:39:57,940 --> 00:40:02,160
+أصغر من سفر هيه اللي هو زاوية منفرجة اتحول لF'
+
+422
+00:40:02,540 --> 00:40:06,820
+أكبر من سفر هيه بيعطينا عندي ايش ماله relative
+
+423
+00:40:06,820 --> 00:40:14,040
+minimum طيب ده عندي A و عندي BM مبرهنة خليني نبرهن
+
+424
+00:40:14,040 --> 00:40:18,440
+مين؟ B خليني نشوف مبرهن مع بعض اللي هو B
+
+425
+00:40:29,950 --> 00:40:33,710
+شوفوا صلوا ع النادي عليه الصلاة والسلام الان بيه
+
+426
+00:40:33,710 --> 00:40:39,410
+اللي هو عندي F prime معطيني F prime of X أصغر أو
+
+427
+00:40:39,410 --> 00:40:47,170
+سوى سفر صح لكل X في الفترة C minus Delta وC
+
+428
+00:40:47,170 --> 00:40:49,710
+ومعطيني أيضا
+
+429
+00:40:52,140 --> 00:40:57,040
+اللي تحت أنا بحكي عن اللي تحت بيه ال .. البعدها F
+
+430
+00:40:57,040 --> 00:41:02,980
+prime of X أكبر أشهر و سفر لكل X وين موجودة في
+
+431
+00:41:02,980 --> 00:41:09,740
+الفترة من C لـC زائد إيش؟ لـC زائد دلتا الآن إيش
+
+432
+00:41:09,740 --> 00:41:14,560
+غاية أثبت أن F has a relative minimum؟ يعني بدي
+
+433
+00:41:14,560 --> 00:41:21,340
+أثبت أن F of Xأكبر الـ Minimum أصغر أكبر أو يساوي
+
+434
+00:41:21,340 --> 00:41:28,380
+F of C لكل X وين موجودة في V Delta of C إذا أثبتت
+
+435
+00:41:28,380 --> 00:41:32,200
+هذا الكلام معناته صار عند في Neighborhood حوالي ال
+
+436
+00:41:32,200 --> 00:41:37,140
+F عند ال C أصغر واحدة فيهم يعني في عندها relative
+
+437
+00:41:37,140 --> 00:41:42,560
+minimumاللي هو في المنطقة هذه هقولك إنها بتتحقق
+
+438
+00:41:42,560 --> 00:41:45,700
+هذه وفي المنطقة هذه بتتحقق هذه معناها تقول لكل
+
+439
+00:41:45,700 --> 00:41:49,100
+النقاط اللي هان لكل النقاط اللي هان هذه بتتحقق
+
+440
+00:41:49,100 --> 00:41:54,740
+معايا؟ والكلام إن شاء الله سهل الآن طبيعي إحنا
+
+441
+00:41:54,740 --> 00:41:58,920
+قاعدين في جوه ال mean value theorem فأكيد هذا
+
+442
+00:41:58,920 --> 00:42:02,440
+جايبينه في نفس ال section عساس إنه إحنا نطبق ال
+
+443
+00:42:02,440 --> 00:42:07,220
+mean value theorem عشان اللي هو إن نصل لنتائجنا
+
+444
+00:42:07,770 --> 00:42:14,790
+الان x element in c minus delta of c، مظبوط؟ اذا
+
+445
+00:42:14,790 --> 00:42:18,590
+صار عندى الان طبعا انا غاية ان اجيب اللى هو علاقة
+
+446
+00:42:18,590 --> 00:42:23,610
+بين f of c و f of x فالان ال x في الفترة هذه
+
+447
+00:42:23,610 --> 00:42:28,410
+معناته انه انا بدي اطبق ال mean value theorem على
+
+448
+00:42:28,410 --> 00:42:32,790
+ال x وعلى ال c الفترة اللى فيهم يعني بدي اقولكن
+
+449
+00:42:32,790 --> 00:42:40,940
+consider the interval x و meanOc XOC
+
+450
+00:42:41,980 --> 00:42:47,220
+الـ file الـ data continuous على X وC، ليش؟ لأن
+
+451
+00:42:47,220 --> 00:42:51,120
+continuous على كل ال interval A وB اللي من ضمنها C
+
+452
+00:42:51,120 --> 00:42:55,680
+إذن، هاي أول شرط متحقق لـ mean value theorem
+
+453
+00:42:55,680 --> 00:42:59,900
+وأيضاً F is differentiable على ال open اللي هي
+
+454
+00:42:59,900 --> 00:43:02,940
+مستثنية الـ C بس و اللي بالزمن في ال mean value
+
+455
+00:43:02,940 --> 00:43:08,060
+theorem بس ال open إذن الشرطين المتحققات على هذه
+
+456
+00:43:08,060 --> 00:43:14,470
+الفترة لمين؟للـ Mean Value Theorem إذا نطبق الـ
+
+457
+00:43:14,470 --> 00:43:18,730
+Mean Value Theorem على هذا الانترال لكي نحصل على
+
+458
+00:43:18,730 --> 00:43:27,890
+ذلك اللي هو F there exist CX element in X وC such
+
+459
+00:43:27,890 --> 00:43:38,850
+that F of CX فبرايم of CX بساوي F of C ناقصF of X
+
+460
+00:43:38,850 --> 00:43:48,870
+على X C نقص X اللي هو اللي بالذنب هذا أمور جهزة
+
+461
+00:43:48,870 --> 00:43:57,090
+ليش جهزة من هنا إذا F of C نقص F of X بساوي C
+
+462
+00:43:57,090 --> 00:44:04,760
+minus Xفى F prime of CX الـ CX هى دى وين موجودة فى
+
+463
+00:44:04,760 --> 00:44:10,760
+الـ X و C يعنى هنا مدام هنا حسب المعطى الـ F prime
+
+464
+00:44:10,760 --> 00:44:15,380
+اللى هيش ماله سالبة إذا هذا أصغر أو يساوي سفروهذا
+
+465
+00:44:15,380 --> 00:44:20,260
+أكبر أو يساوي صفر مع بعض إيه شماله؟ أصغر أو يساوي
+
+466
+00:44:20,260 --> 00:44:24,240
+صفر إذا صار عندي هذا أصغر أو يساوي صفر أو بمعنى
+
+467
+00:44:24,240 --> 00:44:31,560
+آخر F of C أصغر أو يساوي F of X لمين؟ لكل X في هذه
+
+468
+00:44:31,560 --> 00:44:36,440
+الفترة لأن X كانت arbitrary فيها Similarly for
+
+469
+00:44:36,440 --> 00:44:43,700
+اللي هو الجزء الثاني بتاخد الآن X وين موجودة؟في
+
+470
+00:44:43,700 --> 00:44:48,520
+الـ C لـ C زائد دلتا و برضه اللي عملته فوق حطب
+
+471
+00:44:48,520 --> 00:44:54,400
+جهانة بس عالميا على الفترة C O X و نفس الحديث
+
+472
+00:44:54,400 --> 00:44:57,580
+الأولاني F is continuous هلأ و differentiable على
+
+473
+00:44:57,580 --> 00:45:01,480
+ال open then there exist اللي هو by mean value
+
+474
+00:45:01,480 --> 00:45:06,780
+theorem there exist C X C prime X element in C O X
+
+475
+00:45:06,780 --> 00:45:20,440
+such that F prime of Cx' بساوة f of x ناقص f of c
+
+476
+00:45:20,440 --> 00:45:27,800
+على x minus c بساوة هذا و زي ما عملنا بالظبط اللي
+
+477
+00:45:27,800 --> 00:45:31,040
+هو بيصير عندي أضرب طرفين في وسطين بيصير هنا x
+
+478
+00:45:31,040 --> 00:45:37,060
+minus c و هذه بتطلع فوق فبصير عندي هذه x minus c
+
+479
+00:45:37,060 --> 00:45:44,680
+موجبة و هذه f prime cxاللي هو موجبة X minus C
+
+480
+00:45:44,680 --> 00:45:49,780
+موجبة مظبوط و F prime اللي هي أكبر أو سواء صفر
+
+481
+00:45:49,780 --> 00:45:53,780
+فبصير هذا المقدر كله على بعض أشماله أكبر أو سواء
+
+482
+00:45:53,780 --> 00:45:58,400
+صفر يعني هذا المقدر أكبر أو سواء إيش صفر يعني من
+
+483
+00:45:58,400 --> 00:46:05,820
+هنا بصير عند F of X أكبر أو سواء F of إيش of C هذا
+
+484
+00:46:05,820 --> 00:46:11,760
+الكلام صحيح برضه لمين لكل X هناالأن في الجهتين أي
+
+485
+00:46:11,760 --> 00:46:16,040
+إن كان طلعت ال F of C أصغر أو يساوي F of X أصغر أو
+
+486
+00:46:16,040 --> 00:46:21,640
+يساوي F of X لكل X وين موجودة هنا و هنا يعني بمعنى
+
+487
+00:46:21,640 --> 00:46:25,980
+آخر لكل X وين موجودة في ال neighborhood يعني صارت
+
+488
+00:46:25,980 --> 00:46:33,020
+F of X أكبر أو يساوي F of C لكل X في ال V Delta of
+
+489
+00:46:33,020 --> 00:46:41,440
+C وهذا يعني أن F has a local minimum at meanالـ C
+
+490
+00:46:41,440 --> 00:46:44,640
+أي سؤال
+
+491
+00:47:03,950 --> 00:47:07,150
+عندي the converse of the theorem اللي فوقني not to
+
+492
+00:47:07,150 --> 00:47:10,690
+be true in general وهذا اللي هو عبارة عن مين عن
+
+493
+00:47:10,690 --> 00:47:15,090
+exercise 9 برضه exercise 9 من ضمن ال homework اللي
+
+494
+00:47:15,090 --> 00:47:18,090
+معاكم اللي هنتحدث فيه ان شاء الله في ال discussion
+
+495
+00:47:18,090 --> 00:47:22,590
+الآن بعض التطبيقات الأخرى على ال mean value
+
+496
+00:47:22,590 --> 00:47:29,600
+theorem وهذا التطبيقبعض التطبيقات اللي هي علمين
+
+497
+00:47:29,600 --> 00:47:34,120
+value theory ممكن اشي اخدنا مشابه له في ال
+
+498
+00:47:34,120 --> 00:47:40,300
+calculus لكن خلينا نشوف هنا يعني اش اللي بيقوله
+
+499
+00:47:40,300 --> 00:47:47,700
+عندنا التطبيق الأول اللي
+
+500
+00:47:47,700 --> 00:47:54,080
+بيقوله ما هي ليه example example اللي هو
+
+501
+00:48:00,540 --> 00:48:12,120
+show that اكبر
+
+502
+00:48:12,120 --> 00:48:18,080
+او ساوي واحد زائد X لكل X element in R هذا واحد
+
+503
+00:48:18,080 --> 00:48:24,500
+اتنين اللي هو نفس ال W او A اللي هو E to the X
+
+504
+00:48:25,160 --> 00:48:33,500
+بساوي واحد زائد X F أندولي F X بتساوي سفر و proof
+
+505
+00:48:33,500 --> 00:48:42,480
+خلّينا نشوف أنبرهن هذا الأمر أو هذا الجزء طيب
+
+506
+00:48:42,480 --> 00:48:45,920
+بنناسبة أن E to the X أكبر بساوية واحد زائد X لكل
+
+507
+00:48:45,920 --> 00:48:47,880
+X way الموجودة في ال R
+
+508
+00:48:51,200 --> 00:48:55,040
+الان عندنا طبيعي حالة الـ X بتساوي صفر الـ Z باخد
+
+509
+00:48:55,040 --> 00:48:57,860
+الـ function مين انا عشان الرأس الدولة يعرف عن مين
+
+510
+00:48:57,860 --> 00:49:03,740
+بنحكي let F of X ايش هتساوي طبعا E to the X طبعا E
+
+511
+00:49:03,740 --> 00:49:09,120
+to the X معرف هو المكان الانفي حالة ال .. ال ..
+
+512
+00:49:09,120 --> 00:49:12,280
+الأول حاجة لو جينا طلعنا لحالة ال X بتساوي سفر
+
+513
+00:49:12,280 --> 00:49:18,000
+المساواة حادثة ولا لا it was a zero ليه واحد بساوي
+
+514
+00:49:18,000 --> 00:49:21,520
+واحد زائد زيرو اللي هو واحد يعني المساواة حادثة في
+
+515
+00:49:21,520 --> 00:49:25,120
+حالة ال .. ال X بتساوي زيرو الان خليني أخد ال X
+
+516
+00:49:25,120 --> 00:49:31,970
+أشمالها لاتساوي أيش؟ سفرالـ X لا تساوي سفر هثبتلك
+
+517
+00:49:31,970 --> 00:49:36,950
+إنه E to the X strictly أكبر من واحد زاد X بكون
+
+518
+00:49:36,950 --> 00:49:42,410
+خلصت هذه و خلصت هذه مع بعض افتمتين بنشوف كيف لأن X
+
+519
+00:49:42,410 --> 00:49:46,930
+لا تساوي Zero عندي حالتين X أكبر من سفر و X أصغر
+
+520
+00:49:46,930 --> 00:49:52,130
+من سفرالان f of x ده اللي هي من الدوال الحلوة اللي
+
+521
+00:49:52,130 --> 00:49:54,230
+differentiable everywhere إذا continuous
+
+522
+00:49:54,230 --> 00:49:56,830
+everywhere إذا ال mean value theorem على أي
+
+523
+00:49:56,830 --> 00:50:01,610
+interval أو sub interval منها من الار متحققة إذا x
+
+524
+00:50:01,610 --> 00:50:04,930
+أكبر من 0 معناته بتتخد أطبق ال mean value theorem
+
+525
+00:50:04,930 --> 00:50:10,910
+من عند ال zero لعند main ال x معايا إذا by mean
+
+526
+00:50:10,910 --> 00:50:16,490
+value theorem there exists element in zero x such
+
+527
+00:50:16,490 --> 00:50:25,720
+thatF prime of C بساوي F of X ناقص F of Zero على X
+
+528
+00:50:25,720 --> 00:50:30,400
+minus Zero، مظبوط ولا لأ؟ هذه اللي هي F prime of C
+
+529
+00:50:30,400 --> 00:50:34,120
+طبعا ال derivative للـE to the X هي نفسها عند الـC
+
+530
+00:50:34,120 --> 00:50:38,560
+في السير E to the Cوهذه اللي هي عبارة عن E to the
+
+531
+00:50:38,560 --> 00:50:41,900
+X F of Zero اللي هي E to the Zero اللي هي واحد
+
+532
+00:50:41,900 --> 00:50:46,860
+ناقص واحد على مين على X اضربوا طرفين في وساطين
+
+533
+00:50:46,860 --> 00:50:52,340
+فبصير E to the X ناقص واحد بساوي X في مين في E
+
+534
+00:50:52,340 --> 00:50:59,020
+أُسي ماشي الحالة الآنهذه الآن ال E to the C C
+
+535
+00:50:59,020 --> 00:51:05,100
+موجبة صح ال E to the C هتكون أكبر أو تساوي واحد ان
+
+536
+00:51:05,100 --> 00:51:09,120
+هي أكبر strictly من واحد في الواقع مظبوط E to the
+
+537
+00:51:09,120 --> 00:51:13,960
+C أكبر strictly من مين من الواحد الآن X في ال E to
+
+538
+00:51:13,960 --> 00:51:20,920
+the C هتكون أكبر strictly من مين من اللي هي ال Xمع
+
+539
+00:51:20,920 --> 00:51:24,880
+بعض الثنتين بيصير عندى E to the X أكبر من واحد
+
+540
+00:51:24,880 --> 00:51:31,860
+زائد مين زائد ال X الآن في حالة ال X أصغر من سفر
+
+541
+00:51:31,860 --> 00:51:36,600
+في حالة ال X أصغر من سفر بدنا نطبق على مين؟ على
+
+542
+00:51:36,600 --> 00:51:43,380
+الفترة اللي هي X و Zero مظبوط؟ على الفترة X و Zero
+
+543
+00:51:47,630 --> 00:51:53,510
+ب .. كله كله ماشي there exists c element in x و
+
+544
+00:51:53,510 --> 00:52:00,930
+zero such that أقول معايا f prime of c بسوى f of
+
+545
+00:52:00,930 --> 00:52:10,010
+zero نقص f of x على zero minus x هو يساوي واحد
+
+546
+00:52:10,010 --> 00:52:19,700
+minus x نقص e to the xعلى سالم X هذا إيش بيساوي E
+
+547
+00:52:19,700 --> 00:52:25,820
+to the C أضرب الطرفين في وسطين بيصير واحد ناقص E
+
+548
+00:52:25,820 --> 00:52:34,760
+to the X بيساوي ناقص X E to the C أضرب اللي هو في
+
+549
+00:52:34,760 --> 00:52:41,840
+ناقص واحد بيصير E to the X ناقص واحد بيساوي X في E
+
+550
+00:52:41,840 --> 00:52:50,110
+to the Cالان ال C سالبة مظبوط فبصير كسر فبصير عند
+
+551
+00:52:50,110 --> 00:52:56,150
+ال E to the C اللي هو ايش ماله اصغر من واحد فالان
+
+552
+00:52:56,150 --> 00:53:01,210
+لما نضرب ال X السالبة فيه ال X سالبة هنا فبصير
+
+553
+00:53:01,210 --> 00:53:08,110
+اكبر من مين من X فبصير
+
+554
+00:53:08,110 --> 00:53:12,350
+عند E to the X اكبر
+
+555
+00:53:13,170 --> 00:53:21,630
+من واحد زائد مين زائد X الآن صار عندي من الجهتين
+
+556
+00:53:21,630 --> 00:53:26,590
+لما X لا تساوي سفر E to the X strictly أكبر من
+
+557
+00:53:26,590 --> 00:53:32,590
+واحد زائد X إذا X لا تساوي سفر بتعني
+
+558
+00:53:43,580 --> 00:53:50,840
+E to the X أكبر strictly من واحد زائد X هذي واحد
+
+559
+00:53:50,840 --> 00:53:58,260
+الآن اتنين في حالة X بتساوي سفر هذا بيعطينا ان E
+
+560
+00:53:58,260 --> 00:54:03,480
+to the X بيساوي واحد زائد X والجهتين هيساوين واحد
+
+561
+00:54:03,480 --> 00:54:09,600
+واحد فE to the Zero واحد الآن الجهتين مع بعض هذولة
+
+562
+00:54:11,120 --> 00:54:15,880
+بتده لمّ الـ x لا تساوي 0 و الـ x بتساوي 0 يعني e
+
+563
+00:54:15,880 --> 00:54:19,740
+to the x بتاخد بعين الاعتبار المساواة هنا بيصير e
+
+564
+00:54:19,740 --> 00:54:24,740
+to the x أكبر و بيساوي 1 زياد x لكل x نين element
+
+565
+00:54:24,740 --> 00:54:29,120
+in R بالظبط في حالة x بتساوي صفر e to the x بتساوي
+
+566
+00:54:29,120 --> 00:54:33,520
+في حالة x لا تساوي صفر e to the x strictly أكبر
+
+567
+00:54:33,520 --> 00:54:40,750
+الآن هذا x بتساوي صفر يؤدي إلى هذاالـ Trivia زي ما
+
+568
+00:54:40,750 --> 00:54:45,510
+قلنا الان conversely بدنا نثبت هذا هذا أثبتناه في
+
+569
+00:54:45,510 --> 00:54:52,290
+الواقع كيف أثبتناه بدنا نقول بد أثبت الآن الاتجاه
+
+570
+00:54:52,290 --> 00:54:57,370
+هذا هذا الاتجاه اللي محيته اتجاه الـ Trivia اللي
+
+571
+00:54:57,370 --> 00:55:02,050
+سهل دل هذا هذا عشان أثبته بقول Contraposition لو
+
+572
+00:55:02,050 --> 00:55:07,480
+كانت X لا تساوي سفربيعطينا الـ E to the X لا يساوي
+
+573
+00:55:07,480 --> 00:55:12,080
+واحد زاد X ليش؟ لأنه أثبتته أنه strictly أكبر مزام
+
+574
+00:55:12,080 --> 00:55:15,920
+strictly أكبر إذا إيه أشماله؟ لا يساوي إذا صار
+
+575
+00:55:15,920 --> 00:55:23,420
+عندي اللي هو هذا وهذا متحققين الإثنين طيب نيجي
+
+576
+00:55:23,420 --> 00:55:30,920
+نيجي الآن لل example الثاني example معهود ومعروف
+
+577
+00:55:30,920 --> 00:55:36,860
+أو ننهي فيه المرة الجاية ان شاء اللهبنكمل باقي
+
+578
+00:55:36,860 --> 00:55:41,360
+اللي هي ال applications المرة دي ان شاء الله هنعمل
+
+579
+00:55:41,360 --> 00:55:46,780
+discussion في 6-1 يعني أصل 6-1 هنيجي ونحلهم و
+
+580
+00:55:46,780 --> 00:55:52,340
+بالإضافة ل .. اللي هو نكمل 6-2 هنكمل 6-2 و نعمل
+
+581
+00:55:52,340 --> 00:55:57,560
+discussion ل 6-1 طيب اللي مصل على سيدنا محمد
+
+582
+00:55:57,560 --> 00:56:05,210
+example التاني example اللي هو show thatshow that
+
+583
+00:56:05,210 --> 00:56:15,930
+show that show that sign ال X أكبر سوى سالب X و
+
+584
+00:56:15,930 --> 00:56:24,410
+أصغر سوى X for every X أكبر سوى 0 الغروف الآن
+
+585
+00:56:24,410 --> 00:56:28,990
+الدالة اللي بناحكيها عندها مين هي sign ال X هذه
+
+586
+00:56:28,990 --> 00:56:35,260
+letf of x بساوة صين الـ x وهذه continuous
+
+587
+00:56:35,260 --> 00:56:41,320
+everywhere و differentiable everywhere عند اللي هي
+
+588
+00:56:41,320 --> 00:56:47,160
+عند ال zero عند x بساوة zero ال trivial هذه بتتحقق
+
+589
+00:56:47,160 --> 00:56:50,900
+كtrivial لإن ده سفر و ده سفر و ده سفر، مظبوط؟ اللي
+
+590
+00:56:50,900 --> 00:56:57,660
+هنخد الآن forx أكبر من 100 من 0 بدنا نعمل
+
+591
+00:56:57,660 --> 00:57:01,560
+application أو we apply the mean value theorem on
+
+592
+00:57:01,560 --> 00:57:08,120
+the interval 0,100 أو x نظرا نتحقق الشروط بيكون
+
+593
+00:57:08,120 --> 00:57:14,500
+عند there exist c element in 0,x such that f prime
+
+594
+00:57:14,500 --> 00:57:24,510
+of c بيساوي f of x ناقص f of 0 على x minus 0F
+
+595
+00:57:24,510 --> 00:57:29,410
+prime of C هي عبارة عن Cos C لأن انتفاضه للـSin
+
+596
+00:57:29,410 --> 00:57:38,890
+Cos ويساوي F of X هي Sin X Sin 00 على X أضرب طرفين
+
+597
+00:57:38,890 --> 00:57:45,870
+في وسطين بيصير عندي Sin X بيساوي X في Cos C لكن
+
+598
+00:57:45,870 --> 00:57:54,420
+الآن أنا عندي Cos C محصورة butcos c محصورة بين
+
+599
+00:57:54,420 --> 00:58:01,660
+ناقص واحد وبين ash الواحد قل عند x أكبر من 0 اضرب
+
+600
+00:58:01,660 --> 00:58:08,600
+الجهتين في x بيصير هان x و هان x و هان x صح؟ الآن
+
+601
+00:58:08,600 --> 00:58:16,550
+x cos c هي sin x إذا صار عند sin xأظهر يساوي X
+
+602
+00:58:16,550 --> 00:58:22,050
+وأكبر أو يساوي سالب X هذا الأن لكل X لاتش أكبر من
+
+603
+00:58:22,050 --> 00:58:26,490
+صفر but for X بتساوي zero this is trivial إذا صار
+
+604
+00:58:26,490 --> 00:58:33,250
+صين ال X بين نقص X و X لكل X أكبر أو يساوي صفر و
+
+605
+00:58:33,250 --> 00:58:34,970
+المرة القادمة ان شاء الله
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4.srt
new file mode 100644
index 0000000000000000000000000000000000000000..243a249a9aac88e8a175c737ed156988c4e28427
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4.srt
@@ -0,0 +1,1593 @@
+1
+00:00:04,930 --> 00:00:09,090
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:09,090 --> 00:00:12,830
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه 
+
+3
+00:00:12,830 --> 00:00:19,690
+أجمعين هذه المحاضرة رقم عشرين في مساق تحليل حقيقي
+
+4
+00:00:19,690 --> 00:00:24,630
+للطلاب وطالبات الجامعة الإسلامية كلية العلوم
+
+5
+00:00:24,630 --> 00:00:32,660
+قسم رياضيات هنستكمل اليوم الحديث في chapter 8 اللي 
+
+6
+00:00:32,660 --> 00:00:38,380
+هو sequences of functions بالظبط هنستكمل الحديث في 
+
+7
+00:00:38,380 --> 00:00:42,800
+الـ section الأول اللي وقفنا عند نقطة فيه اللي هو 
+
+8
+00:00:42,800 --> 00:00:46,680
+الـ section كان تحت عنوان pointwise and uniform
+
+9
+00:00:46,680 --> 00:00:51,660
+convergence وفرّقنا بين اللي هو الـ pointwise و الـ
+
+10
+00:00:51,660 --> 00:00:54,540
+uniform convergence قلنا أي uniform convergence 
+
+11
+00:00:54,540 --> 00:00:58,140
+هيكون pointwise ولكن الـ pointwise convergence لا يجب أن تكون
+
+12
+00:00:58,140 --> 00:01:01,920
+حقيقية في الجامعة و أخدنا بعض النظريات أول شيء
+
+13
+00:01:01,920 --> 00:01:04,980
+عرفنا شو معناه اللي هو الـ pointwise convergence
+
+14
+00:01:04,980 --> 00:01:11,180
+وبعدين أخدنا مثال على اللي هي الـ function g of x 
+
+15
+00:01:11,180 --> 00:01:14,440
+g n of x أو sequence of functions g n of x بيساوي 
+
+16
+00:01:14,440 --> 00:01:18,950
+x أس n وأثبتنا إن هذه pointwise convergence على 
+
+17
+00:01:18,950 --> 00:01:23,190
+الفترة المفتوحة ( -1 , 1 ) للدالة g of x بـالـساوي 0 
+
+18
+00:01:23,190 --> 00:01:29,210
+زي ما هي أمامنا بعد هيك جينا حكينا عن اللي هو 
+
+19
+00:01:29,210 --> 00:01:34,410
+مثال آخر أخدنا limit 1 على n sin n x زائد n قلنا
+
+20
+00:01:34,410 --> 00:01:37,970
+هذه الـ sequence of functions برضه converts to the
+
+21
+00:01:37,970 --> 00:01:44,540
+zero function أو كانت اللي هو هذه مثال آخر على الـ
+
+22
+00:01:44,540 --> 00:01:50,020
+pointwise convergence على كل R الآن إيجينا بعد هيك
+
+23
+00:01:50,020 --> 00:01:55,080
+اللي أخدنا لما وكيف اللي هو أن نرجم شو معناه F n 
+
+24
+00:01:55,080 --> 00:02:00,400
+converges pointwise قلنا الـ K اللي بنلاقيها 
+
+25
+00:02:00,400 --> 00:02:04,720
+بتعتمد على y و x في الـ pointwise لأنها نقطة نقطة 
+
+26
+00:02:04,720 --> 00:02:09,550
+الـ convergence وبعدين اجينا عرفنا شو معناه uniform
+
+27
+00:02:09,550 --> 00:02:12,750
+convergence وقلنا في حالة الـ uniform convergence
+
+28
+00:02:12,750 --> 00:02:16,630
+هيكون عندي اللي هو الـ K اللي بنلاقيها بس بتعتمد
+
+29
+00:02:16,630 --> 00:02:20,730
+على الـ epsilon والباقي زي ما هو يعني بتنفع الـ K 
+
+30
+00:02:20,730 --> 00:02:24,730
+هذه لكل الـ xات اللي في الـ A note عشان هيك بنسمي
+
+31
+00:02:24,730 --> 00:02:28,110
+uniform convergence وقلنا إذا كانت الـ sequence 
+
+32
+00:02:28,110 --> 00:02:31,530
+uniformly convergence أكيد هتكون convergence و
+
+33
+00:02:31,530 --> 00:02:36,230
+بعدين أخدنا الـ criterion أو اللمّة اللي بننفي فيها 
+
+34
+00:02:36,230 --> 00:02:40,170
+أو بنقول شو معناه F n does not converge uniformly
+
+35
+00:02:40,170 --> 00:02:46,400
+to اللي هو function F قلنا إن الـ fn لا يتعامل مع
+
+36
+00:02:46,400 --> 00:02:51,400
+عمليات الـ fn لا يتعامل مع عمليات الـ f إذا كان
+
+37
+00:02:51,400 --> 00:02:58,280
+لجينا إبسلون و Subsequence fnk of fn Subsequence من
+
+38
+00:02:58,280 --> 00:03:04,040
+sequence الـ functions هذه وعندي sequence xk في الـ A
+
+39
+00:03:04,040 --> 00:03:08,600
+Note بحيث تحقق FNK of XK ناقص F of XK أكبر من 
+
+40
+00:03:08,600 --> 00:03:14,000
+epsilon note هذا على بعضه يعني أن الـ Fn does not
+
+41
+00:03:14,000 --> 00:03:17,320
+converge uniformly to الـ function اللي أمامنا
+
+42
+00:03:17,320 --> 00:03:22,320
+اللي هي F هذا حكينا عنه المرة الماضية وأثبتناها و
+
+43
+00:03:22,320 --> 00:03:27,240
+بعدين أخدنا مثال قلنا خد اللي هو Gn of X بيساوي X
+
+44
+00:03:27,240 --> 00:03:31,670
+plus N sequence of functions على الـ R إحنا أثبتنا
+
+45
+00:03:31,670 --> 00:03:35,810
+إن هذه اللي هي الـ sequence قبل بشوية اللي شفناها 
+
+46
+00:03:35,810 --> 00:03:43,610
+إنها converge to G of X هذه 1 0 X بساوي 1 و X 
+
+47
+00:03:43,610 --> 00:03:48,450
+element  ( -1 , 1 ) converge pointwise لهذه الـ 
+
+48
+00:03:48,450 --> 00:03:52,350
+function الآن وأثبتناها برضه وأثبتناها المرة
+
+49
+00:03:52,350 --> 00:03:56,370
+الماضية إن Gm لا يتعامل بشكل عام لهذه الـ function
+
+50
+00:03:56,370 --> 00:04:00,070
+على الفترة وهذا تأكيد على كلامنا إن الـ pointwise
+
+51
+00:04:00,070 --> 00:04:06,110
+convergence لا يتعامل بشكل عام وهي مثال على ذلك 
+
+52
+00:04:06,740 --> 00:04:12,940
+الآن عرفنا شو معناه اللي هو برضه لو كانت عندي
+
+53
+00:04:12,940 --> 00:04:16,280
+function Phi اللي هي bounded function شو معناه 
+
+54
+00:04:16,280 --> 00:04:19,900
+normها قلنا الـ norm لـ function A على اللي هي الـ
+
+55
+00:04:19,900 --> 00:04:24,060
+6A هو عبارة عن الـ supremum لقيمة الـ function Φ(x) x 
+
+56
+00:04:24,060 --> 00:04:27,660
+element A هذا في حالة اللي هي mean الـ bounded
+
+57
+00:04:27,660 --> 00:04:33,100
+function اللي هي اللي هي سميناها Φ بعد هيك جينا
+
+58
+00:04:34,730 --> 00:04:38,450
+وحكينا عن اللمّة اللي أمامنا هذه إنه لو كان اللي
+
+59
+00:04:38,450 --> 00:04:41,450
+هي a sequence of unbounded functions on A subset
+
+60
+00:04:41,450 --> 00:04:44,890
+من R اللي هي the sequence of mean يعني هذا test 
+
+61
+00:04:44,890 --> 00:04:47,170
+للـ uniform convergence أو عدم الـ uniform
+
+62
+00:04:47,170 --> 00:04:51,670
+convergence لعائلة من الدوال أو sequence of 
+
+63
+00:04:51,670 --> 00:04:55,580
+functions اللي هي of unbounded functions
+
+64
+00:04:55,580 --> 00:04:59,680
+إذن هذه بتنفع لمين؟ للـ bounded functions بتعملي 
+
+65
+00:04:59,680 --> 00:05:04,280
+test عليها إنها uniform ولا مش uniform ببساطة بس 
+
+66
+00:05:04,280 --> 00:05:08,680
+بنجيب norm الـ Fn ناقص F وبنشوفه إذا كان norm مع الـ
+
+67
+00:05:08,680 --> 00:05:12,700
+A بيروح للصفر معناه على طول بنقول Fn converges
+
+68
+00:05:12,700 --> 00:05:16,260
+uniformly to مين؟ to F إذا كان الـ norm هذا does
+
+69
+00:05:16,260 --> 00:05:20,160
+not converge to zero معناته كل سنة وأنت طيب مين؟ 
+
+70
+00:05:20,160 --> 00:05:24,360
+اللي هو الـ Fn does not converge uniformly to مين؟ 
+
+71
+00:05:24,360 --> 00:05:29,060
+to F هذه الآن اللي وصلنا عندها اللي هي اللي ما
+
+72
+00:05:29,060 --> 00:05:36,380
+أخدنا مثال برضه عليها اللي هو Gn of Xn قلنا هذا 
+
+73
+00:05:36,380 --> 00:05:40,260
+اللي هو المثال
+
+74
+00:05:40,260 --> 00:05:45,890
+اللي أمامنا كيف نحله على اللي هي الـ .. الـ .. الـ ..
+
+75
+00:05:45,890 --> 00:05:49,550
+النظرية اللي قبل بشوية قلنا خد Gn of X بيساوي X
+
+76
+00:05:49,550 --> 00:05:55,810
+أُس N على الفترة مين؟ (0,1) جينا إن .. إن Gn 
+
+77
+00:05:55,810 --> 00:06:00,510
+converges uniformly لمين؟ لـ G of X على الفترة اللي
+
+78
+00:06:00,510 --> 00:06:05,410
+أمامنا اللي هي هذه اللي هي فترة (0,1) طبعًا بـ
+
+79
+00:06:05,410 --> 00:06:09,180
+واضح إن الدالة هذه bounded على هذه الفترة عشان 
+
+80
+00:06:09,180 --> 00:06:13,220
+هي تنطبق عليها اللي هي النظرية اللي فوق هذه 
+
+81
+00:06:13,220 --> 00:06:17,980
+المرة الماضية برضه حكيناها أو جينا أو وقفنا عند
+
+82
+00:06:17,980 --> 00:06:23,260
+اللي هي المثال الثاني طبعًا اللي هي المثال الثاني
+
+83
+00:06:23,260 --> 00:06:28,300
+لو على السريع ذكرناكم فيها الآن الـ function مين هي 
+
+84
+00:06:28,300 --> 00:06:34,460
+عندي الـ function عبارة عن gn of x أو sequence of
+
+85
+00:06:34,460 --> 00:06:39,410
+functions بيساوي x أس n الآن لو جينا أخدت الـ norm 
+
+86
+00:06:39,410 --> 00:06:44,870
+لـ gn ناقص الـ g هي الـ gn وهي الـ g أخد الـ norm
+
+87
+00:06:44,870 --> 00:06:50,730
+إلها أخد الـ norm إلها معي
+
+88
+00:06:50,730 --> 00:06:54,650
+يا جماعة هي الـ norm إلها لـ gn ناقص g بيساوي اللي 
+
+89
+00:06:54,650 --> 00:06:59,170
+هو عبارة عن الـ supremum X أس n ناقص 0 يعني X أس
+
+90
+00:06:59,170 --> 00:07:05,750
+n لمن؟ للـ x المتغير في (0,1) أو الجزء الثاني X أس n
+
+91
+00:07:05,750 --> 00:07:10,970
+ناقص 1 اللي هي في حالة الـ x بيساوي 1 لكن في 
+
+92
+00:07:10,970 --> 00:07:13,490
+حالة الـ x بيساوي 1 بيصير G n of X n بيتساوي
+
+93
+00:07:13,490 --> 00:07:17,370
+1 1 ناقص 1 بيصير 0 بيصير عندي الـ norm
+
+94
+00:07:17,370 --> 00:07:20,490
+هذا عبارة عن الـ supremum X n X في الفترة (0,1)
+
+95
+00:07:20,490 --> 00:07:24,340
+و0 لما الـ x بيساوي 1 الـ supremum
+
+96
+00:07:24,340 --> 00:07:29,600
+للقيمة هذه عندي أكيد إيش هيساوي؟ عند اللي هي x
+
+97
+00:07:29,600 --> 00:07:34,380
+بتساوي 1 وy ساوي اللي هو عبارة أو قبل اللي هي 
+
+98
+00:07:34,380 --> 00:07:40,620
+الـ supremum لـ الـ Φ هذه طبعًا هنا x element in (0,1) و
+
+99
+00:07:40,620 --> 00:07:44,780
+1 اللي هي not included فبكون عندي الـ supremum
+
+100
+00:07:44,780 --> 00:07:50,060
+للقيم X n في الفترة من (0,1) إيش هيساوي؟ هيساوي
+
+101
+00:07:50,060 --> 00:07:57,150
+1 يعني الـ supremum لـ x n such that x element
+
+102
+00:07:57,150 --> 00:08:05,210
+in (0,1) حتى لو كانت closed هذه واللي هو
+
+103
+00:08:05,210 --> 00:08:08,730
+اتحاد
+
+104
+00:08:08,730 --> 00:08:13,690
+الـ 0 الـ supremum لها طبعًا للـ x n إيش حيساوي في 
+
+105
+00:08:13,690 --> 00:08:17,330
+النهاية حيساوي 1 وممكن إثبات هذا اللي هو عن
+
+106
+00:08:17,330 --> 00:08:21,900
+طريق الـ least upper bound كيف نثبته زي ما كنا نثبت
+
+107
+00:08:21,900 --> 00:08:26,540
+فيه اللي هو تحليل واحد مش ما كنا نثبته لأن فبيصير
+
+108
+00:08:26,540 --> 00:08:30,280
+عندي الـ norm لـ gn ناقص g إيش بيساوي؟ 1 ما دام الـ
+
+109
+00:08:30,280 --> 00:08:33,980
+norm لـ gn بيساوي 1 إذاً الـ norm لـ gn ناقص g
+
+110
+00:08:33,980 --> 00:08:40,920
+does not converge to 0  إذاً الـ gn لا يتعامل مع الـ g
+
+111
+00:08:40,920 --> 00:08:47,040
+إن ناقص gn ناقص gn ناقص gn ناقص gn ناقص gn
+
+112
+00:08:47,040 --> 00:08:54,180
+ناقص gn ناقص gn ناقص gn ناقص gn ناقص gn
+
+113
+00:08:54,180 --> 00:08:54,660
+ناقص gn ناقص gn ناقص gn ناقص gn ناقص gn
+
+114
+00:08:54,660 --> 00:08:55,450
+ناقص gn ناقص gn ناقص gn ناقص gn ناقص
+
+115
+00:08:55,450 --> 00:09:00,190
+الـ g برضه هيروح على مين؟ على القيمة 1 يعني مش 
+
+116
+00:09:00,190 --> 00:09:05,890
+هيروح للصفر يعني إذاً Gn does not converge 
+
+117
+00:09:05,890 --> 00:09:10,850
+uniformly to مين؟ to G وهذا مثال عليه اللي هو does
+
+118
+00:09:10,850 --> 00:09:16,030
+not converge uniformly على الـ G مثال على اللي هو 
+
+119
+00:09:16,030 --> 00:09:20,250
+توظيف هذه النظرية اللي قبل بشوية لإثبات عدم الـ 
+
+120
+00:09:20,250 --> 00:09:23,930
+uniformly convergent أو لو بدنا نثبت إنه formally
+
+121
+00:09:23,930 --> 00:09:29,910
+convergent ممكن نستخدم هذه بس نثبت إن الـ norm بيروح 
+
+122
+00:09:29,910 --> 00:09:33,710
+للصفر بس بحيث إن تكون اللي هي الـ Gn عبارة عن 
+
+123
+00:09:33,710 --> 00:09:38,350
+sequence زي ما قلنا of bounded functions طيب يجي
+
+124
+00:09:38,350 --> 00:09:42,030
+مثال آخر ما علينا سريع لأنه حكينا المرة الماضية يجي 
+
+125
+00:09:42,030 --> 00:09:48,070
+للي هو مثلنا هذانجي لمثالنا fn of x بيساوي x
+
+126
+00:09:48,070 --> 00:09:53,050
+على n مش x أس n x على n for every x element in R
+
+127
+00:09:53,050 --> 00:09:58,950
+اللي هو بيقول لي show that limit fn of x as x بتروح
+
+128
+00:09:58,950 --> 00:10:02,850
+إلى infinity بيساوي f of x لكل x element in R
+
+129
+00:10:02,850 --> 00:10:08,430
+where f of x إيش بتساوي؟ 0 و fn x بتروح لـ f of 
+
+130
+00:10:08,430 --> 00:10:11,610
+x من R هذا معناه اللي هو pointwise convergence
+
+131
+00:10:11,610 --> 00:10:16,330
+يعني بده limit الـ fn of x ناخذ أي x element in R
+
+132
+00:10:16,330 --> 00:10:19,490
+ناخد x element in R نثبت إن limit fn x إيش هتساوي؟
+
+133
+00:10:19,490 --> 00:10:24,830
+هتساوي 0 الآن هذا اللي هو عبارة عن اللي هو الـ
+
+134
+00:10:24,830 --> 00:10:30,110
+pointwise convergence بعد شوية هنثبت إنه عندنا هذا
+
+135
+00:10:30,110 --> 00:10:34,530
+مش هيكون فيه عندي اللي هو uniform convergence على
+
+136
+00:10:34,530 --> 00:10:40,790
+مكان ما في الأرض نرجع لمثالنا اللي هو عندي مثال 
+
+137
+00:10:40,790 --> 00:10:48,880
+اللي هو Fn X بيساوي X على N لـ X element in R عندي F 
+
+138
+00:10:48,880 --> 00:10:52,820
+of X بيساوي 0 لكل X element in R بدنا نثبت إن Fn
+
+139
+00:10:52,820 --> 00:10:56,500
+of X بتروح pointwise لـ F of X لكل X element in R 
+
+140
+00:10:56,500 --> 00:11:01,180
+وبدنا نثبت إن Fn does not converge لـ F uniformly
+
+141
+00:11:01,180 --> 00:11:04,500
+لـ F on R يعني هذا هو عبارة عن pointwise
+
+142
+00:11:04,500 --> 00:11:10,170
+convergence لكن مش uniform convergence بس هنا جيبنا
+
+143
+00:11:10,170 --> 00:11:14,310
+المثال عشان نقول لك إنه مش هنقدر نستخدم اللي هي
+
+144
+00:11:14,310 --> 00:11:17,630
+اللمّة اللي قبل بشوية لإنه عندي sequence of
+
+145
+00:11:17,630 --> 00:11:23,910
+functions Fn على R اللي هي is not bounded يعني لو
+
+146
+00:11:23,910 --> 00:11:29,150
+جينا أخدنا F1 of X بتساوي X على 1 لكل x element in
+
+147
+00:11:29,150 --> 00:11:33,310
+R وكأن f1 of x عبارة عن x طبعًا الـ x is not
+
+148
+00:11:33,310 --> 00:11:36,790
+bounded function f2x التي هي x على 2 برضه is not
+
+149
+00:11:36,790 --> 00:11:40,250
+bounded لأنها تبقى مالانهاية رايحة من الجهتين،
+
+150
+00:11:40,250 --> 00:11:42,510
+واحد مالانهاية وواحد سالب مالانهاية،
+
+151
+00:11:48,910 --> 00:11:54,190
+عشان هيك ما بنقدرش نستخدم اللي هو عشان نثبت الـ Fn
+
+152
+00:11:54,190 --> 00:11:58,150
+does not converge uniformly ما بنقدرش نستخدم norm
+
+153
+00:11:58,150 --> 00:12:03,310
+Fn ناقص F اللي هو بيروح للصفر لأن هذه الـ Fn is not
+
+154
+00:12:03,530 --> 00:12:07,550
+bounded functions are not bounded functions يعني 
+
+155
+00:12:07,550 --> 00:12:11,190
+اللي هو اللمّة لا تنطبق عليها عشان هيك هنضطر نروح 
+
+156
+00:12:11,190 --> 00:12:15,450
+بالاستخدام التعريف أو نفي التعريف للـ uniformly
+
+157
+00:12:15,450 --> 00:12:18,690
+convergence زي ما عملنا في مثال المرة الماضية
+
+158
+00:12:18,690 --> 00:12:25,290
+وخلينا نشوف الحل نيجي الآن الجزء الأول الجزء 
+
+159
+00:12:25,290 --> 00:12:32,700
+الأول سهل على طول اللي هو عندي لكل X element in R 
+
+160
+00:12:32,700 --> 00:12:40,460
+أكيد عندي limit of Fn of X لما N تروح إلى مالانهاية 
+
+161
+00:12:40,460 --> 00:12:47,240
+بساو limit of Fn of X هي عبارة عن X على N as 
+
+162
+00:12:47,240 --> 00:12:50,860
+N goes to infinity ويساوي كلكم بتعرفوا إيش هتساوي؟ 
+
+163
+00:12:50,860 --> 00:12:56,030
+بتساوي 0 يعني بمعنى آخر اللي هي الـ fn of x 
+
+164
+00:12:56,030 --> 00:13:00,990
+converge لمين؟ للـ f اللي هي of x اللي بتساوي 0
+
+165
+00:13:00,990 --> 00:13:04,610
+وهذا الـ convergence عبارة عن pointwise convergence
+
+166
+00:13:04,610 --> 00:13:10,070
+شوف الآن كيف بدنا نثبت خلصنا من إن إحنا نثبت fn x
+
+167
+00:13:10,070 --> 00:13:14,590
+بتروح pointwise لمين؟ للي هي الـ function zero 
+
+168
+00:13:14,590 --> 00:13:18,350
+بدنا نركز الآن نثبت إن fn does not converge
+
+169
+00:13:18,350 --> 00:13:24,080
+uniformly للـ function f on R اللي هو زي ما قلنا الـ 
+
+170
+00:13:24,080 --> 00:13:27,940
+sequence of functions عندي هنا are not bounded
+
+171
+00:13:27,940 --> 00:13:32,620
+functions so we can't use our لمّة اللي قبل بشوية 
+
+172
+00:13:32,620 --> 00:13:36,060
+اللي هي norm of fn ناقص f converges to zero إذاً 
+
+173
+00:13:36,060 --> 00:13:41,100
+هنتضطر نروح للتعريف أو لللمّة قبلها إذا بتتذكروا
+
+174
+00:13:41,100 --> 00:13:44,820
+اللم
+
+201
+00:16:00,800 --> 00:16:06,820
+نختارها بتساوي K أكيد، وين موجودة في R، then F<sub>NK</sub> of 
+
+202
+00:16:06,820 --> 00:16:14,380
+X<sub>K</sub> ايش هتساوي؟ اللي هي F<sub>K</sub> of X<sub>K</sub>، ويساوي اللي هي
+
+203
+00:16:14,380 --> 00:16:20,080
+عبارة عن X<sub>K</sub> على K حسب اللي هي الدالة اللي عندي أو
+
+204
+00:16:20,080 --> 00:16:22,720
+الـ sequence of functions، طب ما هي الـ X<sub>K</sub> ايش أنا 
+
+205
+00:16:22,720 --> 00:16:27,420
+بختارها بتساوي K، فبصير K على K هو يساوي 1، إذا
+
+206
+00:16:27,420 --> 00:16:34,210
+طلعت عندي F<sub>NK</sub> of X<sub>K</sub> بيساوي 1، يحسب للـ F<sub>NK</sub> of X<sub>K</sub>
+
+207
+00:16:34,210 --> 00:16:37,410
+للـ subsequence اللي عملناها، ناقص F of X<sub>K</sub>، إيش
+
+208
+00:16:37,410 --> 00:16:41,770
+هتساوي؟ هذه عبارة عن 1، وهذه دالة دائماً ثابتة، F
+
+209
+00:16:41,770 --> 00:16:45,610
+عند أي قيمة بتساوي صفر، فبصير اللي هو عبارة عن إيش
+
+210
+00:16:45,610 --> 00:16:49,030
+القيمة 1، إيش أختار ε (epsilon) note اللي بدك إياها
+
+211
+00:16:49,030 --> 00:16:54,680
+أي إيش أصغر منيه؟ من الـ 1، وليكن إيش؟ نصف، بصير عندي
+
+212
+00:16:54,680 --> 00:17:00,140
+F<sub>N</sub> K هيها F<sub>K</sub> و ε (epsilon) بيساوي نص، و X<sub>K</sub> بتساوي K، هي
+
+213
+00:17:00,140 --> 00:17:05,460
+subsequence من الـ F<sub>N</sub>، و هي الي هو ε (epsilon) نص، و هي
+
+214
+00:17:05,460 --> 00:17:09,620
+sequence X<sub>K</sub> in R بحيث أن هذا الـ absolute value 
+
+215
+00:17:09,620 --> 00:17:13,660
+أكبر من مين؟ اللي هو طلع 1، اللي هو أكيد أكبر من
+
+216
+00:17:13,660 --> 00:17:17,900
+إيش؟ من نص اللي هي ε (epsilon) نص، وهذا يعني أن F<sub>N</sub> does
+
+217
+00:17:17,900 --> 00:17:24,390
+not converge uniformly للـ F on R، هكذا أثبتنا أن الـ
+
+218
+00:17:24,390 --> 00:17:27,310
+F<sub>N</sub> لا يتعامل بشكل عام بشكل عام بشكل عام بشكل عام
+
+219
+00:17:27,310 --> 00:17:29,790
+بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل
+
+220
+00:17:29,790 --> 00:17:31,210
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+221
+00:17:31,210 --> 00:17:31,710
+بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل
+
+222
+00:17:31,710 --> 00:17:32,470
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+223
+00:17:32,470 --> 00:17:34,290
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+224
+00:17:34,290 --> 00:17:36,760
+بشكل عام بشكل عام بشكل عام بشكل عام أو الثالث في
+
+225
+00:17:36,760 --> 00:17:42,160
+المثال، بدنا نثبت أن الـ F<sub>N</sub> unconverges للـ F 
+
+226
+00:17:42,160 --> 00:17:48,640
+uniformly على الفترة مين؟ على الفترة [0,1]، يعني لما
+
+227
+00:17:48,640 --> 00:17:53,080
+نغيرنا الفترة، هيصير اللي هو الـ pointwise ماقدرش
+
+228
+00:17:53,080 --> 00:17:57,040
+اللي هو الـ uniform convergence على الـ R، ماقدرش
+
+229
+00:17:57,200 --> 00:18:01,420
+اللي هو يكون uniform convergence، لكن لو زغرنا
+
+230
+00:18:01,420 --> 00:18:06,420
+الفترة أو غيرنا الفترة ممكن نحصل على اللي هو الـ
+
+231
+00:18:06,420 --> 00:18:11,480
+uniform convergence، وهذا طبيعي، هنشوف كيف، الآن
+
+232
+00:18:11,480 --> 00:18:16,640
+عشان اللي هو نظريتنا سهلة الاستخدام اللي قبل بشوية
+
+233
+00:18:16,640 --> 00:18:22,580
+اللي هو ||F<sub>N</sub> - F|| بدي أحسبه و أشوفه على
+
+234
+00:18:22,580 --> 00:18:27,240
+الفترة [0,1] اللي هو conversion to 0 ولا لأ؟ آه
+
+235
+00:18:27,240 --> 00:18:30,520
+الآن قبلها بدي أشوف الـ sequence of functions عندي
+
+236
+00:18:30,520 --> 00:18:36,520
+هذه هل bounded على الفترة [0,1]؟ F<sub>N</sub> X بتساوي X على
+
+237
+00:18:36,520 --> 00:18:44,140
+N، يعني صار عندي F<sub>N</sub> of X بيساوي X على N، لوين X في
+
+238
+00:18:44,140 --> 00:18:47,860
+الفترة [0,1]، خلاص الـ X في الفترة [0,1]، جد ما تكون
+
+239
+00:18:47,860 --> 00:18:51,600
+ليها الـ X نفسها محصورة بين 0 و 1، يعني بتضل الـ X
+
+240
+00:18:51,600 --> 00:18:57,040
+على N عبارة عن قيمة محدودة، قيمة محدودة أكيد اللي
+
+241
+00:18:57,040 --> 00:19:00,680
+هو الـ absolute value اللي هي هذه أصغر أو يساوي
+
+242
+00:19:00,680 --> 00:19:05,100
+اللي هو 1 على N، إذا صار عند .. لكل .. طبعاً هذا
+
+243
+00:19:05,100 --> 00:19:10,510
+محدودة بالنسبة لـ X، أه بيصير F<sub>1</sub> أصغر بيساوي 1، F<sub>2</sub> 
+
+244
+00:19:10,510 --> 00:19:15,350
+أصغر بيساوي نص، F<sub>3</sub> أصغر بيساوي تلت، وهكذا، إذن هي
+
+245
+00:19:15,350 --> 00:19:21,790
+عبارة عن F<sub>1</sub>، F<sub>2</sub>، F<sub>3</sub>، sequence of bounded functions on
+
+246
+00:19:21,790 --> 00:19:26,570
+the interval [0,1]، إذن الآن الأمر مشروع ليه أني 
+
+247
+00:19:26,570 --> 00:19:31,970
+أستخدم اللمة اللي حكينا عنها قبل بشوية، و نشوف كيف
+
+248
+00:19:31,970 --> 00:19:36,110
+نثبت الـ uniform convergence، نحسب مع بعض أكيد لكلكم
+
+249
+00:19:36,440 --> 00:19:43,380
+اللي هو ||F<sub>N</sub> - F|| over A بيساوي الـ supremum لـ
+
+250
+00:19:43,380 --> 00:19:50,380
+F<sub>N</sub> of X ناقص F of X، such that X و M هو وجودة في 
+
+251
+00:19:50,380 --> 00:19:57,120
+الفترة [0,1]، ويساوي الـ supremum لـ F<sub>N</sub> of X اللي 
+
+252
+00:19:57,120 --> 00:20:03,270
+هي أشكل عنها عبارة عن X على N، X على N ناقصها دي صفر، 
+
+253
+00:20:03,270 --> 00:20:07,110
+بتضل X على N، such that X where موجودة في الـ zero
+
+254
+00:20:07,110 --> 00:20:10,330
+أو إيش؟ أو 1، الـ supremum أهم إلها زي ما تقولنا
+
+255
+00:20:10,330 --> 00:20:14,790
+أشملها، أصغرشها 1 على N، هتطلع عبارة عن أ 1
+
+256
+00:20:14,790 --> 00:20:21,670
+على N، هذه اللي هي مين؟ الـ norm للـ F<sub>N</sub> - F، طيب
+
+257
+00:20:21,670 --> 00:20:26,530
+واضح الأمر، مدام الـ norm للـ F<sub>N</sub> - F بيساوي حتى
+
+258
+00:20:26,530 --> 00:20:35,220
+الآن إذا ||F<sub>N</sub> - F||<sub>A</sub> goes to zero لأن F<sub>N</sub>
+
+259
+00:20:35,220 --> 00:20:38,880
+goes to infinity، قيمة هذا الواحدة الآن هتروح إلى
+
+260
+00:20:38,880 --> 00:20:42,960
+صفر، يعني ||F<sub>N</sub>|| هذا بيروح للزيرو، وهذا يعني أن F<sub>N</sub> 
+
+261
+00:20:42,960 --> 00:20:49,240
+converges uniformly لـ F on الفترة [0,1]، وبكون
+
+262
+00:20:49,240 --> 00:20:54,180
+هيك أثبتنا اللي هو باستخدام النظرية أن هذا المثال
+
+263
+00:20:54,180 --> 00:20:58,680
+فعلاً اللي هو F<sub>N</sub> unconverged uniformly للـ F On The
+
+264
+00:20:58,680 --> 00:21:03,380
+Interval [0,1]، But It Does Not Converge Uniformly On 
+
+265
+00:21:03,380 --> 00:21:10,120
+R ككل، لكن Pointwise على الـ R To The كل، طيب نيجي
+
+266
+00:21:10,120 --> 00:21:16,300
+الآن اللي هو التعبير الآخر على الـ uniform
+
+267
+00:21:16,300 --> 00:21:19,880
+convergence، عبرنا عن الـ uniform convergence
+
+268
+00:21:19,880 --> 00:21:26,180
+بواسطة اللي هي التعريف، عبرنا عنه للعائلة of
+
+269
+00:21:26,180 --> 00:21:30,890
+bounded functions بواسطة الـ norm للـ F<sub>N</sub> اللي هو
+
+270
+00:21:30,890 --> 00:21:37,190
+ناقص الـ F بيروح للـ 0، الآن بدنا نعرف عنه .. نعبر
+
+271
+00:21:37,190 --> 00:21:40,070
+عنه عن طريق .. برضه للـ sequence of bounded
+
+272
+00:21:40,070 --> 00:21:43,230
+functions عن طريق حاجة اسمها الـ Cauchy criterion
+
+273
+00:21:43,230 --> 00:21:47,270
+for uniform convergence، وكأنه بدنا ندخل المفهوم
+
+274
+00:21:47,270 --> 00:21:53,590
+الـ Cauchy اللي هي sequence في هذا اللي هو النظرية
+
+275
+00:21:53,590 --> 00:21:57,470
+نشوف Cauchy Criteria لتحويل المستوى المستوى المستوى
+
+276
+00:21:57,470 --> 00:21:58,350
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+277
+00:21:58,350 --> 00:22:03,130
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+278
+00:22:03,130 --> 00:22:06,950
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+279
+00:22:06,950 --> 00:22:10,330
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+280
+00:22:10,330 --> 00:22:10,350
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+281
+00:22:10,350 --> 00:22:10,410
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+282
+00:22:10,410 --> 00:22:10,430
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+283
+00:22:10,430 --> 00:22:15,740
+المستوى المستF، يعني بتكون F<sub>N</sub> unconverted لـ bounded
+
+284
+00:22:15,740 --> 00:22:20,220
+function F، F and only F لكل Y أكبر من 0، there
+
+285
+00:22:20,220 --> 00:22:23,680
+exists H(Y) للـ N such that for every M و N أكبر
+
+286
+00:22:23,680 --> 00:22:28,180
+يساوي H(Y)، حيكون الـ F<sub>M</sub> مقصلة F<sub>N</sub> أصغر أو يساوي Y،
+
+287
+00:22:28,180 --> 00:22:32,940
+وهذا هيذكرنا اللي هو بالـ Cauchy sequences إذا نسيناها،
+
+288
+00:22:33,270 --> 00:22:37,270
+يعني وكأنه خلّيني نقوله roughly بيقول لـ F<sub>N</sub> تحت
+
+289
+00:22:37,270 --> 00:22:42,950
+الظروف اللي موجودة بتروح للـ F uniformly، يكافئ أنه
+
+290
+00:22:42,950 --> 00:22:47,510
+لكل ε (epsilon) أكبر من 0، هذا الـ N الجزء هنا هو اللي
+
+291
+00:22:47,510 --> 00:22:51,070
+يكوش criterion في الواقع، for every ε (epsilon) أكبر من 
+
+292
+00:22:51,070 --> 00:22:56,950
+0، there exists H(ε) element in N such that 
+
+293
+00:22:56,950 --> 00:23:03,680
+for every N و M element in N أو اللي هي أكبر أو
+
+294
+00:23:03,680 --> 00:23:08,140
+يساوي H(ε)، حتكون عندك الـ ε (epsilon) norm للـ 
+
+295
+00:23:08,140 --> 00:23:17,100
+F<sub>M</sub> ناقص F<sub>N</sub> أصغر من 100 أو يساوي اللي هو ε (epsilon)، هذه
+
+296
+00:23:17,100 --> 00:23:20,180
+اللي هي عبارة عن الـ Cauchy criterion، يعني أحيانًا 
+
+297
+00:23:20,180 --> 00:23:24,160
+نحن نعرف .. نكونش عارفين الـ F<sub>N</sub> لمين رايحة، لكن
+
+298
+00:23:24,160 --> 00:23:27,380
+متأكدين أنها converges uniformly to some function
+
+299
+00:23:27,380 --> 00:23:32,600
+F، طب الآن في هذه الحالة اللي بصعفنا في الاستخدام
+
+300
+00:23:32,600 --> 00:23:38,480
+اللي هو أنه ناخد اللي هو بدلالات الـ terms اللي هي
+
+301
+00:23:38,480 --> 00:23:42,440
+الـ F<sub>N</sub> و الـ F<sub>M</sub>، و نبعد عن الـ F لأنه احنا مش عارفينها
+
+302
+00:23:42,440 --> 00:23:45,260
+عشان هيك بنقول F<sub>N</sub> converges uniformly to some
+
+303
+00:23:45,260 --> 00:23:50,340
+function F، يعني هيك تصرفها إذا و فقط إذا اللي هو
+
+304
+00:23:50,340 --> 00:23:53,920
+لكل ε (epsilon) أكبر من صفر، الـ ε (epsilon) element in N
+
+305
+00:23:53,920 --> 00:23:56,300
+such that forever N و M أكبر من صفر الـ ε (epsilon)
+
+306
+00:23:56,300 --> 00:23:59,680
+بيعطينا ||F<sub>N</sub> - F<sub>M</sub>|| أصغر من minimum ε (epsilon)، و كأنه
+
+307
+00:23:59,680 --> 00:24:03,650
+بنقول اللي هو الـ function F<sub>N</sub> is a Cauchy sequence
+
+308
+00:24:03,650 --> 00:24:08,250
+roughly in the space of functions اللي هي الـ norm
+
+309
+00:24:08,250 --> 00:24:11,470
+بيساو الـ supremum اللي هو هذا الموضوع طبعاً مش
+
+310
+00:24:11,470 --> 00:24:15,170
+موضوعنا، موضوع ناخده بالتفصيل في اللي هو الـ
+
+311
+00:24:15,170 --> 00:24:19,870
+functional analysis، طيب شوفوا علينا، نيجي اللي هو
+
+312
+00:24:19,870 --> 00:24:26,710
+انبرهن النظرية، بدنا نفترض ان الـ F<sub>N</sub> الآن بدنا نفترض 
+
+313
+00:24:26,710 --> 00:24:31,970
+أنه ||F<sub>N</sub>|| أو الـ F<sub>N</sub> unconverge uniformly to some
+
+314
+00:24:31,970 --> 00:24:37,790
+function F و نصل لهذا اللي هو الجزء سهل إن شاء
+
+315
+00:24:37,790 --> 00:24:44,450
+الله، شوفوا ركزولنا الآن، suppose that F<sub>N</sub> converge
+
+316
+00:24:44,450 --> 00:24:49,390
+uniformly لـ F on A، إذا الآن من التعريف تبع الـ
+
+317
+00:24:49,390 --> 00:24:54,480
+uniform convergence، لكل Y أكبر من 0، There exists K
+
+318
+00:24:54,480 --> 00:24:58,420
+(Y) على 2 element in N such that for every N أكبر
+
+319
+00:24:58,420 --> 00:25:03,180
+أو يساوي K(Y)، ولكل X element in A، برضه بظبط اللي
+
+320
+00:25:03,180 --> 00:25:08,180
+هو F<sub>N</sub> of X ناقص F of X أصغر من Y على 2، هذا
+
+321
+00:25:08,180 --> 00:25:11,000
+باستخدام التعريف الـ N، باستخدام النظرية اللي قبل 
+
+322
+00:25:11,000 --> 00:25:14,720
+بشوية لأنه بنفع، لإن النظرية احنا قلنا F<sub>N</sub> converges
+
+323
+00:25:14,720 --> 00:25:19,540
+uniformly، إيش معناه هذا؟ معناه ||F<sub>N</sub> - F|| goes to 0 over mean over A، هذه صار 
+
+324
+00:25:19,540 --> 00:25:25,720
+تعريف عادي للـ limit، إيش معناه هذا؟ لكل ε (epsilon)
+
+325
+00:25:25,720 --> 00:25:30,160
+أكبر من صفر، there exists اللي هو مسميها هو إيش
+
+326
+00:25:30,160 --> 00:25:35,980
+مسميها K(ε) على 2 element in M such that
+
+327
+00:25:35,980 --> 00:25:42,440
+اللي هو ||F<sub>N</sub> - F|| هيكون أصغر من ε (epsilon)
+
+328
+00:25:43,150 --> 00:25:50,370
+على اتنين، for every N أكبر أو يساوي مين؟ K(ε) 
+
+329
+00:25:50,370 --> 00:25:54,630
+على اتنين، اللي عندك هذا اللي هو تعريف الدهانية
+
+330
+00:25:54,630 --> 00:25:57,770
+العادية اللي هو sequence O السيكوانس العادية هذه
+
+331
+00:25:57,770 --> 00:26:01,590
+الـ sequence converts to zero، لأنها أعداد اللي هي
+
+332
+00:26:01,590 --> 00:26:04,440
+converge to zero، إذا لكل Y أكبر من صفر، there
+
+333
+00:26:06,560 --> 00:26:10,740
+exists K بحيث أنه ||F<sub>N</sub> - F|| أصغر من Y على اتنين 
+
+334
+00:26:10,740 --> 00:26:19,460
+لكل N أكبر أو سوء K(Y) على اتنين، طيب الآن عندي
+
+335
+00:26:19,460 --> 00:26:25,520
+الـ .. الـ .. الـ F هذه بمقدر نترجمها اللي هو على طول
+
+336
+00:26:25,520 --> 00:26:31,660
+مباشرة إذا اللي هي عندي الـ F مدام هذا أصغر يساوي
+
+337
+00:26:31,660 --> 00:26:35,880
+Y على 2، يعني الـ supremum أصغر يساوي Y على 2، وهذا
+
+338
+00:26:35,880 --> 00:26:40,360
+نفسه أكبر أو يساوي إيش عارفين أنتو؟ F<sub>N</sub> of X ناقص
+
+339
+00:26:40,360 --> 00:26:44,280
+F of X، مظبوط؟ لأن هذا الـ supremum ملهن، إذا صار
+
+341
+00:26:44,280 --> 00:26:53,530
+عندي إذا ||F<sub>N</sub> - F|| of X صار نفسه أصغر من مين؟ من Y
+
+342
+00:26:53,530 --> 00:27:00,550
+على 2، خليها في الذاكرة، الآن عندي احسبلي بناء عليه 
+
+343
+00:27:00,550 --> 00:27:09,430
+احسبلي F<sub>N</sub> of X ناقص F<sub>M</sub> of X، هذا لمين؟ هذا حسب ليه
+
+344
+00:27:09,430 --> 00:27:16,230
+لمين؟ لكل أنه طبعاً هذا الكلام صحيح لكل X، وين؟ لكل
+
+345
+00:27:16,230 --> 00:27:22,340
+X في الـ A تبعتنا، احسبلي لكل N و M أكبر من مين؟ من
+
+346
+00:27:22,340 --> 00:27:26,620
+الـ K اللي لجيها هنا، K(ε) على 2، F<sub>N</sub> of X
+
+347
+00:27:26,620 --> 00:27:30,580
+ناقص F<sub>M</sub> of X أكيد أصغر أو يساوي بالـ Triangle
+
+348
+00:27:30,580 --> 00:27:38,940
+Inequality F<sub>N</sub> of X ناقص F of X زائد F<sub>M</sub> of X ناقص F
+
+349
+00:27:38,940 --> 00:27:43,560
+of X، واضح عشان سويت ضفت الـ F of X وطرحتها وبعدين
+
+350
+00:27:43,560 --> 00:27:48,460
+أخدت Triangle Inequality، الآن هذا صحيح أصغر من Y على 
+
+351
+00:27:48,460 --> 00:27:55,590
+2، لكل N لكل N أكبر من K، طب ما أنا مختار الـ N و الـ M
+
+352
+00:27:55,590 --> 00:27:59,830
+أكبر من K، إذن بنطبق عليها هذا سواء لهذه أو لهذه، إذن
+
+353
+00:27:59,830 --> 00:28:04,630
+هذه صارت أصغر من Y على 2، وهذه أصغر من Y على 2، يعني
+
+354
+00:28:04,630 --> 00:28:10,530
+صار الـ F<sub>N</sub> of X ناقص F<sub>MA</sub> of X أشماله أصغر من Y، أصغر
+
+355
+00:28:10,530 --> 00:28:15,110
+من مين؟ من Y، يعني الآن اللي وصلت إليه، ما يلي يا 
+
+356
+00:28:15,110 --> 00:28:19,590
+جماعة، بدأت بالـ F<sub>N</sub> بتروح للـ F اللي هو uniformly
+
+357
+00:28:20,520 --> 00:28:25,460
+ووصلت إلى مالي؟ وصلت إنه لكل ε (epsilon) أكبر من 0، لجيت 
+
+358
+00:28:25,460 --> 00:28:30,480
+K(ε) على 2 element in N such that for 
+
+359
+00:28:30,480 --> 00:28:35,680
+every N و M أكبر أو يساوي K(ε) على 2، لجيت
+
+360
+00:28:35,680 --> 00:28:46,360
+إن الـ F<sub>N</sub> of X ناقص F<sub>M</sub> of X أصغر من مين؟ من ε (epsilon)، هذا 
+
+361
+00:28:46,360 --> 00:28:52,420
+لكل X وين موجودة في الـ A، ماشي الحال، إذا مدام هذا
+
+362
+00:28:52,420 --> 00:28:57,140
+أصغر من ε (epsilon) لكل X element in A، صارت الـ ε (epsilon) 
+
+363
+00:28:57,140 --> 00:29:01,030
+عبارة عن upper bound لهذه، إذاً الـ ε (epsilon) هتكون
+
+364
+
+401
+00:32:13,110 --> 00:32:16,550
+اللي بتطلع بتعتمد على الـ X لأنه Fn of X واحد لها
+
+402
+00:32:16,550 --> 00:32:20,770
+قيمة، Fn of X اتنين لها قيمة ك sequence، اتكلمت ك
+
+403
+00:32:20,770 --> 00:32:25,510
+sequence الآن صارت عندي هذه الـ limit exist و
+
+404
+00:32:25,510 --> 00:32:32,710
+بتساوي رقم الآن منها بدي أقولك define F من عند الـ
+
+405
+00:32:32,710 --> 00:32:38,010
+a اللي هي اللي لكل x element in a هذا صحيح اللي fn
+
+406
+00:32:38,010 --> 00:32:42,710
+of x converge و لها قيمة define f من a لعند ال R
+
+407
+00:32:42,710 --> 00:32:49,740
+by the limit الـ F N of X as N goes to infinity
+
+408
+00:32:49,740 --> 00:32:56,760
+بيساوي F of X طيب الـ N بدأش احنا رايحين يا
+
+409
+00:32:56,760 --> 00:33:02,440
+جماعة رايحين نثبت انه F N converts uniformly للـ F
+
+410
+00:33:02,440 --> 00:33:08,880
+on A أو بمعنى آخر رايحين نثبت ان F N ناقص F over A
+
+411
+00:33:08,880 --> 00:33:15,800
+converges لـ 0 بتنشوف مع بعض الـ N بالتالي كما قلنا
+
+412
+00:33:15,800 --> 00:33:20,200
+هذا أصغر من Y ومنه اللي هو إذا تنتجنا FN is a
+
+413
+00:33:20,200 --> 00:33:24,360
+Cauchy sequence in R ثم By theorem 3.5.4 اللي هو
+
+414
+00:33:24,360 --> 00:33:27,780
+اللي بتقول إن الـ Cauchy sequence .. a sequence is
+
+415
+00:33:27,780 --> 00:33:31,590
+Cauchy if and only if it converges 
+
+416
+00:33:31,590 --> 00:33:34,930
+sequence define f من a لعند r by f of x سواء اللي
+
+417
+00:33:34,930 --> 00:33:38,590
+هي limit f of n of x زي ما قلنا لأن if we let n in a
+
+418
+00:33:38,590 --> 00:33:42,430
+star goes to infinity الآن بدنا نودي n إلى وين هنا
+
+419
+00:33:42,430 --> 00:33:47,170
+إلى ما لا نهاية مع الاحتفاظ بال m زي ما هي عندي
+
+420
+00:33:47,170 --> 00:33:50,170
+بقدر اعمل هيك اه بقدر اعمل هيك لأنه بقدر اخد ال n
+
+421
+00:33:50,630 --> 00:33:56,570
+أكبر من M والنتين اللي هي أكبر أو يساوي H of
+
+422
+00:33:56,570 --> 00:33:59,610
+Epsilon فالان لما ال N تروح الى ماله نهاية هي حرة
+
+423
+00:33:59,610 --> 00:34:03,370
+بتروح الى ماله نهاية M بتظلها قاعدة زي ما هي الان
+
+424
+00:34:03,370 --> 00:34:11,750
+زي ما قلنا نيجي لمين نودي
+
+425
+00:34:11,750 --> 00:34:15,290
+N الى ماله نهاية وين في ال star هي ال star هي ال
+
+426
+00:34:15,290 --> 00:34:15,630
+star
+
+427
+00:34:20,280 --> 00:34:24,360
+ودي ال M إلى ما لا نهاية لما تودي ال M إلى ما لا نهاية
+
+428
+00:34:24,360 --> 00:34:28,700
+هذه بتروح limit ال M ال FN of X هتروح ل .. اللي هي
+
+429
+00:34:28,700 --> 00:34:34,060
+ال F of X بصير عندي الآن طب امبسلون ثابت مالهاش
+
+430
+00:34:34,060 --> 00:34:38,620
+علاقة إذا صار عندي الآن هذه إيش بتعطيني .. بتعطيني
+
+431
+00:34:38,620 --> 00:34:43,420
+F M of X limit ال FN of X لأنه بتدخل على الـ .. ال
+
+432
+00:34:43,420 --> 00:34:45,520
+absolute value لأن ال limit الـ .. ال absolute
+
+433
+00:34:45,520 --> 00:34:48,300
+value is a continuous function إذا بتدخل عليه إذا
+
+434
+00:34:48,300 --> 00:34:52,760
+Fm of x ناقص limit fn of x as n goes to infinity
+
+435
+00:34:52,760 --> 00:34:55,780
+أصغر أو يسوى مين إبسلون سيبك من هذه ماعلاش علاقة
+
+436
+00:34:55,780 --> 00:34:59,460
+فيها نحكي عن هذه و هذه الآن ال limit هذا إيش سمناه
+
+437
+00:34:59,460 --> 00:35:04,220
+احنا سميناه هنا عبارة عن f of x إذا صار عندي الآن
+
+438
+00:35:04,220 --> 00:35:09,300
+اللي هو f m of x ناقص f of x أصغر أو يسوى إيش
+
+439
+00:35:09,300 --> 00:35:14,110
+إبسلون إذا صار عندي الآن اللي حصلت عليه لكل ي أكبر
+
+440
+00:35:14,110 --> 00:35:19,010
+من صفر there exists h of ي بحيث أن fm of x ناقص f
+
+441
+00:35:19,010 --> 00:35:24,310
+of x أصغر سواء ي لكل m أكبر أو سواء مين؟ h of ي
+
+442
+00:35:24,310 --> 00:35:30,800
+لأن هذا إيش معناته؟ هذا صحيح لكل x إذا صار عندي زي
+
+443
+00:35:30,800 --> 00:35:34,280
+ما قلنا قبل شوية الـ Epsilon Upper Bound لهذه مدام
+
+444
+00:35:34,280 --> 00:35:37,700
+Epsilon Upper Bound إذا اللي حيصير ال Epsilon أكبر
+
+445
+00:35:37,700 --> 00:35:40,900
+أو يساوي الـ least upper bound مين هو الـ least
+
+446
+00:35:40,900 --> 00:35:45,480
+upper bound الـ norm للـ FM نقص FM يعني بمعنى آخر من
+
+447
+00:35:45,480 --> 00:35:49,880
+هنا حصلت على norm للـ FM نقص FM اللي هو اللي هو
+
+448
+00:35:49,880 --> 00:35:54,740
+أصغر أو يساوي Epsilon هذا لكل مين لكل M أكبر أو
+
+449
+00:35:54,740 --> 00:35:59,870
+يساوي اللي هو H of Epsilon وهذا مع بعض الذي حصلنا
+
+450
+00:35:59,870 --> 00:36:03,150
+عليه لكل y أكبر من 0 there exists h of epsilon
+
+451
+00:36:03,150 --> 00:36:07,290
+such that for every m أكبر أو يساوي h of epsilon فم
+
+452
+00:36:07,290 --> 00:36:13,470
+ناقص f أصغر أو يساوي y هذا معناه limit الـ F النورم
+
+453
+00:36:13,470 --> 00:36:18,690
+الـ FM ناقص الـ F over A as M goes to infinity
+
+454
+00:36:18,690 --> 00:36:25,450
+بيساوي Zero و هذا حسب اللي هو النظرية الـ FM أو الـ
+
+455
+00:36:25,450 --> 00:36:31,230
+FM converges uniformly to F وبهيك بنكون
+
+456
+00:36:31,230 --> 00:36:37,850
+حصلنا على اللي هو برهان النظرية و أثبتنا إنه لما
+
+457
+00:36:37,850 --> 00:36:44,080
+نكون اللي هي الـ condition الكوشي كريتيريا المتحقق
+
+458
+00:36:44,080 --> 00:36:51,920
+هذا بكون عندي اللي هي الـ FM أو الـ FN converges
+
+459
+00:36:51,920 --> 00:36:57,800
+uniformly to ال function اللي عندنا اللي هي F
+
+460
+00:36:57,800 --> 00:37:02,740
+و بهيك بنكون احنا انتهينا من ال section الأول
+
+461
+00:37:02,740 --> 00:37:08,000
+ثمانية واحد وإن شاء الله بنكمل المحاضرة القادمة
+
+462
+00:37:08,000 --> 00:37:09,320
+وإلى لقاء آخر
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..0bee0d37823e4f2256f61857e5e5e8cb5a5c3301
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_postprocess.srt
@@ -0,0 +1,1848 @@
+1
+00:00:04,930 --> 00:00:09,090
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:09,090 --> 00:00:12,830
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه
+
+3
+00:00:12,830 --> 00:00:19,690
+أجمعين هذه المحاضرة رقم عشرين في مساق تحليل حقيقه
+
+4
+00:00:19,690 --> 00:00:24,630
+اتنين لطلاب وطالبات الجامعة الإسلامية كلية العلوم
+
+5
+00:00:24,630 --> 00:00:32,660
+قسم رياضياتهنستكمل اليوم الحديث في chapter 8 اللي
+
+6
+00:00:32,660 --> 00:00:38,380
+هو sequences of functions بالظبط هنستكمل الحديث في
+
+7
+00:00:38,380 --> 00:00:42,800
+ال section الأول اللي وجفنا عند نقطة فيه اللي هو
+
+8
+00:00:42,800 --> 00:00:46,680
+ال section كان تحت عنوان born twice and uniform
+
+9
+00:00:46,680 --> 00:00:51,660
+convergenceوفرّجنا بين اللي هو الـ pointwise و ال
+
+10
+00:00:51,660 --> 00:00:54,540
+uniform convergence قلنا أي uniform convergence
+
+11
+00:00:54,540 --> 00:00:58,140
+هيكون pointwise ولكن ال convergence لا يجب أن تكون
+
+12
+00:00:58,140 --> 00:01:01,920
+حقيقية في الجامعة و أخدنا بعض النظريات أول اشي
+
+13
+00:01:01,920 --> 00:01:04,980
+أعرفنا شو معناته اللي هو ال pointwise convergence
+
+14
+00:01:04,980 --> 00:01:11,180
+و بعدين أخدنا مثال على اللي هي ال function g of a
+
+15
+00:01:11,180 --> 00:01:14,440
+g n of x او sequence of functions g n of x بيستوي
+
+16
+00:01:14,440 --> 00:01:18,950
+x أس انو أثبتنا إن هذه pointwise convergence على
+
+17
+00:01:18,950 --> 00:01:23,190
+الفترة المفتوحة بالنقص 1 1 لدلة g of x بالساوية 0
+
+18
+00:01:23,190 --> 00:01:29,210
+1 زي ما هي أمامنا بعد هيك جينا حكينا عن اللي هو
+
+19
+00:01:29,210 --> 00:01:34,410
+مثال آخر أخدنا limit 1 على n sin n x زائد n قلنا
+
+20
+00:01:34,410 --> 00:01:37,970
+هذه ال sequence of functions برضه converts to the
+
+21
+00:01:37,970 --> 00:01:44,540
+zero function أو كانت اللي هو هذه مثال آخرعلى الـ
+
+22
+00:01:44,540 --> 00:01:50,020
+Point Convergence على كل R الآن إيجينا بعد هيك
+
+23
+00:01:50,020 --> 00:01:55,080
+اللي أخدنا لمّا و كيف اللي هو ان ترجم شو معناته Fn
+
+24
+00:01:55,080 --> 00:02:00,400
+Converges Point Twice قولنا الـ K اللي بيلاقيها
+
+25
+00:02:00,400 --> 00:02:04,720
+بتعتمد على Y و X في ال Point Twice لإنه نقطة نقطة
+
+26
+00:02:04,720 --> 00:02:09,550
+ال convergenceو بعدين اجينا عرفنا شو معناه uniform
+
+27
+00:02:09,550 --> 00:02:12,750
+convergence و قولنا في حالة ال uniform convergence
+
+28
+00:02:12,750 --> 00:02:16,630
+هيكون عندي اللي هو ال K اللي بنلاقيها بس بتعتمد
+
+29
+00:02:16,630 --> 00:02:20,730
+على ال epsilon و الباقي زي ما هو يعني بتنفع ال K
+
+30
+00:02:20,730 --> 00:02:24,730
+هذه لكل ال Xات اللي في ال A note عشان هيك بنسمي
+
+31
+00:02:24,730 --> 00:02:28,110
+uniform convergenceوقلنا إذا كانت الـ sequence
+
+32
+00:02:28,110 --> 00:02:31,530
+uniformly convergence أكيد هتكون convergence و
+
+33
+00:02:31,530 --> 00:02:36,230
+بعدين أخدنا ال criterion أو اللمّة اللي بننفي فيها
+
+34
+00:02:36,230 --> 00:02:40,170
+أو بنقول شو معناه Fn does not converge uniformly
+
+35
+00:02:40,170 --> 00:02:46,400
+to اللي هو function Fقلنا ان الـ fn لا يتعامل مع
+
+36
+00:02:46,400 --> 00:02:51,400
+عمليات الـ fn لا يتعامل مع عمليات الـ f إذا كان
+
+37
+00:02:51,400 --> 00:02:58,280
+لجينا إبسلون و سبسيكوينس fnk of fn سبسيكوينس من
+
+38
+00:02:58,280 --> 00:03:04,040
+سيكوينس الـ functions هذه وعندي سيكوينس xkفي الـ A
+
+39
+00:03:04,040 --> 00:03:08,600
+Note بحيث تحقق F N K of X K ناقص F of X K أكبر شهر
+
+40
+00:03:08,600 --> 00:03:14,000
+Y Note هذا على بعضه يعني أن الـ F N does not
+
+41
+00:03:14,000 --> 00:03:17,320
+converge uniformly to الـ function اللي أمامنا
+
+42
+00:03:17,320 --> 00:03:22,320
+اللي هي F هذا حكينا عنه المرة الماضية و أثبتناها و
+
+43
+00:03:22,320 --> 00:03:27,240
+بعدين أخدنا مثال قلنا خد اللي هو G N of X بيساوي X
+
+44
+00:03:27,240 --> 00:03:31,670
+plus Nsequence of functions على الـ R أحنا أثبتنا
+
+45
+00:03:31,670 --> 00:03:35,810
+إنه هذه اللي هي الـ sequence قبل بشوية اللي شفناها
+
+46
+00:03:35,810 --> 00:03:43,610
+إنها converge to G of X هذه 1 0 X بساوة 1 و X
+
+47
+00:03:43,610 --> 00:03:48,450
+element نقص 1 1 converge point-wise لهذه الـ
+
+48
+00:03:48,450 --> 00:03:52,350
+functionالان و أثبتناها برضه و أثبتناها المرة
+
+49
+00:03:52,350 --> 00:03:56,370
+الماضية ان GM لا تتعامل بشكل عام لهذه الـ function
+
+50
+00:03:56,370 --> 00:04:00,070
+على الفترة وهذا تأكيد على كلامنا ان الـ point wise
+
+51
+00:04:00,070 --> 00:04:06,110
+convergence لا يتعامل بشكل عام و هي مثال على ذلك
+
+52
+00:04:06,740 --> 00:04:12,940
+الآن أعرفنا شو معنات اللي هو برضه لو كانت عندي
+
+53
+00:04:12,940 --> 00:04:16,280
+function Phi اللي هي bounded function شو معنات
+
+54
+00:04:16,280 --> 00:04:19,900
+normها قولنا ال norm لـ function A على اللي هي ال
+
+55
+00:04:19,900 --> 00:04:24,060
+6A هو عبارة عن ال supremum لقيمة ال function 5X X
+
+56
+00:04:24,060 --> 00:04:27,660
+element A هذا في حالة اللي هي mean ال bounded
+
+57
+00:04:27,660 --> 00:04:33,100
+function اللي هي اللي هي سمنها Phi بعد هيك جينا
+
+58
+00:04:34,730 --> 00:04:38,450
+وحكينا عن اللمّة اللي أمامنا هذه أنه لو كان اللي
+
+59
+00:04:38,450 --> 00:04:41,450
+هي a sequence of unbounded functions on A subset
+
+60
+00:04:41,450 --> 00:04:44,890
+من R اللي هي the sequence of meal يعني هذا test
+
+61
+00:04:44,890 --> 00:04:47,170
+للـ uniform convergence أو عدم الـ uniform
+
+62
+00:04:47,170 --> 00:04:51,670
+convergence لعائلة من الدوال أو sequence of
+
+63
+00:04:51,670 --> 00:04:55,580
+functions اللي هي of unbounded functionfunctions
+
+64
+00:04:55,580 --> 00:04:59,680
+إذن هذه بتنفع لمين للـbounded functions بتعملي
+
+65
+00:04:59,680 --> 00:05:04,280
+test عليها إنها uniform ولا مش uniform ببساطة بس
+
+66
+00:05:04,280 --> 00:05:08,680
+بنجيب norm ال FN نقص F وبنشوفه إذا كان norm مع ال
+
+67
+00:05:08,680 --> 00:05:12,700
+A بروح للسفر معناته على طول بنقول FN converges
+
+68
+00:05:12,700 --> 00:05:16,260
+uniformly to مين؟ To F إذا كان ال norm هذا does
+
+69
+00:05:16,260 --> 00:05:20,160
+not converge to zero معناته كل سنة و انت تسأل مين؟
+
+70
+00:05:20,160 --> 00:05:24,360
+اللي هو ال FN does not converge uniformly to مين؟
+
+71
+00:05:24,360 --> 00:05:29,060
+To Fهذه الآن اللي وصلنا عندها اللي هي اللي ما
+
+72
+00:05:29,060 --> 00:05:36,380
+واخدنا مثال برضه عليها اللي هو GN of XN قلنا هذا
+
+73
+00:05:36,380 --> 00:05:40,260
+اللي هو المثال
+
+74
+00:05:40,260 --> 00:05:45,890
+اللي أمامناكيف نحله على اللي هي ال .. ال .. ال ..
+
+75
+00:05:45,890 --> 00:05:49,550
+النظرية اللي قبل بشوية قولنا خد GN of X بالساوية X
+
+76
+00:05:49,550 --> 00:05:55,810
+أُس N على الفترة مين؟ سفر واحد لجينا انه .. انه GN
+
+77
+00:05:55,810 --> 00:06:00,510
+converts uniformly لمين؟ لـG of X على الفترة اللي
+
+78
+00:06:00,510 --> 00:06:05,410
+أمامنا اللي هي هذه اللي هي فترة Zero وواحد طبعا با
+
+79
+00:06:05,410 --> 00:06:09,180
+.. واضح انه الدالة هذهbounded على هذه الفترة عشان
+
+80
+00:06:09,180 --> 00:06:13,220
+هي تنفي عن السخدم اللي هي النظرية اللي فوق هذه
+
+81
+00:06:13,220 --> 00:06:17,980
+المرة الماضية برضه حكيناها أو جينا أو جفنا عند
+
+82
+00:06:17,980 --> 00:06:23,260
+اللي هي المثال التاني طبعا اللي هي المثال الفجاني
+
+83
+00:06:23,260 --> 00:06:28,300
+لو على السريع ذكرناكم فيها الان ال function مين هي
+
+84
+00:06:28,300 --> 00:06:34,460
+عندى ال function عبارة عن g n of x او sequence of
+
+85
+00:06:34,460 --> 00:06:39,410
+functions بسوء x أس nالان لو جي ده أخدت ال norm
+
+86
+00:06:39,410 --> 00:06:44,870
+لجي ان ناقص الجي هي ال جي ان وهي الجي خد ال norm
+
+87
+00:06:44,870 --> 00:06:50,730
+إلها خد ال norm إلها معايا
+
+88
+00:06:50,730 --> 00:06:54,650
+يا جماعة هي ال norm إلها لجي ان ناقص جي بساوي اللي
+
+89
+00:06:54,650 --> 00:06:59,170
+هو عبارة عن ال supremum X أس ان ناقص سفر يعني X أس
+
+90
+00:06:59,170 --> 00:07:05,750
+ان لمن؟ لل X المتزير واحد او الجزء الثاني X أس ان
+
+91
+00:07:05,750 --> 00:07:10,970
+ناقص واحداللي هي في حالة ال X بالساوية واحد لكن في
+
+92
+00:07:10,970 --> 00:07:13,490
+حالة ال X بالساوية واحد بيصير G and X and بيتساوية
+
+93
+00:07:13,490 --> 00:07:17,370
+واحد واحد نقص واحد بيصير سفر بيصير عندي ال norm
+
+94
+00:07:17,370 --> 00:07:20,490
+هذا عبارة عن ال supremum X and X في الفترة Zero
+
+95
+00:07:20,490 --> 00:07:24,340
+واحد و Zero لما ال X بالساوية واحدالـ Supremum
+
+96
+00:07:24,340 --> 00:07:29,600
+للقيمة هذه عندي أكيد إيش هيساوي عند اللي هي X
+
+97
+00:07:29,600 --> 00:07:34,380
+بتساوي واحد و Y ساوي اللي هو عبارة أو قبل اللي هي
+
+98
+00:07:34,380 --> 00:07:40,620
+ال Supremum ل ال 6 هذه طبعا هنا X element in 01 و
+
+99
+00:07:40,620 --> 00:07:44,780
+1 اللي هي not included فبكون عندي ال Supremum
+
+100
+00:07:44,780 --> 00:07:50,060
+للقيم X أسن في الفترة من 01 إيش هيساوي؟ هيساوي
+
+101
+00:07:50,060 --> 00:07:57,150
+واحد يعني ال Supremumلأ x أسن such that x element
+
+102
+00:07:57,150 --> 00:08:05,210
+in zero و واحد حتى لو كانت closed هذه و اللي هو
+
+103
+00:08:05,210 --> 00:08:08,730
+اتحاد
+
+104
+00:08:08,730 --> 00:08:13,690
+ال zero ال supreme لها طبعا لل x أسن ايش حيساو في
+
+105
+00:08:13,690 --> 00:08:17,330
+النهاية حيساو واحد و ممكن اثبات هذا اللي هو عن
+
+106
+00:08:17,330 --> 00:08:21,900
+طريق ال least upper boundكيف نثبته زي ما كنا نثبت
+
+107
+00:08:21,900 --> 00:08:26,540
+فيه اللي هو تحليل واحد مش ماكانوا نثبته لان فبصير
+
+108
+00:08:26,540 --> 00:08:30,280
+عندي ال norm لجي عن نقص جي اش بساوي واحدما دام الـ
+
+109
+00:08:30,280 --> 00:08:33,980
+norm لجي أن بيساوي واحد إذا ال norm لجي أن نقص جي
+
+110
+00:08:33,980 --> 00:08:40,920
+does not converge as if الجي أن لا يتعامل مع الجي
+
+111
+00:08:40,920 --> 00:08:47,040
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+112
+00:08:47,040 --> 00:08:54,180
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+113
+00:08:54,180 --> 00:08:54,660
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+114
+00:08:54,660 --> 00:08:55,450
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقنقص
+
+115
+00:08:55,450 --> 00:09:00,190
+الـ G برضه هيروح على مين؟ على القيمة واحد يعني مش
+
+116
+00:09:00,190 --> 00:09:05,890
+هيروح للسفر يعني إذاً GN does not converge
+
+117
+00:09:05,890 --> 00:09:10,850
+uniformly to مين؟ to G وهذا مثال عليه اللي هو does
+
+118
+00:09:10,850 --> 00:09:16,030
+not converge uniformly على الـ G مثال على اللي هو
+
+119
+00:09:16,030 --> 00:09:20,250
+توظيف هذه النظرية الجابلة بشوية لإثبات عدم ال
+
+120
+00:09:20,250 --> 00:09:23,930
+uniformlyconvergent أو لو بدنا نثبت انه formally
+
+121
+00:09:23,930 --> 00:09:29,910
+convergent ممكن نستخدم هذه بس نثبت ان ال norm بروح
+
+122
+00:09:29,910 --> 00:09:33,710
+لصفر بس بحيث انه تكون اللي هي ال GN عبارة عن
+
+123
+00:09:33,710 --> 00:09:38,350
+sequence زي ما قلنا of bounded functions طيب ميجي
+
+124
+00:09:38,350 --> 00:09:42,030
+مثال آخر ماعليش سريع لأنه حكينا المرة الماضية ميجي
+
+125
+00:09:42,030 --> 00:09:48,070
+لا اللي هو مثلنا هذانجي لمثالنا fn of x بيساوي x
+
+126
+00:09:48,070 --> 00:09:53,050
+على n مش x أسان x على n for every x element in R
+
+127
+00:09:53,050 --> 00:09:58,950
+اللي هم بيقوللي show that limit fn of x as x بتروح
+
+128
+00:09:58,950 --> 00:10:02,850
+إلى infinity بيساوي f of xلأ كل x element in R
+
+129
+00:10:02,850 --> 00:10:08,430
+where f of x ايش بتساوي سفر and fn x بتروح ل f of
+
+130
+00:10:08,430 --> 00:10:11,610
+x من R هذا معناته اللي هو pointwise convergence
+
+131
+00:10:11,610 --> 00:10:16,330
+يعني بده limit ال fn of x ناخد اي x element in R
+
+132
+00:10:16,330 --> 00:10:19,490
+ناخد x element in R نثبت ان limit fn x ايش هتساوي
+
+133
+00:10:19,490 --> 00:10:24,830
+هتساوي سفرالان هذا اللي هو عبارة عن اللي هو الـ
+
+134
+00:10:24,830 --> 00:10:30,110
+pointwise convergence بعد شوية هنثبت انه عندنا هذا
+
+135
+00:10:30,110 --> 00:10:34,530
+مش هيكون فيه عندى اللي هو uniform convergence على
+
+136
+00:10:34,530 --> 00:10:40,790
+مكان ما في الأرض نرجع لمثالنا اللي هو عندى مثال
+
+137
+00:10:40,790 --> 00:10:48,880
+اللي هو Fn X بسوية X على Nلـ X element in R عندي F
+
+138
+00:10:48,880 --> 00:10:52,820
+of X بيساوي 0 لكل X element in R بدنا نثبت أن F N
+
+139
+00:10:52,820 --> 00:10:56,500
+of X بتروح pointwise ل F of X لكل X element in R
+
+140
+00:10:56,500 --> 00:11:01,180
+وبدنا نثبت أن F N does not converge لـ F uniformly
+
+141
+00:11:01,180 --> 00:11:04,500
+لـ F on R يعني هذا هو عبارة عن pointwise
+
+142
+00:11:04,500 --> 00:11:10,170
+convergence لكن مش uniform convergenceبس هنا جيبنا
+
+143
+00:11:10,170 --> 00:11:14,310
+المثال عشان نقولك انه مش هنقدر نستخدم اللي هي
+
+144
+00:11:14,310 --> 00:11:17,630
+اللمّة اللي قبل بشوية لإنه عندي sequence of
+
+145
+00:11:17,630 --> 00:11:23,910
+functions Fn على R اللي هي is not bounded يعني لو
+
+146
+00:11:23,910 --> 00:11:29,150
+جينا أخدنا F1 of X بتساوي X على 1لكل x element in
+
+147
+00:11:29,150 --> 00:11:33,310
+R، وكأن f1 of x عبارة عن x، طبعا الـ x is not
+
+148
+00:11:33,310 --> 00:11:36,790
+bounded function، f2x التي هي x على 2 برضه is not
+
+149
+00:11:36,790 --> 00:11:40,250
+bounded، لأنها تبقى مالا نهاية رايحة من الجهتين،
+
+150
+00:11:40,250 --> 00:11:42,510
+واحد مالا نهاية و واحد سالب مالا نهاية،
+
+151
+00:11:48,910 --> 00:11:54,190
+عشان هيك ما بنقدرش نستخدم اللي هو عشان نثبت ال FN
+
+152
+00:11:54,190 --> 00:11:58,150
+does not converge uniformly بنقدرش نستخدم normal
+
+153
+00:11:58,150 --> 00:12:03,310
+FN ناقص F اللي هو يروح للسفر لإن هذه ال FN is not
+
+154
+00:12:03,530 --> 00:12:07,550
+bounded functions are not bounded functions يعني
+
+155
+00:12:07,550 --> 00:12:11,190
+اللي هو اللمّة لا تنطبق عليها عشان هيك هنضطر نروح
+
+156
+00:12:11,190 --> 00:12:15,450
+بالاستخدام التعريف أو نفي التعريف لل uniformly
+
+157
+00:12:15,450 --> 00:12:18,690
+convergence زي ما عملنا في مثال المرة الماضية
+
+158
+00:12:18,690 --> 00:12:25,290
+وخلّينا نشوف الحل نيجي الآن الجزء الأول الجزء
+
+159
+00:12:25,290 --> 00:12:32,700
+الأول سهل على طول اللي هو عندي لكل XElement in R
+
+160
+00:12:32,700 --> 00:12:40,460
+أكيد عندي limit of N of X لما N تروح إلى ماله
+
+161
+00:12:40,460 --> 00:12:47,240
+نهاية بساوي limit of N of X هي عبارة عن X على N as
+
+162
+00:12:47,240 --> 00:12:50,860
+N goes to infinity ويساوي كلكم بعرفيش هتساوي،
+
+163
+00:12:50,860 --> 00:12:56,030
+بتساوي سفريعني بمعنى آخر اللي هي الـ fn of x
+
+164
+00:12:56,030 --> 00:13:00,990
+converge لمين للـ f اللي هي of x اللي بتساوي سفر
+
+165
+00:13:00,990 --> 00:13:04,610
+وهذا ال convergence عبارة عن pointwise convergence
+
+166
+00:13:04,610 --> 00:13:10,070
+شوف الآن كيف بدنا نثبت خلصنا من انه احنا نثبت fn x
+
+167
+00:13:10,070 --> 00:13:14,590
+بتروح pointwise لمين لاللي هي ال function zero
+
+168
+00:13:14,590 --> 00:13:18,350
+بدنا نركز الآن نثبت ان fn does not converge
+
+169
+00:13:18,350 --> 00:13:24,080
+uniformly لل function f on Rاللي هو زي ما قلنا الـ
+
+170
+00:13:24,080 --> 00:13:27,940
+sequence of functions عندي هنا are not bounded
+
+171
+00:13:27,940 --> 00:13:32,620
+functions so we can't use our لمّة اللي قبل بشوية
+
+172
+00:13:32,620 --> 00:13:36,060
+اللي هي normal of a نقص of a converges to zero إذا
+
+173
+00:13:36,060 --> 00:13:41,100
+هنتضرر نروح للتعريف أو للمّة قبلها إذا بتتذكروا
+
+174
+00:13:41,100 --> 00:13:44,820
+اللمّة قلنا of a that does not convergeUniform للـ
+
+175
+00:13:44,820 --> 00:13:48,620
+F إيش بدنا نلاقي؟ بدنا نلاقي Epsilon Note أكبر من
+
+176
+00:13:48,620 --> 00:13:57,080
+0 و Sub-Sequence FNK و Sequence XK in R بحيث أنه
+
+177
+00:13:57,080 --> 00:14:03,060
+اللي هو هيها بدنا نلاقي Epsilon Note و نلاقي Sub
+
+178
+00:14:03,060 --> 00:14:06,440
+-Sequence FNK و نلاقي Sequence XK in R بحيث أنه
+
+179
+00:14:06,440 --> 00:14:14,340
+الـ Norm أو الـ Absolute Value للـ FNK of XK-F of
+
+180
+00:14:14,340 --> 00:14:21,180
+XK يكون أكبر من مين من Y note اللي عندنا اللي
+
+181
+00:14:21,180 --> 00:14:25,240
+بنقول لجميعها إذا عملنا زيك بيكون معناته أن F does
+
+182
+00:14:25,240 --> 00:14:28,900
+not converge uniformly to مين، to F خلّينا نيجي
+
+183
+00:14:28,900 --> 00:14:34,100
+إلى اللي هو ونشوف كيف نسويزي ما عملنا المرة
+
+184
+00:14:34,100 --> 00:14:39,940
+الماضية الفكرة انه الان ال fnk of xk نخليها عبارة
+
+185
+00:14:39,940 --> 00:14:44,640
+عن ثابت لو خلناها ثابت وهذه طبعا سفر بيصير الثابت
+
+186
+00:14:44,640 --> 00:14:49,600
+أكبر بيطلع ثابت خمسة ستة سبعة واحد اتنين بنختار
+
+187
+00:14:49,600 --> 00:14:52,600
+ابسلونوت اللي هي اصغر منه بيطلع حد اكبر من
+
+188
+00:14:52,600 --> 00:14:58,660
+الابسلونوت بيكون خلصناه ده نشوف كيف عندي fn of x
+
+189
+00:14:58,660 --> 00:15:04,320
+ايش بالساوي x على nلو جينا اخترنا اللي هو الـ NK
+
+190
+00:15:04,320 --> 00:15:08,060
+الـ NK الـ subsequence يعني اللي هي الـ NK
+
+191
+00:15:08,060 --> 00:15:13,540
+اخترناها الـ K بيصير اللي هي FNK اللي هي of X
+
+192
+00:15:13,540 --> 00:15:18,620
+بيساوي FK of X بيصير هذا K X على K طب ال X ايش
+
+193
+00:15:18,620 --> 00:15:25,240
+نختارها عشان نخلي هذا ثابت اختارهاK , 2K , 3K زي
+
+194
+00:15:25,240 --> 00:15:30,480
+ما بدك كله حيكون وين موجود في R وكله حيثبت لمين
+
+195
+00:15:30,480 --> 00:15:33,820
+اللي هي ال function اللي عندى و تصير أموري تمام
+
+196
+00:15:33,820 --> 00:15:39,840
+تمام ايش عامل هنا بقول ليه let F and K او let NK
+
+197
+00:15:39,840 --> 00:15:47,440
+هي let NK بساوي K يعني بمعنى اخر اخد ال F and K هو
+
+198
+00:15:47,440 --> 00:15:53,360
+او بتعبير اخر F خد ال F and K هي ال FK اختار الـ
+
+199
+00:15:53,360 --> 00:15:57,320
+FNK المميزة بـ Sequence اللي هي FK و Epsilon
+
+200
+00:15:57,320 --> 00:16:00,800
+خلّينا بعدين نختارها و XK زي ما قلنا ايش راح
+
+201
+00:16:00,800 --> 00:16:06,820
+نختارها بتساوي K أكيد وين موجودة في R then FNK of
+
+202
+00:16:06,820 --> 00:16:14,380
+XK ايش هتساوي اللي هي FK of XK ويساوي اللي هي
+
+203
+00:16:14,380 --> 00:16:20,080
+عبارة عن XKعلى k حسب اللي هي الدقلة اللي عندى او
+
+204
+00:16:20,080 --> 00:16:22,720
+ال sequence of functions طب ما هي ال xk ايش انا
+
+205
+00:16:22,720 --> 00:16:27,420
+بختارها بتساوي k فبصير k على k هو يساوي واحد اذا
+
+206
+00:16:27,420 --> 00:16:34,210
+طلعت عندى fnk of xk بيساوي واحدهيحسب للـ fnk of xk
+
+207
+00:16:34,210 --> 00:16:37,410
+للـ subsequence اللي عملناها ناقص f of xk إيش
+
+208
+00:16:37,410 --> 00:16:41,770
+هتساوي هذه عبارة عن واحد وهذه دالة دايما ثابتة f
+
+209
+00:16:41,770 --> 00:16:45,610
+عند أي قيمة بتساوي سفر فبصير اللي هو عبارة عن إيش
+
+210
+00:16:45,610 --> 00:16:49,030
+القيمة واحد إيش أختار epsilon note اللي بدك إياها
+
+211
+00:16:49,030 --> 00:16:54,680
+أي إيش أصغر منيه من الواحد وليكن إيش نصفبصير عندى
+
+212
+00:16:54,680 --> 00:17:00,140
+FN K هيها FK و Epsilon بساوة نص و XK بساوة K هي
+
+213
+00:17:00,140 --> 00:17:05,460
+sub sequence من ال FN و هي الي هو Epsilon نص و هي
+
+214
+00:17:05,460 --> 00:17:09,620
+sequence XK in R بحيث أن هذا ال absolute value
+
+215
+00:17:09,620 --> 00:17:13,660
+أكبر من مين؟ اللي هو طلع واحد اللي هو أكيد أكبر من
+
+216
+00:17:13,660 --> 00:17:17,900
+إيش؟ من نص اللي هي Epsilon نص وهذا يعني أن FN does
+
+217
+00:17:17,900 --> 00:17:24,390
+not converge uniformly للF on Rهكذا أثبتنا أن الـ
+
+218
+00:17:24,390 --> 00:17:27,310
+Fn لا يتعامل بشكل عام بشكل عام بشكل عام بشكل عام
+
+219
+00:17:27,310 --> 00:17:29,790
+بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل
+
+220
+00:17:29,790 --> 00:17:31,210
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+221
+00:17:31,210 --> 00:17:31,710
+بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل
+
+222
+00:17:31,710 --> 00:17:32,470
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+223
+00:17:32,470 --> 00:17:34,290
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+224
+00:17:34,290 --> 00:17:36,760
+بشكل عام بشكل عام بشكل عام بشكل عامأو الثالث في
+
+225
+00:17:36,760 --> 00:17:42,160
+المثال، بدنا نثبت أن الـ F Unconverges للـ F
+
+226
+00:17:42,160 --> 00:17:48,640
+Uniformly على الفترة مين؟ على الفترة 01، يعني لما
+
+227
+00:17:48,640 --> 00:17:53,080
+نغيرنا الفترة، هيصير اللي هو الـ Pointwise ماقدرش
+
+228
+00:17:53,080 --> 00:17:57,040
+اللي هو الـ Uniform convergence على الـ R، ماقدرش
+
+229
+00:17:57,200 --> 00:18:01,420
+اللي هو يكون uniform convergence لكن لو زغرنا
+
+230
+00:18:01,420 --> 00:18:06,420
+الفترة أو غيرنا الفترة ممكن نحصل على اللي هو ال
+
+231
+00:18:06,420 --> 00:18:11,480
+uniform convergence وهذا طبيعي، هنشوف كيف، الان
+
+232
+00:18:11,480 --> 00:18:16,640
+عشان اللي هو نظريتنا سهلة الاستخدام اللي قبل بشوية
+
+233
+00:18:16,640 --> 00:18:22,580
+اللي هو normal Fن ناقص F بدي أحسبه و أشوفه على
+
+234
+00:18:22,580 --> 00:18:27,240
+الفترة 0 و 1 اللي هو conversion to 0 ولا لأ؟ آه
+
+235
+00:18:27,240 --> 00:18:30,520
+الآن قبلها بدي أشوف ال sequence of functions عندي
+
+236
+00:18:30,520 --> 00:18:36,520
+هذه هل bounded على الفترة 0 و 1؟ Fn X بتساوي X على
+
+237
+00:18:36,520 --> 00:18:44,140
+N يعني صار عندي FN of X بيساوي X على N لوين X في
+
+238
+00:18:44,140 --> 00:18:47,860
+الفترة 0 و 1 خلاص الـ X في الفترة 0 و 1 جد ما تكون
+
+239
+00:18:47,860 --> 00:18:51,600
+ليها الـ X نفسها محصورة بين 0 و 1 يعني بتضل الـ X
+
+240
+00:18:51,600 --> 00:18:57,040
+على N عبارة عن قيمة محدودة قيمة محدودة أكيد اللي
+
+241
+00:18:57,040 --> 00:19:00,680
+هو الـ absolute value اللي هي هذه أصغر أو يساوي
+
+242
+00:19:00,680 --> 00:19:05,100
+اللي هو 1 على N إذا صار عند .. لكل .. طبعا هذا
+
+243
+00:19:05,100 --> 00:19:10,510
+محدودة بالنسبة لـ Xأه بيصير F1 أصغر بيساوي 1 F2
+
+244
+00:19:10,510 --> 00:19:15,350
+أصغر بيساوي نص F3 أصغر بيساوي تلت و هكذا إذن هي
+
+245
+00:19:15,350 --> 00:19:21,790
+عبارة عن F1 F2 F3 sequence of bounded functions on
+
+246
+00:19:21,790 --> 00:19:26,570
+the interval 0 و 1 إذن الآن الأمر مشروع ليه أني
+
+247
+00:19:26,570 --> 00:19:31,970
+أستخدم اللمّة اللي حكينا عنها قبل بشوية و نشوف كيف
+
+248
+00:19:31,970 --> 00:19:36,110
+نثبت ال uniform convergence نحسب مع بعض أكيد لكلكم
+
+249
+00:19:36,440 --> 00:19:43,380
+اللي هو Fn ناقص الـ F over A بساوي الـ Supremum لـ
+
+250
+00:19:43,380 --> 00:19:50,380
+Fn of X ناقص F of X Such that X و M هو وجودة في
+
+251
+00:19:50,380 --> 00:19:57,120
+الفترة 0 و 1 و يساوي الـ Supremum لـ Fn of X اللي
+
+252
+00:19:57,120 --> 00:20:03,270
+هي أشكل عنها عبارة عن X على Nx على n ناقصها دي سفر
+
+253
+00:20:03,270 --> 00:20:07,110
+بتضل x على n such that x where موجودة في ال zero
+
+254
+00:20:07,110 --> 00:20:10,330
+أو إيش أو واحد ال suprem أهم إلها زي ما تقولنا
+
+255
+00:20:10,330 --> 00:20:14,790
+أشملها أصغرشها واحد على n هتطلع عبارة عن أ واحد
+
+256
+00:20:14,790 --> 00:20:21,670
+على nهذه اللي هي مين ال norm للأف أن ناقص أف طيب
+
+257
+00:20:21,670 --> 00:20:26,530
+واضح الأمر مدام ال norm للأف أن ناقص أف بسوى حتى
+
+258
+00:20:26,530 --> 00:20:35,220
+الأن إذا norm للأف أن ناقص أفA goes to zero لأن FN
+
+259
+00:20:35,220 --> 00:20:38,880
+goes to infinity قيمة هذا الواحدة الان هتروح إلى
+
+260
+00:20:38,880 --> 00:20:42,960
+سفر يعني norm هذا بروح للزيرو وهذا يعني أن FN
+
+261
+00:20:42,960 --> 00:20:49,240
+converts uniformly ل F on الفترة زيرو وواحد وبكون
+
+262
+00:20:49,240 --> 00:20:54,180
+هيك أثبتنا اللي هو باستخدام النظرية أن هذا المثال
+
+263
+00:20:54,180 --> 00:20:58,680
+فعلااللي هو F Unconverged uniformly للـ F On The
+
+264
+00:20:58,680 --> 00:21:03,380
+Interval 01 But It Does Not Converge Uniformly On
+
+265
+00:21:03,380 --> 00:21:10,120
+R ككل لكن Pointwise على الـ R To The كل طيب نيجي
+
+266
+00:21:10,120 --> 00:21:16,300
+الآن اللي هوالتعبير آخر على الـ uniform
+
+267
+00:21:16,300 --> 00:21:19,880
+convergence عبرنا عن الـ uniform convergence
+
+268
+00:21:19,880 --> 00:21:26,180
+بواسطة اللي هي التعريف عبرنا عنه للعائلة of
+
+269
+00:21:26,180 --> 00:21:30,890
+bounded functions بواسطة الـ norm للأفئناللي هو
+
+270
+00:21:30,890 --> 00:21:37,190
+ناقص الـ F بروح للـ 0 الآن بدنا نعرف عنه .. نعبر
+
+271
+00:21:37,190 --> 00:21:40,070
+عنه عن طريق .. برضه للـ sequence of bounded
+
+272
+00:21:40,070 --> 00:21:43,230
+functions عن طريق حاجة اسمها الـ Cauchy criterion
+
+273
+00:21:43,230 --> 00:21:47,270
+for uniform convergence وكأنه بدنا ندخل المفهوم
+
+274
+00:21:47,270 --> 00:21:53,590
+الـ Cauchy اللي هي sequence في هذا اللي هو النظرية
+
+275
+00:21:53,590 --> 00:21:57,470
+نشوفكوشي كريتيريا لتحويل المستوى المستوى المستوى
+
+276
+00:21:57,470 --> 00:21:58,350
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+277
+00:21:58,350 --> 00:22:03,130
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+278
+00:22:03,130 --> 00:22:06,950
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+279
+00:22:06,950 --> 00:22:10,330
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+280
+00:22:10,330 --> 00:22:10,350
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+281
+00:22:10,350 --> 00:22:10,410
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+282
+00:22:10,410 --> 00:22:10,430
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+283
+00:22:10,430 --> 00:22:15,740
+المستوى المستF يعني بتكون F unconverted لـ bounded
+
+284
+00:22:15,740 --> 00:22:20,220
+function F F and only F لكل Y أكبر من 0 there
+
+285
+00:22:20,220 --> 00:22:23,680
+exists H of Y للـ N such that for every M و N أكبر
+
+286
+00:22:23,680 --> 00:22:28,180
+يساوي H of Y حيكون الـ FM مقصلة F أصغر أو يساوي Y
+
+287
+00:22:28,180 --> 00:22:32,940
+وهذا هيذكرنا اللي هو بالكوشي sequences إذا نسيناها
+
+288
+00:22:33,270 --> 00:22:37,270
+يعني وكأنه خلّيني نقوله roughly بيقول لـ F N تحت
+
+289
+00:22:37,270 --> 00:22:42,950
+الظروف اللي موجودة بتروح للـ F uniformly يكافئ أنه
+
+290
+00:22:42,950 --> 00:22:47,510
+لكل إبسلون أكبر من 0 هذا الـ N الجزء هنا هو اللي
+
+291
+00:22:47,510 --> 00:22:51,070
+يكوش criterion في الواقع for every إبسلون أكبر من
+
+292
+00:22:51,070 --> 00:22:56,950
+0 there exists H of إبسلون element in N such that
+
+293
+00:22:56,950 --> 00:23:03,680
+for every N و M element in Nأو اللي هي أكبر أو
+
+294
+00:23:03,680 --> 00:23:08,140
+يساوي اتشوف إبسلون حيكون عندك الـ epsilon norm لل
+
+295
+00:23:08,140 --> 00:23:17,100
+FM ناقص FM أصغر من 100 أو يساوي اللي هو إبسلونهذه
+
+296
+00:23:17,100 --> 00:23:20,180
+اللي هي عبارة عن الـ Cauchy criterion يعني أحيانًا
+
+297
+00:23:20,180 --> 00:23:24,160
+نحن نعرف .. نكونش عارفين الـ Fn لمين رايحة لكن
+
+298
+00:23:24,160 --> 00:23:27,380
+متأكدين أنها converges uniformly to some function
+
+299
+00:23:27,380 --> 00:23:32,600
+F طب الآن في هذه الحالة اللي بصعفنا في الاستخدام
+
+300
+00:23:32,600 --> 00:23:38,480
+اللي هو أنه ناخد اللي هو بدلالات ال terms اللي هي
+
+301
+00:23:38,480 --> 00:23:42,440
+ال Fn و ال Fm و نبعد عن ال F لأنه احنا مش عارفينها
+
+302
+00:23:42,440 --> 00:23:45,260
+عشان هيك بنقول Fn converges uniformly to some
+
+303
+00:23:45,260 --> 00:23:50,340
+function Fيعني هيك تصرفها إذا و فقط إذا اللي هو
+
+304
+00:23:50,340 --> 00:23:53,920
+لكل إبسلون أكبر من سفر الـ Epsilon element in N
+
+305
+00:23:53,920 --> 00:23:56,300
+such that forever N و M أكبر من سفر الـ Epsilon
+
+306
+00:23:56,300 --> 00:23:59,680
+بيعطينا norm هذه أصغر من minimum إبسلون و كأنه
+
+307
+00:23:59,680 --> 00:24:03,650
+بنقول اللي هو ال function FNis a Cauchy sequence
+
+308
+00:24:03,650 --> 00:24:08,250
+roughly in the space of functions اللي هي ال norm
+
+309
+00:24:08,250 --> 00:24:11,470
+بساوي ال supremum اللي هو هذا الموضوع طبعا مش
+
+310
+00:24:11,470 --> 00:24:15,170
+موضوعنا موضوع ناخده بالتفصيل في اللي هو ال
+
+311
+00:24:15,170 --> 00:24:19,870
+functional analysis طيب شوفوا علينا نيجي اللي هو
+
+312
+00:24:19,870 --> 00:24:26,710
+انبرهن النظرية بدنا نفترض ان ال FNالان بدنا نفترض
+
+313
+00:24:26,710 --> 00:24:31,970
+انه norm أو الـ F un converge uniformly to some
+
+314
+00:24:31,970 --> 00:24:37,790
+function F و نصل لهذا اللي هو الجزء سهل ان شاء
+
+315
+00:24:37,790 --> 00:24:44,450
+الله شوفوا ركزولنا الان suppose that F un converge
+
+316
+00:24:44,450 --> 00:24:49,390
+uniformly ل F on A إذا الان من التعريف تبع ال
+
+317
+00:24:49,390 --> 00:24:54,480
+uniform convergence لكل Y أكبر من 0There exists K
+
+318
+00:24:54,480 --> 00:24:58,420
+of Y على 2 element in N such that for every N أكبر
+
+319
+00:24:58,420 --> 00:25:03,180
+أو يساوي K of Y ولكل X element in A برضه بظبط اللي
+
+320
+00:25:03,180 --> 00:25:08,180
+هو FN of X ناقص F of X أصغر من Y على 2 هذا
+
+321
+00:25:08,180 --> 00:25:11,000
+باستخدام التعريف الـ N باستخدام النظرية اللي قبل
+
+322
+00:25:11,000 --> 00:25:14,720
+بشوية لإنه بنفع لإن النظرية احنا قلنا FN converges
+
+323
+00:25:14,720 --> 00:25:19,540
+uniformly إيش معناته هذا؟معناته more من الـ F and
+
+324
+00:25:19,540 --> 00:25:25,720
+ناقص الـ F goes to 0 over mean over a هذه صار
+
+325
+00:25:25,720 --> 00:25:30,160
+تعريف عادي للـ limit إيش معناته هذا لكل إبسلون
+
+326
+00:25:30,160 --> 00:25:35,980
+أكبر من سفر there exists اللي هو مسميها هو إيش
+
+327
+00:25:35,980 --> 00:25:42,440
+مسميها K of إبسلون على 2 element in M such that
+
+328
+00:25:43,150 --> 00:25:50,370
+اللي هو norm الـ Fn ناقص الـ F هيكون أصغر من إبسل
+
+329
+00:25:50,370 --> 00:25:54,630
+على اتنين for every N أكبر أو يساوي مين K of إبسل
+
+330
+00:25:54,630 --> 00:25:57,770
+على اتنين اللي عندك هذا اللي هو تعريف الدهانية
+
+331
+00:25:57,770 --> 00:26:01,590
+العادية اللي هو sequence O السيكوانس العادية هذه
+
+332
+00:26:01,590 --> 00:26:04,440
+الـ sequence converts to zero لأنها أعداداللي هي
+
+333
+00:26:04,440 --> 00:26:06,560
+converse to zero إذا لكل ي أكبر من صفر there
+
+334
+00:26:06,560 --> 00:26:10,740
+exists K بحيث أنه F أن ناقص F أصغر من ي على اتنين
+
+335
+00:26:10,740 --> 00:26:19,460
+لكل N أكبر أو سوء K of ي على اتنين طيب الآن عندي
+
+336
+00:26:19,460 --> 00:26:25,520
+ال .. ال .. ال F هذه بمقدر نترجمها اللي هو على طول
+
+337
+00:26:25,520 --> 00:26:31,660
+مباشرة إذا اللي هي عندي ال Fما دام هذا أصغر يساوي
+
+338
+00:26:31,660 --> 00:26:35,880
+Y على 2 يعني ال suprem أصغر يساوي Y على 2 وهذا
+
+339
+00:26:35,880 --> 00:26:40,360
+نفسه أكبر أوي يساوي ايش عارفين انتوا Fn of X ناقص
+
+340
+00:26:40,360 --> 00:26:44,280
+F of X، مظبوط؟ لأن هذا ال suprem ملهن، إذا صار
+
+341
+00:26:44,280 --> 00:26:53,530
+عندي إذا Fnنقص f of x صار نفسه أصغر من مين من y
+
+342
+00:26:53,530 --> 00:27:00,550
+على 2 خليها في الذاكرة الآن عندي احسبلي بناء عليه
+
+343
+00:27:00,550 --> 00:27:09,430
+احسبلين of X ناقص FM of X هذا لمين؟ هذا حسب ليه
+
+344
+00:27:09,430 --> 00:27:16,230
+لمين؟ لكل أنه طبعا هذا الكلام صحيح لكل X وين؟ لكل
+
+345
+00:27:16,230 --> 00:27:22,340
+X في الـA تبعتنااحسبلي لكل N و M أكبر من مين من
+
+346
+00:27:22,340 --> 00:27:26,620
+الـ K اللي لجيها هنا K of Epsilon على 2 F N of X
+
+347
+00:27:26,620 --> 00:27:30,580
+ناقص F M of X أكيد أصغر أو يساوي بالـ Triangling
+
+348
+00:27:30,580 --> 00:27:38,940
+Quality F N of X ناقص F of X زائد F M of X ناقص F
+
+349
+00:27:38,940 --> 00:27:43,560
+of X واضح عشان سويت ضفت الـ F of X وطرحتها و بعدين
+
+350
+00:27:43,560 --> 00:27:48,460
+أخدت Triangling Quality الآن هذاصحيح أصغر من Y على
+
+351
+00:27:48,460 --> 00:27:55,590
+2 لكل Nلكل N أكبر من K طب ما أنا مختار ال N و ال M
+
+352
+00:27:55,590 --> 00:27:59,830
+أكبر من K إذن بنطبق عليها هذا سواء لهذه أو هذه إذن
+
+353
+00:27:59,830 --> 00:28:04,630
+هذه صارت أصغر من Y على 2 وهذه أصغر من Y على 2 يعني
+
+354
+00:28:04,630 --> 00:28:10,530
+صار ال FN of X ناقص FMA of X أشماله أصغر من Y أصغر
+
+355
+00:28:10,530 --> 00:28:15,110
+من مين؟ من Y يعني الآن اللي وصلت إليه ما يالي يا
+
+356
+00:28:15,110 --> 00:28:19,590
+جماعة بدأت بال FN بتروح لل F اللي هو uniformly
+
+357
+00:28:20,520 --> 00:28:25,460
+ووصلت إلى مالي وصلت إنه لكل epsilon أكبر من 0 لجيت
+
+358
+00:28:25,460 --> 00:28:30,480
+K of epsilon على 2 element in N such that for
+
+359
+00:28:30,480 --> 00:28:35,680
+every N و M أكبر أو يساوي K of epsilon على 2 لجيت
+
+360
+00:28:35,680 --> 00:28:46,360
+إن ال FN of Xنقص fm of x أصغر من مين من إبسلون هذا
+
+361
+00:28:46,360 --> 00:28:52,420
+لكل x وين موجودة في الـA ماشي الحال إذا مدام هذا
+
+362
+00:28:52,420 --> 00:28:57,140
+أصغر من إبسلون لكل x element in A صارت الإبسلون
+
+363
+00:28:57,140 --> 00:29:01,030
+عبارة عن upper bound لهذهإذاً الـ Epsilon هتكون
+
+364
+00:29:01,030 --> 00:29:05,050
+أكبر أو تساوي ال least upper bound إذاً يعني بمعنى
+
+365
+00:29:05,050 --> 00:29:09,850
+آخر إذا ال least upper bound اللي هو FN ناقص FM
+
+366
+00:29:09,850 --> 00:29:15,190
+نفسه أصغر من مين أصغر أو يساوي Epsilon وهذا يعني
+
+367
+00:29:15,190 --> 00:29:19,920
+إنه احنافعلاً لجينا أنه في حالة الـ F N conversion
+
+368
+00:29:19,920 --> 00:29:23,960
+form للـ F لجينا لكل Y أكبر من 0 there exists K
+
+369
+00:29:23,960 --> 00:29:29,260
+بحيث لكل N أ أو M أكبر سوء K يطلع عندي Normal FN
+
+370
+00:29:29,260 --> 00:29:34,080
+نقص FM أصغر من مين من إبسلون وهو المطلوب وطبعاً في
+
+371
+00:29:34,080 --> 00:29:37,360
+الآخر جالك ت choose اللي هي K of Epsilon هي اسمها
+
+372
+00:29:37,360 --> 00:29:40,880
+H of Epsilon كان ممكن الأول نسميها H of Epsilon
+
+373
+00:29:40,880 --> 00:29:45,880
+كان مجرد تسميةالان نيجي نمين اللي هو the converse
+
+374
+00:29:45,880 --> 00:29:49,460
+of the theorem conversely بدنا نفترض ان ال
+
+375
+00:29:49,460 --> 00:29:52,820
+condition متحقق و نثبت اللي هو ال uniform
+
+376
+00:29:52,820 --> 00:30:08,320
+convergence الان
+
+377
+00:30:08,320 --> 00:30:13,390
+بدنا نفترضإن سبب ذات لكل Y أكبر من 0 there exists
+
+378
+00:30:13,390 --> 00:30:18,730
+H of Y بحيث إنه لما M و N تكون أكبر من H of Y
+
+379
+00:30:18,730 --> 00:30:24,710
+هيعطيني ال F ماطس أفن أصغر سوى Y هذا على بعض بدي
+
+380
+00:30:24,710 --> 00:30:32,290
+أثب من خلاله إنه لازم ال F أن تروح لل F uniformly
+
+381
+00:30:32,290 --> 00:30:36,230
+شوف كيف بده يعمل وهو الفكرة سهلة إن شاء الله
+
+382
+00:30:36,230 --> 00:30:42,630
+وبسيطة والبرهان كمان سلسانشوف ايش بقول البرهان
+
+383
+00:30:42,630 --> 00:30:47,370
+تبعنا بقول ما يليه الان
+
+384
+00:30:48,590 --> 00:30:53,150
+ما دام هذا الشرط هيوه، هذا الشرط متحقق، إذا لكل X
+
+385
+00:30:53,150 --> 00:30:59,790
+element إيه؟ عندي أكيد الـFM of X ناقص FN X أصغر
+
+386
+00:30:59,790 --> 00:31:03,910
+أو ساوي FM ناقص FN، لأن هذا هو الـSupremum لهدولة
+
+387
+00:31:03,910 --> 00:31:07,990
+كلهم، إذا صار عندي هذا أكبر أو ساوي هذا، إذا صار
+
+388
+00:31:07,990 --> 00:31:13,870
+هذا أصغر من مين؟ من إبسلونإيش معناته هذا؟ صارت
+
+389
+00:31:13,870 --> 00:31:18,270
+الآن الـ F .. بنحكي عن لكل X element in A X
+
+390
+00:31:18,270 --> 00:31:22,990
+arbitrarily back-fixed صارت عندي الـ F M of X بناء
+
+391
+00:31:22,990 --> 00:31:26,130
+على هذا الحديث مذاب الـ Absolute Value اللي قالها
+
+392
+00:31:26,130 --> 00:31:30,690
+الـ F M ناقص F N أصغر من Y يعني بمعنى آخر لكل Y
+
+393
+00:31:30,690 --> 00:31:34,710
+أكبر من 0 لجينا H of Y بحيث لما تكون M و N أكبر من
+
+394
+00:31:34,710 --> 00:31:39,330
+H of Y طلعلي الـ F M of X ناقص F N of X أصغر من
+
+395
+00:31:39,330 --> 00:31:44,990
+100 من Yصارت الـ F M of X أو F N of X زي ما بدك
+
+396
+00:31:44,990 --> 00:31:49,850
+تسميها صارت عبارة عن كوشي سيكوانس مدام كوشي
+
+397
+00:31:49,850 --> 00:31:55,470
+سيكوانس إذا حسب كوشي سيكوانس of numbers إذا أكيد
+
+398
+00:31:55,470 --> 00:32:01,250
+الـ F N of X converts يعني السيكوانس هذه إيش مالها
+
+399
+00:32:01,250 --> 00:32:09,130
+converts خلّينا نسمي limitالـ Fn of X بيساوي F of
+
+400
+00:32:09,130 --> 00:32:13,110
+X as N goes to infinity ليش F of X؟ لأنه القيمة
+
+401
+00:32:13,110 --> 00:32:16,550
+اللي بتطلع بتعتمد على الـ X لأنه Fn of X واحد لها
+
+402
+00:32:16,550 --> 00:32:20,770
+قيمة، Fn of X اتنين لها قيمة ك sequence، اتكلمت ك
+
+403
+00:32:20,770 --> 00:32:25,510
+sequence الآن صارت عندي هذه ال limit exist و
+
+404
+00:32:25,510 --> 00:32:32,710
+بتساوي رقم الآن منها بدي أقولك define Fمن عند الـ
+
+405
+00:32:32,710 --> 00:32:38,010
+a اللي هي اللي لكل x element in a هذا صحيح اللي fn
+
+406
+00:32:38,010 --> 00:32:42,710
+of x converge ويلها قيمة define f من a لعند ال R
+
+407
+00:32:42,710 --> 00:32:49,740
+by limitالـ F N of X as N goes to infinity
+
+408
+00:32:49,740 --> 00:32:56,760
+بيساومين F of X طيب الـ N بدأاش احنا رايحين يا
+
+409
+00:32:56,760 --> 00:33:02,440
+جماعة رايحين نثبت انه F N converts uniformly للـ F
+
+410
+00:33:02,440 --> 00:33:08,880
+on A او بمعنى اخر رايحين نثبت ان F N ناقص F over A
+
+411
+00:33:08,880 --> 00:33:15,800
+converts لـ 0 بتنشوف مع بعض الـ Nبالتالي كما قلنا
+
+412
+00:33:15,800 --> 00:33:20,200
+هذا أصغر من Y ومنه اللي هو إذا تنتجنا FN is a
+
+413
+00:33:20,200 --> 00:33:24,360
+Cauchy sequence in R ثم By theorem 3.5.4 اللي هو
+
+414
+00:33:24,360 --> 00:33:27,780
+اللي بتقول إن الـ Cauchy sequence .. a sequence is
+
+415
+00:33:27,780 --> 00:33:31,590
+Cauchy فندولي if it convergesis convergent
+
+416
+00:33:31,590 --> 00:33:34,930
+sequence define f من a لعنى r by f of x سواء اللي
+
+417
+00:33:34,930 --> 00:33:38,590
+ميت f of n of x زي ما قلنا لأن if we let n in a
+
+418
+00:33:38,590 --> 00:33:42,430
+star goes to infinity الآن بدنا نودي n إلى وين هنا
+
+419
+00:33:42,430 --> 00:33:47,170
+إلى ما لا نهاية مع الاحتفاظ بال m زي ما هي عندي
+
+420
+00:33:47,170 --> 00:33:50,170
+بقدر اعمل هيك اه بقدر اعمل هيك لأنه بقدر اخد ال n
+
+421
+00:33:50,630 --> 00:33:56,570
+أكبر من M والتنتين اللي هى أكبر أو يساوي H of
+
+422
+00:33:56,570 --> 00:33:59,610
+Epsilon فالان لما ال N تروح الى ماله نهاية هى حرة
+
+423
+00:33:59,610 --> 00:34:03,370
+بتروح الى ماله نهاية M بتظلها جاعدة زى ما هى الان
+
+424
+00:34:03,370 --> 00:34:11,750
+زى ما قلنا نيجى لمين نودي
+
+425
+00:34:11,750 --> 00:34:15,290
+N الى ماله نهاية وين في ال star هى ال star هى ال
+
+426
+00:34:15,290 --> 00:34:15,630
+star
+
+427
+00:34:20,280 --> 00:34:24,360
+ودي ال M إلى ما لنهاية لما تودي ال M إلى ما لنهاية
+
+428
+00:34:24,360 --> 00:34:28,700
+هذه بتروح limit ال M ال FN of X هتروح ل .. اللي هي
+
+429
+00:34:28,700 --> 00:34:34,060
+ال F of X بصير عندي الآن طب امبسلون ثابت مالهاش
+
+430
+00:34:34,060 --> 00:34:38,620
+علاقة إذا صار عندي الآن هذه إيش بتعطيني .. بتعطيني
+
+431
+00:34:38,620 --> 00:34:43,420
+F M of X limit ال FN of X لأنه بتدخل على ال .. ال
+
+432
+00:34:43,420 --> 00:34:45,520
+absolute value لأن ال limit ال .. ال absolute
+
+433
+00:34:45,520 --> 00:34:48,300
+value is a continuous function إذا بتدخل عليه إذا
+
+434
+00:34:48,300 --> 00:34:52,760
+Fm of x ناقص limit fn of x as n goes to infinity
+
+435
+00:34:52,760 --> 00:34:55,780
+أصغر أو يسوى مين إبسلون سيبك من هذه ماعلاش علاقة
+
+436
+00:34:55,780 --> 00:34:59,460
+فيها نحكي عن هذه و هذه الآن ال limit هذا إيش سمناه
+
+437
+00:34:59,460 --> 00:35:04,220
+إحنا سمناه هنا عبارة عن f of x إذا صار عندي الآن
+
+438
+00:35:04,220 --> 00:35:09,300
+اللي هو f m of x ناقص f of x أصغر أو يسوى إيش
+
+439
+00:35:09,300 --> 00:35:14,110
+إبسلونإذا صار عندي الآن اللي حصلت عليه لكل ي أكبر
+
+440
+00:35:14,110 --> 00:35:19,010
+من سفر there exists h of ي بحيث أن fm of x ناقص f
+
+441
+00:35:19,010 --> 00:35:24,310
+of x أصغر سواء ي لكل m أكبر أو سواء مين؟ h of ي
+
+442
+00:35:24,310 --> 00:35:30,800
+لأن هذا إيش معناته؟ هذا صحيح لكل xإذا صار عندى زى
+
+443
+00:35:30,800 --> 00:35:34,280
+ما قلنا قبل شوية الـ Epsilon Upper Bound لهذه مدام
+
+444
+00:35:34,280 --> 00:35:37,700
+Epsilon Upper Bound إذا اللى حيصير ال Epsilon أكبر
+
+445
+00:35:37,700 --> 00:35:40,900
+أو يساوي ال least upper bound مين هو ال least
+
+446
+00:35:40,900 --> 00:35:45,480
+upper bound ال norm لل FM نقص FM يعني بمعنى آخر من
+
+447
+00:35:45,480 --> 00:35:49,880
+هنا حصلت على norm لل FM نقص FM اللى هو اللى هو
+
+448
+00:35:49,880 --> 00:35:54,740
+أصغر أو يساوي Epsilon هذا لكل مين لكل M أكبر أو
+
+449
+00:35:54,740 --> 00:35:59,870
+يساوي اللى هو H of Epsilon وهذا مع بعضالذي حصلنا
+
+450
+00:35:59,870 --> 00:36:03,150
+عليه لكل y أكبر من 0 there exists h of epsilon
+
+451
+00:36:03,150 --> 00:36:07,290
+such that for every m أكبر أو سوى h of epsilon فم
+
+452
+00:36:07,290 --> 00:36:13,470
+ناقص f أصغر أو سوى y هذا معناه limit ال Fالنورم
+
+453
+00:36:13,470 --> 00:36:18,690
+الـ FM ناقص الـ F over A as M goes to infinity
+
+454
+00:36:18,690 --> 00:36:25,450
+بساوة Zero و هذا حسب اللي هو النظرية الـ FM أو الـ
+
+455
+00:36:25,450 --> 00:36:31,230
+FM converts uniformly to mean to F وبهيك بنكون
+
+456
+00:36:31,230 --> 00:36:37,850
+حصلنا على اللي هو برهان النظرية و أثبتنا إنه لما
+
+457
+00:36:37,850 --> 00:36:44,080
+نكون اللي هيالـ condition الكوشي كريتيريا المتحقق
+
+458
+00:36:44,080 --> 00:36:51,920
+هذا بكون عندي اللي هي الـ FM أو الـ FN converts
+
+459
+00:36:51,920 --> 00:36:57,800
+uniformly to ال function اللي عندنا اللي هي F
+
+460
+00:36:57,800 --> 00:37:02,740
+وبهيك بنكون احنا انتهينا من ال section الأول
+
+461
+00:37:02,740 --> 00:37:08,000
+تمانية واحد وإن شاء الله بنكمل المحاضرة القادمة
+
+462
+00:37:08,000 --> 00:37:09,320
+وإلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..e2af649a3ade124a430992787219f4f8dfd3916d
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2593, "start": 4.93, "end": 25.93, "text": "بسم الله الرحمن الرحيم الحمد لله رب العالمين والصلاة والسلام على سيدنا محمد وعلى آله وصحبه أجمعين هذه المحاضرة رقم عشرين في مساق تحليل حقيقه اتنين لطلاب وطالبات الجامعة الإسلامية كلية العلوم قسم رياضيات", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21542, 2304, 3215, 24976, 3224, 12602, 3555, 18863, 45340, 9957, 16070, 9381, 15040, 3660, 16070, 3794, 37440, 15844, 8608, 25708, 8315, 3714, 35571, 3215, 4032, 3615, 23942, 19753, 43761, 4032, 9381, 5016, 3555, 3224, 5551, 7435, 2304, 3615, 9957, 29538, 9673, 5016, 46958, 25720, 12602, 4587, 2304, 6225, 46309, 9957, 8978, 47524, 995, 4587, 6055, 5016, 20292, 1211, 11331, 38436, 4587, 3224, 1975, 2655, 1863, 9957, 5296, 9566, 15040, 3555, 4032, 9566, 6027, 3555, 9307, 25724, 10943, 27884, 33688, 3794, 37440, 10632, 28242, 10632, 18863, 1211, 20498, 12174, 38251, 12602, 1829, 46958, 1829, 9307], "avg_logprob": -0.06588041216421586, "compression_ratio": 1.942408376963351, "no_speech_prob": 9.870529174804688e-05, "words": [{"start": 4.93, "end": 5.53, "word": "بسم", "probability": 0.892822265625}, {"start": 5.53, "end": 5.73, "word": " الله", "probability": 0.95703125}, {"start": 5.73, "end": 6.17, "word": " الرحمن", "probability": 0.9729817708333334}, {"start": 6.17, "end": 6.89, "word": " الرحيم", "probability": 0.9944661458333334}, {"start": 6.89, "end": 8.03, "word": " الحمد", "probability": 0.9361979166666666}, {"start": 8.03, "end": 8.35, "word": " لله", "probability": 0.94677734375}, {"start": 8.35, "end": 8.55, "word": " رب", "probability": 0.96875}, {"start": 8.55, "end": 9.09, "word": " العالمين", "probability": 0.99169921875}, {"start": 9.09, "end": 9.63, "word": " والصلاة", "probability": 0.8983154296875}, {"start": 9.63, "end": 10.11, "word": " والسلام", "probability": 0.9674479166666666}, {"start": 10.11, "end": 10.39, "word": " على", "probability": 0.62939453125}, {"start": 10.39, "end": 10.93, "word": " سيدنا", "probability": 0.99462890625}, {"start": 10.93, "end": 11.45, "word": " محمد", "probability": 0.9817708333333334}, {"start": 11.45, "end": 11.73, "word": " وعلى", "probability": 0.853515625}, {"start": 11.73, "end": 12.13, "word": " آله", "probability": 0.95703125}, {"start": 12.13, "end": 12.83, "word": " وصحبه", "probability": 0.962109375}, {"start": 12.83, "end": 13.79, "word": " أجمعين", "probability": 0.97138671875}, {"start": 13.79, "end": 14.55, "word": " هذه", "probability": 0.70263671875}, {"start": 14.55, "end": 15.69, "word": " المحاضرة", "probability": 0.8692626953125}, {"start": 15.69, "end": 16.05, "word": " رقم", "probability": 0.9542643229166666}, {"start": 16.05, "end": 16.75, "word": " عشرين", "probability": 0.9353841145833334}, {"start": 16.75, "end": 17.45, "word": " في", "probability": 0.8974609375}, {"start": 17.45, "end": 18.13, "word": " مساق", "probability": 0.9625651041666666}, {"start": 18.13, "end": 19.15, "word": " تحليل", "probability": 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هنستكمل الحديث في ال section الأول اللي وجفنا عند نقطة فيه اللي هو ال section كان تحت عنوان born twice and uniform convergence", "tokens": [3224, 1863, 14851, 24793, 1211, 45595, 20498, 21542, 16254, 12984, 8978, 7187, 1649, 13672, 1829, 31439, 22978, 295, 6828, 20666, 19913, 3555, 9566, 8032, 1863, 14851, 24793, 1211, 21542, 16254, 12984, 8978, 2423, 3541, 16247, 12610, 13672, 1829, 49610, 5172, 8315, 43242, 8717, 47432, 3660, 8978, 3224, 13672, 1829, 31439, 2423, 3541, 25961, 6055, 33753, 18871, 2407, 7649, 4232, 6091, 293, 9452, 32181], "avg_logprob": -0.13195800315588713, "compression_ratio": 1.6022099447513811, "no_speech_prob": 4.76837158203125e-07, "words": [{"start": 27.36, "end": 28.4, "word": "هنستكمل", "probability": 0.876171875}, {"start": 28.4, "end": 29.1, "word": " اليوم", "probability": 0.936767578125}, {"start": 29.1, "end": 30.08, "word": " الحديث", "probability": 0.98779296875}, {"start": 30.08, "end": 30.28, "word": " في", "probability": 0.9638671875}, {"start": 30.28, "end": 30.6, "word": " chapter", "probability": 0.29052734375}, {"start": 30.6, "end": 31.12, "word": " 8", "probability": 0.71435546875}, {"start": 31.12, "end": 32.66, "word": " اللي", "probability": 0.70361328125}, {"start": 32.66, "end": 32.84, "word": " هو", "probability": 0.97900390625}, {"start": 32.84, "end": 33.32, "word": " sequences", "probability": 0.81787109375}, {"start": 33.32, "end": 34.04, "word": " of", "probability": 0.98095703125}, {"start": 34.04, "end": 34.72, "word": " functions", "probability": 0.85791015625}, {"start": 34.72, "end": 35.7, "word": " بالظبط", "probability": 0.9705810546875}, {"start": 35.7, "end": 37.1, "word": " هنستكمل", "probability": 0.96123046875}, {"start": 37.1, "end": 37.92, "word": " الحديث", "probability": 0.98828125}, {"start": 37.92, "end": 38.38, "word": " في", "probability": 0.9482421875}, {"start": 38.38, "end": 38.52, "word": " ال", "probability": 0.90673828125}, {"start": 38.52, "end": 38.84, "word": " section", 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49.42, "end": 49.82, "word": " هو", "probability": 0.9853515625}, {"start": 49.82, "end": 50.54, "word": " الـ", "probability": 0.711669921875}, {"start": 50.54, "end": 51.34, "word": " pointwise", "probability": 0.643310546875}, {"start": 51.34, "end": 51.58, "word": " و", "probability": 0.49658203125}, {"start": 51.58, "end": 51.66, "word": " ال", "probability": 0.66650390625}, {"start": 51.66, "end": 51.94, "word": " uniform", "probability": 0.63671875}, {"start": 51.94, "end": 52.68, "word": " convergence", "probability": 0.9501953125}, {"start": 52.68, "end": 53.12, "word": " قلنا", "probability": 0.7389322916666666}, {"start": 53.12, "end": 53.44, "word": " أي", "probability": 0.669921875}, {"start": 53.44, "end": 53.82, "word": " uniform", "probability": 0.74755859375}, {"start": 53.82, "end": 54.54, "word": " convergence", "probability": 0.9658203125}, {"start": 54.54, "end": 55.0, "word": " هيكون", "probability": 0.8642578125}, {"start": 55.0, "end": 55.72, "word": " pointwise", "probability": 0.968994140625}, {"start": 55.72, "end": 56.26, "word": " ولكن", "probability": 0.801025390625}, {"start": 56.26, "end": 56.8, "word": " ال", "probability": 0.84228515625}, {"start": 56.8, "end": 57.16, "word": " convergence", "probability": 0.12152099609375}, {"start": 57.16, "end": 57.68, "word": " لا", "probability": 0.445068359375}, {"start": 57.68, "end": 57.88, "word": " يجب", "probability": 0.683349609375}, {"start": 57.88, "end": 57.88, "word": " أن", "probability": 0.60546875}, {"start": 57.88, "end": 58.14, "word": " تكون", "probability": 0.95166015625}, {"start": 58.14, "end": 58.5, "word": " حقيقية", "probability": 0.8807373046875}, {"start": 58.5, "end": 58.98, "word": " في", "probability": 0.51220703125}, {"start": 58.98, "end": 59.34, "word": " الجامعة", "probability": 0.454193115234375}, {"start": 59.34, "end": 60.02, "word": " و", "probability": 0.5615234375}, {"start": 60.02, "end": 60.36, "word": " أخدنا", "probability": 0.82861328125}, {"start": 60.36, "end": 60.64, "word": " بعض", "probability": 0.988525390625}, {"start": 60.64, "end": 61.44, "word": " النظريات", "probability": 0.9727783203125}, {"start": 61.44, "end": 61.7, "word": " أول", "probability": 0.819580078125}, {"start": 61.7, "end": 61.92, "word": " اشي", "probability": 0.6765950520833334}, {"start": 61.92, "end": 62.38, "word": " أعرفنا", "probability": 0.77508544921875}, {"start": 62.38, "end": 62.58, "word": " شو", "probability": 0.8271484375}, {"start": 62.58, "end": 63.04, "word": " معناته", "probability": 0.9148763020833334}, {"start": 63.04, "end": 63.22, "word": " اللي", "probability": 0.96484375}, {"start": 63.22, "end": 63.48, "word": " هو", "probability": 0.98681640625}, {"start": 63.48, "end": 63.62, "word": " ال", "probability": 0.5361328125}, {"start": 63.62, "end": 64.3, "word": " pointwise", "probability": 0.9375}, {"start": 64.3, "end": 64.98, "word": " convergence", "probability": 0.986328125}, {"start": 64.98, "end": 65.98, "word": " و", "probability": 0.52734375}, {"start": 65.98, "end": 66.68, "word": " بعدين", "probability": 0.9404296875}, {"start": 66.68, "end": 67.66, "word": " أخدنا", "probability": 0.963134765625}, {"start": 67.66, "end": 68.48, "word": " مثال", "probability": 0.978759765625}, {"start": 68.48, "end": 69.52, "word": " على", "probability": 0.5419921875}, {"start": 69.52, "end": 69.88, "word": " اللي", "probability": 0.930908203125}, {"start": 69.88, "end": 70.04, "word": " هي", "probability": 0.90234375}, {"start": 70.04, "end": 70.18, "word": " ال", "probability": 0.96923828125}, {"start": 70.18, "end": 70.54, "word": " function", "probability": 0.78369140625}, {"start": 70.54, "end": 70.82, "word": " g", "probability": 0.376953125}, {"start": 70.82, "end": 70.98, "word": " of", "probability": 0.2900390625}, {"start": 70.98, "end": 71.18, "word": " a", "probability": 0.93017578125}, {"start": 71.18, "end": 71.44, "word": " g", "probability": 0.25537109375}, {"start": 71.44, "end": 71.72, "word": " n", "probability": 0.2236328125}, {"start": 71.72, "end": 72.02, "word": " of", "probability": 0.95166015625}, {"start": 72.02, "end": 72.3, "word": " x", "probability": 0.931640625}, {"start": 72.3, "end": 72.44, "word": " او", "probability": 0.6832275390625}, {"start": 72.44, "end": 72.78, "word": " sequence", "probability": 0.65087890625}, {"start": 72.78, "end": 73.06, "word": " of", "probability": 0.9638671875}, {"start": 73.06, "end": 73.46, "word": " functions", "probability": 0.767578125}, {"start": 73.46, "end": 73.66, "word": " g", "probability": 0.79833984375}, {"start": 73.66, "end": 73.84, "word": " n", "probability": 0.7255859375}, {"start": 73.84, "end": 73.96, "word": " of", "probability": 0.9609375}, {"start": 73.96, "end": 74.14, "word": " x", "probability": 0.97412109375}, {"start": 74.14, "end": 74.44, "word": " بيستوي", "probability": 0.61376953125}, {"start": 74.44, "end": 74.66, "word": " x", "probability": 0.84375}, {"start": 74.66, "end": 74.8, "word": " أس", "probability": 0.4548797607421875}, {"start": 74.8, "end": 75.04, "word": " ان", "probability": 0.260986328125}], "temperature": 1.0}, {"id": 4, "seek": 10341, "start": 75.73, "end": 103.41, "text": "و أثبتنا إن هذه pointwise convergence على الفترة المفتوحة بالنقص 1 1 لدلة g of x بالساوية 0 1 زي ما هي أمامنا بعد هيك جينا حكينا عن اللي هو مثال آخر أخدنا limit 1 على n sin n x زائد n قلنا هذه ال sequence of functions برضه converts to the zero function أو كانت اللي هو هذه مثال آخر", "tokens": [2407, 5551, 12984, 3555, 2655, 8315, 36145, 29538, 935, 3711, 32181, 15844, 27188, 2655, 25720, 9673, 5172, 2655, 2407, 5016, 3660, 20666, 1863, 4587, 9381, 502, 502, 5296, 3215, 37977, 290, 295, 2031, 20666, 3794, 995, 2407, 10632, 1958, 502, 30767, 1829, 19446, 39896, 5551, 2304, 10943, 8315, 39182, 39896, 4117, 10874, 1829, 8315, 11331, 4117, 1829, 8315, 18871, 13672, 1829, 31439, 50113, 6027, 19753, 34740, 5551, 9778, 3215, 8315, 4948, 502, 15844, 297, 3343, 297, 2031, 30767, 16373, 3215, 297, 12174, 1211, 8315, 29538, 2423, 8310, 295, 6828, 4724, 43042, 3224, 38874, 281, 264, 4018, 2445, 34051, 25961, 2655, 13672, 1829, 31439, 29538, 50113, 6027, 19753, 34740], "avg_logprob": -0.24383600917431192, "compression_ratio": 1.61003861003861, "no_speech_prob": 0.0, "words": [{"start": 75.73, "end": 76.11, "word": "و", "probability": 0.6123046875}, {"start": 76.11, "end": 76.69, "word": " أثبتنا", "probability": 0.89921875}, {"start": 76.69, "end": 76.91, "word": " إن", "probability": 0.5732421875}, {"start": 76.91, "end": 77.37, "word": " هذه", "probability": 0.5966796875}, {"start": 77.37, "end": 78.19, "word": " pointwise", "probability": 0.336669921875}, {"start": 78.19, "end": 78.73, "word": " convergence", "probability": 0.9736328125}, {"start": 78.73, "end": 78.95, "word": " على", "probability": 0.6494140625}, {"start": 78.95, "end": 79.25, "word": " الفترة", "probability": 0.94482421875}, {"start": 79.25, "end": 79.69, "word": " المفتوحة", "probability": 0.994140625}, {"start": 79.69, "end": 80.13, "word": " بالنقص", "probability": 0.81793212890625}, {"start": 80.13, "end": 80.37, "word": " 1", "probability": 0.403076171875}, {"start": 80.37, "end": 80.85, "word": " 1", "probability": 0.25146484375}, {"start": 80.85, "end": 82.03, "word": " لدلة", "probability": 0.6570638020833334}, {"start": 82.03, "end": 82.25, "word": " g", "probability": 0.347412109375}, {"start": 82.25, "end": 82.39, "word": " of", "probability": 0.233642578125}, {"start": 82.39, "end": 82.55, "word": " x", "probability": 0.6845703125}, {"start": 82.55, "end": 82.95, "word": " بالساوية", "probability": 0.62236328125}, {"start": 82.95, "end": 83.19, "word": " 0", "probability": 0.42333984375}, {"start": 83.19, "end": 83.81, "word": " 1", "probability": 0.90234375}, {"start": 83.81, "end": 84.31, "word": " زي", "probability": 0.925537109375}, {"start": 84.31, "end": 84.43, "word": " ما", "probability": 0.95556640625}, {"start": 84.43, "end": 84.73, "word": " هي", "probability": 0.86962890625}, {"start": 84.73, "end": 85.41, "word": " أمامنا", "probability": 0.94677734375}, {"start": 85.41, "end": 86.17, "word": " بعد", "probability": 0.94921875}, {"start": 86.17, "end": 86.61, "word": " هيك", "probability": 0.73193359375}, {"start": 86.61, "end": 87.05, "word": " جينا", "probability": 0.8291015625}, {"start": 87.05, "end": 87.93, "word": " حكينا", "probability": 0.985595703125}, {"start": 87.93, "end": 88.45, "word": " عن", "probability": 0.97998046875}, {"start": 88.45, "end": 88.81, "word": " اللي", "probability": 0.8466796875}, {"start": 88.81, "end": 89.21, "word": " هو", "probability": 0.99267578125}, {"start": 89.21, "end": 90.09, "word": " مثال", "probability": 0.994140625}, {"start": 90.09, "end": 90.59, "word": " آخر", "probability": 0.95458984375}, {"start": 90.59, "end": 91.11, "word": " أخدنا", "probability": 0.96728515625}, {"start": 91.11, "end": 91.41, "word": " limit", "probability": 0.97900390625}, {"start": 91.41, "end": 91.77, "word": " 1", "probability": 0.83447265625}, {"start": 91.77, "end": 92.07, "word": " على", "probability": 0.82421875}, {"start": 92.07, "end": 92.37, "word": " n", "probability": 0.60888671875}, {"start": 92.37, "end": 92.99, "word": " sin", "probability": 0.638671875}, {"start": 92.99, "end": 93.31, "word": " n", "probability": 0.87060546875}, {"start": 93.31, "end": 93.63, "word": " x", "probability": 0.7421875}, {"start": 93.63, "end": 93.93, "word": " زائد", "probability": 0.8323567708333334}, {"start": 93.93, "end": 94.09, "word": " n", "probability": 0.84619140625}, {"start": 94.09, "end": 94.41, "word": " قلنا", "probability": 0.8126627604166666}, {"start": 94.41, "end": 94.71, "word": " هذه", "probability": 0.595703125}, {"start": 94.71, "end": 94.85, "word": " ال", "probability": 0.9013671875}, {"start": 94.85, "end": 95.15, "word": " sequence", "probability": 0.75341796875}, {"start": 95.15, "end": 95.37, "word": " of", "probability": 0.9052734375}, {"start": 95.37, "end": 95.81, "word": " functions", "probability": 0.83349609375}, {"start": 95.81, "end": 96.29, "word": " برضه", "probability": 0.9602864583333334}, {"start": 96.29, "end": 97.03, "word": " converts", "probability": 0.326416015625}, {"start": 97.03, "end": 97.71, "word": " to", "probability": 0.96142578125}, {"start": 97.71, "end": 97.97, "word": " the", "probability": 0.85791015625}, {"start": 97.97, "end": 98.35, "word": " zero", "probability": 0.53271484375}, {"start": 98.35, "end": 99.03, "word": " function", "probability": 0.72607421875}, {"start": 99.03, "end": 100.83, "word": " أو", "probability": 0.252197265625}, {"start": 100.83, "end": 101.69, "word": " كانت", "probability": 0.98583984375}, {"start": 101.69, "end": 101.81, "word": " اللي", "probability": 0.79443359375}, {"start": 101.81, "end": 102.03, "word": " هو", "probability": 0.9541015625}, {"start": 102.03, "end": 102.25, "word": " هذه", "probability": 0.5126953125}, {"start": 102.25, "end": 102.93, "word": " مثال", "probability": 0.732666015625}, {"start": 102.93, "end": 103.41, "word": " آخر", "probability": 0.8701171875}], "temperature": 1.0}, {"id": 5, "seek": 12526, "start": 103.56, "end": 125.26, "text": "على الـ Point Convergence على كل R الآن إيجينا بعد هيك اللي أخدنا لمّا و كيف اللي هو ان ترجم شو معناته Fn Converges Point Twice قولنا الـ K اللي بيلاقيها بتعتمد على Y و X في ال Point Twice لإنه نقطة نقطة ال convergence", "tokens": [3615, 23942, 2423, 39184, 12387, 2656, 331, 15260, 15844, 28242, 497, 6024, 48506, 11933, 1829, 7435, 1829, 8315, 39182, 39896, 4117, 13672, 1829, 5551, 9778, 3215, 8315, 32767, 11703, 995, 4032, 9122, 33911, 13672, 1829, 31439, 16472, 6055, 47341, 2304, 13412, 2407, 20449, 8315, 47395, 479, 77, 2656, 331, 2880, 12387, 46964, 12174, 12610, 8315, 2423, 39184, 591, 13672, 1829, 4724, 1829, 15040, 38436, 11296, 39894, 34268, 2304, 3215, 15844, 398, 4032, 1783, 8978, 2423, 12387, 46964, 5296, 28814, 1863, 3224, 8717, 47432, 3660, 8717, 47432, 3660, 2423, 32181], "avg_logprob": -0.29618055125077564, "compression_ratio": 1.6019417475728155, "no_speech_prob": 1.3709068298339844e-06, "words": [{"start": 103.56, "end": 104.2, "word": "على", "probability": 0.808837890625}, {"start": 104.2, "end": 104.54, "word": " الـ", "probability": 0.7880859375}, {"start": 104.54, "end": 104.84, "word": " Point", "probability": 0.2364501953125}, {"start": 104.84, "end": 105.96, "word": " Convergence", "probability": 0.814453125}, {"start": 105.96, "end": 106.58, "word": " على", "probability": 0.82666015625}, {"start": 106.58, "end": 107.12, "word": " كل", "probability": 0.9736328125}, {"start": 107.12, "end": 107.64, "word": " R", "probability": 0.71875}, {"start": 107.64, "end": 108.88, "word": " الآن", "probability": 0.53985595703125}, {"start": 108.88, "end": 109.32, "word": " إيجينا", "probability": 0.7367919921875}, {"start": 109.32, "end": 109.56, "word": " بعد", "probability": 0.86572265625}, {"start": 109.56, "end": 110.02, "word": " هيك", "probability": 0.934326171875}, {"start": 110.02, "end": 110.7, "word": " اللي", "probability": 0.5606689453125}, {"start": 110.7, "end": 111.18, "word": " أخدنا", "probability": 0.9688720703125}, {"start": 111.18, "end": 111.72, "word": " لمّا", "probability": 0.6785481770833334}, {"start": 111.72, "end": 111.98, "word": " و", "probability": 0.75634765625}, {"start": 111.98, "end": 112.3, "word": " كيف", "probability": 0.707275390625}, {"start": 112.3, "end": 112.52, "word": " اللي", "probability": 0.8779296875}, {"start": 112.52, "end": 112.88, "word": " هو", "probability": 0.97998046875}, {"start": 112.88, "end": 113.54, "word": " ان", "probability": 0.55224609375}, {"start": 113.54, "end": 113.98, "word": " ترجم", "probability": 0.97412109375}, {"start": 113.98, "end": 114.2, "word": " شو", "probability": 0.850830078125}, {"start": 114.2, "end": 114.62, "word": " معناته", "probability": 0.8626302083333334}, {"start": 114.62, "end": 115.08, "word": " Fn", "probability": 0.3416748046875}, {"start": 115.08, "end": 115.72, "word": " Converges", "probability": 0.6830240885416666}, {"start": 115.72, "end": 116.86, "word": " Point", "probability": 0.67236328125}, {"start": 116.86, "end": 117.46, "word": " Twice", "probability": 0.66015625}, {"start": 117.46, "end": 118.34, "word": " قولنا", "probability": 0.7420247395833334}, {"start": 118.34, "end": 118.62, "word": " الـ", "probability": 0.817138671875}, {"start": 118.62, "end": 118.8, "word": " K", "probability": 0.403076171875}, {"start": 118.8, "end": 119.96, "word": " اللي", "probability": 0.943603515625}, {"start": 119.96, "end": 120.4, "word": " بيلاقيها", "probability": 0.721337890625}, {"start": 120.4, "end": 120.96, "word": " بتعتمد", "probability": 0.957763671875}, {"start": 120.96, "end": 121.2, "word": " على", "probability": 0.892578125}, {"start": 121.2, "end": 121.5, "word": " Y", "probability": 0.70166015625}, {"start": 121.5, "end": 121.8, "word": " و", "probability": 0.53955078125}, {"start": 121.8, "end": 122.22, "word": " X", "probability": 0.8095703125}, {"start": 122.22, "end": 122.74, "word": " في", "probability": 0.88623046875}, {"start": 122.74, "end": 122.88, "word": " ال", "probability": 0.8369140625}, {"start": 122.88, "end": 123.24, "word": " Point", "probability": 0.38623046875}, {"start": 123.24, "end": 123.66, "word": " Twice", "probability": 0.88623046875}, {"start": 123.66, "end": 124.04, "word": " لإنه", "probability": 0.82257080078125}, {"start": 124.04, "end": 124.36, "word": " نقطة", "probability": 0.974609375}, {"start": 124.36, "end": 124.72, "word": " نقطة", "probability": 0.97705078125}, {"start": 124.72, "end": 124.8, "word": " ال", "probability": 0.92041015625}, {"start": 124.8, "end": 125.26, "word": " convergence", "probability": 0.5791015625}], "temperature": 1.0}, {"id": 6, "seek": 14591, "start": 126.07, "end": 145.91, "text": "و بعدين اجينا عرفنا شو معناه uniform convergence و قولنا في حالة ال uniform convergence هيكون عندي اللي هو ال K اللي بنلاقيها بس بتعتمد على ال epsilon و الباقي زي ما هو يعني بتنفع ال K هذه لكل ال Xات اللي في ال A note عشان هيك بنسمي uniform convergence", "tokens": [2407, 39182, 9957, 1975, 7435, 1829, 8315, 6225, 28480, 8315, 13412, 2407, 20449, 8315, 3224, 9452, 32181, 4032, 12174, 12610, 8315, 8978, 11331, 6027, 3660, 2423, 9452, 32181, 39896, 30544, 18871, 16254, 13672, 1829, 31439, 2423, 591, 13672, 1829, 44945, 15040, 38436, 11296, 4724, 3794, 39894, 34268, 2304, 3215, 15844, 2423, 17889, 4032, 29739, 995, 38436, 30767, 1829, 19446, 31439, 37495, 22653, 39894, 1863, 5172, 3615, 2423, 591, 29538, 5296, 28820, 2423, 1783, 9307, 13672, 1829, 8978, 2423, 316, 3637, 6225, 8592, 7649, 39896, 4117, 44945, 38251, 1829, 9452, 32181], "avg_logprob": -0.25085852041349305, "compression_ratio": 1.7741935483870968, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 126.07, "end": 126.25, "word": "و", "probability": 0.344482421875}, {"start": 126.25, "end": 126.87, "word": " بعدين", "probability": 0.888916015625}, {"start": 126.87, "end": 127.47, "word": " اجينا", "probability": 0.6446533203125}, {"start": 127.47, "end": 128.77, "word": " عرفنا", "probability": 0.8802083333333334}, {"start": 128.77, "end": 128.95, "word": " شو", "probability": 0.948974609375}, {"start": 128.95, "end": 129.25, "word": " معناه", "probability": 0.8079427083333334}, {"start": 129.25, "end": 129.55, "word": " uniform", "probability": 0.66845703125}, {"start": 129.55, "end": 130.21, "word": " convergence", "probability": 0.96484375}, {"start": 130.21, "end": 130.45, "word": " و", "probability": 0.8349609375}, {"start": 130.45, "end": 130.85, "word": " قولنا", "probability": 0.7538248697916666}, {"start": 130.85, "end": 131.37, "word": " في", "probability": 0.716796875}, {"start": 131.37, "end": 131.69, "word": " حالة", "probability": 0.9908854166666666}, {"start": 131.69, "end": 131.77, "word": " ال", "probability": 0.91259765625}, {"start": 131.77, "end": 132.01, "word": " uniform", "probability": 0.7451171875}, {"start": 132.01, "end": 132.75, "word": " convergence", "probability": 0.97802734375}, {"start": 132.75, "end": 133.23, "word": " هيكون", "probability": 0.73095703125}, {"start": 133.23, "end": 133.65, "word": " عندي", "probability": 0.6156005859375}, {"start": 133.65, "end": 133.87, "word": " اللي", "probability": 0.916015625}, {"start": 133.87, "end": 134.19, "word": " هو", "probability": 0.94580078125}, {"start": 134.19, "end": 134.79, "word": " ال", "probability": 0.9287109375}, {"start": 134.79, "end": 135.01, "word": " K", "probability": 0.431884765625}, {"start": 135.01, "end": 135.17, "word": " اللي", "probability": 0.956298828125}, {"start": 135.17, "end": 135.67, "word": " بنلاقيها", "probability": 0.75567626953125}, {"start": 135.67, "end": 135.95, "word": " بس", "probability": 0.92529296875}, {"start": 135.95, "end": 136.63, "word": " بتعتمد", "probability": 0.9537353515625}, {"start": 136.63, "end": 136.85, "word": " على", "probability": 0.75439453125}, {"start": 136.85, "end": 137.01, "word": " ال", "probability": 0.9296875}, {"start": 137.01, "end": 137.39, "word": " epsilon", "probability": 0.241455078125}, {"start": 137.39, "end": 137.97, "word": " و", "probability": 0.63037109375}, {"start": 137.97, "end": 138.37, "word": " الباقي", "probability": 0.7879231770833334}, {"start": 138.37, "end": 138.65, "word": " زي", "probability": 0.98876953125}, {"start": 138.65, "end": 138.75, "word": " ما", "probability": 0.9755859375}, {"start": 138.75, "end": 139.13, "word": " هو", "probability": 0.990234375}, {"start": 139.13, "end": 139.91, "word": " يعني", "probability": 0.916259765625}, {"start": 139.91, "end": 140.45, "word": " بتنفع", "probability": 0.9609375}, {"start": 140.45, "end": 140.61, "word": " ال", "probability": 0.6845703125}, {"start": 140.61, "end": 140.73, "word": " K", "probability": 0.9248046875}, {"start": 140.73, "end": 140.95, "word": " هذه", "probability": 0.64306640625}, {"start": 140.95, "end": 141.43, "word": " لكل", "probability": 0.954833984375}, {"start": 141.43, "end": 141.61, "word": " ال", "probability": 0.76171875}, {"start": 141.61, "end": 142.27, "word": " Xات", "probability": 0.453125}, {"start": 142.27, "end": 142.91, "word": " اللي", "probability": 0.991455078125}, {"start": 142.91, "end": 143.13, "word": " في", "probability": 0.9619140625}, {"start": 143.13, "end": 143.31, "word": " ال", "probability": 0.96240234375}, {"start": 143.31, "end": 143.49, "word": " A", "probability": 0.58935546875}, {"start": 143.49, "end": 143.69, "word": " note", "probability": 0.33642578125}, {"start": 143.69, "end": 144.01, "word": " عشان", "probability": 0.9591471354166666}, {"start": 144.01, "end": 144.31, "word": " هيك", "probability": 0.84375}, {"start": 144.31, "end": 144.73, "word": " بنسمي", "probability": 0.7305094401041666}, {"start": 144.73, "end": 145.11, "word": " uniform", "probability": 0.62060546875}, {"start": 145.11, "end": 145.91, "word": " convergence", "probability": 0.96337890625}], "temperature": 1.0}, {"id": 7, "seek": 16338, "start": 146.57, "end": 163.39, "text": "وقلنا إذا كانت الـ sequence uniformly convergence أكيد هتكون convergence و بعدين أخدنا ال criterion أو اللمّة اللي بننفي فيها أو بنقول شو معناه Fn does not converge uniformly to اللي هو function F", "tokens": [2407, 4587, 1211, 8315, 11933, 15730, 25961, 2655, 2423, 39184, 8310, 48806, 32181, 5551, 4117, 25708, 8032, 2655, 30544, 32181, 4032, 39182, 9957, 5551, 9778, 3215, 8315, 2423, 46691, 34051, 13672, 2304, 11703, 3660, 13672, 1829, 44945, 1863, 5172, 1829, 8978, 11296, 34051, 44945, 39648, 13412, 2407, 20449, 8315, 3224, 479, 77, 775, 406, 41881, 48806, 281, 13672, 1829, 31439, 2445, 479], "avg_logprob": -0.27926588531524416, "compression_ratio": 1.4891304347826086, "no_speech_prob": 0.0, "words": [{"start": 146.57, "end": 147.17, "word": "وقلنا", "probability": 0.61865234375}, {"start": 147.17, "end": 147.43, "word": " إذا", "probability": 0.742431640625}, {"start": 147.43, "end": 147.77, "word": " كانت", 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does not converge uniformly to مين؟ to G وهذا مثال عليه اللي هو does not converge uniformly على الـ G مثال على اللي هو توظيف هذه النظرية الجابلة بشوية لإثبات عدم ال uniformly", "tokens": [1863, 4587, 9381, 2423, 39184, 460, 4724, 43042, 3224, 39896, 32887, 5016, 15844, 3714, 9957, 22807, 15844, 25062, 32640, 3660, 36764, 24401, 37495, 22653, 37893, 39896, 32887, 5016, 24976, 3794, 5172, 2288, 37495, 22653, 11933, 15730, 14111, 46411, 775, 406, 41881, 48806, 281, 3714, 9957, 22807, 281, 460, 37037, 15730, 50113, 6027, 47356, 13672, 1829, 31439, 775, 406, 41881, 48806, 15844, 2423, 39184, 460, 50113, 6027, 15844, 13672, 1829, 31439, 33427, 19913, 33911, 29538, 28239, 19913, 2288, 10632, 25724, 16758, 37977, 4724, 8592, 2407, 10632, 5296, 28814, 12984, 3555, 9307, 6225, 40448, 2423, 48806], "avg_logprob": -0.2054276356571599, "compression_ratio": 1.7533632286995515, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 534.83, "end": 535.45, "word": "نقص", "probability": 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{"start": 737.73, "end": 738.15, "word": " المرة", "probability": 0.97509765625}, {"start": 738.15, "end": 738.69, "word": " الماضية", "probability": 0.9939778645833334}, {"start": 738.69, "end": 739.59, "word": " وخلّينا", "probability": 0.8094075520833334}, {"start": 739.59, "end": 740.27, "word": " نشوف", "probability": 0.9825846354166666}, {"start": 740.27, "end": 740.83, "word": " الحل", "probability": 0.98779296875}, {"start": 740.83, "end": 741.73, "word": " نيجي", "probability": 0.920166015625}, {"start": 741.73, "end": 742.67, "word": " الآن", "probability": 0.676025390625}, {"start": 742.67, "end": 743.41, "word": " الجزء", "probability": 0.9588216145833334}, {"start": 743.41, "end": 744.05, "word": " الأول", "probability": 0.94482421875}, {"start": 744.05, "end": 745.29, "word": " الجزء", "probability": 0.7696940104166666}, {"start": 745.29, "end": 745.61, "word": " الأول", "probability": 0.98388671875}, {"start": 745.61, "end": 746.21, "word": " سهل", "probability": 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{"start": 784.61, "end": 785.37, "word": " شوف", "probability": 0.90869140625}, {"start": 785.37, "end": 785.71, "word": " الآن", "probability": 0.733154296875}, {"start": 785.71, "end": 785.91, "word": " كيف", "probability": 0.92724609375}, {"start": 785.91, "end": 786.17, "word": " بدنا", "probability": 0.95751953125}, {"start": 786.17, "end": 786.89, "word": " نثبت", "probability": 0.9962158203125}, {"start": 786.89, "end": 787.91, "word": " خلصنا", "probability": 0.969482421875}, {"start": 787.91, "end": 788.29, "word": " من", "probability": 0.95166015625}, {"start": 788.29, "end": 788.71, "word": " انه", "probability": 0.40533447265625}, {"start": 788.71, "end": 788.93, "word": " احنا", "probability": 0.9153645833333334}, {"start": 788.93, "end": 789.39, "word": " نثبت", "probability": 0.9947509765625}, {"start": 789.39, "end": 789.77, "word": " fn", "probability": 0.922607421875}, {"start": 789.77, "end": 790.07, "word": " x", "probability": 0.466552734375}, {"start": 790.07, 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ان", "probability": 0.7705078125}, {"start": 796.87, "end": 797.33, "word": " fn", "probability": 0.798583984375}, {"start": 797.33, "end": 797.57, "word": " does", "probability": 0.5400390625}, {"start": 797.57, "end": 797.83, "word": " not", "probability": 0.95751953125}, {"start": 797.83, "end": 798.35, "word": " converge", "probability": 0.9873046875}, {"start": 798.35, "end": 799.01, "word": " uniformly", "probability": 0.97998046875}, {"start": 799.01, "end": 799.67, "word": " لل", "probability": 0.6796875}, {"start": 799.67, "end": 800.07, "word": " function", "probability": 0.56201171875}, {"start": 800.07, "end": 800.55, "word": " f", "probability": 0.89599609375}, {"start": 800.55, "end": 800.93, "word": " on", "probability": 0.912109375}, {"start": 800.93, "end": 801.19, "word": " R", "probability": 0.92529296875}], "temperature": 1.0}, {"id": 36, "seek": 82294, "start": 801.44, "end": 822.94, "text": "اللي هو زي ما قلنا الـ sequence of functions عندي هنا are not bounded functions so we can't use our لمّة اللي قبل بشوية اللي هي normal of a نقص of a converges to zero إذا هنتضرر نروح للتعريف أو للمّة قبلها إذا بتتذكروا اللمّة قلنا of a that does not converge", "tokens": [6027, 20292, 31439, 30767, 1829, 19446, 12174, 1211, 8315, 2423, 39184, 8310, 295, 6828, 18871, 16254, 34105, 366, 406, 37498, 6828, 370, 321, 393, 380, 764, 527, 32767, 11703, 3660, 13672, 1829, 12174, 36150, 4724, 8592, 2407, 10632, 13672, 1829, 39896, 2710, 295, 257, 8717, 4587, 9381, 295, 257, 9652, 2880, 281, 4018, 11933, 15730, 8032, 29399, 11242, 2288, 2288, 8717, 32887, 5016, 24976, 2655, 3615, 16572, 5172, 34051, 5296, 19528, 11703, 3660, 12174, 36150, 11296, 11933, 15730, 39894, 2655, 8848, 37983, 14407, 13672, 2304, 11703, 3660, 12174, 1211, 8315, 295, 257, 300, 775, 406, 41881], "avg_logprob": -0.28785439127499296, "compression_ratio": 1.5384615384615385, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 801.44, "end": 801.72, "word": "اللي", "probability": 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{"start": 895.2, "end": 895.48, "word": " نشوف", "probability": 0.95751953125}, {"start": 895.48, "end": 895.8, "word": " كيف", "probability": 0.97900390625}, {"start": 895.8, "end": 896.66, "word": " عندي", "probability": 0.7763671875}, {"start": 896.66, "end": 898.04, "word": " fn", "probability": 0.852783203125}, {"start": 898.04, "end": 898.38, "word": " of", "probability": 0.94482421875}, {"start": 898.38, "end": 898.66, "word": " x", "probability": 0.9326171875}, {"start": 898.66, "end": 898.86, "word": " ايش", "probability": 0.8017578125}, {"start": 898.86, "end": 899.42, "word": " بالساوي", "probability": 0.7249755859375}, {"start": 899.42, "end": 899.9, "word": " x", "probability": 0.61474609375}, {"start": 899.9, "end": 900.12, "word": " على", "probability": 0.77734375}, {"start": 900.12, "end": 900.38, "word": " n", "probability": 0.8369140625}], "temperature": 1.0}, {"id": 40, "seek": 92134, "start": 901.02, "end": 921.34, "text": "لو جينا اخترنا اللي هو الـ NK الـ NK الـ subsequence يعني اللي هي الـ NK اخترناها الـ K بيصير اللي هي FNK اللي هي of X بيساوي FK of X بيصير هذا K X على K طب ال X ايش نختارها عشان نخلي هذا ثابت اختارها", "tokens": [1211, 2407, 10874, 1829, 8315, 1975, 46456, 2288, 8315, 13672, 1829, 31439, 2423, 39184, 426, 42, 2423, 39184, 426, 42, 2423, 39184, 13924, 655, 37495, 22653, 13672, 1829, 39896, 2423, 39184, 426, 42, 1975, 46456, 2288, 8315, 11296, 2423, 39184, 591, 4724, 1829, 9381, 13546, 13672, 1829, 39896, 479, 45, 42, 13672, 1829, 39896, 295, 1783, 4724, 1829, 3794, 995, 45865, 479, 42, 295, 1783, 4724, 1829, 9381, 13546, 23758, 591, 1783, 15844, 591, 23032, 3555, 2423, 1783, 1975, 1829, 8592, 8717, 46456, 9640, 11296, 6225, 8592, 7649, 8717, 9778, 20292, 23758, 38637, 16758, 2655, 1975, 46456, 9640, 11296], "avg_logprob": -0.1793750010430813, "compression_ratio": 1.8994082840236686, "no_speech_prob": 4.827976226806641e-06, "words": [{"start": 901.02, "end": 901.4, "word": "لو", "probability": 0.92333984375}, {"start": 901.4, 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هي", "probability": 0.80615234375}, {"start": 907.58, "end": 907.78, "word": " الـ", "probability": 0.9384765625}, {"start": 907.78, "end": 908.06, "word": " NK", "probability": 0.92041015625}, {"start": 908.06, "end": 908.72, "word": " اخترناها", "probability": 0.92109375}, {"start": 908.72, "end": 908.92, "word": " الـ", "probability": 0.86669921875}, {"start": 908.92, "end": 909.18, "word": " K", "probability": 0.78466796875}, {"start": 909.18, "end": 910.42, "word": " بيصير", "probability": 0.8509521484375}, {"start": 910.42, "end": 910.58, "word": " اللي", "probability": 0.977783203125}, {"start": 910.58, "end": 910.88, "word": " هي", "probability": 0.806640625}, {"start": 910.88, "end": 912.52, "word": " FNK", "probability": 0.684326171875}, {"start": 912.52, "end": 912.96, "word": " اللي", "probability": 0.77587890625}, {"start": 912.96, "end": 913.08, "word": " هي", "probability": 0.72509765625}, {"start": 913.08, "end": 913.3, "word": " of", "probability": 0.7490234375}, {"start": 913.3, "end": 913.54, "word": " X", "probability": 0.94775390625}, {"start": 913.54, "end": 914.08, "word": " بيساوي", "probability": 0.798828125}, {"start": 914.08, "end": 914.86, "word": " FK", "probability": 0.929443359375}, {"start": 914.86, "end": 915.1, "word": " of", "probability": 0.96484375}, {"start": 915.1, "end": 915.52, "word": " X", "probability": 0.9970703125}, {"start": 915.52, "end": 916.7, "word": " بيصير", "probability": 0.9603271484375}, {"start": 916.7, "end": 916.9, "word": " هذا", "probability": 0.360107421875}, {"start": 916.9, "end": 917.22, "word": " K", "probability": 0.796875}, {"start": 917.22, "end": 917.56, "word": " X", "probability": 0.615234375}, {"start": 917.56, "end": 917.76, "word": " على", "probability": 0.8251953125}, {"start": 917.76, "end": 918.02, "word": " K", "probability": 0.97412109375}, {"start": 918.02, "end": 918.18, "word": " طب", "probability": 0.837158203125}, {"start": 918.18, "end": 918.3, "word": " ال", "probability": 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and K هي ال F", "tokens": [42, 3182, 568, 42, 3182, 805, 42, 30767, 1829, 19446, 47525, 4117, 28242, 3224, 11331, 1829, 30544, 4032, 9957, 3714, 29245, 23328, 8978, 497, 4032, 28820, 3224, 11331, 1829, 12984, 3555, 2655, 32767, 9957, 13672, 1829, 39896, 2423, 2445, 13672, 1829, 43242, 7578, 4032, 6055, 9381, 13546, 5551, 2304, 13063, 1829, 46811, 10943, 46811, 10943, 1975, 1829, 8592, 6225, 10943, 1211, 34105, 4724, 39648, 32239, 3224, 718, 479, 293, 591, 1975, 2407, 718, 426, 42, 39896, 718, 426, 42, 4724, 3794, 995, 45865, 591, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 1975, 34740, 1975, 9778, 3215, 2423, 479, 293, 591, 31439, 1975, 2407, 39894, 3615, 3555, 13546, 1975, 34740, 479, 16490, 3215, 2423, 479, 293, 591, 39896, 2423, 479], "avg_logprob": -0.2744791540006796, "compression_ratio": 1.7219730941704037, "no_speech_prob": 0.47216796875, "words": [{"start": 921.96, "end": 922.38, "word": "K", "probability": 0.1309814453125}, {"start": 922.38, "end": 923.12, "word": " ,", 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{"start": 928.6, "end": 929.26, "word": " وكله", "probability": 0.7781575520833334}, {"start": 929.26, "end": 929.92, "word": " حيثبت", "probability": 0.870703125}, {"start": 929.92, "end": 930.48, "word": " لمين", "probability": 0.70361328125}, {"start": 930.48, "end": 930.82, "word": " اللي", "probability": 0.88818359375}, {"start": 930.82, "end": 930.96, "word": " هي", "probability": 0.88623046875}, {"start": 930.96, "end": 931.08, "word": " ال", "probability": 0.97802734375}, {"start": 931.08, "end": 931.34, "word": " function", "probability": 0.77197265625}, {"start": 931.34, "end": 931.58, "word": " اللي", "probability": 0.975830078125}, {"start": 931.58, "end": 931.98, "word": " عندى", "probability": 0.7041015625}, {"start": 931.98, "end": 932.32, "word": " و", "probability": 0.642578125}, {"start": 932.32, "end": 932.72, "word": " تصير", "probability": 0.7706705729166666}, {"start": 932.72, "end": 933.26, "word": " أموري", "probability": 0.8770751953125}, {"start": 933.26, 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"probability": 0.904296875}, {"start": 939.24, "end": 939.84, "word": " NK", "probability": 0.55810546875}, {"start": 939.84, "end": 940.26, "word": " هي", "probability": 0.904296875}, {"start": 940.26, "end": 941.12, "word": " let", "probability": 0.81591796875}, {"start": 941.12, "end": 941.72, "word": " NK", "probability": 0.937255859375}, {"start": 941.72, "end": 942.32, "word": " بساوي", "probability": 0.7406005859375}, {"start": 942.32, "end": 942.66, "word": " K", "probability": 0.9375}, {"start": 942.66, "end": 943.38, "word": " يعني", "probability": 0.923583984375}, {"start": 943.38, "end": 943.84, "word": " بمعنى", "probability": 0.99404296875}, {"start": 943.84, "end": 944.36, "word": " اخر", "probability": 0.755859375}, {"start": 944.36, "end": 945.26, "word": " اخد", "probability": 0.8460286458333334}, {"start": 945.26, "end": 945.46, "word": " ال", "probability": 0.927734375}, {"start": 945.46, "end": 945.94, "word": " F", "probability": 0.84033203125}, {"start": 945.94, "end": 946.42, "word": " and", "probability": 0.92626953125}, {"start": 946.42, "end": 946.88, "word": " K", "probability": 0.9736328125}, {"start": 946.88, "end": 947.44, "word": " هو", "probability": 0.982421875}, {"start": 947.44, "end": 948.36, "word": " او", "probability": 0.9404296875}, {"start": 948.36, "end": 948.94, "word": " بتعبير", "probability": 0.93896484375}, {"start": 948.94, "end": 949.4, "word": " اخر", "probability": 0.888427734375}, {"start": 949.4, "end": 949.74, "word": " F", "probability": 0.4501953125}, {"start": 949.74, "end": 949.96, "word": " خد", "probability": 0.84765625}, {"start": 949.96, "end": 950.1, "word": " ال", "probability": 0.84912109375}, {"start": 950.1, "end": 950.28, "word": " F", "probability": 0.98193359375}, {"start": 950.28, "end": 950.5, "word": " and", "probability": 0.81396484375}, {"start": 950.5, "end": 950.6, "word": " K", "probability": 0.93701171875}, {"start": 950.6, "end": 950.84, "word": " هي", "probability": 0.90771484375}, {"start": 950.84, "end": 951.06, "word": " ال", "probability": 0.7919921875}, {"start": 951.06, "end": 951.44, "word": " F", "probability": 0.96728515625}], "temperature": 1.0}, {"id": 42, "seek": 97612, "start": 951.98, "end": 976.12, "text": "K اختار الـ FNK المميزة بـ Sequence اللي هي FK و Epsilon خلّينا بعدين نختارها و XK زي ما قلنا ايش راح نختارها بتساوي K أكيد وين موجودة في R then FNK of XK ايش هتساوي اللي هي FK of XK ويساوي اللي هي عبارة عن XK", "tokens": [42, 1975, 46456, 9640, 2423, 39184, 479, 45, 42, 9673, 2304, 1829, 11622, 3660, 4724, 39184, 46859, 655, 13672, 1829, 39896, 479, 42, 4032, 462, 16592, 16490, 1211, 11703, 1829, 8315, 39182, 9957, 8717, 46456, 9640, 11296, 4032, 1783, 42, 30767, 1829, 19446, 12174, 1211, 8315, 1975, 1829, 8592, 12602, 39319, 8717, 46456, 9640, 11296, 39894, 3794, 995, 45865, 591, 5551, 4117, 25708, 4032, 9957, 3714, 29245, 23328, 3660, 8978, 497, 550, 479, 45, 42, 295, 1783, 42, 1975, 1829, 8592, 8032, 2655, 3794, 995, 45865, 13672, 1829, 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في ال zero أو إيش أو واحد ال suprem أهم إلها زي ما تقولنا أشملها أصغرشها واحد على n هتطلع عبارة عن أ واحد على n", "tokens": [87, 15844, 297, 8717, 995, 4587, 9381, 11296, 11778, 1829, 8608, 5172, 2288, 39894, 11242, 1211, 2031, 15844, 297, 1270, 300, 2031, 689, 3714, 29245, 23328, 3660, 8978, 2423, 4018, 34051, 11933, 1829, 8592, 34051, 36764, 24401, 2423, 23710, 5551, 16095, 11933, 1211, 11296, 30767, 1829, 19446, 6055, 39648, 8315, 5551, 8592, 42213, 11296, 5551, 9381, 17082, 2288, 8592, 11296, 36764, 24401, 15844, 297, 8032, 2655, 9566, 1211, 3615, 6225, 3555, 9640, 3660, 18871, 5551, 36764, 24401, 15844, 297], "avg_logprob": -0.42890624739229677, "compression_ratio": 1.536723163841808, "no_speech_prob": 2.562999725341797e-06, "words": [{"start": 1201.09, "end": 1201.59, "word": "x", "probability": 0.11029052734375}, {"start": 1201.59, "end": 1201.81, "word": " على", "probability": 0.1761474609375}, {"start": 1201.81, "end": 1202.11, "word": " n", "probability": 0.7265625}, 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"probability": 0.970703125}, {"start": 1503.38, "end": 1503.9, "word": " FN", "probability": 0.5054931640625}, {"start": 1503.9, "end": 1504.12, "word": " of", "probability": 0.94091796875}, {"start": 1504.12, "end": 1504.4, "word": " X", "probability": 0.98828125}, {"start": 1504.4, "end": 1504.78, "word": " ناقص", "probability": 0.8695068359375}, {"start": 1504.78, "end": 1505.0, "word": " F", "probability": 0.9169921875}, {"start": 1505.0, "end": 1505.14, "word": " of", "probability": 0.79345703125}, {"start": 1505.14, "end": 1505.58, "word": " X", "probability": 0.99658203125}, {"start": 1505.58, "end": 1506.34, "word": " أصغر", "probability": 0.982666015625}, {"start": 1506.34, "end": 1506.5, "word": " من", "probability": 0.9921875}, {"start": 1506.5, "end": 1506.76, "word": " Y", "probability": 0.88427734375}, {"start": 1506.76, "end": 1507.06, "word": " على", "probability": 0.90478515625}, {"start": 1507.06, "end": 1507.4, "word": " 2", "probability": 0.9892578125}, {"start": 1507.4, "end": 1508.18, "word": " هذا", "probability": 0.86376953125}, {"start": 1508.18, "end": 1508.84, "word": " باستخدام", "probability": 0.913330078125}, {"start": 1508.84, "end": 1509.32, "word": " التعريف", "probability": 0.990478515625}, {"start": 1509.32, "end": 1509.54, "word": " الـ", "probability": 0.6619873046875}, {"start": 1509.54, "end": 1509.58, "word": " N", "probability": 0.59814453125}, {"start": 1509.58, "end": 1510.22, "word": " باستخدام", "probability": 0.9829915364583334}, {"start": 1510.22, "end": 1510.68, "word": " النظرية", "probability": 0.9615478515625}, {"start": 1510.68, "end": 1510.82, "word": " اللي", "probability": 0.93408203125}, {"start": 1510.82, "end": 1511.0, "word": " قبل", "probability": 0.855224609375}, {"start": 1511.0, "end": 1511.42, "word": " بشوية", "probability": 0.9195556640625}, {"start": 1511.42, "end": 1511.74, "word": " لإنه", "probability": 0.8350830078125}, {"start": 1511.74, "end": 1512.26, "word": " بنفع", "probability": 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over mean over a هذه صار تعريف عادي للـ limit إيش معناته هذا لكل إبسلون أكبر من سفر there exists اللي هو مسميها هو إيش مسميها K of إبسلون على 2 element in M such that", "tokens": [2304, 3615, 8315, 47395, 544, 9154, 2423, 39184, 479, 293, 8717, 995, 4587, 9381, 2423, 39184, 479, 1709, 281, 1958, 670, 914, 670, 257, 29538, 20328, 9640, 37279, 16572, 5172, 6225, 995, 16254, 24976, 39184, 4948, 11933, 1829, 8592, 20449, 8315, 47395, 23758, 5296, 28820, 11933, 3555, 3794, 1211, 11536, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 456, 8198, 13672, 1829, 31439, 3714, 38251, 1829, 11296, 31439, 11933, 1829, 8592, 3714, 38251, 1829, 11296, 591, 295, 11933, 3555, 3794, 1211, 11536, 15844, 568, 4478, 294, 376, 1270, 300], "avg_logprob": -0.3305828758839811, "compression_ratio": 1.497584541062802, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1517.5, "end": 1518.14, "word": "معناته", "probability": 0.89892578125}, {"start": 1518.14, "end": 1518.66, "word": " more", "probability": 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1523.88, "end": 1524.22, "word": " over", "probability": 0.5107421875}, {"start": 1524.22, "end": 1524.54, "word": " a", "probability": 0.342041015625}, {"start": 1524.54, "end": 1525.4, "word": " هذه", "probability": 0.296875}, {"start": 1525.4, "end": 1525.72, "word": " صار", "probability": 0.7099609375}, {"start": 1525.72, "end": 1526.78, "word": " تعريف", "probability": 0.802978515625}, {"start": 1526.78, "end": 1527.1, "word": " عادي", "probability": 0.9015299479166666}, {"start": 1527.1, "end": 1527.36, "word": " للـ", "probability": 0.68798828125}, {"start": 1527.36, "end": 1527.6, "word": " limit", "probability": 0.7431640625}, {"start": 1527.6, "end": 1527.9, "word": " إيش", "probability": 0.7070719401041666}, {"start": 1527.9, "end": 1528.36, "word": " معناته", "probability": 0.9801432291666666}, {"start": 1528.36, "end": 1528.74, "word": " هذا", "probability": 0.64404296875}, {"start": 1528.74, "end": 1529.58, "word": " لكل", "probability": 0.740478515625}, {"start": 1529.58, "end": 1530.16, "word": " إبسلون", "probability": 0.863916015625}, {"start": 1530.16, "end": 1531.98, "word": " أكبر", "probability": 0.9664713541666666}, {"start": 1531.98, "end": 1532.14, "word": " من", "probability": 0.99560546875}, {"start": 1532.14, "end": 1532.6, "word": " سفر", "probability": 0.7291666666666666}, {"start": 1532.6, "end": 1533.34, "word": " there", "probability": 0.68017578125}, {"start": 1533.34, "end": 1534.16, "word": " exists", "probability": 0.517578125}, {"start": 1534.16, "end": 1534.8, "word": " اللي", "probability": 0.766845703125}, {"start": 1534.8, "end": 1534.94, "word": " هو", "probability": 0.958984375}, {"start": 1534.94, "end": 1535.52, "word": " مسميها", "probability": 0.870361328125}, {"start": 1535.52, "end": 1535.72, "word": " هو", "probability": 0.85986328125}, {"start": 1535.72, "end": 1535.98, "word": " إيش", "probability": 0.7305501302083334}, {"start": 1535.98, "end": 1536.66, "word": " مسميها", "probability": 0.912841796875}, 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إبسل على اتنين اللي عندك هذا اللي هو تعريف الدهانية العادية اللي هو sequence O السيكوانس العادية هذه الـ sequence converts to zero لأنها أعداد", "tokens": [6027, 20292, 31439, 2026, 2423, 39184, 479, 77, 8717, 995, 4587, 9381, 2423, 39184, 479, 8032, 1829, 30544, 5551, 9381, 17082, 2288, 9154, 11933, 3555, 3794, 1211, 15844, 1975, 2655, 1863, 9957, 337, 633, 426, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 3714, 9957, 591, 295, 11933, 3555, 3794, 1211, 15844, 1975, 2655, 1863, 9957, 13672, 1829, 43242, 4117, 23758, 13672, 1829, 31439, 37279, 16572, 5172, 32748, 3224, 7649, 10632, 18863, 18513, 10632, 13672, 1829, 31439, 8310, 422, 21136, 1829, 4117, 2407, 7649, 3794, 18863, 18513, 10632, 29538, 2423, 39184, 8310, 38874, 281, 4018, 5296, 33456, 11296, 5551, 22488, 18513], "avg_logprob": -0.2737438748864567, "compression_ratio": 1.7136363636363636, "no_speech_prob": 1.6093254089355469e-06, "words": [{"start": 1543.15, "end": 1543.51, "word": "اللي", "probability": 0.6796875}, 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4032, 28814, 1863, 13412, 16606, 21984, 44945, 24793, 1211, 9673, 5016, 46958, 25720, 25062, 18513, 46007, 4032, 28814, 23942, 5296, 4587, 16606, 19753, 34740], "avg_logprob": -0.23204787218190254, "compression_ratio": 1.7264573991031391, "no_speech_prob": 0.00018417835235595703, "words": [{"start": 2200.04, "end": 2200.56, "word": "الـ", "probability": 0.66650390625}, {"start": 2200.56, "end": 2201.22, "word": " condition", "probability": 0.7158203125}, {"start": 2201.22, "end": 2202.64, "word": " الكوشي", "probability": 0.7504730224609375}, {"start": 2202.64, "end": 2203.2, "word": " كريتيريا", "probability": 0.77314453125}, {"start": 2203.2, "end": 2204.08, "word": " المتحقق", "probability": 0.98291015625}, {"start": 2204.08, "end": 2204.86, "word": " هذا", "probability": 0.802734375}, {"start": 2204.86, "end": 2206.0, "word": " بكون", "probability": 0.620849609375}, {"start": 2206.0, "end": 2206.68, "word": " عندي", "probability": 0.5400390625}, {"start": 2206.68, "end": 2206.98, "word": " اللي", "probability": 0.926513671875}, {"start": 2206.98, "end": 2207.48, "word": " هي", "probability": 0.939453125}, {"start": 2207.48, "end": 2208.26, "word": " الـ", "probability": 0.666748046875}, {"start": 2208.26, "end": 2208.7, "word": " FM", "probability": 0.362060546875}, {"start": 2208.7, "end": 2209.86, "word": " أو", "probability": 0.64990234375}, {"start": 2209.86, "end": 2210.06, "word": " الـ", "probability": 0.807861328125}, {"start": 2210.06, "end": 2210.66, "word": " FN", "probability": 0.593505859375}, {"start": 2210.66, "end": 2211.92, "word": " converts", "probability": 0.287353515625}, {"start": 2211.92, "end": 2212.76, "word": " uniformly", "probability": 0.96435546875}, {"start": 2212.76, "end": 2213.6, "word": " to", "probability": 0.822265625}, {"start": 2213.6, "end": 2214.14, "word": " ال", "probability": 0.896484375}, {"start": 2214.14, "end": 2214.78, "word": " function", "probability": 0.475830078125}, {"start": 2214.78, "end": 2215.54, "word": " اللي", "probability": 0.9755859375}, {"start": 2215.54, "end": 2216.08, "word": " عندنا", "probability": 0.921875}, {"start": 2216.08, "end": 2216.44, "word": " اللي", "probability": 0.94921875}, {"start": 2216.44, "end": 2216.84, "word": " هي", "probability": 0.9189453125}, {"start": 2216.84, "end": 2217.8, "word": " F", "probability": 0.939453125}, {"start": 2217.8, "end": 2219.12, "word": " وبهيك", "probability": 0.7529296875}, {"start": 2219.12, "end": 2219.6, "word": " بنكون", "probability": 0.857177734375}, {"start": 2219.6, "end": 2220.08, "word": " احنا", "probability": 0.8665364583333334}, {"start": 2220.08, "end": 2221.62, "word": " انتهينا", "probability": 0.970458984375}, {"start": 2221.62, "end": 2221.82, "word": " من", "probability": 0.99267578125}, {"start": 2221.82, "end": 2222.0, "word": " ال", "probability": 0.90380859375}, {"start": 2222.0, "end": 2222.3, "word": " section", "probability": 0.65869140625}, {"start": 2222.3, "end": 2222.74, "word": " الأول", "probability": 0.84375}, {"start": 2222.74, "end": 2223.18, "word": " تمانية", "probability": 0.8400065104166666}, {"start": 2223.18, "end": 2223.76, "word": " واحد", "probability": 0.96484375}, {"start": 2223.76, "end": 2224.6, "word": " وإن", "probability": 0.76171875}, {"start": 2224.6, "end": 2224.84, "word": " شاء", "probability": 0.984619140625}, {"start": 2224.84, "end": 2225.22, "word": " الله", "probability": 0.9365234375}, {"start": 2225.22, "end": 2226.66, "word": " بنكمل", "probability": 0.8818359375}, {"start": 2226.66, "end": 2227.4, "word": " المحاضرة", "probability": 0.989013671875}, {"start": 2227.4, "end": 2228.0, "word": " القادمة", "probability": 0.9905598958333334}, {"start": 2228.0, "end": 2228.52, "word": " وإلى", "probability": 0.8805338541666666}, {"start": 2228.52, "end": 2228.9, "word": " لقاء", "probability": 0.9342447916666666}, {"start": 2228.9, "end": 2229.32, "word": " آخر", "probability": 0.796875}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2230.0676875, "duration_after_vad": 2203.481874999997}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..24c69ea1078d4077a7c63afd6e6f838ec81f5561
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/Xd6TZqtP5_4_raw.srt
@@ -0,0 +1,1864 @@
+1
+00:00:04,930 --> 00:00:09,090
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:09,090 --> 00:00:12,830
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه
+
+3
+00:00:12,830 --> 00:00:19,690
+أجمعين هذه المحاضرة رقم عشرين في مساق تحليل حقيقه
+
+4
+00:00:19,690 --> 00:00:24,630
+اتنين لطلاب وطالبات الجامعة الإسلامية كلية العلوم
+
+5
+00:00:24,630 --> 00:00:32,660
+قسم رياضياتهنستكمل اليوم الحديث في chapter 8 اللي
+
+6
+00:00:32,660 --> 00:00:38,380
+هو sequences of functions بالظبط هنستكمل الحديث في
+
+7
+00:00:38,380 --> 00:00:42,800
+ال section الأول اللي وجفنا عند نقطة فيه اللي هو
+
+8
+00:00:42,800 --> 00:00:46,680
+ال section كان تحت عنوان born twice and uniform
+
+9
+00:00:46,680 --> 00:00:51,660
+convergenceوفرّجنا بين اللي هو الـ pointwise و ال
+
+10
+00:00:51,660 --> 00:00:54,540
+uniform convergence قلنا أي uniform convergence
+
+11
+00:00:54,540 --> 00:00:58,140
+هيكون pointwise ولكن ال convergence لا يجب أن تكون
+
+12
+00:00:58,140 --> 00:01:01,920
+حقيقية في الجامعة و أخدنا بعض النظريات أول اشي
+
+13
+00:01:01,920 --> 00:01:04,980
+أعرفنا شو معناته اللي هو ال pointwise convergence
+
+14
+00:01:04,980 --> 00:01:11,180
+و بعدين أخدنا مثال على اللي هي ال function g of a
+
+15
+00:01:11,180 --> 00:01:14,440
+g n of x او sequence of functions g n of x بيستوي
+
+16
+00:01:14,440 --> 00:01:18,950
+x أس انو أثبتنا إن هذه pointwise convergence على
+
+17
+00:01:18,950 --> 00:01:23,190
+الفترة المفتوحة بالنقص 1 1 لدلة g of x بالساوية 0
+
+18
+00:01:23,190 --> 00:01:29,210
+1 زي ما هي أمامنا بعد هيك جينا حكينا عن اللي هو
+
+19
+00:01:29,210 --> 00:01:34,410
+مثال آخر أخدنا limit 1 على n sin n x زائد n قلنا
+
+20
+00:01:34,410 --> 00:01:37,970
+هذه ال sequence of functions برضه converts to the
+
+21
+00:01:37,970 --> 00:01:44,540
+zero function أو كانت اللي هو هذه مثال آخرعلى الـ
+
+22
+00:01:44,540 --> 00:01:50,020
+Point Convergence على كل R الآن إيجينا بعد هيك
+
+23
+00:01:50,020 --> 00:01:55,080
+اللي أخدنا لمّا و كيف اللي هو ان ترجم شو معناته Fn
+
+24
+00:01:55,080 --> 00:02:00,400
+Converges Point Twice قولنا الـ K اللي بيلاقيها
+
+25
+00:02:00,400 --> 00:02:04,720
+بتعتمد على Y و X في ال Point Twice لإنه نقطة نقطة
+
+26
+00:02:04,720 --> 00:02:09,550
+ال convergenceو بعدين اجينا عرفنا شو معناه uniform
+
+27
+00:02:09,550 --> 00:02:12,750
+convergence و قولنا في حالة ال uniform convergence
+
+28
+00:02:12,750 --> 00:02:16,630
+هيكون عندي اللي هو ال K اللي بنلاقيها بس بتعتمد
+
+29
+00:02:16,630 --> 00:02:20,730
+على ال epsilon و الباقي زي ما هو يعني بتنفع ال K
+
+30
+00:02:20,730 --> 00:02:24,730
+هذه لكل ال Xات اللي في ال A note عشان هيك بنسمي
+
+31
+00:02:24,730 --> 00:02:28,110
+uniform convergenceوقلنا إذا كانت الـ sequence
+
+32
+00:02:28,110 --> 00:02:31,530
+uniformly convergence أكيد هتكون convergence و
+
+33
+00:02:31,530 --> 00:02:36,230
+بعدين أخدنا ال criterion أو اللمّة اللي بننفي فيها
+
+34
+00:02:36,230 --> 00:02:40,170
+أو بنقول شو معناه Fn does not converge uniformly
+
+35
+00:02:40,170 --> 00:02:46,400
+to اللي هو function Fقلنا ان الـ fn لا يتعامل مع
+
+36
+00:02:46,400 --> 00:02:51,400
+عمليات الـ fn لا يتعامل مع عمليات الـ f إذا كان
+
+37
+00:02:51,400 --> 00:02:58,280
+لجينا إبسلون و سبسيكوينس fnk of fn سبسيكوينس من
+
+38
+00:02:58,280 --> 00:03:04,040
+سيكوينس الـ functions هذه وعندي سيكوينس xkفي الـ A
+
+39
+00:03:04,040 --> 00:03:08,600
+Note بحيث تحقق F N K of X K ناقص F of X K أكبر شهر
+
+40
+00:03:08,600 --> 00:03:14,000
+Y Note هذا على بعضه يعني أن الـ F N does not
+
+41
+00:03:14,000 --> 00:03:17,320
+converge uniformly to الـ function اللي أمامنا
+
+42
+00:03:17,320 --> 00:03:22,320
+اللي هي F هذا حكينا عنه المرة الماضية و أثبتناها و
+
+43
+00:03:22,320 --> 00:03:27,240
+بعدين أخدنا مثال قلنا خد اللي هو G N of X بيساوي X
+
+44
+00:03:27,240 --> 00:03:31,670
+plus Nsequence of functions على الـ R أحنا أثبتنا
+
+45
+00:03:31,670 --> 00:03:35,810
+إنه هذه اللي هي الـ sequence قبل بشوية اللي شفناها
+
+46
+00:03:35,810 --> 00:03:43,610
+إنها converge to G of X هذه 1 0 X بساوة 1 و X
+
+47
+00:03:43,610 --> 00:03:48,450
+element نقص 1 1 converge point-wise لهذه الـ
+
+48
+00:03:48,450 --> 00:03:52,350
+functionالان و أثبتناها برضه و أثبتناها المرة
+
+49
+00:03:52,350 --> 00:03:56,370
+الماضية ان GM لا تتعامل بشكل عام لهذه الـ function
+
+50
+00:03:56,370 --> 00:04:00,070
+على الفترة وهذا تأكيد على كلامنا ان الـ point wise
+
+51
+00:04:00,070 --> 00:04:06,110
+convergence لا يتعامل بشكل عام و هي مثال على ذلك
+
+52
+00:04:06,740 --> 00:04:12,940
+الآن أعرفنا شو معنات اللي هو برضه لو كانت عندي
+
+53
+00:04:12,940 --> 00:04:16,280
+function Phi اللي هي bounded function شو معنات
+
+54
+00:04:16,280 --> 00:04:19,900
+normها قولنا ال norm لـ function A على اللي هي ال
+
+55
+00:04:19,900 --> 00:04:24,060
+6A هو عبارة عن ال supremum لقيمة ال function 5X X
+
+56
+00:04:24,060 --> 00:04:27,660
+element A هذا في حالة اللي هي mean ال bounded
+
+57
+00:04:27,660 --> 00:04:33,100
+function اللي هي اللي هي سمنها Phi بعد هيك جينا
+
+58
+00:04:34,730 --> 00:04:38,450
+وحكينا عن اللمّة اللي أمامنا هذه أنه لو كان اللي
+
+59
+00:04:38,450 --> 00:04:41,450
+هي a sequence of unbounded functions on A subset
+
+60
+00:04:41,450 --> 00:04:44,890
+من R اللي هي the sequence of meal يعني هذا test
+
+61
+00:04:44,890 --> 00:04:47,170
+للـ uniform convergence أو عدم الـ uniform
+
+62
+00:04:47,170 --> 00:04:51,670
+convergence لعائلة من الدوال أو sequence of
+
+63
+00:04:51,670 --> 00:04:55,580
+functions اللي هي of unbounded functionfunctions
+
+64
+00:04:55,580 --> 00:04:59,680
+إذن هذه بتنفع لمين للـbounded functions بتعملي
+
+65
+00:04:59,680 --> 00:05:04,280
+test عليها إنها uniform ولا مش uniform ببساطة بس
+
+66
+00:05:04,280 --> 00:05:08,680
+بنجيب norm ال FN نقص F وبنشوفه إذا كان norm مع ال
+
+67
+00:05:08,680 --> 00:05:12,700
+A بروح للسفر معناته على طول بنقول FN converges
+
+68
+00:05:12,700 --> 00:05:16,260
+uniformly to مين؟ To F إذا كان ال norm هذا does
+
+69
+00:05:16,260 --> 00:05:20,160
+not converge to zero معناته كل سنة و انت تسأل مين؟
+
+70
+00:05:20,160 --> 00:05:24,360
+اللي هو ال FN does not converge uniformly to مين؟
+
+71
+00:05:24,360 --> 00:05:29,060
+To Fهذه الآن اللي وصلنا عندها اللي هي اللي ما
+
+72
+00:05:29,060 --> 00:05:36,380
+واخدنا مثال برضه عليها اللي هو GN of XN قلنا هذا
+
+73
+00:05:36,380 --> 00:05:40,260
+اللي هو المثال
+
+74
+00:05:40,260 --> 00:05:45,890
+اللي أمامناكيف نحله على اللي هي ال .. ال .. ال ..
+
+75
+00:05:45,890 --> 00:05:49,550
+النظرية اللي قبل بشوية قولنا خد GN of X بالساوية X
+
+76
+00:05:49,550 --> 00:05:55,810
+أُس N على الفترة مين؟ سفر واحد لجينا انه .. انه GN
+
+77
+00:05:55,810 --> 00:06:00,510
+converts uniformly لمين؟ لـG of X على الفترة اللي
+
+78
+00:06:00,510 --> 00:06:05,410
+أمامنا اللي هي هذه اللي هي فترة Zero وواحد طبعا با
+
+79
+00:06:05,410 --> 00:06:09,180
+.. واضح انه الدالة هذهbounded على هذه الفترة عشان
+
+80
+00:06:09,180 --> 00:06:13,220
+هي تنفي عن السخدم اللي هي النظرية اللي فوق هذه
+
+81
+00:06:13,220 --> 00:06:17,980
+المرة الماضية برضه حكيناها أو جينا أو جفنا عند
+
+82
+00:06:17,980 --> 00:06:23,260
+اللي هي المثال التاني طبعا اللي هي المثال الفجاني
+
+83
+00:06:23,260 --> 00:06:28,300
+لو على السريع ذكرناكم فيها الان ال function مين هي
+
+84
+00:06:28,300 --> 00:06:34,460
+عندى ال function عبارة عن g n of x او sequence of
+
+85
+00:06:34,460 --> 00:06:39,410
+functions بسوء x أس nالان لو جي ده أخدت ال norm
+
+86
+00:06:39,410 --> 00:06:44,870
+لجي ان ناقص الجي هي ال جي ان وهي الجي خد ال norm
+
+87
+00:06:44,870 --> 00:06:50,730
+إلها خد ال norm إلها معايا
+
+88
+00:06:50,730 --> 00:06:54,650
+يا جماعة هي ال norm إلها لجي ان ناقص جي بساوي اللي
+
+89
+00:06:54,650 --> 00:06:59,170
+هو عبارة عن ال supremum X أس ان ناقص سفر يعني X أس
+
+90
+00:06:59,170 --> 00:07:05,750
+ان لمن؟ لل X المتزير واحد او الجزء الثاني X أس ان
+
+91
+00:07:05,750 --> 00:07:10,970
+ناقص واحداللي هي في حالة ال X بالساوية واحد لكن في
+
+92
+00:07:10,970 --> 00:07:13,490
+حالة ال X بالساوية واحد بيصير G and X and بيتساوية
+
+93
+00:07:13,490 --> 00:07:17,370
+واحد واحد نقص واحد بيصير سفر بيصير عندي ال norm
+
+94
+00:07:17,370 --> 00:07:20,490
+هذا عبارة عن ال supremum X and X في الفترة Zero
+
+95
+00:07:20,490 --> 00:07:24,340
+واحد و Zero لما ال X بالساوية واحدالـ Supremum
+
+96
+00:07:24,340 --> 00:07:29,600
+للقيمة هذه عندي أكيد إيش هيساوي عند اللي هي X
+
+97
+00:07:29,600 --> 00:07:34,380
+بتساوي واحد و Y ساوي اللي هو عبارة أو قبل اللي هي
+
+98
+00:07:34,380 --> 00:07:40,620
+ال Supremum ل ال 6 هذه طبعا هنا X element in 01 و
+
+99
+00:07:40,620 --> 00:07:44,780
+1 اللي هي not included فبكون عندي ال Supremum
+
+100
+00:07:44,780 --> 00:07:50,060
+للقيم X أسن في الفترة من 01 إيش هيساوي؟ هيساوي
+
+101
+00:07:50,060 --> 00:07:57,150
+واحد يعني ال Supremumلأ x أسن such that x element
+
+102
+00:07:57,150 --> 00:08:05,210
+in zero و واحد حتى لو كانت closed هذه و اللي هو
+
+103
+00:08:05,210 --> 00:08:08,730
+اتحاد
+
+104
+00:08:08,730 --> 00:08:13,690
+ال zero ال supreme لها طبعا لل x أسن ايش حيساو في
+
+105
+00:08:13,690 --> 00:08:17,330
+النهاية حيساو واحد و ممكن اثبات هذا اللي هو عن
+
+106
+00:08:17,330 --> 00:08:21,900
+طريق ال least upper boundكيف نثبته زي ما كنا نثبت
+
+107
+00:08:21,900 --> 00:08:26,540
+فيه اللي هو تحليل واحد مش ماكانوا نثبته لان فبصير
+
+108
+00:08:26,540 --> 00:08:30,280
+عندي ال norm لجي عن نقص جي اش بساوي واحدما دام الـ
+
+109
+00:08:30,280 --> 00:08:33,980
+norm لجي أن بيساوي واحد إذا ال norm لجي أن نقص جي
+
+110
+00:08:33,980 --> 00:08:40,920
+does not converge as if الجي أن لا يتعامل مع الجي
+
+111
+00:08:40,920 --> 00:08:47,040
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+112
+00:08:47,040 --> 00:08:54,180
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+113
+00:08:54,180 --> 00:08:54,660
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+114
+00:08:54,660 --> 00:08:54,660
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+115
+00:08:54,660 --> 00:08:54,660
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقص جي
+
+116
+00:08:54,660 --> 00:08:55,450
+أن نقص جي أن نقص جي أن نقص جي أن نقص جي أن نقنقص
+
+117
+00:08:55,450 --> 00:09:00,190
+الـ G برضه هيروح على مين؟ على القيمة واحد يعني مش
+
+118
+00:09:00,190 --> 00:09:05,890
+هيروح للسفر يعني إذاً GN does not converge
+
+119
+00:09:05,890 --> 00:09:10,850
+uniformly to مين؟ to G وهذا مثال عليه اللي هو does
+
+120
+00:09:10,850 --> 00:09:16,030
+not converge uniformly على الـ G مثال على اللي هو
+
+121
+00:09:16,030 --> 00:09:20,250
+توظيف هذه النظرية الجابلة بشوية لإثبات عدم ال
+
+122
+00:09:20,250 --> 00:09:23,930
+uniformlyconvergent أو لو بدنا نثبت انه formally
+
+123
+00:09:23,930 --> 00:09:29,910
+convergent ممكن نستخدم هذه بس نثبت ان ال norm بروح
+
+124
+00:09:29,910 --> 00:09:33,710
+لصفر بس بحيث انه تكون اللي هي ال GN عبارة عن
+
+125
+00:09:33,710 --> 00:09:38,350
+sequence زي ما قلنا of bounded functions طيب ميجي
+
+126
+00:09:38,350 --> 00:09:42,030
+مثال آخر ماعليش سريع لأنه حكينا المرة الماضية ميجي
+
+127
+00:09:42,030 --> 00:09:48,070
+لا اللي هو مثلنا هذانجي لمثالنا fn of x بيساوي x
+
+128
+00:09:48,070 --> 00:09:53,050
+على n مش x أسان x على n for every x element in R
+
+129
+00:09:53,050 --> 00:09:58,950
+اللي هم بيقوللي show that limit fn of x as x بتروح
+
+130
+00:09:58,950 --> 00:10:02,850
+إلى infinity بيساوي f of xلأ كل x element in R
+
+131
+00:10:02,850 --> 00:10:08,430
+where f of x ايش بتساوي سفر and fn x بتروح ل f of
+
+132
+00:10:08,430 --> 00:10:11,610
+x من R هذا معناته اللي هو pointwise convergence
+
+133
+00:10:11,610 --> 00:10:16,330
+يعني بده limit ال fn of x ناخد اي x element in R
+
+134
+00:10:16,330 --> 00:10:19,490
+ناخد x element in R نثبت ان limit fn x ايش هتساوي
+
+135
+00:10:19,490 --> 00:10:24,830
+هتساوي سفرالان هذا اللي هو عبارة عن اللي هو الـ
+
+136
+00:10:24,830 --> 00:10:30,110
+pointwise convergence بعد شوية هنثبت انه عندنا هذا
+
+137
+00:10:30,110 --> 00:10:34,530
+مش هيكون فيه عندى اللي هو uniform convergence على
+
+138
+00:10:34,530 --> 00:10:40,790
+مكان ما في الأرض نرجع لمثالنا اللي هو عندى مثال
+
+139
+00:10:40,790 --> 00:10:48,880
+اللي هو Fn X بسوية X على Nلـ X element in R عندي F
+
+140
+00:10:48,880 --> 00:10:52,820
+of X بيساوي 0 لكل X element in R بدنا نثبت أن F N
+
+141
+00:10:52,820 --> 00:10:56,500
+of X بتروح pointwise ل F of X لكل X element in R
+
+142
+00:10:56,500 --> 00:11:01,180
+وبدنا نثبت أن F N does not converge لـ F uniformly
+
+143
+00:11:01,180 --> 00:11:04,500
+لـ F on R يعني هذا هو عبارة عن pointwise
+
+144
+00:11:04,500 --> 00:11:10,170
+convergence لكن مش uniform convergenceبس هنا جيبنا
+
+145
+00:11:10,170 --> 00:11:14,310
+المثال عشان نقولك انه مش هنقدر نستخدم اللي هي
+
+146
+00:11:14,310 --> 00:11:17,630
+اللمّة اللي قبل بشوية لإنه عندي sequence of
+
+147
+00:11:17,630 --> 00:11:23,910
+functions Fn على R اللي هي is not bounded يعني لو
+
+148
+00:11:23,910 --> 00:11:29,150
+جينا أخدنا F1 of X بتساوي X على 1لكل x element in
+
+149
+00:11:29,150 --> 00:11:33,310
+R، وكأن f1 of x عبارة عن x، طبعا الـ x is not
+
+150
+00:11:33,310 --> 00:11:36,790
+bounded function، f2x التي هي x على 2 برضه is not
+
+151
+00:11:36,790 --> 00:11:40,250
+bounded، لأنها تبقى مالا نهاية رايحة من الجهتين،
+
+152
+00:11:40,250 --> 00:11:42,510
+واحد مالا نهاية و واحد سالب مالا نهاية،
+
+153
+00:11:48,910 --> 00:11:54,190
+عشان هيك ما بنقدرش نستخدم اللي هو عشان نثبت ال FN
+
+154
+00:11:54,190 --> 00:11:58,150
+does not converge uniformly بنقدرش نستخدم normal
+
+155
+00:11:58,150 --> 00:12:03,310
+FN ناقص F اللي هو يروح للسفر لإن هذه ال FN is not
+
+156
+00:12:03,530 --> 00:12:07,550
+bounded functions are not bounded functions يعني
+
+157
+00:12:07,550 --> 00:12:11,190
+اللي هو اللمّة لا تنطبق عليها عشان هيك هنضطر نروح
+
+158
+00:12:11,190 --> 00:12:15,450
+بالاستخدام التعريف أو نفي التعريف لل uniformly
+
+159
+00:12:15,450 --> 00:12:18,690
+convergence زي ما عملنا في مثال المرة الماضية
+
+160
+00:12:18,690 --> 00:12:25,290
+وخلّينا نشوف الحل نيجي الآن الجزء الأول الجزء
+
+161
+00:12:25,290 --> 00:12:32,700
+الأول سهل على طول اللي هو عندي لكل XElement in R
+
+162
+00:12:32,700 --> 00:12:40,460
+أكيد عندي limit of N of X لما N تروح إلى ماله
+
+163
+00:12:40,460 --> 00:12:47,240
+نهاية بساوي limit of N of X هي عبارة عن X على N as
+
+164
+00:12:47,240 --> 00:12:50,860
+N goes to infinity ويساوي كلكم بعرفيش هتساوي،
+
+165
+00:12:50,860 --> 00:12:56,030
+بتساوي سفريعني بمعنى آخر اللي هي الـ fn of x
+
+166
+00:12:56,030 --> 00:13:00,990
+converge لمين للـ f اللي هي of x اللي بتساوي سفر
+
+167
+00:13:00,990 --> 00:13:04,610
+وهذا ال convergence عبارة عن pointwise convergence
+
+168
+00:13:04,610 --> 00:13:10,070
+شوف الآن كيف بدنا نثبت خلصنا من انه احنا نثبت fn x
+
+169
+00:13:10,070 --> 00:13:14,590
+بتروح pointwise لمين لاللي هي ال function zero
+
+170
+00:13:14,590 --> 00:13:18,350
+بدنا نركز الآن نثبت ان fn does not converge
+
+171
+00:13:18,350 --> 00:13:24,080
+uniformly لل function f on Rاللي هو زي ما قلنا الـ
+
+172
+00:13:24,080 --> 00:13:27,940
+sequence of functions عندي هنا are not bounded
+
+173
+00:13:27,940 --> 00:13:32,620
+functions so we can't use our لمّة اللي قبل بشوية
+
+174
+00:13:32,620 --> 00:13:36,060
+اللي هي normal of a نقص of a converges to zero إذا
+
+175
+00:13:36,060 --> 00:13:41,100
+هنتضرر نروح للتعريف أو للمّة قبلها إذا بتتذكروا
+
+176
+00:13:41,100 --> 00:13:44,820
+اللمّة قلنا of a that does not convergeUniform للـ
+
+177
+00:13:44,820 --> 00:13:48,620
+F إيش بدنا نلاقي؟ بدنا نلاقي Epsilon Note أكبر من
+
+178
+00:13:48,620 --> 00:13:57,080
+0 و Sub-Sequence FNK و Sequence XK in R بحيث أنه
+
+179
+00:13:57,080 --> 00:14:03,060
+اللي هو هيها بدنا نلاقي Epsilon Note و نلاقي Sub
+
+180
+00:14:03,060 --> 00:14:06,440
+-Sequence FNK و نلاقي Sequence XK in R بحيث أنه
+
+181
+00:14:06,440 --> 00:14:14,340
+الـ Norm أو الـ Absolute Value للـ FNK of XK-F of
+
+182
+00:14:14,340 --> 00:14:21,180
+XK يكون أكبر من مين من Y note اللي عندنا اللي
+
+183
+00:14:21,180 --> 00:14:25,240
+بنقول لجميعها إذا عملنا زيك بيكون معناته أن F does
+
+184
+00:14:25,240 --> 00:14:28,900
+not converge uniformly to مين، to F خلّينا نيجي
+
+185
+00:14:28,900 --> 00:14:34,100
+إلى اللي هو ونشوف كيف نسويزي ما عملنا المرة
+
+186
+00:14:34,100 --> 00:14:39,940
+الماضية الفكرة انه الان ال fnk of xk نخليها عبارة
+
+187
+00:14:39,940 --> 00:14:44,640
+عن ثابت لو خلناها ثابت وهذه طبعا سفر بيصير الثابت
+
+188
+00:14:44,640 --> 00:14:49,600
+أكبر بيطلع ثابت خمسة ستة سبعة واحد اتنين بنختار
+
+189
+00:14:49,600 --> 00:14:52,600
+ابسلونوت اللي هي اصغر منه بيطلع حد اكبر من
+
+190
+00:14:52,600 --> 00:14:58,660
+الابسلونوت بيكون خلصناه ده نشوف كيف عندي fn of x
+
+191
+00:14:58,660 --> 00:15:04,320
+ايش بالساوي x على nلو جينا اخترنا اللي هو الـ NK
+
+192
+00:15:04,320 --> 00:15:08,060
+الـ NK الـ subsequence يعني اللي هي الـ NK
+
+193
+00:15:08,060 --> 00:15:13,540
+اخترناها الـ K بيصير اللي هي FNK اللي هي of X
+
+194
+00:15:13,540 --> 00:15:18,620
+بيساوي FK of X بيصير هذا K X على K طب ال X ايش
+
+195
+00:15:18,620 --> 00:15:25,240
+نختارها عشان نخلي هذا ثابت اختارهاK , 2K , 3K زي
+
+196
+00:15:25,240 --> 00:15:30,480
+ما بدك كله حيكون وين موجود في R وكله حيثبت لمين
+
+197
+00:15:30,480 --> 00:15:33,820
+اللي هي ال function اللي عندى و تصير أموري تمام
+
+198
+00:15:33,820 --> 00:15:39,840
+تمام ايش عامل هنا بقول ليه let F and K او let NK
+
+199
+00:15:39,840 --> 00:15:47,440
+هي let NK بساوي K يعني بمعنى اخر اخد ال F and K هو
+
+200
+00:15:47,440 --> 00:15:53,360
+او بتعبير اخر F خد ال F and K هي ال FK اختار الـ
+
+201
+00:15:53,360 --> 00:15:57,320
+FNK المميزة بـ Sequence اللي هي FK و Epsilon
+
+202
+00:15:57,320 --> 00:16:00,800
+خلّينا بعدين نختارها و XK زي ما قلنا ايش راح
+
+203
+00:16:00,800 --> 00:16:06,820
+نختارها بتساوي K أكيد وين موجودة في R then FNK of
+
+204
+00:16:06,820 --> 00:16:14,380
+XK ايش هتساوي اللي هي FK of XK ويساوي اللي هي
+
+205
+00:16:14,380 --> 00:16:20,080
+عبارة عن XKعلى k حسب اللي هي الدقلة اللي عندى او
+
+206
+00:16:20,080 --> 00:16:22,720
+ال sequence of functions طب ما هي ال xk ايش انا
+
+207
+00:16:22,720 --> 00:16:27,420
+بختارها بتساوي k فبصير k على k هو يساوي واحد اذا
+
+208
+00:16:27,420 --> 00:16:34,210
+طلعت عندى fnk of xk بيساوي واحدهيحسب للـ fnk of xk
+
+209
+00:16:34,210 --> 00:16:37,410
+للـ subsequence اللي عملناها ناقص f of xk إيش
+
+210
+00:16:37,410 --> 00:16:41,770
+هتساوي هذه عبارة عن واحد وهذه دالة دايما ثابتة f
+
+211
+00:16:41,770 --> 00:16:45,610
+عند أي قيمة بتساوي سفر فبصير اللي هو عبارة عن إيش
+
+212
+00:16:45,610 --> 00:16:49,030
+القيمة واحد إيش أختار epsilon note اللي بدك إياها
+
+213
+00:16:49,030 --> 00:16:54,680
+أي إيش أصغر منيه من الواحد وليكن إيش نصفبصير عندى
+
+214
+00:16:54,680 --> 00:17:00,140
+FN K هيها FK و Epsilon بساوة نص و XK بساوة K هي
+
+215
+00:17:00,140 --> 00:17:05,460
+sub sequence من ال FN و هي الي هو Epsilon نص و هي
+
+216
+00:17:05,460 --> 00:17:09,620
+sequence XK in R بحيث أن هذا ال absolute value
+
+217
+00:17:09,620 --> 00:17:13,660
+أكبر من مين؟ اللي هو طلع واحد اللي هو أكيد أكبر من
+
+218
+00:17:13,660 --> 00:17:17,900
+إيش؟ من نص اللي هي Epsilon نص وهذا يعني أن FN does
+
+219
+00:17:17,900 --> 00:17:24,390
+not converge uniformly للF on Rهكذا أثبتنا أن الـ
+
+220
+00:17:24,390 --> 00:17:27,310
+Fn لا يتعامل بشكل عام بشكل عام بشكل عام بشكل عام
+
+221
+00:17:27,310 --> 00:17:29,790
+بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل
+
+222
+00:17:29,790 --> 00:17:31,210
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+223
+00:17:31,210 --> 00:17:31,710
+بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل
+
+224
+00:17:31,710 --> 00:17:32,470
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+225
+00:17:32,470 --> 00:17:32,470
+بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل
+
+226
+00:17:32,470 --> 00:17:34,290
+عام بشكل عام بشكل عام بشكل عام بشكل عام بشكل عام
+
+227
+00:17:34,290 --> 00:17:36,760
+بشكل عام بشكل عام بشكل عام بشكل عامأو الثالث في
+
+228
+00:17:36,760 --> 00:17:42,160
+المثال، بدنا نثبت أن الـ F Unconverges للـ F
+
+229
+00:17:42,160 --> 00:17:48,640
+Uniformly على الفترة مين؟ على الفترة 01، يعني لما
+
+230
+00:17:48,640 --> 00:17:53,080
+نغيرنا الفترة، هيصير اللي هو الـ Pointwise ماقدرش
+
+231
+00:17:53,080 --> 00:17:57,040
+اللي هو الـ Uniform convergence على الـ R، ماقدرش
+
+232
+00:17:57,200 --> 00:18:01,420
+اللي هو يكون uniform convergence لكن لو زغرنا
+
+233
+00:18:01,420 --> 00:18:06,420
+الفترة أو غيرنا الفترة ممكن نحصل على اللي هو ال
+
+234
+00:18:06,420 --> 00:18:11,480
+uniform convergence وهذا طبيعي، هنشوف كيف، الان
+
+235
+00:18:11,480 --> 00:18:16,640
+عشان اللي هو نظريتنا سهلة الاستخدام اللي قبل بشوية
+
+236
+00:18:16,640 --> 00:18:22,580
+اللي هو normal Fن ناقص F بدي أحسبه و أشوفه على
+
+237
+00:18:22,580 --> 00:18:27,240
+الفترة 0 و 1 اللي هو conversion to 0 ولا لأ؟ آه
+
+238
+00:18:27,240 --> 00:18:30,520
+الآن قبلها بدي أشوف ال sequence of functions عندي
+
+239
+00:18:30,520 --> 00:18:36,520
+هذه هل bounded على الفترة 0 و 1؟ Fn X بتساوي X على
+
+240
+00:18:36,520 --> 00:18:44,140
+N يعني صار عندي FN of X بيساوي X على N لوين X في
+
+241
+00:18:44,140 --> 00:18:47,860
+الفترة 0 و 1 خلاص الـ X في الفترة 0 و 1 جد ما تكون
+
+242
+00:18:47,860 --> 00:18:51,600
+ليها الـ X نفسها محصورة بين 0 و 1 يعني بتضل الـ X
+
+243
+00:18:51,600 --> 00:18:57,040
+على N عبارة عن قيمة محدودة قيمة محدودة أكيد اللي
+
+244
+00:18:57,040 --> 00:19:00,680
+هو الـ absolute value اللي هي هذه أصغر أو يساوي
+
+245
+00:19:00,680 --> 00:19:05,100
+اللي هو 1 على N إذا صار عند .. لكل .. طبعا هذا
+
+246
+00:19:05,100 --> 00:19:10,510
+محدودة بالنسبة لـ Xأه بيصير F1 أصغر بيساوي 1 F2
+
+247
+00:19:10,510 --> 00:19:15,350
+أصغر بيساوي نص F3 أصغر بيساوي تلت و هكذا إذن هي
+
+248
+00:19:15,350 --> 00:19:21,790
+عبارة عن F1 F2 F3 sequence of bounded functions on
+
+249
+00:19:21,790 --> 00:19:26,570
+the interval 0 و 1 إذن الآن الأمر مشروع ليه أني
+
+250
+00:19:26,570 --> 00:19:31,970
+أستخدم اللمّة اللي حكينا عنها قبل بشوية و نشوف كيف
+
+251
+00:19:31,970 --> 00:19:36,110
+نثبت ال uniform convergence نحسب مع بعض أكيد لكلكم
+
+252
+00:19:36,440 --> 00:19:43,380
+اللي هو Fn ناقص الـ F over A بساوي الـ Supremum لـ
+
+253
+00:19:43,380 --> 00:19:50,380
+Fn of X ناقص F of X Such that X و M هو وجودة في
+
+254
+00:19:50,380 --> 00:19:57,120
+الفترة 0 و 1 و يساوي الـ Supremum لـ Fn of X اللي
+
+255
+00:19:57,120 --> 00:20:03,270
+هي أشكل عنها عبارة عن X على Nx على n ناقصها دي سفر
+
+256
+00:20:03,270 --> 00:20:07,110
+بتضل x على n such that x where موجودة في ال zero
+
+257
+00:20:07,110 --> 00:20:10,330
+أو إيش أو واحد ال suprem أهم إلها زي ما تقولنا
+
+258
+00:20:10,330 --> 00:20:14,790
+أشملها أصغرشها واحد على n هتطلع عبارة عن أ واحد
+
+259
+00:20:14,790 --> 00:20:21,670
+على nهذه اللي هي مين ال norm للأف أن ناقص أف طيب
+
+260
+00:20:21,670 --> 00:20:26,530
+واضح الأمر مدام ال norm للأف أن ناقص أف بسوى حتى
+
+261
+00:20:26,530 --> 00:20:35,220
+الأن إذا norm للأف أن ناقص أفA goes to zero لأن FN
+
+262
+00:20:35,220 --> 00:20:38,880
+goes to infinity قيمة هذا الواحدة الان هتروح إلى
+
+263
+00:20:38,880 --> 00:20:42,960
+سفر يعني norm هذا بروح للزيرو وهذا يعني أن FN
+
+264
+00:20:42,960 --> 00:20:49,240
+converts uniformly ل F on الفترة زيرو وواحد وبكون
+
+265
+00:20:49,240 --> 00:20:54,180
+هيك أثبتنا اللي هو باستخدام النظرية أن هذا المثال
+
+266
+00:20:54,180 --> 00:20:58,680
+فعلااللي هو F Unconverged uniformly للـ F On The
+
+267
+00:20:58,680 --> 00:21:03,380
+Interval 01 But It Does Not Converge Uniformly On
+
+268
+00:21:03,380 --> 00:21:10,120
+R ككل لكن Pointwise على الـ R To The كل طيب نيجي
+
+269
+00:21:10,120 --> 00:21:16,300
+الآن اللي هوالتعبير آخر على الـ uniform
+
+270
+00:21:16,300 --> 00:21:19,880
+convergence عبرنا عن الـ uniform convergence
+
+271
+00:21:19,880 --> 00:21:26,180
+بواسطة اللي هي التعريف عبرنا عنه للعائلة of
+
+272
+00:21:26,180 --> 00:21:30,890
+bounded functions بواسطة الـ norm للأفئناللي هو
+
+273
+00:21:30,890 --> 00:21:37,190
+ناقص الـ F بروح للـ 0 الآن بدنا نعرف عنه .. نعبر
+
+274
+00:21:37,190 --> 00:21:40,070
+عنه عن طريق .. برضه للـ sequence of bounded
+
+275
+00:21:40,070 --> 00:21:43,230
+functions عن طريق حاجة اسمها الـ Cauchy criterion
+
+276
+00:21:43,230 --> 00:21:47,270
+for uniform convergence وكأنه بدنا ندخل المفهوم
+
+277
+00:21:47,270 --> 00:21:53,590
+الـ Cauchy اللي هي sequence في هذا اللي هو النظرية
+
+278
+00:21:53,590 --> 00:21:57,470
+نشوفكوشي كريتيريا لتحويل المستوى المستوى المستوى
+
+279
+00:21:57,470 --> 00:21:58,350
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+280
+00:21:58,350 --> 00:22:03,130
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+281
+00:22:03,130 --> 00:22:06,950
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+282
+00:22:06,950 --> 00:22:06,950
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+283
+00:22:06,950 --> 00:22:10,330
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+284
+00:22:10,330 --> 00:22:10,350
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+285
+00:22:10,350 --> 00:22:10,410
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+286
+00:22:10,410 --> 00:22:10,430
+المستوى المستوى المستوى المستوى المستوى المستوى
+
+287
+00:22:10,430 --> 00:22:15,740
+المستوى المستF يعني بتكون F unconverted لـ bounded
+
+288
+00:22:15,740 --> 00:22:20,220
+function F F and only F لكل Y أكبر من 0 there
+
+289
+00:22:20,220 --> 00:22:23,680
+exists H of Y للـ N such that for every M و N أكبر
+
+290
+00:22:23,680 --> 00:22:28,180
+يساوي H of Y حيكون الـ FM مقصلة F أصغر أو يساوي Y
+
+291
+00:22:28,180 --> 00:22:32,940
+وهذا هيذكرنا اللي هو بالكوشي sequences إذا نسيناها
+
+292
+00:22:33,270 --> 00:22:37,270
+يعني وكأنه خلّيني نقوله roughly بيقول لـ F N تحت
+
+293
+00:22:37,270 --> 00:22:42,950
+الظروف اللي موجودة بتروح للـ F uniformly يكافئ أنه
+
+294
+00:22:42,950 --> 00:22:47,510
+لكل إبسلون أكبر من 0 هذا الـ N الجزء هنا هو اللي
+
+295
+00:22:47,510 --> 00:22:51,070
+يكوش criterion في الواقع for every إبسلون أكبر من
+
+296
+00:22:51,070 --> 00:22:56,950
+0 there exists H of إبسلون element in N such that
+
+297
+00:22:56,950 --> 00:23:03,680
+for every N و M element in Nأو اللي هي أكبر أو
+
+298
+00:23:03,680 --> 00:23:08,140
+يساوي اتشوف إبسلون حيكون عندك الـ epsilon norm لل
+
+299
+00:23:08,140 --> 00:23:17,100
+FM ناقص FM أصغر من 100 أو يساوي اللي هو إبسلونهذه
+
+300
+00:23:17,100 --> 00:23:20,180
+اللي هي عبارة عن الـ Cauchy criterion يعني أحيانًا
+
+301
+00:23:20,180 --> 00:23:24,160
+نحن نعرف .. نكونش عارفين الـ Fn لمين رايحة لكن
+
+302
+00:23:24,160 --> 00:23:27,380
+متأكدين أنها converges uniformly to some function
+
+303
+00:23:27,380 --> 00:23:32,600
+F طب الآن في هذه الحالة اللي بصعفنا في الاستخدام
+
+304
+00:23:32,600 --> 00:23:38,480
+اللي هو أنه ناخد اللي هو بدلالات ال terms اللي هي
+
+305
+00:23:38,480 --> 00:23:42,440
+ال Fn و ال Fm و نبعد عن ال F لأنه احنا مش عارفينها
+
+306
+00:23:42,440 --> 00:23:45,260
+عشان هيك بنقول Fn converges uniformly to some
+
+307
+00:23:45,260 --> 00:23:50,340
+function Fيعني هيك تصرفها إذا و فقط إذا اللي هو
+
+308
+00:23:50,340 --> 00:23:53,920
+لكل إبسلون أكبر من سفر الـ Epsilon element in N
+
+309
+00:23:53,920 --> 00:23:56,300
+such that forever N و M أكبر من سفر الـ Epsilon
+
+310
+00:23:56,300 --> 00:23:59,680
+بيعطينا norm هذه أصغر من minimum إبسلون و كأنه
+
+311
+00:23:59,680 --> 00:24:03,650
+بنقول اللي هو ال function FNis a Cauchy sequence
+
+312
+00:24:03,650 --> 00:24:08,250
+roughly in the space of functions اللي هي ال norm
+
+313
+00:24:08,250 --> 00:24:11,470
+بساوي ال supremum اللي هو هذا الموضوع طبعا مش
+
+314
+00:24:11,470 --> 00:24:15,170
+موضوعنا موضوع ناخده بالتفصيل في اللي هو ال
+
+315
+00:24:15,170 --> 00:24:19,870
+functional analysis طيب شوفوا علينا نيجي اللي هو
+
+316
+00:24:19,870 --> 00:24:26,710
+انبرهن النظرية بدنا نفترض ان ال FNالان بدنا نفترض
+
+317
+00:24:26,710 --> 00:24:31,970
+انه norm أو الـ F un converge uniformly to some
+
+318
+00:24:31,970 --> 00:24:37,790
+function F و نصل لهذا اللي هو الجزء سهل ان شاء
+
+319
+00:24:37,790 --> 00:24:44,450
+الله شوفوا ركزولنا الان suppose that F un converge
+
+320
+00:24:44,450 --> 00:24:49,390
+uniformly ل F on A إذا الان من التعريف تبع ال
+
+321
+00:24:49,390 --> 00:24:54,480
+uniform convergence لكل Y أكبر من 0There exists K
+
+322
+00:24:54,480 --> 00:24:58,420
+of Y على 2 element in N such that for every N أكبر
+
+323
+00:24:58,420 --> 00:25:03,180
+أو يساوي K of Y ولكل X element in A برضه بظبط اللي
+
+324
+00:25:03,180 --> 00:25:08,180
+هو FN of X ناقص F of X أصغر من Y على 2 هذا
+
+325
+00:25:08,180 --> 00:25:11,000
+باستخدام التعريف الـ N باستخدام النظرية اللي قبل
+
+326
+00:25:11,000 --> 00:25:14,720
+بشوية لإنه بنفع لإن النظرية احنا قلنا FN converges
+
+327
+00:25:14,720 --> 00:25:19,540
+uniformly إيش معناته هذا؟معناته more من الـ F and
+
+328
+00:25:19,540 --> 00:25:25,720
+ناقص الـ F goes to 0 over mean over a هذه صار
+
+329
+00:25:25,720 --> 00:25:30,160
+تعريف عادي للـ limit إيش معناته هذا لكل إبسلون
+
+330
+00:25:30,160 --> 00:25:35,980
+أكبر من سفر there exists اللي هو مسميها هو إيش
+
+331
+00:25:35,980 --> 00:25:42,440
+مسميها K of إبسلون على 2 element in M such that
+
+332
+00:25:43,150 --> 00:25:50,370
+اللي هو norm الـ Fn ناقص الـ F هيكون أصغر من إبسل
+
+333
+00:25:50,370 --> 00:25:54,630
+على اتنين for every N أكبر أو يساوي مين K of إبسل
+
+334
+00:25:54,630 --> 00:25:57,770
+على اتنين اللي عندك هذا اللي هو تعريف الدهانية
+
+335
+00:25:57,770 --> 00:26:01,590
+العادية اللي هو sequence O السيكوانس العادية هذه
+
+336
+00:26:01,590 --> 00:26:04,440
+الـ sequence converts to zero لأنها أعداداللي هي
+
+337
+00:26:04,440 --> 00:26:06,560
+converse to zero إذا لكل ي أكبر من صفر there
+
+338
+00:26:06,560 --> 00:26:10,740
+exists K بحيث أنه F أن ناقص F أصغر من ي على اتنين
+
+339
+00:26:10,740 --> 00:26:19,460
+لكل N أكبر أو سوء K of ي على اتنين طيب الآن عندي
+
+340
+00:26:19,460 --> 00:26:25,520
+ال .. ال .. ال F هذه بمقدر نترجمها اللي هو على طول
+
+341
+00:26:25,520 --> 00:26:31,660
+مباشرة إذا اللي هي عندي ال Fما دام هذا أصغر يساوي
+
+342
+00:26:31,660 --> 00:26:35,880
+Y على 2 يعني ال suprem أصغر يساوي Y على 2 وهذا
+
+343
+00:26:35,880 --> 00:26:40,360
+نفسه أكبر أوي يساوي ايش عارفين انتوا Fn of X ناقص
+
+344
+00:26:40,360 --> 00:26:44,280
+F of X، مظبوط؟ لأن هذا ال suprem ملهن، إذا صار
+
+345
+00:26:44,280 --> 00:26:53,530
+عندي إذا Fnنقص f of x صار نفسه أصغر من مين من y
+
+346
+00:26:53,530 --> 00:27:00,550
+على 2 خليها في الذاكرة الآن عندي احسبلي بناء عليه
+
+347
+00:27:00,550 --> 00:27:09,430
+احسبلين of X ناقص FM of X هذا لمين؟ هذا حسب ليه
+
+348
+00:27:09,430 --> 00:27:16,230
+لمين؟ لكل أنه طبعا هذا الكلام صحيح لكل X وين؟ لكل
+
+349
+00:27:16,230 --> 00:27:22,340
+X في الـA تبعتنااحسبلي لكل N و M أكبر من مين من
+
+350
+00:27:22,340 --> 00:27:26,620
+الـ K اللي لجيها هنا K of Epsilon على 2 F N of X
+
+351
+00:27:26,620 --> 00:27:30,580
+ناقص F M of X أكيد أصغر أو يساوي بالـ Triangling
+
+352
+00:27:30,580 --> 00:27:38,940
+Quality F N of X ناقص F of X زائد F M of X ناقص F
+
+353
+00:27:38,940 --> 00:27:43,560
+of X واضح عشان سويت ضفت الـ F of X وطرحتها و بعدين
+
+354
+00:27:43,560 --> 00:27:48,460
+أخدت Triangling Quality الآن هذاصحيح أصغر من Y على
+
+355
+00:27:48,460 --> 00:27:55,590
+2 لكل Nلكل N أكبر من K طب ما أنا مختار ال N و ال M
+
+356
+00:27:55,590 --> 00:27:59,830
+أكبر من K إذن بنطبق عليها هذا سواء لهذه أو هذه إذن
+
+357
+00:27:59,830 --> 00:28:04,630
+هذه صارت أصغر من Y على 2 وهذه أصغر من Y على 2 يعني
+
+358
+00:28:04,630 --> 00:28:10,530
+صار ال FN of X ناقص FMA of X أشماله أصغر من Y أصغر
+
+359
+00:28:10,530 --> 00:28:15,110
+من مين؟ من Y يعني الآن اللي وصلت إليه ما يالي يا
+
+360
+00:28:15,110 --> 00:28:19,590
+جماعة بدأت بال FN بتروح لل F اللي هو uniformly
+
+361
+00:28:20,520 --> 00:28:25,460
+ووصلت إلى مالي وصلت إنه لكل epsilon أكبر من 0 لجيت
+
+362
+00:28:25,460 --> 00:28:30,480
+K of epsilon على 2 element in N such that for
+
+363
+00:28:30,480 --> 00:28:35,680
+every N و M أكبر أو يساوي K of epsilon على 2 لجيت
+
+364
+00:28:35,680 --> 00:28:46,360
+إن ال FN of Xنقص fm of x أصغر من مين من إبسلون هذا
+
+365
+00:28:46,360 --> 00:28:52,420
+لكل x وين موجودة في الـA ماشي الحال إذا مدام هذا
+
+366
+00:28:52,420 --> 00:28:57,140
+أصغر من إبسلون لكل x element in A صارت الإبسلون
+
+367
+00:28:57,140 --> 00:29:01,030
+عبارة عن upper bound لهذهإذاً الـ Epsilon هتكون
+
+368
+00:29:01,030 --> 00:29:05,050
+أكبر أو تساوي ال least upper bound إذاً يعني بمعنى
+
+369
+00:29:05,050 --> 00:29:09,850
+آخر إذا ال least upper bound اللي هو FN ناقص FM
+
+370
+00:29:09,850 --> 00:29:15,190
+نفسه أصغر من مين أصغر أو يساوي Epsilon وهذا يعني
+
+371
+00:29:15,190 --> 00:29:19,920
+إنه احنافعلاً لجينا أنه في حالة الـ F N conversion
+
+372
+00:29:19,920 --> 00:29:23,960
+form للـ F لجينا لكل Y أكبر من 0 there exists K
+
+373
+00:29:23,960 --> 00:29:29,260
+بحيث لكل N أ أو M أكبر سوء K يطلع عندي Normal FN
+
+374
+00:29:29,260 --> 00:29:34,080
+نقص FM أصغر من مين من إبسلون وهو المطلوب وطبعاً في
+
+375
+00:29:34,080 --> 00:29:37,360
+الآخر جالك ت choose اللي هي K of Epsilon هي اسمها
+
+376
+00:29:37,360 --> 00:29:40,880
+H of Epsilon كان ممكن الأول نسميها H of Epsilon
+
+377
+00:29:40,880 --> 00:29:45,880
+كان مجرد تسميةالان نيجي نمين اللي هو the converse
+
+378
+00:29:45,880 --> 00:29:49,460
+of the theorem conversely بدنا نفترض ان ال
+
+379
+00:29:49,460 --> 00:29:52,820
+condition متحقق و نثبت اللي هو ال uniform
+
+380
+00:29:52,820 --> 00:30:08,320
+convergence الان
+
+381
+00:30:08,320 --> 00:30:13,390
+بدنا نفترضإن سبب ذات لكل Y أكبر من 0 there exists
+
+382
+00:30:13,390 --> 00:30:18,730
+H of Y بحيث إنه لما M و N تكون أكبر من H of Y
+
+383
+00:30:18,730 --> 00:30:24,710
+هيعطيني ال F ماطس أفن أصغر سوى Y هذا على بعض بدي
+
+384
+00:30:24,710 --> 00:30:32,290
+أثب من خلاله إنه لازم ال F أن تروح لل F uniformly
+
+385
+00:30:32,290 --> 00:30:36,230
+شوف كيف بده يعمل وهو الفكرة سهلة إن شاء الله
+
+386
+00:30:36,230 --> 00:30:42,630
+وبسيطة والبرهان كمان سلسانشوف ايش بقول البرهان
+
+387
+00:30:42,630 --> 00:30:47,370
+تبعنا بقول ما يليه الان
+
+388
+00:30:48,590 --> 00:30:53,150
+ما دام هذا الشرط هيوه، هذا الشرط متحقق، إذا لكل X
+
+389
+00:30:53,150 --> 00:30:59,790
+element إيه؟ عندي أكيد الـFM of X ناقص FN X أصغر
+
+390
+00:30:59,790 --> 00:31:03,910
+أو ساوي FM ناقص FN، لأن هذا هو الـSupremum لهدولة
+
+391
+00:31:03,910 --> 00:31:07,990
+كلهم، إذا صار عندي هذا أكبر أو ساوي هذا، إذا صار
+
+392
+00:31:07,990 --> 00:31:13,870
+هذا أصغر من مين؟ من إبسلونإيش معناته هذا؟ صارت
+
+393
+00:31:13,870 --> 00:31:18,270
+الآن الـ F .. بنحكي عن لكل X element in A X
+
+394
+00:31:18,270 --> 00:31:22,990
+arbitrarily back-fixed صارت عندي الـ F M of X بناء
+
+395
+00:31:22,990 --> 00:31:26,130
+على هذا الحديث مذاب الـ Absolute Value اللي قالها
+
+396
+00:31:26,130 --> 00:31:30,690
+الـ F M ناقص F N أصغر من Y يعني بمعنى آخر لكل Y
+
+397
+00:31:30,690 --> 00:31:34,710
+أكبر من 0 لجينا H of Y بحيث لما تكون M و N أكبر من
+
+398
+00:31:34,710 --> 00:31:39,330
+H of Y طلعلي الـ F M of X ناقص F N of X أصغر من
+
+399
+00:31:39,330 --> 00:31:44,990
+100 من Yصارت الـ F M of X أو F N of X زي ما بدك
+
+400
+00:31:44,990 --> 00:31:49,850
+تسميها صارت عبارة عن كوشي سيكوانس مدام كوشي
+
+401
+00:31:49,850 --> 00:31:55,470
+سيكوانس إذا حسب كوشي سيكوانس of numbers إذا أكيد
+
+402
+00:31:55,470 --> 00:32:01,250
+الـ F N of X converts يعني السيكوانس هذه إيش مالها
+
+403
+00:32:01,250 --> 00:32:09,130
+converts خلّينا نسمي limitالـ Fn of X بيساوي F of
+
+404
+00:32:09,130 --> 00:32:13,110
+X as N goes to infinity ليش F of X؟ لأنه القيمة
+
+405
+00:32:13,110 --> 00:32:16,550
+اللي بتطلع بتعتمد على الـ X لأنه Fn of X واحد لها
+
+406
+00:32:16,550 --> 00:32:20,770
+قيمة، Fn of X اتنين لها قيمة ك sequence، اتكلمت ك
+
+407
+00:32:20,770 --> 00:32:25,510
+sequence الآن صارت عندي هذه ال limit exist و
+
+408
+00:32:25,510 --> 00:32:32,710
+بتساوي رقم الآن منها بدي أقولك define Fمن عند الـ
+
+409
+00:32:32,710 --> 00:32:38,010
+a اللي هي اللي لكل x element in a هذا صحيح اللي fn
+
+410
+00:32:38,010 --> 00:32:42,710
+of x converge ويلها قيمة define f من a لعند ال R
+
+411
+00:32:42,710 --> 00:32:49,740
+by limitالـ F N of X as N goes to infinity
+
+412
+00:32:49,740 --> 00:32:56,760
+بيساومين F of X طيب الـ N بدأاش احنا رايحين يا
+
+413
+00:32:56,760 --> 00:33:02,440
+جماعة رايحين نثبت انه F N converts uniformly للـ F
+
+414
+00:33:02,440 --> 00:33:08,880
+on A او بمعنى اخر رايحين نثبت ان F N ناقص F over A
+
+415
+00:33:08,880 --> 00:33:15,800
+converts لـ 0 بتنشوف مع بعض الـ Nبالتالي كما قلنا
+
+416
+00:33:15,800 --> 00:33:20,200
+هذا أصغر من Y ومنه اللي هو إذا تنتجنا FN is a
+
+417
+00:33:20,200 --> 00:33:24,360
+Cauchy sequence in R ثم By theorem 3.5.4 اللي هو
+
+418
+00:33:24,360 --> 00:33:27,780
+اللي بتقول إن الـ Cauchy sequence .. a sequence is
+
+419
+00:33:27,780 --> 00:33:31,590
+Cauchy فندولي if it convergesis convergent
+
+420
+00:33:31,590 --> 00:33:34,930
+sequence define f من a لعنى r by f of x سواء اللي
+
+421
+00:33:34,930 --> 00:33:38,590
+ميت f of n of x زي ما قلنا لأن if we let n in a
+
+422
+00:33:38,590 --> 00:33:42,430
+star goes to infinity الآن بدنا نودي n إلى وين هنا
+
+423
+00:33:42,430 --> 00:33:47,170
+إلى ما لا نهاية مع الاحتفاظ بال m زي ما هي عندي
+
+424
+00:33:47,170 --> 00:33:50,170
+بقدر اعمل هيك اه بقدر اعمل هيك لأنه بقدر اخد ال n
+
+425
+00:33:50,630 --> 00:33:56,570
+أكبر من M والتنتين اللي هى أكبر أو يساوي H of
+
+426
+00:33:56,570 --> 00:33:59,610
+Epsilon فالان لما ال N تروح الى ماله نهاية هى حرة
+
+427
+00:33:59,610 --> 00:34:03,370
+بتروح الى ماله نهاية M بتظلها جاعدة زى ما هى الان
+
+428
+00:34:03,370 --> 00:34:11,750
+زى ما قلنا نيجى لمين نودي
+
+429
+00:34:11,750 --> 00:34:15,290
+N الى ماله نهاية وين في ال star هى ال star هى ال
+
+430
+00:34:15,290 --> 00:34:15,630
+star
+
+431
+00:34:20,280 --> 00:34:24,360
+ودي ال M إلى ما لنهاية لما تودي ال M إلى ما لنهاية
+
+432
+00:34:24,360 --> 00:34:28,700
+هذه بتروح limit ال M ال FN of X هتروح ل .. اللي هي
+
+433
+00:34:28,700 --> 00:34:34,060
+ال F of X بصير عندي الآن طب امبسلون ثابت مالهاش
+
+434
+00:34:34,060 --> 00:34:38,620
+علاقة إذا صار عندي الآن هذه إيش بتعطيني .. بتعطيني
+
+435
+00:34:38,620 --> 00:34:43,420
+F M of X limit ال FN of X لأنه بتدخل على ال .. ال
+
+436
+00:34:43,420 --> 00:34:45,520
+absolute value لأن ال limit ال .. ال absolute
+
+437
+00:34:45,520 --> 00:34:48,300
+value is a continuous function إذا بتدخل عليه إذا
+
+438
+00:34:48,300 --> 00:34:52,760
+Fm of x ناقص limit fn of x as n goes to infinity
+
+439
+00:34:52,760 --> 00:34:55,780
+أصغر أو يسوى مين إبسلون سيبك من هذه ماعلاش علاقة
+
+440
+00:34:55,780 --> 00:34:59,460
+فيها نحكي عن هذه و هذه الآن ال limit هذا إيش سمناه
+
+441
+00:34:59,460 --> 00:35:04,220
+إحنا سمناه هنا عبارة عن f of x إذا صار عندي الآن
+
+442
+00:35:04,220 --> 00:35:09,300
+اللي هو f m of x ناقص f of x أصغر أو يسوى إيش
+
+443
+00:35:09,300 --> 00:35:14,110
+إبسلونإذا صار عندي الآن اللي حصلت عليه لكل ي أكبر
+
+444
+00:35:14,110 --> 00:35:19,010
+من سفر there exists h of ي بحيث أن fm of x ناقص f
+
+445
+00:35:19,010 --> 00:35:24,310
+of x أصغر سواء ي لكل m أكبر أو سواء مين؟ h of ي
+
+446
+00:35:24,310 --> 00:35:30,800
+لأن هذا إيش معناته؟ هذا صحيح لكل xإذا صار عندى زى
+
+447
+00:35:30,800 --> 00:35:34,280
+ما قلنا قبل شوية الـ Epsilon Upper Bound لهذه مدام
+
+448
+00:35:34,280 --> 00:35:37,700
+Epsilon Upper Bound إذا اللى حيصير ال Epsilon أكبر
+
+449
+00:35:37,700 --> 00:35:40,900
+أو يساوي ال least upper bound مين هو ال least
+
+450
+00:35:40,900 --> 00:35:45,480
+upper bound ال norm لل FM نقص FM يعني بمعنى آخر من
+
+451
+00:35:45,480 --> 00:35:49,880
+هنا حصلت على norm لل FM نقص FM اللى هو اللى هو
+
+452
+00:35:49,880 --> 00:35:54,740
+أصغر أو يساوي Epsilon هذا لكل مين لكل M أكبر أو
+
+453
+00:35:54,740 --> 00:35:59,870
+يساوي اللى هو H of Epsilon وهذا مع بعضالذي حصلنا
+
+454
+00:35:59,870 --> 00:36:03,150
+عليه لكل y أكبر من 0 there exists h of epsilon
+
+455
+00:36:03,150 --> 00:36:07,290
+such that for every m أكبر أو سوى h of epsilon فم
+
+456
+00:36:07,290 --> 00:36:13,470
+ناقص f أصغر أو سوى y هذا معناه limit ال Fالنورم
+
+457
+00:36:13,470 --> 00:36:18,690
+الـ FM ناقص الـ F over A as M goes to infinity
+
+458
+00:36:18,690 --> 00:36:25,450
+بساوة Zero و هذا حسب اللي هو النظرية الـ FM أو الـ
+
+459
+00:36:25,450 --> 00:36:31,230
+FM converts uniformly to mean to F وبهيك بنكون
+
+460
+00:36:31,230 --> 00:36:37,850
+حصلنا على اللي هو برهان النظرية و أثبتنا إنه لما
+
+461
+00:36:37,850 --> 00:36:44,080
+نكون اللي هيالـ condition الكوشي كريتيريا المتحقق
+
+462
+00:36:44,080 --> 00:36:51,920
+هذا بكون عندي اللي هي الـ FM أو الـ FN converts
+
+463
+00:36:51,920 --> 00:36:57,800
+uniformly to ال function اللي عندنا اللي هي F
+
+464
+00:36:57,800 --> 00:37:02,740
+وبهيك بنكون احنا انتهينا من ال section الأول
+
+465
+00:37:02,740 --> 00:37:08,000
+تمانية واحد وإن شاء الله بنكمل المحاضرة القادمة
+
+466
+00:37:08,000 --> 00:37:09,320
+وإلى لقاء آخر
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA.srt
new file mode 100644
index 0000000000000000000000000000000000000000..d3c24a9c499ab503ecc6baed467a3813eab8bf53
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA.srt
@@ -0,0 +1,2421 @@
+1
+00:00:04,820 --> 00:00:10,480
+بسم الله الرحمن الرحيم المحاضرة الثانية في مساق
+
+2
+00:00:10,480 --> 00:00:18,480
+تحليل رياضي 2 أو تحليل حقيقي 2  تحدثنا 
+
+3
+00:00:18,480 --> 00:00:24,100
+المرة الماضية عن الـ differentiation بدأنا بشروط
+
+4
+00:00:24,100 --> 00:00:28,200
+ستة بدأنا في الـ section الأول 6.1 اللي هو الـ
+
+5
+00:00:28,200 --> 00:00:32,600
+derivative أو اللي هو الاشتقاق أو المشتق الأولى
+
+6
+00:00:32,600 --> 00:00:37,840
+وعرفنا شو معناه اللي هو الاشتقاق عرفنا شو معناه أن 
+
+7
+00:00:37,840 --> 00:00:44,910
+الدالة تكون differentiable عند نقطة اللي هي في
+
+8
+00:00:44,910 --> 00:00:49,970
+interval I حيث أخذنا F عبارة عن الـ function من I
+
+9
+00:00:49,970 --> 00:00:55,230
+إلى R وقلنا C element in I بتكون الدالة عندها قابلة
+
+10
+00:00:55,230 --> 00:01:01,490
+للاشتقاق يعني F prime of C exist if the following
+
+11
+00:01:01,490 --> 00:01:08,230
+limit exist اللي هو limit f of x ناقص f of c على x
+
+12
+00:01:08,230 --> 00:01:14,230
+ناقص c as x goes to c exist إذا كانت هذه الـ
+
+13
+00:01:14,230 --> 00:01:20,210
+limit exist بنقول عن الـ f برايم exist و بتكون هي 
+
+14
+00:01:20,210 --> 00:01:26,650
+قيمة هذه الـ limit الآن وبعد ذلك مررنا على اللي
+
+15
+00:01:26,650 --> 00:01:34,600
+هو بعض الخواص قلنا الخاصية الأولى عندنا اللي هي 
+
+16
+00:01:34,600 --> 00:01:41,540
+أن لو كانت الدالة اللي هي differentiable at C
+
+17
+00:01:41,540 --> 00:01:45,520
+بتكون continuous برضه وين؟ عند C يعني أي 
+
+18
+00:01:45,520 --> 00:01:50,080
+differentiable function must be continuous قلنا إن 
+
+19
+00:01:50,080 --> 00:01:53,960
+الـ converse need not to be true in general  وأخذنا
+
+20
+00:01:53,960 --> 00:01:58,020
+مثال قلنا f of x بتساوي الـ absolute value للـ x
+
+21
+00:01:58,020 --> 00:02:01,780
+اللي هي عبارة عن continuous function لكن مش
+
+22
+00:02:01,780 --> 00:02:05,460
+differentiable عند الـ zero continuous everywhere
+
+23
+00:02:05,460 --> 00:02:09,340
+but it is not differentiable at zero وبرهنّا الكلام
+
+24
+00:02:09,340 --> 00:02:13,140
+هذا وأخذنا مثال آخر على f of x بتساوي الـ summation
+
+25
+00:02:13,140 --> 00:02:17,760
+لـ x ناقص c<sub>i</sub> من عند واحد لعند n وقلنا إن هذا
+
+26
+00:02:17,760 --> 00:02:22,600
+الدالة برضه is continuous everywhere but it is not
+
+27
+00:02:22,600 --> 00:02:29,440
+differentiable at c<sub>1</sub>, c<sub>2</sub>, c<sub>3</sub> لعند c<sub>n</sub> بعد ذلك سألنا
+
+28
+00:02:29,440 --> 00:02:34,100
+وقلنا هل ممكن أن نجد دالة تكون continuous وين
+
+29
+00:02:34,100 --> 00:02:39,790
+ما كانت مش differentiable ولا في مكان قلنا اه و
+
+30
+00:02:39,790 --> 00:02:42,470
+قلنا أخذ الدالة f of x بساعة الـ summation 1 على
+
+31
+00:02:42,470 --> 00:02:47,350
+2<sup>n</sup> x 3<sup>n</sup> x هذه الدالة is 
+
+32
+00:02:47,350 --> 00:02:51,850
+continuous everywhere but it is not differentiable
+
+33
+00:02:51,850 --> 00:02:56,310
+anywhere وبعد هيك أتينا
+
+34
+00:03:04,880 --> 00:03:11,100
+حكينا عن بعض اللي هو قوانين الاشتقاق قلنا لو كانت
+
+35
+00:03:11,100 --> 00:03:16,520
+الدالة F والدالة G التنتين من نفس الـ domain
+
+36
+00:03:16,520 --> 00:03:22,000
+والتنتين قابلتان للاشتقاق عند نقطة اسمها C سميناها
+
+37
+00:03:22,000 --> 00:03:26,560
+قلنا اللي هو مجموع الدالتين هيكون مشتقتهم موجودة
+
+38
+00:03:26,560 --> 00:03:31,660
+عند هذه النقطة وحاصل ضربهما وحاصل قسمتهما أيضا
+
+39
+00:03:31,660 --> 00:03:35,860
+provided that G of C اللي في المقام لا تساوي صفر
+
+40
+00:03:35,860 --> 00:03:41,950
+وأيضا الـ α في الـ F  constant برضه α في F قابلة
+
+41
+00:03:41,950 --> 00:03:46,290
+للاشتقاق وأثبتنا
+
+42
+00:03:46,290 --> 00:03:53,330
+هذا الكلام وبرهنّاه ووصلنا لعند اللي هو التعميم
+
+43
+00:03:53,330 --> 00:04:00,030
+قلنا لو كانت F<sub>1</sub> و F<sub>2</sub> و F<sub>3</sub> لعند F<sub>n</sub> كلهم دوال قابلة
+
+44
+00:04:00,030 --> 00:04:04,710
+للاشتقاق حيكون حاصل جمعها قابل للاشتقاق وحاصل ضربهما
+
+45
+00:04:04,710 --> 00:04:09,510
+أيضا قابل للاشتقاق وفق القوانين اللي ذكرناها سابقا
+
+46
+00:04:10,150 --> 00:04:14,370
+ووقفنا عند اللي هو النقطة اللي هي the chain rule أو
+
+47
+00:04:14,370 --> 00:04:20,650
+قاعدة السلسلة قاعدة السلسلة طبعا ليست غريبة عنكم
+
+48
+00:04:20,650 --> 00:04:29,240
+مرت علينا في تفاضل 1 ومرت علينا في الـ complex أو
+
+49
+00:04:29,240 --> 00:04:32,780
+مرت علينا اللي هو استخدمناها في الـ differential
+
+50
+00:04:32,780 --> 00:04:38,220
+equations وفي مساقات أخرى الآن نظرة تحليلية
+
+51
+00:04:38,220 --> 00:04:44,060
+فهنشوف اللي هو كيف أثبتنا أو كيف بدنا نثبت اللي هو
+
+52
+00:04:44,060 --> 00:04:48,840
+قاعدة السلسلة أي سؤال بتقدر تسأله يعني اللي بدكم
+
+53
+00:04:48,840 --> 00:04:53,960
+إياه وتتفاعلوا بالطريقة اللي بتلاقوها مناسبة let i
+
+54
+00:04:53,960 --> 00:04:56,200
+و j be intervals and let g
+
+55
+00:04:59,060 --> 00:05:06,800
+عبارة عن دالة دي
+
+56
+00:05:06,800 --> 00:05:14,660
+عبارة عن دالة من I لعند مين؟ لعند R والـ F عبارة عن
+
+57
+00:05:14,660 --> 00:05:21,700
+دالة من عند J لعند R حيث I و J اللي هو أشمل عبارة
+
+58
+00:05:21,700 --> 00:05:26,440
+عن intervals الآن أنا اللي هي الـ chain rule هي في
+
+59
+00:05:26,440 --> 00:05:33,380
+الواقع عملية فحص إمكانية أنها تكون الـ composition
+
+60
+00:05:33,380 --> 00:05:37,780
+of two functions is differentiable ولا لأ؟ وإيش
+
+61
+00:05:37,780 --> 00:05:41,200
+الشروط اللي بتخلي اللي هو g composite of f
+
+62
+00:05:41,200 --> 00:05:45,620
+differentiable؟ إذا أنا بدي أحكي عن g composite of
+
+63
+00:05:45,620 --> 00:05:49,590
+اللي هو الـ derivative إيه لها؟ ما دام بدي أحكي عن g
+
+64
+00:05:49,590 --> 00:05:56,690
+composite of f إذا أنا الآن domain الـ g الـ domain
+
+65
+00:05:56,690 --> 00:06:03,850
+الـ f هيكون من عند g لعند مين؟ لعند r معناته أن f of
+
+66
+00:06:03,850 --> 00:06:09,310
+g عشان تكون g معرفة لازم تكون من ضمن مين؟ اللي هو
+
+67
+00:06:09,310 --> 00:06:16,680
+domain الـ I إذا أنا بدي أفترض أن f of J جزئية من I
+
+68
+00:06:16,680 --> 00:06:22,020
+على أساس أن هتكون الـ G composite F is defined
+
+69
+00:06:22,020 --> 00:06:29,960
+وهتكون defined من وين؟ من عند J لعند R عشان هيك
+
+70
+00:06:29,960 --> 00:06:34,400
+احنا لما بدنا نيجي للّي هو .. اللي نفترض .. هنفترض
+
+71
+00:06:34,400 --> 00:06:40,220
+G من I لعند R و F من عند G لعند R دوال ونقول F of I
+
+72
+00:06:40,220 --> 00:06:43,640
+subset أو F of G subset من I عشان تصير اللي هو 
+
+73
+00:06:43,640 --> 00:06:46,980
+اللي بنحكي عنه قاعدة السلسلية اللي هي بمعنى آخر
+
+74
+00:06:46,980 --> 00:06:50,040
+التفاضل الـ composition of two functions is well
+
+75
+00:06:50,040 --> 00:06:54,420
+defined أو is defined اللي هم let C element in مين؟
+
+76
+00:06:54,420 --> 00:07:00,120
+in G؟ بدنا ناخد الـ C element in G اللي هم بيقولي C
+
+77
+00:07:00,120 --> 00:07:06,430
+element in G عشان أقدر أحكي عن g composite of f
+
+78
+00:07:06,430 --> 00:07:12,530
+prime of c تكون exist بده شرطين الشرط الأول بده الـ
+
+79
+00:07:12,530 --> 00:07:17,190
+f تكون differentiable عند مين؟ عند الـ c
+
+80
+00:07:17,190 --> 00:07:22,150
+والشرط الثاني أن الـ g تكون differentiable عند الـ f
+
+81
+00:07:22,150 --> 00:07:30,530
+of c واضح؟ إذن الآن إذا كان الـ F مميز لـ C والـ G
+
+82
+00:07:30,530 --> 00:07:36,710
+مميز لـ F of C فإن مفهوم الـ G مميز لـ F يجب أن
+
+83
+00:07:36,710 --> 00:07:42,410
+يكون مميز لـ C وبكون عنده الـ G مميز لـ F' of C
+
+84
+00:07:42,410 --> 00:07:48,710
+إيش حيساوي؟ حيساوي G' of F of C في F' لمن؟ للـ F
+
+85
+00:07:48,710 --> 00:07:54,350
+of C اللي كنا نقول عنه في التفاضل أنه تفاضل اللي هو
+
+86
+00:07:54,350 --> 00:08:01,320
+القوس ثم تفاضل ما فيه داخل القوس واضح؟ أو تفاضل
+
+87
+00:08:01,320 --> 00:08:05,280
+الدالة في تفاضل ما بداخل الدالة هو طبعا في الواقع
+
+88
+00:08:05,280 --> 00:08:07,940
+اللي هو عبارة عن تفاضل لـ composite of two
+
+89
+00:08:07,940 --> 00:08:12,200
+functions أو اللي هو الـ chain rule وكنا إذا
+
+90
+00:08:12,200 --> 00:08:16,680
+بتتذكروا كانت صورة أخرى للـ chain rule غير الصورة
+
+91
+00:08:16,680 --> 00:08:23,100
+هذه اللي هي g prime of f of c في f prime of c كنا 
+
+92
+00:08:23,100 --> 00:08:29,900
+نأخذ صورة أخرى نقول dy by dx اللي هو عبارة عن ..
+
+93
+00:08:29,900 --> 00:08:38,960
+عبارة عن dy مثلا by du في du by d إيش؟ dx اللي هو 
+
+94
+00:08:38,960 --> 00:08:44,620
+الـ y وكأنها بتتمثل اللي هي الـ g تفاضل الـ g والـ u
+
+95
+00:08:44,620 --> 00:08:48,600
+بتكون اللي هي مين؟ اللي هي الـ F اللي هي F برايم 
+
+96
+00:08:48,600 --> 00:08:52,180
+فكانت اللي هي هذه الصورة أو هذه الصورة احنا الآن
+
+97
+00:08:52,180 --> 00:08:59,630
+خلينا نحكي عن هذه الصورة ونبرهن اللي هيكيف اللي
+
+98
+00:08:59,630 --> 00:09:07,290
+هو نثبت أن هذا الكلام صحيح الآن بس خلينا نحط ..
+
+99
+00:09:07,290 --> 00:09:13,730
+نشوف .. نشوف عندنا الآن نشوف إيش اللي بدنا نثبته
+
+100
+00:09:13,730 --> 00:09:16,770
+احنا في الواقع اللي بدنا نثبته .. بدنا نجو ..
+
+101
+00:09:16,770 --> 00:09:22,290
+بدنا نثبت أن G composite F a prime of C exist عشان 
+
+102
+00:09:22,290 --> 00:09:29,500
+نثبت أنّه exist بإيش بتلزمني؟ بدي آخذ limit G of
+
+103
+00:09:29,500 --> 00:09:39,520
+F, G circle F اللي هي GF of X ناقص G of F of C على
+
+104
+00:09:39,520 --> 00:09:46,960
+X ناقص C لما X تروح للـ C نثبت أن هذا إيش ماله؟
+
+105
+00:09:46,960 --> 00:09:51,920
+exist واضح؟ عشان اللي بدنا نسويه؟ بدنا نثبت أنه
+
+106
+00:09:51,920 --> 00:09:55,800
+exist عشان نثبت أنّه هذا exist بتلزمني عاملين
+
+107
+00:09:55,800 --> 00:10:01,020
+مساعدات الآن هنشوف كيف نعمل هذول العاملين طبعا 
+
+108
+00:10:01,020 --> 00:10:05,200
+طبيعي طبيعي أنا معطيين أن هو الـ F' موجودة والـ G'
+
+109
+00:10:05,480 --> 00:10:10,600
+موجودة إذا أكيد أنا هعمل حسابي أن أستخدمهم على 
+
+110
+00:10:10,600 --> 00:10:14,280
+أساس أن أصل للهدف اللي بدي إياه الآن خلينا نيجي
+
+111
+00:10:14,280 --> 00:10:14,900
+للبرهان
+
+112
+00:10:20,860 --> 00:10:26,680
+الحين الـ proof بدنا
+
+113
+00:10:26,680 --> 00:10:36,080
+نثبت أنّه اللي هو الـ G بنثبت إنه لو كانت G و F 
+
+114
+00:10:36,080 --> 00:10:39,980
+differentiable الأول عند F of C والثاني عند C
+
+115
+00:10:39,980 --> 00:10:45,840
+بيعطينا هذا أنّه G composed of F prime of C exist
+
+116
+00:10:45,840 --> 00:10:50,420
+ويساوي G prime of F of C في F prime of C هذا ما 
+
+117
+00:10:50,420 --> 00:10:58,000
+بنثبته شوف كيف يعني
+
+118
+00:10:58,000 --> 00:11:06,450
+بقى بدنا نبرهن النظرية خليني أعرّف الآن Define G
+
+119
+00:11:06,450 --> 00:11:14,850
+Capital من I لعند الـ R By شوف كيف بدي .. اللي هي
+
+120
+00:11:14,850 --> 00:11:21,050
+يعني أعرّف الـ G بطريقة توصّلني لحساب D composed of
+
+121
+00:11:21,050 --> 00:11:24,890
+F prime of إيش؟ of C جبل بشوية قلنا D composed of
+
+122
+00:11:24,890 --> 00:11:30,590
+F prime of C المتوقع لها أنّه تكون عبارة عن إذا كانت
+
+123
+00:11:30,590 --> 00:11:34,850
+تكون موجودة هي عبارة عن الـ limit لـ G F of X نقص G 
+
+124
+00:11:34,850 --> 00:11:41,270
+F of C على X ناقص C أو G of Y نقص G F of C ماشي
+
+125
+00:11:41,270 --> 00:11:47,250
+الحال الآن بدي أعرّف G على هذا الأساس G of Y بتساوي 
+
+126
+00:11:47,250 --> 00:12:01,160
+اللي هو G of F of Y عند ناقص g f of c على y ناقص
+
+127
+00:12:01,160 --> 00:12:06,480
+f of c الآن قسمت على y ناقص f of c أكيد اللي هو 
+
+128
+00:12:06,480 --> 00:12:15,600
+الـ y أنا بديّش إياها تساوي f of c الآن g of y إيش
+
+129
+00:12:15,600 --> 00:12:23,980
+بديّها؟ أنا بديّها المتوقع الـ limit لهذه الـ limit 
+
+130
+00:12:23,980 --> 00:12:30,300
+لهذه لما y تروح للـ F of C اللي هو وكأنه عبارة عن
+
+131
+00:12:30,300 --> 00:12:37,540
+D prime of مين؟ of F of C فالآن بتديها D prime of F
+
+132
+00:12:37,540 --> 00:12:44,180
+of C ليش هالـ gate بتشوف ليش اللي بنحكيّها الآن عرّفت
+
+133
+00:12:44,180 --> 00:12:49,140
+اللي هي الـ function طبعا هذا الكلام مستقى من
+
+134
+00:12:49,140 --> 00:12:54,120
+اللي هو f prime of c إيش متوقع limitه تساوي؟ هذا
+
+135
+00:12:54,120 --> 00:12:58,200
+مستقى .. مستقى منه بعد شوية هتلاقيني هذه كتبت على
+
+136
+00:12:58,200 --> 00:13:02,980
+صورة هذه في شغل ثانية برضه موجودة اللي هي f prime
+
+137
+00:13:02,980 --> 00:13:07,260
+of c اه
+
+138
+00:13:07,260 --> 00:13:16,350
+آسف g of y الآن نشوف هذه الدالة لو جيت للدالة اللي 
+
+139
+00:13:16,350 --> 00:13:26,010
+عندي هذه خذ limit g of y لما y تروح للـ F of C إيش
+
+140
+00:13:26,010 --> 00:13:32,500
+هتلاقيّها بتساوي؟ بتساوي limit G of Y بعوض جاعد Y 
+
+141
+00:13:32,500 --> 00:13:38,080
+تروح للـ F of C يعني Y أشملها لا تساوي F of C ناقص
+
+142
+00:13:38,080 --> 00:13:45,340
+G of F of C على مين؟ على Y ناقص F of C لما Y تروح
+
+143
+00:13:45,340 --> 00:13:51,560
+لمين؟ للـ F of C وهذه بالضبط هي تعريف مين؟ تعريف الـ
+
+144
+00:13:51,560 --> 00:13:56,480
+derivative للـ G عند النقطة مين؟ النقطة اسمها F of
+
+145
+00:13:56,480 --> 00:14:01,280
+C واضح؟ لأن الـ C من الـ A .. من الـ J الـ F of C
+
+146
+00:14:01,280 --> 00:14:08,260
+وين هتكون؟ من مين؟ من الـ I واضح؟ فالآن هذه عبارة
+
+147
+00:14:08,260 --> 00:14:16,920
+عن G prime للـ F of C ليش هذا؟ because .. G is
+
+148
+00:14:16,920 --> 00:14:25,200
+differentiable at mean at F of C هذا معطى طيب الآن
+
+149
+00:14:25,200 --> 00:14:30,160
+الـ limit لـ G of Y طلعت لما y تروح لـ F of C أش طلعت؟
+
+150
+00:14:30,160 --> 00:14:34,520
+G prime لـ F of C أصلا هذا السبب اللي أنا عارف أنّه
+
+151
+00:14:34,520 --> 00:14:37,440
+هتطلع حاجة السبب اللي خلّاني أعرّف الـ G of Y أش 
+
+152
+00:14:37,440 --> 00:14:44,510
+متساوي G prime F of C عشان تطلع لي منه هذا بيساوي G
+
+153
+00:14:44,510 --> 00:14:52,510
+عند مين؟ عند Y بتساوي F of C يعني هذا سبب لأنّه الـ 
+
+154
+00:14:52,510 --> 00:14:57,610
+G هذه تصير إيش ماله؟ is continuous عند مين؟ عند الـ F 
+
+155
+00:14:57,610 --> 00:15:07,490
+of C إذا hence G is continuous at F of C طب ليش
+
+156
+00:15:07,490 --> 00:15:15,570
+هذا؟ بعد شوية هنشوف وين هنستخدمّها الآن limit g of y
+
+157
+00:15:15,570 --> 00:15:22,890
+طالع عندي بيساوي g of f of c اللي هي هذه g of f of
+
+158
+00:15:22,890 --> 00:15:29,550
+c بيساوي g prime f of c نيجي الآن أنا إيش اللي بدي
+
+159
+00:15:29,550 --> 00:15:35,910
+ا .. بدي الآن أثبت لك أن g composite f prime of c
+
+160
+00:15:35,910 --> 00:15:48,970
+بيساوي limit g of F of X ناقص G of F of C على X 
+
+161
+00:15:48,970 --> 00:15:55,410
+ناقص C لما X تروح للـ C بدي أثبت أنّه exist بظبط؟
+
+162
+00:15:55,410 --> 00:16:02,610
+واضح هاه؟ طيب بناءً عليه خليني أكتب هذه بدلالة إشي
+
+163
+00:16:02,610 --> 00:16:11,060
+بوصلني للي بديّها شوف كيف الآن نحسب هذه هذه لمين؟ for
+
+164
+00:16:11,060 --> 00:16:22,260
+x element in اللي هي j أو x لا تساوي مين؟ لا تساوي c
+
+165
+00:16:22,260 --> 00:16:30,120
+معايا اه الآن احسبولي هذه هذه عبارة عن تساوي g of 
+
+166
+00:16:30,120 --> 00:16:37,650
+f of x اللي هنا نقص g f of c بدي أكتبّها بطريقة g f
+
+167
+00:16:37,650 --> 00:16:49,250
+of x نقص g of f of c على f of x نقص f of c في f of
+
+168
+00:16:49,250 --> 00:16:58,630
+x نقص f of c على x ناقص c دخّلت هذا المقدار في الـ
+
+169
+00:16:58,630 --> 00:17:04,410
+bust وهنا في المقام اللي هو على أساس أن أحصل على
+
+170
+00:17:04,410 --> 00:17:10,530
+هذه مضروبة في هذه اللي أنا عندي معلومات عن هذه و
+
+171
+00:17:10,530 --> 00:17:20,290
+عن هذه واضح؟ أي سؤال؟ ويساوي؟ الآن هذا كلّه على 
+
+172
+00:17
+
+201
+00:20:41,000 --> 00:20:46,740
+لذلك الآن أثبتنا أنه لو كانت F is differentiable
+
+202
+00:20:46,740 --> 00:20:53,080
+أو G is differentiable ويكون الـ composite  is 
+
+203
+00:20:53,080 --> 00:20:57,900
+differentiable طبعا ما أخذ بعين الاعتبار أنه يكون
+
+204
+00:20:57,900 --> 00:21:00,600
+G composite F is defined
+
+205
+00:21:16,030 --> 00:21:25,010
+في عندنا اللي هو المثال الأول مثال
+
+206
+00:21:25,010 --> 00:21:30,750
+يعني بسيط كلنا بنعرف اشتقاقه بس بنشوف كيف احنا
+
+207
+00:21:30,750 --> 00:21:35,530
+بنستخدم الـ الـ الـ الـ chain rule الصورة
+
+208
+00:21:35,530 --> 00:21:39,470
+اللي احنا كتبناها في اللي هو إيجاد الـ derivative
+
+209
+00:21:39,470 --> 00:21:46,520
+جاي اللي عند في example جاي اللي أخذ Example جاي لنت
+
+210
+00:21:46,520 --> 00:21:56,300
+F من عند I لعند R بيكون a differentiable function من
+
+211
+00:21:56,300 --> 00:22:00,080
+هنا نفترض أن الـ F أشبه لها يا جماعة عبارة عندها
+
+212
+00:22:00,080 --> 00:22:05,860
+قابلة للاشتقاق عند نقطة وين على الـ I أو عند الـ I
+
+213
+00:22:05,860 --> 00:22:13,230
+كلها مش مشكلة ماشي الحال بقول لك اثبت لي بس قبل ما 
+
+214
+00:22:13,230 --> 00:22:20,750
+بتخلينا نقول تقول احنا بتوجد لنا الـ Fn لأن في عندي
+
+215
+00:22:20,750 --> 00:22:24,890
+دالة اسمها Fn يعني F مرفوعة للقوة أنا بدي أوجد لها 
+
+216
+00:22:24,890 --> 00:22:30,140
+الـ derivative طبعا الـ Fn كله prime of X من
+
+217
+00:22:30,140 --> 00:22:36,040
+معلوماتنا في الـ Calculus كنا نقول أن تفاضل القوس
+
+218
+00:22:36,040 --> 00:22:41,560
+كله هو N في F N ناقص 1 في تفاضل ما بداخل القوس
+
+219
+00:22:41,560 --> 00:22:47,990
+اللي هو عبارة عن F prime of X هذا تفضل الآن بدنا
+
+220
+00:22:47,990 --> 00:22:54,650
+نبرهن هذا الكلام باستخدام نظرية الـ chain rule
+
+221
+00:22:54,650 --> 00:22:58,050
+يعني هذا شغل المكان زي ما بقوله الآن بدنا نشوف كيف
+
+222
+00:22:58,050 --> 00:23:01,670
+بتشتغل المكينة أو ايش اللي شغل المكينة طيب الـ
+
+223
+00:23:01,670 --> 00:23:13,110
+proof الـ proof لندن كده بدي أعرف function let g
+
+224
+00:23:13,110 --> 00:23:23,530
+from some interval J لعند R defined by G يعني أصلا
+
+225
+00:23:23,530 --> 00:23:30,330
+ممكن أعرفها من R حتى G of Y بيساوي Y أس N element
+
+226
+00:23:30,330 --> 00:23:35,070
+in N يعني حتى لو عرفتها من R ما بعرفها الدالة
+
+227
+00:23:35,070 --> 00:23:46,200
+معايا الآن بناء على هيك صار عندي الآن اللي هو الـ G
+
+228
+00:23:46,200 --> 00:23:56,700
+composite F of X إيش هي هتطلع؟ G of F of X يعني 
+
+229
+00:23:56,700 --> 00:24:05,340
+هتطلع FN of X صح ولا لأ؟ إذن هاي دالتنا اللي بدنا
+
+230
+00:24:05,340 --> 00:24:12,000
+نفضلها كتبتها على صورة مين؟ G composite F والـ F of 
+
+231
+00:24:12,000 --> 00:24:17,520
+X الـ F of X اللي هي الـ F of X من وين جاي بترمي 
+
+232
+00:24:17,520 --> 00:24:22,680
+في الـ R يعني بترمي في مين؟ في domain الـ G واضح
+
+233
+00:24:22,680 --> 00:24:27,840
+أه إذا كل أموري تمام وهذه الدالة الدالة الأسية is 
+
+234
+00:24:27,840 --> 00:24:30,980
+differentiable وهو جاي اللي أن الـ F is
+
+235
+00:24:30,980 --> 00:24:33,780
+differentiable إذا G composite F حسب الـ channel
+
+236
+00:24:33,780 --> 00:24:38,140
+أشمالها is differentiable والـ derivative لها G 
+
+237
+00:24:38,140 --> 00:24:42,900
+composite F prime of X حسب الـ channel اللي حكيناها
+
+238
+00:24:42,900 --> 00:24:49,560
+اللي برهناها بيساوي G prime of F of X في F prime of
+
+239
+00:24:49,560 --> 00:24:53,580
+X ويساوي and G of Y بيساوي Y أس N الـ derivative
+
+240
+00:24:53,580 --> 00:24:59,420
+الهنين هي and Y أس N ناقص 1 يعني and Y أس N 
+
+241
+00:24:59,420 --> 00:25:04,220
+ناقص 1 في F prime of X لكن احنا بنفاضل عند مين
+
+242
+00:25:04,220 --> 00:25:08,860
+عند F of X فبنشيل الـ Y بنحط مكانها قيمتها اللي هي 
+
+243
+00:25:08,860 --> 00:25:14,240
+F of X فبتطلع and F of X N ناقص 1 في F prime of
+
+244
+00:25:14,240 --> 00:25:19,610
+X هذه اللي هي اللي احنا بنشتغلها وبدون ما طبعا 
+
+245
+00:25:19,610 --> 00:25:24,930
+احنا هذا تبرير لاستخدام اللي هو chain rule
+
+246
+00:25:24,930 --> 00:25:28,150
+it's اللي احنا بنعرفها زي ما نيجي لما كنا نقول لكم 
+
+247
+00:25:28,150 --> 00:25:34,810
+مثلا أوجد الـ derivative Y لعندك Y بيساوي Sin X 
+
+248
+00:25:34,810 --> 00:25:39,460
+تكعيب مثلا ما كنت تسوي على طول ماتيش ايش تقول والله
+
+249
+00:25:39,460 --> 00:25:44,040
+بيضاف of X بيساوي X تكعيب و G of Y تساوي Sin
+
+250
+00:25:44,040 --> 00:25:47,600
+Y وتقول إذا صارت هذه عبارة عن composite of two
+
+251
+00:25:47,600 --> 00:25:53,240
+functions خلاص بنتجاوز المرحلة هذه وعرفنا وين أصل
+
+252
+00:25:53,240 --> 00:25:58,160
+الأمر اللي بيقول لنا Y prime بتساوي تفاضل اللي
+
+253
+00:25:58,160 --> 00:26:03,760
+بره اللي هو Cosine X تكعيب في تفاضل اللي جوا 3 
+
+254
+00:26:03,760 --> 00:26:08,100
+X تربيع هذا في الواقع هو عبارة عن استخدام mean للـ
+
+255
+00:26:08,100 --> 00:26:19,080
+chain rule واضح أي سؤال؟ طيب هذا عندنا في عندنا 
+
+256
+00:26:19,080 --> 00:26:31,850
+مثال آخر المثال نعم المثال اللي هو يمكن مرّشك 
+
+257
+00:26:31,850 --> 00:26:36,330
+للمثال حتى معكم في تحليل 1 لكن كان هدفه اللي 
+
+258
+00:26:36,330 --> 00:26:42,910
+هو في حينه الـ continuity إذا أعطيك F of X بتساوي
+
+259
+00:26:42,910 --> 00:26:52,400
+هاي B مثال B F of X بتساوي اللي هو X تربيع Sin 1 
+
+260
+00:26:52,400 --> 00:26:59,800
+على X لما X لا تساوي صفر أو صفر لما X ايش بتساوي
+
+261
+00:26:59,800 --> 00:27:07,140
+صفر الآن بيقول لي بيعطيني طبعا هو بيسأل سؤال جابه 
+
+262
+00:27:07,140 --> 00:27:10,640
+تعليق سؤال في الكتاب بيقول لي لو كانت الـ F is
+
+263
+00:27:10,640 --> 00:27:14,040
+differentiable ركز معاه لو كانت الـ F is
+
+264
+00:27:14,040 --> 00:27:17,800
+differentiable الشرط أن الـ derivative تماما تطلع
+
+265
+00:27:17,800 --> 00:27:23,590
+continuous هذا ليس شرطًا، هذا بس ينفي الـ R لما
+
+266
+00:27:23,590 --> 00:27:26,430
+نحكي عن هو نحكي عن الـ real valued functions
+
+267
+00:27:26,430 --> 00:27:33,210
+from R to R كمان لكن في الـ complex الأمر بيختلف الـ
+
+268
+00:27:33,210 --> 00:27:36,610
+complex أو الـ derivative مرة وجيتها معناته اللي
+
+269
+00:27:36,610 --> 00:27:39,210
+بعيدها واللي بعيدها واللي بعيدها هتظلها أشمالها
+
+270
+00:27:39,210 --> 00:27:43,830
+موجودة يعني if F prime exists then F double prime
+
+271
+00:27:43,830 --> 00:27:48,630
+and F triple F كذا إذا هتطلع مجرد ما تكون الدالة
+
+272
+00:27:48,630 --> 00:27:53,230
+analytic أو differentiable هتظلها على طول
+
+273
+00:27:53,230 --> 00:27:56,750
+differentiable و continuous و و و الآخرين هذا وين
+
+274
+00:27:56,750 --> 00:28:01,890
+في الـ complex أما في الـ real الوضع ليس كذلك وهي
+
+275
+00:28:01,890 --> 00:28:07,970
+مثال على F is differentiable everywhere طبعا is 
+
+276
+00:28:07,970 --> 00:28:11,310
+differentiable عند نقطة zero بالذات لكن its
+
+277
+00:28:11,310 --> 00:28:15,450
+derivative is not continuous at this point واضح
+
+278
+00:28:15,450 --> 00:28:20,330
+اليوم اه طيب الآن show
+
+279
+00:28:23,010 --> 00:28:30,750
+that F prime of X exists everywhere
+
+280
+00:28:30,750 --> 00:28:41,270
+but F prime is not continuous at zero و proof
+
+281
+00:28:41,270 --> 00:28:45,470
+البرهن طبعا البرهان كامل موجود هنا بس خلينا
+
+282
+00:28:45,470 --> 00:28:50,670
+نبرهن لك نوضح ايش ال ايش البرهان نشوفوا صلى
+
+283
+00:28:50,670 --> 00:28:56,670
+الله عليه الصلاة والسلام الآن F prime exist طبيعي
+
+284
+00:28:56,670 --> 00:29:03,310
+for X لا يساوي 0  مباشر سهل مجرد تفاضل عادي F prime 
+
+285
+00:29:03,310 --> 00:29:07,290
+of X لأن هذه الدالة قابلة للاشتقاق عند كل النقاط ما
+
+286
+00:29:07,290 --> 00:29:11,430
+عدا عند الـ X بيساوي 0 هذه بحكي عن هذه لحالها F
+
+287
+00:29:11,430 --> 00:29:19,020
+prime of X بيساوي 2X Sin 1 على X لأن X تربيع في
+
+288
+00:29:19,020 --> 00:29:24,360
+Sin تفاضل الـ Sin Cos وتفاضل اللي جوا سالب X
+
+289
+00:29:24,360 --> 00:29:30,440
+تربيع X ناقص 2 بتروح حد ما حد فبيصير سالب Cos 1
+
+290
+00:29:30,440 --> 00:29:36,480
+على X الآن هذا for X لا يساوي صفر طب for X بيساوي
+
+291
+00:29:36,480 --> 00:29:39,660
+صفر for X بيساوي صفر الوضع اللي لأن بيختلف اللي هو
+
+292
+00:29:39,660 --> 00:29:43,880
+فيه عندي اللي هو الدالة أصلا مش معرفة عند الـ Zero
+
+293
+00:29:43,880 --> 00:29:47,620
+لكن معرفة الدالة ككل عند الـ Zero عند مين بتساوي
+
+294
+00:29:47,620 --> 00:29:52,060
+صفر فمضطر أنا أستخدم التعريف لإثبات أو فحص إذا
+
+295
+00:29:52,060 --> 00:29:56,680
+كانت الدالة الها مشتقة عند الصفر ولا لأ فباجي بقول
+
+296
+00:29:56,680 --> 00:30:03,810
+الآن بتبدأ فحص limit اللي هو F of X ناقص F of Zero
+
+297
+00:30:03,810 --> 00:30:08,910
+على X ناقص Zero as X بتروح للزيرو بدي أشوف هل هذه
+
+298
+00:30:08,910 --> 00:30:13,250
+الـ limit موجودة ولا مش موجودة ماشي الحال Y بيساوي 
+
+299
+00:30:13,250 --> 00:30:16,930
+هذا
+
+300
+00:30:16,930 --> 00:30:20,370
+عشان خاطر أوجد مين أشوف الـ F prime عند الـ Zero
+
+301
+00:30:20,370 --> 00:30:25,630
+موجودة أو لا بيساوي limit F of X مين هي عبارة عن X
+
+302
+00:30:25,630 --> 00:30:29,030
+تربيع لأن X بتروح للزيرو، لأن X مش صفر، اللي هي
+
+303
+00:30:29,030 --> 00:30:34,450
+عبارة عن X تربيع Sin 1 على X ناقص F of Zero اللي
+
+304
+00:30:34,450 --> 00:30:39,310
+هي عبارة عن صفر على X ناقص صفر لما X تروح لمين
+
+305
+00:30:39,310 --> 00:30:46,870
+للصفر وبيساويل limit X term X Sin 1 على X لما X تروح
+
+306
+00:30:46,870 --> 00:30:50,670
+لمين للصفر هذه اللي هي by sandwich theorem حلقيها
+
+307
+00:30:50,670 --> 00:30:56,650
+ايش بتساوي صفر كيف by sandwich theorem عندك X Sin
+
+308
+00:30:56,650 --> 00:31:00,470
+1 على X absolute value أصغر أو يساوي الـ absolute
+
+309
+00:31:00,470 --> 00:31:05,050
+value للـ X وهذه الطبيعي أكبر أو يساوي مين صفر as X
+
+310
+00:31:05,050 --> 00:31:07,870
+goes to zero هذه بتروح للصفر وهذه بتروح للصفر
+
+311
+00:31:07,870 --> 00:31:10,570
+إذا اللي في النص هذه لازم الـ limit هيروح لمين 
+
+312
+00:31:10,570 --> 00:31:15,840
+للصفر ومن ثم للـ continuity لل بسبب الـ continuity
+
+313
+00:31:15,840 --> 00:31:18,040
+لل للـ absolute value بتطلع الـ limit اللي جوا
+
+314
+00:31:18,040 --> 00:31:20,740
+برضه أشمالها بتساوي 0 لأن الـ limit الـ absolute
+
+315
+00:31:20,740 --> 00:31:23,380
+value بيساوي absolute value للـ limit إذا صار عندي
+
+316
+00:31:23,380 --> 00:31:32,120
+limit اللي جوا بيساوي 0 معايا؟ طيب
+
+317
+00:31:32,120 --> 00:31:35,940
+بما أنه الـ limit هذه موجودة بتساوي 0 إذا بنقول
+
+318
+00:31:35,940 --> 00:31:42,580
+بعيدها since كده بيساوي كده then F prime of 0
+
+319
+00:31:42,580 --> 00:31:50,520
+بتساوي صفر إذا اللي أثبتنا احنا أنه F prime exist
+
+320
+00:31:50,520 --> 00:31:55,460
+وين ما كان وا ايش الـ F prime اللي أثبتناها كمان
+
+321
+00:31:55,460 --> 00:32:01,160
+أثبتنا إنه الآن F prime of X أثبتناها إنها بتساوي
+
+322
+00:32:02,250 --> 00:32:08,870
+2X Sin 1 على X ناقص Cos 1 على X لما X ايش معناه لأ
+
+323
+00:32:08,870 --> 00:32:13,070
+تساوي صفر واتبعتناها بتساوي صفر لما X بتساوي صفر
+
+324
+00:32:13,070 --> 00:32:16,990
+إذا صارت الـ F is functional everywhere الآن بدنا
+
+325
+00:32:16,990 --> 00:32:23,350
+نثبت إنه is not this F prime is not continuous at
+
+326
+00:32:23,350 --> 00:32:27,150
+zero شوف
+
+327
+00:32:27,150 --> 00:32:29,410
+الآن عندي
+
+328
+00:32:31,690 --> 00:32:41,570
+بدي أشوف limit اللي هي 2X Sin 1 على X ناقص Sin 1
+
+329
+00:32:41,570 --> 00:32:49,080
+على X لما X تروح للزيرو exist ولا لأ الآن limit الـ
+
+330
+00:32:49,080 --> 00:32:53,540
+أولى 2X Sin 1 على X مضمون أنه exist قبل مش
+
+331
+00:32:53,540 --> 00:32:57,560
+ويعملناها نفسها باي اللي هو sandwich theorem هتطلع
+
+332
+00:32:57,560 --> 00:33:03,720
+صفر إذا الآن أول حاجة اكتب لها since الآن limit
+
+333
+00:33:03,720 --> 00:33:09,420
+Sin 1 على X بيساوي صفر exist الآن 
+
+334
+00:33:09,420 --> 00:33:19,480
+but but limit Cos 1 على X لما X تروح لصفر does not
+
+335
+00:33:19,480 --> 00:33:24,360
+exist وهلّجت بنشوف ليش does not exist but this
+
+336
+00:33:24,360 --> 00:33:26,140
+limit أشمالها does not exist
+
+337
+00:33:28,950 --> 00:33:33,830
+ما دام هذا exist وهذا does not exist حاصل جمع أو
+
+338
+00:33:33,830 --> 00:33:38,290
+طرح أكيد ايش ماله does not exist لأنه لو exist
+
+339
+00:33:38,290 --> 00:33:42,810
+معناه تصير هذا اللي يطلع وهذا exist بيصير مجموع 
+
+340
+00:33:42,810 --> 00:33:45,170
+أو طرح بيساوي هذا أيضًا صار هذا exist وهذا
+
+341
+00:33:45,170 --> 00:33:51,550
+contradiction فاهمين اه الآن hence limit 2X 
+
+342
+00:33:51,550 --> 00:33:56,410
+Sin 1 على X ناقص Cosine 1 على X لما X تروح
+
+343
+00:33:56,410 --> 00:34:05,340
+للصفر اللي هو does not exist دلّ علي بس أوضح اللي
+
+344
+00:34:05,340 --> 00:34:12,220
+هو ليش هذا does not exist بده يوضح ليش هذا does
+
+345
+00:34:12,220 --> 00:34:18,740
+not exist أخذتوا أنتم أخذتوا اللي هو ايش نظرية أن 
+
+346
+00:34:18,740 --> 00:34:28,240
+لو كانت limit F of X لما X بتروح لـ A بيساوي L إذا أي 
+
+347
+00:34:28,240 --> 00:34:34,560
+sequence XN بتروح للـ A لازم يعطيك اللي هو F of XN
+
+348
+00:34:34,560 --> 00:34:39,780
+برضه يروح لمين للـ L بصبوط؟ هذه كـ continuity اللي
+
+349
+00:34:39,780 --> 00:34:43,530
+بتحكي عنها اللي كده كانت limit F of X لما X تروح
+
+350
+00:34:43,530 --> 00:34:48,930
+للـ A بيساوي L و if تأخدت أي sequence XN بتروح للـ
+
+351
+00:34:48,930 --> 00:34:54,670
+A هيطلع limit F of XN مش بيروح بيساوي L ماشي هذه 
+
+352
+00:34:54,670 --> 00:35:00,780
+اللي هي النظرية بناء على هذه النظرية كنا نقول أنه
+
+353
+00:35:00,780 --> 00:35:07,180
+لو جيت two sequences على الأقل يكون بيروح للـ A
+
+354
+00:35:07,180 --> 00:35:13,500
+لكن بيروح لـ limits مختلفة أو واحدة من limits does 
+
+355
+00:35:13,500 --> 00:35:18,600
+not exist بكفالك أن الـ limit هذه أشمالها does not
+
+356
+00:35:18,600 --> 00:35:25,170
+exist واضح؟ الآن إذا بدي أعتمد على هذه في إثبات إنه
+
+357
+00:35:25,170 --> 00:35:29,690
+limit Cosine 1 على X does not exist أكيد كلكم
+
+358
+00:35:29,690 --> 00:35:40,730
+جال take the sequences أو the two sequences XN و
+
+359
+00:35:40,730 --> 00:35:50,290
+بتساوي 1 على 2N و by and YN و بتساوي 1
+
+360
+00:35:50,290 --> 00:35:57,030
+على 2N زائد 1 في Pi مظبوط أكيد قلته هيك 
+
+361
+00:35:57,030 --> 00:36:05,290
+where XN goes to zero and YN goes to zero
+
+362
+00:36:05,290 --> 00:36:13,130
+but Cosine 1 على ايه؟ Cosine 1 على XN 
+
+363
+00:36:13,130 --> 00:36:23,100
+اللي هو بتساوي أو goes to 1 و Cosine 1 على YN 
+
+364
+00:36:23,100 --> 00:36:30,500
+goes to ناقص 1 أوضحها 
+
+365
+00:36:30,500 --> 00:36:35,600
+أوضحها أذكركم الآن 
+
+366
+00:36:35,600 --> 00:36:39,980
+هذا بيصير Cosine مين 2N و by 2N و by ايش 
+
+367
+00:36:39,980 --> 00:36:45,470
+مالها 1 2N زائد 1 لكل في باقي ايش
+
+368
+00:36:45,470 --> 00:36:49,030
+بتصير اللي هو سالب 1 فبيصير هذا الـ Cosine
+
+369
+00:36:49,030 --> 00:36:53,450
+converts 1 وهذا الـ Cosine converts لمين؟ لسالب 
+
+370
+00:36:53,450 --> 00:36:58,490
+1 إذا حسب اللي حكيناها قبل شوية hence hence 
+
+371
+00:36:58,490 --> 00:37:05,910
+hence limit Cosine 1 على X لما X تروح لـ Zero
+
+372
+00:37:05,910 --> 00:37:17,800
+does not exist هذا توضيح توضيح
+
+401
+00:39:59,680 --> 00:40:09,720
+Monotone F من عند هذه theorem أنا ما كتبتش ال theorem
+
+402
+00:40:09,720 --> 00:40:19,060
+تمام لأنه موجود هنا theorem 618 عند F من I لعند R
+
+403
+00:40:19,060 --> 00:40:25,440
+غير عنها strictly monotone
+
+404
+00:40:25,440 --> 00:40:36,900
+ماشي الحال و continuous and continuous الآن لاحظ
+
+405
+00:40:36,900 --> 00:40:41,100
+على ال function هذام دام حكيت عنها strictly
+
+406
+00:40:41,100 --> 00:40:44,540
+monotone يعني strictly increasing أو strictly
+
+407
+00:40:44,540 --> 00:40:49,240
+decreasing معناته صارت automatically الدالة أصبحت
+
+408
+00:40:49,240 --> 00:40:53,800
+one to one ليش؟ لأنه لو كانت strictly increasing
+
+409
+00:40:53,800 --> 00:40:59,060
+معناته x1 أصغر من x2 اللي بتعطيني f of x1 أصغر
+
+410
+00:40:59,060 --> 00:41:03,840
+من f of x2 هذا في حالة إن كانت strictly
+
+411
+00:41:03,840 --> 00:41:09,540
+increasing  أه مش ممكنية للمساواة الآن في حالة
+
+412
+00:41:09,540 --> 00:41:14,520
+strictly decreasing X1 أكبر من X2 بيعطيني F of X1
+
+413
+00:41:14,520 --> 00:41:21,220
+أصغر من F of X2 هذا strictly decreasing الآن سواء
+
+414
+00:41:21,220 --> 00:41:25,060
+كانت هذه الحالة هذه أو الحالة هذه هيكون عندي ال
+
+415
+00:41:25,060 --> 00:41:29,200
+function one to one غصب عنها، ليش؟ لأن لو كانت X1
+
+416
+00:41:29,200 --> 00:41:35,640
+لا تساوي X2 يا أصغر يا أكبر، ما زال مقتصر  أصغرهتكون
+
+417
+00:41:35,640 --> 00:41:40,680
+strictly أصغر يعني بمعنى آخر f of x1 لا تساوي f of
+
+418
+00:41:40,680 --> 00:41:45,460
+x2 ولو كانت هذه أكبر هتكون f of x1 أكبر برضه هتكون
+
+419
+00:41:45,460 --> 00:41:49,060
+أصبحت لا تساوي و similarly في حالة ال decreasing
+
+420
+00:41:49,850 --> 00:41:56,250
+إذن واضح إن X1 ≤ X2 بيعطينا F X1 ≤ F X2 أو بمعنى
+
+421
+00:41:56,250 --> 00:42:03,730
+آخر F X1 هذه هي ال contraposition ل F X1 ≤ F X2
+
+422
+00:42:03,730 --> 00:42:10,010
+بيعطينا X1 ≤ X2 وهذا يعني ال F أصبحت is one to
+
+423
+00:42:10,010 --> 00:42:15,810
+one يعني الآن أنا بوضح ليش أعطاني F strictly
+
+424
+00:42:15,810 --> 00:42:20,790
+monotone؟ عشان شكله أصلا احنا قاعدين .. هذه النظرية
+
+425
+00:42:20,790 --> 00:42:25,030
+الحاكية عن ال inverse بدنا نشوف العلاقة بين ال
+
+426
+00:42:25,030 --> 00:42:27,950
+derivative لل inverse و ال derivative للدالة
+
+427
+00:42:27,950 --> 00:42:32,520
+الأصلية إذا بدأنا للدالة الأصلية أن يكون لها
+
+428
+00:42:32,520 --> 00:42:36,640
+inverse عشان يكون لها inverse لازم أفترض أن الدالة
+
+429
+00:42:36,640 --> 00:42:40,740
+أصبحت one to one فالآن فرضنا إنها strictly
+
+430
+00:42:40,740 --> 00:42:43,980
+monotone يعني strictly increasing أو strictly
+
+431
+00:42:43,980 --> 00:42:50,920
+decreasing الآن دام  one to one إذا في إمكانية
+
+432
+00:42:50,920 --> 00:42:58,230
+نحكي عن inverse لها F inverse لها اللي هو بده
+
+433
+00:42:58,230 --> 00:43:03,290
+يسميه هو الآن بده يسميه G هو بده يسميه G هياخد ال
+
+434
+00:43:03,290 --> 00:43:10,950
+G من وين من ال range لل F يعني من ال F of I لعند
+
+435
+00:43:10,950 --> 00:43:17,150
+مين لعند ال R ماشي ال F of I هذي مسميها مين هو
+
+436
+00:43:17,150 --> 00:43:25,150
+مسميها هو J يعني ما يأخذ ال G من J لعند R الـ G هي
+
+437
+00:43:25,150 --> 00:43:30,750
+عبارة عن مين؟ عن الـ F inverse معايا؟ قولي يا محمد
+
+438
+00:43:30,750 --> 00:43:42,570
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+439
+00:43:42,570 --> 00:43:42,670
+R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R
+
+440
+00:43:42,670 --> 00:43:43,910
+.. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R ..
+
+441
+00:43:43,910 --> 00:43:44,630
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+442
+00:43:44,630 --> 00:43:47,530
+R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R
+
+443
+00:43:47,530 --> 00:43:47,590
+.. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R ..
+
+444
+00:43:47,590 --> 00:43:47,610
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+445
+00:43:47,610 --> 00:43:53,500
+R .. لـ R .. لـ R .. لـ بس لما نكتب I R ايش برضه
+
+446
+00:43:53,500 --> 00:43:57,420
+صحيح بس على أساس أنه .. خلاني أقول المعلومات بت ..
+
+447
+00:43:57,420 --> 00:44:01,780
+بت .. بتأهلني أن أكون أدق لعند مين لعند I خلّاني
+
+448
+00:44:01,780 --> 00:44:06,600
+صحيح طيب الآن عشان ما نبعدش عن اللي بدنا يا F من I
+
+449
+00:44:06,600 --> 00:44:12,080
+ل R strictly monotone و G من F of I ل عند I أصلا
+
+450
+00:44:12,080 --> 00:44:15,140
+إذا كانت ال F من I ل عند R strictly monotone و
+
+451
+00:44:15,140 --> 00:44:19,960
+continuous هتكون الـ G اللي هي ال F inverse من F of
+
+452
+00:44:19,960 --> 00:44:24,400
+I ل I برضه strictly monotone و إيش و continuous
+
+453
+00:44:24,400 --> 00:44:27,940
+strictly monotone و continuous إذن الآن ال F عند
+
+454
+00:44:27,940 --> 00:44:30,660
+ها strictly monotone و continuous و ال F inverse
+
+455
+00:44:30,660 --> 00:44:41,160
+برضه strictly monotone و إيش و continuous معايا؟
+
+456
+00:44:41,160 --> 00:44:49,090
+طيب الآن لحد الآن بعتبر أنه احنا بنحكي عن معطيات
+
+457
+00:44:49,090 --> 00:44:54,030
+حتى لو كنا استنتجنا جزء منها من بعض الآن بيقول لي
+
+458
+00:44:54,030 --> 00:44:58,970
+اللي هو موضوعنا if f is differentiable at c if
+
+459
+00:44:58,970 --> 00:45:12,350
+الآن if if f is differentiable at c element in I
+
+460
+00:45:13,160 --> 00:45:17,380
+ماشي الحال يعني F differentiable عند نقطة C
+
+461
+00:45:17,380 --> 00:45:24,600
+element وين؟ in I بيقول لك then طبعا A و F prime of
+
+462
+00:45:24,600 --> 00:45:28,340
+C أصبحت لاتساوي صفر لأنه هيلزمنا القسم عليها
+
+463
+00:45:28,340 --> 00:45:32,420
+عشان هيك لازم اللي هو عشان نحكي عن ال inverse لل
+
+464
+00:45:32,420 --> 00:45:34,880
+derivative تكون ال F prime عند ال C أصبحت
+
+465
+00:45:34,880 --> 00:45:42,730
+لا تساوي صفر الآن then اللي هي الـ G الـ G اللي
+
+466
+00:45:42,730 --> 00:45:47,810
+سميناها F inverse اللي سميناها F inverse الـ G is
+
+467
+00:45:47,810 --> 00:45:51,150
+differentiable
+
+468
+00:45:51,150 --> 00:46:00,490
+is differentiable at اللي هو F of C is
+
+469
+00:46:00,490 --> 00:46:04,610
+differentiable at إيش؟ at F of C اللي هو مسميها
+
+470
+00:46:04,610 --> 00:46:11,800
+إيش اسمها دي الآن  الواضح أنه لو كانت F
+
+471
+00:46:11,800 --> 00:46:15,960
+differentiable عند الـC و F' للـC لا تساوي صفر
+
+472
+00:46:15,960 --> 00:46:20,400
+هتكون الـF inverse differentiable عند مين؟ عند
+
+473
+00:46:20,400 --> 00:46:34,300
+الـF of C اه and .. and F inverse prime at F of C
+
+474
+00:46:35,750 --> 00:46:46,830
+At F of C هتساوي واحد على F prime لمين؟ Let's see
+
+475
+00:46:46,830 --> 00:46:59,390
+يعني وكأنه وكأنه بيقول F inverse prime لل Y بتساوي
+
+476
+00:46:59,390 --> 00:47:07,690
+واحد على F prime of X where العلاقة بين Y و X اللي
+
+477
+00:47:07,690 --> 00:47:16,070
+هي Y بتساوي F of X واضح، اه، الكلام واضح وسهل اللي
+
+478
+00:47:16,070 --> 00:47:21,190
+هي f prime of c بدي أكتبها بطريقة ثانية بيساوي واحد
+
+479
+00:47:21,190 --> 00:47:30,010
+على f prime of الآن هو سمّى ال f of c أصبحت D يعني
+
+480
+00:47:30,010 --> 00:47:37,080
+ال c بتساوي f inverse لمين؟ ل D صح؟ يعني بمعنى آخر
+
+481
+00:47:37,080 --> 00:47:45,060
+هو بيقول F inverse prime عند مين؟ عند ال D بساوي واحد
+
+482
+00:47:45,060 --> 00:47:50,580
+على F prime مكان ال C ايش بده يحط؟ F inverse
+
+483
+00:47:50,580 --> 00:48:01,420
+of D يعني واحد على F prime فلأ  بكفي اللي هي ماشي
+
+484
+00:48:01,420 --> 00:48:07,720
+الحال G of D لأ لأ اللي هو كاتبها يعني لإنه كاتب هو
+
+485
+00:48:07,720 --> 00:48:15,000
+بدل ال F inverse كاتب مين الـG واضح واضحة طبعا هذه
+
+486
+00:48:15,000 --> 00:48:22,200
+اللي هي اللي احنا بدنا نثبته نيجي للإثبات
+
+487
+00:48:27,580 --> 00:48:32,740
+بدنا نطبطب روحنا شوية عرفناش إننا نطبطب اه عشان اللي
+
+488
+00:48:32,740 --> 00:48:41,260
+خاطر أمسح وبدأ ابدأ أشتغل من فوق النظرية
+
+489
+00:48:41,260 --> 00:48:45,720
+حفظناها اه أو خلينا نقول هي موجودة يعني اه بس
+
+490
+00:48:45,720 --> 00:48:49,160
+بدنا نطلع عالميا على البرهان ال proof
+
+491
+00:48:56,060 --> 00:49:04,920
+بدنا نثبت احنا اللي هو g prime of
+
+492
+00:49:04,920 --> 00:49:14,460
+D بساوي واحد على F prime of C هاي اللي بدنا نثبته
+
+493
+00:49:14,460 --> 00:49:17,820
+أو واحد على F prime .. الـ G prime هذا طالع من
+
+494
+00:49:17,820 --> 00:49:22,140
+اليمين الـ G عبارة عن F inverse ماشي؟ شوفي الآن
+
+495
+00:49:25,040 --> 00:49:33,140
+بدي أعرف ده اللي برضه مساعدة لأن أخذ Y لا تساوي D و
+
+496
+00:49:33,140 --> 00:49:39,760
+Y طبعا وين موجودة؟ Y element in J في domain من ال
+
+497
+00:49:39,760 --> 00:49:43,640
+F inverse متذكرين؟
+
+498
+00:49:43,640 --> 00:49:52,360
+ال F من I لعند R و ال G اللي هي ال F inverse من F
+
+499
+00:49:52,360 --> 00:49:59,500
+of I اللي سميناها J لعند مين؟ لعند الـ I مظبوط؟ طيب
+
+500
+00:49:59,500 --> 00:50:10,320
+الآن هأعرف ال H of Y H of Y بتساوي اللي هو F F
+
+501
+00:50:10,320 --> 00:50:20,250
+inverse of Y اللي هي F of G of Y ما نقص F of G of
+
+502
+00:50:20,250 --> 00:50:25,430
+مين؟ of D نحطها على جنب ايش اللي بنعني فيه F of C
+
+503
+00:50:25,430 --> 00:50:33,110
+بتساوي D أو G of D اللي هي F inverse استذكروا أنه
+
+504
+00:50:33,110 --> 00:50:36,390
+أنا الـ G هي مين هي F inverse عشان تسهل عليكم
+
+505
+00:50:36,390 --> 00:50:41,630
+عملية الحسابات ايش بتساوي؟ C ماشي الآن F of G of Y
+
+506
+00:50:41,630 --> 00:50:48,750
+ناقص F of G of D على G of Y ناقص G of D
+
+507
+00:50:51,950 --> 00:50:56,890
+معايا؟ واضح اه؟ الآن أنا مفترض أن ال Y لا تساوي D
+
+508
+00:50:56,890 --> 00:51:01,290
+طبعا ال functions سبعنا سواء ال F أو ال G كلهم
+
+509
+00:51:01,290 --> 00:51:05,870
+one to one مؤكد ال G of Y ايش مالها لا تساوي G of
+
+510
+00:51:05,870 --> 00:51:09,990
+D إذن أنا مرتاح أنا أخذت Y لا تساوي Z إذن أنا
+
+511
+00:51:09,990 --> 00:51:13,450
+automatically هذه is well defined يعني G of Y ايش
+
+512
+00:51:13,450 --> 00:51:19,270
+مالها لا تساوي مين؟ G of D واضح اه؟ أي سؤال؟
+
+513
+00:51:22,880 --> 00:51:28,700
+الآن الـ H of Y اللي هي بتساوي اللي هو F of G of Y
+
+514
+00:51:28,700 --> 00:51:33,120
+ناقص F وكذا على G of Y minus G of D is اللي هي well
+
+515
+00:51:33,120 --> 00:51:38,640
+defined طبعا الآن اللي فوق بناء على ما كتبناه
+
+516
+00:51:38,640 --> 00:51:45,220
+المستعملينه هنا كرموز هذه نفسها ممكن نطلع إليها
+
+517
+00:51:45,220 --> 00:51:50,520
+كصيغة أخرى أفما هي inverse تبعدها دول فبيصير عبارة
+
+518
+00:51:50,520 --> 00:51:57,980
+عن Y ناقص D صح ولا لأ على G of Y ناقص مين G of D
+
+519
+00:51:57,980 --> 00:52:04,840
+هذا نظرة أخرى لمين للH of Y اللي عندنا أي سؤال؟
+
+520
+00:52:04,840 --> 00:52:10,740
+طبعا الـH of Y واضح أنها مش صفر لأن الـY أصبحت
+
+521
+00:52:10,740 --> 00:52:15,500
+لا تساوي D لأن بعد شوية بدي أقسم عليها عشان هيك بدي
+
+522
+00:52:15,500 --> 00:52:22,000
+أكد لكم أن الـY الـH of Y أصبحت لا تساوي صفر
+
+523
+00:52:22,000 --> 00:52:31,170
+واضح لحد الآن، ماشي الحال، الآن بدي أستخدم اللي هو
+
+524
+00:52:31,170 --> 00:52:35,090
+التعريف في الوصول ال Epsilon Delta Definition في
+
+525
+00:52:35,090 --> 00:52:42,570
+الوصول للي بديه اللي هو G prime of D بيساوي limit G
+
+526
+00:52:42,570 --> 00:52:48,770
+of Y نقص G of D على Y minus D الـY بتروح للـD اللي
+
+527
+00:52:48,770 --> 00:52:51,490
+هو exist .. that .. that .. that .. that مثلك أنه
+
+528
+00:52:51,490 --> 00:52:54,470
+لكل أكبر من صفر يوجد Delta أو Gamma أو اللي هي
+
+529
+00:52:54,470 --> 00:52:58,490
+such that لما تكون مسافة بين Y و D أصغر من Delta
+
+530
+00:52:58,490 --> 00:53:01,900
+يعطينا هدف أصغر من مين؟ من إبسيلون هذا كيف بده
+
+531
+00:53:01,900 --> 00:53:07,060
+يثبته؟ بالاستعانة بيه اللي هو أنا بعرف أن الـ F is
+
+532
+00:53:07,060 --> 00:53:10,320
+differentiable و بالاستعانة بالدالة اللي عندي هذه
+
+533
+00:53:10,320 --> 00:53:14,080
+السؤال أي سؤال حتى الآن حتى الآن احنا اللي أثبتنا
+
+534
+00:53:14,080 --> 00:53:17,540
+أن H و Y اللي ركبناها دي ايش مالها لا تساوي صفر
+
+535
+00:53:17,540 --> 00:53:27,380
+الآن عندي since F is differentiable at مين معطيني؟
+
+536
+00:53:27,380 --> 00:53:34,100
+at C إذن then for every epsilon أكبر من صفر بتستخدم
+
+537
+00:53:34,100 --> 00:53:38,920
+على طول اللي هو limit واضح then for every epsilon
+
+538
+00:53:38,920 --> 00:53:42,660
+أكبر من صفر there exist نستخدم نفس التعبير دلتا
+
+539
+00:53:42,660 --> 00:53:47,840
+أكبر من صفر أو نفس الرموز such that X minus C أكبر
+
+540
+00:53:47,840 --> 00:53:54,020
+من صفر وأصغر من دلتا implies F of X ناقص F of C
+
+541
+00:53:54,020 --> 00:53:59,240
+على X minus C ناقص اللي هو ال derivative اللي هي F
+
+542
+00:53:59,240 --> 00:54:06,500
+prime of C صح؟ أصغر من مين؟ من إبسيلون وسمي هذه ايش
+
+543
+00:54:06,500 --> 00:54:17,000
+مسميها هو؟ هو مسميها واحد هذا سمّي واحد واضح
+
+544
+00:54:17,000 --> 00:54:21,360
+اه؟ في معنى وكده الآن f is differentiable in c
+
+545
+00:54:21,360 --> 00:54:25,580
+than لكل y نقل المستقبل اللي يعني استخدمنا limit f
+
+546
+00:54:25,580 --> 00:54:30,470
+of x نقص f of c على x minus c سوى f prime of c طبعا
+
+547
+00:54:30,470 --> 00:54:34,630
+أنا بعيد على أساس اللي بتابعنا مش سواعب أنكم مش
+
+548
+00:54:34,630 --> 00:54:42,350
+سواعبته واضح؟ ماشي الحاجة الآن بدنا نستخدم ال
+
+549
+00:54:42,350 --> 00:54:47,410
+continuity لمين؟ ل G G is continuous طبعا G is
+
+550
+00:54:47,410 --> 00:54:51,870
+differentiable أصلا G is differentiable عند مين؟
+
+551
+00:54:51,870 --> 00:54:56,690
+عند اللي هو D يعني G is differentiable عند F of C
+
+552
+00:54:56,690 --> 00:55:03,740
+إذا continuous مظبوط؟ إذن الآن since .. since G is
+
+553
+00:55:03,740 --> 00:55:12,540
+continuous at D بتساوي F of C إذن يعني صار عندي
+
+554
+00:55:12,540 --> 00:55:24,060
+limit G of Y لما Y تروح ل D بساوي مين؟ G of D اه
+
+555
+00:55:25,340 --> 00:55:29,760
+واضح ها؟ واضح .. واضح ها؟ خليني أعمل .. على أساس
+
+556
+00:55:29,760 --> 00:55:32,980
+ما نستخدم الرموز هذه نربطها باللي بقى بالها ..
+
+557
+00:55:32,980 --> 00:55:37,900
+بنغير بعدين معايا؟ طيب .. هذه إيش معناتها by
+
+558
+00:55:37,900 --> 00:55:41,460
+epsilon delta definition؟ أنه لكل .. الآن بدي
+
+559
+00:55:41,460 --> 00:55:45,780
+أستخدم الـ delta هذه لإن بدي أستخدم الاتنين مع بعض
+
+560
+00:55:45,780 --> 00:55:49,800
+الوصول للي بديها واضح؟ هذه لكل إبسيلون في الدنيا،
+
+561
+00:55:49,800 --> 00:55:53,140
+أي إن كانت من ضمن إنّي الـDelta اللي لجيناها،
+
+562
+00:55:53,140 --> 00:55:57,300
+واضح اللي بأحكيه؟ الآن لكل إبسيلون أكبر من 0، there
+
+563
+00:55:57,300 --> 00:56:01,160
+exists Gamma اللي هي .. ابحثت إنّه لما يكون Y minus
+
+564
+00:56:01,160 --> 00:56:05,100
+D أصغر من Gamma، يعطيني هذا أصغر، اللي هو من اللي
+
+565
+00:56:05,100 --> 00:56:08,180
+بديه .. من اللي بدي أنا أسلك دموع، اللي هو
+
+566
+00:56:08,180 --> 00:56:16,030
+الـDelta، الآن continuous كده then for every
+
+567
+00:56:16,030 --> 00:56:18,930
+إبسيلون الإبسيلون اللي بده يستخدمها اللي من ضمن
+
+568
+00:56:18,930 --> 00:56:21,290
+الإبسيلون هي اللي بتنفع ايه اللي بتنفع على ال delta
+
+569
+00:56:21,290 --> 00:56:25,970
+هذه then for delta أكبر من صفر there exist gamma
+
+570
+00:56:25,970 --> 00:56:32,670
+أكبر من صفر such that لما y minus d يكون أكبر من
+
+571
+00:56:32,670 --> 00:56:37,170
+صفر وأصغر من الجامعة اللي لجيتها هذا بيعطيني اللي
+
+572
+00
+
+601
+00:58:52,230 --> 00:58:59,750
+أستخدم واحد إذا صار عندي then هذا متحقق الآن then
+
+602
+00:58:59,750 --> 00:59:04,830
+بدي أربط هذا الكلام بمين؟ باللي فوق اللي هو H of Y
+
+603
+00:59:04,830 --> 00:59:12,450
+H of Y ناقص F prime of C شوف كيف بده F مثلك الآن
+
+604
+00:59:12,450 --> 00:59:19,780
+اللي هو بساوي H of Y مين؟ H of Y اللي هو عبارة عن Y
+
+605
+00:59:19,780 --> 00:59:27,960
+ناقص D على G of Y ناقص G of D، مظبوط؟ اللي هو بيه
+
+606
+00:59:27,960 --> 00:59:30,960
+عبارة عن مين؟ F of G of Y ناقص كده، إلى آخره
+
+607
+00:59:30,960 --> 00:59:42,000
+بيصير بساوي اللي هو F of G of Y ناقص F of G of D
+
+608
+00:59:42,000 --> 00:59:52,730
+على G of Y ناقص G of D اللي هو ناقص F prime
+
+609
+00:59:52,730 --> 01:00:00,150
+of C أصغر من مين؟ من Y عارفين ليش؟ لأن الـ G of Y
+
+610
+01:00:00,150 --> 01:00:04,880
+مين هي؟ الـ X و الـ G of D اللي هي مين؟ الـ C
+
+611
+01:00:04,880 --> 01:00:10,480
+فاهمين؟ أنا قصدت أكتبها بالصورة هذه عشان أقول لكم أيش
+
+612
+01:00:10,480 --> 01:00:14,580
+لو سُئلتُ: H of Y ناقص F prime of C بساوي هذا
+
+613
+01:00:14,580 --> 01:00:21,500
+المقدار ناقص هذا متحقق كمان مرة يا جماعة اللي
+
+614
+01:00:21,500 --> 01:00:26,220
+بيحقق هذا غصب عنه بيحقق هذا، واللي بيحقق هذا بيحقق
+
+615
+01:00:26,220 --> 01:00:31,240
+هذا ماشي الحال. أنا الآن أخذت عندي Y element in J
+
+616
+01:00:31,920 --> 01:00:36,960
+و الـ Y لا تساوي D، وأخذت هذا يكون أصغر من دلتا تحت
+
+617
+01:00:36,960 --> 01:00:41,320
+هذا الشرط. الآن أنا بشتغل تحت هذا الشرط الآن. هذا
+
+618
+01:00:41,320 --> 01:00:46,140
+أعطاني حسب اللي هي اتنين أن هذا ناقص هذا أصغر من
+
+619
+01:00:46,140 --> 01:00:51,090
+دلتا. هذا اللي هو X وهذا اللي هو mean الـ C الآن
+
+620
+01:00:51,090 --> 01:00:54,790
+مادام أصغر من دلتا إذا أعطاني واحد. إذا أعطتني هذه
+
+621
+01:00:54,790 --> 01:00:59,510
+أصغر من هذه. هذه في الواقع مين هي؟ F of X هي عبارة
+
+622
+01:00:59,510 --> 01:01:03,830
+عن F of G of Y لأن G of Y هي الـ X، و C ليه F of D
+
+623
+01:01:03,830 --> 01:01:08,690
+صار هذا ناقص هذا أصغر من مين؟ من إبسلون. من جهة أخرى
+
+624
+01:01:08,690 --> 01:01:13,650
+هذا المقدار مين هو؟ عبارة عن H of Y وهذا اللي هي F
+
+625
+01:01:13,650 --> 01:01:18,550
+prime of C. إذا القالة اللي وصلت إليه أنه لكل إبسلون
+
+626
+01:01:18,550 --> 01:01:22,670
+أكبر من 0 وجدت Gamma، لأن Gamma بين إيديا واجدتها
+
+627
+01:01:22,670 --> 01:01:27,050
+من الأصل. إلى لكل إبسلون أكبر من 0 لجيت Gamma، ابحث
+
+628
+01:01:27,050 --> 01:01:30,290
+أنه لما يكون هذه المسافة بينها أصغر من Gamma
+
+629
+01:01:30,290 --> 01:01:33,610
+يعطينا المسافة أصغر من إبسلون، يعني المسافة هذه
+
+630
+01:01:33,610 --> 01:01:40,760
+أصغر من إبسلون. أو بمعنى آخر limit H of Y لما الـ Y
+
+631
+01:01:40,760 --> 01:01:50,320
+تروح للـ D بساوي F prime of مين؟ of C. تصيرنا
+
+632
+01:01:50,320 --> 01:01:55,660
+قريبين من اللي بدنا إياه. عند الـ H of Y limitها بساوي
+
+633
+01:01:55,660 --> 01:02:02,980
+limit هذا بساوي F prime of C. هذا عبارة عن مين؟ عن
+
+634
+01:02:02,980 --> 01:02:10,910
+G prime مقلوب G prime. إذا الآن خلصنا تقريباً، نمسح
+
+635
+01:02:10,910 --> 01:02:16,130
+حدّه، و نمسك
+
+636
+01:02:16,130 --> 01:02:28,150
+بهذه مع هذه. صار عندي limit H of Y لما الـ Y تؤول
+
+637
+01:02:28,150 --> 01:02:34,960
+إلى D. limit H of Y لما Y تؤول إلى D طلع عندي عبارة
+
+638
+01:02:34,960 --> 01:02:40,960
+عن F prime of C. هي تحت. من جهة أخرى هذا بيساوي مين؟
+
+639
+01:02:40,960 --> 01:02:48,720
+limit Y ناقص D على G of Y ناقص G of D لما Y تروح
+
+640
+01:02:48,720 --> 01:02:55,980
+لمين؟ للـ D صح؟ هذا عبارة عن limit أو واحد على
+
+641
+01:02:55,980 --> 01:03:03,130
+limit G of Y ناقص G of D على Y ناقص D لما Y تروح لـ
+
+642
+01:03:03,130 --> 01:03:10,310
+D. 1 على الـ limit هذا فهي
+
+643
+01:03:10,310 --> 01:03:16,390
+ساوي 1 على G prime of D. إذا صارت عندي F prime of
+
+644
+01:03:16,390 --> 01:03:23,710
+C بساوي 1 على G prime of D. أو بمعنى آخر F prime
+
+645
+01:03:23,710 --> 01:03:31,800
+of C بساوي 1 على F G prime اللي هي F inverse prime
+
+646
+01:03:31,800 --> 01:03:42,000
+of D. أو بتعبير آخر F inverse prime of D بساوي
+
+647
+01:03:42,000 --> 01:03:55,700
+واحد على F prime of C وهو المطلوب. طيب
+
+648
+01:04:01,020 --> 01:04:02,360
+خليني أجيب عن نقطة
+
+649
+01:04:19,990 --> 01:04:25,110
+هي نفس اللي فوق بس كانت اللي فوق عندها نقطة هنا in
+
+650
+01:04:25,110 --> 01:04:29,510
+general. الآن لو كانت على كل الـ domain، يعني الآن
+
+651
+01:04:29,510 --> 01:04:33,150
+بقولي لو كانت I subset من R بيه an interval and F
+
+652
+01:04:33,150 --> 01:04:37,510
+من I لعند R strictly monotone on I ماشي ..
+
+653
+01:04:37,510 --> 01:04:42,750
+strictly monotone و let J بيساوي F of I and let G
+
+654
+01:04:42,750 --> 01:04:47,410
+من J لعند R بيه the function inverse of F ماشي ..
+
+655
+01:04:47,980 --> 01:04:52,360
+لأن إذا كانت F as differential على الـ I كلها، إذا
+
+656
+01:04:52,360 --> 01:04:55,860
+أكيد هتكون continuous على الـ I كلها. مدام هتكون
+
+657
+01:04:55,860 --> 01:04:58,300
+continuous على الـ I كلها و strictly monotone
+
+658
+01:04:58,300 --> 01:05:02,180
+هتكون الـ G العكس برضه strictly monotone و
+
+659
+01:05:02,180 --> 01:05:06,870
+continuous. إذا شروط اللي فوق اتحققت. إذا الـ J ..
+
+660
+01:05:06,870 --> 01:05:10,030
+طبعاً الـ F' of X معطيني إياها تساوي X على كل الـ
+
+661
+01:05:10,030 --> 01:05:13,350
+I، إذا أكيد حسب النظرية اللي قبل شوية G is
+
+662
+01:05:13,350 --> 01:05:17,230
+differentiable على كل الـ J و الـ G' بساوي واحدة
+
+663
+01:05:17,230 --> 01:05:22,210
+على F' لأنها composite مين؟ composite G واضح؟ يعني
+
+664
+01:05:22,210 --> 01:05:28,130
+.. كمان مرة؟ إذا كانت الـ F strictly monotone و
+
+665
+01:05:28,130 --> 01:05:32,890
+كانت اللي هي الـ J بتساوي كل الـ F of I، وكانت الـ G
+
+666
+01:05:32,890 --> 01:05:37,310
+هي عبارة عن الـ inverse للـ F، وكانت الـ F
+
+667
+01:05:37,310 --> 01:05:39,570
+differentiable على الـ I. مدام F differentiable على
+
+668
+01:05:39,570 --> 01:05:42,210
+الـ I إذا صارت continuous على كل الـ I. مدام
+
+669
+01:05:42,210 --> 01:05:44,710
+continuous على الـ I و strictly monotone هتكون الـ
+
+670
+01:05:44,710 --> 01:05:47,710
+inverse العكس اللي هي الـ G برضه continuous و
+
+671
+01:05:47,710 --> 01:05:51,700
+strictly monotone. إذا انتحققت شروط النظرية السابقة
+
+672
+01:05:51,700 --> 01:05:55,680
+إذا هتكون الـ F، وطبعاً الـ F' طبعاً لها تساوي صفر ع
+
+673
+01:05:55,680 --> 01:05:59,020
+كل الـ I. إذا صارت الـ G الـ differential بالع كل الـ J
+
+674
+01:05:59,020 --> 01:06:02,220
+و العلاقة بين التنتين G' بساوي واحد على F'
+
+675
+01:06:02,680 --> 01:06:09,040
+composite، mean composite J. أو زي ما بنقول G' of Y
+
+676
+01:06:09,040 --> 01:06:16,220
+بساوي واحد على F' of mean of X. تحت العلاقة X بتساوي
+
+677
+01:06:16,220 --> 01:06:25,320
+F of Y أو Y بس هوقف، و فكس. Y بس هوقف، و فكس. طبعاً هذا
+
+678
+01:06:25,320 --> 01:06:32,220
+البرهان كله حكيناه. الـ remark اللي عندنا. الـ remark
+
+679
+01:06:32,220 --> 01:06:37,100
+اللي عندنا الـ
+
+680
+01:06:37,100 --> 01:06:43,460
+remark
+
+681
+01:06:43,460 --> 01:06:51,180
+اللي عندنا هي اللي حكيتها الآن. G' of Y بساوي واحد على F' of X
+
+682
+01:06:51,180 --> 01:06:53,660
+of X
+
+683
+01:06:57,210 --> 01:06:59,530
+اللي قبل بشوية حكيناها G prime of Y بساوي واحد
+
+684
+01:06:59,530 --> 01:07:03,890
+على F prime composite G ماشي الحال. الآن صارت عندي
+
+685
+01:07:03,890 --> 01:07:08,950
+هذه G prime composite composite F of X بساوي G prime F
+
+686
+01:07:08,950 --> 01:07:12,290
+of X بساوي واحد على F prime composite G أو Y
+
+687
+01:07:12,290 --> 01:07:15,470
+بياخد نظرية مرتين و بساوي واحد على F prime of X
+
+688
+01:07:15,470 --> 01:07:18,370
+التانيين مع بعض. G prime بساوي واحد على F prime
+
+689
+01:07:18,370 --> 01:07:24,310
+of X. و المرة الجاية اللي بنكمل بنعيد هذا اللي إذا
+
+690
+01:07:24,310 --> 01:07:26,470
+كان حكيناها على السريع
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..f63a72489568d62f6deecf3052c2484bc237f0a4
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_postprocess.srt
@@ -0,0 +1,2760 @@
+1
+00:00:04,820 --> 00:00:10,480
+بسم الله الرحمن الرحيم المحاضرة الثانية في مساق
+
+2
+00:00:10,480 --> 00:00:18,480
+تحليل رياضي 2 أو تحليل حقيقي 2 اتحدثنا
+
+3
+00:00:18,480 --> 00:00:24,100
+المرة الماضية عنالـ differentiation بدأنا بشرتر
+
+4
+00:00:24,100 --> 00:00:28,200
+ستة بدأنا في ال section الأول ستة واحد اللي هو ال
+
+5
+00:00:28,200 --> 00:00:32,600
+derivative او اللي هو الاشتقاق او المشتق الأولى
+
+6
+00:00:32,600 --> 00:00:37,840
+وعرفنا شو معناه اللي هو الاشتقاق عرفنا شو معناه ان
+
+7
+00:00:37,840 --> 00:00:44,910
+الدولة تكون differentiable عند نقطة اللي هيفي
+
+8
+00:00:44,910 --> 00:00:49,970
+interval I حيث أخدنا F عبارة عن الـ function من I
+
+9
+00:00:49,970 --> 00:00:55,230
+ل R وقلنا C element in I بتكون الدالة عندها قابلة
+
+10
+00:00:55,230 --> 00:01:01,490
+للاشتقاق يعني F prime of C exist if the following
+
+11
+00:01:01,490 --> 00:01:08,230
+limit exist اللي هو limitf of x ناقص f of c على x
+
+12
+00:01:08,230 --> 00:01:14,230
+minus c as x goes to c exist إذا كانت هذه الـ
+
+13
+00:01:14,230 --> 00:01:20,210
+limit exist بنقول عن الـ f برايب exist و بتكون هي
+
+14
+00:01:20,210 --> 00:01:26,650
+قيمة هذه ال limit الآن و بعد ذلك مرّينا على اللي
+
+15
+00:01:26,650 --> 00:01:34,600
+هو بعض الخواصوقلنا الخاصية الأولى عندنا اللى هى
+
+16
+00:01:34,600 --> 00:01:41,540
+انه لو كانت الدالة اللى هى differentiable at C
+
+17
+00:01:41,540 --> 00:01:45,520
+بتكون continuous برضه وين عند C يعني اي
+
+18
+00:01:45,520 --> 00:01:50,080
+differentiable function must be continuousقلنا إن
+
+19
+00:01:50,080 --> 00:01:53,960
+الـ converse need not to be true in general واخدنا
+
+20
+00:01:53,960 --> 00:01:58,020
+مثال قلنا f of x بتساوي ال absolute value لل x
+
+21
+00:01:58,020 --> 00:02:01,780
+اللي هي عبارة عن continuous function لكن مش
+
+22
+00:02:01,780 --> 00:02:05,460
+differentiable عند ال zero continuous everywhere
+
+23
+00:02:05,460 --> 00:02:09,340
+but it is not differentiable at zeroوبرهننا الكلام
+
+24
+00:02:09,340 --> 00:02:13,140
+هذا واخدنا مثال آخر على f of x بتساوي ال summation
+
+25
+00:02:13,140 --> 00:02:17,760
+ل x minus cii من عند واحد لعند n وقلنا إن هذا
+
+26
+00:02:17,760 --> 00:02:22,600
+الدالة برضه is continuous everywhere but it is not
+
+27
+00:02:22,600 --> 00:02:29,440
+differentiable at c1, c2, c3 لعند cn بعد ذلك سألنا
+
+28
+00:02:29,440 --> 00:02:34,100
+وقلنا هل ممكن اللي جينا دالة تكون continuous وين
+
+29
+00:02:34,100 --> 00:02:39,790
+ما كانو مش differentiable ولا في مكان قلنا اه و
+
+30
+00:02:39,790 --> 00:02:42,470
+قلنا اخد الدالة f of x بساعة ال summation واحد على
+
+31
+00:02:42,470 --> 00:02:47,350
+اتنين أس ان كسين تلاتة أس ان اكس هذه الدالة is
+
+32
+00:02:47,350 --> 00:02:51,850
+continuous everywhere but it is not differentiable
+
+33
+00:02:51,850 --> 00:02:56,310
+anywhere وبعد هيك اتينا
+
+34
+00:03:04,880 --> 00:03:11,100
+حكينا عن بعض اللي هو قوانين الاشتقاق قولنا لو كانت
+
+35
+00:03:11,100 --> 00:03:16,520
+الدالةF والدالة G التنتين من نفس الـ domain
+
+36
+00:03:16,520 --> 00:03:22,000
+والتنتين قابلات للاشتقاق عند نقطة اسمها C سمناها
+
+37
+00:03:22,000 --> 00:03:26,560
+قلنا اللي هو مجموع الدالتين هيكون مشتقتهم موجودة
+
+38
+00:03:26,560 --> 00:03:31,660
+عند هذه النقطة وحاصل ضربهم وحاصل قسمتهم أيضا
+
+39
+00:03:31,660 --> 00:03:35,860
+provided that G of C اللي في المقام لا تساوي سفر
+
+40
+00:03:35,860 --> 00:03:41,950
+وأيضا الالف في ال FA constant برضه ألف في أف قابلة
+
+41
+00:03:41,950 --> 00:03:46,290
+للاشتقاء وأثبتنا
+
+42
+00:03:46,290 --> 00:03:53,330
+هذا الكلام وبرهنناه ووصلنا لعند اللي هو التعميم
+
+43
+00:03:53,330 --> 00:04:00,030
+قلنا لو كانت FF1 وF2 وF3 عند F1 كلهم دوال قابلة
+
+44
+00:04:00,030 --> 00:04:04,710
+للاشتقاق حيكون حاصل جمحا قابل للاشتقاق وحاصل ضربا
+
+45
+00:04:04,710 --> 00:04:09,510
+أيضا قابل للاشتقاق وفق القوانين اللي ذكرناها سابقا
+
+46
+00:04:10,150 --> 00:04:14,370
+وجفنا عند اللي هو النقطة اللي هي the chain rule أو
+
+47
+00:04:14,370 --> 00:04:20,650
+قاعدة السلسلة قاعدة السلسلة طبعا ليست غريبة عنكم
+
+48
+00:04:20,650 --> 00:04:29,240
+مرت علينا في تفاضل الف ومرت علينا في ال complexأو
+
+49
+00:04:29,240 --> 00:04:32,780
+مرت علينا اللي هو استخدمناها في الـ differential
+
+50
+00:04:32,780 --> 00:04:38,220
+equations وفي مساقات أخرى الآن نظرة نظرة تحليلية
+
+51
+00:04:38,220 --> 00:04:44,060
+فهنشوف اللي هو كيف أثبتنا أو كيف بدنا نثبت اللي هو
+
+52
+00:04:44,060 --> 00:04:48,840
+قاعدة السلسلة أي سؤال بتقدر تسأله يعني اللي بدكم
+
+53
+00:04:48,840 --> 00:04:53,960
+إياه و تتفعلوا بالطريقة اللي بتلاقوها مناسبة let i
+
+54
+00:04:53,960 --> 00:04:56,200
+و j be intervals and let g
+
+55
+00:04:59,060 --> 00:05:06,800
+عبارة عن دالة دي
+
+56
+00:05:06,800 --> 00:05:14,660
+عبارة عن دالة من I لعند مين لعند R و ال F عبارة عن
+
+57
+00:05:14,660 --> 00:05:21,700
+دالة من عند J لعند R حيث I و J اللي هو أشمل عبارة
+
+58
+00:05:21,700 --> 00:05:26,440
+عن intervalsالان انا اللي هي الـ chain rule هي في
+
+59
+00:05:26,440 --> 00:05:33,380
+الواقع عملية فحص إمكانية إنها تكون ال composition
+
+60
+00:05:33,380 --> 00:05:37,780
+of two functions is differentiable ولا لأ؟ وإيش
+
+61
+00:05:37,780 --> 00:05:41,200
+الشروط اللي بتخلي اللي هو g composite of is
+
+62
+00:05:41,200 --> 00:05:45,620
+differentiable؟ إذا أنا بدي أحكي عن g composite of
+
+63
+00:05:45,620 --> 00:05:49,590
+اللي هو ال derivative إيه لها؟ما دام بدي احكي عن g
+
+64
+00:05:49,590 --> 00:05:56,690
+composite of f اذا انا الان domain ال g ال domain
+
+65
+00:05:56,690 --> 00:06:03,850
+ال f هيكون من عند g لعند مين لعند r معناته ان f of
+
+66
+00:06:03,850 --> 00:06:09,310
+g عشان تكون g معرفة لازم تكون من ضمن مين اللي هو
+
+67
+00:06:09,310 --> 00:06:16,680
+domain ال I اذا انا بدي افترض ان fof J جزئية من I
+
+68
+00:06:16,680 --> 00:06:22,020
+على أساس ان هتكون الـ G composite F is defined
+
+69
+00:06:22,020 --> 00:06:29,960
+وهتكون defined من وين؟ من عند J لعند R عشان هيك
+
+70
+00:06:29,960 --> 00:06:34,400
+احنا لما بدنا نيجياللي هو .. اللي نفترض .. هنفترض
+
+71
+00:06:34,400 --> 00:06:40,220
+G من I لعند R وF من عند G لعند R دوال ونقول F of I
+
+72
+00:06:40,220 --> 00:06:43,640
+subset او F of G subset من I عشان تصير اللي هو
+
+73
+00:06:43,640 --> 00:06:46,980
+اللي بنحكي عنه قاعدة السلسلية اللي هي بمعنى اخر
+
+74
+00:06:46,980 --> 00:06:50,040
+التفاضل ال composition of two functions is well
+
+75
+00:06:50,040 --> 00:06:54,420
+defined او is defined اللي هم let C element in مين
+
+76
+00:06:54,420 --> 00:07:00,120
+in G؟ بدنا ناخد الـ C element in G اللي هم بقولي C
+
+77
+00:07:00,120 --> 00:07:06,430
+element in Gعشان أقدر أحكي عن g composite of f
+
+78
+00:07:06,430 --> 00:07:12,530
+prime of c تكون exist بده شرطين الشرط الأول بده ال
+
+79
+00:07:12,530 --> 00:07:17,190
+g ال f تكون differentiable عند مين؟ عند ال c
+
+80
+00:07:17,190 --> 00:07:22,150
+والشرط التاني أن ال g تكون differentiable عند ال f
+
+81
+00:07:22,150 --> 00:07:30,530
+of c واضح؟ إذن الآنإذا كان الـ F مميز لـ C و الـ G
+
+82
+00:07:30,530 --> 00:07:36,710
+مميز لـ F of C فإن مفهوم الـ G مميز لـ F يجب أن
+
+83
+00:07:36,710 --> 00:07:42,410
+يكون مميز لـ C وبكون عنده الـ G مميز لـ F' of C
+
+84
+00:07:42,410 --> 00:07:48,710
+أيش حيساوي؟ حيساوي G' of F of C في F' لمن؟ للـ F
+
+85
+00:07:48,710 --> 00:07:54,350
+of C اللي كنا نقول عنه في التفاضلأنه تفاضل اللي هو
+
+86
+00:07:54,350 --> 00:08:01,320
+القوس ثم تفاضل ما فيه داخل القوس واضح؟أو تفاضل
+
+87
+00:08:01,320 --> 00:08:05,280
+الدالة في تفاضل ما بداخل الدالة هو طبعا في الواقع
+
+88
+00:08:05,280 --> 00:08:07,940
+اللي هو عبارة عن تفاضل ل composite of two
+
+89
+00:08:07,940 --> 00:08:12,200
+functions أو اللي هو ال chain rule وكنا إذا
+
+90
+00:08:12,200 --> 00:08:16,680
+بتتذكروا كانت صورة أخرى لل chain rule غير الصورة
+
+91
+00:08:16,680 --> 00:08:23,100
+هذه اللي هي g prime of f of c في f prime of c كنا
+
+92
+00:08:23,100 --> 00:08:29,900
+ناخد صورة أخرى نقول dy by dxاللي هو عبارة عن ..
+
+93
+00:08:29,900 --> 00:08:38,960
+عبارة عن dy مثلا by du في du by d إيش؟ dx اللي هو
+
+94
+00:08:38,960 --> 00:08:44,620
+ال y و كأنها بتتمثل اللي هي ال g تفاضل ال g و ال u
+
+95
+00:08:44,620 --> 00:08:48,600
+بتكون اللي هي مين؟ اللي هي ال F اللي هي F إبراى
+
+96
+00:08:48,600 --> 00:08:52,180
+فكانت اللي هي هذه الصورة أو هذه الصورة احنا الآن
+
+97
+00:08:52,180 --> 00:08:59,630
+خلينا نحكي عن هذه الصورة و نبرهن اللي هيكيف اللي
+
+98
+00:08:59,630 --> 00:09:07,290
+هو نثبت ان هذا الكلام صحيح الان بس خلّينا نحط ..
+
+99
+00:09:07,290 --> 00:09:13,730
+نشوف .. نشوف عندنا الان نشوف ايش اللي بدنا نثبته
+
+100
+00:09:13,730 --> 00:09:16,770
+احنا في الواقع اللي بدنا نثبته .. بدنا نجوه ..
+
+101
+00:09:16,770 --> 00:09:22,290
+بدنا نثبت ان G composite F a prime of C exist عشان
+
+102
+00:09:22,290 --> 00:09:29,500
+نثبت انها exist بإيش بتبلزمني؟ بدي اخد limitG of
+
+103
+00:09:29,500 --> 00:09:39,520
+F, G circle F اللي هي GF of X ناقص G of F of C على
+
+104
+00:09:39,520 --> 00:09:46,960
+X minus C لما X تروح للـ C نثبت أن هذا إش ماله
+
+105
+00:09:46,960 --> 00:09:51,920
+existواضح عشان اللي بدنا نسويه؟ بدنا نثبت انه
+
+106
+00:09:51,920 --> 00:09:55,800
+exist عشان نثبت انه هذا exist بلزمني عاملين
+
+107
+00:09:55,800 --> 00:10:01,020
+مساعدات الآن هنشوف كيف نعمل هذول العاملين طبعا
+
+108
+00:10:01,020 --> 00:10:05,200
+طبيعي طبيعي انا معطين ان هو ال F' موجودة وD'
+
+109
+00:10:05,480 --> 00:10:10,600
+موجودة اذا اكيد انا هعمل حسابي ان استخدمهم على
+
+110
+00:10:10,600 --> 00:10:14,280
+اساس ان اصل الهدف اللي بديه الان خلينا نيجي
+
+111
+00:10:14,280 --> 00:10:14,900
+للبرهان
+
+112
+00:10:20,860 --> 00:10:26,680
+الحين الـ proof بدنا
+
+113
+00:10:26,680 --> 00:10:36,080
+نثبت انه اللي هو الـ Gبنثبت إنه لو كانت G و F
+
+114
+00:10:36,080 --> 00:10:39,980
+differentiable الأول عند F of C والتاني عند C
+
+115
+00:10:39,980 --> 00:10:45,840
+بيعطينا هذا إنه G composed of F prime of C exist
+
+116
+00:10:45,840 --> 00:10:50,420
+ويساوي G prime of F of C في F prime of C هذا ما
+
+117
+00:10:50,420 --> 00:10:58,000
+بنثبته شوف كيف يعني
+
+118
+00:10:58,000 --> 00:11:06,450
+بقى بدنا نبرهن النظريةخلّيني أعرّف الآن Define G
+
+119
+00:11:06,450 --> 00:11:14,850
+Capital من I لعند ال R By شوف كيف بدي .. اللي هي
+
+120
+00:11:14,850 --> 00:11:21,050
+يعني أعرف الـ G بطريقة توصّلني لحساب D composed of
+
+121
+00:11:21,050 --> 00:11:24,890
+F prime of إيش؟ of C جبل بشوية قلنا D composed of
+
+122
+00:11:24,890 --> 00:11:30,590
+F prime of C المتوقع لهاإنه تكون عبارة عن إذا كانت
+
+123
+00:11:30,590 --> 00:11:34,850
+تكون موجودة هي عبارة عن ال limit ل G F of X نقص G
+
+124
+00:11:34,850 --> 00:11:41,270
+F of C على X minus C أو G of Y نقص G F of C ماشي
+
+125
+00:11:41,270 --> 00:11:47,250
+الحال الآن بدي أعرف G على هذا الأساس G of Y بتساوي
+
+126
+00:11:47,250 --> 00:12:01,160
+اللي هو G of F of Yالعند ناقص g f of c على y minus
+
+127
+00:12:01,160 --> 00:12:06,480
+f of c الان قسمت على y minus f of c اكيد اللي هو
+
+128
+00:12:06,480 --> 00:12:15,600
+ال y انا بديش ايها تساومين f of c الان g of y ايش
+
+129
+00:12:15,600 --> 00:12:23,980
+بديها انا بديها المتوقع ال limit لهذهالـ limit
+
+130
+00:12:23,980 --> 00:12:30,300
+لهذه لما y تروح لل F of C اللي هو و كأنه عبارة عن
+
+131
+00:12:30,300 --> 00:12:37,540
+D prime of مين of F of C فالان بتديها D prime of F
+
+132
+00:12:37,540 --> 00:12:44,180
+of C ليش هال gate بتشوف ليش اللي بنحكيهاالان عرفت
+
+133
+00:12:44,180 --> 00:12:49,140
+اللي هي الـ function طبعاً هذا الكلام مستقى من
+
+134
+00:12:49,140 --> 00:12:54,120
+اللي هو f prime of c إيش متوقع limitه تساوي هذا
+
+135
+00:12:54,120 --> 00:12:58,200
+مستقى .. مستقى منه بعد شوية هتلاقيني هذه كتبت على
+
+136
+00:12:58,200 --> 00:13:02,980
+صورة هذه في شغل تانية برضه موجودة اللي هي f prime
+
+137
+00:13:02,980 --> 00:13:07,260
+of c اه
+
+138
+00:13:07,260 --> 00:13:16,350
+أسف g of yالان نشوف هذه الدالة لو جيت للدالة اللى
+
+139
+00:13:16,350 --> 00:13:26,010
+عندى هذه خد limit g of y لما y تروح لل F of C ايش
+
+140
+00:13:26,010 --> 00:13:32,500
+هتلاقيها بتساوي؟ بتساوي limitG of Y بعوض جاعد Y
+
+141
+00:13:32,500 --> 00:13:38,080
+تروح لل F of C يعني Y أشمالها لاتساوي F of C ناقص
+
+142
+00:13:38,080 --> 00:13:45,340
+G of F of C على مين على Y minus F of C لما Y تروح
+
+143
+00:13:45,340 --> 00:13:51,560
+لمين لل F of C وهذه بالظبط هي تعريف مينتعريف الـ
+
+144
+00:13:51,560 --> 00:13:56,480
+derivative للـ G عند النقطة مين؟ النقطة اسمها F of
+
+145
+00:13:56,480 --> 00:14:01,280
+C واضح؟ لأن الـ C من الـ A .. من الـ J الـ F of C
+
+146
+00:14:01,280 --> 00:14:08,260
+وين هتكون؟ من مين؟ من الـ I واضح؟ فالان هذه عبارة
+
+147
+00:14:08,260 --> 00:14:16,920
+عن G prime للـ F of C ليش هذا؟ because ..G is
+
+148
+00:14:16,920 --> 00:14:25,200
+differentiable at mean at F of C هذا معطى طيب الآن
+
+149
+00:14:25,200 --> 00:14:30,160
+ال limit لـ G of Y طلعت لما يتروح ل F of C أش طلعت
+
+150
+00:14:30,160 --> 00:14:34,520
+G prime ل F of C أصلا هذا السبب اللي أنا عارف أنه
+
+151
+00:14:34,520 --> 00:14:37,440
+هتطلع حاجة السبب اللي خلاني أعرف الـ G of Y أش
+
+152
+00:14:37,440 --> 00:14:44,510
+متساوي G prime F of C عشان تطلعلي منه هذا بساويG
+
+153
+00:14:44,510 --> 00:14:52,510
+عند مين عند Y بتساوي F of C يعني هذا سبب لإنه الـ
+
+154
+00:14:52,510 --> 00:14:57,610
+G هذه تصير إيش ماله is continuous عند مين عند ال F
+
+155
+00:14:57,610 --> 00:15:07,490
+of C إذا hence G is continuous at F of C طب ليش
+
+156
+00:15:07,490 --> 00:15:15,570
+هذا بعد شوية هنشوف وين هنستخدمهاالآن limit g of y
+
+157
+00:15:15,570 --> 00:15:22,890
+طالع عندي بيساوي g of f of c اللي هي هذه g of f of
+
+158
+00:15:22,890 --> 00:15:29,550
+c بيساوي g برايم f of c نيجي الآن انا ايش اللي بدي
+
+159
+00:15:29,550 --> 00:15:35,910
+ا .. بدي الآن اثبتلك ان g composite f برايم of c
+
+160
+00:15:35,910 --> 00:15:48,970
+بيساوي limit gof F of X ناقص G of F of C على X
+
+161
+00:15:48,970 --> 00:15:55,410
+minus C لما X تروح لل C بدي أثبت أنه exist بظبط؟
+
+162
+00:15:55,410 --> 00:16:02,610
+واضح هاه؟ طيب بناءً عليه خليني أكتب هذه بدلالة إشي
+
+163
+00:16:02,610 --> 00:16:11,060
+بوصلني للي بديها شوف كيفالان نحسب هذه هذه لمين for
+
+164
+00:16:11,060 --> 00:16:22,260
+x element n اللي هي j أو x لا تساوي مين لا تساوي c
+
+165
+00:16:22,260 --> 00:16:30,120
+معايا اه الان احسبولي هذه هذه عبارة عن تساوي g of
+
+166
+00:16:30,120 --> 00:16:37,650
+f of x اللي هونقص g f of c بدي اكتبها بطريقة g f
+
+167
+00:16:37,650 --> 00:16:49,250
+of x نقص g of f of c على f of x نقص f of c في f of
+
+168
+00:16:49,250 --> 00:16:58,630
+x نقص f of c على x minus cدخلت هذا المقدار في الـ
+
+169
+00:16:58,630 --> 00:17:04,410
+bust و هنا في المقام اللي هو على أساس ان احصل على
+
+170
+00:17:04,410 --> 00:17:10,530
+هذه مضروبة في هذه اللي انا عندي معلومات عن هذه و
+
+171
+00:17:10,530 --> 00:17:20,290
+عن هذه واضح؟ اي سؤال؟ و يساوي؟ الان هذا كله على
+
+172
+00:17:20,290 --> 00:17:33,910
+بعضه مين هو عبارة عن جيcapital of F of X اه F
+
+173
+00:17:33,910 --> 00:17:40,510
+of X اللي هي G of Y بدل ال Y ايش اسمها F of X واضح
+
+174
+00:17:40,510 --> 00:17:46,790
+مضروبة في مين هذا المقدار طلع هذا مضروب في F of X
+
+175
+00:17:48,050 --> 00:17:54,710
+نقص f of c على x minus c خد الآن ال limit للجهتين
+
+176
+00:17:54,710 --> 00:18:01,550
+ال limit للجهتين بيصير ال limit لهذا المقدار ال
+
+177
+00:18:01,550 --> 00:18:09,430
+limit لهذا المقدار معايا يا شباب لما x تروح لل c
+
+178
+00:18:09,430 --> 00:18:18,040
+بساوي خلينا أمسح هان بهمشبعد اذنكم اثبتنا احنا
+
+179
+00:18:18,040 --> 00:18:22,940
+الـG كابيتال continuous خليني انجل هذا الكلام فوق
+
+180
+00:18:22,940 --> 00:18:32,700
+اذا صارت عندي limit الـG of F of X ناقص G of F of
+
+181
+00:18:32,700 --> 00:18:42,160
+C على X minus C لما X تروح للـC بساوي limit اللي
+
+182
+00:18:42,160 --> 00:18:54,190
+هو G of F of Xلما X تروح للـ C أه فيه limit اللي
+
+183
+00:18:54,190 --> 00:19:00,710
+هو F of X ناقص F of C على X minus C لما X تروح
+
+184
+00:19:00,710 --> 00:19:07,820
+لمين للـ Cطبعا طبيعي لما X تروح لل C ال F of X
+
+185
+00:19:07,820 --> 00:19:13,660
+نفسها هتروح لمين؟ لل F of C، مظبوط؟ لما X تروح لل
+
+186
+00:19:13,660 --> 00:19:18,960
+C، ال F of X هتروح لمين؟ لل F of C، لإن أصلا F is
+
+187
+00:19:18,960 --> 00:19:23,240
+differential، بالإذن continuous، معايا؟ طيب، إذن
+
+188
+00:19:23,240 --> 00:19:29,380
+الآن عندي بدي أستخدم ال continuity لمين؟ لل G،
+
+189
+00:19:29,380 --> 00:19:35,050
+limit G of F of Cلما .. لو سمينا هذه كلها على بعض
+
+190
+00:19:35,050 --> 00:19:39,770
+Y قلنا لما X تروح لل C ال Y اللي هي F of X بتروح
+
+191
+00:19:39,770 --> 00:19:45,430
+لمين؟ لل F of C بتصير هذه و كأنها اسمها G of Y و
+
+192
+00:19:45,430 --> 00:19:52,190
+بدل هذه بنحط Y بتروح لمين؟ لل F of C والـ G
+
+193
+00:19:52,190 --> 00:19:56,450
+capital is continuous عند ال F of C إذا على طول
+
+194
+00:19:56,450 --> 00:20:05,010
+limitها بتساوي G of F of Cمظبوط؟ مضروبة في مين؟ في
+
+195
+00:20:05,010 --> 00:20:08,010
+limit F of X نقص F of C على X minus C لما X تروح
+
+196
+00:20:08,010 --> 00:20:14,190
+لمين؟ لـC ليها دي تعريف مين المشتقة الأولى للـF
+
+197
+00:20:14,190 --> 00:20:19,310
+عند الـC نرجع الآن G of F of C إيش بتساوي حسب
+
+198
+00:20:19,310 --> 00:20:22,610
+التعريف الأولاني؟ بساوي G prime
+
+199
+00:20:26,140 --> 00:20:36,180
+للـ F of C بدل هذه في F prime of C وهو المطلوب أي
+
+200
+00:20:36,180 --> 00:20:41,000
+سؤال؟
+
+201
+00:20:41,000 --> 00:20:46,740
+لذالك الآن أثبتنا أنه لو كانت F is differentiable
+
+202
+00:20:46,740 --> 00:20:53,080
+أو G is differentiableو يكون الـ composite الهن is
+
+203
+00:20:53,080 --> 00:20:57,900
+differentiable طبعا ما أخد بعين الاعتبار انه يكون
+
+204
+00:20:57,900 --> 00:21:00,600
+جي كمبوسيت F is defined
+
+205
+00:21:16,030 --> 00:21:25,010
+في عنا اللي هو المثال الأول مثال
+
+206
+00:21:25,010 --> 00:21:30,750
+يعني بسيط كلنا بنعرف انفاضه بس بنشوف كيف احنا
+
+207
+00:21:30,750 --> 00:21:35,530
+بنستخدم ال .. ال .. ال .. ال chain rule الصورة
+
+208
+00:21:35,530 --> 00:21:39,470
+اللي احنا كتبناها في اللي هو ايجاد ال derivative
+
+209
+00:21:39,470 --> 00:21:46,520
+جاي اللي عند في example جاي اللي خدExample جاي لنت
+
+210
+00:21:46,520 --> 00:21:56,300
+F من عند I لعند R بيه a differentiable function من
+
+211
+00:21:56,300 --> 00:22:00,080
+هنا نفترض ان ال F أشبه لها يا جماعة عبارة عندها
+
+212
+00:22:00,080 --> 00:22:05,860
+لقابلة للاشتقاق عند نقطة وين على ال I أو عند ال I
+
+213
+00:22:05,860 --> 00:22:13,230
+كلها مش مشكلة ماشي الحال بقولك اثبتليبس قبل ما
+
+214
+00:22:13,230 --> 00:22:20,750
+بتخلينا نقل تقول احنا بتوجدولنا الـ Fn لان في عندي
+
+215
+00:22:20,750 --> 00:22:24,890
+دالة اسمها Fn يعني F مرفوعة للقوة انا بدي اوجدلها
+
+216
+00:22:24,890 --> 00:22:30,140
+ال derivative طبعا الـ Fn كله prime of Xمن
+
+217
+00:22:30,140 --> 00:22:36,040
+معلوماتنا في الـ Calculus كنا نقول أن تفاضل القوس
+
+218
+00:22:36,040 --> 00:22:41,560
+كله نهو N في F N ناقص واحد في تفاضل ما بداخل القوس
+
+219
+00:22:41,560 --> 00:22:47,990
+اللي هو عبارة عن F prime of Xهذا تفضل الآن بدنا
+
+220
+00:22:47,990 --> 00:22:54,650
+نبرهن هذا الكلام باستخدام نظرية الـ chain rule
+
+221
+00:22:54,650 --> 00:22:58,050
+يعني هذا شغل المكان زي ما بقوله الآن بدنا نشوف كيف
+
+222
+00:22:58,050 --> 00:23:01,670
+بتشتغل المكينة أو ايش اللي شغل المكينة طيب ال
+
+223
+00:23:01,670 --> 00:23:13,110
+proof ال proof لندن كده بدي اعرف function let g
+
+224
+00:23:13,110 --> 00:23:23,530
+fromsome interval J لعند R defined by G يعني أصلا
+
+225
+00:23:23,530 --> 00:23:30,330
+ممكن أعرفها من R حتى G of Y بيسوا Y أساش N element
+
+226
+00:23:30,330 --> 00:23:35,070
+in N يعني حتى لو أعرفتها من R مابعرفها الدالة
+
+227
+00:23:35,070 --> 00:23:46,200
+معايا الآن بناء على هيك صار عندي الآن اللي هوالـ G
+
+228
+00:23:46,200 --> 00:23:56,700
+composite F of X إيش هي هتطلع؟ G of F of X يعني
+
+229
+00:23:56,700 --> 00:24:05,340
+هتطلع FN of X صح ولا لأ؟ إذن هاي دالتنا اللي بدنا
+
+230
+00:24:05,340 --> 00:24:12,000
+نفضلها كتبتها على صورة مين؟ G composite Fوالـ F of
+
+231
+00:24:12,000 --> 00:24:17,520
+X الـ F of X اللي هي الـ F of X من وين جاي بترمي
+
+232
+00:24:17,520 --> 00:24:22,680
+في الـ R يعني بترمي في مين؟ في domain الـ G واضح
+
+233
+00:24:22,680 --> 00:24:27,840
+أهإذا كل أموري تمام وهذه الدالة الدالة الأسية is
+
+234
+00:24:27,840 --> 00:24:30,980
+differentiable وهو جاي اللي أن ال F is
+
+235
+00:24:30,980 --> 00:24:33,780
+differentiable إذا جي كومبوزيت F حسب ال channel
+
+236
+00:24:33,780 --> 00:24:38,140
+أشمالها is differentiable و ال derivative لها جي
+
+237
+00:24:38,140 --> 00:24:42,900
+كومبوزيت F prime of X حسب ال channel اللي حكيناها
+
+238
+00:24:42,900 --> 00:24:49,560
+اللي برهنناها بساوت G prime of F of Xفي f prime of
+
+239
+00:24:49,560 --> 00:24:53,580
+x وي ساون and g of y بيساوي y أس 2 ال derivative
+
+240
+00:24:53,580 --> 00:24:59,420
+الهنين هي and y أس n ناقص واحد يعني and y أس n
+
+241
+00:24:59,420 --> 00:25:04,220
+ناقص واحد في f prime of x لكن احنا بنفاضل عند مين
+
+242
+00:25:04,220 --> 00:25:08,860
+عند f of x فبنشيل ال y بنحكم مكانها قيمتها اللي هي
+
+243
+00:25:08,860 --> 00:25:14,240
+f of x فبتطلع and f of x n ناقص واحد في f prime of
+
+244
+00:25:14,240 --> 00:25:19,610
+xهذه اللي هي اللي احنا بنشتغلها وبدون ما .. طبعا
+
+245
+00:25:19,610 --> 00:25:24,930
+احنا هذا تشريع لاستخدام اللي هو Kedemstrain rule
+
+246
+00:25:24,930 --> 00:25:28,150
+it's اللي احنا بنعرفها زي ما نيجي لما كنا نقولكم
+
+247
+00:25:28,150 --> 00:25:34,810
+مثلا اوجد ال derivative y لعندك y بالساوى sin x
+
+248
+00:25:34,810 --> 00:25:39,460
+تكعيب مثلاكنتش تسوي على طول ماتيش ايش تقول والله
+
+249
+00:25:39,460 --> 00:25:44,040
+بيضاف رضاف of X بيساوي X تتعيب و G of Y تساوي Sin
+
+250
+00:25:44,040 --> 00:25:47,600
+Y و تقول اذا صارت هذه عبارة عن composite of two
+
+251
+00:25:47,600 --> 00:25:53,240
+functions خلاص بنتجاوز المرحلة هذهوعرفنا وين أصل
+
+252
+00:25:53,240 --> 00:25:58,160
+الأمر اللى بيقوللنا why prime بتساوي تفاضل اللى
+
+253
+00:25:58,160 --> 00:26:03,760
+برا اللى هو cosine x كعيب فى تفاضل اللى جوا تلاتة
+
+254
+00:26:03,760 --> 00:26:08,100
+x تربية هذا فى الواقع هو عبارة عن استخدام mean لل
+
+255
+00:26:08,100 --> 00:26:19,080
+chain rule واضح أي سؤال؟ طيب هذا عندنا فى عندنا
+
+256
+00:26:19,080 --> 00:26:31,850
+مثال آخرالمثال .. نعم المثال اللي هو .. يمكن مرشك
+
+257
+00:26:31,850 --> 00:26:36,330
+للمثال حتى معاكم في تحليل واحد لكن كان هدفه اللي
+
+258
+00:26:36,330 --> 00:26:42,910
+هو في حينه ال continuity اذا اعطيك F of X بتساوي
+
+259
+00:26:42,910 --> 00:26:52,400
+هاي B مثال B F of X بتساوي اللي هو X تربيعSin 1
+
+260
+00:26:52,400 --> 00:26:59,800
+على X لما X لا تساوي سفر أو سفر لما X ايش بتساوي
+
+261
+00:26:59,800 --> 00:27:07,140
+سفر الان بيقول لي بيعطيني طبعا هو بيسأل سؤال جابله
+
+262
+00:27:07,140 --> 00:27:10,640
+تعليق سؤال في الكتاب بيقول لي لو كانت ال F is
+
+263
+00:27:10,640 --> 00:27:14,040
+differentiable ركز معاه لو كانت ال F is
+
+264
+00:27:14,040 --> 00:27:17,800
+differentiable الشرط ان ال derivative تماما تطلع
+
+265
+00:27:17,800 --> 00:27:23,590
+continuousهذا ليس شرطًا، هذا بس وينفي الـ R لما
+
+266
+00:27:23,590 --> 00:27:26,430
+نحكي عن الهو نحكي عن الـ real valued functions
+
+267
+00:27:26,430 --> 00:27:33,210
+from R to R كمان لكن في ال complex الأمر بيختلف ال
+
+268
+00:27:33,210 --> 00:27:36,610
+complex أو ال derivative مرة لجيتها معناته اللي
+
+269
+00:27:36,610 --> 00:27:39,210
+بعيدها و اللي بعيدها و اللي بعيدها هتظلها إشمالها
+
+270
+00:27:39,210 --> 00:27:43,830
+موجودة يعني if F prime exists then F double prime
+
+271
+00:27:43,830 --> 00:27:48,630
+and F triple F كذا إذا هتطلع مجرد ما تكون الدالة
+
+272
+00:27:48,630 --> 00:27:53,230
+analytic أو differentiableهتظلها على طول
+
+273
+00:27:53,230 --> 00:27:56,750
+differentiable و continuous و و و الاخرين هذا وين
+
+274
+00:27:56,750 --> 00:28:01,890
+في ال complex أما في ال real الوضع ليس كذلك وهي
+
+275
+00:28:01,890 --> 00:28:07,970
+مثال على F is differentiable everywhere طبعا is
+
+276
+00:28:07,970 --> 00:28:11,310
+differentiable عنده نقطة zero بالذات لكن its
+
+277
+00:28:11,310 --> 00:28:15,450
+derivative is not continuous at this point باضح
+
+278
+00:28:15,450 --> 00:28:20,330
+اليوم اه طيب الآن show
+
+279
+00:28:23,010 --> 00:28:30,750
+that F prime of X exists everywhere
+
+280
+00:28:30,750 --> 00:28:41,270
+but F prime is not continuous at zero و proof
+
+281
+00:28:41,270 --> 00:28:45,470
+البرهن .. طبعا البرهن كامل موجود هنا بس خلينا
+
+282
+00:28:45,470 --> 00:28:50,670
+نبرهن لك .. نوضح إيش ال .. إيش البرهنشوفوا صلى
+
+283
+00:28:50,670 --> 00:28:56,670
+الله عليه الصلاة والسلام الان f prime exist طبيعي
+
+284
+00:28:56,670 --> 00:29:03,310
+for x لا يساوي 0 مقابر سهل مجرد تفاضل عادي f prime
+
+285
+00:29:03,310 --> 00:29:07,290
+of x لأن هذا الدلة قابلة للاشتقاء عند كل النقطة ما
+
+286
+00:29:07,290 --> 00:29:11,430
+عدا عند ال x بالساوية 0 هدى بحكي عن هدى لحالها f
+
+287
+00:29:11,430 --> 00:29:19,020
+prime of x بالساوية2x sin 1 على x لأن x تربيع في
+
+288
+00:29:19,020 --> 00:29:24,360
+sin تفاضل الـ sin cos و تفاضل اللي جوا سالب x
+
+289
+00:29:24,360 --> 00:29:30,440
+تربيع x minus 2 بتروح حد ما حد فبصير سالب cos 1
+
+290
+00:29:30,440 --> 00:29:36,480
+على xالان هذا for x لا تساوي صفر طب for x متساوي
+
+291
+00:29:36,480 --> 00:29:39,660
+صفر for x متساوي صفر الوضع اللي لان بختلف اللي هو
+
+292
+00:29:39,660 --> 00:29:43,880
+فيه عندي اللي هو الدالة أصلا مش معرفة عند ال zero
+
+293
+00:29:43,880 --> 00:29:47,620
+لكن معرفة الدالة ككل عند ال zero عند مين بتساوي
+
+294
+00:29:47,620 --> 00:29:52,060
+صفر فمضطر أنا أستخدم التعريف لإثبات أو فحص إذا
+
+295
+00:29:52,060 --> 00:29:56,680
+كانت الدالة إلها مشتقة عند الصفر ولا لأ فباجي بقول
+
+296
+00:29:56,680 --> 00:30:03,810
+الان بتدا فحص limitاللي هو F of X ناقص F of Zero
+
+297
+00:30:03,810 --> 00:30:08,910
+على X minus Zero as X بتروح للزيرو بدي أشوف هل هذه
+
+298
+00:30:08,910 --> 00:30:13,250
+ال limit موجودة ولا مش موجودة ماشي الحال Y ساوي
+
+299
+00:30:13,250 --> 00:30:16,930
+هذا
+
+300
+00:30:16,930 --> 00:30:20,370
+عشان خاطر أوجد مين أشوف ال F prime عند ال Zero
+
+301
+00:30:20,370 --> 00:30:25,630
+موجودة أو لا بساوي limitF of X مين هي عبارة عن X
+
+302
+00:30:25,630 --> 00:30:29,030
+تربيع .. لأن X بتروح للزيرو، لأن X مش سفر، اللي هي
+
+303
+00:30:29,030 --> 00:30:34,450
+عبارة عن X تربيع صين واحد على X نقص F of Zero اللي
+
+304
+00:30:34,450 --> 00:30:39,310
+هي عبارة عن سفر على X minus سفر لما X تروح لمين
+
+305
+00:30:39,310 --> 00:30:46,870
+للسفر ويساوي limitx term x sin 1 على x لما x تروح
+
+306
+00:30:46,870 --> 00:30:50,670
+لمين للصفر هذه اللي هي by sandwich theorem حلقيها
+
+307
+00:30:50,670 --> 00:30:56,650
+إيش بتساوي سفر كيف by sandwich theorem عندك x sin
+
+308
+00:30:56,650 --> 00:31:00,470
+1 على x absolute value أصغر أو ساوي ال absolute
+
+309
+00:31:00,470 --> 00:31:05,050
+value لل x وهذه الطبيعة أكبر أو ساوي مين سفر as x
+
+310
+00:31:05,050 --> 00:31:07,870
+goes to zero هذه بتروح للزيرو وهذه بتروح للزيرو
+
+311
+00:31:07,870 --> 00:31:10,570
+إذا اللي في النص هذه لازم ال limit هيروح لمين
+
+312
+00:31:10,570 --> 00:31:15,840
+للصفرومن ثم لل continuity لل .. بسبب ال continuity
+
+313
+00:31:15,840 --> 00:31:18,040
+لل .. لل absolute value بتطلع ال limit اللي جوا
+
+314
+00:31:18,040 --> 00:31:20,740
+برضه أشمالها بتساوي 0 لأن ال limit ال absolute
+
+315
+00:31:20,740 --> 00:31:23,380
+value بيساوي absolute value لل limit إذا صار عندى
+
+316
+00:31:23,380 --> 00:31:32,120
+limit اللي جوا بيساوي 0 معايا؟ طيب
+
+317
+00:31:32,120 --> 00:31:35,940
+بما أنه ال limit هذه موجودة بتساوي 0 إذا بنقول
+
+318
+00:31:35,940 --> 00:31:42,580
+بعيدها since كده بيساوي كده thenF prime of 0
+
+319
+00:31:42,580 --> 00:31:50,520
+بتساوي صفر إذا اللي أثبتنا إحنا أنه F prime exist
+
+320
+00:31:50,520 --> 00:31:55,460
+وين ما كان وإيش ال F prime اللي أثبتناها كمان
+
+321
+00:31:55,460 --> 00:32:01,160
+أثبتنا إنه الآن F prime of X أثبتناها إنها بتساوي
+
+322
+00:32:02,250 --> 00:32:08,870
+2x sin 1 على x نقص cos 1 على x لما x ايش معناه لأ
+
+323
+00:32:08,870 --> 00:32:13,070
+تساوي سفر واتبعتناها بتساوي سفر لما x بتساوي سفر
+
+324
+00:32:13,070 --> 00:32:16,990
+إذا صارت ال F is functional everywhere الآن بدنا
+
+325
+00:32:16,990 --> 00:32:23,350
+نثبت إنه is not this F prime is not continuous at
+
+326
+00:32:23,350 --> 00:32:27,150
+zero شوف
+
+327
+00:32:27,150 --> 00:32:29,410
+الآن عندي
+
+328
+00:32:31,690 --> 00:32:41,570
+بدي أشوف limit اللي هي 2x sin 1 على x ناقصه sin 1
+
+329
+00:32:41,570 --> 00:32:49,080
+على x لما x تروح للزيرو exist ولا لأالان limit ال
+
+330
+00:32:49,080 --> 00:32:53,540
+أولى 2x sin 1 على x مضمون انه exist قبل مش
+
+331
+00:32:53,540 --> 00:32:57,560
+ويعملناها نفسها باي اللي هو sandwich theorem هتطلع
+
+332
+00:32:57,560 --> 00:33:03,720
+سفر اذا الان اول حاجة اكتب لها since الان limit
+
+333
+00:33:03,720 --> 00:33:09,420
+sin 1 على x بساوي سفر exist الان
+
+334
+00:33:09,420 --> 00:33:19,480
+but butlimit cos 1 على x لما x تروح لصفر does not
+
+335
+00:33:19,480 --> 00:33:24,360
+exist وهلجت بنشوف ليش does not exist but this
+
+336
+00:33:24,360 --> 00:33:26,140
+limit اشمالها does not exist
+
+337
+00:33:28,950 --> 00:33:33,830
+ما دام هذا exist و هذا does not exist حاصل جمح أو
+
+338
+00:33:33,830 --> 00:33:38,290
+طرح أكيد ايش ماله does not exist لأنه لو exist
+
+339
+00:33:38,290 --> 00:33:42,810
+معناه تصير هذا اللي يطلع و هذا exist بيصير مجموح
+
+340
+00:33:42,810 --> 00:33:45,170
+أو طرح يساوي هذا ايضا صار هذا exist و هذا
+
+341
+00:33:45,170 --> 00:33:51,550
+contradiction فاهمين اه الان hence limit اتنين x
+
+342
+00:33:51,550 --> 00:33:56,410
+sin واحد على x نقص cosine واحد على x لما x تروح
+
+343
+00:33:56,410 --> 00:34:05,340
+للسفر اللي هو does notexist دل عليا بس اوضح اللي
+
+344
+00:34:05,340 --> 00:34:12,220
+هو ليش هذا does not exist بده اوضح ليش هذا does
+
+345
+00:34:12,220 --> 00:34:18,740
+not exist اخدتوا انتوا اخدتوا اللي هو ايش نظرية ان
+
+346
+00:34:18,740 --> 00:34:28,240
+لو كانت limitF of X لما X بتروح ل A بساوي L إذا أي
+
+347
+00:34:28,240 --> 00:34:34,560
+sequence XN بتروح لل A لازم يعطيك اللي هو F of XN
+
+348
+00:34:34,560 --> 00:34:39,780
+برضه يروح لمين لل L، بصبوط؟ هذه كcontinuity اللي
+
+349
+00:34:39,780 --> 00:34:43,530
+بتحكي عنهااللي كده كانت limit F of X لما X تروح
+
+350
+00:34:43,530 --> 00:34:48,930
+للـ A بساوي L و IG تأخدت أي sequence XN بتروح للـ
+
+351
+00:34:48,930 --> 00:34:54,670
+A هيطلع limit F of XN مش بيروح بساوي لL ماشي هذه
+
+352
+00:34:54,670 --> 00:35:00,780
+اللي هي النظريةبناء على هذه النظرية كنا نقول انه
+
+353
+00:35:00,780 --> 00:35:07,180
+لو وجيت two sequences على الأقل ايكون بروح للـA
+
+354
+00:35:07,180 --> 00:35:13,500
+لكن بروح لـLimits مختلفات او واحدة من Limits does
+
+355
+00:35:13,500 --> 00:35:18,600
+not exist بكفالك ان الـLimit هذه اشمالها does not
+
+356
+00:35:18,600 --> 00:35:25,170
+exist واضح؟ الان اذا بدي اعتمد على هذهفي إثبات إنه
+
+357
+00:35:25,170 --> 00:35:29,690
+limit cosine واحد على X does not exist أكيد كلكم
+
+358
+00:35:29,690 --> 00:35:40,730
+جال take the sequences أو the two sequences X and
+
+359
+00:35:40,730 --> 00:35:50,290
+بتساوي واحد على اتنين and by and Y and بتساوي واحد
+
+360
+00:35:50,290 --> 00:35:57,030
+على اتنين and زائد واحد في byمظبوط أكيد قلته هيك
+
+361
+00:35:57,030 --> 00:36:05,290
+where x and goes to zero and y and goes to zero
+
+362
+00:36:05,290 --> 00:36:13,130
+but cosine واحد على ايه؟ cosine واحد على x and
+
+363
+00:36:13,130 --> 00:36:23,100
+اللي هو بتساوي او goes to واحد and cosine1 على I
+
+364
+00:36:23,100 --> 00:36:30,500
+and goes to ناقص واحد اوضحها
+
+365
+00:36:30,500 --> 00:36:35,600
+اوضحها اذكركم الان
+
+366
+00:36:35,600 --> 00:36:39,980
+هذا بيصير cosine مين اتنين and by اتنين and by ايش
+
+367
+00:36:39,980 --> 00:36:45,470
+مالها واحدا اتنين ان زاد واحد لكل في باقي ايش
+
+368
+00:36:45,470 --> 00:36:49,030
+بتصير؟ اللي هو سالب واحد فبصير هاد ال cosine
+
+369
+00:36:49,030 --> 00:36:53,450
+converts واحد و هاد ال cosine converts لمين؟ لسالب
+
+370
+00:36:53,450 --> 00:36:58,490
+واحد اذا حسب اللي حكيناها قبل شوية hence hence
+
+371
+00:36:58,490 --> 00:37:05,910
+hence limit cosine واحد على x لما x تروح ل zero
+
+372
+00:37:05,910 --> 00:37:17,800
+does not existهذا توضيح .. توضيح لماذا اللي هو cos
+
+373
+00:37:17,800 --> 00:37:24,000
+1 على x does not exist لاحظ التالي على جهة لكن لو
+
+374
+00:37:24,000 --> 00:37:28,440
+كانت .. بالمناسبة لو كانت هذه does not exist
+
+375
+00:37:28,440 --> 00:37:33,640
+limitها و هذه does not exist بقدرش أؤكد على إن ال
+
+376
+00:37:33,640 --> 00:37:40,460
+limit للطرح أو الجامعه does not exist صح؟ في مثال؟
+
+377
+00:37:45,330 --> 00:37:50,850
+أه و خروجتها واحدة لأن واحدة لأن بينهين سالم بيصير
+
+378
+00:37:50,850 --> 00:37:53,450
+limit لها دي موجودة أو limit لها دي موجودة .. آسف
+
+379
+00:37:53,450 --> 00:37:55,890
+limit لها دي مش موجودة أو limit لها دي مش موجودة
+
+380
+00:37:55,890 --> 00:38:01,490
+لكن اللي هو حصل إيش طرح موجود أو لو بدك جمحن بتحط
+
+381
+00:38:01,490 --> 00:38:04,890
+اللي هو للتانية تاخد واحدة لأن ونقص واحدة لأن و
+
+382
+00:38:04,890 --> 00:38:10,490
+هكذا أي سؤال؟
+
+383
+00:38:10,490 --> 00:38:11,330
+طيب
+
+384
+00:38:15,770 --> 00:38:21,510
+اللي ييجي الأن إلى
+
+385
+00:38:21,510 --> 00:38:37,290
+النقطة اللي بعيدها أين
+
+386
+00:38:37,290 --> 00:38:38,330
+رحلت أنا؟
+
+387
+00:38:48,090 --> 00:38:53,250
+طيب انتبهونا خلينا لهذه النظرية نحكي عن نصها أو
+
+388
+00:38:53,250 --> 00:39:00,150
+الإشي بالتفصيل و بعدين نيجي نلخصلكم كل المعلومة
+
+389
+00:39:00,150 --> 00:39:05,130
+اللي .. اللي في الأول هنحس إنها بتغلب في اللي هو
+
+390
+00:39:05,130 --> 00:39:13,470
+سفر و من ثم نحاول نبرهنها أه
+
+391
+00:39:13,470 --> 00:39:18,970
+.. أه .. أه .. لأ .. أه ..لت I subsequent R be an
+
+392
+00:39:18,970 --> 00:39:19,610
+interval
+
+393
+00:39:28,670 --> 00:39:34,370
+لأنه احنا بتتذكروا في real واحد خلاص اللي هي ال ..
+
+394
+00:39:34,370 --> 00:39:37,010
+ال .. ال .. ال .. الاشي اللي بده شغل اللي هو
+
+395
+00:39:37,010 --> 00:39:39,890
+النقطة اللي بتكون في ال interval اللي بتكون
+
+396
+00:39:39,890 --> 00:39:42,250
+cluster point اللي مش cluster point automatic
+
+397
+00:39:42,250 --> 00:39:45,970
+هتكون ال limit exist فمش داعي لأصلا لدراسة الموضوع
+
+398
+00:39:45,970 --> 00:39:52,440
+لإنه بيصير كتير من الأمور trivialعشان هي احنا
+
+399
+00:39:52,440 --> 00:39:55,740
+بنشتغل على الـ Intervals let I subset من R بيكون
+
+400
+00:39:55,740 --> 00:39:59,680
+Interval and let F من I لعند R بيكون strictly
+
+401
+00:39:59,680 --> 00:40:09,720
+Monotone F من عند هذه theorem أنا مكتبش ال theorem
+
+402
+00:40:09,720 --> 00:40:19,060
+تمام لأنه موجود هنا theorem 618 عند F من Iلعند R
+
+403
+00:40:19,060 --> 00:40:25,440
+غير عنها strictly monotone
+
+404
+00:40:25,440 --> 00:40:36,900
+ماشي الحال و continuous and continuous الان لاحظ
+
+405
+00:40:36,900 --> 00:40:41,100
+على ال function هذهمدام حكالة عنها strictly
+
+406
+00:40:41,100 --> 00:40:44,540
+monotone يعني strictly increasing او strictly
+
+407
+00:40:44,540 --> 00:40:49,240
+decreasing معناته صارت automatic اتدالة اشمالها
+
+408
+00:40:49,240 --> 00:40:53,800
+one to one ليش؟ لأنه لو كانت strictly increasing
+
+409
+00:40:53,800 --> 00:40:59,060
+معناته x1 أصغر من x2 اللي بتعطيني f of x1 أصغر
+
+410
+00:40:59,060 --> 00:41:03,840
+strictly من f of x2 هذا في حالة نين strictly
+
+411
+00:41:03,840 --> 00:41:09,540
+increasing أه مش ممكنية للمساواةهاالأن في حالة
+
+412
+00:41:09,540 --> 00:41:14,520
+strictly decreasing X1 أكبر من X2 بيعطيني F of X1
+
+413
+00:41:14,520 --> 00:41:21,220
+أصغر من F of X2 هذا strictly decreasing الآن سواء
+
+414
+00:41:21,220 --> 00:41:25,060
+هنا الحالة هذه أو الحالة هذه هيكون عندي ال
+
+415
+00:41:25,060 --> 00:41:29,200
+function one to one غصب عنها، ليش؟ لأن لو كانت X1
+
+416
+00:41:29,200 --> 00:41:35,640
+لا تساوي X2 يا أصغر يا أكبر، ما زال مقت أصغرهتكون
+
+417
+00:41:35,640 --> 00:41:40,680
+strictly أصغر يعني بمعنى أخر f of x1 لا تساوي f of
+
+418
+00:41:40,680 --> 00:41:45,460
+x2 ولو كانت هذه أكبر هتكون f of x1 أكبر برضه هتكون
+
+419
+00:41:45,460 --> 00:41:49,060
+إشمالها لا تساوي و similarly في حالة ال decreasing
+
+420
+00:41:49,850 --> 00:41:56,250
+إذن واضح إن X1 ≤ X2 بيعطينا F X1 ≤ F X2 أو بمعنى
+
+421
+00:41:56,250 --> 00:42:03,730
+آخر F X1 هذه هي الcontraposition ل F X1 ≤ F X2
+
+422
+00:42:03,730 --> 00:42:10,010
+بيعطينا X1 ≤ X2 وهذا يعني ال F أشمالها is one to
+
+423
+00:42:10,010 --> 00:42:15,810
+one يعني الآن أنا بوضحليش أعطاني F strictly
+
+424
+00:42:15,810 --> 00:42:20,790
+monotone؟ عشان شكله أصلا احنا قاعد .. هذه النظرية
+
+425
+00:42:20,790 --> 00:42:25,030
+الحاكية عن ال inverse بدنا نشوف العلاقة بين ال
+
+426
+00:42:25,030 --> 00:42:27,950
+derivative لل inverse و ال derivative للدالة
+
+427
+00:42:27,950 --> 00:42:32,520
+الأصليةإذا بدأ أهل الدالة الأصلية أن يكون لها
+
+428
+00:42:32,520 --> 00:42:36,640
+inverse عشان يكون لها inverse لازم أفترض أن الدالة
+
+429
+00:42:36,640 --> 00:42:40,740
+إشمالها one to one فالآن فرضنا إنها strictly
+
+430
+00:42:40,740 --> 00:42:43,980
+monotone يعني strictly increasing أو strictly
+
+431
+00:42:43,980 --> 00:42:50,920
+decreasing الآن مدامة one to one إذا في إمكانية
+
+432
+00:42:50,920 --> 00:42:58,230
+نحكيعن inverse إلها F inverse إلها اللي هو بده
+
+433
+00:42:58,230 --> 00:43:03,290
+يسميه هو الآن بده يسميه G هو بده يسميه G هياخد ال
+
+434
+00:43:03,290 --> 00:43:10,950
+G من وين من ال range لل F يعني من ال F of I لعند
+
+435
+00:43:10,950 --> 00:43:17,150
+مين لعند ال R ماشي ال F of I هذي مسميها مين هو
+
+436
+00:43:17,150 --> 00:43:25,150
+مسميها هو J يعني ماياخد ال Gمن G لعند R الـ G هي
+
+437
+00:43:25,150 --> 00:43:30,750
+عبارة عن مين؟ عن الـ F inverse معايا؟ قولي يا محمد
+
+438
+00:43:30,750 --> 00:43:42,570
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+439
+00:43:42,570 --> 00:43:42,670
+R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R
+
+440
+00:43:42,670 --> 00:43:43,910
+.. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R ..
+
+441
+00:43:43,910 --> 00:43:44,630
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+442
+00:43:44,630 --> 00:43:47,530
+R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R
+
+443
+00:43:47,530 --> 00:43:47,590
+.. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R ..
+
+444
+00:43:47,590 --> 00:43:47,610
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+445
+00:43:47,610 --> 00:43:53,500
+R .. لـ R .. لـ R .. لـبس لما نكتب I R ايش برضه
+
+446
+00:43:53,500 --> 00:43:57,420
+صحيح بس على أساس انه .. خلاني أقول المعلومات بت ..
+
+447
+00:43:57,420 --> 00:44:01,780
+بت .. بتأهلني ان اكون أدق لعند مين لعند I خلان
+
+448
+00:44:01,780 --> 00:44:06,600
+صحيح طيب الآن عشان ما نبعدش عن اللي بدنايا F من I
+
+449
+00:44:06,600 --> 00:44:12,080
+ل R strictly monotone و G من F of I ل عند I أصلا
+
+450
+00:44:12,080 --> 00:44:15,140
+إذا كانت ال F من I ل عند R strictly monotone و
+
+451
+00:44:15,140 --> 00:44:19,960
+continuousهتكون الـ G اللي هي ال F inverse من F of
+
+452
+00:44:19,960 --> 00:44:24,400
+I ل I برضه strictly monotone و إيش و continuous
+
+453
+00:44:24,400 --> 00:44:27,940
+strictly monotone و continuous إذن الآن ال F عند
+
+454
+00:44:27,940 --> 00:44:30,660
+me strictly monotone و continuous و ال F inverse
+
+455
+00:44:30,660 --> 00:44:41,160
+برضه strictly monotone و إيش و continuous معايا ؟
+
+456
+00:44:41,160 --> 00:44:49,090
+طيبالان لحد الان بعتبر انه احنا بنحكي عن معطيات
+
+457
+00:44:49,090 --> 00:44:54,030
+حتى لو كنا استنتجنا جزء منها من بعض الان بقوللي
+
+458
+00:44:54,030 --> 00:44:58,970
+اللي هو موضوعنا if f is differentiable at c if
+
+459
+00:44:58,970 --> 00:45:12,350
+الان if if f is differentiable at c element in I
+
+460
+00:45:13,160 --> 00:45:17,380
+ماشي الحال يعني F differentiable عنده نقطة C
+
+461
+00:45:17,380 --> 00:45:24,600
+element وين؟ ن I بقول لك then طبعاً A و F prime of
+
+462
+00:45:24,600 --> 00:45:28,340
+C أشمالها لاتساوي سفر لأنه هيلزمنا القسم عليها
+
+463
+00:45:28,340 --> 00:45:32,420
+عشان هيك لازم اللي هو عشان نحكي عن ال inverse لال
+
+464
+00:45:32,420 --> 00:45:34,880
+derivative تكون ال F prime عند ال C أشمالها
+
+465
+00:45:34,880 --> 00:45:42,730
+لاتساوي سفر الآن thenاللي هي الـ G الـ G اللي
+
+466
+00:45:42,730 --> 00:45:47,810
+سميناها F inverse اللي سميناها F inverse الـ G is
+
+467
+00:45:47,810 --> 00:45:51,150
+differentiable
+
+468
+00:45:51,150 --> 00:46:00,490
+is differentiable at اللي هو F of C is
+
+469
+00:46:00,490 --> 00:46:04,610
+differentiable at إيش؟ at F of C اللي هو مسميها
+
+470
+00:46:04,610 --> 00:46:11,800
+إيش اسمها ديالان الإعلان الواضح أنه لو كانت F
+
+471
+00:46:11,800 --> 00:46:15,960
+differentiable عند الـC و F' للـC لاتساوي سفر
+
+472
+00:46:15,960 --> 00:46:20,400
+هتكون الـF inverse differentiable عند مين؟ عند
+
+473
+00:46:20,400 --> 00:46:34,300
+الـF of C اه and .. and F inverse prime at F of C
+
+474
+00:46:35,750 --> 00:46:46,830
+At F of C هتساوي واحد على F prime لمين؟ Let's see
+
+475
+00:46:46,830 --> 00:46:59,390
+يعني وكأنه وكأنه بمجول F inverse prime لل Yبتساوي
+
+476
+00:46:59,390 --> 00:47:07,690
+واحد على F prime of X where العلاقة بين Y و X اللي
+
+477
+00:47:07,690 --> 00:47:16,070
+هي Y بتساوي F of X واضح، اه، الكلام واضح وسهلاللي
+
+478
+00:47:16,070 --> 00:47:21,190
+هي f prime of c بدي ابتبها طريقة ثانية بيساوي واحد
+
+479
+00:47:21,190 --> 00:47:30,010
+على f prime of الان هو سم ال f of c أشمها D يعني
+
+480
+00:47:30,010 --> 00:47:37,080
+ال c بتساوي f انفر سلاميين ل Dصح؟ يعني بمعنى آخر
+
+481
+00:47:37,080 --> 00:47:45,060
+هو بقول F inverse prime عند مين؟ عند ال D بسوء
+
+482
+00:47:45,060 --> 00:47:50,580
+واحد على F prime مكان ال C إيش ودي يحط؟ F inverse
+
+483
+00:47:50,580 --> 00:48:01,420
+of D يعني واحد على F prime فلأ بكفي اللي هى ماشي
+
+484
+00:48:01,420 --> 00:48:07,720
+الحل G of D لأ لأاللي هو كاتبها يعني لإنه كاتب هو
+
+485
+00:48:07,720 --> 00:48:15,000
+بدل ال F inverse كاتب مين الـG واضح واضحة طبعا هذه
+
+486
+00:48:15,000 --> 00:48:22,200
+اللي هي اللي احنا بدنا نثبته نيجي للإثبات
+
+487
+00:48:27,580 --> 00:48:32,740
+بدنا نطبل روحنا شوية عرفناش اننا نطبط اه عشان اللي
+
+488
+00:48:32,740 --> 00:48:41,260
+خاطر امسح وبدأ ابدأ اشتغل من فوق النظرية
+
+489
+00:48:41,260 --> 00:48:45,720
+احفظناها اه او خلينا نقول هي موجودة يعني اه بس
+
+490
+00:48:45,720 --> 00:48:49,160
+بدنا نطلع عالميا على البرهان ال proof
+
+491
+00:48:56,060 --> 00:49:04,920
+بدنا نثبت احنا اللي هو g prime of
+
+492
+00:49:04,920 --> 00:49:14,460
+D بساوة واحد على F prime of C هاي اللي بدنا نثبته
+
+493
+00:49:14,460 --> 00:49:17,820
+او واحد على F prime .. الـ G prime هذا طالع من
+
+494
+00:49:17,820 --> 00:49:22,140
+اليمين الـ G عبارة عن F inverse ماشي؟ شوفي الآن
+
+495
+00:49:25,040 --> 00:49:33,140
+بدي اعرف ده اللي برضه مساعدة لان خد Y لا تساوي D و
+
+496
+00:49:33,140 --> 00:49:39,760
+Y طبعا وين موجودة؟ Y element in J في domain من ال
+
+497
+00:49:39,760 --> 00:49:43,640
+F inverse متذكرين؟
+
+498
+00:49:43,640 --> 00:49:52,360
+ال F من I لعند R و ال G اللي هي ال F inverseمن F
+
+499
+00:49:52,360 --> 00:49:59,500
+of I اللي سمناها J لعند مين؟ لعند الـ I مظبوط؟ طيب
+
+500
+00:49:59,500 --> 00:50:10,320
+الآن هعرف ال H of Y H of Y بتساوي اللي هو F F
+
+501
+00:50:10,320 --> 00:50:20,250
+inverse of Y اللي هي F of G of Y ماقص F of Gof
+
+502
+00:50:20,250 --> 00:50:25,430
+مين؟ of D نحطها على جنب ايش اللي بنعني فيه F of C
+
+503
+00:50:25,430 --> 00:50:33,110
+بتساوي D او G of D اللي هي F inverse استذكروا انه
+
+504
+00:50:33,110 --> 00:50:36,390
+انا الـ G هي مين هي F inverse عشان تسهي عليكم
+
+505
+00:50:36,390 --> 00:50:41,630
+عملية الحسابات ايش بتساوي؟ C ماشي الان F of G of Y
+
+506
+00:50:41,630 --> 00:50:48,750
+ناقص F of G of D على G of Y ناقص G of D
+
+507
+00:50:51,950 --> 00:50:56,890
+معايا؟ واضح اه؟ الان انا مفترض ان ال Y لا تساوي D
+
+508
+00:50:56,890 --> 00:51:01,290
+طبعا ال functions سبعنا سواء ال F او ال G تمتين
+
+509
+00:51:01,290 --> 00:51:05,870
+one to one مأكيد ال G of Y ايش مالها لا تساوي G of
+
+510
+00:51:05,870 --> 00:51:09,990
+D اذا ان انا مرتاح انا ماخد Y لا تساوي Z اذا ان
+
+511
+00:51:09,990 --> 00:51:13,450
+automatic هذه is well defined يعني G of Y ايش
+
+512
+00:51:13,450 --> 00:51:19,270
+مالها لا تساوي مين؟ G of D واضح اه؟ اي سؤال؟
+
+513
+00:51:22,880 --> 00:51:28,700
+الان الـ H of Y اللي هي بتساوي اللي هو F of G of Y
+
+514
+00:51:28,700 --> 00:51:33,120
+نقص F وكذا على G of Y minus G of D is اللي هي well
+
+515
+00:51:33,120 --> 00:51:38,640
+defined طبعا الان اللي فوقبناء على ما كتبناه
+
+516
+00:51:38,640 --> 00:51:45,220
+المستعملينه هنا كرموز هذه نفسها ممكن نطلع إليها
+
+517
+00:51:45,220 --> 00:51:50,520
+كصيغة أخرى أفما هي inverse تبعدها دول فبصير عبارة
+
+518
+00:51:50,520 --> 00:51:57,980
+عن Y ناقص D صح ولا لأ على G of Y ناقص مين G of D
+
+519
+00:51:57,980 --> 00:52:04,840
+هذا نظرة أخرى لمين للH of Y اللي عندناأي سؤال؟
+
+520
+00:52:04,840 --> 00:52:10,740
+طبعا الـH of Y واضح أنها مش سفر لأن الـY أشمالها
+
+521
+00:52:10,740 --> 00:52:15,500
+لاتساوي D لأن بعد شوية بدي أجسم عليها عشان هيك بدي
+
+522
+00:52:15,500 --> 00:52:22,000
+أكتد لكم أن الـY الـH of Y أشمالها لاتساوي سفر
+
+523
+00:52:22,000 --> 00:52:31,170
+واضح لحد الآن، ماشي الحال، الآن بدي أستخدماللي هو
+
+524
+00:52:31,170 --> 00:52:35,090
+التعريف في الوصول الـ Epsilon Delta Definition في
+
+525
+00:52:35,090 --> 00:52:42,570
+الوصول للي بديهاللي هو G prime of D بيساوي limit G
+
+526
+00:52:42,570 --> 00:52:48,770
+of Y نقص G of D على Y minus D الـY بتروح للـD اللي
+
+527
+00:52:48,770 --> 00:52:51,490
+هو exist .. that .. that .. that .. that مثلك انه
+
+528
+00:52:51,490 --> 00:52:54,470
+لكل يكبر من صفر يوجد Delta او Gamma او اللي هي
+
+529
+00:52:54,470 --> 00:52:58,490
+such that لما تكون مسافة بين Y و D أصغر من Delta
+
+530
+00:52:58,490 --> 00:53:01,900
+يعطينا هدف أصغر من مين؟ من يبسلونهذا كيف بده
+
+531
+00:53:01,900 --> 00:53:07,060
+أثبته؟ بالاستعانة بيه اللي هو أنا بعرف أن الـ F is
+
+532
+00:53:07,060 --> 00:53:10,320
+differentiable و بالاستعانة بتدلة اللي عندى هذه
+
+533
+00:53:10,320 --> 00:53:14,080
+السؤال أي سؤال حتى الآن حتى الآن احنا اللي أثبتنا
+
+534
+00:53:14,080 --> 00:53:17,540
+ان H و Y اللي ركّبناها دي إيش مانع لا تساوى سفر
+
+535
+00:53:17,540 --> 00:53:27,380
+الان عندى since F is differentiable at مين معطينى؟
+
+536
+00:53:27,380 --> 00:53:34,100
+at Cاذن thenfor every epsilon أكبر من صفر بتستخدم
+
+537
+00:53:34,100 --> 00:53:38,920
+على طول اللي هو limit واضح then for every epsilon
+
+538
+00:53:38,920 --> 00:53:42,660
+أكبر من صفر there exist نستخدم نفس التعبير دلتا
+
+539
+00:53:42,660 --> 00:53:47,840
+أكبر من صفر أو نفس الرموز such that X minus C أكبر
+
+540
+00:53:47,840 --> 00:53:54,020
+من صفر وأصغر من دلتا implies F of X ناقص F of C
+
+541
+00:53:54,020 --> 00:53:59,240
+على X minus C ناقص اللي هو ال derivative اللي هي F
+
+542
+00:53:59,240 --> 00:54:06,500
+prime of C صح؟أصغر من مين؟ من أبسل وسمي هذه ايش
+
+543
+00:54:06,500 --> 00:54:17,000
+مسميها هو؟ همسميها واحد هذا سمي واحد واضح
+
+544
+00:54:17,000 --> 00:54:21,360
+أه؟ في معنى و كده الان f is differentiable in c
+
+545
+00:54:21,360 --> 00:54:25,580
+than لكل y نقل المستقبل اللي يعني استخدمنا limit f
+
+546
+00:54:25,580 --> 00:54:30,470
+of x نقص f of c على x minus c سوى f prime of cطبعا
+
+547
+00:54:30,470 --> 00:54:34,630
+أنا بعيد على أساس اللي بتابعنا مش سواعب انكم مش
+
+548
+00:54:34,630 --> 00:54:42,350
+سواعبته واضح؟ ماشي الحاجة الآن بدنا نستخدم ال
+
+549
+00:54:42,350 --> 00:54:47,410
+continuity لمين؟ لجي جي is continuous طبعا جي is
+
+550
+00:54:47,410 --> 00:54:51,870
+differentiable أصلا جي is differentiable عند مين؟
+
+551
+00:54:51,870 --> 00:54:56,690
+عند اللي هو D يعني جي is differentiable عند F of C
+
+552
+00:54:56,690 --> 00:55:03,740
+إذا continuousمظبوط؟ اذا الان since .. since G is
+
+553
+00:55:03,740 --> 00:55:12,540
+continuous at D بتساوي F of C اذا يعني صار عندى
+
+554
+00:55:12,540 --> 00:55:24,060
+limit G of Y لما Y تروح ل D بساوة مين؟ G of D اه
+
+555
+00:55:25,340 --> 00:55:29,760
+واضح ها؟ واضح .. واضح ها؟ خلّيني نعمل .. على أساس
+
+556
+00:55:29,760 --> 00:55:32,980
+ما نستخدم الرموز هذه نربطها باللي بقى بالها ..
+
+557
+00:55:32,980 --> 00:55:37,900
+بنغير بعدين معايا؟ طيب .. هذه إيش معناتها by
+
+558
+00:55:37,900 --> 00:55:41,460
+epsilon delta definition؟ أنه لكل .. الآن بدي
+
+559
+00:55:41,460 --> 00:55:45,780
+أستخدم الـ delta هذه لإن بدي أستخدم الاتنين مع بعض
+
+560
+00:55:45,780 --> 00:55:49,800
+الوصول اللي بديهاواضح؟ هذه لكل إبسلون في الدنيا،
+
+561
+00:55:49,800 --> 00:55:53,140
+أي إن كانت من ضمن إنيين الـDelta اللي لاجيناها،
+
+562
+00:55:53,140 --> 00:55:57,300
+واضح اللي بحكيه؟ الآن لكل إبسلون أكبر من 0، there
+
+563
+00:55:57,300 --> 00:56:01,160
+exists Gamma اللي هي .. ابحثت إنه لما يكون Y minus
+
+564
+00:56:01,160 --> 00:56:05,100
+D أصغر من Gamma، يعطيني هذا أصغر، اللي هو من اللي
+
+565
+00:56:05,100 --> 00:56:08,180
+بديه .. من اللي بدي أن أناسك دموع، اللي هو
+
+566
+00:56:08,180 --> 00:56:16,030
+الـDelta، الآنcontinuous ات كده then for every
+
+567
+00:56:16,030 --> 00:56:18,930
+ابسلون الابسلون اللي بده استخدمها اللي من ضمن
+
+568
+00:56:18,930 --> 00:56:21,290
+الابسلون هي اللي بتنفع ايه اللي بتنفع على ال delta
+
+569
+00:56:21,290 --> 00:56:25,970
+هذه then for delta أكبر من سفر there exist gamma
+
+570
+00:56:25,970 --> 00:56:32,670
+أكبر من سفر such that لما y minus d كون أكبر من
+
+571
+00:56:32,670 --> 00:56:37,170
+سفر وأصغر من الجامعة اللي لجيتها هذا بيعطيني اللي
+
+572
+00:56:37,170 --> 00:56:46,080
+هو ال Gof Y نقص الـ G of D يكون أصغر من مين من الـ
+
+573
+00:56:46,080 --> 00:56:51,820
+Epsilon اللي هي سميتها H Delta أي سؤال هذا بده
+
+574
+00:56:51,820 --> 00:57:01,100
+سميه H اسمه بده سميه اتنين اه هساخد من ال N طيب
+
+575
+00:57:01,100 --> 00:57:07,080
+شوفوا
+
+576
+00:57:07,080 --> 00:57:17,060
+الآنأنا في الآخر بدي اللي هو أصل لإنه اللي هي ال
+
+577
+00:57:17,060 --> 00:57:21,680
+derivative لل F prime اللي هو واحد ال F inverse
+
+578
+00:57:21,680 --> 00:57:29,840
+prime واحد على ال F prime ماشي طيب لما
+
+579
+00:57:29,840 --> 00:57:37,150
+تسيري الآن ال Yطبعا ال y .. ال .. ال .. ال .. ال
+
+580
+00:57:37,150 --> 00:57:37,190
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+581
+00:57:37,190 --> 00:57:37,210
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+582
+00:57:37,210 --> 00:57:37,750
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+583
+00:57:37,750 --> 00:57:37,770
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+584
+00:57:37,770 --> 00:57:38,230
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+585
+00:57:38,230 --> 00:57:46,550
+.. ال ..
+
+586
+00:57:46,550 --> 00:57:47,090
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+587
+00:57:47,090 --> 00:57:47,550
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+588
+00:57:47,550 --> 00:57:48,510
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+589
+00:57:48,510 --> 00:57:48,870
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+590
+00:57:48,870 --> 00:57:49,850
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+591
+00:57:49,850 --> 00:57:57,250
+ال ..
+
+592
+00:57:57,250 --> 00:57:58,030
+ال
+
+593
+00:58:03,070 --> 00:58:09,070
+إذا حيطلع عند الـ GY نقص GY دي إيش معلها؟ أقل من
+
+594
+00:58:09,070 --> 00:58:14,570
+مين؟ من دلتا اللي هي by using اتنين G of Y إيش
+
+595
+00:58:14,570 --> 00:58:19,730
+اسمها؟ اللي هي G of Y اللي هي Y إيش بتسوى F of X
+
+596
+00:58:19,730 --> 00:58:25,580
+عن G of Y بتسوى Xأه اللي هي عبارة عن X minus G of
+
+597
+00:58:25,580 --> 00:58:31,900
+D الـ D اللي هي عبارة عن مين؟ اللي هي F of C و الـ
+
+598
+00:58:31,900 --> 00:58:35,820
+G inverse للـ F إذا هتطلع عبارة عن مين؟ عن C أصغر
+
+599
+00:58:35,820 --> 00:58:45,140
+من مين؟ من دلتا واضح طبيعي وأكبر أكيد أكيد أكيد
+
+600
+00:58:45,140 --> 00:58:52,230
+أكيدواضح لحد الآن الان هذا بده يخليني أستخدم مين
+
+601
+00:58:52,230 --> 00:58:59,750
+أستخدم واحد إذا صار عندي then هذا متحقق الان then
+
+602
+00:58:59,750 --> 00:59:04,830
+بدي أربط هذا الكلام بمين باللي فوق اللي هو H of Y
+
+603
+00:59:04,830 --> 00:59:12,450
+H of Y ناقص F prime of C شوف كيف بده F مثلك الان
+
+604
+00:59:12,450 --> 00:59:19,780
+اللي هو بساويH of Y مين؟ H of Y اللي هو عبارة عن Y
+
+605
+00:59:19,780 --> 00:59:27,960
+ناقص D على G of Y ناقص G of D، مظبوط؟ اللي هو بيه
+
+606
+00:59:27,960 --> 00:59:30,960
+عبارة عن مين؟ F of G of Y ناقص كده، إلى آخرين
+
+607
+00:59:30,960 --> 00:59:42,000
+بيصير بساوي اللي هو F of G of Y ناقص F of G of D
+
+608
+00:59:42,000 --> 00:59:52,730
+على Yعلى g of y ناقص g of d اللي هو ناقص f prime
+
+609
+00:59:52,730 --> 01:00:00,150
+of c أصغر من مين؟ من y عارفين ليش؟ لأن الـ g of y
+
+610
+01:00:00,150 --> 01:00:04,880
+مين هي؟ الـ xو الـ G of D اللي هي مين الـ C
+
+611
+01:00:04,880 --> 01:00:10,480
+فاهمين؟ أنا قصد أكتبها بالصورة هذه عشان أقولكم أيش
+
+612
+01:00:10,480 --> 01:00:14,580
+لو سُلتله H of I ناقص أفرو برايم of C بساوي هذا
+
+613
+01:00:14,580 --> 01:00:21,500
+المقدار ناقص هذا متحقق كمان مرة يا جماعة اللي
+
+614
+01:00:21,500 --> 01:00:26,220
+بيحقق هذا غصب عنه بيحقق هذا و اللي بيحقق هذا بيحقق
+
+615
+01:00:26,220 --> 01:00:31,240
+هذا ماشي الحال أنا الأن أخدت عندي Y element in J
+
+616
+01:00:31,920 --> 01:00:36,960
+والـ Y لا تساوي D واخدت هذا يكون أصغر من دولتا تحت
+
+617
+01:00:36,960 --> 01:00:41,320
+هذا الشرط الآن أنا بشتغل تحت هذا الشرط الآن هذا
+
+618
+01:00:41,320 --> 01:00:46,140
+أدالي حسب اللي هي اتنين ان هذا ناقص هذا أصغر من
+
+619
+01:00:46,140 --> 01:00:51,090
+دولتا هذا اللي هو X وهذا اللي هو mean الـCالان
+
+620
+01:00:51,090 --> 01:00:54,790
+مادام أصغر من دلتا إذا أعطاني واحد إذا أعطاتني هذه
+
+621
+01:00:54,790 --> 01:00:59,510
+أصغر من هذه هذه في الواقع مين هي F of X هي عبارة
+
+622
+01:00:59,510 --> 01:01:03,830
+عن F of G of Y لأن G of Y هي ال X و C ليه F of D
+
+623
+01:01:03,830 --> 01:01:08,690
+صار هذا ناقص هذا أصغر من مين من إبسلون من جهة أخرى
+
+624
+01:01:08,690 --> 01:01:13,650
+هذا المقدار مين هو عبارة عن H of Y وهذا اللي هي F
+
+625
+01:01:13,650 --> 01:01:18,550
+prime of Cإذا القالة اللي وصلت إليه أنه لكل إبسلون
+
+626
+01:01:18,550 --> 01:01:22,670
+أكبر من 0 وجدت Gamma لأن Gamma بين إيديا واجدتها
+
+627
+01:01:22,670 --> 01:01:27,050
+من الأصل إلى لكل إبسلون أكبر من 0 لجيت Gamma ابحث
+
+628
+01:01:27,050 --> 01:01:30,290
+أنه لما يكون هذه المسافة بينها أصغر من Gamma
+
+629
+01:01:30,290 --> 01:01:33,610
+يعطينا المسافة أصغر من إبسلون يعني المسافة هذه
+
+630
+01:01:33,610 --> 01:01:40,760
+أصغر من إبسلون أو بمعنى آخر limitH of Y لما الـ Y
+
+631
+01:01:40,760 --> 01:01:50,320
+تروح للـ D بساوة F prime of مين؟ of C تصيرنا
+
+632
+01:01:50,320 --> 01:01:55,660
+جريبين من اللي بدنايا عند الـ H of Y limitها بساوة
+
+633
+01:01:55,660 --> 01:02:02,980
+limit هذا بساوة F prime of C هذا عبارة عن مين؟ عن
+
+634
+01:02:02,980 --> 01:02:10,910
+G prime مقلوق G primeإذا الآن خلصنا تقريبا نمسح
+
+635
+01:02:10,910 --> 01:02:16,130
+حده و نمسك
+
+636
+01:02:16,130 --> 01:02:28,150
+بهدي مع هدي صار عندي limit h of y لما ال y تقول
+
+637
+01:02:28,150 --> 01:02:34,960
+الى d limitH of Y لما Y تقول إلى D طلع عندي عبارة
+
+638
+01:02:34,960 --> 01:02:40,960
+عن F prime of C هي تحت من جهة أخرى هذا بيساوي مين؟
+
+639
+01:02:40,960 --> 01:02:48,720
+limit Y نقص D على G of Y نقص G of D لما Y تروح
+
+640
+01:02:48,720 --> 01:02:55,980
+لمين؟ للـ D صح؟ هذا عبارة عن limit أو واحد على
+
+641
+01:02:55,980 --> 01:03:03,130
+limitG of Y نقص G of D على Y minus D لما Y تروح لـ
+
+642
+01:03:03,130 --> 01:03:10,310
+D1 على الـ limit هذا فهي
+
+643
+01:03:10,310 --> 01:03:16,390
+ساوية 1 على g prime of D إذا صارت عندي F prime of
+
+644
+01:03:16,390 --> 01:03:23,710
+C بساوية 1 على g prime of D أو بمعنى آخر F prime
+
+645
+01:03:23,710 --> 01:03:31,800
+of C بساوية 1 على F G prime اللي هي Finverse prime
+
+646
+01:03:31,800 --> 01:03:42,000
+of D أو بتعبير آخر F inverse prime of D بساوية
+
+647
+01:03:42,000 --> 01:03:55,700
+واحد على F of C وهو المطلوب طيب
+
+648
+01:04:01,020 --> 01:04:02,360
+خدّيني أجف عن نقطة
+
+649
+01:04:19,990 --> 01:04:25,110
+هي نفس اللي فوق بس كانت اللي فوق عنده نقطة هنا in
+
+650
+01:04:25,110 --> 01:04:29,510
+general الان لو كانت على كل ال domain يعني الان
+
+651
+01:04:29,510 --> 01:04:33,150
+بقولي لو كانت I subset من R بيه an interval and F
+
+652
+01:04:33,150 --> 01:04:37,510
+من I لعند R strictly monotone on I ماشي ..
+
+653
+01:04:37,510 --> 01:04:42,750
+strictly monotone و let J بيسوء F of I and let G
+
+654
+01:04:42,750 --> 01:04:47,410
+من I لعند R بيه the function inverse of F ماشي ..
+
+655
+01:04:47,980 --> 01:04:52,360
+لأن إذا كانت F as differential على الـ I كلها إذا
+
+656
+01:04:52,360 --> 01:04:55,860
+أكيد هتكون continuous على الـ I كلها مدام هتكون
+
+657
+01:04:55,860 --> 01:04:58,300
+continuous على الـ I كلها و strictly monotone
+
+658
+01:04:58,300 --> 01:05:02,180
+هتكون الـ G العجبالها برضه strictly monotone و
+
+659
+01:05:02,180 --> 01:05:06,870
+continuous إذا شروط اللي فوق اتحققتإذا الـ J ..
+
+660
+01:05:06,870 --> 01:05:10,030
+طبعا ال F' of X معطينيها إشمالها تسوى X على كل ال
+
+661
+01:05:10,030 --> 01:05:13,350
+I إذا أكيد حسب النظر اللي قبل شوية G is
+
+662
+01:05:13,350 --> 01:05:17,230
+differentiable على كل ال J و ال G' بساوية واحدة
+
+663
+01:05:17,230 --> 01:05:22,210
+على F' لإنه composite مين؟ composite G واضح؟ يعني
+
+664
+01:05:22,210 --> 01:05:28,130
+.. كمان مرة؟ إذا كانت ال F strictly monotone و
+
+665
+01:05:28,130 --> 01:05:32,890
+كانت اللي هي ال J بتساوي كل ال F of Iوكانت الـ G
+
+666
+01:05:32,890 --> 01:05:37,310
+هي عبارة عن ال inverse للـ F وكانت ال F
+
+667
+01:05:37,310 --> 01:05:39,570
+differentiable على ال I مدام F differentiable على
+
+668
+01:05:39,570 --> 01:05:42,210
+ال I إذا صارت continuous على كل ال I مدام
+
+669
+01:05:42,210 --> 01:05:44,710
+continuous على ال I و strictly monotone هتكون ال
+
+670
+01:05:44,710 --> 01:05:47,710
+inverse سبعتها اللي هي ال G برضه continuous و
+
+671
+01:05:47,710 --> 01:05:51,700
+strictly monotoneإذا انتحققت شروط النظرية السابقة
+
+672
+01:05:51,700 --> 01:05:55,680
+إذا هتكون ال F و طبعا ال F' طبعا لها تساوي صفر ع
+
+673
+01:05:55,680 --> 01:05:59,020
+كل ال I إذا صارت ال G ال differential بالع كل ال J
+
+674
+01:05:59,020 --> 01:06:02,220
+و العلاقة بين التنتين G' بسوء واحد على F'
+
+675
+01:06:02,680 --> 01:06:09,040
+composite mean composite J أو زي ما بنقول G' of Y
+
+676
+01:06:09,040 --> 01:06:16,220
+بسوء واحد على F' of mean of X تحت العلاقة X بتساوي
+
+677
+01:06:16,220 --> 01:06:25,320
+F of Yأو Y بس هوقف و فكس Y بس هوقف و فكس طبعا هذا
+
+678
+01:06:25,320 --> 01:06:32,220
+البرهان كله حكيناه ال remark اللي عندنا ال remark
+
+679
+01:06:32,220 --> 01:06:37,100
+اللي عندنا ال
+
+680
+01:06:37,100 --> 01:06:43,460
+remark
+
+681
+01:06:43,460 --> 01:06:51,180
+اللي عندنا هي اللي حكيتها الآنD'Y بسوء واحد عالميا
+
+682
+01:06:51,180 --> 01:06:53,660
+على F'X
+
+683
+01:06:57,210 --> 01:06:59,530
+التي قبل بشوية حكيناها جي برايم أو اي بيسوا واحد
+
+684
+01:06:59,530 --> 01:07:03,890
+على اف برايم كومبز الجي ماشي الحال الآن صارت عندي
+
+685
+01:07:03,890 --> 01:07:08,950
+هذه جي برايم كومبز كومبز اف of X بيسوا جي برايم اف
+
+686
+01:07:08,950 --> 01:07:12,290
+of X بيسوا واحد على اف برايم كومبز الجي أو اي
+
+687
+01:07:12,290 --> 01:07:15,470
+بياخد نظرية مرتين و يسوا واحد على اف برايم of X
+
+688
+01:07:15,470 --> 01:07:18,370
+التانيين مع بعض جي برايم بيسوا واحد على اف برايم
+
+689
+01:07:18,370 --> 01:07:24,310
+of X و المرة الجاهزة اللي بنكمل بنعيد هذا اللي إذا
+
+690
+01:07:24,310 --> 01:07:26,470
+كان حكيناها على السريع
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..e0cc13ed05c1eab9ef7914952d87e2c3a4d08162
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2076, "start": 4.82, "end": 20.76, "text": "بسم الله الرحمن الرحيم المحاضرة الثانية في مساق تحليل رياضي 2 أو تحليل حقيقي 2 اتحدثنا المرة الماضية عن", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 9673, 5016, 46958, 25720, 6024, 104, 7649, 10632, 8978, 47524, 995, 4587, 6055, 5016, 20292, 1211, 12602, 1829, 46958, 1829, 568, 34051, 6055, 5016, 20292, 1211, 11331, 38436, 4587, 1829, 568, 1975, 2655, 24401, 12984, 8315, 9673, 25720, 9673, 46958, 10632, 18871], "avg_logprob": -0.08165565262047145, "compression_ratio": 1.6460176991150441, "no_speech_prob": 6.157159805297852e-05, "words": [{"start": 4.82, "end": 5.16, "word": "بسم", "probability": 0.88330078125}, {"start": 5.16, "end": 5.32, "word": " الله", "probability": 0.95361328125}, {"start": 5.32, "end": 5.66, "word": " الرحمن", "probability": 0.9597981770833334}, {"start": 5.66, "end": 6.22, "word": " الرحيم", "probability": 0.994140625}, {"start": 6.22, "end": 8.02, "word": " 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و", "probability": 0.69873046875}, {"start": 83.63, "end": 84.21, "word": " بعد", "probability": 0.953125}, {"start": 84.21, "end": 84.91, "word": " ذلك", "probability": 0.9873046875}, {"start": 84.91, "end": 85.91, "word": " مرّينا", "probability": 0.78330078125}, {"start": 85.91, "end": 86.31, "word": " على", "probability": 0.9169921875}, {"start": 86.31, "end": 86.65, "word": " اللي", "probability": 0.92724609375}, {"start": 86.65, "end": 87.11, "word": " هو", "probability": 0.994140625}, {"start": 87.11, "end": 88.73, "word": " بعض", "probability": 0.93017578125}, {"start": 88.73, "end": 89.55, "word": " الخواص", "probability": 0.8126627604166666}], "temperature": 1.0}, {"id": 5, "seek": 10838, "start": 90.36, "end": 108.38, "text": "وقلنا الخاصية الأولى عندنا اللى هى انه لو كانت الدالة اللى هى differentiable at C بتكون continuous برضه وين عند C يعني اي differentiable function must be continuous", "tokens": [2407, 4587, 1211, 8315, 33962, 33546, 10632, 16247, 12610, 7578, 43242, 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104.88, "end": 105.1, "word": " C", "probability": 0.8798828125}, {"start": 105.1, "end": 105.34, "word": " يعني", "probability": 0.703369140625}, {"start": 105.34, "end": 105.52, "word": " اي", "probability": 0.5390625}, {"start": 105.52, "end": 106.16, "word": " differentiable", "probability": 0.54901123046875}, {"start": 106.16, "end": 106.58, "word": " function", "probability": 0.97265625}, {"start": 106.58, "end": 106.9, "word": " must", "probability": 0.70263671875}, {"start": 106.9, "end": 107.32, "word": " be", "probability": 0.95361328125}, {"start": 107.32, "end": 108.38, "word": " continuous", "probability": 0.90087890625}], "temperature": 1.0}, {"id": 6, "seek": 12728, "start": 109.42, "end": 127.28, "text": "قلنا إن الـ converse need not to be true in general واخدنا مثال قلنا f of x بتساوي ال absolute value لل x اللي هي عبارة عن continuous function لكن مش differentiable عند ال zero continuous everywhere but it is not differentiable at zero", "tokens": [4587, 1211, 8315, 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0.98876953125}, {"start": 151.02, "end": 151.2, "word": " اللي", "probability": 0.4072265625}, {"start": 151.2, "end": 151.66, "word": " جينا", "probability": 0.7150065104166666}, {"start": 151.66, "end": 152.42, "word": " دالة", "probability": 0.984375}, {"start": 152.42, "end": 153.2, "word": " تكون", "probability": 0.74853515625}, {"start": 153.2, "end": 153.78, "word": " continuous", "probability": 0.96533203125}, {"start": 153.78, "end": 154.1, "word": " وين", "probability": 0.821533203125}, {"start": 154.1, "end": 154.28, "word": " ما", "probability": 0.55322265625}, {"start": 154.28, "end": 154.7, "word": " كان", "probability": 0.794921875}], "temperature": 1.0}, {"id": 8, "seek": 17631, "start": 155.23, "end": 176.31, "text": "و مش differentiable ولا في مكان قلنا اه و قلنا اخد الدالة f of x بساعة ال summation واحد على اتنين أس ان كسين تلاتة أس ان اكس هذه الدالة is continuous everywhere but it is not differentiable anywhere وبعد هيك اتينا", "tokens": [2407, 37893, 819, 9364, 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{"id": 9, "seek": 19178, "start": 184.88, "end": 191.78, "text": "حكينا عن بعض اللي هو قوانين الاشتقاق قولنا لو كانت الدالة", "tokens": [5016, 4117, 1829, 8315, 18871, 45030, 11242, 13672, 1829, 31439, 12174, 2407, 7649, 9957, 2423, 33599, 2655, 4587, 995, 4587, 12174, 12610, 8315, 45164, 25961, 2655, 32748, 6027, 3660], "avg_logprob": -0.19661458482344946, "compression_ratio": 1.2839506172839505, "no_speech_prob": 8.940696716308594e-07, "words": [{"start": 184.88, "end": 186.2, "word": "حكينا", "probability": 0.834716796875}, {"start": 186.2, "end": 186.5, "word": " عن", "probability": 0.974609375}, {"start": 186.5, "end": 187.6, "word": " بعض", "probability": 0.908935546875}, {"start": 187.6, "end": 187.84, "word": " اللي", "probability": 0.60302734375}, {"start": 187.84, "end": 188.06, "word": " هو", "probability": 0.96923828125}, {"start": 188.06, "end": 188.72, "word": " قوانين", "probability": 0.9842529296875}, {"start": 188.72, "end": 189.66, "word": " الاشتقاق", "probability": 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296.2, "text": "أو مرت علينا اللي هو استخدمناها في الـ differential equations وفي مساقات أخرى الآن نظرة نظرة تحليلية فهنشوف اللي هو كيف أثبتنا أو كيف بدنا نثبت اللي هو قاعدة السلسلة أي سؤال بتقدر تسأله يعني اللي بدكم إياه و تتفعلوا بالطريقة اللي بتلاقوها مناسبة let i و j be intervals and let g", "tokens": [10721, 2407, 3714, 43500, 25894, 8315, 13672, 1829, 31439, 44713, 9778, 40448, 8315, 11296, 8978, 2423, 39184, 15756, 11787, 4032, 41185, 47524, 995, 4587, 9307, 5551, 34740, 7578, 6024, 48506, 8717, 19913, 25720, 8717, 19913, 25720, 6055, 5016, 20292, 1211, 10632, 6156, 3224, 1863, 8592, 38688, 13672, 1829, 31439, 9122, 33911, 5551, 12984, 3555, 2655, 8315, 34051, 9122, 33911, 47525, 8315, 8717, 12984, 3555, 2655, 13672, 1829, 31439, 12174, 995, 22488, 3660, 21136, 1211, 3794, 37977, 36632, 8608, 33604, 6027, 39894, 28543, 2288, 6055, 3794, 10721, 43761, 37495, 22653, 13672, 1829, 47525, 24793, 11933, 25528, 3224, 4032, 6055, 2655, 5172, 30241, 14407, 20666, 9566, 16572, 28671, 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133174, "start": 1304.0, "end": 1331.74, "text": "Example جاي لنت F من عند I لعند R بيه a differentiable function من هنا نفترض ان ال F أشبه لها يا جماعة عبارة عندها لقابلة للاشتقاق عند نقطة وين على ال I أو عند ال I كلها مش مشكلة ماشي الحال بقولك اثبتلي", "tokens": [11149, 335, 781, 10874, 47302, 5296, 29399, 479, 9154, 43242, 286, 5296, 3615, 41260, 497, 4724, 1829, 3224, 257, 819, 9364, 2445, 9154, 34105, 8717, 5172, 2655, 43042, 16472, 2423, 479, 5551, 8592, 3555, 3224, 5296, 11296, 35186, 10874, 15042, 27884, 6225, 3555, 9640, 3660, 43242, 11296, 5296, 4587, 16758, 37977, 5296, 15040, 8592, 2655, 4587, 995, 4587, 43242, 8717, 47432, 3660, 4032, 9957, 15844, 2423, 286, 34051, 43242, 2423, 286, 28242, 11296, 37893, 37893, 28820, 3660, 3714, 33599, 1829, 21542, 6027, 4724, 39648, 4117, 1975, 12984, 3555, 2655, 20292], "avg_logprob": -0.2671703303253258, "compression_ratio": 1.5330188679245282, "no_speech_prob": 3.6656856536865234e-05, "words": [{"start": 1304.0, "end": 1304.96, "word": 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"probability": 0.9677734375}, {"start": 1325.86, "end": 1326.32, "word": " كلها", "probability": 0.957275390625}, {"start": 1326.32, "end": 1326.5, "word": " مش", "probability": 0.91162109375}, {"start": 1326.5, "end": 1327.06, "word": " مشكلة", "probability": 0.9676106770833334}, {"start": 1327.06, "end": 1328.02, "word": " ماشي", "probability": 0.9384765625}, {"start": 1328.02, "end": 1328.42, "word": " الحال", "probability": 0.804443359375}, {"start": 1328.42, "end": 1330.1, "word": " بقولك", "probability": 0.8758138020833334}, {"start": 1330.1, "end": 1331.74, "word": " اثبتلي", "probability": 0.9072265625}], "temperature": 1.0}, {"id": 56, "seek": 134891, "start": 1332.47, "end": 1348.91, "text": "بس قبل ما بتخلينا نقل تقول احنا بتوجدولنا الـ Fn لان في عندي دالة اسمها Fn يعني F مرفوعة للقوة انا بدي اوجدلها ال derivative طبعا الـ Fn كله prime of X", "tokens": [3555, 3794, 12174, 36150, 19446, 39894, 9778, 20292, 8315, 8717, 4587, 1211, 6055, 39648, 1975, 5016, 8315, 39894, 29245, 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"word": " احنا", "probability": 0.7145589192708334}, {"start": 1335.47, "end": 1337.55, "word": " بتوجدولنا", "probability": 0.731787109375}, {"start": 1337.55, "end": 1339.23, "word": " الـ", "probability": 0.540283203125}, {"start": 1339.23, "end": 1339.81, "word": " Fn", "probability": 0.3800048828125}, {"start": 1339.81, "end": 1340.23, "word": " لان", "probability": 0.5096435546875}, {"start": 1340.23, "end": 1340.47, "word": " في", "probability": 0.46923828125}, {"start": 1340.47, "end": 1340.75, "word": " عندي", "probability": 0.79833984375}, {"start": 1340.75, "end": 1341.09, "word": " دالة", "probability": 0.9475911458333334}, {"start": 1341.09, "end": 1341.49, "word": " اسمها", "probability": 0.9684244791666666}, {"start": 1341.49, "end": 1342.13, "word": " Fn", "probability": 0.88232421875}, {"start": 1342.13, "end": 1342.89, "word": " يعني", "probability": 0.890625}, {"start": 1342.89, "end": 1343.13, "word": " F", "probability": 0.74853515625}, {"start": 1343.13, "end": 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but this limit اشمالها does not exist", "tokens": [4197, 270, 3792, 502, 15844, 2031, 5296, 15042, 2031, 6055, 32887, 5016, 5296, 9381, 5172, 2288, 775, 406, 2514, 37037, 1211, 7435, 2655, 44945, 8592, 38688, 32239, 8592, 775, 406, 2514, 457, 341, 4948, 1975, 8592, 2304, 6027, 11296, 775, 406, 2514], "avg_logprob": -0.4124273241952408, "compression_ratio": 1.26890756302521, "no_speech_prob": 6.616115570068359e-06, "words": [{"start": 1993.3, "end": 1994.06, "word": "limit", "probability": 0.537139892578125}, {"start": 1994.06, "end": 1994.72, "word": " cos", "probability": 0.473876953125}, {"start": 1994.72, "end": 1995.56, "word": " 1", "probability": 0.5322265625}, {"start": 1995.56, "end": 1995.94, "word": " على", "probability": 0.043609619140625}, {"start": 1995.94, "end": 1996.2, "word": " x", "probability": 0.66357421875}, {"start": 1996.2, "end": 1996.46, "word": " لما", "probability": 0.6142578125}, {"start": 1996.46, "end": 1996.7, "word": " x", "probability": 0.8154296875}, 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"probability": 0.701171875}, {"start": 2088.93, "end": 2089.07, "word": " A", "probability": 0.85693359375}, {"start": 2089.07, "end": 2089.75, "word": " هيطلع", "probability": 0.780029296875}, {"start": 2089.75, "end": 2090.01, "word": " limit", "probability": 0.88916015625}, {"start": 2090.01, "end": 2090.21, "word": " F", "probability": 0.9013671875}, {"start": 2090.21, "end": 2090.37, "word": " of", "probability": 0.9169921875}, {"start": 2090.37, "end": 2090.73, "word": " XN", "probability": 0.89599609375}, {"start": 2090.73, "end": 2090.87, "word": " مش", "probability": 0.10577392578125}, {"start": 2090.87, "end": 2091.43, "word": " بيروح", "probability": 0.868896484375}, {"start": 2091.43, "end": 2092.23, "word": " بساوي", "probability": 0.916748046875}, {"start": 2092.23, "end": 2092.59, "word": " لL", "probability": 0.4432373046875}, {"start": 2092.59, "end": 2094.13, "word": " ماشي", "probability": 0.7257486979166666}, {"start": 2094.13, "end": 2094.67, "word": " هذه", "probability": 0.53271484375}, {"start": 2094.67, "end": 2094.81, "word": " اللي", "probability": 0.9736328125}, {"start": 2094.81, "end": 2095.05, "word": " هي", "probability": 0.8828125}, {"start": 2095.05, "end": 2095.83, "word": " النظرية", "probability": 0.962646484375}], "temperature": 1.0}, {"id": 89, "seek": 212296, "start": 2097.22, "end": 2122.96, "text": "بناء على هذه النظرية كنا نقول انه لو وجيت two sequences على الأقل ايكون بروح للـA لكن بروح لـLimits مختلفات او واحدة من Limits does not exist بكفالك ان الـLimit هذه اشمالها does not exist واضح؟ الان اذا بدي اعتمد على هذه", "tokens": [3555, 1863, 16606, 15844, 29538, 28239, 19913, 2288, 10632, 9122, 8315, 8717, 39648, 16472, 3224, 45164, 49610, 36081, 732, 22978, 15844, 16247, 4587, 1211, 1975, 1829, 30544, 4724, 32887, 5016, 24976, 39184, 32, 44381, 4724, 32887, 5016, 5296, 39184, 43, 332, 1208, 3714, 46456, 46538, 9307, 1975, 2407, 36764, 24401, 3660, 9154, 16406, 1208, 775, 406, 2514, 4724, 4117, 5172, 6027, 4117, 16472, 2423, 39184, 43, 332, 270, 29538, 1975, 8592, 2304, 6027, 11296, 775, 406, 2514, 4032, 46958, 5016, 22807, 2423, 7649, 1975, 15730, 4724, 16254, 1975, 34268, 2304, 3215, 15844, 29538], "avg_logprob": -0.3226396295618504, "compression_ratio": 1.6121495327102804, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2097.22, "end": 2097.76, "word": "بناء", "probability": 0.9423828125}, {"start": 2097.76, "end": 2098.0, "word": " على", "probability": 0.7373046875}, {"start": 2098.0, "end": 2098.22, "word": " هذه", "probability": 0.8115234375}, {"start": 2098.22, "end": 2098.88, "word": " النظرية", "probability": 0.9542236328125}, {"start": 2098.88, "end": 2099.34, "word": " كنا", "probability": 0.784423828125}, {"start": 2099.34, "end": 2099.96, "word": " نقول", "probability": 0.942626953125}, {"start": 2099.96, "end": 2100.78, "word": " انه", "probability": 0.55712890625}, {"start": 2100.78, "end": 2101.1, "word": " لو", "probability": 0.8388671875}, {"start": 2101.1, "end": 2101.96, "word": " وجيت", "probability": 0.408935546875}, {"start": 2101.96, "end": 2102.96, "word": " two", "probability": 0.6396484375}, {"start": 2102.96, "end": 2104.18, "word": " sequences", "probability": 0.86669921875}, {"start": 2104.18, "end": 2104.8, "word": " على", "probability": 0.5859375}, {"start": 2104.8, "end": 2105.4, "word": " الأقل", "probability": 0.9558919270833334}, {"start": 2105.4, "end": 2106.0, "word": " ايكون", "probability": 0.66943359375}, {"start": 2106.0, "end": 2106.52, "word": " بروح", "probability": 0.9000651041666666}, {"start": 2106.52, "end": 2107.18, "word": " للـA", "probability": 0.377685546875}, {"start": 2107.18, "end": 2108.14, "word": " لكن", "probability": 0.76318359375}, {"start": 2108.14, "end": 2108.78, "word": " بروح", "probability": 0.9933268229166666}, {"start": 2108.78, "end": 2109.64, "word": " لـLimits", "probability": 0.585595703125}, {"start": 2109.64, "end": 2111.0, "word": " مختلفات", "probability": 0.837890625}, {"start": 2111.0, "end": 2112.0, "word": " او", "probability": 0.7081298828125}, {"start": 2112.0, "end": 2112.4, "word": " واحدة", "probability": 0.9772135416666666}, {"start": 2112.4, "end": 2112.58, "word": " من", "probability": 0.9912109375}, {"start": 2112.58, "end": 2113.06, "word": " Limits", "probability": 0.612548828125}, {"start": 2113.06, "end": 2113.5, "word": " does", "probability": 0.1541748046875}, {"start": 2113.5, "end": 2113.84, "word": " not", "probability": 0.96484375}, {"start": 2113.84, "end": 2114.4, "word": " exist", "probability": 0.96240234375}, {"start": 2114.4, "end": 2115.76, "word": " بكفالك", "probability": 0.744921875}, {"start": 2115.76, "end": 2116.0, "word": " ان", "probability": 0.8779296875}, {"start": 2116.0, "end": 2116.44, "word": " الـLimit", "probability": 0.6369140625}, {"start": 2116.44, "end": 2116.66, "word": " هذه", "probability": 0.392822265625}, {"start": 2116.66, "end": 2117.4, "word": " اشمالها", "probability": 0.8484375}, {"start": 2117.4, "end": 2118.08, "word": " does", "probability": 0.73583984375}, {"start": 2118.08, "end": 2118.6, "word": " not", "probability": 0.9638671875}, {"start": 2118.6, "end": 2119.42, "word": " exist", "probability": 0.9697265625}, {"start": 2119.42, "end": 2120.68, "word": " واضح؟", "probability": 0.7655029296875}, {"start": 2120.68, "end": 2121.16, "word": " الان", "probability": 0.657470703125}, {"start": 2121.16, "end": 2121.46, "word": " اذا", "probability": 0.827880859375}, {"start": 2121.46, "end": 2121.72, "word": " بدي", "probability": 0.5670166015625}, {"start": 2121.72, "end": 2122.16, "word": " اعتمد", "probability": 0.96630859375}, {"start": 2122.16, "end": 2122.4, "word": " على", "probability": 0.8388671875}, {"start": 2122.4, "end": 2122.96, "word": " هذه", "probability": 0.87890625}], "temperature": 1.0}, {"id": 90, "seek": 215296, "start": 2123.57, "end": 2152.97, "text": "في إثبات إنه limit cosine واحد على X does not exist أكيد كلكم جال take the sequences أو the two sequences X and بتساوي واحد على اتنين and by and Y and بتساوي واحد على اتنين and زائد واحد في by", "tokens": [41185, 11933, 12984, 3555, 9307, 36145, 3224, 4948, 23565, 36764, 24401, 15844, 1783, 775, 406, 2514, 5551, 4117, 25708, 28242, 24793, 10874, 6027, 747, 264, 22978, 34051, 264, 732, 22978, 1783, 293, 39894, 3794, 995, 45865, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 293, 538, 293, 398, 293, 39894, 3794, 995, 45865, 36764, 24401, 15844, 1975, 2655, 1863, 9957, 293, 30767, 16373, 3215, 36764, 24401, 8978, 538], "avg_logprob": -0.2734375030678861, "compression_ratio": 1.6242424242424243, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2123.57, "end": 2123.89, "word": "في", "probability": 0.67236328125}, {"start": 2123.89, "end": 2124.71, "word": " إثبات", "probability": 0.86346435546875}, {"start": 2124.71, "end": 2125.17, "word": " إنه", "probability": 0.4854736328125}, {"start": 2125.17, "end": 2125.75, "word": " limit", "probability": 0.69482421875}, {"start": 2125.75, "end": 2127.05, "word": " cosine", "probability": 0.313720703125}, {"start": 2127.05, "end": 2127.57, "word": " واحد", "probability": 0.668212890625}, {"start": 2127.57, "end": 2127.75, "word": " على", "probability": 0.5546875}, {"start": 2127.75, "end": 2128.01, "word": " X", "probability": 0.58154296875}, {"start": 2128.01, "end": 2128.23, "word": " does", "probability": 0.30224609375}, {"start": 2128.23, "end": 2128.49, "word": " not", "probability": 0.9638671875}, {"start": 2128.49, "end": 2128.87, "word": " exist", "probability": 0.9541015625}, {"start": 2128.87, "end": 2129.25, "word": " أكيد", "probability": 0.904296875}, {"start": 2129.25, "end": 2129.69, "word": " كلكم", "probability": 0.76953125}, {"start": 2129.69, "end": 2130.19, "word": " جال", "probability": 0.893798828125}, {"start": 2130.19, "end": 2131.93, "word": " take", "probability": 0.2098388671875}, {"start": 2131.93, "end": 2134.31, "word": " the", "probability": 0.64892578125}, {"start": 2134.31, "end": 2135.53, "word": " sequences", "probability": 0.5224609375}, {"start": 2135.53, "end": 2136.49, "word": " أو", "probability": 0.4775390625}, {"start": 2136.49, "end": 2136.61, "word": " the", "probability": 0.306640625}, {"start": 2136.61, "end": 2137.11, "word": " two", "probability": 0.9150390625}, {"start": 2137.11, "end": 2138.95, "word": " sequences", "probability": 0.96044921875}, {"start": 2138.95, "end": 2140.21, "word": " X", "probability": 0.763671875}, {"start": 2140.21, "end": 2140.73, "word": " and", "probability": 0.434326171875}, {"start": 2140.73, "end": 2141.93, "word": " بتساوي", "probability": 0.816650390625}, {"start": 2141.93, "end": 2142.95, "word": " واحد", "probability": 0.929443359375}, {"start": 2142.95, "end": 2143.77, "word": " على", "probability": 0.7763671875}, {"start": 2143.77, "end": 2144.33, "word": " اتنين", "probability": 0.9508056640625}, {"start": 2144.33, "end": 2144.61, "word": " and", "probability": 0.9462890625}, {"start": 2144.61, "end": 2145.15, "word": " by", "probability": 0.355224609375}, {"start": 2145.15, "end": 2146.69, "word": " and", "probability": 0.91943359375}, {"start": 2146.69, "end": 2148.03, "word": " Y", "probability": 0.7880859375}, {"start": 2148.03, "end": 2148.71, "word": " and", "probability": 0.95361328125}, {"start": 2148.71, "end": 2149.73, "word": " بتساوي", "probability": 0.96484375}, {"start": 2149.73, "end": 2150.29, "word": " واحد", "probability": 0.97314453125}, {"start": 2150.29, "end": 2150.51, "word": " على", "probability": 0.84912109375}, {"start": 2150.51, "end": 2150.95, "word": " اتنين", "probability": 0.975830078125}, {"start": 2150.95, "end": 2151.19, "word": " and", "probability": 0.9404296875}, {"start": 2151.19, "end": 2151.57, "word": " زائد", "probability": 0.9130859375}, {"start": 2151.57, "end": 2152.15, "word": " واحد", "probability": 0.97802734375}, {"start": 2152.15, "end": 2152.69, "word": " في", "probability": 0.9228515625}, {"start": 2152.69, "end": 2152.97, "word": " by", "probability": 0.4033203125}], "temperature": 1.0}, {"id": 91, "seek": 218087, "start": 2153.65, "end": 2180.87, "text": "مظبوط أكيد قلته هيك where x and goes to zero and y and goes to zero but cosine واحد على ايه؟ cosine واحد على x and اللي هو بتساوي او goes to واحد and cosine", "tokens": [2304, 19913, 3555, 2407, 9566, 5551, 4117, 25708, 12174, 1211, 47395, 39896, 4117, 689, 2031, 293, 1709, 281, 4018, 293, 288, 293, 1709, 281, 4018, 457, 23565, 36764, 24401, 15844, 1975, 1829, 3224, 22807, 23565, 36764, 24401, 15844, 2031, 293, 13672, 1829, 31439, 39894, 3794, 995, 45865, 1975, 2407, 1709, 281, 36764, 24401, 293, 23565], "avg_logprob": -0.31277902690427645, "compression_ratio": 1.4857142857142858, "no_speech_prob": 5.960464477539062e-07, "words": [{"start": 2153.65, "end": 2154.37, "word": "مظبوط", "probability": 0.59599609375}, {"start": 2154.37, "end": 2155.45, "word": " أكيد", "probability": 0.7209065755208334}, {"start": 2155.45, "end": 2155.79, "word": " قلته", "probability": 0.70458984375}, {"start": 2155.79, "end": 2157.03, "word": " هيك", "probability": 0.85400390625}, {"start": 2157.03, "end": 2157.87, "word": " where", "probability": 0.67041015625}, {"start": 2157.87, "end": 2160.43, "word": " x", "probability": 0.412109375}, {"start": 2160.43, "end": 2160.75, "word": " and", "probability": 0.3642578125}, {"start": 2160.75, "end": 2161.67, "word": " goes", "probability": 0.5751953125}, {"start": 2161.67, "end": 2161.93, "word": " to", "probability": 0.9833984375}, {"start": 2161.93, "end": 2162.43, "word": " zero", "probability": 0.5791015625}, {"start": 2162.43, "end": 2163.09, "word": " and", "probability": 0.7431640625}, {"start": 2163.09, "end": 2163.91, "word": " y", "probability": 0.888671875}, {"start": 2163.91, "end": 2164.23, "word": " and", "probability": 0.95361328125}, {"start": 2164.23, "end": 2164.53, "word": " goes", "probability": 0.88720703125}, {"start": 2164.53, "end": 2164.75, "word": " to", "probability": 0.97607421875}, {"start": 2164.75, "end": 2165.29, "word": " zero", "probability": 0.89794921875}, {"start": 2165.29, "end": 2166.49, "word": " but", "probability": 0.84619140625}, {"start": 2166.49, "end": 2168.15, "word": " cosine", "probability": 0.58251953125}, {"start": 2168.15, "end": 2169.75, "word": " واحد", "probability": 0.897216796875}, {"start": 2169.75, "end": 2170.13, "word": " على", "probability": 0.69140625}, {"start": 2170.13, "end": 2170.81, "word": " ايه؟", "probability": 0.572601318359375}, {"start": 2170.81, "end": 2171.39, "word": " cosine", "probability": 0.72607421875}, {"start": 2171.39, "end": 2172.11, "word": " واحد", "probability": 0.987060546875}, {"start": 2172.11, "end": 2172.37, "word": " على", "probability": 0.84033203125}, {"start": 2172.37, "end": 2172.71, "word": " x", "probability": 0.9091796875}, {"start": 2172.71, "end": 2173.13, "word": " and", "probability": 0.92822265625}, {"start": 2173.13, "end": 2174.01, "word": " اللي", "probability": 0.849365234375}, {"start": 2174.01, "end": 2174.35, "word": " هو", "probability": 0.958984375}, {"start": 2174.35, "end": 2176.11, "word": " بتساوي", "probability": 0.94677734375}, {"start": 2176.11, "end": 2176.69, "word": " او", "probability": 0.845458984375}, {"start": 2176.69, "end": 2177.19, "word": " goes", "probability": 0.92138671875}, {"start": 2177.19, "end": 2177.43, "word": " to", "probability": 0.94873046875}, {"start": 2177.43, "end": 2178.01, "word": " واحد", "probability": 0.982177734375}, {"start": 2178.01, "end": 2179.15, "word": " and", "probability": 0.95947265625}, {"start": 2179.15, "end": 2180.87, "word": " cosine", "probability": 0.8232421875}], "temperature": 1.0}, {"id": 92, "seek": 220146, "start": 2181.92, "end": 2201.46, "text": "1 على I and goes to ناقص واحد اوضحها اوضحها اذكركم الان هذا بيصير cosine مين اتنين and by اتنين and by ايش مالها واحد", "tokens": [16, 15844, 286, 293, 1709, 281, 8717, 995, 4587, 9381, 36764, 24401, 1975, 2407, 11242, 5016, 11296, 1975, 2407, 11242, 5016, 11296, 1975, 8848, 37983, 24793, 2423, 7649, 23758, 4724, 1829, 9381, 13546, 23565, 3714, 9957, 1975, 2655, 1863, 9957, 293, 538, 1975, 2655, 1863, 9957, 293, 538, 1975, 1829, 8592, 3714, 6027, 11296, 36764, 24401], "avg_logprob": -0.28207236842105265, "compression_ratio": 1.3969465648854962, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 2181.92, "end": 2182.42, "word": "1", "probability": 0.2349853515625}, {"start": 2182.42, "end": 2182.72, "word": " على", "probability": 0.1448974609375}, {"start": 2182.72, "end": 2183.1, "word": " I", "probability": 0.58544921875}, {"start": 2183.1, "end": 2183.48, "word": " and", "probability": 0.69189453125}, {"start": 2183.48, "end": 2184.98, "word": " goes", "probability": 0.6201171875}, {"start": 2184.98, "end": 2185.78, "word": " to", "probability": 0.97265625}, {"start": 2185.78, "end": 2186.52, "word": " ناقص", "probability": 0.6875}, {"start": 2186.52, "end": 2187.02, "word": " واحد", "probability": 0.851318359375}, {"start": 2187.02, "end": 2190.5, "word": " اوضحها", "probability": 0.8796875}, {"start": 2190.5, "end": 2191.04, "word": " اوضحها", "probability": 0.8671875}, {"start": 2191.04, "end": 2191.8, "word": " اذكركم", "probability": 0.8162841796875}, {"start": 2191.8, "end": 2195.6, "word": " الان", "probability": 0.6771240234375}, {"start": 2195.6, "end": 2195.8, "word": " هذا", "probability": 0.322021484375}, {"start": 2195.8, "end": 2196.16, "word": " بيصير", "probability": 0.8707275390625}, {"start": 2196.16, "end": 2196.5, "word": " cosine", "probability": 0.64892578125}, {"start": 2196.5, "end": 2197.02, "word": " مين", "probability": 0.56475830078125}, {"start": 2197.02, "end": 2198.12, "word": " اتنين", "probability": 0.9371337890625}, {"start": 2198.12, "end": 2198.38, "word": " and", "probability": 0.955078125}, {"start": 2198.38, "end": 2198.72, "word": " by", "probability": 0.79150390625}, {"start": 2198.72, "end": 2199.26, "word": " اتنين", "probability": 0.947021484375}, {"start": 2199.26, "end": 2199.52, "word": " and", "probability": 0.9658203125}, {"start": 2199.52, "end": 2199.76, "word": " by", "probability": 0.98779296875}, {"start": 2199.76, "end": 2199.98, "word": " ايش", "probability": 0.7976888020833334}, {"start": 2199.98, "end": 2200.54, "word": " مالها", "probability": 0.67333984375}, {"start": 2200.54, "end": 2201.46, "word": " واحد", "probability": 0.74267578125}], "temperature": 1.0}, {"id": 93, "seek": 222979, "start": 2202.17, "end": 2229.79, "text": "ا اتنين ان زاد واحد لكل في باقي ايش بتصير؟ اللي هو سالب واحد فبصير هاد ال cosine converts واحد و هاد ال cosine converts لمين؟ لسالب واحد اذا حسب اللي حكيناها قبل شوية hence hence hence limit cosine واحد على x لما x تروح ل zero does not exist", "tokens": [995, 1975, 2655, 1863, 9957, 16472, 30767, 18513, 36764, 24401, 5296, 28820, 8978, 4724, 995, 38436, 1975, 1829, 8592, 39894, 9381, 13546, 22807, 13672, 1829, 31439, 8608, 6027, 3555, 36764, 24401, 6156, 3555, 9381, 13546, 8032, 18513, 2423, 23565, 38874, 36764, 24401, 4032, 8032, 18513, 2423, 23565, 38874, 5296, 2304, 9957, 22807, 5296, 3794, 6027, 3555, 36764, 24401, 1975, 15730, 11331, 35457, 13672, 1829, 11331, 4117, 1829, 8315, 11296, 12174, 36150, 13412, 2407, 10632, 16678, 16678, 16678, 4948, 23565, 36764, 24401, 15844, 2031, 5296, 15042, 2031, 6055, 32887, 5016, 5296, 4018, 775, 406, 2514], "avg_logprob": -0.3348684216800489, "compression_ratio": 1.7871287128712872, "no_speech_prob": 4.231929779052734e-06, "words": [{"start": 2202.17, "end": 2202.45, "word": "ا", "probability": 0.078125}, {"start": 2202.45, "end": 2203.67, "word": " اتنين", "probability": 0.76654052734375}, {"start": 2203.67, "end": 2203.99, "word": " ان", "probability": 0.1522216796875}, {"start": 2203.99, "end": 2204.35, "word": " زاد", "probability": 0.681396484375}, {"start": 2204.35, "end": 2204.59, "word": " واحد", "probability": 0.98291015625}, {"start": 2204.59, "end": 2204.85, "word": " لكل", "probability": 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"probability": 0.98193359375}, {"start": 2214.59, "end": 2214.75, "word": " اللي", "probability": 0.974365234375}, {"start": 2214.75, "end": 2215.19, "word": " حكيناها", "probability": 0.9052734375}, {"start": 2215.19, "end": 2215.39, "word": " قبل", "probability": 0.9375}, {"start": 2215.39, "end": 2215.93, "word": " شوية", "probability": 0.7610677083333334}, {"start": 2215.93, "end": 2217.01, "word": " hence", "probability": 0.7470703125}, {"start": 2217.01, "end": 2218.49, "word": " hence", "probability": 0.342041015625}, {"start": 2218.49, "end": 2221.21, "word": " hence", "probability": 0.85009765625}, {"start": 2221.21, "end": 2222.01, "word": " limit", "probability": 0.93896484375}, {"start": 2222.01, "end": 2223.59, "word": " cosine", "probability": 0.7568359375}, {"start": 2223.59, "end": 2224.11, "word": " واحد", "probability": 0.980224609375}, {"start": 2224.11, "end": 2224.29, "word": " على", "probability": 0.65576171875}, {"start": 2224.29, "end": 2224.57, "word": " x", "probability": 0.53271484375}, {"start": 2224.57, "end": 2224.81, "word": " لما", "probability": 0.874755859375}, {"start": 2224.81, "end": 2225.07, "word": " x", "probability": 0.81005859375}, {"start": 2225.07, "end": 2225.33, "word": " تروح", "probability": 0.9903971354166666}, {"start": 2225.33, "end": 2225.47, "word": " ل", "probability": 0.85986328125}, {"start": 2225.47, "end": 2225.91, "word": " zero", "probability": 0.5400390625}, {"start": 2225.91, "end": 2227.09, "word": " does", "probability": 0.80224609375}, {"start": 2227.09, "end": 2228.11, "word": " not", "probability": 0.96875}, {"start": 2228.11, "end": 2229.79, "word": " exist", "probability": 0.97412109375}], "temperature": 1.0}, {"id": 94, "seek": 226046, "start": 2231.04, "end": 2260.46, "text": "هذا توضيح .. توضيح لماذا اللي هو cos 1 على x does not exist لاحظ التالي على جهة لكن لو كانت .. بالمناسبة لو كانت هذه does not exist limitها و هذه does not exist بقدرش أؤكد على إن ال limit للطرح أو الجامعه does not exist صح؟ في مثال؟", "tokens": [3224, 15730, 33427, 11242, 1829, 5016, 4386, 33427, 11242, 1829, 5016, 32767, 45636, 995, 13672, 1829, 31439, 3792, 502, 15844, 2031, 775, 406, 2514, 20193, 5016, 19913, 16712, 6027, 1829, 15844, 10874, 3224, 3660, 44381, 45164, 25961, 2655, 4386, 20666, 2304, 8315, 35457, 3660, 45164, 25961, 2655, 29538, 775, 406, 2514, 4948, 11296, 4032, 29538, 775, 406, 2514, 4724, 28543, 2288, 8592, 5551, 33604, 4117, 3215, 15844, 36145, 2423, 4948, 24976, 9566, 2288, 5016, 34051, 25724, 10943, 3615, 3224, 775, 406, 2514, 20328, 5016, 22807, 8978, 50113, 6027, 22807], "avg_logprob": -0.2229166724615627, "compression_ratio": 1.7487437185929648, "no_speech_prob": 8.940696716308594e-07, "words": [{"start": 2231.04, "end": 2231.4, "word": "هذا", "probability": 0.833984375}, {"start": 2231.4, "end": 2232.36, "word": " توضيح", "probability": 0.9510498046875}, {"start": 2232.36, "end": 2232.76, "word": " ..", "probability": 0.336181640625}, {"start": 2232.76, "end": 2234.76, 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التالي", "probability": 0.9466145833333334}, {"start": 2241.62, "end": 2241.84, "word": " على", "probability": 0.8974609375}, {"start": 2241.84, "end": 2242.26, "word": " جهة", "probability": 0.9015299479166666}, {"start": 2242.26, "end": 2243.62, "word": " لكن", "probability": 0.373046875}, {"start": 2243.62, "end": 2244.0, "word": " لو", "probability": 0.97705078125}, {"start": 2244.0, "end": 2245.04, "word": " كانت", "probability": 0.99658203125}, {"start": 2245.04, "end": 2245.1, "word": " ..", "probability": 0.4873046875}, {"start": 2245.1, "end": 2246.06, "word": " بالمناسبة", "probability": 0.9654296875}, {"start": 2246.06, "end": 2246.78, "word": " لو", "probability": 0.68115234375}, {"start": 2246.78, "end": 2247.28, "word": " كانت", "probability": 0.9853515625}, {"start": 2247.28, "end": 2247.56, "word": " هذه", "probability": 0.77392578125}, {"start": 2247.56, "end": 2247.76, "word": " does", "probability": 0.61328125}, {"start": 2247.76, "end": 2248.04, "word": " not", 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سالم بيصير limit لها دي موجودة أو limit لها دي موجودة .. آسف limit لها دي مش موجودة أو limit لها دي مش موجودة لكن اللي هو حصل إيش طرح موجود أو لو بدك جمحن بتحط اللي هو للتانية تاخد واحدة لأن ونقص واحدة لأن و هكذا أي سؤال؟ طيب", "tokens": [10721, 3224, 4032, 16490, 2288, 29245, 2655, 11296, 36764, 24401, 3660, 5296, 33456, 36764, 24401, 3660, 5296, 33456, 49374, 3224, 9957, 8608, 45340, 4724, 1829, 9381, 13546, 4948, 5296, 11296, 11778, 1829, 3714, 29245, 23328, 3660, 34051, 4948, 5296, 11296, 11778, 1829, 3714, 29245, 23328, 3660, 4386, 19753, 3794, 5172, 4948, 5296, 11296, 11778, 1829, 37893, 3714, 29245, 23328, 3660, 34051, 4948, 5296, 11296, 11778, 1829, 37893, 3714, 29245, 23328, 3660, 44381, 13672, 1829, 31439, 11331, 36520, 11933, 1829, 8592, 23032, 2288, 5016, 3714, 29245, 23328, 34051, 45164, 47525, 4117, 10874, 2304, 5016, 1863, 39894, 5016, 9566, 13672, 1829, 31439, 24976, 2655, 7649, 10632, 6055, 47283, 3215, 36764, 24401, 3660, 5296, 33456, 4032, 1863, 4587, 9381, 36764, 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273834, "start": 2713.16, "end": 2738.34, "text": "ماشي الحال يعني F differentiable عنده نقطة C element وين؟ ن I بقول لك then طبعاً A و F prime of C أشمالها لاتساوي سفر لأنه هيلزمنا القسم عليها عشان هيك لازم اللي هو عشان نحكي عن ال inverse لال derivative تكون ال F prime عند ال C أشمالها لاتساوي سفر الآن then", "tokens": [2304, 33599, 1829, 21542, 6027, 37495, 22653, 479, 819, 9364, 43242, 3224, 8717, 47432, 3660, 383, 4478, 4032, 9957, 22807, 8717, 286, 4724, 39648, 5296, 4117, 550, 23032, 3555, 3615, 995, 14111, 316, 4032, 479, 5835, 295, 383, 5551, 8592, 2304, 6027, 11296, 5296, 9307, 3794, 995, 45865, 8608, 5172, 2288, 5296, 33456, 3224, 39896, 1211, 11622, 2304, 8315, 25062, 38251, 25894, 11296, 6225, 8592, 7649, 39896, 4117, 5296, 31377, 2304, 13672, 1829, 31439, 6225, 8592, 7649, 8717, 5016, 4117, 1829, 18871, 2423, 17340, 5296, 6027, 13760, 6055, 30544, 2423, 479, 5835, 43242, 2423, 383, 5551, 8592, 2304, 6027, 11296, 5296, 9307, 3794, 995, 45865, 8608, 5172, 2288, 6024, 48506, 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ال", "probability": 0.71337890625}, {"start": 2734.08, "end": 2734.18, "word": " C", "probability": 0.912109375}, {"start": 2734.18, "end": 2734.88, "word": " أشمالها", "probability": 0.9923828125}, {"start": 2734.88, "end": 2735.92, "word": " لاتساوي", "probability": 0.9197265625}, {"start": 2735.92, "end": 2736.36, "word": " سفر", "probability": 0.98974609375}, {"start": 2736.36, "end": 2737.7, "word": " الآن", "probability": 0.6705322265625}, {"start": 2737.7, "end": 2738.34, "word": " then", "probability": 0.82763671875}], "temperature": 1.0}, {"id": 114, "seek": 276641, "start": 2740.63, "end": 2766.41, "text": "اللي هي الـ G الـ G اللي سميناها F inverse اللي سميناها F inverse الـ G is differentiable is differentiable at اللي هو F of C is differentiable at إيش؟ at F of C اللي هو مسميها إيش اسمها دي", "tokens": [6027, 20292, 39896, 2423, 39184, 460, 2423, 39184, 460, 13672, 1829, 8608, 2304, 1829, 8315, 11296, 479, 17340, 13672, 1829, 8608, 2304, 1829, 8315, 11296, 479, 17340, 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15730, 25961, 11331, 4117, 1829, 8315, 11296, 15844, 21136, 16572, 3615], "avg_logprob": -0.2590725907394963, "compression_ratio": 2.6419753086419755, "no_speech_prob": 2.86102294921875e-06, "words": [{"start": 4017.21, "end": 4017.47, "word": "التي", "probability": 0.4805908203125}, {"start": 4017.47, "end": 4017.83, "word": " قبل", "probability": 0.920166015625}, {"start": 4017.83, "end": 4018.11, "word": " بشوية", "probability": 0.7115478515625}, {"start": 4018.11, "end": 4018.55, "word": " حكيناها", "probability": 0.68515625}, {"start": 4018.55, "end": 4018.73, "word": " جي", "probability": 0.54193115234375}, {"start": 4018.73, "end": 4018.91, "word": " برايم", "probability": 0.8689778645833334}, {"start": 4018.91, "end": 4019.01, "word": " أو", "probability": 0.3291015625}, {"start": 4019.01, "end": 4019.09, "word": " اي", "probability": 0.632080078125}, {"start": 4019.09, "end": 4019.33, "word": " بيسوا", "probability": 0.7554931640625}, {"start": 4019.33, "end": 4019.53, "word": " واحد", "probability": 0.8642578125}, {"start": 4019.53, "end": 4019.61, "word": " على", "probability": 0.64599609375}, {"start": 4019.61, "end": 4019.75, "word": " اف", "probability": 0.707763671875}, {"start": 4019.75, "end": 4019.97, "word": " برايم", "probability": 0.9698893229166666}, {"start": 4019.97, "end": 4020.31, "word": " كومبز", "probability": 0.5664825439453125}, {"start": 4020.31, "end": 4020.65, "word": " الجي", "probability": 0.7021484375}, {"start": 4020.65, "end": 4021.49, "word": " ماشي", "probability": 0.7369791666666666}, {"start": 4021.49, "end": 4021.85, "word": " الحال", "probability": 0.96728515625}, {"start": 4021.85, "end": 4022.73, "word": " الآن", "probability": 0.57965087890625}, {"start": 4022.73, "end": 4023.55, "word": " صارت", "probability": 0.9430338541666666}, {"start": 4023.55, "end": 4023.89, "word": " عندي", "probability": 0.906982421875}, {"start": 4023.89, "end": 4024.73, "word": " هذه", "probability": 0.69873046875}, {"start": 4024.73, "end": 4025.67, "word": " جي", "probability": 0.78564453125}, {"start": 4025.67, "end": 4025.97, "word": " برايم", "probability": 0.982421875}, {"start": 4025.97, "end": 4026.61, "word": " كومبز", "probability": 0.8922119140625}, {"start": 4026.61, "end": 4027.51, "word": " كومبز", "probability": 0.80194091796875}, {"start": 4027.51, "end": 4027.73, "word": " اف", "probability": 0.38958740234375}, {"start": 4027.73, "end": 4027.89, "word": " of", "probability": 0.5849609375}, {"start": 4027.89, "end": 4028.05, "word": " X", "probability": 0.395751953125}, {"start": 4028.05, "end": 4028.29, "word": " بيسوا", "probability": 0.931884765625}, {"start": 4028.29, "end": 4028.49, "word": " جي", "probability": 0.954833984375}, {"start": 4028.49, "end": 4028.73, "word": " برايم", "probability": 0.9833984375}, {"start": 4028.73, "end": 4028.95, "word": " اف", "probability": 0.91064453125}, {"start": 4028.95, "end": 4029.09, "word": " of", "probability": 0.97607421875}, {"start": 4029.09, "end": 4029.27, "word": " X", "probability": 0.99365234375}, {"start": 4029.27, "end": 4029.49, "word": " بيسوا", "probability": 0.9791259765625}, {"start": 4029.49, "end": 4029.85, "word": " واحد", "probability": 0.980224609375}, {"start": 4029.85, "end": 4030.51, "word": " على", "probability": 0.82763671875}, {"start": 4030.51, "end": 4030.77, "word": " اف", "probability": 0.92529296875}, {"start": 4030.77, "end": 4031.21, "word": " برايم", "probability": 0.955078125}, {"start": 4031.21, "end": 4031.71, "word": " كومبز", "probability": 0.989013671875}, {"start": 4031.71, "end": 4032.03, "word": " الجي", "probability": 0.645751953125}, {"start": 4032.03, "end": 4032.17, "word": " أو", "probability": 0.73779296875}, {"start": 4032.17, "end": 4032.29, "word": " اي", "probability": 0.485107421875}, {"start": 4032.29, "end": 4032.69, "word": " بياخد", "probability": 0.57208251953125}, {"start": 4032.69, "end": 4033.01, "word": " نظرية", "probability": 0.9210205078125}, {"start": 4033.01, "end": 4033.43, "word": " مرتين", "probability": 0.9933268229166666}, {"start": 4033.43, "end": 4033.93, "word": " و", "probability": 0.76220703125}, {"start": 4033.93, "end": 4034.17, "word": " يسوا", "probability": 0.7513834635416666}, {"start": 4034.17, "end": 4034.37, "word": " واحد", "probability": 0.98291015625}, {"start": 4034.37, "end": 4034.49, "word": " على", "probability": 0.88916015625}, {"start": 4034.49, "end": 4034.67, "word": " اف", "probability": 0.96240234375}, {"start": 4034.67, "end": 4034.97, "word": " برايم", "probability": 0.9676106770833334}, {"start": 4034.97, "end": 4035.13, "word": " of", "probability": 0.444580078125}, {"start": 4035.13, "end": 4035.47, "word": " X", "probability": 0.990234375}, {"start": 4035.47, "end": 4036.19, "word": " التانيين", "probability": 0.49615478515625}, {"start": 4036.19, "end": 4036.41, "word": " مع", "probability": 0.986328125}, {"start": 4036.41, "end": 4036.67, "word": " بعض", "probability": 0.992919921875}, {"start": 4036.67, "end": 4036.87, "word": " جي", "probability": 0.95703125}, {"start": 4036.87, "end": 4037.03, "word": " برايم", "probability": 0.9918619791666666}, {"start": 4037.03, "end": 4037.31, "word": " بيسوا", "probability": 0.972900390625}, {"start": 4037.31, "end": 4037.55, "word": " واحد", "probability": 0.99365234375}, {"start": 4037.55, "end": 4037.67, "word": " على", "probability": 0.9375}, {"start": 4037.67, "end": 4037.87, "word": " اف", "probability": 0.941162109375}, {"start": 4037.87, "end": 4038.37, "word": " برايم", "probability": 0.984375}, {"start": 4038.37, "end": 4039.27, "word": " of", "probability": 0.9345703125}, {"start": 4039.27, "end": 4039.61, "word": " X", "probability": 0.9619140625}, {"start": 4039.61, "end": 4040.69, "word": " و", "probability": 0.6181640625}, {"start": 4040.69, "end": 4040.95, "word": " المرة", "probability": 0.814208984375}, {"start": 4040.95, "end": 4041.29, "word": " الجاهزة", "probability": 0.722265625}, {"start": 4041.29, "end": 4041.43, "word": " اللي", "probability": 0.5963134765625}, {"start": 4041.43, "end": 4042.03, "word": " بنكمل", "probability": 0.77294921875}, {"start": 4042.03, "end": 4043.49, "word": " بنعيد", "probability": 0.9305013020833334}, {"start": 4043.49, "end": 4043.81, "word": " هذا", "probability": 0.9580078125}, {"start": 4043.81, "end": 4044.09, "word": " اللي", "probability": 0.917724609375}, {"start": 4044.09, "end": 4044.31, "word": " إذا", "probability": 0.76318359375}, {"start": 4044.31, "end": 4044.67, "word": " كان", "probability": 0.9892578125}, {"start": 4044.67, "end": 4045.95, "word": " حكيناها", "probability": 0.833203125}, {"start": 4045.95, "end": 4046.03, "word": " على", "probability": 0.71875}, {"start": 4046.03, "end": 4046.47, "word": " السريع", "probability": 0.96484375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 4047.88825, "duration_after_vad": 3872.6581249999795}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..62d2f6963bddce0bae3a04c4f8df07bf109dc9b2
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/YaYohwowYOA_raw.srt
@@ -0,0 +1,2768 @@
+1
+00:00:04,820 --> 00:00:10,480
+بسم الله الرحمن الرحيم المحاضرة الثانية في مساق
+
+2
+00:00:10,480 --> 00:00:18,480
+تحليل رياضي 2 أو تحليل حقيقي 2 اتحدثنا
+
+3
+00:00:18,480 --> 00:00:24,100
+المرة الماضية عنالـ differentiation بدأنا بشرتر
+
+4
+00:00:24,100 --> 00:00:28,200
+ستة بدأنا في ال section الأول ستة واحد اللي هو ال
+
+5
+00:00:28,200 --> 00:00:32,600
+derivative او اللي هو الاشتقاق او المشتق الأولى
+
+6
+00:00:32,600 --> 00:00:37,840
+وعرفنا شو معناه اللي هو الاشتقاق عرفنا شو معناه ان
+
+7
+00:00:37,840 --> 00:00:44,910
+الدولة تكون differentiable عند نقطة اللي هيفي
+
+8
+00:00:44,910 --> 00:00:49,970
+interval I حيث أخدنا F عبارة عن الـ function من I
+
+9
+00:00:49,970 --> 00:00:55,230
+ل R وقلنا C element in I بتكون الدالة عندها قابلة
+
+10
+00:00:55,230 --> 00:01:01,490
+للاشتقاق يعني F prime of C exist if the following
+
+11
+00:01:01,490 --> 00:01:08,230
+limit exist اللي هو limitf of x ناقص f of c على x
+
+12
+00:01:08,230 --> 00:01:14,230
+minus c as x goes to c exist إذا كانت هذه الـ
+
+13
+00:01:14,230 --> 00:01:20,210
+limit exist بنقول عن الـ f برايب exist و بتكون هي
+
+14
+00:01:20,210 --> 00:01:26,650
+قيمة هذه ال limit الآن و بعد ذلك مرّينا على اللي
+
+15
+00:01:26,650 --> 00:01:34,600
+هو بعض الخواصوقلنا الخاصية الأولى عندنا اللى هى
+
+16
+00:01:34,600 --> 00:01:41,540
+انه لو كانت الدالة اللى هى differentiable at C
+
+17
+00:01:41,540 --> 00:01:45,520
+بتكون continuous برضه وين عند C يعني اي
+
+18
+00:01:45,520 --> 00:01:50,080
+differentiable function must be continuousقلنا إن
+
+19
+00:01:50,080 --> 00:01:53,960
+الـ converse need not to be true in general واخدنا
+
+20
+00:01:53,960 --> 00:01:58,020
+مثال قلنا f of x بتساوي ال absolute value لل x
+
+21
+00:01:58,020 --> 00:02:01,780
+اللي هي عبارة عن continuous function لكن مش
+
+22
+00:02:01,780 --> 00:02:05,460
+differentiable عند ال zero continuous everywhere
+
+23
+00:02:05,460 --> 00:02:09,340
+but it is not differentiable at zeroوبرهننا الكلام
+
+24
+00:02:09,340 --> 00:02:13,140
+هذا واخدنا مثال آخر على f of x بتساوي ال summation
+
+25
+00:02:13,140 --> 00:02:17,760
+ل x minus cii من عند واحد لعند n وقلنا إن هذا
+
+26
+00:02:17,760 --> 00:02:22,600
+الدالة برضه is continuous everywhere but it is not
+
+27
+00:02:22,600 --> 00:02:29,440
+differentiable at c1, c2, c3 لعند cn بعد ذلك سألنا
+
+28
+00:02:29,440 --> 00:02:34,100
+وقلنا هل ممكن اللي جينا دالة تكون continuous وين
+
+29
+00:02:34,100 --> 00:02:39,790
+ما كانو مش differentiable ولا في مكان قلنا اه و
+
+30
+00:02:39,790 --> 00:02:42,470
+قلنا اخد الدالة f of x بساعة ال summation واحد على
+
+31
+00:02:42,470 --> 00:02:47,350
+اتنين أس ان كسين تلاتة أس ان اكس هذه الدالة is
+
+32
+00:02:47,350 --> 00:02:51,850
+continuous everywhere but it is not differentiable
+
+33
+00:02:51,850 --> 00:02:56,310
+anywhere وبعد هيك اتينا
+
+34
+00:03:04,880 --> 00:03:11,100
+حكينا عن بعض اللي هو قوانين الاشتقاق قولنا لو كانت
+
+35
+00:03:11,100 --> 00:03:16,520
+الدالةF والدالة G التنتين من نفس الـ domain
+
+36
+00:03:16,520 --> 00:03:22,000
+والتنتين قابلات للاشتقاق عند نقطة اسمها C سمناها
+
+37
+00:03:22,000 --> 00:03:26,560
+قلنا اللي هو مجموع الدالتين هيكون مشتقتهم موجودة
+
+38
+00:03:26,560 --> 00:03:31,660
+عند هذه النقطة وحاصل ضربهم وحاصل قسمتهم أيضا
+
+39
+00:03:31,660 --> 00:03:35,860
+provided that G of C اللي في المقام لا تساوي سفر
+
+40
+00:03:35,860 --> 00:03:41,950
+وأيضا الالف في ال FA constant برضه ألف في أف قابلة
+
+41
+00:03:41,950 --> 00:03:46,290
+للاشتقاء وأثبتنا
+
+42
+00:03:46,290 --> 00:03:53,330
+هذا الكلام وبرهنناه ووصلنا لعند اللي هو التعميم
+
+43
+00:03:53,330 --> 00:04:00,030
+قلنا لو كانت FF1 وF2 وF3 عند F1 كلهم دوال قابلة
+
+44
+00:04:00,030 --> 00:04:04,710
+للاشتقاق حيكون حاصل جمحا قابل للاشتقاق وحاصل ضربا
+
+45
+00:04:04,710 --> 00:04:09,510
+أيضا قابل للاشتقاق وفق القوانين اللي ذكرناها سابقا
+
+46
+00:04:10,150 --> 00:04:14,370
+وجفنا عند اللي هو النقطة اللي هي the chain rule أو
+
+47
+00:04:14,370 --> 00:04:20,650
+قاعدة السلسلة قاعدة السلسلة طبعا ليست غريبة عنكم
+
+48
+00:04:20,650 --> 00:04:29,240
+مرت علينا في تفاضل الف ومرت علينا في ال complexأو
+
+49
+00:04:29,240 --> 00:04:32,780
+مرت علينا اللي هو استخدمناها في الـ differential
+
+50
+00:04:32,780 --> 00:04:38,220
+equations وفي مساقات أخرى الآن نظرة نظرة تحليلية
+
+51
+00:04:38,220 --> 00:04:44,060
+فهنشوف اللي هو كيف أثبتنا أو كيف بدنا نثبت اللي هو
+
+52
+00:04:44,060 --> 00:04:48,840
+قاعدة السلسلة أي سؤال بتقدر تسأله يعني اللي بدكم
+
+53
+00:04:48,840 --> 00:04:53,960
+إياه و تتفعلوا بالطريقة اللي بتلاقوها مناسبة let i
+
+54
+00:04:53,960 --> 00:04:56,200
+و j be intervals and let g
+
+55
+00:04:59,060 --> 00:05:06,800
+عبارة عن دالة دي
+
+56
+00:05:06,800 --> 00:05:14,660
+عبارة عن دالة من I لعند مين لعند R و ال F عبارة عن
+
+57
+00:05:14,660 --> 00:05:21,700
+دالة من عند J لعند R حيث I و J اللي هو أشمل عبارة
+
+58
+00:05:21,700 --> 00:05:26,440
+عن intervalsالان انا اللي هي الـ chain rule هي في
+
+59
+00:05:26,440 --> 00:05:33,380
+الواقع عملية فحص إمكانية إنها تكون ال composition
+
+60
+00:05:33,380 --> 00:05:37,780
+of two functions is differentiable ولا لأ؟ وإيش
+
+61
+00:05:37,780 --> 00:05:41,200
+الشروط اللي بتخلي اللي هو g composite of is
+
+62
+00:05:41,200 --> 00:05:45,620
+differentiable؟ إذا أنا بدي أحكي عن g composite of
+
+63
+00:05:45,620 --> 00:05:49,590
+اللي هو ال derivative إيه لها؟ما دام بدي احكي عن g
+
+64
+00:05:49,590 --> 00:05:56,690
+composite of f اذا انا الان domain ال g ال domain
+
+65
+00:05:56,690 --> 00:06:03,850
+ال f هيكون من عند g لعند مين لعند r معناته ان f of
+
+66
+00:06:03,850 --> 00:06:09,310
+g عشان تكون g معرفة لازم تكون من ضمن مين اللي هو
+
+67
+00:06:09,310 --> 00:06:16,680
+domain ال I اذا انا بدي افترض ان fof J جزئية من I
+
+68
+00:06:16,680 --> 00:06:22,020
+على أساس ان هتكون الـ G composite F is defined
+
+69
+00:06:22,020 --> 00:06:29,960
+وهتكون defined من وين؟ من عند J لعند R عشان هيك
+
+70
+00:06:29,960 --> 00:06:34,400
+احنا لما بدنا نيجياللي هو .. اللي نفترض .. هنفترض
+
+71
+00:06:34,400 --> 00:06:40,220
+G من I لعند R وF من عند G لعند R دوال ونقول F of I
+
+72
+00:06:40,220 --> 00:06:43,640
+subset او F of G subset من I عشان تصير اللي هو
+
+73
+00:06:43,640 --> 00:06:46,980
+اللي بنحكي عنه قاعدة السلسلية اللي هي بمعنى اخر
+
+74
+00:06:46,980 --> 00:06:50,040
+التفاضل ال composition of two functions is well
+
+75
+00:06:50,040 --> 00:06:54,420
+defined او is defined اللي هم let C element in مين
+
+76
+00:06:54,420 --> 00:07:00,120
+in G؟ بدنا ناخد الـ C element in G اللي هم بقولي C
+
+77
+00:07:00,120 --> 00:07:06,430
+element in Gعشان أقدر أحكي عن g composite of f
+
+78
+00:07:06,430 --> 00:07:12,530
+prime of c تكون exist بده شرطين الشرط الأول بده ال
+
+79
+00:07:12,530 --> 00:07:17,190
+g ال f تكون differentiable عند مين؟ عند ال c
+
+80
+00:07:17,190 --> 00:07:22,150
+والشرط التاني أن ال g تكون differentiable عند ال f
+
+81
+00:07:22,150 --> 00:07:30,530
+of c واضح؟ إذن الآنإذا كان الـ F مميز لـ C و الـ G
+
+82
+00:07:30,530 --> 00:07:36,710
+مميز لـ F of C فإن مفهوم الـ G مميز لـ F يجب أن
+
+83
+00:07:36,710 --> 00:07:42,410
+يكون مميز لـ C وبكون عنده الـ G مميز لـ F' of C
+
+84
+00:07:42,410 --> 00:07:48,710
+أيش حيساوي؟ حيساوي G' of F of C في F' لمن؟ للـ F
+
+85
+00:07:48,710 --> 00:07:54,350
+of C اللي كنا نقول عنه في التفاضلأنه تفاضل اللي هو
+
+86
+00:07:54,350 --> 00:08:01,320
+القوس ثم تفاضل ما فيه داخل القوس واضح؟أو تفاضل
+
+87
+00:08:01,320 --> 00:08:05,280
+الدالة في تفاضل ما بداخل الدالة هو طبعا في الواقع
+
+88
+00:08:05,280 --> 00:08:07,940
+اللي هو عبارة عن تفاضل ل composite of two
+
+89
+00:08:07,940 --> 00:08:12,200
+functions أو اللي هو ال chain rule وكنا إذا
+
+90
+00:08:12,200 --> 00:08:16,680
+بتتذكروا كانت صورة أخرى لل chain rule غير الصورة
+
+91
+00:08:16,680 --> 00:08:23,100
+هذه اللي هي g prime of f of c في f prime of c كنا
+
+92
+00:08:23,100 --> 00:08:29,900
+ناخد صورة أخرى نقول dy by dxاللي هو عبارة عن ..
+
+93
+00:08:29,900 --> 00:08:38,960
+عبارة عن dy مثلا by du في du by d إيش؟ dx اللي هو
+
+94
+00:08:38,960 --> 00:08:44,620
+ال y و كأنها بتتمثل اللي هي ال g تفاضل ال g و ال u
+
+95
+00:08:44,620 --> 00:08:48,600
+بتكون اللي هي مين؟ اللي هي ال F اللي هي F إبراى
+
+96
+00:08:48,600 --> 00:08:52,180
+فكانت اللي هي هذه الصورة أو هذه الصورة احنا الآن
+
+97
+00:08:52,180 --> 00:08:59,630
+خلينا نحكي عن هذه الصورة و نبرهن اللي هيكيف اللي
+
+98
+00:08:59,630 --> 00:09:07,290
+هو نثبت ان هذا الكلام صحيح الان بس خلّينا نحط ..
+
+99
+00:09:07,290 --> 00:09:13,730
+نشوف .. نشوف عندنا الان نشوف ايش اللي بدنا نثبته
+
+100
+00:09:13,730 --> 00:09:16,770
+احنا في الواقع اللي بدنا نثبته .. بدنا نجوه ..
+
+101
+00:09:16,770 --> 00:09:22,290
+بدنا نثبت ان G composite F a prime of C exist عشان
+
+102
+00:09:22,290 --> 00:09:29,500
+نثبت انها exist بإيش بتبلزمني؟ بدي اخد limitG of
+
+103
+00:09:29,500 --> 00:09:39,520
+F, G circle F اللي هي GF of X ناقص G of F of C على
+
+104
+00:09:39,520 --> 00:09:46,960
+X minus C لما X تروح للـ C نثبت أن هذا إش ماله
+
+105
+00:09:46,960 --> 00:09:51,920
+existواضح عشان اللي بدنا نسويه؟ بدنا نثبت انه
+
+106
+00:09:51,920 --> 00:09:55,800
+exist عشان نثبت انه هذا exist بلزمني عاملين
+
+107
+00:09:55,800 --> 00:10:01,020
+مساعدات الآن هنشوف كيف نعمل هذول العاملين طبعا
+
+108
+00:10:01,020 --> 00:10:05,200
+طبيعي طبيعي انا معطين ان هو ال F' موجودة وD'
+
+109
+00:10:05,480 --> 00:10:10,600
+موجودة اذا اكيد انا هعمل حسابي ان استخدمهم على
+
+110
+00:10:10,600 --> 00:10:14,280
+اساس ان اصل الهدف اللي بديه الان خلينا نيجي
+
+111
+00:10:14,280 --> 00:10:14,900
+للبرهان
+
+112
+00:10:20,860 --> 00:10:26,680
+الحين الـ proof بدنا
+
+113
+00:10:26,680 --> 00:10:36,080
+نثبت انه اللي هو الـ Gبنثبت إنه لو كانت G و F
+
+114
+00:10:36,080 --> 00:10:39,980
+differentiable الأول عند F of C والتاني عند C
+
+115
+00:10:39,980 --> 00:10:45,840
+بيعطينا هذا إنه G composed of F prime of C exist
+
+116
+00:10:45,840 --> 00:10:50,420
+ويساوي G prime of F of C في F prime of C هذا ما
+
+117
+00:10:50,420 --> 00:10:58,000
+بنثبته شوف كيف يعني
+
+118
+00:10:58,000 --> 00:11:06,450
+بقى بدنا نبرهن النظريةخلّيني أعرّف الآن Define G
+
+119
+00:11:06,450 --> 00:11:14,850
+Capital من I لعند ال R By شوف كيف بدي .. اللي هي
+
+120
+00:11:14,850 --> 00:11:21,050
+يعني أعرف الـ G بطريقة توصّلني لحساب D composed of
+
+121
+00:11:21,050 --> 00:11:24,890
+F prime of إيش؟ of C جبل بشوية قلنا D composed of
+
+122
+00:11:24,890 --> 00:11:30,590
+F prime of C المتوقع لهاإنه تكون عبارة عن إذا كانت
+
+123
+00:11:30,590 --> 00:11:34,850
+تكون موجودة هي عبارة عن ال limit ل G F of X نقص G
+
+124
+00:11:34,850 --> 00:11:41,270
+F of C على X minus C أو G of Y نقص G F of C ماشي
+
+125
+00:11:41,270 --> 00:11:47,250
+الحال الآن بدي أعرف G على هذا الأساس G of Y بتساوي
+
+126
+00:11:47,250 --> 00:12:01,160
+اللي هو G of F of Yالعند ناقص g f of c على y minus
+
+127
+00:12:01,160 --> 00:12:06,480
+f of c الان قسمت على y minus f of c اكيد اللي هو
+
+128
+00:12:06,480 --> 00:12:15,600
+ال y انا بديش ايها تساومين f of c الان g of y ايش
+
+129
+00:12:15,600 --> 00:12:23,980
+بديها انا بديها المتوقع ال limit لهذهالـ limit
+
+130
+00:12:23,980 --> 00:12:30,300
+لهذه لما y تروح لل F of C اللي هو و كأنه عبارة عن
+
+131
+00:12:30,300 --> 00:12:37,540
+D prime of مين of F of C فالان بتديها D prime of F
+
+132
+00:12:37,540 --> 00:12:44,180
+of C ليش هال gate بتشوف ليش اللي بنحكيهاالان عرفت
+
+133
+00:12:44,180 --> 00:12:49,140
+اللي هي الـ function طبعاً هذا الكلام مستقى من
+
+134
+00:12:49,140 --> 00:12:54,120
+اللي هو f prime of c إيش متوقع limitه تساوي هذا
+
+135
+00:12:54,120 --> 00:12:58,200
+مستقى .. مستقى منه بعد شوية هتلاقيني هذه كتبت على
+
+136
+00:12:58,200 --> 00:13:02,980
+صورة هذه في شغل تانية برضه موجودة اللي هي f prime
+
+137
+00:13:02,980 --> 00:13:07,260
+of c اه
+
+138
+00:13:07,260 --> 00:13:16,350
+أسف g of yالان نشوف هذه الدالة لو جيت للدالة اللى
+
+139
+00:13:16,350 --> 00:13:26,010
+عندى هذه خد limit g of y لما y تروح لل F of C ايش
+
+140
+00:13:26,010 --> 00:13:32,500
+هتلاقيها بتساوي؟ بتساوي limitG of Y بعوض جاعد Y
+
+141
+00:13:32,500 --> 00:13:38,080
+تروح لل F of C يعني Y أشمالها لاتساوي F of C ناقص
+
+142
+00:13:38,080 --> 00:13:45,340
+G of F of C على مين على Y minus F of C لما Y تروح
+
+143
+00:13:45,340 --> 00:13:51,560
+لمين لل F of C وهذه بالظبط هي تعريف مينتعريف الـ
+
+144
+00:13:51,560 --> 00:13:56,480
+derivative للـ G عند النقطة مين؟ النقطة اسمها F of
+
+145
+00:13:56,480 --> 00:14:01,280
+C واضح؟ لأن الـ C من الـ A .. من الـ J الـ F of C
+
+146
+00:14:01,280 --> 00:14:08,260
+وين هتكون؟ من مين؟ من الـ I واضح؟ فالان هذه عبارة
+
+147
+00:14:08,260 --> 00:14:16,920
+عن G prime للـ F of C ليش هذا؟ because ..G is
+
+148
+00:14:16,920 --> 00:14:25,200
+differentiable at mean at F of C هذا معطى طيب الآن
+
+149
+00:14:25,200 --> 00:14:30,160
+ال limit لـ G of Y طلعت لما يتروح ل F of C أش طلعت
+
+150
+00:14:30,160 --> 00:14:34,520
+G prime ل F of C أصلا هذا السبب اللي أنا عارف أنه
+
+151
+00:14:34,520 --> 00:14:37,440
+هتطلع حاجة السبب اللي خلاني أعرف الـ G of Y أش
+
+152
+00:14:37,440 --> 00:14:44,510
+متساوي G prime F of C عشان تطلعلي منه هذا بساويG
+
+153
+00:14:44,510 --> 00:14:52,510
+عند مين عند Y بتساوي F of C يعني هذا سبب لإنه الـ
+
+154
+00:14:52,510 --> 00:14:57,610
+G هذه تصير إيش ماله is continuous عند مين عند ال F
+
+155
+00:14:57,610 --> 00:15:07,490
+of C إذا hence G is continuous at F of C طب ليش
+
+156
+00:15:07,490 --> 00:15:15,570
+هذا بعد شوية هنشوف وين هنستخدمهاالآن limit g of y
+
+157
+00:15:15,570 --> 00:15:22,890
+طالع عندي بيساوي g of f of c اللي هي هذه g of f of
+
+158
+00:15:22,890 --> 00:15:29,550
+c بيساوي g برايم f of c نيجي الآن انا ايش اللي بدي
+
+159
+00:15:29,550 --> 00:15:35,910
+ا .. بدي الآن اثبتلك ان g composite f برايم of c
+
+160
+00:15:35,910 --> 00:15:48,970
+بيساوي limit gof F of X ناقص G of F of C على X
+
+161
+00:15:48,970 --> 00:15:55,410
+minus C لما X تروح لل C بدي أثبت أنه exist بظبط؟
+
+162
+00:15:55,410 --> 00:16:02,610
+واضح هاه؟ طيب بناءً عليه خليني أكتب هذه بدلالة إشي
+
+163
+00:16:02,610 --> 00:16:11,060
+بوصلني للي بديها شوف كيفالان نحسب هذه هذه لمين for
+
+164
+00:16:11,060 --> 00:16:22,260
+x element n اللي هي j أو x لا تساوي مين لا تساوي c
+
+165
+00:16:22,260 --> 00:16:30,120
+معايا اه الان احسبولي هذه هذه عبارة عن تساوي g of
+
+166
+00:16:30,120 --> 00:16:37,650
+f of x اللي هونقص g f of c بدي اكتبها بطريقة g f
+
+167
+00:16:37,650 --> 00:16:49,250
+of x نقص g of f of c على f of x نقص f of c في f of
+
+168
+00:16:49,250 --> 00:16:58,630
+x نقص f of c على x minus cدخلت هذا المقدار في الـ
+
+169
+00:16:58,630 --> 00:17:04,410
+bust و هنا في المقام اللي هو على أساس ان احصل على
+
+170
+00:17:04,410 --> 00:17:10,530
+هذه مضروبة في هذه اللي انا عندي معلومات عن هذه و
+
+171
+00:17:10,530 --> 00:17:20,290
+عن هذه واضح؟ اي سؤال؟ و يساوي؟ الان هذا كله على
+
+172
+00:17:20,290 --> 00:17:33,910
+بعضه مين هو عبارة عن جيcapital of F of X اه F
+
+173
+00:17:33,910 --> 00:17:40,510
+of X اللي هي G of Y بدل ال Y ايش اسمها F of X واضح
+
+174
+00:17:40,510 --> 00:17:46,790
+مضروبة في مين هذا المقدار طلع هذا مضروب في F of X
+
+175
+00:17:48,050 --> 00:17:54,710
+نقص f of c على x minus c خد الآن ال limit للجهتين
+
+176
+00:17:54,710 --> 00:18:01,550
+ال limit للجهتين بيصير ال limit لهذا المقدار ال
+
+177
+00:18:01,550 --> 00:18:09,430
+limit لهذا المقدار معايا يا شباب لما x تروح لل c
+
+178
+00:18:09,430 --> 00:18:18,040
+بساوي خلينا أمسح هان بهمشبعد اذنكم اثبتنا احنا
+
+179
+00:18:18,040 --> 00:18:22,940
+الـG كابيتال continuous خليني انجل هذا الكلام فوق
+
+180
+00:18:22,940 --> 00:18:32,700
+اذا صارت عندي limit الـG of F of X ناقص G of F of
+
+181
+00:18:32,700 --> 00:18:42,160
+C على X minus C لما X تروح للـC بساوي limit اللي
+
+182
+00:18:42,160 --> 00:18:54,190
+هو G of F of Xلما X تروح للـ C أه فيه limit اللي
+
+183
+00:18:54,190 --> 00:19:00,710
+هو F of X ناقص F of C على X minus C لما X تروح
+
+184
+00:19:00,710 --> 00:19:07,820
+لمين للـ Cطبعا طبيعي لما X تروح لل C ال F of X
+
+185
+00:19:07,820 --> 00:19:13,660
+نفسها هتروح لمين؟ لل F of C، مظبوط؟ لما X تروح لل
+
+186
+00:19:13,660 --> 00:19:18,960
+C، ال F of X هتروح لمين؟ لل F of C، لإن أصلا F is
+
+187
+00:19:18,960 --> 00:19:23,240
+differential، بالإذن continuous، معايا؟ طيب، إذن
+
+188
+00:19:23,240 --> 00:19:29,380
+الآن عندي بدي أستخدم ال continuity لمين؟ لل G،
+
+189
+00:19:29,380 --> 00:19:35,050
+limit G of F of Cلما .. لو سمينا هذه كلها على بعض
+
+190
+00:19:35,050 --> 00:19:39,770
+Y قلنا لما X تروح لل C ال Y اللي هي F of X بتروح
+
+191
+00:19:39,770 --> 00:19:45,430
+لمين؟ لل F of C بتصير هذه و كأنها اسمها G of Y و
+
+192
+00:19:45,430 --> 00:19:52,190
+بدل هذه بنحط Y بتروح لمين؟ لل F of C والـ G
+
+193
+00:19:52,190 --> 00:19:56,450
+capital is continuous عند ال F of C إذا على طول
+
+194
+00:19:56,450 --> 00:20:05,010
+limitها بتساوي G of F of Cمظبوط؟ مضروبة في مين؟ في
+
+195
+00:20:05,010 --> 00:20:08,010
+limit F of X نقص F of C على X minus C لما X تروح
+
+196
+00:20:08,010 --> 00:20:14,190
+لمين؟ لـC ليها دي تعريف مين المشتقة الأولى للـF
+
+197
+00:20:14,190 --> 00:20:19,310
+عند الـC نرجع الآن G of F of C إيش بتساوي حسب
+
+198
+00:20:19,310 --> 00:20:22,610
+التعريف الأولاني؟ بساوي G prime
+
+199
+00:20:26,140 --> 00:20:36,180
+للـ F of C بدل هذه في F prime of C وهو المطلوب أي
+
+200
+00:20:36,180 --> 00:20:41,000
+سؤال؟
+
+201
+00:20:41,000 --> 00:20:46,740
+لذالك الآن أثبتنا أنه لو كانت F is differentiable
+
+202
+00:20:46,740 --> 00:20:53,080
+أو G is differentiableو يكون الـ composite الهن is
+
+203
+00:20:53,080 --> 00:20:57,900
+differentiable طبعا ما أخد بعين الاعتبار انه يكون
+
+204
+00:20:57,900 --> 00:21:00,600
+جي كمبوسيت F is defined
+
+205
+00:21:16,030 --> 00:21:25,010
+في عنا اللي هو المثال الأول مثال
+
+206
+00:21:25,010 --> 00:21:30,750
+يعني بسيط كلنا بنعرف انفاضه بس بنشوف كيف احنا
+
+207
+00:21:30,750 --> 00:21:35,530
+بنستخدم ال .. ال .. ال .. ال chain rule الصورة
+
+208
+00:21:35,530 --> 00:21:39,470
+اللي احنا كتبناها في اللي هو ايجاد ال derivative
+
+209
+00:21:39,470 --> 00:21:46,520
+جاي اللي عند في example جاي اللي خدExample جاي لنت
+
+210
+00:21:46,520 --> 00:21:56,300
+F من عند I لعند R بيه a differentiable function من
+
+211
+00:21:56,300 --> 00:22:00,080
+هنا نفترض ان ال F أشبه لها يا جماعة عبارة عندها
+
+212
+00:22:00,080 --> 00:22:05,860
+لقابلة للاشتقاق عند نقطة وين على ال I أو عند ال I
+
+213
+00:22:05,860 --> 00:22:13,230
+كلها مش مشكلة ماشي الحال بقولك اثبتليبس قبل ما
+
+214
+00:22:13,230 --> 00:22:20,750
+بتخلينا نقل تقول احنا بتوجدولنا الـ Fn لان في عندي
+
+215
+00:22:20,750 --> 00:22:24,890
+دالة اسمها Fn يعني F مرفوعة للقوة انا بدي اوجدلها
+
+216
+00:22:24,890 --> 00:22:30,140
+ال derivative طبعا الـ Fn كله prime of Xمن
+
+217
+00:22:30,140 --> 00:22:36,040
+معلوماتنا في الـ Calculus كنا نقول أن تفاضل القوس
+
+218
+00:22:36,040 --> 00:22:41,560
+كله نهو N في F N ناقص واحد في تفاضل ما بداخل القوس
+
+219
+00:22:41,560 --> 00:22:47,990
+اللي هو عبارة عن F prime of Xهذا تفضل الآن بدنا
+
+220
+00:22:47,990 --> 00:22:54,650
+نبرهن هذا الكلام باستخدام نظرية الـ chain rule
+
+221
+00:22:54,650 --> 00:22:58,050
+يعني هذا شغل المكان زي ما بقوله الآن بدنا نشوف كيف
+
+222
+00:22:58,050 --> 00:23:01,670
+بتشتغل المكينة أو ايش اللي شغل المكينة طيب ال
+
+223
+00:23:01,670 --> 00:23:13,110
+proof ال proof لندن كده بدي اعرف function let g
+
+224
+00:23:13,110 --> 00:23:23,530
+fromsome interval J لعند R defined by G يعني أصلا
+
+225
+00:23:23,530 --> 00:23:30,330
+ممكن أعرفها من R حتى G of Y بيسوا Y أساش N element
+
+226
+00:23:30,330 --> 00:23:35,070
+in N يعني حتى لو أعرفتها من R مابعرفها الدالة
+
+227
+00:23:35,070 --> 00:23:46,200
+معايا الآن بناء على هيك صار عندي الآن اللي هوالـ G
+
+228
+00:23:46,200 --> 00:23:56,700
+composite F of X إيش هي هتطلع؟ G of F of X يعني
+
+229
+00:23:56,700 --> 00:24:05,340
+هتطلع FN of X صح ولا لأ؟ إذن هاي دالتنا اللي بدنا
+
+230
+00:24:05,340 --> 00:24:12,000
+نفضلها كتبتها على صورة مين؟ G composite Fوالـ F of
+
+231
+00:24:12,000 --> 00:24:17,520
+X الـ F of X اللي هي الـ F of X من وين جاي بترمي
+
+232
+00:24:17,520 --> 00:24:22,680
+في الـ R يعني بترمي في مين؟ في domain الـ G واضح
+
+233
+00:24:22,680 --> 00:24:27,840
+أهإذا كل أموري تمام وهذه الدالة الدالة الأسية is
+
+234
+00:24:27,840 --> 00:24:30,980
+differentiable وهو جاي اللي أن ال F is
+
+235
+00:24:30,980 --> 00:24:33,780
+differentiable إذا جي كومبوزيت F حسب ال channel
+
+236
+00:24:33,780 --> 00:24:38,140
+أشمالها is differentiable و ال derivative لها جي
+
+237
+00:24:38,140 --> 00:24:42,900
+كومبوزيت F prime of X حسب ال channel اللي حكيناها
+
+238
+00:24:42,900 --> 00:24:49,560
+اللي برهنناها بساوت G prime of F of Xفي f prime of
+
+239
+00:24:49,560 --> 00:24:53,580
+x وي ساون and g of y بيساوي y أس 2 ال derivative
+
+240
+00:24:53,580 --> 00:24:59,420
+الهنين هي and y أس n ناقص واحد يعني and y أس n
+
+241
+00:24:59,420 --> 00:25:04,220
+ناقص واحد في f prime of x لكن احنا بنفاضل عند مين
+
+242
+00:25:04,220 --> 00:25:08,860
+عند f of x فبنشيل ال y بنحكم مكانها قيمتها اللي هي
+
+243
+00:25:08,860 --> 00:25:14,240
+f of x فبتطلع and f of x n ناقص واحد في f prime of
+
+244
+00:25:14,240 --> 00:25:19,610
+xهذه اللي هي اللي احنا بنشتغلها وبدون ما .. طبعا
+
+245
+00:25:19,610 --> 00:25:24,930
+احنا هذا تشريع لاستخدام اللي هو Kedemstrain rule
+
+246
+00:25:24,930 --> 00:25:28,150
+it's اللي احنا بنعرفها زي ما نيجي لما كنا نقولكم
+
+247
+00:25:28,150 --> 00:25:34,810
+مثلا اوجد ال derivative y لعندك y بالساوى sin x
+
+248
+00:25:34,810 --> 00:25:39,460
+تكعيب مثلاكنتش تسوي على طول ماتيش ايش تقول والله
+
+249
+00:25:39,460 --> 00:25:44,040
+بيضاف رضاف of X بيساوي X تتعيب و G of Y تساوي Sin
+
+250
+00:25:44,040 --> 00:25:47,600
+Y و تقول اذا صارت هذه عبارة عن composite of two
+
+251
+00:25:47,600 --> 00:25:53,240
+functions خلاص بنتجاوز المرحلة هذهوعرفنا وين أصل
+
+252
+00:25:53,240 --> 00:25:58,160
+الأمر اللى بيقوللنا why prime بتساوي تفاضل اللى
+
+253
+00:25:58,160 --> 00:26:03,760
+برا اللى هو cosine x كعيب فى تفاضل اللى جوا تلاتة
+
+254
+00:26:03,760 --> 00:26:08,100
+x تربية هذا فى الواقع هو عبارة عن استخدام mean لل
+
+255
+00:26:08,100 --> 00:26:19,080
+chain rule واضح أي سؤال؟ طيب هذا عندنا فى عندنا
+
+256
+00:26:19,080 --> 00:26:31,850
+مثال آخرالمثال .. نعم المثال اللي هو .. يمكن مرشك
+
+257
+00:26:31,850 --> 00:26:36,330
+للمثال حتى معاكم في تحليل واحد لكن كان هدفه اللي
+
+258
+00:26:36,330 --> 00:26:42,910
+هو في حينه ال continuity اذا اعطيك F of X بتساوي
+
+259
+00:26:42,910 --> 00:26:52,400
+هاي B مثال B F of X بتساوي اللي هو X تربيعSin 1
+
+260
+00:26:52,400 --> 00:26:59,800
+على X لما X لا تساوي سفر أو سفر لما X ايش بتساوي
+
+261
+00:26:59,800 --> 00:27:07,140
+سفر الان بيقول لي بيعطيني طبعا هو بيسأل سؤال جابله
+
+262
+00:27:07,140 --> 00:27:10,640
+تعليق سؤال في الكتاب بيقول لي لو كانت ال F is
+
+263
+00:27:10,640 --> 00:27:14,040
+differentiable ركز معاه لو كانت ال F is
+
+264
+00:27:14,040 --> 00:27:17,800
+differentiable الشرط ان ال derivative تماما تطلع
+
+265
+00:27:17,800 --> 00:27:23,590
+continuousهذا ليس شرطًا، هذا بس وينفي الـ R لما
+
+266
+00:27:23,590 --> 00:27:26,430
+نحكي عن الهو نحكي عن الـ real valued functions
+
+267
+00:27:26,430 --> 00:27:33,210
+from R to R كمان لكن في ال complex الأمر بيختلف ال
+
+268
+00:27:33,210 --> 00:27:36,610
+complex أو ال derivative مرة لجيتها معناته اللي
+
+269
+00:27:36,610 --> 00:27:39,210
+بعيدها و اللي بعيدها و اللي بعيدها هتظلها إشمالها
+
+270
+00:27:39,210 --> 00:27:43,830
+موجودة يعني if F prime exists then F double prime
+
+271
+00:27:43,830 --> 00:27:48,630
+and F triple F كذا إذا هتطلع مجرد ما تكون الدالة
+
+272
+00:27:48,630 --> 00:27:53,230
+analytic أو differentiableهتظلها على طول
+
+273
+00:27:53,230 --> 00:27:56,750
+differentiable و continuous و و و الاخرين هذا وين
+
+274
+00:27:56,750 --> 00:28:01,890
+في ال complex أما في ال real الوضع ليس كذلك وهي
+
+275
+00:28:01,890 --> 00:28:07,970
+مثال على F is differentiable everywhere طبعا is
+
+276
+00:28:07,970 --> 00:28:11,310
+differentiable عنده نقطة zero بالذات لكن its
+
+277
+00:28:11,310 --> 00:28:15,450
+derivative is not continuous at this point باضح
+
+278
+00:28:15,450 --> 00:28:20,330
+اليوم اه طيب الآن show
+
+279
+00:28:23,010 --> 00:28:30,750
+that F prime of X exists everywhere
+
+280
+00:28:30,750 --> 00:28:41,270
+but F prime is not continuous at zero و proof
+
+281
+00:28:41,270 --> 00:28:45,470
+البرهن .. طبعا البرهن كامل موجود هنا بس خلينا
+
+282
+00:28:45,470 --> 00:28:50,670
+نبرهن لك .. نوضح إيش ال .. إيش البرهنشوفوا صلى
+
+283
+00:28:50,670 --> 00:28:56,670
+الله عليه الصلاة والسلام الان f prime exist طبيعي
+
+284
+00:28:56,670 --> 00:29:03,310
+for x لا يساوي 0 مقابر سهل مجرد تفاضل عادي f prime
+
+285
+00:29:03,310 --> 00:29:07,290
+of x لأن هذا الدلة قابلة للاشتقاء عند كل النقطة ما
+
+286
+00:29:07,290 --> 00:29:11,430
+عدا عند ال x بالساوية 0 هدى بحكي عن هدى لحالها f
+
+287
+00:29:11,430 --> 00:29:19,020
+prime of x بالساوية2x sin 1 على x لأن x تربيع في
+
+288
+00:29:19,020 --> 00:29:24,360
+sin تفاضل الـ sin cos و تفاضل اللي جوا سالب x
+
+289
+00:29:24,360 --> 00:29:30,440
+تربيع x minus 2 بتروح حد ما حد فبصير سالب cos 1
+
+290
+00:29:30,440 --> 00:29:36,480
+على xالان هذا for x لا تساوي صفر طب for x متساوي
+
+291
+00:29:36,480 --> 00:29:39,660
+صفر for x متساوي صفر الوضع اللي لان بختلف اللي هو
+
+292
+00:29:39,660 --> 00:29:43,880
+فيه عندي اللي هو الدالة أصلا مش معرفة عند ال zero
+
+293
+00:29:43,880 --> 00:29:47,620
+لكن معرفة الدالة ككل عند ال zero عند مين بتساوي
+
+294
+00:29:47,620 --> 00:29:52,060
+صفر فمضطر أنا أستخدم التعريف لإثبات أو فحص إذا
+
+295
+00:29:52,060 --> 00:29:56,680
+كانت الدالة إلها مشتقة عند الصفر ولا لأ فباجي بقول
+
+296
+00:29:56,680 --> 00:30:03,810
+الان بتدا فحص limitاللي هو F of X ناقص F of Zero
+
+297
+00:30:03,810 --> 00:30:08,910
+على X minus Zero as X بتروح للزيرو بدي أشوف هل هذه
+
+298
+00:30:08,910 --> 00:30:13,250
+ال limit موجودة ولا مش موجودة ماشي الحال Y ساوي
+
+299
+00:30:13,250 --> 00:30:16,930
+هذا
+
+300
+00:30:16,930 --> 00:30:20,370
+عشان خاطر أوجد مين أشوف ال F prime عند ال Zero
+
+301
+00:30:20,370 --> 00:30:25,630
+موجودة أو لا بساوي limitF of X مين هي عبارة عن X
+
+302
+00:30:25,630 --> 00:30:29,030
+تربيع .. لأن X بتروح للزيرو، لأن X مش سفر، اللي هي
+
+303
+00:30:29,030 --> 00:30:34,450
+عبارة عن X تربيع صين واحد على X نقص F of Zero اللي
+
+304
+00:30:34,450 --> 00:30:39,310
+هي عبارة عن سفر على X minus سفر لما X تروح لمين
+
+305
+00:30:39,310 --> 00:30:46,870
+للسفر ويساوي limitx term x sin 1 على x لما x تروح
+
+306
+00:30:46,870 --> 00:30:50,670
+لمين للصفر هذه اللي هي by sandwich theorem حلقيها
+
+307
+00:30:50,670 --> 00:30:56,650
+إيش بتساوي سفر كيف by sandwich theorem عندك x sin
+
+308
+00:30:56,650 --> 00:31:00,470
+1 على x absolute value أصغر أو ساوي ال absolute
+
+309
+00:31:00,470 --> 00:31:05,050
+value لل x وهذه الطبيعة أكبر أو ساوي مين سفر as x
+
+310
+00:31:05,050 --> 00:31:07,870
+goes to zero هذه بتروح للزيرو وهذه بتروح للزيرو
+
+311
+00:31:07,870 --> 00:31:10,570
+إذا اللي في النص هذه لازم ال limit هيروح لمين
+
+312
+00:31:10,570 --> 00:31:15,840
+للصفرومن ثم لل continuity لل .. بسبب ال continuity
+
+313
+00:31:15,840 --> 00:31:18,040
+لل .. لل absolute value بتطلع ال limit اللي جوا
+
+314
+00:31:18,040 --> 00:31:20,740
+برضه أشمالها بتساوي 0 لأن ال limit ال absolute
+
+315
+00:31:20,740 --> 00:31:23,380
+value بيساوي absolute value لل limit إذا صار عندى
+
+316
+00:31:23,380 --> 00:31:32,120
+limit اللي جوا بيساوي 0 معايا؟ طيب
+
+317
+00:31:32,120 --> 00:31:35,940
+بما أنه ال limit هذه موجودة بتساوي 0 إذا بنقول
+
+318
+00:31:35,940 --> 00:31:42,580
+بعيدها since كده بيساوي كده thenF prime of 0
+
+319
+00:31:42,580 --> 00:31:50,520
+بتساوي صفر إذا اللي أثبتنا إحنا أنه F prime exist
+
+320
+00:31:50,520 --> 00:31:55,460
+وين ما كان وإيش ال F prime اللي أثبتناها كمان
+
+321
+00:31:55,460 --> 00:32:01,160
+أثبتنا إنه الآن F prime of X أثبتناها إنها بتساوي
+
+322
+00:32:02,250 --> 00:32:08,870
+2x sin 1 على x نقص cos 1 على x لما x ايش معناه لأ
+
+323
+00:32:08,870 --> 00:32:13,070
+تساوي سفر واتبعتناها بتساوي سفر لما x بتساوي سفر
+
+324
+00:32:13,070 --> 00:32:16,990
+إذا صارت ال F is functional everywhere الآن بدنا
+
+325
+00:32:16,990 --> 00:32:23,350
+نثبت إنه is not this F prime is not continuous at
+
+326
+00:32:23,350 --> 00:32:27,150
+zero شوف
+
+327
+00:32:27,150 --> 00:32:29,410
+الآن عندي
+
+328
+00:32:31,690 --> 00:32:41,570
+بدي أشوف limit اللي هي 2x sin 1 على x ناقصه sin 1
+
+329
+00:32:41,570 --> 00:32:49,080
+على x لما x تروح للزيرو exist ولا لأالان limit ال
+
+330
+00:32:49,080 --> 00:32:53,540
+أولى 2x sin 1 على x مضمون انه exist قبل مش
+
+331
+00:32:53,540 --> 00:32:57,560
+ويعملناها نفسها باي اللي هو sandwich theorem هتطلع
+
+332
+00:32:57,560 --> 00:33:03,720
+سفر اذا الان اول حاجة اكتب لها since الان limit
+
+333
+00:33:03,720 --> 00:33:09,420
+sin 1 على x بساوي سفر exist الان
+
+334
+00:33:09,420 --> 00:33:19,480
+but butlimit cos 1 على x لما x تروح لصفر does not
+
+335
+00:33:19,480 --> 00:33:24,360
+exist وهلجت بنشوف ليش does not exist but this
+
+336
+00:33:24,360 --> 00:33:26,140
+limit اشمالها does not exist
+
+337
+00:33:28,950 --> 00:33:33,830
+ما دام هذا exist و هذا does not exist حاصل جمح أو
+
+338
+00:33:33,830 --> 00:33:38,290
+طرح أكيد ايش ماله does not exist لأنه لو exist
+
+339
+00:33:38,290 --> 00:33:42,810
+معناه تصير هذا اللي يطلع و هذا exist بيصير مجموح
+
+340
+00:33:42,810 --> 00:33:45,170
+أو طرح يساوي هذا ايضا صار هذا exist و هذا
+
+341
+00:33:45,170 --> 00:33:51,550
+contradiction فاهمين اه الان hence limit اتنين x
+
+342
+00:33:51,550 --> 00:33:56,410
+sin واحد على x نقص cosine واحد على x لما x تروح
+
+343
+00:33:56,410 --> 00:34:05,340
+للسفر اللي هو does notexist دل عليا بس اوضح اللي
+
+344
+00:34:05,340 --> 00:34:12,220
+هو ليش هذا does not exist بده اوضح ليش هذا does
+
+345
+00:34:12,220 --> 00:34:18,740
+not exist اخدتوا انتوا اخدتوا اللي هو ايش نظرية ان
+
+346
+00:34:18,740 --> 00:34:28,240
+لو كانت limitF of X لما X بتروح ل A بساوي L إذا أي
+
+347
+00:34:28,240 --> 00:34:34,560
+sequence XN بتروح لل A لازم يعطيك اللي هو F of XN
+
+348
+00:34:34,560 --> 00:34:39,780
+برضه يروح لمين لل L، بصبوط؟ هذه كcontinuity اللي
+
+349
+00:34:39,780 --> 00:34:43,530
+بتحكي عنهااللي كده كانت limit F of X لما X تروح
+
+350
+00:34:43,530 --> 00:34:48,930
+للـ A بساوي L و IG تأخدت أي sequence XN بتروح للـ
+
+351
+00:34:48,930 --> 00:34:54,670
+A هيطلع limit F of XN مش بيروح بساوي لL ماشي هذه
+
+352
+00:34:54,670 --> 00:35:00,780
+اللي هي النظريةبناء على هذه النظرية كنا نقول انه
+
+353
+00:35:00,780 --> 00:35:07,180
+لو وجيت two sequences على الأقل ايكون بروح للـA
+
+354
+00:35:07,180 --> 00:35:13,500
+لكن بروح لـLimits مختلفات او واحدة من Limits does
+
+355
+00:35:13,500 --> 00:35:18,600
+not exist بكفالك ان الـLimit هذه اشمالها does not
+
+356
+00:35:18,600 --> 00:35:25,170
+exist واضح؟ الان اذا بدي اعتمد على هذهفي إثبات إنه
+
+357
+00:35:25,170 --> 00:35:29,690
+limit cosine واحد على X does not exist أكيد كلكم
+
+358
+00:35:29,690 --> 00:35:40,730
+جال take the sequences أو the two sequences X and
+
+359
+00:35:40,730 --> 00:35:50,290
+بتساوي واحد على اتنين and by and Y and بتساوي واحد
+
+360
+00:35:50,290 --> 00:35:57,030
+على اتنين and زائد واحد في byمظبوط أكيد قلته هيك
+
+361
+00:35:57,030 --> 00:36:05,290
+where x and goes to zero and y and goes to zero
+
+362
+00:36:05,290 --> 00:36:13,130
+but cosine واحد على ايه؟ cosine واحد على x and
+
+363
+00:36:13,130 --> 00:36:23,100
+اللي هو بتساوي او goes to واحد and cosine1 على I
+
+364
+00:36:23,100 --> 00:36:30,500
+and goes to ناقص واحد اوضحها
+
+365
+00:36:30,500 --> 00:36:35,600
+اوضحها اذكركم الان
+
+366
+00:36:35,600 --> 00:36:39,980
+هذا بيصير cosine مين اتنين and by اتنين and by ايش
+
+367
+00:36:39,980 --> 00:36:45,470
+مالها واحدا اتنين ان زاد واحد لكل في باقي ايش
+
+368
+00:36:45,470 --> 00:36:49,030
+بتصير؟ اللي هو سالب واحد فبصير هاد ال cosine
+
+369
+00:36:49,030 --> 00:36:53,450
+converts واحد و هاد ال cosine converts لمين؟ لسالب
+
+370
+00:36:53,450 --> 00:36:58,490
+واحد اذا حسب اللي حكيناها قبل شوية hence hence
+
+371
+00:36:58,490 --> 00:37:05,910
+hence limit cosine واحد على x لما x تروح ل zero
+
+372
+00:37:05,910 --> 00:37:17,800
+does not existهذا توضيح .. توضيح لماذا اللي هو cos
+
+373
+00:37:17,800 --> 00:37:24,000
+1 على x does not exist لاحظ التالي على جهة لكن لو
+
+374
+00:37:24,000 --> 00:37:28,440
+كانت .. بالمناسبة لو كانت هذه does not exist
+
+375
+00:37:28,440 --> 00:37:33,640
+limitها و هذه does not exist بقدرش أؤكد على إن ال
+
+376
+00:37:33,640 --> 00:37:40,460
+limit للطرح أو الجامعه does not exist صح؟ في مثال؟
+
+377
+00:37:45,330 --> 00:37:50,850
+أه و خروجتها واحدة لأن واحدة لأن بينهين سالم بيصير
+
+378
+00:37:50,850 --> 00:37:53,450
+limit لها دي موجودة أو limit لها دي موجودة .. آسف
+
+379
+00:37:53,450 --> 00:37:55,890
+limit لها دي مش موجودة أو limit لها دي مش موجودة
+
+380
+00:37:55,890 --> 00:38:01,490
+لكن اللي هو حصل إيش طرح موجود أو لو بدك جمحن بتحط
+
+381
+00:38:01,490 --> 00:38:04,890
+اللي هو للتانية تاخد واحدة لأن ونقص واحدة لأن و
+
+382
+00:38:04,890 --> 00:38:10,490
+هكذا أي سؤال؟
+
+383
+00:38:10,490 --> 00:38:11,330
+طيب
+
+384
+00:38:15,770 --> 00:38:21,510
+اللي ييجي الأن إلى
+
+385
+00:38:21,510 --> 00:38:37,290
+النقطة اللي بعيدها أين
+
+386
+00:38:37,290 --> 00:38:38,330
+رحلت أنا؟
+
+387
+00:38:48,090 --> 00:38:53,250
+طيب انتبهونا خلينا لهذه النظرية نحكي عن نصها أو
+
+388
+00:38:53,250 --> 00:39:00,150
+الإشي بالتفصيل و بعدين نيجي نلخصلكم كل المعلومة
+
+389
+00:39:00,150 --> 00:39:05,130
+اللي .. اللي في الأول هنحس إنها بتغلب في اللي هو
+
+390
+00:39:05,130 --> 00:39:13,470
+سفر و من ثم نحاول نبرهنها أه
+
+391
+00:39:13,470 --> 00:39:18,970
+.. أه .. أه .. لأ .. أه ..لت I subsequent R be an
+
+392
+00:39:18,970 --> 00:39:19,610
+interval
+
+393
+00:39:28,670 --> 00:39:34,370
+لأنه احنا بتتذكروا في real واحد خلاص اللي هي ال ..
+
+394
+00:39:34,370 --> 00:39:37,010
+ال .. ال .. ال .. الاشي اللي بده شغل اللي هو
+
+395
+00:39:37,010 --> 00:39:39,890
+النقطة اللي بتكون في ال interval اللي بتكون
+
+396
+00:39:39,890 --> 00:39:42,250
+cluster point اللي مش cluster point automatic
+
+397
+00:39:42,250 --> 00:39:45,970
+هتكون ال limit exist فمش داعي لأصلا لدراسة الموضوع
+
+398
+00:39:45,970 --> 00:39:52,440
+لإنه بيصير كتير من الأمور trivialعشان هي احنا
+
+399
+00:39:52,440 --> 00:39:55,740
+بنشتغل على الـ Intervals let I subset من R بيكون
+
+400
+00:39:55,740 --> 00:39:59,680
+Interval and let F من I لعند R بيكون strictly
+
+401
+00:39:59,680 --> 00:40:09,720
+Monotone F من عند هذه theorem أنا مكتبش ال theorem
+
+402
+00:40:09,720 --> 00:40:19,060
+تمام لأنه موجود هنا theorem 618 عند F من Iلعند R
+
+403
+00:40:19,060 --> 00:40:25,440
+غير عنها strictly monotone
+
+404
+00:40:25,440 --> 00:40:36,900
+ماشي الحال و continuous and continuous الان لاحظ
+
+405
+00:40:36,900 --> 00:40:41,100
+على ال function هذهمدام حكالة عنها strictly
+
+406
+00:40:41,100 --> 00:40:44,540
+monotone يعني strictly increasing او strictly
+
+407
+00:40:44,540 --> 00:40:49,240
+decreasing معناته صارت automatic اتدالة اشمالها
+
+408
+00:40:49,240 --> 00:40:53,800
+one to one ليش؟ لأنه لو كانت strictly increasing
+
+409
+00:40:53,800 --> 00:40:59,060
+معناته x1 أصغر من x2 اللي بتعطيني f of x1 أصغر
+
+410
+00:40:59,060 --> 00:41:03,840
+strictly من f of x2 هذا في حالة نين strictly
+
+411
+00:41:03,840 --> 00:41:09,540
+increasing أه مش ممكنية للمساواةهاالأن في حالة
+
+412
+00:41:09,540 --> 00:41:14,520
+strictly decreasing X1 أكبر من X2 بيعطيني F of X1
+
+413
+00:41:14,520 --> 00:41:21,220
+أصغر من F of X2 هذا strictly decreasing الآن سواء
+
+414
+00:41:21,220 --> 00:41:25,060
+هنا الحالة هذه أو الحالة هذه هيكون عندي ال
+
+415
+00:41:25,060 --> 00:41:29,200
+function one to one غصب عنها، ليش؟ لأن لو كانت X1
+
+416
+00:41:29,200 --> 00:41:35,640
+لا تساوي X2 يا أصغر يا أكبر، ما زال مقت أصغرهتكون
+
+417
+00:41:35,640 --> 00:41:40,680
+strictly أصغر يعني بمعنى أخر f of x1 لا تساوي f of
+
+418
+00:41:40,680 --> 00:41:45,460
+x2 ولو كانت هذه أكبر هتكون f of x1 أكبر برضه هتكون
+
+419
+00:41:45,460 --> 00:41:49,060
+إشمالها لا تساوي و similarly في حالة ال decreasing
+
+420
+00:41:49,850 --> 00:41:56,250
+إذن واضح إن X1 ≤ X2 بيعطينا F X1 ≤ F X2 أو بمعنى
+
+421
+00:41:56,250 --> 00:42:03,730
+آخر F X1 هذه هي الcontraposition ل F X1 ≤ F X2
+
+422
+00:42:03,730 --> 00:42:10,010
+بيعطينا X1 ≤ X2 وهذا يعني ال F أشمالها is one to
+
+423
+00:42:10,010 --> 00:42:15,810
+one يعني الآن أنا بوضحليش أعطاني F strictly
+
+424
+00:42:15,810 --> 00:42:20,790
+monotone؟ عشان شكله أصلا احنا قاعد .. هذه النظرية
+
+425
+00:42:20,790 --> 00:42:25,030
+الحاكية عن ال inverse بدنا نشوف العلاقة بين ال
+
+426
+00:42:25,030 --> 00:42:27,950
+derivative لل inverse و ال derivative للدالة
+
+427
+00:42:27,950 --> 00:42:32,520
+الأصليةإذا بدأ أهل الدالة الأصلية أن يكون لها
+
+428
+00:42:32,520 --> 00:42:36,640
+inverse عشان يكون لها inverse لازم أفترض أن الدالة
+
+429
+00:42:36,640 --> 00:42:40,740
+إشمالها one to one فالآن فرضنا إنها strictly
+
+430
+00:42:40,740 --> 00:42:43,980
+monotone يعني strictly increasing أو strictly
+
+431
+00:42:43,980 --> 00:42:50,920
+decreasing الآن مدامة one to one إذا في إمكانية
+
+432
+00:42:50,920 --> 00:42:58,230
+نحكيعن inverse إلها F inverse إلها اللي هو بده
+
+433
+00:42:58,230 --> 00:43:03,290
+يسميه هو الآن بده يسميه G هو بده يسميه G هياخد ال
+
+434
+00:43:03,290 --> 00:43:10,950
+G من وين من ال range لل F يعني من ال F of I لعند
+
+435
+00:43:10,950 --> 00:43:17,150
+مين لعند ال R ماشي ال F of I هذي مسميها مين هو
+
+436
+00:43:17,150 --> 00:43:25,150
+مسميها هو J يعني ماياخد ال Gمن G لعند R الـ G هي
+
+437
+00:43:25,150 --> 00:43:30,750
+عبارة عن مين؟ عن الـ F inverse معايا؟ قولي يا محمد
+
+438
+00:43:30,750 --> 00:43:42,570
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+439
+00:43:42,570 --> 00:43:42,670
+R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R
+
+440
+00:43:42,670 --> 00:43:43,910
+.. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R ..
+
+441
+00:43:43,910 --> 00:43:44,630
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+442
+00:43:44,630 --> 00:43:47,530
+R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R
+
+443
+00:43:47,530 --> 00:43:47,590
+.. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R ..
+
+444
+00:43:47,590 --> 00:43:47,610
+لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ R .. لـ
+
+445
+00:43:47,610 --> 00:43:53,500
+R .. لـ R .. لـ R .. لـبس لما نكتب I R ايش برضه
+
+446
+00:43:53,500 --> 00:43:57,420
+صحيح بس على أساس انه .. خلاني أقول المعلومات بت ..
+
+447
+00:43:57,420 --> 00:44:01,780
+بت .. بتأهلني ان اكون أدق لعند مين لعند I خلان
+
+448
+00:44:01,780 --> 00:44:06,600
+صحيح طيب الآن عشان ما نبعدش عن اللي بدنايا F من I
+
+449
+00:44:06,600 --> 00:44:12,080
+ل R strictly monotone و G من F of I ل عند I أصلا
+
+450
+00:44:12,080 --> 00:44:15,140
+إذا كانت ال F من I ل عند R strictly monotone و
+
+451
+00:44:15,140 --> 00:44:19,960
+continuousهتكون الـ G اللي هي ال F inverse من F of
+
+452
+00:44:19,960 --> 00:44:24,400
+I ل I برضه strictly monotone و إيش و continuous
+
+453
+00:44:24,400 --> 00:44:27,940
+strictly monotone و continuous إذن الآن ال F عند
+
+454
+00:44:27,940 --> 00:44:30,660
+me strictly monotone و continuous و ال F inverse
+
+455
+00:44:30,660 --> 00:44:41,160
+برضه strictly monotone و إيش و continuous معايا ؟
+
+456
+00:44:41,160 --> 00:44:49,090
+طيبالان لحد الان بعتبر انه احنا بنحكي عن معطيات
+
+457
+00:44:49,090 --> 00:44:54,030
+حتى لو كنا استنتجنا جزء منها من بعض الان بقوللي
+
+458
+00:44:54,030 --> 00:44:58,970
+اللي هو موضوعنا if f is differentiable at c if
+
+459
+00:44:58,970 --> 00:45:12,350
+الان if if f is differentiable at c element in I
+
+460
+00:45:13,160 --> 00:45:17,380
+ماشي الحال يعني F differentiable عنده نقطة C
+
+461
+00:45:17,380 --> 00:45:24,600
+element وين؟ ن I بقول لك then طبعاً A و F prime of
+
+462
+00:45:24,600 --> 00:45:28,340
+C أشمالها لاتساوي سفر لأنه هيلزمنا القسم عليها
+
+463
+00:45:28,340 --> 00:45:32,420
+عشان هيك لازم اللي هو عشان نحكي عن ال inverse لال
+
+464
+00:45:32,420 --> 00:45:34,880
+derivative تكون ال F prime عند ال C أشمالها
+
+465
+00:45:34,880 --> 00:45:42,730
+لاتساوي سفر الآن thenاللي هي الـ G الـ G اللي
+
+466
+00:45:42,730 --> 00:45:47,810
+سميناها F inverse اللي سميناها F inverse الـ G is
+
+467
+00:45:47,810 --> 00:45:51,150
+differentiable
+
+468
+00:45:51,150 --> 00:46:00,490
+is differentiable at اللي هو F of C is
+
+469
+00:46:00,490 --> 00:46:04,610
+differentiable at إيش؟ at F of C اللي هو مسميها
+
+470
+00:46:04,610 --> 00:46:11,800
+إيش اسمها ديالان الإعلان الواضح أنه لو كانت F
+
+471
+00:46:11,800 --> 00:46:15,960
+differentiable عند الـC و F' للـC لاتساوي سفر
+
+472
+00:46:15,960 --> 00:46:20,400
+هتكون الـF inverse differentiable عند مين؟ عند
+
+473
+00:46:20,400 --> 00:46:34,300
+الـF of C اه and .. and F inverse prime at F of C
+
+474
+00:46:35,750 --> 00:46:46,830
+At F of C هتساوي واحد على F prime لمين؟ Let's see
+
+475
+00:46:46,830 --> 00:46:59,390
+يعني وكأنه وكأنه بمجول F inverse prime لل Yبتساوي
+
+476
+00:46:59,390 --> 00:47:07,690
+واحد على F prime of X where العلاقة بين Y و X اللي
+
+477
+00:47:07,690 --> 00:47:16,070
+هي Y بتساوي F of X واضح، اه، الكلام واضح وسهلاللي
+
+478
+00:47:16,070 --> 00:47:21,190
+هي f prime of c بدي ابتبها طريقة ثانية بيساوي واحد
+
+479
+00:47:21,190 --> 00:47:30,010
+على f prime of الان هو سم ال f of c أشمها D يعني
+
+480
+00:47:30,010 --> 00:47:37,080
+ال c بتساوي f انفر سلاميين ل Dصح؟ يعني بمعنى آخر
+
+481
+00:47:37,080 --> 00:47:45,060
+هو بقول F inverse prime عند مين؟ عند ال D بسوء
+
+482
+00:47:45,060 --> 00:47:50,580
+واحد على F prime مكان ال C إيش ودي يحط؟ F inverse
+
+483
+00:47:50,580 --> 00:48:01,420
+of D يعني واحد على F prime فلأ بكفي اللي هى ماشي
+
+484
+00:48:01,420 --> 00:48:07,720
+الحل G of D لأ لأاللي هو كاتبها يعني لإنه كاتب هو
+
+485
+00:48:07,720 --> 00:48:15,000
+بدل ال F inverse كاتب مين الـG واضح واضحة طبعا هذه
+
+486
+00:48:15,000 --> 00:48:22,200
+اللي هي اللي احنا بدنا نثبته نيجي للإثبات
+
+487
+00:48:27,580 --> 00:48:32,740
+بدنا نطبل روحنا شوية عرفناش اننا نطبط اه عشان اللي
+
+488
+00:48:32,740 --> 00:48:41,260
+خاطر امسح وبدأ ابدأ اشتغل من فوق النظرية
+
+489
+00:48:41,260 --> 00:48:45,720
+احفظناها اه او خلينا نقول هي موجودة يعني اه بس
+
+490
+00:48:45,720 --> 00:48:49,160
+بدنا نطلع عالميا على البرهان ال proof
+
+491
+00:48:56,060 --> 00:49:04,920
+بدنا نثبت احنا اللي هو g prime of
+
+492
+00:49:04,920 --> 00:49:14,460
+D بساوة واحد على F prime of C هاي اللي بدنا نثبته
+
+493
+00:49:14,460 --> 00:49:17,820
+او واحد على F prime .. الـ G prime هذا طالع من
+
+494
+00:49:17,820 --> 00:49:22,140
+اليمين الـ G عبارة عن F inverse ماشي؟ شوفي الآن
+
+495
+00:49:25,040 --> 00:49:33,140
+بدي اعرف ده اللي برضه مساعدة لان خد Y لا تساوي D و
+
+496
+00:49:33,140 --> 00:49:39,760
+Y طبعا وين موجودة؟ Y element in J في domain من ال
+
+497
+00:49:39,760 --> 00:49:43,640
+F inverse متذكرين؟
+
+498
+00:49:43,640 --> 00:49:52,360
+ال F من I لعند R و ال G اللي هي ال F inverseمن F
+
+499
+00:49:52,360 --> 00:49:59,500
+of I اللي سمناها J لعند مين؟ لعند الـ I مظبوط؟ طيب
+
+500
+00:49:59,500 --> 00:50:10,320
+الآن هعرف ال H of Y H of Y بتساوي اللي هو F F
+
+501
+00:50:10,320 --> 00:50:20,250
+inverse of Y اللي هي F of G of Y ماقص F of Gof
+
+502
+00:50:20,250 --> 00:50:25,430
+مين؟ of D نحطها على جنب ايش اللي بنعني فيه F of C
+
+503
+00:50:25,430 --> 00:50:33,110
+بتساوي D او G of D اللي هي F inverse استذكروا انه
+
+504
+00:50:33,110 --> 00:50:36,390
+انا الـ G هي مين هي F inverse عشان تسهي عليكم
+
+505
+00:50:36,390 --> 00:50:41,630
+عملية الحسابات ايش بتساوي؟ C ماشي الان F of G of Y
+
+506
+00:50:41,630 --> 00:50:48,750
+ناقص F of G of D على G of Y ناقص G of D
+
+507
+00:50:51,950 --> 00:50:56,890
+معايا؟ واضح اه؟ الان انا مفترض ان ال Y لا تساوي D
+
+508
+00:50:56,890 --> 00:51:01,290
+طبعا ال functions سبعنا سواء ال F او ال G تمتين
+
+509
+00:51:01,290 --> 00:51:05,870
+one to one مأكيد ال G of Y ايش مالها لا تساوي G of
+
+510
+00:51:05,870 --> 00:51:09,990
+D اذا ان انا مرتاح انا ماخد Y لا تساوي Z اذا ان
+
+511
+00:51:09,990 --> 00:51:13,450
+automatic هذه is well defined يعني G of Y ايش
+
+512
+00:51:13,450 --> 00:51:19,270
+مالها لا تساوي مين؟ G of D واضح اه؟ اي سؤال؟
+
+513
+00:51:22,880 --> 00:51:28,700
+الان الـ H of Y اللي هي بتساوي اللي هو F of G of Y
+
+514
+00:51:28,700 --> 00:51:33,120
+نقص F وكذا على G of Y minus G of D is اللي هي well
+
+515
+00:51:33,120 --> 00:51:38,640
+defined طبعا الان اللي فوقبناء على ما كتبناه
+
+516
+00:51:38,640 --> 00:51:45,220
+المستعملينه هنا كرموز هذه نفسها ممكن نطلع إليها
+
+517
+00:51:45,220 --> 00:51:50,520
+كصيغة أخرى أفما هي inverse تبعدها دول فبصير عبارة
+
+518
+00:51:50,520 --> 00:51:57,980
+عن Y ناقص D صح ولا لأ على G of Y ناقص مين G of D
+
+519
+00:51:57,980 --> 00:52:04,840
+هذا نظرة أخرى لمين للH of Y اللي عندناأي سؤال؟
+
+520
+00:52:04,840 --> 00:52:10,740
+طبعا الـH of Y واضح أنها مش سفر لأن الـY أشمالها
+
+521
+00:52:10,740 --> 00:52:15,500
+لاتساوي D لأن بعد شوية بدي أجسم عليها عشان هيك بدي
+
+522
+00:52:15,500 --> 00:52:22,000
+أكتد لكم أن الـY الـH of Y أشمالها لاتساوي سفر
+
+523
+00:52:22,000 --> 00:52:31,170
+واضح لحد الآن، ماشي الحال، الآن بدي أستخدماللي هو
+
+524
+00:52:31,170 --> 00:52:35,090
+التعريف في الوصول الـ Epsilon Delta Definition في
+
+525
+00:52:35,090 --> 00:52:42,570
+الوصول للي بديهاللي هو G prime of D بيساوي limit G
+
+526
+00:52:42,570 --> 00:52:48,770
+of Y نقص G of D على Y minus D الـY بتروح للـD اللي
+
+527
+00:52:48,770 --> 00:52:51,490
+هو exist .. that .. that .. that .. that مثلك انه
+
+528
+00:52:51,490 --> 00:52:54,470
+لكل يكبر من صفر يوجد Delta او Gamma او اللي هي
+
+529
+00:52:54,470 --> 00:52:58,490
+such that لما تكون مسافة بين Y و D أصغر من Delta
+
+530
+00:52:58,490 --> 00:53:01,900
+يعطينا هدف أصغر من مين؟ من يبسلونهذا كيف بده
+
+531
+00:53:01,900 --> 00:53:07,060
+أثبته؟ بالاستعانة بيه اللي هو أنا بعرف أن الـ F is
+
+532
+00:53:07,060 --> 00:53:10,320
+differentiable و بالاستعانة بتدلة اللي عندى هذه
+
+533
+00:53:10,320 --> 00:53:14,080
+السؤال أي سؤال حتى الآن حتى الآن احنا اللي أثبتنا
+
+534
+00:53:14,080 --> 00:53:17,540
+ان H و Y اللي ركّبناها دي إيش مانع لا تساوى سفر
+
+535
+00:53:17,540 --> 00:53:27,380
+الان عندى since F is differentiable at مين معطينى؟
+
+536
+00:53:27,380 --> 00:53:34,100
+at Cاذن thenfor every epsilon أكبر من صفر بتستخدم
+
+537
+00:53:34,100 --> 00:53:38,920
+على طول اللي هو limit واضح then for every epsilon
+
+538
+00:53:38,920 --> 00:53:42,660
+أكبر من صفر there exist نستخدم نفس التعبير دلتا
+
+539
+00:53:42,660 --> 00:53:47,840
+أكبر من صفر أو نفس الرموز such that X minus C أكبر
+
+540
+00:53:47,840 --> 00:53:54,020
+من صفر وأصغر من دلتا implies F of X ناقص F of C
+
+541
+00:53:54,020 --> 00:53:59,240
+على X minus C ناقص اللي هو ال derivative اللي هي F
+
+542
+00:53:59,240 --> 00:54:06,500
+prime of C صح؟أصغر من مين؟ من أبسل وسمي هذه ايش
+
+543
+00:54:06,500 --> 00:54:17,000
+مسميها هو؟ همسميها واحد هذا سمي واحد واضح
+
+544
+00:54:17,000 --> 00:54:21,360
+أه؟ في معنى و كده الان f is differentiable in c
+
+545
+00:54:21,360 --> 00:54:25,580
+than لكل y نقل المستقبل اللي يعني استخدمنا limit f
+
+546
+00:54:25,580 --> 00:54:30,470
+of x نقص f of c على x minus c سوى f prime of cطبعا
+
+547
+00:54:30,470 --> 00:54:34,630
+أنا بعيد على أساس اللي بتابعنا مش سواعب انكم مش
+
+548
+00:54:34,630 --> 00:54:42,350
+سواعبته واضح؟ ماشي الحاجة الآن بدنا نستخدم ال
+
+549
+00:54:42,350 --> 00:54:47,410
+continuity لمين؟ لجي جي is continuous طبعا جي is
+
+550
+00:54:47,410 --> 00:54:51,870
+differentiable أصلا جي is differentiable عند مين؟
+
+551
+00:54:51,870 --> 00:54:56,690
+عند اللي هو D يعني جي is differentiable عند F of C
+
+552
+00:54:56,690 --> 00:55:03,740
+إذا continuousمظبوط؟ اذا الان since .. since G is
+
+553
+00:55:03,740 --> 00:55:12,540
+continuous at D بتساوي F of C اذا يعني صار عندى
+
+554
+00:55:12,540 --> 00:55:24,060
+limit G of Y لما Y تروح ل D بساوة مين؟ G of D اه
+
+555
+00:55:25,340 --> 00:55:29,760
+واضح ها؟ واضح .. واضح ها؟ خلّيني نعمل .. على أساس
+
+556
+00:55:29,760 --> 00:55:32,980
+ما نستخدم الرموز هذه نربطها باللي بقى بالها ..
+
+557
+00:55:32,980 --> 00:55:37,900
+بنغير بعدين معايا؟ طيب .. هذه إيش معناتها by
+
+558
+00:55:37,900 --> 00:55:41,460
+epsilon delta definition؟ أنه لكل .. الآن بدي
+
+559
+00:55:41,460 --> 00:55:45,780
+أستخدم الـ delta هذه لإن بدي أستخدم الاتنين مع بعض
+
+560
+00:55:45,780 --> 00:55:49,800
+الوصول اللي بديهاواضح؟ هذه لكل إبسلون في الدنيا،
+
+561
+00:55:49,800 --> 00:55:53,140
+أي إن كانت من ضمن إنيين الـDelta اللي لاجيناها،
+
+562
+00:55:53,140 --> 00:55:57,300
+واضح اللي بحكيه؟ الآن لكل إبسلون أكبر من 0، there
+
+563
+00:55:57,300 --> 00:56:01,160
+exists Gamma اللي هي .. ابحثت إنه لما يكون Y minus
+
+564
+00:56:01,160 --> 00:56:05,100
+D أصغر من Gamma، يعطيني هذا أصغر، اللي هو من اللي
+
+565
+00:56:05,100 --> 00:56:08,180
+بديه .. من اللي بدي أن أناسك دموع، اللي هو
+
+566
+00:56:08,180 --> 00:56:16,030
+الـDelta، الآنcontinuous ات كده then for every
+
+567
+00:56:16,030 --> 00:56:18,930
+ابسلون الابسلون اللي بده استخدمها اللي من ضمن
+
+568
+00:56:18,930 --> 00:56:21,290
+الابسلون هي اللي بتنفع ايه اللي بتنفع على ال delta
+
+569
+00:56:21,290 --> 00:56:25,970
+هذه then for delta أكبر من سفر there exist gamma
+
+570
+00:56:25,970 --> 00:56:32,670
+أكبر من سفر such that لما y minus d كون أكبر من
+
+571
+00:56:32,670 --> 00:56:37,170
+سفر وأصغر من الجامعة اللي لجيتها هذا بيعطيني اللي
+
+572
+00:56:37,170 --> 00:56:46,080
+هو ال Gof Y نقص الـ G of D يكون أصغر من مين من الـ
+
+573
+00:56:46,080 --> 00:56:51,820
+Epsilon اللي هي سميتها H Delta أي سؤال هذا بده
+
+574
+00:56:51,820 --> 00:57:01,100
+سميه H اسمه بده سميه اتنين اه هساخد من ال N طيب
+
+575
+00:57:01,100 --> 00:57:07,080
+شوفوا
+
+576
+00:57:07,080 --> 00:57:17,060
+الآنأنا في الآخر بدي اللي هو أصل لإنه اللي هي ال
+
+577
+00:57:17,060 --> 00:57:21,680
+derivative لل F prime اللي هو واحد ال F inverse
+
+578
+00:57:21,680 --> 00:57:29,840
+prime واحد على ال F prime ماشي طيب لما
+
+579
+00:57:29,840 --> 00:57:37,150
+تسيري الآن ال Yطبعا ال y .. ال .. ال .. ال .. ال
+
+580
+00:57:37,150 --> 00:57:37,150
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+581
+00:57:37,150 --> 00:57:37,190
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+582
+00:57:37,190 --> 00:57:37,210
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+583
+00:57:37,210 --> 00:57:37,750
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+584
+00:57:37,750 --> 00:57:37,770
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+585
+00:57:37,770 --> 00:57:38,230
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+586
+00:57:38,230 --> 00:57:46,550
+.. ال ..
+
+587
+00:57:46,550 --> 00:57:47,090
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+588
+00:57:47,090 --> 00:57:47,550
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+589
+00:57:47,550 --> 00:57:48,510
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+590
+00:57:48,510 --> 00:57:48,510
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+591
+00:57:48,510 --> 00:57:48,870
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+592
+00:57:48,870 --> 00:57:49,850
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+593
+00:57:49,850 --> 00:57:57,250
+ال ..
+
+594
+00:57:57,250 --> 00:57:58,030
+ال
+
+595
+00:58:03,070 --> 00:58:09,070
+إذا حيطلع عند الـ GY نقص GY دي إيش معلها؟ أقل من
+
+596
+00:58:09,070 --> 00:58:14,570
+مين؟ من دلتا اللي هي by using اتنين G of Y إيش
+
+597
+00:58:14,570 --> 00:58:19,730
+اسمها؟ اللي هي G of Y اللي هي Y إيش بتسوى F of X
+
+598
+00:58:19,730 --> 00:58:25,580
+عن G of Y بتسوى Xأه اللي هي عبارة عن X minus G of
+
+599
+00:58:25,580 --> 00:58:31,900
+D الـ D اللي هي عبارة عن مين؟ اللي هي F of C و الـ
+
+600
+00:58:31,900 --> 00:58:35,820
+G inverse للـ F إذا هتطلع عبارة عن مين؟ عن C أصغر
+
+601
+00:58:35,820 --> 00:58:45,140
+من مين؟ من دلتا واضح طبيعي وأكبر أكيد أكيد أكيد
+
+602
+00:58:45,140 --> 00:58:52,230
+أكيدواضح لحد الآن الان هذا بده يخليني أستخدم مين
+
+603
+00:58:52,230 --> 00:58:59,750
+أستخدم واحد إذا صار عندي then هذا متحقق الان then
+
+604
+00:58:59,750 --> 00:59:04,830
+بدي أربط هذا الكلام بمين باللي فوق اللي هو H of Y
+
+605
+00:59:04,830 --> 00:59:12,450
+H of Y ناقص F prime of C شوف كيف بده F مثلك الان
+
+606
+00:59:12,450 --> 00:59:19,780
+اللي هو بساويH of Y مين؟ H of Y اللي هو عبارة عن Y
+
+607
+00:59:19,780 --> 00:59:27,960
+ناقص D على G of Y ناقص G of D، مظبوط؟ اللي هو بيه
+
+608
+00:59:27,960 --> 00:59:30,960
+عبارة عن مين؟ F of G of Y ناقص كده، إلى آخرين
+
+609
+00:59:30,960 --> 00:59:42,000
+بيصير بساوي اللي هو F of G of Y ناقص F of G of D
+
+610
+00:59:42,000 --> 00:59:52,730
+على Yعلى g of y ناقص g of d اللي هو ناقص f prime
+
+611
+00:59:52,730 --> 01:00:00,150
+of c أصغر من مين؟ من y عارفين ليش؟ لأن الـ g of y
+
+612
+01:00:00,150 --> 01:00:04,880
+مين هي؟ الـ xو الـ G of D اللي هي مين الـ C
+
+613
+01:00:04,880 --> 01:00:10,480
+فاهمين؟ أنا قصد أكتبها بالصورة هذه عشان أقولكم أيش
+
+614
+01:00:10,480 --> 01:00:14,580
+لو سُلتله H of I ناقص أفرو برايم of C بساوي هذا
+
+615
+01:00:14,580 --> 01:00:21,500
+المقدار ناقص هذا متحقق كمان مرة يا جماعة اللي
+
+616
+01:00:21,500 --> 01:00:26,220
+بيحقق هذا غصب عنه بيحقق هذا و اللي بيحقق هذا بيحقق
+
+617
+01:00:26,220 --> 01:00:31,240
+هذا ماشي الحال أنا الأن أخدت عندي Y element in J
+
+618
+01:00:31,920 --> 01:00:36,960
+والـ Y لا تساوي D واخدت هذا يكون أصغر من دولتا تحت
+
+619
+01:00:36,960 --> 01:00:41,320
+هذا الشرط الآن أنا بشتغل تحت هذا الشرط الآن هذا
+
+620
+01:00:41,320 --> 01:00:46,140
+أدالي حسب اللي هي اتنين ان هذا ناقص هذا أصغر من
+
+621
+01:00:46,140 --> 01:00:51,090
+دولتا هذا اللي هو X وهذا اللي هو mean الـCالان
+
+622
+01:00:51,090 --> 01:00:54,790
+مادام أصغر من دلتا إذا أعطاني واحد إذا أعطاتني هذه
+
+623
+01:00:54,790 --> 01:00:59,510
+أصغر من هذه هذه في الواقع مين هي F of X هي عبارة
+
+624
+01:00:59,510 --> 01:01:03,830
+عن F of G of Y لأن G of Y هي ال X و C ليه F of D
+
+625
+01:01:03,830 --> 01:01:08,690
+صار هذا ناقص هذا أصغر من مين من إبسلون من جهة أخرى
+
+626
+01:01:08,690 --> 01:01:13,650
+هذا المقدار مين هو عبارة عن H of Y وهذا اللي هي F
+
+627
+01:01:13,650 --> 01:01:18,550
+prime of Cإذا القالة اللي وصلت إليه أنه لكل إبسلون
+
+628
+01:01:18,550 --> 01:01:22,670
+أكبر من 0 وجدت Gamma لأن Gamma بين إيديا واجدتها
+
+629
+01:01:22,670 --> 01:01:27,050
+من الأصل إلى لكل إبسلون أكبر من 0 لجيت Gamma ابحث
+
+630
+01:01:27,050 --> 01:01:30,290
+أنه لما يكون هذه المسافة بينها أصغر من Gamma
+
+631
+01:01:30,290 --> 01:01:33,610
+يعطينا المسافة أصغر من إبسلون يعني المسافة هذه
+
+632
+01:01:33,610 --> 01:01:40,760
+أصغر من إبسلون أو بمعنى آخر limitH of Y لما الـ Y
+
+633
+01:01:40,760 --> 01:01:50,320
+تروح للـ D بساوة F prime of مين؟ of C تصيرنا
+
+634
+01:01:50,320 --> 01:01:55,660
+جريبين من اللي بدنايا عند الـ H of Y limitها بساوة
+
+635
+01:01:55,660 --> 01:02:02,980
+limit هذا بساوة F prime of C هذا عبارة عن مين؟ عن
+
+636
+01:02:02,980 --> 01:02:10,910
+G prime مقلوق G primeإذا الآن خلصنا تقريبا نمسح
+
+637
+01:02:10,910 --> 01:02:16,130
+حده و نمسك
+
+638
+01:02:16,130 --> 01:02:28,150
+بهدي مع هدي صار عندي limit h of y لما ال y تقول
+
+639
+01:02:28,150 --> 01:02:34,960
+الى d limitH of Y لما Y تقول إلى D طلع عندي عبارة
+
+640
+01:02:34,960 --> 01:02:40,960
+عن F prime of C هي تحت من جهة أخرى هذا بيساوي مين؟
+
+641
+01:02:40,960 --> 01:02:48,720
+limit Y نقص D على G of Y نقص G of D لما Y تروح
+
+642
+01:02:48,720 --> 01:02:55,980
+لمين؟ للـ D صح؟ هذا عبارة عن limit أو واحد على
+
+643
+01:02:55,980 --> 01:03:03,130
+limitG of Y نقص G of D على Y minus D لما Y تروح لـ
+
+644
+01:03:03,130 --> 01:03:10,310
+D1 على الـ limit هذا فهي
+
+645
+01:03:10,310 --> 01:03:16,390
+ساوية 1 على g prime of D إذا صارت عندي F prime of
+
+646
+01:03:16,390 --> 01:03:23,710
+C بساوية 1 على g prime of D أو بمعنى آخر F prime
+
+647
+01:03:23,710 --> 01:03:31,800
+of C بساوية 1 على F G prime اللي هي Finverse prime
+
+648
+01:03:31,800 --> 01:03:42,000
+of D أو بتعبير آخر F inverse prime of D بساوية
+
+649
+01:03:42,000 --> 01:03:55,700
+واحد على F of C وهو المطلوب طيب
+
+650
+01:04:01,020 --> 01:04:02,360
+خدّيني أجف عن نقطة
+
+651
+01:04:19,990 --> 01:04:25,110
+هي نفس اللي فوق بس كانت اللي فوق عنده نقطة هنا in
+
+652
+01:04:25,110 --> 01:04:29,510
+general الان لو كانت على كل ال domain يعني الان
+
+653
+01:04:29,510 --> 01:04:33,150
+بقولي لو كانت I subset من R بيه an interval and F
+
+654
+01:04:33,150 --> 01:04:37,510
+من I لعند R strictly monotone on I ماشي ..
+
+655
+01:04:37,510 --> 01:04:42,750
+strictly monotone و let J بيسوء F of I and let G
+
+656
+01:04:42,750 --> 01:04:47,410
+من I لعند R بيه the function inverse of F ماشي ..
+
+657
+01:04:47,980 --> 01:04:52,360
+لأن إذا كانت F as differential على الـ I كلها إذا
+
+658
+01:04:52,360 --> 01:04:55,860
+أكيد هتكون continuous على الـ I كلها مدام هتكون
+
+659
+01:04:55,860 --> 01:04:58,300
+continuous على الـ I كلها و strictly monotone
+
+660
+01:04:58,300 --> 01:05:02,180
+هتكون الـ G العجبالها برضه strictly monotone و
+
+661
+01:05:02,180 --> 01:05:06,870
+continuous إذا شروط اللي فوق اتحققتإذا الـ J ..
+
+662
+01:05:06,870 --> 01:05:10,030
+طبعا ال F' of X معطينيها إشمالها تسوى X على كل ال
+
+663
+01:05:10,030 --> 01:05:13,350
+I إذا أكيد حسب النظر اللي قبل شوية G is
+
+664
+01:05:13,350 --> 01:05:17,230
+differentiable على كل ال J و ال G' بساوية واحدة
+
+665
+01:05:17,230 --> 01:05:22,210
+على F' لإنه composite مين؟ composite G واضح؟ يعني
+
+666
+01:05:22,210 --> 01:05:28,130
+.. كمان مرة؟ إذا كانت ال F strictly monotone و
+
+667
+01:05:28,130 --> 01:05:32,890
+كانت اللي هي ال J بتساوي كل ال F of Iوكانت الـ G
+
+668
+01:05:32,890 --> 01:05:37,310
+هي عبارة عن ال inverse للـ F وكانت ال F
+
+669
+01:05:37,310 --> 01:05:39,570
+differentiable على ال I مدام F differentiable على
+
+670
+01:05:39,570 --> 01:05:42,210
+ال I إذا صارت continuous على كل ال I مدام
+
+671
+01:05:42,210 --> 01:05:44,710
+continuous على ال I و strictly monotone هتكون ال
+
+672
+01:05:44,710 --> 01:05:47,710
+inverse سبعتها اللي هي ال G برضه continuous و
+
+673
+01:05:47,710 --> 01:05:51,700
+strictly monotoneإذا انتحققت شروط النظرية السابقة
+
+674
+01:05:51,700 --> 01:05:55,680
+إذا هتكون ال F و طبعا ال F' طبعا لها تساوي صفر ع
+
+675
+01:05:55,680 --> 01:05:59,020
+كل ال I إذا صارت ال G ال differential بالع كل ال J
+
+676
+01:05:59,020 --> 01:06:02,220
+و العلاقة بين التنتين G' بسوء واحد على F'
+
+677
+01:06:02,680 --> 01:06:09,040
+composite mean composite J أو زي ما بنقول G' of Y
+
+678
+01:06:09,040 --> 01:06:16,220
+بسوء واحد على F' of mean of X تحت العلاقة X بتساوي
+
+679
+01:06:16,220 --> 01:06:25,320
+F of Yأو Y بس هوقف و فكس Y بس هوقف و فكس طبعا هذا
+
+680
+01:06:25,320 --> 01:06:32,220
+البرهان كله حكيناه ال remark اللي عندنا ال remark
+
+681
+01:06:32,220 --> 01:06:37,100
+اللي عندنا ال
+
+682
+01:06:37,100 --> 01:06:43,460
+remark
+
+683
+01:06:43,460 --> 01:06:51,180
+اللي عندنا هي اللي حكيتها الآنD'Y بسوء واحد عالميا
+
+684
+01:06:51,180 --> 01:06:53,660
+على F'X
+
+685
+01:06:57,210 --> 01:06:59,530
+التي قبل بشوية حكيناها جي برايم أو اي بيسوا واحد
+
+686
+01:06:59,530 --> 01:07:03,890
+على اف برايم كومبز الجي ماشي الحال الآن صارت عندي
+
+687
+01:07:03,890 --> 01:07:08,950
+هذه جي برايم كومبز كومبز اف of X بيسوا جي برايم اف
+
+688
+01:07:08,950 --> 01:07:12,290
+of X بيسوا واحد على اف برايم كومبز الجي أو اي
+
+689
+01:07:12,290 --> 01:07:15,470
+بياخد نظرية مرتين و يسوا واحد على اف برايم of X
+
+690
+01:07:15,470 --> 01:07:18,370
+التانيين مع بعض جي برايم بيسوا واحد على اف برايم
+
+691
+01:07:18,370 --> 01:07:24,310
+of X و المرة الجاهزة اللي بنكمل بنعيد هذا اللي إذا
+
+692
+01:07:24,310 --> 01:07:26,470
+كان حكيناها على السريع
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gEcweaCqzlo_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gEcweaCqzlo_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..ea76a79a4769aeec008de4782f198e0c5458f18e
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gEcweaCqzlo_postprocess.srt
@@ -0,0 +1,1516 @@
+1
+00:00:04,930 --> 00:00:08,030
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:08,030 --> 00:00:13,610
+وبركاته ان شاء الله اليوم هنكون في مناقشة الفصل
+
+3
+00:00:13,610 --> 00:00:18,610
+الأول اللي كان يعني مقدمة عامة في الكيمياء في
+
+4
+00:00:18,610 --> 00:00:24,340
+مساقق الكيمياء العامة للعلوم الصحية والهندسيةطبعا
+
+5
+00:00:24,340 --> 00:00:29,320
+احنا اتكلمنا فيه عن المواد و عن المحاليل و فكرة
+
+6
+00:00:29,320 --> 00:00:32,340
+عامة عن أنواع المحاليل و شوية عن ال significant
+
+7
+00:00:32,340 --> 00:00:36,560
+figures و درجات الحرارة و التحويل بينها كل هذه
+
+8
+00:00:36,560 --> 00:00:40,900
+الأمور ليش لأن هو في نهاية فصل يعني مقدمة في
+
+9
+00:00:40,900 --> 00:00:46,370
+الكيمياالان لو جينا نشوف بعض الأسئلة في هذا الفصل
+
+10
+00:00:46,370 --> 00:00:53,090
+السؤال 113 بيقولنا classify يعني صنف each of the
+
+11
+00:00:53,090 --> 00:00:57,330
+following as a pure substance or a mixture يعني
+
+12
+00:00:57,330 --> 00:01:03,620
+صنف كل من الآت كمادة نقية أو كخليطF a mixed jar
+
+13
+00:01:03,620 --> 00:01:08,040
+وإذا كانت خليط indicate whether it is homogeneous
+
+14
+00:01:08,040 --> 00:01:12,540
+or heterogeneous واضحليها هل هو خليط متجانس أو غير
+
+15
+00:01:12,540 --> 00:01:17,760
+متجانس الان بيقول لي rice pudding rice pudding
+
+16
+00:01:17,760 --> 00:01:25,100
+يعني اللي بنسميها احنا المهلبية بالرز يعني مع
+
+17
+00:01:25,100 --> 00:01:29,980
+الحليب الرز و الحليب ايش بنعتبره مركب نقي ولا خليط
+
+18
+00:01:29,980 --> 00:01:35,380
+it is a mixed jarهل هو homogenous؟ لأ طبعا لأنه
+
+19
+00:01:35,380 --> 00:01:40,760
+منفصل بيكون مبين الرز و مبين باقي المحلول فهو
+
+20
+00:01:40,760 --> 00:01:46,180
+عبارة عن خليط غير متجانس هو عبارة عن heterogeneous
+
+21
+00:01:46,180 --> 00:01:53,680
+mixtureالفرع بيه ال sea water ماء البحر ماء البحر
+
+22
+00:01:53,680 --> 00:01:58,200
+من وين ما بتشرب فيه بتحسه مالح بنفس الطريقة هو
+
+23
+00:01:58,200 --> 00:02:03,640
+عبارة عن ماء فيه أملاح كتير إذا هو mixture و it is
+
+24
+00:02:03,640 --> 00:02:08,600
+a homogeneous mixture خليط متجانس يعني مابفرجش من
+
+25
+00:02:08,600 --> 00:02:12,440
+أول البحر ولا من نص البحر تدوج الطعم هتلاقي نفس
+
+26
+00:02:12,440 --> 00:02:19,930
+الاشي بنفس النسبة من الأملاح في الميهالفرع C
+
+27
+00:02:19,930 --> 00:02:25,550
+ماجنيسيام المجنيسيام عبارة عن متل، معدن، يكون نقي،
+
+28
+00:02:25,550 --> 00:02:31,860
+it is a pure substanceNumber D Crushed Ice Crushed
+
+29
+00:02:31,860 --> 00:02:36,500
+يعني الثلج المجروش يعني بتكون حبات التلج في تلاجة
+
+30
+00:02:36,500 --> 00:02:41,920
+خصوصة بتعمل هذا ال crushed ice بيلزم في المختبرات
+
+31
+00:02:41,920 --> 00:02:47,760
+الكيميا للتبريد بيطلع حبيبات ثلج تقريبا مثل حبة
+
+32
+00:02:47,760 --> 00:02:53,320
+الحمصالان هذا التلج هو عبارة عن ماء ماء نقي
+
+33
+00:02:53,320 --> 00:03:00,700
+المفروض وانه موجود على شكل تلج مكسر فهو عبارة عن
+
+34
+00:03:00,700 --> 00:03:04,220
+pure substance يعني في نهاية الثلج هو عبارة عن
+
+35
+00:03:04,220 --> 00:03:13,540
+الماء الماء النقيالسؤال الاربع عشان نفس
+
+36
+00:03:13,540 --> 00:03:16,920
+الفكرة
+
+37
+00:03:16,920 --> 00:03:21,260
+صن فيها الأشياء إما إنها pure mixture أو a pure
+
+38
+00:03:21,260 --> 00:03:24,580
+substance أو a mixture if a mixture indicate
+
+39
+00:03:24,580 --> 00:03:28,730
+whether it is homogeneous or heterogeneousيعني يا
+
+40
+00:03:28,730 --> 00:03:34,250
+مادة نقية يا مخلوط مخلوط بيكون homogeneous متجانس
+
+41
+00:03:34,250 --> 00:03:38,630
+أو heterogeneous غير متجانس بيسألني السؤال الأول
+
+42
+00:03:38,630 --> 00:03:45,960
+air الهواء الهواء عبارة عنمجموعة من الغازات
+
+43
+00:03:45,960 --> 00:03:49,720
+الموجودة بنسبة أكبر النيتروجين كأنه solvent فيه
+
+44
+00:03:49,720 --> 00:03:54,600
+أكسجين فيه ثانو أكسيد كربون فيه غازات ملوثات أخرى
+
+45
+00:03:54,600 --> 00:03:59,740
+لكن كله موجود بنفس النسبة وبناء على ذلك it is a
+
+46
+00:03:59,740 --> 00:04:06,710
+homogeneous mixture خليط متجانسفرع بيه tomato
+
+47
+00:04:06,710 --> 00:04:12,110
+juice عصير الطماطم طبعا عصير الطماطم احنا بنشوف
+
+48
+00:04:12,110 --> 00:04:16,450
+فيه البذور والمواد الصلبة وبناء على ذلك هو خليط
+
+49
+00:04:16,450 --> 00:04:20,830
+الحقيقة ليس مادة نقية خليط من مواد وهذا الخليط
+
+50
+00:04:20,830 --> 00:04:26,830
+heterogeneous mixture خليط غير متجانسالانسان بيقدر
+
+51
+00:04:26,830 --> 00:04:31,330
+يميز بذور الطماطم بيقدر يميز الأشياء الصلبة القشور
+
+52
+00:04:31,330 --> 00:04:37,180
+فعصير الطماطم هو عبارة عن خليط غير متجانسأيوديني
+
+53
+00:04:37,180 --> 00:04:41,740
+كريستال بلورات اليود هذا عبارة عن مادة نقية pure
+
+54
+00:04:41,740 --> 00:04:47,260
+substance ال sand ال sand على فرض أنه فيه أشياء
+
+55
+00:04:47,260 --> 00:04:51,280
+كتيرة غير ثانو أكسيد السيليكون يعني الإنسان بده
+
+56
+00:04:51,280 --> 00:04:55,480
+يفكر هل هو بيقصد بس ثانو أكسيد السيليكون ولا
+
+57
+00:04:55,480 --> 00:05:00,110
+الصخور والمركبات الكتيرة غالبا معظم الرملفي كتير
+
+58
+00:05:00,110 --> 00:05:04,710
+ملوثات ومركبات كتيرة وعشان هي الأصح نعتبره mixture
+
+59
+00:05:04,710 --> 00:05:09,450
+و heterogeneous mixture يعني احنا مابنكونش ناخد بس
+
+60
+00:05:09,450 --> 00:05:13,930
+اللي هي ثانو أكسيد السيليكون الأساسي لكن الرمل
+
+61
+00:05:13,930 --> 00:05:18,750
+مخلوط من أشياء كتيرة أو أوساخ وما إلى ذلك فخليني
+
+62
+00:05:18,750 --> 00:05:25,190
+اعتبره heterogeneous mixture سؤال
+
+63
+00:05:25,190 --> 00:05:32,730
+واحد وعشرينLabel each of the following as either a
+
+64
+00:05:32,730 --> 00:05:37,610
+physical process or a chemical process طبعا ال
+
+65
+00:05:37,610 --> 00:05:41,190
+chemical process اللي بيحصل فيه تفاعل بيغير المادة
+
+66
+00:05:41,190 --> 00:05:45,110
+ال physical process المادة بتتغير ممكن من حالة
+
+67
+00:05:45,110 --> 00:05:50,770
+لحالة من صلب لسائل لغاز ممكن تسخن ممكن تبرد لكن
+
+68
+00:05:50,770 --> 00:05:57,010
+مافيش تغير في جوهر المادة واضح؟الان التفاعل هو
+
+69
+00:05:57,010 --> 00:06:02,690
+يغير جوهر المادة ومكون المادة الأساسي فبيقول ليه؟
+
+70
+00:06:02,690 --> 00:06:05,670
+شوف لهذه التغيرات هل هي chemical أو physical
+
+71
+00:06:05,670 --> 00:06:13,410
+processRusting of a metal can يعني عملية صدأ علبة
+
+72
+00:06:13,410 --> 00:06:20,040
+أو عبوة من الصفيحطبعا الصدق هو عبارة عن تفاعل
+
+73
+00:06:20,040 --> 00:06:26,320
+كيميائي هل الحديد بيبقى حديد بكل خواص المعدن ولا
+
+74
+00:06:26,320 --> 00:06:30,280
+الحديد اللي بيلمع وكويس بيصير عبارة عن مادة اكسيد
+
+75
+00:06:30,280 --> 00:06:35,540
+الحديد ومادة هشة بتختلف اذا هنا عبارة عن تفاعل
+
+76
+00:06:35,540 --> 00:06:40,040
+حديد صار اكسيد حديد وبناء على ذلك هذه عبارة عن
+
+77
+00:06:40,040 --> 00:06:46,020
+chemical processالفرع بـ Boiling a cup of water
+
+78
+00:06:46,020 --> 00:06:49,880
+تسخين فنجان من الميّه يا ترى لو الميّه كانت
+
+79
+00:06:49,880 --> 00:06:55,000
+حرارتها عشرين و سخنت انا لخمسة و تلاتين هل اتغير
+
+80
+00:06:55,000 --> 00:06:59,620
+جوهر الميّه؟ لأ عشان ايه كان ده physical process
+
+81
+00:06:59,620 --> 00:07:03,860
+مش chemical process الميّه ما زالت كمية ميّه لكن
+
+82
+00:07:03,860 --> 00:07:11,060
+سخنةالفرع C is Pulverizing an Aspirin يعني لو جيبت
+
+83
+00:07:11,060 --> 00:07:15,760
+حبة Aspirin و طحنتها و خلتها ك powder الآن ال
+
+84
+00:07:15,760 --> 00:07:19,580
+Aspirin ما زال Aspirin ما تغير تركيبه الكيميائي
+
+85
+00:07:19,580 --> 00:07:24,360
+فهذا فرع C عبارة أيضا عن physical process
+
+86
+00:07:26,730 --> 00:07:33,630
+فرع دي Digesting a candy bar عملية الهضم عملية
+
+87
+00:07:33,630 --> 00:07:38,850
+بيتحول المركبات و ببدأ اللي هو التمثيل الغذائي
+
+88
+00:07:38,850 --> 00:07:42,810
+وبناء على ذلك عملية الهضم هي عملية كيميائية
+
+89
+00:07:43,330 --> 00:07:46,650
+المُركّب نفسه بتفكك لأشياء تانية أو بتحول لأشياء
+
+90
+00:07:46,650 --> 00:07:51,690
+تانية فالـ Digesting a candy bar يعني إنسان مسك
+
+91
+00:07:51,690 --> 00:08:00,330
+يعني نوع من الحلويات قطعة من الحلويات وهضمها هذا
+
+92
+00:08:00,330 --> 00:08:04,270
+عبارة عن chemical processExploding of
+
+93
+00:08:04,270 --> 00:08:08,390
+Nitroglycerin طبعاً في الانفجار المادة المتفجرة
+
+94
+00:08:08,390 --> 00:08:11,830
+بتتحول إلى غازات و هي اللي بتعمل الضغط و بتسبب
+
+95
+00:08:11,830 --> 00:08:16,230
+الهدم طبعاً مادة صلبة صارت غازات إذا أنا ده عبارة
+
+96
+00:08:16,230 --> 00:08:20,170
+عن التحول كينيائي it is a chemical process
+
+97
+00:08:22,670 --> 00:08:28,110
+نشوف سؤال 22 نفس الاشي نفس الفكرة بيسأل عن نفس
+
+98
+00:08:28,110 --> 00:08:32,110
+الموضوع ال physical و ال chemical process بيقول a
+
+99
+00:08:32,110 --> 00:08:37,250
+match is lit يعني ولعنا عود ثقاب أو عود كبريد زي
+
+100
+00:08:37,250 --> 00:08:41,870
+ما بنسميه and held under a cold piece of metal
+
+101
+00:08:41,870 --> 00:08:46,690
+يعني ولعنا عود ثقاب وحطنا تحت قطعة من المعدن لأن
+
+102
+00:08:46,690 --> 00:08:51,390
+قطعة المعدن بدأت تسخن التسخين هذاعبارة عن physical
+
+103
+00:08:51,390 --> 00:08:54,590
+process ما تغيرت المادة يعني الحديد لما بيسخن
+
+104
+00:08:54,590 --> 00:08:59,710
+بيعود يبرد ما بيتغير كمادة واضح؟ the following
+
+105
+00:08:59,710 --> 00:09:03,150
+observations are made صارت عند الملاحظات التالية
+
+106
+00:09:03,150 --> 00:09:07,690
+the match burns عوض الكبريت لما ينحرج ويتحول إلى
+
+107
+00:09:07,690 --> 00:09:12,910
+مواد أخرى وإلى دخان وإلى ثانو أكسيد الكربون عملية
+
+108
+00:09:12,910 --> 00:09:18,400
+الاحتراق هذه is a chemical processنمبر بيه المعدن
+
+109
+00:09:18,400 --> 00:09:24,580
+يتحرر المعدن المعدنية
+
+110
+00:09:24,580 --> 00:09:29,220
+سخنت التسخين عملية فيزيائية لم تتغير فيها المادة
+
+111
+00:09:29,220 --> 00:09:36,500
+عشانك بيه هو اتشانج فيزيائي المعدن يتحرر المعدن لو
+
+112
+00:09:36,500 --> 00:09:45,470
+كان عندى ماء واتكثف على سطح المعدنفي النهاية سواء
+
+113
+00:09:45,470 --> 00:09:50,630
+كان بخار سواء كان سائل سواء كان صلب ثلج يعني ice
+
+114
+00:09:50,630 --> 00:09:55,950
+هو نفس المادة فأي تغير بين الحالات is a physical
+
+115
+00:09:55,950 --> 00:10:00,790
+process تغير سواء بالانصحار أو بالتكثيف هو عبارة
+
+116
+00:10:00,790 --> 00:10:07,310
+عن physical processنمبر دي سوط اللي هو الدخان
+
+117
+00:10:07,310 --> 00:10:12,530
+الأسود اللي بيطلع هذا اللي هو ال carbon و بترسب و
+
+118
+00:10:12,530 --> 00:10:16,470
+دايما تشوفوا لما بنسخن بيصير عندنا سواد و بالذات
+
+119
+00:10:16,470 --> 00:10:21,870
+لما بنشوى بالفحم و غيرهSoot اللي هو عبارة عن
+
+120
+00:10:21,870 --> 00:10:26,370
+carbon is deposited on the metal يعني لما انا ولعت
+
+121
+00:10:26,370 --> 00:10:31,730
+عودي الكابريت لاحظت ان المعدن صار عليه سواد عبارة
+
+122
+00:10:31,730 --> 00:10:39,400
+عن carbon من عودي الاثقاب هذاالان هذا ال carbon هو
+
+123
+00:10:39,400 --> 00:10:44,120
+عبارة عن ناتج عملية احتراق ناتج تفاعل كيميائي يعني
+
+124
+00:10:44,120 --> 00:10:49,680
+عود الخشب ماكنش أساسا carbon كان هو عبارة عن الخشب
+
+125
+00:10:49,680 --> 00:10:54,900
+اللي هو ال cellulose وما إلى ذلك بالاحتراق فقد
+
+126
+00:10:54,900 --> 00:10:58,700
+الماء وصار عبارة عن carbon إذا هذا عبارة عن
+
+127
+00:10:58,700 --> 00:11:00,600
+chemical process
+
+128
+00:11:07,000 --> 00:11:11,860
+السؤال 31 الحقيقة بيعتمد على جدول الاختصارات
+
+129
+00:11:11,860 --> 00:11:15,380
+بإمكان أي طالب يرجع بسهولة للجدول اللي موجود في
+
+130
+00:11:15,380 --> 00:11:19,920
+الشرح ويشوف ايش معنى هذه الاختصارات what
+
+131
+00:11:19,920 --> 00:11:22,380
+exponential notation do the following
+
+132
+00:11:22,380 --> 00:11:28,080
+abbreviations represent يعني ال D هي عبارة عن ال
+
+133
+00:11:28,080 --> 00:11:35,290
+DC ال C الصنتي ال F الفيمتوالـ Mu الـ Micro عشرة
+
+134
+00:11:35,290 --> 00:11:38,950
+أقص ناقص ستة الـ M الميجا عشرة أقص ستة الـ K
+
+135
+00:11:38,950 --> 00:11:42,970
+الكيلو عشرة أقص تلاتة الـ N النانو عشرة أقص ناقص
+
+136
+00:11:42,970 --> 00:11:47,930
+تسعة الـ M الميلي عشرة أقص ناقص تلاتة الـ P البيكو
+
+137
+00:11:47,930 --> 00:11:52,250
+عشرة أقص ناقص اتناشر هذه كل الاختصارات وإيش بتساوي
+
+138
+00:11:52,250 --> 00:11:55,810
+موجود الحقيقة في الجدول اللي عندكم
+
+139
+00:11:58,600 --> 00:12:03,480
+بعد ذلك يأتي بسؤالنا بيقولنا use appropriate
+
+140
+00:12:03,480 --> 00:12:06,800
+metric prefixes to write the following
+
+141
+00:12:06,800 --> 00:12:12,260
+measurements without use of exponents يعني هو بده
+
+142
+00:12:12,260 --> 00:12:18,960
+يعطينا مقياس ممكن ناخد واحد منهم هذا المقياس فيه
+
+143
+00:12:18,960 --> 00:12:23,500
+exponent يعني فيه مرفوع للقوة كذا يعني مثلا لو
+
+144
+00:12:23,500 --> 00:12:28,360
+احنا اجينا اخدنا الفرع ايه؟ two point threeفي عشرة
+
+145
+00:12:28,360 --> 00:12:35,760
+اص ناقص عشرة من ال letter الان انا بدي اختصر هذه
+
+146
+00:12:35,760 --> 00:12:39,840
+من خلال ان استعمل احد الاختصارات اللي هنا و احصل
+
+147
+00:12:39,840 --> 00:12:43,900
+على جواب مافيش فيه هذا ال exponent اللي هي العشرة
+
+148
+00:12:43,900 --> 00:12:51,480
+اص ناقص عشرة من ال letter الانUse Appropriate
+
+149
+00:12:51,480 --> 00:12:54,880
+Metric Prefixes استعمل اختصارات معينة من اللي
+
+150
+00:12:54,880 --> 00:13:00,960
+اتعلمناها بحيث تكتب هذا القياس بدون هذا الكلام
+
+151
+00:13:00,960 --> 00:13:06,220
+الان باختصار احنا بدنا نيجي نقول ايش اقرب اشي
+
+152
+00:13:06,220 --> 00:13:13,090
+للعشرة او ناقص عشرةأقرب اشي للعشرة اصناقس عشرة
+
+153
+00:13:13,090 --> 00:13:18,330
+اللي هي مثلا ممكن نقول العشرة اصناقس تسعة لما بد
+
+154
+00:13:18,330 --> 00:13:26,020
+العشرة اصناقس تسعة انا بقول ال nano ال nanoلتر هو
+
+155
+00:13:26,020 --> 00:13:32,280
+عبارة عن واحد في عشر أقص ناقص تسعة من اللتر واضح
+
+156
+00:13:32,280 --> 00:13:37,680
+فلو أنا استعملت زي هيك وعملت عملية ضرب بسيطة هتروح
+
+157
+00:13:37,680 --> 00:13:40,820
+العشر أقص ناقص تسعة مع جزء من ال hand بيبقى العشر
+
+158
+00:13:40,820 --> 00:13:46,580
+أقص ناقص واحد بصفة الجواب point two three وشطبنا
+
+159
+00:13:46,580 --> 00:13:50,860
+ال letter مع ال letter point two three nano liter
+
+160
+00:13:52,380 --> 00:13:58,420
+وكلهم من ناحية المبدأ بنحلهم بنفس الطريقة يعني انا
+
+161
+00:13:58,420 --> 00:14:03,340
+بدي اشوف اي conversion factor يشطب لهذا الأس هنا
+
+162
+00:14:03,340 --> 00:14:07,440
+ماكنش عندي بالظبط عشرة أس ناقص عشرة في عند العشرة
+
+163
+00:14:07,440 --> 00:14:11,320
+أس ناقص تسعة اللي هي ال nano liter بساوي واحد في
+
+164
+00:14:11,320 --> 00:14:14,680
+عشرة أس ناقص تسعة من ال liter شطبنا ال liter مع ال
+
+165
+00:14:14,680 --> 00:14:17,520
+liter عشرة أس ناقص تسعة مع عشرة أس ناقص تسعة من
+
+166
+00:14:17,520 --> 00:14:25,320
+هنا ونجلنا الفاصلة درجةبطلع الجواب نشوف كمان واحدة
+
+167
+00:14:25,320 --> 00:14:30,520
+منهم مثلا
+
+168
+00:14:30,520 --> 00:14:38,410
+لو بدوا الـ 4.7 في 10 أصناق الـ 6 جرام4.7 في عشرة
+
+169
+00:14:38,410 --> 00:14:43,290
+أصناق الستة جرام الان ايش القياس اللي انا عندى
+
+170
+00:14:43,290 --> 00:14:48,930
+بالنسبة للعشرة أصناق الستة اه انا عندى الواحد
+
+171
+00:14:48,930 --> 00:14:55,850
+micro gram بساوي عشرة أصناق الستة milligram واضح
+
+172
+00:14:56,580 --> 00:15:00,540
+بساوي عشرة أصناق الستة ميلي جرام بناء على ذلك لو
+
+173
+00:15:00,540 --> 00:15:04,140
+احنا شطبنا جرام مع جرام العشرة أصناق الستة مع
+
+174
+00:15:04,140 --> 00:15:10,830
+العشرة أصناق الستة هذه عبارة عن نفس 4.74.7
+
+175
+00:15:10,830 --> 00:15:16,910
+microgram يعني نفس الـ 4.7 في عشرة أصناق الستة
+
+176
+00:15:16,910 --> 00:15:21,310
+جرام أنا ممكن أعبر عنها بطريقة أجمل أقول هي نفسها
+
+177
+00:15:21,310 --> 00:15:27,270
+4.7 microgram كلهم من ناحية المبدأ بنحلم زيك بنشوف
+
+178
+00:15:27,270 --> 00:15:32,070
+إيش ال conversion factor معامل التحويل اللي موجودة
+
+179
+00:15:32,070 --> 00:15:37,350
+بنخليه في المقام وبطلع الجواب المحول عندنا زي ما
+
+180
+00:15:37,350 --> 00:15:41,270
+هو موجودحل باقي الفروع
+
+181
+00:15:49,740 --> 00:15:54,420
+هذه المسألة تلاتة و تلاتين بتحكي عن اللي هو عملية
+
+182
+00:15:54,420 --> 00:15:58,660
+تحويل أو التحويل بين درجات الحرارة make the
+
+183
+00:15:58,660 --> 00:16:03,080
+following convergence من فهرين هايت ل degree
+
+184
+00:16:03,080 --> 00:16:05,960
+centigrade من degree centigrade ل فهرين هايت من
+
+185
+00:16:05,960 --> 00:16:10,600
+degree centigrade ل Kelvin طبعا انا بعرف ان ال
+
+186
+00:16:10,600 --> 00:16:11,140
+Kelvin
+
+187
+00:16:15,730 --> 00:16:20,570
+بساوي 273.15 زائد مقدار الحرارة بال degree
+
+188
+00:16:20,570 --> 00:16:24,010
+centigrade يعني هاي التحويل بين Kelvin و degree
+
+189
+00:16:24,010 --> 00:16:27,710
+centigrade الان بالنسبة للتحويل بين degree
+
+190
+00:16:27,710 --> 00:16:31,910
+centigrade و Fahrenheit الناس اللي متذكرين معناها
+
+191
+00:16:31,910 --> 00:16:35,430
+وهنا بدنا نفرجيهم معاها موجودة وين بالظبط في عندى
+
+192
+00:16:35,430 --> 00:16:41,540
+علاقة تحويلية اللي هيالـ degrees Fahrenheit بتساوي
+
+193
+00:16:41,540 --> 00:16:47,540
+تسعة على خمسة في ال degree centigrade زائد اتنين
+
+194
+00:16:47,540 --> 00:16:52,700
+وتلاتين واضح؟ هذه العلاقة اللي بنحول من خلالها ما
+
+195
+00:16:52,700 --> 00:16:56,740
+بين ال Fahrenheit وال degree centigrade الآن إذا
+
+196
+00:16:56,740 --> 00:17:01,620
+هذه العلاقات كنا احنا مستحضرينها
+
+197
+00:17:03,500 --> 00:17:11,720
+معنى ذلك مجرد أننا نطبق تطبيق بسيط يعني على سبيل
+
+198
+00:17:11,720 --> 00:17:18,120
+المثال لو أحنا بدنا ناخد من الفرع B موجود عنده
+
+199
+00:17:18,120 --> 00:17:25,080
+اتنين و سبعين ديجريز فهرين هايت ديجريز فهرين هايت
+
+200
+00:17:25,080 --> 00:17:28,880
+انا بقول اتنين و سبعين ايش بتساوي في ديجريز
+
+201
+00:17:28,880 --> 00:17:32,980
+سنتيجريت اتنين و سبعين بتساوي تسعة على خمسةفي ال
+
+202
+00:17:32,980 --> 00:17:38,000
+degree centigrade زائد اتنين و تلاتين لو عملنا هذه
+
+203
+00:17:38,000 --> 00:17:40,780
+الحسابات نجلنا الاتنين و التلاتين هنا ضربناها
+
+204
+00:17:40,780 --> 00:17:44,340
+بخمسة جسمناها على تسعة بتطلع ال degree centigrade
+
+205
+00:17:44,340 --> 00:17:48,600
+اتنين و عشرين و اتنين من عشرة واضح؟ و كله بنفس
+
+206
+00:17:48,600 --> 00:17:53,420
+المنطق و ال Kelvin ل degree centigrade أسهل كتير و
+
+207
+00:17:53,420 --> 00:17:59,770
+الإجابات موجودة عندنا نفس الشيء تماماالسؤال اللي
+
+208
+00:17:59,770 --> 00:18:05,410
+بعده بيقول لي the temperature of a warm summer day
+
+209
+00:18:05,410 --> 00:18:10,490
+يعني يوم صيف حار الحرارة سبعة و ثمانين degrees
+
+210
+00:18:10,490 --> 00:18:13,790
+Fahrenheit بدي اعوضها ان سبعة و تمانين degrees
+
+211
+00:18:13,790 --> 00:18:18,750
+Fahrenheit تسعة على خمسة degrees centigrade زي
+
+212
+00:18:18,750 --> 00:18:23,350
+اتنين و تلاتين جداش بتكون هذا اليوم حرارته هذا
+
+213
+00:18:23,350 --> 00:18:27,010
+اليوم حرارته thirty point six degrees centigrade
+
+214
+00:18:27,010 --> 00:18:31,520
+يعني لما نعوضبيظل عندى degrees centigrade وعشان
+
+215
+00:18:31,520 --> 00:18:35,880
+هيك بدوا الإنسان يكون عارف يعني احنا في بلادنا لما
+
+216
+00:18:35,880 --> 00:18:40,660
+بنقول الحرارة تلاتين تقريبا بتكون بامريكا اللى
+
+217
+00:18:40,660 --> 00:18:44,760
+بيستعملوا مقياس ال Fahrenheit سبعة و تمانين
+
+218
+00:18:44,760 --> 00:18:48,240
+degrees Fahrenheit what is the temperature in
+
+219
+00:18:48,240 --> 00:18:50,840
+degrees centigrade حسبناها السبعة و تمانين هي
+
+220
+00:18:50,840 --> 00:18:57,040
+التلاتين الآن بيسألنا سؤال تاني بيهمعظم المعلومات
+
+221
+00:18:57,040 --> 00:19:01,120
+العلمية تم تفجيرها لـ 25 دقائق دقائق دقائق دقائق
+
+222
+00:19:01,120 --> 00:19:02,040
+دقائق دقائق دقائق دقائق دقائق دقائق دقائق دقائق
+
+223
+00:19:02,040 --> 00:19:03,740
+دقائق دقائق دقائق دقائق دقائق دقائق دقائق دقائق
+
+224
+00:19:03,740 --> 00:19:06,040
+دقائق دقائق دقائق دقائق دقائق دقائق دقائق دقائق
+
+225
+00:19:06,040 --> 00:19:08,380
+دقائق دقائق دقائق دقائق دقائق دقائق دقائق دقائق
+
+226
+00:19:08,380 --> 00:19:13,400
+دقائق دقائق
+
+227
+00:19:13,400 --> 00:19:13,760
+دقائق دقائق دقائق دقائق دقائق دقائق دقائق دقائق
+
+228
+00:19:13,760 --> 00:19:20,980
+دقائق دقائق دقائق دقائق دقائق
+
+229
+00:19:22,920 --> 00:19:27,920
+بنقدر نحسب جداش بتساوي بالفهرين هايت بتطلع سبعة و
+
+230
+00:19:27,920 --> 00:19:31,900
+سبعين فهرين هايت الان الخمسة و عشرين نفسها لما
+
+231
+00:19:31,900 --> 00:19:36,120
+يقولياها بالكلفن مائتين و تلاتة و سبعين زائد خمسة
+
+232
+00:19:36,120 --> 00:19:41,200
+و عشرين مائتين و تمانية وتسعين فالعمليات سهلة و
+
+233
+00:19:41,200 --> 00:19:47,980
+نفس الشيء تحويلات الفرع C و D هم هدولة العلاقتين
+
+234
+00:19:47,980 --> 00:19:51,580
+بدنا نستعملهم أكتر من مرة عشان نتدرب على عملية
+
+235
+00:19:51,580 --> 00:19:52,620
+التحويل
+
+236
+00:19:55,820 --> 00:20:03,600
+سؤال خمسة وأربعينيبدو هذا السؤال في موضوع الـ
+
+237
+00:20:03,600 --> 00:20:30,960
+Significant Figures
+
+238
+00:20:31,060 --> 00:20:37,000
+فرع C الصفر اللي في الوسط والصفر اللي على يمين
+
+239
+00:20:37,000 --> 00:20:41,160
+الفاصلة يحتسب اذا واحد اتنين تلاتة اربعة خمس خمس
+
+240
+00:20:41,160 --> 00:20:46,560
+significant figures الفرع D الصفر اللي على لتحديد
+
+241
+00:20:46,560 --> 00:20:50,780
+موقع الفاصلة لا يحتسب اللي في النص يحتسب اذا دول
+
+242
+00:20:50,780 --> 00:20:56,940
+تلاتة significant figures الفرع Eالسفر اللي في
+
+243
+00:20:56,940 --> 00:21:00,420
+النص و اللي على يمين الفاصلة كله يحتسب اذا انا
+
+244
+00:21:00,420 --> 00:21:02,960
+اقول واحد اتنين تلاتة اربعة خمس significant
+
+245
+00:21:02,960 --> 00:21:08,220
+figures الاربعمية الحقيقة طالما مافيش فاصلة بس
+
+246
+00:21:08,220 --> 00:21:12,600
+بنحتسب one significant figure لو كتبلي فاصلة بصير
+
+247
+00:21:12,600 --> 00:21:16,960
+احتسب three significant figures لو كتبلي فاصلة و
+
+248
+00:21:16,960 --> 00:21:20,220
+بعدها سفر بصير احتسب four significant figures لكن
+
+249
+00:21:20,220 --> 00:21:24,560
+بهذا الشكل بدون فاصلة بس بعتبر one significant
+
+250
+00:21:24,560 --> 00:21:24,980
+figure
+
+251
+00:21:30,690 --> 00:21:37,040
+السؤال اللي بعدهبقول لي اعمل تقريب معين around
+
+252
+00:21:37,040 --> 00:21:40,020
+each of the following numbers to four significant
+
+253
+00:21:40,020 --> 00:21:42,300
+figures يعني بدي يكون عندي four significant
+
+254
+00:21:42,300 --> 00:21:45,640
+figures and express the result in standard
+
+255
+00:21:45,640 --> 00:21:48,520
+exponential notation لما يقول لي standard
+
+256
+00:21:48,520 --> 00:21:51,900
+exponential notation يعني انا بدي احط رقم واحد
+
+257
+00:21:51,900 --> 00:21:56,520
+بعدين فاصلة و بعدين اضرب في عشر أقص سواء موجب او
+
+258
+00:21:56,520 --> 00:22:06,810
+سالمفلما نجي مثلا على الفرع A 102.53070
+
+259
+00:22:06,810 --> 00:22:12,750
+لان هذا بداية لما دخلي بال scientific notation بدي
+
+260
+00:22:12,750 --> 00:22:19,630
+أنجل بعد رقم واحد بدي أقول 1.0253070
+
+261
+00:22:19,630 --> 00:22:23,810
+في عشر أقصى اتنين طالما اتحركت للشمال بدي أضرب في
+
+262
+00:22:23,810 --> 00:22:26,690
+عشر أقصى اتنين الان هو يطالب مني كام significant
+
+263
+00:22:26,690 --> 00:22:32,400
+figureأربعة هي واحد اتنين تلاتة اربعة انا بدي
+
+264
+00:22:32,400 --> 00:22:36,800
+لحدها انه الان لو جيت اقرب هذا مابتقربش اذا الجواب
+
+265
+00:22:36,800 --> 00:22:43,600
+حيصير عندي one point oh two five في عشرة أس اتنين
+
+266
+00:22:43,600 --> 00:22:52,740
+واضح ناخد كمان مثال C مثلا والباقي موجود حلهم لو
+
+267
+00:22:52,740 --> 00:22:57,220
+كنا بدنا نشوف مثال C point
+
+268
+00:22:58,550 --> 00:23:10,750
+OO 8 5 4 3 2 1 0 لان هذا لما انا بدي أصير
+
+269
+00:23:10,750 --> 00:23:16,260
+scientific notation بدي أتحرك بهذا الشكلواضح بدا
+
+270
+00:23:16,260 --> 00:23:23,840
+اتحرك بهذا الشكل بناء على ذلك حاسين 8.543210
+
+271
+00:23:23,840 --> 00:23:28,100
+طبعا هضربه في عشرة أس ناقص تلاتة لما اتحرك على
+
+272
+00:23:28,100 --> 00:23:31,700
+اليمين هضرب في عشرة أس ناقص تلاتة على اليسار يعني
+
+273
+00:23:31,700 --> 00:23:35,720
+في ال positive الان هو بدوياها ل four significant
+
+274
+00:23:35,720 --> 00:23:41,630
+figures هاي one two three fourلما بنعمل تقريب
+
+275
+00:23:41,630 --> 00:23:50,990
+هدولة بيروحوا إذا الجواب 8.543 x 10⁻³ وباقي الأمور
+
+276
+00:23:50,990 --> 00:24:02,990
+موجودة بنفس الطريقة لو
+
+277
+00:24:02,990 --> 00:24:12,370
+إجينا لسؤال 49 مثله سؤال 50 هو استعمال نتائج الـ
+
+278
+00:24:12,370 --> 00:24:15,010
+significant figures carry out the following
+
+279
+00:24:15,010 --> 00:24:18,330
+operations and express the answers with the
+
+280
+00:24:18,330 --> 00:24:23,790
+appropriate number of significant figures طبعا
+
+281
+00:24:23,790 --> 00:24:29,050
+الان احنا بنتذكر ان في عندنا قاعدتين اساسيتين
+
+282
+00:24:29,560 --> 00:24:35,960
+القاعدة الأولى للجامع أنه أنا بدي أمشي مع النتيجة
+
+283
+00:24:35,960 --> 00:24:43,260
+الأقل دقة بمعنى مع العدد الأقل من الخاناتهذه قاعدة
+
+284
+00:24:43,260 --> 00:24:46,680
+الجمع انا ممكن اشوفها من هذا الرقم يعني لما انا
+
+285
+00:24:46,680 --> 00:24:51,860
+بدي اقول fourteen point three five oh five بدي
+
+286
+00:24:51,860 --> 00:24:57,380
+اجمعها مع two point six five الان وين الدقة اقل
+
+287
+00:24:57,380 --> 00:25:00,940
+بدي امشي معها وين عدد الأرقاب بعد الفاصلة اقل بدي
+
+288
+00:25:00,940 --> 00:25:06,180
+امشي معها اذا هذا الجواب اللي هيكون عندى فلما بدي
+
+289
+00:25:06,180 --> 00:25:10,960
+اجمع من هنا هيكون عندى five oh zero
+
+290
+00:25:23,310 --> 00:25:26,410
+القائدة الجامع والطريح
+
+291
+00:25:30,370 --> 00:25:37,050
+التانية قاعدة الضرب والقسمة قاعدة الضرب والقسمة
+
+292
+00:25:37,050 --> 00:25:41,230
+بتقول أنا بعد number of significant figures بعد
+
+293
+00:25:41,230 --> 00:25:45,710
+number of significant figures وباخد على العدد
+
+294
+00:25:45,710 --> 00:25:51,490
+الأقل من ال significant figures بعد number of
+
+295
+00:25:51,490 --> 00:25:55,210
+significant figures وباخد على العدد الأقل من ال
+
+296
+00:25:55,210 --> 00:25:59,390
+significant figures يعني مثلالو انا بدي اشوف
+
+297
+00:25:59,390 --> 00:26:03,910
+الفارع C بدي اسألها انا 3.29 كام significant
+
+298
+00:26:03,910 --> 00:26:10,030
+figure عنده؟ تلاتة هذه ما بتدخل مضروبة في 0.2501
+
+299
+00:26:10,030 --> 00:26:13,850
+كام significant figure عنده؟ أربعة الجواب اللي
+
+300
+00:26:13,850 --> 00:26:19,550
+عنده لازم يكون فيه تلاتة significant figuresواضح
+
+301
+00:26:19,550 --> 00:26:23,010
+باخد على العدد الأقل فلما أضربهم على ال calculator
+
+302
+00:26:23,010 --> 00:26:26,450
+باخد بس تلاتة significant figures
+
+303
+00:26:30,840 --> 00:26:37,520
+في كل الأحوال باقي المسائل كلهم بدنا بهدوء انحلهم
+
+304
+00:26:37,520 --> 00:26:43,160
+بنفس الطريقة لا شك ان الطالب بنفسه لازم يتدرب لكن
+
+305
+00:26:43,160 --> 00:26:48,340
+هو مقاعدتين للدرب والقسمة بدي أشوف العدد الأقل من
+
+306
+00:26:48,340 --> 00:26:52,900
+ال significant figures وللجمع والطرح بدي أشوف ال
+
+307
+00:26:52,900 --> 00:26:57,360
+uncertainty الأقل أو اللي هو العدد الأقل من
+
+308
+00:26:57,360 --> 00:27:00,660
+الأرقام بعد الفاصلة بألتزم فيه
+
+309
+00:27:04,130 --> 00:27:09,550
+الآن سؤال 53 بيقول لي using your knowledge of
+
+310
+00:27:09,550 --> 00:27:13,370
+metric units English units and the information on
+
+311
+00:27:13,370 --> 00:27:18,130
+the backs inside cover يعني كل المعلومات اللي أنت
+
+312
+00:27:18,130 --> 00:27:24,640
+بتعرفهابالتحويل ما بين وحدات القياس المختلفة write
+
+313
+00:27:24,640 --> 00:27:27,960
+down the conversion factors needed to convert
+
+314
+00:27:27,960 --> 00:27:35,200
+millimeter to nanometer مثلا الان لو انا كنت بدى
+
+315
+00:27:35,200 --> 00:27:40,780
+احول ال millimeter الى ال nanometer ال millimeter
+
+316
+00:27:40,780 --> 00:27:48,840
+اكم nanometer طبعا انا بقدر اعرف انهالـ Millimeter
+
+317
+00:27:48,840 --> 00:27:57,160
+عشرة أصناق أس تلاتة متر الـ Millimeter
+
+318
+00:27:57,160 --> 00:28:01,340
+عشرة أصناق أس تلاتة متر الـ Millimeter عشرة أصناق
+
+319
+00:28:01,340 --> 00:28:02,940
+أس تلاتة متر الـ Millimeter عشرة أصناق أس تلاتة
+
+320
+00:28:02,940 --> 00:28:06,880
+متر الـ Millimeter عشرة أصناق أس تلاتة متر الـ
+
+321
+00:28:06,880 --> 00:28:07,100
+Millimeter عشرة أصناق أس تلاتة متر الـ Millimeter
+
+322
+00:28:07,100 --> 00:28:07,840
+عشرة أصناق أس تلاتة متر الـ Millimeter عشرة أصناق
+
+323
+00:28:07,840 --> 00:28:09,140
+أس تلاتة متر الـ Millimeter عشرة أصناق أس تلاتة
+
+324
+00:28:09,140 --> 00:28:09,900
+متر الـ Millimeter عشرة أصناق أس تلاتة متر الـ
+
+325
+00:28:09,900 --> 00:28:13,940
+Millimeter عشرة أصناق أس تلاتة متر الـ Millimeter
+
+326
+00:28:13,940 --> 00:28:20,710
+عشرة أصناق أس تلطبعا واضح تماما ان ال nanometer او
+
+327
+00:28:20,710 --> 00:28:26,550
+الواحد nanometer هو عبارة عن عشر أقص ناقص تسعة من
+
+328
+00:28:26,550 --> 00:28:32,830
+المتر واضح يعني بتخيل الوحدات بهذا الشكل ال
+
+329
+00:28:32,830 --> 00:28:36,770
+millimeter عشر أقص ناقص تلاتة من المتر و ال
+
+330
+00:28:36,770 --> 00:28:42,850
+nanometer عشر أقص ناقص تسعة من المتر
+
+331
+00:28:44,290 --> 00:28:52,610
+بناء على ذلك عندما أكمل التحويل سيظهر الميليميتر
+
+332
+00:28:52,610 --> 00:29:01,430
+عبارة عن عشر أقص ناقص ستة من عشر أقص ستة من
+
+333
+00:29:01,430 --> 00:29:09,170
+نانوميتر يعني الواحد ميليميتر هو عبارة عن عشر أقص
+
+334
+00:29:09,170 --> 00:29:15,270
+ناقصعشرة او ستة من المتر طبعا اذا انا عندى متر
+
+335
+00:29:15,270 --> 00:29:19,470
+احسن ان هذا ال conversion factor اكتبه ان ال
+
+336
+00:29:19,470 --> 00:29:25,830
+nanometer هو عبارة عن عشر اقص ناقص تسعة من المتر
+
+337
+00:29:25,830 --> 00:29:29,710
+بدي اشتب من هان المتر المتر وانا عشرة اقص ناقص
+
+338
+00:29:29,710 --> 00:29:34,930
+تلاتة مع عشرة اقص ناقص تسعة بيصير عشرة اقص ناقص6 و
+
+339
+00:29:34,930 --> 00:29:39,510
+بناء على ذلك لما بحسبها بده يطلع عند ال millimeter
+
+340
+00:29:39,510 --> 00:29:44,050
+هو عبارة عن عشرة أس ستة ليش لأنه هذا مش معادلة
+
+341
+00:29:44,050 --> 00:29:49,310
+تساوي فال millimeter هو عبارة عن عشرة أس ستة من ال
+
+342
+00:29:49,310 --> 00:29:54,630
+nanometer طبعا ال millimeter أكبر اكتير من ال
+
+343
+00:29:54,630 --> 00:30:01,830
+nanometerوكلهم من ناحية المبدأ بنحلهم بنفس الطريقة
+
+344
+00:30:01,830 --> 00:30:06,050
+كلهم من ناحية المبدأ بنحلهم بنفس الطريقة يعني لو
+
+345
+00:30:06,050 --> 00:30:11,910
+بده ييجي يسألني ال milligram to kilogram كيف بدي
+
+346
+00:30:11,910 --> 00:30:15,730
+أحسبها؟ أنا بعرف أن الواحد جرام عشرة أس تلاتة
+
+347
+00:30:15,730 --> 00:30:20,790
+milligram والواحد جرام عشرة أس ناقص تلاتة kilogram
+
+348
+00:30:20,790 --> 00:30:26,600
+بناء على ذلك ال milligram شو بده يطلع؟يعني لو أنا
+
+349
+00:30:26,600 --> 00:30:34,160
+أخدتهم حسب كبرهم أصغر واحد فيهم ال milligram بعدين
+
+350
+00:30:34,160 --> 00:30:41,400
+في عندي ال gram بعدين في عندي ال kilogram الآن ال
+
+351
+00:30:41,400 --> 00:30:45,700
+gram عبارة عن الف milligram و ال kilogram
+
+352
+00:30:48,540 --> 00:30:54,440
+الكيلو جرام عبارة عن الف جرام واضح بناء على ذلك لو
+
+353
+00:30:54,440 --> 00:30:59,400
+انا بدي اجي اسأل الان الكيلو جرام اكم ميلي جرام
+
+354
+00:30:59,400 --> 00:31:05,560
+عشرة او ستة ميلي جرام طب الميلي جرام اكم كيلو جرام
+
+355
+00:31:05,560 --> 00:31:11,240
+واضح يعني زي هيك احنا بنقول الكيلو جرام عشرة او
+
+356
+00:31:11,240 --> 00:31:18,490
+ستة ميلي جرام الميلي جرام اقل بديكون الميلي جرامهك
+
+357
+00:31:18,490 --> 00:31:23,190
+بدنا نقول بيساوي عشرة أس ناقص ستة من الكلجرام لو
+
+358
+00:31:23,190 --> 00:31:29,070
+كنت بدي ال kilogram بدقول او اخلي زي ما هو مكتوب
+
+359
+00:31:29,070 --> 00:31:36,610
+ال kilogram بيساوي عشرة أس ناقص ال kilogram عشرة
+
+360
+00:31:36,610 --> 00:31:42,960
+أس ستة منالـ milligram الـ kilogram عشرة أقص ناقص
+
+361
+00:31:42,960 --> 00:31:45,920
+ستة milligram أو ال milligram عشرة أقص ناقص ستة
+
+362
+00:31:45,920 --> 00:31:51,060
+kilogram العلاقات هي علاقات متتالية ال milligram
+
+363
+00:31:51,060 --> 00:31:54,360
+عشرة أقص ناقص تلاتة gram أو ال gram عشرة أقص تلاتة
+
+364
+00:31:54,360 --> 00:31:59,360
+milligram ال kilogram عشرة أقص تلاتة gram أو الف
+
+365
+00:31:59,360 --> 00:32:05,060
+gramفال kilogram لل milligram هذا الأكبر هذا
+
+366
+00:32:05,060 --> 00:32:08,140
+الأصغر بيكون عشر أقص ستة ال milligram لل kilogram
+
+367
+00:32:08,140 --> 00:32:14,100
+بيكون عشر أقص ناقص ستة وكل هذه التحويلات بدنا
+
+368
+00:32:14,100 --> 00:32:27,960
+نعملها بنفس الطريقة طبعا هو كمان بيسألني عن فرع
+
+369
+00:32:27,960 --> 00:32:34,580
+إيشC مثلا كيلومتر to foot الان احنا من الجداول
+
+370
+00:32:34,580 --> 00:32:42,520
+بنشوف ان الكيلومتر 3280 فوت مافي مشكلة انش كيوب to
+
+371
+00:32:42,520 --> 00:32:47,200
+centimeter كيوب هذه مهمة ان الواحد انش الواحد انش
+
+372
+00:32:47,200 --> 00:32:50,760
+اتنين واربع وخمسين من مائة سنتمتر فلو انا بدي ال
+
+373
+00:32:50,760 --> 00:32:56,000
+انش كيوبمقارنة بالسنتمتر كيوب بدي اخلي الـ 2.54
+
+374
+00:32:56,000 --> 00:33:02,580
+كيوب والسنتمتر كيوب فالانش كيوب is 16.39 سنتمتر
+
+375
+00:33:02,580 --> 00:33:08,940
+كيوب ننتبه ان انا احط هذه لما نخلي هذا كيوب هذه
+
+376
+00:33:08,940 --> 00:33:12,120
+بدأت تصير كيوب وهذه بدأت تصير كيوب التنتين مش
+
+377
+00:33:12,120 --> 00:33:12,680
+واحدة
+
+378
+00:33:15,840 --> 00:33:22,460
+وبهيك بننتهي من مناقشة بعض المسائل المتعلقة بالفصل
+
+379
+00:33:22,460 --> 00:33:25,240
+الأول وشكرا لكم
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gz91ZmeweRY_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gz91ZmeweRY_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..eeec959b448e0cbdb4c0ecd0be08782f1047f6e5
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gz91ZmeweRY_postprocess.srt
@@ -0,0 +1,1820 @@
+1
+00:00:05,030 --> 00:00:07,750
+بسم الله الرحمن الرحيم الحمد لله رب العالمين
+
+2
+00:00:07,750 --> 00:00:10,710
+والصلاة والسلام على سيدنا محمد وعلى آله وصحبه
+
+3
+00:00:10,710 --> 00:00:16,270
+أجمعين هذه هي المحاضرة رقم 26 مساق تحليل حقيقه 2
+
+4
+00:00:16,270 --> 00:00:20,290
+طلاب طالبات الجامعة الإسلامية قسم رياضيات كلية
+
+5
+00:00:20,290 --> 00:00:24,650
+العلوم اليوم هنكمل حديثنا في شطر 9 ولكن في section
+
+6
+00:00:24,650 --> 00:00:29,050
+جديد ال section هنبدأ فيه اللي هو tests for
+
+7
+00:00:29,050 --> 00:00:36,950
+absolute convergenceاختبارات الـ Series التي تحتوي
+
+8
+00:00:36,950 --> 00:00:40,490
+على الـ Positive Terms أو اللي هو بمفهوم الـ
+
+9
+00:00:40,490 --> 00:00:46,010
+Absolute Convergence نأتي الآن للأولاني المشهور
+
+10
+00:00:46,010 --> 00:00:51,130
+921بنسميه ال comparison test او اللي هو اختبار
+
+11
+00:00:51,130 --> 00:00:55,490
+المقارن ال let x يعني اللي بيكون عندنا فيه C ال
+
+12
+00:00:55,490 --> 00:00:58,910
+series و بنقارنها ب series أخرى بنعبجها عارفين
+
+13
+00:00:58,910 --> 00:01:02,930
+اللي هو converged ولا diverse وبناء عليه بنحكم على
+
+14
+00:01:02,930 --> 00:01:07,250
+ال series اللي بيبقى انقارنها اللي هي انها
+
+15
+00:01:07,250 --> 00:01:11,900
+converged او diverseLed X بيساوي XN وY بيساوي YN
+
+16
+00:01:11,900 --> 00:01:17,420
+بـReal Sequences Suppose that for some natural
+
+17
+00:01:17,420 --> 00:01:22,340
+numbers K اللي هو XN أكبر أو أصغر أو يساوي YN يعني
+
+18
+00:01:22,340 --> 00:01:26,400
+بمعنى آخر أننا بنحكي عن Sequence of Positive Terms
+
+19
+00:01:26,400 --> 00:01:32,160
+ماشي إذا هذه اللي هي عبارة عن Testلـ Series مكوّنة
+
+20
+00:01:32,160 --> 00:01:36,420
+الـ Terms تبعتها من Positive Terms for N أكبر و
+
+21
+00:01:36,420 --> 00:01:40,320
+ساوة و K then the convergence of summation yn
+
+22
+00:01:40,320 --> 00:01:44,320
+implies the convergence of xn and the divergence
+
+23
+00:01:44,320 --> 00:01:48,440
+of xn implies the divergence of summation yn اللي
+
+24
+00:01:48,440 --> 00:01:53,680
+أنا بقول ليه ال test بيقول بيحكيلي معايا ليه ان لو
+
+25
+00:01:53,680 --> 00:01:56,820
+كان في عندي two sequences xn و yn هم اللي بيكون
+
+26
+00:01:56,820 --> 00:02:01,920
+جسم اللي هو ال seriesباجي إذا كان عندي Xn أصغر
+
+27
+00:02:01,920 --> 00:02:05,700
+سواء و Yn لكل انه طبعا و اتنتين positive باجي بقول
+
+28
+00:02:05,700 --> 00:02:10,800
+إذا كان اللي هي الكبيرة اللي هي converts ال Yn
+
+29
+00:02:10,800 --> 00:02:14,280
+أكيد من باب أولى الصغيرة أشمالها converts يعني ال
+
+30
+00:02:14,280 --> 00:02:18,520
+convergence لل Yn بيعطي ال convergence لمن؟ لل Xn
+
+31
+00:02:19,050 --> 00:02:23,830
+إذا كانت الصغيرة diverges من باب أولاد كل كبيرة
+
+32
+00:02:23,830 --> 00:02:27,750
+إيش ما لها، diverges خلّينا نبرهن الآن Suppose
+
+33
+00:02:27,750 --> 00:02:33,730
+that yn is convergent، ماشي الحال؟suppose that
+
+34
+00:02:33,730 --> 00:02:37,090
+summation yn converges نفترض لكبير أنها converges
+
+35
+00:02:37,090 --> 00:02:41,050
+then for every ي أكبر من صفر there exists m of
+
+36
+00:02:41,050 --> 00:02:45,850
+epsilon element in n such that for every m أكبر من
+
+37
+00:02:45,850 --> 00:02:50,330
+n و أكبر من م يبسلون اللي هو ال الكوشي criterion
+
+38
+00:02:50,330 --> 00:02:55,530
+اللي هو y واحد زي ym ناقص y واحد زي yn أصغر من مين
+
+39
+00:02:55,530 --> 00:03:02,870
+من إبسلون ماشي الحال الان هذا الكلاماللي هو عندي
+
+40
+00:03:02,870 --> 00:03:06,710
+صحيح لكل N أكبر وسوء ASK يعني مش من بداية ال
+
+41
+00:03:06,710 --> 00:03:10,970
+sequence مش كتير مهم أنا عند بداية ال sequence ان
+
+42
+00:03:10,970 --> 00:03:12,950
+اللي بحكم على ال convergence و ال divergence اللي
+
+43
+00:03:12,950 --> 00:03:17,330
+هو mean اللي هو ال terms من عند حد إلى ما لنهاية
+
+44
+00:03:17,330 --> 00:03:20,630
+اللي هو اللي هيأثر على المجموع انه اللي هو موجود
+
+45
+00:03:20,630 --> 00:03:26,630
+او مش موجود طيب الآن هذه اللي هي صحيحة لكل N و M
+
+46
+00:03:26,630 --> 00:03:32,180
+أكبر من M اللي لجناهاوعندي هذه صحيحة لكل mean N
+
+47
+00:03:32,180 --> 00:03:37,840
+أكبر وساوة K عشان أستخدم واحد و أستخدم star لازم
+
+48
+00:03:37,840 --> 00:03:42,520
+أختار أقول from one and star we have for every
+
+49
+00:03:42,520 --> 00:03:46,480
+epsilon there exists N عشان أستخدم واحد و star بدي
+
+50
+00:03:46,480 --> 00:03:50,500
+أخد N هي ال maximum بين ال K و بين ال M of epsilon
+
+51
+00:03:50,500 --> 00:03:54,720
+such thatلما تكون ال M أكبر من N و أكبر من ال K
+
+52
+00:03:54,720 --> 00:03:58,140
+الماكسيموم معناته أكبر من K و أكبر من هذه معناته
+
+53
+00:03:58,140 --> 00:04:02,880
+هذه صحيحة و هذه صحيحة يعني بحج اللي الناس تخدمها
+
+54
+00:04:02,880 --> 00:04:09,920
+we have X1 زيد X2 زيد XM اللي هي ال partial sum ل
+
+55
+00:04:09,920 --> 00:04:17,290
+X و هذه ناقص اللي هي X1 لعند XNأترحى من بعض اللي
+
+56
+00:04:17,290 --> 00:04:21,670
+هو حيساوى الـ XM زائد واحد لعند الـ XM الـ XN زائد
+
+57
+00:04:21,670 --> 00:04:25,650
+واحد أكيد اللي هو أصغر أشهر من YN زائد واحد الـ XN
+
+58
+00:04:25,650 --> 00:04:28,150
+زائد اتنين أصغر من YN زائد اتنين لما أصل الـ XM
+
+59
+00:04:28,150 --> 00:04:33,770
+أصغر أشهر من الـ YMالان هذه عبارة عن مين؟ عن هذه
+
+60
+00:04:33,770 --> 00:04:38,470
+ناقص هذه اللي هي Y1 زي Ym ناقص Y1 زي Yn اللي هي
+
+61
+00:04:38,470 --> 00:04:42,610
+أصغر من وين من Y لإنه converges اللي هي ال Yn بناء
+
+62
+00:04:42,610 --> 00:04:49,190
+عليه اللي هو أثبتنا إن ال SM ناقص ال SN بالنسبة لل
+
+63
+00:04:49,190 --> 00:04:53,570
+X اللي هي satisfies كوشي كراي كتير إن أنا أصغر من
+
+64
+00:04:53,570 --> 00:04:58,510
+Y معناه تصميش لل XN إيش ما لها convergesالعنوان
+
+65
+00:04:58,510 --> 00:05:03,110
+الثاني هو Contrapositive للعنوان الأول لماذا؟ لأن
+
+66
+00:05:03,110 --> 00:05:09,870
+لو كانت الـ summation Xn مختلفة وليس مرتبطة، فأكيد
+
+67
+00:05:09,870 --> 00:05:14,390
+ستعطي الـ summation Yn أشمالها التي ليست مرتبطة
+
+68
+00:05:14,650 --> 00:05:19,810
+يعني diverse يعني الجزء الثاني هذا ما هو إلا عبارة
+
+69
+00:05:19,810 --> 00:05:23,650
+عن اللي هو الجزء الأول بس شماله موفود بال
+
+70
+00:05:23,650 --> 00:05:27,890
+contraposition يعني B implies Q يعني not Q implies
+
+71
+00:05:27,890 --> 00:05:32,730
+not B طيبالان نيجي هذا اللي هو الـ Comparison Test
+
+72
+00:05:32,730 --> 00:05:38,070
+اللي هو المقارنة المباشرة بين ال terms تبعات مين
+
+73
+00:05:38,070 --> 00:05:41,890
+تبعات اللي هو الـ sequences أحياناً عملياً هذه مش
+
+74
+00:05:41,890 --> 00:05:46,830
+كتير الناس بترغبها في test تاني comparison test
+
+75
+00:05:46,830 --> 00:05:50,530
+لكن هذا ال test اللي هو بنسميه limit comparison
+
+76
+00:05:50,530 --> 00:05:53,910
+test وهذا أسهل في الاستخدام وأوسع في الاستخدام
+
+77
+00:05:53,910 --> 00:05:59,310
+والناس بتنيل عادة لاستخدامهاماشي الحال الانصاب
+
+78
+00:05:59,310 --> 00:06:02,450
+both that X بالساوية XN وY بالساوية YN are
+
+79
+00:06:02,450 --> 00:06:07,570
+positive serial sequences if the relation limit XN
+
+80
+00:06:07,570 --> 00:06:12,730
+على YN لان limit XN على YN لا يساوي سفر يعني عبارة
+
+81
+00:06:12,730 --> 00:06:17,400
+عن عدد هذايعني عبارة عن عدد يعني سير ال XN و سير
+
+82
+00:06:17,400 --> 00:06:23,260
+ال YN جريب من بعض لما نقول بطلع عدد limit XN على
+
+83
+00:06:23,260 --> 00:06:29,000
+YN يعني سيرهم جريب من بعض لدرجة أنه في الآخر بطلع
+
+84
+00:06:29,000 --> 00:06:32,280
+عدد يعني لا هو طلع مالة نهاية نقول اللي فوق أسرع
+
+85
+00:06:32,280 --> 00:06:37,180
+كتير ولا طلع سفر نقول اللي تحت أسرع كتير لأ اللي
+
+86
+00:06:37,180 --> 00:06:42,770
+هو limit XN على YN طلعلي عدد عشان هيكلما هذا يحدث
+
+87
+00:06:42,770 --> 00:06:45,970
+if the relation limit xn على yn لا يساوي سفر hold
+
+88
+00:06:45,970 --> 00:06:51,570
+then ال summation xn is convergent if and only if
+
+89
+00:06:51,570 --> 00:06:56,650
+ال summation ال yn أشمال is convergent لأن سيرهما
+
+90
+00:06:56,650 --> 00:07:01,770
+متقارب عشان هيك بنقول يا التنتين converges يا
+
+91
+00:07:01,770 --> 00:07:08,570
+التنتين diverse لكن if the limit in two هنا is
+
+92
+00:07:08,570 --> 00:07:13,540
+zeroمعناته اللي تحت Star طير على المالة نهاية يعني
+
+93
+00:07:13,540 --> 00:07:18,400
+صار كله على إيش بساوي سفر فبكون الآن ال YN إذا
+
+94
+00:07:18,400 --> 00:07:22,840
+الكبير Convergent من باب أولى لصغير إيش ماله؟
+
+95
+00:07:22,840 --> 00:07:27,000
+Convergent لأنه هذا صار سفر معناته هذا أسرع من هذا
+
+96
+00:07:27,000 --> 00:07:32,340
+بكتير معناته لو كان اللي تحت Converged أكيد أكيد
+
+97
+00:07:32,340 --> 00:07:36,600
+أكيد أكيدهيكون اللي فوق اشمله convergence لأن اللي
+
+98
+00:07:36,600 --> 00:07:42,420
+فوق بيكون أبطأ أو أقل قيمة يعني نحو الكبر يعني
+
+99
+00:07:42,420 --> 00:07:46,040
+معناته انه اللي هو ال summation لل YN ال
+
+100
+00:07:46,040 --> 00:07:49,120
+convergence له على طول بيعطي اللي هو ال
+
+101
+00:07:49,120 --> 00:07:52,980
+convergence لل XN هذه حول اللي هو نص النظرية نيجي
+
+102
+00:07:52,980 --> 00:07:59,670
+الآنللبرهان نأتي لبرهان نثبت أنه نفترض ان ال
+
+103
+00:07:59,670 --> 00:08:02,810
+summation نثبت ان ال summation XN is convergent
+
+104
+00:08:02,810 --> 00:08:05,510
+فندهلي في ال summation YN اللي هو convergent الان
+
+105
+00:08:05,510 --> 00:08:09,790
+بدنا نأتي للبرهان قلنا عندي limit ال XN على ال YN
+
+106
+00:08:09,790 --> 00:08:13,430
+ايش بيساوي؟ بيساوي L لأ تساوي سفرإذاً حسب التعريف
+
+107
+00:08:13,430 --> 00:08:16,050
+الـ Epsilon Delta Definition أو الـ Epsilon K
+
+108
+00:08:16,050 --> 00:08:19,650
+Definition for Epsilon بساوة L على 2 أخدت لأي
+
+109
+00:08:19,650 --> 00:08:22,550
+Epsilon أخدت من ضمنها L على 2 للحسابات there
+
+110
+00:08:22,550 --> 00:08:27,210
+exists K such that اللي هو XN على YN ناقص L أصغر
+
+111
+00:08:27,210 --> 00:08:30,470
+من Epsilon اللي هي L على 2 لكل N أكبر أو يساوي K
+
+112
+00:08:30,470 --> 00:08:35,350
+خلّيني نفرضها هذه بيصير عند XN على YN أكبر من L
+
+113
+00:08:35,350 --> 00:08:40,280
+ناقص L على 2 وأصغر من L زائد من L على 2الان صار
+
+114
+00:08:40,280 --> 00:08:43,560
+عندى الان اللى هى ال .. ال .. ال .. ال .. ال .. ال
+
+115
+00:08:43,560 --> 00:08:44,080
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+116
+00:08:44,080 --> 00:08:44,380
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+117
+00:08:44,380 --> 00:08:44,700
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+118
+00:08:44,700 --> 00:08:45,400
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+119
+00:08:45,400 --> 00:08:45,680
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+120
+00:08:45,680 --> 00:08:45,940
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+121
+00:08:45,940 --> 00:08:46,620
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+122
+00:08:46,620 --> 00:08:47,900
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+123
+00:08:47,900 --> 00:08:48,420
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+124
+00:08:48,420 --> 00:08:50,320
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+125
+00:08:50,320 --> 00:08:59,540
+ال .. ال .. ال .. ال .. ال ..
+
+126
+00:08:59,540 --> 00:09:01,420
+ال .. ال ..
+
+127
+00:09:06,850 --> 00:09:11,550
+بال test اللي قبل بشوية إذا ال summation yn مش هذا
+
+128
+00:09:11,550 --> 00:09:16,790
+اللي هنستخدمه هنستخدم اللي هو xn أكبر من yn في L
+
+129
+00:09:16,790 --> 00:09:19,750
+على 2 إذا كانت هذي convergent هتكون هذي إيش مهلها
+
+130
+00:09:19,750 --> 00:09:24,410
+convergent وبما أن yn في L على 2 convergent هتطلع
+
+131
+00:09:24,410 --> 00:09:28,410
+عندي اللي هي ال summation yn إيش مهلها is
+
+132
+00:09:28,410 --> 00:09:34,660
+convergentماشي طيب الان conversely بدي افترض اللي
+
+133
+00:09:34,660 --> 00:09:39,480
+هو ال summation ل Yn ايش ماله is convergent ال
+
+134
+00:09:39,480 --> 00:09:41,920
+summation ل Yn is convergent مدام ال summation ل
+
+135
+00:09:41,920 --> 00:09:46,680
+Yn is convergent هيكون ال summation اللي هي
+
+136
+00:09:46,680 --> 00:09:50,500
+التلاتة في L على 2 في Yn is convergent لأن هذا
+
+137
+00:09:50,500 --> 00:09:54,000
+عبارة عن ثابتهذا لما انفرضنا الـ yn is summation
+
+138
+00:09:54,000 --> 00:09:57,320
+الـ yn convergent إذا هيكون ال summation الكبير
+
+139
+00:09:57,320 --> 00:10:00,940
+هذا convergent من باب أو لا ال summation للـ xn
+
+140
+00:10:00,940 --> 00:10:06,200
+إيش ماله is convergent يعني اتبعوا اللي حكيته هان
+
+141
+00:10:06,200 --> 00:10:14,030
+هذه اللي هي ال convergenceللـ Xn اللي هي الكبيرة
+
+142
+00:10:14,030 --> 00:10:17,850
+بيعطي ال convergence لهذه هو من ثم ال convergence
+
+143
+00:10:17,850 --> 00:10:22,710
+لهذه لأن في الجزء الثاني ال convergence لل Yn
+
+144
+00:10:22,710 --> 00:10:26,430
+بيعطي ال convergence لهذه ال convergence لهذه اللي
+
+145
+00:10:26,430 --> 00:10:31,930
+هو الكبير بيعطي ال convergence لمن؟ لل Xn طيب نيجي
+
+146
+00:10:31,930 --> 00:10:35,910
+الآن طبعا
+
+147
+00:10:35,910 --> 00:10:42,080
+ال .. ال .. ال .. المطلوب بي ..بدنا نفترض
+
+148
+00:10:44,730 --> 00:10:51,270
+suppose that limit xn على yn بساوة سفر ونفترض أن
+
+149
+00:10:51,270 --> 00:10:54,130
+الـ yn أشمالها is convergent بالذات بت الـ xn
+
+150
+00:10:54,130 --> 00:10:59,330
+أشمالها is convergent لأن شو معناه limit xn على yn
+
+151
+00:10:59,330 --> 00:11:03,230
+بساوة سفر لكل y أكبر من سفر؟ there exists k
+
+152
+00:11:03,230 --> 00:11:06,810
+element in n such that xn على yn أصغر من مين من
+
+153
+00:11:06,810 --> 00:11:10,810
+إبسلون لكل n أكبر وسوء k هذا اللي هو تعريف ال
+
+154
+00:11:10,810 --> 00:11:15,700
+limit لل xn على yn بساوة سفرومنه هذا دروب طرفين في
+
+155
+00:11:15,700 --> 00:11:20,680
+وسطين XN أصلا من YN في Y ن إحنا مفترضين أن ال YN
+
+156
+00:11:20,680 --> 00:11:24,240
+is convergent يعني ال Y في ال YN ثابتة ما هي ال Y
+
+157
+00:11:24,240 --> 00:11:27,620
+is convergent إذا كان الكبيرة convergent من باب
+
+158
+00:11:27,620 --> 00:11:32,680
+أولى الصماش لل XN إيش هتكون اللي هي is convergent
+
+159
+00:11:32,680 --> 00:11:39,020
+by اللي هو comparison test الأصليالان اذا خلصنا من
+
+160
+00:11:39,020 --> 00:11:42,300
+ال testين اللي هو ال comparison test وال limit
+
+161
+00:11:42,300 --> 00:11:45,880
+comparison test وكانت ال testين عبارة عن على
+
+162
+00:11:45,880 --> 00:11:50,520
+positive sequences الان بدنا نحكي عن حاجة اسمها
+
+163
+00:11:50,520 --> 00:11:53,700
+the root and the ratio test نشوف ال root و ال
+
+164
+00:11:53,700 --> 00:11:57,260
+ratio test ال test الأولان اللي هو ال root test
+
+165
+00:11:57,260 --> 00:12:03,010
+ايه؟ fx بسوي xnis a sequence in A and R and there
+
+166
+00:12:03,010 --> 00:12:07,410
+exists a positive number R أصغر من واحد and a
+
+167
+00:12:07,410 --> 00:12:11,590
+natural number K such that يعني الخاصية هذه بدوا
+
+168
+00:12:11,590 --> 00:12:16,590
+إياها X N أسواحد على N أصغر أو ساوي R تتحقق لكل N
+
+169
+00:12:16,590 --> 00:12:20,550
+أكبر أو يساوي K من عند K وطالع مش مشكلةلأن أنا
+
+170
+00:12:20,550 --> 00:12:22,670
+اللي في ال divergence و ال convergence مين اللي
+
+171
+00:12:22,670 --> 00:12:26,990
+بتحكم اللي هو ال terms اللي لما .. لإنها بتروح إلى
+
+172
+00:12:26,990 --> 00:12:29,890
+مالة نهاية يعني لو كان أول مليون term أو أول متين
+
+173
+00:12:29,890 --> 00:12:33,970
+مليون أو جد ما يكون في الأول اللي هو لا تحقق
+
+174
+00:12:33,970 --> 00:12:38,250
+الخاصية اللي بديها أنا بهم مني اللي فيما بعد يحقق
+
+175
+00:12:38,250 --> 00:12:40,550
+الخاصية اللي بديها اللي بدا تحكم على ال
+
+176
+00:12:40,550 --> 00:12:44,950
+convergence أو ال divergence إذن الآن لو كان عندي
+
+177
+00:12:44,950 --> 00:12:49,520
+X أقص .. X أقص واحد أنا الأصغر يستخدم Rلكل n أكبر
+
+178
+00:12:49,520 --> 00:12:51,020
+يساوي كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+179
+00:12:51,020 --> 00:12:53,220
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+180
+00:12:53,220 --> 00:13:03,200
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+181
+00:13:03,200 --> 00:13:05,720
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+182
+00:13:05,720 --> 00:13:15,340
+كتابة كتابة كتابة كتابة كتابة كتابة كتابة
+
+183
+00:13:15,340 --> 00:13:22,200
+كتنفترض أنه تلاتة holes يعني نفترض أنه XN أسواحد
+
+184
+00:13:22,200 --> 00:13:26,500
+على N أصغر أو يساوي R لكل N أكبر أو يساوي K الان
+
+185
+00:13:26,500 --> 00:13:29,760
+منها هيكون عند ال absolute value XN أكبر أو يساوي
+
+186
+00:13:29,760 --> 00:13:35,680
+0 أصغر أو يساوي 100 RN لكل N أكبر أو يساوي K الان
+
+187
+00:13:37,230 --> 00:13:42,990
+الـ summation للـ R أُس عن R نحن ناخدها أصغر من
+
+188
+00:13:42,990 --> 00:13:45,950
+واحد و positive طبعاً أصغر من واحد معناته ال
+
+189
+00:13:45,950 --> 00:13:50,250
+summation للـ R أن converge مظبوط؟ مدام الـ R أن
+
+190
+00:13:50,250 --> 00:13:55,310
+converge فال summation لل XN is converge يعني
+
+191
+00:13:55,310 --> 00:13:58,150
+بمعنى أخر صارت ال summation لل XN is absolutely
+
+192
+00:13:58,150 --> 00:14:06,100
+convergent طيبsuppose that أربعة holds اللي هي مين
+
+193
+00:14:06,100 --> 00:14:10,840
+أربعة هذا أكبر أو ساوء واحد لأن لكل N أكبر أو ساوء
+
+194
+00:14:10,840 --> 00:14:17,920
+K معناته اللي هو ال XN اللي هي أس واحد على N لو
+
+195
+00:14:17,920 --> 00:14:23,700
+جيت أخدته limitها limit اللي هي ال XN هيكون عند ال
+
+196
+00:14:23,700 --> 00:14:28,600
+absolute value لل XNأكبر أو ساوء واحد يعني limit
+
+197
+00:14:28,600 --> 00:14:33,340
+ال absolute value ل Xn as n goes to infinity أكيد
+
+198
+00:14:33,340 --> 00:14:37,640
+أكبر أو ساوء واحد إذا exist يعني بتسويش سفر مدام
+
+199
+00:14:37,640 --> 00:14:41,920
+بتسويش سفر إذا نصممش لل Xn بال test اللي هو
+
+200
+00:14:41,920 --> 00:14:46,780
+الأولان اللي أخدناهالان مدام الـ summation xn الـ
+
+201
+00:14:46,780 --> 00:14:51,260
+absolute value xn does not converge to 0 إذا هتكون
+
+202
+00:14:51,260 --> 00:14:55,580
+الـ xn does not converge to 0 ومن ثم ال summation
+
+203
+00:14:55,580 --> 00:15:00,560
+للـ xn أشملها diverse حسب اللي هو اللي قلنا عنه ال
+
+204
+00:15:00,560 --> 00:15:03,420
+divergence test اللي هو ال summation للـ xn
+
+205
+00:15:03,420 --> 00:15:06,820
+converges بلاقاء ال limit للـ xn بساوة 0 لكن عندي
+
+206
+00:15:06,820 --> 00:15:11,360
+طلعت limit للـ xn لا تساوة 0 طيب corollary لللي
+
+207
+00:15:11,360 --> 00:15:15,870
+فوقالأن الـ Corollary بتقول ما هي ليه؟ let X
+
+208
+00:15:15,870 --> 00:15:20,230
+بيساوي Xn بيبقى Sequence اند R and let R بيساوي
+
+209
+00:15:20,230 --> 00:15:25,350
+Limit Xn أس واحدة لان هناك كانت اللي هي بتاخد ال
+
+210
+00:15:25,350 --> 00:15:29,830
+and roots لمن؟ لل terms و بتشوف اللي هنا إذا كان
+
+211
+00:15:29,830 --> 00:15:33,410
+ال and roots لل terms اللي هي أكبر أو يساوي R من
+
+212
+00:15:33,410 --> 00:15:43,630
+عند K و طالعالان بتحكم ليه؟ الان بتحكم ليه؟ الان
+
+213
+00:15:43,630 --> 00:15:48,470
+بتحكم ليه؟ الان بتحكم ليه؟ الان بتحكم ليه؟ الان
+
+214
+00:15:48,470 --> 00:15:48,530
+بتحكم ليه؟ الان بتحكم ليه؟ الان بتحكم ليه؟ الان
+
+215
+00:15:48,530 --> 00:15:48,750
+بتحكم ليه؟ الان بتحكم ليه؟ الان بتحكم ليه؟ الان
+
+216
+00:15:48,750 --> 00:15:49,210
+بتحكم ليه؟ الان بتحكم ليه؟ الان بتحكم ليه؟ الان
+
+217
+00:15:49,210 --> 00:15:53,220
+بتحكم ليه؟ الان بتحكم ليه؟نفترض إن Xn بيكون
+
+218
+00:15:53,220 --> 00:15:57,500
+سيكوينس في R وR بيسوى Limit Xn plus 1 لـ N نفترض
+
+219
+00:15:57,500 --> 00:16:02,840
+إن هذا الـ Limit موجود ثم الصممش للـ Xn هو
+
+220
+00:16:02,840 --> 00:16:10,360
+بالتأكيد مرتبط لما R أصغر من 1 وهو مرتبط عندما R
+
+221
+00:16:10,360 --> 00:16:13,140
+أكبر من 1 وهذا ال test انتوا عارفينه يعني بمعنى
+
+222
+00:16:13,140 --> 00:16:18,040
+آخر باجي بقول باخد Limit Xn
+
+223
+00:16:18,970 --> 00:16:23,270
+أُس واحدة لأن أس أن قصة infinity طلع عندي ايش
+
+224
+00:16:23,270 --> 00:16:28,550
+بيساوي بيساوي R احنا مفترضين ان ال limit exist اذا
+
+225
+00:16:28,550 --> 00:16:34,550
+R أكبر من واحد خلاص ال series xn divers اذا R أصغر
+
+226
+00:16:34,550 --> 00:16:39,030
+من واحد ال series ايش مالها converge و absolutely
+
+227
+00:16:39,030 --> 00:16:44,370
+convergentلكن إذا كانت R بتساوي واحد بنقول ال test
+
+228
+00:16:44,370 --> 00:16:47,650
+fail يعني ممكن يكون convergence وممكن تكون إيش ما
+
+229
+00:16:47,650 --> 00:16:53,650
+لها divergence وفيه أمثلة واضح؟ طيب الآن بدنا نيجي
+
+230
+00:16:53,650 --> 00:16:59,570
+للبرهان suppose that limit Xn أس واحد على N بساوي
+
+231
+00:16:59,570 --> 00:17:05,790
+R وR إيش ما لها أصغر من واحد نفترض إنه limit Xn أس
+
+232
+00:17:05,790 --> 00:17:07,670
+واحد على N بساوي R أصغر من واحد
+
+233
+00:17:10,860 --> 00:17:22,100
+طيب عندى لما اقول limit xn أس واحدة لان بساوى واحد
+
+234
+00:17:22,100 --> 00:17:29,260
+او بساوى R اللى هى أصغر من واحد ايش معناته هذا هذا
+
+235
+00:17:29,260 --> 00:17:35,490
+معناته لكل ي أكبر من سفرThere exist k element in N
+
+236
+00:17:35,490 --> 00:17:39,550
+such that for every N أكبر وسوء k اللي هو بيكون
+
+237
+00:17:39,550 --> 00:17:50,390
+عند X N أسواحدة ل N ناقص R ناقص R أصغر من مين؟ من
+
+238
+00:17:50,390 --> 00:17:58,340
+إبسلونأصغر من إبسلون خد الآن إبسلون إبسلون بتساوي
+
+239
+00:17:58,340 --> 00:18:02,780
+عند واحد ناقص R أكبر من سفر لغرض الحسابات بتاخد
+
+240
+00:18:02,780 --> 00:18:06,720
+إبسلون عبارة عن واحد ناقص R على اتنين واحد ناقص R
+
+241
+00:18:06,720 --> 00:18:09,640
+على اتنين أكيد أكبر من سفر إذا هتصير أصغر من واحد
+
+242
+00:18:09,640 --> 00:18:14,420
+ناقص R على اتنين على اتنين صار عندي الآن Xn أسواحد
+
+243
+00:18:14,420 --> 00:18:20,120
+على N أصغر من واحد ناقص R على اتنين زائد R وأكبر
+
+244
+00:18:20,120 --> 00:18:27,300
+منR ناقص .. اللي هو واحد ناقص R على مين؟ على اتنين
+
+245
+00:18:27,300 --> 00:18:32,080
+انا بتهيب من المنطقة هذه ماشي الحال صار عندى اللي
+
+246
+00:18:32,080 --> 00:18:40,540
+هو ال XL أسواحد على N أصغر من اللي هو واحد
+
+247
+00:18:42,070 --> 00:18:47,950
+1 الان هذه اتنين R اللي هي واحد زائد R على مين على
+
+248
+00:18:47,950 --> 00:18:52,210
+اتنين ماشي؟ لذا صارت ال Xn أسواحد على N absolute
+
+249
+00:18:52,210 --> 00:18:57,790
+value أصغر من واحد زائد R على اتنين لكل N أكبر أو
+
+250
+00:18:57,790 --> 00:19:06,720
+يساوي مين K لذا حصلت على هذه هذه سموليها R'R' أو
+
+251
+00:19:06,720 --> 00:19:20,880
+يسمونها R واحد هذي R واحد اكبر او
+
+252
+00:19:20,880 --> 00:19:30,580
+اكبر اكبر اكبر اكبر اكبر اكبر اكبر اكبر اكبر
+
+253
+00:19:30,580 --> 00:19:33,070
+اكبر اكبر اكبر اكبر اكبرR أصغر من واحد يعني عبارة
+
+254
+00:19:33,070 --> 00:19:38,010
+عن هدولة عبارة عن R يعني R واحد أصغر من مين من R
+
+255
+00:19:38,010 --> 00:19:44,070
+في نفس الوقت هتكون أصغر من واحداللي وصلتله انا
+
+256
+00:19:44,070 --> 00:19:48,030
+suppose that اللي مش مشروح انا شرحته هناك limit x
+
+257
+00:19:48,030 --> 00:19:51,070
+minus واحد على n بساوي R أصغر من واحد then there
+
+258
+00:19:51,070 --> 00:19:55,650
+exists R واحد such that R واحد أكبر من R وأصغر من
+
+259
+00:19:55,650 --> 00:20:02,090
+مين من واحد R واحد أكبر من R وأكيد أصغر من واحد
+
+260
+00:20:02,090 --> 00:20:07,590
+زائد واحد على اتنين ويساوي واحد اللي هي andاللي هو
+
+261
+00:20:07,590 --> 00:20:11,270
+إيش هيكون عنده a natural number K اللي لجيته such
+
+262
+00:20:11,270 --> 00:20:15,210
+that XN plus 1 الان أصغر يسوء R1 لكل N أكبر أو
+
+263
+00:20:15,210 --> 00:20:21,930
+يسوء K والها وهيني فصلتلكم إيها هان الان صار جاهز
+
+264
+00:20:21,930 --> 00:20:25,250
+اللي هو النظرية اللي احنا هتذكرولها لها ال
+
+265
+00:20:25,250 --> 00:20:30,990
+comparison test اللي هو عندي summation لل R1 هذه
+
+266
+00:20:30,990 --> 00:20:37,810
+صارت الآن XN plus 1 الان أصغر من R1 يعنيبمهادر تبه
+
+267
+00:20:37,810 --> 00:20:44,030
+عليها ده خلينا نعملها برضه أفصلها صار عندي الآن يا
+
+268
+00:20:44,030 --> 00:20:53,890
+جماعة عندي اللي هو x and هيها x and absolute value
+
+269
+00:20:53,890 --> 00:21:01,550
+other أو ساوي R واحد أس انماشي هذي الان ال R1 أصغر
+
+270
+00:21:01,550 --> 00:21:07,590
+من واحد مدام أصغر واحد وهذه ال series converts
+
+271
+00:21:07,590 --> 00:21:12,030
+الكبيرة إذا بقى بأول هذه ال series الشمالة تكون
+
+272
+00:21:12,030 --> 00:21:16,690
+converts then by comparison testالآن اللي هي
+
+273
+00:21:16,690 --> 00:21:21,350
+summation R1Rn converges implies بواسطة هذه ان ال
+
+274
+00:21:21,350 --> 00:21:26,350
+summation لل Xn is absolutely convergent الان لو
+
+275
+00:21:26,350 --> 00:21:30,530
+فرضنا ان limit Xn أس واحدة بساوي R أكبر من واحد
+
+276
+00:21:31,090 --> 00:21:34,410
+بنفس الطريقة اللى جيتها قبل بشوية بس هاخد الـY
+
+277
+00:21:34,410 --> 00:21:39,010
+اللى هو هستخدم الـY definition بس هاخد الـY R ناقص
+
+278
+00:21:39,010 --> 00:21:42,250
+واحد على اتنين و هطبّك زي ما طبّجت بالظبط قبل
+
+279
+00:21:42,250 --> 00:21:47,650
+بشوية هلاجي natural number K such that ال X أنقص
+
+280
+00:21:47,650 --> 00:21:52,490
+واحد على N أكبر من واحد ولا لأ أنا بلزمش حتىعلى
+
+281
+00:21:52,490 --> 00:21:56,150
+طول limit xn أسواه دلان اللي هو بسوء r أكبر من
+
+282
+00:21:56,150 --> 00:21:59,010
+واحد هيطلع عندي xn أسواه دلان أكبر من واحد فهو r
+
+283
+00:21:59,010 --> 00:22:03,290
+أكبر سواء k ومنه على طول بما أن هذه اللي هو أكبر
+
+284
+00:22:03,290 --> 00:22:05,590
+من واحد حسب ال test اللي قبل بشوية ال summation لل
+
+285
+00:22:05,590 --> 00:22:11,290
+xn is divergent وده ماعرفش أعمل أعملها زي ما
+
+286
+00:22:11,290 --> 00:22:15,830
+عملتها قبل بشوية أنا none no conclusion وهذا
+
+287
+00:22:15,830 --> 00:22:21,070
+حكيتها no conclusionWhen R بيساوي واحد either
+
+288
+00:22:21,070 --> 00:22:28,830
+convergence or divergence is possible في حالة لما
+
+289
+00:22:28,830 --> 00:22:32,730
+جينا .. أخدنا ال limit و لجينا ال limit لان
+
+290
+00:22:32,730 --> 00:22:36,750
+through ال test بيساوي واحد معناته انه يمكن يكون
+
+291
+00:22:36,750 --> 00:22:43,190
+convergence و ممكن يكون divergence طيب نيجي الآن
+
+292
+00:22:43,190 --> 00:22:49,470
+لل ratio test ال ratio test مشابهأخدنا اللي هو
+
+293
+00:22:49,470 --> 00:22:53,090
+الان comparison test بشقيه العادي و ال limit و
+
+294
+00:22:53,090 --> 00:22:55,890
+أخدنا الان through ال test بشقيه العادي و ال limit
+
+295
+00:22:55,890 --> 00:22:59,630
+العادي اللي هو اللي قصدنا فيه اللي هو X أنوس واحدة
+
+296
+00:22:59,630 --> 00:23:05,910
+أكبر من واحد اللي هو أو أصغر اللي كانت الأولى و
+
+297
+00:23:05,910 --> 00:23:08,350
+اللي بعدها limit comparison test نفس الأشياء أكبر
+
+298
+00:23:08,350 --> 00:23:13,440
+أو أصغر طيب ال ratio testFx بيساوي xn is a
+
+299
+00:23:13,440 --> 00:23:18,280
+sequence of non-zero terms هذه ميزتها إنها
+
+300
+00:23:18,280 --> 00:23:21,760
+بيلزمهاش طبعا زي اللي جابلها بيلزمها الأولى
+
+301
+00:23:21,760 --> 00:23:25,920
+بيلزمها ال comparison test بيلزمها تقارن بسيريز
+
+302
+00:23:25,920 --> 00:23:30,920
+أخرى الان الانثرويد test وال ratio test تقارن
+
+303
+00:23:30,920 --> 00:23:35,790
+بنفسهايعني من داخل نفسها بتقدر تعرف اللي هو ..
+
+304
+00:23:35,790 --> 00:23:38,810
+اللي هي بطريقة معينة تعرف إن هي converged ولا
+
+305
+00:23:38,810 --> 00:23:42,810
+diverse يعني الآن Fx بسوء Xn is a sequence of non
+
+306
+00:23:42,810 --> 00:23:43,910
+-zero elements
+
+307
+00:23:46,750 --> 00:23:51,010
+of R and there is a positive number R أصغر من واحد
+
+308
+00:23:51,010 --> 00:23:55,730
+and a natural number K such that Xn زائد واحد على
+
+309
+00:23:55,730 --> 00:23:59,950
+Xn أصغر سوء R for N أكبر سوء K then the summation
+
+310
+00:23:59,950 --> 00:24:04,030
+Xn is absolutely convergent يعني الآن جيت انت لجيت
+
+311
+00:24:04,030 --> 00:24:08,310
+Xn زائد واحد على Xnاللي جيتها دايماً أصغر أوي ساوي
+
+312
+00:24:08,310 --> 00:24:12,130
+R والـ R أصغر strictly من مين؟ من واحد يعني دايماً
+
+313
+00:24:12,130 --> 00:24:15,830
+ال ratio هذا اللي بعده على اللي جابله اللي بعد على
+
+314
+00:24:15,830 --> 00:24:19,870
+اللي جاب اللي جيته أصغر معناته أنه صارت اللي هي
+
+315
+00:24:19,870 --> 00:24:23,890
+أشمالها عمالها بتصغر ال sequence كل ما ماشيت قدام
+
+316
+00:24:23,890 --> 00:24:27,330
+بيصير اللي ورا ع اللي قدام أصغر من واحد اللي ورا ع
+
+317
+00:24:27,330 --> 00:24:31,570
+اللي قدام أصغر من واحد يعني واضح أنه في اللي هو في
+
+318
+00:24:31,570 --> 00:24:36,410
+ال submission للـ XN is absolutely convergentلأن
+
+319
+00:24:36,410 --> 00:24:39,570
+بيقول if there exists a natural number k such that
+
+320
+00:24:39,570 --> 00:24:44,650
+بعده بيصير تصرفها ان xn زي 1 على xn هيكون أكبر
+
+321
+00:24:44,650 --> 00:24:48,350
+ويسوء 1 مدام أكبر ويسوء 1 معناته صار اللي هو كل
+
+322
+00:24:48,350 --> 00:24:51,890
+مامشيت قدام ال term بيكبر مثلا بيكبر ال term نفسه
+
+323
+00:24:51,890 --> 00:24:55,170
+إذا ال limit مش هيروح للصفر إذا معناته انه صارت ال
+
+324
+00:24:55,170 --> 00:25:01,530
+series divergent طيب ال proof خلينا نبرهن ايهif 6
+
+325
+00:25:01,530 --> 00:25:06,550
+holds يعني لو هو كانت هذا متحقق then by induction
+
+326
+00:25:06,550 --> 00:25:12,590
+one can see that for M أكبر أوي ساوية واحد K X K M
+
+327
+00:25:12,590 --> 00:25:18,890
+أصغر أوي ساوية أرقص M X K بتصور مش صعب إثبات هذه
+
+328
+00:25:18,890 --> 00:25:24,370
+for عندي خد X ال induction عالميا for M أكبر أوي
+
+329
+00:25:24,370 --> 00:25:32,360
+ساوية واحد على Mاللي هان بيصير xk زائد واحد اللي
+
+330
+00:25:32,360 --> 00:25:37,820
+هو xk زائد واحد هنا لإنها بتتحقق من k وطالع k زائد
+
+331
+00:25:37,820 --> 00:25:45,920
+واحد بيصير على ال xk يعني أصغر ساوي R في xk درم
+
+332
+00:25:45,920 --> 00:25:52,980
+التفرق الطرفين في وسطين xk إذن هذه صحيحةR M في
+
+333
+00:25:52,980 --> 00:25:56,020
+حالة الواحد صحيحة في حالة ال M ساو واحد الان
+
+334
+00:25:56,020 --> 00:26:01,420
+suppose that this true for M بدا تبت صحيتها for me
+
+335
+00:26:01,420 --> 00:26:08,360
+in M زائد واحد X K زائد M زائد واحد بدا تبت صحيتها
+
+336
+00:26:08,360 --> 00:26:12,940
+في حالة ال M زائد واحد اللي هو حسب اللي هنا اللي
+
+337
+00:26:12,940 --> 00:26:21,080
+أصغر أو ساوي ال Rثمين في اللي هي هنا N زائد واحد S
+
+338
+00:26:21,080 --> 00:26:28,160
+زائد واحد K M K زائد M K زائد M اذا R في absolute
+
+339
+00:26:28,160 --> 00:26:34,280
+value X K زائد M ماشي لان و بما ان هذا true for M
+
+340
+00:26:34,280 --> 00:26:38,840
+يعني هذا متحققه اذا من هنا ب hypothesis induction
+
+341
+00:26:38,840 --> 00:26:47,440
+بيصير R في R أس Mفي XK Y ساوية R أقصى M زايد واحد
+
+342
+00:26:47,440 --> 00:26:54,500
+في XK يعني صارت هذه صحيحة for M زايد واحد إذا صارت
+
+343
+00:26:54,500 --> 00:26:59,440
+صحيحة لكل M إذا هذه دائما صحيحة لأي M أكبر وساوية
+
+344
+00:26:59,440 --> 00:27:03,120
+واحد في ضوء العلاقة اللي هي 6 صحيحة
+
+345
+00:27:06,690 --> 00:27:13,130
+إذاً هذا لكل M أكبر أو يساوي واحد عندي لكل N أكبر
+
+346
+00:27:13,130 --> 00:27:17,350
+أو يساوي K الان
+
+347
+00:27:17,350 --> 00:27:27,330
+شيلي ال K زائد M وحط مكانها N يعني N بتساوي K زائد
+
+348
+00:27:27,330 --> 00:27:36,860
+M بصير هذه X Nأثر أو يساوي الـ M بتساوي N ناقص K و
+
+349
+00:27:36,860 --> 00:27:42,100
+X K زي ما هي X K لأنه شيلت ال N و حطيت مكانها K
+
+350
+00:27:42,100 --> 00:27:46,100
+زائد M و ال N طبعا اللي هي بتساوي K زائد M أكيد
+
+351
+00:27:46,100 --> 00:27:50,400
+أكبر أو يساوي K إذن هذه اللي هي صحيحة إذن صار عندي
+
+352
+00:27:50,400 --> 00:27:56,340
+الآن اللي عشان بعد شوية هتستخدمها صار عندي الآن ال
+
+353
+00:27:56,340 --> 00:28:03,180
+absolute valueللـ Xn أصغر أو يساوي R أُس N ناقص K
+
+354
+00:28:03,180 --> 00:28:10,180
+في X K ماشي الحال هذا لكل N أكبر أو يساوي إيش K
+
+355
+00:28:10,180 --> 00:28:19,740
+طيب نكمل للبرهان الـ N summation لل R أُس N
+
+356
+00:28:19,740 --> 00:28:20,660
+converts
+
+357
+00:28:23,610 --> 00:28:29,250
+لأن الار مالها بين صفر و واحد مادام هذي متغيرت
+
+358
+00:28:29,250 --> 00:28:33,450
+طبعا هذا term مجرد عدد يعنى term لأنه عند ك كهذا
+
+359
+00:28:33,450 --> 00:28:37,390
+هو اشي fixed يعني هو عبارة عن عدد والمتغير من الان
+
+360
+00:28:37,390 --> 00:28:44,510
+هذه اللي أنا ال summation لها للار الناقص كيعني بس
+
+361
+00:28:44,510 --> 00:28:48,670
+بنشيل K من ال terms مش مشكلة بتضل اللي هو converge
+
+362
+00:28:48,670 --> 00:28:52,270
+اللي هي summation R أُس N converge ثم summation
+
+363
+00:28:52,270 --> 00:28:56,430
+أُس R N ناقص K كيفيكس number عبارة عن converge ثم
+
+364
+00:28:56,430 --> 00:28:59,530
+صارت هذه ال series converge من باب أولى لل series
+
+365
+00:28:59,530 --> 00:29:02,670
+اللي أصغر منها اللي هتكون converge ثم صارت عندي
+
+366
+00:29:02,670 --> 00:29:05,070
+summation absolute value converge او بمعنى اخر
+
+367
+00:29:05,070 --> 00:29:12,450
+summation XN is absolutely convergent الان نيجيلأ
+
+368
+00:29:12,450 --> 00:29:18,050
+الجزء الثاني الان إذا كانت اللي هي سبعة holes سبعة
+
+369
+00:29:18,050 --> 00:29:21,970
+هي سبعة شايفينها هي سبعة اللي هو ال XN زائد واحد
+
+370
+00:29:21,970 --> 00:29:28,990
+على XN أكبر أو ساوى واحد لكل N أكبر أو ساوى مين
+
+371
+00:29:28,990 --> 00:29:36,420
+أكبر أو ساوى K كلام مشابهكلام مشابه if 7 holds
+
+372
+00:29:36,420 --> 00:29:41,900
+then by induction one can see that xkm أكبر أو
+
+373
+00:29:41,900 --> 00:29:46,880
+يساوي xk for m بساوي واحد و بعدين it is true for m
+
+374
+00:29:46,880 --> 00:29:49,480
+then it is true for m زائد واحد و زي ما عملت قبل
+
+375
+00:29:49,480 --> 00:29:54,170
+بشوية هتلاجوها بكل سهولة تطلعالان خد ال limit هذا
+
+376
+00:29:54,170 --> 00:29:58,690
+الان هذا صحيح لكل M أكبر أو يساوي واحد الان خد ال
+
+377
+00:29:58,690 --> 00:30:01,630
+limit as M goes to infinity بيصير أكبر أو يساوي
+
+378
+00:30:01,630 --> 00:30:05,090
+اللي هو المقدار هذا والمقدار هذا strictly أكبر من
+
+379
+00:30:05,090 --> 00:30:09,970
+100 من 0 لأن ال terms اللي عندي كلها لاتساوي 0 من
+
+380
+00:30:09,970 --> 00:30:14,930
+عند K وطالع اذا انتر عندي اللي هو limit ال Xn اللي
+
+381
+00:30:14,930 --> 00:30:20,490
+هو أكبر من 0 and so summation لل Xn is divergent
+
+382
+00:30:21,110 --> 00:30:26,670
+طيب، نريد الان لإيه الـ Corollary الأخيرة في
+
+383
+00:30:26,670 --> 00:30:32,050
+محاضرة اليوم اللي هي بتقول ليت X بسوء XN بيه
+
+384
+00:30:32,050 --> 00:30:36,790
+Sequence in R and Setواضح انه بيعطي اللي هو عادي و
+
+385
+00:30:36,790 --> 00:30:38,990
+بعدين limit عادي و بعدين limit و في ال ratio test
+
+386
+00:30:38,990 --> 00:30:42,410
+نفس الاشي بده ياخد limit ال exant زائد واحد على ال
+
+387
+00:30:42,410 --> 00:30:47,410
+exant بساوي R و الناس هتميل في الحلول في معظمها
+
+388
+00:30:47,410 --> 00:30:51,270
+اللي هي limits whenever the limit exist إذا كانت
+
+389
+00:30:51,270 --> 00:30:55,230
+ال limit هذه exist بنيجيthen the series summation
+
+390
+00:30:55,230 --> 00:30:58,630
+xn converges أو absolutely convergent إذا كان آر
+
+391
+00:30:58,630 --> 00:31:02,210
+أصغر من واحد وdivergent إذا كانت يشملها أكبر من
+
+392
+00:31:02,210 --> 00:31:06,110
+واحد يعني بناخد limit xn زائد واحد على xn absolute
+
+393
+00:31:06,110 --> 00:31:09,110
+value طبعا إذا كانت positive terms على طول بدون
+
+394
+00:31:09,110 --> 00:31:13,780
+absolute values طلع أن دي limit موجودما دام اطلع
+
+395
+00:31:13,780 --> 00:31:18,300
+ال limit موجود في عندي حكم إذا كان اللي هو أكبر من
+
+396
+00:31:18,300 --> 00:31:21,780
+واحد diverse إذا كان أصغر من واحد converge إذا
+
+397
+00:31:21,780 --> 00:31:26,220
+بيساوي واحد معناته the test fails وبنبحث اللي عن
+
+398
+00:31:26,220 --> 00:31:30,440
+test آخر طيب الآن suppose that limit xn زاد واحد
+
+399
+00:31:30,440 --> 00:31:33,300
+على xn as n goes to infinity بيساوي R أصغر من مين
+
+400
+00:31:33,300 --> 00:31:38,410
+من واحدبنفس الطريقة اللي عملناها قبل بشوية في
+
+401
+00:31:38,410 --> 00:31:44,230
+الأول let R1 satisfies R1 أكبر من R أصغر من 1 then
+
+402
+00:31:44,230 --> 00:31:48,810
+اللي هو لأن طبيعي between any two real numbers فيه
+
+403
+00:31:48,810 --> 00:31:53,770
+real number سميته مين؟ R1 الآن بما أن ال limit هذا
+
+404
+00:31:53,770 --> 00:31:57,370
+بيساوي R يدل كل إيبسلون أكبر من 0 من ضمن مين
+
+405
+00:31:57,370 --> 00:32:02,760
+الإيبسلون اللي بدأ أخدها؟ R1-RThen for epsilon
+
+406
+00:32:02,760 --> 00:32:07,280
+بساوة R واحد ناقص R أكبر من سفر there exists K
+
+407
+00:32:07,280 --> 00:32:10,260
+elements in N such that ال absolute value لهذا
+
+408
+00:32:10,260 --> 00:32:13,160
+ناقص هذا أصغر من مين من ال epsilon هي R واحد ناقص
+
+409
+00:32:13,160 --> 00:32:18,180
+مين ناقص R هذا صحيح لكل مين لكل N أكبر أو ساوة K
+
+410
+00:32:18,800 --> 00:32:23,600
+الآن نفردها هذه بس بتاخد الجهة اللي على اليمين هي
+
+411
+00:32:23,600 --> 00:32:27,420
+اللي بتمنى اللي هو صار عندى xn زائد واحد على xn
+
+412
+00:32:27,420 --> 00:32:31,540
+أصغر من r واحد ناقص r ولمّا تجيب هذا على الجهة
+
+413
+00:32:31,540 --> 00:32:34,240
+الثانية بعد ما فكرا absolute value بصير r ويساوي
+
+414
+00:32:34,240 --> 00:32:39,400
+ايش بتساوي؟ ويساوي بساوي r واحدالان و تساوي R واحد
+
+415
+00:32:39,400 --> 00:32:44,620
+اذا اللي هي .. اللي هيكون عندى اللي بال ..
+
+416
+00:32:44,620 --> 00:32:47,480
+بالنظرية اللي هي تسعة اتنين خمسها هتطلع عندى
+
+417
+00:32:47,480 --> 00:32:51,280
+الاصابيش ال absolute value اكسن اشمالها is
+
+418
+00:32:51,280 --> 00:32:56,110
+absolutely convergentهيطلع عندي summation للـ xn
+
+419
+00:32:56,110 --> 00:32:59,350
+is absolutely convergent اللي هي من نظرية تسعة
+
+420
+00:32:59,350 --> 00:33:02,450
+اتنين خمسة لإنه صار عندي ال absolute value لـ xn
+
+421
+00:33:02,450 --> 00:33:07,750
+زائد واحد على xn أصغر من رقم أصغر من واحد لكل n
+
+422
+00:33:07,750 --> 00:33:11,710
+أكبر سوى k من عند k وطالع بمعنى آخر صار عندي اللي
+
+423
+00:33:11,710 --> 00:33:15,530
+هي ال series is convergent لإنه هذا أصلا corollary
+
+424
+00:33:15,530 --> 00:33:21,830
+للي جابلها الآنودناخد اللي هو الجزء الثاني من اللي
+
+425
+00:33:21,830 --> 00:33:25,470
+هي ال .. ال .. ال .. البرهان اللي هو اللي بتعلق في
+
+426
+00:33:25,470 --> 00:33:28,330
+الجزء الثاني من ال Corollary بدنا نفترض انه limit
+
+427
+00:33:28,330 --> 00:33:33,390
+لل Xn زائد واحد ع ال Xn اللي هو بساوي R أكبر من
+
+428
+00:33:33,390 --> 00:33:37,910
+واحد وبدنا نشوف اللي هو كيف diverse يعني بدأ افترض
+
+429
+00:33:37,910 --> 00:33:40,670
+انه limit limit
+
+430
+00:33:42,720 --> 00:33:46,740
+Xn زائد واحد على Xn as n جثه infinity اللي هو
+
+431
+00:33:46,740 --> 00:33:51,940
+إجماله عبارة عن رقم اللي هو أكبر من واحد نسميها R
+
+432
+00:33:51,940 --> 00:33:58,260
+يساوي R أكبر من واحد ونشوف كيف أن هو عندي اللي هي
+
+433
+00:33:58,260 --> 00:34:04,040
+diverse الآنخد الـ Epsilon حلو خد Epsilon for
+
+434
+00:34:04,040 --> 00:34:09,560
+Epsilon for Epsilon بساوي R ناقص واحد أكبر من سفر
+
+435
+00:34:09,560 --> 00:34:13,220
+there exists K element in N such that for every N
+
+436
+00:34:13,220 --> 00:34:19,000
+أكبر وسوء K بطلع عنده هو XN زائد واحد على XN
+
+437
+00:34:19,510 --> 00:34:25,650
+absolute value ناقص الـ R أكبر من أصغر من مين من
+
+438
+00:34:25,650 --> 00:34:30,190
+إبسلون اللي هي عبارة عن R ناقص واحد هي ال إبسلون
+
+439
+00:34:30,190 --> 00:34:33,130
+اللي أخدت R ناقص واحد أو زي ما هو عاملها هان أصغر
+
+440
+00:34:33,130 --> 00:34:36,550
+من هيها أصغر من R minus واحد اللي هي الإبسلون اللي
+
+441
+00:34:36,550 --> 00:34:40,570
+هان فوق فوق اللي هان مش بيهمنيش الجهة اللي هان
+
+442
+00:34:40,570 --> 00:34:43,850
+بيهمني الجهة الثانية عكس اللي أولانياللي عن حيط
+
+443
+00:34:43,850 --> 00:34:47,790
+لعندي xn زائد واحد على xn أكبر من واحد اللي هي r
+
+444
+00:34:47,790 --> 00:34:50,870
+ناقص r minus واحد اللي هي ايش بتطلع واحد صار هذا
+
+445
+00:34:50,870 --> 00:34:54,030
+المقدار أكبر من واحد و بال test اللي قبل بشوية ال
+
+446
+00:34:54,030 --> 00:34:57,790
+theorem تسعة اتنين خمسة بما أنه صار هذا من عند n
+
+447
+00:34:57,790 --> 00:35:01,670
+أكبر يساوي k وطالع من عند k وطالع طلع عندي هذا
+
+448
+00:35:01,670 --> 00:35:05,050
+أكبر من واحد معناته صارت ال information هذه is
+
+449
+00:35:05,050 --> 00:35:10,340
+divergentوطبعا اللي دايما بنقولها دايما اللي هي no
+
+450
+00:35:10,340 --> 00:35:14,320
+conclusion و ال R بتساوي واحد اللي هو ممكن تكون
+
+451
+00:35:14,320 --> 00:35:17,080
+conversion و ممكن تكون diversion عشان هذا الأساس
+
+452
+00:35:17,080 --> 00:35:22,160
+بنقول ال test fail في حالة ال R بتساوي واحد و نبحث
+
+453
+00:35:22,160 --> 00:35:26,900
+عن test آخر و الآن بنيجي ل اللي هو المرة الجاية ان
+
+454
+00:35:26,900 --> 00:35:30,440
+شاء الله بناخد ال integral test و إلى لقاء آخر
+
+455
+00:35:30,440 --> 00:35:31,400
+والسلام عليكم و رحمه الله
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gz91ZmeweRY_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gz91ZmeweRY_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..5990570f6d3b9fb6a9863fad14c2871e72d6cfa6
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/gz91ZmeweRY_raw.json
@@ -0,0 +1 @@
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25.91, "end": 26.19, "word": " section", "probability": 0.86474609375}, {"start": 26.19, "end": 26.71, "word": " هنبدأ", "probability": 0.8819580078125}, {"start": 26.71, "end": 26.89, "word": " فيه", "probability": 0.91357421875}, {"start": 26.89, "end": 26.93, "word": " اللي", "probability": 0.59295654296875}, {"start": 26.93, "end": 27.07, "word": " هو", "probability": 0.95556640625}, {"start": 27.07, "end": 27.41, "word": " tests", "probability": 0.91748046875}, {"start": 27.41, "end": 29.05, "word": " for", "probability": 0.94091796875}, {"start": 29.05, "end": 29.47, "word": " absolute", "probability": 0.87353515625}, {"start": 29.47, "end": 30.11, "word": " convergence", "probability": 0.9443359375}], "temperature": 1.0}, {"id": 2, "seek": 4655, "start": 30.81, "end": 46.55, "text": "اختبارات الـ Series التي تحتوي على الـ Positive Terms أو اللي هو بمفهوم الـ Absolute Convergence نأتي الآن للأولاني المشهور 921", "tokens": [47283, 2655, 3555, 9640, 9307, 2423, 39184, 13934, 38392, 6055, 33753, 45865, 15844, 2423, 39184, 46326, 19835, 82, 34051, 13672, 1829, 31439, 4724, 2304, 5172, 3224, 20498, 2423, 39184, 43965, 1169, 2656, 331, 15260, 8717, 10721, 31371, 6024, 48506, 24976, 10721, 12610, 7649, 1829, 9673, 8592, 3224, 13063, 1722, 4436], "avg_logprob": -0.31035539566301834, "compression_ratio": 1.2781456953642385, "no_speech_prob": 1.6689300537109375e-06, "words": [{"start": 30.81, "end": 31.95, "word": "اختبارات", "probability": 0.9146484375}, {"start": 31.95, "end": 32.17, "word": " الـ", "probability": 0.659423828125}, {"start": 32.17, "end": 32.65, "word": " Series", "probability": 0.496337890625}, {"start": 32.65, "end": 33.15, "word": " التي", "probability": 0.414794921875}, {"start": 33.15, "end": 36.95, "word": " تحتوي", "probability": 0.8619791666666666}, {"start": 36.95, "end": 37.17, "word": " على", "probability": 0.85205078125}, {"start": 37.17, "end": 37.29, "word": " الـ", "probability": 0.57757568359375}, {"start": 37.29, "end": 37.67, "word": " 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62.67, "end": 62.93, "word": " على", "probability": 0.8681640625}, {"start": 62.93, "end": 63.21, "word": " ال", "probability": 0.92724609375}, {"start": 63.21, "end": 63.49, "word": " series", "probability": 0.95947265625}, {"start": 63.49, "end": 63.65, "word": " اللي", "probability": 0.958740234375}, {"start": 63.65, "end": 63.99, "word": " بيبقى", "probability": 0.595166015625}, {"start": 63.99, "end": 66.29, "word": " انقارنها", "probability": 0.84267578125}, {"start": 66.29, "end": 66.89, "word": " اللي", "probability": 0.75146484375}, {"start": 66.89, "end": 67.03, "word": " هي", "probability": 0.8095703125}, {"start": 67.03, "end": 67.25, "word": " انها", "probability": 0.930419921875}, {"start": 67.25, "end": 67.79, "word": " converged", "probability": 0.88671875}, {"start": 67.79, "end": 68.01, "word": " او", "probability": 0.859130859375}, {"start": 68.01, "end": 68.73, "word": " diverse", "probability": 0.837890625}], "temperature": 1.0}, {"id": 4, "seek": 8944, "start": 69.44, "end": 89.44, "text": "Led X بيساوي XN وY بيساوي YN بـReal Sequences Suppose that for some natural numbers K اللي هو XN أكبر أو أصغر أو يساوي YN يعني بمعنى آخر أننا بنحكي عن Sequence of Positive Terms ماشي إذا هذه اللي هي عبارة عن Test", "tokens": [43, 292, 1783, 4724, 1829, 3794, 995, 45865, 1783, 45, 4032, 56, 4724, 1829, 3794, 995, 45865, 398, 45, 4724, 39184, 45261, 46859, 2667, 21360, 300, 337, 512, 3303, 3547, 591, 13672, 1829, 31439, 1783, 45, 5551, 4117, 26890, 34051, 5551, 9381, 17082, 2288, 34051, 7251, 3794, 995, 45865, 398, 45, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 19753, 34740, 14739, 8315, 44945, 5016, 4117, 1829, 18871, 46859, 655, 295, 46326, 19835, 82, 3714, 33599, 1829, 11933, 15730, 29538, 13672, 1829, 39896, 6225, 3555, 9640, 3660, 18871, 9279], "avg_logprob": -0.272823020313563, "compression_ratio": 1.429951690821256, "no_speech_prob": 8.344650268554688e-07, "words": [{"start": 69.44, "end": 69.76, "word": "Led", "probability": 0.23431396484375}, {"start": 69.76, "end": 69.98, "word": " X", "probability": 0.77294921875}, {"start": 69.98, "end": 70.36, "word": " بيساوي", "probability": 0.657696533203125}, {"start": 70.36, "end": 70.78, "word": " XN", "probability": 0.591796875}, {"start": 70.78, "end": 71.16, "word": " وY", "probability": 0.625732421875}, {"start": 71.16, "end": 71.52, "word": " بيساوي", "probability": 0.9662109375}, {"start": 71.52, "end": 71.9, "word": " YN", "probability": 0.9443359375}, {"start": 71.9, "end": 72.68, "word": " بـReal", "probability": 0.6492513020833334}, {"start": 72.68, "end": 74.72, "word": " Sequences", "probability": 0.9365234375}, {"start": 74.72, "end": 75.62, "word": " Suppose", "probability": 0.298095703125}, {"start": 75.62, "end": 76.24, "word": " that", "probability": 0.85302734375}, {"start": 76.24, "end": 76.58, "word": " for", "probability": 0.9208984375}, {"start": 76.58, "end": 76.92, "word": " some", "probability": 0.9091796875}, {"start": 76.92, "end": 77.42, "word": " natural", "probability": 0.87646484375}, {"start": 77.42, "end": 78.02, "word": " numbers", "probability": 0.7509765625}, {"start": 78.02, "end": 78.42, "word": " K", "probability": 0.7333984375}, {"start": 78.42, "end": 79.26, "word": " اللي", "probability": 0.785888671875}, {"start": 79.26, "end": 79.44, "word": " هو", "probability": 0.91064453125}, {"start": 79.44, "end": 79.9, "word": " XN", "probability": 0.88037109375}, {"start": 79.9, "end": 80.2, "word": " أكبر", "probability": 0.9680989583333334}, {"start": 80.2, "end": 80.36, "word": " أو", "probability": 0.470703125}, {"start": 80.36, "end": 81.24, "word": " أصغر", "probability": 0.8458251953125}, {"start": 81.24, "end": 81.34, "word": " أو", "probability": 0.7802734375}, {"start": 81.34, "end": 81.62, "word": " يساوي", "probability": 0.83587646484375}, {"start": 81.62, "end": 82.08, "word": " YN", "probability": 0.783447265625}, {"start": 82.08, "end": 82.34, "word": " يعني", "probability": 0.784423828125}, {"start": 82.34, "end": 82.72, "word": " بمعنى", "probability": 0.9861328125}, {"start": 82.72, "end": 83.04, "word": " آخر", "probability": 0.7880859375}, {"start": 83.04, "end": 83.3, "word": " أننا", "probability": 0.428466796875}, {"start": 83.3, "end": 83.84, "word": " بنحكي", "probability": 0.852783203125}, {"start": 83.84, "end": 84.0, "word": " عن", "probability": 0.97998046875}, {"start": 84.0, "end": 84.6, "word": " Sequence", "probability": 0.82080078125}, {"start": 84.6, "end": 84.76, "word": " of", "probability": 0.841796875}, {"start": 84.76, "end": 85.22, "word": " Positive", "probability": 0.6005859375}, {"start": 85.22, "end": 86.4, "word": " Terms", "probability": 0.834228515625}, {"start": 86.4, "end": 86.82, "word": " ماشي", "probability": 0.8040364583333334}, {"start": 86.82, "end": 87.06, "word": " إذا", "probability": 0.5701904296875}, {"start": 87.06, "end": 87.38, "word": " هذه", "probability": 0.492431640625}, {"start": 87.38, "end": 88.08, "word": " اللي", "probability": 0.89501953125}, {"start": 88.08, "end": 88.42, "word": " هي", "probability": 0.89794921875}, {"start": 88.42, "end": 88.84, "word": " عبارة", "probability": 0.9952392578125}, {"start": 88.84, "end": 89.06, "word": " عن", "probability": 0.98974609375}, {"start": 89.06, "end": 89.44, "word": " Test", "probability": 0.6298828125}], "temperature": 1.0}, {"id": 5, "seek": 11884, "start": 90.08, "end": 118.84, "text": "لـ Series مكوّنة الـ Terms تبعتها من Positive Terms for N أكبر و ساوة و K then the convergence of summation yn implies the convergence of xn and the divergence of xn implies the divergence of summation yn اللي أنا بقول ليه ال test بيقول بيحكيلي معايا ليه ان لو كان في عندي two sequences xn و yn هم اللي بيكون جسم اللي هو ال series", "tokens": [1211, 39184, 13934, 3714, 4117, 2407, 11703, 1863, 3660, 2423, 39184, 19835, 82, 6055, 3555, 34268, 11296, 9154, 46326, 19835, 82, 337, 426, 5551, 4117, 26890, 4032, 8608, 995, 2407, 3660, 4032, 591, 550, 264, 32181, 295, 28811, 17861, 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convergence", "probability": 0.9482421875}, {"start": 99.14, "end": 99.44, "word": " of", "probability": 0.94921875}, {"start": 99.44, "end": 99.86, "word": " summation", "probability": 0.673828125}, {"start": 99.86, "end": 100.32, "word": " yn", "probability": 0.281494140625}, {"start": 100.32, "end": 101.22, "word": " implies", "probability": 0.67724609375}, {"start": 101.22, "end": 101.46, "word": " the", "probability": 0.6611328125}, {"start": 101.46, "end": 101.96, "word": " convergence", "probability": 0.93896484375}, {"start": 101.96, "end": 102.38, "word": " of", "probability": 0.9716796875}, {"start": 102.38, "end": 102.98, "word": " xn", "probability": 0.720947265625}, {"start": 102.98, "end": 103.68, "word": " and", "probability": 0.89208984375}, {"start": 103.68, "end": 103.88, "word": " the", "probability": 0.72998046875}, {"start": 103.88, "end": 104.32, "word": " divergence", "probability": 0.9755859375}, {"start": 104.32, "end": 104.64, "word": " of", "probability": 0.97021484375}, {"start": 104.64, "end": 105.1, "word": " xn", "probability": 0.97705078125}, {"start": 105.1, "end": 105.48, "word": " implies", "probability": 0.94921875}, {"start": 105.48, "end": 105.74, "word": " the", "probability": 0.88818359375}, {"start": 105.74, "end": 106.14, "word": " divergence", "probability": 0.974609375}, {"start": 106.14, "end": 106.54, "word": " of", "probability": 0.9755859375}, {"start": 106.54, "end": 107.08, "word": " summation", "probability": 0.90283203125}, {"start": 107.08, "end": 107.68, "word": " yn", "probability": 0.96728515625}, {"start": 107.68, "end": 108.44, "word": " اللي", "probability": 0.715087890625}, {"start": 108.44, "end": 108.56, "word": " أنا", "probability": 0.54345703125}, {"start": 108.56, "end": 108.86, "word": " بقول", "probability": 0.968017578125}, {"start": 108.86, "end": 109.2, "word": " ليه", "probability": 0.6455078125}, {"start": 109.2, "end": 109.7, "word": " ال", "probability": 0.53466796875}, {"start": 109.7, "end": 109.94, "word": " test", "probability": 0.8720703125}, {"start": 109.94, "end": 110.44, "word": " بيقول", "probability": 0.6241861979166666}, {"start": 110.44, "end": 112.5, "word": " بيحكيلي", "probability": 0.8102213541666666}, {"start": 112.5, "end": 112.84, "word": " معايا", "probability": 0.830078125}, {"start": 112.84, "end": 113.4, "word": " ليه", "probability": 0.93359375}, {"start": 113.4, "end": 113.52, "word": " ان", "probability": 0.2198486328125}, {"start": 113.52, "end": 113.68, "word": " لو", "probability": 0.97412109375}, {"start": 113.68, "end": 113.9, "word": " كان", "probability": 0.9658203125}, {"start": 113.9, "end": 114.04, "word": " في", "probability": 0.826171875}, {"start": 114.04, "end": 114.32, "word": " عندي", "probability": 0.82177734375}, {"start": 114.32, "end": 114.52, "word": " two", "probability": 0.6318359375}, {"start": 114.52, "end": 115.18, "word": " sequences", "probability": 0.92724609375}, {"start": 115.18, "end": 115.66, "word": " xn", "probability": 0.935302734375}, {"start": 115.66, "end": 115.76, "word": " و", "probability": 0.98779296875}, {"start": 115.76, "end": 116.0, "word": " yn", "probability": 0.6640625}, {"start": 116.0, "end": 116.34, "word": " هم", "probability": 0.572509765625}, {"start": 116.34, "end": 116.48, "word": " اللي", "probability": 0.988037109375}, {"start": 116.48, "end": 116.82, "word": " بيكون", "probability": 0.9239908854166666}, {"start": 116.82, "end": 117.52, "word": " جسم", "probability": 0.8779296875}, {"start": 117.52, "end": 118.26, "word": " اللي", "probability": 0.98583984375}, {"start": 118.26, "end": 118.46, "word": " هو", "probability": 0.9912109375}, {"start": 118.46, "end": 118.58, "word": " ال", "probability": 0.935546875}, {"start": 118.58, "end": 118.84, "word": " series", "probability": 0.52001953125}], "temperature": 1.0}, {"id": 6, "seek": 13852, "start": 119.54, "end": 138.52, "text": "باجي إذا كان عندي Xn أصغر سواء و Yn لكل انه طبعا و اتنتين positive باجي بقول إذا كان اللي هي الكبيرة اللي هي converts ال Yn أكيد من باب أولى الصغيرة أشمالها converts يعني ال convergence لل Yn بيعطي ال convergence لمن؟ لل Xn", "tokens": [3555, 26108, 1829, 11933, 15730, 25961, 18871, 16254, 1783, 77, 5551, 9381, 17082, 2288, 8608, 2407, 16606, 4032, 398, 77, 5296, 28820, 16472, 3224, 23032, 3555, 3615, 995, 4032, 1975, 2655, 29399, 9957, 3353, 4724, 26108, 1829, 4724, 39648, 11933, 15730, 25961, 13672, 1829, 39896, 33251, 3555, 48923, 13672, 1829, 39896, 38874, 2423, 398, 77, 5551, 4117, 25708, 9154, 4724, 16758, 5551, 12610, 7578, 31767, 17082, 48923, 5551, 8592, 2304, 6027, 11296, 38874, 37495, 22653, 2423, 32181, 24976, 398, 77, 4724, 40228, 9566, 1829, 2423, 32181, 5296, 27842, 22807, 24976, 1783, 77], "avg_logprob": -0.23874327700625184, "compression_ratio": 1.7386934673366834, "no_speech_prob": 2.086162567138672e-06, "words": [{"start": 119.54, "end": 120.0, "word": "باجي", "probability": 0.69921875}, {"start": 120.0, "end": 120.32, "word": " إذا", "probability": 0.797607421875}, {"start": 120.32, "end": 120.56, "word": " كان", "probability": 0.98828125}, {"start": 120.56, "end": 120.98, "word": " عندي", "probability": 0.7059326171875}, {"start": 120.98, "end": 121.58, "word": " Xn", "probability": 0.36566162109375}, {"start": 121.58, "end": 121.92, "word": " أصغر", "probability": 0.97119140625}, {"start": 121.92, "end": 122.22, "word": " سواء", "probability": 0.5725911458333334}, {"start": 122.22, "end": 122.3, "word": " و", "probability": 0.7021484375}, {"start": 122.3, "end": 122.62, "word": " Yn", "probability": 0.826171875}, {"start": 122.62, "end": 122.96, "word": " لكل", "probability": 0.95947265625}, {"start": 122.96, "end": 123.24, "word": " انه", "probability": 0.3026123046875}, {"start": 123.24, "end": 123.46, "word": " طبعا", "probability": 0.9453125}, {"start": 123.46, "end": 123.58, "word": " و", "probability": 0.626953125}, {"start": 123.58, "end": 123.94, "word": " اتنتين", "probability": 0.647308349609375}, {"start": 123.94, "end": 124.48, "word": " positive", "probability": 0.81494140625}, {"start": 124.48, "end": 125.36, "word": " باجي", "probability": 0.9508463541666666}, {"start": 125.36, "end": 125.7, "word": " بقول", "probability": 0.974365234375}, {"start": 125.7, "end": 126.08, "word": " إذا", "probability": 0.92919921875}, {"start": 126.08, "end": 126.58, "word": " كان", "probability": 0.9951171875}, {"start": 126.58, "end": 126.92, "word": " اللي", "probability": 0.939453125}, {"start": 126.92, "end": 127.2, "word": " هي", "probability": 0.92138671875}, {"start": 127.2, "end": 128.14, "word": " الكبيرة", "probability": 0.9210611979166666}, {"start": 128.14, "end": 129.14, "word": " اللي", "probability": 0.964599609375}, {"start": 129.14, "end": 129.36, "word": " هي", "probability": 0.86181640625}, {"start": 129.36, "end": 130.08, "word": " converts", "probability": 0.381591796875}, {"start": 130.08, "end": 130.32, "word": " ال", "probability": 0.8798828125}, {"start": 130.32, "end": 130.8, "word": " Yn", "probability": 0.805419921875}, {"start": 130.8, "end": 131.18, "word": " أكيد", "probability": 0.9215494791666666}, {"start": 131.18, "end": 131.36, "word": " من", "probability": 0.98876953125}, {"start": 131.36, "end": 131.58, "word": " باب", "probability": 0.973876953125}, {"start": 131.58, "end": 131.88, "word": " أولى", "probability": 0.9220377604166666}, {"start": 131.88, "end": 132.32, "word": " الصغيرة", "probability": 0.9002278645833334}, {"start": 132.32, "end": 133.02, "word": " أشمالها", "probability": 0.85478515625}, {"start": 133.02, "end": 133.82, "word": " converts", "probability": 0.61328125}, {"start": 133.82, "end": 134.16, "word": " يعني", "probability": 0.9208984375}, {"start": 134.16, "end": 134.28, "word": " ال", "probability": 0.88134765625}, {"start": 134.28, "end": 134.74, "word": " convergence", "probability": 0.9814453125}, {"start": 134.74, "end": 135.08, "word": " لل", "probability": 0.7001953125}, {"start": 135.08, "end": 135.66, "word": " Yn", "probability": 0.830078125}, {"start": 135.66, "end": 136.46, "word": " بيعطي", "probability": 0.8851318359375}, {"start": 136.46, "end": 136.54, "word": " ال", "probability": 0.9677734375}, {"start": 136.54, "end": 136.98, "word": " convergence", "probability": 0.97998046875}, {"start": 136.98, "end": 137.72, "word": " لمن؟", "probability": 0.61474609375}, {"start": 137.72, "end": 137.98, "word": " لل", "probability": 0.81494140625}, {"start": 137.98, "end": 138.52, "word": " Xn", "probability": 0.944580078125}], "temperature": 1.0}, {"id": 7, "seek": 15191, "start": 139.05, "end": 151.91, "text": "إذا كانت الصغيرة diverges من باب أولاد كل كبيرة إيش ما لها، diverges خلّينا نبرهن الآن Suppose that yn is convergent، ماشي الحال؟", "tokens": [28814, 15730, 25961, 2655, 31767, 17082, 48923, 18558, 2880, 9154, 4724, 16758, 5551, 12610, 18513, 28242, 9122, 3555, 48923, 11933, 1829, 8592, 19446, 5296, 11296, 12399, 18558, 2880, 16490, 1211, 11703, 1829, 8315, 8717, 26890, 3224, 1863, 6024, 48506, 21360, 300, 17861, 307, 9652, 6930, 12399, 3714, 33599, 1829, 21542, 6027, 22807], "avg_logprob": -0.29967570304870605, "compression_ratio": 1.2745098039215685, "no_speech_prob": 4.172325134277344e-07, "words": [{"start": 139.05, "end": 139.37, "word": "إذا", "probability": 0.787353515625}, {"start": 139.37, "end": 139.77, "word": " كانت", "probability": 0.979248046875}, {"start": 139.77, "end": 140.43, "word": " الصغيرة", "probability": 0.9099934895833334}, {"start": 140.43, "end": 141.67, "word": " diverges", "probability": 0.560546875}, {"start": 141.67, "end": 142.53, "word": " من", "probability": 0.89599609375}, {"start": 142.53, "end": 142.83, "word": " باب", "probability": 0.96337890625}, {"start": 142.83, "end": 143.19, "word": " أولاد", "probability": 0.6339518229166666}, {"start": 143.19, "end": 143.41, "word": " كل", "probability": 0.88525390625}, {"start": 143.41, "end": 143.83, "word": " كبيرة", "probability": 0.9658203125}, {"start": 143.83, "end": 144.09, "word": " إيش", "probability": 0.7693684895833334}, {"start": 144.09, "end": 144.27, "word": " ما", "probability": 0.7177734375}, {"start": 144.27, "end": 144.93, "word": " لها،", "probability": 0.7012125651041666}, {"start": 144.93, "end": 145.69, "word": " diverges", "probability": 0.56854248046875}, {"start": 145.69, "end": 146.33, "word": " خلّينا", "probability": 0.769970703125}, {"start": 146.33, "end": 146.71, "word": " نبرهن", "probability": 0.9334716796875}, {"start": 146.71, "end": 147.09, "word": " الآن", "probability": 0.74365234375}, {"start": 147.09, "end": 147.75, "word": " Suppose", "probability": 0.2548828125}, {"start": 147.75, "end": 148.41, "word": " that", "probability": 0.92041015625}, {"start": 148.41, "end": 148.99, "word": " yn", "probability": 0.3681640625}, {"start": 148.99, "end": 149.95, "word": " is", "probability": 0.9404296875}, {"start": 149.95, "end": 150.99, "word": " convergent،", "probability": 0.8349609375}, {"start": 150.99, "end": 151.29, "word": " ماشي", "probability": 0.9069010416666666}, {"start": 151.29, "end": 151.91, "word": " الحال؟", "probability": 0.9099934895833334}], "temperature": 1.0}, {"id": 8, "seek": 18041, "start": 152.89, "end": 180.41, "text": "suppose that summation yn converges نفترض لكبير أنها converges then for every ي أكبر من صفر there exists m of epsilon element in n such that for every m أكبر من n و أكبر من م يبسلون اللي هو ال الكوشي criterion اللي هو y واحد زي ym ناقص y واحد زي yn أصغر من مين من إبسلون ماشي الحال الان هذا الكلام", "tokens": [36622, 541, 300, 28811, 17861, 9652, 2880, 8717, 5172, 2655, 43042, 5296, 4117, 3555, 13546, 14739, 11296, 9652, 2880, 550, 337, 633, 7251, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 456, 8198, 275, 295, 17889, 4478, 294, 297, 1270, 300, 337, 633, 275, 5551, 4117, 26890, 9154, 297, 4032, 5551, 4117, 26890, 9154, 3714, 7251, 3555, 3794, 1211, 11536, 13672, 1829, 31439, 2423, 2423, 4117, 2407, 8592, 1829, 46691, 13672, 1829, 31439, 288, 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"word": " أنها", "probability": 0.83544921875}, {"start": 156.43, "end": 157.09, "word": " converges", "probability": 0.677490234375}, {"start": 157.09, "end": 157.45, "word": " then", "probability": 0.669921875}, {"start": 157.45, "end": 157.69, "word": " for", "probability": 0.95556640625}, {"start": 157.69, "end": 157.93, "word": " every", "probability": 0.78173828125}, {"start": 157.93, "end": 158.17, "word": " ي", "probability": 0.236083984375}, {"start": 158.17, "end": 158.47, "word": " أكبر", "probability": 0.94921875}, {"start": 158.47, "end": 158.61, "word": " من", "probability": 0.99267578125}, {"start": 158.61, "end": 158.91, "word": " صفر", "probability": 0.7272135416666666}, {"start": 158.91, "end": 159.11, "word": " there", "probability": 0.64892578125}, {"start": 159.11, "end": 159.61, "word": " exists", "probability": 0.496337890625}, {"start": 159.61, "end": 160.41, "word": " m", "probability": 0.41162109375}, {"start": 160.41, "end": 161.05, "word": " of", "probability": 0.8466796875}, {"start": 161.05, "end": 161.41, "word": " epsilon", "probability": 0.267822265625}, {"start": 161.41, "end": 162.17, "word": " element", "probability": 0.330322265625}, {"start": 162.17, "end": 162.53, "word": " in", "probability": 0.72412109375}, {"start": 162.53, "end": 162.69, "word": " n", "probability": 0.6279296875}, {"start": 162.69, "end": 163.01, "word": " such", "probability": 0.9111328125}, {"start": 163.01, "end": 163.49, "word": " that", "probability": 0.95263671875}, {"start": 163.49, "end": 164.15, "word": " for", "probability": 0.9482421875}, {"start": 164.15, "end": 164.57, "word": " every", "probability": 0.8203125}, {"start": 164.57, "end": 164.95, "word": " m", "probability": 0.82080078125}, {"start": 164.95, "end": 165.69, "word": " أكبر", "probability": 0.9677734375}, {"start": 165.69, "end": 165.85, "word": " من", "probability": 0.99169921875}, {"start": 165.85, "end": 166.11, "word": " n", "probability": 0.58349609375}, {"start": 166.11, "end": 166.21, "word": " و", "probability": 0.5927734375}, {"start": 166.21, "end": 166.45, "word": " أكبر", "probability": 0.9571940104166666}, {"start": 166.45, "end": 166.57, "word": " من", "probability": 0.9697265625}, {"start": 166.57, "end": 166.85, "word": " م", "probability": 0.351318359375}, {"start": 166.85, "end": 167.53, "word": " يبسلون", "probability": 0.759521484375}, {"start": 167.53, "end": 168.19, "word": " اللي", "probability": 0.630126953125}, {"start": 168.19, "end": 168.47, "word": " هو", "probability": 0.9765625}, {"start": 168.47, "end": 168.85, "word": " ال", "probability": 0.8271484375}, {"start": 168.85, "end": 169.65, "word": " الكوشي", "probability": 0.63873291015625}, {"start": 169.65, "end": 170.33, "word": " criterion", "probability": 0.841796875}, {"start": 170.33, "end": 171.05, "word": " اللي", "probability": 0.7216796875}, {"start": 171.05, "end": 171.25, "word": " هو", "probability": 0.96142578125}, {"start": 171.25, "end": 171.41, "word": " y", "probability": 0.74609375}, {"start": 171.41, "end": 171.83, "word": " واحد", "probability": 0.6246337890625}, {"start": 171.83, "end": 172.07, "word": " زي", "probability": 0.738525390625}, {"start": 172.07, "end": 172.63, "word": " ym", "probability": 0.4814453125}, {"start": 172.63, "end": 173.07, "word": " ناقص", "probability": 0.950439453125}, {"start": 173.07, "end": 173.33, "word": " y", "probability": 0.9296875}, {"start": 173.33, "end": 173.69, "word": " واحد", "probability": 0.9375}, {"start": 173.69, "end": 173.93, "word": " زي", "probability": 0.957763671875}, {"start": 173.93, "end": 174.41, "word": " yn", "probability": 0.857421875}, {"start": 174.41, "end": 174.87, "word": " أصغر", "probability": 0.97119140625}, {"start": 174.87, "end": 175.09, "word": " من", "probability": 0.99072265625}, {"start": 175.09, "end": 175.53, "word": " مين", "probability": 0.5965576171875}, {"start": 175.53, "end": 176.21, "word": " من", "probability": 0.9345703125}, {"start": 176.21, "end": 176.81, "word": " إبسلون", "probability": 0.859765625}, {"start": 176.81, "end": 177.33, "word": " ماشي", "probability": 0.8297526041666666}, {"start": 177.33, "end": 177.69, "word": " الحال", "probability": 0.87841796875}, {"start": 177.69, "end": 178.73, "word": " الان", "probability": 0.6065673828125}, {"start": 178.73, "end": 179.87, "word": " هذا", "probability": 0.91162109375}, {"start": 179.87, "end": 180.41, "word": " الكلام", "probability": 0.9658203125}], "temperature": 1.0}, {"id": 9, "seek": 20859, "start": 181.61, "end": 208.59, "text": "اللي هو عندي صحيح لكل N أكبر وسوء ASK يعني مش من بداية ال sequence مش كتير مهم أنا عند بداية ال sequence ان اللي بحكم على ال convergence و ال divergence اللي هو mean اللي هو ال terms من عند حد إلى ما لنهاية اللي هو اللي هيأثر على المجموع انه اللي هو موجود او مش موجود طيب الآن هذه اللي هي صحيحة لكل N و M أكبر من M اللي لجناها", "tokens": [6027, 20292, 31439, 18871, 16254, 20328, 5016, 1829, 5016, 5296, 28820, 426, 5551, 4117, 26890, 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204.17, "end": 204.75, "word": " صحيحة", "probability": 0.97275390625}, {"start": 204.75, "end": 205.13, "word": " لكل", "probability": 0.987548828125}, {"start": 205.13, "end": 205.57, "word": " N", "probability": 0.75390625}, {"start": 205.57, "end": 206.29, "word": " و", "probability": 0.888671875}, {"start": 206.29, "end": 206.63, "word": " M", "probability": 0.81689453125}, {"start": 206.63, "end": 207.19, "word": " أكبر", "probability": 0.93798828125}, {"start": 207.19, "end": 207.41, "word": " من", "probability": 0.99072265625}, {"start": 207.41, "end": 207.75, "word": " M", "probability": 0.98095703125}, {"start": 207.75, "end": 207.95, "word": " اللي", "probability": 0.991455078125}, {"start": 207.95, "end": 208.59, "word": " لجناها", "probability": 0.8193359375}], "temperature": 1.0}, {"id": 10, "seek": 23114, "start": 209.16, "end": 231.14, "text": "وعندي هذه صحيحة لكل mean N أكبر وساوة K عشان أستخدم واحد و أستخدم star لازم أختار أقول from one and star we have for every epsilon there exists N عشان أستخدم واحد و star بدي أخد N هي ال maximum بين ال K و بين ال M of epsilon such that", "tokens": [2407, 3615, 1863, 16254, 29538, 20328, 5016, 1829, 5016, 3660, 5296, 28820, 914, 426, 5551, 4117, 26890, 4032, 3794, 995, 2407, 3660, 591, 6225, 8592, 7649, 5551, 14851, 9778, 40448, 36764, 24401, 4032, 5551, 14851, 9778, 40448, 3543, 5296, 31377, 2304, 5551, 46456, 9640, 5551, 39648, 490, 472, 293, 3543, 321, 362, 337, 633, 17889, 456, 8198, 426, 6225, 8592, 7649, 5551, 14851, 9778, 40448, 36764, 24401, 4032, 3543, 4724, 16254, 5551, 9778, 3215, 426, 39896, 2423, 6674, 49374, 2423, 591, 4032, 49374, 2423, 376, 295, 17889, 1270, 300], "avg_logprob": -0.25225693219237855, "compression_ratio": 1.5373831775700935, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 209.16, "end": 209.9, "word": "وعندي", "probability": 0.753173828125}, {"start": 209.9, "end": 210.32, "word": " هذه", "probability": 0.49462890625}, {"start": 210.32, "end": 211.06, "word": " 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"probability": 0.78564453125}, {"start": 319.09, "end": 319.81, "word": " عبارة", "probability": 0.9949951171875}, {"start": 319.81, "end": 320.03, "word": " عن", "probability": 0.998046875}, {"start": 320.03, "end": 320.63, "word": " اللي", "probability": 0.853515625}, {"start": 320.63, "end": 320.85, "word": " هو", "probability": 0.994140625}, {"start": 320.85, "end": 321.31, "word": " الجزء", "probability": 0.9723307291666666}, {"start": 321.31, "end": 321.91, "word": " الأول", "probability": 0.98583984375}, {"start": 321.91, "end": 322.31, "word": " بس", "probability": 0.899658203125}, {"start": 322.31, "end": 323.11, "word": " شماله", "probability": 0.9010009765625}, {"start": 323.11, "end": 323.53, "word": " موفود", "probability": 0.744140625}, {"start": 323.53, "end": 323.65, "word": " بال", "probability": 0.79345703125}, {"start": 323.65, "end": 324.37, "word": " contraposition", "probability": 0.69189453125}, {"start": 324.37, "end": 325.07, "word": " يعني", "probability": 0.861328125}, {"start": 325.07, "end": 325.25, "word": " B", "probability": 0.23974609375}, {"start": 325.25, "end": 325.63, "word": " implies", "probability": 0.88525390625}, {"start": 325.63, "end": 326.07, "word": " Q", "probability": 0.92626953125}, {"start": 326.07, "end": 326.47, "word": " يعني", "probability": 0.939453125}, {"start": 326.47, "end": 326.73, "word": " not", "probability": 0.783203125}, {"start": 326.73, "end": 327.17, "word": " Q", "probability": 0.89208984375}, {"start": 327.17, "end": 327.89, "word": " implies", "probability": 0.9736328125}, {"start": 327.89, "end": 328.75, "word": " not", "probability": 0.9208984375}, {"start": 328.75, "end": 329.29, "word": " B", "probability": 0.94775390625}, {"start": 329.29, "end": 329.77, "word": " طيب", "probability": 0.9152018229166666}], "temperature": 1.0}, {"id": 16, "seek": 35711, "start": 330.33, "end": 357.11, "text": "الان نيجي هذا اللي هو الـ Comparison Test اللي هو المقارنة المباشرة بين ال terms تبعات مين تبعات اللي هو الـ sequences أحياناً عملياً هذه مش كتير الناس بترغبها في test تاني comparison test لكن هذا ال test اللي هو بنسميه limit comparison test وهذا أسهل في الاستخدام وأوسع في الاستخدام والناس بتنيل عادة لاستخدامها", "tokens": [6027, 7649, 8717, 1829, 7435, 1829, 23758, 13672, 1829, 31439, 2423, 39184, 2432, 2181, 2770, 9279, 13672, 1829, 31439, 9673, 4587, 9640, 1863, 3660, 9673, 3555, 33599, 25720, 49374, 2423, 2115, 6055, 3555, 3615, 9307, 3714, 9957, 6055, 3555, 3615, 9307, 13672, 1829, 31439, 2423, 39184, 22978, 5551, 5016, 1829, 7649, 995, 14111, 6225, 2304, 20292, 995, 14111, 29538, 37893, 9122, 2655, 13546, 2423, 8315, 3794, 39894, 2288, 17082, 3555, 11296, 8978, 1500, 6055, 7649, 1829, 9660, 1500, 44381, 23758, 2423, 1500, 13672, 1829, 31439, 44945, 38251, 1829, 3224, 4948, 9660, 1500, 37037, 15730, 5551, 3794, 3224, 1211, 8978, 42963, 14851, 9778, 3215, 10943, 36725, 41779, 3615, 8978, 42963, 14851, 9778, 3215, 10943, 16070, 8315, 3794, 39894, 22653, 1211, 6225, 18513, 3660, 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المقارنة", "probability": 0.98662109375}, {"start": 335.07, "end": 335.99, "word": " المباشرة", "probability": 0.9952392578125}, {"start": 335.99, "end": 336.31, "word": " بين", "probability": 0.962890625}, {"start": 336.31, "end": 336.51, "word": " ال", "probability": 0.9111328125}, {"start": 336.51, "end": 336.99, "word": " terms", "probability": 0.43017578125}, {"start": 336.99, "end": 337.69, "word": " تبعات", "probability": 0.895263671875}, {"start": 337.69, "end": 338.07, "word": " مين", "probability": 0.57513427734375}, {"start": 338.07, "end": 338.65, "word": " تبعات", "probability": 0.8751220703125}, {"start": 338.65, "end": 338.77, "word": " اللي", "probability": 0.758056640625}, {"start": 338.77, "end": 338.91, "word": " هو", "probability": 0.9521484375}, {"start": 338.91, "end": 339.07, "word": " الـ", "probability": 0.661865234375}, {"start": 339.07, "end": 339.61, "word": " sequences", "probability": 0.5126953125}, {"start": 339.61, "end": 340.87, "word": " أحياناً", 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"probability": 0.91796875}, {"start": 347.61, "end": 347.99, "word": " هذا", "probability": 0.9736328125}, {"start": 347.99, "end": 348.15, "word": " ال", "probability": 0.96435546875}, {"start": 348.15, "end": 348.51, "word": " test", "probability": 0.919921875}, {"start": 348.51, "end": 348.69, "word": " اللي", "probability": 0.9912109375}, {"start": 348.69, "end": 348.81, "word": " هو", "probability": 0.96142578125}, {"start": 348.81, "end": 349.53, "word": " بنسميه", "probability": 0.9224853515625}, {"start": 349.53, "end": 349.87, "word": " limit", "probability": 0.57763671875}, {"start": 349.87, "end": 350.53, "word": " comparison", "probability": 0.9169921875}, {"start": 350.53, "end": 350.93, "word": " test", "probability": 0.9609375}, {"start": 350.93, "end": 351.33, "word": " وهذا", "probability": 0.883544921875}, {"start": 351.33, "end": 351.83, "word": " أسهل", "probability": 0.98681640625}, {"start": 351.83, "end": 351.99, "word": " في", "probability": 0.96484375}, {"start": 351.99, "end": 352.63, "word": " الاستخدام", "probability": 0.9548828125}, {"start": 352.63, "end": 353.21, "word": " وأوسع", "probability": 0.781005859375}, {"start": 353.21, "end": 353.35, "word": " في", "probability": 0.98388671875}, {"start": 353.35, "end": 353.91, "word": " الاستخدام", "probability": 0.98798828125}, {"start": 353.91, "end": 354.85, "word": " والناس", "probability": 0.9129231770833334}, {"start": 354.85, "end": 355.59, "word": " بتنيل", "probability": 0.9251302083333334}, {"start": 355.59, "end": 356.17, "word": " عادة", "probability": 0.9886067708333334}, {"start": 356.17, "end": 357.11, "word": " لاستخدامها", "probability": 0.919189453125}], "temperature": 1.0}, {"id": 17, "seek": 37369, "start": 357.53, "end": 373.69, "text": "ماشي الحال الانصاب both that X بالساوية XN وY بالساوية YN are positive serial sequences if the relation limit XN على YN لان limit XN على YN لا يساوي سفر يعني عبارة عن عدد هذا", "tokens": [2304, 33599, 1829, 21542, 6027, 2423, 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360.17, "end": 360.61, "word": " بالساوية", "probability": 0.63450927734375}, {"start": 360.61, "end": 360.97, "word": " XN", "probability": 0.5963134765625}, {"start": 360.97, "end": 361.33, "word": " وY", "probability": 0.641845703125}, {"start": 361.33, "end": 361.71, "word": " بالساوية", "probability": 0.9529296875}, {"start": 361.71, "end": 362.11, "word": " YN", "probability": 0.91748046875}, {"start": 362.11, "end": 362.45, "word": " are", "probability": 0.2919921875}, {"start": 362.45, "end": 363.07, "word": " positive", "probability": 0.8671875}, {"start": 363.07, "end": 363.79, "word": " serial", "probability": 0.767578125}, {"start": 363.79, "end": 364.67, "word": " sequences", "probability": 0.90625}, {"start": 364.67, "end": 365.47, "word": " if", "probability": 0.5810546875}, {"start": 365.47, "end": 365.77, "word": " the", "probability": 0.8369140625}, {"start": 365.77, "end": 366.43, "word": " relation", "probability": 0.9716796875}, {"start": 366.43, "end": 366.85, "word": " limit", "probability": 0.89892578125}, {"start": 366.85, "end": 367.57, "word": " XN", "probability": 0.909912109375}, {"start": 367.57, "end": 368.19, "word": " على", "probability": 0.58642578125}, {"start": 368.19, "end": 368.79, "word": " YN", "probability": 0.93017578125}, {"start": 368.79, "end": 369.11, "word": " لان", "probability": 0.4974365234375}, {"start": 369.11, "end": 369.37, "word": " limit", "probability": 0.9453125}, {"start": 369.37, "end": 369.85, "word": " XN", "probability": 0.94189453125}, {"start": 369.85, "end": 370.01, "word": " على", "probability": 0.82080078125}, {"start": 370.01, "end": 370.41, "word": " YN", "probability": 0.971923828125}, {"start": 370.41, "end": 370.65, "word": " لا", "probability": 0.734375}, {"start": 370.65, "end": 371.21, "word": " يساوي", "probability": 0.77386474609375}, {"start": 371.21, "end": 371.65, "word": " سفر", "probability": 0.7900390625}, {"start": 371.65, "end": 372.29, "word": " يعني", "probability": 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{"start": 558.63, "end": 558.99, "word": " هتكون", "probability": 0.8792317708333334}, {"start": 558.99, "end": 559.27, "word": " هذي", "probability": 0.818359375}, {"start": 559.27, "end": 559.45, "word": " إيش", "probability": 0.681396484375}, {"start": 559.45, "end": 559.75, "word": " مهلها", "probability": 0.664093017578125}, {"start": 559.75, "end": 560.39, "word": " convergent", "probability": 0.832275390625}, {"start": 560.39, "end": 561.11, "word": " وبما", "probability": 0.7423502604166666}, {"start": 561.11, "end": 561.31, "word": " أن", "probability": 0.44775390625}, {"start": 561.31, "end": 561.93, "word": " yn", "probability": 0.94189453125}, {"start": 561.93, "end": 562.17, "word": " في", "probability": 0.912109375}, {"start": 562.17, "end": 562.35, "word": " L", "probability": 0.9853515625}, {"start": 562.35, "end": 562.53, "word": " على", "probability": 0.9072265625}, {"start": 562.53, "end": 562.79, "word": " 2", "probability": 0.9892578125}, {"start": 562.79, "end": 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"start": 569.92, "end": 591.52, "text": "ماشي طيب الان conversely بدي افترض اللي هو ال summation ل Yn ايش ماله is convergent ال summation ل Yn is convergent مدام ال summation ل Yn is convergent هيكون ال summation اللي هي التلاتة في L على 2 في Yn is convergent لأن هذا عبارة عن ثابت", "tokens": [2304, 33599, 1829, 23032, 1829, 3555, 2423, 7649, 2615, 736, 4724, 16254, 1975, 5172, 2655, 43042, 13672, 1829, 31439, 2423, 28811, 5296, 398, 77, 1975, 1829, 8592, 3714, 6027, 3224, 307, 9652, 6930, 2423, 28811, 5296, 398, 77, 307, 9652, 6930, 3714, 3215, 10943, 2423, 28811, 5296, 398, 77, 307, 9652, 6930, 39896, 30544, 2423, 28811, 13672, 1829, 39896, 16712, 1211, 9307, 3660, 8978, 441, 15844, 568, 8978, 398, 77, 307, 9652, 6930, 5296, 33456, 23758, 6225, 3555, 9640, 3660, 18871, 38637, 16758, 2655], "avg_logprob": -0.2694852969225715, "compression_ratio": 1.8076923076923077, "no_speech_prob": 1.9073486328125e-06, "words": [{"start": 569.92, "end": 570.44, "word": "ماشي", "probability": 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0.6359049479166666}, {"start": 577.18, "end": 577.72, "word": " ماله", "probability": 0.8533528645833334}, {"start": 577.72, "end": 578.08, "word": " is", "probability": 0.54345703125}, {"start": 578.08, "end": 578.82, "word": " convergent", "probability": 0.8896484375}, {"start": 578.82, "end": 579.48, "word": " ال", "probability": 0.65380859375}, {"start": 579.48, "end": 579.8, "word": " summation", "probability": 0.95458984375}, {"start": 579.8, "end": 579.98, "word": " ل", "probability": 0.921875}, {"start": 579.98, "end": 580.3, "word": " Yn", "probability": 0.937255859375}, {"start": 580.3, "end": 580.42, "word": " is", "probability": 0.78076171875}, {"start": 580.42, "end": 580.98, "word": " convergent", "probability": 0.92919921875}, {"start": 580.98, "end": 581.34, "word": " مدام", "probability": 0.69091796875}, {"start": 581.34, "end": 581.44, "word": " ال", "probability": 0.953125}, {"start": 581.44, "end": 581.72, "word": " summation", "probability": 0.96630859375}, 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على", "probability": 0.669921875}, {"start": 588.22, "end": 588.66, "word": " 2", "probability": 0.88037109375}, {"start": 588.66, "end": 588.9, "word": " في", "probability": 0.93212890625}, {"start": 588.9, "end": 589.32, "word": " Yn", "probability": 0.96728515625}, {"start": 589.32, "end": 589.4, "word": " is", "probability": 0.457275390625}, {"start": 589.4, "end": 590.0, "word": " convergent", "probability": 0.953369140625}, {"start": 590.0, "end": 590.26, "word": " لأن", "probability": 0.74951171875}, {"start": 590.26, "end": 590.5, "word": " هذا", "probability": 0.64794921875}, {"start": 590.5, "end": 590.78, "word": " عبارة", "probability": 0.9892578125}, {"start": 590.78, "end": 590.94, "word": " عن", "probability": 0.9970703125}, {"start": 590.94, "end": 591.52, "word": " ثابت", "probability": 0.9925130208333334}], "temperature": 1.0}, {"id": 27, "seek": 60926, "start": 591.96, "end": 609.26, "text": "هذا لما انفرضنا الـ yn is summation الـ yn convergent إذا هيكون ال summation الكبير هذا convergent من باب أو لا ال summation للـ xn إيش ماله is convergent يعني اتبعوا اللي حكيته هان هذه اللي هي ال convergence", "tokens": [3224, 15730, 5296, 15042, 16472, 5172, 43042, 8315, 2423, 39184, 17861, 307, 28811, 2423, 39184, 17861, 9652, 6930, 11933, 15730, 39896, 30544, 2423, 28811, 33251, 3555, 13546, 23758, 9652, 6930, 9154, 4724, 16758, 34051, 20193, 2423, 28811, 24976, 39184, 2031, 77, 11933, 1829, 8592, 3714, 6027, 3224, 307, 9652, 6930, 37495, 22653, 1975, 2655, 3555, 3615, 14407, 13672, 1829, 11331, 4117, 36081, 3224, 8032, 7649, 29538, 13672, 1829, 39896, 2423, 32181], "avg_logprob": -0.350260429084301, "compression_ratio": 1.7241379310344827, "no_speech_prob": 4.76837158203125e-07, "words": [{"start": 591.96, "end": 592.22, "word": "هذا", "probability": 0.5618896484375}, {"start": 592.22, "end": 592.4, "word": " لما", "probability": 0.6383056640625}, {"start": 592.4, "end": 592.84, "word": " انفرضنا", "probability": 0.789825439453125}, {"start": 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597.58, "word": " هذا", "probability": 0.89990234375}, {"start": 597.58, "end": 598.4, "word": " convergent", "probability": 0.798828125}, {"start": 598.4, "end": 598.92, "word": " من", "probability": 0.97216796875}, {"start": 598.92, "end": 599.28, "word": " باب", "probability": 0.975341796875}, {"start": 599.28, "end": 599.54, "word": " أو", "probability": 0.7626953125}, {"start": 599.54, "end": 599.74, "word": " لا", "probability": 0.8759765625}, {"start": 599.74, "end": 599.92, "word": " ال", "probability": 0.81298828125}, {"start": 599.92, "end": 600.32, "word": " summation", "probability": 0.88671875}, {"start": 600.32, "end": 600.64, "word": " للـ", "probability": 0.4693603515625}, {"start": 600.64, "end": 600.94, "word": " xn", "probability": 0.46533203125}, {"start": 600.94, "end": 601.2, "word": " إيش", "probability": 0.6858723958333334}, {"start": 601.2, "end": 601.7, "word": " ماله", "probability": 0.8194986979166666}, {"start": 601.7, "end": 602.42, "word": " is", "probability": 0.85009765625}, {"start": 602.42, "end": 603.54, "word": " convergent", "probability": 0.940673828125}, {"start": 603.54, "end": 604.26, "word": " يعني", "probability": 0.902099609375}, {"start": 604.26, "end": 605.06, "word": " اتبعوا", "probability": 0.96259765625}, {"start": 605.06, "end": 605.26, "word": " اللي", "probability": 0.937744140625}, {"start": 605.26, "end": 605.74, "word": " حكيته", "probability": 0.9842529296875}, {"start": 605.74, "end": 606.2, "word": " هان", "probability": 0.733642578125}, {"start": 606.2, "end": 607.02, "word": " هذه", "probability": 0.58740234375}, {"start": 607.02, "end": 607.44, "word": " اللي", "probability": 0.970703125}, {"start": 607.44, "end": 607.86, "word": " هي", "probability": 0.9267578125}, {"start": 607.86, "end": 608.6, "word": " ال", "probability": 0.82421875}, {"start": 608.6, "end": 609.26, "word": " convergence", "probability": 0.91162109375}], "temperature": 1.0}, {"id": 28, "seek": 63969, "start": 610.05, "end": 639.69, "text": "للـ Xn اللي هي الكبيرة بيعطي ال convergence لهذه هو من ثم ال convergence لهذه لأن في الجزء الثاني ال convergence لل Yn بيعطي ال convergence لهذه ال convergence لهذه اللي هو الكبير بيعطي ال convergence لمن؟ لل Xn طيب نيجي الآن طبعا ال .. ال .. ال .. المطلوب بي ..", "tokens": [1211, 1211, 39184, 1783, 77, 13672, 1829, 39896, 33251, 3555, 48923, 4724, 40228, 9566, 1829, 2423, 32181, 46740, 24192, 31439, 9154, 38637, 2304, 2423, 32181, 46740, 24192, 5296, 33456, 8978, 25724, 11622, 38207, 6024, 104, 7649, 1829, 2423, 32181, 24976, 398, 77, 4724, 40228, 9566, 1829, 2423, 32181, 46740, 24192, 2423, 32181, 46740, 24192, 13672, 1829, 31439, 33251, 3555, 13546, 4724, 40228, 9566, 1829, 2423, 32181, 5296, 27842, 22807, 24976, 1783, 77, 23032, 1829, 3555, 8717, 1829, 7435, 1829, 6024, 48506, 23032, 3555, 3615, 995, 2423, 4386, 2423, 4386, 2423, 4386, 9673, 9566, 1211, 37746, 4724, 1829, 4386], "avg_logprob": -0.15806502916596152, "compression_ratio": 2.3392857142857144, "no_speech_prob": 1.1324882507324219e-06, "words": [{"start": 610.05, "end": 610.91, "word": "للـ", "probability": 0.718017578125}, {"start": 610.91, "end": 611.67, "word": " Xn", "probability": 0.571533203125}, {"start": 611.67, "end": 612.25, "word": " اللي", "probability": 0.6685791015625}, {"start": 612.25, "end": 613.35, "word": " هي", "probability": 0.908203125}, {"start": 613.35, "end": 614.03, "word": " الكبيرة", "probability": 0.974609375}, {"start": 614.03, "end": 615.01, "word": " بيعطي", "probability": 0.854736328125}, {"start": 615.01, "end": 615.11, "word": " ال", "probability": 0.86474609375}, {"start": 615.11, "end": 615.51, "word": " convergence", "probability": 0.26611328125}, {"start": 615.51, "end": 616.13, "word": " لهذه", "probability": 0.901611328125}, {"start": 616.13, "end": 616.95, "word": " هو", "probability": 0.66162109375}, {"start": 616.95, "end": 617.15, "word": " من", "probability": 0.96630859375}, {"start": 617.15, "end": 617.35, "word": " ثم", "probability": 0.9501953125}, {"start": 617.35, "end": 617.47, "word": " ال", "probability": 0.96044921875}, {"start": 617.47, "end": 617.85, "word": " convergence", "probability": 0.88134765625}, {"start": 617.85, "end": 618.43, "word": " لهذه", "probability": 0.9912109375}, {"start": 618.43, "end": 619.01, "word": " لأن", "probability": 0.83056640625}, {"start": 619.01, "end": 619.19, "word": " في", "probability": 0.921875}, {"start": 619.19, "end": 619.65, "word": " الجزء", "probability": 0.9812825520833334}, {"start": 619.65, "end": 620.25, "word": " الثاني", "probability": 0.955078125}, {"start": 620.25, "end": 620.85, "word": " ال", "probability": 0.95263671875}, {"start": 620.85, "end": 621.39, "word": " convergence", "probability": 0.93212890625}, {"start": 621.39, "end": 621.93, "word": " لل", "probability": 0.74462890625}, {"start": 621.93, "end": 622.71, "word": " Yn", "probability": 0.69873046875}, {"start": 622.71, "end": 623.67, "word": " بيعطي", "probability": 0.959716796875}, {"start": 623.67, "end": 623.77, "word": " ال", "probability": 0.9716796875}, {"start": 623.77, "end": 624.19, "word": " convergence", "probability": 0.98974609375}, {"start": 624.19, "end": 624.87, "word": " لهذه", "probability": 0.99462890625}, {"start": 624.87, "end": 625.45, "word": " ال", "probability": 0.9609375}, {"start": 625.45, "end": 625.89, "word": " convergence", "probability": 0.9853515625}, {"start": 625.89, "end": 626.29, "word": " لهذه", "probability": 0.978759765625}, {"start": 626.29, "end": 626.43, "word": " اللي", "probability": 0.979248046875}, {"start": 626.43, "end": 626.57, "word": " هو", "probability": 0.94384765625}, {"start": 626.57, "end": 627.11, "word": " الكبير", "probability": 0.9654947916666666}, {"start": 627.11, "end": 627.49, "word": " بيعطي", "probability": 0.9713134765625}, {"start": 627.49, "end": 627.61, "word": " ال", "probability": 0.98486328125}, {"start": 627.61, "end": 628.01, "word": " convergence", "probability": 0.9873046875}, {"start": 628.01, "end": 629.05, "word": " لمن؟", "probability": 0.6730143229166666}, {"start": 629.05, "end": 629.31, "word": " لل", "probability": 0.60107421875}, {"start": 629.31, "end": 630.25, "word": " Xn", "probability": 0.93408203125}, {"start": 630.25, "end": 631.07, "word": " طيب", "probability": 0.9033203125}, {"start": 631.07, "end": 631.93, "word": " نيجي", "probability": 0.85693359375}, {"start": 631.93, "end": 632.41, "word": " الآن", "probability": 0.779541015625}, {"start": 632.41, "end": 635.91, "word": " طبعا", "probability": 0.9144287109375}, {"start": 635.91, "end": 636.35, "word": " ال", "probability": 0.7412109375}, {"start": 636.35, "end": 636.61, "word": " ..", "probability": 0.7021484375}, {"start": 636.61, "end": 638.13, "word": " ال", "probability": 0.73974609375}, {"start": 638.13, "end": 638.13, "word": " ..", "probability": 0.96826171875}, {"start": 638.13, "end": 638.49, "word": " ال", "probability": 0.55908203125}, {"start": 638.49, "end": 638.49, "word": " ..", "probability": 0.91015625}, {"start": 638.49, "end": 639.15, "word": " المطلوب", "probability": 0.974609375}, {"start": 639.15, "end": 639.49, "word": " بي", "probability": 0.833740234375}, {"start": 639.49, "end": 639.69, "word": " ..", "probability": 0.748046875}], "temperature": 1.0}, {"id": 29, "seek": 64208, "start": 641.0, "end": 642.08, "text": "بدنا نفترض", "tokens": [3555, 3215, 8315, 8717, 5172, 2655, 43042], "avg_logprob": -0.6728515774011612, "compression_ratio": 0.6333333333333333, "no_speech_prob": 0.0, "words": [{"start": 641.0, "end": 641.46, "word": "بدنا", "probability": 0.24833170572916666}, {"start": 641.46, "end": 642.08, "word": " نفترض", "probability": 0.99169921875}], "temperature": 1.0}, {"id": 30, "seek": 67373, "start": 644.73, "end": 673.73, "text": "suppose that limit xn على yn بساوة سفر ونفترض أن الـ yn أشمالها is convergent بالذات بت الـ xn أشمالها is convergent لأن شو معناه limit xn على yn بساوة سفر لكل y أكبر من سفر؟ there exists k element in n such that xn على yn أصغر من مين من إبسلون لكل n أكبر وسوء k هذا اللي هو تعريف ال limit لل xn على yn بساوة سفر", "tokens": [36622, 541, 300, 4948, 2031, 77, 15844, 17861, 4724, 3794, 995, 2407, 3660, 8608, 5172, 2288, 4032, 1863, 5172, 2655, 43042, 14739, 2423, 39184, 17861, 5551, 8592, 2304, 6027, 11296, 307, 9652, 6930, 20666, 8848, 9307, 39894, 2423, 39184, 2031, 77, 5551, 8592, 2304, 6027, 11296, 307, 9652, 6930, 5296, 33456, 13412, 2407, 20449, 8315, 3224, 4948, 2031, 77, 15844, 17861, 4724, 3794, 995, 2407, 3660, 8608, 5172, 2288, 5296, 28820, 288, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 22807, 456, 8198, 350, 4478, 294, 297, 1270, 300, 2031, 77, 15844, 17861, 5551, 9381, 17082, 2288, 9154, 3714, 9957, 9154, 11933, 3555, 3794, 1211, 11536, 5296, 28820, 297, 5551, 4117, 26890, 46952, 2407, 38207, 350, 23758, 13672, 1829, 31439, 37279, 16572, 5172, 2423, 4948, 24976, 2031, 77, 15844, 17861, 4724, 3794, 995, 2407, 3660, 8608, 5172, 2288], "avg_logprob": 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{"start": 651.49, "end": 651.75, "word": " yn", "probability": 0.61474609375}, {"start": 651.75, "end": 652.45, "word": " أشمالها", "probability": 0.8720703125}, {"start": 652.45, "end": 652.65, "word": " is", "probability": 0.90771484375}, {"start": 652.65, "end": 653.19, "word": " convergent", "probability": 0.94482421875}, {"start": 653.19, "end": 653.51, "word": " بالذات", "probability": 0.6131998697916666}, {"start": 653.51, "end": 653.67, "word": " بت", "probability": 0.29248046875}, {"start": 653.67, "end": 653.81, "word": " الـ", "probability": 0.6256103515625}, {"start": 653.81, "end": 654.13, "word": " xn", "probability": 0.970703125}, {"start": 654.13, "end": 654.87, "word": " أشمالها", "probability": 0.985546875}, {"start": 654.87, "end": 655.87, "word": " is", "probability": 0.89111328125}, {"start": 655.87, "end": 656.91, "word": " convergent", "probability": 0.966796875}, {"start": 656.91, "end": 657.73, "word": " لأن", "probability": 0.563720703125}, {"start": 657.73, "end": 657.91, "word": " شو", "probability": 0.865234375}, {"start": 657.91, "end": 658.19, "word": " معناه", "probability": 0.7516276041666666}, {"start": 658.19, "end": 658.49, "word": " limit", "probability": 0.8486328125}, {"start": 658.49, "end": 658.89, "word": " xn", "probability": 0.983154296875}, {"start": 658.89, "end": 659.05, "word": " على", "probability": 0.88330078125}, {"start": 659.05, "end": 659.33, "word": " yn", "probability": 0.982421875}, {"start": 659.33, "end": 659.73, "word": " بساوة", "probability": 0.97958984375}, {"start": 659.73, "end": 660.03, "word": " سفر", "probability": 0.9881184895833334}, {"start": 660.03, "end": 660.41, "word": " لكل", "probability": 0.97265625}, {"start": 660.41, "end": 660.69, "word": " y", "probability": 0.331787109375}, {"start": 660.69, "end": 661.09, "word": " أكبر", "probability": 0.9599609375}, {"start": 661.09, "end": 661.25, "word": " من", "probability": 0.99658203125}, {"start": 661.25, "end": 662.19, "word": " سفر؟", "probability": 0.7864990234375}, {"start": 662.19, "end": 662.41, "word": " there", "probability": 0.73876953125}, {"start": 662.41, "end": 662.93, "word": " exists", "probability": 0.5986328125}, {"start": 662.93, "end": 663.23, "word": " k", "probability": 0.736328125}, {"start": 663.23, "end": 663.47, "word": " element", "probability": 0.84814453125}, {"start": 663.47, "end": 663.77, "word": " in", "probability": 0.95068359375}, {"start": 663.77, "end": 663.93, "word": " n", "probability": 0.734375}, {"start": 663.93, "end": 664.17, "word": " such", "probability": 0.8779296875}, {"start": 664.17, "end": 664.51, "word": " that", "probability": 0.9462890625}, {"start": 664.51, "end": 664.97, "word": " xn", "probability": 0.950439453125}, {"start": 664.97, "end": 665.19, "word": " على", "probability": 0.94384765625}, {"start": 665.19, "end": 665.65, "word": " yn", "probability": 0.9873046875}, {"start": 665.65, "end": 666.11, "word": " أصغر", "probability": 0.961669921875}, {"start": 666.11, "end": 666.29, "word": " من", "probability": 0.99560546875}, {"start": 666.29, "end": 666.61, "word": " مين", "probability": 0.76123046875}, {"start": 666.61, "end": 666.81, "word": " من", "probability": 0.96826171875}, {"start": 666.81, "end": 667.33, "word": " إبسلون", "probability": 0.819482421875}, {"start": 667.33, "end": 668.01, "word": " لكل", "probability": 0.968994140625}, {"start": 668.01, "end": 668.23, "word": " n", "probability": 0.8046875}, {"start": 668.23, "end": 668.57, "word": " أكبر", "probability": 0.9817708333333334}, {"start": 668.57, "end": 668.91, "word": " وسوء", "probability": 0.64111328125}, {"start": 668.91, "end": 669.11, "word": " k", "probability": 0.62158203125}, {"start": 669.11, "end": 669.43, "word": " هذا", "probability": 0.71337890625}, {"start": 669.43, "end": 669.67, "word": " اللي", "probability": 0.969482421875}, {"start": 669.67, "end": 670.01, "word": " هو", "probability": 0.990234375}, {"start": 670.01, "end": 670.61, "word": " تعريف", "probability": 0.98876953125}, {"start": 670.61, "end": 670.81, "word": " ال", "probability": 0.8017578125}, {"start": 670.81, "end": 671.67, "word": " limit", "probability": 0.65185546875}, {"start": 671.67, "end": 671.91, "word": " لل", "probability": 0.58544921875}, {"start": 671.91, "end": 672.33, "word": " xn", "probability": 0.818603515625}, {"start": 672.33, "end": 672.51, "word": " على", "probability": 0.90478515625}, {"start": 672.51, "end": 672.89, "word": " yn", "probability": 0.36181640625}, {"start": 672.89, "end": 673.37, "word": " بساوة", "probability": 0.980078125}, {"start": 673.37, "end": 673.73, "word": " سفر", "probability": 0.9934895833333334}], "temperature": 1.0}, {"id": 31, "seek": 69614, "start": 674.2, "end": 696.14, "text": "ومنه هذا دروب طرفين في وسطين XN أصلا من YN في Y ن إحنا مفترضين أن ال YN is convergent يعني ال Y في ال YN ثابتة ما هي ال Y is convergent إذا كان الكبيرة convergent من باب أولى الصماش لل XN إيش هتكون اللي هي is convergent by اللي هو comparison test الأصلي", "tokens": [2407, 27842, 3224, 23758, 11778, 2288, 37746, 23032, 28480, 9957, 8978, 46952, 9566, 9957, 1783, 45, 5551, 9381, 15040, 9154, 398, 45, 8978, 398, 8717, 11933, 5016, 8315, 3714, 5172, 2655, 43042, 9957, 14739, 2423, 398, 45, 307, 9652, 6930, 37495, 22653, 2423, 398, 8978, 2423, 398, 45, 38637, 16758, 2655, 3660, 19446, 39896, 2423, 398, 307, 9652, 6930, 11933, 15730, 25961, 33251, 3555, 48923, 9652, 6930, 9154, 4724, 16758, 5551, 12610, 7578, 31767, 2304, 33599, 24976, 1783, 45, 11933, 1829, 8592, 8032, 2655, 30544, 13672, 1829, 39896, 307, 9652, 6930, 538, 13672, 1829, 31439, 9660, 1500, 16247, 9381, 20292], "avg_logprob": -0.2888304467248444, "compression_ratio": 1.7293577981651376, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 674.2, "end": 674.72, "word": "ومنه", "probability": 0.7090657552083334}, {"start": 674.72, "end": 675.0, "word": " هذا", "probability": 0.73486328125}, {"start": 675.0, "end": 675.24, "word": " دروب", "probability": 0.33935546875}, {"start": 675.24, "end": 675.58, "word": " طرفين", "probability": 0.9353841145833334}, {"start": 675.58, "end": 675.7, "word": " في", "probability": 0.73388671875}, {"start": 675.7, "end": 676.24, "word": " وسطين", "probability": 0.9786783854166666}, {"start": 676.24, "end": 676.7, "word": " XN", "probability": 0.35955810546875}, {"start": 676.7, "end": 677.0, "word": " أصلا", "probability": 0.5891927083333334}, {"start": 677.0, "end": 677.12, "word": " من", "probability": 0.93896484375}, {"start": 677.12, "end": 677.5, "word": " YN", "probability": 0.867431640625}, {"start": 677.5, "end": 677.56, "word": " في", "probability": 0.344482421875}, {"start": 677.56, "end": 677.74, "word": " Y", "probability": 0.83544921875}, {"start": 677.74, "end": 677.98, "word": " ن", "probability": 0.5498046875}, {"start": 677.98, "end": 678.96, "word": " إحنا", "probability": 0.6789143880208334}, {"start": 678.96, "end": 679.74, "word": " مفترضين", "probability": 0.9623046875}, {"start": 679.74, "end": 679.9, "word": " أن", "probability": 0.42333984375}, {"start": 679.9, "end": 680.1, "word": " ال", "probability": 0.405517578125}, {"start": 680.1, "end": 680.68, "word": " YN", "probability": 0.76708984375}, {"start": 680.68, "end": 680.9, "word": " is", "probability": 0.8046875}, {"start": 680.9, "end": 681.7, "word": " convergent", "probability": 0.871826171875}, {"start": 681.7, "end": 682.08, "word": " يعني", "probability": 0.89501953125}, {"start": 682.08, "end": 682.22, "word": " ال", "probability": 0.62939453125}, {"start": 682.22, "end": 682.44, "word": " Y", "probability": 0.67724609375}, {"start": 682.44, "end": 682.7, "word": " في", "probability": 0.384765625}, {"start": 682.7, "end": 682.82, "word": " ال", "probability": 0.50048828125}, {"start": 682.82, "end": 683.18, "word": " YN", "probability": 0.74462890625}, {"start": 683.18, "end": 683.74, "word": " ثابتة", "probability": 0.9237060546875}, {"start": 683.74, "end": 683.8, "word": " ما", "probability": 0.591796875}, {"start": 683.8, "end": 683.92, "word": " هي", "probability": 0.88232421875}, {"start": 683.92, "end": 684.06, "word": " ال", "probability": 0.888671875}, {"start": 684.06, "end": 684.24, "word": " Y", "probability": 0.8818359375}, {"start": 684.24, "end": 684.54, "word": " is", "probability": 0.7490234375}, {"start": 684.54, "end": 685.24, "word": " convergent", "probability": 0.8359375}, {"start": 685.24, "end": 685.88, "word": " إذا", "probability": 0.926025390625}, {"start": 685.88, "end": 686.06, "word": " كان", "probability": 0.96484375}, {"start": 686.06, "end": 686.48, "word": " الكبيرة", "probability": 0.8011067708333334}, {"start": 686.48, "end": 687.16, "word": " convergent", "probability": 0.883544921875}, {"start": 687.16, "end": 687.34, "word": " من", "probability": 0.98095703125}, {"start": 687.34, "end": 687.62, "word": " باب", "probability": 0.98876953125}, {"start": 687.62, "end": 687.94, "word": " أولى", "probability": 0.84765625}, {"start": 687.94, "end": 688.38, "word": " الصماش", "probability": 0.7112630208333334}, {"start": 688.38, "end": 688.56, "word": " لل", "probability": 0.53125}, {"start": 688.56, "end": 688.96, "word": " XN", "probability": 0.800048828125}, {"start": 688.96, "end": 689.12, "word": " إيش", "probability": 0.8333333333333334}, {"start": 689.12, "end": 689.72, "word": " هتكون", "probability": 0.8888346354166666}, {"start": 689.72, "end": 690.52, "word": " اللي", "probability": 0.7095947265625}, {"start": 690.52, "end": 690.98, "word": " هي", "probability": 0.859375}, {"start": 690.98, "end": 691.76, "word": " is", "probability": 0.78369140625}, {"start": 691.76, "end": 692.68, "word": " convergent", "probability": 0.949462890625}, {"start": 692.68, "end": 693.12, "word": " by", "probability": 0.9130859375}, {"start": 693.12, "end": 693.68, "word": " اللي", "probability": 0.851318359375}, {"start": 693.68, "end": 693.84, "word": " هو", "probability": 0.9853515625}, {"start": 693.84, "end": 694.5, "word": " comparison", "probability": 0.8955078125}, {"start": 694.5, "end": 695.34, "word": " test", "probability": 0.79052734375}, {"start": 695.34, "end": 696.14, "word": " الأصلي", "probability": 0.8855794270833334}], "temperature": 1.0}, {"id": 32, "seek": 71942, "start": 696.78, "end": 719.42, "text": "الان اذا خلصنا من ال testين اللي هو ال comparison test وال limit comparison test وكانت ال testين عبارة عن على positive sequences الان بدنا نحكي عن حاجة اسمها the root and the ratio test نشوف ال root و ال ratio test ال test الأولان اللي هو ال root test ايه؟ fx بسوي xn", "tokens": [6027, 7649, 1975, 15730, 16490, 1211, 9381, 8315, 9154, 2423, 1500, 9957, 13672, 1829, 31439, 2423, 9660, 1500, 16070, 4948, 9660, 1500, 4032, 41361, 2655, 2423, 1500, 9957, 6225, 3555, 9640, 3660, 18871, 15844, 3353, 22978, 2423, 7649, 47525, 8315, 8717, 5016, 4117, 1829, 18871, 11331, 26108, 3660, 24525, 2304, 11296, 264, 5593, 293, 264, 8509, 1500, 8717, 8592, 38688, 2423, 5593, 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بيسوى Limit Xn plus 1 لـ N نفترض إن هذا الـ Limit موجود ثم الصممش للـ Xn هو بالتأكيد مرتبط لما R أصغر من 1 وهو مرتبط عندما R أكبر من 1 وهذا ال test انتوا عارفينه يعني بمعنى آخر باجي بقول باخد Limit Xn", "tokens": [1863, 5172, 2655, 43042, 36145, 1783, 77, 4724, 1829, 30544, 8608, 1829, 4117, 2407, 9957, 3794, 8978, 497, 4032, 49, 4724, 1829, 3794, 2407, 7578, 16406, 270, 1783, 77, 1804, 502, 5296, 39184, 426, 8717, 5172, 2655, 43042, 36145, 23758, 2423, 39184, 16406, 270, 3714, 29245, 23328, 38637, 2304, 2423, 9381, 2304, 2304, 8592, 24976, 39184, 1783, 77, 31439, 20666, 2655, 10721, 4117, 25708, 3714, 43500, 3555, 9566, 5296, 15042, 497, 5551, 9381, 17082, 2288, 9154, 502, 4032, 3224, 2407, 3714, 43500, 3555, 9566, 43242, 15042, 497, 5551, 4117, 26890, 9154, 502, 37037, 15730, 2423, 1500, 16472, 2655, 14407, 6225, 9640, 5172, 9957, 3224, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 19753, 34740, 4724, 26108, 1829, 4724, 39648, 4724, 47283, 3215, 16406, 270, 1783, 77], "avg_logprob": -0.4092261845630313, "compression_ratio": 1.6462882096069869, "no_speech_prob": 2.9802322387695312e-06, "words": [{"start": 950.48, "end": 951.1, "word": "نفترض", "probability": 0.9053955078125}, {"start": 951.1, "end": 951.48, "word": " إن", "probability": 0.2403564453125}, {"start": 951.48, "end": 952.9, "word": " Xn", "probability": 0.4427490234375}, {"start": 952.9, "end": 953.22, "word": " بيكون", "probability": 0.6724446614583334}, {"start": 953.22, "end": 953.72, "word": " سيكوينس", "probability": 0.69732666015625}, {"start": 953.72, "end": 953.84, "word": " في", "probability": 0.293212890625}, {"start": 953.84, "end": 954.16, "word": " R", "probability": 0.744140625}, {"start": 954.16, "end": 954.6, "word": " وR", "probability": 0.673095703125}, {"start": 954.6, "end": 954.9, "word": " بيسوى", "probability": 0.4678466796875}, {"start": 954.9, "end": 955.18, "word": " Limit", "probability": 0.59527587890625}, {"start": 955.18, "end": 955.5, "word": " Xn", "probability": 0.912109375}, {"start": 955.5, "end": 955.68, "word": " plus", "probability": 0.028961181640625}, {"start": 955.68, "end": 955.94, "word": " 1", "probability": 0.79443359375}, {"start": 955.94, "end": 956.28, "word": " لـ", "probability": 0.5150146484375}, {"start": 956.28, "end": 956.4, "word": " N", "probability": 0.490234375}, {"start": 956.4, "end": 957.5, "word": " نفترض", "probability": 0.9505615234375}, {"start": 957.5, "end": 957.92, "word": " إن", "probability": 0.82470703125}, {"start": 957.92, "end": 958.66, "word": " هذا", "probability": 0.5537109375}, {"start": 958.66, "end": 958.7, "word": " الـ", "probability": 0.3507080078125}, {"start": 958.7, "end": 959.02, "word": " Limit", "probability": 0.78662109375}, {"start": 959.02, "end": 959.46, "word": " موجود", "probability": 0.7199300130208334}, {"start": 959.46, "end": 960.4, "word": " ثم", "probability": 0.633544921875}, {"start": 960.4, "end": 961.08, "word": " الصممش", "probability": 0.5524658203125}, {"start": 961.08, "end": 961.42, "word": " للـ", "probability": 0.5994873046875}, {"start": 961.42, "end": 962.02, "word": " Xn", "probability": 0.953125}, {"start": 962.02, "end": 962.84, "word": " هو", "probability": 0.43115234375}, {"start": 962.84, "end": 963.56, "word": " بالتأكيد", "probability": 0.6137939453125}, {"start": 963.56, "end": 964.3, "word": " مرتبط", "probability": 0.634307861328125}, {"start": 964.3, "end": 964.86, "word": " لما", "probability": 0.768310546875}, {"start": 964.86, "end": 966.14, "word": " R", "probability": 0.65380859375}, {"start": 966.14, "end": 966.62, "word": " أصغر", "probability": 0.9761962890625}, {"start": 966.62, "end": 966.74, "word": " من", "probability": 0.99169921875}, {"start": 966.74, "end": 967.22, "word": " 1", "probability": 0.94970703125}, {"start": 967.22, "end": 968.52, "word": " وهو", "probability": 0.6503499348958334}, {"start": 968.52, "end": 969.08, "word": " مرتبط", "probability": 0.860107421875}, {"start": 969.08, "end": 969.66, "word": " عندما", "probability": 0.7666015625}, {"start": 969.66, "end": 970.36, "word": " R", "probability": 0.91943359375}, {"start": 970.36, "end": 970.7, "word": " أكبر", "probability": 0.9690755208333334}, {"start": 970.7, "end": 970.84, "word": " من", "probability": 0.99609375}, {"start": 970.84, "end": 971.14, "word": " 1", "probability": 0.98876953125}, {"start": 971.14, "end": 971.72, "word": " وهذا", "probability": 0.857421875}, {"start": 971.72, "end": 971.96, "word": " ال", "probability": 0.724609375}, {"start": 971.96, "end": 971.96, "word": " test", "probability": 0.31005859375}, {"start": 971.96, "end": 972.2, "word": " انتوا", "probability": 0.6427408854166666}, {"start": 972.2, "end": 972.58, "word": " عارفينه", "probability": 0.96171875}, {"start": 972.58, "end": 972.82, "word": " يعني", "probability": 0.778564453125}, {"start": 972.82, "end": 973.14, "word": " بمعنى", "probability": 0.990625}, {"start": 973.14, "end": 973.42, "word": " آخر", "probability": 0.804443359375}, {"start": 973.42, "end": 973.74, "word": " باجي", "probability": 0.8684895833333334}, {"start": 973.74, "end": 974.12, "word": " بقول", "probability": 0.94921875}, {"start": 974.12, "end": 975.0, "word": " باخد", "probability": 0.966796875}, {"start": 975.0, "end": 975.66, "word": " Limit", "probability": 0.819091796875}, {"start": 975.66, "end": 978.04, "word": " Xn", "probability": 0.764892578125}], "temperature": 1.0}, {"id": 44, "seek": 99973, "start": 978.97, "end": 999.73, "text": "أُس واحدة لأن أس أن قصة infinity طلع عندي ايش بيساوي بيساوي R احنا مفترضين ان ال limit exist اذا R أكبر من واحد خلاص ال series xn divers اذا R أصغر من واحد ال series ايش مالها converge و absolutely convergent", "tokens": [10721, 10859, 3794, 36764, 24401, 3660, 5296, 10721, 1863, 5551, 3794, 14739, 12174, 9381, 3660, 13202, 23032, 1211, 3615, 18871, 16254, 1975, 1829, 8592, 4724, 1829, 3794, 995, 45865, 4724, 1829, 3794, 995, 45865, 497, 1975, 5016, 8315, 3714, 5172, 2655, 43042, 9957, 16472, 2423, 4948, 2514, 1975, 15730, 497, 5551, 4117, 26890, 9154, 36764, 24401, 16490, 1211, 33546, 2423, 2638, 2031, 77, 6111, 1975, 15730, 497, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 2423, 2638, 1975, 1829, 8592, 19446, 1211, 11296, 41881, 4032, 3122, 9652, 6930], "avg_logprob": -0.3753551204096187, "compression_ratio": 1.5692307692307692, "no_speech_prob": 1.0132789611816406e-06, "words": [{"start": 978.97, "end": 979.33, "word": "أُس", "probability": 0.3835652669270833}, {"start": 979.33, "end": 979.75, "word": " واحدة", "probability": 0.7657063802083334}, {"start": 979.75, "end": 980.19, "word": " لأن", "probability": 0.6934407552083334}, {"start": 980.19, "end": 980.91, "word": " أس", "probability": 0.35308837890625}, {"start": 980.91, "end": 981.03, "word": " أن", "probability": 0.30615234375}, {"start": 981.03, "end": 981.33, "word": " قصة", "probability": 0.5535481770833334}, {"start": 981.33, "end": 981.57, "word": " infinity", "probability": 0.1522216796875}, {"start": 981.57, "end": 982.05, "word": " طلع", "probability": 0.5614827473958334}, {"start": 982.05, "end": 982.47, "word": " عندي", "probability": 0.72021484375}, {"start": 982.47, "end": 983.27, "word": " ايش", "probability": 0.6212972005208334}, {"start": 983.27, "end": 983.81, "word": " بيساوي", "probability": 0.7947265625}, {"start": 983.81, "end": 984.61, "word": " بيساوي", "probability": 0.88720703125}, {"start": 984.61, "end": 984.85, "word": " R", "probability": 0.55615234375}, {"start": 984.85, "end": 986.35, "word": " احنا", "probability": 0.8159993489583334}, {"start": 986.35, "end": 986.83, "word": " مفترضين", "probability": 0.97666015625}, {"start": 986.83, "end": 986.93, "word": " ان", "probability": 0.73681640625}, {"start": 986.93, "end": 986.99, "word": " ال", "probability": 0.677734375}, {"start": 986.99, "end": 987.19, "word": " limit", "probability": 0.892578125}, {"start": 987.19, "end": 987.63, "word": " exist", "probability": 0.7451171875}, {"start": 987.63, "end": 988.55, "word": " اذا", "probability": 0.767333984375}, {"start": 988.55, "end": 988.93, "word": " R", "probability": 0.9140625}, {"start": 988.93, "end": 989.63, "word": " أكبر", "probability": 0.9054361979166666}, {"start": 989.63, "end": 989.83, "word": " من", "probability": 0.994140625}, {"start": 989.83, "end": 990.41, "word": " واحد", "probability": 0.8251953125}, {"start": 990.41, "end": 991.01, "word": " خلاص", "probability": 0.9720052083333334}, {"start": 991.01, "end": 991.21, "word": " ال", "probability": 0.84326171875}, {"start": 991.21, "end": 991.53, "word": " series", "probability": 0.91552734375}, {"start": 991.53, "end": 992.29, "word": " xn", "probability": 0.23541259765625}, {"start": 992.29, "end": 992.75, "word": " divers", "probability": 0.345947265625}, {"start": 992.75, "end": 993.75, "word": " اذا", "probability": 0.844970703125}, {"start": 993.75, "end": 994.03, "word": " R", "probability": 0.943359375}, {"start": 994.03, "end": 994.55, "word": " أصغر", "probability": 0.955810546875}, {"start": 994.55, "end": 994.75, "word": " من", "probability": 0.9970703125}, {"start": 994.75, "end": 995.31, "word": " واحد", "probability": 0.9697265625}, {"start": 995.31, "end": 995.99, "word": " ال", "probability": 0.73779296875}, {"start": 995.99, "end": 996.37, "word": " series", "probability": 0.93505859375}, {"start": 996.37, "end": 996.69, "word": " ايش", "probability": 0.86376953125}, {"start": 996.69, "end": 997.15, "word": " مالها", "probability": 0.6559244791666666}, {"start": 997.15, "end": 998.31, "word": " converge", "probability": 0.75048828125}, {"start": 998.31, "end": 998.49, "word": " و", "probability": 0.91650390625}, {"start": 998.49, "end": 999.03, "word": " absolutely", "probability": 0.441650390625}, {"start": 999.03, "end": 999.73, "word": " convergent", "probability": 0.96337890625}], "temperature": 1.0}, {"id": 45, "seek": 102767, "start": 1000.75, "end": 1027.67, "text": "لكن إذا كانت R بتساوي واحد بنقول ال test fail يعني ممكن يكون convergence وممكن تكون إيش ما لها divergence وفيه أمثلة واضح؟ طيب الآن بدنا نيجي للبرهان suppose that limit Xn أس واحد على N بساوي R وR إيش ما لها أصغر من واحد نفترض إنه limit Xn أس واحد على N بساوي R أصغر من واحد", "tokens": [1211, 19452, 11933, 15730, 25961, 2655, 497, 39894, 3794, 995, 45865, 36764, 24401, 44945, 39648, 2423, 1500, 3061, 37495, 22653, 3714, 43020, 7251, 30544, 32181, 4032, 2304, 43020, 6055, 30544, 11933, 1829, 8592, 19446, 5296, 11296, 47387, 4032, 41185, 3224, 5551, 2304, 12984, 37977, 4032, 46958, 5016, 22807, 23032, 1829, 3555, 6024, 48506, 47525, 8315, 8717, 1829, 7435, 1829, 24976, 26890, 3224, 7649, 7297, 300, 4948, 1783, 77, 5551, 3794, 36764, 24401, 15844, 426, 4724, 3794, 995, 45865, 497, 4032, 49, 11933, 1829, 8592, 19446, 5296, 11296, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 8717, 5172, 2655, 43042, 36145, 3224, 4948, 1783, 77, 5551, 3794, 36764, 24401, 15844, 426, 4724, 3794, 995, 45865, 497, 5551, 9381, 17082, 2288, 9154, 36764, 24401], "avg_logprob": -0.1937756153648017, "compression_ratio": 1.7883817427385893, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 1000.75, "end": 1001.23, "word": "لكن", "probability": 0.89111328125}, {"start": 1001.23, "end": 1001.49, "word": " إذا", "probability": 0.84765625}, {"start": 1001.49, "end": 1001.95, "word": " كانت", "probability": 0.98046875}, {"start": 1001.95, "end": 1002.17, "word": " R", "probability": 0.67138671875}, {"start": 1002.17, "end": 1002.75, "word": " بتساوي", "probability": 0.766510009765625}, {"start": 1002.75, "end": 1003.25, "word": " واحد", "probability": 0.7335205078125}, {"start": 1003.25, "end": 1003.83, "word": " بنقول", "probability": 0.7376708984375}, {"start": 1003.83, "end": 1004.05, "word": " ال", "probability": 0.857421875}, {"start": 1004.05, "end": 1004.37, "word": " test", "probability": 0.90478515625}, {"start": 1004.37, "end": 1004.91, "word": " fail", "probability": 0.56494140625}, {"start": 1004.91, "end": 1005.27, "word": " يعني", "probability": 0.837646484375}, {"start": 1005.27, "end": 1005.61, "word": " ممكن", "probability": 0.972412109375}, {"start": 1005.61, "end": 1005.89, "word": " يكون", "probability": 0.95703125}, {"start": 1005.89, "end": 1006.43, "word": " convergence", "probability": 0.5693359375}, {"start": 1006.43, "end": 1006.87, "word": " وممكن", "probability": 0.9070638020833334}, {"start": 1006.87, "end": 1007.19, "word": " تكون", "probability": 0.796630859375}, {"start": 1007.19, "end": 1007.43, "word": " إيش", "probability": 0.5849609375}, {"start": 1007.43, "end": 1007.65, "word": " ما", "probability": 0.830078125}, {"start": 1007.65, "end": 1007.91, "word": " لها", "probability": 0.902587890625}, {"start": 1007.91, "end": 1008.95, "word": " divergence", "probability": 0.50537109375}, {"start": 1008.95, "end": 1009.31, "word": " وفيه", "probability": 0.77294921875}, {"start": 1009.31, "end": 1009.79, "word": " أمثلة", "probability": 0.9188232421875}, {"start": 1009.79, "end": 1011.75, "word": " واضح؟", "probability": 0.851318359375}, {"start": 1011.75, "end": 1012.09, "word": " طيب", "probability": 0.986328125}, {"start": 1012.09, "end": 1013.03, "word": " الآن", "probability": 0.6273193359375}, {"start": 1013.03, "end": 1013.37, "word": " بدنا", "probability": 0.888916015625}, {"start": 1013.37, "end": 1013.65, "word": " نيجي", "probability": 0.97509765625}, {"start": 1013.65, "end": 1014.67, "word": " للبرهان", "probability": 0.852294921875}, {"start": 1014.67, "end": 1015.91, "word": " suppose", "probability": 0.415771484375}, {"start": 1015.91, "end": 1016.33, "word": " that", "probability": 0.8798828125}, {"start": 1016.33, "end": 1016.67, "word": " limit", "probability": 0.919921875}, {"start": 1016.67, "end": 1017.35, "word": " Xn", "probability": 0.385986328125}, {"start": 1017.35, "end": 1017.61, "word": " أس", "probability": 0.713623046875}, {"start": 1017.61, "end": 1017.97, "word": " واحد", "probability": 0.888427734375}, {"start": 1017.97, "end": 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الشمالة تكون converts then by comparison test", "tokens": [2304, 33599, 1829, 8032, 8848, 1829, 2423, 7649, 2423, 497, 16, 5551, 9381, 17082, 2288, 9154, 36764, 24401, 3714, 3215, 10943, 5551, 9381, 17082, 2288, 36764, 24401, 37037, 24192, 2423, 2638, 38874, 33251, 3555, 48923, 11933, 15730, 4724, 4587, 7578, 4724, 10721, 12610, 29538, 2423, 2638, 25124, 2304, 6027, 3660, 6055, 30544, 38874, 550, 538, 9660, 1500], "avg_logprob": -0.4242995612580201, "compression_ratio": 1.4487179487179487, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1257.57, "end": 1258.41, "word": "ماشي", "probability": 0.6263834635416666}, {"start": 1258.41, "end": 1259.33, "word": " هذي", "probability": 0.4849853515625}, {"start": 1259.33, "end": 1259.95, "word": " الان", "probability": 0.718994140625}, {"start": 1259.95, "end": 1260.33, "word": " ال", "probability": 0.7978515625}, {"start": 1260.33, "end": 1260.95, "word": " R1", "probability": 0.56787109375}, {"start": 1260.95, "end": 1261.55, 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"probability": 0.983154296875}, {"start": 1738.81, "end": 1739.13, "word": " أولى", "probability": 0.8603515625}, {"start": 1739.13, "end": 1739.27, "word": " لل", "probability": 0.94873046875}, {"start": 1739.27, "end": 1739.53, "word": " series", "probability": 0.94384765625}, {"start": 1739.53, "end": 1739.75, "word": " اللي", "probability": 0.953125}, {"start": 1739.75, "end": 1740.09, "word": " أصغر", "probability": 0.985107421875}, {"start": 1740.09, "end": 1740.49, "word": " منها", "probability": 0.994384765625}, {"start": 1740.49, "end": 1740.95, "word": " اللي", "probability": 0.9423828125}, {"start": 1740.95, "end": 1741.29, "word": " هتكون", "probability": 0.9072265625}, {"start": 1741.29, "end": 1741.83, "word": " converge", "probability": 0.85791015625}, {"start": 1741.83, "end": 1742.13, "word": " ثم", "probability": 0.9326171875}, {"start": 1742.13, "end": 1742.45, "word": " صارت", "probability": 0.9228515625}, {"start": 1742.45, "end": 1742.67, "word": " عندي", "probability": 0.715087890625}, {"start": 1742.67, "end": 1743.03, "word": " summation", "probability": 0.8759765625}, {"start": 1743.03, "end": 1743.41, "word": " absolute", "probability": 0.888671875}, {"start": 1743.41, "end": 1743.79, "word": " value", "probability": 0.9765625}, {"start": 1743.79, "end": 1744.29, "word": " converge", "probability": 0.89697265625}, {"start": 1744.29, "end": 1744.53, "word": " او", "probability": 0.6571044921875}, {"start": 1744.53, "end": 1744.83, "word": " بمعنى", "probability": 0.96748046875}, {"start": 1744.83, "end": 1745.07, "word": " اخر", "probability": 0.7291259765625}, {"start": 1745.07, "end": 1745.73, "word": " summation", "probability": 0.81298828125}, {"start": 1745.73, "end": 1746.79, "word": " XN", "probability": 0.22442626953125}, {"start": 1746.79, "end": 1746.97, "word": " is", "probability": 0.865234375}, {"start": 1746.97, "end": 1747.59, "word": " absolutely", "probability": 0.734375}, {"start": 1747.59, "end": 1749.03, "word": " convergent", "probability": 0.969970703125}, {"start": 1749.03, "end": 1750.01, "word": " الان", "probability": 0.6610107421875}, {"start": 1750.01, "end": 1751.27, "word": " نيجي", "probability": 0.9432373046875}], "temperature": 1.0}, {"id": 77, "seek": 177231, "start": 1752.03, "end": 1772.31, "text": "لأ الجزء الثاني الان إذا كانت اللي هي سبعة holes سبعة هي سبعة شايفينها هي سبعة اللي هو ال XN زائد واحد على XN أكبر أو ساوى واحد لكل N أكبر أو ساوى مين أكبر أو ساوى K كلام مشابه", "tokens": [1211, 10721, 25724, 11622, 38207, 6024, 104, 7649, 1829, 2423, 7649, 11933, 15730, 25961, 2655, 13672, 1829, 39896, 8608, 3555, 27884, 8118, 8608, 3555, 27884, 39896, 8608, 3555, 27884, 13412, 995, 33911, 9957, 11296, 39896, 8608, 3555, 27884, 13672, 1829, 31439, 2423, 1783, 45, 30767, 16373, 3215, 36764, 24401, 15844, 1783, 45, 5551, 4117, 26890, 34051, 8608, 995, 2407, 7578, 36764, 24401, 5296, 28820, 426, 5551, 4117, 26890, 34051, 8608, 995, 2407, 7578, 3714, 9957, 5551, 4117, 26890, 34051, 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"probability": 0.9658203125}, {"start": 1761.97, "end": 1764.37, "word": " على", "probability": 0.69189453125}, {"start": 1764.37, "end": 1765.03, "word": " XN", "probability": 0.90283203125}, {"start": 1765.03, "end": 1765.87, "word": " أكبر", "probability": 0.9357096354166666}, {"start": 1765.87, "end": 1765.97, "word": " أو", "probability": 0.98046875}, {"start": 1765.97, "end": 1766.19, "word": " ساوى", "probability": 0.6453857421875}, {"start": 1766.19, "end": 1766.57, "word": " واحد", "probability": 0.980224609375}, {"start": 1766.57, "end": 1767.03, "word": " لكل", "probability": 0.98291015625}, {"start": 1767.03, "end": 1767.39, "word": " N", "probability": 0.76318359375}, {"start": 1767.39, "end": 1767.97, "word": " أكبر", "probability": 0.9720052083333334}, {"start": 1767.97, "end": 1768.15, "word": " أو", "probability": 0.8955078125}, {"start": 1768.15, "end": 1768.51, "word": " ساوى", "probability": 0.9326171875}, {"start": 1768.51, "end": 1768.99, "word": " مين", "probability": 0.802978515625}, {"start": 1768.99, "end": 1769.75, "word": " أكبر", "probability": 0.8751627604166666}, {"start": 1769.75, "end": 1769.87, "word": " أو", "probability": 0.9208984375}, {"start": 1769.87, "end": 1770.21, "word": " ساوى", "probability": 0.936767578125}, {"start": 1770.21, "end": 1770.51, "word": " K", "probability": 0.80029296875}, {"start": 1770.51, "end": 1771.45, "word": " كلام", "probability": 0.8984375}, {"start": 1771.45, "end": 1772.31, "word": " مشابه", "probability": 0.98974609375}], "temperature": 1.0}, {"id": 78, "seek": 179156, "start": 1773.24, "end": 1791.56, "text": "كلام مشابه if 7 holds then by induction one can see that xkm أكبر أو يساوي xk for m بساوي واحد و بعدين it is true for m then it is true for m زائد واحد و زي ما عملت قبل بشوية هتلاجوها بكل سهولة تطلع", "tokens": [28820, 10943, 37893, 16758, 3224, 498, 1614, 9190, 550, 538, 33371, 472, 393, 536, 300, 2031, 17950, 5551, 4117, 26890, 34051, 7251, 3794, 995, 45865, 2031, 74, 337, 275, 4724, 3794, 995, 45865, 36764, 24401, 4032, 39182, 9957, 309, 307, 2074, 337, 275, 550, 309, 307, 2074, 337, 275, 30767, 16373, 3215, 36764, 24401, 4032, 30767, 1829, 19446, 6225, 42213, 2655, 12174, 36150, 4724, 8592, 2407, 10632, 8032, 2655, 15040, 7435, 2407, 11296, 4724, 28820, 8608, 3224, 12610, 3660, 6055, 9566, 1211, 3615], "avg_logprob": -0.19884673293147767, "compression_ratio": 1.4866310160427807, "no_speech_prob": 1.7881393432617188e-06, "words": [{"start": 1773.2399999999998, "end": 1773.8799999999999, "word": "كلام", "probability": 0.76611328125}, {"start": 1773.8799999999999, "end": 1774.52, "word": " مشابه", "probability": 0.97900390625}, {"start": 1774.52, "end": 1775.3, "word": " if", "probability": 0.292236328125}, {"start": 1775.3, "end": 1775.64, "word": " 7", "probability": 0.49951171875}, {"start": 1775.64, "end": 1776.42, "word": " holds", "probability": 0.74951171875}, {"start": 1776.42, "end": 1777.56, "word": " then", "probability": 0.701171875}, {"start": 1777.56, "end": 1777.88, "word": " by", "probability": 0.95458984375}, {"start": 1777.88, "end": 1778.48, "word": " induction", "probability": 0.970703125}, {"start": 1778.48, "end": 1778.8, "word": " one", "probability": 0.86279296875}, {"start": 1778.8, "end": 1779.14, "word": " can", "probability": 0.9697265625}, {"start": 1779.14, "end": 1779.48, "word": " see", "probability": 0.90380859375}, {"start": 1779.48, "end": 1780.04, "word": " that", "probability": 0.9248046875}, {"start": 1780.04, "end": 1781.22, "word": " xkm", "probability": 0.5205078125}, {"start": 1781.22, "end": 1781.74, "word": " أكبر", "probability": 0.869140625}, {"start": 1781.74, "end": 1781.9, "word": " أو", "probability": 0.85888671875}, {"start": 1781.9, "end": 1782.28, "word": " يساوي", "probability": 0.8997802734375}, {"start": 1782.28, "end": 1782.88, "word": " xk", "probability": 0.87109375}, {"start": 1782.88, "end": 1783.08, "word": " for", "probability": 0.382080078125}, {"start": 1783.08, "end": 1783.26, "word": " m", "probability": 0.75048828125}, {"start": 1783.26, "end": 1783.64, "word": " بساوي", "probability": 0.64874267578125}, {"start": 1783.64, "end": 1783.94, "word": " واحد", "probability": 0.820556640625}, {"start": 1783.94, "end": 1784.04, "word": " و", "probability": 0.74072265625}, {"start": 1784.04, "end": 1784.34, "word": " بعدين", "probability": 0.6387939453125}, {"start": 1784.34, "end": 1784.46, "word": " it", "probability": 0.509765625}, {"start": 1784.46, "end": 1784.62, "word": " is", "probability": 0.62890625}, {"start": 1784.62, "end": 1785.7, "word": " true", "probability": 0.810546875}, {"start": 1785.7, "end": 1785.96, "word": " for", "probability": 0.953125}, {"start": 1785.96, "end": 1786.88, "word": " m", "probability": 0.83203125}, {"start": 1786.88, "end": 1787.14, "word": " then", "probability": 0.57666015625}, {"start": 1787.14, "end": 1787.42, "word": " it", "probability": 0.9287109375}, {"start": 1787.42, "end": 1787.56, "word": " is", "probability": 0.93701171875}, {"start": 1787.56, "end": 1787.78, "word": " true", "probability": 0.98828125}, {"start": 1787.78, "end": 1787.98, "word": " for", "probability": 0.96875}, {"start": 1787.98, "end": 1788.14, "word": " m", "probability": 0.955078125}, {"start": 1788.14, "end": 1788.42, "word": " زائد", "probability": 0.8837890625}, {"start": 1788.42, "end": 1788.74, "word": " واحد", "probability": 0.952880859375}, {"start": 1788.74, "end": 1788.82, "word": " و", "probability": 0.86865234375}, {"start": 1788.82, "end": 1788.9, "word": " زي", "probability": 0.83056640625}, {"start": 1788.9, "end": 1789.0, "word": " ما", "probability": 0.91455078125}, {"start": 1789.0, "end": 1789.26, "word": " عملت", "probability": 0.9342447916666666}, {"start": 1789.26, "end": 1789.48, "word": " قبل", "probability": 0.90625}, {"start": 1789.48, "end": 1789.88, "word": " بشوية", "probability": 0.910400390625}, {"start": 1789.88, "end": 1790.36, "word": " هتلاجوها", "probability": 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" واحد", "probability": 0.949951171875}, {"start": 1865.33, "end": 1865.47, "word": " على", "probability": 0.4208984375}, {"start": 1865.47, "end": 1865.81, "word": " xn", "probability": 0.7412109375}, {"start": 1865.81, "end": 1866.11, "word": " absolute", "probability": 0.8037109375}, {"start": 1866.11, "end": 1866.45, "word": " value", "probability": 0.9892578125}, {"start": 1866.45, "end": 1866.81, "word": " طبعا", "probability": 0.9735107421875}, {"start": 1866.81, "end": 1867.39, "word": " إذا", "probability": 0.870361328125}, {"start": 1867.39, "end": 1867.63, "word": " كانت", "probability": 0.978759765625}, {"start": 1867.63, "end": 1867.83, "word": " positive", "probability": 0.9541015625}, {"start": 1867.83, "end": 1868.23, "word": " terms", "probability": 0.9326171875}, {"start": 1868.23, "end": 1868.39, "word": " على", "probability": 0.93896484375}, {"start": 1868.39, "end": 1868.65, "word": " طول", "probability": 0.9892578125}, {"start": 1868.65, "end": 1869.11, "word": " بدون", "probability": 0.970703125}, {"start": 1869.11, "end": 1869.41, "word": " absolute", "probability": 0.951171875}, {"start": 1869.41, "end": 1869.87, "word": " values", "probability": 0.73291015625}, {"start": 1869.87, "end": 1871.09, "word": " طلع", "probability": 0.7234700520833334}, {"start": 1871.09, "end": 1871.27, "word": " أن", "probability": 0.300048828125}, {"start": 1871.27, "end": 1871.41, "word": " دي", "probability": 0.5823974609375}, {"start": 1871.41, "end": 1871.65, "word": " limit", "probability": 0.87890625}, {"start": 1871.65, "end": 1872.21, "word": " موجود", "probability": 0.9920247395833334}], "temperature": 1.0}, {"id": 83, "seek": 189430, "start": 1872.92, "end": 1894.3, "text": "ما دام اطلع ال limit موجود في عندي حكم إذا كان اللي هو أكبر من واحد diverse إذا كان أصغر من واحد converge إذا بيساوي واحد معناته the test fails وبنبحث اللي عن test آخر طيب الآن suppose that limit xn زاد واحد على xn as n goes to infinity بيساوي R أصغر من مين من واحد", "tokens": 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"word": " that", "probability": 0.923828125}, {"start": 1889.32, "end": 1889.58, "word": " limit", "probability": 0.958984375}, {"start": 1889.58, "end": 1889.92, "word": " xn", "probability": 0.719482421875}, {"start": 1889.92, "end": 1890.2, "word": " زاد", "probability": 0.5968017578125}, {"start": 1890.2, "end": 1890.44, "word": " واحد", "probability": 0.89013671875}, {"start": 1890.44, "end": 1890.6, "word": " على", "probability": 0.55029296875}, {"start": 1890.6, "end": 1890.96, "word": " xn", "probability": 0.92919921875}, {"start": 1890.96, "end": 1891.08, "word": " as", "probability": 0.406494140625}, {"start": 1891.08, "end": 1891.22, "word": " n", "probability": 0.341064453125}, {"start": 1891.22, "end": 1891.4, "word": " goes", "probability": 0.939453125}, {"start": 1891.4, "end": 1891.5, "word": " to", "probability": 0.939453125}, {"start": 1891.5, "end": 1891.7, "word": " infinity", "probability": 0.84130859375}, {"start": 1891.7, "end": 1892.1, "word": " بيساوي", "probability": 0.88203125}, {"start": 1892.1, "end": 1892.34, "word": " R", "probability": 0.671875}, {"start": 1892.34, "end": 1892.72, "word": " أصغر", "probability": 0.9921875}, {"start": 1892.72, "end": 1892.88, "word": " من", "probability": 0.9951171875}, {"start": 1892.88, "end": 1893.3, "word": " مين", "probability": 0.857177734375}, {"start": 1893.3, "end": 1893.88, "word": " من", "probability": 0.666015625}, {"start": 1893.88, "end": 1894.3, "word": " واحد", "probability": 0.9775390625}], "temperature": 1.0}, {"id": 84, "seek": 192063, "start": 1895.71, "end": 1920.63, "text": "بنفس الطريقة اللي عملناها قبل بشوية في الأول let R1 satisfies R1 أكبر من R أصغر من 1 then اللي هو لأن طبيعي between any two real numbers فيه real number سميته مين؟ R1 الآن بما أن ال limit هذا بيساوي R يدل كل إيبسلون أكبر من 0 من ضمن مين الإيبسلون اللي بدأ أخدها؟ R1-R", "tokens": [3555, 1863, 36178, 41950, 16572, 28671, 13672, 1829, 6225, 42213, 8315, 11296, 12174, 36150, 4724, 8592, 2407, 10632, 8978, 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1897.05, "end": 1897.45, "word": " عملناها", "probability": 0.9683837890625}, {"start": 1897.45, "end": 1897.69, "word": " قبل", "probability": 0.965576171875}, {"start": 1897.69, "end": 1898.27, "word": " بشوية", "probability": 0.8961181640625}, {"start": 1898.27, "end": 1898.41, "word": " في", "probability": 0.81884765625}, {"start": 1898.41, "end": 1898.91, "word": " الأول", "probability": 0.939453125}, {"start": 1898.91, "end": 1899.85, "word": " let", "probability": 0.40234375}, {"start": 1899.85, "end": 1900.55, "word": " R1", "probability": 0.495361328125}, {"start": 1900.55, "end": 1901.37, "word": " satisfies", "probability": 0.52734375}, {"start": 1901.37, "end": 1901.91, "word": " R1", "probability": 0.93798828125}, {"start": 1901.91, "end": 1902.25, "word": " أكبر", "probability": 0.8955078125}, {"start": 1902.25, "end": 1902.41, "word": " من", "probability": 0.9892578125}, {"start": 1902.41, "end": 1902.61, "word": " R", "probability": 0.93212890625}, {"start": 1902.61, 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1913.77, "word": " هذا", "probability": 0.91357421875}, {"start": 1913.77, "end": 1914.17, "word": " بيساوي", "probability": 0.804345703125}, {"start": 1914.17, "end": 1914.61, "word": " R", "probability": 0.958984375}, {"start": 1914.61, "end": 1915.33, "word": " يدل", "probability": 0.7591145833333334}, {"start": 1915.33, "end": 1915.61, "word": " كل", "probability": 0.92724609375}, {"start": 1915.61, "end": 1915.97, "word": " إيبسلون", "probability": 0.70904541015625}, {"start": 1915.97, "end": 1916.21, "word": " أكبر", "probability": 0.9656575520833334}, {"start": 1916.21, "end": 1916.37, "word": " من", "probability": 0.99609375}, {"start": 1916.37, "end": 1916.63, "word": " 0", "probability": 0.464599609375}, {"start": 1916.63, "end": 1916.87, "word": " من", "probability": 0.96630859375}, {"start": 1916.87, "end": 1917.09, "word": " ضمن", "probability": 0.9853515625}, {"start": 1917.09, "end": 1917.37, "word": " مين", "probability": 0.859619140625}, {"start": 1917.37, "end": 1917.85, "word": " الإيبسلون", "probability": 0.9593098958333334}, {"start": 1917.85, "end": 1917.95, "word": " اللي", "probability": 0.87158203125}, {"start": 1917.95, "end": 1918.15, "word": " بدأ", "probability": 0.639892578125}, {"start": 1918.15, "end": 1919.07, "word": " أخدها؟", "probability": 0.69462890625}, {"start": 1919.07, "end": 1919.77, "word": " R1", "probability": 0.91943359375}, {"start": 1919.77, "end": 1920.63, "word": "-R", "probability": 0.50531005859375}], "temperature": 1.0}, {"id": 85, "seek": 193818, "start": 1921.62, "end": 1938.18, "text": "Then for epsilon بساوة R واحد ناقص R أكبر من سفر there exists K elements in N such that ال absolute value لهذا ناقص هذا أصغر من مين من ال epsilon هي R واحد ناقص مين ناقص R هذا صحيح لكل مين لكل N أكبر أو ساوة K", "tokens": [51, 2932, 337, 17889, 4724, 3794, 995, 2407, 3660, 497, 36764, 24401, 8717, 995, 4587, 9381, 497, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 456, 8198, 591, 4959, 294, 426, 1270, 300, 2423, 8236, 2158, 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"word": " و", "probability": 0.93359375}, {"start": 2121.82, "end": 2122.16, "word": " نبحث", "probability": 0.9471028645833334}, {"start": 2122.16, "end": 2122.32, "word": " عن", "probability": 0.98779296875}, {"start": 2122.32, "end": 2122.56, "word": " test", "probability": 0.94970703125}, {"start": 2122.56, "end": 2123.02, "word": " آخر", "probability": 0.782958984375}, {"start": 2123.02, "end": 2123.56, "word": " و", "probability": 0.830078125}, {"start": 2123.56, "end": 2124.04, "word": " الآن", "probability": 0.7119140625}, {"start": 2124.04, "end": 2124.74, "word": " بنيجي", "probability": 0.9281005859375}, {"start": 2124.74, "end": 2124.98, "word": " ل", "probability": 0.796875}, {"start": 2124.98, "end": 2125.24, "word": " اللي", "probability": 0.678955078125}, {"start": 2125.24, "end": 2125.66, "word": " هو", "probability": 0.98876953125}, {"start": 2125.66, "end": 2126.42, "word": " المرة", "probability": 0.968994140625}, {"start": 2126.42, "end": 2126.76, "word": " الجاية", "probability": 0.8678385416666666}, {"start": 2126.76, "end": 2126.9, "word": " ان", "probability": 0.5703125}, {"start": 2126.9, "end": 2127.1, "word": " شاء", "probability": 0.98388671875}, {"start": 2127.1, "end": 2127.28, "word": " الله", "probability": 0.95703125}, {"start": 2127.28, "end": 2127.86, "word": " بناخد", "probability": 0.9114583333333334}, {"start": 2127.86, "end": 2127.94, "word": " ال", "probability": 0.990234375}, {"start": 2127.94, "end": 2128.34, "word": " integral", "probability": 0.62939453125}, {"start": 2128.34, "end": 2128.84, "word": " test", "probability": 0.95263671875}, {"start": 2128.84, "end": 2129.32, "word": " و", "probability": 0.92919921875}, {"start": 2129.32, "end": 2129.56, "word": " إلى", "probability": 0.875}, {"start": 2129.56, "end": 2130.08, "word": " لقاء", "probability": 0.7337239583333334}, {"start": 2130.08, "end": 2130.44, "word": " آخر", "probability": 0.929443359375}, {"start": 2130.44, "end": 2130.74, "word": " والسلام", "probability": 0.8465169270833334}, {"start": 2130.74, "end": 2131.04, "word": " عليكم", "probability": 0.8876953125}, {"start": 2131.04, "end": 2131.2, "word": " و", "probability": 0.90576171875}, {"start": 2131.2, "end": 2131.2, "word": " رحمه", "probability": 0.8108723958333334}, {"start": 2131.2, "end": 2131.4, "word": " الله", "probability": 0.974609375}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2131.6383125, "duration_after_vad": 2100.5887499999967}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0.srt
new file mode 100644
index 0000000000000000000000000000000000000000..4af3de27ca6c27f6743c3d4ac33ae08c97ffcbe8
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0.srt
@@ -0,0 +1,1671 @@
+1
+00:00:04,890 --> 00:00:08,010
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:08,010 --> 00:00:13,130
+وبركاته، بنستمر إن شاء الله اليوم مع طلاب الكيمياء
+
+3
+00:00:13,130 --> 00:00:19,310
+العامة للعلوم الصحية والهندسية، وموضوعنا اليوم في 
+
+4
+00:00:19,310 --> 00:00:30,180
+درس مناقشة للفصل الثالث عشر، الآن هنتقي مجموعة محددة 
+
+5
+00:00:30,180 --> 00:00:36,760
+من المسائل الموجودة عند الطلاب أيضًا، ونحاول أن نشرحها
+
+6
+00:00:36,760 --> 00:00:43,800
+السؤال الأول اللي بدنا نشوفه سؤال 13-15 
+
+7
+00:00:43,800 --> 00:00:49,040
+من الكتاب، بيقول: indicate the type of solute 
+
+8
+00:00:49,040 --> 00:00:54,350
+solvent interaction، يعني وضح لماذا تتم عملية 
+
+9
+00:00:54,350 --> 00:01:01,590
+الذوبان، ما هو نوع التفاعل أو الـ interaction ما 
+
+10
+00:01:01,590 --> 00:01:07,350
+بين المذاب والمذيب، الـ solute والـ solvent، هذا يجب 
+
+11
+00:01:07,350 --> 00:01:12,690
+أن يكون أهم نوع 
+
+12
+00:01:12,690 --> 00:01:15,930
+من الـ intermolecular forces بين الـ solute 
+
+13
+00:01:15,930 --> 00:01:22,260
+والـ solvent، الآن لو بدنا نأخذ الـ CCl4 اللي بنسميه 
+
+14
+00:01:22,260 --> 00:01:28,960
+carbon tetrachloride في البنزين، طبعًا واضح أن هذول 
+
+15
+00:01:28,960 --> 00:01:37,140
+الجزيئين non polar molecules، الـ CCl4 جزء بهذا
+
+16
+00:01:37,140 --> 00:01:46,080
+الشكل، وهو بالكيمياء تترا هيدرال، مجموع البولاريتي له
+
+17
+00:01:46,080 --> 00:01:53,810
+صفر، والبنزين، البنزين كيميائيًا طبعًا هو هذا البنزين 
+
+18
+00:01:53,810 --> 00:02:00,170
+الحلقة اللي فيها ستة hydrogens، C6H6، الآن هذا non
+
+19
+00:02:00,170 --> 00:02:05,470
+polar وهذا non polar، بناءً على ذلك نوع القوة
+
+20
+00:02:05,470 --> 00:02:10,570
+between non polar molecules is dispersion forces.
+
+21
+00:02:10,570 --> 00:02:19,860
+الآن لو سألنا الفرع B، أي نوع القوة بين الميثانول 
+
+22
+00:02:19,860 --> 00:02:23,260
+إذا احنا بنتطلع على الميثانول، بنلاحظ أنه فيه H
+
+23
+00:02:23,260 --> 00:02:27,880
+متصلة بـ O، إذا هو جاهز يعمل hydrogen bonding، لو
+
+24
+00:02:27,880 --> 00:02:33,180
+بدأنا، أجي أتطلع على المية طبعًا الكل حافظ شكلها H2O
+
+25
+00:02:33,180 --> 00:02:38,360
+برضه فيه H متصلة بـ O، إذا هنا وهنا حيكون في عندنا
+
+26
+00:02:38,360 --> 00:02:43,940
+hydrogen bonding ما بين الميثانول وما بين المية
+
+27
+00:02:43,940 --> 00:02:50,800
+hydrogen bonding، الآن لو طلعنا على الفرع C، KBr،
+
+28
+00:02:50,800 --> 00:02:55,420
+KBr مركب أيوني، يعني في النهاية حيصير K+ 
+
+29
+00:02:55,420 --> 00:03:00,000
+وحيصير Br−، الأيونات بدأت تنفصل عن بعض
+
+30
+00:03:00,000 --> 00:03:06,660
+الآن هذه الأيونات بدأت تحاط بجزيئات الماء بحيث أنه
+
+31
+00:03:06,660 --> 00:03:14,440
+الجزء السالب من المية بيكون قرب الجزء الموجب أو الـ
+
+32
+00:03:14,440 --> 00:03:19,980
+أيون الموجب، وبنفس الوقت لو كنا بنشوف حوالين الـ Br
+
+33
+00:03:19,980 --> 00:03:23,760
+حيكون الجزء الموجب من المية اللي هو الـ hydrogen
+
+34
+00:03:23,760 --> 00:03:27,860
+هنا، الجزء السالب من المية هو الـ oxygen، الجزء
+
+35
+00:03:27,860 --> 00:03:31,800
+الموجب هو الـ hydrogen، فاللي موجود عندي هو اللي أنا
+
+36
+00:03:31,800 --> 00:03:36,460
+بسميه أيون، هذا هو الـ ion-dipole، ليش لأنه هنا في
+
+37
+00:03:36,460 --> 00:03:40,040
+عندي أنا Partially Positive و Partially Negative
+
+38
+00:03:40,040 --> 00:03:44,940
+هذا dipole وهذا أيون، أيون-dipole، وهذا أيون برضه
+
+39
+00:03:44,940 --> 00:03:50,060
+نفس الفكرة، أيون-dipole، وأيون-dipole، فنوع الـ bonding
+
+40
+00:03:50,060 --> 00:03:59,400
+هنا عبارة عن أيون-dipole، bonding،الفرق الأخير، CH3
+
+41
+00:03:59,400 --> 00:04:06,980
+أو HCl، طبعًا الـ HCl لو احنا بدنا نتخيل شكله هذا هو 
+
+42
+00:04:06,980 --> 00:04:12,140
+شكل الـ HCl، فاللي بنلاحظه أن الـ Cl partially
+
+43
+00:04:12,140 --> 00:04:16,580
+negative والـ H partially positive، بسبب فرق
+
+44
+00:04:16,580 --> 00:04:24,620
+السالبية، هذا هو الـ HCl، نفس الشيء الـ Acetonitrile
+
+45
+00:04:24,620 --> 00:04:27,960
+لما احنا بنتطلع على CH3 
+
+46
+00:04:47,170 --> 00:04:51,150
+بناءً على ذلك، لو احنا بدنا نتخيل نوع الـ bonding 
+
+47
+00:04:51,150 --> 00:04:57,040
+حيكون ما بين هذا الطرف، اللي موجود عندنا هنا السالب
+
+48
+00:04:57,040 --> 00:05:01,840
+من الكلور، والطرف الموجب هنا بدو يصير الـ bonding 
+
+49
+00:05:01,840 --> 00:05:07,140
+فهذا عبارة عن dipole وهذا عبارة عن dipole، فالـ
+
+50
+00:05:07,140 --> 00:05:11,700
+attraction dipole-dipole attraction، dipole
+
+51
+00:05:11,700 --> 00:05:16,900
+dipole attraction، هذا 
+
+52
+00:05:16,900 --> 00:05:24,020
+بالنسبة للفرع الأول اللي بيحكي عن نوع الـ bonding، لو
+
+53
+00:05:24,020 --> 00:05:28,640
+شوفنا سؤال 13-23، بيحكينا عن الـ 
+
+54
+00:05:28,640 --> 00:05:34,140
+solubility of a chromium nitrate 9H2O in water is 
+
+55
+00:05:34,140 --> 00:05:40,360
+208 g per 100 g of water at 15 degrees
+
+56
+00:05:40,360 --> 00:05:44,760
+centigrade، طبعًا هنا في نقطة كثير مهمة أن الـ
+
+57
+00:05:44,760 --> 00:05:48,320
+solubility زي ما احنا بتعلمنا بتعتمد على درجة 
+
+58
+00:05:48,320 --> 00:05:55,390
+الحرارة، يعني كأنه عند 15 درجة مئوية اللي هي الـ
+
+59
+00:05:55,390 --> 00:06:04,630
+temperature، الـ solubility كانت عبارة عن 208
+
+60
+00:06:04,630 --> 00:06:11,850
+g، الـ solubility لكل 100 مل، 208 جرام، الآن لو زدنا 
+
+61
+00:06:11,850 --> 00:06:17,650
+الحرارة، الـ solubility بدأت تزيد، هذا اللي صار، فلما زدنا
+
+62
+00:06:17,650 --> 00:06:23,950
+الحرارة عند 35 درجة مئوية، الـ solubility زادت بدل ما 
+
+63
+00:06:23,950 --> 00:06:32,070
+كانت 208 صارت 324، بمعنى أنه في كمية إضافية مقدرة 
+
+64
+00:06:32,070 --> 00:06:40,410
+حوالي 116 جرام إضافي ذابوا، نتيجة إيش؟ نتيجة زيادة
+
+65
+00:06:40,410 --> 00:06:47,770
+درجة الحرارة، يعني مش شيء بسيط، +116 g كل
+
+66
+00:06:47,770 --> 00:06:51,430
+هذه الذائبية، ازاي ده؟ نتيجة إيش؟ زيادة درجة الحرارة
+
+67
+00:06:51,430 --> 00:06:57,270
+الآن، when this solution، هذا اللي هو على الحرارة 
+
+68
+00:06:57,270 --> 00:07:01,070
+العالية، when this solution is slowly cooled to 15
+
+69
+00:07:01,070 --> 00:07:05,710
+degree centigrade، لو عودنا بردنا، إيش المفروض اللي
+
+70
+00:07:05,710 --> 00:07:09,730
+يصير في هذه الحالة؟ المفروض اللي يصير أنه من هنا
+
+71
+00:07:09,730 --> 00:07:16,610
+يترسب -116 g، يترسب واللي ذاب نتيجة التسخين
+
+72
+00:07:16,610 --> 00:07:22,510
+يترسب نتيجة التبريد، اللي لاحظوا أنه عملنا هالكلام 
+
+73
+00:07:22,510 --> 00:07:29,050
+طبعًا للأسف الشديد ما صار الترسيب، بمعنى صار في عندي
+
+74
+00:07:29,050 --> 00:07:33,350
+بهذه الحالة صار في عندي، بهذه الحالة أنه أنا بردت لـ
+
+75
+00:07:33,350 --> 00:07:44,400
+15 درجة ومازال ذائب الـ 324 g، طبعًا هذا أكثر من
+
+76
+00:07:44,400 --> 00:07:48,300
+احتمال المحلول عند 15 درجة، لأنه المحلول عند
+
+77
+00:07:48,300 --> 00:07:52,540
+15 درجة كان بيتشبع بـ 208، وبنقول عنه
+
+78
+00:07:52,540 --> 00:07:58,500
+saturated، والمحلول عند 35 درجة بيتشبع بـ 324
+
+79
+00:07:58,500 --> 00:08:03,600
+g، وبنقول عنه saturated، بس 324 g عند 15 درجة
+
+80
+00:08:03,600 --> 00:08:08,420
+مئوية هذا بيحمل أكثر من طاقته، يعني عند 15 طاقته
+
+81
+00:08:08,920 --> 00:08:13,720
+208 g، الآن اللي موجود أنه ذائب فعلاً فيه
+
+82
+00:08:13,720 --> 00:08:17,420
+324 g، إذا لما أنا بدي أوصف
+
+83
+00:08:17,420 --> 00:08:22,120
+هذا الكلام، what term describes this solution؟ شو
+
+84
+00:08:22,120 --> 00:08:28,300
+بيكون هذا المحلول؟ هذا بنسميه احنا supersaturated
+
+85
+00:08:28,300 --> 00:08:34,280
+هذا المحلول فوق المشبع، واضح أنه احنا لما بردنا 
+
+86
+00:08:34,280 --> 00:08:40,180
+الأصل يترسب، لكن للأسف الشديد ما ترسبش، الآن بيقولنا 
+
+87
+00:08:40,180 --> 00:08:45,320
+what action؟ شو ممكن تعمل أنت عشان تخليه يترسب؟ طبعًا
+
+88
+00:08:45,320 --> 00:08:51,750
+احنا أخذنا تقنيات معينة للمحلول الـ supersaturated
+
+89
+00:08:51,750 --> 00:08:56,270
+كيف ممكن يترسب؟ أحد هذه التقنيات أنه نعمل seeding
+
+90
+00:08:56,270 --> 00:09:01,790
+بمعنى أن نجيب كمية صغيرة زي بذرة أو نوى، نجيب كمية 
+
+91
+00:09:01,790 --> 00:09:05,730
+صغيرة من المادة ونضيفها على المحلول، فبيصير يترسب
+
+92
+00:09:05,730 --> 00:09:10,530
+حول هذه الأشياء الصغيرة اللي احنا أضفناها أو 
+
+93
+00:09:10,530 --> 00:09:14,690
+البذور الصغيرة اللي احنا أضفناها، أو بنعمل 
+
+94
+00:09:14,690 --> 00:09:21,050
+scratching، بنحك جدار الـ flask فبتنفصل جزيئات زجاج
+
+95
+00:09:21,050 --> 00:09:26,050
+بسيطة جدًا، بتتم الترسيب حولها، فالتقنيات أما بدنا 
+
+96
+00:09:26,050 --> 00:09:32,230
+نعمل عشان نسرع ترسب المادة seeding، نجيب شوية صغيرة
+
+97
+00:09:32,230 --> 00:09:36,620
+منها ونحطها، أو scratching أن نحك الجدار الزجاجي 
+
+98
+00:09:36,620 --> 00:09:40,180
+scratching the glass walls of the container، عادة 
+
+99
+00:09:40,180 --> 00:09:43,900
+بنستعمل glass rod، يعني بيكون فينا قضيب من الزجاج
+
+100
+00:09:43,900 --> 00:09:49,940
+بنحك الزجاج بالزجاج، بتنفصل يعني أشياء very very
+
+101
+00:09:49,940 --> 00:09:52,920
+small، بتتم الترسب حولها
+
+102
+00:10:01,910 --> 00:10:07,510
+السؤال 29 بيقولنا: common laboratory
+
+103
+00:10:07,510 --> 00:10:11,870
+solvents، يعني من الـ solvents، المذيبات المشهورة جدًا
+
+104
+00:10:11,870 --> 00:10:17,470
+في المختبر include أسيتون منها، الأسيتون والأسيتون 
+
+105
+00:10:17,470 --> 00:10:22,310
+polar solvent، ميثانول، الميثانول أيضًا polar solvent
+
+106
+00:10:22,310 --> 00:10:26,790
+وبيعمل hydrogen bonding، التولوين، التولوين أحد مشتقات
+
+107
+00:10:26,790 --> 00:10:32,730
+البنزين اللي موجود بهذا الشكل، حلقة بنزين عليها CH3
+
+108
+00:10:32,730 --> 00:10:40,200
+هذا اللي احنا بنسميه إيه؟ تولوين، التولوين والمياه 
+
+109
+00:10:40,200 --> 00:10:44,920
+معروفة H2O، which of these is the best solvent for
+
+110
+00:10:44,920 --> 00:10:48,360
+non-polar solutes؟ يعني هو أعطاني أربعة
+
+111
+00:10:48,360 --> 00:10:52,660
+solvents، جالي لو عندي أنا solute non-polar، مين
+
+112
+00:10:52,660 --> 00:10:57,440
+بنسبته؟ طبعًا القاعدة non-polar بده non-polar، الآن الـ
+
+113
+00:10:57,440 --> 00:11:01,680
+acetone لأنه فيه C=O، يعني هذا هو شكل 
+
+114
+00:11:01,680 --> 00:11:05,540
+الـ acetone، هذا هو شكل الـ acetone
+
+115
+00:11:08,200 --> 00:11:14,340
+هنا في عندنا C=O، فالـ O طرف سالب والـ
+
+116
+00:11:14,340 --> 00:11:18,820
+C طرف موجب، إذا هذا polar، واضح الأسيتون is polar
+
+117
+00:11:20,370 --> 00:11:24,790
+والميثانول بولر بيعمل hydrogen bonding، والماء بولر
+
+118
+00:11:24,790 --> 00:11:25,350
+بيعمل hydrogen bonding، والماء بولر بيعمل hydrogen
+
+119
+00:11:25,350 --> 00:11:25,610
+bonding، والماء بولر بيعمل hydrogen bonding، والماء
+
+120
+00:11:25,610 --> 00:11:25,630
+بولر بيعمل hydrogen bonding، والماء بولر بيعمل
+
+121
+00:11:25,630 --> 00:11:25,790
+بولر بيعمل hydrogen bonding، والماء بولر بيعمل
+
+122
+00:11:25,790 --> 00:11:25,830
+hydrogen bonding، والماء بولر بيعمل hydrogen
+
+123
+00:11:25,830 --> 00:11:26,670
+bonding، والماء بولر بيعمل hydrogen bonding، والماء
+
+124
+00:11:26,670 --> 00:11:27,750
+بولر بيعمل hydrogen bonding، والماء بولر بيعمل
+
+125
+00:11:27,750 --> 00:11:29,550
+hydrogen bonding، والماء بولر بيعمل hydrogen
+
+126
+00:11:29,550 --> 00:11:33,790
+bonding، والماء بولر بيعمل hydrogen bonding، والماء
+
+127
+00:11:33,790 --> 00:11:40,690
+بولر بيعمل hydrogen bonding، والماء بولر
+
+128
+00:11:40,690 --> 00:11:44,550
+بيعمل hydrogen bonding، والماء بـ
+
+129
+00:11:53,840 --> 00:11:58,380
+السؤال 39 بيحكينا عن وحدات
+
+130
+00:11:58,380 --> 00:12:06,420
+التركيز، تذاب مجموعة صوديوم كبريتية في محلول يحتوي على 
+
+131
+00:12:06,420 --> 00:12:13,110
+10.6 جرام صوديوم كبريتية في 483 جرام ماء، يعني ذوّبنا 
+
+132
+00:12:13,110 --> 00:12:18,090
+Sodium Sulfate كمية معينة في المية، إيش الـ mass 
+
+133
+00:12:18,090 --> 00:12:22,670
+percentage؟ طبعًا الـ mass percentage احنا بنعرفها هي
+
+134
+00:12:22,670 --> 00:12:26,190
+weight على الـ total weight × 100، الآن الـ weight
+
+135
+00:12:26,190 --> 00:12:31,230
+10.6، الـ total weight هو عبارة عن مجموع هذول
+
+136
+00:12:31,230 --> 00:12:36,440
+الاثنين، الـ 10.6 والمية، هذا هو الـ total weight، لما
+
+137
+00:12:36,440 --> 00:12:41,220
+بنجمعهم بيطلع 493.6، weight على total weight ×
+
+138
+00:12:41,220 --> 00:12:46,680
+100، هذا الـ mass percent، بيطلع النسبة 2.15%، هذا 
+
+139
+00:12:46,680 --> 00:12:53,140
+تركيز المحلول في الـ mass percent، هذا الفرع A، الآن
+
+140
+00:12:53,140 --> 00:13:00,180
+الفرع B بيقول أن or، يعني خامة، لما بيطولوا المعادل 
+
+141
+00:13:00,180 --> 00:13:05,060
+من التراب ومن غيرها، بيكون عندهم خامات an ore
+
+142
+00:13:05,060 --> 00:13:12,080
+contains 2.86 g of silver per ton of ore، يعني
+
+143
+00:13:12,080 --> 00:13:19,480
+في عندنا خامة رمل موجود فيه فضة، الآن هذا الرمل في كل 
+
+144
+00:13:19,480 --> 00:13:24,440
+طن منه 2.86 g من الفضة
+
+145
+00:13:24,440 --> 00:13:28,140
+بيقول لي إيش الـ concentration للفضة بالـ part per
+
+146
+00:13:28,140 --> 00:13:32,720
+million؟ طبعًا الـ concentration حيكون الـ weight على 
+
+147
+00:13:32,720 --> 00:13:36,500
+الـ total weight، لما بده يابي الـ ppm بده اضرب في 
+
+148
+00:13:36,500 --> 00:13:41,280
+10⁶، هيك تعريف الـ ppm، فإذا قلنا الـ weight 2
+
+149
+00:13:41,280 --> 00:13:48,780
+.86 g، الآن الـ ton هنا بدنا ننتبه 1000 kg
+
+150
+00:13:48,780 --> 00:13:55,220
+1000 kg، فلما أنا بدي أقول الـ ton بذاته 1000، و
+
+151
+00:13:55,220 --> 00:14:02,140
+الـ kilogram نفسه 1000 g، بناءً على ذلك لما أنا بدي
+
+152
+00:14:02,140 --> 00:14:05,860
+أقول الـ ton a chemical عبارة عن 1000 × 1000 يعني
+
+153
+00:14:05,860 --> 00:14:14,520
+10⁶، واضح، فلما بدي احسب الـ mass
+
+154
+00:14:14,520 --> 00:14:22,490
+percent في هذه الحالة بدي أقول الـ weight على الـ
+
+155
+00:14:22,490 --> 00:14:27,710
+total weight × 10⁶، طبعًا الـ total
+
+156
+00:14:27,710 --> 00:14:34,070
+weight الحقيقة هو عبارة عن الـ 1000000 g اللي هم 
+
+157
+00:14:34,070 --> 00:14:39,530
+الـ ton، أو what is the concentration of the silver؟ لأ 
+
+158
+00:14:39,530 --> 00:14:44,510
+هو الـ ore بتحتوي على هذا الوزن، بناءً على ذلك يعني
+
+159
+00:14:44,510 --> 00:14:49,830
+داخل فيها، يعني الخامة نفسها، الوزن الكل، الطن هذا من
+
+160
+00:14:49,830 --> 00:14:56,390
+ضمنه 2.86 g، إذا بناخدها زي ما هي، 2.86، الـ
+
+161
+00:14:56,390 --> 00:15:01,010
+total weight 10⁶، طبعًا عشان بده ياها
+
+162
+00:15:01,010 --> 00:15:06,980
+بالـ ppm، بده اضرب في 10⁶، واضح؟ ولما نحسب
+
+163
+00:15:06,980 --> 00:15:10,900
+هذا الكلام، weight على الـ total weight × 10⁶ 
+
+164
+00:15:10,900 --> 00:15:17,620
+عشان نخليه بالـ parts per million، بيطلع 2.86، واضح؟
+
+165
+00:15:17,620 --> 00:15:27,980
+هذا هو التركيز في وحدة الـ part per million، المسائل
+
+166
+00:15:27,980 --> 00:15:33,140
+التالية برضه على التركيز، سؤال 41، الـ
+
+167
+00:15:33,140 --> 00:15:36,820
+solution is made containing 14.6
+
+168
+00:15:36,820 --> 00:15:44,380
+g of water in 184 g، 14.6 g
+
+169
+00:15:44,380 --> 00:15:51,520
+ميثانول، و184 g ميثانول، الـ mole fraction الميثانول، 
+
+170
+00:15:51,520 --> 00:15:55,880
+طبعًا الـ mole fraction بيساوي الـ number of moles of
+
+171
+00:15:55,880 --> 00:16:00,000
+methanol على الـ total moles، الآن أنا عندي الوزن،
+
+201
+00:18:32,550 --> 00:18:36,810
+الـ liter لو أنا بدي أقبل وزن من خلال علاقة الكثافة
+
+202
+00:18:36,810 --> 00:18:40,930
+الـ density بتسوّيها على الـ volume بقول إن وزن الـ liter
+
+203
+00:18:40,930 --> 00:18:50,270
+1329 gram هذا وزن الـ liter واضح؟ ضربت بالـ 1000 الآن
+
+204
+00:18:50,270 --> 00:18:53,350
+بيقول الـ calculate the mass percent الـ mass
+
+205
+00:18:53,350 --> 00:18:58,130
+percent هو وزن حمض الكبريتيك على وزن الميّة الآن if
+
+206
+00:18:58,130 --> 00:19:07,230
+the total mass التوتال هو وزن كل المحلول 1329 ووزن
+
+207
+00:19:07,230 --> 00:19:16,450
+الـ H2SO4 لحاله كان 571.6 إذن الفرق بينهم هو وزن
+
+208
+00:19:16,450 --> 00:19:21,210
+الماء واضح؟ وعشان ذلك نستطيع أن نحصل على مولز مياه
+
+209
+00:19:21,210 --> 00:19:25,510
+الماء الـ Total Mass minus Mass of Sulphuric Acid
+
+210
+00:19:25,510 --> 00:19:32,780
+757.4 الآن لو أنا عندي وزن حمض الكبريتيك بقسم على
+
+211
+00:19:32,780 --> 00:19:36,860
+الـ molecular weight بطلع الـ number of moles وزي ما
+
+212
+00:19:36,860 --> 00:19:41,880
+قلت الـ mass الـ بتساوي 1329 من الكثافة لو بدي الـ
+
+213
+00:19:41,880 --> 00:19:46,210
+mass of water هي الفرق ما بين الوزن الكل ووزن حمض
+
+214
+00:19:46,210 --> 00:19:53,610
+الكبريتيك 757 لو بدي الـ moles of water 757.4 على 18
+
+215
+00:19:53,610 --> 00:19:57,090
+على الـ molecular weight صار عندي الـ mass of water
+
+216
+00:19:57,090 --> 00:20:00,670
+و الـ moles of water الـ mass of sulfuric acid و الـ
+
+217
+00:20:00,670 --> 00:20:04,850
+moles of sulfuric acid الآن ايش بدّه أسئلة بكون سهل
+
+218
+00:20:04,850 --> 00:20:10,450
+عليّ أنا أجاوبه؟ ماذا هو الـ Mass Percent؟ الـ Mass
+
+219
+00:20:10,450 --> 00:20:13,430
+Percent هي عبارة عن الـ Mass of Sulphuric Acid على
+
+220
+00:20:13,430 --> 00:20:19,880
+الـ Total Mass ضرب 100 بتطلع 43.01 What is the mole
+
+221
+00:20:19,880 --> 00:20:23,180
+fraction of sulphuric acid؟ هو عبارة عن الـ moles of
+
+222
+00:20:23,180 --> 00:20:27,320
+sulphuric acid على الـ total moles أنا هنا عندي الـ
+
+223
+00:20:27,320 --> 00:20:31,260
+moles of sulphuric acid حسبتها والـ moles of water
+
+224
+00:20:31,260 --> 00:20:37,300
+حسبتها الـ moles على الـ total moles ضرب بدون ما نضرب
+
+225
+00:20:37,300 --> 00:20:41,480
+بـ 100 لإنّه بدّه fraction يعني بدون ما تنضرب بـ 100 moles
+
+226
+00:20:41,480 --> 00:20:45,760
+على الـ total moles هاي الـ mole fraction بنوجده
+
+227
+00:20:46,520 --> 00:20:52,040
+الفرع C المولاليّة المولاليّة الـ number of moles اللي
+
+228
+00:20:52,040 --> 00:20:57,520
+أنا برضه أوجدتها للسلفريك 5.83 على الـ mass of
+
+229
+00:20:57,520 --> 00:21:01,380
+solvent in kilogram هاي الـ mass of water اللي
+
+230
+00:21:01,380 --> 00:21:08,020
+حسبناها بالغرام لما ندّق بالـ kilogram 0.7574 فبتطلع
+
+231
+00:21:08,020 --> 00:21:15,060
+المولاليّة 7.69 مولال الآن لو بدّوا الـ molarity الفرع
+
+232
+00:21:15,060 --> 00:21:21,140
+دي الـ number of moles على الـ total volume هو أساسًا
+
+233
+00:21:21,140 --> 00:21:25,640
+الـ total volume بيقول 1 لتر يعني موجود في 1 لتر هي
+
+234
+00:21:25,640 --> 00:21:28,040
+الـ number of moles of sulfuric acid و الـ total
+
+235
+00:21:28,040 --> 00:21:34,180
+volume is one liter فالمولاريّة 5.83 مولار
+
+236
+00:21:47,440 --> 00:22:15,480
+سؤال 65 سؤال
+
+237
+00:22:15,480 --> 00:22:16,040
+65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+238
+00:22:16,040 --> 00:22:16,060
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+239
+00:22:16,090 --> 00:22:21,670
+دوبته بالميّة دوبته 22.5 جرام of lactose في 200
+
+240
+00:22:21,670 --> 00:22:28,590
+جرام ميّة عند درجة حرارة 338 kelvin الآن طبعًا
+
+241
+00:22:28,590 --> 00:22:32,430
+هيلزمنا الـ vapor pressure لـ pure water يعني كأنّه
+
+242
+00:22:32,430 --> 00:22:40,170
+هذه المسألة إن أنا لو كان في عندي ميّة عاديّة water
+
+243
+00:22:40,170 --> 00:22:44,850
+لحالها في عندي اللي بنسميه الـ pure vapor pressure
+
+244
+00:22:47,180 --> 00:22:48,900
+Pure Vapor Pressure
+
+245
+00:22:55,870 --> 00:23:00,990
+الآن هذا الضغط موجود في جداول .. هذا الضغط موجود
+
+246
+00:23:00,990 --> 00:23:05,770
+في جداول لما إحنا بدنا نشوف ضغط الماء عند ضغط بخار
+
+247
+00:23:05,770 --> 00:23:14,090
+الماء بنلاقيه من الجدول 187.5 تور واضح؟ طيّب هذا
+
+248
+00:23:14,090 --> 00:23:18,370
+ضغط الماء لما يكون نقيّ ..الآن هذا الماء لو إحنا
+
+249
+00:23:18,370 --> 00:23:25,090
+اجينا ودوبنا فيه .. ودوبنا فيه لكتوز طبعًا اللاكتوز
+
+250
+00:23:25,090 --> 00:23:33,150
+يقلّل ضغط البخار لإنّه صار جزء على سطح الماء
+
+251
+00:23:33,150 --> 00:23:38,350
+اللاكتوز يمنع التبخر بناءً على ذلك كمية التبخر في
+
+252
+00:23:38,350 --> 00:23:45,050
+وجود اللاكتوز تقلّ واخذنا معادلة كيف نحسب الـ vapor
+
+253
+00:23:45,050 --> 00:23:46,010
+pressure
+
+254
+00:23:54,440 --> 00:23:58,460
+الآن المعلومات اللي موجودة عندنا اللاكتوز إلى
+
+255
+00:23:58,460 --> 00:24:01,580
+molecular weight 342 و بناءً
+
+256
+00:24:01,580 --> 00:24:05,520
+على ذلك لما بدّي أنا الـ number of moles اللي 22.5 على 342 point oh
+
+257
+00:24:05,520 --> 00:24:08,600
+six six mole
+
+258
+00:24:08,600 --> 00:24:09,840
+six
+
+259
+00:24:16,930 --> 00:24:22,630
+بناءً على ذلك عشان أحسب الـ vapor pressure للمحلول
+
+260
+00:24:22,630 --> 00:24:27,190
+بدّي الـ mole fraction للميّه اللي هي الـ number of moles of
+
+261
+00:24:27,190 --> 00:24:32,390
+water على الـ total number of moles بتطلع point nine
+
+262
+00:24:32,390 --> 00:24:36,490
+nine four الآن إذا كنت أنا عارف الـ pressure بدّي
+
+263
+00:24:36,490 --> 00:24:41,630
+أضرب الـ mole fraction في الـ vapor pressure لـ pure
+
+264
+00:24:41,630 --> 00:24:44,730
+water point nine nine four في 187.5
+
+265
+00:24:44,730 --> 00:24:51,150
+الجواب 186.4 لاحظوا إنّ الـ vapor pressure
+
+266
+00:24:51,150 --> 00:24:56,110
+بسبب إذابة اللاكتوز قلّ يعني لما بدّي أقارن بدون
+
+267
+00:24:56,110 --> 00:25:01,310
+لاكتوز كان الـ vapor pressure 187.5 بوجود اللاكتوز
+
+268
+00:25:01,310 --> 00:25:09,750
+بيقلّ صار 186.4 هذا بالنسبة للفرع A
+
+269
+00:25:12,130 --> 00:25:18,090
+الفرع بيه فرع على محلول ثاني غير اللاكتوز والميّة
+
+270
+00:25:18,090 --> 00:25:24,890
+اللي هو اسمه البروبيلين غلايكول والبروبيلين
+
+271
+00:25:24,890 --> 00:25:30,650
+غلايكول برضه هو عبارة عن مركب بيكون عادة non
+
+272
+00:25:30,650 --> 00:25:37,040
+volatile يعني غير متطاير مركب غير متطاير هذا
+
+273
+00:25:37,040 --> 00:25:43,280
+البروبيلين جليكول صيغته العامة C3H8O3
+
+274
+00:25:43,280 --> 00:25:50,060
+وصيغته بهذا الشكل التفصيلي للناس اللي بيهتمّوا هنا
+
+275
+00:25:50,060 --> 00:26:01,120
+في عندي OH وهنا في عندي OH عشان كان C3H8O2
+
+276
+00:26:01,120 --> 00:26:08,110
+هذا شكل البروبيلين غلايكول الآن بقول احسب كتلة
+
+277
+00:26:08,110 --> 00:26:13,330
+البروبيلين غلايكول اللي يجب أن يتمّ إضافة لـ 0.340
+
+278
+00:26:13,330 --> 00:26:22,010
+كيلو غرام ميّة لتقليل البريشر الفيبر بـ 2.88 تور في
+
+279
+00:26:22,010 --> 00:26:29,090
+40 درجة مئوية واضح؟ طبعًا لو بدنا نبدأ نشوف
+
+280
+00:26:29,090 --> 00:26:33,430
+المعطيات اللي عندنا الـ molecular weight لهذا
+
+281
+00:26:33,430 --> 00:26:38,650
+المركب اللي هو البروبيلين غلايكول بنحسبه 76 الـ
+
+282
+00:26:38,650 --> 00:26:43,230
+pressure للميّة عند 40 degree centigrade 55.3 الآن هو بيقول الـ pressure هذا بينخفض
+
+283
+00:26:43,230 --> 00:26:48,210
+لما أنا بدوّب شوية من البروبيلين غلايكول في الميّة
+
+284
+00:26:48,210 --> 00:26:52,870
+بينخفض الـ pressure يعني الـ new pressure بيصير عبارة
+
+285
+00:26:52,870 --> 00:26:58,900
+عن الـ 55.3 مطروح منها الـ 2.88 اللي هو مقدار
+
+286
+00:26:58,900 --> 00:27:04,200
+الانخفاض بيصير 52.42 الآن إحنا كلنا عارفين إنّ الـ
+
+287
+00:27:04,200 --> 00:27:12,620
+pressure بيساوي الـ X في الـ P node اللي هو الـ
+
+288
+00:27:12,620 --> 00:27:16,520
+pressure لـ الـ pure water الآن هذا الـ pressure صار
+
+289
+00:27:16,520 --> 00:27:20,340
+موجود عندي 52.42 الـ X الـ mole fraction مجهولة و الـ
+
+290
+00:27:20,340 --> 00:27:27,530
+pressure الأصلي 55.3 pressure of pure water بناءً
+
+291
+00:27:27,530 --> 00:27:32,170
+على ذلك أنا بقدر أحسب قيمة الـ X ايش الـ mole
+
+292
+00:27:32,170 --> 00:27:36,710
+fraction للميّة الآن لما أنا بدّي آجي أحسب ايش هو الـ
+
+293
+00:27:36,710 --> 00:27:43,970
+mole fraction للميّة أنا عندي الـ number of moles of
+
+294
+00:27:43,970 --> 00:27:46,910
+water هذا الـ mole fraction للميّة هو عبارة عن
+
+295
+00:27:46,910 --> 00:27:51,190
+الـ number of moles of water على الـ total moles طبعًا
+
+296
+00:27:51,190 --> 00:27:55,550
+الـ number of moles of water طالما الميّة 340
+
+297
+00:27:56,210 --> 00:27:59,310
+وأربعين بقسمها على 18 هاي الـ number of moles
+
+298
+00:27:59,310 --> 00:28:02,410
+of water الآن طيّب الـ number of moles of propylene
+
+299
+00:28:02,410 --> 00:28:07,970
+glycol أنا ما عنديش كم مول of propylene glycol إذا
+
+300
+00:28:07,970 --> 00:28:12,110
+صار المجهول اللي عندي هنا هو الـ number of moles of
+
+301
+00:28:12,110 --> 00:28:16,550
+propylene glycol هاي الـ moles of water على الـ
+
+302
+00:28:16,550 --> 00:28:20,970
+total moles اللي هي عبارة عن moles of water زائد
+
+303
+00:28:20,970 --> 00:28:23,810
+moles of propylene glycol فبحسب الـ y اللي هي الـ
+
+304
+00:28:23,810 --> 00:28:27,810
+moles of البروبيلين غلايكول بتطلع الـ y 1.04 mole
+
+305
+00:28:27,810 --> 00:28:34,150
+هو الآن مش طالب الـ number of moles هو الآن مش طالب
+
+306
+00:28:34,150 --> 00:28:38,830
+الـ number of moles للبروبيلين غلايكول طالب الـ mass
+
+307
+00:28:38,830 --> 00:28:44,110
+وأنا بعرف إنّه عندي علاقة إنّ الـ number of moles بيساوي
+
+308
+00:28:44,110 --> 00:28:49,970
+weight على الـ molecular weight فلما أنا يكون عندي الـ
+
+309
+00:28:49,970 --> 00:28:54,720
+number of moles صرت حاسبه للبروبيلين جلايكول و الـ
+
+310
+00:28:54,720 --> 00:29:00,260
+molecular weight بعرفه من الـ formula و لو حسبته 76
+
+311
+00:29:00,260 --> 00:29:03,820
+شو بيظلّ عندي الـ weight الـ weight بيصير بيساوي الـ
+
+312
+00:29:03,820 --> 00:29:07,840
+number of moles ضرب الـ molecular weight هاي هذا
+
+313
+00:29:07,840 --> 00:29:12,320
+اللي عملناه إنّه الـ mass فعلاً لـ C3H8O2 بيساوي الـ
+
+314
+00:29:12,320 --> 00:29:17,980
+number of moles 1.04 في الـ molecular weight 76
+
+315
+00:29:17,980 --> 00:29:22,980
+بيطلع وزن للبروبيلين جليكول 79.0 جرام
+
+317
+00:29:47,280 --> 00:29:57,650
+ناخذ سؤال 75 سؤال 75 بيحكي عن انخفاض في نقطة التجمد
+
+318
+00:29:57,650 --> 00:30:01,770
+يعني الماء لما بنضيف عليه non volatile solute
+
+319
+00:30:01,770 --> 00:30:05,990
+اتفقنا إنّه نقطة تجمده بتنخفض يعني الماء في الوضع
+
+320
+00:30:05,990 --> 00:30:12,190
+الطبيعي نقطة تجمده صفر بتنخفض بتصير بالـ minus الآن
+
+321
+00:30:12,190 --> 00:30:16,110
+بيجي بيسألنا السؤال how many grams of ethylene
+
+322
+00:30:16,110 --> 00:30:21,930
+glycol ومعطيني شكله C2H6O2 وبرضه الـ Ethylene
+
+323
+00:30:21,930 --> 00:30:25,110
+Glycol اللي اللي بيحبّوا يعرفوا شكله هو عبارة عن
+
+324
+00:30:25,110 --> 00:30:29,130
+CH2CH2OHOH
+
+325
+00:30:29,130 --> 00:30:35,030
+هذا اسمه Ethylene Glycol الآن كم جرام من الإيثيلين
+
+326
+00:30:35,030 --> 00:30:39,930
+جلايكول يجب أن يتمّ إضافة إلى 1 كيلو غرام من
+
+327
+00:30:39,930 --> 00:30:46,110
+الماء لإنتاج حلوى تتجمّد في نصف درجة مئوية طبعًا
+
+328
+00:30:46,110 --> 00:30:51,890
+الأصل إنّه إحنا بنعرف إنّ الميّة عادة الوضع الطبيعي
+
+329
+00:30:51,890 --> 00:30:56,690
+تبعها بتتجمد في صفر درجة مئوية فلما تصير إنّها ناقص
+
+330
+00:30:56,690 --> 00:31:04,180
+5 معنا ذلك دلتا تي بتساوي 5 درجات يعني كم انخفضت
+
+331
+00:31:04,180 --> 00:31:08,760
+نقطة تجمد الميّة من صفر لـ ناقص 5 انخفضت 5 درجات الآن
+
+332
+00:31:08,760 --> 00:31:13,420
+بقدر الانخفاض في نقطة التجمد بيساوي ثابت Kf في الـ
+
+333
+00:31:13,420 --> 00:31:18,080
+molality الآن الميّة الثابت تبعها موجود في جداول
+
+334
+00:31:18,080 --> 00:31:25,220
+وموجود عندنا في الكتاب 1.86 دلتا تي بتساوي الـ 5
+
+335
+00:31:25,220 --> 00:31:27,840
+بتساوي 1.86 في الـ morality الـ
+
+336
+00:31:27,840 --> 00:31:31,580
+molality بتساوي 2.69 الآن الـ
+
+337
+00:31:31,580 --> 00:31:35,260
+molality بتساوي الـ number of moles على الـ mass of
+
+338
+00:31:35,260 --> 00:31:40,560
+solvent أنا الـ mass of solvent عندي واضح؟ 1
+
+339
+00:31:40,560 --> 00:31:45,460
+kilogram of water الـ mass of solvent وبناءً على ذلك
+
+340
+00:31:45,460 --> 00:31:49,420
+لما أنا بدّي أقول الـ morality بتساوي الـ number of
+
+341
+00:31:49,420 --> 00:31:54,580
+moles على الـ kilogram ممكن كمان بالمرة أنا أقول إنّ
+
+342
+00:31:54,580 --> 00:31:59,040
+الـ number of moles نفسها لـ الإيثالين جلايكول بتسوّي
+
+343
+00:31:59,040 --> 00:32:01,180
+weight على الـ molecular weight يعني هذا الـ weight على
+
+344
+00:32:01,180 --> 00:32:08,860
+الـ molecular weight لمن؟ للإيثالين جلايكول واضح؟
+
+345
+00:32:10,730 --> 00:32:12,210
+مثلًا الـ weight على الـ molecular weight
+
+346
+00:32:12,210 --> 00:32:17,370
+للإيثالين جلايكول على الـ mass of solvent in
+
+347
+00:32:17,370 --> 00:32:22,510
+kilogram لو قلت الآن إنّ الـ molality end معروفة و
+
+348
+00:32:22,510 --> 00:32:25,710
+الـ molecular weight للإيثالين جلايكول معروف و الـ
+
+349
+00:32:25,710 --> 00:32:30,910
+mass of solvent الميّة معروفة 1 kilogram ما فيش
+
+350
+00:32:30,910 --> 00:32:35,560
+الحقيقة مجهول غير الـ weight وبناءً على ذلك بقدر أحسب
+
+351
+00:32:35,560 --> 00:32:43,980
+الـ weight 166.7 دمجنا عمليتين لو أي طالب ما بدّوش
+
+352
+00:32:43,980 --> 00:32:49,560
+هذا الكلام ممكن يجي يقول الـ morality بتساوي moles
+
+353
+00:32:49,560 --> 00:32:55,200
+على الـ mass of solvent in kilogram وبالتالي الـ
+
+354
+00:32:55,200 --> 00:33:00,520
+morality عنده موجودة بيعوّض و الـ mass موجودة بيحسب
+
+355
+00:33:00,520 --> 00:33:04,840
+الـ number of moles وبعد ما يحسب الـ number of moles
+
+356
+00:33:04,840 --> 00:33:08,460
+بيقول هذا الـ number of moles بتسوّي على الـ molecular
+
+357
+00:33:08,460 --> 00:33:12,380
+weight يعني ممكن الخطوتين يدخلهم في معادلة واحدة و
+
+358
+00:33:12,380 --> 00:33:15,860
+ممكن يفصلّهم يعني أي طالب ممكن يقول الـ morality
+
+359
+00:33:15,860 --> 00:33:21,760
+بتساوي moles على الـ mass الـ morality موجودة و الـ
+
+360
+00:33:21,760 --> 00:33:26,900
+mass موجودة طبعًا الـ molality إحنا حسبناها من نقطة
+
+361
+00:33:26,900 --> 00:33:31,500
+التجمد و الـ mass of solvent موجودة هذه الـ mass هي
+
+362
+00:33:31,500 --> 00:33:34,600
+الـ mass of solvent in kilogram بيحسب الـ number of
+
+363
+00:33:34,600 --> 00:33:37,940
+moles الـ number of moles of ethylene glycol بتساوي
+
+364
+00:33:37,940 --> 00:33:41,040
+weight على الـ molecular weight الـ number of moles صار
+
+365
+00:33:41,040 --> 00:33:44,820
+موجود الـ molecular weight موجود بيقدر يحسب الـ
+
+366
+00:33:44,820 --> 00:33:48,940
+weight الآن إحنا هنا في نفس المعادلة عوّضنا عن
+
+367
+00:33:48,940 --> 00:33:52,010
+الـ number of moles بـ weight على الـ molecular weight عملناهم
+
+368
+00:33:52,010 --> 00:33:59,510
+خطوة واحدة بأي طريقة الطالب بحلّ بيصل لإيه؟ الجواب
+
+369
+00:33:59,510 --> 00:34:07,110
+سؤال 78 بيقول لنا sea water يعني ميّ البحر
+
+370
+00:34:07,110 --> 00:34:12,210
+contains
+
+401
+00:36:26,620 --> 00:36:32,030
+اضربه في اثنين وعشان هيك الـ final osmotic pressure
+
+402
+00:36:32,030 --> 00:36:39,610
+in fact is 2.80 atmosphere لو كان المركب non-ionic
+
+403
+00:36:39,610 --> 00:36:42,810
+بخليه زي ما هو الـ osmotic pressure اللي طلع عندي
+
+404
+00:36:42,810 --> 00:36:46,490
+لكن لو كان بيقسم لـ أيونين بدي اضربه في اثنين لو
+
+405
+00:36:46,490 --> 00:36:51,170
+كان بيقسم لـ ثلاثة مثلا بدي اضربه في ثلاثة وهكذا
+
+406
+00:37:01,450 --> 00:37:06,930
+نشوف سؤال عن موضوع الـ Osmotic Pressure What is 
+
+407
+00:37:06,930 --> 00:37:12,390
+the osmotic pressure formed by dissolving 44.2
+
+408
+00:37:12,390 --> 00:37:18,390
+milligram of aspirin in 0.358 liter of water؟ طبعاً
+
+409
+00:37:18,390 --> 00:37:21,810
+واضح أنه دي مسألة بسيطة احنا عارفين الـ Osmotic
+
+410
+00:37:21,810 --> 00:37:25,450
+Pressure باي بتساوي Molarity في R في T إذا لازم
+
+411
+00:37:25,450 --> 00:37:29,710
+المولاريتي لما بيعطيني وزن الأسبرين اللي دوّبته 
+
+412
+00:37:29,710 --> 00:37:35,880
+في المية 44.2 ميلي جرام بقدر أحسب الـ number of moles في
+
+413
+00:37:35,880 --> 00:37:40,440
+الأسبرين الأسبرين صيغته العامة C9H8O4
+
+414
+00:37:40,440 --> 00:37:51,340
+الـ molecular weight 9×12 8×1 4×16 180 موليكيلار
+
+415
+00:37:51,340 --> 00:37:51,720
+ويت
+
+416
+00:37:56,390 --> 00:38:00,090
+بنفعش نقسم milligram على gram الـ molecular weight
+
+417
+00:38:00,090 --> 00:38:05,170
+جرام لكل مول بناءً على ذلك بقدر أحول milligram إلى 
+
+418
+00:38:05,170 --> 00:38:09,210
+gram من خلال أنه اضرب في عشرة أس ناقص ثلاثة اللي
+
+419
+00:38:09,210 --> 00:38:14,310
+هو معامل التحويل بين milligram و gram صارت هذه
+
+420
+00:38:14,310 --> 00:38:19,500
+gram وصارت هذه gram لكل مول بنشطب لـ gram مع الـ gram
+
+421
+00:38:19,500 --> 00:38:28,120
+بيطلع الجواب 2.46×10-4 مول الآن number of moles 
+
+422
+00:38:28,120 --> 00:38:33,480
+على الـ total volume هو الـ molarity بنقسم 2.46×10-4
+
+423
+00:38:33,480 --> 00:38:40,020
+على الـ total volume بتطلع الـ molarity 6.86×10-4
+
+424
+00:38:40,020 --> 00:38:44,520
+مولار لما بدي الـ Pi هي عبارة عن الـ molarity اللي
+
+425
+00:38:44,520 --> 00:38:50,250
+أنا جيبتها الآن في الـ R 0.0821 في الـ temperature in
+
+426
+00:38:50,250 --> 00:38:55,150
+Kelvin بتطلع الـ R بتطلع الـ Pi اللي هي الـ Osmotic
+
+427
+00:38:55,150 --> 00:39:01,610
+pressure in this case in atmosphere 0.0168 فلو كان 
+
+428
+00:39:01,610 --> 00:39:06,190
+طالب مني أن أوجد هذه القيمة بالـ torr اللي هي 
+
+429
+00:39:06,190 --> 00:39:12,420
+الـ millimeter زئبق بدي اضرب في 760 واضح؟ عشان أحول
+
+430
+00:39:12,420 --> 00:39:20,260
+من atmosphere إلى torr بدي اضرب بـ 760 12.7 torr
+
+431
+00:39:20,260 --> 00:39:24,580
+الـ osmotic pressure لهذا المحلول من الأسبرين في
+
+432
+00:39:24,580 --> 00:39:25,520
+الماء
+
+433
+00:39:29,700 --> 00:39:36,040
+آخر سؤال نحله على موضوع الـ Osmotic Pressure
+
+434
+00:39:36,040 --> 00:39:42,260
+بيحكينا عن Lysosome is an enzyme هذا إنزيم اسمه
+
+435
+00:39:42,260 --> 00:39:47,140
+Lysosome that breaks bacterial cell walls بيكسر
+
+436
+00:39:47,140 --> 00:39:53,480
+جدران اللي هي خلية البكتيريا solution containing 0
+
+437
+00:39:53,480 --> 00:39:59,140
+.150 gram of this enzyme يعني في عندي محلول دوبت
+
+438
+00:39:59,140 --> 00:40:03,500
+أنا 0.150 gram of this enzyme in two
+
+439
+00:40:03,500 --> 00:40:10,340
+hundred ten milliliter of solution في حجم نهائي 
+
+440
+00:40:10,340 --> 00:40:15,760
+كان 210 mil has an osmotic pressure of
+
+441
+00:40:15,760 --> 00:40:17,380
+0.953
+
+442
+00:40:24,840 --> 00:40:33,160
+طبعاً واضح هنا أنه بيستعمل المعادلة تبعت الـ osmotic
+
+443
+00:40:33,160 --> 00:40:38,400
+pressure Pi بتساوي molecular weight عفواً Pi بتساوي 
+
+444
+00:40:38,400 --> 00:40:44,330
+molarity في R في T اللي هو الـ Osmotic Pressure
+
+445
+00:40:44,330 --> 00:40:48,810
+معروفة لكن رجاءً بدنا ننتبه أن هذا الـ Osmotic
+
+446
+00:40:48,810 --> 00:40:52,570
+Pressure بتور الـ Osmotic Pressure بهذه المعادلة
+
+447
+00:40:52,570 --> 00:40:56,770
+لأن الـ R فيها أتمسفير لازم يكون بالأتمسفير بناءً
+
+448
+00:40:56,770 --> 00:41:00,510
+على ذلك الـ Osmotic Pressure لازم نحوله لـ أتمسفير
+
+449
+00:41:00,510 --> 00:41:04,330
+ابتداءً بدنا نقول 0.953 نقسم على 
+
+450
+00:41:04,330 --> 00:41:07,730
+760 1.25 في عشر أقص ناقص
+
+451
+00:41:07,730 --> 00:41:14,240
+ثلاثة أتمسفير فالـ Pi بهذه الحالة لو عوضنا فيها بعد
+
+452
+00:41:14,240 --> 00:41:20,340
+القيمة المولاريتي مجهولة الـ R معلومة 0.0821
+
+453
+00:41:20,340 --> 00:41:25,000
+أتموسفيل لتر لكل مول كيلوفر الـ temperature معلومة
+
+454
+00:41:25,000 --> 00:41:31,960
+298 كيلوفر بنطلع المولاريتي اللي هي 5.12 في 10 أس
+
+455
+00:41:31,960 --> 00:41:38,000
+ناقص 5 مولار الآن المولاريتي بتساوي number of 
+
+456
+00:41:38,000 --> 00:41:42,680
+moles على الـ volume و الـ number of moles اللي موجودة
+
+457
+00:41:42,680 --> 00:41:46,720
+في معادلة الـ molarity بتساوي weight على molecular
+
+458
+00:41:46,720 --> 00:41:52,160
+weight الآن كأنه صارت الـ molarity بتساوي weight
+
+459
+00:41:52,160 --> 00:41:55,680
+على molecular weight كل هذا مجسوم على الـ volume
+
+460
+00:41:55,680 --> 00:42:00,340
+طيب إذا احنا صار في عندنا الـ molarity معروفة يعني
+
+461
+00:42:00,340 --> 00:42:07,430
+نعوضنا عنها الـ weight اللي احنا أخدناه معروف الـ
+
+462
+00:42:07,430 --> 00:42:11,730
+Volume معروف، لم يبقى لدي مجهول غير الـ Molecular
+
+463
+00:42:11,730 --> 00:42:16,010
+Weight بقدر أحسب الـ Molecular Weight وهذه الحقيقة
+
+464
+00:42:16,010 --> 00:42:19,910
+هي طريقة لحساب الـ Molecular Weight لجزيئات
+
+465
+00:42:19,910 --> 00:42:23,090
+الكبيرة الـ Macromolecules زي الإنزايم وغيرها
+
+466
+00:42:23,090 --> 00:42:27,860
+باستعمال الـ Osmotic Pressure فاحنا في البداية
+
+467
+00:42:27,860 --> 00:42:31,260
+أوجدنا الـ molarity من معادلة Pi بتساوي molarity في
+
+468
+00:42:31,260 --> 00:42:36,880
+R في T و بنعرف احنا أن الـ molarity هي عبارة عن
+
+469
+00:42:36,880 --> 00:42:39,760
+moles على volume أو weight على molecular weight هذا
+
+470
+00:42:39,760 --> 00:42:42,720
+الكلام كله هو number of moles weight على molecular
+
+471
+00:42:42,720 --> 00:42:48,770
+weight كله مقسوم على volume و بنقدر نحسب قيمة الـ
+
+472
+00:42:48,770 --> 00:42:52,070
+molecular weight لو أي حد بده ييجي يقول والله الـ
+
+473
+00:42:52,070 --> 00:42:55,450
+molarity moles على volume ويحسب الـ moles بشكل
+
+474
+00:42:55,450 --> 00:42:59,190
+منفصل ويطلع قيمتها وبعدين ييجي يقول الـ moles
+
+475
+00:42:59,190 --> 00:43:02,450
+بتساوي weight على molecular weight يعني يروح حاله 
+
+476
+00:43:02,450 --> 00:43:05,250
+بتالي ما فيش أي مشكلة
+
+477
+00:43:13,260 --> 00:43:18,440
+بهيك بنكون انتهينا من مناقشة مسائل الفصل الثالث
+
+478
+00:43:18,440 --> 00:43:22,580
+عشرة وإلى لقاء آخر السلام عليكم ورحمة الله وبركاته 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..7d1b04c0cedc43680423ee0c3c992cbf46ceca4a
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_postprocess.srt
@@ -0,0 +1,1912 @@
+1
+00:00:04,890 --> 00:00:08,010
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:08,010 --> 00:00:13,130
+وبركاته بنستمر ان شاء الله اليوم مع طلاب الكيمياق
+
+3
+00:00:13,130 --> 00:00:19,310
+العامة للعلوم الصحية والهندسية وموضوعنا اليوم في
+
+4
+00:00:19,310 --> 00:00:30,180
+درس مناقشة للفصل الثالث عشرالان هنتقي مجموعة محددة
+
+5
+00:00:30,180 --> 00:00:36,760
+من المسائل موجودة عند الطلاب ايضا ونحاول ان نقشها
+
+6
+00:00:36,760 --> 00:00:43,800
+السؤال الأول اللي بدنا نشوفه سؤال تلتاشر خمستاشر
+
+7
+00:00:43,800 --> 00:00:49,040
+من الكتاب بيقول القاتي indicate the type of solute
+
+8
+00:00:49,040 --> 00:00:54,350
+solvent interaction يعني وضح ليهلما تتم عملية
+
+9
+00:00:54,350 --> 00:01:01,590
+الذوبان، ما هو نوع التفاعل أو الـ interaction ما
+
+10
+00:01:01,590 --> 00:01:07,350
+بين المذاب والمذيب، الـ solute والسولفان، هذا يجب
+
+11
+00:01:07,350 --> 00:01:12,690
+أن يكون أهم نوع
+
+12
+00:01:12,690 --> 00:01:15,930
+من الـ intermolecular forces بين الـ solute
+
+13
+00:01:15,930 --> 00:01:22,260
+والسولفانالان لو بدنا ناخد ال CCL4 اللى بنسميه
+
+14
+00:01:22,260 --> 00:01:28,960
+carbon tetrachloride في البنزين طبعا واضح ان هدول
+
+15
+00:01:28,960 --> 00:01:37,140
+الجزيئين non polar moleculesالـ CCL4 جزء بهذا
+
+16
+00:01:37,140 --> 00:01:46,080
+الشكل وهو بالكيميا تترا هيدرال مجموع البولاريتي له
+
+17
+00:01:46,080 --> 00:01:53,810
+صفر والبنزينالبنزين الكيميائي طبعا هو هذا البنزين
+
+18
+00:01:53,810 --> 00:02:00,170
+الحلقة اللى فيها ستة hydrogens C6H6 الان هذا non
+
+19
+00:02:00,170 --> 00:02:05,470
+polar وهذا non polar بناء على ذلك نوع القوة
+
+20
+00:02:05,470 --> 00:02:10,570
+between non polar molecules is dispersion forces
+
+21
+00:02:10,570 --> 00:02:19,860
+الان لو سألنا الفرع Bأي نوع القوة بين الميثانول
+
+22
+00:02:19,860 --> 00:02:23,260
+إذا احنا بنتطلع على الميثانول بنلاحظ انه فيه H
+
+23
+00:02:23,260 --> 00:02:27,880
+متصلة ب O إذا هو جاهز يعمل hydrogen bonding لو
+
+24
+00:02:27,880 --> 00:02:33,180
+بدأنا أجي أتطلع على الميه طبعا الكل حافظ شكلها H2O
+
+25
+00:02:33,180 --> 00:02:38,360
+برضه فيه H متصلة بO إذا هنا و هنا حيكون في عندنا
+
+26
+00:02:38,360 --> 00:02:43,940
+hydrogen bonding ما بين الميثانول و ما بين الميه
+
+27
+00:02:43,940 --> 00:02:50,800
+hydrogen bondingالان لو طلعنا على الفرق C كي بي ار
+
+28
+00:02:50,800 --> 00:02:55,420
+كي بي ار مركب ايوني يعني في نهاية حيصير كي بلس
+
+29
+00:02:55,420 --> 00:03:00,000
+وحيصير بي ار ماينوس الايونات بدأت تنفصل عن بعض
+
+30
+00:03:00,000 --> 00:03:06,660
+الان هذه الايونات بدأت تحاطب جزيئات الماء بحيث انه
+
+31
+00:03:06,660 --> 00:03:14,440
+الجزء السالب من الميه بيكون قربالجزء الموجب او ال
+
+32
+00:03:14,440 --> 00:03:19,980
+ايون الموجب و بنفس الوقت لو كنا بنشوف حوالين ال PR
+
+33
+00:03:19,980 --> 00:03:23,760
+حيكون الجزء الموجب من الميه اللي هو ال hydrogen
+
+34
+00:03:23,760 --> 00:03:27,860
+هنا الجزء السالب من الميه هو ال oxygen الجزء
+
+35
+00:03:27,860 --> 00:03:31,800
+الموجب هو ال hydrogen فاللي موجود عندى هو اللي انا
+
+36
+00:03:31,800 --> 00:03:36,460
+بسميه ايون هذا هو ال ايون دايبول ليش لأنه هنا في
+
+37
+00:03:36,460 --> 00:03:40,040
+عندى اناPartially Positive و Partially Negative
+
+38
+00:03:40,040 --> 00:03:44,940
+هذا دايبول و هذا ايون ايون دايبول و هذا ايون برضه
+
+39
+00:03:44,940 --> 00:03:50,060
+نفس الفكرة ايون دايبول و ايون دايبول فنوع البندق
+
+40
+00:03:50,060 --> 00:03:59,400
+هنا عبارة عن ايون دايبول بندقالفرق الأخير دي CH3
+
+41
+00:03:59,400 --> 00:04:06,980
+أو HCL طبعا الـ HCL لو احنا بدنا نتخيل شكله هذا هو
+
+42
+00:04:06,980 --> 00:04:12,140
+شكل الـ HCL فاللي بنلاحظه ان الـ CL partially
+
+43
+00:04:12,140 --> 00:04:16,580
+negative والـ H partially positive بسبب فرق
+
+44
+00:04:16,580 --> 00:04:24,620
+السالبيةهذا هو الـ HCl نفس الشيء الـ Acetonitrile
+
+45
+00:04:24,620 --> 00:04:27,960
+لما احنا بنتطلع على CH3
+
+46
+00:04:47,170 --> 00:04:51,150
+بناء على ذلك لو احنا بدنا نتخيل نوع ال bonding
+
+47
+00:04:51,150 --> 00:04:57,040
+هيكون ما بين هذا الطرفاللي موجود عندنا هنا السالب
+
+48
+00:04:57,040 --> 00:05:01,840
+من الكلور والطرف الموجب هنا بدو يصير ال bonding
+
+49
+00:05:01,840 --> 00:05:07,140
+فهذا عبارة عن die ball وهذا عبارة عن die ball فال
+
+50
+00:05:07,140 --> 00:05:11,700
+attraction die ball die ball attraction die ball
+
+51
+00:05:11,700 --> 00:05:16,900
+die ball attraction هذا
+
+52
+00:05:16,900 --> 00:05:24,020
+بالنسبة للفرع الأول اللي بيحكي عن نوع ال bondingلو
+
+53
+00:05:24,020 --> 00:05:28,640
+شوفنا سؤال تلتاشر تلاتة و عشرين بيحكينا عن الـ
+
+54
+00:05:28,640 --> 00:05:34,140
+solubility of a chromium nitrate 9H2O in water is
+
+55
+00:05:34,140 --> 00:05:40,360
+208 per gram per 100 gram of water at 15 degrees
+
+56
+00:05:40,360 --> 00:05:44,760
+centigrade طبعا هنا في نقطة كتير مهمة ان ال
+
+57
+00:05:44,760 --> 00:05:48,320
+solubility زي ما احنا بتعلمنا بتعتمد على درجة
+
+58
+00:05:48,320 --> 00:05:55,390
+الحرارة يعني كأنه عند خمستاشردرجة مئوية اللي هي ال
+
+59
+00:05:55,390 --> 00:06:04,630
+temperature السليوبيليت كانت عبارة عن 208
+
+60
+00:06:04,630 --> 00:06:11,850
+السليوبيليت لكل 100 مل 208 اجرام الان لو زدنا
+
+61
+00:06:11,850 --> 00:06:17,650
+الحرارة السليوبيليت بدأ تزيد هذا اللي صارفلما زدنا
+
+62
+00:06:17,650 --> 00:06:23,950
+الحرارة عند 35 درجة مئوية السليوبيليت زادت بدل ما
+
+63
+00:06:23,950 --> 00:06:32,070
+كانت 208 صارت 324 بمعنى انه في كمية إضافية مقدرة
+
+64
+00:06:32,070 --> 00:06:40,410
+حوالي 116 جرام إضافي دابوا نتيجة إيش نتيجة زيادة
+
+65
+00:06:40,410 --> 00:06:47,770
+درجة الحرارة يعني مش مش شيء بسيطplus 116 gram كل
+
+66
+00:06:47,770 --> 00:06:51,430
+هذه الذائبية ازاي ده نتيجة ايش زيادة درجة الحرارة
+
+67
+00:06:51,430 --> 00:06:57,270
+الان when this solution هذا اللي هو على الحرارة
+
+68
+00:06:57,270 --> 00:07:01,070
+العالية when this solution is slowly cold to 15
+
+69
+00:07:01,070 --> 00:07:05,710
+degree centigrade لو عودنا بردنا ايش المفروض اللي
+
+70
+00:07:05,710 --> 00:07:09,730
+يصير في هذه الحالة المفروض اللي يصير انه من هنا
+
+71
+00:07:09,730 --> 00:07:16,610
+يترسب ناقص 116 gramيترثب و اللي ذاب نتيجة التسخين
+
+72
+00:07:16,610 --> 00:07:22,510
+يترثب نتيجة التبريد اللي لاحظوا انه عملنا هالكلام
+
+73
+00:07:22,510 --> 00:07:29,050
+طبعا للأسف الشديد ماصارش الترثيب بمعنى صار في عندى
+
+74
+00:07:29,050 --> 00:07:33,350
+بهذه الحالة صار في عندى بهذه الحالة انه انا بردت ل
+
+75
+00:07:33,350 --> 00:07:44,400
+15 درجة و ما زال ذائب الـ324 جرامطبعا هذا اكتر من
+
+76
+00:07:44,400 --> 00:07:48,300
+احتمال المحلول عند 15 درجة مقوية لان المحلول عند
+
+77
+00:07:48,300 --> 00:07:52,540
+15 درجة مقوية كان بتشبع بـ 208 و بنقول عنه
+
+78
+00:07:52,540 --> 00:07:58,500
+saturated و المحلول عند 35 درجة مقوية بتشبع ب324
+
+79
+00:07:58,500 --> 00:08:03,600
+جرام و بنقول عنه saturated بس 324 جرام عند 15 درجة
+
+80
+00:08:03,600 --> 00:08:08,420
+مقوية هذا بيحمل اكتر من طاقته يعني عند 15 طاقته
+
+81
+00:08:08,920 --> 00:08:13,720
+متين و تمانية الان اللى موجود انه دايب فعلا فيه
+
+82
+00:08:13,720 --> 00:08:17,420
+تلاتمية و اربعة عشرين جرام اذا لما انا بدى اوصف
+
+83
+00:08:17,420 --> 00:08:22,120
+هذا الكلام what term describes this solution شو
+
+84
+00:08:22,120 --> 00:08:28,300
+بيكون هذا المحلول هذا بنسميه احنا super saturated
+
+85
+00:08:28,300 --> 00:08:34,280
+هذا المحلول فوق المشبعواضح انه احنا لما بردنا
+
+86
+00:08:34,280 --> 00:08:40,180
+الاصل يترسب لكن للأسف الشديد ما ترسبش الان بيقولنا
+
+87
+00:08:40,180 --> 00:08:45,320
+what action شو ممكن تعمل انت عشان تخليه يترسب طبعا
+
+88
+00:08:45,320 --> 00:08:51,750
+احنا أخدناتقنيات معينة للمحليل ال super saturated
+
+89
+00:08:51,750 --> 00:08:56,270
+كيف ممكن تترسب احد هذه التقنيات انه نعمل seeding
+
+90
+00:08:56,270 --> 00:09:01,790
+بمعنى ان نجيب كمية صغيرة زي بذرة او نواء نجيب كمية
+
+91
+00:09:01,790 --> 00:09:05,730
+صغيرة من المادة و نضيفها على المحلول فبصير يترسب
+
+92
+00:09:05,730 --> 00:09:10,530
+حول هذه الاشياء الصغيرة اللي احنا اضفناها او
+
+93
+00:09:10,530 --> 00:09:14,690
+البذور الصغيرة اللي احنا اضفناهاأو بنعمل
+
+94
+00:09:14,690 --> 00:09:21,050
+scratching بنحك جدار ال flask فبتنفصل جزيئات زجاج
+
+95
+00:09:21,050 --> 00:09:26,050
+بسيطة جدا بتم الترسيب حولها فالتقنيات اما بدنا
+
+96
+00:09:26,050 --> 00:09:32,230
+نعمل عشان نسرع ترسب المادة seeding نجيب شوية صغيرة
+
+97
+00:09:32,230 --> 00:09:36,620
+منها و نحطهاأو scratching انحق الجدار الزجاجي
+
+98
+00:09:36,620 --> 00:09:40,180
+scratching the glass walls of the container عادة
+
+99
+00:09:40,180 --> 00:09:43,900
+بستعمل glass rod يعني بيكون فينا قضيب من الزجاج
+
+100
+00:09:43,900 --> 00:09:49,940
+بنحق الزجاج بالزجاج تنفصل يعني أشياء very very
+
+101
+00:09:49,940 --> 00:09:52,920
+small بتم الترسب حولها
+
+102
+00:10:01,910 --> 00:10:07,510
+السؤال التسعة و عشرين بيقولناcommon laboratory
+
+103
+00:10:07,510 --> 00:10:11,870
+solvents يعني من ال solvents المذيبات المشهورة جدا
+
+104
+00:10:11,870 --> 00:10:17,470
+في المختبر include أسيتون منها الأسيتون والأسيتون
+
+105
+00:10:17,470 --> 00:10:22,310
+polar solvent ميثانول الميثانول أيضا polar solvent
+
+106
+00:10:22,310 --> 00:10:26,790
+وبيعمل hydrogen bonding الطلوين الطلوين أحد مشتقات
+
+107
+00:10:26,790 --> 00:10:32,730
+البنزين اللي موجود بهذا الشكل حلقة بنزين عليها CH3
+
+108
+00:10:32,730 --> 00:10:40,200
+هذا اللي احنا بنسمي ايه طلوينالطلوين والمياه
+
+109
+00:10:40,200 --> 00:10:44,920
+معروفة H2O which of these is the best solvent for
+
+110
+00:10:44,920 --> 00:10:48,360
+non-polar solutes يعني هو انا اعطاني اربعة
+
+111
+00:10:48,360 --> 00:10:52,660
+solvents جالي لو عندي انا solute non-polar مين
+
+112
+00:10:52,660 --> 00:10:57,440
+بنسبه طبعا القاعدة non-polar بده non-polar الان ال
+
+113
+00:10:57,440 --> 00:11:01,680
+acetone لأنه فيه C double bond O يعني هذا هو شكل
+
+114
+00:11:01,680 --> 00:11:05,540
+ال acetone هذا هو شكل ال acetone
+
+115
+00:11:08,200 --> 00:11:14,340
+هنا في عندنا C double bond O فالـ O طرف سالب والـ
+
+116
+00:11:14,340 --> 00:11:18,820
+C طرف موجب إذا هذا polar واضح الاسيتون is polar
+
+117
+00:11:20,370 --> 00:11:24,790
+والميثانول بولر بيعمل hydrogen bonding والماء بولر
+
+118
+00:11:24,790 --> 00:11:25,350
+بيعمل hydrogen bonding والماء بولر بيعمل hydrogen
+
+119
+00:11:25,350 --> 00:11:25,610
+bonding والماء بولر بيعمل hydrogen bonding والماء
+
+120
+00:11:25,610 --> 00:11:25,630
+بولر بيعمل hydrogen bonding والماء بولر بيعمل
+
+121
+00:11:25,630 --> 00:11:25,790
+بولر بيعمل hydrogen bonding والماء بولر بيعمل
+
+122
+00:11:25,790 --> 00:11:25,830
+hydrogen bonding والماء بولر بيعمل hydrogen
+
+123
+00:11:25,830 --> 00:11:26,670
+bonding والماء بولر بيعمل hydrogen bonding والماء
+
+124
+00:11:26,670 --> 00:11:27,750
+بولر بيعمل hydrogen bonding والماء بولر بيعمل
+
+125
+00:11:27,750 --> 00:11:29,550
+hydrogen bonding والماء بولر بيعمل hydrogen
+
+126
+00:11:29,550 --> 00:11:33,790
+bonding والماء بولر بيعمل hydrogen bonding والماء
+
+127
+00:11:33,790 --> 00:11:40,690
+بولر بيعمل hydrogen bonding والماء بولر
+
+128
+00:11:40,690 --> 00:11:44,550
+بيعمل hydrogen bonding والماء ب
+
+129
+00:11:53,840 --> 00:11:58,380
+السؤال التالف تسعة و تلاتين بيحكينا عن وحدات
+
+130
+00:11:58,380 --> 00:12:06,420
+التركيز تخطط مجموعة صودم صلفية في حلقة تحتوي على
+
+131
+00:12:06,420 --> 00:12:13,110
+10.6 جرام صودم صلفية في 483 جرام ماءيعني دوّبنا
+
+132
+00:12:13,110 --> 00:12:18,090
+Sodium Sulfate كمية معينة في الميّه ايش ال mass
+
+133
+00:12:18,090 --> 00:12:22,670
+percentage طبعا ال mass percentage احنا بنعرفها هي
+
+134
+00:12:22,670 --> 00:12:26,190
+weight على ال total weight ضارب 100 الان ال weight
+
+135
+00:12:26,190 --> 00:12:31,230
+10.6 ال total weight هو عبارة عن مجموع هدول
+
+136
+00:12:31,230 --> 00:12:36,440
+التنتين ال 10.6 والميّه هذا هو ال total weightلما
+
+137
+00:12:36,440 --> 00:12:41,220
+بنجمعهم بيطلع 493.6 weight على total weight ضارب
+
+138
+00:12:41,220 --> 00:12:46,680
+100 هذا ال mass percent بيطلع النسبة 2.15% هذا
+
+139
+00:12:46,680 --> 00:12:53,140
+تركيز المحلول في ال mass percent هذا الفرق A الان
+
+140
+00:12:53,140 --> 00:13:00,180
+الفرق B بيقول ان or يعني خامةلما بيطولوا المعادل
+
+141
+00:13:00,180 --> 00:13:05,060
+من التراب و من غيرها بيكون عندهم خمات an ore
+
+142
+00:13:05,060 --> 00:13:12,080
+contains 2.86 gram of silver per ton of ore يعني
+
+143
+00:13:12,080 --> 00:13:19,480
+في عندنا خامة رمل موجود في فضةالان هذا الرمل في كل
+
+144
+00:13:19,480 --> 00:13:24,440
+طن منه اتنين و ستة و ثمانين من مائة جرام من الفضة
+
+145
+00:13:24,440 --> 00:13:28,140
+بيقول لي ايش ال concentration للفضة بال part per
+
+146
+00:13:28,140 --> 00:13:32,720
+million طبعا ال concentration هيكون ال weight على
+
+147
+00:13:32,720 --> 00:13:36,500
+ال total weight لما بده ياب ال BPM بده اضرب في
+
+148
+00:13:36,500 --> 00:13:41,280
+عشرة او ستةهيك تعريف ال bpm فإذا قلنا ال weight 2
+
+149
+00:13:41,280 --> 00:13:48,780
+.86 جرام الآن ال ton هنا بدنا ننتبه ألف كيلو جرام
+
+150
+00:13:48,780 --> 00:13:55,220
+ألف كيلو جرام فلما أنا بدي أقول ال ton بذاته ألف و
+
+151
+00:13:55,220 --> 00:14:02,140
+ال kilogram نفسه ألف جرامبناء على ذلك لما انا بدي
+
+152
+00:14:02,140 --> 00:14:05,860
+اقول ال tone a chemical عبارة عن الف في الف يعني
+
+153
+00:14:05,860 --> 00:14:14,520
+واحد في عشر أقص ستة واضح فلما بدي احسب ال mass
+
+154
+00:14:14,520 --> 00:14:22,490
+percent في هذه الحالة بدي اقول ال weight علىالـ
+
+155
+00:14:22,490 --> 00:14:27,710
+total weight ضرب عشرة أقل ستة طبعا الـ total
+
+156
+00:14:27,710 --> 00:14:34,070
+weight الحقيقة هو عبارة عن المليون جرام اللي هم
+
+157
+00:14:34,070 --> 00:14:39,530
+الارأو what is the concentration of the silver لأ
+
+158
+00:14:39,530 --> 00:14:44,510
+هو ال ore بتحتوي على هذا الوزن بناء على ذلك يعني
+
+159
+00:14:44,510 --> 00:14:49,830
+داخل فيها يعني الخامة نفسها الوزن الكل الطن هذا من
+
+160
+00:14:49,830 --> 00:14:56,390
+ضمنه ل 82 من 100 جرام اذا بناخدها زي ما هي 82 ال
+
+161
+00:14:56,390 --> 00:15:01,010
+total weight واحد في عشرة او ستة طبعا عشان بدهيا
+
+162
+00:15:01,010 --> 00:15:06,980
+بال BPM بده اضرب في عشرة او ستة واضح؟و لما نحسب
+
+163
+00:15:06,980 --> 00:15:10,900
+هذا الكلام weight على ال total weight ضرب 10 رقم 6
+
+164
+00:15:10,900 --> 00:15:17,620
+عشان نخليه بال parts per million بيطلع 2.86 واضح
+
+165
+00:15:17,620 --> 00:15:27,980
+هذا هو التركيز في وحدة ال part per million المسائل
+
+166
+00:15:27,980 --> 00:15:33,140
+التالية برضه على التركيزسؤال واحد واربعين الـ
+
+167
+00:15:33,140 --> 00:15:36,820
+solution is made containing fourteen point six
+
+168
+00:15:36,820 --> 00:15:44,380
+gram of water in one eight four gram14.6 جرام
+
+169
+00:15:44,380 --> 00:15:51,520
+ميثانول و 184 جرام ميثانول المولي فراكشن الميثانول
+
+170
+00:15:51,520 --> 00:15:55,880
+طبعا المولي فراكشن بيساوي ال number of moles of
+
+171
+00:15:55,880 --> 00:16:00,000
+methanol على ال total moles الان انا عند الوزن ال
+
+172
+00:16:00,000 --> 00:16:03,400
+number of moles weight على molecular weightالماية
+
+173
+00:16:03,400 --> 00:16:06,200
+عند الوزن ال number of moles برضه weight على
+
+174
+00:16:06,200 --> 00:16:18,420
+molecular weight المولي فراكشن المولي
+
+175
+00:16:18,420 --> 00:16:22,180
+فراكشن هو عبارة عن number of moles of methanol على
+
+176
+00:16:22,180 --> 00:16:28,190
+ال total molesواضح ال number of moles على ال total
+
+177
+00:16:28,190 --> 00:16:32,770
+mole اللي هي 0.46 زي 2.2 بيطلع ال mole fraction 0
+
+178
+00:16:32,770 --> 00:16:40,790
+.043 هذا الفرع A بعد ذلك يحتاج ال mass percent of
+
+179
+00:16:40,790 --> 00:16:45,780
+methanolالماس بيرسنت اخدنا هو عبارة عن الماس على
+
+180
+00:16:45,780 --> 00:16:50,100
+ال total mass ضارب مية بطلع seven point three five
+
+181
+00:16:50,100 --> 00:16:54,900
+percent الفرع C بده ال molality الآن بدنا نكون
+
+182
+00:16:54,900 --> 00:16:58,260
+عارفين شو ال molality ال molality بتساوي ال number
+
+183
+00:16:58,260 --> 00:17:02,040
+of moles على ال mass of solvent in kilogram ال
+
+184
+00:17:02,040 --> 00:17:05,740
+number of moles للميثانول point four six ال mass
+
+185
+00:17:05,740 --> 00:17:11,570
+of solvent184 لكن لازم تكون بالكيلوجرام اذا point
+
+186
+00:17:11,570 --> 00:17:15,590
+one eight four فالمولز على ال massive solvent in
+
+187
+00:17:15,590 --> 00:17:21,430
+kilogram بتطلع المحلول او بيطلع المحلول تركيزه two
+
+188
+00:17:21,430 --> 00:17:26,010
+point five molal و احنا اتفقنا ان ال molality
+
+189
+00:17:26,010 --> 00:17:32,150
+بنعطيها الرمز small m و ال molarity بنعطيها الرمز
+
+190
+00:17:32,150 --> 00:17:33,530
+capital M
+
+191
+00:17:40,050 --> 00:17:46,170
+نشوف سؤال 47 برضه على وحدات التركيز a sulfuric
+
+192
+00:17:46,170 --> 00:17:53,330
+acid solution containing 571.6 gram of sulfuric
+
+193
+00:17:53,330 --> 00:17:57,310
+acid per liter of solution لاحظوا هنا اعطاني
+
+194
+00:17:57,310 --> 00:18:02,390
+اكمجرام عندى من ال sulfuric acid على الحجم الحجم
+
+195
+00:18:02,390 --> 00:18:08,560
+المحلول الان هذا المحلولit has a density of 1.326
+
+196
+00:18:08,560 --> 00:18:12,440
+يعني ال density لحمض الكبريتيك اتجيلة الانسان لما
+
+197
+00:18:12,440 --> 00:18:16,600
+بيحمل بُتل فيه حمض الكبريتيك بيحسه ادجل من انه
+
+198
+00:18:16,600 --> 00:18:21,120
+يكون بُتل ميه يعني ال liter من الميه وزنه الف جرام
+
+199
+00:18:21,120 --> 00:18:26,580
+الان هنا من هذه ال density واضح ان ال one liter
+
+200
+00:18:26,580 --> 00:18:32,550
+وزنه الف وتلاتمية وتسعة وعشرين جرامواضح؟ اذا هذا
+
+201
+00:18:32,550 --> 00:18:36,810
+ال liter لو انا بدي اقبل وزن من خلال علاقة الكثافة
+
+202
+00:18:36,810 --> 00:18:40,930
+ال density بتسويت على volume بقول ان وزن ال liter
+
+203
+00:18:40,930 --> 00:18:50,270
+1329 gram هذا وزن ال liter واضح؟ ضربت بال الف الان
+
+204
+00:18:50,270 --> 00:18:53,350
+بيقول ال calculate the mass percent ال mass
+
+205
+00:18:53,350 --> 00:18:58,130
+percent هو وزن حمض الكبريتيك على وزن الميه الان if
+
+206
+00:18:58,130 --> 00:19:07,230
+the total massالتوتر هو وزن كل المحلول 1329 ووزن
+
+207
+00:19:07,230 --> 00:19:16,450
+الـ H2SO4 لحاله كان 571.6 إذن الفرق بينهم هو وزن
+
+208
+00:19:16,450 --> 00:19:21,210
+الماء واضح؟ وعشان ذلك نستطيع أن نحصل على مولز مياه
+
+209
+00:19:21,210 --> 00:19:25,510
+الماء الـ Total Mass minus Mass of Sulphuric Acid
+
+210
+00:19:25,510 --> 00:19:32,780
+757.4الان لو انا عندي وزن حمض الكابريتيك بقسم على
+
+211
+00:19:32,780 --> 00:19:36,860
+ال molecular weight بطلع ال number of moles وزي ما
+
+212
+00:19:36,860 --> 00:19:41,880
+قولت ال mass الو بتساوي 1329 من الكثافة لو بد ال
+
+213
+00:19:41,880 --> 00:19:46,210
+mass of waterهي الفرق ما بين الوزن الكل ووزن حمض
+
+214
+00:19:46,210 --> 00:19:53,610
+الكبريتيك 757 لو بد ال moles of water 757.4 على 18
+
+215
+00:19:53,610 --> 00:19:57,090
+على ال molecular weight صار عندي ال mass of water
+
+216
+00:19:57,090 --> 00:20:00,670
+و ال moles of water ال mass of sulfuric acid و ال
+
+217
+00:20:00,670 --> 00:20:04,850
+moles of sulfuric acid الان ايش بده اسئلة بكون سهل
+
+218
+00:20:04,850 --> 00:20:10,450
+علي أنا اجاوبه؟ماذا هو الـ Mass Percent؟ الـ Mass
+
+219
+00:20:10,450 --> 00:20:13,430
+Percent هي عبارة عن الـ Mass of Sulphuric Acid على
+
+220
+00:20:13,430 --> 00:20:19,880
+الـ Total Mass ضارب 100 تطلع 43.01What is the mole
+
+221
+00:20:19,880 --> 00:20:23,180
+fraction of sulphuric acid؟هو عبارة عن ال moles of
+
+222
+00:20:23,180 --> 00:20:27,320
+sulphuric acid على ال total moles انا هنا عندي ال
+
+223
+00:20:27,320 --> 00:20:31,260
+moles of sulphuric acid حسبتها وال moles of water
+
+224
+00:20:31,260 --> 00:20:37,300
+حسبتها المoles على ال total moles ضرب بدون ما نضرب
+
+225
+00:20:37,300 --> 00:20:41,480
+بمية لش بده fraction يعني بدون ما تنضرب مية moles
+
+226
+00:20:41,480 --> 00:20:45,760
+على ال total moles هاي ال mole fraction بنوجده
+
+227
+00:20:46,520 --> 00:20:52,040
+الفرع C المولالية المولالية number of moles اللي
+
+228
+00:20:52,040 --> 00:20:57,520
+انا برضه اوجدتها للسلفريك 5.83 على ال mass of
+
+229
+00:20:57,520 --> 00:21:01,380
+solvent in kilogram هاي ال mass of water اللي
+
+230
+00:21:01,380 --> 00:21:08,020
+حسبناها بالجرام لما ندق بال kilogram 0.7574 فبتطلع
+
+231
+00:21:08,020 --> 00:21:15,060
+المولالية 7.69 مولالالان لو بدوا ال مولاريتي الفرع
+
+232
+00:21:15,060 --> 00:21:21,140
+دي ال number of moles على ال total volume هو أساسا
+
+233
+00:21:21,140 --> 00:21:25,640
+ال total volume بيقول بير لتر يعني موجود فى لتر هى
+
+234
+00:21:25,640 --> 00:21:28,040
+ال number of moles of sulfuric acid و ال total
+
+235
+00:21:28,040 --> 00:21:34,180
+volume is one liter فالمولاريتي 5.83 مولار
+
+236
+00:21:47,440 --> 00:22:15,480
+سؤال 65 سؤال
+
+237
+00:22:15,480 --> 00:22:16,040
+65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+238
+00:22:16,040 --> 00:22:16,060
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+239
+00:22:16,090 --> 00:22:21,670
+دوبته بالميه دوبته 22.5 جرام of lactose في 200
+
+240
+00:22:21,670 --> 00:22:28,590
+جرام ميه عند درجة حرارة 338 kelvin الان طبعا
+
+241
+00:22:28,590 --> 00:22:32,430
+هيلزمنا ال vapor pressure ل pure water يعني كأنه
+
+242
+00:22:32,430 --> 00:22:40,170
+هذه المسألة ان انا لو كان في عندى ميه عادية water
+
+243
+00:22:40,170 --> 00:22:44,850
+لحالها في عندى اللي بنسميه ال pure vapor pressure
+
+244
+00:22:47,180 --> 00:22:48,900
+Pure Vapor Pressure
+
+245
+00:22:55,870 --> 00:23:00,990
+الان هذا الضغط موجود في جداول .. هذا الضغط موجود
+
+246
+00:23:00,990 --> 00:23:05,770
+في جداول لما احنا بدنا نشوف ضغط الماء عند ضغط بخار
+
+247
+00:23:05,770 --> 00:23:14,090
+الماء بنلاقيه من الجدول 187.5 تور واضح؟ طيب هذا
+
+248
+00:23:14,090 --> 00:23:18,370
+ضغط الماء لما يكون نقي ..الان هذا الماء لو احنا
+
+249
+00:23:18,370 --> 00:23:25,090
+اجينا ودوبنا فيه .. ودوبنا فيه لكتوزطبعاً اللاكتوز
+
+250
+00:23:25,090 --> 00:23:33,150
+يقلل ضغط البخار لأنه صار جزء على سطح الماء
+
+251
+00:23:33,150 --> 00:23:38,350
+اللاكتوز يمنع التبخر بناء على ذلك كمية التبخر في
+
+252
+00:23:38,350 --> 00:23:45,050
+وجود اللاكتوز تقل واخدنا معادلة كيف نحسب ال vapor
+
+253
+00:23:45,050 --> 00:23:46,010
+pressure
+
+254
+00:23:54,440 --> 00:23:58,460
+الان المعلومات اللى موجودة عندنا اللاكتوز الى
+
+255
+00:23:58,460 --> 00:24:01,580
+molecular weight تلت مية و اتنين و اربعين و بناء
+
+256
+00:24:01,580 --> 00:24:05,520
+على ذلك لما بدي انا number of moles الى اتنين و
+
+257
+00:24:05,520 --> 00:24:08,600
+عشرين و نص على تلت مية و اتنين و اربعين point oh
+
+258
+00:24:08,600 --> 00:24:09,840
+six six mole
+
+259
+00:24:16,930 --> 00:24:22,630
+بناء على ذلك عشان احسب ال vapor pressure للمحلول
+
+260
+00:24:22,630 --> 00:24:27,190
+بدي المول fraction للميهاللي هي number of moles of
+
+261
+00:24:27,190 --> 00:24:32,390
+water على total number of moles بتطلع point nine
+
+262
+00:24:32,390 --> 00:24:36,490
+nine four الان اذا كنت انا عارف ال pressure بدي
+
+263
+00:24:36,490 --> 00:24:41,630
+اضرب ال mole fraction في ال vapor pressure لبور
+
+264
+00:24:41,630 --> 00:24:44,730
+water point nine nine four في مية و سبعة و ثمانية
+
+265
+00:24:44,730 --> 00:24:51,150
+point fiveالجواب 186.4 لاحظوا ان ال vapor pressure
+
+266
+00:24:51,150 --> 00:24:56,110
+بسبب إذابة اللاكتوز قل يعني لما بدأ اقارن بدون
+
+267
+00:24:56,110 --> 00:25:01,310
+لاكتوز كان ال vapor pressure 187.5 بوجود اللاكتوز
+
+268
+00:25:01,310 --> 00:25:09,750
+بقل صار 186.4 هذا بالنسبة للفرع A
+
+269
+00:25:12,130 --> 00:25:18,090
+الفرع بيه فرع على محلول تاني غير اللاكتوز والمية
+
+270
+00:25:18,090 --> 00:25:24,890
+اللي هو اسمه البروبيلين غلايكول والبروبيلين
+
+271
+00:25:24,890 --> 00:25:30,650
+غلايكول برضه هو عبارة عن مركب بيكون عادة non
+
+272
+00:25:30,650 --> 00:25:37,040
+volatile يعني غير متطاير مركب غير متطايرهذا
+
+273
+00:25:37,040 --> 00:25:43,280
+البروبيلين جليكول صيغته العامة C3H8O
+
+274
+00:25:43,280 --> 00:25:50,060
+وصيغته بهذا الشكل التفصيلي للناس اللي بيهتموا هنا
+
+275
+00:25:50,060 --> 00:26:01,120
+في عندي OH وهنا في عندي OH عشان كان C3H3328C3H8O2
+
+276
+00:26:01,120 --> 00:26:08,110
+هذا شكل البروبيلين جليكولالان بقول احسب كتلة
+
+277
+00:26:08,110 --> 00:26:13,330
+البروبيلين غلايكول اللي يجب ان يتم اضافه لـ 0.340
+
+278
+00:26:13,330 --> 00:26:22,010
+كيلو جرام ماء لتقلي البريشار الفيبر بي 2.88 تور في
+
+279
+00:26:22,010 --> 00:26:29,090
+40 درجة سنتيجريد واضح؟ طبعا لو بدنا نبدأ نشوف
+
+280
+00:26:29,090 --> 00:26:33,430
+المعطيات اللي عندنا ال molecular weight لهذا
+
+281
+00:26:33,430 --> 00:26:38,650
+المُركّب اللي هو البروبيلين جليكول بنحسبه 76 ال
+
+282
+00:26:38,650 --> 00:26:43,230
+pressure للميّه عند 40 degree centigrade خمسة و
+
+283
+00:26:43,230 --> 00:26:48,210
+خمسين و تلاتة الان هو بيقول ال pressure هذا بينخفض
+
+284
+00:26:48,210 --> 00:26:52,870
+لما أنا بدوّب شوية من البروبيلين جليكول في الميه
+
+285
+00:26:52,870 --> 00:26:58,900
+بينخفض ال pressure يعني ال new pressure بيصيرعبارة
+
+286
+00:26:58,900 --> 00:27:04,200
+عن الـ 55.3 مطروح منها الـ 2.88 اللي هو مقدار
+
+287
+00:27:04,200 --> 00:27:12,620
+الانخفاض بيصير 52.42 الان احنا كلنا عارفين ان ال
+
+288
+00:27:12,620 --> 00:27:16,520
+pressure بيساوي ال X في ال B node اللي هو ال
+
+289
+00:27:16,520 --> 00:27:20,340
+pressure ل ال pure water الان هذا ال pressure صار
+
+290
+00:27:20,340 --> 00:27:27,530
+موجود عندى 52.42 ال X المولي fraction مجهولةو ال
+
+291
+00:27:27,530 --> 00:27:32,170
+pressure الأصلي 55.3 pressure of pure water بناء
+
+292
+00:27:32,170 --> 00:27:36,710
+على ذلك انا بقدر احسب قيمة ال X ايش ال mole
+
+293
+00:27:36,710 --> 00:27:43,970
+fraction للميه الان لما انا بدي اجي احسب ايش هو ال
+
+294
+00:27:43,970 --> 00:27:46,910
+mole fraction للميه انا عندي number of moles of
+
+295
+00:27:46,910 --> 00:27:51,190
+water هذا ال mole fraction للميه هو عبارة عن
+
+296
+00:27:51,190 --> 00:27:55,550
+number of moles of water على ال total moles طبعا
+
+297
+00:27:56,210 --> 00:27:59,310
+الـ number of moles of water طالما الميّه تلاتمية
+
+298
+00:27:59,310 --> 00:28:02,410
+وأربعين باجسمها ع تمانتاش هاي ال number of moles
+
+299
+00:28:02,410 --> 00:28:07,970
+of water الان طيب ال number of moles of propylene
+
+300
+00:28:07,970 --> 00:28:12,110
+glycol انا ماعنديش كام مول of propylene glycol اذا
+
+301
+00:28:12,110 --> 00:28:16,550
+صار المجهول اللي عندي هنا هو ال number of moles of
+
+302
+00:28:16,550 --> 00:28:20,970
+propylene glycol هاي ال moles of water على ال
+
+303
+00:28:20,970 --> 00:28:23,810
+total moles اللي هي عبارة عن moles of water زاد
+
+304
+00:28:23,810 --> 00:28:27,810
+moles of propylene glycolفبحسب ال y اللى هى ال
+
+305
+00:28:27,810 --> 00:28:34,150
+moles of البروبيلين غلايكول بتطلع ال y 1.04 mole
+
+306
+00:28:34,150 --> 00:28:38,830
+هو الان مش طالب ال number of moles هو الان مش طالب
+
+307
+00:28:38,830 --> 00:28:44,110
+ال number of moles للبروبيلين غلايكول طالب ال mass
+
+308
+00:28:44,110 --> 00:28:49,970
+وانا بعرف انه عندى علاقة ان number of moles بساوي
+
+309
+00:28:49,970 --> 00:28:54,720
+weightعلى molecular weight فلما انا يكون عندي ال
+
+310
+00:28:54,720 --> 00:29:00,260
+number of moles صرت حاسبه لبروبيلين جلايكل و ال
+
+311
+00:29:00,260 --> 00:29:03,820
+molecular weight بعرفه من ال formula و لو حسبته 76
+
+312
+00:29:03,820 --> 00:29:07,840
+شو بيظل عندى ال weight ال weight بيصير بساوي ال
+
+313
+00:29:07,840 --> 00:29:12,320
+number of moles ضرب ال molecular weight هاي هذا
+
+314
+00:29:12,320 --> 00:29:17,980
+اللي عملناه انه ال mass فعلا لل C3H8O2 بيساوي ال
+
+315
+00:29:17,980 --> 00:29:22,980
+number of moles 1.04 في ال molecular weight 76
+
+316
+00:29:22,980 --> 00:29:29,800
+بيطلع وزن لبروبيلين جليكول 79.0 إجرام
+
+317
+00:29:47,280 --> 00:29:57,650
+ناخد سؤال 75 سؤال 75 بيحكي عن انخفاض فينقطة التجمد
+
+318
+00:29:57,650 --> 00:30:01,770
+يعني الماء لما بنضيف عليه non volatile solute
+
+319
+00:30:01,770 --> 00:30:05,990
+اتفقنا انه نقطة تجمده بتنخفض يعني الماء في الوضع
+
+320
+00:30:05,990 --> 00:30:12,190
+الطبيعي نقطة تجمده سفر بتنخفض بتصير بال minus الآن
+
+321
+00:30:12,190 --> 00:30:16,110
+بيجي بيسألنا السؤال how many grams of ethylene
+
+322
+00:30:16,110 --> 00:30:21,930
+glycol ومعطيني شكله C2H6O2وبرضه الـ Ethylene
+
+323
+00:30:21,930 --> 00:30:25,110
+Glycol اللي اللي بيحبوا يعرفوا شكله هو عبارة عن
+
+324
+00:30:25,110 --> 00:30:29,130
+CH2CH2OHOH
+
+325
+00:30:29,130 --> 00:30:35,030
+هذا اسمه Ethylene Glycolالان كم جرام من الاتلينيج
+
+326
+00:30:35,030 --> 00:30:39,930
+لايكول يجب ان يتم اضافه إلى واحد كيلو جرام من
+
+327
+00:30:39,930 --> 00:30:46,110
+الماء لإنتاج حلوة تتجمد في نصف درجة سنتيجريت طبعا
+
+328
+00:30:46,110 --> 00:30:51,890
+الاصل انه احنا بنعرف ان الميه عادة الوضع الطبيعي
+
+329
+00:30:51,890 --> 00:30:56,690
+تبعها بتتجمد في نصف درجة سنتيجريت فلما تصير انناقس
+
+330
+00:30:56,690 --> 00:31:04,180
+خمسة معنا ذلك دلتا تيبتساوي 5 درجات يعني كم انخفض
+
+331
+00:31:04,180 --> 00:31:08,760
+نقطة تجمد الميا من 0 لا ناقص 5 انخفضت 5 درجات الان
+
+332
+00:31:08,760 --> 00:31:13,420
+بقدر الانخفاض في نقطة التجمد بيساوي ثابت Kf في ال
+
+333
+00:31:13,420 --> 00:31:18,080
+morality الان الميا الثابت تبعها موجود في جداول
+
+334
+00:31:18,080 --> 00:31:25,220
+وموجود عندنا في الكتاب 1.86دلتا تي بتساوي ال five
+
+335
+00:31:25,220 --> 00:31:27,840
+بتساوي one point eight six في ال morality ال
+
+336
+00:31:27,840 --> 00:31:31,580
+morality بتساوي two point six nine الان ال
+
+337
+00:31:31,580 --> 00:31:35,260
+morality بتساوي number of moles على ال mass of
+
+338
+00:31:35,260 --> 00:31:40,560
+solvent انا ال mass of solvent عندياها واضح؟ one
+
+339
+00:31:40,560 --> 00:31:45,460
+kilogram of water ال mass of solvent وبناء على ذلك
+
+340
+00:31:45,460 --> 00:31:49,420
+لما انا بدي اقول ال morality بتساويالـ number of
+
+341
+00:31:49,420 --> 00:31:54,580
+moles على ال kilogram ممكن كمان بالمرة انا اقول ان
+
+342
+00:31:54,580 --> 00:31:59,040
+number of moles نفسها لال إيثالين جليكول بتسوية
+
+343
+00:31:59,040 --> 00:32:01,180
+على molecular weight يعني هذا ال weight على
+
+344
+00:32:01,180 --> 00:32:08,860
+molecular weight لمن؟ للإيثالين جليكول واضح؟
+
+345
+00:32:10,730 --> 00:32:12,210
+مثلًا الـ weight على molecular weight
+
+346
+00:32:12,210 --> 00:32:17,370
+للإيثالينيجليكول على ال mass of solvent in
+
+347
+00:32:17,370 --> 00:32:22,510
+kilogram لو قلت الآن ان الـ molality end معروفة و
+
+348
+00:32:22,510 --> 00:32:25,710
+ال molecular weight للإيثالينيجليكول معروف و ال
+
+349
+00:32:25,710 --> 00:32:30,910
+mass of solvent الميّه معروفة one kilogram مافيش
+
+350
+00:32:30,910 --> 00:32:35,560
+الحقيقة مجهول غير ال weightوبناء على ذلك بقدر احسب
+
+351
+00:32:35,560 --> 00:32:43,980
+ال weight 166.7 دمجنا عمليتين لو اي طالب ما بدوش
+
+352
+00:32:43,980 --> 00:32:49,560
+هذا الكلام ممكن يجي يقول ال morality بتساوي moles
+
+353
+00:32:49,560 --> 00:32:55,200
+على ال mass of solvent in kilogram وبالتالي ال
+
+354
+00:32:55,200 --> 00:33:00,520
+morality عنده موجودة بيعوض و ال mass موجودة بيحسب
+
+355
+00:33:00,520 --> 00:33:04,840
+ال numberof moles و بعد ما يحسب ال number of moles
+
+356
+00:33:04,840 --> 00:33:08,460
+بيقول هذا ال number of moles بتسويت على molecular
+
+357
+00:33:08,460 --> 00:33:12,380
+weight يعني ممكن الخطوتين يدخلهم في معادلة واحدة و
+
+358
+00:33:12,380 --> 00:33:15,860
+ممكن يفصلهم يعني اي طالب ممكن يقول ال morality
+
+359
+00:33:15,860 --> 00:33:21,760
+بتسوي moles على ال mass ال morality موجودةو ال
+
+360
+00:33:21,760 --> 00:33:26,900
+mass موجودة طبعا ال molality احنا حسبناها من نقطة
+
+361
+00:33:26,900 --> 00:33:31,500
+التجمد و ال mass of solvent موجودة هذه ال mass هي
+
+362
+00:33:31,500 --> 00:33:34,600
+ال mass of solvent in kilogram بيحسب number of
+
+363
+00:33:34,600 --> 00:33:37,940
+moles ال number of moles of ethylene glycol بتساوي
+
+364
+00:33:37,940 --> 00:33:41,040
+weight على molecular weight ال number of moles صار
+
+365
+00:33:41,040 --> 00:33:44,820
+موجود ال molecular weight موجود بيقدر يحسب ال
+
+366
+00:33:44,820 --> 00:33:48,940
+weight الان احنا هنا في نفس المعادلة عوضنا عن
+
+367
+00:33:48,940 --> 00:33:52,010
+number of moles weight على molecular weightعملناهم
+
+368
+00:33:52,010 --> 00:33:59,510
+خطوة واحدة بأي طريقة الطالب بحل بيصل لإيه الجواب
+
+369
+00:33:59,510 --> 00:34:07,110
+سؤال تمانية و سبعين بيقولنا sea water يعني ميط
+
+370
+00:34:07,110 --> 00:34:12,210
+البحر contains 3.4 gram of salts for every liter
+
+371
+00:34:12,210 --> 00:34:15,690
+of solution يعني في كل لتر من ميط البحر في اندي
+
+372
+00:34:15,690 --> 00:34:19,990
+تلاتة او اربعة من عشرة gram من الملحAssuming that
+
+373
+00:34:19,990 --> 00:34:24,410
+the solute consists entirely of NaCl خلّينا ناخد
+
+374
+00:34:24,410 --> 00:34:28,910
+فرضية أن هذا الملح هو ملح الطعام الـ NaCl الـ
+
+375
+00:34:28,910 --> 00:34:32,790
+Sodium Chloride هو عادة بيكون أكثر من 90% نفترض أن
+
+376
+00:34:32,790 --> 00:34:36,850
+كل الملح ملح الطعام بيقول Calculate the osmotic
+
+377
+00:34:36,850 --> 00:34:41,440
+pressure of sea waterطبعا احنا لما بدنا نقول الـ
+
+378
+00:34:41,440 --> 00:34:45,460
+Osmotic Pressure له معادلة المعادلة لل Osmotic
+
+379
+00:34:45,460 --> 00:34:51,840
+Pressure باي بتسوي مولاريتي في R في T وبناء على
+
+380
+00:34:51,840 --> 00:34:54,740
+ذلك عشان احسب ال Osmotic Pressure بدوا يلزمني
+
+381
+00:34:54,740 --> 00:34:58,500
+المولاريتي المولاريتي هي عبارة عن number of moles
+
+382
+00:34:58,500 --> 00:35:02,760
+of NaCl اذا عندى الوزن و ال molecular weight بحسب
+
+383
+00:35:02,760 --> 00:35:06,800
+ال number of moles مقسوم على ال volumeالان انا
+
+384
+00:35:06,800 --> 00:35:11,000
+عندى 3 او 4 من 10 جرام حولتهم الى مول يعني انا
+
+385
+00:35:11,000 --> 00:35:19,820
+عندى 0.058 مول موجودين في 1 لتر اذا انقسمت هذا ال
+
+386
+00:35:19,820 --> 00:35:24,040
+number of moles على ال total volume 1 لتر بتطلع ال
+
+387
+00:35:24,040 --> 00:35:28,480
+molarity بتسوء 0.058 molar الان لما بدون ال
+
+388
+00:35:28,480 --> 00:35:32,720
+osmotic pressure بيسوء ال molarity 0.058
+
+389
+00:35:34,990 --> 00:35:39,250
+مول لكل لتر الار بوينت او ات و ات و ات و ات و ات و
+
+390
+00:35:39,250 --> 00:35:40,790
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+391
+00:35:40,790 --> 00:35:44,930
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+392
+00:35:44,930 --> 00:35:49,450
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+393
+00:35:49,450 --> 00:35:50,690
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+394
+00:35:50,690 --> 00:35:50,790
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+395
+00:35:50,790 --> 00:35:50,810
+ات و ات و ات و ات و ات
+
+396
+00:36:00,200 --> 00:36:08,440
+بدنا ننتبه ان ال NACL هو مركب ايوني بينقسم ل two
+
+397
+00:36:08,440 --> 00:36:13,460
+ions و هذه الخواص Colligative Properties بتعتمد
+
+398
+00:36:13,460 --> 00:36:17,520
+على العدد يعني انا حقيقة كأنه عندي NA plus و كأنه
+
+399
+00:36:17,520 --> 00:36:22,560
+عندي CL minus وبناء على ذلك الجواب اللي بيطلع عنده
+
+400
+00:36:22,560 --> 00:36:26,620
+لان هذا المركب ايوني الواحد منه بيقسم لاتنين لازم
+
+401
+00:36:26,620 --> 00:36:32,030
+اضربه في اتنينوعشان هيك ال final osmotic pressure
+
+402
+00:36:32,030 --> 00:36:39,610
+in fact is 2.80 atmosphere لو كان المركب non-ionic
+
+403
+00:36:39,610 --> 00:36:42,810
+بخليه زي ما هو ال osmotic pressure اللي طلع عندي
+
+404
+00:36:42,810 --> 00:36:46,490
+لكن لو كان بيقسم له ايونين بدي اضربه في اتنين لو
+
+405
+00:36:46,490 --> 00:36:51,170
+كان بيقسم له تلاتة مثلا بدي اضربه في تلاتة و هكذا
+
+406
+00:37:01,450 --> 00:37:06,930
+نشوف سؤال عن موضوع الـ Osmotic Pressure What is
+
+407
+00:37:06,930 --> 00:37:12,390
+the osmotic pressure formed by dissolving 44.2
+
+408
+00:37:12,390 --> 00:37:18,390
+milligram of aspirin in 0.358 liter of water؟طبعاً
+
+409
+00:37:18,390 --> 00:37:21,810
+واضح انه دي مسألة بسيطة احنا عارفين الـ Osmotic
+
+410
+00:37:21,810 --> 00:37:25,450
+Pressure باي بيساوي Molarity في R في T اذا لازم
+
+411
+00:37:25,450 --> 00:37:29,710
+المولاريتي لما بيعطيني وزن الاسبيرين اللي دوّبته
+
+412
+00:37:29,710 --> 00:37:35,880
+في الميه 44.2 ميلي جرام بقدر احسبالنمبر المولز في
+
+413
+00:37:35,880 --> 00:37:40,440
+الاسبيرين الاسبيرين صيغته العامة C9H8O4
+
+414
+00:37:40,440 --> 00:37:51,340
+الموليكيلار ويت 9×12 8×1 4×16 180 موليكيلار
+
+415
+00:37:51,340 --> 00:37:51,720
+ويت
+
+416
+00:37:56,390 --> 00:38:00,090
+بنفعش اقسم milligram على gram ال molecular weight
+
+417
+00:38:00,090 --> 00:38:05,170
+جرام لكل مول بناء على ذلك بقدر احول milligram إلى
+
+418
+00:38:05,170 --> 00:38:09,210
+gram من خلال انه اضرب في عشرة أس ناقص تلاتة اللي
+
+419
+00:38:09,210 --> 00:38:14,310
+هو معامل التحويل بين milligram و gram صارت هذه
+
+420
+00:38:14,310 --> 00:38:19,500
+gram وصارت هذه gramلكل مول بنشطب لجرام مع الجرام
+
+421
+00:38:19,500 --> 00:38:28,120
+بيطلع الجواب 2.46×10-4 مول الان number of moles
+
+422
+00:38:28,120 --> 00:38:33,480
+على ال total volume هو ال molarity بنقسم 2.46×10-4
+
+423
+00:38:33,480 --> 00:38:40,020
+على ال total volume بتطلع ال molarity 6.86×10-4
+
+424
+00:38:40,020 --> 00:38:44,520
+مولار لما بدي ال buy هي عبارة عن ال molarity اللي
+
+425
+00:38:44,520 --> 00:38:50,250
+أنا جيبتها الآنفي الـR 0.0821 في الـtemperature in
+
+426
+00:38:50,250 --> 00:38:55,150
+Kelvin بتطلع الـR بتطلع الـPi اللي هي الـOsmotic
+
+427
+00:38:55,150 --> 00:39:01,610
+pressure in this case in atmosphere 0.0168 فلو كان
+
+428
+00:39:01,610 --> 00:39:06,190
+طالب مني ان اوجد هذه القيمة بالتور اللي هي
+
+429
+00:39:06,190 --> 00:39:12,420
+الmillimeter زئبق بدي اضرب في 760واضح؟ عشان احول
+
+430
+00:39:12,420 --> 00:39:20,260
+من atmosphere إلى tour بدي اضرب بـ 760 12.7 tour
+
+431
+00:39:20,260 --> 00:39:24,580
+الـ osmotic pressure لهذا المحلول من الاسبيرين في
+
+432
+00:39:24,580 --> 00:39:25,520
+الميّه
+
+433
+00:39:29,700 --> 00:39:36,040
+أخر سؤال نحله على الموضوع الـ Osmotic Pressure
+
+434
+00:39:36,040 --> 00:39:42,260
+بيحكينا عن Lysosome is an enzyme هذا إنزيم اسمه
+
+435
+00:39:42,260 --> 00:39:47,140
+Lysosome that breaks bacterial cell walls بيكسر
+
+436
+00:39:47,140 --> 00:39:53,480
+جدران اللي هي خلية البكتيريا solution containing 0
+
+437
+00:39:53,480 --> 00:39:59,140
+.150 gram of this enzymeيعني في عندي محلول دوبت
+
+438
+00:39:59,140 --> 00:40:03,500
+أنا point one five zero gram of this enzyme in two
+
+439
+00:40:03,500 --> 00:40:10,340
+hundred ten milliliter of solution في حجم نهائي
+
+440
+00:40:10,340 --> 00:40:15,760
+كان متين و عشرة mil has an osmotic pressure of
+
+441
+00:40:15,760 --> 00:40:17,380
+point nine five three
+
+442
+00:40:24,840 --> 00:40:33,160
+طبعا واضح هنا انه بيستعمل المعادلة تبعت ال ازموتك
+
+443
+00:40:33,160 --> 00:40:38,400
+بريشار باي بتساوي molecular weight عفوا باي بتساوي
+
+444
+00:40:38,400 --> 00:40:44,330
+molarity في R في Tاللي هو الـ Osmotic Pressure
+
+445
+00:40:44,330 --> 00:40:48,810
+معروفة لكن رجاءً بدنا ننتبه أن هذا الـ Osmotic
+
+446
+00:40:48,810 --> 00:40:52,570
+Pressure بتور الـ Osmotic Pressure بهذه المعادلة
+
+447
+00:40:52,570 --> 00:40:56,770
+لأن الـR فيها أتمسفير لازم يكون بالأتمسفير بناءً
+
+448
+00:40:56,770 --> 00:41:00,510
+على ذلك الـ Osmotic Pressure لازم نحوله لأتمسفير
+
+449
+00:41:00,510 --> 00:41:04,330
+ابتداءً بدنا نقول point nine five three نقسم على
+
+450
+00:41:04,330 --> 00:41:07,730
+سبعمية وستين one point two five في عشر أقص ناقص
+
+451
+00:41:07,730 --> 00:41:14,240
+تلاتة أتمسفير فالـ Pi بهذه الحالةلو عوضنا فيها بعد
+
+452
+00:41:14,240 --> 00:41:20,340
+القيمة المولاريتي مجهولة الـR معلومة 0.0821
+
+453
+00:41:20,340 --> 00:41:25,000
+أتموسفيل لتر لكل مول كيلوفر التمبراتور المعروفة
+
+454
+00:41:25,000 --> 00:41:31,960
+298 كيلوفر بنطلع المولاريتي اللي هي 5.12 في 10 أس
+
+455
+00:41:31,960 --> 00:41:38,000
+ناقص 5 مولار الآن المولاريتي بتساوي number of
+
+456
+00:41:38,000 --> 00:41:42,680
+moles على ال volumeوال number of moles اللي موجودة
+
+457
+00:41:42,680 --> 00:41:46,720
+في معادلة ال molarity بتساوي weight على molecular
+
+458
+00:41:46,720 --> 00:41:52,160
+weight الان كأنه صارت ال molarity بتساوي weight
+
+459
+00:41:52,160 --> 00:41:55,680
+على molecular weight كل هذا مجسوم على ال volume
+
+460
+00:41:55,680 --> 00:42:00,340
+طيب إذا احنا صار في عندنا ال molarity معروفة يعني
+
+461
+00:42:00,340 --> 00:42:07,430
+نعوضنا عنها ال weight اللي احنا أخدناه معروفالـ
+
+462
+00:42:07,430 --> 00:42:11,730
+Volume معروف، لم أبقى لدي مجهول غير الـ Molecular
+
+463
+00:42:11,730 --> 00:42:16,010
+Weight بقدر أحسب الـ Molecular Weight وهذه الحقيقة
+
+464
+00:42:16,010 --> 00:42:19,910
+هي طريقة لحساب الـ Molecular Weight لجزيئات
+
+465
+00:42:19,910 --> 00:42:23,090
+الكبيرة الـ Macromolecules زي الإنزايم وغيرها
+
+466
+00:42:23,090 --> 00:42:27,860
+باستعمال الـ Osmotic Pressureفاحنا في البداية
+
+467
+00:42:27,860 --> 00:42:31,260
+اوجدنا ال molarity من معادلة pi بتساوي molarity في
+
+468
+00:42:31,260 --> 00:42:36,880
+R في T و بنعرف احنا ان ال molarity هي عبارة عن
+
+469
+00:42:36,880 --> 00:42:39,760
+moles ع volume او weight على molecular weight هذا
+
+470
+00:42:39,760 --> 00:42:42,720
+الكلام كله هو number of moles weight ع molecular
+
+471
+00:42:42,720 --> 00:42:48,770
+weight كله مقسوم ع volume و بنقدر نحسبقيمة ال
+
+472
+00:42:48,770 --> 00:42:52,070
+molecular weight لو اي حد بده ييجي يقول والله ال
+
+473
+00:42:52,070 --> 00:42:55,450
+molarity moles على volume ويحسب ال moles بشكل
+
+474
+00:42:55,450 --> 00:42:59,190
+منفصل ويطلع قيمتها و بعدين ييجي يقول ال moles
+
+475
+00:42:59,190 --> 00:43:02,450
+بتساوي weight ع molecular weight يعني يرايح حاله
+
+476
+00:43:02,450 --> 00:43:05,250
+بتالي مافيش اي مشكلة
+
+477
+00:43:13,260 --> 00:43:18,440
+بهيك بنكون انتهينا من مناقشة مسائل الفصل الثالث
+
+478
+00:43:18,440 --> 00:43:22,580
+عشر وإلى لقاء آخر السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..4fa0bb29523aab97e33de4843746a8cbc4944654
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 2429, "start": 4.89, "end": 24.29, "text": "بسم الله الرحمن الرحيم السلام عليكم ورحمة الله وبركاته بنستمر ان شاء الله اليوم مع طلاب الكيمياق العامة للعلوم الصحية والهندسية وموضوعنا اليوم في درس مناقشة للفصل الثالث عشر", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224, 44945, 14851, 29973, 16472, 13412, 16606, 21984, 45595, 20498, 20449, 23032, 15040, 3555, 33251, 32640, 25528, 4587, 18863, 10943, 3660, 24976, 30241, 20498, 31767, 5016, 10632, 16070, 3224, 41260, 3794, 10632, 4032, 2304, 2407, 11242, 45367, 8315, 45595, 20498, 8978, 11778, 2288, 3794, 3714, 8315, 4587, 8592, 3660, 24976, 5172, 36520, 6024, 104, 6027, 12984, 6225, 46309], "avg_logprob": -0.09934413580246913, "compression_ratio": 1.904191616766467, "no_speech_prob": 7.152557373046875e-07, "words": [{"start": 4.89, "end": 5.23, "word": "بسم", "probability": 0.881591796875}, 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{"start": 62.47, "end": 64.13, "word": " المذاب", "probability": 0.7696126302083334}, {"start": 64.13, "end": 64.99, "word": " والمذيب،", "probability": 0.8439534505208334}, {"start": 64.99, "end": 65.09, "word": " الـ", "probability": 0.5128173828125}, {"start": 65.09, "end": 65.43, "word": " solute", "probability": 0.6959228515625}, {"start": 65.43, "end": 66.87, "word": " والسولفان،", "probability": 0.7129313151041666}, {"start": 66.87, "end": 67.07, "word": " هذا", "probability": 0.270751953125}, {"start": 67.07, "end": 67.35, "word": " يجب", "probability": 0.77392578125}, {"start": 67.35, "end": 67.37, "word": " أن", "probability": 0.72265625}, {"start": 67.37, "end": 67.55, "word": " يكون", "probability": 0.974853515625}, {"start": 67.55, "end": 68.15, "word": " أهم", "probability": 0.839599609375}, {"start": 68.15, "end": 72.69, "word": " نوع", "probability": 0.56268310546875}, {"start": 72.69, "end": 72.93, "word": " من", "probability": 0.7314453125}, {"start": 72.93, "end": 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"word": " في", "probability": 0.50244140625}, {"start": 84.7, "end": 85.56, "word": " البنزين", "probability": 0.978759765625}, {"start": 85.56, "end": 86.8, "word": " طبعا", "probability": 0.942138671875}, {"start": 86.8, "end": 87.74, "word": " واضح", "probability": 0.9630533854166666}, {"start": 87.74, "end": 88.08, "word": " ان", "probability": 0.6943359375}, {"start": 88.08, "end": 88.96, "word": " هدول", "probability": 0.8538411458333334}, {"start": 88.96, "end": 89.96, "word": " الجزيئين", "probability": 0.839453125}, {"start": 89.96, "end": 90.48, "word": " non", "probability": 0.453369140625}, {"start": 90.48, "end": 91.0, "word": " polar", "probability": 0.73046875}, {"start": 91.0, "end": 91.78, "word": " molecules", "probability": 0.7529296875}], "temperature": 1.0}, {"id": 5, "seek": 10906, "start": 93.44, "end": 109.06, "text": "الـ CCL4 جزء بهذا الشكل وهو بالكيميا تترا هيدرال مجموع البولاريتي له صفر والبنزين", "tokens": [6027, 39184, 12630, 43, 19, 10874, 11622, 38207, 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"start": 258.68, "end": 267.96, "text": "هذا هو الـ HCl نفس الشيء الـ Acetonitrile لما احنا بنتطلع على CH3", "tokens": [3224, 15730, 31439, 2423, 39184, 389, 9966, 8717, 36178, 25124, 1829, 38207, 2423, 39184, 5097, 26342, 270, 470, 306, 5296, 15042, 1975, 5016, 8315, 4724, 29399, 9566, 1211, 3615, 15844, 5995, 18], "avg_logprob": -0.3037405303030303, "compression_ratio": 1.0526315789473684, "no_speech_prob": 0.0, "words": [{"start": 258.68, "end": 259.04, "word": "هذا", "probability": 0.700927734375}, {"start": 259.04, "end": 259.26, "word": " هو", "probability": 0.90087890625}, {"start": 259.26, "end": 259.68, "word": " الـ", "probability": 0.691650390625}, {"start": 259.68, "end": 260.46, "word": " HCl", "probability": 0.62451171875}, {"start": 260.46, "end": 261.68, "word": " نفس", "probability": 0.855712890625}, {"start": 261.68, "end": 262.6, "word": " الشيء", "probability": 0.96826171875}, {"start": 262.6, "end": 262.98, "word": " الـ", "probability": 0.708984375}, {"start": 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{"start": 389.67, "end": 390.25, "word": " كمية", "probability": 0.9498697916666666}, {"start": 390.25, "end": 391.09, "word": " إضافية", "probability": 0.930908203125}, {"start": 391.09, "end": 392.07, "word": " مقدرة", "probability": 0.9376627604166666}, {"start": 392.07, "end": 393.29, "word": " حوالي", "probability": 0.96044921875}, {"start": 393.29, "end": 394.77, "word": " 116", "probability": 0.915771484375}, {"start": 394.77, "end": 395.35, "word": " جرام", "probability": 0.8805338541666666}, {"start": 395.35, "end": 396.71, "word": " إضافي", "probability": 0.945068359375}, {"start": 396.71, "end": 397.77, "word": " دابوا", "probability": 0.88623046875}, {"start": 397.77, "end": 398.63, "word": " نتيجة", "probability": 0.990478515625}, {"start": 398.63, "end": 399.05, "word": " إيش", "probability": 0.7096354166666666}, {"start": 399.05, "end": 399.65, "word": " نتيجة", "probability": 0.927978515625}, {"start": 399.65, "end": 400.41, "word": " زيادة", "probability": 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{"start": 831.36, "end": 831.78, "word": " جرام", "probability": 0.9827473958333334}, {"start": 831.78, "end": 832.6, "word": " فلما", "probability": 0.9178059895833334}, {"start": 832.6, "end": 832.86, "word": " أنا", "probability": 0.57421875}, {"start": 832.86, "end": 833.24, "word": " بدي", "probability": 0.4814453125}, {"start": 833.24, "end": 833.48, "word": " أقول", "probability": 0.91455078125}, {"start": 833.48, "end": 833.68, "word": " ال", "probability": 0.81298828125}, {"start": 833.68, "end": 833.88, "word": " ton", "probability": 0.75}, {"start": 833.88, "end": 834.52, "word": " بذاته", "probability": 0.8935546875}, {"start": 834.52, "end": 834.9, "word": " ألف", "probability": 0.83935546875}, {"start": 834.9, "end": 835.22, "word": " و", "probability": 0.477294921875}, {"start": 835.22, "end": 835.38, "word": " ال", "probability": 0.85107421875}, {"start": 835.38, "end": 835.9, "word": " kilogram", "probability": 0.96044921875}, {"start": 835.9, "end": 837.42, "word": " نفسه", "probability": 0.9970703125}, {"start": 837.42, "end": 837.96, "word": " ألف", "probability": 0.94140625}, {"start": 837.96, "end": 838.46, "word": " جرام", "probability": 0.98193359375}], "temperature": 1.0}, {"id": 37, "seek": 86148, "start": 839.68, "end": 861.48, "text": "بناء على ذلك لما انا بدي اقول ال tone a chemical عبارة عن الف في الف يعني واحد في عشر أقص ستة واضح فلما بدي احسب ال mass percent في هذه الحالة بدي اقول ال weight على", "tokens": [3555, 1863, 16606, 15844, 29910, 23275, 5296, 15042, 1975, 8315, 4724, 16254, 1975, 39648, 2423, 8027, 257, 7313, 6225, 3555, 9640, 3660, 18871, 27188, 8978, 27188, 37495, 22653, 36764, 24401, 8978, 6225, 46309, 5551, 4587, 9381, 8608, 2655, 3660, 4032, 46958, 5016, 6156, 1211, 15042, 4724, 16254, 1975, 5016, 35457, 2423, 2758, 3043, 8978, 29538, 21542, 6027, 3660, 4724, 16254, 1975, 39648, 2423, 3364, 15844], "avg_logprob": -0.2803030429464398, "compression_ratio": 1.5621301775147929, "no_speech_prob": 0.0, "words": [{"start": 839.68, "end": 840.16, "word": "بناء", "probability": 0.9112955729166666}, {"start": 840.16, "end": 840.54, "word": " على", "probability": 0.68701171875}, {"start": 840.54, "end": 841.26, "word": " ذلك", "probability": 0.98291015625}, {"start": 841.26, "end": 841.68, "word": " لما", "probability": 0.54779052734375}, {"start": 841.68, "end": 841.92, "word": " انا", "probability": 0.77392578125}, {"start": 841.92, "end": 842.14, "word": " بدي", "probability": 0.72021484375}, {"start": 842.14, "end": 842.44, "word": " اقول", "probability": 0.9150390625}, {"start": 842.44, "end": 842.64, "word": " ال", "probability": 0.69580078125}, {"start": 842.64, "end": 842.84, "word": " tone", "probability": 0.170166015625}, {"start": 842.84, "end": 843.08, "word": " a", "probability": 0.52099609375}, {"start": 843.08, "end": 843.38, "word": " chemical", "probability": 0.305419921875}, {"start": 843.38, "end": 844.58, "word": " عبارة", "probability": 0.862060546875}, {"start": 844.58, "end": 844.78, "word": " عن", "probability": 0.943359375}, {"start": 844.78, "end": 845.08, "word": " الف", "probability": 0.55810546875}, {"start": 845.08, "end": 845.24, "word": " في", "probability": 0.72412109375}, {"start": 845.24, "end": 845.46, "word": " الف", "probability": 0.87109375}, {"start": 845.46, "end": 845.86, "word": " يعني", "probability": 0.82421875}, {"start": 845.86, "end": 846.5, "word": " واحد", "probability": 0.90283203125}, {"start": 846.5, "end": 846.64, "word": " في", "probability": 0.90234375}, {"start": 846.64, "end": 846.94, "word": " عشر", "probability": 0.804443359375}, {"start": 846.94, "end": 847.42, "word": " أقص", "probability": 0.7952473958333334}, {"start": 847.42, "end": 849.0, "word": " ستة", "probability": 0.92822265625}, {"start": 849.0, "end": 849.86, "word": " واضح", "probability": 0.8776041666666666}, {"start": 849.86, "end": 851.42, "word": " فلما", "probability": 0.7779947916666666}, {"start": 851.42, "end": 851.7, "word": " بدي", "probability": 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{"id": 38, "seek": 87533, "start": 862.01, "end": 875.33, "text": "الـ total weight ضرب عشرة أقل ستة طبعا الـ total weight الحقيقة هو عبارة عن المليون جرام اللي هم الار", "tokens": [6027, 39184, 3217, 3364, 48812, 25513, 6225, 8592, 25720, 5551, 4587, 1211, 8608, 2655, 3660, 23032, 3555, 3615, 995, 2423, 39184, 3217, 3364, 21542, 38436, 28671, 31439, 6225, 3555, 9640, 3660, 18871, 9673, 20292, 11536, 10874, 2288, 10943, 13672, 1829, 8032, 2304, 2423, 9640], "avg_logprob": -0.3017361111111111, "compression_ratio": 1.376068376068376, "no_speech_prob": 5.960464477539062e-07, "words": [{"start": 862.01, "end": 862.49, "word": "الـ", "probability": 0.4229736328125}, {"start": 862.49, "end": 862.83, "word": " total", "probability": 0.32470703125}, {"start": 862.83, "end": 863.53, "word": " weight", "probability": 0.92626953125}, {"start": 863.53, "end": 864.25, "word": " ضرب", "probability": 0.6693115234375}, {"start": 864.25, "end": 865.53, "word": " عشرة", "probability": 0.6904296875}, {"start": 865.53, "end": 865.73, "word": " أقل", "probability": 0.656005859375}, {"start": 865.73, "end": 866.27, "word": " ستة", "probability": 0.96435546875}, {"start": 866.27, "end": 867.13, "word": " طبعا", "probability": 0.777923583984375}, {"start": 867.13, "end": 867.49, "word": " الـ", "probability": 0.6251220703125}, {"start": 867.49, "end": 867.71, "word": " total", "probability": 0.8681640625}, {"start": 867.71, "end": 868.09, "word": " weight", "probability": 0.9443359375}, {"start": 868.09, "end": 868.97, "word": " الحقيقة", "probability": 0.9632161458333334}, {"start": 868.97, "end": 869.61, "word": " هو", "probability": 0.60009765625}, {"start": 869.61, "end": 870.17, "word": " عبارة", "probability": 0.992431640625}, {"start": 870.17, "end": 870.91, "word": " عن", "probability": 0.9951171875}, {"start": 870.91, "end": 872.71, "word": " المليون", "probability": 0.8636067708333334}, {"start": 872.71, "end": 873.33, "word": " جرام", "probability": 0.8639322916666666}, 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{"start": 960.28, "end": 960.52, "word": " of", "probability": 0.966796875}, {"start": 960.52, "end": 960.84, "word": " moles", "probability": 0.88623046875}, {"start": 960.84, "end": 961.24, "word": " weight", "probability": 0.6796875}, {"start": 961.24, "end": 961.5, "word": " على", "probability": 0.66650390625}, {"start": 961.5, "end": 961.92, "word": " molecular", "probability": 0.716796875}, {"start": 961.92, "end": 962.5, "word": " weight", "probability": 0.919921875}], "temperature": 1.0}, {"id": 43, "seek": 98340, "start": 962.94, "end": 983.4, "text": "الماية عند الوزن ال number of moles برضه weight على molecular weight المولي فراكشن المولي فراكشن هو عبارة عن number of moles of methanol على ال total moles", "tokens": [6027, 15042, 10632, 43242, 2423, 2407, 11622, 1863, 2423, 1230, 295, 34286, 4724, 43042, 3224, 3364, 15844, 19046, 3364, 9673, 12610, 1829, 6156, 23557, 4117, 8592, 1863, 9673, 12610, 1829, 6156, 23557, 4117, 8592, 1863, 31439, 6225, 3555, 9640, 3660, 18871, 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"word": " كان", "probability": 0.9951171875}, {"start": 2206.73, "end": 2207.23, "word": " بيقسم", "probability": 0.9954833984375}, {"start": 2207.23, "end": 2207.39, "word": " له", "probability": 0.9521484375}, {"start": 2207.39, "end": 2208.61, "word": " تلاتة", "probability": 0.9671630859375}, {"start": 2208.61, "end": 2209.19, "word": " مثلا", "probability": 0.96923828125}, {"start": 2209.19, "end": 2209.69, "word": " بدي", "probability": 0.925537109375}, {"start": 2209.69, "end": 2210.07, "word": " اضربه", "probability": 0.8782958984375}, {"start": 2210.07, "end": 2210.15, "word": " في", "probability": 0.95947265625}, {"start": 2210.15, "end": 2210.61, "word": " تلاتة", "probability": 0.984619140625}, {"start": 2210.61, "end": 2210.69, "word": " و", "probability": 0.76708984375}, {"start": 2210.69, "end": 2211.17, "word": " هكذا", "probability": 0.8961588541666666}], "temperature": 1.0}, {"id": 94, "seek": 223767, "start": 2221.45, "end": 2237.67, "text": "نشوف سؤال عن موضوع الـ Osmotic Pressure What is the osmotic pressure formed by dissolving 44.2 milligram of aspirin in 0.358 liter of water؟", "tokens": [1863, 8592, 38688, 8608, 33604, 6027, 18871, 3714, 2407, 11242, 45367, 2423, 39184, 8875, 76, 9411, 6776, 540, 708, 307, 264, 3003, 76, 9411, 3321, 8693, 538, 15840, 798, 16408, 13, 17, 38298, 295, 20003, 259, 294, 1958, 13, 8794, 23, 2733, 295, 1281, 22807], "avg_logprob": -0.2802309879790182, "compression_ratio": 1.1041666666666667, "no_speech_prob": 0.0, "words": [{"start": 2221.4500000000003, "end": 2222.53, "word": "نشوف", "probability": 0.5048014322916666}, {"start": 2222.53, "end": 2223.61, "word": " سؤال", "probability": 0.7932942708333334}, {"start": 2223.61, "end": 2223.83, "word": " عن", "probability": 0.5693359375}, {"start": 2223.83, "end": 2224.37, "word": " موضوع", "probability": 0.90966796875}, {"start": 2224.37, "end": 2224.51, "word": " الـ", "probability": 0.6090087890625}, {"start": 2224.51, "end": 2225.05, "word": " Osmotic", 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milligram", "probability": 0.467041015625}, {"start": 2232.85, "end": 2233.29, "word": " of", "probability": 0.95556640625}, {"start": 2233.29, "end": 2234.17, "word": " aspirin", "probability": 0.849853515625}, {"start": 2234.17, "end": 2234.43, "word": " in", "probability": 0.83642578125}, {"start": 2234.43, "end": 2234.67, "word": " 0", "probability": 0.87353515625}, {"start": 2234.67, "end": 2235.95, "word": ".358", "probability": 0.9669596354166666}, {"start": 2235.95, "end": 2236.33, "word": " liter", "probability": 0.50390625}, {"start": 2236.33, "end": 2236.59, "word": " of", "probability": 0.94287109375}, {"start": 2236.59, "end": 2237.67, "word": " water؟", "probability": 0.89208984375}], "temperature": 1.0}, {"id": 95, "seek": 225351, "start": 2237.79, "end": 2253.51, "text": "طبعاً واضح انه دي مسألة بسيطة احنا عارفين الـ Osmotic Pressure باي بيساوي Molarity في R في T اذا لازم المولاريتي لما بيعطيني وزن الاسبيرين اللي دوّبته في الميه 44.2 ميلي جرام بقدر احسب", "tokens": 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0.99609375}, {"start": 2462.27, "end": 2462.55, "word": " point", "probability": 0.359130859375}, {"start": 2462.55, "end": 2462.89, "word": " nine", "probability": 0.73828125}, {"start": 2462.89, "end": 2463.25, "word": " five", "probability": 0.83154296875}, {"start": 2463.25, "end": 2463.69, "word": " three", "probability": 0.9248046875}, {"start": 2463.69, "end": 2464.17, "word": " نقسم", "probability": 0.9871419270833334}, {"start": 2464.17, "end": 2464.33, "word": " على", "probability": 0.422607421875}, {"start": 2464.33, "end": 2464.85, "word": " سبعمية", "probability": 0.7783203125}, {"start": 2464.85, "end": 2465.45, "word": " وستين", "probability": 0.94091796875}, {"start": 2465.45, "end": 2465.97, "word": " one", "probability": 0.55322265625}, {"start": 2465.97, "end": 2466.23, "word": " point", "probability": 0.96630859375}, {"start": 2466.23, "end": 2466.45, "word": " two", "probability": 0.94140625}, {"start": 2466.45, "end": 2466.77, "word": " five", "probability": 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مجهولة الـR معلومة 0.0821 أتموسفيل لتر لكل مول كيلوفر التمبراتور المعروفة 298 كيلوفر بنطلع المولاريتي اللي هي 5.12 في 10 أس ناقص 5 مولار الآن المولاريتي بتساوي number of moles على ال volume", "tokens": [1211, 2407, 6225, 2407, 11242, 8315, 8978, 11296, 39182, 25062, 32640, 3660, 9673, 12610, 9640, 1829, 31371, 3714, 7435, 3224, 12610, 3660, 2423, 39184, 49, 20449, 1211, 20498, 3660, 1958, 13, 16133, 4436, 5551, 39237, 41779, 5172, 26895, 5296, 2655, 2288, 5296, 28820, 3714, 12610, 9122, 26895, 38688, 2288, 16712, 2304, 26890, 9307, 13063, 9673, 3615, 32887, 5172, 3660, 9413, 23, 9122, 26895, 38688, 2288, 44945, 9566, 1211, 3615, 9673, 12610, 9640, 1829, 31371, 13672, 1829, 39896, 1025, 13, 4762, 8978, 1266, 5551, 3794, 8717, 995, 4587, 9381, 1025, 3714, 12610, 9640, 6024, 48506, 9673, 12610, 9640, 1829, 31371, 39894, 3794, 995, 45865, 1230, 295, 34286, 15844, 2423, 5523], "avg_logprob": -0.25965907736258076, "compression_ratio": 1.6711111111111112, "no_speech_prob": 0.0, "words": 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"probability": 0.640625}, {"start": 2497.64, "end": 2498.0, "word": " of", "probability": 0.9736328125}, {"start": 2498.0, "end": 2498.44, "word": " moles", "probability": 0.87890625}, {"start": 2498.44, "end": 2498.74, "word": " على", "probability": 0.76123046875}, {"start": 2498.74, "end": 2498.92, "word": " ال", "probability": 0.822265625}, {"start": 2498.92, "end": 2499.36, "word": " volume", "probability": 0.91845703125}], "temperature": 1.0}, {"id": 106, "seek": 252550, "start": 2500.2, "end": 2525.5, "text": "وال number of moles اللي موجودة في معادلة ال molarity بتساوي weight على molecular weight الان كأنه صارت ال molarity بتساوي weight على molecular weight كل هذا مجسوم على ال volume طيب إذا احنا صار في عندنا ال molarity معروفة يعني نعوضنا عنها ال weight اللي احنا أخدناه معروف", "tokens": [2407, 6027, 1230, 295, 34286, 13672, 1829, 3714, 29245, 23328, 3660, 8978, 20449, 18513, 37977, 2423, 8015, 17409, 39894, 3794, 995, 45865, 3364, 15844, 19046, 3364, 2423, 7649, 9122, 33456, 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2534.85, "word": " Weight", "probability": 0.96484375}, {"start": 2534.85, "end": 2535.55, "word": " وهذه", "probability": 0.76416015625}, {"start": 2535.55, "end": 2536.01, "word": " الحقيقة", "probability": 0.9503580729166666}, {"start": 2536.01, "end": 2536.17, "word": " هي", "probability": 0.98095703125}, {"start": 2536.17, "end": 2536.65, "word": " طريقة", "probability": 0.9791666666666666}, {"start": 2536.65, "end": 2537.33, "word": " لحساب", "probability": 0.95703125}, {"start": 2537.33, "end": 2537.63, "word": " الـ", "probability": 0.8564453125}, {"start": 2537.63, "end": 2538.05, "word": " Molecular", "probability": 0.91845703125}, {"start": 2538.05, "end": 2538.55, "word": " Weight", "probability": 0.94287109375}, {"start": 2538.55, "end": 2539.91, "word": " لجزيئات", "probability": 0.8292643229166666}, {"start": 2539.91, "end": 2540.43, "word": " الكبيرة", "probability": 0.7765299479166666}, {"start": 2540.43, "end": 2540.65, "word": " الـ", "probability": 0.6044921875}, {"start": 2540.65, "end": 2541.57, "word": " Macromolecules", "probability": 0.8382568359375}, {"start": 2541.57, "end": 2541.89, "word": " زي", "probability": 0.836669921875}, {"start": 2541.89, "end": 2542.57, "word": " الإنزايم", "probability": 0.77099609375}, {"start": 2542.57, "end": 2543.09, "word": " وغيرها", "probability": 0.943115234375}, {"start": 2543.09, "end": 2544.67, "word": " باستعمال", "probability": 0.96884765625}, {"start": 2544.67, "end": 2545.03, "word": " الـ", "probability": 0.713134765625}, {"start": 2545.03, "end": 2545.65, "word": " Osmotic", "probability": 0.9054361979166666}, {"start": 2545.65, "end": 2546.45, "word": " Pressure", "probability": 0.91015625}], "temperature": 1.0}, {"id": 108, "seek": 256636, "start": 2546.64, "end": 2566.36, "text": "فاحنا في البداية اوجدنا ال molarity من معادلة pi بتساوي molarity في R في T و بنعرف احنا ان ال molarity هي عبارة عن moles ع volume او weight على molecular weight هذا الكلام كله هو number of moles weight ع molecular weight كله مقسوم ع volume و بنقدر نحسب", "tokens": [5172, 39319, 8315, 8978, 29739, 28259, 10632, 1975, 29245, 3215, 8315, 2423, 8015, 17409, 9154, 20449, 18513, 37977, 3895, 39894, 3794, 995, 45865, 8015, 17409, 8978, 497, 8978, 314, 4032, 44945, 3615, 28480, 1975, 5016, 8315, 16472, 2423, 8015, 17409, 39896, 6225, 3555, 9640, 3660, 18871, 34286, 6225, 5523, 1975, 2407, 3364, 15844, 19046, 3364, 23758, 2423, 28820, 10943, 28242, 3224, 31439, 1230, 295, 34286, 3364, 6225, 19046, 3364, 28242, 3224, 3714, 4587, 3794, 20498, 6225, 5523, 4032, 44945, 28543, 2288, 8717, 5016, 35457], "avg_logprob": -0.25606618628782385, "compression_ratio": 1.714975845410628, "no_speech_prob": 0.0, "words": [{"start": 2546.64, "end": 2547.1, "word": "فاحنا", "probability": 0.5145263671875}, {"start": 2547.1, "end": 2547.22, "word": " في", "probability": 0.716796875}, {"start": 2547.22, "end": 2547.86, "word": " البداية", "probability": 0.9876302083333334}, {"start": 2547.86, "end": 2548.44, "word": " اوجدنا", "probability": 0.76580810546875}, {"start": 2548.44, "end": 2548.58, "word": " ال", "probability": 0.84814453125}, {"start": 2548.58, "end": 2549.14, "word": " molarity", "probability": 0.541015625}, {"start": 2549.14, "end": 2549.42, "word": " من", "probability": 0.9384765625}, {"start": 2549.42, "end": 2550.02, "word": " معادلة", "probability": 0.97509765625}, {"start": 2550.02, "end": 2550.28, "word": " pi", "probability": 0.1346435546875}, {"start": 2550.28, "end": 2550.64, "word": " بتساوي", "probability": 0.680908203125}, {"start": 2550.64, "end": 2551.1, "word": " molarity", "probability": 0.8251953125}, {"start": 2551.1, "end": 2551.26, "word": " في", "probability": 0.892578125}, {"start": 2551.26, "end": 2551.46, "word": " R", "probability": 0.426513671875}, {"start": 2551.46, "end": 2551.62, "word": " في", "probability": 0.2276611328125}, {"start": 2551.62, "end": 2551.86, "word": " T", "probability": 0.939453125}, {"start": 2551.86, "end": 2552.64, "word": " و", "probability": 0.7578125}, {"start": 2552.64, "end": 2554.16, "word": " بنعرف", "probability": 0.7882486979166666}, {"start": 2554.16, "end": 2554.64, "word": " احنا", "probability": 0.8943684895833334}, {"start": 2554.64, "end": 2555.16, "word": " ان", "probability": 0.90966796875}, {"start": 2555.16, "end": 2555.34, "word": " ال", "probability": 0.82763671875}, {"start": 2555.34, "end": 2555.96, "word": " molarity", "probability": 0.875732421875}, {"start": 2555.96, "end": 2556.26, "word": " هي", "probability": 0.8896484375}, {"start": 2556.26, "end": 2556.66, "word": " عبارة", "probability": 0.987060546875}, {"start": 2556.66, "end": 2556.88, "word": " عن", "probability": 0.95361328125}, {"start": 2556.88, "end": 2557.22, "word": " moles", "probability": 0.82958984375}, {"start": 2557.22, "end": 2557.38, "word": " ع", "probability": 0.79296875}, {"start": 2557.38, "end": 2557.76, "word": " volume", "probability": 0.91064453125}, {"start": 2557.76, "end": 2558.1, "word": " او", "probability": 0.81396484375}, {"start": 2558.1, "end": 2558.42, "word": " weight", "probability": 0.962890625}, {"start": 2558.42, "end": 2558.64, "word": " على", "probability": 0.5615234375}, {"start": 2558.64, "end": 2559.06, "word": " molecular", "probability": 0.837890625}, {"start": 2559.06, "end": 2559.48, "word": " weight", "probability": 0.93212890625}, {"start": 2559.48, "end": 2559.76, "word": " هذا", "probability": 0.6533203125}, {"start": 2559.76, "end": 2560.22, "word": " الكلام", "probability": 0.9348958333333334}, {"start": 2560.22, "end": 2560.52, "word": " كله", "probability": 0.97705078125}, {"start": 2560.52, "end": 2560.68, "word": " هو", "probability": 0.96435546875}, {"start": 2560.68, "end": 2560.98, "word": " number", "probability": 0.67919921875}, {"start": 2560.98, "end": 2561.22, "word": " of", "probability": 0.96533203125}, {"start": 2561.22, "end": 2561.52, "word": " moles", "probability": 0.90380859375}, {"start": 2561.52, "end": 2562.3, "word": " weight", "probability": 0.77880859375}, {"start": 2562.3, "end": 2562.4, "word": " ع", "probability": 0.7333984375}, {"start": 2562.4, "end": 2562.72, "word": " molecular", "probability": 0.861328125}, {"start": 2562.72, "end": 2563.12, "word": " weight", "probability": 0.93994140625}, {"start": 2563.12, "end": 2563.44, "word": " كله", "probability": 0.93212890625}, {"start": 2563.44, "end": 2563.82, "word": " مقسوم", "probability": 0.74652099609375}, {"start": 2563.82, "end": 2563.96, "word": " ع", "probability": 0.92724609375}, {"start": 2563.96, "end": 2564.44, "word": " volume", "probability": 0.9794921875}, {"start": 2564.44, "end": 2565.2, "word": " و", "probability": 0.79833984375}, {"start": 2565.2, "end": 2565.76, "word": " بنقدر", "probability": 0.90771484375}, {"start": 2565.76, "end": 2566.36, "word": " نحسب", "probability": 0.9943033854166666}], "temperature": 1.0}, {"id": 109, "seek": 258525, "start": 2568.17, "end": 2585.25, "text": "قيمة ال molecular weight لو اي حد بده ييجي يقول والله ال molarity moles على volume ويحسب ال moles بشكل منفصل ويطلع قيمتها و بعدين ييجي يقول ال moles بتساوي weight ع molecular weight يعني يرايح حاله بتالي مافيش اي مشكلة", "tokens": [4587, 32640, 3660, 2423, 19046, 3364, 45164, 1975, 1829, 11331, 3215, 47525, 3224, 7251, 1829, 7435, 1829, 7251, 39648, 16070, 43761, 2423, 8015, 17409, 34286, 15844, 5523, 4032, 1829, 5016, 35457, 2423, 34286, 4724, 8592, 28820, 9154, 5172, 36520, 4032, 1829, 9566, 1211, 3615, 12174, 32640, 2655, 11296, 4032, 39182, 9957, 7251, 1829, 7435, 1829, 7251, 39648, 2423, 34286, 39894, 3794, 995, 45865, 3364, 6225, 19046, 3364, 37495, 22653, 7251, 23557, 1829, 5016, 11331, 6027, 3224, 39894, 6027, 1829, 19446, 41185, 8592, 1975, 1829, 37893, 28820, 3660], "avg_logprob": -0.26864346116781235, "compression_ratio": 1.6974358974358974, "no_speech_prob": 4.76837158203125e-07, "words": [{"start": 2568.17, "end": 2568.67, "word": "قيمة", "probability": 0.6985270182291666}, {"start": 2568.67, "end": 2568.77, "word": " ال", "probability": 0.7626953125}, {"start": 2568.77, "end": 2569.15, "word": " molecular", "probability": 0.394287109375}, {"start": 2569.15, "end": 2569.49, "word": " weight", "probability": 0.9326171875}, {"start": 2569.49, "end": 2569.97, "word": " لو", "probability": 0.61376953125}, {"start": 2569.97, "end": 2570.25, "word": " اي", "probability": 0.4735107421875}, {"start": 2570.25, "end": 2570.59, "word": " حد", "probability": 0.774169921875}, {"start": 2570.59, "end": 2571.15, "word": " بده", "probability": 0.360107421875}, {"start": 2571.15, "end": 2571.43, "word": " ييجي", "probability": 0.7391357421875}, {"start": 2571.43, "end": 2571.67, "word": " يقول", "probability": 0.98095703125}, {"start": 2571.67, "end": 2571.93, "word": " والله", "probability": 0.6055908203125}, {"start": 2571.93, "end": 2572.07, "word": " ال", "probability": 0.91650390625}, {"start": 2572.07, "end": 2572.59, "word": " molarity", "probability": 0.5731201171875}, {"start": 2572.59, "end": 2573.07, "word": " moles", "probability": 0.333251953125}, {"start": 2573.07, "end": 2573.37, "word": " على", "probability": 0.75146484375}, {"start": 2573.37, "end": 2573.85, "word": " volume", "probability": 0.88916015625}, {"start": 2573.85, "end": 2574.37, "word": " ويحسب", "probability": 0.930908203125}, {"start": 2574.37, "end": 2574.55, "word": " ال", "probability": 0.666015625}, {"start": 2574.55, "end": 2574.93, "word": " moles", "probability": 0.95849609375}, {"start": 2574.93, "end": 2575.45, "word": " بشكل", "probability": 0.982421875}, {"start": 2575.45, "end": 2576.11, "word": " منفصل", "probability": 0.9464518229166666}, {"start": 2576.11, "end": 2576.65, "word": " ويطلع", "probability": 0.93662109375}, {"start": 2576.65, "end": 2577.33, "word": " قيمتها", "probability": 0.9869384765625}, {"start": 2577.33, "end": 2577.79, "word": " و", "probability": 0.59912109375}, {"start": 2577.79, "end": 2578.23, "word": " بعدين", "probability": 0.7449951171875}, {"start": 2578.23, "end": 2578.51, "word": " ييجي", "probability": 0.945556640625}, {"start": 2578.51, "end": 2578.71, "word": " يقول", "probability": 0.995361328125}, {"start": 2578.71, "end": 2578.89, "word": " ال", "probability": 0.80078125}, {"start": 2578.89, "end": 2579.19, "word": " moles", "probability": 0.96435546875}, {"start": 2579.19, "end": 2579.71, "word": " بتساوي", "probability": 0.8563232421875}, {"start": 2579.71, "end": 2580.15, "word": " weight", "probability": 0.462158203125}, {"start": 2580.15, "end": 2580.29, "word": " ع", "probability": 0.56298828125}, {"start": 2580.29, "end": 2580.69, "word": " molecular", "probability": 0.70849609375}, {"start": 2580.69, "end": 2581.13, "word": " weight", "probability": 0.943359375}, {"start": 2581.13, "end": 2581.33, "word": " يعني", "probability": 0.948486328125}, {"start": 2581.33, "end": 2581.77, "word": " يرايح", "probability": 0.84423828125}, {"start": 2581.77, "end": 2582.45, "word": " حاله", "probability": 0.9791666666666666}, {"start": 2582.45, "end": 2583.17, "word": " بتالي", "probability": 0.5357259114583334}, {"start": 2583.17, "end": 2583.63, "word": " مافيش", "probability": 0.9152018229166666}, {"start": 2583.63, "end": 2584.01, "word": " اي", "probability": 0.841064453125}, {"start": 2584.01, "end": 2585.25, "word": " مشكلة", "probability": 0.9855143229166666}], "temperature": 1.0}, {"id": 110, "seek": 260258, "start": 2593.26, "end": 2602.58, "text": "بهيك بنكون انتهينا من مناقشة مسائل الفصل الثالث عشر وإلى لقاء آخر السلام عليكم ورحمة الله وبركاته", "tokens": [3555, 3224, 1829, 4117, 44945, 30544, 16472, 47395, 1829, 8315, 9154, 3714, 8315, 4587, 8592, 3660, 47524, 16373, 1211, 27188, 36520, 6024, 104, 6027, 12984, 6225, 46309, 4032, 28814, 23942, 5296, 4587, 16606, 19753, 34740, 21136, 37440, 25894, 24793, 4032, 2288, 35571, 3660, 21984, 4032, 26890, 4117, 9307, 3224], "avg_logprob": -0.1585937437415123, "compression_ratio": 1.4471544715447155, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 2593.26, "end": 2593.94, "word": "بهيك", "probability": 0.68359375}, {"start": 2593.94, "end": 2594.72, "word": " بنكون", "probability": 0.6751708984375}, {"start": 2594.72, "end": 2595.34, "word": " انتهينا", "probability": 0.9493408203125}, {"start": 2595.34, "end": 2595.56, "word": " من", "probability": 0.9833984375}, {"start": 2595.56, "end": 2596.48, "word": " مناقشة", "probability": 0.9771484375}, {"start": 2596.48, "end": 2597.3, "word": " مسائل", "probability": 0.9153645833333334}, {"start": 2597.3, "end": 2597.96, "word": " الفصل", "probability": 0.859375}, {"start": 2597.96, "end": 2598.44, "word": " الثالث", "probability": 0.8843994140625}, {"start": 2598.44, "end": 2598.76, "word": " عشر", "probability": 0.97021484375}, {"start": 2598.76, "end": 2599.2, "word": " وإلى", "probability": 0.7527669270833334}, {"start": 2599.2, "end": 2599.84, "word": " لقاء", "probability": 0.92822265625}, {"start": 2599.84, "end": 2600.32, "word": " آخر", "probability": 0.817138671875}, {"start": 2600.32, "end": 2600.92, "word": " السلام", "probability": 0.8369140625}, {"start": 2600.92, "end": 2601.32, "word": " عليكم", "probability": 0.9580078125}, {"start": 2601.32, "end": 2601.74, "word": " ورحمة", "probability": 0.9693603515625}, {"start": 2601.74, "end": 2601.86, "word": " الله", "probability": 0.96826171875}, {"start": 2601.86, "end": 2602.58, "word": " وبركاته", "probability": 0.869384765625}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2602.9569375, "duration_after_vad": 2485.099999999994}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..20b837bad456a24f127a85802acf61f9e9a69ba3
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/hGqFGvjkFm0_raw.srt
@@ -0,0 +1,1952 @@
+1
+00:00:04,890 --> 00:00:08,010
+بسم الله الرحمن الرحيم السلام عليكم ورحمة الله
+
+2
+00:00:08,010 --> 00:00:13,130
+وبركاته بنستمر ان شاء الله اليوم مع طلاب الكيمياق
+
+3
+00:00:13,130 --> 00:00:19,310
+العامة للعلوم الصحية والهندسية وموضوعنا اليوم في
+
+4
+00:00:19,310 --> 00:00:30,180
+درس مناقشة للفصل الثالث عشرالان هنتقي مجموعة محددة
+
+5
+00:00:30,180 --> 00:00:36,760
+من المسائل موجودة عند الطلاب ايضا ونحاول ان نقشها
+
+6
+00:00:36,760 --> 00:00:43,800
+السؤال الأول اللي بدنا نشوفه سؤال تلتاشر خمستاشر
+
+7
+00:00:43,800 --> 00:00:49,040
+من الكتاب بيقول القاتي indicate the type of solute
+
+8
+00:00:49,040 --> 00:00:54,350
+solvent interaction يعني وضح ليهلما تتم عملية
+
+9
+00:00:54,350 --> 00:01:01,590
+الذوبان، ما هو نوع التفاعل أو الـ interaction ما
+
+10
+00:01:01,590 --> 00:01:07,350
+بين المذاب والمذيب، الـ solute والسولفان، هذا يجب
+
+11
+00:01:07,350 --> 00:01:12,690
+أن يكون أهم نوع
+
+12
+00:01:12,690 --> 00:01:15,930
+من الـ intermolecular forces بين الـ solute
+
+13
+00:01:15,930 --> 00:01:22,260
+والسولفانالان لو بدنا ناخد ال CCL4 اللى بنسميه
+
+14
+00:01:22,260 --> 00:01:28,960
+carbon tetrachloride في البنزين طبعا واضح ان هدول
+
+15
+00:01:28,960 --> 00:01:37,140
+الجزيئين non polar moleculesالـ CCL4 جزء بهذا
+
+16
+00:01:37,140 --> 00:01:46,080
+الشكل وهو بالكيميا تترا هيدرال مجموع البولاريتي له
+
+17
+00:01:46,080 --> 00:01:53,810
+صفر والبنزينالبنزين الكيميائي طبعا هو هذا البنزين
+
+18
+00:01:53,810 --> 00:02:00,170
+الحلقة اللى فيها ستة hydrogens C6H6 الان هذا non
+
+19
+00:02:00,170 --> 00:02:05,470
+polar وهذا non polar بناء على ذلك نوع القوة
+
+20
+00:02:05,470 --> 00:02:10,570
+between non polar molecules is dispersion forces
+
+21
+00:02:10,570 --> 00:02:19,860
+الان لو سألنا الفرع Bأي نوع القوة بين الميثانول
+
+22
+00:02:19,860 --> 00:02:23,260
+إذا احنا بنتطلع على الميثانول بنلاحظ انه فيه H
+
+23
+00:02:23,260 --> 00:02:27,880
+متصلة ب O إذا هو جاهز يعمل hydrogen bonding لو
+
+24
+00:02:27,880 --> 00:02:33,180
+بدأنا أجي أتطلع على الميه طبعا الكل حافظ شكلها H2O
+
+25
+00:02:33,180 --> 00:02:38,360
+برضه فيه H متصلة بO إذا هنا و هنا حيكون في عندنا
+
+26
+00:02:38,360 --> 00:02:43,940
+hydrogen bonding ما بين الميثانول و ما بين الميه
+
+27
+00:02:43,940 --> 00:02:50,800
+hydrogen bondingالان لو طلعنا على الفرق C كي بي ار
+
+28
+00:02:50,800 --> 00:02:55,420
+كي بي ار مركب ايوني يعني في نهاية حيصير كي بلس
+
+29
+00:02:55,420 --> 00:03:00,000
+وحيصير بي ار ماينوس الايونات بدأت تنفصل عن بعض
+
+30
+00:03:00,000 --> 00:03:06,660
+الان هذه الايونات بدأت تحاطب جزيئات الماء بحيث انه
+
+31
+00:03:06,660 --> 00:03:14,440
+الجزء السالب من الميه بيكون قربالجزء الموجب او ال
+
+32
+00:03:14,440 --> 00:03:19,980
+ايون الموجب و بنفس الوقت لو كنا بنشوف حوالين ال PR
+
+33
+00:03:19,980 --> 00:03:23,760
+حيكون الجزء الموجب من الميه اللي هو ال hydrogen
+
+34
+00:03:23,760 --> 00:03:27,860
+هنا الجزء السالب من الميه هو ال oxygen الجزء
+
+35
+00:03:27,860 --> 00:03:31,800
+الموجب هو ال hydrogen فاللي موجود عندى هو اللي انا
+
+36
+00:03:31,800 --> 00:03:36,460
+بسميه ايون هذا هو ال ايون دايبول ليش لأنه هنا في
+
+37
+00:03:36,460 --> 00:03:40,040
+عندى اناPartially Positive و Partially Negative
+
+38
+00:03:40,040 --> 00:03:44,940
+هذا دايبول و هذا ايون ايون دايبول و هذا ايون برضه
+
+39
+00:03:44,940 --> 00:03:50,060
+نفس الفكرة ايون دايبول و ايون دايبول فنوع البندق
+
+40
+00:03:50,060 --> 00:03:59,400
+هنا عبارة عن ايون دايبول بندقالفرق الأخير دي CH3
+
+41
+00:03:59,400 --> 00:04:06,980
+أو HCL طبعا الـ HCL لو احنا بدنا نتخيل شكله هذا هو
+
+42
+00:04:06,980 --> 00:04:12,140
+شكل الـ HCL فاللي بنلاحظه ان الـ CL partially
+
+43
+00:04:12,140 --> 00:04:16,580
+negative والـ H partially positive بسبب فرق
+
+44
+00:04:16,580 --> 00:04:24,620
+السالبيةهذا هو الـ HCl نفس الشيء الـ Acetonitrile
+
+45
+00:04:24,620 --> 00:04:27,960
+لما احنا بنتطلع على CH3
+
+46
+00:04:47,170 --> 00:04:51,150
+بناء على ذلك لو احنا بدنا نتخيل نوع ال bonding
+
+47
+00:04:51,150 --> 00:04:57,040
+هيكون ما بين هذا الطرفاللي موجود عندنا هنا السالب
+
+48
+00:04:57,040 --> 00:05:01,840
+من الكلور والطرف الموجب هنا بدو يصير ال bonding
+
+49
+00:05:01,840 --> 00:05:07,140
+فهذا عبارة عن die ball وهذا عبارة عن die ball فال
+
+50
+00:05:07,140 --> 00:05:11,700
+attraction die ball die ball attraction die ball
+
+51
+00:05:11,700 --> 00:05:16,900
+die ball attraction هذا
+
+52
+00:05:16,900 --> 00:05:24,020
+بالنسبة للفرع الأول اللي بيحكي عن نوع ال bondingلو
+
+53
+00:05:24,020 --> 00:05:28,640
+شوفنا سؤال تلتاشر تلاتة و عشرين بيحكينا عن الـ
+
+54
+00:05:28,640 --> 00:05:34,140
+solubility of a chromium nitrate 9H2O in water is
+
+55
+00:05:34,140 --> 00:05:40,360
+208 per gram per 100 gram of water at 15 degrees
+
+56
+00:05:40,360 --> 00:05:44,760
+centigrade طبعا هنا في نقطة كتير مهمة ان ال
+
+57
+00:05:44,760 --> 00:05:48,320
+solubility زي ما احنا بتعلمنا بتعتمد على درجة
+
+58
+00:05:48,320 --> 00:05:55,390
+الحرارة يعني كأنه عند خمستاشردرجة مئوية اللي هي ال
+
+59
+00:05:55,390 --> 00:06:04,630
+temperature السليوبيليت كانت عبارة عن 208
+
+60
+00:06:04,630 --> 00:06:11,850
+السليوبيليت لكل 100 مل 208 اجرام الان لو زدنا
+
+61
+00:06:11,850 --> 00:06:17,650
+الحرارة السليوبيليت بدأ تزيد هذا اللي صارفلما زدنا
+
+62
+00:06:17,650 --> 00:06:23,950
+الحرارة عند 35 درجة مئوية السليوبيليت زادت بدل ما
+
+63
+00:06:23,950 --> 00:06:32,070
+كانت 208 صارت 324 بمعنى انه في كمية إضافية مقدرة
+
+64
+00:06:32,070 --> 00:06:40,410
+حوالي 116 جرام إضافي دابوا نتيجة إيش نتيجة زيادة
+
+65
+00:06:40,410 --> 00:06:47,770
+درجة الحرارة يعني مش مش شيء بسيطplus 116 gram كل
+
+66
+00:06:47,770 --> 00:06:51,430
+هذه الذائبية ازاي ده نتيجة ايش زيادة درجة الحرارة
+
+67
+00:06:51,430 --> 00:06:57,270
+الان when this solution هذا اللي هو على الحرارة
+
+68
+00:06:57,270 --> 00:07:01,070
+العالية when this solution is slowly cold to 15
+
+69
+00:07:01,070 --> 00:07:05,710
+degree centigrade لو عودنا بردنا ايش المفروض اللي
+
+70
+00:07:05,710 --> 00:07:09,730
+يصير في هذه الحالة المفروض اللي يصير انه من هنا
+
+71
+00:07:09,730 --> 00:07:16,610
+يترسب ناقص 116 gramيترثب و اللي ذاب نتيجة التسخين
+
+72
+00:07:16,610 --> 00:07:22,510
+يترثب نتيجة التبريد اللي لاحظوا انه عملنا هالكلام
+
+73
+00:07:22,510 --> 00:07:29,050
+طبعا للأسف الشديد ماصارش الترثيب بمعنى صار في عندى
+
+74
+00:07:29,050 --> 00:07:33,350
+بهذه الحالة صار في عندى بهذه الحالة انه انا بردت ل
+
+75
+00:07:33,350 --> 00:07:44,400
+15 درجة و ما زال ذائب الـ324 جرامطبعا هذا اكتر من
+
+76
+00:07:44,400 --> 00:07:48,300
+احتمال المحلول عند 15 درجة مقوية لان المحلول عند
+
+77
+00:07:48,300 --> 00:07:52,540
+15 درجة مقوية كان بتشبع بـ 208 و بنقول عنه
+
+78
+00:07:52,540 --> 00:07:58,500
+saturated و المحلول عند 35 درجة مقوية بتشبع ب324
+
+79
+00:07:58,500 --> 00:08:03,600
+جرام و بنقول عنه saturated بس 324 جرام عند 15 درجة
+
+80
+00:08:03,600 --> 00:08:08,420
+مقوية هذا بيحمل اكتر من طاقته يعني عند 15 طاقته
+
+81
+00:08:08,920 --> 00:08:13,720
+متين و تمانية الان اللى موجود انه دايب فعلا فيه
+
+82
+00:08:13,720 --> 00:08:17,420
+تلاتمية و اربعة عشرين جرام اذا لما انا بدى اوصف
+
+83
+00:08:17,420 --> 00:08:22,120
+هذا الكلام what term describes this solution شو
+
+84
+00:08:22,120 --> 00:08:28,300
+بيكون هذا المحلول هذا بنسميه احنا super saturated
+
+85
+00:08:28,300 --> 00:08:34,280
+هذا المحلول فوق المشبعواضح انه احنا لما بردنا
+
+86
+00:08:34,280 --> 00:08:40,180
+الاصل يترسب لكن للأسف الشديد ما ترسبش الان بيقولنا
+
+87
+00:08:40,180 --> 00:08:45,320
+what action شو ممكن تعمل انت عشان تخليه يترسب طبعا
+
+88
+00:08:45,320 --> 00:08:51,750
+احنا أخدناتقنيات معينة للمحليل ال super saturated
+
+89
+00:08:51,750 --> 00:08:56,270
+كيف ممكن تترسب احد هذه التقنيات انه نعمل seeding
+
+90
+00:08:56,270 --> 00:09:01,790
+بمعنى ان نجيب كمية صغيرة زي بذرة او نواء نجيب كمية
+
+91
+00:09:01,790 --> 00:09:05,730
+صغيرة من المادة و نضيفها على المحلول فبصير يترسب
+
+92
+00:09:05,730 --> 00:09:10,530
+حول هذه الاشياء الصغيرة اللي احنا اضفناها او
+
+93
+00:09:10,530 --> 00:09:14,690
+البذور الصغيرة اللي احنا اضفناهاأو بنعمل
+
+94
+00:09:14,690 --> 00:09:21,050
+scratching بنحك جدار ال flask فبتنفصل جزيئات زجاج
+
+95
+00:09:21,050 --> 00:09:26,050
+بسيطة جدا بتم الترسيب حولها فالتقنيات اما بدنا
+
+96
+00:09:26,050 --> 00:09:32,230
+نعمل عشان نسرع ترسب المادة seeding نجيب شوية صغيرة
+
+97
+00:09:32,230 --> 00:09:36,620
+منها و نحطهاأو scratching انحق الجدار الزجاجي
+
+98
+00:09:36,620 --> 00:09:40,180
+scratching the glass walls of the container عادة
+
+99
+00:09:40,180 --> 00:09:43,900
+بستعمل glass rod يعني بيكون فينا قضيب من الزجاج
+
+100
+00:09:43,900 --> 00:09:49,940
+بنحق الزجاج بالزجاج تنفصل يعني أشياء very very
+
+101
+00:09:49,940 --> 00:09:52,920
+small بتم الترسب حولها
+
+102
+00:10:01,910 --> 00:10:07,510
+السؤال التسعة و عشرين بيقولناcommon laboratory
+
+103
+00:10:07,510 --> 00:10:11,870
+solvents يعني من ال solvents المذيبات المشهورة جدا
+
+104
+00:10:11,870 --> 00:10:17,470
+في المختبر include أسيتون منها الأسيتون والأسيتون
+
+105
+00:10:17,470 --> 00:10:22,310
+polar solvent ميثانول الميثانول أيضا polar solvent
+
+106
+00:10:22,310 --> 00:10:26,790
+وبيعمل hydrogen bonding الطلوين الطلوين أحد مشتقات
+
+107
+00:10:26,790 --> 00:10:32,730
+البنزين اللي موجود بهذا الشكل حلقة بنزين عليها CH3
+
+108
+00:10:32,730 --> 00:10:40,200
+هذا اللي احنا بنسمي ايه طلوينالطلوين والمياه
+
+109
+00:10:40,200 --> 00:10:44,920
+معروفة H2O which of these is the best solvent for
+
+110
+00:10:44,920 --> 00:10:48,360
+non-polar solutes يعني هو انا اعطاني اربعة
+
+111
+00:10:48,360 --> 00:10:52,660
+solvents جالي لو عندي انا solute non-polar مين
+
+112
+00:10:52,660 --> 00:10:57,440
+بنسبه طبعا القاعدة non-polar بده non-polar الان ال
+
+113
+00:10:57,440 --> 00:11:01,680
+acetone لأنه فيه C double bond O يعني هذا هو شكل
+
+114
+00:11:01,680 --> 00:11:05,540
+ال acetone هذا هو شكل ال acetone
+
+115
+00:11:08,200 --> 00:11:14,340
+هنا في عندنا C double bond O فالـ O طرف سالب والـ
+
+116
+00:11:14,340 --> 00:11:18,820
+C طرف موجب إذا هذا polar واضح الاسيتون is polar
+
+117
+00:11:20,370 --> 00:11:24,790
+والميثانول بولر بيعمل hydrogen bonding والماء بولر
+
+118
+00:11:24,790 --> 00:11:25,350
+بيعمل hydrogen bonding والماء بولر بيعمل hydrogen
+
+119
+00:11:25,350 --> 00:11:25,610
+bonding والماء بولر بيعمل hydrogen bonding والماء
+
+120
+00:11:25,610 --> 00:11:25,630
+بولر بيعمل hydrogen bonding والماء بولر بيعمل
+
+121
+00:11:25,630 --> 00:11:25,630
+hydrogen bonding والماء بولر بيعمل hydrogen
+
+122
+00:11:25,630 --> 00:11:25,630
+bonding والماء بولر بيعمل hydrogen bonding والماء
+
+123
+00:11:25,630 --> 00:11:25,790
+بولر بيعمل hydrogen bonding والماء بولر بيعمل
+
+124
+00:11:25,790 --> 00:11:25,830
+hydrogen bonding والماء بولر بيعمل hydrogen
+
+125
+00:11:25,830 --> 00:11:26,670
+bonding والماء بولر بيعمل hydrogen bonding والماء
+
+126
+00:11:26,670 --> 00:11:27,750
+بولر بيعمل hydrogen bonding والماء بولر بيعمل
+
+127
+00:11:27,750 --> 00:11:29,550
+hydrogen bonding والماء بولر بيعمل hydrogen
+
+128
+00:11:29,550 --> 00:11:33,790
+bonding والماء بولر بيعمل hydrogen bonding والماء
+
+129
+00:11:33,790 --> 00:11:40,690
+بولر بيعمل hydrogen bonding والماء بولر
+
+130
+00:11:40,690 --> 00:11:44,550
+بيعمل hydrogen bonding والماء ب
+
+131
+00:11:53,840 --> 00:11:58,380
+السؤال التالف تسعة و تلاتين بيحكينا عن وحدات
+
+132
+00:11:58,380 --> 00:12:06,420
+التركيز تخطط مجموعة صودم صلفية في حلقة تحتوي على
+
+133
+00:12:06,420 --> 00:12:13,110
+10.6 جرام صودم صلفية في 483 جرام ماءيعني دوّبنا
+
+134
+00:12:13,110 --> 00:12:18,090
+Sodium Sulfate كمية معينة في الميّه ايش ال mass
+
+135
+00:12:18,090 --> 00:12:22,670
+percentage طبعا ال mass percentage احنا بنعرفها هي
+
+136
+00:12:22,670 --> 00:12:26,190
+weight على ال total weight ضارب 100 الان ال weight
+
+137
+00:12:26,190 --> 00:12:31,230
+10.6 ال total weight هو عبارة عن مجموع هدول
+
+138
+00:12:31,230 --> 00:12:36,440
+التنتين ال 10.6 والميّه هذا هو ال total weightلما
+
+139
+00:12:36,440 --> 00:12:41,220
+بنجمعهم بيطلع 493.6 weight على total weight ضارب
+
+140
+00:12:41,220 --> 00:12:46,680
+100 هذا ال mass percent بيطلع النسبة 2.15% هذا
+
+141
+00:12:46,680 --> 00:12:53,140
+تركيز المحلول في ال mass percent هذا الفرق A الان
+
+142
+00:12:53,140 --> 00:13:00,180
+الفرق B بيقول ان or يعني خامةلما بيطولوا المعادل
+
+143
+00:13:00,180 --> 00:13:05,060
+من التراب و من غيرها بيكون عندهم خمات an ore
+
+144
+00:13:05,060 --> 00:13:12,080
+contains 2.86 gram of silver per ton of ore يعني
+
+145
+00:13:12,080 --> 00:13:19,480
+في عندنا خامة رمل موجود في فضةالان هذا الرمل في كل
+
+146
+00:13:19,480 --> 00:13:24,440
+طن منه اتنين و ستة و ثمانين من مائة جرام من الفضة
+
+147
+00:13:24,440 --> 00:13:28,140
+بيقول لي ايش ال concentration للفضة بال part per
+
+148
+00:13:28,140 --> 00:13:32,720
+million طبعا ال concentration هيكون ال weight على
+
+149
+00:13:32,720 --> 00:13:36,500
+ال total weight لما بده ياب ال BPM بده اضرب في
+
+150
+00:13:36,500 --> 00:13:41,280
+عشرة او ستةهيك تعريف ال bpm فإذا قلنا ال weight 2
+
+151
+00:13:41,280 --> 00:13:48,780
+.86 جرام الآن ال ton هنا بدنا ننتبه ألف كيلو جرام
+
+152
+00:13:48,780 --> 00:13:55,220
+ألف كيلو جرام فلما أنا بدي أقول ال ton بذاته ألف و
+
+153
+00:13:55,220 --> 00:14:02,140
+ال kilogram نفسه ألف جرامبناء على ذلك لما انا بدي
+
+154
+00:14:02,140 --> 00:14:05,860
+اقول ال tone a chemical عبارة عن الف في الف يعني
+
+155
+00:14:05,860 --> 00:14:14,520
+واحد في عشر أقص ستة واضح فلما بدي احسب ال mass
+
+156
+00:14:14,520 --> 00:14:22,490
+percent في هذه الحالة بدي اقول ال weight علىالـ
+
+157
+00:14:22,490 --> 00:14:27,710
+total weight ضرب عشرة أقل ستة طبعا الـ total
+
+158
+00:14:27,710 --> 00:14:34,070
+weight الحقيقة هو عبارة عن المليون جرام اللي هم
+
+159
+00:14:34,070 --> 00:14:39,530
+الارأو what is the concentration of the silver لأ
+
+160
+00:14:39,530 --> 00:14:44,510
+هو ال ore بتحتوي على هذا الوزن بناء على ذلك يعني
+
+161
+00:14:44,510 --> 00:14:49,830
+داخل فيها يعني الخامة نفسها الوزن الكل الطن هذا من
+
+162
+00:14:49,830 --> 00:14:56,390
+ضمنه ل 82 من 100 جرام اذا بناخدها زي ما هي 82 ال
+
+163
+00:14:56,390 --> 00:15:01,010
+total weight واحد في عشرة او ستة طبعا عشان بدهيا
+
+164
+00:15:01,010 --> 00:15:06,980
+بال BPM بده اضرب في عشرة او ستة واضح؟و لما نحسب
+
+165
+00:15:06,980 --> 00:15:10,900
+هذا الكلام weight على ال total weight ضرب 10 رقم 6
+
+166
+00:15:10,900 --> 00:15:17,620
+عشان نخليه بال parts per million بيطلع 2.86 واضح
+
+167
+00:15:17,620 --> 00:15:27,980
+هذا هو التركيز في وحدة ال part per million المسائل
+
+168
+00:15:27,980 --> 00:15:33,140
+التالية برضه على التركيزسؤال واحد واربعين الـ
+
+169
+00:15:33,140 --> 00:15:36,820
+solution is made containing fourteen point six
+
+170
+00:15:36,820 --> 00:15:44,380
+gram of water in one eight four gram14.6 جرام
+
+171
+00:15:44,380 --> 00:15:51,520
+ميثانول و 184 جرام ميثانول المولي فراكشن الميثانول
+
+172
+00:15:51,520 --> 00:15:55,880
+طبعا المولي فراكشن بيساوي ال number of moles of
+
+173
+00:15:55,880 --> 00:16:00,000
+methanol على ال total moles الان انا عند الوزن ال
+
+174
+00:16:00,000 --> 00:16:03,400
+number of moles weight على molecular weightالماية
+
+175
+00:16:03,400 --> 00:16:06,200
+عند الوزن ال number of moles برضه weight على
+
+176
+00:16:06,200 --> 00:16:18,420
+molecular weight المولي فراكشن المولي
+
+177
+00:16:18,420 --> 00:16:22,180
+فراكشن هو عبارة عن number of moles of methanol على
+
+178
+00:16:22,180 --> 00:16:28,190
+ال total molesواضح ال number of moles على ال total
+
+179
+00:16:28,190 --> 00:16:32,770
+mole اللي هي 0.46 زي 2.2 بيطلع ال mole fraction 0
+
+180
+00:16:32,770 --> 00:16:40,790
+.043 هذا الفرع A بعد ذلك يحتاج ال mass percent of
+
+181
+00:16:40,790 --> 00:16:45,780
+methanolالماس بيرسنت اخدنا هو عبارة عن الماس على
+
+182
+00:16:45,780 --> 00:16:50,100
+ال total mass ضارب مية بطلع seven point three five
+
+183
+00:16:50,100 --> 00:16:54,900
+percent الفرع C بده ال molality الآن بدنا نكون
+
+184
+00:16:54,900 --> 00:16:58,260
+عارفين شو ال molality ال molality بتساوي ال number
+
+185
+00:16:58,260 --> 00:17:02,040
+of moles على ال mass of solvent in kilogram ال
+
+186
+00:17:02,040 --> 00:17:05,740
+number of moles للميثانول point four six ال mass
+
+187
+00:17:05,740 --> 00:17:11,570
+of solvent184 لكن لازم تكون بالكيلوجرام اذا point
+
+188
+00:17:11,570 --> 00:17:15,590
+one eight four فالمولز على ال massive solvent in
+
+189
+00:17:15,590 --> 00:17:21,430
+kilogram بتطلع المحلول او بيطلع المحلول تركيزه two
+
+190
+00:17:21,430 --> 00:17:26,010
+point five molal و احنا اتفقنا ان ال molality
+
+191
+00:17:26,010 --> 00:17:32,150
+بنعطيها الرمز small m و ال molarity بنعطيها الرمز
+
+192
+00:17:32,150 --> 00:17:33,530
+capital M
+
+193
+00:17:40,050 --> 00:17:46,170
+نشوف سؤال 47 برضه على وحدات التركيز a sulfuric
+
+194
+00:17:46,170 --> 00:17:53,330
+acid solution containing 571.6 gram of sulfuric
+
+195
+00:17:53,330 --> 00:17:57,310
+acid per liter of solution لاحظوا هنا اعطاني
+
+196
+00:17:57,310 --> 00:18:02,390
+اكمجرام عندى من ال sulfuric acid على الحجم الحجم
+
+197
+00:18:02,390 --> 00:18:08,560
+المحلول الان هذا المحلولit has a density of 1.326
+
+198
+00:18:08,560 --> 00:18:12,440
+يعني ال density لحمض الكبريتيك اتجيلة الانسان لما
+
+199
+00:18:12,440 --> 00:18:16,600
+بيحمل بُتل فيه حمض الكبريتيك بيحسه ادجل من انه
+
+200
+00:18:16,600 --> 00:18:21,120
+يكون بُتل ميه يعني ال liter من الميه وزنه الف جرام
+
+201
+00:18:21,120 --> 00:18:26,580
+الان هنا من هذه ال density واضح ان ال one liter
+
+202
+00:18:26,580 --> 00:18:32,550
+وزنه الف وتلاتمية وتسعة وعشرين جرامواضح؟ اذا هذا
+
+203
+00:18:32,550 --> 00:18:36,810
+ال liter لو انا بدي اقبل وزن من خلال علاقة الكثافة
+
+204
+00:18:36,810 --> 00:18:40,930
+ال density بتسويت على volume بقول ان وزن ال liter
+
+205
+00:18:40,930 --> 00:18:50,270
+1329 gram هذا وزن ال liter واضح؟ ضربت بال الف الان
+
+206
+00:18:50,270 --> 00:18:53,350
+بيقول ال calculate the mass percent ال mass
+
+207
+00:18:53,350 --> 00:18:58,130
+percent هو وزن حمض الكبريتيك على وزن الميه الان if
+
+208
+00:18:58,130 --> 00:19:07,230
+the total massالتوتر هو وزن كل المحلول 1329 ووزن
+
+209
+00:19:07,230 --> 00:19:16,450
+الـ H2SO4 لحاله كان 571.6 إذن الفرق بينهم هو وزن
+
+210
+00:19:16,450 --> 00:19:21,210
+الماء واضح؟ وعشان ذلك نستطيع أن نحصل على مولز مياه
+
+211
+00:19:21,210 --> 00:19:25,510
+الماء الـ Total Mass minus Mass of Sulphuric Acid
+
+212
+00:19:25,510 --> 00:19:32,780
+757.4الان لو انا عندي وزن حمض الكابريتيك بقسم على
+
+213
+00:19:32,780 --> 00:19:36,860
+ال molecular weight بطلع ال number of moles وزي ما
+
+214
+00:19:36,860 --> 00:19:41,880
+قولت ال mass الو بتساوي 1329 من الكثافة لو بد ال
+
+215
+00:19:41,880 --> 00:19:46,210
+mass of waterهي الفرق ما بين الوزن الكل ووزن حمض
+
+216
+00:19:46,210 --> 00:19:53,610
+الكبريتيك 757 لو بد ال moles of water 757.4 على 18
+
+217
+00:19:53,610 --> 00:19:57,090
+على ال molecular weight صار عندي ال mass of water
+
+218
+00:19:57,090 --> 00:20:00,670
+و ال moles of water ال mass of sulfuric acid و ال
+
+219
+00:20:00,670 --> 00:20:04,850
+moles of sulfuric acid الان ايش بده اسئلة بكون سهل
+
+220
+00:20:04,850 --> 00:20:10,450
+علي أنا اجاوبه؟ماذا هو الـ Mass Percent؟ الـ Mass
+
+221
+00:20:10,450 --> 00:20:13,430
+Percent هي عبارة عن الـ Mass of Sulphuric Acid على
+
+222
+00:20:13,430 --> 00:20:19,880
+الـ Total Mass ضارب 100 تطلع 43.01What is the mole
+
+223
+00:20:19,880 --> 00:20:23,180
+fraction of sulphuric acid؟هو عبارة عن ال moles of
+
+224
+00:20:23,180 --> 00:20:27,320
+sulphuric acid على ال total moles انا هنا عندي ال
+
+225
+00:20:27,320 --> 00:20:31,260
+moles of sulphuric acid حسبتها وال moles of water
+
+226
+00:20:31,260 --> 00:20:37,300
+حسبتها المoles على ال total moles ضرب بدون ما نضرب
+
+227
+00:20:37,300 --> 00:20:41,480
+بمية لش بده fraction يعني بدون ما تنضرب مية moles
+
+228
+00:20:41,480 --> 00:20:45,760
+على ال total moles هاي ال mole fraction بنوجده
+
+229
+00:20:46,520 --> 00:20:52,040
+الفرع C المولالية المولالية number of moles اللي
+
+230
+00:20:52,040 --> 00:20:57,520
+انا برضه اوجدتها للسلفريك 5.83 على ال mass of
+
+231
+00:20:57,520 --> 00:21:01,380
+solvent in kilogram هاي ال mass of water اللي
+
+232
+00:21:01,380 --> 00:21:08,020
+حسبناها بالجرام لما ندق بال kilogram 0.7574 فبتطلع
+
+233
+00:21:08,020 --> 00:21:15,060
+المولالية 7.69 مولالالان لو بدوا ال مولاريتي الفرع
+
+234
+00:21:15,060 --> 00:21:21,140
+دي ال number of moles على ال total volume هو أساسا
+
+235
+00:21:21,140 --> 00:21:25,640
+ال total volume بيقول بير لتر يعني موجود فى لتر هى
+
+236
+00:21:25,640 --> 00:21:28,040
+ال number of moles of sulfuric acid و ال total
+
+237
+00:21:28,040 --> 00:21:34,180
+volume is one liter فالمولاريتي 5.83 مولار
+
+238
+00:21:47,440 --> 00:22:15,480
+سؤال 65 سؤال
+
+239
+00:22:15,480 --> 00:22:16,040
+65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+240
+00:22:16,040 --> 00:22:16,040
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+241
+00:22:16,040 --> 00:22:16,040
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+242
+00:22:16,040 --> 00:22:16,040
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+243
+00:22:16,040 --> 00:22:16,060
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+244
+00:22:16,060 --> 00:22:16,060
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+245
+00:22:16,060 --> 00:22:16,060
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+246
+00:22:16,060 --> 00:22:16,060
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+247
+00:22:16,060 --> 00:22:16,060
+سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65 سؤال 65
+
+248
+00:22:16,090 --> 00:22:21,670
+دوبته بالميه دوبته 22.5 جرام of lactose في 200
+
+249
+00:22:21,670 --> 00:22:28,590
+جرام ميه عند درجة حرارة 338 kelvin الان طبعا
+
+250
+00:22:28,590 --> 00:22:32,430
+هيلزمنا ال vapor pressure ل pure water يعني كأنه
+
+251
+00:22:32,430 --> 00:22:40,170
+هذه المسألة ان انا لو كان في عندى ميه عادية water
+
+252
+00:22:40,170 --> 00:22:44,850
+لحالها في عندى اللي بنسميه ال pure vapor pressure
+
+253
+00:22:47,180 --> 00:22:48,900
+Pure Vapor Pressure
+
+254
+00:22:55,870 --> 00:23:00,990
+الان هذا الضغط موجود في جداول .. هذا الضغط موجود
+
+255
+00:23:00,990 --> 00:23:05,770
+في جداول لما احنا بدنا نشوف ضغط الماء عند ضغط بخار
+
+256
+00:23:05,770 --> 00:23:14,090
+الماء بنلاقيه من الجدول 187.5 تور واضح؟ طيب هذا
+
+257
+00:23:14,090 --> 00:23:18,370
+ضغط الماء لما يكون نقي ..الان هذا الماء لو احنا
+
+258
+00:23:18,370 --> 00:23:25,090
+اجينا ودوبنا فيه .. ودوبنا فيه لكتوزطبعاً اللاكتوز
+
+259
+00:23:25,090 --> 00:23:33,150
+يقلل ضغط البخار لأنه صار جزء على سطح الماء
+
+260
+00:23:33,150 --> 00:23:38,350
+اللاكتوز يمنع التبخر بناء على ذلك كمية التبخر في
+
+261
+00:23:38,350 --> 00:23:45,050
+وجود اللاكتوز تقل واخدنا معادلة كيف نحسب ال vapor
+
+262
+00:23:45,050 --> 00:23:46,010
+pressure
+
+263
+00:23:54,440 --> 00:23:58,460
+الان المعلومات اللى موجودة عندنا اللاكتوز الى
+
+264
+00:23:58,460 --> 00:24:01,580
+molecular weight تلت مية و اتنين و اربعين و بناء
+
+265
+00:24:01,580 --> 00:24:05,520
+على ذلك لما بدي انا number of moles الى اتنين و
+
+266
+00:24:05,520 --> 00:24:08,600
+عشرين و نص على تلت مية و اتنين و اربعين point oh
+
+267
+00:24:08,600 --> 00:24:09,840
+six six mole
+
+268
+00:24:16,930 --> 00:24:22,630
+بناء على ذلك عشان احسب ال vapor pressure للمحلول
+
+269
+00:24:22,630 --> 00:24:27,190
+بدي المول fraction للميهاللي هي number of moles of
+
+270
+00:24:27,190 --> 00:24:32,390
+water على total number of moles بتطلع point nine
+
+271
+00:24:32,390 --> 00:24:36,490
+nine four الان اذا كنت انا عارف ال pressure بدي
+
+272
+00:24:36,490 --> 00:24:41,630
+اضرب ال mole fraction في ال vapor pressure لبور
+
+273
+00:24:41,630 --> 00:24:44,730
+water point nine nine four في مية و سبعة و ثمانية
+
+274
+00:24:44,730 --> 00:24:51,150
+point fiveالجواب 186.4 لاحظوا ان ال vapor pressure
+
+275
+00:24:51,150 --> 00:24:56,110
+بسبب إذابة اللاكتوز قل يعني لما بدأ اقارن بدون
+
+276
+00:24:56,110 --> 00:25:01,310
+لاكتوز كان ال vapor pressure 187.5 بوجود اللاكتوز
+
+277
+00:25:01,310 --> 00:25:09,750
+بقل صار 186.4 هذا بالنسبة للفرع A
+
+278
+00:25:12,130 --> 00:25:18,090
+الفرع بيه فرع على محلول تاني غير اللاكتوز والمية
+
+279
+00:25:18,090 --> 00:25:24,890
+اللي هو اسمه البروبيلين غلايكول والبروبيلين
+
+280
+00:25:24,890 --> 00:25:30,650
+غلايكول برضه هو عبارة عن مركب بيكون عادة non
+
+281
+00:25:30,650 --> 00:25:37,040
+volatile يعني غير متطاير مركب غير متطايرهذا
+
+282
+00:25:37,040 --> 00:25:43,280
+البروبيلين جليكول صيغته العامة C3H8O
+
+283
+00:25:43,280 --> 00:25:50,060
+وصيغته بهذا الشكل التفصيلي للناس اللي بيهتموا هنا
+
+284
+00:25:50,060 --> 00:26:01,120
+في عندي OH وهنا في عندي OH عشان كان C3H3328C3H8O2
+
+285
+00:26:01,120 --> 00:26:08,110
+هذا شكل البروبيلين جليكولالان بقول احسب كتلة
+
+286
+00:26:08,110 --> 00:26:13,330
+البروبيلين غلايكول اللي يجب ان يتم اضافه لـ 0.340
+
+287
+00:26:13,330 --> 00:26:22,010
+كيلو جرام ماء لتقلي البريشار الفيبر بي 2.88 تور في
+
+288
+00:26:22,010 --> 00:26:29,090
+40 درجة سنتيجريد واضح؟ طبعا لو بدنا نبدأ نشوف
+
+289
+00:26:29,090 --> 00:26:33,430
+المعطيات اللي عندنا ال molecular weight لهذا
+
+290
+00:26:33,430 --> 00:26:38,650
+المُركّب اللي هو البروبيلين جليكول بنحسبه 76 ال
+
+291
+00:26:38,650 --> 00:26:43,230
+pressure للميّه عند 40 degree centigrade خمسة و
+
+292
+00:26:43,230 --> 00:26:48,210
+خمسين و تلاتة الان هو بيقول ال pressure هذا بينخفض
+
+293
+00:26:48,210 --> 00:26:52,870
+لما أنا بدوّب شوية من البروبيلين جليكول في الميه
+
+294
+00:26:52,870 --> 00:26:58,900
+بينخفض ال pressure يعني ال new pressure بيصيرعبارة
+
+295
+00:26:58,900 --> 00:27:04,200
+عن الـ 55.3 مطروح منها الـ 2.88 اللي هو مقدار
+
+296
+00:27:04,200 --> 00:27:12,620
+الانخفاض بيصير 52.42 الان احنا كلنا عارفين ان ال
+
+297
+00:27:12,620 --> 00:27:16,520
+pressure بيساوي ال X في ال B node اللي هو ال
+
+298
+00:27:16,520 --> 00:27:20,340
+pressure ل ال pure water الان هذا ال pressure صار
+
+299
+00:27:20,340 --> 00:27:27,530
+موجود عندى 52.42 ال X المولي fraction مجهولةو ال
+
+300
+00:27:27,530 --> 00:27:32,170
+pressure الأصلي 55.3 pressure of pure water بناء
+
+301
+00:27:32,170 --> 00:27:36,710
+على ذلك انا بقدر احسب قيمة ال X ايش ال mole
+
+302
+00:27:36,710 --> 00:27:43,970
+fraction للميه الان لما انا بدي اجي احسب ايش هو ال
+
+303
+00:27:43,970 --> 00:27:46,910
+mole fraction للميه انا عندي number of moles of
+
+304
+00:27:46,910 --> 00:27:51,190
+water هذا ال mole fraction للميه هو عبارة عن
+
+305
+00:27:51,190 --> 00:27:55,550
+number of moles of water على ال total moles طبعا
+
+306
+00:27:56,210 --> 00:27:59,310
+الـ number of moles of water طالما الميّه تلاتمية
+
+307
+00:27:59,310 --> 00:28:02,410
+وأربعين باجسمها ع تمانتاش هاي ال number of moles
+
+308
+00:28:02,410 --> 00:28:07,970
+of water الان طيب ال number of moles of propylene
+
+309
+00:28:07,970 --> 00:28:12,110
+glycol انا ماعنديش كام مول of propylene glycol اذا
+
+310
+00:28:12,110 --> 00:28:16,550
+صار المجهول اللي عندي هنا هو ال number of moles of
+
+311
+00:28:16,550 --> 00:28:20,970
+propylene glycol هاي ال moles of water على ال
+
+312
+00:28:20,970 --> 00:28:23,810
+total moles اللي هي عبارة عن moles of water زاد
+
+313
+00:28:23,810 --> 00:28:27,810
+moles of propylene glycolفبحسب ال y اللى هى ال
+
+314
+00:28:27,810 --> 00:28:34,150
+moles of البروبيلين غلايكول بتطلع ال y 1.04 mole
+
+315
+00:28:34,150 --> 00:28:38,830
+هو الان مش طالب ال number of moles هو الان مش طالب
+
+316
+00:28:38,830 --> 00:28:44,110
+ال number of moles للبروبيلين غلايكول طالب ال mass
+
+317
+00:28:44,110 --> 00:28:49,970
+وانا بعرف انه عندى علاقة ان number of moles بساوي
+
+318
+00:28:49,970 --> 00:28:54,720
+weightعلى molecular weight فلما انا يكون عندي ال
+
+319
+00:28:54,720 --> 00:29:00,260
+number of moles صرت حاسبه لبروبيلين جلايكل و ال
+
+320
+00:29:00,260 --> 00:29:03,820
+molecular weight بعرفه من ال formula و لو حسبته 76
+
+321
+00:29:03,820 --> 00:29:07,840
+شو بيظل عندى ال weight ال weight بيصير بساوي ال
+
+322
+00:29:07,840 --> 00:29:12,320
+number of moles ضرب ال molecular weight هاي هذا
+
+323
+00:29:12,320 --> 00:29:17,980
+اللي عملناه انه ال mass فعلا لل C3H8O2 بيساوي ال
+
+324
+00:29:17,980 --> 00:29:22,980
+number of moles 1.04 في ال molecular weight 76
+
+325
+00:29:22,980 --> 00:29:29,800
+بيطلع وزن لبروبيلين جليكول 79.0 إجرام
+
+326
+00:29:47,280 --> 00:29:57,650
+ناخد سؤال 75 سؤال 75 بيحكي عن انخفاض فينقطة التجمد
+
+327
+00:29:57,650 --> 00:30:01,770
+يعني الماء لما بنضيف عليه non volatile solute
+
+328
+00:30:01,770 --> 00:30:05,990
+اتفقنا انه نقطة تجمده بتنخفض يعني الماء في الوضع
+
+329
+00:30:05,990 --> 00:30:12,190
+الطبيعي نقطة تجمده سفر بتنخفض بتصير بال minus الآن
+
+330
+00:30:12,190 --> 00:30:16,110
+بيجي بيسألنا السؤال how many grams of ethylene
+
+331
+00:30:16,110 --> 00:30:21,930
+glycol ومعطيني شكله C2H6O2وبرضه الـ Ethylene
+
+332
+00:30:21,930 --> 00:30:25,110
+Glycol اللي اللي بيحبوا يعرفوا شكله هو عبارة عن
+
+333
+00:30:25,110 --> 00:30:29,130
+CH2CH2OHOH
+
+334
+00:30:29,130 --> 00:30:35,030
+هذا اسمه Ethylene Glycolالان كم جرام من الاتلينيج
+
+335
+00:30:35,030 --> 00:30:39,930
+لايكول يجب ان يتم اضافه إلى واحد كيلو جرام من
+
+336
+00:30:39,930 --> 00:30:46,110
+الماء لإنتاج حلوة تتجمد في نصف درجة سنتيجريت طبعا
+
+337
+00:30:46,110 --> 00:30:51,890
+الاصل انه احنا بنعرف ان الميه عادة الوضع الطبيعي
+
+338
+00:30:51,890 --> 00:30:56,690
+تبعها بتتجمد في نصف درجة سنتيجريت فلما تصير انناقس
+
+339
+00:30:56,690 --> 00:31:04,180
+خمسة معنا ذلك دلتا تيبتساوي 5 درجات يعني كم انخفض
+
+340
+00:31:04,180 --> 00:31:08,760
+نقطة تجمد الميا من 0 لا ناقص 5 انخفضت 5 درجات الان
+
+341
+00:31:08,760 --> 00:31:13,420
+بقدر الانخفاض في نقطة التجمد بيساوي ثابت Kf في ال
+
+342
+00:31:13,420 --> 00:31:18,080
+morality الان الميا الثابت تبعها موجود في جداول
+
+343
+00:31:18,080 --> 00:31:25,220
+وموجود عندنا في الكتاب 1.86دلتا تي بتساوي ال five
+
+344
+00:31:25,220 --> 00:31:27,840
+بتساوي one point eight six في ال morality ال
+
+345
+00:31:27,840 --> 00:31:31,580
+morality بتساوي two point six nine الان ال
+
+346
+00:31:31,580 --> 00:31:35,260
+morality بتساوي number of moles على ال mass of
+
+347
+00:31:35,260 --> 00:31:40,560
+solvent انا ال mass of solvent عندياها واضح؟ one
+
+348
+00:31:40,560 --> 00:31:45,460
+kilogram of water ال mass of solvent وبناء على ذلك
+
+349
+00:31:45,460 --> 00:31:49,420
+لما انا بدي اقول ال morality بتساويالـ number of
+
+350
+00:31:49,420 --> 00:31:54,580
+moles على ال kilogram ممكن كمان بالمرة انا اقول ان
+
+351
+00:31:54,580 --> 00:31:59,040
+number of moles نفسها لال إيثالين جليكول بتسوية
+
+352
+00:31:59,040 --> 00:32:01,180
+على molecular weight يعني هذا ال weight على
+
+353
+00:32:01,180 --> 00:32:08,860
+molecular weight لمن؟ للإيثالين جليكول واضح؟
+
+354
+00:32:10,730 --> 00:32:12,210
+مثلًا الـ weight على molecular weight
+
+355
+00:32:12,210 --> 00:32:17,370
+للإيثالينيجليكول على ال mass of solvent in
+
+356
+00:32:17,370 --> 00:32:22,510
+kilogram لو قلت الآن ان الـ molality end معروفة و
+
+357
+00:32:22,510 --> 00:32:25,710
+ال molecular weight للإيثالينيجليكول معروف و ال
+
+358
+00:32:25,710 --> 00:32:30,910
+mass of solvent الميّه معروفة one kilogram مافيش
+
+359
+00:32:30,910 --> 00:32:35,560
+الحقيقة مجهول غير ال weightوبناء على ذلك بقدر احسب
+
+360
+00:32:35,560 --> 00:32:43,980
+ال weight 166.7 دمجنا عمليتين لو اي طالب ما بدوش
+
+361
+00:32:43,980 --> 00:32:49,560
+هذا الكلام ممكن يجي يقول ال morality بتساوي moles
+
+362
+00:32:49,560 --> 00:32:55,200
+على ال mass of solvent in kilogram وبالتالي ال
+
+363
+00:32:55,200 --> 00:33:00,520
+morality عنده موجودة بيعوض و ال mass موجودة بيحسب
+
+364
+00:33:00,520 --> 00:33:04,840
+ال numberof moles و بعد ما يحسب ال number of moles
+
+365
+00:33:04,840 --> 00:33:08,460
+بيقول هذا ال number of moles بتسويت على molecular
+
+366
+00:33:08,460 --> 00:33:12,380
+weight يعني ممكن الخطوتين يدخلهم في معادلة واحدة و
+
+367
+00:33:12,380 --> 00:33:15,860
+ممكن يفصلهم يعني اي طالب ممكن يقول ال morality
+
+368
+00:33:15,860 --> 00:33:21,760
+بتسوي moles على ال mass ال morality موجودةو ال
+
+369
+00:33:21,760 --> 00:33:26,900
+mass موجودة طبعا ال molality احنا حسبناها من نقطة
+
+370
+00:33:26,900 --> 00:33:31,500
+التجمد و ال mass of solvent موجودة هذه ال mass هي
+
+371
+00:33:31,500 --> 00:33:34,600
+ال mass of solvent in kilogram بيحسب number of
+
+372
+00:33:34,600 --> 00:33:37,940
+moles ال number of moles of ethylene glycol بتساوي
+
+373
+00:33:37,940 --> 00:33:41,040
+weight على molecular weight ال number of moles صار
+
+374
+00:33:41,040 --> 00:33:44,820
+موجود ال molecular weight موجود بيقدر يحسب ال
+
+375
+00:33:44,820 --> 00:33:48,940
+weight الان احنا هنا في نفس المعادلة عوضنا عن
+
+376
+00:33:48,940 --> 00:33:52,010
+number of moles weight على molecular weightعملناهم
+
+377
+00:33:52,010 --> 00:33:59,510
+خطوة واحدة بأي طريقة الطالب بحل بيصل لإيه الجواب
+
+378
+00:33:59,510 --> 00:34:07,110
+سؤال تمانية و سبعين بيقولنا sea water يعني ميط
+
+379
+00:34:07,110 --> 00:34:12,210
+البحر contains 3.4 gram of salts for every liter
+
+380
+00:34:12,210 --> 00:34:15,690
+of solution يعني في كل لتر من ميط البحر في اندي
+
+381
+00:34:15,690 --> 00:34:19,990
+تلاتة او اربعة من عشرة gram من الملحAssuming that
+
+382
+00:34:19,990 --> 00:34:24,410
+the solute consists entirely of NaCl خلّينا ناخد
+
+383
+00:34:24,410 --> 00:34:28,910
+فرضية أن هذا الملح هو ملح الطعام الـ NaCl الـ
+
+384
+00:34:28,910 --> 00:34:32,790
+Sodium Chloride هو عادة بيكون أكثر من 90% نفترض أن
+
+385
+00:34:32,790 --> 00:34:36,850
+كل الملح ملح الطعام بيقول Calculate the osmotic
+
+386
+00:34:36,850 --> 00:34:41,440
+pressure of sea waterطبعا احنا لما بدنا نقول الـ
+
+387
+00:34:41,440 --> 00:34:45,460
+Osmotic Pressure له معادلة المعادلة لل Osmotic
+
+388
+00:34:45,460 --> 00:34:51,840
+Pressure باي بتسوي مولاريتي في R في T وبناء على
+
+389
+00:34:51,840 --> 00:34:54,740
+ذلك عشان احسب ال Osmotic Pressure بدوا يلزمني
+
+390
+00:34:54,740 --> 00:34:58,500
+المولاريتي المولاريتي هي عبارة عن number of moles
+
+391
+00:34:58,500 --> 00:35:02,760
+of NaCl اذا عندى الوزن و ال molecular weight بحسب
+
+392
+00:35:02,760 --> 00:35:06,800
+ال number of moles مقسوم على ال volumeالان انا
+
+393
+00:35:06,800 --> 00:35:11,000
+عندى 3 او 4 من 10 جرام حولتهم الى مول يعني انا
+
+394
+00:35:11,000 --> 00:35:19,820
+عندى 0.058 مول موجودين في 1 لتر اذا انقسمت هذا ال
+
+395
+00:35:19,820 --> 00:35:24,040
+number of moles على ال total volume 1 لتر بتطلع ال
+
+396
+00:35:24,040 --> 00:35:28,480
+molarity بتسوء 0.058 molar الان لما بدون ال
+
+397
+00:35:28,480 --> 00:35:32,720
+osmotic pressure بيسوء ال molarity 0.058
+
+398
+00:35:34,990 --> 00:35:39,250
+مول لكل لتر الار بوينت او ات و ات و ات و ات و ات و
+
+399
+00:35:39,250 --> 00:35:40,790
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+400
+00:35:40,790 --> 00:35:44,930
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+401
+00:35:44,930 --> 00:35:49,450
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+402
+00:35:49,450 --> 00:35:50,690
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+403
+00:35:50,690 --> 00:35:50,790
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+404
+00:35:50,790 --> 00:35:50,790
+ات و ات و ات و ات و ات و ات و ات و ات و ات و ات و
+
+405
+00:35:50,790 --> 00:35:50,810
+ات و ات و ات و ات و ات
+
+406
+00:36:00,200 --> 00:36:08,440
+بدنا ننتبه ان ال NACL هو مركب ايوني بينقسم ل two
+
+407
+00:36:08,440 --> 00:36:13,460
+ions و هذه الخواص Colligative Properties بتعتمد
+
+408
+00:36:13,460 --> 00:36:17,520
+على العدد يعني انا حقيقة كأنه عندي NA plus و كأنه
+
+409
+00:36:17,520 --> 00:36:22,560
+عندي CL minus وبناء على ذلك الجواب اللي بيطلع عنده
+
+410
+00:36:22,560 --> 00:36:26,620
+لان هذا المركب ايوني الواحد منه بيقسم لاتنين لازم
+
+411
+00:36:26,620 --> 00:36:32,030
+اضربه في اتنينوعشان هيك ال final osmotic pressure
+
+412
+00:36:32,030 --> 00:36:39,610
+in fact is 2.80 atmosphere لو كان المركب non-ionic
+
+413
+00:36:39,610 --> 00:36:42,810
+بخليه زي ما هو ال osmotic pressure اللي طلع عندي
+
+414
+00:36:42,810 --> 00:36:46,490
+لكن لو كان بيقسم له ايونين بدي اضربه في اتنين لو
+
+415
+00:36:46,490 --> 00:36:51,170
+كان بيقسم له تلاتة مثلا بدي اضربه في تلاتة و هكذا
+
+416
+00:37:01,450 --> 00:37:06,930
+نشوف سؤال عن موضوع الـ Osmotic Pressure What is
+
+417
+00:37:06,930 --> 00:37:12,390
+the osmotic pressure formed by dissolving 44.2
+
+418
+00:37:12,390 --> 00:37:18,390
+milligram of aspirin in 0.358 liter of water؟طبعاً
+
+419
+00:37:18,390 --> 00:37:21,810
+واضح انه دي مسألة بسيطة احنا عارفين الـ Osmotic
+
+420
+00:37:21,810 --> 00:37:25,450
+Pressure باي بيساوي Molarity في R في T اذا لازم
+
+421
+00:37:25,450 --> 00:37:29,710
+المولاريتي لما بيعطيني وزن الاسبيرين اللي دوّبته
+
+422
+00:37:29,710 --> 00:37:35,880
+في الميه 44.2 ميلي جرام بقدر احسبالنمبر المولز في
+
+423
+00:37:35,880 --> 00:37:40,440
+الاسبيرين الاسبيرين صيغته العامة C9H8O4
+
+424
+00:37:40,440 --> 00:37:51,340
+الموليكيلار ويت 9×12 8×1 4×16 180 موليكيلار
+
+425
+00:37:51,340 --> 00:37:51,720
+ويت
+
+426
+00:37:56,390 --> 00:38:00,090
+بنفعش اقسم milligram على gram ال molecular weight
+
+427
+00:38:00,090 --> 00:38:05,170
+جرام لكل مول بناء على ذلك بقدر احول milligram إلى
+
+428
+00:38:05,170 --> 00:38:09,210
+gram من خلال انه اضرب في عشرة أس ناقص تلاتة اللي
+
+429
+00:38:09,210 --> 00:38:14,310
+هو معامل التحويل بين milligram و gram صارت هذه
+
+430
+00:38:14,310 --> 00:38:19,500
+gram وصارت هذه gramلكل مول بنشطب لجرام مع الجرام
+
+431
+00:38:19,500 --> 00:38:28,120
+بيطلع الجواب 2.46×10-4 مول الان number of moles
+
+432
+00:38:28,120 --> 00:38:33,480
+على ال total volume هو ال molarity بنقسم 2.46×10-4
+
+433
+00:38:33,480 --> 00:38:40,020
+على ال total volume بتطلع ال molarity 6.86×10-4
+
+434
+00:38:40,020 --> 00:38:44,520
+مولار لما بدي ال buy هي عبارة عن ال molarity اللي
+
+435
+00:38:44,520 --> 00:38:50,250
+أنا جيبتها الآنفي الـR 0.0821 في الـtemperature in
+
+436
+00:38:50,250 --> 00:38:55,150
+Kelvin بتطلع الـR بتطلع الـPi اللي هي الـOsmotic
+
+437
+00:38:55,150 --> 00:39:01,610
+pressure in this case in atmosphere 0.0168 فلو كان
+
+438
+00:39:01,610 --> 00:39:06,190
+طالب مني ان اوجد هذه القيمة بالتور اللي هي
+
+439
+00:39:06,190 --> 00:39:12,420
+الmillimeter زئبق بدي اضرب في 760واضح؟ عشان احول
+
+440
+00:39:12,420 --> 00:39:20,260
+من atmosphere إلى tour بدي اضرب بـ 760 12.7 tour
+
+441
+00:39:20,260 --> 00:39:24,580
+الـ osmotic pressure لهذا المحلول من الاسبيرين في
+
+442
+00:39:24,580 --> 00:39:25,520
+الميّه
+
+443
+00:39:29,700 --> 00:39:36,040
+أخر سؤال نحله على الموضوع الـ Osmotic Pressure
+
+444
+00:39:36,040 --> 00:39:42,260
+بيحكينا عن Lysosome is an enzyme هذا إنزيم اسمه
+
+445
+00:39:42,260 --> 00:39:47,140
+Lysosome that breaks bacterial cell walls بيكسر
+
+446
+00:39:47,140 --> 00:39:53,480
+جدران اللي هي خلية البكتيريا solution containing 0
+
+447
+00:39:53,480 --> 00:39:59,140
+.150 gram of this enzymeيعني في عندي محلول دوبت
+
+448
+00:39:59,140 --> 00:40:03,500
+أنا point one five zero gram of this enzyme in two
+
+449
+00:40:03,500 --> 00:40:10,340
+hundred ten milliliter of solution في حجم نهائي
+
+450
+00:40:10,340 --> 00:40:15,760
+كان متين و عشرة mil has an osmotic pressure of
+
+451
+00:40:15,760 --> 00:40:17,380
+point nine five three
+
+452
+00:40:24,840 --> 00:40:33,160
+طبعا واضح هنا انه بيستعمل المعادلة تبعت ال ازموتك
+
+453
+00:40:33,160 --> 00:40:38,400
+بريشار باي بتساوي molecular weight عفوا باي بتساوي
+
+454
+00:40:38,400 --> 00:40:44,330
+molarity في R في Tاللي هو الـ Osmotic Pressure
+
+455
+00:40:44,330 --> 00:40:48,810
+معروفة لكن رجاءً بدنا ننتبه أن هذا الـ Osmotic
+
+456
+00:40:48,810 --> 00:40:52,570
+Pressure بتور الـ Osmotic Pressure بهذه المعادلة
+
+457
+00:40:52,570 --> 00:40:56,770
+لأن الـR فيها أتمسفير لازم يكون بالأتمسفير بناءً
+
+458
+00:40:56,770 --> 00:41:00,510
+على ذلك الـ Osmotic Pressure لازم نحوله لأتمسفير
+
+459
+00:41:00,510 --> 00:41:04,330
+ابتداءً بدنا نقول point nine five three نقسم على
+
+460
+00:41:04,330 --> 00:41:07,730
+سبعمية وستين one point two five في عشر أقص ناقص
+
+461
+00:41:07,730 --> 00:41:14,240
+تلاتة أتمسفير فالـ Pi بهذه الحالةلو عوضنا فيها بعد
+
+462
+00:41:14,240 --> 00:41:20,340
+القيمة المولاريتي مجهولة الـR معلومة 0.0821
+
+463
+00:41:20,340 --> 00:41:25,000
+أتموسفيل لتر لكل مول كيلوفر التمبراتور المعروفة
+
+464
+00:41:25,000 --> 00:41:31,960
+298 كيلوفر بنطلع المولاريتي اللي هي 5.12 في 10 أس
+
+465
+00:41:31,960 --> 00:41:38,000
+ناقص 5 مولار الآن المولاريتي بتساوي number of
+
+466
+00:41:38,000 --> 00:41:42,680
+moles على ال volumeوال number of moles اللي موجودة
+
+467
+00:41:42,680 --> 00:41:46,720
+في معادلة ال molarity بتساوي weight على molecular
+
+468
+00:41:46,720 --> 00:41:52,160
+weight الان كأنه صارت ال molarity بتساوي weight
+
+469
+00:41:52,160 --> 00:41:55,680
+على molecular weight كل هذا مجسوم على ال volume
+
+470
+00:41:55,680 --> 00:42:00,340
+طيب إذا احنا صار في عندنا ال molarity معروفة يعني
+
+471
+00:42:00,340 --> 00:42:07,430
+نعوضنا عنها ال weight اللي احنا أخدناه معروفالـ
+
+472
+00:42:07,430 --> 00:42:11,730
+Volume معروف، لم أبقى لدي مجهول غير الـ Molecular
+
+473
+00:42:11,730 --> 00:42:16,010
+Weight بقدر أحسب الـ Molecular Weight وهذه الحقيقة
+
+474
+00:42:16,010 --> 00:42:19,910
+هي طريقة لحساب الـ Molecular Weight لجزيئات
+
+475
+00:42:19,910 --> 00:42:23,090
+الكبيرة الـ Macromolecules زي الإنزايم وغيرها
+
+476
+00:42:23,090 --> 00:42:27,860
+باستعمال الـ Osmotic Pressureفاحنا في البداية
+
+477
+00:42:27,860 --> 00:42:31,260
+اوجدنا ال molarity من معادلة pi بتساوي molarity في
+
+478
+00:42:31,260 --> 00:42:36,880
+R في T و بنعرف احنا ان ال molarity هي عبارة عن
+
+479
+00:42:36,880 --> 00:42:39,760
+moles ع volume او weight على molecular weight هذا
+
+480
+00:42:39,760 --> 00:42:42,720
+الكلام كله هو number of moles weight ع molecular
+
+481
+00:42:42,720 --> 00:42:48,770
+weight كله مقسوم ع volume و بنقدر نحسبقيمة ال
+
+482
+00:42:48,770 --> 00:42:52,070
+molecular weight لو اي حد بده ييجي يقول والله ال
+
+483
+00:42:52,070 --> 00:42:55,450
+molarity moles على volume ويحسب ال moles بشكل
+
+484
+00:42:55,450 --> 00:42:59,190
+منفصل ويطلع قيمتها و بعدين ييجي يقول ال moles
+
+485
+00:42:59,190 --> 00:43:02,450
+بتساوي weight ع molecular weight يعني يرايح حاله
+
+486
+00:43:02,450 --> 00:43:05,250
+بتالي مافيش اي مشكلة
+
+487
+00:43:13,260 --> 00:43:18,440
+بهيك بنكون انتهينا من مناقشة مسائل الفصل الثالث
+
+488
+00:43:18,440 --> 00:43:22,580
+عشر وإلى لقاء آخر السلام عليكم ورحمة الله وبركاته
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U.srt
new file mode 100644
index 0000000000000000000000000000000000000000..9919a8cd8b22b6ed37e7c3ee87fffb9d96e9f262
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U.srt
@@ -0,0 +1,2261 @@
+1
+00:00:05,150 --> 00:00:09,890
+بسم الله الرحمن الرحيم اليوم إن شاء الله عندنا
+
+2
+00:00:09,890 --> 00:00:15,270
+اللي هو المحاضرة رقم عشرة في مساق تحليل حقيقي
+
+3
+00:00:15,270 --> 00:00:21,130
+2 لطلبة الجامعة الإسلامية كلية العلوم قسم
+
+4
+00:00:21,130 --> 00:00:27,370
+الرياضيات اليوم إن شاء الله هنبدأ في chapter سبعة
+
+5
+00:00:28,110 --> 00:00:33,150
+اللي هو الجزء الثاني من المادة أو النص الثاني
+
+6
+00:00:33,150 --> 00:00:37,130
+للمادة اللي هو حكينا عن ال derivatives اليوم نحكي 
+
+7
+00:00:37,130 --> 00:00:42,170
+عن اللي هو the Riemann integral أو تكامل ريمان هيتم
+
+8
+00:00:42,170 --> 00:00:45,730
+الحديث اليوم إن شاء الله في ال section الأول هنحكي
+
+9
+00:00:45,730 --> 00:00:51,450
+عن ال Riemann integrability أو قابلية تكامل الدالة
+
+10
+00:00:51,450 --> 00:00:56,270
+على اللي هو بالنسبة لريمان في البداية هنحكي عن بعض
+
+11
+00:00:56,270 --> 00:01:00,410
+التعريف اللي هتوصلنا للي بدنا إياه في الموضوع اللي
+
+12
+00:01:00,410 --> 00:01:03,910
+هو الـ Riemann Integrability الآن هاخد لنا function
+
+13
+00:01:03,910 --> 00:01:11,090
+F F من some interval I نفترض أن الدالة اللي عند F
+
+14
+00:01:11,090 --> 00:01:18,850
+من A و B لعند R is a bounded function الآن
+
+15
+00:01:18,850 --> 00:01:22,050
+بدنا نعرف بعض التعريفات اللي بتاخدنا إلى الـ
+
+16
+00:01:22,050 --> 00:01:26,950
+Riemann Integrability أول شيء بدنا نعرف شو معناه
+
+17
+00:01:26,950 --> 00:01:30,750
+partition لو كانت هذه الدالة مثلا هي Y بيساوي F
+
+18
+00:01:30,750 --> 00:01:36,170
+of X على الفترة المذكورة وكانت من A ل B اللي هو ال
+
+19
+00:01:36,170 --> 00:01:41,650
+domain بدنا الآن نيجي نجزئ المنطقة اللي هي الفترة 
+
+20
+00:01:41,650 --> 00:01:47,210
+من A ل B اللي هو خلّي نسمي هذا X0 هذا نسميه x1
+
+21
+00:01:47,210 --> 00:01:51,750
+هذا نسميه x2 لعند ما أصل للفترة النموذجية اللي
+
+22
+00:01:51,750 --> 00:01:58,270
+بتسميها xk-1 لعند ال xk وهذا أسميها اللي هي xn
+
+23
+00:01:58,270 --> 00:02:06,110
+الآن الفترة من a لb أنا جزأتها بx0 وx1 وx2 لعند xn
+
+24
+00:02:06,110 --> 00:02:13,370
+هذه التجزئة بتحطها في 6 وسميها P بتساوي x0و X1 و
+
+25
+00:02:13,370 --> 00:02:19,410
+X2 لعند Xn هذا اللي بسميه is a partition of the
+
+26
+00:02:19,410 --> 00:02:26,510
+interval A و B إذا عندي I subset من R closed
+
+27
+00:02:26,510 --> 00:02:30,430
+bounded interval ال ordered set هذه اللي حكيتها
+
+28
+00:02:30,430 --> 00:02:34,410
+كتبتها أن هي عبارة عن X0 و X1 لعند Xn بنسميها a
+
+29
+00:02:34,410 --> 00:02:39,390
+partition of I بحيث أن A X0 أصغر من X1 لعند ما
+
+30
+00:02:39,390 --> 00:02:45,060
+أقول أصغر من Xn الآن الـ function عندي F من I لعند
+
+31
+00:02:45,060 --> 00:02:49,980
+R bounded function on I الآن بدي أعرف بعض
+
+32
+00:02:49,980 --> 00:02:54,660
+التعريفات اللي هتلزمني برضه في اللي هو الوصول إلى
+
+33
+00:02:54,660 --> 00:02:59,160
+الـ Remain Integrability أول تعريف بدي آخده الآن
+
+34
+00:02:59,160 --> 00:03:07,000
+أسمي M1 هو عبارة عن الـ supremum للـ function في
+
+35
+00:03:07,000 --> 00:03:12,990
+المنطقة هذه الـ M1 بدي أسميه عبارة عن الـ Supremum
+
+36
+00:03:12,990 --> 00:03:16,810
+للدالة F of X والـ Supremum موجودة لأن الـ F
+
+37
+00:03:16,810 --> 00:03:21,870
+أشمالها is bounded السوب للـ F of X حيث الـ X
+
+38
+00:03:21,870 --> 00:03:29,310
+موجودة في الفترة من X0 لعند X1 هذه اللي هي بدي
+
+39
+00:03:29,310 --> 00:03:34,570
+أسميها M1 Capital M1 Small اللي هي عبارة عن ال
+
+40
+00:03:34,570 --> 00:03:38,960
+Infimum لـ .. لـ الـ function hand اللي عبارة عن
+
+41
+00:03:38,960 --> 00:03:45,400
+الـ infimum لـ F of X حيث X تنتمي إلى الفترة X
+
+42
+00:03:45,400 --> 00:03:50,440
+0 و X واحد in general .. in general في الفترة
+
+43
+00:03:50,440 --> 00:03:55,280
+النموذجية بدي أسمي Mk هي عبارة عن .. الآن في
+
+44
+00:03:55,280 --> 00:03:59,850
+الفترة هذه على الفترة هذه الـ supremum للـ function
+
+45
+00:03:59,850 --> 00:04:06,170
+هنا بدي اسميها Mk K بسوء F of X such that X
+
+46
+00:04:06,170 --> 00:04:13,030
+element in Xk-1 لعند Xk و ال Mk small k
+
+47
+00:04:13,030 --> 00:04:19,330
+هسميها اللي هي بنفس اللي هسميت X واحد اللي هي ال
+
+48
+00:04:19,330 --> 00:04:23,370
+infimum لل F of X في الفترة من Xk-1 لعند
+
+49
+00:04:23,370 --> 00:04:30,300
+Xk على سبيل المثال الـ Mk يعني هيها هذا اللي هي M
+
+50
+00:04:30,300 --> 00:04:36,340
+k Small والـ Mk Capital على سبيل المثال هيها
+
+51
+00:04:36,340 --> 00:04:42,640
+هي قابتها هذا الـ Mk Capital إذن الآن عندي سميت
+
+52
+00:04:42,640 --> 00:04:45,560
+اللي هي الـ Mk Capital هي الـ Supremum والـ Mk
+
+53
+00:04:45,560 --> 00:04:49,740
+Small هي عبارة عن الـ Infimum وهل جيت هنشوف إيش
+
+54
+00:04:49,740 --> 00:04:55,560
+دواعي هذه التسمية أو ليش سمينا هيك الآن بدنا نعرف
+
+55
+00:04:55,560 --> 00:05:00,380
+اللي هو .. بدنا 
+
+56
+00:05:00,380 --> 00:05:04,380
+نعرف الآن اللي هو حاجة اسمها lower sum و حاجة
+
+57
+00:05:04,380 --> 00:05:12,540
+اسمها إيش ال upper sum الآن لو جيت بعد ما عملت ال
+
+58
+00:05:12,540 --> 00:05:21,020
+partition اللي عندي هذا و جيت أخذت اللي هي 
+
+59
+00:05:28,550 --> 00:05:32,770
+أخذت الـ partition P اللي عندي هذا هذا ال
+
+60
+00:05:32,770 --> 00:05:40,110
+partition P هو عبارة عن X0 X1 X2 اللي عند Xn جيت
+
+61
+00:05:40,110 --> 00:05:44,590
+أنا أخذت اللي هو الـ supremum للدالة في هذه
+
+62
+00:05:44,590 --> 00:05:51,550
+المنطقة والتي تكون طبعا هنا شكلها اللي هي سميتها
+
+63
+00:05:51,550 --> 00:06:02,170
+M1 في X1 ناقص X0 و جيت ضفت لها برضه ال supremum
+
+64
+00:06:02,170 --> 00:06:08,030
+للدالة في المنطقة هذه اللي هي زائد M2 سميت ال
+
+65
+00:06:08,030 --> 00:06:15,230
+supremum اللي هو M2 في X2-X0 لما وصلت لآخر واحدة
+
+66
+00:06:15,230 --> 00:06:20,750
+بنفس الأسلوب أخدت اللي هو ال supremum اللي هي ال
+
+67
+00:06:20,750 --> 00:06:26,870
+supremum طبعا M1 عندي التي هي طول الريانج اللي عندي
+
+68
+00:06:26,870 --> 00:06:32,850
+مضروب في مين اللي هو x1-x0 فبتطلع هذه المنطقة
+
+69
+00:06:32,850 --> 00:06:41,710
+بالضبط مساحة اللي عندي الآن بضل أجمع بنفس الأسلوب
+
+70
+00:06:41,710 --> 00:06:45,830
+بنجمع بنفس الأسلوب عندي 
+
+71
+00:06:53,690 --> 00:06:58,050
+هذا المجموع اللي بالازرق هو اللي بنسميه هسميه ال
+
+72
+00:06:58,050 --> 00:07:03,170
+upper sum هذا ال upper sum اللي هو ال upper لل F
+
+73
+00:07:03,170 --> 00:07:08,090
+للدالة F بواسطة
+
+74
+00:07:08,090 --> 00:07:12,830
+مين ال partition P إذا ال upper sum هذا اللي هو
+
+75
+00:07:12,830 --> 00:07:17,830
+عبارة عن اللي هو لل function بواسطة ال partition P
+
+76
+00:07:17,830 --> 00:07:20,410
+يعني ال upper sum هذا يعتمد على ال F ويعتمد على
+
+77
+00:07:20,410 --> 00:07:25,690
+مين؟ على ال partition اللي عندي الآن similarly بدي
+
+78
+00:07:25,690 --> 00:07:31,190
+آجي أعرف اللي هو ال lower sum إذا ال upper sum
+
+79
+00:07:31,190 --> 00:07:38,890
+بيساوي ال summation ل Xk ناقص Xk-1 مضروب
+
+80
+00:07:38,890 --> 00:07:43,950
+في ال Mk capital k من عند واحد لعند N هذه اللي
+
+81
+00:07:43,950 --> 00:07:48,930
+بيسميها ال upper sum similarly بدي أعرف مين اللي
+
+82
+00:07:48,930 --> 00:07:56,170
+هو ال lower sum للـ F و لل D هيساوي الآن بدل ما أخد
+
+83
+00:07:56,170 --> 00:07:59,330
+اللي هو ال suprem أعلى قيمة بتاخد اللي هو ال
+
+84
+00:07:59,330 --> 00:08:05,390
+infimum بيصير عندي اللي هو summation Xk ناقص Xk
+
+85
+00:08:05,390 --> 00:08:10,790
+-1 في ال Mk small k من عند واحد لعند N
+
+86
+00:08:10,790 --> 00:08:15,050
+هذا اللي هو عبارة عن ال lower sum في الواقع هو
+
+87
+00:08:15,050 --> 00:08:21,350
+هيكون المنطقة اللي بالأحمر هذه بناء على إني باخد
+
+88
+00:08:21,350 --> 00:08:27,670
+أنا اللي هو الارتفاع هو عبارة عن ال Mk Small اللي
+
+89
+00:08:27,670 --> 00:08:33,410
+هو ال minimum لقيمة الدالة على كل المنطقة وهكذا
+
+90
+00:08:33,410 --> 00:08:38,030
+لما أصل لآخر واحدة فبيصير عندي هذا ال summation
+
+91
+00:08:38,030 --> 00:08:43,890
+لهذه اللي هي المستطيلات
+
+92
+00:08:43,890 --> 00:08:51,510
+بناء على ال partition اللي عندي هي عبارة عن مين؟ عن
+
+93
+00:08:51,510 --> 00:08:56,190
+اللي هو ال lower sum طبعا ناخد الدالة اللي هي في
+
+94
+00:08:56,190 --> 00:09:00,670
+منطقة موجبة بأساس إنه اللي هي يكون أكثر توضيحا طبعا
+
+95
+00:09:00,670 --> 00:09:04,650
+لو منطقة سالبة برضه باخد نفس الأسلوب لكن بيصير
+
+96
+00:09:04,650 --> 00:09:08,810
+اللي هو بدل مساحة بيصير سالب مساحة أو مخلوطة بين
+
+97
+00:09:08,810 --> 00:09:13,070
+مساحة موجبة مساحة سالبة و بت cancel بعض إلى أن أنا
+
+98
+00:09:13,070 --> 00:09:16,350
+أخدتها على فترة موجبة بأساس اللي هو تسهل عملية
+
+99
+00:09:16,820 --> 00:09:22,620
+اللي هي التوضيح الآن خلينا تحت شوية إذا اللي حكيته
+
+100
+00:09:22,620 --> 00:09:28,420
+الآن عرفت ال lower sum زي ما أنتم شايفين عند ال
+
+101
+00:09:28,420 --> 00:09:32,780
+lower sum هيه هذا ال lower sum يعتمد في مجموعه على
+
+102
+00:09:32,780 --> 00:09:37,300
+ال Mk اللي هي عبارة عن ال infimum للدالة على
+
+103
+00:09:37,300 --> 00:09:43,960
+الفترة Xk-1 لعند Xk و -xk-1 هي عبارة عن
+
+104
+00:09:43,960 --> 00:09:49,040
+قيم ال partition اللي عندي إذا ال lower sum أو حتى
+
+105
+00:09:49,040 --> 00:09:53,820
+ال upper sum الاثنتين باعتمدين عالميا على اللي هو
+
+106
+00:09:53,820 --> 00:09:57,940
+ال partition وعلى الدالة اللي بنعملها أو لو وجدتها
+
+107
+00:09:57,940 --> 00:10:03,160
+ال lower sum و ال upper sum هما واضح أن ال lower
+
+108
+00:10:03,160 --> 00:10:09,780
+sum ال lower sum واضح أنه اللي هو دائما اللي هو لل
+
+109
+00:10:09,780 --> 00:10:13,480
+partition المعين هيكون أصغر من مين من ال upper sum
+
+110
+00:10:13,480 --> 00:10:21,380
+واضح أه؟ الآن اللي بنحكي if F من I ل R is bounded
+
+111
+00:10:21,380 --> 00:10:27,520
+and P is any partition of I then ال lower sum لل P
+
+112
+00:10:27,520 --> 00:10:32,020
+و ال F أصغر أو يساوي ال upper sum لل P و ال F و برهانه
+
+113
+00:10:32,020 --> 00:10:39,830
+سهل وسريع لو جئنا و أخذنا الـ partition x0 x1
+
+114
+00:10:39,830 --> 00:10:40,550
+عند xn
+
+115
+00:10:43,460 --> 00:10:48,540
+و قررنا بين الـMk والـMk Capital أكيد الـMk Small
+
+116
+00:10:48,540 --> 00:10:50,540
+أصغر من الـMk Capital لأنه هذا عبارة عن
+
+117
+00:10:50,540 --> 00:10:54,820
+الـSupremum للفانكشن على الفترة أو الـSubinterval
+
+118
+00:10:54,820 --> 00:10:59,780
+Xk-1 عند Xk وهذا الـSupremum للفانكشن، هذا
+
+119
+00:10:59,780 --> 00:11:02,860
+الـInfimum وهذا الـSupremum فأكيد هذا أكبر أو
+
+120
+00:11:02,860 --> 00:11:07,080
+يساوي ما هو الـMk Small وهذا الكلام صحيح على كل
+
+121
+00:11:07,080 --> 00:11:11,380
+الـSubintervals يعني صحيح لكل K من 1 إلى 2 عند N
+
+122
+00:11:11,490 --> 00:11:14,650
+وفي نفس الوجهة الـ xk هو الـ xk-1 أكبر من
+
+123
+00:11:14,650 --> 00:11:18,610
+صفر إذا بيصير عندي حاصل ضرب هادي في طول الفترة
+
+124
+00:11:18,610 --> 00:11:22,570
+وهادي في طول الفترة بتظل ال inequality صحيحة لأنه
+
+125
+00:11:22,570 --> 00:11:26,730
+طول الفترة موجبة فبيصير mk في هذا المقدار أصغر أو يساوي
+
+126
+00:11:26,730 --> 00:11:29,950
+mk capital في هذا المقدار بنعمل summation على الكل
+
+127
+00:11:29,950 --> 00:11:32,770
+وهذا ال summation على الكل بيصير عندي هذا هو عبارة
+
+128
+00:11:32,770 --> 00:11:36,150
+عن ال upper sum وهذا ال lower sum يعني اللي وصلنا
+
+129
+00:11:36,150 --> 00:11:44,110
+إلى أن دائما ال lower sum لأن ال lower sum هو أسفل
+
+130
+00:11:44,110 --> 00:11:48,230
+من ال upper sum في القيام فبيكون أكيد ال lower sum
+
+131
+00:11:48,230 --> 00:11:56,870
+أصغر أو يساوي ال upper sum اطلع لفهم الآن احنا لو
+
+132
+00:11:56,870 --> 00:12:01,590
+اجينا لو 
+
+133
+00:12:01,590 --> 00:12:06,170
+اجينا عرفنا .. بنعرف حاجة اللي في عندي P is a
+
+134
+00:12:06,170 --> 00:12:10,890
+partition لو ضفت على الـ partition أي قيمة جديدة
+
+135
+00:12:10,890 --> 00:12:15,370
+بالإضافة للقيم الموجودة بيصير عندي هذا ال .. ال ..
+
+136
+00:12:15,370 --> 00:12:20,710
+ال .. الجديد عبارة عن refinement لمين للـ P يعني
+
+137
+00:12:20,710 --> 00:12:26,290
+تحسين لل partition يعني الآن بنقول عن أن الـ P الـ
+
+138
+00:12:26,290 --> 00:12:30,190
+Q is a refinement of P إذا كانت الـ P subset من
+
+139
+00:12:30,190 --> 00:12:34,990
+مين؟ من الـ Q طيب، الآن نشوف إيش اللي بنقوله
+
+140
+00:12:45,570 --> 00:12:57,310
+الآن لو أخذنا الفترة اللي هي A وB و جزأناها X0 X1
+
+141
+00:12:57,310 --> 00:13:03,460
+لعند XN و الفترة النموذجية اللي هنشتغل عليها xk-
+
+142
+00:13:03,460 --> 00:13:08,640
+1 أو ال sub interval لعند مين لعند xk الآن ال P
+
+143
+00:13:08,640 --> 00:13:13,520
+هذا هو عبارة عن ال partition x0 x1 لعند
+
+144
+00:13:13,520 --> 00:13:17,400
+ال xn وهذا ordered يعني نكتبها على صورة ordered
+
+145
+00:13:17,400 --> 00:13:20,640
+لما نكتبها على هذا متفقين على أن إيش اللي هو احنا
+
+146
+00:13:20,640 --> 00:13:25,380
+نعمله ordered ال order بالنسبة له إيش ماله مهم لأن
+
+147
+00:13:25,380 --> 00:13:28,080
+هذا معناه تبدأ بال x0 وبعدين x1 بعدين
+
+148
+00:13:28,080 --> 00:13:35,240
+اللي هو ال xn لو جينا أخذنا اللي هو أضفنا أي نقطة
+
+149
+00:13:35,240 --> 00:13:39,060
+هنا طبعا أضفنا أي نقطة أو أي رقم بين الـA و بين
+
+150
+00:13:39,060 --> 00:13:45,320
+مين الـB الرقم المضاف اللي هو أضفنا لهذه وسميناها
+
+151
+00:13:45,320 --> 00:13:50,930
+Q فبيكون الـQ هي عبارة عن refinement للـP يعني بمعنى
+
+152
+00:13:50,930 --> 00:13:55,030
+آخر Q refinement ل ال P معناته أن في Q فيها
+
+153
+00:13:55,030 --> 00:14:02,790
+نقاط اللي هي زيادة عن نقطة P و اسمه تحسين لأنه فعلا
+
+154
+00:14:02,790 --> 00:14:08,870
+بيحسن وضع ال lower sum و ال upper sum الآن ال lower
+
+155
+00:14:08,870 --> 00:14:14,250
+sum عرفناه لل P و ال F قلنا بعتمد عليهم بسوء ال
+
+156
+00:14:14,250 --> 00:14:22,130
+summation اللي هو xk ناقص xk-1 في ال mk
+
+157
+00:14:22,130 --> 00:14:25,090
+اللي هي mk small هي عبارة عن قلنا اللي هي عبارة عن
+
+158
+00:14:25,090 --> 00:14:29,050
+ال infimum على العهد الفترة k من عند واحد لعند n
+
+159
+00:14:29,050 --> 00:14:34,770
+والآن ال upper sum ب U, P وF اللي هو بيساوي ال
+
+160
+00:14:34,770 --> 00:14:40,890
+summation لل xk ناقص xk-1 Mk capital k
+
+161
+00:14:40,890 --> 00:14:47,930
+من عند واحد لعند n وقلنا إن الـ lower للـ P و الـ F
+
+162
+00:14:47,930 --> 00:14:52,390
+دايما أصغر أو يساوي ال upper للـ P و الـ F فده اللي حتى
+
+163
+00:14:52,390 --> 00:14:58,470
+الآن اللي حكيناه الآن نيجي ناخد الملاحظة اللي عندي
+
+164
+00:14:58,470 --> 00:15:04,510
+if Q above is a refinement of P لو قلنا أن Q عبارة
+
+165
+00:15:04,510 --> 00:15:09,390
+عن إيه شمالها؟ refinement of P refinement معناته
+
+166
+00:15:09,390 --> 00:15:14,690
+أضفت اللي هو نقطة أو أكثر من نقطة لنقاط ال
+
+167
+00:15:14,690 --> 00:15:20,810
+partition اللي عندي إذا أي فترة أي فترة اللي هي xk
+
+168
+00:15:20,810 --> 00:15:27,550
+-1 لعند xk اللي هو في ال partition P  
+
+169
+00:15:27,550 --> 00:15:34,060
+هقدر أكتبها على صورة مين اللي هو yj-1 yj اتحاد ال
+
+170
+00:15:34,060 --> 00:15:37,940
+yj yj+1 اتحاد ده ما أصل عند اتحاد yn-1
+
+171
+00:15:37,940 --> 00:15:44,040
+لعند مين لعند ال yn ماشي الحال طبعا الآن هذول عدد
+
+172
+00:15:44,040 --> 00:15:47,740
+الفترات على حسب الإضافة اللي هنالو ضفنا نقطة واحدة
+
+173
+00:15:47,740 --> 00:15:52,240
+بين هذول الاثنين معناته هيصير عندي من الـ yj-1 هي
+
+174
+00:15:52,240 --> 00:15:56,460
+عبارة عن xk-1 والنقطة المضافة اتحاد النقطة
+
+175
+00:15:56,460 --> 00:15:59,780
+المضافة اللي هي مع مين مع النقطة هذه بيكون عندي بس
+
+176
+00:15:59,780 --> 00:16:03,690
+فترة ثانية لو كان فيه عندي نقطتين مضاف
+
+201
+00:18:02,150 --> 00:18:07,210
+نقاط الـ Partition اللي هو هذا اللي هي النقاط
+
+202
+00:18:07,210 --> 00:18:12,630
+بتُجلل الجزء الزائد فالآن فعلًا بيكون تحسين على 
+
+203
+00:18:12,630 --> 00:18:15,330
+الأولاني لأنه أصلاً أنا بعد شوية هتلاقي غاية في
+
+204
+00:18:15,330 --> 00:18:21,140
+الوصول أن أصل اللي هو اللي هو أصغر واحد في ال upper
+
+205
+00:18:21,140 --> 00:18:25,720
+sums يصير زي أكبر واحد في ال lower sums يوم ما
+
+206
+00:18:25,720 --> 00:18:29,820
+يلتقيان مع بعض بيكون فعلًا اللي هو ال upper sum وال
+
+207
+00:18:29,820 --> 00:18:35,000
+lower sum اللي هو تجاوزًا بيصير بساوي بالظبط اللي
+
+208
+00:18:35,000 --> 00:18:40,080
+هو المساحة تحت المنحنى أو المنطقة اللي هنسميها في
+
+209
+00:18:40,080 --> 00:18:43,340
+حينها أن هي عبارة عن Riemann integral بل وقيمة ال
+
+210
+00:18:43,340 --> 00:18:47,660
+Riemann integral هو عبارة عن هذه المنطقة اللي صارت من
+
+211
+00:18:47,660 --> 00:18:53,540
+اللي هو أنه ناخد نُزَغّر اللي هو الأبر و نرفع ال
+
+212
+00:18:53,540 --> 00:18:57,540
+lower و نشوف كيف بنزغّر الأبر و كيف بنرفع ال lower
+
+213
+00:18:57,540 --> 00:19:01,580
+اللي هو معنى اللي هيوصل لنا معنى ال integrability
+
+214
+00:19:02,250 --> 00:19:06,710
+إذا الـ Refinement فعلًا Refinement تحسين لما أقول
+
+215
+00:19:06,710 --> 00:19:13,390
+أن بي إي كيو زادت عن هذه الـ Q إيش هتعمل؟ هتُجلل
+
+216
+00:19:13,390 --> 00:19:19,630
+الجزء الزائد في الـ Upper لما أزيد النقاط و هتزيد
+
+217
+00:19:19,630 --> 00:19:24,950
+للجزء المفقود وين في ال lower ماشي الحال ففعلًا
+
+218
+00:19:24,950 --> 00:19:28,310
+هيكون إيه بقى عبارة عن refinement أو تحسين أنتَ
+
+219
+00:19:28,310 --> 00:19:32,470
+نشوف الآن نشوف لمتنا اللي عندنا اللي هو معنى اللي
+
+220
+00:19:32,470 --> 00:19:38,350
+بقوله أنا أنه عندي لو كانت Q refinement لل B بيكون
+
+221
+00:19:38,350 --> 00:19:43,790
+ال lower لجديد عليها أكبر أو يساوي الـ Lower القديم
+
+222
+00:19:43,790 --> 00:19:48,390
+يعني بمعنى آخر اقترب إلى اللي بدنا إياه اللي متوقعين
+
+223
+00:19:48,390 --> 00:19:53,110
+أنه يكون هو التحسين النهائي اللي هنسميه اللي هو ال
+
+224
+00:19:53,110 --> 00:19:58,670
+integration ماشي؟ وال Upper هنا Upper UQ واضح أنه
+
+225
+00:19:58,670 --> 00:20:03,510
+بيصغر اللي هو عن الأصل لأنه صار فيه تحسين واقترب من
+
+226
+00:20:03,510 --> 00:20:08,070
+الحقيقة اقترب من اللي هو القيمة اللي احنا هنسميها
+
+227
+00:20:08,070 --> 00:20:12,370
+في النهاية integration طيب، بدنا نثبت الكلام هذا،
+
+228
+00:20:12,370 --> 00:20:16,810
+شوف كيف بدنا نثبته، اللي أنا عندي بيأخذت I اللي هو
+
+229
+00:20:16,810 --> 00:20:21,290
+ال partition ال arbitrary واخدت اللي هو sigma I،
+
+230
+00:20:21,290 --> 00:20:26,310
+بد أشتغل على إضافة نقطة ومن ثم أقول إضافة النقطة
+
+231
+00:20:26,310 --> 00:20:31,340
+تؤدي إلى اللي هو باقي النقطة، نشوف إيش بقول عند بي
+
+232
+00:20:31,340 --> 00:20:35,080
+هي اللي هو الـ partition أخدت z element in I و الـ
+
+233
+00:20:35,080 --> 00:20:39,100
+z هذي و أنا بدأ أحطها في الفترة xk minus واحد لعند
+
+234
+00:20:39,100 --> 00:20:42,640
+مين لعند الـ xk يعني الآن ال partition اللي بي
+
+235
+00:20:42,640 --> 00:20:51,200
+عندي هو عبارة عن x naught لعند xn xk minus واحد xk
+
+236
+00:20:51,200 --> 00:20:55,620
+indeed في المنطقة هذه ضفت نقطة وسميت بها prime و
+
+237
+00:20:55,620 --> 00:21:03,380
+عبارة عن x naught xk minus واحد Z X K وظلّت الخلايا
+
+238
+00:21:03,380 --> 00:21:07,720
+كله زي ما هو لعد مين لعد X بس اللي غيرت فيه أو
+
+239
+00:21:07,720 --> 00:21:12,420
+اللي ضفت له بس ضفت نقطة هنا اسمها Z بدي أشوف تأثير
+
+240
+00:21:12,420 --> 00:21:16,640
+هذا النقطة إيش بده يعمل بين ال partition القديم
+
+241
+00:21:16,640 --> 00:21:20,520
+وال partition الجديد ومن ثم نستنتج أي refinement
+
+242
+00:21:20,520 --> 00:21:26,990
+على اللي هو ال B القديمة الأمر سهل إن شاء الله الآن
+
+243
+00:21:26,990 --> 00:21:30,290
+الـ .. الـ .. الـ .. الاضافة الجديدة صار عند بي برايم
+
+244
+00:21:30,290 --> 00:21:35,870
+إيش هو X Note X واحد وهي ال Z ضفتها اللي اختلف عن
+
+245
+00:21:35,870 --> 00:21:43,640
+فوق بس ال Z اللي هنا بناء عليه الآن ال MK ال mk'
+
+246
+00:21:43,860 --> 00:21:49,020
+هي عبارة عن الـ supremum للـ function f وين في
+
+247
+00:21:49,020 --> 00:21:58,920
+المنطقة بين zk-1 و مين و z الآن في الفترة بين xk-1
+
+248
+00:21:58,920 --> 00:22:04,800
+و بين xk ضفت z الآن الـ z اللي ضفت هنا هذه المنطقة
+
+249
+00:22:04,800 --> 00:22:11,150
+بدي أسميه ال supremum اللي فيها mk' اللي هنا اللي
+
+250
+00:22:11,150 --> 00:22:16,190
+ضلت نسميها إيش مالها MK double prime مع العلم إنه
+
+251
+00:22:16,190 --> 00:22:19,890
+الـ supremum الأصلية في ال partition الأصلي على كل
+
+252
+00:22:19,890 --> 00:22:23,250
+الفترة من XK minus واحد عند XK اسمها إيش مسمينا
+
+253
+00:22:23,250 --> 00:22:31,130
+احنا MK capital ماشي الحال طيب ركزوا معي الآن
+
+254
+00:22:31,130 --> 00:22:34,970
+عندي الـ MK' هي الـ supremum على اللي هو الـ
+
+255
+00:22:34,970 --> 00:22:39,610
+subinterval هذه الـ MKW' هي ال supremum على اللي
+
+256
+00:22:39,610 --> 00:22:43,910
+هي ال subinterval اللي جنبها اللي هي من Z لعند XK
+
+257
+00:22:43,910 --> 00:22:49,890
+واضح أن ال supremum على الكبيرة هذه من XK minus 1
+
+258
+00:22:49,890 --> 00:22:53,790
+لعند XK أكيد أكبر أو يساوي ال supremum على الصغيرة
+
+259
+00:22:54,620 --> 00:22:58,580
+لأن هذه أشمَلها جزء من 100 من هذه كلها فال supremum
+
+260
+00:22:58,580 --> 00:23:01,940
+لل function عالي كبيرة أكبر أو يساوي ال supremum
+
+261
+00:23:01,940 --> 00:23:06,020
+على مين؟ على الصغيرة و similarly للتانية إذا ال MK
+
+262
+00:23:06,020 --> 00:23:12,180
+أكبر أو يساوي MK' Similarly MK أكبر يساوي MKW' لأن
+
+263
+00:23:12,180 --> 00:23:15,220
+هذه عبارة عن ال supremum على الجزء الثاني منها
+
+264
+00:23:15,220 --> 00:23:20,520
+اللي إلى جاي، واضح؟ إذا صار عندي therefore الآن بدأ
+
+265
+00:23:20,520 --> 00:23:24,840
+أصل لحاجة معينة اللي هي mk في xk ناقص xk minus
+
+266
+00:23:24,840 --> 00:23:31,310
+واحد ضربت اللي هي Mk في هذا هذه الفترة طبعًا اللي هي
+
+267
+00:23:31,310 --> 00:23:40,730
+بتساوي Zk-Zk-Xk-1 زائد Xk-Zk الآن صارت هذه Mk في
+
+268
+00:23:40,730 --> 00:23:45,410
+المقدار هذا بساوي Mk في هذه زائد هذه ماشي الحال
+
+269
+00:23:45,410 --> 00:23:50,820
+الآن ال Mk أكبر أو يساوي mean mk' إذا إن صار المقدار
+
+270
+00:23:50,820 --> 00:23:58,780
+هذا أكبر أو يساوي mk' في zk z ناقص xk minus واحد و
+
+271
+00:23:58,780 --> 00:24:04,820
+هذه ال mk برضه أكبر أو يساوي mk double prime of xk
+
+272
+00:24:04,820 --> 00:24:10,380
+ناقص z ماشي الحالة الآن احنا بين ال two partitions
+
+273
+00:24:10,380 --> 00:24:14,580
+اللي حدث بس تغيير واحد زي ما قلنا بس على المنطقة
+
+274
+00:24:14,580 --> 00:24:23,090
+هذه لاحظ أنتَ الآن ال lower sum ال lower sum لل b و
+
+275
+00:24:23,090 --> 00:24:30,350
+ال f ال b prime و ال f بساوي لو حسبنا للأولاني أول
+
+276
+00:24:30,350 --> 00:24:38,350
+إشي هو عبارة عن m1 في x0 او x1 نقص x0 زائد لما أصل
+
+277
+00:24:38,350 --> 00:24:51,620
+عند mk في xK-XK-1 زائد لما أصل لعند MN في XN-XN-1
+
+278
+00:24:51,620 --> 00:24:57,800
+هذا إيش هو يا جماعة؟ هو عبارة عن الأبرثان الأبرثان
+
+279
+00:24:57,800 --> 00:25:03,120
+لمن؟ لـB و لـF متفجر، ماشي الحالة على مين؟ على ال
+
+280
+00:25:03,120 --> 00:25:12,270
+partition B الآن هذا بساوي m1 في x1 ناقص x0 زائد
+
+281
+00:25:12,270 --> 00:25:18,010
+الآن هذه جزئناها جزئين بعد شوية زائد زائد هذا نفسه
+
+282
+00:25:18,010 --> 00:25:25,690
+ماتغيرش هذا اللي هو عبارة عن mk لسه xk ناقص z
+
+283
+00:25:25,690 --> 00:25:34,850
+مظبوط زائد اللي هو mk خليني أقول ال
+
+284
+00:25:34,850 --> 00:25:46,170
+mk اللي هو Z ناقص كحسابات ناقص X كمينوس واحد صحيح؟
+
+285
+00:25:46,170 --> 00:25:53,130
+واضح زائد الباقي زي ما هو زائد M N في X N ناقص X N
+
+286
+00:25:53,130 --> 00:26:00,980
+ناقص واحد الآن هذا ما تغيرش نفسه زائد هذا ال N هو
+
+287
+00:26:00,980 --> 00:26:05,540
+اللي اتغير وهنا في ال inequality هنا لقينا عبارة عن
+
+288
+00:26:05,540 --> 00:26:12,220
+هذا المقدار اللي هو أكبر أو يساوي اللي هو اللي ما
+
+289
+00:26:12,220 --> 00:26:15,420
+اتغيرش هذا نسميه اللي هو M1
+
+290
+00:26:18,760 --> 00:26:25,640
+معي يا شباب زائد اللي هو mk شرطة في xk ناقص z
+
+291
+00:26:25,640 --> 00:26:31,520
+زائد mk شرطين في z ناقص xk ماينوس واحد زائد دا
+
+292
+00:26:31,520 --> 00:26:38,570
+كما ما تتغيرش زي ما هو mn في xn نقص XN نقص 1 هذا في
+
+293
+00:26:38,570 --> 00:26:42,690
+الواقع مين هو أصلاً هذا هو عبارة عن اللي هو ال
+
+294
+00:26:42,690 --> 00:26:48,510
+upper sum لل partition مين ال B' هو بساوي ال upper
+
+295
+00:26:48,510 --> 00:26:55,270
+sum لل B' والF يعني اللي جناه أنه لما نضفنا اللي
+
+296
+00:26:55,270 --> 00:26:59,830
+هو نقطة على اللي هو الـ partition صار عند الـ
+
+297
+00:26:59,830 --> 00:27:03,590
+partition اللي هو الأبر للـ partition الجديد أصغر
+
+298
+00:27:03,590 --> 00:27:08,360
+أو يساوي الأبر للـ partition من القديم الآن اللي
+
+299
+00:27:08,360 --> 00:27:11,760
+حدث على نقطة ممكن يحدث على نقطتين يعني الآن ممكن
+
+300
+00:27:11,760 --> 00:27:15,640
+نقطة مثلًا انضافت هنا ممكن انضافت كمان مرة هنا كمان
+
+301
+00:27:15,640 --> 00:27:20,560
+نقطة هيصير لجديد بنفس الأسلوب أكبر ويساوي تلت نقاط
+
+302
+00:27:20,560 --> 00:27:24,040
+بيصير أكبر ويساوي أربع نقاط بيصير أكبر ويساوي يعني
+
+303
+00:27:24,040 --> 00:27:29,500
+in general مدام ال refinement UQ اللي محتوي بيه في
+
+304
+00:27:29,500 --> 00:27:34,820
+النهاية عدد الإضافات finite إذا حيكون اللي هو كل
+
+305
+00:27:34,820 --> 00:27:39,300
+مرة بعدد الإضافات بيصير sub intervals بنفس الأسلوب
+
+306
+00:27:39,300 --> 00:27:43,940
+اللي هنا حيصير عنده القبر القديم أكبر أو يساوي
+
+307
+00:27:43,940 --> 00:27:48,620
+القبر الجديد يعني بمعنى آخر in general القبر للـB
+
+308
+00:27:48,620 --> 00:27:54,080
+والـF هيكون أكبر أو يساوي القبر للـQ ال refinement
+
+309
+00:27:54,080 --> 00:27:59,160
+وMain والـF وطبعًا زي ما قلنا هذا عبارة عن تحسين لإن
+
+310
+00:27:59,160 --> 00:28:03,960
+القبر هذا بيصير يصغر يقترب من المساحة الحقيقية
+
+311
+00:28:03,960 --> 00:28:08,360
+اللي تحت المنحنى في حالة اللي هو الدال ايش الموجبة
+
+312
+00:28:08,360 --> 00:28:15,860
+واضح طيب الآن similarly for lower الآن ال lower
+
+313
+00:28:15,860 --> 00:28:20,940
+بيصير اللي هو في ال lower بدل ما يصير عندي لو كان
+
+314
+00:28:20,940 --> 00:28:25,700
+عندي بنحكي عن ال lower احنا ال L بيوقف و بنحكي عن
+
+315
+00:28:25,700 --> 00:28:30,220
+ال M 1 small و ال M 2 small و M K small و M N
+
+316
+00:28:30,220 --> 00:28:36,660
+small بيصير عندي M 1 small و M K small و M K small
+
+317
+00:28:36,660 --> 00:28:44,520
+و mk small و mn small بيصير الآن اللي هي المن هنا بدل
+
+318
+00:28:44,520 --> 00:28:49,680
+ما هي أكبر إيش هتصير على إيش أصغر لإنه بيحكي عن ال
+
+319
+00:28:49,680 --> 00:28:54,810
+infimum من إيش صغير على إيش صغير إلى infimum على
+
+320
+00:28:54,810 --> 00:28:59,570
+إيش إيش اللي هو أصغر فمعناته إيش صار عنده بيصير له
+
+321
+00:28:59,570 --> 00:29:04,910
+infimum اللي هو ال M1 أصغر من ال M1' أو ال Mk
+
+322
+00:29:04,910 --> 00:29:11,730
+Small أصغر أو يساوي ال Mk' Small واضح أه؟ ومنه
+
+323
+00:29:11,730 --> 00:29:17,870
+بنستنتج أنه هيكون عند ال LBF أصغر أو يساوي 
+
+324
+00:29:21,260 --> 00:29:28,950
+الـ LQF هنصل أصغر و ساوي ال QF وفعلًا أيضًا هذا هو
+
+325
+00:29:28,950 --> 00:29:35,410
+تحسين لإنه اللي هي زاد اللي هو ال lower sum بحيث
+
+326
+00:29:35,410 --> 00:29:41,550
+أنه بده يصل أو يروح ناحية مين اللي هو المساحة
+
+327
+00:29:41,550 --> 00:29:46,310
+الحقيقية تحت المنحنى في حالة التقاء الجهتين بيصير
+
+328
+00:29:46,310 --> 00:29:52,670
+عندي اللي هو إنه القيمة إذا يعني أخدنا بمعنى آخر
+
+329
+00:29:52,670 --> 00:29:58,540
+ندخل للموضوع مدام هيك أنه لو أخدنا اللي هو صار عندي
+
+330
+00:29:58,540 --> 00:30:03,140
+ال lower يبدأ يكبر يعني لو أخدت له ال suprem لكل ال
+
+331
+00:30:03,140 --> 00:30:06,960
+lowers على كل ال partitions اللي في الدنيا و أخدت
+
+332
+00:30:06,960 --> 00:30:10,980
+ال infimum على كل كبار اللي هو و اتساون مع بعض ال
+
+333
+00:30:10,980 --> 00:30:14,820
+infimum يعني عالمين على ال upper sums بيصير ال
+
+334
+00:30:14,820 --> 00:30:17,500
+suprem مع ال lower sums و ال infimum مع ال lower
+
+335
+00:30:17,500 --> 00:30:22,780
+sums يوم ما يتساونه هو اللي بنسميه اللي هو الـ
+
+336
+00:30:22,780 --> 00:30:26,920
+Riemann Integrable و بنسمي اللي هي الـ Riemann
+
+337
+00:30:26,920 --> 00:30:30,440
+Integrable في هذه الحالة بساوي هذه القيمة المشتركة
+
+338
+00:30:30,440 --> 00:30:35,720
+اللي أصبحت متساوية نيجي للّمة اللي عندي هذه، لّمة
+
+339
+00:30:35,720 --> 00:30:40,720
+برضه يعني برهانها سهل، نشوف إيش بتقول اللّمة، اللّمة
+
+340
+00:30:40,720 --> 00:30:45,610
+بتقول let F from I to R be bounded if B1 وB2 are
+
+341
+00:30:45,610 --> 00:30:49,470
+any two partitions يعني مش نفس ال partition على أي
+
+342
+00:30:49,470 --> 00:30:54,090
+two partitions دائمًا اللاور على أي partition في
+
+343
+00:30:54,090 --> 00:30:57,150
+الدنيا أصغر أو يساوي الأبر على أي partition ثاني
+
+344
+00:30:57,150 --> 00:30:59,970
+مش مشكلة إيش هو في الأول قلنا على نفس ال partition
+
+345
+00:30:59,970 --> 00:31:05,510
+arbitrarily والإثبات هنستخدم اللّمة الأولى و
+
+346
+00:31:05,510 --> 00:31:09,010
+اللّمة الثانية اللي أخذناها و هنصلي هذه بكل سهولة
+
+347
+00:31:09,010 --> 00:31:13,760
+شوف كده خلّينا ناخد الـ Q إيش بتساوي بي واحد اتحاد
+
+348
+00:31:13,760 --> 00:31:17,240
+بي اثنين صارت الـ Q هذه refinement للأولى و
+
+349
+00:31:17,240 --> 00:31:21,420
+refinement لمين؟ للتانية تحسين للتانتين بناء عليه
+
+350
+00:31:21,420 --> 00:31:25,380
+.. بناء على اللي هي اللّمة اللي أخذناها سبعة واحد
+
+351
+00:31:25,380 --> 00:31:28,660
+واحد وسبعة واحد اثنين احنا حصول على ناية ليه
+
+352
+00:31:28,660 --> 00:31:33,000
+احسبلي ال بي واحد و اف صار refinement مزام صار ..
+
+353
+00:31:33,000 --> 00:31:37,620
+صار له refinement أو تحسين بده يكبر اللاور صار
+
+354
+00:31:37,620 --> 00:31:42,750
+أصغر أو ساوي ال Q F من جهة أخرى قبل بشوية أثبتنا أن
+
+355
+00:31:42,750 --> 00:31:48,310
+الـQF أصغر يساوي UQF for any اللي هو specific mean
+
+356
+00:31:48,310 --> 00:31:53,070
+الـQ لنفس الـQ لأن برضه اللي هو الـQ refinement
+
+357
+00:31:53,070 --> 00:31:57,790
+لمين؟ للـB2 مازال refinement إذا الـQ بدي يصير
+
+358
+00:31:57,790 --> 00:32:03,050
+أصغر أو يساوي مين الـUB2F زي اللي بواسطة أثبتناها
+
+359
+00:32:03,050 --> 00:32:07,470
+قبل بشوية يعني in general in general الـ B1 و F
+
+360
+00:32:07,470 --> 00:32:15,810
+أصغر أو ساوي و B2 و F بدون يعني بغض النظر عن B1 و
+
+361
+00:32:15,810 --> 00:32:21,130
+B2 أنهم كونوا زي بعض أو مختلفات لأي two partitions
+
+362
+00:32:21,130 --> 00:32:26,370
+of the interval I طيب الـ definition الآن اللي
+
+363
+00:32:26,370 --> 00:32:28,830
+حكيناها قبل شوية قولناها قبل شوية بدنا نعرف حاجة
+
+364
+00:32:28,830 --> 00:32:33,940
+اسمها lower integral ال Lower Integral وحاجة اسمها
+
+365
+00:32:33,940 --> 00:32:38,760
+ال Upper Integral أخذنا
+
+366
+00:32:38,760 --> 00:32:42,740
+الـ function F bounded طبعًا معروف ليش bounded عشان
+
+367
+00:32:42,740 --> 00:32:47,880
+يكون الـ M1 و الـ M2 و الـ Mn كلهم موجودات سواء
+
+368
+00:32:47,880 --> 00:32:51,040
+الـ M capital أو الـ M small عشان هما عبارة عن
+
+369
+00:32:51,040 --> 00:32:54,260
+supremum و infimum فلازم تكون bounded ال function
+
+370
+00:32:54,260 --> 00:32:58,280
+الآن فرضنا الـ F عبارة عن bounded function على ال
+
+371
+00:32:58,280 --> 00:33
+
+401
+00:35:39,750 --> 00:35:45,570
+هذا واحد من مين من الـ lower sums، بظبط؟ وهذا أصلا
+
+402
+00:35:45,570 --> 00:35:49,170
+الـ supremum للـ lower sums فهيكون الـ supremum أكبر
+
+403
+00:35:49,170 --> 00:35:52,670
+يساوي مين؟ واحد منهم، هذا زي ما أقول أنها lower
+
+404
+00:35:52,670 --> 00:35:58,450
+sum على partition في A و B. Similarly, M I في الـ B
+
+405
+00:35:58,450 --> 00:36:03,590
+minus A أكبر أو يساوي الـ other sum of F، ليه؟ لأن
+
+406
+00:36:03,590 --> 00:36:07,470
+هذا عبارة عن واحد من الـ other sums وهذا عبارة عن
+
+407
+00:36:07,470 --> 00:36:10,210
+الـ N في المملكة وليه الـ other sums فطبيعي هيكون
+
+408
+00:36:10,210 --> 00:36:14,260
+الـ U of F أكبر أو يساوي B minus A. To be more
+
+409
+00:36:14,260 --> 00:36:18,360
+specific, for every B element in B of I، نحن لدينا M
+
+410
+00:36:18,360 --> 00:36:22,140
+I في B minus A أصغر أو يساوي L, B و F لأن هذا واحد
+
+411
+00:36:22,140 --> 00:36:28,800
+من الـ partitions قلنا اللي هي لفترة I. نفس الشيء
+
+412
+00:36:28,800 --> 00:36:34,800
+فبصير عند الـ L of B و A F أصغر أو يساوي اللي هو الـ
+
+413
+00:36:34,800 --> 00:36:40,060
+U, B و F أصغر أو يساوي M I في B minus A. معروف هذا
+
+414
+00:36:40,060 --> 00:36:43,580
+أنا مش حاجة لزمنه أصلاً فبصير mi أصغر سواء ل و بف
+
+415
+00:36:43,580 --> 00:36:48,060
+و ل و بف أصغر سواء mi في الـ b-a اللي وضحته هناك
+
+416
+00:36:48,060 --> 00:36:54,640
+أوضح من اللي مكتوب هناك. واضح أه. نيجي لنظرية اللي
+
+417
+00:36:54,640 --> 00:37:00,630
+عندي هذه Theorem بتقول لت I بساوية A وB، and لت F من
+
+418
+00:37:00,630 --> 00:37:05,490
+I لـ R بيه bounded function وفهمنا ليش احنا بنقول
+
+419
+00:37:05,490 --> 00:37:08,410
+bounded, then the lower integral L of F and the
+
+420
+00:37:08,410 --> 00:37:12,090
+upper integral U of F on I exist. Moreover, ال L of
+
+421
+00:37:12,090 --> 00:37:15,330
+F أشمالها أصغر ساوية U of F. طبعا الـ existence سبب
+
+422
+00:37:15,330 --> 00:37:18,650
+الـ existence كله اللي هو انه الـ function bounded
+
+423
+00:37:18,650 --> 00:37:21,110
+مثلا bounded. كل اللي هنا الحسابات اللي على أساس الـ
+
+424
+00:37:21,110 --> 00:37:25,350
+supremum information موجودة لأن اللي قال الـ U
+
+425
+00:37:25,350 --> 00:37:29,170
+of F أصغر ساوي مين؟ U بقى، U of F. كلام سهل، شوف
+
+426
+00:37:29,170 --> 00:37:32,210
+إيش بدنا ناخد؟ بتاخد بي واحد و بي اتنين. بي أي two
+
+427
+00:37:32,210 --> 00:37:36,210
+partitions of I، ماشي؟ لأن then by لمّا اللي حكينا
+
+428
+00:37:36,210 --> 00:37:40,150
+قبل شوية عنها، اللي قال بي واحد و أف أصغر ساوي U
+
+429
+00:37:40,150 --> 00:37:44,350
+بي اتنين of F، ماشي الحال؟ لأي two partitions،
+
+430
+00:37:44,350 --> 00:37:48,270
+ثبتلي واحدة منهم الآن. خلي اللي هو بي اتنين of is
+
+431
+00:37:48,270 --> 00:37:52,560
+fixed. ماشي؟ صارت هذه .. هذه القيمة صارت .. بنحكي عن
+
+432
+00:37:52,560 --> 00:37:56,980
+عدد محدد. ماشي بـ 2 arbitrarily صح؟ لكن الآن أنا بدي
+
+433
+00:37:56,980 --> 00:38:01,020
+ثابته وهذه المتغيرة. صار عند الآن الـ U B 2 و F
+
+434
+00:38:01,020 --> 00:38:07,940
+عبارة عن upper bound of the set of all اللي هي مين
+
+435
+00:38:07,940 --> 00:38:13,790
+الـ B I و F على أي partition. طيب هذا upper bound
+
+436
+00:38:13,790 --> 00:38:19,270
+فبنقله في قفل هذه اللي هو الـ minimum الهن الـ
+
+437
+00:38:19,270 --> 00:38:22,470
+supremum الهن مظبوط اللي هو الـ least upper bound
+
+438
+00:38:22,470 --> 00:38:26,350
+إذا الـ least upper bound أكيد أصغر أو شومين الـ
+
+439
+00:38:26,350 --> 00:38:32,340
+upper bound كمان مرة. طيب، الآن therefore U B 2 و F
+
+440
+00:38:32,340 --> 00:38:37,820
+is an upper bound of هذه الست ماشي؟ وهذه الست الـ
+
+441
+00:38:37,820 --> 00:38:42,500
+supremum لها إيش اسمه؟ L of F إذا أكيد اللي هو الـ
+
+442
+00:38:42,500 --> 00:38:45,220
+supremum اللي هو الـ least upper bound اللي هو الـ L
+
+443
+00:38:45,220 --> 00:38:51,620
+of F أصغر أو يساوي واحد من الـ upper bounds، مظبوط؟
+
+444
+00:38:51,620 --> 00:38:58,990
+أي سؤال؟ واضح ها؟ الآن خلصها من هذه. الآن similarly
+
+445
+00:38:58,990 --> 00:39:04,390
+since B2 was arbitrarily partition of I. صارت الآن
+
+446
+00:39:04,390 --> 00:39:10,230
+عندي L of F أصغر أو يساوي UB2 of F. يعني صارت هذه
+
+447
+00:39:10,230 --> 00:39:17,530
+عدد أصغر أو يساوي الـ UB2 of F for any partition B2
+
+448
+00:39:18,500 --> 00:39:22,560
+الآن صارت هذه عبارة عن lower bound لها لهذه الـ
+
+449
+00:39:22,560 --> 00:39:26,200
+collection. lower bound لها. إذا هذه lower bound
+
+450
+00:39:26,200 --> 00:39:30,160
+أصغر أو يساوي الـ greatest lower bound. مين الـ
+
+451
+00:39:30,160 --> 00:39:35,200
+greatest lower bound لهدول اللي هو الـ L اللي هو الـ
+
+452
+00:39:35,200 --> 00:39:45,700
+U of F إذا صار عند الـ L of F أصغر أو يساوي U of F
+
+453
+00:39:45,700 --> 00:39:48,360
+أي سؤال؟
+
+454
+00:39:51,810 --> 00:39:56,530
+واضحة؟ إن في سؤال كان في ريال واحد زي هيك بنحل على
+
+455
+00:39:56,530 --> 00:39:59,690
+هيك اتنين أربع اتنين أربع اتنين أربع اتنين أربع
+
+456
+00:39:59,690 --> 00:40:00,510
+اتنين أربع اتنين أربع اتنين أربع اتنين أربع اتنين 
+
+457
+00:40:00,510 --> 00:40:01,130
+أربع اتنين أربع اتنين أربع اتنين أربع اتنين أربع
+
+458
+00:40:01,130 --> 00:40:01,850
+اتنين أربع اتنين أربع اتنين أربع اتنين أربع اتنين
+
+459
+00:40:01,850 --> 00:40:06,550
+اتنين أربع اتنين أربع اتنين أربع اتنين أربع اتنين
+
+460
+00:40:06,550 --> 00:40:13,540
+أربع اتنين أربع أعداد. هذا هو اللي هو اللي اللي اللي
+
+461
+00:40:13,540 --> 00:40:17,340
+اللي المتغير. صار عند العدد هذا أصغر أو شاوي كل
+
+462
+00:40:17,340 --> 00:40:22,080
+اللي هو أعضاء العائلة اللي بتحكي عنها. إذا صار الآن
+
+463
+00:40:22,080 --> 00:40:27,580
+هذا الـ الوفاف عبارة عن lower bound of the collection
+
+464
+00:40:27,580 --> 00:40:32,600
+of all U, P, T و F عالميا على الـ partitions هذه
+
+465
+00:40:32,600 --> 00:40:36,260
+مادام هذا صار عبارة عن lower bound إذا هذا lower
+
+466
+00:40:36,260 --> 00:40:40,580
+bound أصغر أو يساوي الـ greatest lower bound اللي
+
+467
+00:40:40,580 --> 00:40:43,860
+هو اللي بنسميه الـ infimum إذا صارت الـ L و F أصغر
+
+468
+00:40:43,860 --> 00:40:47,000
+أو يساوي الـ infimum لهدول. الـ infimum لهدول مين هو
+
+469
+00:40:47,000 --> 00:40:52,940
+الـ U و F. الـ U of F هو الـ infimum لكل الـ U, B, T,
+
+470
+00:40:53,280 --> 00:40:58,040
+2 و F such that B2 element in the set of all
+
+471
+00:40:58,040 --> 00:41:07,220
+partitions. نيجي الآن لبعده الـ Definition الآن ..
+
+472
+00:41:07,220 --> 00:41:16,080
+الآن احنا زغرنا الـ other sums وكبرنا الـ other
+
+473
+00:41:16,080 --> 00:41:22,680
+sums. يوم ما التجهن مع بعض بنسمي اللي هو الالتقاء هذا
+
+474
+00:41:22,680 --> 00:41:26,320
+أو الرقم اللي التقن فيه هو قيمة الـ Remain
+
+475
+00:41:26,320 --> 00:41:30,800
+Integral للفانكشن F على الفترة المذكورة وهذا هو
+
+476
+00:41:30,800 --> 00:41:35,040
+التعريف. Let I بسوء A و B and let F من I لعند R be
+
+477
+00:41:35,040 --> 00:41:39,060
+a bounded function, then F is said to be Remain
+
+478
+00:41:39,060 --> 00:41:46,480
+Integrable on I if L of F اللي هو عبارة عن الـ
+
+479
+00:41:46,480 --> 00:41:54,140
+Supremum للـ Lower Bounds للـ Lower Sums بساوي الـ
+
+480
+00:41:54,140 --> 00:42:00,320
+Infimum للـ Upper Sums. يوم ما يتساون هذولة بنسمي
+
+481
+00:42:00,320 --> 00:42:03,780
+الـ Function is Remain Integrable وفي هذه الحالة
+
+482
+00:42:03,780 --> 00:42:06,860
+بيكون القيمة المشتركة هذه هي اللي بنسميها الـ
+
+483
+00:42:06,860 --> 00:42:13,740
+Integration من A ل B F of X D of X. دكتور إن الملحنة
+
+484
+00:42:13,740 --> 00:42:17,420
+احنا بدنا نجيب مساحة كذا كذا. احنا الـ .. المساحة
+
+485
+00:42:17,420 --> 00:42:21,480
+إذا كان فوق الآن إذا كان تحت بيكون سالب المساحة
+
+486
+00:42:21,480 --> 00:42:24,680
+إذا كان مخلوط بيصير .. إذا كان فوق و تحت معناته
+
+487
+00:42:24,680 --> 00:42:28,180
+بيصير جزء مساحة وجزء سالب مساحة عشان نجد مساحته
+
+488
+00:42:28,180 --> 00:42:30,980
+بالظبط تاخد الـ absolute بتقطع لذاك وتاخد الـ
+
+489
+00:42:30,980 --> 00:42:34,340
+absolute value اللي بيطلع اللي هو المساحة عشان
+
+490
+00:42:34,340 --> 00:42:39,580
+نجيبوا مساحته ونحسب الملحنة. هذا لمستطيلة صح؟
+
+491
+00:42:41,500 --> 00:42:46,400
+كمستطيل وبيجيبوا مجموع مستطلاق بدين المنحنى بديك
+
+492
+00:42:46,400 --> 00:42:51,760
+مساحة تحت المنحنى. لما تاخد الـ upper sums و الـ
+
+493
+00:42:51,760 --> 00:42:55,260
+lower sums وتاخد الـ supremum وتاخد الـ infimum ويكون
+
+494
+00:42:55,260 --> 00:42:59,080
+الـ supremum بساوي الـ infimum بتلح هو المساحة
+
+495
+00:42:59,080 --> 00:43:04,990
+تحت المنحنى في حالة الـ positive functions. في حالة
+
+496
+00:43:04,990 --> 00:43:08,710
+الـ negative المخلوق بيصير integration بيصير قيمة
+
+497
+00:43:08,710 --> 00:43:12,170
+integration. قيمة الـ integration لا تعبر عن المساحة
+
+498
+00:43:12,170 --> 00:43:16,670
+اللي هو بالضبط إلا في حالة الـ other إلا في حالة الـ
+
+499
+00:43:16,670 --> 00:43:19,970
+positive الـ function. في حالة الـ negative بتعبر على
+
+500
+00:43:19,970 --> 00:43:24,170
+سلب المساحة. في حالة إشي فوج وإشي تحت بيصير اللي
+
+501
+00:43:24,170 --> 00:43:26,950
+هو .. لأ المساحات بت cancel بعضها وبيصير قيمة
+
+502
+00:43:26,950 --> 00:43:30,930
+integration. بيدك المساحات من وسط الـ integration
+
+503
+00:43:30,930 --> 00:43:33,110
+إنها تستخدم الـ integration في إيجاد المساحة
+
+504
+00:43:33,110 --> 00:43:37,570
+بتجزئها لإيش؟ إلى أجزاء وين بتقطع محور السينات يعني
+
+505
+00:43:37,570 --> 00:43:40,550
+بتحدد وين موجبة وين سالبة. بتاخد الـ absolute
+
+506
+00:43:40,550 --> 00:43:43,410
+value حتى لو ماكنشت يعرف مين الموجبة ومين السالبة
+
+507
+00:43:43,410 --> 00:43:46,570
+وبيطلع اللي هو عن طريق الـ integration بتوجد قيمة
+
+508
+00:43:46,570 --> 00:43:52,290
+مين المساحة. لكن في حالة الدالة موجبة على كل الفترة
+
+509
+00:43:52,290 --> 00:43:59,410
+الـ integration هو بالضبط المساحة. طيب صلّي على النبي
+
+510
+00:43:59,410 --> 00:44:01,990
+عليه الصلاة والسلام. عرفنا الـ Romantic Grability
+
+511
+00:44:01,990 --> 00:44:06,470
+وقلنا اللي هو .. اللي هو الاشياء المشتركة اللي
+
+512
+00:44:06,470 --> 00:44:11,290
+حصلنا عليها. الآن بنعرف برضه كتعريف من A ل B F of X
+
+513
+00:44:11,290 --> 00:44:15,930
+DX بساوي ناقص من B ل A F of X DX وبنعرف كتعريف أن
+
+514
+00:44:15,930 --> 00:44:19,830
+الـ integration من A ل A F of X DX H بساوي صفر. اطلع
+
+515
+00:44:19,830 --> 00:44:25,120
+فوق. خليني أخد مثال وننهي بهذا المثال. الآن .. وهذا
+
+516
+00:44:25,120 --> 00:44:30,080
+المثال احنا .. يعني .. كنا ناخد .. نفس المثال اللي
+
+517
+00:44:30,080 --> 00:44:33,760
+ناخده في الـ calculus خلينا نحكي عنه بالتفصيل الآن. 
+
+518
+00:44:33,760 --> 00:44:37,480
+Example: show that g of x بساوة x is integrable on
+
+519
+00:44:37,480 --> 00:44:41,700
+0 و 1. بدنا نثبت أنه الدالة اللي هي g of x اللي
+
+520
+00:44:41,700 --> 00:44:45,220
+بنعرفها اللي هي المثال البسيط بساوة x عبارة عن
+
+521
+00:44:45,220 --> 00:44:52,130
+integrable عالميا على اللي هو الفترة 0 و 1. الآن أنا
+
+522
+00:44:52,130 --> 00:44:56,270
+لتسهيل عملية الحسابات هاخد عائلة من الـ partitions
+
+523
+00:44:56,270 --> 00:45:03,050
+معينة. هذه العائلة بتسهيل الحسابات. طب بنفع في ضوء
+
+524
+00:45:03,050 --> 00:45:07,910
+العلاقات اللي هنشوفها كيف توصلنا بالظبط إن الـ L of
+
+525
+00:45:07,910 --> 00:45:12,830
+F بساوة الـ U of F، ولخصوصية الحسابات ده نشوف كيف
+
+526
+00:45:12,830 --> 00:45:21,760
+عندي الـ F of X بساوة X. عنديها بديها من 0 لعند مين؟
+
+527
+00:45:21,760 --> 00:45:28,580
+لعند 1. هذه y بساوة x على الفترة 0 و 1. إيش بدنا
+
+528
+00:45:28,580 --> 00:45:34,580
+نسوي؟ بدنا نيجي اللي هو إن نعمل partition بين هذا
+
+529
+00:45:34,580 --> 00:45:39,540
+الـ partition. بديه يسهل الإحسابات فأسهل شيء بدي
+
+530
+00:45:39,540 --> 00:45:44,640
+أجزء هذه الفترة من 0 ل1 لـ N من الأجزاء وبديها تكون
+
+531
+00:45:44,640 --> 00:45:49,430
+كلها متساوية. بنفع، أه، هينفع، هتشوف كيف اللي هو هذا
+
+532
+00:45:49,430 --> 00:45:52,650
+نفع في ضوء العلاقات بين اللي هو الـ Supremum و
+
+533
+00:45:52,650 --> 00:45:57,050
+الانفموع المجموعة الأكبر والمجموعة الأصغر، وأيضًا
+
+534
+00:45:57,050 --> 00:46:01,130
+في ضوء الـ limits المعينة اللي هان اللي هتيجي تخلي
+
+535
+00:46:01,130 --> 00:46:05,010
+اللي هو علاقة توصّل إلى اللي بدنيها. تفهم علش؟
+
+536
+00:46:05,010 --> 00:46:10,610
+ماقولهم كده. هذا صار عندي، هذا Zero أما ده بدأت
+
+537
+00:46:10,610 --> 00:46:13,510
+تجزئها. طولها واحد. تجزئها الآن من الفتراء sub
+
+538
+00:46:13,510 --> 00:46:16,130
+intervals إذا كل واحدة واحدة على الآن. هذا التاني
+
+539
+00:46:16,130 --> 00:46:20,130
+اتنين على الآن. التالتة تلاتة على الآن. تلاتة على
+
+540
+00:46:20,130 --> 00:46:24,770
+الآن لما نصل للفترة النموذجية xk اللي هي تلقى k
+
+541
+00:46:24,770 --> 00:46:30,610
+minus واحد على ان وk على ان. هذا اللي هي xk minus
+
+542
+00:46:30,610 --> 00:46:36,670
+واحد وهذا الـ xk وهذا xنوت واد اكس واحد الى اخرى
+
+543
+00:46:36,670 --> 00:46:39,790
+لان ما اصل الاخر واحدة اللى هى n على n اللى هى
+
+544
+00:46:39,790 --> 00:46:45,150
+واحدة. ماشي الحال؟ هذا اللى عندى اللى .. لو اجيت
+
+545
+00:46:45,150 --> 00:46:48,190
+حسبت الآن مدى احسب هذا الـ partitions اللي صار عندي
+
+546
+00:46:48,190 --> 00:46:53,930
+عبارة عن zero والواحدة الآن و2 على n. لما اصل
+
+547
+00:46:53,930 --> 00:46:58,490
+لعند الـ n على n اللي هي إيش الواحدة. هذه عائلة الـ
+
+548
+00:46:58,490 --> 00:47:04,620
+partitions اللي اختارتها. ماشي؟ شوفوا عليها الـ n. لو
+
+549
+00:47:04,620 --> 00:47:09,460
+جيت حسبت اللي هو طول الـ interval واحدة. لأن لو بده
+
+550
+00:47:09,460 --> 00:47:16,960
+أحسب الآن L للـ partition هذا اللي هو مين هو L الـ F
+
+551
+00:47:16,960 --> 00:47:23,840
+عندي والـ P N هتساوي عبارة عن الـ summation. عارفين
+
+552
+00:47:23,840 --> 00:47:29,780
+انتوا الـ summation اللي هو M small K في X K ناقص X
+
+553
+00:47:29,780 --> 00:47:36,290
+K minus واحد. K من عند واحد عند N ويساوي. الآن الـ MK
+
+554
+00:47:36,290 --> 00:47:41,890
+في الفترة الأولى إيش هيكون عنده؟ صفر. في الفترة
+
+555
+00:47:41,890 --> 00:47:47,430
+التانية إيش هيكون؟ واحد على الانكريزين جدا اللي
+
+556
+00:47:47,430 --> 00:47:50,150
+بعدها اتنين على انواع وهكذا.
+
+557
+00:47:52,920 --> 00:47:57,520
+الـ Mk in general إيش هيصير عنده الـ Mk وإيش
+
+558
+00:47:57,520 --> 00:48:03,020
+المساحة هي K minus واحد على مين على هذا اللي هو
+
+559
+00:48:03,020 --> 00:48:08,280
+طول تبع مين الـ Mk. واضحة؟ لأن الـ infimum على
+
+560
+00:48:08,280 --> 00:48:12,760
+الفترة هذه النموذجية اللي هو الـ .. الـ .. الـ
+
+561
+00:48:12,760 --> 00:48:17,920
+infimum عليها اللي هو أول واحدة لإنها increasing
+
+562
+00:48:17,920 --> 00:48:22,630
+واضحة؟ تصور فلان هذه ثابتة أنا اختارتها أصلاً ثابتة
+
+563
+00:48:22,630 --> 00:48:26,430
+طول الفترة ثابتة اللي هو 1 على n بيصير هذا إيش n
+
+564
+00:48:26,430 --> 00:48:32,670
+تربيع كامل عند واحد لعند مين لعند n. واضح أه، ويساوي
+
+565
+00:48:32,670 --> 00:48:38,110
+1 على n تربيع فيه الـ summation هذا اللي هو عبارة
+
+566
+00:48:38,110 --> 00:48:42,830
+عن بيبدأ من 0 زائد 1 ولمّا يصل لعند مين لعند n
+
+567
+00:48:42,830 --> 00:48:49,760
+ناقص 1. هذا بعرف ما هو مجموعه، ويساوي 1 الآن تربيع في
+
+568
+00:48:49,760 --> 00:48:54,760
+هذا ان لما نكون من واحد لعند ان. إن فان زائد واحد
+
+569
+00:48:54,76
+
+601
+00:51:44,820 --> 00:51:45,160
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+602
+00:51:45,160 --> 00:51:48,400
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+603
+00:51:48,400 --> 00:51:48,420
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+604
+00:51:48,420 --> 00:51:50,120
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+605
+00:51:50,120 --> 00:51:50,220
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+606
+00:51:50,220 --> 00:51:50,340
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+607
+00:51:50,340 --> 00:51:59,480
+.. ال .. ال .. ال .. ال .. ال ..
+
+608
+00:51:59,480 --> 00:52:02,920
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+609
+00:52:02,920 --> 00:52:09,060
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. such that
+
+610
+00:52:09,060 --> 00:52:14,120
+N element in N هذه اللي هي main the collection of 
+
+611
+00:52:14,120 --> 00:52:19,260
+all partitions the B1 وB2 وB3 عند BN لو جينا أخدنا
+
+612
+00:52:19,260 --> 00:52:24,740
+the supremum لكل القيم هذه هيكون أصغر أو يساوي الـ
+
+613
+00:52:24,740 --> 00:52:32,320
+Supremum لمين؟ للـ L of F و B such that B element in
+
+614
+00:52:32,320 --> 00:52:36,620
+B and B of I لأن the collection of all partitions هذه
+
+615
+00:52:36,620 --> 00:52:41,940
+اللي هو تحتوى ال partitions هدولة واضح الحمد لله
+
+616
+00:52:41,940 --> 00:52:45,500
+الآن the supremum على القيم هذه اللي هي مين القيم
+
+617
+00:52:45,500 --> 00:52:51,230
+كانت عنده اللي هو عبارة عن نص في واحد ناقص ميل واحد
+
+618
+00:52:51,230 --> 00:52:55,590
+علان the supremum لهذه عبارة عن zero إذن بتظل هي
+
+619
+00:52:55,590 --> 00:53:00,590
+إيش بتساوي؟ نص the supremum للنص في واحد ناقص واحد
+
+620
+00:53:00,590 --> 00:53:05,070
+علان معايا؟
+
+621
+00:53:05,070 --> 00:53:16,470
+أي سؤال؟ طيب إذا هذا مين هو؟ هذا عبارة عن the أو ما 
+
+622
+00:53:16,470 --> 00:53:21,170
+فاهم ولا لأ؟ إذا صار عند ال L و ال F مثبتها هذه ال
+
+623
+00:53:21,170 --> 00:53:29,390
+L و ال F هيصير أكبر أو يساوي النص نحك طيب
+
+624
+00:53:29,390 --> 00:53:35,650
+similarly for me and for you الآن عندي حصلت على ال
+
+625
+00:53:35,650 --> 00:53:41,970
+U, F و B أن لهذه العائلة الآن الآن the set of all
+
+626
+00:53:41,970 --> 00:53:50,880
+U, F و B أن such that N element in N هذه العائلة
+
+627
+00:53:50,880 --> 00:53:53,900
+مجموعة جزئية من the العائلة الكبيرة the B of I the
+
+628
+00:53:53,900 --> 00:54:00,940
+partitions للـ I إذن هذه الـ infimum لها هيصير
+
+629
+00:54:00,940 --> 00:54:10,580
+أكبر أو يساوي الـ infimum لـ U, F و B such that B
+
+630
+00:54:10,580 --> 00:54:18,550
+element in B of I the collection هذه أصغر من the
+
+631
+00:54:18,550 --> 00:54:21,730
+collection هذه، بظبط؟ إذا صار الـ infimum على هذه
+
+632
+00:54:21,730 --> 00:54:25,610
+أكبر و ساوي الـ infimum على هذه الـ infimum لهذه
+
+633
+00:54:25,610 --> 00:54:27,470
+the collection و اللي هو الـ infimum في الواقع
+
+634
+00:54:27,470 --> 00:54:31,110
+لمين؟ الـ infimum لهذه الـ infimum لهذه ليش
+
+635
+00:54:31,110 --> 00:54:35,910
+هيساوي؟ نص فبصير عندي نص أكبر و ساوي الـ infimum
+
+636
+00:54:35,910 --> 00:54:38,430
+هذا اللي هو أكبر و ساوي هذا اللي هو هذا اللي هو
+
+637
+00:54:38,430 --> 00:54:44,010
+مين؟ the U of F إذا حصلنا على كمان حاجة إن الـ U
+
+638
+00:54:44,010 --> 00:54:52,790
+of F أصغر أو يساوي نص اللخص المعلومتين هدولة ونحصل
+
+639
+00:54:52,790 --> 00:54:57,810
+على اللي بدناه من المعلومتين من المعلومتين اللي
+
+640
+00:54:57,810 --> 00:55:06,860
+صار عندي اللي هو L of F اللي هو اكبر او يساوي نص
+
+641
+00:55:06,860 --> 00:55:10,920
+و أنا بعرف ان دائما ال L of F ايش ماله؟ اصغر او يساوي
+
+642
+00:55:10,920 --> 00:55:15,180
+ال U of F و ال U of F ايش حصلته؟ اصغر او يساوي من
+
+643
+00:55:15,180 --> 00:55:20,360
+نص إذا هذا اللي خلى اللي هو the partition اللي عندي
+
+644
+00:55:20,360 --> 00:55:23,540
+اللي سهل اللي اخترته او the set of partitions انها
+
+645
+00:55:23,540 --> 00:55:28,580
+توصلني للنتيجة اللي بديها بيصير عندي ال L of F اصغر
+
+646
+00:55:28,580 --> 00:55:33,220
+او يساوي ال U of F و هذا بين النص التنتين اذا L of F
+
+647
+00:55:33,220 --> 00:55:40,650
+اف بساوي U of F بساوي نص و هذا معناه ان the function
+
+648
+00:55:40,650 --> 00:55:45,770
+is reversible and integrable and the integration من
+
+649
+00:55:45,770 --> 00:55:52,650
+zero ل واحد لل x dx بساوي نص و الله يعطيك العافية
+
+650
+00:55:52,650 --> 00:55:54,590
+و المرة القادمة نكملها
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..a09f05fedfb0e4f455c826b0adeb2bea6ca636d9
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U_postprocess.srt
@@ -0,0 +1,2600 @@
+1
+00:00:05,150 --> 00:00:09,890
+بسم الله الرحمن الرحيم اليوم ان شاء الله عندنا
+
+2
+00:00:09,890 --> 00:00:15,270
+اللي هو المحاضرة رقم عشرة في مساق تحليل حقيقه
+
+3
+00:00:15,270 --> 00:00:21,130
+اتنين لطلبة الجامعة الإسلامية كلية العلوم قسم
+
+4
+00:00:21,130 --> 00:00:27,370
+الرياضيات اليوم ان شاء الله هنبدأ في chapter سبعة
+
+5
+00:00:28,110 --> 00:00:33,150
+اللي هو الجزء الثاني من المادة أو العصب الثاني
+
+6
+00:00:33,150 --> 00:00:37,130
+للمادة اللي هو حكينا عن ال derivatives اليوم نحكي
+
+7
+00:00:37,130 --> 00:00:42,170
+عن اللي هو the reman integral أو تكامل رمان هيتم
+
+8
+00:00:42,170 --> 00:00:45,730
+الحديث اليوم ان شاء الله في ال section الأول هنحكي
+
+9
+00:00:45,730 --> 00:00:51,450
+عن ال reman integrability أو قابلية تكامل الدالة
+
+10
+00:00:51,450 --> 00:00:56,270
+على اللي هو بالنسبة لرمانفي البداية هنحكي عن بعض
+
+11
+00:00:56,270 --> 00:01:00,410
+التعريف اللي هتوصلنا للي بدناياه في الموضوع اللي
+
+12
+00:01:00,410 --> 00:01:03,910
+هو الـ Riemann Integrability الان هاخدلنا function
+
+13
+00:01:03,910 --> 00:01:11,090
+F F من some interval I نفترض ان الدالة اللي عند F
+
+14
+00:01:11,090 --> 00:01:18,850
+من A و B لعند R is a bounded function الان
+
+15
+00:01:18,850 --> 00:01:22,050
+بدنا نعرف بعض التعريفات اللي بتخودنا الى الـ
+
+16
+00:01:22,050 --> 00:01:26,950
+Riemann Integrabilityأول اشي بدنا نعرف شو معناه
+
+17
+00:01:26,950 --> 00:01:30,750
+partition لو كانت هذه الدانلة مثلا هي Y بالساوية F
+
+18
+00:01:30,750 --> 00:01:36,170
+of X على الفترة المذكورة وكانت من A ل B اللي هو ال
+
+19
+00:01:36,170 --> 00:01:41,650
+domain بدنا الآن نيجي نجزق المنطقة اللي هي الفترة
+
+20
+00:01:41,650 --> 00:01:47,210
+من A ل B اللي هو خلّي نسمي هذا X noteهذا نسميه x1
+
+21
+00:01:47,210 --> 00:01:51,750
+هذا نسميه x2 لعند ما أصل للفترة النموذجية اللي
+
+22
+00:01:51,750 --> 00:01:58,270
+بتسميها xk-1 لعند ال xk وهذا أسميها اللي هي xn
+
+23
+00:01:58,270 --> 00:02:06,110
+الان الفترة من a لb أنا جزقتها بx0 وx1 وx2 لعند xn
+
+24
+00:02:06,110 --> 00:02:13,370
+هذه التجزية بتحطها في 6 وسميها بي بتساوي x0و X1 و
+
+25
+00:02:13,370 --> 00:02:19,410
+X2 لعند Xn هذا اللي بسميه is a partition of the
+
+26
+00:02:19,410 --> 00:02:26,510
+interval A و B إذا عندي I subset من R closed
+
+27
+00:02:26,510 --> 00:02:30,430
+bounded interval ال ordered set هذه اللي حكيتها
+
+28
+00:02:30,430 --> 00:02:34,410
+كتبتها ان هي عبارة عن X0 و X1 لعند Xn بنسميها a
+
+29
+00:02:34,410 --> 00:02:39,390
+partition of I بحيث انه A X0 أصغر من X1 لعند ما
+
+30
+00:02:39,390 --> 00:02:45,060
+أقول أصغر من Xnالان الـ function عندى F من I لعند
+
+31
+00:02:45,060 --> 00:02:49,980
+R bounded function on I الان بدى أعرف بعض
+
+32
+00:02:49,980 --> 00:02:54,660
+التعريفات اللى هتلزمني برضه في اللى هو الوصول إلى
+
+33
+00:02:54,660 --> 00:02:59,160
+الـ Remain Integrability أول تعريف بدى أخد الآن
+
+34
+00:02:59,160 --> 00:03:07,000
+أسمي M1 هو عبارة عن الـ supremum للـ function في
+
+35
+00:03:07,000 --> 00:03:12,990
+المنطقة هذهالـ M1 بدي أسميه عبارة عن الـ Supremum
+
+36
+00:03:12,990 --> 00:03:16,810
+للدالة F of X والـ Supremum موجودة لأن الـ F
+
+37
+00:03:16,810 --> 00:03:21,870
+أشمالها is bounded السوب للـ F of X حيث الـ X
+
+38
+00:03:21,870 --> 00:03:29,310
+موجودة في الفترة من X note لعند X1 هذه اللي هي بدي
+
+39
+00:03:29,310 --> 00:03:34,570
+أسميها M1 Capital M1 Small اللي هي عبارة عن ال
+
+40
+00:03:34,570 --> 00:03:38,960
+Infimumلـ .. لـ الـ function hand اللي عبارة عن
+
+41
+00:03:38,960 --> 00:03:45,400
+الـ infimum لـ F of X حيث X تنتمي إلى الفترة X
+
+42
+00:03:45,400 --> 00:03:50,440
+naught و X واحد in general .. in general في الفترة
+
+43
+00:03:50,440 --> 00:03:55,280
+النموذجية بدي أسمي M K هي عبارة عن .. الآن في
+
+44
+00:03:55,280 --> 00:03:59,850
+الفترة هذهعالفترة هذه الـ supremum للـ function
+
+45
+00:03:59,850 --> 00:04:06,170
+هنا بدي اسميها M K K بسوء F of X such that X
+
+46
+00:04:06,170 --> 00:04:13,030
+element in X K minus واحد لعند X K و ال M small K
+
+47
+00:04:13,030 --> 00:04:19,330
+هسميها اللي هي بنفس اللي هسميت X واحد اللي هي ال
+
+48
+00:04:19,330 --> 00:04:23,370
+infimum لل F of X في الفترة من X K minus واحد لعند
+
+49
+00:04:23,370 --> 00:04:30,300
+X K على سبيل المثالالـ M K يعني هيها هذا اللي هي M
+
+50
+00:04:30,300 --> 00:04:36,340
+K Small والـ M K Capital على سبيل المثال هيها
+
+51
+00:04:36,340 --> 00:04:42,640
+هيقبتها هذا الـ M K Capital إذن الآن عندي سميت
+
+52
+00:04:42,640 --> 00:04:45,560
+اللي هي الـ M K Capital هي الـ Supremum والـ M K
+
+53
+00:04:45,560 --> 00:04:49,740
+Small هي عبارة عن الـ Infimum وهلجيت حنشوف إيش
+
+54
+00:04:49,740 --> 00:04:55,560
+دواعي هذه التسمية أو ليش سمينا هيكالان بدنا نعرف
+
+55
+00:04:55,560 --> 00:05:00,380
+اللي هو .. بدنا
+
+56
+00:05:00,380 --> 00:05:04,380
+نعرف الآن اللي هو حاجة اسمها lower sum و حاجة
+
+57
+00:05:04,380 --> 00:05:12,540
+اسمها ايش ال upper sum الان لو جيت بعد ما عملت ال
+
+58
+00:05:12,540 --> 00:05:21,020
+partition اللي عندي هذا و جيت أخدت اللي هي
+
+59
+00:05:28,550 --> 00:05:32,770
+أخدت الـ partition B اللي عندي هذا هذا ال
+
+60
+00:05:32,770 --> 00:05:40,110
+partition B هو عبارة عن X0 X1 X2 اللي عند XN جيت
+
+61
+00:05:40,110 --> 00:05:44,590
+أنا أخدت اللي هو الـ supremum للدالة في هذه
+
+62
+00:05:44,590 --> 00:05:51,550
+المنطقة والتي تكون طبعا هنا شكلها اللي هي سميتها
+
+63
+00:05:51,550 --> 00:06:02,170
+M1 في X1 نقص X0و جيت ضفت لها برضه ال supremum
+
+64
+00:06:02,170 --> 00:06:08,030
+للدالة في المنطقة هذه اللي هي زائد M2 سميت ال
+
+65
+00:06:08,030 --> 00:06:15,230
+supremum اللي هو M2 في X2-X0 لما نوصلت لآخر واحدة
+
+66
+00:06:15,230 --> 00:06:20,750
+بنفس الأسلوب أخدت اللي هو ال supremum اللي هي ال
+
+67
+00:06:20,750 --> 00:06:26,870
+supremum طبعا M1 عندىالتي هي طول الريانج اللي عندى
+
+68
+00:06:26,870 --> 00:06:32,850
+مضروب في مين اللي هو x1-x0 فبتطلع هذه المنطقة
+
+69
+00:06:32,850 --> 00:06:41,710
+بالظبط مساحة اللي عندى الان بضل أجمع بنفس الأسلوب
+
+70
+00:06:41,710 --> 00:06:45,830
+بجمحن بنفس الأسلوب عندى
+
+71
+00:06:53,690 --> 00:06:58,050
+هذا المجموع اللي بالازرق هو اللي بنسميه هسميه ال
+
+72
+00:06:58,050 --> 00:07:03,170
+other sum هذا ال other sum اللي هو ال other لل F
+
+73
+00:07:03,170 --> 00:07:08,090
+للدالة F بواسطة
+
+74
+00:07:08,090 --> 00:07:12,830
+مين ال partition B إذا ال other sum هذا اللي هو
+
+75
+00:07:12,830 --> 00:07:17,830
+عبارة عن اللي هو لل function بواسطة ال partition B
+
+76
+00:07:17,830 --> 00:07:20,410
+يعني ال other sum هذا يعتمد على ال F ويعتمد على
+
+77
+00:07:20,410 --> 00:07:25,690
+مين؟ على ال partition اللي عنديالأن similarly بدي
+
+78
+00:07:25,690 --> 00:07:31,190
+آجي أعرف اللي هو ال lower sum إذا ال upper sum
+
+79
+00:07:31,190 --> 00:07:38,890
+بيساوي ال summation ل X K نقص X K minus واحد مضروب
+
+80
+00:07:38,890 --> 00:07:43,950
+في ال M K capital K من عند واحد لعند N هذه اللي
+
+81
+00:07:43,950 --> 00:07:48,930
+بيسميها ال upper sum similarly بدي أعرف مين اللي
+
+82
+00:07:48,930 --> 00:07:56,170
+هو ال lower sumللـ F و لل D هيساوي الآن بدل ما أخد
+
+83
+00:07:56,170 --> 00:07:59,330
+اللي هو ال suprem أعلى قيمة بتاخد اللي هو ال
+
+84
+00:07:59,330 --> 00:08:05,390
+infimum بيصير عندي اللي هو summation X K minus X K
+
+85
+00:08:05,390 --> 00:08:10,790
+minus واحد في ال M K small K من عند واحد لعند N
+
+86
+00:08:10,790 --> 00:08:15,050
+هذا اللي هو عبارة عن ال lower sum في الواقع هو
+
+87
+00:08:15,050 --> 00:08:21,350
+هيكون المنطقة اللي بالأحمر هذهبناء على إني باخد
+
+88
+00:08:21,350 --> 00:08:27,670
+أنا اللي هو الارتفاع هو عبارة عن ال M K Small اللي
+
+89
+00:08:27,670 --> 00:08:33,410
+هو ال minimum لقيمة الدالة على كل المنطقة وهكذا
+
+90
+00:08:33,410 --> 00:08:38,030
+لما أصل لآخر واحدة فبصير عندى هذا ال summation
+
+91
+00:08:38,030 --> 00:08:43,890
+لهذه اللي هي المستطيلات
+
+92
+00:08:43,890 --> 00:08:51,510
+بناء على ال partition اللي عندىهي عبارة عن مين؟ عن
+
+93
+00:08:51,510 --> 00:08:56,190
+اللي هو اللا وارسم طبعا ناخد الدالة اللي هي في
+
+94
+00:08:56,190 --> 00:09:00,670
+منطقة موجة بأساس إنه اللي هي يكون أكثر توضيحا طبعا
+
+95
+00:09:00,670 --> 00:09:04,650
+لو منطقة سالبة برضه باخد نفس الأسلوب لكن بيصير
+
+96
+00:09:04,650 --> 00:09:08,810
+اللي هو بديل مساحة بيصير سالب مساحة أو مخلوطة بين
+
+97
+00:09:08,810 --> 00:09:13,070
+مساحة موجة مساحة سالبة و بت cancel بعض إلى أن أنا
+
+98
+00:09:13,070 --> 00:09:16,350
+أخدتها على فترة موجة بأساس اللي هو تسهل مسعملية
+
+99
+00:09:16,820 --> 00:09:22,620
+اللي هي التوضيح الآن خلينا تحت شوية إذا اللي حكيته
+
+100
+00:09:22,620 --> 00:09:28,420
+الآن أعرفت ال lower sum زي ما أنتوا شايفين عند ال
+
+101
+00:09:28,420 --> 00:09:32,780
+lower sum هيه هذا ال lower sum يعتمد في مجموعه على
+
+102
+00:09:32,780 --> 00:09:37,300
+ال M K اللي هي عبارة عن ال infimum لدالة على
+
+103
+00:09:37,300 --> 00:09:43,960
+الفترة X K minus واحد لعند X Kو-xk-1 هي عبارة عن
+
+104
+00:09:43,960 --> 00:09:49,040
+قيم ال partition اللي عندي إذا ال lower sum أو حتى
+
+105
+00:09:49,040 --> 00:09:53,820
+ال upper sum التمتين باعتمدين عالميا على اللي هو
+
+106
+00:09:53,820 --> 00:09:57,940
+ال partition وعلى الدالة اللي بنعملها أو لو وجدلها
+
+107
+00:09:57,940 --> 00:10:03,160
+ال lower sum و ال upper sum الهن واضح أن ال lower
+
+108
+00:10:03,160 --> 00:10:09,780
+sum ال lower sum واضح أنه اللي هو دائمااللي هو لل
+
+109
+00:10:09,780 --> 00:10:13,480
+partition المعين هيكون أصغر من مين من ال upper sum
+
+110
+00:10:13,480 --> 00:10:21,380
+واضح أه؟ الان اللي بنحكي if F من I ل R is bounded
+
+111
+00:10:21,380 --> 00:10:27,520
+and B is any partition of I then ال lower sum لل B
+
+112
+00:10:27,520 --> 00:10:32,020
+و ال F أصغر سوى ال upper sum لل B و ال F وبرهانه
+
+113
+00:10:32,020 --> 00:10:39,830
+سهل وسريعلو جئنا و أخذنا الـ bi-partition x0 x1
+
+114
+00:10:39,830 --> 00:10:40,550
+عند xn
+
+115
+00:10:43,460 --> 00:10:48,540
+وقررنا بين الـMK والـMK Capital أكيد الـMK Small
+
+116
+00:10:48,540 --> 00:10:50,540
+أصغر من الـMK Capital لإنه هذا عبارة عن
+
+117
+00:10:50,540 --> 00:10:54,820
+الـSupremum للفункشن على الفترة أو الـSubinterval
+
+118
+00:10:54,820 --> 00:10:59,780
+XK minus 1 عند XK وهذا الـSupremum للفункشن، هذا
+
+119
+00:10:59,780 --> 00:11:02,860
+الـInfimum وهذا الـSupremum فأكيد هذا أكبر أو
+
+120
+00:11:02,860 --> 00:11:07,080
+يساوي ما هو الـMK Small وهذا الكلام صحيح على كل
+
+121
+00:11:07,080 --> 00:11:11,380
+الـSubintervals يعني صحيح لكل K من 1 إلى 2 عند N
+
+122
+00:11:11,490 --> 00:11:14,650
+وفي نفس الوجهة الـ xk هو الـ xk minus واحد أكبر من
+
+123
+00:11:14,650 --> 00:11:18,610
+سفر إذا بيصير عندي حاصل ضرب هادي فى طول الفترة
+
+124
+00:11:18,610 --> 00:11:22,570
+وهادي فى طول الفترة بتظل ال inequality صحيحة لإنه
+
+125
+00:11:22,570 --> 00:11:26,730
+طول الفترة موجبة فبصير mk فى هذا المقدار أصغر ساوى
+
+126
+00:11:26,730 --> 00:11:29,950
+mk capital فى هذا المقدار بنعمل summation على الكل
+
+127
+00:11:29,950 --> 00:11:32,770
+وهذا ال summation على الكل بيصير عندى هذا هو عبارة
+
+128
+00:11:32,770 --> 00:11:36,150
+عن ال upper sum وهذا ال lower sum يعني اللى وصلنا
+
+129
+00:11:36,150 --> 00:11:44,110
+إلى انه دائما ال lower sumلأن ال lower sum هو أسفل
+
+130
+00:11:44,110 --> 00:11:48,230
+من ال upper sum في القيام فبكون أكيد ال lower sum
+
+131
+00:11:48,230 --> 00:11:56,870
+أصغر أو يساوي ال upper sum اطلع لفهم الان احنا لو
+
+132
+00:11:56,870 --> 00:12:01,590
+اجينا لو
+
+133
+00:12:01,590 --> 00:12:06,170
+اجينا اعرفنا .. بنعرف حاجة اللي في عندي P is a
+
+134
+00:12:06,170 --> 00:12:10,890
+partitionلو ضفت على الـ partition أي قيمة جديدة
+
+135
+00:12:10,890 --> 00:12:15,370
+بالاضافة للقيم الموجودة بيصير عندى هذا ال .. ال ..
+
+136
+00:12:15,370 --> 00:12:20,710
+ال .. الجديد عبارة عن refinement لمين للـ B يعني
+
+137
+00:12:20,710 --> 00:12:26,290
+تحسين لل partition يعني الآن بنقول عن أن الـ B الـ
+
+138
+00:12:26,290 --> 00:12:30,190
+Q is a refinement of B إذا كانت الـ B subset من
+
+139
+00:12:30,190 --> 00:12:34,990
+مين؟ من الـ Q طيب، الآن نشوف إيش اللي بنقوله
+
+140
+00:12:45,570 --> 00:12:57,310
+الأن لو أخدنا الفترة اللي هي A وB و جزّقناها X0 X1
+
+141
+00:12:57,310 --> 00:13:03,460
+لعند XNوالفترة النموذجية اللي نشرع عليها xk minus
+
+142
+00:13:03,460 --> 00:13:08,640
+واحد أو ال sub interval لعند مين لعند xk الان ال b
+
+143
+00:13:08,640 --> 00:13:13,520
+هذا هو عبارة عن ال partition x naught x واحد لعند
+
+144
+00:13:13,520 --> 00:13:17,400
+ال xn وهذا ordered يعني نكتبها على صورة ordered
+
+145
+00:13:17,400 --> 00:13:20,640
+لما نكتبها على هذا متفقين على انه ايش اللي هو احنا
+
+146
+00:13:20,640 --> 00:13:25,380
+نعمله ordered ال order بالنسبة له ايش ماله مهم لأن
+
+147
+00:13:25,380 --> 00:13:28,080
+هذا معناه تبدأ بال x naught وبعدين x واحد بعدين
+
+148
+00:13:28,080 --> 00:13:35,240
+اللي هو ال xnلو جينا أخدنا اللي هو أضفنا أي نقطة
+
+149
+00:13:35,240 --> 00:13:39,060
+هنا طبعاً أضفنا أي نقطة أو أي رقم بين الـA و بين
+
+150
+00:13:39,060 --> 00:13:45,320
+مين الـB الرقم المضاف اللي هو أضفنا لهذه وسميناها
+
+151
+00:13:45,320 --> 00:13:50,930
+Q فبكون الـQ هي عبارة عن refinement للـBيعني بمعنى
+
+152
+00:13:50,930 --> 00:13:55,030
+اخر اكيو refinement ل ال B معناته انه في اكيو فيها
+
+153
+00:13:55,030 --> 00:14:02,790
+نقاط اللي هي زيادة عن نقطة B واسمه تحسين لانه فعلا
+
+154
+00:14:02,790 --> 00:14:08,870
+بحسن وضع ال lower sum و ال upper sum الان ال lower
+
+155
+00:14:08,870 --> 00:14:14,250
+sum عرفناه لل B و ال F قلنا بعتمد عليهم بسوء ال
+
+156
+00:14:14,250 --> 00:14:22,130
+summationاللي هو xk ناقص xk minus واحد في ال mk
+
+157
+00:14:22,130 --> 00:14:25,090
+اللي هي mk small هي عبارة عن قلنا اللي هي عبارة عن
+
+158
+00:14:25,090 --> 00:14:29,050
+ال infimum لدى العهد الفترة k من عند واحد لعند n
+
+159
+00:14:29,050 --> 00:14:34,770
+والان ال upper sum ب U, B وF اللي هو بساوى ال
+
+160
+00:14:34,770 --> 00:14:40,890
+summation لل xk minus xk minus واحد mk capital k
+
+161
+00:14:40,890 --> 00:14:47,930
+من عند واحد لعند nوقلنا إن الـ lower للـ B و الـ F
+
+162
+00:14:47,930 --> 00:14:52,390
+دايما أصغر سوى ال upper للـ B و الـ F فده اللي حتى
+
+163
+00:14:52,390 --> 00:14:58,470
+الآن اللي حكيناه الآن نيجي ناخد الملاحظة اللي عندي
+
+164
+00:14:58,470 --> 00:15:04,510
+if Q above is a refinement of B لو قلنا إن Q عبارة
+
+165
+00:15:04,510 --> 00:15:09,390
+عن إيه شمالها؟ refinement of Brefinement معناته
+
+166
+00:15:09,390 --> 00:15:14,690
+اضفت اللي هو نقطة او اكتر من نقطة لنقاط ال
+
+167
+00:15:14,690 --> 00:15:20,810
+partition اللي عندى اذا اي فترة اي فترة اللي هي xk
+
+168
+00:15:20,810 --> 00:15:27,550
+minus واحد لعند xk اللي هو في ال partition بيه
+
+169
+00:15:27,550 --> 00:15:34,060
+هقدر اكتبها على صورة ميناللي هو yj-1 yj اتحاد ال
+
+170
+00:15:34,060 --> 00:15:37,940
+yj yj زاد 1 اتحاد ده ما اصل عند اتحاد yn نقص 1
+
+171
+00:15:37,940 --> 00:15:44,040
+لعند مين لعند ال yn ماشي الحال طبعا الان هذول عدد
+
+172
+00:15:44,040 --> 00:15:47,740
+الفترات على حسب الإضافة اللي هنالو ضفنا نقطة واحدة
+
+173
+00:15:47,740 --> 00:15:52,240
+بين هذول التنتين معناته هيصير عندى من الـ yj-1 هي
+
+174
+00:15:52,240 --> 00:15:56,460
+عبارة عن xk-1 والنقطة المضافة اتحاده النقطة
+
+175
+00:15:56,460 --> 00:15:59,780
+المضافة اللي هي مع مين مع النقطة هذه بيكون عندى بس
+
+176
+00:15:59,780 --> 00:16:03,690
+فترة تانية لو كان فيه عندىنقطين مضافات بيصير عندي
+
+177
+00:16:03,690 --> 00:16:06,710
+اللي هو للنقطة الأولى و بعدين من النقطة الأولى
+
+178
+00:16:06,710 --> 00:16:12,030
+لنقطة ثانية و بعدين من النقطة الثانية لـ XK فبيصير
+
+179
+00:16:12,030 --> 00:16:16,030
+عندي أكم فترة انضاف تلت .. بيصير تلت فترات و هكذا
+
+180
+00:16:16,030 --> 00:16:19,970
+إذن و لو مافيش إضافة في هذه الفترة بالمرة بيصير
+
+181
+00:16:19,970 --> 00:16:24,460
+عندي بس اللي هي ال interval الأولىأه إذا صار عندى
+
+182
+00:16:24,460 --> 00:16:29,640
+الآن في حالة اللى هو الـQ refinement للـB أي فترة
+
+183
+00:16:29,640 --> 00:16:35,040
+في الـB الأصلية بقدر أكتبها على صورة اللى هى هذا
+
+184
+00:16:35,040 --> 00:16:40,000
+الاتحاد وهذا الاتحاد يحتمل اللى هو الفترة الحاله
+
+185
+00:16:40,000 --> 00:16:44,460
+أو فترتين أو تلاته إلى آخره على حسب اللى هو النقاط
+
+186
+00:16:44,460 --> 00:16:50,060
+المضافة واضح اللى ما اللى هي سبعة واحد اتنينif f
+
+187
+00:16:50,060 --> 00:16:53,960
+من I عند R is bounded and B is any partition of I
+
+188
+00:16:53,960 --> 00:16:57,680
+and Q is a refinement of B فرضنا ان Q ايش ما هي
+
+189
+00:16:57,680 --> 00:17:03,180
+refinement لـ B إذا ال lower sum الاعتماد على
+
+190
+00:17:03,180 --> 00:17:06,820
+partition B و ال function F أصغر يساوي ال lower
+
+191
+00:17:06,820 --> 00:17:12,780
+sum لـ Q و ال F و ال U ال Q و ال F أصغر يساوي ال U
+
+192
+00:17:12,780 --> 00:17:17,900
+و ال B و ال F هو أصلا واقعا ال refinement هو فعلا
+
+193
+00:17:17,900 --> 00:17:25,250
+تحسين أصلاأحنا لما نجينا دربنا المثال في الأول و
+
+194
+00:17:25,250 --> 00:17:28,690
+رسمنا الرسمة على سبيل المثال هذه الرسمة اللي عندي
+
+195
+00:17:28,690 --> 00:17:36,290
+و أخدنا من a لعند b مثلا اللي هو الفترة و كان في
+
+196
+00:17:36,290 --> 00:17:42,150
+عندي اللي هو اللي عملنا تجزء x note x1 الاخره x2
+
+197
+00:17:42,150 --> 00:17:49,420
+x3 لعند الاخيرة xn مثلاعلى سبيل المثال واخدنا
+
+198
+00:17:49,420 --> 00:17:52,860
+القبر صم القبر صم هيكون عبارة عن المنطقة هذه
+
+199
+00:17:52,860 --> 00:17:57,240
+والقبر صم عبارة عن المنطقة هذه والقبر صم بيكون هذه
+
+200
+00:17:57,240 --> 00:18:02,150
+المنطقة كلها اللي بالأزرقالآن لما نزيد اللي هو
+
+201
+00:18:02,150 --> 00:18:07,210
+نقاط الـ Partition اللي هو هذا اللي هي النقاط
+
+202
+00:18:07,210 --> 00:18:12,630
+بتجلل الجزء الزائد فالآن فعلًا بيكون تحسين على
+
+203
+00:18:12,630 --> 00:18:15,330
+الأولاني لأنه أصلا أنا بعد شوية هتلاقي غاية في
+
+204
+00:18:15,330 --> 00:18:21,140
+الوصول أن أصل اللي هو اللي هو أصغر واحدفي ال upper
+
+205
+00:18:21,140 --> 00:18:25,720
+sums يصير زي أكبر واحد في ال lower sums يوم ما
+
+206
+00:18:25,720 --> 00:18:29,820
+يلتجوا مع بعض بيكون فعلا اللي هو ال upper sum وال
+
+207
+00:18:29,820 --> 00:18:35,000
+lower sum اللي هو تجاوزا بيصير بساوي بالظبط اللي
+
+208
+00:18:35,000 --> 00:18:40,080
+هو المساحة تحت المنحنة أو المنطقة اللي هنسميها في
+
+209
+00:18:40,080 --> 00:18:43,340
+حينه ان هي عبارة عن remen integral بل وقيمة ال
+
+210
+00:18:43,340 --> 00:18:47,660
+remen integral هو عبارة عن هذه المنطقة اللي صارتمن
+
+211
+00:18:47,660 --> 00:18:53,540
+اللي هو انه ناخد نزغر اللي هو الأبر و نرفع ال
+
+212
+00:18:53,540 --> 00:18:57,540
+lower و نشوف كيف بنزغر الأبر و كيف بنرفع ال lower
+
+213
+00:18:57,540 --> 00:19:01,580
+اللي هو معنى اللي هيوصل لنا معنى ال integrability
+
+214
+00:19:02,250 --> 00:19:06,710
+إذا الـ Refinement فعلاً Refinement تحسين لما أقول
+
+215
+00:19:06,710 --> 00:19:13,390
+أن بي إي كيو زادت عن هذه الـ Q إيش هتعمل؟ هتتجللل
+
+216
+00:19:13,390 --> 00:19:19,630
+الجزء الزائد في الـ Upper لما أزيد النقاط و هتزيد
+
+217
+00:19:19,630 --> 00:19:24,950
+للجزء المفقود وين في ال lowerماشي الحال ففعلا
+
+218
+00:19:24,950 --> 00:19:28,310
+هيكون ايه بقى عبارة عن refinement او تحسين انت
+
+219
+00:19:28,310 --> 00:19:32,470
+نشوف الآن نشوف لمتنا اللي عندنا اللي هو معنى اللي
+
+220
+00:19:32,470 --> 00:19:38,350
+بقوله انا انه عندي لو كانت Q refinement لل B بكون
+
+221
+00:19:38,350 --> 00:19:43,790
+الlower لجديدعليه أكبر أو يساوي الـ Lower القديم
+
+222
+00:19:43,790 --> 00:19:48,390
+يعني بمعنى آخر اقترب إلى اللي بدناهي اللي متوقعين
+
+223
+00:19:48,390 --> 00:19:53,110
+انه يكون هو التحسين النهائي اللي هنسميه اللي هو ال
+
+224
+00:19:53,110 --> 00:19:58,670
+integration ماشي؟ وال Upper هنا Upper UQ واضح انه
+
+225
+00:19:58,670 --> 00:20:03,510
+بصغر اللي هو عن الأصل لأنه صار فيه تحسين واقترب من
+
+226
+00:20:03,510 --> 00:20:08,070
+الحقيقة اقترب من اللي هو القيمة اللي احنا هنسميها
+
+227
+00:20:08,070 --> 00:20:12,370
+في النهاية integrationطيب، بدنا نثبت الكلام هذا،
+
+228
+00:20:12,370 --> 00:20:16,810
+شوف كيف بدنا نثبته، اللي أنا عندي بيأخدت I اللي هو
+
+229
+00:20:16,810 --> 00:20:21,290
+ال partition ال arbitrary واخدت اللي هو sigma I،
+
+230
+00:20:21,290 --> 00:20:26,310
+بد أشتغل على إضافة نقطة ومن ثم أقول إضافة النقطة
+
+231
+00:20:26,310 --> 00:20:31,340
+تؤدي إلى اللي هو باقي النقطة، نشوف إيش بقولعند بي
+
+232
+00:20:31,340 --> 00:20:35,080
+هي اللي هو الـ partition أخدت z element in I و الـ
+
+233
+00:20:35,080 --> 00:20:39,100
+z هذي و أنا بدأ أحطها في الفترة xk minus واحد لعند
+
+234
+00:20:39,100 --> 00:20:42,640
+مين لعند الـ xk يعني الآن ال partition اللي بي
+
+235
+00:20:42,640 --> 00:20:51,200
+عندي هو عبارة عن x naught لعند xn xk minus واحد xk
+
+236
+00:20:51,200 --> 00:20:55,620
+indeed في المنطقة هذه ضفت نقطة وسميت بيها prime و
+
+237
+00:20:55,620 --> 00:21:03,380
+عبارة عن x naught xk minus واحدZ X K وظل الخلايت
+
+238
+00:21:03,380 --> 00:21:07,720
+كله زي ما هو لعد مين لعد X بس اللي غيرت فيه او
+
+239
+00:21:07,720 --> 00:21:12,420
+اللي ضفتله بس ضفت نقطة هنا اسمها Z بدى اشوف تأثير
+
+240
+00:21:12,420 --> 00:21:16,640
+هذا النقطة ايش بده يعمل بين ال partition القديم
+
+241
+00:21:16,640 --> 00:21:20,520
+وال partition الجديد ومن ثم نستنتج اي refinement
+
+242
+00:21:20,520 --> 00:21:26,990
+على اللي هو ال B القديمةالأمر سهل ان شاء الله الان
+
+243
+00:21:26,990 --> 00:21:30,290
+ال .. ال .. ال .. الاضافة الجديدة صار عند بي برايم
+
+244
+00:21:30,290 --> 00:21:35,870
+ايش هو X Note X واحد وهي ال Z ضفتهان اللي اختلف عن
+
+245
+00:21:35,870 --> 00:21:43,640
+فوق بس ال Z اللي هان بناء عليه الان ال M Kالـ mk'
+
+246
+00:21:43,860 --> 00:21:49,020
+هي عبارة عن الـ supremum للـ function f وين في
+
+247
+00:21:49,020 --> 00:21:58,920
+المنطقة بين zk-1 و مين و z الآن في الفترة بين xk-1
+
+248
+00:21:58,920 --> 00:22:04,800
+و بين xk ضفت z الآن الـ z اللي ضفت هنا هذه المنطقة
+
+249
+00:22:04,800 --> 00:22:11,150
+بدي أسميه ال supremum اللي فيها mk' اللي هنااللي
+
+250
+00:22:11,150 --> 00:22:16,190
+ضالت نسميها إيش مالها MK double prime مع العلم إنه
+
+251
+00:22:16,190 --> 00:22:19,890
+الـ supremum الأصلية في ال partition الأصلي على كل
+
+252
+00:22:19,890 --> 00:22:23,250
+الفترة من XK minus واحد عند XK اسمها إيش مسمينا
+
+253
+00:22:23,250 --> 00:22:31,130
+احنا MK capital باشي الحال طيب ركزوا معاياالآن
+
+254
+00:22:31,130 --> 00:22:34,970
+عندي الـ MK' هي الـ supremum على اللي هو الـ
+
+255
+00:22:34,970 --> 00:22:39,610
+subinterval هذه الـ MKW' هي ال supremum على اللي
+
+256
+00:22:39,610 --> 00:22:43,910
+هي ال subinterval اللي جنبها اللي هي من Z لعند XK
+
+257
+00:22:43,910 --> 00:22:49,890
+واضح أن ال supremum على الكبيرة هذه من XK minus 1
+
+258
+00:22:49,890 --> 00:22:53,790
+لعند XK أكيد أكبر أو يساوي ال supremum على الصغيرة
+
+259
+00:22:54,620 --> 00:22:58,580
+لأن هذه أشملها جزء من 100 من هذه كلها فال supremum
+
+260
+00:22:58,580 --> 00:23:01,940
+لل function عالي كبيرة أكبر أو يساوي ال supremum
+
+261
+00:23:01,940 --> 00:23:06,020
+على مين؟ على الصغيرة و similarly للتانية إذا ال MK
+
+262
+00:23:06,020 --> 00:23:12,180
+أكبر أو يساوي MK' Similarly MK أكبر يساوي MKW' لأن
+
+263
+00:23:12,180 --> 00:23:15,220
+هذه عبارة عن ال supremum على الجزء الثاني منها
+
+264
+00:23:15,220 --> 00:23:20,520
+اللي إلى جاي، واضح؟إذا صار عندي therefore الآن بدأ
+
+265
+00:23:20,520 --> 00:23:24,840
+أصل لحاجة معينة اللي هي mk في xk ناقص xk minus
+
+266
+00:23:24,840 --> 00:23:31,310
+واحددربت اللي هي Mk في هذا هذه الفترة طبعا اللي هي
+
+267
+00:23:31,310 --> 00:23:40,730
+بتساوي Zk-Zk-Xk-1 زائد Xk-Zk الان صارت هذه Mk في
+
+268
+00:23:40,730 --> 00:23:45,410
+المقدار هذا بساوي Mk في هذه زائد هذه ماشي الحال
+
+269
+00:23:45,410 --> 00:23:50,820
+الان ال Mkأكبر أو يساوي mean mk' إذا انصار المقدار
+
+270
+00:23:50,820 --> 00:23:58,780
+هذا أكبر أو يساوي mk' في zk z ناقص xk minus واحد و
+
+271
+00:23:58,780 --> 00:24:04,820
+هذه ال mk برضه أكبر أو يساوي mk double prime of xk
+
+272
+00:24:04,820 --> 00:24:10,380
+ناقص z ماشي الحالة الآن احنا بين ال two partitions
+
+273
+00:24:10,380 --> 00:24:14,580
+اللي حدث بس تغيير واحد زي ما قلنا بس على المنطقة
+
+274
+00:24:14,580 --> 00:24:23,090
+هذه لاحظ انتالان ال lower sum ال lower sum لل b و
+
+275
+00:24:23,090 --> 00:24:30,350
+ال f ال b prime و ال f بساوي لو حسبنا للأولاني أول
+
+276
+00:24:30,350 --> 00:24:38,350
+إشي هو عبارة عن m1 في x0 او x1 نقص x0 زاد لما أصل
+
+277
+00:24:38,350 --> 00:24:51,620
+عند mk في xK-XK-1 زائد لما أصل لعند MN في XN-XN-1
+
+278
+00:24:51,620 --> 00:24:57,800
+هذا إيش هو يا جماعة؟ هو عبارة عن الأبرثان الأبرثان
+
+279
+00:24:57,800 --> 00:25:03,120
+لمن؟ لـB و لـF متفجر، ماشي الحالة على مين؟ على ال
+
+280
+00:25:03,120 --> 00:25:12,270
+partition B الآن هذا بساويم1 في x1 ناقص x0 زائد
+
+281
+00:25:12,270 --> 00:25:18,010
+الان هذه جزئناها جزئين بعد شوية زائد زائد هذا نفسه
+
+282
+00:25:18,010 --> 00:25:25,690
+ماتغيرش هذا اللي هو عبارة عن mk لسه xk ناقص z
+
+283
+00:25:25,690 --> 00:25:34,850
+مظبوط زائد اللي هو mk خليني اقول ال
+
+284
+00:25:34,850 --> 00:25:46,170
+mkاللي هو Z ناقص كحسابات ناقص X كمينوس واحد صحيح؟
+
+285
+00:25:46,170 --> 00:25:53,130
+باضح زائد الباقي زي ما هو زائد M N في X N ناقص X N
+
+286
+00:25:53,130 --> 00:26:00,980
+ناقص واحد الآن هذا ماتغيرش نفسهزائد هذا الـ N هو
+
+287
+00:26:00,980 --> 00:26:05,540
+اللي اتغير وهنا في ال inequality هنا لقنا عبارة عن
+
+288
+00:26:05,540 --> 00:26:12,220
+هذا المقدار اللي هو أكبر أو يساوي اللي هو اللي ما
+
+289
+00:26:12,220 --> 00:26:15,420
+اتغيرش هذا نسميه اللي هو M1
+
+290
+00:26:18,760 --> 00:26:25,640
+معايا يا شباب زائد اللي هو mk شرطة في xk ناقص z
+
+291
+00:26:25,640 --> 00:26:31,520
+زائد mk شرطين في z ناقص xk ماينوس واحد زائدة دا
+
+292
+00:26:31,520 --> 00:26:38,570
+كما تتغيرش زي ما هو mn في xnنقص XN نقص 1 هذا في
+
+293
+00:26:38,570 --> 00:26:42,690
+الواقع مين هو أصلاً هذا هو عبارة عن اللي هو ال
+
+294
+00:26:42,690 --> 00:26:48,510
+upper sum لل partition مين ال B' هو بساوي ال upper
+
+295
+00:26:48,510 --> 00:26:55,270
+sum لل B' والFيعني اللي لجناه أنه لما نضفنا اللي
+
+296
+00:26:55,270 --> 00:26:59,830
+هو نقطة على اللي هو الـ partition صار عند الـ
+
+297
+00:26:59,830 --> 00:27:03,590
+partition اللي هو الأبر للـ partition الجديد أصغر
+
+298
+00:27:03,590 --> 00:27:08,360
+أو يساوي الأبر للـ partition من القديمالان اللي
+
+299
+00:27:08,360 --> 00:27:11,760
+حدث على نقطة ممكن يحدث على نقطين يعني الان ممكن
+
+300
+00:27:11,760 --> 00:27:15,640
+نقطة مثلا انضافت هنا ممكن انضافت كمان مرة هنا كمان
+
+301
+00:27:15,640 --> 00:27:20,560
+نقطة هيصير لجديد بنفس الأسلوب أكبر ويساوي تلت نقاط
+
+302
+00:27:20,560 --> 00:27:24,040
+بيصير أكبر ويساوي أربع نقاط بيصير أكبر ويساوي يعني
+
+303
+00:27:24,040 --> 00:27:29,500
+in general مدام ال refinement UQ اللي محتوي بيه في
+
+304
+00:27:29,500 --> 00:27:34,820
+النهاية عدد الإضافات finiteإذا حيكون اللي هو كل
+
+305
+00:27:34,820 --> 00:27:39,300
+مرة بعدد الإضافات بيصير sub intervals بنفس الأسلوب
+
+306
+00:27:39,300 --> 00:27:43,940
+اللي هان حيصير عنده القبر القديم أكبر أو يساوي
+
+307
+00:27:43,940 --> 00:27:48,620
+القبر الجديد يعني بمعنى أخر in general القبر للـB
+
+308
+00:27:48,620 --> 00:27:54,080
+والـF هيكون أكبر أو يساوي القبر للـQ ال refinement
+
+309
+00:27:54,080 --> 00:27:59,160
+وMain والـFوطبعا زي ما قلنا هذا عبارة عن تحسين لإن
+
+310
+00:27:59,160 --> 00:28:03,960
+القبر هذا بيصير يصغر يقترب من المساحة الحقيقية
+
+311
+00:28:03,960 --> 00:28:08,360
+اللي تحت المنحنة في حالة اللي هو الدل ايش الموجبة
+
+312
+00:28:08,360 --> 00:28:15,860
+واضح طيب الان similarly for lower الان الlower
+
+313
+00:28:15,860 --> 00:28:20,940
+بيصير اللي هوفي الـ lower بدل ما يصير عندي لو كان
+
+314
+00:28:20,940 --> 00:28:25,700
+عندي بنحكي عن ال lower احنا ال L بيوقف و بنحكي عن
+
+315
+00:28:25,700 --> 00:28:30,220
+ال M 1 small و ال M 2 small و M K small و M N
+
+316
+00:28:30,220 --> 00:28:36,660
+small بيصير عندي M 1 small و M K small و M KSmall
+
+317
+00:28:36,660 --> 00:28:44,520
+ومكSmall ومنSmall بيصير الان اللي هي المن هنا بدل
+
+318
+00:28:44,520 --> 00:28:49,680
+ما هي أكبر إيش هتصير على إيش أصغر لإنه بيحكي عن ال
+
+319
+00:28:49,680 --> 00:28:54,810
+infimum منأيش صغير على أيش صغير إلى infimum على
+
+320
+00:28:54,810 --> 00:28:59,570
+أيش أيش اللي هو أصغر فمعناته أيش صار عنده بيصير له
+
+321
+00:28:59,570 --> 00:29:04,910
+infimum اللي هو ال M1 أصغر من ال M1' أو ال Mk
+
+322
+00:29:04,910 --> 00:29:11,730
+Small أصغر أو يساوي ال Mk' Small واضح أه؟ ومنه
+
+323
+00:29:11,730 --> 00:29:17,870
+بنستنتج أنه هيكون عند ال LBF أصغر أو يساوي
+
+324
+00:29:21,260 --> 00:29:28,950
+الـ LBF هنصل أصغر و ساويالـ QF وفعلًا أيضًا هذا هو
+
+325
+00:29:28,950 --> 00:29:35,410
+تحسين لإنه اللي هي زاد اللي هو ال lower sum بحيث
+
+326
+00:29:35,410 --> 00:29:41,550
+إنه بده يصل أو يروح ناحية مين اللي هو المساحة
+
+327
+00:29:41,550 --> 00:29:46,310
+الحقيقية تحت المنحنة في حالة التقاء الجهتين بيصير
+
+328
+00:29:46,310 --> 00:29:52,670
+عندي اللي هو إنه القيمة إذا يعني أخدنا بمعنى أخر
+
+329
+00:29:52,670 --> 00:29:58,540
+ندخل للموضوع مدام هيكإنه لو أخدنا اللي هو صار عندى
+
+330
+00:29:58,540 --> 00:30:03,140
+ال lower يبدأ يكبر يعني لو أخدتله ال suprem لكل ال
+
+331
+00:30:03,140 --> 00:30:06,960
+lowers على كل ال partitions اللي في الدنيا و أخدت
+
+332
+00:30:06,960 --> 00:30:10,980
+ال infimum على كل كبار اللي هو و اتساون مع بعض ال
+
+333
+00:30:10,980 --> 00:30:14,820
+infimum يعني عالمين على ال upper sums بيصير ال
+
+334
+00:30:14,820 --> 00:30:17,500
+suprem مع ال lower sums و ال infimum مع ال lower
+
+335
+00:30:17,500 --> 00:30:22,780
+sums يوم ما يتساونهو اللي بنسميه اللي هو الـ
+
+336
+00:30:22,780 --> 00:30:26,920
+Remain Integrable وبنسمي اللي هي الـ Remain
+
+337
+00:30:26,920 --> 00:30:30,440
+Integrable في هذه الحالة بساوي هذه القيمة المشتركة
+
+338
+00:30:30,440 --> 00:30:35,720
+اللي أصبحت متساوية نيجي لللمة اللي عندي هذه، لمة
+
+339
+00:30:35,720 --> 00:30:40,720
+برضه يعني برهانها سهل، نشوف إيش بتقول اللمة، اللمة
+
+340
+00:30:40,720 --> 00:30:45,610
+بتقول let F from I to R be boundedif B1 وB2 are
+
+341
+00:30:45,610 --> 00:30:49,470
+any two partitions يعني مش نفس ال partition على أي
+
+342
+00:30:49,470 --> 00:30:54,090
+two partitions دائما اللاور على أي partition في
+
+343
+00:30:54,090 --> 00:30:57,150
+الدنيا أصغر أو يساوي الأبر على أي partition تاني
+
+344
+00:30:57,150 --> 00:30:59,970
+مش مشكلة إيش هو في الأول قلنا على نفس ال partition
+
+345
+00:30:59,970 --> 00:31:05,510
+arbitrarily و الإثبات هنستخدم اللمّة الأولى و
+
+346
+00:31:05,510 --> 00:31:09,010
+اللمّة الثانية اللي أخدناها و هنصلي هذه بكل سهولة
+
+347
+00:31:09,010 --> 00:31:13,760
+شوف كدهخلّينا ناخد الـ Q ايش بتساوي بي واحد اتحاد
+
+348
+00:31:13,760 --> 00:31:17,240
+دي اتنين صارت الـ Q هذه refinement للأولى و
+
+349
+00:31:17,240 --> 00:31:21,420
+refinement لمين؟ للتانية تحسين للتانتين بناء عليه
+
+350
+00:31:21,420 --> 00:31:25,380
+.. بناء على اللي هي اللمّة اللي أخدناها سبعة واحد
+
+351
+00:31:25,380 --> 00:31:28,660
+واحد وسبعة واحد اتنين احنا حصول على ناية ليه
+
+352
+00:31:28,660 --> 00:31:33,000
+احسبلي ال بي واحد و أف صار refinement مزام صار ..
+
+353
+00:31:33,000 --> 00:31:37,620
+صار له refinement او تحسين بده يكبر اللاور صار
+
+354
+00:31:37,620 --> 00:31:42,750
+أصغر أو ساوي ال Q Fمن جهة أخرى قبل بشوية أثبتنا أن
+
+355
+00:31:42,750 --> 00:31:48,310
+الـQF أصغر يساوي UQF for any اللي هو specific mean
+
+356
+00:31:48,310 --> 00:31:53,070
+الـQ لنفس الـQ لأن برضه اللي هو الـQ refinement
+
+357
+00:31:53,070 --> 00:31:57,790
+لمين؟ للـB2 مازال refinement إذا الـQ بدي يصير
+
+358
+00:31:57,790 --> 00:32:03,050
+أصغر أو يساوي مين الـUB2F زي اللي بواسطة أثبتناها
+
+359
+00:32:03,050 --> 00:32:07,470
+قبل بشوية يعني in generalin general الـ B1 و F
+
+360
+00:32:07,470 --> 00:32:15,810
+أصغر أو ساوي و B2 و F بدون يعني بغض النظر عن B1 و
+
+361
+00:32:15,810 --> 00:32:21,130
+B2 أنهم كونوا زي بعض أو مختلفات لأي two partitions
+
+362
+00:32:21,130 --> 00:32:26,370
+of the interval I طيب الـ definition الآن اللي
+
+363
+00:32:26,370 --> 00:32:28,830
+حكيناها قبل شوية قولناها قبل شوية بدنا نعرف حاجة
+
+364
+00:32:28,830 --> 00:32:33,940
+اسمها lower integralالـ Lower Integral وحاجة اسمها
+
+365
+00:32:33,940 --> 00:32:38,760
+الـ Upper Integral أخدنا
+
+366
+00:32:38,760 --> 00:32:42,740
+الـ function F bounded طبعا معروف ليش bounded عشان
+
+367
+00:32:42,740 --> 00:32:47,880
+يكون الـ M1 و الـ M2 و الـ Mn كلهم موجودات سواء
+
+368
+00:32:47,880 --> 00:32:51,040
+الـ M capital أو الـ M small عشان هما عبارة عن
+
+369
+00:32:51,040 --> 00:32:54,260
+supremum و infimum فلازم تكون bounded ال function
+
+370
+00:32:54,260 --> 00:32:58,280
+الآن فرضنا الـ F عبارة عن bounded function على ال
+
+371
+00:32:58,280 --> 00:33:01,520
+close bounded interval اللي عندى هذهبنعرف الـ
+
+372
+00:33:01,520 --> 00:33:06,320
+lower integral of F on I اللي هو المعتمد عالميا
+
+373
+00:33:06,320 --> 00:33:11,680
+على ال lower sums ال lower sums قولنا انحسن انحسن
+
+374
+00:33:11,680 --> 00:33:16,960
+انحسن انحسن احسن تحسين اذا جهز التعبير اللي هو ال
+
+375
+00:33:16,960 --> 00:33:21,640
+supremum لكل مين لكل ال lower sums على كل ال
+
+376
+00:33:21,640 --> 00:33:25,040
+partitions اللي في الدنيا هذول ايه الشمال اللي
+
+377
+00:33:25,040 --> 00:33:31,260
+بنسميه L of F اللي بنسميه lower sumأخد الـ
+
+378
+00:33:31,260 --> 00:33:35,400
+supremum هي ال lower sums تحت المنحنة لما أخد اللي
+
+379
+00:33:35,400 --> 00:33:40,220
+هنا ال supremum اللي هو هذا ال supremum بدي أسميه
+
+380
+00:33:40,220 --> 00:33:46,300
+ال lower integral الآن ال upper sums بدي أخد اللي
+
+381
+00:33:46,300 --> 00:33:52,020
+هنا ال infimum يعني بدي أزغرهن اللي هو و أسميه ال
+
+382
+00:33:52,020 --> 00:33:57,760
+upper integral إذا صار عندي lowerIntegral هو عبارة
+
+383
+00:33:57,760 --> 00:34:02,080
+عن الـ Supremum للـ Lower Sums و ال .. واضح يا
+
+384
+00:34:02,080 --> 00:34:06,060
+عيسى و ال Upper Integral هو عبارة عن ال Infimum
+
+385
+00:34:06,060 --> 00:34:10,620
+لمن؟ لل Upper Sums متوقع أنه يقول لنا أنه احنا
+
+386
+00:34:10,620 --> 00:34:16,810
+بنقول عنهافوق الصفحة التانية متوقع يقولنا انه عنا
+
+387
+00:34:16,810 --> 00:34:20,970
+الآن بنسميها remain integrable اذا تساوت ايش
+
+388
+00:34:20,970 --> 00:34:26,130
+اتجهتين ماشي جابيه لبس خلينا نبرهن او نتطلع على
+
+389
+00:34:26,130 --> 00:34:33,430
+هذه اللي أمامنا برضه ملاحظة سهلة الملاحظة بتقولي
+
+390
+00:34:33,430 --> 00:34:42,780
+لو الان جيناسمّينا الـ M I small عبارة عن الـ
+
+391
+00:34:42,780 --> 00:34:49,860
+infimum للدالة على الفترة كلها الـ infimum يعني
+
+392
+00:34:49,860 --> 00:34:57,460
+الـ M I سماها عبارة عن الـ infimum للدالة F of X
+
+393
+00:34:57,460 --> 00:35:02,860
+حيث X موجودة على I وكأنه في الواقع أخذ الـ
+
+394
+00:35:02,860 --> 00:35:07,520
+partition هو عبارة عن نقطيناللي عبارة عن a و مين و
+
+395
+00:35:07,520 --> 00:35:12,800
+b ال partition بس a و b في هذا الحلق و جالك اللي
+
+396
+00:35:12,800 --> 00:35:18,020
+هو .. ليش .. هلجيت تشوفوا ال mi capital عبارة عن
+
+397
+00:35:18,020 --> 00:35:21,860
+ال supreme لل f of x such that x element in I
+
+398
+00:35:21,860 --> 00:35:27,780
+بقولك الآن إذن .. إذن mi ال mi small مضروبة في طول
+
+399
+00:35:27,780 --> 00:35:32,120
+الفترة يعني ال partition واحد لأن ال partition a و
+
+400
+00:35:32,120 --> 00:35:39,750
+b بس أصغر أو يساوياللي هو اللوفاف طبيعي طبيعي لأن
+
+401
+00:35:39,750 --> 00:35:45,570
+هذا واحد من مين من ال lower sums، بظبط؟ وهذا أصلا
+
+402
+00:35:45,570 --> 00:35:49,170
+ال supremum لل lower sums فهيكون ال supremum أكبر
+
+403
+00:35:49,170 --> 00:35:52,670
+يساوي مين؟ واحد منهم، هذا زي ما أقول أنها lower
+
+404
+00:35:52,670 --> 00:35:58,450
+sum على partition في A و BSimilarly, M I في الـ B
+
+405
+00:35:58,450 --> 00:36:03,590
+minus A أكبر أو ساوي ال other sum of F، ليه؟ لأن
+
+406
+00:36:03,590 --> 00:36:07,470
+هذا عبارة عن واحد من ال other sums و هذا عبارة عن
+
+407
+00:36:07,470 --> 00:36:10,210
+ال N في المملكة و ليه ال other sums فطبيعي هيكون
+
+408
+00:36:10,210 --> 00:36:14,260
+ال U of F أكبر أو ساوي B minus Ato be more
+
+409
+00:36:14,260 --> 00:36:18,360
+specific for every B element in B of I نحن لدينا M
+
+410
+00:36:18,360 --> 00:36:22,140
+I في B minus A أصغر أو يساوي L, B و F لأن هذا واحد
+
+411
+00:36:22,140 --> 00:36:28,800
+من ال partitions قلنا اللي هي لفترة I نفس الشيء
+
+412
+00:36:28,800 --> 00:36:34,800
+فبصير عند ال L of B و A F أصغر أو يساوي اللي هو ال
+
+413
+00:36:34,800 --> 00:36:40,060
+U, B و F أصغر أو يساوي M I في B minus Aمعروف هذا
+
+414
+00:36:40,060 --> 00:36:43,580
+أنا مش حاجة لزمنه أصلاً فبصير mi أصغر سواء ل و بف
+
+415
+00:36:43,580 --> 00:36:48,060
+و ل و بف أصغر سواء mi في ال b-a اللي وضحته هناك
+
+416
+00:36:48,060 --> 00:36:54,640
+أوضح من اللي مكتوب هناك واضح أه نيجي لنظرية اللي
+
+417
+00:36:54,640 --> 00:37:00,630
+عندى هذهTheorem بتقول لت I بساوة A وB and لت F من
+
+418
+00:37:00,630 --> 00:37:05,490
+I لR بيه bounded function وفهمنا ليش احنا بنقول
+
+419
+00:37:05,490 --> 00:37:08,410
+bounded then the lower integral L of F and the
+
+420
+00:37:08,410 --> 00:37:12,090
+upper integral U of F on I exist moreover ال L of
+
+421
+00:37:12,090 --> 00:37:15,330
+F أشمالها أصغر ساوية U of F طبعا ال existence سبب
+
+422
+00:37:15,330 --> 00:37:18,650
+ال existence كله اللي هو انه ال function bounded
+
+423
+00:37:18,650 --> 00:37:21,110
+مثلا bounded كل اللي هنا الحسابات اللي على أساس ال
+
+424
+00:37:21,110 --> 00:37:25,350
+supernormal information موجودةلأن اللي قال الـ U
+
+425
+00:37:25,350 --> 00:37:29,170
+of F أصغر ساوي مين؟ U بقى، U of F كلام سهل، شوف
+
+426
+00:37:29,170 --> 00:37:32,210
+إيش بدنا ناخد بتاخد بي واحد و بي اتنين بي أي two
+
+427
+00:37:32,210 --> 00:37:36,210
+partitions of I، ماشي؟ لأن then by لمّا اللي حكينا
+
+428
+00:37:36,210 --> 00:37:40,150
+قبل شوية عنها، اللي قال بي واحد و أف أصغر ساوي U
+
+429
+00:37:40,150 --> 00:37:44,350
+بي اتنين of F، ماشي الحال؟ لأي two partitions،
+
+430
+00:37:44,350 --> 00:37:48,270
+ثبتلي واحدة منهم الآن، خلّي اللي هو بي اتنين of is
+
+431
+00:37:48,270 --> 00:37:52,560
+fixedماشي صارت هذه .. هذه القيمة صارت .. بنحكي عن
+
+432
+00:37:52,560 --> 00:37:56,980
+عدد محدد ماشي بـ 2 arbitrarily صح لكن الان انا بدي
+
+433
+00:37:56,980 --> 00:38:01,020
+ثابته وهذه المتغيرة صار عند الان ال U B 2 و F
+
+434
+00:38:01,020 --> 00:38:07,940
+عبارة عن upper bound of the set of all اللي هي مين
+
+435
+00:38:07,940 --> 00:38:13,790
+ال B I و F على أي partitionطيب هذا upper bound
+
+436
+00:38:13,790 --> 00:38:19,270
+فبنقله في قفل هذه اللي هو ال minimum الهن ال
+
+437
+00:38:19,270 --> 00:38:22,470
+supremum الهن مظبوط اللي هو ال least upper bound
+
+438
+00:38:22,470 --> 00:38:26,350
+إذا ال least upper bound أكيد أصغر أو شومين ال
+
+439
+00:38:26,350 --> 00:38:32,340
+upper bound كمان مرةطيب، الآن therefore U B 2 و F
+
+440
+00:38:32,340 --> 00:38:37,820
+is an upper bound of هذه الست ماشي؟ و هذه الست ال
+
+441
+00:38:37,820 --> 00:38:42,500
+supremum لها إيش اسمه؟ L of F إذا أكيد اللي هو ال
+
+442
+00:38:42,500 --> 00:38:45,220
+supremum اللي هو ال least upper bound اللي هو ال L
+
+443
+00:38:45,220 --> 00:38:51,620
+of F أصغر أو يساوي واحد من ال upper bounds، مظبوط؟
+
+444
+00:38:51,620 --> 00:38:58,990
+أي سؤال؟واضح ها؟ الان خلصها من هذه الان similarly
+
+445
+00:38:58,990 --> 00:39:04,390
+since B2 was arbitrarily partition of I صارت الان
+
+446
+00:39:04,390 --> 00:39:10,230
+عند L of F أصغر أو يساوي UB2 of F يعني صارت هذه
+
+447
+00:39:10,230 --> 00:39:17,530
+عدد أصغر أو يساوي ال UB2 of F for any partition B2
+
+448
+00:39:18,500 --> 00:39:22,560
+الان صارت هذه عبارة عن lower bound لها لهذه ال
+
+449
+00:39:22,560 --> 00:39:26,200
+collection lower bound لها اذا هذه lower bound
+
+450
+00:39:26,200 --> 00:39:30,160
+اصغر او يساوي ال greatest lower bound مين ال
+
+451
+00:39:30,160 --> 00:39:35,200
+greatest lower bound لهدول اللي هو ال L اللي هو ال
+
+452
+00:39:35,200 --> 00:39:45,700
+U of F اذا صار عند ال L of F اصغر او يساوي U of F
+
+453
+00:39:45,700 --> 00:39:48,360
+اي سؤال؟
+
+454
+00:39:51,810 --> 00:39:56,530
+واضحة؟ ان في سؤال كان في ريال واحد زي هيك بنحل على
+
+455
+00:39:56,530 --> 00:39:59,690
+هيك اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+456
+00:39:59,690 --> 00:40:00,510
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+457
+00:40:00,510 --> 00:40:01,130
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+458
+00:40:01,130 --> 00:40:01,850
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+459
+00:40:01,850 --> 00:40:06,550
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+460
+00:40:06,550 --> 00:40:13,540
+اربع اتنين اربع اعدد هذا هو اللي هو اللي اللي اللي
+
+461
+00:40:13,540 --> 00:40:17,340
+اللي المتغير صار عند العدد هذا أصغر أو شاوي كل
+
+462
+00:40:17,340 --> 00:40:22,080
+اللي هو أعضاء العائلة اللي بتحكي عنها إذا صار الآن
+
+463
+00:40:22,080 --> 00:40:27,580
+هذا الالوفاف عبارة عن lower bound ofthe collection
+
+464
+00:40:27,580 --> 00:40:32,600
+of all U, P, T و F عالميا على ال partitions هذه
+
+465
+00:40:32,600 --> 00:40:36,260
+مادام هذا صار عبارة عن lower bound إذا هذا lower
+
+466
+00:40:36,260 --> 00:40:40,580
+bound أصغر أو يساوي ال greatest lower bound اللي
+
+467
+00:40:40,580 --> 00:40:43,860
+هو اللي بنسميه ال infimum إذا صارت ال L و F أصغر
+
+468
+00:40:43,860 --> 00:40:47,000
+أو يساوي ال infimum لهدول ال infimum لهدول مين هو
+
+469
+00:40:47,000 --> 00:40:52,940
+ال U و Fالـ U of F هو الـ infimum لكل الـ U, B, T,
+
+470
+00:40:53,280 --> 00:40:58,040
+2 و F such that B2 element in the set of all
+
+471
+00:40:58,040 --> 00:41:07,220
+partitions نيجي الآن لبعده الـ Definition الان ..
+
+472
+00:41:07,220 --> 00:41:16,080
+الان احنا زغرنا ال other sums و كبرنا ال other
+
+473
+00:41:16,080 --> 00:41:22,680
+sumsيوم مالتجهن مع بعض بنسمي اللي هو الالتقاء هذا
+
+474
+00:41:22,680 --> 00:41:26,320
+أو الرقم اللي التقن فيه هو قيمة الـ Remain
+
+475
+00:41:26,320 --> 00:41:30,800
+Integral للفункشن F على الفترة المذكورة وهذا هو
+
+476
+00:41:30,800 --> 00:41:35,040
+التعريف let I بسوء A و B and let F من I لعند R be
+
+477
+00:41:35,040 --> 00:41:39,060
+a bounded function then F is said to be Remain
+
+478
+00:41:39,060 --> 00:41:46,480
+Integrable on I if L of Fاللي هو عبارة عن الـ
+
+479
+00:41:46,480 --> 00:41:54,140
+Supremum للـ Lower Bounds للـ Lower Sums بساوي ال
+
+480
+00:41:54,140 --> 00:42:00,320
+Infimum لل Upper Sums يوم ما يتساون هذولة بنسمي
+
+481
+00:42:00,320 --> 00:42:03,780
+الـ Function is Remain Integrable وفي هذه الحالة
+
+482
+00:42:03,780 --> 00:42:06,860
+بيكون القيمة المشتركة هذه هي اللي بنسميها الـ
+
+483
+00:42:06,860 --> 00:42:13,740
+Integration من A ل B F of X D of Xدكتور ان الملحنة
+
+484
+00:42:13,740 --> 00:42:17,420
+احنا بدنا نجيب مساحة كذا كذا احنا ال .. المساحة
+
+485
+00:42:17,420 --> 00:42:21,480
+إذا كان فوق الآن إذا كان تحت بيكون سالب المساحة
+
+486
+00:42:21,480 --> 00:42:24,680
+إذا كان مخلوط بيصير .. إذا كان فوق و تحت معناته
+
+487
+00:42:24,680 --> 00:42:28,180
+بيصير جزء مساحة و جزء سالب مساحة عشان نجد مساحته
+
+488
+00:42:28,180 --> 00:42:30,980
+بالظبط تاخد ال absolute بتقطع لذاك و تاخد ال
+
+489
+00:42:30,980 --> 00:42:34,340
+absolute value اللي بيطلع اللي هو المساحة عشان
+
+490
+00:42:34,340 --> 00:42:39,580
+نجيبوا مساحته و نجسم الملحنة هذا لمستطيلة صح؟
+
+491
+00:42:41,500 --> 00:42:46,400
+كمستطيل و بيجيبوا مجموع مستطلاق بدين المنحنى بديك
+
+492
+00:42:46,400 --> 00:42:51,760
+مساحة تحت المنحنى لما تاخد ال upper sums و ال
+
+493
+00:42:51,760 --> 00:42:55,260
+lower sums و تاخد ال supremum و تاخد ال infimum و
+
+494
+00:42:55,260 --> 00:42:59,080
+يكون ال supremum بساوي ال infimum بتلح هو المساحة
+
+495
+00:42:59,080 --> 00:43:04,990
+تحت المنحنى في حالة ال positive functionsفي حالة
+
+496
+00:43:04,990 --> 00:43:08,710
+الـ negative المخلوق بيصير integration بيصير قيمة
+
+497
+00:43:08,710 --> 00:43:12,170
+integration قيمة ال integration لا تعبر عن المساحة
+
+498
+00:43:12,170 --> 00:43:16,670
+اللي هو بالضبط إلا في حالة ال other إلا في حالة ال
+
+499
+00:43:16,670 --> 00:43:19,970
+positive ال function في حالة ال negative بتعبر على
+
+500
+00:43:19,970 --> 00:43:24,170
+سلب المساحة في حالة إشي فوج و إشي تحت بيصير اللي
+
+501
+00:43:24,170 --> 00:43:26,950
+هو .. لأ المساحات بت cancel بعضها و بيصير قيمة
+
+502
+00:43:26,950 --> 00:43:30,930
+integrationبيدك المساحات من وسط ال integration
+
+503
+00:43:30,930 --> 00:43:33,110
+انها تستخدم ال integration في إيجاد المساحة
+
+504
+00:43:33,110 --> 00:43:37,570
+بتجزئها لإيش إلى أجزاء وين بتقطع محور السينات يعني
+
+505
+00:43:37,570 --> 00:43:40,550
+بتحدد وين موجبة و وين سالبة بتاخد ال absolute
+
+506
+00:43:40,550 --> 00:43:43,410
+value حتى لو ماكنشت يعرف مين الموجبة ومين السالبة
+
+507
+00:43:43,410 --> 00:43:46,570
+وبيطلع اللي هو عن طريق ال integration بتوجد قيمة
+
+508
+00:43:46,570 --> 00:43:52,290
+مين المساحة لكن في حالة الدالة موجبة على كل الفترة
+
+509
+00:43:52,290 --> 00:43:59,410
+ال integration هو بالظبط المساحة طيبصلى على النبي
+
+510
+00:43:59,410 --> 00:44:01,990
+عليه الصلاة والسلام عرفنا الـ Romantic Grability
+
+511
+00:44:01,990 --> 00:44:06,470
+وقلنا اللي هو .. اللي هو الاشياء المشتركة اللي
+
+512
+00:44:06,470 --> 00:44:11,290
+حصلنا عليها الآن بنعرف برضه كتعريف من A ل B F of X
+
+513
+00:44:11,290 --> 00:44:15,930
+DX بساوي ناقص من B ل A F of X DX وبنعرف كتعريف أن
+
+514
+00:44:15,930 --> 00:44:19,830
+ال integration من A ل A F of X DX H بساوي سفر اطلع
+
+515
+00:44:19,830 --> 00:44:25,120
+فوق خليني أخد مثال وننهي بهذا المثالالآن .. و هذا
+
+516
+00:44:25,120 --> 00:44:30,080
+المثال احنا .. يعني .. كنا ناخد .. نفس المثال اللي
+
+517
+00:44:30,080 --> 00:44:33,760
+ناخده في ال calculus خلّينا نحكي عنه بالتفصيل الان
+
+518
+00:44:33,760 --> 00:44:37,480
+example show that g of x بساوة x is integrable on
+
+519
+00:44:37,480 --> 00:44:41,700
+0 و 1 بدنا نثبت أنه الدالة اللي هي g of x اللي
+
+520
+00:44:41,700 --> 00:44:45,220
+بنعرفها اللي هي المثال البسيط بساوة x عبارة عن
+
+521
+00:44:45,220 --> 00:44:52,130
+integrable عالميا على اللي هو الفترة 0 و 1الان انا
+
+522
+00:44:52,130 --> 00:44:56,270
+لتسهيل عملية الحسابات هاخد عائلة من ال partitions
+
+523
+00:44:56,270 --> 00:45:03,050
+معينة هذه العائلة بتسهيل الحسابات طب بنفع في ضوء
+
+524
+00:45:03,050 --> 00:45:07,910
+العلاقات اللي هنشوفها كيف توصلنا بالظبط ان ال L of
+
+525
+00:45:07,910 --> 00:45:12,830
+F بساوة ال U of F و لخصوصية الحسابات ده نشوف كيف
+
+526
+00:45:12,830 --> 00:45:21,760
+عندى ال F of X بساوة Xعنديها بديها من 0 لعند مين؟
+
+527
+00:45:21,760 --> 00:45:28,580
+لعند 1 هذه y بساوة x على الفترة 0 و 1 ايش بدنا
+
+528
+00:45:28,580 --> 00:45:34,580
+نسوي؟ بدنا نيجي اللي هو ان نعمل partition بين هذا
+
+529
+00:45:34,580 --> 00:45:39,540
+ال partition بديه يسهل الإحسابات فأسهل شيء بدي
+
+530
+00:45:39,540 --> 00:45:44,640
+أجزء هذه الفترة من 0 ل1 لN من الأجزاء و بديها تكون
+
+531
+00:45:44,640 --> 00:45:49,430
+كلها متساويةبنفع، أه، هينفع، هتشوف كيف اللي هو هذا
+
+532
+00:45:49,430 --> 00:45:52,650
+نفع في ضوء العلاقات بين اللي هو الـ Supremum و
+
+533
+00:45:52,650 --> 00:45:57,050
+الانفموع المجموعة الأكبر و المجموعة الأصغر و أيضًا
+
+534
+00:45:57,050 --> 00:46:01,130
+في ضوء ال limits المعينة اللي هان اللي هتيجي تخلي
+
+535
+00:46:01,130 --> 00:46:05,010
+اللي هو علاقة توصّل إلى اللي بدنيها، تفهم علش؟
+
+536
+00:46:05,010 --> 00:46:10,610
+ماقولهم كده، هذا صار عندي، هذا Zeroأما ده بدأت
+
+537
+00:46:10,610 --> 00:46:13,510
+تجزئها طولها واحد تجزئها الان من الفتراء sub
+
+538
+00:46:13,510 --> 00:46:16,130
+intervals إذا كل واحدة واحدة على الان هذا التاني
+
+539
+00:46:16,130 --> 00:46:20,130
+اتنين على الان التالتة تلاتة على الان تلاتة على
+
+540
+00:46:20,130 --> 00:46:24,770
+الان لما نصل للفترة النموذجية xk اللي هي تلقى k
+
+541
+00:46:24,770 --> 00:46:30,610
+minus واحد على ان وk على ان هذا اللي هي xk minus
+
+542
+00:46:30,610 --> 00:46:36,670
+واحد وهذا ال xk وهذا xنوت و اد اكس واحد الى اخرى
+
+543
+00:46:36,670 --> 00:46:39,790
+لان ما اصل الاخر واحدة اللى هى n على n اللى هى
+
+544
+00:46:39,790 --> 00:46:45,150
+واحدة ماشي الحالة هذا اللى عندى اللى .. لو اجيت
+
+545
+00:46:45,150 --> 00:46:48,190
+حسبت الان مدى احسب هذا ال partitions اللى صار عندى
+
+546
+00:46:48,190 --> 00:46:53,930
+عبارة عن zero و الواحدة الان و اتنين على n لما اصل
+
+547
+00:46:53,930 --> 00:46:58,490
+لعند ال n على n اللى هى ايش الواحدة هذه عائلة ال
+
+548
+00:46:58,490 --> 00:47:04,620
+partitions اللى اختارتها ماشي شوفوا عليها ال nلو
+
+549
+00:47:04,620 --> 00:47:09,460
+جيت حسبت اللي هو طول الـ interval واحدة لان لو بده
+
+550
+00:47:09,460 --> 00:47:16,960
+أحسب الآن L لل partition هذا اللي هو مين هو L ال F
+
+551
+00:47:16,960 --> 00:47:23,840
+عندي و ال P N هتساوي عبارة عن ال summation عارفين
+
+552
+00:47:23,840 --> 00:47:29,780
+انتوا ال summation اللي هو M small K في X K ناقص X
+
+553
+00:47:29,780 --> 00:47:36,290
+K minus واحد K من عند واحد عند N ويساويالان ال MK
+
+554
+00:47:36,290 --> 00:47:41,890
+في الفترة الأولى ايش هيكون عنده؟ صفر في الفترة
+
+555
+00:47:41,890 --> 00:47:47,430
+التانية ايش هيكون؟ واحدة على الانكريزين جدا اللي
+
+556
+00:47:47,430 --> 00:47:50,150
+بعدها اتنين على انواع و هكذا
+
+557
+00:47:52,920 --> 00:47:57,520
+الـ Mk in general إيش هيصير عنده الـ Mk وإيش
+
+558
+00:47:57,520 --> 00:48:03,020
+المساحة هي K minus واحد على مين على هذا اللي هو
+
+559
+00:48:03,020 --> 00:48:08,280
+طول تبع مين الـ Mk واضحة؟ لأن الـ infimum على
+
+560
+00:48:08,280 --> 00:48:12,760
+الفترة هذه النموذجية اللي هو ال .. ال .. ال
+
+561
+00:48:12,760 --> 00:48:17,920
+infimum عليها اللي هو أول واحدة لإنها increasing
+
+562
+00:48:17,920 --> 00:48:22,630
+واضحة تصورفلان هذه ثابتة أنا اختارتها أصلا ثابتة
+
+563
+00:48:22,630 --> 00:48:26,430
+طول الفترة ثابتة اللي هو 1 على n بيصير هذا إيش n
+
+564
+00:48:26,430 --> 00:48:32,670
+تربيع كامل عند واحد لعند مين لعند n واضح أه ويساوي
+
+565
+00:48:32,670 --> 00:48:38,110
+1 على n تربيع فيه ال summation هذا اللي هو عبارة
+
+566
+00:48:38,110 --> 00:48:42,830
+عن بيبدأ من 0 زائد 1 ولمّا يصل لعند مين لعند n
+
+567
+00:48:42,830 --> 00:48:49,760
+ناقص 1هذا بعرف ما هو مجموعه ويساوي 1 الان تربيع في
+
+568
+00:48:49,760 --> 00:48:54,760
+هذا ان لما نكون من واحد لعند ان ان فان زاد واحد
+
+569
+00:48:54,760 --> 00:48:58,440
+الان على اتنين الان هذا من وين واحد لعند ان ناقص
+
+570
+00:48:58,440 --> 00:49:03,540
+واحد فان ناقص واحد لكل عالمين يا جماعة على اتنين
+
+571
+00:49:03,540 --> 00:49:09,580
+ويساوي الان اعملي هذا بيطلع نص عندي ويساوي نص في
+
+572
+00:49:09,580 --> 00:49:15,150
+اللي هو ان تربيع ناقص من انعلى أن تربيعه بيصير
+
+573
+00:49:15,150 --> 00:49:22,650
+واحد ناقص واحدة لان صح ولا لأ طيب similarly for me
+
+574
+00:49:22,650 --> 00:49:29,430
+لأن لما بدي أخد اللي قاله و أفله اه ال .. ال ..
+
+575
+00:49:29,430 --> 00:49:36,930
+بعدين شوف عليهم الآن القبر للأف و بي أن بنفس
+
+576
+00:49:36,930 --> 00:49:44,250
+السلوب بيصير عندي عبارة عن summation mkفي xk-xk-1
+
+577
+00:49:44,250 --> 00:49:49,630
+k من عند 1 لعند n الان ال mk ال m واحد هيه لان
+
+578
+00:49:49,630 --> 00:49:54,010
+المتنين هيه إذا ال mk هي عبارة عن مين؟ k على n
+
+579
+00:49:54,010 --> 00:49:59,070
+لأنه increasing زي ما قلنا ويساوي صماشen k على n
+
+580
+00:49:59,070 --> 00:50:04,800
+في 1 على nK من عند واحد لعند ان ويساوي summation
+
+581
+00:50:04,800 --> 00:50:07,640
+واحدة لان تربية في summation لK من واحد لان لان
+
+582
+00:50:07,640 --> 00:50:12,800
+اللي ان فان زائد واحد الان هو الكل مكسوب على اتنين
+
+583
+00:50:12,800 --> 00:50:17,400
+ويساوي نص في اللي هو بطلع عندي هذا عبارة عن واحد
+
+584
+00:50:17,400 --> 00:50:23,920
+زائد واحدة لان، مظبوط؟ طيب إذا انصار عندي الآن
+
+585
+00:50:23,920 --> 00:50:28,410
+حصلت على التاليزي ما شوفنا حسابيا جميعا من هنا
+
+586
+00:50:28,410 --> 00:50:34,530
+اطلع لفوق صار عندى الان LPNG او F عبارة عن نصف
+
+587
+00:50:34,530 --> 00:50:40,230
+واحد نقص واحدة الان UPNJ ايش بيساوي نصف واحد زائد
+
+588
+00:50:40,230 --> 00:50:48,330
+واحد على مين على الان واضح الان الان اه الان الان
+
+589
+00:50:48,330 --> 00:50:53,100
+الان انت بي .. عندى ال collection هذهالـ
+
+590
+00:50:53,100 --> 00:50:57,280
+collection هذه partitions B1 is a partition اللي
+
+591
+00:50:57,280 --> 00:51:03,740
+هو عبارة عن Zero و واحد B2 Zero نص و واحد B3 Zero
+
+592
+00:51:03,740 --> 00:51:09,160
+و تلت و تلتين و واحد و هكذا إذا هذه عائلة من ال
+
+593
+00:51:09,160 --> 00:51:14,060
+partitionsجزء من مين؟ من الـ partitions العامة
+
+594
+00:51:14,060 --> 00:51:17,100
+يعني ال .. ال part .. هي the set of all partitions
+
+595
+00:51:17,100 --> 00:51:22,220
+هذه أكيد جزئية من the set of all partitions لمين؟
+
+596
+00:51:22,220 --> 00:51:25,980
+لل I اللي عنده اللي هي الفترة من مين؟ من 01 إذا
+
+597
+00:51:25,980 --> 00:51:31,500
+هذه مجموعة جزئية من هذه، مظبوط؟ طيب، الآن لو أجينا
+
+598
+00:51:31,500 --> 00:51:35,820
+أخدنا اللي هو ال set of partitions هذه أكيد ال
+
+599
+00:51:35,820 --> 00:51:39,580
+supreme على هذهالان ال .. ال .. ال .. ال .. ال ..
+
+600
+00:51:39,580 --> 00:51:44,820
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+601
+00:51:44,820 --> 00:51:45,160
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+602
+00:51:45,160 --> 00:51:48,400
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+603
+00:51:48,400 --> 00:51:48,420
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+604
+00:51:48,420 --> 00:51:50,120
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+605
+00:51:50,120 --> 00:51:50,220
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+606
+00:51:50,220 --> 00:51:50,340
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+607
+00:51:50,340 --> 00:51:59,480
+.. ال .. ال .. ال .. ال .. ال ..
+
+608
+00:51:59,480 --> 00:52:02,920
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+609
+00:52:02,920 --> 00:52:09,060
+ال .. ال .. ال .. ال .. ال .. ال .. ال ..such that
+
+610
+00:52:09,060 --> 00:52:14,120
+N element in N هذه اللي هي main ال collection of
+
+611
+00:52:14,120 --> 00:52:19,260
+all partitions ال B1 وB2 وB3 عند BN لو جينا أخدنا
+
+612
+00:52:19,260 --> 00:52:24,740
+ال supremumلكل القيم هذه هيكون أصغر أو يساوي الـ
+
+613
+00:52:24,740 --> 00:52:32,320
+Supremum لمين لل L of F و B such that B element in
+
+614
+00:52:32,320 --> 00:52:36,620
+B and B of I لأن collection of all partitions هذه
+
+615
+00:52:36,620 --> 00:52:41,940
+اللي هو تحتوى ال partitions هدولة واضح الحمد الله
+
+616
+00:52:41,940 --> 00:52:45,500
+الان ال Supremum على القيم هذه اللي هي مين القيم
+
+617
+00:52:45,500 --> 00:52:51,230
+كانت عندهاللي هو عبارة عن نص في واحد ناقص ميل واحد
+
+618
+00:52:51,230 --> 00:52:55,590
+علان ال supremum لهذه عبارة عن zero إذن بتظل هي
+
+619
+00:52:55,590 --> 00:53:00,590
+إيش بتساوي نص ال supremum للنص في واحد ناقص واحد
+
+620
+00:53:00,590 --> 00:53:05,070
+علان معايا؟
+
+621
+00:53:05,070 --> 00:53:16,470
+أي سؤال؟ طيب إذا هذا مين هو؟ هذا عبارة عن الأو ما
+
+622
+00:53:16,470 --> 00:53:21,170
+فاهم ولا لأ إذا صار عند ال L و ال F مثبتها هذه ال
+
+623
+00:53:21,170 --> 00:53:29,390
+L و ال F هيصير أكبر أو يساوي النص نحك طيب
+
+624
+00:53:29,390 --> 00:53:35,650
+similarly for me and for you الآن عندي حصلت على ال
+
+625
+00:53:35,650 --> 00:53:41,970
+U, F و B أن لهذه العائلة الان الان the set of all
+
+626
+00:53:41,970 --> 00:53:50,880
+U, F و B أنsuch that N element in N هذه العائلة
+
+627
+00:53:50,880 --> 00:53:53,900
+مجموعة جزئية من العائلة الكبيرة الـ B of I الـ
+
+628
+00:53:53,900 --> 00:54:00,940
+partitions للـ I إذن هذه الـ infimum لها هيصير
+
+629
+00:54:00,940 --> 00:54:10,580
+أكبر أو يساوي الـ infimum لـ U, F و B such that B
+
+630
+00:54:10,580 --> 00:54:18,550
+element in B of Iالـ collection هذه أصغر من الـ
+
+631
+00:54:18,550 --> 00:54:21,730
+collection هذه، بظبط؟ إذا صار الـ infimum على هذه
+
+632
+00:54:21,730 --> 00:54:25,610
+أكبر و ساوي الـ infimum على هذه الـ infimum لهذه
+
+633
+00:54:25,610 --> 00:54:27,470
+الـ collection و اللي هو الـ infimum في الواقع
+
+634
+00:54:27,470 --> 00:54:31,110
+لمين؟ الـ infimum لهذه الـ infimum لهذه ليش
+
+635
+00:54:31,110 --> 00:54:35,910
+هيساوي؟ نص فبصير عندي نص أكبر و ساوي الـ infimum
+
+636
+00:54:35,910 --> 00:54:38,430
+هذا اللي هو أكبر و ساوي هذا اللي هو هذا اللي هو
+
+637
+00:54:38,430 --> 00:54:44,010
+مين؟ الـ U of F إذا حصلنا على كمان حاجة إن الـ U
+
+638
+00:54:44,010 --> 00:54:52,790
+of Fأصغر أو يساوي نص اللخص المعلومتين هدولة ونحصل
+
+639
+00:54:52,790 --> 00:54:57,810
+على اللي بدناه من المعلومتين من المعلومتين اللي
+
+640
+00:54:57,810 --> 00:55:06,860
+صار عندى اللي هو L of Fاللي هو اكبر او يساوي نص
+
+641
+00:55:06,860 --> 00:55:10,920
+وانا بعرف ان دائما ال الوف ايش ماله اصغر او يساوي
+
+642
+00:55:10,920 --> 00:55:15,180
+ال يوف اف و ال يوف اف ايش حصلته اصغر او يساوي من
+
+643
+00:55:15,180 --> 00:55:20,360
+نص اذا هذا اللي خلى اللي هو ال partition اللي عندى
+
+644
+00:55:20,360 --> 00:55:23,540
+اللي سهل اللي اخترته او ال set of partitions انها
+
+645
+00:55:23,540 --> 00:55:28,580
+توصلني للنتيجة اللي بديها بيصير عندى ال الوف اصغر
+
+646
+00:55:28,580 --> 00:55:33,220
+او يساوي ال يوف اف و هذا بين النص التنتين اذا الوف
+
+647
+00:55:33,220 --> 00:55:40,650
+افبساوية وف اف بساوية نص وهذا معناه ان ال function
+
+648
+00:55:40,650 --> 00:55:45,770
+is reversible and integrable and ال integration من
+
+649
+00:55:45,770 --> 00:55:52,650
+zero ل واحد لل x dx بساوية نص و الله يعطيك العافية
+
+650
+00:55:52,650 --> 00:55:54,590
+و المرة القادمة نكملها
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..1336de6334f1983fce3687e08cbc9d07799932da
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rKngzYKfO2U_raw.srt
@@ -0,0 +1,2616 @@
+1
+00:00:05,150 --> 00:00:09,890
+بسم الله الرحمن الرحيم اليوم ان شاء الله عندنا
+
+2
+00:00:09,890 --> 00:00:15,270
+اللي هو المحاضرة رقم عشرة في مساق تحليل حقيقه
+
+3
+00:00:15,270 --> 00:00:21,130
+اتنين لطلبة الجامعة الإسلامية كلية العلوم قسم
+
+4
+00:00:21,130 --> 00:00:27,370
+الرياضيات اليوم ان شاء الله هنبدأ في chapter سبعة
+
+5
+00:00:28,110 --> 00:00:33,150
+اللي هو الجزء الثاني من المادة أو العصب الثاني
+
+6
+00:00:33,150 --> 00:00:37,130
+للمادة اللي هو حكينا عن ال derivatives اليوم نحكي
+
+7
+00:00:37,130 --> 00:00:42,170
+عن اللي هو the reman integral أو تكامل رمان هيتم
+
+8
+00:00:42,170 --> 00:00:45,730
+الحديث اليوم ان شاء الله في ال section الأول هنحكي
+
+9
+00:00:45,730 --> 00:00:51,450
+عن ال reman integrability أو قابلية تكامل الدالة
+
+10
+00:00:51,450 --> 00:00:56,270
+على اللي هو بالنسبة لرمانفي البداية هنحكي عن بعض
+
+11
+00:00:56,270 --> 00:01:00,410
+التعريف اللي هتوصلنا للي بدناياه في الموضوع اللي
+
+12
+00:01:00,410 --> 00:01:03,910
+هو الـ Riemann Integrability الان هاخدلنا function
+
+13
+00:01:03,910 --> 00:01:11,090
+F F من some interval I نفترض ان الدالة اللي عند F
+
+14
+00:01:11,090 --> 00:01:18,850
+من A و B لعند R is a bounded function الان
+
+15
+00:01:18,850 --> 00:01:22,050
+بدنا نعرف بعض التعريفات اللي بتخودنا الى الـ
+
+16
+00:01:22,050 --> 00:01:26,950
+Riemann Integrabilityأول اشي بدنا نعرف شو معناه
+
+17
+00:01:26,950 --> 00:01:30,750
+partition لو كانت هذه الدانلة مثلا هي Y بالساوية F
+
+18
+00:01:30,750 --> 00:01:36,170
+of X على الفترة المذكورة وكانت من A ل B اللي هو ال
+
+19
+00:01:36,170 --> 00:01:41,650
+domain بدنا الآن نيجي نجزق المنطقة اللي هي الفترة
+
+20
+00:01:41,650 --> 00:01:47,210
+من A ل B اللي هو خلّي نسمي هذا X noteهذا نسميه x1
+
+21
+00:01:47,210 --> 00:01:51,750
+هذا نسميه x2 لعند ما أصل للفترة النموذجية اللي
+
+22
+00:01:51,750 --> 00:01:58,270
+بتسميها xk-1 لعند ال xk وهذا أسميها اللي هي xn
+
+23
+00:01:58,270 --> 00:02:06,110
+الان الفترة من a لb أنا جزقتها بx0 وx1 وx2 لعند xn
+
+24
+00:02:06,110 --> 00:02:13,370
+هذه التجزية بتحطها في 6 وسميها بي بتساوي x0و X1 و
+
+25
+00:02:13,370 --> 00:02:19,410
+X2 لعند Xn هذا اللي بسميه is a partition of the
+
+26
+00:02:19,410 --> 00:02:26,510
+interval A و B إذا عندي I subset من R closed
+
+27
+00:02:26,510 --> 00:02:30,430
+bounded interval ال ordered set هذه اللي حكيتها
+
+28
+00:02:30,430 --> 00:02:34,410
+كتبتها ان هي عبارة عن X0 و X1 لعند Xn بنسميها a
+
+29
+00:02:34,410 --> 00:02:39,390
+partition of I بحيث انه A X0 أصغر من X1 لعند ما
+
+30
+00:02:39,390 --> 00:02:45,060
+أقول أصغر من Xnالان الـ function عندى F من I لعند
+
+31
+00:02:45,060 --> 00:02:49,980
+R bounded function on I الان بدى أعرف بعض
+
+32
+00:02:49,980 --> 00:02:54,660
+التعريفات اللى هتلزمني برضه في اللى هو الوصول إلى
+
+33
+00:02:54,660 --> 00:02:59,160
+الـ Remain Integrability أول تعريف بدى أخد الآن
+
+34
+00:02:59,160 --> 00:03:07,000
+أسمي M1 هو عبارة عن الـ supremum للـ function في
+
+35
+00:03:07,000 --> 00:03:12,990
+المنطقة هذهالـ M1 بدي أسميه عبارة عن الـ Supremum
+
+36
+00:03:12,990 --> 00:03:16,810
+للدالة F of X والـ Supremum موجودة لأن الـ F
+
+37
+00:03:16,810 --> 00:03:21,870
+أشمالها is bounded السوب للـ F of X حيث الـ X
+
+38
+00:03:21,870 --> 00:03:29,310
+موجودة في الفترة من X note لعند X1 هذه اللي هي بدي
+
+39
+00:03:29,310 --> 00:03:34,570
+أسميها M1 Capital M1 Small اللي هي عبارة عن ال
+
+40
+00:03:34,570 --> 00:03:38,960
+Infimumلـ .. لـ الـ function hand اللي عبارة عن
+
+41
+00:03:38,960 --> 00:03:45,400
+الـ infimum لـ F of X حيث X تنتمي إلى الفترة X
+
+42
+00:03:45,400 --> 00:03:50,440
+naught و X واحد in general .. in general في الفترة
+
+43
+00:03:50,440 --> 00:03:55,280
+النموذجية بدي أسمي M K هي عبارة عن .. الآن في
+
+44
+00:03:55,280 --> 00:03:59,850
+الفترة هذهعالفترة هذه الـ supremum للـ function
+
+45
+00:03:59,850 --> 00:04:06,170
+هنا بدي اسميها M K K بسوء F of X such that X
+
+46
+00:04:06,170 --> 00:04:13,030
+element in X K minus واحد لعند X K و ال M small K
+
+47
+00:04:13,030 --> 00:04:19,330
+هسميها اللي هي بنفس اللي هسميت X واحد اللي هي ال
+
+48
+00:04:19,330 --> 00:04:23,370
+infimum لل F of X في الفترة من X K minus واحد لعند
+
+49
+00:04:23,370 --> 00:04:30,300
+X K على سبيل المثالالـ M K يعني هيها هذا اللي هي M
+
+50
+00:04:30,300 --> 00:04:36,340
+K Small والـ M K Capital على سبيل المثال هيها
+
+51
+00:04:36,340 --> 00:04:42,640
+هيقبتها هذا الـ M K Capital إذن الآن عندي سميت
+
+52
+00:04:42,640 --> 00:04:45,560
+اللي هي الـ M K Capital هي الـ Supremum والـ M K
+
+53
+00:04:45,560 --> 00:04:49,740
+Small هي عبارة عن الـ Infimum وهلجيت حنشوف إيش
+
+54
+00:04:49,740 --> 00:04:55,560
+دواعي هذه التسمية أو ليش سمينا هيكالان بدنا نعرف
+
+55
+00:04:55,560 --> 00:05:00,380
+اللي هو .. بدنا
+
+56
+00:05:00,380 --> 00:05:04,380
+نعرف الآن اللي هو حاجة اسمها lower sum و حاجة
+
+57
+00:05:04,380 --> 00:05:12,540
+اسمها ايش ال upper sum الان لو جيت بعد ما عملت ال
+
+58
+00:05:12,540 --> 00:05:21,020
+partition اللي عندي هذا و جيت أخدت اللي هي
+
+59
+00:05:28,550 --> 00:05:32,770
+أخدت الـ partition B اللي عندي هذا هذا ال
+
+60
+00:05:32,770 --> 00:05:40,110
+partition B هو عبارة عن X0 X1 X2 اللي عند XN جيت
+
+61
+00:05:40,110 --> 00:05:44,590
+أنا أخدت اللي هو الـ supremum للدالة في هذه
+
+62
+00:05:44,590 --> 00:05:51,550
+المنطقة والتي تكون طبعا هنا شكلها اللي هي سميتها
+
+63
+00:05:51,550 --> 00:06:02,170
+M1 في X1 نقص X0و جيت ضفت لها برضه ال supremum
+
+64
+00:06:02,170 --> 00:06:08,030
+للدالة في المنطقة هذه اللي هي زائد M2 سميت ال
+
+65
+00:06:08,030 --> 00:06:15,230
+supremum اللي هو M2 في X2-X0 لما نوصلت لآخر واحدة
+
+66
+00:06:15,230 --> 00:06:20,750
+بنفس الأسلوب أخدت اللي هو ال supremum اللي هي ال
+
+67
+00:06:20,750 --> 00:06:26,870
+supremum طبعا M1 عندىالتي هي طول الريانج اللي عندى
+
+68
+00:06:26,870 --> 00:06:32,850
+مضروب في مين اللي هو x1-x0 فبتطلع هذه المنطقة
+
+69
+00:06:32,850 --> 00:06:41,710
+بالظبط مساحة اللي عندى الان بضل أجمع بنفس الأسلوب
+
+70
+00:06:41,710 --> 00:06:45,830
+بجمحن بنفس الأسلوب عندى
+
+71
+00:06:53,690 --> 00:06:58,050
+هذا المجموع اللي بالازرق هو اللي بنسميه هسميه ال
+
+72
+00:06:58,050 --> 00:07:03,170
+other sum هذا ال other sum اللي هو ال other لل F
+
+73
+00:07:03,170 --> 00:07:08,090
+للدالة F بواسطة
+
+74
+00:07:08,090 --> 00:07:12,830
+مين ال partition B إذا ال other sum هذا اللي هو
+
+75
+00:07:12,830 --> 00:07:17,830
+عبارة عن اللي هو لل function بواسطة ال partition B
+
+76
+00:07:17,830 --> 00:07:20,410
+يعني ال other sum هذا يعتمد على ال F ويعتمد على
+
+77
+00:07:20,410 --> 00:07:25,690
+مين؟ على ال partition اللي عنديالأن similarly بدي
+
+78
+00:07:25,690 --> 00:07:31,190
+آجي أعرف اللي هو ال lower sum إذا ال upper sum
+
+79
+00:07:31,190 --> 00:07:38,890
+بيساوي ال summation ل X K نقص X K minus واحد مضروب
+
+80
+00:07:38,890 --> 00:07:43,950
+في ال M K capital K من عند واحد لعند N هذه اللي
+
+81
+00:07:43,950 --> 00:07:48,930
+بيسميها ال upper sum similarly بدي أعرف مين اللي
+
+82
+00:07:48,930 --> 00:07:56,170
+هو ال lower sumللـ F و لل D هيساوي الآن بدل ما أخد
+
+83
+00:07:56,170 --> 00:07:59,330
+اللي هو ال suprem أعلى قيمة بتاخد اللي هو ال
+
+84
+00:07:59,330 --> 00:08:05,390
+infimum بيصير عندي اللي هو summation X K minus X K
+
+85
+00:08:05,390 --> 00:08:10,790
+minus واحد في ال M K small K من عند واحد لعند N
+
+86
+00:08:10,790 --> 00:08:15,050
+هذا اللي هو عبارة عن ال lower sum في الواقع هو
+
+87
+00:08:15,050 --> 00:08:21,350
+هيكون المنطقة اللي بالأحمر هذهبناء على إني باخد
+
+88
+00:08:21,350 --> 00:08:27,670
+أنا اللي هو الارتفاع هو عبارة عن ال M K Small اللي
+
+89
+00:08:27,670 --> 00:08:33,410
+هو ال minimum لقيمة الدالة على كل المنطقة وهكذا
+
+90
+00:08:33,410 --> 00:08:38,030
+لما أصل لآخر واحدة فبصير عندى هذا ال summation
+
+91
+00:08:38,030 --> 00:08:43,890
+لهذه اللي هي المستطيلات
+
+92
+00:08:43,890 --> 00:08:51,510
+بناء على ال partition اللي عندىهي عبارة عن مين؟ عن
+
+93
+00:08:51,510 --> 00:08:56,190
+اللي هو اللا وارسم طبعا ناخد الدالة اللي هي في
+
+94
+00:08:56,190 --> 00:09:00,670
+منطقة موجة بأساس إنه اللي هي يكون أكثر توضيحا طبعا
+
+95
+00:09:00,670 --> 00:09:04,650
+لو منطقة سالبة برضه باخد نفس الأسلوب لكن بيصير
+
+96
+00:09:04,650 --> 00:09:08,810
+اللي هو بديل مساحة بيصير سالب مساحة أو مخلوطة بين
+
+97
+00:09:08,810 --> 00:09:13,070
+مساحة موجة مساحة سالبة و بت cancel بعض إلى أن أنا
+
+98
+00:09:13,070 --> 00:09:16,350
+أخدتها على فترة موجة بأساس اللي هو تسهل مسعملية
+
+99
+00:09:16,820 --> 00:09:22,620
+اللي هي التوضيح الآن خلينا تحت شوية إذا اللي حكيته
+
+100
+00:09:22,620 --> 00:09:28,420
+الآن أعرفت ال lower sum زي ما أنتوا شايفين عند ال
+
+101
+00:09:28,420 --> 00:09:32,780
+lower sum هيه هذا ال lower sum يعتمد في مجموعه على
+
+102
+00:09:32,780 --> 00:09:37,300
+ال M K اللي هي عبارة عن ال infimum لدالة على
+
+103
+00:09:37,300 --> 00:09:43,960
+الفترة X K minus واحد لعند X Kو-xk-1 هي عبارة عن
+
+104
+00:09:43,960 --> 00:09:49,040
+قيم ال partition اللي عندي إذا ال lower sum أو حتى
+
+105
+00:09:49,040 --> 00:09:53,820
+ال upper sum التمتين باعتمدين عالميا على اللي هو
+
+106
+00:09:53,820 --> 00:09:57,940
+ال partition وعلى الدالة اللي بنعملها أو لو وجدلها
+
+107
+00:09:57,940 --> 00:10:03,160
+ال lower sum و ال upper sum الهن واضح أن ال lower
+
+108
+00:10:03,160 --> 00:10:09,780
+sum ال lower sum واضح أنه اللي هو دائمااللي هو لل
+
+109
+00:10:09,780 --> 00:10:13,480
+partition المعين هيكون أصغر من مين من ال upper sum
+
+110
+00:10:13,480 --> 00:10:21,380
+واضح أه؟ الان اللي بنحكي if F من I ل R is bounded
+
+111
+00:10:21,380 --> 00:10:27,520
+and B is any partition of I then ال lower sum لل B
+
+112
+00:10:27,520 --> 00:10:32,020
+و ال F أصغر سوى ال upper sum لل B و ال F وبرهانه
+
+113
+00:10:32,020 --> 00:10:39,830
+سهل وسريعلو جئنا و أخذنا الـ bi-partition x0 x1
+
+114
+00:10:39,830 --> 00:10:40,550
+عند xn
+
+115
+00:10:43,460 --> 00:10:48,540
+وقررنا بين الـMK والـMK Capital أكيد الـMK Small
+
+116
+00:10:48,540 --> 00:10:50,540
+أصغر من الـMK Capital لإنه هذا عبارة عن
+
+117
+00:10:50,540 --> 00:10:54,820
+الـSupremum للفункشن على الفترة أو الـSubinterval
+
+118
+00:10:54,820 --> 00:10:59,780
+XK minus 1 عند XK وهذا الـSupremum للفункشن، هذا
+
+119
+00:10:59,780 --> 00:11:02,860
+الـInfimum وهذا الـSupremum فأكيد هذا أكبر أو
+
+120
+00:11:02,860 --> 00:11:07,080
+يساوي ما هو الـMK Small وهذا الكلام صحيح على كل
+
+121
+00:11:07,080 --> 00:11:11,380
+الـSubintervals يعني صحيح لكل K من 1 إلى 2 عند N
+
+122
+00:11:11,490 --> 00:11:14,650
+وفي نفس الوجهة الـ xk هو الـ xk minus واحد أكبر من
+
+123
+00:11:14,650 --> 00:11:18,610
+سفر إذا بيصير عندي حاصل ضرب هادي فى طول الفترة
+
+124
+00:11:18,610 --> 00:11:22,570
+وهادي فى طول الفترة بتظل ال inequality صحيحة لإنه
+
+125
+00:11:22,570 --> 00:11:26,730
+طول الفترة موجبة فبصير mk فى هذا المقدار أصغر ساوى
+
+126
+00:11:26,730 --> 00:11:29,950
+mk capital فى هذا المقدار بنعمل summation على الكل
+
+127
+00:11:29,950 --> 00:11:32,770
+وهذا ال summation على الكل بيصير عندى هذا هو عبارة
+
+128
+00:11:32,770 --> 00:11:36,150
+عن ال upper sum وهذا ال lower sum يعني اللى وصلنا
+
+129
+00:11:36,150 --> 00:11:44,110
+إلى انه دائما ال lower sumلأن ال lower sum هو أسفل
+
+130
+00:11:44,110 --> 00:11:48,230
+من ال upper sum في القيام فبكون أكيد ال lower sum
+
+131
+00:11:48,230 --> 00:11:56,870
+أصغر أو يساوي ال upper sum اطلع لفهم الان احنا لو
+
+132
+00:11:56,870 --> 00:12:01,590
+اجينا لو
+
+133
+00:12:01,590 --> 00:12:06,170
+اجينا اعرفنا .. بنعرف حاجة اللي في عندي P is a
+
+134
+00:12:06,170 --> 00:12:10,890
+partitionلو ضفت على الـ partition أي قيمة جديدة
+
+135
+00:12:10,890 --> 00:12:15,370
+بالاضافة للقيم الموجودة بيصير عندى هذا ال .. ال ..
+
+136
+00:12:15,370 --> 00:12:20,710
+ال .. الجديد عبارة عن refinement لمين للـ B يعني
+
+137
+00:12:20,710 --> 00:12:26,290
+تحسين لل partition يعني الآن بنقول عن أن الـ B الـ
+
+138
+00:12:26,290 --> 00:12:30,190
+Q is a refinement of B إذا كانت الـ B subset من
+
+139
+00:12:30,190 --> 00:12:34,990
+مين؟ من الـ Q طيب، الآن نشوف إيش اللي بنقوله
+
+140
+00:12:45,570 --> 00:12:57,310
+الأن لو أخدنا الفترة اللي هي A وB و جزّقناها X0 X1
+
+141
+00:12:57,310 --> 00:13:03,460
+لعند XNوالفترة النموذجية اللي نشرع عليها xk minus
+
+142
+00:13:03,460 --> 00:13:08,640
+واحد أو ال sub interval لعند مين لعند xk الان ال b
+
+143
+00:13:08,640 --> 00:13:13,520
+هذا هو عبارة عن ال partition x naught x واحد لعند
+
+144
+00:13:13,520 --> 00:13:17,400
+ال xn وهذا ordered يعني نكتبها على صورة ordered
+
+145
+00:13:17,400 --> 00:13:20,640
+لما نكتبها على هذا متفقين على انه ايش اللي هو احنا
+
+146
+00:13:20,640 --> 00:13:25,380
+نعمله ordered ال order بالنسبة له ايش ماله مهم لأن
+
+147
+00:13:25,380 --> 00:13:28,080
+هذا معناه تبدأ بال x naught وبعدين x واحد بعدين
+
+148
+00:13:28,080 --> 00:13:35,240
+اللي هو ال xnلو جينا أخدنا اللي هو أضفنا أي نقطة
+
+149
+00:13:35,240 --> 00:13:39,060
+هنا طبعاً أضفنا أي نقطة أو أي رقم بين الـA و بين
+
+150
+00:13:39,060 --> 00:13:45,320
+مين الـB الرقم المضاف اللي هو أضفنا لهذه وسميناها
+
+151
+00:13:45,320 --> 00:13:50,930
+Q فبكون الـQ هي عبارة عن refinement للـBيعني بمعنى
+
+152
+00:13:50,930 --> 00:13:55,030
+اخر اكيو refinement ل ال B معناته انه في اكيو فيها
+
+153
+00:13:55,030 --> 00:14:02,790
+نقاط اللي هي زيادة عن نقطة B واسمه تحسين لانه فعلا
+
+154
+00:14:02,790 --> 00:14:08,870
+بحسن وضع ال lower sum و ال upper sum الان ال lower
+
+155
+00:14:08,870 --> 00:14:14,250
+sum عرفناه لل B و ال F قلنا بعتمد عليهم بسوء ال
+
+156
+00:14:14,250 --> 00:14:22,130
+summationاللي هو xk ناقص xk minus واحد في ال mk
+
+157
+00:14:22,130 --> 00:14:25,090
+اللي هي mk small هي عبارة عن قلنا اللي هي عبارة عن
+
+158
+00:14:25,090 --> 00:14:29,050
+ال infimum لدى العهد الفترة k من عند واحد لعند n
+
+159
+00:14:29,050 --> 00:14:34,770
+والان ال upper sum ب U, B وF اللي هو بساوى ال
+
+160
+00:14:34,770 --> 00:14:40,890
+summation لل xk minus xk minus واحد mk capital k
+
+161
+00:14:40,890 --> 00:14:47,930
+من عند واحد لعند nوقلنا إن الـ lower للـ B و الـ F
+
+162
+00:14:47,930 --> 00:14:52,390
+دايما أصغر سوى ال upper للـ B و الـ F فده اللي حتى
+
+163
+00:14:52,390 --> 00:14:58,470
+الآن اللي حكيناه الآن نيجي ناخد الملاحظة اللي عندي
+
+164
+00:14:58,470 --> 00:15:04,510
+if Q above is a refinement of B لو قلنا إن Q عبارة
+
+165
+00:15:04,510 --> 00:15:09,390
+عن إيه شمالها؟ refinement of Brefinement معناته
+
+166
+00:15:09,390 --> 00:15:14,690
+اضفت اللي هو نقطة او اكتر من نقطة لنقاط ال
+
+167
+00:15:14,690 --> 00:15:20,810
+partition اللي عندى اذا اي فترة اي فترة اللي هي xk
+
+168
+00:15:20,810 --> 00:15:27,550
+minus واحد لعند xk اللي هو في ال partition بيه
+
+169
+00:15:27,550 --> 00:15:34,060
+هقدر اكتبها على صورة ميناللي هو yj-1 yj اتحاد ال
+
+170
+00:15:34,060 --> 00:15:37,940
+yj yj زاد 1 اتحاد ده ما اصل عند اتحاد yn نقص 1
+
+171
+00:15:37,940 --> 00:15:44,040
+لعند مين لعند ال yn ماشي الحال طبعا الان هذول عدد
+
+172
+00:15:44,040 --> 00:15:47,740
+الفترات على حسب الإضافة اللي هنالو ضفنا نقطة واحدة
+
+173
+00:15:47,740 --> 00:15:52,240
+بين هذول التنتين معناته هيصير عندى من الـ yj-1 هي
+
+174
+00:15:52,240 --> 00:15:56,460
+عبارة عن xk-1 والنقطة المضافة اتحاده النقطة
+
+175
+00:15:56,460 --> 00:15:59,780
+المضافة اللي هي مع مين مع النقطة هذه بيكون عندى بس
+
+176
+00:15:59,780 --> 00:16:03,690
+فترة تانية لو كان فيه عندىنقطين مضافات بيصير عندي
+
+177
+00:16:03,690 --> 00:16:06,710
+اللي هو للنقطة الأولى و بعدين من النقطة الأولى
+
+178
+00:16:06,710 --> 00:16:12,030
+لنقطة ثانية و بعدين من النقطة الثانية لـ XK فبيصير
+
+179
+00:16:12,030 --> 00:16:16,030
+عندي أكم فترة انضاف تلت .. بيصير تلت فترات و هكذا
+
+180
+00:16:16,030 --> 00:16:19,970
+إذن و لو مافيش إضافة في هذه الفترة بالمرة بيصير
+
+181
+00:16:19,970 --> 00:16:24,460
+عندي بس اللي هي ال interval الأولىأه إذا صار عندى
+
+182
+00:16:24,460 --> 00:16:29,640
+الآن في حالة اللى هو الـQ refinement للـB أي فترة
+
+183
+00:16:29,640 --> 00:16:35,040
+في الـB الأصلية بقدر أكتبها على صورة اللى هى هذا
+
+184
+00:16:35,040 --> 00:16:40,000
+الاتحاد وهذا الاتحاد يحتمل اللى هو الفترة الحاله
+
+185
+00:16:40,000 --> 00:16:44,460
+أو فترتين أو تلاته إلى آخره على حسب اللى هو النقاط
+
+186
+00:16:44,460 --> 00:16:50,060
+المضافة واضح اللى ما اللى هي سبعة واحد اتنينif f
+
+187
+00:16:50,060 --> 00:16:53,960
+من I عند R is bounded and B is any partition of I
+
+188
+00:16:53,960 --> 00:16:57,680
+and Q is a refinement of B فرضنا ان Q ايش ما هي
+
+189
+00:16:57,680 --> 00:17:03,180
+refinement لـ B إذا ال lower sum الاعتماد على
+
+190
+00:17:03,180 --> 00:17:06,820
+partition B و ال function F أصغر يساوي ال lower
+
+191
+00:17:06,820 --> 00:17:12,780
+sum لـ Q و ال F و ال U ال Q و ال F أصغر يساوي ال U
+
+192
+00:17:12,780 --> 00:17:17,900
+و ال B و ال F هو أصلا واقعا ال refinement هو فعلا
+
+193
+00:17:17,900 --> 00:17:25,250
+تحسين أصلاأحنا لما نجينا دربنا المثال في الأول و
+
+194
+00:17:25,250 --> 00:17:28,690
+رسمنا الرسمة على سبيل المثال هذه الرسمة اللي عندي
+
+195
+00:17:28,690 --> 00:17:36,290
+و أخدنا من a لعند b مثلا اللي هو الفترة و كان في
+
+196
+00:17:36,290 --> 00:17:42,150
+عندي اللي هو اللي عملنا تجزء x note x1 الاخره x2
+
+197
+00:17:42,150 --> 00:17:49,420
+x3 لعند الاخيرة xn مثلاعلى سبيل المثال واخدنا
+
+198
+00:17:49,420 --> 00:17:52,860
+القبر صم القبر صم هيكون عبارة عن المنطقة هذه
+
+199
+00:17:52,860 --> 00:17:57,240
+والقبر صم عبارة عن المنطقة هذه والقبر صم بيكون هذه
+
+200
+00:17:57,240 --> 00:18:02,150
+المنطقة كلها اللي بالأزرقالآن لما نزيد اللي هو
+
+201
+00:18:02,150 --> 00:18:07,210
+نقاط الـ Partition اللي هو هذا اللي هي النقاط
+
+202
+00:18:07,210 --> 00:18:12,630
+بتجلل الجزء الزائد فالآن فعلًا بيكون تحسين على
+
+203
+00:18:12,630 --> 00:18:15,330
+الأولاني لأنه أصلا أنا بعد شوية هتلاقي غاية في
+
+204
+00:18:15,330 --> 00:18:21,140
+الوصول أن أصل اللي هو اللي هو أصغر واحدفي ال upper
+
+205
+00:18:21,140 --> 00:18:25,720
+sums يصير زي أكبر واحد في ال lower sums يوم ما
+
+206
+00:18:25,720 --> 00:18:29,820
+يلتجوا مع بعض بيكون فعلا اللي هو ال upper sum وال
+
+207
+00:18:29,820 --> 00:18:35,000
+lower sum اللي هو تجاوزا بيصير بساوي بالظبط اللي
+
+208
+00:18:35,000 --> 00:18:40,080
+هو المساحة تحت المنحنة أو المنطقة اللي هنسميها في
+
+209
+00:18:40,080 --> 00:18:43,340
+حينه ان هي عبارة عن remen integral بل وقيمة ال
+
+210
+00:18:43,340 --> 00:18:47,660
+remen integral هو عبارة عن هذه المنطقة اللي صارتمن
+
+211
+00:18:47,660 --> 00:18:53,540
+اللي هو انه ناخد نزغر اللي هو الأبر و نرفع ال
+
+212
+00:18:53,540 --> 00:18:57,540
+lower و نشوف كيف بنزغر الأبر و كيف بنرفع ال lower
+
+213
+00:18:57,540 --> 00:19:01,580
+اللي هو معنى اللي هيوصل لنا معنى ال integrability
+
+214
+00:19:02,250 --> 00:19:06,710
+إذا الـ Refinement فعلاً Refinement تحسين لما أقول
+
+215
+00:19:06,710 --> 00:19:13,390
+أن بي إي كيو زادت عن هذه الـ Q إيش هتعمل؟ هتتجللل
+
+216
+00:19:13,390 --> 00:19:19,630
+الجزء الزائد في الـ Upper لما أزيد النقاط و هتزيد
+
+217
+00:19:19,630 --> 00:19:24,950
+للجزء المفقود وين في ال lowerماشي الحال ففعلا
+
+218
+00:19:24,950 --> 00:19:28,310
+هيكون ايه بقى عبارة عن refinement او تحسين انت
+
+219
+00:19:28,310 --> 00:19:32,470
+نشوف الآن نشوف لمتنا اللي عندنا اللي هو معنى اللي
+
+220
+00:19:32,470 --> 00:19:38,350
+بقوله انا انه عندي لو كانت Q refinement لل B بكون
+
+221
+00:19:38,350 --> 00:19:43,790
+الlower لجديدعليه أكبر أو يساوي الـ Lower القديم
+
+222
+00:19:43,790 --> 00:19:48,390
+يعني بمعنى آخر اقترب إلى اللي بدناهي اللي متوقعين
+
+223
+00:19:48,390 --> 00:19:53,110
+انه يكون هو التحسين النهائي اللي هنسميه اللي هو ال
+
+224
+00:19:53,110 --> 00:19:58,670
+integration ماشي؟ وال Upper هنا Upper UQ واضح انه
+
+225
+00:19:58,670 --> 00:20:03,510
+بصغر اللي هو عن الأصل لأنه صار فيه تحسين واقترب من
+
+226
+00:20:03,510 --> 00:20:08,070
+الحقيقة اقترب من اللي هو القيمة اللي احنا هنسميها
+
+227
+00:20:08,070 --> 00:20:12,370
+في النهاية integrationطيب، بدنا نثبت الكلام هذا،
+
+228
+00:20:12,370 --> 00:20:16,810
+شوف كيف بدنا نثبته، اللي أنا عندي بيأخدت I اللي هو
+
+229
+00:20:16,810 --> 00:20:21,290
+ال partition ال arbitrary واخدت اللي هو sigma I،
+
+230
+00:20:21,290 --> 00:20:26,310
+بد أشتغل على إضافة نقطة ومن ثم أقول إضافة النقطة
+
+231
+00:20:26,310 --> 00:20:31,340
+تؤدي إلى اللي هو باقي النقطة، نشوف إيش بقولعند بي
+
+232
+00:20:31,340 --> 00:20:35,080
+هي اللي هو الـ partition أخدت z element in I و الـ
+
+233
+00:20:35,080 --> 00:20:39,100
+z هذي و أنا بدأ أحطها في الفترة xk minus واحد لعند
+
+234
+00:20:39,100 --> 00:20:42,640
+مين لعند الـ xk يعني الآن ال partition اللي بي
+
+235
+00:20:42,640 --> 00:20:51,200
+عندي هو عبارة عن x naught لعند xn xk minus واحد xk
+
+236
+00:20:51,200 --> 00:20:55,620
+indeed في المنطقة هذه ضفت نقطة وسميت بيها prime و
+
+237
+00:20:55,620 --> 00:21:03,380
+عبارة عن x naught xk minus واحدZ X K وظل الخلايت
+
+238
+00:21:03,380 --> 00:21:07,720
+كله زي ما هو لعد مين لعد X بس اللي غيرت فيه او
+
+239
+00:21:07,720 --> 00:21:12,420
+اللي ضفتله بس ضفت نقطة هنا اسمها Z بدى اشوف تأثير
+
+240
+00:21:12,420 --> 00:21:16,640
+هذا النقطة ايش بده يعمل بين ال partition القديم
+
+241
+00:21:16,640 --> 00:21:20,520
+وال partition الجديد ومن ثم نستنتج اي refinement
+
+242
+00:21:20,520 --> 00:21:26,990
+على اللي هو ال B القديمةالأمر سهل ان شاء الله الان
+
+243
+00:21:26,990 --> 00:21:30,290
+ال .. ال .. ال .. الاضافة الجديدة صار عند بي برايم
+
+244
+00:21:30,290 --> 00:21:35,870
+ايش هو X Note X واحد وهي ال Z ضفتهان اللي اختلف عن
+
+245
+00:21:35,870 --> 00:21:43,640
+فوق بس ال Z اللي هان بناء عليه الان ال M Kالـ mk'
+
+246
+00:21:43,860 --> 00:21:49,020
+هي عبارة عن الـ supremum للـ function f وين في
+
+247
+00:21:49,020 --> 00:21:58,920
+المنطقة بين zk-1 و مين و z الآن في الفترة بين xk-1
+
+248
+00:21:58,920 --> 00:22:04,800
+و بين xk ضفت z الآن الـ z اللي ضفت هنا هذه المنطقة
+
+249
+00:22:04,800 --> 00:22:11,150
+بدي أسميه ال supremum اللي فيها mk' اللي هنااللي
+
+250
+00:22:11,150 --> 00:22:16,190
+ضالت نسميها إيش مالها MK double prime مع العلم إنه
+
+251
+00:22:16,190 --> 00:22:19,890
+الـ supremum الأصلية في ال partition الأصلي على كل
+
+252
+00:22:19,890 --> 00:22:23,250
+الفترة من XK minus واحد عند XK اسمها إيش مسمينا
+
+253
+00:22:23,250 --> 00:22:31,130
+احنا MK capital باشي الحال طيب ركزوا معاياالآن
+
+254
+00:22:31,130 --> 00:22:34,970
+عندي الـ MK' هي الـ supremum على اللي هو الـ
+
+255
+00:22:34,970 --> 00:22:39,610
+subinterval هذه الـ MKW' هي ال supremum على اللي
+
+256
+00:22:39,610 --> 00:22:43,910
+هي ال subinterval اللي جنبها اللي هي من Z لعند XK
+
+257
+00:22:43,910 --> 00:22:49,890
+واضح أن ال supremum على الكبيرة هذه من XK minus 1
+
+258
+00:22:49,890 --> 00:22:53,790
+لعند XK أكيد أكبر أو يساوي ال supremum على الصغيرة
+
+259
+00:22:54,620 --> 00:22:58,580
+لأن هذه أشملها جزء من 100 من هذه كلها فال supremum
+
+260
+00:22:58,580 --> 00:23:01,940
+لل function عالي كبيرة أكبر أو يساوي ال supremum
+
+261
+00:23:01,940 --> 00:23:06,020
+على مين؟ على الصغيرة و similarly للتانية إذا ال MK
+
+262
+00:23:06,020 --> 00:23:12,180
+أكبر أو يساوي MK' Similarly MK أكبر يساوي MKW' لأن
+
+263
+00:23:12,180 --> 00:23:15,220
+هذه عبارة عن ال supremum على الجزء الثاني منها
+
+264
+00:23:15,220 --> 00:23:20,520
+اللي إلى جاي، واضح؟إذا صار عندي therefore الآن بدأ
+
+265
+00:23:20,520 --> 00:23:24,840
+أصل لحاجة معينة اللي هي mk في xk ناقص xk minus
+
+266
+00:23:24,840 --> 00:23:31,310
+واحددربت اللي هي Mk في هذا هذه الفترة طبعا اللي هي
+
+267
+00:23:31,310 --> 00:23:40,730
+بتساوي Zk-Zk-Xk-1 زائد Xk-Zk الان صارت هذه Mk في
+
+268
+00:23:40,730 --> 00:23:45,410
+المقدار هذا بساوي Mk في هذه زائد هذه ماشي الحال
+
+269
+00:23:45,410 --> 00:23:50,820
+الان ال Mkأكبر أو يساوي mean mk' إذا انصار المقدار
+
+270
+00:23:50,820 --> 00:23:58,780
+هذا أكبر أو يساوي mk' في zk z ناقص xk minus واحد و
+
+271
+00:23:58,780 --> 00:24:04,820
+هذه ال mk برضه أكبر أو يساوي mk double prime of xk
+
+272
+00:24:04,820 --> 00:24:10,380
+ناقص z ماشي الحالة الآن احنا بين ال two partitions
+
+273
+00:24:10,380 --> 00:24:14,580
+اللي حدث بس تغيير واحد زي ما قلنا بس على المنطقة
+
+274
+00:24:14,580 --> 00:24:23,090
+هذه لاحظ انتالان ال lower sum ال lower sum لل b و
+
+275
+00:24:23,090 --> 00:24:30,350
+ال f ال b prime و ال f بساوي لو حسبنا للأولاني أول
+
+276
+00:24:30,350 --> 00:24:38,350
+إشي هو عبارة عن m1 في x0 او x1 نقص x0 زاد لما أصل
+
+277
+00:24:38,350 --> 00:24:51,620
+عند mk في xK-XK-1 زائد لما أصل لعند MN في XN-XN-1
+
+278
+00:24:51,620 --> 00:24:57,800
+هذا إيش هو يا جماعة؟ هو عبارة عن الأبرثان الأبرثان
+
+279
+00:24:57,800 --> 00:25:03,120
+لمن؟ لـB و لـF متفجر، ماشي الحالة على مين؟ على ال
+
+280
+00:25:03,120 --> 00:25:12,270
+partition B الآن هذا بساويم1 في x1 ناقص x0 زائد
+
+281
+00:25:12,270 --> 00:25:18,010
+الان هذه جزئناها جزئين بعد شوية زائد زائد هذا نفسه
+
+282
+00:25:18,010 --> 00:25:25,690
+ماتغيرش هذا اللي هو عبارة عن mk لسه xk ناقص z
+
+283
+00:25:25,690 --> 00:25:34,850
+مظبوط زائد اللي هو mk خليني اقول ال
+
+284
+00:25:34,850 --> 00:25:46,170
+mkاللي هو Z ناقص كحسابات ناقص X كمينوس واحد صحيح؟
+
+285
+00:25:46,170 --> 00:25:53,130
+باضح زائد الباقي زي ما هو زائد M N في X N ناقص X N
+
+286
+00:25:53,130 --> 00:26:00,980
+ناقص واحد الآن هذا ماتغيرش نفسهزائد هذا الـ N هو
+
+287
+00:26:00,980 --> 00:26:05,540
+اللي اتغير وهنا في ال inequality هنا لقنا عبارة عن
+
+288
+00:26:05,540 --> 00:26:12,220
+هذا المقدار اللي هو أكبر أو يساوي اللي هو اللي ما
+
+289
+00:26:12,220 --> 00:26:15,420
+اتغيرش هذا نسميه اللي هو M1
+
+290
+00:26:18,760 --> 00:26:25,640
+معايا يا شباب زائد اللي هو mk شرطة في xk ناقص z
+
+291
+00:26:25,640 --> 00:26:31,520
+زائد mk شرطين في z ناقص xk ماينوس واحد زائدة دا
+
+292
+00:26:31,520 --> 00:26:38,570
+كما تتغيرش زي ما هو mn في xnنقص XN نقص 1 هذا في
+
+293
+00:26:38,570 --> 00:26:42,690
+الواقع مين هو أصلاً هذا هو عبارة عن اللي هو ال
+
+294
+00:26:42,690 --> 00:26:48,510
+upper sum لل partition مين ال B' هو بساوي ال upper
+
+295
+00:26:48,510 --> 00:26:55,270
+sum لل B' والFيعني اللي لجناه أنه لما نضفنا اللي
+
+296
+00:26:55,270 --> 00:26:59,830
+هو نقطة على اللي هو الـ partition صار عند الـ
+
+297
+00:26:59,830 --> 00:27:03,590
+partition اللي هو الأبر للـ partition الجديد أصغر
+
+298
+00:27:03,590 --> 00:27:08,360
+أو يساوي الأبر للـ partition من القديمالان اللي
+
+299
+00:27:08,360 --> 00:27:11,760
+حدث على نقطة ممكن يحدث على نقطين يعني الان ممكن
+
+300
+00:27:11,760 --> 00:27:15,640
+نقطة مثلا انضافت هنا ممكن انضافت كمان مرة هنا كمان
+
+301
+00:27:15,640 --> 00:27:20,560
+نقطة هيصير لجديد بنفس الأسلوب أكبر ويساوي تلت نقاط
+
+302
+00:27:20,560 --> 00:27:24,040
+بيصير أكبر ويساوي أربع نقاط بيصير أكبر ويساوي يعني
+
+303
+00:27:24,040 --> 00:27:29,500
+in general مدام ال refinement UQ اللي محتوي بيه في
+
+304
+00:27:29,500 --> 00:27:34,820
+النهاية عدد الإضافات finiteإذا حيكون اللي هو كل
+
+305
+00:27:34,820 --> 00:27:39,300
+مرة بعدد الإضافات بيصير sub intervals بنفس الأسلوب
+
+306
+00:27:39,300 --> 00:27:43,940
+اللي هان حيصير عنده القبر القديم أكبر أو يساوي
+
+307
+00:27:43,940 --> 00:27:48,620
+القبر الجديد يعني بمعنى أخر in general القبر للـB
+
+308
+00:27:48,620 --> 00:27:54,080
+والـF هيكون أكبر أو يساوي القبر للـQ ال refinement
+
+309
+00:27:54,080 --> 00:27:59,160
+وMain والـFوطبعا زي ما قلنا هذا عبارة عن تحسين لإن
+
+310
+00:27:59,160 --> 00:28:03,960
+القبر هذا بيصير يصغر يقترب من المساحة الحقيقية
+
+311
+00:28:03,960 --> 00:28:08,360
+اللي تحت المنحنة في حالة اللي هو الدل ايش الموجبة
+
+312
+00:28:08,360 --> 00:28:15,860
+واضح طيب الان similarly for lower الان الlower
+
+313
+00:28:15,860 --> 00:28:20,940
+بيصير اللي هوفي الـ lower بدل ما يصير عندي لو كان
+
+314
+00:28:20,940 --> 00:28:25,700
+عندي بنحكي عن ال lower احنا ال L بيوقف و بنحكي عن
+
+315
+00:28:25,700 --> 00:28:30,220
+ال M 1 small و ال M 2 small و M K small و M N
+
+316
+00:28:30,220 --> 00:28:36,660
+small بيصير عندي M 1 small و M K small و M KSmall
+
+317
+00:28:36,660 --> 00:28:44,520
+ومكSmall ومنSmall بيصير الان اللي هي المن هنا بدل
+
+318
+00:28:44,520 --> 00:28:49,680
+ما هي أكبر إيش هتصير على إيش أصغر لإنه بيحكي عن ال
+
+319
+00:28:49,680 --> 00:28:54,810
+infimum منأيش صغير على أيش صغير إلى infimum على
+
+320
+00:28:54,810 --> 00:28:59,570
+أيش أيش اللي هو أصغر فمعناته أيش صار عنده بيصير له
+
+321
+00:28:59,570 --> 00:29:04,910
+infimum اللي هو ال M1 أصغر من ال M1' أو ال Mk
+
+322
+00:29:04,910 --> 00:29:11,730
+Small أصغر أو يساوي ال Mk' Small واضح أه؟ ومنه
+
+323
+00:29:11,730 --> 00:29:17,870
+بنستنتج أنه هيكون عند ال LBF أصغر أو يساوي
+
+324
+00:29:21,260 --> 00:29:28,950
+الـ LBF هنصل أصغر و ساويالـ QF وفعلًا أيضًا هذا هو
+
+325
+00:29:28,950 --> 00:29:35,410
+تحسين لإنه اللي هي زاد اللي هو ال lower sum بحيث
+
+326
+00:29:35,410 --> 00:29:41,550
+إنه بده يصل أو يروح ناحية مين اللي هو المساحة
+
+327
+00:29:41,550 --> 00:29:46,310
+الحقيقية تحت المنحنة في حالة التقاء الجهتين بيصير
+
+328
+00:29:46,310 --> 00:29:52,670
+عندي اللي هو إنه القيمة إذا يعني أخدنا بمعنى أخر
+
+329
+00:29:52,670 --> 00:29:58,540
+ندخل للموضوع مدام هيكإنه لو أخدنا اللي هو صار عندى
+
+330
+00:29:58,540 --> 00:30:03,140
+ال lower يبدأ يكبر يعني لو أخدتله ال suprem لكل ال
+
+331
+00:30:03,140 --> 00:30:06,960
+lowers على كل ال partitions اللي في الدنيا و أخدت
+
+332
+00:30:06,960 --> 00:30:10,980
+ال infimum على كل كبار اللي هو و اتساون مع بعض ال
+
+333
+00:30:10,980 --> 00:30:14,820
+infimum يعني عالمين على ال upper sums بيصير ال
+
+334
+00:30:14,820 --> 00:30:17,500
+suprem مع ال lower sums و ال infimum مع ال lower
+
+335
+00:30:17,500 --> 00:30:22,780
+sums يوم ما يتساونهو اللي بنسميه اللي هو الـ
+
+336
+00:30:22,780 --> 00:30:26,920
+Remain Integrable وبنسمي اللي هي الـ Remain
+
+337
+00:30:26,920 --> 00:30:30,440
+Integrable في هذه الحالة بساوي هذه القيمة المشتركة
+
+338
+00:30:30,440 --> 00:30:35,720
+اللي أصبحت متساوية نيجي لللمة اللي عندي هذه، لمة
+
+339
+00:30:35,720 --> 00:30:40,720
+برضه يعني برهانها سهل، نشوف إيش بتقول اللمة، اللمة
+
+340
+00:30:40,720 --> 00:30:45,610
+بتقول let F from I to R be boundedif B1 وB2 are
+
+341
+00:30:45,610 --> 00:30:49,470
+any two partitions يعني مش نفس ال partition على أي
+
+342
+00:30:49,470 --> 00:30:54,090
+two partitions دائما اللاور على أي partition في
+
+343
+00:30:54,090 --> 00:30:57,150
+الدنيا أصغر أو يساوي الأبر على أي partition تاني
+
+344
+00:30:57,150 --> 00:30:59,970
+مش مشكلة إيش هو في الأول قلنا على نفس ال partition
+
+345
+00:30:59,970 --> 00:31:05,510
+arbitrarily و الإثبات هنستخدم اللمّة الأولى و
+
+346
+00:31:05,510 --> 00:31:09,010
+اللمّة الثانية اللي أخدناها و هنصلي هذه بكل سهولة
+
+347
+00:31:09,010 --> 00:31:13,760
+شوف كدهخلّينا ناخد الـ Q ايش بتساوي بي واحد اتحاد
+
+348
+00:31:13,760 --> 00:31:17,240
+دي اتنين صارت الـ Q هذه refinement للأولى و
+
+349
+00:31:17,240 --> 00:31:21,420
+refinement لمين؟ للتانية تحسين للتانتين بناء عليه
+
+350
+00:31:21,420 --> 00:31:25,380
+.. بناء على اللي هي اللمّة اللي أخدناها سبعة واحد
+
+351
+00:31:25,380 --> 00:31:28,660
+واحد وسبعة واحد اتنين احنا حصول على ناية ليه
+
+352
+00:31:28,660 --> 00:31:33,000
+احسبلي ال بي واحد و أف صار refinement مزام صار ..
+
+353
+00:31:33,000 --> 00:31:37,620
+صار له refinement او تحسين بده يكبر اللاور صار
+
+354
+00:31:37,620 --> 00:31:42,750
+أصغر أو ساوي ال Q Fمن جهة أخرى قبل بشوية أثبتنا أن
+
+355
+00:31:42,750 --> 00:31:48,310
+الـQF أصغر يساوي UQF for any اللي هو specific mean
+
+356
+00:31:48,310 --> 00:31:53,070
+الـQ لنفس الـQ لأن برضه اللي هو الـQ refinement
+
+357
+00:31:53,070 --> 00:31:57,790
+لمين؟ للـB2 مازال refinement إذا الـQ بدي يصير
+
+358
+00:31:57,790 --> 00:32:03,050
+أصغر أو يساوي مين الـUB2F زي اللي بواسطة أثبتناها
+
+359
+00:32:03,050 --> 00:32:07,470
+قبل بشوية يعني in generalin general الـ B1 و F
+
+360
+00:32:07,470 --> 00:32:15,810
+أصغر أو ساوي و B2 و F بدون يعني بغض النظر عن B1 و
+
+361
+00:32:15,810 --> 00:32:21,130
+B2 أنهم كونوا زي بعض أو مختلفات لأي two partitions
+
+362
+00:32:21,130 --> 00:32:26,370
+of the interval I طيب الـ definition الآن اللي
+
+363
+00:32:26,370 --> 00:32:28,830
+حكيناها قبل شوية قولناها قبل شوية بدنا نعرف حاجة
+
+364
+00:32:28,830 --> 00:32:33,940
+اسمها lower integralالـ Lower Integral وحاجة اسمها
+
+365
+00:32:33,940 --> 00:32:38,760
+الـ Upper Integral أخدنا
+
+366
+00:32:38,760 --> 00:32:42,740
+الـ function F bounded طبعا معروف ليش bounded عشان
+
+367
+00:32:42,740 --> 00:32:47,880
+يكون الـ M1 و الـ M2 و الـ Mn كلهم موجودات سواء
+
+368
+00:32:47,880 --> 00:32:51,040
+الـ M capital أو الـ M small عشان هما عبارة عن
+
+369
+00:32:51,040 --> 00:32:54,260
+supremum و infimum فلازم تكون bounded ال function
+
+370
+00:32:54,260 --> 00:32:58,280
+الآن فرضنا الـ F عبارة عن bounded function على ال
+
+371
+00:32:58,280 --> 00:33:01,520
+close bounded interval اللي عندى هذهبنعرف الـ
+
+372
+00:33:01,520 --> 00:33:06,320
+lower integral of F on I اللي هو المعتمد عالميا
+
+373
+00:33:06,320 --> 00:33:11,680
+على ال lower sums ال lower sums قولنا انحسن انحسن
+
+374
+00:33:11,680 --> 00:33:16,960
+انحسن انحسن احسن تحسين اذا جهز التعبير اللي هو ال
+
+375
+00:33:16,960 --> 00:33:21,640
+supremum لكل مين لكل ال lower sums على كل ال
+
+376
+00:33:21,640 --> 00:33:25,040
+partitions اللي في الدنيا هذول ايه الشمال اللي
+
+377
+00:33:25,040 --> 00:33:31,260
+بنسميه L of F اللي بنسميه lower sumأخد الـ
+
+378
+00:33:31,260 --> 00:33:35,400
+supremum هي ال lower sums تحت المنحنة لما أخد اللي
+
+379
+00:33:35,400 --> 00:33:40,220
+هنا ال supremum اللي هو هذا ال supremum بدي أسميه
+
+380
+00:33:40,220 --> 00:33:46,300
+ال lower integral الآن ال upper sums بدي أخد اللي
+
+381
+00:33:46,300 --> 00:33:52,020
+هنا ال infimum يعني بدي أزغرهن اللي هو و أسميه ال
+
+382
+00:33:52,020 --> 00:33:57,760
+upper integral إذا صار عندي lowerIntegral هو عبارة
+
+383
+00:33:57,760 --> 00:34:02,080
+عن الـ Supremum للـ Lower Sums و ال .. واضح يا
+
+384
+00:34:02,080 --> 00:34:06,060
+عيسى و ال Upper Integral هو عبارة عن ال Infimum
+
+385
+00:34:06,060 --> 00:34:10,620
+لمن؟ لل Upper Sums متوقع أنه يقول لنا أنه احنا
+
+386
+00:34:10,620 --> 00:34:16,810
+بنقول عنهافوق الصفحة التانية متوقع يقولنا انه عنا
+
+387
+00:34:16,810 --> 00:34:20,970
+الآن بنسميها remain integrable اذا تساوت ايش
+
+388
+00:34:20,970 --> 00:34:26,130
+اتجهتين ماشي جابيه لبس خلينا نبرهن او نتطلع على
+
+389
+00:34:26,130 --> 00:34:33,430
+هذه اللي أمامنا برضه ملاحظة سهلة الملاحظة بتقولي
+
+390
+00:34:33,430 --> 00:34:42,780
+لو الان جيناسمّينا الـ M I small عبارة عن الـ
+
+391
+00:34:42,780 --> 00:34:49,860
+infimum للدالة على الفترة كلها الـ infimum يعني
+
+392
+00:34:49,860 --> 00:34:57,460
+الـ M I سماها عبارة عن الـ infimum للدالة F of X
+
+393
+00:34:57,460 --> 00:35:02,860
+حيث X موجودة على I وكأنه في الواقع أخذ الـ
+
+394
+00:35:02,860 --> 00:35:07,520
+partition هو عبارة عن نقطيناللي عبارة عن a و مين و
+
+395
+00:35:07,520 --> 00:35:12,800
+b ال partition بس a و b في هذا الحلق و جالك اللي
+
+396
+00:35:12,800 --> 00:35:18,020
+هو .. ليش .. هلجيت تشوفوا ال mi capital عبارة عن
+
+397
+00:35:18,020 --> 00:35:21,860
+ال supreme لل f of x such that x element in I
+
+398
+00:35:21,860 --> 00:35:27,780
+بقولك الآن إذن .. إذن mi ال mi small مضروبة في طول
+
+399
+00:35:27,780 --> 00:35:32,120
+الفترة يعني ال partition واحد لأن ال partition a و
+
+400
+00:35:32,120 --> 00:35:39,750
+b بس أصغر أو يساوياللي هو اللوفاف طبيعي طبيعي لأن
+
+401
+00:35:39,750 --> 00:35:45,570
+هذا واحد من مين من ال lower sums، بظبط؟ وهذا أصلا
+
+402
+00:35:45,570 --> 00:35:49,170
+ال supremum لل lower sums فهيكون ال supremum أكبر
+
+403
+00:35:49,170 --> 00:35:52,670
+يساوي مين؟ واحد منهم، هذا زي ما أقول أنها lower
+
+404
+00:35:52,670 --> 00:35:58,450
+sum على partition في A و BSimilarly, M I في الـ B
+
+405
+00:35:58,450 --> 00:36:03,590
+minus A أكبر أو ساوي ال other sum of F، ليه؟ لأن
+
+406
+00:36:03,590 --> 00:36:07,470
+هذا عبارة عن واحد من ال other sums و هذا عبارة عن
+
+407
+00:36:07,470 --> 00:36:10,210
+ال N في المملكة و ليه ال other sums فطبيعي هيكون
+
+408
+00:36:10,210 --> 00:36:14,260
+ال U of F أكبر أو ساوي B minus Ato be more
+
+409
+00:36:14,260 --> 00:36:18,360
+specific for every B element in B of I نحن لدينا M
+
+410
+00:36:18,360 --> 00:36:22,140
+I في B minus A أصغر أو يساوي L, B و F لأن هذا واحد
+
+411
+00:36:22,140 --> 00:36:28,800
+من ال partitions قلنا اللي هي لفترة I نفس الشيء
+
+412
+00:36:28,800 --> 00:36:34,800
+فبصير عند ال L of B و A F أصغر أو يساوي اللي هو ال
+
+413
+00:36:34,800 --> 00:36:40,060
+U, B و F أصغر أو يساوي M I في B minus Aمعروف هذا
+
+414
+00:36:40,060 --> 00:36:43,580
+أنا مش حاجة لزمنه أصلاً فبصير mi أصغر سواء ل و بف
+
+415
+00:36:43,580 --> 00:36:48,060
+و ل و بف أصغر سواء mi في ال b-a اللي وضحته هناك
+
+416
+00:36:48,060 --> 00:36:54,640
+أوضح من اللي مكتوب هناك واضح أه نيجي لنظرية اللي
+
+417
+00:36:54,640 --> 00:37:00,630
+عندى هذهTheorem بتقول لت I بساوة A وB and لت F من
+
+418
+00:37:00,630 --> 00:37:05,490
+I لR بيه bounded function وفهمنا ليش احنا بنقول
+
+419
+00:37:05,490 --> 00:37:08,410
+bounded then the lower integral L of F and the
+
+420
+00:37:08,410 --> 00:37:12,090
+upper integral U of F on I exist moreover ال L of
+
+421
+00:37:12,090 --> 00:37:15,330
+F أشمالها أصغر ساوية U of F طبعا ال existence سبب
+
+422
+00:37:15,330 --> 00:37:18,650
+ال existence كله اللي هو انه ال function bounded
+
+423
+00:37:18,650 --> 00:37:21,110
+مثلا bounded كل اللي هنا الحسابات اللي على أساس ال
+
+424
+00:37:21,110 --> 00:37:25,350
+supernormal information موجودةلأن اللي قال الـ U
+
+425
+00:37:25,350 --> 00:37:29,170
+of F أصغر ساوي مين؟ U بقى، U of F كلام سهل، شوف
+
+426
+00:37:29,170 --> 00:37:32,210
+إيش بدنا ناخد بتاخد بي واحد و بي اتنين بي أي two
+
+427
+00:37:32,210 --> 00:37:36,210
+partitions of I، ماشي؟ لأن then by لمّا اللي حكينا
+
+428
+00:37:36,210 --> 00:37:40,150
+قبل شوية عنها، اللي قال بي واحد و أف أصغر ساوي U
+
+429
+00:37:40,150 --> 00:37:44,350
+بي اتنين of F، ماشي الحال؟ لأي two partitions،
+
+430
+00:37:44,350 --> 00:37:48,270
+ثبتلي واحدة منهم الآن، خلّي اللي هو بي اتنين of is
+
+431
+00:37:48,270 --> 00:37:52,560
+fixedماشي صارت هذه .. هذه القيمة صارت .. بنحكي عن
+
+432
+00:37:52,560 --> 00:37:56,980
+عدد محدد ماشي بـ 2 arbitrarily صح لكن الان انا بدي
+
+433
+00:37:56,980 --> 00:38:01,020
+ثابته وهذه المتغيرة صار عند الان ال U B 2 و F
+
+434
+00:38:01,020 --> 00:38:07,940
+عبارة عن upper bound of the set of all اللي هي مين
+
+435
+00:38:07,940 --> 00:38:13,790
+ال B I و F على أي partitionطيب هذا upper bound
+
+436
+00:38:13,790 --> 00:38:19,270
+فبنقله في قفل هذه اللي هو ال minimum الهن ال
+
+437
+00:38:19,270 --> 00:38:22,470
+supremum الهن مظبوط اللي هو ال least upper bound
+
+438
+00:38:22,470 --> 00:38:26,350
+إذا ال least upper bound أكيد أصغر أو شومين ال
+
+439
+00:38:26,350 --> 00:38:32,340
+upper bound كمان مرةطيب، الآن therefore U B 2 و F
+
+440
+00:38:32,340 --> 00:38:37,820
+is an upper bound of هذه الست ماشي؟ و هذه الست ال
+
+441
+00:38:37,820 --> 00:38:42,500
+supremum لها إيش اسمه؟ L of F إذا أكيد اللي هو ال
+
+442
+00:38:42,500 --> 00:38:45,220
+supremum اللي هو ال least upper bound اللي هو ال L
+
+443
+00:38:45,220 --> 00:38:51,620
+of F أصغر أو يساوي واحد من ال upper bounds، مظبوط؟
+
+444
+00:38:51,620 --> 00:38:58,990
+أي سؤال؟واضح ها؟ الان خلصها من هذه الان similarly
+
+445
+00:38:58,990 --> 00:39:04,390
+since B2 was arbitrarily partition of I صارت الان
+
+446
+00:39:04,390 --> 00:39:10,230
+عند L of F أصغر أو يساوي UB2 of F يعني صارت هذه
+
+447
+00:39:10,230 --> 00:39:17,530
+عدد أصغر أو يساوي ال UB2 of F for any partition B2
+
+448
+00:39:18,500 --> 00:39:22,560
+الان صارت هذه عبارة عن lower bound لها لهذه ال
+
+449
+00:39:22,560 --> 00:39:26,200
+collection lower bound لها اذا هذه lower bound
+
+450
+00:39:26,200 --> 00:39:30,160
+اصغر او يساوي ال greatest lower bound مين ال
+
+451
+00:39:30,160 --> 00:39:35,200
+greatest lower bound لهدول اللي هو ال L اللي هو ال
+
+452
+00:39:35,200 --> 00:39:45,700
+U of F اذا صار عند ال L of F اصغر او يساوي U of F
+
+453
+00:39:45,700 --> 00:39:48,360
+اي سؤال؟
+
+454
+00:39:51,810 --> 00:39:56,530
+واضحة؟ ان في سؤال كان في ريال واحد زي هيك بنحل على
+
+455
+00:39:56,530 --> 00:39:59,690
+هيك اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+456
+00:39:59,690 --> 00:40:00,510
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+457
+00:40:00,510 --> 00:40:01,130
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+458
+00:40:01,130 --> 00:40:01,850
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+459
+00:40:01,850 --> 00:40:01,850
+اربع اتنين اربع اتنين اربع اتنين اربع اتنين اربع
+
+460
+00:40:01,850 --> 00:40:06,550
+اتنين اربع اتنين اربع اتنين اربع اتنين اربع اتنين
+
+461
+00:40:06,550 --> 00:40:13,540
+اربع اتنين اربع اعدد هذا هو اللي هو اللي اللي اللي
+
+462
+00:40:13,540 --> 00:40:17,340
+اللي المتغير صار عند العدد هذا أصغر أو شاوي كل
+
+463
+00:40:17,340 --> 00:40:22,080
+اللي هو أعضاء العائلة اللي بتحكي عنها إذا صار الآن
+
+464
+00:40:22,080 --> 00:40:27,580
+هذا الالوفاف عبارة عن lower bound ofthe collection
+
+465
+00:40:27,580 --> 00:40:32,600
+of all U, P, T و F عالميا على ال partitions هذه
+
+466
+00:40:32,600 --> 00:40:36,260
+مادام هذا صار عبارة عن lower bound إذا هذا lower
+
+467
+00:40:36,260 --> 00:40:40,580
+bound أصغر أو يساوي ال greatest lower bound اللي
+
+468
+00:40:40,580 --> 00:40:43,860
+هو اللي بنسميه ال infimum إذا صارت ال L و F أصغر
+
+469
+00:40:43,860 --> 00:40:47,000
+أو يساوي ال infimum لهدول ال infimum لهدول مين هو
+
+470
+00:40:47,000 --> 00:40:52,940
+ال U و Fالـ U of F هو الـ infimum لكل الـ U, B, T,
+
+471
+00:40:53,280 --> 00:40:58,040
+2 و F such that B2 element in the set of all
+
+472
+00:40:58,040 --> 00:41:07,220
+partitions نيجي الآن لبعده الـ Definition الان ..
+
+473
+00:41:07,220 --> 00:41:16,080
+الان احنا زغرنا ال other sums و كبرنا ال other
+
+474
+00:41:16,080 --> 00:41:22,680
+sumsيوم مالتجهن مع بعض بنسمي اللي هو الالتقاء هذا
+
+475
+00:41:22,680 --> 00:41:26,320
+أو الرقم اللي التقن فيه هو قيمة الـ Remain
+
+476
+00:41:26,320 --> 00:41:30,800
+Integral للفункشن F على الفترة المذكورة وهذا هو
+
+477
+00:41:30,800 --> 00:41:35,040
+التعريف let I بسوء A و B and let F من I لعند R be
+
+478
+00:41:35,040 --> 00:41:39,060
+a bounded function then F is said to be Remain
+
+479
+00:41:39,060 --> 00:41:46,480
+Integrable on I if L of Fاللي هو عبارة عن الـ
+
+480
+00:41:46,480 --> 00:41:54,140
+Supremum للـ Lower Bounds للـ Lower Sums بساوي ال
+
+481
+00:41:54,140 --> 00:42:00,320
+Infimum لل Upper Sums يوم ما يتساون هذولة بنسمي
+
+482
+00:42:00,320 --> 00:42:03,780
+الـ Function is Remain Integrable وفي هذه الحالة
+
+483
+00:42:03,780 --> 00:42:06,860
+بيكون القيمة المشتركة هذه هي اللي بنسميها الـ
+
+484
+00:42:06,860 --> 00:42:13,740
+Integration من A ل B F of X D of Xدكتور ان الملحنة
+
+485
+00:42:13,740 --> 00:42:17,420
+احنا بدنا نجيب مساحة كذا كذا احنا ال .. المساحة
+
+486
+00:42:17,420 --> 00:42:21,480
+إذا كان فوق الآن إذا كان تحت بيكون سالب المساحة
+
+487
+00:42:21,480 --> 00:42:24,680
+إذا كان مخلوط بيصير .. إذا كان فوق و تحت معناته
+
+488
+00:42:24,680 --> 00:42:28,180
+بيصير جزء مساحة و جزء سالب مساحة عشان نجد مساحته
+
+489
+00:42:28,180 --> 00:42:30,980
+بالظبط تاخد ال absolute بتقطع لذاك و تاخد ال
+
+490
+00:42:30,980 --> 00:42:34,340
+absolute value اللي بيطلع اللي هو المساحة عشان
+
+491
+00:42:34,340 --> 00:42:39,580
+نجيبوا مساحته و نجسم الملحنة هذا لمستطيلة صح؟
+
+492
+00:42:41,500 --> 00:42:46,400
+كمستطيل و بيجيبوا مجموع مستطلاق بدين المنحنى بديك
+
+493
+00:42:46,400 --> 00:42:51,760
+مساحة تحت المنحنى لما تاخد ال upper sums و ال
+
+494
+00:42:51,760 --> 00:42:55,260
+lower sums و تاخد ال supremum و تاخد ال infimum و
+
+495
+00:42:55,260 --> 00:42:59,080
+يكون ال supremum بساوي ال infimum بتلح هو المساحة
+
+496
+00:42:59,080 --> 00:43:04,990
+تحت المنحنى في حالة ال positive functionsفي حالة
+
+497
+00:43:04,990 --> 00:43:08,710
+الـ negative المخلوق بيصير integration بيصير قيمة
+
+498
+00:43:08,710 --> 00:43:12,170
+integration قيمة ال integration لا تعبر عن المساحة
+
+499
+00:43:12,170 --> 00:43:16,670
+اللي هو بالضبط إلا في حالة ال other إلا في حالة ال
+
+500
+00:43:16,670 --> 00:43:19,970
+positive ال function في حالة ال negative بتعبر على
+
+501
+00:43:19,970 --> 00:43:24,170
+سلب المساحة في حالة إشي فوج و إشي تحت بيصير اللي
+
+502
+00:43:24,170 --> 00:43:26,950
+هو .. لأ المساحات بت cancel بعضها و بيصير قيمة
+
+503
+00:43:26,950 --> 00:43:30,930
+integrationبيدك المساحات من وسط ال integration
+
+504
+00:43:30,930 --> 00:43:33,110
+انها تستخدم ال integration في إيجاد المساحة
+
+505
+00:43:33,110 --> 00:43:37,570
+بتجزئها لإيش إلى أجزاء وين بتقطع محور السينات يعني
+
+506
+00:43:37,570 --> 00:43:40,550
+بتحدد وين موجبة و وين سالبة بتاخد ال absolute
+
+507
+00:43:40,550 --> 00:43:43,410
+value حتى لو ماكنشت يعرف مين الموجبة ومين السالبة
+
+508
+00:43:43,410 --> 00:43:46,570
+وبيطلع اللي هو عن طريق ال integration بتوجد قيمة
+
+509
+00:43:46,570 --> 00:43:52,290
+مين المساحة لكن في حالة الدالة موجبة على كل الفترة
+
+510
+00:43:52,290 --> 00:43:59,410
+ال integration هو بالظبط المساحة طيبصلى على النبي
+
+511
+00:43:59,410 --> 00:44:01,990
+عليه الصلاة والسلام عرفنا الـ Romantic Grability
+
+512
+00:44:01,990 --> 00:44:06,470
+وقلنا اللي هو .. اللي هو الاشياء المشتركة اللي
+
+513
+00:44:06,470 --> 00:44:11,290
+حصلنا عليها الآن بنعرف برضه كتعريف من A ل B F of X
+
+514
+00:44:11,290 --> 00:44:15,930
+DX بساوي ناقص من B ل A F of X DX وبنعرف كتعريف أن
+
+515
+00:44:15,930 --> 00:44:19,830
+ال integration من A ل A F of X DX H بساوي سفر اطلع
+
+516
+00:44:19,830 --> 00:44:25,120
+فوق خليني أخد مثال وننهي بهذا المثالالآن .. و هذا
+
+517
+00:44:25,120 --> 00:44:30,080
+المثال احنا .. يعني .. كنا ناخد .. نفس المثال اللي
+
+518
+00:44:30,080 --> 00:44:33,760
+ناخده في ال calculus خلّينا نحكي عنه بالتفصيل الان
+
+519
+00:44:33,760 --> 00:44:37,480
+example show that g of x بساوة x is integrable on
+
+520
+00:44:37,480 --> 00:44:41,700
+0 و 1 بدنا نثبت أنه الدالة اللي هي g of x اللي
+
+521
+00:44:41,700 --> 00:44:45,220
+بنعرفها اللي هي المثال البسيط بساوة x عبارة عن
+
+522
+00:44:45,220 --> 00:44:52,130
+integrable عالميا على اللي هو الفترة 0 و 1الان انا
+
+523
+00:44:52,130 --> 00:44:56,270
+لتسهيل عملية الحسابات هاخد عائلة من ال partitions
+
+524
+00:44:56,270 --> 00:45:03,050
+معينة هذه العائلة بتسهيل الحسابات طب بنفع في ضوء
+
+525
+00:45:03,050 --> 00:45:07,910
+العلاقات اللي هنشوفها كيف توصلنا بالظبط ان ال L of
+
+526
+00:45:07,910 --> 00:45:12,830
+F بساوة ال U of F و لخصوصية الحسابات ده نشوف كيف
+
+527
+00:45:12,830 --> 00:45:21,760
+عندى ال F of X بساوة Xعنديها بديها من 0 لعند مين؟
+
+528
+00:45:21,760 --> 00:45:28,580
+لعند 1 هذه y بساوة x على الفترة 0 و 1 ايش بدنا
+
+529
+00:45:28,580 --> 00:45:34,580
+نسوي؟ بدنا نيجي اللي هو ان نعمل partition بين هذا
+
+530
+00:45:34,580 --> 00:45:39,540
+ال partition بديه يسهل الإحسابات فأسهل شيء بدي
+
+531
+00:45:39,540 --> 00:45:44,640
+أجزء هذه الفترة من 0 ل1 لN من الأجزاء و بديها تكون
+
+532
+00:45:44,640 --> 00:45:49,430
+كلها متساويةبنفع، أه، هينفع، هتشوف كيف اللي هو هذا
+
+533
+00:45:49,430 --> 00:45:52,650
+نفع في ضوء العلاقات بين اللي هو الـ Supremum و
+
+534
+00:45:52,650 --> 00:45:57,050
+الانفموع المجموعة الأكبر و المجموعة الأصغر و أيضًا
+
+535
+00:45:57,050 --> 00:46:01,130
+في ضوء ال limits المعينة اللي هان اللي هتيجي تخلي
+
+536
+00:46:01,130 --> 00:46:05,010
+اللي هو علاقة توصّل إلى اللي بدنيها، تفهم علش؟
+
+537
+00:46:05,010 --> 00:46:10,610
+ماقولهم كده، هذا صار عندي، هذا Zeroأما ده بدأت
+
+538
+00:46:10,610 --> 00:46:13,510
+تجزئها طولها واحد تجزئها الان من الفتراء sub
+
+539
+00:46:13,510 --> 00:46:16,130
+intervals إذا كل واحدة واحدة على الان هذا التاني
+
+540
+00:46:16,130 --> 00:46:20,130
+اتنين على الان التالتة تلاتة على الان تلاتة على
+
+541
+00:46:20,130 --> 00:46:24,770
+الان لما نصل للفترة النموذجية xk اللي هي تلقى k
+
+542
+00:46:24,770 --> 00:46:30,610
+minus واحد على ان وk على ان هذا اللي هي xk minus
+
+543
+00:46:30,610 --> 00:46:36,670
+واحد وهذا ال xk وهذا xنوت و اد اكس واحد الى اخرى
+
+544
+00:46:36,670 --> 00:46:39,790
+لان ما اصل الاخر واحدة اللى هى n على n اللى هى
+
+545
+00:46:39,790 --> 00:46:45,150
+واحدة ماشي الحالة هذا اللى عندى اللى .. لو اجيت
+
+546
+00:46:45,150 --> 00:46:48,190
+حسبت الان مدى احسب هذا ال partitions اللى صار عندى
+
+547
+00:46:48,190 --> 00:46:53,930
+عبارة عن zero و الواحدة الان و اتنين على n لما اصل
+
+548
+00:46:53,930 --> 00:46:58,490
+لعند ال n على n اللى هى ايش الواحدة هذه عائلة ال
+
+549
+00:46:58,490 --> 00:47:04,620
+partitions اللى اختارتها ماشي شوفوا عليها ال nلو
+
+550
+00:47:04,620 --> 00:47:09,460
+جيت حسبت اللي هو طول الـ interval واحدة لان لو بده
+
+551
+00:47:09,460 --> 00:47:16,960
+أحسب الآن L لل partition هذا اللي هو مين هو L ال F
+
+552
+00:47:16,960 --> 00:47:23,840
+عندي و ال P N هتساوي عبارة عن ال summation عارفين
+
+553
+00:47:23,840 --> 00:47:29,780
+انتوا ال summation اللي هو M small K في X K ناقص X
+
+554
+00:47:29,780 --> 00:47:36,290
+K minus واحد K من عند واحد عند N ويساويالان ال MK
+
+555
+00:47:36,290 --> 00:47:41,890
+في الفترة الأولى ايش هيكون عنده؟ صفر في الفترة
+
+556
+00:47:41,890 --> 00:47:47,430
+التانية ايش هيكون؟ واحدة على الانكريزين جدا اللي
+
+557
+00:47:47,430 --> 00:47:50,150
+بعدها اتنين على انواع و هكذا
+
+558
+00:47:52,920 --> 00:47:57,520
+الـ Mk in general إيش هيصير عنده الـ Mk وإيش
+
+559
+00:47:57,520 --> 00:48:03,020
+المساحة هي K minus واحد على مين على هذا اللي هو
+
+560
+00:48:03,020 --> 00:48:08,280
+طول تبع مين الـ Mk واضحة؟ لأن الـ infimum على
+
+561
+00:48:08,280 --> 00:48:12,760
+الفترة هذه النموذجية اللي هو ال .. ال .. ال
+
+562
+00:48:12,760 --> 00:48:17,920
+infimum عليها اللي هو أول واحدة لإنها increasing
+
+563
+00:48:17,920 --> 00:48:22,630
+واضحة تصورفلان هذه ثابتة أنا اختارتها أصلا ثابتة
+
+564
+00:48:22,630 --> 00:48:26,430
+طول الفترة ثابتة اللي هو 1 على n بيصير هذا إيش n
+
+565
+00:48:26,430 --> 00:48:32,670
+تربيع كامل عند واحد لعند مين لعند n واضح أه ويساوي
+
+566
+00:48:32,670 --> 00:48:38,110
+1 على n تربيع فيه ال summation هذا اللي هو عبارة
+
+567
+00:48:38,110 --> 00:48:42,830
+عن بيبدأ من 0 زائد 1 ولمّا يصل لعند مين لعند n
+
+568
+00:48:42,830 --> 00:48:49,760
+ناقص 1هذا بعرف ما هو مجموعه ويساوي 1 الان تربيع في
+
+569
+00:48:49,760 --> 00:48:54,760
+هذا ان لما نكون من واحد لعند ان ان فان زاد واحد
+
+570
+00:48:54,760 --> 00:48:58,440
+الان على اتنين الان هذا من وين واحد لعند ان ناقص
+
+571
+00:48:58,440 --> 00:49:03,540
+واحد فان ناقص واحد لكل عالمين يا جماعة على اتنين
+
+572
+00:49:03,540 --> 00:49:09,580
+ويساوي الان اعملي هذا بيطلع نص عندي ويساوي نص في
+
+573
+00:49:09,580 --> 00:49:15,150
+اللي هو ان تربيع ناقص من انعلى أن تربيعه بيصير
+
+574
+00:49:15,150 --> 00:49:22,650
+واحد ناقص واحدة لان صح ولا لأ طيب similarly for me
+
+575
+00:49:22,650 --> 00:49:29,430
+لأن لما بدي أخد اللي قاله و أفله اه ال .. ال ..
+
+576
+00:49:29,430 --> 00:49:36,930
+بعدين شوف عليهم الآن القبر للأف و بي أن بنفس
+
+577
+00:49:36,930 --> 00:49:44,250
+السلوب بيصير عندي عبارة عن summation mkفي xk-xk-1
+
+578
+00:49:44,250 --> 00:49:49,630
+k من عند 1 لعند n الان ال mk ال m واحد هيه لان
+
+579
+00:49:49,630 --> 00:49:54,010
+المتنين هيه إذا ال mk هي عبارة عن مين؟ k على n
+
+580
+00:49:54,010 --> 00:49:59,070
+لأنه increasing زي ما قلنا ويساوي صماشen k على n
+
+581
+00:49:59,070 --> 00:50:04,800
+في 1 على nK من عند واحد لعند ان ويساوي summation
+
+582
+00:50:04,800 --> 00:50:07,640
+واحدة لان تربية في summation لK من واحد لان لان
+
+583
+00:50:07,640 --> 00:50:12,800
+اللي ان فان زائد واحد الان هو الكل مكسوب على اتنين
+
+584
+00:50:12,800 --> 00:50:17,400
+ويساوي نص في اللي هو بطلع عندي هذا عبارة عن واحد
+
+585
+00:50:17,400 --> 00:50:23,920
+زائد واحدة لان، مظبوط؟ طيب إذا انصار عندي الآن
+
+586
+00:50:23,920 --> 00:50:28,410
+حصلت على التاليزي ما شوفنا حسابيا جميعا من هنا
+
+587
+00:50:28,410 --> 00:50:34,530
+اطلع لفوق صار عندى الان LPNG او F عبارة عن نصف
+
+588
+00:50:34,530 --> 00:50:40,230
+واحد نقص واحدة الان UPNJ ايش بيساوي نصف واحد زائد
+
+589
+00:50:40,230 --> 00:50:48,330
+واحد على مين على الان واضح الان الان اه الان الان
+
+590
+00:50:48,330 --> 00:50:53,100
+الان انت بي .. عندى ال collection هذهالـ
+
+591
+00:50:53,100 --> 00:50:57,280
+collection هذه partitions B1 is a partition اللي
+
+592
+00:50:57,280 --> 00:51:03,740
+هو عبارة عن Zero و واحد B2 Zero نص و واحد B3 Zero
+
+593
+00:51:03,740 --> 00:51:09,160
+و تلت و تلتين و واحد و هكذا إذا هذه عائلة من ال
+
+594
+00:51:09,160 --> 00:51:14,060
+partitionsجزء من مين؟ من الـ partitions العامة
+
+595
+00:51:14,060 --> 00:51:17,100
+يعني ال .. ال part .. هي the set of all partitions
+
+596
+00:51:17,100 --> 00:51:22,220
+هذه أكيد جزئية من the set of all partitions لمين؟
+
+597
+00:51:22,220 --> 00:51:25,980
+لل I اللي عنده اللي هي الفترة من مين؟ من 01 إذا
+
+598
+00:51:25,980 --> 00:51:31,500
+هذه مجموعة جزئية من هذه، مظبوط؟ طيب، الآن لو أجينا
+
+599
+00:51:31,500 --> 00:51:35,820
+أخدنا اللي هو ال set of partitions هذه أكيد ال
+
+600
+00:51:35,820 --> 00:51:39,580
+supreme على هذهالان ال .. ال .. ال .. ال .. ال ..
+
+601
+00:51:39,580 --> 00:51:44,820
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+602
+00:51:44,820 --> 00:51:45,160
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+603
+00:51:45,160 --> 00:51:48,400
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+604
+00:51:48,400 --> 00:51:48,420
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+605
+00:51:48,420 --> 00:51:50,120
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+606
+00:51:50,120 --> 00:51:50,220
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+607
+00:51:50,220 --> 00:51:50,340
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+608
+00:51:50,340 --> 00:51:59,480
+.. ال .. ال .. ال .. ال .. ال ..
+
+609
+00:51:59,480 --> 00:52:02,920
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+610
+00:52:02,920 --> 00:52:02,920
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+611
+00:52:02,920 --> 00:52:02,920
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+612
+00:52:02,920 --> 00:52:02,920
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+613
+00:52:02,920 --> 00:52:09,060
+ال .. ال .. ال .. ال .. ال .. ال .. ال ..such that
+
+614
+00:52:09,060 --> 00:52:14,120
+N element in N هذه اللي هي main ال collection of
+
+615
+00:52:14,120 --> 00:52:19,260
+all partitions ال B1 وB2 وB3 عند BN لو جينا أخدنا
+
+616
+00:52:19,260 --> 00:52:24,740
+ال supremumلكل القيم هذه هيكون أصغر أو يساوي الـ
+
+617
+00:52:24,740 --> 00:52:32,320
+Supremum لمين لل L of F و B such that B element in
+
+618
+00:52:32,320 --> 00:52:36,620
+B and B of I لأن collection of all partitions هذه
+
+619
+00:52:36,620 --> 00:52:41,940
+اللي هو تحتوى ال partitions هدولة واضح الحمد الله
+
+620
+00:52:41,940 --> 00:52:45,500
+الان ال Supremum على القيم هذه اللي هي مين القيم
+
+621
+00:52:45,500 --> 00:52:51,230
+كانت عندهاللي هو عبارة عن نص في واحد ناقص ميل واحد
+
+622
+00:52:51,230 --> 00:52:55,590
+علان ال supremum لهذه عبارة عن zero إذن بتظل هي
+
+623
+00:52:55,590 --> 00:53:00,590
+إيش بتساوي نص ال supremum للنص في واحد ناقص واحد
+
+624
+00:53:00,590 --> 00:53:05,070
+علان معايا؟
+
+625
+00:53:05,070 --> 00:53:16,470
+أي سؤال؟ طيب إذا هذا مين هو؟ هذا عبارة عن الأو ما
+
+626
+00:53:16,470 --> 00:53:21,170
+فاهم ولا لأ إذا صار عند ال L و ال F مثبتها هذه ال
+
+627
+00:53:21,170 --> 00:53:29,390
+L و ال F هيصير أكبر أو يساوي النص نحك طيب
+
+628
+00:53:29,390 --> 00:53:35,650
+similarly for me and for you الآن عندي حصلت على ال
+
+629
+00:53:35,650 --> 00:53:41,970
+U, F و B أن لهذه العائلة الان الان the set of all
+
+630
+00:53:41,970 --> 00:53:50,880
+U, F و B أنsuch that N element in N هذه العائلة
+
+631
+00:53:50,880 --> 00:53:53,900
+مجموعة جزئية من العائلة الكبيرة الـ B of I الـ
+
+632
+00:53:53,900 --> 00:54:00,940
+partitions للـ I إذن هذه الـ infimum لها هيصير
+
+633
+00:54:00,940 --> 00:54:10,580
+أكبر أو يساوي الـ infimum لـ U, F و B such that B
+
+634
+00:54:10,580 --> 00:54:18,550
+element in B of Iالـ collection هذه أصغر من الـ
+
+635
+00:54:18,550 --> 00:54:21,730
+collection هذه، بظبط؟ إذا صار الـ infimum على هذه
+
+636
+00:54:21,730 --> 00:54:25,610
+أكبر و ساوي الـ infimum على هذه الـ infimum لهذه
+
+637
+00:54:25,610 --> 00:54:27,470
+الـ collection و اللي هو الـ infimum في الواقع
+
+638
+00:54:27,470 --> 00:54:31,110
+لمين؟ الـ infimum لهذه الـ infimum لهذه ليش
+
+639
+00:54:31,110 --> 00:54:35,910
+هيساوي؟ نص فبصير عندي نص أكبر و ساوي الـ infimum
+
+640
+00:54:35,910 --> 00:54:38,430
+هذا اللي هو أكبر و ساوي هذا اللي هو هذا اللي هو
+
+641
+00:54:38,430 --> 00:54:44,010
+مين؟ الـ U of F إذا حصلنا على كمان حاجة إن الـ U
+
+642
+00:54:44,010 --> 00:54:52,790
+of Fأصغر أو يساوي نص اللخص المعلومتين هدولة ونحصل
+
+643
+00:54:52,790 --> 00:54:57,810
+على اللي بدناه من المعلومتين من المعلومتين اللي
+
+644
+00:54:57,810 --> 00:55:06,860
+صار عندى اللي هو L of Fاللي هو اكبر او يساوي نص
+
+645
+00:55:06,860 --> 00:55:10,920
+وانا بعرف ان دائما ال الوف ايش ماله اصغر او يساوي
+
+646
+00:55:10,920 --> 00:55:15,180
+ال يوف اف و ال يوف اف ايش حصلته اصغر او يساوي من
+
+647
+00:55:15,180 --> 00:55:20,360
+نص اذا هذا اللي خلى اللي هو ال partition اللي عندى
+
+648
+00:55:20,360 --> 00:55:23,540
+اللي سهل اللي اخترته او ال set of partitions انها
+
+649
+00:55:23,540 --> 00:55:28,580
+توصلني للنتيجة اللي بديها بيصير عندى ال الوف اصغر
+
+650
+00:55:28,580 --> 00:55:33,220
+او يساوي ال يوف اف و هذا بين النص التنتين اذا الوف
+
+651
+00:55:33,220 --> 00:55:40,650
+افبساوية وف اف بساوية نص وهذا معناه ان ال function
+
+652
+00:55:40,650 --> 00:55:45,770
+is reversible and integrable and ال integration من
+
+653
+00:55:45,770 --> 00:55:52,650
+zero ل واحد لل x dx بساوية نص و الله يعطيك العافية
+
+654
+00:55:52,650 --> 00:55:54,590
+و المرة القادمة نكملها
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU.srt
new file mode 100644
index 0000000000000000000000000000000000000000..238a299ca65184b0999312bef290a8a75ee88807
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU.srt
@@ -0,0 +1,2009 @@
+1
+00:00:05,650 --> 00:00:13,670
+بسم الله الرحمن الرحيم سنبدأ اليوم في مساق تحليل
+
+2
+00:00:13,670 --> 00:00:22,140
+رياضي 2 أو تحليل حقيقي 2، متطلب تخصص لكلية العلوم
+
+3
+00:00:22,140 --> 00:00:29,800
+لقسم الرياضيات، نبدأ الآن بالـ course outline أو خطة 
+
+4
+00:00:29,800 --> 00:00:36,840
+المساق، لمساق real analysis two، math four three one
+
+5
+00:00:36,840 --> 00:00:44,350
+zero، الـ course outline كما هو أمامكم الآن، الـ text
+
+6
+00:00:44,350 --> 00:00:48,090
+book هو نفس اللي هو الـ text book اللي أخدناه في 
+
+7
+00:00:48,090 --> 00:00:52,650
+real analysis واحد، يعني انتداد لـ اللي هي المادة
+
+8
+00:00:52,650 --> 00:00:55,750
+اللي موجودة في real analysis واحد، الكتاب طبعًا
+
+9
+00:00:55,750 --> 00:01:02,090
+introduction to real analysis لـ bartel and sherbert
+
+10
+00:01:02,590 --> 00:01:11,230
+الـ course outline اللي هيتمثل في الوصف الآن للمادة
+
+11
+00:01:11,230 --> 00:01:17,630
+ومن ثم طريقة التقييم وبعض الـ references اللي ممكن
+
+12
+00:01:17,630 --> 00:01:18,750
+نستعين بها
+
+13
+00:01:21,640 --> 00:01:27,800
+في البداية هنبدأ، نهينا لعند نهاية تقريبًا أو جزء من
+
+14
+00:01:27,800 --> 00:01:31,620
+chapter 5، المفروض في real analysis 1، الآن هنبدأ في 
+
+15
+00:01:31,620 --> 00:01:34,880
+chapter 6، chapter 6 اللي هو بيبدأ من عند الـ
+
+16
+00:01:34,880 --> 00:01:40,980
+derivatives، الـ derivative أو الاشتقاق، هنلاقي نلاحظ 
+
+17
+00:01:40,980 --> 00:01:46,100
+أنه Real Analysis 2 بتمثل يعني أو بيميزه إنه معظم
+
+18
+00:01:46,100 --> 00:01:51,060
+المعلومات العامة، الأسماء مرت علينا سواء في الـ
+
+19
+00:01:51,060 --> 00:01:55,900
+calculus أو اللي هو مرت في بعض اللي هي حتى في الـ
+
+20
+00:01:55,900 --> 00:02:03,930
+complex أو بعض المساقات، فالعناوين مألوفة، ولكن طبعًا
+
+21
+00:02:03,930 --> 00:02:09,810
+سنتحدث في أثناء الفصل عن اللي هي المعلومات أو
+
+22
+00:02:09,810 --> 00:02:16,590
+الأمور التحليلية وخصوصا براهين النظريات وإعطاء
+
+23
+00:02:16,590 --> 00:02:20,370
+الأمثلة اللي كنا قبل هيك في الـ calculus ما نبرهنهاش
+
+24
+00:02:20,370 --> 00:02:27,280
+فالآن هنبرهن هذه اللي هي النظريات، عنا في الـ 
+
+25
+00:02:27,280 --> 00:02:29,780
+Chapter 6، هنحكي عن الـ Derivative، هنحكي عن الـ
+
+26
+00:02:29,780 --> 00:02:36,160
+Mean Value Theorem، هنحكي عن الـ .. Lobitals Rule، عن
+
+27
+00:02:36,160 --> 00:02:42,960
+Taylor's Theorem، هنحكي عن اللي هو في الـ Chapter
+
+28
+00:02:42,960 --> 00:02:47,000
+السابع، هي كما نقول خلصنا الـ Chapter السادس، و هنبدأ 
+
+29
+00:02:47,000 --> 00:02:49,040
+في الـ Chapter السابع اللي هو الـ Remain
+
+30
+00:02:49,040 --> 00:02:54,520
+Integrability أو اللي هو قابلية التكامل with
+
+31
+00:02:54,520 --> 00:02:59,200
+respect to remand بالنسبة لـ remand، و هنتحدث أيضًا
+
+32
+00:02:59,200 --> 00:03:03,600
+على بعض خواص أو properties of the remand integral
+
+33
+00:03:03,600 --> 00:03:07,100
+متوقع
+
+34
+00:03:07,100 --> 00:03:13,960
+أن هذه الموضوعات، أن نخلص منها خلال شهر و نص إن شاء
+
+35
+00:03:13,960 --> 00:03:18,700
+الله أو أقل شوية، هيكون عنا بكون أوصلنا لـ اللي هو الـ
+
+36
+00:03:18,700 --> 00:03:23,130
+first exam، الحناخد الـ first exam، إذا في chapter 6
+
+37
+00:03:23,130 --> 00:03:28,230
+وجزء من chapter 7، الآن الجزء الثاني من المادة
+
+38
+00:03:28,230 --> 00:03:32,450
+هنكمل chapter 7، هنحكي عن the fundamental theorem
+
+39
+00:03:32,450 --> 00:03:37,730
+of calculus على الشقين، وحنحكي عن the integral as a
+
+40
+00:03:37,730 --> 00:03:41,750
+limit، الوصول للـ integration عن طريق الـ limit، في
+
+41
+00:03:41,750 --> 00:03:44,750
+الأول هنحكي عن وصول للـ integration أو مفهوم الـ
+
+42
+00:03:44,750 --> 00:03:47,930
+integration عن طريق اللي هو الـ upper sum و الـ
+
+43
+00:03:47,930 --> 00:03:51,890
+lower sum أو عن طريق اللي هي الـ supremum لـ الـ
+
+44
+00:03:51,890 --> 00:03:55,810
+upper sum و الـ infimum أو الـ supremum لـ الـ least
+
+45
+00:03:55,810 --> 00:04:00,110
+upper bound و الـ greatest lower bound، الجزء الثاني
+
+46
+00:04:00,110 --> 00:04:04,250
+اللي هنتحدث فيه اللي هو .. بعيد قاعد الجزء .. 
+
+47
+00:04:04,250 --> 00:04:06,870
+الجزء الثاني اللي هنتحدث فيه اللي هو the
+
+48
+00:04:06,870 --> 00:04:10,970
+fundamental theorem of calculus، و the integral as a
+
+49
+00:04:10,970 --> 00:04:14,650
+limit، وقلنا اللي هو الوصول للـ integration عن طريق
+
+50
+00:04:14,650 --> 00:04:20,930
+الـ limit للـ riman sum أو عن طريق اللي هو الـ upper
+
+51
+00:04:20,930 --> 00:04:26,510
+sum والـ lower sum، وحنشوف إيش اللي هو كيف حنصل لـ
+
+52
+00:04:26,510 --> 00:04:31,250
+اللي هو الـ integration من الطريقين إلى نفس اللي هو
+
+53
+00:04:31,250 --> 00:04:34,890
+الـ riman integral، يعني هيكون they are equivalent
+
+54
+00:04:35,870 --> 00:04:42,330
+الجزء الآخر من chapter 7، هنحكي عن approximate
+
+55
+00:04:42,330 --> 00:04:48,570
+integration، الآن احنا خدنا في تفاوض الـ B اللي هو 
+
+56
+00:04:48,570 --> 00:04:53,070
+الـ techniques of integrations، لأن أحيانا بعض
+
+57
+00:04:53,070 --> 00:04:57,790
+التكاملات بتكون non elementary، فبضطر إن أنا مش
+
+58
+00:04:57,790 --> 00:05:03,510
+قادر أوجد لها exact اللي هو solution أو exact
+
+59
+00:05:03,510 --> 00:05:09,110
+value، فبضطر إن أروح باتجاه الـ approximate of
+
+60
+00:05:09,110 --> 00:05:14,610
+integration أو approximation للـ integration، بكون هيك
+
+61
+00:05:14,610 --> 00:05:21,290
+احنا نهينا chapter 7، وحنبدأ في chapter 8 اللي هو
+
+62
+00:05:21,290 --> 00:05:24,110
+هناخد فيه sectionين، الـ section الأول اللي هو
+
+63
+00:05:24,110 --> 00:05:28,790
+pointwise، pointwise and uniform convergence
+
+64
+00:05:29,990 --> 00:05:33,150
+ومتوقعين طبعًا الأسبوع ياخد منها، وثاني الـ
+
+65
+00:05:33,150 --> 00:05:37,670
+interchange of limits، يعني هناخد نوعين من الـ
+
+66
+00:05:37,670 --> 00:05:39,490
+convergence، هنا حاجة اسمها الـ pointwise
+
+67
+00:05:39,490 --> 00:05:42,830
+convergence، وحاجة اسمها الـ uniform convergence
+
+68
+00:05:42,830 --> 00:05:49,150
+وحنقوله في حينه إيش معناه اللي هو كل نوع وما
+
+69
+00:05:49,150 --> 00:05:55,680
+علاقته بالنوع الآخر، وأيضًا ما تأثير الـ Pointwise
+
+70
+00:05:55,680 --> 00:06:00,620
+أو الـ Uniform Convergence على إدخال اللي هو الـ
+
+71
+00:06:00,620 --> 00:06:04,800
+Limit حتى للـ Integration أو اللي هو اللي هو الـ
+
+72
+00:06:04,800 --> 00:06:06,480
+Limit للـ Integration يساوي الـ Integration للـ
+
+73
+00:06:06,480 --> 00:06:11,120
+Limit، هنعمل اللي هو discussion لهذا الأمر ونشوف
+
+74
+00:06:11,120 --> 00:06:14,840
+وين اللي هو متى بندخل الـ Limit ومتى مابنقدرش 
+
+75
+00:06:14,840 --> 00:06:23,110
+ندخلها، أو إيش بالظبط من ظروف للحصول على هذه
+
+76
+00:06:23,110 --> 00:06:26,710
+الخاصية، حيكون في حينه إن شاء الله، بكون هيك إنه
+
+77
+00:06:26,710 --> 00:06:31,770
+احنا انهينا الجزء الثاني من المادة، وحيكون فيه عندي
+
+78
+00:06:31,770 --> 00:06:36,540
+اللي هو الـ second، اللي هو our exam، بعد هيك هيجينا
+
+79
+00:06:36,540 --> 00:06:41,860
+الجزء الثالث من المادة، الجزء الثالث يمكن اللي هو 
+
+80
+00:06:41,860 --> 00:06:48,900
+الأول مرة إنه احنا بنتحدث عن هذا الجزء اللي هو في 
+
+81
+00:06:48,900 --> 00:06:55,780
+العنوانين الأول والثاني، وأنا حبيت إنه يكون الطالب 
+
+82
+00:06:56,980 --> 00:07:01,320
+في صورة إنه كل شغله، طبعًا كل شغلنا سواء فكرة الـ calculus
+
+83
+00:07:01,320 --> 00:07:04,920
+ألف أو با أو جيم أو حتى الـ complex أو حتى الـ
+
+84
+00:07:04,920 --> 00:07:07,380
+differential equations اللي بنشتغل فيها الـ
+
+85
+00:07:07,380 --> 00:07:11,440
+ordinary أو اللي هو الـ partial، وحتى في الإحصاء أو 
+
+86
+00:07:11,440 --> 00:07:17,900
+الاحتمالات، كله واضح إنه إحنا بنشتغل على R أو على C
+
+87
+00:07:17,900 --> 00:07:25,160
+واللي بنشتغل على R أو على C، بلزمه اللي هو زي ما
+
+88
+00:07:25,160 --> 00:07:28,840
+بتعرفوا topology، الآن بدي أشوف أنا هذا الـ topology
+
+89
+00:07:28,840 --> 00:07:34,840
+وين موقعه بالضبط في اللي هو الـ topological spaces
+
+90
+00:07:34,840 --> 00:07:38,520
+الـ metric spaces، الـ normed spaces، الـ Herbart
+
+91
+00:07:38,520 --> 00:07:44,520
+spaces، باختصار في عندنا احنا الشيء اللي .. اللي ..
+
+92
+00:07:44,520 --> 00:07:45,820
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+93
+00:07:45,820 --> 00:07:47,020
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+94
+00:07:47,020 --> 00:07:47,180
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+95
+00:07:47,180 --> 00:07:47,580
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+96
+00:07:47,580 --> 00:07:48,260
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+97
+00:07:48,260 --> 00:07:48,280
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+98
+00:07:48,280 --> 00:07:48,320
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+99
+00:07:48,320 --> 00:07:56,260
+اللي .. اللي .. اللي .. اللي
+
+100
+00:07:56,260 --> 00:08:00,450
+.. اللي داخل الـ topological spaces، هنحكي الـ 
+
+101
+00:08:00,450 --> 00:08:05,740
+metric spaces، المقصود أن أي metric space هيكون
+
+102
+00:08:05,740 --> 00:08:09,300
+topological space، يعني هيولد topology داخل الـ 
+
+103
+00:08:09,300 --> 00:08:13,360
+metric space، هيكون فيه عنده normed space و الـ 
+
+104
+00:08:13,360 --> 00:08:16,940
+normed space هيكون اللي هو any normed space اللي
+
+105
+00:08:16,940 --> 00:08:22,760
+هو بولد metric و الـ metric نفسه بولد topology، لأن
+
+106
+00:08:22,760 --> 00:08:25,520
+الـ normed spaces هيكون فيه عنده topology اللي هو
+
+107
+00:08:25,520 --> 00:08:28,840
+induced by this norm، وفي داخل الـ Normed Spaces
+
+108
+00:08:28,840 --> 00:08:32,940
+حيكون فيه عند الـ Hilbert Spaces، طبعًا لو بين
+
+109
+00:08:32,940 --> 00:08:37,920
+الجهتين فيه اللي هو issue topological مش metric و
+
+110
+00:08:37,920 --> 00:08:40,780
+issue metric مش normed و issue normed مش Hilbert
+
+111
+00:08:41,200 --> 00:08:46,560
+وإلى آخره، ومن ثم هنقعد نشوف حالنا وين احنا قاعدين
+
+112
+00:08:46,560 --> 00:08:49,200
+مع الـ R اللي هنشتغل عليها، اللي بنشتغل عليها كل 
+
+113
+00:08:49,200 --> 00:08:53,720
+شغلنا هنا على الـ R اللي هو الـ real numbers، ومعاه
+
+114
+00:08:53,720 --> 00:08:56,180
+فيه topology اللي هو اللي كنتوا تسموه في الـ
+
+115
+00:08:56,180 --> 00:08:59,440
+topology اللي بيسميه usual topology، الـ usual
+
+116
+00:08:59,440 --> 00:09:02,590
+topology، الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+117
+00:09:02,590 --> 00:09:02,850
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+118
+00:09:02,850 --> 00:09:03,490
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+119
+00:09:03,490 --> 00:09:03,730
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+120
+00:09:03,730 --> 00:09:05,910
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+121
+00:09:05,910 --> 00:09:12,950
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+122
+00:09:12,950 --> 00:09:15,070
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+123
+00:09:15,070 --> 00:09:15,490
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+124
+00:09:15,490 --> 00:09:15,650
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+125
+00:09:15,650 --> 00:09:19,710
+الـ الـ الـ
+
+126
+00:09:19,710 --> 00:09:24,760
+الـ الـ الـ، اللي هو و normed space و طبيعي metrical
+
+127
+00:09:24,760 --> 00:09:28,500
+space و topological space، فبكون عرفنا موقعنا
+
+128
+00:09:28,500 --> 00:09:32,940
+بالظبط، و هنجيل الكلام هذا إن شاء الله بالتفصيل، أي
+
+129
+00:09:32,940 --> 00:09:40,530
+سؤال؟ أحكي اه لسه جينا ..الآن احنا بعد ما نكون
+
+130
+00:09:40,530 --> 00:09:45,710
+خلصنا اللي هي الجزء هذا، طبعًا الجزء هذا و الجزء هذا
+
+131
+00:09:45,710 --> 00:09:52,510
+هيكون في عندي two references هتروح على اللي هو الـ
+
+132
+00:09:52,510 --> 00:09:55,470
+reference two و three اللي هنحكي عنها من بعد شوية
+
+133
+00:09:56,110 --> 00:10:01,270
+الآن هنحكي في النهاية عن الـ infinite series، بما
+
+134
+00:10:01,270 --> 00:10:05,890
+نستطيع من خلال وقت هذا الـ course، هذه تقريبًا اللي
+
+135
+00:10:05,890 --> 00:10:13,590
+هو وصف المساق أو وصف ما نريد أن ندرس في هذا المساق
+
+136
+00:10:13,590 --> 00:10:18,210
+الـ normed spaces و الـ topological spaces و الـ 
+
+137
+00:10:18,210 --> 00:10:21,030
+metric spaces و الـ Hilbert spaces، فيه two
+
+138
+00:10:21,030 --> 00:10:24,150
+references، و في عندي slides كاملة اللي موجودة
+
+139
+00:10:24,150 --> 00:10:28,590
+تلخيص كامل للمادة، موجودة عندي topology without
+
+140
+00:10:28,590 --> 00:10:34,570
+tiers، اللي هو الكتاب اللي احنا عادة بندرسه في اللي
+
+141
+00:10:34,570 --> 00:10:40,530
+هو علم في الجامعة، وموجود يعني، وهيكون المتطلب منه
+
+142
+00:10:40,530 --> 00:10:45,850
+لهذا الـ course، موجود slides، الآن الجزء اللي بعده
+
+143
+00:10:46,580 --> 00:10:49,200
+اللي هو introductory functional analysis with
+
+144
+00:10:49,200 --> 00:10:55,240
+applications لـ grazing، هذا الجزء هيتعلق، هناخد منه
+
+145
+00:10:55,240 --> 00:11:00,220
+اللي هو ما يتعلق بالـ normed spaces and اللي هو 
+
+146
+00:11:00,220 --> 00:11:10,330
+Hilbert spaces، الآن هذه تقريبًا اللي هو الوصف
+
+147
+00:11:10,330 --> 00:11:14,230
+بالإضافة للمراجع اللي موجودة، بالإضافة للـ textbook
+
+148
+00:11:14,230 --> 00:11:19,150
+الأصلي، وأي اللي هو كتاب real analysis برضه بيكون
+
+149
+00:11:19,150 --> 00:11:26,110
+مفيد لهذه المادة، طريقة التقييم زي ما اتعودنا هيكون
+
+150
+00:11:26,110 --> 00:11:33,260
+في عنده اللي هو three exams، two hour exams، and one
+
+151
+00:11:33,260 --> 00:11:38,020
+final exam، the two بالـ term exams اللي هنا خمسين في
+
+152
+00:11:38,020 --> 00:11:43,100
+المية، الـ final exam برضه خمسين في المية، طبعًا و
+
+153
+00:11:43,100 --> 00:11:47,260
+طريقة الحساب يعني إن شاء الله بنتفق عليها غالبًا
+
+154
+00:11:47,260 --> 00:11:50,940
+هناخد أحسن واحد من الامتحانين ونحسب عليه تلاتين، و
+
+155
+00:11:50,940 --> 00:11:55,340
+الثاني هنحسب عليه عشرين، و الـ final بكون خمسين، أي
+
+156
+00:11:55,340 --> 00:12:01,090
+سؤال أو أي تعليق، نبدأ الآن بـ Chapter 6 اللي هو
+
+157
+00:12:01,090 --> 00:12:05,150
+بداية المادة، هنحكي عن الـ differentiation أو
+
+158
+00:12:05,150 --> 00:12:12,010
+الاشتقاق، طبعًا الاشتقاق مر علينا ومفهومه مر علينا
+
+159
+00:12:12,010 --> 00:12:17,910
+نذكركم بمعنى الاشتقاق، الآن احنا هناخد اللي هو الـ
+
+160
+00:12:17,910 --> 00:12:23,650
+function F، نفترض F عبارة عن دالة من I لعند R، حيث I
+
+161
+00:12:23,650 --> 00:12:27,530
+is an interval، اللي بنحكي عنها، I is an interval
+
+162
+00:12:28,790 --> 00:12:38,650
+وبناخد C element in I، بنقول أن الـ F prime عند C
+
+163
+00:12:38,650 --> 00:12:45,330
+exist أو بمعنى آخر المشتقة الأولى للدالة F موجودة
+
+164
+00:12:45,330 --> 00:12:51,020
+إذا كان F prime of C اللي بنسميه F prime of C، إذا
+
+165
+00:12:51,020 --> 00:12:57,220
+كان ما يلي موجود، اللي هو limit f of x ناقص f of c
+
+166
+00:12:57,220 --> 00:13:02,880
+على x minus c لما x تروح للـ c، exist، إذا كان exist
+
+167
+00:13:02,880 --> 00:13:08,420
+وبساوي قيمة سمناها مثلًا الـ، بنسمي الـ هذه هي الـ
+
+168
+00:13:08,420 --> 00:13:12,740
+derivative لمين؟ للـ F، و بنرمز لها للـ F prime of إيش
+
+169
+00:13:12,740 --> 00:13:17,080
+of c، طبعًا الـ X هذه أكيد الـ X من وين هنكون ماخدينها
+
+170
+00:13:17,080 --> 00:13:24,060
+برضه من I، و الـ X أشمالهالا تساوي C، لأن X تذهب أو
+
+171
+00:13:24,060 --> 00:13:29,920
+تقول إلى C وليس تساوي C، إذن ممكن نعبر عن هذا
+
+172
+00:13:29,920 --> 00:13:40,520
+الكلام بطريقة أخرى، نقول F has a derivative at
+
+173
+00:13:40,520 --> 00:13:48,820
+C element in I، if for every epsilon أكبر من 0، there
+
+174
+00:13:48,820 --> 00:13:53,040
+exists delta أكبر من سفر، has a derivative مثلًا
+
+175
+00:13:53,040 --> 00:13:58,220
+نسميها L، has a derivative L at C limited on I، if
+
+176
+00:13:58,220 --> 00:14:00,440
+for every Y أكبر من سفر، there exists delta أكبر من
+
+177
+00:14
+
+201
+00:17:00,530 --> 00:17:05,350
+نذكركم بس، شو معنى الـ Dialogue هتكون متصلة عن الـ
+
+202
+00:17:05,350 --> 00:17:15,130
+C؟ بيكون دي limit F of X لما X تروح للـ C بيساوي F 
+
+203
+00:17:15,130 --> 00:17:18,790
+of C إذا أثبتنا هذا الكلام، بنكون خلصنا و أثبتنا 
+
+204
+00:17:18,790 --> 00:17:27,950
+أن الدالة إيش مالها؟ متصلة إذن أول شيء، since F is
+
+205
+00:17:27,950 --> 00:17:34,220
+differentiable at c .. حسب التعريف اللي جابناه قبل 
+
+206
+00:17:34,220 --> 00:17:43,040
+قليل limit f of x ناقص f of c على x minus c لما x 
+
+207
+00:17:43,040 --> 00:17:49,490
+تروح للـ c اللي هو existence بيساوي f prime of c
+
+208
+00:17:49,490 --> 00:17:54,450
+exist لأنه احنا مفترضين انها إيش مالها موجودة طبعا 
+
+209
+00:17:54,450 --> 00:17:58,850
+الـ X احنا بنقول الـ إيش مالها لا تساوي الـ C و الـ X 
+
+210
+00:17:58,850 --> 00:18:05,290
+موجودة وين في الـ interval I طيب، الآن بدنا نثبت أن 
+
+211
+00:18:05,290 --> 00:18:13,050
+limit f of x يساوي f of c شوف، سهل، عند f of x- f
+
+212
+00:18:13,050 --> 00:18:17,170
+of c على x minus c أكيد هيستخدمها و لا ليش كانت 
+
+213
+00:18:17,170 --> 00:18:20,030
+المعطيلة الـ function is differentiable الآن لو
+
+214
+00:18:20,030 --> 00:18:29,230
+ضربناها الآن for x element in I و x لا تساوي c f
+
+215
+00:18:29,230 --> 00:18:31,950
+of x نقص f of c على x minus c لو ضربناها في x 
+
+216
+00:18:31,950 --> 00:18:37,550
+minus c هيصير عندي F of X minus F of C طب لو ضفنا
+
+217
+00:18:37,550 --> 00:18:42,510
+لها الـ F of C ثابت الـ F of C هذا المقدار كمقدار
+
+218
+00:18:42,510 --> 00:18:48,150
+إيش هيساوي؟ الـ F of X الآن أنا ضامن إن الـ limit
+
+219
+00:18:48,150 --> 00:18:52,700
+هنا موجودة لأن إيش مالها F براها، دي عبارة عن F براهم
+
+220
+00:18:52,700 --> 00:18:56,220
+of C which exists وهدي ال limit موجودة، بولنوميا 
+
+221
+00:18:56,220 --> 00:19:00,200
+العادية و X بتروح للـC موجودة و ال F of C ثابت، 
+
+222
+00:19:00,200 --> 00:19:03,540
+إذن كل ال limit موجودة للجهتين، إذا بقدر أوزع ال
+
+223
+00:19:03,540 --> 00:19:09,940
+limit على الجهتين، بيصير عندي، إذا limit F of X
+
+224
+00:19:09,940 --> 00:19:18,510
+لما X تروح للـC بيساوي limit F of X نقص F of C على X 
+
+225
+00:19:18,510 --> 00:19:24,790
+minus C لما X تروح للـ C في limit X minus C لما X
+
+226
+00:19:24,790 --> 00:19:30,830
+تروح للـ C زائد F of C لأنه ثابت ال limit هو نفسه 
+
+227
+00:19:30,830 --> 00:19:37,930
+ويساوي الآن يا جماعة عندي هذا ال limit هو عبارة عن 
+
+228
+00:19:37,930 --> 00:19:42,310
+مين عبارة عن عدد مضمون أنه موجود اسمه F prime of C
+
+229
+00:19:43,220 --> 00:19:47,120
+في مضروب هذا ال limit لما X تروح لـ C هذا المقدار
+
+230
+00:19:47,120 --> 00:19:54,000
+عشان بتكون سفر بولنوميا سهلة Linear زائد F of C في 
+
+231
+00:19:54,000 --> 00:19:57,680
+الآخر إيش حيساوي إذا صارت limit F of X لما X تروح
+
+232
+00:19:57,680 --> 00:20:03,720
+لـ C بيساوي F of C وهذا يعني hence أو therefore F 
+
+233
+00:20:03,720 --> 00:20:11,740
+is continuous at C أي 
+
+234
+00:20:11,740 --> 00:20:12,240
+سؤال
+
+235
+00:20:19,280 --> 00:20:23,920
+طبعا الـ .. برضه اللي هنحكي فيه كنا على طول في 
+
+236
+00:20:23,920 --> 00:20:26,220
+الـ calculation نقول إذا كانت F is differentiable or 
+
+237
+00:20:26,220 --> 00:20:30,140
+continuous كنا نسأل طب لو كانت continuous ليس شرط 
+
+238
+00:20:30,140 --> 00:20:34,060
+أن تكون differentiable والمثال المشهود والمعروف 
+
+239
+00:20:34,060 --> 00:20:37,140
+اللي هو عبارة عن الـ corners اللي هي عندنا هنا
+
+240
+00:20:37,140 --> 00:20:41,440
+عبارة عن القيمة المطلقة الـ absolute value عند الـ 
+
+241
+00:20:41,440 --> 00:20:44,960
+corner الـ function is continuous لكن مش 
+
+242
+00:20:44,960 --> 00:20:46,320
+differentiable
+
+243
+00:20:56,540 --> 00:21:03,380
+الآن الـ converse of
+
+244
+00:21:03,380 --> 00:21:18,220
+the above theorem need not be true in general for
+
+245
+00:21:18,220 --> 00:21:27,090
+example f of x سوى الـ absolute value للـ x is
+
+246
+00:21:27,090 --> 00:21:33,230
+continuous everywhere طبعا مضمن إن الـ صفر is 
+
+247
+00:21:33,230 --> 00:21:43,270
+continuous at zero but it is not differentiable at
+
+248
+00:21:43,270 --> 00:21:47,530
+zero طبعا إن أنتم عارفين اسمها عبارة عن corner 
+
+249
+00:21:47,530 --> 00:21:51,330
+point الـ corner point هنا is not differentiable 
+
+250
+00:21:51,330 --> 00:21:54,950
+لأنه ميل المماس من اليمين وميل المماس من اليسار
+
+251
+00:21:54,950 --> 00:22:03,370
+حتى جهة المختلفات طيب نشوف هذا الكلام نبرهنه عندي
+
+252
+00:22:03,370 --> 00:22:11,280
+الـ continuity سهلة limit f of x لما X تروح للصفر
+
+253
+00:22:11,280 --> 00:22:15,120
+بيساوى limit لـ absolute value X لما X تروح للصفر
+
+254
+00:22:15,120 --> 00:22:19,780
+هو يساوى إيه شمالها؟ Zero واضح إن ال limit لهذه أو 
+
+255
+00:22:19,780 --> 00:22:23,060
+limit لهذه كلها وين بروح وين بصبن لما X تروح للـC
+
+256
+00:22:23,060 --> 00:22:27,820
+من اليمين أو من اليسار بصبن وين؟ عند الصفر الآن
+
+257
+00:22:27,820 --> 00:22:32,720
+بدنا نشوف كيف إنها is not differentiable at zero 
+
+258
+00:22:32,720 --> 00:22:41,320
+طيب خد الآن نحسب لـ limit f of x نقص f of 0 على x 
+
+259
+00:22:41,320 --> 00:22:45,000
+minus 0 لما x تروح للصفر بدا أثبت لك إن هذا الـ 
+
+260
+00:22:45,000 --> 00:22:48,200
+limit does not exist مذام does not exist إذا الـ 
+
+261
+00:22:48,200 --> 00:22:54,480
+function مالها is not differentiable at 0 بوي ساوى
+
+262
+00:22:54,480 --> 00:22:58,920
+الآن limit الـ f of x اللي هو absolute value لـ x
+
+263
+00:22:58,920 --> 00:23:05,980
+f of 0، 0 على الـ x نقص صفر لما x تروح للصفر y ساول
+
+264
+00:23:05,980 --> 00:23:10,640
+limit absolute value of x على الـ x لما x تروح
+
+265
+00:23:10,640 --> 00:23:16,380
+للصفر الآل عندي عشان تكون ال limit هذه exist لأن
+
+266
+00:23:16,380 --> 00:23:19,180
+في absolute value فوق فوق فمخوفاني يعني لذا 
+
+267
+00:23:19,180 --> 00:23:21,980
+بتروحلها من اليمين و بيروحلها من وين؟ من اليسار
+
+268
+00:23:21,980 --> 00:23:29,850
+الآل لما أقول limit absolute value ل X على X لما X 
+
+269
+00:23:29,850 --> 00:23:35,730
+تروح للصفر من وين من اليمين يعني قيم X موجبة مدام
+
+270
+00:23:35,730 --> 00:23:41,490
+موجبة إذا قيمة ال absolute value هي X وهنا على X 
+
+271
+00:23:41,490 --> 00:23:46,570
+لما X تروح للصفر من اليمين ويساوي جداش واحد but
+
+272
+00:23:46,570 --> 00:23:56,900
+but limit limit ال absolute value ل X على x لما x 
+
+273
+00:23:56,900 --> 00:24:00,000
+تروح للصفر من وين؟ من اليسار قيمة الـ absolute 
+
+274
+00:24:00,000 --> 00:24:04,940
+value لل x للقيم السالبة سلبها مظبوط و يساوى limit
+
+275
+00:24:04,940 --> 00:24:10,540
+سالب x على x لما x تروح للصفر من اليسار و يساوي
+
+276
+00:24:10,540 --> 00:24:14,760
+سالب و أكيد الواحد مش هو السالب واحد إذا ال limit 
+
+277
+00:24:14,760 --> 00:24:17,780
+من اليمين مستويش ال limit من اليسار إذا صار عندي 
+
+278
+00:24:17,780 --> 00:24:22,080
+هذه ال function هذه ال limit إجمالها does not 
+
+279
+00:24:22,080 --> 00:24:27,580
+exist إذا صارت عندي الـ function f of x بساعته الـ 
+
+280
+00:24:27,580 --> 00:24:33,840
+absolute value of x is continuous is continuous at 
+
+281
+00:24:33,840 --> 00:24:40,700
+zero but is not differentiable at zero أي سؤال 
+
+282
+00:24:40,700 --> 00:24:51,450
+الآن لو كررنا أمورنا لو قلنا ناخد دالة مثال ثاني لو
+
+283
+00:24:51,450 --> 00:24:54,890
+جينا أخدنا نقول لأ خد ال absolute value ل X minus 
+
+284
+00:24:54,890 --> 00:24:55,610
+C1
+
+285
+00:24:59,510 --> 00:25:03,070
+هذا واضح إنه عند الـ c1 مالها is not 
+
+286
+00:25:03,070 --> 00:25:07,070
+differentiable لكن continuous لو ضفنا لها كمان
+
+287
+00:25:07,070 --> 00:25:12,470
+absolute value x minus c2 برضه هتكون عند الـ c2 is
+
+288
+00:25:12,470 --> 00:25:16,610
+not differentiable بقى continuous لو زدنا كمان 
+
+289
+00:25:16,610 --> 00:25:21,790
+بنفس الأسلوب لعند ما أصل ل x minus cn اللي هي هذه 
+
+290
+00:25:21,790 --> 00:25:27,530
+الـ function صارت continuous everywhere but it is
+
+291
+00:25:27,530 --> 00:25:33,370
+not differentiable عند c1 و c2 و c3 لعند مين لعند 
+
+292
+00:25:33,370 --> 00:25:40,150
+cn وطبعا وبرهانها سهل بنفس الأسلوب هنا عند c1 عند
+
+293
+00:25:40,150 --> 00:25:44,730
+c2 عند c3 عند cn بنفس الأسلوب اللي قبل بشوية الآن
+
+294
+00:25:44,730 --> 00:25:53,590
+مش هيك في دالة أغرب من هيك دالة بتقولك أنه أنا 
+
+295
+00:25:54,690 --> 00:26:01,650
+عبارة عن continuous function وين ما كان لكن is not
+
+296
+00:26:01,650 --> 00:26:06,450
+differentiable anywhere يعني تخيل إنه .. طبعا
+
+297
+00:26:06,450 --> 00:26:09,050
+تخيلها يمكن صعبة تخيل إنه هذا عبارة عن كلها 
+
+298
+00:26:09,050 --> 00:26:13,950
+corners دالة كلها corners فش منطقة أو فش نقطة 
+
+299
+00:26:13,950 --> 00:26:18,070
+بتكون عندها الدالة إيش مالها is differentiable
+
+300
+00:26:18,070 --> 00:26:21,550
+هذه الدالة اللي هي f of x
+
+301
+00:26:24,880 --> 00:26:33,180
+بتساوي summation 1 على 2 أُس N في cosine تلاتة أُس 
+
+302
+00:26:33,180 --> 00:26:38,800
+N X N من صفر إلى مالة نهاية هذه للمعرفة بس لأنه 
+
+303
+00:26:38,800 --> 00:26:43,300
+مرهانها خارج النطاق اللي بنحكي فيه F of X بيساوى 
+
+304
+00:26:43,300 --> 00:26:48,180
+summation 1 على 2 أُس N cosine تلاتة أُس N في X 
+
+305
+00:26:48,180 --> 00:26:52,660
+هذه is continuous everywhere but is not 
+
+306
+00:26:52,660 --> 00:26:54,420
+differentiable anywhere
+
+307
+00:26:59,710 --> 00:27:04,890
+هذه علاقة الإتصال بالاشتقاق 
+
+308
+00:27:04,890 --> 00:27:09,610
+الـ relation between continuity and 
+
+309
+00:27:09,610 --> 00:27:15,710
+differentiability one implies the other but the
+
+310
+00:27:15,710 --> 00:27:21,110
+other need not implies the first I mean by the
+
+311
+00:27:21,110 --> 00:27:25,890
+first differentiability and the other is
+
+312
+00:27:25,890 --> 00:27:30,750
+continuity طيب صلى على النبي عليه الصلاة والسلام 
+
+313
+00:27:30,750 --> 00:27:36,370
+يمكن الـ .. الـ .. الـ .. في البداية كان نحكي عن .. 
+
+314
+00:27:36,370 --> 00:27:41,870
+بس ملاحظة ضايقة بسيطة يعني اللي بنحكيها في الـ 
+
+315
+00:27:41,870 --> 00:27:45,710
+calculus دائما بس هتذكركم فيها اللي هي المعنى 
+
+316
+00:27:45,710 --> 00:27:50,890
+الهندسي لل differentiability الآن لما نقول إن
+
+317
+00:27:50,890 --> 00:27:59,250
+الدالة في إلها مشتقة عند الـ C الآن لو جيت أخدت .. 
+
+318
+00:27:59,250 --> 00:28:03,150
+روحت للـ C من اليمين أو من اليسار و بدي أجيلها من 
+
+319
+00:28:03,150 --> 00:28:07,990
+اليمين هذه اللي هي النقطة X نقطة X عاملة بتروح على 
+
+320
+00:28:07,990 --> 00:28:14,530
+اللي هي الـ C هذا اللي هو القاطع اللي بين الـ X 
+
+321
+00:28:14,530 --> 00:28:20,550
+والـ C هذه من hand ل hand هذه المنطقة عبارة عن F 
+
+322
+00:28:20,550 --> 00:28:26,660
+of X ناقص F of C صح ولا لا؟ وهذه من هنا لها عبارة
+
+323
+00:28:26,660 --> 00:28:32,900
+عن x minus c ميل القاطع هذا اللي هو عبارة عن f of
+
+324
+00:28:32,900 --> 00:28:40,990
+x ناقص f of c على x minus c هذا ميل القاطع الـ 
+
+325
+00:28:40,990 --> 00:28:45,590
+limit له نهاية ميل القاطع، نهاية ميل القاطع لما X
+
+326
+00:28:45,590 --> 00:28:49,870
+تروح للـ C هذا القاطع لما X تروح للـ C بتجرب، بتجرب
+
+327
+00:28:49,870 --> 00:28:54,810
+لما نسير في النهاية عبارة عن إيش؟ عن مماس، بيصير
+
+328
+00:28:54,810 --> 00:28:59,870
+هو عبارة عن اللي هي مين؟ F prime of C أو ميل
+
+329
+00:28:59,870 --> 00:29:06,630
+المماس عند نقطة مين؟ عند نقطة C طبعا أخدت أنا من
+
+330
+00:29:06,630 --> 00:29:10,410
+اليمين ومن اليسار بنفس الأسلوب يعني بيصير طبعا 
+
+331
+00:29:10,410 --> 00:29:14,990
+على اعتبار أنه من اليمين ومن اليسار متساوين فبيكون
+
+332
+00:29:14,990 --> 00:29:19,510
+اللي هو الـ F prime موجودة وميل القاطع نهايته في
+
+333
+00:29:19,510 --> 00:29:27,110
+نهاية بيساوى ميل المماس ع ده ده المقطع أي سؤال؟ نيجي
+
+334
+00:29:27,110 --> 00:29:32,670
+الآن للقوانين برضه قوانين احنا أخدناها في الـ 
+
+335
+00:29:32,670 --> 00:29:40,090
+calculus النظرية بتجاوب على أنه لو كان في عندي
+
+336
+00:29:40,090 --> 00:29:47,230
+دالتين مشتقته الموجودة هيكون مجموع .. مجموع ..
+
+337
+00:29:47,230 --> 00:29:53,250
+مجموع مشتقته موجودة أو حاصل ضربه المشتقته موجودة 
+
+338
+00:29:53,250 --> 00:29:56,650
+وحاصل قسمته المشتقته موجودة provided that أن
+
+339
+00:29:56,650 --> 00:30:01,390
+اللي هو اللي في المقام قيمة الـ الـ G عنده نقطة لا 
+
+340
+00:30:01,390 --> 00:30:11,470
+يساوى 0 وأيضًا الثابت في الدالة برضه مشتقته موجودة 
+
+341
+00:30:11,470 --> 00:30:16,030
+نشوف الكلام اللي بحكيه بالتفصيل عبر النظرية اللي
+
+342
+00:30:16,030 --> 00:30:22,870
+أمامنا ونبرهن النظرية اللي عندنا let I be subset 
+
+343
+00:30:22,870 --> 00:30:26,850
+من R بيكون interval يعني ماخد I عبارة عن interval
+
+344
+00:30:26,850 --> 00:30:33,680
+وماخد C element of I ومفترض إن F من I لR وG من I لR
+
+345
+00:30:33,680 --> 00:30:38,100
+الاثنتين قابلات للاشتقاق عند نقطة مين C they are
+
+346
+00:30:38,100 --> 00:30:43,440
+differentiable at C إذن the following holds إيش ال
+
+347
+00:30:43,440 --> 00:30:46,860
+following أول واحدة Alpha في F is differentiable
+
+348
+00:30:46,860 --> 00:30:51,510
+at C يعني اللي هو الدالة الجديدة اللي هي اسمها
+
+349
+00:30:51,510 --> 00:30:56,330
+Alpha في F Alpha ثابت قابل الاشتقاق عن C وطبعا زي
+
+350
+00:30:56,330 --> 00:31:00,010
+ما أنتم عارفين الاشتقاق للAlpha في F prime of C
+
+351
+00:31:00,010 --> 00:31:07,500
+اللي هي عبارة عن Alpha في F prime of C الآن F زائد 
+
+352
+00:31:07,500 --> 00:31:09,500
+G is differentiable at C ما زالت الـ F
+
+353
+00:31:09,500 --> 00:31:11,540
+differentiable و G differentiable مجموع الـ
+
+354
+00:31:11,540 --> 00:31:14,620
+differentiable و F زائد G كل prime of C بيساوى F
+
+355
+00:31:14,620 --> 00:31:18,500
+prime of C زائد G prime of C أيضًا بالنسبة للضرب
+
+356
+00:31:18,500 --> 00:31:21,860
+القاعدة مشهورة أيضًا F في G هتكون differentiable
+
+357
+00:31:21,860 --> 00:31:26,180
+عند C و F في G prime هو عبارة عن F prime في G زائد
+
+358
+00:31:26,180 --> 00:31:31,800
+G prime في F الآن الشرط اللي بعده الـ D اللي هي 
+
+359
+00:31:31,800 --> 00:31:36,600
+بتفترض أنه G of C إيش مالها مش صفر مش مسترطلك على 
+
+360
+00:31:36,600 --> 00:31:41,200
+كل الـ X صفر G of X بيساوى صفر مش مسترطلك بس أنه
+
+361
+00:31:41,200 --> 00:31:46,480
+الـ G عند الـ C مش صفر و بقولك إذا F على G prime
+
+362
+00:31:46,480 --> 00:31:52,020
+of C هتساوي F prime في G ناقص F في G prime على 
+
+363
+00:31:52,020 --> 00:31:56,560
+اللي هو G تربيع عارفين إنه مقام تربيع؟ وفوق اللي 
+
+364
+00:31:56,560 --> 00:32:00,580
+هو مشتقة البسط في المقام نقص مشتقة المقام في البسط 
+
+365
+00:32:00,580 --> 00:32:09,100
+واضحة؟ هذه الآن اللي هي theorem 6.1.3 اللي الآن 
+
+366
+00:32:09,100 --> 00:32:18,760
+بدنا نبرهن اللي هو COD شوفوا خلينا الآن اللي .. 
+
+367
+00:32:18,760 --> 00:32:25,550
+اللي بدنا نبرهنه لو كانت F و D is differentiable At 
+
+368
+00:32:25,550 --> 00:32:32,230
+C بيعطينا أن F في G is differentiable at C وخلونا
+
+369
+00:32:32,230 --> 00:32:35,010
+نقول الـ differentiable لت اللي قلنا عنه F في G
+
+370
+00:32:35,010 --> 00:32:38,150
+الكل برايم هو عبارة عن F برايم في G زي G برايم في
+
+371
+00:32:38,150 --> 00:32:49,050
+F طيب، نشوف كيف، سميلي هذه let H بتساوي F في G عشان
+
+372
+00:32:49,050 --> 00:32:52,
+
+401
+00:35:33,510 --> 00:35:37,910
+الـ exist؟ لأن الـ limit لهذا هو عبارة عن f prime of 
+
+402
+00:35:37,910 --> 00:35:41,570
+c لأنه معطيني إياه طب الـ limit لك g of x لما x تروح 
+
+403
+00:35:41,570 --> 00:35:44,770
+للـ c؟ أه موجودة لأن الـ differential بالـيدا
+
+404
+00:35:44,770 --> 00:35:48,970
+المتصلة، الـيدا المتصلة إذا الـ limit g of x لما x
+
+405
+00:35:48,970 --> 00:35:53,790
+تروح للـ c موجود ويساوي g of c كمان طب هنا هذه ثابتة
+
+406
+00:35:53,790 --> 00:35:57,450
+بتطلع وهذه اللي هي قيمة مشتقة الـ G إذا كل أموري
+
+407
+00:35:57,450 --> 00:36:00,670
+تمام التمام يعني أنا واخد الـ limit للطرفين باخد و
+
+408
+00:36:00,670 --> 00:36:06,940
+بوزع و أنا مرتاح إذا بدي أقول limit لهذه لما X تروح
+
+409
+00:36:06,940 --> 00:36:11,280
+للـC بيساوي limit لهذه لما X تروح للـC بيساوي
+
+410
+00:36:11,280 --> 00:36:14,740
+limit لهذه لما X تروح للـC بيساوي limit لهذه لما X 
+
+411
+00:36:14,740 --> 00:36:18,380
+تروح للـC أو أنت بعد ما أثبتت هذه بيساوي هذه بتاخد 
+
+412
+00:36:18,380 --> 00:36:21,420
+limit الاتنين على جهة و بتقول limit هذه بيساوي
+
+413
+00:36:21,420 --> 00:36:29,480
+limit هذه و بتبدأ تكمل ويساوي limit
+
+414
+00:36:29,480 --> 00:36:40,590
+G of X لما X تروح للـC في limit f of x ناقص f of c
+
+415
+00:36:40,590 --> 00:36:46,810
+على x ناقص c لما x تروح للـc زائد f of c لأنها 
+
+416
+00:36:46,810 --> 00:36:53,480
+ثابتة تطلع برة الـ limit في limit g of x ناقص G of C 
+
+417
+00:36:53,480 --> 00:36:59,840
+على X ناقص C لما X تروح للـ C لأن الـ function is 
+
+418
+00:36:59,840 --> 00:37:02,520
+differentiable الـ G is a continuous فده
+
+419
+00:37:02,520 --> 00:37:06,000
+continuous إذا إيه إيش اللي هو هذا بيساوي G of C
+
+420
+00:37:06,000 --> 00:37:11,420
+وهذا إيش بتساوي التعريف اللي هو الـ F prime of C
+
+421
+00:37:11,420 --> 00:37:17,300
+زائد هذه F of C في G prime of C إذا صار عندك الـ 
+
+422
+00:37:17,300 --> 00:37:22,400
+derivative هذه اللي هي بتمثل مين؟ H prime of C
+
+423
+00:37:22,400 --> 00:37:28,640
+اللي هي مين؟ قولنا عنها FG prime of C لأنه أنا 
+
+424
+00:37:28,640 --> 00:37:33,120
+بأسمي الـ H هي مين؟ الـ FG صارت عندي الآن FG prime 
+
+425
+00:37:33,120 --> 00:37:43,440
+of C exist and equal كذا أي سؤال؟
+
+426
+00:37:43,440 --> 00:37:51,780
+طيب، الآن احنا أثبتنا C، بدنا نثبت الآن زي ما
+
+427
+00:37:51,780 --> 00:37:57,960
+أثبتنا الضرب، بدنا نثبت القسمة القسمة مشابهة، بس في
+
+428
+00:37:57,960 --> 00:38:05,480
+في البداية يعني أمر تحليلي بشرط اللي هو
+
+429
+00:38:09,520 --> 00:38:13,900
+القسمة على g of x لإكسات معينة عشان أسلك طريقي
+
+430
+00:38:13,900 --> 00:38:19,500
+للوصول إلى انه اللي هو ال f على g لكل prime exist
+
+431
+00:38:19,500 --> 00:38:24,240
+تشوف ايش اللي بقوله أول حاجة إيش اللي بدأت بتواجه 
+
+432
+00:38:24,240 --> 00:38:26,740
+معاه بدأت بتواجه أنه لو كانت ال f و ال g
+
+433
+00:38:26,740 --> 00:38:31,760
+differentiable at c و g of c بس لا تساوي صفر
+
+434
+00:38:31,760 --> 00:38:38,940
+بيعطيني اللي f على g ال prime exist وزي ما قلنا إيش 
+
+435
+00:38:38,940 --> 00:38:45,360
+كيف exist قيمته بالظبط شوفوا كيفه صلى على النبي
+
+436
+00:38:45,360 --> 00:38:50,400
+عليه الصلاة والسلام و proof أول حاجة بس خليني أعمل 
+
+437
+00:38:50,400 --> 00:38:56,340
+fresh بحيث أن أقدر أستخدم اللي بدي إياه اللي بلزمني
+
+438
+00:38:56,340 --> 00:39:07,320
+أول إشي بدي أذكركوا بس بنظرية في chapter 4 النظرية 
+
+439
+00:39:07,320 --> 00:39:11,780
+أربعة اتنين سبعة حاجة زي هيك المهم إيش بتقول بتقول
+
+440
+00:39:11,780 --> 00:39:17,680
+لو كانت الـ limit لل F of X لما X بتروح لل C أكبر
+
+441
+00:39:17,680 --> 00:39:23,360
+من صفر أكبر من صفر إذا يوجد there exists an
+
+442
+00:39:23,360 --> 00:39:30,080
+epsilon of C يعني يوجد جوار بحيث أنه اللي هو F of 
+
+443
+00:39:30,080 --> 00:39:38,110
+X بتكون أكبر من صفر لكل X في هذا الجوار أه يعني 
+
+444
+00:39:38,110 --> 00:39:42,770
+بلاقي جوار حوالي الـ C قيمة الدالة إيه شمالها 
+
+445
+00:39:42,770 --> 00:39:47,870
+تكون أكبر من 0 strictly لما كانت limit f of x لما 
+
+446
+00:39:47,870 --> 00:39:52,570
+x تروح للـ C أكبر من 0 أكيد في جوار حواليها هتكون 
+
+447
+00:39:52,570 --> 00:39:56,290
+الـ limit إيه شمالها أكبر من 0 وهذه النظرية إنبرهنة
+
+448
+00:39:56,290 --> 00:40:02,390
+لو كان أصغر من 0 نفس الشيء بلاقي جوار بحيث أن f of 
+
+449
+00:40:02,390 --> 00:40:06,410
+x أصغر من 100 من 0 في هذا الجوار واضح
+
+450
+00:40:09,850 --> 00:40:16,470
+الآن عندي الـ G differentiable عند الـ C مدام الـ 
+
+451
+00:40:16,470 --> 00:40:21,050
+G differentiable عند الـ C إذا continuous عند الـ
+
+452
+00:40:21,050 --> 00:40:30,390
+C يعني بما أن G is differentiable at C then it is
+
+453
+00:40:30,390 --> 00:40:36,990
+continuous at C صح ولا لأ؟ إيش معناه هذا الكلام؟
+
+454
+00:40:36,990 --> 00:40:42,880
+معناه limit G of X لما X تروح للـ C بيساوي G of 
+
+455
+00:40:42,880 --> 00:40:48,980
+إيش؟ of C الـ G of C معناه إيش ماله؟ هذا عدد لا
+
+456
+00:40:48,980 --> 00:40:53,400
+يساوي صفر هيك جاي اللي هو مدام لا يساوي صفر معناه
+
+457
+00:40:53,400 --> 00:40:58,960
+يا أكبر من صفر يا إيش؟ يا أصغر من صفر يعني بمعنى 
+
+458
+00:40:58,960 --> 00:41:03,440
+آخر there exist حسب النظرية هذه there exist
+
+459
+00:41:03,440 --> 00:41:09,800
+neighborhood جوار حوالي الـC سميّه J هذا جوار لإيه؟
+
+460
+00:41:09,800 --> 00:41:14,060
+حوالي الـC there exist جوار حوالي الـC طبعًا 
+
+461
+00:41:14,060 --> 00:41:17,500
+subset من مين هيكون؟ من الـinterval اللي معرفة
+
+462
+00:41:17,500 --> 00:41:23,260
+عليها مين أنا F و G صح ولا لأ؟ there exist G of C 
+
+463
+00:41:23,260 --> 00:41:29,830
+subset من I such that G of X إذا كانت هذا أكبر من 
+
+464
+00:41:29,830 --> 00:41:33,570
+صفر تكون أكبر من صفر إذا كان أصغر من صفر تكون 
+
+465
+00:41:33,570 --> 00:41:40,130
+أشماله أصغر من صفر قلنا there exist gc اللي هو 
+
+466
+00:41:40,130 --> 00:41:43,690
+عبارة عن neighborhood حوالي الـC جزء طبعًا من I
+
+467
+00:41:43,690 --> 00:41:49,070
+اللي هي الـ interval معرفة عليها G such that الـ g
+
+468
+00:41:49,070 --> 00:41:54,450
+of x أشماله لا يساوي أيضًا صفر طبعًا أنا جعت ليش 
+
+469
+00:41:54,450 --> 00:41:58,790
+أعمل تهيئك عشان أشرع لحالي بعد شوية القسم على g of
+
+470
+00:41:58,790 --> 00:42:04,110
+x لأن دالة اللي أنا بحكي عليها مين هي f على g
+
+471
+00:42:04,110 --> 00:42:09,370
+فلازم أشغل في مكان تكون الـ g مش صفر
+
+472
+00:42:12,500 --> 00:42:16,300
+when there exists J C subset من I سجدات G of X لا
+
+473
+00:42:16,300 --> 00:42:21,700
+يساوي صفر لأن هذا الجوار اللي من خلاله بده تروح لـ
+
+474
+00:42:21,700 --> 00:42:26,620
+C X اللي فيه بده تروح لـ C فهذا الجوار مضمون الآن 
+
+475
+00:42:26,620 --> 00:42:33,240
+تكون الأفعال جيمالها معرفة فيه واضح؟ طيب، الآن 
+
+476
+00:42:33,240 --> 00:42:37,000
+فرشنا الفرش هذا، بدنا نيجي لموضوع الحسابات اللي
+
+477
+00:42:37,000 --> 00:42:43,070
+هيوصلنا إليه لأ اللي بديها شوف الآن عند دالة مين 
+
+478
+00:42:43,070 --> 00:42:48,570
+هذا let اللي هو زي ما سميها Q إيش متساوي F على G 
+
+479
+00:42:48,570 --> 00:42:55,010
+بدي أثبت أنه اللي هو Q is differential اللي عند ال
+
+480
+00:42:55,010 --> 00:42:59,730
+C بنفس الطريقة اللي قبل بشوية إيش بقول هنحسب
+
+481
+00:42:59,730 --> 00:43:06,470
+الحساب اللي عند Q of X ناقص Q of C على X ناقص C 
+
+482
+00:43:07,020 --> 00:43:11,220
+ماشي الحال، هذه الـ Xات مين؟ حريص أنا أنها تكون 
+
+483
+00:43:11,220 --> 00:43:17,660
+الـ X لا تساوي C و الـ X تبعتنا وين موجودة؟ في
+
+484
+00:43:17,660 --> 00:43:21,240
+الـجوار C، أنا بسلم في الجوار، جوار عشان أروح له جوار
+
+485
+00:43:21,240 --> 00:43:25,340
+حتى مهما كان صغير، بدي أروح من خلاله على الـ C 
+
+486
+00:43:25,340 --> 00:43:29,580
+احنا اللي بتاخد limit بعد شوية الـ Q of X ناقص Q 
+
+487
+00:43:29,580 --> 00:43:35,210
+of Z على X ناقص C، احسب ليها، بيساوي اللي هو F of x
+
+488
+00:43:35,210 --> 00:43:42,050
+على g of x ناقص f of c على g of c الكل مجسوم على x
+
+489
+00:43:42,050 --> 00:43:50,030
+ناقص c ماشي وحد المقامات الآن ويساوي عبارة عن x
+
+490
+00:43:50,030 --> 00:43:58,350
+ناقص c مضروب g of x g of c وهنا بيصير f of x g of 
+
+491
+00:43:58,350 --> 00:44:06,120
+c ناقص f of c g of x مظبوط؟ بس وحدنا المقامات و 
+
+492
+00:44:06,120 --> 00:44:11,620
+يساوي الآن هذا عبارة عن زي ما عملت قبل شوية بضيف 
+
+493
+00:44:11,620 --> 00:44:17,780
+term و أطرح term اللي يخدمني الآن f of x g of c 
+
+494
+00:44:17,780 --> 00:44:23,060
+الآن بدي أطلع اللي هو g of c عن المشتركة إذا بيصير 
+
+495
+00:44:23,060 --> 00:44:33,700
+مزق ناقص g of c في f of c صح؟ زائد اللي هو g of cF
+
+496
+00:44:33,700 --> 00:44:42,520
+of C ناقص F of C في G of X على X ناقص C في G of X 
+
+497
+00:44:42,520 --> 00:44:47,560
+في G of C شيء اللي سويته بس ضفت هذا المقدار وطرحته
+
+498
+00:44:47,560 --> 00:44:52,020
+مش معنى هو لأن أنا حسبنا في علنا إنه ناخد الـ G of
+
+499
+00:44:52,020 --> 00:44:54,940
+C عامل مشترك فبتظل F of X ناقص F of C وهديك
+
+500
+00:44:54,940 --> 00:44:59,360
+هتعمل لي برضه شيء محترم زيّه سبحان الله كل شيء بتناسق
+
+501
+00:44:59,360 --> 00:45:01,000
+ويساوي
+
+502
+00:45:04,550 --> 00:45:11,890
+الآن أخد اللي هو f of x ولا g of c عامل مشترك فيه
+
+503
+00:45:11,890 --> 00:45:24,390
+عندي f of x ناقص f of c على x ناقص c وخلّي ليه؟
+
+504
+00:45:24,390 --> 00:45:29,030
+خلّيلي هدول أطلّحهم برة بالمرة اللي هي عبارة عن 
+
+505
+00:45:29,030 --> 00:45:35,000
+واحد على g of x في g of c طلعت هدولة بالمرة برة ال
+
+506
+00:45:35,000 --> 00:45:40,380
+... الـ summation برة ... برة الكثير كله زائد ... هذه
+
+507
+00:45:40,380 --> 00:45:46,420
+الآن أخدت هدول مع بعض هنا على x ناقص c هيهم لأن 
+
+508
+00:45:46,420 --> 00:45:52,410
+أخد هذا برضه على هذا زائد اللي هو f of c برة هي
+
+509
+00:45:52,410 --> 00:45:56,130
+اللي بدي أخده أخد من الـ f of c عامل مشترك في اللي
+
+510
+00:45:56,130 --> 00:46:01,110
+هو f خليني أقول ... خليني أخد سالب بالمرة بعد إذنكم 
+
+511
+00:46:01,110 --> 00:46:06,410
+أخد سالب بالمرة بعد إذنكم فبيصير f of c فبيصير هذا
+
+512
+00:46:06,410 --> 00:46:11,290
+عبارة عن g of x ناقص g of c طبعًا واضح عشان سويته
+
+513
+00:46:11,290 --> 00:46:17,530
+على x ناقص c كل هذا مضروب في مين؟ في واحد هذا g
+
+514
+00:46:17,530 --> 00:46:23,960
+of x في g of c الآن إيش هسوي يا جماعة زي ما عملت
+
+515
+00:46:23,960 --> 00:46:27,060
+قبل بشوية أنا غرض مين اللي هو الـ derivative يعني
+
+516
+00:46:27,060 --> 00:46:31,360
+غرض أخد الـ limit لهذه ماشي إذا أنصار عندي الـ limit
+
+517
+00:46:31,360 --> 00:46:35,860
+لهذه لما x تروح لل c بيساوي الـ limit لهذا المقدار
+
+518
+00:46:35,860 --> 00:46:43,540
+كله لما x تروح لمين لل c هذا بيخلص الآن عشان
+
+519
+00:46:43,540 --> 00:46:47,480
+تركزوا على الكلمتين هدول هدول خلصت منهم بظبط
+
+520
+00:46:47,480 --> 00:46:54,110
+اطلعوا لللي ضال عندي هنا الآن الـ limit هذه بقدر
+
+521
+00:46:54,110 --> 00:46:57,450
+أدخلها زي ما بدي ليش؟ أول شيء الـ limit ... طبعًا
+
+522
+00:46:57,450 --> 00:46:59,670
+لاحظوا أن الـ g of x أنا بشتغل فيه و أنا مرتاح لإنه
+
+523
+00:46:59,670 --> 00:47:02,950
+مش 0 لإنه فرقة ثبتناها وخلصنا خدنا الـ x's اللي في
+
+524
+00:47:02,950 --> 00:47:09,310
+ال ... في ال ... في الـ jc ... jc اللي فيها الـ g of x 
+
+525
+00:47:09,310 --> 00:47:15,230
+لاتساوي 0 طيب الـG الـdifferential اللي كنت نفسي عند 
+
+526
+00:47:15,230 --> 00:47:19,030
+الـC إذا الـlimit G of X بيساوي G of C إذا هذه 
+
+527
+00:47:19,030 --> 00:47:23,690
+الـlimit اللي بدخل الليمتين وبدخل الليمتين وبدخل 
+
+528
+00:47:23,690 --> 00:47:27,930
+الليمتين وهذه بتطلع لأنها ثابت بيصير عبارة عن إيش
+
+529
+00:47:27,930 --> 00:47:34,970
+هذا؟ واحد على G of C الكل تربيع، مظبوط؟ لأن في
+
+530
+00:47:34,970 --> 00:47:39,370
+اللي جوا اللي هو عبارة عن G of C مين هذا؟ في
+
+531
+00:47:39,370 --> 00:47:46,670
+الـprime of C ناقص F of C في G prime of C وهو 
+
+532
+00:47:46,670 --> 00:47:50,550
+المطلوب اللي هي G of C في F prime ناقص F of C في G 
+
+533
+00:47:50,550 --> 00:47:54,410
+prime على مين؟ على G of C الكل تربيع وبكون هيك
+
+534
+00:47:54,410 --> 00:47:58,090
+أثبتنا أنه لو كانت الـ F و الـ G الـ differential 
+
+535
+00:47:58,090 --> 00:48:02,770
+اللي عندنا نقطة C و G of C لا تساوي صفر فعندنا الـ 
+
+536
+00:48:02,770 --> 00:48:10,390
+F على G prime of C exist وقيمتها اللي أمامنا أي
+
+537
+00:48:10,390 --> 00:48:20,890
+سؤال؟ طيب عندنا آخر نقطة طولت عليكم بهمش تفاصيل 
+
+538
+00:48:20,890 --> 00:48:28,010
+التلخيص هي موجودة هان اللي
+
+539
+00:48:28,010 --> 00:48:31,030
+... اللي هي ت ... اللي خلينا نقول تعميم للجبل بشوية
+
+540
+00:48:31,030 --> 00:48:35,790
+لو كانت F1 F1 
+
+541
+00:48:37,710 --> 00:48:41,150
+و F2 عند FN are functions on an interval I subset
+
+542
+00:48:41,150 --> 00:48:44,250
+من R that are differentiable on C element in I
+
+543
+00:48:44,250 --> 00:48:48,390
+then F1 زائد F2 زائد FN برضه differentiable عند الـ C
+
+544
+00:48:48,390 --> 00:48:53,050
+و بتبع induction وبكون عند F1 زائد FN لكل prime of C
+
+545
+00:48:53,050 --> 00:48:56,190
+بيساوي F1 prime of C زائد F2 prime of C لما نصل عند
+
+546
+00:48:56,190 --> 00:49:02,180
+FN prime of C similarly بي ... برضه تعميم F1 في F2
+
+547
+00:49:02,180 --> 00:49:06,220
+في FN الـdifferentiable at C إذا حيكون حاصل الضرب 
+
+548
+00:49:06,220 --> 00:49:10,400
+برضه الـdifferentiable و F prime F1 prime of C F2
+
+549
+00:49:10,400 --> 00:49:15,440
+of C FN of C و بعدين بنثبت الأولى و بنشتق الثانية
+
+550
+00:49:15,440 --> 00:49:19,100
+و هكذا لما نصل لآخر واحدة اللي هي F1 of C F2 of C
+
+551
+00:49:19,100 --> 00:49:24,680
+لعند FN prime of C وبهيك نكون ... نهينا المحاضرة
+
+552
+00:49:24,680 --> 00:49:29,040
+الأولى وإن شاء الله المرة القادمة بنبدأ بالمحاضرة
+
+553
+00:49:29,040 --> 00:49:33,280
+الثانية وحنبدأ من عند The Chain Rule الله يعطيكوا 
+
+554
+00:49:33,280 --> 00:49:33,720
+العافية 
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..97388c4adcd4a5582788715eaa5fe22d916c161e
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU_raw.json
@@ -0,0 +1 @@
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معناته اللي هو كل نوع وما علاقته بالنوع الآخر", "tokens": [20498, 2655, 30543, 3615, 9957, 23032, 3555, 3615, 995, 16247, 35457, 45367, 7251, 47283, 3215, 9154, 11296, 4032, 12984, 7649, 1829, 2423, 39184, 30358, 295, 10406, 37495, 22653, 8032, 1863, 47283, 3215, 8717, 45367, 9957, 9154, 2423, 32181, 34105, 11331, 26108, 3660, 24525, 2304, 11296, 2423, 935, 3711, 32181, 4032, 5016, 26108, 3660, 24525, 2304, 11296, 2423, 9452, 32181, 4032, 5016, 1863, 39648, 3224, 8978, 11331, 9957, 3224, 11933, 1829, 8592, 20449, 8315, 47395, 13672, 1829, 31439, 28242, 8717, 45367, 4032, 15042, 11203, 995, 38149, 3224, 20666, 1863, 45367, 6024, 95, 34740], "avg_logprob": -0.14448924603000765, "compression_ratio": 1.6711711711711712, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 329.99, "end": 330.81, "word": "ومتوقعين", "probability": 0.868359375}, {"start": 330.81, "end": 331.13, "word": " طبعا", "probability": 0.8922119140625}, {"start": 331.13, "end": 331.51, "word": " الأسبوع", 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{"start": 945.87, "end": 946.99, "word": " قابلة", "probability": 0.9908854166666666}, {"start": 946.99, "end": 948.21, "word": " للاشتقاق", "probability": 0.8749186197916666}, {"start": 948.21, "end": 948.81, "word": " إذن", "probability": 0.8006998697916666}, {"start": 948.81, "end": 949.27, "word": " أكيد", "probability": 0.9775390625}, {"start": 949.27, "end": 950.13, "word": " إشمالها", "probability": 0.87861328125}, {"start": 950.13, "end": 950.57, "word": " هي", "probability": 0.284912109375}, {"start": 950.57, "end": 951.39, "word": " متصلة", "probability": 0.9226888020833334}, {"start": 951.39, "end": 952.43, "word": " الآن", "probability": 0.65087890625}, {"start": 952.43, "end": 953.95, "word": " هنحكي", "probability": 0.9447265625}, {"start": 953.95, "end": 954.81, "word": " بالظبط", "probability": 0.9866943359375}], "temperature": 1.0}, {"id": 43, "seek": 98575, "start": 958.23, "end": 985.75, "text": "قابلية الاشتقاق عند نقطة تعني أن الدالة متصلة عند هذه النقطة نشوف theorem if F من I لعند R is differentiable at", "tokens": [4587, 16758, 1211, 10632, 2423, 33599, 2655, 4587, 995, 4587, 43242, 8717, 47432, 3660, 37279, 22653, 14739, 32748, 6027, 3660, 44650, 36520, 3660, 43242, 29538, 28239, 47432, 3660, 8717, 8592, 38688, 20904, 498, 479, 9154, 286, 5296, 3615, 41260, 497, 307, 819, 9364, 412], "avg_logprob": -0.13671875529819064, "compression_ratio": 1.2666666666666666, "no_speech_prob": 1.5497207641601562e-06, "words": [{"start": 958.23, "end": 959.19, "word": "قابلية", "probability": 0.9381103515625}, {"start": 959.19, "end": 960.51, "word": " الاشتقاق", "probability": 0.8356119791666666}, {"start": 960.51, "end": 960.93, "word": " عند", "probability": 0.98828125}, {"start": 960.93, "end": 961.91, "word": " نقطة", "probability": 0.9861653645833334}, {"start": 961.91, "end": 963.31, "word": " تعني", "probability": 0.6754150390625}, {"start": 963.31, "end": 965.03, "word": " أن", "probability": 0.7998046875}, {"start": 965.03, "end": 965.65, 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هذه ال function هذه ال limit إجمالها does not exist", "tokens": [3615, 23942, 2031, 5296, 15042, 2031, 6055, 32887, 5016, 24976, 9381, 5172, 2288, 9154, 4032, 9957, 22807, 9154, 45595, 3794, 9640, 12174, 32640, 3660, 2423, 39184, 8236, 2158, 24976, 2031, 24976, 4587, 32640, 21136, 6027, 49401, 8608, 46152, 11296, 3714, 19913, 3555, 2407, 9566, 4032, 7251, 3794, 995, 45865, 4948, 8608, 6027, 3555, 2031, 15844, 2031, 5296, 15042, 2031, 6055, 32887, 5016, 24976, 9381, 5172, 2288, 9154, 45595, 3794, 9640, 4032, 7251, 3794, 995, 45865, 8608, 6027, 3555, 4032, 5551, 4117, 25708, 2423, 14407, 24401, 37893, 31439, 21136, 6027, 3555, 36764, 24401, 11933, 15730, 2423, 4948, 9154, 45595, 2304, 9957, 3714, 14851, 45865, 8592, 2423, 4948, 9154, 45595, 3794, 9640, 11933, 15730, 20328, 9640, 18871, 16254, 29538, 2423, 2445, 29538, 2423, 4948, 11933, 7435, 2304, 6027, 11296, 775, 406, 2514], "avg_logprob": -0.23163167574933466, "compression_ratio": 2.0330578512396693, "no_speech_prob": 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1.744047619047619, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 2409.8500000000004, "end": 2411.09, "word": "الآن", "probability": 0.69921875}, {"start": 2411.09, "end": 2412.33, "word": " عندي", "probability": 0.60321044921875}, {"start": 2412.33, "end": 2413.19, "word": " الـ", "probability": 0.9267578125}, {"start": 2413.19, "end": 2413.33, "word": " G", "probability": 0.61376953125}, {"start": 2413.33, "end": 2414.17, "word": " differentiable", "probability": 0.82666015625}, {"start": 2414.17, "end": 2414.93, "word": " عند", "probability": 0.95703125}, {"start": 2414.93, "end": 2415.15, "word": " الـ", "probability": 0.82373046875}, {"start": 2415.15, "end": 2415.35, "word": " C", "probability": 0.79638671875}, {"start": 2415.35, "end": 2416.33, "word": " مدام", "probability": 0.7965494791666666}, {"start": 2416.33, "end": 2416.47, "word": " الـ", "probability": 0.881591796875}, {"start": 2416.47, "end": 2416.57, "word": " G", "probability": 0.94384765625}, {"start": 2416.57, "end": 2417.23, "word": " differentiable", "probability": 0.901123046875}, {"start": 2417.23, "end": 2418.33, "word": " عند", "probability": 0.8173828125}, {"start": 2418.33, "end": 2418.55, "word": " الـ", "probability": 0.93310546875}, {"start": 2418.55, "end": 2418.81, "word": " C", "probability": 0.94873046875}, {"start": 2418.81, "end": 2419.59, "word": " إذا", "probability": 0.64599609375}, {"start": 2419.59, "end": 2420.29, "word": " continuous", "probability": 0.8642578125}, {"start": 2420.29, "end": 2420.79, "word": " عند", "probability": 0.97705078125}, {"start": 2420.79, "end": 2421.05, "word": " الـ", "probability": 0.93115234375}, {"start": 2421.05, "end": 2421.29, "word": " C", "probability": 0.9130859375}, {"start": 2421.29, "end": 2422.65, "word": " يعني", "probability": 0.755615234375}, {"start": 2422.65, "end": 2422.97, "word": " بما", "probability": 0.92919921875}, {"start": 2422.97, "end": 2423.19, "word": " أن", "probability": 0.6845703125}, {"start": 2423.19, "end": 2423.79, "word": " G", "probability": 0.30322265625}, {"start": 2423.79, "end": 2424.71, "word": " is", "probability": 0.91650390625}, {"start": 2424.71, "end": 2426.45, "word": " differentiable", "probability": 0.972900390625}, {"start": 2426.45, "end": 2426.97, "word": " at", "probability": 0.9599609375}, {"start": 2426.97, "end": 2427.61, "word": " C", "probability": 0.97119140625}, {"start": 2427.61, "end": 2428.49, "word": " then", "probability": 0.76025390625}, {"start": 2428.49, "end": 2429.47, "word": " it", "probability": 0.95166015625}, {"start": 2429.47, "end": 2430.39, "word": " is", "probability": 0.939453125}, {"start": 2430.39, "end": 2431.77, "word": " continuous", "probability": 0.90576171875}, {"start": 2431.77, "end": 2432.71, "word": " at", "probability": 0.962890625}, {"start": 2432.71, "end": 2434.35, "word": " C", "probability": 0.44482421875}, {"start": 2434.35, "end": 2434.69, "word": " صح", "probability": 0.714599609375}, {"start": 2434.69, "end": 2434.85, "word": " ولا", "probability": 0.908203125}, {"start": 2434.85, "end": 2435.47, "word": " لأ؟", "probability": 0.77587890625}, {"start": 2435.47, "end": 2435.65, "word": " إيش", "probability": 0.8268229166666666}, {"start": 2435.65, "end": 2436.03, "word": " معناته", "probability": 0.96044921875}, {"start": 2436.03, "end": 2436.23, "word": " هذا", "probability": 0.9560546875}, {"start": 2436.23, "end": 2436.99, "word": " الكلام؟", "probability": 0.76708984375}, {"start": 2436.99, "end": 2437.39, "word": " معناته", "probability": 0.9597981770833334}, {"start": 2437.39, "end": 2437.73, "word": " limit", "probability": 0.62255859375}], "temperature": 1.0}, {"id": 101, "seek": 246404, "start": 2438.64, "end": 2464.04, "text": "G of X لما X تروح للـ C بيساوي G of إيش؟ of C الـ G of C معناته إيش ماله؟ هذا عدد لا يساوي سفر هيك جاي اللي هو مدام لا يساوي سفر معناته يا أكبر من سفر يا إيش؟ يا أصغر من سفر يعني بمعنى آخر there exist حسب النظرية هذه there exist neighborhood", "tokens": [38, 295, 1783, 5296, 15042, 1783, 6055, 32887, 5016, 24976, 39184, 383, 4724, 1829, 3794, 995, 45865, 460, 295, 11933, 1829, 8592, 22807, 295, 383, 2423, 39184, 460, 295, 383, 20449, 8315, 47395, 11933, 1829, 8592, 3714, 6027, 3224, 22807, 23758, 6225, 3215, 3215, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 39896, 4117, 10874, 47302, 13672, 1829, 31439, 3714, 3215, 10943, 20193, 7251, 3794, 995, 45865, 8608, 5172, 2288, 20449, 8315, 47395, 35186, 5551, 4117, 26890, 9154, 8608, 5172, 2288, 35186, 11933, 1829, 8592, 22807, 35186, 5551, 9381, 17082, 2288, 9154, 8608, 5172, 2288, 37495, 22653, 4724, 2304, 3615, 1863, 7578, 19753, 34740, 456, 2514, 11331, 35457, 28239, 19913, 2288, 10632, 29538, 456, 2514, 7630], "avg_logprob": -0.16960469626972818, "compression_ratio": 1.7627906976744185, "no_speech_prob": 6.616115570068359e-06, "words": [{"start": 2438.64, "end": 2439.04, "word": "G", "probability": 0.51708984375}, {"start": 2439.04, "end": 2439.3, "word": " of", "probability": 0.38720703125}, {"start": 2439.3, "end": 2439.72, "word": " X", "probability": 0.94091796875}, {"start": 2439.72, "end": 2440.52, "word": " لما", "probability": 0.842041015625}, {"start": 2440.52, "end": 2440.76, "word": " X", "probability": 0.94921875}, {"start": 2440.76, "end": 2441.1, "word": " تروح", "probability": 0.9698893229166666}, {"start": 2441.1, "end": 2441.44, "word": " للـ", "probability": 0.6650390625}, {"start": 2441.44, "end": 2441.66, "word": " C", "probability": 0.5595703125}, {"start": 2441.66, "end": 2442.32, "word": " بيساوي", "probability": 0.809814453125}, {"start": 2442.32, "end": 2442.62, "word": " G", "probability": 0.8828125}, {"start": 2442.62, "end": 2442.88, "word": " of", "probability": 0.947265625}, {"start": 2442.88, "end": 2444.22, "word": " إيش؟", "probability": 0.7840576171875}, {"start": 2444.22, "end": 2444.56, "word": " of", "probability": 0.37890625}, {"start": 2444.56, "end": 2444.84, "word": " C", "probability": 0.9404296875}, {"start": 2444.84, "end": 2445.14, "word": " الـ", "probability": 0.58270263671875}, {"start": 2445.14, "end": 2445.28, "word": " G", "probability": 0.88916015625}, {"start": 2445.28, "end": 2445.4, "word": " of", "probability": 0.97314453125}, {"start": 2445.4, "end": 2445.52, "word": " C", "probability": 0.9794921875}, {"start": 2445.52, "end": 2445.98, "word": " معناته", "probability": 0.9710286458333334}, {"start": 2445.98, "end": 2446.2, "word": " إيش", "probability": 0.91650390625}, {"start": 2446.2, "end": 2447.12, "word": " ماله؟", "probability": 0.8641357421875}, {"start": 2447.12, "end": 2447.52, "word": " هذا", "probability": 0.30908203125}, {"start": 2447.52, "end": 2448.1, "word": " عدد", "probability": 0.9791666666666666}, {"start": 2448.1, "end": 2448.98, "word": " لا", "probability": 0.51513671875}, {"start": 2448.98, "end": 2449.54, "word": " يساوي", "probability": 0.9652099609375}, {"start": 2449.54, "end": 2449.88, "word": " سفر", "probability": 0.7704264322916666}, {"start": 2449.88, "end": 2450.16, "word": " هيك", "probability": 0.67041015625}, {"start": 2450.16, "end": 2450.36, "word": " جاي", "probability": 0.5487060546875}, {"start": 2450.36, "end": 2450.58, "word": " اللي", "probability": 0.963134765625}, {"start": 2450.58, "end": 2451.02, "word": " هو", "probability": 0.99755859375}, {"start": 2451.02, "end": 2452.14, "word": " مدام", "probability": 0.7809244791666666}, {"start": 2452.14, "end": 2452.28, "word": " لا", "probability": 0.84619140625}, {"start": 2452.28, "end": 2452.58, "word": " يساوي", "probability": 0.9749755859375}, {"start": 2452.58, "end": 2452.9, "word": " سفر", "probability": 0.9931640625}, {"start": 2452.9, "end": 2453.4, "word": " معناته", "probability": 0.9358723958333334}, {"start": 2453.4, "end": 2453.62, "word": " يا", "probability": 0.88232421875}, {"start": 2453.62, "end": 2454.02, "word": " أكبر", "probability": 0.98046875}, {"start": 2454.02, "end": 2454.24, "word": " من", "probability": 0.998046875}, {"start": 2454.24, "end": 2454.7, "word": " سفر", "probability": 0.9850260416666666}, {"start": 2454.7, "end": 2455.34, "word": " يا", "probability": 0.6435546875}, {"start": 2455.34, "end": 2456.6, "word": " إيش؟", "probability": 0.9293212890625}, {"start": 2456.6, "end": 2456.74, "word": " يا", "probability": 0.9501953125}, {"start": 2456.74, "end": 2457.14, "word": " أصغر", "probability": 0.99072265625}, {"start": 2457.14, "end": 2457.38, "word": " من", "probability": 0.998046875}, {"start": 2457.38, "end": 2457.82, "word": " سفر", "probability": 0.9920247395833334}, {"start": 2457.82, "end": 2458.48, "word": " يعني", "probability": 0.941650390625}, {"start": 2458.48, "end": 2458.96, "word": " بمعنى", "probability": 0.9931640625}, {"start": 2458.96, "end": 2459.34, "word": " آخر", "probability": 0.81298828125}, {"start": 2459.34, "end": 2459.6, "word": " there", "probability": 0.58935546875}, {"start": 2459.6, "end": 2460.36, "word": " exist", "probability": 0.73974609375}, {"start": 2460.36, "end": 2461.38, "word": " حسب", "probability": 0.956787109375}, {"start": 2461.38, "end": 2461.92, "word": " النظرية", "probability": 0.9747314453125}, {"start": 2461.92, "end": 2462.36, "word": " هذه", "probability": 0.94189453125}, {"start": 2462.36, "end": 2462.98, "word": " there", "probability": 0.86767578125}, {"start": 2462.98, "end": 2463.44, "word": " exist", "probability": 0.69091796875}, {"start": 2463.44, "end": 2464.04, "word": " neighborhood", "probability": 0.45849609375}], "temperature": 1.0}, {"id": 102, "seek": 248750, "start": 2465.12, "end": 2487.5, "text": "جوار حوالين الـC سميه J هذا جوار لإيه؟ حوالين الـC there exist جوار حوالين الـC طبعاً subset من مين هيكون؟ من الـinterval اللي معرف عليها مين أنا F و G صح ولا لأ؟ there exist G of C subset من I such that G of X", "tokens": [7435, 2407, 9640, 11331, 2407, 6027, 9957, 2423, 39184, 34, 8608, 2304, 1829, 3224, 508, 23758, 10874, 2407, 9640, 5296, 28814, 1829, 3224, 22807, 11331, 2407, 6027, 9957, 2423, 39184, 34, 456, 2514, 10874, 2407, 9640, 11331, 2407, 6027, 9957, 2423, 39184, 34, 23032, 3555, 3615, 995, 14111, 25993, 9154, 3714, 9957, 39896, 30544, 22807, 9154, 2423, 39184, 5106, 3337, 13672, 1829, 20449, 28480, 25894, 11296, 3714, 9957, 41850, 479, 4032, 460, 20328, 5016, 49429, 5296, 10721, 22807, 456, 2514, 460, 295, 383, 25993, 9154, 286, 1270, 300, 460, 295, 1783], "avg_logprob": -0.19667120051124823, "compression_ratio": 1.6774193548387097, "no_speech_prob": 6.80088996887207e-05, "words": [{"start": 2465.12, "end": 2465.98, "word": "جوار", "probability": 0.8670247395833334}, {"start": 2465.98, "end": 2466.84, "word": " حوالين", "probability": 0.9149169921875}, {"start": 2466.84, "end": 2467.14, "word": " الـC", "probability": 0.6878255208333334}, {"start": 2467.14, "end": 2467.74, "word": " سميه", "probability": 0.637451171875}, {"start": 2467.74, "end": 2467.94, "word": " J", "probability": 0.7626953125}, {"start": 2467.94, "end": 2468.84, "word": " هذا", "probability": 0.53759765625}, {"start": 2468.84, "end": 2469.34, "word": " جوار", "probability": 0.9676106770833334}, {"start": 2469.34, "end": 2469.8, "word": " لإيه؟", "probability": 0.6853515625}, {"start": 2469.8, "end": 2470.24, "word": " حوالين", "probability": 0.9031982421875}, {"start": 2470.24, "end": 2470.6, "word": " الـC", "probability": 0.96142578125}, {"start": 2470.6, "end": 2471.24, "word": " there", "probability": 0.4375}, {"start": 2471.24, "end": 2471.68, "word": " exist", "probability": 0.6904296875}, {"start": 2471.68, "end": 2472.22, "word": " جوار", "probability": 0.9580078125}, {"start": 2472.22, "end": 2472.72, "word": " حوالين", "probability": 0.9793701171875}, {"start": 2472.72, "end": 2473.24, "word": " الـC", "probability": 0.9775390625}, {"start": 2473.24, "end": 2474.06, "word": " طبعاً", "probability": 0.8455078125}, {"start": 2474.06, "end": 2474.34, "word": " subset", "probability": 0.85888671875}, {"start": 2474.34, "end": 2474.58, "word": " من", "probability": 0.9951171875}, {"start": 2474.58, "end": 2474.78, "word": " مين", "probability": 0.96240234375}, {"start": 2474.78, "end": 2476.02, "word": " هيكون؟", "probability": 0.6904296875}, {"start": 2476.02, "end": 2476.2, "word": " من", "probability": 0.9873046875}, {"start": 2476.2, "end": 2476.9, "word": " الـinterval", "probability": 0.798828125}, {"start": 2476.9, "end": 2477.08, "word": " اللي", "probability": 0.947998046875}, {"start": 2477.08, "end": 2477.5, "word": " معرف", "probability": 0.931396484375}, {"start": 2477.5, "end": 2477.9, "word": " عليها", "probability": 0.958251953125}, {"start": 2477.9, "end": 2478.14, "word": " مين", "probability": 0.9765625}, {"start": 2478.14, "end": 2478.56, "word": " أنا", "probability": 0.90869140625}, {"start": 2478.56, "end": 2479.64, "word": " F", "probability": 0.271484375}, {"start": 2479.64, "end": 2479.82, "word": " و", "probability": 0.80859375}, {"start": 2479.82, "end": 2480.08, "word": " G", "probability": 0.490478515625}, {"start": 2480.08, "end": 2480.74, "word": " صح", "probability": 0.9208984375}, {"start": 2480.74, "end": 2480.9, "word": " ولا", "probability": 0.94189453125}, {"start": 2480.9, "end": 2481.94, "word": " لأ؟", "probability": 0.8929036458333334}, {"start": 2481.94, "end": 2482.12, "word": " there", "probability": 0.8720703125}, {"start": 2482.12, "end": 2482.46, "word": " exist", "probability": 0.82177734375}, {"start": 2482.46, "end": 2482.76, "word": " G", "probability": 0.71044921875}, {"start": 2482.76, "end": 2482.96, "word": " of", "probability": 0.84130859375}, {"start": 2482.96, "end": 2483.26, "word": " C", "probability": 0.9658203125}, {"start": 2483.26, "end": 2483.74, "word": " subset", "probability": 0.94775390625}, {"start": 2483.74, "end": 2484.0, "word": " من", "probability": 0.9931640625}, {"start": 2484.0, "end": 2484.36, "word": " I", "probability": 0.98828125}, {"start": 2484.36, "end": 2485.16, "word": " such", "probability": 0.89599609375}, {"start": 2485.16, "end": 2485.74, "word": " that", "probability": 0.9599609375}, {"start": 2485.74, "end": 2486.8, "word": " G", "probability": 0.4296875}, {"start": 2486.8, "end": 2487.06, "word": " of", "probability": 0.927734375}, {"start": 2487.06, "end": 2487.5, "word": " X", "probability": 0.9931640625}], "temperature": 1.0}, {"id": 103, "seek": 250637, "start": 2488.49, "end": 2506.37, "text": "إذا كانت هذا أكبر من صفر تكون أكبر من صفر إذا كان أصغر من صفر تكون أشماله أصغر من صفر قلنا there exist gc اللي هو عبارة عن neighborhood حوالين الـC جزء طبعا من I اللي هي ال interval معرفة عليها G", "tokens": [28814, 15730, 25961, 2655, 23758, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 6055, 30544, 5551, 4117, 26890, 9154, 20328, 5172, 2288, 11933, 15730, 25961, 5551, 9381, 17082, 2288, 9154, 20328, 5172, 2288, 6055, 30544, 5551, 8592, 2304, 6027, 3224, 5551, 9381, 17082, 2288, 9154, 20328, 5172, 2288, 12174, 1211, 8315, 456, 2514, 290, 66, 13672, 1829, 31439, 6225, 3555, 9640, 3660, 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\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU_raw.srt
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index 0000000000000000000000000000000000000000..6a9bb6284fcf323c5441ef97ca7c57e3abe5ab33
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+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/rynkk8jX4rU_raw.srt
@@ -0,0 +1,2220 @@
+1
+00:00:05,650 --> 00:00:13,670
+بسم الله الرحمن الرحيم سنبدأ اليوم في مساق تحليل
+
+2
+00:00:13,670 --> 00:00:22,140
+رياضي 2 أو تحليل حقيقي 2متطلب تخصص لكلية العلوم
+
+3
+00:00:22,140 --> 00:00:29,800
+لقسم الرياضيات نبدأ الآن بال course outline أو خطة
+
+4
+00:00:29,800 --> 00:00:36,840
+المساق لمساق real analysis two math four three one
+
+5
+00:00:36,840 --> 00:00:44,350
+zero ال course outline كما هو أمامكم الآنال text
+
+6
+00:00:44,350 --> 00:00:48,090
+book هو نفس اللي هو ال text book اللي أخدناه في
+
+7
+00:00:48,090 --> 00:00:52,650
+real analysis واحد يعني انتداد لاللي هي المادة
+
+8
+00:00:52,650 --> 00:00:55,750
+اللي موجودة في real analysis واحد الكتاب طبعا
+
+9
+00:00:55,750 --> 00:01:02,090
+introduction to real analysis لbartel and sherbert
+
+10
+00:01:02,590 --> 00:01:11,230
+الcourse outline اللي هيتمثل في الوصف الآن للمادة
+
+11
+00:01:11,230 --> 00:01:17,630
+ومن ثم طريقة التقييم وبعض ال references اللي ممكن
+
+12
+00:01:17,630 --> 00:01:18,750
+نستعين بها
+
+13
+00:01:21,640 --> 00:01:27,800
+في البداية هنبدأ نهينا لعند نهاية تقريبا أو جزء من
+
+14
+00:01:27,800 --> 00:01:31,620
+chapter 5 المفروض في real analysis 1 الآن هنبدأ في
+
+15
+00:01:31,620 --> 00:01:34,880
+chapter 6 chapter 6 اللي هو بيبدأ من عند ال
+
+16
+00:01:34,880 --> 00:01:40,980
+derivatives ال derivative أو الاشتقاقحنلاقي نلاحظ
+
+17
+00:01:40,980 --> 00:01:46,100
+إنه Real Analysis 2 بتمثل يعني أو بيميزه إنه معظم
+
+18
+00:01:46,100 --> 00:01:51,060
+المعلومات العامة الأسماء مرت علينا سواء في ال
+
+19
+00:01:51,060 --> 00:01:55,900
+calculus أو اللي هو مرت في بعض اللي هي حتى في ال
+
+20
+00:01:55,900 --> 00:02:03,930
+complex أو بعض المساقات فالعناوين مألوفةولكن طبعا
+
+21
+00:02:03,930 --> 00:02:09,810
+سنتحدث في أثناء الفصل عن اللي هي المعلومات أو
+
+22
+00:02:09,810 --> 00:02:16,590
+الأمور التحليلية وخصوصا براهين النظريات وإعطاء
+
+23
+00:02:16,590 --> 00:02:20,370
+الأمثل اللي كنا قبل هيك في ال calculus ما نبرهنهاش
+
+24
+00:02:20,370 --> 00:02:27,280
+فالآن هنبرهن هذه اللي هي النظرياتعنا في الـ
+
+25
+00:02:27,280 --> 00:02:29,780
+Chapter 6 هنحكي عن الـ Derivative هنحكي عن الـ
+
+26
+00:02:29,780 --> 00:02:36,160
+Mean Value Theorem هنحكي عن ال .. Lobitals Rule عن
+
+27
+00:02:36,160 --> 00:02:42,960
+Taylor's Theorem هنحكي عن اللي هو في الـ Chapter
+
+28
+00:02:42,960 --> 00:02:47,000
+السابع هي كما نقول خلصنا ال Chapter السادس و هنبدأ
+
+29
+00:02:47,000 --> 00:02:49,040
+في ال Chapter السابع اللي هو الـ Remain
+
+30
+00:02:49,040 --> 00:02:54,520
+Integrability أو اللي هو قابلية التكاملwith
+
+31
+00:02:54,520 --> 00:02:59,200
+respect to remand بالنسبة لـ remand و هنتحدث أيضًا
+
+32
+00:02:59,200 --> 00:03:03,600
+على بعض خواص أو properties of the remand integral
+
+33
+00:03:03,600 --> 00:03:07,100
+متوقع
+
+34
+00:03:07,100 --> 00:03:13,960
+أن هذه الموضوعات أن نخلص منها خلال شهر و نص ان شاء
+
+35
+00:03:13,960 --> 00:03:18,700
+الله أو أقل شوية هيكون عنا بكون أوصلنا لاللي هو ال
+
+36
+00:03:18,700 --> 00:03:23,130
+first examالحناخد ال first exam إذا في chapter 6
+
+37
+00:03:23,130 --> 00:03:28,230
+وجزء من chapter 7 الآن الجزء الثاني من المادة
+
+38
+00:03:28,230 --> 00:03:32,450
+هنكمل chapter 7 هنحكي عن the fundamental theorem
+
+39
+00:03:32,450 --> 00:03:37,730
+of calculus على الشقين وحنحكي عن the integral as a
+
+40
+00:03:37,730 --> 00:03:41,750
+limit الوصول لل integration عن طريق ال limitفي
+
+41
+00:03:41,750 --> 00:03:44,750
+الأول هنحكي عن وصول لل integration أو مفهوم ال
+
+42
+00:03:44,750 --> 00:03:47,930
+integration عن طريق اللي هو ال upper sum و ال
+
+43
+00:03:47,930 --> 00:03:51,890
+lower sum أو عن طريق اللي هي ال supremum ل ال
+
+44
+00:03:51,890 --> 00:03:55,810
+upper sum و ال infimum أو ال supremum ل ال least
+
+45
+00:03:55,810 --> 00:04:00,110
+upper bound و ال greatest lower bound الجزء الثاني
+
+46
+00:04:00,110 --> 00:04:04,250
+اللي هنتحدث فيه اللي هو .. بعيد قاعد الجزء ..
+
+47
+00:04:04,250 --> 00:04:06,870
+الجزء الثاني اللي هنتحدث فيه اللي هو the
+
+48
+00:04:06,870 --> 00:04:10,970
+fundamental theorem of calculusو the integral as a
+
+49
+00:04:10,970 --> 00:04:14,650
+limit وقلنا اللي هو الوصول لل integration عن طريق
+
+50
+00:04:14,650 --> 00:04:20,930
+ال limit للriman sum أو عن طريق اللي هو ال upper
+
+51
+00:04:20,930 --> 00:04:26,510
+sum والlower sum وحنشوف إيش اللي هو كيف حنصل ل
+
+52
+00:04:26,510 --> 00:04:31,250
+اللي هو ال integration من الطريقين إلى نفس اللي هو
+
+53
+00:04:31,250 --> 00:04:34,890
+الriman integral يعني هيكون they are equivalent
+
+54
+00:04:35,870 --> 00:04:42,330
+الجزء الآخر من chapter 7 هنحكي عن approximate
+
+55
+00:04:42,330 --> 00:04:48,570
+integrationالآن احنا خدنا في تفاوض الـ B اللي هو
+
+56
+00:04:48,570 --> 00:04:53,070
+الـ techniques of integrations لأن أحيانا بعض
+
+57
+00:04:53,070 --> 00:04:57,790
+التكاملات بتكون non elementary فبضطر إن أنا مش
+
+58
+00:04:57,790 --> 00:05:03,510
+قادر أوجد لها exact اللي هو solution أو exact
+
+59
+00:05:03,510 --> 00:05:09,110
+value فبضطر إن أروح باتجاه ال approximate of
+
+60
+00:05:09,110 --> 00:05:14,610
+integration أو approximation للintegrationبكون هيك
+
+61
+00:05:14,610 --> 00:05:21,290
+احنا نهينا chapter 7 وحنبدأ في chapter 8 اللي هو
+
+62
+00:05:21,290 --> 00:05:24,110
+هناخد فيه sectionين ال section الأول اللي هو
+
+63
+00:05:24,110 --> 00:05:28,790
+pointwise pointwise and uniform convergence
+
+64
+00:05:29,990 --> 00:05:33,150
+ومتوقعين طبعا الأسبوع ياخد منها وثاني الـ
+
+65
+00:05:33,150 --> 00:05:37,670
+interchange of limits يعني هناخد نوعين من ال
+
+66
+00:05:37,670 --> 00:05:39,490
+convergence هنا حاجة اسمها ال pointwise
+
+67
+00:05:39,490 --> 00:05:42,830
+convergence وحاجة اسمها ال uniform convergence
+
+68
+00:05:42,830 --> 00:05:49,150
+وحنقوله في حينه إيش معناته اللي هو كل نوع وما
+
+69
+00:05:49,150 --> 00:05:55,680
+علاقته بالنوع الآخروأيضًا ما تأثير الـ Pointwise
+
+70
+00:05:55,680 --> 00:06:00,620
+أو الـ Uniform Convergence على إدخال اللي هو الـ
+
+71
+00:06:00,620 --> 00:06:04,800
+Limit حتى للـ Integration أو اللي هو اللي هو الـ
+
+72
+00:06:04,800 --> 00:06:06,480
+Limit للـ Integration يساوي الـ Integration للـ
+
+73
+00:06:06,480 --> 00:06:11,120
+Limit هنعمل اللي هو discussion لهذا الأمر ونشوف
+
+74
+00:06:11,120 --> 00:06:14,840
+وين اللي هو متى بندخل الـ Limit ومتى مابنقدرش
+
+75
+00:06:14,840 --> 00:06:23,110
+ندخلهاأو ايش بالزمنة من ظروف للحصول على هذه
+
+76
+00:06:23,110 --> 00:06:26,710
+الخاصية حيكون في حينه ان شاء الله بكون هيك انه
+
+77
+00:06:26,710 --> 00:06:31,770
+احنا انهينا الجزء الثاني من المادة وحيكون فيه عندى
+
+78
+00:06:31,770 --> 00:06:36,540
+اللي هو ال secondاللي هو our exam بعد هيك هيجينا
+
+79
+00:06:36,540 --> 00:06:41,860
+الجزء الثالث من المادة الجزء الثالث يمكن اللي هو
+
+80
+00:06:41,860 --> 00:06:48,900
+الأول مرة انه احنا بنتحدث عن هذا الجزء اللي هو في
+
+81
+00:06:48,900 --> 00:06:55,780
+العنوانين الأول والثاني وانا حبيت انه يكون الطالب
+
+82
+00:06:56,980 --> 00:07:01,320
+في صورة إنه كل شغله طبعا كل شغلنا سواء فكرة القلص
+
+83
+00:07:01,320 --> 00:07:04,920
+ألف أو با أو جيم أو حتى ال complex أو حتى ال
+
+84
+00:07:04,920 --> 00:07:07,380
+differential equations اللي بنشتغل فيها ال
+
+85
+00:07:07,380 --> 00:07:11,440
+ordinary أو اللي هو ال partial وحتى في الإحصاء أو
+
+86
+00:07:11,440 --> 00:07:17,900
+الإحتمالات كله واضح إنه إحنا بنشتغل على R أو على C
+
+87
+00:07:17,900 --> 00:07:25,160
+واللي بنشتغل على R أو على C بلزمه اللي هو زي ما
+
+88
+00:07:25,160 --> 00:07:28,840
+بتعرفوا topologyالآن بدي أشوف أنا هذا الـ topology
+
+89
+00:07:28,840 --> 00:07:34,840
+وين موقعه بالظبط في اللي هو الـ topological spaces
+
+90
+00:07:34,840 --> 00:07:38,520
+الـ metric spaces الـ normed spaces الـ Herbart
+
+91
+00:07:38,520 --> 00:07:44,520
+spaces باختصار في عندنا احنا الشيء اللي .. اللي ..
+
+92
+00:07:44,520 --> 00:07:45,820
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+93
+00:07:45,820 --> 00:07:47,020
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+94
+00:07:47,020 --> 00:07:47,020
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+95
+00:07:47,020 --> 00:07:47,180
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+96
+00:07:47,180 --> 00:07:47,580
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+97
+00:07:47,580 --> 00:07:48,260
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+98
+00:07:48,260 --> 00:07:48,280
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+99
+00:07:48,280 --> 00:07:48,320
+اللي .. اللي .. اللي .. اللي .. اللي .. اللي ..
+
+100
+00:07:48,320 --> 00:07:56,260
+اللي .. اللي .. اللي .. اللي
+
+101
+00:07:56,260 --> 00:08:00,450
+.. اللفي داخل الـ topological spaces هنحكي الـ
+
+102
+00:08:00,450 --> 00:08:05,740
+metric spacesالمقصود أن أي metric space هيكون
+
+103
+00:08:05,740 --> 00:08:09,300
+topological space يعني هيولد topology داخل ال
+
+104
+00:08:09,300 --> 00:08:13,360
+metric space هيكون فيه عنده normed space و ال
+
+105
+00:08:13,360 --> 00:08:16,940
+normed space هيكون اللي هو any normed space اللي
+
+106
+00:08:16,940 --> 00:08:22,760
+هو بولد metric و ال metric نفسه بولد topology لأن
+
+107
+00:08:22,760 --> 00:08:25,520
+ال normed spaces هيكون فيه عنده topology اللي هو
+
+108
+00:08:25,520 --> 00:08:28,840
+induced by this normوفي داخل الـ Normed Spaces
+
+109
+00:08:28,840 --> 00:08:32,940
+حيكون فيه عند الـ Hilbert Spaces طبعا لو بين
+
+110
+00:08:32,940 --> 00:08:37,920
+الجهتين فيه اللي هو issue topological مش metric و
+
+111
+00:08:37,920 --> 00:08:40,780
+issue metric مش normed و issue normed مش Hilbert
+
+112
+00:08:41,200 --> 00:08:46,560
+وإلى آخره ومن ثم هنقعد نشوف حالنا وين احنا جاعدين
+
+113
+00:08:46,560 --> 00:08:49,200
+مع ال R اللي هنشتغل عليها اللي بنشتغل عليها كل
+
+114
+00:08:49,200 --> 00:08:53,720
+شغلنا هنا على ال R اللي هو ال real numbers ومعاه
+
+115
+00:08:53,720 --> 00:08:56,180
+فيه topology اللي هو اللي كنتوا تسموه في ال
+
+116
+00:08:56,180 --> 00:08:59,440
+topology اللي بيسميه usual topology ال usual
+
+117
+00:08:59,440 --> 00:09:02,590
+topologyالـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+118
+00:09:02,590 --> 00:09:02,850
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+119
+00:09:02,850 --> 00:09:03,490
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+120
+00:09:03,490 --> 00:09:03,730
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+121
+00:09:03,730 --> 00:09:05,910
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+122
+00:09:05,910 --> 00:09:12,950
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+123
+00:09:12,950 --> 00:09:15,070
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+124
+00:09:15,070 --> 00:09:15,490
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+125
+00:09:15,490 --> 00:09:15,650
+الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ الـ
+
+126
+00:09:15,650 --> 00:09:19,710
+الـ الـ الـ
+
+127
+00:09:19,710 --> 00:09:24,760
+الـ الـ الـاللي هو و normed space و طبيعي metrical
+
+128
+00:09:24,760 --> 00:09:28,500
+space و topological space فبكون عرفنا موقعنا
+
+129
+00:09:28,500 --> 00:09:32,940
+بالظبط و هنجيل الكلام هذا ان شاء الله بالتفصيل أي
+
+130
+00:09:32,940 --> 00:09:40,530
+سؤال؟أحكي اه لسه جينا ..الان احنا بعد ما نكون
+
+131
+00:09:40,530 --> 00:09:45,710
+خلصنا اللي هي الجزء هذا طبعا الجزء هذا و الجزء هذا
+
+132
+00:09:45,710 --> 00:09:52,510
+هيكون في عندى two references هتروح على اللي هو ال
+
+133
+00:09:52,510 --> 00:09:55,470
+reference two و three اللي هنحكي عنها من بعد شوية
+
+134
+00:09:56,110 --> 00:10:01,270
+الأن هنحكي في النهاية عن ال infinite series بما
+
+135
+00:10:01,270 --> 00:10:05,890
+نستطيع من خلال وقت هذا ال course هذه تقريبا اللي
+
+136
+00:10:05,890 --> 00:10:13,590
+هو وصف المساق أو وصف ما نريد أن ندرس في هذا المساق
+
+137
+00:10:13,590 --> 00:10:18,210
+ال normed spaces و ال topological spaces و ال
+
+138
+00:10:18,210 --> 00:10:21,030
+metric spaces و ال Hilbert spaces فيه two
+
+139
+00:10:21,030 --> 00:10:24,150
+references و في عندي slice كاملات اللي موجودة
+
+140
+00:10:24,150 --> 00:10:28,590
+تلخيص كامل للمادة موجودةعندي topology without
+
+141
+00:10:28,590 --> 00:10:34,570
+tiers اللي هو الكتاب اللي احنا عادة بندرسه في اللي
+
+142
+00:10:34,570 --> 00:10:40,530
+هو علم في الجامعة وموجود يعني وهيكون المتطلب منه
+
+143
+00:10:40,530 --> 00:10:45,850
+لهذا ال course موجود slides الآن الجزء اللي بعده
+
+144
+00:10:46,580 --> 00:10:49,200
+اللي هو introductory functional analysis with
+
+145
+00:10:49,200 --> 00:10:55,240
+applications ل-grazing هذا الجزء هيتعلق، هناخد منه
+
+146
+00:10:55,240 --> 00:11:00,220
+اللي هو ما يتعلق بال normed spaces and اللي هو
+
+147
+00:11:00,220 --> 00:11:10,330
+Hilbert spacesالأن هذه تقريبًا اللي هو الوصف
+
+148
+00:11:10,330 --> 00:11:14,230
+بالإضافة للمراجع اللي موجودة بالإضافة لل textbook
+
+149
+00:11:14,230 --> 00:11:19,150
+الأصلي وأي اللي هو كتاب real analysis برضه بيكون
+
+150
+00:11:19,150 --> 00:11:26,110
+مفيد لهذه المادة طريقة التقييم زي ما اتعودنا هيكون
+
+151
+00:11:26,110 --> 00:11:33,260
+في عنده اللي هو three exams two hour examsand one
+
+152
+00:11:33,260 --> 00:11:38,020
+final exam the two بالterm exams اللي هنا خمسين في
+
+153
+00:11:38,020 --> 00:11:43,100
+المية ال final exam برضه خمسين في المية طبعا و
+
+154
+00:11:43,100 --> 00:11:47,260
+طريقة الحساب يعني ان شاء الله بنتفق عليها غالبا
+
+155
+00:11:47,260 --> 00:11:50,940
+هناخد أحسن واحد من الامتحانين و نحسب عليه تلاتين و
+
+156
+00:11:50,940 --> 00:11:55,340
+الثاني هنحسب عليه عشرين و ال final بكون خمسين أي
+
+157
+00:11:55,340 --> 00:12:01,090
+سؤال أو أي تعليقنبدأ الآن بـ Chapter 6 اللي هو
+
+158
+00:12:01,090 --> 00:12:05,150
+بداية المادة هنحكي عن الـ differentiation أو
+
+159
+00:12:05,150 --> 00:12:12,010
+الاشتقاق طبعاً الاشتقاق مر علينا ومفهومه مر علينا
+
+160
+00:12:12,010 --> 00:12:17,910
+نذكركم بمعنى الاشتقاق الآن احنا هناخد اللي هو الـ
+
+161
+00:12:17,910 --> 00:12:23,650
+function F نفترض F عبارة عن دالة من I لعند R حيث I
+
+162
+00:12:23,650 --> 00:12:27,530
+is an interval اللي بنحكي عنها I is an interval
+
+163
+00:12:28,790 --> 00:12:38,650
+وبناخد C element in I بنقول أن الـ F prime عند C
+
+164
+00:12:38,650 --> 00:12:45,330
+exist أو بمعنى آخر المشتقة الأولى للدالة F موجودة
+
+165
+00:12:45,330 --> 00:12:51,020
+إذا كان F prime of C اللي بنسميه F prime of Cإذا
+
+166
+00:12:51,020 --> 00:12:57,220
+كان ما يلي موجود اللي هو limit f of x ناقص f of c
+
+167
+00:12:57,220 --> 00:13:02,880
+على x minus c لما x تروح للـ c exist إذا كان exist
+
+168
+00:13:02,880 --> 00:13:08,420
+و بساوء قيمة سمناها مثلا ال بنسمي ال هذه هي ال
+
+169
+00:13:08,420 --> 00:13:12,740
+derivative لمين للف و بنرمز لها للف prime of إيش
+
+170
+00:13:12,740 --> 00:13:17,080
+of c طبعا ال X هذه أكيد ال X من وين هنكون ماخدينها
+
+171
+00:13:17,080 --> 00:13:24,060
+برضه من I و ال X أشمالهالا تساوي C لأن X تذهب أو
+
+172
+00:13:24,060 --> 00:13:29,920
+تقول إلى C وليس تساوي C إذن ممكن نعبر عن هذا
+
+173
+00:13:29,920 --> 00:13:40,520
+الكلام بطريقة أخرى نقول F has a derivative at
+
+174
+00:13:40,520 --> 00:13:48,820
+C element in I if for every epsilon أكبر من 0there
+
+175
+00:13:48,820 --> 00:13:53,040
+exists delta أكبر من سفر has a derivative مثلا
+
+176
+00:13:53,040 --> 00:13:58,220
+نسميها L has a derivative L at C limited on I if
+
+177
+00:13:58,220 --> 00:14:00,440
+for every Y أكبر من سفر there exists delta أكبر من
+
+178
+00:14:00,440 --> 00:14:09,140
+سفر such that X minus C أكبر من سفر وأصغر من Delta
+
+179
+00:14:09,140 --> 00:14:17,000
+implies اللي هو مينأف of X ناقص أف of C على X
+
+180
+00:14:17,000 --> 00:14:24,280
+minus C ناقص L أصغر من مين من Y هذا طبعا في مجموعه
+
+181
+00:14:24,280 --> 00:14:28,600
+معناه إنه limit أف of X ناقص أف of C على X minus C
+
+182
+00:14:28,600 --> 00:14:31,760
+لما X تروح للـC بساوي اللي هي الـL اللي سميناها F
+
+183
+00:14:31,760 --> 00:14:37,910
+prime of C if such limit existsWe say that F has a
+
+184
+00:14:37,910 --> 00:14:44,210
+derivative which is denoted by F prime of C هذا
+
+185
+00:14:44,210 --> 00:14:49,990
+اللي هو معناته اللي هو الـ F لها مشتقة أولى عند
+
+186
+00:14:49,990 --> 00:14:59,610
+النقطة C الآن، مدام عندي الـ function F عندي من I
+
+187
+00:14:59,610 --> 00:15:00,730
+لعند R
+
+188
+00:15:03,790 --> 00:15:11,090
+والدالة و F' موجودة عند C اللي من وين جاية؟ من I
+
+189
+00:15:11,090 --> 00:15:18,550
+فطبيعي domain الـ F' هيكون جزء من مين؟ من الـ I
+
+190
+00:15:18,550 --> 00:15:21,670
+فهذه
+
+191
+00:15:21,670 --> 00:15:25,470
+الملاحظة الأولى إذا كانت F من I عند R دين domain
+
+192
+00:15:25,470 --> 00:15:33,130
+الـ F' subset من Iالأن نيجي للنظرية المشهورة اللي
+
+193
+00:15:33,130 --> 00:15:41,670
+بتعرفوها و اللي مرت علينا في أكثر من .. من مساق
+
+194
+00:15:41,670 --> 00:15:48,810
+اللي هي إنه إذا كانت الدالة قابلة للاشتقاق إذن
+
+195
+00:15:48,810 --> 00:15:54,810
+أكيد إشمالها هي متصلة الآن هنحكي بالظبط
+
+196
+00:15:58,230 --> 00:16:06,590
+قابلية الاشتقاق عند نقطة تعني أن الدالة متصلة عند
+
+197
+00:16:06,590 --> 00:16:22,350
+هذه النقطة نشوف theorem if F من I لعند R is
+
+198
+00:16:22,350 --> 00:16:34,170
+differentiable atC element in I then it must be
+
+199
+00:16:34,170 --> 00:16:41,230
+continuous at this C لو كانت الدالة قبل الاشتقاق
+
+200
+00:16:41,230 --> 00:16:46,030
+عند نقطة C أكيد هتكون متصلة عند هذه النقطة
+
+201
+00:16:46,030 --> 00:16:54,070
+والبرهان برضه برهان بسيط ومر عليكم قبل هيك
+
+202
+00:17:00,530 --> 00:17:05,350
+نذكركم بس، شو معناة الـ Dialogue هتكون متصل عن الـ
+
+203
+00:17:05,350 --> 00:17:15,130
+C؟ بيكون دي limit F of X لما X تروح للـ C بيساوي F
+
+204
+00:17:15,130 --> 00:17:18,790
+of C إذا أثبتنا هذا الكلام، بنكون خلصنا و أثبتنا
+
+205
+00:17:18,790 --> 00:17:27,950
+أن الدالة إيش مالها؟ متصلة إذن أول شيء، since F is
+
+206
+00:17:27,950 --> 00:17:34,220
+differentiableat c .. حسب التعريف اللي جابناه قبل
+
+207
+00:17:34,220 --> 00:17:43,040
+قليل limit f of x ناقص f of c على x minus c لما x
+
+208
+00:17:43,040 --> 00:17:49,490
+تروح للـ c اللي هو existence بساوة 100f prime of c
+
+209
+00:17:49,490 --> 00:17:54,450
+exist لأنه احنا مفترضين انها اش مالها موجودة طبعا
+
+210
+00:17:54,450 --> 00:17:58,850
+ال X احنا بنقول الاش مالها لا تساوي ال C و ال X
+
+211
+00:17:58,850 --> 00:18:05,290
+موجودة وين في ال interval I طيب، الان بدنا نثبت ان
+
+212
+00:18:05,290 --> 00:18:13,050
+limit f of x يساوي f of c شوف، سهل، عند f of x- f
+
+213
+00:18:13,050 --> 00:18:17,170
+of c على x minus c أكيد هيستخدمها و لا ليش كانت
+
+214
+00:18:17,170 --> 00:18:20,030
+المعطيلة الـ function is differentiable الان لو
+
+215
+00:18:20,030 --> 00:18:29,230
+ضربناها الان for x element in I و x لا تساوي c f
+
+216
+00:18:29,230 --> 00:18:31,950
+of x نقص f of c على x minus c لو ضربناها في x
+
+217
+00:18:31,950 --> 00:18:37,550
+minus cهيصير عندي F of X minus F of C طب لو ضفنا
+
+218
+00:18:37,550 --> 00:18:42,510
+لها الـ F of C ثابت الـ F of C هذا المقدار كمقدار
+
+219
+00:18:42,510 --> 00:18:48,150
+إيش هيساوي؟ الـ F of X الآن أنا ضامن إن الـ limit
+
+220
+00:18:48,150 --> 00:18:52,700
+هنا موجودةلأن أشملها F براها، دي عبارة عن F براهم
+
+221
+00:18:52,700 --> 00:18:56,220
+of C which exists وهدي ال limit موجودة، بولنوميا
+
+222
+00:18:56,220 --> 00:19:00,200
+العادية و X بتروح للـC موجودة و ال F of C ثابت،
+
+223
+00:19:00,200 --> 00:19:03,540
+إذن كل ال limit موجودة للجهتين، إذا بقدر أوزع ال
+
+224
+00:19:03,540 --> 00:19:09,940
+limit على الجهتين، بيصير عندي، إذا limit F of X
+
+225
+00:19:09,940 --> 00:19:18,510
+لما X تروح للـC بساوي limitF of X نقص F of C على X
+
+226
+00:19:18,510 --> 00:19:24,790
+minus C لما X تروح للـ C في limit X minus C لما X
+
+227
+00:19:24,790 --> 00:19:30,830
+تروح للـ C زائد F of C لأنه ثابت ال limit هو نفسه
+
+228
+00:19:30,830 --> 00:19:37,930
+ويساوي الآن يا جماعة عندي هذا ال limit هو عبارة عن
+
+229
+00:19:37,930 --> 00:19:42,310
+مين عبارة عن عدد مضمون أنه موجود اسمه F prime of C
+
+230
+00:19:43,220 --> 00:19:47,120
+فى مضروب هذا ال limit لما X تروح لـ C هذا المقدار
+
+231
+00:19:47,120 --> 00:19:54,000
+عشان بتكون سفر بولنوميا سهلة Linear زائد F of C في
+
+232
+00:19:54,000 --> 00:19:57,680
+الآخر إيش حيساوي إذا صارت limit F of X لما X تروح
+
+233
+00:19:57,680 --> 00:20:03,720
+لـ C بساوي F of C وهذا يعني hence أو therefore F
+
+234
+00:20:03,720 --> 00:20:11,740
+is continuous at C أي
+
+235
+00:20:11,740 --> 00:20:12,240
+سؤال
+
+236
+00:20:19,280 --> 00:20:23,920
+طبعا ال .. برضه اللي هنحكي فيه كنا على طول في
+
+237
+00:20:23,920 --> 00:20:26,220
+الcalculation نقول إذا كانت F is differentiable or
+
+238
+00:20:26,220 --> 00:20:30,140
+continuous كنا نسأل طب لو كانت continuous ليس شرط
+
+239
+00:20:30,140 --> 00:20:34,060
+أن تكون differentiable والمثال المشهود والمعروف
+
+240
+00:20:34,060 --> 00:20:37,140
+اللي هو عبارة عن ال corners اللي هي عندنا هنا
+
+241
+00:20:37,140 --> 00:20:41,440
+عبارة عن القيمة المطلقة ال absolute value عند ال
+
+242
+00:20:41,440 --> 00:20:44,960
+corner ال function is continuous لكن مش
+
+243
+00:20:44,960 --> 00:20:46,320
+differentiable
+
+244
+00:20:56,540 --> 00:21:03,380
+الأن الـ converse of
+
+245
+00:21:03,380 --> 00:21:18,220
+the above theorem need not be true in general for
+
+246
+00:21:18,220 --> 00:21:27,090
+example f of x سوى الـ absolute value للـ xis
+
+247
+00:21:27,090 --> 00:21:33,230
+continuous everywhere طبعا مضمن ان السفر is
+
+248
+00:21:33,230 --> 00:21:43,270
+continuous at zero but it is not differentiable at
+
+249
+00:21:43,270 --> 00:21:47,530
+zero طبعا ان انت عارفين اسمها عبارة عن corner
+
+250
+00:21:47,530 --> 00:21:51,330
+pointالـ corner point هنا is not differentiable
+
+251
+00:21:51,330 --> 00:21:54,950
+لأنه ميل المماس من اليمين وميل المماس من اليسار
+
+252
+00:21:54,950 --> 00:22:03,370
+حتى جهة المختلفات طيب نشوف هذا الكلام نبرهنه عندي
+
+253
+00:22:03,370 --> 00:22:11,280
+ال continuity سهلة limit f of xلما X تروح للصفر
+
+254
+00:22:11,280 --> 00:22:15,120
+بساوي limit لـ absolute value X لما X تروح للصفر
+
+255
+00:22:15,120 --> 00:22:19,780
+هو يساوي إيه شمالها؟ Zero واضح إن ال limit لهذه أو
+
+256
+00:22:19,780 --> 00:22:23,060
+limit لهذه كلها وين بروح وين بصبن لما X تروح للـC
+
+257
+00:22:23,060 --> 00:22:27,820
+من اليمين أو من اليسار بصبن وين؟ عند الصفر الآن
+
+258
+00:22:27,820 --> 00:22:32,720
+بدنا نشوف كيف إنها is not differentiable and zero
+
+259
+00:22:32,720 --> 00:22:41,320
+طيب خد الآنحسب لـ limit f of x نقص f of 0 على x
+
+260
+00:22:41,320 --> 00:22:45,000
+minus 0 لما x تروح للصفر بدا أثبت لك إن هذا الـ
+
+261
+00:22:45,000 --> 00:22:48,200
+limit does not exist مذام does not exist إذا الـ
+
+262
+00:22:48,200 --> 00:22:54,480
+function مالها is not differentiable at 0 بوي ساوى
+
+263
+00:22:54,480 --> 00:22:58,920
+الآن limit الـ f of x اللي هو absolute value لـ x
+
+264
+00:22:58,920 --> 00:23:05,980
+f of 0، 0على الـ x نقص صفر لما x تروح للصفر y ساول
+
+265
+00:23:05,980 --> 00:23:10,640
+limit absolute value of x على الـ x لما x تروح
+
+266
+00:23:10,640 --> 00:23:16,380
+للصفر الآل عندي عشان تكون ال limit هذه exist لأن
+
+267
+00:23:16,380 --> 00:23:19,180
+في absolute value فوق فوق فمخوفاني يعني لذا
+
+268
+00:23:19,180 --> 00:23:21,980
+بتروحلها من اليمين و بيروحلها من وين؟ من اليسار
+
+269
+00:23:21,980 --> 00:23:29,850
+الآل لما أقول limitabsolute value ل X على X لما X
+
+270
+00:23:29,850 --> 00:23:35,730
+تروح للصفر من وين من اليمين يعني قيم X موجبة مدام
+
+271
+00:23:35,730 --> 00:23:41,490
+موجبة إذا قيمة ال absolute value هي X وهنا على X
+
+272
+00:23:41,490 --> 00:23:46,570
+لما X تروح للصفر من اليمين ويساوي جداش واحد but
+
+273
+00:23:46,570 --> 00:23:56,900
+but limit limit ال absolute value ل Xعلى x لما x
+
+274
+00:23:56,900 --> 00:24:00,000
+تروح للصفر من وين؟ من اليسار قيمة الـ absolute
+
+275
+00:24:00,000 --> 00:24:04,940
+value لل x للقيم السالبة سلبها مظبوط و يساوي limit
+
+276
+00:24:04,940 --> 00:24:10,540
+سالب x على x لما x تروح للصفر من اليسار و يساوي
+
+277
+00:24:10,540 --> 00:24:14,760
+سالب و أكيد الواحد مش هو السالب واحد إذا ال limit
+
+278
+00:24:14,760 --> 00:24:17,780
+من اليمين مستويش ال limit من اليسار إذا صار عندي
+
+279
+00:24:17,780 --> 00:24:22,080
+هذه ال function هذه ال limit إجمالها does not
+
+280
+00:24:22,080 --> 00:24:27,580
+existإذا صارت عندي الـ function f of x بساعته الـ
+
+281
+00:24:27,580 --> 00:24:33,840
+weight value of x is continuous is continuous at
+
+282
+00:24:33,840 --> 00:24:40,700
+zero but is not differentiable at zero أي سؤال
+
+283
+00:24:40,700 --> 00:24:51,450
+الآن لو كررنا أمورنا لو قلنا ناخد دالةمثال ثاني لو
+
+284
+00:24:51,450 --> 00:24:54,890
+جينا أخدنا نقول لأ خد ال absolute value ل X minus
+
+285
+00:24:54,890 --> 00:24:55,610
+C1
+
+286
+00:24:59,510 --> 00:25:03,070
+هذا واضح إنه عند الـ c1 مالها is not
+
+287
+00:25:03,070 --> 00:25:07,070
+differentiable لكن continuous لو ضفنا لها كمان
+
+288
+00:25:07,070 --> 00:25:12,470
+absolute value x minus c2 برضه هتكون عند الـ c2 is
+
+289
+00:25:12,470 --> 00:25:16,610
+not differentiable بقى continuous لو زدنا كمان
+
+290
+00:25:16,610 --> 00:25:21,790
+بنفس الأسلوب لعند ما أصل ل x minus cn اللي هي هذه
+
+291
+00:25:21,790 --> 00:25:27,530
+الـ function صارت continuous everywherebut it is
+
+292
+00:25:27,530 --> 00:25:33,370
+not differentiable عند c1 و c2 و c3 لعند مين لعند
+
+293
+00:25:33,370 --> 00:25:40,150
+cn وطبعا وبرهانها سهل بنفس الأسلوب هنا عند c1 عند
+
+294
+00:25:40,150 --> 00:25:44,730
+c2 عند c3 عند cn بنفس الأسلوب اللي قبل بشوية الآن
+
+295
+00:25:44,730 --> 00:25:53,590
+مش هيك في دالة أغرب من هيك دالة بتقولك أنه أنا
+
+296
+00:25:54,690 --> 00:26:01,650
+عبارة عن continuous function وين ما كان لكن is not
+
+297
+00:26:01,650 --> 00:26:06,450
+differentiable anywhere يعني تخيل إنه .. طبعا
+
+298
+00:26:06,450 --> 00:26:09,050
+تخيلها يمكن صعبة تخيل إنه هذا عبارة عن كلها
+
+299
+00:26:09,050 --> 00:26:13,950
+corners دالة كلها corners فش منطقة أو فش نقطة
+
+300
+00:26:13,950 --> 00:26:18,070
+بتكون عندها الدالة إيه شمالها is differentiable
+
+301
+00:26:18,070 --> 00:26:21,550
+هذه الدالة اللي هي f of x
+
+302
+00:26:24,880 --> 00:26:33,180
+بتساوي summation 1 على 2 أُس N في cosine تلاتة أُس
+
+303
+00:26:33,180 --> 00:26:38,800
+N X N من صفر إلى مالة نهاية هذه للمعرفة بس لأنه
+
+304
+00:26:38,800 --> 00:26:43,300
+مرهانها خارج النطاق اللي بنحكي فيه F of X بساوي
+
+305
+00:26:43,300 --> 00:26:48,180
+summation 1 على 2 أُس N cosine تلاتة أُس N في X
+
+306
+00:26:48,180 --> 00:26:52,660
+هذه is continuous everywhere but is not
+
+307
+00:26:52,660 --> 00:26:54,420
+differentiable anywhere
+
+308
+00:26:59,710 --> 00:27:04,890
+هذه علاقة الإتصال بالاشتقام
+
+309
+00:27:04,890 --> 00:27:09,610
+الـ relation between continuity and
+
+310
+00:27:09,610 --> 00:27:15,710
+differentiability one implies the other but the
+
+311
+00:27:15,710 --> 00:27:21,110
+other need not implies the first I mean by the
+
+312
+00:27:21,110 --> 00:27:25,890
+first differentiability and the other is
+
+313
+00:27:25,890 --> 00:27:30,750
+continuityطيب صلى على النبي عليه الصلاة والسلام
+
+314
+00:27:30,750 --> 00:27:36,370
+يمكن ال .. ال .. ال .. في البداية كان نحكي عن ..
+
+315
+00:27:36,370 --> 00:27:41,870
+بس ملاحظة ضايقة بسيطة يعني اللي بنحكيها في ال
+
+316
+00:27:41,870 --> 00:27:45,710
+calculus دائما بس هتذكركم فيها اللي هي المعنى
+
+317
+00:27:45,710 --> 00:27:50,890
+الهندسي لل differentiability الآن لما نقول إن
+
+318
+00:27:50,890 --> 00:27:59,250
+الدالة في إلها مشتقة عند الـCالآن لو جيت أخدت ..
+
+319
+00:27:59,250 --> 00:28:03,150
+روحت للـ C من اليمين أو من اليسار و بدي أجيلها من
+
+320
+00:28:03,150 --> 00:28:07,990
+اليمين هذه اللي هي النقطة X نقطة X عاملا بتروح على
+
+321
+00:28:07,990 --> 00:28:14,530
+اللي هي الـ C هذا اللي هو القاطع اللي بين الـ X
+
+322
+00:28:14,530 --> 00:28:20,550
+والـ C هذه من hand ل hand هذه المنطقة عبارة عن F
+
+323
+00:28:20,550 --> 00:28:26,660
+of X ناقص F of C صح ولا لا؟وهذه من هنا لها عبارة
+
+324
+00:28:26,660 --> 00:28:32,900
+عن x minus c ميل القاطع هذا اللي هو عبارة عن f of
+
+325
+00:28:32,900 --> 00:28:40,990
+x ناقص f of c على x minus c هذا ميل القاطعالـ
+
+326
+00:28:40,990 --> 00:28:45,590
+limit له نهاية ميل القاطع، نهاية ميل القاطع لما X
+
+327
+00:28:45,590 --> 00:28:49,870
+تروح للـC هذا القاطع لما X تروح للـC بتجرب، بتجرب
+
+328
+00:28:49,870 --> 00:28:54,810
+لما نسير في النهاية عبارة عن إيش؟ عن مماس، بيصير
+
+329
+00:28:54,810 --> 00:28:59,870
+هو عبارة عن اللي هي مين؟ F prime of C أو ميل
+
+330
+00:28:59,870 --> 00:29:06,630
+المماس عنده نقطة مين؟ عنده نقطة Cطبعاً أخدت أنا من
+
+331
+00:29:06,630 --> 00:29:10,410
+اليمين ومن اليسار بنفس الأسلوب يعني بيصير طبعاً
+
+332
+00:29:10,410 --> 00:29:14,990
+على اعتبار أنه من اليمين ومن اليسار متساوين فبيكون
+
+333
+00:29:14,990 --> 00:29:19,510
+اللي هو الـ F prime موجودة وميل القاطع نهايته في
+
+334
+00:29:19,510 --> 00:29:27,110
+نهاية بساوة ميل المماسع ده ده المقطع أي سؤال؟ نيجي
+
+335
+00:29:27,110 --> 00:29:32,670
+الآن للقوانين برضه قوانين احنا أخدناها في ال
+
+336
+00:29:32,670 --> 00:29:40,090
+calculusالنظرية بتجاوب على أنه لو كان في عندي
+
+337
+00:29:40,090 --> 00:29:47,230
+دالتين مشتقته الموجودة هيكون مجموح .. مجموح ..
+
+338
+00:29:47,230 --> 00:29:53,250
+مجموح مشتقته موجودة أو حاصل ضربه المشتقته الموجودة
+
+339
+00:29:53,250 --> 00:29:56,650
+وحاصل قسمته المشتقته الموجودة provided that أن
+
+340
+00:29:56,650 --> 00:30:01,390
+اللي هو اللي في المقام قيمة الـ الـ G عنده نقطة لا
+
+341
+00:30:01,390 --> 00:30:11,470
+يساوي 0وأيضًا الثابت في الدالة برضه مشتقته موجودة
+
+342
+00:30:11,470 --> 00:30:16,030
+نشوف الكلام اللي بحكيه بالتفصيل عبر النظرية اللي
+
+343
+00:30:16,030 --> 00:30:22,870
+أمامنا ونبرهن النظرية اللي عندنا let I be subset
+
+344
+00:30:22,870 --> 00:30:26,850
+من R بيكون interval يعني ماخد I عبارة عن interval
+
+345
+00:30:26,850 --> 00:30:33,680
+وماخد C element of Iومفترض إن F من I لR وG من I لR
+
+346
+00:30:33,680 --> 00:30:38,100
+التنتين قابلات للاشتقاق عند نقطة مين C they are
+
+347
+00:30:38,100 --> 00:30:43,440
+differentiable at C إذن the following holds إيش ال
+
+348
+00:30:43,440 --> 00:30:46,860
+following أول واحدة Alpha في F is differentiable
+
+349
+00:30:46,860 --> 00:30:51,510
+at Cيعني اللي هو الدالة الجديدة اللي هي اسمها
+
+350
+00:30:51,510 --> 00:30:56,330
+Alpha في F Alpha ثابت قابل الاشتقاق عن C وطبعا زي
+
+351
+00:30:56,330 --> 00:31:00,010
+ما انتوا عارفين الاشتقاق للAlpha في F prime of C
+
+352
+00:31:00,010 --> 00:31:07,500
+اللي هي عبارة عن Alpha في F prime of Cالآن F ذائد
+
+353
+00:31:07,500 --> 00:31:09,500
+G is differentiable at C ما زالت الـ F
+
+354
+00:31:09,500 --> 00:31:11,540
+differentiable و G differentiable مجموح الـ
+
+355
+00:31:11,540 --> 00:31:14,620
+differentiable و F ذائد G لكل prime of C بسوأ F
+
+356
+00:31:14,620 --> 00:31:18,500
+prime of C ذائد G prime of C أيضًا بالنسبة للضرب
+
+357
+00:31:18,500 --> 00:31:21,860
+القاعدة مشهورة أيضًا F في G هتكون differentiable
+
+358
+00:31:21,860 --> 00:31:26,180
+عند C و F في G prime هو عبارة عن F prime في G ذائد
+
+359
+00:31:26,180 --> 00:31:31,800
+G prime في Fالآن الشرط اللي بعده الـ D اللي هي
+
+360
+00:31:31,800 --> 00:31:36,600
+بتفترض أنه G of C إيش مالها مش سفر مش مسترطلك على
+
+361
+00:31:36,600 --> 00:31:41,200
+كل ال X سفر G of X بيسويش سفر مش مسترطلك بس أنه
+
+362
+00:31:41,200 --> 00:31:46,480
+الـ G عند الـ C مش سفر و بقولك إذا F على G prime
+
+363
+00:31:46,480 --> 00:31:52,020
+of C هتساوي F prime في G ناقص F في G prime على
+
+364
+00:31:52,020 --> 00:31:56,560
+اللي هو G تربيع عارفين إنه مقام تربيع؟و فوق اللي
+
+365
+00:31:56,560 --> 00:32:00,580
+هو مشتقة البصد في المقام نقص مشتقة المقام في البصد
+
+366
+00:32:00,580 --> 00:32:09,100
+واضحة؟ هذه الآن اللي هي theorem 6.1.3 اللي الآن
+
+367
+00:32:09,100 --> 00:32:18,760
+بدنا نبرهن اللي هو COD شوفوا خلينا الآن اللي ..
+
+368
+00:32:18,760 --> 00:32:25,550
+اللي بدنا نبرهنه لو كانت F و D is differentiableAt
+
+369
+00:32:25,550 --> 00:32:32,230
+C بيعطينا أن F في G is differentiable at C وخلّونا
+
+370
+00:32:32,230 --> 00:32:35,010
+نقول الـ differentiable لت اللي قلنا عنه F في G
+
+371
+00:32:35,010 --> 00:32:38,150
+الكل برايم هو عبارة عن F برايم في G زي G برايم في
+
+372
+00:32:38,150 --> 00:32:49,050
+F طيب، نشوف كيف، سميلي هذه let H بتساوي F في Gعشان
+
+373
+00:32:49,050 --> 00:32:52,630
+أثبت إن الـ F في G is differentiable بدي أثبتلك
+
+374
+00:32:52,630 --> 00:32:59,510
+إنه خد الآن X element in I و X لا تساوي C بدي
+
+375
+00:32:59,510 --> 00:33:08,010
+أثبتلك إنه limit الـ F of X الـ H of X ناقص
+
+376
+00:33:08,800 --> 00:33:13,980
+H of C على X minus C بدأ أثبت لك إنه limit لما X
+
+377
+00:33:13,980 --> 00:33:18,300
+تروح إلى C exist و بساوة الكميريا اللي أنا بحكي
+
+378
+00:33:18,300 --> 00:33:23,320
+عنها أو بدعيها، معايا؟ هذا في حالة وجوده، ممكن
+
+379
+00:33:23,320 --> 00:33:29,300
+خلّصناه، وجود limitه الآن عبّولي هذه، خلّيني أنعوض
+
+380
+00:33:29,300 --> 00:33:35,640
+هذه عبارة عن H of X عبارة عن F of X في G of X صح
+
+381
+00:33:35,640 --> 00:33:42,410
+ولا لأ؟عوضت مكان الأشقبت اللي أنا فرضها FG ناقص F
+
+382
+00:33:42,410 --> 00:33:48,770
+of C في G of C الكل مقسم عالمياً على X minus C
+
+383
+00:33:48,770 --> 00:33:54,380
+ويساويالآن أنا مفترض أنه f is differentiable و g
+
+384
+00:33:54,380 --> 00:33:58,300
+is differentiable يعني أصلا أنا متوقع أني بدي
+
+385
+00:33:58,300 --> 00:34:02,640
+أستخدم limit f of x ناقص f of c على x minus c يعني
+
+386
+00:34:02,640 --> 00:34:07,560
+بديها تظهر f of x ناقص f of c عشان تظهر بدي أضيف
+
+387
+00:34:07,560 --> 00:34:12,400
+term و أطرح term و ال term فيه f of x أو g of x و
+
+388
+00:34:12,400 --> 00:34:16,200
+التاني و التاني زيه بس واحد فيه c و واحد مافيش فيه
+
+389
+00:34:16,200 --> 00:34:23,180
+c إيش اللي بقصده؟ بقصده بقول f of xG of X ناقص F
+
+390
+00:34:23,180 --> 00:34:33,540
+of .. F of C G of X عارفين ليش؟ زائد اللي هو F of
+
+391
+00:34:33,540 --> 00:34:38,320
+C G of X اللي ضيفته بتضيفه أو اللي طرحته بتضيفه
+
+392
+00:34:38,320 --> 00:34:45,400
+ناقص F of C في G of C على X-C ويساوي ليش أعملك
+
+393
+00:34:45,400 --> 00:34:49,520
+هيك؟ عساس أني بدي أخد الـF أو الـG of X عام
+
+394
+00:34:49,520 --> 00:34:55,200
+المشترك بساوي G of X فيه و بدي أجزقه جزء هنا وجزء
+
+395
+00:34:55,200 --> 00:35:00,840
+هنا فيه اللي هو اللي بيظل عندي هنا مين هو F of X
+
+396
+00:35:00,840 --> 00:35:06,610
+تنقص F of C لكن كله مجسم على مين؟ X minus Cزائد
+
+397
+00:35:06,610 --> 00:35:10,470
+اللي بعدها بنفس الاسلوب باخد مين الآن عاملوا شترك
+
+398
+00:35:10,470 --> 00:35:17,330
+F of C في G of X ناقص G of C الكل مجسوم على X
+
+399
+00:35:17,330 --> 00:35:23,770
+minus C الآن أموري جاهزة زي ما بدي أشبك صوتنا نشوف
+
+400
+00:35:23,770 --> 00:35:29,670
+الآن هذه اللي هو بدي أخد limit للطرف هذا و limit
+
+401
+00:35:29,670 --> 00:35:33,510
+للطرف هذا طيب انت ضابط ان limit exist؟ أه ليش دوام
+
+402
+00:35:33,510 --> 00:35:37,910
+ال exist؟لأن ال limit لهذا هو عبارة عن f prime of
+
+403
+00:35:37,910 --> 00:35:41,570
+c لأنه معطني إياه طب ال limit لك g of x لما x تروح
+
+404
+00:35:41,570 --> 00:35:44,770
+لل c؟ أه موجودة لأن ال differential باليدا
+
+405
+00:35:44,770 --> 00:35:48,970
+المتصلة، الميدا المتصلة إذا ال limit g of x لما x
+
+406
+00:35:48,970 --> 00:35:53,790
+تروح لل c موجود ويسوى g of c كمانطب هنا هذه ثابتة
+
+407
+00:35:53,790 --> 00:35:57,450
+بتطلع وهذه اللي هي قيمة مشتقة الـ G إذا كل أموري
+
+408
+00:35:57,450 --> 00:36:00,670
+تمام التمام يعني أنا ماخد ال limit للطرفين باخد و
+
+409
+00:36:00,670 --> 00:36:06,940
+بوزع و أنا مرتاح إذا بدي أقول limit لهذهلما X تروح
+
+410
+00:36:06,940 --> 00:36:11,280
+للـC بيساوي limit لهذه لما X تروح للـC بيساوي
+
+411
+00:36:11,280 --> 00:36:14,740
+limit لهذه لما X تروح للـC بيساوي limit لهذه لما X
+
+412
+00:36:14,740 --> 00:36:18,380
+تروح للـC أو انت بعد ما اثبتت هذه بيساوي هذه بتاخد
+
+413
+00:36:18,380 --> 00:36:21,420
+limit الاتنين على جهة و بتقول limit هذه بيساوي
+
+414
+00:36:21,420 --> 00:36:29,480
+limit هذه و بتبدأ تكمل ويساوي limit
+
+415
+00:36:29,480 --> 00:36:40,590
+G of X لما X تروح للـCفي limit f of x نقص f of c
+
+416
+00:36:40,590 --> 00:36:46,810
+على x minus c لما x تروح للc زائد f of c لأنها
+
+417
+00:36:46,810 --> 00:36:53,480
+ثابت تطلع برا ال limit في limit g of xنقص G of C
+
+418
+00:36:53,480 --> 00:36:59,840
+على X minus C لما X تروح للـ C لأن الـ function is
+
+419
+00:36:59,840 --> 00:37:02,520
+differentiable الـ G is a continuous فده
+
+420
+00:37:02,520 --> 00:37:06,000
+continuous إذا إيه إيش اللي هو هذا بيساوي G of C
+
+421
+00:37:06,000 --> 00:37:11,420
+وهذا إيش بتساويالتعريف اللي هو الـ F prime of C
+
+422
+00:37:11,420 --> 00:37:17,300
+زائد هذه F of C في G prime of C إذا صار عندك الـ
+
+423
+00:37:17,300 --> 00:37:22,400
+derivative هذه اللي هي بتمثل مين؟ H prime of C
+
+424
+00:37:22,400 --> 00:37:28,640
+اللي هي مين؟ قولنا عنها FG prime of C لأنه أنا
+
+425
+00:37:28,640 --> 00:37:33,120
+بسمي الـ H هي مين؟ الـ FG صارت عندي الآن FG prime
+
+426
+00:37:33,120 --> 00:37:43,440
+of C exist and equal كذاأي سؤال؟
+
+427
+00:37:43,440 --> 00:37:51,780
+طيب، الآن احنا أثبتنا C، بدنا نثبت الآن زي ما
+
+428
+00:37:51,780 --> 00:37:57,960
+أثبتنا الضرب، بدنا نثبت القسمة القسمة مشابه، بس في
+
+429
+00:37:57,960 --> 00:38:05,480
+في البداية يعني أمر تحليلي بشرّعلي اللي هو
+
+430
+00:38:09,520 --> 00:38:13,900
+القسمة على g of x لإكسات معينة عشان أسلك طريقي
+
+431
+00:38:13,900 --> 00:38:19,500
+للوصول إلى انه اللي هو ال f على g لكل prime exist
+
+432
+00:38:19,500 --> 00:38:24,240
+تشوف ايش اللي بقوله اول حاجة ايش اللي بدأت بتواجه
+
+433
+00:38:24,240 --> 00:38:26,740
+معاه بدأت بتواجه انه لو كانت ال f و ال g
+
+434
+00:38:26,740 --> 00:38:31,760
+differentiable at c و g of c بس لا تساوي سفر
+
+435
+00:38:31,760 --> 00:38:38,940
+بيعطيني اللي f على g ال primeexist وزي ما قلنا إيش
+
+436
+00:38:38,940 --> 00:38:45,360
+كيف exist قيمته بالظبط شوفوا كيفه صلى على النبي
+
+437
+00:38:45,360 --> 00:38:50,400
+عليه الصلاة والسلام و proof أول حاجة بس خليني أعمل
+
+438
+00:38:50,400 --> 00:38:56,340
+fresh بحيث أن أقدر أستخدم اللي بديه اللي بلزمني
+
+439
+00:38:56,340 --> 00:39:07,320
+أول إشي بدي أذكركوا بس بنظرية في chapter 4النظرية
+
+440
+00:39:07,320 --> 00:39:11,780
+أربعة اتنين سبعة حاجة زي هيك المهم إيش بتقول بتقول
+
+441
+00:39:11,780 --> 00:39:17,680
+لو كانت ال limit لل F of X لما X بتروح لل C أكبر
+
+442
+00:39:17,680 --> 00:39:23,360
+من سفر أكبر من سفر إذا يوجد there exists an
+
+443
+00:39:23,360 --> 00:39:30,080
+epsilon of C يعني يوجد جوار بحيث أنه اللي هو F of
+
+444
+00:39:30,080 --> 00:39:38,110
+X بتكون أكبر من سفر لكل X في هذا الجوارأه يعني
+
+445
+00:39:38,110 --> 00:39:42,770
+بلاقي جوار حوالين الـ C قيمة الدالة إيه شمالها
+
+446
+00:39:42,770 --> 00:39:47,870
+تكون أكبر من 0 strictly لما كانت limit f of x لما
+
+447
+00:39:47,870 --> 00:39:52,570
+x تروح للـ C أكبر من 0 أكيد في جوار حواليها هتكون
+
+448
+00:39:52,570 --> 00:39:56,290
+ال limit إيه شماله أكبر من 0 وهذه النظرية انبرهنة
+
+449
+00:39:56,290 --> 00:40:02,390
+لو كان أصغر من 0 نفس الأشي بلاقي جوار بحيث أن f of
+
+450
+00:40:02,390 --> 00:40:06,410
+x أصغر من 100 من 0 في هذا الجوار واضح
+
+451
+00:40:09,850 --> 00:40:16,470
+الآن عندي الـ G differentiable عند الـ C مدام الـ
+
+452
+00:40:16,470 --> 00:40:21,050
+G differentiable عند الـ C إذا continuous عند الـ
+
+453
+00:40:21,050 --> 00:40:30,390
+C يعني بما أن G is differentiable at C then it is
+
+454
+00:40:30,390 --> 00:40:36,990
+continuous at C صح ولا لأ؟ إيش معناته هذا الكلام؟
+
+455
+00:40:36,990 --> 00:40:42,880
+معناته limitG of X لما X تروح للـ C بيساوي G of
+
+456
+00:40:42,880 --> 00:40:48,980
+إيش؟ of C الـ G of C معناته إيش ماله؟ هذا عدد لا
+
+457
+00:40:48,980 --> 00:40:53,400
+يساوي سفر هيك جاي اللي هو مدام لا يساوي سفر معناته
+
+458
+00:40:53,400 --> 00:40:58,960
+يا أكبر من سفر يا إيش؟ يا أصغر من سفر يعني بمعنى
+
+459
+00:40:58,960 --> 00:41:03,440
+آخر there exist حسب النظرية هذه there exist
+
+460
+00:41:03,440 --> 00:41:09,800
+neighborhoodجوار حوالين الـC سميه J هذا جوار لإيه؟
+
+461
+00:41:09,800 --> 00:41:14,060
+حوالين الـC there exist جوار حوالين الـC طبعاً
+
+462
+00:41:14,060 --> 00:41:17,500
+subset من مين هيكون؟ من الـinterval اللي معرف
+
+463
+00:41:17,500 --> 00:41:23,260
+عليها مين أنا F و G صح ولا لأ؟ there exist G of C
+
+464
+00:41:23,260 --> 00:41:29,830
+subset من I such that G of Xإذا كانت هذا أكبر من
+
+465
+00:41:29,830 --> 00:41:33,570
+صفر تكون أكبر من صفر إذا كان أصغر من صفر تكون
+
+466
+00:41:33,570 --> 00:41:40,130
+أشماله أصغر من صفر قلنا there exist gc اللي هو
+
+467
+00:41:40,130 --> 00:41:43,690
+عبارة عن neighborhood حوالين الـC جزء طبعا من I
+
+468
+00:41:43,690 --> 00:41:49,070
+اللي هي ال interval معرفة عليها Gsuch that الـ g
+
+469
+00:41:49,070 --> 00:41:54,450
+of x أشماله لا يسوى أيضًا سفر طبعًا أنا جاعت ليش
+
+470
+00:41:54,450 --> 00:41:58,790
+أعمل تهيأك عشان أشرع لحالي بعد شوية القسم على g of
+
+471
+00:41:58,790 --> 00:42:04,110
+x لأن دالة اللي أنا بحكي عليها مين هي f على g
+
+472
+00:42:04,110 --> 00:42:09,370
+فلازم أشغل في مكان تكون الـ g مش سفر
+
+473
+00:42:12,500 --> 00:42:16,300
+when there exists J C subset من I سجدات G of X لا
+
+474
+00:42:16,300 --> 00:42:21,700
+يساوي سفر لأن هذا الجوار اللي من خلاله بده تروح لـ
+
+475
+00:42:21,700 --> 00:42:26,620
+C X اللي فيه بده تروح لـ C فهذا الجوار مضمون الآن
+
+476
+00:42:26,620 --> 00:42:33,240
+تكون الأفعال جيمالها معرفة فيه واضح؟ طيب، الآن
+
+477
+00:42:33,240 --> 00:42:37,000
+فرشنا الفرش هذا، بدنا نيجي لموضوع الحسابات اللي
+
+478
+00:42:37,000 --> 00:42:43,070
+هيوصلنا إليهلأ اللي بديها شوف الآن عند ادالة مين
+
+479
+00:42:43,070 --> 00:42:48,570
+هذا let اللي هو زي ما سميها Q ايش متساوي F على G
+
+480
+00:42:48,570 --> 00:42:55,010
+بدي اثبت انه اللي هو Q is differential اللي عند ال
+
+481
+00:42:55,010 --> 00:42:59,730
+C بنفس الطريقة اللي قبل بشوية ايش بقول هنحسب
+
+482
+00:42:59,730 --> 00:43:06,470
+الحساب اللي عند Q of X ناقص Q of C على X minus C
+
+483
+00:43:07,020 --> 00:43:11,220
+ماشي الحال، هذه الـ Xات مين؟ حريص أنا أنها تكون
+
+484
+00:43:11,220 --> 00:43:17,660
+الـ X لا تساوي C و الـ X تبعتنا وين موجودة؟ في
+
+485
+00:43:17,660 --> 00:43:21,240
+الجهة C، أنا بسلم في الجوار، جوار عشان أروحله جوار
+
+486
+00:43:21,240 --> 00:43:25,340
+حتى مهما كان صغير، بدي أروح من خلاله على الـ C
+
+487
+00:43:25,340 --> 00:43:29,580
+أحنا اللي بتاخد limit بعد شوية الـ Q of X ناقص Q
+
+488
+00:43:29,580 --> 00:43:35,210
+of Z على X minus C، احسبليها، بساوي اللي هو Fof x
+
+489
+00:43:35,210 --> 00:43:42,050
+على g of x نقص f of c على g of c الكل مجسوم على x
+
+490
+00:43:42,050 --> 00:43:50,030
+minus c ماشي وحد المقامات الآن ويساوي عبارة عن x
+
+491
+00:43:50,030 --> 00:43:58,350
+minus c مضروب g of x g of c وهنا بيصير f of x g of
+
+492
+00:43:58,350 --> 00:44:06,120
+c نقص f of c g of x مظبوط؟بس وحدنا المقنعات و
+
+493
+00:44:06,120 --> 00:44:11,620
+يساوي الآن هذا عبارة عن زي ما عملت قبل شوية بضيف
+
+494
+00:44:11,620 --> 00:44:17,780
+term و أطرح term اللي يخدمني الان f of x g of c
+
+495
+00:44:17,780 --> 00:44:23,060
+الآن بدي أطلع اللي هو g of c عن المشتركة إذا بيصير
+
+496
+00:44:23,060 --> 00:44:33,700
+مزق ناقص g of c في f of c صح؟ زائد اللي هو g of cF
+
+497
+00:44:33,700 --> 00:44:42,520
+of C ناقص F of C في G of X على X minus C في G of X
+
+498
+00:44:42,520 --> 00:44:47,560
+في G of C أشي اللي سويته بس ضفت هذا المقدار وطرحته
+
+499
+00:44:47,560 --> 00:44:52,020
+مش معنى هو لإن أنا حسبنا في علنا إنه ناخد الـ G of
+
+500
+00:44:52,020 --> 00:44:54,940
+C عامل مشترك فبتظل F of X ناقص F of C وهديك
+
+501
+00:44:54,940 --> 00:44:59,360
+هتعملّي برضه إشي محترم زيه سبحان الله كل شي بتناسق
+
+502
+00:44:59,360 --> 00:45:01,000
+ويساوي
+
+503
+00:45:04,550 --> 00:45:11,890
+الأن أخد اللي هو f of x ولا g of c عامل مشترك فيه
+
+504
+00:45:11,890 --> 00:45:24,390
+عندي f of x ناقص f of c على x minus c وخلّي ليه؟
+
+505
+00:45:24,390 --> 00:45:29,030
+خلّيلي هدول أطلّحهم برا بالمرة اللي هي عبارة عن
+
+506
+00:45:29,030 --> 00:45:35,000
+واحد على g of xفي g of c طلعت هدولة بالمرة برا ال
+
+507
+00:45:35,000 --> 00:45:40,380
+.. ال summation برا .. برا الكثير كله زائد .. هذه
+
+508
+00:45:40,380 --> 00:45:46,420
+الأن أخدت هدول مع بعض هنا على x minus c هيهم لأن
+
+509
+00:45:46,420 --> 00:45:52,410
+أخد هذا برضه على هذا زائد اللي هو f of c براهي
+
+510
+00:45:52,410 --> 00:45:56,130
+اللي بدي أخده أخد من الـ f of c عامل مشترك في اللي
+
+511
+00:45:56,130 --> 00:46:01,110
+هو f خليني أقول .. خليني أخد سالب بالمرة بعد أذنكم
+
+512
+00:46:01,110 --> 00:46:06,410
+أخد سالب بالمرة بعد أذنكم فبصير f of c فبصير هذا
+
+513
+00:46:06,410 --> 00:46:11,290
+عبارة عن g of x ناقص g of c طبعا واضح عشان سويته
+
+514
+00:46:11,290 --> 00:46:17,530
+على x minus c كل هذا مضروب في مين؟ في واحد هذا g
+
+515
+00:46:17,530 --> 00:46:23,960
+of x في g of cالأن إيش هسوي يا جماعة زي ما عملت
+
+516
+00:46:23,960 --> 00:46:27,060
+قبل بشوية أنا غرض مين اللي هو ال derivative يعني
+
+517
+00:46:27,060 --> 00:46:31,360
+غرض أخد ال limit لهذه ماشي إذا انصار عندي ال limit
+
+518
+00:46:31,360 --> 00:46:35,860
+لهذه لما x تروح لل c بساوي ال limit لهذا المقدار
+
+519
+00:46:35,860 --> 00:46:43,540
+كله لما x تروح لمين لل c هذا بيخلص الآن عشان
+
+520
+00:46:43,540 --> 00:46:47,480
+تركزوا على الكلمتين هدولة هدولة خلصت منهم بظبط
+
+521
+00:46:47,480 --> 00:46:54,110
+اتطلعوا لللي ضال عندي هناالان ال limit هذه بقدر
+
+522
+00:46:54,110 --> 00:46:57,450
+ادخلها زي ما بدي ليش؟ اول اشي ال limit .. طبعا
+
+523
+00:46:57,450 --> 00:46:59,670
+راحظوا ان ال g of x أنا بشتغل فيه وانا مرتاح لإنه
+
+524
+00:46:59,670 --> 00:47:02,950
+مش 0 لإنه فرقة ثبتناها وخلصنا خدنا ال x's اللي في
+
+525
+00:47:02,950 --> 00:47:09,310
+ال .. في ال .. في ال jc .. jc اللي فيها ال g of x
+
+526
+00:47:09,310 --> 00:47:15,230
+لاتشهر 0 طيبالـG الـdifferential اللي كنت نفسي عند
+
+527
+00:47:15,230 --> 00:47:19,030
+الـC إذا الـlimit G of X بيساوي G of C إذا هذه
+
+528
+00:47:19,030 --> 00:47:23,690
+الـlimit اللي بدخل الليمتين وبدخل الليمتين وبدخل
+
+529
+00:47:23,690 --> 00:47:27,930
+الليمتين وهذه بتطلع لإنها ثابت بيصير عبارة عن إيش
+
+530
+00:47:27,930 --> 00:47:34,970
+هذا؟ واحد على G of C الكل تربيع، مظبوط؟ لأن في
+
+531
+00:47:34,970 --> 00:47:39,370
+اللي جوا اللي هو عبارة عن G of C مين هذا؟ في
+
+532
+00:47:39,370 --> 00:47:46,670
+الـprime of Cناقص F of C في G prime of C وهو
+
+533
+00:47:46,670 --> 00:47:50,550
+المطلوب اللي هي G of C في F prime ناقص F of C في G
+
+534
+00:47:50,550 --> 00:47:54,410
+prime على مين؟ على G of C الكل تربيع وبكون هيك
+
+535
+00:47:54,410 --> 00:47:58,090
+أثبتنا أنه لو كانت الـ F و الـ G الـ differential
+
+536
+00:47:58,090 --> 00:48:02,770
+اللي عندنا دقطة C و G of C لا تساوي سفر فعندنا الـ
+
+537
+00:48:02,770 --> 00:48:10,390
+F على G prime of C exist وقيمتها اللي أمامنا أي
+
+538
+00:48:10,390 --> 00:48:20,890
+سؤال؟طيب عندنا أخر نقطة طولت عليكم بهمش تفاصيل
+
+539
+00:48:20,890 --> 00:48:28,010
+التلخيص هي موجودة هان اللي
+
+540
+00:48:28,010 --> 00:48:31,030
+.. اللي هي ت .. اللي خلينا نقول تعميم للجبل بشوية
+
+541
+00:48:31,030 --> 00:48:35,790
+لو كانت F1 F1
+
+542
+00:48:37,710 --> 00:48:41,150
+وF2 عند FN are functions on an interval I subset
+
+543
+00:48:41,150 --> 00:48:44,250
+من R that are differentiable on C element in I
+
+544
+00:48:44,250 --> 00:48:48,390
+then F1 زد F2 زد FN برضه differentiable عند الـ C
+
+545
+00:48:48,390 --> 00:48:53,050
+وبتبع induction وبكون عند F1 زد FN لكل prime of C
+
+546
+00:48:53,050 --> 00:48:56,190
+بيساوي F1 prime of C زد F2 prime of C لما نصل عند
+
+547
+00:48:56,190 --> 00:49:02,180
+FN prime of C similarly بي ..برضه تعميم F1 في F2
+
+548
+00:49:02,180 --> 00:49:06,220
+في FN الـdifferentiable at C إذا حيكون حاصل الضرب
+
+549
+00:49:06,220 --> 00:49:10,400
+برضه الـdifferentiable و F prime F1 prime of C F2
+
+550
+00:49:10,400 --> 00:49:15,440
+of C FN of C و بعدين بنثبت الأولة و بنشتق الثانية
+
+551
+00:49:15,440 --> 00:49:19,100
+و هكذا لما نصل لآخر واحدة اللي هي F1 of C F2 of C
+
+552
+00:49:19,100 --> 00:49:24,680
+لعند FN prime of Cوبهيك نكون .. نهينا المحاضرة
+
+553
+00:49:24,680 --> 00:49:29,040
+الأولى وإن شاء الله المرة القادمة بنبدأ بالمحاضرة
+
+554
+00:49:29,040 --> 00:49:33,280
+التانية وحنبدأ من عند The Chain Rule الله يعطيكوا
+
+555
+00:49:33,280 --> 00:49:33,720
+العافية
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..260e3c008b4cbe27fb572cc0c338615c84c2a739
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_postprocess.srt
@@ -0,0 +1,1704 @@
+1
+00:00:05,480 --> 00:00:11,800
+بسم الله الرحمن الرحيم محاضرة رقم تسعة مساق تحليل
+
+2
+00:00:11,800 --> 00:00:19,160
+حقيقي اتنين طلاب الجامعة الإسلامية كلية العلوم قسم
+
+3
+00:00:19,160 --> 00:00:24,470
+الرياضياتالمحاضرة اليوم على جزئين الجزء الأول
+
+4
+00:00:24,470 --> 00:00:29,350
+هنتحدث عن applications of taylor's theorem والجزء
+
+5
+00:00:29,350 --> 00:00:34,250
+الثاني هنتحدث عن اللي هو discussion لsection 6.2
+
+6
+00:00:34,250 --> 00:00:37,890
+و6.3 اللي هو discussion لل mean value theorem وإذا
+
+7
+00:00:37,890 --> 00:00:43,270
+في وجهة بنعمل discussion ل اللي هو lobitals rules
+
+8
+00:00:43,960 --> 00:00:48,240
+الان خلّينا نكمل اللي اتحدثنا عنه المرة الماضية
+
+9
+00:00:48,240 --> 00:00:53,200
+ولكن على صورة تطبيقات ليه؟ اللي هو Taylor's
+
+10
+00:00:53,200 --> 00:00:58,660
+theorem وكيف بدنا اللي هو نستغل Taylor's theoremفي
+
+11
+00:00:58,660 --> 00:01:03,360
+إثبات بعض النظريات المهمة في موضوع اللي هو ال
+
+12
+00:01:03,360 --> 00:01:08,080
+maximum و ال minimum أو اللي هو ال test لإن اللي
+
+13
+00:01:08,080 --> 00:01:12,140
+عندنا هل هي local maximum أو local minimum ولا لا
+
+14
+00:01:12,140 --> 00:01:15,860
+هي local maximum ولا هي local minimum كنا اتحدثنا
+
+15
+00:01:15,860 --> 00:01:20,560
+سابقا على اللي هو ال first derivative test لإن
+
+16
+00:01:20,560 --> 00:01:23,200
+النقطة اللي تكون الداخلية هل في عندها local
+
+17
+00:01:23,200 --> 00:01:27,540
+maximum أو local minimumالان بدنا نتحدث عن الـ
+
+18
+00:01:27,540 --> 00:01:31,140
+Higher Derivative Test احنا يمكن نمر عنا اللي هو
+
+19
+00:01:31,140 --> 00:01:33,720
+الـ Second Derivative Test اللي هو في ال .. في
+
+20
+00:01:33,720 --> 00:01:36,900
+الCalculus اللي قابلة نتحدث على ال Higher
+
+21
+00:01:36,900 --> 00:01:40,080
+Derivative Test و اللي هي .. اللي هي ال Second
+
+22
+00:01:40,080 --> 00:01:44,090
+Derivative Test جزء من هذا اللي هو ال testنتطلع
+
+23
+00:01:44,090 --> 00:01:48,030
+على الـ test إيش بقول اللي هي عبارة عن theorem 644
+
+24
+00:01:48,030 --> 00:01:52,310
+let I be an interval نفترض أن I عبارة عن interval
+
+25
+00:01:52,310 --> 00:01:57,510
+و X note اللي هي نقطة اللي احنا بدنا نفحص عندها هل
+
+26
+00:01:57,510 --> 00:02:02,770
+في maximum أو في minimum أو لا في maximum ولا في
+
+27
+00:02:02,770 --> 00:02:06,470
+minimum X note اللي هي عبارة عن interior point
+
+28
+00:02:06,470 --> 00:02:10,890
+لمن؟ لل interval I يعني نقطة داخليةونفترض الـ N
+
+29
+00:02:10,890 --> 00:02:14,570
+تبعتنا أكبر أو ساوى 2 لأن الـ N أكبر أو ساوى 2
+
+30
+00:02:14,570 --> 00:02:17,690
+يعني الـ Second Derivative Test أو أعلى هو اللي
+
+31
+00:02:17,690 --> 00:02:20,910
+بده يحددنا أنه عندنا Maximum أو Minimum أو مافيش
+
+32
+00:02:20,910 --> 00:02:25,450
+لا Maximum ولا Minimum Suppose ده ال derivatives
+
+33
+00:02:25,450 --> 00:02:31,120
+اللي هي F' و F'' لعند FN existsexist and are
+
+34
+00:02:31,120 --> 00:02:34,040
+continuous in a neighborhood of x naught نفترض إن
+
+35
+00:02:34,040 --> 00:02:39,140
+هدولة continuous وموجودة في اللي هو
+
+36
+00:02:41,860 --> 00:02:44,760
+اللي هي في neighborhood حوالي مين الـ X Note
+
+37
+00:02:44,760 --> 00:02:49,920
+ونفترض أن F' عند X Note لعند F N ناقص واحد prime
+
+38
+00:02:49,920 --> 00:02:54,060
+of X Note إيش بيساوي؟ سفر لكن عند F N of X Note
+
+39
+00:02:54,060 --> 00:02:59,140
+إيش مالها؟ لاتساوي سفر إذا ال test إيش بيصير؟ أول
+
+40
+00:02:59,140 --> 00:03:06,120
+إشيبنشوف f prime نقطة x note هذه نقطة متوقع عندها
+
+41
+00:03:06,120 --> 00:03:10,340
+maximum أو minimum أو مافيش شيء من النقاط اللي هي
+
+42
+00:03:10,340 --> 00:03:13,520
+بدنا نفحص عندها maximum أو minimum إيش بدنا نسوي
+
+43
+00:03:13,520 --> 00:03:18,520
+في هذا ال test بنجيب f prime عند ال x note إذا
+
+44
+00:03:18,520 --> 00:03:23,500
+لجيناها سفر بنيجي لل f double prime عند ال x note
+
+45
+00:03:23,500 --> 00:03:29,130
+إذا لجيناها سفراللي هو بنيجي للـ F W' عند الـ X0
+
+46
+00:03:29,130 --> 00:03:34,830
+وين بنصل للـ F W' .. وين بنصل للـ F N of X0 لا
+
+47
+00:03:34,830 --> 00:03:39,210
+تساوي سفر؟ بيبدأ الخطوات تبعات ال test ايش ال
+
+48
+00:03:39,210 --> 00:03:43,710
+test؟ اللي هي الأول إذا كانت N even طلعت عندنا N
+
+49
+00:03:43,710 --> 00:03:47,850
+هادئية أول واحدة ال derivative عندها عند ال X0 مش
+
+50
+00:03:47,850 --> 00:03:53,930
+سفروكانت ال Fn of X0 أكبر من 0 بتكون ال F أيه
+
+51
+00:03:53,930 --> 00:03:57,470
+شمالها؟ اللي هي relative minimum زي ال second
+
+52
+00:03:57,470 --> 00:04:01,710
+derivative test كنا لما نقول ال FW' عند ال X0 أكبر
+
+53
+00:04:01,710 --> 00:04:06,170
+من 0 على طول نحكم أنه في local minimum هنا الآن لأ
+
+54
+00:04:06,170 --> 00:04:08,510
+احنا على ال higher derivative و هتحللنا المشاكل
+
+55
+00:04:08,510 --> 00:04:10,290
+اللي كانت ماتتحللاش جامعي اللي هي second
+
+56
+00:04:10,290 --> 00:04:16,670
+derivative test الان Fn is evenو ال F N of X0 أصغر
+
+57
+00:04:16,670 --> 00:04:21,510
+من 0 إذا Fها زي إيه شمالها relative maximum at X0
+
+58
+00:04:21,510 --> 00:04:25,670
+إذا لو طلعت عندي N is odd إيش يعني N is odd؟ يعني
+
+59
+00:04:25,670 --> 00:04:31,130
+أول .. أول derivative ال F N عندها لاتساوي 0 طلعت
+
+60
+00:04:31,130 --> 00:04:35,390
+عندي هذه oddطبعا أكبر إذا هو 2، 3 أو 5 أو 7 إلى
+
+61
+00:04:35,390 --> 00:04:39,490
+آخره على طول احنا بنحكم then F has neither
+
+62
+00:04:39,490 --> 00:04:43,610
+relative minimum nor إيش ما لها relative maximum
+
+63
+00:04:43,610 --> 00:04:50,790
+at X note بتصور هيك النظرية واضحة إيش نصها بنجيبF
+
+64
+00:04:50,790 --> 00:04:56,570
+prime of X0 و F double prime of X0 إلى آخره إذا
+
+65
+00:04:56,570 --> 00:05:01,170
+هذا بيساوي سفر بهذا بيساوي سفر بضلي ماشي لعند F N
+
+66
+00:05:01,170 --> 00:05:08,550
+of X0 لاتساوي سفر و اگ تست تصل لاتساوي سفر باجي لل
+
+67
+00:05:08,550 --> 00:05:17,620
+test اللي هو uneven عند حالتينالحالة الأولى FN of
+
+68
+00:05:17,620 --> 00:05:27,280
+X0 أكبر من 0 Local MinimumFn of X0 أصغر من 0 Local
+
+69
+00:05:27,280 --> 00:05:33,180
+maximum لكن إذا طلعت n is odd يعني أول واحدة بعد
+
+70
+00:05:33,180 --> 00:05:37,720
+التنين اللي هو بيطلع عند F ال derivative لها Fn of
+
+71
+00:05:37,720 --> 00:05:43,200
+X0 لا تساوي 0 طلعت عند n is odd إذا بحكم على طول F
+
+72
+00:05:43,200 --> 00:05:52,030
+has neither Local maximum nor Local minimumتعالوا
+
+73
+00:05:52,030 --> 00:05:58,670
+اللي هو نبرهم اللي هو النظرية ونشوف إنه حجر الأساس
+
+74
+00:05:58,670 --> 00:06:02,470
+للبرهان اللي هي taylor's theorem هي taylor's
+
+75
+00:06:02,470 --> 00:06:12,430
+theorem تنشوف كيف الآن عندي بدنا نطبق taylor's
+
+76
+00:06:12,430 --> 00:06:15,820
+theoremعلى .. طبعا احنا بنشتغل على الـ interval من
+
+77
+00:06:15,820 --> 00:06:21,520
+I بناخد اللي هو X element من I و بنطبقلها من الـ
+
+78
+00:06:21,520 --> 00:06:26,120
+Taylor's theorem عند النقطة اللي احنا بدنا نختبر
+
+79
+00:06:26,120 --> 00:06:29,840
+عند X0 فهي اللي بده أجيب Taylor's theorem حواليها
+
+80
+00:06:29,840 --> 00:06:36,160
+اذا حسب Taylor's theorem ان there exists C between
+
+81
+00:06:36,160 --> 00:06:45,630
+اللي هي X and X0 such thatاللي هو F of X F of X
+
+82
+00:06:45,630 --> 00:06:52,110
+بساوي بي أن ناقص واحد بدا أقبّقها اللي بحيث أن ال
+
+83
+00:06:52,110 --> 00:06:56,250
+polynomial ما أن ناقص واحد ليش؟ لإنه أنا ماعطيني
+
+84
+00:06:56,250 --> 00:07:01,110
+هو عنده ال derivative اللي هي لعند F أن ناقص واحد
+
+85
+00:07:01,110 --> 00:07:05,870
+كلها سفار واللي مش سفر اللي هي مين F n of X not
+
+86
+00:07:06,550 --> 00:07:11,390
+فعند اللي هي R نقص واحد هطبق عشان تحكم لل FN of X
+
+87
+00:07:11,390 --> 00:07:15,310
+node على إنه عندني local maximum أو local minimum
+
+88
+00:07:15,310 --> 00:07:19,750
+أو neither كيف؟ خلّينا نطبق ال Taylor's theorem
+
+89
+00:07:19,750 --> 00:07:28,350
+زائد اللي هو أشملها R N نقص واحد of X هذه عندي
+
+90
+00:07:28,350 --> 00:07:33,230
+Taylor's theorem الآن ال BN نقص واحد اللي هي عبارة
+
+91
+00:07:33,230 --> 00:07:40,820
+عن إياش Fof x naught زائد f prime of x naught في x
+
+92
+00:07:40,820 --> 00:07:45,440
+minus x naught اللي هو على واحد factorial زائد لما
+
+93
+00:07:45,440 --> 00:07:51,400
+أصل لآخر واحد f n ناقص واحد of x naught في x minus
+
+94
+00:07:51,400 --> 00:07:55,540
+x naught قس n ناقص واحد على n ناقص واحد لكل
+
+95
+00:07:55,540 --> 00:08:02,700
+factorial زائد main ال remainder اللي هو main اللي
+
+96
+00:08:02,700 --> 00:08:09,810
+هو عبارة عن xنقص x note أس n على n factorial في f
+
+97
+00:08:09,810 --> 00:08:18,850
+n of mean of عندي c و ال c مالها between x note
+
+98
+00:08:18,850 --> 00:08:27,170
+and mean and x واضح و يساوي الأن كل هذول أصفار صح
+
+99
+00:08:27,170 --> 00:08:33,500
+ولا لأ إذا حيصير عندي هذا بيساوي لإنه معطيليالـ F
+
+100
+00:08:33,500 --> 00:08:41,120
+prime of X note لعند FN نقص واحد of X note كله
+
+101
+00:08:41,120 --> 00:08:46,700
+نصفار إذا صلي عندي هذا بيساوي F of X note لأن هذا
+
+102
+00:08:46,700 --> 00:08:55,740
+كله بتصفر زاد X minus X note أس N على N factorial
+
+103
+00:08:55,740 --> 00:09:09,290
+في FNof c واضح اه اي سؤال طيب عندى انا اللى هو
+
+104
+00:09:09,290 --> 00:09:19,230
+معطينى ال ال f n of x naught اللى هى لاتساوى سفر
+
+105
+00:09:19,230 --> 00:09:25,170
+اذا ال f n of x naught يا اكبر من سفر يا اصغر من h
+
+106
+00:09:25,170 --> 00:09:36,570
+من سفرإذا أكيد عندي ال FN of X note و ال FN of C
+
+107
+00:09:36,570 --> 00:09:43,810
+الهن أيش الهن the same sign ماشي اه الهن نفس نين
+
+108
+00:09:43,810 --> 00:09:50,770
+نفس اللي هو الإشارة طبعا السبب أن FN أشمالها is
+
+109
+00:09:50,770 --> 00:09:55,950
+continuous انوضحلكم إياها تتذكروا
+
+110
+00:09:57,370 --> 00:10:02,330
+قلنا خلّينا ناخد حالة نفترض أن هذا أكبر من مين من
+
+111
+00:10:02,330 --> 00:10:12,690
+سفر احنا كنا نقول limit أي حاجة g of x لما x تروح
+
+112
+00:10:12,690 --> 00:10:18,970
+لل c مثلا أكبر من سفر معناته في جوار في جوار
+
+113
+00:10:18,970 --> 00:10:23,210
+حوالين اللي هو ال c ال g of x نفسها إيش هتكون أكبر
+
+114
+00:10:23,210 --> 00:10:28,110
+من سفر لكل x في هذا الجواريعني هيكون عندي إذاً g
+
+115
+00:10:28,110 --> 00:10:37,710
+of x أكبر من 0 لكل x في sum جوار delta of mean of
+
+116
+00:10:37,710 --> 00:10:41,690
+z هذا اللي بدأ أستخدمها الآن في الوصول للي بديها
+
+117
+00:10:41,690 --> 00:10:45,590
+خلّيني نشوف كيف بدأ نستخدم هذه المعلومة الآن أنا
+
+118
+00:10:45,590 --> 00:10:55,750
+عندي الـ Fn is continuous نفسها إذاً limit Fnof X
+
+119
+00:10:55,750 --> 00:11:03,290
+أفقًا of X اللي هو لما الـ X تروح للـ X0 اللي هو
+
+120
+00:11:03,290 --> 00:11:10,070
+أش متساوي أفقًا of X0 وبرضه continuous عند مين؟
+
+121
+00:11:11,690 --> 00:11:16,810
+وبرضه continuous عند مين؟ عند الـ C معايا وعند الـ
+
+122
+00:11:16,810 --> 00:11:21,030
+C إيش مالها؟ continuous أنا اللي هيلزمني مين؟ هنا
+
+123
+00:11:21,030 --> 00:11:26,930
+الـ X note مظبوط ولا لأ؟ عندي هتلزمني مين؟ اللي هي
+
+124
+00:11:26,930 --> 00:11:35,950
+ال X note إذا صار عنديفن of X note هي اللي هتحددلي
+
+125
+00:11:35,950 --> 00:11:45,090
+من هنا اللي هي إشارة مين فن of C واضح الآن من هذه
+
+126
+00:11:45,090 --> 00:11:54,310
+إذا كانت F of X note مثلا أكبر من سفر هتطلع حد
+
+127
+00:11:54,310 --> 00:11:58,870
+أكبر من سفر خلينا نشوفها إذا الآن خليني أبدأ في
+
+128
+00:11:58,870 --> 00:12:06,270
+مينفي I بدي افترض مين الان fn of x not أكبر من أيش
+
+129
+00:12:06,270 --> 00:12:13,010
+من سفر و افترض ان الان عند ايش معنها is even بدي
+
+130
+00:12:13,010 --> 00:12:19,950
+أنطلق من هذه اذا الان fn of x not أكبر من سفر و n
+
+131
+00:12:19,950 --> 00:12:25,730
+is even بناء على هذه for every y أكبر من سفر there
+
+132
+00:12:25,730 --> 00:12:32,440
+exists such that المهم بيطلع عندىاللي هو F هذه ايش
+
+133
+00:12:32,440 --> 00:12:37,880
+مفترضين احنا اكبر من ايه ايش من سفر اذا حسب اللي
+
+134
+00:12:37,880 --> 00:12:45,450
+حكيته قبل بشوية there existV Delta of إيش؟ Of X0
+
+135
+00:12:45,450 --> 00:12:53,590
+حوالي مين؟ الـ X0 Such that اللي هو FN of X زي
+
+136
+00:12:53,590 --> 00:12:59,290
+إشارة هذه اللي هي أكبر من 0 لكل X وين في الـ V
+
+137
+00:12:59,290 --> 00:13:06,880
+Delta of مين؟ Of X0ماشي الحال واللي من ضمن هدوله
+
+138
+00:13:06,880 --> 00:13:15,060
+مين؟ C إذا صارت عندي الان FN of C برضه إيه شمالها؟
+
+139
+00:13:15,060 --> 00:13:21,860
+أكبر من سفر الان اتضحت الصورة الان صارت عندي من
+
+140
+00:13:21,860 --> 00:13:26,420
+الأصل F of X بسوء F of X note زائد هذا المقدار بدا
+
+141
+00:13:26,420 --> 00:13:31,020
+أقول الآن بما أن هذه أكبر من سفر وان أنا مفترضها
+
+142
+00:13:31,020 --> 00:13:36,490
+إنها evenإذا حيصير عندى هذا أكبر من صفر وهذا أكبر
+
+143
+00:13:36,490 --> 00:13:41,570
+من صفر إذا الـ remainder كله إيش ماله أكبر من صفر
+
+144
+00:13:41,570 --> 00:13:47,710
+مدام الـ remainder أكبر من صفر واضح اللي بحكي FN
+
+145
+00:13:47,710 --> 00:13:53,010
+of C أكبر من صفر وهذا المقدر أكبر من صفر لأنه N is
+
+146
+00:13:53,010 --> 00:13:58,710
+even إذا صار كل هذا أكبر من صفر إذا automatic لما
+
+147
+00:13:58,710 --> 00:14:04,660
+أشيلههيصغر المقدار، بظبط ولا لأ؟ فهيصير عند ال F
+
+148
+00:14:04,660 --> 00:14:12,040
+of X اللي هو أكبر من مين من ال F of X naught لأن F
+
+149
+00:14:12,040 --> 00:14:17,680
+of X بساوي هذا زائد هذا فأكيد عند ال F of X أكبر
+
+150
+00:14:17,680 --> 00:14:22,160
+من مين من F of X naught في وين، لكل X وين موجودة
+
+151
+00:14:22,160 --> 00:14:30,530
+في الجوار اللي حواليها، إذا صار عند Fhas local
+
+152
+00:14:30,530 --> 00:14:40,870
+minimum at x0 similarly لو
+
+153
+00:14:40,870 --> 00:14:53,630
+بدنا نفترض ان المقدار هذا عندي fn of x0 أصغر من 0
+
+154
+00:14:54,540 --> 00:15:00,740
+وبدأ أبدأ أحكي عن مين؟ عن I I بيصير عندي الآن حسب
+
+155
+00:15:00,740 --> 00:15:07,320
+الحديث هنا ال FN of C اللي هي أصغر من 0 لإنه هتكون
+
+156
+00:15:07,320 --> 00:15:13,600
+ال FN of X لكل X في هذا الدوار the same sign as FN
+
+157
+00:15:13,600 --> 00:15:18,120
+of X note أصغر من 0 إذا صارت عندي FN of C أصغر من
+
+158
+00:15:18,120 --> 00:15:24,050
+0هذه صارت سالبة وهذه موجبة لإنه and is even إذا
+
+159
+00:15:24,050 --> 00:15:28,290
+صار المقدار هذا كله على بعض إيه شماله؟ سالب في كل
+
+160
+00:15:28,290 --> 00:15:33,070
+هذا الجوار ما ده صار سالب معناته إنه لما أنا أشيله
+
+161
+00:15:33,070 --> 00:15:39,330
+بكبر اللي بظهر يعني بيصير F of X note بناء عليه
+
+162
+00:15:39,330 --> 00:15:46,630
+أكبر من F of X مظبوط؟وهذا يعني انه صار ال F في هذه
+
+163
+00:15:46,630 --> 00:15:56,010
+الحالة عندها local maximum local maximum واضح ؟ اي
+
+164
+00:15:56,010 --> 00:16:01,510
+سؤال ؟ طيب، ضال عند مين الحالة الأخيرة، هذي
+
+165
+00:16:01,510 --> 00:16:09,170
+تذكرينها لأنه برضه بدي اشتغل نفس الشيء في الحالة
+
+166
+00:16:09,170 --> 00:16:15,680
+الأخيرة او اشي مشابهعند الان بدي افترض mean ان
+
+167
+00:16:15,680 --> 00:16:25,980
+الحالة الثالثة بدي افترض ان ال is mean of of زي
+
+168
+00:16:25,980 --> 00:16:35,810
+ما عملنا قبل بشوية إذا كانت اللي هي Fفن of x not
+
+169
+00:16:35,810 --> 00:16:40,890
+أكبر من سفر زي ما عملنا قبل شوية هيعطيني فن of c
+
+170
+00:16:40,890 --> 00:16:49,990
+أكبر من إيش؟ من سفر معايا؟ طيب الآن هذا الكلام لو
+
+171
+00:16:49,990 --> 00:16:56,930
+جيت أخدت عندي هي عندي هذه اللي هي x not وهنا
+
+172
+00:16:56,930 --> 00:17:01,650
+الجوار اللي حواليهاالجوار اللي حواليها اللي لاجئت
+
+173
+00:17:01,650 --> 00:17:09,230
+في كل الجوار زي ما عملت قبل شوية fn of x هيكون
+
+174
+00:17:09,230 --> 00:17:15,830
+اللي هو هنا اللي هو هنا و هنا موجب ماشي الحال الآن
+
+175
+00:17:15,830 --> 00:17:21,850
+مدام موجب هنا و هنا اللي هيتحكم في الموضوع اليمين
+
+176
+00:17:21,850 --> 00:17:27,770
+x minus x naught أسن لكن لسوء الحظ لو كانت x هنا
+
+177
+00:17:28,970 --> 00:17:35,350
+بصير عندى و هذا طبعا لكل X وين على يمين ال X node
+
+178
+00:17:35,350 --> 00:17:41,450
+هيطلع عندى ال X minus X node أس N وهذه إيه شمالها
+
+179
+00:17:41,450 --> 00:17:49,110
+odd is أكبر من أيش من سفر فصارت الآن هذه أكبر من
+
+180
+00:17:49,110 --> 00:17:54,330
+سفر وهذه أكبر من سفر التنتير مع بعض أكبر من سفر
+
+181
+00:17:55,160 --> 00:18:01,380
+ماشي فصارت عندي ال F of X أكبر من مين من ال F of X
+
+182
+00:18:01,380 --> 00:18:07,600
+naught ماشي الحال في المنطقة هذه هذه المنطقة هذه
+
+183
+00:18:07,600 --> 00:18:16,060
+صارت عندي ال F of X أكبر من F of X naught فاهمين؟
+
+184
+00:18:16,060 --> 00:18:23,240
+لأن الجزء الثاني ال Xفي المنطقة اللي هان أي x في
+
+185
+00:18:23,240 --> 00:18:27,880
+المنطقة اللي هان عند الـ x minus x naught أصغر من
+
+186
+00:18:27,880 --> 00:18:32,480
+مين من سفر فبصير عند هذا المقدار أصغر من سفر وهذا
+
+187
+00:18:32,480 --> 00:18:37,020
+أكبر من سفر حاصل الدرب أصغر من سفر أصغر من سفر بقد
+
+188
+00:18:37,020 --> 00:18:42,200
+إلى أن أف of x في المنطقة هذه تصير أصغر من أف of x
+
+189
+00:18:42,200 --> 00:18:47,250
+naught يعني لو أتيت لأي جوارحوالين الـ X Note
+
+190
+00:18:47,250 --> 00:18:51,770
+هتلاقي اللي هو مرة الـ F of X Note أصغر من F of X
+
+191
+00:18:51,770 --> 00:18:56,810
+ومرة F of X Note أكبر من مين من F of X يعني بمعنى
+
+192
+00:18:56,810 --> 00:19:02,270
+آخر لا في عندي Local Maximum ولا Local Minimum في
+
+193
+00:19:02,270 --> 00:19:08,450
+هذه المنطقة Similarly وبنفس الأسلوب لو كانت FN of
+
+194
+00:19:08,450 --> 00:19:14,000
+X Note أصغر من مينمن صفر معايا بيصير هذه أصغر من
+
+195
+00:19:14,000 --> 00:19:19,020
+صفر و بيصير بس بتنقلب بيصير هذه في المنطقة هذه
+
+196
+00:19:19,020 --> 00:19:23,840
+أصغر و هذه في المنطقة هذه ايش اكبر اذا in both
+
+197
+00:19:23,840 --> 00:19:30,160
+cases عندي if and is odd اذا we have neither اللي
+
+198
+00:19:30,160 --> 00:19:37,660
+هو relative maximum nor relative minimum اي سؤال
+
+199
+00:19:37,660 --> 00:19:45,760
+واضح اه؟اللي بعده اللي بعده الان بدنا نعرف حاجة
+
+200
+00:19:45,760 --> 00:19:51,140
+اسمها اللي هي مين ال convex function تعالي على
+
+201
+00:19:51,140 --> 00:19:55,640
+الرسم يا بو محمد عندي بدنا نعرف مين ال convex
+
+202
+00:19:55,640 --> 00:19:59,860
+function الان ال convex function إيش يكون cave
+
+203
+00:19:59,860 --> 00:20:05,200
+اللي هو إيش up خليني نشوف إيش هذي اللي هو بعبر شو
+
+204
+00:20:05,200 --> 00:20:07,980
+معناه ال F is convex
+
+205
+00:20:10,750 --> 00:20:14,990
+خاصية الـ convex أو الـ convexity بنحكي فيها على
+
+206
+00:20:14,990 --> 00:20:22,090
+interval يعني بنقول F من عند I لعند R بنقول عنها F
+
+207
+00:20:22,090 --> 00:20:28,170
+is convex on إيش؟ on I on وين؟ ع ال interval I إذا
+
+208
+00:20:28,170 --> 00:20:35,970
+كان شوف كيف خدلي الآن X1 أي نقطة في داخل الـ I و
+
+209
+00:20:35,970 --> 00:20:41,970
+X2 أي نقطة في داخل الـ Iوخدلي ال parameter T بين
+
+210
+00:20:41,970 --> 00:20:45,830
+السفر و بين الواحد الان النقطة ال linear
+
+211
+00:20:45,830 --> 00:20:51,230
+combination بين X1 و X2 بواسطة 1-T X1 زائد T X2
+
+212
+00:20:51,230 --> 00:20:56,530
+هتكون بين X1 و بين X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+213
+00:20:56,530 --> 00:20:56,950
+X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+214
+00:20:56,950 --> 00:21:06,250
+X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+215
+00:21:06,250 --> 00:21:14,050
+X2 X2 X2 X2 X2 X2 X2لو إجيت ل F of X1 هيها وهذا F
+
+216
+00:21:14,050 --> 00:21:20,650
+of X2 لو ضربت ال F of X1 و 1 minus T و جمعتها ل T
+
+217
+00:21:20,650 --> 00:21:25,170
+في F of X2 هتلاقيها بين اللي هو مين على اللي هو
+
+218
+00:21:25,170 --> 00:21:32,010
+الخط المستقيم القاطع بين X1 و F of X1 و بين X2 و F
+
+219
+00:21:32,010 --> 00:21:37,970
+of X2 دائما هتلاقي هذه النقطةالصورة هنا أعلى من
+
+220
+00:21:37,970 --> 00:21:42,910
+هذه وهذا معنى اللي هو ال convex وعن عرف ال convex
+
+221
+00:21:42,910 --> 00:21:50,390
+على هذا الأساس بيقول F is convex on I طلعلي على
+
+222
+00:21:50,390 --> 00:21:54,650
+التعريف إذا كان أخدنا أي X واحد و X اتنين element
+
+223
+00:21:54,650 --> 00:22:00,690
+in I و أخدنا T اللي هو element in 01 يعني بين 01
+
+224
+00:22:00,690 --> 00:22:13,020
+بيحققلي F اللي هوfor any كده f
+
+225
+00:22:13,020 --> 00:22:19,140
+is a convex in I if for any كده وكده we have
+
+226
+00:22:19,140 --> 00:22:25,940
+فاهمين عليا f of واحد minus T X واحد زائد T في X
+
+227
+00:22:25,940 --> 00:22:32,520
+اتنين ايش ماله هيسير؟أصغر أو يساوي واحد ناقص T في
+
+228
+00:22:32,520 --> 00:22:38,000
+X واحد زائد T في F of X اتنين
+
+229
+00:22:40,410 --> 00:22:44,370
+طبعاً اللي هو ال .. ال .. ال .. ال .. ال convex
+
+230
+00:22:44,370 --> 00:22:48,330
+functions في عليها دراسات كثيرة ودراسات حديثة كمان
+
+231
+00:22:48,330 --> 00:22:55,070
+حتى يعني في اللي هو real analysis بما يتعلق بال ..
+
+232
+00:22:55,070 --> 00:22:58,950
+مش بال integration العادي كمان باللي هو .. و
+
+233
+00:22:58,950 --> 00:23:03,310
+علاقتها بال integration في اللي هو ال .. ال big
+
+234
+00:23:03,310 --> 00:23:08,760
+integra كمانوطبعاً وإليها اللي هو خلّيني أقول
+
+235
+00:23:08,760 --> 00:23:14,020
+تطبيقات أو استخدامات في اللي هو optimization اللي
+
+236
+00:23:14,020 --> 00:23:20,350
+هي theoryالان احنا هنتحك .. هنحكي فيها بما يتعلق
+
+237
+00:23:20,350 --> 00:23:24,470
+في هذا الجزء من ال chapter اللي هو اللي بتعلق ك ..
+
+238
+00:23:24,470 --> 00:23:28,350
+ك .. كمين؟ ك Taylor's theorem وكيف بدنا نطبق ال
+
+239
+00:23:28,350 --> 00:23:32,590
+Taylor's theorem في إثبات نظرية تعرف لنا ال test
+
+240
+00:23:32,590 --> 00:23:37,730
+إيه لها؟ قول يا محمد هل
+
+241
+00:23:37,730 --> 00:23:46,670
+جيت و تشوفها؟ في ال calculus في ال calculusكنا
+
+242
+00:23:46,670 --> 00:23:52,670
+ناخد اللي هو test لل increasing وال decreasing
+
+243
+00:23:52,670 --> 00:23:59,380
+باستخدام ال first derivativeإذا بتتذكروا كنا ناخد
+
+244
+00:23:59,380 --> 00:24:04,220
+أيضا اللي هو عشان نعرف الدالة concave up أو
+
+245
+00:24:04,220 --> 00:24:08,340
+concave down بوسط تمين ال second derivative test
+
+246
+00:24:08,340 --> 00:24:11,880
+ماشي الحال إذا كانت F double prime أكبر من صفر
+
+247
+00:24:11,880 --> 00:24:15,360
+كانت اللي هي concave up إذا كانت F double prime
+
+248
+00:24:15,360 --> 00:24:20,060
+أصغر من صفر كانت أشمالها concave down الآن هتشوفوا
+
+249
+00:24:20,060 --> 00:24:25,000
+النظرية اللي بتشابه اللي بنحكي و هتستنتج أيش اللي
+
+250
+00:24:25,000 --> 00:24:32,080
+بحكيهبجاوب على سؤالك شوف الآن الان عندي واضح انه
+
+251
+00:24:32,080 --> 00:24:36,940
+من التعريف ان نفحص الدالة انها convex ولا مش
+
+252
+00:24:36,940 --> 00:24:41,920
+convex بالشكل اللي حكيناه بالتعريف اللي حكيناه
+
+253
+00:24:41,920 --> 00:24:46,380
+بغلب وبعض الدول مش هنقدر او مش هنعرف اللي هو
+
+254
+00:24:46,380 --> 00:24:49,980
+لصعوبة الحسابات ان نثبت انها convex او مش convex
+
+255
+00:24:49,980 --> 00:24:58,330
+فاكيد بدنا testيسهّلنا عملية إيجاد أو معرفة إن هذه
+
+256
+00:24:58,330 --> 00:25:03,070
+الدالة is convex ولا لأ واضح اللي بحكيه؟ يلّا نشوف
+
+257
+00:25:03,070 --> 00:25:07,810
+مع بعض الان قبلها بيعطينا بس counter example
+
+258
+00:25:07,810 --> 00:25:15,110
+بقولّي إن ال convex function ممكن ما تكونش
+
+259
+00:25:15,110 --> 00:25:18,540
+differentiable على الفترة اللي فيها convexيعني
+
+260
+00:25:18,540 --> 00:25:20,660
+ممكن اللي هي ال .. ال .. ال .. احنا اللي جابنا قبل
+
+261
+00:25:20,660 --> 00:25:24,800
+شوية اللي جيبناها الدالة هي محلاها في انسياب وفي
+
+262
+00:25:24,800 --> 00:25:27,860
+differentiability و ال short أعلى من ال .. من
+
+263
+00:25:27,860 --> 00:25:31,260
+الدالة نفسها و كده و ال function differentiable
+
+264
+00:25:31,260 --> 00:25:34,440
+لكن هذا الكلام مش دايما حاصل انه ال convex
+
+265
+00:25:34,440 --> 00:25:37,100
+function تكون differentiable و أشهر منه و مثال
+
+266
+00:25:37,100 --> 00:25:41,460
+مشهور وسهل اللي هو F of X بسوء ال absolute value
+
+267
+00:25:41,460 --> 00:25:45,720
+لل X F of X بسوء ال absolute value لل X هذا اللي
+
+268
+00:25:45,720 --> 00:25:52,270
+هوis أكيد كلكم عارف إنها not differentiable at 0
+
+269
+00:25:52,270 --> 00:26:01,530
+but it is convex on any interval I containing إيه
+
+270
+00:26:01,530 --> 00:26:05,370
+إيش الـ0، بظبط؟ ال .. ال .. it is not
+
+271
+00:26:05,370 --> 00:26:08,030
+differentiable اللي عارفينها ال .. ال .. ال .. ال
+
+272
+00:26:08,030 --> 00:26:13,730
+.. نثبت إنها convex let على interval I X1 و X2
+
+273
+00:26:13,730 --> 00:26:19,800
+element in Iو T element in 0 و 1 اخسب اللي لان F
+
+274
+00:26:19,800 --> 00:26:29,780
+of 1 minus T في X1 زي T في X2 بساوية عبارة عن ال
+
+275
+00:26:29,780 --> 00:26:36,240
+absolute value واحد نقص T X1 زي T في X2 وهذا من
+
+276
+00:26:36,240 --> 00:26:39,700
+خلاص ال absolute value اصغر بساوية واحد نقص T في
+
+277
+00:26:39,700 --> 00:26:46,100
+X1زاية T في X2 وهذا طبعا مين هو؟ هو عبارة عن 1 نقص
+
+278
+00:26:46,100 --> 00:26:53,120
+T F of X1 زاية T في F of X2 إذا صار عندي ال F
+
+279
+00:26:53,120 --> 00:26:56,240
+أشمالها is convex but it is not differentiable
+
+280
+00:26:56,240 --> 00:27:00,620
+اطلع لفوق الان يجيلا اللي هو ال test اللي معطينيا
+
+281
+00:27:00,620 --> 00:27:07,060
+اللي بقوللي اللي بجاوب على سؤالك يا محمد let I be
+
+282
+00:27:08,590 --> 00:27:13,190
+And open interval and let f من I لR have a second
+
+283
+00:27:13,190 --> 00:27:18,890
+derivative on I نفترض أن الـ F'' إيش ما لها موجودة
+
+284
+00:27:18,890 --> 00:27:25,600
+على I إذا من رأس الدور مفترضين الـ F'' موجودةالـ
+
+285
+00:27:25,600 --> 00:27:30,140
+test هو كما قالي f is convex on i if and only if f
+
+286
+00:27:30,140 --> 00:27:34,740
+double prime a شمالها أكبر من 0 لكل x when element
+
+287
+00:27:34,740 --> 00:27:39,300
+in I بصور واضحة الأمر الآن بدي أستخدم في المرهان
+
+288
+00:27:39,300 --> 00:27:44,520
+اللي هو سؤال اللي هو 16 في ال exercises و هنحله
+
+289
+00:27:44,520 --> 00:27:48,380
+وهذا من ضمن ال exercises اللي معاكم هو Morgue و
+
+290
+00:27:48,380 --> 00:27:53,910
+هنحله في حينهالان ايش اللي هو ال .. ال .. ال .. ال
+
+291
+00:27:53,910 --> 00:27:58,390
+.. السؤال بيقول اذا كانت f double prime موجودةثم
+
+292
+00:27:58,390 --> 00:28:02,570
+اثبت لي ان f double prime can be written as limit
+
+293
+00:28:02,570 --> 00:28:06,470
+f of a زاد h نقص اتنين f of a زاد f of a minus h
+
+294
+00:28:06,470 --> 00:28:11,850
+على مين على h تربية ماشي الحال هذا بدأ اعتمد على
+
+295
+00:28:11,850 --> 00:28:17,950
+انه اللي هو مثبت في اثبات النظرية ونشوف كيف ناخد
+
+296
+00:28:17,950 --> 00:28:23,540
+الاتجاه الاول اول اشي بده اللي هو بنفترض انهعندي
+
+297
+00:28:23,540 --> 00:28:29,080
+ال .. اللي هو ال F is convex ومنها نصل لإنه ال
+
+298
+00:28:29,080 --> 00:28:34,300
+derivative ايه شمالها أكبر أو يساوي سفر ماشي الحال
+
+299
+00:28:34,300 --> 00:28:43,480
+خليني نشوف مع بعض كيف اللي هو نثبت
+
+300
+00:28:43,480 --> 00:28:55,740
+انه if F is convex thenF double prime of A أكبر
+
+301
+00:28:55,740 --> 00:29:01,720
+شهر السفر لما نقول suppose that
+
+302
+00:29:01,720 --> 00:29:11,780
+proof فوضيح بس .. suppose that أن F is convex
+
+303
+00:29:11,780 --> 00:29:16,880
+لنثبت أن F double prime of A أكبر شهر السفرالان
+
+304
+00:29:16,880 --> 00:29:21,240
+انا ال F is convex وين على ال I و differentiable
+
+305
+00:29:21,240 --> 00:29:25,560
+وين second derivative لها على ال I خليني اختار
+
+306
+00:29:25,560 --> 00:29:31,280
+الان اللي هو ال H بحيث بدي اختار H بحيث انه A
+
+307
+00:29:31,280 --> 00:29:36,440
+minus H و A زائد H التانين يكون هنا وين موجودات في
+
+308
+00:29:36,440 --> 00:29:39,600
+ال interval I اللي عليها شغل اللي انا بدي اشتغل
+
+309
+00:29:39,600 --> 00:29:49,760
+عليها الان خدلي الان Fof A واكتبها على صورة F of
+
+310
+00:29:49,760 --> 00:29:59,450
+نص في A زاد H زاد نص في A minus Hلأن هذه عبارة عن
+
+311
+00:29:59,450 --> 00:30:03,330
+نص A و نص A هي الـ A و نص H ناقص نص H بروحهم مع
+
+312
+00:30:03,330 --> 00:30:06,730
+بعض ده منرجع لمين لل F و فيها كده تعرفي ليش عملتيل
+
+313
+00:30:06,730 --> 00:30:12,450
+طبعا هذه هتشتغل بمثابة واحد ناقص T و هذه T إذا T
+
+314
+00:30:12,450 --> 00:30:16,370
+بين ال Zero و الواحد صحيحة و بما أن F is convex و
+
+315
+00:30:16,370 --> 00:30:20,750
+هذولة اتتين نقطين عبارة عن نقطين في I لأن هيك
+
+316
+00:30:20,750 --> 00:30:26,520
+اختار ال H هنا هيكون هذا أصغر أو يساوينص F of A
+
+317
+00:30:26,520 --> 00:30:35,620
+زائد H زائد نص F of A minus H واضح؟
+
+318
+00:30:35,620 --> 00:30:43,000
+أي سؤال؟ واضح أه؟ جيبلي هذا على الجهة التانية
+
+319
+00:30:43,000 --> 00:30:51,620
+بيصير عندي نص F of A زائد H زائد نص F of A minus H
+
+320
+00:30:51,620 --> 00:30:59,230
+ناقص F of Aكل و إيه شماله أكبر أو يساوي إيش أكبر
+
+321
+00:30:59,230 --> 00:31:05,010
+أو يساوي سفر وضرب الجهتين في اتنين بيصير عندي
+
+322
+00:31:05,010 --> 00:31:10,550
+مضربت في اتنين بيصير هذه اللي هو النص بيصير واحد و
+
+323
+00:31:10,550 --> 00:31:14,490
+النص هذا بيصير واحد و هذا بيصير إيه شماله ناقص
+
+324
+00:31:14,490 --> 00:31:22,650
+اتنين of of ايه اقسم لليه الآن على اتش تربيعوH
+
+325
+00:31:22,650 --> 00:31:26,370
+تربيع موجة بقى، إذا بيضلوا هذا أشمال المفضل أكبر
+
+326
+00:31:26,370 --> 00:31:32,030
+أو يساوي سفر خد ال limit للجهتين limit as H goes
+
+327
+00:31:32,030 --> 00:31:40,440
+to zero بيضلوا أكبر أو يساوي سفر، مظبوط؟ و هذاBy
+
+328
+00:31:40,440 --> 00:31:44,920
+ال exercise اللي بنحكي عنه هو عبارة عن مين؟ F
+
+329
+00:31:44,920 --> 00:31:48,100
+double prime of A فطلع عندي F double prime of A
+
+330
+00:31:48,100 --> 00:31:52,320
+أكبر أساوي الصفر طبعا ماقدرتش أستخدم هذا أنه بنكتب
+
+331
+00:31:52,320 --> 00:31:56,840
+على صورة هذا إلا في ضوء أن هو أصلا حاكي لي ال F
+
+332
+00:31:56,840 --> 00:32:03,940
+double prime exist on I واضح أه؟أي سؤال إذا الان
+
+333
+00:32:03,940 --> 00:32:08,580
+اطلع لفوق و بنكون احنا خلصنا الجزء الأول و بنكون
+
+334
+00:32:08,580 --> 00:32:11,860
+بدنا نبدأ بالجزء الثاني اللي هو بال prime أكبر
+
+335
+00:32:11,860 --> 00:32:17,600
+سواء سفر و نصل لمين لل F is convex أوضحت هذه خليني
+
+336
+00:32:17,600 --> 00:32:24,100
+بس أنسحها و أقول ما يليه
+
+337
+00:32:24,100 --> 00:32:27,560
+نفترض
+
+338
+00:32:27,560 --> 00:32:30,060
+لأن صبوذ
+
+339
+00:32:32,450 --> 00:32:39,370
+that f double prime of a أشمالها أكبر أو يساوي سفر
+
+340
+00:32:39,370 --> 00:32:45,330
+ماشي الحال .. بدنا نثبت لك أن f is convex كيف بدنا
+
+341
+00:32:45,330 --> 00:32:52,130
+نثبت ان f is convex؟ بدنا نقول let x1 و x2 بيه أي
+
+342
+00:32:52,130 --> 00:32:57,630
+element in مين؟ ان i و t element in zero واش و
+
+343
+00:32:57,630 --> 00:33:06,820
+واحدوسميلي X not هي عبارة عن واحد minus T في X
+
+344
+00:33:06,820 --> 00:33:13,580
+واحد زائد T في مين في X اتنين الآن بدي اطبق اللي
+
+345
+00:33:13,580 --> 00:33:16,480
+هو taylor's theorem اول اش طبعا taylor's theorem
+
+346
+00:33:16,480 --> 00:33:21,960
+قابل للتطبيق اللي عند مينعند and بتساوي واحد بس
+
+347
+00:33:21,960 --> 00:33:25,640
+لأن المشتقة الأولى والتانية موجودة بعد هيك ماحدش
+
+348
+00:33:25,640 --> 00:33:29,480
+حكالي فإذا بيحجلي ان اطبق Taylor's theorem بس على
+
+349
+00:33:29,480 --> 00:33:34,260
+وين على اللي هو and بتساوي سفر او and بتساوي واحد
+
+350
+00:33:34,260 --> 00:33:36,800
+and بتساوي سفر بيصير main value theorem and بتساوي
+
+351
+00:33:36,800 --> 00:33:41,000
+واحد اللي هو اللي بتطبق عليها و هتخدمني ماشي الحال
+
+352
+00:33:41,000 --> 00:33:46,470
+نشوفالان apply taylor's theorem ع الفترة على اللي
+
+353
+00:33:46,470 --> 00:33:51,970
+هو ال X واحد وال X note اللي عندى there exist اللي
+
+354
+00:33:51,970 --> 00:34:00,270
+هو C واحد between X واحد and X note such that اه
+
+355
+00:34:00,270 --> 00:34:08,370
+اللي هو F of X note مظبوط او F of .. ده طبقه على X
+
+356
+00:34:08,370 --> 00:34:15,010
+واحد اول شي F of X واحد بساوياللي هو F of X not
+
+357
+00:34:15,010 --> 00:34:22,550
+زائد F prime of X not في X واحد نقص X not زائد مين
+
+358
+00:34:22,550 --> 00:34:26,950
+الآن ال remainder زائد ال remainder ال remainder
+
+359
+00:34:26,950 --> 00:34:33,950
+إيش هو F double prime عند ال C واحد اللي لاجيناها
+
+360
+00:34:33,950 --> 00:34:41,390
+في X minus X not الكل تربيع على 2 فكتورياأصغر من
+
+361
+00:34:41,390 --> 00:34:48,530
+اللي هو similarly there exist c2 بتطبقها على x1
+
+362
+00:34:48,530 --> 00:34:57,090
+على x2 و x0 there exist a c2 between x2 and x0
+
+363
+00:34:57,090 --> 00:35:07,070
+such thatof x2 بيساوي f of x0 زاد f prime of x0 في
+
+364
+00:35:07,070 --> 00:35:17,690
+x2 ناقص x0 زاد f double prime of c2 في x minus x0
+
+365
+00:35:17,690 --> 00:35:24,950
+الكل تربيع على 2 factorial واضح اللي بعمله
+
+366
+00:35:28,900 --> 00:35:37,440
+فهي عندى اللى هى هذه مع هذه هى اللى بدى استخدمه
+
+367
+00:35:37,440 --> 00:35:43,500
+هنا الآن فى الوصول لإنه F is convex يعنى F of هذا
+
+368
+00:35:43,500 --> 00:35:49,180
+المقدار يكون أصغر أو يساوي اللى هو F 1 minus T
+
+369
+00:35:49,180 --> 00:35:56,920
+فهذه زائد T فهذه إذا بدى أحسب الآن 1 minus T ف F
+
+370
+00:35:56,920 --> 00:36:05,420
+of X 1زائد T في F of إيش X2 بدي أحسبها و أوصلها
+
+371
+00:36:05,420 --> 00:36:09,080
+لعلاقتها مع مين مع F of X note اللي هي F of هذه
+
+372
+00:36:09,080 --> 00:36:20,940
+بساوي اضربلي بيصير T واحد نقص T في F of X note
+
+373
+00:36:20,940 --> 00:36:31,310
+معايا يا شباب زائد واحد minus Tف F prime of X note
+
+374
+00:36:31,310 --> 00:36:40,590
+في X واحد minus X note زائد اللي هو mean ده دروفة
+
+375
+00:36:40,590 --> 00:36:47,390
+هادة اللي هو واحد minus T في F double prime of C
+
+376
+00:36:47,390 --> 00:36:52,970
+واحد في X minus X note اللي كل تربيع على اتنين
+
+377
+00:36:52,970 --> 00:37:02,060
+factorial اللي هو اتنين معايا يا شبابزائد T .. T
+
+378
+00:37:02,060 --> 00:37:10,500
+.. F of X2 هيها اللي هي F of X note زائد T في F
+
+379
+00:37:10,500 --> 00:37:15,860
+prime of X note بضرب فادي قاعد في X2 نقص X note
+
+380
+00:37:15,860 --> 00:37:26,720
+زائد اللي هو T في F double prime of C2fx-x0 الكل
+
+381
+00:37:26,720 --> 00:37:34,100
+تربيه على مين؟ على اتنين نيجي الان و يساوي اطلعلي
+
+382
+00:37:34,100 --> 00:37:41,720
+على هذه الان واحد minus T في F of X0 وT في F of X0
+
+383
+00:37:41,720 --> 00:37:51,520
+بيصير عبارة عن F of X0 مظبوط؟ زائدالان هذا f prime
+
+384
+00:37:51,520 --> 00:37:55,420
+of x naught مضروبة في T مضروبة في هذه إذا عبارة عن
+
+385
+00:37:55,420 --> 00:38:03,660
+واحد minus T زائد T اه اللي هو مية المية بيصير
+
+386
+00:38:03,660 --> 00:38:10,300
+عندي الآن خد اللي هي x واحد ناقص x naught وخلينا
+
+387
+00:38:10,300 --> 00:38:19,510
+نجمحها واحد minus يعني خلينا نضيع ال Tفيها نقص
+
+388
+00:38:19,510 --> 00:38:24,730
+اتنين و X واحد او X او X واحد نفردها واحد minus T
+
+389
+00:38:24,730 --> 00:38:32,970
+في F ناخد عن المشترك يمكن أسهلنا F prime of X note
+
+390
+00:38:32,970 --> 00:38:43,330
+في واحد minus T في X واحد ناقص X note معايا شباب
+
+391
+00:38:43,330 --> 00:38:51,670
+زائد باخد بين هذه هي عمالو بين هذه أخدت ال F prime
+
+392
+00:38:51,670 --> 00:38:57,430
+of X naught عام نشترك بينهم زائد T في X2 ناقص X
+
+393
+00:38:57,430 --> 00:39:06,290
+naught زائد هذا المقدار كله بعد إذنكم هي اللي
+
+394
+00:39:06,290 --> 00:39:12,390
+بلزرج هذا اللي حطيته بلزرج سميني يا إيش R واضح
+
+395
+00:39:12,390 --> 00:39:21,890
+زائد Rالان نطلع على اللي عندي هدا واحد ناقص T اول
+
+396
+00:39:21,890 --> 00:39:27,670
+اشي ناقص T في ناقص X Note بيصير اللي هو زائد T في
+
+397
+00:39:27,670 --> 00:39:31,850
+X Note مع ناقص T في X Note بروحين مظبوط؟ اذا هذا
+
+398
+00:39:31,850 --> 00:39:38,410
+الان بروح وهذا
+
+399
+00:39:38,410 --> 00:39:44,290
+برضه بروح صح ولا لأ؟ نتأكد في اللي عملناه
+
+400
+00:39:48,090 --> 00:40:00,190
+واضح اه بظل x .. x note اه بظل ناقص اللي هو زائد x
+
+401
+00:40:00,190 --> 00:40:07,950
+note ناقص x note ناقص x note عملها كمان مرة بظل
+
+402
+00:40:07,950 --> 00:40:15,160
+ناقص x note من هناو من هنا بيظهر مين ناقص X note
+
+403
+00:40:15,160 --> 00:40:21,640
+لأنه بتظهر تبعت اللي هنا تظهر ناقص X note واضح أه؟
+
+404
+00:40:21,640 --> 00:40:28,760
+الان هذه كلها على بعض مين هي؟ هذه إيش اسمها؟ كلها
+
+405
+00:40:28,760 --> 00:40:32,900
+اسمها X note أه؟
+
+406
+00:40:32,900 --> 00:40:36,960
+فبيصير X note ناقص X note إيش بيصير؟ سفر فبيصير
+
+407
+00:40:36,960 --> 00:40:41,560
+هذا المقدار عبارة عن Fof x naught زائد ال
+
+408
+00:40:41,560 --> 00:40:45,380
+remainder مين؟ R نتطلع على الـ R هذا أو هو R لما
+
+409
+00:40:45,380 --> 00:40:48,000
+نقول remainder؟ الـ R مجموع الـ remainder دلين
+
+410
+00:40:48,000 --> 00:40:58,960
+الان ملاحظ هو ماعطيني عندي الان أفضل براءة عندي
+
+411
+00:40:58,960 --> 00:41:05,340
+هذا X نقص X واحد كل تربيع على اتنين موجبة و X نقص
+
+412
+00:41:05,340 --> 00:41:10,330
+X اتنين نوت على اتنين اشمالها موجبةوالان ال f
+
+413
+00:41:10,330 --> 00:41:15,510
+double prime عندى اللى هتطلع أشمالها موجبة بنفس
+
+414
+00:41:15,510 --> 00:41:21,550
+الطريقة السابقة زى ما عملنا قبل شوية عندى f double
+
+415
+00:41:21,550 --> 00:41:27,310
+prime أكبر يساوي سفر هيطلع عندى في الجوار اللى
+
+416
+00:41:27,310 --> 00:41:30,610
+بطبق عليه برضه أكبر يساوي سفر فبطلع عندى هذا موجب
+
+417
+00:41:30,610 --> 00:41:35,690
+و هذا موجب فبطلع عندى هذا المقدار اللى بدأنا فيه
+
+418
+00:41:36,380 --> 00:41:41,720
+هذا المقدار اللي بدأنا فيه طلع بساوي هذا المقدار
+
+419
+00:41:41,720 --> 00:41:49,760
+اللي هو هذا أكبر أو يساوي F of X naught اللي هي
+
+420
+00:41:49,760 --> 00:41:57,540
+mean F of X naught اللي عبارة عن F معين واحد نقص T
+
+421
+00:41:57,540 --> 00:42:07,040
+في X واحد زائد T في X اتنين هاي اللي وصلنا إليهمع
+
+422
+00:42:07,040 --> 00:42:21,020
+هذا وهذا تعريف أيه؟ F is convex F is convex أي
+
+423
+00:42:21,020 --> 00:42:21,680
+سؤال؟
+
+424
+00:42:24,290 --> 00:42:29,090
+و بهيك نكون خلصنا ال section اللي الأخير في
+
+425
+00:42:29,090 --> 00:42:34,250
+chapter اللي هو 6 خلصنا الجزء الأول من المحاضرة و
+
+426
+00:42:34,250 --> 00:42:37,250
+الآن بدنا نبدأ في الجزء الثاني من المحاضرة
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..161fd9660da40193822583527186577c3d2a0a14
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_raw.json
@@ -0,0 +1 @@
+{"segments": [{"id": 1, "seek": 1989, "start": 5.48, "end": 19.9, "text": "بسم الله الرحمن الرحيم محاضرة رقم تسعة مساق تحليل حقيقي اتنين طلاب الجامعة الإسلامية كلية العلوم قسم الرياضيات", "tokens": [3555, 38251, 21984, 34892, 5016, 27842, 34892, 5016, 32640, 3714, 5016, 46958, 25720, 12602, 4587, 2304, 6055, 3794, 27884, 47524, 995, 4587, 6055, 5016, 20292, 1211, 11331, 38436, 38436, 1975, 2655, 1863, 9957, 23032, 15040, 3555, 25724, 10943, 27884, 33688, 3794, 37440, 10632, 9122, 20292, 3660, 18863, 1211, 20498, 12174, 38251, 2423, 16572, 46958, 1829, 9307], "avg_logprob": -0.07709703633659765, "compression_ratio": 1.6370967741935485, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 5.48, "end": 5.92, "word": "بسم", "probability": 0.8251953125}, {"start": 5.92, "end": 6.12, "word": " الله", "probability": 0.9580078125}, {"start": 6.12, "end": 6.52, "word": " الرحمن", "probability": 0.9698893229166666}, {"start": 6.52, "end": 7.06, "word": " الرحيم", "probability": 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differentiable و أشهر منه و مثال مشهور وسهل اللي هو F of X بسوء ال absolute value لل X F of X بسوء ال absolute value لل X هذا اللي هو", "tokens": [40228, 22653, 3714, 43020, 13672, 1829, 39896, 2423, 4386, 2423, 4386, 2423, 4386, 1975, 5016, 8315, 13672, 1829, 10874, 16758, 8315, 12174, 36150, 13412, 2407, 10632, 13672, 1829, 10874, 1829, 3555, 8315, 11296, 32748, 6027, 3660, 39896, 3714, 5016, 15040, 11296, 8978, 16472, 3794, 1829, 16758, 4032, 41185, 819, 72, 2310, 4032, 2423, 2099, 5551, 3615, 23942, 9154, 2423, 4386, 9154, 32748, 6027, 3660, 8717, 36178, 11296, 4032, 9122, 3215, 3224, 4032, 2423, 2445, 819, 9364, 44381, 23758, 2423, 28820, 10943, 37893, 11778, 47302, 15042, 11331, 33546, 1211, 16472, 3224, 2423, 42432, 2445, 6055, 30544, 819, 9364, 4032, 5551, 8592, 3224, 2288, 9154, 3224, 4032, 50113, 6027, 37893, 3224, 13063, 46952, 3224, 1211, 13672, 1829, 31439, 479, 295, 1783, 4724, 3794, 2407, 38207, 2423, 8236, 2158, 24976, 1783, 479, 295, 1783, 4724, 3794, 2407, 38207, 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T .. 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"end": 2518.64, "word": " X", "probability": 0.9716796875}, {"start": 2518.64, "end": 2519.26, "word": " واحد", "probability": 0.96728515625}, {"start": 2519.26, "end": 2520.1, "word": " زائد", "probability": 0.833984375}, {"start": 2520.1, "end": 2520.54, "word": " T", "probability": 0.97412109375}, {"start": 2520.54, "end": 2521.36, "word": " في", "probability": 0.96044921875}, {"start": 2521.36, "end": 2521.58, "word": " X", "probability": 0.9970703125}, {"start": 2521.58, "end": 2522.04, "word": " اتنين", "probability": 0.873779296875}, {"start": 2522.04, "end": 2523.22, "word": " هاي", "probability": 0.666259765625}, {"start": 2523.22, "end": 2523.44, "word": " اللي", "probability": 0.987548828125}, {"start": 2523.44, "end": 2523.96, "word": " وصلنا", "probability": 0.9837239583333334}, {"start": 2523.96, "end": 2524.46, "word": " إليه", "probability": 0.6553548177083334}], "temperature": 1.0}, {"id": 103, "seek": 254167, "start": 2526.66, "end": 2541.68, "text": "مع هذا وهذا تعريف أيه؟ F is convex F is convex أي سؤال؟", "tokens": [2304, 3615, 23758, 37037, 15730, 37279, 16572, 5172, 36632, 3224, 22807, 479, 307, 42432, 479, 307, 42432, 36632, 8608, 33604, 6027, 22807], "avg_logprob": -0.29483694615571393, "compression_ratio": 1.1111111111111112, "no_speech_prob": 4.76837158203125e-07, "words": [{"start": 2526.66, "end": 2527.04, "word": "مع", "probability": 0.931396484375}, {"start": 2527.04, "end": 2527.44, "word": " هذا", "probability": 0.96875}, {"start": 2527.44, "end": 2528.46, "word": " وهذا", "probability": 0.81640625}, {"start": 2528.46, "end": 2528.98, "word": " تعريف", "probability": 0.98779296875}, {"start": 2528.98, "end": 2530.18, "word": " أيه؟", "probability": 0.5109456380208334}, {"start": 2530.18, "end": 2531.18, "word": " F", "probability": 0.3740234375}, {"start": 2531.18, "end": 2532.8, "word": " is", "probability": 0.90380859375}, {"start": 2532.8, "end": 2535.18, "word": " convex", "probability": 0.75830078125}, {"start": 2535.18, "end": 2536.72, "word": " F", "probability": 0.32568359375}, {"start": 2536.72, "end": 2537.2, "word": " is", "probability": 0.9482421875}, {"start": 2537.2, "end": 2537.9, "word": " convex", "probability": 0.9658203125}, {"start": 2537.9, "end": 2541.02, "word": " أي", "probability": 0.7578125}, {"start": 2541.02, "end": 2541.68, "word": " سؤال؟", "probability": 0.9713134765625}], "temperature": 1.0}, {"id": 104, "seek": 255725, "start": 2544.29, "end": 2557.25, "text": "و بهيك نكون خلصنا ال section اللي الأخير في chapter اللي هو 6 خلصنا الجزء الأول من المحاضرة و الآن بدنا نبدأ في الجزء الثاني من المحاضرة", "tokens": [2407, 39627, 1829, 4117, 8717, 30544, 16490, 1211, 9381, 8315, 2423, 3541, 13672, 1829, 16247, 9778, 13546, 8978, 7187, 13672, 1829, 31439, 1386, 16490, 1211, 9381, 8315, 25724, 11622, 38207, 16247, 12610, 9154, 9673, 5016, 46958, 25720, 4032, 6024, 48506, 47525, 8315, 8717, 44510, 10721, 8978, 25724, 11622, 38207, 6024, 104, 7649, 1829, 9154, 9673, 5016, 46958, 25720], 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"word": " اللي", "probability": 0.919189453125}, {"start": 2549.91, "end": 2550.15, "word": " هو", "probability": 0.98779296875}, {"start": 2550.15, "end": 2550.53, "word": " 6", "probability": 0.59619140625}, {"start": 2550.53, "end": 2551.87, "word": " خلصنا", "probability": 0.872314453125}, {"start": 2551.87, "end": 2552.23, "word": " الجزء", "probability": 0.8341471354166666}, {"start": 2552.23, "end": 2552.61, "word": " الأول", "probability": 0.979736328125}, {"start": 2552.61, "end": 2552.77, "word": " من", "probability": 0.9853515625}, {"start": 2552.77, "end": 2553.45, "word": " المحاضرة", "probability": 0.98828125}, {"start": 2553.45, "end": 2554.25, "word": " و", "probability": 0.70556640625}, {"start": 2554.25, "end": 2554.77, "word": " الآن", "probability": 0.6883544921875}, {"start": 2554.77, "end": 2555.15, "word": " بدنا", "probability": 0.917724609375}, {"start": 2555.15, "end": 2555.63, "word": " نبدأ", "probability": 0.9943033854166666}, {"start": 2555.63, "end": 2555.81, "word": " في", "probability": 0.91748046875}, {"start": 2555.81, "end": 2556.31, "word": " الجزء", "probability": 0.9876302083333334}, {"start": 2556.31, "end": 2556.83, "word": " الثاني", "probability": 0.9619140625}, {"start": 2556.83, "end": 2556.95, "word": " من", "probability": 0.607421875}, {"start": 2556.95, "end": 2557.25, "word": " المحاضرة", "probability": 0.962158203125}], "temperature": 1.0}], "language": "ar", "language_probability": 1.0, "duration": 2557.49225, "duration_after_vad": 2500.3318749999944}
\ No newline at end of file
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_raw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_raw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..200776e43e2b0ead4cb1f0a1eddba56a9d2016e8
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/uaBBi7JsFSQ_raw.srt
@@ -0,0 +1,1712 @@
+1
+00:00:05,480 --> 00:00:11,800
+بسم الله الرحمن الرحيم محاضرة رقم تسعة مساق تحليل
+
+2
+00:00:11,800 --> 00:00:19,160
+حقيقي اتنين طلاب الجامعة الإسلامية كلية العلوم قسم
+
+3
+00:00:19,160 --> 00:00:24,470
+الرياضياتالمحاضرة اليوم على جزئين الجزء الأول
+
+4
+00:00:24,470 --> 00:00:29,350
+هنتحدث عن applications of taylor's theorem والجزء
+
+5
+00:00:29,350 --> 00:00:34,250
+الثاني هنتحدث عن اللي هو discussion لsection 6.2
+
+6
+00:00:34,250 --> 00:00:37,890
+و6.3 اللي هو discussion لل mean value theorem وإذا
+
+7
+00:00:37,890 --> 00:00:43,270
+في وجهة بنعمل discussion ل اللي هو lobitals rules
+
+8
+00:00:43,960 --> 00:00:48,240
+الان خلّينا نكمل اللي اتحدثنا عنه المرة الماضية
+
+9
+00:00:48,240 --> 00:00:53,200
+ولكن على صورة تطبيقات ليه؟ اللي هو Taylor's
+
+10
+00:00:53,200 --> 00:00:58,660
+theorem وكيف بدنا اللي هو نستغل Taylor's theoremفي
+
+11
+00:00:58,660 --> 00:01:03,360
+إثبات بعض النظريات المهمة في موضوع اللي هو ال
+
+12
+00:01:03,360 --> 00:01:08,080
+maximum و ال minimum أو اللي هو ال test لإن اللي
+
+13
+00:01:08,080 --> 00:01:12,140
+عندنا هل هي local maximum أو local minimum ولا لا
+
+14
+00:01:12,140 --> 00:01:15,860
+هي local maximum ولا هي local minimum كنا اتحدثنا
+
+15
+00:01:15,860 --> 00:01:20,560
+سابقا على اللي هو ال first derivative test لإن
+
+16
+00:01:20,560 --> 00:01:23,200
+النقطة اللي تكون الداخلية هل في عندها local
+
+17
+00:01:23,200 --> 00:01:27,540
+maximum أو local minimumالان بدنا نتحدث عن الـ
+
+18
+00:01:27,540 --> 00:01:31,140
+Higher Derivative Test احنا يمكن نمر عنا اللي هو
+
+19
+00:01:31,140 --> 00:01:33,720
+الـ Second Derivative Test اللي هو في ال .. في
+
+20
+00:01:33,720 --> 00:01:36,900
+الCalculus اللي قابلة نتحدث على ال Higher
+
+21
+00:01:36,900 --> 00:01:40,080
+Derivative Test و اللي هي .. اللي هي ال Second
+
+22
+00:01:40,080 --> 00:01:44,090
+Derivative Test جزء من هذا اللي هو ال testنتطلع
+
+23
+00:01:44,090 --> 00:01:48,030
+على الـ test إيش بقول اللي هي عبارة عن theorem 644
+
+24
+00:01:48,030 --> 00:01:52,310
+let I be an interval نفترض أن I عبارة عن interval
+
+25
+00:01:52,310 --> 00:01:57,510
+و X note اللي هي نقطة اللي احنا بدنا نفحص عندها هل
+
+26
+00:01:57,510 --> 00:02:02,770
+في maximum أو في minimum أو لا في maximum ولا في
+
+27
+00:02:02,770 --> 00:02:06,470
+minimum X note اللي هي عبارة عن interior point
+
+28
+00:02:06,470 --> 00:02:10,890
+لمن؟ لل interval I يعني نقطة داخليةونفترض الـ N
+
+29
+00:02:10,890 --> 00:02:14,570
+تبعتنا أكبر أو ساوى 2 لأن الـ N أكبر أو ساوى 2
+
+30
+00:02:14,570 --> 00:02:17,690
+يعني الـ Second Derivative Test أو أعلى هو اللي
+
+31
+00:02:17,690 --> 00:02:20,910
+بده يحددنا أنه عندنا Maximum أو Minimum أو مافيش
+
+32
+00:02:20,910 --> 00:02:25,450
+لا Maximum ولا Minimum Suppose ده ال derivatives
+
+33
+00:02:25,450 --> 00:02:31,120
+اللي هي F' و F'' لعند FN existsexist and are
+
+34
+00:02:31,120 --> 00:02:34,040
+continuous in a neighborhood of x naught نفترض إن
+
+35
+00:02:34,040 --> 00:02:39,140
+هدولة continuous وموجودة في اللي هو
+
+36
+00:02:41,860 --> 00:02:44,760
+اللي هي في neighborhood حوالي مين الـ X Note
+
+37
+00:02:44,760 --> 00:02:49,920
+ونفترض أن F' عند X Note لعند F N ناقص واحد prime
+
+38
+00:02:49,920 --> 00:02:54,060
+of X Note إيش بيساوي؟ سفر لكن عند F N of X Note
+
+39
+00:02:54,060 --> 00:02:59,140
+إيش مالها؟ لاتساوي سفر إذا ال test إيش بيصير؟ أول
+
+40
+00:02:59,140 --> 00:03:06,120
+إشيبنشوف f prime نقطة x note هذه نقطة متوقع عندها
+
+41
+00:03:06,120 --> 00:03:10,340
+maximum أو minimum أو مافيش شيء من النقاط اللي هي
+
+42
+00:03:10,340 --> 00:03:13,520
+بدنا نفحص عندها maximum أو minimum إيش بدنا نسوي
+
+43
+00:03:13,520 --> 00:03:18,520
+في هذا ال test بنجيب f prime عند ال x note إذا
+
+44
+00:03:18,520 --> 00:03:23,500
+لجيناها سفر بنيجي لل f double prime عند ال x note
+
+45
+00:03:23,500 --> 00:03:29,130
+إذا لجيناها سفراللي هو بنيجي للـ F W' عند الـ X0
+
+46
+00:03:29,130 --> 00:03:34,830
+وين بنصل للـ F W' .. وين بنصل للـ F N of X0 لا
+
+47
+00:03:34,830 --> 00:03:39,210
+تساوي سفر؟ بيبدأ الخطوات تبعات ال test ايش ال
+
+48
+00:03:39,210 --> 00:03:43,710
+test؟ اللي هي الأول إذا كانت N even طلعت عندنا N
+
+49
+00:03:43,710 --> 00:03:47,850
+هادئية أول واحدة ال derivative عندها عند ال X0 مش
+
+50
+00:03:47,850 --> 00:03:53,930
+سفروكانت ال Fn of X0 أكبر من 0 بتكون ال F أيه
+
+51
+00:03:53,930 --> 00:03:57,470
+شمالها؟ اللي هي relative minimum زي ال second
+
+52
+00:03:57,470 --> 00:04:01,710
+derivative test كنا لما نقول ال FW' عند ال X0 أكبر
+
+53
+00:04:01,710 --> 00:04:06,170
+من 0 على طول نحكم أنه في local minimum هنا الآن لأ
+
+54
+00:04:06,170 --> 00:04:08,510
+احنا على ال higher derivative و هتحللنا المشاكل
+
+55
+00:04:08,510 --> 00:04:10,290
+اللي كانت ماتتحللاش جامعي اللي هي second
+
+56
+00:04:10,290 --> 00:04:16,670
+derivative test الان Fn is evenو ال F N of X0 أصغر
+
+57
+00:04:16,670 --> 00:04:21,510
+من 0 إذا Fها زي إيه شمالها relative maximum at X0
+
+58
+00:04:21,510 --> 00:04:25,670
+إذا لو طلعت عندي N is odd إيش يعني N is odd؟ يعني
+
+59
+00:04:25,670 --> 00:04:31,130
+أول .. أول derivative ال F N عندها لاتساوي 0 طلعت
+
+60
+00:04:31,130 --> 00:04:35,390
+عندي هذه oddطبعا أكبر إذا هو 2، 3 أو 5 أو 7 إلى
+
+61
+00:04:35,390 --> 00:04:39,490
+آخره على طول احنا بنحكم then F has neither
+
+62
+00:04:39,490 --> 00:04:43,610
+relative minimum nor إيش ما لها relative maximum
+
+63
+00:04:43,610 --> 00:04:50,790
+at X note بتصور هيك النظرية واضحة إيش نصها بنجيبF
+
+64
+00:04:50,790 --> 00:04:56,570
+prime of X0 و F double prime of X0 إلى آخره إذا
+
+65
+00:04:56,570 --> 00:05:01,170
+هذا بيساوي سفر بهذا بيساوي سفر بضلي ماشي لعند F N
+
+66
+00:05:01,170 --> 00:05:08,550
+of X0 لاتساوي سفر و اگ تست تصل لاتساوي سفر باجي لل
+
+67
+00:05:08,550 --> 00:05:17,620
+test اللي هو uneven عند حالتينالحالة الأولى FN of
+
+68
+00:05:17,620 --> 00:05:27,280
+X0 أكبر من 0 Local MinimumFn of X0 أصغر من 0 Local
+
+69
+00:05:27,280 --> 00:05:33,180
+maximum لكن إذا طلعت n is odd يعني أول واحدة بعد
+
+70
+00:05:33,180 --> 00:05:37,720
+التنين اللي هو بيطلع عند F ال derivative لها Fn of
+
+71
+00:05:37,720 --> 00:05:43,200
+X0 لا تساوي 0 طلعت عند n is odd إذا بحكم على طول F
+
+72
+00:05:43,200 --> 00:05:52,030
+has neither Local maximum nor Local minimumتعالوا
+
+73
+00:05:52,030 --> 00:05:58,670
+اللي هو نبرهم اللي هو النظرية ونشوف إنه حجر الأساس
+
+74
+00:05:58,670 --> 00:06:02,470
+للبرهان اللي هي taylor's theorem هي taylor's
+
+75
+00:06:02,470 --> 00:06:12,430
+theorem تنشوف كيف الآن عندي بدنا نطبق taylor's
+
+76
+00:06:12,430 --> 00:06:15,820
+theoremعلى .. طبعا احنا بنشتغل على الـ interval من
+
+77
+00:06:15,820 --> 00:06:21,520
+I بناخد اللي هو X element من I و بنطبقلها من الـ
+
+78
+00:06:21,520 --> 00:06:26,120
+Taylor's theorem عند النقطة اللي احنا بدنا نختبر
+
+79
+00:06:26,120 --> 00:06:29,840
+عند X0 فهي اللي بده أجيب Taylor's theorem حواليها
+
+80
+00:06:29,840 --> 00:06:36,160
+اذا حسب Taylor's theorem ان there exists C between
+
+81
+00:06:36,160 --> 00:06:45,630
+اللي هي X and X0 such thatاللي هو F of X F of X
+
+82
+00:06:45,630 --> 00:06:52,110
+بساوي بي أن ناقص واحد بدا أقبّقها اللي بحيث أن ال
+
+83
+00:06:52,110 --> 00:06:56,250
+polynomial ما أن ناقص واحد ليش؟ لإنه أنا ماعطيني
+
+84
+00:06:56,250 --> 00:07:01,110
+هو عنده ال derivative اللي هي لعند F أن ناقص واحد
+
+85
+00:07:01,110 --> 00:07:05,870
+كلها سفار واللي مش سفر اللي هي مين F n of X not
+
+86
+00:07:06,550 --> 00:07:11,390
+فعند اللي هي R نقص واحد هطبق عشان تحكم لل FN of X
+
+87
+00:07:11,390 --> 00:07:15,310
+node على إنه عندني local maximum أو local minimum
+
+88
+00:07:15,310 --> 00:07:19,750
+أو neither كيف؟ خلّينا نطبق ال Taylor's theorem
+
+89
+00:07:19,750 --> 00:07:28,350
+زائد اللي هو أشملها R N نقص واحد of X هذه عندي
+
+90
+00:07:28,350 --> 00:07:33,230
+Taylor's theorem الآن ال BN نقص واحد اللي هي عبارة
+
+91
+00:07:33,230 --> 00:07:40,820
+عن إياش Fof x naught زائد f prime of x naught في x
+
+92
+00:07:40,820 --> 00:07:45,440
+minus x naught اللي هو على واحد factorial زائد لما
+
+93
+00:07:45,440 --> 00:07:51,400
+أصل لآخر واحد f n ناقص واحد of x naught في x minus
+
+94
+00:07:51,400 --> 00:07:55,540
+x naught قس n ناقص واحد على n ناقص واحد لكل
+
+95
+00:07:55,540 --> 00:08:02,700
+factorial زائد main ال remainder اللي هو main اللي
+
+96
+00:08:02,700 --> 00:08:09,810
+هو عبارة عن xنقص x note أس n على n factorial في f
+
+97
+00:08:09,810 --> 00:08:18,850
+n of mean of عندي c و ال c مالها between x note
+
+98
+00:08:18,850 --> 00:08:27,170
+and mean and x واضح و يساوي الأن كل هذول أصفار صح
+
+99
+00:08:27,170 --> 00:08:33,500
+ولا لأ إذا حيصير عندي هذا بيساوي لإنه معطيليالـ F
+
+100
+00:08:33,500 --> 00:08:41,120
+prime of X note لعند FN نقص واحد of X note كله
+
+101
+00:08:41,120 --> 00:08:46,700
+نصفار إذا صلي عندي هذا بيساوي F of X note لأن هذا
+
+102
+00:08:46,700 --> 00:08:55,740
+كله بتصفر زاد X minus X note أس N على N factorial
+
+103
+00:08:55,740 --> 00:09:09,290
+في FNof c واضح اه اي سؤال طيب عندى انا اللى هو
+
+104
+00:09:09,290 --> 00:09:19,230
+معطينى ال ال f n of x naught اللى هى لاتساوى سفر
+
+105
+00:09:19,230 --> 00:09:25,170
+اذا ال f n of x naught يا اكبر من سفر يا اصغر من h
+
+106
+00:09:25,170 --> 00:09:36,570
+من سفرإذا أكيد عندي ال FN of X note و ال FN of C
+
+107
+00:09:36,570 --> 00:09:43,810
+الهن أيش الهن the same sign ماشي اه الهن نفس نين
+
+108
+00:09:43,810 --> 00:09:50,770
+نفس اللي هو الإشارة طبعا السبب أن FN أشمالها is
+
+109
+00:09:50,770 --> 00:09:55,950
+continuous انوضحلكم إياها تتذكروا
+
+110
+00:09:57,370 --> 00:10:02,330
+قلنا خلّينا ناخد حالة نفترض أن هذا أكبر من مين من
+
+111
+00:10:02,330 --> 00:10:12,690
+سفر احنا كنا نقول limit أي حاجة g of x لما x تروح
+
+112
+00:10:12,690 --> 00:10:18,970
+لل c مثلا أكبر من سفر معناته في جوار في جوار
+
+113
+00:10:18,970 --> 00:10:23,210
+حوالين اللي هو ال c ال g of x نفسها إيش هتكون أكبر
+
+114
+00:10:23,210 --> 00:10:28,110
+من سفر لكل x في هذا الجواريعني هيكون عندي إذاً g
+
+115
+00:10:28,110 --> 00:10:37,710
+of x أكبر من 0 لكل x في sum جوار delta of mean of
+
+116
+00:10:37,710 --> 00:10:41,690
+z هذا اللي بدأ أستخدمها الآن في الوصول للي بديها
+
+117
+00:10:41,690 --> 00:10:45,590
+خلّيني نشوف كيف بدأ نستخدم هذه المعلومة الآن أنا
+
+118
+00:10:45,590 --> 00:10:55,750
+عندي الـ Fn is continuous نفسها إذاً limit Fnof X
+
+119
+00:10:55,750 --> 00:11:03,290
+أفقًا of X اللي هو لما الـ X تروح للـ X0 اللي هو
+
+120
+00:11:03,290 --> 00:11:10,070
+أش متساوي أفقًا of X0 وبرضه continuous عند مين؟
+
+121
+00:11:11,690 --> 00:11:16,810
+وبرضه continuous عند مين؟ عند الـ C معايا وعند الـ
+
+122
+00:11:16,810 --> 00:11:21,030
+C إيش مالها؟ continuous أنا اللي هيلزمني مين؟ هنا
+
+123
+00:11:21,030 --> 00:11:26,930
+الـ X note مظبوط ولا لأ؟ عندي هتلزمني مين؟ اللي هي
+
+124
+00:11:26,930 --> 00:11:35,950
+ال X note إذا صار عنديفن of X note هي اللي هتحددلي
+
+125
+00:11:35,950 --> 00:11:45,090
+من هنا اللي هي إشارة مين فن of C واضح الآن من هذه
+
+126
+00:11:45,090 --> 00:11:54,310
+إذا كانت F of X note مثلا أكبر من سفر هتطلع حد
+
+127
+00:11:54,310 --> 00:11:58,870
+أكبر من سفر خلينا نشوفها إذا الآن خليني أبدأ في
+
+128
+00:11:58,870 --> 00:12:06,270
+مينفي I بدي افترض مين الان fn of x not أكبر من أيش
+
+129
+00:12:06,270 --> 00:12:13,010
+من سفر و افترض ان الان عند ايش معنها is even بدي
+
+130
+00:12:13,010 --> 00:12:19,950
+أنطلق من هذه اذا الان fn of x not أكبر من سفر و n
+
+131
+00:12:19,950 --> 00:12:25,730
+is even بناء على هذه for every y أكبر من سفر there
+
+132
+00:12:25,730 --> 00:12:32,440
+exists such that المهم بيطلع عندىاللي هو F هذه ايش
+
+133
+00:12:32,440 --> 00:12:37,880
+مفترضين احنا اكبر من ايه ايش من سفر اذا حسب اللي
+
+134
+00:12:37,880 --> 00:12:45,450
+حكيته قبل بشوية there existV Delta of إيش؟ Of X0
+
+135
+00:12:45,450 --> 00:12:53,590
+حوالي مين؟ الـ X0 Such that اللي هو FN of X زي
+
+136
+00:12:53,590 --> 00:12:59,290
+إشارة هذه اللي هي أكبر من 0 لكل X وين في الـ V
+
+137
+00:12:59,290 --> 00:13:06,880
+Delta of مين؟ Of X0ماشي الحال واللي من ضمن هدوله
+
+138
+00:13:06,880 --> 00:13:15,060
+مين؟ C إذا صارت عندي الان FN of C برضه إيه شمالها؟
+
+139
+00:13:15,060 --> 00:13:21,860
+أكبر من سفر الان اتضحت الصورة الان صارت عندي من
+
+140
+00:13:21,860 --> 00:13:26,420
+الأصل F of X بسوء F of X note زائد هذا المقدار بدا
+
+141
+00:13:26,420 --> 00:13:31,020
+أقول الآن بما أن هذه أكبر من سفر وان أنا مفترضها
+
+142
+00:13:31,020 --> 00:13:36,490
+إنها evenإذا حيصير عندى هذا أكبر من صفر وهذا أكبر
+
+143
+00:13:36,490 --> 00:13:41,570
+من صفر إذا الـ remainder كله إيش ماله أكبر من صفر
+
+144
+00:13:41,570 --> 00:13:47,710
+مدام الـ remainder أكبر من صفر واضح اللي بحكي FN
+
+145
+00:13:47,710 --> 00:13:53,010
+of C أكبر من صفر وهذا المقدر أكبر من صفر لأنه N is
+
+146
+00:13:53,010 --> 00:13:58,710
+even إذا صار كل هذا أكبر من صفر إذا automatic لما
+
+147
+00:13:58,710 --> 00:14:04,660
+أشيلههيصغر المقدار، بظبط ولا لأ؟ فهيصير عند ال F
+
+148
+00:14:04,660 --> 00:14:12,040
+of X اللي هو أكبر من مين من ال F of X naught لأن F
+
+149
+00:14:12,040 --> 00:14:17,680
+of X بساوي هذا زائد هذا فأكيد عند ال F of X أكبر
+
+150
+00:14:17,680 --> 00:14:22,160
+من مين من F of X naught في وين، لكل X وين موجودة
+
+151
+00:14:22,160 --> 00:14:30,530
+في الجوار اللي حواليها، إذا صار عند Fhas local
+
+152
+00:14:30,530 --> 00:14:40,870
+minimum at x0 similarly لو
+
+153
+00:14:40,870 --> 00:14:53,630
+بدنا نفترض ان المقدار هذا عندي fn of x0 أصغر من 0
+
+154
+00:14:54,540 --> 00:15:00,740
+وبدأ أبدأ أحكي عن مين؟ عن I I بيصير عندي الآن حسب
+
+155
+00:15:00,740 --> 00:15:07,320
+الحديث هنا ال FN of C اللي هي أصغر من 0 لإنه هتكون
+
+156
+00:15:07,320 --> 00:15:13,600
+ال FN of X لكل X في هذا الدوار the same sign as FN
+
+157
+00:15:13,600 --> 00:15:18,120
+of X note أصغر من 0 إذا صارت عندي FN of C أصغر من
+
+158
+00:15:18,120 --> 00:15:24,050
+0هذه صارت سالبة وهذه موجبة لإنه and is even إذا
+
+159
+00:15:24,050 --> 00:15:28,290
+صار المقدار هذا كله على بعض إيه شماله؟ سالب في كل
+
+160
+00:15:28,290 --> 00:15:33,070
+هذا الجوار ما ده صار سالب معناته إنه لما أنا أشيله
+
+161
+00:15:33,070 --> 00:15:39,330
+بكبر اللي بظهر يعني بيصير F of X note بناء عليه
+
+162
+00:15:39,330 --> 00:15:46,630
+أكبر من F of X مظبوط؟وهذا يعني انه صار ال F في هذه
+
+163
+00:15:46,630 --> 00:15:56,010
+الحالة عندها local maximum local maximum واضح ؟ اي
+
+164
+00:15:56,010 --> 00:16:01,510
+سؤال ؟ طيب، ضال عند مين الحالة الأخيرة، هذي
+
+165
+00:16:01,510 --> 00:16:09,170
+تذكرينها لأنه برضه بدي اشتغل نفس الشيء في الحالة
+
+166
+00:16:09,170 --> 00:16:15,680
+الأخيرة او اشي مشابهعند الان بدي افترض mean ان
+
+167
+00:16:15,680 --> 00:16:25,980
+الحالة الثالثة بدي افترض ان ال is mean of of زي
+
+168
+00:16:25,980 --> 00:16:35,810
+ما عملنا قبل بشوية إذا كانت اللي هي Fفن of x not
+
+169
+00:16:35,810 --> 00:16:40,890
+أكبر من سفر زي ما عملنا قبل شوية هيعطيني فن of c
+
+170
+00:16:40,890 --> 00:16:49,990
+أكبر من إيش؟ من سفر معايا؟ طيب الآن هذا الكلام لو
+
+171
+00:16:49,990 --> 00:16:56,930
+جيت أخدت عندي هي عندي هذه اللي هي x not وهنا
+
+172
+00:16:56,930 --> 00:17:01,650
+الجوار اللي حواليهاالجوار اللي حواليها اللي لاجئت
+
+173
+00:17:01,650 --> 00:17:09,230
+في كل الجوار زي ما عملت قبل شوية fn of x هيكون
+
+174
+00:17:09,230 --> 00:17:15,830
+اللي هو هنا اللي هو هنا و هنا موجب ماشي الحال الآن
+
+175
+00:17:15,830 --> 00:17:21,850
+مدام موجب هنا و هنا اللي هيتحكم في الموضوع اليمين
+
+176
+00:17:21,850 --> 00:17:27,770
+x minus x naught أسن لكن لسوء الحظ لو كانت x هنا
+
+177
+00:17:28,970 --> 00:17:35,350
+بصير عندى و هذا طبعا لكل X وين على يمين ال X node
+
+178
+00:17:35,350 --> 00:17:41,450
+هيطلع عندى ال X minus X node أس N وهذه إيه شمالها
+
+179
+00:17:41,450 --> 00:17:49,110
+odd is أكبر من أيش من سفر فصارت الآن هذه أكبر من
+
+180
+00:17:49,110 --> 00:17:54,330
+سفر وهذه أكبر من سفر التنتير مع بعض أكبر من سفر
+
+181
+00:17:55,160 --> 00:18:01,380
+ماشي فصارت عندي ال F of X أكبر من مين من ال F of X
+
+182
+00:18:01,380 --> 00:18:07,600
+naught ماشي الحال في المنطقة هذه هذه المنطقة هذه
+
+183
+00:18:07,600 --> 00:18:16,060
+صارت عندي ال F of X أكبر من F of X naught فاهمين؟
+
+184
+00:18:16,060 --> 00:18:23,240
+لأن الجزء الثاني ال Xفي المنطقة اللي هان أي x في
+
+185
+00:18:23,240 --> 00:18:27,880
+المنطقة اللي هان عند الـ x minus x naught أصغر من
+
+186
+00:18:27,880 --> 00:18:32,480
+مين من سفر فبصير عند هذا المقدار أصغر من سفر وهذا
+
+187
+00:18:32,480 --> 00:18:37,020
+أكبر من سفر حاصل الدرب أصغر من سفر أصغر من سفر بقد
+
+188
+00:18:37,020 --> 00:18:42,200
+إلى أن أف of x في المنطقة هذه تصير أصغر من أف of x
+
+189
+00:18:42,200 --> 00:18:47,250
+naught يعني لو أتيت لأي جوارحوالين الـ X Note
+
+190
+00:18:47,250 --> 00:18:51,770
+هتلاقي اللي هو مرة الـ F of X Note أصغر من F of X
+
+191
+00:18:51,770 --> 00:18:56,810
+ومرة F of X Note أكبر من مين من F of X يعني بمعنى
+
+192
+00:18:56,810 --> 00:19:02,270
+آخر لا في عندي Local Maximum ولا Local Minimum في
+
+193
+00:19:02,270 --> 00:19:08,450
+هذه المنطقة Similarly وبنفس الأسلوب لو كانت FN of
+
+194
+00:19:08,450 --> 00:19:14,000
+X Note أصغر من مينمن صفر معايا بيصير هذه أصغر من
+
+195
+00:19:14,000 --> 00:19:19,020
+صفر و بيصير بس بتنقلب بيصير هذه في المنطقة هذه
+
+196
+00:19:19,020 --> 00:19:23,840
+أصغر و هذه في المنطقة هذه ايش اكبر اذا in both
+
+197
+00:19:23,840 --> 00:19:30,160
+cases عندي if and is odd اذا we have neither اللي
+
+198
+00:19:30,160 --> 00:19:37,660
+هو relative maximum nor relative minimum اي سؤال
+
+199
+00:19:37,660 --> 00:19:45,760
+واضح اه؟اللي بعده اللي بعده الان بدنا نعرف حاجة
+
+200
+00:19:45,760 --> 00:19:51,140
+اسمها اللي هي مين ال convex function تعالي على
+
+201
+00:19:51,140 --> 00:19:55,640
+الرسم يا بو محمد عندي بدنا نعرف مين ال convex
+
+202
+00:19:55,640 --> 00:19:59,860
+function الان ال convex function إيش يكون cave
+
+203
+00:19:59,860 --> 00:20:05,200
+اللي هو إيش up خليني نشوف إيش هذي اللي هو بعبر شو
+
+204
+00:20:05,200 --> 00:20:07,980
+معناه ال F is convex
+
+205
+00:20:10,750 --> 00:20:14,990
+خاصية الـ convex أو الـ convexity بنحكي فيها على
+
+206
+00:20:14,990 --> 00:20:22,090
+interval يعني بنقول F من عند I لعند R بنقول عنها F
+
+207
+00:20:22,090 --> 00:20:28,170
+is convex on إيش؟ on I on وين؟ ع ال interval I إذا
+
+208
+00:20:28,170 --> 00:20:35,970
+كان شوف كيف خدلي الآن X1 أي نقطة في داخل الـ I و
+
+209
+00:20:35,970 --> 00:20:41,970
+X2 أي نقطة في داخل الـ Iوخدلي ال parameter T بين
+
+210
+00:20:41,970 --> 00:20:45,830
+السفر و بين الواحد الان النقطة ال linear
+
+211
+00:20:45,830 --> 00:20:51,230
+combination بين X1 و X2 بواسطة 1-T X1 زائد T X2
+
+212
+00:20:51,230 --> 00:20:56,530
+هتكون بين X1 و بين X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+213
+00:20:56,530 --> 00:20:56,950
+X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+214
+00:20:56,950 --> 00:21:06,250
+X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+215
+00:21:06,250 --> 00:21:06,250
+X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+216
+00:21:06,250 --> 00:21:06,250
+X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2 X2
+
+217
+00:21:06,250 --> 00:21:14,050
+X2 X2 X2 X2 X2 X2 X2لو إجيت ل F of X1 هيها وهذا F
+
+218
+00:21:14,050 --> 00:21:20,650
+of X2 لو ضربت ال F of X1 و 1 minus T و جمعتها ل T
+
+219
+00:21:20,650 --> 00:21:25,170
+في F of X2 هتلاقيها بين اللي هو مين على اللي هو
+
+220
+00:21:25,170 --> 00:21:32,010
+الخط المستقيم القاطع بين X1 و F of X1 و بين X2 و F
+
+221
+00:21:32,010 --> 00:21:37,970
+of X2 دائما هتلاقي هذه النقطةالصورة هنا أعلى من
+
+222
+00:21:37,970 --> 00:21:42,910
+هذه وهذا معنى اللي هو ال convex وعن عرف ال convex
+
+223
+00:21:42,910 --> 00:21:50,390
+على هذا الأساس بيقول F is convex on I طلعلي على
+
+224
+00:21:50,390 --> 00:21:54,650
+التعريف إذا كان أخدنا أي X واحد و X اتنين element
+
+225
+00:21:54,650 --> 00:22:00,690
+in I و أخدنا T اللي هو element in 01 يعني بين 01
+
+226
+00:22:00,690 --> 00:22:13,020
+بيحققلي F اللي هوfor any كده f
+
+227
+00:22:13,020 --> 00:22:19,140
+is a convex in I if for any كده وكده we have
+
+228
+00:22:19,140 --> 00:22:25,940
+فاهمين عليا f of واحد minus T X واحد زائد T في X
+
+229
+00:22:25,940 --> 00:22:32,520
+اتنين ايش ماله هيسير؟أصغر أو يساوي واحد ناقص T في
+
+230
+00:22:32,520 --> 00:22:38,000
+X واحد زائد T في F of X اتنين
+
+231
+00:22:40,410 --> 00:22:44,370
+طبعاً اللي هو ال .. ال .. ال .. ال .. ال convex
+
+232
+00:22:44,370 --> 00:22:48,330
+functions في عليها دراسات كثيرة ودراسات حديثة كمان
+
+233
+00:22:48,330 --> 00:22:55,070
+حتى يعني في اللي هو real analysis بما يتعلق بال ..
+
+234
+00:22:55,070 --> 00:22:58,950
+مش بال integration العادي كمان باللي هو .. و
+
+235
+00:22:58,950 --> 00:23:03,310
+علاقتها بال integration في اللي هو ال .. ال big
+
+236
+00:23:03,310 --> 00:23:08,760
+integra كمانوطبعاً وإليها اللي هو خلّيني أقول
+
+237
+00:23:08,760 --> 00:23:14,020
+تطبيقات أو استخدامات في اللي هو optimization اللي
+
+238
+00:23:14,020 --> 00:23:20,350
+هي theoryالان احنا هنتحك .. هنحكي فيها بما يتعلق
+
+239
+00:23:20,350 --> 00:23:24,470
+في هذا الجزء من ال chapter اللي هو اللي بتعلق ك ..
+
+240
+00:23:24,470 --> 00:23:28,350
+ك .. كمين؟ ك Taylor's theorem وكيف بدنا نطبق ال
+
+241
+00:23:28,350 --> 00:23:32,590
+Taylor's theorem في إثبات نظرية تعرف لنا ال test
+
+242
+00:23:32,590 --> 00:23:37,730
+إيه لها؟ قول يا محمد هل
+
+243
+00:23:37,730 --> 00:23:46,670
+جيت و تشوفها؟ في ال calculus في ال calculusكنا
+
+244
+00:23:46,670 --> 00:23:52,670
+ناخد اللي هو test لل increasing وال decreasing
+
+245
+00:23:52,670 --> 00:23:59,380
+باستخدام ال first derivativeإذا بتتذكروا كنا ناخد
+
+246
+00:23:59,380 --> 00:24:04,220
+أيضا اللي هو عشان نعرف الدالة concave up أو
+
+247
+00:24:04,220 --> 00:24:08,340
+concave down بوسط تمين ال second derivative test
+
+248
+00:24:08,340 --> 00:24:11,880
+ماشي الحال إذا كانت F double prime أكبر من صفر
+
+249
+00:24:11,880 --> 00:24:15,360
+كانت اللي هي concave up إذا كانت F double prime
+
+250
+00:24:15,360 --> 00:24:20,060
+أصغر من صفر كانت أشمالها concave down الآن هتشوفوا
+
+251
+00:24:20,060 --> 00:24:25,000
+النظرية اللي بتشابه اللي بنحكي و هتستنتج أيش اللي
+
+252
+00:24:25,000 --> 00:24:32,080
+بحكيهبجاوب على سؤالك شوف الآن الان عندي واضح انه
+
+253
+00:24:32,080 --> 00:24:36,940
+من التعريف ان نفحص الدالة انها convex ولا مش
+
+254
+00:24:36,940 --> 00:24:41,920
+convex بالشكل اللي حكيناه بالتعريف اللي حكيناه
+
+255
+00:24:41,920 --> 00:24:46,380
+بغلب وبعض الدول مش هنقدر او مش هنعرف اللي هو
+
+256
+00:24:46,380 --> 00:24:49,980
+لصعوبة الحسابات ان نثبت انها convex او مش convex
+
+257
+00:24:49,980 --> 00:24:58,330
+فاكيد بدنا testيسهّلنا عملية إيجاد أو معرفة إن هذه
+
+258
+00:24:58,330 --> 00:25:03,070
+الدالة is convex ولا لأ واضح اللي بحكيه؟ يلّا نشوف
+
+259
+00:25:03,070 --> 00:25:07,810
+مع بعض الان قبلها بيعطينا بس counter example
+
+260
+00:25:07,810 --> 00:25:15,110
+بقولّي إن ال convex function ممكن ما تكونش
+
+261
+00:25:15,110 --> 00:25:18,540
+differentiable على الفترة اللي فيها convexيعني
+
+262
+00:25:18,540 --> 00:25:20,660
+ممكن اللي هي ال .. ال .. ال .. احنا اللي جابنا قبل
+
+263
+00:25:20,660 --> 00:25:24,800
+شوية اللي جيبناها الدالة هي محلاها في انسياب وفي
+
+264
+00:25:24,800 --> 00:25:27,860
+differentiability و ال short أعلى من ال .. من
+
+265
+00:25:27,860 --> 00:25:31,260
+الدالة نفسها و كده و ال function differentiable
+
+266
+00:25:31,260 --> 00:25:34,440
+لكن هذا الكلام مش دايما حاصل انه ال convex
+
+267
+00:25:34,440 --> 00:25:37,100
+function تكون differentiable و أشهر منه و مثال
+
+268
+00:25:37,100 --> 00:25:41,460
+مشهور وسهل اللي هو F of X بسوء ال absolute value
+
+269
+00:25:41,460 --> 00:25:45,720
+لل X F of X بسوء ال absolute value لل X هذا اللي
+
+270
+00:25:45,720 --> 00:25:52,270
+هوis أكيد كلكم عارف إنها not differentiable at 0
+
+271
+00:25:52,270 --> 00:26:01,530
+but it is convex on any interval I containing إيه
+
+272
+00:26:01,530 --> 00:26:05,370
+إيش الـ0، بظبط؟ ال .. ال .. it is not
+
+273
+00:26:05,370 --> 00:26:08,030
+differentiable اللي عارفينها ال .. ال .. ال .. ال
+
+274
+00:26:08,030 --> 00:26:13,730
+.. نثبت إنها convex let على interval I X1 و X2
+
+275
+00:26:13,730 --> 00:26:19,800
+element in Iو T element in 0 و 1 اخسب اللي لان F
+
+276
+00:26:19,800 --> 00:26:29,780
+of 1 minus T في X1 زي T في X2 بساوية عبارة عن ال
+
+277
+00:26:29,780 --> 00:26:36,240
+absolute value واحد نقص T X1 زي T في X2 وهذا من
+
+278
+00:26:36,240 --> 00:26:39,700
+خلاص ال absolute value اصغر بساوية واحد نقص T في
+
+279
+00:26:39,700 --> 00:26:46,100
+X1زاية T في X2 وهذا طبعا مين هو؟ هو عبارة عن 1 نقص
+
+280
+00:26:46,100 --> 00:26:53,120
+T F of X1 زاية T في F of X2 إذا صار عندي ال F
+
+281
+00:26:53,120 --> 00:26:56,240
+أشمالها is convex but it is not differentiable
+
+282
+00:26:56,240 --> 00:27:00,620
+اطلع لفوق الان يجيلا اللي هو ال test اللي معطينيا
+
+283
+00:27:00,620 --> 00:27:07,060
+اللي بقوللي اللي بجاوب على سؤالك يا محمد let I be
+
+284
+00:27:08,590 --> 00:27:13,190
+And open interval and let f من I لR have a second
+
+285
+00:27:13,190 --> 00:27:18,890
+derivative on I نفترض أن الـ F'' إيش ما لها موجودة
+
+286
+00:27:18,890 --> 00:27:25,600
+على I إذا من رأس الدور مفترضين الـ F'' موجودةالـ
+
+287
+00:27:25,600 --> 00:27:30,140
+test هو كما قالي f is convex on i if and only if f
+
+288
+00:27:30,140 --> 00:27:34,740
+double prime a شمالها أكبر من 0 لكل x when element
+
+289
+00:27:34,740 --> 00:27:39,300
+in I بصور واضحة الأمر الآن بدي أستخدم في المرهان
+
+290
+00:27:39,300 --> 00:27:44,520
+اللي هو سؤال اللي هو 16 في ال exercises و هنحله
+
+291
+00:27:44,520 --> 00:27:48,380
+وهذا من ضمن ال exercises اللي معاكم هو Morgue و
+
+292
+00:27:48,380 --> 00:27:53,910
+هنحله في حينهالان ايش اللي هو ال .. ال .. ال .. ال
+
+293
+00:27:53,910 --> 00:27:58,390
+.. السؤال بيقول اذا كانت f double prime موجودةثم
+
+294
+00:27:58,390 --> 00:28:02,570
+اثبت لي ان f double prime can be written as limit
+
+295
+00:28:02,570 --> 00:28:06,470
+f of a زاد h نقص اتنين f of a زاد f of a minus h
+
+296
+00:28:06,470 --> 00:28:11,850
+على مين على h تربية ماشي الحال هذا بدأ اعتمد على
+
+297
+00:28:11,850 --> 00:28:17,950
+انه اللي هو مثبت في اثبات النظرية ونشوف كيف ناخد
+
+298
+00:28:17,950 --> 00:28:23,540
+الاتجاه الاول اول اشي بده اللي هو بنفترض انهعندي
+
+299
+00:28:23,540 --> 00:28:29,080
+ال .. اللي هو ال F is convex ومنها نصل لإنه ال
+
+300
+00:28:29,080 --> 00:28:34,300
+derivative ايه شمالها أكبر أو يساوي سفر ماشي الحال
+
+301
+00:28:34,300 --> 00:28:43,480
+خليني نشوف مع بعض كيف اللي هو نثبت
+
+302
+00:28:43,480 --> 00:28:55,740
+انه if F is convex thenF double prime of A أكبر
+
+303
+00:28:55,740 --> 00:29:01,720
+شهر السفر لما نقول suppose that
+
+304
+00:29:01,720 --> 00:29:11,780
+proof فوضيح بس .. suppose that أن F is convex
+
+305
+00:29:11,780 --> 00:29:16,880
+لنثبت أن F double prime of A أكبر شهر السفرالان
+
+306
+00:29:16,880 --> 00:29:21,240
+انا ال F is convex وين على ال I و differentiable
+
+307
+00:29:21,240 --> 00:29:25,560
+وين second derivative لها على ال I خليني اختار
+
+308
+00:29:25,560 --> 00:29:31,280
+الان اللي هو ال H بحيث بدي اختار H بحيث انه A
+
+309
+00:29:31,280 --> 00:29:36,440
+minus H و A زائد H التانين يكون هنا وين موجودات في
+
+310
+00:29:36,440 --> 00:29:39,600
+ال interval I اللي عليها شغل اللي انا بدي اشتغل
+
+311
+00:29:39,600 --> 00:29:49,760
+عليها الان خدلي الان Fof A واكتبها على صورة F of
+
+312
+00:29:49,760 --> 00:29:59,450
+نص في A زاد H زاد نص في A minus Hلأن هذه عبارة عن
+
+313
+00:29:59,450 --> 00:30:03,330
+نص A و نص A هي الـ A و نص H ناقص نص H بروحهم مع
+
+314
+00:30:03,330 --> 00:30:06,730
+بعض ده منرجع لمين لل F و فيها كده تعرفي ليش عملتيل
+
+315
+00:30:06,730 --> 00:30:12,450
+طبعا هذه هتشتغل بمثابة واحد ناقص T و هذه T إذا T
+
+316
+00:30:12,450 --> 00:30:16,370
+بين ال Zero و الواحد صحيحة و بما أن F is convex و
+
+317
+00:30:16,370 --> 00:30:20,750
+هذولة اتتين نقطين عبارة عن نقطين في I لأن هيك
+
+318
+00:30:20,750 --> 00:30:26,520
+اختار ال H هنا هيكون هذا أصغر أو يساوينص F of A
+
+319
+00:30:26,520 --> 00:30:35,620
+زائد H زائد نص F of A minus H واضح؟
+
+320
+00:30:35,620 --> 00:30:43,000
+أي سؤال؟ واضح أه؟ جيبلي هذا على الجهة التانية
+
+321
+00:30:43,000 --> 00:30:51,620
+بيصير عندي نص F of A زائد H زائد نص F of A minus H
+
+322
+00:30:51,620 --> 00:30:59,230
+ناقص F of Aكل و إيه شماله أكبر أو يساوي إيش أكبر
+
+323
+00:30:59,230 --> 00:31:05,010
+أو يساوي سفر وضرب الجهتين في اتنين بيصير عندي
+
+324
+00:31:05,010 --> 00:31:10,550
+مضربت في اتنين بيصير هذه اللي هو النص بيصير واحد و
+
+325
+00:31:10,550 --> 00:31:14,490
+النص هذا بيصير واحد و هذا بيصير إيه شماله ناقص
+
+326
+00:31:14,490 --> 00:31:22,650
+اتنين of of ايه اقسم لليه الآن على اتش تربيعوH
+
+327
+00:31:22,650 --> 00:31:26,370
+تربيع موجة بقى، إذا بيضلوا هذا أشمال المفضل أكبر
+
+328
+00:31:26,370 --> 00:31:32,030
+أو يساوي سفر خد ال limit للجهتين limit as H goes
+
+329
+00:31:32,030 --> 00:31:40,440
+to zero بيضلوا أكبر أو يساوي سفر، مظبوط؟ و هذاBy
+
+330
+00:31:40,440 --> 00:31:44,920
+ال exercise اللي بنحكي عنه هو عبارة عن مين؟ F
+
+331
+00:31:44,920 --> 00:31:48,100
+double prime of A فطلع عندي F double prime of A
+
+332
+00:31:48,100 --> 00:31:52,320
+أكبر أساوي الصفر طبعا ماقدرتش أستخدم هذا أنه بنكتب
+
+333
+00:31:52,320 --> 00:31:56,840
+على صورة هذا إلا في ضوء أن هو أصلا حاكي لي ال F
+
+334
+00:31:56,840 --> 00:32:03,940
+double prime exist on I واضح أه؟أي سؤال إذا الان
+
+335
+00:32:03,940 --> 00:32:08,580
+اطلع لفوق و بنكون احنا خلصنا الجزء الأول و بنكون
+
+336
+00:32:08,580 --> 00:32:11,860
+بدنا نبدأ بالجزء الثاني اللي هو بال prime أكبر
+
+337
+00:32:11,860 --> 00:32:17,600
+سواء سفر و نصل لمين لل F is convex أوضحت هذه خليني
+
+338
+00:32:17,600 --> 00:32:24,100
+بس أنسحها و أقول ما يليه
+
+339
+00:32:24,100 --> 00:32:27,560
+نفترض
+
+340
+00:32:27,560 --> 00:32:30,060
+لأن صبوذ
+
+341
+00:32:32,450 --> 00:32:39,370
+that f double prime of a أشمالها أكبر أو يساوي سفر
+
+342
+00:32:39,370 --> 00:32:45,330
+ماشي الحال .. بدنا نثبت لك أن f is convex كيف بدنا
+
+343
+00:32:45,330 --> 00:32:52,130
+نثبت ان f is convex؟ بدنا نقول let x1 و x2 بيه أي
+
+344
+00:32:52,130 --> 00:32:57,630
+element in مين؟ ان i و t element in zero واش و
+
+345
+00:32:57,630 --> 00:33:06,820
+واحدوسميلي X not هي عبارة عن واحد minus T في X
+
+346
+00:33:06,820 --> 00:33:13,580
+واحد زائد T في مين في X اتنين الآن بدي اطبق اللي
+
+347
+00:33:13,580 --> 00:33:16,480
+هو taylor's theorem اول اش طبعا taylor's theorem
+
+348
+00:33:16,480 --> 00:33:21,960
+قابل للتطبيق اللي عند مينعند and بتساوي واحد بس
+
+349
+00:33:21,960 --> 00:33:25,640
+لأن المشتقة الأولى والتانية موجودة بعد هيك ماحدش
+
+350
+00:33:25,640 --> 00:33:29,480
+حكالي فإذا بيحجلي ان اطبق Taylor's theorem بس على
+
+351
+00:33:29,480 --> 00:33:34,260
+وين على اللي هو and بتساوي سفر او and بتساوي واحد
+
+352
+00:33:34,260 --> 00:33:36,800
+and بتساوي سفر بيصير main value theorem and بتساوي
+
+353
+00:33:36,800 --> 00:33:41,000
+واحد اللي هو اللي بتطبق عليها و هتخدمني ماشي الحال
+
+354
+00:33:41,000 --> 00:33:46,470
+نشوفالان apply taylor's theorem ع الفترة على اللي
+
+355
+00:33:46,470 --> 00:33:51,970
+هو ال X واحد وال X note اللي عندى there exist اللي
+
+356
+00:33:51,970 --> 00:34:00,270
+هو C واحد between X واحد and X note such that اه
+
+357
+00:34:00,270 --> 00:34:08,370
+اللي هو F of X note مظبوط او F of .. ده طبقه على X
+
+358
+00:34:08,370 --> 00:34:15,010
+واحد اول شي F of X واحد بساوياللي هو F of X not
+
+359
+00:34:15,010 --> 00:34:22,550
+زائد F prime of X not في X واحد نقص X not زائد مين
+
+360
+00:34:22,550 --> 00:34:26,950
+الآن ال remainder زائد ال remainder ال remainder
+
+361
+00:34:26,950 --> 00:34:33,950
+إيش هو F double prime عند ال C واحد اللي لاجيناها
+
+362
+00:34:33,950 --> 00:34:41,390
+في X minus X not الكل تربيع على 2 فكتورياأصغر من
+
+363
+00:34:41,390 --> 00:34:48,530
+اللي هو similarly there exist c2 بتطبقها على x1
+
+364
+00:34:48,530 --> 00:34:57,090
+على x2 و x0 there exist a c2 between x2 and x0
+
+365
+00:34:57,090 --> 00:35:07,070
+such thatof x2 بيساوي f of x0 زاد f prime of x0 في
+
+366
+00:35:07,070 --> 00:35:17,690
+x2 ناقص x0 زاد f double prime of c2 في x minus x0
+
+367
+00:35:17,690 --> 00:35:24,950
+الكل تربيع على 2 factorial واضح اللي بعمله
+
+368
+00:35:28,900 --> 00:35:37,440
+فهي عندى اللى هى هذه مع هذه هى اللى بدى استخدمه
+
+369
+00:35:37,440 --> 00:35:43,500
+هنا الآن فى الوصول لإنه F is convex يعنى F of هذا
+
+370
+00:35:43,500 --> 00:35:49,180
+المقدار يكون أصغر أو يساوي اللى هو F 1 minus T
+
+371
+00:35:49,180 --> 00:35:56,920
+فهذه زائد T فهذه إذا بدى أحسب الآن 1 minus T ف F
+
+372
+00:35:56,920 --> 00:36:05,420
+of X 1زائد T في F of إيش X2 بدي أحسبها و أوصلها
+
+373
+00:36:05,420 --> 00:36:09,080
+لعلاقتها مع مين مع F of X note اللي هي F of هذه
+
+374
+00:36:09,080 --> 00:36:20,940
+بساوي اضربلي بيصير T واحد نقص T في F of X note
+
+375
+00:36:20,940 --> 00:36:31,310
+معايا يا شباب زائد واحد minus Tف F prime of X note
+
+376
+00:36:31,310 --> 00:36:40,590
+في X واحد minus X note زائد اللي هو mean ده دروفة
+
+377
+00:36:40,590 --> 00:36:47,390
+هادة اللي هو واحد minus T في F double prime of C
+
+378
+00:36:47,390 --> 00:36:52,970
+واحد في X minus X note اللي كل تربيع على اتنين
+
+379
+00:36:52,970 --> 00:37:02,060
+factorial اللي هو اتنين معايا يا شبابزائد T .. T
+
+380
+00:37:02,060 --> 00:37:10,500
+.. F of X2 هيها اللي هي F of X note زائد T في F
+
+381
+00:37:10,500 --> 00:37:15,860
+prime of X note بضرب فادي قاعد في X2 نقص X note
+
+382
+00:37:15,860 --> 00:37:26,720
+زائد اللي هو T في F double prime of C2fx-x0 الكل
+
+383
+00:37:26,720 --> 00:37:34,100
+تربيه على مين؟ على اتنين نيجي الان و يساوي اطلعلي
+
+384
+00:37:34,100 --> 00:37:41,720
+على هذه الان واحد minus T في F of X0 وT في F of X0
+
+385
+00:37:41,720 --> 00:37:51,520
+بيصير عبارة عن F of X0 مظبوط؟ زائدالان هذا f prime
+
+386
+00:37:51,520 --> 00:37:55,420
+of x naught مضروبة في T مضروبة في هذه إذا عبارة عن
+
+387
+00:37:55,420 --> 00:38:03,660
+واحد minus T زائد T اه اللي هو مية المية بيصير
+
+388
+00:38:03,660 --> 00:38:10,300
+عندي الآن خد اللي هي x واحد ناقص x naught وخلينا
+
+389
+00:38:10,300 --> 00:38:19,510
+نجمحها واحد minus يعني خلينا نضيع ال Tفيها نقص
+
+390
+00:38:19,510 --> 00:38:24,730
+اتنين و X واحد او X او X واحد نفردها واحد minus T
+
+391
+00:38:24,730 --> 00:38:32,970
+في F ناخد عن المشترك يمكن أسهلنا F prime of X note
+
+392
+00:38:32,970 --> 00:38:43,330
+في واحد minus T في X واحد ناقص X note معايا شباب
+
+393
+00:38:43,330 --> 00:38:51,670
+زائد باخد بين هذه هي عمالو بين هذه أخدت ال F prime
+
+394
+00:38:51,670 --> 00:38:57,430
+of X naught عام نشترك بينهم زائد T في X2 ناقص X
+
+395
+00:38:57,430 --> 00:39:06,290
+naught زائد هذا المقدار كله بعد إذنكم هي اللي
+
+396
+00:39:06,290 --> 00:39:12,390
+بلزرج هذا اللي حطيته بلزرج سميني يا إيش R واضح
+
+397
+00:39:12,390 --> 00:39:21,890
+زائد Rالان نطلع على اللي عندي هدا واحد ناقص T اول
+
+398
+00:39:21,890 --> 00:39:27,670
+اشي ناقص T في ناقص X Note بيصير اللي هو زائد T في
+
+399
+00:39:27,670 --> 00:39:31,850
+X Note مع ناقص T في X Note بروحين مظبوط؟ اذا هذا
+
+400
+00:39:31,850 --> 00:39:38,410
+الان بروح وهذا
+
+401
+00:39:38,410 --> 00:39:44,290
+برضه بروح صح ولا لأ؟ نتأكد في اللي عملناه
+
+402
+00:39:48,090 --> 00:40:00,190
+واضح اه بظل x .. x note اه بظل ناقص اللي هو زائد x
+
+403
+00:40:00,190 --> 00:40:07,950
+note ناقص x note ناقص x note عملها كمان مرة بظل
+
+404
+00:40:07,950 --> 00:40:15,160
+ناقص x note من هناو من هنا بيظهر مين ناقص X note
+
+405
+00:40:15,160 --> 00:40:21,640
+لأنه بتظهر تبعت اللي هنا تظهر ناقص X note واضح أه؟
+
+406
+00:40:21,640 --> 00:40:28,760
+الان هذه كلها على بعض مين هي؟ هذه إيش اسمها؟ كلها
+
+407
+00:40:28,760 --> 00:40:32,900
+اسمها X note أه؟
+
+408
+00:40:32,900 --> 00:40:36,960
+فبيصير X note ناقص X note إيش بيصير؟ سفر فبيصير
+
+409
+00:40:36,960 --> 00:40:41,560
+هذا المقدار عبارة عن Fof x naught زائد ال
+
+410
+00:40:41,560 --> 00:40:45,380
+remainder مين؟ R نتطلع على الـ R هذا أو هو R لما
+
+411
+00:40:45,380 --> 00:40:48,000
+نقول remainder؟ الـ R مجموع الـ remainder دلين
+
+412
+00:40:48,000 --> 00:40:58,960
+الان ملاحظ هو ماعطيني عندي الان أفضل براءة عندي
+
+413
+00:40:58,960 --> 00:41:05,340
+هذا X نقص X واحد كل تربيع على اتنين موجبة و X نقص
+
+414
+00:41:05,340 --> 00:41:10,330
+X اتنين نوت على اتنين اشمالها موجبةوالان ال f
+
+415
+00:41:10,330 --> 00:41:15,510
+double prime عندى اللى هتطلع أشمالها موجبة بنفس
+
+416
+00:41:15,510 --> 00:41:21,550
+الطريقة السابقة زى ما عملنا قبل شوية عندى f double
+
+417
+00:41:21,550 --> 00:41:27,310
+prime أكبر يساوي سفر هيطلع عندى في الجوار اللى
+
+418
+00:41:27,310 --> 00:41:30,610
+بطبق عليه برضه أكبر يساوي سفر فبطلع عندى هذا موجب
+
+419
+00:41:30,610 --> 00:41:35,690
+و هذا موجب فبطلع عندى هذا المقدار اللى بدأنا فيه
+
+420
+00:41:36,380 --> 00:41:41,720
+هذا المقدار اللي بدأنا فيه طلع بساوي هذا المقدار
+
+421
+00:41:41,720 --> 00:41:49,760
+اللي هو هذا أكبر أو يساوي F of X naught اللي هي
+
+422
+00:41:49,760 --> 00:41:57,540
+mean F of X naught اللي عبارة عن F معين واحد نقص T
+
+423
+00:41:57,540 --> 00:42:07,040
+في X واحد زائد T في X اتنين هاي اللي وصلنا إليهمع
+
+424
+00:42:07,040 --> 00:42:21,020
+هذا وهذا تعريف أيه؟ F is convex F is convex أي
+
+425
+00:42:21,020 --> 00:42:21,680
+سؤال؟
+
+426
+00:42:24,290 --> 00:42:29,090
+و بهيك نكون خلصنا ال section اللي الأخير في
+
+427
+00:42:29,090 --> 00:42:34,250
+chapter اللي هو 6 خلصنا الجزء الأول من المحاضرة و
+
+428
+00:42:34,250 --> 00:42:37,250
+الآن بدنا نبدأ في الجزء الثاني من المحاضرة
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw.srt
new file mode 100644
index 0000000000000000000000000000000000000000..6be641ef5b2f2da9a42519a050e4c0544356020e
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw.srt
@@ -0,0 +1,2201 @@
+1
+00:00:05,620 --> 00:00:11,560
+بسم الله الرحمن الرحيم المحاضرة الثالثة في مساق
+
+2
+00:00:11,560 --> 00:00:19,660
+تحليل رياضي 2 أو تحليل حقيقي 2 كلية العلوم قسم
+
+3
+00:00:19,660 --> 00:00:28,520
+الرياضيات بالجامعة الإسلامية مراجعة سريعة لما تم 
+
+4
+00:00:28,520 --> 00:00:34,380
+في المحاضرة السابقة والتي قبلها حكينا عن شو معناته 
+
+5
+00:00:34,380 --> 00:00:37,500
+اللي هو الـ derivative قلنا f prime of c عند نقطة 
+
+6
+00:00:37,500 --> 00:00:41,760
+c لو كانت f function من i لعند ال r هو عبارة عن 
+
+7
+00:00:41,760 --> 00:00:49,200
+limit f of x نقص f of c على x minus c لما x تروح 
+
+8
+00:00:49,200 --> 00:00:53,460
+أو تقول ل c قلنا لو كانت ال limit exist هذه بنقول 
+
+9
+00:00:53,460 --> 00:00:56,740
+إنه ال derivative موجودة بنرمزلها بالرمز f prime
+
+10
+00:00:56,740 --> 00:01:02,600
+of c  أخدنا بعيدة اللي هو نظرية قولنا لو كانت f is
+
+11
+00:01:02,600 --> 00:01:11,860
+differentiable at c then it is continuous at c
+
+12
+00:01:11,860 --> 00:01:15,320
+قولنا the converse need not to be true in general
+
+13
+00:01:15,320 --> 00:01:20,100
+وأخدنا مثال على هيك قولنا خد ال f of x بسوء ال
+
+14
+00:01:20,100 --> 00:01:23,560
+absolute value لل x is continuous اللي هو 
+
+15
+00:01:23,560 --> 00:01:26,880
+everywhere but it is not differentiable at zero
+
+16
+00:01:27,430 --> 00:01:30,590
+واخدنا مثال آخر قولنا f of x بيساوي ال summation
+
+17
+00:01:30,590 --> 00:01:36,090
+لل x ناقص ci i من عند 1 لعند n قولنا هذا ال
+
+18
+00:01:36,090 --> 00:01:39,210
+function is continuous everywhere but it is not 
+
+19
+00:01:39,210 --> 00:01:43,250
+differentiable at the corner points which are c1, 
+
+20
+00:01:43,350 --> 00:01:49,170
+c2 لعند ال cl و بعدين أخدنا مثال f of x بيساوي ال
+
+21
+00:01:49,170 --> 00:01:56,590
+summation 1 على 2 to the n cosine cos 3 أُس N X N 
+
+22
+00:01:56,590 --> 00:02:01,690
+من واحد إلى ملا نهاية قلنا إن الـ function is 
+
+23
+00:02:01,690 --> 00:02:07,750
+continuous everywhere but it is not differentiable
+
+24
+00:02:07,750 --> 00:02:16,850
+anywhere الـ function متصلة دائما لكن ليس لها أي 
+
+25
+00:02:16,850 --> 00:02:22,760
+derivative عند أي نقطة من النقاط دومينابعد هيك
+
+26
+00:02:22,760 --> 00:02:28,500
+أخدنا اللي هو .. اللي هو نظرية اللي هي بنحكي فيها 
+
+27
+00:02:28,500 --> 00:02:34,950
+عن خواص اللي هي التفاضل قلنا لو كانت ال Fوالـ G
+
+28
+00:02:34,950 --> 00:02:39,370
+are differentiable اللي هو كان من I لعند R
+
+29
+00:02:39,370 --> 00:02:44,730
+differentiable عند نقطة C بيكون مجموعة Alpha F حيث
+
+30
+00:02:44,730 --> 00:02:49,570
+Alpha اللي هو عبارة عن constant Alpha F' موجودة
+
+31
+00:02:49,570 --> 00:02:57,230
+وبتساوي Alpha في F' عند نقطة Cو بعدين قلنا أن f زي 
+
+32
+00:02:57,230 --> 00:03:03,290
+g أو نقص prime of c برضه exist بسوء f prime of c
+
+33
+00:03:03,290 --> 00:03:09,390
+زي أو نقص g prime of c هذه من الخواصل هو ال 
+
+34
+00:03:09,390 --> 00:03:15,040
+derivative و أيضا قلنااللي هو الخواص الأخرى أنه
+
+35
+00:03:15,040 --> 00:03:21,000
+الـ F في G الكل prime عنده نقطة C حاصل ضرر دلتين 
+
+36
+00:03:21,000 --> 00:03:28,400
+بساوي اللي هو F prime of c في g of c زائد f of c في
+
+37
+00:03:28,400 --> 00:03:33,320
+g prime of c ومن ثم أخدنا أيضًا اللي هو 
+
+38
+00:03:54,080 --> 00:03:59,480
+و من ثم أيضا حكينا عن اللي هو اخدنا اللي هو ال 
+
+39
+00:03:59,480 --> 00:04:04,540
+chain rule theorem ال chain rule لو كانت F من I 
+
+40
+00:04:04,540 --> 00:04:12,080
+لعند ال R و G من عند ال J لعند ال R حيث ال F of I
+
+41
+00:04:12,080 --> 00:04:15,780
+subset من ال J عشان نعرف اللي هو ال composition
+
+42
+00:04:15,780 --> 00:04:19,480
+قلنا لو كانت F is differentiable و G is 
+
+43
+00:04:19,480 --> 00:04:23,890
+differentiable يعني F is differentiable عند Cis
+
+44
+00:04:23,890 --> 00:04:32,470
+differentiable at c and g is differentiable at f 
+
+45
+00:04:32,470 --> 00:04:39,210
+of c then the composition اللي هو ال general يعني 
+
+46
+00:04:39,210 --> 00:04:44,490
+التفاضل ال composition the composition تفاضله كما 
+
+47
+00:04:44,490 --> 00:04:55,650
+يلي then f composite G prime of C بساوة F prime عند
+
+48
+00:04:55,650 --> 00:05:02,070
+G of C في G prime of C هذا اللي حكينا عنه المرة
+
+49
+00:05:02,070 --> 00:05:06,830
+الماضية وأيضا أخدنا طبعا أمثلة على هيك وأخدنا
+
+50
+00:05:06,830 --> 00:05:15,030
+نظرية أخرى اللي هي لو كانت عند F من عند I لعند R
+
+51
+00:05:15,030 --> 00:05:17,730
+قلنا is strictly 
+
+52
+00:05:25,190 --> 00:05:31,450
+monotone and continuous
+
+53
+00:05:31,450 --> 00:05:41,770
+وكانت اللي هو G من عند J لعند I برضه strictly is 
+
+54
+00:05:41,770 --> 00:05:55,620
+strictly monotone and continuous قلنا هنا at C وهنا
+
+55
+00:05:55,620 --> 00:06:06,340
+continuous at F of C حيث اللي هو الـ G بتساوي F of
+
+56
+00:06:06,340 --> 00:06:12,180
+mean of I عشان نقدر نعرف اللي هو تكون inverse is 
+
+57
+00:06:12,180 --> 00:06:17,640
+strictly monotone and continuous where G
+
+58
+00:06:19,450 --> 00:06:28,790
+is the inverse of F، مظبوط؟ طبعاً، اللي بضماننا إن 
+
+59
+00:06:28,790 --> 00:06:31,070
+ال inverse موجود إن هو strictly إيش؟ معناها
+
+60
+00:06:31,070 --> 00:06:33,490
+monotone، strictly monotone سواء increasing أو
+
+61
+00:06:33,490 --> 00:06:37,090
+decreasing، معناته one to one معناته إن هو اللي هي
+
+62
+00:06:37,090 --> 00:06:41,030
+ال inverse موجودة من اللي هو F of I اللي هي J،
+
+63
+00:06:41,030 --> 00:06:44,610
+لعند مين؟ لعند ال I، وهذه ال J في الواقع كل عنها
+
+64
+00:06:44,610 --> 00:06:52,800
+مين؟ هي ال F inverse الآن الشرط أنه if F is
+
+65
+00:06:52,800 --> 00:07:03,640
+differentiable at C at C then اللي هي G اللي هي 
+
+66
+00:07:03,640 --> 00:07:07,280
+الـ F inverse دي في الواقع F inverse اللي سمناها 
+
+67
+00:07:07,280 --> 00:07:14,020
+مين؟ F inverse لازم تكون differentiable at F of C
+
+68
+00:07:14,020 --> 00:07:15,660
+and
+
+69
+00:07:18,190 --> 00:07:23,390
+اللي سمناها F of C في حين وده بتتذكره D اللي هي
+
+70
+00:07:23,390 --> 00:07:35,170
+and F prime أو F inverse prime at this D بتساوي
+
+71
+00:07:35,170 --> 00:07:44,930
+واحد على F prime of C أو بمعنى آخر بتساوي واحد على
+
+72
+00:07:44,930 --> 00:07:53,880
+F prime F of C بيساوي D يعني C بيساوي G of D أو F
+
+73
+00:07:53,880 --> 00:07:59,600
+inverse of D اللي هي يعني composite G الكل أفمين
+
+74
+00:07:59,600 --> 00:08:13,970
+of C واضح هاه F of مين of D قلنا بعدين اللي هو in 
+
+75
+00:08:13,970 --> 00:08:18,530
+general لو كانت اللي هي هذه تحت الشروط هذه بس الآن 
+
+76
+00:08:18,530 --> 00:08:24,430
+مش عند اللي هو نقطة لو F is differentiable على كل 
+
+77
+00:08:24,430 --> 00:08:29,470
+ال I هتكون ال G is differentiable على ال J و
+
+78
+00:08:29,470 --> 00:08:36,110
+العلاقة بينهم زي ما حكينا المرة الماضية هتكون اللي 
+
+79
+00:08:36,110 --> 00:08:43,620
+هوطب هالمرة F is differentiable on I then G is 
+
+80
+00:08:43,620 --> 00:08:47,280
+differentiable on J اللي هي بساوة F of I طبعا تحت
+
+81
+00:08:47,280 --> 00:08:55,140
+هذا الشروط and هيكون F inverse prime of Y بساوة 1
+
+82
+00:08:55,140 --> 00:09:01,330
+على F prime of X حيث العلاقة بين Y و X اللي هي Y
+
+83
+00:09:01,330 --> 00:09:08,050
+بتساوي F of X أو بتعبير آخر بتساوي 1 على F prime
+
+84
+00:09:08,050 --> 00:09:13,630
+of composite G اللي هي بتسميها F inverse كله of
+
+85
+00:09:13,630 --> 00:09:23,790
+mean of Y هذا اللي حكينا المرة الماضية نيجي 
+
+86
+00:09:23,790 --> 00:09:26,510
+الآن نكمل
+
+87
+00:09:27,590 --> 00:09:33,710
+وصلنا عند المثال اللي هو 6 1 10 وقلنا لو كانت عند
+
+88
+00:09:33,710 --> 00:09:41,270
+g of y example وقفنا عند ال example التالي g of y 
+
+89
+00:09:41,270 --> 00:09:48,710
+بتساوي y أسواحدة ل n for y element in j اللي هي 
+
+90
+00:09:48,710 --> 00:09:54,970
+عبارة عن ال interval من zero ل infinity الآن بقول 
+
+91
+00:09:54,970 --> 00:10:03,260
+show that G prime of Y بتساوي 1 على N Y أس 1 على N
+
+92
+00:10:03,260 --> 00:10:12,580
+ناقص 1 حيث N is even solution نيجي الأن نحل تطبيق 
+
+93
+00:10:12,580 --> 00:10:20,240
+على ال theorem الأخيرة اللي هو ناخد inverse لهذه
+
+94
+00:10:20,240 --> 00:10:25,750
+طبعا واضح أن ال inverse لو قلنا let F of X بيساوي X 
+
+95
+00:10:25,750 --> 00:10:30,230
+أُس N هذا عبارة عن polynomial أو اللي هو دالة 
+
+96
+00:10:30,230 --> 00:10:35,170
+أُسية أو جزء من polynomial عند X أُس N طبعاً لأي X 
+
+97
+00:10:35,170 --> 00:10:39,350
+في الفترة 01 هذا is differentiable ومش هيك كمان
+
+98
+00:10:39,350 --> 00:10:42,910
+ومدامة الـ N even في الفترة هذه strictly 
+
+99
+00:10:42,910 --> 00:10:46,580
+increasing ده الـ Strict increase in continuous ده 
+
+100
+00:10:46,580 --> 00:10:50,860
+كل .. كل أمورها متحققة اللي قبل بشوية في النظرية
+
+101
+00:10:50,860 --> 00:10:56,000
+فالآن و هذه عبارة عن ال inverse لمين؟ للـ G فبكون
+
+102
+00:10:56,000 --> 00:11:04,540
+الآن G prime of Y إيش هتساوي؟ بتساوي 1 على F prime
+
+103
+00:11:04,540 --> 00:11:11,410
+عند G of Y مظبوط ولا لا؟ Y ساوي الان ال F prime هنا
+
+104
+00:11:11,410 --> 00:11:15,930
+واضح اللي هي N X أس N نقص واحد يعني عبارة عن واحد 
+
+105
+00:11:15,930 --> 00:11:22,070
+على بدل ال X إيش هحط؟ G of Y لأن هذا F prime of X
+
+106
+00:11:22,070 --> 00:11:26,290
+بساوي N X N نقص واحد طب F prime of G Y هي عبارة عن
+
+107
+00:11:26,290 --> 00:11:33,870
+G of Y أس N نقص واحد مضروبة في مين؟ في ال N Y ساوي
+
+108
+00:11:35,600 --> 00:11:43,600
+1 على N في G of Y إيش هي عبارة عن Y أس 1 على N
+
+109
+00:11:43,600 --> 00:11:49,640
+فبيصير اللي هو هذي بالمرة نرفحها بعد إذنكم G of Y 
+
+110
+00:11:49,640 --> 00:11:59,620
+أس 1 ناقص N ويساوي إن عوض 1 على N في Yأس واحد على 
+
+111
+00:11:59,620 --> 00:12:07,260
+N أس واحد ناقص N ويساوي واحد على N Y أس واحد على N
+
+112
+00:12:07,260 --> 00:12:12,660
+ناقص واحد وهو المطلوب هذه اللي هي تطبيق مباشرة على 
+
+113
+00:12:12,660 --> 00:12:17,580
+النظرية السابقة الآن تطبيق آخر مثال آخر أيضا مثال
+
+114
+00:12:17,580 --> 00:12:20,380
+يعني شبيه بالأمثل اللي كنتوا تاخدوه في ال calculus
+
+115
+00:12:20,380 --> 00:12:29,610
+المثال دي اللي بناخده اللي هو Y بتساوي صين ال X و
+
+116
+00:12:29,610 --> 00:12:37,590
+الـ x في ناقص πي على اتنين لعين باي على اتنين الآن
+
+117
+00:12:37,590 --> 00:12:43,470
+بقوللي اثبت انه sign inverse prove that اللي هو
+
+118
+00:12:43,470 --> 00:12:54,110
+show that sign inverse لل Y بتساوي واحد على جدر
+
+119
+00:12:54,110 --> 00:12:57,750
+التربيع لواحد ناقص مين Y تربيع
+
+120
+00:13:00,300 --> 00:13:04,160
+prime اه اه ال derivative لل sign inverse ايش
+
+121
+00:13:04,160 --> 00:13:06,840
+بتساوي واحدة الجدر تربيعه لو واحد نقص Y تربيعه
+
+122
+00:13:06,840 --> 00:13:10,940
+برضه تطبيق مباشر خلينا نشوف كيف نطبق على النظرية 
+
+123
+00:13:10,940 --> 00:13:20,000
+ناخد let F of X اللي هو بيساوي sign ال X عشان عند
+
+124
+00:13:20,000 --> 00:13:24,500
+G of Y او عند F of X بيساوي sign ال X اللي بتسميها
+
+125
+00:13:24,500 --> 00:13:31,610
+Y يعني الآن بتصير عند G of Y ايش بتساوي Sin Inverse
+
+126
+00:13:31,610 --> 00:13:37,790
+لمن؟ لـ Y معايا يا شباب، أه؟ لأن حسب الـ F of X 
+
+127
+00:13:37,790 --> 00:13:41,510
+بساعة و شوية إن الـ X الـ X في الفترة ناقص Y على 2
+
+128
+00:13:41,510 --> 00:13:47,690
+و Y على 2 الـ Y القيمة المقابلة إليها هتكون بين 
+
+129
+00:13:47,690 --> 00:13:54,560
+ناقص 1 و 1 لكن لاحظ الآن لما أشتق المشتقة لازم تكون 
+
+130
+00:13:54,560 --> 00:13:58,380
+لا تساوي صفر في الفترة هذه ففي الفترة هذه اللي هي
+
+131
+00:13:58,380 --> 00:14:02,900
+المشتقة طبعا لهذه اللي هي عبارة عن cosine فهتكون 
+
+132
+00:14:02,900 --> 00:14:06,900
+اللي هي ال X بس في ال open interval من ناقص y على
+
+133
+00:14:06,900 --> 00:14:11,540
+2 على y على 2 عساس ما نجسمش عالميا على الصفر إذا 
+
+134
+00:14:11,540 --> 00:14:17,620
+الآن صار عندي G prime of Y نفسي اللي عملناه فوق،
+
+135
+00:14:17,620 --> 00:14:23,720
+إيش بتساوي؟ واحد على F prime of G of Y Y ساوي،
+
+136
+00:14:23,720 --> 00:14:28,880
+واحد F .. اللي هي مين الأن F prime؟ اللي هي cosine
+
+137
+00:14:28,880 --> 00:14:36,320
+اللي هو G of Y، مظبوط؟ وال cosine احنا إيش؟ عبارة
+
+138
+00:14:36,320 --> 00:14:39,160
+عن إيش؟ عبارة عن إيش تكون؟ عبارة عن الجدر 
+
+139
+00:14:39,160 --> 00:14:44,080
+التربيعي؟ لا، طبعا ال cosine X إيش بتساوي؟ هذا 
+
+140
+00:14:44,080 --> 00:14:52,100
+عبارة عن X، مظبوط؟ هو الجدر التربيعي؟ لا، واحد ناقص
+
+141
+00:14:52,100 --> 00:14:57,980
+sign تربيع الـ X اللي هي G of Y، مظبوط؟ معايا يا 
+
+142
+00:14:57,980 --> 00:15:03,200
+شباب Y ساوي واحد على جدر التربيع إلى واحد ناقص Y
+
+143
+00:15:03,200 --> 00:15:08,940
+تربيع و Y في الفترة ناقص باي على اتنين و باي على
+
+144
+00:15:08,940 --> 00:15:13,800
+اتنين اللي هي الـ X و اللي موجودة أسفل X في ناقص 
+
+145
+00:15:13,800 --> 00:15:16,560
+باي على اتنين و باي على اتنين و الـ Y في الفترة 
+
+146
+00:15:16,560 --> 00:15:21,640
+ناقص واحد و واحد حتى نتجنب القسمة عالميا على اللي 
+
+147
+00:15:21,640 --> 00:15:25,380
+هو الصفر هذا التطبيق على النظرية 
+
+148
+00:15:27,410 --> 00:15:30,770
+ال homework عندنا تلاتة أو أربعة أو سبعة أو تسعة 
+
+149
+00:15:30,770 --> 00:15:34,250
+أو تلاتاش من section 6-1 ان شاء الله المرة الجاية
+
+150
+00:15:34,250 --> 00:15:39,510
+أو اللي بعدها بنحاول نعمل discussion فيا ريت اللي 
+
+151
+00:15:39,510 --> 00:15:44,410
+هو أنكم اتحلوا الأسئلة و اللي بتعرفوش اتحلوا او
+
+152
+00:15:44,410 --> 00:15:50,880
+بدكم نذاك بنحلو ان شاء الله نيجي الآن لصفحة
+
+153
+00:15:50,880 --> 00:15:53,800
+اللي بعدي ده اللي هو ال section التاني ممكن هيكنا
+
+154
+00:15:53,800 --> 00:15:58,460
+هنا ستة واحد الآن هنبدأ في ال section التاني اللي
+
+155
+00:15:58,460 --> 00:16:02,380
+هو ال main value theorem وهو محاضرة اليوم بشكل
+
+156
+00:16:02,380 --> 00:16:06,740
+أساسي عالمي على the main value theorem ومتعلقات
+
+157
+00:16:06,740 --> 00:16:12,870
+هذه النظرية هنلاقي جدّاش مهمة هذه النظرية وجدّاش
+
+158
+00:16:12,870 --> 00:16:17,650
+فيه عندنا نتائج تتعلق إيه بها وإحنا يعني فعلا 
+
+159
+00:16:17,650 --> 00:16:25,530
+بستعملها كثير وفعلا بتساعدنا اللي هو في اللي ده 
+
+160
+00:16:25,530 --> 00:16:30,270
+يعني إذا رح أطلب لما حتى أخدنا ال Calculus كانت
+
+161
+00:16:30,270 --> 00:16:37,010
+اللي هي إلها تطبيقات واسعة وحنشوف الآن وين أهمية
+
+162
+00:16:37,680 --> 00:16:42,520
+هذه نظرية نظرية القيم المتوسطة من خلال الحديث أو
+
+163
+00:16:42,520 --> 00:16:47,080
+شرح حادي المادة الآن في الأول زي ما قلنا ال
+
+164
+00:16:47,080 --> 00:16:49,420
+section 6.2 the mean value theorem نيجي لل
+
+165
+00:16:49,420 --> 00:16:53,420
+definition الأول ال definition اللي هو مر علينا 
+
+166
+00:16:53,420 --> 00:16:58,380
+أشي زيه أو مر علينا اللي هو تعريف ال local 
+
+167
+00:16:58,380 --> 00:17:04,500
+extremum أو اللي هو relative extremum أو اللي هو
+
+168
+00:17:04,500 --> 00:17:07,460
+الـ local maximum أو الـ local minimum أو الـ
+
+169
+00:17:07,460 --> 00:17:10,840
+relative extremum أو الـ relative اللي هو الـ 
+
+170
+00:17:10,840 --> 00:17:17,900
+minimum طبعاً بتعرفوا أن أنا بدي أعرف شو معناته دي
+
+171
+00:17:17,900 --> 00:17:24,480
+اللي هي النقطة عنده النقطة C في عند local .. local 
+
+172
+00:17:24,480 --> 00:17:29,860
+maximum مثلا أو relative maximum بنقول في عند C
+
+173
+00:17:30,870 --> 00:17:35,370
+عندي function f من some interval I لعند R هاي الـ 
+
+174
+00:17:35,370 --> 00:17:39,490
+function اللي عندي هنا من هنا لهنا بنقول C element
+
+175
+00:17:39,490 --> 00:17:48,430
+in I at C element in I we have a local أو relative
+
+176
+00:17:48,430 --> 00:18:00,820
+maximum when اللي هو there exists V
+
+201
+00:20:30,120 --> 00:20:33,720
+الأولى إحنا كنا في الـ calculus إذا بتتذكر ونسأل
+
+202
+00:20:33,720 --> 00:20:43,110
+سؤال لما بدنا نبحث في مسألة اللي هو الـ maximum و
+
+203
+00:20:43,110 --> 00:20:46,410
+الـ minimum سواء local maximum و local minimum أو
+
+204
+00:20:46,410 --> 00:20:50,230
+absolute maximum و absolute minimum كنا نسأل حالنا
+
+205
+00:20:50,230 --> 00:20:54,390
+سؤال طبعًا مش معقول أنا بدي آجي كل نقاط الـ domain 
+
+206
+00:20:54,390 --> 00:20:57,990
+أفحص فيها عند maximum ولا minimum لأن نقاط الـ
+
+207
+00:20:57,990 --> 00:21:01,170
+domain عدد لا نهائي uncountable number of points
+
+208
+00:21:01,170 --> 00:21:07,180
+في غالبها فإيش كنت حصرها؟ كنا نقول أين يمكن إيجاد
+
+209
+00:21:07,180 --> 00:21:12,480
+الـ local maximum أو الـ local minimum بعد ما نحدد
+
+210
+00:21:12,480 --> 00:21:17,100
+النقاط هذه بنقول يأتي الآن الفحص إنه هل هي local
+
+211
+00:21:17,100 --> 00:21:21,660
+maximum ولا local minimum إذن مسألة اللي هو إيجاد
+
+212
+00:21:21,660 --> 00:21:25,180
+الـ maximum و الـ minimum بتتم عبر خطوتين الخطوة
+
+213
+00:21:25,180 --> 00:21:31,380
+الأولى وين ممكن نلاقيها والخطوة الثانية اللي هي هل
+
+214
+00:21:31,380 --> 00:21:36,220
+هي فعلاً local maximum ولا local minimum ولا لا
+
+215
+00:21:36,220 --> 00:21:40,500
+local maximum ولا local minimum اللي بده يشرعلنا
+
+216
+00:21:40,500 --> 00:21:45,130
+الإجابة الأولى هي النظرية هذه النظرية هذه كنا كمان
+
+217
+00:21:45,130 --> 00:21:49,390
+نجاوب نقول الإجابة توجد الـ local maximum أو 
+
+218
+00:21:49,390 --> 00:21:53,030
+الـ local minimum في أحد ثلاث مناطق إما F prime
+
+219
+00:21:53,030 --> 00:21:58,070
+بساوي صفر أو F prime does not exist كنقط داخلية
+
+220
+00:21:58,070 --> 00:22:03,410
+أو عند الـ endpoints فكنا نحصرها في ثلاث أنهي مناطق
+
+221
+00:22:03,410 --> 00:22:07,530
+فبيصير عدد النقاط اللي بدنا نفحص الـ local max أو
+
+222
+00:22:07,530 --> 00:22:11,490
+الـ local minimum عددها محدود وقليل بيصير السؤال
+
+223
+00:22:11,490 --> 00:22:15,630
+مقدور عليه الآن النظرية اللي بتشرح لنا إياها دي إيش
+
+224
+00:22:15,630 --> 00:22:23,390
+النظرية بتقول بتقول ما يلي اللي بتقوله الـ theorem 
+
+225
+00:22:23,390 --> 00:22:29,120
+اللي هي حتى اسمها اللي هي interior extremum theorem
+
+226
+00:22:29,120 --> 00:22:32,920
+interior extremum theorem يعني هنحكي عن الـ
+
+227
+00:22:32,920 --> 00:22:38,020
+interior point بتقول لي نفترض إن C عبارة عن نقطة
+
+228
+00:22:38,020 --> 00:22:43,400
+داخلية لمين؟ لـ some interval I C اللي هو إشماله
+
+229
+00:22:43,400 --> 00:22:49,140
+نقطة داخلية لـ I الـ interval اللي عندي وعند الـ
+
+230
+00:22:49,140 --> 00:22:55,040
+function F من I لعند R لو فرضنا أن الـ function F
+
+231
+00:22:55,040 --> 00:23:02,000
+has a relative extremum at let's say element كده
+
+232
+00:23:02,000 --> 00:23:18,480
+if F has a relative extremum at C element in I عند
+
+233
+00:23:18,480 --> 00:23:23,520
+C an interior point of the interval I at which F
+
+234
+00:23:23,520 --> 00:23:27,640
+has a relative extremum يعني مفترضين إنه في عندي
+
+235
+00:23:27,640 --> 00:23:33,140
+يا relative maximum يا relative minimum عند هذه
+
+236
+00:23:33,140 --> 00:23:43,660
+النقطة الداخلية ونفترض كمان أن F prime of C exist
+
+237
+00:23:43,660 --> 00:23:51,610
+تحت كل هذه الشروط لازم تطلع عند F prime of C إيش
+
+238
+00:23:51,610 --> 00:23:56,050
+بتساوي؟ بتساوي صفر يعني بمعنى آخر لو كانت النقطة
+
+239
+00:23:56,050 --> 00:24:00,690
+داخلية و النقطة هذه الـ derivative عندها موجودة
+
+240
+00:24:01,820 --> 00:24:05,280
+وكانت عندها relative maximum أو relative minimum
+
+241
+00:24:05,280 --> 00:24:09,280
+قطعًا هتكون الـ F prime إليها إيش بتساوي؟ بتساوي
+
+242
+00:24:09,280 --> 00:24:12,960
+صفر يعني عند .. عند النقاط الداخلية اللي الـ F
+
+243
+00:24:12,960 --> 00:24:17,280
+prime إليها موجودة و بتساوي صفر انساها و ماتبحثش
+
+244
+00:24:17,280 --> 00:24:20,780
+فيها قطعًا فيش فيها لا relative maximum ولا
+
+245
+00:24:20,780 --> 00:24:25,750
+relative إيش .. minimum الآن هذه النظرية وكأنه
+
+246
+00:24:25,750 --> 00:24:31,810
+بتحكي لنا في داخلها أن النقاط اللي بتكون داخلية
+
+247
+00:24:31,810 --> 00:24:36,590
+واحتمال يكون عندها relative maximum و relative
+
+248
+00:24:36,590 --> 00:24:42,410
+minimum نوعين يا F prime مش موجودة أو الـ F' إيش
+
+249
+00:24:42,410 --> 00:24:48,190
+بتساوي؟ بتساوي 0 لأنه إذا كانت F' موجودة وقفقلها
+
+250
+00:24:48,190 --> 00:24:51,910
+relative maximum أو relative minimum وكانت النقطة
+
+251
+00:24:51,910 --> 00:24:56,710
+داخلية لازم الـ F' إيش مالها؟ تساوي 0 ولاحظ هنا إن
+
+252
+00:24:56,710 --> 00:25:02,190
+الحديث عن مين؟ عن النقاط الداخلية يعني من فحوى
+
+253
+00:25:02,190 --> 00:25:06,110
+الحديث هنا بنجاوب على سؤالنا، هذه اللي بتجاوبنا
+
+254
+00:25:06,110 --> 00:25:09,860
+على سؤالنا بتقول اللي هو الـ relative maximum أو
+
+255
+00:25:09,860 --> 00:25:13,880
+الـ relative minimum هتكون في أحد ثلاث مناطق يا F
+
+256
+00:25:13,880 --> 00:25:18,060
+prime does not exist يا F prime بساوي صفر يا
+
+257
+00:25:18,060 --> 00:25:27,960
+end point واضح؟ أي سؤال؟ واضح أه؟ طيب، دي يجي
+
+258
+00:25:27,960 --> 00:25:36,550
+تبرهن النظرية الآن هو البرهان اللي مكتوب هنا we
+
+259
+00:25:36,550 --> 00:25:41,170
+prove for the case that F has relative maximum at
+
+260
+00:25:41,170 --> 00:25:45,150
+C هي مكتوب أنا ما بديش أبرهن الـ maximum بتبرهنكم
+
+261
+00:25:45,150 --> 00:25:52,290
+مين؟ الـ minimum إذا الآن بدنا نفترض نفترض الـ proof
+
+262
+00:25:52,290 --> 00:25:57,870
+خليني أكتب هنا الـ proof بدي
+
+263
+00:25:57,870 --> 00:26:04,820
+أفترض C نقطة داخلية ونفترض إن F أشبهلها have a
+
+264
+00:26:04,820 --> 00:26:08,560
+relative minimum relative maximum محلولة لأن
+
+265
+00:26:08,560 --> 00:26:12,640
+suppose that
+
+266
+00:26:12,640 --> 00:26:21,020
+F has a relative maximum
+
+267
+00:26:21,020 --> 00:26:28,120
+at C element in I ماشي؟
+
+268
+00:26:37,000 --> 00:26:41,080
+نعم نفترض إنها minimum لأن الـ case تبعت الـ maximum
+
+269
+00:26:41,080 --> 00:26:47,860
+محلولة واضحة؟ الآن لو كان F as relative extremum
+
+270
+00:26:47,860 --> 00:26:52,160
+بدنا نصل الآن extreme يا بتكون maximum يا بتكون
+
+271
+00:26:52,160 --> 00:26:56,440
+minimum case الـ maximum هي مكتوبة بتجروها لحالكم
+
+272
+00:26:56,440 --> 00:27:01,980
+نفس الفكرة ده أكتف بت mean F as a relative H
+
+273
+00:27:01,980 --> 00:27:08,360
+minimum الآن بدي أفترض لك أن الـ F' أي شمالها لا
+
+274
+00:27:08,360 --> 00:27:11,700
+تساوي صفر و أسألك لـ contradiction إذا الآن بدي
+
+275
+00:27:11,700 --> 00:27:18,780
+أفترض لك و بدي أقول لك suppose that F prime of C
+
+276
+00:27:18,780 --> 00:27:25,620
+لا تساوي إيش؟ صفر، ماشي الحال، إذا عندي two cases
+
+277
+00:27:25,620 --> 00:27:34,360
+الـ case الأول أن F prime of C أكبر من إيش؟ من صفر
+
+278
+00:27:34,360 --> 00:27:43,580
+ماشي الحال إذا الآن عندي limit f of x ناقص f of c
+
+279
+00:27:43,580 --> 00:27:50,340
+على x minus c as x بتروح لـ c أشماله اللي هو هذا f
+
+280
+00:27:50,340 --> 00:27:57,040
+prime of c حسب المعطى إيش هيكون؟ أكبر من صفر ماشي
+
+281
+00:27:57,040 --> 00:28:03,480
+الحال طيب الآن مدام الـ f prime of c أكبر من صفر إذا
+
+282
+00:28:03,480 --> 00:28:07,700
+حسب نظرية قلنا قبل هيك إنه لو كانت الـ limit للـ
+
+283
+00:28:07,700 --> 00:28:12,800
+function أكبر من 0 هنلاقي جوار حوالين النقطة اللي
+
+284
+00:28:12,800 --> 00:28:16,660
+بدنا نروح لها يكون اللي هو قيمة اللي جوا الـ limit
+
+285
+00:28:16,660 --> 00:28:21,780
+أكبر من 100 من 0 لأن كما قالي مدام الـ limit هذا
+
+286
+00:28:21,780 --> 00:28:29,260
+أكبر من 0 then there exists Delta أكبر من صفر أو V
+
+287
+00:28:29,260 --> 00:28:35,380
+Delta أكبر من صفر Such that اللي هو F of X نقص F
+
+288
+00:28:35,380 --> 00:28:43,400
+of C على X minus C أكبر من صفر لكل X وين موجودة في
+
+289
+00:28:43,400 --> 00:28:48,240
+الـ V Delta of C صح؟
+
+290
+00:28:48,240 --> 00:28:53,820
+هاي عندي إحنا مفترضين إن هي أشملها relative
+
+291
+00:28:53,820 --> 00:28:57,940
+minimum خليني أشتغل على هذه المنطقة أساسي أن
+
+292
+00:28:57,940 --> 00:29:02,260
+تشوفوها على الرسم
+
+293
+00:29:02,260 --> 00:29:13,880
+يمكن تكون أوضح عندي هي النقطة C وهذا C ناقص Delta
+
+294
+00:29:13,880 --> 00:29:21,060
+وهذا C زائد Delta، واضح؟ الـ F of X ناقص F of Z
+
+295
+00:29:21,060 --> 00:29:24,000
+على X minus C أكبر من مين؟ من صفر لكل X و الموجودة
+
+296
+00:29:24,000 --> 00:29:32,040
+في الـ V Delta of C أه، الآن .. الآن عندي هو .. بدي
+
+297
+00:29:32,040 --> 00:29:37,020
+.. عندي أنا مفترض إنه اللي هو F هذا relative A
+
+298
+00:29:37,020 --> 00:29:45,580
+شماله minimum إذا في جوار .. في جوار بتكون F of C
+
+299
+00:29:45,580 --> 00:29:52,700
+أصغر أو يساوي F of X لكل X في هذا الجوار اللي هو V
+
+300
+00:29:52,700 --> 00:29:59,440
+Delta Prime of إيش؟ Of C، ماشي الحال الآن الـ Delta
+
+301
+00:29:59,440 --> 00:30:03,820
+هذه فيه Delta أكبر من 0 باخدها الـ Delta أصغر من
+
+302
+00:30:03,820 --> 00:30:08,840
+هذه بحكم فيها، فده بتنفع للـ Delta أكبر من هذه إن
+
+303
+00:30:08,840 --> 00:30:11,880
+هذا الكون المقدر لكل X في الـ V Delta يكون أكبر من
+
+304
+00:30:11,880 --> 00:30:17,100
+0، إذا بديجه، برضه بنفع، إذا بختار الـ Delta هذه
+
+305
+00:30:17,100 --> 00:30:20,420
+بناء على اختيار لهذه، فبتكون الـ Delta هذه أنا
+
+306
+00:30:20,420 --> 00:30:24,770
+بختارها أصغر أو يساوي من الـ Delta prime ماشي الحال
+
+307
+00:30:24,770 --> 00:30:28,770
+طيب إذا صار عندي F of X نقص F of C على X minus C
+
+308
+00:30:28,770 --> 00:30:34,970
+أكبر من 100 من 0 لكل X limited V Delta O of C
+
+309
+00:30:34,970 --> 00:30:38,090
+أذكركم مش بده أنا أنا بده أصل يكون contradiction
+
+310
+00:30:38,090 --> 00:30:45,050
+يعني بده أصل لنقاط في المكان هذا تتحققش هذه معايا؟
+
+311
+00:30:46,850 --> 00:30:48,790
+بدي أسأل الـ contradiction كيف أسأل الـ
+
+312
+00:30:48,790 --> 00:30:53,210
+contradiction؟ بدي أسأل ألاقي نقاط في المنطقة هذه
+
+313
+00:30:53,210 --> 00:30:59,750
+و هذه لا تتحقق شوف الآن عندي هذا المقدار إيش ماله؟
+
+314
+00:30:59,750 --> 00:31:07,730
+موجب، مظبوط ولا لأ؟ ماشي، الآن اللي بدي أخلي اللي
+
+315
+00:31:07,730 --> 00:31:13,830
+هي F of X minus F of C تظل أكبر من 0 لأن لو خليتها
+
+316
+00:31:13,830 --> 00:31:19,210
+أكبر من صفر بيصير F أصغر من صفر بيصير F of X لما
+
+317
+00:31:19,210 --> 00:31:22,030
+يصير أصغر من صفر بيصير أصغر من F of C بتبطّل أش
+
+318
+00:31:22,030 --> 00:31:28,650
+مالها عبارة عن minimum بتخلي
+
+319
+00:31:28,650 --> 00:31:35,130
+هذا المقدار أصغر من أش من صفر بدون هذا أه؟ بيصير F
+
+320
+00:31:35,130 --> 00:31:41,120
+of X أصغر من F of C بيصير F of Z مش minimum واضحة؟
+
+321
+00:31:41,120 --> 00:31:45,560
+طيب إيش اللي بده يخلي لي هذا أصغر من 0 هذا المقدار
+
+322
+00:31:45,560 --> 00:31:52,560
+أصغر من 0 أني أضرب أعمل choose x بحيث إنه يكون هذا
+
+323
+00:31:52,560 --> 00:31:57,380
+سالب وإن هيكون سالب x minus c بختار اللي هي x
+
+324
+00:31:57,380 --> 00:32:01,680
+minus c بختار x وإن ما أقول بقول choose x element
+
+325
+00:32:01,680 --> 00:32:07,240
+إن c نقص دلتة و c فماشي الحال then
+
+326
+00:32:09,590 --> 00:32:15,310
+اللي هو لما اخترت الـ X هنا هيكون الـ X minus C
+
+327
+00:32:15,310 --> 00:32:20,570
+أصغر من 0 مدام صار الـ X minus C أصغر من 0 وهذا
+
+328
+00:32:20,570 --> 00:32:26,630
+أكبر من 0 معطى من التنتين مع بعض بيصير عند X minus
+
+329
+00:32:26,630 --> 00:32:37,630
+C في F of X minus F of C على X minus C أصغر من 0،
+
+330
+00:32:37,630 --> 00:32:43,370
+اللي هو هذا في الواقع مين؟ هو عبارة عن F of X ناقص
+
+331
+00:32:43,370 --> 00:32:47,210
+F of C صار الـ F of X minus F of C، لأن هذا بتروح
+
+332
+00:32:47,210 --> 00:32:50,690
+محدد، لأنه مش صفر، بيصير F of X ناقص F of C أصغر من
+
+333
+00:32:50,690 --> 00:32:57,150
+0، معناته صار F of X أصغر من F of C، هذا لمين؟
+
+334
+00:32:57,150 --> 00:33:02,050
+للنقطة X اللي في هذا الجوار، يعني اللي لجأت نقطة
+
+335
+00:33:02,050 --> 00:33:04,790
+في هذا الجوار، لا تحقق هذا
+
+336
+00:33:07,600 --> 00:33:15,160
+هذا contradiction طيب اللي ما وضحت له هذه خلينا
+
+337
+00:33:15,160 --> 00:33:19,780
+نوضحها في الحالة الثانية هذا في حالة مين؟ أه في
+
+338
+00:33:19,780 --> 00:33:24,440
+prime و في c أشماله أكبر من صفر الآن بدأت أقول
+
+339
+00:33:24,440 --> 00:33:31,000
+suppose that الحالة الثانية كاس اتنين يعني اللي
+
+340
+00:33:31,000 --> 00:33:37,320
+بقوله في حالة فرضت F prime لا تساوي صفر هأصل دائمًا
+
+341
+00:33:37,320 --> 00:33:41,200
+لتناقض F prime أكبر من صفر حصل لتناقض وصلت لتناقض
+
+342
+00:33:41,200 --> 00:33:47,060
+والآن حالة F prime of C أصغر من صفر برضه حصل لمين؟
+
+343
+00:33:47,460 --> 00:33:51,860
+لـ تناقض ماشي الحال الآن إيش معناه F prime of c أكبر
+
+344
+00:33:51,860 --> 00:33:58,040
+من 0 بنفس الأسلوب limit f of x ناقص f of c على x
+
+345
+00:33:58,040 --> 00:34:03,860
+minus c as x بتروح للـ c برضه إيش ماله؟ أصغر من 0
+
+346
+00:34:03,860 --> 00:34:09,820
+إذا في جوار there exist v delta of x و باخد الـ
+
+347
+00:34:09,820 --> 00:34:13,500
+delta اللي هي delta prime بقدر أخدها و أصغر يساوي
+
+348
+00:34:13,500 --> 00:34:20,950
+الـ delta prime by such that اللي هو F of x ناقص F
+
+349
+00:34:20,950 --> 00:34:27,310
+of c على x minus c أصغر من مين؟ من 0 الآن اللي
+
+350
+00:34:27,310 --> 00:34:31,430
+بيوصلني لتناقض هتلاقيها مش choosing x هنا و أنا
+
+351
+00:34:31,430 --> 00:34:35,450
+هاختارها في الجهة الثانية في الجهة بتشوف كيف لأن
+
+352
+00:34:35,450 --> 00:34:43,570
+بيقول choose x element in c و c زائد delta then
+
+353
+00:34:47,030 --> 00:34:54,130
+هيكون عندي الآن الـ X minus C إيش ماله؟ أكبر من 0
+
+354
+00:34:54,130 --> 00:35:00,610
+واضح أه؟ لأن هذا أكبر من صفر وهذا أكبر أصغر من صفر
+
+355
+00:35:00,610 --> 00:35:07,110
+ثم اللي هي F of X ناقص F of C اللي هي بتساوي X
+
+356
+00:35:07,110 --> 00:35:10,670
+minus C اللي هي أكبر من صفر في المقدار اللي هو F
+
+357
+00:35:10,670 --> 00:35:15,390
+of X ناقص F of C على X minus C اللي هو أصغر من صفر
+
+358
+00:35:15,390 --> 00:35:19,430
+حصل أصغر من صفر وأكبر من صفر هيكون أصغر من صفر
+
+359
+00:35:19,430 --> 00:35:26,680
+ومنه hence F of X أصغر من F of C لنقطة في هذا
+
+360
+00:35:26,680 --> 00:35:30,720
+الجوار اللي هو باخدها very close to C يعني معناته
+
+361
+00:35:30,720 --> 00:35:35,040
+هتكون في داخل هذا الجوار لجهة نقطة بحيث أنه F of C
+
+362
+00:35:35,040 --> 00:35:41,540
+أكبر من F of X which is a contradiction therefore F
+
+363
+00:35:41,540 --> 00:35:47,240
+prime of C must equal zero إذن بيكون أثبتنا إن لو
+
+364
+00:35:47,240 --> 00:35:52,000
+كان في عندي local minimum عند الـ interior point C
+
+365
+00:35:52,000 --> 00:35:59,200
+إذن اللي هي بتكون الـ F prime of C بساوي صفر طبعًا
+
+366
+00:35:59,200 --> 00:36:04,660
+لاحظوا أنا كل الجوارات اللي أخدت هنا الـ V Delta و
+
+367
+00:36:04,660 --> 00:36:09,010
+V Delta prime كل في داخل الـ interval عشان هيك هو
+
+368
+00:36:09,010 --> 00:36:12,850
+طبعًا هو كاتب تقاطع الـ .. بكتب تقاطع الـ subset من
+
+369
+00:36:12,850 --> 00:36:17,450
+الـ interval لأن كل اللي بشتغله وين بشتغله؟ جوات الـ
+
+370
+00:36:17,450 --> 00:36:25,510
+interval أي سؤال؟ اللي هو؟
+
+371
+00:36:25,510 --> 00:36:33,890
+تكملة
+
+372
+00:36:33,890 --> 00:36:38,810
+يعني F of c أقل من F of x دقيقة بس هنكتب اللي
+
+373
+00:36:38,810 --> 00:36:45,870
+بتقوله عشان اللي بيشوف تصويرنا و بيسمع كل معاناته
+
+374
+00:36
+
+401
+00:38:55,380 --> 00:38:58,200
+لو يعني لاحظت في الـ .. في الـ .. طبعا احنا ما
+
+402
+00:38:58,200 --> 00:39:00,220
+ستخدمناش the limit من اليمين و the limit من اليسار،
+
+403
+00:39:00,220 --> 00:39:08,700
+ستخدمنا the limit مباشرة، بصير، بصير، فهم صحيح هو
+
+404
+00:39:08,700 --> 00:39:13,640
+الحل اللي أنت بتعرفه و يعني خليني أقوله مرتاح له
+
+405
+00:39:13,640 --> 00:39:17,500
+هو الأسهل آه يعني أنت ممكن تشوف هذا أسهل و هذا
+
+406
+00:39:17,500 --> 00:39:24,260
+أسهل كلام صحيح يلا اللي بعده الآن corollary منيح
+
+407
+00:39:24,260 --> 00:39:26,620
+محمد اللي بعده
+
+408
+00:39:33,180 --> 00:39:36,640
+the corollary اللي ممكن حكيتها the corollary هذه أنا
+
+409
+00:39:36,640 --> 00:39:40,380
+اللي هي من المعلومات اللي قبل بشوية لو كانت الـ
+
+410
+00:39:40,380 --> 00:39:44,420
+function f اللي هو has اللي هي f من I من I اللي 
+
+411
+00:39:44,420 --> 00:39:47,980
+عندها continuous و non-interval and suppose that f
+
+412
+00:39:47,980 --> 00:39:51,420
+has a relative extremum at an interior point c of
+
+413
+00:39:51,420 --> 00:39:58,540
+I then either the derivative of f exists or f بسوة
+
+414
+00:39:58,540 --> 00:40:02,780
+سفرعادت صيغة للي قبل بشوية أنه لو كانت الـ
+
+415
+00:40:02,780 --> 00:40:08,720
+function عندي من I لعند R continuous وكانت اللي هي 
+
+416
+00:40:08,720 --> 00:40:13,540
+الـ C interior point عندها relative maximum أو 
+
+417
+00:40:13,540 --> 00:40:17,920
+relative minimum هذا لازم يعطينا أن الـ F prime يا
+
+418
+00:40:17,920 --> 00:40:21,860
+does not exist يا إما zero طبعا و هذي مباشرة لو
+
+419
+00:40:21,860 --> 00:40:28,150
+جلنا نفترض أن F prime existبنحصل عالميين على الـ 0
+
+420
+00:40:28,150 --> 00:40:32,190
+من النظرية اللي قبله شوية وهو المطلوب واضح نجي
+
+421
+00:40:32,190 --> 00:40:35,230
+للنظرية البعيدة اللي هي إيش اسمها النظرية المشهورة
+
+422
+00:40:35,230 --> 00:40:41,090
+اللي هي اسمها Rolle's theorem Rolle's theorem يا جماعة
+
+423
+00:40:41,090 --> 00:40:44,370
+طبعا مرت علينا في الـ calculus ويمكن كمان صارت في
+
+424
+00:40:44,370 --> 00:40:49,690
+الثانوية يدرسوها النظرية طبعا معناها الهندسي كمان
+
+425
+00:40:49,690 --> 00:40:53,910
+بمر معاكم أنه لو كانت عند الـ function is
+
+426
+00:40:53,910 --> 00:41:05,950
+continuous على F من A و B لعند R continuous on A و
+
+427
+00:41:05,950 --> 00:41:15,830
+B و differentiable on A و B open وعندي F of A .. F
+
+428
+00:41:15,830 --> 00:41:22,960
+of A هيها و F of B هيهابصير عندي معطيني كمان f of
+
+429
+00:41:22,960 --> 00:41:26,900
+a بساوي f of b ماعطينيهاش بساوي، بساوي صفر، لو
+
+430
+00:41:26,900 --> 00:41:30,260
+كانت بتساوي ثابت برضه بتظبط، بتكون نفسها، تعميم
+
+431
+00:41:30,260 --> 00:41:35,500
+بسيط لها الـ f of a بساوي f of b بساوي صفر، بقوللي
+
+432
+00:41:35,500 --> 00:41:46,320
+إذا هالاجي نقطة then هذا معطى if this holds then
+
+433
+00:41:47,250 --> 00:41:52,110
+there exists C وحدة، تنتين، تلات، أو .. المهم على 
+
+434
+00:41:52,110 --> 00:41:55,790
+الأقل في واحدة there exists C بين الـ A و الـ B such
+
+435
+00:41:55,790 --> 00:42:01,290
+that F prime of C بيساوي صفر ماذا يعني؟ يعني بالله
+
+436
+00:42:01,290 --> 00:42:06,230
+بمعنى آخر نقطة C أو أكثر في بين الـ A و الـ B بحيث
+
+437
+00:42:06,230 --> 00:42:11,390
+أن ميل المماس عندها أو المماس اللي عندها بوازى 
+
+438
+00:42:11,390 --> 00:42:15,570
+محور السينات طبعا هنا في واحد هاي اثنين هاي ثلاثة
+
+439
+00:42:16,090 --> 00:42:21,070
+طبعا حسب .. حسب الدالة اللي هو أكيد بتلاقي إذا 
+
+440
+00:42:21,070 --> 00:42:26,130
+تحققت الشروط أقل في نقطة عندها اللي هو المماس
+
+441
+00:42:26,130 --> 00:42:35,870
+بوزى محور السينات أي سؤال؟ the proof the proof برضه
+
+442
+00:42:35,870 --> 00:42:39,570
+احنا كنا أحيانا نبرهنه في الـ calculus خلينا 
+
+443
+00:42:39,570 --> 00:42:46,150
+نبرهنوها، أكيد هنبرهنوها عندك لو كانت الـ F هي 
+
+444
+00:42:46,150 --> 00:42:51,550
+الدالة الصفرية يعني F بتساوي 0 it is a trivial أنه
+
+445
+00:42:51,550 --> 00:42:54,390
+مش حلها جنوبها كل نقاطها الـ F prime of C هي إيش
+
+446
+00:42:54,390 --> 00:42:59,770
+بتتساوي؟ بتساوي 0 إذا الـ case F بتساوي 0 is a
+
+447
+00:42:59,770 --> 00:43:07,590
+trivial case الآن case 2 case 1 الآن F ليست الدالة
+
+448
+00:43:07,590 --> 00:43:14,870
+الصفرية في الاحتمالين أنه بدأ عملها أن F assumes
+
+449
+00:43:14,870 --> 00:43:25,310
+positive values الاحتمال الثاني F assumes
+
+450
+00:43:25,310 --> 00:43:33,730
+negative values أنا بدعي أن هذولا بيجيبن كل الحلال
+
+451
+00:43:34,430 --> 00:43:38,270
+لو كانت F assumes positive value ممكن تاخد 
+
+452
+00:43:38,270 --> 00:43:40,830
+positive وتاخد negative وممكن تاخد positive لحالها
+
+453
+00:43:40,830 --> 00:43:46,250
+بظبط ولا لأ الـ F assume negative values إذا هتاخد
+
+454
+00:43:46,250 --> 00:43:49,530
+حالة F assumes negative values لحالها أو negative
+
+455
+00:43:49,530 --> 00:43:56,780
+values مع مين؟ مع positive values واضح؟ طيب، الآن
+
+456
+00:43:56,780 --> 00:44:00,520
+بدي أثبتلك في الحالة الأولى على طول هتطلع F prime
+
+457
+00:44:00,520 --> 00:44:06,700
+of C == 0 وفي الحالة الثانية similarly، معايا؟ الآن
+
+458
+00:44:06,700 --> 00:44:10,780
+في عندنا نظرية في تحليل واحد أو في الـ calculus حتى
+
+459
+00:44:10,780 --> 00:44:16,640
+كنا نقول if F is continuous on a closed bounded
+
+460
+00:44:16,640 --> 00:44:22,640
+interval then it must attain its maximum and its
+
+461
+00:44:22,640 --> 00:44:28,140
+minimum in this absolute كمان in its interval،
+
+462
+00:44:28,140 --> 00:44:35,130
+مظبوط؟ يعني بمعنى آخر عندي بما أن F continuous على
+
+463
+00:44:35,130 --> 00:44:39,830
+closed bounded interval then its absolute maximum
+
+464
+00:44:39,830 --> 00:44:43,650
+and absolute minimum is attained here يعني بمعنى
+
+465
+00:44:43,650 --> 00:44:50,470
+آخر there exists C element in A وB such that F of
+
+466
+00:44:50,470 --> 00:44:55,900
+C هي الـ maximum إيه في الـ maximum يعني؟ يعني الـ
+
+467
+00:44:55,900 --> 00:45:01,040
+supremum أو الـ maximum لكل F of X حيث X وين موجودة
+
+468
+00:45:01,040 --> 00:45:08,400
+في الـ A و الـ B، مظبوط؟ إذا صارت عندي في C في الـ A
+
+469
+00:45:08,400 --> 00:45:12,180
+و الـ B بحيث أن F of C بالساوى the supremum هذا الـ
+
+470
+00:45:12,180 --> 00:45:17,780
+supremum يا جماعة أكيد أكبر strictly من صفر
+
+471
+00:45:18,660 --> 00:45:23,920
+ليش؟ لأن احنا تحت أي عنوان بنشتغل F assumes
+
+472
+00:45:23,920 --> 00:45:27,640
+positive values إذا في على أقل قيمة واحدة positive
+
+473
+00:45:27,640 --> 00:45:33,440
+إذا F of C هيكون أكبر strictly من مين؟ من صفر طيب
+
+474
+00:45:33,440 --> 00:45:41,920
+بقول برضه F of C صارت عبارة عن maximum point الـ C
+
+475
+00:45:41,920 --> 00:45:50,520
+مظبوط؟ هذا واحد اثنين الـ C قطعاً لا هي الـ A ولا
+
+476
+00:45:50,520 --> 00:45:58,060
+هي الـ B ليش؟ لأنه بما أن F of A بساوي F of B 
+
+477
+00:45:58,060 --> 00:46:06,240
+بساوي 0 إذا أكيد C هتكون نقطة داخلية لا هتكون الـ
+
+478
+00:46:06,240 --> 00:46:10,120
+A ولا هتكون الـ B إذا صار في عندي شغلتين أنه C
+
+479
+00:46:10,120 --> 00:46:16,880
+نقطة داخلية و C عندها maximum إذا وكانت وعندي الـ F
+
+480
+00:46:16,880 --> 00:46:19,800
+is differentiable على الـ A و الـ B على الـ open
+
+481
+00:46:19,800 --> 00:46:24,260
+interval إذا حسب النظرية اللي قبله شوية لازم تكون
+
+482
+00:46:24,260 --> 00:46:30,800
+F prime of C عشان تساوي صفر then F prime of C
+
+483
+00:46:30,800 --> 00:46:40,530
+تساوي صفر واضح آه طيب الآن الحالة الثانية F assumes
+
+484
+00:46:40,530 --> 00:46:44,670
+negative values مدام F assumes negative values إذا 
+
+485
+00:46:44,670 --> 00:46:53,930
+ناقص الـ F assumes positive values صح ولا لا؟ الآن
+
+486
+00:46:53,930 --> 00:46:56,870
+اللي بنتبج على الـ F اللي فوق صار ينتبج على ناقص F
+
+487
+00:46:56,870 --> 00:46:57,390
+كله
+
+488
+00:47:05,040 --> 00:47:08,060
+differentiable على الـ A و الـ B و continuous على 
+
+489
+00:47:08,060 --> 00:47:12,620
+الـ closed interval و هذا هيتحقق على مين بس؟ على
+
+490
+00:47:12,620 --> 00:47:19,480
+ناقص F إذا من الـ case الثانية ناقص F of C prime
+
+491
+00:47:19,480 --> 00:47:26,300
+هتساوي صفر أو بمعنى آخر F prime of C هيساوي صفر
+
+492
+00:47:26,300 --> 00:47:32,380
+for some C element in A و B هذه برهان الـ Rolle's
+
+493
+00:47:32,380 --> 00:47:45,410
+theory أي سؤال اللي بعده الآن أوصلنا لإن نقدر الآن
+
+494
+00:47:45,410 --> 00:47:50,890
+نحكي عن اللي هو نظرية القيمة المتوسطة أو الـ Mean
+
+495
+00:47:50,890 --> 00:47:54,990
+Value Theorem نظرية القيمة المتوسطة أو الـ Mean
+
+496
+00:47:54,990 --> 00:47:57,510
+Value Theorem برضه
+
+497
+00:47:59,060 --> 00:48:03,900
+اللي هو مرت عليكم في calculus ألف و بتمر أيضًا في 
+
+498
+00:48:03,900 --> 00:48:09,000
+اللي هو ملهجة ثانوية إيش الـ mean value theorem؟ هي 
+
+499
+00:48:09,000 --> 00:48:15,840
+عبارة عن تعميم لمين للـ Rolle's theorem إيش بتقول الـ
+
+500
+00:48:15,840 --> 00:48:27,580
+mean value theorem؟ mean value theorem if F من A و 
+
+501
+00:48:27,580 --> 00:48:38,570
+B لعند R is continuous and differentiable on A وB
+
+502
+00:48:38,570 --> 00:48:47,130
+then there exists C element in A وB such that F
+
+503
+00:48:47,130 --> 00:48:54,010
+prime of C تساوي F of B ناقص F of A على B minus A
+
+504
+00:48:54,010 --> 00:48:57,590
+طبعا معناها الهندسية برضه مارة عليكم خليني أقول
+
+505
+00:48:57,590 --> 00:49:05,120
+كمان مرة إيش اللي بتقوله النظرية؟ بتقول كمان يا ليه؟
+
+506
+00:49:05,120 --> 00:49:14,860
+لإن مش مزايد بس ماشي
+
+507
+00:49:14,860 --> 00:49:22,960
+مفترض إن هي اللي عندي الـ A وهي عند مين الـ B الآن F
+
+508
+00:49:22,960 --> 00:49:27,180
+of B ناقص F of A على B minus A اللي هو عبارة عن ميل
+
+509
+00:49:27,180 --> 00:49:34,540
+عن ميل القاطع هذا ها .. هايه هذا اللي هو عبارة عن
+
+510
+00:49:34,540 --> 00:49:43,760
+النقطة B و F of B و هذا النقطة A و F of A طبعا ميل
+
+511
+00:49:43,760 --> 00:49:50,380
+القاطع زي ما أنتم عارفين هو F of B هنا ناقص F of
+
+512
+00:49:50,380 --> 00:49:57,060
+A هذه فبكون هذا على اللي هو B كلها هذه ناقص A
+
+513
+00:49:57,060 --> 00:50:02,360
+فبكون اللي هو ميل القاطع هذا اللي هو F of B ناقص F
+
+514
+00:50:02,360 --> 00:50:06,900
+of A على B minus A و الـ F prime of C اللي هو عبارة
+
+515
+00:50:06,900 --> 00:50:10,860
+عن إيش؟ عن ميل المماس عند النقطة C يعني وكأنه
+
+516
+00:50:10,860 --> 00:50:14,400
+نظرية بتقولنا إنه لو كانت F is continuous على
+
+517
+00:50:14,400 --> 00:50:17,840
+closed interval A وB و differentiable على الـ open
+
+518
+00:50:17,840 --> 00:50:22,340
+interval A وB بلاقي اللي هو مما .. وهي عند القاطع
+
+519
+00:50:22,340 --> 00:50:29,040
+هيه بلاقي مماس لدالة مماس واحد أو اثنين أو ثلاثة
+
+520
+00:50:29,040 --> 00:50:37,560
+بوزى مين يعني عند نقطة C بلاقي اللي هو مماس بوازى
+
+521
+00:50:37,560 --> 00:50:41,260
+.. بوازى القاطع في Rolle's theorem كانت اللي هو الـ A
+
+522
+00:50:41,260 --> 00:50:45,740
+و الـ B صورة .. صورة انتحار على الـ X axis فطبيعي ما 
+
+523
+00:50:45,740 --> 00:50:49,980
+اللي هو قطح هو الـ X axis من القاطع اللي هو صفر
+
+524
+00:50:50,790 --> 00:50:56,170
+فاهمين؟ فبكون اللي هو الـ F' اللي هو طبيعي بوازى
+
+525
+00:50:56,170 --> 00:51:00,890
+أو صفر أو بوازى الـ X-axis يعني بمعنى آخر و الـ
+
+526
+00:51:00,890 --> 00:51:05,750
+theorem هي حالة خاصة من الـ mean value theorem
+
+527
+00:51:05,750 --> 00:51:12,490
+واضح؟ طيب، الآن الـ proof the proof طبعا أكيد هنعتمد
+
+528
+00:51:13,290 --> 00:51:16,690
+على اللي هو الـ mean على الـ Rolle's theorem في 
+
+529
+00:51:16,690 --> 00:51:21,950
+الإثبات بدنا اللي هو نركب اللي هو الـ function
+
+530
+00:51:21,950 --> 00:51:27,210
+تكون بتحقق الـ mean value theorem و بتجيبلنا اللي 
+
+531
+00:51:27,210 --> 00:51:33,750
+هي الصورة هذه لو أخدنا Φ(x) بتساوي F of x ناقص F of a
+
+532
+00:51:33,750 --> 00:51:38,110
+ناقص اللي هو الميل اللي بديه F of b ناقص F of a
+
+533
+00:51:38,110 --> 00:51:45,970
+على b minus a مضروبة في X minus A هلجيت بتشوف ليش
+
+534
+00:51:45,970 --> 00:51:52,910
+أخدنا الدالة الدالة الآن هذه لو جيت حسبة Φ(A)
+
+535
+00:51:52,910 --> 00:51:57,630
+لها يعني consider this function Φ(A) هيساوي F
+
+536
+00:51:57,630 --> 00:52:02,730
+of A ناقص F of A صفر و A ناقص A صفر فبصير بصفر هذا
+
+537
+00:52:02,730 --> 00:52:10,040
+إذن بيساوي صفر لو جيت حسبة Φ(B) Φ(B) هيصير Φ
+
+538
+00:52:10,040 --> 00:52:14,280
+of B بيساوي F of B ناقص F of A خليني عندك و B minus
+
+539
+00:52:14,280 --> 00:52:18,260
+A بتروح هذه محادي هذه ناقص لنفسها بيطلع برضه إيش
+
+540
+00:52:18,260 --> 00:52:23,000
+صفر إذا الآن Φ(A) بتساوي Φ(B) بتساوي صفر هاي
+
+541
+00:52:23,000 --> 00:52:27,600
+واحد اثنين عندي هذه الدالة is continuous على
+
+542
+00:52:27,600 --> 00:52:33,120
+closed bounded interval وهذه ثوابت و X minus A X
+
+543
+00:52:33,120 --> 00:52:36,120
+minus A ده اللي خطية continuous وين مكان و
+
+544
+00:52:36,120 --> 00:52:39,520
+differentiable وين مكان إذا صارت هذه الدالة كلها
+
+545
+00:52:39,520 --> 00:52:43,360
+على بعض اللي هي Φ(X) is continuous وين مكان
+
+546
+00:52:43,360 --> 00:52:45,900
+بما فيها الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+547
+00:52:45,900 --> 00:52:46,040
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ
+
+548
+00:52:46,040 --> 00:52:46,320
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+549
+00:52:46,320 --> 00:52:46,420
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+550
+00:52:46,420 --> 00:52:48,560
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+551
+00:52:48,560 --> 00:52:48,600
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+552
+00:52:48,600 --> 00:52:49,020
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+553
+00:52:49,020 --> 00:52:49,700
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+554
+00:52:49,700 --> 00:52:50,180
+الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ .. الـ 
+
+555
+00:52:50,180 --> 00:52:50,420
+الـ .. الـ .. الـ .. الـ .. الـ
+
+556
+00:52:59,650 --> 00:53:05,310
+و differentiable on a و b و أيضا اللي هو Φ(a)
+
+557
+00:53:05,310 --> 00:53:10,410
+بيساوي Φ(b) بيساوي صفر إذا .. إذا by Rolle's
+
+558
+00:53:10,410 --> 00:53:16,810
+theorem there exists c element in a و b such that
+
+559
+00:53:16,810 --> 00:53:22,230
+Φ prime of c ايش بتساويك؟ بتساوي صفر طيب صفر
+
+560
+00:53:22,230 --> 00:53:26,490
+يعني بتساوي Φ prime of c إيش Φ prime of c؟
+
+561
+00:53:26,490 --> 00:53:31,900
+خلينا نفرض لهناهي عبارة عن F prime of C مكان الـ
+
+562
+00:53:31,900 --> 00:53:36,840
+X وهذه صفر وهذه اللي هي تفاضل الـ X وهذه صفر
+
+563
+00:53:36,840 --> 00:53:42,640
+طبعا مشتقة الـ X هي شمالها واحد بصير ناقص F of B
+
+564
+00:53:42,640 --> 00:53:49,300
+فاهمين؟ F of A على B minus A هذا بيساوى صفر وهذا
+
+565
+00:53:49,300 --> 00:53:56,140
+يعني أن F prime of C بيساوى F of B ناقص F of A على
+
+566
+00:53:56,140 --> 00:54:06,450
+B minus A وهو المطلوب، مظبوط؟ نيجي الآن لبعض
+
+567
+00:54:06,450 --> 00:54:14,290
+النتائج اللي احنا بنستعملها كتير اللي 
+
+568
+00:54:14,290 --> 00:54:20,630
+هي في الواقع برهانة عن طريق الـ Mean Value Theorem
+
+569
+00:54:20,630 --> 00:54:26,120
+النظرية اللي بتقول ليه اللي هي اللي باختصار بنقول
+
+570
+00:54:26,120 --> 00:54:33,900
+أنه لو كانت المشتقة الأولى لدالة بتساوي صفر إذن هي
+
+571
+00:54:33,900 --> 00:5
+
+601
+00:57:58,000 --> 00:58:02,800
+لكل X في الفترة A و B بما فيها Min الـ A لأنه
+
+602
+00:58:02,800 --> 00:58:12,520
+نفسها بتساوي إذا صارت عندي hence F is constant أي
+
+603
+00:58:12,520 --> 00:58:20,890
+سؤال طيب اللي بعدها corollary
+
+604
+00:58:20,890 --> 00:58:24,470
+الـ corollary برضه من الأشياء المشهورة اللي احنا
+
+605
+00:58:24,470 --> 00:58:29,190
+بنستخدمه اللي هو لو كان دالتين بتحققوا عن الشروط
+
+606
+00:58:29,190 --> 00:58:34,910
+طبعا لأنها تفاضلية زي بعض إذن هيكون اللي هو هن نفس
+
+607
+00:58:34,910 --> 00:58:42,990
+الشيء بس باختلف عن بعض بالكونستانت ثابت طيب الآن ال
+
+608
+00:58:42,990 --> 00:58:50,450
+corollary كما يلي الـ Corollary بتقول suppose that
+
+609
+00:58:50,450 --> 00:58:56,970
+F and G are continuous on I, A وB then that they
+
+610
+00:58:56,970 --> 00:59:01,190
+are differentiable on A وB يعني نفترض أن الدالتين
+
+611
+00:59:01,190 --> 00:59:05,230
+F وG continuous على الـ closed اللي هي bounded
+
+612
+00:59:05,230 --> 00:59:08,910
+interval A وB و differentiable على الـ open interval
+
+613
+00:59:08,910 --> 00:59:12,070
+A وB والـ derivative للأولى بتساوي الـ derivative
+
+614
+00:59:12,070 --> 00:59:15,690
+للثانية عند كل نقطة من نقاط الـ A و الـ B إذا 
+
+615
+00:59:15,690 --> 00:59:18,910
+there exists a constant C such that F بتساوي G
+
+616
+00:59:18,910 --> 00:59:25,350
+زائد C زائد H طبعا البرهان سهل و أكيد يمكن بعضكم 
+
+617
+00:59:25,350 --> 00:59:34,390
+برهنه في راسه let H بتساوي F ناقص G مظبوط؟ الآن
+
+618
+00:59:34,390 --> 00:59:43,530
+then F و G continuous إذن H is continuous on A و B
+
+619
+00:59:43,530 --> 00:59:51,210
+و أيضًا F و G differentiable على A و B and H is
+
+620
+00:59:51,210 --> 00:59:55,210
+differentiable on A و B moreover
+
+621
+00:59:57,700 --> 01:00:02,620
+عندي H برايم اللي هو بتساوي F برايم ناقص G برايم
+
+622
+01:00:02,620 --> 01:00:08,340
+طبعا عند ال X عند الـ X of X of X of X بتساوي صفر
+
+623
+01:00:08,340 --> 01:00:16,340
+لكل X موجودة في الـ A و الـ B then by the above
+
+624
+01:00:16,340 --> 01:00:20,300
+theorem اللي بنحكي عنها اللي هي theorem by theorem
+
+625
+01:00:20,300 --> 01:00:31,180
+625 H of X بتساوي C for every X element in A وB and
+
+626
+01:00:31,180 --> 01:00:35,300
+hence F
+
+627
+01:00:35,300 --> 01:00:44,040
+of X بتساوي G of X زائد some constant C
+
+628
+01:00:50,890 --> 01:00:54,570
+بكونوا احنا وصلنا لعند theorem 627 إن شاء الله
+
+629
+01:00:54,570 --> 01:01:01,010
+المرة القادمة بنكمل والسلام عليكم ورحمة الله
+
+630
+01:01:01,010 --> 01:01:01,670
+وبركاته
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw_postprocess.srt b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw_postprocess.srt
new file mode 100644
index 0000000000000000000000000000000000000000..c1e85864e3b21a086ad8029934fffc2ed442d229
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw_postprocess.srt
@@ -0,0 +1,2520 @@
+1
+00:00:05,620 --> 00:00:11,560
+بسم الله الرحمن الرحيم المحاضرة الثالثة في مساق
+
+2
+00:00:11,560 --> 00:00:19,660
+تحليل رياضي 2 أو تحليل حقيقي 2 كلية العلوم قسم
+
+3
+00:00:19,660 --> 00:00:28,520
+الرياضيات بالجامعة الإسلامية مراجعة سريعة لما تم
+
+4
+00:00:28,520 --> 00:00:34,380
+في المحاضرة السابقة والتي قبلهاحكينا عن شو معناته
+
+5
+00:00:34,380 --> 00:00:37,500
+اللي هو الـ derivative قلنا f prime of c عند نقطة
+
+6
+00:00:37,500 --> 00:00:41,760
+c لو كانت f function من i لعند ال r هو عبارة عن
+
+7
+00:00:41,760 --> 00:00:49,200
+limit f of x نقص f of c على x minus c لما x تروح
+
+8
+00:00:49,200 --> 00:00:53,460
+أو تقول ل c قلنا لو كانت ال limit exist هذه بنقول
+
+9
+00:00:53,460 --> 00:00:56,740
+إنه ال derivative موجودة بنرمزلها بالرمز f prime
+
+10
+00:00:56,740 --> 00:01:02,600
+of cأخدنا بعيدة اللي هو نظرية قولنا لو كانت f is
+
+11
+00:01:02,600 --> 00:01:11,860
+differentiable at c then it is continuous at c
+
+12
+00:01:11,860 --> 00:01:15,320
+قولنا the converse need not to be true in general
+
+13
+00:01:15,320 --> 00:01:20,100
+وأخدنا مثال على هيك قولنا خد ال f of x بسوء ال
+
+14
+00:01:20,100 --> 00:01:23,560
+absolute value لل x is continuous اللي هو
+
+15
+00:01:23,560 --> 00:01:26,880
+everywhere but it is not differentiable at zero
+
+16
+00:01:27,430 --> 00:01:30,590
+واخدنا مثال آخر قولنا f of x بيساوي ال summation
+
+17
+00:01:30,590 --> 00:01:36,090
+لل x ناقص ci i من عند 1 لعند n قولنا هذا ال
+
+18
+00:01:36,090 --> 00:01:39,210
+function is continuous everywhere but it is not
+
+19
+00:01:39,210 --> 00:01:43,250
+differentiable at the corner points which are c1,
+
+20
+00:01:43,350 --> 00:01:49,170
+c2 لعند ال cl و بعدين أخدنا مثال f of x بيساوي ال
+
+21
+00:01:49,170 --> 00:01:56,590
+summation 1 على 2 to the n cosinecos 3 أُس N X N
+
+22
+00:01:56,590 --> 00:02:01,690
+من واحد إلى ملا نهاية قلنا إن الـ function is
+
+23
+00:02:01,690 --> 00:02:07,750
+continuous everywhere but it is not differentiable
+
+24
+00:02:07,750 --> 00:02:16,850
+anywhere الـ function متصلة دائما لكن ليس لها أي
+
+25
+00:02:16,850 --> 00:02:22,760
+derivative عند أي نقطة من النقاط دومينابعد هيك
+
+26
+00:02:22,760 --> 00:02:28,500
+أخدنا اللي هو .. اللي هو نظرية اللي هي بنحكي فيها
+
+27
+00:02:28,500 --> 00:02:34,950
+عن خواص اللي هي التفاضل قلنا لو كانت ال Fوالـ G
+
+28
+00:02:34,950 --> 00:02:39,370
+are differentiable اللي هو كان من I لعند R
+
+29
+00:02:39,370 --> 00:02:44,730
+differentiable عند نقطة C بيكون مجموعة Alpha F حيث
+
+30
+00:02:44,730 --> 00:02:49,570
+Alpha اللي هو عبارة عن constant Alpha F' موجودة
+
+31
+00:02:49,570 --> 00:02:57,230
+وبتساوي Alpha في F' عند نقطة Cو بعدين قلنا أن f زي
+
+32
+00:02:57,230 --> 00:03:03,290
+g أو نقص prime of c برضه exist بسوء f prime of c
+
+33
+00:03:03,290 --> 00:03:09,390
+زي أو نقص g prime of c هذه من الخواصل هو ال
+
+34
+00:03:09,390 --> 00:03:15,040
+derivative و أيضا قلنااللي هو الخواص الأخرى أنه
+
+35
+00:03:15,040 --> 00:03:21,000
+الـ F في G الكل prime عنده نقطة C حاصل ضرر دلتين
+
+36
+00:03:21,000 --> 00:03:28,400
+بساوي اللي هو F primeof c في g of c زائد f of c في
+
+37
+00:03:28,400 --> 00:03:33,320
+g prime of c ومن ثم أخدنا أيضًا اللي هو
+
+38
+00:03:54,080 --> 00:03:59,480
+و من ثم أيضا حكينا عن اللي هو اخدنا اللي هو ال
+
+39
+00:03:59,480 --> 00:04:04,540
+chain rule theorem ال chain rule لو كانت F من I
+
+40
+00:04:04,540 --> 00:04:12,080
+لعند ال R و G من عند ال J لعند ال R حيث ال F of I
+
+41
+00:04:12,080 --> 00:04:15,780
+subset من ال J عشان نعرف اللي هو ال composition
+
+42
+00:04:15,780 --> 00:04:19,480
+قلنا لو كانت F is differentiable و G is
+
+43
+00:04:19,480 --> 00:04:23,890
+differentiable يعني F is differentiable عند Cis
+
+44
+00:04:23,890 --> 00:04:32,470
+differentiable at c and g is differentiable at f
+
+45
+00:04:32,470 --> 00:04:39,210
+of c then the composition اللي هو ال general يعني
+
+46
+00:04:39,210 --> 00:04:44,490
+التفاضل ال composition the composition تفاضله كما
+
+47
+00:04:44,490 --> 00:04:55,650
+يلي then fcomposite G prime of C بساوة F prime عند
+
+48
+00:04:55,650 --> 00:05:02,070
+G of C في G prime of C هذا اللي حكينا عنه المرة
+
+49
+00:05:02,070 --> 00:05:06,830
+الماضية وأيضا أخدنا طبعا أمثلة على هيك وأخدنا
+
+50
+00:05:06,830 --> 00:05:15,030
+نظرية أخرى اللي هي لو كانت عند F من عند I لعند R
+
+51
+00:05:15,030 --> 00:05:17,730
+قلنا is strictly
+
+52
+00:05:25,190 --> 00:05:31,450
+monotone and continuous
+
+53
+00:05:31,450 --> 00:05:41,770
+وكانت اللي هو G من عند J لعند I برضه strictly is
+
+54
+00:05:41,770 --> 00:05:55,620
+strictly monotone andcontinuous قلنا هنا at C وهنا
+
+55
+00:05:55,620 --> 00:06:06,340
+continuous at F of C حيث اللي هو الـ G بتساوي F of
+
+56
+00:06:06,340 --> 00:06:12,180
+mean of I عشان نقدر نعرف اللي هو تكون inverse is
+
+57
+00:06:12,180 --> 00:06:17,640
+strictly monotone and continuous where G
+
+58
+00:06:19,450 --> 00:06:28,790
+is the inverse of F، مظبوط؟ طبعاً، اللي بضماننا إن
+
+59
+00:06:28,790 --> 00:06:31,070
+ال inverse موجود إن هو strictly إيش؟ معناها
+
+60
+00:06:31,070 --> 00:06:33,490
+monotone، strictly monotone سواء increasing أو
+
+61
+00:06:33,490 --> 00:06:37,090
+decreasing، معناته one to one معناته إن هو اللي هي
+
+62
+00:06:37,090 --> 00:06:41,030
+ال inverse موجودة من اللي هو F of I اللي هي J،
+
+63
+00:06:41,030 --> 00:06:44,610
+لعند مين؟ لعند ال I، وهذه ال J في الواقع كل عنها
+
+64
+00:06:44,610 --> 00:06:52,800
+مين؟ هي ال F inverseالآن الشرط أنه if F is
+
+65
+00:06:52,800 --> 00:07:03,640
+differentiable at C at C then اللي هي G اللي هي
+
+66
+00:07:03,640 --> 00:07:07,280
+الـ F inverse دي في الواقع F inverse اللي سمناها
+
+67
+00:07:07,280 --> 00:07:14,020
+مين؟ F inverse لازم تكون differentiable at F of C
+
+68
+00:07:14,020 --> 00:07:15,660
+and
+
+69
+00:07:18,190 --> 00:07:23,390
+اللي سمناها F of C في حين وده بتتذكره D اللي هي
+
+70
+00:07:23,390 --> 00:07:35,170
+and F prime أو F inverse prime at this D بتساوي
+
+71
+00:07:35,170 --> 00:07:44,930
+واحد على F prime of C أو بمعنى آخر بتساوي واحد على
+
+72
+00:07:44,930 --> 00:07:53,880
+F primeF of C بيساوي D يعني C بيساوي G of D أو F
+
+73
+00:07:53,880 --> 00:07:59,600
+inverse of D اللي هي يعني composite G الكل أفمين
+
+74
+00:07:59,600 --> 00:08:13,970
+of C واضح هاه F of مين of Dقلنا بعدين اللي هو in
+
+75
+00:08:13,970 --> 00:08:18,530
+general لو كانت اللي هي هذه تحت الشروط هذه بس الآن
+
+76
+00:08:18,530 --> 00:08:24,430
+مش عند اللي هو نقطة لو F is differentiable على كل
+
+77
+00:08:24,430 --> 00:08:29,470
+ال I هتكون ال G is differentiable على ال J و
+
+78
+00:08:29,470 --> 00:08:36,110
+العلاقة بينهم زي ما حكينا المرة الماضية هتكون اللي
+
+79
+00:08:36,110 --> 00:08:43,620
+هوطب هالمرة F is differentiable on I then G is
+
+80
+00:08:43,620 --> 00:08:47,280
+differentiable on J اللي هي بساوة F of I طبعا تحت
+
+81
+00:08:47,280 --> 00:08:55,140
+هذا الشروط and هيكون F inverse prime of Y بساوة 1
+
+82
+00:08:55,140 --> 00:09:01,330
+على F prime of Xحيث العلاقة بين Y و X اللي هي Y
+
+83
+00:09:01,330 --> 00:09:08,050
+بتساوي F of X أو بتعبير آخر بتساوي 1 على F prime
+
+84
+00:09:08,050 --> 00:09:13,630
+of composite G اللي هي بتسميها F inverse كله of
+
+85
+00:09:13,630 --> 00:09:23,790
+mean of Y هذا اللي حكينا المرة الماضية نيجي
+
+86
+00:09:23,790 --> 00:09:26,510
+الآن نكمل
+
+87
+00:09:27,590 --> 00:09:33,710
+وصلنا عند المثال اللي هو 6 1 10 وقلنا لو كانت عند
+
+88
+00:09:33,710 --> 00:09:41,270
+g of y example وقفنا عند ال example التالي g of y
+
+89
+00:09:41,270 --> 00:09:48,710
+بتساوي y أسواحدة ل n for y element in j اللي هي
+
+90
+00:09:48,710 --> 00:09:54,970
+عبارة عن ال interval من zero ل infinity الآن بقول
+
+91
+00:09:54,970 --> 00:10:03,260
+show thatG prime of Y بتساوي 1 على N Y أس 1 على N
+
+92
+00:10:03,260 --> 00:10:12,580
+ناغس 1 حيث N is even solution نيجي الأن نحل تطبيق
+
+93
+00:10:12,580 --> 00:10:20,240
+على ال theorem الأخيرة اللي هو ناخد inverse لهذه
+
+94
+00:10:20,240 --> 00:10:25,750
+طبعا واضح أن ال inverse لو قلنا let F of Xبساوي X
+
+95
+00:10:25,750 --> 00:10:30,230
+أُس N هذا عبارة عن polynomial أو اللي هو دالة
+
+96
+00:10:30,230 --> 00:10:35,170
+أُسية أو جزء من polynomial عند X أُس N طبعاً لأي X
+
+97
+00:10:35,170 --> 00:10:39,350
+في الفترة 01 هذا is differentiable ومش هيك كمان
+
+98
+00:10:39,350 --> 00:10:42,910
+ومدامة الـ N even في الفترة هذه strictly
+
+99
+00:10:42,910 --> 00:10:46,580
+increasingده الـ Strict increase in continuous ده
+
+100
+00:10:46,580 --> 00:10:50,860
+كل .. كل أمورها متحققة اللي قبل بشوية في النظرية
+
+101
+00:10:50,860 --> 00:10:56,000
+فالآن و هذه عبارة عن ال inverse لمين؟ للـ G فبكون
+
+102
+00:10:56,000 --> 00:11:04,540
+الآن G prime of Y إيش هتساوي؟ بتساوي 1 على F prime
+
+103
+00:11:04,540 --> 00:11:11,410
+عند G of Yمظبوط ولا لا؟ Y ساوي الان ال F prime هنا
+
+104
+00:11:11,410 --> 00:11:15,930
+واضح اللي هي N X أس N نقص واحد يعني عبارة عن واحد
+
+105
+00:11:15,930 --> 00:11:22,070
+على بدل ال X إيش هحط؟ G of Y لأن هذا F prime of X
+
+106
+00:11:22,070 --> 00:11:26,290
+بساوي N X N نقص واحد طب F prime of G Y هي عبارة عن
+
+107
+00:11:26,290 --> 00:11:33,870
+G of Y أس N نقص واحد مضروبة في مين؟ في ال N Y ساوي
+
+108
+00:11:35,600 --> 00:11:43,600
+1 على N في G of Y إيش هي عبارة عن Y أس 1 على N
+
+109
+00:11:43,600 --> 00:11:49,640
+فبيصير اللي هو هذي بالمرة نرفحها بعد إذنكم G of Y
+
+110
+00:11:49,640 --> 00:11:59,620
+أس 1 ناقص N ويساوي إن عوض 1 على N في Yأس واحد على
+
+111
+00:11:59,620 --> 00:12:07,260
+N أس واحد ناقص N ويساوي واحد على N Y أس واحد على N
+
+112
+00:12:07,260 --> 00:12:12,660
+ناقص واحد وهو المطلوب هذه اللي هي تطبيق مباشرة على
+
+113
+00:12:12,660 --> 00:12:17,580
+النظرية السابقة الآن تطبيق آخر مثال آخر أيضا مثال
+
+114
+00:12:17,580 --> 00:12:20,380
+يعني شبيه بالأمثل اللي كنتوا تاخدوه في ال calculus
+
+115
+00:12:20,380 --> 00:12:29,610
+المثال دي اللي بناخده اللي هو Y بتساوي صين ال Xو
+
+116
+00:12:29,610 --> 00:12:37,590
+الـ x في ناقص πي على اتنين لعين باي على اتنين الآن
+
+117
+00:12:37,590 --> 00:12:43,470
+بقوللي اثبت انه sign inverse prove that اللي هو
+
+118
+00:12:43,470 --> 00:12:54,110
+show that sign inverse لل Y بتساوي واحد على جدر
+
+119
+00:12:54,110 --> 00:12:57,750
+التربيع لواحد ناقص مين Y تربيع
+
+120
+00:13:00,300 --> 00:13:04,160
+prime اه اه ال derivative لل sign inverse ايش
+
+121
+00:13:04,160 --> 00:13:06,840
+بتساوي واحدة الجدر تربيعه لو واحد نقص Y تربيعه
+
+122
+00:13:06,840 --> 00:13:10,940
+برضه تطبيق مباشر خلينا نشوف كيف نطبق على النظرية
+
+123
+00:13:10,940 --> 00:13:20,000
+ناخد let F of X اللي هو بيساوي sign ال X عشان عند
+
+124
+00:13:20,000 --> 00:13:24,500
+G of Y او عند F of X بيساوي sign ال X اللي بتسميها
+
+125
+00:13:24,500 --> 00:13:31,610
+Y يعني الآن بتصير عند G of Y ايش بتساويSin Inverse
+
+126
+00:13:31,610 --> 00:13:37,790
+لمن؟ لـ Y معايا يا شباب، أه؟ لأن حسب الـ F of X
+
+127
+00:13:37,790 --> 00:13:41,510
+بساعة و شوية إن الـ X الـ X في الفترة ناقص Y على 2
+
+128
+00:13:41,510 --> 00:13:47,690
+و Y على 2 الـ Y القيمة المقابلة إليها هتكون بين
+
+129
+00:13:47,690 --> 00:13:54,560
+ناقص 1 و 1لكن لاحظ الآن لما أشتق المشتقة لازم تكون
+
+130
+00:13:54,560 --> 00:13:58,380
+لا تساوي صفر في الفترة هذه ففي الفترة هذه اللي هي
+
+131
+00:13:58,380 --> 00:14:02,900
+المشتقة طبعا لهذه اللي هي عبارة عن cosine فهتكون
+
+132
+00:14:02,900 --> 00:14:06,900
+اللي هي ال X بس في ال open interval من ناقص y على
+
+133
+00:14:06,900 --> 00:14:11,540
+2 على y على 2 عساس ما نجسمش عالميا على الصفر إذا
+
+134
+00:14:11,540 --> 00:14:17,620
+الآن صار عندي G prime of Yنفسي اللي عملناه فوق،
+
+135
+00:14:17,620 --> 00:14:23,720
+إيش بتساوي؟ واحد على F prime of G of Y Y ساوي،
+
+136
+00:14:23,720 --> 00:14:28,880
+واحد F .. اللي هي مين الأن F prime؟ اللي هي cosine
+
+137
+00:14:28,880 --> 00:14:36,320
+اللي هو G of Y، مظبوط؟ وال cosine احنا إيش؟ عبارة
+
+138
+00:14:36,320 --> 00:14:39,160
+عن إيش؟ عبارة عن إيش تكون؟ عبارة عن الجدر
+
+139
+00:14:39,160 --> 00:14:44,080
+التربيعي؟ لا، طبعا ال cosine X إيش بتساوي؟ هذا
+
+140
+00:14:44,080 --> 00:14:52,100
+عبارة عن X، مظبوط؟هو الجدر التربيعي؟ لا، واحد ناقص
+
+141
+00:14:52,100 --> 00:14:57,980
+sign تربيع الـ X اللي هي G of Y، مظبوط؟ معايا يا
+
+142
+00:14:57,980 --> 00:15:03,200
+شبابY ساوي واحد على جدر التربيع إلى واحد ناقص Y
+
+143
+00:15:03,200 --> 00:15:08,940
+تربيع و Y في الفترة ناقص باي على اتنين و باي على
+
+144
+00:15:08,940 --> 00:15:13,800
+اتنين اللي هي الـ X و اللي موجودة أسفل X في ناقص
+
+145
+00:15:13,800 --> 00:15:16,560
+باي على اتنين و باي على اتنين و الـ Y في الفترة
+
+146
+00:15:16,560 --> 00:15:21,640
+ناقص واحد و واحد حتى نتجنب القسمة عالميا على اللي
+
+147
+00:15:21,640 --> 00:15:25,380
+هو الصفر هذا التطبيق على النظرية
+
+148
+00:15:27,410 --> 00:15:30,770
+ال homework عندنا تلاتة أو أربعة أو سبعة أو تسعة
+
+149
+00:15:30,770 --> 00:15:34,250
+أو تلاتاش من section 6-1 ان شاء الله المرة الجاية
+
+150
+00:15:34,250 --> 00:15:39,510
+أو اللي بعدها بنحاول نعمل discussion فيا ريت اللي
+
+151
+00:15:39,510 --> 00:15:44,410
+هو أنكم اتحلوا الأسئلة و اللي بتعرفوش اتحلوا او
+
+152
+00:15:44,410 --> 00:15:50,880
+بدكم نذاك بنحلو ان شاء اللهنيجي الآن لصفحة
+
+153
+00:15:50,880 --> 00:15:53,800
+اللي بعدي ده اللي هو ال section التاني ممكن هيكنا
+
+154
+00:15:53,800 --> 00:15:58,460
+هنا ستة واحد الآن هنبدأ في ال section التاني اللي
+
+155
+00:15:58,460 --> 00:16:02,380
+هو ال main value theorem وهو محاضرة اليوم بشكل
+
+156
+00:16:02,380 --> 00:16:06,740
+أساسي عالمي على the main value theorem ومتعلقات
+
+157
+00:16:06,740 --> 00:16:12,870
+هذه النظريةهنلاقي جدّاش مهمة هذه النظرية وجدّاش
+
+158
+00:16:12,870 --> 00:16:17,650
+فيه عندنا نتائج تتعلق إيه بها وإحنا يعني فعلا
+
+159
+00:16:17,650 --> 00:16:25,530
+بستعملها كثير وفعلا بتساعدنا اللي هو في اللي ده
+
+160
+00:16:25,530 --> 00:16:30,270
+يعني إذا رح أطلب لما حتى أخدنا ال Calculus كانت
+
+161
+00:16:30,270 --> 00:16:37,010
+اللي هي إلها تطبيقات واسعة وحنشوف الآن وين أهمية
+
+162
+00:16:37,680 --> 00:16:42,520
+هذه نظرية نظرية القيم المتوسطة من خلال الحديث أو
+
+163
+00:16:42,520 --> 00:16:47,080
+شرح حادي المادة الآن في الأول زي ما قلنا ال
+
+164
+00:16:47,080 --> 00:16:49,420
+section 6.2 the mean value theorem نيجي لل
+
+165
+00:16:49,420 --> 00:16:53,420
+definition الأول ال definition اللي هو مر علينا
+
+166
+00:16:53,420 --> 00:16:58,380
+اشي زيه أو مر علينا اللي هو تعريف ال local
+
+167
+00:16:58,380 --> 00:17:04,500
+extremum أو اللي هو relative extremumأو اللي هو
+
+168
+00:17:04,500 --> 00:17:07,460
+الـ local maximum أو الـ local minimum أو الـ
+
+169
+00:17:07,460 --> 00:17:10,840
+relative extremum أو الـ relative اللي هو الـ
+
+170
+00:17:10,840 --> 00:17:17,900
+minimum طبعاً بتعرفوا أن أنا بدي أعرف شو معناته دي
+
+171
+00:17:17,900 --> 00:17:24,480
+اللي هي النقطة عنده النقطة C في عند local .. local
+
+172
+00:17:24,480 --> 00:17:29,860
+maximum مثلا أو relative maximum بنقول في عند C
+
+173
+00:17:30,870 --> 00:17:35,370
+عندي function f من some interval I لعند R هاي الـ
+
+174
+00:17:35,370 --> 00:17:39,490
+function اللي عندي هنا من هنا لهنا بنقول C element
+
+175
+00:17:39,490 --> 00:17:48,430
+in I at C element in I we have a local أو relative
+
+176
+00:17:48,430 --> 00:18:00,820
+maximum when اللي هو there existsV Delta of C يعني
+
+177
+00:18:00,820 --> 00:18:05,800
+بنلاقي جوار حوالين الـ C بنصف قطر Delta C نقص
+
+178
+00:18:05,800 --> 00:18:12,060
+Delta of C زائد Delta في هذا الجوار such that
+
+179
+00:18:12,060 --> 00:18:20,320
+بتكون F of C هي أكبر قيمة للدالة F of C أكبر أو
+
+180
+00:18:20,320 --> 00:18:27,920
+سوى F of X لكل X في هذا الجوارإذا تحقق هذا الشرط
+
+181
+00:18:27,920 --> 00:18:35,840
+بنقول إن F has a relative maximum
+
+182
+00:18:35,840 --> 00:18:36,780
+at C
+
+183
+00:18:39,910 --> 00:18:45,290
+v delta neighborhood يعني جوار بنصف قطر دلتا بحيث
+
+184
+00:18:45,290 --> 00:18:50,590
+يتحقق أن ال f of c أكبر شو f of x لكل x في هذا
+
+185
+00:18:50,590 --> 00:19:01,850
+الجوار similarly بنقول at d element in I we have a
+
+186
+00:19:01,850 --> 00:19:11,870
+relative minimum whenthere exist أي جوار آخر بتكون
+
+187
+00:19:11,870 --> 00:19:17,390
+تسموه V Delta Prime أو V Delta of D such that يعني
+
+188
+00:19:17,390 --> 00:19:20,870
+بنلاقي .. بنقول هذه النقطة في عندها relative
+
+189
+00:19:20,870 --> 00:19:27,970
+minimum إذا كان لقينا جوار حواليهاهذه النقطة C D
+
+190
+00:19:27,970 --> 00:19:33,810
+وهذه D زائد Delta وهذه D نقص Delta في جوار حواليها
+
+191
+00:19:33,810 --> 00:19:38,670
+بتكون قيمة الدالة أصغر من كل القيم اللي حواليها
+
+192
+00:19:38,670 --> 00:19:45,210
+يعني اللي هو there exists V Delta of D such that F
+
+193
+00:19:45,990 --> 00:19:53,190
+of D أصغر أو ساوى F of X لكل X موجودة في V Delta
+
+194
+00:19:53,190 --> 00:20:01,370
+of D فهذه نسميها نقطة صغيرة محلية وهذه نسميها نقطة
+
+195
+00:20:01,370 --> 00:20:06,460
+عظمة محليةهذا معناته اللي هو الـ local maximum و
+
+196
+00:20:06,460 --> 00:20:10,340
+الـ local minimum أو زي ما هو مصطلح عليها ال
+
+197
+00:20:10,340 --> 00:20:14,580
+relative maximum و relative minimum لما بنقصد و
+
+198
+00:20:14,580 --> 00:20:20,880
+بنقول local extremum معناته يا local maximum أو
+
+199
+00:20:20,880 --> 00:20:24,520
+local minimum أو relative extremum يا relative
+
+200
+00:20:24,520 --> 00:20:30,120
+maximum يا relative minimumنجي الآن لنظريتنا
+
+201
+00:20:30,120 --> 00:20:33,720
+الأولى احنا كنا في ال calculus إذا بتتذكر و نسأل
+
+202
+00:20:33,720 --> 00:20:43,110
+سؤاللما بدنا نبحث في مسألة اللي هو الـ maximum و
+
+203
+00:20:43,110 --> 00:20:46,410
+الـ minimum سواء local maximum و local minimum أو
+
+204
+00:20:46,410 --> 00:20:50,230
+absolute maximum و absolute minimum كنا نسأل حالنا
+
+205
+00:20:50,230 --> 00:20:54,390
+سؤال طبعا مش معجول أنا بدي آجي كل نقاط ال domain
+
+206
+00:20:54,390 --> 00:20:57,990
+أفحص فيه عند maximum ولا minimum لأن نقاط ال
+
+207
+00:20:57,990 --> 00:21:01,170
+domain عدد لا نهائي uncountable number of points
+
+208
+00:21:01,170 --> 00:21:07,180
+في غالبهافإيش كنت حصرها؟ كنا نقول أين يمكن إيجاد
+
+209
+00:21:07,180 --> 00:21:12,480
+الـlocal maximum أو الـlocal minimum بعد ما نحدد
+
+210
+00:21:12,480 --> 00:21:17,100
+النقاط هذه بنقول يأتي الآن الفحص إنه هل هي local
+
+211
+00:21:17,100 --> 00:21:21,660
+maximum ولا local minimum إذن مسألة اللي هو إيجاد
+
+212
+00:21:21,660 --> 00:21:25,180
+ال maximum و ال minimum تبرع عبر خطوتين الخطوة
+
+213
+00:21:25,180 --> 00:21:31,380
+الأولى وين ممكن نلاقيهاوالخطوة الثانية اللي هي هل
+
+214
+00:21:31,380 --> 00:21:36,220
+هي فعلا local maximum ولا local minimum ولا لا
+
+215
+00:21:36,220 --> 00:21:40,500
+local maximum ولا local minimum اللي بده يشرعلنا
+
+216
+00:21:40,500 --> 00:21:45,130
+الإجابة الأولى هي النظرية هذهالنظرية هذه كنا كمان
+
+217
+00:21:45,130 --> 00:21:49,390
+نجاوب نقول الإجابة توجد الـlocal maximum أو
+
+218
+00:21:49,390 --> 00:21:53,030
+الـlocal minimum في أحد ثلاث مناطق إما F prime
+
+219
+00:21:53,030 --> 00:21:58,070
+بساوي سفر أو F prime does not exist كنقات داخلية
+
+220
+00:21:58,070 --> 00:22:03,410
+أو عند الـendpoints فكنا نحصرها في تلت أنهي مناطق
+
+221
+00:22:03,410 --> 00:22:07,530
+فبصير عدد النقاط اللي بدنا نفحص الـlocal max أو
+
+222
+00:22:07,530 --> 00:22:11,490
+الـlocal minimum عددها محدود وقليل بصير السؤال
+
+223
+00:22:11,490 --> 00:22:15,630
+مقدور عليهالان النظرية اللي بتتشرح علناها دي ايش
+
+224
+00:22:15,630 --> 00:22:23,390
+النظرية بتقول بتقول ما يليه اللي بتقوله ال theorem
+
+225
+00:22:23,390 --> 00:22:29,120
+اللي هيحتى اسمها اللي هي interior extremum theorem
+
+226
+00:22:29,120 --> 00:22:32,920
+interior extremum theorem يعني هنحكي عن ال
+
+227
+00:22:32,920 --> 00:22:38,020
+interior point بتقولي نفترض إن C عبارة عن نقطة
+
+228
+00:22:38,020 --> 00:22:43,400
+داخلية لمين ل some interval I C اللي هو إشمالة
+
+229
+00:22:43,400 --> 00:22:49,140
+نقطة داخلية ل I ال interval اللي عنديوعند الـ
+
+230
+00:22:49,140 --> 00:22:55,040
+function F من I لعند R لو فرضنا أن الـ function F
+
+231
+00:22:55,040 --> 00:23:02,000
+has a relative extremum at let's say element كده
+
+232
+00:23:02,000 --> 00:23:18,480
+if F has a relative extremum at C element in I0عند
+
+233
+00:23:18,480 --> 00:23:23,520
+C an interior point of the interval I at which F
+
+234
+00:23:23,520 --> 00:23:27,640
+has a relative extremum يعني مفترضين أنه في عندي
+
+235
+00:23:27,640 --> 00:23:33,140
+يا relative maximum يا relative minimum عند هذه
+
+236
+00:23:33,140 --> 00:23:43,660
+النقطة الداخلية ونفرض كمان أن F prime of C exist
+
+237
+00:23:43,660 --> 00:23:51,610
+تحت كل هذه الشروطلازم تطلع عند F prime of C ايش
+
+238
+00:23:51,610 --> 00:23:56,050
+بتساوي؟ بتساوي سفر يعني بمعنى اخر لو كانت النقطة
+
+239
+00:23:56,050 --> 00:24:00,690
+داخلية و النقطة هذه الـ derivative عندها موجودة
+
+240
+00:24:01,820 --> 00:24:05,280
+وكانت عندها relative maximum أو relative minimum
+
+241
+00:24:05,280 --> 00:24:09,280
+قطعاً هتكون الـ F prime إليها إيش بتساوي .. بتساوي
+
+242
+00:24:09,280 --> 00:24:12,960
+سفر يعني عند .. عند النقاط الداخلية اللي الـ F
+
+243
+00:24:12,960 --> 00:24:17,280
+prime إليها موجودة و بتشاويش سفر انساها و ماتبحثش
+
+244
+00:24:17,280 --> 00:24:20,780
+فيها قطعا فيش فيها لا relative maximum ولا
+
+245
+00:24:20,780 --> 00:24:25,750
+relative إيش .. minimumالآن هذه النظرية وكأنه
+
+246
+00:24:25,750 --> 00:24:31,810
+بتحكيلنا في داخلها أن النقاط اللي بتكون داخلية
+
+247
+00:24:31,810 --> 00:24:36,590
+واحتمال يكون عندها relative maximum و relative
+
+248
+00:24:36,590 --> 00:24:42,410
+minimum نوعين يا f prime مش موجودةأو الـ F' أيش
+
+249
+00:24:42,410 --> 00:24:48,190
+بتساوي؟ بتساوي 0 لأنه إذا كانت F' موجودة وقفقلها
+
+250
+00:24:48,190 --> 00:24:51,910
+relative maximum أو relative minimum وكانت النقطة
+
+251
+00:24:51,910 --> 00:24:56,710
+داخلية لازم الـ F' أيش ما لها تساوي 0 ولاحظ هنا إن
+
+252
+00:24:56,710 --> 00:25:02,190
+الحديث عن مين؟ عن النقاط الداخلية يعني من فحوة
+
+253
+00:25:02,190 --> 00:25:06,110
+الحديث هنا بنجاوب على سؤالنا، هذه اللي بتجاوبنا
+
+254
+00:25:06,110 --> 00:25:09,860
+على سؤالنا بتقولاللي هو الـ relative maximum أو
+
+255
+00:25:09,860 --> 00:25:13,880
+الـ relative minimum هتكون في أحد ثلاث مناطق يا F
+
+256
+00:25:13,880 --> 00:25:18,060
+prime does not exist يا F prime بالساوية سفر يا
+
+257
+00:25:18,060 --> 00:25:27,960
+end point واضح؟ أي سؤال؟ واضح أه؟ طيب، دي يجي
+
+258
+00:25:27,960 --> 00:25:36,550
+تبرهن النظريةالآن هو البروحان اللي مكتوب هنا we
+
+259
+00:25:36,550 --> 00:25:41,170
+prove for the case that F has relative maximum at
+
+260
+00:25:41,170 --> 00:25:45,150
+C هيه مكتوب أنا مابديش أبرهن ال maximum بتبرهنكم
+
+261
+00:25:45,150 --> 00:25:52,290
+مين ال minimum إذا الآن بدنا نفترض نفترض ال proof
+
+262
+00:25:52,290 --> 00:25:57,870
+خليني أكتب هنا ال proof بدي
+
+263
+00:25:57,870 --> 00:26:04,820
+أفترضC نقطة داخلية ونفترض إن F أشبهلها have a
+
+264
+00:26:04,820 --> 00:26:08,560
+relative minimum relative maximum محلولة لأن
+
+265
+00:26:08,560 --> 00:26:12,640
+suppose that
+
+266
+00:26:12,640 --> 00:26:21,020
+F has a relative maximum
+
+267
+00:26:21,020 --> 00:26:28,120
+at C element in I0 ماشي؟
+
+268
+00:26:37,000 --> 00:26:41,080
+نعم نفترض إنها minimum لأن ال case تبعت ال maximum
+
+269
+00:26:41,080 --> 00:26:47,860
+محلولةواضح؟ الان لو كان F as relative extremum
+
+270
+00:26:47,860 --> 00:26:52,160
+بدنا نصل الان extreme يابتكون maximum يابتكون
+
+271
+00:26:52,160 --> 00:26:56,440
+minimum case ال maximum هي مكتوبة بتجروها لحالكوا
+
+272
+00:26:56,440 --> 00:27:01,980
+نفس الفكرة ده أكتف بت mean F as a relative H
+
+273
+00:27:01,980 --> 00:27:08,360
+minimumالآن بدي أفترض لك أن الـ F' أي شمالها لا
+
+274
+00:27:08,360 --> 00:27:11,700
+تساوي صفر و أسألك لـ contradiction إذا الآن بدي
+
+275
+00:27:11,700 --> 00:27:18,780
+أفترض لك و بدي أقول لك suppose that F prime of C
+
+276
+00:27:18,780 --> 00:27:25,620
+لا تساوي أيش؟ صفر، ماشي الحال، إذا عندي two cases
+
+277
+00:27:25,620 --> 00:27:34,360
+ال case الأول أن F prime of C أكبر من أيش؟من صفر
+
+278
+00:27:34,360 --> 00:27:43,580
+ماشي الحال إذا الآن عندي limit f of x ناقص f of c
+
+279
+00:27:43,580 --> 00:27:50,340
+على x minus c as x بتروح ل c أشماله اللي هو هذا f
+
+280
+00:27:50,340 --> 00:27:57,040
+prime of c حسب المعطى إيش هيكون أكبر من صفر ماشي
+
+281
+00:27:57,040 --> 00:28:03,480
+الحال طيب الآن مدام ال f prime of c أكبر من صفرإذا
+
+282
+00:28:03,480 --> 00:28:07,700
+حسب نظرية قلنا قبل هيك إنه لو كانت ال limit لل
+
+283
+00:28:07,700 --> 00:28:12,800
+function أكبر من 0 هنلاقي جوار حوالين النقطة اللي
+
+284
+00:28:12,800 --> 00:28:16,660
+بدنا نروحلها يكون اللي هو قيمة اللي جوا ال limit
+
+285
+00:28:16,660 --> 00:28:21,780
+أكبر من 100 من 0 لأن كما قالي مدام ال limit هذا
+
+286
+00:28:21,780 --> 00:28:29,260
+أكبر من 0 then there exists Deltaأكبر من سفر أو V
+
+287
+00:28:29,260 --> 00:28:35,380
+Delta أكبر من سفر Such that اللي هو F of X نقص F
+
+288
+00:28:35,380 --> 00:28:43,400
+of C على X minus C أكبر من سفر لكل X وين موجودة في
+
+289
+00:28:43,400 --> 00:28:48,240
+الـ V Delta of C صح؟
+
+290
+00:28:48,240 --> 00:28:53,820
+هاي عندي احنا مفترضين إن هي أشملها relative
+
+291
+00:28:53,820 --> 00:28:57,940
+minimum خلّيني أشتغل على هذه المنطقةأساسي أن
+
+292
+00:28:57,940 --> 00:29:02,260
+تشوفوها على الرسم
+
+293
+00:29:02,260 --> 00:29:13,880
+يمكن تكون أوضح عندي هي النقطة C وهذا C ناقص Delta
+
+294
+00:29:13,880 --> 00:29:21,060
+وهذا C زائد Delta، راضح؟ الـ F of X ناقص F of Z
+
+295
+00:29:21,060 --> 00:29:24,000
+على X minus C أكبر من مين من سفر لكل X و الموجودة
+
+296
+00:29:24,000 --> 00:29:32,040
+في الـ V Delta of Cأه، الآن .. الآن عندي هو .. بدي
+
+297
+00:29:32,040 --> 00:29:37,020
+.. عندي أنا مفترض أنه اللي هو F هذا relative A
+
+298
+00:29:37,020 --> 00:29:45,580
+شماله minimum إذا في جوار .. في جوار بتكون F of C
+
+299
+00:29:45,580 --> 00:29:52,700
+أصغر أو ساوة F of X لكل X في هذا الجوار اللي هو V
+
+300
+00:29:52,700 --> 00:29:59,440
+Delta Prime of إيش؟ Of C، ماشي الحالالآن الـ Delta
+
+301
+00:29:59,440 --> 00:30:03,820
+هذه فيه Delta أكبر من 0 باخدها الـ Delta أصغر من
+
+302
+00:30:03,820 --> 00:30:08,840
+هذه بحكم فيها، فده بتنفع للـ Delta أكبر من هذه إن
+
+303
+00:30:08,840 --> 00:30:11,880
+هذا الكون المقدر لكل X في الـ V Delta يكون أكبر من
+
+304
+00:30:11,880 --> 00:30:17,100
+0، إذا بديجه، برضه بنفع، إذا بختار الـ Delta هذه
+
+305
+00:30:17,100 --> 00:30:20,420
+بناء على اختيار لهذه، فبتكون الـ Delta هذه أنا
+
+306
+00:30:20,420 --> 00:30:24,770
+بختارها أصغر أو سوء من الـ Delta primeماشي الحال
+
+307
+00:30:24,770 --> 00:30:28,770
+طيب إذا صار عندي F of X نقص F of C على X minus C
+
+308
+00:30:28,770 --> 00:30:34,970
+أكبر من 100 من 0 لكل X limited V Delta O of C
+
+309
+00:30:34,970 --> 00:30:38,090
+أذكركم مش بده أنا أنا بده أصل يكون contradiction
+
+310
+00:30:38,090 --> 00:30:45,050
+يعني بده أصل لنقاط في المكان هذا تتحققش هذه معايا؟
+
+311
+00:30:46,850 --> 00:30:48,790
+بدي أسأل ال contradiction كيف أسأل ال
+
+312
+00:30:48,790 --> 00:30:53,210
+contradiction؟ بدي أسأل ألاقي نقاط في المنطقة هذه
+
+313
+00:30:53,210 --> 00:30:59,750
+و هذه لا تتحقق شوف الآن عندي هذا المقدار إيش ماله؟
+
+314
+00:30:59,750 --> 00:31:07,730
+موجب، مظبوط ولا لأ؟ ماشي، الآن اللي بدي أخلي اللي
+
+315
+00:31:07,730 --> 00:31:13,830
+هي F of X minus F of C تظل أكبر من 0لأن لو خلّيتها
+
+316
+00:31:13,830 --> 00:31:19,210
+أكبر من صفر بيصير F أصغر من صفر بيصير F of X لما
+
+317
+00:31:19,210 --> 00:31:22,030
+صير أصغر من صفر بيصير أصغر من F of C بتبطّل أش
+
+318
+00:31:22,030 --> 00:31:28,650
+مالها عبارة عن minimum بتخلي
+
+319
+00:31:28,650 --> 00:31:35,130
+هذا المقدار أصغر من أش من صفر بدون هذا أه؟ بيصير F
+
+320
+00:31:35,130 --> 00:31:41,120
+of X أصغر من F of C بيصير F of Z مش minimumواضح؟
+
+321
+00:31:41,120 --> 00:31:45,560
+طيب إيش اللي بده يخليلي هذا أصغر من 0 هذا المقدار
+
+322
+00:31:45,560 --> 00:31:52,560
+أصغر من 0 أني أضرب أعمل choose x بحيث أنه يكون هذا
+
+323
+00:31:52,560 --> 00:31:57,380
+سالب وإن هيكون سالب x minus c بختار اللي هي x
+
+324
+00:31:57,380 --> 00:32:01,680
+minus c بختار x وإن ما أقول بقول choose x element
+
+325
+00:32:01,680 --> 00:32:07,240
+أن c نقص دلتة وc فاشي الحال then
+
+326
+00:32:09,590 --> 00:32:15,310
+اللي هو لما اخترت الـ X هنا هيكون الـ X minus C
+
+327
+00:32:15,310 --> 00:32:20,570
+أصغر من 0 مدام صار الـ X minus C أصغر من 0 وهذا
+
+328
+00:32:20,570 --> 00:32:26,630
+أكبر من 0 معطى من التنتين مع بعض بيصير عند X minus
+
+329
+00:32:26,630 --> 00:32:37,630
+C في F of X minus F of C على X minus Cأصغر من 0،
+
+330
+00:32:37,630 --> 00:32:43,370
+اللي هو هذا في الواقع مين؟ هو عبارة عن F of X ناقص
+
+331
+00:32:43,370 --> 00:32:47,210
+F of C صار الـ F of X minus F of C، لأن هذا بتروح
+
+332
+00:32:47,210 --> 00:32:50,690
+محدد، لأنه مش سفر، بصير F of X ناقص F of C أصغر من
+
+333
+00:32:50,690 --> 00:32:57,150
+0، معناته صار F of X أصغر من F of C، هذا لمين؟
+
+334
+00:32:57,150 --> 00:33:02,050
+للنقطة X اللي في هذا الجوار، يعني اللي لجأت نقطة
+
+335
+00:33:02,050 --> 00:33:04,790
+في هذا الجوار، لا تحقق هذا
+
+336
+00:33:07,600 --> 00:33:15,160
+هذا contradiction طيب اللي ما وضحتلوش هذه خلّينا
+
+337
+00:33:15,160 --> 00:33:19,780
+نوضحها في الحالة التانية هذا في حالة مين أه في
+
+338
+00:33:19,780 --> 00:33:24,440
+prime و في c أشمالها أكبر من سفر الآن بدأ أقول
+
+339
+00:33:24,440 --> 00:33:31,000
+suppose that الحالة التانية كاس اتنينيعني اللي
+
+340
+00:33:31,000 --> 00:33:37,320
+بقوله في حالة فرضت F prime لا تساوي سفر هأصل دائما
+
+341
+00:33:37,320 --> 00:33:41,200
+لتناقض F prime أكبر من سفر حصل لتناقض وصلت لتناقض
+
+342
+00:33:41,200 --> 00:33:47,060
+والآن حالة F prime of C أصغر من سفر برضه حصل لمين؟
+
+343
+00:33:47,460 --> 00:33:51,860
+لطلاقط ماشي الحال الان ايش معناه f prime of c أكبر
+
+344
+00:33:51,860 --> 00:33:58,040
+من 0 بنفس الأسلوب limit f of x ناقص f of c على x
+
+345
+00:33:58,040 --> 00:34:03,860
+minus c as x بتروح لل c برضه ايش ماله أصغر من 0
+
+346
+00:34:03,860 --> 00:34:09,820
+إذا في جوار there exist v delta of x و باخد ال
+
+347
+00:34:09,820 --> 00:34:13,500
+delta اللي هي delta prime بقدر أخدها و أصغر يساوي
+
+348
+00:34:13,500 --> 00:34:20,950
+ال delta prime by such thatاللي هو f of x ناقص f
+
+349
+00:34:20,950 --> 00:34:27,310
+of c على x minus c أصغر من مين؟ من 0 الآن اللي
+
+350
+00:34:27,310 --> 00:34:31,430
+بيوصلني لتناقض هتلاقيها مش choosing x هنا و انا
+
+351
+00:34:31,430 --> 00:34:35,450
+هاختارها في الجهة التانية في الجهة بتشوف كيف لأن
+
+352
+00:34:35,450 --> 00:34:43,570
+بيقول choose x element in c و c زائد delta then
+
+353
+00:34:47,030 --> 00:34:54,130
+هيكون عندى الآن الـ X minus C إيش ماله أكبر من 0
+
+354
+00:34:54,130 --> 00:35:00,610
+واضح أه؟لأن هذا أكبر من صفر وهذا أكبر أصغر من صفر
+
+355
+00:35:00,610 --> 00:35:07,110
+ثم اللي هي F of X ناقص F of C اللي هي بتساوي X
+
+356
+00:35:07,110 --> 00:35:10,670
+minus C اللي هي أكبر من صفر في المقدار اللي هو F
+
+357
+00:35:10,670 --> 00:35:15,390
+of X ناقص F of C على X minus C اللي هو أصغر من صفر
+
+358
+00:35:15,390 --> 00:35:19,430
+حصل أصغر من صفر وأكبر من صفر حيكون أصغر من صفر
+
+359
+00:35:19,430 --> 00:35:26,680
+ومنه henceF of X أصغر من F of C لنقطة في هذا
+
+360
+00:35:26,680 --> 00:35:30,720
+الجوار اللي هو باخدها very close to C يعني معناته
+
+361
+00:35:30,720 --> 00:35:35,040
+هتكون في داخل هذا الجوار لجهة نقطة بحيث أنه F of C
+
+362
+00:35:35,040 --> 00:35:41,540
+أكبر من F of X which is a contradiction thereforeF
+
+363
+00:35:41,540 --> 00:35:47,240
+prime of C must equal zero إذن بيكون أثبتنا إن لو
+
+364
+00:35:47,240 --> 00:35:52,000
+كان في عندي local minimum عند ال interior point C
+
+365
+00:35:52,000 --> 00:35:59,200
+إذن اللي هي بتكون ال F prime of C بالساعة سفر طبعا
+
+366
+00:35:59,200 --> 00:36:04,660
+لاحظوا أنا كل الجوارات اللي أخدت هنا ال V Delta و
+
+367
+00:36:04,660 --> 00:36:09,010
+V Delta prime كل في داخل الintervalعشان هيك هو
+
+368
+00:36:09,010 --> 00:36:12,850
+طبعا هو كاتب تقاطع ال .. بكتب تقاطع ال subset من
+
+369
+00:36:12,850 --> 00:36:17,450
+ال interval لأن كل اللي بشتغله وين بشتغله؟ جوات ال
+
+370
+00:36:17,450 --> 00:36:25,510
+interval أي سؤال؟ اللي هو؟
+
+371
+00:36:25,510 --> 00:36:33,890
+تكمّل
+
+372
+00:36:33,890 --> 00:36:38,810
+يعني f of c أقل من f of xدقيقة بس هنكتب اللي
+
+373
+00:36:38,810 --> 00:36:45,870
+بتقوله عشان اللي بشوف تصويرنا و بسمع كل معاناه
+
+374
+00:36:45,870 --> 00:36:49,050
+أيوة
+
+375
+00:36:49,050 --> 00:36:54,330
+يا محمد قول أصغر
+
+376
+00:36:54,330 --> 00:37:02,390
+يسوي F of X لكل X في V Delta of Z F
+
+377
+00:37:02,390 --> 00:37:12,690
+of C معقص F of Xأصغر من 0 مش
+
+378
+00:37:12,690 --> 00:37:22,070
+مشكلة limit
+
+379
+00:37:22,070 --> 00:37:31,690
+f of x نقص f of c على x minus c لما x تروح ل c من
+
+380
+00:37:31,690 --> 00:37:36,360
+اليسارمعناته الـ X أكبر من الـ C إذاً هذه صارت
+
+381
+00:37:36,360 --> 00:37:41,080
+موجبة هذه موجبة و هذه موجبة ده بتكون أشماله أكبر
+
+382
+00:37:41,080 --> 00:37:48,060
+من سفر أكبر أو يساوي سفر similarly limit F of X
+
+383
+00:37:48,060 --> 00:37:54,380
+ناقص F of C على X minus C لما X تروح ل C من اليسار
+
+384
+00:37:54,380 --> 00:38:00,110
+اللي هو هذه اللي هي أصغر من سفروهذه تصبح أصغر أو
+
+385
+00:38:00,110 --> 00:38:06,970
+ساوى 0 ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+386
+00:38:06,970 --> 00:38:07,570
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+387
+00:38:07,570 --> 00:38:08,030
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+388
+00:38:08,030 --> 00:38:08,130
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+389
+00:38:08,130 --> 00:38:08,690
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+390
+00:38:08,690 --> 00:38:10,510
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+391
+00:38:10,510 --> 00:38:11,170
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+392
+00:38:11,170 --> 00:38:19,550
+.. ال .. ال .. ال ..F of X ناقص F of C على X minus
+
+393
+00:38:19,550 --> 00:38:24,130
+C as X بتروح للـ C exist إذا ال limit من اليمين
+
+394
+00:38:24,130 --> 00:38:26,010
+تساوي ال limit من اليسار مش ال continuity ال
+
+395
+00:38:26,010 --> 00:38:28,850
+differentiability إذا ال limit من اليمين تساوي ال
+
+396
+00:38:28,850 --> 00:38:32,610
+limit من اليسار هذا لا يمكن أن يحدث في هذه الحالة
+
+397
+00:38:32,610 --> 00:38:38,490
+إلا غير لما تكون كلام صحيح بدك تستخدم الصحيح بنفع
+
+398
+00:38:38,490 --> 00:38:42,210
+بنفع بنفع
+
+399
+00:38:45,000 --> 00:38:51,520
+بدونها .. بدونها .. بدونها بنفع اه
+
+400
+00:38:51,520 --> 00:38:55,380
+بنفع اه .. اه ما هو by contradiction هو انت لو ..
+
+401
+00:38:55,380 --> 00:38:58,200
+لو يعني لاحظت في ال .. في ال .. طبعا احنا ما
+
+402
+00:38:58,200 --> 00:39:00,220
+ساخدمناش limit من اليمين و limit من اليسار،
+
+403
+00:39:00,220 --> 00:39:08,700
+ساخدمنا ال limit مباشرة، بصير، بصير، فهم صحيحهو
+
+404
+00:39:08,700 --> 00:39:13,640
+الحل اللي انت بتعرفه و يعني خليني أقوله مرتاح له
+
+405
+00:39:13,640 --> 00:39:17,500
+هو الأسهل اه يعني انت ممكن تشوف هذا أسهل و هذا
+
+406
+00:39:17,500 --> 00:39:24,260
+أسهل كلام صحيح يلا اللي بعده الان corollary منيحي
+
+407
+00:39:24,260 --> 00:39:26,620
+محمد اللي بعده
+
+408
+00:39:33,180 --> 00:39:36,640
+corollary اللي ممكن حكيتها الـ corollary هذه أنا
+
+409
+00:39:36,640 --> 00:39:40,380
+اللي هي من المعلومات اللي قبل بشوية لو كانت ال
+
+410
+00:39:40,380 --> 00:39:44,420
+function f اللي هو has اللي هي f من I من I اللي
+
+411
+00:39:44,420 --> 00:39:47,980
+عندها continuous و non-interval and suppose that f
+
+412
+00:39:47,980 --> 00:39:51,420
+has a relative extremum of an interior point c of
+
+413
+00:39:51,420 --> 00:39:58,540
+I then either the derivative of f exists or f بسوة
+
+414
+00:39:58,540 --> 00:40:02,780
+سفرعادت صيغة للي قبل بشوية أنه لو كانت الـ
+
+415
+00:40:02,780 --> 00:40:08,720
+function عندي من I لعند R continuous وكانت اللي هي
+
+416
+00:40:08,720 --> 00:40:13,540
+الـ C interior point عندها relative maximum أو
+
+417
+00:40:13,540 --> 00:40:17,920
+relative minimum هذا لازم يعطينا أن الـ F prime يا
+
+418
+00:40:17,920 --> 00:40:21,860
+does not exist يا إما zero طبعا و هذي مباشرة لو
+
+419
+00:40:21,860 --> 00:40:28,150
+جلنا نفترض أن F prime existبنحصل عالميين على الـ 0
+
+420
+00:40:28,150 --> 00:40:32,190
+من النظرية اللي قبله شوية وهو المطلوب واضح نجي
+
+421
+00:40:32,190 --> 00:40:35,230
+للنظرية البعيدة اللي هي إيش اسمها النظرية المشهورة
+
+422
+00:40:35,230 --> 00:40:41,090
+اللي هي اسمها Rawls theorem Rawls theorem يا جماعة
+
+423
+00:40:41,090 --> 00:40:44,370
+طبعا مرت علينا في ال calculus ويمكن كمان صارت في
+
+424
+00:40:44,370 --> 00:40:49,690
+الثانوية يدرسوها النظرية طبعا معناها الهندسي كمان
+
+425
+00:40:49,690 --> 00:40:53,910
+بمر معاكم أنه لو كانت عند ال function is
+
+426
+00:40:53,910 --> 00:41:05,950
+continuousعلى F من A و B لعند R continuous on A و
+
+427
+00:41:05,950 --> 00:41:15,830
+B و differentiable on A و B open وعندي F of A .. F
+
+428
+00:41:15,830 --> 00:41:22,960
+of A هيها و F of B هيهابصير عندي معطيني كمان f of
+
+429
+00:41:22,960 --> 00:41:26,900
+a بساوي f of b ماعطينيهاش بساوي، بساوي سفر، لو
+
+430
+00:41:26,900 --> 00:41:30,260
+كانت بتساوي ثابت برضه بتظبط، بتكون نفسها، تعميم
+
+431
+00:41:30,260 --> 00:41:35,500
+بسيط لها ال f of a بساوي f of b بساوي سفر، بقوللي
+
+432
+00:41:35,500 --> 00:41:46,320
+إذا هالاجي نقطة then هذا معطى if this holds then
+
+433
+00:41:47,250 --> 00:41:52,110
+there exists C وحدة، تنتين، تلات، او .. المهم على
+
+434
+00:41:52,110 --> 00:41:55,790
+الأقل في واحدة there exists C المتل A و B such
+
+435
+00:41:55,790 --> 00:42:01,290
+that F prime of C بيساوز فماذا يعني؟ يعني بالله
+
+436
+00:42:01,290 --> 00:42:06,230
+بمعنى اخر نقطة C او اكثر في بين ال A و ال B بحيث
+
+437
+00:42:06,230 --> 00:42:11,390
+انه ميل المماس عندها او المماس اللي عندها بوازمين
+
+438
+00:42:11,390 --> 00:42:15,570
+محور السناد طبعا هنا في واحد هاي اتنين هاي تلاتة
+
+439
+00:42:16,090 --> 00:42:21,070
+طبعا حسب .. حسب الدالة اللي هو أكيد بتلاقي إذا
+
+440
+00:42:21,070 --> 00:42:26,130
+تحققت الشروط الأقل في نقطة عندها اللي هو المماس
+
+441
+00:42:26,130 --> 00:42:35,870
+بوازي محور السينات أي سؤال؟ ال proof ال proof برضه
+
+442
+00:42:35,870 --> 00:42:39,570
+احنا كنا أحيانا نبرهنه في ال calculus خلينا
+
+443
+00:42:39,570 --> 00:42:46,150
+نبرهنوها، أكيد هندرهنوها عنديلو كانت الـ F هي
+
+444
+00:42:46,150 --> 00:42:51,550
+الدالة الصفرية يعني F بتساوي 0 it is a trivial أنه
+
+445
+00:42:51,550 --> 00:42:54,390
+مش حلها جنوبها كل نقاطها الـ F prime of C هي إيش
+
+446
+00:42:54,390 --> 00:42:59,770
+بتتساوي؟ بتساوي 0 إذا ال case F بتساوي 0 is a
+
+447
+00:42:59,770 --> 00:43:07,590
+trivial case الآن case 2 case 1 الآن F ليست الدالة
+
+448
+00:43:07,590 --> 00:43:14,870
+الصفرية في الاحتمالين أنه بدأ عملها أن Fassumes
+
+449
+00:43:14,870 --> 00:43:25,310
+positive values الاحتمال التانوي F assumes
+
+450
+00:43:25,310 --> 00:43:33,730
+negative values أنا بدعي ان هذولا بيجيبن كل الحلال
+
+451
+00:43:34,430 --> 00:43:38,270
+لو كانت F assumes positive value ممكن تاخد
+
+452
+00:43:38,270 --> 00:43:40,830
+positive وتاخد negative وممكن تاخد positive لحالها
+
+453
+00:43:40,830 --> 00:43:46,250
+بظبط ولا لأ ال F assume negative values إذا هتاخد
+
+454
+00:43:46,250 --> 00:43:49,530
+حالة F assumes negative values لحالها أو negative
+
+455
+00:43:49,530 --> 00:43:56,780
+values مع مين؟ مع positive values واضح؟طيب، الآن
+
+456
+00:43:56,780 --> 00:44:00,520
+بدي أثبتلك في الحالة الأولى على طول هتطلع F prime
+
+457
+00:44:00,520 --> 00:44:06,700
+of C == 0 وفي الحالة التانية similarly، معايا؟الآن
+
+458
+00:44:06,700 --> 00:44:10,780
+في عندنا نظرية في تحليل واحد أو في الكالكولاست حتى
+
+459
+00:44:10,780 --> 00:44:16,640
+كنا نقول if F is continuous on a closed bounded
+
+460
+00:44:16,640 --> 00:44:22,640
+interval then it must attain its maximum and its
+
+461
+00:44:22,640 --> 00:44:28,140
+minimum in this absolute كمان in its interval،
+
+462
+00:44:28,140 --> 00:44:35,130
+مظبوط؟ يعني بمعنى آخرعندي بما أن F continuous على
+
+463
+00:44:35,130 --> 00:44:39,830
+closed bounded interval then its absolute maximum
+
+464
+00:44:39,830 --> 00:44:43,650
+and absolute minimum is attained here يعني بمعنى
+
+465
+00:44:43,650 --> 00:44:50,470
+أخر there exists C element in A وB such that F of
+
+466
+00:44:50,470 --> 00:44:55,900
+C هي ال maximumإيه في ال maximum يعني؟ يعني ال
+
+467
+00:44:55,900 --> 00:45:01,040
+supremum أو ال maximum لكل F of X حيث X وين موجودة
+
+468
+00:45:01,040 --> 00:45:08,400
+في ال A و ال B، مظبوط؟ إذا صارت عندي في C في ال A
+
+469
+00:45:08,400 --> 00:45:12,180
+و ال B بحيث أن F of C بالساوى ال supremum هذا ال
+
+470
+00:45:12,180 --> 00:45:17,780
+supremum يا جماعة أكيد A شماله أكبر strictly من A
+
+471
+00:45:18,660 --> 00:45:23,920
+من صفر ليش؟ لأن احنا تحت أي عنوان بنشتغل F assumes
+
+472
+00:45:23,920 --> 00:45:27,640
+positive values إذا في على أقل قيمة واحدة positive
+
+473
+00:45:27,640 --> 00:45:33,440
+إذا F of C هيكون strictly أكبر من مين؟ من صفر طيب
+
+474
+00:45:33,440 --> 00:45:41,920
+بقول برضه F of C صارت عبارة عن maximum point الـ C
+
+475
+00:45:41,920 --> 00:45:50,520
+مظبوط؟ هذا واحد اتنين الـ Cقطعاً لا هي الـ A ولا
+
+476
+00:45:50,520 --> 00:45:58,060
+هي الـ B ليش؟ لأنه بما أن أف of A بساوي أف of B
+
+477
+00:45:58,060 --> 00:46:06,240
+بساوي 0 إذا أكيد C هتكون نقطة داخلية لا هتكون الـ
+
+478
+00:46:06,240 --> 00:46:10,120
+A ولا هتكون الـ B إذا صار في عندي شغلتين إنه C
+
+479
+00:46:10,120 --> 00:46:16,880
+نقطة داخلية وC عندها maximumإذا وكانت وعندي ال F
+
+480
+00:46:16,880 --> 00:46:19,800
+is differentiable على ال A و ال B على ال open
+
+481
+00:46:19,800 --> 00:46:24,260
+interval إذا حسب النظرية اللي قبله شوية لازم تكون
+
+482
+00:46:24,260 --> 00:46:30,800
+F prime of C عشان تساوي ساو سفر then F prime of C
+
+483
+00:46:30,800 --> 00:46:40,530
+ساو سفر واضح أه طيب الآنالحالة الثانية F assumes
+
+484
+00:46:40,530 --> 00:46:44,670
+negative values مدام F assumes negative values إذا
+
+485
+00:46:44,670 --> 00:46:53,930
+ناقص الـ F assumes positive values صح ولا لا؟ الآن
+
+486
+00:46:53,930 --> 00:46:56,870
+اللي بنتبج على الـ F اللي فوق صار ينتبج على ناقص F
+
+487
+00:46:56,870 --> 00:46:57,390
+كله
+
+488
+00:47:05,040 --> 00:47:08,060
+differentiable على الـ A و الـ B و continuous على
+
+489
+00:47:08,060 --> 00:47:12,620
+الـ closed interval و هذا هيتحقق على مين بس؟ على
+
+490
+00:47:12,620 --> 00:47:19,480
+ناقص F إذا من ال case التانية ناقص F of C prime
+
+491
+00:47:19,480 --> 00:47:26,300
+هتساوي سفر أو بمعنى آخر F prime of C هيساوي سفر
+
+492
+00:47:26,300 --> 00:47:32,380
+for some C element in A و B هذه برهان ال roles
+
+493
+00:47:32,380 --> 00:47:45,410
+theoryأي سؤال اللي بعده الآن أوصلنا لإن نقدر الآن
+
+494
+00:47:45,410 --> 00:47:50,890
+نحكي عن اللي هو نظرية القيمة المتوسطة أو الـ Mean
+
+495
+00:47:50,890 --> 00:47:54,990
+Value Theorem نظرية القيمة المتوسطة أو الـ Mean
+
+496
+00:47:54,990 --> 00:47:57,510
+Value Theorem برضه
+
+497
+00:47:59,060 --> 00:48:03,900
+اللي هو مرت عليكم في calculus ألف و بتمر أيضًا في
+
+498
+00:48:03,900 --> 00:48:09,000
+اللي هو ملهجة ثانوية إيش ال mean value theorem؟ هي
+
+499
+00:48:09,000 --> 00:48:15,840
+عبارة عن تعميم لمين لل roles theorem إيش بتقول ال
+
+500
+00:48:15,840 --> 00:48:27,580
+mean value theorem؟ mean value theorem if F من A و
+
+501
+00:48:27,580 --> 00:48:38,570
+Bالعلب R is continuous and differentiable on A وB
+
+502
+00:48:38,570 --> 00:48:47,130
+then there exists C element in A وB such that F
+
+503
+00:48:47,130 --> 00:48:54,010
+prime of C سواء F of B ناقص F of A على B minus A
+
+504
+00:48:54,010 --> 00:48:57,590
+طبعا معناها الأهندسية برضه مارة عليكم خلّيني أقول
+
+505
+00:48:57,590 --> 00:49:05,120
+كمان مرةإيش اللي بتقوله النظرية؟ بتقول كمايا ليه؟
+
+506
+00:49:05,120 --> 00:49:14,860
+لإن مش مزايد بس ماشي
+
+507
+00:49:14,860 --> 00:49:22,960
+مفترض إن هي اللي عندي الـA وهي عند مين الـBالان F
+
+508
+00:49:22,960 --> 00:49:27,180
+of B نقص F of A على B minus A اللي هو عبارة عن ميل
+
+509
+00:49:27,180 --> 00:49:34,540
+عن ميل القاطع هذا ها .. هايه هذا اللي هو عبارة عن
+
+510
+00:49:34,540 --> 00:49:43,760
+النقطة B و F of B و هذا النقطة A و F of A طبعا ميل
+
+511
+00:49:43,760 --> 00:49:50,380
+القاطع زي ما انتوا عارفين هو F of B لهناناقص F of
+
+512
+00:49:50,380 --> 00:49:57,060
+A هذه فبكون هذا على اللي هو B كلها هذه ناقص A
+
+513
+00:49:57,060 --> 00:50:02,360
+فبكون اللي هو ميل القاطع هذا اللي هو F of B ناقص F
+
+514
+00:50:02,360 --> 00:50:06,900
+of A على B minus Aوالـ F prime of C اللي هو عبارة
+
+515
+00:50:06,900 --> 00:50:10,860
+عن إيش؟ عن ميل المماس عنده النقطة C يعني وكأنه
+
+516
+00:50:10,860 --> 00:50:14,400
+نظرية بتقولنا إنه لو كانت F is continuous على
+
+517
+00:50:14,400 --> 00:50:17,840
+closed interval A وB و differentiable على ال open
+
+518
+00:50:17,840 --> 00:50:22,340
+interval A وB بلاجي اللي هو مما .. وهي عند القاطع
+
+519
+00:50:22,340 --> 00:50:29,040
+هيه بلاجي مماس لدالة مماس واحد او اتنين او تلاتة
+
+520
+00:50:29,040 --> 00:50:37,560
+موازن مينيعني عند نقطة C بلاق اللي هو مماس يوازمين
+
+521
+00:50:37,560 --> 00:50:41,260
+.. يوازم القاطع في rule theorem كانت اللي هو ال A
+
+522
+00:50:41,260 --> 00:50:45,740
+و ال B صورة .. صورة انتحار على ال X axis فطبيعي ما
+
+523
+00:50:45,740 --> 00:50:49,980
+اللي هو قطح هو ال X axis من القاطع اللي هو سفر
+
+524
+00:50:50,790 --> 00:50:56,170
+فاهمين؟ فبكون اللي هو الـ F' اللي هو طبيعي موازي
+
+525
+00:50:56,170 --> 00:51:00,890
+أو صفر أو موازي للـ X-axis يعني بمعنى أخر والـ
+
+526
+00:51:00,890 --> 00:51:05,750
+theorem هي حالة خاصة من الـ mean value theorem
+
+527
+00:51:05,750 --> 00:51:12,490
+واضح؟ طيب، الآن ال proof ال proof طبعا أكيد هنعتمد
+
+528
+00:51:13,290 --> 00:51:16,690
+على اللي هو الـ mean على الـ Rolle's theorem في
+
+529
+00:51:16,690 --> 00:51:21,950
+الإثبات بدنا اللي هو نركب اللي هو الـ function
+
+530
+00:51:21,950 --> 00:51:27,210
+تكون بتحقق الـ mean value theorem و بتجيبلنا اللي
+
+531
+00:51:27,210 --> 00:51:33,750
+هي الصورة هذه لو أخدنا 5x بساوي f of x ناقص f of a
+
+532
+00:51:33,750 --> 00:51:38,110
+ناقص اللي هو المال اللي بديه f of b ناقص f of a
+
+533
+00:51:38,110 --> 00:51:45,970
+على b minus a مضروبةفي X minus A هلجيت بتشوف ليش
+
+534
+00:51:45,970 --> 00:51:52,910
+أخدنا الدالة الدالة الأن هذه لو جيت حسبة Phi of A
+
+535
+00:51:52,910 --> 00:51:57,630
+لها يعني consider this function Phi of A هيساوي F
+
+536
+00:51:57,630 --> 00:52:02,730
+of A نقص F of A سفر وA نقص A سفر فبصير بالسفر هذا
+
+537
+00:52:02,730 --> 00:52:10,040
+إذن بيساوي سفر لو جيت حسبة Phi of B5 of B هيصير 5
+
+538
+00:52:10,040 --> 00:52:14,280
+of B بيساوي F of B نقص F of A خليني عندك و B minus
+
+539
+00:52:14,280 --> 00:52:18,260
+A بتروح هذه محادي هذه ناقص لنفسها بيطلع برضه أياش
+
+540
+00:52:18,260 --> 00:52:23,000
+سفر إذا الأن 5 of A بيساوي 5 of B بيساوي سفر هاي
+
+541
+00:52:23,000 --> 00:52:27,600
+واحد اتنين عندي هذه الدالة is continuous على
+
+542
+00:52:27,600 --> 00:52:33,120
+closed bounded intervalوهذه ثوابت و X minus A X
+
+543
+00:52:33,120 --> 00:52:36,120
+minus A ده اللي خطية continuous وين مكان و
+
+544
+00:52:36,120 --> 00:52:39,520
+differentiable وين مكان إذا صارت هذه الدالة كلها
+
+545
+00:52:39,520 --> 00:52:43,360
+على بعض اللي هي A Phi of X is continuous وين مكان
+
+546
+00:52:43,360 --> 00:52:45,900
+بما فيها اللي .. ال .. ال .. ال .. ال .. ال .. ال
+
+547
+00:52:45,900 --> 00:52:46,040
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+548
+00:52:46,040 --> 00:52:46,320
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+549
+00:52:46,320 --> 00:52:46,420
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+550
+00:52:46,420 --> 00:52:48,560
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+551
+00:52:48,560 --> 00:52:48,600
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+552
+00:52:48,600 --> 00:52:49,020
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+553
+00:52:49,020 --> 00:52:49,700
+.. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال ..
+
+554
+00:52:49,700 --> 00:52:50,180
+ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال .. ال
+
+555
+00:52:50,180 --> 00:52:50,420
+.. ال .. ال .. ال .. ال .. ال
+
+556
+00:52:59,650 --> 00:53:05,310
+و differentiable on a و b و أيضا اللي هو five a
+
+557
+00:53:05,310 --> 00:53:10,410
+بيساوي five b بيساوي سفر اذا .. اذا by Rolle's
+
+558
+00:53:10,410 --> 00:53:16,810
+theorem there exists c element in a و b such that
+
+559
+00:53:16,810 --> 00:53:22,230
+five prime of c ايش بتساويك؟ بتساوي سفر طيب سفر
+
+560
+00:53:22,230 --> 00:53:26,490
+يعني بتساوي five prime of c ايش five prime of c؟
+
+561
+00:53:26,490 --> 00:53:31,900
+خلينا نفرض لهناهي عبارة عن F prime of C مكان الـ
+
+562
+00:53:31,900 --> 00:53:36,840
+XC وهذه سفر وهذه اللي هي تفاضل الـ X وهذه سفر
+
+563
+00:53:36,840 --> 00:53:42,640
+طبعاً ضال الـ X هي شمالها واحد بصير ناقص F of B
+
+564
+00:53:42,640 --> 00:53:49,300
+فاهمين؟ F of A على B minus A هذا بيسوء سفر وهذا
+
+565
+00:53:49,300 --> 00:53:56,140
+يعني أن F prime of C بيسوء F of B ناقص F of A على
+
+566
+00:53:56,140 --> 00:54:06,450
+B minus Aوهو المطلوب، مظبوط؟ نيجي الآن لبعض
+
+567
+00:54:06,450 --> 00:54:14,290
+النتائج اللي احنا بستعملها كتير اللي
+
+568
+00:54:14,290 --> 00:54:20,630
+هي في الواقع برهانة عن طريق الـ Mean Value Theorem
+
+569
+00:54:20,630 --> 00:54:26,120
+النظرية اللي بتقول ليهاللي هي اللي باختصار بنقول
+
+570
+00:54:26,120 --> 00:54:33,900
+أنه لو كانت المشتقة الأولى لدالة بتساوي سفر إذن هي
+
+571
+00:54:33,900 --> 00:54:41,340
+اللي هتكون مين هي المشتقة الثابتة طيب نشوف بالظبط
+
+572
+00:54:41,340 --> 00:54:46,960
+أيش اللي بقوله بقولوا .. نفترض أن F continuous F
+
+573
+00:54:46,960 --> 00:54:53,280
+عبارة عن دالة من A و Bلعيد R continuous ماشي
+
+574
+00:54:53,280 --> 00:54:57,060
+continuous على closed اللي هو open closed bounded
+
+575
+00:54:57,060 --> 00:55:02,060
+interval الان او كانت differentiable و F
+
+576
+00:55:02,060 --> 00:55:08,080
+differentiable على ال A و ال B باشي الحال بالاضافة
+
+577
+00:55:08,080 --> 00:55:15,300
+معطيك and F prime of X بساوة سفر لكل X element in
+
+578
+00:55:15,300 --> 00:55:19,600
+A و B هذا بقوللي على بعضه then
+
+579
+00:55:22,250 --> 00:55:29,470
+then اللي هو f of x بساوي c لكل x element in a و b
+
+580
+00:55:29,470 --> 00:55:39,910
+for some c constant for some constant c طيب الآن
+
+581
+00:55:39,910 --> 00:55:46,370
+ال proof البرهانة عمبرا عن exercise صغير بسيط
+
+582
+00:55:46,370 --> 00:55:56,110
+نبرهنه مع بعض الآن خد xlet x element in a و b ماشي
+
+583
+00:55:56,110 --> 00:56:00,530
+و خلّينا ناخد ال x أكبر من مين من a يعني x في أين
+
+584
+00:56:00,530 --> 00:56:04,410
+موجود في ال open .. في ال half open interval من a
+
+585
+00:56:04,410 --> 00:56:11,610
+و b let x be fixed but arbitrary ماشي الآن بدي
+
+586
+00:56:11,610 --> 00:56:17,570
+أطبّق المي .. طبعا هي ال a و ال b هي ال a و هي ال
+
+587
+00:56:17,570 --> 00:56:21,960
+bوأخدت X أي نقطة هنا هنا .. هنا .. هنا .. هنا ..
+
+588
+00:56:21,960 --> 00:56:26,160
+هنا .. هنا .. حتى لو كانت ال .. إيش ال B إذا صارت
+
+589
+00:56:26,160 --> 00:56:36,020
+ال F عندي ال F صارت ال F continuous on اللي هو X و
+
+590
+00:56:36,020 --> 00:56:43,300
+B صح؟ continuous عليها دي أكيد و لا بناخدها على
+
+591
+00:56:43,300 --> 00:56:53,420
+الفترة A و X أه عشان A و Xو differentiable on a و
+
+592
+00:56:53,420 --> 00:57:01,100
+x صح ولا لأ اذا by الهو mean value theorem there
+
+593
+00:57:01,100 --> 00:57:09,660
+exist c element in a و x such that f prime of c
+
+594
+00:57:09,660 --> 00:57:21,390
+بساوي f of x ناقص f of a على x minus aصحيح؟ حيث C
+
+595
+00:57:21,390 --> 00:57:27,630
+في A و B طب احنا مدام C في A و X بين A و X و احنا
+
+596
+00:57:27,630 --> 00:57:31,830
+مفترضين ان F prime of X بيساوي صفر لكل X في A و B
+
+597
+00:57:31,830 --> 00:57:38,930
+معايا اه؟ اه، اذا هتصير هذا اشمالها صفر، اذا صارت
+
+598
+00:57:38,930 --> 00:57:46,680
+عندي F هذا بيساوي صفر، اذا صارت F of X بتساوي Fof
+
+599
+00:57:46,680 --> 00:57:53,220
+A لكل X وين موجودة الآن for all such X وهذه كانت X
+
+600
+00:57:53,220 --> 00:57:58,000
+fixed but arbitrary إذا صار F of X بيساوي F of A
+
+601
+00:57:58,000 --> 00:58:02,800
+لكل X في الفترة A و B بما فيها ميني الـ A لأنه
+
+602
+00:58:02,800 --> 00:58:12,520
+نفسها بتساوي إذا صارت عندي hence F is constant أي
+
+603
+00:58:12,520 --> 00:58:20,890
+سؤالطيب اللي بعدها corollary
+
+604
+00:58:20,890 --> 00:58:24,470
+الـ corollary برضه من الاشي المشهور اللي احنا
+
+605
+00:58:24,470 --> 00:58:29,190
+بنستخدمه اللي هو لو كان دالتين بحاجة عن الشروط
+
+606
+00:58:29,190 --> 00:58:34,910
+طبعا لأنها تفاضلن زي بعض إذن هيكون اللي هو هن نفس
+
+607
+00:58:34,910 --> 00:58:42,990
+الاشي بس باختلف عن بعض بالأكترين ثابت طيب الآن ال
+
+608
+00:58:42,990 --> 00:58:50,450
+corollary كما يليهالـ Corollary بتقول suppose that
+
+609
+00:58:50,450 --> 00:58:56,970
+F and G are continuous on I, A وB then that they
+
+610
+00:58:56,970 --> 00:59:01,190
+are differentiable on A وB يعني نفترض أن الدالتين
+
+611
+00:59:01,190 --> 00:59:05,230
+F وG continuous على ال closed اللي هي bounded
+
+612
+00:59:05,230 --> 00:59:08,910
+interval A وB وdifferentiable على ال open interval
+
+613
+00:59:08,910 --> 00:59:12,070
+A وBوالـ derivative للأولى بساوي ال derivative
+
+614
+00:59:12,070 --> 00:59:15,690
+للثانية عند كل نقطة من النقاط الـ A و الـ B إذا
+
+615
+00:59:15,690 --> 00:59:18,910
+there exists a constant C such that F بتساوي G
+
+616
+00:59:18,910 --> 00:59:25,350
+زائد H زائد C طبعا البرهان سهل و أكيد يمكن بعضكم
+
+617
+00:59:25,350 --> 00:59:34,390
+برهنه في راسه let H بتساوي F ناقص G مظبوط؟ الآن
+
+618
+00:59:34,390 --> 00:59:43,530
+thenF و G continuous إذن H is continuous on A و B
+
+619
+00:59:43,530 --> 00:59:51,210
+و أيضًا F و G differentiable على A و B and H is
+
+620
+00:59:51,210 --> 00:59:55,210
+differentiable on A و B moreover
+
+621
+00:59:57,700 --> 01:00:02,620
+عندي H برايم اللي هو بساوي F برايم نقص G برايم
+
+622
+01:00:02,620 --> 01:00:08,340
+طبعا عند ال X عند ال X of X of X of X بساوي صفر
+
+623
+01:00:08,340 --> 01:00:16,340
+لكل X موجودة في ال A و ال B then by the above
+
+624
+01:00:16,340 --> 01:00:20,300
+theorem اللي بنحكي عنها اللي هي theorem by theorem
+
+625
+01:00:20,300 --> 01:00:31,180
+625 Hof X بتساوي C for every X element in A وB and
+
+626
+01:00:31,180 --> 01:00:35,300
+hence F
+
+627
+01:00:35,300 --> 01:00:44,040
+of X بساوي G of X زاد some constant C
+
+628
+01:00:50,890 --> 01:00:54,570
+بكونوا احنا وصلنا لعين theorem 627 ان شاء الله
+
+629
+01:00:54,570 --> 01:01:01,010
+المرة القادمة بنكمل والسلام عليكم ورحمة الله
+
+630
+01:01:01,010 --> 01:01:01,670
+وبركاته
+
diff --git a/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw_raw.json b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw_raw.json
new file mode 100644
index 0000000000000000000000000000000000000000..ed0c92337235d517407b3aacf01ba0887fa3ad67
--- /dev/null
+++ b/PL9fwy3NUQKwbxj_Y7aSVHQBzhMp9X9OOe/xgo_ibuk2Tw_raw.json
@@ -0,0 +1 @@
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404.61, "end": 404.81, "word": " مين؟", "probability": 0.9248046875}, {"start": 404.81, "end": 404.97, "word": " هي", "probability": 0.7421875}, {"start": 404.97, "end": 405.13, "word": " ال", "probability": 0.94775390625}, {"start": 405.13, "end": 405.43, "word": " F", "probability": 0.96533203125}, {"start": 405.43, "end": 406.21, "word": " inverse", "probability": 0.84130859375}], "temperature": 1.0}, {"id": 17, "seek": 43566, "start": 407.18, "end": 435.66, "text": "الآن الشرط أنه if F is differentiable at C at C then اللي هي G اللي هي الـ F inverse دي في الواقع F inverse اللي سمناها مين؟ F inverse لازم تكون differentiable at F of C and", "tokens": [6027, 148, 48506, 25124, 2288, 9566, 14739, 3224, 498, 479, 307, 819, 9364, 412, 383, 412, 383, 550, 13672, 1829, 39896, 460, 13672, 1829, 39896, 2423, 39184, 479, 17340, 11778, 1829, 8978, 2423, 14407, 4587, 3615, 479, 17340, 13672, 1829, 8608, 2304, 8315, 11296, 3714, 9957, 22807, 479, 17340, 5296, 31377, 2304, 6055, 30544, 819, 9364, 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local أو relative maximum when اللي هو there exists", "tokens": [3615, 1863, 16254, 2445, 283, 9154, 512, 15035, 286, 5296, 3615, 41260, 497, 8032, 47302, 2423, 39184, 2445, 13672, 1829, 18871, 16254, 34105, 9154, 34105, 5296, 3224, 8315, 44945, 39648, 383, 4478, 294, 286, 412, 383, 4478, 294, 286, 321, 362, 257, 2654, 34051, 4972, 6674, 562, 13672, 1829, 31439, 456, 8198], "avg_logprob": -0.2713738252531807, "compression_ratio": 1.4625, "no_speech_prob": 0.0, "words": [{"start": 1050.87, "end": 1051.27, "word": "عندي", "probability": 0.6962076822916666}, {"start": 1051.27, "end": 1051.63, "word": " function", "probability": 0.693359375}, {"start": 1051.63, "end": 1052.13, "word": " f", "probability": 0.68017578125}, {"start": 1052.13, "end": 1052.51, "word": " من", "probability": 0.869140625}, {"start": 1052.51, "end": 1052.77, "word": " some", "probability": 0.57666015625}, {"start": 1052.77, "end": 1053.25, "word": " interval", "probability": 0.9716796875}, {"start": 1053.25, "end": 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للدالة F of C أكبر أو سوى F of X لكل X في هذا الجوار", "tokens": [53, 18183, 295, 383, 37495, 22653, 44945, 15040, 38436, 10874, 2407, 9640, 11331, 2407, 6027, 9957, 2423, 39184, 383, 44945, 9381, 5172, 12174, 9566, 2288, 18183, 383, 8717, 4587, 9381, 18183, 295, 383, 30767, 16373, 3215, 18183, 8978, 23758, 25724, 2407, 9640, 1270, 300, 39894, 30544, 479, 295, 383, 39896, 5551, 4117, 26890, 12174, 32640, 3660, 24976, 3215, 6027, 3660, 479, 295, 383, 5551, 4117, 26890, 34051, 8608, 2407, 7578, 479, 295, 1783, 5296, 28820, 1783, 8978, 23758, 25724, 2407, 9640], "avg_logprob": -0.14300686375397006, "compression_ratio": 1.6184971098265897, "no_speech_prob": 7.212162017822266e-06, "words": [{"start": 1078.6, "end": 1079.24, "word": "V", "probability": 0.42529296875}, {"start": 1079.24, "end": 1079.82, "word": " Delta", "probability": 0.2064208984375}, {"start": 1079.82, "end": 1080.18, "word": " of", "probability": 0.74853515625}, {"start": 1080.18, "end": 1080.46, "word": " C", 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{"id": 46, "seek": 116555, "start": 1148.27, "end": 1165.55, "text": "there exist أي جوار آخر بتكون تسموه V Delta Prime أو V Delta of D such that يعني بنلاقي .. بنقول هذه النقطة في عندها relative minimum إذا كان لقينا جوار حواليها", "tokens": [15456, 2514, 36632, 10874, 2407, 9640, 19753, 34740, 39894, 30544, 6055, 38251, 2407, 3224, 691, 18183, 9655, 34051, 691, 18183, 295, 413, 1270, 300, 37495, 22653, 44945, 15040, 38436, 4386, 44945, 39648, 29538, 28239, 47432, 3660, 8978, 43242, 11296, 4972, 7285, 11933, 15730, 25961, 5296, 4587, 9957, 995, 10874, 2407, 9640, 11331, 2407, 6027, 1829, 11296], "avg_logprob": -0.23985745614035087, "compression_ratio": 1.3072625698324023, "no_speech_prob": 1.8477439880371094e-06, "words": [{"start": 1148.27, "end": 1148.71, "word": "there", "probability": 0.2462158203125}, {"start": 1148.71, "end": 1149.57, "word": " exist", "probability": 0.6259765625}, {"start": 1149.57, "end": 1150.69, "word": " أي", "probability": 0.81103515625}, {"start": 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{"start": 1213.62, "end": 1214.44, "word": " بنقصد", "probability": 0.898193359375}, {"start": 1214.44, "end": 1214.58, "word": " و", "probability": 0.4111328125}, {"start": 1214.58, "end": 1215.12, "word": " بنقول", "probability": 0.94384765625}, {"start": 1215.12, "end": 1216.26, "word": " local", "probability": 0.796875}, {"start": 1216.26, "end": 1217.58, "word": " extremum", "probability": 0.84521484375}, {"start": 1217.58, "end": 1218.6, "word": " معناته", "probability": 0.93701171875}, {"start": 1218.6, "end": 1219.12, "word": " يا", "probability": 0.80517578125}, {"start": 1219.12, "end": 1219.54, "word": " local", "probability": 0.89794921875}, {"start": 1219.54, "end": 1220.24, "word": " maximum", "probability": 0.9658203125}, {"start": 1220.24, "end": 1220.88, "word": " أو", "probability": 0.59423828125}, {"start": 1220.88, "end": 1221.38, "word": " local", "probability": 0.87744140625}, {"start": 1221.38, "end": 1222.12, "word": " minimum", "probability": 0.98046875}, 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12610, 7578, 1975, 5016, 8315, 9122, 8315, 8978, 2423, 33400, 11933, 15730, 39894, 2655, 8848, 37983, 4032, 8717, 3794, 10721, 1211, 8608, 33604, 6027], "avg_logprob": -0.2358530469842859, "compression_ratio": 1.198019801980198, "no_speech_prob": 5.960464477539063e-08, "words": [{"start": 1227.42, "end": 1227.9, "word": "نجي", "probability": 0.57861328125}, {"start": 1227.9, "end": 1228.26, "word": " الآن", "probability": 0.6734619140625}, {"start": 1228.26, "end": 1230.12, "word": " لنظريتنا", "probability": 0.9013671875}, {"start": 1230.12, "end": 1230.92, "word": " الأولى", "probability": 0.9889322916666666}, {"start": 1230.92, "end": 1231.58, "word": " احنا", "probability": 0.80908203125}, {"start": 1231.58, "end": 1231.84, "word": " كنا", "probability": 0.99072265625}, {"start": 1231.84, "end": 1231.98, "word": " في", "probability": 0.90478515625}, {"start": 1231.98, "end": 1232.08, "word": " ال", "probability": 0.95263671875}, {"start": 1232.08, "end": 1232.38, "word": " calculus", "probability": 0.36376953125}, {"start": 1232.38, "end": 1232.7, "word": " إذا", "probability": 0.6474609375}, {"start": 1232.7, "end": 1233.24, "word": " بتتذكر", "probability": 0.930908203125}, {"start": 1233.24, "end": 1233.36, "word": " و", "probability": 0.701171875}, {"start": 1233.36, "end": 1233.72, "word": " نسأل", "probability": 0.80352783203125}, {"start": 1233.72, "end": 1234.18, "word": " سؤال", "probability": 0.97314453125}], "temperature": 1.0}, {"id": 51, "seek": 126181, "start": 1235.77, "end": 1261.81, "text": "لما بدنا نبحث في مسألة اللي هو الـ maximum و الـ minimum سواء local maximum و local minimum أو absolute maximum و absolute minimum كنا نسأل حالنا سؤال طبعا مش معجول أنا بدي آجي كل نقاط ال domain أفحص فيه عند maximum ولا minimum لأن نقاط ال domain عدد لا نهائي uncountable number of points في غالبها", "tokens": [1211, 15042, 47525, 8315, 8717, 49628, 12984, 8978, 47524, 10721, 37977, 13672, 1829, 31439, 2423, 39184, 6674, 4032, 2423, 39184, 7285, 8608, 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طبعا", "probability": 0.9033203125}, {"start": 1251.29, "end": 1251.43, "word": " مش", "probability": 0.8369140625}, {"start": 1251.43, "end": 1251.89, "word": " معجول", "probability": 0.6904296875}, {"start": 1251.89, "end": 1252.13, "word": " أنا", "probability": 0.6875}, {"start": 1252.13, "end": 1252.47, "word": " بدي", "probability": 0.7099609375}, {"start": 1252.47, "end": 1252.91, "word": " آجي", "probability": 0.715087890625}, {"start": 1252.91, "end": 1253.47, "word": " كل", "probability": 0.9775390625}, {"start": 1253.47, "end": 1253.93, "word": " نقاط", "probability": 0.9392903645833334}, {"start": 1253.93, "end": 1254.09, "word": " ال", "probability": 0.84326171875}, {"start": 1254.09, "end": 1254.39, "word": " domain", "probability": 0.57666015625}, {"start": 1254.39, "end": 1254.95, "word": " أفحص", "probability": 0.91259765625}, {"start": 1254.95, "end": 1255.17, "word": " فيه", "probability": 0.98779296875}, {"start": 1255.17, "end": 1255.35, "word": " عند", "probability": 0.951171875}, {"start": 1255.35, "end": 1255.85, "word": " maximum", "probability": 0.6552734375}, {"start": 1255.85, "end": 1256.17, "word": " ولا", "probability": 0.493896484375}, {"start": 1256.17, "end": 1256.59, "word": " minimum", "probability": 0.93701171875}, {"start": 1256.59, "end": 1257.37, "word": " لأن", "probability": 0.758544921875}, {"start": 1257.37, "end": 1257.83, "word": " نقاط", "probability": 0.95849609375}, {"start": 1257.83, "end": 1257.99, "word": " ال", "probability": 0.943359375}, {"start": 1257.99, "end": 1258.35, "word": " domain", "probability": 0.9716796875}, {"start": 1258.35, "end": 1258.81, "word": " عدد", "probability": 0.9768880208333334}, {"start": 1258.81, "end": 1259.01, "word": " لا", "probability": 0.71923828125}, {"start": 1259.01, "end": 1259.49, "word": " نهائي", "probability": 0.884765625}, {"start": 1259.49, "end": 1260.23, "word": " uncountable", "probability": 0.7154947916666666}, {"start": 1260.23, "end": 1260.59, "word": " number", "probability": 0.97607421875}, {"start": 1260.59, "end": 1260.79, "word": " of", "probability": 0.9794921875}, {"start": 1260.79, "end": 1261.17, "word": " points", "probability": 0.93505859375}, {"start": 1261.17, "end": 1261.31, "word": " في", "probability": 0.80615234375}, {"start": 1261.31, "end": 1261.81, "word": " غالبها", "probability": 0.808349609375}], "temperature": 1.0}, {"id": 52, "seek": 128794, "start": 1262.26, "end": 1287.94, "text": "فإيش كنت حصرها؟ كنا نقول أين يمكن إيجاد الـlocal maximum أو الـlocal minimum بعد ما نحدد النقاط هذه بنقول يأتي الآن الفحص إنه هل هي local maximum ولا local minimum إذن مسألة اللي هو إيجاد ال maximum و ال minimum تبرع عبر خطوتين الخطوة الأولى وين ممكن نلاقيها", "tokens": [5172, 28814, 1829, 8592, 9122, 29399, 11331, 9381, 2288, 11296, 22807, 9122, 8315, 8717, 39648, 5551, 9957, 7251, 43020, 11933, 1829, 7435, 18513, 2423, 39184, 5842, 304, 6674, 34051, 2423, 39184, 5842, 304, 7285, 39182, 19446, 8717, 24401, 3215, 28239, 4587, 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"probability": 0.98193359375}, {"start": 1278.7, "end": 1279.58, "word": " إذن", "probability": 0.6885579427083334}, {"start": 1279.58, "end": 1280.06, "word": " مسألة", "probability": 0.9763997395833334}, {"start": 1280.06, "end": 1280.24, "word": " اللي", "probability": 0.9287109375}, {"start": 1280.24, "end": 1280.58, "word": " هو", "probability": 0.9033203125}, {"start": 1280.58, "end": 1281.66, "word": " إيجاد", "probability": 0.844970703125}, {"start": 1281.66, "end": 1281.78, "word": " ال", "probability": 0.921875}, {"start": 1281.78, "end": 1282.12, "word": " maximum", "probability": 0.5322265625}, {"start": 1282.12, "end": 1282.34, "word": " و", "probability": 0.69287109375}, {"start": 1282.34, "end": 1282.38, "word": " ال", "probability": 0.5703125}, {"start": 1282.38, "end": 1282.64, "word": " minimum", "probability": 0.9013671875}, {"start": 1282.64, "end": 1283.1, "word": " تبرع", "probability": 0.8805338541666666}, {"start": 1283.1, "end": 1283.34, "word": " عبر", "probability": 0.889892578125}, {"start": 1283.34, "end": 1284.74, "word": " خطوتين", "probability": 0.84320068359375}, {"start": 1284.74, "end": 1285.18, "word": " الخطوة", "probability": 0.9658203125}, {"start": 1285.18, "end": 1285.76, "word": " الأولى", "probability": 0.99609375}, {"start": 1285.76, "end": 1286.56, "word": " وين", "probability": 0.887451171875}, {"start": 1286.56, "end": 1287.2, "word": " ممكن", "probability": 0.9931640625}, {"start": 1287.2, "end": 1287.94, "word": " نلاقيها", "probability": 0.866943359375}], "temperature": 1.0}, {"id": 53, "seek": 130300, "start": 1288.3, "end": 1303.0, "text": "والخطوة الثانية اللي هي هل هي فعلا local maximum ولا local minimum ولا لا local maximum ولا local minimum اللي بده يشرعلنا الإجابة الأولى هي النظرية هذه", "tokens": [2407, 6027, 9778, 9566, 2407, 3660, 6024, 104, 7649, 10632, 13672, 1829, 39896, 8032, 1211, 39896, 6156, 3615, 15040, 2654, 6674, 49429, 2654, 7285, 49429, 20193, 2654, 6674, 49429, 2654, 7285, 13672, 1829, 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{"start": 1293.12, "end": 1293.72, "word": " ولا", "probability": 0.76513671875}, {"start": 1293.72, "end": 1294.08, "word": " local", "probability": 0.9169921875}, {"start": 1294.08, "end": 1294.62, "word": " minimum", "probability": 0.98388671875}, {"start": 1294.62, "end": 1295.5, "word": " ولا", "probability": 0.82763671875}, {"start": 1295.5, "end": 1296.22, "word": " لا", "probability": 0.6904296875}, {"start": 1296.22, "end": 1296.66, "word": " local", "probability": 0.91845703125}, {"start": 1296.66, "end": 1297.26, "word": " maximum", "probability": 0.95263671875}, {"start": 1297.26, "end": 1297.7, "word": " ولا", "probability": 0.958984375}, {"start": 1297.7, "end": 1298.08, "word": " local", "probability": 0.90478515625}, {"start": 1298.08, "end": 1298.58, "word": " minimum", "probability": 0.9755859375}, {"start": 1298.58, "end": 1299.3, "word": " اللي", "probability": 0.960205078125}, {"start": 1299.3, "end": 1299.64, "word": " بده", "probability": 0.728271484375}, {"start": 1299.64, "end": 1300.5, "word": " يشرعلنا", "probability": 0.8372802734375}, {"start": 1300.5, "end": 1301.52, "word": " الإجابة", "probability": 0.967041015625}, {"start": 1301.52, "end": 1301.98, "word": " الأولى", "probability": 0.9845377604166666}, {"start": 1301.98, "end": 1302.1, "word": " هي", "probability": 0.9638671875}, {"start": 1302.1, "end": 1302.7, "word": " النظرية", "probability": 0.834716796875}, {"start": 1302.7, "end": 1303.0, "word": " هذه", "probability": 0.91943359375}], "temperature": 1.0}, {"id": 54, "seek": 133237, "start": 1303.61, "end": 1332.37, "text": "النظرية هذه كنا كمان نجاوب نقول الإجابة توجد الـlocal maximum أو الـlocal minimum في أحد ثلاث مناطق إما F prime بساوي سفر أو F prime does not exist كنقات داخلية أو عند الـendpoints فكنا نحصرها في تلت أنهي مناطق فبصير عدد النقاط اللي بدنا نفحص الـlocal max أو الـlocal minimum عددها محدود وقليل بصير السؤال مقدور عليه", "tokens": [6027, 1863, 19913, 2288, 10632, 29538, 9122, 8315, 9122, 2304, 7649, 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1367.9, "text": "حتى اسمها اللي هي interior extremum theorem interior extremum theorem يعني هنحكي عن ال interior point بتقولي نفترض إن C عبارة عن نقطة داخلية لمين ل some interval I C اللي هو إشمالة نقطة داخلية ل I ال interval اللي عندي", "tokens": [33753, 7578, 24525, 2304, 11296, 13672, 1829, 39896, 10636, 4040, 449, 20904, 10636, 4040, 449, 20904, 37495, 22653, 8032, 1863, 5016, 4117, 1829, 18871, 2423, 10636, 935, 39894, 39648, 1829, 8717, 5172, 2655, 43042, 36145, 383, 6225, 3555, 9640, 3660, 18871, 8717, 47432, 3660, 11778, 47283, 1211, 10632, 32767, 9957, 5296, 512, 15035, 286, 383, 13672, 1829, 31439, 11933, 8592, 2304, 6027, 3660, 8717, 47432, 3660, 11778, 47283, 1211, 10632, 5296, 286, 2423, 15035, 13672, 1829, 18871, 16254], "avg_logprob": -0.19363132722770113, "compression_ratio": 1.7486033519553073, "no_speech_prob": 3.5762786865234375e-07, "words": [{"start": 1345.56, "end": 1346.12, "word": "حتى", "probability": 0.945556640625}, {"start": 1346.12, "end": 1346.58, "word": 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146270, "start": 1441.82, "end": 1462.7, "text": "وكانت عندها relative maximum أو relative minimum قطعاً هتكون الـ F prime إليها إيش بتساوي .. بتساوي سفر يعني عند .. عند النقاط الداخلية اللي الـ F prime إليها موجودة و بتشاويش سفر انساها و ماتبحثش فيها قطعا فيش فيها لا relative maximum ولا relative إيش .. minimum", "tokens": [2407, 41361, 2655, 43242, 11296, 4972, 6674, 34051, 4972, 7285, 12174, 9566, 3615, 995, 14111, 8032, 2655, 30544, 2423, 39184, 479, 5835, 11933, 20292, 11296, 11933, 1829, 8592, 39894, 3794, 995, 45865, 4386, 39894, 3794, 995, 45865, 8608, 5172, 2288, 37495, 22653, 43242, 4386, 43242, 28239, 4587, 41193, 32748, 47283, 1211, 10632, 13672, 1829, 2423, 39184, 479, 5835, 11933, 20292, 11296, 3714, 29245, 23328, 3660, 4032, 39894, 8592, 995, 45865, 8592, 8608, 5172, 2288, 16472, 3794, 995, 11296, 4032, 3714, 9307, 49628, 12984, 8592, 8978, 11296, 12174, 9566, 3615, 995, 8978, 8592, 8978, 11296, 20193, 4972, 6674, 49429, 4972, 11933, 1829, 8592, 4386, 7285], 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{"start": 1502.85, "end": 1503.29, "word": " هنا", "probability": 0.97314453125}, {"start": 1503.29, "end": 1504.47, "word": " بنجاوب", "probability": 0.9381103515625}, {"start": 1504.47, "end": 1504.63, "word": " على", "probability": 0.76904296875}, {"start": 1504.63, "end": 1505.23, "word": " سؤالنا،", "probability": 0.8673828125}, {"start": 1505.23, "end": 1505.49, "word": " هذه", "probability": 0.431396484375}, {"start": 1505.49, "end": 1505.61, "word": " اللي", "probability": 0.988525390625}, {"start": 1505.61, "end": 1506.11, "word": " بتجاوبنا", "probability": 0.97900390625}, {"start": 1506.11, "end": 1506.23, "word": " على", "probability": 0.875}, {"start": 1506.23, "end": 1506.55, "word": " سؤالنا", "probability": 0.9429931640625}, {"start": 1506.55, "end": 1506.97, "word": " بتقول", "probability": 0.901611328125}], "temperature": 1.0}, {"id": 63, "seek": 153038, "start": 1507.24, "end": 1530.38, "text": "اللي هو الـ relative maximum أو الـ relative minimum هتكون في أحد ثلاث مناطق يا F prime does not exist يا F prime بالساوية سفر يا end point واضح؟ أي سؤال؟ واضح أه؟ طيب، دي يجي تبرهن النظرية", "tokens": [6027, 20292, 31439, 2423, 39184, 4972, 6674, 34051, 2423, 39184, 4972, 7285, 8032, 2655, 30544, 8978, 5551, 24401, 38637, 1211, 5718, 104, 9154, 41193, 4587, 35186, 479, 5835, 775, 406, 2514, 35186, 479, 5835, 20666, 3794, 995, 2407, 10632, 8608, 5172, 2288, 35186, 917, 935, 4032, 46958, 5016, 22807, 36632, 8608, 33604, 6027, 22807, 4032, 46958, 5016, 5551, 3224, 22807, 23032, 1829, 3555, 12399, 11778, 1829, 7251, 7435, 1829, 6055, 26890, 3224, 1863, 28239, 19913, 2288, 10632], "avg_logprob": -0.20773236701885858, "compression_ratio": 1.4607329842931938, "no_speech_prob": 1.9073486328125e-06, "words": [{"start": 1507.24, "end": 1507.82, "word": "اللي", "probability": 0.704833984375}, {"start": 1507.82, "end": 1508.06, "word": " هو", "probability": 0.97607421875}, {"start": 1508.06, "end": 1508.66, "word": " الـ", "probability": 0.742919921875}, {"start": 1508.66, "end": 1508.98, "word": " relative", "probability": 0.80810546875}, {"start": 1508.98, "end": 1509.58, "word": " maximum", "probability": 0.953125}, {"start": 1509.58, "end": 1509.86, "word": " أو", "probability": 0.84521484375}, {"start": 1509.86, "end": 1509.98, "word": " الـ", "probability": 0.76904296875}, {"start": 1509.98, "end": 1510.2, "word": " relative", "probability": 0.9775390625}, {"start": 1510.2, "end": 1510.56, "word": " minimum", "probability": 0.9833984375}, {"start": 1510.56, "end": 1511.1, "word": " هتكون", "probability": 0.7933756510416666}, {"start": 1511.1, "end": 1511.28, "word": " في", "probability": 0.96826171875}, {"start": 1511.28, "end": 1511.48, "word": " أحد", "probability": 0.927978515625}, {"start": 1511.48, "end": 1511.8, "word": " ثلاث", "probability": 0.9642333984375}, {"start": 1511.8, "end": 1512.34, "word": " مناطق", "probability": 0.96826171875}, {"start": 1512.34, "end": 1513.16, "word": " يا", "probability": 0.6171875}, {"start": 1513.16, "end": 1513.88, "word": " F", "probability": 0.529296875}, {"start": 1513.88, "end": 1514.28, "word": " prime", "probability": 0.2744140625}, {"start": 1514.28, "end": 1514.5, "word": " does", "probability": 0.72314453125}, {"start": 1514.5, "end": 1514.7, "word": " not", "probability": 0.966796875}, {"start": 1514.7, "end": 1515.24, "word": " exist", "probability": 0.9501953125}, {"start": 1515.24, "end": 1516.0, "word": " يا", "probability": 0.75927734375}, {"start": 1516.0, "end": 1516.24, "word": " F", "probability": 0.98876953125}, {"start": 1516.24, "end": 1516.54, "word": " prime", "probability": 0.81201171875}, {"start": 1516.54, "end": 1516.94, "word": " بالساوية", "probability": 0.7162109375}, {"start": 1516.94, "end": 1517.36, "word": " سفر", "probability": 0.8170572916666666}, {"start": 1517.36, "end": 1518.06, "word": " يا", "probability": 0.57861328125}, {"start": 1518.06, "end": 1518.56, "word": " end", "probability": 0.7275390625}, {"start": 1518.56, "end": 1519.3, "word": " point", "probability": 0.634765625}, {"start": 1519.3, "end": 1522.0, "word": " واضح؟", "probability": 0.876953125}, {"start": 1522.0, "end": 1522.72, "word": " أي", "probability": 0.80224609375}, {"start": 1522.72, "end": 1524.96, "word": " سؤال؟", "probability": 0.9730224609375}, {"start": 1524.96, "end": 1526.04, "word": " واضح", "probability": 0.9871419270833334}, {"start": 1526.04, "end": 1527.18, "word": " أه؟", "probability": 0.6534016927083334}, {"start": 1527.18, "end": 1527.64, "word": " طيب،", "probability": 0.76947021484375}, {"start": 1527.64, "end": 1527.78, "word": " دي", "probability": 0.9208984375}, {"start": 1527.78, "end": 1527.96, "word": " يجي", "probability": 0.7312825520833334}, {"start": 1527.96, "end": 1528.4, "word": " تبرهن", "probability": 0.8951416015625}, {"start": 1528.4, "end": 1530.38, "word": " النظرية", "probability": 0.9403076171875}], "temperature": 1.0}, {"id": 64, "seek": 155839, "start": 1530.93, "end": 1558.39, "text": "الآن هو البروحان اللي مكتوب هنا we prove for the case that F has relative maximum at C هيه مكتوب أنا مابديش أبرهن ال maximum بتبرهنكم مين ال minimum إذا الآن بدنا نفترض نفترض ال proof خليني أكتب هنا ال proof بدي أفترض", "tokens": [6027, 148, 48506, 31439, 2423, 26890, 2407, 5016, 7649, 13672, 1829, 3714, 4117, 2655, 37746, 34105, 321, 7081, 337, 264, 1389, 300, 479, 575, 4972, 6674, 412, 383, 39896, 3224, 3714, 4117, 2655, 37746, 41850, 3714, 16758, 16254, 8592, 5551, 26890, 3224, 1863, 2423, 6674, 39894, 26890, 3224, 1863, 24793, 3714, 9957, 2423, 7285, 11933, 15730, 6024, 48506, 47525, 8315, 8717, 5172, 2655, 43042, 8717, 5172, 2655, 43042, 2423, 8177, 16490, 1211, 9957, 1829, 5551, 4117, 2655, 3555, 34105, 2423, 8177, 4724, 16254, 5551, 5172, 2655, 43042], "avg_logprob": -0.2745028402317654, "compression_ratio": 1.6927083333333333, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 1530.93, "end": 1531.55, "word": "الآن", "probability": 0.71728515625}, {"start": 1531.55, "end": 1533.91, "word": " هو", "probability": 0.876953125}, {"start": 1533.91, "end": 1535.05, "word": " البروحان", "probability": 0.86904296875}, {"start": 1535.05, "end": 1535.19, "word": " اللي", "probability": 0.707763671875}, {"start": 1535.19, "end": 1535.55, "word": " مكتوب", "probability": 0.9898681640625}, {"start": 1535.55, "end": 1535.97, "word": " هنا", "probability": 0.97900390625}, {"start": 1535.97, "end": 1536.55, "word": " we", "probability": 0.390625}, {"start": 1536.55, "end": 1537.01, "word": " prove", "probability": 0.8935546875}, {"start": 1537.01, "end": 1538.53, "word": " for", "probability": 0.720703125}, {"start": 1538.53, "end": 1538.77, "word": " the", "probability": 0.88330078125}, {"start": 1538.77, "end": 1539.11, "word": " case", "probability": 0.94677734375}, {"start": 1539.11, "end": 1539.45, "word": " that", "probability": 0.92919921875}, {"start": 1539.45, "end": 1539.65, "word": " F", "probability": 0.5712890625}, {"start": 1539.65, "end": 1539.89, "word": " has", "probability": 0.90869140625}, {"start": 1539.89, "end": 1540.29, "word": " relative", "probability": 0.90185546875}, {"start": 1540.29, "end": 1540.91, "word": " maximum", "probability": 0.94677734375}, {"start": 1540.91, "end": 1541.17, "word": " at", "probability": 0.50439453125}, {"start": 1541.17, "end": 1541.37, "word": " C", "probability": 0.61083984375}, {"start": 1541.37, "end": 1541.91, "word": " هيه", "probability": 0.3560791015625}, {"start": 1541.91, "end": 1542.41, "word": " مكتوب", "probability": 0.9825439453125}, {"start": 1542.41, "end": 1543.09, "word": " أنا", "probability": 0.767578125}, {"start": 1543.09, "end": 1543.51, "word": " مابديش", "probability": 0.70440673828125}, {"start": 1543.51, "end": 1543.95, "word": " أبرهن", "probability": 0.85986328125}, {"start": 1543.95, "end": 1544.05, "word": " ال", "probability": 0.84716796875}, {"start": 1544.05, "end": 1544.37, "word": " maximum", "probability": 0.939453125}, {"start": 1544.37, "end": 1545.15, "word": " بتبرهنكم", "probability": 0.7531494140625}, {"start": 1545.15, "end": 1545.49, "word": " مين", "probability": 0.5552978515625}, {"start": 1545.49, "end": 1546.03, "word": " ال", "probability": 0.8935546875}, {"start": 1546.03, "end": 1546.39, "word": " minimum", "probability": 0.97119140625}, {"start": 1546.39, "end": 1546.85, "word": " إذا", "probability": 0.7685546875}, {"start": 1546.85, "end": 1547.41, "word": " الآن", "probability": 0.761474609375}, {"start": 1547.41, "end": 1548.43, "word": " بدنا", "probability": 0.942138671875}, {"start": 1548.43, "end": 1549.81, "word": " نفترض", "probability": 0.9906005859375}, {"start": 1549.81, "end": 1551.73, "word": " نفترض", "probability": 0.9288330078125}, {"start": 1551.73, "end": 1551.89, "word": " ال", "probability": 0.182373046875}, {"start": 1551.89, "end": 1552.29, "word": " proof", "probability": 0.8857421875}, {"start": 1552.29, "end": 1552.69, "word": " خليني", "probability": 0.7745361328125}, {"start": 1552.69, "end": 1552.93, "word": " أكتب", "probability": 0.874267578125}, {"start": 1552.93, "end": 1553.29, "word": " هنا", "probability": 0.97705078125}, {"start": 1553.29, "end": 1554.23, "word": " ال", "probability": 0.72265625}, {"start": 1554.23, "end": 1554.79, "word": " proof", "probability": 0.98828125}, {"start": 1554.79, "end": 1557.87, "word": " بدي", "probability": 0.4412841796875}, {"start": 1557.87, "end": 1558.39, "word": " أفترض", "probability": 0.8985595703125}], "temperature": 1.0}, {"id": 65, "seek": 158812, "start": 1559.3, "end": 1588.12, "text": "C نقطة داخلية ونفترض إن F أشبهلها have a relative minimum relative maximum محلولة لأن suppose that F has a relative maximum at C element in I0 ماشي؟", "tokens": [34, 8717, 47432, 3660, 11778, 47283, 1211, 10632, 4032, 1863, 5172, 2655, 43042, 36145, 479, 5551, 8592, 3555, 3224, 1211, 11296, 362, 257, 4972, 7285, 4972, 6674, 3714, 5016, 1211, 12610, 3660, 5296, 33456, 7297, 300, 479, 575, 257, 4972, 6674, 412, 383, 4478, 294, 286, 15, 3714, 33599, 1829, 22807], "avg_logprob": -0.30618990843112653, "compression_ratio": 1.230263157894737, "no_speech_prob": 1.1920928955078125e-07, "words": [{"start": 1559.3, "end": 1559.76, "word": "C", "probability": 0.625}, {"start": 1559.76, "end": 1560.5, "word": " نقطة", "probability": 0.89697265625}, {"start": 1560.5, "end": 1561.18, "word": " داخلية", "probability": 0.9940185546875}, {"start": 1561.18, "end": 1562.6, "word": " ونفترض", "probability": 0.95107421875}, {"start": 1562.6, "end": 1562.76, "word": " إن", "probability": 0.2734375}, {"start": 1562.76, "end": 1563.0, "word": " F", "probability": 0.6982421875}, {"start": 1563.0, "end": 1563.84, "word": " أشبهلها", "probability": 0.7257080078125}, {"start": 1563.84, "end": 1564.56, "word": " have", "probability": 0.251708984375}, {"start": 1564.56, "end": 1564.82, "word": " a", "probability": 0.8427734375}, {"start": 1564.82, "end": 1565.14, "word": " relative", "probability": 0.9736328125}, {"start": 1565.14, "end": 1565.56, "word": " minimum", "probability": 0.7353515625}, {"start": 1565.56, "end": 1566.04, "word": " relative", "probability": 0.71826171875}, {"start": 1566.04, "end": 1566.42, "word": " maximum", "probability": 0.97802734375}, {"start": 1566.42, "end": 1568.02, "word": " محلولة", "probability": 0.8470703125}, {"start": 1568.02, "end": 1568.56, "word": " لأن", "probability": 0.3917236328125}, {"start": 1568.56, "end": 1569.46, "word": " suppose", "probability": 0.79736328125}, {"start": 1569.46, "end": 1572.64, "word": " that", "probability": 0.8896484375}, {"start": 1572.64, "end": 1574.32, "word": " F", "probability": 0.8720703125}, {"start": 1574.32, "end": 1576.08, "word": " has", "probability": 0.93798828125}, {"start": 1576.08, "end": 1577.2, "word": " a", "probability": 0.8193359375}, {"start": 1577.2, "end": 1578.0, "word": " relative", "probability": 0.9599609375}, {"start": 1578.0, "end": 1581.02, "word": " maximum", "probability": 0.9677734375}, {"start": 1581.02, "end": 1582.6, "word": " at", "probability": 0.9638671875}, {"start": 1582.6, "end": 1583.38, "word": " C", "probability": 0.8583984375}, {"start": 1583.38, "end": 1584.0, "word": " element", "probability": 0.83935546875}, {"start": 1584.0, "end": 1584.4, "word": " in", "probability": 0.59814453125}, {"start": 1584.4, "end": 1586.46, "word": " I0", "probability": 0.3023681640625}, {"start": 1586.46, "end": 1588.12, "word": " ماشي؟", "probability": 0.88427734375}], "temperature": 1.0}, {"id": 66, "seek": 160188, "start": 1597.0, "end": 1601.88, "text": "نعم نفترض إنها minimum لأن ال case تبعت ال maximum محلولة", "tokens": [1863, 25957, 8717, 5172, 2655, 43042, 36145, 11296, 7285, 5296, 33456, 2423, 1389, 6055, 3555, 34268, 2423, 6674, 3714, 5016, 1211, 12610, 3660], "avg_logprob": -0.359375, "compression_ratio": 1.011764705882353, "no_speech_prob": 2.3245811462402344e-06, "words": [{"start": 1596.9999999999998, "end": 1597.6799999999998, "word": "نعم", "probability": 0.78125}, {"start": 1597.6799999999998, "end": 1598.36, "word": " نفترض", "probability": 0.6769866943359375}, {"start": 1598.36, "end": 1598.62, "word": " إنها", "probability": 0.639404296875}, {"start": 1598.62, "end": 1599.02, "word": " minimum", "probability": 0.85498046875}, {"start": 1599.02, "end": 1599.84, "word": " لأن", "probability": 0.671630859375}, {"start": 1599.84, "end": 1599.98, "word": " ال", "probability": 0.306396484375}, {"start": 1599.98, "end": 1600.28, "word": " case", "probability": 0.810546875}, {"start": 1600.28, "end": 1600.62, "word": " تبعت", "probability": 0.76904296875}, {"start": 1600.62, "end": 1600.74, "word": " ال", "probability": 0.4853515625}, {"start": 1600.74, "end": 1601.08, "word": " maximum", "probability": 0.95703125}, {"start": 1601.08, "end": 1601.88, "word": " محلولة", "probability": 0.97841796875}], "temperature": 1.0}, {"id": 67, "seek": 162266, "start": 1602.38, "end": 1622.66, "text": "واضح؟ الان لو كان F as relative extremum بدنا نصل الان extreme يابتكون maximum يابتكون minimum case ال maximum هي مكتوبة بتجروها لحالكوا نفس الفكرة ده أكتف بت mean F as a relative H minimum", "tokens": [2407, 46958, 5016, 22807, 2423, 7649, 45164, 25961, 479, 382, 4972, 4040, 449, 47525, 8315, 8717, 36520, 2423, 7649, 8084, 7251, 16758, 2655, 30544, 6674, 7251, 16758, 2655, 30544, 7285, 1389, 2423, 6674, 39896, 3714, 4117, 2655, 37746, 3660, 39894, 7435, 32887, 11296, 5296, 5016, 6027, 4117, 14407, 8717, 36178, 27188, 4117, 25720, 11778, 3224, 5551, 4117, 2655, 5172, 39894, 914, 479, 382, 257, 4972, 389, 7285], "avg_logprob": -0.3384650792269146, "compression_ratio": 1.5549132947976878, "no_speech_prob": 2.980232238769531e-07, "words": [{"start": 1602.38, "end": 1603.62, "word": "واضح؟", "probability": 0.926513671875}, {"start": 1603.62, "end": 1604.12, "word": " الان", "probability": 0.53082275390625}, {"start": 1604.12, "end": 1605.72, "word": " لو", "probability": 0.68505859375}, {"start": 1605.72, "end": 1606.02, "word": " كان", "probability": 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minus c as x بتروح لل c برضه ايش ماله أصغر من 0 إذا في جوار there exist v delta of x و باخد ال delta اللي هي delta prime بقدر أخدها و أصغر يساوي ال delta prime by such that", "tokens": [1211, 9566, 15040, 4587, 9566, 3714, 33599, 1829, 21542, 6027, 2423, 7649, 1975, 1829, 8592, 20449, 8315, 3224, 283, 5835, 295, 269, 5551, 4117, 26890, 9154, 1958, 44945, 36178, 16247, 3794, 1211, 37746, 4948, 283, 295, 2031, 8717, 995, 4587, 9381, 283, 295, 269, 15844, 2031, 3175, 269, 382, 2031, 39894, 32887, 5016, 24976, 269, 4724, 43042, 3224, 1975, 1829, 8592, 3714, 6027, 3224, 5551, 9381, 17082, 2288, 9154, 1958, 11933, 15730, 8978, 10874, 2407, 9640, 456, 2514, 371, 8289, 295, 2031, 4032, 4724, 47283, 3215, 2423, 8289, 13672, 1829, 39896, 8289, 5835, 4724, 28543, 2288, 5551, 9778, 3215, 11296, 4032, 5551, 9381, 17082, 2288, 7251, 3794, 995, 45865, 2423, 8289, 5835, 538, 1270, 300], "avg_logprob": -0.2591594771064561, "compression_ratio": 1.598360655737705, "no_speech_prob": 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{"start": 2043.38, "end": 2043.54, "word": " من", "probability": 0.99755859375}, {"start": 2043.54, "end": 2043.86, "word": " 0", "probability": 0.876953125}, {"start": 2043.86, "end": 2044.8, "word": " إذا", "probability": 0.595703125}, {"start": 2044.8, "end": 2045.06, "word": " في", "probability": 0.86328125}, {"start": 2045.06, "end": 2045.76, "word": " جوار", "probability": 0.9202473958333334}, {"start": 2045.76, "end": 2046.58, "word": " there", "probability": 0.75439453125}, {"start": 2046.58, "end": 2047.12, "word": " exist", "probability": 0.53515625}, {"start": 2047.12, "end": 2048.22, "word": " v", "probability": 0.53564453125}, {"start": 2048.22, "end": 2048.62, "word": " delta", "probability": 0.65478515625}, {"start": 2048.62, "end": 2048.9, "word": " of", "probability": 0.9775390625}, {"start": 2048.9, "end": 2049.28, "word": " x", "probability": 0.98095703125}, {"start": 2049.28, "end": 2049.4, "word": " و", "probability": 0.55908203125}, {"start": 2049.4, "end": 2049.68, "word": " باخد", "probability": 0.9405924479166666}, {"start": 2049.68, "end": 2049.82, "word": " ال", "probability": 0.74951171875}, {"start": 2049.82, "end": 2050.08, "word": " delta", "probability": 0.529296875}, {"start": 2050.08, "end": 2050.32, "word": " اللي", "probability": 0.90673828125}, {"start": 2050.32, "end": 2050.66, "word": " هي", "probability": 0.9072265625}, {"start": 2050.66, "end": 2051.4, "word": " delta", "probability": 0.7802734375}, {"start": 2051.4, "end": 2051.94, "word": " prime", "probability": 0.8740234375}, {"start": 2051.94, "end": 2052.4, "word": " بقدر", "probability": 0.8896484375}, {"start": 2052.4, "end": 2052.82, "word": " أخدها", "probability": 0.9259033203125}, {"start": 2052.82, "end": 2053.02, "word": " و", "probability": 0.328125}, {"start": 2053.02, "end": 2053.26, "word": " أصغر", "probability": 0.9803466796875}, {"start": 2053.26, "end": 2053.5, "word": " يساوي", "probability": 0.821533203125}, {"start": 2053.5, "end": 2053.6, "word": " ال", "probability": 0.73095703125}, {"start": 2053.6, "end": 2053.74, "word": " delta", "probability": 0.86474609375}, {"start": 2053.74, "end": 2054.18, "word": " prime", "probability": 0.87158203125}, {"start": 2054.18, "end": 2055.12, "word": " by", "probability": 0.236083984375}, {"start": 2055.12, "end": 2055.56, "word": " such", "probability": 0.9716796875}, {"start": 2055.56, "end": 2055.98, "word": " that", "probability": 0.94873046875}], "temperature": 1.0}, {"id": 84, "seek": 208357, "start": 2057.23, "end": 2083.57, "text": "اللي هو f of x ناقص f of c على x minus c أصغر من مين؟ من 0 الآن اللي بيوصلني لتناقض هتلاقيها مش choosing x هنا و انا هاختارها في الجهة التانية في الجهة بتشوف كيف لأن بيقول choose x element in c و c زائد delta then", "tokens": [6027, 20292, 31439, 283, 295, 2031, 8717, 995, 4587, 9381, 283, 295, 269, 15844, 2031, 3175, 269, 5551, 9381, 17082, 2288, 9154, 3714, 9957, 22807, 9154, 1958, 6024, 48506, 13672, 1829, 4724, 1829, 2407, 36520, 22653, 5296, 2655, 8315, 4587, 11242, 8032, 2655, 15040, 38436, 11296, 37893, 10875, 2031, 34105, 4032, 1975, 8315, 8032, 47283, 2655, 9640, 11296, 8978, 25724, 3224, 3660, 16712, 7649, 10632, 8978, 25724, 3224, 3660, 39894, 8592, 38688, 9122, 33911, 5296, 33456, 4724, 1829, 39648, 2826, 2031, 4478, 294, 269, 4032, 269, 30767, 16373, 3215, 8289, 550], "avg_logprob": -0.255264942412791, "compression_ratio": 1.5233644859813085, "no_speech_prob": 1.6570091247558594e-05, "words": [{"start": 2057.23, "end": 2057.51, "word": "اللي", "probability": 0.726318359375}, {"start": 2057.51, "end": 2057.73, "word": " هو", "probability": 0.96630859375}, {"start": 2057.73, "end": 2058.01, "word": " f", "probability": 0.354248046875}, {"start": 2058.01, "end": 2058.19, "word": " of", "probability": 0.46240234375}, {"start": 2058.19, "end": 2058.65, "word": " x", "probability": 0.763671875}, {"start": 2058.65, "end": 2060.71, "word": " ناقص", "probability": 0.80755615234375}, {"start": 2060.71, "end": 2060.95, 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"probability": 0.6885986328125}, {"start": 2067.17, "end": 2067.31, "word": " اللي", "probability": 0.94287109375}, {"start": 2067.31, "end": 2067.93, "word": " بيوصلني", "probability": 0.759814453125}, {"start": 2067.93, "end": 2068.59, "word": " لتناقض", "probability": 0.9298828125}, {"start": 2068.59, "end": 2069.23, "word": " هتلاقيها", "probability": 0.87958984375}, {"start": 2069.23, "end": 2069.49, "word": " مش", "probability": 0.884765625}, {"start": 2069.49, "end": 2069.99, "word": " choosing", "probability": 0.828125}, {"start": 2069.99, "end": 2070.39, "word": " x", "probability": 0.88720703125}, {"start": 2070.39, "end": 2070.77, "word": " هنا", "probability": 0.8916015625}, {"start": 2070.77, "end": 2071.33, "word": " و", "probability": 0.94384765625}, {"start": 2071.33, "end": 2071.43, "word": " انا", "probability": 0.39508056640625}, {"start": 2071.43, "end": 2072.27, "word": " هاختارها", "probability": 0.832958984375}, {"start": 2072.27, "end": 2072.87, "word": " في", "probability": 0.9541015625}, {"start": 2072.87, "end": 2073.19, "word": " الجهة", "probability": 0.9845377604166666}, {"start": 2073.19, "end": 2073.47, "word": " التانية", "probability": 0.9500325520833334}, {"start": 2073.47, "end": 2073.59, "word": " في", "probability": 0.1221923828125}, {"start": 2073.59, "end": 2073.85, "word": " الجهة", "probability": 0.8935546875}, {"start": 2073.85, "end": 2074.47, "word": " بتشوف", "probability": 0.7584635416666666}, {"start": 2074.47, "end": 2074.79, "word": " كيف", "probability": 0.9765625}, {"start": 2074.79, "end": 2075.45, "word": " لأن", "probability": 0.6796875}, {"start": 2075.45, "end": 2075.79, "word": " بيقول", "probability": 0.8924153645833334}, {"start": 2075.79, "end": 2077.17, "word": " choose", "probability": 0.87939453125}, {"start": 2077.17, "end": 2078.69, "word": " x", "probability": 0.85302734375}, {"start": 2078.69, "end": 2079.39, "word": " element", "probability": 0.84423828125}, {"start": 2079.39, "end": 2079.77, "word": " in", "probability": 0.89208984375}, {"start": 2079.77, "end": 2080.13, "word": " c", "probability": 0.63916015625}, {"start": 2080.13, "end": 2080.37, "word": " و", "probability": 0.9326171875}, {"start": 2080.37, "end": 2080.75, "word": " c", "probability": 0.75732421875}, {"start": 2080.75, "end": 2081.41, "word": " زائد", "probability": 0.79931640625}, {"start": 2081.41, "end": 2081.83, "word": " delta", "probability": 0.3720703125}, {"start": 2081.83, "end": 2083.57, "word": " then", "probability": 0.292724609375}], "temperature": 1.0}, {"id": 85, "seek": 209719, "start": 2087.03, "end": 2097.19, "text": "هيكون عندى الآن الـ X minus C إيش ماله أكبر من 0 واضح أه؟", "tokens": [3224, 1829, 30544, 43242, 7578, 6024, 48506, 2423, 39184, 1783, 3175, 383, 11933, 1829, 8592, 3714, 6027, 3224, 5551, 4117, 26890, 9154, 1958, 4032, 46958, 5016, 5551, 3224, 22807], "avg_logprob": -0.29244790474573773, "compression_ratio": 1.0108695652173914, "no_speech_prob": 1.7881393432617188e-07, "words": [{"start": 2087.03, "end": 2087.53, "word": "هيكون", "probability": 0.7618001302083334}, {"start": 2087.53, "end": 2087.93, "word": " عندى", "probability": 0.8251953125}, {"start": 2087.93, "end": 2088.31, "word": " الآن", "probability": 0.6416015625}, {"start": 2088.31, "end": 2088.71, "word": " الـ", "probability": 0.89306640625}, {"start": 2088.71, "end": 2088.97, "word": " X", "probability": 0.488037109375}, {"start": 2088.97, "end": 2089.41, "word": " minus", "probability": 0.33203125}, {"start": 2089.41, "end": 2089.99, "word": " C", "probability": 0.95556640625}, {"start": 2089.99, "end": 2091.25, "word": " إيش", "probability": 0.6885579427083334}, {"start": 2091.25, "end": 2091.79, "word": " ماله", "probability": 0.8185221354166666}, {"start": 2091.79, "end": 2093.71, "word": " أكبر", "probability": 0.8929036458333334}, {"start": 2093.71, "end": 2093.87, "word": " من", "probability": 0.9970703125}, {"start": 2093.87, "end": 2094.13, "word": " 0", "probability": 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"end": 2562.53, "text": "طبعا حسب .. حسب الدالة اللي هو أكيد بتلاقي إذا تحققت الشروط الأقل في نقطة عندها اللي هو المماس بوازي محور السينات أي سؤال؟ ال proof ال proof برضه احنا كنا أحيانا نبرهنه في ال calculus خلينا نبرهنوها، أكيد هندرهنوها عندي", "tokens": [9566, 3555, 3615, 995, 11331, 35457, 4386, 11331, 35457, 32748, 6027, 3660, 13672, 1829, 31439, 5551, 4117, 25708, 39894, 15040, 38436, 11933, 15730, 6055, 5016, 4587, 38149, 25124, 32887, 9566, 16247, 4587, 1211, 8978, 8717, 47432, 3660, 43242, 11296, 13672, 1829, 31439, 9673, 15042, 3794, 4724, 2407, 31377, 1829, 3714, 5016, 13063, 21136, 1829, 8315, 2655, 36632, 8608, 33604, 6027, 22807, 2423, 8177, 2423, 8177, 4724, 43042, 3224, 1975, 5016, 8315, 9122, 8315, 5551, 5016, 1829, 7649, 995, 8717, 26890, 3224, 1863, 3224, 8978, 2423, 33400, 16490, 20292, 8315, 8717, 26890, 3224, 1863, 2407, 11296, 12399, 5551, 4117, 25708, 8032, 41260, 2288, 3224, 1863, 2407, 11296, 18871, 16254], "avg_logprob": -0.14936927207019352, 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"probability": 0.76904296875}, {"start": 2850.88, "end": 2851.8, "word": " برهان", "probability": 0.8477783203125}, {"start": 2851.8, "end": 2851.94, "word": " ال", "probability": 0.90087890625}, {"start": 2851.94, "end": 2852.38, "word": " roles", "probability": 0.25537109375}, {"start": 2852.38, "end": 2853.14, "word": " theory", "probability": 0.49462890625}], "temperature": 1.0}, {"id": 116, "seek": 287751, "start": 2854.07, "end": 2877.51, "text": "أي سؤال اللي بعده الآن أوصلنا لإن نقدر الآن نحكي عن اللي هو نظرية القيمة المتوسطة أو الـ Mean Value Theorem نظرية القيمة المتوسطة أو الـ Mean Value Theorem برضه", "tokens": [10721, 1829, 8608, 33604, 6027, 13672, 1829, 39182, 3224, 6024, 48506, 34051, 36520, 8315, 5296, 28814, 1863, 8717, 28543, 2288, 6024, 48506, 8717, 5016, 4117, 1829, 18871, 13672, 1829, 31439, 8717, 19913, 2288, 10632, 25062, 32640, 3660, 9673, 2655, 41779, 9566, 3660, 34051, 2423, 39184, 12302, 39352, 440, 37956, 8717, 19913, 2288, 10632, 25062, 32640, 3660, 9673, 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3257.15, "end": 3257.49, "word": " طريق", "probability": 0.9949544270833334}, {"start": 3257.49, "end": 3257.89, "word": " الـ", "probability": 0.48779296875}, {"start": 3257.89, "end": 3257.89, "word": " Mean", "probability": 0.454345703125}, {"start": 3257.89, "end": 3259.81, "word": " Value", "probability": 0.95458984375}, {"start": 3259.81, "end": 3260.63, "word": " Theorem", "probability": 0.84033203125}, {"start": 3260.63, "end": 3262.73, "word": " النظرية", "probability": 0.95751953125}, {"start": 3262.73, "end": 3262.83, "word": " اللي", "probability": 0.98486328125}, {"start": 3262.83, "end": 3263.21, "word": " بتقول", "probability": 0.980224609375}, {"start": 3263.21, "end": 3263.57, "word": " ليه", "probability": 0.794189453125}], "temperature": 1.0}, {"id": 132, "seek": 328954, "start": 3264.16, "end": 3289.54, "text": "اللي هي اللي باختصار بنقول أنه لو كانت المشتقة الأولى لدالة بتساوي سفر إذن هي اللي هتكون مين هي المشتقة الثابتة طيب نشوف بالظبط أيش اللي بقوله بقولوا .. نفترض أن F continuous F عبارة عن دالة من A و B", "tokens": [6027, 20292, 39896, 13672, 1829, 4724, 47283, 2655, 9381, 9640, 44945, 39648, 14739, 3224, 45164, 25961, 2655, 9673, 8592, 2655, 28671, 16247, 12610, 7578, 5296, 3215, 6027, 3660, 39894, 3794, 995, 45865, 8608, 5172, 2288, 11933, 8848, 1863, 39896, 13672, 1829, 8032, 2655, 30544, 3714, 9957, 39896, 9673, 8592, 2655, 28671, 6024, 104, 16758, 2655, 3660, 23032, 1829, 3555, 8717, 8592, 38688, 20666, 19913, 3555, 9566, 36632, 8592, 13672, 1829, 4724, 39648, 3224, 4724, 39648, 14407, 4386, 8717, 5172, 2655, 43042, 14739, 479, 10957, 479, 6225, 3555, 9640, 3660, 18871, 11778, 6027, 3660, 9154, 316, 4032, 363], "avg_logprob": -0.18016581784705726, "compression_ratio": 1.71, "no_speech_prob": 1.6689300537109375e-06, "words": [{"start": 3264.16, "end": 3264.48, "word": "اللي", "probability": 0.880615234375}, {"start": 3264.48, "end": 3264.62, "word": " هي", "probability": 0.5859375}, {"start": 3264.62, "end": 3264.82, "word": " اللي", 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0.91552734375}, {"start": 3532.65, "end": 3532.99, "word": " I,", "probability": 0.9775390625}, {"start": 3533.07, "end": 3533.19, "word": " A", "probability": 0.82568359375}, {"start": 3533.19, "end": 3533.51, "word": " وB", "probability": 0.6767578125}, {"start": 3533.51, "end": 3534.53, "word": " then", "probability": 0.73046875}, {"start": 3534.53, "end": 3536.53, "word": " that", "probability": 0.82861328125}, {"start": 3536.53, "end": 3536.97, "word": " they", "probability": 0.9404296875}, {"start": 3536.97, "end": 3537.27, "word": " are", "probability": 0.92333984375}, {"start": 3537.27, "end": 3537.95, "word": " differentiable", "probability": 0.9404296875}, {"start": 3537.95, "end": 3538.19, "word": " on", "probability": 0.82666015625}, {"start": 3538.19, "end": 3538.41, "word": " A", "probability": 0.9755859375}, {"start": 3538.41, "end": 3538.59, "word": " وB", "probability": 0.939208984375}, {"start": 3538.59, "end": 3538.81, "word": " يعني", "probability": 0.82861328125}, {"start": 3538.81, "end": 3539.43, "word": " نفترض", "probability": 0.9796142578125}, {"start": 3539.43, "end": 3540.19, "word": " أن", "probability": 0.5673828125}, {"start": 3540.19, "end": 3541.19, "word": " الدالتين", "probability": 0.724578857421875}, {"start": 3541.19, "end": 3541.39, "word": " F", "probability": 0.9296875}, {"start": 3541.39, "end": 3541.79, "word": " وG", "probability": 0.854248046875}, {"start": 3541.79, "end": 3542.99, "word": " continuous", "probability": 0.92333984375}, {"start": 3542.99, "end": 3543.23, "word": " على", "probability": 0.60791015625}, {"start": 3543.23, "end": 3543.33, "word": " ال", "probability": 0.783203125}, {"start": 3543.33, "end": 3543.69, "word": " closed", "probability": 0.380615234375}, {"start": 3543.69, "end": 3543.97, "word": " اللي", "probability": 0.89501953125}, {"start": 3543.97, "end": 3544.15, "word": " هي", "probability": 0.78759765625}, {"start": 3544.15, "end": 3545.23, "word": " bounded", "probability": 0.525390625}, {"start": 3545.23, "end": 3545.69, "word": " interval", "probability": 0.984375}, {"start": 3545.69, "end": 3545.97, "word": " A", "probability": 0.90576171875}, {"start": 3545.97, "end": 3546.29, "word": " وB", "probability": 0.96044921875}, {"start": 3546.29, "end": 3547.99, "word": " وdifferentiable", "probability": 0.86572265625}, {"start": 3547.99, "end": 3548.15, "word": " على", "probability": 0.8857421875}, {"start": 3548.15, "end": 3548.27, "word": " ال", "probability": 0.7626953125}, {"start": 3548.27, "end": 3548.43, "word": " open", "probability": 0.78662109375}, {"start": 3548.43, "end": 3548.91, "word": " interval", "probability": 0.97802734375}, {"start": 3548.91, "end": 3549.19, "word": " A", "probability": 0.9853515625}, {"start": 3549.19, "end": 3549.43, "word": " وB", "probability": 0.992431640625}], "temperature": 1.0}, {"id": 142, "seek": 357533, "start": 3550.11, "end": 3575.33, "text": "والـ derivative للأولى بساوي ال derivative للثانية عند كل نقطة من النقاط الـ A و الـ B إذا there exists a constant C such that F بتساوي G زائد H زائد C طبعا البرهان سهل و أكيد يمكن بعضكم برهنه في راسه let H بتساوي F ناقص G مظبوط؟ الآن then", "tokens": [2407, 6027, 39184, 13760, 24976, 10721, 12610, 7578, 4724, 3794, 995, 45865, 2423, 13760, 24976, 12984, 7649, 10632, 43242, 28242, 8717, 47432, 3660, 9154, 28239, 4587, 41193, 2423, 39184, 316, 4032, 2423, 39184, 363, 11933, 15730, 456, 8198, 257, 5754, 383, 1270, 300, 479, 4724, 2655, 3794, 995, 45865, 460, 30767, 16373, 3215, 389, 30767, 16373, 3215, 383, 23032, 3555, 3615, 995, 2423, 26890, 3224, 7649, 8608, 3224, 1211, 4032, 5551, 4117, 25708, 7251, 43020, 45030, 11242, 24793, 4724, 2288, 3224, 1863, 3224, 8978, 12602, 32277, 3224, 718, 389, 4724, 2655, 3794, 995, 45865, 479, 8717, 995, 4587, 9381, 460, 3714, 19913, 3555, 2407, 9566, 22807, 6024, 48506, 550], "avg_logprob": -0.2410511404275894, "compression_ratio": 1.5381355932203389, "no_speech_prob": 5.364418029785156e-07, "words": [{"start": 3550.11, "end": 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3554.03, "end": 3554.25, "word": " الـ", "probability": 0.701171875}, {"start": 3554.25, "end": 3554.35, "word": " A", "probability": 0.33251953125}, {"start": 3554.35, "end": 3554.49, "word": " و", "probability": 0.66357421875}, {"start": 3554.49, "end": 3554.61, "word": " الـ", "probability": 0.69921875}, {"start": 3554.61, "end": 3554.87, "word": " B", "probability": 0.78369140625}, {"start": 3554.87, "end": 3555.69, "word": " إذا", "probability": 0.719970703125}, {"start": 3555.69, "end": 3556.25, "word": " there", "probability": 0.5966796875}, {"start": 3556.25, "end": 3556.77, "word": " exists", "probability": 0.5546875}, {"start": 3556.77, "end": 3556.95, "word": " a", "probability": 0.98974609375}, {"start": 3556.95, "end": 3557.37, "word": " constant", "probability": 0.96533203125}, {"start": 3557.37, "end": 3557.65, "word": " C", "probability": 0.8232421875}, {"start": 3557.65, "end": 3557.91, "word": " such", "probability": 0.84375}, {"start": 3557.91, "end": 3558.19, "word": " that", "probability": 0.9580078125}, {"start": 3558.19, "end": 3558.37, "word": " F", "probability": 0.7548828125}, {"start": 3558.37, "end": 3558.73, "word": " بتساوي", "probability": 0.7263671875}, {"start": 3558.73, "end": 3558.91, "word": " G", "probability": 0.7119140625}, {"start": 3558.91, "end": 3559.29, "word": " زائد", "probability": 0.7783203125}, {"start": 3559.29, "end": 3559.73, "word": " H", "probability": 0.26220703125}, {"start": 3559.73, "end": 3560.61, "word": " زائد", "probability": 0.9192708333333334}, {"start": 3560.61, "end": 3560.85, "word": " C", "probability": 0.9384765625}, {"start": 3560.85, "end": 3561.73, "word": " طبعا", "probability": 0.969482421875}, {"start": 3561.73, "end": 3562.77, "word": " البرهان", "probability": 0.8865966796875}, {"start": 3562.77, "end": 3563.47, "word": " سهل", "probability": 0.9977213541666666}, {"start": 3563.47, "end": 3564.27, "word": " و", "probability": 0.5087890625}, {"start": 3564.27, "end": 3564.59, "word": " أكيد", "probability": 0.8751627604166666}, {"start": 3564.59, "end": 3564.89, "word": " يمكن", "probability": 0.940673828125}, {"start": 3564.89, "end": 3565.35, "word": " بعضكم", "probability": 0.9734700520833334}, {"start": 3565.35, "end": 3565.79, "word": " برهنه", "probability": 0.777685546875}, {"start": 3565.79, "end": 3565.89, "word": " في", "probability": 0.7919921875}, {"start": 3565.89, "end": 3566.33, "word": " راسه", "probability": 0.87353515625}, {"start": 3566.33, "end": 3567.47, "word": " let", "probability": 0.67529296875}, {"start": 3567.47, "end": 3568.93, "word": " H", "probability": 0.8330078125}, {"start": 3568.93, "end": 3569.55, "word": " بتساوي", "probability": 0.847021484375}, {"start": 3569.55, "end": 3570.07, "word": " F", "probability": 0.98974609375}, {"start": 3570.07, "end": 3571.47, "word": " ناقص", "probability": 0.9217529296875}, {"start": 3571.47, "end": 3571.73, "word": " G", "probability": 0.9853515625}, {"start": 3571.73, "end": 3573.71, "word": " مظبوط؟", "probability": 0.7209879557291666}, {"start": 3573.71, "end": 3574.39, "word": " الآن", "probability": 0.6629638671875}, {"start": 3574.39, "end": 3575.33, "word": " then", "probability": 0.6259765625}], "temperature": 1.0}, {"id": 143, "seek": 359521, "start": 3576.67, "end": 3595.21, "text": "F و G continuous إذن H is continuous on A و B و أيضًا F و G differentiable على A و B and H is differentiable on A و B moreover", "tokens": [37, 4032, 460, 10957, 11933, 8848, 1863, 389, 307, 10957, 322, 316, 4032, 363, 4032, 36632, 11242, 14111, 995, 479, 4032, 460, 819, 9364, 15844, 316, 4032, 363, 293, 389, 307, 819, 9364, 322, 316, 4032, 363, 544, 3570], "avg_logprob": -0.2289062563329935, "compression_ratio": 1.3883495145631068, "no_speech_prob": 2.384185791015625e-07, "words": [{"start": 3576.67, "end": 3577.07, "word": "F", "probability": 0.459228515625}, {"start": 3577.07, "end": 3577.27, "word": " و", "probability": 0.79638671875}, {"start": 3577.27, "end": 3577.37, "word": " G", "probability": 0.62841796875}, {"start": 3577.37, "end": 3578.11, "word": " continuous", "probability": 0.6259765625}, {"start": 3578.11, "end": 3578.57, "word": " إذن", "probability": 0.5338541666666666}, {"start": 3578.57, "end": 3578.97, "word": " H", "probability": 0.87353515625}, {"start": 3578.97, "end": 3579.53, "word": " is", "probability": 0.7978515625}, {"start": 3579.53, "end": 3581.11, "word": " continuous", "probability": 0.9150390625}, {"start": 3581.11, "end": 3582.35, "word": " on", "probability": 0.91162109375}, {"start": 3582.35, "end": 3582.89, "word": " A", "probability": 0.91357421875}, {"start": 3582.89, "end": 3583.27, "word": " و", "probability": 0.95556640625}, {"start": 3583.27, "end": 3583.53, "word": " B", "probability": 0.517578125}, {"start": 3583.53, "end": 3584.47, "word": " و", "probability": 0.69091796875}, {"start": 3584.47, "end": 3585.05, "word": " أيضًا", "probability": 0.8682861328125}, {"start": 3585.05, "end": 3585.65, "word": " F", "probability": 0.88916015625}, {"start": 3585.65, "end": 3585.83, "word": " و", "probability": 0.98095703125}, {"start": 3585.83, "end": 3585.93, "word": " G", "probability": 0.95654296875}, {"start": 3585.93, "end": 3586.83, "word": " differentiable", "probability": 0.859375}, {"start": 3586.83, "end": 3587.23, "word": " على", "probability": 0.73388671875}, {"start": 3587.23, "end": 3587.43, "word": " A", "probability": 0.8828125}, {"start": 3587.43, "end": 3587.51, "word": " و", "probability": 0.98583984375}, {"start": 3587.51, "end": 3587.83, "word": " B", "probability": 0.814453125}, {"start": 3587.83, "end": 3588.93, "word": " and", "probability": 0.57373046875}, {"start": 3588.93, "end": 3590.17, "word": " H", "probability": 0.9560546875}, {"start": 3590.17, "end": 3591.21, "word": " is", "probability": 0.96533203125}, {"start": 3591.21, "end": 3592.23, "word": " differentiable", "probability": 0.973388671875}, {"start": 3592.23, "end": 3593.41, "word": " on", "probability": 0.94873046875}, {"start": 3593.41, "end": 3593.91, "word": " A", "probability": 0.99169921875}, {"start": 3593.91, "end": 3594.07, "word": " و", "probability": 0.99072265625}, {"start": 3594.07, "end": 3594.27, "word": " B", "probability": 0.87158203125}, {"start": 3594.27, "end": 3595.21, "word": " moreover", "probability": 0.827392578125}], "temperature": 1.0}, {"id": 144, "seek": 362386, "start": 3597.7, "end": 3623.86, "text": "عندي H برايم اللي هو بساوي F برايم نقص G برايم طبعا عند ال X عند ال X of X of X of X بساوي صفر لكل X موجودة في ال A و ال B then by the above theorem اللي بنحكي عنها اللي هي theorem by theorem 625 H", "tokens": [3615, 1863, 16254, 389, 4724, 23557, 32640, 13672, 1829, 31439, 4724, 3794, 995, 45865, 479, 4724, 23557, 32640, 8717, 4587, 9381, 460, 4724, 23557, 32640, 23032, 3555, 3615, 995, 43242, 2423, 1783, 43242, 2423, 1783, 295, 1783, 295, 1783, 295, 1783, 4724, 3794, 995, 45865, 20328, 5172, 2288, 5296, 28820, 1783, 3714, 29245, 23328, 3660, 8978, 2423, 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