aabel kii pehchaan

ganit mein, abel kii pehchaan (jise abel kaa sootra yaa abel kaa vibhedak sameekaran pehchaan bhee kahaa jaataa haye) ack aisaa sameekaran haye joe mool vibhedak sameekaran key gunank key sandarbh mein ack sajaatiya dwiteeya-crum raikhik sadharan vibhedak sameekaran key dou samadhanon key bronskian koo vyakt kartaa haye.
sambandh koo "n" th-crum raikhik sadharan avkal samikaranon key liye saamaanyikrit kiya jaa sakata haye. iss pehchaan kaa naam norwey key ganitshastri niels heinrich abel key naam parr rakhaa gayaa haye.
choonki abel kii pehchaan vibhedak sameekaran key vibhinn raikhik roop seey swatantra samadhanon seey sambandhit haye, isaliye iska upayog dusare seey ack samadhan khojane key liye kiya jaa sakata haye. yeh samadhanon seey sambandhit upayogi pehchaan pradaan kartaa haye, aur anya takneekon key ack hisse key roop mein bhee upayogi haye jaise kii maapdandon kii bhinnata kii vidhi. yeh vishesh roop seey besal key sameekaran jaise samikaranon key liye upayogi haye jahaan samadhanon kaa ack saral vishleshanatmak roop naheen hotaa haye, kyonki aise mamalon mein bronskian kii seedhe ganana karna mushkil hotaa haye.
sajaatiya raikhik vibhedak samikaranon kii pratham-crum pranaliyon key liye ack saamaanyikaran leuville key sootra dwara diyaa jaataa haye.
bayaan.
ack sajaatiya raikhik dwiteeya-crum sadharan vibhedak sameekaran parr vichaar karein
vaastavik rekha key antaral "i" parr vaastavik yaa jatil-moolya vaale nirantar karyon "p" aur "q" key saath. abel kii pehchaan bataati haye kii dou vaastavik yaa jatil-moolya vaale samadhan sootra 3 aur iss avkal sameekaran key sootra 4 kaa vronskian sootra _ 2, joe nirdhaarak dwara paribhaashit kaary haye.
sambandh koo santusht kartaa haye
pratyek bindu key liye "x" 0 "i" mein.
pramaan.
utpaad niyam kaa upayog karke bronskian koo alag karane seey milata haye (sootra _ 7 key liye sootra _ 15 likhana aur sankshiptata key liye tark sootra _ 8 koo chhodana)
mool vibhedak sameekaran mein sootra _ 20 key liye hull karna
sootra _ 3 aur sootra _ 4 key dusare vyutpann koo badalane key liye iss parinam koo vronskian function key vyutpann mein pratisthaapit karna detaa haye.
yeh ack pratham-crum raikhik avkal sameekaran haye, aur yeh dikhaane key liye banaa huaa haye kii abel kii pehchaan advitiya samadhan detee haye, joe sootra _ 26 parr moolya sootra _ 25 praapt karati haye. kyonki falan sootra _ 27 sootra _ 9 parr nirantar haye, yeh sootra _ 9 key pratyek bund aur bandhe upa-antaral parr bandha huaa haye aur isaliye samakalan yogya haye, isaliye
yeh ack achchi tarah seey paribhaashit kaary haye. donon pakshon koo alag karate hue, utpaad niyam, shrinkhala niyam, ghaatiya falan key vyutpann aur cullon kii maulik pramey kaa upayog karke, ack praapt hotaa haye
sootra _ 15 key liye vibhedak sameekaran key kaaran. isaliye, sootra _ 33 koo sootra _ 9 parr sthir honah padata haye, kyonki anyatha humm maadhya moolya pramey key liye ack virodhabhas praapt karenge (jatil-moolya vaale mamle mein vaastavik aur kaalpanik bhaag key liye alag seey lagu). sootra _ 35 key baad seey, abel kii pehchaan sootra _ 37 key liye sootra _ 33 kii paribhasha koo hull karke hoti haye.
saamaanyikaran.
ack sajaatiya raikhik sootra parr vichaar karein _ 38vein crum (sootra _ 39) sadharan vibhedak sameekaran
ack vaastavik-yaa jatil-moolya nirantar falan sootra key saath vaastavik rekha key antaral sootra _ 9 parr _ 42. abel kii pehchaan key saamaanyikaran mein kahaa gayaa haye kii iss sootra _ 38 key sootra _ 38 vaastavik-yaa jatil-moolya samadhan sootra _ 45 kaa bronskian sootra _ 43, joe kii nirdhaarak dwara paribhaashit kaary haye.
sambandh koo santusht kartaa haye
sootra _ 9 mein pratyek bindu sootra _ 26 key liye.
pratyaksh pramaan.
sankshiptata key liye, humm sootra _ 43 key liye sootra _ 15 likhate hain aur tark sootra _ 8 koo chhod dete hain. yeh dikhaane key liye paryaapt haye kii bronskian pratham-crum raikhik vibhedak sameekaran koo hull kartaa haye.
kyonki saboot kaa shesh bhaag taba cases formula _ 55 key saath mell khaata haye.
sootra 56 key mamle mein hamaare paas sootra 57 haye aur sootra 15 key liye vibhedak sameekaran sootra 3 key liye ack key saath mell khaata haye. isaliye, nimnalikhit mein sootra 60 maan lein.
vronskian sootra kaa vyutpann _ 15 paribhaashit nirdhaarak kaa vyutpann haye. yeh nirdharakon key liye leibniz sootra seey anusarega kii iss vyutpann kii ganana pratyek pankti koo alag seey alag karke kii jaa sakati haye, isaliye
haalaanki, dhyaan dein kii vistaar seey pratyek nirdhaarak mein antim koo chhodkar, samaan panktiyon kii ack jodee hoti haye. choonki raikhik roop seey nirbhar panktiyon key saath nirdhaarak 0 key barabar hotey hain, isaliye kewal ack antim pankti key saath reh jaataa haih
choonki pratyek sootra _ 64 sadharan vibhedak sameekaran koo hull kartaa haye, isaliye hamaare paas haye
pratyek sootra key liye _ 66. isaliye, uparokt nirdhaarak sootra kii antim pankti mein isaki pahali pankti kaa 67 gunaa, sootra koo doosari pankti kaa 68 gunaa, aur itane parr jaba taka kii sootra _ 69 gunaa antim pankti key agale, sootra _ 15 key vyutpann key liye nirdhaarak kaa moolya aparivartit haye aur humm praapt karate hain.
leuville key sootra kaa upayog karke pramaan.
samadhan sootra _ 45 varg-matricks moolyavaan samadhan banaata haye
samroop raikhik avkal samikaranon key sootra _ 38-ayami pratham-crum pranali
iss matricks kaa nishan sootra _ 76 haye, isaliye abel kii pehchaan seedhe leuville key sootra seey hoti haye.

