
projecteevaayi vipuleekaricha riyal line

yadhaartha vishakalanathil, projecteevaayi extended riyal line (riyal laininte vann-pooyint compactification ennum vilikkunnu), soochippicha oru pooyint upayogichu formula _ 1 enna riyal nambarukalude koottatthinte extensionaanu. athinaal ithu saadhyamaakunnidathellaam vipuleekaricha staanderd ganitha pravarthanangalulla settu formula _ 2 aanu, chilappol formula _ 3 allengil formula _ 4 kondu soochippikkunnu, cherttha pooyintine ananthathayile pooyint ennu vilikkunnu, kaaranam ithu yadhaartha variyude iruvashathinteyum ayalkkaaranaayi kanakkaakkappedunnu. kooduthal krithyamaayi paranjaal, ananthathayile pooyint ennathu paramamaya moolyangal vardhikkukayum paridhiyillaathavayumaaya yadhaartha samkhyakalude oro shreniyudeyum paridhiyaan.
projecteevaayi vipuleekaricha riyal lainine oru yadhaartha projective line upayogichu thirichariyaan kazhiyum, athil moonnu pooyintukal nirddishta moolyangal nalkiyittundu, koodaathe. projecteevaayi extended riyal nambar line, extended riyal nambar lainil ninnu vyathyasthamaan, athil avaa vyathyasthamaan.
poojyathaal harikkuka.
samkhyakalude mikka ganithashaasthra maathrkakalil ninnum vyathyasthamayi, eee ghadana poojyathaal vibhajikkaan anuvadikkunnu
poojyamallaatha "a". prathyekichum, eee ghadanayil reciprocal functione oru motham functionakkunnu. ennirunnaalum, ghadana oru feeld allah, koodaathe bainari ganitha pravarthanangalonnum motham allah-udaaharanathinu, parasparabandham motham aanenkilum nirvachikkappettitla. ennirunnaalum ithinu upayogayogyamaaya vyaakhyaanangalundu-udaaharanathinu, jyamithiyil, oru lambarekhayude charivu.
yadhaartha variyude vipuleekaranangal.
riman golam sankeernna samkhyakalude feeld vyapippikkunnathupole, parambaraagathamaayi vilikkunna orotta pooyint cherthu, projecteevaayi vipuleekaricha riyal line riyal nambarukalude feeld vipuleekarikkunnu.
ithinu vipareethamaayi, vipuleekaricha yadhaartha samkhya rekha (yadhaartha rekhayude rund pooyint compactification ennum vilikkunnu) aandu thammilulla vyathyasam vyakthamaakkunnu.
oordar.
oordar bandham formula _ 6 lake ardhavathaaya reethiyil vyaapippikkaan kazhiyilla. oru samkhya nalkiyaal, randennam nirvachikkaan bodhyappeduthunna vaadangalonnumilla. mattu gadakangalumaayi thaarathamyam cheyyaan kazhiyaathathinal, eee bandham formula 6-ll nilanirthunnathil ardhamilla. ennirunnaalum, formula 6-leeii nirvachanagalil formula 1-leeii kramam upayogikkunnu.
jyamiti.
mattethoru pooyintil ninnum vyathyasthamalla" enna aashayathinte adisthaanamaanu yadhaartha projective line oru ekeekritha sthalamaaya reethi, vaasthavathil oru vrithathinu homomorphic aanu. udaaharanathinu, 2x2 riyal invertible maatriksukalude general leanier groupil oru traansitteevu aakshan undu. leanier fraakshanal traansformeshante denominator aayirikkumbol, chithram mobius transformation (leanier fraakshanal transformation ennum vilikkunnu) aayirikkam group aakshan prakadippikkunnathu.
pravartthanatthinte vishadamaaya vishakalanam kaanikkunnath, eathenkilum moonnu vyathyastha pooyintukalaaya "p", "q", "r" ennivaykku, "p" muthal 0 vareyum "q" muthal 1 vareyum "r" vareyum edukkunna oru leanier fraakshanal parivartthanam undu, athaayath, leanier fraakshanal parivartthanangalude group yadhaartha projective lainil moonnu madangu traansitteevu aanu. cross anupatham maattamillaathathinaal ithu 4-tupple poyintukalilekku vyaapippikkaan kazhiyilla.
padavali projective line uchithamaan, kaaranam pooyintukal formula _ 10-nte oru dimensional leanier subspaceukalumaayi 1-muthal-1 vare yojikkunnu.
ganithashaasthraparamaaya pravartthanangal.
ganithashaasthra pravarthanangalkkulla prachodanam.
eee sthalathe ganitha pravartthanangal realsil athe pravartthanangalude vipuleekaranamaanu. puthiya nirvachanagalkkulla prachodanam yadhaartha samkhyakalude pravartthanangalude paridhikalaan.
nirvachikkapetta ganitha pravartthanangal.
formula _ 6-nte subset formula _ 1-leeii staanderd pravarthanangalkku purame, soochippichathupole ozhivaakkalukalode formula _ 13-naayi inipparayunna pravartthanangal nirvachikkappedunnu
nirvachikkappedaathe avasheshikkunna ganitha pravartthanangal.
yadhaartha pravartthanangalude paridhikal pariganikkunnathiloode inipparayunna padaprayogangal prachodippikkaan kazhiyilla, koodaathe avayude oru nirvachanavum ellah nirvachikkapetta kesukalilum staanderd beejaganita gunangalude prasthavana roopathil maattamillathe nilanirthaan anuvadikkunnilla.</kanakk>, staanderd niyamangal pariharikkuka wheel sidhaantham kaanuka.}} thathfalamaayi, avaa nirvachikkappedaathe avasheshikkunnu
exponential function formula _ 16 formula _ 6 lake vipuleekarikkaan kazhiyilla.
beejaganita gunangal.
thaazhepparayunna samathvangal ardhamaakkunnathu "onnukil iru vashangalum nirvachikkappettitla, allengil iru vashangalum nirvachikkappedukayum thulyavumaanu."eathu soothravaakyathinum ithu sariyaanu _ 18
eathenkilum soothravaakyathinu, ulppettirikkunna padaprayogangal nirvachikkumbozhellaam inipparayunnava sariyaanu _ 18
pothuve, formula _ 1 nuu saadhuthayulla ganithatthinte ellah niyamangalum formula _ 6 nuu saadhuthayullavayaanu, sambhavikkunna ellah padaprayogangalum nirvachikkappedumbozhale.
edavelakalum topology.
oru edavela enna ashayam formula _ 6 lake vipuleekarikkaam. ennirunnaalum, ithu oru oordar cheytha settallathathinaal, edavelaykku alpam vyathyasthamaaya ardhamundu. adacha edavelakalkulla nirvachanangal inipparayunnavayaanu (ithu anumaanikkappedunnu 
soothravaakyam _ 25):
end pooyintukal thulyamaakumbol ozhike, athath end pooyintukal neekkam cheythaanu anubandha ooppan, haff ooppan edavelakal nirvachikkunnath. 0 adangiya edavela kondu harikkumbol edavela ganithathil eee punarnirvacanam upayogapradamaanu.
oru adithara enna nilayil thuranna edavelakal formula _ 6-ll oru topology nirvachikkunnu. oru aditharaykku mathiyakum formula _ 1-leeii parimithamaaya thuranna edavelakalum formula _ 37-leeii ellah formulakalkkum edavela formula _ 36 umm formula _ 38 poleyaanu.
paranjathupole, topology oru vrithathinu homomorphic aanu. athinaal ithu eee vrithathile saadhaarana metrik (nere allengil vrithathil alakkunnu) anusarichulla (oru nischitha homeomorphisathinu) alakkaavunna alavaanu. formula _ 39-ll saadhaarana metrik-nte vipuleekaranamaaya metrik illa.
edavela ganithashaasthram.
edavela ganithashaasthram formula 1 ll ninnu formula 6 lake vyaapikkunnu. oru bainari pravarthanathodulla edavelakalil poruthappedaattha moolyangal adangumbol ozhike, edavelakalil oru ganitha pravartthanatthinte phalam ellaayppozhum oru edavelayaanu, ithu nirvachikkappedaatha falatthilekku nayikkunnu. prathyekichum, oro soothravaakyathinum namukk undu
rund edavelakalilum ulppedunnundo ennathu pariganikkathe thanne.
calculus.
formula _ 6-nte pravartthanangal vishakalanam cheyyaan kaalkkulasinte upakaranangal upayogikkaam. nirvachanangal eee sthalathinte topologyil ninnu prachodanam ulkkondathaanu.
ayalpakkangal.
formula _ 45 umm formula _ 46 umm aakatte.
parimithikal.
parimithikalude adisthaana nirvachanangal.
formula _ 53 formula _ 54 umm formula _ 55 umm aakatte.
f" & "hairsp; (" x ") ennathinte paridhi" p "sameepikkumbol" l "ennaanu soochippikkunnathu.
l" nte oro ayalpakkathinum "a" ennathinu "p" yude panchar cheytha ayalpakkamaaya "b" undengil maathram, aaa formula _ 57 formula _ 58 soochippikkunnu.
valathuvashath (idathu) ninnu "x" "p" neey sameepikkumbol "f" & hairsp; ("x") nte ekapaksheeya paridhi "l" ennaanu soochippikkunnathu.
l" nte oro ayalpakkathinum "a" enna soothravaakyam soochippikkunnathupole "p" nte valathuvashathulla (idathu vashamulla) panchar cheytha ayalpakkamaaya "b" undengil maathram
formula 63 umm formula 64 umm undengil maathrame aaa formula 56 kaanikkan kazhiyoo.
formula _ 1 leeii paridhikalumayi thaarathamyam cheyyuka.
mukalil nalkiyirikkunna nirvachanangal yadhaartha pravartthanangalude paridhikalude saadhaarana nirvachanangalumayi thaarathamyam cheyyaam. thaazhepparayunna prasthaavanakalil, formula _ 66-ll aadya paridhi mukalil nirvachichathupole aanu, randaamathe paridhi saadhaarana ardhathilaanu
parimithikalude vipuleekritha nirvachanam.
soothravaakyam _ 46 ennu karuthuka. appol "p" ennathu "a" yude oru paridhi poyintaanu, "p" yude oro ayalpakkavum oru pooyint formula _ 80 ulppeduthiyaal maathram, athaayath soothravaakyam _ 81.
soothravaakyam _ 82, "p" ennathu "a" yude oru paridhi pooyint ayirikkatte. f" & "hairsp; (" x ") ennathinte paridhi" a "ennathiloode" p "ennathilekku adukkumbol" l "aanu," l "ennathinte oro ayalpakkathinum" b "ennathinte panchar cheytha ayalpakkamaaya" c "undengil maathram, athaayath formula _ 83 formula _ 84 soochippikkunnu.
ithu formula _ 85 leeii subspace topologyilum formula _ 86 lekkulla "f" niyanthranatthilum prayogikkunna thudarchayude pathiv topological nirvachanavumaayi poruthappedunnu.
thudarcha.
pravartthanam
ivide nirvachichittundenkil maathram thudarchayaayirikkum
soothravaakyam _ 89 aanenkil function
oro soothravaakyathinum _ 91 ll nirvachichittundenkil maathrame thudarchayulloo, koodaathe formulayude paridhi _ 92 formula _ 93 aanu
polynomialukalaaya oro yukthisahamaya function, formula 6 muthal formula 6 vareyulla oru functionilekku savisheshamaaya reethiyil neettaan kazhiyum, athu formula 4 ll thudarchayaayi thudarunnu, ithu polynomial functionukalude kaaryamaan, avaa formula 98 ll moolya formula 97 edukkunnu.
koodaathe, taanjant function formula _ 99 vipuleekarichaal
appol formula 99 formula 102 ll thudarchayaayirikkunnu, pakshe formula 4 ll thudarchayaaya oru functionilekku kooduthal neettaan kazhiyilla.
104 enna soothravaakyathil thudarchayaayirikkunna niravadhi praadhamika functionukal 105 enna soothravaakyathil thudarchayaaya functionukalilekku neettaan kazhiyilla, udaaharanathinu, exponential functionteyum ellah trikonamitheeya functionukaludeyum kaaryamaanith. udaaharanathinu, saine function formula _ 102 ll thudarchayaayirikkunnu, pakshe mukalil kaanunnathupole formula _ 107 ll thudarchayaayi cheyyaan kazhiyilla, taanjant function formula _ 102 ll thudarchayaaya oru functionilekku neettaan kazhiyum, ennaal eee function formula _ 107 ll thudarchayaayi nadatthaan kazhiyilla.
codomaine formula _ 6 lake vipuleekarikkumbol thudarchayaayi maarunna niravadhi discondinues functionukal, codomaine afini extended riyal nambar sisttam formula _ 111 lake vipuleekarichaal discondinues aayi thudarunnu, ithu maruvashathu function formula _ 112 nte kaaryamaan, formula _ 104 ll thudarchayaayi pravartthikkunna chila functionukal, domaine formula _ 111 lake vipuleekarichaal formula _ 114 ll discondinues cheiyunnathu thudarchayaayi maarunnu.
oru projective rench enna nilayil.
yadhaartha projective line yadhaartha projective plaininte pashchaathalatthil pariganikkumbol, disargues sidhaanthathinte anantharafalangal avyakthamaanu. prathyekichum, pooyintukal thammilulla projective harmonic konjugettu bandhatthinte nirmmaanam yadhaartha projective laininte gadanayude bhagamaanu. udaaharanathinu, eathenkilum jody pooyintukal nalkiyaal, ananthathayile pooyint avayude madhyabinduvinte projective harmonic konjugettaanu.
projectivittikal harmonic bandham samrakshikkunnathinaal, avaa yadhaartha projective laininte automorphisangal roopappeduthunnu. oru valayathinu mukalilulla oru projective laininte pothuvaaya nirmmaanamanusarichu, yadhaartha samkhyakal oru valayam roopappeduthunnathinaal projectivittikale beejaganithaparamaayi homographikalaayi vivarikkunnu. avar koottaayi pii. jii. ell (2, aarr) enna group roopeekarikkunnu.
swantham vipareethangalaaya projectivittikale involsionukal ennu vilikkunnu. oru hyperbolic involsion rund nischitha poyintukalundu. ivayil randennam yadhaartha projective lainile praadhamika, ganitha pravarthanangalumaayi poruthappedunnu nishedhavum paraspara vinimayavum. vaasthavathil, 0 umm φ umm nishedhathinu keezhil sthireekarikkappettirukku, athesamayam 1 umm −1 umm paraspara vinimayathinu keezhil sthirappeduthappettiru.

