
saamaanyavalkkaricha stooks sidhaantham

vekttar calculusilum deferential jyaamithiyilum, vekttar calculusil ninnulla niravadhi sidhaanthangale lalithamaakkukayum saamanyavalkkarikkukayum cheiyunna maanifoldukalile deferential roopangalude samyojanathekkurichulla oru prasthaavanayaan stooks-carton sidhaantham ennum vilikkappedunna saamaanyavalkkaricha stooks sidhaantham (chilappol stoxinte sidhaantham allengil stoxinte sidhaantham enna nilayil apostrophy). prathyekichum, kaalkkulasinte adisthaana sidhaantham oru line segmentaaya manifold, green sidhaanthavum stooks sidhaanthavum formula _ 1 allengil formula _ 2 leeii oru uparithalathinte kesukalum divergence sidhaantham formula _ 3 leeii voliyathinte kesumayathinal, sidhaanthathe chilappol multiveriyettu kaalkkulasinte adisthaana sidhaantham ennu vilikkunnu.
chila ooriyantabil manifold formula _ 6-nte athirthi formula _ 5-nuu mukalilulla oru deferential phoom formula _ 4-nte integral, formula _ 6, i-yude muzhuvan formula athinte eksteeriyar derivative formula _ 7-nte integralinu thulyamaanennu stoxinte sidhaantham parayunnu.iii.,
formula _ 9
vittoo voltera, edward gorsaatt, henry poincare ennivar vekttar kaalkkulasinte sidhaanthangalude saamaanyavalkkaranathekkurathu munkaala pravarthanangaletthudarnna 1945 ll elii carton stoxinte sidhaantham athinte aadhunika roopathil roopappeduthi.
1850 juulai 2-nuu loard kelvin jorjju stoxinu ayacha kathil paranja classical phalatthinte vishalamaaya saamaanyavalkkaranamaanu stoxinte sidhaanthathinte eee aadhunika roopam. stooks eee sidhaanthathe 1854-leeii smithinte sammana pareekshayil oru chodyamaayi sajjamakki, athu addehathinte perulla falatthilekku nayichu. 1861ll herman hankel aanu ithu aadyamaayi prasidheekarichathu. eee classical cases euclidean three-spacil oru uparithalathinu mukalilulla (athaayath, formulayude fluxes) oru vekttar feeld formulayude kelnte uparithala avibhaajyaghadakathe uparithala athirthikku mukalilulla vekttar feeldinte line integralumaayi bandhappettirikkunnu.
aamukham.
kaalkkulasinte randaamathe adisthaana sidhaantham prasthaavikkunnath, interwell formula 13 nuu mukalilulla oru function formulayude avibhaajyaghadakam formula 12 nte oru antiderivative formula _ 14 kandethunnathiloode kanakkakkaam
soothravaakyam _ 16
stoxinte sidhaantham thaazhepparayunna ardhathil eee sidhaanthathinte vishalamaaya saamaanyavalkkaranamaanu. 
lalithamaayi paranjaal, oraalkk pooyintukale vakrangalude athirukalaayi kanakkakkaam, athaayath 1-dimensional maanifoldukalude 0-dimensional athirukal. athinaal, 0-dimensional boundarikalile (formula _ 43) aanti-derivative (formula _ 14) pariganichukondu oru dimensional manifoldinu (formula _ 13) mukalilulla oru integralinte (formula _ 40) moolyam kandethaan kazhiyunnathupole, maanifoldinte formula _ dimensional boundarikalile (formula _ 5) antiderivative (formula _ 4) pariganichukondu formula _ 45-dimensional maanifoldukal (formula _ 6) ennathinekkaal integralukalude (formula _ 7) moolyam kaikaaryam cheiyunnathinu, kurachu adhika munnariyippukalode oraalkk kaalkkulasinte adisthaana sidhaantham saamaanyavalkkarikkam.
athinaal adisthaana sidhaantham ingane vaayikkunnu
soothravaakyam _ 50
athirthiyulla minusamarnna maanifoldukalkkulla formulation.
formula _ 6 athirthiyulla dimension formula _ 45-nte oru ooriyantad minusamarnna manifold ayirikkatte, formula _ 53 oru minusamarnna formula _ 45 deferential roopamaayirikkatte, athu formula _ 6-ll kompaakt aayi pinthunaykkunnu. aadyam, formula _ 53 orotta, ooriyantad coordinate chart formula _ 57-nte domainil kompaakt aayi pinthunaykkunnuvennu karuthuka. eee saahacharyathil, formula _ 53 ovar formula _ 6-nte integral nammal nirvachikkunnu
soothravaakyam _ 60
aii.iii., formula _ 53 muthal formula _ 62 vareyulla pinvalikkal vazhi.
kooduthal pothuve, formula _ 6 nuu mukalilulla formula _ 53-nte avibhaajyaghadakam inipparayunna reethiyil nirvachikkappedunnu formula _ 65 (sthiramaayi ooriyantad) coordinate chaarttukalude praadeshikamaayi parimithamaaya cover formula _ 66-maayi bandhapetta aikyathinte oru vibhajanamayirikkatte, thudarnnu avibhaajyaghadakam nirvachikkuka.
soothravaakyam _ 67
mukalil vivarichathupole formula _ 62 lake thirike valichukondu thukayile oro padavum vilayiruthunnu. eee alavu nannayi nirvachikkappettirikkunnu; athaayath, ithu coordinate chaarttukalude thiranjeduppineyo aikyathinte vibhajanatheyo aasrayichirikkunnilla.
saamaanyavalkkaricha stooks sidhaantham iprakaaramaanu
ivide formula _ 69 ennathu eksteeriyar derivative aanu, ithu manifold ghadana upayogichu maathram nirvachikkappedunnu. 48-manifold formulakku athirthiyilla enna vastutha oonnipparayaan valathuvashathe chilappol formula 70 ennu ezhuthunnu. (eee vastutha stoxinte sidhaanthathinte oru soochana koodiyaanu, kaaranam thannirikkunna minusamarnna soothravaakyam _ 45-dimensional manifold formula _ 6-nuu, sidhaanthathinte prayogam randuthavana eathenkilum soothravaakyam _ 76-phoom formula _ 4-nuu soothravaakyam _ 75 nalkunnu, ithu soothravaakyam _ 78-neey soochippikkunnu) samavaakyathinte valathuvasham palappozhum "avibhaajya" niyamangal roopappeduthaan upayogikkunnu; idathu vasham thudarnnu thulyamaaya "deferential" formulationukalilekku nayikkunnu (chuvade kaanuka).
formula _ 6 ennathu formula _ 4 nirvachichirikkunna chila valia maanifoldukalude ulchertha ooriyantad sub manifold aayirikkunna saahacharyangalil sidhaantham palappozhum upayogikkunnu, palappozhum formula _ 80.
topological preliminarykal; shringalakalkku mukalilulla samyojanam.
oru minusamarnna palamadangu ayirikkatte. eiii (smooth) singular-simplexes inn ennathu staanderd simplexes muthal inn vare oru smooth mappaayi nirvachikkappedunnu. ekavachana-shringalakalude koottatthe ekavachana-lalithathakalude koottatthile swatantra abelian groupaayi nirvachikkunnu. eee grouppukal, athirthi bhoopadathodoppam, oru cheyin complexes nirvachikkunnu. anubandha homology (rep. cohomology) group saadhaarana singular homology groupinu (rep. ekavachana cohomology group), minusamarnna lalithathakalekkaal thudarchayaayi nirvachichirikkunnu.
maruvashathu, connecting mapp enna nilayil, baahya derivetteevaaya deferential roopangal oru kocheyin complexes roopappeduthunnu, ithu dii raam cohomology grouppukalude formula _ 82 nirvachikkunnu.
deferential-foomukal oru-simplexil swaabhaavika reethiyil samyojippikkaan kazhiyum, athilekku thirike valikkunnathiloode. lineerity vazhi vikasippikkunnath shringalakalkku mukaliloode samyojippikkaan oraale anuvadikkunnu. ithu-foomukal muthal ekavachana kocheyinukalude th group vare, leanier functionalukal vare oru leanier mapp nalkunnu. mattoru vidhathil paranjaal, oru-phoom oru pravartthanatthe nirvachikkunnu.
soothravaakyam _ 83
shringalakalil. ithu yadhaartha gunakangalulla dii raam cohomology muthal singular cohomology vareyulla oru shringhala bhoopadamaanennu stoxinte sidhaantham parayunnu; baahya derivative, roopangalude "duval" pole pravartthikkunnu. ithu dii raam cohomology muthal singular cohomology vare oru homomorphism nalkunnu. roopangalude thalatthil, ithu ardhamaakkunnathu
eee homomorphism yadhaarthathil oru isomorphisamaanennu dii raaminte sidhaantham kaanikkunnu. athinaal mukalil paranja 1,2 ennivaykku vipareetham sariyaanu. mattoru vidhathil paranjaal, th homology groupine srishtikkunna chakrangalaanenkil, bandhapetta eathenkilum yadhaartha samkhyakalkku, oru clossed phoom undu, athaayath
soothravaakyam _ 84
eee roopam krithyamaaya roopangal vare savisheshamaan.
minusamarnna maanifoldukalekkurichulla stoxinte sidhaantham minusamarnna maanifoldukalile shringalakalkkaayulla stoxinte sidhaanthathil ninnu uruthirinjekkaam, thirichum. audyogikamaayi prasthaavicha randaamathethu iprakaaramaanu
adisthaana thathwam.
eee topological vaadangal lalithamaakkunnathin, alavukalkku oru udaaharanam pariganichukondu adisthaana thathwam parishodhikkunnath moolyavathaanu. idathuvashathulla rekhaachithrathiloode avashya ashayam manassilakkaan kazhiyum, ithu kaanikkunnath, oru maanifoldinte ooriyantad tilingle, interior pathakal vipareeta dishakaliloode sanjarikkunnu; paathu integralilekkulla avayude sambhavanakal parasparam jody thirichaayi raddaakkunnu. thalphalamaayi, athirthiyil ninnulla sambhavana maathrame avasheshikkunnulloo. athinaal, saadhaaranayaayi budhimuttullathallaatha, mathiyaya nerttha cherivukalkku (allengil, thulyamaayi, lalithangalkk) stoxinte sidhaantham theliyikkaan ithu mathiyakum.
classical vekttar vishakalana udaaharanam.
soothravaakyam _ 85 oru peacewise minusamarnna jordan playin karvu ayirikkatte. jordan karvu sidhaantham soochippikkunnathu formula 86 formula 1 neey rund gadakangalaayi vibhajikkunnu, onnu kompaakt, mattonn kompaakt allaathathaan. formula _ 88 formula _ 86 kondu parimithappedutthiyirikku compact bhaagathe soochippikkatte, formula _ 90 minusamaarnnathaanennu karuthuka, formula _ 91. formula _ 92 formula _ 93 nirvachichirikkunna space karvum formula _ 10 formula _ 95 leeii minusamarnna vekttar feeldumaanengil, pinne
soothravaakyam _ 96
eee classical prasthavana, vekttar feeld 1-formilum athinte kell rund formilum thiricharinju kazhinjaal pothuvaaya formulationte oru prathyeka kesaanu. 
soothravaakyam _ 97
soothravaakyam _ 98
ruff settukalilekkulla saamaanyavalkkaranam.
6 enna soothravaakyam athirthiyulla minusamarnna manifold aayirikkunna mukalilulla formulation pala prayogangalilum paryaapthamalla. udaaharanathinu, ekeekaranathinte domaine rund soothravaakyangalkkum 100-kordinettukalkkum rund functionukalude graafukalkkum edayilulla thalam mekhalayaayi nirvachikkappettal, domaine konukal undakaan saadhyathayund. atharamoru saahacharyathil, corner pooyintukal ardhamaakkunnathu formula _ 6 athirthiyulla minusamarnna manifold allah, athinaal mukalil nalkiyirikkunna stoxinte sidhaanthathinte prasthavana baadhakamalla. ennirunnaalum, stoxinte sidhaanthathinte nigamanam ippozhum shariyaano ennu parisodhikkaan kazhiyum. formula _ 6 umm athinte athirthiyum oru cheria koottam pooyintukalil ninnu (oru alavu poojyam settu) nannayi perumaarunnathaan ithinu kaaranam.
parukkanthakkaan anuvadikkunna stoxinte sidhaanthathinte oru pathipp vitni theliyichu. formula _ 88 formula _ 62-nte bandhippicha parimithamaaya ooppan upavibhaagamaanennu karuthuka. formula _ 88 ennu vilikkuka, athu inipparayunna property thripthippeduthukayaanel formula _ 107-nte oru upavibhaga formula _ 106 nilavilundu, formula _ 107-ll thurannirikkunnu, athinte formula _ 107-leeii poorakatthinu hasdorff formula _ 48-alavu poojyamundu; koodaathe formula _ 106-nte oro pooyintinum oru "saamaanyavalkkaricha saadhaarana vekttar" undu. ithu oru vekttar formula _ 112 aanu, athinaal oru coordinate sisttam thiranjedukkukayaanenkil formula _ 112 aadya adisthaana vekttaraan, appol, formula _ 100 nuu chuttumulla oru thuranna ayalpakkathil, oru smooth function formula _ 115 nilavilundu, athaayath formula _ 106 ennathu graph formula _ 117 umm formula _ 88 ennathu reejian formula _ 119 umm aanu. oru staanderd domaininte athirthi seero hasdorff formula _ 48 mesharinte oru koottatthinte samyojanamaanennum smooth formula _ 48 maanifoldukalude parimithamo kanakkakkaavunnatho aaya samyojanamaanennum vitni abhipraayappedunnu, avayil oronninum oru vashath domaine maathrameyulloo. formula _ 88 formula _ 62 leeii oru staanderd domainanengil, formula _ 4 ennathu formula _ 126 ll nirvachikkappettathum thudarchayaayathum parimithappeduthiyathumaaya oru formula _ 48 roopamaanennum formula _ 88 ll minusamaarnnathum formula _ 106 ll samyojippikkavunnathumaaya formula _ 7 formula _ 88 ll samyojippikkaavunnathaane addeham theliyikkunnu, appol stoxinte sidhaantham nilanilkkunnu, athaayath,
formula _ 131
parukkan settukalude alavukol-saidhaanthika gunangalekkurichulla patanam jyaamitheeya alavukol sidhaanthathilekku nayikkunnu. stoxinte sidhaanthathinte kooduthal pothuvaaya pathippukal federer harrison theliyichittundu.
prathyeka kesukal.
deferential roopangal upayogikkunna stooks sidhaanthathinte pothuvaaya roopam prathyeka kesukalekkaal kooduthal shakthavum upayogikkaan eluppavumaanu. deferential jyaamithiyude yanthrangalillaathe cartician kordinettukal upayogichu parambaraagatha pathippukal roopappeduthaam, athinaal avaa kooduthal axus cheyyaavunnathaanu. koodaathe, avarkk praayamaayathinaal avarude perukal kooduthal parichithamaanu. parambaraagatha roopangal palappozhum shaasthranjarum enchineeyarmaarum parisheelikkumbol kooduthal soukaryapradamaayi kanakkaakkappedunnu, ennaal mattu coordinate sisttangal upayogikkumbol, golaakrithiyilullatho silindar kordinettukal poleyulla parichithamaayava polum parambaraagatha formulationte swaabhaavikathayillaayma prakadamaakunnu. perukal prayogikkunna reethiyilum eratta formulationukalude upayogathilum aashayakkuzhappathin saadhyathayund.
classical (vekttar calculus) cases.
ithu oru (duvalised) (1 + 1)-dimensional cases aanu, oru 1-foominu (duvalised kaaranam ithu vekttar feeldukalekkurichulla oru prasthaavanayaan). eee prathyeka cases palappozhum pala aamukha yooniversitti vekttar calculus kozhsukalilum "stooks sidhaantham" ennu vilikkappedunnu, ithu bhauthikashaasthrathilum enchineeyaringilum upayogikkunnu. ithu chilappol kell sidhaantham ennum ariyappedunnu.
classical stooks sidhaantham euclidean three-spacil oru uparithala soothravaakyathinu mukalilulla oru vekttar feeldinte kelnte uparithala avibhaajyaghadakathe athinte athirthikku mukalilulla vekttar feeldinte line avibhaajyaghadakavumaayi bandhappettirikkunnu. euclidean 3-speysile metrik upayogichu 1-roopamulla oru vekttar feeld thiricharinjaal ithu general stooks sidhaanthathinte (formula 133 ulla) oru prathyeka kesaanu. line integralinte vakramaaya formula 134 nuu positteev ooriyanteshan undaayirikkanam, athaayath uparithala normal formula 45 viewerilekku choondikkaanikkumbol formula 134 gadikaaradishayil ethirdishayil pooyint cheiyunnu.
eee sidhaanthathinte oru anantharafalamaanu poojyamaaya churunda oru vekttar feeldinte feeld lineukalkku clossed kondoorukal aakaan kazhiyilla ennathaanu. formula ingane ezhuthaam
green sidhaantham.
mukalil udharichathum,, ennivayude adisthaanathil integralil iruvashangaludeyum moonnaamathe integraantaayi green sidhaantham udan thirichariyaan kazhiyum.
vaidyuthakaanthikathayil.
naalu maiks well samavakyangalil randennam 3-dii vekttar feeldukalude churulukal ulkkollunnu, avayude deferential, integral roopangal stoxinte sidhaanthathinte prathyeka trimana (vekttar calculus) kesumaayi bandhappettirikkunnu. neengunna athirthikalulla kesukal ozhivakkaan jagrata paalikkanam bhagika samaya derivateevukal atharam kesukal ozhivakkaan uddeshichullathaanu. chalikkunna athirthikal ulppeduthiyittundel, samyojanathinteyum deferentiationteyum kaimaattam chuvadeyulla phalangalil ulppeduthiyittillaathi athirthi chalanavumaayi bandhapetta nibandhanakal avatharippikkunnu (avibhaajya chinnathinu keezhilulla deferentiation kaanuka):
maiks vellinte samavaakyangalude mukalil listtucheytha upavibhagangal sii yoonittukalil prakadippikkunna vaidyuthakaanthika mandalangalkk saadhuthayullathaanu. sii. jii. ezz allengil gausiyan unitukal polulla yoonittukalude mattu samvidaanangalil, padangalude scaling gadakangal vyathyasthamaan. udaaharanathinu, gausiyan yoonittukalil, faaradeyude induction niyamavum aambiyarinte niyamavum roopangal edukkunnu
soothravaakyam _ 137
yathaakramam, shoonyathayil prakaashatthinte vegatha evideyaanu?
vyathichalana sidhaantham.
athupole, vyathichalana sidhaantham
formula _ 138
euclidean vooliyam roopavumaayi vekttar feeld churukki _ 48 enna soothravaakyam upayogichu oru vekttar feeld thirichariyukayaanenkil ithu oru prathyeka kesaanu. ithinte prayogam cases formula _ 140 aanu, ivide formula _ 141 oru ekapaksheeyamaaya sthiramaaya vektaraanu. uthppannathinte vyathiyaanam kandethunnathu nalkunnu
formula _ 142
ithu ellah formulakalkkum baadhakamaayathinaal _ 141 nammal kandethunnu.
formula _ 144
scaler feeldinte gradiantinte vooliyam integral.
formula _ 145 oru scaler feeld ayirikkatte. appol
soothravaakyam _ 146
ivide oru nischitha ghattathil uparithala soothravaakyamaaya 148-nte saadhaarana vektaraanu formula _ 147.
thelivu
141 enna soothravaakyam oru vector ayirikkatte. appol
soothravaakyam _ 150
ithu 141 enna soothravaakyathinu baadhakamaayathinaal (prathyekichu, oro adisthaana vektarinum), phalam pinthudarunnu.