veliyetta tensor

nyoottante guruthwaakarshana sidhaanthathilum pothu aapekshikatha polulla guruthwaakarshanathinte vividha aapekshika classical sidhaanthangalilum veliyetta tensor prathinidheekarikkunnu. 
veliyetta tensor ananthamaaya dooram kondu verthirikkapetta rund testt pindangalude guruthwaakarshanam moolamulla aapekshika thwaranathe prathinidheekarikkunnu. ghadaka soothravaakyam _ 1 ennathu soothravaakyam _ 3 dishayilulla sthaanachalanathiloode uthppaadippikkunna soothravaakyam _ 2 dishayile aapekshika thwaranathe prathinidheekarikkunnu.
oru golaakrithiyilulla sareerathinu veliyetta tensor.
veliyettangalude ettavum saadhaaranamaaya udaaharanam oru golaakrithiyilulla sareerathinu chuttumulla veliyetta sakthiyaanu ("iii.jii.", oru graham allengil oru chandran).
ivide nammal oru otappetta golaakrithiyilulla samamiti pindamulla vasthuvin purathulla guruthwaakarshana mandalathinulla veliyetta tensor kanakkakkunnu. nyoottante guruthwaakarshana niyamam anusarichu, oru kendra pindamaya "m" ll ninnu "r" akaleyulla "a" thvaranam 
namukk nammude trimana euclidean sthalathinaayi dhruva kordinettukalile frame sweekarikkaam, koodaathe radial, asimuthal dishakalile ananthamaaya sthaanachalanangal, formula _ 5, formula _ 6 enniva pariganikkaam, avaa yathaakramam 1,2,3 ennee subscriptukal nalkiyirikkunnu.
eee framil prakadippichirikkunna tidal tensorinte oro gadakavum nammal nerittu kanakkakkum.
aadyam, kendra pindatthil ninnu akale ore radial lainil kidakkunna aduthulla rund vasthukkalude guruthwaakarshana shakthikale "h" enna doorathaal vyathyaasappettirikkunnu
kaaranam, tensarukalekkurichu charcha cheyyumbol nammal multilenear aaljibrayumaayi edapedumbol, nammal phastt oordar padangal maathrame nilanirthunnulloo, athinaal formula 9. radial dishayilulla sthaanachalanathethudarnnu formula 10 allengil formula 11 dishayil aaksilareshan illaathathinaal, mattu radial padangal poojyamaanu formula 12.
athupole, ore aaram formula kidakkunna rund aduthulla nireekshakarile guruthwaakarshana sakthiye namukk thaarathamyam cheyyaam, ennaal formula (ananthamaaya) dooram "h" upayogichu sthaanabhramsham varuthaam. chila praadhamika trikonamithiyum cheria konukalude ekadeshavum upayogichu, fozhsu vektarukal oru vector taanjant upayogichu vyapthiyulla golavumaayi vyathyasappettirikkunnavu nammal kandethunnu.
small angle ekadesha kanakk upayogikkunnathiloode, oordar formula _ 17-nte ellah nibandhanakalum nammal avaganichu, athinaal tangential gadakangal formula _ 18 aanu. veendum, asimuthal dishakalilonnil sthaanachalanangal kaaranam radial dishayil aaksilareshan illaathathinaal, mattu asimuthal padangal poojyamaanu formula _ 19.
eee vivarangal samyojippichu, veliyetta tensor frame gadakangalulla diagonal aanennu njangal kandethunnu.
soothravaakyam _ 20
neutonian bhauthikashaasthrathile golaakrithiyilulla samamiti kendra shakti mandalangalude savisheshathayaaya "coolomb phoom" aanith.
hessian formulation.
pindam golaakrithiyilulla oru samamita kendra vasthuvallaatha kooduthal pothuvaaya saahacharyathil, poyson samavaakyam anusarikkunna guruthwaakarshana pottensial formula _ 21 ll ninnu tidal tensor uruthirinjekkaam
ivide formula _ 23 ennathu nilavilulla eathenkilum dravyathinte pinda saandrathayaanu, koodaathe formula _ 24 laplace oopparettaraanu. eee samavaakyam soochippikkunnathu oru vaakvam parihaaratthil pottensial oru harmonic function maathramaanennu shradhikkuka.
tidal tensor" nalkunnathu "trassless bhagam" aanu." 
hessian
ivide nammal e3-naayi euclidean metrik tenserinoppam staanderd "cartician chart" upayogikkunnu.
vekttar calculusile staanderd phalangal upayogichu, ithu "pollar golaakrithiyilulla chart" polulla mattu coordinate chaarttukalil saadhuvaaya padaprayogangalaayi eluppathil parivartthanam cheyyappedunnu."
golaakrithiyilulla samamiti feeld.
udaaharanathinu, hessian upayogichu oru golaakrithiyilulla shareerathinulla veliyetta tensor namukk kanakkakkaam. aduthathaayi, namukk guruthwaakarshana pottensial formula _ 30 hessian ennathil cherkkaam. mukalilulla padaprayogam dhruva golaakrithiyilulla kordinettukalil saadhuvaaya onnaayi parivartthanam cheyyaam, allengil plug inn cheiyunnathinu munbu namukk pottensial cartician kordinettukalaayi parivartthanam cheyyaam. randaamathe kozhs sweekarichu, namukk formula _ 31 undu, athu nalkunnu
dhruva golaakrithiyilulla kordinettukalumaayi poruthappedunna nammude framinte bhramanathinu sesham, eee padaprayogam nammude muun falavumaayi yojikkunnu. ithu kaananulla ettavum eluppamaarggam formula _ 33 poojyamaayi sajjamaakkuka ennathaanu, angane oof-diagonal padangal aprathyakshamaakukayum formula _ 34 aakukayum thudarnnu golaakrithiyilulla samamiti abhyarthikkukayum cheiyunnu.
pothuvaaya aapekshikathayil.
pothuvaaya aapekshikathayil, tidal tensorine riman karvechar tensor saamaanyavalkkarikkunnu. durbalamaaya feeld paridhiyil, tidal tensor nalkunnathu vakrata tensorinte ghadaka soothravaakyam _ 35 aanu.

