bahya sahparivartan vyutpann

vibhedak bhoomitichya ganiteey kshetrat, bahya sahparivartan vyutpann haah vibhedak mukhya bundle kinwa vector bandalchya jodanishi sambandhit bahya vyutpann kalpanecha vistar aahe.
vyakhya.
jee' haah eka khote gut asoo dya aanee gulgulit 'emm' varr mukhya 'jee'-bundle asoo dya. samaja "pii" varr eka jodani aahe; yamule aadvya aanee ubhya upasthananmadhye pratyek sparshikechya jaageche naisargik thet berries vighatan sootra _ 1 milate. sootra _ 2 heey aadvya upkshetrache prakshepan asoo dya.
jarr 'φ' heey 'p' varr 'k'-swaroop asel aanee vector space 'v' madhye moolye asateel, tarr tyache bahya sahparivartan vyutpann 'dφ' heey eka asey swaroop aahe jyachi vyakhya
jethe "v" "i" heey "u" varr "p" sathee sparshika vahak aahet.
samaja teey vector spacever 'jee' chey pratinidhitva aahe. jarr 'φ' yaa arthane samaroop asel tarr
jethe sootra _ 5 aahe, tevha 'dφ' heey 'ρ' prakarchya 'p' varr eka tensorial-form aaheh teey sampariman aanee aadve aahe (jarr 'si' heey form aadve asel tarr.)
sanketancha gairavapar karoon, olakh ghatkamadhil 'ρ' chaa farak punha 'ρ' dware darshavila jaau shakatoh
sootra _ 7 heey jodaniche eka-swaroop asoo dya aanee sootra _ 8 heey sootra _ 9 madheel jodaniche pratinidhitva aahe, mhanje sootra _ 8 heey eka sootra _ 11-moolya asalele swaroop aahe, jeey aadvya upkshetravar nahise hott aahe. jarr "φ" heey "ρ" prakarache tensorial "k"-swaroop asel, tarr
jethe, sanketankanantar, "aamhi lihile
nehamichya bahya vyutpann, jeey 0 paryant varg karate, tyachya ulat bahya sahparivartan vyutpann hott nahee. sarvasadharanpane, eka, tensorial shoonya-form "φ" sathee, aahe.
vakratechya done-swarupachya ω chya sootra _ 11 madheel pratinidhitva kuthe aahe? vidyutchumbakiyatemadhye tee bajavat asalelya bhoomikechya sadrushyat, eff yaa swarupala kadhikadhi kshetra shakti tensor mhanoon sambodhale jaate. lakshat ghya qee sapat jodanisathi 'dii' 2 nahisa hoto (i.eae. kevha).
jarr, tarr konee lihu shakel
jethe sootra _ 17 haah matricks aahe jyamadhye-th praveshavar 1 aanee itar praveshanvar shoonya aahe. matricks sootra _ 18 jyachya nondi "pii" varr 2-form aahet tyanna vakrata matricks mhanatat.
vector bundlesathi.
jodani aanee shrenisah eka gulgulit vastavik vahak bundle dilyas, bahya sahparivartan vyutpann heey vector-moolya asalelya vibhedak swarupanvar eka vastavik-rekheey nakasha aahe jyache moolya khalilpramane aaheh
sahparivartan vyutpann haah assa nakasha aahe. bahya sahparivartan vyutpann heey nakashe sarvasadharanparyant vistartat. yaa vastuchi vyakhya karnyasathi anek samatulya marg aaheth
kshullak vastavik reshechya bandalchya babatit tyachya manak jodanisah, vector-moolya vibhedak rupe aanee vibhedak rupe naisargikaritya ekmekanshi olakhali jaau shakatat aanee vareel pratyek vyakhya manak bahya vyutpattishi julate.
mukhya bundle dile tarr, rachana gatache konatehi resheey pratinidhitva sambandhit bundle paribhashit karate aanee mukhya bandalvaril konatehi connection sambandhit vector bandalvar connection prerit karate. vector bandalmadhye moolya asalele vibhedak prakar naisargikaritya mukhya bandalchya ekoon jaaget purnapane samamiteey-virodhi tensorial swarupansah olakhale jaau shakatat. yaa olakh antargat, mukhya bandalsathi aanee vector bandalsathi bahya sahparivartan vyutpattichya kalpana ekmekanshi julatat.
vector bandalvaril jodanichi vakrata hii done bahya sahparivartan vyutpannanchi rachana mhanoon paribhashit kelee jaau shakate aanee jenekarun tee vastavik-rekheey nakasha mhanoon paribhashit kelee jaeel. heey eka moolbhut parantu lagech spasht naan honare tathya aahe jeey keval tyavar avalambun aahe aanee teey reshiyapane karate. tyamule vakrata haah eka ghatak manla jaau shakato. bahya sahparivartan vyutpann kase tayar kele jaate yavar avalambun, vakratechya vividh paryayi parantu samatulya vyakhya (kahee bahya bhinnatechya bhasheshivay) midavata yetat.
heey eka suprasiddha satya aahe qee swatahsah manak bahya vyutpattichi rachana shoonya aaheh sadhyachya sandarbhat, heey asey mhatale jaau shakate qee kshullak reshechya bandalvaril manak jodanimadhye shoonya vakrata aahe.

