
moolbhut vector kshetra

ganeet aanee visheshtah vibhedak bhoomitichya abhyasat, moolbhut vector kshetra heey eka asey sadhan aahe jeey gulgulit bahugunit varr gulgulit khote gut krutichya anant vartanache varnan karate. asha sadish kshetranna khote siddhant, samaroop bhoomiti aanee hamiltonian gut krutinchya abhyasat mahattvache upayog aadhaltat.
prerana.
ganeet aanee bhautikshastratil anuprayogansathi mahattvachi mhanje bahuvidh pravahanchi kalpana. visheshtah, jarr sootra _ 1 heey eka gulgulit bahugunit aanee sootra _ 2 heey eka gulgulit vector kshetra asel, tarr sootra _ 3 chey avibhajya vakr shodhanyat swarasya aahe. adhik tantotant, sootra _ 4 dile tarr tyala vakr sootra _ 5 madhye rus aahe, jenekarun
jyasathi sthanik upayanchi hummy samanya vibhedak samikarananchya astitva aanee advitiyatechya prameyadvare dilee jaate. jarr sootra _ 3 heey aanakhi eka sampoorn vector kshetra asel, tarr sootra _ 3 chaa pravah, sootra _ 3 sathee serve avibhajya vakrancha sangrah mhanoon paribhashit keela jaato, too sootra _ 10 chaa defiomorphism aahe. sootra _ 12 dware dilele pravah sootra _ 11 hii pratyakshat sootra _ 10 vareel yogatmak khote gut sootra _ 13 chee kriya aahe.
yaulat, pratyek gulgulit krutee sootra _ 15 heey samikaranadvare sampoorn sadish kshetra sootra _ 3 paribhashit karate.
tyanantar haah eka sopa parinam aahe qee sootra _ 1 vareel sootra _ 18 kriya aanee sootra _ 1 vareel sampoorn vector kshetranmadhye eka bijective patravyavahar aahe.
pravah siddhantachya bhashet, sadish kshetra sootra _ 3 laa "anant janitra" mhanatat. antardnyanane, pratyek binduvaril pravahache vartan vector kshetrane darshavalelya "disheshi" sambandhit aahe. sootra _ 1 varr vector feeled aanee adhik aniyantrit khote gut kriya yanchyat samaan sanvad sthapit karata yeil qaa, assa prashna vicharane swabhavik aahe.
vyakhya.
sootra _ 23 haah sambandhit khote bijganit sootra _ 24 asalela khote gut asoo dya. shivay, sootra _ 1 heey gulgulit krutee sootra _ 26 saha sampann eka gulgulit bahugunit asoo dya. nakasha sootra _ 27 asey darshava qee sootra _ 28, jyala "sootra _ 29 chaa kakshiy nakasha sootra _ 30 shhee sambandhit" mhanatat. sootra _ 31 sathee, sootra _ 3 shhee sambandhit moolbhut vector kshetra sootra _ 32 hii khalilpaiki konatihi samatulya vyakhya aaheh
jethe sootra _ 37 heey gulgulit nakashache vibhedak aahe aanee sootra _ 38 heey vector space sootra _ 39 madheel shoonya vector aahe.
tyanantar nakashache sootra _ 40 heey khote bijganit samarupata asalyache darshavile jaau shakate.
anuprayog.
khote gut.
khote gut sootra _ 23 chey khote bijganit heey sootra _ 23 vareel ektar davya kinwa ujavya-bhinn vector kshetransah olakhale jaau shakate. haah eka suprasiddha parinam aahe qee ashee vector kshetre sootra _ 43, olakh asalelya sparshikechya jagesathi samaroop aahet. kharey tarr, jarr aapan sootra _ 23 laa ujavya-gunakaradvare swatahvar karya karoo dile, tarr sambandhit moolbhut vector kshetra tantotant davya-bhinn vector kshetranche asatat.
hamiltonian gut krutee.
prernet, heey darshavile gele qee gulgulit sootra _ 18 kriya aanee sampoorn vector kshetranmadhye eka bijective patravyavahar aahe. tyachpramane, symplectic kriya (prerit defiomorphism heey serve symplectomorphism aahet) aanee sampoorn symplectic vector fields yanchyat eka bijective patravyavahar aahe.
hamiltonian vector kshetrachi eka jawalchi sambandhit kalpana aahe. eka samamiteey bahugunit sootra _ 46 dile tarr, aapan mhanato qee jarr eka gulgulit karya sootra _ 48 asel tarr sootra _ 47 heey hamiltonian vector kshetra aahe.
jethe nakasha sootra _ 50 heey antargat utpadan aahe. heey 'hamiltonian gut krutee' chee vyakhya khalilpramane prerit karateh jarr sootra _ 23 haah khote bijganit sootra _ 24 asalela khote gut asel aanee sootra _ 53 hii gulgulit bahugunit sootra _ 1 varr sootra _ 23 chee gut kriya asel, tarr aapan mhanato qee sootra _ 56 hii hamiltonian gut krutee aahe, jarr kshan nakasha sootra _ 57 astitvat asel tarr pratyek sutrasathi _ 31, 
jethe sootra _ 60 aanee sootra _ 32 heey sootra _ 3 chey moolbhut vector kshetra aahe.