
gatox vyutpann

ganitat, gatox vibhedak kinwa gatox vyutpann heey vibhedak gananat dishatmak vyutpann sankalpaneche samanyikaran aahe. pahilya mahayuddhat lahan asatana maran pavalelya rene gatox yaa french ganitshastradnyachya navavarun heey naav denyat aale aahe, tyachi vyakhya sthanik patalivar bahirmukh topological vector spaces, jase qee banach spaces, yanchyatil karyansathi kelee jaate. banach jagetil frechet vyutpannapramane, gatox vibhedak sahasa bhinnata aanee bhautikshastratil gananat samanyatah vaparlya jaanarya karyatmak vyutpannala aupcharik karnyasathi vaparale jaate.
derivativeschya itar prakaranpramane, karyacha gatox differential nonlinear asoo shakato. tathapi, anekada gatox vibhedachya vyakhyesathi dekhil teey satat resheey parivartan asane aavashyak asate. kahee lekhak, jase qee, gatox vibhedak (jeey nonlinear asoo shakate) aanee gatox vyutpann (jeey teey resheey manatat) yanchyat aanakhi farak karatat. bahutek anuprayoganmadhye, satat reshatmakata hii kahee adhik adim sthitipasun yete jee vishisht mandanisathi naisargik aahe, jase qee anant aayami holomorphychya sandarbhat jatil bhinnata ladane kinwa araikhik vishleshanatil satat bhinnata.
vyakhya.
samaja sootra _ 1 aanee sootra _ 2 heey sthanik patalivar bahirmukh topological vector space aahet (udaharanarth, banach space), sootra _ 3 ughade aahe aanee sootra _ 4 heey sootra _ 6 chey gatox vibhedak sootra _ 5 heey sootra _ 7 varr, disha sootra _ 8 madhye asey paribhashit kele aahe
jarr serve sootra _ 9 sathee maryada astitvat asel tarr sootra _ 6 heey sootra _ 11 varr gatox differentiable aahe asey mhanatat.
jarr sootra _ 1 aanee sootra _ 2 heey vastavik topological vector space asateel tarr () madhye disanari maryada sootra _ 12 chya topologychya sapeksh ghetali jaate, tarr dusarikade vastavik sootra _ 15 sathee maryada ghetali jaate, jarr sootra _ 1 aanee sootra _ 2 heey jatil topological vector space asateel, tarr vareel maryada samanyatah jatil pratalamadhye sootra _ 18 mhanoon ghetali jaate, jase qee jatil bhinnatechya vyakhyet. kahee prakarananmadhye, majboot maryadeaivaji kamkuvat maryada ghetali jaate, jyamule kamkuvat gatox vyutpattichi kalpana yete.
reshatmakata aanee satatya.
pratyek bindu sootra _ 19 varr gatox vibhedak karya paribhashit karate
sootra _ 20
heey karya yaa arthane eksandh aahe qee serve scalar sutransathi _ 21
sootra _ 22
tathapi, heey karya yojak asanyachi aavashyakta nahee, jenekarun gatox vibhedak freckett vyutpannapeksha resheey honyat ayashasvi houu shakate. jaree resheey asale taree, sootra _ 1 aanee sootra _ 2 heey anant aayami asalyas teey sootra _ 23 varr satat avalambun rahnyas ayashasvi houu shakate. yavyatirikt, sootra _ 26 madhye rekheey aanee nirantar asalelya gatox bhinnatesathi tyanchi nirantar bhinnata tayar karnyache anek asman marg aahet.
udaharanarth, done vastavik chalanchya vastavik-moolya asalelya karya sutracha vichar qaraa
sootra _ 28
heey sootra _ 29 varr gatox differentiable aahe aanee tethe tyache differential aahe
sootra _ 30
tathapi, heey anant parimananmadhye yuktivad sootra _ 31 madhye satat aahe parantu resheey nahee, sootra _ 1 vareel konatehi visangat resheey karyatmak gatox differentiable aahe, parantu sootra _ 33 varr tyache gatox differential resheey aahe parantu satat nahee.
freckett vyutpattishi sambandh
jarr sootra _ 6 heey freckett differentiable asel, tarr teey gatox differentiable dekhil aahe aanee tyache freckett aanee gatox derivative sahmat aahet. gatox vyutpann resheey kinwa satat asanyas ayashasvi houu shakate, tyamule ulat spashtapane kharey nahee. kharey tarr, gatox vyutpann resheey aanee satat asane dekhil shakya aahe parantu frechet vyutpann astitvat nasane.
tathapi, banach space formula _ 1 pasoon dusarya jatil banach space formula _ 37 paryantchya functions formula _ 6 sathee gatox vyutpann (jethe jatil sutravar maryada ghetali jaate aanee jatil bhinnatechya vyakhyenusar shunyakade valate) svayamchalitapane resheey asate, jeey tyache pramey aahe. yavyatirikt, jarr sootra _ 6 heey (jatil) gatox asel tarr pratyek sutravar _ 7 vyutpannasah bhinn karata yeil
sootra _ 41
mug sootra _ 6 heey sootra _ 43 varr frechet vyutpann sootra _ 44 saha frechet differentiable aahe. heey moolbhut jatil vishleshanachya parinamashi samaan aahe qee jarr ekhade karya khulya sanchamadhye klisht bhinn karnyayogya asel tarr teey vishleshanatmak aahe aanee anant aayami holomorphychya abhyasatil eka moolbhut parinam aahe.
satatchi bhinnata
satatchya gatoxchya bhinnatechi vyakhya done asman marganni kelee jaau shakate. samaja, heey sootra _ 45 heey khulya sanch sootrachya pratyek binduvar gatox differentiabil aahe, 46 sutratil satat differentiabilitychya eka kalpanesathi utpadan jagevaril mapping aavashyak aahe.
sootra _ 48
satatyapurn raaha. reshatmakata gruheet dharanyachi garaj naheeh jarr sootra _ 1 aanee sootra _ 2 heey freckett space asateel, tarr sootra _ 51 serve sootra _ 52 sathee svayamchalitapane baddh aanee resheey aahe.
satatchya bhinnatechi majboot kalpana aavashyak aahe qee
sootra _ 53
satatche nakashe tayar qaraa
sootra _ 54
sootra _ 43 pasoon sootra _ 1 teey sootra _ 12 paryantchya satat resheey karyanchya jageparyant lakshat ghya qee heey aadhich sootra _ 58 chee reshatmakata purvavat karate.
tantrik soyichi baab mhanoon, satat bhinnatechi hii nantarchi kalpana vaishishtyapurn aahe (parantu sarvatrik nahee) jevha rikt jaaga sootra _ 1 aanee sootra _ 2 banach asatat, kaaran sootra _ 61 dekhil banach asate aanee nantar karyatmak vishleshanatun manak parinam vaparale jaau shakatat. pahili hii nonlinear vishleshanachya kshetranmadhye adhik samanya vyakhya aahe jithe samavisht asalelya karya jaaga aavashyakpane banach jaaga nasatat. udaharanarth, freckett spacemadhil vibhedache nash-moser inverse function prameyasarkhe anuprayog aahet jyamadhye swarasya asalelya function spacemadhye baryachada anek patanvar gulgulit functions asatat.
uchch vyutpann.
uchch kramache freckett vyutpann hii naisargikaritya punaravruttidvare bahupakshiy karye mhanoon paribhashit kelee jaatat, tarr samarupata sootra _ 62 uchch kramache gatox vyutpann vaparoon asha prakare paribhashit kele jaau shakat nahee. tyaeivaji _ 65 yaa dishene sootra asalelya _ 64 yaa karya sutrache sootra _ 63 vya kramankache gatox vyutpann heey paribhashit kele aahe
bahurekheey karyapeksha, heey tyaeivaji sootra _ 67 madheel degree sootra _ 63 chey eksandh karya aahe.
uchch kram vyutpattichya vyakhyesathi aanakhi eka umedvar aahe, karya
heey naisargikaritya badalanchya gananet sootra _ 68 chaa dusra badal mhanoon udbhavate, kiman tyaa vishesh prakaranat jethe sootra _ 6 heey scalar-moolya aahe. tathapi, sootra _ 65 aanee sootra _ 71 madhye swatantrapane ekroop asanyakherij, sootra _ 72 heey sootra _ 65 aanee sootra _ 74 chey samamiteey dwillingi karya aahe aanee teey sootra _ 75 chya dhruvikaranashi sahmat aahe yachee khatri karnyasathi puresha atty asane isht aahe.
udaharanarth, khaleel pureshi aut lagoo hote. samaja sootra _ 6 heey sootra _ 77 aahe yaa arthane qee mapping
sootra _ 78
utpadan topologymadhye satatyapurn aahe aanee yavyatirikt () dware paribhashit kelele dusare vyutpann dekhil yaa arthane satatyapurn aahe qee
sootra _ 79
satat aahe. mug sootra _ 72 heey sootra _ 65 madhye dwillingi aanee samamiteey aahe aanee sootra _ 71 heey dwilingitemule dhruvikaran olakh dharan karate. 
sootra _ 83
dusya kramankachya vyutpann sutras _ 84shhee sambandhit vibhedak sutras _ 85shhee sambandhit nishkarsh uchch kramankachya vyutpannansathi asatat.
malmatta.
gananechya moolbhut prameyatil eka aavrutti sootra _ 68 chya gatox vyutpattisathi dharan karate, jarr sootra _ 6 purese satat bhinn asalyache manale jaate. visheshatah
vyutpannache itar anek parichit gunadharm hyatoon anusaratat, jase qee uchch-kram vyutpannanchi bahurekheeyata aanee samyavad. moolbhut siddhantache parinam mhanoon pudhil gunadharmanmadhye khaleel goshtincha samavesh hotoh
udaharanadakhal.
sootra _ 1 heey euclidean space sutratil lebsgue mojata yenyajogya sanch sutravar chauras-ekatmik karyanchi hilbert jaaga asoo dya _ 105 karyatmak
sootra _ 107
sootra _ 108
jethe sootra _ 6 heey vastavik chalche vastavik-moolya asalele karya aahe aanee sootra _ 52 heey vastavik moolyansah sootra _ 105 varr paribhashit kele aahe, tethe gatox vyutpann aahe.
sootra _ 112
kharey tarr, vareel maryada sootra _ 18 aahe
sootra _ 114