aparivartaniy (ganeet)

ganitat, aparivartak haah ganiteey vastoocha (kinwa ganiteey vastooncha varg) eka gunadharm aahe joou objectover vishisht prakarchya operations kinwa parivartan lagoo kelyanantar badalat rahato. vastooncha vishisht varg aanee parivartanancha prakar sahasa jyaa sandarbhat hii sandnya vaparali jaate tyaa sandarbhadvare darshavila jaato. udaharanarth, euclidean pratalachya samamitichya sandarbhat trikonache kshetrafal aparivartaniy asate. antargat aparivartaniy' aanee 'parivartanasathi aparivartaniy' yaa donhi vakyanshancha vapar keela jaato. adhik samanyatah, samatulya sambandhanchya sandarbhat aparivartaniy haah eka gunadharm aahe joou pratyek samatulya vargavar sthir asato.
ganitatil bhoomiti, topology, bijganit aanee swatantra ganeet yasarkhya vividh kshetranmadhye aparivartaniy ghatak vaparale jaatat. parivartananche kahee mahattvache varg teey badalat nasalelya aparivartkadvare paribhashit kele jaatat. udaharanarth, anuroop nakashe heey coone jatan jatan pratalache parivartan mhanoon paribhashit kele jaatat. ganiteey vastunche vargikaran karnyachya prakriyetil eka mahattvachi payari mhanje aparivartanancha shodh.
udaharane.
aparivartanache eka sadhe udaharan aaplya mojanyachya kshamatemadhye vyakt kele jaate. konatyahi prakarchya vastoonchya maryadit sanchasathi, eka sankhya asate jichyavar aapan nehmi pohochato, mug aapan sanchatil vastoo konatya kramane mojato yachee parva naan karata. praman-eka mukhya sankhya-sanchasah sambandhit aahe aanee mojanichya prakriyet aparivartaniy aahe.
olakh heey eka asey samikaran aahe jeey tyachya chalanchya serve mulyansathi kharey rahate. tyanchya chalanchi moolye badalat asatana dekhil asamantadekhil satya rahatat.
donhi sankhyanmadhye samaan praman jodoon, sankhya reshevaril done bindunmadhil antar badalat nahee. dusarikade, gunakaramadhye haah samaan gunadharm nasato, kaaran gunakarat antar aparivartaniy nasate.
scaling, rotation, translation aanee reflectionmadhye coone aanee antarache gunottar aparivartaniy asatat. heey parivartan samaan aakar tayar karatat, joou trikonamiticha aadhar aahe. yaulat, eksaman nasalelya scalingmadhye (jase qee stretching) coone aanee gunottar aparivartaniy nasatat. vareel serve kriyanmadhye trikonachya aateel konanchi (180°) berries aparivartaniy asate. dusare udaharan mhanoon, serve vartule samaan aaheth tyanche ekmekanmadhye rupantar kele jaau shakate aanee parigh aanee vyasache gunottar aparivartaniy aahe (greek akshar π (π) dware darshavile jaate).
kahee adhik klisht udaharane -
muu kode.
myu kode heey tarkik samasyeche eka changle udaharan aahe, jethe aparivartaniyata nishchit karane heey ashakyatechya puravyasathi upayukt aahe. kode ekhadyala mea yaa shabdane suruvat karnyas aanee tyache myu yaa shabdat rupantar karnyas sangate, pratyek tappyat khaleel parivartanachya niyamanpaiki eka vaparteh
eka udaharan vyutpatti (lagoo niyam niyam superscripts) aahe
yaa parshvabhumivar, keval yaa chaar parivartanachya niyamancha vapar karoon meeche myumadhye rupantar karane shakya aahe qaa assa prashna ekhadyala padala asel. heey parivartan niyam taranvar lagoo karnyat konihi anek taas ghalavu shakato. tathapi, serve niyamanmadhye aparivartaniy asaleli malmatta shodhane jalad asoo shakate (i.eae. teey tyanchyapaiki konihi badallele nahee) aanee teey darshavate qee myuparyant pohochane ashakya aahe. tarkik drushtikonatun kode pahoon, ekhadyala heey lakshat yeil qee konatyahi aay pasoon mukt honyacha ekmev marg mhanje stringmadhye salag teen aay asane. yamule khaleel aparivartaniy babincha vichar karane manoranjak tharateh
pratyek parivartanachya niyamansathi khalilpramane asel tarr heey samasyesathi aparivartaniy aaheh niyam lagoo karnyapurvi aparivartaniy thevalyas, too lagoo kelyanantarhi teey tickel. aay aanee yoochya sankhyevar niyam lagoo karnyacha nivval parinam pahata, heey serve niyamanchya babatit pratyakshat aahe heey konihi pahoo shakateh
vareel takta spashtapane darshavito qee pratyek sambhavya parivartan niyamansathi aparivartaniy aahe, jyacha arth assa aahe qee niyam lagoo karnyapurvi jarr i chee sankhya teenchi gunakar nasel tarr, joou niyam konee nivdel, too konatyahi sthiteet asel, tarr too nantarhi nasel.
suruvatichya stringmadhye eka i aahe aanee joou teencha gunank nahee, heey lakshat gheta, nantar ekhada assa nishkarsh kadhu shakato qee mi pasoon mu paryant jaane ashakya aahe (kaaran i chee sankhya kadhihi teenchi gunank honar nahee).
aparivartaniy sanch.
mapping "tii" chya "yuu" domencha upasanchah "yuu" → "yuu" haah mappingkhali eka aparivartaniy sanch aahe jevha sootra _ 5 nondavate qee "ess" chey ghatak nishchit naheet, jaree sanch "ess" haah "yuu" chya power setmadhye nishchit aahe. (kahee lekhak 'setwise inverent' viruddha 'setwise inverent' hii shabdavali vapartat. "yaa prakarananmadhye farak karnyasathi, "binduroop aparivartaniy".)
udaharanarth, vartul haah vartulachya kendrasthani firtana pratalacha eka aparivartaniy upasanch asato. pudhe, shankuchya aakaracha prushthbhag jagechya samarupatekhali sanch mhanoon aparivartaniy asato.
operation 'tii' chaa eka aparivartaniy sanch dekhil 'tii' antargat sthir asalyache mhatale jaate. udaharanarth, gut siddhantat itake mahattvache asalele samanya upasamuh heey asey upasamuh aahet jeey sabhovatalachya gatachya aateel automorphism antargat sthir asatat.
rekheey bijganitemadhye, jarr rekheey parivartan "t" madhye eigenwector v asel, tarr 0 aanee v madhoon jaanari resha "t" antargat eka aparivartaniy sanch aahe, asha paristhitit eigenwector eka aparivartaniy upkshetra vyapatat jeey "t" antargat sthir asate.
jevha "tii" heey screw visthapan asate, tevha screw aksh hii eka aparivartaniy resha asate, jaree jarr khelpatti shunyetar asel tarr "tii" laa konatehi nishchit bindu nasatat.
aupcharik nivedan.
ganitamadhye, samooh krutee, sadarikaran aanee vikrutee yandware, aparivartanachi kalpana teen vegvegalya prakare aupcharik kelee jaate.
samooh krutee antargat badal keela jaat nahee.
pratham, jarr ekhadya ganiti vastuvar (kinwa vastoonchya sanchavar) 'x' varr karya karnara 'g' gut asel, tarr ekhada vyakti vicharoo shakato qee konate bindu 'x' badallele naheet, gut krutee antargat 'aparivartaniy' aahet kinwa gatachya 'g' ghatkakhali aahet.
anekada 'x' sanchavar karya karnara gut asel, joou 'sambandhit' sanch 'f' ('x') madheel konatya vastoo aparivartaniy aahet heey tharvanyasathi ekala sodato. udaharanarth, eka bindubaddalachya pratalatil paribhramanamule too jyaa bindubaddal firto too bindu aparivartaniy rahato, tarr pratalatil anuvad konatyahi bindunna aparivartaniy thevat nahee, parantu serve resha anuvadachya dishene samantar resha mhanoon aparivartaniy sodato. aupcharikpane, pratalatil reshancha sanch "pii" laa "ell" ("pii") mhanoon paribhashit qaraa; nantar pratalchi kathor gatee reshanna reshanmadhye ghete-kathor halchalincha gut reshanchya sanchavar karya karato-aanee ekhadi vyakti vicharoo shakate qee krutidvare konatya resha badalat naheet.
sarvat mahattvache mhanje, ekhadi vyakti sanchavaril 'karya' paribhashit karoo shakate, jase qee 'pratalatil vartulachi trijya' aanee nantar heey karya samooh krutee antargat aparivartaniy aahe qaa, jase qee kathor halchali.
aparivartaniyatechya kalpaneshi duheri mhanje 'sahparivartanshil', jyanna 'kaksh' asehi mhanatat, jeey ekarupatechya kalpanela aupcharik roop deteh jyaa vastoo samooh krutidvare ekmekankade nelya jaau shakatat. udaharanarth, pratalachya kathor halchalinchya gatantargat, trikonachi parimiti aparivartaniy asate, tarr dilelya trikonachya samaroop trikonancha sanch sahparivartanshil asato.
heey khalilpramane jodalele aaheth sahparivartananvar aparivartaniy sthir asatat (udaharanarth, sarvasamaveshak trikonanchi parimiti samaan asate), tarr eka aparivartanachya moolyat sahmat asalelya done vastoo sarvasam asoo shakatat kinwa nasoo shakatat (udaharanarth, samaan parimiti asalelya done trikon sarvasam asane aavashyak nahee). vargikaran samasyanmadhye, ekhadi vyakti aparivartanancha sampoorn sanch shodhanyacha prayatna karoo shakate, jase qee jarr done vastunchi yaa aparivartananchya sanchasathi samaan moolye asateel tarr tee sarvasamatv aahet.
udaharanarth, tinhi bajoo samaan aahet asey trikon kathor halchalinmadhye, ess. ess. ess. sarvasamatedvare sarvasam aahet aanee asha prakare tinhi bajunchi lambi trikonansathi aparivartanancha sampoorn sanch tayar karate. trikonache teen coone mojmap dekhil kathor halchalinmadhye aparivartaniy asatat, parantu sampoorn sanch tayar karat naheet kaaran visangat trikon samaan coone mojmap samayik karoo shakatat. tathapi, jarr ekhadyane kathor halchalinvyatirikt scalingla parvanagi dilee, tarr aaa samanatecha nikash darshavito qee haah aparivartanancha sampoorn sanch aahe.
sadarikaranapasun swatantra.
dusare mhanje, ekhadya ganiteey vastuche kahee sadarikaran kinwa vighatan yaa sandarbhat karya paribhashit kele jaau shakate; udaharanarth, peshi sankulache yular vaishishtya pratyek parimanatil peshinchya sankhyechi paryayi berries mhanoon paribhashit kele jaate. ekhadi vyakti peshi sankul rachana visaroo shakate aanee keval antarnihit topological spacekade (manifold) pahoo shakate-kaaran vegvegalya peshi sankule samaan antarnihit manifold detat, ekhadi vyakti vicharoo shakate qee karya "sadarikaran" chya nivadipasoon "swatantra" aahe qaa, asha paristhitit teey "aantarikpane" paribhashit aparivartaniy aahe. yoolarchya vaishishtyachya babatit heey aahe aanee dilelya sadarikaranasathi tyanna paribhashit karane aanee nantar teey sadarikaranachya nivadipasoon swatantra aahet heey darshavine hii aparivartaniy paribhashit karanyachi aanee ganana karanyachi eka samanya paddhat aahe. yaa arthane samoohik krutichi kalpana nahee heey lakshat ghya.
sarvat samanya udaharane ashee aaheth
vikshobhakhali badal keela jaat nahee.
tisare mhanje, jarr ekhadi vyakti kutumbat badalat asalelya ekhadya vastoocha abhyas karat asel, jase qee bijganitiy bhoomiti aanee vibhedak bhoomitimadhye samanya aahe, tarr ekhadi vyakti vicharoo shakate qee vikshobhat gunadharm badallela nahee qaa (udaharanarth, ekhadi vastoo kutumbanvar sthir asel kinwa metrikchya badalat aparivartaniy asel).
sanganak vidnyanatil aparivartaniy.
sanganak vidnyanat, eka aparivartaniy haah eka tarkik dawa aahe joou sanganak programchya amalbajavanichya vishisht tappyadarmyan nehmich kharaa manla jaato. udaharanarth, loop inverent hii eka aut aahe jee loopchya pratyek punaravruttichya suruvatila aanee shevati kharee asate.
sanganakachya programchya achukatebaddal tark kartana aparivartaniy goshti visheshtah upayukt tharatat. sankalkanna anukool karnyacha siddhant, kararanusar rachanechi karyapaddhati aanee karyakramachi achookata nishchit karnyasathichya aupcharik paddhati, heey serve aparivartananvar mothya pramanat avalambun asatat.
programmer anekada aparivartaniy goshti spasht karnyasathi tyanchya kodmadhil dave vapartat. kahee object oriented programming bhashanmadhye varg aparivartak nirdisht karnyasathi vishesh vakyarachana asate.
atyavashyak karyakramanmadhye svayanchalit aparivartaniy shodh.
amoort arth lavnyachi sadhane dilelya anivarya sanganak programchya sadhya aparivartananchi ganana karoo shakatat. konatya prakarche gunadharm aadhalu shakatat heey vaparlelya amoort kshetranvar avalambun asate. vishisht udaharan gunadharm mhanje cocksey _ 1 sarkhya eka purnank chull shreni, cocksey _ 2 sarkhya anek chalanmadhil sambandh aanee cocksey _ 3 sarkhya modulus mahitee. shaikshanik sanshodhan prototype dekhil nirdeshak sanrachananchya sadhya gunadharmancha vichar karatat.
adhik atyadhunik aparivartak samanyatah hatane puravale pahijet.
visheshtah, hoar calculus vaparoon anivarya programchi padtalani kartana, programmadhil pratyek loopsathi loop invarient swahaste pradan karava lagato, jeey bahutek programsathi haah drushtikon samanyatah avyavharik asanyache eka kaaran aahe.
vareel myu kode udaharananchya sandarbhat, sadhya asey konatehi samanya svayanchalit sadhan nahee jeey fakt 1-4 niyam vaparoon mea teey myu paryant vyutpann karane ashakya aahe heey shodhu shakate. tathapi, ekada stringmadhun tyachya "aay" chya sankhyeparyantche amoort hatane kele gele qee, udaharanarth, pudhil see programkade, eka amoort vyakhya sadhan heey shodhu shakel qee conco _ 4 0 asoo shakat nahee aanee mhanoon "vile"-loop kadhihi samapt honar nahee.
shoonya mubalak (shoonya) {
 asthir antarvarti yadruchchik;
 int icount = 1, eucount = 0;
 tarr (aay. see. akaunt% 3!= 0)// naan sampanari loop
 badal (yadruchchik niyam) {
 prakaran 1: yuu count + = 1; break;
 prakaran 2: icounts * = 2; eucounts * = 2; break;
 prakaran 3: icount-= 3; ucont + = 1; break;
 prakaran 4: yuu count-= 2; break;
 }// sanganit aparivartaniyah icount% 3 = = 1 .. icount% 3 = = 2

