avakal jyaamiti

ek atiprabalayaakaar pairaabolvaayad mein sthit ek tribhuj aur do atismaantar rekhaaeain

anukram

avakal jyaamiti (maapeeya)

avakal jyaamiti (Differential geometry) ganit ki ek vidha (discipline) hai jo kailakulas tatha rekheeya tatha bahurekheeya beejaganit (multilinear algebra) ka upayog karke jyaamiteeya samasyaaon ka adhyayan karti hai. ismein un talon aur bahugunon (maineefolds) ke gunon k adhyayan kiya jaata hai jo apne kisi alpaansh (elimeint) ke sameep sthit hon jaise kisi vakr athva tal ke gunon ka adhyayan, uske kisi bindu ke pados mein. maapeeya avakal jyamiti ka sambandh un gunon se hai jinmein naapane ki kriya nihit ho.

shaastreeya avakal jyaamiti mein aise vakron aur talon ka adhyayan kiya jaata hai jo trivimeeya yooklideeya avakaash (space) mein sthit hon. ismein avakal kalan (difreinshiyl kailkyulas) aur samaakalan (inategral kailkyulas) ki vidhiyon ka prayog hota hai; ya yo kahiye ki is vidya mein ham vakron aur talon ke un gunon ka adhyayan karte hain, jo trivistaari gatiyon mein bhi nishchal (inavairiyant) rahate hain.

avakal jyaamiti (prakshepeeya)

vikshepaatmak avakal jyaamiti (projektiv difreshiyl jyometri) mein ham kisi jyaamiteeya aakruti ke kisi saarvik alpaansh (jenaral elimeint) ke sameep uske un gunon ka adhyayan karte hain jinmein kisi saarvik vikshepaatmak roopaantar (traisafaarmeshan) se koi vikaar naheen hota. jaise kisi vakr ke ye gun ki uske kisi bindu par sparsh rekha athva aashleshan samatal (oskyuleting plane) ka astitv hai athva naheen, vikshepaatmak avakaleeya gun hain kintu kisi tal ka yeh gun ki usapar alpaantari (jiodesik) ka astitv hai ya naheen, vikshepaatmak naheen hai, kyonki ismein lanbaai ka bhaav nihit hai jo vikshepaatmak naheen hai.

aakrutiyon ke vikshepaatmak avakal gunon ke adhyayan ki kam se kam teen vidhiyaan nikal chuki hain jo is prakaar hain:

(1) avakal sameekaran,

(2) ghaati-shreni-prasaar (power seeraaj ekspainshan) aur

(3) kisi bindu ke vikshep nirdeshaankon (projektiv koordinets) ka ek praachal (pairaameetar) athva avakal roopon (difreinshiyl faurms) ke padon mein prasaar.

pehli aur teesari vidhiyon mein pradish kalan (teinsar kailkyulas) ka prayog kiya ja sakta hai.

upayukt nirdesh tribhuj (traaiaingil ov refreins) chunane se, jiske chunaav ka dhang adviteeya hoga, kisi samatal vakr ka sameekaran is roop mein dhaala ja sakta hai:

(sameekaran, imej ke roop mein)

is ghaat shreni ke samast gunaank saarvik vikshep roopaantar ke antargat, vakr ke param nishchal (aibasolyoot inavairiyant) hain, at: ve moolabindu par vakr ke samast vikshepaatmak avakal gunon ko vyakt karte hain. kisi vakr ke kisi bindu par ke sparshi ka bhaav suparichit hai. maan leejiye ki ham kisi vakr ke bindu pa ke sameep chaar anya bindu lete hai. jab ye chaaron bindu pa ki or agrasar hote hain, tab in paanchon binduon dvaara kheenche gaye shaankav (kaunik) ki jo seemaasthiti hogi, use vakr ke bindu pa par, aashleshan shaankav (oskayuletig kaunik) kehte hain. isi prakaar ek samatal trighaati (plane kyoobik) ke is gun ki sahaayata se ki usaka nirdhaaran nau svechha (aabitrairi) binduon se hota hai, ham aashleshan trighaati (oskyuletig kyoobik) ki paribhaasha de sakte hain. is adhyayan mein, seema (limit) ke prayog ke kaaran, kalan (kailkyulas) bahut kaam mein aata hai.

saadhaaranataya trivistaari vikshepaatmak avakaash (three-daaimeinshanal projektiv space) mein anantasparshi vakro (aisimpatotik karvj) ke do ekapraachal parivaar (van-pairaameetar faimileej) hote hain. yadi do se kam parivaar hon to tal (sarfes) vikaasya (divelapebul) hoga. yadi do se adhik hon to tal ek samatal (plane) hoga. yadi vikaasya talon or samatalon ko chhod diya jaae aur anantasparshi rekhaaon ko tal ke praachaleeya vakr maan liya jaae to samaghaat nirdeshaank (homojeeniys koaadinets) is prakaar chune ja sakte hain ki ve avakal sameekaranon ki nimnalikhit sanhati (system) ko santusht karein :

inhein fyabin ke avakal sameekaran (difreshiyl ikveshans) kehte hain. gunaakan u, oo p f tal ke nishchal hain.

kisi tal ke vikshepaatmak gunon mein se ek gun hota hai usaka kisi anya tal se sparshakram (order ov kaunataikt). visheshakar, dvighaat talon ka ek tripraachal parivaar hota hai jiska tal (prushth) pru se kisi bindoo moo par dviteeya kram ka sparsh hota hai. yadi dvighaati (kvaudriks) is prakaar chune jaaeain ki moo par, pratichhed vakr ke sparshi, moo ke anantaspashiyon ke prati abhidhruvi (aipolar) ho to dvighaatiyon ko daarbo dvighaati (kvaudriks)3-bindu sparshiyon ko daarbo sparshi kehte hain. pru ke pratyek bindu par daarbo dvighaatiyon ka ek ekapraachal parivaar hota hai. ismein se bahut vishesh prakaar ke dvighaati hote hain. kadaachit li dvighaati (kvaudriks) sabse rochak hote hain. inka vivran is prakaar diya ja sakta hai: moo ke anantasparshi vakr va par do sameepasth bindu pa aur paa2 lekar teenon binduon par anantasparshi vakr ke sparshi kheencho. ye teen sparshi ek dvighaati ka nirdhaaran karte hain. jab pa aur paa2 vakr va ke anudish moo ki or agrasar hote hain, tab ukt dvighaati ki seemaasthiti ko li dvighaati kehte hain.

rekhaaon ke kisi dvipraachal parivaar ko sarvaangasamata (kaungueains) kehte hain. udaaharanat: kisi tal ke maapaatmak abhilanb (motrik naarmals) ek sarvaangasamata banaate hain. yadi pru ke kisi bindu moo ka saahacharya (aisosiyeshan) ek rekha se hai jiski sthiti moo ke saath saath badalti rahati hai to aisi rekhaaon ke sangrah se ek sarvaangasamata ka nirmaan hota hai. jab moo tal pru ke kisi upayukt vakr par chalta hai tab sarvaangasamata ki sahachar rekha vakr ko sparsh karti hai aur is prakaar ek vikaasya tal ka srujan karti hai. saadhaaranat: kisi tal par aise vakron ke do ekapraachal parivaar hote hain. sarvaangasamata ke vikaasya talon se inki sangati baithati hai. ab maan leejiye ki ek sarvaangasamata ka nirmaan tal pru ke binduon ke madhya se jaanevaali aisi rekhaaon se hota hai jo un binduon par kheenche gaye pru ke sparshatalon par sthit naheen hain, to kisi bhi daarbo dvighaati ke prati in rekhaaon ki vyutkram dhruviyaaain (resiprokal polars) ek sarvaagasamata ka nirmaan karti hain jiski rekhaaeain pru ke sparshasamatalon par sthit hoti hain, kintu unke sparshabinduo mein se hokar naheen jaateen. sarvaangasamataaon ke aise jodon ko vyutkram sarvaangasamataaeain (resiprokal kaunagrueainsej) kehte hain. aaj tak vyutkram sarvaangasamataaon ke bahut se jodon ka adhyayan ho chuka hai. inheen mein se ek yugm viljiski ki niyat sarvaangasamataaon (daairektris kaunagraeainsej) ka hai. inki paribhaasha is prakaar di ja sakti hai: yadi t ki vyutkram sarvaangasamataaon ki ek jodi ke vikaasyon ke sangat vakron ke do kulak (sets) abhinn (koinsideint) ho jaaeain to ukt sarvaangasamataaon ko vinljiski ki niyat sarvaangasamataaeain kehte hain.

yeh jaanane ke liye ki vikshep jyaamiti mein sarvaagamataaon ka kya mahatva hai, sanyugmi jaalon (kaunajuget nets) ki kalpana ko bhi samajh lena aavashyak hai. inki paribhaasha ham is prakaar de sakte hain:

maan leejiye, kisi tal pru ke kisi bindu ke madhya se anantasparshi vakr kheenche gaye hain, to is bindu ka sparshi aur ukt vakron par us bindu par kheenche gaye sparshiyon ke prati usaka haraatmak sanyugmi (haarmonik kaunajuget), ye donon milkar sanyugmi sparshi kahalaate hain. yadi sanyugmi sparshiyon ke kisi jode mein se ek ko kisi ekapraachal vakraparivaar ke ek vakr ka sparshi maan liya jaae to jode ka doosra sparshi ek anya ekapraachal vakraparivaar ka sparshi ho jaaega.

vakron ke aise do kulakon se sanyugmi jaal ka nirmaan hota hai. sanyugmi jaalon ka ek anya laakshanik gun (kairektaristik prauparti) in shabdon mein vayakt ho sakta hai: jab koi bindu moo sanyugmi jaal ke ek vakr par chalta hai tab jaal ke doosare vakr par bindu moo par kheenche gaye sparshi ek vikaasya tal ka srujan karte hain. jab ek bindu tal t ke kisi vakr par chalta hai, to usaka maapaatmak abhilanb ek rijurekhaj (roold) tal ka srujan karta hai. yadi vakr ke sthaan mein vakrataarekha (line ov karvechar) lein to yeh rijurekhaj tal vikaasya ho jaata hai. vakrataarekhaaon dvaara nirmit jaal ek sanyugmi jaal hota hai aur maapaatmak abhilanb sarvaangasamata (metriknaurmal kaunagrueains) se usaki sangati (kauresapaundeins) baithati hai. ham isi baat ko is prakaar vyakt karte hain ki maapaatmak abhilanb sarvaangasamata tal se sanyugmi hai.

vikshepaatmak avakal jyamiti mein bahut si sarvaangasamataaeain aisi hain jo saarveekrut abhilanb sarvaangasamataaeain (jenarailaaijd naurmal kaunagraeainsej) kahala sakti hain, kyonki sarvaagasamata ka nirdhaaran tal se hota hai aur vah tal se sanyugmi rahati hai. inheen mein se ek yathaakathit green-fayoobini vikshep abhilanb (projektiv naurmal) bhi hai.

vah vakr jiske sparshi ek vikaasya tal ka nirmaan karte hain, tal ki nishit kor (kaspidl ej) kahalaata hai. moo ke sanyugmi sparshiyon ke laakshanik gun se yeh nishkarsh niklata hai ki jode me se pratyek sparshi rashmibindu (re point) par nishit kor ka sparshi hota hai. is prakaar jo do rashmibindu praapt hote hain ve moo ke jaal ki ek rashmi ka nirdhaaran karte hain. jaal ke vakron ke bindu moo par ke aashleshan samatalon ki pratichhed rekha jaal ka aksh hoti hai. rashmi tatha aksh aur unke dvaara janit sarvaagaasamataaon ka adhyayan bahut se vyaktiyon ne kiya hai.

kuchh logon ne alpaantariyon ki kalpana ka, yeh dekhkar ki inka maapaatmak avakal jyaamiti mein kitna mahatva hai, vikshep jyaamiti mein prayog karne ka prayatn kiya hai. pratham to nishchal anukal ke baahmajon (ekstreemals) ko vikshep alpaantari kehte hain. samast vikshep alpaantariyon ke aashleshan samatal kaksha 3 ka ek shanku (kon) banaate hain. ukt shanku ka nishit aksh green aur fyoobini ka vikshep abhilanb hota hai. alpikaaon ka ek anya saarveekaran sarvaagasamata ke sanyog vakr (union karv) mein milta hain. ukt vakr tal pru ka ek aisa vakr hota hai, jiske pratyek bindu ka aashleshan samatal us bindu ki sarvaagasamata rekha (line ov kaunagrueains) ke madhya se jaata hai.

sandarbh granth

  • leson sur la thiori jeneraal de surafaas, 4 khand (peris, 1887-96);
  • len E.pi. : 1. projektiv difreshial jiometri ov karvj aind sarfesej (Chicago,1932); 2. A treeteej on projektiv difreinshial jiaametri (Chicago, 1942);
  • ji. fyoobini aur sekh : jiometriya proiettiva difretsiaal 2. khand (bolonya, 1926-27);
  • viljiski, E.ji. : projektiv difreinshial jiometri ov karvj aind roold sarfesej (laaipajig 1906). (ra.bi.)

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