Teorema pro neyavnu funkciyu zagalna nazva dlya teorem sho garantuyut lokalne isnuvannya i opisuyut vlastivosti neyavnoyi funkciyi tobto funkciyi y f x f X Y displaystyle y f x quad f colon X to Y zadanoyi rivnyannyam F x y 0 F X Y Z displaystyle F x y 0 quad F colon X times Y to Z U matematici tochnishe v analizi funkcij bagatoh zminnih teorema pro neyavnu funkciyu a ye instrumentom yakij dozvolyaye perehoditi vid spivvidnoshen do funkcij kilkoh dijsnih zminnih en Ce dosyagayetsya shlyahom predstavlennya spivvidnoshen u viglyadi grafika funkciyi Mozhe ne isnuvati yedinoyi funkciyi grafik yakoyi mozhe predstavlyati vse spivvidnoshennya ale mozhe isnuvati taka funkciya pri obmezhenni na oblast viznachennya spivvidnoshennya Teorema pro neyavnu funkciyu daye dostatnyu umovu isnuvannya takoyi funkciyi Tochnishe dlya zadanoyi sistemi z m displaystyle m rivnyan f i x 1 x n y 1 y m 0 displaystyle f i x 1 dots x n y 1 dots y m 0 i 1 m displaystyle i 1 dots m chasto skorocheno zapisuyemo yiyi yak F x y 0 displaystyle F x y 0 teorema stverdzhuye sho za nezhorstkih umov na chastinni pohidni vidnosno zminnih y i displaystyle y i u tochci m displaystyle m zminnih y i displaystyle y i ye diferencijovnimi funkciyami zminnih x j displaystyle x j v deyakomu okoli tochki Hocha ci funkciyi u zagalnomu vipadku ne mozhut buti predstavleni v zamknenomu viglyadi en ale voni neyavno viznachayutsya rivnyannyami i ce motivuvalo nazvu teoremi 1 Inshimi slovami za nezhorstkih umov na chastinni pohidni mnozhina nuliv sistemi rivnyan lokalno en ye grafikom funkciyi Zmist 1 Istoriya 2 Pershij priklad 3 Oznachennya 4 Formulyuvannya teoremi 4 1 Pohidni vishih poryadkiv 5 Odnovimirnij vipadok 6 Bagatovimirnij vipadok 7 Dovedennya v dvovimirnomu vipadku 8 Priklad kola 9 Zastosuvannya zmina koordinat 9 1 Priklad polyarni koordinati 10 Uzagalnennya 10 1 Vipadok banahovogo prostoru 10 2 Vipadok nediferencijovanih funkcij 11 Divis takozh 12 Vinoski 13 Primitki 14 LiteraturaIstoriya RedaguvatiOgyustenu Luyi Koshi 1789 1857 pripisuyut pershe chitke formulyuvannya teoremi pro neyavnu funkciyu Uliss Dini 1845 1918 uzagalniv variant teoremi pro neyavnu funkciyu na vipadok funkcij bud yakoyi kilkosti dijsnih zminnih 2 Pershij priklad Redaguvati nbsp Odinichne kolo mozhna zadati u viglyadi liniyi rivnya f x y 1 displaystyle f x y 1 nbsp funkciyi f x y x 2 y 2 displaystyle f x y x 2 y 2 nbsp V okoli tochki A displaystyle A nbsp zminnu y displaystyle y nbsp mozhna viraziti yak funkciyu y x displaystyle y x nbsp U comu prikladi cyu funkciyu mozhna zapisati v yavnomu viglyadi yak g 1 x 1 x 2 displaystyle g 1 x sqrt 1 x 2 nbsp U bagatoh vipadkah takogo yavnogo spivvidnoshennya ne isnuye ale vse she mozhna vikoristovuvati termin neyavnoyi funkciyi y x displaystyle y x nbsp V okoli tochki B displaystyle B nbsp takoyi funkciyi nemaye Yaksho viznachiti funkciyu f x y x 2 y 2 displaystyle f x y x 2 y 2 nbsp to rivnyannya f x y 1 displaystyle f x y 1 nbsp virizaye odinichne kolo yak mnozhinu rivnya x y f x y 1 displaystyle x y f x y 1 nbsp Nemaye mozhlivosti predstaviti odinichne kolo u viglyadi grafika funkciyi odniyeyi zminnoyi y g x displaystyle y g x nbsp oskilki dlya kozhnogo x 1 1 displaystyle x in 1 1 nbsp ye dva varianti dlya zminnoyi y displaystyle y nbsp a same 1 x 2 displaystyle pm sqrt 1 x 2 nbsp Odnak mozhna predstaviti chastinu kola u viglyadi grafika funkciyi odniyeyi zminnoyi Yaksho vizmemo g 1 x 1 x 2 displaystyle g 1 x sqrt 1 x 2 nbsp pri 1 x 1 displaystyle 1 leq x leq 1 nbsp to grafik y g 1 x displaystyle y g 1 x nbsp viznachaye verhnyu polovinu kola Analogichno yaksho g 1 x 1 x 2 displaystyle g 1 x sqrt 1 x 2 nbsp todi grafik y g 2 x displaystyle y g 2 x nbsp viznachaye nizhnyu polovinu kola Cil teoremi pro neyavnu funkciyu rozpovisti pro isnuvannya takih funkcij yak g 1 x displaystyle g 1 x nbsp i g 2 x displaystyle g 2 x nbsp navit u vipadkah koli nemozhlivo zapisati yavni formuli Ce garantuye sho g 1 x displaystyle g 1 x nbsp i g 2 x displaystyle g 2 x nbsp ye diferencijovanimi i ce pracyuye navit u vipadkah koli nemaye formuli dlya f x y displaystyle f x y nbsp Oznachennya RedaguvatiNehaj f R n m R m displaystyle f colon mathbb R n m rightarrow mathbb R m nbsp neperervno diferencijovana funkciya Rozglyanemo R n m displaystyle mathbb R n m nbsp yak dekartiv dobutok prostoriv R n R m displaystyle mathbb R n times mathbb R m nbsp i zapishemo tochku cogo dobutku yak x y x 1 x n y 1 y m displaystyle bf x bf y x 1 dots x n y 1 dots y m nbsp Startuyuchi z danoyi funkciyi f displaystyle f nbsp nasha meta polyagaye v pobudovi funkciyi g R n R m displaystyle g colon mathbb R n rightarrow mathbb R m nbsp grafik yakoyi x g x displaystyle bf x g bf x nbsp ye same mnozhinoyu vsih tochok x y displaystyle bf x bf y nbsp takih sho f x y 0 displaystyle f bf x bf y bf 0 nbsp Yak zaznachili vishe ce ne zavzhdi mozhlivo Tomu zafiksuyemo tochku a b a 1 a n b 1 b m displaystyle textbf a textbf b a 1 dots a n b 1 dots b m nbsp yaka zadovolnyaye rivnyannya f a b 0 displaystyle f textbf a textbf b textbf 0 nbsp i znajdemo funkciyu g displaystyle g nbsp viznachenu v okoli tochki a b displaystyle textbf a textbf b nbsp Inshimi slovami znahodimo vidkritu mnozhinu U R n displaystyle U subset mathbb R n nbsp sho mistit tochku a displaystyle textbf a nbsp ta vidkritu mnozhinu V R m displaystyle V subset mathbb R m nbsp sho mistit tochku b displaystyle textbf b nbsp i funkciyu g U V displaystyle g colon U rightarrow V nbsp taku sho grafik funkciyi g displaystyle g nbsp zadovolnyaye spivvidnoshennya f a b 0 displaystyle f textbf a textbf b textbf 0 nbsp na U V displaystyle U times V nbsp i sho zhodni inshi tochki z U V displaystyle U times V nbsp ne zadovolnyayut cyu umovu U simvolnij formi x g x x U x y U V f x y 0 displaystyle textbf x g textbf x textbf x in U textbf x textbf y in U times V f textbf x textbf y textbf 0 nbsp Shob sformulyuvati teoremu pro neyavnu funkciyu neobhidne ponyattya matrici Yakobi dlya funkciyi f displaystyle f nbsp yaka ye matriceyu chastinnih pohidnih funkciyi f displaystyle f nbsp Vvivshi poznachennya a 1 a n b 1 b m displaystyle a 1 dots a n b 1 dots b m nbsp yak a b displaystyle textbf a textbf b nbsp matricyu Yakobi mozhna zapisati u viglyadi D f a b f 1 x 1 a b f 1 x n a b f m x 1 a b f m x n a b f 1 y 1 a b f 1 y m a b f m y 1 a b f m y m a b X Y displaystyle begin aligned amp Df mathbf a mathbf b left begin matrix dfrac partial f 1 partial x 1 mathbf a mathbf b amp cdots amp dfrac partial f 1 partial x n mathbf a mathbf b vdots amp ddots amp vdots dfrac partial f m partial x 1 mathbf a mathbf b amp cdots amp dfrac partial f m partial x n mathbf a mathbf b end matrix right left begin matrix dfrac partial f 1 partial y 1 mathbf a mathbf b amp cdots amp dfrac partial f 1 partial y m mathbf a mathbf b vdots amp ddots amp vdots dfrac partial f m partial y 1 mathbf a mathbf b amp cdots amp dfrac partial f m partial y m mathbf a mathbf b end matrix right X Y end aligned nbsp de X displaystyle X nbsp matricya chastinnih pohidnih za zminnimi x i displaystyle x i nbsp Y displaystyle Y nbsp matricya chastinnih pohidnih za zminnimi y j displaystyle y j nbsp Teorema pro neyavnu funkciyu stverdzhuye sho yaksho Y displaystyle Y nbsp ye oborotna matricya to isnuyut vidpovidno vidkriti mnozhini U displaystyle U nbsp V displaystyle V nbsp ta funkciya g displaystyle g nbsp Zapisavshi usi gipotezi razom otrimuyemo nastupne tverdzhennya Formulyuvannya teoremi RedaguvatiNehaj Matricya Yakobi f R n m R m displaystyle f colon mathbb R n m to mathbb R m nbsp neperervno diferencijovana funkciya i nehaj R n m displaystyle mathbb R n m nbsp maye koordinati x y displaystyle textbf x textbf y nbsp Zafiksuyemo tochku a b a 1 a n b 1 b m displaystyle textbf a textbf b a 1 dots a n b 1 dots b m nbsp taku sho f a b 0 displaystyle f textbf a textbf b mathbf 0 nbsp de 0 R m displaystyle mathbf 0 in mathbb R m nbsp nulovij vektor Yaksho matricya Yakobi prava chastina matrici Yakobi z poperednogo punktu J f y a b f i y j a b displaystyle J f mathbf y mathbf a mathbf b left frac partial f i partial y j mathbf a mathbf b right nbsp ye oborotnoyu to isnuye vidkrita mnozhina U R n displaystyle U subset mathbb R n nbsp yaka mistit tochku a displaystyle textbf a nbsp taka sho isnuye yedina neperervno diferencijovana funkciya g U R m displaystyle g colon U to mathbb R m nbsp taka sho g a b displaystyle g mathbf a mathbf b nbsp i f x g x 0 displaystyle f mathbf x g mathbf x mathbf 0 nbsp dlya vsih tochok x U displaystyle mathbf x in U nbsp Bilsh togo yaksho poznachiti livu chastinu matrici Yakobi z poperednogo punktu yak J f x a b f i x j a b displaystyle J f mathbf x mathbf a mathbf b left frac partial f i partial x j mathbf a mathbf b right nbsp to matricya Yakobi chastinnih pohidnih funkciyi g displaystyle g nbsp v U displaystyle U nbsp viznachayetsya za dopomogoyu dobutku matric 3 g i x j x m n J f y x g x m m 1 J f x x g x m n displaystyle left frac partial g i partial x j mathbf x right m times n left J f mathbf y mathbf x g mathbf x right m times m 1 left J f mathbf x mathbf x g mathbf x right m times n nbsp Pohidni vishih poryadkiv Redaguvati Yaksho krim togo funkciya f displaystyle f nbsp ye analitichnoyu abo k displaystyle k nbsp raziv neperervno diferencijovanoyu v okoli tochki a b displaystyle textbf a textbf b nbsp todi mozhna vibrati vidkritu mnozhinu U displaystyle U nbsp dlya togo shob te same vikonuvalos dlya funkciyi g displaystyle g nbsp vseredini U displaystyle U nbsp 4 Vidpovidna teorema v analitichnomu vipadku nazivayetsya teoremoyu pro neyavnu analitichnu funkciyu Odnovimirnij vipadok RedaguvatiProsta teorema pro neyavnu funkciyu polyagaye v nastupnomu Yaksho funkciya F R R R displaystyle F colon mathbb R times mathbb R to mathbb R nbsp neperervna u deyakomu okoli tochki x 0 y 0 displaystyle x 0 y 0 nbsp F x 0 y 0 0 displaystyle F x 0 y 0 0 nbsp i pri fiksovanomu x displaystyle x nbsp funkciya F x y displaystyle F x y nbsp strogo monotonna po y displaystyle y nbsp u danomu okoli todi u deyakomu dvovimirnomu promizhku I I x I y displaystyle I I x times I y nbsp sho ye okolom tochki x 0 y 0 displaystyle x 0 y 0 nbsp i taka neperervna funkciya f I x I y displaystyle f colon I x to I y nbsp sho dlya bud yakoyi tochki x y I displaystyle x y in I nbsp F x y 0 y f x displaystyle F x y 0 Leftrightarrow y f x nbsp Zvichajno dodatkovo peredbachayetsya sho funkciya F displaystyle F nbsp neperervno diferencijovna v comu vipadku umova monotonnosti viplivaye z togo sho F y x 0 y 0 0 displaystyle F y x 0 y 0 neq 0 nbsp tut F y displaystyle F y nbsp poznachaye chastkovu pohidnu F displaystyle F nbsp po y displaystyle y nbsp Bilsh togo v comu vipadku pohidna funkciyi f displaystyle f nbsp mozhe buti obchislena za formuloyu f x F x x f x F y x f x displaystyle f x dfrac F x x f x F y x f x nbsp Bagatovimirnij vipadok RedaguvatiNehaj R n displaystyle mathbb R n nbsp i R m displaystyle mathbb R m nbsp n displaystyle n nbsp i m displaystyle m nbsp vimirni evklidovi prostori z fiksovanimi sistemami koordinat tochki yakih vidpovidno x x 1 x n displaystyle x x 1 dots x n nbsp i y y 1 y m displaystyle y y 1 dots y m nbsp Nehaj F displaystyle F nbsp vidobrazhaye deyakij okil W displaystyle W nbsp tochki x 0 y 0 R n R m displaystyle x 0 y 0 in mathbb R n times mathbb R m nbsp u prostir R m displaystyle mathbb R m nbsp i F 1 F 2 F m displaystyle F 1 F 2 ldots F m nbsp koordinatni funkciyi vid zminnih x 1 x n y 1 y m displaystyle x 1 dots x n y 1 dots y m nbsp vidobrazhennya F displaystyle F nbsp tobto F F 1 F 2 F m displaystyle F F 1 F 2 ldots F m nbsp Pripustimo sho F x 0 y 0 0 displaystyle F x 0 y 0 0 nbsp i vidobrazhennya F displaystyle F nbsp neperervno diferencijovne v okoli W displaystyle W nbsp a yakobian vidobrazhennya y F x 0 y displaystyle y mapsto F x 0 y nbsp ne rivnij nulyu v tochci y 0 displaystyle y 0 nbsp tobto viznachnik matrici F y x 0 y 0 displaystyle frac partial F partial y x 0 y 0 nbsp ne rivnij nulyu Todi isnuyut okoli U displaystyle U nbsp i V displaystyle V nbsp tochok x 0 displaystyle x 0 nbsp i y 0 displaystyle y 0 nbsp vidpovidno v prostorah R n displaystyle mathbb R n nbsp i R m displaystyle mathbb R m nbsp prichomu U V W displaystyle U times V subset W nbsp i yedine vidobrazhennya f U V displaystyle f colon U to V nbsp take sho dlya vsih x U displaystyle x in U nbsp vikonuyetsya totozhnist F x f x 0 displaystyle F x f x 0 nbsp Pri comu f x 0 y 0 displaystyle f x 0 y 0 nbsp i vidobrazhennya f displaystyle f nbsp ye k displaystyle k nbsp raz neperervno diferencijovnim na U displaystyle U nbsp Yaksho funkciya F displaystyle F nbsp ye nepererfno diferencijovnoyu do poryadku k displaystyle k nbsp v mnozhini U V displaystyle U times V nbsp to takoyu zh ye i funkciya f displaystyle f nbsp u mnozhini U displaystyle U nbsp i vikonuyetsya d f d x j x F y x f x 1 F x j x displaystyle frac rm d f rm d x j x left frac partial F partial y x f x right 1 frac partial F partial x j x nbsp Dovedennya v dvovimirnomu vipadku RedaguvatiNehaj funkciya F R 2 R displaystyle F colon mathbb R 2 to mathbb R nbsp neperervno diferencijovana funkciya sho viznachaye krivu F r F x y 0 displaystyle F bf r F x y 0 nbsp Nehaj x 0 y 0 displaystyle x 0 y 0 nbsp tochka na cij krivij Tverdzhennya vishenavedenoyi teoremi mozhna perepisati dlya cogo prostogo vipadku nastupnim chinom Teorema Yaksho F y x 0 y 0 0 displaystyle left frac partial F partial y right x 0 y 0 neq 0 nbsp to dlya krivoyi v okoli tochki x 0 y 0 displaystyle x 0 y 0 nbsp mozhna zapisati y f x displaystyle y f x nbsp de f displaystyle f nbsp dijsnoznachna funkciya Dovedennya Oskilki funkciya F displaystyle F nbsp ye diferencijovanoyu to zapisuyemo diferencial funkciyi F displaystyle F nbsp cherez chastinni pohidni d F grad F d r F x d x F y d y displaystyle rm d F operatorname grad F cdot rm d r frac partial F partial x rm d x frac partial F partial y rm d y nbsp Oskilki obmezhuyemosya ruhom po krivij d F 0 displaystyle rm d F 0 nbsp i za pripushennyam F y 0 displaystyle dfrac partial F partial y neq 0 nbsp v okoli tochki x 0 y 0 displaystyle x 0 y 0 nbsp oskilki F y displaystyle dfrac partial F partial y nbsp neperervna u tochci x 0 y 0 displaystyle x 0 y 0 nbsp i F y x 0 y 0 0 displaystyle left dfrac partial F partial y right x 0 y 0 neq 0 nbsp to otrimuyemo zvichajne diferencialne rivnyannya pershogo poryadku x F d x y F d y 0 y x 0 y 0 displaystyle partial x F rm d x partial y F rm d y 0 quad y x 0 y 0 nbsp Dali shukayemo rozv yazok cogo zvichajnogo diferencialnogo rivnyannya na vidkritomu intervali v okoli tochki x 0 y 0 displaystyle x 0 y 0 nbsp dlya yakogo v kozhnij jogo tochci y F 0 displaystyle partial y F neq 0 nbsp Oskilki funkciya F displaystyle F nbsp neperervno diferencijovana i z pripushennya otrimayemo x F lt y F lt y F 0 displaystyle partial x F lt infty quad partial y F lt infty quad partial y F neq 0 nbsp Z cogo viplivaye sho funkciya x F y F displaystyle dfrac partial x F partial y F nbsp ye neperervnoyu i obmezhenoyu na oboh kincyah intervalu Zvidsi funkciya x F y F displaystyle dfrac partial x F partial y F nbsp ye neperervnoyu za Lipshicom yak vidnosno zminnoyi x displaystyle x nbsp tak i vidnosno zminnoyi y displaystyle y nbsp Otzhe za teoremoyu Pikara Lipshica isnuye yedina funkciya y x displaystyle y x nbsp yaka ye rozv yazkom zadanogo difencialnogo rivnyannya z pochatkovimi umovami Sho j treba bulo dovesti Priklad kola RedaguvatiPovernemosya do prikladu odinichnogo kola U comu vipadku n m 1 displaystyle n m 1 nbsp i f x y x 2 y 2 1 displaystyle f x y x 2 y 2 1 nbsp Matricya chastinnih pohidnih 1 2 displaystyle 1 times 2 nbsp matricya sho zadayetsya formuloyu D f a b f x a b f y a b 2 a 2 b displaystyle Df a b begin bmatrix dfrac partial f partial x a b amp dfrac partial f partial y a b end bmatrix begin bmatrix 2a amp 2b end bmatrix nbsp Takim chinom komponenta Y displaystyle Y nbsp u formulyuvanni teoremi ye prosto chislom 2 b displaystyle 2b nbsp viznachene tak linijne vidobrazhennya ye obernenim Todi j lishe todi todi j lishe todi koli b 0 displaystyle b neq 0 nbsp Z teoremi pro neyavnu funkciyu viplivaye sho mozhna lokalno zapisati kolo u viglyadi funkciyi y g x displaystyle y g x nbsp dlya vsih tochok de y 0 displaystyle y neq 0 nbsp Dlya tochok 1 0 displaystyle pm 1 0 nbsp stikayemosya z problemami yak zaznachalosya vishe Teoremu pro neyavnu funkciyu vse she mozhna zastosuvati do cih dvoh tochok zapisavshi x displaystyle x nbsp yak funkciyu zminnoyi y displaystyle y nbsp tobto x h y displaystyle x h y nbsp teper grafik funkciyi bude mati viglyad h y y displaystyle left h y y right nbsp oskilki pri b 0 displaystyle b 0 nbsp mayemo sho a 1 displaystyle a 1 nbsp i umovi lokalnogo predstavlennya funkciyi v takomu viglyadi vikonuyutsya Neyavnu pohidnu vid funkciyi y displaystyle y nbsp za zminnoyi x displaystyle x nbsp ta vid funkciyi x displaystyle x nbsp za zminnoyu y displaystyle y nbsp mozhna znajti shlyahom povnogo diferenciyuvannya neyavnoyi funkciyi x 2 y 2 1 displaystyle x 2 y 2 1 nbsp i pririvnyuvannya do 0 displaystyle 0 nbsp 2 x d x 2 y d y 0 displaystyle 2x rm d x 2y rm d y 0 nbsp U rezultati d y d x x y i d x d y y x displaystyle dfrac rm d y rm d x dfrac x y quad text i quad dfrac rm d x rm d y dfrac y x nbsp Zastosuvannya zmina koordinat RedaguvatiNehaj zadano m displaystyle m nbsp vimirnij prostir parametrizovanij naborom koordinat x 1 x m displaystyle x 1 dots x m nbsp Vvedemo novu sistemu koordinat x 1 x m displaystyle x 1 dots x m nbsp za dopomogoyu m displaystyle m nbsp funkcij h 1 h m displaystyle h 1 dots h m nbsp kozhna z yakih ye neperervno diferencijovanoyu Ci funkciyi dozvolyayut obchisliti novi koordinati x 1 x m displaystyle x 1 dots x m nbsp tochki z urahuvannyam starih koordinat tochki x 1 x m displaystyle x 1 dots x m nbsp za dopomogoyu formul x 1 h 1 x 1 x m x m h m x 1 x m displaystyle x 1 h 1 x 1 dots x m dots x m h m x 1 dots x m nbsp Mozhna pereviriti chi mozhlive protilezhne zadano koordinati x 1 x m displaystyle x 1 dots x m nbsp chi mozhna povernutisya i obchisliti vihidni koordinati tiyeyi zh tochki x 1 x m displaystyle x 1 dots x m nbsp Teorema pro neyavnu funkciyu daye vidpovid na ce pitannya Koordinati novi ta stari x 1 x m x 1 x m displaystyle x 1 dots x m x 1 dots x m nbsp pov yazani za dopomogoyu formuli f 0 displaystyle f 0 nbsp de f x 1 x m x 1 x m h 1 x 1 x m x 1 h m x 1 x m x m displaystyle f x 1 dots x m x 1 dots x m h 1 x 1 dots x m x 1 dots h m x 1 dots x m x m nbsp Teper matricya Yakobi funkciyi f displaystyle f nbsp u pevnij tochci a b displaystyle a b nbsp de a x 1 x m displaystyle a x 1 dots x m nbsp b x 1 x m displaystyle b x 1 dots x m nbsp viznachayetsya yak D f a b 1 0 0 1 h 1 x 1 b h 1 x m b h m x 1 b h m x m b I m J displaystyle Df a b left begin matrix 1 amp cdots amp 0 vdots amp ddots amp vdots 0 amp cdots amp 1 end matrix left begin matrix dfrac partial h 1 partial x 1 b amp cdots amp dfrac partial h 1 partial x m b vdots amp ddots amp vdots dfrac partial h m partial x 1 b amp cdots amp dfrac partial h m partial x m b end matrix right right I m J nbsp de I m displaystyle I m nbsp m m displaystyle m times m nbsp odinichna matricya J displaystyle J nbsp m m displaystyle m times m nbsp matricya chastinnih pohidnih obchislenih v a b displaystyle a b nbsp Vishe ci bloki poznachalisya X displaystyle X nbsp i Y displaystyle Y nbsp U comu konkretnomu zastosuvanni teoremi zhodna z matric ne zalezhit vid a displaystyle a nbsp Teorema pro neyavnu funkciyu teper stverdzhuye sho mozhna lokalno viraziti x 1 x m displaystyle x 1 dots x m nbsp yak funkciya vid x 1 x m displaystyle x 1 dots x m nbsp yaksho matricya J displaystyle J nbsp ye oborotnoyu Vimoga oborotnosti matrici J displaystyle J nbsp ekvivalentna umovi det J 0 displaystyle det J neq 0 nbsp Otzhe mozhna povernutisya vid shtrihovanih koordinat do neshtrihovanih yaksho viznachnik yakobiana J displaystyle J nbsp vidminnij vid nulya Ce tverdzhennya takozh vidome yak teorema pro obernenu funkciyu Priklad polyarni koordinati Redaguvati V yakosti prostogo zastosuvannya rozglyanemo ploshinu parametrizovanu polyarnimi koordinatami R 8 displaystyle R theta nbsp Perejdemo do novoyi sistemi koordinat dekartovih koordinat viznachivshi funkciyi x R 8 R cos 8 displaystyle x R theta R cos theta nbsp i y R 8 R sin 8 displaystyle y R theta R sin theta nbsp Ce daye zmogu dlya bud yakoyi tochki R 8 displaystyle R theta nbsp znajti vidpovidni dekartovi koordinati x y displaystyle x y nbsp Koli mozhna povernutisya nazad tobto perejti vid dekartovih koordinat do polyarnih Zgidno poperednogo prikladu dlya cogo dostatno vikonannya umovi det J 0 displaystyle det J neq 0 nbsp de J x R 8 R x R 8 8 y R 8 R y R 8 8 cos 8 R sin 8 sin 8 R cos 8 displaystyle J begin bmatrix dfrac partial x R theta partial R amp dfrac partial x R theta partial theta dfrac partial y R theta partial R amp dfrac partial y R theta partial theta end bmatrix begin bmatrix cos theta amp R sin theta sin theta amp R cos theta end bmatrix nbsp Oskilki det J R displaystyle det J R nbsp to peretvorennya nazad do polyarnih koordinat mozhlive yaksho R 0 displaystyle R neq 0 nbsp Otzhe zalishilosya pereviriti vipadok R 0 displaystyle R 0 nbsp Legko pomititi sho u vipadku R 0 displaystyle R 0 nbsp nashe peretvorennya koordinat ne ye oborotnim u pochatku koordinat znachennya 8 displaystyle theta nbsp ne ye odnoznachno viznachenim Uzagalnennya RedaguvatiVipadok banahovogo prostoru Redaguvati Na osnovi teoremi pro obernenu funkciyu v banahovih prostorah mozhna uzagalniti teoremu pro neyavnu funkciyu na vidobrazhennya zi znachennyami v banahovih prostorah 5 6 Nehaj X displaystyle X nbsp Y displaystyle Y nbsp Z displaystyle Z nbsp banahovi prostori a vidobrazhennya f X Y Z displaystyle f colon X times Y rightarrow Z nbsp ye neperervno diferencijovane za Freshe Yaksho x 0 y 0 X Y displaystyle x 0 y 0 in X times Y nbsp f x 0 y 0 0 displaystyle f x 0 y 0 0 nbsp a y D f x 0 y 0 0 y displaystyle y mapsto Df x 0 y 0 0 y nbsp izomorfizm banahovih prostoriv Y displaystyle Y nbsp i Z displaystyle Z nbsp todi isnuye okil U displaystyle U nbsp tochki x 0 displaystyle x 0 nbsp i okil V displaystyle V nbsp tochki y 0 displaystyle y 0 nbsp i diferencijovana za Freshe vidobrazhennya g U V displaystyle g colon U rightarrow V nbsp take sho f x g x 0 displaystyle f x g x 0 nbsp i f x y 0 displaystyle f x y 0 nbsp todi j lishe todi koli y g x displaystyle y g x nbsp dlya vsih x y U V displaystyle x y in U times V nbsp Vipadok nediferencijovanih funkcij Redaguvati U vipadku nediferencijovanoyi funkciyi f displaystyle f nbsp mayut misce rizni formulyuvannya teoremi pro neyavnu funkciyu Standartnim ye te sho v odnovimirnomu vipadku dostatno lokalnoyi strogoyi monotonnosti 7 Nastupne bilsh zagalne formulyuvannya bulo dovedene Kumagayem na osnovi sposterezhennya Dzhittorntuma 8 9 Rozglyanemo neperervne vidobrazhennya f R n R m R n displaystyle f colon mathbb R n times mathbb R m to mathbb R n nbsp take sho f x 0 y 0 0 displaystyle f x 0 y 0 0 nbsp Isnuyut vidkriti okoli A R n displaystyle A subset mathbb R n nbsp i B R m displaystyle B subset mathbb R m nbsp vidpovidno tochok x 0 displaystyle x 0 nbsp i y 0 displaystyle y 0 nbsp taki sho dlya vsih y displaystyle y nbsp z B displaystyle B nbsp vidobrazhennya f y A R n displaystyle f cdot y colon A to mathbb R n nbsp ye lokalnoyu biyekciyeyu todi i lishe todi koli isnuyut vidkriti okoli A 0 R n displaystyle A 0 in mathbb R n nbsp i B 0 R m displaystyle B 0 in mathbb R m nbsp tochok x 0 displaystyle x 0 nbsp i y 0 displaystyle y 0 nbsp taki sho dlya vsih y B 0 displaystyle y in B 0 nbsp rivnyannya f x y 0 displaystyle f x y 0 nbsp maye yedinij rozv yazok x g y A 0 displaystyle x g y in A 0 nbsp de g displaystyle g nbsp neperervna funkciya z B 0 displaystyle B 0 nbsp v A 0 displaystyle A 0 nbsp Divis takozh RedaguvatiTeorema pro obernenu funkciyu Teorema pro postijnij rang teoremu pro neyavnu funkciyu i teoremu pro obernenu funkciyu mozhna rozglyadati yak chastinni vipadki teoremi pro postijnij rang Vinoski Redaguvati U pizanskij matematichnij shkoli yiyi nazivali teoremoyu Dini V anglomovnij literaturi teorema Dini insha teorema matematichnogo analizu Primitki Redaguvati Chiang Alpha C 1984 Fundamental Methods of Mathematical Economics vid 3rd McGraw Hill s 204 206 ISBN 0 07 010813 7 Krantz Steven Parks Harold 2003 The Implicit Function Theorem Modern Birkhauser Classics Birkhauser ISBN 0 8176 4285 4 de Oliveira Oswaldo 2013 The Implicit and Inverse Function Theorems Easy Proofs Real Anal Exchange 39 1 214 216 doi 10 14321 realanalexch 39 1 0207 Fritzsche K Grauert H 2002 From Holomorphic Functions to Complex Manifolds Springer s 34 ISBN 9780387953953 Lang Serge 1999 Fundamentals of Differential Geometry Graduate Texts in Mathematics New York Springer s 15 21 ISBN 0 387 98593 X Edwards Charles Henry 1994 1973 1994 Advanced Calculus of Several Variables Mineola New York Dover Publications pp 417 418 ISBN 0 486 68336 2 Kudryavtsev Lev Dmitrievich 2001 Implicit function U Hazewinkel Michiel Encyclopedia of Mathematics Springer ISBN 978 1 55608 010 4 Jittorntrum K 1978 An Implicit Function Theorem Journal of Optimization Theory and Applications 25 4 575 577 doi 10 1007 BF00933522 Kumagai S 1980 An implicit function theorem Comment Journal of Optimization Theory and Applications 31 2 285 288 doi 10 1007 BF00934117 Literatura RedaguvatiAllendoerfer Carl B 1974 Theorems about Differentiable Functions Calculus of Several Variables and Differentiable Manifolds New York Macmillan s 54 88 ISBN 0 02 301840 2 Binmore K G 1983 Implicit Functions Calculus New York Cambridge University Press s 198 211 ISBN 0 521 28952 1 Loomis Lynn H Sternberg Shlomo 1990 Advanced Calculus vid Revised Boston Jones and Bartlett s 164 171 ISBN 0 86720 122 3 Protter Murray H Morrey Charles B Jr 1985 Implicit Function Theorems Jacobians Intermediate Calculus vid 2nd New York Springer s 390 420 ISBN 0 387 96058 9 Grigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1100 s ukr Fihtengolc G M Kurs differencialnogo i integralnogo ischisleniya Moskva Nauka 1962 T 1 607 s ros Zorich V A Matematicheskij analiz 10 e Moskva MCNMO 2019 T 1 564 s ISBN 978 5 4439 4029 8 Ilin V A Poznyak E G Osnovy matematicheskogo analiza 7 e Moskva Fizmat lit 2004 T 1 644 s ISBN 5 9221 0536 1 Kolmogorov A N Fomin S V Elementy teorii funkcij i funkcionalnogo analiza 4 e izd Moskva Nauka 1976 544 s ISBN 5 9221 0266 4 ros Otrimano z https uk wikipedia org w index php title Teorema pro neyavnu funkciyu amp oldid 39885145
