Rivnya nnya Lapla sa odnoridne linijne rivnyannya v chastkovih pohidnih drugogo poryadku eliptichnogo tipu D u 2 u 2 u x 2 2 u y 2 2 u z 2 0 displaystyle Delta u nabla 2 u frac partial 2 u partial x 2 frac partial 2 u partial y 2 frac partial 2 u partial z 2 0 Funkciyi yaki zadovolnyayut rivnyannyu Laplasa nazivayutsya garmonichnimi Vidpovidne neodnoridne rivnyannya nazivayetsya rivnyannyam Puassona Zmist 1 Inshi formi zapisu rivnyannya Laplasa 2 Zastosuvannya u fizici 3 Div takozh 4 PosilannyaInshi formi zapisu rivnyannya Laplasa RedaguvatiV cilindrichnih koordinatah D f 1 r r r f r 1 r 2 2 f ϕ 2 2 f z 2 0 displaystyle Delta f frac 1 r frac partial partial r left r frac partial f partial r right frac 1 r 2 frac partial 2 f partial phi 2 frac partial 2 f partial z 2 0 nbsp Vivedennya Cilindrichni i pryamokutni koordinati pov yazani tak x r cos 8 y r sin 8 z z displaystyle x r cos theta y r sin theta z z nbsp Mi takozh mozhemo zapisati x 2 y 2 r 2 displaystyle x 2 y 2 r 2 nbsp i 8 tan 1 y x displaystyle theta tan 1 left frac y x right nbsp Pripustimo u x y z displaystyle u x y z nbsp ye neperervnoyu z neperervnimi pershoyu i drugoyu chastkovimi pohidnimi v deyakij oblasti R displaystyle R nbsp Mi takozh mozhemo dumati pro u displaystyle u nbsp yak pro funkciyu vid r 8 z displaystyle r theta z nbsp Z lancyugovogo pravila i poperednih rivnyan mi otrimuyemo u x u r r x u 8 8 x u z z x displaystyle frac partial u partial x frac partial u partial r frac partial r partial x frac partial u partial theta frac partial theta partial x frac partial u partial z frac partial z partial x nbsp x x 2 y 2 u r y x 2 y 2 u 8 displaystyle frac x sqrt x 2 y 2 frac partial u partial r frac y x 2 y 2 frac partial u partial theta nbsp x r u r y r 2 u 8 displaystyle frac x r frac partial u partial r frac y r 2 frac partial u partial theta nbsp Tut mi skoristalis tim sho x displaystyle x nbsp i z displaystyle z nbsp nezalezhni i zapisali z x 0 displaystyle frac partial z partial x 0 nbsp Podibno u y y r u r x r 2 u 8 displaystyle frac partial u partial y frac y r frac partial u partial r frac x r 2 frac partial u partial theta nbsp Shob obchisliti drugu pohidnu diferenciyuyemo dali vidpovidno shodo x displaystyle x nbsp i y displaystyle y nbsp 2 u x 2 u r x x r u 8 x y r 2 x r x u r y r 2 x u 8 displaystyle frac partial 2 u partial x 2 frac partial u partial r frac partial partial x left frac x r right frac partial u partial theta frac partial partial x left frac y r 2 right frac x r frac partial partial x left frac partial u partial r right frac y r 2 frac partial partial x left frac partial u partial theta right nbsp Zauvazhimo sho u x 8 u 8 x displaystyle u x theta u theta x nbsp Todi zastosovuyuchi lancyugove pravilo shodo dvoh ostannih dodankiv otrimuyemo 2 u x 2 displaystyle frac partial 2 u partial x 2 nbsp U sferichnih koordinatah 1 r 2 r r 2 f r 1 r 2 sin 8 8 sin 8 f 8 1 r 2 sin 2 8 2 f f 2 0 displaystyle 1 over r 2 partial over partial r left r 2 partial f over partial r right 1 over r 2 sin theta partial over partial theta left sin theta partial f over partial theta right 1 over r 2 sin 2 theta partial 2 f over partial varphi 2 0 nbsp Zastosuvannya u fizici RedaguvatiRivnyannya Laplasa opisuye elektrostatichne pole v prostori bez elektrichnih zaryadiv Rivnyannyam Laplasa opisuyetsya stacionarnij rozpodil temperaturi u prostorovomu tili Rivnyannyu Laplasa zadovolnyaye potencial gravitacijnih hvil na poverhni ridini Div takozh RedaguvatiOperator Laplasa GradiyentPosilannya RedaguvatiWeisstein Eric W Rivnyannya Laplasa angl na sajti Wolfram MathWorld Cya stattya ne mistit posilan na dzherela Vi mozhete dopomogti polipshiti cyu stattyu dodavshi posilannya na nadijni avtoritetni dzherela Material bez dzherel mozhe buti piddano sumnivu ta vilucheno listopad 2015 nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Rivnyannya Laplasa amp oldid 27744722
