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Ovi korisnici su zahvalili autoru
Gandalf za post:
Daniel
Reputacija: 4.55%
od Gandalf » Utorak, 09. Februar 2016, 16:17
@Onomatopeja
U pravu si, ovo jesu u pitanju parcijalni izvodi, moja greška pa nisam dobro ukucao u Latex-u. I svih ovih dana razbijanja glave kako da ga uradim, mislim da sam pronašao rješenje.
Mislim da si u pravu, zadatak je banalan samim time i zvuči glupo; potrebno je samo "prebaciti" nepoznatu funkciju [inlmath]f(x,y)=f\bigl(x(u,v),y(u,v)\bigr)=\Phi(u,v)[/inlmath] u [inlmath]x,y[/inlmath] oblik.
E sad, ja sam to radio po definiciji za parcijalne izvode višeg reda
[dispmath]\frac{\partial^2\Phi}{\partial u\partial v}=\frac{\partial}{\partial u}\left[\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}\right]=\\
=\frac{\partial^2f}{\partial x^2}\frac{\partial x}{\partial u}\frac{\partial x}{\partial v}+\frac{\partial^2f}{\partial y\partial x}\frac{\partial y}{\partial u}\frac{\partial x}{\partial v}+\frac{\partial f}{\partial x}\frac{\partial^2x}{\partial u\partial v}+\frac{\partial^2f}{\partial x\partial y}\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}+\frac{\partial^2f}{\partial y^2}\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial^2y}{\partial u\partial v}[/dispmath]
Dalje za ovaj drugi član sam radio ovako
[dispmath]\frac{\partial^2\Phi}{\partial v^2}=\frac{\partial^2f}{\partial y^2}\left(\frac{\partial y}{\partial v}\right)^2+\frac{\partial^2f}{\partial x\partial y}\frac{\partial x}{\partial v}\frac{\partial y}{\partial v}+\frac{\partial f}{\partial y}\frac{\partial^2y}{\partial u^2}+\frac{\partial^2f}{\partial y\partial x}\frac{\partial y}{\partial v}\frac{\partial x}{\partial v}+\frac{\partial^2f}{\partial x^2}\left(\frac{\partial x}{\partial v}\right)^2+\frac{\partial f}{\partial x}\frac{\partial^2x}{\partial v^2}[/dispmath]
Treći član bi izgledao ovako
[dispmath]\frac{\partial\Phi}{\partial v}=\frac{\partial f}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial f}{\partial x}\frac{\partial x}{\partial v}[/dispmath]
Nakon toga bi išao račun i to je to ako se ne varam.
Voila.
PS. Naravno provjerite ako sam pogrešio.