Vi har en funksjon w = f(u,v) som tilfredstiller Laplace-likningen fuu + fvv = 0, [tex]u = \frac{x^2-y^2}{2}[/tex], [tex]v = xy[/tex]. Hvis da at w tilfredstiller Laplace-likningen w_xx + w_yy = 0.
[tex]\frac{\partial{w}}{\partial{x}} = \frac{\partial{w}}{\partial{u}}x + \frac{\partial{w}}{\partial{v}}y[/tex]
[tex]\frac{\partial^2{w}}{\partial{x}^2} = \frac{\partial^2{w}}{\partial{u}^2}x + \frac{\partial{w}}{\partial{u}} + \frac{\partial^2{w}}{\partial{u}^2}y[/tex]
[tex]\frac{\partial{w}}{\partial{y}} = -\frac{\partial{w}}{\partial{u}}y + \frac{\partial{w}}{\partial{v}}x[/tex]
[tex]\frac{\partial^2{w}}{\partial{y}^2} = -\frac{\partial^2{w}}{\partial{u}^2}y - \frac{\partial{w}}{\partial{u}} + \frac{\partial^2{w}}{\partial{v}^2}x[/tex]
Gir:
[tex]w_{xx}+w_{yy} = x(f_{uu} + f_{vv}) + y(f_{vv}-f_{uu})[/tex]
Hvor er fuck-up'en min?
![Razz :P](./images/smilies/icon_razz.gif)