I've a question and it is:
Evaluate ${\partial^2z \over \partial u^2}+{\partial^2z \over \partial v^2}$, if ${\partial^2z \over \partial x^2}+{\partial^2z \over \partial y^2}=0$ and $z=z(x,y)$, $x=e^u cos v$ and $y=e^u sin v$ I can't solve it because I don't know how to begin and which one I derive first.
By the chain rule,
$$\begin{align} \frac{\partial z}{\partial u}&=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u}\\ &=(e^u \cos v)\frac{\partial z}{\partial x}+(e^u \sin v)\frac{\partial z}{\partial y} \end{align}$$
$$\begin{align} \frac{\partial^2 z}{\partial u^2}&=\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u}\right)\\ &=\frac{\partial }{\partial u}\left(\frac{\partial z}{\partial x}\right)\frac{\partial x}{\partial u}+ \left(\frac{\partial z}{\partial x}\right)\left(\frac{\partial x}{\partial u}\right)^2+ \frac{\partial }{\partial u}\left(\frac{\partial z}{\partial y}\right)\frac{\partial y}{\partial u}+ \left(\frac{\partial z}{\partial y}\right)\left(\frac{\partial y}{\partial u}\right)^2 \end{align}$$
Can you take it from here?