How do I express $\dfrac{\partial^2z}{\partial\theta^2}$ in terms of Cartesian coordinates given that $(x,y)$ are Cartesian coordinates and $(r,\theta)$ are polar coordinates.
Attempt:
$$
\frac{\partial^2z}{\partial\theta^2} =
\frac{\partial }{\partial \theta}\left[\frac{\partial z }{\partial x}\frac{\partial x}{\partial \theta}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial \theta}\right]
$$
I'm not entirely sure if I am on the right track because z isn't specified so simplifying that expression down isn't possible.
Best Answer
Yes, you are on the right track.
If $z=z(x,y)$, then $$ \frac{\partial z}{\partial\theta}=\frac{\partial z }{\partial x}\frac{\partial x}{\partial \theta}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial \theta} =(-y)\frac{\partial z }{\partial x} + x\frac{\partial z}{\partial y}:=w $$ Now you have $w=w(x,y)=(-y)\frac{\partial z }{\partial x} + x\frac{\partial z}{\partial y}$. And you simply repeat what you did for $z$ to $w$. Note that $$ \frac{\partial^2 z}{\partial\theta^2}=\frac{\partial w}{\partial\theta}. $$