$U=\{(u,v)\in {R}^2:u>0\}$
Define a function $F:U \rightarrow R^2$ by $F(u,v)= (u\cos(v),u\sin(v))= (x,y)$
a) Show $F$ is an open mapping on $U$. [I've done this.]
b) Calculate $\Large \frac{\partial u}{\partial x},\frac{\partial u}{\partial y},\frac{\partial v}{\partial x},\frac{\partial v}{\partial y}$.
Partial result: I think I can write $u$ in terms of $x$ and $y$ by doing the following: $x^2+y^2=u^2(\sin^2(v)+\cos^2(v))$ Thus we have $u=\sqrt{x^2+y^2}$. Then I could differentiate with respect to $x$ of course.
However, I think there is a way to compute these partials directly from the derivative of $F$ using the Inverse Function Theorem, but I don't know how.