The task is this: given the following pair of functions:
\begin{cases} u = e^x cos(y) \\ v = e^x sin(y) \end{cases}
Determine the inverse functions, and compute the Jacobian of the inverse functions of these functions.
Am I understanding correctly that what the Jacobian matrix is:
\begin{bmatrix} \frac{\partial u^{-1}_x}{\partial x} & \frac{\partial u^{-1}_y}{\partial y} \\ \frac{\partial v^{-1}_x}{\partial x} & \frac{\partial v^{-1}_y}{\partial y} \end{bmatrix}
Just to finish off what Simon S said, to get the inverse functions:
Note that $u^2+v^2=e^{2x} \rightarrow x(u,v) = \frac{\ln (u^2+v^2)}{2}$
and $\frac v u= \tan y \rightarrow y(u,v) = \arctan \frac v u$.
Now, the Jacobian is $\begin{vmatrix} x_u & x_v \\ y_u & y_v \end{vmatrix} = \begin{vmatrix} \frac{u}{u^2+v^2} & \frac{v}{u^2+v^2} \\ \frac{-u}{u^2+v^2} & \frac{v}{u^2+v^2} \end{vmatrix} = \frac{2uv}{(u^2+v^2)^2}$
(of course there are certain restrictions on domains and ranges from the arctan and the sines and cosines but whatever)