I try to understand change of variables for double integrals.
For this reason I try to calculate $\int_0^{\pi/2}\int_0^{\pi/2}\cos\theta\cos{\phi}\,d\theta d\phi$ by using this change $\alpha=\theta+\phi$, $\beta=\theta-\phi$ and $\cos\theta\cos\phi=\frac{1}{2}(\cos\alpha+\cos\beta)$. I repeat I know how to calculate the first integral and this is just an exercise for me. I found that the Jacobian is $J=-\frac{1}{2}$ but I even don't found on which surface I have to integrate!
Does somebody have an idea?