I'm having trouble finding the Jacobian when trying to compute a distribution. If $(X,Y)$ is a point on a unit disk with radius $1$, I'd like to find the density of the distance between the point and the centre of the disk.
So the joint density function of $X$ and $Y$ is
$$ f_{XY}(x,y) = \begin{cases} 1/\pi, & x^2 + y^2 \leq 1,\\ 0, & \text{otherwise} \end{cases}. $$
I put $U = \sqrt{X^2+Y^2}$ and use an auxiliary random variable $V = \mathrm{arctan}(Y/X).$ So there is a 2-to-1 mapping.
Now I wonder what the Jacobian might be. Or, well, the book tells me it's $u$, but I don't know how to get there myself. Using wolfram alpha (admittedly I'm probably just using it wrong) gets me the answer $0$, not $u$.
Normally, I would work with simpler auxiliary variables, which have helped me compute the Jacobian for other problems, but in this case I'm adviced to use $\mathrm{arctan}(Y/X)$ and that could be what makes this so hard for me.