This is an example from my textbook.
A man throws darts at a circular target with radius 1. The darts are uniformally spread over the board, meaning that $f_{(x,y)}= \begin{cases} \frac{1}{\pi} & \text{if } x^2+y^2\leq 1 \\ 0 & \text{otherwise} \end{cases}$.
We are interested in the distribution of $U=\sqrt{X^2+Y^2}$, i.e. the distance from the center of the target.
Using the "Transformation Theorem" we introduce the auxillary variable $V = \arctan(Y/X)$ and get that the joint distribution of $(U,V)$ is
$f_{U,V}(u,v)=\begin{cases} \frac{2}{\pi}\cdot u & \text{if } 0 < u < 1, \frac{-\pi}{2} < v < \frac{\pi}{2} \\ 0 & \text{otherwise} \end{cases}$
I follow along untill here.
From that it follows, according to the book, that $f_{U}(u)=2u$ for $0<u<1$, and that U and V are independent (and presumably $f_V(v)=\frac{1}{\pi}$).
But how do we know that this is indeed the distribution of the variables, instead of some different distributions that give the same joint distribution (for instance, $f_{U}(u)=\frac{2u}{\pi}$ and $f_V(v)=1$)?