Chapter 6, Problem 6.52P

Let X and Y denote the coordinates of a point uniformly chosen in the circle of radius I centered at the origin. That is, their joint density is $f\left(x,y\right)=\frac{1}{\pi }{x}^2+y^2\le 1$

Find the joint density function of the polar coordinates $R={\left(X^2+Y^2\right)}^{\frac{1}{2}}$ and $\varnothing ={\tan }^{-1}\frac{Y}{X}$.