In general, the conversion function between two distributions $g(x)$ and $h(y)$ is,

$g(x)dx=h(y)dy$

By integration on both side, we have

$G(x) = H(y) \Rightarrow y=H^{-1}(G(x))=f(x)$

We are going to show a generation of uniform distribution on a Disk first. We set the distribution $g(x) = 1$ be a uniform distribution form $(0,1)$. From the polar coordinate, we have,

$dx dy=rdrd\theta$

the right side is the unit area element in polar coordinate. We can set $dy = d\theta$, $h(r) = r$, thus,

$H(r) = r^2/2 \Rightarrow x=r^2/2 \Rightarrow f(x)=\sqrt(2x)$

in Mathematica,

n=10000;
θ=RandomReal[{0,2 pi},n];
f[x_]:=Sqrt[x];
r=Table[f[RandomReal[{0,1}]],{i,1,n}];
disk=Table[r[[i]]{Cos[θ[[i]]],Sin[θ[[i]]]},{i,1,n}]
ListPlot[disk]

Similarly, For a uniform sphere,

$dxdydz=r^2dr sin(\theta)d\theta d\phi$

Thus we have,

$f_{r}(x) = (3x)^{1/3}, f_{\theta}(y)=Cos^{-1}(2y-1), f_{\phi}(z)=z$