In general, the conversion function between two distributions and is,

By integration on both side, we have

We are going to show a generation of uniform distribution on a Disk first. We set the distribution be a uniform distribution form . From the polar coordinate, we have,

the right side is the unit area element in polar coordinate. We can set , , thus,

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,

Thus we have,

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