If we are given the probability density function of a multidimensional random variable, how to find the density of the corresponding normalized random variable?
In other words, given $f_{X_1,X_2,\dots,X_n}(x_1,x_2,\dots,x_n)$ to be pdf of $(X_1,X_2,\dots,X_n)$, how to find the density of $\left(\frac{X_1}{\sqrt{X_1^2+X_2^2+\dots+X_n^2}},\frac{X_2}{\sqrt{X_1^2+X_2^2+\dots+X_n^2}},\dots,\frac{X_n}{\sqrt{X_1^2+X_2^2+\dots+X_n^2}}\right)$?
I was thinking of defining $\frac{X_i}{\sqrt{X_1^2+X_2^2+\dots+X_n^2}}=Y_i$ for all $i$, then writing $(X_1,X_2,\dots,X_n) = (rY_1,rY_2,\dots,rY_n)$ and then using Change of variable formula. But that didn't work because the Jacobian matrix becomes rectangular then.
I think you're heading to the right direction there. You get rectangular Jacobian because once $n-1$ of the $Y_{i}$ are determined, the last $Y_{i}$ is fixed (because their sum of squares equals one).
I would define $Y_{1}=\sqrt{1-\sum_{i=2}^{n}{Y_{i}^{2}}}$, find the density function of $\left(r,Y_{2},Y_{3},...,Y_{n}\right)$ using Change of Variable and then integrate w.r.t $r$ from $0$ to $\infty$.