Let $(X,Y,Z)$ be distributed according to the pdf $p(x,y,z)$. What would be the pdf $q(x,y,z)$ of the multivariate random variable $(X',Y',Z') = (X,Y,Z)/\sqrt{X^2+Y^2+Z^2}$? The method that I have studied in school with the Jacobian and the inverse of the transformation doesn't seem to apply to this case (the function doesn't have a well-defined inverse since we lose the length information). Is there any generic method to deal with cases like this?
I would also appreciate any recommendation for a good reference (preferably) for engineers that deals with problems like this too, and not only with the standard problems studied in an engineering degree not focused on mathematics.
I found a reference which solves this issue: https://en.wikibooks.org/wiki/Probability/Transformation_of_Probability_Densities