I have been following the proof of the Box-Muller transform, (eg Box-Muller Transform Normality) and I have a question about the process of finding the pdf of the transformed variables. Doesn’t the transformation have to be monotonic in order to use this formula:
$$f_{U, T}(u,t) = f_{X,Y}(x, y) |\delta(x,y) / \delta(u, t) |$$