Some time ago I was asked the question: do you know how to do a change of variables in a multi-dimensional definite integral?  I thought I knew, but I was wrong.  I only know how to do a change of variable in a multi-dimensional indefinite integral. 

This question really intrigued me.  Of course, this is such an obvious question, I just started pulling out textbook after textbook off my bookshelf, expecting to find the answer in one of them.  Much to my surprise, not a single one of them covered this topic!  The indefinite case was covered in detail, but the definite case was either not mentionned or glossed over.  But if one is trying to implement an algorithm to perform such a change of variable, the details are crucial!  Wikipedia actually has the best coverage that I have found.  But it is still nowhere good enough to write a piece of code.

I have been able to figure out some cases (via pen and paper), but anytime I try to generalize my ideas, things fall apart.  I won't spell out what I have here (yet), as I think this is kind of a fun mathematical problem that some of you might enjoy puzzling out.  I won't even show you the neat counter-examples I have to overly simplistic approaches to the problem.  I am rather hoping that no one will crack it too fast!  In that case, I think it would be fun if we could work out the details here on primes.