Is there a general algorithm to determine the new region of integration upon a multivariable change of variables (where the old variables are a function of all the new variables). I have to do a 6-dimensional integral of an ugly integrand over a simple region. I found a change of variables that transforms the integrand to something really nice, but I have no idea how to get the new integration ranges.
Actually, the change of variables is complex, which means the new integrals are actually contour integrals in their respective complex planes. So I need the new integration contours.
The only method I know of to find the new integration ranges is from what I learned in school: draw pictures, and look at the corners, edges, and faces of the new region. Well, I can't do that in a 6-dimensional space.
I tried to change the variables sequentially, but failed because each integration variable is a function of all the new integration variables.
What can I do?