Expressions used in change of variables

[eccefeles]

Hi. I know the title is not very informative. Here's what I'm trying to do:
I have f(x,y). I want to perform a change of variables to obtain a pre-defined g(u,v). How can I work out the actual expressions u(x,y) and v(x,y) so that it works out (including the Jacobian as well)?

I have a vague feeling that there may not be a general approach but only techniques for specific forms of f and g. I have Googled and searched for "change of variables" on this forum. 9 pages later, I think no one has asked this before. Some directions (even Google keywords) would be much appreciated.

 

[HallsofIvy]

Actually, you can't! There may not be any u(x,y), v(x,y) that converts a given f(x,y) into a given g(u,v)!

 

[eccefeles]

OK, I accept there is no general approach. However, what about some "special" forms of f(x, y) and g(u, v)?

I started pondering this question after I read about the Box-Muller method of generating random deviates with a Gaussian distribution from uniform deviates. (http://en.wikipedia.org/wiki/Box-Muller_transform" ) It made me wonder if there are systematic methods of working out alternative transforms that would achieve the same purpose and be superior in terms of computing efficiency.