Inverse function of a function of two variables

[closet mathemetician]

If I have z=f(x,y), then how would I go about finding the inverse function?

More specifically, say I have a parametric function of the form

f(u,v) = (x(u,v), y(u,v),0)

which is a coordinate transformation. How do I find the inverse of this function?

All references I can find on inverse functions deal with single variable functions.

Say I choose various 2-variable functions as coordinate charts that I can map to a 2-dimensional manifold. I can pick various functions easily enough, but I'm having trouble figuring out how to get their inverses.

I'm trying to understand this whole idea of contraviant and covariant vectors and raising and lowering indicies and I want to play with some examples, but I'm running into the problem above.

 

[mathwonk]

for invertible functions from 2 dimensions to 2 dimensions, try polar coordinatyes, x = rcos(t), y = rsin(t).

then r^2 = x^2 + y^2, and t = arctan(y/x).

on appropriate domains.