A theorem which effectively describes how lengths, areas, volumes, and generalized n-dimensional volumes (contents) are distorted by differentiable functions. In particular, the change of variables theorem reduces the whole problem of figuring out the distortion of the content to understanding the infinitesimal distortion, i.e., the distortion of the derivative (a linear map), which is given by the linear map's determinant. So f:R^n->R^n is an area-preserving linear transformation iff left bracketing bar det(f') right bracketing bar = 1, and in more generality, if S is any subset of R^n, the content of its image is given by left bracketing bar det(f') right bracketing bar times the content of the original. The change of variables theorem takes this infinitesimal knowledge, and applies calculus by breaking up the domain into small pieces and adds up the change in area, bit by bit.

implicit function theorem | Jacobian