change of variables

Sometimes a region is awkward in one coordinate system but tidy in another — a disk is a mess in x and y, but a clean rectangle in polar coordinates. Change of variables lets you re-map the region to the friendlier one, but you must pay a price: a stretching factor that accounts for how the new coordinates compress or expand area or volume locally.

Precisely, let T be a C^1 bijection from an open set in R^n onto another, with everywhere-nonzero Jacobian determinant. Then for an integrable f, the integral of f over T(U) equals the integral over U of f(T(u)) times the absolute value of the Jacobian determinant of T. That extra factor, |det DT|, is the local volume-scaling of the substitution; it is the multivariable generalization of the one-variable du = (du/dx) dx.

The Jacobian determinant — not the matrix, the determinant, and in absolute value — is the whole point. Forgetting it is the single most common mistake. In polar coordinates the factor is r; in spherical coordinates it is rho^2 sin(phi). The hypotheses (bijectivity, C^1, nonvanishing Jacobian) can be relaxed to hold off a measure-zero set, which suffices in practice.