change-of-variables theorem

In a single integral, the substitution rule lets you trade x for a new variable u, paying the price dx = (dx/du) du — that little factor dx/du stretches or compresses the axis to keep the area honest. The change-of-variables theorem is exactly this idea in two or three dimensions. You replace the original coordinates by new ones that fit the region better (polar instead of Cartesian for a disk, say), and you pay a stretch factor that accounts for how the coordinate change distorts areas or volumes. That stretch factor is the Jacobian.

Precisely: suppose a transformation takes new coordinates (u, v) to old coordinates (x, y) = (x(u, v), y(u, v)), mapping a tidy region S in the uv-plane onto your messy region R in the xy-plane. Then the double integral over R of f(x, y) dA equals the double integral over S of f(x(u, v), y(u, v)) times the absolute value of J, dudv, where J is the Jacobian determinant — the determinant of the matrix of partial derivatives of (x, y) with respect to (u, v). The absolute value of J measures how much a tiny uv-rectangle's area is scaled when mapped into the xy-plane. The same statement holds in three dimensions with a 3-by-3 Jacobian. Polar, cylindrical, and spherical coordinate formulas are all special cases: their Jacobians are r, r, and rho^2 sin(phi).

This theorem is the engine behind nearly every multiple integral worth doing. It is why integrating over a disk in polar coordinates introduces the famous extra r, why the spherical volume element carries rho^2 sin(phi), and why the multidimensional Gaussian integral can be cracked by switching to polar coordinates. The honest fine print: the map should be smooth and one-to-one on the interior (a bijection up to a measure-zero set), with a non-vanishing Jacobian there — at points where the Jacobian is zero, like the origin in polar coordinates, the map degenerates, but a single such point or curve does not spoil the integral.