Change of variables in multiple integrals

Statement

Let $U \subseteq \mathbb{R}^n$ be open, and let $\varphi \colon U \to \mathbb{R}^n$ be injective and of class C^1 with $\det D\varphi(u) \ne 0$ for each $u \in U$. Let $f \colon \varphi(U) \to \mathbb{R}$ be continuous with compact support in $\varphi(U)$. Then $$\int_{\varphi(U)} f(v) \, dv = \int_U f(\varphi(u)) \, \lvert \det D\varphi(u) \rvert \, du.$$

Remarks

Here $\det D\varphi(u)$ is the Jacobian determinant of $\varphi$ at $u$. Under these hypotheses, $\varphi(U)$ is open and $\varphi$ is a diffeomorphism from $U$ onto $\varphi(U)$. The factor $\lvert \det D\varphi(u) \rvert$ is the ratio by which $\varphi$ scales small volumes near $u$. A linear map $A$ maps the unit cube to a parallelepiped of volume $\lvert \det A \rvert$. The formula extends to Lebesgue integrable $f$ on $\varphi(U)$, and to maps that fail the hypotheses only on a set of measure zero. Important cases are integration in polar coordinates, cylindrical coordinates and spherical coordinates.

Explanations

L1 human

In one variable, substitution needs the factor $\varphi'(u)$. In several variables, the Jacobian determinant plays this role. It corrects for the way the change of coordinates stretches or squeezes small regions.