The Change of Variables Theorem

The change of variables theorem (also known as the substitution formula for multiple integrals) generalizes $u$ -substitution to higher dimensions using the Jacobian determinant.

The General Theorem

Theorem9.6Change of Variables in $\mathbb{R}^n$

Let $T : G \to D$ be a $C^1$ bijection between open regions in $\mathbb{R}^n$ with $\det J_T \neq 0$ on $G$ . Then for any integrable function $f$ on $D$ : $\int_D f(\mathbf{x})\,d\mathbf{x} = \int_G f(T(\mathbf{u})) \, |\det J_T(\mathbf{u})| \, d\mathbf{u}$ where $J_T$ is the Jacobian matrix of $T$ .

The absolute value of the Jacobian determinant $|\det J_T|$ accounts for how the transformation $T$ locally stretches or compresses volume. Intuitively, $|\det J_T(\mathbf{u})| \, d\mathbf{u}$ is the "true volume" element in the new coordinates.

Proof Idea and Key Examples

ExampleThe Gaussian integral via polar coordinates

We compute $I = \int_{-\infty}^{\infty} e^{-x^2} dx$ by first evaluating $I^2$ : $I^2 = \iint_{\mathbb{R}^2} e^{-(x^2+y^2)}\,dx\,dy = \int_0^{2\pi}\int_0^\infty e^{-r^2} r\,dr\,d\theta = 2\pi \cdot \frac{1}{2} = \pi$ Therefore $I = \sqrt{\pi}$ , establishing the fundamental Gaussian integral $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$ .

ExampleVolume of the $n$-ball

By induction and change of variables, the volume of the unit ball $B^n = \{x_1^2 + \cdots + x_n^2 \leq 1\}$ is $V_n = \frac{\pi^{n/2}}{\Gamma(n/2 + 1)}$ giving $V_1 = 2$ , $V_2 = \pi$ , $V_3 = \frac{4\pi}{3}$ , $V_4 = \frac{\pi^2}{2}$ . Notably, $V_n \to 0$ as $n \to \infty$ : high-dimensional spheres have vanishingly small volume.

Generalized Coordinates

Theorem9.7Integration in Curvilinear Coordinates

For a general curvilinear coordinate system $(q_1, \ldots, q_n)$ with position vector $\mathbf{r}(q_1, \ldots, q_n)$ , the volume element is $dV = \sqrt{|\det g_{ij}|} \, dq_1 \cdots dq_n$ where $g_{ij} = \frac{\partial \mathbf{r}}{\partial q_i} \cdot \frac{\partial \mathbf{r}}{\partial q_j}$ is the metric tensor. For orthogonal coordinates, this simplifies to $dV = h_1 h_2 \cdots h_n \, dq_1 \cdots dq_n$ where $h_i = \left\|\frac{\partial \mathbf{r}}{\partial q_i}\right\|$ .

RemarkThe Jacobian and orientation

The sign of $\det J_T$ indicates whether $T$ preserves or reverses orientation. We take the absolute value in the change of variables formula because we are computing unsigned volume. In the theory of differential forms, the signed Jacobian appears naturally and orientation becomes a fundamental concept.