Change of Variables in Multiple Integrals

Coordinate transformations simplify the evaluation of multiple integrals by choosing coordinates adapted to the geometry of the domain, with the Jacobian determinant accounting for the distortion of area or volume.

The Change of Variables Formula

Definition

A coordinate transformation (or change of variables) is a differentiable bijection $T : G \to D$ from a region $G$ in $(u,v)$ -space to a region $D$ in $(x,y)$ -space, given by $x = x(u,v)$ , $y = y(u,v)$ . The Jacobian of $T$ is $\frac{\partial(x,y)}{\partial(u,v)} = \det \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix}$ Its absolute value $\left|\frac{\partial(x,y)}{\partial(u,v)}\right|$ measures the local area scaling factor of the transformation.

Theorem9.2Change of Variables (Double Integral)

If $T : G \to D$ is a $C^1$ bijection with $\frac{\partial(x,y)}{\partial(u,v)} \neq 0$ on $G$ , then $\iint_D f(x,y)\,dx\,dy = \iint_G f(x(u,v), y(u,v)) \left|\frac{\partial(x,y)}{\partial(u,v)}\right| du\,dv$

Standard Coordinate Systems

ExamplePolar, cylindrical, and spherical coordinates

Polar ( $x = r\cos\theta$ , $y = r\sin\theta$ ): Jacobian $= r$ $\iint_D f(x,y)\,dA = \int_{\theta_1}^{\theta_2}\int_{r_1(\theta)}^{r_2(\theta)} f(r\cos\theta, r\sin\theta)\, r\,dr\,d\theta$

Cylindrical ( $x = r\cos\theta$ , $y = r\sin\theta$ , $z = z$ ): Jacobian $= r$

Spherical ( $x = \rho\sin\phi\cos\theta$ , $y = \rho\sin\phi\sin\theta$ , $z = \rho\cos\phi$ ): Jacobian $= \rho^2\sin\phi$ $\iiint_E f\,dV = \int\!\!\int\!\!\int f(\rho,\phi,\theta)\,\rho^2 \sin\phi\,d\rho\,d\phi\,d\theta$

ExampleArea of a disk using polar coordinates

$A = \iint_{x^2+y^2 \leq R^2} dA = \int_0^{2\pi}\int_0^R r\,dr\,d\theta = \int_0^{2\pi} \frac{R^2}{2}\,d\theta = \pi R^2$

RemarkChoosing the right coordinates

The key to effective use of coordinate transformations is matching the coordinates to the symmetry of the domain. Circular domains suggest polar coordinates, cylindrical domains suggest cylindrical coordinates, and spherical domains suggest spherical coordinates. In general, choose coordinates in which the boundary of the domain takes a simple form.