Change of Variables in Multiple Integrals — Mathematics Knowledge System

Overview

Change of variables in multiple integrals generalizes u-substitution to R^n. The key formula: integral_R f(x,y) dA = integral_S f(x(u,v), y(u,v)) |J| du dv, where J = det(Jacobian) is the local area scaling factor. The most important cases are polar, cylindrical (dV = r dr dtheta dz), and spherical coordinates (dV = rho^2 sin phi drho dphi dtheta).

Intuition

In 1D, u-substitution: integral f(g(x))g'(x)dx = integral f(u)du. The g'(x) is the 'stretch factor.' In 2D, if you map (u,v) -> (x,y) via x=x(u,v), y=y(u,v), the Jacobian determinant J tells you how areas stretch. A tiny parallelogram in (u,v)-space maps to a parallelogram in (x,y)-space scaled by |J|. So dA_{xy} = |J| dA_{uv}.

Formal Definition

Definition

If G: D* -> D is a C^1 bijection with G(u,v)=(x(u,v),y(u,v)), then integral_D f(x,y) dA = integral_{D*} f(G(u,v)) |det J_G(u,v)| dA*. Polar: x=r cos t, y=r sin t, |J|=r. Cylindrical: x=r cos t, y=r sin t, z=z, |J|=r. Spherical: x=rho sin phi cos t, y=rho sin phi sin t, z=rho cos phi, |J|=rho^2 sin phi.

\iint_D f(x,y)\,dA = \iint_{D^*} f(G(u,v))\,|\det J_G|\,dA^*

Change of variables (2D)

dV = r\,dr\,d\theta\,dz \text{ (cylindrical)}

Cylindrical volume element

dV = \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta \text{ (spherical)}

Spherical volume element

Notation

Notation	Meaning
$\|\det J_G\|$	Absolute Jacobian determinant (area/volume scaling)
$\rho, \phi, \theta$	Spherical coordinates: radial distance, polar angle, azimuthal angle

Theorems

Theorem 1: Change of Variables Theorem

For a C^1 bijection G: D* -> D with nonzero Jacobian: integral_D f dA = integral_{D*} f(G) |det J_G| dA*. Applicable to all coordinate changes (polar, cylindrical, spherical, etc.).

Worked Examples

1
Integrate over 0 <= rho <= R, 0 <= phi <= pi, 0 <= theta <= 2pi.
$V = \int_0^{2\pi}\!\int_0^\pi\!\int_0^R \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta$
2
Integrate rho: [rho^3/3]_0^R = R^3/3.
$= \frac{R^3}{3}\int_0^{2\pi}d\theta \int_0^\pi \sin\phi\,d\phi$
3
Integrate phi: [-cos phi]_0^pi = 2. Integrate theta: 2pi.
$= \frac{R^3}{3} \cdot 2\pi \cdot 2 = \frac{4\pi R^3}{3}$

✓ Answer

V = (4/3)*pi*R^3.

Practice Problems

Mediumapplication

Evaluate the integral of e^{-(x^2+y^2)} over all of R^2 using polar coordinates.

Common Mistakes

Common Mistake

Forgetting the absolute value of the Jacobian.

Quiz

In cylindrical coordinates (r, theta, z), the volume element dV equals:

dr dtheta dzr dr dtheta dzr^2 dr dtheta dzsin(theta) r dr dtheta dz

Historical Background

The change-of-variables formula for multiple integrals was developed in the 18th and 19th centuries as integral calculus was extended to higher dimensions. The role of the Jacobian determinant was clarified by Jacobi himself. The formula is the multivariable generalization of the substitution rule, and its geometric meaning (Jacobian as volume scaling) makes it intuitive.

1841
Jacobi systematizes the change-of-variables formula with his functional determinants
Carl Gustav Jacob Jacobi

Summary

Change of variables: integral_D f dA = integral_{D*} f(G)|det J_G| dA*.
Polar: dA = r dr dtheta. Cylindrical: dV = r dr dtheta dz. Spherical: dV = rho^2 sin(phi) drho dphi dtheta.
Choose coordinates to simplify the region of integration or the integrand.

References

BookStewart, J. Multivariable Calculus. 8th ed. Brooks/Cole, 2015.