change of variables in integral on n

Theorem 1.

Let g:XY be a diffeomorphismMathworldPlanetmath between open subsets X and Y of Rn. Then for any measurable functionMathworldPlanetmath f:YR, and any measurable setMathworldPlanetmath EX,


Also, if one of these integrals does not exist, then neither does the other.

This theorem is a generalizationPlanetmathPlanetmath of the substitution rule for integrals from one-variable calculus.

To go from the left-hand side to the right-hand side or vice versa, we can perform the formal substitutions:


The volume scalingMathworldPlanetmath factor |detDg(x)| is sometimes called the Jacobian or Jacobian determinant.

Theorem 1 is typically applied when integrating over 2 using polar coordinates, or when integrating over 3 using cylindrical or spherical coordinatesMathworldPlanetmath.

Intuitively speaking, the image of a small cube centered at x, under a differentiable map g is approximately the parallelogramMathworldPlanetmath resulting from the linear mapping Dg(x) applied on that cube. If the volume of the original cube is dx, then the volume of the image parallelogram is dy=|detDg(x)|dx. The integral formulaMathworldPlanetmathPlanetmath in Theorem 1 follows for an arbitrary set by approximating it by many numbers of small cubes, and taking limits.

Figure 1: Illustration of linear approximation to g(Q) by x+Dg(x)(Q-x). program in Python for diagram

Proofs of Theorem 1 can be obtained by making this procedure rigorous; see [7], [1], or [3].

A slightly stronger version of the theorem that does not require g to be a diffeomorphism (i.e. that g is a bijectionMathworldPlanetmath and has non-singular derivativeMathworldPlanetmathPlanetmath) is:

Theorem 2.

Let g:XRn be continuously differentiable on an open subset X of Rn. Then for any measurable function f:YR, and any measurable set EX,


where #g|E-1(y){1,2,,} counts the number of pre-images in E of y.

Observe that Theorem 2 (as well as its proof) includes a special case of Sard’s Theorem.

The idea of Theorem 2 is that we may ignore those pieces of the set E that transform to zero volumes, and if the map g is not one-to-one, then some pieces of the image g(E) may be counted multiple times in the left-hand integral.

These formulas can also be generalized for Hausdorff measuresMathworldPlanetmath ( on n, and non-differentiable, but LipschitzPlanetmathPlanetmath, functions g. See [4] or other geometric measure theory books for details.


  • 1 T. M. Flett. “On TransformationsMathworldPlanetmath in n and a Theorem of Sard”. American Mathematical Monthly, Vol. 71, No. 6 (Jun–Jul 1964), p. 623–629.
  • 2 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
  • 3 Miguel De Guzman. “Change-of-Variables Formula Without Continuity”. American Mathematical Monthly, Vol. 87, No. 9 (Nov 1980), p. 736–739.
  • 4 Frank Morgan. Geometric Measure Theory: A Beginner’s Guide, second ed. Academic Press, 1995.
  • 5 James R. Munkres. Analysis on Manifolds. Westview Press, 1991.
  • 6 Arthur Sard. “ MeasureMathworldPlanetmath of the Critical Values of Differentiable Maps”. Bulletins of the American Mathematical Society, Vol. 48 (1942), No. 12, p. 883-890.
  • 7 J. Schwartz. “The Formula for Change in Variables in a Multiple Integral”. American Mathematical Monthly, Vol. 61, No. 2 (Feb 1954), p. 81–95.
  • 8 Michael Spivak. Calculus on Manifolds. Perseus Books, 1998.
Title change of variables in integral on n
Canonical name ChangeOfVariablesInIntegralOnmathbbRn
Date of creation 2013-03-22 15:29:32
Last modified on 2013-03-22 15:29:32
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 8
Author stevecheng (10074)
Entry type Theorem
Classification msc 28A25
Classification msc 26B15
Classification msc 26B10
Synonym integral substitution formula
Synonym integral substitution rule
Synonym change-of-variables formula
Related topic JacobiDeterminant
Related topic LebesgueMeasure
Related topic AreaFormula
Related topic PotentialOfHollowBall
Related topic ExampleOfRiemannTripleIntegral
Related topic ExampleOfRiemannDoubleIntegral