PlanetMath: change of variables in integral on $\mathbb{R}^n$

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change of variables in integral on $\mathbb{R}^n$

(Theorem)

Theorem 1 Let $g\colon X \to Y$ be a diffeomorphism between open subsets and of $\mathbb{R}^n$ . Then for any measurable function $f\colon Y \to \mathbb{R}$ , and any measurable set $E \subseteq X$ ,

$\displaystyle \int_E f(g(x)) \lvert\det {\mathrm{D}}g(x) \rvert dx = \int_{g(E)} f(y) dy .$

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

This theorem is a generalization 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:

$\displaystyle y = g(x) , \quad dy = g(dx) = \lvert \det {\mathrm{D}}g(x) \rvert dx .$

The volume scaling factor $\lvert\det {\mathrm{D}}g(x)\rvert$ is sometimes called the Jacobian or Jacobian determinant.

Theorem 1 is typically applied when integrating over $\mathbb{R}^2$ using polar coordinates, or when integrating over $\mathbb{R}^3$ using cylindrical or spherical coordinates.

Intuitively speaking, the image of a small cube centered at , under a differentiable map is approximately the parallelogram resulting from the linear mapping ${\mathrm{D}}g(x)$ applied on that cube. If the volume of the original cube is , then the volume of the image parallelogram is $dy = \lvert\det {\mathrm{D}}g(x)\rvert dx$ . The integral formula in Theorem 1 follows for an arbitrary set by approximating it by many numbers of small cubes, and taking limits.

**Figure:** Illustration of linear approximation to by $x + {\mathrm{D}}g(x) (Q-x)$ . Source program in Python for diagram
$\includegraphics{jacobian.eps}$

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 to be a diffeomorphism (i.e. that is a bijection and has non-singular derivative) is:

Theorem 2 Let $g\colon X \to \mathbb{R}^n$ be continuously differentiable on an open subset of $\mathbb{R}^n$ . Then for any measurable function $f\colon Y \to \mathbb{R}$ , and any measurable set $E \subseteq X$ ,

$\displaystyle \int_E f(g(x)) \lvert\det {\mathrm{D}}g(x) \rvert dx = \int_{g(E)} f(y) \char93 g\vert _E^{-1}(y) dy ,$

where $\char93 g\vert _E^{-1}(y) \in \{ 1, 2, \dotsc, \infty \}$ counts the number of pre-images in of .

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 that transform to zero volumes, and if the map is not one-to-one, then some pieces of the image may be counted multiple times in the left-hand integral.

These formulas can also be generalized for Hausdorff measures on $\mathbb{R}^n$ , and non-differentiable, but Lipschitz, functions . See [4] or other geometric measure theory books for details.

Bibliography

1: T. M. Flett. ``On Transformations in $\mathbb{R}^n$ 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. ``The Measure 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.

"change of variables in integral on $\mathbb{R}^n$ " is owned by stevecheng.

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Other names:

integral substitution formula, integral substitution rule, change-of-variables formula

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Cross-references: theory, measure, functions, Lipschitz, multiple, one-to-one, map, Transform, Sard's theorem, derivative, non-singular, bijection, stronger, proofs, limits, numbers, formula, linear mapping, parallelogram, differentiable map, cube, image, spherical coordinates, polar coordinates, determinant, Jacobian, factor, scaling, volume, substitutions, side, Calculus, substitution rule, theorem, integrals, measurable set, measurable function, open subsets, diffeomorphism
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This is version 5 of change of variables in integral on $\mathbb{R}^n$ , born on 2005-08-28, modified 2007-08-03.
Object id is 7349, canonical name is ChangeOfVariablesInIntegralOnMathbbRn.
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Classification:

AMS MSC:	26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables)
	26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume)
	28A25 (Measure and integration :: Classical measure theory :: Integration with respect to measures and other set functions)

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