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

###### Theorem 1.

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

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

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:

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

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

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 bijection and has non-singular derivative) is:

###### Theorem 2.

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

 $\int_{E}f(g(x))\,\lvert\det\operatorname{D}g(x)\rvert\,dx=\int_{g(E)}f(y)\,\#g% |_{E}^{-1}(y)\,dy\,,$

where $\#g|_{E}^{-1}(y)\in\{1,2,\ldots,\infty\}$ 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 measures (http://planetmath.org/AreaFormula) on $\mathbb{R}^{n}$, and non-differentiable, but Lipschitz, functions $g$. See [4] or other geometric measure theory books for details.

## References

• 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. “http://www.ams.org/bull/1942-48-12/S0002-9904-1942-07812-8/S0002-9904-1942-07812-8.pdfThe 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.
 Title change of variables in integral on $\mathbb{R}^{n}$ 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