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# 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 $E\subseteq X$,

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

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:

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

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

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 $E\subseteq X$,

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

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 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. “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.

## Mathematics Subject Classification

28A25*no label found*26B15

*no label found*26B10

*no label found*

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