change of variables in integral on ${\mathbb{R}}^{n}$
Theorem 1.
Let $g\mathrm{:}X\mathrm{\to}Y$ be a diffeomorphism^{} between open subsets $X$ and $Y$ of ${\mathrm{R}}^{n}$. Then for any measurable function^{} $f\mathrm{:}Y\mathrm{\to}\mathrm{R}$, and any measurable set^{} $E\mathrm{\subseteq}X$,
$${\int}_{E}f(g(x))|det\mathrm{D}g(x)|\mathit{d}x={\int}_{g(E)}f(y)\mathit{d}y.$$ |
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:
$$y=g(x),dy=g(dx)=|det\mathrm{D}g(x)|dx.$$ |
The volume scaling^{} factor $|det\mathrm{D}g(x)|$ 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 $\mathrm{D}g(x)$ applied on that cube. If the volume of the original cube is $dx$, then the volume of the image parallelogram is $dy=|det\mathrm{D}g(x)|dx$. 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 $g\mathrm{:}X\mathrm{\to}{\mathrm{R}}^{n}$ be continuously differentiable on an open subset $X$ of ${\mathrm{R}}^{n}$. Then for any measurable function $f\mathrm{:}Y\mathrm{\to}\mathrm{R}$, and any measurable set $E\mathrm{\subseteq}X$,
$${\int}_{E}f(g(x))|det\mathrm{D}g(x)|\mathit{d}x={{\int}_{g(E)}f(y)\mathrm{\#}g|}_{E}^{-1}(y)dy,$$ |
where ${\mathrm{\#}\mathit{}g\mathrm{|}}_{E}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{(}y\mathrm{)}\mathrm{\in}\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{\dots}\mathrm{,}\mathrm{\infty}\mathrm{\}}$ 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 |