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

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 $E\mathrm{\subseteq }X$,

$${\int }_{E}f(g(x))|det\mathrm{D}g(x)|𝑑x={\int }_{g(E)}f(y)𝑑y.$$

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:

$$y=g(x),dy=g(dx)=|det\mathrm{D}g(x)|dx.$$

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 ${ℝ}^2$ using polar coordinates, or when integrating over ${ℝ}^3$ 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.

Figure 1: Illustration of linear approximation to $g(Q)$ by $x+\mathrm{D}g(x)(Q-x)$. http://aux.planetmath.org/files/objects/7349/jacobian.pySource 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 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 $E\mathrm{\subseteq }X$,

$${\int }_{E}f(g(x))|det\mathrm{D}g(x)|𝑑x={{\int }_{g(E)}f(y)\mathrm{\#}g|}_E^{-1}(y)dy,$$

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 ${ℝ}^n$, and non-differentiable, but Lipschitz, functions $g$. See [4] or other geometric measure theory books for details.

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