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

Theorem 1.

Let $g\mathrm{:}X\mathrm{\to }Y$ be a diffeomorphism betweenopen subsets $X$ and $Y$ of ${\mathrm{R}}^n$.Then for any measurable function $f\mathrm{:}Y\mathrm{\to }\mathrm{R}$, and any measurableset $E\mathrm{\subseteq }X$,

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

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

This theorem is a generalization of the substitution rulefor 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 calledthe Jacobian or Jacobian determinant.

Theorem 1 is typically appliedwhen 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 approximatelythe 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 anarbitrary set by approximating it by many numbers of small cubes,and taking limits.

Figure 1: Illustration oflinear 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 obtainedby 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 differentiableon an open subset $X$ of ${\mathrm{R}}^n$.Then for any measurable function $f\mathrm{:}Y\mathrm{\to }\mathrm{R}$, andany 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) includesa 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 timesin the left-hand integral.

These formulas can also be generalized forHausdorff 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.