(Theorem) Change of variables

For functions of a single variable

Change of variables for a single variable ( $u$-substitution)

Let $f:[a,b]\to \mathbb{R}$ be continuous and $g:[c,d]\to \mathbb{R}$ be continuously differentiable with $g([c,d])\le [a,b]$. Then

$${\int }_c^{d}f(g(x))g^{\prime }(x)dx={\int }_{g(c)}^{g(d)}f(y)dy.$$

Proof from Modern Analysis I.

Define $F:[a,b]\to \mathbb{R}$ by

$$F(x)={\int }_a^{b}f(t)dt.$$

Since $f$ is continuous on all of $[a,b]$, we know by FTC I that $F$ is differentiable on $[a,b]$ and $F’(x)=f(x)$. Now consider the composition $F\circ g:[c,d]\to \mathbb{R}$. By the chain rule, we have

$$\begin{array}{rl}(F\circ g{)}^{\prime }(x)& =F^{\prime }(g(x))g^{\prime }(x)=f(g(x))g^{\prime }(x).\end{array}$$

Applying FTC II, we obtain the claimed equality:

$$\begin{array}{rl}{\int }_c^{d}f(g(x))g^{\prime }(x)dx& =(F\circ g)(d)-(F\circ g)(c)\\ & ={\int }_a^{g(d)}f(t)dt-{\int }_a^{g(c)}f(t)dt\\ & ={\int }_{g(c)}^{g(d)}f(t)dt.\end{array}$$

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For functions of multiple variables

Change of variables in higher dimensions

Suppose $U\subseteq {\mathbb{R}}^n$ is open, $D\subseteq U$ is closed and bounded, and $\phi :D\to {\mathbb{R}}^n$ is $C^2$ or continuously differentiable. Set $D’=\phi (D)$. Further, suppose there exists a volume-0 subset $C\subseteq D$ such that:

• (i) The dervative map $d{\phi }_{\vec{x}}$ is invertible for $\vec{x}\in D\backslash C$;
• (ii) $\phi$ is injective on $D\backslash C$;

that is, $\phi$ is “essentially” an invertible $C^2$ mapping $D\to D’$ except on a volume 0 subset, which does not affect the integral.

In this situation, if $f:D’\to \mathbb{R}$ is integrable on $D’=\phi (D)$, then

$${\int }_{D^{\prime }}f(\vec{y})d{\text{vol}}_n(\vec{y})={\int }_{D}f(\phi (\vec{x})|\text{det}(d{\phi }_{\vec{x}}|d{\text{vol}}_n(\vec{x}).$$

In particular, the claim of the theorem is that the right-hand side of the theorem is defined and can be used to compute the expression on the left-hand side.

Proof from Honors Mathematics B. wip