Change of Variables Formula
1. The Setup and Statement of the Theorem
Throughout these notes, \(U\) and \(V\) will be open sets in \({\mathbb R}^n\) and \(\Phi : U \rightarrow V\) will be a \(C^1\) bijection from \(U\) to \(V\). Furthermore, \(D_x \Phi\) will be assumed to be invertible at every point of \(U\). We use the notation \(\Delta_\Phi(x)\) to denote the determinant of \(D_x \Phi\) at \(x \in U\).
We say that a Jordan region \(E \subset U\) is a safe Jordan region in \(U\) when its closure is contained in \(U\) as well. From the notes on Jordan regions, for any safe Jordan region \(E \subset U\), its image via \(\Phi\), which we denote by \(\Phi(E)\), will be a Jordan region in \(V\). Because \(\overline{E}\) is itself a safe Jordan region in \(U\), \(\Phi(\overline{E})\) will be a Jordan region in \(V\) as well. Since \(\overline{E}\) is compact (Jordan regions must be bounded), \(\Phi(\overline{E})\) is compact, as it is a continuous image of a compact set. Thus the closure of \(\Phi(E)\) must be contained in \(\Phi(\overline{E})\), which is contained in \(V\). Therefore \(\Phi(E)\) will also be a safe Jordan region in \(V\). By the Inverse Function Theorem, \(\Phi^{-1}\) will be a \(C^1\) bijection from \(V\) to \(U\) with invertible total derivative at every point, and so it is also true that if \(F \subset V\) is a safe Jordan region, then \(\Phi^{-1}(F)\) is a safe Jordan region in \(U\).
The goal is the following:
Theorem (Change of Variables Formula)
Suppose \(U\) and \(V\) are open subsets of \({\mathbb R}^n\) and that \(\Phi : U \rightarrow V\) is a \(C^1\) bijection with the property that \(\Delta_\Phi(x) \neq 0\) for all \(x \in U\). If \(E\) is a safe Jordan region in \(V\) and if \(f\) is Riemann integrable on \(E\), then \(f \circ \Phi\) is Riemann integrable on \(\Phi^{-1}(E)\) and
\[ \int_{E} f(y) dy = \int_{\Phi^{-1}(E)} f (\Phi(x)) |\Delta_\Phi(x)| dx.\]
2. Initial Consideration
The following result is used many times during the proof:
Proposition
Suppose that \(E_1,\ldots,E_N\) are Jordan regions in \({\mathbb R}^n\) such that \(E_i \cap E_j\) has Jordan content zero when \(i \neq j\). Then
\[ \int_{\bigcup_{i=1}^N E_j} f dx = \sum_{j=1}^N \int_{E_j} f dx \]
for any \(f\) which is Riemann-integrable on \(\bigcup_{j=1}^N E_j\). In particular, if \(E \subset \bigcup_{j=1}^N E_j\) is a Jordan region, then
\[ \operatorname{Vol}\left(E\right) = \sum_{j=1}^N \operatorname{Vol}\left(E \cap E_j\right). \]
3. The Plan of Approach
The proof is long and technical. It's easiest to accomplish by breaking into a series of smaller steps which increasingly refine strength of the result.
Point (Why the proof is organized this way)
The proof below is not the shortest or most efficient proof known to exist. In particular, there exist relatively short proofs that cut straight to Claim 5 by an argument involving the Implicit Function Theorem and induction on dimension. The present proof has been chosen because it highlights more hands-on, robust techniques that we've already used many times now. Proving theorems as sophisticated as this one can be accomplished systematically with the right perspective and doesn't require special degrees of cleverness or insight.
Claim (1. How Linear Transformations Affect Volume)
For any convex Jordan region \(E \subset {\mathbb R}^n\), if we regard its points as vectors and apply an invertible matrix \(M \in {\mathbb R}^{n \times n}\), the set \(M E\) is a convex Jordan region and
\[ \operatorname{Vol}\left(ME\right) = |\det M| \operatorname{Vol}\left(E\right).\]
In other words, applying a linear transformation \(M\) to a convex Jordan region yields another region of the same sort whose volume is simply the volume of the initial region times the absolute value of the determinant of \(M\).
Proof
Claim (2. How Approximately Linear Maps Affect Volumes of Small Cubes)
For each \(x \in {\mathbb R}^n\) and each \(r > 0\), let \(x + Q_r\) be the cube centered at \(x\) with side length \(r\). For any \(\epsilon > 0\) and any compact set \(K \subset U\), there is an \(r_0 > 0\) such that for all \(x \in K\) and all \(r \in (0,r_0)\),
\[{}(1-\epsilon)^n |\Delta_\Phi(x)| \operatorname{Vol}\left(x + Q_r\right){}\]
\[{}\leq \operatorname{Vol}\left(\Phi(x + Q_r)\right){}\]
\[{}\leq (1 + \epsilon)^n |\Delta_\Phi(x) | \operatorname{Vol}\left(x + Q_r\right).{}\]
Proof
- It suffices to show that under the hypotheses of the claim, \(\Phi(x+ Q_r)\) contains a subset of volume at least \((1-\epsilon)^n |\Delta_\Phi(x)| \operatorname{Vol}\left(x + Q_r\right)\) and is contained in a set of volume at most \((1 + \epsilon)^n |\Delta_\Phi(x) | \operatorname{Vol}\left(x + Q_r\right)\). We will show\[{}\Phi(x) + D_x \Phi(Q_{(1-\epsilon)r}){}\]\[{}\subset \Phi(x+Q_r){}\]\[{}\subset \Phi(x) + D_x \Phi(Q_{(1+\epsilon)r}){}\]for all \(x \in K\) and all \(r < r_0\).
Figure. \(C^1\) images can be bounded by linear map images - Let \(||\cdot||\) denote the \(\ell^\infty\) norm on \({\mathbb R}^n\). Let \(r_1\) be half the distance from \(K\) to \(\partial U\). The value of \(r_0\) will never exceed \(r_1\). Let \(K_1\) be the compact set consisting of all points at most distance \(r_1\) to the set \(K\).
- If \(x \in K\), \(r < r_1\), and \(y \in x + Q_r\), the line segment joining \(x\) and \(y\) is contained in \(x + Q_r\) and in \(K_1 \subset U\). By the Mean Value Theorem, any convex function \(\varphi\) admits some \(\xi \in K_1\) such that\[ \varphi( \Phi(x) - \Phi(y)) \leq \varphi(D_\xi \Phi(x-y)). \]We can specifically choose \(\varphi(z) := || -x + y + (D_x \Phi)^{-1} z||_{\infty}\). It follows that\[{}|| - x + y + (D_x \Phi)^{-1}(\Phi(x) - \Phi(y))||{}\]\[{}\leq || (-I + (D_x \Phi)^{-1} D_\xi \Phi)(x-y)||.{}\]Since \(-I + (D_x \Phi)^{-1} D_\xi \Phi\) has continuous entries in \(x\) and \(\xi\) which vanish when \(\xi = x\), uniform continuity on compact sets implies the existence of \(r_2 < r_1\) such that \(||| -I + (D_x \Phi)^{-1} D_\xi \Phi ||| < \epsilon\) whenever \(x \in K_1\) and \(||x - y|| < r_2/2\). This implies\[{}||(D_x \Phi)^{-1}(\Phi(x) - \Phi(y))||{}\]\[{}\leq (1 + \epsilon) ||x-y||{}\]meaning that \(\Phi(x + Q_r) \subset \Phi(x) + D_x \Phi(Q_{(1+\epsilon)r})\) for all \(x \in K\) and all \(r < r_2\).
- As \(\Phi(K_1)\) is compact and contained in \(V\) and because \(D_x \Phi\) is continuous as a function of \(x\), there exists some \(r_3 > 0\) such that \(\Phi(x) + D_x \Phi(Q_{r}) \subset V\) when \(x \in K_1\) and \(r < r_3\). Let \(z = \Phi(x) + D_x \Phi(u)\) for some \(u \in Q_{(1-\epsilon)r}\); for any convex \(\varphi\),\[{}\varphi( \Phi^{-1}(z) - \Phi^{-1}(\Phi(x))){}\]\[{}\leq \varphi(D_\xi (\Phi^{-1})(z - \Phi(x))){}\]where \(\xi\) is some point in \(\Phi(x) + D_x \Phi(Q_{(1-\epsilon)r})\). This time let \(\varphi(w) := || w - u||_\infty\). This implies\[{}|| (\Phi^{-1}(z) - x) - u||{}\]\[{}\leq || ((D_{\Phi^{-1}(\xi)} \Phi)^{-1} D_x \Phi - I) u||.{}\]Once again, uniform continuity combined with the fact that the matrix vanishes when \(\xi = x\) implies the existence of some \(r_3\) such that \(|||((D_{\Phi^{-1}(\xi)} \Phi)^{-1} D_x \Phi - I)||| \leq \epsilon\) when \(x \in K_1\) and \(\xi \in \Phi(x) + D_x \Phi(Q_{(1-\epsilon)r})\) for any \(r < r_3\). The triangle inequality implies\[ || \Phi^{-1}(z) - x|| \leq (1 + \epsilon) ||u|| \]when \(r < \min \{r_1,r_2,r_3\} =: r_0\) and \(u \in Q_{(1-\epsilon)r}\) This means that \(\Phi^{-1}\) maps \(\Phi(x) + D_x \Phi(Q_{(1-\epsilon)r})\) into \(x + Q_{(1+\epsilon)(1-\epsilon)r}\), and since \((1+\epsilon)(1-\epsilon) = 1-\epsilon^2 < 1\), it follows that \(\Phi(x + Q_r)\) must contain \(\Phi(x) + D_x \Phi(Q_{(1-\epsilon)r})\).
Claim (3. Increasing Precision of the Previous Claim)
For any compact set \(K \subset U\), there is an \(r_0 > 0\) such that for any \(x \in K\) and all \(r \in (0,r_0)\), there is a point \(x' \in x + Q_r\) such that
\[{}\operatorname{Vol}\left(\Phi(x + Q_r)\right) = |\Delta_\Phi({x'})| \operatorname{Vol}\left(x + Q_r\right).{}\]
Proof
Claim (4. Increasing Precision of Previous Claim and Removing Smallness Restriction)
For any (closed) cube \(Q \subset U\),
\[ \int_Q |\Delta_\Phi(x)| dx = \operatorname{Vol}\left(\Phi(Q)\right). \]
Proof
Claim (5. Moving from Cubes to Boxes)
For any (closed) box \(R \subset U\),
\[ \int_R |\Delta_\Phi(x)| dx = \operatorname{Vol}\left(\Phi(R)\right). \]
Proof
Claim (6. General Volume Statement for Jordan Regions and Nonlinear Maps)
For any safe Jordan region \(E \subset U\),
\[ \operatorname{Vol}\left(\Phi(E)\right) = \int_E |\Delta_\Phi(x)| dx. \]
Proof
4. Later Stages of Proof
Claim (7. From Jordan Regions to Integration)
Let \(R\) be a box contained entirely in \(V\) and suppose that \(f\) is a Riemann integrable function on \(R\). Then \(f (\Phi(x)) |\Delta_\Phi(x)|\) is integrable on \(\Phi^{-1}(R)\) and
\[ \int_R f(y) dy = \int_{\Phi^{-1}(R)} f(\Phi(x)) |\Delta_\Phi(x)| dx. \]
Proof
Let \({\mathcal P}\) be a partition of \(R\). For every \(R' \in {\mathcal P}\), use Claim 6 on the set \(E := \Phi^{-1}(R')\) to conclude that
\[ |R'| = \int_{\Phi^{-1}(R')} |\Delta_\Phi(x)| dx \]
for every \(R' \in \mathcal P\). We substitute this identity into the formula for the upper sum of \(f\) on \({\mathcal P}\):
\[{}\mathcal U(f,{\mathcal P}){}\]
\[{}= \sum_{R' \in {\mathcal P}} |R'| \sup_{y \in R'} f(y){}\]
\[{}= \sum_{R' \in {\mathcal P}} \int_{\Phi^{-1}(R')} |\Delta_\Phi(x)| \left( \sup_{y \in R'} f(y) \right) dx{}\]
\[{}= \int_{\Phi^{-1}(R)} |\Delta_\Phi(x)| {}\]
\[{}\cdot \left[ \sum_{R' \in {\mathcal P}} \chi_{\Phi^{-1}(R')}(x) \sup_{y \in R'} f(y) \right] dx{}\]
\[{}\geq(U) \int |\Delta_\Phi(x)| \chi_{\Phi^{-1}(R)} (x) f (\Phi(x)) dx.{}\]
Similarly, applying the previous claim to the image of the interior of \(R'\) via \(\Phi^{-1}\) gives that
\[{}\mathcal L(f,{\mathcal P}){}\]
\[{}= \sum_{R' \in {\mathcal P}} \int_{\Phi^{-1}({R'}^\circ)} \inf_{y \in R'} f(y) dx{}\]
\[{}= \sum_{R' \in {\mathcal P}} \int_{\Phi^{-1}({R'}^\circ)} |\Delta_\Phi(x)| \left( \inf_{y \in R'} f(y) \right) dx{}\]
\[{}= \int_{\Phi^{-1}(R)} |\Delta_\Phi(x)| {}\]
\[{}\cdot \left[ \sum_{R' \in {\mathcal P}} \chi_{\Phi^{-1}({R'}^\circ)}(x) \inf_{y \in R'} f(y) \right] dx{}\]
\[{}\leq (L) \int |\Delta_\Phi(x)| \chi_{\Phi^{-1}(R)} (x) f (\Phi(x)) dx.{}\]
Proof of the Theorem Let \(\mathcal P\) be a collection of nonoverlapping boxes covering \(E\) and entirely contained in \(V\); since \(E\) is a safe Jordan region, its distance to the boundary of \(V\) is strictly positive and consequently we can guarantee that every box in the covering of \(E\) is entirely contained in \(V\) if we simply subdivide the boxes so that the diameter is sufficiently small and then throw out any boxes not intersecting \(E\). Then by Claim 7, for every \(R' \in {\mathcal P}\), we have that \(f (\Phi(x)) \chi_E(\Phi(x)) |\Delta_\Phi(x)|\) is Riemann integrable on \(\Phi^{-1}(R')\) and
\[{}\int f(y) \chi_{E \cap R'}(y) dy = \int_{R'} f(y) \chi_{E}(y) dy{}\]
\[{}= \int_{\Phi^{-1}(R')} f(\Phi(y)) \chi_E(\Phi(x)) |\Delta_\Phi(x)| dx{}\]
\[{}= \int f(\Phi(x)) \chi_{\Phi^{-1}(E \cap R')}(x) |\Delta_\Phi (x)| dx.{}\]
Summing over \(R' \in {\mathcal P}\) gives that
\[{}\int f(y) \chi_{E}(y) dy{}\]
\[{}= \sum_{R' \in {\mathcal P}} \int f(y) \chi_{E \cap R'}(y) dy{}\]
\[{}= \sum_{R' \in {\mathcal P}}\int f(\Phi(x)) \chi_{\Phi^{-1}(E \cap R')}(x) dx{}\]
\[{}= \int \! f(\Phi(x)) |\Delta_\Phi(x)| {}\]
\[{}\cdot \sum_{R' \in {\mathcal P}} \chi_{\Phi^{-1}(E \cap R')}(x) dx{}\]
\[{}= \int_{\Phi^{-1}(E)} f(\Phi(x))|\Delta_\Phi(x)| dx{}\]
by virtue of the fact that all the boxes \(R'\) are nonoverlapping (and \(\Phi^{-1}\) preserves nonoverlapping-ness because it sends sets of content zero to sets of content zero).