Change of Variables in a Multiple Integral

Derives Lemma A44.1. Write \((AB)^{i}{}_{j}=\sum_{k}A^{i}{}_{k}B^{k}{}_{j}\) and expand Equation (9.19), distributing the product over \(j\):

the interchange being a finite rearrangement. The inner sum is, by Equation (9.19) again, the determinant of the matrix whose \(j\)-th column is the \(k_{j}\)-th column of \(A\). If two of the \(k_{j}\) coincide that matrix has a repeated column and the determinant vanishes, so only the tuples with \(k_{j}=\tau(j)\) for a permutation \(\tau\) survive; and bringing the columns back into their natural order costs \(\sgn(\tau)\), by the antisymmetry recorded after Equation (9.19). Hence the inner sum equals \(\sgn(\tau)\det A\) and

Derives Lemma A44.3. (1) Given \(\varepsilon>0\), cover \(Z\) by finitely many open cubes of total volume less than \(\varepsilon\) and let \(K\) be the complement in \(R\) of their union: \(K\) is closed and bounded, hence compact (Theorem 10.12). Note that \(f\) is continuous, as a function on \(R\), at every point of \(K\).

We claim there is \(\lambda>0\) such that any subset of \(R\) of diameter less than \(\lambda\) which meets \(K\) carries an oscillation of \(f\) of at most \(\varepsilon\). If not, then for each \(j\) there are \(\vect{x}_{j}\in K\) and \(\vect{y}_{j},\vect{z}_{j}\in R\) within \(1/j\) of \(\vect{x}_{j}\) with \(\abs{f(\vect{y}_{j})-f(\vect{z}_{j})}>\varepsilon\); by compactness a subsequence \(\vect{x}_{j}\) converges to some \(\vect{x}\in K\), the corresponding \(\vect{y}_{j}\) and \(\vect{z}_{j}\) converge to \(\vect{x}\) as well, and the continuity of \(f\) at \(\vect{x}\) makes both \(f(\vect{y}_{j})\) and \(f(\vect{z}_{j})\) tend to \(f(\vect{x})\) — a contradiction.

Now partition \(R\) with mesh smaller than \(\lambda/\sqrt{N}\). Writing \(A\) for the union of the covering cubes, every sub-box either lies inside \(A\) or meets \(K=R\setminus A\). Those of the first kind have total volume at most \(\operatorname{vol}(A)<\varepsilon\) and contribute at most \(2M\varepsilon\) to the gap between the upper and the lower sum; on those of the second kind the oscillation of \(f\) is at most \(\varepsilon\), contributing at most \(\varepsilon\operatorname{vol}(R)\). The gap can therefore be made as small as desired, and \(f\) is integrable (Definition 11.125).

(2) By (1), \(f\) is integrable, since it is continuous (with value zero) off \(Z\). Cover \(Z\) by finitely many cubes of total volume less than \(\varepsilon\) and partition \(R\) with a mesh small compared with the smallest of those cubes. Every sub-box on which \(f\) is not identically zero meets \(Z\), hence lies in the union of the cubes enlarged by one mesh on every side, whose volume is then at most \(2\varepsilon\); so \(\abs{\int_{R}f}\le2M\varepsilon\) for every \(\varepsilon>0\).

(3) The function \(\chi_{E}-\sum_{i}\chi_{E_{i}}\) vanishes off \(\bigcup_{i<j}\left(E_{i}\cap E_{j}\right)\), a set of zero content, and is bounded by \(m\); multiplying by \(f\) and applying (2) gives the statement.

Derives Lemma A44.4. (1) Let \(\varepsilon>0\). By uniform continuity (Theorem 11.25) there is \(\delta>0\) such that \(g\) varies by less than \(\varepsilon\) on any subset of \(B\) of diameter less than \(\delta\). Cover \(B\) by cubes of side \(h<\min\set{\delta/\sqrt{N-1},\varepsilon}\), in number at most \(C_{B}h^{-(N-1)}\) with \(C_{B}\) depending on \(B\) alone. Over each such cube the graph lies in a box of base that cube and height \(\varepsilon\), which is covered by \(\lceil\varepsilon/h\rceil\) cubes of side \(h\), of total volume at most \(\left(\varepsilon+h\right)h^{N-1}\le2\varepsilon h^{N-1}\). Summing gives a total volume at most \(2C_{B}\varepsilon\). A face of a box is the graph of a constant function.

(2) Cover \(Z\) by finitely many closed balls contained in \(W\); on each, the entries of \(D\vect{F}\) are bounded, and the mean value theorem (Theorem 11.35) applied to each component along the segment joining two points of the ball — which stays in the ball, the ball being convex — gives a constant \(L\) with \(\abs{\vect{F}(\vect{x})-\vect{F}(\vect{y})}\le L\abs{\vect{x}-\vect{y}}\) there. Take \(L\) to be the largest of the finitely many constants. Now cover \(Z\) by cubes of side \(h\) and total volume less than \(\varepsilon\), refining \(h\) so small that each such cube meeting \(Z\) lies in one of the balls. A cube of side \(h\) has diameter \(\sqrt{N}h\), so its image has diameter at most \(L\sqrt{N}h\) and lies in a cube of that side, of volume \(\left(L\sqrt{N}\right)^{N}h^{N}\). The images therefore cover \(\vect{F}(Z)\) with total volume at most \(\left(L\sqrt{N}\right)^{N}\varepsilon\).

Derives Lemma A44.5. Write \(F(\vect{u}')=\int_{a}^{b}f(\vect{u}',t)\,\dd t\). Let \(P'\) be a partition of \(R'\) into sub-boxes \(S'\) and \(P_{t}\) a partition \(a=t_{0}<\cdots<t_{n}=b\); together they partition \(R\) into the sub-boxes \(S=S'\times[t_{i-1},t_{i}]\). Fix \(\vect{u}'\in S'\). For each \(i\) and every \(t\in[t_{i-1},t_{i}]\) one has \(f(\vect{u}',t)\ge\inf_{S}f\), so

the right-hand side being independent of \(\vect{u}'\in S'\); hence it is a lower bound for \(\inf_{S'}F\). Multiplying by \(\operatorname{vol}(S')\) and summing over \(S'\) gives \(L(F,P')\ge L(f,P)\), where \(L\) denotes the lower Darboux sum Equation (11.22). The mirror argument gives \(U(F,P')\le U(f,P)\). Since \(f\) is integrable, the outer sums squeeze: for every \(\varepsilon>0\) a partition \(P\) with \(U(f,P)-L(f,P)<\varepsilon\) yields \(U(F,P')-L(F,P')<\varepsilon\), so \(F\) is integrable on \(R'\), and both \(\int_{R'}F\) and \(\int_{R}f\) lie between \(L(f,P)\) and \(U(f,P)\); letting \(\varepsilon\to0\) makes them equal. Relabelling the coordinates puts any chosen one in the role of \(t\).

Derives Corollary A44.6. Take a box \(R\) containing the support in its interior. A continuous function is integrable on \(R\) and on every slice (Theorem 11.40), so Lemma A44.5 applies with any chosen coordinate innermost; the inner integral is again continuous, by the uniform continuity of \(f\) on the compact \(R\) (Theorem 11.25), so the lemma may be applied again to it, and \(N\) applications reduce \(\int_{R}f\) to the iterated integral in the chosen order. All \(N!\) orders therefore give the same number. Finally \(f\circ\sigma\) is continuous with compact support, and its iterated integral in the order \(u^{1},\ldots,u^{N}\) becomes, on renaming the integration variable of the \(j\)-th integration from \(u^{j}\) to \(u^{\sigma(j)}\), the iterated integral of \(f\) in the order \(u^{\sigma(1)},\ldots,u^{\sigma(N)}\), which is \(\int_{\R^{N}}f\).

Derives Lemma A44.8. Let \(g\in C_{c}(W)\). Then \(f:=\left(g\circ\vect{\Psi}\right)\abs{\det D\vect{\Psi}}\) lies in \(C_{c}(V)\), and applying Equation (A44.5) first to \(\vect{\Psi}\) and then to \(\vect{\Phi}\),

By the chain rule (Proposition 11.104), \(D\left(\vect{\Psi}\circ\vect{\Phi}\right)(\vect{u}) =D\vect{\Psi}\left(\vect{\Phi}(\vect{u})\right)D\vect{\Phi}(\vect{u})\), and Lemma A44.1 turns the product of the two moduli into \(\abs{\det D\left(\vect{\Psi}\circ\vect{\Phi}\right)}\).

Derives Lemma A44.9. For each \(\vect{x}\in K\) choose \(j(\vect{x})\) with \(\vect{x}\in W_{j(\vect{x})}\) and \(r(\vect{x})>0\) with the closed ball of radius \(2r(\vect{x})\) about \(\vect{x}\) inside \(W_{j(\vect{x})}\). The open balls of radius \(r(\vect{x})\) cover \(K\); extract a finite subcover (Definition 10.9) with centres \(\vect{x}_{1},\ldots,\vect{x}_{p}\) and let \(K_{j}\) be the union of the closed balls \(\overline{B}_{r(\vect{x}_{i})}(\vect{x}_{i})\) with \(j(\vect{x}_{i})=j\). Each \(K_{j}\) is compact and contained in \(W_{j}\), and \(K\subseteq\bigcup_{j}K_{j}\). Let \(\rho_{j}>0\) be smaller than the distance from \(K_{j}\) to the complement of \(W_{j}\) — positive because \(K_{j}\) is compact and the complement closed — and put

which is continuous, non-negative, strictly positive exactly on the open set \(\set{\operatorname{dist}(\cdot,K_{j})<\rho_{j}}\), and supported in its closure, a compact subset of \(W_{j}\). On the open set \(O=\set{\sum_{i}\psi_{i}>0}\), which contains \(K\), set \(\lambda_{j}=\psi_{j}/\sum_{i}\psi_{i}\); off \(O\) set \(\lambda_{j}=0\). Each \(\lambda_{j}\) is continuous on \(O\) and vanishes on a neighbourhood of every point of \(\R^{N}\setminus\operatorname{supp}\psi_{j}\), so it is continuous everywhere; its support lies in that of \(\psi_{j}\); and \(\sum_{j}\lambda_{j}=1\) on \(O\).

Derives Lemma A44.10. Let \(f\in C_{c}(V)\) with \(K=\operatorname{supp}f\). The set \(\vect{\Phi}^{-1}(K)\) is compact, so finitely many \(N_{\vect{p}_{1}},\ldots,N_{\vect{p}_{m}}\) cover it, and the open sets \(W_{j}=\vect{\Phi}(N_{\vect{p}_{j}})\subseteq V\) cover \(K\). Take \(\lambda_{1},\ldots,\lambda_{m}\) as in Lemma A44.9. Each \(\lambda_{j}f\) is continuous, vanishes off \(\operatorname{supp}\lambda_{j}\cap K\) — a compact subset of \(W_{j}\) — and hence lies in \(C_{c}(W_{j})\); and \(\sum_{j}\lambda_{j}f=f\), since the two sides agree on the neighbourhood of \(K\) where \(\sum_{j}\lambda_{j}=1\) and both vanish off \(K\). Applying the substitution property of \(\vect{\Phi}|_{N_{\vect{p}_{j}}}\) to \(\lambda_{j}f\) and summing over \(j\),

the first and last steps by the linearity of the integral over a box containing every support involved.

Derives Lemma A44.11. Its matrix has exactly one \(1\) in each row and column, so Equation (9.19) leaves a single term and the determinant is \(\sgn(\sigma)=\pm1\); hence \(\abs{\det}=1\) and Equation (A44.5) reduces to \(\int f=\int f\circ\sigma\), which is Corollary A44.6. This is the only step of the whole derivation at which the sign of the determinant is disposed of, and it is why Equation (11.152) carries a modulus while Equation (9.19) does not.

Derives Lemma A44.13. Let \(f\in C_{c}(V)\) and write \(g=\left(f\circ\vect{\Phi}\right)\abs{\pp_{t}\Phi^{N}}\in C_{c}(U)\). Both \(f\) and \(g\), extended by zero, are continuous with compact support, so by Corollary A44.6 each integral is the iterated integral with \(t\) innermost. It therefore suffices to prove, for every fixed \(\vect{u}'\in\R^{N-1}\),

Fix \(\vect{u}'\) and put \(U_{\vect{u}'}=\set{t\mid(\vect{u}',t)\in U}\) and \(V_{\vect{u}'}=\set{s\mid(\vect{u}',s)\in V}\), both open in \(\R\). Because \(\vect{\Phi}\) leaves \(\vect{u}'\) untouched and is a bijection of \(U\) onto \(V\), the map \(\varphi(t)=\Phi^{N}(\vect{u}',t)\) is a bijection of \(U_{\vect{u}'}\) onto \(V_{\vect{u}'}\); it is \(C^{1}\) with \(\varphi'=\pp_{t}\Phi^{N}\neq0\), hence strictly monotone on each connected component of \(U_{\vect{u}'}\), mapping that component onto a component of \(V_{\vect{u}'}\).

The integrand on the right of Equation (A44.10) vanishes off the compact set \(\vect{\Phi}^{-1}(\operatorname{supp}f)\), whose slice at \(\vect{u}'\) is a compact subset \(T\) of \(U_{\vect{u}'}\). The components of \(U_{\vect{u}'}\) are open and disjoint and cover \(T\), so only finitely many of them, \(I_{1},\ldots,I_{q}\), meet \(T\); choose in each a compact interval \([\alpha_{r},\beta_{r}]\subset I_{r}\) containing \(T\cap I_{r}\), with \(\alpha_{r}<\beta_{r}\). On \(I_{r}\) the substitution rule Equation (11.27), applied to the \(C^{1}\) function \(\varphi\) and the continuous \(s\mapsto f(\vect{u}',s)\), gives

If \(\varphi\) increases on \(I_{r}\) then \(\varphi'=\abs{\varphi'}\) and the right-hand side is the integral over \(\left[\varphi(\alpha_{r}),\varphi(\beta_{r})\right]\); if \(\varphi\) decreases then \(\varphi'=-\abs{\varphi'}\) and the right-hand side is minus the integral over \(\left[\varphi(\beta_{r}),\varphi(\alpha_{r})\right]\). In both cases

Outside \(\left[\alpha_{r},\beta_{r}\right]\) the left integrand vanishes, so the left-hand side is the integral over the whole of \(I_{r}\); and \(f(\vect{u}',\cdot)\) vanishes on \(\varphi(I_{r})\setminus \varphi\left(\left[\alpha_{r},\beta_{r}\right]\right)\), since a point \(s\) there has \(\varphi^{-1}(s)\notin T\), so the right-hand side is the integral over the component \(\varphi(I_{r})\) of \(V_{\vect{u}'}\). Summing over \(r=1,\ldots,q\) and noting that \(f(\vect{u}',\cdot)\) vanishes on the components of \(V_{\vect{u}'}\) not among the \(\varphi(I_{r})\), and off \(V_{\vect{u}'}\) altogether, gives Equation (A44.10).

Derives Lemma A44.14. Say that a \(C^{1}\) diffeomorphism \(\vect{\Theta}\) defined near a point is \(m\)-adapted there, \(1\le m\le N+1\), if \(\Theta^{i}(\vect{u})=u^{i}\) for every \(i<m\) throughout a neighbourhood. Every diffeomorphism is \(1\)-adapted (the condition is empty), and an \((N+1)\)-adapted one is the identity. We show that an \(m\)-adapted \(\vect{\Theta}\) is, near the point, the composition of an \((m+1)\)-adapted diffeomorphism with one primitive map and one transposition of coordinates; \(N\) applications then prove the lemma, since a primitive map in the \(m\)-th coordinate is a transposition conjugate of one in the last coordinate.

Let \(\vect{\Theta}\) be \(m\)-adapted at \(\vect{q}\). Its Jacobian there has the block form

with \(\identity\) of size \(m-1\) and \(C\) the matrix \(\pp\Theta^{i}/\pp u^{j}\) with \(i,j\ge m\); expanding the determinant along the first \(m-1\) rows shows \(\det C=\det D\vect{\Theta}\neq0\), so the rows of \(C\) are independent and in particular the row \(\left(\pp\Theta^{m}/\pp u^{j}\right)_{j\ge m}\) is non-zero: there is \(j\ge m\) with \(\pp\Theta^{m}/\pp u^{j}(\vect{q})\neq0\).

Let \(\tau\) be the transposition of the coordinates \(m\) and \(j\), and set

a map changing only the \(m\)-th coordinate. At \(\vect{q}'=\tau(\vect{q})\) its \(m\)-th partial derivative in the \(m\)-th coordinate is \(\pp\Theta^{m}/\pp u^{j}(\vect{q})\neq0\), so \(\det D\vect{\varsigma}(\vect{q}')\neq0\) and, by the inverse function theorem (Corollary 11.113), \(\vect{\varsigma}\) is a \(C^{1}\) diffeomorphism of a neighbourhood of \(\vect{q}'\) onto an open set; it is a primitive map in the \(m\)-th coordinate.

Put \(\vect{\Xi}=\vect{\Theta}\circ\tau\circ\vect{\varsigma}^{-1}\), so that \(\vect{\Theta}=\vect{\Xi}\circ\vect{\varsigma}\circ\tau\) near \(\vect{q}\) (\(\tau\) being its own inverse). For \(i<m\), neither \(\tau\) nor \(\vect{\varsigma}\) — hence nor \(\vect{\varsigma}^{-1}\) — touches the \(i\)-th coordinate, and \(\vect{\Theta}\) preserves it, so \(\Xi^{i}(\vect{v})=v^{i}\). For \(i=m\): writing \(\vect{u}=\vect{\varsigma}^{-1}(\vect{v})\) one has \(v^{m}=\varsigma^{m}(\vect{u})=\Theta^{m}\left(\tau(\vect{u})\right) =\Xi^{m}(\vect{v})\). Hence \(\vect{\Xi}\) is \((m+1)\)-adapted, as claimed.

Derives Proposition A44.15. By Lemma A44.14 each point of \(U\) has a neighbourhood on which \(\vect{\Phi}\) is a composition of primitive maps and permutations; each factor has the substitution property by Lemmas A44.11 and A44.13, so the composition has it by Lemma A44.8. Thus \(\vect{\Phi}\) has the property locally, and Lemma A44.10 globalises it.

Derives Lemma A44.16. It is enough to treat one graph \(\Gamma\), of a continuous \(g\) over a box \(B\subset\R^{N-1}\), and to bound the content of its \(\delta\)-neighbourhood. By uniform continuity (Theorem 11.25) split \(B\) into cubes of side \(h\) on each of which \(g\) varies by less than \(\delta\). A point within distance \(\delta\) of \(\Gamma\) and lying over such a cube has last coordinate within \(2\delta\) of the value of \(g\) at the cube's centre, and lies over the cube enlarged by \(\delta\); so the neighbourhood is covered by boxes of total volume at most \(4\delta\left(\operatorname{vol}(B)+C\delta\right)\) with \(C\) depending on \(B\) alone. Letting first \(h\to0\) and then \(\delta\to0\) gives the claim; the finitely many graphs are summed.

Derives Theorem A44.17. Existence. Extend \(f\) by zero to a box \(R\supseteq\overline{V}\). The extension is bounded and continuous off \(\pp V\), which has zero content by Lemma A44.4(1); so it is integrable by Lemma A44.3(1). The same argument applies on the \(U\) side: \(\left(f\circ\vect{\Phi}\right) \abs{\det D\vect{\Phi}}\) is bounded there, because \(D\vect{\Phi}\) extends continuously to the compact \(\overline{U}\), and continuous off \(\pp U\).

Equality. Let \(M\) bound \(\abs{f}\) and \(J\) bound \(\abs{\det D\vect{\Phi}}\) on \(\overline{U}\). For \(\delta>0\) put

a continuous function, zero where the distance is at most \(\delta\) and one where it is at least \(2\delta\). The set where \(\eta_{\delta}\neq0\) has closure inside \(\set{\operatorname{dist}(\cdot,\R^{N}\setminus V)\ge\delta}\), which is closed and bounded, hence a compact subset of \(V\); so \(f_{\delta}=\eta_{\delta}f\) lies in \(C_{c}(V)\) and Proposition A44.15 gives

On the left, \(f-f_{\delta}\) vanishes outside \(V_{2\delta}\) and is bounded by \(M\), so the difference of the two integrals is at most \(M\) times the content of \(V_{2\delta}\), which tends to zero by Lemma A44.16. On the right, the integrands differ only on \(\vect{\Phi}^{-1}(V_{2\delta})\) and by at most \(MJ\). The extension of \(\vect{\Phi}^{-1}\) is \(C^{1}\) on a neighbourhood of the compact \(\overline{V}\), hence Lipschitz there with some constant \(L\) by the argument of Lemma A44.4(2); covering \(V_{2\delta}\) by cubes of total volume \(\varepsilon\) and taking images covers \(\vect{\Phi}^{-1}(V_{2\delta})\) with total volume at most \(\left(L\sqrt{N}\right)^{N}\varepsilon\), so its content also tends to zero. Letting \(\delta\to0\) in Equation (A44.18) gives Equation (11.152).

Derives Corollary A44.18. \(U\setminus Z\) is open, and \(\vect{\Phi}|_{U\setminus Z}\) is a \(C^{1}\) diffeomorphism onto the open set \(V\setminus\vect{\Phi}(Z)\); note that \(\vect{\Phi}(Z\cap K)\) has zero content for every compact \(K\subset\overline{U}\), by Lemma A44.4. Apply Proposition A44.15, or Theorem A44.17, to that restriction. Both integrals change by nothing when the sets \(Z\) and \(\vect{\Phi}(Z)\) are restored, because the integrands are bounded and the two sets have zero content (Lemma A44.3(2)).