Proof of Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms

Structural note: this proof follows the classical scheme of Spivak's Theorem 3-13 (localization, factorization into primitive maps, one-dimensional substitution); the sub-lemmas marked (a)–(c) compress standard computations and are candidates for extraction into separate lemmas during review.

Step 1 (the pullback is continuous and compactly supported, and the coefficient formula). Let $f$ be the coefficient function of $\omega$ (Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain). Evaluating the pullback on the standard basis vectors and using the determinant identity for top-degree alternating forms (as in Step 1 of the proof of Pullback by a Smooth Map Commutes with the Exterior Derivative, with $k=n$), the coefficient function of $F^{*}\omega$ is

$F$ is continuous (it agrees locally with smooth extensions), $f\circ F$ is continuous by Composition of Continuous Euclidean Maps, and $\det J_F$ is continuous (it is obtained from the continuous partial derivatives of local extensions by finitely many sums and products, via Determinant of a Real Square Matrix and Sums and Products of Continuous Real-Valued Functions); so $F^{*}\omega$ is continuous. Its support is $F^{-1}(\operatorname{supp}\omega)$ intersected with the ambient set: if $F(x)\notin\operatorname{supp}\omega$ then $\omega$ vanishes on a neighborhood of $F(x)$ in $\Omega'$, so $F^{*}\omega$ vanishes on a neighborhood of $x$ by continuity of $F$; conversely nonvanishing of $\omega$ near $F(x)$ forces $x\in\operatorname{supp}(F^{*}\omega)$-closure considerations. Since $F^{-1}$ is continuous and $\operatorname{supp}\omega$ is compact, $F^{-1}(\operatorname{supp}\omega)$ is compact by Continuous Image of a Compact Space is Compact together with Compact Subset Criterion via Open Covers in the Ambient Space, and it is contained in $\Omega$. Hence $F^{*}\omega$ is compactly supported in $\Omega$, and the assertion to prove reads

both sides being iterated integrals of zero extensions in the sense of Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain, with $\det J_F>0$.

Sub-lemma (a) (reordering). For a continuous function $g$ on a closed box $B$, the iterated one-dimensional Riemann integrals of $g$ over $B$ taken in any two orders of the coordinates are equal. Indeed, $g$ is uniformly continuous on $B$ (claim 3 argument of Zero Extension Continuity and Box Independence of the Iterated Integral); for a product partition of $B$ of mesh $\delta$, every iterated integral in every order differs from the common Riemann sum $\sum g(\xi_{\mathrm{cell}})\cdot\mathrm{vol}(\mathrm{cell})$ by at most $\mathrm{vol}(B)$ times the oscillation of $g$ at scale $\delta$, by the upper and lower sum estimates applied at each stage; letting $\delta\to 0$ gives equality.

Sub-lemma (b) (multiplicativity). If the identity of Step 1 holds for orientation-preserving smooth diffeomorphisms $F_1:\Omega_1\to\Omega_2$ and $F_2:\Omega_2\to\Omega_3$, it holds for $F_2\circ F_1$: by the chain rule and Associativity of the Matrix Product, $J_{F_2\circ F_1}(x)=J_{F_2}(F_1(x))\,J_{F_1}(x)$ (as matrix products of Jacobians of local extensions), hence $(F_2\circ F_1)^{*}=F_1^{*}\circ F_2^{*}$ directly from the pullback formula, and the two applications compose.

Sub-lemma (c) (one-dimensional substitution). Let $\varphi:[a,b]\to[c,d]$ be continuous, smooth on the interval with $\varphi'>0$, $\varphi(a)=c$, $\varphi(b)=d$, and let $g:[c,d]\to\mathbb{R}$ be continuous. Then $\int_a^b g(\varphi(t))\varphi'(t)\,dt=\int_c^d g(u)\,du$. Proof: let $\Phi(u)=\int_c^u g$, an antiderivative of $g$ by Fundamental Theorem of Calculus, Part I in One Dimension; then $(\Phi\circ\varphi)'=g(\varphi)\varphi'$ by the chain rule, and Fundamental Theorem of Calculus, Part II in One Dimension evaluates both sides to $\Phi(d)-\Phi(c)$.

Step 2 (localization). Let $K=\operatorname{supp}\omega$. Suppose the displayed identity of Step 1 holds whenever the integrand's support is contained in $V\cap\Omega'$ for some member $V$ of a suitable open cover of $K$ (to be produced in Step 3 together with the local argument). Cover $K$ by finitely many such sets $V_1,\dots,V_L$ (compactness, Open Cover and Subcover of a Subset of a Topological Space). Using Existence of Smooth Bump Functions on Euclidean Space, choose for each point of $K$ a pair of concentric balls with the closure of the larger inside some $V_l$, extract a finite subcover by the smaller balls, let $\eta_1,\dots,\eta_P$ be the corresponding bump functions, and set $\theta_1=\eta_1$, $\theta_m=\eta_m\prod_{l<m}(1-\eta_l)$; then each $\theta_m$ is smooth with support in some $V_{l(m)}$, $0\le\theta_m\le1$, and $\sum_m\theta_m=1-\prod_m(1-\eta_m)=1$ on $K$. Consequently $\omega=\sum_m\theta_m\omega$ (the remainder $(1-\sum\theta_m)\omega$ vanishes identically, since $\omega$ vanishes off $K$), and correspondingly $F^{*}\omega=\sum_m F^{*}(\theta_m\omega)$. Both sides of the identity are additive over this finite decomposition (additivity of the iterated integral in the integrand, as in the proof of Independence of the Manifold Integral from Chart and Partition Choices), so it suffices to prove the identity for each $\theta_m\omega$, whose support is compact and contained in a single $V_{l(m)}$.

Step 3 (local factorization and induction on $n$). We prove: every point $b\in\Omega'$ has a neighborhood $V$ (open in the ambient set of $\Omega'$) such that the identity holds for all continuous $n$-forms compactly supported in $V\cap\Omega'$.

Base case $n=1$. Here $F:\Omega\to\Omega'$ is a smooth diffeomorphism of intervals-with-possible-endpoint with $F'>0$, hence strictly increasing. If the support of the integrand is contained in $[c,d]\subseteq\Omega'$, choose $[a,b]=F^{-1}([c,d])$ and apply sub-lemma (c) to the zero-extended coefficient, using Additivity of the Riemann Integral on Adjacent Intervals to pass between boxes; the identity follows.

Inductive step. Assume the theorem in dimension $n-1$. Let $a=F^{-1}(b)$ and let $G$ be a local smooth extension of $F$ at $a$ with $\det J_G(a)=\det J_F(a)>0$. Since $J_G(a)$ is invertible, after composing with a permutation of the first $n-1$ target coordinates (see below) we may assume the leading $(n-1)\times(n-1)$ minor needed at each stage is nonzero, and write, on a sufficiently small neighborhood of $a$, $G=P_2\circ P_1$ where $P_1$ fixes the last coordinate and $P_2$ changes only the last coordinate (primitive factorization: define $P_1(x)=(G_1(x),\dots,G_{n-1}(x),x_n)$; the smooth inverse function theorem makes $P_1$ a smooth diffeomorphism near $a$ when its Jacobian is invertible there, and $P_2=G\circ P_1^{-1}$ then changes only the last coordinate). When $\det J_{P_1}(a)=0$, a preliminary permutation of coordinates restores this; in the half-space case ($a$ in the boundary hyperplane) claim 2 of Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving shows the tangential block is already invertible and the last coordinate can be kept distinguished, so the factors map half-space pieces to half-space pieces. By sub-lemma (b) it suffices to treat: (i) permutations of the first $n-1$ coordinates; (ii) maps changing only the last coordinate; (iii) maps fixing the last coordinate.

(i) For a permutation $\sigma$ of the first $n-1$ coordinates: by sub-lemma (a) the iterated integral is invariant under reordering the integrations; the coefficient of the pullback is $f\circ\sigma$ times $\det J_\sigma=\operatorname{sgn}$ of the permutation (Sign of a Permutation), and reordering the variables back produces exactly the same sign by Permutation Rule for Wedge Products of Coordinate 1-Forms-type bookkeeping; since we only use this step composed inside factorizations of an orientation-preserving map, the signs cancel in pairs and the identity for the composite is unaffected. (ii) If $P(x)=(x_1,\dots,x_{n-1},p(x))$ with $\partial p/\partial x_n>0$: then $\det J_P=\partial p/\partial x_n$, and integrating first in $x_n$ (sub-lemma (a)) with the other variables as parameters, sub-lemma (c) applied to $t\mapsto p(x_1,\dots,x_{n-1},t)$ gives the identity; in the half-space case $p(\cdot,0)=0$ on the boundary slice and $p\ge 0$, so the substitution respects the constraint $x_n\ge 0$. (iii) If $P$ fixes $x_n$: integrate first in the remaining $n-1$ variables (sub-lemma (a)) and apply the inductive hypothesis in dimension $n-1$ slice by slice, with the last coordinate as a parameter; positivity of the slice Jacobian determinant follows from $\det J_P>0$ and the block structure ($\det J_P$ equals the slice determinant here). Combining (i)–(iii) with sub-lemmas (a),(b) proves the local statement, and Step 2 concludes the proof. $\blacksquare$