Proof of Stokes Theorem for Compact Oriented Smooth Manifolds with Boundary

Fix the chosen oriented smooth atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$ of $M$ and write $\Omega_\alpha=\varphi_\alpha(U_\alpha)$.

Step 1 (reduction to a single chart). By Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary choose $N$, indices $\alpha_1,\dots,\alpha_N$, and a smooth partition of unity $\chi_1,\dots,\chi_N$ subordinate to the chart domains. For each chart $\beta$ and each coordinate direction $l$, the one-dimensional product rule applied to coordinate slices of local smooth extensions gives, for the coefficient functions in the coordinate expansion,

$$\frac{\partial(\chi_{i,\beta}\,a)}{\partial x_l}=\chi_{i,\beta}\frac{\partial a}{\partial x_l}+a\frac{\partial\chi_{i,\beta}}{\partial x_l},$$

so from Exterior Derivative of a Smooth Differential Form on a Smooth Manifold with Boundary and Exterior Derivative of a C^1 Differential Form on a Euclidean Open Set, summing over $i$ and using $\sum_i\chi_{i,\beta}=1$ and $\sum_i\partial\chi_{i,\beta}/\partial x_l=0$ (finite sums), we get $\sum_{i=1}^{N}d(\chi_i\omega)=d\omega$ chart by chart. Likewise $\iota^{*}$ is additive in the form directly from the formula in Restriction of a Smooth Differential Form to the Boundary, so $\sum_i\iota^{*}(\chi_i\omega)=\iota^{*}\omega$. Since $\int_M$ and $\int_{\partial M}$ are additive over finite sums of smooth forms (additivity of the iterated integral in the integrand, as in the proofs of Independence of the Manifold Integral from Chart and Partition Choices), it suffices to prove

$$\int_M d\eta=\int_{\partial M}\iota^{*}\eta$$

for $\eta=\chi_i\omega$, a smooth $(n-1)$-form whose support (in the sense used in Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary) is compact and contained in a single chart domain $U_\gamma$, with $\varphi=\varphi_\gamma$, $\Omega=\Omega_\gamma$, and $K=\varphi(\operatorname{supp}\eta)\subseteq\Omega$ compact (as in claim 1 of Independence of the Manifold Integral from Chart and Partition Choices).

Step 2 (single-chart evaluation of integrals over $M$). If a smooth $n$-form $\sigma$ on $M$ has compact support inside one chart domain $U_\gamma$, then $\int_M\sigma=\int_\Omega\sigma_\gamma$: computing $\int_M\sigma$ by Integral of a Smooth n-Form over a Compact Oriented Smooth Manifold with Boundary with any partition of unity $\chi'_k$, each term $\int_{\Omega_{\beta_k}}(\chi'_k\sigma)_{\beta_k}$ equals $\int_{\Omega}(\chi'_k\sigma)_{\gamma}$ by exactly the transport argument of claim 3 of Independence of the Manifold Integral from Chart and Partition Choices (restriction to the overlap image, compatibility, and Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms); summing over $k$ and using $\sum_k\chi'_k=1$ with additivity gives the claim. We apply this to $\sigma=d\eta$ (its support is contained in that of $\eta$: where $\eta$ vanishes on an open set, so do the exterior derivatives of its local extensions).

Write the representative in the chart as

$$\eta_\gamma=\sum_{j=1}^{n}a_j\,dx_1\wedge\cdots\wedge\widehat{dx_j}\wedge\cdots\wedge dx_n,$$

with the hat denoting omission; the coefficients $a_j:\Omega\to\mathbb{R}$ agree locally with smooth functions on open sets. By Active Coordinate Coefficient Formula for the Exterior Derivative applied to local extensions (with the full index set $1,\dots,n$), the coefficient of $(d\eta)_\gamma$ is

$$c=\sum_{j=1}^{n}(-1)^{j-1}\frac{\partial a_j}{\partial x_j}.$$

Choose $R>0$ so large that $K$ is contained in the interior of the box $B=[-R,R]^{n-1}\times[0,R]$ relative to the ambient set (claim 2 of Zero Extension Continuity and Box Independence of the Iterated Integral, enlarging $R$ by $1$; if $\Omega$ is treated as open in $\mathbb{R}^n$ and $K$ has points with negative last coordinate, the boundary case below is vacuous and the argument is the same with $B=[-R,R]^n$). By Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain and additivity, $\int_M d\eta=\sum_j(-1)^{j-1}I_j$, where $I_j$ is the iterated integral over $B$ of the zero extension of $\partial a_j/\partial x_j$. By the reordering sub-lemma (a) in the proof of Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms, we may integrate the $j$th variable first. Fix the other variables and consider $g(t)=\tilde a_j(\dots,t,\dots)$, the zero extension of $a_j$ along the $j$th coordinate slice. Every point of the slice lies either in $\Omega$, where $g$ agrees locally with a smooth extension of $a_j$ and is differentiable with derivative $\partial a_j/\partial x_j$ (Derivative of a Coordinate Slice of a C^1 Function on a Euclidean Open Set), or outside $K$, where $g$ vanishes on a neighborhood and has derivative $0$; so $g$ is an antiderivative of the zero-extended integrand on the whole interval. By Fundamental Theorem of Calculus, Part II in One Dimension: for $j<n$, the inner integral equals $g(R)-g(-R)=0-0=0$ since $K$ avoids the faces $x_j=\pm R$; hence $I_j=0$. For $j=n$, the inner integral equals $g(R)-g(0)=-\tilde a_n(t',0)$, where $t'=(t_1,\dots,t_{n-1})$ and $\tilde a_n(t',0)$ is the zero extension of $a_n$ at $(t',0)$. Therefore

$$\int_M d\eta=(-1)^{n-1}\cdot\Bigl(-\int_{[-R,R]^{n-1}}\tilde a_n(t',0)\,dt'\Bigr)=(-1)^{n}\,I,\qquad I=\int_{[-R,R]^{n-1}}\tilde a_n(t',0)\,dt',$$

with the $(n-1)$-fold iterated integral on the right.

Step 3 (the boundary side). If $U_\gamma\cap\partial M=\varnothing$, then $\Omega\cap\partial H^n=\varnothing$ by claim 3 of Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 (or $\partial M=\varnothing$ altogether), so $\tilde a_n(t',0)=0$ for all $t'$ and $I=0$; also $\iota^{*}\eta$ is the zero form on $\partial M$ (its defining formula in Restriction of a Smooth Differential Form to the Boundary evaluates $\eta$'s representatives at boundary points, where they vanish since the corresponding manifold points lie outside the support of $\eta$), so both sides vanish and the identity holds — including the convention when $\partial M=\varnothing$.

Otherwise, consider the map $\vartheta$ on $U_\gamma\cap\partial M$ given by $\vartheta=T_c\circ\pi\circ\varphi$ when $n$ is even and $\vartheta=\rho\circ T_c\circ\pi\circ\varphi$ when $n$ is odd, where $\pi$, $T_c$, $\rho$ are as in Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1 and Induced Orientation on the Boundary of an Oriented Smooth Manifold with Boundary, and $c$ is chosen so that the image of the compact set $\pi(K\cap\partial H^n)$ lands in the region with positive last coordinate. The transports between $\vartheta$-coordinates and the charts of the induced orientation atlas are smooth diffeomorphisms of admissible domains with positive Jacobian determinant: this is the computation of claims 2–3 of Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving, which applies verbatim to $\vartheta$ (built from the oriented-atlas chart $\varphi$ by the same recipe as the induced boundary charts, with the same parity convention for $\rho$). Hence, computing $\int_{\partial M}\iota^{*}\eta$ by Integral of a Smooth n-Form over a Compact Oriented Smooth Manifold with Boundary with a partition of unity on $\partial M$ and transporting every term into $\vartheta$-coordinates via Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms (as in Step 2), we obtain

$$\int_{\partial M}\iota^{*}\eta=\int_{P}(\iota^{*}\eta)_\vartheta,$$

where $P$ is the $\vartheta$-image, an open subset of $\mathbb{R}^{n-1}$.

It remains to compute $(\iota^{*}\eta)_\vartheta$ from the pullback formula of Restriction of a Smooth Differential Form to the Boundary, with the affine map $j$ satisfying $j(y)=\varphi(\vartheta^{-1}(y))$. The Jacobian matrix $J_j$ maps $\mathbb{R}^{n-1}$ into the boundary hyperplane (its last row is zero), so for every basis $(n-1)$-form of $\eta_\gamma$ containing a factor $dx_n$ — that is, every term $a_j$ with $j<n$ — the evaluation on $J_jv_1,\dots,J_jv_{n-1}$ contains a zero row in the corresponding determinant and vanishes. Only the $a_n$ term survives. When $n$ is even, $j(y)=(T_c^{-1}(y),0)$ and the surviving pullback is $a_n(T_c^{-1}(y),0)\,dy_1\wedge\cdots\wedge dy_{n-1}$; the translation $T_c$ is an orientation-preserving smooth diffeomorphism, so by Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms (or directly, since translating a box translates the iterated integrals),

$$\int_P(\iota^{*}\eta)_\vartheta=\int_{[-R,R]^{n-1}}\tilde a_n(t',0)\,dt'=I=(-1)^n I=\int_M d\eta,$$

using $(-1)^n=1$. When $n$ is odd, $j(y)=(T_c^{-1}(\rho(y)),0)$, and the linear part of $j$ composes the insertion with $\rho$, whose determinant on the first $n-1$ coordinates is $-1$; the surviving pullback is $-\,a_n(T_c^{-1}(\rho(y)),0)\,dy_1\wedge\cdots\wedge dy_{n-1}$. Substituting $y_1\mapsto-y_1$ in the first integration (the one-dimensional substitution of sub-lemma (c) in the proof of Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms, applied to the affine map $t\mapsto-t$ with the limits exchanged) shows

$$\int_P(\iota^{*}\eta)_\vartheta=-\int_{[-R,R]^{n-1}}\tilde a_n(t',0)\,dt'=-I=(-1)^n I=\int_M d\eta,$$

using $(-1)^n=-1$. In both parities $\int_{\partial M}\iota^{*}\eta=\int_M d\eta$, completing Step 3 and, with Step 1, the proof. $\blacksquare$