The General Stokes Theorem for Differential Forms

Contents

Manifolds with boundary
Orientation and the integral of an $n$-form
Bump functions and partitions of unity
Statement and the half-space computation
Proof of the theorem
The instance stated in the chapter
What is quoted, and what is not

This appendix proves the theorem that Equation (17.231) of Differentiable Manifolds, Tensors, and Curvature is an instance of: for a compact oriented $n$-manifold $M$ with boundary and a smooth $(n-1)$-form $\omega$ on it,

\begin{equation}\tag{A29.1} \int_{M}\dd\omega = \int_{\pp M}\omega\ep \end{equation}

It is the single statement of which Green's theorem (Theorem 11.131), the classical Stokes theorem (Theorem 11.132), Gauss's theorem (Theorem 11.133) and the antisymmetric-tensor form Equation (17.231) are the low-dimensional faces, and Remark 11.134 names it as the formulation the treatise was still owed.

There is no circularity to create. Real Analysis proves its three classical theorems directly from the fundamental theorem of calculus over simple regions and does not lean on the present section; the present section does not use them. What it does use from Real Analysis is the multiple integral (Definition 11.125), the three properties of it collected as declared inputs in Remark 11.128 — integrability of a continuous function, additivity, and reduction to an iterated integral — and the fundamental theorem of calculus (Theorem 11.43). From Differentiable Manifolds, Tensors, and Curvature it uses the whole exterior calculus: Definition 17.98, Proposition 17.100, Definition 17.102, Definition 17.103 and Equation (17.246). From this appendix it uses the pullback of a $k$-form (Definition A12.4 and Lemma A12.5) and the inverse function theorem (Corollary A28.7). One input is quoted, the change-of-variables formula for multiple integrals; it is stated precisely as Theorem A29.6 and discussed in Remark A29.19.

Manifolds with boundary

Definition A29.1 (Half-space; smoothness on it).

The closed half-space is

\begin{equation}\tag{A29.2} \mathbb{H}^{n} = \set{x=(x^{1},\ldots,x^{n})\in\R^{n}\mid x^{n}\le 0}\ec \qquad \pp\mathbb{H}^{n} = \set{x\in\mathbb{H}^{n}\mid x^{n}=0}\ec \end{equation}

carrying the subspace topology of $\R^{n}$ (Definition 10.1). A map defined on a subset $S\subseteq\mathbb{H}^{n}$ that is open in $\mathbb{H}^{n}$ is called smooth if it extends to a smooth map on some open subset of $\R^{n}$ containing $S$; all partial derivatives are then defined at points of $\pp\mathbb{H}^{n}$ as well, and take there the values of the one-sided limits, so that every identity of the differential calculus of Section 11.10.2 continues to hold on $S$. Rests on Definitions 10.1 and 11.98.

Definition A29.2 (Manifold with boundary).

An $n$-manifold with boundary is defined exactly as in Definition 17.48, with one change: the charts $\varphi:U\longrightarrow\varphi(U)$ of the atlas (Definition 17.44) are homeomorphisms onto subsets of $\mathbb{H}^{n}$ open in $\mathbb{H}^{n}$, rather than onto open subsets of $\R^{n}$, and the transition maps are smooth in the sense of Definition A29.1. A point $P\in M$ is a boundary point if $\varphi(P)\in\pp\mathbb{H}^{n}$ for some chart around it, and an interior point otherwise; the set of boundary points is written $\pp M$. Rests on Definitions 17.44, 17.48 and A29.1.

Lemma A29.3 (The boundary is well defined, and is a manifold).

No point is a boundary point in one chart and an interior point in another. Moreover $\pp M$, with the charts obtained by dropping the last coordinate of a boundary chart, is an $(n-1)$-manifold without boundary. Rests on Definition A29.2 and Corollary A28.7.

Proof.

Derives Lemma A29.3. Suppose $P$ had charts $\varphi$ on $U$ and $\psi$ on $V$ with $z_{0}=\varphi(P)$ satisfying $z_{0}^{n}<0$ and $\psi(P)\in\pp\mathbb{H}^{n}$. Because $z_{0}^{n}<0$, the set $\varphi(U\cap V)$ contains an open subset $O\subseteq\R^{n}$ around $z_{0}$. Write $\tau=\psi\circ\varphi^{-1}$ on $O$ and let $\varsigma$ be a smooth extension to an open subset of $\R^{n}$, as in Definition A29.1, of the transition map $\varphi\circ\psi^{-1}$ near $\psi(P)$. Shrinking $O$ so that $\tau(O)$ lies in the domain of $\varphi\circ\psi^{-1}$, we have $\varsigma\circ\tau=\id$ on $O$, so the chain rule Equation (11.91) gives $D\varsigma\bigl(\tau(z_{0})\bigr)D\tau(z_{0})=\identity$ and $\det\left[D\tau(z_{0})\right]\ne0$. By the inverse function theorem Corollary A28.7, $\tau$ maps some neighbourhood of $z_{0}$ onto an open subset of $\R^{n}$ containing $\tau(z_{0})=\psi(P)\in\pp\mathbb{H}^{n}$. Every open subset of $\R^{n}$ containing a point of $\pp\mathbb{H}^{n}$ contains points with $x^{n}>0$, which do not lie in $\mathbb{H}^{n}$ — contradicting that $\psi$ takes values in $\mathbb{H}^{n}$.

For the second claim, let $\varphi=(x^{1},\ldots,x^{n})$ be a chart with $\varphi(U)\cap\pp\mathbb{H}^{n}\neq\varnothing$. By the first claim, $\varphi(U\cap\pp M) = \varphi(U)\cap\pp\mathbb{H}^{n}$, which is an open subset of $\R^{n-1}=\pp\mathbb{H}^{n}$, and $\varphi'=(x^{1},\ldots,x^{n-1})$ is a homeomorphism of $U\cap\pp M$ onto it. Two such charts are smoothly related, because a transition map of $M$ carries $\pp\mathbb{H}^{n}$ into $\pp\mathbb{H}^{n}$ — by the first claim again — so its first $n-1$ components restricted to $x^{n}=0$ are the transition map of the boundary charts, and they are smooth. Hence Definition 17.44 is met with model space $\R^{n-1}$.

∎

Orientation and the integral of an $n$-form

Lemma A29.4 (Transformation of the top form).

Let $\varphi=(x^{1},\ldots,x^{n})$ and $\psi=(y^{1},\ldots,y^{n})$ be two charts on a common domain. Then

\begin{equation}\tag{A29.3} \dd x^{1}\wedge\cdots\wedge\dd x^{n} = \det\left[\pdv{x^{i}}{y^{j}}\right]\, \dd y^{1}\wedge\cdots\wedge\dd y^{n}\ep \end{equation}

Consequently, if an $n$-form reads $f\,\dd x^{1}\wedge\cdots\wedge\dd x^{n}$ and $g\,\dd y^{1}\wedge\cdots\wedge\dd y^{n}$ in the two charts, then $g = f\,\det\left[\pp x/\pp y\right]$. Rests on Definition 17.102, Equation (17.241) and Proposition 17.100.

Proof.

Derives Lemma A29.4. By the chain rule, $\dd x^{i} = \left(\pp x^{i}/\pp y^{j}\right)\dd y^{j}$. Substituting into the left-hand side and expanding the wedge, every term in which two of the $y$-indices coincide vanishes (Equation (17.241)), so only the $n!$ terms indexed by a permutation $\rho$ of $(1,\ldots,n)$ survive, and reordering $\dd y^{\rho(1)}\wedge\cdots\wedge\dd y^{\rho(n)}$ into increasing order costs $\sgn\rho$:

\begin{equation*} \dd x^{1}\wedge\cdots\wedge\dd x^{n} = \left(\sum_{\rho}\sgn(\rho)\, \pdv{x^{1}}{y^{\rho(1)}}\cdots\pdv{x^{n}}{y^{\rho(n)}}\right) \dd y^{1}\wedge\cdots\wedge\dd y^{n}\ec \end{equation*}

and the bracket is the Leibniz expansion of the determinant (Linear Algebra and Representation Theory). The last sentence follows because $\Lambda^{n}$ is one-dimensional (Equation (17.235)).

∎

Definition A29.5 (Orientation).

An atlas of $M$ is oriented if every transition map between two of its charts has everywhere positive Jacobian determinant, $\det\left[\pp x/\pp y\right]>0$. $M$ is orientable if it admits such an atlas, and an oriented manifold is a manifold together with a maximal oriented atlas; a chart belongs to it, and is called positively oriented, if its transitions with every chart of the atlas have positive Jacobian determinant. Rests on Definition 17.44 and Lemma A29.4.

Theorem A29.6 (Change of variables for multiple integrals; quoted).

Let $\tau: O\longrightarrow O'$ be a diffeomorphism between open subsets of $\R^{n}$ and let $u$ be continuous with compact support in $O'$. Then $u\circ\tau\,\abs{\det D\tau}$ has compact support in $O$ and

\begin{equation}\tag{A29.4} \int_{O'}u(z)\,\dd z^{1}\cdots\dd z^{n} = \int_{O}u\bigl(\tau(w)\bigr)\, \abs{\det D\tau(w)}\,\dd w^{1}\cdots\dd w^{n}\ec \end{equation}

both integrals being multiple integrals in the sense of Definition 11.125. Rests on Definition 11.125 and Remark 11.128.

Definition A29.7 (Integral over a chart).

Let $M$ be an oriented $n$-manifold with boundary, let $\varphi=(x^{1},\ldots,x^{n})$ be a positively oriented chart on $U$, and let $\alpha$ be a continuous $n$-form on $M$ whose support is a compact subset of $U$. Writing $\alpha = f\,\dd x^{1}\wedge\cdots\wedge\dd x^{n}$ on $U$, set

\begin{equation}\tag{A29.5} \int_{M}\alpha = \int_{\varphi(U)}\left(f\circ\varphi^{-1}\right)(x)\, \dd x^{1}\cdots\dd x^{n}\ep \end{equation}

Rests on Definition A29.5, Definition 11.125 and Proposition 17.100.

Lemma A29.8 (The chart integral is well defined).

The value Equation (A29.5) does not depend on which positively oriented chart containing the support is used. Rests on Definition A29.7, Theorem A29.6 and Lemma A29.4.

Proof.

Derives Lemma A29.8. Let $\psi=(y^{1},\ldots,y^{n})$ be a second positively oriented chart containing $\operatorname{supp}\alpha$, and let $\tau=\varphi\circ\psi^{-1}$, a bijection between the images of the common domain.

Reduction to open sets. By Lemma A29.3, $\tau$ carries interior points to interior points and boundary points to boundary points, so it restricts to a diffeomorphism between the two open subsets of $\R^{n}$ obtained by deleting the face $x^{n}=0$; and deleting that face changes neither multiple integral, because a bounded portion of a hyperplane contributes nothing to an upper–lower Darboux gap — the argument of Remark 11.128 for the graph of a continuous function, applied to the constant function $0$. Both sides of the claimed equality may therefore be computed over the open sets, where Theorem A29.6 applies as stated.

By Lemma A29.4 the component functions are related by $g = f\,\det\left[\pp x/\pp y\right] = f\,\det D\tau$, and $\det D\tau>0$ by Definition A29.5, so $\abs{\det D\tau}=\det D\tau$. Then Equation (A29.4) with $u=f\circ\varphi^{-1}$ gives

\begin{equation*} \int_{\varphi(U)}\left(f\circ\varphi^{-1}\right)\dd x^{1}\cdots\dd x^{n} = \int_{\psi(U)}\left(f\circ\varphi^{-1}\circ\tau\right) \det D\tau\ \dd y^{1}\cdots\dd y^{n} = \int_{\psi(U)}\left(g\circ\psi^{-1}\right)\dd y^{1}\cdots\dd y^{n}\ec \end{equation*}

since $f\circ\varphi^{-1}\circ\tau = f\circ\psi^{-1}$.

∎

Bump functions and partitions of unity

The construction below is elementary and is carried out in full, because the treatise does not have it: nothing here is quoted.

Lemma A29.9 (The standard smooth bump).

The function

\begin{equation}\tag{A29.6} \lambda(t) = \ee^{-1/t}\ \ (t>0)\ec\qquad \lambda(t) = 0\ \ (t\le 0) \end{equation}

is smooth on $\R$. Consequently, for $0<a<b$,

\begin{equation}\tag{A29.7} \beta(t) = \frac{\lambda(b-t)}{\lambda(b-t)+\lambda(t-a)} \end{equation}

is smooth, takes values in $[0,1]$, equals $1$ for $t\le a$ and $0$ for $t\ge b$; and for $0<r_{1}<r_{2}$ the function

\begin{equation}\tag{A29.8} \chi(x) = \beta\left(\abs{x}^{2}\right)\ec \qquad a = r_{1}^{2}\ec\quad b = r_{2}^{2}\ec \end{equation}

is smooth on $\R^{n}$, equals $1$ on $\overline{B}_{r_{1}}(0)$, and vanishes off $B_{r_{2}}(0)$. Rests on Definition 11.53, Theorem 11.37 and Proposition 11.104.

Proof.

Derives Lemma A29.9. Smoothness of $\lambda$. On $t>0$ an induction gives $\lambda^{(k)}(t)=P_{k}(1/t)\,\ee^{-1/t}$ with $P_{k}$ a polynomial: this holds for $k=0$ with $P_{0}=1$, and differentiating $P_{k}(1/t)\ee^{-1/t}$ gives $\left[-t^{-2}P_{k}'(1/t)+t^{-2}P_{k}(1/t)\right]\ee^{-1/t}$, of the same shape with $P_{k+1}(s)=s^{2}\left(P_{k}(s)-P_{k}'(s)\right)$. On $t<0$ every derivative is $0$. At $t=0$ we show by induction that $\lambda^{(k)}(0)$ exists and is $0$. Assume it for $k$; the left difference quotient is $0$, and the right one is $P_{k}(1/t)\ee^{-1/t}/t$. Put $s=1/t\to+\infty$; the quotient is $s\,P_{k}(s)\,\ee^{-s}$, and for every $N$ the exponential series (Equation (11.31)) gives $\ee^{s}>s^{N}/N!$, i.e.\ $\ee^{-s}<N!\,s^{-N}$, so choosing $N$ larger than $1+\deg P_{k}$ makes $s\,P_{k}(s)\ee^{-s}\to0$. Hence $\lambda^{(k+1)}(0)=0$. Each $\lambda^{(k)}$ is then continuous at $0$ by the same estimate, so $\lambda$ is smooth.

$\beta$. The denominator of Equation (A29.7) never vanishes: for $t<b$ the first term is positive, and for $t\ge b>a$ the second is. So $\beta$ is smooth, and it lies in $[0,1]$ because both terms are non-negative. For $t\le a$ the second term vanishes and $\beta=1$; for $t\ge b$ the first vanishes and $\beta=0$.

$\chi$. $x\longmapsto\abs{x}^{2}$ is a polynomial, hence smooth, so $\chi$ is smooth by the chain rule Proposition 11.104; $\abs{x}\le r_{1}$ gives $\abs{x}^{2}\le a$ and $\chi=1$, and $\abs{x}\ge r_{2}$ gives $\chi=0$.

∎

Lemma A29.10 (Partition of unity on a compact manifold).

Let $M$ be a compact $n$-manifold with boundary and $\set{U_{a}}$ an open cover of $M$. Then there are finitely many smooth functions $\rho_{1},\ldots,\rho_{N}:M\longrightarrow[0,1]$ such that each $\operatorname{supp}\rho_{i}$ is a compact set contained in the domain of a single chart which is itself contained in some $U_{a}$, and

\begin{equation}\tag{A29.9} \sum_{i=1}^{N}\rho_{i}(P) = 1\qquad\text{for every }P\in M\ep \end{equation}

Rests on Lemma A29.9, Definition 10.9 and Definition A29.2.

Proof.

Derives Lemma A29.10. Let $P\in M$. Choose a chart $\varphi_{P}$ on a domain $V_{P}\ni P$ with $V_{P}$ contained in some $U_{a}$ and $\varphi_{P}(P)=0$ (translate the chart). Since $\varphi_{P}(V_{P})$ is open in $\mathbb{H}^{n}$ there is $r_{P}>0$ with $\overline{B}_{r_{P}}(0)\cap\mathbb{H}^{n}\subseteq\varphi_{P}(V_{P})$. Let $\chi_{P}$ be the function Equation (A29.8) with $r_{1}=r_{P}/2$ and $r_{2}=r_{P}$, and set

\begin{equation*} \widetilde{\chi}_{P} = \chi_{P}\circ\varphi_{P}\ \text{on }V_{P}\ec \qquad \widetilde{\chi}_{P} = 0\ \text{on } M\setminus V_{P}\ep \end{equation*}

This is smooth on $M$: it is smooth on $V_{P}$, and it vanishes identically on the open set $M\setminus\varphi_{P}^{-1}\left(\overline{B}_{r_{P}}(0)\cap \mathbb{H}^{n}\right)$, whose complement in $M$ is the compact set $\varphi_{P}^{-1}\left(\overline{B}_{r_{P}}(0)\cap\mathbb{H}^{n}\right) \subseteq V_{P}$; the two open sets cover $M$ and the definitions agree on the overlap. The same compact set contains $\operatorname{supp}\widetilde{\chi}_{P}$.

Put $W_{P}=\varphi_{P}^{-1}\left(B_{r_{P}/2}(0)\cap\mathbb{H}^{n}\right)$, an open neighbourhood of $P$ on which $\widetilde{\chi}_{P}=1$. The family $\set{W_{P}}$ is an open cover of the compact $M$ (Definition 10.9), so finitely many $W_{P_{1}},\ldots,W_{P_{N}}$ cover it. Then $S=\sum_{i}\widetilde{\chi}_{P_{i}}$ is smooth and satisfies $S\ge1$ everywhere, since each point lies in some $W_{P_{i}}$ where the corresponding term is $1$ and no term is negative. Setting $\rho_{i}=\widetilde{\chi}_{P_{i}}/S$ gives smooth functions with the stated supports and Equation (A29.9).

∎

Definition A29.11 (Integral over the manifold).

Let $M$ be a compact oriented $n$-manifold with boundary and $\alpha$ a continuous $n$-form on $M$. Choose a partition of unity $\set{\rho_{i}}$ subordinate to a cover by domains of positively oriented charts (Lemma A29.10) and set

\begin{equation}\tag{A29.10} \int_{M}\alpha = \sum_{i=1}^{N}\int_{M}\rho_{i}\alpha\ec \end{equation}

each summand being a chart integral (Definition A29.7). The value does not depend on the partition: if $\set{\sigma_{j}}$ is another one, then by linearity of the multiple integral and Lemma A29.8 — each $\rho_{i}\sigma_{j}\alpha$ being supported in one chart —

\begin{equation*} \sum_{i}\int_{M}\rho_{i}\alpha = \sum_{i,j}\int_{M}\rho_{i}\sigma_{j}\alpha = \sum_{j}\int_{M}\sigma_{j}\alpha\ep \end{equation*}

Rests on Lemmas A29.8 and A29.10.

Definition A29.12 (The induced orientation of the boundary).

Let $M$ be oriented and $\varphi=(x^{1},\ldots,x^{n})$ a positively oriented boundary chart. The induced orientation of $\pp M$ is the one in which the restricted chart $\varphi'=(x^{1},\ldots,x^{n-1})$ of Lemma A29.3 is positively oriented when $n$ is odd and negatively oriented when $n$ is even; equivalently, the chart $\left((-1)^{n-1}x^{1},x^{2},\ldots,x^{n-1}\right)$ is positively oriented. This is consistent: the transition map between two boundary charts has, at $x^{n}=0$, Jacobian determinant equal to $\left(\pp x^{n}/\pp y^{n}\right)$ times that of the restricted transition, and $\pp x^{n}/\pp y^{n}>0$ because both charts map into $\mathbb{H}^{n}$ and both send $\pp M$ to $x^{n}=0$; hence the restricted transitions inherit positive determinants, and multiplying every one of them by the same sign $(-1)^{n-1}$ leaves them positive. Rests on Definition A29.5 and Lemma A29.3.

Remark A29.13 (Outward normal first).

The sign $(-1)^{n-1}$ is not a convention chosen to make Equation (A29.1) come out right; it is the outward-normal-first rule counted honestly. In a boundary chart the outward direction is that of increasing $x^{n}$, since $\mathbb{H}^{n}=\set{x^{n}\le0}$, so the rule declares $(v_{1},\ldots,v_{n-1})$ positively oriented in $T_{\pp M}$ exactly when $(\pp_{n},v_{1},\ldots,v_{n-1})$ is positively oriented in $T_{M}$. Moving $\pp_{n}$ from the front of $(\pp_{n},\pp_{1},\ldots,\pp_{n-1})$ to the back costs $n-1$ transpositions, so $(\pp_{1},\ldots,\pp_{n-1})$ is positively oriented precisely when $(-1)^{n-1}=+1$, which is Definition A29.12. For $n=3$ — a surface in space, the case of Equation (17.231) — the sign is $+1$ and the rule reduces to the familiar right-hand relation between the normal of $S$ and the sense of circulation on $\pp S$.

Statement and the half-space computation

Theorem A29.14 (General Stokes theorem).

Let $M$ be a compact oriented $n$-manifold with boundary, $n\ge1$, let $\pp M$ carry the induced orientation, and let $\omega$ be a smooth $(n-1)$-form on $M$. Then

\begin{equation}\tag{A29.11} \int_{M}\dd\omega = \int_{\pp M}\iota^{\ast}\omega\ec \end{equation}

where $\iota:\pp M\longrightarrow M$ is the inclusion. If $\pp M=\varnothing$ the right-hand side is $0$. Rests on Definitions 17.103, A29.11 and A29.12.

Lemma A29.15 (The computation in one chart).

Let $\omega$ be a smooth $(n-1)$-form on $\mathbb{H}^{n}$ with compact support. Write

\begin{equation}\tag{A29.12} \omega = \sum_{i=1}^{n}(-1)^{i-1}f_{i}\; \dd x^{1}\wedge\cdots\wedge\widehat{\dd x^{i}}\wedge\cdots \wedge\dd x^{n}\ec \end{equation}

the hat marking the omitted factor; this is possible, with smooth $f_{i}$, because those $n$ products form a basis of $\Lambda^{n-1}$ (Equation (17.235)). Then

\begin{equation}\tag{A29.13} \int_{\mathbb{H}^{n}}\dd\omega = \int_{\R^{n-1}}f_{n}\left(x^{1},\ldots,x^{n-1},0\right) \dd x^{1}\cdots\dd x^{n-1} = \int_{\pp\mathbb{H}^{n}}\iota^{\ast}\omega\ec \end{equation}

the last integral taken with the induced orientation Definition A29.12. If the support of $\omega$ lies in the open half-space $\set{x^{n}<0}$, both sides vanish. Rests on Definition 17.103, Theorem 11.43 and Remark 11.128.

Proof.

Derives Lemma A29.15. The exterior derivative. Write $\theta_{i}=\dd x^{1}\wedge\cdots\wedge\widehat{\dd x^{i}} \wedge\cdots\wedge\dd x^{n}$. Carrying $\dd x^{i}$ leftwards past the $i-1$ factors $\dd x^{1},\ldots,\dd x^{i-1}$ using Equation (17.241),

\begin{equation}\tag{A29.14} \dd x^{i}\wedge\theta_{i} = (-1)^{i-1}\,\dd x^{1}\wedge\cdots\wedge\dd x^{n}\ep \end{equation}

By the Leibniz rule Equation (17.246) and $\dd(\dd x^{j})=0$,

\begin{equation*} \dd\omega = \sum_{i}(-1)^{i-1}\,\dd f_{i}\wedge\theta_{i} = \sum_{i,j}(-1)^{i-1}\pp_{j}f_{i}\,\dd x^{j}\wedge\theta_{i}\ec \end{equation*}

and $\dd x^{j}\wedge\theta_{i}=0$ unless $j=i$, since otherwise $\dd x^{j}$ is repeated. With Equation (A29.14),

\begin{equation}\tag{A29.15} \dd\omega = \left(\sum_{i=1}^{n}\pp_{i}f_{i}\right) \dd x^{1}\wedge\cdots\wedge\dd x^{n}\ep \end{equation}

The integral. Choose $R>0$ so large that $\operatorname{supp}\omega$ is contained in the box $Q=[-R,R]^{n-1}\times[-R,0]$, and note that each $f_{i}$ vanishes identically outside it. By Equation (A29.15), Definition A29.7 and additivity, the left-hand side of Equation (A29.13) is $\sum_{i}\int_{Q}\pp_{i}f_{i}$. Reduce each term to an iterated integral, integrating first in $x^{i}$ (Remark 11.128) and applying the fundamental theorem of calculus Theorem 11.43. For $i<n$,

\begin{equation*} \int_{-R}^{R}\pp_{i}f_{i}\,\dd x^{i} = f_{i}\big|_{x^{i}=-R}^{x^{i}=+R} = 0\ec \end{equation*}

both endpoint values lying outside $Q$; hence those $n-1$ terms vanish. For $i=n$ the $x^{n}$-integration runs only up to $0$:

\begin{equation*} \int_{-R}^{0}\pp_{n}f_{n}\,\dd x^{n} = f_{n}\left(x^{1},\ldots,x^{n-1},0\right) - f_{n}\left(x^{1},\ldots,x^{n-1},-R\right) = f_{n}\left(x^{1},\ldots,x^{n-1},0\right)\ec \end{equation*}

which gives the first equality in Equation (A29.13). If $\operatorname{supp}\omega\subseteq\set{x^{n}<0}$ then $f_{n}(\cdot,0)=0$ and the expression vanishes.

The boundary integral. The inclusion is $\iota(x^{1},\ldots,x^{n-1})=(x^{1},\ldots,x^{n-1},0)$, so by Equation (A12.7) $\iota^{\ast}\dd x^{j}=\dd x^{j}$ for $j<n$ while $\iota^{\ast}\dd x^{n}=\dd\left(x^{n}\circ\iota\right)=0$. By Equation (A12.4) the pullback distributes over the wedge, so every $\theta_{i}$ with $i<n$ — each of which still contains the factor $\dd x^{n}$ — pulls back to zero, and

\begin{equation}\tag{A29.16} \iota^{\ast}\omega = (-1)^{n-1}\,f_{n}\left(x^{1},\ldots,x^{n-1},0\right)\, \dd x^{1}\wedge\cdots\wedge\dd x^{n-1}\ep \end{equation}

The induced orientation is $(-1)^{n-1}$ times the one in which $(x^{1},\ldots,x^{n-1})$ is positively oriented (Definition A29.12), and reversing an orientation reverses the sign of the integral Equation (A29.5): replacing $x^{1}$ by $-x^{1}$ multiplies the component function by $-1$ (Lemma A29.4, the Jacobian determinant of the reflection being $-1$), while the multiple integral over the reflected region is unchanged by Equation (A29.4), whose factor $\abs{\det D\tau}$ is $1$ for a reflection. Hence

\begin{equation*} \int_{\pp\mathbb{H}^{n}}\iota^{\ast}\omega = (-1)^{n-1}\int_{\R^{n-1}}(-1)^{n-1} f_{n}\left(x',0\right)\dd x^{1}\cdots\dd x^{n-1} = \int_{\R^{n-1}}f_{n}\left(x',0\right)\dd x^{1}\cdots\dd x^{n-1}\ec \end{equation*}

which is the second equality in Equation (A29.13).

∎

Proof of the theorem

Proof of Theorem A29.14. Derives Theorem A29.14. Cover $M$ by domains of positively oriented charts and let $\set{\rho_{1},\ldots,\rho_{N}}$ be a subordinate partition of unity (Lemma A29.10). Since $\sum_{i}\rho_{i}=1$ is constant, $\sum_{i}\dd\rho_{i}=\dd(1)=0$, so by the Leibniz rule Equation (17.246)

\begin{equation}\tag{A29.17} \sum_{i=1}^{N}\dd\left(\rho_{i}\omega\right) = \sum_{i}\dd\rho_{i}\wedge\omega + \left(\sum_{i}\rho_{i}\right)\dd\omega = \dd\omega\ep \end{equation}

Each $\rho_{i}\omega$ is a smooth $(n-1)$-form whose support is a compact subset of the domain $U_{i}$ of a positively oriented chart $\varphi_{i}$; transporting it by $\varphi_{i}$ and extending by zero gives a smooth compactly supported $(n-1)$-form on the whole of $\mathbb{H}^{n}$, smooth because the two definitions — the transported one on the open set $\varphi_{i}(U_{i})$, and zero on the complement of the compact support — agree on the overlap of two open sets covering $\mathbb{H}^{n}$. The transport carries the two sides of Equation (A29.18) to the two sides of Equation (A29.13): the chart integral Equation (A29.5) is by definition the multiple integral of the chart representative, and pullback commutes with $\dd$ (Equation (A12.5)), so the chart representative of $\dd(\rho_{i}\omega)$ is the exterior derivative of the chart representative of $\rho_{i}\omega$. Hence Lemma A29.15 applies and gives

\begin{equation}\tag{A29.18} \int_{M}\dd\left(\rho_{i}\omega\right) = \int_{\pp M}\iota^{\ast}\left(\rho_{i}\omega\right)\ec \end{equation}

where the right-hand side is $0$ when $U_{i}$ misses $\pp M$, that being the last clause of Lemma A29.15. Summing Equation (A29.18) over $i$, using Equation (A29.17) on the left and, on the right, that $\set{\iota^{\ast}\rho_{i}}$ restricts to a partition of unity on $\pp M$ — $\sum_{i}\rho_{i}=1$ there too, and each $\iota^{\ast}\rho_{i}$ is supported in a boundary chart — gives Equation (A29.11) by Definition A29.11. If $\pp M=\varnothing$, every chart misses the boundary and every term on the right vanishes.

∎

The instance stated in the chapter

Corollary A29.16 (Stokes' theorem for an antisymmetric tensor field).

Let $S\subset E_{3}$ be a compact oriented two-dimensional submanifold with boundary $\pp S$, carrying the induced orientation, and let $H_{i}$ be a smooth covector field defined near $S$, with rotor tensor

\begin{equation}\tag{A29.19} H_{ij} = \pp_{i}H_{j}-\pp_{j}H_{i}\ep \end{equation}

Then, with $\dd f^{ij}$ the antisymmetric surface element of $S$,

\begin{equation}\tag{A29.20} \frac{1}{2}\int_{S}H_{ij}\,\dd f^{ij} = \oint_{\pp S}H_{i}\,\dd x^{i}\ec \end{equation}

which is Equation (17.231). Rests on Theorem A29.14, Equation (17.232) and Definition 11.126.

Proof.

Derives Corollary A29.16. Let $H=H_{i}\,\dd x^{i}$ be the one-form with the given components. By Definition 17.103,

\begin{equation}\tag{A29.21} \dd H = \pp_{j}H_{i}\,\dd x^{j}\wedge\dd x^{i} = \frac{1}{2}\left(\pp_{i}H_{j}-\pp_{j}H_{i}\right) \dd x^{i}\wedge\dd x^{j} = \frac{1}{2}H_{ij}\,\dd x^{i}\wedge\dd x^{j}\ec \end{equation}

the middle step by antisymmetrizing the coefficient against $\dd x^{j}\wedge\dd x^{i}=-\dd x^{i}\wedge\dd x^{j}$ (Equation (17.241)). Theorem A29.14 applied to $M=S$ and $\omega=\iota_{S}^{\ast}H$, the restriction of $H$ to $S$, gives

\begin{equation}\tag{A29.22} \int_{S}\iota_{S}^{\ast}\left(\dd H\right) = \oint_{\pp S}\iota_{\pp S}^{\ast}H\ec \end{equation}

where pullback and $\dd$ were exchanged by Equation (A12.5).

It remains to recognize the two sides. Let $\vect{r}:D\longrightarrow S$ be a positively oriented parametrization of a piece of $S$, with parameters $(u,v)$. The two-form $\dd x^{i}\wedge\dd x^{j}$ has components $\delta^{i}_{\ a}\delta^{j}_{\ b}-\delta^{i}_{\ b}\delta^{j}_{\ a}$ (Definition 17.102), so Equation (A12.3) gives

\begin{equation}\tag{A29.23} \iota_{S}^{\ast}\left(\dd x^{i}\wedge\dd x^{j}\right) = \left(\pp_{u}r^{i}\,\pp_{v}r^{j} - \pp_{v}r^{i}\,\pp_{u}r^{j}\right)\dd u\wedge\dd v\ep \end{equation}

That bracket is the antisymmetric surface element. Indeed, Remark 17.97 writes $\dd f^{ij}=\varepsilon^{ijk}\dd f_{k}$ with $\dd f_{k}$ the oriented surface element Equation (11.147) of Definition 11.126, whose components are $\dd f_{k}=\varepsilon_{klm}\,\pp_{u}r^{l}\,\pp_{v}r^{m}\,\dd u\,\dd v$; contracting the two Levi-Civita symbols (Linear Algebra and Representation Theory),

\begin{equation}\tag{A29.24} \dd f^{ij} = \varepsilon^{ijk}\varepsilon_{klm}\, \pp_{u}r^{l}\,\pp_{v}r^{m}\,\dd u\,\dd v = \left(\delta^{i}_{\ l}\delta^{j}_{\ m} -\delta^{i}_{\ m}\delta^{j}_{\ l}\right) \pp_{u}r^{l}\,\pp_{v}r^{m}\,\dd u\,\dd v\ec \end{equation}

which is the bracket of Equation (A29.23) times $\dd u\,\dd v$. Hence, by Definition A29.7 and Equation (A29.21),

\begin{equation*} \int_{S}\iota_{S}^{\ast}\left(\dd H\right) = \frac{1}{2}\iint_{D}H_{ij}\left(\vect{r}(u,v)\right) \left(\pp_{u}r^{i}\pp_{v}r^{j}-\pp_{v}r^{i}\pp_{u}r^{j}\right) \dd u\,\dd v = \frac{1}{2}\int_{S}H_{ij}\,\dd f^{ij}\ec \end{equation*}

and, parametrizing $\pp S$ by arc parameter $t$, $\iota_{\pp S}^{\ast}H = H_{i}(\vect{r}(t))\,\dv{r^{i}}{t}\,\dd t$, whose integral is the line integral Equation (11.145) of $H_{i}\dd x^{i}$ over $\pp S$. Substituting both into Equation (A29.22) gives Equation (A29.20).

∎

Remark A29.17 (The reading under which the identity is false).

Remark 17.97 records that Equation (17.231) holds for the rotor tensor Equation (17.232) and is false if $H_{i}$ is instead read as the dual pseudo-vector $\tfrac12\varepsilon_{ijk}H_{jk}$ of Equation (17.224). The derivation above respects that, and shows where the hypothesis enters: the only step in which the relation between $H_{ij}$ and $H_{i}$ is used is Equation (A29.21), which asserts that the two-form $\tfrac12 H_{ij}\dd x^{i}\wedge\dd x^{j}$ is exact, with the one-form $H=H_{i}\dd x^{i}$ as its primitive. Under the dual reading there is no such relation: $\tfrac12 H_{ij}\dd x^{i}\wedge\dd x^{j}$ would be an arbitrary two-form and $H_{i}\dd x^{i}$ an unrelated one-form, so Theorem A29.14 says nothing connecting their integrals, and the left-hand side of Equation (17.231) would be a flux and the right-hand side a circulation of the same vector field — two quantities with no reason to agree. Writing $\dd f^{ij}=\varepsilon^{ijk}\dd f_{k}$ turns Equation (A29.20) into $\int_{S}\left(\nabla\times\vect{H}\right)\cdot\dd\vect{f} =\oint_{\pp S}\vect{H}\cdot\dd\vect{l}$, the classical statement Theorem 11.132 — which Real Analysis proves directly and independently, so the two derivations check each other rather than one resting on the other.

Remark A29.18 (The SI reading of the three-dimensional instance).

The theorem is an identity between integrals of forms and is unit-homogeneous by construction: if $H_{i}$ carries an SI unit $\left[H\right]$ then $H_{i}\dd x^{i}$ carries $\left[H\right]\cdot\mathrm{m}$, $H_{ij}$ carries $\left[H\right]/\mathrm{m}$, and $\dd f^{ij}$ carries $\mathrm{m}^{2}$, so both sides of Equation (A29.20) carry $\left[H\right]\cdot\mathrm{m}$. The physical instance the treatise meets first is the magnetic vector potential: with $H_{i}=A_{i}$ in $\mathrm{T}\,\mathrm{m}$ and $H_{ij}=\pp_{i}A_{j}-\pp_{j}A_{i}$ in $\mathrm{T}$, the identity reads

\begin{equation}\tag{A29.25} \int_{S}\vect{B}\cdot\dd\vect{f} = \oint_{\pp S}\vect{A}\cdot \dd\vect{l}\ec \end{equation}

both sides in $\mathrm{Wb}=\mathrm{T}\,\mathrm{m}^{2}$: the magnetic flux through a surface equals the circulation of the potential round its edge.

What is quoted, and what is not

Remark A29.19 (What is quoted here).

Exactly one theorem is assumed without proof: Theorem A29.6, the change-of-variables formula for multiple integrals. It is used once, in Lemma A29.8, to show that the chart integral Equation (A29.5) does not depend on the chart; every later step is built on that one fact about the integral. This treatise does not prove it, and the omission is not new here: Remark 11.134 of Real Analysis already records the same formula as a quoted input, needed there for the independence of a flux integral from its parametrization, alongside the three properties of the multiple integral collected in Remark 11.128. A citation for it is owed and no entry of the bibliography currently carries it, so it is stated here rather than referenced. Nothing else is quoted: the partition of unity, which is the input a reader would most expect to be imported, is constructed outright in Lemmas A29.9 and A29.10, and the topology of the boundary is settled in Lemma A29.3 from the inverse function theorem proved in The Implicit Function Theorem.

Remark A29.20 (Where compactness and orientability are used).

Compactness enters once, in Lemma A29.10, to make the partition of unity finite — so that the sum Equation (A29.17) has finitely many terms and no convergence question arises. It may be traded for the weaker hypothesis that $\omega$ have compact support, at the cost of building a locally finite partition of unity. Orientability enters twice, both times essentially: in Lemma A29.8, where $\det D\tau>0$ is what removes the absolute value from Equation (A29.4) and lets the chart integrals agree rather than differ in sign; and in Definition A29.12, where the induced orientation is what makes the sign $(-1)^{n-1}$ produced by the half-space computation cancel against the sign in Equation (A29.16). On a non-orientable manifold no consistent $\int_{M}$ of an $n$-form exists at all, and Equation (A29.11) has no meaning.

Remark A29.21.

The General Stokes Theorem for Differential Forms discharges the derivation owed at Equation (17.231) of Differentiable Manifolds, Tensors, and Curvature, and with it the general formulation that Remark 11.134 of Real Analysis names as the common source of Theorems 11.131, 11.132 and 11.133. The chapter's statement is recovered as Corollary A29.16 under the hypothesis that Remark 17.97 makes explicit, and the general theorem Theorem A29.14 is what Section 17.6 was building towards: the exterior derivative and the boundary operator are adjoint, so that $\dd^{2}=0$ (Equation (17.245)) and $\pp\pp=\varnothing$ are the same statement seen from the two sides of Equation (A29.11).