nLab Stokes theorem

Idea

The Stokes theorem (also Stokes' theorem or Stokes's theorem) asserts that the integral of an exterior differential form on the boundary of an oriented manifold with boundary (or submanifold or chain of such) equals the integral of the de Rham differential of the form on the manifold itself. (The theorem also applies to exterior pseudoforms on a chain of pseudoriented submanifolds.)

Statement

Traditional statement

Let

$$\Delta_{Diff} : \Delta \to Diff$$

be the cosimplicial object of standard $k$-simplices in SmoothMfd: in degree $k$ this is the standard $k$-simplex $\Delta^k_{Diff} \subset \mathbb{R}^k$ regarded as a smooth manifold with boundary and corners. This may be parameterized as

$$\Delta^k = \{ t^1, \cdots, t^k \in \mathbb{R}_{\geq 0} | \sum_i t^i \leq 1\} \subset \mathbb{R}^k \,.$$

In this parameterization the coface maps of $\Delta_{Diff}$ are

$$\partial_i : (t^1, \cdots, t^{k-1}) \mapsto \left\{ \array{ (t^1, \cdots, t^{i-1}, t^{i+1} , \cdots, t^{k-1}) & | i \gt 0 \\ (1- \sum_{i=1}^{k-1} t^i, t^1, \cdots, t^{k-1}) } \right. \,.$$

For $X$ any smooth manifold a smooth $k$-simplex in $X$ is a smooth function

$$\sigma : \Delta^k \to X \,.$$

The boundary of this simplex in $X$ is the chain (formal linear combination of smooth $(k-1)$-simplices)

$$\partial \sigma = \sum_{i = 0}^k (-1)^i \sigma \circ \partial_i \,.$$

Let $\omega \in \Omega^{k-1}(X)$ be a degree $(k-1)$-differential form on $X$.

Theorem

(Stokes theorem)

The integral of $\omega$ over the boundary of the simplex equals the integral of its de Rham differential over the simplex itself

$$\int_{\partial \sigma} \omega = \int_\sigma d \omega \,.$$

It follows that for $C$ any $k$-chain in $X$ and $\partial C$ its boundary $(k-1)$-chain, we have

$$\int_{\partial C} \omega = \int_{C} d \omega \,.$$

More generally:

Proposition

(Stokes theorem for fiber integration)

If $U$ is any smooth manifold and $\omega \in \Omega^\bullet(U \times \Sigma)$ is a differential form on the Cartesian product, then with respect to fiber-wise integration of differential forms

$$\int_\Sigma \;\colon\; \Omega^{\bullet + dim(\Sigma)}(U \times \Sigma) \longrightarrow \Omega^\bullet(U)$$

along $U \times \Sigma \overset{pr_1}{\to} U$ we have

$$\int_\Sigma d \omega \;=\; \int_{\partial_\Sigma} \omega + (-1)^{dim(\Sigma)} d \int_\Sigma \omega \,.$$

(e.g. Gomi & Terashima 2000, remark 3.1)

Abstract formulation in cohesive homotopy-type theory

We discuss here a general abstract formulation of differential forms, their integration and Stokes theorem in the axiomatics of cohesive homotopy type theory (following Bunke-Nikolaus-Völkl 13, theorem 3.2).

Let $\mathbf{H}$ be a cohesive (∞,1)-topos and write $T \mathbf{H}$ for its tangent cohesive (∞,1)-topos.

Assume that there is an interval object

$$\ast \cup \ast \stackrel{(i_0, i_1)}{\longrightarrow} \Delta^1$$

“exhibiting the cohesion” (see at continuum) in that there is a (chosen) equivalence between the shape modality $\Pi$ and the localization $L_{\Delta^1}$ at the the projection maps out of Cartesian products with this line $\Delta^1\times (-) \to (-)$

$$\Pi \simeq L_{\Delta^1} \,.$$

This is the case for instance for the “standard continuum”, the real line in $\mathbf{H} =$ Smooth∞Grpd.

It follows in particular that there is a chosen equivalence of (∞,1)-categories

$$\flat(\mathbf{H})\simeq L_{\Delta^1}\mathbf{H}$$

between the flat modal homotopy-types and the $\Delta^1$-homotopy invariant homotopy-types.

Given a stable homotopy type $\hat E \in Stab(\mathbf{H})\hookrightarrow T \mathbf{H}$ cohesion provides two objects

$$\Pi_{dR} \Omega \hat E \,,\;\; \flat_{dR}\Sigma \hat E \;\; \in Stab(\mathbf{H})$$

which may be interpreted as de Rham complexes with coefficients in $\Pi(\flat_{dR} \Sigma \hat E)$, the first one restricted to negative degree, the second to non-negative degree. Moreover, there is a canonical map

$$\array{ \Pi_{dR}\Omega \hat E && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{dR}\Sigma \hat E \\ & {}_{\mathllap{\iota}}\searrow && \nearrow_{\mathrlap{\theta_{\hat E}}} \\ && \hat E }$$

which interprets as the de Rham differential $\mathbf{d}$. See at differential cohomology diagram for details.

Throughout in the following we leave the “inclusion” $\iota$ of “differential forms regarded as $\hat E$-connections on trivial $E$-bundles” implicit.

Definition

Integration of differential forms is the map

$$\int_{\Delta^1} \;\colon\; [\Delta^1, \flat_{dR}\Sigma \hat E] \longrightarrow \Pi_{dR}\Omega \hat E$$

which is induced via the homotopy cofiber property of $\flat_{dR}\Omega \hat E$ from the counit naturality square of the flat modality on $[(\ast \coprod \ast \stackrel{(i_0, i_1)}{\to} \Delta ^1 ), -]$, using that this square exhibits a null homotopy due to the $\Delta^1$-homotopy invariance of $\flat \hat E$.

Proposition

Stokes’ theorem holds:

$$\int_{\Delta^1} \circ \mathbf{d} \;\simeq\; i_1^\ast - i_0^\ast \,.$$

(Bunke-Nikolaus-Völkl 13, theorem 3.2)

Classical forms

In early 20th-century vector analysis? (and even today in undergraduate Calculus courses), the Stokes theorem took various classical forms about vector fields in the Cartesian space $\mathbb{R}^n$:

• if $n = 1$ and $k = 1$, then this is the second Fundamental Theorem of Calculus: $\int_{[a,b]} f' = f(b) - f(a)$, where $a \leq b$ are real numbers and $f$ is a continuously differentiable function on a neighbourhood of the interval $[a,b]$;

• if $k = 1$ more generally, then this is a generalized form of the FTC: $\int_C grad f \cdot \mathbf{T} = f(Q) - f(P)$, where $C$ is a continuously differentiable oriented curve in $\mathbb{R}^n$, $P$ and $Q$ are the beginning and ending points (respectively) of $C$, $\mathbf{T}$ is the unit vector field on $C$ tangent to $C$ in the direction given by its orientation, and $f$ is a continuously differentiable function on a neighbourhood of $C$;

• if $n = 2$ and $k = 2$, then this is Green's Theorem (see there for other forms; stated without proof by Cauchy in 1846, proved by Riemann in 1851): $\int\int_R (\partial{v}/\partial{x} - \partial{u}/\partial{y}) = \int_C (u \,\mathrm{d}x + v \,\mathrm{d}y)$, where $C$ is a continuously differentiable simple closed curve in $\mathbb{R}^2$ (oriented using the standard orientation on $\mathbb{R}^2$), $R$ is the region that it encloses (guaranteed by the Jordan Curve Theorem), and $u$ and $v$ are continuously differentiable functions of the coordinates $x$ and $y$ on a neighbourhood of $R$;

• if $n = 3$ and $k = 2$, then this is the Kelvin–Stokes Theorem or Curl Theorem (stated without proof by Stokes? in 1854, proved by Hankel? in 1861): $\int\int_R curl \mathbf{F} \cdot \mathbf{n} = \int_C \mathbf{F} \cdot \mathbf{T}$, where $R$ is a continuously differentiable pseudooriented surface in $\mathbb{R}^3$ with a continuously differentiable boundary $C$ (oriented to match the pseudoorientation of $R$ using the standard orientation on $\mathbb{R}^3$), $\mathbf{n}$ is the unit normal vector field on $R$ in the direction given by the pseudoorientation of $R$, $\mathbf{T}$ is the unit tangent vector field on $C$, and $\mathbf{F}$ is a continuously differentiable vector field on a neighbourhood of $R$;

• if $n = 3$ and $k = 3$, then this is the Ostrogradsky–Gauss Theorem or Divergence Theorem (special cases were proved by Gauss in 1813, the general case was proved by Ostrogradsky in 1826): $\int\int\int_D div \mathbf{F} = \int\int_R \mathbf{F} \cdot \mathbf{n}$, where $R$ is a continuously differentiable closed surface in $\mathbb{R}^3$, $D$ is the region that it encloses (guaranteed by the Jordan–Brouwer Separation Theorem), $\mathbf{n}$ is the outward-pointing unit normal vector field on $R$, and $\mathbf{F}$ is a continuously differentiable vector field on a neighbourhood of $D$;

• if $k = n$ more generally, then this is the generalized Divergence Theorem (proved by Ostrogradsky in 1836): $\int_D div \mathbf{F} = \int_R \mathbf{F} \cdot \mathbf{n}$, where $R$ is a continuously differentiable closed hypersurface in $\mathbb{R}^n$, $D$ is the region that it encloses, $\mathbf{n}$ is the outward-pointing unit normal vector field on $R$, and $\mathbf{F}$ is a continuously differentiable vector field on a neighbourhood of $D$.

Related concepts

• Poincaré lemma

• Stokes theorem

  • nonabelian Stokes theorem

• de Rham theorem

• Hodge theorem

• a special case is Cauchy's integral theorem

References

The basic statement:

• Raoul Bott, Loring Tu, Theorem 3.5 in: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/978-1-4757-3951-0)

In the generality of manifolds with corners:

• John Lee, Theorem 10.32 in: Introduction to Smooth Manifolds, Springer 2012 (doi:10.1007/978-1-4419-9982-5, pdf)

• Brian Conrad, Stokes’ theorem with corners (pdf, pdf)

Statement of the fiberwise Stokes theorem:

• Kiyonori Gomi, Yuji Terashima, Remark 3.1 in: A Fiber Integration Formula for the Smooth Deligne Cohomology, International Mathematics Research Notices 2000, No. 13 (pdf, pdf, doi:10.1155/S1073792800000386)

• Liviu Nicolaescu, Theorem 3.4.54 of: Lectures on the Geometry of Manifolds, 2018 (pdf, pdf)

and specifically for simplicial differential forms:

• Dupont, Ljungman, Theorem 5.10 in: Integration of simllicial forms and Deligne cohomology (arXiv:math/0402059)

Statement of the Stokes theorem in the full generality of fiberwise integration over fibers with corners:

• Alberto Cattaneo, Nima Moshayedi, Konstantin Wernli, equation (65) in: Globalization for Perturbative Quantization of Nonlinear Split AKSZ Sigma Models on Manifolds with Boundary (arXiv:1807.11782)

Discussion of chains of smooth singular simplices

• Stokes’ theorem on chains (pdf)

Discussion for manifolds with more general singularities on the boundary is in

• Friedrich Sauvigny, Partial Differential Equations: Vol. 1 Foundations and Integral Representations

Discussion in cohesive homotopy type theory is in

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)