nLab Stokes theorem





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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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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.)


Traditional statement


Δ Diff:ΔDiff \Delta_{Diff} : \Delta \to Diff

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

Δ k={t 1,,t k 0| it i1} k. \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 Δ Diff\Delta_{Diff} are

i:(t 1,,t k1){(t 1,,t i1,t i+1,,t k1) |i>0 (1 i=1 k1t i,t 1,,t k1). \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 XX any smooth manifold a smooth kk-simplex in XX is a smooth function

σ:Δ kX. \sigma : \Delta^k \to X \,.

The boundary of this simplex in XX is the chain (formal linear combination of smooth (k1)(k-1)-simplices)

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

Let ωΩ k1(X)\omega \in \Omega^{k-1}(X) be a degree (k1)(k-1)-differential form on XX.


(Stokes theorem)

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

σω= σdω. \int_{\partial \sigma} \omega = \int_\sigma d \omega \,.

It follows that for CC any kk-chain in XX and C\partial C its boundary (k1)(k-1)-chain, we have

Cω= Cdω. \int_{\partial C} \omega = \int_{C} d \omega \,.

More generally:


(Stokes theorem for fiber integration)

If UU is any smooth manifold and ωΩ (U×Σ)\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

Σ:Ω +dim(Σ)(U×Σ)Ω (U) \int_\Sigma \;\colon\; \Omega^{\bullet + dim(\Sigma)}(U \times \Sigma) \longrightarrow \Omega^\bullet(U)

along U×Σpr 1UU \times \Sigma \overset{pr_1}{\to} U we have

Σdω= Σω+(1) dim(Σ)d Σω. \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 H\mathbf{H} be a cohesive (∞,1)-topos and write THT \mathbf{H} for its tangent cohesive (∞,1)-topos.

Assume that there is an interval object

**(i 0,i 1)Δ 1 \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 Π\Pi and the localization L Δ 1L_{\Delta^1} at the the projection maps out of Cartesian products with this line Δ 1×()()\Delta^1\times (-) \to (-)

ΠL Δ 1. \Pi \simeq L_{\Delta^1} \,.

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

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

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

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

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

Π dRΩE^, dRΣE^Stab(H) \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 Π( dRΣE^)\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

Π dRΩE^ d dRΣE^ ι θ E^ E^ \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 d\mathbf{d}. See at differential cohomology diagram for details.

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


Integration of differential forms is the map

Δ 1:[Δ 1, dRΣE^]Π dRΩE^ \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 dRΩE^\flat_{dR}\Omega \hat E from the counit naturality square of the flat modality on [(**(i 0,i 1)Δ 1),][(\ast \coprod \ast \stackrel{(i_0, i_1)}{\to} \Delta ^1 ), -], using that this square exhibits a null homotopy due to the Δ 1\Delta^1-homotopy invariance of E^\flat \hat E.


Stokes’ theorem holds:

Δ 1di 1 *i 0 *. \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 n\mathbb{R}^n:

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

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

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

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

  • if n=3n = 3 and k=3k = 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): DdivF= RFn\int\int\int_D div \mathbf{F} = \int\int_R \mathbf{F} \cdot \mathbf{n}, where RR is a continuously differentiable closed surface in 3\mathbb{R}^3, DD is the region that it encloses (guaranteed by the Jordan–Brouwer Separation Theorem), n\mathbf{n} is the outward-pointing unit normal vector field on RR, and F\mathbf{F} is a continuously differentiable vector field on a neighbourhood of DD;

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


The basic statement:

In the generality of manifolds with corners:

Statement of the fiberwise Stokes theorem:

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

Last revised on November 18, 2023 at 14:59:35. See the history of this page for a list of all contributions to it.