# nLab Stokes theorem

cohomology

### Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(ʃ \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$ʃ_{dR} \dashv \flat_{dR}$

• tangent cohesion

• differential cohomology diagram
• differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

$(\Re \dashv \Im \dashv \&)$

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

• 

\array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& &#643; &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

Stokes theorem asserts that the integration of differential forms of the de Rham differential of a differential form over a domain equals the integral of the form itself over the boundary of the domain.

## 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

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

### 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:

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

## References

A standard account is for instance in

• Reyer Sjamaar, Manifolds and differential forms, pdf

Discussion of chains of smooth singular simplices

• Stokes’ theorem on chains (pdf)

Discussion of Stokes theorem on manifolds with corners is in

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 October 26, 2017 at 11:29:33. See the history of this page for a list of all contributions to it.