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Stokes theorem

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cohesion

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graded differential cohesion

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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

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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

Traditional statement

Let

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.

Theorem

(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:

Proposition

(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 00, 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.

Definition

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.

Proposition

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)

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

Revised on October 26, 2017 11:29:33 by Urs Schreiber (195.37.234.86)