nLab
Stokes theorem

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

The 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

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 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 \,.

References

A standard account is for instance in

  • Reyer Sjamaar, Manifolds and differential forms, 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

Revised on December 19, 2012 20:38:47 by Urs Schreiber (131.174.41.124)