nLab
Stokes theorem

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

be the cosimplicial object of standard k-simplices in SmoothMfd: in degree k this is the standard k-simplex Δ Diff k⊂ℝ k regarded as a smooth manifold with boundary and corners. This may be parameterized as

Δ k={t 1,⋯,t k∈ℝ ≥0∣∑ it i≤1}⊂ℝ 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 are

∂ i:(t 1,⋯,t k−1)↦{(t 1,⋯,t i−1,t i+1,⋯,t k−1) ∣i>0 (1−∑ i=1 k−1t i,t 1,⋯,t k−1).\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

σ:Δ k→X.\sigma : \Delta^k \to X \,.

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

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

Let ω∈Ω k−1(X) be a degree (k−1)-differential form on X.

Theorem

(Stokes theorem)

The integral of ω 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 C any k-chain in X and ∂C its boundary (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)
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