nLab
Stokes theorem

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

$${\Delta }_{\mathrm{Diff}}:\Delta \to \mathrm{Diff}$$\Delta_{Diff} : \Delta \to Diff

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

$${\Delta }^k=\{t^1,\cdots ,t^k\in {ℝ}_{\ge 0}\mid \sum _i{t}^i\le 1\}\subset {ℝ}^k.$$\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 ${\Delta }_{\mathrm{Diff}}$ are

$${\partial }_i:(t^1,\cdots ,t^{k-1})\mapsto \{\begin{array}{cc}(t^1,\cdots ,t^{i-1},t^{i+1},\cdots ,t^{k-1})& \mid i>0\\ (1-\sum _{i=1}^{k-1}{t}^i,t^1,\cdots ,t^{k-1})\end{array}.$$\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

$$\sigma :{\Delta }^k\to X.$$\sigma : \Delta^k \to X \,.

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

$$\partial \sigma =\sum _{i=0}^k(-1{)}^i\sigma \circ {\partial }_i.$$\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

$${\int }_{\partial \sigma }\omega ={\int }_{\sigma }d\omega .$$\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

$${\int }_{\partial C}\omega ={\int }_{C}d\omega .$$\int_{\partial C} \omega = \int_{C} d \omega \,.

Related concepts

• Poincaré lemma

• Stokes theorem

  • nonabelian Stokes theorem

• de Rham theorem

• Hodge theorem

• a special case is Cauchy's integral theorem

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

• Brian Conrad, Differential geometry Math 396. Stokes’ theorem with corners (pdf)

Discussion for manifolds with more general singularities on the boundary is in

• Friedrich Sauvigny, Partial Differential Equations: Vol. 1 Foundations and Integral Representations