# nLab Stokes theorem

cohomology

### Theorems

#### Differential geometry

differential geometry

synthetic differential geometry

# 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

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

${\Delta }^{k}=\left\{{t}^{1},\cdots ,{t}^{k}\in {ℝ}_{\ge 0}\mid \sum _{i}{t}^{i}\le 1\right\}\subset {ℝ}^{k}\phantom{\rule{thinmathspace}{0ex}}.$\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

${\partial }_{i}:\left({t}^{1},\cdots ,{t}^{k-1}\right)↦\left\{\begin{array}{cc}\left({t}^{1},\cdots ,{t}^{i-1},{t}^{i+1},\cdots ,{t}^{k-1}\right)& \mid i>0\\ \left(1-\sum _{i=1}^{k-1}{t}^{i},{t}^{1},\cdots ,{t}^{k-1}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\sigma :{\Delta }^{k}\to X\phantom{\rule{thinmathspace}{0ex}}.$\sigma : \Delta^k \to X \,.

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

$\partial \sigma =\sum _{i=0}^{k}\left(-1{\right)}^{i}\sigma \circ {\partial }_{i}\phantom{\rule{thinmathspace}{0ex}}.$\partial \sigma = \sum_{i = 0}^k (-1)^i \sigma \circ \partial_i \,.

Let $\omega \in {\Omega }^{k-1}\left(X\right)$ be a degree $\left(k-1\right)$-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

${\int }_{\partial \sigma }\omega ={\int }_{\sigma }d\omega \phantom{\rule{thinmathspace}{0ex}}.$\int_{\partial \sigma} \omega = \int_\sigma d \omega \,.

It follows that for $C$ any $k$-chain in $X$ and $\partial C$ its boundary $\left(k-1\right)$-chain, we have

${\int }_{\partial C}\omega ={\int }_{C}d\omega \phantom{\rule{thinmathspace}{0ex}}.$\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)