# 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

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

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

$\Delta^k = \{ t^1, \cdots, t^k \in \mathbb{R}_{\geq 0} | \sum_i t^i \leq 1\} \subset \mathbb{R}^k \,.$

In this parameterization the coface maps of $\Delta_{Diff}$ are

$\partial_i : (t^1, \cdots, t^{k-1}) \mapsto \left\{ \array{ (t^1, \cdots, t^{i-1}, t^{i+1} , \cdots, t^{k-1}) & | i \gt 0 \\ (1- \sum_{i=1}^{k-1} t^i, t^1, \cdots, t^{k-1}) } \right. \,.$

For $X$ any smooth manifold a smooth $k$-simplex in $X$ is a smooth function

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

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

$\partial \sigma = \sum_{i = 0}^k (-1)^i \sigma \circ \partial_i \,.$

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

$\int_{\partial \sigma} \omega = \int_\sigma d \omega \,.$

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

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

## 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)