Special and general types
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.
be the cosimplicial object of standard -simplices in SmoothMfd: in degree this is the standard -simplex regarded as a smooth manifold with boundary and corners. This may be parameterized as
In this parameterization the coface maps of are
For any smooth manifold a smooth -simplex in is a smooth function
The boundary of this simplex in is the the chain (formal linear combination of smooth -simplices)
Let be a degree -differential form on .
The integral of over the boundary of the simplex equals the integral of its de Rham differential over the simplex itself
It follows that for any -chain in and its boundary -chain, we have
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