group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
Let
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
In this parameterization the coface maps of $\Delta_{Diff}$ are
For $X$ any smooth manifold a smooth $k$-simplex in $X$ is a smooth function
The boundary of this simplex in $X$ is the the chain (formal linear combination of smooth $(k-1)$-simplices)
Let $\omega \in \Omega^{k-1}(X)$ be a degree $(k-1)$-differential form on $X$.
(Stokes theorem)
The integral of $\omega$ over the boundary of the simplex equals the integral of its de Rham differential over the simplex itself
It follows that for $C$ any $k$-chain in $X$ and $\partial C$ its boundary $(k-1)$-chain, we have
Stokes theorem
a special case is Cauchy's integral theorem
A standard account is for instance in
Discussion of Stokes theorem on manifolds with corners is in
Discussion for manifolds with more general singularities on the boundary is in