This theorem can be thought of as an expression of duality between the homology of chains and the de Rham cohomology, and it is used extensively in the proof of the Poincare-Hopf theorem.
A non-Abelian version of this theorem has been used to show the relationship between the Wilson loop and magnetic monopole in QCD. This theorem can be thought of as an expression of duality between the homology of chains and the de Rham cohomology, and it is used extensively in the proof of the Poincare-Hopf theorem. It is the real-plane generalization of Cauchy's integral theorem. The most general form of this theorem relates the integral of a differential form over a manifold boundary to the integral of the external derivative of that manifold, and it can also be reduced to the Ostrogradsky-Gauss theorem. FTP, name this theorem whose best known form relates the integral around a boundary of a vector field to the surface integral of the curl of that vector field, a three-dimensional analog of Green's theorem.
A non-Abelian version of this theorem has been used to show the relationship between the Wilson loop and magnetic monopole in QCD.
Stokes' Theorem