Collection: Generalized Stokes Theorem

Mathematics is a very practical subject but it also has its aesthetic elements. One of the most beautiful topics is the Generalized Stokes Theorem.

This beauty comes from bringing together a variety of topics: integration, differentiation, manifolds and boundaries.

In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem,is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.

Stokes' theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω, i.e.,

The mathematical theory behind this moment is a few steps past calculus and fairly deep into analysis, so instead of focusing on a rigorous definition, just take a moment to enjoy this really tiny formula: