Seth Greendale

The classical Stokes’ Theorem is a familiar result from vector calculus,relating the integral over a surface inR3to the integral over the1-dimensional boundary of that surface. A series of other classicaltheorems offer similar relationships between the integralover a region andan integral over its boundary, including Green’s Theorem, the DivergenceTheorem, and even the Fundamental Theorem of Calculus. The GeneralStokes’ Theorem extends this relationship to k-dimensionalmanifolds-with-boundary. In order to understand this generalization, weintroduce the differential form as a new type of integrand. Differentialforms offer a coordinate-free approach to vector calculus,which in turnwill allow us to define integration over manifolds and ultimately to definethe General Stokes’ Theorem. This talk should be accessibleto studentswith a background in vector calculus and linear algebra.