Abstract:

The objective of this short course is to state and prove a version of Stokes' Theorem in Euclidean space with minimal mathematical prerequisites, making this version of Stokes' Theorem available to those who do not know the smooth manifold theory.  The content of this talk is based on Chapter 4 of M. Spivak's Calculus on Manifolds (1965).

We begin with a brief discussion of the definition and construction of differential forms in Euclidean space as maps that send points  p in space to alternating multilinear functions to the tangent space at p.  We then define singular cubes and chains in Euclidean space and their boundaries, the integral of a differential form on a chain, we prove a version of Stokes' Theorem in this setting, which is only a calculation because of the way forms and chains were defined. From this Stokes' Theorem we deduce versions of the classical Green's, Stokes', and Divergence Theorems.  We end with a fun example, computing a 4-dimensional line integral using our new forms and chains. 

The only prerequisites for this talk are basic knowledge of linear transformations, and of course, some multivariable calculus.   It is nice that we get a precise and effective version of Stokes' Theorem, considering that the work to obtain it is quite easy.

Meeting ID: 931 9296 4149 | Password: GAUSS

We will have milk and cookies! Remember to bring your own mugs.