San José State University 

appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley USA 


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. The Generalized Stokes Theorem can be stated quite succinctly:
In this statement ω^{p} is a differential pform and the dimension of the manifold B is p+1. The theorem says that the integral of the diffential of ω^{p}, itself a differential (p+1)form, over the manifold B is equal to the integral of ω^{p} over the oriented boundary of B, denoted as ∂B,the dimension of which is p1. The nature of differential forms, pforms, is explained in more detail elsewhere. Here the nature of the Generalized Stokes Theorem will be illustrated.
Several important theorems are simply special cases of the Generalized Stokes Theorem.
The Fundamental Theorem of Calculus:
In terms of the above statement of the GST, the manifold B is the line segment from x=a to x=b. Its boundary is the pair of points x=a and x=b. The ω^{0} is F(x) and its integral over the boundary is [F(b)−F(a)].
More explicitly the integral over the boundary is h(b)F(b)+h(a)F(a), where h(b) is the orientation of the boundary point x=b and h(a) is that of x=a. The orientation of b is +1 and that a is −1, hence F(b)−F(a).
Stokes Theorem:
Stokes' Theorem: The integral of a vector function F(x, y, z) around a
directed closed curve ∂B, which is the oriented boundary of an oriented
surface B is equal to
the integral of the curl of F over the surface B. Restated this is
Green's Theorem: Let B be a region in a plane that is enclosed by a positively oriented smooth simple closed curve ∂B. Then
Three Dimensions:
Gauss' Theorem: The integral of a vector function F(x,y,z) over the surface of a closed three dimensional volume B is equal to the integral of the divergence of F(x,y,z) over the volume B. Restated in symbols this is:
HOME PAGE OF appletmagic.com
HOME PAGE OF Thayer Watkins 
