|San José State University|
& Tornado Alley
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 ωp is a differential p-form and the dimension of the manifold B is p+1. The theorem says that the integral of the diffential of ωp, itself a differential (p+1)-form, over the manifold B is equal to the integral of ωp over the oriented boundary of B, denoted as ∂B,the dimension of which is p-1. The nature of differential forms, p-forms, 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 ω0 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: 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
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 applet-magic.com
HOME PAGE OF Thayer Watkins