Vector calculus

Date: Fri, 18 Nov 1994 08:06:29 -0800 (PST) 
From: Uma Chandavarkar

Hello! I am a senior at Monta Vista High School, and am currently 
enrolled in Calculus. I have been studying vector calculus, and I was 
interested in finding more about the principles of the divergence theorem 
and Stokes' theorem, which relate to flux and curl, respectively. Thank you.

                                - Uma Chandavarkar

Date: Fri, 18 Nov 1994 11:52:04 -0500 (EST)
From: Dr. Ken
Subject: Re: Vector Calculus

Hello Uma!

Ah, Stokes' theorem.  A beautiful theorem.  I never could figure out,
though, why they don't just call all those kinds of theorems "The
Fundamental Theorem of Calculus" and leave it at that, rather than giving
them all kinds of different names like Green's theorem, Gauss's theorem, 
and whatever all the rest are.  I guess it's the engineers and the physicists.

Anyway, if you've seen Stokes' theorem, you've probably been told that it's
the n-dimensional analogue of the Fundamental theorem of Calculus you 
saw in regular one-dimensional calculus.  The integral of a differential form 
over the boundary of a closed region is equal to the integral of the
differential of that form over the region itself.  Nice stuff.

In one dimensional work, you do it like so:  you've got an integral of a
one-form F (is this terminology you've seen before?) over an interval in R1,
and that's equal to the integral (i.e. the sum) of a form K which has F as
its differential over the boundary of the interval (i.e. the endpoints of
the interval).  In practice, you just say K is an antiderivative of F.

This is a pretty deep topic, and Multivariable Calculus is essentially the
study of trying to understand it.  So if you're taking the course, you're in
the right place.

-Ken "Dr." Math