Surface Integrals of Vector Calculus

Date: 01/05/99 at 16:29:46
From: David McLaren
Subject: Vector Calculus - Divergence Theorem

I am struggling generally with the question below. I have tried taking  
divF and integrating over the surface, but I think that I am going way 
wrong with the integration.

Find the SURFACE_INT(F dS) of the vector field F = (x^3, y^3, z^3) 
through the surface of the solid circular cylinder of radius r and 
length L, given parametrically by m(a,b,c) = (a cos(b), a sin(b), c) 
0 <= a <= r, 0 <= b <= 2pi, 0 <= c <= L, using the divergence theorem.

Help!

Date: 01/05/99 at 19:17:38
From: Doctor Schwa
Subject: Re: Vector Calculus - Divergence Theorem

The divergence theorem says:

The surface integral of F dS on the boundary of a region is equal to
the volume integral of the divergence of F on the inside.

So the divergence theorem turns your problem into finding the 
divergence of F, then doing a triple integral of that (cylindrical 
coordinates are probably easiest ...)

That is, you were right to take div F, but you're supposed to integrate 
it over the VOLUME, not the surface.

If you need more of a hint, please write back!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/