From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: Gauss' divergence theorem: restrictions on surface?
Date: Fri, 05 Nov 1999 14:08:52 GMT
Newsgroups: sci.math
Keywords: Stokes' theorem with singularities

In article <7vulb2$jj0$1@nnrp1.deja.com>,
  David Bernier <bernier@my-deja.com> wrote:
> When I learnt about the 3-D (classical) divergence
> theorem, the necessary and/or sufficient conditions
> on the surface S over which the surface integral is
> done were not mentioned.  I believe S being the
> boundary of [0,1]^3 is ok.  What about more exotic
> surfaces S with "spikes" such as obtained by revolving
> the graph of
>   y= x^n * (1-x) around the x-axis where n=1,2,3, etc ?
>   [0 <=x <= 1 ]
>
> or  doing same with the graph of:
>    y= exp(-1/x) * (1-x)  [ y(0):= 0 ]  x \in [0,1]  ?
>
> Do we still have
>   \int_{"inside" S}(div F dx dy dz) = \int_{S)( F . \vec dS)
> for "nice" vector fields F?

In his book Differential Manifolds, Lang discusses Stokes with
singularities. I don't have my copy to hand, but the upshot is that
it's OK as long as the singular points on the boundary hav measure
zero in an appropriate sense.

--
Robin Chapman
 http://www.maths.ex.ac.uk/~rjc/rjc.html
  "`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
    Greg Egan, _Teranesia_


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