If I had to guess, I'd say the closed surface has to enclose a region without a singularity. So if you added to your unit sphere a little sphere around (0,0,0) to make the region enclosed no longer include (0,0,0), the net flux would be 0. --ms On 26-Mar-13 15:46, Dan Asimov wrote:
This question is directed to everyone whose initials are not E.S. or V.E.
Assume any vector fields or surface mentioned is at least C^2. Or even real analytic. The symbol == means "equals for all values of the domain".
Under Divergenceless Field[1] at MathWorld, the first sentence asserts that
(*) div(V) == 0 implies that there exists a vector field W such that V == curl(W).
But their entry for Curl Theorem[2] states that the flux of a vector field of the form curl(W) through a surface S is equal to the line integral of W around its boundary bd(S).
In particular, this implies: (**) The flux of curl(W) through a closed surface must be 0, for any vector field W.
Now consider the vector field V given by V(x,y,z) = (x,y,z) / (x^2 + y^2 + z^2)^(3/2), (x,y,z) unequal to (0,0,0).
It's easy to check that div(V) == 0.
By (*) there exists a W such that V == curl(W).
But it's also easy to check that the flux of V through the unit sphere x^2 + y^2 + z^2 = 1 is 4pi. This contradicts (**).
PUZZLE: Explain how this paradox is possible.
--Dan _____________________________________________________________ [1] < http://mathworld.wolfram.com/DivergencelessField.html > [2] < http://mathworld.wolfram.com/CurlTheorem.html > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun