On Tue, Mar 26, 2013 at 3:46 PM, Dan Asimov <dasimov@earthlink.net> 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".
Since V is discontinuous at the origin, we'll have to work in some domain that doesn't include the origin. For example, we could use the region epsilon < r < 1.
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).
Well, if we're working in 3 dimensions, it's a surface integral, not a line integral. You mean
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.
But the unit sphere is only one of the two components of the boundary of our region. No matter how small we make the other component of the boundary, the small sphere around the origin, the flux of V through that other component wil be -4pi. The divergence measures the "sourceness" of the vector field at a point, but V has a "point source" at the origin. Andy Latto But
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
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