[math-fun] Volume-preserving vector field paradox
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 >
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
I agree with Mike's second statement, also stated by Andy: IF we drew a parallel sphere (say) smaller than the unit sphere, then the two spheres would bound a region and the net flux across the two spheres of the vector field V would be 0. But I'm not sure how this bears on the resolution of the paradox. --Dan On 2013-03-26, at 1:31 PM, Mike Speciner wrote:
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 27/03/2013 17:46, Dan Asimov wrote:
I agree with Mike's second statement, also stated by Andy: IF we drew a parallel sphere (say) smaller than the unit sphere, then the two spheres would bound a region and the net flux across the two spheres of the vector field V would be 0.
But I'm not sure how this bears on the resolution of the paradox. ...
The theorem that says that div V = 0 ==> V = curl(something) applies only when V is defined on a simply-connected region. The V you chose is singular at the origin, and the theorem doesn't apply to it. (Or, if you prefer to repair the singularity by patching in some particular value at the origin, your V doesn't have div V = 0 throughout the relevant region because that doesn't hold at the origin.) -- g
But the region is simply connected. So, is it the second homptopy group that applies here? -- Gene
________________________________ From: Gareth McCaughan <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Sent: Wednesday, March 27, 2013 12:03 PM Subject: Re: [math-fun] Volume-preserving vector field paradox
On 27/03/2013 17:46, Dan Asimov wrote:
I agree with Mike's second statement, also stated by Andy: IF we drew a parallel sphere (say) smaller than the unit sphere, then the two spheres would bound a region and the net flux across the two spheres of the vector field V would be 0.
But I'm not sure how this bears on the resolution of the paradox. ...
The theorem that says that div V = 0 ==> V = curl(something) applies only when V is defined on a simply-connected region. The V you chose is singular at the origin, and the theorem doesn't apply to it.
(Or, if you prefer to repair the singularity by patching in some particular value at the origin, your V doesn't have div V = 0 throughout the relevant region because that doesn't hold at the origin.)
-- g
participants (4)
-
Dan Asimov -
Eugene Salamin -
Gareth McCaughan -
Mike Speciner