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