May 1st, 2009, 07:23 PM  #1 
Newbie Joined: May 2009 Posts: 1 Thanks: 0  vector field
A vector field in the usual torus T is obtained by parametrizing the meridians of T by arc length and defining w(p) as the velocity vector of the meridian through p. Prove that the vector field is differentiable.

July 2nd, 2009, 03:44 PM  #2 
Member Joined: Aug 2007 Posts: 42 Thanks: 4  Re: vector field
Consider a point on , a coordinate neighbourhood of and (a section of a) meridian contained in .If is the tangent vector at , then . Along , there is an obvious choice of orthonormal bases for the tangent spaces . Consider now a differentiable function . If is differentiable at for all such , then is differentiable. To this end, express . Since is differentiable, we have for some differentiable at functions . Therefore , which proves that is differentiable at . 

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