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May 1st, 2009, 07:23 PM   #1
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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.
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July 2nd, 2009, 03:44 PM   #2
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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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