Date: Jan 18, 2013 12:34 AM
Author: Bacle H
Subject: How to Define Derivative of a Vector Field in this Case ( Curve in R^3)?

Hi, All:

 Let C:I-->R^3 be a smooth curve, and let Z(s) be a vector field along the curve,
 parametrized by arc-length.

 We define the derivative of a vector field Z along the curve to be the quotient:

 Lim_ds->0   [Z(s+ds)-Z(s)]/ds

  Now, I don't know how to make sense of the difference in the numerator:

 The two vectors Z(s+ds) and Z(s) , are in different tangent spaces --

 tangent space at s+ds and s respectively -- and , AFAIK, the difference

 of vectors in different tangent spaces is not defined, except for cases

 where there is a natural isomorphism between the tangent spaces, as in the

 case where the tangent spaces are those in R^n itself. Any suggestions,

 please?