> On Thu, 17 Jan 2013, cku wrote: > > > 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: > > Z(s + ds) and Z(s) are two vectors in R^3, as is > their difference.
That is incorrect. Z(s+ds) is an element of the tangent space at point C(s+ds), whereas Z(s) is an element of the tangent space at C(s). They cannot be added or subtracted, since they belong to different vector spaces.