Hi, All: It's been a long time since I did this, and I have some basic doubts; please bear with me: In Lee's Riemannian Mflds, p.48, he states that , given a parametrization : γ:(a,b)→M, of a curve , "the velocity vector γ'(t) has a coordinate-independent meaning for each t in M" (this should be for each t in (a,b). Now, Lee goes on to give an example of two parametrizations of S1 , one of which is γ(t)=(cost,sint), and the other is the polar-coordinate expression: γ 2(t)=(r(t),θ(t))=(1,t) . Now, in the first parametrization, the velocity is given by: γ'(t)=(-sint, cost) , while in the second one, we get: γ'2(t)=(0,1) So, in what sense is the velocity coordinate-independent then? Thanks. EDIT: Moreover, the reason given for the "ambiguity" in defining acceleration seems to apply to the definition of velocity too: In the difference quotient LimΔt→0 [f(x+Δt)-f(x)]/Δt the vectors x and x+Δt live in tangent spaces that are not naturally isomorphic to each other, right? Lastly --hope this is not too long of a question-- I understand at an informal level that a connection is a device used to define/select a choice of isomorphism between vector spaces that are not naturally-isomorphic to each other, but I do not see anywhere in this chapter where/how those isomorphisms are defined. Any Ideas/Suggestions? Thanks.