I have a question concerning the tangent space. Consider a manifold Mn and take Mn to be ℝn with the Euclidean metric for the purposes of this question. The directional derivative of a function in the direction of a vector v is (a) vf = ∑ vi(∂f/∂xi) where the sum runs from 1 to n. The vector v is then given as (b) v=∑ vi(∂/∂xi). Furthermore the claim is that the space of derivations at p is isomorphic to the space of geometric vectors at p. Thus, we can make the identifications: (c) (∂/∂xi)p <-----------> ei And I can see this to be true if we let the operator in (c) operate on the coordinate functions, and indeed that is what the books do. But to me, there seems to be a slight of hand going on here because in (a) the operator operated on an arbitrary function then suddenly in (b) and (c) the assumption is made that the operator operates on coordinate functions and not arbitrary functions. All books that I have read make this change from the vector operating on a arbitrary function to operating on the coordinate functions without justifying it. So my question is, what is the justification for specifying a specific set of functions in (b) and (c)?