I have been working through Spivak's fine book, but the part about differential forms and tangent spaces has left me confused. In particular, Spivak defines the Tangent Space [itex]\mathbb R^n_p[/itex] of [itex]\mathbb R^n[/itex] at the point p as the set of tuples [itex](p,x),x\in\mathbb R^n[/itex]. Afterwards, Vector fields are defined as functions F on [itex]\mathbb R^n[/itex], such that [itex]F(p) \in \mathbb R^n_p \ \forall p \in \mathbb R^n[/itex]. My analysis professor had defined a vector field to simply be a function [itex] f: \mathbb R^n \to \mathbb R^n [/itex]. Now it appears to me that the definition according to Spivak is way more elegant in the sense that it maches the geometric intuition behind a vector field. But at the same time, as I see it, neither definition includes more "information" than the other. And then the actually confusion comes around: A differential k-form [itex]\omega[/itex] is defined to be a function with [itex] \omega (p) \in \bigwedge^k(\mathbb R^n_p) [/itex].([itex]\bigwedge^k(\mathbb R^n_p) [/itex] in Spivak's book corresponds to [itex]\bigwedge^k(\mathbb R^n_p)^*[/itex] in other books). Now the question is the following: Would the definition [itex] \omega (p) \in \bigwedge^k(\mathbb R^n) [/itex] not suffice? For example, I find the Notation [itex] dx^i(p)(v_p) = v^i , v_p=(p,v) [/itex] confusing. Why not just define it as [itex] dx^i(p)(v) = v^i [/itex] (so that in fact [itex] dx^i [/itex]would not depend on p and be a constant function)? What am I missing here?