# Confusion regarding differential forms and tangent space (Spivak,Calc. on Manifolds)

 P: 9 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 $\mathbb R^n_p$ of $\mathbb R^n$ at the point p as the set of tuples $(p,x),x\in\mathbb R^n$. Afterwards, Vector fields are defined as functions F on $\mathbb R^n$, such that $F(p) \in \mathbb R^n_p \ \forall p \in \mathbb R^n$. My analysis professor had defined a vector field to simply be a function $f: \mathbb R^n \to \mathbb R^n$. 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 $\omega$ is defined to be a function with $\omega (p) \in \bigwedge^k(\mathbb R^n_p)$.($\bigwedge^k(\mathbb R^n_p)$ in Spivak's book corresponds to $\bigwedge^k(\mathbb R^n_p)^*$ in other books). Now the question is the following: Would the definition $\omega (p) \in \bigwedge^k(\mathbb R^n)$ not suffice? For example, I find the Notation $dx^i(p)(v_p) = v^i , v_p=(p,v)$ confusing. Why not just define it as $dx^i(p)(v) = v^i$ (so that in fact $dx^i$would not depend on p and be a constant function)? What am I missing here?
 Sci Advisor P: 1,720 On any given tangent space, a differential form is a multi-linear function. But this multi-linear function is not constant. It varies from point to point. That is, it is a different multilinear map on different tangent planes. So for instance dx$^{i}$ is not a constant function but is a collection of linear maps parameterized by R$^{n}$. Usually though the point at which the form is evaluated is understood and is not explicitly n=mentioned unless necessary for clarity.
 P: 9 Thank you for your reply. I realize that on every point it is a different function. The condition $\omega (p) \in \bigwedge^k(\mathbb R^n)$ would express this fact, except that the new function acts on the actual vector space and not on the tangent space. But if my analogy to that of vector fields is correct, then these two formulations should be equivalent. Or am I missing something again?
PF Gold
P: 1,622
Confusion regarding differential forms and tangent space (Spivak,Calc. on Manifolds)

 Quote by madshiver Or am I missing something again?
You are missing the big picture. Spivak sets things up in this way because it parallels how you will construct these objects on manifolds.
 Quote by madshiver Thank you for your reply. I realize that on every point it is a different function. The condition $\omega (p) \in \bigwedge^k(\mathbb R^n)$ would express this fact, except that the new function acts on the actual vector space and not on the tangent space. But if my analogy to that of vector fields is correct, then these two formulations should be equivalent. Or am I missing something again?