vector fields in $\mathbb{R}^n$ I am seeing the notion of a vector field for the first time, and I am quite confused. The author (of Gauge Fields, Knots and Gravity) has defined a vector field to be any linear function $v: C^\infty(M) \to C^\infty(M)$ such that $v(fg) = v(f)g + fv(g)$, where $M$ is any manifold. I'm OK with this. He denotes the collection of all vector fields on $M$ by $Vect(M)$ and remarks that this has a $C^\infty(M)$ module structure. So good so far.
Then, he points out that if $v \in Vect(\mathbb{R}^n)$, then $v = \sum\limits_{i=1}^n v_i \partial_i$ for some $v_i \in C^\infty(M)$. I am guessing that the author means that $\partial_i : C^\infty(\mathbb{R}^n) \to C^\infty(\mathbb{R}^n)$ is the partial derivative operator at the $i$th coordinate.
This seems terribly odd to me, and it seems to suggest that the only vector fields on $\mathbb{R}^n$ are directional derivatives (in fact, this is what it's saying!)
If someone could let me know why this is true, then that would be helpful.
[Note: I am aware of why the directional derivative in $\mathbb{R}^n$ has this property].
 A: Given any point $p\in\mathbb{R}^n$ you can show pretty easily that $\displaystyle \left\{\frac{\partial}{\partial x_i}\mid_p\right\}$ (partial derivative evaluated at $p$) is a basis for $T_p \mathbb{R}^n$. Thus, for every $p$ there must exist constant $v_i(p)$ such that $\displaystyle v(p)=\sum_{i=1}^{n}v_i(p)\frac{\partial}{\partial x_i}\mid _p$. Letting $\displaystyle \partial_i$ denote the vector field $\displaystyle p\mapsto \frac{\partial}{\partial x_i}\mid p\in T_p M$ then the above shows that there are functions $v_1,\cdots,v_n$ such that $\displaystyle v(p)=\sum_{i=1}^{n}v_i(p)\partial_i(p)$. Thus, with the $C^\infty(\mathbb{R}^n)$ structure $\displaystyle v=\sum_{i=1}^{n}v_i \partial_i$. Note the maps $v_i$ are smooth basically by definition that $v$ is a smooth map $\mathbb{R}^n\to T\mathbb{R}^n$ (these are basically the local coordinates on $T\mathbb{R}^n$!).
The confusion I think was in the definition of $\partial_i$--this isn't the ''indefinite derivative operator'' but a vector field which assigns to each point the ''$i$-th partial and then evaluate at $p$'' tangent vector in $T_p M$.
Note you can do basically the same thing on a general manifold $M$ within a given chart neighborhood $(U,\varphi)$ where the only difference is that now you will have the pushforward $(\varphi^{-1})_\ast(\partial_i)$ of the $\partial_i$ operators (note this is solid because $\varphi$ is a diffeomorphism so pushforwards of vector fields make sense). The global nature of the above construction for $\mathbb{R}^n$ really just came from the fact that $\mathbb{R}^n$ has a global chart--the identity map.
