Given $v,p \in \mathbb{R}^n$, I will write $T_p(v)$ for $v$ viewed as a vector based at $p$. So: $$ T_p(v) \in T_p\mathbb{R}^n$$
Question 0. Is there an accepted notation for what I'm denoting $T_p(v)$?
Now define a function $f : \mathbb{R} \rightarrow T\mathbb{R}$ as follows:
$$f(p) = T_p(1)$$
This can be viewed as a vectorfield on $\mathbb{R},$ since it is a section of the canonical projection $T\mathbb{R} \rightarrow \mathbb{R}$.
Question 1. Is there an accepted notation for what I'm denoting $f$?
More generally, given a natural number $n$, we get, for each natural $i<n$, a special vector field $$f_i : \mathbb{R}^n \rightarrow T\mathbb{R}^n$$ given by $$f_i(p) = T_p(e_i)$$ where $e_i$ is the $i$th vector in the distinguished basis of $\mathbb{R}^n$.
Question 2. Is there an accepted notation for what I'm denoting $f_i$?
Edit. I had an idea. A vectorfield on $\mathbb{R}^n$ can be defined as a section $\mathbb{R}^n \rightarrow T\mathbb{R}^n$. It can also be defined as a derivation $\mathcal{C}^\infty(\mathbb{R}^n,\mathbb{R}) \rightarrow \mathcal{C}^\infty(\mathbb{R}^n,\mathbb{R}).$ What's really going on here is that we can interconvert between these objects. So in particular, if we're given a derivation $X : \mathcal{C}^\infty(\mathbb{R}^n,\mathbb{R}) \rightarrow \mathcal{C}^\infty(\mathbb{R}^n,\mathbb{R}),$ we get a corresponding section $[X] : \mathbb{R}^n \rightarrow T\mathbb{R}^n$ defined as follows:
$$[X](p)(f) = X(f)(p)$$
So all I really need is to find a standard notation for $[X]$, at which point the function $f_i : \mathbb{R}^n \rightarrow T\mathbb{R}^n$ can be denoted $[\partial/\partial x_i]$.
In maximum generality, then, what's needed is a standard notation for the transformation that turns a function $A \rightarrow (B \rightarrow C)$ into a function $B \rightarrow (A \rightarrow C)$. The notation can then be used to transform back-and-forth between vectorfields-as-sections and vectorfields-as-derivations.