On Vector fields as differential operators. I am having trouble with a simple statement in some lecture notes, they say:
take a vector field $\sum_{i=1}^n a_i(x) \frac{\partial}{\partial x_i}$.
I do not know how this differential operator (?) represents a vector field (or is this meant to be a differential form? if it is a differential form I have studied them but never made the connection ).
My naive understanding of a vector field is a vector valued function that assigns to each point of a plane a value in $\mathbb{R}^2$, as an example I have seen them described in a physics course as
$$F(x,y) = x i + yj$$
Where $i,j$ represent the standard basis in $\mathbb{R}^2$. This is a very simple object that I know how to draw while I have no idea how to represent this vector field in the form $\sum_{i=1}^n a_i(x) \frac{\partial}{\partial x_i}$(how would this be done?).
 A: In differential geometry, to each point $p$ of $\mathbb R^n$ you associate another copy of $\mathbb R^n$, containing all vectors that originate at $p$. This is called the "tangent space at $p$" and denoted by $T_p\mathbb R^n$. A vector $v_p\in T_p\mathbb R^n$ can be considered either as a $n$-uple $v_p=(v_p^1, v_p^2,\ldots, v_p^n)$ (these are not exponents) or, equivalently, as the derivative operator defined as
$$ 
D_{v_p} f = \lim_{h\to 0} \frac{ f(p+hv_p)-f(p)}{h}, $$
where $f$ is an arbitrary smooth function. This can be also written as
$$\tag{1}
D_{v_p}=v_p^1\frac{\partial}{\partial x_1} + v_p^2\frac{\partial}{\partial x_2} +\ldots+ v_p^n \frac{\partial}{\partial x_n}, $$
as you know from calculus.
A vector field, as the name implies, is the assignment of a vector for every point of the space. If you represent vectors with (1), a vector field $V$ is given by
$$
V(p)=\sum_{j=1}^n V^j(p)\frac{\partial}{\partial x_j}.$$
Here I have written $V(p)$ instead of $V_p$ to match the notation of your lecture notes. Both notations are in use. Let me remark that, for each $p\in\mathbb R^n$, $V(p)$ is an operator that takes an arbitrary function and acts on it, while the $V^j$ are functions defined on $\mathbb R^n$, usually assumed to be smooth.
