"Restriction" of a Differential Operator on a Vector Bundle to the Space of Local Sections? I'm trying to understand  the definition of differential operators on vector bundles.
The material I'm following
http://www.mat.univie.ac.at/~stein/research/talks/nhops.pdf
starts with the following construction:
Let $(E, \pi_E, M)$ and $(F, \pi_F, M)$ be two smooth vector bundles and let $\Gamma(E)$ and $\Gamma(F)$ be the space of smooth sections of $E$ and $F$. We say a linear operator $$P:\Gamma(E)\longrightarrow \Gamma(F),$$ is local if $$\textrm{supp}(Pu)\subseteq \textrm{supp}(u),$$ for all $u\in \Gamma(E)$. 
With this in mind, if $U\subseteq M$ is open we might define a new operator $$P|_{\Gamma(U)}:\Gamma(U)\longrightarrow \Gamma(U),$$ as follows $$P|_{\Gamma(U)}(u)=(Pv)|_u,$$ where $v\in \Gamma(E)$ is such that $v|_U=u$. Of course, we must verify $P|_{\Gamma(U)}$ is well defined. If $w\in \Gamma(E)$ is another section such that $w|_U=u$ then $$\textrm{supp}(Pv-Pw)=\textrm{supp}(P(v-w))\subseteq \textrm{supp}(v-w)\subseteq M\setminus U,$$ hence if $x\in U$ then,$$Pv(x)-Pw(x)=(Pv-Pw)(x)=0\Rightarrow (Pv)|_U=(Pw)|_U.$$
I have some questions concearning this:
1- Why is there some $v\in \Gamma(E)$ such that $v|_U=u$?
2- When I write $P|_{\Gamma(U)}$ this sugests this is the restriction of $P$ to a subspace of $\Gamma(E)$. But $\Gamma(U)$ is a not a subspace of $\Gamma(E)$, so why to invoke the notation $P|_{\Gamma(U)}$? Further, what is the relation between $\Gamma(U)$ and $\Gamma(E)$?
Thanks 
 A: 1 - Generally there may not be a smooth global section $v$ that restricts to $u$ on $U$, unless $u$ is nice enough. Think, for example, of the trivial bundle $\mathbb{R} \times \mathbb{R}$ over $\mathbb{R}$, whose sections are real-valued functions. $u(x) = \frac{1}{x}$ is a smooth function on the open set $U = \{ x > 0 \} \subset \mathbb{R}$, but $u$ does not extend to a smooth function on all of $\mathbb{R}$.
If I'm understanding this correctly, on p. 5 of the notes you linked to, the author deals with this problem by introducing a smaller set $V \subset U$ and using a cutoff function $\chi$.
2 - You're correct that the notation $P|_{\Gamma(U)}$ is a bit misleading, since $\Gamma(U)$ is not a subspace of $\Gamma(E)$. (Maybe $P|_U$ would be better?) Here's one simple thing that can be said about the relationship between $\Gamma(U)$ and $\Gamma(E)$: there is a restriction map $\Gamma(E) \to \Gamma(U)$ given by the obvious restriction of global sections to $U$. Unfortunately, this map is generally neither surjective (for the reason in my first paragraph) nor injective (since there may exist distinct global sections that agree on $U$).
You may be interested in reading about sheaves, e.g., on Wikipedia.
