restricting a vector field to an open set

Question

I have a vector field $X\in\mathfrak{X}(M)$ on a manifold $M$ and an open set $U$ of $M$. What does it mean to restrict $X$ to $U$? How do we define the following?

$$X\vert_U:C^\infty(U)\rightarrow C^\infty(U)$$

Answer 1

Recall that a vector field over a smooth manifold $M$ is a section of the tangent bundle $TM$. So it is a map $$ X : M \to TM $$ such that for every $p \in M$ it holds $X(p) \in T_pM$. Restricting $X$ to $U$ means nothing else but restricting the domain of $X$, as you would do in general when you consider the restriction of a function.