For instance consider the smooth vector field
$$
\phi: \mathscr{M} \longrightarrow T\mathscr{M}
$$
that maps every point on a smooth manifold $\mathscr{M}$ to an element in the tangent space of $\mathscr{M}$.
A lot of books, after defining the idea of a smooth vector field, move on to talk about the space of all such vector fields $\Gamma(T\mathscr{M})$. It's further expressed that this space forms a $C^{\infty}(\mathscr{M})$ – module but in order to show this, they mysteriously talk about the tangent vector fields as functions from $C^{\infty}(\mathscr{M})$ to $C^{\infty}(\mathscr{M})$. Am I missing a step here? I thought the vector fields are functions on the manifold, i,e, the argument of the function is a point on the manifold. How can a vector field of this sort act on a smooth function? It doesn't sense to me. Can someone please clarify this for me. An elaborate answer would be much appreciated as I'm just a beginner in this subject.
Best Answer
Think about what a tangent vector $v$ at a point $x$ does: it takes in a function $f:M \to \mathbb{R}$ and computes the 'directional derivative' $v(f)$ in the direction of $v$ at the point $x$. In this sense, $v$ can be thought of as a function $v:C^\infty(M)\to \mathbb{R}$ (Which satisfies the product rule at $x$, and 'really' acts on smooth functions locally defined near $x$)
A map $V:M \to TM$, then, is a collection of these maps from $C^\infty(M)$ to $\mathbb{R}$; the natural question is then how these maps vary as we move around the manifold. That is, if $f\in C^\infty(M)$ is a function and $x,y\in M$, the values of the vector field $V_x$ and $V_y$ are tangent vectors at $x$ and $y$ respectively. Then naturally we can apply both of them to $f$; what should the relationship be between $V_x(f)$ and $V_y(f)$ be? A priori there's no reason to believe there really should be any relationship between them at all; but one way to define a smooth vector field (or, if you're using another definition of a smooth vector field, a reasonably straightforward theorem) is to say that the map $x\mapsto V_x (f)$, with $V$ and $f$ fixed and the point on $M$ allowed to vary, should be a smooth function. It's this function that we call $V(f)$, and this defines a map $V:C^\infty(M) \to C^\infty(M)$.