Action of vector field on Smooth Function

Question

Let $F$ be a $C^\infty$ function from a $n$ dimensional manifold $M$ to a Hilbert space $H$. Let $X$ be a vector field defined on $M$. How can one define the action of $X$ on $F$?

Answer 1

The usual way that vector fields act on functions is by taking derivatives in the direction of the vector field at each point. One way to say this is that $XF:M\to H$ is the function defined by $(XF)(p)=(F\circ \gamma)'(0)$ where $\gamma:(-\epsilon,\epsilon)\to M$ is some smooth curve with $\gamma(0)=p$ and $\gamma'(0)=X(p)$.