What is the map $Xf$, i.e. a section of the vector bundle apply to a smooth function on a smooth manifold $M$?

Question

In the definition of connections, we have a defining property $D_X (fY) = (Xf)Y + fD_X Y$, where $X, Y$ are two $C^\infty$ sections of vector bundles of smooth manifold $M$ and $f$ is a $C^\infty$ real valued function on $M$. I wonder what is the map $Xf$?

Answer 1

$Xf$ is a $C^\infty$ function on $M$. In local coordinates $x^i$, if $X=X^i \frac{\partial}{\partial x^i}$, then $Xf$ is exactly what you would expect it to be: $$ Xf(p)=\left(\left(X^i \frac{\partial}{\partial x^i}\right) f\right) (p)= X^i \frac{\partial f}{\partial x^i}(p). $$ I always find things the easiest to understand in local coordinates...indeed, this is how a manifold is defined in the first place