I have a subtle doubt about the physical interpretation of the mathematical definition of vector field as a derivation. In basic physics we understand a vector quantity as a quantity that needs more than magnitude to be fully specified, in other words, quantities with the notion of direction. This goes very well with the also basic mathematical definition that a vector is an equivalence class of oriented line segments.
However, when we go to the study of manifolds, we see that a better definition of vector is to say that a vector at a point is a derivation on the algebra of smooth functions on that point. But then, we represent forces for instance with vectors, what's the interpretation of representing one force acting on a point by a derivation on the smooth functions on the point? I imagine that there must be some interpretation for that, but I didn't find what's it.
Sorry if this question seems silly. I'm just trying to bring together those concepts. And thanks you all in advance.