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.
Answer
The definition of a vector as a derivation is only really necessary on a curved manifold. If the manifold is flat - and particularly if it is a vector space, as in newtonian mechanics - then its tangent space is canonically equivalent to said vector space, and talk of derivations is simply fancy talk for simple objects.
In that setting I would counter that the real answer to "what is a vector?" is provided by linear algebra. "Who cares what is a vector?", they say: the important thing is how it behaves. Thus a vector is something that obeys the vector space axioms. (And more physically, we also ask that they transform according to the proper representation of $O(3)$ under rotations.) Since forces, velocities, accelerations, electric fields, and so on, obey those axioms, we call them vectors.
Talk of derivations is definitely important, though, on a curved manifold, which is the case in general relativity and closely related areas. Vector objects play a centre-stage role there, but I think the norm is to just handle them appropriately and not really talk about what they are. (There's not really any need! Again, it's much more important how mathematical objects behave than what they're defined as.)
Like all good mathematics, though, they definitely have sound physical interpretations. Velocity is obviously a derivation: $v(f)$ is the rate of change of $f$ along the worldline with velocity $v$, with respect to the particle's proper time. Force, which you ask about, is actually a covector, the derivative of the potential energy $U$: this is evident classically from the formula $\mathbf{F}=-\nabla U$, or from the differential $$dU=\mathbf{F}\cdot d\mathbf{r}.$$
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