How to motivate vectors as derivations?

Question

In a manifold it's easy to motivate the definition of vectors as equivalence classes of curves. On the other hand the definition as derivations is harder to motivate. I know how to show that the space obtained with derivations is isomorphic to the space obtained with curves, so I know that defining with derivations works.

My doubt it is: how would I motivate the usage of derivations? This question is, how can it be intuitively visible that linearity and Leibniz rule alone are sufficient for assuring that the operator is a directional derivative?

I've seem some discussion about it on the "Applied Differential Geometry" book from William Burke where he tries to motivate this saying that a differential operator measures terms in the Taylor series and that Leibniz rule assures the operator is sensible only to first order terms. But I didn't get this idea yet.

Answer 1

The idea is, as always when dealing with local notions on manifolds, to pass to a chart and see what happens. On a chart (i.e. Euclidean space) you have an obvious correspondence between vectors and directional derivatives, in the sense that derivation in direction $v$ is given by $Df\cdot v$. If you write this down component-wise and lift it up to the manifold, you'll see the correspondence between tangent vectors and derivations on the manifold.

Answer 2

How can it be intuitively visible that linearity and Leibniz rule alone are sufficient for assuring that the operator is a directional derivative?

If we define a derivation by these conditions, we can prove that the set of derivations at a point is an $n$-dimensional vector space - this theorem should be found in any textbook covering abstract manifolds. Thus these conditions already give us a space of the correct size to be the space of directional derivatives - any extra conditions would either be superfluous, would decrease the dimension of the space or would make it fail to be a vector space at all.