It turns out that the tangent bundle construction is actually a functor. Given a smooth map between smooth manifolds, we will get a smooth map . Yes, we’d usually write for a functor’s action on a map, but the notation is pretty classical.
So if we’re given a tangent vector we want to get a tangent vector . And since we already have sending points of to points of , it only makes sense to ask that . That is, in terms of the tangent bundle projection functions, we can write . In other words, the projection will be a natural transformation from the tangent bundle functor to the identity functor.
Anyway, this means that for each we’ll get a map . Since these are both vector spaces, it only stands to reason that we’d have a linear map. We haven’t yet established the connection between our “tangent vectors” and the geometric notion, but we do have a notion from multivariable calculus of a linear map that takes tangent vectors to tangent vectors: the Jacobian, which we saw as a certain extension of the notion of the derivative. We will find that our map is the analogue of the same concept on manifolds, and so we will call it the derivative of .
So here’s our definition: if is a differentiable map in some open set and if , then we define our map by
where is any smooth function on a neighborhood of . That is, is a linear functional on ; if represents a germ at we can compose it with to represent a germ at , and then we can apply itself to this germ. It should be immediately clear that this construction is linear in .