Let’s take the derivative and see what it looks like in terms of coordinates. Say we have a smooth manifold and a smooth map from an open subset of to another smooth manifold . If is any point, we define the derivative as before.
Now, if is a coordinate patch — even if there isn’t a single coordinate patch on the whole domain of we can restrict down to a coordinate patch containing — we get a basis of coordinate vectors at . Similarly, if is a coordinate patch around we get a basis of coordinate vectors at . We want to write down the matrix of in terms of these two bases.
So, the obvious path is to take one of the coordinate vectors at , hit it with , and write the result out in terms of the coordinate vectors at . The generic problem, then, is to calculate the th component — the one corresponding to — of . But we know that this coefficient comes from sticking into this vector and seeing what pops out!
We’re taking the th partial derivative of the th component of the function , which goes from the open set into , where and are the dimensions of and , respectively. Like we saw for coordinate transforms in place, this is just the Jacobian again.
So if we want to write out the derivative in terms of local coordinates, we first write out our local coordinate version of as a function from one Euclidean space to another, and then we take the Jacobian of that function at the appropriate point.
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 .