Derivatives in Coordinates
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.