Given a coordinate patch in a neighborhood of a point in an -dimensional manifold , we get coordinate vectors which form a basis for the tangent space . But this is true of any coordinate patch! If we have another patch , we can get another basis . Today we’ll examine how these bases are related.
As is usual, we can use the two bases to come up with a change of basis matrix. This can be used to take the components of any vector written out in terms of the basis and get the components of the same vector written out in terms of the basis. And we come up with the matrix by asking how to write out each of the latter basis vectors in terms of the former.
So, how did we find the components of the tangent vector in terms of the basis? We evaluated . Let’s stick into and see what we get:
But all this means is that we take the transition function , take the th component, and take the th partial derivative of that function. And this is precisely the definition of the Jacobian of this transition function!
This basic fact starts showing us how everything we did when talking about multivariable calculus is on the one hand a special case of the concepts of differential geometry we’re coming up with now, while on the other hand they’re exactly the groundwork we need to build up the more general tools. What we did before in the simple Euclidean spaces is the model for all the more complicated manifolds we study.