Consider a point belonging to the intersection of the coordinate charts .
Let be smooth and denote , .
Now, the map is a diffeomorphism, and let be its Jacobian at . Now, by using the chain rule on ,
, or .
I try to show this equation:
Let M be a manifold. For a tangent space and a coordinate system (U, ) at , we have a basis for [LaTeX ERROR: Convert failed] whereas
Now i want to prove the coordinate change, in the case, where we have two different coordinate systems in m. The Formula i want to proove is in the book of warner at page 15 Remark 1.20(c) see below. I hope you can see it in the link. Have you an Idea how to proof the statement? Some Hint perhaps...
Thanks in advance!
Foundations of differentiable ... - Google Bücher