Hello,

I try to show this equation:

Let M be a manifold. For a tangent space $\displaystyle T_mM$ and a coordinate system (U,$\displaystyle \phi$) at $\displaystyle m \in M$, we have a basis for $\displaystyle T_mM: B=\{\frac{d}{d\phi_i}: i=1,...,d\}$ whereas

$\displaystyle \frac{d}{d\phi_i}(f)=\frac{d(f\circ\phi^{-1})}{dx_i}(\phi(m))$

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