Basis of Tangent space

**Sogan** · December 12th 2010, 07:22 AM

Hello,

I try to show this equation:
Let M be a manifold. For a tangent space $T_mM$ and a coordinate system (U, $\phi$ ) at $m \in M$ , we have a basis for [LaTeX ERROR: Convert failed] whereas
$\frac{d}{d\phi_i}(f)=\frac{d(f\circ\phi^{-1})}{dx_i}(\phi(m))$

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

**Rebesques** · December 12th 2010, 01:54 PM

Consider a point $p\in M$ belonging to the intersection of the coordinate charts $(U,\phi),\ (V,\psi)$ .
Let $f:M\rightarrow\mathbb{R}^{n}$ be smooth and denote $\frac{\partial}{\partial x_i}(f)=\frac{\partial}{\partial x_i}(f\circ\phi^{-1})(\phi(p))$ , $\frac{\partial}{\partial y_j}(f)=\frac{\partial}{\partial y_j}(f\circ\psi^{-1})(\psi(p))$ .
Now, the map $y=\psi\circ\phi^{-1}:\phi(U)\rightarrow\psi(V)$ is a diffeomorphism, and let $J(p)$ be its Jacobian at $p$ . Now, by using the chain rule on $\mathbb{R}^n$ ,
$\frac{\partial}{\partial x_i}(f)=\frac{\partial}{\partial x_i}(f\circ\psi^{-1}\circ\psi\circ\phi^{-1})(\phi(p))=\sum_j\frac{\partial}{\partial y_j}(f)\frac{\partial y_j}{\partial x_i}$ , or $\frac{\partial}{\partial x_i}=J(p)\frac{\partial}{\partial y_j}$ .

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