# Thread: Basis of Tangent space

1. ## Basis of Tangent space

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...

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