Let \((x^i)\) be a coordinate map corresponding to a local chart \((U, \varphi)\). Then \((\partial/\partial x^i)_p\) (\(1\le i\le n)\) defined in (0x66a8d586) are independent since \((\partial/\partial x^i)_p(x_j)=\delta_{ij}\) and therefore they form a basis.
Then, we have for \(v\in T_pM\)
\begin{equation*} v(f)= v^i \frac{\partial f}{\partial x^i} (p). \end{equation*}The basis elements \((\partial/\partial x^i)_p\) are called coordinate basis of \(T_pM\) and \((v^1,\ldots,v^n)\) are the components of \(v\in T_pM\). [@lee2013smooth_manifolds]
Remark
- That is why \(v(f)\) may be identified with the directional derivative of \(f\) into the direction \(v\).
- An upper index in the denominator is considered as a lower index, therefore the Einsteins sum convention holds.