speed of a curve on a Manifold

Question

For a curve, $$c:I\rightarrow M$$ onto a Manifold $M$ and for a chart $(U, \phi)$, and $\phi \circ c = (x^1,...,x^m) $, how does one arrive at the following expression for the derivative: $$\dot{c}=\sum_j \dot{x}^j(\frac{\partial}{\partial\phi^i})\circ c$$

Answer 1

It is an application of the chain rule \begin{equation}c'(0)f:=(f\circ c)'(0)=\left((f\circ \phi^{-1})\circ (\phi \circ c)\right)'(0)=\sum_{i=1}^m \frac{d}{dt}(\phi \circ c)^i \frac{\partial (f\circ \phi^{-1})}{\partial u^i}=\sum_{i=1}^n\dot{x}^i \frac{\partial}{\partial \phi^i}f, \end{equation} where in the last equality the definition of $\frac{\partial}{\partial \phi^i}$ was used. To get your formula just think the $f$ away.