Consider a smooth manifold $M$ and let $E_1, \cdots, E_n$ be a local frame on $M$ (let's say in an open set $U$).
Now assume that we have a smooth curve $\gamma:(-\varepsilon, \varepsilon) \to M$ such that $\gamma$ is contained in $U$.
We now can look at the differential of $\gamma$, and we would get that $$\gamma'(t) = \sum_{i=1}^n A^i(\gamma(t))E_i|_{\gamma(t)}. $$
My question is: is there a way to compute the functions $A^1, \cdots, A^n$? If the frame would come from a chart, then it would be easy, as we would just evaluate the above vector fields on each coordinate function of the chart. However, the above local frame may not come from a chart, so I don't really know what to do in this case.