I have a question about the following (this is from Tu's book "Introduction to topological manifolds").
From what I understand, if our chart $(U, \phi=(x^1, \dots, x^n))$ is given, then Tu concludes that
$$(x^i \circ c)'(t) = a^i(c(t))$$
Am I interpreting this correctly?
If yes, how does this differential equation imply that we have a unique solution? From what I know, we must be able to write the DE in the form $$y' = f(x,y)$$ to be able to conclude. Clearly we need $y = c$ here but then it is not in the right form. What am I missing?
Edit After a discussion in the comments, the nature of the confusion has become clear. This answer tries to be very precise about the use of the smooth chart to use Picard-Lindelöf. We have a smooth chart $(U,\phi)$, which consists of an open set $U\subset M$ and a diffeomorphism $\phi:U\rightarrow V\subset \mathbb R^m,$ where $V$ is open in $\mathbb R^m$. The vector field is a map $X:M\rightarrow TM$ such that $X_p\in T_pM$ for all $p\in M$. Since we have a map $\phi:U\rightarrow V$ and its derivative $D\phi:TU\rightarrow TV,$ we may define a new vector field $\phi^*X:V\rightarrow TV$ by $f=D\phi\circ X\circ\phi^{-1}.$ But $TV=V\times \mathbb R^m,$ so $\phi^*X(p)=(p,v)$ can just be described by a map $f:p\mapsto v$, i.e. the unique function $f:V\rightarrow \mathbb R^m$ such that $\phi^*X(p)=\big(p,f(p)\big)$. Since this is a vector-valued function on an open subset on $\mathbb R^m$, we may apply the Picard-Lindelöf theorem to find some $\epsilon>0$ and a function $y:\mathbb (-\epsilon,\epsilon)\rightarrow \mathbb R^m$ satisfying $y'(t)=f\circ y(t)$ and $y(0)=\phi(c(0))$. Then we define $c:(-\epsilon,\epsilon)\rightarrow M$ by $c(t)=\phi^{-1}\circ y(t).$ This clearly satisfies the prescribed initial conditions. Moreover, you may check that the derivative $c':\mathbb R\rightarrow TM$ satisfies $c'(t)=X_{c(t)}.$ This completes the solution of the differential equation on the manifold $M$.
While it is good to be comfortable with the formality of such arguments, I think that differential geometers usually treat smooth charts in a more hand-waving style. For example, since we know that ODEs of the form $y'=f(y,t)$ can be solved on open subsets of $\mathbb R^m$ and smooth manifolds "look like" open subsets of $\mathbb R^m$, such ODEs can be solved locally on manifolds. But since Picard-Lindelöf also guarantees uniqueness, these local solutions piece together globally (once you move on from integral curves to global flow). I think your wish for formality will serve you well, so long as you also aim to understand things intuitively. I hope the rest of Tu's book goes well for you!