I am reading the proof for a theorem about connections on a manifold, but I'm not comfortable with the fancy language of vector bundles and flows of vector fields I think. I wonder if there's an easy way of understand this calculation using just multivariable calculus:
So, we have a fiber bundle $\pi: E \to M$ and for any $\gamma(t) \in M$ we have a family of diffeomorphisms that makes parallel transport possible: $P^t_{\gamma}: E_{x_0} \to E_{\gamma(t)}$. It says that $P^t_{\gamma}$ is the flow of some vector field $Y(t,p)$ on the total space of the pullback bundle $\gamma^* E$ (what is that?) Choosing any section $s: M \to E$ with $s(x_0)=p_0$ and writing $F(t,p) = (P^t_{\gamma})^{-1}(p)$, we have:
$$\nabla_{\dot \gamma(0)}s = \left.\frac{d}{dt}F(t,s(\gamma(t))) \right|_{t=0} = \frac{\partial F}{\partial t}(0,p_0)+ D_2F(0,p_0) \circ Ts(\dot \gamma(0))=-Y(0,p_0)+Ts(\dot \gamma(0))$$
So, here is where I get confused. How does this differentiation work? It's easy to differentiate the first component of $F$ with respect to $t$ but the second one is complicated and I can't figure out how that works. Moreover, why $\frac{\partial F}{\partial t} (0,p_o) = -Y(t,p)$?