Problem with differentiating an identity regarding one-family of diffeomorphisms

Question

I really dont see how the differentiation with respect to t has been done. Before and after the third equation I am ok.

Hopefully this is enough information.

Answer 1

It's good you're ok after that equation, because I have no idea who $q$ is!

This sort of calculation is best understood by applying the chain rule to a composition. Set $$F(s,t) = f_s\circ\Phi_t$$ and consider $F\circ g$, where $g(t) = (t,t)$. Now you'll agree that the usual chain rule tells us that \begin{align*} \frac{\partial F}{\partial s} &= \frac{\partial f_s}{\partial s}\circ\Phi_t, \text{ and} \\ \frac{\partial F}{\partial t} &= (Df_s\circ\Phi_t)\big(\frac{\partial\Phi_t}{\partial t}\big). \end{align*} Now, putting this together, we have $$\dfrac d{dt}F(t,t) = \big(\dfrac{\partial F}{\partial s}+\dfrac{\partial F}{\partial t}\big)\Big|_{s=t}= \frac{\partial f_t}{\partial s}\circ\Phi_t + (Df_t\circ\Phi_t)\big(\frac{\partial\Phi_t}{\partial t}\big),$$ which is your desired equation.