In this question I learned that the total derivative not-just-at-a-point can be thought of as:
a map $Df:\mathbf R^n\to\bigsqcup_{p\in\mathbf R^n}\text{Hom}(T_p\mathbf R^n, T_{f(p)}\mathbf R^m)$ that maps each $p$ to $D_pf$
a map $Df:T\mathbf R^n\to T\mathbf R^m$, that when restricted to $T_p\mathbf R^n$ recovers the original derivative $D_pf$
where if the first map is $F_1$ and the second map is $F_2$, then $F_2(p,x)=(f(p),[F_1(p)](x))$.
I was wondering how the symbol-pattern extends to second and third derivatives. I'm guessing we have both:
a map $D^2f:\mathbf R^n\to\bigsqcup_{p\in\mathbf R^n}\text{Hom}(T_p\mathbf R^n, \text{Hom}(T_p\mathbf R^n, T_{f(p)}\mathbf R^m))$ that maps each $p$ to $D_p(D_pf)$
a map $D^2f:TT\mathbf R^n\to TT\mathbf R^m$.
Call the first map $G_1$ and the second map $G_2$.
After doing some reading, I found the formula $$G_2((p, x), (w_1, w_2)) = (F_2(p, x), ([F_1(p)](w_1), [F_1(p)](w_2) + [F_1([F_1(p)](x))](w_1)))$$
but I don't know if this is correct or how to connect it to $G_1$.
