Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function. Then we know that its differential $df: \mathbb{R}^2 \rightarrow Hom(\mathbb{R}^2,\mathbb{R}^2)$ maps vectors to matrices/linear maps.
But how do we define $D^2 f$? Is it defined like $D^2f: \mathbb{R}^2 \rightarrow Hom(\mathbb{R}^2, Hom(\mathbb{R}^2, \mathbb{R}^2))$ (and in this case: how do I write it explicitly?) or is it defined like $D^2 f(y)(z,w) := D(Df(y)(w))(z)$ or like something else? In the solutions of the problem 5.2 of this problem sheet there's this (screenshot).
I don't understand why does $D^2f$ have two arguments, $w$ and $z$.
Is it possible to generalize the whole thing for $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$?
Thank you!
I am not sure about the geometric interpretation, but this might be a way to find your $D^2$.
Similarly to your intuition, think of $D^2f(x)(v,w)$ as the derivative of $$x\mapsto Df(x)v=:G_v(x)$$ in direction $w$. Then you can easily check by hand that the two definitions agree.
Of course this easily generalizes to maps $f:\mathbb{R}^n\rightarrow \mathbb{R}^m$ via the same definition, and similarly $$D^kf(x)(v_1,\dots,v_k)$$ can be defined recursively in the same way.