I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points $Df(a)$ and $Df(b)$ which are matrices! Now,
$$D[Df(a)Df(b)] = D^2f(a)Df(b)+Df(a)D^2f(b).$$
My question is, what is $D^2f(a)$? How can I treat this? I imagine is something identifiable with $\mathbb{R}^{n\times n \times n}$. In such a case, if I wish to compute the "matrix" norm of $D[Df(a)Df(b)]$ (as the sum of all entries) is this then the sum of all possible combinations of
$$\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j} \frac{\partial}{\partial x_k} f(a) \ ?$$
Thank you very much for your help!