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It is generally known that the first derivative of $f:\Bbb R^m \to \Bbb R^n$ at $x$ is a linear map $Df(\mathbf x)\in L(\Bbb R^m,\Bbb R^n)$ satisfying $$ \lim_{|\mathbf h|\to 0} \frac{||f(\mathbf x+\mathbf h)-f(\mathbf x)-Df(\mathbf x)h||}{\mathbf |h|} = 0, $$ so technically $Df:\Bbb R^m\to L(\Bbb R^m,\Bbb R^n)$. With the use of partial derivatives, we can express $Df$ w.r.t. the natural coordinates by $$Df= \begin{bmatrix} D_1f_1 & D_2f_1 & \dots & D_mf_1\\ D_1f_2 & D_2f_2 & \dots & D_mf_2\\ \vdots & \vdots & \ddots &\vdots\\ D_1f_n & D_2f_n &\dots & D_mf_n \end{bmatrix} $$ where $f(\mathbf x)=(f_1(\mathbf x),\dots,f_n(\mathbf x))^T$, $\mathbf x=(x_1,\dots,x_m)$.

Now my question is

What's $D^2f$?
What is a nice way to represent it?

Well, I know that we'd need $D^2f:\Bbb R^m \to L(\Bbb R^m,L(\Bbb R^m,\Bbb R^n))$ but what are some nice ways to think about $L(\Bbb R^m,L(\Bbb R^m,\Bbb R^n))$?

Intuitively I kind of see what $D^2f$ do, but I since my background in Differential Geometry and Multi-variable Calculus is quite limited, I've never come across any book that offer a nice symbolic way to work with $D^2f$. I suspect that it has something to do with tensor notation that I'm not familiar with.

Any help is very appreciated, especially if you can point me some nice books/articles treating the topic. My background is mostly in functional analysis so I'm not scared off by terms like Frechet Derivative and such, please feel free to use them.

This is a related thing I found. The answers were good but did not address my question, which is the representation of $D^2f$.

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    See http://math.stackexchange.com/questions/2038218/what-is-the-second-derivative-part-i/2038258#2038258 – 2017-02-15
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    Indeed, the second derivative at a point $p \in \mathbb{R}^m$ is a bilinear map $\mathbb{R}^m \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ and then it becomes comfortable to use tensor notation to work with such objects. Such bilinear maps correspond bijectively to triply-indexed sequences $a_{ij}^k$ where $1 \leq i,j \leq m$ and $1 \leq k \leq n$. – 2017-02-15
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    More explicitly, if you fix a basis $v_1,\dots,v_m$ for $\mathbb{R}^m$, a basis $w_1,\dots,w_n$ for $\mathbb{R}^n$ and denote by $v^1,\dots,v^m$ the corresponding dual basis (and similarly for $w^1, \dots,w^m$) then any sequence defines a bilinear map $a_{ij}^k v^i \otimes v^j \otimes w_k$ (where an Einstein summation convention is used) and any such map is of this form. – 2017-02-15
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    @levap Excellent! I've also read your answer in the link you provided and it was very good. Thank you very much. I have 2 question though, if you wouldn't mind. – 2017-02-15
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    Sure, go ahead. Edit the question if it is appropriate or ask them here. – 2017-02-15
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    Firstly, I don't fully understand when you said that $D^2f_p$ can be interpreted in term of directional derivative. Could you please explain a little be more on why should it be the case? Lastly, do you have a name of any book that treat this matter the same way you just did? – 2017-02-15
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    If you wouldn't mind, I want you to write a short answer addressing my little first question so that I can upvote and accept it. – 2017-02-15
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    No problem, I'll do it soon. – 2017-02-15

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