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I wonder what is the meaning of the second derivative or what kind of object it is when we have a function $f:\mathbb{R}^n \rightarrow \mathbb{R}^m$.

The first derivative is the Jacobian matrix, but then, what is the second derivative? How can I treat them when I write $f''$ or $D^2 f$?

Thanks a lot for your help!

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The second derivative is something very big and complicated! :-) –  Giuseppe Negro Dec 12 '12 at 13:24
    
More seriously, the second derivative of a mapping like that is an object with 3 indices, a kind of "cubic matrix". Tensor algebra could help in giving it a more precise name, but still, it is a complicated object. –  Giuseppe Negro Dec 12 '12 at 13:25
    
Roughly speaking, the second derivative is a tensor. –  Siminore Dec 12 '12 at 13:26

2 Answers 2

$f:\mathbb{R}^n \rightarrow \mathbb{R}^m$ is differentiable iff its increment has the form $$f(x+h)-f(x)=Df(x)h+\alpha(x,\,h),$$ where $Df(x)$ is linear mapping $Df(x)\colon \; \mathbb{R}^n \to \mathbb{R}^m$ and $\alpha$ satisfies $$\| \alpha(x,\,h) \|_{\mathbb{R}^m}=o(\|h\|_{\mathbb{R}^n}).$$ Analogously, the second order derivative is bilinear mapping $D^{2}f(x)\colon \; \mathbb{R}^n \to \mathbb{R}^m,$ which acts on a pair of vectors $(h_1,\,h_2), \quad h_1, \,h_2\in \mathbb{R}^n.$ Derivative of $k^{th}$ order is polylinear (more precisely, $k$-linear) mapping from $ \mathbb{R}^n $ to $\mathbb{R}^m$.

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The first thing that comes to mind is the Hessian matrix which is the special case for $m=1$:

$$f: \mathbb{R}^n \rightarrow \mathbb{R}$$

The case you asked about occurs in differential geometry, where it is involved in defining the covariant derivative (see this) and related to the connection coefficients.

In a rough sense it is a curvature. Much like the second derivative of a single variabled function tells you how quickly the curve is changing.

As for notation, I usually write this tensor as

$$\left[\frac{\partial^2 f^k(\xi)}{\partial x^i \partial x^j}\right] \;\;\;\ \text{or more simply} \;\;\; \left[\partial_i \partial_j f^k(x)\right]$$

If writing the entries individually, would probably write a Hessian for each $k$.

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