What kind of object is the second derivative of a function $f:\mathbb R^n\to\mathbb R^m$?

Question

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!

Answer 1

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