Let $F: R^n \to R^m $ be a vector valued function with components $F = (f_1 ,f_2 ,..., f_m).$ What is the most proper and universal accepted definition for the function $F$ be twice differentiable at the point $x$? Does that simply mean $f_i$ be twice-differentiable at point $x$ for all $i = 1,2,...,m$. I know this is the case when we are talking about a function being $C^2$ (see What does it mean for a vector function to be twice differentiable? ).
Or does it mean $F$ be differentiable around $x$ and the derivative of $F$ which is a matrix valued function, denoted by $\nabla F : U \to R^m \times R^n $ be differentiable (in some sense) at the point $x$.
Any reference about that would be appreciated .
The derivative of $F$ at $x \in \mathbb{R}^n$ is a linear map $dF(x) : \mathbb{R}^n \to \mathbb{R}^m$. It is characterized as the best linear approximation of $F$ of $x$. See for example my answer to Higher dimensional total derivative.
If $F$ is differentiable in an open neighborhood $U$ if $x$, we obtain a map $dF : U \to \mathcal{L} = \mathcal{L}(\mathbb{R}^n, \mathbb{R}^m)$ = vector space of all linear maps $\mathbb{R}^n \to \mathbb{R}^m$. This is a finite-dimensional vector space which can be identified with $\mathbb{R}^{n\cdot m}$. We call $F$ twice differentiable at $x$ is $dF$ is differentiable at $x$.
Note that it essential that $dF(x')$ exists in a neighborhood of $x$.