Total derivative (Fréchet derivative in ℝ^k)

Let $U\subseteq\mathbb{R}^k$ be open and let $f:U\to\mathbb{R}^m$ . The function $f$ is (Fréchet) differentiable at $a\in U$ if there exists a linear map $A:\mathbb{R}^k\to\mathbb{R}^m$ such that

\lim_{h\to 0}\frac{\|f(a+h)-f(a)-A h\|_2}{\|h\|_2}=0.

The map $A$ is uniquely determined when it exists and is called the (total) derivative of $f$ at $a$ , denoted $Df(a)$ .

This definition captures the best linear approximation of $f$ near $a$ . In coordinates, $Df(a)$ is represented by the Jacobian matrix $J_f(a)$ when $f$ has continuous partial derivatives.

Examples

If $f(x)=Ax+b$ is affine (with $A$ an $m\times k$ matrix), then $Df(a)=A$ for all $a$ .
If $f:\mathbb{R}^2\to\mathbb{R}$ , $f(x,y)=x^2+y^2$ , then $Df(a)$ is the linear map $h\mapsto 2\langle a,h\rangle$ (equivalently, gradient dot $h$ ).
Existence of all partial derivatives at $a$ does not necessarily imply existence of $Df(a)$ (Fréchet differentiability).