Differentiable map - Knowlpedia

Let $U\subseteq\mathbb R^n$ be open, let $a\in U$ , and let $f:U\to\mathbb R^m$ . The map $f$ is differentiable at $a$ if there is a linear map $L:\mathbb R^n\to\mathbb R^m$ such that

\lim_{h\to 0}\frac{\|f(a+h)-f(a)-Lh\|}{\|h\|}=0,

where $h$ is small enough that $a+h\in U$ , and $\|\cdot\|$ is the Euclidean norm.

The map $L$ , which is necessarily unique, is the Fréchet derivative $Df(a)$ . In standard coordinates it is represented by the Jacobian matrix.

Examples

Any affine map $f(x)=Ax+b$ is differentiable everywhere, with derivative $Df(x)h=Ah$ .
The map $f:\mathbb{R}^2\to \mathbb{R}^3$ given by $f(x,y)=(x^2,xy,\sin y)$ is differentiable everywhere.