Differentiable map

A differentiable map at a point $a$ is a map $f:U\to \mathbb{R}^m$ (with $U\subseteq \mathbb{R}^n$) for which there exists a linear map $L:\mathbb{R}^n\to \mathbb{R}^m$ such that

where $\|\cdot\|$ is the Euclidean norm .

In this case $L$ is the Fréchet derivative of $f$ at $a$, and (when it exists) it is represented in coordinates by the Jacobian matrix . Maps that are differentiable at every point of their domain are the basic objects of multivariable differentiability in higher dimensions, and higher smoothness is recorded by C^k maps .

Examples:

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