Differentiable map - Imported Knowlpedia Content

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

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

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.