Fréchet derivative

A Fréchet derivative of a function $f:U\to \mathbb{R}^m$ (with $U\subseteq \mathbb{R}^n$) at a point $a\in U$ is a linear map $Df(a):\mathbb{R}^n\to \mathbb{R}^m$ such that

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

If such a map exists, it is unique; it is the linear part of the first-order approximation $f(a+h)=f(a)+Df(a)h+o(\|h\|)$. When $f$ has partial derivatives, $Df(a)$ is represented by the Jacobian matrix at $a$, and existence of $Df(a)$ is the defining condition for a differentiable map at $a$.

Examples:

• If $f(x)=Ax$ for a fixed matrix $A$, then $Df(a)h=Ah$ for every $a$.
• For $f:\mathbb{R}^2\to \mathbb{R}^2$ given by $f(x,y)=(x^2y,\,x+y)$, the derivative at $(a,b)$ is the linear map represented by the matrix $\begin{pmatrix} 2ab & a^2\\ 1 & 1 \end{pmatrix}.$