The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder

Let $m,n\in \mathbb{N}$, let $U\subseteq {\mathbb{R}}^m$ be open in the metric induced by its Euclidean norm, let $a\in U$, and let $f:U\to {\mathbb{R}}^n$. The map $f$ is totally differentiable at $a$ when there is a linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ (A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates) such that for every $\epsilon >0$ there is $\delta >0$ for which

$\mathrm{\parallel }f(a+h)-f(a)-Lh{\mathrm{\parallel }}_2\le \epsilon \mathrm{\parallel }h{\mathrm{\parallel }}_2\text{whenever }a+h\in U\text{ and }\mathrm{\parallel }h{\mathrm{\parallel }}_2<\delta \mathrm{.}$

The map $L$, when it exists, is denoted $Df(a)$ and called the total derivative. Equivalently, $f(a+h)=f(a)+Df(a)h+r(h)$ with the displayed remainder estimate for every $\epsilon >0$. At $h=0$ the remainder is zero because $L0=0$.

For $m\ge 1$, openness of $U$ provides admissible nonzero increments arbitrarily near zero, and dividing the estimate by $\mathrm{\parallel }h{\mathrm{\parallel }}_2>0$ gives the usual notation

${\lim }_{h\to 0}\frac{\mathrm{\parallel }f(a+h)-f(a)-Lh{\mathrm{\parallel }}_2}{\mathrm{\parallel }h{\mathrm{\parallel }}_2}=0,$

where the quotient is considered for $h\mathrm{\ne }0$ with $a+h\in U$. Conversely this punctured normwise limit gives the estimate above (using a smaller positive tolerance if necessary). Thus the positive-dimensional definition is unchanged. This is the multivariate normwise remainder condition; no limit at an isolated point is asserted.

When $m=0$, the Euclidean zero space consists only of $0$, so $a=0$ and $U=\{0\}$; its Euclidean norm and induced metric exist by A norm on a real vector space, the induced metric, and the dictionary with the metric axioms. The unique linear map $L:{\mathbb{R}}^0\to {\mathbb{R}}^n$ sends $0$ to $0$, and its remainder is identically zero. It is therefore the total derivative by the displayed estimate, with any $\delta >0$. When $n=0$, the unique map $f$ and the unique linear map into ${\mathbb{R}}^0$ likewise have zero remainder, in every source dimension.

A linear map $L:{\mathbb{R}}^m\to {\mathbb{R}}^n$ in Euclidean coordinates

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms