Let $n,m \in \mathbb{N}$, let $U \subset \mathbb{R}^n$ be open, let $f: U \to \mathbb{R}^m$ satisfy $f \in C^1(U;\mathbb{R}^m)$, and let $a \in U$. Let $\mathcal{L}(\mathbb{R}^n,\mathbb{R}^m)$ denote the [vector space](/page/Vector%20Space) of linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$, equipped with the operator norm induced by the Euclidean norms. Let $Df_a: \mathbb{R}^n \to \mathbb{R}^m$ denote the total derivative of $f$ at $a$, and define

\begin{align*} V := \{h \in \mathbb{R}^n : a+h \in U\}. \end{align*}

Then there exists a map $r: V \to \mathbb{R}^m$ such that, for every $h \in V$,

\begin{align*} f(a+h)=f(a)+Df_a(h)+r(h), \end{align*}

and

\begin{align*} \frac{|r(h)|}{|h|}\to 0 \end{align*}

as $h \to 0$ with $h \in V \setminus \{0\}$.