Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be a map. Assume there exists an open neighbourhood $V \subset U$ of $a$ such that $f$ is continuously differentiable on $V$ and the derivative map $F: V \to \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$, defined by $F(x)=Df_x$ and equipped with the operator norm $\|\cdot\|_{\mathrm{op}}$ on $\mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$, is Fréchet differentiable at $a$. Let $A:=DF_a \in \mathcal{L}(\mathbb{R}^m,\mathcal{L}(\mathbb{R}^m,\mathbb{R}^n))$, and define the [second derivative](/page/Second%20Derivative) bilinear map $D^2f_a: \mathbb{R}^m \times \mathbb{R}^m \to \mathbb{R}^n$ by $D^2f_a(h,k)=A(h)(k)$. Then, as $h \to 0$ in $\mathbb{R}^m$ with $a+h \in U$,

\begin{align*} f(a+h)=f(a)+Df_a(h)+\frac{1}{2}D^2f_a(h,h)+o(|h|^2). \end{align*}