Let $n,m\in\mathbb{N}$. Let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$, and let $a=(a_1,\dots,a_n)\in U$. Suppose that for every $j\in\{1,\dots,m\}$ and every $i\in\{1,\dots,n\}$ the partial derivative \ref{def:partial-derivative-coordinate-map-2026a} $\frac{\partial f_j}{\partial x_i}(a)$ exists. The matrix

is called the Jacobian matrix of $f$ at $a$. We say that $f$ is differentiable at $a$ if for every $\varepsilon>0$ there exists $\delta>0$ with the following property: whenever $h=(h_1,\dots,h_n)\in\mathbb{R}^n$ satisfies

and $a+h\in U$, one has