Jacobian Matrix of a Map Between Euclidean Spaces

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, and let $U$ be an open subset of Euclidean space $\mathbb{R}^n$. Let $f:U\to\mathbb{R}^m$, and for $k$ a natural number with $1\le k\le m$ let $f_k:U\to\mathbb{R}$ be the $k$th coordinate function of $f$, so that $f_k(x)$ is the $k$th coordinate of the point $f(x)$ of $\mathbb{R}^m$. Let $a\in U$, and suppose that for all natural numbers $k$ and $i$ with $1\le k\le m$ and $1\le i\le n$ the partial derivative of $f_k$ with respect to the $i$th variable exists at $a$, written $\partial f_k/\partial x_i(a)$ as in that definition.

The Jacobian matrix of $f$ at $a$, denoted $Df(a)$, is the real matrix with $m$ rows and $n$ columns whose entry in row $k$ and column $i$ is

$$\bigl(Df(a)\bigr)_{ki}=\frac{\partial f_k}{\partial x_i}(a).$$