Jacobian Determinant on a Euclidean Open Set

Let $n$ be a natural number, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$, let $f:U\to\mathbb{R}^{n}$, and let $a\in U$. Suppose that the Jacobian matrix $Df(a)$ is defined; it is then a real matrix with $n$ rows and $n$ columns.

The Jacobian determinant of $f$ at $a$ is the determinant of $Df(a)$, written $\det Df(a)$.

If $Df(x)$ is defined for every $x\in U$, the Jacobian determinant function of $f$ is the function from $U$ to $\mathbb{R}$ whose value at $x$ is $\det Df(x)$.