Let $n\in\mathbb{N}$, let $U\subseteq\mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $F:U\to\mathbb{R}^n$, and let $a\in U$. Suppose that $F$ is differentiable at $a$ in the sense of \ref{def:differentiable-map-at-point-euclidean-2026a}, so that the Jacobian matrix

is defined. The determinant of this matrix, computed using \ref{def:determinant-real-square-matrix-2026a}, is denoted by

and is called the Jacobian determinant of $F$ at $a$.

If $F$ is \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth} on $U$, then the function

from $U$ to $\mathbb{R}$ is called the Jacobian determinant function of $F$.