Jacobian Determinant of a Differentiable Map Between Euclidean Open Sets

Let $n\in\mathbb{N}$, let $U\subseteq\mathbb{R}^n$ be open, let $F:U\to\mathbb{R}^n$, and let $a\in U$. Suppose that $F$ is differentiable at $a$ in the sense of Differentiability at a Point and Jacobian Matrix for Maps Between Euclidean Spaces, so that the Jacobian matrix

$$J_F(a)$$

is defined. The determinant of this matrix, computed using Determinant of a Real Square Matrix, is denoted by

$$\det J_F(a)$$

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

If $F$ is smooth on $U$, then the function

$$x\longmapsto \det J_F(x)$$

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