Jacobian determinant

Let $U\subseteq\mathbb{R}^k$ be open and let $f:U\to\mathbb{R}^k$. If $f$ has a Jacobian matrix $J_f(a)$ at $a\in U$, the Jacobian determinant of $f$ at $a$ is

When $f$ is $C^1$ , $\det J_f(a)\neq 0$ is the nondegeneracy condition in the inverse function theorem. In integration, $|\det J_f|$ appears in the change-of-variables formula as the local volume scaling factor.

Examples:

• If $f(x)=Ax$ is linear on $\mathbb{R}^k$, then $\det J_f(a)=\det A$ for all $a$.
• For $f(x,y)=(2x,3y)$, $J_f=\begin{pmatrix}2&0\\0&3\end{pmatrix}$ so $\det J_f=6$.
• For a rotation in $\mathbb{R}^2$, $\det J_f=1$ (orientation-preserving and area-preserving).