Jacobian determinant

A Jacobian determinant of a differentiable map $f:U\to \mathbb{R}^n$ (with $U\subseteq \mathbb{R}^n$) at a point $a\in U$ is

the determinant of the Jacobian matrix at $a$.

The Jacobian determinant controls local invertibility and local volume scaling: nonvanishing of $\det Jf(a)$ is the hypothesis of the inverse function theorem , and it appears in the change of variables formula for integrals.

Examples:

• For the linear map $f(x,y)=(2x,3y)$, one has $\det Jf(x,y)=6$ for all $(x,y)$.
• For $f(r,\theta)=(r\cos\theta,r\sin\theta)$, the Jacobian determinant is $\det Jf(r,\theta)=r$.