I have a smooth function $f:\mathbb{R}^m\to\mathbb{R}^m$ and I want to know if $f$ is volume-preserving in some neighborhood of $x\in\mathbb{R}^m$.
One criterion to test is that $f$ has unit (in absolute value) Jacobian determinant : $$|\mathrm{det}(\nabla f(x))| = 1.$$ Suppose that I know $f$ to be smooth, but computing its Jacobian is not available. I could approximate the Jacobian via finite (central) differences: \begin{align} \partial_{x_i} f(x) \approx \frac{f(x+\delta e_i / 2) - f(x-\delta e_i / 2)}{\delta }. \end{align} However, this might be computationally expensive if $m$ is large, since I would then need to find the determinant of a $m\times m$ matrix.
I am wondering if there is some other criterion I could efficiently compute to give me some indication that my map is volume-preserving?