Phase-space flow Jacobian ($J(t,z)=\exp {\int }_0^t\mathrm{div}b(s,{\Phi }_s(z))ds$)

The Jacobian determinant of a differentiable characteristic flow map measures its change of volume. The variational equation and the Liouville formula for a fundamental matrix give the displayed formula for a complete smooth field $b$. For kinetic advection with $b=(v,F)$, its divergence is ${\nabla }_v\cdot F$. A zero divergence preserves phase-space Lebesgue measure.