Phase-space flow Jacobian
= Phase-space flow Jacobian
{title2=$J(t,z)=\exp\int_0^t\operatorname{div}b(s,\Phi_s(z))\,ds$}
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>.