= Invariant volume criterion for a smooth flow
{title2=$\operatorname{div}_{\Omega}F=Fu\iff\mathcal L_F(e^{-u}\Omega)=0$}
For a <volume form> $\Omega$, define divergence by $\mathcal L_F\Omega=(\operatorname{div}_{\Omega}F)\Omega$. The product rule for the <Lie derivative> gives $\operatorname{div}_{e^h\Omega}F=\operatorname{div}_{\Omega}F+Fh$. Consequently the flow preserves a smooth positive volume precisely when its divergence in any reference volume is a smooth <flow coboundary>. The displayed density has the minus sign dictated by this product rule.
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