Casimir invariant of a transport equation ($C_F(f)=\int F(f)dxdv$)

For an incompressible transport flow, every integrable function of the transported scalar has constant integral: the scalar is unchanged along characteristics and the flow's Jacobian determinant equals one. For a finite integral on infinite phase space, require integrability; $F(0)=0$ and compactly supported data are one common choice. For a whole-space conservation statement also require no escape or boundary flux at infinity.