Morrison, Rev.Mod.Phys. 70 (1998) Sect.III C

p.481L Eq.(94) $ \rho(\vec r,t)=\int_D\delta(\vec r-\vec q(\vec a,t))\rho(\vec a,0)d^3\vec q

$ s : entropy per unit mass, $ \sigma = \rho s : entropy per unit volume; p.481R Eq.(98) $ s(q(a,t),t) = s(q(a,0),0)

Eq.(96)+(98) $ \omega^3_S(q(a,t),t) = s(q(a,t),t)\rho(q(a,t),t)d^3q(a,t) = s(a,0)\rho(a,0)d^3a = \omega^3_S(a,0)

Eq.(101) $ V[q]=\int_D\rho(a,0)\,U\left(s(a,0),\frac{\rho(a,0)}{\displaystyle\left|{\partial q^i}/{\partial a^i}\right|}\right) d^3\vec a ←ここを3-form$ U （$ \delta U = T \delta S - p \delta V ）に出来ないか？

Eq.(100) $ v(r,t)=\dot q(a,t)|_{a=q^{-1}(r,t)} ×→$ v^i(q(a,t),t)=\dot q^i(a,t) $ \because v(r,t)=v^i(r,t)\underbrace{\frac{\partial}{\partial r^i}} \ne \dot q(a,t)=q^i(a,t)\underbrace{\frac{\partial}{\partial a^i}}