A similar question has already been asked here: Derivation of Dirac-Delta with complicated argument $\delta(f(x))$
But now I want to do the following. Given the Hamiltonian $H=H(q(t),p(t),t)=E$ is a constant we can define its phase space volume as: $$\int \delta(H(q(t),p(t),t)-E)dqdp$$ How can one evaluate the rate of change of the phase space volume, considering that q and p are functions of time? $$\frac{d}{dt}\int \delta(H(q(t),p(t),t)-E)dqdp$$