Lemma Let $\phi_t(x)$ denote the solution of $\dot{y}=f(y)$ from initial condition $x$. Then the following identity holds: $$\det \frac{{\rm d}\phi_t(x)}{{\rm d} x}=e^{\int_0^t \nabla\cdot f(\phi_s(x)){\rm ds}}$$
I have found the above Lemma, without any proof given. It is simply stated that ``it follows from standard ODE techniques''. It seems to be strictly related to Jacobi's formula, however, I did not manage to prove it.