This is a theorem of Moser:
Assume $\omega_0, \omega_1$ are two volume forms (with the same total
a compact manifold. Then there is a diffeomorphism $\phi$ on $M$
so that $\phi^*\omega_1=\omega_0$.
Since $\omega_0$ and $\omega_1$ has the same total mass, they are
in the same cohomology class.
So there is an $n-1$ form $\eta$ so that
Observe, (this is most easily seen when writing all the forms
in a local coordinate system $(x_1,...,x_n)$,
there is a unique vector field $X_s$ so that
Let $\phi_s$ is the one parameter group of diffeomorphism that is
generated by $X_s$.
Compute, at time $s=t$,
Here notice, $(\phi_t)_*X_t=X_t$ as a vector field.