Invariant Phase space volume under canonical transfromation

Consider a phase space volume $domega = prod dq^idp_i$. Time evolution can be viewed as the unfolding of canonical transformations, hence, we shall evolve this system $trightarrow t+dt$. We wish to show invariance of the measure under this canonical transformation $(q,p) rightarrow (Q,P)$ where: $$ Q^i = q^i + dot q^id t, qquad P^i = p_i + dot p_idt $$ Compute the new volume $domega' = prod dQ^idP_i$:

begin{align} domega ' &= prod d( q^i + dot q^id t, p_i + dot p_id t) & = prod ( dq^i + frac{partial dot q^i}{partial q^i}dt, dp_i + frac{partial dot p_i}{partial p_i}d t) end{align} Let us now expand this and retain terms linear in $dt$ we obtain $$ domega' approx sum_i bigg( frac{partial dot q^i}{partial q^i} + frac{partial dot p_i}{partial p_i}bigg) = bigg{frac{partial }{partial q^i}frac{partial H}{partial p_i} - frac{partial }{partial p_i}frac{partial H}{partial q^i}bigg} = 0 $$ Therefore, we have shown invariance of the measure under a canonical transformation.