Reynolds transport theorem ($\frac{d}{dt}{\int }_{V(t)}fdV={\int }_{V(t)}{\partial }_{t}fdV+{\int }_{\partial V(t)}f{v}_b\cdot ndS$)

For a smoothly moving volume whose boundary has normal velocity $v_b\cdot n$, the derivative of an integral equals the local time-change integral plus the boundary-transport flux. The boundary term comes from the thin layer swept by the moving surface during a short time interval. Taking $f=1$ gives $d|V|/dt=\int v_b\cdot ndS$. If the boundary moves with the fluid, this combines with the divergence theorem to give the integral of the material derivative plus $f$ times the velocity divergence.