= Reynolds transport theorem
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{title2=$\frac d{dt}\int_{V(t)}f\,dV=\int_{V(t)}\partial_tf\,dV+\int_{\partial V(t)}f v_b\cdot n\,dS$}
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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 $d|V|/dt=\int v_b\cdot n\,dS$. 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.
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