Consider integrating f = f (x, t) over the time-dependent region Ω(t) that has boundary ∂Ω(t), then taking the derivative with respect to time: d d t ∫ ∫ Ω Ω ( t ) f d V . If we wish to move the derivative into the integral, there are two issues: the time dependence of f, and the introduction of and removal of space from Ω due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.