Let an arbitrary region in a continuous medium be denoted by Ω and the bounding closed surface of this region be continuous and denoted by Γ. Let each point on the bounding surface move with the...

Let an arbitrary region in a continuous medium be denoted by Ω and the bounding closed surface of this region be continuous and denoted by Γ. Let each point on the bounding surface move with the velocity v. It can be shown that the time derivative of the volume integral over some continuous function Q(x, t) is given by

This expression for the differentiation of a volume integral with variable limits is sometimes known as the three-dimensional Leibniz rule. The material derivative operator d/dt corresponds to changes with respect to a fixed mass, that is ρ dv is constant with respect to this operator. Show formally by means of Leibniz’s rule, the divergence theorem, and conservation of mass that