$$\frac{\mathrm d}{\mathrm dt^i} \underset{\large V(t)}{\iiint} \Psi \,\mathrm dV= \underset{\large V(t)}{\iiint} \left({\frac{\partial \Psi}{\partial t}+\nabla\cdot\Psi\mathbf v_i}\right)\, \mathrm dV$$
The expression shown above is a condition.
Prove that if $V=\text{constant}$ then the second part in the parentheses after the integral sign is equal to $0$.
The book says the equation within the expression is a form of Reynold's transport theorem
You can pretty much ignore the $i$ there.
reference: http://mathworld.wolfram.com/ReynoldsTransportTheorem.html