To develop the integral equation that represents mass conservation for an arbitrarily moving control volume \( V^*(t) \) with surface \( A^*(t) \), \(\boxed{ \int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV + \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA = 0} \) must be modified to involve integrations over \( V^*(t) \) and \( A^*(t) \)
This modification is motivated by the frequent need to conserve mass within a volume that is not a material volume, for example a stationary control volume
The first step in this modification is to set \( F = \rho \) in Reynolds Transport Theorem \[ \frac{d}{dt} \int_{V^*(t)} F(\mathbf{x}, t) \, dV = \int_{V^*(t)} \frac{\partial F(\mathbf{x}, t)}{\partial t} \, dV + \int_{A^*(t)} F(\mathbf{x}, t) \, \mathbf{b} \cdot \mathbf{n} \, dA \] to obtain \[ \frac{d}{dt} \int_{V^*(t)} \rho(\mathbf{x}, t) dV - \int_{V^*(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV - \int_{A^*(t)} \rho(\mathbf{x}, t) \mathbf{b} \cdot \mathbf{n} \, dA = 0 \]