Gauss’ Theorem: \( \iiint_V \frac{\partial Q}{\partial x_i} \, dV = \iint_A n_i Q \, dA \)
The differential form of mass conservation can be obtained directly by applying Gauss’ divergence theorem to the surface integration in \( \int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV + \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA = 0 \)
\[ \int_{V(t)} \frac{\partial \rho(\mathbf{x}, t)}{\partial t} \, dV + \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \, dA = \int_{V(t)} \left\{ \frac{\partial \rho(\mathbf{x}, t)}{\partial t} + \nabla \cdot \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) \right\} dV = 0 \]
The final equality \(\boxed{\int_{V(t)} \left\{ \frac{\partial \rho(\mathbf{x}, t)}{\partial t} + \nabla \cdot \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) \right\} dV = 0}\) can only be possible if the integrand vanishes at every point in space. If the integrand did not vanish at every point in space, then integrating the differential equation that represents mass conservation in a small volume around a point where the integrand is nonzero would produce a nonzero integral
\[ \frac{\partial \rho(\mathbf{x}, t)}{\partial t} + \nabla \cdot \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) = 0 \quad \text{or in index notation:} \quad \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i} (\rho u_i) = 0 \] This is called the continuity equation
It expresses the principle of conservation of mass in differential form, but is insufficient for fully determining flow fields because it is a single equation that involves two field quantities, \( \rho \) and \( \mathbf{u} \), and \( \mathbf{u} \) is a vector with three components