Convert two surface integrals in the box to volume integrals using Gauss’ theorem \(\iiint_V \frac{\partial Q}{\partial x_i} \, dV = \iint_A n_i Q \, dA \) \[ \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \left( \mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n} \right) dA = \int_{V(t)} \nabla \cdot \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV = \int_{V(t)} \frac{\partial}{\partial x_i} (\rho u_i u_j) \, dV \quad \textcolor{#7DCDF4}{(a)} \] \[ \int_{A(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA = \int_{A(t)} n_i T_{ij} \, dA = \int_{V(t)} \frac{\partial}{\partial x_i}(T_{ij}) \, dV \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \textcolor{#7DCDF4}{(b)} \] The explicit listing of the independent variables has been dropped upon moving to index notation. Substituting \(\textcolor{#7DCDF4}{(a)}\)\(\textcolor{#7DCDF4}{(b)}\) into \(\boxed{\int_{V(t)} \frac{\partial}{\partial t} \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) dV + \int_{A(t)} \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \, (\mathbf{u}(\mathbf{x}, t) \cdot \mathbf{n}) \, dA = \int_{V(t)} \rho(\mathbf{x}, t) \, \mathbf{g} \, dV + \int_{A(t)} \mathbf{f}(\mathbf{n}, \mathbf{x}, t) \, dA}\) and collecting all the terms on one side of the equation into the same volume integration produces \[ \int_{V(t)} \left\{ \frac{\partial}{\partial t}(\rho u_j) + \frac{\partial}{\partial x_i}(\rho u_i u_j) - \rho g_j - \frac{\partial}{\partial x_i}(T_{ij}) \right\} dV = 0 \] The integral in the above equation can only be zero for any material volume if the integrand vanishes at every point in space; thus it requires \( \frac{\partial}{\partial t}(\rho u_j) + \frac{\partial}{\partial x_i}(\rho u_i u_j) = \rho g_j + \frac{\partial}{\partial x_i}(T_{ij}) \). This equation can be put into a more standard form \( \frac{\partial}{\partial t}(\rho u_j) + \frac{\partial}{\partial x_i}(\rho u_i u_j) = \rho \frac{\partial u_j}{\partial t} + u_j \left[ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i}(\rho u_i) \right] + \rho u_i \frac{\partial u_j}{\partial x_i} = \rho \frac{D u_j}{D t} \) by expanding. The contents of the \([\,]\)-brackets are zero because of the continuity equation \(\boxed{ \frac{\partial \rho(\mathbf{x}, t)}{\partial t} + \nabla \cdot \left( \rho(\mathbf{x}, t) \, \mathbf{u}(\mathbf{x}, t) \right) = 0 \text{ or } \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i}(\rho u_i) = 0} \), and using the definition of the material derivative \(D/Dt\) from \(\boxed{ \frac{D F}{D t} \equiv \frac{\partial F}{\partial t} + \mathbf{u} \cdot \nabla F, \text{ or } \frac{D F}{D t} \equiv \frac{\partial F}{\partial t} + u_i \frac{\partial F}{\partial x_i}}\). The final result is