Balance of linear momentum ($\rho D\mathbf{v}/Dt=\nabla \cdot \mathbf{\sigma }+\rho \mathbf{b}$)

For a body occupying $V$, the rate of change of ${\int }_V\rho \mathbf{v}dV$ equals the body forces and surface tractions. With the Cauchy stress tensor, the local form is $\rho D\mathbf{v}/Dt=\nabla \cdot \mathbf{\sigma }+\rho \mathbf{b}$.