The continuity equation expresses the conservation of mass. It is a direct consequence of Gauss theorem.
If the mass enclosed in a region is , by definition of mass density :
The net loss of matter in this region must be caused by an outward flow across its boundary:
According to Gauss theorem,
Since the region is a general one, and it does not change with time, the resulting equation is
As a direct consequence an incompressible fluid, with constant , implies a solenoidal velocity field: .