Conservation of Mass Density

Consider conservation of mass density, p(x, y, z, t). The mass continuity equation comes from consideration of the balance of the mass density in the volume V and its flux through the surface surrounding the volume with subsequent application of the Gauss theorem  

Here the velocity v(jc, y, z, t) of a liquid is measured in a particular fixed point in space x, y, z. It is not the velocity of a small unit volume of moving liquid. Sources mean the presence of sources and sinks in the volume discussed. If we are not interested in propagation of sound i.e. ignore a local compression and dilatation we may put 8p = 0 and p = const. Then, it is the case of incompressible liquid.  

For the subsequent discussion let us write down the continuity equation in the 

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