Equating the two expressions for the rate of flow out of a small volume, we have
A collection of small volumes can be built which fill a finite volume, and this result can be extended to volumes of any size. In that case, the result is known as the divergence theorem.
The surface integral above is known as the net flux of the momentum
density field, 
. As we saw, the net flux of the momentum density field is just
the net rate of mass outflow, through the surface.
To understand the physical significance of
, we shrink the volume until that quantity is essentially constant
over the entire volume. Then "moves outside" the integral and we have
This says that the divergence of a field in a small region is a measure of the flow out of that region. We can think of divergence of a (vector) field as the net (outward) flux per unit volume for the field.
In our example, if the divergence of the momentum density is positive, then the integral of the momentum density across the surface is positive: The mass inside the volume decreases.
If a velocity field shows a positive divergence at a point in space, then fluid flows out of a volume around that point. This requires a source of fluid. A high pressure tank of gas with its valve open acts like such a source: The gas velocity diverges away from the valve. (Beware: If this experiment is done without chaining the tank to a strong wall, the tank turns into a torpedo, changing the character of the experiment.) On the other hand, an evacuated tank with its valve open acts like a sink for gas: The velocity field near the valve has negative divergence, as gas flows into the valve.
In electromagnetic theory, the flux of electric (and of magnetic fields) is a useful quantity. The name flux indicates flow, and derives from the fluid version of the divergence theorem shown above. The electric field around a positive charge has a positive divergence, and the electric field around a negative charge has a negative divergence. The positive charge looks like a "source" of electric field. The negative charge looks like a "sink."
THE EQUATION OF CONTINUITY
We now return to the equation for mass decrease in terms of the volume integral,
.
The left side can be written in terms of the density, using
.
Since the volume is time-independent, we may write
Combining these under a single volume integral we have
The integrand must be zero for this equation to hold in general, so
This is called the equation of continuity. It reflects the assumption that mass is conserved, and must enter or exit a volume by flowing across the boundaries of the volume.
The equation of continuity describes the change in mass in a fixed volume due to flow across the surface of the volume. In the case of the burning oil, the mass of oil in a unit volume is decreasing because some if it is burned each second. The divergence of the momentum density is positive, consistent with the continuity equation.
The continuity equation is elevated from "description" to "principle" when we assert that the only way the mass can change is via flow across the surface. That is matter cannot be created or destroyed within the volume.
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Go forward to viscosity.