Let the vector field U of ℝ³ be interpreted, as in the remark of the parent entry, as the velocity field of a stationary flow of a liquid. Then the flux

of U through a closed surface a expresses how much more liquid per time-unit it comes from inside of a to outside than contrarily. Since for a usual incompressible liquid, the outwards flow and the inwards flow are equal, we must think in the case that the flux differs from 0 either that the flowing liquid is suitably compressible or that there are inside the surface some sources creating liquid and sinks annihilating liquid. Ordinarily, one uses the latter idea. Both the sources and the sinks may be called sources, when the sinks are negative sources. The flux of the vector U through a is called the productivity or the strength of the sources inside a.

For example, the sources and sinks of an electric field (E) are the locations containing positive and negative charges, respectively. The Gravitational Field has only sinks, which are the locations containing mass.

The expression

where Δv means a region in the vector field and also its volume, is the productivity of the sources in Δv per a volume-unit. When we let Δv to shrink towards a point P in it, to an infinitesimal volume-element dv, we get the limiting value

ϱ := ∮ ∂dvU ⋅ da,

(1)

called the source density in P. Thus the productivity of the source in P is ϱdv. If ϱ = 0, there is in P neither a source, nor a sink.

The Gauss’s theorem

applied to dv says that

∇⋅U = ∮ ∂dvU ⋅ da.

(2)

Accordingly,

ϱ = ∇⋅U

(3)

and

∮ aU ⋅ da = ∫ vϱdv.

(4)

This last formula can be read that the flux of the vector through a closed surface equals to the total productivity of the sources inside the surface. For example, if U is the electric flux density D, (4) means that the electric flux through a closed surface equals to the total charge inside.