Editing Divergence theorem (section)

== Explanation using liquid flow ==
{{See also|Sources and sinks}}
[[Vector field]]s are often illustrated using the example of the [[velocity]] field of a [[fluid]], such as a gas or liquid. A moving liquid has a velocity—a speed and a direction—at each point, which can be represented by a [[Vector (mathematics and physics)|vector]], so that the velocity of the liquid at any moment forms a vector field.  Consider an imaginary closed surface ''S'' inside a body of liquid, enclosing a volume of liquid.  The [[flux]] of liquid out of the volume at any time is equal to the volume rate of fluid crossing this surface, i.e., the [[surface integral]] of the velocity over the surface.

Since liquids are incompressible, the amount of liquid inside a closed volume is constant; if there are no [[sources and sinks|sources or sinks]] inside the volume then the flux of liquid out of ''S'' is zero.  If the liquid is moving, it may flow into the volume at some points on the surface ''S'' and out of the volume at other points, but the amounts flowing in and out at any moment are equal, so the ''net'' flux of liquid out of the volume is zero.

However, if a ''source'' of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions.  This will cause a net outward flow through the surface ''S''.  The flux outward through ''S'' equals the volume rate of flow of fluid into ''S'' from the pipe.  Similarly if there is a ''sink'' or drain inside ''S'', such as a pipe which drains the liquid off, the external pressure of the liquid will cause a velocity throughout the liquid directed inward toward the location of the drain.  The volume rate of flow of liquid inward through the surface ''S'' equals the rate of liquid removed by the sink.

If there are multiple sources and sinks of liquid inside ''S'', the flux through the surface can be calculated by adding up the volume rate of liquid added by the sources and subtracting the rate of liquid drained off by the sinks.  The volume rate of flow of liquid through a source or sink (with the flow through a sink given a negative sign) is equal to the ''[[divergence]]'' of the velocity field at the pipe mouth, so adding up (integrating) the divergence of the liquid throughout the volume enclosed by ''S'' equals the volume rate of flux through ''S''. This is the divergence theorem.<ref>{{cite book |author1=R. G. Lerner |author1-link=Rita G. Lerner|author2=G. L. Trigg |edition = 2nd | title = Encyclopaedia of Physics | publisher = VHC | year = 1994 | isbn = 978-3-527-26954-9 }}</ref>

The divergence theorem is employed in any [[conservation law]] which states that the total volume of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.<ref>{{Citation | last1 = Byron | first1 = Frederick | last2 = Fuller | first2 = Robert | author2-link = Robert W. Fuller | title = Mathematics of Classical and Quantum Physics | publisher = Dover Publications | year = 1992 | page = [https://archive.org/details/mathematicsofcla00byro/page/22 22] | isbn = 978-0-486-67164-2 | url = https://archive.org/details/mathematicsofcla00byro/page/22 }}</ref>