Editing Divergence (section)

== Definition ==
[[Image:Definition of divergence.svg|thumb|The divergence at a point {{math|'''x'''}} is the limit of the ratio of the flux <math>\Phi</math> through the surface {{math|''S''<sub>''i''</sub>}} ''(red arrows)'' to the volume <math>|V_i|</math> for any sequence of closed regions {{math|''V''<sub>1</sub>, ''V''<sub>2</sub>, ''V''<sub>3</sub>, …}} enclosing {{math|'''x'''}} that approaches zero volume:<br/> <math>\operatorname{div} \mathbf{F} = \lim_{|V_i| \to 0} \frac{\Phi(S_i)}{|V_i|}</math>]]

The divergence of a vector field {{math|'''F'''('''x''')}} at a point {{math|'''x'''<sub>0</sub>}} is defined as the [[limit (mathematics)|limit]] of the ratio of the [[surface integral]] of {{math|'''F'''}} out of the closed surface of a volume {{math|''V''}} enclosing {{math|'''x'''<sub>0</sub>}} to the volume of {{math|''V''}}, as {{math|''V''}} shrinks to zero
:{{oiint
| preintegral = <math>\left. \operatorname{div} \mathbf{F} \right|_\mathbf{x_0} = \lim_{V \to 0} \frac{1}{|V|}</math>
| intsubscpt = <math>\scriptstyle S(V)</math>
| integrand = <math>\mathbf{F} \cdot \mathbf{\hat n} \, dS</math>
}}
where {{math|{{abs|''V''}}}} is the volume of {{math|''V''}}, {{math|''S''(''V'')}} is the boundary of {{math|''V''}}, and  <math>\mathbf{\hat n}</math> is the outward [[normal vector|unit normal]] to that surface.  It can be shown that the above limit always converges to the same value for any sequence of volumes that contain {{math|'''x'''<sub>0</sub>}} and approach zero volume.  The result, {{math|div '''F'''}}, is a scalar function of {{math|'''x'''}}.

Since this definition is coordinate-free, it shows that the divergence is the same in any [[coordinate system]].  However the above definition is not often used practically to calculate divergence; when the vector field is given in a coordinate system the coordinate definitions below are much simpler to use.

A vector field with zero divergence everywhere is called ''[[solenoidal vector field|solenoidal]]'' – in which case any closed surface has no net flux across it. This is the same as saying that the (flow of the) vector field preserves volume: The volume of any region does not change after it has been transported by the flow for any period of time.