Editing Divergence (section)

== Decomposition theorem==
{{Main|Helmholtz decomposition}}
It can be shown that any stationary flux {{math|'''v'''('''r''')}} that is twice continuously differentiable in {{math|'''R'''<sup>3</sup>}} and vanishes sufficiently fast for {{math|{{abs|'''r'''}} → ∞}} can be decomposed uniquely into an ''irrotational part'' {{math|'''E'''('''r''')}} and a ''source-free part'' {{math|'''B'''('''r''')}}. Moreover, these parts are explicitly determined by the respective ''source densities'' (see above) and ''circulation densities'' (see the article [[Curl (mathematics)|Curl]]):

For the irrotational part one has

:<math>\mathbf E=-\nabla \Phi(\mathbf r),</math>
with
:<math>\Phi (\mathbf{r})=\int_{\mathbb R^3}\,d^3\mathbf r'\;\frac{\operatorname{div} \mathbf{v}(\mathbf{r}')}{4\pi\left|\mathbf{r}-\mathbf{r}'\right|}.</math>

The source-free part, {{math|'''B'''}}, can be similarly written: one only has to replace the ''scalar potential'' {{math|Φ('''r''')}} by a ''vector potential'' {{math|'''A'''('''r''')}} and the terms {{math|−∇Φ}} by {{math|+∇ × '''A'''}}, and the source density {{math|div '''v'''}}
by the circulation density {{math|∇ × '''v'''}}.

This "decomposition theorem" is a by-product of the stationary case of [[electrodynamics]]. It is a special case of the more general [[Helmholtz decomposition]], which works in dimensions greater than three as well.