It can be shown that any stationary flux '''v'''('''r''') which is at least two times continuously differentiable in <math> {\mathbb R}^3</math> and vanishes sufficiently fast for {{nowrap|{{!}}'''r'''{{!}} → ∞}} can be decomposed into an ''irrotational part'' '''E'''('''r''') and a ''source-free part'' '''B'''('''r'''). Moreover, these parts are explicitly determined by the respective ''source-densities'' (see above) and ''circulation densities'' (see the article [[Curl (mathematics)|Curl]]):
The source-free part, '''B''', can be similarly written: one only has to replace the ''scalar potential'' Φ('''r''') by a ''vector potential'' '''A'''('''r''') and the terms −'''∇'''Φ by +'''∇'''×'''A''', and finally the source-density {{nowrap|div '''v'''}}
by the circulation-density '''∇'''×'''v'''.
This "decomposition theorem" is in fact 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.
