Editing Solenoidal vector field (section)

==Properties==
The [[divergence theorem]] gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:
{{block indent|em=1.6|text={{oiint | integrand=<math>\;\; \mathbf{v} \cdot \, d\mathbf{S} = 0 ,</math>}}}}
where <math>d\mathbf{S}</math> is the outward normal to each surface element.

The [[Helmholtz decomposition|fundamental theorem of vector calculus]] states that any vector field can be expressed as the sum of an [[irrotational]] and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field '''v''' has only a [[vector potential]] component, because the definition of the vector potential '''A''' as:
<math display="block">\mathbf{v} = \nabla \times \mathbf{A}</math>
automatically results in the [[Vector calculus identities|identity]] (as can be shown, for example, using Cartesian coordinates):
<math display="block">\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0.</math>
The [[converse (logic)|converse]] also holds: for any solenoidal '''v''' there exists a vector potential '''A''' such that <math>\mathbf{v} = \nabla \times \mathbf{A}.</math> (Strictly speaking, this holds subject to certain technical conditions on '''v''', see [[Helmholtz decomposition]].)