Solenoidal vector field

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In vector calculus a solenoidal vector field is a vector field v with divergence zero:

\nabla \cdot \mathbf{v} = 0.\,

This condition is satisfied whenever v has a vector potential, because if

\mathbf{v} = \nabla \times \mathbf{A}

then

\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0.

The converse also holds: for any solenoidal v there exists a vector potential A such that \mathbf{v} = \nabla \times \mathbf{A}. (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decomposition.)

Gauss's theorem, gives the equivalent integral definition of a solonoidal field; namely that for any closed surface S, the net total flux out through the surface must be zero:

\iint_S \mathbf{v} \cdot \, d\mathbf{s} = 0,

where d\mathbf{s} is the outward normal to each surface element.

[edit] Examples

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