Solenoidal vector field

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

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

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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.)

The divergence theorem, gives the equivalent integral definition of a solenoidal field; namely that for any closed surface $S$ , the net total flux through the surface must be zero:

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