Solenoidal vector field

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

<math> \nabla \cdot \mathbf{v} = 0.\, </math>

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

<math>\mathbf{v} = \nabla \times \mathbf{A}</math>

then

<math>\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0.</math>

The 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 only subject to certain technical conditions on v, see Helmholtz decomposition.)

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