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 clearly 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 holds: for any solenoidal v there exists a vector potential A such that <math>\mathbf{v} = \nabla \times \mathbf{A}<math>. (Strictly, this holds only subject to certain technical conditions on v.)
Examples:
- one of Maxwell's equations states that the magnetic field B is solenoidal;
- the velocity field of an incompressible fluid flow is solenoidal.