Chemistry - 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 clearly 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 holds: for any solenoidal v there exists a vector potential A such that $\mathbf{v} = \nabla \times \mathbf{A}$ . (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.

Categories: Multivariate calculus | Fluid dynamics