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Solenoidal vector field



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

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

Additional recommended knowledge

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,

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

Examples

  • the magnetic field B is solenoidal (see Maxwell's equations);
  • the velocity field of an incompressible fluid flow is solenoidal;
  • the electric field in regions where ρe = 0;
  • the current density, J, if əρe/ət = 0.
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Solenoidal_vector_field". A list of authors is available in Wikipedia.
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