Beltrami vector field

In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that
$\mathbf\times (\nabla\times\mathbf)=0.$

Thus $\mathbf$ and $\nabla\times\mathbf$ are parallel vectors in other words, $\nabla\times\mathbf = \lambda \mathbf$.

If $\mathbf$ is solenoidal - that is, if $\nabla \cdot \mathbf = 0$ such as for an incompressible fluid or a magnetic field, the identity $\nabla \times (\nabla \times \mathbf) \equiv -\nabla^2 \mathbf + \nabla (\nabla \cdot \mathbf)$ becomes $\nabla \times (\nabla \times \mathbf) \equiv -\nabla^2 \mathbf$ and this leads to
$-\nabla^2 \mathbf = \nabla \times(\lambda \mathbf)$
and if we further assume that $\lambda$ is a constant, we arrive at the simple form
$\nabla^2 \mathbf = -\lambda^2 \mathbf.$

Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions.

The vector field
$\mathbf = -\frac\mathbf + \frac\mathbf$
is a multiple of the standard contact structure −''z'' i + j, and furnishes an example of a Beltrami vector field.

See also

*Beltrami flow
*Complex lamellar vector field
* Conservative vector field

References

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Vector calculus

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