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

$\mathbf{F}\times (\nabla\times\mathbf{F})=0.$

If $\mathbf{F}$ is solenoidal - that is, if $\nabla \cdot \mathbf{F} = 0$ such as for an incompressible fluid or a magnetic field, we may examine $\nabla \times (\nabla \times \mathbf{F}) \equiv -\nabla^2 \mathbf{F} + \nabla (\nabla \cdot \mathbf{F})$ and apply this identity twice to find that

$-\nabla^2 \mathbf{F} = \nabla \times(\lambda \mathbf{F})$

and if we further assume that $\lambda$ is a constant, we arrive at the simple form

$\nabla^2 \mathbf{F} = -\lambda^2 \mathbf{F}.$

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

The vector field

$\mathbf{F} = -\frac{z}{\sqrt{1+z^2}}\mathbf{i} + \frac{1}{\sqrt{1+z^2}}\mathbf{j}$

is a multiple of the standard contact structure −z i + j, and furnishes an example of a Beltrami vector field.

See also

• Complex lamellar vector field
• Conservative vector field

References

• Aris, Rutherford (1989), Vectors, tensors, and the basic equations of fluid mechanics, Dover, ISBN 0-486-66110-5
• Lakhtakia, Akhlesh (1994), Beltrami fields in chiral media, World Scientific, ISBN 981-02-1403-0
• Etnyre, J.; Ghrist, R. (2000), "Contact topology and hydrodynamics. I. Beltrami fields and the Seifert conjecture", Nonlinearity 13 (2): 441–448, Bibcode:2000Nonli..13..441E, doi:10.1088/0951-7715/13/2/306.