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.

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 .

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Beltrami vector field

See also

References