# 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

Thus and are parallel vectors in other words, .

If is solenoidal - that is, if such as for an incompressible fluid or a magnetic field, the identity becomes and this leads to

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

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

The vector field

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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