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 zi + j, and furnishes an example of a Beltrami vector field.

See also


  • 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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