Beltrami vector field
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.
- 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.