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
\( {\mathbf {F}}\times (\nabla \times {\mathbf {F}})=0. \)
Thus \( \mathbf {F} \) and \( {\displaystyle \nabla \times \mathbf {F} } \) are parallel vectors in other words, \( {\displaystyle \nabla \times \mathbf {F} =\lambda \mathbf {F} }. \)
If \( \mathbf {F} \) is solenoidal - that is, if \( \nabla \cdot {\mathbf {F}}=0 \) such as for an incompressible fluid or a magnetic field, the identity \( \nabla \times (\nabla \times {\mathbf {F}})\equiv -\nabla ^{2}{\mathbf {F}}+\nabla (\nabla \cdot {\mathbf {F}}) \) becomes \( {\displaystyle \nabla \times (\nabla \times \mathbf {F} )\equiv -\nabla ^{2}\mathbf {F} } \) and this leads to
\( -\nabla ^{2}{\mathbf {F}}=\nabla \times (\lambda {\mathbf {F}}) \)
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
Beltrami flow
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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