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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 (mathematics)|curl]]. That is, '''F''' is a Beltrami vector field provided that
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:<math>\mathbf{F}\times (\nabla\times\mathbf{F})=0.</math>
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If <math>\mathbf{F}</math> is solenoidal - that is, if <math>\nabla . \mathbf{F} = 0</math> such as for an incompressible fluid or a magnetic field, we may examine <math>\nabla \times (\nabla \times \mathbf{F}) \equiv -\nabla^2 \mathbf{F} + \nabla \nabla . \mathbf{F}</math> and apply this identity twice to find that
 
::<math>-\nabla^2 \mathbf{F} = \nabla \times(\lambda \mathbf{F})</math>
 
and if we further assume that <math>\lambda</math> is a constant, we arrive at the simple form
 
:<math>\nabla^2 \mathbf{F} = -\lambda^2 \mathbf{F}.</math>
 
Beltrami vector fields with nonzero curl correspond to Euclidean [[contact form]]s in three dimensions.
 
The vector field
:<math>\mathbf{F} = -\frac{z}{\sqrt{1+z^2}}\mathbf{i} + \frac{1}{\sqrt{1+z^2}}\mathbf{j}</math>
is a multiple of the standard contact structure &minus;''z''&thinsp;'''i'''&nbsp;+&nbsp;'''j''', and furnishes an example of a Beltrami vector field.
 
==See also==
*[[Complex lamellar vector field]]
*[[Conservative vector field]]
 
==References==
*{{citation | title=Vectors, tensors, and the basic equations of fluid mechanics | authorlink=Rutherford Aris | first=Rutherford | last=Aris | publisher=Dover | year=1989 | isbn=0-486-66110-5 }}
*{{citation | title=Beltrami fields in chiral media | authorlink=Akhlesh Lakhtakia | first=Akhlesh | last=Lakhtakia | publisher=World Scientific | year=1994 | isbn=981-02-1403-0 }}
*{{citation|last1=Etnyre|first1=J.|last2=Ghrist|first2=R.|title=Contact topology and hydrodynamics. I. Beltrami fields and the Seifert conjecture|journal=Nonlinearity|year=2000|volume=13|pages=441&ndash;448|doi=10.1088/0951-7715/13/2/306|bibcode = 2000Nonli..13..441E|issue=2 }}.
 
[[Category:Vector calculus]]
 
{{geometry-stub}}

Latest revision as of 09:01, 25 September 2014

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