“Beltrami vector field”的意思、由来-开放百科全书

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Beltrami vector field

释义

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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. 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

{{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=441–448|doi=10.1088/0951-7715/13/2/306|bibcode = 2000Nonli..13..441E|issue=2 }}.

1 : Vector calculus

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