Complex lamellar vector field: Difference between revisions
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In [[vector calculus]], a '''complex lamellar vector field''' is a [[vector field]] |
In [[vector calculus]], a '''complex lamellar vector field''' is a [[vector field]] which is orthogonal to a family of surfaces. In the broader context of [[differential geometry]], complex lamellar vector fields are more often called '''hypersurface-orthogonal vector fields.''' They can be characterized in a number of different ways, many of which involve the [[curl (mathematics)|curl]]. A '''lamellar vector field''' is a special case given by vector fields with zero curl. |
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⚫ | The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The ''lamellae'' to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field. This language is particularly popular with authors in [[rational mechanics]].{{sfnm|1a1=Panton|1y=2013|1p=434}} |
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==Complex lamellar vector fields== |
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Complex lamellar vector fields are precisely those that are normal to a family of surfaces. A special case are [[irrotational vector field]]s, satisfying |
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In [[vector calculus]], a '''complex lamellar vector field''' is a [[vector field]] in three dimensions which is [[orthogonality|orthogonal]] to its own [[curl (mathematics)|curl]].{{sfnm|1a1=Aris|1y=1962|1p=64|2a1=Panton|2y=2013|2loc=Section 17.4}} That is, |
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The term '''lamellar vector field''' is sometimes used as a synonym for the special case of an [[irrotational vector field]], meaning that{{sfnm|1a1=Aris|1y=1962|1p=64}} |
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:<math>\nabla\times\mathbf{F}=\mathbf{0}.</math> |
:<math>\nabla\times\mathbf{F}=\mathbf{0}.</math> |
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An irrotational vector field is locally the [[gradient]] of a function, and is therefore orthogonal to the family of [[level surface]]s (the [[equipotential surface]]s). |
Complex lamellar vector fields are precisely those that are normal to a family of surfaces. An irrotational vector field is locally the [[gradient]] of a function, and is therefore orthogonal to the family of [[level surface]]s (the [[equipotential surface]]s).{{sfnm|1a1=Aris|1y=1962|1p=66}} Any vector field can be decomposed as the sum of an irrotational vector field and a complex lamellar field.{{sfnm|1a1=Aris|1y=1962|1p=72|2a1=Panton|2y=2013|2loc=Section 17.4}} |
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==Hypersurface-orthogonal vector fields== |
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In greater generality, a vector field {{math|'''F'''}} on a [[pseudo-Riemannian manifold]] is said to be '''hypersurface-orthogonal''' if through an arbitrary point there is a [[embedded submanifold|smoothly embedded hypersurface]] which, at all of its points, is orthogonal to the vector field. By the [[Frobenius theorem (differential topology)|Frobenius theorem]] this is equivalent to requiring that the [[Lie bracket of vector fields|Lie bracket]] of any smooth vector fields orthogonal to {{math|'''F'''}} is still orthogonal to {{math|'''F'''}}.{{sfnm|1a1=O'Neill|1y=1983|1loc=Proposition 12.30}} |
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The condition of hypersurface-orthogonality can be rephrased in terms of the [[differential form|differential 1-form]] {{math|ω}} which is dual to {{math|'''F'''}}. The previously given Lie bracket condition can be reworked to require that the [[exterior derivative]] {{math|dω}}, when evaluated on any two tangent vectors which are orthogonal to {{math|'''F'''}}, is zero.{{sfnm|1a1=O'Neill|1y=1983|1loc=Proposition 12.30}} This may also be phrased as the requirement that there is a smooth 1-form whose [[wedge product]] with {{math|ω}} equals {{math|dω}}.{{sfnm|1a1=Lee|1y=2013|1loc=Lemma 19.6}} |
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Alternatively, this may be written as the condition that the [[differential form|differential 3-form]] {{math|ω ∧ dω}} is zero. This can also be phrased, in terms of the [[Levi-Civita connection]] defined by the metric, as requiring that the totally anti-symmetric part of the 3-tensor field {{math|ω<sub>''i''</sub>∇<sub>''j'' </sub>ω<sub>''k''</sub>}} is zero.{{sfnm|1a1=Wald|1y=1984|1loc=Appendix B.3}} Using a different formulation of the Frobenius theorem, it is also equivalent to require that {{math|ω}} is locally expressible as {{math|λ d''u''}} for some functions {{math|λ}} and {{mvar|u}}.{{sfnm|1a1=Flanders|1y=1989|1pp=96–97|2a1=Stephani|2a2=Kramer|2a3=MacCallum|2a4=Hoenselaers|2y=2003|2p=68}} |
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In the special case of vector fields on [[Euclidean space|three-dimensional Euclidean space]], the hypersurface-orthogonal condition is equivalent to the complex lamellar condition, as seen by rewriting {{math|ω ∧ dω}} in terms of the [[Hodge star operator]] as {{math|∗⟨ω, ∗dω⟩}}, with {{math|∗dω}} being the 1-form dual to the curl vector field.{{sfnm|1a1=Choquet-Bruhat|1a2=DeWitt-Morette|1a3=Dillard-Bleick|1y=1982|1p=247}} |
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Hypersurface-orthogonal vector fields are particularly important in [[general relativity]], where (among other reasons) the existence of a [[Killing vector field]] which is hypersurface-orthogonal is one of the requirements of a [[static spacetime]].{{sfnm|1a1=O'Neill|1y=1983|1p=360|2a1=Stephani|2a2=Kramer|2a3=MacCallum|2a4=Hoenselaers|2y=2003|3a1=Wald|3y=1984|3loc=Section 6.1}} In this context, hypersurface-orthogonality is sometimes called ''irrotationality'', although this is in conflict with the standard usage in three dimensions.{{sfnm|1a1=O'Neill|1y=1983|1p=358}} Another name is '''rotation-freeness'''.{{sfnm|1a1=Misner|1a2=Thorne|1a3=Wheeler|1y=1973|1pp=123–124}} |
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An even more general notion, in the language of [[Pfaffian system]]s, is that of a [[Integrability conditions for differential systems|completely integrable 1-form]] {{math|ω}}, which amounts to the condition {{math|ω ∧ dω {{=}} 0}} as given above.{{sfnm|1a1=Choquet-Bruhat|1a2=DeWitt-Morette|1a3=Dillard-Bleick|1y=1982|1loc=Section IV.C.6}} In this context, there is no metric and so there is no notion of "orthogonality". |
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==See also== |
==See also== |
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==References== |
==References== |
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{{refbegin}} |
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*{{ |
*{{cite book | title=Vectors, tensors, and the basic equations of fluid mechanics | first=Rutherford | last=Aris | authorlink=Rutherford Aris | location=Englewood Cliffs, NJ|publisher=[[Prentice-Hall, Inc.]] | year=1962|others=Reprinted in 1989 | isbn=0-486-66110-5 | url-access=registration | url=https://archive.org/details/vectorstensorsba00aris|zbl=0123.41502 }} |
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*{{cite book|mr=0685274|last1=Choquet-Bruhat|first1=Yvonne|last2=DeWitt-Morette|first2=Cécile|last3=Dillard-Bleick|first3=Margaret|title=Analysis, manifolds and physics|edition=Second edition of 1977 original|publisher=[[North-Holland Publishing Company|North-Holland Publishing Co.]]|location=Amsterdam–New York|year=1982|isbn=0-444-86017-7|author-link1=Yvonne Choquet-Bruhat|author-link2=Cécile DeWitt-Morette|zbl=0492.58001}} |
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*{{cite book|last1=Flanders|first1=Harley|title=Differential forms with applications to the physical sciences|edition=Second edition of 1963 original|series=Dover Books on Advanced Mathematics|publisher=[[Dover Publications, Inc.]]|location=New York|year=1989|isbn=0-486-66169-5|mr=1034244|author-link1=Harley Flanders}} |
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*{{cite book|last1=Lee|first1=John M.|title=Introduction to smooth manifolds|edition=Second edition of 2003 original|series=[[Graduate Texts in Mathematics]]|volume=218|publisher=[[Springer Publishing|Springer]]|location=New York|year=2013|isbn=978-1-4419-9981-8|doi=10.1007/978-1-4419-9982-5|mr=2954043|author-link1=John M. Lee|zbl=1258.53002}} |
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*{{cite book|last1=Misner|first1=Charles W.|last2=Thorne|first2=Kip S.|last3=Wheeler|first3=John Archibald|title=Gravitation|publisher=[[W. H. Freeman and Co.]]|location=San Francisco, CA|year=1973|author-link1=Charles Misner|author-link2=Kip Thorne|author-link3=John A. Wheeler|mr=0418833|isbn=978-0-691-17779-3|zbl=1375.83002}} |
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* {{cite book|last1=O'Neill|first1=Barrett|author-link1=Barrett O'Neill|title=Semi-Riemannian geometry. With applications to relativity|series=Pure and Applied Mathematics|volume=103|publisher=[[Academic Press|Academic Press, Inc.]]|location=New York|year=1983|isbn=0-12-526740-1|mr=0719023|zbl=0531.53051|doi=10.1016/s0079-8169(08)x6002-7 }} |
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*{{cite book|last1=Panton|first1=Ronald L.|title=Incompressible flow|edition=Fourth revised, expanded, and updated edition of 1984 original|zbl=1275.76001|location=Hoboken, NJ|publisher=[[John Wiley & Sons]]|isbn=978-1-118-01343-4|year=2013}} |
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*{{cite book|mr=2003646|last1=Stephani|first1=Hans|last2=Kramer|first2=Dietrich|last3=MacCallum|first3=Malcolm|last4=Hoenselaers|first4=Cornelius|last5=Herlt|first5=Eduard|title=Exact solutions of Einstein's field equations|edition=Second edition of 1980 original|series=Cambridge Monographs on Mathematical Physics|publisher=[[Cambridge University Press]]|location=Cambridge|year=2003|isbn=0-521-46136-7|doi=10.1017/CBO9780511535185|author-link1=Hans Stephani|zbl=1057.83004}} |
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*{{cite encyclopedia|last1=Truesdell|first1=C.|last2=Toupin|first2=R.|title=The classical field theories|others=With an appendix on tensor fields by [[Jerald Ericksen|J. L. Ericksen]]|encyclopedia=Principles of Classical Mechanics and Field Theory|series=Encyclopedia of Physics|volume=III/1|pages=226–858|publisher=[[Springer Publishing|Springer]]|location=Berlin|year=1960|mr=0118005|editor-last1=Flügge|editor-first1=S.|doi=10.1007/978-3-642-45943-6_2|zbl=0118.39702|isbn=978-3-540-02547-4|author-link1=Clifford Truesdell|editor-link1=Siegfried Flügge}} |
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*{{cite book|mr=0757180|author-link1=Robert Wald|last1=Wald|first1=Robert M.|title=General relativity|publisher=[[University of Chicago Press]]|location=Chicago, IL|year=1984|isbn=0-226-87032-4|doi=10.7208/chicago/9780226870373.001.0001|title-link=General Relativity (book)|zbl=0549.53001}} |
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{{refend}} |
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[[Category:Vector calculus]] |
[[Category:Vector calculus]] |
Revision as of 07:52, 27 February 2022
In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl. A lamellar vector field is a special case given by vector fields with zero curl.
The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The lamellae to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field. This language is particularly popular with authors in rational mechanics.[1]
Complex lamellar vector fields
In vector calculus, a complex lamellar vector field is a vector field in three dimensions which is orthogonal to its own curl.[2] That is,
The term lamellar vector field is sometimes used as a synonym for the special case of an irrotational vector field, meaning that[3]
Complex lamellar vector fields are precisely those that are normal to a family of surfaces. An irrotational vector field is locally the gradient of a function, and is therefore orthogonal to the family of level surfaces (the equipotential surfaces).[4] Any vector field can be decomposed as the sum of an irrotational vector field and a complex lamellar field.[5]
Hypersurface-orthogonal vector fields
In greater generality, a vector field F on a pseudo-Riemannian manifold is said to be hypersurface-orthogonal if through an arbitrary point there is a smoothly embedded hypersurface which, at all of its points, is orthogonal to the vector field. By the Frobenius theorem this is equivalent to requiring that the Lie bracket of any smooth vector fields orthogonal to F is still orthogonal to F.[6]
The condition of hypersurface-orthogonality can be rephrased in terms of the differential 1-form ω which is dual to F. The previously given Lie bracket condition can be reworked to require that the exterior derivative dω, when evaluated on any two tangent vectors which are orthogonal to F, is zero.[6] This may also be phrased as the requirement that there is a smooth 1-form whose wedge product with ω equals dω.[7]
Alternatively, this may be written as the condition that the differential 3-form ω ∧ dω is zero. This can also be phrased, in terms of the Levi-Civita connection defined by the metric, as requiring that the totally anti-symmetric part of the 3-tensor field ωi∇j ωk is zero.[8] Using a different formulation of the Frobenius theorem, it is also equivalent to require that ω is locally expressible as λ du for some functions λ and u.[9]
In the special case of vector fields on three-dimensional Euclidean space, the hypersurface-orthogonal condition is equivalent to the complex lamellar condition, as seen by rewriting ω ∧ dω in terms of the Hodge star operator as ∗⟨ω, ∗dω⟩, with ∗dω being the 1-form dual to the curl vector field.[10]
Hypersurface-orthogonal vector fields are particularly important in general relativity, where (among other reasons) the existence of a Killing vector field which is hypersurface-orthogonal is one of the requirements of a static spacetime.[11] In this context, hypersurface-orthogonality is sometimes called irrotationality, although this is in conflict with the standard usage in three dimensions.[12] Another name is rotation-freeness.[13]
An even more general notion, in the language of Pfaffian systems, is that of a completely integrable 1-form ω, which amounts to the condition ω ∧ dω = 0 as given above.[14] In this context, there is no metric and so there is no notion of "orthogonality".
See also
Notes
- ^ Panton 2013, p. 434.
- ^ Aris 1962, p. 64; Panton 2013, Section 17.4.
- ^ Aris 1962, p. 64.
- ^ Aris 1962, p. 66.
- ^ Aris 1962, p. 72; Panton 2013, Section 17.4.
- ^ a b O'Neill 1983, Proposition 12.30.
- ^ Lee 2013, Lemma 19.6.
- ^ Wald 1984, Appendix B.3.
- ^ Flanders 1989, pp. 96–97; Stephani et al. 2003, p. 68.
- ^ Choquet-Bruhat, DeWitt-Morette & Dillard-Bleick 1982, p. 247.
- ^ O'Neill 1983, p. 360; Stephani et al. 2003; Wald 1984, Section 6.1.
- ^ O'Neill 1983, p. 358.
- ^ Misner, Thorne & Wheeler 1973, pp. 123–124.
- ^ Choquet-Bruhat, DeWitt-Morette & Dillard-Bleick 1982, Section IV.C.6.
References
- Aris, Rutherford (1962). Vectors, tensors, and the basic equations of fluid mechanics. Reprinted in 1989. Englewood Cliffs, NJ: Prentice-Hall, Inc. ISBN 0-486-66110-5. Zbl 0123.41502.
- Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile; Dillard-Bleick, Margaret (1982). Analysis, manifolds and physics (Second edition of 1977 original ed.). Amsterdam–New York: North-Holland Publishing Co. ISBN 0-444-86017-7. MR 0685274. Zbl 0492.58001.
- Flanders, Harley (1989). Differential forms with applications to the physical sciences. Dover Books on Advanced Mathematics (Second edition of 1963 original ed.). New York: Dover Publications, Inc. ISBN 0-486-66169-5. MR 1034244.
- Lee, John M. (2013). Introduction to smooth manifolds. Graduate Texts in Mathematics. Vol. 218 (Second edition of 2003 original ed.). New York: Springer. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9981-8. MR 2954043. Zbl 1258.53002.
- Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco, CA: W. H. Freeman and Co. ISBN 978-0-691-17779-3. MR 0418833. Zbl 1375.83002.
- O'Neill, Barrett (1983). Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics. Vol. 103. New York: Academic Press, Inc. doi:10.1016/s0079-8169(08)x6002-7. ISBN 0-12-526740-1. MR 0719023. Zbl 0531.53051.
- Panton, Ronald L. (2013). Incompressible flow (Fourth revised, expanded, and updated edition of 1984 original ed.). Hoboken, NJ: John Wiley & Sons. ISBN 978-1-118-01343-4. Zbl 1275.76001.
- Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003). Exact solutions of Einstein's field equations. Cambridge Monographs on Mathematical Physics (Second edition of 1980 original ed.). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511535185. ISBN 0-521-46136-7. MR 2003646. Zbl 1057.83004.
- Truesdell, C.; Toupin, R. (1960). "The classical field theories". In Flügge, S. (ed.). Principles of Classical Mechanics and Field Theory. Encyclopedia of Physics. Vol. III/1. With an appendix on tensor fields by J. L. Ericksen. Berlin: Springer. pp. 226–858. doi:10.1007/978-3-642-45943-6_2. ISBN 978-3-540-02547-4. MR 0118005. Zbl 0118.39702.
- Wald, Robert M. (1984). General relativity. Chicago, IL: University of Chicago Press. doi:10.7208/chicago/9780226870373.001.0001. ISBN 0-226-87032-4. MR 0757180. Zbl 0549.53001.