128:
181:
393:
225:
51:
to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field.
1024:
916:
776:
186:
Complex lamellar vector fields are precisely those that are normal to a family of surfaces. An irrotational vector field is locally the
85:
1076:
958:
877:
828:
728:
687:
650:
680:
147:
756:
229:
1060:
942:
213:
1054:
820:
417:
138:
1112:
721:
198:). Any vector field can be decomposed as the sum of an irrotational vector field and a complex lamellar field.
672:
346:, the hypersurface-orthogonal condition is equivalent to the complex lamellar condition, as seen by rewriting
810:
296:
668:
412:
195:
29:
320:
is zero. Using a different formulation of the
Frobenius theorem, it is also equivalent to require that
941:. Cambridge Monographs on Mathematical Physics (Second edition of 1980 original ed.). Cambridge:
1004:
908:
370:
221:
1000:
988:
760:
643:
351:
261:
403:
as given above. In this context, there is no metric and so there is no notion of "orthogonality".
984:
814:
366:
76:
37:
854:
299:
defined by the metric, as requiring that the totally anti-symmetric part of the 3-tensor field
1072:
1020:
954:
912:
873:
824:
772:
724:
683:
646:
289:
248:
1068:
1090:
1064:
1038:
1012:
972:
946:
922:
891:
865:
842:
790:
764:
720:. Dover Books on Advanced Mathematics (Second edition of 1963 original ed.). New York:
701:
656:
381:, although this is in conflict with the standard usage in three dimensions. Another name is
374:
1086:
1034:
968:
887:
838:
786:
738:
697:
273:, is zero. This may also be phrased as the requirement that there is a smooth 1-form whose
1094:
1082:
1042:
1030:
996:
976:
964:
926:
895:
883:
846:
834:
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734:
713:
705:
693:
660:
631:
389:
343:
60:
17:
1008:
907:(Fourth revised, expanded, and updated edition of 1984 original ed.). Hoboken, NJ:
47:
The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The
862:
802:
937:; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius; Herlt, Eduard (2003).
869:
1106:
934:
636:
274:
191:
72:
68:
36:
They can be characterized in a number of different ways, many of which involve the
25:
995:. Encyclopedia of Physics. Vol. III/1. With an appendix on tensor fields by
260:. The previously given Lie bracket condition can be reworked to require that the
1050:
1016:
746:
806:
768:
950:
247:
The condition of hypersurface-orthogonality can be rephrased in terms of the
614:
558:
502:
500:
187:
224:
which, at all of its points, is orthogonal to the vector field. By the
28:
which is orthogonal to a family of surfaces. In the broader context of
759:. Vol. 218 (Second edition of 2003 original ed.). New York:
750:
365:
Hypersurface-orthogonal vector fields are particularly important in
267:, when evaluated on any two tangent vectors which are orthogonal to
574:
546:
377:. In this context, hypersurface-orthogonality is sometimes called
373:
which is hypersurface-orthogonal is one of the requirements of a
123:{\displaystyle \mathbf {F} \cdot (\nabla \times \mathbf {F} )=0.}
679:(Second edition of 1977 original ed.). Amsterdam–New York:
718:
Differential forms with applications to the physical sciences
288:
Alternatively, this may be written as the condition that the
638:
Vectors, tensors, and the basic equations of fluid mechanics
190:
of a function, and is therefore orthogonal to the family of
137:
is sometimes used as a synonym for the special case of an
861:. Pure and Applied Mathematics. Vol. 103. New York:
859:
Semi-Riemannian geometry. With applications to relativity
44:
is a special case given by vector fields with zero curl.
987:; Toupin, R. (1960). "The classical field theories". In
615:
Choquet-Bruhat, DeWitt-Morette & Dillard-Bleick 1982
559:
Choquet-Bruhat, DeWitt-Morette & Dillard-Bleick 1982
176:{\displaystyle \nabla \times \mathbf {F} =\mathbf {0} .}
32:, complex lamellar vector fields are more often called
150:
88:
602:
295:is zero. This can also be phrased, in terms of the
993:Principles of Classical Mechanics and Field Theory
635:
175:
122:
369:, where (among other reasons) the existence of a
388:An even more general notion, in the language of
362:being the 1-form dual to the curl vector field.
939:Exact solutions of Einstein's field equations
8:
642:. Reprinted in 1989. Englewood Cliffs, NJ:
590:
570:
506:
232:of any smooth vector fields orthogonal to
228:this is equivalent to requiring that the
220:if through an arbitrary point there is a
165:
157:
149:
106:
89:
87:
542:
342:In the special case of vector fields on
428:
1069:10.7208/chicago/9780226870373.001.0001
491:
451:
435:
206:In greater generality, a vector field
34:hypersurface-orthogonal vector fields.
202:Hypersurface-orthogonal vector fields
7:
578:
530:
487:
475:
463:
447:
675:; Dillard-Bleick, Margaret (1982).
518:
151:
100:
14:
603:Misner, Thorne & Wheeler 1973
399:, which amounts to the condition
344:three-dimensional Euclidean space
752:Introduction to smooth manifolds
166:
158:
107:
90:
677:Analysis, manifolds and physics
222:smoothly embedded hypersurface
111:
97:
55:Complex lamellar vector fields
1:
870:10.1016/s0079-8169(08)x6002-7
757:Graduate Texts in Mathematics
71:in three dimensions which is
65:complex lamellar vector field
22:complex lamellar vector field
681:North-Holland Publishing Co.
394:completely integrable 1-form
1061:University of Chicago Press
1017:10.1007/978-3-642-45943-6_2
1129:
943:Cambridge University Press
903:Panton, Ronald L. (2013).
324:is locally expressible as
214:pseudo-Riemannian manifold
821:W. H. Freeman and Company
769:10.1007/978-1-4419-9982-5
418:Conservative vector field
139:irrotational vector field
951:10.1017/CBO9780511535185
722:Dover Publications, Inc.
811:Wheeler, John Archibald
238:is still orthogonal to
218:hypersurface-orthogonal
673:DeWitt-Morette, Cécile
669:Choquet-Bruhat, Yvonne
297:Levi-Civita connection
196:equipotential surfaces
177:
124:
909:John Wiley & Sons
819:. San Francisco, CA:
413:Beltrami vector field
178:
135:lamellar vector field
125:
42:lamellar vector field
30:differential geometry
1003:. pp. 226–858.
863:Academic Press, Inc.
575:Stephani et al. 2003
547:Stephani et al. 2003
509:, Proposition 12.30.
401:ω ∧ dω = 0
371:Killing vector field
148:
86:
1009:1960HDP.....2..226T
905:Incompressible flow
644:Prentice-Hall, Inc.
605:, pp. 123–124.
356:∗⟨ω, ∗dω⟩
352:Hodge star operator
331:for some functions
290:differential 3-form
262:exterior derivative
249:differential 1-form
1056:General relativity
803:Misner, Charles W.
545:, pp. 96–97;
367:general relativity
173:
120:
1026:978-3-540-02547-4
918:978-1-118-01343-4
778:978-1-4419-9981-8
617:, Section IV.C.6.
383:rotation-freeness
254:which is dual to
226:Frobenius theorem
1120:
1098:
1046:
980:
930:
899:
855:O'Neill, Barrett
850:
798:
742:
714:Flanders, Harley
709:
664:
641:
632:Aris, Rutherford
618:
612:
606:
600:
594:
588:
582:
568:
562:
556:
550:
540:
534:
528:
522:
516:
510:
504:
495:
485:
479:
473:
467:
461:
455:
445:
439:
433:
402:
398:
390:Pfaffian systems
375:static spacetime
361:
357:
350:in terms of the
349:
348:ω ∧ dω
338:
334:
330:
323:
319:
294:
293:ω ∧ dω
284:
280:
272:
266:
259:
253:
243:
237:
211:
182:
180:
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169:
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129:
127:
126:
121:
110:
93:
1128:
1127:
1123:
1122:
1121:
1119:
1118:
1117:
1113:Vector calculus
1103:
1102:
1101:
1079:
1059:. Chicago, IL:
1051:Wald, Robert M.
1049:
1027:
983:
961:
933:
919:
902:
880:
853:
831:
801:
779:
745:
731:
712:
690:
667:
653:
630:
626:
621:
613:
609:
601:
597:
589:
585:
573:, p. 360;
569:
565:
557:
553:
541:
537:
533:, Appendix B.3.
529:
525:
517:
513:
505:
498:
494:, Section 17.4.
486:
482:
474:
470:
462:
458:
454:, Section 17.4.
446:
442:
434:
430:
426:
409:
400:
396:
392:, is that of a
379:irrotationality
359:
355:
347:
336:
332:
325:
321:
318:
312:
306:
300:
292:
282:
278:
268:
264:
255:
251:
239:
233:
207:
204:
146:
145:
141:, meaning that
84:
83:
61:vector calculus
57:
18:vector calculus
12:
11:
5:
1126:
1124:
1116:
1115:
1105:
1104:
1100:
1099:
1077:
1047:
1025:
997:J. L. Ericksen
981:
959:
935:Stephani, Hans
931:
917:
900:
878:
851:
829:
807:Thorne, Kip S.
799:
777:
743:
729:
710:
688:
665:
651:
627:
625:
622:
620:
619:
607:
595:
593:, p. 358.
583:
581:, Section 6.1.
563:
561:, p. 247.
551:
535:
523:
511:
496:
490:, p. 72;
480:
468:
456:
450:, p. 64;
440:
438:, p. 434.
427:
425:
422:
421:
420:
415:
408:
405:
314:
308:
302:
216:is said to be
203:
200:
192:level surfaces
184:
183:
172:
168:
164:
160:
156:
153:
131:
130:
119:
116:
113:
109:
105:
102:
99:
96:
92:
56:
53:
13:
10:
9:
6:
4:
3:
2:
1125:
1114:
1111:
1110:
1108:
1096:
1092:
1088:
1084:
1080:
1078:0-226-87032-4
1074:
1070:
1066:
1062:
1058:
1057:
1052:
1048:
1044:
1040:
1036:
1032:
1028:
1022:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
985:Truesdell, C.
982:
978:
974:
970:
966:
962:
960:0-521-46136-7
956:
952:
948:
944:
940:
936:
932:
928:
924:
920:
914:
910:
906:
901:
897:
893:
889:
885:
881:
879:0-12-526740-1
875:
871:
867:
864:
860:
856:
852:
848:
844:
840:
836:
832:
830:0-7503-0948-2
826:
822:
818:
817:
812:
808:
804:
800:
796:
792:
788:
784:
780:
774:
770:
766:
762:
758:
754:
753:
748:
744:
740:
736:
732:
730:0-486-66169-5
726:
723:
719:
715:
711:
707:
703:
699:
695:
691:
689:0-444-86017-7
685:
682:
678:
674:
670:
666:
662:
658:
654:
652:0-486-66110-5
648:
645:
640:
639:
633:
629:
628:
623:
616:
611:
608:
604:
599:
596:
592:
587:
584:
580:
576:
572:
567:
564:
560:
555:
552:
549:, p. 68.
548:
544:
543:Flanders 1989
539:
536:
532:
527:
524:
521:, Lemma 19.6.
520:
515:
512:
508:
503:
501:
497:
493:
489:
484:
481:
478:, p. 66.
477:
472:
469:
466:, p. 64.
465:
460:
457:
453:
449:
444:
441:
437:
432:
429:
423:
419:
416:
414:
411:
410:
406:
404:
395:
391:
386:
384:
380:
376:
372:
368:
363:
353:
345:
340:
329:
317:
311:
305:
298:
291:
286:
276:
275:wedge product
271:
263:
258:
250:
245:
242:
236:
231:
227:
223:
219:
215:
210:
201:
199:
197:
193:
189:
170:
162:
154:
144:
143:
142:
140:
136:
117:
114:
103:
94:
82:
81:
80:
78:
74:
70:
66:
62:
54:
52:
50:
45:
43:
39:
35:
31:
27:
23:
19:
1055:
992:
938:
904:
858:
815:
751:
747:Lee, John M.
717:
676:
637:
610:
598:
591:O'Neill 1983
586:
571:O'Neill 1983
566:
554:
538:
526:
514:
507:O'Neill 1983
483:
471:
459:
443:
431:
387:
382:
378:
364:
341:
327:
315:
309:
303:
287:
269:
256:
246:
240:
234:
217:
208:
205:
185:
134:
132:
69:vector field
64:
58:
48:
46:
41:
33:
26:vector field
21:
15:
816:Gravitation
492:Panton 2013
452:Panton 2013
436:Panton 2013
230:Lie bracket
79:. That is,
75:to its own
1095:0549.53001
1043:0118.39702
999:. Berlin:
989:Flügge, S.
977:1057.83004
927:1275.76001
896:0531.53051
847:1375.83002
795:1258.53002
706:0492.58001
661:0123.41502
624:References
73:orthogonal
579:Wald 1984
531:Wald 1984
488:Aris 1962
476:Aris 1962
464:Aris 1962
448:Aris 1962
155:×
152:∇
133:The term
104:×
101:∇
95:⋅
1107:Category
1053:(1984).
1001:Springer
857:(1983).
813:(1973).
761:Springer
749:(2013).
716:(1989).
634:(1962).
519:Lee 2013
407:See also
360:∗dω
326:λ d
188:gradient
49:lamellae
1087:0757180
1035:0118005
1005:Bibcode
991:(ed.).
969:2003646
888:0719023
839:0418833
787:2954043
739:1034244
698:0685274
358:, with
283:dω
281:equals
265:dω
1093:
1085:
1075:
1041:
1033:
1023:
975:
967:
957:
925:
915:
894:
886:
876:
845:
837:
827:
793:
785:
775:
737:
727:
704:
696:
686:
659:
649:
397:ω
333:λ
322:ω
313:ω
301:ω
279:ω
252:ω
424:Notes
277:with
212:on a
194:(the
67:is a
24:is a
1073:ISBN
1021:ISBN
955:ISBN
913:ISBN
874:ISBN
825:ISBN
773:ISBN
725:ISBN
684:ISBN
647:ISBN
335:and
77:curl
63:, a
40:. A
38:curl
20:, a
1091:Zbl
1065:doi
1039:Zbl
1013:doi
973:Zbl
947:doi
923:Zbl
892:Zbl
866:doi
843:Zbl
791:Zbl
765:doi
702:Zbl
657:Zbl
354:as
59:In
16:In
1109::
1089:.
1083:MR
1081:.
1071:.
1063:.
1037:.
1031:MR
1029:.
1019:.
1011:.
971:.
965:MR
963:.
953:.
945:.
921:.
911:.
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882:.
872:.
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783:MR
781:.
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735:MR
733:.
700:.
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671:;
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385:.
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118:0.
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337:u
328:u
316:k
310:j
307:∇
304:i
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257:F
241:F
235:F
209:F
171:.
167:0
163:=
159:F
115:=
112:)
108:F
98:(
91:F
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