211:
100:
90:
69:
36:
201:
174:
27:
332:. Now, what is this page trying to achieve? Is it going to state the lemma, in effect, in vector calculus terms, so as not to frighten the horses? Is it going to state some valid special case? Is it going to try to prove (better, sketch a proof of) something? Anyway these points seem bound up with trying to get our necessary and our sufficient conditions clearer.
1024:
think "combing pipes so they're always connected". I'm guessing this is a 19th century idea. If you read James Clerk
Maxwell's diaries, they're filled with these esoteric thoughts of mechanistic fluids sloshing around in cycles. The concept of a vector field, clearly popular in the 19th cent, its hard to imagine that its much older than that.
1023:
It's "solenoidal" because of the curl. You can imagine each vector in the field as being a tiny pipe, and the pipes must always be connected so that there's just as much coming in as there is going out: the pipes don't leak. So instead of the conventional "combing hair without flat spots or peaks",
982:
These fields are almost certainly called solenoidal because of their relationship to solenoidal magnetic fields (i.e. magnetic fields generated from pipe-like coils of current-carrying wires). This seems to me to be a folk etymology (and a quite bizarre one at that), it definitely needs a source.
548:
As far as I understand the definition of a vector field that is solenoidal on a domain Omega (using the term domain loosely here) is that there is no net flow through any closed surface (again using that term loosely) contained in Omega. For many types of domains, such as e.g. simply connected or
850:. Michiel Hazewinkel "Encyclopaedia of Mathematics" vol. 9 p402 (viewable in Google Books) gives a stronger surface-integral definition, and notes that divergencelessness does not imply solenoidality in domains with holes. I can't immediately find any other book giving this stronger definition.
16:
Is there any reason at all not to edit this down so that it just says (1) solenoidal means zero divergence and (2) this is equivalent to having a vector potential? I don't think there's any other content on this page that isn't either incorrect or irrelevant. Perhaps it's worth adding that
734:
552:
However, this analogy does not hold in all cases. A typical example, that has some relevance e.g. in the Earth
Sciences, are annuli and thick spherical shells. For such domains a field can by divergence-free, but non-solenoidal. An example is given by the field
350:
an anon contribution, and hit "Return" before finishing my comment. I meant to say that the proof in the article is rigurious. Even if the curl is not the same as the cross-product, they obey the same laws.
382:
in a number of articles. It's clearly incorrect in this case (I reckon), so I've reverted it. Please discuss the matter here if you disagree. Also, it'd be helpful if you registered a user name. Thanks,
522:
F such that f = curl F is only supposed be true on specific domains (I'm not sure whether it's star-like or simply connected one implying the other anyway). Don't you think it should be added ?
623:
156:
779:
614:
1070:
1100:
279:
184:
471:
Actually B is now considered the
Magnetic Field by physicists, so please never use Magnetic Field to mean H. Special Contributions 83.131.29.96 is totally correct on that one!
302:
True, most of the article seems to go on and on and the essence of the subject gets lost. What I find odd is that the article starts by saying that a solenoidal vector field
1009:
confirms that such a word was used in
Ancient Greek. How it came to be applied to the vector field I don't know, though your explanation does seem rather more plausible. -
1075:
1060:
802:
267:
1105:
40:
1065:
1085:
146:
1080:
1055:
1095:
257:
122:
210:
1090:
855:
We should follow the sources; start with the 'elementary' definition, then follow with the 'advanced' definition, noting how they differ. --
1002:
811:
does only constitute one part of the boundary of the volume enclosed by it, the other one being the inner boundary of the domain itself.
1050:
948:
1025:
529:
371:
233:
113:
74:
1006:
1110:
824:
G. Backus, Poloidal and
Torroidal Fields in Geomagnetic Field Modelling, Reviews of Geophysics, no. 1, vol. 24, 1986, pp. 75-109
487:
729:{\displaystyle \Omega _{(\alpha ,\beta )}={\Big \{}r\in \mathbb {R} ^{3}\,{\Big |}\,0<\alpha <|r|<\beta {\Big \}}}
224:
179:
49:
846:
most commonly defined as divergenceless - however theses are largely elementary texts which confine their attention to
430:
549:
star-shaped this is the case (for a continuously differentiable field) if and only if the field is divergence-free.
943:
a common way of expressing the property of a vector field's having zero divergence. It is not a way at all. It is
446:"Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density."
356:
745:
832:
558:
26:
1014:
952:
533:
819:
1029:
828:
449:
375:
335:
322:
55:
99:
461:
738:
This field can be shown to be divergence-free on this domain. However, the net flow through a sphere
525:
475:
352:
984:
963:
856:
483:
384:
890:. Solenoidal implies divergenceless, but the converse is true only in some specific domains, like
232:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
121:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
1010:
988:
967:
915:
860:
314:
but that's too much for me. A rewrite (leaving out the talk of cars highways and gears) would be
105:
89:
68:
329:
311:
887:
934:
A common way of expressing this property is to say that the field has no sources or sinks.
784:
479:
453:
422:
414:
379:
216:
1044:
911:
402:
394:
1033:
1018:
992:
971:
956:
919:
864:
836:
537:
511:
491:
464:
441:
387:
360:
508:
200:
173:
118:
315:
206:
95:
998:
978:
There is no way solenoidal comes from Greek "constrained as if in a pipe"
229:
807:
This does not contradict the divergence theorem, since the sphere
906:)=0). It is false even in some simply connected domains, like
20:
457:
347:
328:
While here - there is certainly a connection with the
787:
748:
626:
561:
228:, a collaborative effort to improve the coverage of
117:, a collaborative effort to improve the coverage of
796:
773:
728:
608:
436:, and it is not itself an magnetic field, just as
929:The introductory section includes this sentence:
721:
681:
654:
1071:Knowledge level-5 vital articles in Mathematics
886:should be restricted to vector fields having a
1101:Start-Class physics articles of Mid-importance
8:
894:or star-shaped domains (in general: domains
306:when div v = 0 and ends by saying that is
24:
774:{\displaystyle \Omega _{(\alpha ,\beta )}}
168:
63:
1076:Start-Class vital articles in Mathematics
786:
753:
747:
720:
719:
708:
700:
680:
679:
671:
667:
666:
653:
652:
631:
625:
609:{\displaystyle v(r)={\frac {r}{|r|^{3}}}}
597:
592:
583:
577:
560:
882:) vector fields is misleading. The term
1061:Knowledge vital articles in Mathematics
1001:says the word comes from Ancient Greek
962:What counter-examples had you in mind?
686:
677:
170:
65:
18:a velocity field with zero divergence.
842:Flicking through a few books, I find
7:
222:This article is within the scope of
111:This article is within the scope of
1106:Start-Class fluid dynamics articles
54:It is of interest to the following
1066:Start-Class level-5 vital articles
750:
628:
393:What does "divless" mean? Just as
372:Special:Contributions/83.131.29.96
310:div v = 0. Maybe mathematics is a
14:
1086:Mid-priority mathematics articles
444:of an field is the field itself (
440:is not itself an electric field.
131:Knowledge:WikiProject Mathematics
1081:Start-Class mathematics articles
1056:Knowledge level-5 vital articles
1007:Liddell & Scott's dictionary
947:, as the simplest examples show.
429:is just an magnetic analogue of
209:
199:
172:
134:Template:WikiProject Mathematics
98:
88:
67:
34:
25:
17:incompressible fluid flow =: -->
1096:Mid-importance physics articles
820:Lecture Notes in Geomathematics
262:This article has been rated as
151:This article has been rated as
766:
754:
709:
701:
644:
632:
593:
584:
571:
565:
512:13:46, 12 September 2007 (UTC)
1:
993:21:24, 20 February 2023 (UTC)
972:13:08, 31 December 2019 (UTC)
957:10:35, 31 December 2019 (UTC)
920:15:36, 20 February 2014 (UTC)
865:17:53, 28 December 2009 (UTC)
837:11:05, 28 December 2009 (UTC)
361:22:55, 13 November 2005 (UTC)
318:. --12:27, 16 Mar 2005 (UTC)
298:Much of it should go methinks
277:This article is supported by
242:Knowledge:WikiProject Physics
236:and see a list of open tasks.
125:and see a list of open tasks.
1091:Start-Class physics articles
939:No, that is most definitely
245:Template:WikiProject Physics
431:electric displacement field
321:OK: done, rather brutally.
1127:
1051:Start-Class vital articles
1034:16:23, 22 March 2024 (UTC)
1019:00:18, 23 March 2023 (UTC)
544:Correctness of definition?
409:over an area, and just as
268:project's importance scale
492:00:34, 17 June 2014 (UTC)
338:17:03, 20 Mar 2005 (UTC)
325:14:11, 2005 Mar 20 (UTC)
276:
261:
194:
150:
83:
62:
538:08:48, 21 May 2008 (UTC)
465:10:46, 8 July 2007 (UTC)
388:17:04, 7 July 2007 (UTC)
280:Fluid Dynamics Taskforce
157:project's priority scale
1111:Fluid dynamics articles
517:Simply connected domain
114:WikiProject Mathematics
798:
775:
730:
610:
925:Terrible introduction
799:
797:{\displaystyle 4\pi }
776:
731:
611:
450:magnetic flux density
376:magnetic flux density
41:level-5 vital article
785:
746:
624:
559:
401:over an area, so is
137:mathematics articles
342:My recent reversion
225:WikiProject Physics
794:
771:
726:
687:
678:
606:
366:B is divless not H
106:Mathematics portal
50:content assessment
604:
540:
528:comment added by
503:Why is it called
495:
478:comment added by
295:
294:
291:
290:
287:
286:
167:
166:
163:
162:
1118:
888:vector potential
803:
801:
800:
795:
780:
778:
777:
772:
770:
769:
735:
733:
732:
727:
725:
724:
712:
704:
685:
684:
676:
675:
670:
658:
657:
648:
647:
615:
613:
612:
607:
605:
603:
602:
601:
596:
587:
578:
523:
521:div f = 0 =: -->
494:
472:
458:more than a year
336:Charles Matthews
323:Gareth McCaughan
250:
249:
248:physics articles
246:
243:
240:
219:
214:
213:
203:
196:
195:
190:
187:
176:
169:
139:
138:
135:
132:
129:
108:
103:
102:
92:
85:
84:
79:
71:
64:
47:
38:
37:
30:
29:
21:
1126:
1125:
1121:
1120:
1119:
1117:
1116:
1115:
1041:
1040:
1005:"pipe-shaped";
980:
927:
901:
818:Carsten Mayer,
783:
782:
749:
744:
743:
665:
627:
622:
621:
591:
582:
557:
556:
546:
519:
501:
473:
397:is integral of
368:
353:Oleg Alexandrov
344:
300:
247:
244:
241:
238:
237:
215:
208:
188:
182:
136:
133:
130:
127:
126:
104:
97:
77:
48:on Knowledge's
45:
35:
12:
11:
5:
1124:
1122:
1114:
1113:
1108:
1103:
1098:
1093:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1053:
1043:
1042:
1039:
1038:
1037:
1036:
979:
976:
975:
974:
926:
923:
899:
880:incompressible
876:divergengeless
868:
867:
852:
851:
829:InfoBroker2020
826:
825:
822:
793:
790:
768:
765:
762:
759:
756:
752:
723:
718:
715:
711:
707:
703:
699:
696:
693:
690:
683:
674:
669:
664:
661:
656:
651:
646:
643:
640:
637:
634:
630:
618:on the domain
600:
595:
590:
586:
581:
576:
573:
570:
567:
564:
545:
542:
518:
515:
500:
497:
470:
468:
467:
454:magnetic field
423:magnetic field
415:electric field
380:magnetic field
370:I notice that
367:
364:
343:
340:
330:Poincaré lemma
299:
296:
293:
292:
289:
288:
285:
284:
275:
272:
271:
264:Mid-importance
260:
254:
253:
251:
234:the discussion
221:
220:
217:Physics portal
204:
192:
191:
189:Mid‑importance
185:Fluid Dynamics
177:
165:
164:
161:
160:
149:
143:
142:
140:
123:the discussion
110:
109:
93:
81:
80:
72:
60:
59:
53:
31:
13:
10:
9:
6:
4:
3:
2:
1123:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1052:
1049:
1048:
1046:
1035:
1031:
1027:
1022:
1021:
1020:
1016:
1012:
1011:LaetusStudiis
1008:
1004:
1000:
997:
996:
995:
994:
990:
986:
977:
973:
969:
965:
961:
960:
959:
958:
954:
950:
949:50.205.142.35
946:
942:
937:
935:
930:
924:
922:
921:
917:
913:
909:
905:
897:
893:
889:
885:
881:
877:
873:
866:
862:
858:
854:
853:
849:
845:
841:
840:
839:
838:
834:
830:
823:
821:
817:
816:
815:
812:
810:
805:
791:
788:
763:
760:
757:
742:contained in
741:
736:
716:
713:
705:
697:
694:
691:
688:
672:
662:
659:
649:
641:
638:
635:
619:
616:
598:
588:
579:
574:
568:
562:
554:
550:
543:
541:
539:
535:
531:
527:
516:
514:
513:
510:
506:
498:
496:
493:
489:
485:
481:
477:
466:
463:
459:
455:
452:redirects to
451:
447:
443:
439:
435:
432:
428:
424:
420:
416:
412:
408:
404:
403:magnetic flux
400:
396:
395:electric flux
392:
391:
390:
389:
386:
381:
377:
373:
365:
363:
362:
358:
354:
349:
341:
339:
337:
333:
331:
326:
324:
319:
317:
313:
309:
305:
297:
282:
281:
274:
273:
269:
265:
259:
256:
255:
252:
235:
231:
227:
226:
218:
212:
207:
205:
202:
198:
197:
193:
186:
181:
178:
175:
171:
158:
154:
148:
145:
144:
141:
124:
120:
116:
115:
107:
101:
96:
94:
91:
87:
86:
82:
76:
73:
70:
66:
61:
57:
51:
43:
42:
32:
28:
23:
22:
19:
1026:67.198.37.16
981:
944:
940:
938:
933:
931:
928:
907:
903:
895:
891:
883:
879:
875:
871:
869:
847:
843:
827:
813:
808:
806:
739:
737:
620:
617:
555:
551:
547:
530:81.245.62.57
520:
504:
502:
474:— Preceding
469:
445:
442:Flux density
437:
433:
426:
418:
410:
406:
405:integral of
398:
374:has changed
369:
345:
334:
327:
320:
307:
303:
301:
278:
263:
223:
153:Mid-priority
152:
112:
78:Mid‑priority
56:WikiProjects
39:
15:
1003:σωληνοειδής
524:—Preceding
462:83.131.8.91
128:Mathematics
119:mathematics
75:Mathematics
46:Start-class
1045:Categories
999:Wiktionary
884:solenoidal
872:solenoidal
844:solenoidal
505:solenoidal
460:anyway. --
316:good thing
814:See e.g.
480:Jwkeohane
312:tautology
44:is rated
985:INLegred
964:catslash
945:not true
912:Txebixev
910:-{0}. --
898:having H
870:Calling
857:catslash
526:unsigned
488:contribs
476:unsigned
417:, so is
385:catslash
348:reverted
448:), and
346:I just
266:on the
239:Physics
230:Physics
180:Physics
155:on the
509:Abdull
413:is an
52:scale.
33:This
1030:talk
1015:talk
989:talk
968:talk
953:talk
916:talk
878:(or
874:the
861:talk
833:talk
714:<
698:<
692:<
534:talk
507:? --
499:Name
484:talk
456:for
357:talk
941:not
781:is
421:an
378:to
308:why
258:Mid
147:Mid
1047::
1032:)
1017:)
991:)
970:)
955:)
936:"
918:)
900:dR
863:)
835:)
804:.
792:π
764:β
758:α
751:Ω
717:β
695:α
663:∈
642:β
636:α
629:Ω
536:)
490:)
486:•
425:.
383:--
359:)
304:is
183::
1028:(
1013:(
987:(
966:(
951:(
932:"
914:(
908:R
904:U
902:(
896:U
892:R
859:(
848:R
831:(
809:S
789:4
767:)
761:,
755:(
740:S
722:}
710:|
706:r
702:|
689:0
682:|
673:3
668:R
660:r
655:{
650:=
645:)
639:,
633:(
599:3
594:|
589:r
585:|
580:r
575:=
572:)
569:r
566:(
563:v
532:(
482:(
438:D
434:D
427:H
419:B
411:E
407:B
399:E
355:(
283:.
270:.
159:.
58::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
