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is a column vector of the partial derivative operators, so should be always emboldened if we are doing that to vectors, even though as an operator ββ’ returns a scalar. Either way, this is not confusing, as the symbols are just grungy shorthand notation, and what they mean is obvious and unambiguous.
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There is a tendency for physicists to use \nabla^2 and mathematicians to use \Delta (a tendency only, I've seen counterexamples on both sides). I think people should be aware that there are two different notations, and it's reasonable for the article to be neutral about the choice, but I think it's
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Knowledge has any standard on this? We see the triangle up a lot (an instance this very page) but the triangle down squared is also there (wave equation, for example). I like the triangle squared, and have a particular dislike for the box representing dalembertian (notably absent in wave equation)
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is just a vector of operators, and β really is that squared, the appropriate product (vector dot product). The author of your link gets a bit attached to his idea of thinking of his notation for operators in terms of what they return, not what they are themselves. So, he states that ββ’ is "never"
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shouldn't really be its own article. The
Laplacian on scalars (as originally defined and as described here) is essentially the same as on vectors, just what we apply it to is different. It feels natural to me that we should have one article to represent the Laplacian as a single concept, which
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notation universally used for
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of those among its eigenvalues that satisfy the special condition described by the phrase "for which there is a corresponding eigenfunction ...".
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Something seems to be missing here: "... the wave equation describes wave propagation, and the SchrΓΆdinger equation in quantum mechanics."
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unnecessarily confusing how the article seems to randomly jump back and forth between the two notations.
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But it does not belong as the first thing that this article says about the definition of the
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on 31 October 2020. For the contribution history and old versions of the redirected page, please see
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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1410:{\displaystyle \displaystyle \Delta _{M}f:=\mathrm {tr} \nabla df\in {\mathsf {C}}^{\infty }(M)}
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So there should be, at least, a hint that the definition is used but not necessarily correct.
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Give an explanation of what the
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Although heavily used by physicists the notion is misleading, not to say wrong: On a
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401:(Disclaimer: I'm neither an experienced Knowledge editor nor a mathematician). The
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1606:{\displaystyle \displaystyle \Delta _{M}f(x)=\sum _{i=1}^{n}\nabla df(E_{i},E_{i})}
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It's not that bad really; no worse than anything else the physicists come up with.
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Of course, this definition filters though to the case of the special manifold
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This may be fun notation to serve as a mnemonic for the
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is compared to this one. It would be no great addition to merge the contents.
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You can add a footnote if you want, but I can't see a particular need.β
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Because of how it is written, it sounds as if the spectrum consists
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http://physics.ucsd.edu/~emichels/FunkyQuantumConcepts.pdf#page=118
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Add a reference explaining the confusing (but universally used)
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there is no operator that can be applied twice to the function
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Quick example of the
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applies equally to scalars and vectors (and tensors, too).
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Regardless of the attempt to justify this at the end, the
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where the latter notations derive from formally writing:
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More precisely, this means for any orthonormal basis
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1729:Merge with Vector Laplacian?
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2382:{\displaystyle \Delta }
2329:{\displaystyle \Delta }
567:{\displaystyle \nabla }
511:16:30, 1 May 2015 (UTC)
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1806:
1805:
1804:
1803:
1802:
1761:
1730:
1727:
1712:
1711:
1699:
1694:
1690:
1685:
1682:
1678:
1675:
1670:
1666:
1639:
1634:
1629:
1626:
1615:
1614:
1601:
1596:
1592:
1588:
1583:
1579:
1575:
1572:
1569:
1566:
1561:
1556:
1553:
1550:
1546:
1542:
1539:
1536:
1533:
1530:
1525:
1521:
1496:
1491:
1485:
1460:
1456:
1452:
1449:
1446:
1443:
1440:
1435:
1431:
1419:
1418:
1405:
1402:
1399:
1394:
1388:
1382:
1379:
1376:
1373:
1369:
1366:
1362:
1359:
1354:
1350:
1321:
1306:
1305:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1264:
1260:
1256:
1253:
1225:
1222:
1217:
1211:
1205:
1202:
1197:
1191:
1185:
1181:
1177:
1174:
1143:
1140:
1137:
1134:
1123:
1122:
1110:
1107:
1102:
1097:
1094:
1089:
1084:
1080:
1076:
1073:
1068:
1064:
1043:
1040:
1035:
1030:
1026:
1022:
1019:
999:
996:
991:
987:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
939:
935:
931:
908:
905:
902:
897:
893:
889:
886:
866:
863:
858:
853:
849:
845:
842:
822:
817:
814:
811:
808:
802:
796:
793:
788:
782:
776:
771:
767:
763:
760:
757:
737:
717:
712:
705:
701:
697:
694:
689:
684:
681:
678:
658:
634:
631:
628:
625:
600:
596:
593:
590:
587:
563:
540:
537:
534:
531:
528:
514:
513:
475:
474:
415:
411:
398:
384:
380:
368:
345:
342:
337:
336:
324:
321:
320:
315:
311:
309:
306:
305:
302:
301:
291:
290:
285:
279:
272:
271:
268:
267:
264:
258:
213:
211:
204:
203:
176:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
2454:
2443:
2440:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2408:
2405:
2403:
2400:
2399:
2397:
2390:
2368:
2363:
2360:
2343:
2315:
2310:
2308:
2303:
2289:
2286:
2283:
2280:
2277:
2271:
2264:
2260:
2259:
2258:eigenfunction
2253:
2252:
2248:
2241:
2239:
2231:
2229:
2227:
2223:
2220:
2206:
2196:
2181:
2174:
2165:
2158:
2156:
2142:
2137:
2129:
2126:
2116:
2115:
2112:
2110:
2105:
2091:
2087:
2078:
2074:
2061:
2058:
2055:
2047:
2043:
2029:
2025:
2015:
2007:
2000:
1998:
1984:
1978:
1972:
1969:
1964:
1956:
1953:
1943:
1942:
1939:
1938:
1924:
1916:
1913:
1897:
1875:
1864:
1848:
1837:
1836:
1832:
1829:-dimensional
1826:
1824:
1817:
1815:
1812:Thee section
1807:
1801:
1797:
1793:
1790:
1788:
1771:
1770:
1769:
1766:
1762:
1759:
1755:
1754:
1753:
1752:
1748:
1744:
1739:
1736:
1728:
1726:
1725:
1721:
1717:
1697:
1692:
1676:
1673:
1668:
1655:
1654:
1653:
1637:
1627:
1624:
1594:
1590:
1586:
1581:
1577:
1570:
1567:
1559:
1554:
1551:
1548:
1544:
1540:
1534:
1528:
1523:
1510:
1509:
1508:
1494:
1489:
1458:
1454:
1450:
1447:
1444:
1441:
1438:
1433:
1429:
1400:
1380:
1377:
1374:
1360:
1357:
1352:
1339:
1338:
1337:
1335:
1319:
1311:
1291:
1288:
1282:
1279:
1270:
1267:
1262:
1254:
1251:
1220:
1215:
1203:
1200:
1195:
1175:
1172:
1157:
1156:
1155:
1141:
1138:
1135:
1132:
1105:
1095:
1092:
1074:
1071:
1066:
1038:
1020:
1017:
994:
989:
978:
975:
969:
966:
960:
957:
951:
942:
937:
922:
921:
920:
903:
895:
887:
884:
861:
843:
840:
820:
812:
806:
791:
786:
774:
769:
761:
758:
735:
715:
703:
699:
692:
682:
679:
676:
656:
648:
632:
629:
626:
623:
615:
591:
588:
585:
577:
554:
535:
532:
529:
519:
512:
508:
504:
503:Kirisutogomen
499:
498:
497:
495:
491:
487:
483:
473:
469:
465:
460:
456:
451:
447:
446:
445:
444:
440:
436:
432:
413:
382:
367:
365:
361:
357:
353:
343:
341:
333:
328:
323:
322:
308:
307:
300:
297:
296:
293:
292:
288:
283:
278:
277:
265:
262:
261:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
229:
227:
223:
212:
210:
209:
201:
200:its talk page
196:
191:
187:
183:
182:
177:
170:
169:
153:
149:
148:High-priority
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
73:Highβpriority
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
2366:
2364:
2361:
2313:
2311:
2306:
2304:
2262:
2255:
2244:
2242:
2237:
2236:The section
2235:
2225:
2224:
2221:
2194:
2172:
2170:
2159:
2108:
2106:
2013:
2012:
2001:
1889:
1828:
1820:
1818:
1813:
1811:
1785:
1740:
1732:
1713:
1616:
1420:
1307:
1124:
575:
515:
486:201.9.205.23
480:β Preceding
476:
458:
454:
449:
400:
356:92.12.18.195
350:βΒ Preceding
347:
344:Poor wording
340:
326:
286:
259:
252:
246:
240:
234:
220:
214:
179:
147:
107:
51:WikiProjects
34:
2251:eigenvalues
1656:To sum up:
1334:contraction
195:its history
123:Mathematics
114:mathematics
70:Mathematics
2396:Categories
1835:divergence
1814:Definition
1507:, we have
1154:, we call
222:To-do list
184:page were
2195:to obtain
2111:equation
1861:) of the
1789:complete.
1765:earfluffy
551:with its
397:notation?
299:Archive 1
39:is rated
2247:spectrum
1890:Thus if
1863:gradient
1054:), i.e.
482:unsigned
464:Kan8eDie
453:written
352:unsigned
287:Archives
1825:in the
1792:Klbrain
1743:Ramzuiv
645:as the
327:90 days
250:refresh
238:history
150:on the
41:C-class
1787:Merger
186:merged
47:scale.
2367:which
2263:with:
2109:first
1910:is a
728:over
576:trace
244:watch
188:into
28:This
2307:only
2245:The
1796:talk
1747:talk
1720:talk
1716:Wueb
1473:for
1308:the
1247:Hess
1168:Hess
507:talk
490:talk
468:talk
439:talk
360:talk
232:edit
224:for
142:High
1888:).
1652:.
1312:of
649:of
616:of
2398::
2377:Ξ
2359:?
2347:Ξ
2344:β
2324:Ξ
2302:"
2284:Ξ»
2275:Ξ
2272:β
2228:.
2219:.
2204:Ξ
2134:β
2124:Ξ
2104:"
2071:β
2067:β
2059:β¦
2040:β
2036:β
2023:β
1982:β
1979:β
1976:β
1961:β
1951:Ξ
1873:β
1849:β
1846:β
1798:)
1763:--
1749:)
1722:)
1689:β
1677:β
1665:β
1565:β
1545:β
1520:Ξ
1393:β
1381:β
1372:β
1361::=
1349:Ξ
1286:β
1283:β‘
1277:β
1274:β
1271:β‘
1259:β
1255::=
1237:,
1216:β
1204:β
1196:β
1180:Ξ
1176:β
1133:Ξ±
1096:β
1079:Ξ
1075:β
1072:Ξ±
1063:β
1025:Ξ
1021:β
986:β
979:Ξ±
976:β
967:Ξ±
958::=
943:Ξ±
934:β
892:Ξ©
888:β
885:Ξ±
848:Ξ
844:β
795:β
704:β
696:β
624:Ξ±
595:β
562:β
509:)
492:)
470:)
441:)
410:β
379:β
362:)
2290:.
2287:f
2281:=
2278:f
2261:f
2254:Ξ»
2243:"
2207:f
2182:f
2164:)
2162:1
2160:(
2143:f
2138:2
2130:=
2127:f
2092:.
2088:)
2079:n
2075:x
2062:,
2056:,
2048:1
2044:x
2030:(
2026:=
2006:)
2004:1
2002:(
1985:f
1973:=
1970:f
1965:2
1957:=
1954:f
1925:f
1898:f
1876:f
1865:(
1838:(
1827:n
1819:"
1794:(
1783:Y
1745:(
1741:-
1718:(
1710:.
1698:f
1693:2
1684:r
1681:t
1674:f
1669:2
1638:n
1633:R
1628:=
1625:M
1613:.
1600:)
1595:i
1591:E
1587:,
1582:i
1578:E
1574:(
1571:f
1568:d
1560:n
1555:1
1552:=
1549:i
1541:=
1538:)
1535:x
1532:(
1529:f
1524:M
1495:M
1490:x
1484:T
1459:n
1455:E
1451:,
1448:.
1445:.
1442:.
1439:,
1434:1
1430:E
1417:.
1404:)
1401:M
1398:(
1387:C
1378:f
1375:d
1368:r
1365:t
1358:f
1353:M
1320:f
1304:,
1292:f
1289:d
1280:f
1268:f
1263:2
1252:f
1224:)
1221:M
1210:T
1201:M
1190:T
1184:(
1173:f
1142:f
1139:d
1136:=
1121:.
1109:)
1106:M
1101:T
1093:M
1088:T
1083:(
1067:A
1042:)
1039:M
1034:T
1029:(
1018:B
1010:(
998:)
995:B
990:A
982:(
973:)
970:B
964:(
961:A
955:)
952:B
949:(
946:)
938:A
930:(
907:)
904:M
901:(
896:1
865:)
862:M
857:T
852:(
841:A
821:N
816:)
813:x
810:(
807:f
801:T
792:M
787:x
781:T
775::
770:x
766:)
762:f
759:d
756:(
736:M
716:N
711:T
700:f
693:M
688:T
683::
680:f
677:d
657:f
633:f
630:d
627:=
599:R
592:M
589::
586:f
539:)
536:g
533:,
530:M
527:(
505:(
488:(
466:(
459:β
455:β
450:β
437:(
414:2
383:2
358:(
253:Β·
247:Β·
241:Β·
235:Β·
228::
202:.
154:.
53::
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