968:. In this case, left and right multiplication are simply translation. By post-composing the manifold-type derivative with the tangent space trivialization, for each point in the domain we obtain a linear map from the tangent space at the domain point to the Lie algebra of
1160:
288:
1165:
Since the tangent spaces involved are one-dimensional, this linear map is just multiplication by some scalar. (This scalar can change depending on what basis we use for the vector spaces, but the
1824:
1199:
857:
676:
1597:
2070:
1388:
1024:
1016:
1467:
1881:
600:
360:
149:
90:
795:
The reason that one might call the
Darboux derivative a more natural generalization of the derivative of single-variable calculus is this. In single-variable calculus, the
1221:
988:
966:
920:
890:
28:
is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable
1348:
785:
710:
125:
1321:
1255:
450:
2019:
859:
assigns to each point in the domain a single number. According to the more general manifold ideas of derivatives, the derivative assigns to each point in the domain a
505:
2093:
1921:
821:
1967:
1686:
1493:
1414:
413:
317:
387:
863:
from the tangent space at the domain point to the tangent space at the image point. This derivative encapsulates two pieces of data: the image of the domain point
867:
the linear map. In single-variable calculus, we drop some information. We retain only the linear map, in the form of a scalar multiplying agent (i.e. a number).
2160:
2140:
2113:
1987:
1941:
1901:
1847:
1660:
1640:
1620:
1517:
1283:
940:
750:
730:
620:
576:
552:
532:
470:
189:
169:
62:
1849:-form on any smooth manifold, all the terms in this equation make sense, so for any such form we can ask whether or not it satisfies this structural equation.
870:
One way to justify this convention of retaining only the linear map aspect of the derivative is to appeal to the (very simple) Lie group structure of
2206:
2171:
682:
1853:
201:
29:
2249:
1694:
1223:
gives a canonical choice of basis, and hence a canonical choice of scalar.) This scalar is what we usually denote by
1172:
826:
628:
1545:
2027:
1353:
192:
2183:
2177:
1155:{\displaystyle v\in T_{x}\mathbb {R} \mapsto (T_{f(x)}L_{f(x)})^{-1}\circ (T_{x}f)v\in T_{0}\mathbb {R} .}
993:
1422:
1862:
581:
322:
130:
71:
2254:
1520:
1204:
971:
949:
903:
873:
1536:
1326:
763:
688:
103:
97:
1288:
418:
33:
2228:
2202:
1992:
900:
can be trivialized via left (or right) multiplication. This means that every tangent space in
478:
2078:
1906:
2216:
1946:
1665:
1472:
1393:
392:
296:
1226:
365:
1166:
512:
508:
801:
2221:
2145:
2125:
2098:
1972:
1926:
1886:
1832:
1645:
1625:
1605:
1502:
1268:
925:
893:
735:
715:
605:
561:
537:
517:
455:
174:
154:
47:
2243:
2118:
For a global generalization of the fundamental theorem, one needs to study certain
2219:(1964). "Chapter V, Lie Groups. Section 2, Invariant forms and the Lie algebra.".
943:
93:
860:
796:
2119:
897:
65:
25:
2232:
2199:
Differential
Geometry: Cartan's Generalization of Klein's Erlangen Program
32:
to higher dimensions, in a different vein than the generalization that is
21:
1519:
is of course the analogue of the constant that appears when taking an
1856:
for single-variable calculus has the following local generalization.
2095:
has a primitive defined in a neighborhood of every point of
922:
may be identified with the tangent space at the identity,
2174: – Fundamental construction of differential calculus
283:{\displaystyle \omega _{G}(X_{g})=(T_{g}L_{g})^{-1}X_{g}}
2148:
2128:
2101:
2081:
2030:
1995:
1975:
1949:
1929:
1909:
1889:
1865:
1835:
1697:
1668:
1648:
1628:
1608:
1548:
1505:
1475:
1425:
1396:
1356:
1329:
1291:
1271:
1229:
1207:
1175:
1027:
996:
974:
952:
928:
906:
876:
829:
804:
766:
738:
718:
691:
631:
608:
584:
564:
540:
520:
481:
458:
421:
395:
368:
325:
299:
204:
177:
157:
133:
106:
74:
50:
1943:
satisfies the structural equation, then every point
2220:
2154:
2134:
2107:
2087:
2064:
2013:
1981:
1961:
1935:
1915:
1895:
1875:
1841:
1818:
1680:
1654:
1634:
1614:
1591:
1511:
1487:
1461:
1408:
1382:
1342:
1315:
1277:
1249:
1215:
1193:
1154:
1010:
982:
960:
934:
914:
884:
851:
815:
779:
744:
724:
704:
670:
614:
594:
570:
546:
526:
499:
464:
444:
407:
381:
354:
311:
282:
183:
163:
143:
119:
84:
56:
1819:{\displaystyle (d\omega )_{x}(X_{x},Y_{x})+=0.}
1194:{\displaystyle {\frac {\partial }{\partial t}}}
852:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
671:{\displaystyle \omega _{f}:=f^{*}\omega _{G},}
8:
389:denotes left multiplication by the element
2180: – Mathematical operation in calculus
1592:{\displaystyle d\omega +{\frac {1}{2}}=0.}
2147:
2127:
2100:
2080:
2065:{\displaystyle \omega _{f}=\omega |_{U},}
2053:
2048:
2035:
2029:
1994:
1974:
1948:
1928:
1908:
1888:
1867:
1866:
1864:
1834:
1798:
1785:
1769:
1756:
1737:
1724:
1711:
1696:
1667:
1647:
1627:
1607:
1558:
1547:
1504:
1474:
1424:
1395:
1374:
1361:
1355:
1334:
1328:
1290:
1270:
1228:
1209:
1208:
1206:
1176:
1174:
1145:
1144:
1138:
1116:
1097:
1078:
1059:
1045:
1044:
1038:
1026:
1004:
1003:
995:
976:
975:
973:
954:
953:
951:
927:
908:
907:
905:
878:
877:
875:
845:
844:
837:
836:
828:
803:
771:
765:
737:
717:
696:
690:
659:
649:
636:
630:
607:
586:
585:
583:
563:
539:
519:
480:
457:
436:
426:
420:
394:
373:
367:
343:
330:
324:
298:
274:
261:
251:
241:
222:
209:
203:
176:
156:
135:
134:
132:
111:
105:
76:
75:
73:
49:
1383:{\displaystyle \omega _{f}=\omega _{g}}
1602:This means that for all vector fields
7:
1868:
1527:The fundamental theorem of calculus
587:
136:
77:
1390:, then there exists some constant
1182:
1178:
14:
2223:Lectures in differential geometry
2172:Generalizations of the derivative
1011:{\displaystyle x\in \mathbb {R} }
1462:{\displaystyle f(x)=C\cdot g(x)}
1876:{\displaystyle {\mathfrak {g}}}
1854:fundamental theorem of calculus
595:{\displaystyle {\mathfrak {g}}}
355:{\displaystyle X_{g}\in T_{g}G}
144:{\displaystyle {\mathfrak {g}}}
85:{\displaystyle {\mathfrak {g}}}
30:fundamental theorem of calculus
2049:
2005:
1807:
1804:
1791:
1775:
1762:
1749:
1743:
1717:
1708:
1698:
1580:
1568:
1456:
1450:
1435:
1429:
1307:
1244:
1238:
1125:
1109:
1094:
1088:
1082:
1069:
1063:
1052:
1049:
841:
491:
258:
234:
228:
215:
1:
2186: – Mathematical concept
1216:{\displaystyle \mathbb {R} }
983:{\displaystyle \mathbb {R} }
961:{\displaystyle \mathbb {R} }
915:{\displaystyle \mathbb {R} }
885:{\displaystyle \mathbb {R} }
2201:. Springer-Verlag, Berlin.
1829:For any Lie algebra-valued
1343:{\displaystyle \omega _{f}}
780:{\displaystyle \omega _{f}}
705:{\displaystyle \omega _{G}}
120:{\displaystyle \omega _{G}}
2271:
1316:{\displaystyle f,g:M\to G}
445:{\displaystyle T_{g}L_{g}}
1969:has an open neighborhood
2014:{\displaystyle f:U\to G}
1261:Uniqueness of primitives
500:{\displaystyle f:M\to G}
2088:{\displaystyle \omega }
1916:{\displaystyle \omega }
1323:are both primitives of
990:. In symbols, for each
193:Lie algebra valued form
2178:Logarithmic derivative
2156:
2136:
2109:
2089:
2066:
2015:
1983:
1963:
1962:{\displaystyle p\in M}
1937:
1917:
1897:
1877:
1843:
1820:
1682:
1681:{\displaystyle x\in G}
1656:
1636:
1616:
1593:
1513:
1489:
1488:{\displaystyle x\in M}
1463:
1410:
1409:{\displaystyle C\in G}
1384:
1344:
1317:
1279:
1251:
1217:
1195:
1156:
1012:
984:
962:
936:
916:
886:
853:
817:
781:
746:
726:
706:
672:
616:
596:
572:
548:
528:
501:
466:
446:
409:
408:{\displaystyle g\in G}
383:
356:
313:
312:{\displaystyle g\in G}
284:
185:
165:
145:
121:
86:
58:
2250:Differential calculus
2197:R. W. Sharpe (1996).
2157:
2137:
2110:
2090:
2067:
2016:
1984:
1964:
1938:
1918:
1898:
1878:
1844:
1821:
1683:
1657:
1637:
1617:
1594:
1514:
1490:
1464:
1411:
1385:
1345:
1318:
1280:
1252:
1250:{\displaystyle f'(x)}
1218:
1196:
1157:
1013:
985:
963:
937:
917:
887:
854:
818:
782:
747:
727:
707:
673:
617:
597:
573:
549:
529:
502:
467:
452:is its derivative at
447:
410:
384:
382:{\displaystyle L_{g}}
357:
314:
285:
186:
166:
146:
122:
87:
59:
2146:
2126:
2099:
2079:
2028:
1993:
1973:
1947:
1927:
1907:
1887:
1863:
1833:
1695:
1666:
1646:
1626:
1606:
1546:
1503:
1473:
1423:
1394:
1354:
1327:
1289:
1269:
1227:
1205:
1173:
1025:
994:
972:
950:
926:
904:
892:under addition. The
874:
827:
802:
764:
736:
716:
689:
629:
606:
582:
562:
538:
518:
479:
456:
419:
393:
366:
323:
297:
202:
175:
155:
131:
104:
72:
48:
1533:structural equation
1521:indefinite integral
1018:we look at the map
20:of a map between a
2184:Maurer–Cartan form
2152:
2132:
2105:
2085:
2062:
2011:
1979:
1959:
1933:
1913:
1893:
1873:
1839:
1816:
1678:
1652:
1632:
1612:
1589:
1537:Maurer-Cartan form
1509:
1485:
1459:
1406:
1380:
1340:
1313:
1285:is connected, and
1275:
1247:
1213:
1191:
1152:
1008:
980:
958:
932:
912:
882:
849:
816:{\displaystyle f'}
813:
777:
742:
722:
702:
668:
612:
592:
568:
556:Darboux derivative
544:
524:
497:
462:
442:
405:
379:
352:
309:
280:
181:
161:
141:
117:
98:Maurer-Cartan form
82:
54:
18:Darboux derivative
2227:. Prentice-Hall.
2155:{\displaystyle G}
2135:{\displaystyle M}
2108:{\displaystyle M}
1989:and a smooth map
1982:{\displaystyle U}
1936:{\displaystyle M}
1896:{\displaystyle 1}
1842:{\displaystyle 1}
1655:{\displaystyle G}
1635:{\displaystyle Y}
1615:{\displaystyle X}
1566:
1512:{\displaystyle C}
1278:{\displaystyle M}
1189:
935:{\displaystyle 0}
745:{\displaystyle f}
725:{\displaystyle f}
615:{\displaystyle 1}
571:{\displaystyle f}
547:{\displaystyle G}
527:{\displaystyle M}
465:{\displaystyle g}
184:{\displaystyle G}
164:{\displaystyle 1}
57:{\displaystyle G}
40:Formal definition
2262:
2236:
2226:
2217:Shlomo Sternberg
2212:
2161:
2159:
2158:
2153:
2141:
2139:
2138:
2133:
2114:
2112:
2111:
2106:
2094:
2092:
2091:
2086:
2071:
2069:
2068:
2063:
2058:
2057:
2052:
2040:
2039:
2020:
2018:
2017:
2012:
1988:
1986:
1985:
1980:
1968:
1966:
1965:
1960:
1942:
1940:
1939:
1934:
1922:
1920:
1919:
1914:
1902:
1900:
1899:
1894:
1882:
1880:
1879:
1874:
1872:
1871:
1848:
1846:
1845:
1840:
1825:
1823:
1822:
1817:
1803:
1802:
1790:
1789:
1774:
1773:
1761:
1760:
1742:
1741:
1729:
1728:
1716:
1715:
1687:
1685:
1684:
1679:
1661:
1659:
1658:
1653:
1641:
1639:
1638:
1633:
1621:
1619:
1618:
1613:
1598:
1596:
1595:
1590:
1567:
1559:
1518:
1516:
1515:
1510:
1494:
1492:
1491:
1486:
1468:
1466:
1465:
1460:
1415:
1413:
1412:
1407:
1389:
1387:
1386:
1381:
1379:
1378:
1366:
1365:
1349:
1347:
1346:
1341:
1339:
1338:
1322:
1320:
1319:
1314:
1284:
1282:
1281:
1276:
1265:If the manifold
1256:
1254:
1253:
1248:
1237:
1222:
1220:
1219:
1214:
1212:
1200:
1198:
1197:
1192:
1190:
1188:
1177:
1161:
1159:
1158:
1153:
1148:
1143:
1142:
1121:
1120:
1105:
1104:
1092:
1091:
1073:
1072:
1048:
1043:
1042:
1017:
1015:
1014:
1009:
1007:
989:
987:
986:
981:
979:
967:
965:
964:
959:
957:
941:
939:
938:
933:
921:
919:
918:
913:
911:
891:
889:
888:
883:
881:
858:
856:
855:
850:
848:
840:
822:
820:
819:
814:
812:
786:
784:
783:
778:
776:
775:
751:
749:
748:
743:
731:
729:
728:
723:
711:
709:
708:
703:
701:
700:
677:
675:
674:
669:
664:
663:
654:
653:
641:
640:
621:
619:
618:
613:
601:
599:
598:
593:
591:
590:
577:
575:
574:
569:
553:
551:
550:
545:
533:
531:
530:
525:
506:
504:
503:
498:
471:
469:
468:
463:
451:
449:
448:
443:
441:
440:
431:
430:
414:
412:
411:
406:
388:
386:
385:
380:
378:
377:
361:
359:
358:
353:
348:
347:
335:
334:
318:
316:
315:
310:
289:
287:
286:
281:
279:
278:
269:
268:
256:
255:
246:
245:
227:
226:
214:
213:
190:
188:
187:
182:
170:
168:
167:
162:
150:
148:
147:
142:
140:
139:
127:, is the smooth
126:
124:
123:
118:
116:
115:
91:
89:
88:
83:
81:
80:
63:
61:
60:
55:
2270:
2269:
2265:
2264:
2263:
2261:
2260:
2259:
2240:
2239:
2215:
2209:
2196:
2193:
2168:
2144:
2143:
2124:
2123:
2097:
2096:
2077:
2076:
2047:
2031:
2026:
2025:
1991:
1990:
1971:
1970:
1945:
1944:
1925:
1924:
1905:
1904:
1885:
1884:
1861:
1860:
1831:
1830:
1794:
1781:
1765:
1752:
1733:
1720:
1707:
1693:
1692:
1664:
1663:
1644:
1643:
1624:
1623:
1604:
1603:
1544:
1543:
1529:
1501:
1500:
1471:
1470:
1421:
1420:
1392:
1391:
1370:
1357:
1352:
1351:
1330:
1325:
1324:
1287:
1286:
1267:
1266:
1263:
1230:
1225:
1224:
1203:
1202:
1181:
1171:
1170:
1134:
1112:
1093:
1074:
1055:
1034:
1023:
1022:
992:
991:
970:
969:
948:
947:
942:, which is the
924:
923:
902:
901:
872:
871:
825:
824:
805:
800:
799:
793:
767:
762:
761:
734:
733:
714:
713:
692:
687:
686:
655:
645:
632:
627:
626:
604:
603:
580:
579:
560:
559:
536:
535:
516:
515:
513:smooth manifold
509:smooth function
477:
476:
454:
453:
432:
422:
417:
416:
391:
390:
369:
364:
363:
339:
326:
321:
320:
295:
294:
270:
257:
247:
237:
218:
205:
200:
199:
173:
172:
153:
152:
129:
128:
107:
102:
101:
70:
69:
46:
45:
42:
34:Stokes' theorem
12:
11:
5:
2268:
2266:
2258:
2257:
2252:
2242:
2241:
2238:
2237:
2213:
2207:
2192:
2189:
2188:
2187:
2181:
2175:
2167:
2164:
2151:
2131:
2104:
2084:
2073:
2072:
2061:
2056:
2051:
2046:
2043:
2038:
2034:
2010:
2007:
2004:
2001:
1998:
1978:
1958:
1955:
1952:
1932:
1912:
1892:
1870:
1838:
1827:
1826:
1815:
1812:
1809:
1806:
1801:
1797:
1793:
1788:
1784:
1780:
1777:
1772:
1768:
1764:
1759:
1755:
1751:
1748:
1745:
1740:
1736:
1732:
1727:
1723:
1719:
1714:
1710:
1706:
1703:
1700:
1677:
1674:
1671:
1651:
1631:
1611:
1600:
1599:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1565:
1562:
1557:
1554:
1551:
1528:
1525:
1508:
1499:This constant
1497:
1496:
1484:
1481:
1478:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1405:
1402:
1399:
1377:
1373:
1369:
1364:
1360:
1337:
1333:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1274:
1262:
1259:
1246:
1243:
1240:
1236:
1233:
1211:
1187:
1184:
1180:
1167:canonical unit
1163:
1162:
1151:
1147:
1141:
1137:
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978:
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931:
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894:tangent bundle
880:
847:
843:
839:
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832:
823:of a function
811:
808:
792:
789:
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741:
721:
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695:
679:
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611:
589:
578:is the smooth
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2208:0-387-94732-9
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2122:questions in
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1397:
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1331:
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1304:
1301:
1298:
1295:
1292:
1272:
1260:
1258:
1241:
1234:
1231:
1185:
1169:vector field
1168:
1149:
1139:
1135:
1131:
1128:
1122:
1117:
1113:
1106:
1101:
1098:
1085:
1079:
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1056:
1039:
1035:
1031:
1028:
1021:
1020:
1019:
1000:
997:
945:
929:
899:
895:
868:
866:
862:
833:
830:
809:
806:
798:
791:More natural?
790:
788:
772:
768:
759:
755:
752:is called an
739:
719:
697:
693:
684:
665:
660:
656:
650:
646:
642:
637:
633:
625:
624:
623:
609:
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541:
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510:
494:
488:
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482:
473:
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437:
433:
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423:
402:
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374:
370:
349:
344:
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336:
331:
327:
306:
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300:
275:
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265:
262:
252:
248:
242:
238:
231:
223:
219:
210:
206:
198:
197:
196:
195:) defined by
194:
178:
158:
112:
108:
99:
95:
67:
51:
39:
37:
35:
31:
27:
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19:
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1858:
1851:
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1264:
1164:
869:
864:
794:
757:
753:
680:
555:
474:
292:
43:
17:
15:
1416:such that
944:Lie algebra
554:. Then the
94:Lie algebra
2255:Lie groups
2244:Categories
2191:References
2021:such that
1852:The usual
1688:, we have
861:linear map
797:derivative
732:. The map
511:between a
68:, and let
2120:monodromy
2083:ω
2045:ω
2033:ω
2006:→
1954:∈
1911:ω
1783:ω
1754:ω
1705:ω
1673:∈
1578:ω
1572:ω
1553:ω
1480:∈
1445:⋅
1401:∈
1372:ω
1359:ω
1332:ω
1308:→
1183:∂
1179:∂
1132:∈
1107:∘
1099:−
1050:↦
1032:∈
1001:∈
898:Lie group
842:→
769:ω
758:primitive
694:ω
657:ω
651:∗
634:ω
492:→
400:∈
337:∈
304:∈
263:−
207:ω
171:-form on
109:ω
66:Lie group
26:Lie group
2166:See also
1883:-valued
1662:and all
1535:for the
1469:for all
1235:′
810:′
754:integral
683:pullback
602:-valued
293:for all
151:-valued
22:manifold
1350:, i.e.
896:of any
622:-form
362:. Here
92:be its
2233:529176
2231:
2205:
1903:-form
96:. The
24:and a
2075:i.e.
1859:If a
507:be a
191:(cf.
64:be a
2229:OCLC
2203:ISBN
2142:and
1622:and
1539:is:
1531:The
681:the
534:and
475:Let
415:and
319:and
44:Let
16:The
1923:on
1642:on
1201:on
946:of
865:and
760:of
756:or
712:by
685:of
558:of
2246::
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643::=
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235:(
232:=
229:)
224:g
220:X
216:(
211:G
179:G
159:1
137:g
113:G
78:g
52:G
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