1317:
1551:
222:
1123:
1116:
1682:
613:
724:
1128:
832:
935:
739:
Similar definitions can be given for other pairs of indices. As the term "part" suggests, a tensor is the sum of its symmetric part and antisymmetric part for a given pair of indices, as in
1397:
1361:
72:
944:
458:
1733:
1558:
501:
1392:
478:
415:
395:
367:
347:
313:
290:
268:
841:
A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions, for an order 2 covariant tensor
1312:{\displaystyle {\begin{aligned}M_{}&={\frac {1}{2!}}\,\delta _{ab}^{cd}M_{cd},\\T_{}&={\frac {1}{3!}}\,\delta _{abc}^{def}T_{def}.\end{aligned}}}
2313:
516:
742:
1892:
1857:
1830:
627:
848:
2497:
2176:
1956:
1913:
2378:
1698:
Trivially, all scalars and vectors (tensors of order 0 and 1) are totally antisymmetric (as well as being totally symmetric).
2229:
2161:
1368:
2254:
2492:
1364:
2303:
2123:
1747:
1322:
1975:
2477:
2559:
2431:
2138:
1686:
This decomposition is not in general true for tensors of rank 3 or more, which have more complex symmetries.
2529:
2216:
2133:
2103:
2487:
2343:
2298:
1743:
1702:
58:(+/−) when any two indices of the subset are interchanged. The index subset must generally either be all
2569:
2524:
2004:
1949:
1546:{\displaystyle T_{}={\frac {1}{p!}}\delta _{a_{1}\dots a_{p}}^{b_{1}\dots b_{p}}T_{b_{1}\dots b_{p}}.}
217:{\displaystyle T_{ijk\dots }=-T_{jik\dots }=T_{jki\dots }=-T_{kji\dots }=T_{kij\dots }=-T_{ikj\dots }}
2544:
2472:
2358:
2224:
2118:
1760:
2421:
2244:
2234:
2083:
2068:
2024:
1883:
427:
35:
1708:
1555:
In general, every tensor of rank 2 can be decomposed into a symmetric and anti-symmetric pair as:
2554:
2411:
2264:
2078:
2014:
1904:
1775:
370:
55:
1796: – process that converts any function in n variables to a symmetric function in n variables
17:
2549:
2457:
2318:
2293:
2108:
2019:
1999:
1909:
1888:
1853:
1847:
1826:
271:
2600:
2564:
2462:
2239:
2206:
2191:
2073:
1942:
1787:
1769:
1736:
2534:
2482:
2426:
2406:
2308:
2196:
2063:
2034:
1929:
483:
2574:
2539:
2519:
2436:
2269:
2259:
2249:
2171:
2143:
2128:
2113:
2029:
1793:
1781:
1377:
463:
400:
380:
352:
332:
298:
275:
253:
1111:{\displaystyle T_{}={\frac {1}{3!}}(T_{abc}-T_{acb}+T_{bca}-T_{bac}+T_{cab}-T_{cba}).}
2594:
2511:
2416:
2328:
2201:
1878:
1819:
294:, and a completely antisymmetric contravariant tensor field may be referred to as a
2579:
2383:
2368:
2333:
2181:
2166:
1677:{\displaystyle T_{ij}={\frac {1}{2}}(T_{ij}+T_{ji})+{\frac {1}{2}}(T_{ij}-T_{ji}).}
248:
244:
1374:
More generally, irrespective of the number of dimensions, antisymmetrization over
2467:
2441:
2363:
2052:
1991:
295:
224:
holds when the tensor is antisymmetric with respect to its first three indices.
31:
2348:
2323:
2274:
608:{\displaystyle U_{(ij)k\dots }={\frac {1}{2}}(U_{ijk\dots }+U_{jik\dots })}
2353:
2338:
2047:
2009:
719:{\displaystyle U_{k\dots }={\frac {1}{2}}(U_{ijk\dots }-U_{jik\dots })}
2373:
1965:
39:
1790: – Tensor invariant under permutations of vectors it acts on
1938:
1784: – Tensor index notation for tensor-based calculations
1934:
1778: – Antisymmetric permutation object acting on tensors
827:{\displaystyle U_{ijk\dots }=U_{(ij)k\dots }+U_{k\dots }.}
27:
Tensor equal to the negative of any of its transpositions
1765:
Pages displaying short descriptions of redirect targets
1711:
1561:
1400:
1380:
1325:
1126:
947:
851:
745:
630:
519:
486:
466:
430:
403:
383:
355:
335:
301:
278:
256:
75:
1798:
Pages displaying wikidata descriptions as a fallback
930:{\displaystyle M_{}={\frac {1}{2!}}(M_{ab}-M_{ba}),}
2510:
2450:
2399:
2392:
2284:
2215:
2152:
2096:
2043:
1990:
1983:
1120:In any 2 and 3 dimensions, these can be written as
1818:
1727:
1676:
1545:
1386:
1355:
1311:
1110:
929:
826:
718:
607:
506:has symmetric and antisymmetric parts defined as:
495:
472:
452:
409:
389:
361:
341:
307:
284:
262:
216:
1846:Juan Ramón Ruíz-Tolosa; Enrique Castillo (2005).
1821:Mathematical methods for physics and engineering
1908:. W.H. Freeman & Co. pp. 85–86, §3.5.
1950:
1902:J.A. Wheeler; C. Misner; K.S. Thorne (1973).
1356:{\displaystyle \delta _{ab\dots }^{cd\dots }}
8:
1930:Antisymmetric Tensor – mathworld.wolfram.com
1817:K.F. Riley; M.P. Hobson; S.J. Bence (2010).
1772: – Algebra of exterior/ wedge products
227:If a tensor changes sign under exchange of
2396:
1987:
1957:
1943:
1935:
1716:
1710:
1659:
1643:
1626:
1611:
1595:
1578:
1566:
1560:
1532:
1519:
1514:
1502:
1489:
1484:
1477:
1464:
1459:
1440:
1426:
1413:
1405:
1399:
1379:
1341:
1330:
1324:
1290:
1274:
1263:
1258:
1243:
1218:
1198:
1185:
1177:
1172:
1157:
1135:
1127:
1125:
1090:
1071:
1052:
1033:
1014:
995:
973:
952:
946:
912:
896:
874:
856:
850:
800:
772:
750:
744:
698:
676:
659:
635:
629:
587:
565:
548:
524:
518:
485:
465:
435:
429:
402:
382:
354:
334:
300:
277:
255:
199:
174:
152:
127:
105:
80:
74:
2314:Covariance and contravariance of vectors
243:. A completely antisymmetric covariant
231:pair of its indices, then the tensor is
1809:
1694:Totally antisymmetric tensors include:
7:
937:and for an order 3 covariant tensor
321:Antisymmetric and symmetric tensors
2177:Tensors in curvilinear coordinates
25:
329:that is antisymmetric on indices
18:Completely antisymmetric tensor
1825:. Cambridge University Press.
1668:
1636:
1620:
1588:
1432:
1406:
1231:
1219:
1145:
1136:
1102:
988:
965:
953:
921:
889:
866:
857:
810:
801:
782:
773:
713:
669:
645:
636:
602:
558:
534:
525:
1:
2230:Exterior covariant derivative
2162:Tensor (intrinsic definition)
1369:Einstein summation convention
453:{\displaystyle U_{ijk\dots }}
377:that is symmetric on indices
2255:Raising and lowering indices
1728:{\displaystyle F_{\mu \nu }}
1394:indices may be expressed as
2493:Gluon field strength tensor
1365:generalized Kronecker delta
2617:
2304:Cartan formalism (physics)
2124:Penrose graphical notation
1748:pseudo-Riemannian manifold
369:has the property that the
1976:Glossary of tensor theory
1972:
1852:. Springer. p. 225.
2560:Gregorio Ricci-Curbastro
2432:Riemann curvature tensor
2139:Van der Waerden notation
1763: – Form of a matrix
270:may be referred to as a
2530:Elwin Bruno Christoffel
2463:Angular momentum tensor
2134:Tetrad (index notation)
2104:Abstract index notation
1849:From Vectors to Tensors
2344:Levi-Civita connection
1744:Riemannian volume form
1729:
1703:electromagnetic tensor
1678:
1547:
1388:
1357:
1313:
1112:
931:
828:
731:(antisymmetric part).
720:
609:
497:
474:
460:and a pair of indices
454:
411:
391:
363:
343:
309:
286:
264:
218:
2570:Jan Arnoldus Schouten
2525:Augustin-Louis Cauchy
2005:Differential geometry
1730:
1679:
1548:
1389:
1358:
1314:
1113:
932:
829:
721:
610:
498:
475:
455:
420:For a general tensor
412:
392:
364:
344:
310:
287:
265:
219:
2545:Carl Friedrich Gauss
2478:stress–energy tensor
2473:Cauchy stress tensor
2225:Covariant derivative
2187:Antisymmetric tensor
2119:Multi-index notation
1761:Antisymmetric matrix
1709:
1559:
1398:
1378:
1323:
1124:
945:
849:
743:
628:
517:
484:
464:
428:
401:
381:
353:
333:
299:
276:
254:
73:
2422:Nonmetricity tensor
2277:(2nd-order tensors)
2245:Hodge star operator
2235:Exterior derivative
2084:Transport phenomena
2069:Continuum mechanics
2025:Multilinear algebra
1884:The Road to Reality
1509:
1352:
1285:
1193:
36:theoretical physics
2555:Tullio Levi-Civita
2498:Metric tensor (GR)
2412:Levi-Civita symbol
2265:Tensor contraction
2079:General relativity
2015:Euclidean geometry
1776:Levi-Civita symbol
1725:
1674:
1543:
1455:
1384:
1353:
1326:
1309:
1307:
1259:
1173:
1108:
927:
824:
716:
605:
496:{\displaystyle j,}
493:
470:
450:
417:is identically 0.
407:
387:
359:
339:
305:
282:
260:
214:
2588:
2587:
2550:Hermann Grassmann
2506:
2505:
2458:Moment of inertia
2319:Differential form
2294:Affine connection
2109:Einstein notation
2092:
2091:
2020:Exterior calculus
2000:Coordinate system
1894:978-0-679-77631-4
1887:. Vintage books.
1859:978-3-540-22887-5
1832:978-0-521-86153-3
1634:
1586:
1453:
1387:{\displaystyle p}
1256:
1170:
986:
887:
735:
734:
667:
620:(symmetric part)
556:
473:{\displaystyle i}
410:{\displaystyle j}
390:{\displaystyle i}
362:{\displaystyle j}
342:{\displaystyle i}
308:{\displaystyle k}
285:{\displaystyle k}
263:{\displaystyle k}
54:if it alternates
16:(Redirected from
2608:
2565:Bernhard Riemann
2397:
2240:Exterior product
2207:Two-point tensor
2192:Symmetric tensor
2074:Electromagnetism
1988:
1959:
1952:
1945:
1936:
1919:
1898:
1865:
1863:
1843:
1837:
1836:
1824:
1814:
1799:
1788:Symmetric tensor
1770:Exterior algebra
1766:
1737:electromagnetism
1734:
1732:
1731:
1726:
1724:
1723:
1683:
1681:
1680:
1675:
1667:
1666:
1651:
1650:
1635:
1627:
1619:
1618:
1603:
1602:
1587:
1579:
1574:
1573:
1552:
1550:
1549:
1544:
1539:
1538:
1537:
1536:
1524:
1523:
1508:
1507:
1506:
1494:
1493:
1483:
1482:
1481:
1469:
1468:
1454:
1452:
1441:
1436:
1435:
1431:
1430:
1418:
1417:
1393:
1391:
1390:
1385:
1362:
1360:
1359:
1354:
1351:
1340:
1318:
1316:
1315:
1310:
1308:
1301:
1300:
1284:
1273:
1257:
1255:
1244:
1235:
1234:
1206:
1205:
1192:
1184:
1171:
1169:
1158:
1149:
1148:
1117:
1115:
1114:
1109:
1101:
1100:
1082:
1081:
1063:
1062:
1044:
1043:
1025:
1024:
1006:
1005:
987:
985:
974:
969:
968:
936:
934:
933:
928:
920:
919:
904:
903:
888:
886:
875:
870:
869:
833:
831:
830:
825:
820:
819:
792:
791:
764:
763:
725:
723:
722:
717:
712:
711:
690:
689:
668:
660:
655:
654:
614:
612:
611:
606:
601:
600:
579:
578:
557:
549:
544:
543:
511:
510:
502:
500:
499:
494:
479:
477:
476:
471:
459:
457:
456:
451:
449:
448:
424:with components
416:
414:
413:
408:
396:
394:
393:
388:
368:
366:
365:
360:
348:
346:
345:
340:
314:
312:
311:
306:
291:
289:
288:
283:
269:
267:
266:
261:
223:
221:
220:
215:
213:
212:
188:
187:
166:
165:
141:
140:
119:
118:
94:
93:
44:antisymmetric on
21:
2616:
2615:
2611:
2610:
2609:
2607:
2606:
2605:
2591:
2590:
2589:
2584:
2535:Albert Einstein
2502:
2483:Einstein tensor
2446:
2427:Ricci curvature
2407:Kronecker delta
2393:Notable tensors
2388:
2309:Connection form
2286:
2280:
2211:
2197:Tensor operator
2154:
2148:
2088:
2064:Computer vision
2057:
2039:
2035:Tensor calculus
1979:
1968:
1963:
1926:
1916:
1901:
1895:
1877:
1874:
1869:
1868:
1860:
1845:
1844:
1840:
1833:
1816:
1815:
1811:
1806:
1797:
1764:
1757:
1712:
1707:
1706:
1692:
1655:
1639:
1607:
1591:
1562:
1557:
1556:
1528:
1515:
1510:
1498:
1485:
1473:
1460:
1445:
1422:
1409:
1401:
1396:
1395:
1376:
1375:
1321:
1320:
1306:
1305:
1286:
1248:
1236:
1214:
1211:
1210:
1194:
1162:
1150:
1131:
1122:
1121:
1086:
1067:
1048:
1029:
1010:
991:
978:
948:
943:
942:
908:
892:
879:
852:
847:
846:
839:
796:
768:
746:
741:
740:
694:
672:
631:
626:
625:
583:
561:
520:
515:
514:
482:
481:
462:
461:
431:
426:
425:
399:
398:
379:
378:
351:
350:
331:
330:
323:
297:
296:
274:
273:
252:
251:
195:
170:
148:
123:
101:
76:
71:
70:
52:an index subset
48:with respect to
28:
23:
22:
15:
12:
11:
5:
2614:
2612:
2604:
2603:
2593:
2592:
2586:
2585:
2583:
2582:
2577:
2575:Woldemar Voigt
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2540:Leonhard Euler
2537:
2532:
2527:
2522:
2516:
2514:
2512:Mathematicians
2508:
2507:
2504:
2503:
2501:
2500:
2495:
2490:
2485:
2480:
2475:
2470:
2465:
2460:
2454:
2452:
2448:
2447:
2445:
2444:
2439:
2437:Torsion tensor
2434:
2429:
2424:
2419:
2414:
2409:
2403:
2401:
2394:
2390:
2389:
2387:
2386:
2381:
2376:
2371:
2366:
2361:
2356:
2351:
2346:
2341:
2336:
2331:
2326:
2321:
2316:
2311:
2306:
2301:
2296:
2290:
2288:
2282:
2281:
2279:
2278:
2272:
2270:Tensor product
2267:
2262:
2260:Symmetrization
2257:
2252:
2250:Lie derivative
2247:
2242:
2237:
2232:
2227:
2221:
2219:
2213:
2212:
2210:
2209:
2204:
2199:
2194:
2189:
2184:
2179:
2174:
2172:Tensor density
2169:
2164:
2158:
2156:
2150:
2149:
2147:
2146:
2144:Voigt notation
2141:
2136:
2131:
2129:Ricci calculus
2126:
2121:
2116:
2114:Index notation
2111:
2106:
2100:
2098:
2094:
2093:
2090:
2089:
2087:
2086:
2081:
2076:
2071:
2066:
2060:
2058:
2056:
2055:
2050:
2044:
2041:
2040:
2038:
2037:
2032:
2030:Tensor algebra
2027:
2022:
2017:
2012:
2010:Dyadic algebra
2007:
2002:
1996:
1994:
1985:
1981:
1980:
1973:
1970:
1969:
1964:
1962:
1961:
1954:
1947:
1939:
1933:
1932:
1925:
1924:External links
1922:
1921:
1920:
1914:
1899:
1893:
1879:Penrose, Roger
1873:
1870:
1867:
1866:
1858:
1838:
1831:
1808:
1807:
1805:
1802:
1801:
1800:
1794:Symmetrization
1791:
1785:
1782:Ricci calculus
1779:
1773:
1767:
1756:
1753:
1752:
1751:
1740:
1722:
1719:
1715:
1699:
1691:
1688:
1673:
1670:
1665:
1662:
1658:
1654:
1649:
1646:
1642:
1638:
1633:
1630:
1625:
1622:
1617:
1614:
1610:
1606:
1601:
1598:
1594:
1590:
1585:
1582:
1577:
1572:
1569:
1565:
1542:
1535:
1531:
1527:
1522:
1518:
1513:
1505:
1501:
1497:
1492:
1488:
1480:
1476:
1472:
1467:
1463:
1458:
1451:
1448:
1444:
1439:
1434:
1429:
1425:
1421:
1416:
1412:
1408:
1404:
1383:
1350:
1347:
1344:
1339:
1336:
1333:
1329:
1304:
1299:
1296:
1293:
1289:
1283:
1280:
1277:
1272:
1269:
1266:
1262:
1254:
1251:
1247:
1242:
1239:
1237:
1233:
1230:
1227:
1224:
1221:
1217:
1213:
1212:
1209:
1204:
1201:
1197:
1191:
1188:
1183:
1180:
1176:
1168:
1165:
1161:
1156:
1153:
1151:
1147:
1144:
1141:
1138:
1134:
1130:
1129:
1107:
1104:
1099:
1096:
1093:
1089:
1085:
1080:
1077:
1074:
1070:
1066:
1061:
1058:
1055:
1051:
1047:
1042:
1039:
1036:
1032:
1028:
1023:
1020:
1017:
1013:
1009:
1004:
1001:
998:
994:
990:
984:
981:
977:
972:
967:
964:
961:
958:
955:
951:
926:
923:
918:
915:
911:
907:
902:
899:
895:
891:
885:
882:
878:
873:
868:
865:
862:
859:
855:
838:
835:
823:
818:
815:
812:
809:
806:
803:
799:
795:
790:
787:
784:
781:
778:
775:
771:
767:
762:
759:
756:
753:
749:
737:
736:
733:
732:
729:
726:
715:
710:
707:
704:
701:
697:
693:
688:
685:
682:
679:
675:
671:
666:
663:
658:
653:
650:
647:
644:
641:
638:
634:
622:
621:
618:
615:
604:
599:
596:
593:
590:
586:
582:
577:
574:
571:
568:
564:
560:
555:
552:
547:
542:
539:
536:
533:
530:
527:
523:
492:
489:
469:
447:
444:
441:
438:
434:
406:
386:
373:with a tensor
358:
338:
322:
319:
304:
281:
259:
211:
208:
205:
202:
198:
194:
191:
186:
183:
180:
177:
173:
169:
164:
161:
158:
155:
151:
147:
144:
139:
136:
133:
130:
126:
122:
117:
114:
111:
108:
104:
100:
97:
92:
89:
86:
83:
79:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2613:
2602:
2599:
2598:
2596:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2521:
2518:
2517:
2515:
2513:
2509:
2499:
2496:
2494:
2491:
2489:
2486:
2484:
2481:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2455:
2453:
2449:
2443:
2440:
2438:
2435:
2433:
2430:
2428:
2425:
2423:
2420:
2418:
2417:Metric tensor
2415:
2413:
2410:
2408:
2405:
2404:
2402:
2398:
2395:
2391:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2329:Exterior form
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2291:
2289:
2283:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2222:
2220:
2218:
2214:
2208:
2205:
2203:
2202:Tensor bundle
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2160:
2159:
2157:
2151:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2101:
2099:
2095:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2061:
2059:
2054:
2051:
2049:
2046:
2045:
2042:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1997:
1995:
1993:
1989:
1986:
1982:
1978:
1977:
1971:
1967:
1960:
1955:
1953:
1948:
1946:
1941:
1940:
1937:
1931:
1928:
1927:
1923:
1917:
1915:0-7167-0344-0
1911:
1907:
1906:
1900:
1896:
1890:
1886:
1885:
1880:
1876:
1875:
1871:
1861:
1855:
1851:
1850:
1842:
1839:
1834:
1828:
1823:
1822:
1813:
1810:
1803:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1762:
1759:
1758:
1754:
1749:
1745:
1741:
1738:
1720:
1717:
1713:
1704:
1700:
1697:
1696:
1695:
1689:
1687:
1684:
1671:
1663:
1660:
1656:
1652:
1647:
1644:
1640:
1631:
1628:
1623:
1615:
1612:
1608:
1604:
1599:
1596:
1592:
1583:
1580:
1575:
1570:
1567:
1563:
1553:
1540:
1533:
1529:
1525:
1520:
1516:
1511:
1503:
1499:
1495:
1490:
1486:
1478:
1474:
1470:
1465:
1461:
1456:
1449:
1446:
1442:
1437:
1427:
1423:
1419:
1414:
1410:
1402:
1381:
1372:
1370:
1366:
1348:
1345:
1342:
1337:
1334:
1331:
1327:
1302:
1297:
1294:
1291:
1287:
1281:
1278:
1275:
1270:
1267:
1264:
1260:
1252:
1249:
1245:
1240:
1238:
1228:
1225:
1222:
1215:
1207:
1202:
1199:
1195:
1189:
1186:
1181:
1178:
1174:
1166:
1163:
1159:
1154:
1152:
1142:
1139:
1132:
1118:
1105:
1097:
1094:
1091:
1087:
1083:
1078:
1075:
1072:
1068:
1064:
1059:
1056:
1053:
1049:
1045:
1040:
1037:
1034:
1030:
1026:
1021:
1018:
1015:
1011:
1007:
1002:
999:
996:
992:
982:
979:
975:
970:
962:
959:
956:
949:
940:
924:
916:
913:
909:
905:
900:
897:
893:
883:
880:
876:
871:
863:
860:
853:
844:
836:
834:
821:
816:
813:
807:
804:
797:
793:
788:
785:
779:
776:
769:
765:
760:
757:
754:
751:
747:
730:
727:
708:
705:
702:
699:
695:
691:
686:
683:
680:
677:
673:
664:
661:
656:
651:
648:
642:
639:
632:
624:
623:
619:
616:
597:
594:
591:
588:
584:
580:
575:
572:
569:
566:
562:
553:
550:
545:
540:
537:
531:
528:
521:
513:
512:
509:
508:
507:
505:
490:
487:
467:
445:
442:
439:
436:
432:
423:
418:
404:
384:
376:
372:
356:
336:
328:
320:
318:
316:
302:
293:
279:
272:differential
257:
250:
246:
242:
241:antisymmetric
238:
234:
230:
225:
209:
206:
203:
200:
196:
192:
189:
184:
181:
178:
175:
171:
167:
162:
159:
156:
153:
149:
145:
142:
137:
134:
131:
128:
124:
120:
115:
112:
109:
106:
102:
98:
95:
90:
87:
84:
81:
77:
69:For example,
67:
65:
64:contravariant
61:
57:
53:
49:
45:
41:
37:
33:
19:
2580:Hermann Weyl
2384:Vector space
2369:Pseudotensor
2334:Fiber bundle
2287:abstractions
2186:
2182:Mixed tensor
2167:Tensor field
1974:
1903:
1882:
1848:
1841:
1820:
1812:
1693:
1685:
1554:
1373:
1119:
938:
842:
840:
738:
503:
421:
419:
374:
326:
324:
245:tensor field
240:
236:
232:
228:
226:
68:
63:
59:
51:
47:
43:
29:
2520:Élie Cartan
2468:Spin tensor
2442:Weyl tensor
2400:Mathematics
2364:Multivector
2155:definitions
2053:Engineering
1992:Mathematics
1905:Gravitation
1864:section §7.
1371:is in use.
371:contraction
32:mathematics
2349:Linear map
2217:Operations
1872:References
1367:, and the
233:completely
2488:EM tensor
2324:Dimension
2275:Transpose
1721:ν
1718:μ
1653:−
1526:…
1496:…
1471:…
1457:δ
1420:…
1349:…
1338:…
1328:δ
1261:δ
1175:δ
1084:−
1046:−
1008:−
906:−
817:…
789:…
761:…
709:…
692:−
687:…
652:…
598:…
576:…
541:…
446:…
325:A tensor
210:…
193:−
185:…
163:…
146:−
138:…
116:…
99:−
91:…
60:covariant
2595:Category
2354:Manifold
2339:Geodesic
2097:Notation
1881:(2007).
1755:See also
1690:Examples
837:Notation
2601:Tensors
2451:Physics
2285:Related
2048:Physics
1966:Tensors
1363:is the
317:field.
315:-vector
237:totally
62:or all
2379:Vector
2374:Spinor
2359:Matrix
2153:Tensor
1912:
1891:
1856:
1829:
1319:where
728:
617:
40:tensor
2299:Basis
1984:Scope
1804:Notes
1746:on a
292:-form
249:order
1910:ISBN
1889:ISBN
1854:ISBN
1827:ISBN
1742:The
1701:The
480:and
397:and
349:and
235:(or
229:each
56:sign
46:(or
38:, a
34:and
1735:in
247:of
42:is
30:In
2597::
1705:,
941:,
845:,
239:)
66:.
50:)
1958:e
1951:t
1944:v
1918:.
1897:.
1862:.
1835:.
1750:.
1739:.
1714:F
1672:.
1669:)
1664:i
1661:j
1657:T
1648:j
1645:i
1641:T
1637:(
1632:2
1629:1
1624:+
1621:)
1616:i
1613:j
1609:T
1605:+
1600:j
1597:i
1593:T
1589:(
1584:2
1581:1
1576:=
1571:j
1568:i
1564:T
1541:.
1534:p
1530:b
1521:1
1517:b
1512:T
1504:p
1500:b
1491:1
1487:b
1479:p
1475:a
1466:1
1462:a
1450:!
1447:p
1443:1
1438:=
1433:]
1428:p
1424:a
1415:1
1411:a
1407:[
1403:T
1382:p
1346:d
1343:c
1335:b
1332:a
1303:.
1298:f
1295:e
1292:d
1288:T
1282:f
1279:e
1276:d
1271:c
1268:b
1265:a
1253:!
1250:3
1246:1
1241:=
1232:]
1229:c
1226:b
1223:a
1220:[
1216:T
1208:,
1203:d
1200:c
1196:M
1190:d
1187:c
1182:b
1179:a
1167:!
1164:2
1160:1
1155:=
1146:]
1143:b
1140:a
1137:[
1133:M
1106:.
1103:)
1098:a
1095:b
1092:c
1088:T
1079:b
1076:a
1073:c
1069:T
1065:+
1060:c
1057:a
1054:b
1050:T
1041:a
1038:c
1035:b
1031:T
1027:+
1022:b
1019:c
1016:a
1012:T
1003:c
1000:b
997:a
993:T
989:(
983:!
980:3
976:1
971:=
966:]
963:c
960:b
957:a
954:[
950:T
939:T
925:,
922:)
917:a
914:b
910:M
901:b
898:a
894:M
890:(
884:!
881:2
877:1
872:=
867:]
864:b
861:a
858:[
854:M
843:M
822:.
814:k
811:]
808:j
805:i
802:[
798:U
794:+
786:k
783:)
780:j
777:i
774:(
770:U
766:=
758:k
755:j
752:i
748:U
714:)
706:k
703:i
700:j
696:U
684:k
681:j
678:i
674:U
670:(
665:2
662:1
657:=
649:k
646:]
643:j
640:i
637:[
633:U
603:)
595:k
592:i
589:j
585:U
581:+
573:k
570:j
567:i
563:U
559:(
554:2
551:1
546:=
538:k
535:)
532:j
529:i
526:(
522:U
504:U
491:,
488:j
468:i
443:k
440:j
437:i
433:U
422:U
405:j
385:i
375:B
357:j
337:i
327:A
303:k
280:k
258:k
207:j
204:k
201:i
197:T
190:=
182:j
179:i
176:k
172:T
168:=
160:i
157:j
154:k
150:T
143:=
135:i
132:k
129:j
125:T
121:=
113:k
110:i
107:j
103:T
96:=
88:k
85:j
82:i
78:T
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
