2599:
1953:
2594:{\displaystyle {\tilde {D}}^{M}={\begin{pmatrix}D_{1111}&D_{1122}&D_{1133}&{\sqrt {2}}D_{1123}&{\sqrt {2}}D_{1113}&{\sqrt {2}}D_{1112}\\D_{2211}&D_{2222}&D_{2233}&{\sqrt {2}}D_{2223}&{\sqrt {2}}D_{2213}&{\sqrt {2}}D_{2212}\\D_{3311}&D_{3322}&D_{3333}&{\sqrt {2}}D_{3323}&{\sqrt {2}}D_{3313}&{\sqrt {2}}D_{3312}\\{\sqrt {2}}D_{2311}&{\sqrt {2}}D_{2322}&{\sqrt {2}}D_{2333}&2D_{2323}&2D_{2313}&2D_{2312}\\{\sqrt {2}}D_{1311}&{\sqrt {2}}D_{1322}&{\sqrt {2}}D_{1333}&2D_{1323}&2D_{1313}&2D_{1312}\\{\sqrt {2}}D_{1211}&{\sqrt {2}}D_{1222}&{\sqrt {2}}D_{1233}&2D_{1223}&2D_{1213}&2D_{1212}\\\end{pmatrix}}.}
1317:
25:
819:
1039:
416:
1476:
1830:
636:
643:
1620:
826:
240:
1330:
1627:
1278:
423:
2635:
with 81 components (3×3×3×3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be
1481:
814:{\displaystyle {\boldsymbol {\epsilon }}={\begin{bmatrix}\epsilon _{xx}&\epsilon _{xy}&\epsilon _{xz}\\\epsilon _{yx}&\epsilon _{yy}&\epsilon _{yz}\\\epsilon _{zx}&\epsilon _{zy}&\epsilon _{zz}\end{bmatrix}}.}
1194:
1034:{\displaystyle {\tilde {\epsilon }}=(\epsilon _{xx},\epsilon _{yy},\epsilon _{zz},\gamma _{yz},\gamma _{xz},\gamma _{xy})\equiv (\epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon _{4},\epsilon _{5},\epsilon _{6}),}
411:{\displaystyle {\boldsymbol {\sigma }}={\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{bmatrix}}.}
2640:
by a 6×6 matrix. However, Voigt's form does not preserve the sum of the squares, which in the case of Hooke's law has geometric significance. This explains why weights are introduced (to make the mapping an
1471:{\displaystyle {\boldsymbol {\sigma }}={\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}}
1825:{\displaystyle {\tilde {\sigma }}:{\tilde {\sigma }}={\tilde {\sigma }}^{M}\cdot {\tilde {\sigma }}^{M}=\sigma _{11}^{2}+\sigma _{22}^{2}+\sigma _{33}^{2}+2\sigma _{23}^{2}+2\sigma _{13}^{2}+2\sigma _{12}^{2}.}
230:
631:{\displaystyle {\tilde {\sigma }}=(\sigma _{xx},\sigma _{yy},\sigma _{zz},\sigma _{yz},\sigma _{xz},\sigma _{xy})\equiv (\sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5},\sigma _{6}).}
1186:
1137:
1088:
161:. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application.
1948:
1890:
1615:{\displaystyle {\tilde {\sigma }}^{M}=\langle \sigma _{11},\sigma _{22},\sigma _{33},{\sqrt {2}}\sigma _{23},{\sqrt {2}}\sigma _{13},{\sqrt {2}}\sigma _{12}\rangle .}
1478:
only six components are distinct, the three on the diagonal and the others being off-diagonal. Thus it can be expressed, in Mandel notation, as the vector
2817:
Peter
Helnwein (February 16, 2001). "Some Remarks on the Compressed Matrix Representation of Symmetric Second-Order and Fourth-Order Tensors".
1273:{\displaystyle {\boldsymbol {\sigma }}\cdot {\boldsymbol {\epsilon }}=\sigma _{ij}\epsilon _{ij}={\tilde {\sigma }}\cdot {\tilde {\epsilon }}}
173:
1950:
has 81 components in three-dimensional space, but only 36 components are distinct. Thus, in Mandel notation, it can be expressed as
3227:
170:
has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus it can be expressed as the vector
640:
The strain tensor, similar in nature to the stress tensor—both are symmetric second-order tensors --, is given in matrix form as
3376:
1624:
The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors, for example:
2801:
2768:
89:
2616:. It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalized
61:
3411:
3090:
68:
2870:
2716:
108:
3292:
1312:
Voigt indexes are numbered consecutively from the starting point to the end (in the example, the numbers in blue).
1142:
1093:
1044:
42:
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46:
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3168:
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A discussion of invariance of Voigt's notation and Mandel's notation can be found in
Helnwein (2001).
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3371:
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3391:
3345:
3052:
1191:
The benefit of using different representations for stress and strain is that the scalar invariance
3443:
3130:
3047:
3017:
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1283:
Likewise, a three-dimensional symmetric fourth-order tensor can be reduced to a 6×6 matrix.
35:
3401:
3257:
3212:
2784:
Maher
Moakher (2009). "The Algebra of Fourth-Order Tensors with Application to Diffusion MRI".
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2918:
2863:
82:
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8:
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3207:
3022:
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2913:
2797:
2764:
2745:
2712:
1299:
Write down the second order tensor in matrix form (in the example, the stress tensor)
141:
by reducing its order. There are a few variants and associated names for this idea:
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3120:
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3183:
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3085:
3042:
3027:
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2732:
Jean Mandel (1965). "Généralisation de la théorie de plasticité de WT Koiter".
2662:
2617:
2609:
1316:
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3115:
2788:. Mathematics and Visualization. Springer Berlin Heidelberg. pp. 57–80.
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24:
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420:
In Voigt notation it is simplified to a 6-dimensional vector:
2848:
1984:
1347:
660:
257:
2819:
Computer
Methods in Applied Mechanics and Engineering
2761:
The Finite
Element Method: Its Basis and Fundamentals
1956:
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1096:
1047:
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426:
243:
225:{\displaystyle \langle x_{11},x_{22},x_{12}\rangle .}
176:
237:The stress tensor (in matrix notation) is given as
49:. Unsourced material may be challenged and removed.
2709:Foundations of anisotropy for exploration seismics
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224:
1308:Go back to the first element along the first row.
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2759:O.C. Zienkiewicz; R.L. Taylor; J.Z. Zhu (2005).
2816:
2763:(6 ed.). Elsevier Butterworth—Heinemann.
2734:International Journal of Solids and Structures
2681:
2864:
2786:Visualization and Processing of Tensor Fields
2783:
1295:for memorizing Voigt notation is as follows:
2706:
1606:
1507:
1181:{\displaystyle \gamma _{zx}=2\epsilon _{zx}}
1132:{\displaystyle \gamma _{yz}=2\epsilon _{yz}}
1083:{\displaystyle \gamma _{xy}=2\epsilon _{xy}}
216:
177:
2731:
1834:A symmetric tensor of rank four satisfying
2871:
2857:
2631:Hooke's law has a symmetric fourth-order
164:For example, a 2×2 symmetric tensor
109:Learn how and when to remove this message
3228:Covariance and contravariance of vectors
823:Its representation in Voigt notation is
2810:
1335:
1207:
1199:
648:
245:
157:is a revival by Helbig of old ideas of
3507:
2608:The notation is named after physicist
1327:For a symmetric tensor of second rank
2852:
2700:
47:adding citations to reliable sources
18:
13:
3091:Tensors in curvilinear coordinates
1322:
14:
3536:
1943:{\displaystyle D_{ijkl}=D_{ijlk}}
1885:{\displaystyle D_{ijkl}=D_{jikl}}
1315:
1286:
23:
2603:
1188:are engineering shear strains.
34:needs additional citations for
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1:
3144:Exterior covariant derivative
3076:Tensor (intrinsic definition)
2839:10.1016/s0045-7825(00)00263-2
2668:
3169:Raising and lowering indices
2746:10.1016/0020-7683(65)90034-x
1305:Continue on the third column
7:
3407:Gluon field strength tensor
2878:
2794:10.1007/978-3-540-88378-4_4
2685:Lehrbuch der Kristallphysik
2658:Vectorization (mathematics)
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10:
3541:
3218:Cartan formalism (physics)
3038:Penrose graphical notation
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3364:
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3010:
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2890:Glossary of tensor theory
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3474:Gregorio Ricci-Curbastro
3346:Riemann curvature tensor
3053:Van der Waerden notation
137:is a way to represent a
3444:Elwin Bruno Christoffel
3377:Angular momentum tensor
3048:Tetrad (index notation)
3018:Abstract index notation
2682:Woldemar Voigt (1910).
2622:finite element analysis
1302:Strike out the diagonal
3258:Levi-Civita connection
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1944:
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1472:
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1133:
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632:
412:
226:
3520:Mathematical notation
3484:Jan Arnoldus Schouten
3439:Augustin-Louis Cauchy
2919:Differential geometry
2707:Klaus Helbig (1994).
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1945:
1887:
1827:
1617:
1473:
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1183:
1134:
1085:
1036:
816:
633:
413:
227:
147:Mandel–Voigt notation
3459:Carl Friedrich Gauss
3392:stress–energy tensor
3387:Cauchy stress tensor
3139:Covariant derivative
3101:Antisymmetric tensor
3033:Multi-index notation
2825:(22–23): 2753–2770.
2614:John Nye (scientist)
1954:
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1838:
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1094:
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827:
644:
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241:
234:As another example:
174:
43:improve this article
16:Mathematical Concept
3336:Nonmetricity tensor
3191:(2nd-order tensors)
3159:Hodge star operator
3149:Exterior derivative
2998:Transport phenomena
2983:Continuum mechanics
2939:Multilinear algebra
2831:2001CMAME.190.2753H
1818:
1797:
1776:
1755:
1737:
1719:
135:multilinear algebra
3469:Tullio Levi-Civita
3412:Metric tensor (GR)
3326:Levi-Civita symbol
3179:Tensor contraction
2993:General relativity
2929:Euclidean geometry
2688:. Teubner, Leipzig
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1804:
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222:
153:are others found.
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3501:
3464:Hermann Grassmann
3420:
3419:
3372:Moment of inertia
3233:Differential form
3208:Affine connection
3023:Einstein notation
3006:
3005:
2934:Exterior calculus
2914:Coordinate system
2803:978-3-540-88377-7
2770:978-0-7506-6431-8
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3479:Bernhard Riemann
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3154:Exterior product
3121:Two-point tensor
3106:Symmetric tensor
2988:Electromagnetism
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2633:stiffness tensor
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139:symmetric tensor
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58:"Voigt notation"
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3525:Solid mechanics
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3449:Albert Einstein
3416:
3397:Einstein tensor
3360:
3341:Ricci curvature
3321:Kronecker delta
3307:Notable tensors
3302:
3223:Connection form
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3194:
3125:
3111:Tensor operator
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2978:Computer vision
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2949:Tensor calculus
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1323:Mandel notation
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1116:
1101:
1097:
1095:
1092:
1091:
1071:
1067:
1052:
1048:
1046:
1043:
1042:
1019:
1015:
1006:
1002:
993:
989:
980:
976:
967:
963:
954:
950:
932:
928:
916:
912:
900:
896:
884:
880:
868:
864:
852:
848:
831:
830:
828:
825:
824:
801:
800:
791:
787:
785:
776:
772:
770:
761:
757:
754:
753:
744:
740:
738:
729:
725:
723:
714:
710:
707:
706:
697:
693:
691:
682:
678:
676:
667:
663:
656:
655:
647:
645:
642:
641:
616:
612:
603:
599:
590:
586:
577:
573:
564:
560:
551:
547:
529:
525:
513:
509:
497:
493:
481:
477:
465:
461:
449:
445:
428:
427:
425:
422:
421:
398:
397:
388:
384:
382:
373:
369:
367:
358:
354:
351:
350:
341:
337:
335:
326:
322:
320:
311:
307:
304:
303:
294:
290:
288:
279:
275:
273:
264:
260:
253:
252:
244:
242:
239:
238:
210:
206:
197:
193:
184:
180:
175:
172:
171:
155:Kelvin notation
143:Mandel notation
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3538:
3528:
3527:
3522:
3517:
3500:
3499:
3497:
3496:
3491:
3489:Woldemar Voigt
3486:
3481:
3476:
3471:
3466:
3461:
3456:
3454:Leonhard Euler
3451:
3446:
3441:
3436:
3430:
3428:
3426:Mathematicians
3422:
3421:
3418:
3417:
3415:
3414:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3368:
3366:
3362:
3361:
3359:
3358:
3353:
3351:Torsion tensor
3348:
3343:
3338:
3333:
3328:
3323:
3317:
3315:
3308:
3304:
3303:
3301:
3300:
3295:
3290:
3285:
3280:
3275:
3270:
3265:
3260:
3255:
3250:
3245:
3240:
3235:
3230:
3225:
3220:
3215:
3210:
3204:
3202:
3196:
3195:
3193:
3192:
3186:
3184:Tensor product
3181:
3176:
3174:Symmetrization
3171:
3166:
3164:Lie derivative
3161:
3156:
3151:
3146:
3141:
3135:
3133:
3127:
3126:
3124:
3123:
3118:
3113:
3108:
3103:
3098:
3093:
3088:
3086:Tensor density
3083:
3078:
3072:
3070:
3064:
3063:
3061:
3060:
3058:Voigt notation
3055:
3050:
3045:
3043:Ricci calculus
3040:
3035:
3030:
3028:Index notation
3025:
3020:
3014:
3012:
3008:
3007:
3004:
3003:
3001:
3000:
2995:
2990:
2985:
2980:
2974:
2972:
2970:
2969:
2964:
2958:
2955:
2954:
2952:
2951:
2946:
2944:Tensor algebra
2941:
2936:
2931:
2926:
2924:Dyadic algebra
2921:
2916:
2910:
2908:
2899:
2895:
2894:
2887:
2884:
2883:
2876:
2875:
2868:
2861:
2853:
2845:
2844:
2809:
2802:
2776:
2769:
2751:
2740:(3): 273–295.
2724:
2717:
2699:
2673:
2672:
2670:
2667:
2666:
2665:
2660:
2653:
2650:
2610:Woldemar Voigt
2605:
2602:
2590:
2585:
2577:
2573:
2569:
2566:
2562:
2558:
2554:
2551:
2547:
2543:
2539:
2536:
2532:
2528:
2522:
2517:
2513:
2509:
2503:
2498:
2494:
2490:
2484:
2479:
2478:
2473:
2469:
2465:
2462:
2458:
2454:
2450:
2447:
2443:
2439:
2435:
2432:
2428:
2424:
2418:
2413:
2409:
2405:
2399:
2394:
2390:
2386:
2380:
2375:
2374:
2369:
2365:
2361:
2358:
2354:
2350:
2346:
2343:
2339:
2335:
2331:
2328:
2324:
2320:
2314:
2309:
2305:
2301:
2295:
2290:
2286:
2282:
2276:
2271:
2270:
2265:
2261:
2255:
2250:
2246:
2242:
2236:
2231:
2227:
2223:
2217:
2212:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2175:
2170:
2166:
2160:
2155:
2151:
2147:
2141:
2136:
2132:
2128:
2122:
2117:
2113:
2109:
2105:
2101:
2097:
2093:
2089:
2085:
2081:
2080:
2075:
2071:
2065:
2060:
2056:
2052:
2046:
2041:
2037:
2033:
2027:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1994:
1990:
1986:
1985:
1983:
1978:
1973:
1966:
1963:
1937:
1934:
1931:
1928:
1924:
1920:
1915:
1912:
1909:
1906:
1902:
1879:
1876:
1873:
1870:
1866:
1862:
1857:
1854:
1851:
1848:
1844:
1821:
1816:
1811:
1807:
1803:
1800:
1795:
1790:
1786:
1782:
1779:
1774:
1769:
1765:
1761:
1758:
1753:
1748:
1744:
1740:
1735:
1730:
1726:
1722:
1717:
1712:
1708:
1704:
1699:
1692:
1689:
1682:
1677:
1670:
1667:
1660:
1654:
1651:
1645:
1639:
1636:
1611:
1608:
1603:
1599:
1593:
1588:
1583:
1579:
1573:
1568:
1563:
1559:
1553:
1548:
1543:
1539:
1535:
1530:
1526:
1522:
1517:
1513:
1509:
1506:
1501:
1494:
1491:
1465:
1457:
1453:
1449:
1445:
1441:
1437:
1433:
1429:
1425:
1424:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1386:
1381:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1349:
1348:
1346:
1341:
1337:
1324:
1321:
1310:
1309:
1306:
1303:
1300:
1288:
1285:
1280:is preserved.
1266:
1263:
1257:
1251:
1248:
1242:
1237:
1234:
1230:
1224:
1221:
1217:
1213:
1209:
1205:
1201:
1175:
1172:
1168:
1164:
1161:
1156:
1153:
1149:
1126:
1123:
1119:
1115:
1112:
1107:
1104:
1100:
1077:
1074:
1070:
1066:
1063:
1058:
1055:
1051:
1030:
1027:
1022:
1018:
1014:
1009:
1005:
1001:
996:
992:
988:
983:
979:
975:
970:
966:
962:
957:
953:
949:
946:
943:
938:
935:
931:
927:
922:
919:
915:
911:
906:
903:
899:
895:
890:
887:
883:
879:
874:
871:
867:
863:
858:
855:
851:
847:
844:
838:
835:
810:
805:
797:
794:
790:
786:
782:
779:
775:
771:
767:
764:
760:
756:
755:
750:
747:
743:
739:
735:
732:
728:
724:
720:
717:
713:
709:
708:
703:
700:
696:
692:
688:
685:
681:
677:
673:
670:
666:
662:
661:
659:
654:
650:
627:
624:
619:
615:
611:
606:
602:
598:
593:
589:
585:
580:
576:
572:
567:
563:
559:
554:
550:
546:
543:
540:
535:
532:
528:
524:
519:
516:
512:
508:
503:
500:
496:
492:
487:
484:
480:
476:
471:
468:
464:
460:
455:
452:
448:
444:
441:
435:
432:
407:
402:
394:
391:
387:
383:
379:
376:
372:
368:
364:
361:
357:
353:
352:
347:
344:
340:
336:
332:
329:
325:
321:
317:
314:
310:
306:
305:
300:
297:
293:
289:
285:
282:
278:
274:
270:
267:
263:
259:
258:
256:
251:
247:
221:
218:
213:
209:
205:
200:
196:
192:
187:
183:
179:
127:Voigt notation
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3537:
3526:
3523:
3521:
3518:
3516:
3513:
3512:
3510:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3470:
3467:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3431:
3429:
3427:
3423:
3413:
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3369:
3367:
3363:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3331:Metric tensor
3329:
3327:
3324:
3322:
3319:
3318:
3316:
3312:
3309:
3305:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3249:
3246:
3244:
3243:Exterior form
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3205:
3203:
3197:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3136:
3134:
3132:
3128:
3122:
3119:
3117:
3116:Tensor bundle
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3092:
3089:
3087:
3084:
3082:
3079:
3077:
3074:
3073:
3071:
3065:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3015:
3013:
3009:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2975:
2973:
2968:
2965:
2963:
2960:
2959:
2956:
2950:
2947:
2945:
2942:
2940:
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2911:
2909:
2907:
2903:
2900:
2896:
2892:
2891:
2885:
2881:
2874:
2869:
2867:
2862:
2860:
2855:
2854:
2851:
2840:
2836:
2832:
2828:
2824:
2820:
2813:
2805:
2799:
2795:
2791:
2787:
2780:
2772:
2766:
2762:
2755:
2747:
2743:
2739:
2735:
2728:
2720:
2718:0-08-037224-4
2714:
2710:
2703:
2687:
2686:
2678:
2674:
2664:
2661:
2659:
2656:
2655:
2649:
2646:
2644:
2639:
2634:
2629:
2627:
2626:Diffusion MRI
2623:
2620:, as well as
2619:
2615:
2611:
2601:
2588:
2583:
2575:
2571:
2567:
2560:
2556:
2552:
2545:
2541:
2537:
2530:
2526:
2520:
2511:
2507:
2501:
2492:
2488:
2482:
2471:
2467:
2463:
2456:
2452:
2448:
2441:
2437:
2433:
2426:
2422:
2416:
2407:
2403:
2397:
2388:
2384:
2378:
2367:
2363:
2359:
2352:
2348:
2344:
2337:
2333:
2329:
2322:
2318:
2312:
2303:
2299:
2293:
2284:
2280:
2274:
2263:
2259:
2253:
2244:
2240:
2234:
2225:
2221:
2215:
2206:
2202:
2194:
2190:
2182:
2178:
2168:
2164:
2158:
2149:
2145:
2139:
2130:
2126:
2120:
2111:
2107:
2099:
2095:
2087:
2083:
2073:
2069:
2063:
2054:
2050:
2044:
2035:
2031:
2025:
2016:
2012:
2004:
2000:
1992:
1988:
1981:
1976:
1971:
1961:
1935:
1932:
1929:
1926:
1922:
1918:
1913:
1910:
1907:
1904:
1900:
1877:
1874:
1871:
1868:
1864:
1860:
1855:
1852:
1849:
1846:
1842:
1832:
1819:
1814:
1809:
1805:
1801:
1798:
1793:
1788:
1784:
1780:
1777:
1772:
1767:
1763:
1759:
1756:
1751:
1746:
1742:
1738:
1733:
1728:
1724:
1720:
1715:
1710:
1706:
1702:
1697:
1687:
1680:
1675:
1665:
1658:
1649:
1643:
1634:
1622:
1609:
1601:
1597:
1591:
1586:
1581:
1577:
1571:
1566:
1561:
1557:
1551:
1546:
1541:
1537:
1533:
1528:
1524:
1520:
1515:
1511:
1504:
1499:
1489:
1463:
1455:
1451:
1443:
1439:
1431:
1427:
1417:
1413:
1405:
1401:
1393:
1389:
1379:
1375:
1367:
1363:
1355:
1351:
1344:
1339:
1320:
1318:
1313:
1307:
1304:
1301:
1298:
1297:
1296:
1294:
1293:mnemonic rule
1287:Mnemonic rule
1284:
1281:
1261:
1255:
1246:
1240:
1235:
1232:
1228:
1222:
1219:
1215:
1211:
1203:
1189:
1173:
1170:
1166:
1162:
1159:
1154:
1151:
1147:
1124:
1121:
1117:
1113:
1110:
1105:
1102:
1098:
1075:
1072:
1068:
1064:
1061:
1056:
1053:
1049:
1028:
1020:
1016:
1012:
1007:
1003:
999:
994:
990:
986:
981:
977:
973:
968:
964:
960:
955:
951:
944:
936:
933:
929:
925:
920:
917:
913:
909:
904:
901:
897:
893:
888:
885:
881:
877:
872:
869:
865:
861:
856:
853:
849:
842:
833:
821:
808:
803:
795:
792:
788:
780:
777:
773:
765:
762:
758:
748:
745:
741:
733:
730:
726:
718:
715:
711:
701:
698:
694:
686:
683:
679:
671:
668:
664:
657:
652:
638:
625:
617:
613:
609:
604:
600:
596:
591:
587:
583:
578:
574:
570:
565:
561:
557:
552:
548:
541:
533:
530:
526:
522:
517:
514:
510:
506:
501:
498:
494:
490:
485:
482:
478:
474:
469:
466:
462:
458:
453:
450:
446:
439:
430:
418:
405:
400:
392:
389:
385:
377:
374:
370:
362:
359:
355:
345:
342:
338:
330:
327:
323:
315:
312:
308:
298:
295:
291:
283:
280:
276:
268:
265:
261:
254:
249:
235:
232:
219:
211:
207:
203:
198:
194:
190:
185:
181:
169:
168:
162:
160:
156:
152:
148:
144:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3494:Hermann Weyl
3298:Vector space
3283:Pseudotensor
3248:Fiber bundle
3201:abstractions
3096:Mixed tensor
3081:Tensor field
3057:
2888:
2822:
2818:
2812:
2785:
2779:
2760:
2754:
2737:
2733:
2727:
2711:. Pergamon.
2708:
2702:
2692:November 29,
2690:. Retrieved
2684:
2677:
2647:
2637:
2630:
2607:
2604:Applications
1833:
1623:
1326:
1314:
1311:
1290:
1282:
1190:
822:
639:
419:
236:
233:
166:
165:
163:
154:
151:Nye notation
150:
146:
142:
130:
126:
120:
105:
99:October 2016
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3434:Élie Cartan
3382:Spin tensor
3356:Weyl tensor
3314:Mathematics
3278:Multivector
3069:definitions
2967:Engineering
2906:Mathematics
2663:Hooke's law
2638:represented
2618:Hooke's law
159:Lord Kelvin
123:mathematics
3509:Categories
3263:Linear map
3131:Operations
2669:References
131:Voigt form
69:newspapers
3402:EM tensor
3238:Dimension
3189:Transpose
1965:~
1806:σ
1785:σ
1764:σ
1743:σ
1725:σ
1707:σ
1691:~
1688:σ
1681:⋅
1669:~
1666:σ
1653:~
1650:σ
1638:~
1635:σ
1607:⟩
1598:σ
1578:σ
1558:σ
1538:σ
1525:σ
1512:σ
1508:⟨
1493:~
1490:σ
1452:σ
1440:σ
1428:σ
1414:σ
1402:σ
1390:σ
1376:σ
1364:σ
1352:σ
1336:σ
1291:A simple
1265:~
1262:ϵ
1256:⋅
1250:~
1247:σ
1229:ϵ
1216:σ
1208:ϵ
1204:⋅
1200:σ
1167:ϵ
1148:γ
1118:ϵ
1099:γ
1069:ϵ
1050:γ
1017:ϵ
1004:ϵ
991:ϵ
978:ϵ
965:ϵ
952:ϵ
945:≡
930:γ
914:γ
898:γ
882:ϵ
866:ϵ
850:ϵ
837:~
834:ϵ
789:ϵ
774:ϵ
759:ϵ
742:ϵ
727:ϵ
712:ϵ
695:ϵ
680:ϵ
665:ϵ
649:ϵ
614:σ
601:σ
588:σ
575:σ
562:σ
549:σ
542:≡
527:σ
511:σ
495:σ
479:σ
463:σ
447:σ
434:~
431:σ
386:σ
371:σ
356:σ
339:σ
324:σ
309:σ
292:σ
277:σ
262:σ
246:σ
217:⟩
178:⟨
3268:Manifold
3253:Geodesic
3011:Notation
2652:See also
2643:isometry
3515:Tensors
3365:Physics
3199:Related
2962:Physics
2880:Tensors
2827:Bibcode
83:scholar
3293:Vector
3288:Spinor
3273:Matrix
3067:Tensor
2800:
2767:
2715:
2624:, and
2612:&
1139:, and
1041:where
85:
78:
71:
64:
56:
3213:Basis
2898:Scope
90:JSTOR
76:books
2798:ISBN
2765:ISBN
2713:ISBN
2694:2016
2576:1212
2561:1213
2546:1223
2531:1233
2512:1222
2493:1211
2472:1312
2457:1313
2442:1323
2427:1333
2408:1322
2389:1311
2368:2312
2353:2313
2338:2323
2323:2333
2304:2322
2285:2311
2264:3312
2245:3313
2226:3323
2207:3333
2195:3322
2183:3311
2169:2212
2150:2213
2131:2223
2112:2233
2100:2222
2088:2211
2074:1112
2055:1113
2036:1123
2017:1133
2005:1122
1993:1111
1892:and
149:and
62:news
2835:doi
2823:190
2790:doi
2742:doi
2645:).
133:in
129:or
121:In
45:by
3511::
2833:.
2821:.
2796:.
2736:.
2628:.
1810:12
1789:13
1768:23
1747:33
1729:22
1711:11
1602:12
1582:13
1562:23
1542:33
1529:22
1516:11
1456:33
1444:32
1432:31
1418:23
1406:22
1394:21
1380:13
1368:12
1356:11
1090:,
212:12
199:22
186:11
145:,
125:,
2872:e
2865:t
2858:v
2841:.
2837::
2829::
2806:.
2792::
2773:.
2748:.
2744::
2738:1
2721:.
2696:.
2589:.
2584:)
2572:D
2568:2
2557:D
2553:2
2542:D
2538:2
2527:D
2521:2
2508:D
2502:2
2489:D
2483:2
2468:D
2464:2
2453:D
2449:2
2438:D
2434:2
2423:D
2417:2
2404:D
2398:2
2385:D
2379:2
2364:D
2360:2
2349:D
2345:2
2334:D
2330:2
2319:D
2313:2
2300:D
2294:2
2281:D
2275:2
2260:D
2254:2
2241:D
2235:2
2222:D
2216:2
2203:D
2191:D
2179:D
2165:D
2159:2
2146:D
2140:2
2127:D
2121:2
2108:D
2096:D
2084:D
2070:D
2064:2
2051:D
2045:2
2032:D
2026:2
2013:D
2001:D
1989:D
1982:(
1977:=
1972:M
1962:D
1936:k
1933:l
1930:j
1927:i
1923:D
1919:=
1914:l
1911:k
1908:j
1905:i
1901:D
1878:l
1875:k
1872:i
1869:j
1865:D
1861:=
1856:l
1853:k
1850:j
1847:i
1843:D
1820:.
1815:2
1802:2
1799:+
1794:2
1781:2
1778:+
1773:2
1760:2
1757:+
1752:2
1739:+
1734:2
1721:+
1716:2
1703:=
1698:M
1676:M
1659:=
1644::
1610:.
1592:2
1587:,
1572:2
1567:,
1552:2
1547:,
1534:,
1521:,
1505:=
1500:M
1464:]
1345:[
1340:=
1241:=
1236:j
1233:i
1223:j
1220:i
1212:=
1174:x
1171:z
1163:2
1160:=
1155:x
1152:z
1125:z
1122:y
1114:2
1111:=
1106:z
1103:y
1076:y
1073:x
1065:2
1062:=
1057:y
1054:x
1029:,
1026:)
1021:6
1013:,
1008:5
1000:,
995:4
987:,
982:3
974:,
969:2
961:,
956:1
948:(
942:)
937:y
934:x
926:,
921:z
918:x
910:,
905:z
902:y
894:,
889:z
886:z
878:,
873:y
870:y
862:,
857:x
854:x
846:(
843:=
809:.
804:]
796:z
793:z
781:y
778:z
766:x
763:z
749:z
746:y
734:y
731:y
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716:y
702:z
699:x
687:y
684:x
672:x
669:x
658:[
653:=
626:.
623:)
618:6
610:,
605:5
597:,
592:4
584:,
579:3
571:,
566:2
558:,
553:1
545:(
539:)
534:y
531:x
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518:z
515:x
507:,
502:z
499:y
491:,
486:z
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475:,
470:y
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459:,
454:x
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406:.
401:]
393:z
390:z
378:y
375:z
363:x
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343:y
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316:x
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299:z
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266:x
255:[
250:=
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208:x
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195:x
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106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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