976:
662:
971:{\displaystyle {\begin{aligned}v=v^{i}e_{i}={\begin{bmatrix}e_{1}&e_{2}&\cdots &e_{n}\end{bmatrix}}{\begin{bmatrix}v^{1}\\v^{2}\\\vdots \\v^{n}\end{bmatrix}}\\w=w_{i}e^{i}={\begin{bmatrix}w_{1}&w_{2}&\cdots &w_{n}\end{bmatrix}}{\begin{bmatrix}e^{1}\\e^{2}\\\vdots \\e^{n}\end{bmatrix}}\end{aligned}}}
597:
vectors, where the position of an index indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when
592:
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. When dealing with
2007:
2456:
2254:
226:
1276:
2888:
2078:
2821:
1632:
1347:
1906:
1699:
2700:
1532:
667:
2535:
582:
1925:
2313:
506:
2350:
2157:
269:
1833:
1737:
1071:
1044:
441:
and should appear only once per term. If such an index does appear, it usually also appears in every other term in an equation. An example of a free index is the "
98:
1797:
1777:
1757:
1218:
2828:
2012:
2761:
1569:
1288:
3082:
1141:
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see
1858:
1396:. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.
1647:
2615:
1481:
2463:
3491:
1142:
626:
618:
594:
2973:
432:" without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).
3640:
2939:. Then, vectors themselves carry upper abstract indices and covectors carry lower abstract indices, as per the example in the
309:
is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see
2259:
1389:
described by a matrix. This led
Einstein to propose the convention that repeated indices imply the summation is to be done.
3675:
3354:
51:
over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of
3134:
3556:
511:
231:
3778:
655:
In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its
358:
is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are
88:), it implies summation of that term over all the values of the index. So where the indices can range over the
3798:
3407:
3339:
3086:
3066:
3793:
3783:
3432:
2733:
1114:
451:
376:
is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are
3670:
3061:
3481:
3301:
2913:
3153:
3635:
1701:
the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.
399:. This should not be confused with a typographically similar convention used to distinguish between
3803:
3737:
3655:
3609:
3316:
3056:
1089:
85:
3707:
3394:
3311:
3281:
2903:
2002:{\displaystyle \mathbf {u} \times \mathbf {v} ={\varepsilon ^{i}}_{jk}u^{j}v^{k}\mathbf {e} _{i}}
1802:
1363:
404:
3665:
3521:
3476:
2557:
1378:
3747:
3702:
3182:
3127:
2451:{\displaystyle \mathbf {C} _{ik}=(\mathbf {A} \mathbf {B} )_{ik}=\sum _{j=1}^{N}A_{ij}B_{jk}}
2326:
1386:
400:
32:
1712:
3722:
3650:
3536:
3402:
3364:
3296:
2985:
2908:
1160:
1049:
1022:
28:
2249:{\displaystyle \mathbf {u} _{i}=(\mathbf {A} \mathbf {v} )_{i}=\sum _{j=1}^{N}A_{ij}v_{j}}
8:
3599:
3422:
3412:
3261:
3246:
3202:
2720:
indices, there is no summation and the indices are not eliminated by the multiplication.
2989:
3732:
3589:
3442:
3256:
3192:
2918:
2091:
1852:
1782:
1762:
1742:
347:
3029:
3727:
3496:
3471:
3197:
3177:
3001:
1135:
1106:
630:
89:
55:; however, it is often used in physics applications that do not distinguish between
3788:
3742:
3417:
3384:
3369:
3251:
3120:
2993:
2936:
1844:
1370:
1131:
1110:
637:
3712:
3660:
3604:
3584:
3486:
3374:
3241:
3212:
2969:
2923:
2101:
1641:
649:
64:
60:
1082:
598:
not considering coordinate vectors), one may choose to use only subscripts; see
3752:
3717:
3614:
3447:
3437:
3427:
3349:
3321:
3306:
3291:
3207:
1410:
1393:
1374:
422:
392:
373:
355:
274:
80:
According to this convention, when an index variable appears twice in a single
52:
24:
3697:
1835:. We can then write the following operations in Einstein notation as follows.
3772:
3689:
3594:
3506:
3379:
2997:
2737:
2578:
2545:
1919:
1848:
648:
They transform contravariantly or covariantly, respectively, with respect to
436:
56:
221:{\displaystyle y=\sum _{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}}
3757:
3561:
3546:
3511:
3359:
3344:
2935:
This applies only for numerical indices. The situation is the opposite for
1400:
396:
282:
1381:. The individual terms in the sum are not. When the basis is changed, the
1271:{\displaystyle {\begin{bmatrix}w_{1}&\cdots &w_{k}\end{bmatrix}}.}
3645:
3619:
3541:
3230:
3169:
1093:
278:
20:
303:(this can occasionally lead to ambiguity). The upper index position in
3526:
3101:
2560:
is the sum of the diagonal elements, hence the sum over a common index
1910:
This can also be calculated by multiplying the covector on the vector.
1414:
1121:
1078:
600:
2883:{\displaystyle g^{\mu \sigma }{T_{\sigma }}^{\alpha }=T^{\mu \alpha }}
2073:{\displaystyle {\varepsilon ^{i}}_{jk}=\delta ^{il}\varepsilon _{ljk}}
3501:
3452:
612:
48:
2816:{\displaystyle g_{\mu \sigma }{T^{\sigma }}_{\beta }=T_{\mu \beta }}
3531:
3516:
1922:
intrinsically involves summations over permutations of components:
1627:{\displaystyle \mathbf {e} ^{i}(\mathbf {e} _{j})=\delta _{j}^{i}.}
1342:{\displaystyle {\begin{bmatrix}v^{1}\\\vdots \\v^{k}\end{bmatrix}}}
641:
3083:"Lecture 10 – Einstein Summation Convention and Vector Identities"
2943:
of this article. Elements of a basis of vectors may carry a lower
3225:
3187:
1399:
The value of the
Einstein convention is that it applies to other
3551:
3143:
2898:
2729:
81:
1855:) is the sum of corresponding components multiplied together:
1373:
is invariant under transformations of basis. In particular, a
1433:
with itself, has a basis consisting of tensors of the form
3112:
1362:
The virtue of
Einstein notation is that it represents the
3006:
1901:{\displaystyle \mathbf {u} \cdot \mathbf {v} =u_{j}v^{j}}
1046:
are each column vectors, and the covector basis elements
601:§ Superscripts and subscripts versus only subscripts
1694:{\displaystyle \operatorname {Hom} (V,W)=V^{*}\otimes W}
1918:
Again using an orthogonal basis (in 3 dimensions), the
403:
and the closely related but distinct basis-independent
1297:
1227:
908:
853:
759:
704:
514:
2831:
2764:
2695:{\displaystyle {A^{i}}_{j}=u^{i}v_{j}={(uv)^{i}}_{j}}
2618:
2466:
2353:
2262:
2160:
2015:
1928:
1861:
1805:
1785:
1765:
1745:
1715:
1704:
1650:
1572:
1527:{\displaystyle \mathbf {T} =T^{ij}\mathbf {e} _{ij}.}
1484:
1291:
1221:
1163:(column vectors), while covectors are represented as
1052:
1025:
665:
454:
234:
101:
2974:"The Foundation of the General Theory of Relativity"
1138:, one has the option to work with only subscripts.
2882:
2815:
2694:
2530:{\displaystyle {C^{i}}_{k}={A^{i}}_{j}{B^{j}}_{k}}
2529:
2450:
2307:
2248:
2072:
2001:
1900:
1827:
1791:
1771:
1751:
1731:
1709:In Einstein notation, the usual element reference
1693:
1626:
1526:
1341:
1270:
1065:
1038:
970:
613:Superscripts and subscripts versus only subscripts
576:
500:
263:
220:
2317:This is a special case of matrix multiplication.
1153:In the above example, vectors are represented as
3770:
291:should be understood as the second component of
2723:
2131:
1019:are its components. The basis vector elements
3128:
577:{\textstyle v_{i}=\sum _{j}(a_{i}b_{j}x^{j})}
3135:
3121:
1285:Contravariant vectors are column vectors:
75:
47:) is a notational convention that implies
1173:When using the column vector convention:
1074:
607:
3492:Covariance and contravariance of vectors
3080:
2968:
2320:
1143:Covariance and contravariance of vectors
619:covariance and contravariance of vectors
16:Shorthand notation for tensor operations
3054:
1913:
1357:
391:In general, indices can range over any
329:would be equivalent to the traditional
3771:
3022:
2962:
2308:{\displaystyle u^{i}={A^{i}}_{j}v^{j}}
1349:Hence the upper index indicates which
1278:Hence the lower index indicates which
636:lower indices represent components of
625:upper indices represent components of
508:, which is equivalent to the equation
435:An index that is not summed over is a
3116:
1392:As for covectors, they change by the
501:{\displaystyle v_{i}=a_{i}b_{j}x^{j}}
228:is simplified by the convention to:
3085:. Oxford University. Archived from
2736:by contracting the tensor with the
1366:quantities with a simple notation.
1204:vectors that have indices that are
1120:A basis gives such a form (via the
13:
3355:Tensors in curvilinear coordinates
1705:Common operations in this notation
1073:are each row covectors. (See also
410:An index that is summed over is a
84:and is not otherwise defined (see
63:. It was introduced to physics by
14:
3815:
3074:
2940:
2749:. For example, taking the tensor
2110:, there is no difference between
2734:raise an index or lower an index
2572:
2381:
2376:
2356:
2185:
2180:
2163:
1989:
1938:
1930:
1871:
1863:
1838:
1590:
1575:
1508:
1486:
311:
277:but are indices of coordinates,
3048:
70:
3102:"Understanding NumPy's einsum"
3081:Rawlings, Steve (2007-02-01).
2676:
2666:
2386:
2372:
2190:
2176:
2104:. Based on this definition of
1669:
1657:
1600:
1585:
587:
571:
538:
425:since any symbol can replace "
350:, a common convention is that
1:
3408:Exterior covariant derivative
3340:Tensor (intrinsic definition)
2955:
2128:but the position of indices.
41:Einstein summation convention
3433:Raising and lowering indices
2724:Raising and lowering indices
2132:Matrix-vector multiplication
1148:
992:are its components (not the
285:. That is, in this context
264:{\displaystyle y=c_{i}x^{i}}
7:
3671:Gluon field strength tensor
3142:
3062:Encyclopedia of Mathematics
2892:
2825:Or one can raise an index:
1828:{\displaystyle {A^{m}}_{n}}
1215:Covectors are row vectors:
1075:§ Abstract description
595:covariant and contravariant
45:Einstein summation notation
10:
3820:
3482:Cartan formalism (physics)
3302:Penrose graphical notation
2914:Penrose graphical notation
2758:, one can lower an index:
1170:matrices (row covectors).
297:rather than the square of
273:The upper indices are not
23:, especially the usage of
3688:
3628:
3577:
3570:
3462:
3393:
3330:
3274:
3221:
3168:
3161:
3154:Glossary of tensor theory
3150:
1124:), hence when working on
3738:Gregorio Ricci-Curbastro
3610:Riemann curvature tensor
3317:Van der Waerden notation
3055:Kuptsov, L. P. (2001) ,
2998:10.1002/andp.19163540702
2929:
2539:
2136:The product of a matrix
1427:, the tensor product of
1385:of a vector change by a
86:Free and bound variables
3708:Elwin Bruno Christoffel
3641:Angular momentum tensor
3312:Tetrad (index notation)
3282:Abstract index notation
2904:Abstract index notation
1115:raise and lower indices
421:". It is also called a
405:abstract index notation
76:Statement of convention
3522:Levi-Civita connection
2884:
2817:
2696:
2531:
2452:
2421:
2309:
2250:
2222:
2074:
2003:
1902:
1829:
1793:
1773:
1753:
1733:
1732:{\displaystyle A_{mn}}
1695:
1628:
1528:
1379:Lorentz transformation
1343:
1272:
1067:
1040:
972:
608:Vector representations
578:
502:
265:
222:
128:
3779:Mathematical notation
3748:Jan Arnoldus Schouten
3703:Augustin-Louis Cauchy
3183:Differential geometry
2885:
2818:
2697:
2581:of the column vector
2532:
2453:
2401:
2321:Matrix multiplication
2310:
2251:
2202:
2145:with a column vector
2075:
2004:
1903:
1830:
1794:
1779:-th column of matrix
1774:
1754:
1734:
1696:
1629:
1566:which obeys the rule
1529:
1387:linear transformation
1377:is invariant under a
1344:
1273:
1088:In the presence of a
1068:
1066:{\displaystyle e^{i}}
1041:
1039:{\displaystyle e_{i}}
973:
627:contravariant vectors
579:
503:
401:tensor index notation
266:
223:
108:
33:differential geometry
3799:Mathematical physics
3723:Carl Friedrich Gauss
3656:stress–energy tensor
3651:Cauchy stress tensor
3403:Covariant derivative
3365:Antisymmetric tensor
3297:Multi-index notation
3030:"Einstein Summation"
2829:
2762:
2616:
2464:
2351:
2260:
2158:
2013:
1926:
1914:Vector cross product
1859:
1803:
1783:
1763:
1743:
1713:
1648:
1570:
1482:
1358:Abstract description
1289:
1219:
1200:variant tensors are
1050:
1023:
1010:is the covector and
663:
512:
452:
315:below). Typically,
232:
99:
39:(also known as the
29:mathematical physics
3794:Riemannian geometry
3784:Multilinear algebra
3600:Nonmetricity tensor
3455:(2nd-order tensors)
3423:Hodge star operator
3413:Exterior derivative
3262:Transport phenomena
3247:Continuum mechanics
3203:Multilinear algebra
3032:. Wolfram Mathworld
2990:1916AnP...354..769E
2947:index and an upper
2100:is the generalized
1620:
1478:can be written as:
1090:non-degenerate form
3733:Tullio Levi-Civita
3676:Metric tensor (GR)
3590:Levi-Civita symbol
3443:Tensor contraction
3257:General relativity
3193:Euclidean geometry
2978:Annalen der Physik
2919:Levi-Civita symbol
2880:
2813:
2692:
2587:by the row vector
2527:
2448:
2305:
2246:
2092:Levi-Civita symbol
2070:
1999:
1898:
1853:vector dot product
1825:
1789:
1769:
1749:
1729:
1691:
1624:
1606:
1524:
1339:
1333:
1268:
1259:
1063:
1036:
986:is the vector and
968:
966:
958:
897:
809:
748:
574:
537:
498:
448:" in the equation
348:general relativity
312:§ Application
261:
218:
3766:
3765:
3728:Hermann Grassmann
3684:
3683:
3636:Moment of inertia
3497:Differential form
3472:Affine connection
3287:Einstein notation
3270:
3269:
3198:Exterior calculus
3178:Coordinate system
1792:{\displaystyle A}
1772:{\displaystyle n}
1752:{\displaystyle m}
1136:orthonormal basis
1107:Riemannian metric
1105:, for instance a
528:
37:Einstein notation
3811:
3743:Bernhard Riemann
3575:
3574:
3418:Exterior product
3385:Two-point tensor
3370:Symmetric tensor
3252:Electromagnetism
3166:
3165:
3137:
3130:
3123:
3114:
3113:
3109:
3097:
3095:
3094:
3069:
3042:
3041:
3039:
3037:
3026:
3020:
3019:
3017:
3016:
3010:
3000:. Archived from
2970:Einstein, Albert
2966:
2937:abstract indices
2909:Bra–ket notation
2889:
2887:
2886:
2881:
2879:
2878:
2863:
2862:
2857:
2856:
2855:
2844:
2843:
2822:
2820:
2819:
2814:
2812:
2811:
2796:
2795:
2790:
2789:
2788:
2777:
2776:
2757:
2748:
2715:
2709:
2701:
2699:
2698:
2693:
2691:
2690:
2685:
2684:
2683:
2660:
2659:
2650:
2649:
2637:
2636:
2631:
2630:
2629:
2611:
2605:
2595:
2586:
2568:
2555:
2536:
2534:
2533:
2528:
2526:
2525:
2520:
2519:
2518:
2507:
2506:
2501:
2500:
2499:
2485:
2484:
2479:
2478:
2477:
2457:
2455:
2454:
2449:
2447:
2446:
2434:
2433:
2420:
2415:
2397:
2396:
2384:
2379:
2368:
2367:
2359:
2346:
2337:
2329:of two matrices
2314:
2312:
2311:
2306:
2304:
2303:
2294:
2293:
2288:
2287:
2286:
2272:
2271:
2255:
2253:
2252:
2247:
2245:
2244:
2235:
2234:
2221:
2216:
2198:
2197:
2188:
2183:
2172:
2171:
2166:
2153:
2144:
2127:
2118:
2109:
2099:
2089:
2079:
2077:
2076:
2071:
2069:
2068:
2053:
2052:
2037:
2036:
2028:
2027:
2026:
2008:
2006:
2005:
2000:
1998:
1997:
1992:
1986:
1985:
1976:
1975:
1966:
1965:
1957:
1956:
1955:
1941:
1933:
1907:
1905:
1904:
1899:
1897:
1896:
1887:
1886:
1874:
1866:
1845:orthogonal basis
1834:
1832:
1831:
1826:
1824:
1823:
1818:
1817:
1816:
1798:
1796:
1795:
1790:
1778:
1776:
1775:
1770:
1758:
1756:
1755:
1750:
1738:
1736:
1735:
1730:
1728:
1727:
1700:
1698:
1697:
1692:
1684:
1683:
1639:
1633:
1631:
1630:
1625:
1619:
1614:
1599:
1598:
1593:
1584:
1583:
1578:
1565:
1559:
1553:
1547:
1541:
1533:
1531:
1530:
1525:
1520:
1519:
1511:
1505:
1504:
1489:
1477:
1467:
1461:
1432:
1426:
1408:
1348:
1346:
1345:
1340:
1338:
1337:
1330:
1329:
1309:
1308:
1277:
1275:
1274:
1269:
1264:
1263:
1256:
1255:
1239:
1238:
1189:ower indices go
1169:
1159:
1132:Euclidean metric
1129:
1111:Minkowski metric
1104:
1081:, below and the
1072:
1070:
1069:
1064:
1062:
1061:
1045:
1043:
1042:
1037:
1035:
1034:
1018:
1009:
1003:
997:
991:
985:
977:
975:
974:
969:
967:
963:
962:
955:
954:
934:
933:
920:
919:
902:
901:
894:
893:
877:
876:
865:
864:
844:
843:
834:
833:
814:
813:
806:
805:
785:
784:
771:
770:
753:
752:
745:
744:
728:
727:
716:
715:
695:
694:
685:
684:
583:
581:
580:
575:
570:
569:
560:
559:
550:
549:
536:
524:
523:
507:
505:
504:
499:
497:
496:
487:
486:
477:
476:
464:
463:
447:
431:
420:
414:, in this case "
386:
368:
342:
328:
308:
302:
296:
290:
270:
268:
267:
262:
260:
259:
250:
249:
227:
225:
224:
219:
217:
216:
207:
206:
194:
193:
184:
183:
171:
170:
161:
160:
148:
147:
138:
137:
127:
122:
94:
61:cotangent spaces
3819:
3818:
3814:
3813:
3812:
3810:
3809:
3808:
3804:Albert Einstein
3769:
3768:
3767:
3762:
3713:Albert Einstein
3680:
3661:Einstein tensor
3624:
3605:Ricci curvature
3585:Kronecker delta
3571:Notable tensors
3566:
3487:Connection form
3464:
3458:
3389:
3375:Tensor operator
3332:
3326:
3266:
3242:Computer vision
3235:
3217:
3213:Tensor calculus
3157:
3146:
3141:
3100:
3092:
3090:
3077:
3057:"Einstein rule"
3051:
3046:
3045:
3035:
3033:
3028:
3027:
3023:
3014:
3012:
3004:
2967:
2963:
2958:
2932:
2924:DeWitt notation
2895:
2871:
2867:
2858:
2851:
2847:
2846:
2845:
2836:
2832:
2830:
2827:
2826:
2804:
2800:
2791:
2784:
2780:
2779:
2778:
2769:
2765:
2763:
2760:
2759:
2755:
2750:
2746:
2741:
2726:
2711:
2705:
2686:
2679:
2675:
2665:
2664:
2655:
2651:
2645:
2641:
2632:
2625:
2621:
2620:
2619:
2617:
2614:
2613:
2607:
2601: ×
2597:
2593:
2588:
2582:
2575:
2566:
2561:
2553:
2548:
2542:
2521:
2514:
2510:
2509:
2508:
2502:
2495:
2491:
2490:
2489:
2480:
2473:
2469:
2468:
2467:
2465:
2462:
2461:
2439:
2435:
2426:
2422:
2416:
2405:
2389:
2385:
2380:
2375:
2360:
2355:
2354:
2352:
2349:
2348:
2344:
2339:
2335:
2330:
2323:
2299:
2295:
2289:
2282:
2278:
2277:
2276:
2267:
2263:
2261:
2258:
2257:
2240:
2236:
2227:
2223:
2217:
2206:
2193:
2189:
2184:
2179:
2167:
2162:
2161:
2159:
2156:
2155:
2151:
2146:
2142:
2137:
2134:
2125:
2120:
2116:
2111:
2105:
2102:Kronecker delta
2095:
2087:
2082:
2058:
2054:
2045:
2041:
2029:
2022:
2018:
2017:
2016:
2014:
2011:
2010:
1993:
1988:
1987:
1981:
1977:
1971:
1967:
1958:
1951:
1947:
1946:
1945:
1937:
1929:
1927:
1924:
1923:
1916:
1892:
1888:
1882:
1878:
1870:
1862:
1860:
1857:
1856:
1841:
1819:
1812:
1808:
1807:
1806:
1804:
1801:
1800:
1784:
1781:
1780:
1764:
1761:
1760:
1744:
1741:
1740:
1720:
1716:
1714:
1711:
1710:
1707:
1679:
1675:
1649:
1646:
1645:
1642:Kronecker delta
1635:
1615:
1610:
1594:
1589:
1588:
1579:
1574:
1573:
1571:
1568:
1567:
1561:
1555:
1549:
1543:
1536:
1512:
1507:
1506:
1497:
1493:
1485:
1483:
1480:
1479:
1469:
1463:
1460:
1451:
1442:
1434:
1428:
1418:
1417:. For example,
1404:
1360:
1332:
1331:
1325:
1321:
1318:
1317:
1311:
1310:
1304:
1300:
1293:
1292:
1290:
1287:
1286:
1258:
1257:
1251:
1247:
1245:
1240:
1234:
1230:
1223:
1222:
1220:
1217:
1216:
1181:per indices go
1164:
1154:
1151:
1125:
1096:
1057:
1053:
1051:
1048:
1047:
1030:
1026:
1024:
1021:
1020:
1016:
1011:
1005:
999:
993:
987:
981:
965:
964:
957:
956:
950:
946:
943:
942:
936:
935:
929:
925:
922:
921:
915:
911:
904:
903:
896:
895:
889:
885:
883:
878:
872:
868:
866:
860:
856:
849:
848:
839:
835:
829:
825:
816:
815:
808:
807:
801:
797:
794:
793:
787:
786:
780:
776:
773:
772:
766:
762:
755:
754:
747:
746:
740:
736:
734:
729:
723:
719:
717:
711:
707:
700:
699:
690:
686:
680:
676:
666:
664:
661:
660:
650:change of basis
615:
610:
590:
565:
561:
555:
551:
545:
541:
532:
519:
515:
513:
510:
509:
492:
488:
482:
478:
472:
468:
459:
455:
453:
450:
449:
442:
426:
415:
412:summation index
395:, including an
377:
359:
330:
316:
304:
298:
292:
286:
255:
251:
245:
241:
233:
230:
229:
212:
208:
202:
198:
189:
185:
179:
175:
166:
162:
156:
152:
143:
139:
133:
129:
123:
112:
100:
97:
96:
92:
78:
73:
65:Albert Einstein
17:
12:
11:
5:
3817:
3807:
3806:
3801:
3796:
3791:
3786:
3781:
3764:
3763:
3761:
3760:
3755:
3753:Woldemar Voigt
3750:
3745:
3740:
3735:
3730:
3725:
3720:
3718:Leonhard Euler
3715:
3710:
3705:
3700:
3694:
3692:
3690:Mathematicians
3686:
3685:
3682:
3681:
3679:
3678:
3673:
3668:
3663:
3658:
3653:
3648:
3643:
3638:
3632:
3630:
3626:
3625:
3623:
3622:
3617:
3615:Torsion tensor
3612:
3607:
3602:
3597:
3592:
3587:
3581:
3579:
3572:
3568:
3567:
3565:
3564:
3559:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3484:
3479:
3474:
3468:
3466:
3460:
3459:
3457:
3456:
3450:
3448:Tensor product
3445:
3440:
3438:Symmetrization
3435:
3430:
3428:Lie derivative
3425:
3420:
3415:
3410:
3405:
3399:
3397:
3391:
3390:
3388:
3387:
3382:
3377:
3372:
3367:
3362:
3357:
3352:
3350:Tensor density
3347:
3342:
3336:
3334:
3328:
3327:
3325:
3324:
3322:Voigt notation
3319:
3314:
3309:
3307:Ricci calculus
3304:
3299:
3294:
3292:Index notation
3289:
3284:
3278:
3276:
3272:
3271:
3268:
3267:
3265:
3264:
3259:
3254:
3249:
3244:
3238:
3236:
3234:
3233:
3228:
3222:
3219:
3218:
3216:
3215:
3210:
3208:Tensor algebra
3205:
3200:
3195:
3190:
3188:Dyadic algebra
3185:
3180:
3174:
3172:
3163:
3159:
3158:
3151:
3148:
3147:
3140:
3139:
3132:
3125:
3117:
3111:
3110:
3106:Stack Overflow
3098:
3076:
3075:External links
3073:
3072:
3071:
3050:
3047:
3044:
3043:
3021:
2960:
2959:
2957:
2954:
2953:
2952:
2931:
2928:
2927:
2926:
2921:
2916:
2911:
2906:
2901:
2894:
2891:
2877:
2874:
2870:
2866:
2861:
2854:
2850:
2842:
2839:
2835:
2810:
2807:
2803:
2799:
2794:
2787:
2783:
2775:
2772:
2768:
2753:
2744:
2725:
2722:
2716:represent two
2689:
2682:
2678:
2674:
2671:
2668:
2663:
2658:
2654:
2648:
2644:
2640:
2635:
2628:
2624:
2591:
2574:
2571:
2564:
2551:
2541:
2538:
2524:
2517:
2513:
2505:
2498:
2494:
2488:
2483:
2476:
2472:
2460:equivalent to
2445:
2442:
2438:
2432:
2429:
2425:
2419:
2414:
2411:
2408:
2404:
2400:
2395:
2392:
2388:
2383:
2378:
2374:
2371:
2366:
2363:
2358:
2342:
2333:
2327:matrix product
2322:
2319:
2302:
2298:
2292:
2285:
2281:
2275:
2270:
2266:
2256:equivalent to
2243:
2239:
2233:
2230:
2226:
2220:
2215:
2212:
2209:
2205:
2201:
2196:
2192:
2187:
2182:
2178:
2175:
2170:
2165:
2149:
2140:
2133:
2130:
2123:
2114:
2085:
2067:
2064:
2061:
2057:
2051:
2048:
2044:
2040:
2035:
2032:
2025:
2021:
1996:
1991:
1984:
1980:
1974:
1970:
1964:
1961:
1954:
1950:
1944:
1940:
1936:
1932:
1915:
1912:
1895:
1891:
1885:
1881:
1877:
1873:
1869:
1865:
1840:
1837:
1822:
1815:
1811:
1788:
1768:
1748:
1726:
1723:
1719:
1706:
1703:
1690:
1687:
1682:
1678:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1623:
1618:
1613:
1609:
1605:
1602:
1597:
1592:
1587:
1582:
1577:
1548:, has a basis
1542:, the dual of
1523:
1518:
1515:
1510:
1503:
1500:
1496:
1492:
1488:
1456:
1447:
1438:
1411:tensor product
1394:inverse matrix
1375:Lorentz scalar
1369:In physics, a
1359:
1356:
1355:
1354:
1336:
1328:
1324:
1320:
1319:
1316:
1313:
1312:
1307:
1303:
1299:
1298:
1296:
1283:
1267:
1262:
1254:
1250:
1246:
1244:
1241:
1237:
1233:
1229:
1228:
1226:
1213:
1194:
1193:eft to right."
1150:
1147:
1060:
1056:
1033:
1029:
1014:
961:
953:
949:
945:
944:
941:
938:
937:
932:
928:
924:
923:
918:
914:
910:
909:
907:
900:
892:
888:
884:
882:
879:
875:
871:
867:
863:
859:
855:
854:
852:
847:
842:
838:
832:
828:
824:
821:
818:
817:
812:
804:
800:
796:
795:
792:
789:
788:
783:
779:
775:
774:
769:
765:
761:
760:
758:
751:
743:
739:
735:
733:
730:
726:
722:
718:
714:
710:
706:
705:
703:
698:
693:
689:
683:
679:
675:
672:
669:
668:
646:
645:
634:
614:
611:
609:
606:
589:
586:
573:
568:
564:
558:
554:
548:
544:
540:
535:
531:
527:
522:
518:
495:
491:
485:
481:
475:
471:
467:
462:
458:
389:
388:
374:Latin alphabet
370:
356:Greek alphabet
258:
254:
248:
244:
240:
237:
215:
211:
205:
201:
197:
192:
188:
182:
178:
174:
169:
165:
159:
155:
151:
146:
142:
136:
132:
126:
121:
118:
115:
111:
107:
104:
77:
74:
72:
69:
53:Ricci calculus
25:linear algebra
15:
9:
6:
4:
3:
2:
3816:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3776:
3774:
3759:
3756:
3754:
3751:
3749:
3746:
3744:
3741:
3739:
3736:
3734:
3731:
3729:
3726:
3724:
3721:
3719:
3716:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3695:
3693:
3691:
3687:
3677:
3674:
3672:
3669:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3633:
3631:
3627:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3601:
3598:
3596:
3595:Metric tensor
3593:
3591:
3588:
3586:
3583:
3582:
3580:
3576:
3573:
3569:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3507:Exterior form
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3469:
3467:
3461:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3400:
3398:
3396:
3392:
3386:
3383:
3381:
3380:Tensor bundle
3378:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3337:
3335:
3329:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3279:
3277:
3273:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3243:
3240:
3239:
3237:
3232:
3229:
3227:
3224:
3223:
3220:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3175:
3173:
3171:
3167:
3164:
3160:
3156:
3155:
3149:
3145:
3138:
3133:
3131:
3126:
3124:
3119:
3118:
3115:
3107:
3103:
3099:
3089:on 2017-01-06
3088:
3084:
3079:
3078:
3068:
3064:
3063:
3058:
3053:
3052:
3031:
3025:
3011:on 2006-08-29
3008:
3003:
2999:
2995:
2991:
2987:
2983:
2979:
2975:
2971:
2965:
2961:
2950:
2946:
2942:
2938:
2934:
2933:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2896:
2890:
2875:
2872:
2868:
2864:
2859:
2852:
2848:
2840:
2837:
2833:
2823:
2808:
2805:
2801:
2797:
2792:
2785:
2781:
2773:
2770:
2766:
2756:
2747:
2739:
2738:metric tensor
2735:
2731:
2721:
2719:
2714:
2708:
2702:
2687:
2680:
2672:
2669:
2661:
2656:
2652:
2646:
2642:
2638:
2633:
2626:
2622:
2610:
2604:
2600:
2594:
2585:
2580:
2579:outer product
2573:Outer product
2570:
2567:
2559:
2554:
2547:
2546:square matrix
2537:
2522:
2515:
2511:
2503:
2496:
2492:
2486:
2481:
2474:
2470:
2458:
2443:
2440:
2436:
2430:
2427:
2423:
2417:
2412:
2409:
2406:
2402:
2398:
2393:
2390:
2369:
2364:
2361:
2345:
2336:
2328:
2318:
2315:
2300:
2296:
2290:
2283:
2279:
2273:
2268:
2264:
2241:
2237:
2231:
2228:
2224:
2218:
2213:
2210:
2207:
2203:
2199:
2194:
2173:
2168:
2152:
2143:
2129:
2126:
2117:
2108:
2103:
2098:
2093:
2088:
2080:
2065:
2062:
2059:
2055:
2049:
2046:
2042:
2038:
2033:
2030:
2023:
2019:
1994:
1982:
1978:
1972:
1968:
1962:
1959:
1952:
1948:
1942:
1934:
1921:
1920:cross product
1911:
1908:
1893:
1889:
1883:
1879:
1875:
1867:
1854:
1850:
1849:inner product
1846:
1839:Inner product
1836:
1820:
1813:
1809:
1786:
1766:
1746:
1724:
1721:
1717:
1702:
1688:
1685:
1680:
1676:
1672:
1666:
1663:
1660:
1654:
1651:
1643:
1638:
1621:
1616:
1611:
1607:
1603:
1595:
1580:
1564:
1558:
1552:
1546:
1539:
1534:
1521:
1516:
1513:
1501:
1498:
1494:
1490:
1476:
1472:
1466:
1462:. Any tensor
1459:
1455:
1450:
1446:
1441:
1437:
1431:
1425:
1421:
1416:
1412:
1407:
1402:
1401:vector spaces
1397:
1395:
1390:
1388:
1384:
1380:
1376:
1372:
1367:
1365:
1352:
1334:
1326:
1322:
1314:
1305:
1301:
1294:
1284:
1281:
1265:
1260:
1252:
1248:
1242:
1235:
1231:
1224:
1214:
1211:
1207:
1203:
1199:
1195:
1192:
1188:
1184:
1180:
1176:
1175:
1174:
1171:
1168:
1162:
1157:
1146:
1144:
1139:
1137:
1133:
1128:
1123:
1118:
1116:
1112:
1108:
1103:
1099:
1095:
1091:
1086:
1084:
1080:
1076:
1058:
1054:
1031:
1027:
1017:
1008:
1002:
996:
990:
984:
978:
959:
951:
947:
939:
930:
926:
916:
912:
905:
898:
890:
886:
880:
873:
869:
861:
857:
850:
845:
840:
836:
830:
826:
822:
819:
810:
802:
798:
790:
781:
777:
767:
763:
756:
749:
741:
737:
731:
724:
720:
712:
708:
701:
696:
691:
687:
681:
677:
673:
670:
658:
653:
651:
643:
639:
635:
632:
628:
624:
623:
622:
620:
605:
603:
602:
596:
585:
566:
562:
556:
552:
546:
542:
533:
529:
525:
520:
516:
493:
489:
483:
479:
473:
469:
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460:
456:
445:
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439:
433:
429:
424:
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406:
402:
398:
394:
384:
380:
375:
371:
366:
362:
357:
353:
352:
351:
349:
344:
340:
337:
334:
326:
323:
320:
314:
313:
307:
301:
295:
289:
284:
283:basis vectors
280:
276:
271:
256:
252:
246:
242:
238:
235:
213:
209:
203:
199:
195:
190:
186:
180:
176:
172:
167:
163:
157:
153:
149:
144:
140:
134:
130:
124:
119:
116:
113:
109:
105:
102:
91:
87:
83:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
3758:Hermann Weyl
3562:Vector space
3547:Pseudotensor
3512:Fiber bundle
3465:abstractions
3360:Mixed tensor
3345:Tensor field
3286:
3152:
3105:
3091:. Retrieved
3087:the original
3060:
3049:Bibliography
3034:. Retrieved
3024:
3013:. Retrieved
3002:the original
2981:
2977:
2964:
2948:
2944:
2941:introduction
2824:
2751:
2742:
2727:
2717:
2712:
2706:
2703:
2608:
2602:
2598:
2589:
2583:
2576:
2562:
2549:
2543:
2459:
2340:
2331:
2324:
2316:
2147:
2138:
2135:
2121:
2112:
2106:
2096:
2083:
2081:
1917:
1909:
1842:
1759:-th row and
1708:
1636:
1562:
1556:
1550:
1544:
1537:
1535:
1474:
1470:
1464:
1457:
1453:
1448:
1444:
1439:
1435:
1429:
1423:
1419:
1405:
1398:
1391:
1382:
1368:
1361:
1350:
1279:
1210:co-row-below
1209:
1205:
1201:
1197:
1190:
1186:
1182:
1178:
1172:
1166:
1155:
1152:
1140:
1134:and a fixed
1126:
1119:
1101:
1097:
1087:
1012:
1006:
1000:
998:th covector
994:
988:
982:
979:
656:
654:
647:
617:In terms of
616:
599:
591:
443:
437:
434:
427:
416:
411:
409:
397:infinite set
393:indexing set
390:
382:
378:
364:
360:
345:
338:
335:
332:
324:
321:
318:
310:
305:
299:
293:
287:
279:coefficients
272:
79:
71:Introduction
44:
40:
36:
18:
3698:Élie Cartan
3646:Spin tensor
3620:Weyl tensor
3578:Mathematics
3542:Multivector
3333:definitions
3231:Engineering
3170:Mathematics
1403:built from
1353:you are in.
1282:you are in.
1113:), one can
1094:isomorphism
588:Application
423:dummy index
21:mathematics
3773:Categories
3527:Linear map
3395:Operations
3093:2008-07-02
3015:2006-09-03
2984:(7): 769.
2956:References
2732:, one can
2596:yields an
1409:using the
1383:components
1165:1 ×
1122:dual basis
657:components
438:free index
3666:EM tensor
3502:Dimension
3453:Transpose
3067:EMS Press
2945:numerical
2876:α
2873:μ
2860:α
2853:σ
2841:σ
2838:μ
2809:β
2806:μ
2793:β
2786:σ
2774:σ
2771:μ
2718:different
2403:∑
2204:∑
2056:ε
2043:δ
2020:ε
1949:ε
1935:×
1868:⋅
1843:Using an
1686:⊗
1681:∗
1655:
1608:δ
1364:invariant
1315:⋮
1243:⋯
1185:to down;
1158:× 1
1149:Mnemonics
940:⋮
881:⋯
791:⋮
732:⋯
659:, as in:
642:covectors
640:vectors (
638:covariant
530:∑
275:exponents
110:∑
93:{1, 2, 3}
67:in 1916.
49:summation
3532:Manifold
3517:Geodesic
3275:Notation
3036:13 April
2972:(1916).
2949:abstract
2893:See also
2728:Given a
1799:becomes
1739:for the
1540: *
1473:⊗
1422:⊗
1161:matrices
1083:examples
3789:Tensors
3629:Physics
3463:Related
3226:Physics
3144:Tensors
2986:Bibcode
2606:matrix
2090:is the
1640:is the
1560:, ...,
1415:duality
1130:with a
1079:duality
631:vectors
604:below.
446:
430:
419:
57:tangent
3557:Vector
3552:Spinor
3537:Matrix
3331:Tensor
2951:index.
2899:Tensor
2730:tensor
2704:Since
2556:, the
2544:For a
2094:, and
2009:where
1847:, the
1634:where
1371:scalar
1280:column
980:where
3477:Basis
3162:Scope
2930:Notes
2558:trace
2540:Trace
1644:. As
1206:below
385:, ...
367:, ...
3038:2011
2710:and
2577:The
2347:is:
2338:and
2325:The
2154:is:
2119:and
1413:and
1092:(an
372:the
354:the
82:term
59:and
31:and
3007:PDF
2994:doi
2982:354
2124:ijk
2086:ijk
1652:Hom
1468:in
1351:row
1212:)."
1202:row
1109:or
1004:),
346:In
281:or
90:set
43:or
27:in
19:In
3775::
3104:.
3065:,
3059:,
2992:.
2980:.
2976:.
2745:μν
2740:,
2612::
2569:.
2343:jk
2334:ij
2141:ij
2115:jk
1554:,
1452:⊗
1443:=
1440:ij
1198:Co
1183:up
1179:Up
1145:.
1117:.
1100:→
1085:)
1077:;
652:.
644:).
633:),
621:,
584:.
407:.
387:),
381:,
369:),
363:,
343:.
95:,
35:,
3136:e
3129:t
3122:v
3108:.
3096:.
3070:.
3040:.
3018:.
3009:)
3005:(
2996::
2988::
2869:T
2865:=
2849:T
2834:g
2802:T
2798:=
2782:T
2767:g
2754:β
2752:T
2743:g
2713:j
2707:i
2688:j
2681:i
2677:)
2673:v
2670:u
2667:(
2662:=
2657:j
2653:v
2647:i
2643:u
2639:=
2634:j
2627:i
2623:A
2609:A
2603:n
2599:m
2592:j
2590:v
2584:u
2565:i
2563:A
2552:j
2550:A
2523:k
2516:j
2512:B
2504:j
2497:i
2493:A
2487:=
2482:k
2475:i
2471:C
2444:k
2441:j
2437:B
2431:j
2428:i
2424:A
2418:N
2413:1
2410:=
2407:j
2399:=
2394:k
2391:i
2387:)
2382:B
2377:A
2373:(
2370:=
2365:k
2362:i
2357:C
2341:B
2332:A
2301:j
2297:v
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2284:i
2280:A
2274:=
2269:i
2265:u
2242:j
2238:v
2232:j
2229:i
2225:A
2219:N
2214:1
2211:=
2208:j
2200:=
2195:i
2191:)
2186:v
2181:A
2177:(
2174:=
2169:i
2164:u
2150:j
2148:v
2139:A
2122:ε
2113:ε
2107:ε
2097:δ
2084:ε
2066:k
2063:j
2060:l
2050:l
2047:i
2039:=
2034:k
2031:j
2024:i
1995:i
1990:e
1983:k
1979:v
1973:j
1969:u
1963:k
1960:j
1953:i
1943:=
1939:v
1931:u
1894:j
1890:v
1884:j
1880:u
1876:=
1872:v
1864:u
1851:(
1821:n
1814:m
1810:A
1787:A
1767:n
1747:m
1725:n
1722:m
1718:A
1689:W
1677:V
1673:=
1670:)
1667:W
1664:,
1661:V
1658:(
1637:δ
1622:.
1617:i
1612:j
1604:=
1601:)
1596:j
1591:e
1586:(
1581:i
1576:e
1563:e
1557:e
1551:e
1545:V
1538:V
1522:.
1517:j
1514:i
1509:e
1502:j
1499:i
1495:T
1491:=
1487:T
1475:V
1471:V
1465:T
1458:j
1454:e
1449:i
1445:e
1436:e
1430:V
1424:V
1420:V
1406:V
1335:]
1327:k
1323:v
1306:1
1302:v
1295:[
1266:.
1261:]
1253:k
1249:w
1236:1
1232:w
1225:[
1208:(
1196:"
1191:l
1187:l
1177:"
1167:n
1156:n
1127:R
1102:V
1098:V
1059:i
1055:e
1032:i
1028:e
1015:i
1013:w
1007:w
1001:v
995:i
989:v
983:v
960:]
952:n
948:e
931:2
927:e
917:1
913:e
906:[
899:]
891:n
887:w
874:2
870:w
862:1
858:w
851:[
846:=
841:i
837:e
831:i
827:w
823:=
820:w
811:]
803:n
799:v
782:2
778:v
768:1
764:v
757:[
750:]
742:n
738:e
725:2
721:e
713:1
709:e
702:[
697:=
692:i
688:e
682:i
678:v
674:=
671:v
629:(
572:)
567:j
563:x
557:j
553:b
547:i
543:a
539:(
534:j
526:=
521:i
517:v
494:j
490:x
484:j
480:b
474:i
470:a
466:=
461:i
457:v
444:i
428:i
417:i
383:j
379:i
365:ν
361:μ
341:)
339:z
336:y
333:x
331:(
327:)
325:x
322:x
319:x
317:(
306:x
300:x
294:x
288:x
257:i
253:x
247:i
243:c
239:=
236:y
214:3
210:x
204:3
200:c
196:+
191:2
187:x
181:2
177:c
173:+
168:1
164:x
158:1
154:c
150:=
145:i
141:x
135:i
131:c
125:3
120:1
117:=
114:i
106:=
103:y
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