987:
673:
982:{\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}}}
608:
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
603:
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
2018:
2467:
2265:
237:
1287:
2899:
2089:
2832:
1643:
1358:
1917:
1710:
2711:
1543:
678:
2546:
593:
1936:
2324:
517:
2361:
2168:
280:
1844:
1748:
1082:
1055:
452:
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 "
109:
1808:
1788:
1768:
1229:
2839:
2023:
2772:
1580:
1299:
3093:
1152:
However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see
1869:
1407:. 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.
1658:
17:
2626:
1492:
2474:
3502:
1153:
637:
629:
605:
2984:
443:" without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).
3651:
2950:. Then, vectors themselves carry upper abstract indices and covectors carry lower abstract indices, as per the example in the
320:
is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see
2270:
1400:
described by a matrix. This led
Einstein to propose the convention that repeated indices imply the summation is to be done.
3686:
3365:
62:
over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of
3145:
3567:
522:
242:
3789:
666:
In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its
369:
is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are
99:), it implies summation of that term over all the values of the index. So where the indices can range over the
3809:
3418:
3350:
3097:
3077:
3804:
3794:
3443:
2744:
1125:
462:
387:
is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are
3681:
3072:
3492:
3312:
2924:
3164:
3646:
1712:
the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.
410:. This should not be confused with a typographically similar convention used to distinguish between
3814:
3748:
3666:
3620:
3327:
3067:
1100:
96:
3718:
3405:
3322:
3292:
2914:
2013:{\displaystyle \mathbf {u} \times \mathbf {v} ={\varepsilon ^{i}}_{jk}u^{j}v^{k}\mathbf {e} _{i}}
1813:
1374:
415:
3676:
3532:
3487:
2568:
1389:
3758:
3713:
3193:
3138:
2462:{\displaystyle \mathbf {C} _{ik}=(\mathbf {A} \mathbf {B} )_{ik}=\sum _{j=1}^{N}A_{ij}B_{jk}}
2337:
1397:
411:
43:
1723:
3733:
3661:
3547:
3413:
3375:
3307:
2996:
2919:
1171:
1060:
1033:
39:
2260:{\displaystyle \mathbf {u} _{i}=(\mathbf {A} \mathbf {v} )_{i}=\sum _{j=1}^{N}A_{ij}v_{j}}
8:
3610:
3433:
3423:
3272:
3257:
3213:
2731:
indices, there is no summation and the indices are not eliminated by the multiplication.
3000:
3743:
3600:
3453:
3267:
3203:
2929:
2102:
1863:
1793:
1773:
1753:
358:
3040:
3738:
3507:
3482:
3208:
3188:
3012:
1146:
1117:
641:
100:
66:; however, it is often used in physics applications that do not distinguish between
3799:
3753:
3428:
3395:
3380:
3262:
3131:
3004:
2947:
1855:
1381:
1142:
1121:
648:
3723:
3671:
3615:
3595:
3497:
3385:
3252:
3223:
2980:
2934:
2112:
1652:
660:
75:
71:
1093:
609:
not considering coordinate vectors), one may choose to use only subscripts; see
3763:
3728:
3625:
3458:
3448:
3438:
3360:
3332:
3317:
3302:
3218:
1421:
1404:
1385:
433:
403:
384:
366:
285:
91:
According to this convention, when an index variable appears twice in a single
63:
35:
3708:
1846:. We can then write the following operations in Einstein notation as follows.
3783:
3700:
3605:
3517:
3390:
3008:
2748:
2589:
2556:
1930:
1859:
659:
They transform contravariantly or covariantly, respectively, with respect to
447:
67:
232:{\displaystyle y=\sum _{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}}
3768:
3572:
3557:
3522:
3370:
3355:
2946:
This applies only for numerical indices. The situation is the opposite for
1411:
407:
293:
1392:. The individual terms in the sum are not. When the basis is changed, the
1282:{\displaystyle {\begin{bmatrix}w_{1}&\cdots &w_{k}\end{bmatrix}}.}
3656:
3630:
3552:
3241:
3180:
1104:
289:
31:
314:(this can occasionally lead to ambiguity). The upper index position in
3537:
3112:
2571:
is the sum of the diagonal elements, hence the sum over a common index
1921:
This can also be calculated by multiplying the covector on the vector.
1425:
1132:
1089:
611:
2894:{\displaystyle g^{\mu \sigma }{T_{\sigma }}^{\alpha }=T^{\mu \alpha }}
2084:{\displaystyle {\varepsilon ^{i}}_{jk}=\delta ^{il}\varepsilon _{ljk}}
3512:
3463:
623:
59:
2827:{\displaystyle g_{\mu \sigma }{T^{\sigma }}_{\beta }=T_{\mu \beta }}
3542:
3527:
1933:
intrinsically involves summations over permutations of components:
1638:{\displaystyle \mathbf {e} ^{i}(\mathbf {e} _{j})=\delta _{j}^{i}.}
1353:{\displaystyle {\begin{bmatrix}v^{1}\\\vdots \\v^{k}\end{bmatrix}}}
652:
3094:"Lecture 10 – Einstein Summation Convention and Vector Identities"
2954:
of this article. Elements of a basis of vectors may carry a lower
3236:
3198:
1410:
The value of the
Einstein convention is that it applies to other
3562:
3154:
2909:
2740:
92:
1866:) is the sum of corresponding components multiplied together:
1384:
is invariant under transformations of basis. In particular, a
1444:
with itself, has a basis consisting of tensors of the form
3123:
1373:
The virtue of
Einstein notation is that it represents the
3017:
1912:{\displaystyle \mathbf {u} \cdot \mathbf {v} =u_{j}v^{j}}
1057:
are each column vectors, and the covector basis elements
612:§ Superscripts and subscripts versus only subscripts
1705:{\displaystyle \operatorname {Hom} (V,W)=V^{*}\otimes W}
1929:
Again using an orthogonal basis (in 3 dimensions), the
414:
and the closely related but distinct basis-independent
1308:
1238:
919:
864:
770:
715:
525:
2842:
2775:
2706:{\displaystyle {A^{i}}_{j}=u^{i}v_{j}={(uv)^{i}}_{j}}
2629:
2477:
2364:
2273:
2171:
2026:
1939:
1872:
1816:
1796:
1776:
1756:
1726:
1715:
1661:
1583:
1538:{\displaystyle \mathbf {T} =T^{ij}\mathbf {e} _{ij}.}
1495:
1302:
1232:
1174:(column vectors), while covectors are represented as
1063:
1036:
676:
465:
245:
112:
2985:"The Foundation of the General Theory of Relativity"
1149:, one has the option to work with only subscripts.
2893:
2826:
2705:
2541:{\displaystyle {C^{i}}_{k}={A^{i}}_{j}{B^{j}}_{k}}
2540:
2461:
2318:
2259:
2083:
2012:
1911:
1838:
1802:
1782:
1762:
1742:
1720:In Einstein notation, the usual element reference
1704:
1637:
1537:
1352:
1281:
1076:
1049:
981:
624:Superscripts and subscripts versus only subscripts
587:
511:
274:
231:
2328:This is a special case of matrix multiplication.
1164:In the above example, vectors are represented as
3781:
302:should be understood as the second component of
2734:
2142:
1030:are its components. The basis vector elements
3139:
588:{\textstyle v_{i}=\sum _{j}(a_{i}b_{j}x^{j})}
3146:
3132:
1296:Contravariant vectors are column vectors:
86:
58:) is a notational convention that implies
1184:When using the column vector convention:
1085:
618:
3503:Covariance and contravariance of vectors
3091:
2979:
2331:
1154:Covariance and contravariance of vectors
630:covariance and contravariance of vectors
27:Shorthand notation for tensor operations
3065:
1924:
1368:
402:In general, indices can range over any
340:would be equivalent to the traditional
14:
3782:
3033:
2973:
2319:{\displaystyle u^{i}={A^{i}}_{j}v^{j}}
1360:Hence the upper index indicates which
1289:Hence the lower index indicates which
647:lower indices represent components of
636:upper indices represent components of
519:, which is equivalent to the equation
446:An index that is not summed over is a
3127:
1403:As for covectors, they change by the
512:{\displaystyle v_{i}=a_{i}b_{j}x^{j}}
239:is simplified by the convention to:
3096:. Oxford University. Archived from
2747:by contracting the tensor with the
1377:quantities with a simple notation.
1215:vectors that have indices that are
1131:A basis gives such a form (via the
24:
3366:Tensors in curvilinear coordinates
1716:Common operations in this notation
1084:are each row covectors. (See also
421:An index that is summed over is a
95:and is not otherwise defined (see
74:. It was introduced to physics by
25:
3826:
3085:
2951:
2760:. For example, taking the tensor
2121:, there is no difference between
2745:raise an index or lower an index
2583:
2392:
2387:
2367:
2196:
2191:
2174:
2000:
1949:
1941:
1882:
1874:
1849:
1601:
1586:
1519:
1497:
322:
288:but are indices of coordinates,
3059:
81:
3113:"Understanding NumPy's einsum"
3092:Rawlings, Steve (2007-02-01).
2687:
2677:
2397:
2383:
2201:
2187:
2115:. Based on this definition of
1680:
1668:
1611:
1596:
598:
582:
549:
436:since any symbol can replace "
361:, a common convention is that
13:
1:
3419:Exterior covariant derivative
3351:Tensor (intrinsic definition)
2966:
2139:but the position of indices.
52:Einstein summation convention
18:Einstein summation convention
3444:Raising and lowering indices
2735:Raising and lowering indices
2143:Matrix-vector multiplication
1159:
1003:are its components (not the
296:. That is, in this context
275:{\displaystyle y=c_{i}x^{i}}
7:
3682:Gluon field strength tensor
3153:
3073:Encyclopedia of Mathematics
2903:
2836:Or one can raise an index:
1839:{\displaystyle {A^{m}}_{n}}
1226:Covectors are row vectors:
1086:§ Abstract description
606:covariant and contravariant
56:Einstein summation notation
10:
3831:
3493:Cartan formalism (physics)
3313:Penrose graphical notation
2925:Penrose graphical notation
2769:, one can lower an index:
1181:matrices (row covectors).
308:rather than the square of
284:The upper indices are not
34:, especially the usage of
3699:
3639:
3588:
3581:
3473:
3404:
3341:
3285:
3232:
3179:
3172:
3165:Glossary of tensor theory
3161:
1135:), hence when working on
3749:Gregorio Ricci-Curbastro
3621:Riemann curvature tensor
3328:Van der Waerden notation
3066:Kuptsov, L. P. (2001) ,
3009:10.1002/andp.19163540702
2940:
2550:
2147:The product of a matrix
1438:, the tensor product of
1396:of a vector change by a
97:Free and bound variables
3719:Elwin Bruno Christoffel
3652:Angular momentum tensor
3323:Tetrad (index notation)
3293:Abstract index notation
2915:Abstract index notation
1126:raise and lower indices
432:". It is also called a
416:abstract index notation
87:Statement of convention
3533:Levi-Civita connection
2895:
2828:
2707:
2542:
2463:
2432:
2320:
2261:
2233:
2085:
2014:
1913:
1840:
1804:
1784:
1764:
1744:
1743:{\displaystyle A_{mn}}
1706:
1639:
1539:
1390:Lorentz transformation
1354:
1283:
1078:
1051:
983:
619:Vector representations
589:
513:
276:
233:
139:
3790:Mathematical notation
3759:Jan Arnoldus Schouten
3714:Augustin-Louis Cauchy
3194:Differential geometry
2896:
2829:
2708:
2592:of the column vector
2543:
2464:
2412:
2332:Matrix multiplication
2321:
2262:
2213:
2156:with a column vector
2086:
2015:
1914:
1841:
1805:
1790:-th column of matrix
1785:
1765:
1745:
1707:
1640:
1577:which obeys the rule
1540:
1398:linear transformation
1388:is invariant under a
1355:
1284:
1099:In the presence of a
1079:
1077:{\displaystyle e^{i}}
1052:
1050:{\displaystyle e_{i}}
984:
638:contravariant vectors
590:
514:
412:tensor index notation
277:
234:
119:
44:differential geometry
3810:Mathematical physics
3734:Carl Friedrich Gauss
3667:stress–energy tensor
3662:Cauchy stress tensor
3414:Covariant derivative
3376:Antisymmetric tensor
3308:Multi-index notation
3041:"Einstein Summation"
2840:
2773:
2627:
2475:
2362:
2271:
2169:
2024:
1937:
1925:Vector cross product
1870:
1814:
1794:
1774:
1754:
1724:
1659:
1581:
1493:
1369:Abstract description
1300:
1230:
1211:variant tensors are
1061:
1034:
1021:is the covector and
674:
523:
463:
326:below). Typically,
243:
110:
50:(also known as the
40:mathematical physics
3805:Riemannian geometry
3795:Multilinear algebra
3611:Nonmetricity tensor
3466:(2nd-order tensors)
3434:Hodge star operator
3424:Exterior derivative
3273:Transport phenomena
3258:Continuum mechanics
3214:Multilinear algebra
3043:. Wolfram Mathworld
3001:1916AnP...354..769E
2958:index and an upper
2111:is the generalized
1631:
1489:can be written as:
1101:non-degenerate form
3744:Tullio Levi-Civita
3687:Metric tensor (GR)
3601:Levi-Civita symbol
3454:Tensor contraction
3268:General relativity
3204:Euclidean geometry
2989:Annalen der Physik
2930:Levi-Civita symbol
2891:
2824:
2703:
2598:by the row vector
2538:
2459:
2316:
2257:
2103:Levi-Civita symbol
2081:
2010:
1909:
1864:vector dot product
1836:
1800:
1780:
1760:
1740:
1702:
1635:
1617:
1535:
1350:
1344:
1279:
1270:
1074:
1047:
997:is the vector and
979:
977:
969:
908:
820:
759:
585:
548:
509:
459:" in the equation
359:general relativity
323:§ Application
272:
229:
3777:
3776:
3739:Hermann Grassmann
3695:
3694:
3647:Moment of inertia
3508:Differential form
3483:Affine connection
3298:Einstein notation
3281:
3280:
3209:Exterior calculus
3189:Coordinate system
1803:{\displaystyle A}
1783:{\displaystyle n}
1763:{\displaystyle m}
1147:orthonormal basis
1118:Riemannian metric
1116:, for instance a
539:
48:Einstein notation
16:(Redirected from
3822:
3754:Bernhard Riemann
3586:
3585:
3429:Exterior product
3396:Two-point tensor
3381:Symmetric tensor
3263:Electromagnetism
3177:
3176:
3148:
3141:
3134:
3125:
3124:
3120:
3108:
3106:
3105:
3080:
3053:
3052:
3050:
3048:
3037:
3031:
3030:
3028:
3027:
3021:
3011:. Archived from
2981:Einstein, Albert
2977:
2948:abstract indices
2920:Bra–ket notation
2900:
2898:
2897:
2892:
2890:
2889:
2874:
2873:
2868:
2867:
2866:
2855:
2854:
2833:
2831:
2830:
2825:
2823:
2822:
2807:
2806:
2801:
2800:
2799:
2788:
2787:
2768:
2759:
2726:
2720:
2712:
2710:
2709:
2704:
2702:
2701:
2696:
2695:
2694:
2671:
2670:
2661:
2660:
2648:
2647:
2642:
2641:
2640:
2622:
2616:
2606:
2597:
2579:
2566:
2547:
2545:
2544:
2539:
2537:
2536:
2531:
2530:
2529:
2518:
2517:
2512:
2511:
2510:
2496:
2495:
2490:
2489:
2488:
2468:
2466:
2465:
2460:
2458:
2457:
2445:
2444:
2431:
2426:
2408:
2407:
2395:
2390:
2379:
2378:
2370:
2357:
2348:
2340:of two matrices
2325:
2323:
2322:
2317:
2315:
2314:
2305:
2304:
2299:
2298:
2297:
2283:
2282:
2266:
2264:
2263:
2258:
2256:
2255:
2246:
2245:
2232:
2227:
2209:
2208:
2199:
2194:
2183:
2182:
2177:
2164:
2155:
2138:
2129:
2120:
2110:
2100:
2090:
2088:
2087:
2082:
2080:
2079:
2064:
2063:
2048:
2047:
2039:
2038:
2037:
2019:
2017:
2016:
2011:
2009:
2008:
2003:
1997:
1996:
1987:
1986:
1977:
1976:
1968:
1967:
1966:
1952:
1944:
1918:
1916:
1915:
1910:
1908:
1907:
1898:
1897:
1885:
1877:
1856:orthogonal basis
1845:
1843:
1842:
1837:
1835:
1834:
1829:
1828:
1827:
1809:
1807:
1806:
1801:
1789:
1787:
1786:
1781:
1769:
1767:
1766:
1761:
1749:
1747:
1746:
1741:
1739:
1738:
1711:
1709:
1708:
1703:
1695:
1694:
1650:
1644:
1642:
1641:
1636:
1630:
1625:
1610:
1609:
1604:
1595:
1594:
1589:
1576:
1570:
1564:
1558:
1552:
1544:
1542:
1541:
1536:
1531:
1530:
1522:
1516:
1515:
1500:
1488:
1478:
1472:
1443:
1437:
1419:
1359:
1357:
1356:
1351:
1349:
1348:
1341:
1340:
1320:
1319:
1288:
1286:
1285:
1280:
1275:
1274:
1267:
1266:
1250:
1249:
1200:ower indices go
1180:
1170:
1143:Euclidean metric
1140:
1122:Minkowski metric
1115:
1092:, below and the
1083:
1081:
1080:
1075:
1073:
1072:
1056:
1054:
1053:
1048:
1046:
1045:
1029:
1020:
1014:
1008:
1002:
996:
988:
986:
985:
980:
978:
974:
973:
966:
965:
945:
944:
931:
930:
913:
912:
905:
904:
888:
887:
876:
875:
855:
854:
845:
844:
825:
824:
817:
816:
796:
795:
782:
781:
764:
763:
756:
755:
739:
738:
727:
726:
706:
705:
696:
695:
594:
592:
591:
586:
581:
580:
571:
570:
561:
560:
547:
535:
534:
518:
516:
515:
510:
508:
507:
498:
497:
488:
487:
475:
474:
458:
442:
431:
425:, in this case "
397:
379:
353:
339:
319:
313:
307:
301:
281:
279:
278:
273:
271:
270:
261:
260:
238:
236:
235:
230:
228:
227:
218:
217:
205:
204:
195:
194:
182:
181:
172:
171:
159:
158:
149:
148:
138:
133:
105:
72:cotangent spaces
21:
3830:
3829:
3825:
3824:
3823:
3821:
3820:
3819:
3815:Albert Einstein
3780:
3779:
3778:
3773:
3724:Albert Einstein
3691:
3672:Einstein tensor
3635:
3616:Ricci curvature
3596:Kronecker delta
3582:Notable tensors
3577:
3498:Connection form
3475:
3469:
3400:
3386:Tensor operator
3343:
3337:
3277:
3253:Computer vision
3246:
3228:
3224:Tensor calculus
3168:
3157:
3152:
3111:
3103:
3101:
3088:
3068:"Einstein rule"
3062:
3057:
3056:
3046:
3044:
3039:
3038:
3034:
3025:
3023:
3015:
2978:
2974:
2969:
2943:
2935:DeWitt notation
2906:
2882:
2878:
2869:
2862:
2858:
2857:
2856:
2847:
2843:
2841:
2838:
2837:
2815:
2811:
2802:
2795:
2791:
2790:
2789:
2780:
2776:
2774:
2771:
2770:
2766:
2761:
2757:
2752:
2737:
2722:
2716:
2697:
2690:
2686:
2676:
2675:
2666:
2662:
2656:
2652:
2643:
2636:
2632:
2631:
2630:
2628:
2625:
2624:
2618:
2612: ×
2608:
2604:
2599:
2593:
2586:
2577:
2572:
2564:
2559:
2553:
2532:
2525:
2521:
2520:
2519:
2513:
2506:
2502:
2501:
2500:
2491:
2484:
2480:
2479:
2478:
2476:
2473:
2472:
2450:
2446:
2437:
2433:
2427:
2416:
2400:
2396:
2391:
2386:
2371:
2366:
2365:
2363:
2360:
2359:
2355:
2350:
2346:
2341:
2334:
2310:
2306:
2300:
2293:
2289:
2288:
2287:
2278:
2274:
2272:
2269:
2268:
2251:
2247:
2238:
2234:
2228:
2217:
2204:
2200:
2195:
2190:
2178:
2173:
2172:
2170:
2167:
2166:
2162:
2157:
2153:
2148:
2145:
2136:
2131:
2127:
2122:
2116:
2113:Kronecker delta
2106:
2098:
2093:
2069:
2065:
2056:
2052:
2040:
2033:
2029:
2028:
2027:
2025:
2022:
2021:
2004:
1999:
1998:
1992:
1988:
1982:
1978:
1969:
1962:
1958:
1957:
1956:
1948:
1940:
1938:
1935:
1934:
1927:
1903:
1899:
1893:
1889:
1881:
1873:
1871:
1868:
1867:
1852:
1830:
1823:
1819:
1818:
1817:
1815:
1812:
1811:
1795:
1792:
1791:
1775:
1772:
1771:
1755:
1752:
1751:
1731:
1727:
1725:
1722:
1721:
1718:
1690:
1686:
1660:
1657:
1656:
1653:Kronecker delta
1646:
1626:
1621:
1605:
1600:
1599:
1590:
1585:
1584:
1582:
1579:
1578:
1572:
1566:
1560:
1554:
1547:
1523:
1518:
1517:
1508:
1504:
1496:
1494:
1491:
1490:
1480:
1474:
1471:
1462:
1453:
1445:
1439:
1429:
1428:. For example,
1415:
1371:
1343:
1342:
1336:
1332:
1329:
1328:
1322:
1321:
1315:
1311:
1304:
1303:
1301:
1298:
1297:
1269:
1268:
1262:
1258:
1256:
1251:
1245:
1241:
1234:
1233:
1231:
1228:
1227:
1192:per indices go
1175:
1165:
1162:
1136:
1107:
1068:
1064:
1062:
1059:
1058:
1041:
1037:
1035:
1032:
1031:
1027:
1022:
1016:
1010:
1004:
998:
992:
976:
975:
968:
967:
961:
957:
954:
953:
947:
946:
940:
936:
933:
932:
926:
922:
915:
914:
907:
906:
900:
896:
894:
889:
883:
879:
877:
871:
867:
860:
859:
850:
846:
840:
836:
827:
826:
819:
818:
812:
808:
805:
804:
798:
797:
791:
787:
784:
783:
777:
773:
766:
765:
758:
757:
751:
747:
745:
740:
734:
730:
728:
722:
718:
711:
710:
701:
697:
691:
687:
677:
675:
672:
671:
661:change of basis
626:
621:
601:
576:
572:
566:
562:
556:
552:
543:
530:
526:
524:
521:
520:
503:
499:
493:
489:
483:
479:
470:
466:
464:
461:
460:
453:
437:
426:
423:summation index
406:, including an
388:
370:
341:
327:
315:
309:
303:
297:
266:
262:
256:
252:
244:
241:
240:
223:
219:
213:
209:
200:
196:
190:
186:
177:
173:
167:
163:
154:
150:
144:
140:
134:
123:
111:
108:
107:
103:
89:
84:
76:Albert Einstein
28:
23:
22:
15:
12:
11:
5:
3828:
3818:
3817:
3812:
3807:
3802:
3797:
3792:
3775:
3774:
3772:
3771:
3766:
3764:Woldemar Voigt
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3729:Leonhard Euler
3726:
3721:
3716:
3711:
3705:
3703:
3701:Mathematicians
3697:
3696:
3693:
3692:
3690:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3643:
3641:
3637:
3636:
3634:
3633:
3628:
3626:Torsion tensor
3623:
3618:
3613:
3608:
3603:
3598:
3592:
3590:
3583:
3579:
3578:
3576:
3575:
3570:
3565:
3560:
3555:
3550:
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3495:
3490:
3485:
3479:
3477:
3471:
3470:
3468:
3467:
3461:
3459:Tensor product
3456:
3451:
3449:Symmetrization
3446:
3441:
3439:Lie derivative
3436:
3431:
3426:
3421:
3416:
3410:
3408:
3402:
3401:
3399:
3398:
3393:
3388:
3383:
3378:
3373:
3368:
3363:
3361:Tensor density
3358:
3353:
3347:
3345:
3339:
3338:
3336:
3335:
3333:Voigt notation
3330:
3325:
3320:
3318:Ricci calculus
3315:
3310:
3305:
3303:Index notation
3300:
3295:
3289:
3287:
3283:
3282:
3279:
3278:
3276:
3275:
3270:
3265:
3260:
3255:
3249:
3247:
3245:
3244:
3239:
3233:
3230:
3229:
3227:
3226:
3221:
3219:Tensor algebra
3216:
3211:
3206:
3201:
3199:Dyadic algebra
3196:
3191:
3185:
3183:
3174:
3170:
3169:
3162:
3159:
3158:
3151:
3150:
3143:
3136:
3128:
3122:
3121:
3117:Stack Overflow
3109:
3087:
3086:External links
3084:
3083:
3082:
3061:
3058:
3055:
3054:
3032:
2971:
2970:
2968:
2965:
2964:
2963:
2942:
2939:
2938:
2937:
2932:
2927:
2922:
2917:
2912:
2905:
2902:
2888:
2885:
2881:
2877:
2872:
2865:
2861:
2853:
2850:
2846:
2821:
2818:
2814:
2810:
2805:
2798:
2794:
2786:
2783:
2779:
2764:
2755:
2736:
2733:
2727:represent two
2700:
2693:
2689:
2685:
2682:
2679:
2674:
2669:
2665:
2659:
2655:
2651:
2646:
2639:
2635:
2602:
2585:
2582:
2575:
2562:
2552:
2549:
2535:
2528:
2524:
2516:
2509:
2505:
2499:
2494:
2487:
2483:
2471:equivalent to
2456:
2453:
2449:
2443:
2440:
2436:
2430:
2425:
2422:
2419:
2415:
2411:
2406:
2403:
2399:
2394:
2389:
2385:
2382:
2377:
2374:
2369:
2353:
2344:
2338:matrix product
2333:
2330:
2313:
2309:
2303:
2296:
2292:
2286:
2281:
2277:
2267:equivalent to
2254:
2250:
2244:
2241:
2237:
2231:
2226:
2223:
2220:
2216:
2212:
2207:
2203:
2198:
2193:
2189:
2186:
2181:
2176:
2160:
2151:
2144:
2141:
2134:
2125:
2096:
2078:
2075:
2072:
2068:
2062:
2059:
2055:
2051:
2046:
2043:
2036:
2032:
2007:
2002:
1995:
1991:
1985:
1981:
1975:
1972:
1965:
1961:
1955:
1951:
1947:
1943:
1926:
1923:
1906:
1902:
1896:
1892:
1888:
1884:
1880:
1876:
1851:
1848:
1833:
1826:
1822:
1799:
1779:
1759:
1737:
1734:
1730:
1717:
1714:
1701:
1698:
1693:
1689:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1634:
1629:
1624:
1620:
1616:
1613:
1608:
1603:
1598:
1593:
1588:
1559:, has a basis
1553:, the dual of
1534:
1529:
1526:
1521:
1514:
1511:
1507:
1503:
1499:
1467:
1458:
1449:
1422:tensor product
1405:inverse matrix
1386:Lorentz scalar
1380:In physics, a
1370:
1367:
1366:
1365:
1347:
1339:
1335:
1331:
1330:
1327:
1324:
1323:
1318:
1314:
1310:
1309:
1307:
1294:
1278:
1273:
1265:
1261:
1257:
1255:
1252:
1248:
1244:
1240:
1239:
1237:
1224:
1205:
1204:eft to right."
1161:
1158:
1071:
1067:
1044:
1040:
1025:
972:
964:
960:
956:
955:
952:
949:
948:
943:
939:
935:
934:
929:
925:
921:
920:
918:
911:
903:
899:
895:
893:
890:
886:
882:
878:
874:
870:
866:
865:
863:
858:
853:
849:
843:
839:
835:
832:
829:
828:
823:
815:
811:
807:
806:
803:
800:
799:
794:
790:
786:
785:
780:
776:
772:
771:
769:
762:
754:
750:
746:
744:
741:
737:
733:
729:
725:
721:
717:
716:
714:
709:
704:
700:
694:
690:
686:
683:
680:
679:
657:
656:
645:
625:
622:
620:
617:
600:
597:
584:
579:
575:
569:
565:
559:
555:
551:
546:
542:
538:
533:
529:
506:
502:
496:
492:
486:
482:
478:
473:
469:
400:
399:
385:Latin alphabet
381:
367:Greek alphabet
269:
265:
259:
255:
251:
248:
226:
222:
216:
212:
208:
203:
199:
193:
189:
185:
180:
176:
170:
166:
162:
157:
153:
147:
143:
137:
132:
129:
126:
122:
118:
115:
88:
85:
83:
80:
64:Ricci calculus
36:linear algebra
26:
9:
6:
4:
3:
2:
3827:
3816:
3813:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3787:
3785:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3706:
3704:
3702:
3698:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3644:
3642:
3638:
3632:
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3612:
3609:
3607:
3606:Metric tensor
3604:
3602:
3599:
3597:
3594:
3593:
3591:
3587:
3584:
3580:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3549:
3546:
3544:
3541:
3539:
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3518:Exterior form
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3480:
3478:
3472:
3465:
3462:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3411:
3409:
3407:
3403:
3397:
3394:
3392:
3391:Tensor bundle
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3348:
3346:
3340:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3290:
3288:
3284:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3250:
3248:
3243:
3240:
3238:
3235:
3234:
3231:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3186:
3184:
3182:
3178:
3175:
3171:
3167:
3166:
3160:
3156:
3149:
3144:
3142:
3137:
3135:
3130:
3129:
3126:
3118:
3114:
3110:
3100:on 2017-01-06
3099:
3095:
3090:
3089:
3079:
3075:
3074:
3069:
3064:
3063:
3042:
3036:
3022:on 2006-08-29
3019:
3014:
3010:
3006:
3002:
2998:
2994:
2990:
2986:
2982:
2976:
2972:
2961:
2957:
2953:
2949:
2945:
2944:
2936:
2933:
2931:
2928:
2926:
2923:
2921:
2918:
2916:
2913:
2911:
2908:
2907:
2901:
2886:
2883:
2879:
2875:
2870:
2863:
2859:
2851:
2848:
2844:
2834:
2819:
2816:
2812:
2808:
2803:
2796:
2792:
2784:
2781:
2777:
2767:
2758:
2750:
2749:metric tensor
2746:
2742:
2732:
2730:
2725:
2719:
2713:
2698:
2691:
2683:
2680:
2672:
2667:
2663:
2657:
2653:
2649:
2644:
2637:
2633:
2621:
2615:
2611:
2605:
2596:
2591:
2590:outer product
2584:Outer product
2581:
2578:
2570:
2565:
2558:
2557:square matrix
2548:
2533:
2526:
2522:
2514:
2507:
2503:
2497:
2492:
2485:
2481:
2469:
2454:
2451:
2447:
2441:
2438:
2434:
2428:
2423:
2420:
2417:
2413:
2409:
2404:
2401:
2380:
2375:
2372:
2356:
2347:
2339:
2329:
2326:
2311:
2307:
2301:
2294:
2290:
2284:
2279:
2275:
2252:
2248:
2242:
2239:
2235:
2229:
2224:
2221:
2218:
2214:
2210:
2205:
2184:
2179:
2163:
2154:
2140:
2137:
2128:
2119:
2114:
2109:
2104:
2099:
2091:
2076:
2073:
2070:
2066:
2060:
2057:
2053:
2049:
2044:
2041:
2034:
2030:
2005:
1993:
1989:
1983:
1979:
1973:
1970:
1963:
1959:
1953:
1945:
1932:
1931:cross product
1922:
1919:
1904:
1900:
1894:
1890:
1886:
1878:
1865:
1861:
1860:inner product
1857:
1850:Inner product
1847:
1831:
1824:
1820:
1797:
1777:
1757:
1735:
1732:
1728:
1713:
1699:
1696:
1691:
1687:
1683:
1677:
1674:
1671:
1665:
1662:
1654:
1649:
1632:
1627:
1622:
1618:
1614:
1606:
1591:
1575:
1569:
1563:
1557:
1550:
1545:
1532:
1527:
1524:
1512:
1509:
1505:
1501:
1487:
1483:
1477:
1473:. Any tensor
1470:
1466:
1461:
1457:
1452:
1448:
1442:
1436:
1432:
1427:
1423:
1418:
1413:
1412:vector spaces
1408:
1406:
1401:
1399:
1395:
1391:
1387:
1383:
1378:
1376:
1363:
1345:
1337:
1333:
1325:
1316:
1312:
1305:
1295:
1292:
1276:
1271:
1263:
1259:
1253:
1246:
1242:
1235:
1225:
1222:
1218:
1214:
1210:
1206:
1203:
1199:
1195:
1191:
1187:
1186:
1185:
1182:
1179:
1173:
1168:
1157:
1155:
1150:
1148:
1144:
1139:
1134:
1129:
1127:
1123:
1119:
1114:
1110:
1106:
1102:
1097:
1095:
1091:
1087:
1069:
1065:
1042:
1038:
1028:
1019:
1013:
1007:
1001:
995:
989:
970:
962:
958:
950:
941:
937:
927:
923:
916:
909:
901:
897:
891:
884:
880:
872:
868:
861:
856:
851:
847:
841:
837:
833:
830:
821:
813:
809:
801:
792:
788:
778:
774:
767:
760:
752:
748:
742:
735:
731:
723:
719:
712:
707:
702:
698:
692:
688:
684:
681:
669:
664:
662:
654:
650:
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643:
639:
635:
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631:
616:
614:
613:
607:
596:
577:
573:
567:
563:
557:
553:
544:
540:
536:
531:
527:
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494:
490:
484:
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476:
471:
467:
456:
451:
450:
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440:
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386:
382:
377:
373:
368:
364:
363:
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360:
355:
351:
348:
345:
337:
334:
331:
325:
324:
318:
312:
306:
300:
295:
294:basis vectors
291:
287:
282:
267:
263:
257:
253:
249:
246:
224:
220:
214:
210:
206:
201:
197:
191:
187:
183:
178:
174:
168:
164:
160:
155:
151:
145:
141:
135:
130:
127:
124:
120:
116:
113:
102:
98:
94:
79:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
3769:Hermann Weyl
3573:Vector space
3558:Pseudotensor
3523:Fiber bundle
3476:abstractions
3371:Mixed tensor
3356:Tensor field
3297:
3163:
3116:
3102:. Retrieved
3098:the original
3071:
3060:Bibliography
3045:. Retrieved
3035:
3024:. Retrieved
3013:the original
2992:
2988:
2975:
2959:
2955:
2952:introduction
2835:
2762:
2753:
2738:
2728:
2723:
2717:
2714:
2619:
2613:
2609:
2600:
2594:
2587:
2573:
2560:
2554:
2470:
2351:
2342:
2335:
2327:
2158:
2149:
2146:
2132:
2123:
2117:
2107:
2094:
2092:
1928:
1920:
1853:
1770:-th row and
1719:
1647:
1573:
1567:
1561:
1555:
1548:
1546:
1485:
1481:
1475:
1468:
1464:
1459:
1455:
1450:
1446:
1440:
1434:
1430:
1416:
1409:
1402:
1393:
1379:
1372:
1361:
1290:
1221:co-row-below
1220:
1216:
1212:
1208:
1201:
1197:
1193:
1189:
1183:
1177:
1166:
1163:
1151:
1145:and a fixed
1137:
1130:
1112:
1108:
1098:
1023:
1017:
1011:
1009:th covector
1005:
999:
993:
990:
667:
665:
658:
628:In terms of
627:
610:
602:
454:
448:
445:
438:
427:
422:
420:
408:infinite set
404:indexing set
401:
393:
389:
375:
371:
356:
349:
346:
343:
335:
332:
329:
321:
316:
310:
304:
298:
290:coefficients
283:
90:
82:Introduction
55:
51:
47:
29:
3709:Élie Cartan
3657:Spin tensor
3631:Weyl tensor
3589:Mathematics
3553:Multivector
3344:definitions
3242:Engineering
3181:Mathematics
1414:built from
1364:you are in.
1293:you are in.
1124:), one can
1105:isomorphism
599:Application
434:dummy index
32:mathematics
3784:Categories
3538:Linear map
3406:Operations
3104:2008-07-02
3026:2006-09-03
2995:(7): 769.
2967:References
2743:, one can
2607:yields an
1420:using the
1394:components
1176:1 ×
1133:dual basis
668:components
449:free index
3677:EM tensor
3513:Dimension
3464:Transpose
3078:EMS Press
2956:numerical
2887:α
2884:μ
2871:α
2864:σ
2852:σ
2849:μ
2820:β
2817:μ
2804:β
2797:σ
2785:σ
2782:μ
2729:different
2414:∑
2215:∑
2067:ε
2054:δ
2031:ε
1960:ε
1946:×
1879:⋅
1854:Using an
1697:⊗
1692:∗
1666:
1619:δ
1375:invariant
1326:⋮
1254:⋯
1196:to down;
1169:× 1
1160:Mnemonics
951:⋮
892:⋯
802:⋮
743:⋯
670:, as in:
653:covectors
651:vectors (
649:covariant
541:∑
286:exponents
121:∑
104:{1, 2, 3}
78:in 1916.
60:summation
3543:Manifold
3528:Geodesic
3286:Notation
3047:13 April
2983:(1916).
2960:abstract
2904:See also
2739:Given a
1810:becomes
1750:for the
1551: *
1484:⊗
1433:⊗
1172:matrices
1094:examples
3800:Tensors
3640:Physics
3474:Related
3237:Physics
3155:Tensors
2997:Bibcode
2617:matrix
2101:is the
1651:is the
1571:, ...,
1426:duality
1141:with a
1090:duality
642:vectors
615:below.
457:
441:
430:
68:tangent
3568:Vector
3563:Spinor
3548:Matrix
3342:Tensor
2962:index.
2910:Tensor
2741:tensor
2715:Since
2567:, the
2555:For a
2105:, and
2020:where
1858:, the
1645:where
1382:scalar
1291:column
991:where
3488:Basis
3173:Scope
2941:Notes
2569:trace
2551:Trace
1655:. As
1217:below
396:, ...
378:, ...
3049:2011
2721:and
2588:The
2358:is:
2349:and
2336:The
2165:is:
2130:and
1424:and
1103:(an
383:the
365:the
93:term
70:and
42:and
3018:PDF
3005:doi
2993:354
2135:ijk
2097:ijk
1663:Hom
1479:in
1362:row
1223:)."
1213:row
1120:or
1015:),
357:In
292:or
101:set
54:or
38:in
30:In
3786::
3115:.
3076:,
3070:,
3003:.
2991:.
2987:.
2756:μν
2751:,
2623::
2580:.
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2152:ij
2126:jk
1565:,
1463:⊗
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1451:ij
1209:Co
1194:up
1190:Up
1156:.
1128:.
1111:→
1096:)
1088:;
663:.
655:).
644:),
632:,
595:.
418:.
398:),
392:,
380:),
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354:.
106:,
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3147:e
3140:t
3133:v
3119:.
3107:.
3081:.
3051:.
3029:.
3020:)
3016:(
3007::
2999::
2880:T
2876:=
2860:T
2845:g
2813:T
2809:=
2793:T
2778:g
2765:β
2763:T
2754:g
2724:j
2718:i
2699:j
2692:i
2688:)
2684:v
2681:u
2678:(
2673:=
2668:j
2664:v
2658:i
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2650:=
2645:j
2638:i
2634:A
2620:A
2614:n
2610:m
2603:j
2601:v
2595:u
2576:i
2574:A
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2534:k
2527:j
2523:B
2515:j
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2504:A
2498:=
2493:k
2486:i
2482:C
2455:k
2452:j
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2442:j
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2435:A
2429:N
2424:1
2421:=
2418:j
2410:=
2405:k
2402:i
2398:)
2393:B
2388:A
2384:(
2381:=
2376:k
2373:i
2368:C
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2285:=
2280:i
2276:u
2253:j
2249:v
2243:j
2240:i
2236:A
2230:N
2225:1
2222:=
2219:j
2211:=
2206:i
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2197:v
2192:A
2188:(
2185:=
2180:i
2175:u
2161:j
2159:v
2150:A
2133:ε
2124:ε
2118:ε
2108:δ
2095:ε
2077:k
2074:j
2071:l
2061:l
2058:i
2050:=
2045:k
2042:j
2035:i
2006:i
2001:e
1994:k
1990:v
1984:j
1980:u
1974:k
1971:j
1964:i
1954:=
1950:v
1942:u
1905:j
1901:v
1895:j
1891:u
1887:=
1883:v
1875:u
1862:(
1832:n
1825:m
1821:A
1798:A
1778:n
1758:m
1736:n
1733:m
1729:A
1700:W
1688:V
1684:=
1681:)
1678:W
1675:,
1672:V
1669:(
1648:δ
1633:.
1628:i
1623:j
1615:=
1612:)
1607:j
1602:e
1597:(
1592:i
1587:e
1574:e
1568:e
1562:e
1556:V
1549:V
1533:.
1528:j
1525:i
1520:e
1513:j
1510:i
1506:T
1502:=
1498:T
1486:V
1482:V
1476:T
1469:j
1465:e
1460:i
1456:e
1447:e
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1435:V
1431:V
1417:V
1346:]
1338:k
1334:v
1317:1
1313:v
1306:[
1277:.
1272:]
1264:k
1260:w
1247:1
1243:w
1236:[
1219:(
1207:"
1202:l
1198:l
1188:"
1178:n
1167:n
1138:R
1113:V
1109:V
1070:i
1066:e
1043:i
1039:e
1026:i
1024:w
1018:w
1012:v
1006:i
1000:v
994:v
971:]
963:n
959:e
942:2
938:e
928:1
924:e
917:[
910:]
902:n
898:w
885:2
881:w
873:1
869:w
862:[
857:=
852:i
848:e
842:i
838:w
834:=
831:w
822:]
814:n
810:v
793:2
789:v
779:1
775:v
768:[
761:]
753:n
749:e
736:2
732:e
724:1
720:e
713:[
708:=
703:i
699:e
693:i
689:v
685:=
682:v
640:(
583:)
578:j
574:x
568:j
564:b
558:i
554:a
550:(
545:j
537:=
532:i
528:v
505:j
501:x
495:j
491:b
485:i
481:a
477:=
472:i
468:v
455:i
439:i
428:i
394:j
390:i
376:ν
372:μ
352:)
350:z
347:y
344:x
342:(
338:)
336:x
333:x
330:x
328:(
317:x
311:x
305:x
299:x
268:i
264:x
258:i
254:c
250:=
247:y
225:3
221:x
215:3
211:c
207:+
202:2
198:x
192:2
188:c
184:+
179:1
175:x
169:1
165:c
161:=
156:i
152:x
146:i
142:c
136:3
131:1
128:=
125:i
117:=
114:y
20:)
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