43:
1390:. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only for
2353:
1999:
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2107:
903:
has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which is
1826:
561:
1676:
1837:
2348:{\displaystyle T_{n}^{m}(V)\cong L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};F)\cong L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};F).}
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in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span the
699:
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2035:
1994:{\displaystyle f(\alpha _{1},\ldots ,\alpha _{m},v_{1},\ldots ,v_{n})=T_{f}(\alpha _{1}\otimes \cdots \otimes \alpha _{m}\otimes v_{1}\otimes \cdots \otimes v_{n})}
72:
1699:
2593:{\displaystyle {\begin{aligned}T_{0}^{1}(V)&\cong L(V^{*};F)\cong V,\\T_{1}^{0}(V)&\cong L(V;F)=V^{*},\\T_{1}^{1}(V)&\cong L(V;V).\end{aligned}}}
1087:
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422:(for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type
2847:
2825:
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3419:
3098:
1821:{\displaystyle T_{f}\in L(\underbrace {V^{*}\otimes \cdots \otimes V^{*}} _{m}\otimes \underbrace {V\otimes \cdots \otimes V} _{n};W)}
2878:
2759:
2737:
2718:
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556:{\displaystyle T_{n}^{m}(V)=\underbrace {V\otimes \dots \otimes V} _{m}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{n}.}
94:
3300:
2815:
1411:
3151:
2680:
813:
3176:
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1228:
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708:
1515: = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from
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55:
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can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function
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1215:. Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of
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1671:{\displaystyle f\in L^{m+n}(\underbrace {V^{*},\ldots ,V^{*}} _{m},\underbrace {V,\ldots ,V} _{n};W)}
1192:
1184:
669:
2641:. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.
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inside the argument of the linear maps, and vice versa. (Note that in the former case, there are
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2007:
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to determine, and low rank decompositions of tensors are sometimes of great practical interest (
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is multilinear if it is linear in each argument. The space of all multilinear mappings from
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or more generally; and the rules for manipulations of tensors arise as an extension of
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The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a
30:
For an introduction to the nature and significance of tensors in a broad context, see
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154:, and then doesn't need to make reference to coordinates at all. The same is true in
1394:, and the "universal" approach carries over more easily to more general situations.
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is then defined to be an element of (i.e., a vector in) a vector space of the form:
3501:
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3255:
3103:
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1173:{\displaystyle T_{ij\dots }^{k\ell \dots }=a_{i}b_{j}\cdots c^{k}d^{\ell }\cdots .}
807:, pp. 4)) is a tensor that can be written as a product of tensors of the form
394:
225:
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147:
330:{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}}
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is the smallest number of such outer products that can be summed to produce it:
900:
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1216:
1071:{\displaystyle A=v_{1}w_{1}^{\mathrm {T} }+\cdots +v_{k}w_{k}^{\mathrm {T} }.}
637:
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803:(also called a tensor of rank one, elementary tensor or decomposable tensor (
17:
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3196:
1203:). In fact, the problem of finding the rank of an order 3 tensor over any
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An example of such a bilinear form may be defined, termed the associated
151:
406:. The tensors of order zero are just the scalars (elements of the field
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for instance. The rank of an order 3 or higher tensor is however often
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127:
concept. Their properties can be derived from their definitions, as
31:
1219:
can be recast as the problem of simultaneously evaluating a set of
881:. Every tensor can be expressed as a sum of simple tensors. The
2860:
2732:, Lecture Notes in Computer Science, vol. 245, Springer,
36:
2382:, and in the latter case vice versa). In particular, one has
2771:"An Introduction to Tensors and Group Theory for Physicists"
912:
vectors from a finite-dimensional vector space of dimension
2856:
162:. The component-free approach is also used extensively in
146:, an intrinsic geometric statement may be described by a
931:. A matrix thus has rank one if it can be written as an
1081:
In indices, a tensor of rank 1 is a tensor of the form
1459:{\displaystyle f:V_{1}\times \cdots \times V_{N}\to F}
190:. An overview of the subject can be found in the main
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410:), those of contravariant order 1 are the vectors in
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is the minimum number of simple tensors that sum to
847:{\displaystyle T=a\otimes b\otimes \cdots \otimes d}
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877:– that is, if the tensor is nonzero and completely
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329:
2362:in the definition of the tensor corresponds to a
1297:{\displaystyle z_{k}=\sum _{ij}T_{ijk}x_{i}y_{j}}
1543:universal characterization of the tensor product
768:{\displaystyle T_{2}^{0}(V)=V^{*}\otimes V^{*}.}
64:but its sources remain unclear because it lacks
2730:Lectures on the Complexity of Bilinear Problems
2082:Using the universal property, it follows, when
200:Definition via tensor products of vector spaces
2691:(2 ed.), Reading, Mass.: Addison-Wesley,
2872:
178:This article assumes an understanding of the
8:
1545:implies that, for each multilinear function
1321:. If a low-rank decomposition of the tensor
629:{\displaystyle T_{1}^{1}(V)=V\otimes V^{*},}
371:in our product, the tensor is said to be of
2840:Tensor Spaces and Numerical Tensor Calculus
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414:, and those of covariant order 1 are the
321:
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278:
95:Learn how and when to remove this message
3236:Covariance and contravariance of vectors
893:
2650:
701:corresponds in a natural way to a type
2817:The Art of Computer Programming vol. 2
1188:
27:Coordinate-free definition of a tensor
2637:is sometimes used as a shorthand for
2073:{\displaystyle \alpha _{i}\in V^{*}.}
1330:
7:
972:{\displaystyle A=vw^{\mathrm {T} }.}
2658:Håstad, Johan (November 15, 1989).
1191:, §51), and can be determined from
123:, expressing some definite type of
3099:Tensors in curvilinear coordinates
2820:(3rd ed.), pp. 145–146,
2711:Elements of Mathematics, Algebra I
1059:
1023:
960:
256:, an element of which is termed a
25:
640:in a natural way to the space of
170:, where tensors arise naturally.
2752:Finite-dimensional Vector Spaces
158:, of tensor fields describing a
41:
694:{\displaystyle V\times V\to F,}
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2511:
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2417:
2411:
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2229:
2141:
2132:
2126:
1988:
1924:
1908:
1844:
1815:
1727:
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1450:
1397:A scalar-valued function on a
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1363:
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727:
682:
601:
595:
465:
459:
1:
3152:Exterior covariant derivative
3084:Tensor (intrinsic definition)
1211:, and over the rationals, is
3177:Raising and lowering indices
2660:"Tensor Rank Is NP-Complete"
1375:{\displaystyle T_{n}^{m}(V)}
1325:is known, then an efficient
115:approach to the theory of a
3415:Gluon field strength tensor
385:and contravariant of order
3539:
3226:Cartan formalism (physics)
3046:Penrose graphical notation
2607:
2030:{\displaystyle v_{i}\in V}
1382:can be characterized by a
923:extends the notion of the
792:
265:tensor on the vector space
29:
2898:Glossary of tensor theory
2894:
2769:Jeevanjee, Nadir (2011),
2728:de Groote, H. F. (1987),
795:Tensor rank decomposition
781:, and is usually denoted
3482:Gregorio Ricci-Curbastro
3354:Riemann curvature tensor
3061:Van der Waerden notation
2689:Foundations of Mechanics
935:of two nonzero vectors:
908:when each product is of
50:This article includes a
3452:Elwin Bruno Christoffel
3385:Angular momentum tensor
3056:Tetrad (index notation)
3026:Abstract index notation
2842:, Springer, p. 4,
662:on a real vector space
389:and covariant of order
79:more precise citations.
3266:Levi-Civita connection
2594:
2349:
2074:
2031:
1995:
1822:
1695:
1672:
1460:
1376:
1298:
1174:
1072:
973:
848:
769:
695:
642:linear transformations
630:
557:
331:
3492:Jan Arnoldus Schouten
3447:Augustin-Louis Cauchy
2927:Differential geometry
2664:Journal of Algorithms
2625:must often deal with
2615:Differential geometry
2595:
2350:
2090:, that the space of (
2075:
2032:
1996:
1823:
1696:
1673:
1461:
1377:
1333:, pp. 506–508).
1299:
1175:
1073:
982:The rank of a matrix
974:
849:
770:
696:
631:
558:
332:
235:, one may form their
144:differential geometry
119:views a tensor as an
3467:Carl Friedrich Gauss
3400:stress–energy tensor
3395:Cauchy stress tensor
3147:Covariant derivative
3109:Antisymmetric tensor
3041:Multi-index notation
2389:
2108:
2041:
2008:
1838:
1708:
1685:
1552:
1412:
1388:multilinear mappings
1345:
1229:
1193:Gaussian elimination
1088:
993:
942:
814:
709:
670:
577:
441:
277:
3344:Nonmetricity tensor
3199:(2nd-order tensors)
3167:Hodge star operator
3157:Exterior derivative
3006:Transport phenomena
2991:Continuum mechanics
2947:Multilinear algebra
2836:Hackbusch, Wolfgang
2787:2012PhT....65d..64P
2713:, Springer-Verlag,
2685:Marsden, Jerrold E.
2548:
2479:
2410:
2125:
2100:natural isomorphism
2098:)-tensors admits a
1405:) of vector spaces
1362:
1327:evaluation strategy
1117:
1064:
1028:
929:range of the matrix
869:are nonzero and in
726:
594:
458:
204:Given a finite set
168:homological algebra
137:multilinear algebra
3477:Tullio Levi-Civita
3420:Metric tensor (GR)
3334:Levi-Civita symbol
3187:Tensor contraction
3001:General relativity
2937:Euclidean geometry
2590:
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2465:
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2088:finite dimensional
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2027:
1991:
1818:
1808:
1801:
1774:
1767:
1691:
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1651:
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1384:universal property
1372:
1348:
1337:Universal property
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969:
896:, II, §7, no. 8).
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580:
569:The space of type
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156:general relativity
52:list of references
3510:
3509:
3472:Hermann Grassmann
3428:
3427:
3380:Moment of inertia
3241:Differential form
3216:Affine connection
3031:Einstein notation
3014:
3013:
2942:Exterior calculus
2922:Coordinate system
2849:978-3-642-28027-6
2827:978-0-201-89684-8
2804:978-0-8176-4714-8
2795:10.1063/PT.3.1523
2707:Bourbaki, Nicolas
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1694:{\displaystyle W}
1630:
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1399:Cartesian product
1307:for given inputs
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507:
505:
473:
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160:physical property
105:
104:
97:
16:(Redirected from
3530:
3487:Bernhard Riemann
3319:
3162:Exterior product
3129:Two-point tensor
3114:Symmetric tensor
2996:Electromagnetism
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2830:
2812:Knuth, Donald E.
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2701:
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2631:smooth manifolds
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925:rank of a matrix
921:rank of a tensor
883:rank of a tensor
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164:abstract algebra
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75:this article by
66:inline citations
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3457:Albert Einstein
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3405:Einstein tensor
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3349:Ricci curvature
3329:Kronecker delta
3315:Notable tensors
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3231:Connection form
3208:
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3119:Tensor operator
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2986:Computer vision
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2957:Tensor calculus
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186:without chosen
121:abstract object
101:
90:
84:
81:
70:
56:related reading
46:
42:
35:
28:
23:
22:
15:
12:
11:
5:
3536:
3534:
3526:
3525:
3515:
3514:
3508:
3507:
3505:
3504:
3499:
3497:Woldemar Voigt
3494:
3489:
3484:
3479:
3474:
3469:
3464:
3462:Leonhard Euler
3459:
3454:
3449:
3444:
3438:
3436:
3434:Mathematicians
3430:
3429:
3426:
3425:
3423:
3422:
3417:
3412:
3407:
3402:
3397:
3392:
3387:
3382:
3376:
3374:
3370:
3369:
3367:
3366:
3361:
3359:Torsion tensor
3356:
3351:
3346:
3341:
3336:
3331:
3325:
3323:
3316:
3312:
3311:
3309:
3308:
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3268:
3263:
3258:
3253:
3248:
3243:
3238:
3233:
3228:
3223:
3218:
3212:
3210:
3204:
3203:
3201:
3200:
3194:
3192:Tensor product
3189:
3184:
3182:Symmetrization
3179:
3174:
3172:Lie derivative
3169:
3164:
3159:
3154:
3149:
3143:
3141:
3135:
3134:
3132:
3131:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3094:Tensor density
3091:
3086:
3080:
3078:
3072:
3071:
3069:
3068:
3066:Voigt notation
3063:
3058:
3053:
3051:Ricci calculus
3048:
3043:
3038:
3036:Index notation
3033:
3028:
3022:
3020:
3016:
3015:
3012:
3011:
3009:
3008:
3003:
2998:
2993:
2988:
2982:
2980:
2978:
2977:
2972:
2966:
2963:
2962:
2960:
2959:
2954:
2952:Tensor algebra
2949:
2944:
2939:
2934:
2932:Dyadic algebra
2929:
2924:
2918:
2916:
2907:
2903:
2902:
2895:
2892:
2891:
2886:
2884:
2883:
2876:
2869:
2861:
2855:
2854:
2848:
2832:
2826:
2808:
2803:
2766:
2760:
2744:
2738:
2725:
2719:
2703:
2697:
2681:Abraham, Ralph
2674:
2673:
2649:
2648:
2646:
2643:
2608:Main article:
2605:
2602:
2601:
2600:
2585:
2582:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2559:
2557:
2554:
2551:
2546:
2541:
2537:
2533:
2532:
2529:
2524:
2520:
2516:
2513:
2510:
2507:
2504:
2501:
2498:
2495:
2492:
2490:
2488:
2485:
2482:
2477:
2472:
2468:
2464:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2440:
2436:
2432:
2429:
2426:
2423:
2421:
2419:
2416:
2413:
2408:
2403:
2399:
2395:
2394:
2356:
2355:
2344:
2341:
2338:
2335:
2330:
2324:
2320:
2317:
2314:
2311:
2308:
2301:
2296:
2290:
2284:
2280:
2276:
2273:
2270:
2265:
2261:
2253:
2248:
2245:
2242:
2238:
2234:
2231:
2228:
2225:
2220:
2214:
2210:
2207:
2204:
2201:
2198:
2191:
2186:
2180:
2174:
2170:
2166:
2163:
2160:
2155:
2151:
2143:
2140:
2137:
2134:
2131:
2128:
2123:
2118:
2114:
2069:
2064:
2060:
2056:
2051:
2047:
2026:
2023:
2018:
2014:
2002:
2001:
1990:
1985:
1981:
1977:
1974:
1971:
1966:
1962:
1958:
1953:
1949:
1945:
1942:
1939:
1934:
1930:
1926:
1921:
1917:
1913:
1910:
1905:
1901:
1897:
1894:
1891:
1886:
1882:
1878:
1873:
1869:
1865:
1862:
1859:
1854:
1850:
1846:
1843:
1829:
1828:
1817:
1814:
1811:
1806:
1800:
1796:
1793:
1790:
1787:
1784:
1777:
1772:
1766:
1760:
1756:
1752:
1749:
1746:
1741:
1737:
1729:
1726:
1723:
1718:
1714:
1690:
1679:
1678:
1667:
1664:
1661:
1656:
1650:
1646:
1643:
1640:
1637:
1634:
1627:
1622:
1616:
1610:
1606:
1602:
1599:
1596:
1591:
1587:
1579:
1574:
1571:
1568:
1564:
1560:
1557:
1503:
1497:
1480:
1474:
1467:
1466:
1455:
1452:
1447:
1443:
1439:
1436:
1433:
1428:
1424:
1420:
1417:
1371:
1368:
1365:
1360:
1355:
1351:
1338:
1335:
1317:
1310:
1305:
1304:
1291:
1287:
1281:
1277:
1271:
1268:
1265:
1261:
1255:
1252:
1248:
1244:
1239:
1235:
1221:bilinear forms
1201:de Groote 1987
1181:
1180:
1169:
1166:
1161:
1157:
1151:
1147:
1143:
1138:
1134:
1128:
1124:
1120:
1115:
1112:
1109:
1104:
1101:
1098:
1094:
1079:
1078:
1067:
1061:
1055:
1051:
1045:
1041:
1037:
1034:
1031:
1025:
1019:
1015:
1009:
1005:
1001:
998:
980:
979:
968:
962:
957:
953:
950:
947:
855:
854:
843:
840:
837:
834:
831:
828:
825:
822:
819:
805:Hackbusch 2012
793:Main article:
790:
787:
764:
759:
755:
751:
746:
742:
738:
735:
732:
729:
724:
719:
715:
690:
687:
684:
681:
678:
675:
625:
620:
616:
612:
609:
606:
603:
600:
597:
592:
587:
583:
564:
563:
552:
547:
541:
535:
531:
527:
524:
521:
516:
512:
504:
499:
493:
489:
486:
483:
480:
477:
470:
467:
464:
461:
456:
451:
447:
338:
337:
324:
320:
316:
313:
310:
305:
301:
297:
294:
291:
288:
285:
282:
250:
243:
237:tensor product
228:over a common
217:
210:
201:
198:
197:
196:
180:tensor product
133:linear algebra
113:component-free
103:
102:
60:external links
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3535:
3524:
3521:
3520:
3518:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3439:
3437:
3435:
3431:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3378:
3377:
3375:
3371:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3339:Metric tensor
3337:
3335:
3332:
3330:
3327:
3326:
3324:
3320:
3317:
3313:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
3251:Exterior form
3249:
3247:
3244:
3242:
3239:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3213:
3211:
3205:
3198:
3195:
3193:
3190:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3144:
3142:
3140:
3136:
3130:
3127:
3125:
3124:Tensor bundle
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3081:
3079:
3073:
3067:
3064:
3062:
3059:
3057:
3054:
3052:
3049:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3023:
3021:
3017:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2983:
2981:
2976:
2973:
2971:
2968:
2967:
2964:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2919:
2917:
2915:
2911:
2908:
2904:
2900:
2899:
2893:
2889:
2882:
2877:
2875:
2870:
2868:
2863:
2862:
2859:
2851:
2845:
2841:
2837:
2833:
2829:
2823:
2819:
2818:
2813:
2809:
2806:
2800:
2796:
2792:
2788:
2784:
2780:
2776:
2775:Physics Today
2772:
2767:
2763:
2761:0-387-90093-4
2757:
2753:
2749:
2745:
2741:
2739:3-540-17205-X
2735:
2731:
2726:
2722:
2720:3-540-64243-9
2716:
2712:
2708:
2704:
2700:
2698:0-201-40840-6
2694:
2690:
2686:
2682:
2678:
2677:
2669:
2665:
2661:
2654:
2651:
2644:
2642:
2640:
2636:
2632:
2628:
2627:tensor fields
2624:
2620:
2616:
2611:
2604:Tensor fields
2603:
2583:
2577:
2574:
2571:
2565:
2562:
2560:
2552:
2544:
2539:
2535:
2527:
2522:
2518:
2514:
2508:
2505:
2502:
2496:
2493:
2491:
2483:
2475:
2470:
2466:
2458:
2455:
2452:
2446:
2443:
2438:
2434:
2427:
2424:
2422:
2414:
2406:
2401:
2397:
2385:
2384:
2383:
2381:
2377:
2373:
2369:
2365:
2361:
2342:
2336:
2333:
2328:
2322:
2318:
2315:
2312:
2309:
2306:
2299:
2294:
2288:
2282:
2278:
2274:
2271:
2268:
2263:
2259:
2246:
2243:
2240:
2236:
2232:
2226:
2223:
2218:
2212:
2208:
2205:
2202:
2199:
2196:
2189:
2184:
2178:
2172:
2168:
2164:
2161:
2158:
2153:
2149:
2138:
2135:
2129:
2121:
2116:
2112:
2104:
2103:
2102:
2101:
2097:
2093:
2089:
2085:
2080:
2067:
2062:
2058:
2054:
2049:
2045:
2024:
2021:
2016:
2012:
1983:
1979:
1975:
1972:
1969:
1964:
1960:
1956:
1951:
1947:
1943:
1940:
1937:
1932:
1928:
1919:
1915:
1911:
1903:
1899:
1895:
1892:
1889:
1884:
1880:
1876:
1871:
1867:
1863:
1860:
1857:
1852:
1848:
1841:
1834:
1833:
1832:
1812:
1809:
1804:
1798:
1794:
1791:
1788:
1785:
1782:
1775:
1770:
1764:
1758:
1754:
1750:
1747:
1744:
1739:
1735:
1724:
1721:
1716:
1712:
1704:
1703:
1702:
1688:
1662:
1659:
1654:
1648:
1644:
1641:
1638:
1635:
1632:
1625:
1620:
1614:
1608:
1604:
1600:
1597:
1594:
1589:
1585:
1572:
1569:
1566:
1562:
1558:
1555:
1548:
1547:
1546:
1544:
1539:
1535:
1531:
1527:
1522:
1518:
1514:
1510:
1506:
1496:
1492:
1488:
1483:
1473:
1453:
1445:
1441:
1437:
1434:
1431:
1426:
1422:
1418:
1415:
1408:
1407:
1406:
1404:
1400:
1395:
1393:
1389:
1385:
1366:
1358:
1353:
1349:
1336:
1334:
1332:
1328:
1324:
1320:
1313:
1289:
1285:
1279:
1275:
1269:
1266:
1263:
1259:
1253:
1250:
1246:
1242:
1237:
1233:
1225:
1224:
1223:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1194:
1190:
1186:
1167:
1164:
1159:
1155:
1149:
1145:
1141:
1136:
1132:
1126:
1122:
1118:
1113:
1110:
1107:
1102:
1099:
1096:
1092:
1084:
1083:
1082:
1065:
1053:
1049:
1043:
1039:
1035:
1032:
1029:
1017:
1013:
1007:
1003:
999:
996:
989:
988:
987:
985:
966:
955:
951:
948:
945:
938:
937:
936:
934:
933:outer product
930:
926:
922:
917:
915:
911:
907:
902:
897:
895:
894:Bourbaki 1989
891:
887:
884:
880:
876:
872:
868:
864:
860:
841:
838:
835:
832:
829:
826:
823:
820:
817:
810:
809:
808:
806:
802:
801:simple tensor
796:
788:
786:
784:
780:
779:
778:metric tensor
762:
757:
753:
749:
744:
740:
736:
730:
722:
717:
713:
688:
685:
679:
676:
673:
665:
661:
660:bilinear form
657:
653:
651:
647:
643:
639:
623:
618:
614:
610:
607:
604:
598:
590:
585:
581:
568:
550:
545:
539:
533:
529:
525:
522:
519:
514:
510:
502:
497:
491:
487:
484:
481:
478:
475:
468:
462:
454:
449:
445:
437:
436:
435:
431:
427:
421:
417:
413:
409:
404:
400:
396:
393:and of total
392:
388:
384:
381:
377:
370:
366:
362:
358:
355:If there are
353:
351:
347:
343:
322:
318:
314:
311:
308:
303:
299:
295:
292:
289:
286:
283:
280:
273:
272:
271:
269:
266:
261:
259:
253:
249:
242:
238:
234:
231:
227:
226:vector spaces
220:
216:
209:
199:
195:
193:
189:
185:
184:vector spaces
181:
177:
173:
172:
171:
169:
165:
161:
157:
153:
149:
145:
140:
138:
134:
130:
126:
122:
118:
114:
111:, the modern
110:
99:
96:
88:
78:
74:
68:
67:
61:
57:
53:
48:
39:
38:
33:
19:
18:Simple tensor
3502:Hermann Weyl
3306:Vector space
3291:Pseudotensor
3256:Fiber bundle
3209:abstractions
3104:Mixed tensor
3089:Tensor field
3083:
2896:
2839:
2816:
2778:
2774:
2754:, Springer,
2751:
2748:Halmos, Paul
2729:
2710:
2688:
2667:
2663:
2653:
2639:tensor field
2638:
2634:
2613:
2610:tensor field
2379:
2375:
2371:
2367:
2363:
2359:
2357:
2095:
2091:
2083:
2081:
2003:
1830:
1680:
1540:
1533:
1529:
1525:
1520:
1516:
1512:
1508:
1501:
1494:
1490:
1486:
1478:
1471:
1468:
1396:
1392:free modules
1386:in terms of
1340:
1322:
1315:
1308:
1306:
1205:finite field
1196:
1182:
1080:
983:
981:
920:
918:
913:
909:
905:
898:
889:
885:
882:
879:factorizable
874:
870:
866:
862:
858:
856:
800:
798:
782:
776:
663:
655:
654:
649:
645:
566:
565:
429:
425:
419:
411:
407:
402:
398:
390:
386:
379:
375:
372:
368:
364:
360:
356:
354:
349:
341:
339:
267:
264:
262:
257:
251:
247:
240:
232:
218:
214:
207:
203:
175:
174:
148:tensor field
141:
116:
106:
91:
85:October 2023
82:
71:Please help
63:
3442:Élie Cartan
3390:Spin tensor
3364:Weyl tensor
3322:Mathematics
3286:Multivector
3077:definitions
2975:Engineering
2914:Mathematics
2633:. The term
2623:engineering
1523:is denoted
1489:is denoted
1217:polynomials
1209:NP-Complete
1189:Halmos 1974
901:zero tensor
789:Tensor rank
434:is denoted
129:linear maps
125:multilinear
109:mathematics
77:introducing
3271:Linear map
3139:Operations
2670:: 644–654.
2645:References
2378:copies of
2370:copies of
1831:such that
1403:direct sum
1341:The space
1331:Knuth 1998
1329:is known (
705:tensor in
656:Example 2.
638:isomorphic
567:Example 1.
367:copies of
359:copies of
346:dual space
3410:EM tensor
3246:Dimension
3197:Transpose
2814:(1998) ,
2781:(4): 64,
2563:≅
2523:∗
2494:≅
2453:≅
2439:∗
2425:≅
2323:⏟
2313:…
2289:⏟
2283:∗
2272:…
2264:∗
2233:≅
2213:⏟
2206:⊗
2203:⋯
2200:⊗
2190:⊗
2179:⏟
2173:∗
2165:⊗
2162:⋯
2159:⊗
2154:∗
2136:≅
2063:∗
2055:∈
2046:α
2022:∈
1976:⊗
1973:⋯
1970:⊗
1957:⊗
1948:α
1944:⊗
1941:⋯
1938:⊗
1929:α
1893:…
1868:α
1861:…
1849:α
1799:⏟
1792:⊗
1789:⋯
1786:⊗
1776:⊗
1765:⏟
1759:∗
1751:⊗
1748:⋯
1745:⊗
1740:∗
1722:∈
1649:⏟
1639:…
1615:⏟
1609:∗
1598:…
1590:∗
1559:∈
1451:→
1438:×
1435:⋯
1432:×
1247:∑
1197:very hard
1165:⋯
1160:ℓ
1142:⋯
1114:…
1111:ℓ
1103:…
1033:⋯
919:The term
839:⊗
836:⋯
833:⊗
827:⊗
758:∗
750:⊗
745:∗
683:→
677:×
619:∗
611:⊗
573:tensors,
540:⏟
534:∗
526:⊗
523:⋯
520:⊗
515:∗
503:⊗
492:⏟
485:⊗
482:⋯
479:⊗
416:one-forms
323:∗
315:⊗
312:⋯
309:⊗
304:∗
296:⊗
290:⊗
287:⋯
284:⊗
3517:Category
3276:Manifold
3261:Geodesic
3019:Notation
2838:(2012),
2750:(1974),
2709:(1989),
2687:(1985),
2004:for all
1511:). When
1477:× ... ×
246:⊗ ... ⊗
194:article.
152:manifold
3523:Tensors
3373:Physics
3207:Related
2970:Physics
2888:Tensors
2783:Bibcode
2619:physics
1681:(where
1507:;
1500:, ...,
1213:NP-Hard
865:, ...,
344:is the
213:, ...,
73:improve
3301:Vector
3296:Spinor
3281:Matrix
3075:Tensor
2846:
2824:
2801:
2758:
2736:
2717:
2695:
2635:tensor
1185:matrix
857:where
703:(0, 2)
571:(1, 1)
374:type (
340:where
258:tensor
192:tensor
117:tensor
32:Tensor
3221:Basis
2906:Scope
2358:Each
644:from
395:order
230:field
188:bases
176:Note:
150:on a
58:, or
2844:ISBN
2822:ISBN
2799:ISBN
2756:ISBN
2734:ISBN
2715:ISBN
2693:ISBN
2621:and
2374:and
2037:and
1541:The
1401:(or
1314:and
899:The
363:and
166:and
2791:doi
2629:on
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1519:to
1485:to
1207:is
873:or
652:.
648:to
636:is
418:in
348:of
224:of
182:of
142:In
135:to
107:In
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2789:,
2779:65
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2683:;
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861:,
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666:,
658:A
428:,
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352:.
263:A
260:.
206:{
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2880:e
2873:t
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2584:.
2581:)
2578:V
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1984:n
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1168:.
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1024:T
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997:A
984:A
967:.
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892:(
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818:T
783:g
763:.
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731:V
728:(
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714:T
689:,
686:F
680:V
674:V
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599:V
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551:.
546:n
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222:}
219:n
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