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3171:
2606:. Thus we can regard covector fields not just as sections of the cotangent bundle, but also linear mappings of vector fields into functions. By the double-dual construction, vector fields can similarly be expressed as mappings of covector fields into functions (namely, we could start "natively" with covector fields and work up from there).
3641:
approach rather than truly a foundational one. Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses. The construction of tensor densities is a 'twisting' at the cocycle level. Geometers have not been in any doubt about the
3874:
times. While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the
Jacobian determinant, or its absolute value. Non-integral powers of the (positive)
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This kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to a function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously.
1550:. (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space
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The concept of a tensor field can be generalized by considering objects that transform differently. An object that transforms as an ordinary tensor field under coordinate transformations, except that it is also multiplied by the determinant of the
3850:
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transition functions of the bundle of densities make sense, so that the weight of a density, in that sense, is not restricted to integer values. Restricting to changes of coordinates with positive
Jacobian determinant is possible on
531:
In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.
3606:
make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion.
1618:
3166:{\displaystyle \underbrace {\Omega ^{1}(M)\times \ldots \times \Omega ^{1}(M)} _{l\ \mathrm {times} }\times \underbrace {{\mathfrak {X}}(M)\times \ldots \times {\mathfrak {X}}(M)} _{k\ \mathrm {times} }\to C^{\infty }(M).}
2328:
288:
1333:{\displaystyle {T_{i_{1}\cdots i_{p}}}^{j_{1}\cdots j_{q}}\mapsto A_{i_{1}}^{i'_{1}}\cdots A_{i_{p}}^{i'_{p}}{T_{i'_{1}\cdots i'_{p}}}^{j'_{1}\cdots j'_{q}}(A^{-1})_{j'_{1}}^{j_{1}}\cdots (A^{-1})_{j'_{q}}^{j_{q}}.}
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could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is
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A special case are the scalar densities. Scalar 1-densities are especially important because it makes sense to define their integral over a manifold. They appear, for instance, in the
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posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to
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552:), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems.
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Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a
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in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of tensor giving rise to tensor fields.
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The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold
2584:{\displaystyle {\tilde {\omega }}(fX)(p)=\omega (p)((fX)(p))=\omega (p)(f(p)X(p))=f(p)\omega (p)(X(p))=(f\omega )(p)(X(p))=(f{\tilde {\omega }})(X)(p)}
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More generally, any tensor density is the product of an ordinary tensor with a scalar density of the appropriate weight. In the language of
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usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (C) vector fields on
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3879:, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of
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in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold
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2106:), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over any
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3523:; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a
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3752:. The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition:
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The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle
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What is usually spoken of as the 'classical' approach to tensors tries to read this backwards – and is therefore a heuristic,
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properties of tensor constructions to the chain rule itself; this is why they also are intrinsic (read, 'natural') concepts.
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to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold
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is an invariant object that does not depend on the coordinate system chosen. More generally, a tensor of valence (
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Now consider more complicated fields. For example, if the manifold is
Riemannian, then it has a metric field
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The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be
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carries over in a routine way – again independently of coordinates, as mentioned in the introduction.
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There is another more abstract (but often useful) way of characterizing tensor fields on a manifold
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concept is independent of any choice of basis, taking the tensor product of two vector bundles on
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In a complete parallel to the construction of ordinary single tensors (not tensor fields!) on
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4203:. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press.
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of the transition functions (in the given class). Likewise, a contravariant vector field
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1753:. Thus a tensor section is not only a linear map on the vector space of sections, but a
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143:. Many mathematical structures called "tensors" are also tensor fields. For example, the
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3845:{\displaystyle \det(g')=\left(\det {\frac {\partial x}{\partial x'}}\right)^{2}\det(g),}
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of the coordinates that, under such an affine transformation undergoes a transformation
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is a weather map showing horizontal wind velocity at each point of the Earth's surface.
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111:(a magnitude and a direction, like velocity), a tensor field is a generalization of a
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taking a pair of vector fields to a vector field, does not define a tensor field on
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is important in physics, and these two tensors are related by
Einstein's theory of
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1644:
1613:{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*},}
123:
that assigns, respectively, a scalar or vector to each point of space. If a tensor
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of sections. This property is used to check, for example, that even though the
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that are infinitely-differentiable. A tensor field is an element of this set.
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in general relativity. The most common example of a scalar 1-density is the
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283:{\displaystyle T\ \in \ \Gamma (M,V^{\otimes p}\otimes (V^{*})^{\otimes q})}
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is the dimension of the space), as opposed to taking their values in just
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A frequent example application of this general rule is showing that the
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548:, the concept of a tensor relies on a concept of a reference frame (or
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3950:" employed in the English translation of Schouten is no longer in use.
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One feature of the bundle of densities (again assuming orientability)
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Because of the pointwise nature of everything involved, the action of
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as multilinear maps on vectors and covectors, we can regard general (
1876:. This should not be confused with the very similar looking notation
963:
This is precisely the requirement needed to ensure that the quantity
2018:
multilinear mappings), though of a different type (although this is
27:
Assignment of a tensor continuously varying across a region of space
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which is the transformation law for a scalar density of weight +2.
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The curvature tensor is discussed in differential geometry and the
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where the fiber is a tensor product of any number of copies of the
172:
72:
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4300:
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849:{\displaystyle \mathbf {e} _{k}\mapsto A_{k}^{i}\mathbf {e} _{i}}
44:
3403:. Even to formulate such equations requires a fresh notion, the
1433:
of the base space, which is a manifold. As such, the fiber is a
797:
transforms as a covector, since under the affine transformation
5307:
4664:
4256:
1931:
Curly (script) letters are sometimes used to denote the set of
362:
56:
3407:. This handles the formulation of variation of a tensor field
677:). A covariant vector, or covector, is a system of functions
107:(a pure number representing a value, for example speed) and a
3883:-forms are distinct. For more on the intrinsic meaning, see
2083:{\displaystyle {\mathfrak {X}}(M):={\mathcal {T}}_{0}^{1}(M)}
704:
that transforms under this affine transformation by the rule
3437:
An extension of the tensor field idea incorporates an extra
3908: – Tensor index notation for tensor-based calculations
3464:
is a bundle of vector spaces of just the same dimension as
2271:{\displaystyle {\tilde {\omega }}(X)(p):=\omega (p)(X(p)).}
4225:
2090:(see the section on notation above) as a single space — a
4014:
Semi-Riemannian
Geometry With Applications to Relativity
3936:
Semi-Riemannian
Geometry With Applications to Relativity
3372:, the electric and magnetic fields are combined into an
2173:{\displaystyle \Omega ^{1}(M)={\mathcal {T}}_{1}^{0}(M)}
99:
in material object, and in numerous applications in the
3916:
Pages displaying short descriptions of redirect targets
3492:. (To be strictly accurate, one should also apply the
2810:), it turns out that it arises from a tensor field on
2014:, which makes tensor fields into honest tensors (i.e.
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Half-densities are applied in areas such as defining
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upstairs indices, with the transformation law being
3425:, led to the isolation of the geometric concept of
3342:{\displaystyle \nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y}
1450:
vector space of one dimension depending on an angle
953:{\displaystyle v^{k}\mapsto (A^{-1})_{j}^{k}v^{j}.}
856:. A contravariant vector is a system of functions
4164:McGraw Hill Encyclopaedia of Physics (2nd Edition)
4161:
3844:
3744:
3626:Abstractly, we can identify the chain rule as a 1-
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3682:th power, is called a tensor density with weight
659:{\displaystyle x^{k}\mapsto A_{j}^{k}x^{j}+a^{k}}
5760:
3827:
3788:
3762:
3734:
3678:of the inverse coordinate transformation to the
3654:argument justifies abstractly the whole theory.
3594:and all the fields are taken to be invariant by
1686:can be considered itself as a tensor section of
4125:C. Misner, K. S. Thorne, J. A. Wheeler (1973),
768:The list of Cartesian coordinate basis vectors
3504:.) For a more traditional explanation see the
1468:, the corresponding field concept is called a
321:Equivalently, it is a collection of elements
4897:
4241:
3184:, which is a mapping of smooth vector fields
535:
151:as it associates a tensor to each point of a
4150:: CS1 maint: multiple names: authors list (
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3432:
2113:As a motivating example, consider the space
758:{\displaystyle v_{k}\mapsto v_{i}A_{k}^{i}.}
4071:
4011:
3717:, which in the presence of a metric tensor
3610:
3468:. This allows one to define the concept of
1714:{\displaystyle \scriptstyle E^{*}\otimes F}
1437:and the tensor bundle is a special kind of
4904:
4890:
4248:
4234:
3870:that can be used to 'twist' other bundles
3519:is well-defined for real number values of
1382:of the coordinates that transforms by the
555:For example, coordinates belonging to the
4084:
3244:
3225:{\displaystyle (X,Y)\mapsto \nabla _{X}Y}
1980:{\displaystyle {\mathcal {T}}_{n}^{m}(M)}
1355:, etc.). A covector field is a function
571:
545:
383:
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4605:Covariance and contravariance of vectors
4195:
4182:
541:
103:. As a tensor is a generalization of a
4105:
4030:
14:
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4159:
4077:Encyclopaedia of Physics (2nd Edition)
4055:Relativity, Gravitation, and Cosmology
3500:– this makes little difference for an
3472:, a 'twisted' type of tensor field. A
2814:if and only if it is multilinear over
1538:We therefore can give a definition of
4885:
4229:
4036:The Geometry of Physics (3rd edition)
3694:such as the (1-dimensional) space of
3547:. In general we can take sections of
995:{\displaystyle v^{k}\mathbf {e} _{k}}
2890:over M is canonically isomorphic to
1862:{\displaystyle T_{0}^{1}(M)=T(M)=TM}
1413:transforms by the inverse Jacobian.
3662:
3092:
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2035:
1533:component-free treatment of tensors
1531:) the whole apparatus explained at
24:
4468:Tensors in curvilinear coordinates
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3615:As an advanced explanation of the
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2755:{\displaystyle {\mathfrak {X}}(M)}
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2663:{\displaystyle {\mathfrak {X}}(M)}
2303:{\displaystyle {\tilde {\omega }}}
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2006:Tensor fields as multilinear forms
1952:
415:, such that given any two vectors
223:
127:is defined on a vector fields set
25:
5800:
2858:-module of tensor fields of type
2721:Now, given any arbitrary mapping
1416:
3862:, the determinant bundle of the
3745:{\displaystyle {\sqrt {\det g}}}
3581:
3452:is the tensor product bundle of
1801:) might sometimes be written as
1678:) from the space of sections of
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836:
806:
790:{\displaystyle \mathbf {e} _{k}}
777:
584:{\displaystyle \mathbb {R} ^{n}}
4088:Applications of Tensor Analysis
3619:concept, one can interpret the
3419:notion, which was later called
3352:
4944:Differentiable/Smooth manifold
4188:Tensor Analysis for Physicists
4057:, Cambridge University Press,
4038:, Cambridge University Press,
3977:
3953:
3940:
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3417:absolute differential calculus
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3270:)-linearity, it satisfies the
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2955:{\displaystyle C^{\infty }(M)}
2949:
2943:
2919:{\displaystyle C^{\infty }(M)}
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2877:
2865:
2851:{\displaystyle C^{\infty }(M)}
2845:
2839:
2795:{\displaystyle \Omega ^{1}(M)}
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2703:{\displaystyle \Omega ^{1}(M)}
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4521:Exterior covariant derivative
4453:Tensor (intrinsic definition)
4201:The Topology of Fibre Bundles
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3236:. This is because it is only
162:
4546:Raising and lowering indices
3374:electromagnetic tensor field
3251:{\displaystyle \mathbb {R} }
1913:{\displaystyle T_{0}^{1}(V)}
1721:if and only if it satisfies
79:. Tensor fields are used in
30:Not to be confused with the
7:
5650:Classification of manifolds
4784:Gluon field strength tensor
4255:
4053:Lambourne , R.J.A. (2010),
3990:Encyclopedia of Mathematics
3962:"Notes on Smooth Manifolds"
3914: – Geometric structure
3890:
2180:of smooth covector fields (
1784:
1745:) and each smooth function
10:
5805:
4595:Cartan formalism (physics)
4415:Penrose graphical notation
3721:is the square root of its
3666:
3476:is the special case where
536:Via coordinate transitions
496:{\displaystyle g_{x}(v,w)}
29:
5726:over commutative algebras
5683:
5642:
5575:
5472:
5368:
5315:
5306:
5142:
5065:
5004:
4924:
4801:
4741:
4690:
4683:
4575:
4506:
4443:
4387:
4334:
4281:
4274:
4267:Glossary of tensor theory
4263:
4190:, Oxford University Press
4131:, W.H. Freeman & Co,
4085:McConnell, A. J. (1957),
4012:O'neill, Barrett (1983).
3690:taking their values in a
3433:Twisting by a line bundle
1990:are the sections of the (
1933:infinitely-differentiable
1650:Given two tensor bundles
1508:is the vector space "at"
461:, their inner product is
5442:Riemann curvature tensor
4851:Gregorio Ricci-Curbastro
4723:Riemann curvature tensor
4430:Van der Waerden notation
3921:
3725:in coordinates, denoted
3611:Cocycles and chain rules
3551:, the tensor product of
3379:It is worth noting that
1460:. Given a vector bundle
1456:or alternatively like a
145:Riemann curvature tensor
32:Tensor product of fields
4821:Elwin Bruno Christoffel
4754:Angular momentum tensor
4425:Tetrad (index notation)
4395:Abstract index notation
3711:Einstein–Hilbert action
3482:densities on a manifold
2322:)-linear map, that is,
1014:downstairs indices and
5789:Functions and mappings
5234:Manifold with boundary
4949:Differential structure
4635:Levi-Civita connection
4184:Schouten, Jan Arnoldus
4108:Relativity DeMystified
4091:, Dover Publications,
4075:; Trigg, G.L. (1991),
3846:
3746:
3576:geometric quantization
3397:differential equations
3343:
3252:
3226:
3182:Levi-Civita connection
3167:
2956:
2920:
2884:
2852:
2796:
2756:
2704:
2664:
2585:
2304:
2272:
2174:
2084:
1981:
1914:
1863:
1715:
1614:
1407:
1376:
1334:
996:
954:
877:
850:
791:
759:
698:
673:-dimensional indices,
660:
593:affine transformations
585:
517:
497:
455:
435:
409:
384:Geometric introduction
319:
284:
5779:Differential topology
5774:Differential geometry
4861:Jan Arnoldus Schouten
4816:Augustin-Louis Cauchy
4296:Differential geometry
4160:Parker, C.B. (1994),
4110:, McGraw Hill (USA),
3885:density on a manifold
3847:
3747:
3561:tensor density fields
3401:differential calculus
3344:
3253:
3227:
3168:
2957:
2921:
2885:
2883:{\displaystyle (k,l)}
2853:
2797:
2757:
2705:
2665:
2586:
2305:
2273:
2175:
2098:of smooth functions,
2085:
1982:
1915:
1864:
1781:built from them are.
1773:are not tensors, the
1716:
1615:
1480:, a choice of vector
1421:A tensor bundle is a
1408:
1406:{\displaystyle v^{k}}
1377:
1375:{\displaystyle v_{k}}
1335:
997:
955:
878:
876:{\displaystyle v^{k}}
851:
792:
760:
699:
697:{\displaystyle v_{k}}
661:
586:
561:real coordinate space
518:
498:
456:
436:
410:
285:
184:
91:, in the analysis of
81:differential geometry
5381:Covariant derivative
4932:Topological manifold
4836:Carl Friedrich Gauss
4769:stress–energy tensor
4764:Cauchy stress tensor
4516:Covariant derivative
4478:Antisymmetric tensor
4410:Multi-index notation
4106:McMahon, D. (2006),
4016:. Elsevier Science.
3877:orientable manifolds
3759:
3729:
3498:transition functions
3405:covariant derivative
3359:stress–energy tensor
3277:
3240:
3188:
2972:
2930:
2894:
2862:
2826:
2770:
2733:
2678:
2641:
2602:and smooth function
2329:
2285:
2199:
2117:
2030:
1946:
1883:
1808:
1771:covariant derivative
1761:)-linear map on the
1737:), for each section
1690:
1557:
1390:
1359:
1025:
967:
890:
860:
801:
772:
711:
681:
602:
566:
507:
465:
445:
419:
399:
208:
5769:Multilinear algebra
5415:Exterior derivative
5017:Atiyah–Singer index
4966:Riemannian manifold
4713:Nonmetricity tensor
4568:(2nd-order tensors)
4536:Hodge star operator
4526:Exterior derivative
4375:Transport phenomena
4360:Continuum mechanics
4316:Multilinear algebra
3502:orientable manifold
3393:theoretical physics
2621:) tensor fields on
2188:and a vector field
2160:
2070:
1998:) tensor bundle on
1967:
1900:
1825:
1472:of the bundle: for
1326:
1312:
1275:
1261:
1225:
1209:
1192:
1176:
1156:
1154:
1121:
1119:
936:
833:
751:
632:
526:Tissot's indicatrix
434:{\displaystyle v,w}
317:of vector bundles.
175:, for instance the
153:Riemannian manifold
59:to each point of a
5721:Secondary calculus
5675:Singularity theory
5630:Parallel transport
5398:De Rham cohomology
5037:Generalized Stokes
4846:Tullio Levi-Civita
4789:Metric tensor (GR)
4703:Levi-Civita symbol
4556:Tensor contraction
4370:General relativity
4306:Euclidean geometry
3960:Claudio Gorodski.
3934:O'Neill, Barrett.
3842:
3742:
3598:by the vectors of
3574:on manifolds, and
3572:integral operators
3486:determinant bundle
3395:and other fields,
3381:differential forms
3363:general relativity
3339:
3262:(in place of full
3248:
3222:
3163:
3137:
3110:
3057:
3030:
2952:
2916:
2880:
2848:
2792:
2752:
2725:from a product of
2700:
2660:
2581:
2300:
2268:
2170:
2142:
2080:
2052:
1977:
1949:
1910:
1886:
1859:
1811:
1711:
1710:
1631:at that point and
1610:
1452:could look like a
1403:
1372:
1330:
1300:
1295:
1249:
1244:
1213:
1197:
1180:
1164:
1142:
1125:
1107:
1090:
992:
950:
922:
873:
846:
819:
787:
755:
737:
694:
656:
618:
581:
513:
493:
451:
431:
405:
280:
139:a tensor field on
89:general relativity
85:algebraic geometry
65:mathematical space
5756:
5755:
5638:
5637:
5403:Differential form
5057:Whitney embedding
4991:Differential form
4879:
4878:
4841:Hermann Grassmann
4797:
4796:
4749:Moment of inertia
4610:Differential form
4585:Affine connection
4400:Einstein notation
4383:
4382:
4311:Exterior calculus
4291:Coordinate system
4210:978-0-691-00548-5
4064:978-0-521-13138-4
4045:978-1-107-60260-1
3814:
3740:
3646:nature of tensor
3527:, the case where
3480:is the bundle of
3118:
3063:
3061:
3038:
2977:
2975:
2964:multilinear forms
2557:
2341:
2297:
2211:
1935:tensor fields on
1779:curvature tensors
1448:. For example, a
675:summation implied
550:coordinate system
516:{\displaystyle g}
454:{\displaystyle x}
408:{\displaystyle g}
342:) for all points
222:
216:
157:topological space
101:physical sciences
16:(Redirected from
5796:
5748:Stratified space
5706:Fréchet manifold
5420:Interior product
5313:
5312:
5010:
4906:
4899:
4892:
4883:
4882:
4856:Bernhard Riemann
4688:
4687:
4531:Exterior product
4498:Two-point tensor
4483:Symmetric tensor
4365:Electromagnetism
4279:
4278:
4250:
4243:
4236:
4227:
4226:
4222:
4199:(5 April 1999).
4197:Steenrod, Norman
4191:
4178:
4167:
4155:
4149:
4141:
4120:
4101:
4080:
4079:, VHC Publishers
4067:
4048:
4027:
3999:
3998:
3985:"Tensor density"
3981:
3975:
3974:
3972:
3971:
3966:
3957:
3951:
3944:
3938:
3932:
3917:
3902:
3851:
3849:
3848:
3843:
3826:
3825:
3820:
3816:
3815:
3813:
3812:
3800:
3792:
3775:
3751:
3749:
3748:
3743:
3741:
3733:
3688:multilinear maps
3663:Tensor densities
3546:
3544:
3543:
3540:
3537:
3490:cotangent bundle
3370:electromagnetism
3348:
3346:
3345:
3340:
3335:
3334:
3289:
3288:
3257:
3255:
3254:
3249:
3247:
3231:
3229:
3228:
3223:
3218:
3217:
3172:
3170:
3169:
3164:
3150:
3149:
3136:
3135:
3116:
3111:
3106:
3096:
3095:
3071:
3070:
3056:
3055:
3036:
3031:
3026:
3016:
3015:
2988:
2987:
2961:
2959:
2958:
2953:
2942:
2941:
2925:
2923:
2922:
2917:
2906:
2905:
2889:
2887:
2886:
2881:
2857:
2855:
2854:
2849:
2838:
2837:
2801:
2799:
2798:
2793:
2782:
2781:
2761:
2759:
2758:
2753:
2742:
2741:
2709:
2707:
2706:
2701:
2690:
2689:
2669:
2667:
2666:
2661:
2650:
2649:
2590:
2588:
2587:
2582:
2559:
2558:
2550:
2343:
2342:
2334:
2309:
2307:
2306:
2301:
2299:
2298:
2290:
2277:
2275:
2274:
2269:
2213:
2212:
2204:
2179:
2177:
2176:
2171:
2159:
2154:
2149:
2148:
2129:
2128:
2108:commutative ring
2089:
2087:
2086:
2081:
2069:
2064:
2059:
2058:
2039:
2038:
1986:
1984:
1983:
1978:
1966:
1961:
1956:
1955:
1919:
1917:
1916:
1911:
1899:
1894:
1868:
1866:
1865:
1860:
1824:
1819:
1720:
1718:
1717:
1712:
1703:
1702:
1645:cotangent bundle
1619:
1617:
1616:
1611:
1606:
1605:
1587:
1586:
1412:
1410:
1409:
1404:
1402:
1401:
1381:
1379:
1378:
1373:
1371:
1370:
1339:
1337:
1336:
1331:
1325:
1324:
1323:
1313:
1308:
1294:
1293:
1274:
1273:
1272:
1262:
1257:
1243:
1242:
1227:
1226:
1221:
1205:
1195:
1194:
1193:
1188:
1172:
1155:
1150:
1140:
1139:
1138:
1120:
1115:
1105:
1104:
1103:
1086:
1085:
1084:
1083:
1071:
1070:
1060:
1059:
1058:
1057:
1056:
1044:
1043:
1001:
999:
998:
993:
991:
990:
985:
979:
978:
959:
957:
956:
951:
946:
945:
935:
930:
921:
920:
902:
901:
882:
880:
879:
874:
872:
871:
855:
853:
852:
847:
845:
844:
839:
832:
827:
815:
814:
809:
796:
794:
793:
788:
786:
785:
780:
764:
762:
761:
756:
750:
745:
736:
735:
723:
722:
703:
701:
700:
695:
693:
692:
665:
663:
662:
657:
655:
654:
642:
641:
631:
626:
614:
613:
590:
588:
587:
582:
580:
579:
574:
546:McConnell (1957)
522:
520:
519:
514:
502:
500:
499:
494:
477:
476:
460:
458:
457:
452:
440:
438:
437:
432:
414:
412:
411:
406:
289:
287:
286:
281:
276:
275:
263:
262:
247:
246:
220:
214:
142:
138:
134:
130:
126:
21:
5804:
5803:
5799:
5798:
5797:
5795:
5794:
5793:
5759:
5758:
5757:
5752:
5691:Banach manifold
5684:Generalizations
5679:
5634:
5571:
5468:
5430:Ricci curvature
5386:Cotangent space
5364:
5302:
5144:
5138:
5097:Exponential map
5061:
5006:
5000:
4920:
4910:
4880:
4875:
4826:Albert Einstein
4793:
4774:Einstein tensor
4737:
4718:Ricci curvature
4698:Kronecker delta
4684:Notable tensors
4679:
4600:Connection form
4577:
4571:
4502:
4488:Tensor operator
4445:
4439:
4379:
4355:Computer vision
4348:
4330:
4326:Tensor calculus
4270:
4259:
4254:
4211:
4176:
4168:, McGraw Hill,
4143:
4142:
4139:
4118:
4099:
4065:
4046:
4024:
4008:
4003:
4002:
3983:
3982:
3978:
3969:
3967:
3964:
3958:
3954:
3945:
3941:
3933:
3929:
3924:
3915:
3900:
3893:
3821:
3805:
3801:
3793:
3791:
3787:
3783:
3782:
3768:
3760:
3757:
3756:
3732:
3730:
3727:
3726:
3671:
3665:
3660:
3658:Generalizations
3650:; this kind of
3613:
3592:Euclidean space
3584:
3559:, and consider
3541:
3538:
3535:
3534:
3532:
3435:
3422:tensor calculus
3415:. The original
3389:
3387:Tensor calculus
3355:
3330:
3326:
3284:
3280:
3278:
3275:
3274:
3243:
3241:
3238:
3237:
3213:
3209:
3189:
3186:
3185:
3145:
3141:
3119:
3112:
3091:
3090:
3066:
3065:
3064:
3062:
3039:
3032:
3011:
3007:
2983:
2979:
2978:
2976:
2973:
2970:
2969:
2937:
2933:
2931:
2928:
2927:
2901:
2897:
2895:
2892:
2891:
2863:
2860:
2859:
2833:
2829:
2827:
2824:
2823:
2777:
2773:
2771:
2768:
2767:
2737:
2736:
2734:
2731:
2730:
2685:
2681:
2679:
2676:
2675:
2645:
2644:
2642:
2639:
2638:
2549:
2548:
2333:
2332:
2330:
2327:
2326:
2289:
2288:
2286:
2283:
2282:
2203:
2202:
2200:
2197:
2196:
2155:
2150:
2144:
2143:
2124:
2120:
2118:
2115:
2114:
2065:
2060:
2054:
2053:
2034:
2033:
2031:
2028:
2027:
2008:
1962:
1957:
1951:
1950:
1947:
1944:
1943:
1895:
1890:
1884:
1881:
1880:
1820:
1815:
1809:
1806:
1805:
1787:
1698:
1694:
1691:
1688:
1687:
1682:to sections of
1666:, a linear map
1637:cotangent space
1601:
1597:
1582:
1578:
1558:
1555:
1554:
1527:(the bundle of
1506:
1495:
1488:
1431:cotangent space
1419:
1397:
1393:
1391:
1388:
1387:
1366:
1362:
1360:
1357:
1356:
1319:
1315:
1314:
1304:
1299:
1286:
1282:
1268:
1264:
1263:
1253:
1248:
1235:
1231:
1217:
1201:
1196:
1184:
1168:
1163:
1159:
1158:
1157:
1146:
1141:
1134:
1130:
1129:
1111:
1106:
1099:
1095:
1094:
1079:
1075:
1066:
1062:
1061:
1052:
1048:
1039:
1035:
1034:
1030:
1029:
1028:
1026:
1023:
1022:
986:
981:
980:
974:
970:
968:
965:
964:
941:
937:
931:
926:
913:
909:
897:
893:
891:
888:
887:
867:
863:
861:
858:
857:
840:
835:
834:
828:
823:
810:
805:
804:
802:
799:
798:
781:
776:
775:
773:
770:
769:
746:
741:
731:
727:
718:
714:
712:
709:
708:
688:
684:
682:
679:
678:
650:
646:
637:
633:
627:
622:
609:
605:
603:
600:
599:
575:
570:
569:
567:
564:
563:
542:Schouten (1951)
538:
508:
505:
504:
472:
468:
466:
463:
462:
446:
443:
442:
420:
417:
416:
400:
397:
396:
386:
359:
348:T : M → V
340:
333:
326:
268:
264:
258:
254:
239:
235:
209:
206:
205:
201:) is a section
177:Euclidean plane
165:
140:
136:
132:
128:
124:
69:Euclidean space
35:
28:
23:
22:
15:
12:
11:
5:
5802:
5792:
5791:
5786:
5781:
5776:
5771:
5754:
5753:
5751:
5750:
5745:
5740:
5735:
5730:
5729:
5728:
5718:
5713:
5708:
5703:
5698:
5693:
5687:
5685:
5681:
5680:
5678:
5677:
5672:
5667:
5662:
5657:
5652:
5646:
5644:
5640:
5639:
5636:
5635:
5633:
5632:
5627:
5622:
5617:
5612:
5607:
5602:
5597:
5592:
5587:
5581:
5579:
5573:
5572:
5570:
5569:
5564:
5559:
5554:
5549:
5544:
5539:
5529:
5524:
5519:
5509:
5504:
5499:
5494:
5489:
5484:
5478:
5476:
5470:
5469:
5467:
5466:
5461:
5456:
5455:
5454:
5444:
5439:
5438:
5437:
5427:
5422:
5417:
5412:
5411:
5410:
5400:
5395:
5394:
5393:
5383:
5378:
5372:
5370:
5366:
5365:
5363:
5362:
5357:
5352:
5347:
5346:
5345:
5335:
5330:
5325:
5319:
5317:
5310:
5304:
5303:
5301:
5300:
5295:
5285:
5280:
5266:
5261:
5256:
5251:
5246:
5244:Parallelizable
5241:
5236:
5231:
5230:
5229:
5219:
5214:
5209:
5204:
5199:
5194:
5189:
5184:
5179:
5174:
5164:
5154:
5148:
5146:
5140:
5139:
5137:
5136:
5131:
5126:
5124:Lie derivative
5121:
5119:Integral curve
5116:
5111:
5106:
5105:
5104:
5094:
5089:
5088:
5087:
5080:Diffeomorphism
5077:
5071:
5069:
5063:
5062:
5060:
5059:
5054:
5049:
5044:
5039:
5034:
5029:
5024:
5019:
5013:
5011:
5002:
5001:
4999:
4998:
4993:
4988:
4983:
4978:
4973:
4968:
4963:
4958:
4957:
4956:
4951:
4941:
4940:
4939:
4928:
4926:
4925:Basic concepts
4922:
4921:
4909:
4908:
4901:
4894:
4886:
4877:
4876:
4874:
4873:
4868:
4866:Woldemar Voigt
4863:
4858:
4853:
4848:
4843:
4838:
4833:
4831:Leonhard Euler
4828:
4823:
4818:
4813:
4807:
4805:
4803:Mathematicians
4799:
4798:
4795:
4794:
4792:
4791:
4786:
4781:
4776:
4771:
4766:
4761:
4756:
4751:
4745:
4743:
4739:
4738:
4736:
4735:
4730:
4728:Torsion tensor
4725:
4720:
4715:
4710:
4705:
4700:
4694:
4692:
4685:
4681:
4680:
4678:
4677:
4672:
4667:
4662:
4657:
4652:
4647:
4642:
4637:
4632:
4627:
4622:
4617:
4612:
4607:
4602:
4597:
4592:
4587:
4581:
4579:
4573:
4572:
4570:
4569:
4563:
4561:Tensor product
4558:
4553:
4551:Symmetrization
4548:
4543:
4541:Lie derivative
4538:
4533:
4528:
4523:
4518:
4512:
4510:
4504:
4503:
4501:
4500:
4495:
4490:
4485:
4480:
4475:
4470:
4465:
4463:Tensor density
4460:
4455:
4449:
4447:
4441:
4440:
4438:
4437:
4435:Voigt notation
4432:
4427:
4422:
4420:Ricci calculus
4417:
4412:
4407:
4405:Index notation
4402:
4397:
4391:
4389:
4385:
4384:
4381:
4380:
4378:
4377:
4372:
4367:
4362:
4357:
4351:
4349:
4347:
4346:
4341:
4335:
4332:
4331:
4329:
4328:
4323:
4321:Tensor algebra
4318:
4313:
4308:
4303:
4301:Dyadic algebra
4298:
4293:
4287:
4285:
4276:
4272:
4271:
4264:
4261:
4260:
4253:
4252:
4245:
4238:
4230:
4224:
4223:
4209:
4193:
4180:
4174:
4157:
4137:
4122:
4116:
4103:
4097:
4082:
4069:
4063:
4050:
4044:
4028:
4022:
4007:
4004:
4001:
4000:
3976:
3952:
3939:
3926:
3925:
3923:
3920:
3919:
3918:
3909:
3906:Ricci calculus
3903:
3892:
3889:
3864:tangent bundle
3860:vector bundles
3853:
3852:
3841:
3838:
3835:
3832:
3829:
3824:
3819:
3811:
3808:
3804:
3799:
3796:
3790:
3786:
3781:
3778:
3774:
3771:
3767:
3764:
3739:
3736:
3715:volume element
3698:-forms (where
3692:density bundle
3669:Tensor density
3667:Main article:
3664:
3661:
3659:
3656:
3612:
3609:
3583:
3580:
3506:tensor density
3494:absolute value
3474:tensor density
3470:tensor density
3434:
3431:
3388:
3385:
3354:
3351:
3338:
3333:
3329:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3287:
3283:
3246:
3221:
3216:
3212:
3208:
3205:
3202:
3199:
3196:
3193:
3174:
3173:
3162:
3159:
3156:
3153:
3148:
3144:
3140:
3134:
3131:
3128:
3125:
3122:
3115:
3109:
3105:
3102:
3099:
3094:
3089:
3086:
3083:
3080:
3077:
3074:
3069:
3060:
3054:
3051:
3048:
3045:
3042:
3035:
3029:
3025:
3022:
3019:
3014:
3010:
3006:
3003:
3000:
2997:
2994:
2991:
2986:
2982:
2951:
2948:
2945:
2940:
2936:
2915:
2912:
2909:
2904:
2900:
2879:
2876:
2873:
2870:
2867:
2847:
2844:
2841:
2836:
2832:
2791:
2788:
2785:
2780:
2776:
2751:
2748:
2745:
2740:
2699:
2696:
2693:
2688:
2684:
2659:
2656:
2653:
2648:
2592:
2591:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2556:
2553:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2340:
2337:
2296:
2293:
2279:
2278:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2210:
2207:
2169:
2166:
2163:
2158:
2153:
2147:
2141:
2138:
2135:
2132:
2127:
2123:
2079:
2076:
2073:
2068:
2063:
2057:
2051:
2048:
2045:
2042:
2037:
2007:
2004:
1988:
1987:
1976:
1973:
1970:
1965:
1960:
1954:
1922:
1921:
1909:
1906:
1903:
1898:
1893:
1889:
1870:
1869:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1823:
1818:
1814:
1786:
1783:
1767:Lie derivative
1709:
1706:
1701:
1697:
1641:tangent bundle
1621:
1620:
1609:
1604:
1600:
1596:
1593:
1590:
1585:
1581:
1577:
1574:
1571:
1568:
1565:
1562:
1542:, namely as a
1529:tangent spaces
1525:tangent bundle
1517:tensor product
1504:
1499:
1498:
1493:
1486:
1418:
1417:Tensor bundles
1415:
1400:
1396:
1369:
1365:
1349:differentiable
1341:
1340:
1329:
1322:
1318:
1311:
1307:
1303:
1298:
1292:
1289:
1285:
1281:
1278:
1271:
1267:
1260:
1256:
1252:
1247:
1241:
1238:
1234:
1230:
1224:
1220:
1216:
1212:
1208:
1204:
1200:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1162:
1153:
1149:
1145:
1137:
1133:
1128:
1124:
1118:
1114:
1110:
1102:
1098:
1093:
1089:
1082:
1078:
1074:
1069:
1065:
1055:
1051:
1047:
1042:
1038:
1033:
989:
984:
977:
973:
961:
960:
949:
944:
940:
934:
929:
925:
919:
916:
912:
908:
905:
900:
896:
870:
866:
843:
838:
831:
826:
822:
818:
813:
808:
784:
779:
766:
765:
754:
749:
744:
740:
734:
730:
726:
721:
717:
691:
687:
667:
666:
653:
649:
645:
640:
636:
630:
625:
621:
617:
612:
608:
578:
573:
537:
534:
512:
492:
489:
486:
483:
480:
475:
471:
450:
430:
427:
424:
404:
385:
382:
374:tangent bundle
368:Often we take
357:
338:
331:
324:
315:tensor product
291:
290:
279:
274:
271:
267:
261:
257:
253:
250:
245:
242:
238:
234:
231:
228:
225:
219:
213:
164:
161:
131:over a module
77:physical space
26:
9:
6:
4:
3:
2:
5801:
5790:
5787:
5785:
5782:
5780:
5777:
5775:
5772:
5770:
5767:
5766:
5764:
5749:
5746:
5744:
5743:Supermanifold
5741:
5739:
5736:
5734:
5731:
5727:
5724:
5723:
5722:
5719:
5717:
5714:
5712:
5709:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5688:
5686:
5682:
5676:
5673:
5671:
5668:
5666:
5663:
5661:
5658:
5656:
5653:
5651:
5648:
5647:
5645:
5641:
5631:
5628:
5626:
5623:
5621:
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5582:
5580:
5578:
5574:
5568:
5565:
5563:
5560:
5558:
5555:
5553:
5550:
5548:
5545:
5543:
5540:
5538:
5534:
5530:
5528:
5525:
5523:
5520:
5518:
5514:
5510:
5508:
5505:
5503:
5500:
5498:
5495:
5493:
5490:
5488:
5485:
5483:
5480:
5479:
5477:
5475:
5471:
5465:
5464:Wedge product
5462:
5460:
5457:
5453:
5450:
5449:
5448:
5445:
5443:
5440:
5436:
5433:
5432:
5431:
5428:
5426:
5423:
5421:
5418:
5416:
5413:
5409:
5408:Vector-valued
5406:
5405:
5404:
5401:
5399:
5396:
5392:
5389:
5388:
5387:
5384:
5382:
5379:
5377:
5374:
5373:
5371:
5367:
5361:
5358:
5356:
5353:
5351:
5348:
5344:
5341:
5340:
5339:
5338:Tangent space
5336:
5334:
5331:
5329:
5326:
5324:
5321:
5320:
5318:
5314:
5311:
5309:
5305:
5299:
5296:
5294:
5290:
5286:
5284:
5281:
5279:
5275:
5271:
5267:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5247:
5245:
5242:
5240:
5237:
5235:
5232:
5228:
5225:
5224:
5223:
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5178:
5175:
5173:
5169:
5165:
5163:
5159:
5155:
5153:
5150:
5149:
5147:
5141:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5103:
5102:in Lie theory
5100:
5099:
5098:
5095:
5093:
5090:
5086:
5083:
5082:
5081:
5078:
5076:
5073:
5072:
5070:
5068:
5064:
5058:
5055:
5053:
5050:
5048:
5045:
5043:
5040:
5038:
5035:
5033:
5030:
5028:
5025:
5023:
5020:
5018:
5015:
5014:
5012:
5009:
5005:Main results
5003:
4997:
4994:
4992:
4989:
4987:
4986:Tangent space
4984:
4982:
4979:
4977:
4974:
4972:
4969:
4967:
4964:
4962:
4959:
4955:
4952:
4950:
4947:
4946:
4945:
4942:
4938:
4935:
4934:
4933:
4930:
4929:
4927:
4923:
4918:
4914:
4907:
4902:
4900:
4895:
4893:
4888:
4887:
4884:
4872:
4869:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4832:
4829:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4809:
4808:
4806:
4804:
4800:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4760:
4757:
4755:
4752:
4750:
4747:
4746:
4744:
4740:
4734:
4731:
4729:
4726:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4708:Metric tensor
4706:
4704:
4701:
4699:
4696:
4695:
4693:
4689:
4686:
4682:
4676:
4673:
4671:
4668:
4666:
4663:
4661:
4658:
4656:
4653:
4651:
4648:
4646:
4643:
4641:
4638:
4636:
4633:
4631:
4628:
4626:
4623:
4621:
4620:Exterior form
4618:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4586:
4583:
4582:
4580:
4574:
4567:
4564:
4562:
4559:
4557:
4554:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4522:
4519:
4517:
4514:
4513:
4511:
4509:
4505:
4499:
4496:
4494:
4493:Tensor bundle
4491:
4489:
4486:
4484:
4481:
4479:
4476:
4474:
4471:
4469:
4466:
4464:
4461:
4459:
4456:
4454:
4451:
4450:
4448:
4442:
4436:
4433:
4431:
4428:
4426:
4423:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4392:
4390:
4386:
4376:
4373:
4371:
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4352:
4350:
4345:
4342:
4340:
4337:
4336:
4333:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4302:
4299:
4297:
4294:
4292:
4289:
4288:
4286:
4284:
4280:
4277:
4273:
4269:
4268:
4262:
4258:
4251:
4246:
4244:
4239:
4237:
4232:
4231:
4228:
4220:
4216:
4212:
4206:
4202:
4198:
4194:
4189:
4185:
4181:
4177:
4175:0-07-051400-3
4171:
4166:
4165:
4158:
4153:
4147:
4140:
4138:0-7167-0344-0
4134:
4130:
4129:
4123:
4119:
4117:0-07-145545-0
4113:
4109:
4104:
4100:
4098:9780486145020
4094:
4090:
4089:
4083:
4078:
4074:
4070:
4066:
4060:
4056:
4051:
4047:
4041:
4037:
4033:
4029:
4025:
4023:9780080570570
4019:
4015:
4010:
4009:
3996:
3992:
3991:
3986:
3980:
3963:
3956:
3949:
3943:
3937:
3931:
3927:
3913:
3910:
3907:
3904:
3898:
3895:
3894:
3888:
3886:
3882:
3878:
3873:
3869:
3865:
3861:
3856:
3839:
3833:
3822:
3817:
3809:
3806:
3797:
3784:
3779:
3772:
3769:
3755:
3754:
3753:
3737:
3724:
3720:
3716:
3712:
3707:
3705:
3701:
3697:
3693:
3689:
3685:
3681:
3677:
3670:
3655:
3653:
3649:
3645:
3640:
3635:
3633:
3629:
3624:
3622:
3618:
3608:
3605:
3601:
3597:
3593:
3589:
3582:The flat case
3579:
3577:
3573:
3568:
3566:
3562:
3558:
3554:
3550:
3530:
3526:
3522:
3518:
3514:
3509:
3507:
3503:
3499:
3495:
3491:
3487:
3484:, namely the
3483:
3479:
3475:
3471:
3467:
3463:
3459:
3455:
3451:
3447:
3443:
3440:
3430:
3428:
3424:
3423:
3418:
3414:
3410:
3406:
3402:
3398:
3394:
3384:
3382:
3377:
3375:
3371:
3366:
3364:
3360:
3350:
3336:
3331:
3323:
3320:
3317:
3311:
3308:
3302:
3296:
3293:
3285:
3273:
3272:Leibniz rule,
3269:
3265:
3261:
3235:
3219:
3214:
3200:
3197:
3194:
3183:
3178:
3160:
3154:
3142:
3113:
3107:
3100:
3087:
3084:
3081:
3075:
3058:
3033:
3027:
3020:
3012:
3004:
3001:
2998:
2992:
2984:
2968:
2967:
2966:
2965:
2946:
2934:
2910:
2898:
2874:
2871:
2868:
2842:
2830:
2821:
2817:
2813:
2809:
2805:
2786:
2778:
2765:
2746:
2728:
2724:
2719:
2717:
2713:
2694:
2686:
2673:
2654:
2636:
2632:
2628:
2624:
2620:
2616:
2612:
2607:
2605:
2601:
2597:
2575:
2566:
2551:
2545:
2539:
2530:
2524:
2515:
2506:
2503:
2497:
2488:
2482:
2473:
2467:
2461:
2455:
2452:
2443:
2437:
2431:
2425:
2416:
2410:
2407:
2398:
2389:
2386:
2374:
2368:
2365:
2359:
2350:
2347:
2335:
2325:
2324:
2323:
2321:
2317:
2313:
2291:
2265:
2256:
2250:
2241:
2235:
2232:
2226:
2217:
2205:
2195:
2194:
2193:
2191:
2187:
2183:
2164:
2156:
2151:
2139:
2133:
2125:
2111:
2109:
2105:
2101:
2097:
2093:
2074:
2066:
2061:
2049:
2043:
2025:
2021:
2017:
2013:
2003:
2001:
1997:
1993:
1971:
1963:
1958:
1942:
1941:
1940:
1938:
1934:
1929:
1927:
1904:
1896:
1891:
1887:
1879:
1878:
1877:
1875:
1856:
1853:
1850:
1844:
1838:
1835:
1829:
1821:
1816:
1812:
1804:
1803:
1802:
1800:
1796:
1792:
1782:
1780:
1776:
1772:
1768:
1764:
1760:
1756:
1752:
1748:
1744:
1740:
1736:
1732:
1728:
1724:
1707:
1704:
1699:
1695:
1685:
1681:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1648:
1646:
1642:
1638:
1634:
1630:
1629:tangent space
1626:
1607:
1602:
1598:
1594:
1591:
1588:
1583:
1579:
1575:
1572:
1569:
1566:
1563:
1560:
1553:
1552:
1551:
1549:
1548:tensor bundle
1545:
1541:
1536:
1534:
1530:
1526:
1522:
1518:
1513:
1511:
1507:
1496:
1489:
1483:
1482:
1481:
1479:
1476:varying over
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1447:
1442:
1440:
1439:vector bundle
1436:
1432:
1428:
1427:tangent space
1424:
1414:
1398:
1394:
1385:
1367:
1363:
1354:
1350:
1346:
1327:
1320:
1316:
1309:
1305:
1301:
1290:
1287:
1283:
1276:
1269:
1265:
1258:
1254:
1250:
1239:
1236:
1232:
1222:
1218:
1214:
1210:
1206:
1202:
1198:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1160:
1151:
1147:
1143:
1135:
1131:
1126:
1122:
1116:
1112:
1108:
1100:
1096:
1091:
1080:
1076:
1072:
1067:
1063:
1053:
1049:
1045:
1040:
1036:
1031:
1021:
1020:
1019:
1017:
1013:
1009:
1005:
987:
975:
971:
947:
942:
938:
932:
927:
917:
914:
910:
898:
894:
886:
885:
884:
868:
864:
841:
829:
824:
820:
811:
782:
752:
747:
742:
738:
732:
728:
719:
715:
707:
706:
705:
689:
685:
676:
672:
651:
647:
643:
638:
634:
628:
623:
619:
610:
606:
598:
597:
596:
594:
576:
562:
559:-dimensional
558:
553:
551:
547:
543:
533:
529:
527:
510:
487:
484:
481:
473:
469:
448:
428:
425:
422:
402:
393:
391:
381:
379:
375:
371:
366:
364:
360:
353:
349:
345:
341:
334:
327:
318:
316:
313:and ⊗ is the
312:
308:
304:
300:
299:vector bundle
296:
272:
269:
259:
255:
248:
243:
240:
236:
232:
229:
217:
211:
204:
203:
202:
200:
196:
192:
188:
183:
181:
178:
174:
170:
160:
158:
155:, which is a
154:
150:
146:
122:
121:
116:
115:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
67:(typically a
66:
62:
58:
54:
50:
46:
42:
37:
33:
19:
5670:Moving frame
5665:Morse theory
5655:Gauge theory
5447:Tensor field
5446:
5376:Closed/Exact
5355:Vector field
5323:Distribution
5264:Hypercomplex
5259:Quaternionic
4996:Vector field
4954:Smooth atlas
4871:Hermann Weyl
4675:Vector space
4660:Pseudotensor
4625:Fiber bundle
4578:abstractions
4473:Mixed tensor
4458:Tensor field
4457:
4265:
4200:
4187:
4163:
4126:
4107:
4087:
4076:
4073:Lerner, R.G.
4054:
4035:
4013:
3988:
3979:
3968:. Retrieved
3955:
3942:
3935:
3930:
3912:Spinor field
3880:
3871:
3857:
3854:
3718:
3708:
3703:
3699:
3695:
3683:
3679:
3672:
3647:
3643:
3638:
3636:
3625:
3616:
3614:
3603:
3599:
3596:translations
3587:
3585:
3569:
3564:
3563:with weight
3560:
3556:
3552:
3548:
3528:
3525:half-density
3524:
3520:
3516:
3512:
3510:
3481:
3477:
3473:
3469:
3465:
3461:
3457:
3453:
3449:
3445:
3441:
3436:
3420:
3416:
3413:vector field
3408:
3390:
3378:
3367:
3356:
3353:Applications
3271:
3267:
3263:
3259:
3233:
3179:
3175:
2819:
2815:
2811:
2807:
2803:
2763:
2726:
2722:
2720:
2715:
2711:
2671:
2634:
2630:
2626:
2622:
2618:
2614:
2610:
2608:
2603:
2599:
2595:
2593:
2319:
2315:
2311:
2280:
2192:, we define
2189:
2185:
2112:
2103:
2099:
2023:
2019:
2015:
2011:
2009:
1999:
1995:
1991:
1989:
1936:
1930:
1925:
1923:
1873:
1871:
1798:
1794:
1790:
1788:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1730:
1726:
1722:
1683:
1679:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1649:
1632:
1624:
1622:
1540:tensor field
1539:
1537:
1520:
1514:
1509:
1502:
1500:
1491:
1484:
1477:
1473:
1469:
1465:
1461:
1454:Möbius strip
1449:
1445:
1443:
1435:vector space
1423:fiber bundle
1420:
1342:
1015:
1011:
1007:
1003:
962:
767:
670:
668:
556:
554:
539:
530:
503:. The field
394:
390:curved space
387:
377:
369:
367:
355:
354:). Elements
351:
347:
343:
336:
329:
322:
320:
306:
302:
294:
292:
198:
194:
191:tensor field
190:
186:
185:
179:
168:
166:
148:
147:is a tensor
120:vector field
118:
114:scalar field
112:
75:) or of the
55:assigning a
49:tensor field
48:
38:
36:
5615:Levi-Civita
5605:Generalized
5577:Connections
5527:Lie algebra
5459:Volume form
5360:Vector flow
5333:Pushforward
5328:Lie bracket
5227:Lie algebra
5192:G-structure
4981:Pushforward
4961:Submanifold
4811:Élie Cartan
4759:Spin tensor
4733:Weyl tensor
4691:Mathematics
4655:Multivector
4446:definitions
4344:Engineering
4283:Mathematics
4128:Gravitation
4032:Frankel, T.
3868:line bundle
3723:determinant
3439:line bundle
3258:-linear in
2926:-module of
1639:. See also
361:are called
187:Definition.
41:mathematics
5763:Categories
5738:Stratifold
5696:Diffeology
5492:Associated
5293:Symplectic
5278:Riemannian
5207:Hyperbolic
5134:Submersion
5042:Hopf–Rinow
4976:Submersion
4971:Smooth map
4640:Linear map
4508:Operations
4006:References
3970:2024-06-24
3946:The term "
3897:Jet bundle
3648:quantities
3632:functorial
3621:chain rule
3427:connection
2822:). Namely
2766:copies of
2729:copies of
2674:copies of
2637:copies of
1515:Since the
540:Following
372:to be the
163:Definition
135:, we call
5620:Principal
5595:Ehresmann
5552:Subbundle
5542:Principal
5517:Fibration
5497:Cotangent
5369:Covectors
5222:Lie group
5202:Hermitian
5145:manifolds
5114:Immersion
5109:Foliation
5047:Noether's
5032:Frobenius
5027:De Rham's
5022:Darboux's
4913:Manifolds
4779:EM tensor
4615:Dimension
4566:Transpose
3995:EMS Press
3803:∂
3795:∂
3644:geometric
3508:article.
3328:∇
3282:∇
3211:∇
3207:↦
3147:∞
3139:→
3108:⏟
3088:×
3085:…
3082:×
3059:×
3028:⏟
3009:Ω
3005:×
3002:…
2999:×
2981:Ω
2939:∞
2903:∞
2835:∞
2775:Ω
2683:Ω
2555:~
2552:ω
2507:ω
2468:ω
2411:ω
2369:ω
2339:~
2336:ω
2295:~
2292:ω
2236:ω
2209:~
2206:ω
2122:Ω
2094:over the
1705:⊗
1700:∗
1603:∗
1595:⊗
1592:⋯
1589:⊗
1584:∗
1576:⊗
1570:⊗
1567:⋯
1564:⊗
1288:−
1277:⋯
1237:−
1211:⋯
1178:⋯
1123:⋯
1088:↦
1073:⋯
1046:⋯
915:−
904:↦
817:↦
725:↦
616:↦
441:at point
270:⊗
260:∗
249:⊗
241:⊗
224:Γ
218:∈
193:of type (
18:Tensorial
5716:Orbifold
5711:K-theory
5701:Diffiety
5425:Pullback
5239:Oriented
5217:Kenmotsu
5197:Hadamard
5143:Types of
5092:Geodesic
4917:Glossary
4645:Manifold
4630:Geodesic
4388:Notation
4219:40734875
4186:(1951),
4146:citation
4034:(2012),
3891:See also
3810:′
3773:′
3676:Jacobian
3639:post hoc
3515:is that
2594:for any
1939:. Thus,
1785:Notation
1546:of some
1458:cylinder
1384:Jacobian
1353:analytic
1310:′
1259:′
1223:′
1207:′
1190:′
1174:′
1152:′
1117:′
173:manifold
73:manifold
53:function
5784:Tensors
5660:History
5643:Related
5557:Tangent
5535:)
5515:)
5482:Adjoint
5474:Bundles
5452:density
5350:Torsion
5316:Vectors
5308:Tensors
5291:)
5276:)
5272:,
5270:Pseudo−
5249:Poisson
5182:Finsler
5177:Fibered
5172:Contact
5170:)
5162:Complex
5160:)
5129:Section
4742:Physics
4576:Related
4339:Physics
4257:Tensors
3997:, 2001
3948:affinor
3652:descent
3628:cocycle
3545:
3533:
3496:to the
3488:of the
3460:, then
2182:1-forms
1775:torsion
1635:is the
1627:is the
1544:section
1470:section
1429:and/or
363:tensors
309:is its
45:physics
5625:Vector
5610:Koszul
5590:Cartan
5585:Affine
5567:Vector
5562:Tensor
5547:Spinor
5537:Normal
5533:Stable
5487:Affine
5391:bundle
5343:bundle
5289:Almost
5212:Kähler
5168:Almost
5158:Almost
5152:Closed
5052:Sard's
5008:(list)
4670:Vector
4665:Spinor
4650:Matrix
4444:Tensor
4217:
4207:
4172:
4135:
4114:
4095:
4061:
4042:
4020:
3617:tensor
3117:
3037:
2092:module
2016:single
1763:module
1674:) → Γ(
1623:where
1501:where
1345:smooth
1010:) has
669:(with
370:V = TM
293:where
221:
215:
117:and a
109:vector
105:scalar
97:strain
93:stress
61:region
57:tensor
5733:Sheaf
5507:Fiber
5283:Rizza
5254:Prime
5085:Local
5075:Curve
4937:Atlas
4590:Basis
4275:Scope
3965:(PDF)
3922:Notes
3866:is a
3590:is a
3586:When
3555:with
3456:with
3448:. If
3409:along
2802:into
2762:and
2710:into
2314:is a
1741:in Γ(
1464:over
344:x ∈ M
297:is a
171:be a
149:field
63:of a
51:is a
47:, a
5600:Form
5502:Dual
5435:flow
5298:Tame
5274:Sub−
5187:Flat
5067:Maps
4215:OCLC
4205:ISBN
4170:ISBN
4152:link
4133:ISBN
4112:ISBN
4093:ISBN
4059:ISBN
4040:ISBN
4018:ISBN
3604:does
2670:and
2625:as
2096:ring
1777:and
1769:and
1729:) =
1670:: Γ(
1658:and
1643:and
1347:(or
544:and
311:dual
167:Let
129:X(M)
95:and
43:and
5522:Jet
3828:det
3789:det
3763:det
3735:det
3444:on
3391:In
3368:In
2718:).
2598:in
2310:on
2020:not
1749:on
1490:in
376:of
350:⊗ (
335:⊗ (
301:on
71:or
39:In
5765::
5513:Co
4213:.
4148:}}
4144:{{
3993:,
3987:,
3887:.
3578:.
3567:.
3531:=
3429:.
3411:a
3376:.
3365:.
2233::=
2110:.
2050::=
2026:,
1928:.
1793:=
1791:TM
1731:fA
1727:fs
1662:→
1654:→
1647:.
1512:.
1441:.
1351:,
595::
528:.
380:.
365:.
305:,
197:,
189:A
182:.
159:.
87:,
83:,
5531:(
5511:(
5287:(
5268:(
5166:(
5156:(
4919:)
4915:(
4905:e
4898:t
4891:v
4249:e
4242:t
4235:v
4221:.
4192:.
4179:.
4156:.
4154:)
4121:.
4102:.
4081:.
4068:.
4049:.
4026:.
3973:.
3881:n
3872:w
3840:,
3837:)
3834:g
3831:(
3823:2
3818:)
3807:x
3798:x
3785:(
3780:=
3777:)
3770:g
3766:(
3738:g
3719:g
3704:R
3700:n
3696:n
3684:w
3680:w
3600:M
3588:M
3565:s
3557:L
3553:V
3549:W
3542:2
3539:/
3536:1
3529:s
3521:s
3517:L
3513:L
3478:L
3466:V
3462:W
3458:L
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3450:W
3446:M
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3337:Y
3332:X
3324:f
3321:+
3318:Y
3315:)
3312:f
3309:X
3306:(
3303:=
3300:)
3297:Y
3294:f
3291:(
3286:X
3268:M
3266:(
3264:C
3260:Y
3245:R
3234:M
3220:Y
3215:X
3204:)
3201:Y
3198:,
3195:X
3192:(
3161:.
3158:)
3155:M
3152:(
3143:C
3133:s
3130:e
3127:m
3124:i
3121:t
3114:k
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3101:M
3098:(
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3079:)
3076:M
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3013:1
2996:)
2993:M
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2944:(
2935:C
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2878:)
2875:l
2872:,
2869:k
2866:(
2846:)
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2840:(
2831:C
2820:M
2818:(
2816:C
2812:M
2808:M
2806:(
2804:C
2790:)
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2779:1
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2739:X
2727:k
2723:T
2716:M
2714:(
2712:C
2698:)
2695:M
2692:(
2687:1
2672:l
2658:)
2655:M
2652:(
2647:X
2635:k
2631:M
2629:(
2627:C
2623:M
2619:l
2617:,
2615:k
2611:M
2604:f
2600:M
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2579:)
2576:p
2573:(
2570:)
2567:X
2564:(
2561:)
2546:f
2543:(
2540:=
2537:)
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2531:p
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2522:(
2519:)
2516:p
2513:(
2510:)
2504:f
2501:(
2498:=
2495:)
2492:)
2489:p
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2483:X
2480:(
2477:)
2474:p
2471:(
2465:)
2462:p
2459:(
2456:f
2453:=
2450:)
2447:)
2444:p
2441:(
2438:X
2435:)
2432:p
2429:(
2426:f
2423:(
2420:)
2417:p
2414:(
2408:=
2405:)
2402:)
2399:p
2396:(
2393:)
2390:X
2387:f
2384:(
2381:(
2378:)
2375:p
2372:(
2366:=
2363:)
2360:p
2357:(
2354:)
2351:X
2348:f
2345:(
2320:M
2318:(
2316:C
2312:X
2266:.
2263:)
2260:)
2257:p
2254:(
2251:X
2248:(
2245:)
2242:p
2239:(
2230:)
2227:p
2224:(
2221:)
2218:X
2215:(
2190:X
2186:ω
2168:)
2165:M
2162:(
2157:0
2152:1
2146:T
2140:=
2137:)
2134:M
2131:(
2126:1
2104:M
2102:(
2100:C
2078:)
2075:M
2072:(
2067:1
2062:0
2056:T
2047:)
2044:M
2041:(
2036:X
2024:M
2012:M
2000:M
1996:n
1994:,
1992:m
1975:)
1972:M
1969:(
1964:m
1959:n
1953:T
1937:M
1926:M
1920:;
1908:)
1905:V
1902:(
1897:1
1892:0
1888:T
1874:M
1857:M
1854:T
1851:=
1848:)
1845:M
1842:(
1839:T
1836:=
1833:)
1830:M
1827:(
1822:1
1817:0
1813:T
1799:M
1797:(
1795:T
1759:M
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