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Assignment of a tensor continuously varying across a region of space
to in the shorter form "tensor". For example, the Riemann curvature tensor refers a tensor field, as it associates a tensor to each point of a Riemannian
Tensor_field
Mathematical object that describes the electromagnetic field in spacetime
electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a tensor that describes
Electromagnetic_tensor
Algebraic object with geometric applications
(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, etc.), and general relativity (stress–energy tensor, curvature tensor, etc.). In
Tensor
Field-equations in general relativity
Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum
Einstein_field_equations
Tensor describing energy momentum density in spacetime
stress-energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity
Stress–energy_tensor
Tensor field in Riemannian geometry
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno
Riemann_curvature_tensor
Structure defining distance on a manifold
numbers), and a metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite
Metric_tensor
Second-rank tensor in quantum chromodynamics
In theoretical particle physics, the gluon field strength tensor is a second-order tensor field characterizing the gluon interaction between quarks. The
Gluon_field_strength_tensor
Tensor used in general relativity
differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature
Einstein_tensor
Tensor index notation for tensor-based calculations
element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements. Tensors and tensor fields can be expressed
Ricci_calculus
Coordinate-free definition of a tensor
mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear
Tensor_(intrinsic_definition)
Physical quantities taking values at each point in space and time
example of a vector field. Strain tensor, representing the deformation of matter caused by stress, is an example of a tensor field. Field theories, mathematical
Field_(physics)
Type of derivative in differential geometry
the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is
Lie_derivative
Tensor in differential geometry
relativity, the Ricci curvature tensor enters the Einstein field equations through the Einstein tensor, formed from the Ricci tensor, the scalar curvature, and
Ricci_curvature
Generalization of tensor fields
geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing
Tensor_density
Mathematical operation on vector spaces
two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense
Tensor_product
Ring produced from two fields
the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must
Tensor_product_of_fields
Specification of a derivative along a tangent vector of a manifold
covariant derivative of a tensor field along a vector field v is again a tensor field of the same type. Explicitly, let T be a tensor field of type (p, q). Consider
Covariant_derivative
Tensor in general relativity
a Killing tensor or Killing tensor field is a generalization of a Killing vector, for symmetric tensor fields instead of just vector fields. It is a concept
Killing_tensor
Operation in mathematics
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. This example
Tensor_contraction
Assignment of numbers to points in space
associate a tensor to every point in space. For example, in general relativity gravitation is associated with the tensor field called Einstein tensor. In Kaluza–Klein
Scalar_field
Field theory in physics that aims to unify the fundamental forces and particles
electromagnetic field, spinor fields whose quanta are fermionic particles such as electrons, and tensor fields such as the metric tensor field that describes
Unified_field_theory
Theory in physics with scalars and tensors both describing a force or interaction
In theoretical physics, a scalar–tensor theory is a field theory that includes both a scalar field and a tensor field to represent a certain interaction
Scalar–tensor_theory
Tensor that describes the 4D geometry of spacetime
manifold M {\displaystyle M} and the metric tensor is given as a covariant, second-degree, symmetric tensor on M {\displaystyle M} , conventionally denoted
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Tensor equal to the negative of any of its transpositions
completely antisymmetric contravariant tensor field may be referred to as a k {\displaystyle k} -vector field. A tensor A that is antisymmetric on indices
Antisymmetric_tensor
Measure of the curvature of a pseudo-Riemannian manifold
Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann
Weyl_tensor
is a vector field. If T {\displaystyle {\boldsymbol {T}}} is a tensor field of order n > 1 then the divergence of the field is a tensor of order n− 1
Tensor derivative (continuum mechanics)
Tensor_derivative_(continuum_mechanics)
Spinning motion in theoretical physics
theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general
Spin_tensor
Vector operator in vector calculus
coordinates at Wolfram Mathworld Gurtin 1981, p. 30. "1.14 Tensor Calculus I: Tensor Fields" (PDF). Foundations of Continuum Mechanics. Archived (PDF)
Divergence
together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle. A tensor field is then defined
Mathematics of general relativity
Mathematics_of_general_relativity
Application of Lagrangian mechanics to field theories
package the E and B fields into what is known as the electromagnetic tensor F μ ν {\displaystyle F_{\mu \nu }} . We define this tensor as F μ ν = ∂ μ A ν
Lagrangian_(field_theory)
Physical theory describing classical fields
the Einstein tensor, G a b = R a b − 1 2 R g a b {\displaystyle G_{ab}\,=R_{ab}-{\frac {1}{2}}Rg_{ab}} written in terms of the Ricci tensor Rab and Ricci
Classical_field_theory
Concept in physics
gravitational field is characterized by a symmetric rank-2 tensor, the metric tensor. The possibility of generalizing the metric tensor has been considered
Nonsymmetric gravitational theory
Nonsymmetric_gravitational_theory
Mathematical identities
Scribner's Sons. pp. 159, 161–162. Kelly, P. (2013). "Chapter 1.14 Tensor Calculus 1: Tensor Fields". Mechanics Lecture Notes Part III: Foundations of Continuum
Vector_calculus_identities
Calculus of vector-valued functions
{\displaystyle q} -fold tensor products of vectors and covectors, respectively. Thus a ( p , q ) {\displaystyle (p,q)} tensor field is a map from a manifold
Vector_calculus
Mathematical operation
{R} } be a multilinear form on W (also known as a tensor – not to be confused with a tensor field – of rank (0, s), where s is the number of factors
Pullback (differential geometry)
Pullback_(differential_geometry)
Physical quantity that expresses internal forces in a continuous material
the first and second Piola–Kirchhoff stress tensors, the Biot stress tensor, and the Kirchhoff stress tensor. Bending Compressive strength Critical plane
Stress_(mechanics)
Object in differential geometry
differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors
Torsion_tensor
Representation of a tensor in Euclidean space
a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from
Cartesian_tensor
Operation that pairs a left and a right R-module into an abelian group
universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and
Tensor_product_of_modules
Mathematical concept
symmetrization over those indices. The Saint-Venant tensor W {\displaystyle W} of a symmetric rank-k tensor field T {\displaystyle T} is defined by W i 1 . .
Saint-Venant's compatibility condition
Saint-Venant's_compatibility_condition
of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory
Glossary_of_tensor_theory
Differential form of degree one or section of a cotangent bundle
coordinate system to another. Thus a one-form is an order 1 covariant tensor field. The most basic non-trivial differential one-form is the "change in angle"
One-form
Curvilinear coordinates can be formulated in tensor calculus, with important applications in physics and engineering, particularly for describing transportation
Tensors in curvilinear coordinates
Tensors_in_curvilinear_coordinates
electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field. The stress–energy tensor describes the flow
Electromagnetic stress–energy tensor
Electromagnetic_stress–energy_tensor
Representation of mechanical stress at every point within a deformed 3D object
Cauchy stress tensor (symbol σ {\displaystyle {\boldsymbol {\sigma }}} , named after Augustin-Louis Cauchy), also called true stress tensor or simply stress
Cauchy_stress_tensor
Electromagnetic stress
electromagnetism, the Maxwell stress tensor (named after James Clerk Maxwell) is the stress tensor of an electromagnetic field. In tensor index notation it is given
Maxwell_stress_tensor
Tensor invariant under permutations of vectors it acts on
In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T (
Symmetric_tensor
Assignment of a vector to each point in a subset of Euclidean space
to the manifold). Vector fields are one kind of tensor field. Given a subset S of Rn, a vector field is represented by a vector-valued function V: S →
Vector_field
Tensor having both covariant and contravariant indices
In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed
Mixed_tensor
Concept in machine learning
learning, the term tensor informally refers to two different concepts: (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data
Tensor_(machine_learning)
In the mathematical field of differential geometry, a Codazzi tensor (named after Delfino Codazzi) is a symmetric 2-tensor whose covariant derivative is
Codazzi_tensor
Proposed theories of gravity
Will and Nordtvedt are both vector–tensor theories. In addition to the metric tensor there is a timelike vector field K μ . {\displaystyle K_{\mu }.} The
Alternatives to general relativity
Alternatives_to_general_relativity
finally arrived at in 1915, general relativity, is a tensor theory, not a scalar theory, with a 2-tensor, the metric, as the potential. Unlike his 1913 scalar
Scalar theories of gravitation
Scalar_theories_of_gravitation
Abbreviation in the fields of special and general relativity
relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime. General four-tensors are usually written in tensor index notation
Four-tensor
Aspect of general relativity
metric tensor, κ {\displaystyle \kappa } is a constant, and T μ ν {\displaystyle T_{\mu \nu }} is the stress–energy tensor. The Einstein field equations
Solutions of the Einstein field equations
Solutions_of_the_Einstein_field_equations
can be generated. As indicated above, the tensor components correspond to gravitational waves. The tensor S T i j {\displaystyle S^{T}{}_{ij}} is gauge
Scalar–vector–tensor decomposition
Scalar–vector–tensor_decomposition
differentiable manifolds of arbitrary dimension. The basic objects are tensor fields and not tensor components in a given vector frame or coordinate chart. This
Sage_Manifolds
Antisymmetric permutation object acting on tensors
to be a tensor density field in two different ways. It may be regarded as a contravariant tensor density of weight +1 or as a covariant tensor density
Levi-Civita_symbol
Quadratic form related to curvatures of surfaces
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional
Second_fundamental_form
Visual aid used in mathematics
shear – of a 3 × 3 {\displaystyle 3\times 3} matrix. It is used for tensor field visualization, where a data-matrix is available at every point in the
Tensor_glyph
Universal construction in multilinear algebra
Let V be a vector space over a field K. For any nonnegative integer k, we define the kth tensor power of V to be the tensor product of V with itself k times:
Tensor_algebra
If (E,p,M) is any vector bundle with the canonical vector field V and a (1,1)-tensor field J that satisfies the properties listed above, with VE in place
Double_tangent_bundle
Circulation density in a vector field
ε denotes the Levi-Civita tensor, ∇ the covariant derivative, g {\displaystyle g} is the determinant of the metric tensor and the Einstein summation
Curl_(mathematics)
these tensor fields should also give rise to specific contributions to the stress–energy tensor T α β {\displaystyle T^{\alpha \beta }} . (A field is described
Exact solutions in general relativity
Exact_solutions_in_general_relativity
Physical condition
strain) tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility
Compatibility_(mechanics)
{\displaystyle (0,1)} -tensor field is a vector field, and a ( 0 , k ) {\displaystyle (0,k)} -tensor field is k {\displaystyle k} -vector field. While differential
Multivector_field
Topics referred to by the same term
field, assignment of a tensor to each point in a mathematical space Vector field, assignment of a vector to each point in a mathematical space Field of
Field
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
the metric tensor along an integral curve generated by the vector field (whose image is parallel to the x-axis). Furthermore, the metric tensor is independent
Killing_vector_field
Tensor operator generalizes the notion of operators which are scalars and vectors
graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which
Tensor_operator
Physics term
{\triangledown }{\mathbf {A} }}} is the upper-convected time derivative of a tensor field A {\displaystyle \mathbf {A} } D D t {\displaystyle {\frac {D}{Dt}}}
Upper-convected time derivative
Upper-convected_time_derivative
Theory of gravity
metric tensor field, both defined in terms of a dynamical tetrad field. The crucial new idea, for Einstein, was the introduction of a tetrad field, i.e
Teleparallelism
Isomorphism between the tangent and cotangent bundles of a manifold
index of an ( r , s ) {\displaystyle (r,s)} tensor gives a ( r − 1 , s + 1 ) {\displaystyle (r-1,s+1)} tensor, while raising an index gives a ( r + 1 ,
Musical_isomorphism
Proposed theory of gravitation
relativity, the source of the gravitational field is considered to be the stress–energy tensor or matter tensor. However, the way in which the immediate
Brans–Dicke_theory
_{W}Y-\nabla _{[W,X]}Y,Z{\Big )}.} With this convention, the Ricci tensor is a (0,2)-tensor field defined by Rjk=gilRijkl and the scalar curvature is defined
Ricci_decomposition
Type of monoidal category
collection of tensors. There are several equivalent alternative ways of defining modular tensor categories. One definition is as follows: a modular tensor category
Modular_tensor_category
Array of numbers describing a metric connection
gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry
Christoffel_symbols
Concept in mathematics
In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold
Tensor_bundle
Quantum field theory enjoying conformal symmetry
each tensor structure. In the case of two scalar fields and a symmetric traceless tensor of rank ℓ {\displaystyle \ell } , there is only one tensor structure
Conformal_field_theory
Process in algebra
In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting
Tensor_decomposition
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
thought of as a tensor, and is written δ j i {\displaystyle \delta _{j}^{i}} . Sometimes the Kronecker delta is called the substitution tensor. When juxtaposition
Kronecker_delta
Hypothetical elementary particle that mediates gravity
stress–energy tensor, a second-order tensor (compared with electromagnetism's spin-1 photon, the source of which is the four-current, a first-order tensor). Additionally
Graviton
{\displaystyle g^{il}W_{ijkl}=0} The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors: R j k = R k j {\displaystyle R_{jk}=R_{kj}}
List of formulas in Riemannian geometry
List_of_formulas_in_Riemannian_geometry
Decomposition in multilinear algebra
multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal
Tensor_rank_decomposition
Physical field surrounding an electric charge
materials the E and D fields are not parallel, and so E and D are related by the permittivity tensor (a 2nd order tensor field), in component form: D
Electric_field
Ways of writing certain laws of physics
electromagnetic tensor is the combination of the electric and magnetic fields into a covariant antisymmetric tensor whose entries are B-field quantities.
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Algebraic operation on coordinate vectors
(single-) dot product between a tensor of order n {\displaystyle n} and a tensor of order m {\displaystyle m} is a tensor of order n + m − 2 {\displaystyle
Dot_product
Topics referred to by the same term
Tensing may refer to: Tenseness (or tensing), a concept in the linguistic fields of phonetics and phonology Ten Sing, a Christian youth program Tenzing
Tensing
Rank-3 tensor in general relativity associated with gauge fields
The Lanczos tensor or Lanczos potential is a rank 3 tensor in general relativity that generates the Weyl tensor. It was first introduced by Cornelius
Lanczos_tensor
Graph-structured representation of high-dimensional tensors and quantum many-body states
compression and contraction are both efficient. Tensor network states are the special case in which a tensor network represents a variational wave function
Tensor_network
Topics referred to by the same term
objects related to a vector space. Tensor may also refer to: Tensor (intrinsic definition) Tensor field Tensor product Tensor (obsolete), the norm used on the
Tensor_(disambiguation)
Tensorial object depending on two points in a manifold
relativity, a bitensor (or bi-tensor) is a tensorial object that depends on two points in a manifold, as opposed to ordinary tensors which depend on a single
Bitensor
Class of mathematical software
similar to MATLAB and GNU Octave, but designed specifically for tensors. Tensor is a tensor package written for the Mathematica system. It provides many
Tensor_software
Mathematical model for describing material deformation under stress
deformation tensors. In 1839, George Green introduced a deformation tensor known as the right Cauchy–Green deformation tensor or Green's deformation tensor (the
Finite_strain_theory
Topics referred to by the same term
angle. Tensor Ring, a closed-loop device (often copper) utilizing cubit-length geometry to generate a stabilized energy column or coherent tensor field. Toroidal
Toroidal
Tensor product of algebras over a field; itself another algebra
the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the
Tensor_product_of_algebras
Measurement of variations in Earth's gravitational field
Earth's gravity field via measurements of the spatial gradient of gravitational acceleration. The gravity gradient tensor is a 3 × 3 tensor; it is given
Gravity_gradiometry
Manifold upon which it is possible to perform calculus
tensor bundle is the direct sum of all tensor products of the tangent bundle and the cotangent bundle. Each element of the bundle is a tensor field,
Differentiable_manifold
Operator in differential topology
derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X {\displaystyle X} . The Lie bracket is
Lie_bracket_of_vector_fields
Theorem in calculus
_{i}n_{i}\,\mathrm {d} S} suggestively, replacing the vector field F with a rank-n tensor field T, this can be generalized to: ∭ V ∂ T i 1 i 2 ⋯ i q ⋯ i n
Divergence_theorem
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TENSOR FIELD
TENSOR FIELD
Surname or Lastname
English
English : perhaps an altered spelling of Janson.Respelling of Danish, Norwegian, and North German Jensen.
Surname or Lastname
French
French : unexplained.English : unexplained.Possibly a respelling of Menter, an unexplained name of German origin.
Surname or Lastname
English
English : patronymic from Penn 3 or Paine 1.English : habitational name from Penson in Devon.
Male
English
English surname transferred to forename use, BENSON means "son of Ben."
Male
Scandinavian
Scandinavian form of Latin Theodorus, TEODOR means "gift of God."
Surname or Lastname
English (mainly Yorkshire)
English (mainly Yorkshire) : nickname for a peasant who gave himself airs and graces, from Anglo-Norman French segneur ‘lord’ (Latin senior ‘elder’).English and Dutch : distinguishing nickname for the elder of two bearers of the same personal name (for example, a father and son or two brothers), from Latin senior ‘elder’.
Surname or Lastname
English
English : unexplained.
Surname or Lastname
English
English : patronymic from the personal name Henn(e), a short form of Henry 1, Hayne (see Hain 2), or Hendy.Irish : Anglicized form of Gaelic Ó hAmhsaigh (see Hampson 2).
Surname or Lastname
English
English : variant spelling of Ensor.
Surname or Lastname
English
English : habitational name for someone from Edensor in Derbyshire, which derives its name from the genitive case of the Old English personal name Ēadhūn (see Eden 1) + Old English ofer ‘ridge’.
Boy/Male
French
Works in iron.
Male
Greek
(ÎœÎντωÏ) Greek name derived from the word menos, MENTOR means "spirit." In mythology, this is the name of the son of Ãlkimos.
Surname or Lastname
English
English : probably a variant of Manser.
Boy/Male
Polish Spanish
Surname or Lastname
English
English : variant of Windsor. This is the spelling used for places so named in Devon and Hampshire.Perhaps also an Americanized spelling of German Winzer.
Surname or Lastname
English
English : patronymic from a reduced form of the personal name Steven.English : habitational name from a place in Derbyshire, recorded in Domesday Book as Steintune, later as Steineston, from the Old Norse personal name Steinn (meaning ‘stone’) + Old English tūn ‘enclosure’, ‘settlement’.Variant of Steenson 2.
Boy/Male
Muslim
Winner
Surname or Lastname
English
English : patronymic from the medieval personal name Benne, a pet form of Benedict (see Benn).English : habitational name from a place in Oxfordshire named Benson, from Old English Benesingtūn ‘settlement (Old English tūn) associated with Benesa’, a personal name of obscure origin, perhaps a derivative of Bana meaning ‘slayer’.Jewish (Ashkenazic) : patronymic composed of a pet form of the personal name Beniamin (see Bien, Benjamin) + German Sohn ‘son’.Scandinavian : altered form of such names as Bengtsson, Bendtsen, patronymics from Bengt, Bendt, etc., Scandinavian forms of Benedict.
Surname or Lastname
German
German : variant of Tanner 2.English : from Old French teneor, teneur, tenor, ‘holder of a tenement’, hence an equivalent of Tennant.
Surname or Lastname
English
English : variant of Tennyson.
TENSOR FIELD
TENSOR FIELD
TENSOR FIELD
TENSOR FIELD
TENSOR FIELD
TENSOR FIELD
TENSOR FIELD
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