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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 tensor
Mixed_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
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
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 in differential geometry
converge. Formally, it is a symmetric rank-two tensor obtained by taking a trace of the Riemann curvature tensor of a Riemannian or pseudo-Riemannian metric
Ricci_curvature
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
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
Tensor field in Riemannian geometry
mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the
Riemann_curvature_tensor
Assignment of a tensor continuously varying across a region of space
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space
Tensor_field
Generalization of tensor fields
differential 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
Structure defining distance on a manifold
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 if g ( v , v ) >
Metric_tensor
Tensor index notation for tensor-based calculations
notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern
Ricci_calculus
Tensor equal to the negative of any of its transpositions
tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor
Antisymmetric_tensor
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
Branch of mathematics
Pseudovector Spinor Tensor Tensor algebra, Free algebra Tensor contraction Symmetric algebra, Symmetric power Symmetric tensor Mixed tensor Pandey, Divyanshu;
Multilinear_algebra
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
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
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
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)
Antisymmetric permutation object acting on tensors
independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms
Levi-Civita_symbol
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
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
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
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)
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
inertia tensor of a body calculated at its center of mass, and R {\displaystyle \mathbf {R} } be the displacement vector of the body. The inertia tensor of
Moment_of_inertia
Universal construction in multilinear algebra
the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any order) with multiplication being the tensor product
Tensor_algebra
Exterior algebraic map taking tensors from p forms to n-p forms
signature of the metric tensor is all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian
Hodge_star_operator
Specification of a derivative along a tangent vector of a manifold
fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction). Given
Covariant_derivative
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
Shorthand notation for tensor operations
the multiplication. Given a tensor, one can raise an index or lower an index by contracting the tensor with the metric tensor, g μ ν {\displaystyle g_{\mu
Einstein_notation
Mathematical operation
A general mixed tensor field will then transform using Φ {\displaystyle \Phi } and Φ − 1 {\displaystyle \Phi ^{-1}} according to the tensor product decomposition
Pullback (differential geometry)
Pullback_(differential_geometry)
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
Vector behavior under coordinate changes
changes in the coordinates. Active and passive transformation Mixed tensor Two-point tensor, a generalization allowing indices to reference multiple vector
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
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
Graphical notation for multilinear algebra calculations
essentially the composition of functions. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting
Penrose_graphical_notation
Algebra associated to any vector space
alternating tensor subspace. On the other hand, the image A ( T ( V ) ) {\displaystyle {\mathcal {A}}(\mathrm {T} (V))} is always the alternating tensor graded
Exterior_algebra
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
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
Matrix operation which flips a matrix over its diagonal
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Transpose
Covariant derivative of the metric tensor
In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It can be interpreted as the failure
Nonmetricity_tensor
Topological space that locally resembles Euclidean space
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Manifold
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
Electromagnetism in general relativity
inverse of the metric tensor g α β {\displaystyle g_{\alpha \beta }} , and g {\displaystyle g} is the determinant of the metric tensor. Notice that A α {\displaystyle
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
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
Theory of gravitation as curved spacetime
stress–energy tensor, which includes both energy and momentum densities as well as stress: pressure and shear. Using the equivalence principle, this tensor is readily
General_relativity
Property of a mathematical space
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Dimension
Method for specifying point positions
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Coordinate_system
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
Second order tensor in vector algebra
mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second-order tensor, written in a notation that fits in with vector algebra. There
Dyadics
energy–momentum tensor and the Petrov classification of the Weyl tensor. There are various methods of classifying these tensors, some of which use tensor invariants
Mathematics of general relativity
Mathematics_of_general_relativity
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
Expression that may be integrated over a region
covariant tensor field of rank k {\displaystyle k} . The differential forms on M {\displaystyle M} are in one-to-one correspondence with such tensor fields
Differential_form
Straight path on a curved surface or a Riemannian manifold
and real trees. In a Riemannian manifold M {\displaystyle M} with metric tensor g {\displaystyle g} , the length L {\displaystyle L} of a continuously differentiable
Geodesic
Continuous surjection satisfying a local triviality condition
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Fiber_bundle
Mathematical function, in linear algebra
linear maps are said to be 1-co- 1-contra-variant objects, or type (1, 1) tensors. A linear transformation between topological vector spaces, for example
Linear_map
Array of numbers
multiplication can be defined with entries objects of a category equipped with a "tensor product" similar to multiplication in a ring, having coproducts similar
Matrix_(mathematics)
Differential form of degree one or section of a cotangent bundle
one 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
One-form
Vector operator in vector calculus
some authors define the divergence of a mixed tensor by using the musical isomorphism ♯: if T is a (p, q)-tensor (p for the contravariant vector and q for
Divergence
Set of vectors used to define coordinates
of redirect targets Spherical basis – Basis used to express spherical tensors Brown, William A. (1991). Matrices and vector spaces. New York: M. Dekker
Basis_(linear_algebra)
Non-tensorial representation of the spin group
distinguished from the tensor representations given by Weyl's construction by the weights. Whereas the weights of the tensor representations are integer
Spinor
Branch of mathematics
where N J {\displaystyle N_{J}} is a tensor of type (2, 1) related to J {\displaystyle J} , called the Nijenhuis tensor (or sometimes the torsion). An almost
Differential_geometry
Array of numbers describing a metric connection
corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γijk are zero
Christoffel_symbols
Ways of writing certain laws of physics
t^{2}}-\nabla ^{2}.} The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is (+ − − −), corresponding
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Mathematical operation on vectors in 3D space
seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form, a (0,3)-tensor, by raising an index
Cross_product
Conserved physical quantity; rotational analogue of linear momentum
as an anti-symmetric second order tensor, with components ωij. The relation between the two anti-symmetric tensors is given by the moment of inertia which
Angular_momentum
Theory of interwoven space and time by Albert Einstein
coordinates are divided by c or factors of c±2 are included in the metric tensor. These numerous conventions can be superseded by using natural units where
Special_relativity
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
field. Tensors also have extensive applications in physics: Electromagnetic tensor (or Faraday's tensor) in electromagnetism Finite deformation tensors for
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Branch of physics which studies the behavior of materials modeled as continuous media
stress tensor, and ρ 0 {\displaystyle \rho _{0}} is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related
Continuum_mechanics
Physics concept
a coordinate system, a tensor defined in this way is independent of the choice of a coordinate system. The notation of a tensor is T ( σ , … , ρ , u ,
Covariant_transformation
Type of derivative in differential geometry
differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field
Lie_derivative
Affine connection on the tangent bundle of a manifold
components of a contravariant vector. This discovery was the real beginning of tensor analysis. In 1906, L. E. J. Brouwer was the first mathematician to consider
Levi-Civita_connection
Mathematical notation for tensors and spinors
between tensor factors of type V {\displaystyle V} and those of type V ∗ {\displaystyle V^{*}} . A general homogeneous tensor is an element of a tensor product
Abstract_index_notation
Tensor quantum field of mixed symmetry
physics, the Curtright field (named after Thomas Curtright) is a tensor quantum field of mixed symmetry, whose gauge-invariant dynamics are dual to those of
Curtright_field
Type of physical quantity
spacetime Tensor – Algebraic object with geometric applications Tensor density – Generalization of tensor fields Tensor field – Assignment of a tensor continuously
Pseudotensor
Mapping from p forms to p-1 forms
generalized dot productPages displaying short descriptions of redirect targets Tensor contraction – Operation in mathematics Tu, Sec 20.5. There is another formula
Interior_product
coordinates. Thus, a two-point tensor will have one capital and one lower-case index; for example, AjM. A conventional tensor can be viewed as a transformation
Two-point_tensor
System of moving vectors in differential geometry
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Parallel_transport
Examples of tensor representations: Not all irreducible representations of G L ( n , C ) {\displaystyle GL(n,\mathbb {C} )} are tensor representations
Representations of classical Lie groups
Representations_of_classical_Lie_groups
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
Nvidia software development kit for deep learning inference
TensorRT is a software development kit (SDK) and inference optimization runtime developed by Nvidia for deploying trained deep learning and machine learning
TensorRT
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
Software library for LLM inference
UINT64 // starting position within the tensor_data block, relative to the start of the block // (n+1)-th tensor ... Llama.cpp supports many large language
Llama.cpp
scalars). The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs
Two-vector
Differential form
absolute value of the determinant of the matrix representation of the metric tensor on the manifold. The volume form is denoted variously by ω = v o l n = ε
Volume_form
Concept in multilinear algebra and representation theory
convention as the deviator of any tensor has by definition zero trace. Furthermore, mixed invariants between pairs of rank two tensors may also be defined. These
Invariants_of_tensors
Basis used to express spherical tensors
a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank k {\displaystyle k} is 1. For
Spherical_basis
Operation on differential forms
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Exterior_derivative
Topics referred to by the same term
Look up mixed in Wiktionary, the free dictionary. Mixed is the past tense of mix. Mixed may refer to: Mixed (United Kingdom ethnicity category), an ethnicity
Mixed
Mathematical Concept
notation is as follows: Write down the second order tensor in matrix form (in the example, the stress tensor) Strike out the diagonal Continue on the third
Voigt_notation
that is 0 whenever arguments are linearly dependent Antisymmetric tensor – Tensor equal to the negative of any of its transpositions Hazewinkel (1990)
Symmetrization
Construct in differenital geometry
the field strength tensor, a classical one using R as the curvature tensor, and the classical notation for the Riemann curvature tensor, most of which can
Metric_connection
Function that is invariant under all permutations of its variables
functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector
Symmetric_function
Mathematical notation
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Multi-index_notation
GPU microarchitecture by Nvidia
estimated to provide 25 Gbit/s per lane. (Disabled for Titan V) Tensor cores: A tensor core is a unit that multiplies two 4×4 FP16 matrices, and then adds
Volta_(microarchitecture)
2025 Android smartphones developed by Google
needed] The custom Google Tensor G5 System-on-Chip (SoC) is a noticeable upgrade over the Tensor G4 and other previous Tensor processors. Instead of using
Pixel_10
Mathematical operation on matrices
specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map
Kronecker_product
Mathematics of smooth surfaces
due to Élie Cartan. In the language of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written
Differential geometry of surfaces
Differential_geometry_of_surfaces
Branch of the trigeminal nerve responsible for the lower face and jaw
Medial pterygoid nerve Medial pterygoid muscle Tensor tympani muscle Tensor veli palatini (via tensor veli palatini branch) Lateral pterygoid nerve Lateral
Mandibular_nerve
Physical quantity that changes sign with improper rotation
yields a bivector which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two
Pseudovector
MIXED TENSOR
MIXED TENSOR
Girl/Female
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Fixed
Girl/Female
Hindu, Indian, Marathi, Sanskrit, Telugu
Immovable; Fixed; Quiet
Girl/Female
Tamil
Dhruvika | தà¯à®°à¯à®µà®¿à®•à®¾
Firmly fixed
Dhruvika | தà¯à®°à¯à®µà®¿à®•à®¾
Boy/Male
Indian, Sanskrit
Firmly Fixed
Boy/Male
French, Indian, Sanskrit
Fat; A Mixed Caste
Boy/Male
Indian, Sanskrit
Well Fixed
Boy/Male
Hindu, Indian, Kannada, Telugu
Fixed
Boy/Male
Indian, Marathi
Mixed with Soil
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sikh
Mixed Sweet
Boy/Male
Indian, Sanskrit
Firmly Fixed
Girl/Female
Hindu, Indian, Marathi
Directed; Fixed
Boy/Male
Bengali, Indian
A Mixed Raag
Girl/Female
Assamese, Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya
Firmly Fixed
Girl/Female
Tamil
Fixed
Boy/Male
Indian, Sanskrit
Fixed
Girl/Female
Bengali, Indian, Kannada, Marathi
Firmly Fixed
Girl/Female
Gujarati, Indian
Firmly Fixed
Girl/Female
Tamil
Fixed
Girl/Female
Hindu
Fixed
Surname or Lastname
English (East Anglia)
English (East Anglia) : unexplained.
MIXED TENSOR
MIXED TENSOR
Male
Danish
, reward of the gods.
Boy/Male
Hindu
Stars
Surname or Lastname
English
English : variant spelling of Thorndike.
Girl/Female
Christian, French, Indian, Italian, Latin
Priest
Boy/Male
Indian, Punjabi, Sikh
One who Remembers God
Boy/Male
Hindu, Indian, Marathi
Happy; Delighted
Boy/Male
German
Famous fighter.
Male
Dutch
, home ruler.
Boy/Male
Muslim
Frankness. Sincerity.
Boy/Male
Sikh
True devotee of God, Light of truth
MIXED TENSOR
MIXED TENSOR
MIXED TENSOR
MIXED TENSOR
MIXED TENSOR
a.
Mixed with opiates.
a.
Gray; grayish; sprinkled or mixed with gray; of a mixed white and black.
imp. & p. p.
of Mix
a.
Mixed; confounded.
imp. & p. p.
of Mire
a.
Capable of being mixed.
n.
Capability of being mixed.
imp. & p. p.
of Fix
n.
Mixed mathematics.
a.
Mixed; of mixed material or color.
n.
One who, or that which, mixes.
a.
Fig.: Mixed.
a.
Formed by mixing; united; mingled; blended. See Mix, v. t. & i.
n.
A compost heap; a dunghill.
n.
Clay mixed with straw.
a.
Securely placed or fastened; settled; established; firm; imovable; unalterable.
adv.
In a mixed manner.
imp. & p. p.
of Mine
a.
Stable; non-volatile.
a.
Capable of being mixed.