AI & ChatGPT searches , social queries for TWO POINT-TENSOR
Search references for TWO POINT-TENSOR. Phrases containing TWO POINT-TENSOR
See searches and references containing TWO POINT-TENSOR!
Search references for TWO POINT-TENSOR. Phrases containing TWO POINT-TENSOR
See searches and references containing TWO POINT-TENSOR!TWO POINT-TENSOR
Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used
Two-point_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
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
Structure defining distance on a manifold
manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying infinitesimal
Metric_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
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
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
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
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
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
Operation in mathematics
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 with two small
Tensor_contraction
Mathematical operation on vector spaces
and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle
Tensor_product
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
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 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
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)
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
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
index by contracting the tensor with the metric tensor, g μ ν {\displaystyle g_{\mu \nu }} . For example, taking the tensor T α β {\displaystyle {T^{\alpha
Einstein_notation
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
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 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
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
Specification of a derivative along a tangent vector of a manifold
arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction). Given a point p ∈ M
Covariant_derivative
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
Topological space that locally resembles Euclidean space
is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold, or n {\displaystyle
Manifold
Isomorphism between the tangent and cotangent bundles of a manifold
metric tensor is symmetric. The trace of an ( r , s ) {\displaystyle (r,s)} tensor can be taken in a similar way, so long as one specifies which two distinct
Musical_isomorphism
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
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
Matrix in computer version
thesis. It is sometimes also referred to as the "bifocal tensor". As a tensor it is a two-point tensor in that it is a bilinear form relating points in distinct
Fundamental matrix (computer vision)
Fundamental_matrix_(computer_vision)
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
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
Stress case in finite deformations
models (for example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant; thus creating problems in
Piola–Kirchhoff stress tensors
Piola–Kirchhoff_stress_tensors
Object in differential geometry
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 X , Y {\displaystyle
Torsion_tensor
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
Property of a mathematical space
of two (2D) because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on
Dimension
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
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
Exterior algebraic map taking tensors from p forms to n-p forms
space L ( V , V ) {\displaystyle L(V,V)} is naturally isomorphic to the tensor product V ∗ ⊗ V ≅ V ⊗ V {\displaystyle V^{*}\!\!\otimes V\cong V\otimes
Hodge_star_operator
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
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
Method for specifying point positions
coordinate since only two are needed to specify a point on the plane, but this system is useful in that it represents any point on the projective plane
Coordinate_system
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
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
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
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
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
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
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
Matrix operation which flips a matrix over its diagonal
matrix is the same as the determinant of its transpose. The dot product of two column vectors a and b can be computed as the single entry of the matrix
Transpose
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
Branch of mathematics
various areas, including: Classical treatment of tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning
Multilinear_algebra
Straight path on a curved surface or a Riemannian manifold
∇ ¯ {\displaystyle \nabla ,{\bar {\nabla }}} are two connections such that the difference tensor D ( X , Y ) = ∇ X Y − ∇ ¯ X Y {\displaystyle D(X,Y)=\nabla
Geodesic
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)
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
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
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
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)
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
electromagnetic field tensor F a b {\displaystyle F^{ab}} , a rank-two antisymmetric tensor. Although the word 'tensor' refers to an object at a point, it is common
Mathematics of general relativity
Mathematics_of_general_relativity
Concept in machine learning
("data tensor"), may be analyzed either by artificial neural networks or tensor methods. Tensor decomposition factors data tensors into smaller tensors. Operations
Tensor_(machine_learning)
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)
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
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
Series of system-on-chip processors
2020. The first-generation Tensor chip debuted on the Pixel 6 smartphone series in 2021, and was succeeded by the Tensor G2 chip in 2022, G3 in 2023
Google_Tensor
Muscle of the thigh
The tensor fasciae latae (or tensor fasciæ latæ or, formerly, tensor vaginae femoris) is a muscle of the thigh. Together with the gluteus maximus, it acts
Tensor_fasciae_latae_muscle
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
Second order tensor in vector algebra
or dyadic tensor is a second-order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors
Dyadics
Continuous surjection satisfying a local triviality condition
construct the associated unit sphere bundle, for which the fiber over a point x {\displaystyle x} is the set of all unit vectors in E x {\displaystyle
Fiber_bundle
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
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
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
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
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
Tensor used in continuum mechanics
The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed
Viscous_stress_tensor
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
Mathematics of smooth surfaces
of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written as H2 − |h|2 = R and the two Codazzi
Differential geometry of surfaces
Differential_geometry_of_surfaces
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
point of a physical space; that is, a vector field. Tensors also have extensive applications in physics: Electromagnetic tensor (or Faraday's tensor)
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
System of moving vectors in differential geometry
keep them "pointing in the same direction" in an intuitive sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the
Parallel_transport
Tensor related to gradients
specified neighborhood around a point and makes the information invariant to the observing coordinates. The structure tensor is often used in image processing
Structure_tensor
Construct allowing differentiation of tangent vector fields of manifolds
known as tensor calculus) by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita between 1880 and the turn of the 20th century. Tensor calculus
Affine_connection
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
Operation on differential forms
flux through an infinitesimal k {\displaystyle k} -parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring
Exterior_derivative
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
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
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
Tensorial object depending on two points in a manifold
(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 point. Bitensors
Bitensor
commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures
Alternative_stress_measures
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
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
Study of curves from a differential point of view
principal normal vectors to these two curves are identical at each corresponding point. In other words, if γ1(t) and γ2(t) are two curves in R 3 {\displaystyle
Differentiable_curve
Math/physics concept
arbitrary E-valued forms, thus regarding it as a differential operator on the tensor product of E with the full exterior algebra of differential forms. Given
Connection_form
Concept in physics
In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the strain (i.e.,
Strain-rate_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
Construct in differenital geometry
two is given by the contorsion tensor. In component notation, the covariant derivative ∇ {\displaystyle \nabla } is compatible with the metric tensor
Metric_connection
Stress-strain relation in a linear elastic material
elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness
Elasticity_tensor
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)
{\displaystyle v^{\alpha }=w^{\alpha }} . Two-point tensor Bivector § Tensors and matrices (but note that the stress–energy tensor is symmetric, not skew-symmetric)
Two-vector
Relative deformation of a physical body
ISO 80000-4 (Mechanics), as a "tensor quantity representing the deformation of matter caused by stress. Strain tensor is symmetric and has three linear
Strain_(mechanics)
TWO POINT-TENSOR
TWO POINT-TENSOR
TWO POINT-TENSOR
TWO POINT-TENSOR
TWO POINT-TENSOR
TWO POINT-TENSOR
TWO POINT-TENSOR
TWO POINT-TENSOR
TWO POINT-TENSOR