# Category: Tensors

Angular velocity
In physics, angular velocity or rotational velocity (ω or Ω), also known as angular frequency vector, is a pseudovector representation of how fast the angular position or orientation of an object chan
Schur–Weyl duality
Schur–Weyl duality is a mathematical theorem in representation theory that relates irreducible finite-dimensional representations of the general linear and symmetric groups. It is named after two pion
Finite strain theory
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate a
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geo
PlaidML
PlaidML is a portable . Tensor compilers bridge the gap between the universal mathematical descriptions of deep learning operations, such as convolution, and the platform and chip specific code needed
Komar superpotential
In general relativity, the Komar superpotential, corresponding to the invariance of the Hilbert–Einstein Lagrangian , is the tensor density: associated with a vector field , and where denotes covarian
Four-tensor
In physics, specifically for special relativity and general relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime.
Voigt notation
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel
Cartesian tensor
In geometry and linear algebra, 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 one such basis
Stress–energy–momentum pseudotensor
In the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor that incorporates th
Laguerre form
In mathematics, the Laguerre form is generally given as a third degree tensor-valued form, that can be written as, . * v * t * e
Lanczos tensor
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 in 1949. The theoretical importance of the
Veronese surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complet
Reynolds stress
In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuatio
Penrose graphical notation
In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A d
Spherical basis
In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis clos
Gyration tensor
In physics, the gyration tensor is a tensor that describes the second moments of position of a collection of particles where is the Cartesian coordinate of the position vector of the particle. The ori
Multivector
In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra Λ(V) of a vector space V. This algebra is graded, associative and alternating, and consis
Tensor Contraction Engine
The Tensor Contraction Engine (TCE) is a compiler for a domain-specific language that allows chemists to specify the computation in a high-level Mathematica-style language. It transforms tensor summat
Tensor product of quadratic forms
In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible (that is, R has
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a
Lisp Algebraic Manipulator
The Lisp Algebraic Manipulator (also known as LAM) was created by , who had written Atlas LISP Algebraic Manipulation (ALAM was designed in 1970). LAM later became the basis for the interactive comput
Symmetric tensor
In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: for every permutation σ of the symbols {1, 2, ..., r}. Alternatively, a symmetric tensor o
Recurrent tensor
In mathematics and physics, a recurrent tensor, with respect to a connection on a manifold M, is a tensor T for which there is a one-form ω on M such that
Raising and lowering indices
In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor express
Tensor (intrinsic definition)
In 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 concept. Their properties can be derived
Characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of
Pseudotensor
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation-preserving coordinate transformation (e.g. a proper rotation) but additionally chang
Kulkarni–Nomizu product
In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two (0, 2)-tensors and gives as a result a (0, 4
Plebanski tensor
The Plebanski tensor is an order 4 tensor in general relativity constructed from the trace-free Ricci tensor. It was first defined by Jerzy Plebański in 1964. Let be the trace-free Ricci tensor: Then
Generalized structure tensor
In image analysis, the generalized structure tensor (GST) is an extension of the Cartesian structure tensor to curvilinear coordinates. It is mainly used to detect and to represent the "direction" par
Tensor software
Tensor software is a class of mathematical software designed for manipulation and calculation with tensors.
List of moments of inertia
Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass (which determines an object's re
Relative scalar
In mathematics, a relative scalar (of weight w) is a scalar-valued function whose transform under a coordinate transform, on an n-dimensional manifold obeys the following equation where that is, the d
Tensor rank decomposition
In multilinear algebra, the tensor rank decomposition, the exact decomposition of a tensor in terms of the minimum terms, is an open problem. Canonical polyadic decomposition (CPD) is a variant of the
Second fundamental form
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 Euclidean space, usually denoted by (read
Jordan and Einstein frames
The Lagrangian in scalar-tensor theory can be expressed in the Jordan frame in which the scalar field or some function of it multiplies the Ricci scalar, or in the Einstein frame in which Ricci scalar
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is
Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, fo
Einstein notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational conve
Structure tensor
In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified ne
Tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical te
Cotton tensor
In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for n = 3 is necessar
Glossary of tensor theory
This is a glossary of tensor theory. For expositions of from different points of view, see: * Tensor * Tensor (intrinsic definition) * Application of tensor theory in engineering science For some h
Calculus of moving surfaces
The calculus of moving surfaces (CMS) is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative whose original definition was put for
Manifest covariance
In general relativity, a manifestly covariant equation is one in which all expressions are tensors. The operations of addition, tensor multiplication, tensor contraction, raising and lowering indices,
Contorsion tensor
The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbei
Total position spread
In physics, the total position-spread (TPS) tensor is a quantity originally introduced in the modern theory of electrical conductivity. In the case of molecular systems, this tensor measures the fluct
Axiality and rhombicity
In physics and mathematics, axiality and rhombicity are two characteristics of a symmetric second-rank tensor in three-dimensional Euclidean space, describing its directional asymmetry. Let A denote a
Two-vector
A two-vector or bivector is a tensor of type and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars). The tens
Segre classification
The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy–momentum tensor (or
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 in continuum mechanics to transform between r
Mixed tensor
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 will be a subscript (covariant) and at leas
Higher-order singular value decomposition
In multilinear algebra, the higher-order singular value decomposition (HOSVD) of a tensor is a specific orthogonal Tucker decomposition. It may be regarded as one generalization of the matrix singular
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum
Invariants of tensors
In mathematics, in the fields of multilinear algebra and representation theory, the principal invariants of the second rank tensor are the coefficients of the characteristic polynomial , where is the
Cotangent space
In differential geometry, the cotangent space is a vector space associated with a point on a smooth (or differentiable) manifold ; one can define a cotangent space for every point on a smooth manifold
Bach tensor
In differential geometry and general relativity, the Bach tensor is a trace-free tensor of rank 2 which is conformally invariant in dimension n = 4. Before 1968, it was the only known conformally inva
Covariant transformation
In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The transformation that describes the new basis vector
Stewart–Walker lemma
The Stewart–Walker lemma provides necessary and sufficient conditions for the linear perturbation of a tensor field to be gauge-invariant. if and only if one of the following holds 1. 2. is a constant
Tensor reshaping
In multilinear algebra, a reshaping of tensors is any bijection between the set of indices of an order- tensor and the set of indices of an order- tensor, where . The use of indices presupposes tensor
Abstract index notation
Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a
Tensor representation
In mathematics, the tensor representations of the general linear group are those that are obtained by taking finitely many tensor products of the fundamental representation and its dual. The irreducib
Contracted Bianchi identities
In general relativity and tensor calculus, the contracted Bianchi identities are: where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation. These identities are named a
Trace diagram
In mathematics, trace diagrams are a graphical means of performing computations in linear and multilinear algebra. They can be represented as (slightly modified) graphs in which some edges are labeled
Tensor decomposition
In multilinear algebra, a tensor decomposition is any scheme for expressing a tensor as a sequence of elementary operations acting on other, often simpler tensors. Many tensor decompositions generaliz
Tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic
Covariance and contravariance of vectors
In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a chan
Nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective
Palatini identity
In general relativity and tensor calculus, the Palatini identity is: where denotes the variation of Christoffel symbols and indicates covariant differentiation. A proof can be found in the entry Einst
Multilinear multiplication
In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.
Schouten tensor
In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for n ≥ 3 by: where Ric is the Ricci tensor (defined by contracting the first and third
Deformation (physics)
In physics, deformation is the continuum mechanics transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particle
Tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregori
Scalar–tensor theory
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. For example, the Brans–Dicke theory of gravita
Codazzi tensor
In the mathematical field of differential geometry, a Codazzi tensor (named after Delfino Codazzi) is a symmetric 2-tensor whose covariant derivative is also symmetric. Such tensors arise naturally in
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vect
Antisymmetric tensor
In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index sub
Fractional anisotropy
Fractional anisotropy (FA) is a scalar value between zero and one that describes the degree of anisotropy of a diffusion process. A value of zero means that diffusion is isotropic, i.e. it is unrestri
Tensor density
In 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 from one coordinate system to
Dyadics
In 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 are numerous ways to multiply two Eu
SHEEP (symbolic computation system)
SHEEP is one of the earliest interactive symbolic computation systems. It is specialized for computations with tensors, and was designed for the needs of researchers working with general relativity an
Diffusion MRI
Diffusion-weighted magnetic resonance imaging (DWI or DW-MRI) is the use of specific MRI sequences as well as software that generates images from the resulting data that uses the diffusion of water mo
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as
Tensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, i
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as
Stress (mechanics)
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as force per unit area. When an object is pulle
Tensor sketch
In statistics, machine learning and algorithms, a tensor sketch is a type of dimensionality reduction that is particularly efficient when applied to vectors that have tensor structure. Such a sketch c
Saint-Venant's compatibility condition
In the mathematical theory of elasticity, Saint-Venant's compatibility condition defines the relationship between the strain and a displacement field by where . Barré de Saint-Venant derived the compa
Metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as
Spin tensor
In mathematics, mathematical physics, and theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The tensor has application in general rel
Multilinear subspace learning
Multilinear subspace learning is an approach to dimensionality reduction. Dimensionality reduction can be performed on a data tensor whose observations have been vectorized and organized into a data t
Pullback (differential geometry)
Suppose that φ : M → N is a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to