This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:

• Tensor
• Classical treatment of tensors
• Tensor (intrinsic definition)
• Intermediate treatment of tensors
• Application of tensor theory in engineering science

For some history of the abstract theory see also Multilinear algebra. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... The following is a component-based classical treatment of tensors. ... In mathematics, the modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept. ... Note: The following is a modern component-based treatment of tensors (sometimes called the classical treatment of tensors). ... Tensor theory is extremely useful in advanced engineering theory. ... In mathematics, multilinear algebra extends the methods of linear algebra. ...

Classical notation

Rank of a tensor Note: This is a fairly abstract mathematical approach to tensors. ...

A tensor written in component form is an indexed array. The rank of a tensor is the number of indices required. In computer programming, an array, also known as a vector or list (for one-dimensional arrays) or a matrix (for two-dimensional arrays), is one of the simplest data structures. ...

Dyadic tensor A dyadic tensor in multilinear algebra is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, i. ...

A dyadic tensor has rank two, and may be represented as a square matrix. The conventions a_ij, a_i^j, and a^ij, do have different meanings, in that the first may represent a quadratic form, the second a linear transformation, and the distinction is important in contexts that require tensors that aren't orthogonal (see below). A dyad is a tensor such as a_ib^j, product component-by-component of rank one tensors. In this case it represents a linear transformation, of rank one in the sense of linear algebra - a clashing terminology that can cause confusion. For the square matrix section, see square matrix. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...

Einstein notation This states that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index. For example if a_ij is a matrix, then under this convention a_ii is its trace. The Einstein convention is generally used in physics and engineering texts, to the extent that if summation is not applied it is normal to note that explicitly. In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

Kronecker delta In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

Levi-Civita symbol The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

Covariant tensor, Contravariant tensor In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ...

The classical interpretation is by components. For example in the differential form a_idx^j the components a_i are a covariant vector. That means all indices are lower; contravariant means all indices are upper.

Mixed tensor In tensor analysis, a mixed tensor is a tensor which is neither covariant nor contravariant. ...

This refers to any tensor with lower and upper indices.

Orthogonal tensor In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

In the presence of a tensor δ_i^j, there is no need to maintain the distinction of upper and lower indices. That is the case given a distinguished set of orthogonal co-ordinates. Orthogonal tensors are also called cartesian tensors.

Contraction of a tensor In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ...

Symmetric tensor A tensor A, with components Aij, is said to be symmetric if Aij = Aji for all i, j. ...

Antisymmetric tensor In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ...

Multiple cross products In mathematics, there are tricks for multiple cross products. ...

Algebraic notation

This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.

Tensor product

If v and w are vectors in vector spaces V and W respectively, then In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...

is a tensor in

.

That is, the operation is a binary operation, but it takes values in a fresh space (it is in a strong sense external). The operation is bilinear; but no other conditions are applied to it. In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

Pure tensor

A pure tensor of is one that is of the form .

It could be written dyadically a_ib_j, or more accurately a_ib_j e_if_j, where the e_i are a basis for V and the f_j a basis for W. Therefore, unless V and W have the same dimension, the array of components need not be square. Such pure tensors are not generic: if both V and W have dimension > 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure. For more see Segre embedding. In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety. ...

Tensor algebra

In the tensor algebra T(V) of a vector space V, the operation 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. ...

becomes a normal (internal) binary operation. This is at the cost of T(V) being of infinite dimension, unless V has dimension 0. The free algebra on a set 'X is for practical purposes the same as the tensor algebra on the vector space with X as basis. In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring. ...

Hodge star operator

In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. ...

Exterior power

The wedge product is the anti-symmetric form of the operation. The quotient space of T(V) on which it becomes an internal operation is the exterior algebra of V; it is a graded algebra, with the graded piece of weight k being called the k-th exterior power of V. In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ...

Symmetric power, symmetric algebra

This is the invariant way of constructing polynomial algebras. In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ...

Applications

Metric tensor In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

Strain tensor The strain tensor [ε] is a symmetric tensor used to quantify the strain of an object undergoing a 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij (i ≠ j) are the...

Stress-energy tensor This article is in need of attention from an expert on the subject. ...

Tensor field theory

Jacobian matrix In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

Tensor field In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ...

Tensor density Tensor field - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

Lie derivative In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

Tensor derivative In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

Differential geometry In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

Abstract algebra

Tensor product of fields In mathematics, the theory of fields in abstract algebra lacks a direct product: the direct product of two fields, considered as ring (mathematics) is never itself a field. ...

This is an operation on fields, that does not always produce a field.

Tensor product of R-algebras In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on RZS then makes it a coproduct in the category of commutative rings. ...

Representations of Clifford algebras In mathematics, the representations of Clifford algebras are also known as Clifford modules. ...

These may be worked out directly, or by a theory of Clifford modules.

Tor functors The Tor functors are the derived functors of the tensor product functor in mathematics. ...

These are the derived functors of the tensor product, and feature strongly in homological algebra. The name comes from the torsion subgroup in abelian group theory. In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

Symbolic method of invariant theory In mathematics, the symbolic method in invariant theory is a highly formal algorithm developed in the 19th century for computing form invariants — invariants of algebraic forms. ...

Derived category Grothendieck's six operations In mathematics, the derived category D(C) of a category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C (which therefore should already be an abelian category). ...

These are highly abstract approaches used in some parts of geometry.

Spinors

See: spin group, spin-c group, spinors, pin group, pinors, spinor field, Killing spinor, spin manifold. In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ... In geometry, a spin structure on a principal SO(N) bundle E over a Riemannian manifold M is a lift of E to a principal Spin(N) bundle. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... ... This article needs to be cleaned up to conform to a higher standard of quality. ... Categories: Wikipedia cleanup | Stub ... In geometry, a spin structure on a principal SO(N) bundle E over a Riemannian manifold M is a lift of E to a principal Spin(N) bundle. ...