- Glossary of tensor theory
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 .**Classical notation****Tensor Rank**A tensor written in component form is an indexed

array . The "rank" of a tensor is the number of indices required.Dyadic tensor 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 aquadratic form , the second alinear 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"_{"i"}"b"^{"j"}, product component-by-component of rank one tensors. In this case it represents a linear transformation, of rank one in the sense oflinear algebra - a clashing terminology that can cause confusion.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"Einstein notation _{"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.Kronecker delta Levi-Civita symbol ,Covariant tensorContravariant tensorThe classical interpretation is by components. For example in the differential form "a"

_{"i"}"dx"^{"j"}the**"components**" "a"_{"i"}are a covariant vector. That means all indices are lower; contravariant means all indices are upper.Mixed tensor This refers to any tensor with lower and upper indices.

**Cartesian tensor**Cartesian tensors are widely used in various branches of

continuum mechanics , such asfluid mechanics and elasticity. In classicalcontinuum mechanics , the space of interest is usually 3-dimensionalEuclidean space , as is the tangent space at each point. If we restrict the local coordinates to beCartesian coordinates with the same scale centered at the point of interest, themetric tensor is theKronecker delta . This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and tensor densities. All cartesian-tensor indices are written as subscripts.Cartesian tensor s achieve considerable computational simplification at the cost of generality and of some theoretical insight.Raising and lowering indices Symmetric tensor Antisymmetric tensor 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 space s "V" and "W" respectively, then:$v\; otimes\; w$

is a tensor in

:$V\; otimes\; W$.

That is, the $otimes$ operation is a

binary operation , but it takes values in a fresh space (it is in a strong sense "external"). The $otimes$ operation isbilinear ; but no other conditions are applied to it.**Pure tensor**A pure tensor of $V\; otimes\; W$ is one that is of the form $v\; otimes\; w$.

It could be written dyadically "a"

_{"i"}"b"_{"j"}, or more accurately "a"_{"i"}"b"_{"j"}e_{"i"}$otimes$f_{"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 seeSegre embedding .Tensor algebra In the tensor algebra "T"("V") of a vector space "V", the operation

:$otimes$

becomes a normal (internal)

binary operation . This is at the cost of "T"("V") being of infinite dimension, unless "V" has dimension 0. Thefree algebra on a set "'X" is for practical purposes the same as the tensor algebra on the vector space with "X" as basis.Hodge star operator Exterior power The

wedge product is the anti-symmetric form of the $otimes$ operation. The quotient space of T(V) on which it becomes an internal operation is the "exterior algebra" of V; it is agraded algebra , with the graded piece of weight "k" being called the "k"-th**exterior power**of V.Symmetric power ,symmetric algebra This is the invariant way of constructing

polynomial algebra s.**Applications**Metric tensor Strain tensor Stress-energy tensor **Tensor field theory**Jacobian matrix Tensor field Tensor density Lie derivative Tensor derivative Differential geometry **Abstract algebra**Tensor product of fields This is an operation on fields, that does not always produce a field.

Tensor product of R-algebras Clifford module A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.

Tor functors These are the

derived functor s of the tensor product, and feature strongly inhomological algebra . The name comes from thetorsion subgroup inabelian group theory.Symbolic method of invariant theory Derived category Grothendieck's six operations These are "highly" abstract approaches used in some parts of geometry.

**pinors**See:

spin group ,spin-c group ,spinor s,pin group ,pinor s,spinor field ,Killing spinor ,spin manifold .

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