Glossary of tensor theory
From Academic Kids

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.
Contents 
Classical notation
Rank of a tensor
A tensor written in component form is an indexed array. The rank of a tensor is the number of indices required.
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_{i}b^{j}, product componentbycomponent 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.
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.
Covariant tensor, Contravariant tensor
The 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.
This refers to any tensor with lower and upper indices.
Orthogonal tensor
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 coordinates. Orthogonal tensors are also called cartesian tensors
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
 <math>v \otimes w<math>
is a tensor in
 <math>V \otimes W<math>.
That is, the <math> \otimes <math> operation is a binary operation, but it takes values in a fresh space (it is in a strong sense external). The <math> \otimes <math> operation is bilinear; but no other conditions are applied to it.
Pure tensor
A pure tensor of <math>V \otimes W<math> is one that is of the form <math>v \otimes w<math>.
It could be written dyadically a_{i}b_{j}, or more accurately a_{i}b_{j} e_{i}<math> \otimes <math>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 nonlinear conditions for a tensor to satisfy, to be pure. For more see Segre embedding.
Tensor algebra
In the tensor algebra T(V) of a vector space V, the operation
 <math> \otimes <math>
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.
Hodge star operator
Exterior power
The wedge product is the antisymmetric form of the <math> \otimes <math> 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 kth exterior power of V.
Symmetric power, symmetric algebra
This is the invariant way of constructing polynomial algebras.
Applications
Tensor field theory
Abstract algebra
This is an operation on fields, that does not always produce a field.
Representations of Clifford algebras
These may be worked out directly, or by a theory of Clifford modules.
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.
Symbolic method of invariant theory
Derived category Grothendieck's six operations
These are highly abstract approaches used in some parts of geometry.
Spinors
See: spin group, spinc group, spinors, pin group, pinors, spinor field, Killing spinor, spin manifold.