Glossary of tensor theory

Classical notation

The earliest foundation of tensor theory – tensor index notation.

The components of a tensor with respect to a basis is an indexed array. The order of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term degree or rank.

The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order.

A dyadic tensor is a tensor of order two, and may be represented as a square matrix. In contrast, a dyad is specifically a dyadic tensor of rank one.

This notation is based on the understanding that in a term in an expression contains a repeated index letter, the default interpretation is that the product is summed over all permitted 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 widely used in physics and engineering texts, to the extent that if summation is not to be applied, it is normal to note that explicitly.

The classical interpretation is by components. For example, in the differential form a_idxⁱ 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 that has both lower and upper indices.

Cartesian tensor

Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity. In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the metric tensor is the Kronecker 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 tensors achieve considerable computational simplification at the cost of generality and of some theoretical insight.

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

is a tensor in

That is, the ⊗ operation is a binary operation, but it takes values into a fresh space (it is in a strong sense external). The ⊗ operation is a bilinear map; but no other conditions are applied to it.

Pure tensor

A pure tensor of V ⊗ W is one that is of the form v ⊗ w

It could be written dyadically aⁱb^j, or more accurately aⁱb^j e_i ⊗ 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 greater than 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.

Tensor algebra

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

becomes a normal (internal) binary operation. A consequence is that T(V) has 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.

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.

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.

A representation of a Clifford algebra which gives a realisation of a Clifford algebra as a matrix algebra.

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.

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

Spinors

See:

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