Tensor

A tensor of type on a finite-dimensional vector space over a field is an element of the vector space tensor product

(1)

where is the dual vector space. Equivalently, using the natural pairing between and , such a tensor can be regarded as a multilinear map from to . This definition is independent of a choice of vector basis.

After a vector basis is chosen, a tensor is represented by an array of components with contravariant and covariant indices. The component transformation rules express the fact that different arrays in different vector bases represent the same tensor. Thus a tensor is not merely an array of numbers, although its components are often the most convenient way to calculate with it. Its tensor rank in this sense is .

Scalars, vectors, covectors, and linear operators are tensors of types , , , and , respectively.

Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity.

The component notation for a tensor is similar to that of a matrix (i.e., ), except that a tensor , , , etc., may have an arbitrary number of indices. In addition, a tensor with rank may be of mixed type , consisting of so-called "contravariant" (upper) indices and "covariant" (lower) indices. Note that the positions of the slots in which contravariant and covariant indices are placed are significant so, for example, is distinct from .

A metric tensor identifies vectors with covectors in any dimension. Their components coincide in an orthonormal basis of Euclidean space, where the metric tensor is the Kronecker delta. Tensors expressed in such bases are Cartesian tensors. In general coordinates, upper and lower components need not coincide, even in two or three dimensions.

A rank-1 tensor may be a vector with components or a covector with components . Rank-2 tensors have component arrays that can be written as matrices, but their transformation laws depend on their type. In particular, a linear operator has type and components , whereas a metric tensor has type and components .

Tensors may be operated on by other tensors (such as metric tensors, the permutation tensor, or the Kronecker delta) or by tensor operators (such as the covariant derivative). The manipulation of tensor indices to produce identities or to simplify expressions is known as index gymnastics, which includes index lowering and index raising as special cases. These can be achieved through multiplication by a metric tensor or its matrix inverse , e.g.,

			(2)
			(3)

(Arfken 1985, p. 159).

Tensor notation can provide a very concise way of writing vector and more general identities. For example, in tensor notation, the dot product is simply written

(4)

where and repeated indices are summed over (Einstein summation). In a positively oriented orthonormal basis of three-dimensional Euclidean space, the cross product can be written as

(5)

where is the permutation tensor.

Under a change of coordinates given by a diffeomorphism, rank-2 contravariant tensors have components that transform as

(6)

Rank-2 covariant tensors have components that transform as

(7)

Rank-2 mixed tensors have components that transform as

(8)

If two tensors and on the same vector space have the same type and index-slot ordering, then they can be added componentwise in the same vector basis,

			(9)
			(10)
			(11)

The generalization of the dot product applied to tensors is called tensor contraction, and consists of setting two unlike indices equal to each other and then summing using the Einstein summation convention. Various types of derivatives can be taken of tensors, the most common being the comma derivative and covariant derivative.

If the components of a tensor of any tensor rank vanish in one coordinate system, they vanish in every coordinate system. A change of coordinates changes the components, not the tensor itself. At each point, the new components are linear homogeneous functions of the old components.

On a manifold , tensors at a point belong to the tensor space formed from the tangent space and its dual vector space. These spaces form a vector bundle. For example,

(12)

is the vector bundle of type tensors, where is the tangent bundle and is the cotangent bundle. A smooth assignment of a tensor to each point is a bundle section of the corresponding vector bundle. In particular, vector fields have type and one-forms have type .

An invertible linear map similarly induces a map given on tensors by

(13)

In component notation, the covector factor transforms by , where is the transpose. The same construction extends to tensors of any type. For a change of coordinates, is the Jacobian matrix at the point in question.