Intermediate treatment of tensors

Note: The following is a modern component-based treatment of tensors (sometimes called the "classical treatment" of tensors). Read the article tensor for a simple description of tensors, or see the component-free treatment of tensors for a more abstract treatment.

Note that the word "tensor" is often used as a shorthand for "tensor field", a concept which defines a tensor value at every point in a manifold. To understand tensor fields, you need to first understand tensors.

A tensor is the mathematical idealization of a geometric or physical quantity whose analytic description, relative to a fixed frame of reference, consists of an array of numbers.

This way of viewing tensors, called tensor analysis, was used by Einstein and is generally prefered by physicists. It is very grossly a generalization of the concept of vectorss and matrices and allows the writing of equations independently of any given coordinate system.

It should be noted that the array of numbers representation of a tensor is not the same thing as the tensor. An image and the object represented by the image are not the same thing. The mass of a stone is not a number. Rather the mass can be described by a number relative to some specified unit mass. Similarly, a given numerical representation of a tensor only makes sense in a particular coordinate system.

Some well known examples of tensors in geometry are quadratic forms, and the curvature tensor. Examples of physical tensors are the energy-momentum tensor and the polarization tensor.

Geometric and physical quantities may be categorized by considering the degrees of freedom inherent in their description. The scalar quantities are those that can be represented by a single number --- speed, mass, temperature, for example. There are also vector-like quantities, such as force, that require a list of numbers for their description. Finally, quantities such as quadratic forms naturally require a multiply indexed array for their representation. These latter quantities can only be conceived of as tensors.

Actually, the tensor notion is quite general, and applies to all of the above examples; scalars and vectors are special kinds of tensors. The feature that distinguishes a scalar from a vector, and distinguishes both of those from a more general tensor quantity is the number of indices in the representing array. This number is called the rank of a tensor. Thus, scalars are rank zero tensors (no indices at all), and vectors are rank one tensors.

It is also necessary to distinguish between two types of indices, depending on whether the corresponding numbers transform covariantly or contravariantly relative to a change in the frame of reference. Contravariant indices are written as superscripts, while the covariant indices are written as subscripts. The valence of a tensor is the pair , where is the number contravariant and the number of covariant indices, respectively.

It is customary to represent the actual tensor, as a stand-alone entity, by a bold-face symbol such as . The corresponding array of numbers for a type tensor is denoted by the symbol

where the superscripts and

subscripts are indices that vary from to . This number , the range of the indices, is called the dimension of the tensor. The total degrees of freedom required for the specification of a particular tensor is a power of the dimension; the exponent is the tensor's rank.

Again, it must be emphasized that the tensor and the representing array are not the same thing. The values of the representing array are given relative to some frame of reference, and undergo a linear transformation when the frame is changed.

Finally, it must be mentioned that most physical and geometric applications are concerned with tensor fields, that is to say tensor valued functions, rather than tensors themselves. Some care is required, because it is common to see a tensor field called simply a tensor. There is a difference, however; the entries of a tensor array

are numbers, whereas the entries

of a tensor field are functions. The present entry treats the purely algebraic aspect of tensors. Tensor field concepts, which typically involved derivatives of some kind, are discussed elsewhere.

Table of contents

1 Definition
2 Transformation rules
3 Further reading

Definition

The formal definition of a tensor quantity begins with a finite-dimensional vector space , which furnishes the uniform "building blocks" for tensors of all valences. In typical applications, is the tangent space at a point of a manifold; the elements of typically represent physical quantities such as velocities and forces. The space of -valent tensors, denoted here by is obtained by taking the tensor product of copies of and copies of the dual vector space . To wit,

In order to represent a tensor by a concrete array of numbers, we require a frame of reference, which is essentially a basis of , say

Every vector in can be "measured" relative to this basis, meaning that for every

there exist unique scalars , such

that (note the use of the Einstein summation convention)

These scalars are called the components of relative to the frame in question.

Let be the corresponding dual basis, i.e.

where the latter is the Kronecker delta array. For every covector

there exists a unique array of components  such

that

More generally, every tensor has a unique representation in terms of components. That is to say, there exists a unique array of scalars such that

Transformation rules

Next, suppose that a change is made to a different frame of reference, say

Any two frames are uniquely related by an invertible transition matrix , having the property that for all values of we have the frame transformation rule

Let be a vector, and let and denote the corresponding component arrays relative to the two frames. From

and from the frame transformation rule we infer the vector transformation rule

where is the matrix inverse of , i.e.

Thus, the transformation rule for a vector's components is contravariant to the transformation rule for the frame of reference. It is for this reason that the superscript indices of a vector are called contravariant.

To establish the transformation rule for vectors, we note that the transformation rule for the dual basis takes the form

and that

while

The transformation rule for covector components is covariant. Let

be a given covector, and let  and
be the corresponding component arrays. Then

The above relation is easily established. We need only remark that

and that

and then use the transformation rule for the frame of reference.

In light of the above discussion, we see that the transformation rule for a general type tensor takes the form