Tensor

In mathematics, a tensor is (in an informal sense) a generalized linear 'quantity' or 'geometrical entity' that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. The rank of a particular tensor is the number of array indices required to describe such a quantity. For example, mass, temperature, and other scalar quantities are tensors of rank 0; force, momentum and other vector-like quantities are tensors of rank 1; a linear transformation such as an anisotropic relationship between force and acceleration vectors is a tensor of rank 2.

This article attempts to provide a non-technical introduction to the idea of tensors, and to provide an introduction to the articles which describe different, complementary treatments of the theory of tensors in detail. For a formal, axiomatic definition of tensor, see the entry entitled tensor (intrinsic definition). The entry Classical treatment of tensors gives an older, less formal definition. The entry intermediate treatment of tensors attempts to bridge the two extremes and to show their relationships.

It should also be noted that many mathematical structures informally called 'tensors' are actually 'tensor fields', tensorial quantities that vary from point to point. However, to better understand tensor fields, one should first understand the basic idea of tensors.

Importance and usage

Tensors are of importance in physics and engineering. In the field of diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the brain. Perhaps the most important engineering examples are the stress tensor and strain tensor, which are both 2nd rank tensors, and are related in a general linear material by a fourth rank elasticity tensor.

Specifically, a 2nd rank tensor quantifying stress in a 3-dimensional/solid object has components which can be conveniently represented as 3x3 array. The three Cartesian faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number (being in three-space). Thus, 3x3, or 9 components are required to describe the stress at this cube-shaped infintesimal segment (which may now be treated as a point). Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, the need for a 2nd order tensor is produced.

While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor theory is to explain further implications of saying that a quantity is a tensor, beyond specifying that it requires a number of indexed components. In particular, tensors behave in specific ways under coordinate transformations. The abstract theory of tensors is a branch of linear algebra, now called multilinear algebra.

History

The word tensor was introduced in 1846 by William Rowan Hamilton^[1] to describe the norm operation in a certain type of algebraic system (eventually known as a Clifford algebra). The word was used in its current meaning by Woldemar Voigt in 1899. The notation was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and was made accessible to many mathematicians by the publication of Tullio Levi-Civita's 1900 classic text of the same name (in Italian; translations followed). In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. General Relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann^[2], or perhaps from Levi-Civita himself. Tensors are used also in other fields such as continuum mechanics.

The choice of approach

There are two ways of approaching the definition of tensors:

The usual physics way of defining tensors, in terms of objects whose components transform according to certain rules, introducing the ideas of covariant or contravariant transformations.

The usual mathematics way, which involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed. Covariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the contravariant vectors.

Physicists and engineers are among the first to recognise that vectors and tensors have a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated. Similarly, mathematicians find there are some tensor relations which are more conveniently derived in a co-ordinate notation.

Examples

A tensor may be expressed as the sequence of values represented by a function with a vector valued domain and a scalar valued range. These vectors in the domain are vectors of counting numbers, and these numbers are called indexes. For example, a rank 3 tensor might have dimensions 2, 5, and 7. Here, the vectors range from <1, 1, 1> through <2, 5, 7>. Here, the tensor would have one value at <1, 1, 1>, another at <1, 1, 2>, and so on for a total of 70 values. (Likewise, vectors may be expressed as a sequence of values represented by a function with a scalar valued domain and a scalar valued range, and the numbers in the domain are counting numbers called indices, and the number of distinct indices is sometimes called the dimension of the vector.)

A tensor field associates a tensor value with every point on a manifold. Thus, instead of simply having 70 values as indicated in the above example, for a rank 3 tensor field with dimensions <2, 5, 7> every point in the space would have 70 values associated with it. In other words, a tensor field means there's some tensor valued function which has the Euclidean space as its domain. Not just any function is allowed here -- see tensor field for more coverage of these requirements.

Not all relationships in nature are linear, but most are differentiable and so may be locally approximated with sums of multilinear maps. Thus most quantities in the physical sciences can be usefully expressed as tensors.

As a simple example, consider a ship in the water. We want to describe its response to an applied force. Force is a vector, and the ship will respond with an acceleration, which is also a vector. The acceleration will in general not be in the same direction as the force, because of the particular shape of the ship's body. However, it turns out that the relationship between force and acceleration is linear in classical mechanics. Such a relationship is described by a tensor of type (1,1) (that is to say, it transforms a vector into another vector). The tensor can be represented as a matrix which when multiplied by a vector results in another vector. Just as the numbers which represent a vector will change if one changes the coordinate system, the numbers in the matrix that represents the tensor will also change when the coordinate system is changed.

In engineering, the stresses inside a rigid body or fluid are also described by a tensor; the word "tensor" is Latin for something that stretches, i.e., causes tension. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2,0), or more precisely by a tensor field of type (2,0) since the stresses may change from point to point.

Some well known examples of tensors in geometry are quadratic forms, and the curvature tensor. Examples of physical tensors are the energy-momentum tensor, the inertia 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 --- pressure, 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 (or the order) of a tensor. Thus, scalars are rank zero tensors (no indices at all), and vectors are rank one tensors.

Another example of a tensor is the Riemann curvature tensor from the theory of General Relativity, which is of rank 4 with dimensions <4, 4, 4, 4> (3 spatial + time = 4 dimensions). It can be treated as matrix (or vector) with 256 components (256 = 4 × 4 × 4 × 4). Only 20 of these components are actually independent of each other, greatly simplifying the matrix (or vector).

Approaches, in detail

There are equivalent approaches to visualizing and working with tensors; that the content is actually the same may only become apparent with some familiarity with the material.

The classical approach

The classical approach views tensors as multidimensional arrays that are n-dimensional generalizations of scalars, 1-dimensional vectors and 2-dimensional matrices. The "components" of the tensor are the values in the array. This idea can then be further generalized to tensor fields, where the elements of the tensor are functions, or even differentials.

However, to count as a tensor, the arrays need to transform correctly when the reference co-ordinate system is changed. This transformation is a generalisation of the relationship which holds for vector components, and is similarly an expression of the independence of the underlying entity from the reference frame in which it is expressed.

The modern approach

The modern (component-free) approach views tensors initially as abstract objects, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. This treatment has attempted to replace the component-based treatment for advanced study, in the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector. You could say that the slogan is 'tensors are elements of some tensor space'. Nevertheless, a component-free approach has not become fully popular, owing to the difficulties involved with giving a geometrical interpretation to higher-rank tensors.

The intermediate treatment of tensors article attempts to bridge the two extremes, and to show their relationships.

In the end the same computational content is expressed, both ways. See glossary of tensor theory for a listing of technical terms.

Tensor densities

It is also possible for a tensor field to have a "density". A tensor with density r transforms as an ordinary tensor under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian to the r^th power. This is best explained, perhaps, using vector bundles: where the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times.

External links

Historic references

^ William Rowan Hamilton, On some Extensions of Quaternions[1]
^ Abraham Pais, Subtle is the Lord

Reference books

Tensors, Differential Forms, and Variational Principles (1989) David Lovelock, Hanno Rund
Tensor Analysis on Manifolds (1981) Richard L Bishop, Samuel I. Goldberg
Introduction to Tensor Calculus, Relativity and Cosmology (2003) D. F. Lawden
Tensor Analysis (2003) L.P. Lebedev, Michael J. Cloud

Tensor software

GRTensorII is a computer algebra package for performing calculations in the general area of differential geometry. GRTensor II is not a stand alone package, the program runs with all versions of Maple V Release 3 through Maple 9.5. A limited version (GRTensorM) has been ported to Mathematica.
Tensorial 3.0 Tensorial is a general purpose tensor calculus package for Mathematica 4.1 or better. Some of its features are: complete freedom in choosing tensor labels and indices; base indices may be any set of integers or symbols; tensor shortcuts for easy entry of tensors; flavored (colored or annotated) indices for different coordinate systems; CircleTimes notation available; easy methods for storing and substituting tensor values; routines for partial, covariant, total, absolute (Intrinsic) and Lie derivatives; There is extensive documentation, with a Help page and numerous examples for each command. In addition there are a number of tutorial and sample application notebooks.

You may wish to check the site occasionally for updates. A section in the Help Introduction now gives a history of the major additions and changes in usage. Load down TensorCalculus3.zip 369KB Package, StyleSheet and Documentation, 11 August 2005. TensorialReadMe.txt 3KB Instructions for installation, 11 October 2003. Related applications TMecanica, TContinuumMechanics, TGeneralRelativity, are also available on the site.

MathTensor is a tensor analysis system written for the Mathematica system. It provides more than 250 functions and objects for elementary and advanced users.
Tensors in Physics is a tensor package written for the Mathematica system. It provides many functions relevant for General Relativity calculations in general Riemann-Cartan geometries.
maxima is a free software computer algebra system which should be usable for making tensor algebra calculations
- tensors in maxima
Ricci is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free.
Tela is a software package similar to Matlab and Octave, but designed specifically for tensors.
Tensor Toolbox Multilinear algebra MATLAB software.

[1] William Rowan Hamilton, On some Extensions of Quaternions[1]

[2] Abraham Pais, Subtle is the Lord

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