Tensor Information & Tensor Links at HealthHaven.com

For component-based "classical" treatment of tensors, see Classical treatment of tensors. See Component-free treatment of tensors for a modern abstract treatment, and Intermediate treatment of tensors for an approach which bridges the two.

Components of stress, a second-order tensor, in three dimensions. The tensor in the image is the row vector $\scriptstyle\sigma = \begin{bmatrix}\mathbf{T}^{(\mathbf{e}_1)} \mathbf{T}^{(\mathbf{e}_2)} \mathbf{T}^{(\mathbf{e}_3)}\end{bmatrix}$ of the forces acting on the X, Y, and Z faces of the cube. Those forces are represented by column vectors. The row and column vectors that make up the tensor can be represented together by a matrix,
$\sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}$

Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, (geometric) vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another. An example is the stress, that takes one vector as input and produces another vector as output and so expresses a relationship between the input and output vectors. Tensors were first conceived by Bernhard Riemann and Elwin Bruno Christoffel, and later developed by Tullio Levi-Civita and Gregorio Ricci-Curbastro, in order to formulate the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

Because they express a relationship between vectors, tensors themselves are independent of a particular choice of coordinate system. It is possible to represent a tensor by examining what it does to a coordinate basis or frame of reference; the resulting quantity is then an organized multi-dimensional array of numerical values. The coordinate-independence of a tensor then takes the form of a "covariant" transformation law that relates the array computed in one coordinate system to that computed in another one. The order (or degree) of a tensor is the dimensionality of the array needed to represent it. Thus a scalar is a zeroth-order tensor: its magnitude is its sole component, so it can be represented as a 0-dimensional array. A vector is a first-order tensor, being representable in coordinates as a 1-dimensional array of components. A matrix is a second-order tensor, being representable as a 2-dimensional array. And so on: an order-k tensor can be represented as a k-dimensional array of components. The order is the number of numerical indices necessary to specify an individual component of a tensor.

The term tensor is slightly ambiguous, which often leads to misunderstandings; roughly speaking, there are different default meanings in mathematics and physics. In the mathematical fields of multilinear algebra and differential geometry, a tensor is first an element of a tensor product of vector spaces. In physics, the same term often means what a mathematician would call a tensor field: an association of a different mathematical tensor with each point of a geometric space, varying continuously with position. This difference of emphasis conceals the agreement on the geometric nature of tensors. Also, in applications of tensors, different types of notation may be used to represent the same underlying calculations or structure.

[edit] Early 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 1898.^[2]

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892 (in volume XVI of the Bulletin des Sciences Mathématiques).

It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications) (Ricci & Levi-Civita 1900) (in French; 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.^[3] Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect, with Einstein at one point writing:^[4]

I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.

– Einstein, to Levi-Civita on tensor analysis

Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the curvature tensor. The exterior algebra of Hermann Grassmann is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus. The work of Élie Cartan made differential forms one of the basic kinds of tensor fields used in mathematics.

[edit] Modern mathematical usage

For more details on this topic, see Tensor (intrinsic definition).

In physics, numbers may be obtained as physical quantities that depend on a basis, and the collection is determined to be a tensor if the quantities transform appropriately under change of basis. 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.

In the modern formal treatment, tensor theory is approached as a topic in multilinear algebra, and the nature of tensors is to be bilinear, trilinear,... multilinear in a word. For example, the Euclidean outer product—a real-valued function of two vectors that is linear in each—is a matrix, while the outer product of three or more vectors is a tensor. Similarly, on a smooth curved surface such as a torus, the metric tensor (field) essentially defines a different inner product of tangent vectors at each point of the surface. Engineering applications do not usually require the full, general theory, which talks about tensors which are n-linear (can be written down as arrays with n components), but theoretical physics now does.

From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology (for example in the Künneth theorem). Correspondingly there are types of tensors at work in many branches of abstract algebra, particularly in homological algebra and representation theory. Multilinear algebra can be developed in greater generality than for scalars coming from a field, but the theory is then certainly less geometric, and computations more technical and less algorithmic.

This leaves 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. Contravariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the covariant vectors.

There is less conflict here than is sometimes made out. Physicists and engineers recognise that vectors and tensors have a physical significance as entities, which must go beyond any arbitrary coordinate system in which their components are enumerated. Similarly, mathematicians do find there are some tensor relations which are more conveniently derived in a coordinate notation.

[edit] Formulation

For more details on this topic, see Glossary of tensor theory.

There are many kinds of tensors, and some technical language is required to formulate precise descriptions of the particular type. Where index notation is used to give tensors by components, you need to know both what the indexation means, and the range over which indices run.

Formally speaking, a tensor has a particular type according to the construction with tensor products that give rise to it. For computational purposes, it may be expressed as the sequence of values represented by a function with a tuple-valued domain and a scalar valued range. Domain values are tuples of counting numbers, and these numbers are called indices. For example, a third-order tensor might have dimensions 2, 5, and 7. Here, the indices range from «1, 1, 1» through «2, 5, 7»; thus the tensor would have one value at «1, 1, 1», another at «1, 1, 2», and so on for a total of 70 values. As a special case, (finite-dimensional) vectors may be expressed as a sequence of values represented by a function with a scalar valued domain and a scalar valued range; the number of distinct indices is the dimension of the vector. Using this approach, the third-order tensor of dimension (2,5,7) can be represented as a 3-dimensional array of size 2 × 5 × 7. In this usage, the number of "dimensions" comprising the array is equivalent to the "order" of the tensor, and the dimensions of the tensor are equivalent to the "size" of each array dimension.

[edit] Tensor valence

In physical applications, array indices are distinguished by being contravariant (superscripts) or covariant (subscripts), depending upon the type of transformation properties. The valence of a particular tensor is the number and type of array indices; tensors with the same tensor order but different valence are not, in general, identical. However, any given covariant index can be transformed into a contravariant one, and vice versa, by applying the metric tensor. This operation is generally known as raising or lowering indices.

[edit] Einstein notation

For more details on this topic, see Einstein notation.

Einstein notation is a convention for writing tensors that dispenses with writing summation signs. It relies on the idea that any repeated index is summed over: if the index i is used twice in a given term of a tensor expression, it means that the values are to be summed over i. Several distinct pairs of indices may be summed this way.

[edit] As opposed to "matrix"

The concept of a tensor is often conflated with that of a matrix. A clear distinction between a matrix and a tensor is usually made when one needs to work with more than second-order objects (as in the trifocal tensor generalizing the fundamental matrix) or when considering transformation properties or representing tensors in non-orthogonal coordinate systems. While it is tempting to say that a square matrix is a second-order tensor (as opposed to other orders of tensors) and that a vector is a column or row matrix, this belies the conceptual power of tensors to capture ideas in science and engineering and to enforce sensible behavior.

[edit] Applications

Tensors are important 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; in this technique tensors are in effect made visible.

[edit] In physics

Perhaps the most important examples are provided by continuum mechanics. The stresses inside a solid body or fluid are described by a tensor. The stress tensor and strain tensor are both 2nd order tensors, and are related in a general linear elastic material by a fourth-order elasticity tensor. In detail, the 2nd order tensor quantifying stress in a 3-dimensional solid object has components which can be conveniently represented as a 3×3 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, 3×3, or 9 components are required to describe the stress at this cube-shaped infinitesimal 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, a 2nd order tensor is needed.

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), in linear elasticity, or more precisely by a tensor field of type (2,0) since the stresses may change from point to point.

[edit] Other examples from physics

The field of nonlinear optics studies the changes to material polarization density under extreme electric fields. The polarization waves generated are related to the generating electric fields through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:

$P_i / \epsilon_0 = \sum_j \chi^{(1)}_{ij} E_j + \sum_{jk} \chi_{ijk}^{(2)} E_j E_k + \sum_{jk\ell} \chi_{ijk\ell}^{(3)} E_j E_k E_\ell + \cdots \!$

Here $χ (1)$ is the linear susceptibility, $χ (2)$ gives the Pockels effect and second harmonic generation, and $χ (3)$ gives the Kerr effect. This expansion shows the way higher-order tensors arise in the subject.

[edit] Generalizations

[edit] Tensor densities

Main article: Tensor density

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. Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle such as the (1-dimensional) space of n-forms (where n is the dimension of the space), as opposed to taking their values in just R. Higher "weights" then just correspond to taking additional tensor products with this space in the range. In the language of vector bundles, the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times.

[edit] Spinors

Main article: Spinor

Starting with an orthonormal coordinate system, a tensor transforms in a certain way when a rotation is applied. However, there is additional structure to the group of rotations that is not exhibited by the transformation law for tensors: see orientation entanglement and plate trick. Mathematically, the rotation group is not simply connected. Spinors are mathematical objects that generalize the transformation law for tensors in a way that is sensitive to this fact.

[edit] See also

Glossary of tensor theory

[edit] Exposition

[edit] Notation

[edit] Foundational

[edit] Applications

[edit] Tensor software

[edit] Standalone open-source software

Cadabra is a computer algebra system (CAS) designed specifically for the solution of problems encountered in field theory. It has extensive functionality for tensor polynomial simplification including multi-term symmetries, fermions and anti-commuting variables, Clifford algebras and Fierz transformations, implicit coordinate dependence, multiple index types and many more. The input format is a subset of TeX. Both a command-line and a graphical interface are available.
Tela is a software package similar to Matlab and Octave, but designed specifically for tensors.

[edit] Open-source packages for use with open-source algebra systems

Maxima is a free open source computer algebra system which can be used for tensor algebra calculations - it is particularly useful for calculations with abstract tensors (i.e. when one wishes to do calculations without defining all components of the tensor explicitly). It comes with three tensor packages: itensor for abstract (indicial) tensor manipulation, ctensor for component-defined tensors, and atensor for algebraic tensor manipulation. The itensor package guide

[edit] Software for use with Mathematica

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.
Ricci is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free.
TTC Tools of Tensor Calculus is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds.
EDC and RGTC "Exterior Differential Calculus" and "Riemannian Geometry & Tensor Calculus" are free Mathematica packages for tensor calculus especially designed but not only for general relativity.
Tensorial "Tensorial 4.0" is a general purpose tensor calculus package for Mathematica.
xAct: Efficient Tensor Computer Algebra for Mathematica. xAct is a collection of packages for fast manipulation of tensor expressions.

[edit] Software for use with Maple

GRTensorII is a computer algebra package for performing calculations in the general area of differential geometry.

[edit] Libraries

FTensor is a high performance tensor library written in C++.
TL is a multi-threaded tensor library implemented in C++ used in Dynare++. The library allows for folded/unfolded, dense/sparse tensor representations, general ranks (symmetries). The library implements Faa Di Bruno formula and is adaptive to available memory. Dynare++ is a standalone package solving higher order Taylor approximations to equilibria of non-linear stochastic models with rational expectations.
Spartns is a Sparse Tensor framework for Common Lisp.

[edit] Other

Tensor Toolbox Multilinear algebra MATLAB software.

[edit] Notes

^ William Rowan Hamilton, On some Extensions of Quaternions
^ Woldemar Voigt, Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung (Leipzig, 1898)
^ Abraham Pais, Subtle is the Lord: The Science and the Life of Albert Einstein
^ [1]

[edit] References

Bishop, Richard L.; Samuel I. Goldberg (1980). Tensor Analysis on Manifolds. Dover. ISBN 978-0-486-64039-6.
Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
Lawden, D. F. (2003). Introduction to Tensor Calculus, Relativity and Cosmology (3/e ed.). Dover. ISBN 978-0-486-42540-5.
Lebedev, Leonid P.; Michael J. Cloud (2003). Tensor Analysis. World Scientific. ISBN 978-981-238-360-0.
Lovelock, David; Hanno Rund (1989). Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications", Mathematische Annalen (Springer) 54 (1–2): 125–201, doi:10.1007/BF01454201, http://www.springerlink.com/content/u21237446l22rgg7/fulltext.pdf

[edit] External links

Weisstein, Eric W., "Tensor" from MathWorld.
Introduction to Vectors and Tensors, Vol 1: Linear and Multilinear Algebra by Ray M. Bowen and C. C. Wang.
Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis by Ray M. Bowen and C. C. Wang.
An Introduction to Tensors for Students of Physics and Engineering, released by NASA
A discussion of the various approaches to teaching tensors, and recommendations of textbooks
A Quick Introduction to Tensor Analysis by R. A. Sharipov.
Introduction to Tensors by Joakim Strandberg.

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

[1] Woldemar Voigt, Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung (Leipzig, 1898)

[2] Abraham Pais, Subtle is the Lord: The Science and the Life of Albert Einstein

[Goodstein-3] [1]

[1]

[2]

[3]

[4]

Contents

[edit] Early history

[edit] Modern mathematical usage

[edit] Formulation

[edit] Tensor valence

[edit] Einstein notation

[edit] As opposed to "matrix"

[edit] Applications

[edit] In physics

[edit] Other examples from physics

[edit] Generalizations

[edit] Tensor densities

[edit] Spinors

[edit] See also

[edit] Exposition

[edit] Notation

[edit] Foundational

[edit] Applications

[edit] Tensor software

[edit] Standalone open-source software

[edit] Open-source packages for use with open-source algebra systems

[edit] Software for use with Mathematica

[edit] Software for use with Maple

[edit] Libraries

[edit] Other

[edit] Notes

[edit] References

[edit] External links