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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 geometric 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 tensor, 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 0th-order tensor is a scalar—its magnitude is its sole component, so it can be represented as a 0-dimensional array. A first-order tensor is a vector, being representable in coordinates as a 1-dimensional array of components. A second-order tensor is representable as a square matrix, which is a 2-dimensional array of components. And so on—an order-k tensor can be represented as a k-dimensional array of components. The order of a tensor is the number of indices necessary to specify an individual component of a tensor.

Contents

Terminology

The term tensor is slightly ambiguous, which often leads to misunderstandings; roughly speaking, there are different default meanings in mathematics and physics. According to the Encyclopedia of Mathematics, tensor calculus is the umbrella term for two meanings, and

is the traditional name of the part of mathematics studying tensors and tensor fields [...] Tensor calculus is divided into tensor algebra (entering as an essential part in multilinear algebra) and tensor analysis, studying differential operators on the algebra of tensor fields.[1]

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 (on the two parts of "tensor calculus") conceals the agreement on the geometric nature of tensors. In applications of tensors, different types of notation may be used to represent the same underlying calculations or structure.

History

A detailed history of the origins of tensors has been written by Karin Reich.[2] This work argues that the concepts of later tensor analysis arose from the work of C. F. Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed in the middle of the nineteenth century. The word "tensor" itself was introduced in 1846 by William Rowan Hamilton[3] to describe something different from what is now meant by a tensor.[4] The contemporary usage was brought in by Woldemar Voigt in 1898.[5]

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892.[6] 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.[7] 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:[8]

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.

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, from the middle of the nineteenth century, 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.

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. Tensors are formulated within category theory by means of the concept of monoidal category, from the 1960s.

Formulation

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 the ranges over which given indices run. A tensor T can be formed from multiplying vectors together through the tensor product, a generalization of the Kronecker product. Valid basic intuitions can be taken from this idea, even if not all tensors are formed in this way, but with extra linear combinations as well. The number of vectors multiplied supplies the rank[9] or order of T, while index notation will require an array with a total number of components formed by multiplying the numbers of components in each of the vectors. In other words the order tells you the number of dimensions in the array, while its size depends on the geometrical application. The most common types correspond to a square matrix, cubic array, and so on, where each index runs over the same range, but this is not a restriction of the mathematical possibilities.

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Tensor valence

In physical applications, array indices are further 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 total tensor order but different valence are not, in general, identical because their geometrical meanings are different. Any given covariant index can however be transformed into a contravariant one, and vice versa, by applying the metric tensor. This geometrical operation is generally described as raising or lowering indices, which refers to the change of notation.

Einstein notation

Einstein notation is a convention for writing tensors that dispenses with writing summation signs by leaving them implicit. 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, but commonly only when each index has the same range, so all the omitted summations are sums from 1 to N for some given N.

Tensors as abstract objects

Tensors can be concretely represented by multi-dimensional arrays of components, by describing how they behave under coordinate transformations. However, tensors are often represented abstractly, independently of their array representation, by defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed. The abstract theory of tensors is a branch of linear algebra, now called multilinear algebra, in which the transformation property that tensors obey under coordinate transformation is built in automatically. The nature of tensors is to be bilinear, trilinear,... n-linear where n is the order of the tensor: multilinear in a word. Covariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the contravariant vectors.

Definition via tensor products of vector spaces

Given V1, ... , Vn, vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn. This is an adequate setting for relating the common notions of tensor.

A tensor on the vector space V is then defined to be an element of a vector space of the form:

V \otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^*

where V* is the dual space of V.[10] In many contexts this is what the term "tensor" refers to.

If there are m copies of V and n copies of V* in our product, the tensor is said to be of type (m, n) and contravariant of order m and covariant order n and total order m + n. There are special cases: tensors of order zero are just the scalars (elements of the field F), those of contravariant order 1 are the vectors in V, and those of covariant order 1 are the one-forms in V* (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (m,n) is denoted

 \begin{matrix} T^m_n(V) & = & \underbrace{ V\otimes \dots \otimes V} & \otimes & \underbrace{ V^*\otimes \dots \otimes V^*} \\ & & m & & n \end{matrix}.

The (1,1) tensors

V \otimes V^* \,

are isomorphic in a natural way to the space of linear transformations from V to V. An inner product of a real vector space V, defined as V × VR corresponds in a natural way to a (0,2) tensor in

V^* \otimes V^*, \,

in some applications called the associated metric.

Compatibility

While the formal mathematical definition of a tensor quantity begins with an abstract finite-dimensional vector space V, which then furnishes the uniform "building blocks" for tensors of all types (i.e. valences). In typical applications, V is the tangent space at a point of a manifold. The elements of V typically represent physical quantities such as velocities or forces. To represent a tensor by a concrete array of numbers, there must be a choice of frame of reference, which amounts to a choice of a basis of V as a vector space,

\mathbf{e}_1,\ldots,\mathbf{e}_n \in V.

Every vector in V can be "measured" relative to this basis, meaning that for every v in V there exist unique scalars vi such that

\mathbf{v} = v^i\mathbf{e}_i,

with the use from now on of the Einstein notation to omit summations. These scalars are called the components of v relative to the frame in question.

Let ε1,...,εnV be the corresponding dual basis: the basis of the dual space V of V such that

\varepsilon^i(\mathbf{e}_j) = \delta^i {}_j,

where the right-hand side is the Kronecker delta array. For every covector α in V there exists a unique array of components αi such that

\alpha = \alpha_i\, \varepsilon^i.

More generally, every tensor T in Tmn(V) has a unique representation in terms of components, meaning that there exists a unique array of scalars Ti1...imj1...in such that

\mathbf{T} = T^{i_1\ldots i_m}_{j_1\ldots j_n}\, \mathbf{e}_{i_1} \otimes \cdots\otimes \mathbf{e}_{i_m} \otimes \varepsilon^{j_1}\otimes\cdots\otimes \varepsilon^{j_n}.

This passage to components forms the bridge between the abstract mathematical notation for tensors, and the way they are commonly written in theoretical physics and engineering texts. More does need to be said, since tensor notation given component-wise only captures part of the idea that a tensor is a "geometric quantity": it covers the aspect of "quantity" but is silent on the geometry. The following explains what happens when a change is made to a different frame of reference, say

\hat{\mathbf{e}}_1,\ldots,\hat{\mathbf{e}}_n\in V.

Any two frames are uniquely related, by an invertible transition matrix Aij, having the property that for all values of j there holds the frame transformation rule

\hat{\mathbf{e}}_j = A^i {}_j\, \mathbf{e}_i.

Let v in V be a vector, and let vi and \hat{v}^i denote the corresponding component arrays relative to the two frames. From

\mathbf{v} = v^i\mathbf{e}_i = \hat{v}^i\hat{\mathbf{e}}_i,

and from the frame transformation rule is inferred the vector transformation rule

 \hat{v}^i = B^i {}_j\, v^j,

where Bij is the matrix inverse of Aij, i.e.,

A^i {}_k B^k {}_j = \delta^i {}_j. \,

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 covectors, use the transformation rule for the dual basis in the form

\hat{\varepsilon}^i = B^i {}_j \, \varepsilon^j.

Then

v^i = \varepsilon^i(\mathbf{v}),

while

\hat{v}^i = \hat{\varepsilon}^i(\mathbf{v}).

The transformation rule for covector components is covariant. What this means is the following: \mathbf{\alpha}\in V^* be a given covector, and let αi and \hat{\alpha}_i be the corresponding component arrays. Then

\hat{\alpha}_j = A^i {}_j \alpha_i.

The above relation is easily established because

\alpha_i = \mathbf{\alpha}(\mathbf{e}_i),

and

\hat{\alpha}_j = \mathbf{\alpha}(\hat{\mathbf{e}}_j);

then use the transformation rule for the frame of reference.

In light of the above discussion, the transformation rule for a general type (m,n) tensor takes the form

\hat{T}^{i_1\ldots i_m}_{\,j_1\ldots j_n} = B^{i_1} {}_{k_1}\cdots B^{i_m} {}_{k_m} A^{l_1} {}_{j_n}\cdots A^{l_n} {}_{j_n} T^{k_1\ldots k_m}_{l_1\ldots l_n}.

In summary, the compatibility of the two standard approaches to tensors means that this component-wise approach, and the abstract mathematics of tensor products introduced in this section, express the same content in two different ways.

Compatibility for tensor fields

For tensor fields, a further reformulation in terms of partial derivatives is

\hat{T}^{i_1\dots i_m}_{j_1\dots j_n} = \frac{\partial \bar{x}^{i_1}}{\partial x^{k_1}} \cdots \frac{\partial \bar{x}^{i_m}}{\partial x^{k_m}} \frac{\partial x^{l_1}}{\partial \bar{x}^{j_1}} \cdots \frac{\partial x^{l_n}}{\partial \bar{x}^{j_n}} T^{k_1\dots k_n}_{l_1\dots l_n}.

This is sometimes termed the tensor transformation law. What is meant is that the components of the Jacobian matrix of a general non-linear coordinate transformation may be used, in the same way as the components of the matrices A and B above, and the resulting geometric "law" is the correct way to recognise tensor fields. This law can be derived in the same way as before: it corresponds mathematically to discussing the tangent bundle rather than the tangent space, and the fact that this law is applied in continuum mechanics supports the previous remark that applications arise from the tangent space as basic model.

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. That application is of a tensor of second order. While such uses of tensors are the most frequent, tensors of higher order also matter in many fields.

Continuum mechanics

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 second order tensors, and are related in a general linear elastic material by a fourth-order elasticity tensor. In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently represented as a 3×3 array. The three 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. Thus, 3×3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment. Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, a second 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 vary from point to point.

Other examples from physics

Common applications include

Applications of tensors of order > 2

The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of computer vision, with the trifocal tensor generalizing the fundamental matrix.

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:

 \frac{P_i}{\varepsilon_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 naturally in the subject matter.

Generalizations

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 rth power.[11] 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. While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value. Non-integral powers of the (positive) transition functions of the bundle of densities make sense, so that the weight of a density, in that sense, is not restricted to integer values.

Restricting to changes of coordinates with positive Jacobian determinant is possible on orientable manifolds, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of n-forms are distinct. For more on the intrinsic meaning, see density on a manifold.)

Spinors

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.

See also

Notation

Foundational

Applications

Notes

  1. ^ Hazewinkel, Michiel, ed. (2001), "Tensor calculus", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/t/t09238.htm 
  2. ^ Karin Reich, Die Entwicklung des Tensorkalküls (1994).
  3. ^ William Rowan Hamilton, On some Extensions of Quaternions
  4. ^ Namely, the norm operation in a certain type of algebraic system (now known as a Clifford algebra).
  5. ^ Woldemar Voigt, Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung (Leipzig, 1898)
  6. ^ In volume XVI of the Bulletin des Sciences Mathématiques.
  7. ^ Abraham Pais, Subtle is the Lord: The Science and the Life of Albert Einstein
  8. ^ Judtih R. Goodstein, The Italian Mathematicians of Relativity (2007)
  9. ^ Rank is ambiguous, since a matrix has a rank of a different sort, and will not be used further.
  10. ^ Hazewinkel, Michiel, ed. (2001), "Affine tensor", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/a/a011120.htm 
  11. ^ Hazewinkel, Michiel, ed. (2001), "Tensor density", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/T/t092390.htm 

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. 
  • Munkres, James, Analysis on Manifolds, Westview Press, 1991. Chapter six gives a "from scratch" introduction to covariant tensors.
  • 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 
  • Kay, David C (1988-04-01). Schaum's Outline of Tensor Calculus. McGraw-Hill. ISBN 978-0070334847. 
  • Schutz, Bernard, Geometrical methods of mathematical physics, Cambridge University Press, 1980.
  • Synge JL, Schild A (1978-07-01). Tensor Calculus. Dover Publications. ISBN 978-0486636122. 

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