Tensor

Note that in common usage, the term tensor is also used to refer to a tensor field.

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values. The order (also degree or rank) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map can be represented by a matrix, a 2-dimensional array, and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a 1st-order tensor. Scalars are single numbers and are thus zeroth-order tensors.

Tensors are used to represent correspondences between sets of geometrical vectors. For example, the stress tensor T takes a direction v as input and produces the stress T(v) on the surface normal to this vector as output and so expresses a relationship between these two vectors. Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. Taking a coordinate basis or frame of reference and applying the tensor to it results in an organized multidimensional array representing the tensor in that basis, or as it looks from that frame of reference. 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. This transformation law is considered to be built in to the notion of a tensor in a geometrical or physical setting, and the precise form of the transformation law determines the type (or valence) of the tensor.

Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as elasticity, fluid mechanics, and general relativity. Tensors were first conceived by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.[1]

Contents

History

The concepts of later tensor analysis arose from the work of Carl 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.[2] 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.[Note 1] The contemporary usage was brought in by Woldemar Voigt in 1898.[4]

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro (also called just Ricci) under the title absolute differential calculus, and originally presented by Ricci in 1892.[5] 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).[6]

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 Riemann 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 generalized within category theory by means of the concept of monoidal category, from the 1960s.

Definition

There are several approaches to defining tensors. Although seemingly different, the approaches just describe the same geometric concept using different languages and at different levels of abstraction.

As multidimensional arrays

Just as a scalar is described by a single number, and a vector with respect to a given basis is described by an array, any tensor with respect to a basis is described by a multidimensional array. The numbers in the array are known as the scalar components of the tensor or simply its components. They are denoted by indices giving their position in the array, in subscript and superscript, after the symbolic name of the tensor. The total number of indices required to uniquely specify each component is equal to the dimension of the array, and is called the order or the rank of the tensor.[Note 2] For example, the entries of an order 2 tensor T would be denoted Tij, where i and j are indices running from 1 to the dimension of the related vector space.

Just like the components of a vector change when we change the basis of the vector space, the entries of a tensor also change under such a transformation. Each tensor comes equipped with a transformation law that details how the components of the tensor respond to a change of basis. The components of a vector can respond in two distinct ways to a change of basis (see covariance and contravariance of vectors),

\mathbf{\hat{e}}_i = \sum_j R^j_i \mathbf{e}_j = R^j_i \mathbf{e}_j,

where Rj
i is a matrix and in the second expression the summation sign was suppressed (a notational convenience introduced by Einstein that will be used throughout this article). The components, vi, of a regular (or column) vector, v, transform with the inverse of the matrix R,

\hat{v}^i = (R^{-1})^i_j v^j,

where the hat denotes the components in the new basis. While the components, wi, of a covector or (row vector), w transform with the matrix R itself,

\hat{w}_i = R_i^j w_j.

The components of a tensor transform in a similar manner with a transformation matrix for each index. If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index. The transformation law for a rank m tensor with n contravariant indices and m−n covariant indices is thus given as,

\hat{T}^{i_1,\ldots,i_n}_{i_{n%2B1},\ldots,i_m}= (R^{-1})^{i_1}_{j_1}\cdots(R^{-1})^{i_n}_{j_n} R^{j_{n%2B1}}_{i_{n%2B1}}\cdots R^{j_{m}}_{i_{m}}T^{j_1,\ldots,j_n}_{j_{n%2B1},\ldots,j_m}.

Such a tensor is said to be of order or type (n,m−n).[Note 3] This discussion motivates the following formal definition:

Definition. A tensor of type (n, m−n) is an assignment of a multidimensional array

T^{i_1\dots i_n}_{i_{n%2B1}\dots i_m}[\mathbf{f}]
to each basis f = (e1,...,eN) such that, if we apply the change of basis
\mathbf{f}\mapsto \mathbf{f}\cdot R = \left( R_1^i \mathbf{e}_i, \dots, R_N^i\mathbf{e}_i\right)
then the multidimensional array obeys the transformation law
T^{i_1\dots i_n}_{i_{n%2B1}\dots i_m}[\mathbf{f}\cdot R] = (R^{-1})^{i_1}_{j_1}\cdots(R^{-1})^{i_n}_{j_n} R^{j_{n%2B1}}_{i_{n%2B1}}\cdots R^{j_{m}}_{i_{m}}T^{j_1,\ldots,j_n}_{j_{n%2B1},\ldots,j_m}[\mathbf{f}].

The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.[1] Nowadays, this definition is still used in physics and engineering text books.[9][10]

Tensor fields

Main article: Tensor field

In many applications, especially in differential geometry and physics, it is natural to consider the components of a tensor to be functions. This was, in fact, the setting of Ricci's original work. In modern mathematical terminology such an object is called a tensor field, but they are often simply referred to as tensors themselves.

In this context the defining transformation law takes a different form. The "basis" for the tensor field is determined by the coordinates of the underlying space, and the defining transformation law is expressed in terms of partial derivatives of the coordinate functions, \bar{x}_i(x_1,\ldots,x_k), defining a coordinate transformation,

\hat{T}^{i_1\dots i_n}_{i_{n%2B1}\dots i_m}(\bar{x}_1,\ldots,\bar{x}_k) = \frac{\partial \bar{x}^{i_1}}{\partial x^{j_1}} \cdots \frac{\partial \bar{x}^{i_n}}{\partial x^{j_n}} \frac{\partial x^{j_{n%2B1}}}{\partial \bar{x}^{i_{n%2B1}}} \cdots \frac{\partial x^{j_m}}{\partial \bar{x}^{i_m}} T^{j_1\dots j_n}_{j_{n%2B1}\dots j_m}(x_1,\ldots,x_k).

As multilinear maps

A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach is to define a tensor as a multilinear map. In that approach a type (n,m) tensor T is defined as a map,

\begin{matrix} T: & \underbrace{ V^* \times\dots\times V^*} & \times & \underbrace{ V \times\dots\times V} &\rightarrow \mathbf{R}, \\ & \text{n copies}& &\text{m copies} & & \end{matrix}

where V is a vector space and V* is the corresponding dual space of covectors, which is linear in each of its arguments.

By applying a multilinear map T of type (n,m) to a basis {ej} for V and a canonical cobasis {εi} for V*,

T^{i_1\dots i_n}_{j_1\dots j_m} \equiv T(\mathbf{\varepsilon}^{i_1},\ldots,\mathbf{\varepsilon}^{i_n},\mathbf{e}_{j_1},\ldots,\mathbf{e}_{j_m}),

a n+m dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realised as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.

This approach, defining tensors as multilinear maps, is used in modern differential geometry textbooks[11] and more mathematically inclined physics textbooks.[12] [13]

Using tensor products

Main article: Tensor (intrinsic definition)

For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property. A type (n,m) tensor is defined in this context as an element of the tensor product of vector spaces,[14]

\begin{matrix} T\in & \underbrace{V \otimes\dots\otimes V} & \otimes & \underbrace{V^* \otimes\dots\otimes V^*}. \\ & \text{n copies}& &\text{m copies} \end{matrix}

If vi is a basis of V and wj is a basis of W, then the tensor product V\otimes W has a natural basis \mathbf{v}_i\otimes \mathbf{w}_j. The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {ei} for V and its dual {εj}, i.e.

T = T^{i_1\dots i_n}_{j_1\dots j_m}\; \mathbf{e}_{i_1}\otimes\cdots\otimes \mathbf{e}_{i_n}\otimes \mathbf{\varepsilon}^{j_1}\otimes\cdots\otimes \mathbf{\varepsilon}^{j_m}.

Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (m,n) tensor. Moreover, the universal property of the tensor product gives a 1-to-1 correspondence between tensors defined in this way and tensors defined as multilinear maps.

Examples

This table shows important examples of tensors, including both tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type (n, m). For example, a bilinear form is the same thing as a (0, 2)-tensor; an inner product is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, N)-entry of the table, N denotes the dimension of the underlying vector space or manifold.

n=0 n=1 n=2 ...
m=0 scalar, e.g. scalar curvature vector bivector, e.g. inverse metric tensor
m=1 covector, linear functional, 1-form linear transformation
m=2 bilinear form, e.g. inner product, metric tensor, Ricci curvature, 2-form, symplectic form e.g. cross product in three dimensions e.g. elasticity tensor
m=3 e.g. 3-form e.g. Riemann curvature tensor
...
m=N e.g. N-form i.e. determinant, volume form
...

Raising an index on an (n, m)-tensor produces an (n + 1, m - 1)-tensor; this can be visualized as moving diagonally up and to the right on the table. Symmetrically, lowering an index can be visualized as moving diagonally down and to the left on the table. Contracting an (n, m)-tensor produces an (n - 1, m - 1)-tensor; this can be visualized as moving diagonally up and to the left on the table.

Notation

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 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.

Abstract index notation

For more details on this topic, see Abstract index notation.

The abstract index notation is a way to write tensors such that the indices are no longer thought of as numerical, but rather are indeterminates. The abstract index notation captures the expressiveness of indices and the basis-independence of index-free notation.

Operations

There are a number of basic operations that may be conducted on tensors that again produce a tensor. The linear nature of tensor implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the scaling of a vector. On components, these operations are simply performed component for component. These operations do not change the type of the tensor, however there also exist operations that change the type of the tensors.

Tensor product

Main article: Tensor product

The tensor product takes two tensors, S and T, and produces a new tensor, S ⊗ T, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.

(S\otimes T)(v_1,\ldots, v_n, v_{n%2B1},\ldots, v_{n%2Bm}) = S(v_1,\ldots, v_n)T( v_{n%2B1},\ldots, v_{n%2Bm}),

which again produces a map that is linear in all its arguments. On components the effect similarly is to multiply the components of the two input tensors, i.e.

(S\otimes T)^{i_1\ldots i_l i_{l%2B1}\ldots i_{l%2Bm}}_{j_1\ldots j_k j_{k%2B1}\ldots j_{k%2Bn}} = S^{i_1\ldots i_l}_{j_1\ldots j_k} T^{i_{l%2B1}\ldots i_{l%2Bm}}_{j_{k%2B1}\ldots j_{k%2Bn}},

If S is of type (k,l) and T is of type (n,m), then the tensor product S ⊗ T has type (k+n,l+m).

Contraction

Main article: Tensor contraction

Tensor contraction is an operation that reduces the total order of a tensor by two. More precisely, it reduces a type (n,m) tensor to a type (n-1,m-1) tensor. In terms of components, the operation is achieved by summing over one contravariant and one covariant index of tensor. For example, a (1,1)-tensor Tij can be contracted to a scalar through

T_i^i.

Where the summation is again implied. When the (1,1)-tensor is interpreted as a linear map, this operation is known as the trace.

The contraction is often used in conjunction with the tensor product to contract an index from each tensor.

The contraction can also be understood in terms of the definition of a tensor as an element of a tensor product of copies of the space V with the space V* by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from V* to a factor from V. For example, a tensor

T \in V\otimes V\otimes V^*

can be written as a linear combination

T=v_1\otimes w_1\otimes \alpha_1 %2B v_2\otimes w_2\otimes \alpha_2 %2B\cdots %2B v_N\otimes w_N\otimes \alpha_N.

The contraction of T on the first and last slots is then the vector

\alpha_1(v_1)w_1 %2B \alpha_2(v_2)w_2%2B\cdots%2B\alpha_N(v_N)w_N.

Raising or lowering an index

Main article: Raising and lowering indices

When a vector space is equipped with an inner product (or metric as it is often called in this context), there exist operations that convert a contravariant (upper) index into a covariant (lower) index and vice versa. A metric itself is a (symmetric) (0,2)-tensor, it is thus possible to contract an upper index of a tensor with one of lower indices of the metric. This produces a new tensor with the same index structure as the previous, but with lower index in the position of the contracted upper index. This operation is quite graphically known as lowering an index.

Conversely, a metric has an inverse which is a (2,0)-tensor. This inverse metric can be contracted with a lower index to produce an upper index. This operation is called raising an index.

Applications

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 %2B \sum_{jk} \chi_{ijk}^{(2)} E_j E_k %2B \sum_{jk\ell} \chi_{ijk\ell}^{(3)} E_j E_k E_\ell %2B \cdots \!

Here \chi^{(1)} is the linear susceptibility, \chi^{(2)} gives the Pockels effect and second harmonic generation, and \chi^{(3)} gives the Kerr effect. This expansion shows the way higher-order tensors arise naturally in the subject matter.

Generalizations

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 rth power.[15] 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

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.

See also

Notation

Foundational

Applications

Notes

  1. ^ Namely, the norm operation in a certain type of algebraic system (now known as a Clifford algebra).
  2. ^ This article will be using the term order, since the term rank has a different meaning in the related context of matrices.
  3. ^ There is a plethora of different terminology for this around. The terms order, type, rank, valence, and degree are in use for the same concept. This article uses the term "order" or "total order" for the total dimension of the array (or its generalisation in other definitions) m in the preceding example, and the term type for the pair giving the number contravariant and covariant indices. A tensor of type (n,m−n) will also be referred to as a "(n,m−n)" tensor for short.

References

General
  • 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) [1975]. 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. 
Specific
  1. ^ a b Kline, Morris (1972). Mathematical thought from ancient to modern times, Vol. 3. Oxford University Press. pp. 1122–1127. 
  2. ^ Reich, Karin (1994). Die Entwicklung des Tensorkalküls. Birkhäuser. ISBN 9783764328146. http://books.google.com/books?id=O6lixBzbc0gC. 
  3. ^ Hamilton, William Rowan (1854–1855). Wilkins, David R.. ed. "On some Extensions of Quaternions". Philosophical Magazine (7–9): 492–499, 125–137, 261–269, 46–51, 280–290. http://www.emis.de/classics/Hamilton/ExtQuat.pdf. 
  4. ^ Voigt, Woldemar (1898). Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung. Von Veit. 
  5. ^ Ricci Curbastro, G. (1892). "Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique". Bulletin des Sciences Mathématiques 2 (16): 167–189. 
  6. ^ (Ricci & Levi-Civita 1900)
  7. ^ Pais, Abraham (2005). Subtle Is the Lord: The Science and the Life of Albert Einstein. Oxford University Press. ISBN 9780192806727. http://books.google.com/books?id=U2mO4nUunuw. 
  8. ^ Goodstein, Judith R (1982). "The Italian Mathematicians of Relativity". Centaurus 26 (3): 241–261. Bibcode 1982Cent...26..241G. doi:10.1111/j.1600-0498.1982.tb00665.x. 
  9. ^ Marion, J.B.; Thornton, S.T. (1995). Classical Dynamics of Particles and Systems (4th ed.). Saunders College Publishing. p. 424. ISBN 9780030989674. 
  10. ^ Griffiths, D.J. (1999). Introduction to Electrodynamics (3 ed.). Prentice Hall. pp. 11–12 and 535--. ISBN 9780138053260. 
  11. ^ Lee, J.M. (1997). Riemannian Manifolds. Springer. p. 12. ISBN 0387983228. 
  12. ^ Carroll, S.M. (2004). Spacetime and Geometry. Addison Wesley. p. 21. ISBN 0805387323. 
  13. ^ Jeevanjee, Nadir (2011). An Introduction to Tensors and Group Theory for Physicists. Birkhauser. ISBN 978-0-8176-4714-8. http://www.springer.com/new+%26+forthcoming+titles+(default)/book/978-0-8176-4714-8. 
  14. ^ Hazewinkel, Michiel, ed. (2001), "Affine tensor", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=a/a011120 
  15. ^ Hazewinkel, Michiel, ed. (2001), "Tensor density", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=T/t092390 

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