Tables for
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 7

Section Definition of a tensor

A. Authiera*

aInstitut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: Definition of a tensor

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For the mathematical definition of tensors, the reader may consult, for instance, Lichnerowicz (1947)[link], Schwartz (1975)[link] or Sands (1995)[link]. Linear forms

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A linear form in the space [E_{n}] is written [T({\bf x}) = t_{i}x^{i},]where [T({\bf x})] is independent of the chosen basis and the [t_{i}]'s are the coordinates of T in the dual basis. Let us consider now a bilinear form in the product space [E_{n} \otimes F_{p}] of two vector spaces with n and p dimensions, respectively: [T({\bf x},{\bf y}) = t_{ij}x^{i}y\hskip1pt^{j}.]

The np quantities [t_{ij}]'s are, by definition, the components of a tensor of rank 2 and the form [T({\bf x},{\bf y})] is invariant if one changes the basis in the space [E_{n} \otimes F_{p}]. The tensor [t_{ij}] is said to be twice covariant. It is also possible to construct a bilinear form by replacing the spaces [E_{n}] and [F_{p}] by their respective conjugates [E^{n}] and [F^{p}]. Thus, one writes [T({\bf x},{\bf y}) = t_{ij}x^{i}y\hskip1pt^{j} = t\hskip1.5pt_{i}^{j} x^{i}y_{j} = t\hskip1pt^{i}_{j}x_{i}y\hskip1pt^{j} = t^{ij}x_{i}y_{j},]where [t^{ij}] is the doubly contravariant form of the tensor, whereas [t\hskip1pt_{i}^{j}] and [t^{i}_{j}] are mixed, once covariant and once contravariant.

We can generalize by defining in the same way tensors of rank 3 or higher by using trilinear or multilinear forms. A vector is a tensor of rank 1, and a scalar is a tensor of rank 0. Tensor product

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Let us consider two vector spaces, [E_{n}] with n dimensions and [F_{p}] with p dimensions, and let there be two linear forms, [T({\bf x})] in [E_{n}] and [S({\bf y})] in [F_{p}]. We shall associate with these forms a bilinear form called a tensor product which belongs to the product space with np dimensions, [E_{n} \otimes F_{p}]: [P({\bf x},{\bf y}) = T({\bf x}) \otimes S({\bf y}).]

This correspondence possesses the following properties:

  • (i) it is distributive from the right and from the left;

  • (ii) it is associative for multiplication by a scalar;

  • (iii) the tensor products of the vectors with a basis [E_{n}] and those with a basis [F_{p}] constitute a basis of the product space.

The analytical expression of the tensor product is then [\left.\matrix{T({\bf x}) = t_{i} x\hskip1pt^{j} \cr S({\bf y}) = s_{j} y^{i}\cr}\right\} P({\bf x},{\bf y}) = p_{ij}x^{i}y\hskip1pt^{j} = t_{i}x^{i}s_{j}y\hskip1pt^{j}= t_{i}s_{j}x^{i}y\hskip1pt^{j}.]One deduces from this that [p_{ij} = t_{i}s_{j}.]

It is a tensor of rank 2. One can equally well envisage the tensor product of more than two spaces, for example, [E_{n} \otimes F_{p} \otimes G_{q}] in npq dimensions. We shall limit ourselves in this study to the case of affine tensors, which are defined in a space constructed from the product of the space [E_{n}] with itself or with its conjugate [E^{n}]. Thus, a tensor product of rank 3 will have [n^{3}] components. The tensor product can be generalized as the product of multilinear forms. One can write, for example, [\left.\matrix{P({\bf x}, {\bf y}, {\bf z}) = T({\bf x},{\bf y}) \otimes S({\bf z})\hfill\cr p\hskip1pt_{ik}^{j}x^{i}y_{j}z^{k} = t\hskip1pt_{i}^{j}x^{i}y_{j}s_{k}z^{k}.\hfill\cr}\right\} \eqno(]


Lichnerowicz, A. (1947). Algèbre et analyse linéaires. Paris: Masson.
Sands, D. E. (1995). Vectors and tensors in crystallography. New York: Dover.
Schwartz, L. (1975). Les tenseurs. Paris: Hermann.

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