Definition of a tensor

Authier, A.

doi:10.1107/97809553602060000628

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

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

Section 1.1.3.1. Definition of a tensor

A. Authier^a ^*

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.3.1. Definition of a tensor

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

1.1.3.1.1. 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.

1.1.3.1.2. 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, $[\left.\matrix{P({\bf x}, {\bf y}, {\bf z}) = T({\bf x},{\bf y}) \otimes S({\bf z})\hfill\cr p\hskip1pt_{ik}^{j}x^{i}y_{j}z^{k} = t\hskip1pt_{i}^{j}x^{i}y_{j}s_{k}z^{k}.\hfill\cr}\right\} \eqno(1.1.3.1)]$

References

Lichnerowicz, A. (1947). Algèbre et analyse linéaires. Paris: Masson.Google Scholar

Sands, D. E. (1995). Vectors and tensors in crystallography. New York: Dover.Google Scholar

Schwartz, L. (1975). Les tenseurs. Paris: Hermann.Google Scholar

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