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

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

## Section 1.1.3.1. 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

#### 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 is written where is independent of the chosen basis and the 's are the coordinates of T in the dual basis. Let us consider now a bilinear form in the product space of two vector spaces with n and p dimensions, respectively:

The np quantities 's are, by definition, the components of a tensor of rank 2 and the form is invariant if one changes the basis in the space . The tensor is said to be twice covariant. It is also possible to construct a bilinear form by replacing the spaces and by their respective conjugates and . Thus, one writes where is the doubly contravariant form of the tensor, whereas and 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, with n dimensions and with p dimensions, and let there be two linear forms, in and in . We shall associate with these forms a bilinear form called a tensor product which belongs to the product space with np dimensions, :

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 and those with a basis constitute a basis of the product space.

The analytical expression of the tensor product is then One deduces from this that

It is a tensor of rank 2. One can equally well envisage the tensor product of more than two spaces, for example, 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 with itself or with its conjugate . Thus, a tensor product of rank 3 will have components. The tensor product can be generalized as the product of multilinear forms. One can write, for example,

### References

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.