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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, p. 9

Section Definition

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

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The tensor defined by [{\bf x} \bigwedge {\bf y} = {\bf x} \otimes {\bf y} - {\bf y} \otimes {\bf x}]is called the outer product of vectors x and y. (Note: The symbol is different from the symbol [\wedge] for the vector product.) The analytical expression of this tensor of rank 2 is [\left. \matrix{{\bf x} & =x^{i}{\bf e}_{i} \cr {\bf y} & =y\hskip1pt^{j}{\bf e}_{j} \cr}\right\} \quad \Longrightarrow \quad{\bf x}\bigwedge {\bf y} = (x^{i}y\hskip1pt^{j}- y^{i}x\hskip1pt^{j})\, {\bf e}_{i} \otimes {\bf e}_{j}.]

The components [p^{ij} = x^{i}y\hskip1pt^{j}- y^{i}x\hskip1pt^{j}] of this tensor satisfy the properties [p^{ij}= - p\hskip1pt^{ji} ;\quad p^{ii}= 0.]It is an antisymmetric tensor of rank 2.

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