Mathematical Tools, Tensor Properties of Crystals, and Geometrical Crystallography

Abstract

Let a point P have the coordinates x₁, x₂, x₃ in a space of three dimensions in an X system of Cartesian coordinate axes. These three coordinates of P, which may be collectively represented in an abbreviated form by x _i , are independent of one another and form a set of linearly independent variables. Let the same point P have the coordinates x _i ′ in a different Cartesian coordinate system called the X′ system, having the same origin but with some or all of their axes not coincident with one another (Fig. 2.1). A rotation of axes and a reflection in a plane are examples of such a pair of Cartesian coordinate systems. Then x _i ′ and x _i are related by

$$\begin{array}{*{20}c} {x'_1 = a_{11} x_1 + a_{12} x_2 + a_{13} x_3 } \\ {x'_2 = a_{21} x_1 + a_{22} x_2 + a_{23} x_3 } \\ {x'_3 = a_{31} x_1 + a_{32} x_2 + a_{33} x_3 } \\ \end{array}$$

(2.1)

where the coefficients a _ij define the direction cosines of X _i ′ with respect to X_i according to the scheme Thus, for example, α₁₁, α₁₂, and α₁₃ are the direction cosines of X₁′ with respect to X₁, X₂, and X₃. Transformation of the coordinates of P from one system to another is called a linear transformation. A rotation of axes and a reflection in a plane will cause such a linear transformation of the components of P from one system to another.