Mathematical Tools, Tensor Properties of Crystals, and Geometrical Crystallography

Abstract

Let a point P have the coordinates x 1, x 2, x 3 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 2.1 TeX\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} 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 X 1′ with respect to X 1, X 2, and X 3. 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.