# Mathematical Tools, Tensor Properties of Crystals, and Geometrical Crystallography

• T. S. Narasimhamurty

## 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 Xi 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.

## Keywords

Point Group Mathematical Tool Direction Cosine Symmetry Operation Symmetry Element
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