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

Chapter

## Abstract

Let a point
where the coefficients

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

(2.1)

*a*_{ ij }define the direction cosines of*X*_{ i }′ with respect to*X*_{i}according to the scheme Thus, for example, α_{11}, α_{12}, and α_{13}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.## Keywords

Point Group Mathematical Tool Direction Cosine Symmetry Operation Symmetry Element
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Plenum Press, New York 1981