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Metric tensor

The scalar product of two vectors r1 and r2 referred to the same base system consisting of the three non-coplanar vectors $\boldsymbol{\tau}_1$,$\boldsymbol{\tau}_2$, $\boldsymbol{\tau}_3$ is defined as:

\begin{displaymath}
\textbf{r}_1 \cdot \textbf{r}_2 = (x_1 \boldsymbol{\tau}_1 +...
 ...bol{\tau}_1 + y_2\boldsymbol{\tau}_2 + z_2\boldsymbol{\tau}_3).\end{displaymath} (1)

In matrix notation it could be written:

\begin{displaymath}
\textbf{r}_1 \cdot \textbf{r}_2 = [x_1y_1z_1]\left[\begin{ar...
 ...ght]
\left[\begin{array}
{c}
x_2\\ y_2\\ z_2\end{array}\right];\end{displaymath} (2)
it is easy to verify that formulae (1) and (2) are equivalent. Relation (2) can be written more briefly as follows:

\begin{displaymath}
\textbf{r}_1 \cdot \textbf{r}_2 = r^t_1Gr_2\end{displaymath} (3)
where r2 is a column matrix

\begin{displaymath}
\left[\begin{array}
{c}x_2\\ y_2\\ z_2\end{array}\right]\end{displaymath}

and rt1 a transposed column matrix [$x_1\,y_1\,z_1$]; 9 is the 3 $\times$ 3 matrix of relation (2) and is called a metric matrix or metric tensor,[*] because its elements $g_{ij} = \boldsymbol{\tau}_i \cdot \boldsymbol{\tau}_j$ are dependent both on the length of the base vectors and on the angles formed by them.

If in (3) we assume r1 = r2, we have:

\begin{displaymath}
\textbf{r}_1 \cdot \textbf{r}_1 = \vert\textbf{r}_1\vert \cdot \vert\textbf{r}_1\vert = r^t_1
Gr_1\end{displaymath} (4)
and therefore:

\begin{displaymath}
\vert\textbf{r}_1\vert = \sqrt{r^t_1Gr_1}.\end{displaymath} (5)

On the other hand, bearing in mind that $\textbf{r}_1 \cdot \textbf{r}_2 =
\vert\textbf{r}_1\vert \vert\textbf{r}_2\vert\cos \phi$, where $\phi$ is the angle between $\textbf{r}_1$ and $\textbf{r}_2$, we have:

\begin{displaymath}
\textbf{r}_1 \cdot \textbf{r}_2 = \vert\textbf{r}_1\vert\vert\textbf{r}_2\vert \cos \phi =
r^t_1Gr_2 \end{displaymath} (6)
and finally, using relation (5), we obtain:

\begin{displaymath}
\cos \phi = \frac{r^t_1Gr_2}{\sqrt{r^t_1Gr_1} \cdot \sqrt{r^t_2Gr_2}}.\end{displaymath} (7)

Equations (5) and (7) are the rules to obtain the vector lengths and the angles between vectors. The space in which the lengths and the angles between vectors are defined, is called metric space. The metric is given by the G matrix.


next up previous
Next: Symmetry operations Up: Introduction Previous: Introduction

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