Subsection 3.2.2 The line element and a metric tensor
The length of arc \(\,ds\) is obtained by \(\,ds^{2} = \,dx^{2}+\,dy^{2}+\,dz^{2}\) in rectangular coordinates and it is
\begin{equation*}
\,ds^{2} = \sum\limits_{p=1}^{3} \sum\limits_{q=1}^{3} g_{pq} \,du_{p}\,du_{q}
\end{equation*}
in curvilinear coordinates. we define the line element $ds$ in the \(n\)- dimensional space to be given by the quadratic form called metric. The quantities \(g_{pq}\) are called metric coefficients and are symmetric, i.e., \(g_{pq} = g_{qp}\text{.}\) If \(g_{pq} = 0\) and \(p \neq q\text{,}\) then the coordinate system is orthogonal. In this case \(g_{11} = h_{1}^{2}{,} g_{22} = h_{2}^{2} {,} g_{33} = h_{3}^{2}\text{.}\) The generalized form of element of length \(\,ds\) between the points \(x^{i}\) and \(x^{i} + \,dx^{i}\) is given by \(\,ds^{2}=g_{ij}\,dx^{i}\,dx^{j}\text{,}\) where \(g_{ij}\) are functions of \(x^{i}\text{,}\) and \(g_{ij} = g_{ji}\text{.}\) (symmetric)
1
Let,
\begin{equation*}
\vec{r}=\vec{r}(u_{1}, u_{2},u_{3}),
\end{equation*}
then,
\begin{equation*}
\,d\vec{r} = \frac{\partial\vec{r}}{\partial u_{1}}\,du_{1}
+\frac{\partial\vec{r}}{\partial u_{2}}\,du_{2}
+ \frac{\partial\vec{r}}{\partial u_{3}}\,du_{3}
\end{equation*}
\begin{equation*}
= \alpha_{1}\,du_{1} + \alpha_{2}\,du_{2} + \alpha_{3}\,du_{3},
\end{equation*}
where,
\begin{equation*}
\alpha_{p}= \frac{\partial\vec{r}}{\partial u_{p}} ;
\end{equation*}
Now,
\begin{equation*}
\,ds^{2} = \vec{\,dr}\cdot\vec{\,dr} = \alpha_{1} \alpha_{1}du^{2}_{1}
+ \alpha_{1}\alpha_{2}\,du_{1}\,du_{2}
\end{equation*}
\begin{equation*}
+ \alpha_{1}\alpha_{3}\,du_{1}\,du_{3}
+\alpha_{2}\alpha_{1}\,du_{2}\,du_{1}
+\alpha_{2}\alpha_{2}\,du^{2}_{2}
\end{equation*}
\begin{equation*}
+ \alpha_{2}\alpha_{3}\,du_{2}\,du_{3}
+\alpha_{3}\alpha_{1}\,du_{3}\,du_{1}
\end{equation*}
\begin{equation*}
+\alpha_{3}\alpha_{2}\,du_{3}\,du_{2}
+\alpha_{3}\alpha_{3}\,du^{2}_{3}
=\sum\limits_{p=1}^{3} \sum\limits_{q=1}^{3} g_{pq} \,du_{p}du_{q}
\end{equation*}
where \(g_{pq}= \alpha_{p}\alpha_{q}\) also in a tensor, \(g_{pq}\rightarrow g_{ij}\) and \(\,du_{p}\rightarrow \,du^{i}, \,du_{q}\rightarrow \,du^{j}\text{.}\)
