Covariant basis¶
Description¶
I consider
\[\pder{\vec{E}_i}{X^j}.\]
By using the relation
\[\fromctog,\]
this is equal to
\[\pder{}{X^j}
\left(
\sum_l
\pder{x^l}{X^i}
\vec{e}_l
\right)
=
\sum_l
\pder{}{X^j}
\left(
\pder{x^l}{X^i}
\right)
\vec{e}_l
+
\sum_l
\pder{x^l}{X^i}
\pder{\vec{e}_l}{X^j}.\]
Since the reference Cartesian basis vectors are constant, the second term disappears and I am left with
\[\begin{split}\sum_l
\pder{}{X^j}
\left(
\pder{x^l}{X^i}
\right)
\vec{e}_l
&
=
\sum_l
\pder{}{X^j}
\left(
\pder{x^l}{X^i}
\right)
\left(
\sum_k
\pder{X^k}{x^l}
\vec{E}_k
\right) \\
&
=
\sum_{kl}
\pder{}{X^j}
\left(
\pder{x^l}{X^i}
\right)
\pder{X^k}{x^l}
\vec{E}_k \\
&
=
\sum_k
\left\{
\sum_l
\pder{}{X^j}
\left(
\pder{x^l}{X^i}
\right)
\pder{X^k}{x^l}
\right\}
\vec{E}_k \\
&
=
\sum_k
\Gamma_{ij}^k
\vec{E}_k,\end{split}\]
where \(\Gamma_{ij}^k\) is the Christoffel symbol of the second kind:
\[\christoffel,\]
and by using the relation between the symbol and the scale factors:
\[\fromchristoffeltoscalefactor,\]
the change of the basis vector is given by
\[\begin{split}\newcommand{\tmp}[3]{
\pder{}{X^{#3}}
\left(
\delta_{#1#2}
H_{#1}
H_{#2}
\right)
}
&
\sum_k
\frac{1}{2 H_k H_k}
\left\{
-
\tmp{i}{j}{k}
+
\tmp{k}{i}{j}
+
\tmp{j}{k}{i}
\right\}
\vec{E}_k \\
=
&
-
\sum_k
\delta_{ij}
\frac{H_j}{2 H_k H_k}
\pder{H_i}{X^k}
\vec{E}_k
-
\sum_k
\delta_{ij}
\frac{H_i}{2 H_k H_k}
\pder{H_j}{X^k}
\vec{E}_k
+
\frac{1}{H_i}
\pder{H_i}{X^j}
\vec{E}_i
+
\frac{1}{H_j}
\pder{H_j}{X^i}
\vec{E}_j,\end{split}\]
and in summary:
\[\dedx{i}{j}{k}.\]
This relation is written down explicitly below for an intuitive understanding.
\[\begin{split}\newcommand{\diag}[1]{
\sum_k
\frac{H_#1}{H_k H_k}
\pder{H_#1}{X^k}
\vec{E}_k
}
\newcommand{\nodiag}[2]{
\frac{1}{H_{#1}}
\pder{H_{#1}}{X^{#2}}
\vec{E}_{#1}
+
\frac{1}{H_{#2}}
\pder{H_{#2}}{X^{#1}}
\vec{E}_{#2}
}
\begin{pmatrix}
\pder{\vec{E}_1}{X^1} & \pder{\vec{E}_2}{X^1} & \pder{\vec{E}_3}{X^1} \\
\pder{\vec{E}_1}{X^2} & \pder{\vec{E}_2}{X^2} & \pder{\vec{E}_3}{X^2} \\
\pder{\vec{E}_1}{X^3} & \pder{\vec{E}_2}{X^3} & \pder{\vec{E}_3}{X^3} \\
\end{pmatrix}
=
&
-
\begin{pmatrix}
\diag{1} & 0 & 0 \\
0 & \diag{2} & 0 \\
0 & 0 & \diag{3} \\
\end{pmatrix} \\
&
+
\begin{pmatrix}
0 & \nodiag{1}{2} & \nodiag{1}{3} \\
sym. & 0 & \nodiag{2}{3} \\
sym. & sym. & 0 \\
\end{pmatrix}.\end{split}\]
Example¶
The changes of the basis vectors
\[\begin{split}\begin{pmatrix}
\pder{\vec{E}_1}{X^1} & \pder{\vec{E}_2}{X^1} & \pder{\vec{E}_3}{X^1} \\
\pder{\vec{E}_1}{X^2} & \pder{\vec{E}_2}{X^2} & \pder{\vec{E}_3}{X^2} \\
\pder{\vec{E}_1}{X^3} & \pder{\vec{E}_2}{X^3} & \pder{\vec{E}_3}{X^3} \\
\end{pmatrix}\end{split}\]
read as follows.
Cylindrical coordinates¶
\[\begin{split}\begin{pmatrix}
0 & \frac{1}{X^1} \vec{E}_2 & 0 \\
\frac{1}{X^1} \vec{E}_2 & - X^1 \vec{E}_1 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}.\end{split}\]
Rectilinear coordinates¶
\[\begin{split}\begin{pmatrix}
- \frac{1}{H_1} \pder{H_1}{X^1} \vec{E}_1 & 0 & 0 \\
0 & - \frac{1}{H_2} \pder{H_2}{X^2} \vec{E}_2 & 0 \\
0 & 0 & - \frac{1}{H_3} \pder{H_3}{X^3} \vec{E}_3 \\
\end{pmatrix}.\end{split}\]
Application¶
\[\begin{split}\begin{pmatrix}
- \frac{1}{H_1} \pder{H_1}{X^1} \vec{E}_1 & \frac{1}{H_2} \pder{H_2}{X^1} \vec{E}_2 & 0 \\
\frac{1}{H_2} \pder{H_2}{X^1} \vec{E}_2 & - \frac{H_2}{H_1 H_1} \pder{H_2}{X^1} \vec{E}_1 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}.\end{split}\]