Normalised basis¶
Description¶
By substituting
\[\vec{E}_i
=
H_i
\vec{\hat{E}}_i\]
into the covariant relation:
\[\dedx{i}{j}{k},\]
I obtain
\[\begin{split}\pder{}{X^j}
\left(
H_i
\vec{\hat{E}}_i
\right)
&
=
\pder{H_i}{X^j}
\vec{\hat{E}}_i
+
H_i
\pder{\hat{E}_i}{X^j} \\
&
=
-
\sum_k
\delta_{ij}
\frac{H_j}{2 H_k}
\pder{H_i}{X^k}
\vec{\hat{E}}_k
-
\sum_k
\delta_{ij}
\frac{H_i}{2 H_k}
\pder{H_j}{X^k}
\vec{\hat{E}}_k
+
\pder{H_i}{X^j}
\vec{\hat{E}}_i
+
\pder{H_j}{X^i}
\vec{\hat{E}}_j,\end{split}\]
or simply (\(j = i\) is used to unify the first two terms)
\[\dehatdx{i}{j}{k}.\]
This relation is written down explicitly below for an intuitive understanding.
\[\begin{split}\newcommand{\diag}[2]{
-
\frac{1}{H_{#2}}
\pder{H_{#1}}{X^{#2}}
\vec{\hat{E}}_{#2}
}
\newcommand{\nodiag}[2]{
\frac{1}{H_{#1}}
\pder{H_{#2}}{X^{#1}}
\vec{\hat{E}}_{#2}
}
\begin{pmatrix}
\pder{\vec{\hat{E}}_1}{X^1} & \pder{\vec{\hat{E}}_2}{X^1} & \pder{\vec{\hat{E}}_3}{X^1} \\
\pder{\vec{\hat{E}}_1}{X^2} & \pder{\vec{\hat{E}}_2}{X^2} & \pder{\vec{\hat{E}}_3}{X^2} \\
\pder{\vec{\hat{E}}_1}{X^3} & \pder{\vec{\hat{E}}_2}{X^3} & \pder{\vec{\hat{E}}_3}{X^3} \\
\end{pmatrix}
=
\begin{pmatrix}
\diag{1}{2}
\diag{1}{3}
&
\nodiag{2}{1}
&
\nodiag{3}{1}
\\
\nodiag{1}{2}
&
\diag{2}{1}
\diag{2}{3}
&
\nodiag{3}{2}
\\
\nodiag{1}{3}
&
\nodiag{2}{3}
&
\diag{3}{1}
\diag{3}{2}
\end{pmatrix}.\end{split}\]
Example¶
In cylindrical coordinates, I have
\[\begin{split}\begin{pmatrix}
\pder{\vec{\hat{E}}_1}{X^1} & \pder{\vec{\hat{E}}_2}{X^1} & \pder{\vec{\hat{E}}_3}{X^1} \\
\pder{\vec{\hat{E}}_1}{X^2} & \pder{\vec{\hat{E}}_2}{X^2} & \pder{\vec{\hat{E}}_3}{X^2} \\
\pder{\vec{\hat{E}}_1}{X^3} & \pder{\vec{\hat{E}}_2}{X^3} & \pder{\vec{\hat{E}}_3}{X^3} \\
\end{pmatrix}
=
\begin{pmatrix}
0 & 0 & 0 \\
\vec{\hat{E}}_2 & - \vec{\hat{E}}_1 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}.\end{split}\]