Normalised basis¶
Description¶
Assigning
\[\vec{u}
=
\sum_i
\hat{U}^i
\vec{\hat{E}}_i\]
yields
\[\pder{}{t}
\left(
\sum_i
\hat{U}^i
\vec{\hat{E}}_i
\right)
+
\sum_j
\frac{\hat{V}^j}{H_j}
\pder{}{X^j}
\left(
\sum_i
\hat{U}^i
\vec{\hat{E}}_i
\right)
=
\sum_i
\pder{\hat{U}^i}{t}
\vec{\hat{E}}_i
+
\sum_{ij}
\frac{\hat{V}^j}{H_j}
\pder{\hat{U}^i}{X^j}
\vec{\hat{E}}_i
+
\sum_{ij}
\frac{\hat{V}^j}{H_j}
\hat{U}^i
\pder{\vec{\hat{E}}_i}{X^j}.\]
By using the relation:
\[\dehatdx{i}{j}{k},\]
the last term leads to the sum of
\[\sum_{ij}
\frac{\hat{V}^j}{H_j}
\hat{U}^i
\frac{1}{H_i}
\pder{H_j}{X^i}
\vec{\hat{E}}_j
=
\sum_{ij}
\frac{\hat{V}^i}{H_i}
\hat{U}^j
\frac{1}{H_j}
\pder{H_i}{X^j}
\vec{\hat{E}}_i\]
and
\[-
\sum_{ijk}
\frac{\hat{V}^j}{H_j}
\hat{U}^i
\delta_{ij}
\frac{1}{H_k}
\pder{H_i}{X^k}
\vec{\hat{E}}_k
=
-
\sum_{ij}
\frac{\hat{V}^j}{H_j}
\hat{U}^j
\frac{1}{H_i}
\pder{H_j}{X^i}
\vec{\hat{E}}_i.\]
The explicit form is the sum of
\[\begin{split}\newcommand{\tmp}[1]{
\pder{\hat{U}^{#1}}{t}
}
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\tmp{1}
\\
\tmp{2}
\\
\tmp{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\newcommand{\tmp}[2]{
\frac{\hat{V}^{#2}}{H_{#2}}
\pder{\hat{U}^{#1}}{X^{#2}}
}
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\tmp{1}{1} + \tmp{1}{2} + \tmp{1}{3}
\\
\tmp{2}{1} + \tmp{2}{2} + \tmp{2}{3}
\\
\tmp{3}{1} + \tmp{3}{2} + \tmp{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\newcommand{\tmp}[2]{
\frac{\hat{V}^{#1}}{H_{#1}}
\hat{U}^{#2}
\frac{1}{H_{#2}}
\pder{H_{#1}}{X^{#2}}
}
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\tmp{1}{1} + \tmp{1}{2} + \tmp{1}{3}
\\
\tmp{2}{1} + \tmp{2}{2} + \tmp{2}{3}
\\
\tmp{3}{1} + \tmp{3}{2} + \tmp{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\newcommand{\tmp}[2]{
\frac{\hat{V}^{#2}}{H_{#2}}
\hat{U}^{#2}
\frac{1}{H_{#1}}
\pder{H_{#2}}{X^{#1}}
}
\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
- \tmp{1}{1} - \tmp{1}{2} - \tmp{1}{3}
\\
- \tmp{2}{1} - \tmp{2}{2} - \tmp{2}{3}
\\
- \tmp{3}{1} - \tmp{3}{2} - \tmp{3}{3}
\end{pmatrix}.\end{split}\]
Example¶
The sum of
\[\vec{\hat{E}}_1
\left(
\pder{\hat{U}^1}{t}
+
\hat{V}^1
\pder{\hat{U}^1}{X^1}
+
\frac{\hat{V}^2}{X^1}
\pder{\hat{U}^1}{X^2}
+
\hat{V}^3
\pder{\hat{U}^1}{X^3}
-
\frac{\hat{V}^2}{X^1}
\hat{U}^2
\right),\]
\[\vec{\hat{E}}_2
\left(
\pder{\hat{U}^2}{t}
+
\hat{V}^1
\pder{\hat{U}^2}{X^1}
+
\frac{\hat{V}^2}{X^1}
\pder{\hat{U}^2}{X^2}
+
\hat{V}^3
\pder{\hat{U}^2}{X^3}
+
\frac{\hat{V}^2}{X^1}
\hat{U}^1
\right),\]
\[\vec{\hat{E}}_3
\left(
\pder{\hat{U}^3}{t}
+
\hat{V}^1
\pder{\hat{U}^3}{X^1}
+
\frac{\hat{V}^2}{X^1}
\pder{\hat{U}^3}{X^2}
+
\hat{V}^3
\pder{\hat{U}^3}{X^3}
\right).\]