Covariant basis¶
Description¶
Assigning
\[\vec{u}
=
\sum_i
U^i
\vec{E}_i\]
yields
\[\pder{}{t}
\left(
\sum_i
U^i
\vec{E}_i
\right)
+
\sum_j
V^j
\pder{}{X^j}
\left(
\sum_i
U^i
\vec{E}_i
\right)
=
\sum_i
\pder{U^i}{t}
\vec{E}_i
+
\sum_{ij}
V^j
\pder{U^i}{X^j}
\vec{E}_i
+
\sum_{ij}
V^j
U^i
\pder{\vec{E}_i}{X^j}.\]
Regarding the last term, by adopting the relation:
\[\dedx{i}{j}{k},\]
the last term is reformulated as
\[\begin{split}&
\sum_{ij}
V^j
U^i
\frac{1}{H_i}
\pder{H_i}{X^j}
\vec{E}_i
+
\sum_{ij}
V^i
U^j
\frac{1}{H_i}
\pder{H_i}{X^j}
\vec{E}_i
-
\sum_{ij}
V^j
U^j
\frac{H_j}{H_i H_i}
\pder{H_j}{X^i}
\vec{E}_i \\
=
&
\sum_{ij}
\left(
V^j
U^i
+
V^i
U^j
\right)
\frac{1}{H_i}
\pder{H_i}{X^j}
\vec{E}_i
-
\sum_{ij}
V^j
U^j
\frac{H_j}{H_i H_i}
\pder{H_j}{X^i}
\vec{E}_i.\end{split}\]
Explicitly, the material derivative is given by the sum of
\[\begin{split}\newcommand{\tmp}[1]{
\pder{U^{#1}}{t}
}
\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
\tmp{1}
\\
\tmp{2}
\\
\tmp{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\newcommand{\tmp}[2]{
V^{#2}
\pder{U^{#1}}{X^{#2}}
}
\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
\tmp{1}{1} + \tmp{1}{2} + \tmp{1}{3}
\\
\tmp{2}{1} + \tmp{2}{2} + \tmp{2}{3}
\\
\tmp{3}{1} + \tmp{3}{2} + \tmp{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\newcommand{\tmp}[2]{
\left(
V^{#2}
U^{#1}
+
V^{#1}
U^{#2}
\right)
\frac{1}{H_{#1}}
\pder{H_{#1}}{X^{#2}}
}
\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
\tmp{1}{1} + \tmp{1}{2} + \tmp{1}{3}
\\
\tmp{2}{1} + \tmp{2}{2} + \tmp{2}{3}
\\
\tmp{3}{1} + \tmp{3}{2} + \tmp{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\newcommand{\tmp}[2]{
V^{#2}
U^{#2}
\frac{H_{#2}}{H_{#1} H_{#1}}
\pder{H_{#2}}{X^{#1}}
}
\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
- \tmp{1}{1} - \tmp{1}{2} - \tmp{1}{3}
\\
- \tmp{2}{1} - \tmp{2}{2} - \tmp{2}{3}
\\
- \tmp{3}{1} - \tmp{3}{2} - \tmp{3}{3}
\end{pmatrix}.\end{split}\]
Example¶
Cylindrical coordinates¶
The sum of
\[\vec{E}_1
\left(
\pder{U^1}{t}
+
V^1 \pder{U^1}{X^1}
+
V^2 \pder{U^1}{X^2}
+
V^3 \pder{U^1}{X^3}
-
V^2 U^2 X^1
\right),\]
\[\vec{E}_2
\left(
\pder{U^2}{t}
+
V^1 \pder{U^2}{X^1}
+
V^2 \pder{U^2}{X^2}
+
V^3 \pder{U^2}{X^3}
+
\frac{V^1 U^2}{X^1}
+
\frac{V^2 U^1}{X^1}
\right),\]
\[\vec{E}_3
\left(
\pder{U^3}{t}
+
V^1 \pder{U^3}{X^1}
+
V^2 \pder{U^3}{X^2}
+
V^3 \pder{U^3}{X^3}
\right).\]
Rectilinear coordinates¶
The sum of
\[\vec{E}_1
\left(
\pder{U^1}{t}
+
V^1 \pder{U^1}{X^1}
+
V^2 \pder{U^1}{X^2}
+
V^3 \pder{U^1}{X^3}
+
V^1 U^1 \frac{1}{H_1} \pder{H_1}{X^1}
\right),\]
\[\vec{E}_2
\left(
\pder{U^2}{t}
+
V^1 \pder{U^2}{X^1}
+
V^2 \pder{U^2}{X^2}
+
V^3 \pder{U^2}{X^3}
+
V^2 U^2 \frac{1}{H_2} \pder{H_2}{X^2}
\right),\]
\[\vec{E}_3
\left(
\pder{U^3}{t}
+
V^1 \pder{U^3}{X^1}
+
V^2 \pder{U^3}{X^2}
+
V^3 \pder{U^3}{X^3}
+
V^3 U^3 \frac{1}{H_3} \pder{H_3}{X^3}
\right).\]
Application¶
The sum of
\[\vec{E}_1
\left(
\pder{U^1}{t}
+
V^1 \pder{U^1}{X^1}
+
V^2 \pder{U^1}{X^2}
+
V^3 \pder{U^1}{X^3}
+
V^1 U^1 \frac{1}{H_1} \pder{H_1}{X^1}
-
V^2 U^2 \frac{H_2}{H_1 H_1} \pder{H_2}{X^1}
\right),\]
\[\vec{E}_2
\left(
\pder{U^2}{t}
+
V^1 \pder{U^2}{X^1}
+
V^2 \pder{U^2}{X^2}
+
V^3 \pder{U^2}{X^3}
+
V^1 U^2 \frac{1}{H_2} \pder{H_2}{X^1}
+
V^2 U^1 \frac{1}{H_2} \pder{H_2}{X^1}
\right),\]
\[\vec{E}_3
\left(
\pder{U^3}{t}
+
V^1 \pder{U^3}{X^1}
+
V^2 \pder{U^3}{X^2}
+
V^3 \pder{U^3}{X^3}
\right).\]