Covariant basis¶
Description¶
The divergence of a second-order tensor reads
\[\left(
\sum_k
\vec{E}^k
\pder{}{X^k}
\right)
\cdot
\left\{
\sum_{ij}
\left(
\vec{E}_j
\otimes
\vec{E}_i
\right)
S^{ij}
\right\},\]
which has three components:
\[\begin{split}\sum_{ijk}
\left(
\vec{E}^k
\cdot
\pder{\vec{E}_j}{X^k}
\right)
\vec{E}_i
S^{ij}
=
&
\sum_{ijkl}
\left(
\vec{E}^k
\cdot
\vec{E}_l
=
\delta_l^k
\right)
\vec{E}_i
\Gamma_{jk}^l
S^{ij} \\
=
&
\sum_{ijk}
\vec{E}_i
\Gamma_{jk}^k
S^{ij} \\
=
&
\sum_{ij}
\vec{E}_i
\frac{1}{J}
\pder{J}{X^j}
S^{ij},\end{split}\]
\[\begin{split}\sum_{ijk}
\left(
\vec{E}^k
\cdot
\vec{E}_j
=
\delta_j^k
\right)
\pder{\vec{E}_i}{X^k}
S^{ij}
=
&
\sum_{ij}
\pder{\vec{E}_i}{X^j}
S^{ij} \\
=
&
\sum_{ij}
\left(
\frac{1}{H_i}
\pder{H_i}{X^j}
\vec{E}_i
+
\frac{1}{H_j}
\pder{H_j}{X^i}
\vec{E}_j
-
\sum_k
\delta_{ij}
\frac{H_j}{2 H_k H_k}
\pder{H_i}{X^k}
\vec{E}_k
-
\sum_k
\delta_{ij}
\frac{H_i}{2 H_k H_k}
\pder{H_j}{X^k}
\vec{E}_k
\right)
S^{ij} \\
=
&
\newcommand{\tmpa}[2]{
\frac{1}{H_{#1}}
\pder{H_{#1}}{X^{#2}}
\left(
S^{#1 #2}
+
S^{#2 #1}
\right)
}
\newcommand{\tmpb}[2]{
\frac{H_{#2}}{H_{#1} H_{#1}}
\pder{H_{#2}}{X^{#1}}
S^{#2 #2}
}
\sum_{ij}
\vec{E}_i
\tmpa{i}{j}
-
\sum_{ij}
\vec{E}_i
\tmpb{i}{j},\end{split}\]
\[\sum_{ijk}
\left(
\vec{E}^k
\cdot
\vec{E}_j
=
\delta_j^k
\right)
\vec{E}_i
\pder{S^{ij}}{X^k}
=
\sum_{ij}
\vec{E}_i
\pder{S^{ij}}{X^j}.\]
The sum of the first and the last elements yields
\[\newcommand{\tmpc}[2]{
\frac{1}{J}
\pder{}{X^{#2}}
\left(
J
S^{#1 #2}
\right)
}
\sum_{ij}
\vec{E}_i
\tmpc{i}{j}.\]
Explicitly, the sum of
\[\begin{split}\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
\tmpc{1}{1}
+
\tmpc{1}{2}
+
\tmpc{1}{3}
\\
\tmpc{2}{1}
+
\tmpc{2}{2}
+
\tmpc{2}{3}
\\
\tmpc{3}{1}
+
\tmpc{3}{2}
+
\tmpc{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
\tmpa{1}{1}
+
\tmpa{1}{2}
+
\tmpa{1}{3}
\\
\tmpa{2}{1}
+
\tmpa{2}{2}
+
\tmpa{2}{3}
\\
\tmpa{3}{1}
+
\tmpa{3}{2}
+
\tmpa{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix}
\vec{E}_1
&
\vec{E}_2
&
\vec{E}_3
\end{pmatrix}
\begin{pmatrix}
-
\tmpb{1}{1}
-
\tmpb{1}{2}
-
\tmpb{1}{3}
\\
-
\tmpb{2}{1}
-
\tmpb{2}{2}
-
\tmpb{2}{3}
\\
-
\tmpb{3}{1}
-
\tmpb{3}{2}
-
\tmpb{3}{3}
\end{pmatrix}.\end{split}\]
Example¶
Cylindrical coordinates¶
The sum of
\[\vec{E}_1
\left(
\frac{1}{X^1}
\pder{X^1 S^{1 1}}{X^1}
+
\pder{S^{1 2}}{X^2}
+
\pder{S^{1 3}}{X^3}
-
X^1
S^{2 2}
\right),\]
\[\vec{E}_2
\left(
\frac{1}{X^1}
\pder{X^1 S^{2 1}}{X^1}
+
\pder{S^{2 2}}{X^2}
+
\pder{S^{2 3}}{X^3}
+
\frac{
S^{2 1}
+
S^{1 2}
}{
X^1
}
\right),\]
\[\vec{E}_3
\left(
\frac{1}{X^1}
\pder{X^1 S^{3 1}}{X^1}
+
\pder{S^{3 2}}{X^2}
+
\pder{S^{3 3}}{X^3}
\right).\]
Rectilinear coordinates¶
The sum of
\[\vec{E}_1
\left(
\frac{1}{J}
\pder{}{X^1}
\left(
J
S^{11}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
S^{12}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
S^{13}
\right)
+
\frac{1}{H_1}
\pder{H_1}{X^1}
S^{11}
\right),\]
\[\vec{E}_2
\left(
\frac{1}{J}
\pder{}{X^1}
\left(
J
S^{21}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
S^{22}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
S^{23}
\right)
+
\frac{1}{H_2}
\pder{H_2}{X^2}
S^{22}
\right),\]
\[\vec{E}_3
\left(
\frac{1}{J}
\pder{}{X^1}
\left(
J
S^{31}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
S^{32}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
S^{33}
\right)
+
\frac{1}{H_3}
\pder{H_3}{X^3}
S^{33}
\right).\]
Application¶
The sum of
\[\vec{E}_1
\left(
\frac{1}{J}
\pder{}{X^1}
\left(
J
S^{11}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
S^{12}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
S^{13}
\right)
+
\frac{1}{H_1}
\pder{H_1}{X^1}
S^{11}
-
\frac{H_2}{H_1 H_1}
\pder{H_2}{X^1}
S^{22}
\right),\]
\[\vec{E}_2
\left(
\frac{1}{J}
\pder{}{X^1}
\left(
J
S^{21}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
S^{22}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
S^{23}
\right)
+
\frac{1}{H_2}
\pder{H_2}{X^1}
S^{21}
+
\frac{1}{H_2}
\pder{H_2}{X^1}
S^{12}
\right),\]
\[\vec{E}_3
\left(
\frac{1}{J}
\pder{}{X^1}
\left(
J
S^{31}
\right)
+
\frac{1}{J}
\pder{}{X^2}
\left(
J
S^{32}
\right)
+
\frac{1}{J}
\pder{}{X^3}
\left(
J
S^{33}
\right)
\right).\]