Normalised basis¶
Description¶
The divergence of a second-order tensor reads
\[\left(
\sum_k
\frac{1}{H_k}
\vec{\hat{E}}_k
\pder{}{X^k}
\right)
\cdot
\left\{
\sum_{ij}
\left(
\vec{\hat{E}}_j
\otimes
\vec{\hat{E}}_i
\right)
\hat{S}^{ij}
\right\},\]
which has the following three components.
The first one is
\[\sum_{ijk}
\left(
\vec{\hat{E}}_k
\cdot
\pder{\vec{\hat{E}}_j}{X^k}
\right)
\frac{1}{H_k}
\vec{\hat{E}}_i
\hat{S}^{ij}.\]
By using the relation:
\[\dehatdx{j}{k}{l},\]
this leads to
\[\begin{split}&
\sum_{ijk}
\left(
\vec{\hat{E}}_k
\cdot
\frac{1}{H_j}
\pder{H_k}{X^j}
\vec{\hat{E}}_k
-
\vec{\hat{E}}_k
\cdot
\sum_l
\delta_{jk}
\frac{1}{H_l}
\pder{H_j}{X^l}
\vec{\hat{E}}_l
\right)
\frac{1}{H_k}
\vec{\hat{E}}_i
\hat{S}^{ij} \\
=
&
\sum_{ijk}
\vec{\hat{E}}_i
\frac{1}{H_j}
\frac{1}{H_k}
\pder{H_k}{X^j}
\hat{S}^{ij}
-
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_j}
\frac{1}{H_j}
\pder{H_j}{X^j}
\hat{S}^{ij} \\
=
&
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_j}
\frac{1}{J}
\pder{J}{X^j}
\hat{S}^{ij}
-
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_j}
\frac{1}{H_j}
\pder{H_j}{X^j}
\hat{S}^{ij},\end{split}\]
where the relation:
\[\sumofchristoffel\]
is adopted.
The second one is
\[\begin{split}&
\sum_{ijk}
\left(
\vec{\hat{E}}_k
\cdot
\vec{\hat{E}}_j
=
\delta_{kj}
\right)
\frac{1}{H_k}
\pder{\vec{\hat{E}}_i}{X^k}
\hat{S}^{ij} \\
=
&
\sum_{ij}
\frac{1}{H_j}
\pder{\vec{\hat{E}}_i}{X^j}
\hat{S}^{ij}.\end{split}\]
By using the relation:
\[\dehatdx{i}{j}{k},\]
this leads to
\[\begin{split}&
\sum_{ij}
\frac{1}{H_j}
\frac{1}{H_i}
\pder{H_j}{X^i}
\vec{\hat{E}}_j
\hat{S}^{ij}
-
\sum_{ijk}
\frac{1}{H_j}
\delta_{ij}
\frac{1}{H_k}
\pder{H_i}{X^k}
\vec{\hat{E}}_k
\hat{S}^{ij} \\
=
&
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_i}
\frac{1}{H_j}
\pder{H_i}{X^j}
\hat{S}^{ji}
-
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_i}
\frac{1}{H_j}
\pder{H_j}{X^i}
\hat{S}^{jj}.\end{split}\]
The third one is
\[\begin{split}&
\sum_{ijk}
\left(
\vec{\hat{E}}_k
\cdot
\vec{\hat{E}}_j
=
\delta_{kj}
\right)
\frac{1}{H_k}
\vec{\hat{E}}_i
\pder{\hat{S}^{ij}}{X^k} \\
=
&
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_j}
\pder{\hat{S}^{ij}}{X^j}.\end{split}\]
The sum of the whole elements yields
\[\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_j}
\frac{1}{J}
\pder{
J
\hat{S}^{ij}
}{
X^j
}
-
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_j}
\frac{1}{H_j}
\pder{H_j}{X^j}
\hat{S}^{ij}
+
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_i}
\frac{1}{H_j}
\pder{H_i}{X^j}
\hat{S}^{ji}
-
\sum_{ij}
\vec{\hat{E}}_i
\frac{1}{H_i}
\frac{1}{H_j}
\pder{H_j}{X^i}
\hat{S}^{jj},\]
where the part of the first element and the last element were unified to yield the Jacobian relation.
Furthermore, the first two terms can be unified to yield
\[\newcommand{\tmpa}[2]{
\frac{1}{J}
\pder{}{X^#2}
\left(
\frac{J}{H_#2}
\hat{S}^{#1 #2}
\right)
}
\newcommand{\tmpb}[2]{
\frac{1}{H_#1}
\frac{1}{H_#2}
\pder{H_#1}{X^#2}
\hat{S}^{#2 #1}
}
\newcommand{\tmpc}[2]{
\frac{1}{H_#1}
\frac{1}{H_#2}
\pder{H_#2}{X^#1}
\hat{S}^{#2 #2}
}
\sum_{ij}
\vec{\hat{E}}_i
\tmpa{i}{j}
+
\sum_{ij}
\vec{\hat{E}}_i
\tmpb{i}{j}
-
\sum_{ij}
\vec{\hat{E}}_i
\tmpc{i}{j}.\]
Explicitly, the sum of
\[\begin{split}\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\tmpa{1}{1}
+
\tmpa{1}{2}
+
\tmpa{1}{3}
\\
\tmpa{2}{1}
+
\tmpa{2}{2}
+
\tmpa{2}{3}
\\
\tmpa{3}{1}
+
\tmpa{3}{2}
+
\tmpa{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
\tmpb{1}{1}
+
\tmpb{1}{2}
+
\tmpb{1}{3}
\\
\tmpb{2}{1}
+
\tmpb{2}{2}
+
\tmpb{2}{3}
\\
\tmpb{3}{1}
+
\tmpb{3}{2}
+
\tmpb{3}{3}
\end{pmatrix},\end{split}\]
\[\begin{split}\begin{pmatrix}
\vec{\hat{E}}_1
&
\vec{\hat{E}}_2
&
\vec{\hat{E}}_3
\end{pmatrix}
\begin{pmatrix}
-
\tmpc{1}{1}
-
\tmpc{1}{2}
-
\tmpc{1}{3}
\\
-
\tmpc{2}{1}
-
\tmpc{2}{2}
-
\tmpc{2}{3}
\\
-
\tmpc{3}{1}
-
\tmpc{3}{2}
-
\tmpc{3}{3}
\end{pmatrix}.\end{split}\]
Example¶
The sum of
\[\vec{\hat{E}}_1
\left(
\frac{1}{X^1}
\pder{X^1 \hat{S}^{1 1}}{X^1}
+
\frac{1}{X^1}
\pder{\hat{S}^{1 2}}{X^2}
+
\pder{\hat{S}^{1 3}}{X^3}
-
\frac{
\hat{S}^{2 2}
}{
X^1
}
\right),\]
\[\vec{\hat{E}}_2
\left(
\frac{1}{X^1}
\pder{X^1 \hat{S}^{2 1}}{X^1}
+
\frac{1}{X^1}
\pder{\hat{S}^{2 2}}{X^2}
+
\pder{\hat{S}^{2 3}}{X^3}
+
\frac{
\hat{S}^{1 2}
}{
X^1
}
\right),\]
\[\vec{\hat{E}}_3
\left(
\frac{1}{X^1}
\pder{X^1 \hat{S}^{3 1}}{X^1}
+
\frac{1}{X^1}
\pder{\hat{S}^{3 2}}{X^2}
+
\pder{\hat{S}^{3 3}}{X^3}
\right).\]