Contravariant component¶
Description¶
The divergence of the velocity vector reads
\[\left(
\sum_j
\vec{E}^j
\pder{}{X^j}
\right)
\cdot
\left(
\sum_i
\vec{E}_i
U^i
\right)
=
\sum_{ij}
\vec{E}^j
\cdot
\pder{\vec{E}_i}{X^j}
U^i
+
\sum_{ij}
\vec{E}^j
\cdot
\vec{E}_i
\pder{U^i}{X^j}.\]
The first term leads to
\[\begin{split}\sum_{ijk}
\left(
\vec{E}^j
\cdot
\vec{E}_k
=
\delta_k^j
\right)
\Gamma_{ij}^k
U^i
&
=
\sum_{ij}
\Gamma_{ij}^j
U^i \\
&
=
\frac{1}{J}
\sum_i
\pder{J}{X^i}
U^i,\end{split}\]
where the relation
\[\sumofchristoffel\]
is used.
The second term is
\[\begin{split}\sum_{ij}
\delta_i^j
\pder{U^i}{X^j}
&
=
\sum_i
\pder{U^i}{X^i} \\
&
=
\frac{1}{J}
\sum_i
J
\pder{U^i}{X^i}.\end{split}\]
As a consequence, the sum yields
\[\frac{1}{J}
\sum_i
\pder{}{X^i}
\left(
J
U^i
\right).\]
Example¶
Cylindrical coordinates¶
\[\frac{1}{X^1}
\pder{X^1 U^1}{X^1}
+
\pder{U^2}{X^2}
+
\pder{U^3}{X^3}.\]
Rectilinear coordinates¶
\[\frac{1}{J}
\pder{
J
U^1
}{
X^1
}
+
\frac{1}{J}
\pder{
J
U^2
}{
X^2
}
+
\frac{1}{J}
\pder{
J
U^3
}{
X^3
}.\]