Normalised component¶
Description¶
The divergence of the velocity vector reads
\[\left(
\sum_j
\frac{1}{H_j}
\vec{\hat{E}}_j
\pder{}{X^j}
\right)
\cdot
\left(
\sum_i
\vec{\hat{E}}_i
\hat{U}^i
\right)
=
\sum_{ij}
\frac{1}{H_j}
\vec{\hat{E}}_j
\cdot
\pder{\vec{\hat{E}}_i}{X^j}
\hat{U}^i
+
\sum_{ij}
\frac{1}{H_j}
\vec{\hat{E}}_j
\cdot
\vec{\hat{E}}_i
\pder{\hat{U}^i}{X^j},\]
which is
\[\sum_i
\frac{1}{H_i}
\pder{\hat{U}^i}{X^i}
+
\sum_{ij}
\frac{1}{H_i}
\frac{1}{H_j}
\pder{H_j}{X^i}
\hat{U}^i
-
\sum_i
\frac{1}{H_i}
\frac{1}{H_i}
\pder{H_i}{X^i}
\hat{U}^i.\]
Writing down explicitly would be helpful to find the resulting relation:
\[\frac{1}{J}
\sum_i
\pder{}{X^i}
\left(
\frac{J}{H_i}
\hat{U}^i
\right).\]
Example¶
In cylindrical coordinates, I have
\[\frac{1}{X^1}
\pder{X^1 U^1}{X^1}
+
\frac{1}{X^1}
\pder{U^2}{X^2}
+
\pder{U^3}{X^3}.\]