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General Orthogonal Coordinates

Let $(x_{1}, x_{2}, x_{3})$ be 3 coordinates with the associated unit vectors ${\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3}$. For example, in Cartesian coordinates $x_{1} = x$ and ${\bf e}_{1} = {\bf i} = \nabla
x$. Let the associated scale factors be $h_{1}, h_{2}, h_{3}$. Then,
\begin{displaymath}
\nabla f ={1\over h_{1}}{\partial f \over \partial x_{1}} {...
...+ {1\over h_{3}}{\partial f \over \partial x_{3}} {\bf e}_{3}.
\end{displaymath} (1.37)


\begin{displaymath}
\nabla \cdot {\bf A} = {1\over h_{1}h_{2}h_{3}}\left \{{\pa...
... \over \partial x_{3}}\left (h_{1}h_{2}A_{3}\right ) \right \}
\end{displaymath} (1.38)


$\displaystyle \nabla \times {\bf A}$ $\textstyle =$ $\displaystyle {1\over h_{2}h_{3}}\left \{{\partial
\over \partial x_{2}}\left (...
...- {\partial \over
\partial x_{3}}\left (h_{2}A_{2}\right )\right \}{\bf e}_{1},$  
    $\displaystyle {1\over h_{1}h_{3}}\left \{{\partial
\over \partial x_{3}}\left (...
...- {\partial \over
\partial x_{1}}\left (h_{3}A_{3}\right )\right \}{\bf e}_{2},$ (1.39)
    $\displaystyle {1\over h_{1}h_{2}}\left \{{\partial
\over \partial x_{1}}\left (...
...- {\partial \over
\partial x_{2}}\left (h_{1}A_{1}\right )\right \}{\bf e}_{3},$  

and $\nabla ^{2}f$ follows from $\nabla \cdot \nabla f$.

Example 1. .10In Cartesian coordinates
$x_{1} = x$ $x_{2} = y$ $x_{3} = z$
$h_{1} = 1$ $h_{2} = 1$ $h_{3} = 1$

In Cylindrical coordinates
$x_{1} = R$ $x_{2} = \phi$ $x_{3} = z$
$h_{1} = 1$ $h_{2} = R$ $h_{3} = 1$

In spherical coordinates
$x_{1} = r$ $x_{2} = \theta$ $x_{3} = \phi$
$h_{1} = 1$ $h_{2} = r$ $h_{3} = r \sin \theta$


next up previous
Next: Fluids and Nonmagnetic Plasmas Up: Non-Cartesian Coordinates Previous: 3-D Cylindrical Polar Coordinates
Prof. Alan Hood
2000-11-06