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,

$$\nabla f ={1\over h_{1}}{\partial f \over \partial x_{1}} {... ...+ {1\over h_{3}}{\partial f \over \partial x_{3}} {\bf e}_{3}.$$ (1.37)

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

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

$${1\over h_{1}h_{3}}\left \{{\partial \over \partial x_{3}}\left (... ...- {\partial \over \partial x_{1}}\left (h_{3}A_{3}\right )\right \}{\bf e}_{2},$$ (1.39)

$${1\over h_{1}h_{2}}\left \{{\partial \over \partial x_{1}}\left (... ...- {\partial \over \partial x_{2}}\left (h_{1}A_{1}\right )\right \}{\bf e}_{3},$$

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

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