Spherical coordinates

Longitude-latitude coordinates $(\lambda,\theta,a)$ are defined with $\lambda\in[-\pi,\pi]$ the longitude, $\theta\in[-\pi/2,\pi/2]$ the latitude, and $a$ the radius of the earth. A new latitude coordinate $\mu$ is introduced:

$$\mu =  \sin\theta \quad,\qquad \cos\theta = \sqrt{1-\mu^2}$$ (3.1)

Table 3.1 lists the coordinate transformations from 2D carthesian to spherical.

Table 3.1: Coordiante transformations from 2D carthesian to spherical.

	carthesian	spherical	spherical, $\mu=\sin\theta$
$\vec{\mathbf{x}}$	$(x,y)$	$(\lambda,\theta)$	$(\lambda,\mu)$
	$\mbox{d}x$	$a \cos\theta \mbox{d}\lambda$
	$\mbox{d}y$	$a \mbox{d}\theta$
$\vec{\mathbf{\nabla}}$	$( \frac{\partial}{\partial x} , \frac{\partial}{\partial y} )$	$( \frac{1}{a\cos\theta}\frac{\partial}{\partial\lambda}  ,  \frac{1}{a}\frac{\partial}{\partial\theta} )$	$\frac{1}{a\cos\theta} (\ \frac{\partial}{\partial\lambda}  ,  (1-\mu^2)\frac{\partial}{\partial\mu} )$
$\vec{\mathbf{\nabla}}\cdot\vec{\mathbf{v}}$	$\frac{\partial v_{x}}{\partial x} + \frac{\partial v_y}{\partial y}$	$\frac{1}{a\cos^2\theta} \{\ \frac{\partial v_{\lambda}\cos\theta}{\partial\lambda}  +  \cos\theta\frac{\partial(v_{\theta}\cos\theta)}{\partial\theta} \}$	$\frac{1}{a\cos^2\theta} \{\ \frac{\partial(v_{\lambda}\cos\theta)}{\partial\lambda}  +  (1-\mu^2)\frac{\partial(v_{\theta}\cos\theta)}{\partial\mu} \}$