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Spherical coordinates

Every point in $ \ensuremath{\mathbb{R}}^3$ can be described in spherical coordinates by a vector $ (r,\vartheta,\varphi)^{\top}$ with the radius $ r\ge 0$ and two angles $ \vartheta \in [0,\pi ]$ , $ \varphi \in [0,2\pi )$ . We denote by $ \S ^2$ the two-dimensional unit sphere embedded into $ \ensuremath{\mathbb{R}}^3$ , i.e.

$\displaystyle \S ^2 := \left\{\mathbf{x} \in \ensuremath{\mathbb{R}}^{3}:\; \Vert\mathbf{x}\Vert _2=1\right\}$    

and identify a point from $ \S ^2$ with the corresponding vector $ (\vartheta,\varphi)^{\top}$ . The spherical coordinate system is illustrated in Figure 3.2.
Figure: The spherical coordinate system in $ \ensuremath{\mathbb{R}}^3$ : Every point $ \boldsymbol {\xi }$ on a sphere with radius $ r$ centred at the origin can be described by angles $ \vartheta \in [0,\pi ]$ , $ \varphi \in [0,2\pi )$ and the radius $ r \in \mathbb{R}^+$ . For $ \vartheta = 0$ or $ \vartheta = \pi $ the point $ \boldsymbol {\xi }$ coincides with the North or the South pole, respectively.
\includegraphics[width=0.60\textwidth]{images/sphere}



Jens Keiner 2006-11-20