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.

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

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.