spherical coordinates

Spherical coordinates are a system of coordinates for ${ℝ}^3$,or more generally ${ℝ}^n$.One coordinate is the distance from the origin,which can be thought of asthe radius of the sphere centred at the origin on which the point lies.The other coordinates are angles that specify the position of the point on this sphere.

In ${ℝ}^3$ the coordinates are given by

$\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right)$$=\left(\begin{array}{c}\hfill r\sin \varphi \cos \theta \hfill \\ \hfill r\sin \varphi \sin \theta \hfill \\ \hfill r\cos \varphi \hfill \end{array}\right),$

where $r$ is the distance from the origin,$\theta$ is the azimuthal angle defined for $\theta \in [0,2\pi )$,and $\varphi \in [0,\pi ]$ is the polar angle.Note that $\varphi =0$ corresponds to the top of the sphere and $\varphi =\pi$ corresponds to the bottom of the sphere.There is a clash between the mathematicians’ and the physicists’ definition of spherical coordinates, interchanging both the direction of $\varphi$ and the choice of names for the two angles (physicists often use $\theta$ as the azimuthal angle and $\varphi$ as the polar one).

Spherical coordinates are a generalization of polar coordinates,and can be further generalized to ${ℝ}^n$,with $n-2$ polar angles ${\varphi }_1,\mathrm{\dots },{\varphi }_{n-2}$ and one azimuthal angle $\theta$:

$\left(\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_k\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill x_{n-1}\hfill \\ \hfill x_n\hfill \end{array}\right)$$=\left(\begin{array}{c}\hfill r\cos {\varphi }_1\hfill \\ \hfill r\sin {\varphi }_1\cos {\varphi }_2\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill r\left({\displaystyle \prod _{i=1}^{k-1}}\sin {\varphi }_i\right)\cos {\varphi }_k\hfill \\ \hfill \mathrm{⋮}\hfill \\ \hfill r\sin {\varphi }_1\sin {\varphi }_2\mathrm{\cdots }\cos \theta \hfill \\ \hfill r\sin {\varphi }_1\sin {\varphi }_2\mathrm{\cdots }\sin {\varphi }_{n-2}\sin \theta .\hfill \end{array}\right).$

These are sometimes called hyperspherical coordinates if $n>3$.