:PROPERTIES: :ID: 77d7976b-b3db-44ed-87b7-cc80d150d68b :mtime: 20231019010745 20231017014715 :ctime: 20231017014710 :END: #+title: spherical coordinates #+filetags: :public:project: * Definition of spherical coordinates spherical coordinates are a type of [[id:d02a62fb-1601-48a4-9cf2-bbb2ade26402][Coordinate System (Vector Space)]]. spherical coordinates are given by the triple $(r , \theta , \phi)$, where - $r$ is the radius from the origin - $\theta$ is the polar or inclination angle from the $z$-axis, - $\phi$ is the azimuthal angle from the $x$-axis * Notational Warning WARNING: some people get confused about notation because in [[id:ea993ee3-14c0-492a-8030-7c89836741c0][cylindrical polar coordinates]], $\theta$ represents the azimuthal angle, but in spherical coordiantes, $\phi$ represents the azimuthal angle. * Coordinate Restrictions We need to restrict things - let's restrict $0 \le r$ - let's restrict $0 \le \theta \le \pi$ - let's restrict $0 \le \phi < 2\pi$ * Converting between rectangular and spherical coordinates | $x=r\sin\theta\cos\phi$ | $y = r\sin\theta\sin\phi$ | $z = r\cos\theta$ | | $r = \sqrt{x^{2}+y^{2}+z^{2}}$ | $\theta = \arccos \frac{z}{r}$ | $\phi = \arctan \frac{y}{x}$ |