:PROPERTIES: :ID: ea993ee3-14c0-492a-8030-7c89836741c0 :mtime: 20231017013103 :ctime: 20231017013047 :END: #+title: cylindrical polar coordinates #+filetags: :public:project: * Definition Cylindrical polar coordinates are a type of [[id:d02a62fb-1601-48a4-9cf2-bbb2ade26402][Coordinate System (Vector Space)]]. Cylindrical coordinates are given by the triple $(r , \theta , z)$, where - $r$ is the radius from the $z$ axis - $\theta$ is the azimuthal angle from the $x$-axis, - $z$ is the distance from the $xy$ plane. * Non-uniqueness of coordinates We should add some restrictions to make sure that representations are unqiue. We should restrict $0 \le \theta < 2\pi$. We should restrict $0 \le r$. In the case of $r = 0$, the vectors along the $z$-axis have an infinite number of polar representations. Other than the case of $r = 0$, almost every point has one and only one cylindrical polar coordinate representation. * Non-cartesianess Cylindrical polar coordinates are not [[id:785f1d20-1537-4c06-9f24-d82141d2b408][cartesian]]. In order to avoid confusion, you should stick with cartesian coordinates. * Orientation The orientation of the basis vectors can cause a problem. The orientation of $\hat{e}_{r}, \hat{e}_{\theta}$ depends on position. * Relationship to [[id:785f1d20-1537-4c06-9f24-d82141d2b408][cartesian coordinate system]] We can convert between $(x,y,z)$ coordinates and $(r , \theta , z)$ coordinates. | $x = r \cos \theta$ | $y = r \sin \theta$ | $z = z$ | | $r = \sqrt{x^2 + y^2}$ | $\theta = \arctan{\frac{y}{x}}$ | $z = z$ |