[b][size=150]Cylindrical and Spherical Coordinates[/size][/b][br][br]In [math]\mathbb{R}^3[/math], the following are the three most commonly-used coordinate systems:[br][br][list=1][*][b]Rectangular (Cartesian) coordinates[/b]: A point is specified by the coordinates [math](x,y,z)[/math], where [math]x,y[/math] and [math]z[/math] are real numbers indicating the projection of the point onto three axes.[/*][br][*][b]Cylindrical coordinates[/b]: A point is specified by the coordinates [math](r,\theta,z)[/math], where [math]z[/math] is the [math]z[/math]-coordinate in rectangular coordinates and [math](r,\theta)[/math] are the polar coordinates of the projection of the point onto xy plane. Note that here we assume [math]r\geq 0[/math] and [math]0\leq \theta \leq 2\pi[/math].[/*][br][*][b]Spherical coordinates[/b]: A point is specified by the coordinates [math](\rho,\theta,\phi)[/math], where [math]\theta[/math] is the [math]\theta[/math]-coordinate in cylindrical coordinates, [math]\rho[/math] is the distance between the point and the origin, and [math]\phi[/math] is the angle measured from [math]z[/math]-axis to the line segment joining the point to the origin. Note that [math]\rho\geq 0[/math], [math]0\leq \theta\leq 2\pi[/math] and [math]0\leq \phi \leq \pi[/math][/*][/list][br][br]The applet below illustrates these three coordinate systems.[br]

The following are the conversions between these three coordinate systems:[br]([math]\textbf{R}=[/math] Rectangular, [math]\textbf{C}=[/math] Cylindrical, [math]\textbf{S}=[/math] Spherical)[br][br][br][math]\textbf{C}\to\textbf{R}[/math]:[br][br][math]x=r\cos\theta[/math][br][math]y=r\sin\theta[/math][br][math]z=z[/math][br][br][math]\textbf{R}\to\textbf{C}[/math]:[br][br][math]r=\sqrt{x^2+y^2}[/math][br][math]\theta=\tan^{-1}\left(\frac yx\right)[/math], where [math]0\leq \theta \leq 2\pi[/math][br][math]z=z[/math][br][br][math]\textbf{S}\to\textbf{R}[/math]:[br][br][math]x=\rho\sin\phi\cos\theta[/math][br][math]y=\rho\sin\phi\sin\theta[/math][br][math]z=\rho\cos\phi[/math][br][br][math]\textbf{R}\to\textbf{S}[/math]:[br][br][math]\rho=\sqrt{x^2+y^2+z^2}[/math][br][math]\theta=\tan^{-1}\left(\frac yx\right)[/math], where [math]0\leq \theta \leq 2\pi[/math][br][math]\phi=\cos^{-1}\left(\frac {z}{\rho}\right)=\cos^{-1}\left(\frac {z}{\sqrt{x^2+y^2+z^2}}\right)[/math], where [math]0\leq \phi \leq \pi[/math][br][br][math]\textbf{C}\to\textbf{S}[/math]:[br][br][math]\rho=\sqrt{r^2+z^2}[/math][br][math]\theta=\theta[/math][br][math]\phi=\tan^{-1}\left(\frac rz\right)[/math], where [math]0\leq \phi \leq \pi[/math][br][br][math]\textbf{S}\to\textbf{C}[/math]:[br][br][math]r=\rho\sin\phi[/math][br][math]\theta=\theta[/math][br][math]z=\rho\cos\phi[/math][br][br]

[u]Surfaces in cylindrical and spherical coordinates[/u][br][br]Here are some examples of equations of various geometric objects in different coordinate systems:[br][br][list=1][*]A unit sphere centered at the origin[br][list][*]Rectangular coordinates: [math]x^2+y^2+z^2=1[/math][/*][*]Cylindrical coordinates: [math]r^2+z^2=1[/math][/*][*]Spherical coordinates: [math]\rho=1[/math][/*][/list][/*][br][*]A cylinder whose cross-section on xy plane is the unit circle centered at the origin[br][list][*]Rectangular coordinates: [math]x^2+y^2=1[/math][/*][*]Cylindrical coordinates: [math]r=1[/math][/*][*]Spherical coordinates: [math]\rho\sin\phi=1[/math][/*][/list][/*][br][*]A inverted cone whose vertex is the origin and its vertical angle is a right angle[br][list][*]Rectangular coordinates: [math]z=\sqrt{x^2+y^2}[/math][/*][*]Cylindrical coordinates: [math]z=r[/math][/*][*]Spherical coordinates: [math]\phi=\frac{\pi}4[/math][/*][/list][/*][br][/list]