Spherical Coordinates

Cartesian to Spherical

\[\begin{align*} x = r \sin{\theta} \cos{\phi} \\ y = r \sin{\theta} \sin{\phi} \\ z = r \cos{\theta} \end{align*}\] \[\begin{align*} \hat{x} = \sin{\theta} \cos{\phi} \, \hat{r} + \cos{\theta} \cos{\phi} \, \hat{\theta} - \sin{\phi} \, \hat{\phi} \\ \hat{y} = \sin{\theta} \sin{\phi} \, \hat{r} + \cos{\theta} \sin{\phi} \, \hat{\theta} + \cos{\phi} \, \hat{\phi} \\ \hat{z} = \cos{\theta} \, \hat{r} - \sin{\theta} \, \hat{\theta} \end{align*}\]

Spherical to Cartesian

\[\begin{align*} r &= \sqrt{x^2 + y^2 + z^2} \\ \theta &= \tan^{-1} \left( \frac{\sqrt{x^2 + y^2}}{z} \right) \\ \phi &= \tan^{-1} \left( \frac{y}{x} \right) \end{align*}\] \[\begin{align*} \hat{r} &= \sin{\theta} \cos{\phi} \, \hat{x} + \sin{\theta} \sin{\phi} \, \hat{y} + \cos{\theta} \, \hat{z} \\ \hat{\theta} &= \cos{\theta} \cos{\phi} \, \hat{x} + \cos{\theta} \sin{\phi} \, \hat{y} - \sin{\theta} \, \hat{z} \\ \hat{\phi} &= - \sin{\phi} \, \hat{x} + \cos{\phi} \, \hat{y} \end{align*}\]

Cylindrical Coordinates

Cartesian to Cylindrical

\[\begin{align*} x = s \cos{\phi} \\ y = s \sin{\phi} \\ z = z \end{align*}\] \[\begin{align*} \hat{x} &= \cos{\phi} \, \hat{s} - \sin{\phi} \, \hat{\phi} \\ \hat{y} &= \sin{\phi} \, \hat{s} + \cos{\phi} \, \hat{\phi} \\ \hat{z} &= \hat{z} \end{align*}\]

Cylindrical to Cartesian

\[\begin{align*} s &= \sqrt{x^2 + y^2} \\ \phi &= \tan^{-1} \left( \frac{y}{x} \right) \\ z &= z \end{align*}\] \[\begin{align*} \hat{s} &= \cos{\phi} \, \hat{x} + \sin{\phi} \, \hat{y} \\ \hat{\phi} &= - \sin{\phi} \, \hat{x} + \cos{\phi} \, \hat{y} \\ \hat{z} &= \hat{z} \end{align*}\]