Converting between Coordinate Systems

Dot product of Cartesian and Spherical Coordinates

	$\rho$	$\phi$	$z$
$x$	$cos(\phi)$	$-sin(\phi)$	$0$
$y$	$sin(\phi)$	$cos(\phi)$	$0$
$z$	$0$	$0$	$1$

Dot product of cylindrical and Spherical Coordinates

	$r$	$\theta$	$\phi$
$x$	$sin \theta cos\phi$	$cos \theta cos \phi$	$-sin \phi$
$y$	$sin\theta sin\phi$	$cos \theta sin \phi$	$cos \phi$
$z$	$cos \theta$	$-sin \theta$	$0$

Dot product of spherical and cylindrical coordinates

	$\rho$	$\phi$	$z$
$r$	$sin \theta$	$0$	$cos \theta$
$\theta$	$cos \theta$	$0$	$-sin\theta$
$\phi$	$0$	$1$	$0$

Conversion

Rectangular to Cylindrical

$$\rho = \sqrt{ x^2 + y^2 }$$ $$\phi = tan^{-1}(\frac{y}{x})$$ $$z = z$$

Cylindrical to Rectangular

$$x = \rho cos \phi$$ $$y = \rho sin \phi$$ $$z = z$$

Rectangular to Spherical

$$r = \sqrt{x^2 + y^2 + z^2}$$ $$\phi = tan^{-1}(\frac{y}{x})$$ $$\theta = cos^{-1}(\frac{z}{\sqrt{x^2+y^2+z^2}})$$

Spherical to Rectangular

$$x = r sin\theta cos\phi$$ $$y = r sin\theta sin\phi$$ $$z = r cos\theta$$

Cylindrical to Spherical

$$r = \sqrt{\rho^2 + z^2}$$ $$\phi = \phi$$ $$\theta = \cos^{-1} (\frac{z}{r}) = \cos^{-1} (\frac{z}{\sqrt{\rho^2 + z^2}})$$

Spherical to Cylindrical

$$\phi = \phi$$ $$\rho = r sin\theta$$ $$z = r cos\theta$$