Question

To convert from cylindrical to spherical coordinates:

$\rho =----,\theta =----,\varphi =----$

Short answer

$\rho =\sqrt{r^2+z^2},\theta =\theta ,\varphi ={\tan }^{-1}\frac{r}{z}$

Step-by-step solution

Given Information

Given $\left(\rho ,\theta ,\varphi \right)$are spherical coordinates and $(r,\theta ,z)$are cylindrical coordinates.

Simplification

The relation between two is:

$\rho =\sqrt{r^2+z^2},\theta =\theta ,\tan \varphi =\frac{r}{z}$

Hence,

$\rho =\sqrt{r^2+z^2}$

$\theta =\theta$

and$\varphi ={\tan }^{-1}\frac{r}{z}$