Q 10. - Calculus

Question

To convert from spherical to rectangular coordinates:

$x=-----, y=-----, z=-----$

Step-by-Step Solution

Answer

$x=\rho \sin \varphi \cos \theta , y=\rho \sin \varphi \sin \theta , z=\rho \cos \varphi$

Step 1: Given Information

Given $\left(x,y,z\right)$ are rectangular coordinates and $\left(\rho ,\theta ,\varphi \right)$ are spherical coordinates.

Step 2: Simplification

$\left(x,y,z\right)$ are rectangular coordinates and $\left(\rho ,\theta ,\varphi \right)$ are spherical coordinates.

The relation is as below:

$x=\rho \sin \varphi \cos \theta , y=\rho \sin \varphi \sin \theta , z=\rho \cos \varphi$

Hence,
$x=\rho \sin \varphi \cos \theta$

$y=\rho \sin \varphi \sin \theta$

and $z=\rho \cos \varphi$