Question

Find an equation in spherical coordinates for the equation given in rectangular coordinates. X^2+y2+z^2-3z=0

Answer 1

We have been given the equation in rectangular coordinates as

`x^2+y^2+z^2-3z=0`

In Spherical coordinates, we know that

`x=\rho \sin \phi \cos \theta\\ y=\rho \sin \phi \sin \theta\\ z=\rho \cos \phi\\ x^2+y^2+z^2=\rho^2`

On substituting these values in the given equation, we get

`(x^2+y^2+z^2)-3z=0`

`\rho^2- 3\rho \cos \phi=0\\ \\ \rho^2= 3\rho \cos \phi\\ \\ \rho = 3 \cos \phi`

Thus, the equation in spherical coordinates is given by

`\rho = 3 \cos \phi`