# How do you convert x^2+y^2=z into spherical and cylindrical form?

Aug 8, 2016

Spherical form+ $r = \cos \phi {\csc}^{2} \theta$.
Cylindrical form: $r = z {\csc}^{2} \theta$

#### Explanation:

The conversion formulas,

Cartesian $\to$ spherical::

$\left(x , y , z\right) = r \left(\sin \phi \cos \theta , \sin \phi \sin \theta , \cos \phi\right) , r = \sqrt{{x}^{2} + {y}^{2} + {z}^{2}}$

Cartesian $\to$ cylindrical:

$\left(x , y , z\right) = \left(\rho \cos \theta , \rho \sin \theta , z\right) , \rho = \sqrt{{x}^{2} + {y}^{2}}$

Substitutions in ${x}^{2} + {y}^{2} = z$ lead to the forms in the answer.

Note the nuances at the origin:

r = 0 is Cartesian (x, y, z) = (0, 0, 0). This is given by

$\left(r , \theta , \phi\right) = \left(0 , \theta , \phi\right)$, in spherical form, and

$\left(\rho , \theta , z\right) = \left(0 , \theta , 0\right)$, in cylindrical form...
.