How to convert cartesian inequalities to spherical inequalities?

Question

I have no clue how this went from $x^2+y^2 <=3$ to whatever that other thing is. I'm looking at conversion this and this is all I'm finding:

And when I replace x with that squared and y with that other thing squared, I'm not getting the same result.

Any ideas how to get from those cartesian inequalities to those spherical inequalities? Thanks.

Answer 1

You can just substitute for $x$ and $y$ and apply trigonometric identities. For $x^2+ y^2 \leq 3$ for example:

$$x^2+y^2 = (\rho \sin \varphi \cos \theta)^2 + (\rho \sin \varphi \sin \theta)^2 = \rho^2 \sin^2 \varphi (\cos^2 \theta + \sin^2 \theta)=\rho^2 \sin^2 \varphi \leq 3$$