# Rewrite equation using cylindrical and spherical coordinates.

I want to rewrite the equation $z=x^2-y^2$ using cylindrical and spherical coordinates.

The cartesian coordinates are of the form $(x,y,z)$.

The spherical coordinates are of the form $(\rho, \theta, \phi)$ where $\rho=\sqrt{x^2+y^2+z^2}, \theta= arc \tan{\left( \frac{y}{x} \right)}, \phi=arc \cos{\left( \frac{z}{r}\right)}$.

The cylindrical coordinates are defined as follows:

$$x= \rho \sin{\phi} \cos{\theta}, y=\rho \sin{\phi} \sin{\theta}, z=\rho \cos{\phi}$$

So $z=x^2-y^2 \Rightarrow \rho \cos{\phi}=\rho^2 \sin^2{\phi} \cos^2{\theta}-\rho^2 \sin^2{\phi} \sin^2{\theta} \\ \Rightarrow \cos{\phi}=\rho \sin^2{\phi} \cos^2{\theta}-\rho \sin^2{\phi} \sin^2{\theta}=\rho \sin^2{\phi}(\cos^2{\theta}-\sin^2{\theta})=\rho \sin^2{\phi} \cos{2 \theta}$

How could we continue? What does this equation represent?

Also the cylindrical coordinated of a point $(x,y,z)$ are defined by the following relations:

$$x=r \cos{\theta}, y=r \sin{\theta}, z=z$$

So we have $z=r^2 \cos^2{\theta}-r^2 \sin^2{\theta}=r^2(\cos^2{\theta}-\sin^2{\theta})=r^2 \cos{(2 \theta)}$

Do we continue or do we let it like that?

What does the equation represent?