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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?

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Your transformations seem fine. You seem confused about what the transformations are though, since you ask

What does this equation represent?

It represents exactly the same as in Cartesian coordinates. By making the coordinate transformation, you have simply changed the representation of the mathematical structure; not the structure itself.

The equation can be visualized here.

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