Yesterday, I asked a question about the critic points of the surface
$$z = (x^2 + y^2)e^{-(x^2 + y^2)}$$ and my question was if I had a easier way to classify the critic points of this surfaces without calculating the determinant of the Hessian matrix. Fortunately, I've find out that surface was a revolution surface, which made my job of classifying the points much easier. Unfortunately, I couldn't conclude it by myself, since I do not know (if there is one) the general formula of a rotation surface around the $z$-axis ($y$ or $x$ too).
I couldn't find much about it in the calculus books that I know and searching about it, I didn't make much progress too.
So, is there any general formula for a surface of revolution?
Thanks in advance!
If the surface is given by $z=f(x^2+y^2)$, then it is a surface of revolution around the $z$ axis because its level curves are circles.
However, it may not be as easy to see that $z=f(x^2+y^2)$ as it is in your example. Moreover, the surface may have some other axis of revolution, which will make it even harder to spot.