Hello,

I try to solve this problem:

$\displaystyle h(t)=(h_1(t),h_3(t))$ some smooth path in $\displaystyle \mathbb{R}^2$, $\displaystyle t \in (0,1), |h'(t)|=1, h_1(t)>0$ and $\displaystyle g(t,\alpha)=(h_1(t)*cos\alpha, h_1(t)*sin\alpha, h_3(t)), \alpha \in [0,2\pi)$ the corresponding surface of revolution.

$\displaystyle h_3$ is a diffeomorphism onto its image.

Find $\displaystyle U \subset \mathbb{R}^3$ and a function f:U->$\displaystyle \mathbb{R}$,s.t. 0 is a regular value of f and Im(g)=$\displaystyle f^{-1}(0)$

I have try to define some function f on a set U, s.t. the Points on the surface are mapped to 0. But i couldn't find it.

My results are quite little, we know that the length of the curve h is 1. Therefore we can find a open set U, which contains the surface.

I try it with some sinus and cosinus relations, like $\displaystyle sin^2+cos^2=1$, but it doesn't help. because in equations like f(x,y,z)=x^2+y^2-... we get $\displaystyle c_1^2$

How can i define f reasonable?

Regards