I want to find the unit normal for a surface of revolution of the form
$F(t,s)=(r(t)cos(s),r(t)sin(s),z(t))$
where $\gamma(t)=(r(t),z(t))$ is a curve with unit speed and $r(t)>0$.
I know that if I have a submanifold $M$ given by some smooth $f$, s.t. $f^{-1}(0)=M$, then the unit normal is $\dfrac{grad \ f}{\Vert grad \ f \Vert}$ so I want to find such an $f$ for the above situation but I can't fix it. Is there a way to do it or do I need to try a different approach?
Referring to, for example, this How to find the normal on a surface of revolution? I can calculate the unit normal to be $(-z'(t)cos(s),-z'(t) sin(s), r'(t)$.(Is that correct? ) But I'm looking for a way to solve this using my above approach.