Let $\Sigma$ be the surface parameterized by $f(u,v)=(u\cos{v}, u\sin{v},u^2)$ for $u\geq 0$ and $0 \leq v \leq 2\pi$. Let $\Sigma_{r}$ the protion of the surface with $0 \leq u \leq r$. What is $\chi(\Sigma_{R})$ ?
The problem asked previously about the geodesic curvature, which I have wrong for some mistake in the computation. I do not know if it's useful here, but my gess is that it's not.