Surface integrals-Important parametrization of a surface

Question

As we know, an elipse is parametrized as $x=ar\cos(\theta)$ and $y=b r\sin(\theta)$, where $r$ is the radius and $a,b$ are some constants.

Well, my question is, how shall I parametrize the surface $z=5-(x^2/2)-y^2$ in terms of the radius($r$) and $\theta$?

Answer 1

We can rewrite the surface as $z=5-r^2$ and put $x=\sqrt{2}rcos{\theta}$ and $y=rsin{\theta}$. We obtain ${\beta(r,\theta)=(\sqrt{2}rcos\theta,rsin\theta,5-r^2)}$