Coordinate Curves of a Parametrization

Question

If I have a paraboloid

$z = 1 - x^{2} - y^2$

with a parametrization of

$\phi(r,\theta) = (rcos\theta, rsin\theta, 1 - r^2)$

I believe to prove $\phi$ as a parametrization I need $\phi$ to be smooth (continously differentiable) and I need $\phi_r \times \phi_\theta$ to not equal $0$ at all points.

I have done this for my homework, but my professor also asks us to describe coordinate curves of this parametrization. Does anyone know how to do this?

Answer 1

Of course, polar coordinates always has a problem at $r=0$, so you'll have an issue at the vertex of of your paraboloid.

Coordinate curves are the images under $\phi$ of the curves $r=\text{constant}$ and $\theta=\text{constant}$.