How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

Question

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above.

I don't know how to do it and would like some help.

Answer 1

For each fixed value of $z$, you have a circle of radius $|z|$ on $x$ and $y$. So, as $z\in[0,1]$, you can parametrize $$ x=z\,\cos t\,\ \ y=z\,\sin t,\ \ z\in[0,1],\ t\in[0,2\pi). $$

If you want to parametrize the volume inside the surface, you can use the same idea to parametrize $$ x=r\,\cos t\,\ \ y=r\,\sin t,\ \ 0\leq r\leq z,\ \ z\in[0,1],\ t\in[0,2\pi). $$

Answer 2

If you want a 1-1 parameter inaction, you can take your domain the closed unit disk and define $(x,y)\mapsto (x,y,\sqrt{x^2+y^2})$.