Parameterizing a circle Suppose I wanted to parameterize $S = \{x^2 + y^2 \leq 1, 0 \leq z \leq 1\}$. 
Would this parameterization be given by $G(u,v) = (u \cos(v), u \sin(v), u)$ for $0 \leq u \leq 1$ and $0 \leq v \leq 2\pi$?
The confusion I am having is with regards to $x^2 + y^2 \leq 1$. If this were simply one, it would be parameterizing the unit circle, but being $\leq 1$ throws me off here.
Any help appreciated.
Following suggestions, I assume this would then be $G(u,v, z) = (u \cos(v), u \sin(v), z)$ for $0 \leq u \leq 1$ and $0 \leq v \leq 2\pi, 0 \leq z \leq 1$

In order to parameterize the solid $S$ in $\mathbb{R}^3$, you have to have 3 parameters. In your current parameterization, your $z$ coordinate depends also on the radius of the circle. This would cause it have a cone-like shape. If you wanted to parameterize the solid it would be something like:
$g(u,v,z) = (u\cos{v},u\sin{v},z)$ where $0\leq z\leq1$ and $u,v$ have the same restrictions.
If you wanted to parameterize only the surface of the cylinder, than, we can parameterize in 2 variables with the parameterization: 
$g(u,v) = (\cos{v},\sin{v},u)$ where $v$ has the usual restriction and $u$ acts as the $z$-coordinate so its restriction is similar to $z$, namely: $0\leq u\leq 1$.
