How to show $x-y^2+z^2=0$ and $y^2+z^2<4$ is a smooth manifold?

Question

If $L$ is the set in $\mathbb{R}^3$ defined by $x-y^2+z^2=0$ and $y^2+z^2<4$ how would I show that $L$ is a smooth manifold?

Answer 1

Take the determinant of the hessian of your first equation. If determinant equals 0, then you have a smooth manifold