I am trying to show that
$$\{(x, y, z)\mid z=x^2+y^2, z\leq 2\}$$
is a manifold.
I am trying to express it as a set where $f(x, y, z)\geq 0$ for some smooth $f$ on an open set, but as the set $\{(x, y, z)\mid z=x^2+y^2\}$ is closed I don't know how to approach this.
This is actually a manifold with boundary. Denote the set $\{ (x,y,z) \big| z =x^2 +y^2, z\leq 2 \}$ by $S$. It's clear that $(x,y,z) \in S$ if and only if $(x,y) \in B $ where $B$ denotes the closed ball whose center is the origin and has radius $\sqrt{2}$ in $R^2$. Define $\phi : B \longrightarrow S$ by $(x,y) \mapsto (x,y,x^2+y^2)$. $\phi$ has the projection $p : S \longrightarrow B$ as its inverse. So you see $S$ and $B$ are homeomorphic and $\phi$ already gives you the chart for $\text{int}S$. The reason why $S$ is a manifold with boundary is then the same as $B$.