Let the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the boundary curve of $U$. Sketch $S, U $ & $ C$.
I've tried to use slices to draw $S$. So $z = xy$. Letting $x = 1$, then S in the $zy$-plane is the whole plain. Letting $x = 1$ then the $zy$-plane $S$ is the whole plane again; and letting $z = 1$ then $xy$-plane is occupied a series of hyperbolas. But how do I graph these together? Also, could someone please explain the concept of a boundary to me? Guessing - is it the set of inequalities that bound a graph?
EDIT: Does it look like this?
