Let the surface $S \subset \mathbb{R}^3$ be the solutions of the equation $g(x, y, z)$ $ = 1$ where $g(x,y,z)=x^2 +y^2 +4z^2$. Let $U$ be the finite region of S satisfying $z > 0$ and let $C$ be the boundary of $U$. Sketch the surface $S$, the region $U$ and boundary $C$.
So I've trouble visualising this. I know I've to sketch $x^2+y^2+4z^2 = 1$, so I thought I should let $y=0$. Then $x^2+4z^2 = 1$ is an ellipse, which I can draw. Similarly, $y^2+4z^2 = 1$ is an ellipse. Letting $z = 0$ then $x^2+y^2 = 1$a circle of radius $1$. However, I'm unable to put this together in 3-D.
Also, what's the definition of boundary? The set of inequalities that bound that part of the surface?
EDIT: If sketch all three in 3D axis, I get this:
The two ellipses are inside the circle, which is wrong.
EDIT 2:
EDIT 3: Should it look like this?
But in that case I would like to ask what happened to the circle in the $xy$ plane?