
\vspace*{.2in}
\noindent{\bf Supplementary problem 1: }
Since $C$ is defined implicitly as the $0$-level set of a function
$f:\R^3\to\R$, we will have that $C$ is a codimension one (i.e. dimension 2)
$C^1$ submanifold of $\R^3$ at all points where $f'$ has maximal rank $1$.
Specifically,
$$
f'(x,y,z) = (2x,-2y,-2z)
$$
has rank one unless all three entries vanish---i.e. unless $(x,y,z) = (0,0,0)$
So $C$ is a two dimensional submanifold of $\R^3$ except at the origin (which,
incidentally, actually does lie in $C$).  

The picture you draw should show a circular cone generated by rotating the
line $x=y$ about the $x$ axis.
