Is there a function of two variables whose z=0 level curve consists of the circles x^2+y^2=4 and x^2+y^2=10? If so, what is an example? If not, why not?

Question

anonymous · Answer

@Callisto

anonymous · Answer

@genius12

anonymous · Answer

$$f(x,y)=\left\{
\begin{array}{lr}
0, & (x^2+y^2==4) \vee (x^2+y^2==10) \
1, & (x^2+y^2\not=4) \wedge (x^2+y^2\not=10)
\end{array}
\right.$$
Take a look at $f(x)=x^4-4x^2+k,~~~0<k<4$.

anonymous · Answer

The first one is almost cheating, but it works.
The second seems that , with some manipulation, and the introduction of a second independent variable $y$, it could possible form two circles on the intersection of $z=0$ and $z=f(x,y)$.

anonymous · Answer

Yeah the first one definitely seems that way

anonymous · Answer

but i can see how that works though

anonymous · Answer

thank you