No possible Level Surface?

Question

Given a function of three variables, is it possible to not have a level surface at all?

Ex: I'm working on a problem that tells me to describe the level set (level surface) for $p(x,y,z) = e^{-x^2-y^2-4z^2}$. I manipulated the function to be $ln(k)=-x^2-y^2-4z^2$. I'm thinking the level surface must be a type of ellipsoid but I'm having trouble graphing it both by myself and my program doesn't seem to be able to graph it. Am I doing something wrong or is this just an odd case?

Ex: I'm plugging in a number for the constant and trying to graph it. Should I possibly be moving z over and having x and y equal to z?

Answer 1

Level surface are defined as

$$p(x,y,z) = e^{(-x^2-y^2-4z^2)}=k\implies -x^2-y^2-4z^2=\log k<0 \implies k\in(0,1)$$

thus set $-\log k=c\in (0,+\infty)$ and the level surface are expressed by

$$x^2+y^2+4z^2=c$$

that, fixed c, is precisely an ellipsoid. To visualize you can fix $c$ and then consider the plot for $z=0$ and for $x or y=0$.

Answer 2

To answer the conceptual question, it's perfectly possible for a function to not have a level set (more precisely, to have an empty level set) for a partcular value of $k$. This will happen for every $k$ that is in the range of $f$ but not the image of $f$. For example, if $f:\mathbb{R^3}\to\mathbb R$ is given by $f(x,y,z)=(xyz)^2$ then the function will never take on the value $-7$ and so the level set for $k=-7$ is $\emptyset$.

It is not possible for the function to have an empty level set for every $k$ when the function is reasonably defined. The only way to get an empty level set for every $k$ is for the function to be defined on a domain on which the function is undefined at every point. For example, $f(x)=\frac{1}{(x-2)(x-3)}$ defined on $\{2,3\}$. In practice, this never comes up.