I know how to find the level surfaces for a $2$ variable functions, $z=(x,y)$, by finding the $3$ planes. How would you find the level surfaces for a $3$ variable function, $w=(x,y,z)$. Would you find $4$ traces? $(wxy, wxz, wyz, xyz)$? Here is a sample question of what I'm referring to incase what I am asking is completely incorrect.
Math the functions with the verbal description of the level surfaces.
$$\begin{align} w&=x+2y+3z&\hbox C\\ w&=\sqrt{x+2y+3z}&\hbox F\\ w&=x^2+2y^2+3z^2&\hbox B\\ w&=\sqrt{x^2+2y^2+3z^2}&\hbox B\\ w&=\sqrt{x^2+y^2+z^2}&\hbox E\\ w&=x^2+y^2+z^2&\hbox A\\ w&=x^2-y^2-z^2&\hbox D \end{align}$$
A. a collection of unequally spaced concentric spheres
B. a collection of concentric ellipsoids
C. a collection of equally spaced parallel planes
D. two cones and two collections of hyperboloids
E. a collection of equally spaced concentric spheres
F. a collection of unequally spaced parallel planes
Most of them make sense, except the one with the hyperboloids and cone. (I know it varies when w differs $(w>0, w<0, w=0)$, but would you just have to plug in a bunch of $w$ to see that?
Also can someone explain the plane problems properly. I have a feeling of how to do them, but I would like a set way to look at them and know if equally spaced, or not.
From a quick look these level surfaces maintain their "general" shape no matter the value of $w$. Except for the one answer that is two different surfaces. If you make $ w $ a constant you should be able to recognize the equation of the closed surface/solid or surface it represents.