Tips or steps on how to visualise a multivariable function Question. My doubt is about what steps do you take when given a multivariable function like $z^2 = x^2 - y^2$, to know how it looks like in general. For example if it looks like a bunch of waves, or a mountain, or a ellipsoid, or an sphere, etc. I want to improve in being able to visualize most functions, of course not knowing exactly how they are but a general shape or knowledge about it. I know it's needed a lot of practice but I'm asking for tips/steps not another thing.
 A: Functions $w = f(x,y,z)$ that depend on all three variables in a non-trivial way are hard to visualize, since they form a $3$-dimensional$^*$ hypersurface
$$\{(x,y,z,w)\in\mathbb{R}^4 : w = f(x,y,z)\}\subset\mathbb{R}^4.$$
One way to study this graph is by analyzing its level sets, i.e. by fixing $w_0\in\mathbb{R}$ and looking at the (usually implicitly given) set $$\{(x,y,z)\in\mathbb{R} : w_0 = f(x,y,z)\},$$ which can be seen as a $2$-dimensional$^*$ cross section of said $3$-dimensional hypersurface.
which varies when $w\in\mathbb{R}$ is allowed to move. For example, $f(x,y,z) = x^2+y^2+z^2$, when seen as $w_0 = x^2+y^2+z^2$, describes a sphere of radius $\sqrt{w_0}$ for $w_0\geq 0$, and is empty for $w_0<0$.
Functions like $w_0 = x^2-y^2-z^2$ are a bit harder to analyze, so it is often wise to use plotting programs to get an implicit plot. Or, using above method again, one can analyze the level sets of this function by fixing one variable. For example, fixing $x = x_0$, one gets $y^2+z^2 = x_0^2-w_0$, so varying $x$ yields circles of varying radius. Solving for $y = y_0$ yields $x^2-z^2 = w_0+y^2$, which yields hyperbolas. But of course, these are $1$-dimensional$^*$ cross sections of a fixed $2$-dimensional$^*$ cross section of a $3$-dimensional$^*$ hypersurface, so one has to do quite some mental gymnastics to get a picture of the whole thing.
So, my tip would be for you to try sketching some of these cross sections, and then have a look at a $3$d-plot of it with Maple or Wolfram Alpha, to see how these sections connect.
$^*$ The dimensions can of course vary from point to point, as long as the implicit function theorem does not hold.
