Examples of Level Surfaces of 3d functions.

Warning! This is a lot harder than visualizing level curves of 2d functions. The problem is that the "graph" of the function is in 4d. So you often have to play around with functions for a while before you really understand what's going on. Luckily, computers make this possible.

The Ball Function Revisited

ContourPlot3D[x^2+y^2+z^2,{x,-2,2},{y,-2,2},{z,-2,2},ContourStyle->Directive[Opacity[0.8]]]

Manipulate[ContourPlot3D[x^2+y^2+z^2 == n,{x,-2,2},{y,-2,2},{z,-2,2},ContourStyle->Directive[Opacity[0.8]]],{n,0,8}]

A 3d "saddle surface"

ContourPlot3D[x^2+y^2-z^2,{x,-2,2},{y,-2,2},{z,-2,2},ContourStyle->Directive[Opacity[0.8]]]

Manipulate[ContourPlot3D[x^2+y^2-z^2 == n,{x,-2,2},{y,-2,2},{z,-2,2},ContourStyle->Directive[Opacity[0.8]]],{n,0,8}]

A More Interesting Function

ContourPlot3D[x^4 + y^4 + z^4 - (x^2 + y^2 + z^2)^2 + 3 (x^2 + y^2 + z^2) , {x, -2, 2}, {y, -2, 2}, {z, -2, 2},ContourStyle -> Directive[ Opacity[0.8]],Contours-> 5]

Manipulate[ContourPlot3D[x^4 + y^4 + z^4 - (x^2 + y^2 + z^2)^2 + 3 (x^2 + y^2 + z^2) == n, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},ContourStyle -> Directive[ Opacity[0.8] ]],{n,0,10}]