level surface

The set of points where a three-variable function $f(x,y,z)$ equals a fixed output — a surface floating in space.

The level surface of $f(x,y,z)$ at value $k$ is $\{(x,y,z): f(x,y,z)=k\}$ . Unlike a level curve, a level surface can't be produced by slicing the graph of $f$ with a plane, because the graph of a three-variable function would need a fourth dimension to draw — instead, a level surface IS the entire object being examined, the full set of inputs producing one fixed output. Different values of $k$ produce non-intersecting level surfaces, for the same reason different level curves never cross.

Examples

The level surface of $f(x,y,z)=x^2+y^2+z^2$ at $k=4$ is the sphere $x^2+y^2+z^2=4$ , radius 2.