Level Sets & Contour Surfaces
Text A contour of a function f(x,y,z) is obtained by setting w = k constant where w = f(x,y,z).
The resulting surface f(x,y,z) = k is a contour in the domain of f.
A set of contours in either the two or three variable case is called a level set. Level sets that have singularities are especially interesting since the hypersurface has critical points at these values of k.
Demos
Contour Surface
 
This demo shows some contour surfaces of the fourdimensional graph of the function f(x,y,z) = x^{4} + 2x^{2}  y^{4} + 2y^{2}  z^{4} + 2z^{2} .

Exercises Use the tapedeck controllers to analyze the different level sets of the function f. What can you say about critical points of the hypersurface f(x,y,z)?
