level surfaces?

Question

how do you describe level surfaces for f(x,y, z) = x^2 - y^2

Answer 1

A level surface is the set of points where the function has the same value.

For each value v, f(x,y,z) = v if and only if v = x^2 - y^2.

x^2 - y^2 = v is a conic. To see what kind of conic, check out:

http://en.wikipedia.org/wiki/Conic_section

x^2 - y^2 can take on any real value so there is a level surface for every real value v.

z does not enter into the computation of v so for each v the constant surface is a generalized cylinder with primary axis being z and cross section being a conic section.

For the special case of v = 0 = x^2 - y^2, we have:

x^2 = y^2 or x = +/- y

so the level surface for v = 0 consists of the two planes defined by x = y and x = -y.