Final answer:
The equation x^2 y^2 z^2 = 1 defines a three-dimensional surface known as a rectangular hyperbolic paraboloid, which is a three-dimensional analog to hyperbolas. It consists of two disconnected pieces, one above and one below the xy-plane, and cannot be defined at x=0 or y=0.
Step-by-step explanation:
The equation x^2 y^2 z^2 = 1 defines a surface in three-dimensional space where the product of the squares of the coordinates equals 1. This surface is not a common geometric shape like a sphere or a cylinder. Instead, the surface is a three-dimensional analog to hyperbolas, known as a rectangular hyperbolic paraboloid. This can be seen by rewriting the equation in terms of one of the variables, such as solving for z:
This implies that for every combination of x and y that are not zero, there is a corresponding positive and negative z value. However, the surface is undefined at x=0 or y=0 because you cannot divide by zero. The surface consists of two disconnected pieces, one with positive z values and one with negative z values.