Final answer:
The equation xy = 6 in R³ describes a hyperbolic paraboloid, a saddle-shaped surface extending infinitely in all directions.
Step-by-step explanation:
The surface in R³ represented by the equation xy = 6 is a hyperbolic paraboloid. This type of surface has a saddle shape and is not limited in any direction, which means it extends infinitely in all directions within three-dimensional space. The equation specifies that for any point (x, y, z) on the surface, the product of the x and y coordinates must be 6. This also implies that neither x nor y can be zero on the surface. To visualize the surface, one can consider various constant z-sections. For a fixed z, we would see hyperbolas aligned along the xy-plane, and as z changes, these hyperbolas would shift their positions, creating the saddle shape of the hyperbolic paraboloid.