Paraboloid

Paraboloid

12.1 Planes and surfaces

12.2 Graphs and level curves
14.6 Surface integrals

Note: Section numbers refers to Calculus, Early Transcendentals (2nd edition) by Briggs, Cochran, and Gillett.

The surface defined by the equation

is a Paraboloid. The following 3D print shows a polar coordinate parameterization of a portion of the paraboloid above the xy-plane.

The horizontal curves are the z-level sets. They are all are ellipses, and in the special case a=b, they are circles. 

The parametrization is given by 

 

Thus the horizontal curves are curves of fixed z, and the vertical curves are curves of fixed t.