Introduction:
The
paraboloid is so called because it has parabolic cross sections (see right).
The plot above and the parameterization below describe a circular paraboloid,
because crosssections parallel to the xy plane are circular. It is also
possible to define an elliptic paraboloid whose crosssections are elliptic.
Definition:
Properties:
Partial Derivatives:
Because z is a function of x and y, we can take partial derivatives:

Tangent Planes:
At u = u_{0}, v = v_{0},
the tangent plane to the surface is parameterized by:

Infinitesimal Area:
The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:


Gaussian curvature of the surface.

Surface colored by Gaussian curvature.


Mean Curvature:


Mean curvature of the surface.

Surface colored by Mean curvature.


