Description of and notation for parametric surfaces
Parametric equations are , ,
Planes as parametric surfaces
A plane that contains a point , which corresponds to , and vectors and has equation
Spheres as parametric surfaces
A sphere of radius
and center
is parameterized as
,
,
Functions as parametric surfaces
A surface given by a function
is parametrized as
Identify the surface with the given vector equation.
,
Find a parametric representation for the surface.
The part of the part of the cylinder that lies above the -plane and between the planes and .
The part of the plane that lies inside the cylinder .
Tangent planes
The tangent plane to a surface given by has normal vector
Find an equation of the tangent plane to the given parametric
surface at the specified point.
,
,
;
Surface area
If a smooth parametric surface is given by , and is covered exactly once as covers , then
If , then
For a surface of revolution obtained by rotating
,
about the
-axis,
use
,
,
,
,
so that
Find the area of the part of the surface that lies above the triangle with vertices , , and .