Section 17.6
Parametric Surfaces
“Parametric Equations Defining Surfaces”
In Chapter 14, we discussed how curves could be represented in space
through the use of parametric equations in one variable. In this section,
we consider the same idea for surfaces.
1.
Parametric Surfaces
Suppose that
vector
r
(
u, v
) =
x
(
u, v
)
vector
i
+
y
(
u, v
)
vector
j
+
z
(
u, v
)
vector
k
is a vector valued
function defined on a region
D
in 3space. For each value of (
u, v
), we
associate the point (
x
(
u, v
)
, y
(
u, v
)
, z
(
u, v
)) to the vector
vector
r
(
u, v
). Then
as (
u, v
) vary over
D
, the vector function
vector
r
(
t
) traces out a surface
S
called the parametric surface with parametric equations
vector
r
(
u, v
).
We
illustrate.
Example 1.1.
Identify the surface with parametric equations
vector
r
(
u, v
) =
u
cos (
v
)
vector
i
+
u
sin (
v
)
vector
j
+
u
2
vector
k.
Since
x
=
u
cos (
v
),
y
=
u
sin (
v
) and
z
=
u
2
, at any point on this
surface we have
x
2
+
y
2
=
u
2
=
z
. This is the equation for a parabolic
bowl centered on the
z
axis with vertex at the origin.
Example 1.2.
Identify the surface with parametric equations
vector
rx, ϑ
) =
u
vector
i
+
u
cos (
ϑ
)
vector
j
+
u
sin (
ϑ
)
vector
k.
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 Fall '08
 Stefanov
 Calculus, Equations, Parametric Equations, 2J, Parametric equation, Parametric surface, Vectorvalued function

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