LESSON 21
Parametric Surfaces and Surface Integrals
READ:
Section 17.4
NOTES:
There is a two dimensional analog of line integrals called surface integrals where the region of integration is a
surface rather than a curve. Just as parametric representations of curves are used to compute line integrals,
surfaces will often be represented parametrically to do surface integrals.
One way to describe a surface is with an explicit function
z
=
f
(
x,y
). For example,
z
=
x
2
+ 2
y
2
is
an explicit representation of an elliptic paraboloid. Not every surface can be described with an explicit
function. For example, the surface of the sphere of radius 1 centered at the origin is described by the
equation
x
2
+
y
2
+
z
2
= 1, but of course that cannot be solved for
z
to give an explicit representation since
z
is not a function of
x
and
y
in this case. To get explicit representations of the surface of the sphere, it
would be necessary to look at two separate equations:
z
=
p
1

x
2

y
2
gives the top half of the sphere and
z
=

p
1

x
2

y
2
gives the bottom half of the sphere.
There is a second way to describe a surface that can be more convenient than an explicit representation. In
a
parametric
representation of a surface, there will be three equations which describe
x
,
y
, and
z
in terms of
two parameters
u
and
v
:
x
=
x
(
u,v
)
y
=
y
(
u,v
)
z
=
z
(
u,v
)
.
The variables
u
and
v
take on values in some region
D
of the
u
,
v
plane, and the corresponding points (
x,y,z
)
trace out a surface in 3space. This parametric representation can also be expressed in vector form as
→
r
(
u,v
) =
x
(
u,v
)
→
ı
+
y
(
u,v
)
→
+
z
(
u,v
)
→
k ,
(
u,v
)
∈
D.
As for the parameterization of a curve, a surface can be parameterized in many diﬀerent ways.
Most of the formulas and computations with surface integrals will be expressed in terms of a vector equation
for the surface. If a surface is given by an explicit function, it is always possible to produce a paramet
ric equation for the surface in a simple fashion. Suppose the surface has equation
z
=
f
(
x,y
). For the
parameters, we will use
x
and
y
themselves. Then the surface has vector equation
→
r
(
x,y
) =
x
→
ı
+
y
→
+
f
(
x,y
)
→
k .
For surfaces not given by an explicit function, it will be necessary to select two parameters that can be used
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '11
 JerryMetzger
 Calculus, Integrals, Surface, T D

Click to edit the document details