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