832
CHAPTER 15
Vector Fields
23.
Review Example 5 in Section 15.2 in which it was shown that
∂
∂
y
±
−
y
x
2
+
y
2
²
=
∂
∂
x
±
x
x
2
+
y
2
²
,
for all
(
x
,
y
)
6=
(
0
,
0
)
. Why does this result, together with
that of Exercise 22, not contradict the ±nal assertion in the
remark following Theorem 1?
24.
3
(Winding number)
Let
C
be a piecewise smooth curve in
the
xy
-plane which does not pass through the origin. Let
θ
=
θ(
x
,
y
)
be the polar angle coordinate of the point
P
=
(
x
,
y
)
on
C
, not restricted to an interval of length 2
π
,
but varying continuously as
P
moves from one end of
C
to
the other. As in Example 5 of Section 15.2, it happens that
∇
θ
=−
y
x
2
+
y
2
i
+
x
x
2
+
y
2
j
.
If, in addition,
C
is a closed curve, show that
w(
C
)
=
1
2
π
I
C
xdy
−
ydx
x
2
+
y
2
has an integer value.
w
is called the
winding number
of
C
about the origin.
15.5
Surfaces and Surface Integrals
This section and the next are devoted to integrals of functions de±ned over surfaces in
3-space. Before we can begin, it is necessary to make more precise just what is meant
by the term “surface.” Until now we have been treating surfaces in an intuitive way,
either as the graphs of functions
f
(
x
,
y
)
or as the graphs of equations
f
(
x
,
y
,
z
)
=
0.
A smooth curve is a
one-dimensional
object because points on it can be located
by giving
one coordinate
(for instance, the distance from an endpoint). Therefore, the
curve can be de±ned as the range of a vector-valued function of one real variable. A
surface is a
two-dimensional
object;points on it can be located by using
two coordinates
,
and it can be de±ned as the range of a vector-valued function of two real variables. We
will call certain such functions parametric surfaces.
Parametric Surfaces
DEFINITION
4
A
parametric surface
in 3-space is a continuous function
r
de±ned on some
rectangle
R
given by
a
≤
u
≤
b
,
c
≤
v
≤
d
in the
u
v
-plane and having values in
3-space:
r
(
u
,v)
=
x
(
u
i
+
y
(
u
j
+
z
(
u
k
,(
u
in
R
.
Figure 15.16
A parametric surface
S
de±ned on parameter region
R
.The
contour curves
on
S
correspond to the
rulings of
R
x
y
z
a
b
c
d
v
u
r
(
u
(
u
R
S
Actually, we think of the
range
of the function
r
(
u
as being the parametric surface.
It is a set
S
of points
(
x
,
y
,
z
)
in 3-space whose position vectors are the vectors
r
(
u
for
(
u
in
R
. (See Figure 15.16.) If
r
is one-to-one,then the surface does not intersect
itself. In this case
r
maps the boundary of the rectangle
R
(the four edges) onto a curve
in 3-space, which we call the
boundary of the parametric surface
. The requirement