1156
CHAPTER 14
.
.
Vector Calculus
1442
14.4
GREEN’S THEOREM
HISTORICAL
NOTES
George Green (1793–1841)
English mathematician who
discovered Green’s Theorem.
Green was selftaught, receiving
only two years of schooling before
going to work in his father’s
bakery at age 9. He continued to
work in and eventually took over
the family mill while teaching
himself mathematics. In 1828, he
published an essay in which he
gave potential functions their
name and applied them to the
study of electricity and
magnetism. This littleread essay
introduced Green’s Theorem and
the socalled Green’s functions
used in the study of partial
differential equations. Green was
admitted to Cambridge University
at age 40 and published several
papers before his early death
from illness. The significance of his
original essay remained unknown
until shortly after his death.
In this section, we develop a connection between certain line integrals around a closed curve
in the plane and double integrals over the region enclosed by the curve. At first glance, you
might think this a strange and abstract connection, one that only a mathematician could care
about. Actually, the reverse is true; Green’s Theorem is a significant result with farreaching
implications. It is of fundamental importance in the analysis of fluid flows and in the theories
of electricity and magnetism.
Before stating the main result, we briefly define some terminology. Recall that for a
plane curve
C
defined parametrically by
C
= {
(
x
,
y
)

x
=
g
(
t
)
,
y
=
h
(
t
)
,
a
≤
t
≤
b
}
,
C
is closed if its two endpoints are the same, i.e., (
g
(
a
)
,
h
(
a
))
=
(
g
(
b
)
,
h
(
b
)). A curve
C
is
simple
if it does not intersect itself, except at the endpoints. We illustrate a simple closed
curve in Figure 14.28a and a closed curve that is not simple in Figure 14.28b.
y
x
C
y
x
C
FIGURE 14.28a
FIGURE 14.28b
Simple closed curve
Closed, but not simple curve
We say that a simple closed curve
C
has
positive orientation
if the region
R
enclosed by
C
stays to the left of
C
, as the curve is traversed; a curve has
negative orientation
if the
region
R
stays to the right of
C
. In Figures 14.29a and 14.29b, we illustrate a simple closed
curve with positive orientation and one with negative orientation, respectively.
y
x
C
R
y
x
C
R
FIGURE 14.29a
FIGURE 14.29b
Positive orientation
Negative orientation
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1443
SECTION 14.4
.
.
Green’s Theorem
1157
We use the notation
C
F
(
x
,
y
)
·
d
r
to denote a line integral along a simple closed curve
C
oriented in the positive direction.
y
a
b
x
A
R
B
C
2
:
y
g
2
(
x
)
C
1
:
y
g
1
(
x
)
FIGURE 14.30a
The region
R
We can now state the main result of the section.
THEOREM 4.1
(Green’s Theorem)
Let
C
be a piecewisesmooth, simple closed curve in the plane with positive
orientation and let
R
be the region enclosed by
C
, together with
C
. Suppose that
M
(
x
,
y
) and
N
(
x
,
y
) are continuous and have continuous first partial derivatives in
some open region
D
, with
R
⊂
D
. Then,
C
M
(
x
,
y
)
dx
+
N
(
x
,
y
)
dy
=
R
∂
N
∂
x
−
∂
M
∂
y
dA
.
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 Spring '10
 CARTER
 Manifold, dy, George Green

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