Math 200, Spring 2010
Handout
#28
Section 171
Green’s Theorem
Green’s Theorem
A
simple curv
e is one that does not intersect itself, except possibly at its endpoints.
A
positively oriented closed curve
is one whose enclosed region lies to the left of
the curve as it is traversed.
Green’s Theorem:
Let
C
be a simple closed curve with a positive orientation.
If
P
and
Q
have
continuous partial derivatives on the region
D
enclosed by
C
, then
C
D
Q
P
Pdx
Qdy
dA
x
y
∂
∂
+
=
−
∂
∂
∫
∫∫
Note:
Other notation used for the line integral on a positively oriented closed curve:
C
Pdx
Qdy
+
∫
integralloop
The symbol
C
∫
integralloop
is used to denote a positive orientation of the closed curve
C
.
D
Pdx
Qdy
∂
+
∫
The symbol
D
∂
is used to represent the positively oriented boundary curve of region
D
.
If
,
P Q
=
F
, then
( )
( )
(
)
( )
( )
(
)
,
,
,
,
b
C
a
C
dx
dy
d
P x t
y t
Q x t
y t
dt
Pdx
Qdy
dt
dt
=
=
+
∫
∫
∫
F
s
i
i
integralloop
integralloop
Note:
Green’s Theorem can be thought of as a Fundamental Theorem of Calculus for double integrals.
Comparing
D
D
Q
P
dA
Pdx
Qdy
x
y
∂
∂
∂
−
=
+
∂
∂
∫∫
∫
with
(
)
(
)
(
)
b
a
F
x dx
F b
F a
=
−
′
∫
,
in both cases, the left side involves an integral of a derivative expression and, in both cases, the right
side involves the values on the original functions (
P
,
Q
, and
F
) only on the boundary of the domain.
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.
 Spring '10
 JAMESDCAMPBELL
 Math, Vector Calculus, Manifold, 2 j, closed curve, Qdy

Click to edit the document details