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Green's Theorem(s) In the Planegeneral conditions
ì G
is a simple, piecewise smooth, closed curve in the plane.
[C is defined
parametrically by
<
Ð>Ñß
+ Ÿ > Ÿ ,
Ð+Ñ o Ð,Ñß
ß <
<
and the curve does not intersect itself anywhere else.]
ì H
is the plane region bounded by C.
D contains the points inside and on C.
ì
+
C is parameterized so that the region D is on the left as the parameter increases from
to
counterclockwise orientation)
,Þ Ð
ì TÐBß CÑß UÐBß CÑ
are defined and have continuous partial derivatives on an open region
that contains D.
[You can think of these functions individually or as the components of a
vector field
(
) =
]
→
J
3
4
5
Bß C
TÐBß CÑ UÐBß CÑ !
Green's Theorem
Tangential Component or CirculationCurl Form).
Ð
'
'
'
G
+
+
.
.>
Ð>Ñ
,
,
>Ñ
J † < o
J †
J
† <
J †
m<
m
.
.> o
ÒBÐ>Ñß CÐ>ÑÓ
Ð>Ñ .> o
Ð>Ñ .> o
'
+
,
<
w
w
< Ð
m<
m
w
w
'
'
'
G
G
G
.B
.B
.=
.=
.
.C
.C
J † X
.= o
ÒTÐBß CÑ
UÐBß CÑ
Ó .= o
ÒTÐBß CÑ
UÐBß CÑ
Ó .
t
dt
t
=
=
'
'
G
w
w
ÒTÐBß CÑB Ð>Ñ UÐBß CÑC Ð>ÑÓ .>
T.B U.C o
G
' '
' '
' '
V
V
V
Ð
Ñ .E o
?<6
†
.E o
Ðf ‚
†
.E
`U
`B
`C
`T
J 5
JÑ 5
Notes:
‡
.=
In the line integral,
,
is the scalar tangential component of the field at
'
G
J † X
J † X
a point on the curve and
is the incremental distance along the curve.
If
is a
.=
J
force
field, the resultant integral represents
done by the field in moving along the curve C.
work
If
is a
field, the resultant integral represents
the curve C.
(It
J
velocity
flow
along
represents
if C is a closed curve as it is for Green's Theoroem.)
circulation
‡ Ð
Ñ
ÐBß CÑ
Þ
?<6
ÐBß CÑÞ
is a function of
on R It is the
component of
It may
`U
`B
`C
`T
5
J
also be shown to equal the "circulation density" at
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 Spring '07
 RickRugangYe
 Math, Vector Calculus

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