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18.02 Multivariable Calculus
Fall
200
7
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V5.
SimplyConnected Regions
1.
The Extended Green's Theorem.
In the work on Green's theorem so far, it has been assumed that the region
R
has as
its boundary a single simple closed curve. But this isn't necessary. Suppose the region
has a boundary composed of several simple closed curves, like the
ones pictured. We suppose these boundary curves C1,.
. . ,
C,
all
lie within the domain where
F
is continuously differentiable. Most
importantly, all the curves must be directed so that the normal
n
points away from
R.
Extended Green's Theorem
With the curve orientations as shown,
In
other words, Green's theorem also applies to regions with several boundary curves, pro
vided that we take the line integral over the complete boundary, with each part of the
boundary oriented so the normal
n
points outside
R.
Proof.
We use subdivision; the idea is adequately conveyed by an exam
ple. Consider a region with three boundary curves as shown. The three cuts
illustrated divide up
R
into two regions
R1
and
R2,
each bounded by a single
simple closed curve, and Green's theorem in the usual form can be applied to
each piece. Letting
B1
and
B2
be the boundary curves shown, we have therefore
(2)
f
F
dr
=
u,
curl
F
dA
h2
F
.
dr
=
IS,,
curl
F
dA
B1
'
I
I1
Add these two equations together. The right sides add up to the right side of
(1). The left sides add up to the left side of (1) (for
m
=
2), since over each of
the three cuts, there are two line integrals taken in opposite directions, which
therefore cancel each other out.
2.
Simplyconnected and multiplyconnected regions.
Though Green's theorem is still valid for a region with "holes" like the ones we just
considered, the relation curl
F
=
0
+
F
=
V
f
is not. The reason for this is as follows.
We are trying to show that
curl
F
=
0
+
for any closed curve lying in
R.
We expect to be able to use Green's
theorem. But if the region has a hole, like the one pictured, we cannot ap
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 Spring '08
 Auroux
 Multivariable Calculus, Vector Calculus, Vector field, simple closed curve

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