9.12%20Green

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9.12 Greens Theorem

• Topics to review
• Double integral (9.10)
• Double integral in polar coordinate (9.11)

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Double integral
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Double integral
b
R
a
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Example 1 Evaluate (9.10)
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Positive Direction
We say the positive direction around a simple
closed curve C is that the direction a person
must walk on C, in order to keep the region R
bounded by C to the left.
Positive Direction
Negative Direction
counterclockwise
clockwise
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THEOREM 9.13 Greens Theorem in the plane
NOTE 1)Dirction 2)Relation double 3)conditions
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THEOREM 9.13 Greens Theorem in the plane
How to convert
Double integral
Line integral
George Green (1793-1841) an English mathematician
and physicist.
One of the most important theorem WHY????
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Example 1 using Greens Theorem
(1,1)
R
or -
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Example 1 using Greens Theorem
R
or -
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HW
HW
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• Monday
• HW 9.89.9
• Quiz 9.7 9.8 9.9

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Groups
Write an equivalent statement to statement 1
( as much as you can )
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Example 4 use Greens
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Greens Theorem (Region with Holes)
R
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Greens Theorem (Region with Holes) Proof
d1
R1
d3
d5
d6
R2
d4
d2
E1 closed curve bounding R1 d1 U d5 U
d3 U d6
E2 closed curve bounding R2 d2 U (-d6)
U d4 U (-d5)
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OFFICE HOURS
SAT SUN MON TUE WED
1220 110 1220 110 1220 110
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Example 5
Where C C1 U C2 is the boundary of the blue
region R
R
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FACT replace a complicated closed path with a
path that is simpler
Proof
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Example 6 use Greens (example 4)
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Example 6 use Greens (example 4)
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HW
HW
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HW
HW
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HW
HW