16.4 Green`s Theorem
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Transcript 16.4 Green`s Theorem
Chapter 16 – Vector Calculus
16.4 Green’s Theorem
Objectives:
Understand Green’s
Theorem for various regions
Apply Green’s Theorem in
evaluating a line integral
16.4 Green’s Theorem
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Introduction
George Green
(1793 – 1841)
George Green worked in his father’s bakery
from the age of 9 and taught himself
mathematics from library books.
In 1828 published privately what we know
today as Green’s Theorem. It was not widely
known at the time.
At 40, Green went to Cambridge and died 4
years after graduating.
In 1846, Green’s Theorem was published
again.
Green was the first person to try to formulate
a mathematical theory of electricity and
magnetism.
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Introduction
Green’s Theorem gives the relationship between a
line integral around a simple closed curve C and a
double integral over the plane region D bounded
by C.
◦ We assume that D
consists of all points
inside C as well as
all points on C.
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Introduction
In stating Green’s Theorem, we use the convention:
◦ The positive orientation of a simple closed curve C
refers to a single counterclockwise traversal of C.
◦ Thus, if C is given
by the vector
function r(t),
a ≤ t ≤ b, then the
region D is always
on the left as the
point r(t) traverses
C.
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Green’s Theorem
Let C be a positively oriented, piecewise-smooth,
simple closed curve in the plane and let D be the
region bounded by C.
◦ If P and Q have continuous partial derivatives
on an open region that contains D, then
Q P
P
dx
Q
dy
dA
C
D x y
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Green’s Theorem Notes
The notation
C
P dx Q dy or
C
P dx Q dy
is sometimes used to indicate that the line
integral is calculated using the positive
orientation of the closed curve C.
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Green’s Theorem Notes
Another notation for the positively oriented
boundary curve of D is ∂D.
So, the equation in Green’s Theorem can
be written as equation 1:
Q P
dA
P
dx
Q
dy
D x y D
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Green’s Theorem and FTC
Green’s Theorem should be regarded as the counterpart of
the Fundamental Theorem of Calculus (FTC) for double
integrals.
Compare Equation 1 with the statement of the FTC Part 2
(FTC2), in this equation:
b
a
F '( x) dx F (b) F (a)
In both cases,
◦ There is an integral involving derivatives (F’, ∂Q/∂x, and ∂P/∂y)
on the left side.
◦ The right side involves the values of the original functions
(F, Q, and P) only on the boundary of the domain.
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Area
Green’s Theorem gives the following
formulas for the area of D:
1
A xdy ydx xdy ydx
2C
C
C
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Example 1 – pg. 1060 #6
Use Green’s Theorem to evaluate the line integral
along the given positively oriented curve.
cos ydx x
2
sin ydy
C
C is the rectangle with verticies
0, 0 , 5, 0 , 5, 2 , and 0, 2
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Example 2 – pg. 1060 #8
Use Green’s Theorem to evaluate the line integral
along the given positively oriented curve.
xe
2 x
dx x 2 x y dy
4
2
2
C
C is the boundary of the region between the circles
x y 1 and x y 4
2
2
2
2
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Example 3 – pg. 1060 #12
Use Green’s Theorem to evaluate F dr .
C
Check the orientation of the curve before
applying the theorem.
F( x, y ) y cos x, x 2 y sin x
2
2
C is the triangle from
0, 0 to 2, 6 to 2, 0 to 0, 0
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