3. Green's Theorem
A closed curve is a loop.
A simple, closed curve is a loop that that doesn't
intersect itself.
4. Green's Theorem
A closed curve is a loop.
A simple, closed curve is a loop that that doesn't
intersect itself.
Not Closed
5. Green's Theorem
A closed curve is a loop.
A simple, closed curve is a loop that that doesn't
intersect itself.
Not Closed Closed not Simple
6. Green's Theorem
A closed curve is a loop.
A simple, closed curve is a loop that that doesn't
intersect itself.
Not Closed Closed not Simple Simple and Closed
7. Green's Theorem
A closed curve is a loop.
A simple, closed curve is a loop that that doesn't
intersect itself.
A piecewise smooth
curve is a curve that is
differentiable everywhere
except possibilly at
finitely many points.
Not Closed Closed not Simple Simple and Closed
8. Green's Theorem
A closed curve is a loop.
A simple, closed curve is a loop that that doesn't
intersect itself.
A piecewise smooth
curve is a curve that is
differentiable everywhere
except possibilly at
finitely many points.
Not Closed Closed not Simple Simple and Closed
Simple Closed Pieceswise Smooth
9. Recall the work that is done in a vector field is given
by the formula W =
C(t) = <x(t), y(t)> C'(t) = <x'(t), y'(t)>, with a < t < b.
∫C
F • dC where
Green's Theorem
10. Recall the work that is done in a vector field is given
by the formula W =
The above line integral may be written as two integrals:
C(t) = <x(t), y(t)> C'(t) = <x'(t), y'(t)>, with a < t < b.
F(x, y) = <f(x, y), g(x, y)> = <f(x(t), y(t)), g(x(t), y(t))>
∫C
F • dC = ∫a
F • C' dt =
b
<f(x(t), y(t)), g(x(t), y(t))>•<x'(t), y'(t)> dt∫a
b
∫C
F • dC where
Green's Theorem
11. Recall the work that is done in a vector field is given
by the formula W =
The above line integral may be written as two integrals:
C(t) = <x(t), y(t)> C'(t) = <x'(t), y'(t)>, with a < t < b.
F(x, y) = <f(x, y), g(x, y)> = <f(x(t), y(t)), g(x(t), y(t))>
∫C
F • dC = ∫a
F • C' dt =
b
<f(x(t), y(t)), g(x(t), y(t))>•<x'(t), y'(t)> dt∫a
b
f(x(t), y(t)) x'(t)dt + g(x(t), y(t)) y'(t)dt∫a
b
∫a
b
∫C
F • dC where
Green's Theorem
=
12. Recall the work that is done in a vector field is given
by the formula W =
The above line integral may be written as two integrals:
C(t) = <x(t), y(t)> C'(t) = <x'(t), y'(t)>, with a < t < b.
F(x, y) = <f(x, y), g(x, y)> = <f(x(t), y(t)), g(x(t), y(t))>
∫C
F • dC = ∫a
F • C' dt =
b
<f(x(t), y(t)), g(x(t), y(t))>•<x'(t), y'(t)> dt∫a
b
f(x(t), y(t)) x'(t)dt + g(x(t), y(t)) y'(t)dt or that∫a
b
∫a
b
f(x, y)dx + g(x, y)dy =∫C
∫C
∫C
F • dC where
Green's Theorem
=
where dx = x'(t)dt, dy = y'(t)dt.
∫C
F • dC = ∫fdx + gdy
C
13. Green's Theorem
A simple, closed piecewise smooth curve encloses a
simply connected region (one piece without holes).
14. Green's Theorem
A simple, closed piecewise smooth curve encloses a
simply connected region (one piece without holes).
C
R
15. Green's Theorem
A simple, closed piecewise smooth curve encloses a
simply connected region (one piece without holes).
C
Green's Theorem: Let C be a simple, closed
piecewise smooth curve parameterized counter
clockwisely. Let R be the the region enclosed by C.
R
16. Green's Theorem
A simple, closed piecewise smooth curve encloses a
simply connected region (one piece without holes).
C
Green's Theorem: Let C be a simple, closed
piecewise smooth curve parameterized counter
clockwisely. Let R be the the region enclosed by C.
Let f(x, y) and g(x, y) be two real valued functions
with continuous partial derivatives over R.
R
17. Green's Theorem
A simple, closed piecewise smooth curve encloses a
simply connected region (one piece without holes).
C
Green's Theorem: Let C be a simple, closed
piecewise smooth curve parameterized counter
clockwisely. Let R be the the region enclosed by C.
Let f(x, y) and g(x, y) be two real valued functions
with continuous partial derivatives over R. Then
∫C
f(x, y)dx + g(x, y)dy = ∫R
∫gx – fy dA
R
18. Green's Theorem
A simple, closed piecewise smooth curve encloses a
simply connected region (one piece without holes).
C∫C
f(x, y)dx + g(x, y)dy = ∫R
∫gx – fy dA
RGreen's theorem converts line
integrals of vector fields over C
into double integrals over R
Green's Theorem: Let C be a simple, closed
piecewise smooth curve parameterized counter
clockwisely. Let R be the the region enclosed by C.
Let f(x, y) and g(x, y) be two real valued functions
with continuous partial derivatives over R. Then
19. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
Green's Theorem
(1, 0)
(1, 1)
C
R
20. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
∫C
F • dC
Green's Theorem
(1, 0)
(1, 1)
Given that F(x, y) = <xy, 2x>
the work is
C
R
21. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
(1, 0)
(1, 1)
Given that F(x, y) = <xy, 2x>
the work is
C
R
22. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
(1, 0)
(1, 1)
Given that F(x, y) = <xy, 2x>
the work is
C
∫R
= ∫
R
d(xy)
dy
d2x
dx dA
23. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
(1, 0)
(1, 1)
Given that F(x, y) = <xy, 2x>
the work is
C
∫R
= ∫
R
d(xy)
dy
d2x
dx dA ∫ y=0
= ∫ 2 – x dy dx
x
x=0
1
24. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
(1, 0)
(1, 1)
Given that F(x, y) = <xy, 2x>
the work is
C
∫R
= ∫
R
d(xy)
dy
d2x
dx dA ∫ y=0
= ∫ 2 – x dy dx
x
x=0
1
∫ y=0
= 2y – xy| dx
x
x=0
25. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
(1, 0)
(1, 1)
Given that F(x, y) = <xy, 2x>
the work is
C
∫R
= ∫
R
d(xy)
dy
d2x
dx dA ∫ y=0
= ∫ 2 – x dy dx
x
x=0
1
∫ y=0
= 2y – xy| dx
x
x=0
∫= 2x – x2
dx
x=0
1
= x2
– x3
/3 |
x=0
1
26. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
triangle as shown. Find the amount of work done
using Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
(1, 0)
(1, 1)
Given that F(x, y) = <xy, 2x>
the work is
C
∫R
= ∫
R
d(xy)
dy
d2x
dx dA ∫ y=0
= ∫ 2 – x dy dx
x
x=0
1
∫ y=0
= 2y – xy| dx
x
x=0
∫= 2x – x2
dx
x=0
1
= x2
– x3
/3 |
x=0
1
= 2/3
27. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
unit circle. Find the amount of work done using
Green's theorem.
Green's Theorem
28. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
unit circle. Find the amount of work done using
Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
The work done is
29. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
unit circle. Find the amount of work done using
Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
The work done is
∫= ∫ d(xy)
dy
d2x
dx dA
30. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
unit circle. Find the amount of work done using
Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
The work done is
∫
R
= ∫ d(xy)
dy
d2x
dx dA ∫= ∫ 2 – x dA
31. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
unit circle. Find the amount of work done using
Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
The work done is
∫
R
= ∫ d(xy)
dy
d2x
dx dA ∫= ∫ 2 – x dA
Switch to polar form,
32. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
unit circle. Find the amount of work done using
Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
The work done is
∫
R
= ∫ d(xy)
dy
d2x
dx dA ∫= ∫ 2 – x dA
Switch to polar form,
r=0
∫= ∫ [2 – rcos(θ)]*rdrdθ
1
θ=0
2π
33. Example: A particle travels subjected to the force
F(x, y) = <xy, 2x> counter-clockwisely around the
unit circle. Find the amount of work done using
Green's theorem.
∫C
F • dC ∫C
= xy dx + 2xdy
Green's Theorem
The work done is
∫
R
= ∫ d(xy)
dy
d2x
dx dA ∫= ∫ 2 – x dA
Switch to polar form,
r=0
∫= ∫ [2 – rcos(θ)]*rdrdθ = 4π
1
θ=0
2π
