Math 251-copyright Joe Kahlig, 15A
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Section 14.4: Green’s Theorem
The positive orientation of a simple closed curve C refers to a single counterclockwise traversal of C.
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
ZZ Z
∂Q ∂P
P dx + Qdy =
dA
−
∂x
∂y
D
C
Alternate notations: WhenF = hP, Qi and curve given by r(t)
Z
Z
I
Z
ZZ ∂Q ∂P
F · dr =
P dx + Qdy =
P dx + Qdy =
P dx + Qdy =
dA
−
∂x
∂y
C
C
C
∂D
D
Math 251-copyright Joe Kahlig, 15A
I
Example: Evaluate
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x2 ydx + xdy where C is the triangular path from
C
(0, 0) to (1, 0) to (1, 4) to (0.0).
Example: Suppose a particle travels one revolution
counter-clockwise around the unit circle under the
x
3
3
force field F(x, y) = e − y , cos(y) + x . Find the work done by the field.
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Math 251-copyright Joe Kahlig, 15A
I 1 2
1 3
2
sin(x)
Example: Evaluate
x y+ y +e
dx + xy + x + x − arctan(y) dy where C is the tri2
3
C
angular path from (0, 0) to (1, 0) to (1, 4) to (0.0).
Area Using Line Integrals:
I
ZZ ∂Q ∂P
∂Q ∂P
−
−
=1
Since
P dx + Qdy =
dA , we need
∂x
∂y
∂x
∂y
C
D
Method 1: Let Q = x and P = 0
Method 2: Let P = −y and Q = 0
Method 3: Let P =
1
−1
y and Q = x
2
2
Example: Find the area enclosed by the ellipse
x2 y 2
+
=1
4
9
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Math 251-copyright Joe Kahlig, 15A
Z
Example: Evaluate
∂D
y 2 dx + 3xydy, where D is the region in the upper half-plane between the
circles x2 + y 2 = 4 and x2 + y 2 = 9.