Lecture 24 (Nov. 12)
Green’s Theorem (reading: 16.4)
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 (x, y) and Q(x, y) have continuous
partials on an open region containing D, then
◆
Z
ZZ ✓
@Q @P
(P dx + Qdy) =
.
@x
@y
C
D
Remark: Green’s theorem can be thought of as a version of the FTC for double
integrals.
Proof of Green’s theorem for ‘simple’ regions D (those of both Type I and Type II):
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Example: let C be the triangle Rwith vertices (0, 0), (1, 0), and (0, 2), with counterclockwise orientation. Evaluate C (xydx + x2 y 3 dy) directly, and using Green’s theorem.
(Sketch of ) proof of: @Q/@x = @P/@y in D, simply connected, implies
independent of path in D (hence F = hP, Qi conservative in D).
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R
C
hP, Qi · dr
Example: find the area inside the ellipse x2 /a2 + y 2 /b2 = 1.
Example: let F = h y/(x2 + y 2 ), x/(x2 + y 2 )i. Show
closed path encircling the origin.
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R
C
F · dr = 2⇡ for any simple