James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 16.4 Green’s Theorem 1101
D
y
Cª
C
x
with center the origin and radius a, where a is chosen to be small enough that C9 lies
inside C. (See Figure 11.) Let D be the region bounded by C and C9. Then its positively
oriented boundary is C ø s2C9d and so the general version of Green’s Theorem gives
dA
−yD
y P dx 1 Q dy 1
C
y P dx 1 Q dy −
2C9
y y S −Q
−x 2 −P
D
FIGURE 11
− y y F 2G y 2 2 x 2
sx 2 1 y 2 d 2 y 2 2 x 2
dA − 0
2 sx 2 1 y 2 d
D
Therefore
that is,
y C
P dx 1 Q dy − y C9
P dx 1 Q dy
y C
F dr − y C9
F dr
We now easily compute this last integral using the parametrization given by
rstd − a cos t i 1 a sin t j, 0 < t < 2. Thus
y C
F dr − y C9
F dr − y 2
− y 2
0
0
Fsrstdd r9std dt
s2a sin tds2a sin td 1 sa cos tdsa cos td
a 2 cos 2 t 1 a 2 sin 2 t
dt − y 2
dt − 2
0
■
We end this section by using Green’s Theorem to discuss a result that was stated in the
preceding section.
Sketch of Proof of Theorem 16.3.6 We’re assuming that F − P i 1 Q j is a vector
field on an open simply-connected region D, that P and Q have continuous first-order
partial derivatives, and that
−P
−y − −Q
−x
throughout D
If C is any simple closed path in D and R is the region that C encloses, then Green’s
Theorem gives
y C
F dr − y C
P dx 1 Q dy − yy
R
S −Q
−x 2 −P
dA −
−yD yy 0 dA − 0
A curve that is not simple crosses itself at one or more points and can be broken up
into a number of simple curves. We have shown that the line integrals of F around these
simple curves are all 0 and, adding these integrals, we see that y C
F dr − 0 for any
closed curve C. Therefore y C
F dr is independent of path in D by Theo rem 16.3.3. It
follows that F is a conservative vector field.
■
R
1–4 Evaluate the line integral by two methods: (a) directly and
(b) using Green’s Theorem.
1. y C
y 2 dx 1 x 2 y dy,
C is the rectangle with vertices s0, 0d, s5, 0d, s5, 4d, and s0, 4d
2. y C
y dx 2 x dy,
C is the circle with center the origin and radius 4
3. y C
xy dx 1 x 2 y 3 dy,
C is the triangle with vertices s0, 0d, (1, 0), and (1, 2)
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