Page 399 - Advanced engineering mathematics

12.3 An Extension of Green’s Theorem 379

y

x
K
C

FIGURE 12.8 Case 2 of Example 12.10.

Case 1 If C does not enclose the origin, Green’s theorem applies and

−y x ∂g ∂ f

dx + dy = − dA = 0.
C x + y 2 x + y 2 D ∂x ∂y
2
2
Case 2 If C encloses the origin, then C encloses a point where f and g are not defined. Now
use equation (12.3). Let K be a circle about the origin, with sufficiently small radius r that K
does not intersect C (Figure 12.8). Then

fdx + gdy
C

∂g ∂ f

= fdx + gdy + − dA
∂x ∂y
K D ∗

= fdx + gdy
K
where D is the region between D and K, including both curves. Both of these line integrals
∗
are in the counterclockwise sense. The last line integral is over a circle and can be evaluated
explicitly. Parametrize K by x =r cos(θ), y =r sin(θ) for 0 ≤ θ ≤ 2π. Then

fdx + gdy
K
−r sin(θ) r cos(θ)
2π
= [−r sin(θ)]+ [r cos(θ)] dθ
r 2 r 2
0
2π

= dθ = 2π.
0
We conclude that

0 if C does not enclose the origin
fdx + gdy =
C 2π if C encloses the origin.

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October 14, 2010 14:53 THM/NEIL Page-379 27410_12_ch12_p367-424