Page 397 - Advanced engineering mathematics

12.3 An Extension of Green’s Theorem 377

K 1 K 1

P 1 P 1
P 3 P 3

K 3 K 3
P 2 P 2
K 2 K 2

C
C
FIGURE 12.5 Enclosing points with small cir- FIGURE 12.6 Channels connecting C to K 1 ,
K 1 to K 2 , ···,K n−1 to K n .
cles interior to C.

C *

FIGURE 12.7 The simple closed path C ,
∗
with each P j exterior to C .
∗

from C to K 1 , then from K 1 to K 2 , and so on, until the last channel is drawn from K n−1 to K n .
This is illustrated in Figure 12.6 for n = 3.
∗
Now form the simple closed path C of Figure 12.7, consisting of “most of ” C,“mostof ”
each K j , and the inserted channel lines. By “most of” C, we mean that a small arc of C and each
circle between the channel cuts has been excised in forming C .
∗
Each P j is external to C and f, g, ∂ f/∂y and ∂g/∂x are continuous on and in the inte-
∗
rior of C . The orientation on C is also crucial. If we begin at a point of C just before the
∗
∗
channel to K 1 , we move counterclockwise on C until we reach the first channel cut, then go
along this cut to K 1 , then clockwise around part of K 1 until we reach a channel cut to K 2 ,
then clockwise around K 2 until we reach a cut to K 3 . After going clockwise around part of
K 3 , we reach the other side of the cut from this circle to K 2 , move clockwise around it to the
cut to K 1 , then clockwise around it to the cut back to C, and then continue counterclockwise
around C.

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October 14, 2010 14:53 THM/NEIL Page-377 27410_12_ch12_p367-424