Page 251 - Schaum's Outline of Theory and Problems of Advanced Calculus

242 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

simply connected is called multiply connected. We have shown here that Green’s theorem in the plane
applies to simply connected regions bounded by closed curves. In Problem 10.10 the theorem is extended to
multiply connected regions.
For more complicated simply connected regions, it may be necessary to construct more lines, such as
ST,to establish the theorem.

þ
10.8. Show that the area bounded by a simple closed curve C is given by 1 xdy ydx.
2
C
In Green’s theorem, put P ¼ y; Q ¼ x. Then
þ ðð ðð
@ @
ð yÞ dx dy ¼ 2 dx dy ¼ 2A
xdy ydx ¼ ðxÞ
C @x @y
r r
þ
1 xdy ydx.
where A is the required area. Thus, A ¼ 2
C
10.9. Find the area of the ellipse x ¼ a cos ; y ¼ b sin .
þ ð 2
1 1 ða cos Þðb cos Þ d ðb sin Þð a sin Þ d
2 2
Area ¼ xdy ydx ¼
C 0
ð 2 ð 2
1 2 2 1 ab d ¼ ab
2 2
¼ abðcos þ sin Þ d ¼
0 0
10.10. Show that Green’s theorem in the plane is also valid for a multiply connected region r such as
shown in Fig. 10-11.
The shaded region r,shown in the figure, is multiply
connected since not every closed curve lying in r can be
shrunk to a point without leaving r,asis observed by con-
sidering a curve surrounding DEFGD,for example. The
boundary of r, which consists of the exterior boundary
AHJKLA and the interior boundary DEFGD,is tobe tra-
versed in the positive direction, so that a person traveling in
this direction always has the region on his left. It is seen that
the positive directions are those indicated in the adjoining
figure.
In order to establish the theorem, construct a line, such
as AD,called a cross-cut,connecting the exterior and interior
boundaries. The region bounded by ADEFGDALKJHA is
simply connected, and so Green’s theorem is valid. Then Fig. 10-11
ðð
þ
@Q @P
dx dy
@x @y
Pdx þ Qdy ¼
ADEFGDALKJHA r
But the integral on the left, leaving out the integrand, is equal to
ð ð ð ð ð ð
þ þ þ ¼ þ
AD DEFGD DA ALKJHA DEFGD ALKJHA
ð ð
since ¼ . Thus, if C 1 is the curve ALKJHA, C 2 is the curve DEFGD and C is the boundary of r
AD DA ð ð ð
consisting of C 1 and C 2 (traversed in the positive directions), then þ ¼ and so
C 1 C 2 C
@Q @P
þ ðð
dx dy
Pdx þ Qdy ¼
C @x @y
r