Page 250 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 241

2
2
The plane curves y ¼ x and y ¼ x intersect at ð0; 0Þ and ð1; 1Þ. The positive direction in traversing C
is as shown in Fig. 10-9.
2
Along y ¼ x ,the line integral equals
1 1
ð ð
2 2 2 2 2 3 2 5
fð2xÞðx Þ x g dx þfx þðx Þ g dðx Þ¼ ð2x þ x þ 2x Þ dx ¼ 7=6
x¼0 0
2
Along y ¼ x the line integral equals
ð 0 ð 0
2 2 2 2 2 2 4 5 2
f2ð y Þð yÞ ð y Þ g dð y Þþ f y þ y g dy ¼ ð4y 2y þ 2y Þ dy ¼ 17=15
y¼1 1
Then the required line integral ¼ 7=6 17=15 ¼ 1=30.
ðð ðð
@Q @P @ 2 @ ð2xy x Þ dx dy
2
@x @y dx dy ¼ @x ðx þ y Þ @y
r r
ðð ð 1 ð p ffiffi x
ð1 2xÞ dy dx
¼ ð1 2xÞ dx dy ¼
x¼0 y¼x 2
r
ð 1
x
p ffiffi
y¼x
¼ ð y 2xyÞj 2 dx
x¼0
ð 1
3
2
ðx 1=2 2x 3=2 x þ 2x Þ dx ¼ 1=30
¼
0
Fig. 10-9
Hence, Green’s theorem is verified.
10.7. Extend the proof of Green’s theorem in the plane given in Problem
10.5 to the curves C for which lines parallel to the coordinate axes
may cut C in more than two points.
Consider a closed curve C such as shown in the adjoining Fig. 10-10,
in which lines parallel to the axes may meet C in more than two points.
By constructing line ST the region is divided into two regions r 1 and r 2 ,
which are of the type considered in Problem 10.5 and for which Green’s
theorem applies, i.e.,
@Q @P
ð ðð
dx dy;
@x @y
ð1Þ Pdx þ Qdy ¼
STUS r 1
ð ðð Fig. 10-10
@Q @P
dx dy
@x @y
ð2Þ Pdx þ Qdy ¼
SVTS r 2
Adding the left-hand sides of (1) and (2), we have, omitting the integrand Pdx þ Qdy in each case,
ð ð ð ð ð ð ð ð ð
þ ¼ þ þ þ ¼ þ ¼
STUS SVTS ST TUS SVT TS TUS SVT TUSVT
ð ð
using the fact that ¼ .
ST TS ðð ðð ðð
Adding the right-hand sides of (1) and (2), omitting the integrand, þ ¼ where r consists of
regions r 1 and r 2 .
r 1 r 2 r
ð ðð
@Q @P
Then dx dy and the theorem is proved.
@x @y
Pdx þ Qdy ¼
TUSVT r
A region r such as considered here and in Problem 10.5, for which any closed curve lying in r can be
continuously shrunk to a point without leaving r,iscalled a simply connected region.A region which is not