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Y2 x P x; y jy Y1 x dx
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Similarly let the equations of curves EAF and EBF be x X1 y and x X2 y respectively. ! X2 y f f @Q @Q dx dy dx dy Q X2 ; y Q X1 ; y dy @x y c x x1 y @x c
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Q X2 ; y dy
Q dy
Q dy
Then Adding (1) and (2),
@Q dx dy @x
 P dx Q dy
 @Q @P dx dy @x @y
10.6. Verify Green s theorem in the plane for 2xy x2 dx x y2 dy
where C is the closed curve of the region bounded by y x2 and y2 x.
CHAP. 10]
LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS
The plane curves y x2 and y2 x intersect at 0; 0 and 1; 1 . The positive direction in traversing C is as shown in Fig. 10-9. Along y x2 , the line integral equals 1 1 f 2x x2 x2 g dx fx x2 2 g d x2 2x3 x2 2x5 dx 7=6
x 0 0
Along y2 x the line integral equals 0 0 f2 y2 y y2 2 g d y2 f y2 y2 g dy 4y4 2y5 2y2 dy 17=15
y 1 1
Then the required line integral 7=6 17=15 1=30.  '  & @Q @P @ @ x y2 2xy x2 dx dy dx dy @x @y @x @y r r p
r 1
1 2x dx dy
1 2x dy dx
x 0 y x2 p x y 2xy jy x2 dx
x1=2 2x3=2 x2 2x3 dx 1=30 Fig. 10-9
Hence, Green s theorem is veri ed.
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 r1 and r2 , which are of the type considered in Problem 10.5 and for which Green s theorem applies, i.e.,   @Q @P 1 P dx Q dy dx dy; @x @y STUS r1   @Q @P P dx Q dy dx dy 2 @x @y
SVTS r2
Fig. 10-10
Adding the left-hand sides of (1) and (2), we have, omitting the integrand P dx Q dy in each case,
STUS SVTS ST TUS SVT TS TUS SVT TUSVT
using the fact that
Adding the right-hand sides of (1) and (2), omitting the integrand, where r consists of regions r1 and r2 . r1 r2 r   @Q @P P dx Q dy dx dy and the theorem is proved. Then @x @y
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, is called a simply connected region. A region which is not
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.
10.8. Show that the area bounded by a simple closed curve C is given by
x dy y dx.
In Green s theorem, put P y; Q x. Then   @ @ x y dx dy 2 x dy y dx dx dy 2A @x @y C r r where A is the required area. Thus, A 1 x dy y dx. 2
10.9. Find the area of the ellipse x a cos ; y b sin .
Area 1 2 1 2
x dy y dx 1 2
2
a cos  b cos  d b sin  a sin  d 2
2
ab cos2  sin2  d 1 2
ab d ab
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 gure, is multiply connected since not every closed curve lying in r can be shrunk to a point without leaving r, as is observed by considering a curve surrounding DEFGD, for example. The boundary of r, which consists of the exterior boundary AHJKLA and the interior boundary DEFGD, is to be traversed 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 gure. 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   @Q @P P dx Q dy dx dy @x @y
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