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376 CHAPTER 12 Vector Integral Calculus

SECTION 12.2 PROBLEMS

1. A particle moves once counterclockwise about the tri- (c) Show that the area of D equals
angle with vertices (0,0),(4,0) and (1,6), under the
influence of the force F= xyi+ xj. Calculate the work 1 −ydx + xdy.
done by this force. 2 C
2. A particle moves once counterclockwise around the 13. Let u(x, y) be continuous with continuous first and
circle of radius 6 about the origin, under the influence second partial derivatives on a simple closed path C
x
of the force F = (e − y + x cosh(x))i + (y 3/2 + x)j. and throughout the interior D of C. Show that
Calculate the work done.
2
2
∂u ∂u ∂ u ∂ u
3. A particle moves once counterclockwise about the − ∂y dx + ∂x dy = 2 + ∂y 2 dA.
C D ∂x
rectangle with vertices (1,1),(1,7), (3,1) and (3,7),
4 14. Fill in the details of the following argument to prove
under the influence of the force F = (−cosh(4x ) +
xy)i + (e −y + x)j. Calculate the work done. Green’s theorem under special conditions. Assume
that D can be described in two ways. First, D con-
In each of Problems 4 through 11, use Green’s theorem to sists of all (x, y) with q(x) ≤ y ≤ p(x),for a ≤ x ≤ b.

evaluate F · dR. All curves are oriented positively. This means that D has an upper boundary (graph
C
of y = p(x)) and a lower boundary (y = q(x))for
4. F=2yi− xj and C is the circle of radius 4 about (1,3) a ≤ x ≤ b. Also assume that D consists of all (x, y)
with α(y) ≤ x ≤ β(y), with c ≤ y ≤ d. In this descrip-
2
5. F = x i − 2xyj and C is the triangle with vertices
tion, the graph of x = α(y) is a left boundary of D,
(1,1),(4,1),(2,6)
and the graph of x = β(y) is a right boundary.
2
6. F = (x + y)i + (x − y)j and C is the ellipse x + Using the first description of D, show that
2
4y =1
d c

2
7. F = 8xy j and C is the circle of radius 4 about the g(x, y)dy = g(β(y), y)dy + g(α(y), y)dy
C c d
origin
and
2
3y
8. F = (x − y)i + (cos(2y) − e + 4x)j and C is any
d β(y)
square with sides of length 5 ∂g ∂g
dA = dA
x
x
9. F = e cos(y)i − e sin(y)j and C is any simple closed D ∂x c α(y) ∂x
path in the plane c
= (g(β(y), y) − g(α(y), y))dy.
2
2
10. F = x yi − xy j and C is the boundary of the region c
2
2
x + y ≤ 4, x ≥ 0, y ≥ 0 Thus, conclude that
2
11. F = xyi + (xy − e cos(y) )j and C is the triangle with ∂g
vertices (0,0), (3,0) and (0,5) g(x, y)dy = dA.
C D ∂x
12. Let D be the interior of a positively oriented simple
Now use the other description of D to show that
closed path C.

(a) Show that the area of D equals −ydx. ∂ f
C
f (x, y)dx =− dA.

(b) Show that the area of D equals xdy. C D ∂y
C
12.3 An Extension of Green’s Theorem
There is an extension of Green’s theorem to include the case that there are finitely many points
P 1 ,··· , P n enclosed by C at which f , g, ∂ f/∂y and/or ∂g/∂x are not continuous, or perhaps
not even defined. The idea is to excise these points by enclosing them in small disks which are
thought of as cut out of D.
Enclose each P j with a circle K j of sufficiently small radius that no circle intersects either C
or any of the other circles (Figure 12.5). Draw a channel consisting of two parallel line segments
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October 14, 2010 14:53 THM/NEIL Page-376 27410_12_ch12_p367-424