Calculus 2nd Edition Rogawski

1020 C H A P T E R 18 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS
28. Estimate the circulation of a vector field F around a circle of radius
R = 0.1, assuming that curl z (F) takes the value 4 at the center of the
circle.
∮
〈
29. Estimate F · ds, where F = x + 0.1y 2 ,y− 0.1x 2〉 and C encloses
a small region of area 0.25 containing the point P = (1,
C
1).
T
P
n
F
30. Let F be the velocity field. Estimate the circulation of F around a
circle of radius R = 0.05 with center P , assuming that curl z (F)(P ) =
−3. In which direction would a small paddle placed at P rotate? How
fast would it rotate (in radians per second) if F is expressed in meters
per second?
31. Let C R be the circle of radius R centered at the ∮ origin. Use Green’s
Theorem to find the value of R that maximizes y 3 dx + xdy.
C R
32. Area of a Polygon Green’s Theorem leads to a convenient formula
for the area of a polygon.
(a) Let C be the line segment joining (x 1 ,y 1 ) to (x 2 ,y 2 ). Show that
1
2
∫
−ydx+ xdy= 1
C
2 (x 1y 2 − x 2 y 1 )
(b) Prove that the area of the polygon with vertices (x 1 ,y 1 ), (x 2 ,y 2 ),
...,(x n ,y n ) is equal [where we set (x n+1 ,y n+1 ) = (x 1 ,y 1 )] to
1
2
n∑
(x i y i+1 − x i+1 y i )
i=1
33. Use the result of Exercise 32 to compute the areas of the polygons
in Figure 27. Check your result for the area of the triangle in (A) using
geometry.
FIGURE 28 The flux of F is the integral of the normal component F · n
around the curve.
34. ∮ Show that the flux of F = ⟨P,Q⟩ across C is equal to
Pdy− Qdx.
C
35. Define div(F) = ∂P
∂x + ∂Q . Use Green’s Theorem to prove that
∂y
for any simple closed curve C,
∫∫
Flux across C = div(F)dA 12
D
where D is the region enclosed by C. This is a two-dimensional version
of the Divergence Theorem discussed in Section 18.3.
〈
36. Use Eq. (12) to compute the flux of F = 2x + y 3 , 3y − x 4〉 across
the unit circle.
37. Use Eq. (12) to compute the flux of F = ⟨cos y,sin y⟩ across the
square 0 ≤ x ≤ 2, 0 ≤ y ≤ π 2 .
38. If v is the velocity field of a fluid, the flux of v across C is equal to
the flow rate (amount of fluid flowing across C in m 2 /s). Find the flow
rate across the circle of radius 2 centered at the origin if div(v) = x 2 .
3
1
y
(2, 3)
(2, 1) (5, 1)
2 5
(A)
x
y
(−3, 5)
5
3
(1, 3)
(5, 3)
2
(−1, 1)
1
(3, 2)
x
−3 −1 1 3 5
(B)
FIGURE 27
39. A buffalo (Figure 29) stampede is described by a velocity vector
field F = 〈 xy − y 3 ,x 2 + y 〉 km/h in the region D defined by 2 ≤ x ≤ 3,
2 ≤ y ≤ 3 in units of kilometers (Figure 30). Assuming a density is
ρ = 500 buffalo per square kilometer, use Eq. (12) to determine the net
number of buffalo leaving or entering D per minute (equal to ρ times
the flux of F across the boundary of D).
3
y
Exercises 34–39: In Section 17.2, we defined the flux of F across a
curve C (Figure 28) as the integral of the normal component of F along
C, and we showed that if c(t) = (x(t), y(t)) is a parametrization of C
for a ≤ t ≤ b, then the flux is equal to
where n(t) = ⟨y ′ (t), −x ′ (t)⟩.
∫ b
F(c(t)) · n(t) dt
a
FIGURE 29 Buffalo stampede.
2
2
3
FIGURE 〈 30 The vector field 〉
F = xy − y 3 ,x 2 + y .
x