Michael Corral: Vector Calculus

191
Section 2.7 (p. 100)
( ( )
−4
1. min. √, −2 √
); max. √
4, √5
2
( 5 5
( 5
20
3. min. √
30
, √
); max. − 20
13 13
4. min.
5. 8abc
3 √ 3
Chapter 3
(
−9
√
5
,0,
2 √5
); max.
Section 3.1 (p. 104)
)
√ ,−√ 30
13 13
( √
9
8 , 59
4 ,−1 4
1. 1 3.
7
12
5. 7 6
7. 5 9. 1 2
11. 15
Section 3.2 (p. 109)
1. 1 3. 8ln2−3 5. π 4
6. 1 4
7. 2 9. 1 6
10. 6 5
Section 3.3 (p. 112)
9
1.
2
3. (2cos(π 2 )+π 4 1
− 2)/4 5.
6
7. 6
10. 1 3
Section 3.4 (p. 116)
1. The values should converge to≈1.318.
(Hint: In Java the exponential function
e x can be obtained with Math.exp(x).
Other languages have similar functions,
otherwise use e = 2.7182818284590455 in
your program.)
2.≈ 1.146 3.≈ 0.705 4.≈ 0.168
Section 3.5 (p. 123)
1. 8π 3. 4π 3 (8−33/2 ) 7. 1− sin2
2
9. 2πab
Section 3.6 (p. 127)
1. (1,8/3) 3. (0, 4a
3π
) 5. (0,3π/16)
7. (0,0,5a/12) 9. (7/12,7/12,7/12)
)
Section 3.7 (p. 134)
1. √ π 2. 1 6. Both are
n
(n+1) 2 (n+2)
Chapter 4
Section 4.1 (p. 142)
1. 1/2 3. 23 5. 24π 7.−2π 9. 2π
11. 4π
Section 4.2 (p. 149)
7. 1 n
1. 0 3. No 4. Yes. F(x,y)= x2
2 − y2
2
5. No 9. (b) No. Hint: Think of how F is
defined. 10. Yes. F(x,y)=axy+bx+cy+d
Section 4.3 (p. 155)
1. 16/15 3.−5π 5. Yes. F(x,y)= xy 2 + x 3
7. Yes. F(x,y)=4x 2 y+2y 2 +3x
Section 4.4 (p. 163)
1. 216π 2. 3 3. 12π/5 7. 15/4
Section 4.5 (p. 175)
1. 2 √ 2π 2 2. (17 √ 17−5 √ 5)/3 3. 2/5
4. 1 5. 2π(π−1) 7. 67/15 9. 6
11. Yes 13. No 19. Hint: Think of
howavectorfieldf(x,y)=P(x,y)i+Q(x,y)j
in 2 can be extended in a natural way to
be a vector field in 3 .
Section 4.6 (p. 186)
1. 0 3. 12 √ x 2 +y 2 +z 2 5. 6(x+y+z)
7. 12ρ 8. (4ρ 2 −6)e −ρ2 9.− 2z e
r 3 r + 1 e
r 2 z
11. div f= 2 ρ − sinθ
sinφ +cotφ;
curl f=cotφ cosθe ρ +2e θ −2cosθe φ
25. Hint: Start by showing that e r =
cosθi+sinθj, e θ =−sinθi+cosθj, e z = k.