Solutions to exercises 641
Section 9.6
37.− 21 − 12 t 2 (1 1 − 1 t
2
)
122
38.−(2 1 sin 1 t 1 + 1 3)e
2cost− 3 t39.(−a 1 sin 1 t 1 + 1 b 1 cos 1 t 1 + 1 c) 2 (a 1 cos 1 t 1 + 1 b 1 sin 1 t 1 + 1 ct)
40.(3 1 cos 1 θ 1 − 121 sin 1 θ) 2 (2 1 cos 1 θ 1 + 131 sin 1 θ)
- (i)(1 1 − 1 sin 1 u) 1 cos(u 1 + 1 v),
(ii)−u(1 1 − 1 sin 1 u) 1 cos(u 1 + 1 v)
42.− 5 x
4
y 2 (x
5
1 − 1 cos 1 y)
44.2(y
2
1 − 1 x
2
) 2 y
- (i)
(ii)
- (ii)
(iii)
Section 9.7
53.exact 54.not exact 55.exact
Section 9.8
- 6423 58. 2 59. 20
- 723 62. 31
- (i)− 123 , (ii) 0
Section 9.9
- 24 65.2(1 1 − 1 e
−R
) 23
Section 9.10
- 4023 67.a
5
215
- (ii)− 5122
Section 9.11
- 315 π 28
71.π 28
1
44
1
π
yx
f
x
nx y
f
y
ny
nn n∂
∂
=
∂
∂
=
−−
44
1
,
11
x
nxu
f
y
ny
f
u
f
n∂
∂
=
∂
∂
−
∂
∂
− 1
v
v
yu
f
x
nx
f
u
f
n∂
∂
=
∂
∂
∂
∂
− 1
v
v
∂
∂
=+
∂
∂
=+
f
u
abu
f
22 ab
22 22
() (),
v
v
∂
∂
=
∂
∂
∂
∂
∂
∂
=
∂
∂
−
∂
∂
f
u
a
f
x
b
f
y
f
b
f
x
a
f
y
,
v
xy
xy
xy
cos − siny
3
Vn
p
T
nR
Vnb
,
∂
∂
=
−
Chapter 10
Section 10.2
1.(0, 0 ,1) 2.(0, 2 ,0)
- 4.(1,π 22 ,0)
5.(1,π 22 ,π 2 2) 6.(3,cos
− 1
(2 2 3),tan
− 1
(2))
7.(9,cos
− 1
(− 82 9),tan
− 1
(−4) 1 + 1 π)
8.(7,cos
− 1
(6 2 7),tan
− 1
(3 2 2) 1 + 1 π)
9.(13,cos
− 1
(− 122 13),tan
− 1
(− 42 3) 1 + 1 π)
Section 10.3
10.r
2
1 sin
2
1 θ 1 cos 12 φ 11.tan
2
1 θ
12.r
2
(3 1 cos
2
1 θ 1 − 1 1) 13. (i) 12 r (ii)−x 2 r
3
Section 10.4
- 1 15.a
2
b
3
c
4
224 16. 122
- 1 18. 4 πa
5
25 19. 2 πa
2
23
- 480 π 22. 23. 6 a
0
- 180
Section 10.5
- 2 y
3
z
4
1 + 16 x
2
yz
4
1 + 112 x
2
y
3
z
2
27.e
−r2 r
Section 10.6
- (i)x 1 = 1 a 1 cos 1 t,y 1 = 1 −a 1 sin 1 t,z 1 = 1 bt
(ii)ρ 1 = 1 a,φ 1 = 1 −t,z 1 = 1 bt
39.πa
4
216 40.π(1 1 + 1 R)e
−R
Chapter 11
Section 11.2
- 8.x
4
1 + 1 ax
2
1 + 1 bx 1 + 1 c
−++1
9
cos 3 xaxb
−+
−
e
c
3 x3
x
c
3
3
π
2
() 1 +
−
Re
R
xze
r
−r−
2
1
4
3
1
9
2
3
2
2
3
−−
−
r
re
rsin sinθ φ
nn
r
an
r
are
nar() ()+
−
−
12 1
2
2
1
81
2
π
(,,)−−32 32 1