640 Solutions to exercises
- (i)
Section 8.7
- 125 50. 6265
Chapter 9
Section 9.1
- (i) 0 (ii) 16 (iii) 36
- (i) 1 (ii)− 322 (iii) 2
Section 9.3
- 4 x,− 2 y 4. 2 x 1 − 13 , 4 y 1 + 12 5. 2 z, 3 z
- 2 x 1 cos(x
2
1 − 1 y
2
),− 2 y 1 cos(x
2
1 − 1 y
2
)
9.u
x1 = 16 x 1 + 12 y
3
,u
y1 = 12 y 1 + 16 xy
2
,
u
xx1 = 16 ,u
yx1 = 1 u
xy1 = 16 y
2
,u
yy1 = 121 + 112 xy,
u
yyx1 = 1 u
yxy1 = 1 u
xyy1 = 112 y,u
yyy1 = 112 x,
u
xyyy1 = 1 u
yxyy1 = 1 u
yyxy1 = 1 u
yyyx1 = 112
∂
∂
=+−+
−
z
y
xy xye
xycos( ) sin( ) ,
∂
∂
=+++
−
z
x
xy xye
xysin( ) cos( ) ,
∂
∂∂
=− +=
∂
∂∂
22
4
z
xy
xxy
z
yx
cos( )
∂
∂
=− +
2
2
z
y
cos(xy),
∂
∂
=− +
2
2
4
z
x
yxycos( ),
∂
∂
=− +
z
y
2 xxy
2
sin( ),
∂
∂
=− +
z
x
4sin( ),xy x y
∂
∂∂
=
∂
∂∂∂
=
∂
∂∂
=
3
2
33
2
8
z
xy
z
yxy
z
yx
∂
∂∂
=
∂
∂∂∂
=
∂
∂∂
=−
3
2
33
2
6
z
xy
z
xyx
z
yx
∂
∂∂
=− + =
∂
∂∂
22
68
z
xy
xy
z
yx
∂
∂
=−
∂
∂
=
2
2
2
2
26 8
z
x
y
z
y
, x
∂
∂
=− +
∂
∂
=− +
z
x
xxyy
z
y
26 4 3 8xxy
22
,
(cos sin), sin2
22xxyyxye xe xy
xx−−
12 π
1
1
2
1
1
2
,(),,( ),±±ii −±− 11 i
- (i)
(ii)
(iii)nR 2 (V 1 − 1 nb)
(iv) 2 n
2
a 2 V
3
1 − 1 (p 1 + 12 n
2
a 2 V
2
) 2 (V 1 − 1 nb)
Section 9.4
21.(− 223 , 42 3) 22.(1,2),(− 1 ,2)
24.(− 223 , 42 3): maximum
25.(1,2): minimum; (− 1 ,2) : saddle point
- : saddle points; (1,2) : minimum;
(− 1 ,−2): maximum
27.f 1 = 1 1 at (x,y,z) 1 = 1 (1 22 , 123 , 12 6)
28.f 1 = 1 c
6
2 27 atx
2
1 = 1 y
2
1 = 1 z
2
1 = 1 c
2
23
- (i)f 1 = 1 4 at (x,y,z) 1 = 1 (1 23 , 223 , 22 3);
f 1 = 1 16 at (x,y,z) 1 = 1 (− 123 ,− 223 ,− 22 3)
- (ii)
Section 9.5
- 2 x 1 dx 1 + 12 y 1 dy
32.[6x 1 + 1 cos(x 1 − 1 y)]dx 1 − 1 cos(x 1 − 1 y)dy
- 3 x
2
y
2
dx 1 + 1 (2x
3
y 1 + 112 y)dy
35.sin 1 θ 1 sin 1 φdr 1 + 1 r 1 cos 1 θ 1 sin 1 φdθ 1
- 1 r 1 sin 1 θ 1 cos 1 φdφ
∂
∂
∂
∂
Tpn Tp
V
n
dn
n
V
n
,, ,,
B
A
A
A
B
ddn
B
dV
n
V
T
dT
n
V
p
pn Tn
=
∂
∂
∂
∂
,,,,
AB AB
dp
−
++
++
1
22232
()
()
xyz
xdx ydy zdz
λ=± : ,± ,
()
ab 2121212
λ=: ,,−
()
a 12012 ,
(, )03±
( ,±−−), ( , ), ( , )031212
−− − +
()Vnb p
na
V
nab
V
2
2
3
3
2
nR p
na
V
nab
V
−+
2
2
3
3
2
−
2
2
xyz
yxz
∂
∂
=
∂
∂
=
∂
∂
=
r
x
x
r
r
y
y
r
r
z
z
r
,,
∂
∂∂
=
∂
∂∂
=− +
−
22
2
z
xy
z
yx
xye
xysin( )
∂
∂
=− +
−
2
2
2
z
y
xye
xycos( ) ,
∂
∂
=+
−
2
2
2
z
x
xye
xycos( ) ,