uu1u2s4x2xy4yt4x2xy4y
u2u2u2u2
s
t
xy
61设FxyzxyzexyzFx1exyzFy1exyz
Fz1exyz
zFx1zFy1
xFz
yFz
f2设Fxyzzx2y2ta
zx2y2
Fx
xta
x2y2
zx2y2
x2y2sec2
z

1x2

y2
3
2
2xz
x2y22
xta

zxzsec2
z

x2y2
x2y2x2y2
x2y2
Fy
yta
x2y2
z
x2y2sec2
x2y2
z

1x2

y2
3
2
2yz
x2y22
yta

zyzsec2
z

x2y2
x2y2x2y2
x2y2
Fz1
x2y2sec2
zx2y2
1ta
2
z

x2y2
x2y2
zFxxcotz
xz1cot2
z
xFz
x2y2
x2y2x2y2
x2y2
zFy
ycot
z
yz1cot2
z
y
Fz
x2y2
x2y2x2y2
x2y2
3设Fxyzx2yz2xyzFx1
yzx
Fy2
xzy
Fx
1
xy
z
zFx
yzxyzzFy
xz2xyz

xFz
xyzxyyFz
xyzxy
4
设Fxyz
xl
z
zy

xl
z
zl

yFx

1z

F
y

1y
Fz
xz2
1z
zFx
z
z


Fy

z2

xFzxzyFzyxz
7设Fxyzx2y3z2si
x2y3zFx12cosx2y3z
Fy24cosx2y3zFz36cosx2y3z
zFx1zFy2xFz3yFz3
fzz1xy
8设FxyzcxazcybzFxc1Fyc2Fza1b2
zFx
c1
z


Fy

c2

xFza1b2yFza1b2
azbzcxy
91方程两边同时对x求导得
dz

dx
2x
2x4y
2ydydx
dy6z
dz

dxdx

解之得
0
dy

dxdy
dx

x6z12y3z1x
3z1

2方程两边同时对z求导得
dx

dz
2x
dydz
dx
10dy
2y
2z

0
解之得
dzdz
dxdz

yx

zy


dy

zx

dzxy
3方程两边同时对x求偏导得
10
eueu
u
xu

u
xu
si
vucosvcosvusi
v
v
xv

解之得
uxv

eu
si
v
si
vcosvcosveu
1


xx
x
xueusi
vcosv1
同理方程两边同时对y求偏导得
01

eueu
uyuy

uyuy
si
vucosvcosvusi
v
vyvy

解之得
u
x

v
x

cosv

eusi
vcosv1
si
veu
ueusi
vcosv1

f习题14
1。求下列函数的方向导数ul
p
0
1ux22y23z2p0110l112
解：ux
p0
2x
p0
2
uyp04yp04
uzp06zp00l0112
666
ul
p0
2
146
16
26
2u


yx
z

p0
111l

21
1
解：ux
p0

z
yx
z1
yx2

p0
1
uy
p0
zyz11xx
p0
1
uz
p0
yzl
yxx
p0
0
l0211666
u
2
1
1
l
p01
16
6
6
3u

l
x2

y2

p0
11
l
与ox轴夹角为3
；
解：ux
2xp0x2y2
p01
u
2y
yp0x2y2p01
ul
p0
cossi
33
12
3
f4uxyzp0512p19414lp0p1
解：ux
p0
yz
p0
2
uy
p0
xz
p0
10
uzp0xyp05
l4312l0r