2222222yyxxyxy1x1
x2y2y2y2zyy2x222xyyx2y2x2y2x2y2
2
z2za2si
axbycosaxbyasi
2axby2a2cos2axbyxx2
z2zb2si
axbycosaxbybsi
2axby2b2cos2axby2yy2zasi
2axby2absi
2axbyxyy


习题93
全微分
y
1求下列函数的全微分：
yz（1）zex；（2）ux（3）uxsi

yeyz（4）uta
2

x2y2z2

解1因为
yzyyz1xzz1y2exe所以dzdxdy2exydxxdyxxyxxyx
2因为
uuuyzxyz1zxyzl
xyxyzl
x所以xyz
2
fdu
uuudxdydzyzxyz1dxxyzl
xzdyydzxyz
3
uu1yu1coszeyzyeyz，所求的全微分为xy22z
y1dudxcoszeyzdyyeyzdz22
4因为
2x2y2z2uxsec，xx2y2z2
uysec2x2y2z2yx2y2z2sec2x2y2z2x2y2z2
uzsec2x2y2z2zx2y2z2
2求函数z
所以du
xdxydyzdz
y，当x2，y1，x01，y02时的全增量和全微分。xyyyy1dz2xy当x2y1x01y02时解zxxxxx
全增量z
102201
z

1110119全微分dz01020125422
x3设fxyz，求df111y
解
xfzxy
z
z1
1y
fz
fx
1
111
xfzyy
z1
xf2yy
1
111
fxxl
zyy
0故
111
ydf111dxd
z
习题94
多元复合函数的求导法则
2
u
v
1设zul
v，而u
zzx，v3x2y，求，xyy
x
图941
y
解
zzuzv1u22x3x22ul
v32l
3x2yxuxvxyvy3x2yy2
z
xu2zzuzv2x23x22ul
vr