习题64
1求下列函数的一阶偏导数1zl
xx2y2xy2zxxx2y2
2
1
x1xy2
2

yx2y2zyxx2y22zzxxxy2
2
yx2y2xx2y2

x2xy
223
1x2y2


y2xy
223

zxy3yx2y23zxxl
zxyl
x
y
1zyxy1l
xxy1zyxy1l
xxy1zx1zzxyl
xl
xzxyl
2xzyyxy4zxyxyxzy2y22xxyxyxyyx2zx22yxyxy5zarcsi
xyyzzx2x1xyy2y1x2y6zxexyzzexyxexyyexy1xyx2exyxyyzx7uxyzuy1u1zu1x2xxzyxy2zyz2
f8uxyzuuuyzxyz1xzxyz1xyzl
xyxyz2求下列函数在指定点的偏导数1zxarccosy1y1cosxzz求及x01y011si
xsi
y1
x0
zdxx01dx1si
x
d1si
xxcosxdx1si
x2
y1
1
x0
zdy1y01dy1si
y12z
d1si
y1y1cosy1dy1si
y12
1
y1
2yzz求及ycosxxπ1yπ1
22
z2ysi
xzycosxy2cosx2×22xycosxyycosxycosx2zz20πx1yπ1
22
3fxyzl
xyz求fx210fy210fz210fxxyzyx1fyxyzfzxyzxyzxyzxyz11fx210fy2101fzxyz22
x2y2xy≠003证明函数fxyxy0xy00在00连续但是fx00不存在证fxyx2y2≤xy→0xy→00xyfxy→f000xy→00fxy在00连续x2xxfx00limlim不存在x→0xx→0x
fyzzz4设zxsi
证明xyxxy2证为齐12次函数根据关于齐次函数微分的一个定理立得结论直接计算如下z1yyyzy1si
xcos2xcosx2xxxxyxxxzzxyyyyyxyzysi
coscossi
xy2xxx2x2xx
f5求下列函数的二阶混合偏导数fxy1fxyl
2x3y26fxfxy2x3y2x3y22fxyysi
xexfxycosxexfxycosx3fxyxxy24x3r