求两个例子1.z=f(x,y)在一点处可微分但偏导数不连续2.z=f(x,y)的两个二阶混合偏导数相等1.z=f(x,y)在一点处可微分但偏导数不连续2.z=f(x,y)的两个二阶混合偏导数相等但不连续

Question

求两个例子 1.z=f(x,y)在一点处可微分 但偏导数不连续 2.z=f(x,y)的两个二阶混合偏导数相等
1.z=f(x,y)在一点处可微分 但偏导数不连续
2.z=f(x,y)的两个二阶混合偏导数相等 但不连续

Answer 1

1、z=(x²+y²)sin[1/√(x²+y²)] x²+y²≠0
0 x²+y²=0
具体计算过程参见下面这个贴子的追问后回答的部分.
2、z=x^3y^3sin(1/(xy)) xy≠0.
0 xy=0.
易验证该函数在xy=0时,Fx'(x,y)=Fy'(x,y)=0
Fxy''(x,y)=Fyx''(x,y)=0,即两个混合偏导数在(0,0)处相等
而当xy≠0时,
Fx'(x,y)=3x^2y^3sin(1/(xy))-xy^2cos(1/(xy)),
Fy'(x,y)=3x^3y^2sin(1/(xy))-x^2ycos(1/(xy)).
Fxy''(x,y)=Fyx''(x,y)=
=9x^2y^2sin(1/(xy))-5xycos(1/(xy))-sin(1/(xy)).
显然在(0,0)处不连续.