Question

高数 多元函数微分 求全微分

Answer 1

x²+y²+z²=xe^z
两边微分，得
2xdx+2ydy+2zdz=e^zdx+xe^zdz
∴dz=(2x-e^z)dx/(xe^z-2z）
+2ydy/(xe^z-2z)

Answer 2

Answer 3

Answer 4

对x求偏导：2x + 2zz'x = e^z + xe^z z'x => z'x = (2x-e^z)/(xe^z - 2z)
对y求偏导：2y + 2z z'y = xe^z z'y ==> z'y = 2y/(xe^z-2z)
dz = z'x dx + z'y dy = (2x-e^z)/(xe^z - 2z) dx + 2y/(xe^z-2z) dy

Answer 5

设F(x，y，z)=x²+y²+z²-xe^z=0;
那么 ∂z/∂x=-(∂F/∂x)/(∂F/∂z)=-(2x-e^z)/(2z-xe^z)=(e^z-2x)/(2z-xe^z);
∂z/∂y=-(∂F/∂y)/(∂F/∂z)=-2y/(2z-xe^z)；
故全微分dz=(∂z/∂x)dx+(∂z/∂y)dy=[(e^z-2x)dx-2ydy]/(2z-xe^z);