求曲面e^z+x+y=z在点(1,0,0)处的切面及法线方程

Question

求曲面e^z+x+y=z在点(1,0,0)处的切面及法线方程

Answer 1

F(x,y,z) = x + y - z + e^z = 0
F(x,y,z)的梯度就是法向量
grand(F) = (∂F/∂x ,∂F/∂y ,∂F/∂z)
∂F/∂x = 1
∂F/∂y = 1
∂F/∂z = -1+e^z
可得grand(F)|(1,0,0) = (1,1,0)
法向量即法线的方向向量,可得法线的参数方程为{x=1+t ,y=1+t ,z=0}
而切平面的点法式方程为(x-1)+(y-1)+(z-0)=0,即x+y+z=2