曲面的切平面与法线

castelu

　　设曲面由方程$F(x,y,z)=0$给出，它在点$P_0(x_0,y_0,z_0)$的某邻域内满足隐函数定理条件（这里不妨设$F_z(x_0,y_0,z_0) \ne 0$）。于是方程$F(x,y,z)=0$在点$P_0$附近确定惟一连续可微的隐函数$z=f(x,y)$使得$z_0=f(x_0,y_0)$，且
$$\frac{\partial z}{\partial x}=-\frac{F_x(x,y,z)}{F_z(x,y,z)}，\frac{\partial z}{\partial y}=-\frac{F_y(x,y,z)}{F_z(x,y,z)}。$$
　　由于在点$P_0$附近$F(x,y,z)=0$与$z=f(x,y)$表示同一曲面，从而该曲面在$P_0$处有切平面与法线，它们的方程分别是
$$z-z_0=-\frac{F_x(x_0,y_0,z_0)}{F_z(x_0,y_0,z_0)}(x-x_0)-\frac{F_y(x_0,y_0,z_0)}{F_z(x_0,y_0,z_0)}(y-y_0)$$
　　与
$$\frac{x-x_0}{-\frac{F_x(x_0,y_0,z_0)}{F_z(x_0,y_0,z_0)}}=\frac{y-y_0}{-\frac{F_y(x_0,y_0,z_0)}{F_z(x_0,y_0,z_0)}}=\frac{z-z_0}{-1}。$$
　　它们也可分别写成如下形式：
$$F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0$$
　　与
$$\frac{x-x_0}{F_x(x_0,y_0,z_0)}=\frac{y-y_0}{F_y(x_0,y_0,z_0)}=\frac{z-z_0}{F_z(x_0,y_0,z_0)}$$
　　这种形式对于$F_x(x_0,y_0,z_0) \ne 0$或$F_y(x_0,y_0,z_0) \ne 0$也同样适合。