问题
设曲面 $\Sigma$ 的方程为 $z$ $=$ $f(x, y)$, 则在 $\Sigma$ 上的点 $\left(x_{0}, y_{0}, z_{0}\right)$ 处的法线方程是多少？选项
[A]. $\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(x-x_{0}\right)$ $+$ $\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}\left(y-y_{0}\right)$ $-$ $\left(z-z_{0}\right)$ $=$ $0$[B]. $\frac{x+x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y+y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z+z_{0}}{-1}$
[C]. $\frac{x-x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{1}$
[D]. $\frac{x-x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{-1}$
$\frac{x-x_{0}}{\left.\frac{\partial z}{\partial x}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{y-y_{0}}{\left.\frac{\partial z}{\partial y}\right|_{\left(x_{0}, y_{0}, z_{0}\right)}}$ $=$ $\frac{z-z_{0}}{-1}$