梯度和法向量

过原点的平面

所以梯度向量和法向量也是相对应的

切平面方程

曲面 $z = z (x, y)$ 微分得切平面方程

$z - z_{0} = z_{1}^{'} (x - x_{0}) + z_{2}^{'} (y - y_{0})$

曲面 $z (x, y)$ 的梯度为 $(z_{1}^{'}, z_{2}^{'})$

将该向量扩展到切平面上: $(z_{1}^{'}, z_{2}^{'}, z_{1}^{'2} + z_{2}^{'2})$

$(z_{1}^{'}, z_{2}^{'}, - 1)$

$F (x, y, z) = 0$ 微分得切平面方程

$\frac{\partial F}{\partial x} (x - x_{0}) + \frac{\partial F}{\partial y} (y - y_{0}) + \frac{\partial F}{\partial z} (z - z_{0}) = 0$

梯度向量 $(- \frac{\partial F}{\partial x}, - \frac{\partial F}{\partial y}, \frac{1}{\frac{\partial F}{\partial z}} (\frac{\partial F}{\partial x}^{2} + \frac{\partial F}{\partial y}^{2}))$
法向量 $(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z})$