9.5 多元函数微分的几何应用

空间曲面的切平面与法线

曲面 $F(x,y,z)=0$ 在其上一点 $M(x_0,y_0,z_0)$ 处的法向量为 ${({\displaystyle \frac{\partial F}{\partial x}},{\displaystyle \frac{\partial F}{\partial y}},{\displaystyle \frac{\partial F}{\partial z}})|}_{(x_0,y_0,z_0)}$

有法向量，即可得到切平面与法线方程。

空间曲线的切线与法平面

参数方程表达

曲线 $\{\begin{array}{l}x=x(t)\\ y=y(t)\\ z=z(t)\end{array}$ 在 $P(x_0,y_0,z_0)$ 处的切向量为 $(x^{\prime }(t),y^{\prime }(t),z^{\prime }(t))$

曲面交线表达

曲线 $\{\begin{array}{l}F(x,y,z)=0\\ G(z,y,z)=0\end{array}$ 在 $P(x_0,y_0,z_0)$ 处的切向量为

$${({\displaystyle \frac{\partial F}{\partial x}},{\displaystyle \frac{\partial F}{\partial y}},{\displaystyle \frac{\partial F}{\partial z}})\times ({\displaystyle \frac{\partial G}{\partial x}},{\displaystyle \frac{\partial G}{\partial y}},{\displaystyle \frac{\partial G}{\partial z}})|}_{(x_0,y_0,z_0)}$$

即：

$$\left|\begin{array}{ccc}\mathit{i}& \mathit{j}& \mathit{k}\\ {\displaystyle \frac{\partial F}{\partial x}}& {\displaystyle \frac{\partial F}{\partial y}}& {\displaystyle \frac{\partial F}{\partial z}}\\ {\displaystyle \frac{\partial G}{\partial x}}& {\displaystyle \frac{\partial G}{\partial y}}& {\displaystyle \frac{\partial G}{\partial z}}\end{array}\right|$$

有切向量，即可得到切线方程与法平面方程。