一般来说二元函数$z = f(x,y)$在几何上表示一个曲面，这曲面被平面$z = c$（ $c$是常数）

所截得的曲线$L$的方程为

\[\left\{ \begin{gathered} z = f(x,y) \hfill \\ z = c \hfill \\ \end{gathered} \right.\]

这条曲线$L$在$xOy$面上的投影是一条平面曲线${L^*}$，它在$xOy$平面直角坐标系中的方程为

$$f(x,y) = c$$

对于曲线${L^*}$上的一切点，已知函数的函数值都是$c$，所以我们称平面曲线${L^*}$为函数$z = f(x,y)$的等值线.

若${f_x},{f_y}$不同时为零，则等值线$f(x,y) = c$上任意一点${P_0}({x_0},{y_0})$处的一个单位法向量为

\[\begin{gathered} n = \hfill \\ \frac{{({f_x}({x_0},{y_0}),{f_y}({x_0},{y_0}))}}{{\sqrt {f_x^2({x_0},{y_0}) + f_y^2({x_0},{y_0})} }} \hfill \\ = \frac{{\nabla f({x_0},{y_0})}}{{|f({x_0},{y_0})|}} \hfill \\ \end{gathered} \]

$f(x,y) = c$是曲面$z = f(x,y)$上的一条曲线，设曲线$f(x,y) = c$的参数方程为

\[\begin{gathered} x = \phi (t),y = \psi (t),z = \omega (t), \hfill \\ (\alpha \leqslant t \leqslant \beta )...\left( 1 \right) \hfill \\ \end{gathered} \]

曲面$z = f(x,y)$由隐式方程$F(x,y,z) = 0$给出，则

$$F(\phi (t),\psi (t),\omega (t)) \equiv 0$$

设$F(x,y,z) = 0$在点$({x_0},{y_0},{z_0})$处有连续一阶偏导数，$\phi '({t_0}),\psi '({t_0}),\omega '({t_0})$存在且不全为0，则有

$$\frac{{\text{d}}}{{{\text{d}}t}}[F(\phi (t),\psi (t),\omega (t))]{|_{t = {t_0}}} = 0$$

即有

\[\begin{gathered} {F_x}({x_0},{y_0},{z_0})\phi '({t_0}) + {F_y}({x_0},{y_0},{z_0})\psi '({t_0}) \hfill \\ + {F_z}({x_0},{y_0},{z_0})\omega '({t_0}) = 0 \hfill \\ \end{gathered} \]

向量

\[\begin{gathered} n = \hfill \\ \left( \begin{gathered} {F_x}({x_0},{y_0},{z_0}),{F_y}({x_0},{y_0},{z_0}), \hfill \\ {F_z}({x_0},{y_0},{z_0}) \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \]

即为曲线(1)在点$({x_0},{y_0},{z_0})$处的法线向量.

因为$z = \omega (t) = c$，有${F_z}({x_0},{y_0},{z_0}) = 0$，即等值线$f(x,y) = 0$在点${P_0}({x_0},{y_0})$的法线向量为

$$n = ({f_x}({x_0},{y_0}),{f_y}({x_0},{y_0}))$$