如果空间曲线以

\[\left\{ \begin{gathered} F\left( {x,y,z} \right) = 0 \hfill \\ G\left( {x,y,z} \right) = 0 \hfill \\ \end{gathered} \right.\]

的形式给出，$M\left( {{x_0},{y_0},{z_0}} \right)$是曲线上的一个点，又设$F,G$有对各个变量的连续偏导数，假设曲线方程在点$M\left( {{x_0},{y_0},{z_0}} \right)$的某一邻域内确定了一组函数$y = \varphi \left( x \right),z = \phi \left( x \right)$.在恒等式

\[\left\{ \begin{gathered} F\left[ {x,\varphi \left( x \right),\phi \left( x \right)} \right] = 0, \hfill \\ G\left[ {x,\varphi \left( x \right),\phi \left( x \right)} \right] = 0 \hfill \\ \end{gathered} \right.\]

两边分别对$x$求全导数，得

\[\left\{ \begin{gathered} \frac{{\partial F}}{{\partial x}} + \frac{{\partial F}}{{\partial y}}\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{{\partial F}}{{\partial z}}\frac{{{\text{d}}z}}{{{\text{d}}x}} = 0, \hfill \\ \frac{{\partial G}}{{\partial x}} + \frac{{\partial G}}{{\partial y}}\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{{\partial G}}{{\partial z}}\frac{{{\text{d}}z}}{{{\text{d}}x}} = 0 \hfill \\ \end{gathered} \right.\]

解得

\[\begin{gathered} \frac{{{\text{d}}z}}{{{\text{d}}x}} = \frac{{\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial y}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial y}}}}{{\frac{{\partial F}}{{\partial z}}\frac{{\partial G}}{{\partial y}} - \frac{{\partial G}}{{\partial z}}\frac{{\partial F}}{{\partial y}}}}, \hfill \\ \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial z}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial z}}}}{{\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}}}} \hfill \\ \end{gathered} \]

将分母改写为形式相同，得

\[\begin{gathered} \frac{{{\text{d}}z}}{{{\text{d}}x}} = \frac{{\frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial y}} - \frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial y}}}}{{\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}}}}, \hfill \\ \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial z}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial z}}}}{{\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}}}} \hfill \\ \end{gathered} \]

上面的解要求

\[\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}} \ne 0\]

即

\[{\left. {\frac{{\partial \left( {F,G} \right)}}{{\partial \left( {x,y} \right)}}} \right|_{\left( {{x_0},{y_0},{z_0}} \right)}} \ne 0.\]

因此，只要$F,G$在点$M\left( {{x_0},{y_0},{z_0}} \right)$的某一邻域内有对各个变量的连续偏导数，曲线方程在点$M\left( {{x_0},{y_0},{z_0}} \right)$的某一邻域内就确定了一组连续且具有连续导数的函$y = \varphi \left( x \right),z = \phi \left( x \right)$这和前面隐函数求导得条件是相同的.

于是$T = \left( {1,\varphi '\left( {{x_0}} \right),\phi '\left( {{x_0}} \right)} \right)$是曲线在点M处的一个切向量（之所以说“一个”，是因为还有另一个大小相等方向相反的切向量）为

\[T = \left( \begin{gathered} 1,{\left. {\frac{{\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial z}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial z}}}}{{\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}}}}} \right|_M}, \hfill \\ {\left. {\frac{{\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial y}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial y}}}}{{\frac{{\partial F}}{{\partial z}}\frac{{\partial G}}{{\partial y}} - \frac{{\partial G}}{{\partial z}}\frac{{\partial F}}{{\partial y}}}}} \right|_M} \hfill \\ \end{gathered} \right)\]

(参考空间曲线以参数方程给出时的推导)

把上面的切向量T乘以$\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}} \ne 0$，得

\[T' = \left( \begin{gathered} {\left. {\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}}} \right|_M}, \hfill \\ {\left. {\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial z}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial z}}} \right|_M}, \hfill \\ {\left. {\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial y}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial y}}} \right|_M} \hfill \\ \end{gathered} \right)\]

这也是曲线在点M处的一个切向量，由此在点$M\left( {{x_0},{y_0},{z_0}} \right)$处的切线方程为

\[\begin{gathered} \frac{{x - {x_0}}}{{{{\left. {\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}}} \right|}_M}}} = \hfill \\ \frac{{y - {y_0}}}{{{{\left. {\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial z}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial z}}} \right|}_M}}} = \hfill \\ \frac{{z - {z_0}}}{{{{\left. {\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial y}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial y}}} \right|}_M}}} \hfill \\ \end{gathered} \]

曲线在点$M\left( {{x_0},{y_0},{z_0}} \right)$处的法平面方程为

\[\begin{gathered} {\left. {\frac{{\partial F}}{{\partial y}}\frac{{\partial G}}{{\partial z}} - \frac{{\partial G}}{{\partial y}}\frac{{\partial F}}{{\partial z}}} \right|_M}\left( {x - {x_0}} \right) + \hfill \\ {\left. {\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial z}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial z}}} \right|_M}\left( {y - {y_0}} \right) + \hfill \\ {\left. {\frac{{\partial G}}{{\partial x}}\frac{{\partial F}}{{\partial y}} - \frac{{\partial F}}{{\partial x}}\frac{{\partial G}}{{\partial y}}} \right|_M}\left( {z - {z_0}} \right) = 0 \hfill \\ \end{gathered} \]