定义1
设空间曲线段\(\Gamma\)有参数方程
\[ \left\{ \begin{aligned} & x = x(t), \\ & y = y(t), \\ & z = z(t), \end{aligned} \right. \quad (\alpha \le t \le \beta) \]
或用向量形式表示为
\[ \boldsymbol{r} = \boldsymbol{r}(t) \quad (\alpha \le t \le \beta) \]
其中\(x(t),y(t),z(t)\)都在区间\(I = [\alpha, \beta]\)上连续可导，且满足条件
\[ (x^\prime(t))^2 + (y^\prime(t))^2 + (z^\prime(t))^2 \ne 0 \quad (\alpha \le t \le \beta) \]
称满足这些条件的曲线为光滑曲线。
定理1
设光滑曲线\(\Gamma\)上有一点\(P_0(x(t_0), y(t_0), z(t_0))\)，则曲线\(\Gamma\)在\(P_0\)处的切线方程为
\[ \frac{x - x(t_0)}{x^\prime(t_0)} = \frac{y - y(t_0)}{y^(t_0)} = \frac{z - z(t_0)}{z^\prime(t_0))} \]
证：切线方向的一个向量为
\[
\boldsymbol{r}^\prime(t) = \lim \limits_{t \to 0}
\frac{\boldsymbol{r}(t) - \boldsymbol{r}(t_0)}{t - t_0} =
(x^\prime(t_0), y^\prime(t_0), z^\prime(t_0))
\]
称为曲线\(\Gamma\)在点\(P_0\)出的切向量，又由于切线过点\(P_0\)，所以切线方程为
\[
\frac{x - x(t_0)}{x^\prime(t_0)} = \frac{y - y(t_0)}{y^\prime(t_0)}
= \frac{z - z(t_0)}{z^\prime(t_0))}
\]
Q.E.D.
定理2
光滑曲线的弧长\(s\)也可以作为曲线的参数，即\(\boldsymbol{r} = \boldsymbol{r}(s)\)，且有
\[ \Vert \boldsymbol{r}^\prime(s) \Vert = 1 \]
证明：由微积分在几何上的应用一的定理2可知，在曲线参数为\(t \in [\alpha, \beta]\)时，弧长
\[
s(\Gamma) = \int_\alpha^\beta \Vert \boldsymbol{r}^{\prime}(t) \Vert
\mathrm{d} t
\]
记函数\(s(t)\)为从曲线起点\(\boldsymbol{r}(\alpha)\)沿着曲线到曲线上任意一点\(\boldsymbol{r}(t)\)这一段弧长，则
\[
s(t) = \int_\alpha^t \Vert \boldsymbol{r}^\prime(\tau)\Vert
\mathrm{d} \tau \quad (\alpha \le t \le \beta)
\]
而\(s(t)\)是一个变上限积分，将其对\(t\)求导得
\[
\frac{\partial s}{\partial t} = \Vert \boldsymbol{r}^\prime(t) \Vert
> 0 \tag {1}
\]
这表明\(s(t)\)是关于\(t\)的严格递增函数，意味着可以将\(t\)作为\(s\)得函数反解出来得到\(t =
t(s)\)，且这也是一个严格递增函数，所以\(s\)也可以作为弧长的参数，映射关系为
\[
s \rightarrow t \rightarrow \boldsymbol{r}
\]
记以弧长\(s\)为参数的曲线方程\(\boldsymbol{r} =
\boldsymbol{r}(s)\)，由式\((1)\)可知
\[
1 = \frac{\partial s}{\partial s} = \Vert \boldsymbol{r}^\prime(s)
\Vert
\]
Q.E.D.
定义2：曲率
设\(\Gamma: \boldsymbol{r} = \boldsymbol{r}(t)(\alpha \le t \le \beta)\)是一段光滑曲线，\(\boldsymbol{r}^\prime(t_0)\)与\(\boldsymbol{r}^\prime(t_0 + \Delta t)\)之间的夹角记为\(\Delta \theta\)，\(\boldsymbol{r}(t_0)\)与\(\boldsymbol{r}(t_0 + \Delta t)\)之间的弧长记为\(\Delta s\)，如果\(\lim \limits_{\Delta t \to 0} |\Delta \theta / \Delta s|\)存在，就称此极限为\(\Gamma\)在\(\boldsymbol{r}(t_0)\)处的曲率，记为
\[ k(t_0) = \lim \limits_{\Delta t \to 0} \left|\frac{\Delta \theta}{\Delta s} \right| \]
定理3
设曲线\(\Gamma: \boldsymbol{r} = \boldsymbol{r}(s)(s\)是弧长参数\()\)的每一点处有一个单位向量\(\boldsymbol{s}\)，记\(\boldsymbol{a}(s+\Delta s)\)和\(\boldsymbol{a}(s)\)之间的夹角为\(\Delta \theta\)，如果\(\boldsymbol{a}(s)\)可导，那么
\[ \Vert \boldsymbol{a}^\prime(s) \Vert = \lim \limits_{\Delta s \to 0} \left| \frac{\Delta \theta}{\Delta s} \right| \]
证明：由于
\[
\begin{aligned}
\Vert \boldsymbol{a}^\prime(s) \Vert & = \left\Vert \lim
\limits_{\Delta s \to 0} \frac{\boldsymbol{a}(s + \Delta s) -
\boldsymbol{a}(s)}{\Delta s} \right\Vert \\
& = \lim \limits_{\Delta s \to 0} \frac{\Vert \boldsymbol{a}(s +
\Delta s) - \boldsymbol{a}(s) \Vert}{\Vert \Delta s \Vert} \\
&= \lim \limits_{\Delta s \to 0} \left| \frac{2 \sin (\Delta
\theta / 2)}{\Delta s} \right| = \lim \limits_{\Delta s \to 0} \left|
\frac{ \sin (\Delta \theta / 2)}{\Delta \theta / 2} \right| \left|
\frac{2 \Delta \theta}{2\Delta s} \right| \\
& = \lim \limits_{\Delta s \to 0} \left| \frac{\Delta
\theta}{\Delta s} \right|
\end{aligned}
\]
Q.E.D.
定理4
设\(\Gamma: \boldsymbol{r} = \boldsymbol{r}(s)\)是一条以弧长\(s\)为参数的光滑曲线，且\(\boldsymbol{r}^{\prime\prime}(s)\)存在，那么它的曲率为
\[ k(s) = \Vert \boldsymbol{r}^{\prime\prime}(s) \Vert \]
证明：由定理2可知，曲线在\(s\)处的切线向量为单位向量，令\(\boldsymbol{a}(s) =
\boldsymbol{r}^\prime(s)\)，此时再由微积分在几何上的应用二定理3可知，
\[
k(s) = \lim \limits_{\Delta s \to 0} \left| \frac{\Delta
\theta}{\Delta s} \right| = \Vert \boldsymbol{r}^{\prime\prime}(s) \Vert
\]
Q.E.D.