问题及解答

证明: 对于光滑闭曲线 $C:\vec{r}=\vec{r}(s)$, 有

\[\int_C(\vec{r}\dot{\kappa}+\tau\vec{b})ds=0\]

成立. 这里 $s$ 为弧长参数, $\vec{b}$ 是副法向量(deputy normal vector)

\[\vec{b}=\frac{\dot{r}(s)\times\ddot{r}(s)}{|\dot{r}(s)\times\ddot{r}(s)|}.\]

根据 Frenet 公式

\[
\dot{e}_i(t)=\sum_{j}\omega_{ij}(t)e_j(t),
\]

可参见 问题724的回答.

\[
\dot{e}_2(s)=\omega_{21}(s)e_1(s)+\omega_{23}(s)e_3(s),
\]

其中 $e_1(s)=\dot{r}(s)$, $e_2(s)=\frac{\ddot{r}(s)}{|\ddot{r}(s)|}$, $e_3(s)=\frac{\dot{r}(s)\times\ddot{r}(s)}{|\ddot{r}(s)|}=\frac{\dot{r}(s)\times\ddot{r}(s)}{|\dot{r}(s)\times\ddot{r}(s)|}$. (注意, 有的书上也记为它们为 $\alpha,\beta,\gamma$, 分别称为切向量, 主法向量, 副法向量.) 代入上式, 得

\[
\begin{split}
\frac{d}{ds}\biggl(\frac{\ddot{r}(s)}{|\ddot{r}(s)|}\biggr)&=-\omega_{12}(s)\dot{r}(s)+\omega_{23}(s)\frac{\dot{r}(s)\times\ddot{r}(s)}{|\ddot{r}(s)|}\\
&=-\kappa(s)\dot{r}(s)+\tau(s)\vec{b}(s),
\end{split}
\]

又

\[r(s)\dot{\kappa}(s)+\dot{r}(s)\kappa(s)=(r(s)\kappa(s))\'_s,\]

因此

\[
\begin{split}
\int_C(r(s)\dot{\kappa}(s)+\tau(s)\vec{b}(s))ds&=\int_C\biggl[(r(s)\kappa(s))\'_s+\frac{d}{ds}\biggl(\frac{\ddot{r}(s)}{|\ddot{r}(s)|}\biggr)\biggr]ds\\
&=\int_C\frac{d}{ds}\biggl(r(s)\kappa(s)+\frac{\ddot{r}(s)}{|\ddot{r}(s)|}\biggr)ds\\
&=0.
\end{split}
\]

(或直接根据 Frenet 公式)

由

\[\dot{n}(s)=\dot{e_2}(s)=-\kappa(s)e_1(s)+\tau(s)e_3(s)=-\kappa(s)v(s)+\tau(s)b(s)\]

推出 $\tau(s)\vec{b}(s)=\dot{n}(s)+\kappa\dot{r}(s)$, 代入积分式

\[
\begin{split}
\int_C(r(s)\dot{\kappa}(s)+\tau(s)\vec{b}(s))ds&=\int_C(r(s)\dot{\kappa}(s)+\dot{n}(s)+\kappa(s)\dot{r}(s))ds\\
&=\int_C\frac{d}{ds}(n(s)+\kappa(s)\gamma(s))ds\\
&=0.
\end{split}
\]