6-3

以下の式を証明せよ。

$$\frac{\vn \times \left\{ \left(\vn-\bm{\beta}(t')\right)\times \d... ...\{ \frac{\vn\times \left(\vn \times \bm{\beta}(t')\right)}{\kappa(t')} \right\}$$ (43)

6-3解答

$$\vn\times \left\{ \left(\vn-\bm{\beta}(t')\right)\times \dot{\bm{\beta}}(t')\right\} = -\dot{\bm{\beta}}(t')+ \left(\vn\cdot \bm{\beta}(t')\right)\dot... ...t')+ \left(\vn \cdot \dot{\bm{\beta}}(t')\right)\left(\vn-\bm{\beta}(t')\right)$$

$$\vn\times \left(\vn\times \bm{\beta}(t')\right) =-\bm{\beta}(t')+ \left( \vn \cdot \bm{\beta}(t')\right)\vn$$

$$\dI{t'}\left\{ \frac{\vn\times \left(\vn \times \bm{\beta}(t')\right)}{\kappa(t')} \right\} =\frac{ \vn\times \left(\vn \times \dot{\bm{\beta}}(t')\right)\ka... ...ft\{ \vn \times \left(\vn \times \bm{\beta}(t')\right)\right\} } {\kappa^2(t')}$$

$$= \frac{ \vn\times \left(\vn \times \dot{\bm{\beta}}(t')\right) \... ...ft\{ \vn \times \left(\vn \times \bm{\beta}(t')\right)\right\} } {\kappa^2(t')}$$

$$=\frac{ -\dot{\bm{\beta}}(t') +\left(\vn\cdot \dot{\bm{\beta}}(t'... ...(t') -\left(\vn\cdot \dot{\bm{\beta}}(t')\right)\bm{\beta}(t') } {\kappa^2(t')}$$

$$=\frac{-\dot{\bm{\beta}}(t')+\left(\vn\cdot \bm{\beta}(t')\right)... ...ft(\vn -\bm{\beta}(t')\right)\times \dot{\bm{\beta}}(t')\right\}}{\kappa^2(t')}$$

著者: 茅根裕司 chinone_at_astr.tohoku.ac.jp