User:The Scarlet Letter/Tests

$$ \begin{array}{lcl} \frac{d\mathbf{t}}{ds}(s)=\kappa(s) \mathbf{p}(s) \\ \frac{d\mathbf{p}}{ds}(s)=-\kappa(s)\mathbf{t}(s)+\tau(s)\mathbf{b}(s) \\ \frac{d\mathbf{b}}{ds}(s)=-\tau(s)\mathbf{p}(s)\end{array}\Bigg\}, (\kappa>0).$$

$$ \begin{array}{lcl} \frac{dt_i}{ds}(s)=\kappa(s) p_i(s) \\ \frac{dp_i}{ds}(s)=-\kappa(s) t_i+\tau(s) b_i(s) \\ \frac{db_i}{ds}(s)=-\tau(s) p_i(s)\end{array}\Bigg\}, (i=1,2,3, \kappa>0).$$

$$ \mathbf{t}=\mathbf{v_1}, \mathbf{p}=\mathbf{v_2}, \mathbf{b}=\mathbf{v_3}$$

$$ \frac{d\mathbf{v_j}}{ds}=\sum_{k=1}^3 c_{jk}\mathbf{v_k}, (j=1,2,3).$$

$$ \mathbf{v_j}$$

$$ \frac{dv_j}{ds}=\sum_{k=1}^3 c_{jk}v_k, (j=1,2,3).$$

$$\begin{vmatrix} c_{jk} \end{vmatrix} = \begin{vmatrix} 0      & \kappa & 0    \\ -\kappa & 0     & \tau \\ 0      & -\tau  & 0 \end{vmatrix}$$

$$
 * c_{jk}|\le \frac{k}{3}, k>0.$$

$$v_j(0) = v_j^0,$$

$$|v_j^0|\le 1$$

$$v_j(s)\,\!$$

$$v_j(s)=v_j^0+\int_0^s \sum_{k=1}^3 c_{jk}(\sigma) v_k (\sigma)d\sigma$$

$$v_k\,\!$$

$$v_k^0$$

$$v_j^{(1)}=v_j^0+\int_0^s \sum_{k=1}^3 c_{jk} v_k^0 d\sigma.$$

$$v_j^{(1)}$$

$$v_j^{(2)}$$

$$v_k^{(n-1)}$$

$$v_j^{(n)}=v_j^0+\int_0^s \sum_{k=1}^3 c_{jk} v_k^{(n-1)} d\sigma, (j=1,2,3).$$

$$v_j^{(n)}$$

$$|v_j^{(1)}-v_j^0|\le ks;$$

$$|v_j^{(2)}-v_j^{(1)}|\le k^2\frac{s^2}{2},$$

$$|v_j^{(n)}-v_j^{(n-1)}|\le k^n\frac{n^2}{n!}.$$