備忘録~Frameが関数なときの微分~

　これも備忘録だ．もし証明が必要ならこれをコピペする気だ．

Frameが関数なときの微分

使い方

曲線$\gamma(s)$とせれ・フレネ枠$\{\bm{t},\bm{n},\bm{b}\}$に対して曲線上のベクトル場
$$X(s)=f(s)\bm{t}+g(s)\bm{n}+h(s)\bm{b}$$
の発散は
$$\begin{array}{rcl} \grad\cdot X(s) &=&\d \qty(\frac{\partial \gamma}{\partial x}\frac{d}{ds} + \frac{\partial \gamma}{\partial y}\frac{d}{ds} + \frac{\partial \gamma}{\partial z}\frac{d}{ds})\qty(\qty(\bm{t},\bm{n},\bm{b})\vecTri{f}{g}{h})\\ &=& \d (\bm{e}_1,\bm{e}_2,\bm{e}_3)\qty(\frac{\partial \gamma}{\partial x}, \frac{\partial \gamma}{\partial y}, \frac{\partial \gamma}{\partial z})\cdot\qty{\qty(\bm{t},\bm{n},\bm{b})'\vecTri{f}{g}{h} + \qty(\bm{t},\bm{n},\bm{b}) \vecTri{f}{g}{h}'}\\ &=& \d (\bm{e}_1,\bm{e}_2,\bm{e}_3)\qty(\frac{\partial \gamma}{\partial x}, \frac{\partial \gamma}{\partial y}, \frac{\partial \gamma}{\partial z})\cdot\qty{\qty(\bm{t},\bm{n},\bm{b})\qty(\qty(\begin{array}{ccc} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{array})^t - \qty(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}))\vecTri{f}{g}{h}} \end{array}$$
また$\bm{t} = \bm{a}_1$，$\bm{t} = \bm{a}_2$，$\bm{t} = \bm{a}_3$としたとき(特に意味はない．僕がタイプミスしただけだ)
$$\begin{array}{rcl} \bm{e}_1 &=& (\bm{e}_1\cdot\bm{a}_1)\bm{a}_1+(\bm{e}_1\cdot\bm{a}_2)\bm{a}_2+(\bm{e}_1\cdot\bm{a}_3)\bm{a}_3 \\ \bm{e}_2 &=& (\bm{e}_2\cdot\bm{a}_1)\bm{a}_1+(\bm{e}_2\cdot\bm{a}_2)\bm{a}_2+(\bm{e}_2\cdot\bm{a}_3)\bm{a}_3 \\ \bm{e}_3 &=& (\bm{e}_3\cdot\bm{a}_1)\bm{a}_1+(\bm{e}_3\cdot\bm{a}_2)\bm{a}_2+(\bm{e}_3\cdot\bm{a}_3)\bm{a}_3 \end{array}$$
より
$$\vecTri{\bm{e}_1}{\bm{e}_2}{\bm{e}_3} = \qty(\begin{array}{ccc} (\bm{e}_1\cdot\bm{t}) & (\bm{e}_1\cdot\bm{n}) & (\bm{e}_1\cdot\bm{b})\\ (\bm{e}_2\cdot\bm{t}) & (\bm{e}_2\cdot\bm{n}) & (\bm{e}_2\cdot\bm{b})\\ (\bm{e}_3\cdot\bm{t}) & (\bm{e}_3\cdot\bm{n}) & (\bm{e}_3\cdot\bm{b})\\ \end{array})\vecTri{\bm{t}}{\bm{n}}{\bm{b}} = \vecTri{\bm{e}_1}{\bm{e}_2}{\bm{e}_3}\vecTri{\bm{t}}{\bm{n}}{\bm{b}}({\bm{e}_1,}\ {\bm{e}_2,}\ {\bm{e}_3})$$
よって
$$\begin{array}{rcl} \grad\cdot X(s) &=& \d (\bm{e}_1,\bm{e}_2,\bm{e}_3)\qty(\frac{\partial \gamma}{\partial x}, \frac{\partial \gamma}{\partial y}, \frac{\partial \gamma}{\partial z})\cdot\qty{\qty(\bm{t},\bm{n},\bm{b})\qty(\qty(\begin{array}{ccc} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{array})^t - \qty(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}))\vecTri{f}{g}{h}}\\ &=& \d \qty(\bm{t},\bm{n},\bm{b})\vecTri{\bm{e}_1}{\bm{e}_2}{\bm{e}_3}({\bm{t}},\ {\bm{n}},\ {\bm{b}})\qty(\frac{\partial \gamma}{\partial x}, \frac{\partial \gamma}{\partial y}, \frac{\partial \gamma}{\partial z})\cdot\qty{\qty(\bm{t},\bm{n},\bm{b})\qty(\qty(\begin{array}{ccc} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{array})^t - \qty(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}))\vecTri{f}{g}{h}}\\ &=& \d \vecTri{\bm{e}_1}{\bm{e}_2}{\bm{e}_3}({\bm{t}},\ {\bm{n}},\ {\bm{b}})\qty(\frac{\partial \gamma}{\partial x}, \frac{\partial \gamma}{\partial y}, \frac{\partial \gamma}{\partial z})\cdot\qty{\qty(\qty(\begin{array}{ccc} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{array})^t - \qty(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}))\vecTri{f}{g}{h}} \end{array}$$
以上よりセレ・フレネ行列の固有値が1かが重要ということになる．ちなみにその固有値は$0,\sqrt{-(\kappa^2+\tau^2)}$なためそれは起こらない．