spin tensor
$ \bm w:=\frac12(\bm l-\bm l^\top)={\cal\pmb W}:\bm l
性質
$ {\Large\bm\epsilon}を通じて渦度$ \bm\Omegaと相互変換できる $ \bm\Omega={\Large\bm\epsilon}:\bm w={\Large\bm\epsilon}:\bm l
$ {\Large\bm\epsilon}:{\cal\pmb W}={\Large\bm\epsilon}なのでどちらを使ってもいい
$ \bm w={\Large\bm\epsilon}\cdot\bm\Omega
剛体運動($ \bm U=\bm 0)のとき$ \bm w=\dot\bm R\cdot\bm R^\top
$ \bm Uの主軸が時間変化しないときも$ \bm w=\dot\bm R\cdot\bm R^\top
$ \because (\dot\bm U\cdot\bm U^{-1})^\top=\dot\bm U\cdot\bm U^{-1}
根拠となる導出は後述
極分解$ \bm F=\bm R\cdot\bm Uを使った書き換え $ \bm w={\cal\pmb W}:\left.((\dot\bm R\cdot\bm U+\bm R\cdot\dot\bm U)\cdot(\bm R\cdot\bm U)^{-1})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}
$ ={\cal\pmb W}:\left.((\dot\bm R\cdot\bm U+\bm R\cdot\dot\bm U)\cdot\bm U^{-1}\cdot\bm R^{-1})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}
$ ={\cal\pmb W}:\left.((\dot\bm R\cdot\bm U+\bm R\cdot\dot\bm U)\cdot\bm U^{-1}\cdot\bm R^{-1})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}
$ ={\cal\pmb W}:\left.(\dot\bm R\cdot\bm R^{-1}+\bm R\cdot\dot\bm U\cdot\bm U^{-1}\cdot\bm R^{-1})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}
$ ={\cal\pmb W}:\left.(\dot\bm R\cdot\bm R^\top+\bm R\cdot\dot\bm U\cdot\bm U^{-1}\cdot\bm R^\top)\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}
$ =\left.(\dot\bm R\cdot\bm R^\top+{\cal\pmb W}:(\bm R\cdot\dot\bm U\cdot\bm U^{-1}\cdot\bm R^\top))\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}