速度勾配tensor
\bm l:=\bm\nabla\bm v
性質
\bm l=\left.(\dot{\bm{F}}\cdot{\bm{F}^{-1}})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}
導出
\bm{l}:=\bm{\nabla}\bm{v}
=\bm{\nabla}\dot{\bm{\phi}}(\bm{\phi}^{-1}(\bm{x},t),t)
=\left.\bm{\nabla}\dot{\bm{\phi}}\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}\cdot\bm{\nabla}\bm{\phi}^{-1}(\bm{x},t)
=\dot{\bm{F}}(\bm{\phi}^{-1}(\bm{x},t),t)\cdot{\bm{F}(\bm{\phi}^{-1}(\bm{x},t),t)}^{-1}
=\left.(\dot{\bm{F}}\cdot{\bm{F}^{-1}})\right|_{\bm{X}=\bm{\phi}^{-1}(\bm{x},t)}
\bm l=\bm d+\bm w