変形勾配tensorの時間微分

$ \dot{\bm F}=\frac{\partial\bm F}{\partial t}=\bm\nabla\dot{\bm\phi}

$ \dot{{\bm F}^{-1}}=\frac{\partial\bm F^{-1}}{\partial t}

$ =-{\bm F}^{-1}\cdot\dot{\bm F}\cdot{\bm F}^{-1}

$ =-{\bm F}^{-1}\cdot\left.\bm l\right|_{\bm x=\bm\phi(\bm X,t)}

$ {\dot{\bm F}}^{-1}={\frac{\partial\bm F}{\partial t}}^{-1}\neq\dot{{\bm F}^{-1}}

$ \dot{\bm F^\top}=\dot{\bm\nabla\bm\phi^\top}=\bm\nabla\dot{\bm\phi}^\top=\dot{\bm F}^\top

全微分

$ \mathrm d\dot{\bm\phi}=\dot{\bm F}\cdot\mathrm d\bm X+\ddot{\bm\phi}\mathrm dt

逆函数の微分

$ \frac{\partial\bm\nabla\bm\phi^{-1}}{\partial t}=\frac{\partial}{\partial t}\left(\left.{\bm F}^{-1}\right|_{\bm X=\bm\phi^{-1}(\bm x,t)}\right)

$ = \frac{\partial\bm\phi^{-1}}{\partial t}\cdot\left.\bm\nabla{\bm F}^{-1}\right|_{\bm X=\bm\phi^{-1}(\bm x,t)}+\left.\dot{{\bm F}^{-1}}\right|_{\bm X=\bm\phi^{-1}(\bm x,t)}

$ \bm x=\bm\phi(\bm\phi^{-1}(\bm x,t),t)

$ \bm0=\frac{\partial}{\partial t}\bm\phi(\bm\phi^{-1}(\bm x,t),t)=\left.\bm F\right|_{\bm X=\bm\phi^{-1}(\bm x,t)}\cdot\frac{\partial\bm\phi^{-1}}{\partial t}+\bm v

$ \iff \frac{\partial\bm\phi^{-1}}{\partial t}=-\left.{\bm F}^{-1}\right|_{\bm X=\bm\phi^{-1}(\bm x,t)}\cdot\bm v