面積変化率

導出

その1

$ \left.\mathrm d\bm a\right|_{\bm x=\bm\phi(\bm X,t)}=(\bm\phi\overleftarrow{\bm\nabla}\cdot\mathrm d\bm{X}_0)\times(\bm\phi\overleftarrow{\bm\nabla}\cdot\mathrm d\bm{X}_1)

$ = (\bm F\cdot\mathrm d\bm X_0)\times(\bm F\cdot\mathrm d\bm X_1)

$ \implies\forall\bm n;\mathrm d\bm a\cdot\bm F\cdot\bm n=((\bm F\cdot\mathrm d\bm X_0)\times(\bm F\cdot\mathrm d\bm X_1))\cdot\bm F\cdot\bm n

$ =J(\mathrm d\bm X_0\times \mathrm d\bm X_1)\cdot\bm n

$ =J\mathrm d\bm A\cdot\bm n

$ \implies\left.\mathrm d\bm a\right|_{\bm x=\bm\phi(\bm X,t)}\cdot\bm F=J\mathrm d\bm A

$ \underline{\iff \mathrm d\bm a=J{\bm F^{-1}}^\top\cdot\mathrm d\bm A\quad}_\blacksquare

その2

$ \left.\mathrm d\bm a\right|_{\bm x=\bm\phi(\bm X,t)}\cdot(\bm F\cdot\mathrm d\bm X_2)=\left.\mathrm dv\right|_{\bm x=\bm\phi(\bm X,t)}

$ =J\mathrm dV

$ =J(\mathrm d\bm X_0\times \mathrm d\bm X_1)\cdot\mathrm d\bm X_2

$ =J\mathrm d\bm A\cdot\mathrm d\bm X_2

$ \implies\left.\mathrm d\bm a\right|_{\bm x=\bm\phi(\bm X,t)}\cdot\bm F=J\mathrm d\bm A

$ \underline{\iff \mathrm d\bm a=J{\bm F^{-1}}^\top\cdot\mathrm d\bm A\quad}_\blacksquare

性質

$ \frac{|\mathrm d\bm a|}{|\mathrm d\bm A|}=J\sqrt{\hat{\mathrm d\bm A}\cdot\bm C^{-1}\cdot\hat{\mathrm d\bm A}}

導出

$ |\mathrm d\bm a|^2=\left(J{\bm F^{-1}}\cdot J{\bm F^{-1}}^\top\right):\mathrm d\bm A\mathrm d\bm A

$ =J^2\left(\bm F^\top\cdot\bm F\right)^{-1}:\mathrm d\bm A\mathrm d\bm A

$ = J^2\bm C^{-1}:\mathrm d\bm A\mathrm d\bm A

$ \underline{\implies\frac{|\mathrm d\bm a|}{|\mathrm d\bm A|}=J\sqrt{\hat{\mathrm d\bm A}\cdot\bm C^{-1}\cdot\hat{\mathrm d\bm A}}\quad}_\blacksquare

References