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直交曲線座標での成分展開結果のまとめ

$\begin{displaymath}(\mathop{\hbox{\rm grad}}f)_i = \frac{1}{g_i} \frac{\partial f}{\partial x_i}\end{displaymath}$

$\begin{displaymath}\mathop{\hbox{\rm div}}\hbox{\rm\bf v} = \frac{1}{\eta} \frac{\partial }{\partial x_l} \left(\frac{\eta v_l}{g_l}\right)\end{displaymath}$

$\begin{displaymath}(\mathop{\hbox{\rm rot}}\hbox{\rm\bf v})_i = \frac{\epsilon_{ilm}}{g_l g_m} \frac{\partial (g_m v_m)}{\partial x_l}\end{displaymath}$

$\begin{displaymath}\Delta f = \frac{1}{\eta} \frac{\partial }{\partial x_l} \left(\frac{\eta}{g_l^2} \frac{\partial f}{\partial x_l}\right)\end{displaymath}$

$\begin{eqnarray*}(\Delta\hbox{\rm\bf v})_i &=& \frac{1}{g_i} \frac{\partial }{\... ...ac{\partial }{\partial x_l} \left(\frac{\eta}{g_l}\right)\right) \end{eqnarray*}$

$\begin{displaymath}\mathop{(\hbox{\rm\bf V}\cdot\mathop{\hbox{\rm grad}})}f = \frac{V_l}{g_l} \frac{\partial f}{\partial x_l}\end{displaymath}$

$\begin{displaymath}\{\mathop{(\hbox{\rm\bf V}\cdot\mathop{\hbox{\rm grad}})}\hbo... ...l} - \frac{V_l v_l}{g_i g_l} \frac{\partial g_l}{\partial x_i}\end{displaymath}$

$\begin{displaymath}e_{ij} = \frac{\delta_{ij}}{g_i} \frac{\partial g_i}{\partial... ...ac{\partial }{\partial x_j} \left(\frac{v_i}{g_i}\right)\right)\end{displaymath}$

$\begin{displaymath}\hbox{特に対角成分は}\quad e_{ii} = \frac{1}{g_i} \frac{\part... ...elta_{il} \frac{v_l}{g_i g_l} \frac{\partial g_i}{\partial x_l}\end{displaymath}$

$\begin{displaymath}(\hbox{但し、} \check\delta_{ij} \mathrel{\mathop=\limits^{\scriptscriptstyle def}}1 - \delta_{ij} ) \end{displaymath}$

Ichiro Tamagawa 平成11年9月24日