有限要素法・流体力学による数値計算

\[ \begin{aligned} [\nabla\times\vec{v}] &=\left[\sum_{i=0}^{3}\vec{\delta}_{i}\frac{\partial}{\partial x_i}\times\sum_{j=0}^{3}\vec{\delta}_{j}v_j\right]\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}\left[\vec{\delta}_{i}\frac{\partial}{\partial x_i}\times\vec{\delta}_{j}v_j\right]\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}[\vec{\delta}_{i}\times\vec{\delta}_{j}]\frac{\partial}{\partial x_i}v_j\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}\sum_{k=0}^{3}\varepsilon_{ijk}\vec{\delta}_{k}\frac{\partial v_j}{\partial x_i}\\[6pt] &=\varepsilon_{123}\vec{\delta}_{z}\frac{\partial v_y}{\partial x_x}+\varepsilon_{231}\vec{\delta}_{x}\frac{\partial v_z}{\partial x_y}+\varepsilon_{312}\vec{\delta}_{y}\frac{\partial v_x}{\partial x_z}\\[6pt] &\quad+\varepsilon_{321}\vec{\delta}_{x}\frac{\partial v_y}{\partial x_z}+\varepsilon_{213}\vec{\delta}_{z}\frac{\partial v_x}{\partial x_y}+\varepsilon_{132}\vec{\delta}_{y}\frac{\partial v_z}{\partial x_x}\\[6pt] &=\vec{\delta}_{x}\left(\frac{\partial v_z}{\partial x_y}-\frac{\partial v_y}{\partial x_z}\right)+\vec{\delta}_{y}\left(\frac{\partial v_x}{\partial x_z}-\frac{\partial v_z}{\partial x_x}\right)+\vec{\delta}_{z}\left(\frac{\partial v_y}{\partial x_x}-\frac{\partial v_x}{\partial x_y}\right) \end{aligned} \]