有限要素法・流体力学による数値計算

\[ \begin{aligned} [\nabla\cdot\vec{\tau}] &=\left[\sum_{i=0}^{3}\vec{\delta}_{i}\frac{\partial}{\partial x_i}\cdot\sum_{j=0}^{3}\sum_{k=0}^{3}\vec{\delta}_{j}\vec{\delta}_{k}\tau_{jk}\right]\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}\sum_{k=0}^{3}\left[\vec{\delta}_{i}\frac{\partial}{\partial x_i}\cdot\vec{\delta}_{j}\vec{\delta}_{k}\tau_{jk}\right]\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}\sum_{k=0}^{3}\left[(\vec{\delta}_{i}\cdot\vec{\delta}_{j})\vec{\delta}_{k}\right]\frac{\partial}{\partial x_i}\tau_{jk}\\[6pt] &=\sum_{i=0}^{3}\sum_{j=0}^{3}\sum_{k=0}^{3}\delta_{ij}\vec{\delta}_{k}\frac{\partial\tau_{jk}}{\partial x_i}\\[6pt] &=\sum_{j=0}^{3}\sum_{k=0}^{3}\vec{\delta}_{k}\frac{\partial\tau_{jk}}{\partial x_i}\qquad(i=j)\\[6pt] &=\sum_{k=0}^{3}\vec{\delta}_{k}\sum_{j=0}^{3}\frac{\partial\tau_{jk}}{\partial x_i}\\[6pt] &=\sum_{k=0}^{3}\vec{\delta}_{k}\left(\frac{\partial\tau_{xk}}{\partial x}+\frac{\partial\tau_{yk}}{\partial y}+\frac{\partial\tau_{zk}}{\partial z}\right)\\[6pt] &=\vec{\delta}_{x}\left(\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{\partial z}\right)\\[6pt] &\quad+\vec{\delta}_{y}\left(\frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{zy}}{\partial z}\right)\\[6pt] &\quad+\vec{\delta}_{z}\left(\frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\tau_{zz}}{\partial z}\right) \end{aligned} \]

\[(\nabla\cdot\vec{\tau})\neq(\vec{\tau}\cdot\nabla)\]