return 0;
wmake

张量公式查询

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本文介绍笛卡尔坐标系下的张量操作。

基本操作

\begin{equation} \nabla p = \left[\begin{matrix} \frac{\partial p}{\partial x} \\ \frac{\partial p}{\partial y} \\ \frac{\partial p}{\partial z} \end{matrix} \right] \end{equation}
\begin{equation} \nabla \cdot(\nabla p)=\nabla ^2p=\frac{\partial^2p}{\partial x^2}+\frac{\partial^2p}{\partial y^2}+\frac{\partial^2p}{\partial z^2} \end{equation}
\begin{equation} \mathbf{U} \cdot \mathbf{V} =\mathbf{U}^\rT\cdot\mathbf{V}= [u_1, u_2, u_3] \left[\begin{matrix} v_1 \\ v_2 \\ v_3 \end{matrix} \right]=u_1v_1+u_2v_2+u_3v_3 \end{equation}
\begin{equation} \mathbf{U} \times \mathbf{V}=\left[ \begin{matrix} u_2v_3-u_3v_2\\ u_3v_1-u_1v_3\\ u_1v_2-u_2v_1\\ \end{matrix} \right] \end{equation}
\begin{equation} \nabla \cdot \mathbf{U} = \frac{\partial u_1}{\partial x}+\frac{\partial u_2}{\partial y}+\frac{\partial u_3}{\partial z} \end{equation}
\begin{equation}\label{gradientV} \nabla \mathbf{U} = \left[ \begin{matrix} \frac{\partial u_1}{\partial x} & \frac{\partial u_2}{\partial x} & \frac{\partial u_3}{\partial x}\\ \frac{\partial u_1}{\partial y} & \frac{\partial u_2}{\partial y} & \frac{\partial u_3}{\partial y} \\ \frac{\partial u_1}{\partial z} & \frac{\partial u_2}{\partial z} & \frac{\partial u_3}{\partial z}\\ \end{matrix} \right] \end{equation}
\begin{equation} \nabla \cdot(\nabla \mathbf{U})= \left[ \begin{matrix} \frac{\partial}{\partial x}\left(\frac{\partial u_1}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_1}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_1}{\partial z}\right)\\ \frac{\partial}{\partial x}\left(\frac{\partial u_2}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_2}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_2}{\partial z}\right)\\ \frac{\partial}{\partial x}\left(\frac{\partial u_3}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_3}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_3}{\partial z}\right)\\ \end{matrix} \right] \end{equation}
\begin{equation} \mathbf{U}\mathbf{V}=\mathbf{U}\otimes\mathbf{V}=\mathbf{U}\cdot\mathbf{V}^\rT=\left[ \begin{matrix} u_1 v_1 & u_1 v_2 & u_1 v_3\\ u_2 v_1 & u_2 v_2 & u_2 v_3\\ u_3 v_1 & u_3 v_2 & u_3 v_3 \end{matrix} \right] \end{equation}
\begin{equation} \nabla\times\mathbf{U}=\left[ \begin{matrix} \frac{\partial u_3}{\partial y}-\frac{\partial u_2}{\partial z}\\ \frac{\partial u_1}{\partial z}-\frac{\partial u_3}{\partial x}\\ \frac{\partial u_2}{\partial x}-\frac{\partial u_1}{\partial y}\\ \end{matrix} \right] \end{equation}
\begin{equation} \nabla \cdot \tau = \left[\begin{matrix} \frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{\partial z} \\ \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{zy}}{\partial z} \\ \frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\tau_{zz}}{\partial z} \end{matrix}\right] \end{equation}
\begin{equation} \tau:\tau=\tau_{11}\tau_{11}+\tau_{12}\tau_{21}+\tau_{13}\tau_{31}+ \tau_{21}\tau_{12}+\tau_{22}\tau_{22}+\tau_{23}\tau_{32}+ \tau_{31}\tau_{13}+\tau_{32}\tau_{23}+\tau_{33}\tau_{33} \end{equation}
\begin{equation} \mathrm{tr} \left(\tau\right)=\tau_{xx}+\tau_{yy}+\tau_{zz} \end{equation}
\begin{equation} \mathrm{symm} \left(\tau\right)=\frac{\tau+\tau^T}{2} \end{equation}
\begin{equation} \mathrm{skew} \left(\tau\right)=\frac{\tau-\tau^T}{2} \end{equation}
\begin{equation} \mathrm{dev} \left(\tau\right)=\tau-\frac{1}{3}\mathrm{tr}\left(\tau\right)\mathbf{I} \end{equation}
\begin{equation} \mathrm{dev}2 \left(\tau\right)=\tau-\frac{2}{3}\mathrm{tr}\left(\tau\right)\mathbf{I} \end{equation}
\begin{equation} \mathrm{hyd} \left(\tau\right)=\frac{1}{3}\mathrm{tr}\left(\tau\right)\mathbf{I} \end{equation}
\begin{equation} \tau=\mathrm{hyd}\left(\tau\right)+\mathrm{dev}\left(\tau\right) \end{equation}

张量运算

\begin{equation} \mathbf{U}+\mathbf{V}=\mathbf{V}+\mathbf{U} \end{equation}
\begin{equation} \alpha\mathbf{U}=\mathbf{U}\alpha \end{equation}
\begin{equation} \mathbf{U}\cdot\mathbf{V}=\mathbf{V}\cdot\mathbf{U} \end{equation}
\begin{equation} \mathbf{U}\times\mathbf{V}=-\mathbf{V}\times\mathbf{U} \end{equation}
\begin{equation} \mathbf{U}\times\left(\mathbf{V}\times\mathbf{W}\right)\neq\left(\mathbf{U}\times\mathbf{V}\right)\times\mathbf{W} \end{equation}
\begin{equation} \nabla\cdot\left(\nabla\times\mathbf{U}\right)=0 \end{equation}
\begin{equation} \nabla\times\nabla\alpha=0 \end{equation}
\begin{equation} \nabla (\alpha p)=\alpha\nabla p+p\nabla\alpha \end{equation}
\begin{equation} \nabla \cdot(\alpha \mathbf{U})=\alpha\nabla\cdot \mathbf{U}+\mathbf{U} \cdot \nabla\alpha=\alpha\nabla\cdot\mathbf{U}+\nabla\alpha\cdot\mathbf{U} \end{equation}
\begin{equation} \nabla \times (\alpha \mathbf{U})=\alpha\nabla\times \mathbf{U}+\left(\nabla\alpha\right) \times\mathbf{U} \end{equation}
\begin{equation} \nabla(\mathbf{U}\cdot\mathbf{V})=\mathbf{U}\times(\nabla\times\mathbf{V})+\mathbf{V}\times(\nabla\times\mathbf{U})+(\mathbf{U}\cdot\nabla)\mathbf{V}+(\mathbf{V}\cdot\nabla)\mathbf{U} \end{equation}
\begin{equation} \nabla\cdot(\mathbf{U}\times\mathbf{V})=\mathbf{V}\cdot(\nabla\times\mathbf{U})-\mathbf{U}\cdot(\nabla\times\mathbf{V}) \end{equation}
\begin{equation} \nabla\times(\mathbf{U}\times\mathbf{V})=\mathbf{U}(\nabla\cdot\mathbf{V})-\mathbf{V}(\nabla\cdot\mathbf{U})+(\mathbf{V}\cdot\nabla)\mathbf{U}-(\mathbf{U}\cdot\nabla)\mathbf{V} \end{equation}
\begin{equation} \nabla\times(\nabla\times\mathbf{U})=\nabla(\nabla\cdot\mathbf{U})-\nabla^2\mathbf{U} \end{equation}
\begin{equation} (\nabla\times\mathbf{U})\times\mathbf{U}=\mathbf{U}\cdot(\nabla\mathbf{U})-\nabla(\mathbf{U}\cdot\mathbf{U}) \end{equation}
\begin{equation} \nabla\cdot\nabla\mathbf{U}=\nabla(\nabla\cdot\mathbf{U})-\nabla\times(\nabla\times\mathbf{U}) \end{equation}
\begin{equation} \nabla\cdot(\mathbf{U} \mathbf{U})=\mathbf{U} \cdot \nabla \mathbf{U}+\mathbf{U} \nabla \cdot \mathbf{U} \end{equation}
\begin{equation} \nabla\cdot(\alpha \tau)=\tau \cdot\nabla \alpha + \alpha \nabla \cdot \tau \end{equation}
\begin{equation} \mathrm{tr}\left(\nabla\mathbf{U}\right)=\nabla\cdot\mathbf{U}=\mathrm{tr}\left(\nabla\mathbf{U}^{\mathrm{T}}\right) \end{equation}
\begin{equation} \tau=\frac{1}{2}\left(\tau+\tau^\mathrm{T}\right)+\frac{1}{2}\left(\tau-\tau^\mathrm{T}\right)=\mathrm{symm}\left(\tau\right)+\mathrm{skew}\left(\tau\right) \end{equation}

更新历史
2018.04.20经G8S7指正修正公式\eqref{gradientV},重整页面

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