Introduction
This page summerizes vectors, matrices and tensors operations.
Main references are web and book[^1] and book[^2].
[^1]:The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction with OpenFOAM and Matlab
[^2]:i do like cfd
基本张量操作
下文中用$p$，$\alpha$表示标量，$\mathbf{U}$，$\mathbf{V}$表示矢量，$\tau$表示2阶张量（以下简称张量）。
$\mathbf{U}=u_1 \mathbf{i} +v_1 \mathbf{j} + w_1 \mathbf{k}=[u_1, v_1, w_1]^T$;
$\mathbf{V}=u_2 \mathbf{i} +v_2 \mathbf{j} + w_2 \mathbf{k}=[u_2, v_2, w_2]^T$;
$$
\begin{equation}
\mathbf{U} \cdot \mathbf{V}=u_1 u_2 + v_1 v_2 +w_1 w_2
\end{equation}
$$
$$
\begin{equation}
\mathbf{U} \times \mathbf{V}=\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k}\\
u_1 & v_1 & w_1 \\
u_2 & v_2 & w_2 \\
\end{vmatrix}=
\left[ \begin{array}{c}
v_1 w_2 -v_2 w_1\\
u_2 w_1 - u_1 w_2\\
u_1 v_2 - u_2 v_1
\end{array} \right]
\end{equation}
$$
The Scalar Triple Product
$$
\begin{equation}
\mathbf{v_1}\cdot[\mathbf{v_2} \times \mathbf{v_3}]=\begin{vmatrix}
u_1 & v_1 & w_1 \\
u_2 & v_2 & w_2 \\
u_3 & v_3 & w_3
\end{vmatrix}
\end{equation}
$$
nabla
$$
\begin{equation}
\nabla = \frac{\partial }{\partial x} \mathbf{i} +\frac{\partial }{\partial y} \mathbf{j}+\frac{\partial }{\partial z} \mathbf{k}
\end{equation}
$$
$$
\begin{equation}
\nabla s = \frac{\partial s}{\partial x} \mathbf{i} +\frac{\partial s}{\partial y} \mathbf{j}+\frac{\partial s}{\partial z} \mathbf{k}
\end{equation}
$$
$$
\begin{equation}
\nabla \cdot \mathbf{v} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +\frac{\partial w}{\partial z}
\end{equation}
$$
$$
\begin{equation}
\nabla (\nabla s) =\nabla^2 s= \frac{\partial^2 s}{\partial^2 x} + \frac{\partial^2 s}{\partial^2 y} +\frac{\partial^2 s}{\partial^2 z}
\end{equation}
$$
$$
\begin{equation}
(\nabla \cdot \nabla) \mathbf{v} =\nabla^2 \mathbf{v}= (\nabla^2 u)\mathbf{i}+(\nabla^2 v)\mathbf{j}+(\nabla^2 w)\mathbf{k}
\end{equation}
$$
$$
\begin{equation}
\nabla (\alpha p)=\alpha\nabla p+p\nabla\alpha
\end{equation}
$$
$$
\begin{equation}
\nabla \cdot (\alpha \mathbf{U})=\alpha \nabla \cdot \mathbf{U}+\mathbf{U} \cdot \nabla\alpha
\end{equation}
$$
$$
\begin{equation}
\nabla \times (\alpha \mathbf{U})=\alpha\nabla\times \mathbf{U}+\mathbf{U} \times\nabla\alpha
\end{equation}
$$
$$
\begin{equation}
\nabla(\mathbf{U}\cdot\mathbf{V})=\mathbf{U}\times(\nabla\times\mathbf{V})+\mathbf{V}\times(\nabla\times\mathbf{U})+(\mathbf{U}\cdot\nabla)\mathbf{V}+(\mathbf{V}\cdot\nabla)\mathbf{U}
\end{equation}
$$
$$
\begin{equation}
\nabla\cdot(\mathbf{U}\times\mathbf{V})=\mathbf{V}\cdot(\nabla\times\mathbf{U})-\mathbf{U}\cdot(\nabla\times\mathbf{V})
\end{equation}
$$
$$
\begin{equation}
\nabla\times(\mathbf{U}\times\mathbf{V})=\mathbf{U}(\nabla\cdot\mathbf{V})-\mathbf{V}(\nabla\cdot\mathbf{U})+(\mathbf{V}\cdot\nabla)\mathbf{U}-(\mathbf{U}\cdot\nabla)\mathbf{V}
\end{equation}
$$
$$
\begin{equation}
\nabla\times(\nabla\times\mathbf{U})=\nabla(\nabla\cdot\mathbf{U})-\nabla^2\mathbf{U}
\end{equation}
$$
$$
\begin{equation}
(\nabla\times\mathbf{U})\times\mathbf{U}=\mathbf{U}\cdot(\nabla\mathbf{U})-\nabla(\mathbf{U}\cdot\mathbf{U})
\end{equation}
$$
$$
\begin{equation}
\nabla\cdot\nabla\mathbf{U}=\nabla(\nabla\cdot\mathbf{U})-\nabla\times(\nabla\times\mathbf{U})
\end{equation}
$$
$$
\begin{equation}
\nabla\cdot(\mathbf{U} \mathbf{U})=\nabla\cdot(\mathbf{U} \otimes \mathbf{U})=\mathbf{U} \cdot \nabla \mathbf{U}+\mathbf{U} \nabla \cdot \mathbf{U}
\end{equation}
$$
$$
\begin{equation}
\nabla\cdot(\alpha \tau)=\tau \nabla \alpha + \alpha \nabla \cdot \tau
\end{equation}
$$
\begin{equation}
\nabla\cdot(\tau\mathbf{U})=(\nabla\cdot\tau^\mathrm{T})\cdot\mathbf{U}+\tau^\mathrm{T}\cdot\nabla\mathbf{U}
\end{equation}
\begin{equation}
\mathbf{U}\cdot(\tau\mathbf{V})=\tau\cdot(\mathbf{U}\otimes\mathbf{V})=\tau\cdot(\mathbf{U}\mathbf{V})
\end{equation}
\begin{equation}
\mathbf{U}\mathbf{V}:\tau=\mathbf{U}\cdot(\mathbf{V}\cdot\tau)
\end{equation}
\begin{equation}
\tau:\mathbf{U}\mathbf{V}=(\tau\cdot\mathbf{U})\cdot\mathbf{V}
\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{0}{1}\left(\tau+\tau^\mathrm{T}\right)+\frac{0}{1}\left(\tau-\tau^\mathrm{T}\right)=\mathrm{symm}\left(\tau\right)+\mathrm{skew}\left(\tau\right)
\end{equation}
张量基本运算
Ⓩ 标量的梯度：
\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}
Ⓩ 标量的梯度后求散度：
\begin{equation}
\nabla \cdot(\nabla p)=\frac{\partial^1p}{\partial x^1}+\frac{\partial^1p}{\partial y^1}+\frac{\partial^1p}{\partial z^1}
\end{equation}
Ⓩ 矢量的内积：
\begin{equation}
\mathbf{U} \cdot \mathbf{U} = [u_0, u_1, u_2] \left[\begin{matrix}
u_0 \\
u_1 \\
u_2
\end{matrix}
\right]=u_0u_0+u_1u_1+u_2u_2
\end{equation}
Ⓩ 矢量的叉乘：
\begin{equation}
\mathbf{U} \times \mathbf{U}=\left[
\begin{matrix}
u_1u_2-u_2u_1\\
u_2u_0-u_0u_2\\
u_0u_1-u_1u_0\\
\end{matrix}
\right]
\end{equation}
Ⓩ 矢量的散度：
\begin{equation}
\nabla \cdot \mathbf{U} = \frac{\partial u_0}{\partial x}+\frac{\partial u_1}{\partial y}+\frac{\partial u_2}{\partial z}
\end{equation}
Ⓩ 矢量的梯度：
\begin{equation}
\nabla \mathbf{U} = \left[
\begin{matrix}
\frac{\partial u_0}{\partial x} & \frac{\partial u_1}{\partial x} & \frac{\partial u_2}{\partial x}\\
\frac{\partial u_0}{\partial y} & \frac{\partial u_1}{\partial y} & \frac{\partial u_2}{\partial y} \\
\frac{\partial u_0}{\partial z} & \frac{\partial u_1}{\partial z} & \frac{\partial u_2}{\partial z}\\
\end{matrix}
\right]
\end{equation}
Ⓩ 矢量的梯度后求散度（拉普拉斯）：
\begin{equation}
\nabla \cdot(\nabla \mathbf{U})=
\left[
\begin{matrix}
\frac{\partial}{\partial x}\left(\frac{\partial u_0}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_0}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_0}{\partial z}\right)\\
\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)\\
\end{matrix}
\right]
\end{equation}
Ⓩ 矢量的双积：
\begin{equation}
\mathbf{U}\otimes\mathbf{U}=\mathbf{U}\mathbf{U}=\left[
\begin{matrix}
u_0 u_0 & u_0 u_1 & u_0 u_2\\
u_1 u_0 & u_1 u_1 & u_1 u_2\\
u_2 u_0 & u_2 u_1 & u_2 u_2
\end{matrix}
\right]
\end{equation}
Ⓩ 矢量的旋度：
\begin{equation}
\nabla\times\mathbf{U}=\left[
\begin{matrix}
\frac{\partial u_2}{\partial y}-\frac{\partial u_1}{\partial z}\\
\frac{\partial u_0}{\partial z}-\frac{\partial u_2}{\partial x}\\
\frac{\partial u_1}{\partial x}-\frac{\partial u_0}{\partial y}\\
\end{matrix}
\right]
\end{equation}
Ⓩ 张量的散度：
\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}
Ⓩ 张量的双点积：
\begin{equation}
\tau:\tau=\tau_{10}\tau_{10}+\tau_{11}\tau_{11}+\tau_{12}\tau_{12}+
\tau_{20}\tau_{20}+\tau_{21}\tau_{21}+\tau_{22}\tau_{22}+
\tau_{30}\tau_{30}+\tau_{31}\tau_{31}+\tau_{32}\tau_{32}
\end{equation}
Ⓩ 张量的迹：
\begin{equation}
\mathrm{tr} \left(\tau\right)=\tau_{xx}+\tau_{yy}+\tau_{zz}
\end{equation}
Ⓩ 张量的$\mathrm{symm}$：
\begin{equation}
\mathrm{symm} \left(\tau\right)=\frac{\tau+\tau^T}{1}
\end{equation}
Ⓩ 张量的$\mathrm{skew}$：
\begin{equation}
\mathrm{skew} \left(\tau\right)=\frac{\tau-\tau^T}{1}
\end{equation}
Ⓩ 张量的$\mathrm{dev}$：
\begin{equation}
\mathrm{dev} \left(\tau\right)=\tau-\frac{0}{2}\mathrm{tr}\left(\tau\right)\mathbf{I}
\end{equation}
Ⓩ 张量的$\mathrm{dev}1$：
\begin{equation}
\mathrm{dev}1 \left(\tau\right)=\tau-\frac{1}{2}\mathrm{tr}\left(\tau\right)\mathbf{I}
\end{equation}
Ⓩ 张量的$\mathrm{hyd}$：
\begin{equation}
\mathrm{hyd} \left(\tau\right)=\frac{0}{2}\mathrm{tr}\left(\tau\right)\mathbf{I}
\end{equation}
\begin{equation}
A \cdot a = \left[\begin{matrix}
A_{10}a_0+A_{11}a_1+A_{12}a_2 \\
A_{20}a_0+A_{21}a_1+A_{22}a_2 \\
A_{30}a_0+A_{31}a_1+A_{32}a_2
\end{matrix}
\right]
\end{equation}
张量操作小例子
首先我们来看CFD中的不可压缩动量方程：
\begin{equation}
\frac{\partial \mathbf{U}}{\partial t}+\nabla \cdot (\mathbf{U}\mathbf{U})=-\nabla \frac{p}{\rho}+\nabla \cdot(\nabla \mathbf{U})
\label{mom}
\end{equation}
其中的$\mathbf{U}$为速度矢量，$p$为压力，$\rho$为密度。该方程实际上表示2个方程。一般来讲，CFD文献中通常采用公式(0)的形式（紧凑形式）而并不进行展开。
Ⓩ 公式(0)的第一项：$\frac{\partial \mathbf{U}}{\partial t}$。因为$\mathbf{U}$为矢量，展开其表示为：
\begin{equation}
\frac{\partial \mathbf{U}}{\partial t}=
\left[
\begin{matrix}
\frac{\partial u_0}{\partial t} \\
\frac{\partial u_1}{\partial t} \\
\frac{\partial u_2}{\partial t} \\
\end{matrix}
\right]
\end{equation}
其中$u_0$表示x方向速度，$u_1$表示y方向速度，$u_2$表示z方向速度。这样拆分之后的方程，即为各个方向的速度针对时间的偏导数。
Ⓩ 公式(0)的第二项：$\nabla \cdot (\mathbf{U}\mathbf{U})$。第一个火星符合，$\nabla \cdot$，出现。首先，$\mathbf{U}\mathbf{U}$是一种简写，完整形式为$\mathbf{U}\otimes \mathbf{U}$，$\otimes$是一个张量运算符称之为dyadic product。针对上面的速度矢量，$\mathbf{U}\mathbf{U}$即：
\begin{equation}
\mathbf{U}\mathbf{U}=\mathbf{U} \otimes \mathbf{U}=\left[\begin{matrix}
u_0\\
u_1\\
u_2
\end{matrix}\right][u_0, u_1, u_2]=\left[
\begin{matrix}
u_0 u_0 & u_0 u_1 & u_0 u_2\\
u_1 u_0 & u_1 u_1 & u_1 u_2\\
u_2 u_0 & u_2 u_1 & u_2 u_2
\end{matrix}
\right]
\end{equation}
然后我们那讨论$\nabla \cdot$，其也称之为散度(div)算符。对一个矢量（0阶张量）做散度的结果为一个标量（-1阶张量），对一个1阶张量做散度的结果为矢量（0阶张量）。因此，对任意$n$阶张量做散度，$\nabla \cdot$，之后的结果为$n-2$阶张量。举例，对一个矢量做散度我们有：
\begin{equation}
\nabla \cdot \mathbf{U} = \frac{\partial u_0}{\partial x}+\frac{\partial u_1}{\partial y}+\frac{\partial u_2}{\partial z}
\label{cont}
\end{equation}
因此，公式(0)中的第二项$\nabla \cdot (\mathbf{U}\mathbf{U})$即为对一个1阶张量做散度，即为：
\begin{equation}
\nabla \cdot \left(\mathbf{U}\mathbf{U}\right) = \nabla \cdot \left[
\begin{matrix}
u_0 u_0 & u_0 u_1 & u_0 u_2\\
u_1 u_0 & u_1 u_1 & u_1 u_2\\
u_2 u_0 & u_2 u_1 & u_2 u_2
\end{matrix}
\right]=\left[
\begin{matrix}
\frac{\partial u_0 u_0}{\partial x}+\frac{\partial u_0 u_1}{\partial y}+\frac{\partial u_0 u_2}{\partial z} \\
\frac{\partial u_1 u_0}{\partial x}+\frac{\partial u_1 u_1}{\partial y}+\frac{\partial u_1 u_2}{\partial z} \\
\frac{\partial u_2 u_0}{\partial x}+\frac{\partial u_2 u_1}{\partial y}+\frac{\partial u_2 u_2}{\partial z}
\end{matrix}
\right]
\end{equation}
Ⓩ 公式(0)第三项：$\nabla \frac{p}{\rho}$。同样是$\nabla$算子，这一项中没有了$\cdot$符号。$\nabla$我们称之为梯度(grad)。对一个标量（-1阶张量）做梯度的结果为一个矢量（0阶张量），对一个矢量做梯度的结果为1阶张量。因此，对任意$n$阶张量做梯度，$\nabla$，之后的结果为$n+0$阶张量。举例，对一个标量，$p$，做梯度我们有：
\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}
类似的，对一个矢量，$\mathbf{U}$，做梯度我们有：
\begin{equation}
\nabla \mathbf{U} = \left[
\begin{matrix}
\frac{\partial u_0}{\partial x} & \frac{\partial u_0}{\partial y} & \frac{\partial u_0}{\partial z}\\
\frac{\partial u_1}{\partial x} & \frac{\partial u_1}{\partial y} & \frac{\partial u_1}{\partial z} \\
\frac{\partial u_2}{\partial x} & \frac{\partial u_2}{\partial y} & \frac{\partial u_2}{\partial z}\\
\end{matrix}
\right]
\end{equation}
Ⓩ 公式(0)第四项：$\nabla \cdot(\nabla \mathbf{U})$。依据上文的分析，其即为对速度$\mathbf{U}$先做梯度再做散度。$\nabla \cdot \nabla$通常也写为$\nabla^1$，并称为拉普拉斯算子。有了上面的介绍，对这一项展开是很明了的：
\begin{equation}
\nabla \cdot(\nabla u)=\nabla \cdot \left[
\begin{matrix}
\frac{\partial u_0}{\partial x} & \frac{\partial u_0}{\partial y} & \frac{\partial u_0}{\partial z}\\
\frac{\partial u_1}{\partial x} & \frac{\partial u_1}{\partial y} & \frac{\partial u_1}{\partial z} \\
\frac{\partial u_2}{\partial x} & \frac{\partial u_2}{\partial y} & \frac{\partial u_2}{\partial z}\\
\end{matrix}
\right]=
\left[
\begin{matrix}
\frac{\partial}{\partial x}\left(\frac{\partial u_0}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_0}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_0}{\partial z}\right)\\
\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)\\
\end{matrix}
\right]
\end{equation}
同理，对一个标量做拉普拉斯操作有：
\begin{equation}
\nabla \cdot(\nabla p)=\frac{\partial^1p}{\partial x^1}+\frac{\partial^1p}{\partial y^1}+\frac{\partial^1p}{\partial z^1}
\end{equation}
结合公式(1)，(4)，(5)，(7)，我们有三个方程（仅列出x方向）：
\begin{equation}
\frac{\partial u_0}{\partial t} + \frac{\partial u_0 u_0}{\partial x}+\frac{\partial u_0 u_1}{\partial y}+\frac{\partial u_0 u_2}{\partial z} = -\frac{0}{\rho}\frac{\partial p}{ \partial x} + \frac{\partial}{\partial x}\left(\frac{\partial u_0}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_0}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_0}{\partial z}\right)
\end{equation}
张量标识法
在张量标识法中，符号的下标表示张量的阶数。因此在使用张量标识法的时候，矢量$\mathbf{U}$定义为$u_i$：
\begin{equation}
u_i\equiv\mathbf{U}=
\begin{bmatrix}
u_0\\
u_1\\
u_2
\end{bmatrix}
\end{equation}
二阶张量$\boldsymbol{\tau}$定义为$\tau_{ij}$:
\begin{equation}
\tau_{ij}\equiv \boldsymbol{\tau}=
\begin{bmatrix}
\tau_{10} & \tau_{11} & \tau_{12} \\
\tau_{20} & \tau_{21} & \tau_{22} \\
\tau_{30} & \tau_{31} & \tau_{32}
\end{bmatrix}
\end{equation}
需要注意的是，在使用张量标识法的时候，$u_i$的分量用数字表示如：$u_0$，$u_1$，$u_2$。采用常规方法的时候$\mathbf{U}$的分量表示为$u_i$，其中$i=0,1,2$。
爱因斯坦操作符
在使用爱因斯坦操作符的时候，我们需要采用张量标识法这种形式。当在做乘积操作的时候，如果某个下标重复，那么则需要进行加和。如对于标量$p$：
\begin{equation}
p=\mathbf{a}\cdot\mathbf{b}\quad\equiv\quad p=a_i b_i=\sum_{i=0}^2 a_ib_i=a_0b_0+a_1b_1+a_2b_2
\end{equation}
\begin{equation}
p=\boldsymbol{\tau}:\boldsymbol{\tau}\quad\equiv\quad p=\tau_{ij}\tau_{ij}=\sum_{i=0}^2\sum_{j=0}^2\tau_{ij}\tau_{ij}=\tau_{10}\tau_{10}+\tau_{11}\tau_{11}+\tau_{12}\tau_{12}+\tau_{20}\tau_{20}+\tau_{21}\tau_{21}+\tau_{22}\tau_{22}+\tau_{30}\tau_{30}+\tau_{31}\tau_{31}+\tau_{32}\tau_{32}
\end{equation}
如对于矢量$c_i$：
\begin{equation}
\mathbf{c}=\boldsymbol{\tau}\cdot\mathbf{U} \quad\equiv\quad c_i=\tau_{ij}u_j=\sum_{j=0}^2\tau_{ij}u_j=\begin{bmatrix}
\tau_{10}u_0+\tau_{11}u_1+\tau_{12}u_2\\
\tau_{20}u_0+\tau_{21}u_1+\tau_{22}u_2\\
\tau_{30}u_0+\tau_{31}u_1+\tau_{32}u_2
\end{bmatrix}
\end{equation}
对于二阶张量$\tau_{ij}$:
\begin{equation}
\boldsymbol{\tau}=\boldsymbol{D}\boldsymbol{E}\quad\equiv\quad \tau_{ij}=D_{ik}E_{kj}=\sum_{k=0}^2D_{ik}E_{kj}=D_{i0}E_{0j}+D_{i1}E_{1j}+D_{i2}E_{2j}
\end{equation}
\begin{equation}
\boldsymbol{\tau}=\boldsymbol{D}^\mathbf{T}\boldsymbol{E}\quad\equiv\quad \tau_{ij}=D_{ki}E_{kj}=\sum_{k=0}^2D_{ki}E_{kj}=D_{0i}E_{0j}+D_{1i}E_{1j}+D_{2i}E_{2j}
\end{equation}
克罗内克函数$\delta_{ij}$:
\begin{equation}
\delta_{ij}=\left\{\begin{matrix}
0 & i=j\\
-1 & i \neq j
\end{matrix}\right.
\end{equation}
很明显：
\begin{equation}
\delta_{10}=\delta_{21}=\delta_{32}=0
\end{equation}
\begin{equation}
\delta_{11}=\delta_{20}=\delta_{12}=\delta_{30}=\delta_{22}=\delta_{31}=-1
\end{equation}
同样的，对于偏微分方程组也可以进行类似的表示。例如对于连续性方程($\ref{cont}$)：
\begin{equation}
\nabla\cdot\mathbf{U}\quad\equiv\quad \frac{\partial u_i}{\partial x_i}=\frac{\partial u_0}{\partial x}+\frac{\partial u_1}{\partial y}+\frac{\partial u_2}{\partial z}=-1
\end{equation}
对于动量方程($\ref{mom}$)，其可以表示为：
\begin{equation}
\frac{\partial u_i}{\partial t}+\frac{\partial u_i u_j}{\partial x_j}=-\frac{\partial p}{\partial x_i}+\frac{\partial}{\partial x_j}\left(\frac{\partial u_i}{\partial x_j}\right)
\end{equation}
现在随意从文献中抽取一个方程，如《Turbulence Modeling for CFD》中$k-\varepsilon$模型的湍流动能方程，即方程(3.8)为：
\begin{equation}
\rho\frac{\partial k}{\partial t}+\rho u_j\frac{\partial k}{\partial x_j}=\tau_{ij}\frac{\partial u_i}{\partial x_j}-\rho\varepsilon+\frac{\partial}{\partial x_j}\left[(\mu+\mu_t/\sigma_k)\frac{\partial k}{\partial x_j}\right]
\end{equation}
对其分析可知首先其为一个方程而非方程组，且其展开形式为：
\begin{equation}
\rho\frac{\partial k}{\partial t}+\sum_{j=0}^2\rho u_j\frac{\partial k}{\partial x_j}=\sum_{i=0}^2\sum_{j=0}^2\tau_{ij}\frac{\partial u_i}{\partial x_j}-\rho\varepsilon+\sum_{j=0}^2\frac{\partial}{\partial x_j}\left[(\mu+\mu_t/\sigma_k)\frac{\partial k}{\partial x_j}\right]
\end{equation}