$$
A=
\left\{
\begin{array}{cccc|c}
a & b & c & d & e\\
f & g & h & i & j \\
k & l & m & n & o \\
p & q & r & s & t
\end{array}
\right\}
$$

$$f(n)=n^n \quad \text{n$\in N^\ast$}$$

$$a = \left(1 + 2 + 3 + \cdots\right. \\ \left. n - 2 + n - 1 + n\right.)$$

$$
\frac {\frac{x^2}{2y+z}}{x^3+5x-2y}
$$

应用举例：
$$\sum_{1<=i<=N}{\frac {N-i+1}{i^{N-1}} }$$

$$\sum_{i=1}^{N}{\frac {N-i+1}{i^{N-1}} }$$

$$
\begin{Vmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 4 & 5 & 6 \\ \end{Vmatrix}
$$

$$
\begin{bmatrix}
a_{1,1} & a_{1,2} & a_{1,3} & \cdots & a_{1,n} \\
a_{2,1} & a_{2,2} & a_{2,3} & \cdots & a_{2,n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{m,1} & a_{m,2} & a_{m,3} & \cdots & a_{m,n} \\
\end{bmatrix}
$$

$$
\left \{
\begin{array}{c}
a_1x+b_1y+c_1z=d_1 \\
a_2x+b_2y+c_2z=d_2 \\
a_3x+b_3y+c_3z=d_3
\end{array}
\right.
$$

$$x+y=1  \tag{1} $$
$$x-y=-1  \tag{2}$$
$$联立得，x=2,y=-1$$

$$
\begin{aligned}
&max=2*x1+3*x2;\\
&x1+2*x2<=8;\\
&4*x1<=16;\\
\text{subject to}\quad&s4*x2<=12;
\end{aligned}
$$

$$
\begin{aligned}
&max=2*x1+3*x2;\\
&\text{s.t.}\\
&x1+2*x2<=8;\\
&4*x1<=16;\\
&s4*x2<=12;
\end{aligned}
$$

$$
f(x)=\int_{2}^{+\infty} \frac{dx}{x\cdot \sqrt[3]{x^2-3x+2}}
$$

$$
\lim_{x\rightarrow{\infty}}(1+\frac{1}{x})^{x}=e
$$
$$
\begin{cases}
   y'' &=f(x,y,y') \\
   y|_{x=x_0}&=y_0 \\
   y'|_{x=x_0}&=y'_0
\end{cases}
$$
$$
\left(h\frac{\partial}{\partial x}  + k\frac{\partial}{\partial y} \right) ^m
f(x_0,y_0)  =\sum_{p=0}^{m}C_m^ph^pk^{m-p}
\frac{\partial^mf}{\partial x^p\partial y^{m-p}}\Big|_{x_0-y_0}
$$

Gamma公式展示 $\Gamma(n) = (n-1)!\quad\forall
n\in\mathbb N$ 是通过欧拉积分
$$
\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt\,.
$$

$$A^TA=\left(              
\begin{array}{lcr}   
1 & 2 \\
0 & 0 \\           
0 & 0
\end{array}         
\right)
\left(              
\begin{array}{lcr}   
1 & 0 & 0 \\           
2 & 0 & 0
\end{array}         
\right)=
\left(              
\begin{array}{lcr}   
5 & 0 & 0 \\           
0 & 0 & 0 \\
0 & 0 & 0
\end{array}         
\right)$$，

$$
            \begin{pmatrix}
            1 & a_1 & a_1^2 & \cdots & a_1^n \\
            1 & a_2 & a_2^2 & \cdots & a_2^n \\
            \vdots & \vdots & \vdots & \ddots & \vdots \\
            1 & a_m & a_m^2 & \cdots & a_m^n \\
            \end{pmatrix}
$$

有趣；[backcolor=rgb(238, 238, 238) !important]$ y_k=\varphi(u_k+v_k)$ [backcolor=rgb(238, 238, 238) !important]$J\alpha(x) = \sum{m=0}^\infty \frac{(-1)^m}{m! \Gamma (m + \alpha + 1)} {\left({ \frac{x}{2} }\right)}^{2m + \alpha}$注意下面的写法：(右对齐)[backcolor=rgb(238, 238, 238) !important]$$ y_k=\varphi(u_k+v_k)$$

$$\frac{\partial u}{\partial t}
= h^2 \left( \frac{\partial^2 u}{\partial x^2}
+ \frac{\partial^2 u}{\partial y^2}  
+ \frac{\partial^2 u}{\partial z^2}\right)$$

$$ F^{HLLC}=\left\{
\begin{array}{rcl}
F_L       &      & {0      <      S_L}\\
F^*_L     &      & {S_L \leq 0 < S_M}\\
F^*_R     &      & {S_M \leq 0 < S_R}\\
F_R       &      & {S_R \leq 0}
\end{array} \right. $$

$$
matrix
\left[ \left( \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right)\right]$$
$$
\left[ \chi(\lambda) = \left| \begin{array}{ccc}
\lambda - a & -b & -c \\
-d & \lambda - e & -f \\
-g & -h & \lambda - i \end{array} \right|.\right]
$$

$$
f(x) = \left\{
  \begin{array}{lr}
    x^2 & : x < 0\\
    x^3 & : x \ge 0
  \end{array}
\right.
$$

$$
u(x) =
  \begin{cases}
   \exp{x} & \text{if } x \geq 0 \\
   1       & \text{if } x < 0
  \end{cases}
$$

$$
J(\theta) = \frac{1}{2m}\sum_{i = 0} ^m(y^i - h_\theta (x^i))^2
$$

$$\begin{aligned}
\left.\begin{aligned}
        B'&=-\partial \times E,\\         %加&指定对齐位置
        E'&=\partial \times B - 4\pi j,
       \end{aligned}
\right\}                                                                %加右}
\qquad \text{Maxwell's equations}
\end{aligned}$$

$$
\left(
\begin
{array}{cccc}
{l_{11}} & {l_{12}} & {\cdots} & {l_{1 m}} \\
{l_{21}} & {l_{22}} & {\cdots} & {l_{2 m}} \\
  {\vdots} & {\vdots} & { } & {\vdots} \\
  {l_{m 1}} & {l_{m 2}} & {\cdots} &{l_{mm}}\end{array}\right) \left( \begin{array}{c}{\hat{b}_{1}} \\
  {\hat{b}_{2}} \\
  {\vdots} \\
  {\hat{b}_{m}}\end{array}\right)=\left( \begin{array}{c}{l_{1 y}}\\
  {l_{2 y}} \\
   {\vdots} \\ {l_{m y}}\end{array}\right)
$$

$$
%指定4列，第1列水平居中，第1列和第2列间垂直线，第2、3、4列分别左靠齐、居中、右靠齐
\begin{array}{c|lcr}
n & \text{Left} & \text{Center} & \text{Right} \\
\hline
1 & 0.24 & 1 & 125 \\
2 & -1 & 189 & -8 \\
3 & -20 & 2000 & 1+10i \\
\end{array}
$$

$$ \left[
      \begin{array}{cc|c}
        1&2&3\\
        4&5&6
      \end{array}
    \right]
$$

$$
f(n) =
\begin{cases}
        n/2,  & \text{if $n$ is even} \\
        3n+1, & \text{if $n$ is odd}  \\
\end{cases}
\\%简单的换行
\left.%这个.必须写,这是反括号
\begin{array}{l}
\text{if $n$ is even:}& n/2\\
\text{if $n$ is odd:} & 3n+1
\end{array}
\right\}
=f(n)
$$

$$
\left\{  
             \begin{array}{lr}
             x=\dfrac{3\pi}{2}(1+2t)\cos(\dfrac{3\pi}{2}(1+2t)),&  \\  
             y =s,& 0\leq s\leq L,|t|\leq1.\\  
             z =\dfrac{3\pi}{2}(1+2t)\sin(\dfrac{3\pi}{2}(1+2t)), &   
             \end{array}  
\right.   
$$

$$ \lim_{n \to +\infty} \frac{1}{n(n+1)} \quad and \quad \lim_{x\leftarrow{示例}} \frac{1}{n(n+1)} $$

\[\beta=\sum_\infty^n \omega\]