Mathematical expressions are related

vector , matrix , determinant

Use MathJax Rendering LaTex The mathematical formula , See math.stackexchange.com, stay Markdown To input mathematical formulas in LaTeX Grammar support . Reference link

vector

• Make letters bold in vector form \pmb{x}. Like a vector $x$ Written as $\pmb{x}$, Is shown as $xx$
• Add arrows to letters in vector form \vec{x}, Is shown as $\vec{x}$; Or write it as \overrightarrow{x}, Is shown as $\overrightarrow{x}$( Not recommended )

matrix

Use \begin{matrix} --- \end{matrix|, The dotted line part writes specific matrix elements , Among the elements of the same line & Separate , Use a backslash at the end of each line \\ separate , There is no need to use a backslash on the last line \\

Common grammar

• If the element needs to use parentheses The parcel , Will matrix Replace with pmatrix. Such as ```

$$\begin{pmatrix}
1&1&1\\
1&1&1\\
1&1&1
\end{pmatrix}$$

Is shown as
$$\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)$$

• If the element needs to use brackets The parcel , Will matrix Replace with bmatrix.

$$\begin{bmatrix}
1&1&1\\
1&1&1\\
1&1&1
\end{bmatrix}$$

$$\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]$$

Don't use grammar

• If the element needs to use Curly braces The parcel , Will matrix Replace with Bmatrix.
• If the element needs to use Single vertical line The parcel , Will matrix Replace with vmatrix.
• If the element needs to use Double vertical line The parcel , Will matrix Replace with Vmatrix.

------- Curly braces ----------
$$\begin{Bmatrix}
1&1\\
1&1
\end{Bmatrix}$$
------- Single vertical line ---------- It doesn't seem to apply to multiple lines 
$$\begin{vmatrix}
1&1
\end{vmatrix}$$
------- Double vertical line ---------- It doesn't seem to apply to multiple lines 
$$\begin{Vmatrix}
1&1
\end{Vmatrix}$$

They are displayed as
$$\left\{\begin{array}{cc}1& 1\\ 1& 1\end{array}\right\}$$
$$\left|\begin{array}{cc}1& 1\end{array}\right|$$
$$\Vert \begin{array}{cc}1& 1\end{array}\Vert$$

determinant

With \begin{array}{ Alignment mode }---\end{array} structure , The dotted line part is the specific determinant constituent elements , Between the elements & separate , For each line \\ separate . Package symbol ： The symbol on the left is \begin{array} Write before \left Required symbols ; The symbol on the right is \end{array} Post write \right Required symbols . The alignment is centered with letters c, Left justify with letters l, Right justify with letters r. Default （ The part of the dimension alignment is empty , The curly braces of the package alignment cannot be omitted ） Align for Center
Such as

$$
\left|
\begin{array}{ccc} 
    1  &  0   & 0 \\ 
   -5  &  2   & 3\\ 
    3  &  3   & 5 
\end{array}
\right|
$$

Is shown as
$$\left|\begin{array}{ccc}1& 0& 0\\ -5& 2& 3\\ 3& 3& 5\end{array}\right|$$
The effect of using a single vertical line matrix is as follows
$$\left|\begin{array}{ccc}1& 0& 0\\ -5& 2& 3\\ 3& 3& 5\end{array}\right|$$

array

The basic grammar of the same row column . No package symbol （ I.e. not required \left,\right Symbol ） The difference is that the array sometimes adds some horizontal lines （\hline） A vertical bar （ Write in the alignment , Use the vertical line directly |） Such as

$$
\begin{array}{c|cc} 
    1  &  0   & 0 \\ 
   \hline
   -5  &  2   & 3\\ 
    3  &  3   & 5 
\end{array}
$$

Is shown as ：
$$\begin{array}{ccc}1& 0& 0\\ -5& 2& 3\\ 3& 3& 5\end{array}$$

Ellipsis

Middle horizontal ellipsis ：\cdots $\cdots$ Is shown as $1,2,3,\cdots ,n$
Bottom horizontal ellipsis ：\cdots $\ldots$ Is shown as $1,2,3,\dots ,n$
Put up the ellipsis ：\vdots$\vdots$ Is shown as $⋮$
Oblique ellipsis ：\ddots$\cdots$ Is shown as $\ddots$

Norm and inner product

Use $\parallel x\parallel$ Is shown as
$$\parallel x\parallel$$
Use $\langle x,y \rangle$ Is shown as $$*x,y*$$

Piecewise functions and equations

Use \begin{cases}----\end{cases} Sentence structure , Each expression ends with \\
Such as

$$f(x)=\begin{cases}
1, x>0\\
0, x\leq0
\end{cases}$$

Is shown as ：
$$f(x)=\{\begin{array}{l}1,x>0\\ 0,x\le 0\end{array}$$

$$\begin{cases}
a_{11}x_1+a_{12}x_2=3\\
a_{21}x_1+a_{22}x_2=4
\end{cases}$$

Is shown as ：
$$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2=3\\ a_{21}{x}_1+a_{22}{x}_2=4\end{array}$$

Binomial combination

${ The upper formula \choose The lower formula }$. Example ： from $n$ Of the elements $m$ Elements : $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$

Alphabetic font

• Hollowed out blackbody （Blackboard Bold）：$\mathbb{ Letter }$ , Such as $\mathbb{E}, Is shown as $\mathbb{E}$
• Printer Fonts （Typewriter）：$\mathtt{E}$, Show $\mathtt{E}$
• Sans serif body （Sans Serif）:$\mathsf{E}$, Show $\mathsf{E}$
• In black （boldface）:$\mathbf{E}$, Is shown as $\mathbf{E}$
• roman （roman）：$\mathrm{E}$, Is shown as $\mathrm{E}$
• Calligraphy （calligraphic ）:$\mathcal{E}$ , Is shown as $\mathcal{E}$
• Handwriting (script) ：$\mathscr{E}$, Is shown as $\mathcal{E}$
• German Gothic (Fraktur)：$\mathfrak{E}$, Is shown as $\mathfrak{E}$
• Italics (italic): $\mathit{E}$, Is shown as $E$

Space

The following reference links

Mathematical operations

arithmetic

• Addition and subtraction ：$x\pm y$ Is shown as ：$x\pm y$
• Subtraction and addition operation ：$x\mp y$ Is shown as ：$x\mp y$
• Cross multiplication ：$x\times y$ Is shown as ：$x\times y$
• Dot multiplication ：$x\cdot y$ Is shown as ：$x\cdot y$
• Star multiplication ：$x\ast y$ Is shown as ：$x\ast y$
• Division operations ：$x\div y$ Is shown as ：$x÷y$
• Slash division ：$x/ y$ Is shown as ：$x/y$
• Fractional operation ：$x\over y$ or $\frac{x}{y}$ Is shown as ：$\frac{x}{y}$, $\frac{x}{y}$

Advanced Computing

• Average operation ：$\overline{ Arithmetic expression }$ Is shown as ：$\overline{Arithmetic\; expression}$
• Open quadratic operation ：$\sqrt{ Arithmetic expression }$ Is shown as ：$\sqrt{Arithmetic\; expression}$
• open $n$ Secondary operation ：$\sqrt[n]{ Arithmetic expression }$ Is shown as ：$\sqrt[n]{Arithmetic\; expression}$
• logarithmic ：$\log( Arithmetic expression )$, Is shown as ：$\log (Arithmetic\; expression)$
• Limit operation ：$lim_{ The variable limit written in the lower part }^{ The variable limit written in the upper part }$, Is shown as ：$li{m}_{The\; variable\; limit\; written\; in\; the\; lower\; part}^{The\; variable\; limit\; written\; in\; the\; upper\; part}$
• Write the variable limit in the limit operation of positive up and positive down ：$\displaystyle_{ The variable limit written in the lower part }^{ The variable limit written in the upper part }$, Such as $\displaystyle \lim_{x\to0}^{y\to\infty}$ Is shown as ：$\underset{x\to 0}{\overset{y\to \infty }{\lim }}$
• Integral operations ：${\int }_{The\; variable\; limit\; written\; in\; the\; lower\; part}^{The\; variable\; limit\; written\; in\; the\; upper\; part}expressiondx$, Such as $\int_{x\to 0}^{\infty}xdx$, Is shown as ：${\int }_{x\to 0}^{\infty }xdx$
• The integral operation of writing the limit of variable in positive up and down ： stay \int Before to add \displaystyle. Such as $\displaystyle \int_{x\to 0}^{\infty}xdx$, Is shown as ：${\int }_{x\to 0}^{\infty }xdx$
• Summation ：${\sum }_{The\; variable\; limit\; written\; in\; the\; lower\; part}^{The\; variable\; limit\; written\; in\; the\; upper\; part}expression$, Such as $\sum_{x=0}^{\infty}x$, Is shown as ：${\sum }_{x=0}^{\infty }x$
• Write the variable limit in the sum operation of positive up and positive down ： stay \sum Before to add \displaystyle . Such as $\displaystyle \sum_{x=0}^{\infty}x$, Is shown as ：$\sum _{x=0}^{\infty }x$
• Summation of multiple variable conditions $\sum_{ The upper formula \atop The lower formula }$, Such as $\sum_{i=0\atop j=0}^{i=n\atop j=n}i+j$$\sum _{\genfrac{}{}{0ex}{}{i=0}{j=0}}^{\genfrac{}{}{0ex}{}{i=n}{j=n}}(i+j)$
• differential
• • Ordinary differential
• • • First order ordinary differential
      $\mathrm{d}x$ Is shown as ：$\mathrm{d}x$
      Point differential （ derivative ）$\dot x$ Is shown as ：$\dot{x}$
• • • Second order ordinary differential
      $\ddot x$ Is shown as ：$\ddot{x}$
• • • $n$ Order ordinary differential
      $x^{(n)}$ Is shown as ：$x^{(n)}$
• • Partial differential
• • • First order partial differential
      $\partialx$ Is shown as ：$\partial x$
• • • $n$ Order partial differential
      $\partial^nx$ Is shown as ：${\partial }^{n}x$
• • gradient $\nabla$$\nabla$

Logical symbols

• It's not equal to ：$\neq$,$\ne$
• Greater than or equal to ： $\geq$,$\ge$
• Not greater than or equal to ： $\ngeq$,$\ngeqq$;$\notgeq$,$\ngeqq$
• Less than or equal to ：$\leq$,$\le$
• Not less than or equal to ： $\nleq$,$\nleqq$;$\notleq$,$\nleqq$
• About equal to ：$\approx$$\approx$
• Constant is equal to ：$\equiv$$\equiv$

Set operations

• It belongs to operations $\in$$\in$
• It's not an operation $\notin$ or $\not\in$\notin$
• Subset operations $\subset$$\subset$; $\supset$$\supset$
• Non subset operations $\not\subset$$\not\subset$; $\not\supset$$\not\supset$
• Proper subset operation $\subseteq$$\subseteq$; $\supseteq$$\supseteq$
• Non proper subset operations $\subsetneq$$⊊$; $\supsetneq$$⊋$
• intersection $\cap$$\cap$
• Combine $\cup$$\cup$
• Difference set $\setminus$$\setminus$
• Same as or $\bigodot$$⨀$
• Tongyu $\bigotimes$$⨂$
• An empty set $\emptyset$$\varnothing$

Special symbols

• $\infty$$\infty$
• $\hat{a}$$\hat{a}$
• $\check{a}$$\check{a}$
• $\breve{a}$$\breve{a}$
• $\tilde{a}$$\tilde{a}$
• $\bar{a}$$\bar{a}$
• $\vec{a}$$\vec{a}$
• $\acute{a}$$\acute{a}$
• $\grave{a}$$\grave{a}$
• $\mathring{a}$$\mathring{a}$
• $\uparrow$$\uparrow$
• $\Uparrow$$\Uparrow$
• $\downarrow$$\downarrow$
• $\Downarrow$$\Downarrow$
• $\leftarrow$$\leftarrow$
• $\Leftarrow$$\Leftarrow$
• $\rightarrow$$\to$
• $\Rightarrow$$\Rightarrow$