Latex - 在线数学公式编辑器 - 您的超级在线工具

$ \left(x-1\right)\left(x+3\right) $

$ \sqrt{a^2+b^2} $

$ \left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}} $

$ \frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd} $

$ \frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0 $

$ \sqrt[n]{a^{n}}=\left ( \sqrt[n]{a}\right )^{n} $

$ x ={-b \pm \sqrt{b^2-4ac}\over 2a} $

$ \begin{aligned} a\ge 0,b\ge 0 \\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b} \end{aligned} $

$ \begin{aligned} a\geqslant 0,b\geqslant 0 \\ \sqrt[n]{\frac{a}{b}}= \frac{\sqrt[n]{a}}{\sqrt[n]{a}} \end{aligned} $

$ x = a_0 + \frac{1}{\displaystyle a_1 + \frac{1}{\displaystyle a_2 + \frac{1}{\displaystyle a_3 + a_4}}} $

$ \begin{aligned} \text{若}b_{i}\neq 0,\lambda_{i}\text{为任意常数},i=1,2, \cdots ,n \\ \frac{a_{1}}{b_{1}}= \frac{a_{2}}{b_{2}}= \cdots =\frac{a_{n}}{b_{n}}\\ \Rightarrow \frac{a_{1}}{b_{1}}= \frac{\sum_{n}^{i=1}a_{i}}{\sum_{n}^{i=1}b_{i}} = \frac{\sum_{n}^{i=1}\lambda_{i}a_{i}}{\sum_{n}^{i=1}\lambda_{i}b_{i}} = \frac{\sqrt{\sum_{n}^{i=1}a_{i}^{2}}}{\sqrt{\sum_{n}^{i=1}b_{i}^{2}}} \end{aligned} $

$ \begin{aligned} \text{若}a \neq 0,b \neq 0,c \neq 0,d \neq 0\\ \frac{a}{b}= \frac{c}{d} \Rightarrow \left\{\begin{matrix} ad = bc & \text{交叉积}\\ \frac{b}{a}= \frac{d}{c}& \text{反比}\\ \frac{a}{c}=\frac{b}{d}& \text{更比}\\ \frac{a+b}{b}= \frac{c+d}{d}&\text{合比}\\ \frac{a-b}{b}=\frac{c-d}{d}& \text{分比}\\ \frac{a+b}{a-b}=\frac{c+d}{c-d}&\text{合分比} \end{matrix}\right. \end{aligned} $

$ \overrightarrow{AB} $

$ \overleftrightarrow{AB} $

$ \widehat{AB $

$ \Delta A B C $

$ a \parallel c,b \parallel c \Rightarrow a \parallel b $

$ l \perp \beta ,l \subset \alpha \Rightarrow \alpha \perp \beta $

$ \left.\begin{matrix}a \perp \alpha\\b \perp \alpha\end{matrix}\right\}\Rightarrow a \parallel b $

$ P \in \alpha ,P \in \beta , \alpha \cap \beta =l \Rightarrow P \in l $

$ \alpha \perp \beta , \alpha \cap \beta =l,a \subset \alpha ,a \perp l \Rightarrow a \perp \beta $

$ \left.\begin{matrix}a \subset \beta ,b \subset \beta ,a \cap b=P\\a \parallel \partial ,b \parallel \partial\end{matrix}\right\}\Rightarrow \beta \parallel \alpha $

$ \alpha \parallel \beta , \gamma \cap \alpha =a, \gamma \cap \beta =b \Rightarrow a \parallel b $

$ A \in l,B \in l,A \in \alpha ,B \in \alpha \Rightarrow l \subset \alpha $

$ A,B,C\text{不共线}\Rightarrow A,B,C\text{确定平面}\alpha $

$ \left.\begin{matrix}m \subset \alpha ,n \subset \alpha ,m \cap n=P\\a \perp m,a \perp n\end{matrix}\right\}\Rightarrow a \perp \alpha $

\begin{aligned} \text{直角三角形中,直角边长a,b,斜边边长c}\\a^{2}+b^{2}=c^{2} \end{aligned}

$ \frac{\partial y}{\partial x} $

$ \frac{d}{dx}c^n=nx^{n-1} $

$ \frac{d}{dx}e^{ax}=a\,e^{ax} $

$ \frac{d}{dx}\ln(x)=\frac{1}{x} $

$ \frac{d}{dx}\sin x=\cos x $

$ \frac{d}{dx}\cos x=-\sin x $

$ \int k\text{d}x = kx+C $

$ \frac{d}{dx}\tan x=\sec^2 x $

$ \frac{d}{dx}\cot x=-\csc^2 x $

$ \int \frac{1}{x}\text{d}x= \text{ln}\left| x \right| +C $

$ \int \frac{1}{\sqrt{1-x^{2}}}\text{d}x= \text{arcsin}x+C $

$ \int \frac{1}{1+x^{2}}\text{d}x= \text{arctan}x+C $

$ \int u \frac{dv}{dx}\,dx=uv-\int \frac{du}{dx}v\,dx $

$ f( x ) = \int_{-\infty}^\infty\hat f( x )\xi\,e^{2 \pi i \xi x}\,d\xi $

$ \int x^{\mu}\text{d}x=\frac{x^{\mu +1}}{\mu +1}+C, \left({\mu \neq -1}\right) $

$ \begin{pmatrix}1 & 0 \\0 & 1\end{pmatrix} $

$ \begin{pmatrix}a_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{pmatrix} $

$ \begin{pmatrix}a_{11}& \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{m1}& \cdots & a_{mn} \end{pmatrix} $

$ \begin{aligned}A=A^{T}\\A=-A^{T}\end{aligned} $

$ O = \begin{bmatrix}0 & 0 & \cdots & 0 \\0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & 0\end{bmatrix} $

$ A_{m\times n}=\begin{bmatrix}a_{11}& a_{12}& \cdots & a_{1n}\\a_{21}& a_{22}& \cdots & a_{2n}\\\vdots & \vdots & \ddots & \vdots \\a_{m1}& a_{m2}& \cdots & a_{mn}\end{bmatrix}=\left [ a_{ij}\right ] $

$ \begin{aligned}A={\left[ a_{ij}\right]_{m \times n}},B={\left[ b_{ij}\right]_{n \times s}}\\c_{ij}= \sum \limits_{k=1}^{{n}}a_{ik}b_{kj}\\C=AB=\left[ c_{ij}\right]_{m \times s}= \left[ \sum \limits_{k=1}^{n}a_{ik}b_{kj}\right]_{m \times s}\end{aligned} $

$ \mathbf{V}_1 \times \mathbf{V}_2 =\begin{vmatrix}\mathbf{i}& \mathbf{j}& \mathbf{k}\\\frac{\partial X}{\partial u}& \frac{\partial Y}{\partial u}& 0 \\\frac{\partial X}{\partial v}& \frac{\partial Y}{\partial v}& 0 \\\end{vmatrix} $

$ C_{r}^{n} $

$ \frac{n!}{r!(n-r)!} $

$ \sum_{i=1}^{n}{X_i} $

$ \sum_{i=1}^{n}{X_i^2} $

$ X_1, \cdots,X_n $

$ \frac{x-\mu}{\sigma} $

$ \sum_{i=1}^{n}{(X_i - \overline{X})^2} $

$ \begin{aligned}\text{若}P \left( AB \right) =P \left( A \right) P \left( B \right) \\\text{则}P \left( A \left| B\right. \right) =P \left({B}\right)\end{aligned} $

$ P(E) ={n \choose k}p^k (1-p)^{n-k} $

$ P \left( A \right) = \lim \limits_{n \to \infty}f_{n}\left ( A \right ) $

$ P \left( \bigcup \limits_{i=1}^{+ \infty}A_{i}\right) = \prod \limits_{i=1}^{+ \infty}P{\left( A_{i}\right)} $

$ \begin{aligned}P \left( \emptyset \right) =0 \\ P \left( S \right) =1\end{aligned} $

$ \begin{aligned}\forall A \in S\\ P \left( A \right) \ge 0\end{aligned} $

$ P \left( \bigcup \limits_{i=1}^{n}A_{i}\right) = \prod \limits_{i=1}^{n}P \left( A_{i}\right) $

$ \begin{aligned}S= \binom{N}{n},A_{k}=\binom{M}{k}\cdot \binom{N-M}{n-k}\\ P\left ( A_{k}\right ) = \frac{\binom{M}{k}\cdot \binom{N-M}{n-k}}{\binom{N}{n}}\end{aligned} $