代数

  1. 1a=aa,a≥01a=aa,a≥0\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a\ge 0

  2. {x=a+rcosθy=b+rsinθ\left\{\begin{matrix} x=a + r\text{cos}\theta \\ y=b + r\text{sin}\theta \end{matrix}\right.

  3. 对于方程形如：x3−1=0设:ω=−1+3i2x1=1,x2=ω=−1+3i2x3=ω2=−1−3i2\begin{array}{l} \text{对于方程形如：}x^{3}-1=0 \\ \text{设}\text{:}\omega =\frac{-1+\sqrt{3}i}{2} \\ x_{1}=1,x_{2}= \omega =\frac{-1+\sqrt{3}i}{2} \\ x_{3}= \omega ^{2}=\frac{-1-\sqrt{3}i}{2} \end{array}

几何

  1. α∥β,γ∩α=a,γ∩β=b⇒a∥b\alpha \parallel \beta , \gamma \cap \alpha =a, \gamma \cap \beta =b \Rightarrow a \parallel b

  2. a⊂β,b⊂β,a∩b=Pa∥∂,b∥∂}⇒β∥α\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

  3. 直角三角形中,直角边长a,b,斜边边长ca2+b2=c2\text{直角三角形中,直角边长a,b,斜边边长c} a^{2}+b^{2} = c^{2}

不等式

  1. a>b,c>0⇒ac>bca>b,c<0⇒ac<bca \gt b,c \gt 0 \Rightarrow ac \gt bc \\ a \gt b,c \lt 0 \Rightarrow ac \lt bc

  2. Hn=n∑i=1n1xi=n1x1+1x2+⋯+1xnGn=∏i=1nxin=x1x2⋯xnnAn=1n∑i=1nxi=x1+x2+⋯+xnnQn=∑i=1nxi2=x12+x22+⋯+xn2nHn≤Gn≤An≤QnH_{n}=\frac{n}{\sum \limits_{i=1}^{n}\frac{1}{x_{i}}}= \frac{n}{\frac{1}{x_{1}}+ \frac{1}{x_{2}}+ \cdots + \frac{1}{x_{n}}} \\ G_{n}=\sqrt[n]{\prod \limits_{i=1}^{n}x_{i}}= \sqrt[n]{x_{1}x_{2}\cdots x_{n}} \\ A_{n}=\frac{1}{n}\sum \limits_{i=1}^{n}x_{i}=\frac{x_{1}+ x_{2}+ \cdots + x_{n}}{n} \\ Q_{n}=\sqrt{\sum \limits_{i=1}^{n}x_{i}^{2}}= \sqrt{\frac{x_{1}^{2}+ x_{2}^{2}+ \cdots + x_{n}^{2}}{n}} \\ H_{n}\leq G_{n}\leq A_{n}\leq Q_{n}

积分

  1. ∫kdx=kx+C\int k\mathrm{d}x = kx+C

  2. ∫udvdx dx=uv−∫dudxv dx\int u \frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x=uv-\int \frac{\mathrm{d}u}{\mathrm{d}x}v\,\mathrm{d}x

矩阵

  1. (a11a12a13a21a22a23a31a32a33)\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}

  2. Am×n=[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn]=[aij]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 ]

  3. V1×V2=∣ijk∂X∂u∂Y∂u0∂X∂v∂Y∂v0∣\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}

三角

  1. cos⁡α−cos⁡β=−2sin⁡α+β2sin⁡α−β2\cos \alpha - \cos \beta =-2\sin \frac{\alpha + \beta}{2}\sin \frac{\alpha - \beta}{2}

  2. eiθe^{i \theta}

统计

  1. P(E)=(nk)pk(1−p)n−kP(E) ={n \choose k}p^k (1-p)^{n-k}

  2. P(⋃i=1+∞Ai)=∏i=1+∞P(Ai)P \left( \bigcup \limits_{i=1}^{+ \infty}A_{i}\right) = \prod \limits_{i=1}^{+ \infty}P{\left( A_{i}\right)}

  3. S=(Nn),Ak=(Mk)⋅(N−Mn−k)P(Ak)=(Mk)⋅(N−Mn−k)(Nn)\begin{array}{c} 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{array}

物理

  1. Q=I2RtQ = I ^ { 2 } R \mathrm { t }

  2. y0=Acos⁡(ωt+φ0){y_0} = A \cos ( \omega {t} + { \varphi _0})

  3. ∇⋅D=ρf∇⋅B=0∇×E=−∂B∂t∇×H=Jf+∂D∂t\begin{array}{l} \nabla \cdot \mathbf{D} =\rho _f \\ \nabla \cdot \mathbf{B} = 0 \\ \nabla \times \mathbf{E} = -\cfrac{\partial \mathbf{B}}{\partial t } \\ \nabla \times \mathbf{H} = \mathbf{J}_f + \cfrac{\partial \mathbf{D}}{\partial t } \end{array}

化学

  1. 2H2+O2→n,m2H2O2H_2 + O_2 \xrightarrow{n,m}2H_2O

  2. A⇄100∘C0∘CBA\underset{0^{\circ}C }{\overset{100^{\circ}C}{\rightleftarrows}}B