PURE MATHEMATICS 纯数学

Mensuration 测量学

  • Volume of sphere 球的体积=43πr3= \frac{4}{3} \pi r^3

  • Surface area of sphere 球的表面积=4πr2= 4 \pi r^2

  • Volume of cone or pyramid 圆锥、棱锥的体积=13×bace area 底面积×height 高= \frac{1}{3} \times bace\ area\ 底面积 \times height\ 高

  • Area of curved surface of cone 圆锥斜面面积=πr×slant height 斜高=\pi r\times slant\ height\ 斜高

  • Arc length of circle 扇形的弧长=rθ=r\theta (θ in radians)

  • Area of sector of circle 扇形的面积=12r2θ=\frac{1}{2}r^2\theta (θ in radians)

Algebra 代数

  • For the quadratic equation 二次方程ax2+bx+c=0ax^2+bx+c=0：

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2 -4ac}}{2a}

  • For an arithmetic series 等差数列：

un=a+(n−1)dSn=12n(a+1)=12n{2a+(n−1)d}u_n=a+(n-1)d\quad\quad S_n=\frac{1}{2}n(a+1)=\frac{1}{2}n\{2a+(n-1)d\}

  • For a geometric series 等比数列：

un=arn−1Sn=a(1−rn)1−r(r≠1)S∞=a1−r (∣r∣<1)u_n=ar^{n-1}\quad\quad S_n=\frac{a(1-r^n)}{1-r}(r\neq1)\quad\quad S_\infty=\frac{a}{1-r}\ (\vert r\vert<1)

  • Binomial series 二项式级数：

Where nn is a positive integer 当nn是正整数

(a+b)n=an+(n1)an−1b+(n2)an−2b2+(n3)an−3b3+⋯+bn(a+b)^n=a^n+\binom{n}{1}a^{n-1}b+\binom{n}{2}a^{n-2}b^2+\binom{n}{3}a^{n-3}b^3+\cdots+b^n

And 且

(nr)=n!r!(n−r)!\binom{n}{r}=\frac{n!}{r!(n-r)!}

Where nn is rational and∣x∣<1\vert x\vert<1 当nn为有理数且∣x∣<1\vert x\vert<1

(1+x)n=1+nx+n(n−1)2!x2+n(n−1)(n−2)3!x3+⋯(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\cdots

Trigonometry 三角学

  • Six Fundamental Trigonometric Functions 六种基本三角函数

x=临边y=对边r=斜边x=临边\quad\quad y=对边\quad\quad r=斜边

sin⁡θ=yrcos⁡θ=xrtan⁡θ=yx\sin\theta=\frac{y}{r}\quad\quad\cos\theta=\frac{x}{r}\quad\quad\tan\theta=\frac{y}{x}

cosec⁡θ=rysec⁡θ=rxcot⁡θ=xy\cosec\theta=\frac{r}{y}\quad\quad\sec\theta=\frac{r}{x}\quad\quad\cot\theta=\frac{x}{y}

  • Formulas 公式

tan⁡θ≡sin⁡θcos⁡θ\tan\theta\equiv\frac{\sin\theta}{\cos\theta}

cos⁡2θ+sin⁡2θ≡11+tan⁡2θ≡sec⁡2θcot⁡2θ+1≡cosec⁡2θ\cos^2\theta+\sin^2\theta\equiv1\quad\quad1+\tan^2\theta\equiv\sec^2\theta\quad\quad\cot^2\theta+1\equiv\cosec^2\theta

sin⁡(A±B)≡sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B)\equiv\sin A\cos B\pm\cos A\sin B

cos⁡(A±B)≡cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A\pm B)\equiv\cos A\cos B\mp\sin A\sin B

tan⁡(A±B)≡tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A\pm B)\equiv\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}

sin⁡2A≡2sin⁡Acos⁡A\sin2A\equiv2\sin A\cos A

cos⁡2A≡cos⁡2A−sin⁡2A≡2cos⁡2A−1≡1−2sin⁡2A\cos2A\equiv\cos^2A-\sin^2A\equiv2\cos^2A-1\equiv1-2\sin^2A

tan⁡2A≡2tan⁡A1−tan⁡2A\tan2A\equiv\frac{2\tan A}{1-\tan^2A}

  • Principal values 常见的值

−12π≤sin⁡−1x≤12π0≤cos⁡−1x≤π−12π<tan⁡−1x<12π-\frac{1}{2}\pi\le\sin^{-1}x\le\frac{1}{2}\pi\quad\quad0\le\cos^{-1}x\le\pi\quad\quad-\frac{1}{2}\pi<\tan^{-1}x<\frac{1}{2}\pi

Differentiation 求导/微分

f′(x)f'(x)

Integration 求积分

Vectors 矢量

博主正在学习中

MECHANICS 力学

Uniformly accelerated motion 匀加速运动

vav=v+u2s=(v+u2)ta=v−utv_{av}=\frac{v+u}{2}\quad\quad s=(\frac{v+u}{2})t\quad\quad a=\frac{v-u}{t}

v=u+ats=ut+12at2v2=u2+2asv=u+at\quad\quad s=ut+\frac{1}{2}at^2\quad\quad v^2=u^2+2as

s=vt−12at2s=vt-\frac{1}{2}at^2

PROBABILITY & STATISTICS 概率与统计学

Summary statistics 概括统计量

  • For ungrouped data 对于未分组的数据：

x‾=∑xnstandard deviation 标准差=∑(x−x‾)2n=∑x2n−x‾2\overline x=\frac{\sum x}{n}\quad\quad standard\ deviation\ 标准差=\sqrt{\frac{\sum(x-\overline x)^2}{n}}=\sqrt{\frac{\sum x^2}{n}-\overline x^2}

  • For grouped data 对于分组数据：

x‾=∑xf∑fstandard deviation 标准差=∑(x−x‾)2f∑f=∑x2f∑f−x‾2\overline x=\frac{\sum xf}{\sum f}\quad\quad standard\ deviation\ 标准差=\sqrt{\frac{\sum(x-\overline x)^2f}{\sum f}}=\sqrt{\frac{\sum x^2f}{\sum f}-\overline x^2}

Discrete random variables 离散随机变量

For the binomial distribution B( , ) n p :

For the geometric distribution Geo(p):

For the Poisson distribution Po( ) λ

Continuous random variables 连续随机变量

Sampling and testing 抽样与检测

Unbiased estimators:

Central Limit Theorem:

Approximate distribution of sample proportion: