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Summation 求和
- $$\sum_{i=1}^{n}a=an$$
- $$\sum_{i=1}^{n}ka_{i}=k\sum_{i=1}^{n}a_{i}$$
- $$\sum_{i=1}^{n}(a_{i}+b_{i})=\sum_{i=1}^{n}a_{i}+\sum_{i=1}^{n}b_{i}$$
Binomial Theorem 二項式定理
$$(a+b)^n=C_0^na^n+C_1^na^{n-1}b$$
$$+C_2^na^{n-2}b^2+\cdots+C_r^na^{n-r}b^r$$
$$+\cdots+C_n^nb^n$$
$$=\sum_{r=0}^nC_r^na^{n-r}b^r$$
Trigonometry 三角學
Arc length & Area
$$\begin{aligned}\ell&=r\theta\\A&=\frac12r^2\theta\\&=\frac12r\ell\end{aligned}$$
Trigonometric Functions
$$\begin{aligned}\sin\theta&=\frac{y}{r},\quad\mathrm{csc}\theta=\frac{r}{y};\\\cos\theta&=\frac{x}{r},\quad\mathrm{sec}\theta=\frac{r}{x};\\\tan\theta&=\frac{y}{x},\quad\cot\theta=\frac{x}{y}.\end{aligned}$$
Trigonometric Identities
- $$csc\theta=\frac{1}{\sin\theta}\\\sec\theta=\frac{1}{\cos\theta}\\\cot\theta=\frac{1}{\tan\theta}$$
- $$\tan\theta=\frac{\sin\theta}{\cos\theta},\cot\theta=\frac{\cos\theta}{\sin\theta}$$
- $$\begin{aligned}&\sin^2\theta+\cos^2\theta=1\\&1+\tan^2\theta=\sec^2\theta\\&1+\cot^2\theta=csc^2\theta\end{aligned}$$
Constant e 數學常數 e
- $$\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}$$
- $$e^{x}=1+x+\frac{x^{2}}{2!}+\cdots=\sum_{r=0}^{\infty}\frac{x^{r}}{r!}$$
- $$\begin{aligned}lne^y&=y\\e^{\ln x}&=x\end{aligned}$$
Logarithm 對數
- $$\ln MN=\ln M+\ln N$$
- $$\ln\frac{M}{N}=lnM-lnN$$
- $$lnM^{k}=k\mathrm{ln}M$$
- $$lnN=\frac{\log_{a}N}{\log_{a}e}(a,N>0\mathrm{~and~}a\neq1)$$
Limits 極限
- $$\lim_{x\rightarrow a}k=k$$
- $$\lim_{x\rightarrow a}kf(x)=k\lim_{x\rightarrow a}f\left(x\right)$$
- $$\lim_{x\rightarrow a}[f(x)\pm g(x)]=\lim_{x\rightarrow a}f\left(x\right)\pm\lim_{x\rightarrow a}g\left(x\right)$$
- $$\lim_{x\to a}\bigl[f(x)\cdot g(x)\bigr]=\lim_{x\to a}f\left(x\right)\cdot\lim_{x\to a}g\left(x\right)$$
- $$\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}=\frac{L}{M}$$
- $$\lim_{x\rightarrow a}[f(x)]^{m}=\left[\lim_{x\rightarrow a}f\left(x\right)\right]^{m}$$
- $$\lim_{x\to a}\sqrt[m]{f(x)}=\sqrt[m]{\lim_{x\to a}f\left(x\right)}$$
Special Limits 特別極限
- $$\lim_{\theta\to0}\frac{\sin\theta}{\theta}=1$$
- $$\lim_{x\to0}\frac{e^{x}-1}{x}=1$$
Basic Differentiation 基本微分
First Principle 基本原理
$$\frac{dy}{dx}=f^{\prime}(x)=\lim_{\Delta x\rightarrow0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$
Basic Differentiation Formula
- $$\frac d{dx}(k)=0$$
- $$\frac{d}{dx}x^{n}=nx^{n-1}$$
- $$\frac{d}{dx}(k\cdot u)=k\cdot\frac{du}{dx}$$
- $$\frac{d}{dx}(u+v)=\frac{du}{dx}+\frac{dv}{dx}$$
- $$\frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}$$
- $$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$$
- $$\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$
Differentiation of Trigonometric Functions三角函數
Differentiation of Trigonometric Functions
- $$\frac{d}{dx}\left(\sin x\right)=\cos x$$
- $$\frac{d}{dx}\left(\cos x\right)=-\sin x$$
- $$\frac{d}{dx}(\tan x)=\sec^{2}x$$
Differentiation of Exponential Functions and Logarithmic Functions 指數函數和對數函數
- $$\frac{d}{dx}e^{x}=e^{x}$$
- $$\frac{d}{dx}\ln x=\frac{1}{x}$$
Rate of Change 變率
- $$\text{Equation of the tangent at }P{:}\\y-y_{1}=f^{\prime}(x_{1})(x-x_{1})$$
- $$\text{Instantaneous velocity }v=\frac{ds}{dt}$$
- $$\text{Instantaneous acceleration }\\a=\frac{dv}{dt}=\frac{d^{2}s}{dt^{2}}$$
Indefinite integration 不定積分
Basic Indefinite Integrals
- $$\int f(x)dx=F(x)+C$$
- $$\int kdx=kx+C\text{,where }k\text{ is a constant}$$
- $$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C\text{,where }n\neq-1$$
- $$\int\frac{1}{x}dx=\ln\lvert x\rvert+C,\text{where }x\neq0$$
- $$\int e^xdx=e^x+C$$
- $$\int kf(x)dx=k\int f(x)dx\\\text{where k is a constant}$$
- $$\int\left[f(x)\pm g(x)\right]dx\\=\int f(x)dx\pm\int g(x)dx$$
Trigonometric Integrals
- $$\int\sin xdx=-\cos x+C$$
- $$\int\cos xdx=\sin x+C$$
- $$\int\sec^{2}xdx=\tan x+C$$
Integration by Substitution 代換積分
$$\int f(g(x))g^{\prime}(x)dx=\int(u)du\\\mathrm{where~}g^{\prime}(x)dx=du$$
Integration by Part 分部積分
$$\int f(x)d[g(x)]\\=f(x)g(x)-\int g(x)d[f(x)]$$
Definite Integration 定積分
Basic Definite Integrals
- $$\int_{a}^{b}f(x)dx=\int_{a}^{b}f(u)du\\\text{where }x\text{ and }u\text{ are variables}.$$
- $$\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx$$
- $$\int_{a}^{b}[f(x)\pm g(x)]dx\\=\int_{a}^{b}f(x)dx\pm\int_{a}^{b}g(x)dx$$
- $$\int_{a}^{a}f(x)dx=0$$
- $$\int_{b}^{a}f(x)dx=-\int_{a}^{b}f(x)dx$$
- $$\int_{a}^{b}f(x)dx\\=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx,\text{where }a\leq c\leq b.$$
- $$\int_{a}^{b}f(x)dx=F(b)-F(a)$$
- $$\int_a^bf(g(x))g^{\prime}(x)dx=\int_{g(a)}^{g(b)}f(u)du$$
- $$\int_{a}^{b}f(x)d[g(x)]\\=[f(x)g(x)]_{a}^{b}-\int_{a}^{b}g(x)d[f(x)]$$
Area and Volume by Definite Integration
- $$A=\int_{a}^{b}[f(x)-g(x)]dx$$
- $$V=\pi\int_{a}^{b}\{[g(x)]^{2}-[f(x)]^{2}\}dx$$
Definite Integration with Odd and Even functions
Odd functions
$$\int_{-a}^af(x)dx=0$$
Even functions
$$\int_{-a}^af(x)dx=2\int_{0}^{a}f(x)dx$$
Determinants 行列式
Expanding Determinants
$$\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\\end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}$$
$$\begin{aligned}&\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}\\\\&=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}\end{aligned}$$
$$-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}$$
Rules of Determinants
- $$\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=\begin{vmatrix}a_{11}&a_{21}&a_{31}\\a_{12}&a_{22}&a_{32}\\a_{13}&a_{23}&a_{33}\end{vmatrix}$$
- $$\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=-\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{31}&a_{32}&a_{33}\\a_{21}&a_{22}&a_{23}\end{vmatrix}$$
- $$\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\0&0&0\end{vmatrix}=0$$
- $$\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\ka_{11}&ka_{12}&ka_{13}\\\end{vmatrix}=0$$
- $$\begin{vmatrix}ka_{11}&ka_{12}&ka_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}\\=k\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}$$
- $$\begin{vmatrix}a_{11}+b_{11}&a_{12}+b_{12}&a_{13}+b_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{vmatrix}$$
$$=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{vmatrix}+\begin{vmatrix}b_{11}&b_{12}&b_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{vmatrix}$$ - $$\begin{vmatrix}a_{11}+b_{11}&a_{12}&a_{13}\\a_{21}+b_{21}&a_{22}&a_{23}\\a_{31}+b_{31}&a_{32}&a_{33}\\\end{vmatrix}$$
$$=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}+\begin{vmatrix}b_{11}&a_{12}&a_{13}\\b_{21}&a_{22}&a_{23}\\b_{31}&a_{32}&a_{33}\end{vmatrix}$$ - $$\begin{vmatrix}a_{11}+ka_{21}&a_{12}+ka_{22}&a_{13}+ka_{23}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{vmatrix}$$
$$=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{vmatrix}$$ - $$\begin{vmatrix}a_{11}+ka_{12}&a_{12}&a_{13}\\a_{21}+ka_{22}&a_{22}&a_{23}\\a_{31}+ka_{32}&a_{32}&a_{33}\end{vmatrix}$$
$$=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\\\end{vmatrix}$$ - $$\det(AB)=\det A\cdot\det B$$
- $$\det I=1,\det\mathbf{0}=0$$
Matrix 矩陣
scalar multiplication
$$n\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}\quad=\quad\begin{bmatrix}na&nb&nc\\nd&ne&nf\end{bmatrix}$$
matrix addition
$$\begin{bmatrix}a&b\\c&d\\e&f\end{bmatrix}\quad+\quad\begin{bmatrix}g&h\\i&j\\k&l\end{bmatrix}\quad\\=\quad\begin{bmatrix}a+g&b+h\\c+i&d+j\\e+k&f+1\end{bmatrix}$$
matrix multiplication
$$\begin{bmatrix}a&b&c\\d&e&f\end{bmatrix}\quad\begin{bmatrix}g&h\\i&j\\k&l\end{bmatrix}\quad\\=\quad\begin{bmatrix}ag+bi+ck&ah+bj+cl\\dg+ei+fk&dh+ej+fl\end{bmatrix}$$
other matrix formula
- $$A^{-1}=\frac{1}{\det A}\operatorname{adj}A$$
- $$(A^{-1})^{-1}=A$$
- $$(\lambda A)^{-1}=\frac{1}{\lambda}A^{-1}$$
- $$(A^T)^{-1}=(A^{-1})^T$$
- $$(AB)^{-1}=B^{-1}A^{-1}$$
- $$(A^{n})^{-1}=(A^{-1})^{n}$$
- $$det(A^{-1})=\frac{1}{detA}\mathrm{if}detA\neq0$$
System of Linear Equations 線性方程式
- $$X=A^{-1}B$$
- $$\Delta=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}$$
$$\Delta_{x}=\begin{vmatrix}b_{1}&a_{12}&a_{13}\\b_{2}&a_{22}&a_{23}\\b_{3}&a_{32}&a_{33}\end{vmatrix}$$
$$\Delta_{y}=\begin{vmatrix}a_{11}&b_{1}&a_{13}\\a_{21}&b_{2}&a_{23}\\a_{31}&b_{3}&a_{33}\end{vmatrix}$$
$$\Delta_z=\begin{vmatrix}a_{11}&a_{12}&b_1\\a_{21}&a_{22}&b_2\\a_{31}&a_{32}&b_3\\\end{vmatrix}$$ - $$\text{If }\Delta\neq0\text{, then the system}\\\text{has a unique solution }x$$
$$x=\frac{\Delta_{x}}{\Delta},y=\frac{\Delta_{y}}{\Delta}\mathrm{and}z=\frac{\Delta_{z}}{\Delta}$$
Basic Vector 基本向量
- $$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$$
- $$\mathbf{a}+\mathbf{0}=\mathbf{a}$$
- $$\mathbf{a+b=b+a}$$
- $$\mathbf{a+(b+c)=(a+b)+c}$$
- $$\lambda(\mu\mathbf{a})=(\lambda\mu)\mathbf{a}=\mu(\lambda\mathbf{a})$$
- $$(\lambda+\mu)\mathbf{a}=\lambda\mathbf{a}+\mu\mathbf{a}$$
- $$\lambda\mathbf{(a+b)}=\lambda\mathbf{a}+\lambda\mathbf{b}$$
- $$\begin{aligned}&\overrightarrow{OP}=x\boldsymbol{i}+y\boldsymbol{j}\\&|\overrightarrow{OP}|=\sqrt{x^2+y^2}\end{aligned}$$
- $$\begin{aligned}&\overrightarrow{OP}=a\boldsymbol{i}+b\boldsymbol{j}+c\boldsymbol{k}\\&\left|\overrightarrow{OP}\right|=\sqrt{a^2+b^2+c^2}\end{aligned}$$
Point of Division
$$AP\colon PB=r\colon s\\\text{, then }\overline{OP}=\frac{s\overrightarrow{OA}+r\overrightarrow{OB}}{s+r}$$
Vector Properties
$$a\cdot a=|a|^{2}\geq0$$
$$\boldsymbol{a}\cdot\boldsymbol{0}=0$$
$$a\cdot b=b\cdot a$$
$$a\cdot(\lambda\boldsymbol{b})=\lambda(\boldsymbol{a}\cdot\boldsymbol{b}),\mathrm{where~\lambda~is~a~scalar}$$
$$a\cdot(b+c)=a\cdot b+a\cdot c$$
For non-zero vectors a and b,
$$a\cdot b=0$$
if and only if a and b are perpendicular
Dot Product 點積
- $$a\cdot b=|a|\boldsymbol{b}|\cos\theta $$
- $$i\cdot i=j\cdot j=k\cdot k=1$$
- $$i\cdot j=j\cdot k=k\cdot i=0$$
- $$\text{Let }\boldsymbol{a}=x_{1}\boldsymbol{i}+y_{1}\boldsymbol{j},\boldsymbol{b}=x_{2}\boldsymbol{i}+y_{2}\boldsymbol{j}$$
$$\boldsymbol{a}\cdot\boldsymbol{b}=x_{1}x_{2}+y_{1}y_{2}$$
$$\cos\theta=\frac{a\cdot b}{|a||b|}=\frac{x_{1}x_{2}+y_{1}y_{2}}{\sqrt{x_{1}^{2}+y_{1}^{2}}\sqrt{x_{2}^{2}+y_{2}^{2}}}$$
Cross Product 叉積
- $$\boldsymbol{a}\times\boldsymbol{b}=|a|b|\sin\theta $$
- $$\mathbf{i\times i=j\times j=k\times k=0}$$
- $$\mathbf{i\times j=k,j\times k=i,k\times i=j}$$
- $$a\times a=0$$
- $$a\times\mathbf{0}=\mathbf{0}$$
- $$b\times a=-(a\times b)$$
- $$(a+b)\times c=a\times c+b\times c$$
- $$a\times(b+c)=a\times b+a\times c$$
- $$(\lambda\boldsymbol{a})\times\boldsymbol{b}=\lambda(\boldsymbol{a}\times\boldsymbol{b})=\boldsymbol{a}\times(\lambda\boldsymbol{b})$$
- For non-zero vectors a and b,
$$\boldsymbol{a}\times\boldsymbol{b}=\mathbf{0}$$
if and only if a and b are parallel. - $$(a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k})\times(b_{1}\mathbf{i}+b_{2}\mathbf{j}+b_{3}\mathbf{k})$$
$$=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$$
Area by Cross Product
Area of $$\Delta ABC$$
$$=\frac{1}{2}|\overrightarrow{BA}\parallel\overrightarrow{BC}|\sin\angle ABC$$
$$\begin{gathered}
={\frac{1}{2}}|{\overrightarrow{BA}}\times{\overrightarrow{BC}}| \\
=\frac{1}{2}|\overline{CB}\times\overline{CA}| \\
={\frac{1}{2}}|{\overrightarrow{AC}}\times{\overrightarrow{AB}}|
\end{gathered}$$
Area of parallelogram ABCD
$$=|\overrightarrow{BA}\times\overrightarrow{BC}|$$
Projection of vector 向量投影
$$\text{Projection of }\overrightarrow{AB}\mathrm{~onto~}\overrightarrow{AC}$$
$$=(|\overrightarrow{AB}|\mathrm{cos}\theta)\frac{\overrightarrow{AC}}{|\overrightarrow{AC}|}\\=\left(\frac{\overrightarrow{AB}\cdot\overrightarrow{AC}}{|\overrightarrow{AC}|^{2}}\right)\overrightarrow{AC}$$