数学公式

高数

有些格式懒得调了，凑合着看吧

等价无穷小

$\begin{array}{c}\sin x\sim x,\tan x\sim x,\arcsin x\sim x,\arctan x\sim x,\\ \text{e}^x-1\sim x,\ln(1+x)\sim x,a^x-1=\operatorname{e}^{xln a}-1\sim x\ln a(a>0\operatorname{且}a\neq1),\\ 1-\cos x\sim\dfrac{1}{2}x^2,(1+x)^a-1\sim ax(a\neq0),\end{array}$

泰勒公式

$\begin{array}{l} \mathrm{e}^{x}=1+x+\frac{x^{2}}{2 !}+\cdots+\frac{x^{n}}{n !}+\cdots=\sum_{n=0}^{\infty} \frac{x^{n}}{n !} \\ \sin x=x-\frac{1}{3 !} x^{3}+\cdots+(-1)^{n} \frac{1}{(2 n+1) !} x^{2 n+1}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n+1}}{(2 n+1) !}, \\ \cos x=1-\frac{1}{2 !} x^{2}+\cdots+(-1)^{n} \frac{1}{(2 n) !} x^{2 n}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !} \\ \ln (1+x)=x-\frac{1}{2} x^{2}+\cdots+(-1)^{n-1} \frac{x^{n}}{n}+\cdots=\sum_{n=1}^{\infty}(-1)^{n-1} \frac{x^{n}}{n},-1<x \leqslant 1 \\ \frac{1}{1-x}=1+x+x^{2}+\cdots+x^{n}+\cdots=\sum_{n=0}^{\infty} x^{n},|x|<1 \\ \frac{1}{1+x}=1-x+x^{2}-\cdots+(-1)^{n} x^{n}+\cdots=\sum_{n=0}^{\infty}(-1)^{n} x^{n},|x|<1 \\ (1+x)^{a}=1+\alpha x+\frac{\alpha(\alpha-1)}{2} x^{2}+o\left(x^{2}\right)(x \rightarrow 0, \alpha \neq 0) \\ \tan x=x+\frac{1}{3} x^{3}+o\left(x^{3}\right)(x \rightarrow 0) \\ \arcsin x=x+\frac{1}{6} x^{3}+o\left(x^{3}\right)(x \rightarrow 0) \\ \arctan x=x-\frac{1}{3} x^{3}+o\left(x^{3}\right)(x \rightarrow 0) \end{array}$

导数定义

$左导数 f_{-}^{\prime}\left(x_{0}\right)=\lim _{\Delta x \rightarrow 0^{-}} \frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x} ; \\ 右导数 f_{+}^{\prime}\left(x_{0}\right)=\lim _{\Delta x \rightarrow 0^{+}} \frac{f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)}{\Delta x} . \\ f^{\prime}\left(x_{0}\right) \text { 存在 } \Leftrightarrow f_{-}^{\prime}\left(x_{0}\right)=f_{+}^{\prime}\left(x_{0}\right) . \\ 高阶导数 \quad f^{(n)}\left(x_{0}\right)=\lim _{x \rightarrow x_{0}} \frac{f^{(n-1)}(x)-f^{(n-1)}\left(x_{0}\right)}{x-x_{0}} .$

基本求导公式

$(1) \left(x^{k}\right)^{\prime}=k x^{k-1} ( k 为任意实数).\\ (2) (\ln x)^{\prime}=\frac{1}{x}(x>0) .\\ (3) \left(\mathrm{e}^{x}\right)^{\prime}=\mathrm{e}^{x} ;\left(a^{x}\right)^{\prime}=a^{x} \ln a, a>0, a \neq 1 .\\ (4) (\sin x)^{\prime}=\cos x ;(\cos x)^{\prime}=-\sin x ;\\ (\tan x)^{\prime}=\sec ^{2} x ;(\cot x)^{\prime}=-\csc ^{2} x \\ (\sec x)^{\prime}=\sec x \tan x ;(\csc x)^{\prime}=-\csc x \cot x ; \\ (\ln |\cos x|)^{\prime}=-\tan x ;(\ln |\sin x|)^{\prime}=\cot x ; \\ (\ln |\sec x+\tan x|)^{\prime}=\sec x ;(\ln |\csc x-\cot x|)^{\prime}=\csc x .\\ (5) (\arctan x)^{\prime}=\frac{1}{1+x^{2}} ;(\operatorname{arccot} x)^{\prime}=-\frac{1}{1+x^{2}} .\\ (6) (\arcsin x)^{\prime}=\frac{1}{\sqrt{1-x^{2}}} ;(\arccos x)^{\prime}=-\frac{1}{\sqrt{1-x^{2}}} .\\ (7) \left[\ln \left(x+\sqrt{x^{2}+a^{2}}\right)\right]^{\prime}=\frac{1}{\sqrt{x^{2}+a^{2}}} , 常见 a=1 ;\\ \left[\ln \left(x+\sqrt{x^{2}-a^{2}}\right)\right]^{\prime}=\frac{1}{\sqrt{x^{2}-a^{2}}}(x>a>0) , 常见 a=1 .$

莱布尼兹公式

$设 u=u(x), v=v(x) 均 n 阶可导, 则 (u \pm v)^{(n)}=u^{(n)} \pm v^{(n)} ,\\ \begin{aligned} (u v)^{(n)} &=u^{(n)} v+\mathrm{C}_{n}^{1} u^{(n-1)} v^{\prime}+\mathrm{C}_{n}^{2} u^{(n-2)} v^{\prime \prime}+\cdots+\mathrm{C}_{n}^{k} u^{(n-k)} v^{(k)}+\cdots+\mathrm{C}_{n}^{n-1} u^{\prime} v^{(n-1)}+u v^{(n)} \\ &=\sum_{k=0}^{n} \mathrm{C}_{n}^{k} u^{(n-k)} v^{(k)} . \end{aligned}$

麦克劳林公式

$任何一个无穷阶可导的函数都可写成 y=f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n} ,\\ 或者 y=f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} x^{n} .$

渐近线

$(1) 铅垂渐近线.\\ 若 \lim _{x \rightarrow x_{0}^{+}} f(x)=\infty\left(\right. 或 \left.\lim _{x \rightarrow x_{0}^{-}} f(x)=\infty\right) , 则 x=x_{0} 为一条铅垂渐近线.\\ (2) 水平渐近线.\\ 若 \lim _{x \rightarrow+\infty} f(x)=y_{1} , 则 y=y_{1} 为一条水平渐近线; 若 \lim _{x \rightarrow-\infty} f(x)=y_{2} , 则 y=y_{2} 为一条水平渐 近线;\\ 若 \lim _{x \rightarrow+\infty} f(x)=\lim _{x \rightarrow-\infty} f(x)=y_{0} , 则 y=y_{0} 为一条水平渐近线.\\ (3) 斜渐近线.\\ 若 \lim _{x \rightarrow+\infty} \frac{f(x)}{x}=k_{1}, \lim _{x \rightarrow+\infty}\left[f(x)-k_{1} x\right]=b_{1} ,则 y=k_{1} x+b_{1} 是曲线 y=f(x) 的一条斜渐 近线;\\ 若 \lim _{x \rightarrow-\infty} \frac{f(x)}{x}=k_{2}, \lim _{x \rightarrow-\infty}\left[f(x)-k_{2} x\right]=b_{2} , 则 y=k_{2} x+b_{2} 是曲线 y=f(x) 的一条斜渐 近线;\\ 若 \lim _{x \rightarrow+\infty} \frac{f(x)}{x}=\lim _{x \rightarrow-\infty} \frac{f(x)}{x}=k, \lim _{x \rightarrow+\infty}[f(x)-k x]=\lim _{x \rightarrow-\infty}[f(x)-k x]=b , 则 y=k x+b 是 曲线 y=f(x) 的一条斜渐近线.$

曲率

$\text { 曲率 } k=\frac{\left|y^{\prime \prime}\right|}{\left[1+\left(y^{\prime}\right)^{2}\right]^{\frac{3}{2}}} \text {, 曲率半径 } R=\frac{1}{k} \text {. }$

基本积分公式

$(1) \int x^{k} \mathrm{~d} x=\frac{1}{k+1} x^{k+1}+C, k \neq-1 ;\left\{\begin{array}{l}\int \frac{1}{x^{2}} \mathrm{~d} x=-\frac{1}{x}+C, \\ \int \frac{1}{\sqrt{x}} \mathrm{~d} x=2 \sqrt{x}+C .\end{array}\right.\\ (2) \int \frac{1}{x} \mathrm{~d} x=\ln |x|+C .\\ (3) \int \mathrm{e}^{x} \mathrm{~d} x=\mathrm{e}^{x}+C ; \int a^{x} \mathrm{~d} x=\frac{a^{x}}{\ln a}+C, a>0 且 a \neq 1 .\\ (4) \int \sin x \mathrm{~d} x=-\cos x+C ; \int \cos x \mathrm{~d} x=\sin x+C ;\\ \int \tan x \mathrm{~d} x=-\ln |\cos x|+C ; \int \cot x \mathrm{~d} x=\ln |\sin x|+C ; \\ \int \frac{\mathrm{d} x}{\cos x}=\int \sec x \mathrm{~d} x=\ln |\sec x+\tan x|+C ; \\ \int \frac{\mathrm{d} x}{\sin x}=\int \csc x \mathrm{~d} x=\ln |\csc x-\cot x|+C ; \\ \int \sec ^{2} x \mathrm{~d} x=\tan x+C ; \int \csc ^{2} x \mathrm{~d} x=-\cot x+C ; \\ \int \sec x \tan x \mathrm{~d} x=\sec x+C ; \int \csc x \cot x \mathrm{~d} x=-\csc x+C .\\ \begin{array}{l} \text { (5) }\left\{\begin{array}{l} \int \frac{1}{1+x^{2}} \mathrm{~d} x=\arctan x+C, \\ \int \frac{1}{a^{2}+x^{2}} \mathrm{~d} x=\frac{1}{a} \arctan \frac{x}{a}+C(a>0) . \end{array}\right.\\ \text { (6) }\left\{\begin{array}{l} \int \frac{1}{\sqrt{1-x^{2}}} \mathrm{~d} x=\arcsin x+C, \\ \int \frac{1}{\sqrt{a^{2}-x^{2}}} \mathrm{~d} x=\arcsin \frac{x}{a}+C(a>0) . \end{array}\right.\\ \text { (7) }\left\{\begin{array}{l} \int \frac{1}{\sqrt{x^{2}+a^{2}}} \mathrm{~d} x=\ln \left(x+\sqrt{x^{2}+a^{2}}\right)+C(\text { 常见 } a=1), \\ \int \frac{1}{\sqrt{x^{2}-a^{2}}} \mathrm{~d} x=\ln \left|x+\sqrt{x^{2}-a^{2}}\right|+C(|x|>|a|) . \end{array}\right. \end{array}\\ (8) \int \frac{1}{x^{2}-a^{2}} \mathrm{~d} x=\frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+C\left(\int \frac{1}{a^{2}-x^{2}} \mathrm{~d} x=\frac{1}{2 a} \ln \left|\frac{x+a}{x-a}\right|+C\right) .\\ (9) \int \sqrt{a^{2}-x^{2}} \mathrm{~d} x=\frac{a^{2}}{2} \arcsin \frac{x}{a}+\frac{x}{2} \sqrt{a^{2}-x^{2}}+C(a>|x| \geqslant 0) .\\ (10) \int \sin ^{2} x \mathrm{~d} x=\frac{x}{2}-\frac{\sin 2 x}{4}+C\left(\sin ^{2} x=\frac{1-\cos 2 x}{2}\right) ;\\ \int \cos ^{2} x \mathrm{~d} x=\frac{x}{2}+\frac{\sin 2 x}{4}+C\left(\cos ^{2} x=\frac{1+\cos 2 x}{2}\right) ;\\ \int \tan ^{2} x \mathrm{~d} x=\tan x-x+C\left(\tan ^{2} x=\sec ^{2} x-1\right) ; \\ \int \cot ^{2} x \mathrm{~d} x=-\cot x-x+C\left(\cot ^{2} x=\csc ^{2} x-1\right) .\\$

区间再现

$\begin{array}{c} \int_{a}^{b} f(x) \mathrm{d} x=\int_{a}^{b} f(a+b-x) \mathrm{d} x . \\ \int_{a}^{b} f(x) \mathrm{d} x=\frac{1}{2} \int_{a}^{b}[f(x)+f(a+b-x)] \mathrm{d} x .\\ \int_{-a}^{a} f(x) \mathrm{d} x=\int_{0}^{a}[f(x)+f(-x)] \mathrm{d} x(a>0) . \end{array}$

点火公式

$\begin{array}{l} \int_{0}^{\frac{\pi}{2}} \sin ^{n} x \mathrm{~d} x=\int_{0}^{\frac{\pi}{2}} \cos ^{n} x \mathrm{~d} x\\ =\left\{\begin{array}{ll} \frac{n-1}{n}, \frac{n-3}{n-2}, \cdots \cdot \frac{2}{3} \cdot 1, & n \text { 为大于 } 1 \text { 的奇数, } \\ \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \cdots \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & n \text { 为正偶数. } \end{array}\right.\\ \int_{0}^{\pi} \sin ^{n} x \mathrm{~d} x=\left\{\begin{array}{ll} 2 \cdot \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \cdots \cdot \frac{2}{3} \cdot 1, & n \text { 为大于 } 1 \text { 的奇数, } \\ 2 \cdot \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \cdots \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & n \text { 为正偶数. } \end{array}\right.\\ \int_{0}^{\pi} \cos ^{n} x \mathrm{~d} x=\left\{\begin{array}{ll} 0, & n \text { 为正奇数, } \\ 2 \cdot \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \cdots \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & n \text { 为正偶数. } \end{array}\right.\\ \int_{0}^{2 \pi} \sin ^{n} x \mathrm{~d} x=\left\{\begin{array}{ll} 0, & n \text { 为正奇数， } \\ 4 \cdot \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \cdots \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & n \text { 为正偶数. } \end{array}\right.\\ \int_{0}^{2 \pi} \cos ^{n} x \mathrm{~d} x=\int_{0}^{2 \pi} \sin ^{n} x \mathrm{~d} x=\left\{\begin{array}{ll} 0, & n \text { 为正奇数, } \\ 4 \cdot \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot \cdots \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & n \text { 为正偶数. } \end{array}\right.\\ \int_{0}^{\pi} x f(\sin x) \mathrm{d} x=\frac{\pi}{2} \int_{0}^{\pi} f(\sin x) \mathrm{d} x \text {. }\\ \int_{0}^{\pi} x f(\sin x) \mathrm{d} x=\pi \int_{0}^{\frac{\pi}{2}} f(\sin x) \mathrm{d} x .\\ \int_{0}^{\frac{\pi}{2}} f(\sin x) \mathrm{d} x=\int_{0}^{\frac{\pi}{2}} f(\cos x) \mathrm{d} x .\\ \int_{0}^{\frac{\pi}{2}} f(\sin x, \cos x) \mathrm{d} x=\int_{0}^{\frac{\pi}{2}} f(\cos x, \sin x) \mathrm{d} x . \end{array}$

诱导公式

$(1) \sin (\pi \pm t)=\mp \sin t .\\ (2) \cos (\pi \pm t)=-\cos t .\\ (3) \sin \left(\frac{\pi}{2} \pm t\right)=\cos t .\\ (4) \cos \left(\frac{\pi}{2} \pm t\right)=\mp \sin t .$

全微分

$\mathrm{d} z=\frac{\partial z}{\partial x} \mathrm{~d} x+\frac{\partial z}{\partial y} \mathrm{~d} y$

无条件极值

$无条件极值\\ (1)二元函数取极值的必要条件 (类比一元函数).\\ 设 z=f(x, y) 在点 \left(x_{0}, y_{0}\right)\left\{\begin{array}{l}\text { 一阶偏导数存在, } \\ \text { 取极值, } f_{x}^{\prime}\left(x_{0}, y_{0}\right)=0, f_{y}^{\prime}\left(x_{0}, y_{0}\right)=0 .\end{array}\right. \\ 【注】该必要条件同样适用于三元及三元以上函数.\\ (2) 二元函数取极值的充分条件.\\ 记 \left\{\begin{array}{l}f_{x x}^{\prime \prime}\left(x_{0}, y_{0}\right)=A, \\ f_{x y}^{\prime \prime}\left(x_{0}, y_{0}\right)=B, \text { 则 } \Delta=A C-B^{2} \\ f_{y y}^{\prime \prime}\left(x_{0}, y_{0}\right)=C,\end{array}\left\{\begin{array}{l}>0 \Rightarrow \text { 极值 }\left\{\begin{array}{l}A<0 \Rightarrow \text { 极大值, } \\ A>0 \Rightarrow \text { 极小值, }\end{array}\right. \\ <0 \Rightarrow \text { 非极值, } \\ =0 \Rightarrow \text { 方法失效, 另谋他法. }\end{array}\right.\right.\\ 【注】该充分条件不适用于三元及三元以上函数.$

条件极值和拉格朗日乘数

$条件极值与拉格朗日乘数法\\ 求目标函数 u=f(x, y, z) 在条件 \left\{\begin{array}{l}\varphi(x, y, z)=0, \\ \psi(x, y, z)=0\end{array}\right. 下的最值, 则\\ (1) 构造辅助函数 F(x, y, z, \lambda, \mu)=f(x, y, z)+\lambda \varphi(x, y, z)+\mu \psi(x, y, z) ;\\ (2) 令\\ \left\{\begin{array}{l} F_{x}^{\prime}=f_{x}^{\prime}+\lambda \varphi_{x}^{\prime}+\mu \psi_{x}^{\prime}=0, \\ F_{y}^{\prime}=f_{y}^{\prime}+\lambda \varphi_{y}^{\prime}+\mu \psi_{y}^{\prime}=0, \\ F_{z}^{\prime}=f_{z}^{\prime}+\lambda \varphi_{z}^{\prime}+\mu \psi_{z}^{\prime}=0, \\ F_{\lambda}^{\prime}=\varphi(x, y, z)=0, \\ F_{\mu}^{\prime}=\psi(x, y, z)=0 ; \end{array}\right.\\ (3) 解上述方程组得备选点 P_{i}, i=1,2,3, \cdots, n , 并求 f\left(P_{i}\right) , 取其最大值为 u_{\max } , 最小值为 u_{\min } ;\\ (4)根据实际问题, 必存在最值, 所得即为所求.$

一阶线性微分方程

$y^{\prime}+p(x) y=q(x)\\ y=\mathrm{e}^{-\int p(x) \mathrm{d} x}\left[\int \mathrm{e}^{\int p(x) \mathrm{d} x} \cdot q(x) \mathrm{d} x+C\right] .$

伯努利方程

$(仅数学一) 能写成 y^{\prime}+p(x) y=q(x) y^{n}(n \neq 0,1) (伯努利方程)\\ a. 先变形为 y^{-n} \cdot y^{\prime}+p(x) y^{1-n}=q(x) ;\\ b. 令 z=y^{1-n} , 得 \frac{\mathrm{d} z}{\mathrm{~d} x}=(1-n) y^{-n} \frac{\mathrm{d} y}{\mathrm{~d} x} , 则 \frac{1}{1-n} \frac{\mathrm{d} z}{\mathrm{~d} x}+p(x) z=q(x) ;\\ c. 解此一阶线性微分方程即可.$

欧拉方程

$能写成 x^{2} y^{\prime \prime}+p x y^{\prime}+q y=f(x) (仅数学一)\\ (1) 当 x>0 时, 令 x=\mathrm{e}^{t} , 则 t=\ln x, \frac{\mathrm{d} t}{\mathrm{~d} x}=\frac{1}{x} , 于是\\ \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} \cdot \frac{\mathrm{d} t}{\mathrm{~d} x}=\frac{1}{x} \frac{\mathrm{d} y}{\mathrm{~d} t}, \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{1}{x} \frac{\mathrm{d} y}{\mathrm{~d} t}\right)=-\frac{1}{x^{2}} \frac{\mathrm{d} y}{\mathrm{~d} t}+\frac{1}{x} \frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{\mathrm{d} y}{\mathrm{~d} t}\right)=-\frac{1}{x^{2}} \frac{\mathrm{d} y}{\mathrm{~d} t}+\frac{1}{x^{2}} \frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}},\\ 方程化为 \frac{\mathrm{d}^{2} y}{\mathrm{~d} t^{2}}+(p-1) \frac{\mathrm{d} y}{\mathrm{~d} t}+q y=f\left(\mathrm{e}^{t}\right) ,\\ 即可求解(别忘了用 t=\ln x 回代成 x 的函数).\\ (2) 当 x<0 时, 令 x=-\mathrm{e}^{t} , 同理可得.$

解的结构

$如 y^{\prime \prime \prime}+p_{1} y^{\prime \prime}+p_{2} y^{\prime}+p_{3} y=0 , 写 \lambda^{3}+p_{1} \lambda^{2}+p_{2} \lambda+p_{3}=0 \Rightarrow \lambda_{1,2,3} .\\ (1) 若 \lambda 为单实根, 写 \ C \mathrm{e}^{\lambda x} ;\\ (2) 若 \lambda 为 k 重实根,写\\ \left(C_{1}+C_{2} x+C_{3} x^{2}+\cdots+C_{k} x^{k-1}\right) \mathrm{e}^{\lambda x} ; \\ (3) 若 \lambda 为单复根 \alpha \pm \beta_{1} , 写 \\ \mathrm{e}^{\alpha x}\left(C_{1} \cos \beta x+C_{2} \sin \beta x\right) .$

非齐次线性微分方程的特解

$\text { (1) 当自由项 } f(x)=P_{n}(x) \mathrm{e}^{\alpha x} \text { 时, 特解要设为 } y^{*}=\mathrm{e}^{\alpha x} Q_{n}(x) x^{k} \text {, }\\ \left\{\begin{array}{l} \mathrm{e}^{\alpha x} \text { 照抄, } \\ Q_{n}(x) \text { 为 } x \text { 的 } n \text { 次一般多项式, } \\ k=\left\{\begin{array}{ll} 0, & \alpha \neq \lambda_{1}, \alpha \neq \lambda_{2}, \\ 1, & \alpha=\lambda_{1} \text { 或 } \alpha=\lambda_{2}, \\ 2, & \alpha=\lambda_{1}=\lambda_{2} . \end{array}\right. \end{array}\right.\\ (2) 当自由项 f(x)=\mathrm{e}^{\alpha x}\left[P_{m}(x) \cos \beta x+P_{n}(x) \sin \beta x\right] 时, 特解要设为\\ y^{*}=\mathrm{e}^{\alpha x}\left[Q_{l}^{(1)}(x) \cos \beta x+Q_{l}^{(2)}(x) \sin \beta x\right] x^{k},\\ 其中 \left\{\begin{array}{l}\mathrm{e}^{\alpha x} \text { 照抄, } \\ l=\max \{m, n\}, Q_{l}^{(1)}(x), Q_{l}^{(2)}(x) \text { 分别为 } x \text { 的两个不同的 } l \text { 次一般多项式, } \\ k=\left\{\begin{array}{ll}0, & \alpha \pm \beta \mathrm{i} \text { 不是特征根, } \\ 1, & \alpha \pm \beta \mathrm{i} \text { 是特征根. }\end{array}\right.\end{array}\right.$

无穷级数

$(1) \ln (1+x)=\sum_{n=1}^{\infty}(-1)^{n-1} \cdot \frac{x^{n}}{n},-1<x \leqslant 1 .\\ (2) \frac{1}{2} \ln (1+x)=\sum_{n=1}^{\infty}(-1)^{n-1} \cdot \frac{x^{n}}{2 n},-1<x \leqslant 1 .\\ (3) \arctan x=\sum_{n=0}^{\infty}(-1)^{n} \cdot \frac{x^{2 n+1}}{2 n+1},-1 \leqslant x \leqslant 1 .\\ (4) \mathrm{e}^{x}=\sum_{n=0}^{\infty} \frac{x^{n}}{n !},-\infty<x<+\infty :\\ (5) \frac{\mathrm{e}^{x}+\mathrm{e}^{-x}}{2}=\sum_{n=0}^{\infty} \frac{x^{2 n}}{(2 n) !},-\infty<x<+\infty .\\ (6) \cos x=\sum_{n=0}^{\infty}(-1)^{n} \cdot \frac{x^{2 n}}{(2 n) !},-\infty<x<+\infty .\\ (7) \frac{\mathrm{e}^{x}-\mathrm{e}^{-x}}{2}=\sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !},-\infty<x<+\infty .\\ (8) \sin x=\sum_{n=0}^{\infty}(-1)^{n} \cdot \frac{x^{2 n+1}}{(2 n+1) !},-\infty<x<+\infty .$

p级数/p积分

$(1) 等比级数 \sum_{n=1}^{\infty} a q^{n-1}\left\{\begin{array}{ll}=\frac{a}{1-q}, & |q|<1, \\ \text { 发散, } & |q| \geqslant 1 .\end{array}\right. \\ \text { (2) } p \text { 级数 } \sum_{n=1}^{\infty} \frac{1}{n^{p}}\left\{\begin{array}{ll} \text { 收敛, } & p>1 \text {, } \\ \text { 发散, } & p \leqslant 1 . \end{array}\right.\\ (3) 广义 p 级数 \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\left\{\begin{array}{ll}\text { 收敛, } & p>1, \\ \text { 发散, } & p \leqslant 1 .\end{array}\right. \\ (4) 交错 p 级数 \sum_{n=1}^{\infty}(-1)^{n-1} \frac{1}{n^{p}}\left\{\begin{array}{ll}\text { 绝对收敛, } \quad p>1, \\ \text { 条件收敛, } \quad 0<p \leqslant 1 .\end{array}\right.\\ \text { (5) } p \text { 积分 } \int_{0}^{1} \frac{1}{x^{p}} d x\left\{\begin{array}{ll} \text { 收敛, } & 0<p<1 .\\ \text { 发散, } & p \geqslant 1 \text {, } \end{array}\right.\\ \text { (6) } p \text { 积分 } \int_{1}^{+\infty} \frac{1}{x^{p}} d x\left\{\begin{array}{ll} \text { 收敛, } & p>1 \text {, } \\ \text { 发散, } & p \leqslant 1 . \end{array}\right.\\$

傅里叶

$S(x)=\left\{\begin{array}{ll} f(x), & x \text { 为连续点, } \\ \frac{f(x-0)+f(x+0)}{2}, & x \text { 为间断点, } \\ \frac{f(-l+0)+f(l-0)}{2}, & x=\pm l . \end{array}\right.\\ \\ \begin{array}{l} f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n} \cos \frac{n \pi x}{l}+b_{n} \sin \frac{n \pi x}{l}\right), \\ a_{0}=\frac{1}{l} \int_{-l}^{l} f(x) \mathrm{d} x, \\ a_{n}=\frac{1}{l} \int_{-l}^{l} f(x) \cos \frac{n \pi x}{l} \mathrm{~d} x, n=1,2, \cdots \\ b_{n}=\frac{1}{l} \int_{-l}^{l} f(x) \sin \frac{n \pi x}{l} \mathrm{~d} x, \end{array}$

平面方程

$以下假设平面的法向量 \boldsymbol{n}=(A, B, C) .\\ (1) 一般式: A x+B y+C z+D=0 .\\ (2) 点法式: A\left(x-x_{0}\right)+B\left(y-y_{0}\right)+C\left(z-z_{0}\right)=0 .\\ (3) 三点式: \left|\begin{array}{lll}x-x_{1} & y-y_{1} & z-z_{1} \\ x-x_{2} & y-y_{2} & z-z_{2} \\ x-x_{3} & y-y_{3} & z-z_{3}\end{array}\right|=0\left(\right. 平面过不共线的三点 P_{i}\left(x_{i}, y_{i}, z_{i}\right), i=1,2,3 ).\\ (4) 截距式: \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1 (平面过 (a, 0,0),(0, b, 0),(0,0, c) 三点).\\ (5) 平面束方程.\\ 设 A_{1}, B_{1}, C_{1} 与 A_{2}, B_{2}, C_{2} 不成比例,则 \left\{\begin{array}{l}A_{1} x+B_{1} y+C_{1} z+D_{1}=0, \\ A_{2} x+B_{2} y+C_{2} z+D_{2}=0\end{array}\right. \\ 表示两个不平行平面的交线 L , 则方程\\ \mu\left(A_{1} x+B_{1} y+C_{1} z+D_{1}\right)+\lambda\left(A_{2} x+B_{2} y+C_{2} z+D_{2}\right)=0 \\ 叫作平面束方程.(\mu=1则不包含\left(A_{2} x+B_{2} y+C_{2} z+D_{2}\right)=0)$

直线方程

$以下假设直线的方向向量 \tau=(l, m, n) .\\ (1)一般式: \left\{\begin{array}{l}A_{1} x+B_{1} y+C_{1} z+D_{1}=0, \\ A_{2} x+B_{2} y+C_{2} z+D_{2}=0,\end{array} \boldsymbol{n}_{1}=\left(A_{1}, B_{1}, C_{1}\right), \boldsymbol{n}_{2}=\left(A_{2}, B_{2}, C_{2}\right)\right. , 其中 \boldsymbol{n}_{1} \nparallel \boldsymbol{n}_{2} \\ 【注】其几何背景很直观, 是两个平面的交线, 且该直线的方向向量 \tau=\boldsymbol{n}_{1} \times \boldsymbol{n}_{2} .\\ (2) 点向式(标准式、对称式) : \frac{x-x_{0}}{l}=\frac{y-y_{0}}{m}=\frac{z-z_{0}}{n} .\\ (3) 参数式: \left\{\begin{array}{l}x=x_{0}+l t, \\ y=y_{0}+m t, M\left(x_{0}, y_{0}, z_{0}\right) \text { 为直线上的已知点, } t \text { 为参数. } \\ z=z_{0}+n t,\end{array}\right. \\ (4) 两点式: \frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}} (直线过不同的两点 P_{i}\left(x_{i}, y_{i}, z_{i}\right), i=1,2 ).$

点到平面的距离

$点 P_{0}\left(x_{0}, y_{0}, z_{0}\right) 到平面 A x+B y+C z+D=0 的距离 d=\frac{\left|A x_{0}+B y_{0}+C z_{0}+D\right|}{\sqrt{A^{2}+B^{2}+C^{2}}} .$

梯度 散度 旋度 (修复中)

\left.grad u\right|_{P_{0}}=\left(u_{x}^{\prime}\left(P_{0}\right), u_{y}^{\prime}\left(P_{0}\right), u_{z}^{\prime}\left(P_{0}\right)\right) \\ \\ div \boldsymbol{A}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\ \\ rot \boldsymbol{A}=\left|\begin{array}{ccc} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array}\right|\

极坐标 球坐标

$\left\{\begin{array}{l} x=r \cos \theta \\ y=r \sin \theta \end{array}\right.\\ \iiint_{\Omega} f(x, y, z) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z=\iiint_{\Omega} f(r \cos \theta, r \sin \theta, z) r \mathrm{~d} r \mathrm{~d} \theta \mathrm{d} z .\\ \\ \left\{\begin{array}{l} x=r \sin \varphi \cos \theta \\ y=r \sin \varphi \sin \theta \\ z=r \cos \varphi \end{array}\right.\\ \iiint_{\Omega} f(x, y, z) \mathrm{d} x \mathrm{~d} y \mathrm{~d} z=\iiint_{\Omega} f(r \sin \varphi \cos \theta, r \sin \varphi \sin \theta, r \cos \varphi) r^{2} \sin \varphi \mathrm{d} \theta \mathrm{d} \varphi \mathrm{d} r .$

格林公式 高斯公式

$\oint_{L} P(x, y) \mathrm{d} x+Q(x, y) \mathrm{d} y=\iint_{D}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) \mathrm{d} \sigma\\ \oiint_{\Sigma} P \mathrm{~d} y \mathrm{~d} z+Q \mathrm{~d} z \mathrm{~d} x+R \mathrm{~d} x \mathrm{~d} y=\iiint_{\Omega}\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right) \mathrm{d} v .$

杂项

$椭圆面积：S=π(圆周率)×a×b\\ sinx函数一拱面积：2\\ 圆锥体积：V=1/3(s*h)\\ 等比数列求和：S_{n}=a_{1} \frac{1-q^{n}}{1-q}\\$

线性代数

行列式初等变换

$第一类初等变换（换行换列）使行列式变号，\\ 第二类初等变换（某行或某列乘k倍）使行列式变k倍，\\ 第三类初等变换（某行（列）乘k倍加到另一行（列））使行列式不变。$

副对角线行列式

$副对角线行列式 \begin{array}{l} \left|\begin{array}{cccc} a_{11} & \cdots & a_{1, n-1} & a_{1 n} \\ a_{21} & \cdots & a_{2, n-1} & 0 \\ \vdots & & \vdots & \vdots \\ a_{n 1} & \cdots & 0 & 0 \end{array}\right|=\left|\begin{array}{cccc} 0 & \cdots & 0 & a_{1 n} \\ 0 & \cdots & a_{2, n-1} & a_{2 n} \\ \vdots & & \vdots & \vdots \\ a_{n 1} & \cdots & a_{n, n-1} & a_{nn} \end{array}\right|=\left|\begin{array}{cccc} 0 & \cdots & 0 & a_{1 n} \\ 0 & \cdots & a_{2, n-1} & 0 \\ \vdots & & \vdots & \vdots \\ a_{n 1} & \cdots & 0 & 0 \end{array}\right| \\ =(-1)^{\frac{n(n-1)}{2}} a_{1, n} a_{2, n-1} \cdots a_{n 1} . \\ \end{array}$

拉普拉斯展开

$设 \boldsymbol{A} 为 m 阶矩阵, \boldsymbol{B} 为 n 阶矩阵, 则 \begin{array}{c} \left|\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right|=\left|\begin{array}{ll} \boldsymbol{A} & \boldsymbol{C} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right|=\left|\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{C} & \boldsymbol{B} \end{array}\right|=|\boldsymbol{A}||\boldsymbol{B}|, \\ \left|\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right|=\left|\begin{array}{ll} \boldsymbol{C} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right|=\left|\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{C} \end{array}\right|=(-1)^{m n}|\boldsymbol{A}||\boldsymbol{B}| . \end{array}$

范德蒙德行列式

$\left|\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ x_{1} & x_{2} & \cdots & x_{n} \\ x_{1}^{2} & x_{2}^{2} & \cdots & x_{n}^{2} \\ \vdots & \vdots & & \vdots \\ x_{1}^{n-1} & x_{2}^{n-1} & \cdots & x_{n}^{n-1} \end{array}\right|=\prod_{1 \leqslant i<j \leqslant n}\left(x_{j}-x_{i}\right) .$

克拉默法则

$定理 \ 设有由 \mathrm{n} 个方程组成的 \mathrm{n} 元线性方程组\\ \left\{\begin{array}{l} a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n}=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n}=b_{2} \\ \cdots \cdots \cdots \\ a_{n 1} x_{1}+a_{n 2} x_{2}+\cdots+a_{n n} x_{n}=b_{n} \end{array}\right.\\ 如果方程组的系数行列式 D=\left|a_{i j}\right|_{n} \neq 0 ，则方程组有唯一解 x_{j}=\frac{D_{j}}{D}(j=1,2, \cdots n) ，\\ 其中 D_{j} 是把D中第列换成常数项 b_{1}, b_{2}, \cdots, b_{n} 后所得到的n阶行列式。$

方阵乘积的行列式

$\text { 设 } \boldsymbol{A}, \boldsymbol{B} \text { 是同阶方阵,则 }|\boldsymbol{A B}|=|\boldsymbol{A}||\boldsymbol{B}| \text {. }$

矩阵转置

$\text { (1) }\left(\boldsymbol{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\boldsymbol{A} \text ; \\ (2) (k \boldsymbol{A})^{\mathrm{T}}=k \boldsymbol{A}^{\mathrm{T}} \text ; \\ (3) (\boldsymbol{A}+\boldsymbol{B})^{\mathrm{T}}=\boldsymbol{A}^{\mathrm{T}}+\boldsymbol{B}^{\mathrm{T}} \text ;\\ (4) (\boldsymbol{A} \boldsymbol{B})^{\mathrm{T}}=\boldsymbol{B}^{\mathrm{T}} \boldsymbol{A}^{\mathrm{T}} \text ; \\ (5) 当 m=n \text { 时, }\left|\boldsymbol{A}^{\mathrm{T}}\right|=|\boldsymbol{A}| \text {. }$

逆矩阵

$1. 逆矩阵的定义\\ (1) 定义 \boldsymbol{A}, \boldsymbol{B} 是 n 阶方阵, \boldsymbol{E} 是 n 阶单位矩阵, 若 \boldsymbol{A B}=\boldsymbol{B} \boldsymbol{A}=\boldsymbol{E} ,\\ 则称 \boldsymbol{A} 是可逆矩阵, 并称 \boldsymbol{B} 是 \boldsymbol{A} 的逆 矩阵, 且逆矩阵是唯一的, 记作 \boldsymbol{A}^{-1} .\\ (2) \boldsymbol{A} 可逆的充分必要条件是 |\boldsymbol{A}| \neq 0 . 当 |\boldsymbol{A}| \neq 0 时, \boldsymbol{A} 可逆, 且\\ \boldsymbol{A}^{-1}=\frac{1}{|\boldsymbol{A}|} \boldsymbol{A}^{*} .\\ 2. 逆矩阵的性质与重要公式\\ 设 \boldsymbol{A}, \boldsymbol{B} 是同阶可逆矩阵, 则\\ (1) \left(\boldsymbol{A}^{-1}\right)^{-1}=\boldsymbol{A} ;\\ (2) 若 k \neq 0 , 则 (k \boldsymbol{A})^{-1}=\frac{1}{k} \boldsymbol{A}^{-1} ;\\ (3) \boldsymbol{A B} 也可逆,且 (\boldsymbol{A B})^{-1}=\boldsymbol{B}^{-1} \boldsymbol{A}^{-1} ;\\ (4) \boldsymbol{A}^{\mathrm{T}} 也可逆, 且 \left(\boldsymbol{A}^{\mathrm{T}}\right)^{-1}=\left(\boldsymbol{A}^{-1}\right)^{\mathrm{T}} ;\\ 【注】此处可称为 “穿脱”原则, 即穿衣时先内后外, 脱衣时先外后内.\\ (5) \left|\boldsymbol{A}^{-1}\right|=|\boldsymbol{A}|^{-1} .\\ \text { 【注】 } \boldsymbol{A}+\boldsymbol{B} \text { 不一定可逆, 且 }(\boldsymbol{A}+\boldsymbol{B})^{-1} \neq \boldsymbol{A}^{-1}+\boldsymbol{B}^{-1} \text {. }\\ (6) \left[\begin{array}{ll} \boldsymbol{A} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B} \end{array}\right]^{-1}=\left[\begin{array}{cc} \boldsymbol{A}^{-1} & \boldsymbol{O} \\ \boldsymbol{O} & \boldsymbol{B}^{-1} \end{array}\right], \quad\left[\begin{array}{ll} \boldsymbol{O} & \boldsymbol{A} \\ \boldsymbol{B} & \boldsymbol{O} \end{array}\right]^{-1}=\left[\begin{array}{cc} \boldsymbol{O} & \boldsymbol{B}^{-1} \\ \boldsymbol{A}^{-1} & \boldsymbol{O} \end{array}\right] .$

伴随矩阵

$1. 伴随矩阵的定义\\ 伴随矩阵 将行列式 |\boldsymbol{A}| 的 n^{2} 个元素的代数余子式按如下形式排成的矩阵称为 \boldsymbol{A} 的伴随矩阵, 记作 \boldsymbol{A}^{*} , 即\\ \boldsymbol{A}^{*}=\left[\begin{array}{cccc} A_{11} & A_{21} & \cdots & A_{n 1} \\ A_{12} & A_{22} & \cdots & A_{n 2} \\ \vdots & \vdots & & \vdots \\ A_{1 n} & A_{2 n} & \cdots & A_{nn} \end{array}\right],\\ 且有 \boldsymbol{A A}^{*}=\boldsymbol{A}^{*} \boldsymbol{A}=|\boldsymbol{A}| \boldsymbol{E} .\\ 2. 伴随矩阵的性质与重要公式\\ (1) 对任意 n 阶方阵 \boldsymbol{A} , 都有伴随矩阵 \boldsymbol{A}^{*} , 且有公式\\ \boldsymbol{A A}^{*}=\boldsymbol{A}^{*} \boldsymbol{A}=|\boldsymbol{A}| \boldsymbol{E}, \quad\left|\boldsymbol{A}^{*}\right|=|\boldsymbol{A}|^{n-1} .\\ 当 |\boldsymbol{A}| \neq 0 时, 有\\ \begin{array}{l} \boldsymbol{A}^{*}=|\boldsymbol{A}| \boldsymbol{A}^{-1}, \quad \boldsymbol{A}^{-1}=\frac{1}{|\boldsymbol{A}|} \boldsymbol{A}^{*}, \quad \boldsymbol{A}=|\boldsymbol{A}|\left(\boldsymbol{A}^{*}\right)^{-1} ; \\ (k \boldsymbol{A})(k \boldsymbol{A})^{*}=|k \boldsymbol{A}| \boldsymbol{E} ; \\ \boldsymbol{A}^{\mathrm{T}}\left(\boldsymbol{A}^{\mathrm{T}}\right)^{*}=\left|\boldsymbol{A}^{\mathrm{T}}\right| \boldsymbol{E} ; \\ \boldsymbol{A}^{-1}\left(\boldsymbol{A}^{-1}\right)^{*}=\left|\boldsymbol{A}^{-1}\right| \boldsymbol{E} ; \\ \boldsymbol{A}^{*}\left(\boldsymbol{A}^{*}\right)^{*}=\left|\boldsymbol{A}^{*}\right| \boldsymbol{E} . \end{array}\\ (2) \left(\boldsymbol{A}^{\mathrm{T}}\right)^{*}=\left(\boldsymbol{A}^{*}\right)^{\mathrm{T}},\left(\boldsymbol{A}^{-1}\right)^{*}=\left(\boldsymbol{A}^{*}\right)^{-1},(\boldsymbol{A B})^{*}=\boldsymbol{B}^{*} \boldsymbol{A}^{*},\left(\boldsymbol{A}^{*}\right)^{*}=|\boldsymbol{A}|^{n-2} \boldsymbol{A} .\\ 【注】 (\boldsymbol{A}+\boldsymbol{B})^{*} \neq \boldsymbol{A}^{*}+\boldsymbol{B}^{*} .$

矩阵的秩

$设 \boldsymbol{A} 是 m \times n 矩阵, \boldsymbol{B} 是满足有关矩阵运算要求的矩阵, 则\\ (1) 0 \leqslant r(\boldsymbol{A}) \leqslant \min \{m, n\} ;\\ (2) r(k \boldsymbol{A})=r(\boldsymbol{A})(k \neq 0) ;\\ (3) r(\boldsymbol{A B}) \leqslant \min \{r(\boldsymbol{A}), r(\boldsymbol{B})\} ;\\ (4) r(\boldsymbol{A}+\boldsymbol{B}) \leqslant r(\boldsymbol{A})+r(\boldsymbol{B}) ;\\ (5) r\left(\boldsymbol{A}^{*}\right)=\left\{\begin{array}{ll}n, & r(\boldsymbol{A})=n, \\ 1, & r(\boldsymbol{A})=n-1,其中\boldsymbol{A}为n阶方阵. \\ 0, & r(\boldsymbol{A})<n-1,\end{array}\right.$

施密特正交化

$线性无关向量组 \boldsymbol{\alpha}_{1}, \boldsymbol{\alpha}_{2} 的标准正交化 (又称正交规范化)公式为\\ \begin{array}{l} \boldsymbol{\beta}_{1}=\boldsymbol{\alpha}_{1}, \\ \boldsymbol{\beta}_{2}=\boldsymbol{\alpha}_{2}-\frac{\left(\boldsymbol{\alpha}_{2}, \boldsymbol{\beta}_{1}\right)}{\left(\boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{1}\right)} \boldsymbol{\beta}_{1}, \end{array}\\ 得到的 \boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2} 是正交向量组.\\ 将 \boldsymbol{\beta}_{1}, \boldsymbol{\beta}_{2} 单位化, 得\\ \boldsymbol{\eta}_{1}=\frac{\boldsymbol{\beta}_{1}}{\left\|\boldsymbol{\beta}_{1}\right\|}, \quad \boldsymbol{\eta}_{2}=\frac{\boldsymbol{\beta}_{2}}{\left\|\boldsymbol{\beta}_{2}\right\|},\\ 则 \boldsymbol{\eta}_{1}, \boldsymbol{\eta}_{2} 是标准正交向量组.$

向量空间

$1. 基本概念\\ 若 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n} 是 n 维向量空间 \mathbf{R}^{n} 中的线性无关的有序向量组,\\ 则任一向量 \boldsymbol{\alpha} \in \mathbf{R}^{n} 均可由 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots , \boldsymbol{\xi}_{n} 线性表出,记表出式为\\ \boldsymbol{\alpha}=a_{1} \boldsymbol{\xi}_{1}+a_{2} \boldsymbol{\xi}_{2}+\cdots+a_{n} \boldsymbol{\xi}_{n},\\ 称有序向量组 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n} 是 \mathbf{R}^{n} 的一个基, 基向量的个数 n 称为向量空间的维数,\\ 而 \left[a_{1}, a_{2}, \cdots, a_{n}\right]\left(\left[a_{1}\right.\right. , \left.a_{2}, \cdots, a_{n}\right]^{\mathrm{T}} ) 称为向量 \boldsymbol{\alpha} 在基 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n} 下的坐标, 或称为 \boldsymbol{\alpha} 的坐标行 (列) 向量.\\ 2. 基变换、坐标变换\\ 若 \boldsymbol{\eta}_{1}, \boldsymbol{\eta}_{2}, \cdots, \boldsymbol{\eta}_{n} 和 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n} 是 \mathbf{R}^{n} 中的两个基, 且有关系\\ \left[\boldsymbol{\eta}_{1}, \boldsymbol{\eta}_{2}, \cdots, \boldsymbol{\eta}_{n}\right]=\left[\boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n}\right]\left[\begin{array}{cccc} c_{11} & c_{12} & \cdots & c_{1 n} \\ c_{21} & c_{22} & \cdots & c_{2 n} \\ \vdots & \vdots & & \vdots \\ c_{n 1} & c_{n 2} & \cdots & c_{nn} \end{array}\right]=\left[\boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n}\right] \boldsymbol{C}, \ (*) \\ 则 (*) 式称为由基 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n} 到基 \boldsymbol{\eta}_{1}, \boldsymbol{\eta}_{2}, \cdots, \boldsymbol{\eta}_{n} 的基变换公式, 矩阵 \boldsymbol{C} 称为由基 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n} 到\\基 \boldsymbol{\eta}_{1} , \boldsymbol{\eta}_{2}, \cdots, \boldsymbol{\eta}_{n} 的过渡矩阵, \boldsymbol{C} 的第 i 列即是 \boldsymbol{\eta}_{i} 在基 \boldsymbol{\xi}_{1}, \boldsymbol{\xi}_{2}, \cdots, \boldsymbol{\xi}_{n} 下的坐标, 且过渡矩阵 \boldsymbol{C} 是可逆矩阵.$

线性方程组解的性质

$设 \boldsymbol{\eta}_{1}, \boldsymbol{\eta}_{2}, \boldsymbol{\eta} 是非齐次线性方程组 \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b} 的解, \boldsymbol{\xi} 是对应齐次线性方程组 \boldsymbol{A} \boldsymbol{x}=\boldsymbol{0} 的解, 则: \\ (1) \boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2} 是 \boldsymbol{A} \boldsymbol{x}=\mathbf{0} 的解; (2) k \boldsymbol{\xi}+\boldsymbol{\eta} 是 \boldsymbol{A} \boldsymbol{x}=\boldsymbol{b} 的解.$

特征值的性质

$设 \boldsymbol{A}=\left(a_{i j}\right)_{n \times n}, \lambda_{i}(i=1,2, \cdots, n) 是 \boldsymbol{A} 的特征值,则\\ (1) \sum_{i=1}^{n} \lambda_{i}=\sum_{i=1}^{n} a_{i i}=\operatorname{tr}(\boldsymbol{A}) ;\\ (2) \prod_{i=1}^{n} \lambda_{i}=|\boldsymbol{A}| .$

相似矩阵的性质

$1. 定义\\ 设 \boldsymbol{A}, \boldsymbol{B} 是两个 n 阶方阵, 若存在 n 阶可逆矩阵 \boldsymbol{P} , 使得 \boldsymbol{P}^{-1} \boldsymbol{A} \boldsymbol{P}=\boldsymbol{B} , 则称 \boldsymbol{A} 相似于 \boldsymbol{B} , 记成 \boldsymbol{A} \sim \boldsymbol{B} .\\ 2. 相似矩阵的性质\\ (1) 若 \boldsymbol{A} \sim \boldsymbol{B} , 则有: (1) r(\boldsymbol{A})=r(\boldsymbol{B}) ; (2) |\boldsymbol{A}|=|\boldsymbol{B}| ; (3) |\lambda \boldsymbol{E}-\boldsymbol{A}|=|\lambda \boldsymbol{E}-\boldsymbol{B}| ; (4) \boldsymbol{A}, \boldsymbol{B} 有相同的特征值.\\ (2) 若 \boldsymbol{A} \sim \boldsymbol{B} , 则 \boldsymbol{A}^{m} \sim \boldsymbol{B}^{m}, f(\boldsymbol{A}) \sim f(\boldsymbol{B} ) (其中 f(x) 是多项式) .\\ (3) 若 \boldsymbol{A} \sim \boldsymbol{B} , 且 \boldsymbol{A} 可逆, 则 \boldsymbol{A}^{-1} \sim \boldsymbol{B}^{-1}, f\left(\boldsymbol{A}^{-1}\right) \sim f\left(\boldsymbol{B}^{-1}\right) (其中 f(x) 是多项式).\\ (4) 若 \boldsymbol{A} \sim \boldsymbol{B} , 则 \boldsymbol{A}^{\mathrm{T}} \sim \boldsymbol{B}^{\mathrm{T}} .\\ \text { (5) 若 } \boldsymbol{A} \sim \boldsymbol{B} \text {, 且 } \boldsymbol{A} \text { 可逆, 则 } \boldsymbol{A}^{*} \sim \boldsymbol{B}^{*} \text {. }$

其他

$\text { 实对称矩阵 } \boldsymbol{A} \text { 的属于不同特征值的特征向量相互正交 }\\ \text { 上、下三角矩阵与对角矩阵的特征值就是对角线元素. }\\$

惯性定理

$无论选取什么样的可逆线性变换, 将二次型化成标准形或规范形,\\ 其正项个数 p , 负项个数 q 都是不 变的, p 称为正惯性指数, q 称为负惯性指数.\\ \text { (1) 若二次型的秩为 } r \text {, 则 } r=p+q \text {, 可逆线性变换不改变正、负惯性指数; }\\ \text { (2) 两个二次型（或实对称矩阵）合同的充要条件是有相同的正、负惯性指数，} \\ {或有相同的秩及正（或负）惯性指数. }\\$

正定

$1. 定义\\ n 元二次型 f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\boldsymbol{x}^{\mathrm{T}} \boldsymbol{A x} . 若对任意的 \boldsymbol{x}=\left[x_{1}, x_{2}, \cdots, x_{n}\right]^{\mathrm{T}} \neq \mathbf{0} , 均有 \boldsymbol{x}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{x}>0 , \\ 则称 f 为正 定二次型, 称二次型的对应矩阵 \boldsymbol{A} 为正定矩阵.\\ 2. 二次型正定的充要条件\\ n 元二次型 f=\boldsymbol{x}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{x} 正定 \Leftrightarrow 对任意 \boldsymbol{x} \neq \boldsymbol{0} , 有 \boldsymbol{x}^{\mathrm{T}} \boldsymbol{A} \boldsymbol{x}>0 (定义)\\ \Leftrightarrow f 的正惯性指数 p=n \\ \Leftrightarrow 存在可逆矩阵 \boldsymbol{D} , 使 \boldsymbol{A}=\boldsymbol{D}^{\mathrm{T}} \boldsymbol{D} \\ \Leftrightarrow \boldsymbol{A} \simeq \boldsymbol{E} \\ \Leftrightarrow \boldsymbol{A} 的特征值 \lambda_{i}>0(i=1,2, \cdots, n) \\ \Leftrightarrow \boldsymbol{A} 的全部顺序主子式均大于 0 .\\ 3. 二次型正定的必要条件\\ (1) a_{i i}>0(i=1,2, \cdots, n) .\\ (2) |\boldsymbol{A}|>0 .$

概率论与数理统计

A+B

$P(A+B)=P(A)+P(B)-P(A B)$

全概率公式

$P(B)=\sum_{i=1}^{n} P\left(A_{i}\right) P\left(B \mid A_{i}\right)$

贝叶斯公式

$P\left(A_{j} \mid B\right)=\frac{P\left(A_{j} B\right)}{P(B)}=\frac{P\left(A_{j}\right) P\left(B \mid A_{j}\right)}{\sum_{i=1}^{n} P\left(A_{i}\right) P\left(B \mid A_{i}\right)}, j=1,2, \cdots, n .$

最值

$当遇到与 \max \{X, Y\}, \min \{X, Y\} 有关的事件时, 下面一些关系式是经常要用到的:\\ (1) \{\max \{X, Y\} \leqslant a\}=\{X \leqslant a\} \cap\{Y \leqslant a\} ;\\ (2) \{\max \{X, Y\}>a\}=\{X>a\} \cup\{Y>a\} ;\\ (3) \{\min \{X, Y\} \leqslant a\}=\{X \leqslant a\} \cup\{Y \leqslant a\} ;\\ (4) \{\min \{X, Y\}>a\}=\{X>a\} \cap\{Y>a\} ;\\ (5) \{\max \{X, Y\} \leqslant a\} \subseteq\{\min \{X, Y\} \leqslant a\} ;\\ (6) \{\min \{X, Y\}>a\} \subseteq\{\max \{X, Y\}>a\} .$

二项分布

$X \sim B(n, p)\left\{\begin{array}{l}\text { a. } n \text { 次试验相互独立; } \\ \text { b. } P(A)=p ; \\ \text { c. 只有 } A, \bar{A} \text { 两种结果. }\end{array}\right. \\ 记 X 为 A 发生的次数,则\\ P\{X=k\}=\mathrm{C}_{n}^{k} \cdot p^{k}(1-p)^{n-k}, k=0,1,2, \cdots, n .$

几何分布

$X \sim G(p) 首中即停止 (等待型分布), 记 X 为试验次数, 则\\ P\{X=k\}=p \cdot(1-p)^{k-1}, k=1,2, \cdots .$

超几何分布

$N 件产品中有 M 件正品, 无放回取 n 次, 则取到 k 个正品的概率\\ P\{X=k\}=\frac{\mathrm{C}_{M}^{k} \mathrm{C}_{N-M}^{n-k}}{\mathrm{C}_{N}^{n}}, k \text { 为整数, } \max \{0, n-N+M\} \leqslant k \leqslant \min \{n, M\} \text {. }$

泊松分布

$某单位时间段, 某场合下, 源源不断的随机质点流的个数, 也常用于描述稀有事件的概率.\\ P\{X=k\}=\frac{\lambda^{k}}{k !} \mathrm{e}^{-\lambda}(k=0,1, \cdots ; \lambda>0), \lambda \text { 表示强度 }(E X=\lambda) . \\ 泊松定理 \ 若 X \sim B(n, p) , 当 n 很大, p 很小, \lambda=n p 适中时, 二项分布可用泊松分布近似表示, 即\\ \mathrm{C}_{n}^{k} p^{k}(1-p)^{n-k} \approx \frac{\lambda^{k}}{k !} \mathrm{e}^{-\lambda} .\\ 一般地, 当 n \geqslant 20, p \leqslant 0.05 时, 用泊松近似公式逼近二项分布效果比较好, 特别当 n \geqslant 100, n p \leqslant 10 时,逼近效果更佳.$

指数分布

$如果 X 的概率密度或分布函数分别为\\ f(x)=\left\{\begin{array}{ll} \lambda \mathrm{e}^{-\lambda x}, & x \geqslant 0, \\ 0, & \text { 其他 } \end{array}(\lambda>0), F(x)=\left\{\begin{array}{ll} 1-\mathrm{e}^{-\lambda x}, & x \geqslant 0, \\ 0, & x<0 \end{array}(\lambda>0),\right.\right.\\ 则称 X 服从参数为 \lambda 的指数分布, 记为 X \sim E(\lambda) .\\ 注意: 1^{\circ} P\{X \geqslant t+s \mid X \geqslant t\}=P\{X \geqslant s\} 无记忆性. \\ 2^{\circ} \mathrm{F}(x)=\left\{\begin{array}{ll}1-\mathrm{e}^{-\lambda x}, & x \geqslant 0, \\ 0, & x<0\end{array}(\lambda>0)(\right. 记, 易考).\\ 3^{\circ}\left\{\begin{array}{l}\text { 几何分布 } \Rightarrow \text { 离散型等待分布 } \\ \text { 指数分布 } \Rightarrow \text { 连续型等待分布 }\end{array} \Rightarrow\right. 无记忆性.$

正态分布

$若 \quad X \sim f(x)=\frac{1}{\sqrt{2 \pi} \sigma} \mathrm{e}^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}},-\infty<x<+\infty , 其中 -\infty<\mu<+\infty, \sigma>0 , \\ 则称 X 服从参数为 \left(\mu, \sigma^{2}\right) 的正态分布, 记为 X \sim N\left(\mu, \sigma^{2}\right) .\\ 注意: 1^{\circ} \mu=0, \sigma=1 时的正态分布 N(0,1) 为标准正态分布,\\ X \sim \varphi(x)=\frac{1}{\sqrt{2 \pi}} \mathrm{e}^{-\frac{x^{2}}{2}}, \Phi(x)=\int_{-\infty}^{x} \frac{1}{\sqrt{2 \pi}} \mathrm{e}^{-\frac{t^{2}}{2}} \mathrm{~d} t,\\ 则 X \sim N(0,1) .\\ 2^{\circ} 若 X \sim N\left(\mu, \sigma^{2}\right) , 则\\ \begin{array}{l} \frac{X-\mu}{\sigma} \sim N(0,1) \\ F(x)=P\{X \leqslant x\}=\Phi\left(\frac{x-\mu}{\sigma}\right) \\ P\{a \leqslant X \leqslant b\}=\Phi\left(\frac{b-\mu}{\sigma}\right)-\Phi\left(\frac{a-\mu}{\sigma}\right) \\ P\{\mu-\sigma \leqslant X \leqslant \mu+\sigma\}=2 \Phi(1)-1 \\ P\{\mu-k \sigma \leqslant X \leqslant \mu+k \sigma\}=2 \Phi(k)-1(k>0) \end{array}\\ 3^{\circ} 若 X \sim N(0,1) , 则\\ \begin{array}{c} \Phi(-x)=1-\Phi(x) \\ P\{|X| \leqslant a\}=2 \Phi(a)-1(a>0) \\ P\{|X|>a\}=2[1-\Phi(a)](a>0) \end{array}$

二维正态分布

$如果 (X, Y) 的概率密度为 \\ f(x, y)=\frac{1}{2 \pi \sigma_{1} \sigma_{2} \sqrt{1-\rho^{2}}} \exp \left\{-\frac{1}{2\left(1-\rho^{2}\right)}\left[\left(\frac{x-\mu_{1}}{\sigma_{1}}\right)^{2}-2 \rho\left(\frac{x-\mu_{1}}{\sigma_{1}}\right)\left(\frac{y-\mu_{2}}{\sigma_{2}}\right)+\left(\frac{y-\mu_{2}}{\sigma_{2}}\right)^{2}\right]\right\},\\ 其中 \mu_{1} \in \mathbf{R}, \mu_{2} \in \mathbf{R}, \sigma_{1}>0, \sigma_{2}>0,-1<\rho<1 , \\ 则称 (X, Y) 服从参数为 \mu_{1}, \mu_{2}, \sigma_{1}^{2}, \sigma_{2}^{2}, \rho 的二维正态分布, 记 为 (X, Y) \sim N\left(\mu_{1}, \mu_{2} ; \sigma_{1}^{2}, \sigma_{2}^{2} ; \rho\right) .\\ 【注】有下面 6 条重要结论.\\ (1) 若 \left(X_{1}, X_{2}\right) \sim N\left(\mu_{1}, \mu_{2} ; \sigma_{1}^{2}, \sigma_{2}^{2} ; \rho\right) , 则\\ X_{1} \sim N\left(\mu_{1}, \sigma_{1}^{2}\right), X_{2} \sim N\left(\mu_{2}, \sigma_{2}^{2}\right) .\\ (2) 若 X_{1} \sim N\left(\mu_{1}, \sigma_{1}^{2}\right), X_{2} \sim N\left(\mu_{2}, \sigma_{2}^{2}\right) 且 X_{1}, X_{2} 相互独立,则\\ \left(X_{1}, X_{2}\right) \sim N\left(\mu_{1}, \mu_{2} ; \sigma_{1}^{2}, \sigma_{2}^{2} ; 0\right) .\\ (3) \left(X_{1}, X_{2}\right) \sim N \Rightarrow k_{1} X_{1}+k_{2} X_{2} \sim N\left(k_{1}, k_{2}\right. 是不全为 0 的常数 ) .\\ (4) \left(X_{1}, X_{2}\right) \sim N, Y_{1}=a_{1} X_{1}+a_{2} X_{2}, Y_{2}=b_{1} X_{1}+b_{2} X_{2} , 且\\ \left|\begin{array}{ll} a_{1} & a_{2} \\ b_{1} & b_{2} \end{array}\right| \neq 0 \Rightarrow\left(Y_{1}, Y_{2}\right) \sim N .\\ (5) \left(X_{1}, X_{2}\right) \sim N , 则 X_{1}, X_{2} 相互独立 \Leftrightarrow X_{1}, X_{2} 不相关.\\ 以上 5 条可推广至有限个随机变量的情形.\\ (6) (X, Y) \sim N , 则 f_{X \mid Y}(x \mid y) \sim N, f_{Y \mid X}(y \mid x) \sim N (二维正态分布的条件分布仍是正态分布).$

最值函数的分布

$若 X_{i}(i=1,2, \cdots, n ; n \geqslant 2) 独立同分布, 其分布函数为 F(x) , 概率密度为 f(x) , \\ 记 Y=\min \left\{X_{1}, X_{2}, \cdots\right. , \left.X_{n}\right\}, Z=\max \left\{X_{1}, X_{2}, \cdots, X_{n}\right\} , 则\\ \begin{array}{l} F_{Y}(y)=1-[1-F(y)]^{n}, f_{Y}(y)=n[1-F(y)]^{n-1} f(y) \Rightarrow E Y=\int_{-\infty}^{+\infty} y f_{Y}(y) \mathrm{d} y \\ F_{Z}(z)=[F(z)]^{n}, f_{Z}(z)=n[F(z)]^{n-1} f(z) \Rightarrow E Z=\int_{-\infty}^{+\infty} z f_{Z}(z) \mathrm{d} z \end{array}$

方差

$D X=E\left(X^{2}\right)-(E X)^{2}\\ D(X \pm Y)=D X+D Y \pm 2 \operatorname{Cov}(X, Y)\\$

协方差 相关系数

$\operatorname{Cov}(X, Y)=E[(X-E X)(Y-E Y)]\\ \begin{aligned} \operatorname{Cov}(X, Y) &=E(X Y-X \cdot E Y-E X \cdot Y+E X \cdot E Y) \\ &=E(X Y)-E X \cdot E Y-E X \cdot E Y+E X \cdot E Y \\ &=E(X Y)-E X E Y . \end{aligned}\\ (2) \rho_{X Y} 定义. (相关系数, 表线性相依程度)\\ \rho_{X Y}=\frac{\operatorname{Cov}(X, Y)}{\sqrt{D X} \sqrt{D Y}}\left\{\begin{array}{l}=0 \Leftrightarrow X, Y \text { 不相关, } \\ \neq 0 \Leftrightarrow X, Y \text { 相关. }\end{array}\right. \\ (量纲为 1 , 无单位)\\ (3)性质.\\ (1) \operatorname{Cov}(X, Y)=\operatorname{Cov}(Y, X) .\\ (2) \operatorname{Cov}(a X, b Y)=a b \operatorname{Cov}(X, Y) .\\ (3) \operatorname{Cov}\left(X_{1}+X_{2}, Y\right)=\operatorname{Cov}\left(X_{1}, Y\right)+\operatorname{Cov}\left(X_{2}, Y\right) .\\ (4) \left|\rho_{X Y}\right| \leqslant 1 .\\ (5) \rho_{X Y}=1 \Leftrightarrow P\{Y=a X+b\}=1(a>0) .\\ \rho_{X Y}=-1 \Leftrightarrow P\{Y=a X+b\}=1(a<0) .\\ 考试时, Y=a X+b, a>0 \Rightarrow \rho_{X Y}=1 .\\ Y=a X+b, a<0 \Rightarrow \rho_{X Y}=-1 .\\ (6) 五个充要条件.\\ \begin{aligned} \rho_{X Y}=0 & \Leftrightarrow \operatorname{Cov}(X, Y)=0 \Leftrightarrow E(X Y)=E X \cdot E Y \\ & \Leftrightarrow D(X+Y)=D X+D Y \Leftrightarrow D(X-Y)=D X+D Y \end{aligned}\\ (7) X, Y 独立 \Rightarrow \rho_{X Y}=0 .\\ (8) 若 (X, Y) \sim\left(\mu_{1}, \mu_{2} ; \sigma_{1}^{2}, \sigma_{2}^{2} ; \rho_{X Y}\right) , 则 X, Y 独立 \Leftrightarrow X, Y 不相关 \left(\rho_{X Y}=0\right) .$

常用分布的EX,DX

$(1) 0-1 分布, E X=p, D X=p-p^{2}=(1-p) p .\\ (2) X \sim B(n, p), E X=n p, D X=n p(1-p) .\\ (3) X \sim P(\lambda), E X=\lambda, D X=\lambda .\\ (4) X \sim G(p), E X=\frac{1}{p}, D X=\frac{1-p}{p^{2}} .\\ (5) X \sim U(a, b), E X=\frac{a+b}{2}, D X=\frac{(b-a)^{2}}{12} .\\ (6) X \sim E(\lambda), E X=\frac{1}{\lambda}, D X=\frac{1}{\lambda^{2}} .\\ (7) X \sim N\left(\mu, \sigma^{2}\right), E X=\mu, D X=\sigma^{2} .\\ (8) X \sim \chi^{2}(n), E X=n, D X=2 n .$

正态总体下的常用结论

$设 X_{1}, X_{2}, \cdots, X_{n} 是取自正态总体 N\left(\mu, \sigma^{2}\right) 的一个样本, \bar{X}, S^{2} 分别是样本均值和样本 方差, 则\\ \text { (1) } \bar{X} \sim N\left(\mu, \frac{\sigma^{2}}{n}\right) \text {, 即 } \frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=\frac{\sqrt{n}(\bar{X}-\mu)}{\sigma} \sim N(0,1) \text {; }\\ (2) \frac{1}{\sigma^{2}} \sum_{i=1}^{n}\left(X_{i}-\mu\right)^{2} \sim \chi^{2}(n) ;\\ (3) \frac{(n-1) S^{2}}{\sigma^{2}}=\sum_{i=1}^{n}\left(\frac{X_{i}-\bar{X}}{\sigma}\right)^{2} \sim \chi^{2}(n-1)(\mu 末知, 在“(2)” 中用 \bar{X} 替代 \mu) ;\\ (4) \bar{X} 与 S^{2} 相互独立, \frac{\sqrt{n}(\bar{X}-\mu)}{S} \sim t(n-1) ( \sigma 末知, 在 “(1)” 中用 S 替代 \sigma ). 进一步有 \frac{n(\bar{X}-\mu)^{2}}{S^{2}} \sim F(1, n-1).$

施工中…