董功杭
发布于 2024-05-08 / 141 阅读
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微积分

1.函数与极限

1.1数集

常见数集

\begin{array}{l} \text{自然数集}& \mathbb{N} =\left\{ 0,1,2,...,n,... \right\}\\ \text{整数集}& \mathbb{Z} =\left\{ ...,-n,...,-2,-1,0,1,2,...,n,... \right\}\\ \text{有理数集}& \mathbb{Q} =\left\{ \frac{p}{q}\mid p\in \mathbb{Z} ,\;q\in \mathbb{N} ^+\;\mathrm{and}\;p\bot q \right\}\\ \text{实数集}& \mathbb{R}\\ \text{非零实数集}& \mathbb{R} *\\ \text{正实数集}& \mathbb{R} ^+\\ \end{array}

数集的直积

\mathbb{R} ^2=\mathbb{R} \times \mathbb{R} =\left\{ \left( x,y \right) \mid x\in \mathbb{R} ,y\in \mathbb{R} \right\}

1.2极限

符号对照表

\begin{aligned} C&\phantom{=}\text{常数}\\ f\left( x \right) &\phantom{=}\text{函数}\\ n&\phantom{=}\text{正整数}\\ \left\{ x_n \right\} ,\left\{ y_n \right\} &\phantom{=}\text{数列}\\ \end{aligned}

极限运算法则1

\lim \left[ Cf\left( x \right) \right] =C\left[ \lim f\left( x \right) \right]

极限运算法则2

\lim \left[ f\left( x \right) \right] ^n=\left[ \lim f\left( x \right) \right] ^n

极限运算法则3

\lim_{n\rightarrow +\infty} \left( x_n\pm y_n \right) =\lim_{n\rightarrow +\infty} x_n\pm \lim_{n\rightarrow +\infty} y_n

极限运算法则4

\lim_{n\rightarrow +\infty} \left( x_n\cdot y_n \right) =\lim_{n\rightarrow +\infty} x_n\cdot \lim_{n\rightarrow +\infty} y_n

极限运算法则5

\lim_{n\rightarrow +\infty} \left( \frac{x_n}{y_n} \right) =\frac{\lim_{n\rightarrow +\infty} x_n}{\lim_{n\rightarrow +\infty} y_n},\left( y_n\ne 0,\;n=1,2,3,... \right)

1.3不定式极限

洛必达法则 0/0 型不定式极限

\text{若函数}f\left( x \right) \text{,}g\left( x \right) \text{满足如下条件:} \\ \underline{\begin{array}{l} \lim_{x\rightarrow a} f\left( x \right) =0, \lim_{x\rightarrow a} g\left( x \right) =0\\ \hdashline \text{在点}a\text{的某去心邻域内}f\left( x \right) \text{和}g\left( x \right) \text{都可导,且}g\prime\left( x \right) \ne 0\\ \hdashline \lim_{x\rightarrow a} \frac{f\prime\left( x \right)}{g\prime\left( x \right)}=A\quad \left( A\text{为实数} \right)\\ \end{array}} \\ \text{则有} \\ \lim_{x\rightarrow a} \frac{f\left( x \right)}{g\left( x \right)}=\lim_{x\rightarrow a} \frac{f\prime\left( x \right)}{g\prime\left( x \right)}=A

洛必达法则 ∞/∞ 型不定式极限

\text{若函数}f\left( x \right) \text{,}g\left( x \right) \text{满足如下条件:} \\ \underline{\begin{array}{l} \lim_{x\rightarrow a} f\left( x \right) =\infty , \lim_{x\rightarrow a} g\left( x \right) =\infty\\ \hdashline \text{在点}a\text{的某去心邻域内}f\left( x \right) \text{和}g\left( x \right) \text{都可导,且}g\prime\left( x \right) \ne 0\\ \hdashline \lim_{x\rightarrow a} \frac{f\prime\left( x \right)}{g\prime\left( x \right)}=A\quad \left( A\text{为实数} \right)\\ \end{array}} \\ \text{则有} \\ \lim_{x\rightarrow a} \frac{f\left( x \right)}{g\left( x \right)}=\lim_{x\rightarrow a} \frac{f\prime\left( x \right)}{g\prime\left( x \right)}=A

2.导数与微分

2.1导数

导数的定义

f\prime\left( x_0 \right) =\lim_{\Delta x\rightarrow 0} \frac{\Delta y}{\Delta x}=\lim_{\Delta x\rightarrow 0} \frac{f\left( x_0+\Delta x \right) -f\left( x_0 \right)}{\Delta x}

导数的记法

y\prime\mid_{x=x_0}^{}

导数的牛顿记法 对时间的导数

\dot{y}\mid_{t=t_0}^{}

导数的莱布尼兹记法

\frac{\mathrm{d}y}{\mathrm{d}x}\mid_{x=x_0}^{} \\ \frac{\mathrm{d}f\left( x \right)}{\mathrm{d}x}\mid_{x=x_0}^{}

函数的和、差、积、商的求导法则

\left( u\pm v \right) \prime=u\prime\pm v\prime \\ \left( Cu \right) \prime=Cu\prime \\ \left( uv \right) \prime=u\prime+uv\prime \\ \left( \frac{u}{v} \right) \prime=\frac{u\prime-uv\prime}{v^2}\left( v\ne 0 \right)

反函数的求导法则

\left[ f^{-1}\left( x \right) \right] \prime=\frac{1}{f\prime\left( y \right)}

复合函数的求导法则

y=f\left( u \right) ,\;u=g\left( x \right) \\ \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\cdot \frac{\mathrm{d}u}{\mathrm{d}x}

基本导数1

\left( C \right) \prime=0 \\ \left( x^{\mu} \right) \prime=\mu x^{\mu -1}

基本导数2

\left( \sin x \right) \prime=\cos x \\ \left( \cos x \right) \prime=-\sin x \\ \left( \tan x \right) \prime=\sec ^2x \\ \left( \cot x \right) \prime=-\csc ^2x \\ \left( \sec x \right) \prime=\sec x\tan x \\ \left( \csc x \right) \prime=-\csc x\cot x

基本导数3

\left( a^x \right) \prime=a^x\ln a \\ \left( e^x \right) \prime=e^x \\ \left( \log _ax \right) \prime=\frac{1}{x\ln a} \\ \left( \ln x \right) \prime=\frac{1}{x}

基本导数4

\left( \mathrm{arc}\sin x \right) \prime=\frac{1}{\sqrt{1-x^2}} \\ \left( \mathrm{arc}\cos x \right) \prime=-\frac{1}{\sqrt{1-x^2}} \\ \left( \mathrm{arc}\tan x \right) \prime=\frac{1}{1+x^2} \\ \left( \mathrm{arc}\cot x \right) \prime=-\frac{1}{1+x^2}

2.2微分

函数的和、差、积、商的微分法则

\mathrm{d}\left( u\pm v \right) =\mathrm{d}u\pm \mathrm{d}v \\ \mathrm{d}\left( Cu \right) =C\mathrm{d}u \\ \mathrm{d}\left( uv \right) =v\mathrm{d}u+u\mathrm{d}v \\ \mathrm{d}\left( \frac{u}{v} \right) =\frac{v\mathrm{d}u-u\mathrm{d}v}{v^2}\left( v\ne 0 \right)

复合函数的微分法则 微分形式不变性

y=f\left( u \right) ,\;u=g\left( x \right) \\ \mathrm{d}y=y\prime_x\mathrm{d}x=f\prime\left( u \right) g\prime\left( x \right) \mathrm{d}x

泰勒中值定理 泰勒展开

f\left( x \right) =f\left( x_0 \right) +f\prime\left( x_0 \right) \left( x-x_0 \right) +\frac{f\prime\prime\left( x_0 \right)}{2!}\left( x-x_0 \right) ^2+\cdots +\frac{f^{\left( n \right)}\left( x_0 \right)}{n!}\left( x-x_0 \right) ^n+R_n\left( x \right) \\ =f\left( x_0 \right) +\sum_{k=1}^n{\frac{f^{\left( k \right)}\left( x_0 \right)}{k!}\left( x-x_0 \right) ^k}+R_n\left( x \right)

拉格朗日余项

R_n\left( x \right) =\frac{f^{\left( n+1 \right)}\left( \xi \right)}{\left( n+1 \right) !}\left( x-x_0 \right) ^{n+1},\;x_0\leqslant \xi \leqslant x

3积分

3.1不定积分

基本积分1

\int{k\mathrm{d}x}=kx+C \\ \int{x^{\mu}\mathrm{d}x}=\frac{x^{\mu +1}}{\mu +1}+C\ \ \left( \mu \ne -1 \right) \\ \int{\frac{1}{x}\mathrm{d}x}=\ln \left| x \right|+C \\ \int{\frac{1}{1+x^2}\mathrm{d}x}=\mathrm{arc}\tan x+C \\ \int{\frac{1}{\sqrt{1-x^2}}\mathrm{d}x}=\mathrm{arc}\sin x+C

基本积分2

\int{\cos x\mathrm{d}x}=\sin x+C \\ \int{\sin x\mathrm{d}x}=-\cos x+C \\ \int{\frac{1}{\cos ^2x}\mathrm{d}x}=\int{\sec ^2x\mathrm{d}x}=\tan x+C \\ \int{\frac{1}{\sin ^2x}\mathrm{d}x}=\int{\csc ^2x\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

基本积分3

\int{e^x\mathrm{d}x}=e^x+C \\ \int{a^x\mathrm{d}x}=\frac{a^x}{\ln a}+C \\ \int{\mathrm{sh}x\mathrm{d}x}=\mathrm{ch}x+C \\ \int{\mathrm{ch}x\mathrm{d}x}=\mathrm{sh}x+C

基本积分4

\int{\tan x\mathrm{d}x}=-\ln \left| \cos x \right|+C \\ \int{\cot x\mathrm{d}x}=\ln \left| \sin x \right|+C \\ \int{\sec x\mathrm{d}x}=\ln \left| \sec x+\tan x \right|+C \\ \int{\csc x\mathrm{d}x}=\ln \left| \csc x-\cot x \right|+C

基本积分5

\int{\frac{1}{a^2+x^2}\mathrm{d}x}=\frac{1}{a}\mathrm{arc}\tan \frac{x}{a}+C \\ \int{\frac{1}{x^2-a^2}\mathrm{d}x}=\frac{1}{2a}\ln \left| \frac{x-a}{x+a} \right|+C \\ \int{\frac{1}{\sqrt{a^2-x^2}}\mathrm{d}x}=\mathrm{arc}\sin \frac{x}{a}+C \\ \int{\frac{1}{\sqrt{x^2+a^2}}\mathrm{d}x}=\ln \left( x+\sqrt{x^2+a^2} \right) +C \\ \int{\frac{1}{\sqrt{x^2-a^2}}\mathrm{d}x}=\ln \left( x+\sqrt{x^2a^2} \right) +C

3.2不定积分的性质

不定积分性质 1

\int{\left[ f\left( x \right) +g\left( x \right) \right] \mathrm{d}x}=\int{f\left( x \right) \mathrm{d}x}+\int{g\left( x \right) \mathrm{d}x}

不定积分性质 2

\int{kf\left( x \right) \mathrm{d}x}=k\int{f\left( x \right) \mathrm{d}x}

分部积分法

\int{u\mathrm{d}v}=uv-\int{v\mathrm{d}u}

3.3定积分

牛顿-莱布尼兹公式

\int_a^b{F\prime\left( x \right) \mathrm{d}x}=F\left( b \right) -F\left( a \right)

定积分性质1

a=b\Rightarrow \int_a^b{f\left( x \right) \mathrm{d}x}=0

定积分性质2

\int_a^b{f\left( x \right) \mathrm{d}x}=-\int_b^a{f\left( x \right) \mathrm{d}x}

定积分性质3

\int_a^b{\left[ f\left( x \right) \pm g\left( x \right) \right] \mathrm{d}x}=\int_a^b{f\left( x \right) \mathrm{d}x}\pm \int_a^b{g\left( x \right) \mathrm{d}x}

定积分性质4

\int_a^b{kf\left( x \right) \mathrm{d}x}=k\int_a^b{f\left( x \right) \mathrm{d}x}

定积分性质5

\forall c\in \left( a,b \right) \\ \int_a^b{f\left( x \right) \mathrm{d}x}=\int_a^c{f\left( x \right) \mathrm{d}x}+\int_c^b{f\left( x \right) \mathrm{d}x}

定积分性质6

\int_a^b{\mathrm{d}x}=b-a

定积分性质7

f\left( x \right) \geqslant 0,\;x\in \left[ a,b \right] \Rightarrow \int_a^b{f\left( x \right) \mathrm{d}x}\geqslant 0

定积分性质8

f\left( x \right) \geqslant g\left( x \right) ,\;x\in \left[ a,b \right] \Rightarrow \int_a^b{f\left( x \right) \mathrm{d}x}\geqslant \int_a^b{g\left( x \right) \mathrm{d}x}

定积分性质9

a<b\Rightarrow \left| \int_a^b{f\left( x \right) \mathrm{d}x} \right|\leqslant \int_a^b{\left| f\left( x \right) \right|\mathrm{d}x}

定积分性质10

M=f_{\max}\left( x \right) ,\;m=f_{\min}\left( x \right) ,\;x\in \left[ a,b \right] \\ m\left( b-a \right) \leqslant \int_a^b{f\left( x \right) \mathrm{d}x}\leqslant M\left( b-a \right)

3.4积分中值定理

基本形式

\text{若函数在闭区间}\left[ a,b \right] \text{上连续,则} \\ \exists \xi \in \left[ a,b \right] \\ \int_a^b{f\left( x \right) \mathrm{d}x}=f\left( \xi \right) \left( b-a \right)

推广第一定理

\text{若}f\left( x \right) \text{、}g\left( x \right) \text{在闭区间}\left[ a,b \right] \text{上可积,且}g\left( x \right) \text{在此区间上不变号,则} \\ \exists \xi \in \left[ a,b \right] \\ \int_a^b{f\left( x \right) g\left( x \right) \mathrm{d}x}=f\left( \xi \right) \int_a^b{g\left( x \right) \mathrm{d}x}

推广第二定理1

\text{若}f\left( x \right) \text{、}g\left( x \right) \text{在闭区间}\left[ a,b \right] \text{上可积,}f\left( x \right) \text{为单调函数,则} \\ \exists \xi \in \left[ a,b \right] \\ \int_a^b{f\left( x \right) g\left( x \right) \mathrm{d}x}=f\left( a \right) \int_a^{\xi}{g\left( x \right) \mathrm{d}x}+f\left( b \right) \int_{\xi}^b{g\left( x \right) \mathrm{d}x}

推广第二定理2

\text{若}f\left( x \right) \text{、}g\left( x \right) \text{在闭区间}\left[ a,b \right] \text{上可积,}f\left( x \right) \geqslant 0\text{且为单调递减函数,则} \\ \exists \xi \in \left[ a,b \right] \\ \int_a^b{f\left( x \right) g\left( x \right) \mathrm{d}x}=f\left( a \right) \int_a^{\xi}{g\left( x \right) \mathrm{d}x}

推广第二定理3

\text{若}f\left( x \right) \text{、}g\left( x \right) \text{在闭区间}\left[ a,b \right] \text{上可积,}f\left( x \right) \geqslant 0\text{且为单调递增函数,则} \\ \exists \xi \in \left[ a,b \right] \\ \int_a^b{f\left( x \right) g\left( x \right) \mathrm{d}x}=f\left( b \right) \int_{\xi}^b{g\left( x \right) \mathrm{d}x}

4.多元函数微分

4.1平面点集

符号对照表

\begin{aligned} P&\phantom{=}\text{平面上的点}\\ \mathbb{R} ^2&\phantom{=}\text{二元有序实数集,坐标平面}\\ O&\phantom{=}\text{坐标原点}\\ E&\phantom{=}\text{平面点集}\\ P_i&\phantom{=}\text{坐标平面上的一个点}\\ \delta &\phantom{=}\text{某一正实数}\\ \partial E&\phantom{=}\text{所有边界点的集合,平面点集}E\text{的边界}\\ \end{aligned}

二元有序实数集

\mathbb{R} ^2=\mathbb{R} \times \mathbb{R} =\left\{ \left( x, y \right) \mid x, y\in \mathbb{R} \right\}

平面点集

E=\left\{ \left( x,y \right) \mid \left( x,y \right) \,\,\mathrm{has} \mathrm{property} P \right\} =\left\{ P \right\}

邻域

U\left( P_0, \delta \right) =\left\{ P\mid \left| PP_0 \right|<\delta \right\} =\left\{ \left( x, y \right) \mid \sqrt{\left( x-x_0 \right) ^2+\left( y-y_0 \right) ^2}<\delta \right\}

去心邻域

\mathring{U}\left( P_0, \delta \right) =\left\{ \left( x, y \right) \begin{cases} \sqrt{\left( x-x_0 \right) ^2+\left( y-y_0 \right) ^2}<\delta\\ \sqrt{\left( x-x_0 \right) ^2+\left( y-y_0 \right) ^2}\ne 0\\ \end{cases} \right\}

内点

\mathrm{for} : P_1\in \mathbb{R} ^2, E\subset \mathbb{R} ^2 \\ \exists U\left( P_1, \delta \right) \subset E

边界点 边界

\mathrm{for} : P_2\in \mathbb{R} ^2, E\subset \mathbb{R} ^2, \forall \delta >0 \\ U\left( P_2, \delta \right) \nsubseteq E\,\,\mathrm{and} U\left( P_2, \delta \right) \cap E\ne \varnothing \\ \partial E=\left\{ P_2 \right\}

\nsubset 符号错误 \nsubseteq 不带等于号

外点

\mathrm{for} : P_3\in \mathbb{R} ^2, E\subset \mathbb{R} ^2 \\ \exists U\left( P_3, \delta \right) \cap E=\varnothing

聚点

\forall \delta >0 \\ \mathring{U}\left( P, \delta \right) \cap E\ne \varnothing

开集

\text{若点集}E\text{中的点都是}E\text{的内点}

闭集

\text{若点集}E\text{的边界}\partial E\subset E

连通集

\text{若点集}E\text{内任意两点都可以用折线连接,且折线上的点都属于}E

区域 开区域

\text{连通的开集}

闭区域

\text{开区域及其边界所构成的点集}

有界集

\text{对于平面点集}E\text{,若}\exists r>0\text{,使得} \\ E\subset U\left( O, r \right)

无界集

\text{一个集合若不是有界集}

4.2 多维空间

符号对照表

\begin{aligned} \boldsymbol{x}, \boldsymbol{y}&\phantom{=}n\text{维向量}\\ \mathbb{R} ^n&\phantom{=}n\text{维空间}\\ \lambda &\phantom{=}\text{任意实数}\\ \rho \left( \boldsymbol{x}, \boldsymbol{y} \right) &\phantom{=}n\text{维向量}\boldsymbol{x}, \boldsymbol{y}\text{之间的欧氏距离}\\ P_i&\phantom{=}\text{坐标平面上的一个点}\\ \delta &\phantom{=}\text{某一正实数}\\ \end{aligned}

n维空间

\mathbb{R} ^n=\mathbb{R} \times \mathbb{R} \times \,\,\cdots \,\,\times \mathbb{R} =\left\{ \left( x_1, x_2, \cdots , x_n \right) \mid x_i\in \mathbb{R} , i=1,2,\cdots ,n \right\}

n维向量的线性运算

\mathrm{for} : \boldsymbol{x}=\left( x_1, x_2, \cdots , x_n \right) , \boldsymbol{y}=\left( y_1, y_2, \cdots , y_n \right) \\ \boldsymbol{x}+\boldsymbol{y}=\left( x_1+y_1, x_2+y_2, \cdots , x_n+y_n \right) \\ \lambda x=\left( \lambda x_1, \lambda x_2, \cdots , \lambda x_n \right)

欧氏距离

\rho \left( \boldsymbol{x}, \boldsymbol{y} \right) =\left\| \boldsymbol{x}-\boldsymbol{y} \right\| =\sqrt{\sum_{i=1}^n{\left( x_i-y_i \right) ^2}}

与零元之间欧氏距离

\left\| \boldsymbol{x} \right\| =\rho \left( \boldsymbol{x},\mathbf{0} \right) =\left\| \boldsymbol{x}-\mathbf{0} \right\| =\sqrt{\sum_{i=1}^n{{x_i}^2}}

趋于固定元

\mathrm{for} : \boldsymbol{x}=\left( x_1, x_2, \cdots , x_n \right) , a=\left( a_1, a_2, \cdots , a_n \right) \in \mathbb{R} ^n \\ \left\| \boldsymbol{x}-\boldsymbol{a} \right\| \rightarrow 0\Leftrightarrow \boldsymbol{x}\rightarrow \boldsymbol{a}\Leftrightarrow x_i\rightarrow a_i, i=1,2,3,...,n

4.3多元函数

符号对照表

\begin{aligned} f\left( x,y \right) &\phantom{=}\text{二元函数}\\ \mathbb{R} ^2&\phantom{=}\text{二元有序实数集,坐标平面,二维空间}\\ D&\phantom{=}\text{二元函数}f\left( x,y \right) \text{的定义域}\\ P_0&\phantom{=}\text{定义域}D\text{上的一个聚点}\\ f_{\max}&\phantom{=}f\left( x,y \right) \text{在}D\text{上的最大值}\\ f_{\min}&\phantom{=}f\left( x,y \right) \text{在}D\text{上的最小值}\\ \end{aligned}

二元函数

\mathrm{for} :\ D\subset \mathbb{R} ^2, D\ne \varnothing \\ z=f\left( x,y \right) , \left( x,y \right) \in D

多元函数

\mathrm{for} :\ D\subset \mathbb{R} ^n, D\ne \varnothing \\ z=f\left( \boldsymbol{x} \right) , x=\left( x_1, x_2, \cdots , x_n \right) \in D

二元函数的极限

\text{设}P_0\left( x_0,y_0 \right) \text{为}D\text{的聚点,若存在常数}A\text{,对任意给定整数}\varepsilon \text{,总存在}\delta \text{,使得当点} \\ P\left( x,y \right) \in D\cap \mathring{U}\left( P_0, \delta \right) \text{时,都有} \\ \left| f\left( P \right) -A \right|=\left| f\left( x,y \right) -A \right|<\varepsilon \\ \text{称}A\text{为函数}f\left( x,y \right) \text{当}\left( x,y \right) \rightarrow \left( x_0,y_0 \right) \text{时的极限,记为:} \\ \lim_{\left( x,y \right) \rightarrow \left( x_0,y_0 \right)} f\left( x,y \right)

二元函数的连续性

\lim_{\left( x,y \right) \rightarrow \left( x_0,y_0 \right)} f\left( x,y \right) =f\left( x_0,y_0 \right)

二元函数的有界性

\lim_{\left( x,y \right) \rightarrow \left( x_0,y_0 \right)} f\left( x,y \right) =f\left( x_0,y_0 \right) \Rightarrow f_{\min}\leqslant f\left( x,y \right) \leqslant f_{\max}

二元函数的一致连续性

\text{若}f\left( x,y \right) \text{的定义域为有界闭区域,且}f\left( x,y \right) \text{在}D\text{上连续,则} \\ f\left( x,y \right) \text{在}D\text{上一致连续}

4.4偏导数

偏导数的定义

\frac{\partial f}{\partial x}\mid_{\begin{array}{c} x=x_0\\ y=y_0\\ \end{array}}^{}=f_x\left( x_0,y_0 \right) =\lim_{\Delta x\rightarrow 0} \frac{f\left( x_0+\Delta x,y_0 \right) -f\left( x_0,y_0 \right)}{\Delta x} \\ \frac{\partial f}{\partial y}\mid_{\begin{array}{c} x=x_0\\ y=y_0\\ \end{array}}^{}=f_y\left( x_0,y_0 \right) =\lim_{\Delta y\rightarrow 0} \frac{f\left( x_0,y_0+\Delta y \right) -f\left( x_0,y_0 \right)}{\Delta y}

二阶偏导数

\frac{\partial}{\partial x}\left( \frac{\partial f}{\partial x} \right) =\frac{\partial ^2f}{\partial x^2}=f_{xx}\left( x,y \right) \\ \frac{\partial}{\partial y}\left( \frac{\partial f}{\partial x} \right) =\frac{\partial ^2f}{\partial x\partial y}=f_{xy}\left( x,y \right) \\ \frac{\partial}{\partial x}\left( \frac{\partial f}{\partial y} \right) =\frac{\partial ^2f}{\partial y\partial x}=f_{yx}\left( x,y \right) \\ \frac{\partial}{\partial y}\left( \frac{\partial f}{\partial y} \right) =\frac{\partial ^2f}{\partial y^2}=f_{yy}\left( x,y \right)

定理1

\text{若}\frac{\partial ^2f}{\partial x\partial y}\text{及}\frac{\partial ^2f}{\partial y\partial x}\text{在}f\left( x,y \right) \text{的定义域}D\text{内连续,则必有} \\ \frac{\partial ^2f}{\partial x\partial y}=\frac{\partial ^2f}{\partial y\partial x}

4.5全微分

偏微分定义

f\left( x+\Delta x,y \right) \approx f_x\left( x,y \right) \cdot \Delta x \\ f\left( x,y+\Delta y \right) \approx f_y\left( x,y \right) \cdot \Delta y

全增量

z=f\left( x,y \right) \\ \Delta z=f\left( x+\Delta x,y+\Delta y \right) -f\left( x,y \right)

全微分

\text{若全增量可表示为} \\ \Delta z=A\cdot \Delta x+B\cdot \Delta y+o\left( \rho \right) \\ \text{其中}A\text{、}B\text{不依赖于}\Delta x\text{和}\Delta y\text{而仅仅与}x,y\text{有关,且} \\ \rho =\sqrt{\left( \Delta x \right) ^2+\left( \Delta y \right) ^2} \\ \text{则有全微分} \\ \mathrm{d}z=A\cdot \Delta x+B\cdot \Delta y

定理 1 必要条件

\text{若}z=f\left( x,y \right) \text{在点}\left( x,y \right) \text{可微,则} \\ \mathrm{d}z=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y

定理 2 充分条件

\text{若}z=f\left( x,y \right) \text{的偏导数}\frac{\partial z}{\partial x}\text{,}\frac{\partial z}{\partial y}\text{在点}\left( x,y \right) \text{连续,则函数在该点可微}

二元函数的泰勒展式

f\left( x_0+h, y_0+k \right) =f\left( x_0, y_0 \right) +\left( h\frac{\partial}{\partial x}+k\frac{\partial}{\partial y} \right) f\left( x_0, y_0 \right) +\frac{1}{2 !}\left( h\frac{\partial}{\partial x}+k\frac{\partial}{\partial y} \right) ^2f\left( x_0, y_0 \right) +\cdots +\frac{1}{n\,\,!}\left( h\frac{\partial}{\partial x}+k\frac{\partial}{\partial y} \right) ^nf\left( x_0, y_0 \right) +R_n

拉格朗日型余项

R_n=\frac{1}{\left( n+1 \right) \,\,!}\left( h\frac{\partial}{\partial x}+k\frac{\partial}{\partial y} \right) ^{n+1}f\left( x_0+\theta h, y_0+\theta k \right) , \left( 0<\theta <1 \right)

拉格朗日中值定理

f\left( x_0+h, y_0+k \right) =f\left( x_0, y_0 \right) +\left( h\frac{\partial}{\partial x}+k\frac{\partial}{\partial y} \right) f\left( x_0+\theta h, y_0+\theta k \right) , \left( 0<\theta <1 \right)

4.6 复合函数的求导法则

一元函数和多元函数复合的情形

u=\varphi \left( t \right) , v=\psi \left( t \right) \\ z=f\left( u,v \right) =f\left[ \varphi \left( t \right) , \psi \left( t \right) \right] \\ \text{若}u,v\text{在}t\text{点可导,}z=f\left( u,v \right) \text{在对应点处具有连续偏导数,则} \\ \frac{\mathrm{d}z}{\mathrm{d}t}=\frac{\partial z}{\partial u}\frac{\mathrm{d}u}{\mathrm{d}t}+\frac{\partial z}{\partial v}\frac{\mathrm{d}v}{\mathrm{d}t}

多元函数和多元函数复合的情形

u=\varphi \left( x,y \right) , v=\psi \left( x,y \right) \\ z=f\left( u,v \right) =f\left[ \varphi \left( x,y \right) , \psi \left( x,y \right) \right] \\ \text{若}u,v\text{在}\left( x,y \right) \text{具有对}x\text{及}y\text{的偏导数,}z=f\left( u,v \right) \text{在对应点处具有连续偏导数,则} \\ \frac{\mathrm{d}z}{\mathrm{d}x}=\frac{\partial z}{\partial u}\frac{\mathrm{d}u}{\mathrm{d}x}+\frac{\partial z}{\partial v}\frac{\mathrm{d}v}{\mathrm{d}x} \\ \frac{\mathrm{d}z}{\mathrm{d}y}=\frac{\partial z}{\partial u}\frac{\mathrm{d}u}{\mathrm{d}y}+\frac{\partial z}{\partial v}\frac{\mathrm{d}v}{\mathrm{d}y}

全微分形式不变性

u=\varphi \left( x,y \right) , v=\psi \left( x,y \right) \\ z=f\left( u,v \right) =f\left[ \varphi \left( x,y \right) , \psi \left( x,y \right) \right] \\ \text{若}u,v\text{在}\left( x,y \right) \text{具有对}x\text{及}y\text{的偏导数,}z=f\left( u,v \right) \text{在对应点处具有连续偏导数,则} \\ \mathrm{d}z=\frac{\partial z}{\partial u}\mathrm{d}u+\frac{\partial z}{\partial v}\mathrm{d}v \\ \mathrm{d}z=\frac{\partial z}{\partial x}\mathrm{d}x+\frac{\partial z}{\partial y}\mathrm{d}y \\ \mathrm{d}z\begin{array}{l} =\left( \frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x} \right) \mathrm{d}x+\left( \frac{\partial z}{\partial u}\frac{\partial u}{\partial y}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial y} \right) \mathrm{d}y\\ =\frac{\partial z}{\partial u}\left( \frac{\partial u}{\partial x}\mathrm{d}x+\frac{\partial u}{\partial y}\mathrm{d}y \right) +\frac{\partial z}{\partial v}\left( \frac{\partial v}{\partial x}\mathrm{d}x+\frac{\partial v}{\partial y}\mathrm{d}y \right)\\ =\frac{\partial z}{\partial u}\mathrm{d}u+\frac{\partial z}{\partial v}\mathrm{d}v\\ \end{array}

4.7隐函数的求导公式

隐函数

F\left( x,y \right) =0

隐函数求导公式

\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{F_x}{F_y}

隐函数的二阶导数

\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=-\frac{F_{xx}{F_y}^2-2F_{xy}F_xF_y+F_{yy}{F_x}^2}{{F_y}^3}

雅克比行列式

\begin{cases} F\left( x,y,u,v \right) =0\\ G\left( x,y,u,v \right) =0\\ \end{cases} \\ J=\frac{\partial \left( F,G \right)}{\partial \left( u,v \right)}=\left| \begin{matrix} \frac{\partial F}{\partial u}& \frac{\partial F}{\partial v}\\ \frac{\partial G}{\partial u}& \frac{\partial G}{\partial v}\\ \end{matrix} \right| \\ \frac{\partial u}{\partial x}=-\frac{1}{J}\frac{\partial \left( F,G \right)}{\partial \left( x,v \right)}=-\frac{\left| \begin{matrix} F_x& F_v\\ G_x& G_v\\ \end{matrix} \right|}{\left| \begin{matrix} F_u& F_v\\ G_u& G_v\\ \end{matrix} \right|} \\ \frac{\partial v}{\partial x}=-\frac{1}{J}\frac{\partial \left( F,G \right)}{\partial \left( u,x \right)}=-\frac{\left| \begin{matrix} F_u& F_x\\ G_u& G_x\\ \end{matrix} \right|}{\left| \begin{matrix} F_u& F_v\\ G_u& G_v\\ \end{matrix} \right|} \\ \frac{\partial u}{\partial y}=-\frac{1}{J}\frac{\partial \left( F,G \right)}{\partial \left( y,v \right)}=-\frac{\left| \begin{matrix} F_y& F_v\\ G_y& G_v\\ \end{matrix} \right|}{\left| \begin{matrix} F_u& F_v\\ G_u& G_v\\ \end{matrix} \right|} \\ \frac{\partial v}{\partial y}=-\frac{1}{J}\frac{\partial \left( F,G \right)}{\partial \left( u,x \right)}=-\frac{\left| \begin{matrix} F_u& F_y\\ G_u& G_y\\ \end{matrix} \right|}{\left| \begin{matrix} F_u& F_v\\ G_u& G_v\\ \end{matrix} \right|}

4.8方向导数与梯度

符号对照表

\begin{aligned} l&\phantom{=}\text{函数在某点处的方向}\\ \cos \alpha , \cos \beta &\phantom{=}\text{方向}l\text{的方向余弦}\\ \boldsymbol{i}, \boldsymbol{j}&\phantom{=}\text{底矢,单位方向向量}\\ \end{aligned}

方向导数

\frac{\partial f}{\partial l}\mid_{x_0,y_0}^{}=f_x\left( x_0,y_0 \right) \cos \alpha +f_y\left( x_0,y_0 \right) \cos \beta

梯度

\mathbf{grad}f\left( x_0,y_0 \right) =\mathbf{\nabla }f\left( x_0,y_0 \right) =f_x\left( x_0,y_0 \right) \boldsymbol{i}+f\left( x_0,y_0 \right) \boldsymbol{j}

方向导数与梯度的关系

\frac{\partial f}{\partial l}\mid_{x_0,y_0}^{}=\mathbf{\nabla }f\left( x_0,y_0 \right) \cdot \boldsymbol{e}_l=\left| \mathbf{\nabla }f\left( x_0,y_0 \right) \right|\cos \theta \\ \text{其中} \\ \theta =\left( \widehat{\mathbf{\nabla }f\left( x_0,y_0 \right) , \boldsymbol{e}_l} \right)

4.9多元函数的极值

必要条件

f\left( x,y \right) \text{在}\left( x_0,y_0 \right) \text{处取极值的必要条件} \\ f_x\left( x_0,y_0 \right) =f_y\left( x_0,y_0 \right) =0

充分条件

A=f_{xx}\left( x_0,y_0 \right) , B=f_{xy}\left( x_0,y_0 \right) , C=f_{yy}\left( x_0,y_0 \right) \\ \begin{cases} AC-B^2>0\ \text{时具有极值,且}A>0\text{时具有极小值,}A<0\text{时具有极大值;}\\ AC-B^2<0\ \text{时没有极值;}\\ AC-B^2=0\ \text{时可能有极值也可能没有极值,需另作讨论}\\ \end{cases}

条件极值

\text{要找}z=f\left( x,y \right) \text{在附加条件}\varphi \left( x,y \right) =0\text{下的可能极值点,可先建立函数} \\ L\left( x,y \right) =f\left( x,y \right) +\lambda \varphi \left( x,y \right) \\ \text{该函数称为拉格朗日函数,其中}\lambda \text{为参数,并建立方程组:} \\ \begin{cases} f_x\left( x,y \right) +\lambda \varphi _x\left( x,y \right) =0\\ f_y\left( x,y \right) +\lambda \varphi _y\left( x,y \right) =0\\ \varphi \left( x,y \right) =0\\ \end{cases} \\ \text{由此方程组解出}x,y,\lambda \text{可获得}z=f\left( x,y \right) \text{在附加条件}\varphi \left( x,y \right) =0\text{下的可能极值点}

5.重积分

5.1二重积分

符号对照表

\begin{aligned} D&\phantom{=}\text{有界闭区域,积分区域}\\ \Delta \sigma _i&\phantom{=}\text{第}i\text{个小闭区域}\\ \lambda &\phantom{=}\text{所有小闭区域中直径的最大值}\\ f\left( x,y \right) , g\left( x,y \right) &\phantom{=}\text{定义在}D\text{上的有界函数,被积函数}\\ \sigma &\phantom{=}D\text{的面积}\\ \mathrm{d}\sigma , \mathrm{d}x\mathrm{d}y&\phantom{=}\text{直角坐标系中的微元面积}\\ \alpha , \beta &\phantom{=}\text{常数}\\ m, M&\phantom{=}\text{在}D\text{上}f\left( x,y \right) \text{的最小值和最大值}\\ \end{aligned}

二重积分定义

\iint_D{f\left( x,y \right) \mathrm{d}\sigma}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{f\left( \xi _i,\eta _i \right) \Delta \sigma _i}

二重积分的计算

\iint_D{f\left( x,y \right) \mathrm{d}x\mathrm{d}y}=\int_a^b{\left[ \int_{\varphi _1\left( x \right)}^{\varphi _2\left( x \right)}{f\left( x,y \right) \mathrm{d}y} \right]}\mathrm{d}x

重积分性质 1

\iint_D{\left[ \alpha f\left( x,y \right) +\beta g\left( x,y \right) \right] \mathrm{d}\sigma}=\alpha \iint_D{f\left( x,y \right) \mathrm{d}\sigma}+\beta \iint_D{g\left( x,y \right) \mathrm{d}\sigma}

重积分性质 2

D=D_1+D_2\Rightarrow \iint_D{f\left( x,y \right)}=\iint_{D_1}{f\left( x,y \right) \mathrm{d}\sigma}+\iint_{D_2}{f\left( x,y \right) \mathrm{d}\sigma}

重积分性质 3

f\left( x,y \right) \leqslant g\left( x,y \right) \Rightarrow \iint_D{f\left( x,y \right) \mathrm{d}\sigma}\leqslant \iint_D{g\left( x,y \right) \mathrm{d}\sigma}

重积分性质 4

\left| \iint_D{f\left( x,y \right) \mathrm{d}\sigma} \right|\leqslant \iint_D{\left| f\left( x,y \right) \right|\mathrm{d}\sigma}

重积分性质 5

m\leqslant f\left( x,y \right) \leqslant M\Rightarrow \iint_D{m\mathrm{d}\sigma}\leqslant \iint_D{f\left( x,y \right) \mathrm{d}\sigma}\leqslant \iint_D{M\mathrm{d}\sigma}

重积分性质 6 二重积分中值定理

\exists \left( \xi ,\eta \right) \in D \\ \iint_D{f\left( x,y \right) \mathrm{d}\sigma}=f\left( \xi ,\eta \right) \sigma

二重积分换元法

\text{设有变换} \\ T\,\,: x=x\left( u,v \right) , y=y\left( u,v \right) , D\rightarrow D\prime \\ \text{且}x\left( u,v \right) \text{和}y\left( u,v \right) \text{在}D\prime\text{上具有一阶连续偏导数,并且} \\ J\left( u,v \right) =\frac{\partial \left( x,y \right)}{\partial \left( u,v \right)}\ne 0 \\ \iint_D{f\left( x,y \right) \mathrm{d}x\mathrm{d}y}=\iint_{D\prime}{f\left[ x\left( u,v \right) ,y\left( u,v \right) \right] \left| J\left( u,v \right) \right|\mathrm{d}u\mathrm{d}v}

5.2三重积分

符号对照表

\begin{aligned} \varOmega &\phantom{=}\text{空间有界闭区域,积分区域}\\ \Delta v_i&\phantom{=}\text{第}i\text{个小闭区域}\\ f\left( x,y,z \right) &\phantom{=}\text{定义在}\varOmega \text{上的有界函数,被积函数}\\ v&\phantom{=}\varOmega \text{的体积}\\ \mathrm{d}v, \mathrm{d}x\mathrm{d}y\mathrm{d}z&\phantom{=}\text{直角坐标系中的微元面积}\\ \end{aligned}

三重积分的定义

\iiint_{\varOmega}{f\left( x,y,z \right) \mathrm{d}v}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{f\left( \xi _i,\eta _i,\zeta _i \right) \Delta v_i}

三重积分的计算

\iiint_{\varOmega}{f\left( x,y,z \right) \mathrm{d}x\mathrm{d}y\mathrm{d}z}=\int_a^b{\left\{ \int_{y_1\left( x \right)}^{y_2\left( x \right)}{\left[ \int_{z_1\left( x,y \right)}^{z_2\left( x,y \right)}{f\left( x,y \right) \mathrm{d}z} \right] \mathrm{d}y} \right\}}\mathrm{d}x

6曲线积分

6.1对弧长的曲线积分

符号对照表

\begin{aligned} L&\phantom{=}\text{光滑曲线弧,积分弧段}\\ \Delta s_i&\phantom{=}\text{第}i\text{个小弧段}\\ \lambda &\phantom{=}\text{所有小弧段长度的最大值}\\ f\left( x,y \right) , g\left( x,y \right) &\phantom{=}\text{有界函数,被积函数}\\ \mathrm{d}s&\phantom{=}\text{弧长微元}\\ \alpha , \beta &\phantom{=}\text{常数}\\ \end{aligned}

对弧长的曲线积分

\int_L{f\left( x,y \right) \mathrm{d}s}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{f\left( \xi _i,\eta _i \right) \Delta s_i}

闭曲线上的曲线积分

\oint_L{f\left( x,y \right) \mathrm{d}s}

第一类曲线积分的计算

\begin{cases} x=\varphi \left( t \right)\\ y=\psi \left( t \right)\\ \end{cases}, \alpha \leqslant t\leqslant \beta \\ \int_L{f\left( x,y \right) \mathrm{d}s}=\int_{\alpha}^{\beta}{f\left[ \varphi \left( t \right) ,\psi \left( t \right) \right] \sqrt{\left[ \varphi \prime\left( t \right) \right] ^2+\left[ \psi \prime\left( t \right) \right] ^2}\mathrm{d}t}

第一类曲线积分的性质 1

L=L_1+L_2\Rightarrow \int_L{f\left( x,y \right) \mathrm{d}s}=\int_{L_1}{f\left( x,y \right) \mathrm{d}s}+\int_{L_2}{f\left( x,y \right) \mathrm{d}s}

第一类曲线积分的性质 2

\iint_L{\left[ \alpha f\left( x,y \right) +\beta g\left( x,y \right) \right] \mathrm{d}s}=\alpha \iint_L{f\left( x,y \right) \mathrm{d}s}+\beta \iint_L{g\left( x,y \right) \mathrm{d}s}

第一类曲线积分的性质 3

f\left( x,y \right) \leqslant g\left( x,y \right) \Rightarrow \iint_L{f\left( x,y \right) \mathrm{d}s}\leqslant \iint_L{g\left( x,y \right) \mathrm{d}s}

第一类曲线积分的性质 4

\left| \iint_L{f\left( x,y \right) \mathrm{d}s} \right|\leqslant \iint_L{\left| f\left( x,y \right) \right|\mathrm{d}s}

6.2对坐标的曲面积分

符号对照表

\begin{aligned} L&\phantom{=}\text{光滑曲线弧,积分弧段}\\ L^-&\phantom{=}\text{曲线弧}L\text{的反向曲线弧}\\ \Delta s_i&\phantom{=}\text{第}i\text{个小弧段}\\ \lambda &\phantom{=}\text{所有小弧段长度的最大值}\\ P\left( x,y \right) , Q\left( x,y \right) &\phantom{=}\text{有界函数,被积函数}\\ \mathrm{d}s&\phantom{=}\text{弧长微元}\\ \alpha , \beta &\phantom{=}\text{常数}\\ \end{aligned}

对坐标的曲线积分

\int_L{P\left( x,y \right) \mathrm{d}x}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{P\left( \xi _i,\eta _i \right) \Delta x_i} \\ \int_L{Q\left( x,y \right) \mathrm{d}y}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{Q\left( \xi _i,\eta _i \right) \Delta y_i}

向量函数的曲线积分

\boldsymbol{F}\left( x,y \right) =P\left( x,y \right) \boldsymbol{i}+Q\left( x,y \right) \boldsymbol{j} \\ \mathrm{d}\boldsymbol{r}=\mathrm{d}x\boldsymbol{i}+\mathrm{d}y\boldsymbol{j} \\ \int_L{\boldsymbol{F}\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}=\int_L{P\left( x,y \right) \mathrm{d}x+Q\left( x,y \right) \mathrm{d}y}

第二类曲线积分的计算

\begin{cases} x=\varphi \left( t \right)\\ y=\psi \left( t \right)\\ \end{cases}, \alpha \leqslant t\leqslant \beta \\ \int_L{P\left( x,y \right) \mathrm{d}x+Q\left( x,y \right) \mathrm{d}y}=\int_{\alpha}^{\beta}{\left\{ P\left[ \varphi \left( t \right) ,\psi \left( t \right) \right] \varphi \prime\left( t \right) +Q\left[ \varphi \left( t \right) ,\psi \left( t \right) \right] \psi \prime\left( t \right) \right\}}\mathrm{d}t

第二类曲线积分的性质 1

\int_L{\left[ \alpha \boldsymbol{F}_1\left( x,y \right) +\beta \boldsymbol{F}_2\left( x,y \right) \right] \cdot \mathrm{d}\boldsymbol{r}}=\alpha \int_L{\boldsymbol{F}_1\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}+\beta \int_L{\boldsymbol{F}_2\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}

第二类曲线积分的性质 2

L=L_1+L_2\Rightarrow \int_L{\boldsymbol{F}\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}=\int_{L_1}{\boldsymbol{F}\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}+\int_{L_2}{\boldsymbol{F}\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}

第二类曲线积分的性质 3

\int_{L^-}{\boldsymbol{F}\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}=-\int_L{\boldsymbol{F}\left( x,y \right) \cdot \mathrm{d}\boldsymbol{r}}

6.3两类曲线积分之间的关系

两类曲线积分之间的关系

\mathrm{d}\boldsymbol{r}=\mathrm{d}x\boldsymbol{i}+\mathrm{d}y\boldsymbol{j} \\ \mathrm{d}s=\sqrt{\left( \mathrm{d}x \right) ^2+\left( \mathrm{d}y \right) ^2} \\ \int_L{P\left( x,y \right) \mathrm{d}x+Q\left( x,y \right) \mathrm{d}y}=\int_L{\left[ P\left( x,y \right) \cos \alpha +Q\left( x,y \right) \cos \beta \right] \mathrm{d}s}

6.4格林公式

单连通区域与复连通区域

\text{若平面区域}D\text{中任一闭曲线所围的部分均属于}D \\ \text{平面单连通区域是不含有}“\text{洞}”\text{的区域} \\ \text{否则称为复连通区域}

平面闭区域边界的方向

\text{当观察这沿边界线}L\text{的正向行走时,}D\text{内在他所处的那一部分总在他的左边}

定理 1 格林公式

\text{设闭区域}D\text{有分段光滑的曲线}L\text{围成,} \\ \text{若函数}P\left( x,y \right) \text{和}Q\left( x,y \right) \text{在}D\text{上具有一阶连续偏导数,则有} \\ \iint_D{\left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right) \mathrm{d}x\mathrm{d}y}=\oint_L{P\mathrm{d}x+Q\mathrm{d}y} \\ \text{其中}L\text{是}D\text{的正向边界线}

定理 2 曲线积分与路径无关的充要条件

\text{设区域}G\text{是一个单连通域,} \\ \text{函数}P\left( x,y \right) \text{,}Q\left( x,y \right) \text{在}G\text{内具有一阶连续偏导数,} \\ \text{则曲线积分}\int_L{P\mathrm{d}x+Q\mathrm{d}y}\text{在}G\text{内与路径无关(或} \\ \text{沿}G\text{内任意闭曲线积分为零)的充要条件是} \\ \frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y} \\ \text{在}G\text{内恒成立}

定理 3

\text{设区域}G\text{是一个单连通域,} \\ \text{函数}P\left( x,y \right) \text{,}Q\left( x,y \right) \text{在}G\text{内具有一阶连续偏导数,} \\ \text{则}P\mathrm{d}x+Q\mathrm{d}y\text{在}G\text{内为某一函数}u\left( x,y \right) \text{的全微分的充要条件是} \\ \frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y} \\ \text{在}G\text{内恒成立}

定理 4

\text{设区域}G\text{是一个单连通域,} \\ \text{函数}P\left( x,y \right) \text{,}Q\left( x,y \right) \text{在}G\text{内具有一阶连续偏导数,} \\ \text{则曲线积分}\int_L{P\mathrm{d}x+Q\mathrm{d}y}\text{在}G\text{内与路径无关的充要条件是:} \\ \text{在}G\text{内存在函数}u\left( x,y \right) \text{,使得} \\ \mathrm{d}u=P\mathrm{d}x+Q\mathrm{d}y

定理 5 曲线积分的基本定理

\text{设} \\ \boldsymbol{F}\left( x,y \right) =P\left( x,y \right) \boldsymbol{i}+Q\left( x,y \right) \boldsymbol{j} \\ \text{是平面区域}G\text{内的一个向量场,且}P\left( x,y \right) \text{和}Q\left( x,y \right) \text{都在}G\text{内连续,} \\ \text{且存在一个数量函数}f\left( x,y \right) \text{,使得}\boldsymbol{F}=\nabla f\text{,则曲线积分} \\ \int_L{\boldsymbol{F}\cdot \mathrm{d}\boldsymbol{r}}=f\left( B \right) -f\left( A \right) \\ \text{其中}L\text{是位于}G\text{内起点为}A\text{、终点为}B\text{的任意分段光滑曲线}

7曲线积分

7.1第一类曲面积分

符号对照表

\begin{aligned} \varSigma &\phantom{=}\text{积分曲面}\\ D_{xy}&\phantom{=}\text{积分曲面}\varSigma \text{在}xOy\text{平面上的投影}\\ f\left( x,y,z \right) &\phantom{=}\text{被积函数}\\ \lambda &\phantom{=}\text{小块曲面直径的最大值}\\ \end{aligned}

定义 1 光滑曲面

\text{若曲面上个点均有切平面,且当点在曲面上连续移动时,切平面也连续转动}

定义 2 第一类曲面积分

\text{设}\varSigma \text{曲面是光滑的,函数} \\ f\left( x,y,z \right) \\ \text{在曲面}\varSigma \text{上有界,把}\varSigma \text{分成任意小块}\varDelta S_i\text{(}\varDelta S_i\text{同时代表第}i\text{小块曲面的面积)} \\ \text{设}\left( \xi _i,\eta _i,\zeta _i \right) \text{是}\varDelta S_i\text{上的任意一点,作乘积} \\ f\left( \xi _i,\eta _i,\zeta _i \right) \Delta S_i,i=1,2,3,\cdots n \\ \text{并对其求和} \\ \sum_{i=1}^n{f\left( \xi _i,\eta _i,\zeta _i \right) \Delta S_i} \\ \text{如果当小块曲面的直径最大值}\lambda \rightarrow 0\text{时,这个求和的极限存在} \\ \text{则称此极限为函数}f\left( x,y,z \right) \text{在曲面}\varSigma \text{上对面积的曲面积分或称为} \\ \mathbf{第一类曲面积分} \\ \iint_S{f\left( x,y,z \right) \mathrm{d}S}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{f\left( \xi _i,\eta _i,\zeta _i \right) \Delta S_i} \\ \text{其中}f\left( x,y,z \right) \text{为被积函数,}\varSigma \text{叫做积分曲面}

第一类曲面积分的计算方法

\iint_S{f\left( x,y,z \right) \mathrm{d}S}=\iint_{D_{xy}}{f\left[ x,y,z\left( x,y \right) \right] \sqrt{1+{z_x}^2+{z_y}^2}\mathrm{d}x\mathrm{d}y}

7.2第二类曲面积分

符号对照表

\begin{aligned} \varSigma &\phantom{=}\text{积分曲面,有向曲面}\\ D_{xy}&\phantom{=}\text{积分曲面}\varSigma \text{在}xOy\text{平面上的投影}\\ \left( \Delta S_i \right) _{xy}&\phantom{=}\text{小块曲面}\Delta S_i\text{在}xOy\text{平面上的投影}\\ R\left( x,y,z \right) &\phantom{=}\text{被积函数}\\ \lambda &\phantom{=}\text{小块曲面直径的最大值}\\ \end{aligned}

定义 1 第二类曲面积分

\text{设}\varSigma \text{曲面是光滑的有向曲面,函数} \\ R\left( x,y,z \right) \\ \text{在曲面}\varSigma \text{上有界,把}\varSigma \text{分成任意小块}\varDelta S_i\text{(}\varDelta S_i\text{同时代表第}i\text{小块曲面的面积),} \\ \varDelta S_i\text{在}xOy\text{上的投影为}\left( \varDelta S_i \right) _{xy}\text{,}\left( \xi _i,\eta _i,\zeta _i \right) \text{是}\varDelta S_i\text{上的任意一点} \\ \text{如果当小块曲面的直径最大值}\lambda \rightarrow 0\text{时,} \\ \lim_{\lambda \rightarrow 0} \sum_{i=1}^n{R\left( \xi _i,\eta _i,\zeta _i \right) \left( \Delta S_i \right) _{xy}} \\ \text{存在,则称此极限为函数}R\left( x,y,z \right) \text{在有向曲面}\varSigma \text{上对坐标}x\text{、}y\text{的曲面积分} \\ \iint_{\varSigma}{R\left( x,y,z \right) \mathrm{d}x\mathrm{d}y}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{R\left( \xi _i,\eta _i,\zeta _i \right) \left( \Delta S_i \right) _{xy}} \\ \text{另有} \\ \iint_{\varSigma}{P\left( x,y,z \right) \mathrm{d}y\mathrm{d}z}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{R\left( \xi _i,\eta _i,\zeta _i \right) \left( \Delta S_i \right) _{yz}} \\ \iint_{\varSigma}{Q\left( x,y,z \right) \mathrm{d}z\mathrm{d}x}=\lim_{\lambda \rightarrow 0} \sum_{i=1}^n{R\left( \xi _i,\eta _i,\zeta _i \right) \left( \Delta S_i \right) _{zx}} \\ \text{以上三式称为第二类曲面积分} \\ \text{其中}R\left( x,y,z \right) \text{为被积函数,}\varSigma \text{叫做积分曲面}

第二类曲面积分的合并形式

\iint_{\varSigma}{P\left( x,y,z \right) \mathrm{d}y\mathrm{d}z}+\iint_{\varSigma}{Q\left( x,y,z \right) \mathrm{d}z\mathrm{d}x}+\iint_{\varSigma}{R\left( x,y,z \right) \mathrm{d}x\mathrm{d}y}=\iint_{\varSigma}{P\left( x,y,z \right) \mathrm{d}y\mathrm{d}z+Q\left( x,y,z \right) \mathrm{d}z\mathrm{d}x+R\left( x,y,z \right) \mathrm{d}x\mathrm{d}y}

第二类曲面积分的计算方法

\text{曲面上侧} \\ \iint_{\varSigma ^+}{R\left( x,y,z \right) \mathrm{d}x\mathrm{d}y}=\iint_{D_{xy}}{R\left[ x,y,z\left( x,y \right) \right] \mathrm{d}x\mathrm{d}y} \\ \text{曲面下侧} \\ \iint_{\varSigma ^-}{R\left( x,y,z \right) \mathrm{d}x\mathrm{d}y}=-\iint_{D_{xy}}{R\left[ x,y,z\left( x,y \right) \right] \mathrm{d}x\mathrm{d}y}

7.3两类曲面积分之间的关系

两类曲面积分之间的关系

\cos \alpha ,\cos \beta ,\cos \gamma \text{为有向曲面}\varSigma \text{在点}\left( x,y,z \right) \text{处的方向余弦} \\ \iint_{\varSigma}{P\mathrm{d}y\mathrm{d}z+Q\mathrm{d}z\mathrm{d}x+R\mathrm{d}x\mathrm{d}y}=\iint_{\varSigma}{\left( P\cos \alpha +Q\cos \beta +R\cos \gamma \right) \mathrm{d}S} \\ \text{或} \\ \iint_{\varSigma}{\boldsymbol{A}\cdot \mathrm{d}\boldsymbol{S}}=\iint_{\varSigma}{\boldsymbol{A}\cdot \boldsymbol{n}\mathrm{d}S}=\iint_{\varSigma}{A_n\mathrm{d}S}

7.4高斯公式

高斯公式

\text{设空间闭区域}\varOmega \text{是由分片光滑的闭曲面}\varSigma \text{(外侧)所围成,} \\ \text{函数}P\left( x,y,z \right) \text{,}Q\left( x,y,z \right) \text{,}R\left( x,y,z \right) \text{在}\varOmega \text{上具有一阶连续偏导数,则有} \\ \iiint_{\varOmega}{\left( \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} \right) \mathrm{d}v}=\oiint_{\varSigma}{P\mathrm{d}y\mathrm{d}z+Q\mathrm{d}z\mathrm{d}x+R\mathrm{d}x\mathrm{d}y} \\ \text{或} \\ \iiint_{\varOmega}{\left( \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} \right) \mathrm{d}v}=\oiint_{\varSigma}{\left( P\cos \alpha +Q\cos \beta +R\cos \gamma \right) \mathrm{d}S}

沿任意闭曲面的曲面积分为零的条件

\text{设}G\text{是空间二维单连通域,} \\ \text{函数}P\left( x,y,z \right) \text{,}Q\left( x,y,z \right) \text{,}R\left( x,y,z \right) \text{在}G\text{内具有一阶连续偏导数} \\ \text{则曲面积分} \\ \iint_{\varSigma}{P\mathrm{d}y\mathrm{d}z+Q\mathrm{d}z\mathrm{d}x+R\mathrm{d}x\mathrm{d}y} \\ \text{在}G\text{内与所取曲面}\varSigma \text{无关仅取决于边界曲线的充要条件是} \\ \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}=0 \\ \text{在}G\text{内恒成立}

定义 1 场的通量

\text{对向量场} \\ \boldsymbol{A}\left( x,y,z \right) =P\left( x,y,z \right) \boldsymbol{i}+Q\left( x,y,z \right) \boldsymbol{j}+R\left( x,y,z \right) \boldsymbol{k} \\ \text{其中函数}P\text{、}Q\text{、}R\text{均具有一阶偏导数,} \\ \text{若}\varSigma \text{是场内一片有向曲面,}\boldsymbol{n}\text{是}\varSigma \text{在点}\left( x,y,z \right) \text{处的单位法向量,则积分} \\ \iint_{\varSigma}{\boldsymbol{A}\cdot \boldsymbol{n}\mathrm{d}S} \\ \text{称为通量(流量)}

定义 2 场的散度

\text{在点}M\text{处,对速度场} \\ \boldsymbol{v}\left( x,y,z \right) =P\left( x,y,z \right) \boldsymbol{i}+Q\left( x,y,z \right) \boldsymbol{j}+R\left( x,y,z \right) \boldsymbol{k} \\ \mathbf{\nabla }\cdot \boldsymbol{v}=\mathrm{div}\boldsymbol{v}\left( M \right) =\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}=\lim_{\varOmega \rightarrow M} \frac{1}{V}\oiint_{\varSigma}{v_n\mathrm{d}S} \\ \text{称为散度(通量密度)}

定义 3 无源场

\text{散度处处为零的场}

7.5第二类格林公式

第二类格林公式

\text{设}u\left( x,y,z \right) \text{、}v\left( x,y,z \right) \text{是两个定义在闭区域}\varOmega \text{上的} \\ \text{具有二阶连续偏导数的函数} \\ \frac{\partial u}{\partial n}\text{、}\frac{\partial v}{\partial n} \\ \varSigma \text{是空间闭区域}\varOmega \text{的整个边界曲面} \\ \text{依次表示}u\left( x,y,z \right) \text{、}v\left( x,y,z \right) \text{沿}\varSigma \text{的外法线方向的方向导数} \\ \text{则有} \\ \iiint_{\varOmega}{\left( u\Delta v-v\Delta u \right) \mathrm{d}x\mathrm{d}y\mathrm{d}z}=\oiint_{\varSigma}{\left( u\frac{\partial v}{\partial n}-v\frac{\partial u}{\partial n} \right)}\mathrm{d}S

7.6斯托克斯公式

斯托克斯公式

\text{设}\varGamma \text{为分段光滑的空间有向闭曲线,} \\ \varSigma \text{是以}\varGamma \text{为边界的分片光滑有向曲面} \\ \varGamma \text{的正向与}\varSigma \text{的侧符合右手法则,} \\ \text{函数}P\left( x,y,z \right) \text{,}Q\left( x,y,z \right) \text{,}R\left( x,y,z \right) \text{在}\varSigma \text{(连同边界}\varGamma \text{)上具有一阶连续偏导数} \\ \text{则有} \\ \iint_{\varSigma}{\left( \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z} \right) \mathrm{d}y\mathrm{d}z+\left( \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x} \right) \mathrm{d}z\mathrm{d}x+\left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right) \mathrm{d}x\mathrm{d}y}=\oint_{\varGamma}{P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z} \\ \text{或} \\ \iint_{\varSigma}{\left| \begin{matrix} \mathrm{d}y\mathrm{d}z& \mathrm{d}z\mathrm{d}x& \mathrm{d}x\mathrm{d}y\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ P& Q& R\\ \end{matrix} \right|}=\oint_{\varGamma}{P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z} \\ \text{又或} \\ \iint_{\varSigma}{\left| \begin{matrix} \cos \alpha& \cos \beta& \cos \gamma\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ P& Q& R\\ \end{matrix} \right|}\mathrm{d}S=\oint_{\varGamma}{P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z} \\ \text{其中}\boldsymbol{n}=\left( \cos \alpha ,\cos \beta ,\cos \gamma \right) \text{为有向曲面}\varSigma \text{在点}\left( x,y,z \right) \text{处的单位法向量}

空间闭曲线曲线积分为零的充要条件

\text{设空间区域}G\text{是以为单连通域,} \\ \text{函数}P\left( x,y,z \right) \text{,}Q\left( x,y,z \right) \text{,}R\left( x,y,z \right) \text{在}G\text{内具有一阶连续偏导数} \\ \text{则空间曲线积分} \\ \oint_{\varGamma}{P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z} \\ \text{在}G\text{内与路径无关的充要条件是} \\ \begin{cases} \frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}\\ \frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}\\ \frac{\partial R}{\partial x}=\frac{\partial P}{\partial z}\\ \end{cases} \\ \text{在}G\text{内恒成立}

定义 1 环流量

\text{对向量场} \\ \boldsymbol{A}\left( x,y,z \right) =P\left( x,y,z \right) \boldsymbol{i}+Q\left( x,y,z \right) \boldsymbol{j}+R\left( x,y,z \right) \boldsymbol{k} \\ \text{其中函数}P\text{、}Q\text{、}R\text{均具有一阶偏导数,} \\ \text{若}\varGamma \text{是}\boldsymbol{A}\text{的定义域内的一条分段光滑有向闭曲线。} \\ \boldsymbol{\tau }\text{是}\varGamma \text{在点}\left( x,y,z \right) \text{处的单位切向量,则积分} \\ \iint_{\varGamma}{\boldsymbol{A}\cdot \boldsymbol{\tau }\mathrm{d}s\,\,} \\ \text{称为向量场}\boldsymbol{A}\text{沿}\boldsymbol{\varGamma }\text{的环流量}

定义 2 旋度

\text{对向量场} \\ \boldsymbol{A}\left( x,y,z \right) =P\left( x,y,z \right) \boldsymbol{i}+Q\left( x,y,z \right) \boldsymbol{j}+R\left( x,y,z \right) \boldsymbol{k} \\ \text{其中函数}P\text{、}Q\text{、}R\text{均具有一阶偏导数,} \\ \mathbf{rot}\boldsymbol{A}=\left( \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z} \right) \boldsymbol{i}+\left( \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x} \right) \boldsymbol{j}+\left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right) \boldsymbol{k} \\ \text{称为向量场}\boldsymbol{A}\text{的旋度,另记为:} \\ \mathbf{rot}\boldsymbol{A}=\mathbf{\nabla }\times \boldsymbol{A}=\left| \begin{matrix} \boldsymbol{i}& \boldsymbol{j}& \boldsymbol{k}\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ P& Q& R\\ \end{matrix} \right|

定义 3 无旋场

\text{旋度处处为零的场}

定义 4 调和场

\text{散度、旋度均处处为零的场}

斯托克斯公式的向量形式

\iint_{\varSigma}{\mathbf{rot}\boldsymbol{A}\cdot \boldsymbol{n}\mathrm{d}S}=\oint_{\varGamma}{\boldsymbol{A}\cdot \boldsymbol{\tau }\mathrm{d}s} \\ \text{或} \\ \iint_{\varSigma}{\left( \mathbf{rot}\boldsymbol{A} \right) _n\mathrm{d}S}=\oint_{\varGamma}{\boldsymbol{A}\cdot \boldsymbol{\tau }\mathrm{d}s}

8无穷级数

8.1常数项级数

定义 1 常数项级数

\text{对常数项数列}\left\{ u_n \right\} ,n=1,2,3,\cdots ,\infty \\ \text{记} \\ \sum_{n=1}^{\infty}{u_n}=u_1+u_2+u_3+\cdots +u_n+\cdots \\ \text{为常数项级数,}u_n\text{称为级数的一般项}

等比级数

\sum_{n=0}^{\infty}{aq^n}=a+aq+aq^2+\cdots +aq^n+\cdots \\ \text{其中}q\text{称为公比}

8.2无穷级数的收敛性

定义 1 无穷级数的收敛性

\text{若级数}\sum_{n=1}^{\infty}{u_n}\text{的部分和数列}\left\{ s_n \right\} \text{有极限}s\text{,即} \\ \lim_{n\rightarrow \infty} s_n=s \\ \text{则称无穷级数}\sum_{n=1}^{\infty}{u_n}\text{收敛,}s\text{称为级数的和,否则级数发散}

收敛级数的基本性质 1

\text{若}\sum_{n=1}^{\infty}{u_n}\text{收敛于和}s\text{,则}\sum_{n=1}^{\infty}{ku_n}\text{也收敛,其和为}ks

收敛级数的基本性质 2

\text{若}\sum_{n=1}^{\infty}{u_n}\text{收敛于}s\text{,}\sum_{n=1}^{\infty}{v_n}\text{收敛于}\sigma \text{,则}\sum_{n=1}^{\infty}{\left( u_n\pm v_n \right)}\text{也收敛,其和为}s\pm \sigma

收敛级数的基本性质 3

\text{在级数中去掉或加上有限项,不改变级数的收敛性}

收敛级数的基本性质 4

\text{若级数}\sum_{n=1}^{\infty}{u_n}\text{收敛,则对级数的项任意加括号分类后所成的级数仍收敛,且和不便}

收敛级数的基本性质 5 级数收敛的必要条件

\text{若级数}\sum_{n=1}^{\infty}{u_n}\text{收敛,则其一般项}u_n\text{趋于零}

柯西审敛法

\text{级数}\sum_{n=1}^{\infty}{u_n}\text{收敛的充要条件为} \\ \text{对于任意给定的正数}\varepsilon \text{,总存在正整数}N\text{,使得当} \\ n>N \\ \text{时,对于任意正整数}p\text{,都有} \\ \left| u_{n+1}+u_{n+2}+\cdots +u_{n+p} \right|<\varepsilon \\ \text{成立}

8.3幂级数

符号对照表

\begin{aligned} a_n&\phantom{=}\text{系数}\\ f\left( x \right) &\phantom{=}\text{函数}\\ U\left( x_0 \right) &\phantom{=}\text{点}x_0\text{的某邻域}\\ \end{aligned}

定义 1 幂级数

\sum_{n=0}^{\infty}{a_nx^n}=a_0+a_1x+a_2x^2+\cdots +a_nx^n+\cdots \\ \text{其中}a_n\text{称为幂级数的系数}

泰勒级数

f\left( x \right) =\sum_{n=0}^{\infty}{\frac{1}{n!}f^{\left( n \right)}\left( x_0 \right) \left( x-x_0 \right) ^n}, x\in U\left( x_0 \right)

麦克劳林级数

f\left( x \right) =\sum_{n=0}^{\infty}{\frac{1}{n!}f^{\left( n \right)}\left( 0 \right) x^n}, x\in U\left( 0 \right)

常用幂级数展开

\underline{\begin{array}{c|c} \frac{1}{1-x}=\sum_{n=0}^{\infty}{x^n}& \left( -1<x<1 \right)\\ \hline e^x=\sum_{n=0}^{\infty}{\frac{1}{n!}x^n}& \left( -\infty <x<\infty \right)\\ \hline \sin x=\sum_{k=0}^{\infty}{\frac{\left( -1 \right) ^k}{\left( 2k+1 \right) !}x^{2k+1}}& \left( -\infty <x<\infty \right)\\ \hline \cos x=\sum_{k=0}^{\infty}{\frac{\left( -1 \right) ^k}{\left( 2k \right) !}x^{2k}}& \left( -\infty <x<\infty \right) \,\,\\ \hline \frac{1}{1+x}=\sum_{n=0}^{\infty}{\left( -1 \right) ^nx^n}& \left( -1<x<1 \right)\\ \hline \,\,\ln \left( 1+x \right) =\sum_{n=0}^{\infty}{\frac{\left( -1 \right) ^n}{n+1}x^{n+1}}& \left( -1<x\leqslant 1 \right)\\ \hline a^x=\sum_{n=0}^{\infty}{\frac{\left( \ln a \right) ^n}{n!}x^n}& \left( -\infty <x<\infty \right)\\ \hline \frac{1}{1+x^2}=\sum_{n=0}^{\infty}{\left( -1 \right) ^nx^{2n}}& \left( -1<x<1 \right)\\ \end{array}}

二项展开式

\left( 1+x \right) ^m=1+mx+\frac{m\left( m-1 \right)}{2!}x^2+\cdots +\frac{m\left( m-1 \right) \cdots \left( m-n+1 \right)}{n!}x^n+\cdots \quad \left( -1<x<1 \right) \\ \text{特别的} \\ \sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{2\cdot 4}x^2+\frac{1\cdot 3}{2\cdot 4\cdot 6}x^3-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6\cdot 8}x^4+\cdots \quad \left( -1<x<1 \right) \\ \frac{1}{\sqrt{1+x}}=1-\frac{1}{2}x+\frac{1\cdot 3}{2\cdot 4}x^2-\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}x^3+\frac{1\cdot 3\cdot 5\cdot 7}{2\cdot 4\cdot 6\cdot 8}x^4+\cdots \quad \left( -1<x<1 \right)

利用幂级数求解微分方程

\text{对于二阶齐次线性方程} \\ y''+P\left( x \right) y\prime+Q\left( x \right) y=0 \\ \text{若}P\left( x \right) \text{和}Q\left( x \right) \text{在}-R<x<R\text{内可展开为}x\text{的幂级数,那么} \\ \text{在}-R<x<R\text{范围内,必有形如} \\ y=\sum_{n=1}^{\infty}{a_nx^n} \\ \text{的解}

欧拉公式

e^{ix}=1+ix+\frac{1}{2!}\left( ix \right) ^2+\frac{1}{3!}\left( ix \right) ^3+\cdots \frac{1}{n!}\left( ix \right) ^n+\cdots \\ =1+ix-\frac{1}{2!}x^2-i\frac{1}{3!}x^3+\frac{1}{4!}x^4+i\frac{1}{5!}x^5-\cdots \\ =\left( 1-\frac{1}{2!}x^2+\frac{1}{4!}x^4-\cdots \right) +i\left( x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\cdots \right) \\ =\cos x+i\sin x

8.4三角级数

符号对照表

\begin{aligned} y&\phantom{=}\text{函数,动点位置}\\ \omega &\phantom{=}\text{角频率}\\ A&\phantom{=}\text{振幅}\\ \varphi &\phantom{=}\text{初相}\\ t&\phantom{=}\text{时间}\\ \end{aligned}

简谐振动

y=A\sin \left( \omega t+\varphi \right)

三角级数

\frac{a_0}{2}+\sum_{n=1}^{\infty}{\left( a_n\cos \frac{n\pi t}{l}+b_n\sin \frac{n\pi t}{l} \right)}\xlongequal{x=\frac{\pi t}{l}}\frac{a_0}{2}+\sum_{n=1}^{\infty}{\left( a_n\cos nx+b_n\sin nx \right)}

三角函数系

1,\cos x,\sin x,\cos 2x,\sin 2x,\cdots \cos nx,\sin nx,\cdots

三角函数系的正交性

\text{对于}n=1,2,3,\cdots , k=1,2,3,\cdots , n\ne k, \text{有} \\ \underline{\begin{aligned} 0&=\int_{-\pi}^{\pi}{\cos nx\mathrm{d}x}\\ \hline 0&=\int_{-\pi}^{\pi}{\sin nx\mathrm{d}x}\\ \hline 0&=\int_{-\pi}^{\pi}{\sin kx\cdot \cos nx\mathrm{d}x}\\ \hline 0&=\int_{-\pi}^{\pi}{\sin kx\cdot \sin nx\mathrm{d}x}\\ \hline 0&=\int_{-\pi}^{\pi}{\cos kx\cdot \cos nx\mathrm{d}x}\\ \end{aligned}}

8.5傅里叶级数

傅立叶级数

\text{设}f\left( x \right) \text{是周期为}2\pi \text{的周期函数,可展开为} \\ f\left( x \right) =\frac{a_0}{2}+\sum_{k=1}^{\infty}{\left( a_k\cos kx+b_k\sin kx \right)} \\ a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \cos nx\mathrm{d}x}\quad \left( n=0,1,2,3,\cdots \right) \\ b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}{f\left( x \right) \sin nx\mathrm{d}x}\quad \left( n=0,1,2,3,\cdots \right)

正弦级数

\text{当}f\left( x \right) \text{为奇函数时,傅里叶级数为} \\ f\left( x \right) =\sum_{n=1}^{\infty}{b_n\sin nx}

余弦级数

\text{当}f\left( x \right) \text{为偶函数时,傅里叶级数为} \\ f\left( x \right) =\frac{a_0}{2}+\sum_{n=1}^{\infty}{a_n\cos nx}

傅立叶级数的复数形式

f\left( x \right) =\sum_{n=-\infty}^{\infty}{c_ne^{inx}} \\ c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f\left( x \right) e^{-inx}\mathrm{d}x}\quad \left( n=0,\pm 1,\pm 2,\cdots \right)


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