From 6cfedef2960c13b02eb623d3bdb067784b5b0c19 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 19 Oct 2022 22:43:41 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- advanced-math/exercise/1-limit/limit.pdf | Bin 190990 -> 191030 bytes advanced-math/exercise/1-limit/limit.tex | 2 +- .../exercise/2-function/function.pdf | Bin 142455 -> 171199 bytes .../exercise/2-function/function.tex | 241 +++++++++++++++++- ...ntegal-of-functions-of-single-variable.pdf | Bin 271238 -> 283051 bytes ...ntegal-of-functions-of-single-variable.tex | 123 +++++++-- ...ral-calculus-of-multivariate-functions.pdf | Bin 147813 -> 148623 bytes ...ral-calculus-of-multivariate-functions.tex | 7 + .../applications-of-derivatives.pdf | Bin 157236 -> 158818 bytes .../applications-of-derivatives.tex | 8 +- .../6-quadratic-form/quadratic-form.pdf | Bin 91778 -> 94595 bytes 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a=1,b=e$。 中值定理一般用于判断不等式。 +一般分为连续函数闭区间上的性质(零点定理、介值定理)和微分中值定理(罗尔定理、拉格朗日中值定理、柯西中值定理、泰勒中值定理)。 + +\subsection{介值定理} + +适用于$\exists\xi\in[a,b]$,或函数和$f(a)+f(b)$的形式。 + +\begin{enumerate} + \item 利用最值定理$m\leqslant f(x)\leqslant M$。 + \item 利用介值定理。 +\end{enumerate} + +\textbf{例题:}$f(x)$、$g(x)$在$[a,b]$上连续,且$g(x)>0$,证明第二积分中值定理:存在一点$\xi\in[a,b]$,使得$\int_a^bf(x)g(x)\,\textrm{d}=f(\xi)\int_a^bg(x)\,\textrm{d}x$。 + +解:首先看式子形式,是积分形式,没有导数,所以不会是四大微分中值定理。 + +首先将未知数提到一边:$f(\xi)=\dfrac{\int_a^bf(x)g(x)\,\textrm{d}x}{\int_a^bg(x)\,\textrm{d}x}$,这是一个数。 + +又$f(x)$、$g(x)$在$[a,b]$上连续,由最值定理$m\leqslant f(x)\leqslant M$。 + +由于已知$g(x)>0$,所以就要对$f(x)$进行处理,这样移动$g(x)$时不等式不会变号,如果要移动$f(x)$则要讨论$f(x)$的正负性。 + +由保号性,对其积分$\int_a^bg(x)\,\textrm{d}x>0$,从而$m\cdot g(x)\leqslant f(x)g(x)\leqslant M\cdot g(x)$。 + +再对其积分:$m\int_a^bg(x)\,\textrm{d}x\leqslant\int_a^bf(x)g(x)\,\textrm{d}x\leqslant M\int_a^bg(x)\,\textrm{d}x$。 + +从而$m\leqslant f(\xi)=\dfrac{\int_a^bf(x)g(x)\,\textrm{d}x}{\int_a^bg(x)\,\textrm{d}x}\leqslant M$。 + +\subsection{零点定理} + +适用于$\exists\xi\in[a,b]$,且$f(a)$、$f(b)$已知。 + +\begin{enumerate} + \item 令$\xi=x$,移项得$f(x)=0$。 + \item 令辅助函数$F(x)=f(x)$,验证是否满足零点定理,若满足则得证。 + \item 若不满足,令$F'(x)=f(x)$,对$f(x)$积分取$C=0$,$F(x)=\int f(x)\,\textrm{d}x$,此时需要使用罗尔定理,若满足则得证。 + \item (基本上不会考到)若不满足,令$F''(x)=f(x)$,两次积分得到$F(x)$,将$F(x)$使用泰勒展开为一阶,使用拉格朗日余项。 +\end{enumerate} + +\subsubsection{直接零点定理} + +\textbf{例题:}已知函数$f(x)$在$[0,1]$上连续,且$f(0)=0$,$f(1)=1$,证明$\exists\xi\in(0,1)$使得$f(\xi)=1-\xi$。 + +解:构造辅助函数$F(x)=f(x)+x-1$。 + +$F(0)=f(0)+0-1=-1<0$,$F(1)=f(1)+1-1=1>0$。 + +根据零点定理$\exists\xi\in(0,1)$使得$F(\xi)=0$。 + +即$\exists\xi\in(0,1)$使得$f(\xi)=1-\xi$。 + +\subsubsection{罗尔定理} + +\textbf{例题:}设$f(x)$、$g(x)$在$[a,b]$上连续,证明$\exists\xi\in(a,b)$,使得$g(\xi)\int_a^\xi f(x)\,\textrm{d}x=f(\xi)\int_\xi^bg(x)\,\textrm{d}x$。 + +解:由于不含导数,所以不使用微分中值定理。 + +首先令$\xi=x$,移项$g(x)\int_a^xf(t)\,\textrm{d}t-f(x)\int_x^bg(t)\,\textrm{d}t=0$。 + +令$F(x)=g(x)\int_a^xf(t)\,\textrm{d}t-f(x)\int_x^bg(t)\,\textrm{d}t$,验证是否满足零点定理。 + +$F(a)=-f(a)\int_a^bg(t)\,\textrm{d}t$,$F(b)=g(b)\int_a^bf(t)\,\textrm{d}t$。零点定理无法判断。 + +改令$F'(x)=g(x)\int_a^xf(t)\,\textrm{d}t+f(x)\int_b^xg(t)\,\textrm{d}t$。 + +所以能看出$F(x)=\int_a^xf(t)\,\textrm{d}t\int_b^xg(t)\,\textrm{d}t$ + +$F(a)=0$,$F(b)=0$,满足罗尔定理,所以存在$\xi$使得$F'(\xi)=0$,得证。 + \subsection{罗尔定理} 重点是找到两点$f(a)=f(b)$。 -\subsubsection{判断不等式} +\subsubsection{寻找原函数} -\paragraph{寻找原函数} \leavevmode \medskip +% 即证明什么就构造他的原函数为函数式子。 + +\begin{itemize} + \item 简单型:直接可以看出。 + \item 公式型:$f'(\xi)+g(\xi)f(\xi)=0$,则令$G(x)=\int g(x)\,\textrm{d}x$,$F(x)=e^{G(x)}f(x)$。 +\end{itemize} 通过乘积求导公式$(uv)'=u'v+uv'$的逆运算来构造辅助函数。 如$f(x)f'(x)$,作$F(x)=f^2(x)$,$[f'(x)]^2+f(x)f''(x)$,作$F(x)=f(x)f'(x)$,$f'(x)+f(x)\varphi'(x)$,作$F(x)=f(x)e^{\varphi(x)}$。 -即证明什么就构造他的原函数为函数式子。 +\textbf{例题:}设$f(x)$在$[a,b]$上连续,在$(a,b)$内可导,$f(a)=b$,$f(b)=a$,$ab$同号,证明$\exists\xi\in(a,b)$使得$f'(\xi)=\dfrac{-f(\xi)}{\xi}$。 -\paragraph{介值定理} \leavevmode \medskip +解:对$f'(x)+\dfrac{1}{x}f(x)=0$,即$g(x)=\dfrac{1}{x}$,$G(x)=\int\dfrac{1}{x}\textrm{d}x=\ln x$,辅助函数为$F(x)=e^{\ln x}f(x)=xf(x)$。 + +又$f(a)=b$,$f(b)=a$,$F(a)=af(a)=ab$,$F(b)=bf(b)=ab$,所以$F(a)=F(b)$。根据罗尔定理$\exists\xi\in(a,b)$使得$F'(\xi)=0$,即$f(\xi)+\xi f'(\xi)=0$。 + +又$ab$同号,所以$ab\neq0$,$\xi\neq0$,从而$f'(x)=\dfrac{-f(\xi)}{\xi}$。 + +\textbf{例题:}设函数$f(x)$、$g(x)$在$[a,b]$上二阶可导,且$g''(x)\neq0$,$f(a)=f(b)=g(a)=g(b)=0$,证明: + +(1)在$(a,b)$内$g(x)\neq0$。 + +(2)在$(a,b)$内至少有一点$\xi$,使得$\dfrac{f(\xi)}{g(\xi)}=\dfrac{f''(\xi)}{g''(x)}$。 + +解: + +(1)为了简单使用反证法。 + +假设$\exists c\in(a,b)$,使得$g(c)=0$。对两边区间使用罗尔定理: + +$\exists\xi_1\in(a,c)$使得$g'(\xi_1)=0$,$\exists\xi_2\in(a,c)$使得$g'(\xi_2)=0$。 + +再次使用罗尔定理$\exists\xi\in(\xi_1,\xi_2)\in(a,b)$使得$g''(\xi)=0$,与题设矛盾,则假设不成立。 + +(2)对式子交叉相乘,令$\xi=x$,即证明$f(x)g''(x)-f''(x)g(x)=0$成立。 + +由于公式只针对$f(x)$和$f'(x)$没有二阶导,所以将式子加上一阶导的式子并减去,来补全一阶导。 + +即$f(x)g''(x)+f'(x)g'(x)-f'(x)g'(x)-f''(x)g(x)=0$。将其拆为两个部分$R_1-R_2$。 + +对于$R_1=f(x)g''(x)+f'(x)g'(x)=0$即得$f'(x)+\dfrac{g''(x)}{g'(x)}f(x)=0$。 + +$G_1'(x)=\ln(g'(x))$所以$R_1$原函数为$g'(x)f(x)$。 + +对于$R2=f'(x)g'(x)-f''(x)g(x)=0$即得$f''(x)+\dfrac{g'(x)}{g(x)}f'(x)=0$。 + +令辅助函数为$F(x)=g'(x)f(x)-g(x)f'(x)$。 + +$F(a)=g'(a)f(a)-g(a)f'(a)=0$,$F(b)=g'(b)f(b)-g(b)f'(b)=0$,所以根据罗尔定理$\exists\xi\in(a,b)$,使得$F'(\xi)=0$。 + +即根据上面过程反推回去得证。 + +\subsubsection{导数关系式} + +\paragraph{导数值为常量} \leavevmode \medskip + +即$f^{(n)}(\xi)=0$: + +\begin{enumerate} + \item $f^{(n-1)}(x)$在$x=\xi$处满足费马定理。 + \item $f^{(n-1)}$在包含$x=\xi$于其内的闭区间上满足罗尔定理(最常使用)。 + \item 利用泰勒公式展开(基本不会考)。 +\end{enumerate} \textbf{例题:}设$f(x)$在$[0,3]$上连续,$(0,3)$上可导,且$f(0)+f(1)+f(2)=3$,$f(3)=1$,证明存在$\xi\in(0,3)$,使得$f'(\xi)=0$。 @@ -202,7 +325,35 @@ $\therefore a=1,b=e$。 由介值定理,可得一定存在某点$c\in[0,2]$,使得$f(c)=\dfrac{f(0)+f(1)+f(2)}{3}=1$。 -所以$f(c)=f(3)=1$,根据罗尔定理,得证。 +所以$f(c)=f(3)=1$,根据罗尔定理,$\xi\in(c,3)$上,得证。 + +\textbf{例题:}$f(x)$在$[0,2]$上连续,在$(0,2)$内二阶可导,$f(0)=f(1)$,$f(2)=2\int_1^{\frac{3}{2}}f(x)\,\textrm{d}x$,证明$\exists\xi\in(0,2)$使得$f''(\xi)=0$。 + +解:遇到积分式子首先利用积分中值定理$f(2)=2\int_1^{\frac{3}{2}}f(x)\,\textrm{d}x=2f(\xi')(\dfrac{3}{2}-1)=f(\xi')$($1<\xi'<\dfrac{3}{2}$)。 + +所以根据罗尔定理$\exists\xi_1\in(\xi',2)$使得$f'(\xi_1)=0$。 + +又$f(0)=f(1)$,$\therefore\exists\xi_2\in(0,1)$,罗尔定理$f'(x_2)=0$。 + +$\therefore f'(\xi_1)=f'(\xi_2)$,罗尔定理$\exists\xi\in(\xi_1,\xi_2)$使得$f''(\xi)=0$。 + +\paragraph{导数方程} \leavevmode \medskip + +类似微分方程,代数式由$a$、$b$、$f(a)$、$f(b)$、$f(\xi)$、$f'(\xi)$、$\cdots$、$f^{(n)}(\xi)$构成。 + +若$\xi$是$ab$可分离的式子: + +\begin{itemize} + \item 拉格朗日中值定理:分离出$f(b)-f(a)$或$\dfrac{f(b)-f(a)}{b-a}$。 + \item 柯西中值定理:分离出$\dfrac{f(b)-f(a)}{F(b)-F(a)}$。 +\end{itemize} + +若$\xi$是$ab$不可分离的式子: + +\begin{enumerate} + \item 构造辅助函数。 + \item 利用罗尔定理。 +\end{enumerate} \subsubsection{零点情况} @@ -234,7 +385,11 @@ $\therefore a=1,b=e$。 \subsection{拉格朗日中值定理} -证明不等式最重要的还是找到$f(x)$,即出现差值$f(a)-f(b)$,那么$f(x)$就是我们的目标函数,有时候不等式不存在$f(a)-f(b)$这种式子,就需要我们转换。 +证明不等式最重要的还是找到$f(x)$,即出现差值$f(a)-f(b)$,那么$f(x)$就是我们的目标函数,有时候不等式不存在$f(a)-f(b)$这种式子,就需要我们转换,常用的就是$f(b)=0$。 + +注意$a$和$b$可以是常量也可以是变量,$\xi$依赖$a$和$b$,所以$f'(\xi)$不为常数不能随便提出来。 + +如$f(x)-f(a)=f'(\xi)(x-a)\Rightarrow f(x)=f(a)+f'(\xi)(x-a)\nRightarrow\int_a^bf(x)\,\textrm{d}x=\int_a^bf(a)\,\textrm{d}x+f'(\xi)\int_a^b(x-a)\,\textrm{d}x$。 \subsubsection{式子转换} @@ -310,6 +465,20 @@ $f(a)\geqslant f(a+b)-f(b)$,所以$f(a+b)\leqslant f(a)+f(b)$。 从而$e^x-e=e^\xi(x-1)$,$\xi\in(1,x)$,所以$e^x-e>e(x-1)$,即$e^x>ex$,得证。 +\subsubsection{目标为常数} + +使用拉格朗日中值定理的推论:$f(x)$在$I$上连续且可导,则$I$上$f(x)=C\Leftrightarrow f'(x)\equiv 0$。 + +\textbf{例题:}证明$\arctan e^x+\arctan e^{-x}=\dfrac{\pi}{2}$。 + +解:令$F(x)=\arctan e^x+\arctan e^{-x}-\dfrac{\pi}{2}$。 + +$F'(x)=\dfrac{e^x}{1+e^{2x}}+\dfrac{-e^{-x}}{1+e^{-2x}}=0$。 + +根据拉格朗日中值定理,$F(x)\equiv C$。 + +又$F(0)=\arctan 1+\arctan 1-\dfrac{\pi}{2}=0$,$C=0$,得证。 + \subsection{柯西中值定理} 需要找到两个函数,使得$\dfrac{f(b)-f(a)}{F(b)-F(a)}=\dfrac{f'(\xi)}{F'(\xi)}$。 @@ -322,6 +491,66 @@ $f(b)-f(a)=\xi f'(\xi)\ln\dfrac{b}{a}=\dfrac{f(b)-f(a)}{\ln b-\ln a}=\dfrac{f'(\ 根据柯西中值定理得证。 +\subsection{泰勒中值定理} + +\begin{enumerate} + \item $n$阶可导则只展开到$n-1$阶,第$n$阶留给余项。 + \item 只要给出二阶或二阶以上可导,可以先泰勒展开看看。 +\end{enumerate} + +证明$\exists\xi\in(a,b)$使得$F(\xi,f^{(n)}(\xi))\geqslant0$($n\geqslant2$)使用带拉格朗日余项的泰勒公式。 + +\begin{enumerate} + \item 把$f(x)$在给定信息最多的点上泰勒展开。 + \item 若两点信息一样多,则优先在有导数信息的一点展开。 + \item 若两点都是导数信息或函数信息,则对两点分别展开。 +\end{enumerate} + +\textbf{例题:}$f(x)$在$[a,b]$上二阶可导,$f'(a)=f'(b)=0$,证明$\exists\xi\in(a,b)$,使得$\vert f''(\xi)\vert\geqslant4\dfrac{\vert f(b)-f(a)\vert}{(b-a)^2}$。 + +解:由于$ab$两点信息一样多,所以对两点进行泰勒一阶展开: + +$f(x)=f(a)+\dfrac{f''(\xi_1)}{2!}(x-a)^2$,$f(x)=f(b)+\dfrac{f''(\xi_2)}{2!}(x-b)^2$。 + +因为最后要证明的式子里面$ab$都要有,所以代入$x=\dfrac{a+b}{2}$: + +$f\left(\dfrac{a+b}{2}\right)=f(a)+\dfrac{f''(\xi_1')}{8}(b-a)^2$,$f\left(\dfrac{a+b}{2}\right)=f(b)+\dfrac{f''(\xi_2')}{8}(b-a)^2$($\xi_1$和$\xi_2$是关于$ab$的变量,此时都发生变化了) + +两式相减得$f(b)-f(a)=\dfrac{(b-a)^2}{8}[f''(\xi_1')-f''(\xi_2)]$,取绝对值$\vert f(b)-f(a)\vert=\dfrac{(b-a)^2}{8}\vert f''(\xi_1')-f''(\xi_2)\vert\leqslant\dfrac{(b-a)}{8}\vert f''(\xi_1')\vert+\vert f''(\xi_2')\vert$。 + +取$\xi=\max\{\xi_1',\xi_2'\}$,$\dfrac{(b-a)}{4}\vert f''(\xi')\vert\geqslant\vert f(b)-f(a)\vert$,得证。 + +\subsection{双中值问题} + +即有两个值$\xi$和$\eta$,分离$\xi$和$\eta$,两边都使用一次中值定理: + +\begin{itemize} + \item 不要求$\xi\neq\eta$,对$a,b$使用两次中值定理,要么两次拉格朗日中值定理,要么一次拉格朗日中值定理一次柯西中值定理。 + \item 要求$\xi\neq\eta$,找点$c$,在$(a,c)$、$c,b$区间上分别使用两次拉格朗日中值定理或一次拉格朗日中值定理一次柯西中值定理。对于$c$的取值,要么是第一问给出,要么逆推假设$c$已知根据要证的结论推出$c$。(基本上不会考到) +\end{itemize} + +重点关注中值项,其余不用管。 + +\subsubsection{两次拉格朗日} + +\textbf{例题:}$f(x)$在$[a,b]$上连续,$(a,b)$内可导,且$f(a)=f(b)=1$,证明存在$\xi,\eta\in(a,b)$使得$e^{\eta-\xi}[f(\eta)+f'(\eta)]=1$。 + +解:首先将$\xi$和$\eta$分开:$e^\eta[f(\eta)+f'(\eta)]=e^\xi$。根据拉格朗日中值定理$f'(\xi)=\dfrac{f(b)-f(a)}{b-a}$: + +对于右边$e^\xi=t'(\xi)$,$t(x)=\int e^x\,\textrm{d}x=e^x$,$\therefore e^\xi=\dfrac{e^b-e^a}{b-a}$。 + +对于左边$e^\eta[f(\eta)+f'(\eta)]=[e^xf(x)]'_{x=\eta}$,$\therefore e^\eta[f(\eta)+f'(\eta)]=\dfrac{e^bf(b)-e^af(a)}{b-a}\\=\dfrac{e^b-e^a}{b-a}=e^\xi$,所以等式成立,得证。 + +\subsubsection{一次拉格朗日一次柯西} + +\textbf{例题:}$f(x)$在$[a,b]$上连续,$(a,b)$内可导,且$f'(x)\neq0$,证明存在$\xi,\eta\in(a,b)$使得$\dfrac{f'(\xi)}{f'(\eta)}=\dfrac{e^b-e^a}{b-a}e^{-\eta}$。 + +解:首先将$\xi$和$\eta$分开:$f'(\xi)=\dfrac{e^b-e^a}{b-a}\dfrac{e^{-\eta}}{f'(\eta)}$。 + +对$f'(\xi)$使用拉格朗日,$=\dfrac{f(b)-f(a)}{b-a}$。对$\dfrac{e^{-\eta}}{f'(\eta)}$使用柯西,$=\dfrac{f(b)-f(a)}{e^b-e^a}$。 + +左右相等,得证。 + \section{导数应用} \subsection{单调性} diff --git a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.pdf 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zwfJXCrb%G$QL86O;MfhTD<#n2KT)8e-jZ1oc=dWN%_> c3UhQ}a&&ldWe;WwGB!CjHZuw(B}Gq03TYhpK>z>% diff --git a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex index a9f2205..46b4068 100644 --- a/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex +++ b/advanced-math/exercise/4-integal-of-functions-of-single-variable/integal-of-functions-of-single-variable.tex @@ -975,16 +975,27 @@ $D.\int_0^{1-\cos x}\sqrt{\sin^3t}\,\textrm{d}t$ \section{反常积分} -反常积分就是取极限,基本计算方法一样。 +\begin{itemize} + \item 反常积分就是取极限,基本计算方法一样。反常积分的计算就是判敛,如果存在就计算其值。 + \item 瑕积分的瑕点可能在积分区间内,此时需要根据瑕点分隔积分区间。 + \item 反常积分只有收敛才能使用奇偶性。 +\end{itemize} -反常积分的瑕点可能在积分区间内,此时需要根据瑕点分隔积分区间。 +处理方法: -反常积分的计算就是判敛,如果存在就计算其值。 +\begin{enumerate} + \item 判断$f(x)$有几个可能反常的点。包括$\infty$和无穷间断点。每次研究一个点。 + \item 由于使用等价无穷小替换,或乘以非零常数都不影响敛散性,所以使用这两种方法对$f(x)$化简。 +\end{enumerate} -\subsection{直接计算} +\subsection{定义法} 即可以根据不定积分求出原函数,再对原函数求极限。 +这种方法是定量的方法,所以比较麻烦,而下面的审敛法是定性的方法,所以更简单。 + +如果判断敛散性则优先审敛法。如果是直接求值则用定义法。 + \subsubsection{求值} \textbf{例题:}求$\displaystyle{\int_0^{+\infty}\dfrac{\textrm{d}x}{(1+x)(1+x^2)}}$。\medskip @@ -1043,27 +1054,103 @@ $\therefore I_n=n!$。 \subsection{极限审敛法} -不用计算积分值,直接通过幂函数的极限来代替积分。 +不用计算积分值,直接通过函数的极限来代替积分。 -\subsubsection{等级无穷小} - -\subsubsection{无穷区间} - -对于无限反常积分$\int_a^{+\infty}f(x)\,\textrm{d}x$,$f(x)$在$[a,+\infty)$上连续非负,对于常数$\rho$: +\subsubsection{判断方法} \begin{itemize} - \item 若存在$\rho>1$,使得$\lim\limits_{x\to+\infty}x^\rho f(x)=c<+\infty$,则积分收敛。 - \item 若只存在$\rho=1$,使得$\lim\limits_{x\to+\infty}xf(x)=c>0$或为$-\infty$,则积分发散。 + \item 等价无穷小:对于无穷小,高阶吸收低阶。 + \item 对于每个瑕点或无限区间都要对区域进行划分。 + \item 对于无限反常积分$\int_a^{+\infty}f(x)\,\textrm{d}x$,$f(x)$在$[a,+\infty)$上连续非负,对于常数$\rho$:若存在$\rho>1$,使得$\lim\limits_{x\to+\infty}x^\rho f(x)=c<+\infty$,则积分收敛。若只存在$\rho=1$,使得$\lim\limits_{x\to+\infty}xf(x)=c>0$或为$-\infty$,则积分发散。 + \item 对于瑕积分$\int_a^bf(x)\,\textrm{d}x$,其中$a$为瑕点,$f(x)$在$[a,b]$上连续非负,对于常数$\rho$:若存在$0<\rho<1$,使得$\lim\limits_{x\to a^+}(x-a)^\rho f(x)=c$存在,则积分收敛。若只存在$\rho=1$,使得$\lim\limits_{x\to a^+}(x-a)f(x)=c>0$或为$+\infty$,则积分发散。 \end{itemize} -\subsubsection{无界函数} +\subsubsection{幂函数} -对于瑕积分$\int_a^bf(x)\,\textrm{d}x$,其中$a$为瑕点,$f(x)$在$[a,b]$上连续非负,对于常数$\rho$: +对判断方法进行特殊化: -\begin{itemize} - \item 若存在$0<\rho<1$,使得$\lim\limits_{x\to a^+}(x-a)^\rho f(x)=c$存在,则积分收敛。 - \item 若只存在$\rho=1$,使得$\lim\limits_{x\to a^+}(x-a)f(x)=c>0$或为$+\infty$,则积分发散。 -\end{itemize} +\begin{enumerate} + \item 无限反常积分$\displaystyle{\int_a^{+\infty}\dfrac{1}{x^\rho}\textrm{d}x}$当$\rho>1$时收敛,当$\rho\leqslant1$时发散。 + \item 对于瑕积分$\displaystyle{\int_a^b\dfrac{1}{(x-a)^\rho}\textrm{d}x}$当$\rho<1$时收敛,当$\rho\geqslant1$时发散。 +\end{enumerate} + +\textbf{例题:}判断$\displaystyle{\int_0^{+\infty}\dfrac{1}{\sqrt{x+x^3}}}$的敛散性。 + +解:对于$x^{-\frac{1}{2}}$和$x^{-\frac{3}{2}}$,当$x\to+\infty$时,两者都趋向于0,而$x^{-\frac{3}{2}}$更快,所以更高阶,当$x\to0$时,两者都趋向于$+\infty$,而$x^{-\frac{3}{2}}$更快,所以$x^{-\frac{1}{2}}$更高阶(无穷小的比较是看哪个更快接近0或更慢远离0,具体的方法是取趋向于0的极限然后看值来判断)。 + +$\therefore x\to+\infty$时,$\dfrac{1}{\sqrt{x+x^3}}\sim\dfrac{1}{\sqrt{x^3}}$,$\therefore x\to0$时,$\dfrac{1}{\sqrt{x+x^3}}\sim\dfrac{1}{\sqrt{x}}$。 + +为了方便研究,将积分拆为两个区域:$\int_0^1f(x)\,\textrm{d}x+\int_1^{+\infty}f(x)\,\textrm{d}x$。 + +$\displaystyle{\int_1^{+\infty}\dfrac{1}{x^{\frac{3}{2}}}\,\textrm{d}x}$收敛,故原积分收敛,$\displaystyle{\int_0^1\dfrac{1}{x^{\frac{1}{2}}}\,\textrm{d}x}$收敛,故原积分收敛。 + +故原积分收敛。 + +\textbf{例题:}判断$\displaystyle{\int_1^{+\infty}\dfrac{1}{x\sqrt{x^2-1}}\textrm{d}x}$的敛散性。 + +解:首先判断区域和瑕点,有$x\to+\infty$和$x\to1^+$,所以拆分两个区域$\int_1^2f(x)\,\textrm{d}x$、$\int_2^{+\infty}f(x)\,\textrm{d}x$。 + +对于$x\to+\infty$,无穷区域所以$-1$不影响敛散性,直接去掉,$\dfrac{1}{x\sqrt{x^2-1}}\sim\dfrac{1}{x^2}$,该部分区域收敛。 + +对于$x\to1^+$,根据瑕点将非0部分分离出来,首先因式分解,$\dfrac{1}{x\sqrt{x^2-1}}=\dfrac{1}{x\sqrt{(x+1)(x-1)}}=\dfrac{1}{\sqrt{2}\sqrt{(x-1)}}\textrm{d}x$。 + +常量不影响敛散性,所以$\sim\dfrac{1}{(x-1)^{\frac{1}{2}}}$,所以该区域收敛。 + +故原积分收敛。 + +\subsubsection{指数函数} + +指数函数增长的速度远大于幂函数。所以判断很简单。 + +\textbf{例题:}判断$\displaystyle{\int_{-\infty}^{+\infty}\dfrac{1}{x^2}e^{\frac{1}{x}}\textrm{d}x}$的敛散性。 + +解:根据式子和积分上下限,所以需要关注$x\to+\infty$、$x\to-\infty$、$x\to0$。 + +$x\to+\infty$时,$\sim\dfrac{1}{x^2}$,收敛。 + +$x\to-\infty$时,$\sim\dfrac{1}{x^2}$,收敛。 + +$x\to0^+$时,对于$\dfrac{1}{x^2}\sim+\infty$,$e^{\frac{1}{x}}\sim+\infty$,发散。 + +故原积分发散。 + +\subsubsection{对数函数} + +对数函数增长较慢,比较较困难。若要判断$f(x)$的敛散性,当$x\to+\infty$或$x\to0^+$时,则$f(x)\ln^ax$更容易发散,因为$\ln^ax$增速比$x$慢,$\dfrac{f(x)}{\ln^ax}$更容易收敛,因为$\dfrac{1}{\ln^ax}$增速反而更快了。 + +若趋向于无穷小: + +\begin{enumerate} + \item 当$x\to1$时,$\ln x\sim x-1$。 + \item 当$x\to0^+$时,$\ln(1+x)\sim x$。 +\end{enumerate} + +若趋向于无穷大: + +\begin{enumerate} + \item $\displaystyle{\int_a^{+\infty}\dfrac{1}{\ln^\rho x}\textrm{d}x}$($a>1$)无论$\rho$取多少都发散。因为当$x\to+\infty$时对数函数永远小于幂级数,所以必然发散。 + \item $\displaystyle{\int_0^a\dfrac{1}{\ln^\rho x}\textrm{d}x}$($a<1$)无论$\rho$取多少都必然收敛。因为当$x\to0^+$时对数函数的倒数永远大于幂级数,所以必然收敛。 + \item $\displaystyle{\int_a^{+\infty}\dfrac{\ln^{\rho'}x}{x^\rho}\textrm{d}x\sim\int_a^{+\infty}\dfrac{1}{x^\rho}\textrm{d}x}$($a>0$)。 + \item $\displaystyle{\int_a^{+\infty}\dfrac{1}{x^\rho\ln^{\rho'}x}\textrm{d}x}$,当$\rho\neq1$时$\sim\displaystyle{\int_0^{+\infty}\dfrac{1}{x^\rho}\textrm{d}x}$,当$\rho=1$时,$\rho'>1$收敛,$\rho'\leqslant1$发散。($a>0$且当$x\to1$时$\sim\dfrac{1}{\ln^{\rho'}x}$) +\end{enumerate} + +\textbf{例题:}判断$\displaystyle{\int_0^{+\infty}\dfrac{\ln(1+x)}{x^{\frac{3}{2}}}\textrm{d}x}$的敛散性。 + +解:当$x\to+\infty$时,$\sim\dfrac{\ln x}{x^{\frac{3}{2}}}\sim\dfrac{1}{x^{\frac{3}{2}}}$,收敛。 + +当$x\to0^+$时,$\sim\dfrac{x}{x^{\frac{3}{2}}}\sim\dfrac{1}{x^{\frac{1}{2}}}$,收敛。 + +积分收敛。 + +\textbf{例题:}判断$\displaystyle{\int_0^{+\infty}\dfrac{\sqrt{\ln x}}{x\sqrt{x}-1}\textrm{d}x}$的敛散性。 + +解:当$x\to+\infty$时,$\sim\dfrac{\ln^\frac{1}{2}x}{x^\frac{3}{2}}\sim\dfrac{1}{x^{\frac{3}{2}}}$,收敛。 + +当$x\to0^+$时,$\sim-\ln^\frac{1}{2}x$,收敛。 + +当$x\to1$时,$\sim\dfrac{(x-1)^{\frac{1}{2}}}{(1+x-1)^\frac{3}{2}-1}\sim\dfrac{2(x-1)^{\frac{1}{2}}}{3(x-1)}\sim\dfrac{1}{(x-1)^{\frac{1}{2}}}$,收敛。 + +积分收敛。 % \subsubsection{结论} diff --git a/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf index b4cff7e9b13b71211384ad21e8596b32502a07f3..453c15157fac231154063e72008f1fb96ea725c6 100644 GIT binary patch delta 110816 zcmZ75(_$rx0wmB5I<{@wwr$(Cy<^+9ZQHhO+w9DldAsuu^-!zoBoHz#0@_~Qzvg;UevNTnt^VE*7&}m`O%oTWF)x6Djg5~F z%Gt%q)X)~nWAjF5Cl03#7I3HE;M=5FO_i{$?#&oUWb4mET9Jo!X2@@sgyk9l+F%}*Gg@)+=8}R4-cRQD)m&>(m(~K)w2x%h;QbH_IMkD>?|0bHNOqmG_L}& z(ffIPv^)oxlJ8RdLr8|x^YSpum9FaR*7@mHItHo!H&fH%gyBh`z}UHxl>($-5hwqz 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