From eac3b4d91984588eb664083602015d9be365d07a Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Thu, 24 Jun 2021 23:39:48 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=A7=AF=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../function-and-limit.pdf | Bin 329138 -> 230848 bytes .../function-and-limit.tex | 2 + ...efinite-integral-and-definite-integral.pdf | Bin 254142 -> 176373 bytes ...efinite-integral-and-definite-integral.tex | 42 +++++++++++++++++- linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 119795 -> 119801 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 3 +- 6 files changed, 44 insertions(+), 3 deletions(-) diff --git a/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf b/advanced-math/knowledge/1-function-and-limit/function-and-limit.pdf index ace6f0e71e88f0c40bdcf0372b677f482d501dfe..59de7de2fbb4cacb58e02609df6b6d8c30721a58 100644 GIT binary patch literal 230848 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b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -332,10 +332,22 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo \textcolor{aqua}{\textbf{定理:}}设函数$f(x)$在区间$[a,b]$上有界,且只有有限个间断点,则$f(x)$在该区间上可积。 +\textcolor{aqua}{\textbf{定理:}}若函数$f(x)$是周期函数且周期为$T$,$\int_a^{a+T}f(x)\,\textrm{d}x=\int_0^Tf(x)\,\textrm{d}x$对于$\forall a$成立。 + +证明:根据积分区间可拆性: + +$\int_a^{a+T}f(x)\,\textrm{d}x=\int_0^af(x)\,\textrm{d}x+\int_0^Tf(x)\,\textrm{d}x+\int_T^{a+T}f(x)\,\textrm{d}x$ + \subsection{变限积分} 设$f(x)$在$[a,b]$上连续,且$\Phi(x)=\int_a^xf(t)\,\textrm{d}t(x\in[a,b])$,这个函数就是积分上限函数或叫积分变限函数(如果$\int_x^af(t)\,\textrm{d}t$就是变下限积分或积分下限函数)。 +积分值与积分变量无关,即$\int_a^xf(x)\,\textrm{d}x=\int_a^xf(t)\,\textrm{d}t$。 + +对变限积分$\int_{a}^xf(t)\,\textrm{d}t$求导得到$f(t)$,再求导就得到$f'(t)$。 + +定限积分是一个数值,而变限积分是一个函数。 + \textcolor{aqua}{\textbf{定理:}}设$f(x)$在$[a,b]$上连续,则$\int_a^xf(t)\,\textrm{d}t$在$[a,b]$上$(\int_a^xf(t)\,\textrm{d}t)'=f(x)$。 证明:设$x\in(a,b)$。 @@ -348,8 +360,14 @@ $\dfrac{P}{x^n}=\dfrac{A}{x}+\dfrac{B_0x+B_1}{x^2}+\cdots+\dfrac{N_0x^{n-1}+\cdo 同理当$x=a,\Delta x>0$与$x=b,\Delta x<0$时也同样成立。 +固定的上限或下限都不会影响到最后的变限积分结果,因为他们之间只差了一个常数。所以一般会将$a$取为0,这样更方便计算。 + \textcolor{aqua}{\textbf{定理:}}设$f(x)$在$[a,b]$上连续,则$\int_a^xf(t)\,\textrm{d}t$是$f(x)$在$[a,b]$上的一个原函数。 +\textcolor{aqua}{\textbf{定理:}}$\int_a^{\varphi(x)}f(t)\,\textrm{d}t=\int_a^xf(\varphi(x))\,\textrm{d}(\varphi(x))=\int_a^xf(\varphi(x))\varphi'(x)\,\textrm{d}x$, + +所以$(\int_a^{\varphi(x)}f(t)\,\textrm{d}t)'=f(\varphi(x))\varphi'(x)$。 + \textbf{例题:}求$F(x)=\int_0^{x^2}e^{-t^2}\,\textrm{d}t$的导数。 由定理,可以将式子看作复合函数求导(注意定理中积分上限为$x$,而这里不是$x$,但是对$x$求导,所以必须看作为一个复合函数求导)。 @@ -362,12 +380,34 @@ $\therefore F'_x(x)=F'_u(x)\cdot u'_x=e^{-u^2}\cdot 2x=2xe^{-x^4}$。 也同理,如果上限下限都在变化,则可以利用积分区间的可加性,将这个积分的区间插入一个常数(一般为0),将一个积分式子变为两个积分式子,再分别进行运算。 -所以变限积分\textcolor{aqua}{\textbf{定理:}}若$\phi(x)$与$\psi(x)$都可导,$f(x)$连续,则$\dfrac{\textrm{d}}{\textrm{d}x}\int_{\psi(x)}^{\phi(x)}=f(\psi(x))\psi'(x)-f(\phi(x))\phi'(x)$。 +所以变限积分\textcolor{aqua}{\textbf{定理:}}若$\phi(x)$与$\psi(x)$都可导,$f(x)$连续,则$\dfrac{\textrm{d}\int_{\psi(x)}^{\phi(x)}f(t)\,\textrm{d}t}{\textrm{d}x}=f(\psi(x))\psi'(x)-f(\phi(x))\phi'(x)$。 \textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{\int_0^{\sin^2x}\ln(1+t)\,\textrm{d}t}{x(\sqrt{1+x^3}-1)}$。 原式$=\lim\limits_{x\to 0}\dfrac{\ln(1+\sin^2x)2\sin x\cos x}{x(\sqrt{1+x^3}-1)}=\lim\limits_{x\to 0}\dfrac{x^2\cdot 2x\cdot 1}{\dfrac{4}{3}x^3}=\dfrac{3}{2}$。\smallskip +\textcolor{aqua}{\textbf{定理:}}若函数$f(x)$是连续的偶函数,则其积分只有一个$\int^x_0f(t)\,\textrm{d}t$是奇函数。 + +证明:令$F(x)=\int_0^xf(t)\,\textrm{d}t$,需要证明$F(-x)=-F(x)$。 + +$F(-x)=\int_0^{-x}f(t)\,\textrm{d}t$,令$t=-u$,所以得到$\int_0^xf(-u)\,\textrm{d}(-u)$。 + +又$f(x)$偶函数,所以$f(-x)=f(x)$,从而$=\int_0^xf(-u)\,\textrm{d}(-u)=-\int_0^xf(u)\,\textrm{d}u$ + +$=-\int_0^xf(t)\,\textrm{d}t=-F(x)$。这就是个奇函数。若加上一个常数就不是个奇函数了。 + +\textcolor{aqua}{\textbf{定理:}}若函数$f(x)$是连续的奇函数,则其所有积分$\int^x_af(t)\,\textrm{d}t$都是偶函数。 + +证明:令$F(x)=\int_0^xf(t)\,\textrm{d}t$,需要证明$F(-x)=F(x)$。 + +$F(-x)=\int_0^{-x}f(t)\,\textrm{d}t$,令$t=-u$,所以得到$\int_0^xf(-u)\,\textrm{d}(-u)$。 + +又$f(x)$奇函数,所以$f(-x)=-f(x)$,从而$=\int_0^x-f(u)(-\textrm{d}u)=\int_0^xf(u)\,\textrm{d}u=\int_0^xf(t)\,\textrm{d}t=F(x)$。而$\int_a^xf(t)\,\textrm{d}t=\int_a^0f(t)\,\textrm{d}t+\int_0^xf(t)\,\textrm{d}t$也为偶函数。 + +\textcolor{aqua}{\textbf{定理:}}若函数$f(x)$是周期函数且周期为$T$,虽然其导数也为周期函数且周期,但是其变限积分不一定为周期函数。若$\int_0^Tf(x)\,\textrm{d}x=0$即一个周期上的定积分值为0,则这个函数为周期函数,且周期为$T$。 + + + \subsection{牛顿-莱布尼茨公式} \textcolor{aqua}{\textbf{定理:}}(微积分基本定理/牛顿-莱布尼茨公式)若函数$F(x)$是连续函数$f(x)$在区间$[a,b]$上的一个原函数,则$\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)$。 diff --git a/linear-algebra/knowledge/2-matrix/matrix.pdf b/linear-algebra/knowledge/2-matrix/matrix.pdf index 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zp27^w#A}$1K0J??@TyAspRQ9ZzF*F)N@q@nGq=P3pPMRqK39r;R*Fej?RL8keTW_UArS7jQ4d1Plsx&6K((JWzTcxGkmDW@%Wh(D#Txt7irCjBG zpzV7q?O)u}57Aa0sAf0Wu23y5jI#ZxYZtlus_>_kC#r4nw(oS^W0AIN@fp_R3v9wy g*o<$mRkh=v$ba;@d)t@L(*Ym}Gd2n(B}Gq03MN)}HUIzs diff --git a/linear-algebra/knowledge/2-matrix/matrix.tex b/linear-algebra/knowledge/2-matrix/matrix.tex index 6c87555..3df06bd 100644 --- a/linear-algebra/knowledge/2-matrix/matrix.tex +++ b/linear-algebra/knowledge/2-matrix/matrix.tex @@ -747,5 +747,4 @@ $$(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} 对于任何矩阵都能通过初等列变换变为行阶梯形矩阵和行最简形矩阵,再通过列变换可以变为\textbf{标准形}:左上角是一个单位矩阵,其他元全部是0。 - -\end{document} \ No newline at end of file +\end{document} \ No newline at end of file