From 3b45efd08c6877cac6b964930b25174df07257b7 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sun, 18 Jul 2021 23:03:54 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E4=BA=8C=E9=87=8D=E7=A7=AF?= =?UTF-8?q?=E5=88=86?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...ial-calculus-of-multivariate-functions.pdf | Bin 116867 -> 140323 bytes ...ial-calculus-of-multivariate-functions.tex | 113 +++++++++- ...ral-calculus-of-multivariate-functions.pdf | Bin 13007 -> 106676 bytes ...ral-calculus-of-multivariate-functions.tex | 204 +++++++++++++++++- .../differential-equation.pdf | Bin 145249 -> 103928 bytes .../differential-equation.tex | 15 +- 6 files changed, 317 insertions(+), 15 deletions(-) diff --git a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf 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zjl5vhf5rWpSkaSV?Z&kC-~Wc0F=XFTO8Z<>->JoPC#R%F<#h9tO=_MBN5du~z# zWSzmj8>Fep^+j^SHtll8^jtK9d+eI62c>aCdh8L&^_OxZYVOiz#+I|dE^Q;|MV~Qa z&9Q)c?8s$?;xpq~?n$nPWH=)gZ-ZqczPcg*wT->tctG|AI;iA@0emF~ UBSc`~WMSrkqokBjl!W_#08Ld?LI3~& diff --git a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex index f983798..bb95b14 100644 --- a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex +++ b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.tex @@ -147,10 +147,6 @@ $\textrm{d}z=A\Delta x+B\Delta y=\dfrac{\partial z}{\partial x}\Delta x+\dfrac{\ \item 写出极限$\lim\limits_{\substack{\Delta x\to0\\\Delta y\to0}}\dfrac{\Delta z-(A\Delta x+B\Delta y)}{\sqrt{(\Delta x)^2+(\Delta y)^2}}$,若极限等于0,则$z=f(x,y)$在点$(x_0,y_0)$可微,否则不可微。 \end{enumerate} -% \textbf{例题:}已知函数$z=f(x,y)$的全微分$\textrm{d}z=2x\,\textrm{d}x+\sin y\,\textrm{d}y$且$f(1,0)=2$,求$f(x,y)$。 - -% 解: - \subsection{偏导数连续性} 对$z=f(x,y)$,讨论其在某特殊点$(x_0,y_0)$处偏导数是否连续的步骤: @@ -240,24 +236,123 @@ $\dfrac{\partial^2z}{\partial x\partial y}=\dfrac{\partial(f_1'\cdot e^x\sin y)} 求导路径$f_1'-1-y$和$f_1'-2-y$:$=(f_{11}''e^x\cos y+f_{12}''2y)\cdot e^x\sin y+f_1'\cdot e^x\cos y$。 +其中$\dfrac{\partial(f_2'\cdot2x)}{\partial y}=2x\dfrac{\partial f_2'}{\partial y}$,求导路径$f_2'-1-y$和$f_2'-2-y$:$=2x(f_{21}''\cdot e^x\cos y+f_{22}''\cdot2y)$。 + +$\therefore\dfrac{\partial^2z}{\partial x\partial y}=(f_{11}''e^x\cos y+f_{12}''2y)\cdot e^x\sin y+f_1'\cdot e^x\cos y+2x(f_{21}''\cdot e^x\cos y+f_{22}''\cdot2y)$。 + +若$f$具有二阶连续偏导数,所以可以交换求导顺序,$f_{12}''=f_{21}''$。 + +化简:$=f_1'e^x\cos y+f_{11}''e^{2x}\sin y\cos y+2e^xf_{12}''(y\sin y+x\cos y)+4f_{22}''xy$。 + \subsection{隐函数存在定理} -主要对隐函数的微分。 +主要对隐函数的微分。隐函数的最大问题就是变量纠缠在一起,而公式法所得到的式子中变量都是独立的。 + +若对每个$x\in D$对应的函数值$y$总是唯一的,这样定义的函数为\textbf{单值函数}。若给定一个对应法则,按法则对$x$总有$y$与之对应,但是$y$不唯一,此时就不是函数,而确定一个\textbf{多值函数}。 + +只要满足着定义域的条件下,形如$y=f(x)$的函数就是\textbf{显函数},如$y=\sin x$。由方程$F(x,y)=0$确定的函数为\textbf{隐函数},如$x+y^3-1=0$显式表示为$y=\sqrt[3]{1-x}$。 + +\textcolor{violet}{\textbf{定义:}}设函数$F(x,y)$在点$P(x_0,y_0)$的某一邻域内具有连续偏导数,$F(x_0,y_0)$\\$=0$,$F_y'(x_0,y_0)\neq0$,则方程$F(x,y)=0$在点$(x_0,y_0)$的某一邻域内能唯一确定一个连续且具有连续导数的函数$y=f(x)$,满足$y_0=f(x_0)$。 + +\textcolor{violet}{\textbf{定义:}}二元隐函数求导公式:$\dfrac{\textrm{d}y}{\textrm{d}x}=-\dfrac{F_x'}{F_y'}$。 + +因为$y=y(x)$,所以对$F(x,y)$进行求导:$F_x'\cdot1+F_y'\cdot\dfrac{\textrm{d}y}{\textrm{d}x}=0$,就解出隐函数求导公式,$F_y'(x_0,y_0)\neq0$是定理关键。 + +如给出一个圆的方程$F(x,y)=x^2+y^2-1=0$,$F_x'=2x$,$F_y'=2y$,$F(0,1)=0$,$F'_y(0,1)=2\neq0$。所以在$(0,1)$和$(0,-1)$是单值的,从而能确定一个连续导数的隐函数,而在$(\pm1,0)$的邻域内不存在,因为其切线是竖直的。 + +\textcolor{violet}{\textbf{定义:}}设函数$F(x,y,z)$在点$P(x_0,y_0,z_0)$的某一邻域内具有连续偏导数,$F(x_0,y_0,z_0)=0$,$F_z'(x_0,y_0,z_0)\neq0$,则方程$F(x,y,z)=0$在点$(x_0,y_0,z_0)$的某一邻域内能唯一确定一个连续且具有连续导数的函数$z=f(x,y)$,满足$z_0=f(x_0,y_0)$。 + +\textcolor{violet}{\textbf{定义:}}三元隐函数求导公式:$\dfrac{\partial z}{\partial x}=-\dfrac{F_x'}{F_z'}$,$\dfrac{\partial z}{\partial y}=-\dfrac{F_y'}{F_z'}$。 + +因为$x$是$x$的函数,$y$是$y$的函数,$z$是$xy$的函数。所以$F_x'\cdot1+F_z'\cdot\dfrac{\partial z}{\partial x}=0$,解得$\dfrac{\partial z}{\partial x}=-\dfrac{F_x'}{F_z'}$。同理$F_y'\cdot1+F_z'\cdot\dfrac{\partial z}{\partial y}$也可得。 + +\textbf{例题:}设$z=f(x,y)$是由方程$z-y-x+xe^{z-y-x}=0$所确定的二元函数,求$\textrm{d}z$。 + +解:$\textrm{d}z=\dfrac{\partial z}{\partial x}\textrm{d}x+\dfrac{\partial z}{\partial y}\textrm{d}y$。其中$F_x'=-1+e^{z-y-x}-xe^{z-y-x}$,$F_y'=-1-xe^{z-y-z}$,$F_z'=1+xe^{z-y-x}$。 + +直接代入:$\dfrac{\partial z}{\partial x}\textrm{d}x+\dfrac{\partial z}{\partial y}\textrm{d}y=-\dfrac{F_x'}{F_z'}-\dfrac{F_y'}{F_z'}=\dfrac{1+(x-1)e^{z-y-x}}{1+xe^{z-y-x}}+1$。 + +\textbf{例题:}已知函数$z=f(x,y)$的全微分$\textrm{d}z=2x\,\textrm{d}x+\sin y\,\textrm{d}y$且$f(1,0)=2$,求$f(x,y)$。 + +解:$\because\textrm{d}z=2x\,\textrm{d}x+\sin y\,\textrm{d}y$,$\therefore\dfrac{\partial z}{\partial x}=2x$,$\dfrac{\partial z}{\partial y}=\sin y$。 + +对偏导进行积分:$f(x,y)=x^2+\varphi(y)$,$\dfrac{\partial(x^2+\varphi(y))}{\partial y}=\varphi'(y)=\sin y$。 + +又$\varphi(y)=-\cos y+C$,$f(x,y)=x^2-\cos y+C$,代入$f(1,0)=2$。 + +$C=2$,$f(x,y)=x^2-\cos y+2$。 \section{多元函数极值最值} \subsection{概念} +\textcolor{violet}{\textbf{定义:}}若存在$(x_0,y_0)$的某个邻域,使得在该邻域内的任意一点$(x,y)$均有$f(x,y)\leqslant f(x_0,y_0)$或$f(x,y)\geqslant f(x_0,y_0)$成立,则称$(x_0,y_0)$为$f(x,y)$的\textbf{广义的极大值点/极小值点},$f(x_0,y_0)$为$f(x,y)$的\textbf{广义的极大值/极小值}。 + +\textcolor{violet}{\textbf{定义:}}若存在$(x_0,y_0)$的某个去心邻域,使得在该邻域内的任意一点$(x,y)$均有$f(x,y)f(x_0,y_0)$成立,则称$(x_0,y_0)$为$f(x,y)$的\textbf{真正的极大值点/极小值点},$f(x_0,y_0)$为$f(x,y)$的\textbf{真正的极大值/极小值}。 + +\textcolor{violet}{\textbf{定义:}}设$(x_0,y_0)$为$f(x,y)$定义域内一点,若对于$f(x,y)$的定义域内任意一点$(x,y)$均有$f(x,y)\leqslant f(x_0,y_0)$或$f(x,y)\geqslant f(x_0,y_0)$成立,则称$(x_0,y_0)$为$f(x,y)$的\textbf{广义的最大值点/最小值点},$f(x_0,y_0)$为$f(x,y)$的\textbf{广义的最大值/最小值}。 + +\textcolor{violet}{\textbf{定义:}}设$(x_0,y_0)$为$f(x,y)$定义域内一点,若对于$f(x,y)$的定义域内任意一个异于$(x_0,y_0)$的点$(x,y)$均有$f(x,y)f(x_0,y_0)$成立,则称$(x_0,y_0)$为$f(x,y)$的\textbf{真正的最大值点/最小值点},$f(x_0,y_0)$为$f(x,y)$的\textbf{真正的最大值/最小值}。 + \subsection{无条件极值} -\subsubsection{隐函数} +\textcolor{aqua}{\textbf{定理:}}二元函数取极值的必要条件:设$z=f(x,y)$在点$(x_0,y_0)$一阶偏导数存在且取极值,则$f_x'(x_0,y_0)=0$,$f_y'(x_0,y_0)=0$。三元及以上可以类推。 -\subsubsection{显函数} +\textcolor{aqua}{\textbf{定理:}}二元函数取极值的充分条件:若对函数求二阶偏导$\left\{\begin{array}{l} + f_{xx}''(x_0,y_0)=A \\ + f_{xy}''(x_0,y_0)=B \\ + f_{yy}''(x_0,y_0)=C +\end{array}\right.$,则$\Delta=B^2-AC=\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,使用定义法} +\end{array}\right.$。只适用于二元函数极值。 + +\textbf{例题:}求函数$f(x,y)=x^4+y^4-(x+y)^2$的极值。 + +解:$f_x'=4x^3-2(x+y)=0$,$f_y'=4y^3-2(x+y)=0$,解得$x=y=-1,0,1$。 + +$f_{xx}''=12x^2-2$,$f_{xy}''=-2$,$f_{yy}''=12y^2-2$。各自代入: + +$P_1:A_1=10,B_1=-2,C_1=10,\Delta=B_1^2-A_1C_1=-96<0,A_1>0$,极小。 + +同理$P3$也是极小值点。极小值为-2。 + +$P2:A_2=-2,B_2=-2,C_2=-2,\Delta_2=B_2^2-A_2C_2=0$。该方法失效。 + +取$y=x$的路径,$f(x,y)=f(x,x)=2x^4-4x^2=2x^2(x+\sqrt{2})(x-\sqrt{2})<0$。 + +取$y=-x$的路径,$f(x,y)=f(x,-x)=2x^4>0$。而$f(0,0)=0$。 + +所以不同的路径上有大于该值的也有小于该值的,所以该点不为极值点。 + +% \subsubsection{隐函数} + +% \subsubsection{显函数} \subsection{条件极值与拉格朗日乘数法} -\subsubsection{闭区域边界最值} +求目标函数$u=f(x,y,z)$在一组条件函数$\varphi_1(x,y,z)=0,\varphi_2(x,y,z)=0,\cdots,\varphi_n(x,y,z)=0$下的最值,则: -\subsubsection{闭区域上最值} +\begin{enumerate} + \item 构造辅助函数带$\lambda_i$:$F(x,y,z,\lambda_1,\lambda_2,\cdots,\lambda_n)=f(x,y,z)+\lambda_1\varphi_1(x,y,z)+$\\$\lambda_2\varphi_2(x,y,z)+\lambda_n\varphi_n(x,y,z)$,其中$\lambda_i\varphi_i$为\textbf{拉格朗日乘数}。 + \item 对函数依次对$x,y,z,\lambda_i$求偏导并令为0:$F_x'=f_x'+\lambda_1\varphi_{1x}'+\lambda_2\varphi_{2x}'+\cdots+\lambda_n\varphi_{nx}'=0$,$F_y'=f_y'+\lambda_1\varphi_{1y}'+\lambda_2\varphi_{2y}'+\cdots+\lambda_n\varphi_{ny}'=0$,$F_z'=f_z'+\lambda_1\varphi_{1z}'+\lambda_2\varphi_{2z}'+\cdots+\lambda_n\varphi_{nz}'=0$,$F_{\lambda_i}'=\varphi_i(x,y,z)=0$。一共$3+n$个方程。 + \item 解上述方程组得备选点$P_i$,$i=1,2,\cdots,n$,并求$f(P_i)$并取其最大值$m_{\max}$和最小值$u_{\min}$。 +\end{enumerate} + +\textbf{例题:}求函数$u=xyz$在约束条件$\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=\dfrac{1}{a}$($x>0,y>0,z>0,a>0$)下的最小值。 + +解:令$F(x,y,z,\lambda)=xyz+\lambda\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}-\dfrac{1}{a}\right)$。 + +令$F_x'=yz+-\dfrac{\lambda}{x^2}=0$,$F_y'=xz+-\dfrac{\lambda}{y^2}=0$,$F_z'=xy+-\dfrac{\lambda}{z^2}=0$,$F_\lambda'=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}-\dfrac{1}{a}=0$。 + +解得$x=y=z=3a$,从而最小值为$u_{\min}=27a^3$。 + +% \subsubsection{闭区域边界最值} + +% \subsubsection{闭区域上最值} \end{document} diff --git a/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf b/advanced-math/knowledge/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.pdf index bffbdc50ee3da03723313b351ed51dae2b7a3989..58d4e155d7ed1fbf3b5cbb09b40f5855390de86e 100644 GIT binary patch literal 106676 zcma&NLy#`evaMUTZQHi1R;{va+qTWWY}>YN+r}!}xcfxBcyVzWrWlZhNT`Y*W z*trA*V4Pi?Obu;eJT|X0WaF2Zkiu@iqVNtBSo3}W^8^3`aytocAYugBC5OZQ-#(^| z(h|8W3IvI2w<4M-+!0n|!MbZlo$tzekWWMOau|V*^vWH1ZwLFe_@7R}Pyf%Kp8~H> zWLl6je#_^1wel-0iq9S2r1NBU47;@@99I8~Je# zQ2ZN?+R0<(6+T!)=5|Y#9Q$>Y`NE+mK8&2wdC^EK1zX}$ZDRZJ^Pp1~6bt{Vo1P~Hr&H53=&vXP 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zP@NF+@iWOBBxILXl&{S-eh~+BDWRbcevKJaNiM`-tivQMgi7wYKP*F&uA?oQqEktT z{v0Dg5tRPT;Ds85x4b$c;NM{wFn@X3en9V$dG-3xXy)6J^t^l5b8uiad2bedf{6!) h4MaNr?0$乘上高$f(x,y)$就得到一个小柱体体积,再把所有$D$上的柱体相加起来就是整个曲顶柱体的体积。 + +\subsubsection{性质} + +\begin{itemize} + \item 求区域面积:$\iint\limits_D1\cdot\textrm{d}\sigma=\iint\limits_D\textrm{d}\sigma=A$,其中$A$为$D$的面积。 + \item 可积函数必有界:当$f(x,y)$在有界闭区间$D$上可积时,$f(x,y)$在$D$上必有界。 + \item 积分线性性质:$k_1,k_2$为常数,则$\iint\limits_D[k_1f(x,y)\pm k_2g(x,y)]\,\textrm{d}\sigma=\iint\limits_{D_1}f(x,y)\,\textrm{d}\sigma$\\$\pm k_2\iint\limits_{D_2}f(x,y)\,\textrm{d}\sigma$。 + \item 积分可加性:当$f(x,y)$在有界闭区间$D$上可积时,且$D_1\cup D_2=D$,$D_1\cap U_2=\varnothing$,则$\iint\limits_Df(x,y)\,\textrm{d}\sigma=\iint\limits_{D_1}f(x,y)\,\textrm{d}\sigma+\iint\limits_{D_2}f(x,y)\,\textrm{d}\sigma$。 + \item 积分保号性:当$f(x,y),g(x,y))$在有界闭区间$D$上可积时,若在$D$上有$f(x,y)\leqslant g(x,y)$,则$\iint\limits_Df(x,y)\,\textrm{d}\sigma\leqslant\iint\limits_Dg(x,y)\,\textrm{d}\sigma$,特别$\left\vert\iint\limits_Df(x,y)\,\textrm{d}\sigma\right\vert\leqslant\iint\limits_D\vert f(x,y)\vert\,\textrm{d}\sigma$。 + \item 二重积分估值定理:设$M,m$,分别为$f(x,y)$在有界闭区域$D$上的最大值和最小值,$A$为$D$的面积,则有$mA\leqslant\iint\limits_Df(x,y)\,\textrm{d}\sigma\leqslant MA$。 + \item 二重积分中值定理:设函数$f(x,y)$在有界闭区域$D$上连续,$A$为$D$的面积,则在$D$上至少存在一点$(\xi,\eta)$使得$\iint\limits_Df(x,y)\,\textrm{d}\sigma=f(\xi,\eta)A$。 +\end{itemize} + +\textbf{例题:}设$I_1=\iint\limits_D\cos\sqrt{x^2+y^2}\,\textrm{d}\sigma$,$I_2=\iint\limits_D\cos(x^2+y^2)\,\textrm{d}\sigma$,$I_3=\iint\limits_D\cos(x^2+y^2)^2\,\textrm{d}\sigma$,其中$D=\{(x,y)|x^2+y^2\leqslant1\}$,则()。 + +$A.I_3>I_2>I_1$\qquad$B.I_1>I_2>I_3$\qquad$C.I_2>I_1>I_3$\qquad$D.I_3>I_1>I_2$ + +解:令$x^2+y^2=t$,$\therefore0D)q(qs=+oU%i*r4eu-RO!k(z_!kYJR#2m5-?~?r!OKk2K&HU;riScuUq{1R z1g%k{#?aw7#bB$f$y(-OdB_GeS1h$x+J2g9Sa%LM&IA_M^eB)xWgn^6+3l_WX_;L>y>ktKZd7 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\definecolor{orange}{RGB}{255,127,0} +\definecolor{violet}{RGB}{192,0,255} \definecolor{aqua}{RGB}{0,255,255} \usepackage{geometry} \setcounter{tocdepth}{4} @@ -49,19 +50,23 @@ \section{微分方程基本概念} -表示未知函数、未知函数的导数与自变量之间的关系的方程,即含导数的方程就是微分方程。导数可能是一阶导数也可能是二阶以及以上阶数的导数。 +\subsection{微分方程构成} -微分方程所出现的未知函数的最高阶导数的阶数就是该微分方程的\textbf{阶}。 +\textcolor{violet}{\textbf{定义:}}表示未知函数、未知函数的导数与自变量之间的关系的方程,即含导数的方程就是微分方程。导数可能是一阶导数也可能是二阶以及以上阶数的导数。 + +\textcolor{violet}{\textbf{定义:}}微分方程所出现的未知函数的最高阶导数的阶数就是该微分方程的\textbf{阶}。 $n$阶微分方程的形式是$F(x,y,y',\cdots,y^{(n)})=0$。其中最高阶导数是必须出现的。若能从中解出最高阶导数,则可得微分方程$y^{(n)}=f(x,y,y',\cdots,y^{(n-1)})$。 -若微分方程中的解中含有任意常数,且任意常数的个数与微分方程的阶数相同,则就是微分方程的\textbf{通解}。 +\subsection{微分方程的解} + +\textcolor{violet}{\textbf{定义:}}若微分方程中的解中含有任意常数,且任意常数的个数与微分方程的阶数相同,则就是微分方程的\textbf{通解}。 如若$y''=3$,则$y'=3x+C_1$,$y=\dfrac{3}{2}x^2+C_1x+C_2$,此时含有两个任意常数$C_1C_2$,则微分方程的阶数也为2。 -确定通解中任意常数后,就得到微分方程的\textbf{特解}。 +\textcolor{violet}{\textbf{定义:}}确定通解中任意常数后,就得到微分方程的\textbf{特解}。 -当给出$x=x_0$时$y_0$与$y_0'$的值,那么这些条件就是\textbf{初值条件},如上面的$y''=3$。 +\textcolor{violet}{\textbf{定义:}}当给出$x=x_0$时$y_0$与$y_0'$的值,那么这些条件就是\textbf{初值条件},如上面的$y''=3$。 求微分方程$y'=f(x,y)$满足初值条件$y\vert_{x=x_0}=y_0$的特解这样的问题,就是一阶微分方程的初值问题,记为$\left\{\begin{array}{l} y'=f(x,y) \\