From b0093808c5e82aecb29b24de499bb30ab6b1af0c Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 11 Oct 2021 23:42:43 +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 --- ...ral-calculus-of-multivariate-functions.pdf | Bin 0 -> 44693 bytes ...ral-calculus-of-multivariate-functions.tex | 63 ++++++++++++++++++ ...ral-calculus-of-multivariate-functions.pdf | Bin 164714 -> 164707 bytes ...ral-calculus-of-multivariate-functions.tex | 2 +- linear-algebra/exercise/5-similar/similar.pdf | Bin 112022 -> 81743 bytes linear-algebra/exercise/5-similar/similar.tex | 20 ++++++ .../6-quadratic-form/quadratic-form.pdf | Bin 0 -> 12936 bytes .../6-quadratic-form/quadratic-form.tex | 38 +++++++++++ .../knowledge/5-similar/similar.pdf | Bin 136182 -> 97881 bytes .../knowledge/5-similar/similar.tex | 12 ++++ 10 files changed, 134 insertions(+), 1 deletion(-) create mode 100644 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a/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex new file mode 100644 index 0000000..f153f91 --- /dev/null +++ b/advanced-math/exercise/7-integral-calculus-of-multivariate-functions/integral-calculus-of-multivariate-functions.tex @@ -0,0 +1,63 @@ +\documentclass[UTF8, 12pt]{ctexart} +% UTF8编码,ctexart现实中文 +\usepackage{color} +% 使用颜色 +\usepackage{geometry} +\setcounter{tocdepth}{4} +\setcounter{secnumdepth}{4} +% 设置四级目录与标题 +\geometry{papersize={21cm,29.7cm}} +% 默认大小为A4 +\geometry{left=3.18cm,right=3.18cm,top=2.54cm,bottom=2.54cm} +% 默认页边距为1英尺与1.25英尺 +\usepackage{indentfirst} +\setlength{\parindent}{2.45em} +% 首行缩进2个中文字符 +\usepackage{setspace} +\renewcommand{\baselinestretch}{1.5} +% 1.5倍行距 +\usepackage{amssymb} +% 因为所以 +\usepackage{amsmath} +% 数学公式 +\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} +% 超链接 +\author{Didnelpsun} +\title{多元函数积分学} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} +\section{二重积分} + +\subsection{交换积分次序} + +\subsubsection{直角坐标系} + +\textbf{例题:}交换积分次序$\int_0^1\textrm{d}x\int_0^{x^2}f(x,y)\,\textrm{d}y+\int_1^3\textrm{d}x\int_0^{\frac{1}{2}(3-x)}f(x,y)\,\textrm{d}y$。 + +解:已知积分区域分为两个部分。将$X$型变为$Y$型。画出图形可以知道$y\in(0,1)$,$x$的上下限由$y=x^2$和$y=\dfrac{1}{2}(3-x)$转化为$\sqrt{y}$和$3-2y$。 + +所以转换为$\int_0^1\textrm{d}y\int_{\sqrt{y}}^{3-2y}f(x,y)\,\textrm{d}x$。 + +\subsubsection{极坐标系} + +\subsection{极直互化} + +\textbf{例题:}将$I=\int_0^{\frac{\sqrt{2}}{2}R}e^{-y^2}\textrm{d}y\int_0^ye^{-x^2}\,\textrm{d}x+\int_{\frac{\sqrt{2}}{2}R}^Re^{-y^2}\,\textrm{d}y\int_0^{\sqrt{R^2-y^2}}e^{-x^2}\,\textrm{d}x$转换为极坐标系并计算结果。 + +解:首先根据积分上下限得到积分区域$D=\left\{0\leqslant y\leqslant\dfrac{\sqrt{2}}{2}R,0\leqslant x\leqslant y\right\}\cup\left\{\dfrac{\sqrt{2}}{2}R\leqslant y\leqslant R,0\leqslant x\leqslant\sqrt{R^2-y^2}\right\}$,$D$为一个八分之一圆的扇形。 + +根据$x=r\cos\theta$,$y=r\sin\theta$替换得到$D=\left\{(x,y)\bigg|0\leqslant r\leqslant R,\dfrac{\pi}{4}\leqslant\theta\leqslant\dfrac{\pi}{2}\right\}$。 + +又$e^{-y^2}\cdot e^{-x^2}=e^{-(x^2+y^2)}=e^{-r^2}$。 + +$\therefore I=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\textrm{d}\theta\int_0^Re^{-r^2}r\,\textrm{d}r$。 + +\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 e95d7a64ec7de5b782c76ed701919bac892acad2..63ab8844eefb16e28ecf06dc79a436acf6ce1155 100644 GIT binary patch delta 14713 zcmajFQ*b6s)HOP>ZA|QmZQHhO>xrJ&wmq?J+nLz5?c}`Q|JSMd&egf;>gw7T-D}sz zTD{LhVGlxK>w;N;>@58Jur997W=3|fp6gdSa&d=k2;C_o#YwCW1|ugY@hsd`FCNVhM~JjkyhBi 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I=\dfrac{1}{2}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)\displaystyle{\iint\limits_D(x^2+y^2)\textrm{d}\sigma}$。 -根据公式三转换为极坐标系:$I=\dfrac{1}{2}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)\int_0^{2\pi}\textrm{d}\theta\int)^Rr^2r\,\textrm{d}r$。 +根据公式三转换为极坐标系:$I=\dfrac{1}{2}\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)\int_0^{2\pi}\textrm{d}\theta\int_0^Rr^2r\,\textrm{d}r$。 即$I=\left(\dfrac{1}{a^2}+\dfrac{1}{b^2}\right)\dfrac{\pi R^4}{4}$。 diff --git a/linear-algebra/exercise/5-similar/similar.pdf b/linear-algebra/exercise/5-similar/similar.pdf index cbcad614bdd7d9da1fde33edf7435d259bca6103..86d304e53de08dbe64753e616958f1497cf20312 100644 GIT binary patch delta 79607 zcmZs>Q?M?;5~X?0wr$(CZQHhO{b$>@ZQHhO+qUQSOixVT>5iy}ddhlPk@e+TnQ{p> z`vZbVsvsgp%SguxMS661{RYLv7LP~@U}5ItgK~CpGBvb?^4Pde>yFuEfEj(|3)Ul` zl8i?JMPbKr;LX$aw-E*Bj!r`ug>k_3J zy<}|Pq;`8WpI(1;!S(d=^Vx5o)}BH5M3LBSZYKH49k5Y)aCKey`Q*o-klf?|{Oo$y zKlb8$_as7}u`_|!yv%C-V=n(%!1tLL+Z%%;*vwd6Vi0WbxP`use(SZ6(+?Rw1PzeW 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a/linear-algebra/exercise/5-similar/similar.tex +++ b/linear-algebra/exercise/5-similar/similar.tex @@ -265,4 +265,24 @@ $\therefore AP=PB$,$P^{-1}AP=B$,$A\sim B$。 \subsection{正交相似} +\textbf{例题:}已知$A$是三阶实对称矩阵,若正交矩阵$Q$使得$Q^{-1}AQ=\left[\begin{array}{ccc} + 3 & 0 & 0 \\ + 0 & 3 & 0 \\ + 0 & 0 & 6 +\end{array}\right]$,如果$\alpha_1=(1,0,-1)^T$和$\alpha_2=(0,1,1)^T$是矩阵$A$属于特征值$\lambda=3$的特征向量,求$Q$。 + +解:首先由正交矩阵就可以知道各特征值正交。令$\alpha_3=(x_1,x_2,x_3)^T$。对应的$\lambda_3=6$。 + +$\alpha_3^T\alpha_1=x_1-x_3=0$,$\alpha_3^T\alpha_2=x_2+x_3=0$,求$\lambda_3$的特征值,则不如令$x_3=1$,则解得$\alpha_3=(1,-1,1)^T$。 + +这样$Q=\left[\begin{array}{ccc} + 1 & 0 & 1 \\ + 0 & 1 & -1 \\ + -1 & 1 & 1 +\end{array}\right]$,还需要将$Q$正交单位化。可知$\alpha_3$根据正交规律求出来,一定是正交的,而$\alpha_1^T\alpha_2=-1\neq0$所以需要正交。 + +令$\beta_1=\alpha_1=(1,0,-1)^T$,$\beta_2=\alpha_2-\dfrac{<\alpha_2,\beta_1>}{<\beta_1,\beta_1>}\beta_1=(0,1,1)^T+\dfrac{1}{2}(1,0,-1)^T=(\dfrac{1}{2},1,\dfrac{1}{2})^T$。 + +最后对整个$Q$进行单位化:$\gamma_1=\dfrac{1}{\sqrt{2}}(1,0,-1)^T$,$\gamma_2=\dfrac{1}{\sqrt{6}}(1,2,1)^T$,$\gamma_3=\dfrac{1}{\sqrt{3}}(1,-1,1)^T$。 + \end{document} diff --git a/linear-algebra/exercise/6-quadratic-form/quadratic-form.pdf b/linear-algebra/exercise/6-quadratic-form/quadratic-form.pdf new file mode 100644 index 0000000000000000000000000000000000000000..760c21d80dc7dc4027e1703e02f539c04fa6e152 GIT binary patch literal 12936 zcmbVy1#nzRvaOiel58TIrg?ad6NInwZ-g*wPA`>)0Fc z8|Yc-8_}?HnEa0410;D=j!2t;CFX*m5a-=G;7BF~l7TIcB zjnjLR5`_pZAEdVFKft~X1WRE)dqR+FXFDUP5V{S>>TnKf<2W=g+@F8oZWuYg>7(#E z0g@lO6=oE4Mqvx0RxJ5i%X<@Acw`1*^`(V9L*&68d?F|W zsse`Nhw~m54nDU;xj+Cm!*%jCPa7~SE zCWrNHBdn0yu*~`kb$S};AI!@jSmo57yx1{>3T}fo@a_UPXpJTcb;aNeEcO4f)!)$X z+yI#W6OsTx`ahDS5K#lytBv6Al(nsy4G8gi2NELrV*c{?`c#COf$RP;pp2*5i;C9S z)`tT&iQ3$2CQa}~>*UgABb9M{bOT4#a1_pJqfK2AIU?r_r(M7tyiuJ<~*{%8oxb{^6E5dsuHwHdW#y1ZsQBMH0q1z%6xE-y;+^*(!oD zMZopN!Qy|i$UyfWvB<*uCl+TWLjCAaKsMW?07EolS54gT#8i#G(~eGM9^OE#wI33^ 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a/linear-algebra/knowledge/5-similar/similar.tex +++ b/linear-algebra/knowledge/5-similar/similar.tex @@ -286,6 +286,18 @@ $\therefore\lambda_1=\lambda_2=1$,$\lambda_3=10$。 所以内积是一个数值。 +单位化是保持向量方向不变,将其长度化为1。 + +正交化是指将线性无关向量系转化为正交系的过程。 + +\subsubsection{施密特正交化} + +将一个线性无关向量组变为一个标准正交向量组。 + +对于线性无关向量组$\alpha_1,\alpha_2,\cdots,\alpha_n$,令$\beta_1=\alpha_1$,$\beta_2=\alpha_2-\dfrac{<\alpha_2,\beta_1>}{<\beta_1,\beta_1>}\beta_1$,$\beta_3=\alpha_3-\dfrac{<\alpha_3,\beta_1>}{<\beta_1,\beta_1}\beta_1-\dfrac{<\alpha_3,\beta_2>}{<\beta_2,\beta_2>}\beta_2$,$\cdots$,$\beta_n=\alpha_n-\dfrac{<\alpha_n,\beta_1>}{<\beta_1,\beta_1>}\beta_1-\dfrac{<\alpha_n,\beta_2>}{<\beta_2,\beta_2>}\beta_2-\cdots-\dfrac{<\alpha_n,\beta_{n-1}}{<\beta_{n-1},\beta_{n-1}>}\beta_{n-1}$。其中$$代表$n,n$的内积。 + +最后单位化:$\gamma_i=\dfrac{\beta_i}{\Vert\beta_i\Vert}$。 + \subsubsection{定义} \textcolor{violet}{\textbf{定义:}}$A^T=A$则$A$就是对称矩阵,若$A$的元素都是实数,则$A$是实对称矩阵。

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