From afdd17a3b23ca241c73d5c884d01aa5c4d15d844 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 21 Jul 2021 23:15:05 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E5=90=91=E9=87=8F=E4=BD=99?= =?UTF-8?q?=E8=A7=A3=E6=9E=90=E5=87=A0=E4=BD=95?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...or-algebra-and-space-analytic-geometry.pdf | Bin 14038 -> 91074 bytes ...or-algebra-and-space-analytic-geometry.tex | 244 +++++++++++++++++- ...ial-calculus-of-multivariate-functions.pdf | Bin 140323 -> 151649 bytes ...ial-calculus-of-multivariate-functions.tex | 82 ++++++ .../8-infinite-series/infinite-series.pdf | Bin 144721 -> 153057 bytes .../8-infinite-series/infinite-series.tex | 36 +++ .../differential-equation.pdf | Bin 124187 -> 125449 bytes .../differential-equation.tex | 10 + 8 files changed, 371 insertions(+), 1 deletion(-) diff --git a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.pdf b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.pdf index 7af0cc5df56a105d0e5e074539a20173d149daba..6d9a5e89a3380cdb7da50424040d9b9553fca9f6 100644 GIT binary patch literal 91074 zcma&NL#%Mowypaw+qP}nwr$(CZQHhO+qP|ct-EjX3h(6H!cInECSz1)dS)A6Z?kn0 zd0|l+23i(KlH>cEcSt5ye0qF4LrX|*ZaPs5YiAQjI#Fu_XA@x)BRgXgI%yMIGiP&r z26`rDUS3EiXGaqQ8%X!fo6H&8Eq3^^yDwzV0Q<8fJpKit0Na2bns}{+Ama(1m-;)u zjRd3;Qk#)tOuBhhaOeriag3S2u?6Uq+dWvWdc12umjN;~CAW7|G~SJ~tEaQ2jkl{Y zxVW7>KLfG-Zn7KPI2^*xAIZ8H-%kwy>9a~OqE18#7Hw2VA55X>9g#$!b|V*gs5B3Q%zxpTc8dH*o)jE#|r zwG8cptDH(e6e1Pg0t9i&JTTM55SKfshnl=$bbjnA%kdgGle1>xZ}`y4Qt*7$ZR~-j zP1~slB23G*=o3SQ_Sr~|a{=f!sKA~)6rO_$1Qe zc!BqvQHU&92+E*}^X|F=_&5`kK;a#Jfq}HMCnf8C+|W6?GMO6^&PRJwXmd0RdXkj& z%J0IhOh&b}goMc#Fef}smQ3u4EdSxeXs>%Ra&y6$BLO!!C(_Ql({R{37ef&`(dHo- z^k)j=pI=|t8@!oX`)YY&s*@LQ{}WP_Vm1H?z4{;wGMGTCp+9zYL*M-uDmdwizD|ky z;tLb{OA2i~%RkqE5IT4ZI}|q*AGQp40|MNnl2c6r?c)LxFqABNrdmB$^7puND#7?t z0T^%YR`f8K)}z}U&jmFP(^1FDj6E(Q^8sDQ!VC?hPPV2+mR6ngv^wEYep#T4nN`JF zg^4A5m^?T&emM$SVun!@!GMn8Zcc9Jl;!2)MVAa^vBi{@@0RWjASQChWCv6)JqBZc z%<9H|3oJ%p(+2 z$0GY&&uuEKUwottBoz)zp33`ej_xNoj|n_?vPtMJEXGe2sMg#!MN~vUH7y&q_?(NZ z*)o^1mERMnk_WKX+_!T>8`Z7zwf$V{Q}i0*sMSnH#kls!UG{0h_ttWQPHg3GNWT3( z^2UyN%o7J$_KGPtqGc-{(!FLfW7&OjLmVsI)D6R;YwM)9_-051 zy~48l=2o!XIFYfn!^W`(7)o87r62yvkJ<+)B=VzX3XN{Fj}AZvTjo>hF29Uxw}-;O z1wi5RDG~JpiqN#RlX%;sLqx6R87p()i&{mnbH@7S>_^7_NN=ZckXxe~?Fb!s=SM|O9JS$%Hs`1WhQKu?SS!1sE4VxUvk69p46)W+ zcYaGYC^>@xqfBXYTi8c~sA11x%TE~c@uyRVq_FUOWRESpWFW-K-q!83Q2Figyc(hx z`}h5T7p4=7329<${Qrr8}26sGkd7{}-H42>LW};FSi?`f>Bk zis>Ei?hR#|R}|mzlt*XN&TO>nX7MRMkIoQQUz!u&CXrE8cWdDNH4FcbO?2Mf(O*fcX*XDgxApi@oBYyeFX z3SMo%xm7%}bhS|E_Ji%1aN3gN(X)IMT@psX;aulI=W*uMA3Xo%K+I}u6| zotAaTKI!RnNZuR+PA$OX_ywdI-YAz&burO&ev##DfiYFny#taiWUmj=X*^6knGqq<2vkb9(S2xNhAI%(aEO23cy`*5d=!G%p+rDC{Zn*pD`~F~9 zaQYT*&05@x1Rbw;WSg3%%?mjCZ7t>p#Z~}{_4Q?JeqQfJPhnwC2iyO!c=+MKOowo@ zWb8>p=W}gCvRy~2?0k6FK$5JHX*VUb3?o;y;3Q(9mx#vkLsI#|LB&6s^Z}#A#^tEv zFxT)il1s^KYFfH?ju>zz^1!_#1vBpCT4{(Xh8gt>!mlWJpZ1(LdYjDXXBah?66jJ3$wBo48vJ=pyuhBcc?szylw|4W&N(32K z*{`9GZnM5(jC#G?-1Ii!ht)ff4(id?MVCiWokiiBo(uVh2q|-3lx*UJL*NKqN5lWjh3f2lL=EXTMK3_1gh<*k{=7mzEIvSa~~ zB*0xs!woL)`7D9Kki6?|_f%KN!ob?Y$KhNEi`G|^2yLjtz4eF`zf;>^w-7(n*}(Oc zoAT2UEQVd!u$b*(A>IDE>)8GAvr8&~+*e_^9?m78>z#)T+E!td>H!S=$~%Z+3=j!e zX)&Z9uqXH3t99HR6jGP!*nVBvdJEwgKMR)f*dHqis}uM)qg_|^5T3KM_G*1aMAaH~ zIl3^9D2aMiIaY91oUyelvOjQkC{OnPvl1Qq+Xbv0rRH~TtjI`>7__Uyd}72p($MQx zhJ0l$7j!DZQn-A*dH9`6+CD$M<*E_N+7Ca;`Si&oslfw?n>xyl`>ETn{&@v-f}wfx zt2jMwp!Oui={_tkkMV_UYZOb1etWug5_-D41}ggOEO-rhwrR@w!IJV6t{96T!G|M? 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a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex index 16c2d42..b895f74 100644 --- a/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex +++ b/advanced-math/knowledge/5-vector-algebra-and-space-analytic-geometry/vector-algebra-and-space-analytic-geometry.tex @@ -2,6 +2,9 @@ % UTF8编码,ctexart现实中文 \usepackage{color} % 使用颜色 +\definecolor{orange}{RGB}{255,127,0} +\definecolor{violet}{RGB}{192,0,255} +\definecolor{aqua}{RGB}{0,255,255} \usepackage{geometry} \setcounter{tocdepth}{4} \setcounter{secnumdepth}{4} @@ -34,6 +37,245 @@ \newpage \pagestyle{plain} \setcounter{page}{1} -\section{} + +该部分的内容服务于后面的多元函数积分学。 + +\section{向量代数} + +\subsection{向量及其表达形式} + +\textcolor{violet}{\textbf{定义:}}既有方向又有大小的向量称为\textbf{向量}。 + +向量的相等性体现在大小和方向,与空间位置无关。 + +向量表达形式为$\vec{a}=(a_x,a_y,a_z)=a_x\vec{i}+a_y\vec{j}+a_z\vec{k}$。 + +\subsection{向量运算} + +设$\vec{a}=(a_x,a_y,a_z)$,$\vec{b}=(b_x,b_y,b_z)$,$\vec{c}=(c_x,c_y,c_z)$,$\vec{a},\vec{b},\vec{c}$均不是零向量。 + +\subsubsection{数量积} + +称为内积或点积。 + +\begin{itemize} + \item $\vec{a}\cdot\vec{b}=(a_x,a_y,a_z)\cdot(b_x,b_y,b_z)=a_xb_x+a_yb_y+a_zb_z$。 + \item $\vec{a}\cdot\vec{b}=\vert a\vert\vert b\vert\cos\theta$,则$\cos\theta=\dfrac{a\cdot b}{\vert a\vert\vert b\vert}=\dfrac{a_xb_x+a_yb_y+a_zb_z}{\sqrt{a_x^2+a_y^2+a_z^2}\cdot\sqrt{b_x^2+b_y^2+b_z^2}}$,其中$\theta$为$\vec{a},\vec{b}$夹角。 + \item $a\bot b\Leftrightarrow\theta=\dfrac{\pi}{2}\Leftrightarrow a\cdot b=\vert a\vert\vert b\vert\cos\theta=0\Leftrightarrow a_xb_x+a_yb_y+a_zb_z=0$。 + \item $Prj_ba=\dfrac{a\cdot b}{\vert b\vert}=\dfrac{a_xb_x+a_yb_y+a_zb_z}{\sqrt{b_x^2+b_y^2+b_z^2}}$称$a$在$b$上的\textbf{投影}。 +\end{itemize} + +\subsubsection{向量积} + +也称为外积、叉积。 + +\begin{itemize} + \item $\vec{a}\times\vec{b}=\left\vert\begin{array}{ccc} + \vec{i} & \vec{j} & \vec{k} \\ + a_x & a_y & a_z \\ + b_x & b_y & b_z + \end{array}\right\vert$,其中$\vert a\times b\vert=\vert a\vert\vert b\vert\sin\theta$,用右手规则确定方向(转向角不超过$\pi$),其中$\theta$为$\vec{a},\vec{b}$夹角。 + \item $a//b\Leftrightarrow\theta=0$或$\pi\Leftrightarrow\dfrac{a_x}{b_x}=\dfrac{a_y}{b_y}=\dfrac{a_z}{b_z}$。 +\end{itemize} + +\subsubsection{混合积} + +\begin{itemize} + \item $[\vec{a}\vec{b}\vec{c}]=(a\times b)\cdot c=\left\vert\begin{array}{ccc} + a_x & a_y & a_z \\ + b_x & b_y & b_z \\ + c_x & c_y & c_z \\ + \end{array}\right\vert$。 + \item $\left\vert\begin{array}{ccc} + a_x & a_y & a_z \\ + b_x & b_y & b_z \\ + c_x & c_y & c_z \\ + \end{array}\right\vert=0\Leftrightarrow$三向量共面。 +\end{itemize} + +\subsection{向量方向角与方向余项} + +\begin{itemize} + \item $\vec{a}$与$x$轴、$y$轴、$z$轴正向的夹角$\alpha$、$\beta$、$\gamma$为$\vec{a}$的\textbf{方向角}。 + \item $\cos\alpha$,$\cos\beta$,$\cos\gamma$称为$\vec{a}$的\textbf{方向余弦},且$\cos\alpha=\dfrac{a_x}{\vert\vec{a}\vert}$,$\cos\beta=\dfrac{a_y}{\vert\vec{a}\vert}$,$\cos\gamma=\dfrac{a_z}{\vert\vec{a}\vert}$。 + \item $a^\circ=\dfrac{\vec{a}}{\vert\vec{a}\vert}=(\cos\alpha,\cos\beta,\cos\gamma)$称为向量$\vec{a}$的\textbf{单位向量}(表示方向的向量)。 + \item 任意向量$\vec{r}=x\vec{i}+y\vec{j}+z\vec{k}=(r\cos\alpha,r\cos\beta,r\cos\gamma)=r(\cos\alpha,\cos\beta,\cos\gamma)$,$\cos\alpha$,$\cos\beta$,$\cos\gamma$称为$\vec{r}$的方向余弦,$r$为$\vec{r}$的模,$\cos\alpha=\dfrac{x}{\sqrt{x^2+y^2+z^2}}$,$\cos\beta=\dfrac{y}{\sqrt{x^2+y^2+z^2}}$,$\cos\gamma=\dfrac{z}{\sqrt{x^2+y^2+z^2}}$,$r=\sqrt{x^2+y^2+z^2}$。 +\end{itemize} + +\section{空间解析几何} + +\subsection{平面方程} + +假设平面的法向量$\vec{n}=(A,B,C)$。 + +\begin{itemize} + \item 一般式:$Ax+By+Cz+D=0$。 + \item 点法式:$A(x-x_0)+B(y-y_0)+C(z-z_0)=0$。 + \item 三点式:$\left\vert\begin{array}{ccc} + x-x_1 & y-y_1 & z-z_1 \\ + x-x_2 & y-y_2 & z-z_2 \\ + x-x_3 & y-y_3 & z-z_3 \\ + \end{array}\right\vert=0$(平面过不共线的三点)。 + \item 截距式:$\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1$(平面过$(a,0,0)$,$(0,b,0)$,$(0,0,c)$三点)。 +\end{itemize} + +\subsection{直线方程} + +假设直线的方向向量$\vec{\tau}=(l,m,n)$。 + +\begin{itemize} + \item 一般式:$\left\{\begin{array}{l} + A_1x+B_1y+C_1z+D_1=0,\vec{n}_1=(A_1,B_1,C_1) \\ + A_2x+B_2y+C_2z+D_2=0,\vec{n}_2=(A_2,B_2,C_2) + \end{array}\right.$,其中$\vec{n}_1$不平行于$\vec{n}_2$。(两个平面的交线,该直线方向向量$\vec{\tau}=n_1\times n_2$) + \item 点向式(标准式、对称式):$\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n}$。(直线上一点与方向向量成比例) + \item 参数式:$\left\{\begin{array}{l} + x=x_0+lt \\ + y=y_0+mt \\ + z=z_0+nt + \end{array}\right.$,$M(x_0,y_0,z_0)$为直线上已知点,$t$为参数。 + \item 两点式:$\dfrac{x-x_1}{x_2-x_1}=\dfrac{y-y_1}{y_2-y_1}=\dfrac{z-z_1}{z_2-z_1}$。(直线过不同的两点) +\end{itemize} + +\subsection{位置关系} + +\subsubsection{点到平面距离} + +点$P_0(x_0,y_0,z_0)$到平面$Ax+By+Cz+D=0$的距离为$d=$\\$\dfrac{\vert Ax_0+By_0+Cz_0+D\vert}{\sqrt{A^2+B^2+C^2}}$。 + +\subsubsection{直线关系} + +设$\vec{\tau}_1=(l_1,m_1,n_1)$,$\vec{\tau}_2=(l_2,m_2,n_2)$分别为直线$L_1$,$L_2$的方向向量。 + +\begin{itemize} + \item $L_1\bot L_2\Leftrightarrow\vec{\tau}_1\bot\vec{\tau}_2\Leftrightarrow l_1l_2+m_1m_2+n_1n_2=0$。 + \item $L_1//L_2\Leftrightarrow\vec{\tau}_1//\vec{\tau}_2\Leftrightarrow\dfrac{l_1}{l_2}\Leftrightarrow\dfrac{m_1}{m_2}=\dfrac{n_1}{n_2}$。 +\end{itemize} + +\subsubsection{平面关系} + +设平面$\pi_1$,$\pi_2$的法向量分别为$\vec{n}_1=(A_1,B_1,C_1)$,$\vec{n}_2=(A_2,B_2,C_2)$。 + +\begin{itemize} + \item $\pi_1\bot\pi_2\Leftrightarrow\vec{n}_1\bot\vec{n}_2\Leftrightarrow A_1A_2+B_1B_2+C_1C_2=0$。 + \item $\pi_1//\pi_2\Leftrightarrow\vec{n}_1//\vec{n}_2\Leftrightarrow\dfrac{A_1}{A_2}=\dfrac{B_1}{B_2}=\dfrac{C_1}{C_2}$。 +\end{itemize} + +\subsubsection{直线与平面关系} + +设直线$L$的方向向量为$\tau=(l,m,n)$,平面$\vec{\tau}$的法向量为$\vec{n}=(A,B,C)$。 + +\begin{itemize} + \item $L\bot\pi\Leftrightarrow\vec{\tau}//\vec{n}\Leftrightarrow\dfrac{A}{l}=\dfrac{B}{m}=\dfrac{C}{n}$。 + \item $L//\pi\Leftrightarrow\vec{\tau}\bot\vec{n}\Leftrightarrow Al+Bm+Cn=0$。 +\end{itemize} + +\subsection{空间曲线} + +\subsubsection{表达式} + +\begin{itemize} + \item 一般式:$\varGamma:\left\{\begin{array}{l} + F(x,y,z)=0 \\ + G(x,y,z)=0 + \end{array}\right.$,表示两个曲面的交线。 + \item 参数方程:$\varGamma:\left\{\begin{array}{l} + x=\phi(t) \\ + y=\psi(t) \\ + z=\omega(t) + \end{array}\right.$,$t\in[\alpha,\beta]$。 +\end{itemize} + +\subsubsection{空间曲线在坐标面投影} + +如求曲线$\varOmega$在$xOy$平面上的投影曲线,讲$\varGamma:\left\{\begin{array}{l} + F(x,y,z)=0 \\ + G(x,y,z)=0 +\end{array}\right.$中的$z$消去,得到$\varphi(x,y)=0$,则曲线$\varOmega$在$xOy$面上的投影曲线包含于$\left\{\begin{array}{l} + \varphi(x,y)=0 \\ + z=0 +\end{array}\right.$。 + +\subsection{空间曲面} + +\subsubsection{曲面方程} + +隐式表达式:$F(x,y,z)=0$,显式表达式:$z=z(x,y)$。 + +\subsubsection{二次曲面} + +\begin{itemize} + \item 球面:$x^2+y^2+z^2=r^2$。 + \item 椭球面:$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$。 + \item 单叶双曲面:$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1$。 + \item 双叶双曲面:$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1$。 + \item 椭圆抛物面:$\dfrac{x^2}{2p}+\dfrac{y^2}{2q}=z$($p,q>0$)。(常考旋转抛物面$x^2+y^2=z$) + \item 椭圆锥面:$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\dfrac{z^2}{c^2}$。(常考旋转锥面$z=\sqrt{x^2+y^2}$) + \item 双曲抛物面(马鞍面):$-\dfrac{x^2}{2p}+\dfrac{y^2}{2q}=z$($p,q>0$)。(可能考$z=xy$) +\end{itemize} + +\subsubsection{柱面} + +空间解析几何中一般认为缺少变量的方程为柱面。 + +是动直线沿定曲线平行移动所形成的曲面。 + +\begin{itemize} + \item 椭圆柱面:$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$(当$a=b$为圆柱面)。 + \item 双曲柱面:$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$。 + \item 抛物柱面:$y=ax^2$。 +\end{itemize} + +\subsubsection{旋转曲面} + +绕某轴转,其就不变,把另外一个字母写成另外两个字母的平方和的开方。 + +是曲线$\varGamma$绕一条定直线旋转一周所形成的曲面。 + +给定一条直线$L:\dfrac{x-x_0}{l}=\dfrac{y-y_0}{m}=\dfrac{z-z_0}{n}$,其方向向量为$\vec{\tau}(l,m,n)$,上有一点$P_0(x_0,y_0,z_0)$。 + +现在给定一条曲线$\varGamma:\left\{\begin{array}{l} + F(x,y,z)=0 \\ + G(x,y,z)=0 +\end{array}\right.$。 + +在$\varGamma$上找一点$P_1(x_1,y_1,z_1)$,然后再讲$P_1$绕$L$旋转一周得到一个纬圆,去纬圆上一点$P(x,y,z)$,则$P$为旋转曲面上任意一点。 + +因为$P_1$在曲线$\varGamma$上,所以$F(x_1,y_1,z_1)=0$,$G(x_1,y_1,z_1)=0$。 + +同一个纬圆到$L$上的$P_0$距离相等,既$\vert\overrightarrow{P_1P_0}\vert=\vert\overrightarrow{PP_0}\vert$,即$(x_1-x_0)^2+(y_1-y_0)^2+(z_1-z_0)^2=(x-x_0)^2+(y-y_0)^2+(z-z_0)^2$。 + +每一个纬圆的平面与旋转中心$L$的方向向量$\vec{\tau}$垂直,而$P_1P$在平面上,所以该连线向量$\overrightarrow{P_1P}\bot\vec{\tau}$,即$l(x-x_1)+m(y-y_1)+n(z-z_1)=0$。 + +为了得到旋转曲面面积,需要消去$x_1,y_1,z_1$,得到$H(x,y,z)=0$。 + +\textbf{例题:}求曲线$L:\left\{\begin{array}{l} + x-y+2z-1=0 \\ + x-3y-2z+1=0 +\end{array}\right.$绕$y$轴旋转一周所形成的曲面方程。 + +解:令$P_1(x_1,y_1,z_1)$在曲线上,所以$x_1-y_1+2z_1-1=0$,$x_1-3y_1-2z_1+1=0$。 + +然后任意一点$P(x,y,z)$到$P_0(x_0,y_0,z_0)$的距离与$P_1$到$P_0$距离相同,取$P_0(0,0,0)$,则$x_1^2+y_1^2+z_1^2=x^2+y^2+z^2$。 + +两条连线垂直$y$轴,即$\overrightarrow{P_1P}\bot(0,1,0)$,即$y=y_1$。 + +消去$x_1,y_1,z_1$,根据$y=y_1$,所以$x_1^2+z_1^2=x^2+z^2$。 + +根据交线方程解得$x_1=2y$,$z_1=\dfrac{1}{2}(1-y)$。 + +再代入得到$x^2+z^2=(2y)^2+\dfrac{1}{4}(1-y)^2$,解得$x^2-\dfrac{17}{4}y^2+z^2+\dfrac{y}{2}-\dfrac{1}{4}=0$。 + +\section{场论初步} + +\subsection{方向导数} + +\subsection{梯度} + +\subsection{方向导数与梯度关系} + +\subsection{散度} + +\subsection{旋度} \end{document} diff --git a/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf b/advanced-math/knowledge/6-differential-calculus-of-multivariate-functions/differential-calculus-of-multivariate-functions.pdf index 3ae113fbdaba083f5dc95f8938e81f24e3366404..fccffc3dd79673f1f95385d7ff3198bd589428ac 100644 GIT binary patch delta 112824 zcmZsBQO;^fK5hs?l@xKTI; zwVwx#Or|I*PRB&g219mqcl`##!WxfE3gBX8=jVrUc5yN_w1x55xXzf2IATW&dG&

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+设空间曲线$\varGamma$由参数方程$\left\{\begin{array}{l} + x=\phi(t) \\ + y=\psi(t) \\ + z=\omega(t) +\end{array}\right.$给出,其中$\phi(t),\psi(t),\omega(t)$均可导,$P_0(x_0,y_0,z_0)$为$\varOmega$上的点,且当$t=t_0$时,$\phi'(t_0)$,$\psi'(t_0)$,$\omega'(t_0)$均不为0,则: + +\begin{itemize} + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切向量为$\vec{\tau}=(\phi'(t_0),\psi'(t_0),\omega'(t_0))$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切线方程为$\dfrac{x-x_0}{\phi'(t_0)}=\dfrac{y-y_0}{\psi'(t_0)}=\dfrac{z-z_0}{\omega'(t_0)}$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的法平面(过$P_0$且与切线垂直的平面)方程为$\phi'(t_0)(x-x_0)+\psi'(t_0)(y-y_0)+\omega'(t_0)(z-z_0)=0$。 +\end{itemize} + +\subsubsection{交面式方程} + +设空间曲线$\varGamma$由交面方程$\left\{\begin{array}{l} + F(x,y,z)=0 \\ + G(x,y,z)=0 +\end{array}\right.$给出,则: + +\begin{itemize} + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切向量为\\$\vec{\tau}=\left(\left\vert\begin{array}{cc} + F_y' & F_z' \\ + G_y' & G_z' + \end{array}\right\vert_{P_0},\left\vert\begin{array}{ll} + F_z' & F_x' \\ + G_z' & G_x' + \end{array}\right\vert_{P_0},\left\vert\begin{array}{ll} + F_x' & F_y' \\ + G_x' & G_y' + \end{array}\right\vert_{P_0}\right)$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的切线方程为\\$\dfrac{x-x_0}{\left\vert\begin{array}{cc} + F_y' & F_z' \\ + G_y' & G_z' + \end{array}\right\vert_{P_0}},\dfrac{y-y_0}{\left\vert\begin{array}{ll} + F_z' & F_x' \\ + G_z' & G_x' + \end{array}\right\vert_{P_0}},\dfrac{z-z_0}{\left\vert\begin{array}{ll} + F_x' & F_y' \\ + G_x' & G_y' + \end{array}\right\vert_{P_0}}$。 + \item 曲线$\varGamma$在点$P_0(x_0,y_0,z_0)$处的法平面方程为\\$\left\vert\begin{array}{cc} + F_y' & F_z' \\ + G_y' & G_z' + \end{array}\right\vert_{P_0}(x-x_0)+\left\vert\begin{array}{ll} + F_z' & F_x' \\ + G_z' & G_x' + \end{array}\right\vert_{P_0}(y-y_0)+\left\vert\begin{array}{ll} + F_x' & F_y' \\ + G_x' & G_y' + \end{array}\right\vert_{P_0}(z-z_0)=0$。 +\end{itemize} + +\subsection{空间曲面的切平面与法线} + +\subsubsection{隐式} + +设空间曲面$\varSigma$由方程$F(x,y,z)=0$给出,$P_0(x_0,y_0,z_0)$是$\varSigma$上的点,则: + +\begin{itemize} + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的法向量为$\vec{n}=(F_x'(x_0,y_0,z_0),F_y'(x_0,y_0,z_0),$\\$F_z'(x_0,y_0,z_0))$且法线方程为$\dfrac{x-x_0}{F_x'(x_0,y_0,z_0)}=\dfrac{y-y_0}{F_y'(x_0,y_0,z_0)}=\dfrac{z-z_0}{F_z'(x_0,y_0,z_0}$。 + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的切平面方程为$F_x'(x_0,y_0,z_0)(x-x_0)+F_y'$\\$(x_0,y_0,z_0)(y-y_0)+F_z'(x_0,y_0,z_0)(z-z_0)=0$。 +\end{itemize} + +\subsubsection{显式} + +设空间曲面$\varSigma$由方程$z=f(x,y)$给出,令$F(x,y,z)=f(x,y)-z$,假定法向量的方向向下,即其余$z$轴正向所成的角为钝角,即$z$为-1,则: + +\begin{itemize} + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的法向量为$\vec{n}=(f_x'(x_0,y_0),f_y'(x_0,y_0),-1)$,且法线方程为$\dfrac{x-x_0}{f_x'(x_0,y_0)}=\dfrac{y-y_0}{f_y'(x_0,y_0)}=\dfrac{z-z_0}{-1}$。 + \item 曲面$\varSigma$在点$P_0(x_0,y_0,z_0)$处的切平面方程为$f_x'(x_0,y_0)(x-x_0)+f_y'(x_0,y_0)$\\$(y-y_0)-(z-z_0)=0$。 +\end{itemize} + +若是反之成锐角,则将里面所有的-1都换成1。 + +若用$\alpha$,$\beta$,$\gamma$表示曲面$z=f(x,y)$在点$(x_0,y_0,z_0)$处的法向量的方向角,并这里假定法向量的方向是向上的,即其余$z$轴正向所成的角$\gamma$为锐角,则法向量\textbf{方向余弦}为$\cos\alpha=\dfrac{-f_x}{\sqrt{1+f_x^2+f_y^2}}$,$\cos\beta=\dfrac{-f_y}{\sqrt{1+f_x^2+f_y^2}}$,$\cos\gamma=\dfrac{1}{\sqrt{1+f_x^2+f_y^2}}$,其中$f_x=f_x'(x_0,y_0)$,$f_y=f_y'(x_0,y_0)$。 + \end{document} diff --git a/advanced-math/knowledge/8-infinite-series/infinite-series.pdf b/advanced-math/knowledge/8-infinite-series/infinite-series.pdf index 63408753f8206ec8acccb7c63f3fee41cebefbc6..184a882100f0f038abc29a2961877bc5a8df7c34 100644 GIT binary patch delta 117045 zcmY(qV{|6K(lwf7V%xTD+nm_8?L4tg zxvd{j&fPyg>;%2sK^BcKCNnuuGw56qq;!U#{N5SfaJ`2hr`a&LoI7?)_~2w<{hlt!gagCe%Sx zEQLHB^XqK5A(^R4)nkn?f7gmV;0}=GXmm&FPQ8qu!1s;pd?7mr5Ri4z@$9bFWXpNz z!MAKOPVGLxtqoMTYO*(8ldG;l1TJkEX9D2lb+X=y(z|ZNz_|~t>S0UQaQJbK*>p2p zb(}yWufC#!Q$2PG3?^ZkiUMa7^qf~T;gGk*I+m2gWvBzBP*QOTj@0Vo0i+zi%J}d3 zk1N103xT^$Ad9_$p5)08BtJpJq2_8^=3nqEe7V1kHV 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a/advanced-math/knowledge/8-infinite-series/infinite-series.tex b/advanced-math/knowledge/8-infinite-series/infinite-series.tex index 5e2a8fe..f8bdb03 100644 --- a/advanced-math/knowledge/8-infinite-series/infinite-series.tex +++ b/advanced-math/knowledge/8-infinite-series/infinite-series.tex @@ -427,4 +427,40 @@ $x$的取值指其幂指数的收敛域。第七个幂函数问题较复杂, \section{傅里叶级数} +\subsection{* 三角级数} + +\textcolor{violet}{\textbf{定义:}}将正弦函数$A_n\sin(n\omega t+\varphi_n)$按三角公式变形得到$A_n\sin\varphi_n\cos n\omega t+A_n\cos\varphi_n\sin n\omega t$,令$\dfrac{a_0}{2}=A_0$,$a_n=A\sin\varphi_n$,$b_n=A_n\cos\varphi_n$,$\omega=\dfrac{\pi}{l}$,则$\dfrac{a_0}{2}+\sum\limits_{n=1}^\infty\left(a_n\cos\dfrac{n\pi t}{l}+b_n\sin\dfrac{n\pi t}{l}\right)$。这个级数就是\textbf{三角级数}。 + +\subsection{函数展开为傅里叶级数} + +\textcolor{violet}{\textbf{定义:}}设$f(x)$在$[-l,l]$上连续或只有有限个第一类间断点,且至多只有有限个真正的极值点,则$f(x)$的傅里叶级数处处收敛,记起和函数为$S(x)$,有$S(x)=\dfrac{a_0}{2}+\sum\limits_{n=1}^\infty\left(a_n\cos\dfrac{n\pi}{l}x+b_n\sin\dfrac{n\pi}{l}x\right)$。 + +$\displaystyle{a_n=\dfrac{1}{l}\int_{-l}^lf(x)\cos\dfrac{n\pi}{l}x\,\textrm{d}x}$,$\displaystyle{b_n=\dfrac{1}{l}\int_{-l}^lf(x)\sin\dfrac{n\pi}{l}x\,\textrm{d}x}$,$n=1,2,\cdots$。 + +其中三角函数也可以展开为幂级数,所以最后都能通过幂级数展开。 + +\textbf{例题:}将$f(x)=1-x^2$($-\pi\leqslant x\leqslant\pi$)展开为傅里叶级数。 + +解:$S(x)=\dfrac{a_0}{2}+\sum\limits_{n=1}^\infty\left(a_n\cos\dfrac{n\pi}{l}x+b_n\sin\dfrac{n\pi}{l}x\right)$。 + +其中$a_0=\dfrac{1}{\pi}\int\limits_{-\pi}^\pi(1-x^2)\,\textrm{d}x=\dfrac{2}{\pi}\int\limits_0^\pi(1-x^2)\,\textrm{d}x=2-\dfrac{2}{3}\pi^2$。 + +$a_n=\dfrac{1}{\pi}\int\limits_{-\pi}^\pi(1-x^2)\cos nx\,\textrm{d}x=\dfrac{2}{\pi}(\int\limits_0^\pi\cos nx\,\textrm{d}x-\int\limits_0^\pi x^2\cos nx\,\textrm{d}x)=\dfrac{4}{n^2}(-1)^{n+1}$。 + +$b_n=\dfrac{1}{\pi}\int\limits_{-\pi}^\pi(1-x^2)\sin nx\,\textrm{d}x=0$(奇函数乘偶函数为奇函数,且上下限对称) + +$f(x)\sim S(x)=1-\dfrac{\pi^2}{3}+\sum\limits_{n=1}^\infty\dfrac{4}{n^2}(-1)^{n+1}\cos nx$。 + +\textcolor{violet}{\textbf{定义:}}当$f(x)$是偶函数,则$\sin$被消去,$f(x)\sim S(x)=\dfrac{a_0}{2}+\sum\limits_{n=1}^\infty a_n\cos\dfrac{n\pi}{l}x$,称为\textbf{余弦级数}。 + +\textcolor{violet}{\textbf{定义:}}当$f(x)$是奇函数,则$\cos$被消去,$f(x)\sim S(x)=\dfrac{a_0}{2}+\sum\limits_{n=1}^\infty b_n\sin\dfrac{n\pi}{l}x$,称为\textbf{正弦级数}。 + +若$f(x)$因为定义区间不对称导致无奇偶性,则补充定义域,使其称为奇偶函数。 + +迪利克雷定理\textcolor{violet}{\textbf{定义:}}$f(x)\sim S(x)=\left\{\begin{array}{ll} + f(x), & x\text{为连续点} \\ + \dfrac{f(x-0)+f(x+0)}{2}, & x\text{为间断点} \\ + \dfrac{f(-l+0)+f(l-0)}{2}, & x=\pm l +\end{array}\right.$ + \end{document} diff --git a/advanced-math/knowledge/9-differential-equation/differential-equation.pdf b/advanced-math/knowledge/9-differential-equation/differential-equation.pdf index 265834ac23b3b6ac0eabd97b2d07e6e9743afca4..41236b93f79b85eef47bbb1541feca090beb660b 100644 GIT binary patch delta 75735 zcmY(qV|OlG)3h7gwr$(Cv0~eH&Sb^5ZQHhO+qPD;ulN4+?7wh~(X+a%YGzGBMl3_t zH`oGLnONBb1YlfToz0BwU_3WhbS7d+$B=tIX+5(;2UJEAvvw|n;B^8NQ~5cyAQR8UI1n4StP_Pj}7JbDj6~uD6zc1ekQl?YKi{Y9DCKmyPZnVr-_fDYSB3g`>fM8k+ 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znnkn&sn+z`@xMCG-HJGuMqN{XPF8h1fEd(w;+k2UUj?1kaT;<9F9X9_Xpk!)YWFA#GB-F`}B0$时,令$x=e^t$,则$t=\ln x$,$\dfrac{\textrm{d}t}{\textrm{d}x}=\dfrac{1}{x}$,$\dfrac{\textrm{d}y}{\textrm{d}x}=\dfrac{\textrm{d}y}{\textrm{d}t}\dfrac{\textrm{d}t}{\textrm{d}x}=\dfrac{1}{x}\dfrac{\textrm{d}y}{\textrm{d}t}$,$\dfrac{\textrm{d}^2y}{\textrm{d}x^2}=\dfrac{\textrm{d}}{\textrm{d}x}\left(\dfrac{1}{x}\dfrac{\textrm{d}y}{\textrm{d}t}\right)=-\dfrac{1}{x^2}\dfrac{\textrm{d}y}{\textrm{d}t}+\dfrac{1}{x}\dfrac{\textrm{d}}{\textrm{d}x}\left(\dfrac{\textrm{d}y}{\textrm{d}t}\right)=-\dfrac{1}{x^2}\dfrac{\textrm{d}y}{\textrm{d}t}+\dfrac{1}{x^2}\dfrac{\textrm{d}^2y}{\textrm{d}t^2}$,方程化为$\dfrac{\textrm{d}^2y}{\textrm{d}t^2}+(p-1)\dfrac{\textrm{d}y}{\textrm{d}t}+qy=f(e^t)$,解出结果,组后用$t=\ln x$回代。 当$x<0$是,令$x=-e^t$,同理可得。 +\textbf{例题:}求欧拉方程$x^2\dfrac{\textrm{d}^2y}{\textrm{d}x^2}+4x\dfrac{\textrm{d}y}{\textrm{d}x}+2y=0$($x>0$)的通解。 + +解:可以直接利用公式,变为$\dfrac{\textrm{d}^2y}{\textrm{d}t^2}+3\dfrac{\textrm{d}y}{\textrm{d}t}+2y=0$ + +即$y''+3y'+2y=0$,特征方程$\lambda^2+3\lambda+2=0$,$\lambda_1=-1$,$\lambda_2=-2$。 + +$\therefore y=C_1e^{-x}+C_2e^{-2x}$。代入$x=e^t$,$y=\dfrac{C_1}{x}+\dfrac{C_2}{x^2}$。 + \end{document}