\documentclass[11pt ]{article} \setlength{\textwidth}{14 true cm} \setlength{\textheight}{20 true cm} \usepackage{graphicx} \usepackage{hangul} \usepackage{amscd,amsmath} \usepackage{amsfonts} \usepackage{amssymb,theorem} \usepackage{longtable} \newcommand{\we}{\wedge} \newcommand{\vts}{v_1\otimes\cdot\cdot\cdot\ot v_k} \newcommand{\vs}{V_{1}\times\cdot\cdot\cdot\times V_k} \newcommand{\ot}{\otimes} \newcommand{\pauo}{\frac{\partial}{\partial u_1}} \newcommand{\paui}{\frac{\partial}{\partial u_i}} \newcommand{\pauj}{\frac{\partial}{\partial u_j}} \newcommand{\wid}{\widetilde} \newcommand{\ml}{\mathcal{L}} \newcommand{\hs}{\hspace} \newcommand{\inv}{^{-1}} \newcommand{\vphi}{\varphi} \newcommand{\pax}{\frac{\partial}{\partial x}} \newcommand{\pay}{\frac{\partial}{\partial y}} \newcommand{\paz}{\frac{\partial}{\partial z}} \newcommand{\paxi}{\frac{\partial}{\partial x_i}} \newcommand{\payj}{\frac{\partial}{\partial y_j}} \newcommand{\paxj}{\frac{\partial}{\partial x_j}} \newcommand{\payi}{\frac{\partial}{\partial y_i}} \newcommand{\li}{[X,Y]} \newcommand{\dis}{\displaystyle} \newcommand{\disi}{\displaystyle{\sum_{i=1}^n}} \newcommand{\disj}{\displaystyle{\sum_{j=1}^n}} \newcommand{\pa}{\partial} \newcommand{\Aff}{\mbox{\it Aff}} \newcommand{\aff}{\mbox{\it aff}} \newcommand{\cc}{\mathcal{C}} \newcommand{\dc}{\mathcal{D}} \newcommand{\fc}{\mathcal{F}} \newcommand{\h}{\hspace{1em}} \newcommand{\hh}{\hspace{2em}} \newcommand{\hhh}{\hspace{3em}} \newcommand{\hhhh}{\hspace{4em}} \newcommand{\ccn}{\mathcal{C}^{\infty}} \newcommand{\sbb}{\mathbb{S}} \newcommand{\rb}{\mathbb{R}} \newcommand{\cb}{\mathbb{C}} \newcommand{\hb}{\mathbb{H}} \newcommand{\rc}{\mathcal{R}} \newcommand{\alp}{\alpha} \newcommand{\bet}{\beta} \newcommand{\del}{\delta} \newcommand{\gam}{\gamma} \newcommand{\vep}{\varepsilon} \newcommand{\eps}{\epsilon} \newcommand{\lam}{\lambda} \newcommand{\kap}{\kappa} \newcommand{\sig}{\sigma} \newcommand{\om}{\omega} \newcommand{\Gam}{\Gamma} \newcommand{\Ome}{\Omega} \newcommand{\Sig}{\Sigma} \newcommand{\Del}{\Delta} \newcommand{\Lam}{\Lambda} \newcommand{\pr}{(\,\,,\,\,)} \newtheorem{thm}{정리} \newtheorem{cor}[thm]{따름정리} \newtheorem{lem}[thm]{보조정리} \newtheorem{prop}[thm]{명제} \newtheorem{cl}{Claim} {\theorembodyfont{\rm} \newtheorem{ex}{예} \newtheorem{que}{질문} \newtheorem{notation}{Notation}[section] \newtheorem{defn}{정의} \newtheorem{rem}{주} \newtheorem{note}{Note} } \renewcommand{\thenote}{} \renewcommand{\therem}{} \newenvironment{proof}{{\bf 증명}}{\hfill\framebox[2mm]{}} \newenvironment{proof1}{{\bf 정리증명}}{\hfill\framebox[2mm]{}} \begin{document} \parindent=0cm \section*{ 4. Integration of a function on $M$.} 이 장에서는 manifold위에서의 form의 적분뿐만이 아니라 함수의 적분을 다루기로 한다.\\ {\bf 1. Volume element $\alp$ and $\int_Mf\alp$}\\ Let $\alp$ be a function given on $M^n$ such that $\forall p\in M$, $\alp(p)=|\om_p|$ for some $\om_p\in\Lam^n(T^*_pM)$ where $|\om_p|(X_1,\cdot\cdot\cdot,X_n):=|\om_p(X_1,\cdot\cdot\cdot X_n)|,\,\,X_i\in T_pM.$\\ Such a function $\alp$ is called a {\it volume element} - a way of measuring n-dimensional volume(without sign). $\alp$ is {\it smooth} if it can be written locally as $\alp=g|dx_1\we\cdot\cdot\cdot\we dx_n|$ for $g\in\ccn,\,\,g\geq 0.$ Let $\alp$ be a smooth volume element on $M$. $M$ 이 $\rb^n$일 때 대표적인 volume element로는 $|du_1\we\cdot\cdot\cdot\we du_n|$이 있다. ( $\rb^n$에서는 $\int_{\rb^n}f|du_1\we\cdot\cdot\cdot\we du_n|:=\int_{\rb^n}fdu_1\cdots du_n$ 으로 정의된다.)\\ $<$ Change of variable formula. $>$ Domain $D\subset\rb^n$ 상에서 diffeomorphism $\vphi:D\rightarrow \vphi(D)$에 대해 $\int_D(f\circ\vphi)|det\vphi_*|dt_1\cdot\cdot\cdot dt_n=\int_{\vphi(D)}fdu_1\cdot\cdot\cdot du_n$ (변수변환공식) $\hs{2em}\Rightarrow\dis{\hs{2em}\int_D\vphi^*(f|du_1\we\cdot\cdot\cdot\we du_n |)=\int_{\vphi(D)}f|du_1\we\cdot\cdot\cdot\we du_n|}$ (증명) $\vphi^*|du_1\we\cdot\cdot\cdot du_n|(X_1,\cdot\cdot\cdot X_n )=|du_1\we\cdot\cdot\we du_n|(\vphi_*X_1,\cdot\cdot,\vphi_*X_n)$ $\hs{3em}=|du_1\we\cdot\cdot\cdot\we du_n(\vphi_*X_1,\cdot\cdot\cdot,\vphi_*X_n)|$ $\hs{3em}=|\vphi^* du_1\we\cdot\cdot\cdot\we\vphi^* du_n(X_1,\cdots,X_n)|$ $\hs{3em}=|d\vphi^*u_1\we\cdot\cdot\cdot\we d\vphi^*u_n(X_1,\cdots,X_n)|$ $\hs{3em}=|d(u_1\circ\vphi)\we\cdots\we d(u_n\circ\vphi)(X_1,\cdots,X_n)|$ $\hs{3em}=|det \vphi_*\cdot dt_1\we\cdots\we dt_n(X_1,\cdots,X_n)|$ $\hs{3em}=|det\vphi_*||dt_1\we\cdots\we dt_n|(X_1,\cdots,X_n)$ $\therefore \vphi^*|du_1\we\cdots\we du_n|=|det\vphi_*||dt_1\we\cdots\we dt_n|\hs{8em}\square$\\ For $f$ with $supp\,f\subset(U,x)$, let $\vphi=x\inv$, $\widetilde{U}=x(U)$. Then $\hs{3em}\dis{\int_Mf\alp:=\int_{\widetilde{U}}\vphi^*(f\alp)}$ is well defined as before.\\ Note that if $M$ is orientable and $\om$ is a positive n-form, then $\int_Mf\om=\int_Mf|\om|$.\\ In general, for $f$ with compact support on $(M,\alp)$, choose a finite cover $\{U_1,\cdots,U_k\}$ of $supp\,f$ and a partition of unity subordinate to this cover and define $\dis{\int_Mf\alp:=\sum_{i=1}^k\int_M\rho_if\alp}$. This is well-defined as before independent of choice of a cover and a partition of unity.\\ {\bf 2. Classical examples.}\\ (1) $n=1$ : $\alp=ds=$length element는 $\alp=|"ds"|$ 로 정의된다. (여기서 "ds" 는 Riemannian oriented volume form이다.) i.e., $ds(v_p)=|v_p|$. 그러면 a parametrization $\vphi:[a,b]\rightarrow M\subset\rb^n$에 대해 $\dis{\hs{2em}\int_Mfds=\int_a^bf(\vphi(t))\frac{ds}{dt}dt=\int_{[a,b]}\vphi^*(fds)}$ 가 된다.\\ 실제로 $\vphi^*(fds)=(\vphi^*f)(\vphi^*ds)$이고 $\vphi^*ds$를 구하기 위해 $\vphi^*ds=g|dt|$라 두자. 그러면 $\vphi^*ds(\frac{d}{dt})=g|dt|(\frac{d}{dt})=g|dt(\frac{d}{dt})|=g|1|=g.$ $\therefore \vphi^*ds=\vphi^*ds(\frac{d}{dt})|dt|$ $\hs{3.2em}=ds(\vphi_*\frac{d}{dt})|dt|$ $\hs{3.2em}=ds(\frac{d\vphi}{dt})|dt|$ $\hs{3.2em}=|\vphi'(t)||dt|$\\ $\therefore \vphi^*(fds)=(f\circ\vphi)|\vphi'(t)||dt|=(f\circ\vphi)\frac{ds}{dt}|dt|$.\\ (2) $n=2$ : $\alp=d\sigma=$area element=$|"d\sigma"|$, $d\sigma(X_p,Y_p)=|"d\sigma"(X_p,Y_p)|$= $X_p$와 $Y_p$가 이루는 평행사변형의 넓이가 된다. 실제로 \\ Choose $N=(a_1,a_2,a_3),$ a unit normal on $M$ such that $[(N,X,Y)]=[(X,Y,N)]$ giving the standard orientation of $\rb^3$. Then\\ $\hs{4em}$ area of parallelogram $(X,Y)$ $\hs{4.2em}$= volume of parallelepiped $(N,X,Y)\hs{2em}(\because |N|=1)$ $\hs{4.2em}=|det(N,X,Y)|$ $\hs{4.2em}=|dx\we dy\we dz(N,X,Y)|\hs{1.5em}$ $\hs{4.2em}=|i_N\mu(X,Y)|\hs{1.5em}$ (let $ dx\we dy\we dz= \mu )$ $\hs{4.2em}=|i_N\mu|(X,Y)$ $\hs{4.2em}=|i_N(dx\we dy\we dz)|(X,Y)$ $\hs{4.2em}=|i_N(dx)\we dy\we dz-dx\we i_N(dy)\we dz+dx\we dy\we i_N(dz) |(X,Y)\hs{1.5em}$ Since $a_1=dx(N),a_2=dy(N),a_3=dz(N),$ $\hs{4.2em}=|\underbrace{a_1 dy\we dz-a_2dx\we dz+a_3dx\we dy}|(X,Y)$ $\hs{12em}"d\sigma"$ $\hs{4.2em}=d\sigma(X,Y)$\\ parametrization이 주어졌을 때 실제 면적분을 계산해 보자. $M$에 대한 parametrization 을 $\vphi:D(\subset\rb^2)\rightarrow M$ 이라 두면 $\dis{\int_Mfd\sigma=\int_{D}\vphi^*(fd\sigma)=\int_{D}(f\circ\vphi)\vphi^*(d\sigma)}$ 에서 $\vphi^*(d\sigma)$를 구하기 위해 이를 $f|du\we dv|$라 두자. ($\vphi^*(d\sigma)$는 $D$에서의 measure이므로 이와 같이 둘 수 있다. ) 그러면 $\vphi^*(d\sigma)(\frac{\pa}{\pa u},\frac{\pa}{\pa v})=f$ 이고 따라서 $\vphi^*(d\sigma)=\vphi^*(d\sigma)(\frac{\pa}{\pa u},\frac{\pa}{\pa v})|du\we dv|$ $\hs{3.2em}=d\sigma(\vphi_*\frac{\pa}{\pa u},\vphi_*\frac{\pa}{\pa v})|du\we dv|$ $\hs{3.2em}=d\sigma(\frac{\pa\vphi}{\pa u},\frac{\pa\vphi}{\pa v})|du\we dv|$ $\hs{3.2em}=$area of $(\frac{\pa\vphi}{\pa u},\frac{\pa\vphi}{\pa v})\cdot|du\we dv|$ $\hs{3.2em}=|\frac{\pa\vphi}{\pa u}\times\frac{\pa\vphi}{\pa v}||du\we dv|$\\ $\hs{2em}\therefore\dis{\int_Mfd\sigma=\int_{D}(f\circ\vphi)|\,\frac{\pa\vphi}{\pa u}\times\frac{\pa\vphi}{\pa v}\,|du\,dv}$\\\\ {\bf 3. Riemannian volume element.}\\ Let $g$ be a Riemannian metric and let $\{e_1,\cdots,e_n\}$ be an orthonormal basis at $p\in M$ and $\{\eps_1,\cdots,\eps_n\}$ be its dual basis. The {\it Rimannian volume element}, dvol at $p$ is given by $|\eps_1\we\cdots\we\eps_n|$. ($\nu=\eps_1\we\cdots\we\eps_n$ is called a {\it oriented volume form } if $(e_1,\cdots,e_n)$ is positive.) Locally on $(U,x)$, let $g_{ij}=g(\paxi,\paxj)$ and $G=(g_{ij})$. If we let $\paxj=\sum_{i=1}^na_{ij}e_i$, then $\eps_i=\sum_ja_{ij}dx_j$. $\therefore\eps_1\we \cdots\we\eps_n=(det\,A)dx_1\we\cdots\we dx_n,\,\,where\,\,\,A=(a_{ij})$. 한편 $g_{ij}=g(\paxi,\paxj)=g(\sum_ka_{ki}e_k,\sum_la_{lj}e_l)$ $\hs{3.6em}=\sum_{k,l}a_{ki}a_{lj}\delta_{kl}$ $\hs{3.6em}=\sum_{k}a_{ki}a_{kj}=\sum_k\,^ta_{ik}a_{kj}$ $\therefore G=\,^tAA$ and $det\,G=(det\,A)^2(>0)$. $\therefore dvol=|\eps_1\we\cdots\we \eps_n|=|det\,A||dx_1\we\cdots\we dx_n|=\sqrt{det\,G}|dx_1\we\cdots\we dx_n|$\\ {\bf 예.} $\,\,M^{n-1}\subset\rb^n$ with induced Riemannian metric 에 대해 먼저 $M$이 orientable일 경우, positive orthonormal frame $e_2,\cdots,e_n$과 unit normal $N$을 $(N,e_2,\cdots,e_n)$이 $\rb^n$의 positive orthonormal frame이 되도록 잡는다. 그리고 $e_1=N$이라 두고 $\{e_1,\cdots,e_n\}$의 dual basis $\{\eps_1,\cdots,\eps_n\}$에 대해 $\nu$를 $M$의 Riemannian oriented volume form이라 두자. 그러면 $\nu=\eps_2\we\cdots\we\eps_n=i_N(\eps_1\we\cdots\we\eps_n)$ 이고 ($\,\,\because \,\,\,i_N(\eps_1\we\cdots\we\eps_n)=i_N(\eps_1)\we\eps_2\we\cdots\we\eps_n-\eps_1\we i_N(\eps_2)\we\cdots\we\eps_n+\cdots$ 에서 첫항을 제외한 나머지는 모두 0이 된다.) 또한 $(\frac{\pa}{\pa x_1},\cdots,\frac{\pa}{\pa x_n})$도 $\rb^n$의 orthonormal basis이므로 $\eps_1\we\cdots\we \eps_n=dx_1\we \cdots\we dx_n$임을 알 수 있다. 따라서 $\nu=\eps_2\we\cdots\we\eps_n=i_N(\eps_1\we\cdots\we\eps_n)=i_N(dx_1\we\cdots\we dx_n )$. $\therefore\,\,|\nu|=|\sum(-1)^{i-1}a_i dx_1\we\cdots\we\overset{\we}{dx_i}\we\cdots\we dx_n |\hs{1em}for\,\,\,N=(a_1,\cdots,a_n)$ \\ $M$이 orientable이 아닌 경우 앞의 orientable의 경우를 locally(in fact pointwise) 적용하면 $|\nu|$는 orientation에 관계없이 잘 정의된다.\\ {\bf 예.} $M=\sbb^{n-1}_r(0)=$ sphere of radius $r$ 인 경우 $N=\frac{1}{r}(x_1,\cdots,x_n)$이고 따라서 구의 Riemannian volume form은 다음과 같다. $\nu=\frac{1}{r}\sum(-1)^{i-1}x_idx_i\we\cdots\we\overset{\we}{dx_i}\we\cdots\we dx_n$.\\ $\rb^n$에서 중요하게 쓰이는 또다른 form으로는 solid angle form $"d\theta"$가 있다. (통상적으로 $d\theta$로 쓰나 $\rb^n\backslash\{0\}$상에서 exact form은 아니다.) $\dis{d\theta:=\frac{1}{r^{n-1}}\nu=\frac{1}{r^{n-1}}\sum(-1)^{i-1}x_i dx_1\we\cdots\we \overset{\we}{dx_i}\we\cdots\we dx_n}$\\ {\bf Exercise.} $d\theta$ 가 closed임을 보여라.\\\\ {\bf Exercise.}(See. 김홍종,윤옥경, "미분다양체론 입문")\\ 1. (1) Let $\mu$ be an $m$-form on $M^m$ and $\nu$ be an $n$-form on $N^n$. Let $"\mu\we\nu"=\pi_1^*\mu\we\pi^*_2\nu$ where $\pi_i$ is the canonical projection of $M\times N$ onto $M$ or $N$. Then\\ $\hs{3em}\dis{\int_{M\times N}f(x,y)\mu\we\nu=\int_M(\int_Nf(x,y)\nu)\mu}$.\\ (2) For $f\in \ccn_c(\rb^n) $, $\dis{\int_{\rb^n}fdx_1\we \cdots\we dx_n=\int_{\sbb^{n-1}}(\int_0^{\infty}f(r,\theta)r^{n-1}dr )d\theta}$\\ 2. Volume of $\mathbb{B}^n$ and $\sbb^n$:\\ $\hs{3em}$vol($\sbb^{n-1}$)=$\dis{\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}} \hs{1.5em}\hs{1em}where\,\,\,\Gamma(s)=\dis{\int_0^{\infty}e^{-t}t^{s-1}dt}\,\,\,$ and \\ $\hs{3em}$vol$(\mathbb{B}^n)=\dis{\frac{1}{n}}$vol$(\sbb^{n-1})\,\,\,$.\\ (proof of 2) $\dis{\int_{\rb^n}e^{-r^2}=(\int_{\rb}e^{-x^2}dx)^n}=(\sqrt{\pi})^n$ 에서\\ 위 식의 좌변을 극형식으로 표현하면\\ $\dis{\int_{\sbb^{n-1}}\int_0^{\infty}e^{-r^2}r^{n-1}drd\theta}=$vol$\dis{(\sbb^{n-1})\int_0^{\infty}e^{-r^2}r^{n-1}dr}\hs{3em}$ $\hs{9.2em}$=vol$(\sbb^{n-1})\dis{\int_0^{\infty}e^{-t}t^{\frac{n}{2}-1}\frac{1}{2}dt}$ $\hs{2em}$(Let $r^2=t$ then $2rdr=dt)$ $\hs{9.2em}=$vol$\dis{(\sbb^{n-1})\frac{1}{2}\Gamma(\frac{n}{2})}$\\ $\therefore\,\,\,\dis{(\sqrt{\pi})^n}=$vol$\dis{(\sbb^{n-1})\frac{1}{2}\Gamma(\frac{n}{2})}$. vol$(\mathbb{B}^n)=\dis{\frac{1}{n}}$vol$(\sbb^{n-1})$ 는 Stokes 정리를 쓰면 보일 수 있다. \end{document}