From 0bb864442ff971324333d5800796a8dada0fcb35 Mon Sep 17 00:00:00 2001 From: Robert Date: Wed, 31 Mar 2021 21:54:02 +0200 Subject: [PATCH] Finished measure theory --- chapters/measures_integrals.tex | 1 + chapters/sections/trans_thm.tex | 182 ++++++++++++++++++++++++++++++++ script.pdf | Bin 1081504 -> 1096751 bytes script.tex | 1 + 4 files changed, 184 insertions(+) create mode 100644 chapters/sections/trans_thm.tex diff --git a/chapters/measures_integrals.tex b/chapters/measures_integrals.tex index 1f7f583..6ec57eb 100644 --- a/chapters/measures_integrals.tex +++ b/chapters/measures_integrals.tex @@ -10,4 +10,5 @@ \subfile{sections/integrals.tex} \subfile{sections/int_real_nums.tex} \subfile{sections/product_measures.tex} + \subfile{sections/trans_thm.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/trans_thm.tex b/chapters/sections/trans_thm.tex new file mode 100644 index 0000000..874ce12 --- /dev/null +++ b/chapters/sections/trans_thm.tex @@ -0,0 +1,182 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{The Transformation Theorem} + +\begin{defi} + Let $U, V \subset \realn^n$ be open. A mapping $T: U \rightarrow V$ is said to be a diffeomorphism if it is bijective and if + $T$ and $\inv{T}$ are continuously differentiable. Analogously we define + \begin{gather*} + C^r \text{-diffeomorphism if it is } r \text{-times differentiable} \\ + C^{\infty} \text{-diffeomorphism if it is infinitely differentiable} + \end{gather*} +\end{defi} + +\begin{rem} + \begin{enumerate}[(i)] + \item In physics, $f$ and $f \circ T$ are often denoted with the same symbol + \item We can apply the chain rule to $T \circ \inv{T} = \idf_V$ + \[ + DT(\inv{T}(y)) \cdot D\inv{T}(y) = I_V + \] + Since $\inv{T}$ is surjective, $DT(x)$ is invertible $\forall x \in U$. According to the theorem about inverse functions, + the inverse $\inv{T}$ of a bijective mapping is continuously differentiable if $DT(x)$ is invertible + \item If $T$ is a diffeomorphism, then $\inv{T}$ is one too. + \end{enumerate} +\end{rem} + +\begin{eg} + \begin{enumerate}[(i)] + \item Polar coordinates: + \begin{align*} + T: (0, \infty) \times (0, 2\pi) &\longrightarrow \realn^2 \setminus \set{[0, \infty] \times \set{0}} \\ + (r, \phi) &\longmapsto (r \cos\phi, r \sin\phi) + \end{align*} + + \item Another diffeomorphism would be + \begin{align*} + T: \oball[1](0) &\longrightarrow \realn^n \\ + x &\longmapsto \frac{x}{\sqrt{1 - \norm{x}}} + \end{align*} + + \item An example for a mapping that is no diffeomorphism would be + \begin{align*} + T: \realn &\longrightarrow \realn \\ + x &\longmapsto x^3 + \end{align*} + The Jacobian "matrix" $T'(x) = 3x^2$ is not invertible. + + \item Another counter example would be + \begin{align*} + T: (0, \infty) \times \realn &\longrightarrow \realn^2 \setminus \set{0} \\ + (r, \phi) &\longmapsto (r\cos\phi, r\sin\phi) + \end{align*} + This function is not injective, so it's not a diffeomorphism. + \end{enumerate} +\end{eg} + +\begin{thm}[Transformation Theorem] + Let $U, V \subset \realn^n$ and $T: U \rightarrow V$ a diffeomorphism. + Then $f: V \rightarrow \realn$ is integrable over $V$ if and only if $f \circ T \cdot \abs{\det DT}$ is integrable over $U$. + In this case + \[ + \int_V d \dd{\lambda^n} = \int_U f \circ T \cdot \abs{\det DT} \dd{\lambda^n} + \] +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{eg}[Area of the unit circle] + The area is defined as + \[ + \lambda^2(K_1(0)) = \int_{\realn^2} \charfun_{K_1(0)} \dd{\lambda^2} + \] + We transform into polar coordinates: + \begin{align*} + U &= (0, \infty) \times (0, 2\pi) \\ + V &= \realn^2 \setminus \underbrace{([0, \infty] \times \set{0})}_{\lambda^2-null set} + \end{align*} + We define the transformation + \[ + T: (r, \phi) \longmapsto (r \cos\phi, r \sin\phi) + \] + Which results in + \begin{align*} + \det DT(r, \phi) &= r \\ + \charfun_{K_1(0)} \circ T(r, \phi) &= \charfun_{(0, 1]}(r) + \end{align*} + So we can calculate + \begin{align*} + \lambda^2(K_1(0)) &= \int_B \charfun_{(0, 1]}(x, y) \dd{\lambda^2}(x, y) \\ + &= \int_U \charfun_{(0, 1]}(r) \cdot r \cdot \dd{\lambda^2}(r, \phi) \\ + &= \int_0^{\infty} \int_0^{2\pi} \charfun_{(0, 1]} r \dd{\phi} \dd{r} \\ + &= 2\pi \int_0^{\infty} \charfun_{(0, 1]}(r) \dd{r} = 2\pi \int_0^1 r \dd{r} \\ + &= \pi r^2 = \pi + \end{align*} +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item Consider + \begin{align*} + T: \realn^n &\longrightarrow \realn^n \\ + x &\longmapsto Ax ~~A \in \realn^{n \times n} + \end{align*} + If $\exists \inv{A}$, then $T$ is a diffeomorphism with $DT = A$ + \[ + \implies \int f \dd{\lambda^2} = \abs{\det A} \int f \circ T \dd{\lambda^2} + \] + + \item Let $A$ be an orthogonal matrix (so a rotation/mirroring). + \[ + \det A = \pm 1 \implies \abs{\det A} = 1 + \] + Thus, rotations and mirrorings do not change the volume. + + \item Let $A = \diag(a, a, \cdots, a) ~~a \in (0, \infty)$ (this is a scaling matrix). Then + \[ + \det A = a^n + \] + which means that continuous scaling of a factor $a$ scales the $\lambda^n$-volume by $a^n$. + + \item This is a "generalization" of the substitution rule + \[ + \int_{\realn} f(g(x)) g'(x) \dd{x} = \int_{\realn} f(y) \dd{y} + \] + \end{enumerate} +\end{rem} + +\begin{eg} + We want to compute + \[ + K = \int_{\realn} e^{-x^2} \dd{x} + \] + Consider + \[ + K^2 = \int_{\realn} e^{-x^2} \dd{x} \int_{\realn} e^{-y^2} \dd{y} = \int_{\realn^2} e^{-(x^2 + y^2)} \dd{\lambda^2(x, y)} + \] + By transforming $f = e^{-(x^2 + y^2)}$ into polar coordinates + \begin{align*} + K^2 &= \int_U f \circ T \abs{\det DT} \dd{\lambda^2} \\ + &= \int_V e^{-r^2} \cdot r \dd{\lambda^2(r, \phi)} \\ + &= \int_0^{\infty} \int_0^{2\pi} r e^{-r^2} \dd{r}\dd{\phi} \\ + &= 2\pi \int_0^{\infty} r e^{-r^2} \dd{r} \\ + &= 2\pi \limn\left(-\frac{1}{2} e^{-n^2} + \frac{1}{2}\right) = \pi + \end{align*} + Thus $K = \sqrt{\pi}$. +\end{eg} + +\begin{eg}[Integrability of radial functions] + Let $f: [0, \infty] \rightarrow \realn$ be measureable and set + \begin{align*} + F: \realn^n &\longrightarrow \realn \\ + x &\longmapsto f(\norm{x}) + \end{align*} + $\dnorm$ is the Euclidian norm. Under which conditions is $F$ $\lambda^n$-integrable? + Let $D := (0, \infty) \times \underbrace{(0, \pi)^{n-2} \times (0, 2\pi)}_{D_{\phi}}$. And define + \begin{align*} + T: D &\longrightarrow \realn^n \setminus A \\ + (r, \phi) &\longmapsto \begin{pmatrix} + r \cos\phi_1 \\ + r \sin\phi_1 \cos\phi_2 \\ + r \sin\phi_1 \sin\phi_2 \cos\phi_3 \\ + \vdots \\ + r \sin\phi_1 \cdots \sin\phi_{n-2} \cos\phi_{n-1} \\ + r \sin\phi_1 \cdots \sin\phi_{n-2} \sin\phi_n + \end{pmatrix}^T + \end{align*} + Then $\norm{T(r, \phi)} = r$ and + \[ + \abs{\det DT(r, \phi)} = r^{n-1} \sin^{n-2}\phi_1 \sin^{n-3} \phi_2 \cdots \sin\phi_{n-2} = r^{n-1} A_n(\phi) + \] + Thus + \begin{align*} + \int_{\realn^n} \abs{F(x)} \dd{\lambda^n}(x) &= \int_D \underbrace{\abs{F \circ T(r, \phi)}}_{f(r)} \abs{\det DT(r, \phi)} \dd{\lambda^n}(r, \phi) \\ + &= \int_{D_{\phi}} \int_0^{\infty} r^{n-1} \abs{f(r)} A_n(\phi) \dd{r} \dd{\lambda^{n-1}}(\phi) \\ + &= \int_0^{\infty} r^{n-1} \abs{f(x)} \dd{r} \underbrace{\int_{D_{\phi}} \abs{A_n(\phi)} \dd{\lambda^{n-1}}(\phi)}_{< \infty} + \end{align*} + So $F$ is $\lambda^n$-integrable if $r^{n-1} f(x)$ is integrable over $[0, \infty)$. +\end{eg} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index ce95a2c7021e0ea7fce56b5356f42ada2e8c459e..284e110474e97b3c4d9b8f2e786bf99684dd1c76 100644 GIT binary patch delta 239635 zcmcexN!Ab{x6p$i_h5r2BGqacM?B(*}>+}2l{(+s(?Cxzoz0J(x3Aev<`S!IR zrIlW=nF7syO){0?xen#g>ha2AOKS7354c>e5%iya^q-;hpC{=*57U22=s#?B2)*j3 zOm(Cwduv#giD^w_uS;=PPpNUg+w0DBWp>HTYxmeweKN^|r3q=IRfcR?e=+TMiA>lf zWjWp60KFwHugjn(OP(GvV9=A9ZZA2}*W9?X&&Ov9N0dB0EYt1t@?qPdLrb1|GV^}i zm(Skr_ImjLKJ|YuyEC&-TtPwHprHdx`ad4$_WG0=HOnNw(!W-gN|@qrcy>Wn3PgkY<^4$<@+? z>SXu(=GK|{A^c@7_WgP7JT43(D9aCGkgsgbt10Z1ID|4R$HDC1ieDotA#v$^zEg~|?G(`VPWx4I!>g~HI$J@4N z?;mx0d!G*HEbt;+fO5F0>JIB8{XnA>?`}?Iw{*}{hQrDp!YUG|d+q?ggKX3o` z=Cy-A4BYnUolOUQ{oq>nl!uSbI~mCCU6_5<_2E~K^msC>{*+g%4+?%!H*3}x4ZeK0 z{Vwm0=i6Or`hmeXZcpny_q?3(-CLnA6Yd!B#bdW8{%hWyapwn^rg(PlIQjPDe~zz_ zot8bN`K*}}-7~8nc62+??a2A;woj!F=#XZf{rej=mVeT}^qw8Br+xkWq26P*8b3;_ zfBW`k4fa1g`00BO4E*Ndw|fk|qisS%Q>Qa~n>@4px1=|}SaI;KAO1@|*uUn`e{Wm! z<&A~gXV-tK^ydaGKHE7WtKEnF>b9z3IA^nY$9!{UYQD5`+4$N9gM00k_+!mpyo<8vFn}4+&d+)Ip-N>w2H+|NZS`{`q zJn3d~-=FVXoBQmR=D)uFl(VGe(oHv4b{{^zN5;8+BlneD|NE1_?q60^JE6wP+rFOS z$a`zdS8ra~*SFg94W|@!ZjH!tqK?a0!_t!6oo^mux1dcVKwJ`kMH_%YYm_m(#Ve+z z^Y30ZWm=c-j_v+*!86zVH<4o`r)jhm%qI_aLtN8in`^ZnUYdrC1 zX?3%}H}PwR%z6E2EAPYipKbho-3!BJZ9Q76?p*~1;TQiLTW~GD@$3TS#7MjB4v{5Z zW0oVt#8A+8AtK#vZrmWE?;?^5mgS1U9ppN}9n4qOwyT}%4ylSSq$<1+S9tCall@G0 z$j?O`3Q*bZsRbR?Mrl@fryO)cm))y;Tv$jhPM3<53+-*nL+v3^MO~E3_$3i(oHDMR!p`t>lNP2aps8BgjTwN(Ho~n%OT#ZcqcY;M( z+_p`M&*f#3CpD#yPpE0_XcyFls(Mc>$suQECd8A1U1rOA=RIr8%C9}DlUniC zR7aH6%6Ivw`n7WPTTnp8=f|b+pY$tbrB^wU zAfac5QnwesPbtQrPh>C&g9DMluAb>iYH#&6Q?GR8%%JKfYG3lniLR1ExuaKlQa)G?8-0QzF#3rZ&22_S>g`m<4@ly#Z_ztlS>2-=8Ph<% z^L}ns^4f`%im7~I#q-^qee>FZfiGJ5J6~f&&w%(_mfS`hLYC z`AJf|71WN#e?G&%iQ!)vJg^RK9saZ}N7*nine5M!RlHwPN6L=_(vv(8{uz_Wl_j&udgtrVBUlXKr$EbBjixR!;K{w`<9`EXtC zx7kTQVr^{pF+HM;EvXHcscf$)H!+Gu?ztrADxa1l6I?jSg*R+zifLF9Zojv`WLG>T z8A*B+s8hpB(v#?yYyWIE8$j7I)V@gjW^%XZpXC&#lR-I}wCQa%3QWCsY`Z>xoW73lYdO=g? zHjc@Vf=bz#yDL{C0;*$US0Inb`0mVnKPl}2m9kD1U#)dR!5m-^ zDu;3)q%7%VU;p~cac1fyQ1!;%EYLlmxEEzN;9st>p3Fzk3Q-`)_1^Ru$x6TNIm+%C zcKq*9woFe`s+A@acXROnH4HkGCM)X}+m+yq4CTerBzFBTTz{-yzlZBuORAZfVF|G5 z-vPR|kV_JUI23njdJ_Fs;S1PINok7m_@X3|l zWtB-=6#_qpsWyNG%QKOPm~^wfk}ZAMPp0tYs7E}2p4gzMP4i%GfY1INtW#L zl8f=OV|~`^4Q!GJDkLFQnX#w3-K9qE6|t&kR!2)?i;7F7&YD^VkIO^K#zT!gx+YQf zxRkbs97@gA>B_P-HRzCHvFGZHWSz>vQ_AzJbCY!(NWnW---ZV%&YX&UO=_}R=VkIa zhjP!FG-c^~$sD=U)}$yuV!)95#hMiS6&I_^_f*cO8fJnX*Yc^Ls;alQn_nGMxq{_eCD|g>B?8?JCq$8oJ#dDd!bXgz9CKN z7#ZBQF->_HgNwC{Day!=smiB(#-VK3n4y^7PvmdAv@wI!nUheRI-wp8a)&dNl^<7Q zo~#Ng2}e0c>dYG4oUy4o^{_o|uW}%qV@LW#P(&1-`hJd*xuH5W^f7FVRuV+!*OF8q zwm?$gIbvh1I}()Doows#HjOaL9x(JYDbwkJz1L;9Fh!M##iP=w#g5wScFO-R>`0P4 zl;0Vdl$|@0F{Lcto*Tvc03Y3+tF+#khzF>=N=pPs!VN4d-< zs2rWzk*@eY;=kfh8t+V1o-7{?+?k%F^Jeip)l{$VOjmyUi2f8E9mlBI$B|K;kJ6OE zbi}3T{*Tg?n|qiF!(Yl&*c{xE^L2G*OCqrEKT1_b?qQx9#TC_Oj*runDW3?SE&#pu z6~30<;Z#aLPF1FEudd8Ldgq{sRu@8#*n%h$S1WD;aXnh8^uao;GB;Ny?`1)VZT;(? z{9qv?ii{9F?lWkOgvaNjqSuI%RpVcHR59237r+dW?MmWTFDPH^O+sd}>{O0o-j_^H z3`LMgWmLIipPh{q-`;d(BA;?7Pwh=pw(m>E6N~mXRc^AY!?V{^M9NYT;QWFcy!+KbgR=A}{ps^7oy&5RclOg?nTo4(`}ub%N6ONb z8Aq#Is8is;{C8U3x@(qUnsu(7Jok4~- zkYgAzQP{N7Dw0}DHYSr1_7LeVXx0jUj2W@3U)QG=8?Ko98)9Ov7^yeMW|uwG`KplD zzK7~a$P!X2V(W4!%pP^4p2HC&QRv6*DoJVSH3^OJ-APC?L5k_jYraz`;I2Vt#%#B zQ64(0${hFBCLBq#Gv7wzu6P1LW%ZHV7~gjNNJg?Qf};;)AI(*6oJv%kyJwE36VF^* z=SL$b7bP+#%>>E_LMReFinPq7H+Wqxa-x^fv3}CAPprzw)78m>4<%;`z)=b}6x7

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