From 7ef8cdf485cb9d666bfc4cdce85a4e95666a5970 Mon Sep 17 00:00:00 2001 From: JosuaKugler Date: Wed, 17 Jun 2020 12:17:08 +0200 Subject: [PATCH] ana 14 complete --- ana14.pdf | Bin 254831 -> 289049 bytes ana14.tex | 124 +++++++++++++++++++++++++++++++++++++++++++++++-- ana6.pdf | Bin 194134 -> 273722 bytes ana6.tex | 2 +- ana7.pdf | Bin 220317 -> 296625 bytes ana7.tex | 2 +- analysisII.pdf | Bin 785690 -> 801482 bytes 7 files changed, 122 insertions(+), 6 deletions(-) diff --git a/ana14.pdf b/ana14.pdf index bb7dbab2c7ed50c63aba31e37238d4e76468b1df..d71d5d1cb155b0dabdf58f3152cd0f882bb4acbc 100644 GIT binary patch delta 170165 zcmZs?Ly#_7u&rCRZQHhO+qUh$Y}+)@Cet4$dee`aGVZzqp=&s#`7F7Idj={d)Og=@Z&jSNMDTt05D zbQ0GQ>-y%D<{2iEu$kDV5u5zHTnoWjDr2*J+d4~0!TcFl4qtb`)5D%*!>Puc!l}kg zU#2VX=YiMU?DaIG0iHka@B8sph-o=UQ!=v&R!D?TG+c&V$9Vt&o@)xTbf_4@TvN}x z2LqL|j9o``rJ3MKtTGR8mtjNnRi{z2r>n5#s7-GNFRx)w*4bOX`NO0nrQ**gSZ4|L zFzJGb*%UQN+++ZtGvHxqo{cxG-#<^8BzJcAJ#DQc8(XAvqQ2>0!Iy1}o!Al#0%0!H zRJfH)d89i^y=gWPXJJYq_FdF#&Og@js8bqU^o2Abh^O7su?9BGg~G+jE1VjX&3!ny0gyI@*v5SSzsu@n%ZJb&UEMB=23NL zSg1ya7Z+zk1JdGu^@s-PvOOG5@V2(W$gFZqL%9xA$xsG)5xOLKRvH0@sq6qcvWlqS z;i@&fRa&!-VANe2l4hpm%P<^Uw!j-j1+8OOr4g2@u9>; zTi?&Ky+@CdYdQ=~OeH@kiddoQa_bcuLwL+*m{>8qWS(+ak9(f(qVfQDXJF<%XsuhrN!QJ^-hsJ~_^Y!$-Jeic>en zwuxddU%Chw_4n>(7?eGSs`u84M+p%kjF52zt}DZ47YE>(7u98W!(++u zL7HfmMcCQRp@V@yG{Oc&BbDgI7*AA62DV>@wC@5)8-68U36XZvsR(4Sn>mR`uCY?s z|0$y)BS2I*;o#YEHzx$SuSPug&$&v;scZ|i(6#>X?`wY`o!{Tjyu?~QYkA#{2y}>V zhO%>?`JoCoaS0lYxoO2qiSrdu@`peq7}fW6U4(+P=3c$XJ7fKRfU=^TQ@dJ12h~_*fR#2s* z#1T~lcoimpDu~c)+HlFt!JrLCDt;()&c8Up}=^XO@;GG5gmRnRF(m;S!Z616?U02sO| z2E}RDj>MrE1GI{=e)o9k`hItaOA<{leN14rIKCcbOg`K51AIjOTI+p)!ve{SLd?e% zkaVbtB6p1L0uD7rdULEh=?NX^%O3ou2T~W-v_JQOjKZ|PI^Z@37V?03f*3N8W*AVe z)?xIK~SaM$QD2E?-ZV6uoPF61#ud z)1#xQSwQ)c3(NEq?+HUvV}?aa^$WPO+Be1y^?aGHLCjv~j);^hZ}q-B8*D5+2%>T# z!t-T@ptIc}_EGMko!Sz%1)jctv(bpB4M|TdugPmP)F^jTIER-pJ{zjSmeu^*ieU1R zYYmDLj;QaSd3vPn&T7uWl`gBVl$vUEwC5F~Q>z(cxX@}y#Ihz%lMZ}R-UYz55BHE2 z;X{tNj=~S|3>JC{y8qx2m*ifG+u7%%FGpXugT2;*=m74{g?ufbbZ2bgCxm<|)VygE zkj3OD+k>(ujpAZI6#3^e=YMZ+rGSJQ5tI506zwoWK+{>I-d!AR;@$n?h1$G@0@v0L z+35>onT}esgEi-6^P9ICi3C^#rn{Y^!SIVzVHhOS6gL5_*BGa`FPN4=)NrI#4M^iM zg4z+U?x6Wpcx5l6p1?9JE*DOowktWj$i>XfX>k!rZ{WcgPMHQ*P1Lqm2EW2d-LiU+ zlGjyuS=^l+U9sM&9VzZ^-#$B0S(6?nk2iWBUsJ(Mfw+$+|hb zd@1jG7#!iK#1G-}a`CQ|yUF%^$$j#M8L=j~eI$v2XHYaY&J|_TAD^h0T);@=2^E2l z>hV?ETcw67+ysIvg8+5=+Cfz#ge*j+?rp<0c`%I;YqR=6v zJ@zQOdU=#c8LV`;9X3?boR-2lk=x@+a7H6a$j1>EN#1IvyV%4)*Q|GaC|9;}evvrL z^iiWdJJtnUx1tHLIi)y=OAP_yDx%RNT|Q*me+Fn##`(fl6F$qe8e&eWxftt{PjnF$ zvH8;2zq8J$cL6|P8pcM{X|cA%#9qW=fvB_G&a<7rnM4x)GBIUNbC>4*wQ_DgvO^0Ndqo z^GQJoP_p#0eJ3!d;?Kk~Ak(?GVY4by?R04EVB%%H@TP5g2s3*QZZvS5$bFECoJf$X z1*cB`%)cU))#em%7AwQ6Hif2_l7GI_lxPL;KP|Jtolq>!M<%YFd3VFLtfiv(lTlc!i8rlo}KRa=;QaqepKCb}axLwJc9dcMO`*p7l&n zEldZDT%*8WHWC|8YliA`3^^e7PE{9_Mr9peu~C%+DjV7Wo;jz|HKq&j?cxl$YtnN- ziOZh@JH}4W5e$;47EY7PD#_(Ot#~W}?vA2oTjUjfu*tcoTP0MC&v{?x)?(N0)-Lcs z1Za@}WB+9c-ocQ-(k=}V)IRa2seUIrXTD5XOe9jAg&2K$39f|B0g?ibp-?`&aH)x- z^4^Nk$!6f-Axx5Q!`X_mWOszFkC>7VpY(3B1y`E>wo=xm8*^iAadP{FR(DI#XYNCW z#+*E@eS}2&^MLF>pf$nZ@lS<=eTCRI0e~9sBIe*%-O5RA>xg5^NMLqDxDcNIL4Xq? zrBRuLj3H5_$V{OC>|AI~ zsU5)uILi+)Sqp!KvyUdDruYlo*j<@U(O+iZ3r>muH{{p;6mZ9blRkiZ@JDP<3xLRO zT;jnbT_rbrr-gXk0dMDXs?(A4q!9DI-${Vh7@7OIAcwdL)z;F&dh(ZrohR}$K~2mg zOiqlsFl(1mLu`~rWi(;mMnkY(UhFpGxj!?J z4f>^Hf8~iovBEmYqBUxkl}W^sK26=x2a6Yo#^WLwBA^c9tm{4^HCwB0wzhbchktqPUOBZK;eCj7vCdO{b)$F1ib~nztTJYT&j30Xl43_RSbkL#7Q=XOFO=F zu-Bs`LUTAF8FT=Qac%CPTu&&z|Lw>L0tO&XNw$P7UTohS_oB}73&|eCrvPK_VCL%P zVs32zKb5124Gb$M5fjn>6nuOzjB@42R?gZBhXs@c`2~J&t$3aT>VsihIdPZbN#mtHous$ zs1|n002k2q7#}Lq$@Rle73Hos2le;qKNvBqhv`_PMtU8OH5Op>Y|au|fS)>uNVaO-oA?|F;n-H3khZ6hlb|EcJ@}6wEyo_t%~fqH zj1^@_CeDpM3Sm9neB4jnBC50gAwT0YX) zhR;52s9v(t*3B(BO9GjWrz|^bW+QYyi&=XrRZX-Np=%`nm<~ros||(i32-g`&i!

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Dann gilt : $\nabla f(x) = 0$. \end{satz} \begin{proof} - Für $i = 1,\dots,n$, betrachte $g_i(t)\coloneqq f(x + te_i)$. Da $D$ offen sind alle $g_i$ auf einem Intervall $(-\delta, \delta)$ definiert für ein $\delta > 0$, und differenzierbar. $g_i(t)$ hat in $t = 0$ ein lokales Minimum/Maximum $\implies \dv{g_i(t)}{t}\bigg|_{t=0} = 0 \forall i = 1, \dots, n$. - $f$ total differenzierbar - \[\implies 0 = \dv{g_i(t)}{t}\bigg|_{t=0} \oldstackrel{\text{Kettenregel}}{=} \sum_{j = 1}^{n}\pdv{f(x)}{x_j} \cdot \delta_{ij} = \pdv{f(x)}{x_i} \forall i = 1\dots, n + Für $i = 1,\dots,n$, betrachte $g_i(t)\coloneqq f(x + te_i)$. Da $D$ offen ist, sind alle $g_i$ auf einem Intervall $(-\delta, \delta), \delta > 0$ wohldefiniert und differenzierbar. $g_i(t)$ hat in $t = 0$ ein lokales Minimum/Maximum , also gilt $\forall i = 1, \dots, n$ \[\dv{g_i(t)}{t}\bigg|_{t=0} = 0.\] + Aufgrund der totalen Differenzierbarkeit von $f$ folgt + \[ 0 = \dv{g_i(t)}{t}\bigg|_{t=0} \oldstackrel{\text{Kettenregel}}{=} \sum_{j = 1}^{n}\pdv{f(x)}{x_j} \cdot \delta_{ij} = \pdv{f(x)}{x_i} \qquad \forall i = 1\dots, n \] \end{proof} \begin{bem} @@ -188,7 +188,7 @@ Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung v \[\norm{G(x,y_1) - G(x,y_2)}_2 \oldstackrel{\text{MWS}}{\le} \sup_{(x,y)\in K_s^x(0)\times K_r^y(0)}\norm{D_yG(x,y)}_2 \cdot \norm{y_1-y_2}_2 \le \frac{1}{2}\norm{y_1-y_2}_2.\] Sei $y\in \overline{K_r^y(0)}$ \[\norm{G(x,y)}_2 \le \norm{G(x,y)-G(x,0)}_2 + \norm{G(x,0)}_2 \le \frac{1}{2}\norm{y}_2 + \frac{1}{2}r\le r\] - d.h. $G(x,\cdot)$ ist eine Selbstabbildung der abgeschlossenen Kugler $\overline{K_r^y(0)}$. + d.h. $G(x,\cdot)$ ist eine Selbstabbildung der abgeschlossenen Kugel $\overline{K_r^y(0)}$. Außerdem, $\norm{G(x,y_1) -G(x,y_2)}_2 \le \frac{1}{2}\norm{y_1-y_2}\implies G(x,\cdot)$ ist eine Kontraktion mit Lipschitz-Konstante $L=\frac{1}{2}$. Aus dem Banachschen Fixpunktsatz %add reference? folgt $\forall x \in K_s^x(0) \exists!$ ein Fixpunkt $y(x)\in K_r^y(0)$ von $G(x,\cdot)$: @@ -201,6 +201,122 @@ Für $\mathunderline{blue}{x_0 =1,\; y_0 = 0}$ hingegen gibt es keine Umgebung v Aus der Abschätzung $\norm{y(x)-y^{(k)}(x)}\leq 2^{-k-1}\cdot r,\; x\in K_s^x(0)$ folgt, dass $y^{(k)}(x)\oldstackrel{k\to \infty}{\to y(x)}$ gleichmäßig auf $K_s^x(0)$ konvergiert $\implies y(x)$ stetig. $\implies$ 1. Behauptung für $(\hat x, \hat y) = (0,0)$ mit $U(\hat x)\coloneqq K_s^x(0)$ und $U(\hat y)\coloneqq K_r^y(0)$. \item Die Eindeutigkeit von $y = f(x)$ folgt nun aus dem Banachschen Fixpunktsatz. Für $x\in K_s(\hat x)$ ist der Fixpunkt der Gleichung $G(x,y) = y$ eindeutig bestimmt. + \item Da $F(x,y)$ in $(0,0)$ differenzierbar ist, gilt nach Definition der Differenzierbarkeit + \[ + F(x,y) = \underbrace{F(0,0)}_{=0} + D_xF(0,0)\cdot x + \underbrace{D_yF(0,0)}_{=J_y}\cdot y + \omega(x,y), + \] + wobei $\omega \colon K_s^x(0)\times K_r^y(0) \to \R^m$ die Eigenschaft $\norm{\omega(x,y)}_2 = o(\norm{(x,y)}_2)$ besitzt. Aus dem Beweis von 1. wissen wir, dass $F(x,f(x)) = 0$ für $x\in K_s^x(0)$ gilt. Einsetzen ergibt + \begin{align*} + 0 &= F(x,f(x)) = D_xF(0,0) \cdot x + J_y \cdot f(x) + \omega(x, f(x))\\ + f(x) &= -J_y^{-1}\cdot D_xF(0,0) \cdot x - \underbrace{J_y^{-1} \cdot \omega(x,f(x))}_{\eqqcolon \psi(x)}\\ + &= -J_y^{-1} D_xF(0,0) x + \psi(x) + \end{align*} + Reminder: Def. Differenzierbarkeit: $f(0 +x) = \underbrace{f(0)}_{=0} + Df(x) \cdot x + \psi(x)$ mit $\psi(x) = o(\norm{x}_2)$ + Es genügt also zu zeigen, dass $\psi(x) = o(\norm{x}_2)$, d.h. $\lim\limits_{x\to 0} \frac{\psi(x)}{\norm{x}_2} = 0$. Wir nutzen $\omega(x,y) = o(\norm{(x,y)}_2)$, d.h. \[\frac{\norm{\omega(x,y)}_2}{\norm{(x,y)}_2} \oldstackrel{\norm{(x,y)_2}\to 0}{\to} 0\] + d.h. $\forall \epsilon > 0 \exists \delta_1 \in (0,s),\; \delta_2\in (0,r)$ sodass $\forall x$ mit $\norm{x}_2\le \delta_1$ und $\forall y$ mit $\norm{y}_2 \le \delta_2$ gilt + \[ + \norm{\omega(x,y)}_2 \le \epsilon \norm{(x,y)}_2 \le \epsilon(\norm{x}_2 + \norm{y}_2) + \] + Da $f$ überdies auch stetig ist (siehe Beweis 1.), gibt es ein $\delta \in (0,\delta_1)$ sodass $\forall \norm{x}_2 \le \delta$ gilt $\norm{f(x)}_2\le \delta_2$. Daraus schließen wir für $x$ mit $\norm{x}_2 \le \delta$: + \[ + \norm{\omega(x,f(x))}_2 \le \epsilon(\norm{x}_2 + \norm{f(x)}_2) + \] + Dies können wir nun auf $f(x)$ anwenden. + \begin{salign*} + f(x) &= -J_y^{-1} D_xF(0,0)\cdot x + \Psi(x) + \intertext{Es gilt $\psi(x) = -J_y^{-1}\omega(x,f(x))$} + \norm{f(x)} &\le \underbrace{\norm{J_y^{-1} D_xF(0,0)}_2}_{\eqqcolon c_1} \cdot \norm{x}_2 + \underbrace{\norm{J_y^{-1}}_2}_{\eqqcolon c_2} \cdot \norm{\omega(x,f(x))}_2\\ + &\le c_1 \norm{x}_2 + c_2 \norm{\omega(x,f(x))}_2 + \intertext{Setzen wir nun $\epsilon = \frac{1}{2c_1}$, so erhalten wir für genügend kleines $x$} + &\le c_1\norm{x}_2 + c_2\cdot \frac{1}{2c_1}(\norm{x}_2 + \norm{f(x)}_2)\\ + &= \left(c_1 + \frac{1}{2}\right) \norm{x}_2 + \frac{1}{2}\norm{f(x)}_2\\ + \implies \frac{1}{2}\norm{f(x)}_2 &\le \left(c_1+ \frac{1}{2}\right)\norm{x}_2 + \end{salign*} + Zusammen erhalten wir also: + \begin{salign*} + \norm{\psi(x)}_2 &\le \norm{J_y^{-1}}_2 \cdot \norm{\omega(x,f(x))}_2 + \intertext{Für $\norm{x}_2 \le \delta$ gilt} + &\le c_1 \cdot \epsilon(\norm{x}_2 + (2c_1 +1)\norm{x}_2)\\ + &\le \epsilon \cdot \norm{x}_2 + \intertext{Daraus schließen wir} + \frac{\norm{\psi(x)}_2}{\norm{x}_2} &\le \epsilon c_3 + \intertext{Da $\epsilon$ beliebig ist, folgt} + \frac{\norm{\psi(x)}_2}{\norm{x}_2} &\stackrel{x \to 0}{\to} 0\\ + \implies \frac{\psi(x)}{\norm{x}_2} &\stackrel{x \to 0}{\to} 0 + \end{salign*} + Daraus folgt nun schließlich für das Differential von $f$ in $x = 0$ + \[D_xf(0) = -J_y^{-1} D_xF(0,0)\] + Weiter müssen wir noch zeigen, dass $D_xf(x)$ in $K_s^x(0) \subset K_s^x(0)$ existiert und stetig ist in $x = 0$. Da $D_yF(x,y)$ stetig in $(0,0)$ ist, gibt es ein $\delta > 0$, sodass $\forall (x,y) \in K_s^x(0) \times K_\delta^y(0) \subset K_s^x(0)\times K_r^y(0)$ gilt + \[ + \norm{D_yF(x,y) - D_yF(0,0)}_2 < \frac{1}{\norm{D_yF(0,0)^{-1}}_2} + \] + Nach dem Störungssatz %add reference + ist also $D_yF(x,y)$ regulär, also $\det D_yF(x,y) \neq 0$. Die Elemente von $D_yF(x,y)^{-1}$ sind nach der Cramerschen Regel stetige Funktionen der Elemente von $D_yF(x,y)$. + Daher ist \[(D_yF(x,y))^{-1}D_xF(x,y)\] als Produkt stetiger Funktionen stetig und nach Definition der Stetigkeit gilt + \[ + \norm{(D_yF(x,y))^{-1}D_xF(x,y) - (D_yF(0,0))^{-1}D_xF(0,0)} \oldstackrel{\norm{(x,y)}_2 \to 0}{\to} 0 + \] + Analog wie oben folgern wir + \[f(x+h)-f(x) = -(D_yF(x,f(x)))^{-1} D_xF(x,f(x)) \cdot h + \omega(x,h),\qquad \omega(x,h) = o(\norm{h}_2)\] + Die Ableitung $D_xf(x) = -(D_yF(x,f(x)))^{-1} D_xF(x,f(x))$ von $f$ ist stetig in $x = 0$. \end{enumerate} \end{proof} +\begin{bsp} + $F(x,y) \coloneqq x^2 + y^2 -1$. Dann ist $D_yF(x,y) = 2y$. Nach dem Satz über implizite Funktionen folgt,dass $F(x,y) = 0$ in einer Umgebung von $(\hat x, \hat y)$ mit $\hat x^2 + \hat y^2 -1 = 0,\; \hat y \neq 0$ (d.h. $\hat x \neq \pm 1$) eindeutig durch $y = \sqrt{1-x^2}$ oder $y = -\sqrt{1-x^2}$ nach $y$ auflösbar ist. +\end{bsp} +\begin{bem}[Implizites Differenzieren] + SIF und $F(x,y) = 0 \Leftrightarrow y=f(x),\; F(x,f(x)) = 0$. + Kettenregel: $0 = F(x,f(x))$. + \[ + 0 = \dv{F}{x}(x,f(x)) = \left(\pdv{F}{x} + \pdv{F}{y} \cdot \pdv{f}{x}\right) \implies \pdv{f}{x} = -\left(\pdv{F}{y}\right)^{-1}\pdv{F}{x} + \] + Die zweiten Ableitungen erhält man durch implizites Differenzieren von $\dv{F(x,f(x))}{x} = 0$. +\end{bem} +\begin{bsp} + $F\colon \R^2 \to \R,\; F(x,f(x)) = 0$ + \begin{align*} + 0 &=\pdv{F}{x} + \pdv{F}{y}f'&&\text{1. Ableitung}\\ + 0 &= \dv{}{x}\left(\pdv{F}{x} + \pdv{F}{y}f'\right) &&\text{2. Ableitung}\\ + 0 &= \pdv{^2F}{x^2} + \pdv{^2F}{y\partial x}f' + \left(\pdv{^2F}{x\partial y} + \pdv{^2F}{y^2}f'\right)f' + \pdv{F}{y}f''\\ + 0 &= \pdv{^2F}{x^2} + 2\pdv{^2F}{y\partial x}f' + \pdv{^2F}{y^2}{f'}^2 + \pdv{F}{y}f''\\ + f'' &= -\left(\pdv{F}{y}\right)^{-1}\left(\pdv{^2F}{x^2} + 2\pdv{^2F}{y\partial x}f' + \pdv{^2F}{y^2}{f'}^2\right) + \end{align*} +\end{bsp} +\vspace*{-1cm} +\subsection{Umkehrabbildungen} +Fragestellung: Sei $f \colon D\subset \R^n \to \R^n$. Existiert die Umkehrabbildung $f^{-1}: B_f \to \R^n$? +\begin{definition} + Sei $D\subset \R^n$ offen. Eine Abbildung $f\colon D\to \R^n$ heißt regulär in $\hat x\in D$, wenn $\exists K_\delta(\hat x) \subset D$, sodass $f$ in $K_\delta(\hat x)$ stetig differenzierbar und die Jacobimatrix $J_f(\hat x)$ regulär ist. $f$ heißt regulär in $D$, wenn $f$ in jedem Punkt $\hat x \in D$ regulär ist. +\end{definition} +\begin{satz}[Umkehrabbildung]\label{umkehrfunktion} + Sei $D\subset \R^n$ offen und $f\colon D\to \R^n$ regulär in $\hat x\in D$. Dann $\exists$ eine offene Umgebung $V(\hat x) \subset D$ von $\hat x$ derart, dass $U(\hat y) \coloneqq f(V(\hat x))$ eine offene Umgebung von $\hat y = f(\hat x) $ ist und $f\colon V(\hat x) \to U(\hat y)$ bijektiv. Weiter gilt: Die Umkehrabbildung $f^{-1}: U(\hat y) \to V(\hat x)$ ist regulär in $\hat y$ und + \[J_{f^{-1}}(\hat y) = (J_f(\hat x))^{-1},\; \det J_{f^{-1}}(\hat y) = \frac{1}{\det J_f(\hat x)}\] +\end{satz} +\begin{proof} + Sei $\hat x \in D$ und $\hat y \coloneqq f(\hat x)\in f(D)$. + Betrachte $F \colon \R^n \times D \to \R^n, F(y,x) = y-f(x)$. Für $(\hat x, \hat y)$ gilt $F(\hat y, \hat x) = 0$. Die Jacobimatrix $D_xF(y,x) = -J_f(x)$ ist regulär in $\hat x$. Mit Vertauschung von $x$ und $y$ folgt aus dem Satz über implizite Funktionen, dass Umgebungen $U(\hat y)$, $U(\hat x)$ und genau eine stetig differenzierbare Abbildung $g:U(\hat y) \to U(\hat x)$ existieren, sodass $\forall y \in U(\hat y)$ + \[ + 0 = F(y,g(y)) = y-f(g(y)), \implies \exists! x = g(y)\in U(\hat x) \text{ mit } y= f(x) + \] + Setze dann \[V(\hat x) \coloneqq U(\hat x) \cap f^{-1}(U(\hat y)) = \{ x\in U(\hat x)\mid f(x) \in U(\hat)y\}.\] + Da $U(\hat x)$ und $f^{-1}(U(\hat y))$ offen sind, ist auch $V(\hat x)$ offen. Daher ist $f\colon V(\hat x)\to U(\hat y)$ bijektiv und die Umkehrabbildung $f^{-1}\colon U(\hat y) \to V(\hat x)$ ist gerade $g$. + \begin{align*} + J_{f\circ f^{-1}}(\cdot) = J_{\mathrm{id}}(\cdot) &= \mathbb{I} + \intertext{Mit der Kettenregel folgt} + J_f(\hat x) \cdot J_{f^{-1}}(f(\hat x)) = \mathbb{I}\\ + \implies J_{f^{-1}}(f(\hat x)) = J_f(\hat x)^{-1} + \end{align*} +\end{proof} +\begin{korollar} + Sei $f\colon D\to \R^n,\; D\subset R^n$ regulär und $O \subset D$ offen. Dann ist $f(O)$ offen. +\end{korollar} +\begin{proof} + Sei $O \subset D$ offen und $y\in f(O)$ beliebig mit $y = f(x),\; x\in O$. Nach Satz \ref{umkehrfunktion} existieren $\forall y$ Umgebungen $K_r(y)\subset f(O)$ und $K_s(x)\subset O$ sodass $K_r(y)\subset f(K_s(x))\implies f(O)$ offen. +\end{proof} +\begin{bem} + \begin{enumerate} + \item Satz \ref{umkehrfunktion} garantiert nur die lokale Umkehrbarkeit von $f$. + \item Es seien $U, V$ offene Mengen in $\R^n$ und $f\colon U\to V$ stetig differenzierbar.\\ $f$ heißt \underline{Diffeomorphismus}, falls $f$ bijektiv ist und die Umkehrabbildung $f^{-1}: V \to U$ stetig differenzierbar ist. + \end{enumerate} +\end{bem} \end{document} \ No newline at end of file diff --git a/ana6.pdf b/ana6.pdf index 0107ca9fc96649f67c646936f37268dd843c9db2..fbc9f33849bad8ce89907885ffadb8feb20187dc 100644 GIT binary patch delta 170528 zcmV)HK)t`#>-TG1!~g@3k|-s4B1A!r?w&rrKA^`hU;HSt^X(&^ zoo77z>d_Z3A3c6vfz}bTLKeKXNp;2s+$@* zrb9=Dz>K!pymOzR`&tS5pO2qQmkG_3ChpI&xfX&sFgI768KYW>A}ZO>b0IjNeU)Ec z$@yr7#Q6UU=g+3BPI45&sVy^{ElI7~N1Qwzg<(729i 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\begin{lemma}[Ungleichung von Young] Seien $p,q \in \R, p>1, \ q<\infty, \ \frac{1}{p} + \frac{1}{q} = 1$. Dann gilt - $$|x\cdot y| \leq \frac{|x|^p}{p} + \frac{|y|^q}{q} \quad x,x \in \R$$ + $$|x\cdot y| \leq \frac{|x|^p}{p} + \frac{|y|^q}{q} \quad x,y \in \R$$ \end{lemma} \begin{proof} diff --git a/ana7.pdf b/ana7.pdf index 1cb30f0c44c843575396403f862d28726dd6edab..4a05e745ee1f15c7629e111a4548a6588b5c05c2 100644 GIT binary patch delta 157957 zcmV)CK*GPBxec+65|AVUG%}MBWhj5mTy1mP#u5ImUx9oo)t2Xbe^JIWu_`;0H&2>6 z(@yPlh7KVaHbu82)i_S`-@A(gz&mhBkP>D4Lt_d!+{@nXv(N7B^}(|rz8A&S@`M*x z1uwolIe&KY;EDD{V8KdTJi9EMU|Q+I8PB|x#j|Pge8ibxr(^0dMTcdh2VVjZ8d){CyNr629kL6@?|xf-psGcIX-Zs$<1YaGcB*W>m1Kh3-wJ^*F4Zrg~LB)TdAEM7>lK8y+ zUcCXLIW#lWP3*-C&CD_z#Kz1v`_$gC0OG%^{GuhvWH&viRxpJFEe$%v{k$k& zaHZ$fs(K4*qqT39t*6>fF$tYDy7Gb>zE-}#*UATPn$Pp0K|s1wHI^w+ZzRtQ?*@7d zgTxVLX@5DH&6dPE zBxjy?^74et4Kqrwl|v{{rP1dA&451erzScTsNJspMiPPvO!I%xmrHNS+;-b^Rn1n! z)Rbl>sC=P>PFQAyOVyj|ig~xO%6Ei*0mh1MUVkEb!e1)sNuQ358vROi1-r_VmS9|C z>HO#+rT75n#`Wh%=|7;G^oJ~#SR@!((7lOCNv;;458iJqm!6q*2hHqPYGyf72}-3^ 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