From fa23d205d249bf3006b5cb122a398140e995e25a Mon Sep 17 00:00:00 2001 From: JosuaKugler Date: Tue, 14 Jul 2020 15:21:09 +0200 Subject: [PATCH] added ana20 without images --- ana20.tex | 238 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 1004407 -> 1027966 bytes analysisII.tex | 1 + 3 files changed, 239 insertions(+) create mode 100644 ana20.tex diff --git a/ana20.tex b/ana20.tex new file mode 100644 index 0000000..5cd68ec --- /dev/null +++ b/ana20.tex @@ -0,0 +1,238 @@ +\documentclass{lecture} + +\begin{document} + \chapter{Kurven im \texorpdfstring{$\R^{n}$}{R\unichar{"207F}}} + \section{Kurven} + \begin{definition}[Kurve] + Eine Kurve $\gamma$ im $\R^n$ ist eine stetige Abbildung $\gamma\colon I\to \R^n$, $I$ Intervall (z.B. $I = [a,b]$ oder $I = \R$.) Schreibweise: \[\gamma(t) = \begin{pmatrix} + \gamma_1(t)\\ + \vdots\\ + \gamma_n(t) + \end{pmatrix}.\] + Dabei gilt $\gamma$ stetig $\Leftrightarrow$ $\gamma_i$ stetig $\forall i = 1,\dots, n$. + \end{definition} + \begin{bsp} + \begin{enumerate} + \item Gerade in $\R^n$ durch einen Punkt $a\in \R^n$ in Richtung $v \in \R^n\setminus\{0\}: \gamma(t) = a + tv,\; I = \R$. + \item Kreis in $\R^2$ um $a\in \R^2$ mit Radius $r > 0$ + \[ + \gamma(t) = a + r\begin{pmatrix} + \cos(t)\\ + \sin(t) + \end{pmatrix},\; t\in [0, 2\pi] + \] + \item Helix in $\R^3$ mit $r > 0, c \neq 0$. + \[ + \gamma(t) = \begin{pmatrix} + r\cos(T)\\ + r\sin(t)\\ + c\cdot t + \end{pmatrix} + \] + \end{enumerate} + \end{bsp} + \begin{definition}[Differenzierbarkeit] + \begin{enumerate} + \item $\gamma$ heißt stetig differenzierbar, wenn $\gamma_1, \dots, \gamma_n$ stetig differenzierbar sind. Dabei bezeichnet man + \[\gamma'(t) = \begin{pmatrix} + \gamma_1'(t)\\ + \vdots\\ + \gamma_n'(t) + \end{pmatrix}\] + als Tangential- bzw. Geschwindigkeitsvektor. + \item $\gamma$ heißt regulär, wenn gilt: $\forall t\in I$ gilt $\gamma'(t)\neq 0$. + \item $r(t) \coloneqq \norm{\gamma'(t)}_2\colon I\to \R$ heißt Geschwindigkeit von $\gamma$. + \[ + \norm{\gamma'(t)}_2 = \sqrt{|\gamma_1'(t)|^2 + \dots + |\gamma_n'(t)|^2} + \] + \end{enumerate} + \end{definition} + \begin{bsp} + \begin{enumerate} + \item Gerade: $\gamma(t) = a + v\cdot t$. + \[ + \gamma'(t) = \begin{pmatrix} + v_1\\ + \vdots\\ + v_n + \end{pmatrix} = v,\; r(t) = \norm{v}_2 \xRightarrow{v\neq 0} \gamma \text{ ist regulär} + \] + \item Kreis: $\gamma(t) = a + r\begin{pmatrix} + \cos(t)\\ + \sin(t) + \end{pmatrix}$. + \[ + \gamma'(t) = r\begin{pmatrix} + -\sin(t)\\ + \cos(t) + \end{pmatrix} \xRightarrow{r \neq 0} \gamma'(t) \neq 0\;\forall t, + \] + da $\sin$ und $\cos$ keine gemeinsamen Nullstellen haben. + \[ + r(t) = \norm{\gamma'(t)}_2 = \sqrt{r^2\sin^2(t) + r^2\cos^2(t)} = r + \] + \item Helix: $\gamma(t) = \left(r\cos(t), r\sin(t), ct\right)^T$ + \[ + \gamma'(t) = \begin{pmatrix} + -r\sin(t)\\ + r\cos(t)\\ + c + \end{pmatrix} \neq 0\; (c \neq 0) + \] + Außerdem ist + \[ + r(t) = \sqrt{r^2 + c^2} > 0 + \] + \item Kurven stellen nicht notwendig injektive Abbildungen dar. + \[ + \gamma\colon \R\to \R^2,\quad \gamma(t) = \begin{pmatrix} + t^2-1\\ + t^3-t + \end{pmatrix} + \] + Das Bild von $\gamma$ ist $\gamma(\R) = \{(x,y)\in \R^2\colon y^2 = x^2 + x^3\}$. Dabei ist $\gamma(-1) = \begin{pmatrix} + 0\\0 + \end{pmatrix} = \gamma(1)$. Allerdings ist $\gamma'(t) = \begin{pmatrix} + 2t\\3t^2-1 + \end{pmatrix}$ bei $-1$ gleich $\begin{pmatrix} + -2\\2 + \end{pmatrix}$ und bei $1$ gleich $\begin{pmatrix} + 2\\2 + \end{pmatrix}$. + \item Neilsche Parabel $\gamma\colon \R \to \R^2, \quad \gamma(t) = (t^2, t^3)$. Das Bild von $\gamma$ ist $\gamma(\R) = \{(x,y) \in \R^2\colon x\geq 0, y = \pm \sqrt{x^3}\}$. Es gilt $\gamma'(t) = \begin{pmatrix} + 2t\\3t^2 + \end{pmatrix}$ und daher insbesondere $\gamma'(0) = \begin{pmatrix} + 0\\0 + \end{pmatrix}$. Daher ist $\gamma(t)$ nicht regulär und $t = 0$ ist ein singulärer Punkt. + \end{enumerate} + \end{bsp} + \begin{definition}[Tangente] + Sei $\gamma\in C^1(I;\R^n)$. Sei ein $t_0\in I$ regulär (d.h. $\gamma'(t_0) \neq 0$). Dann ist die Tangente an $\gamma(t_0)$ eine Gerade durch $\gamma(t_0)$ in Richtung $\gamma'(t_0)$ + \[ + \gamma(t_0) + s\gamma'(t_0) \mid s\in \R\}. + \] + \end{definition} + \section{Die Bogenlänge} + Sei $\gamma\colon I \to \R^n$ eine stetige Kurve. Sei $\mathcal{Z} = \{t_0,t_1,\dots, t_M\},\;t_i\in I$ eine Partition des Intervalls $I$. $\mathcal{Z}$ definiert ein Sehnenpolygon von $\gamma$ mit Ecken $\gamma(t_0),\dots, \gamma(t_M)$ der Länge + \[ + S(\mathcal{Z})\coloneqq \sum_{i = 1}^{M}\norm{\gamma(t_i)- \gamma(t_{i-1})}_2,\; S(\mathcal{Z})\in \R. + \] + Sei $\mathcal{Z}^*$ eine weitere Partition des Intervalls $I$, die aus $z$ durch Hinzunahme weiterer Teilungspunkte entsanden ist, dann gilt $S(\mathcal{Z}^*) \geq S(\mathcal{Z})$. + Betrachte Teilintervall $[t_0,t_1]$ und sei $t_0 = s_0 < s_1 < \dots < s_K = t_1$, dann ist + \[ + \norm{\gamma(t_1) - \gamma(t_0)} = \norm{\sum_{i = 1}^{K} \gamma(s_i) - \gamma(s_{i-1})} \leq \sum_{i = 1}^{K}\norm{\gamma(s_i) - \gamma(s_{i-1})} \implies S(\mathcal{Z}) \leq S(\mathcal{Z}^*). + \] + Seien $\mathcal{Z}_1$ und $\mathcal{Z}_2$ zwei Zerlegungen und $\mathcal{Z}^*$ eine gemeinsame Verfeinerung. Dann gilt $S(\mathcal{Z}^*) \geq \max(S(\mathcal{Z}_1), S(\mathcal{Z}_2))$ + \begin{definition}[Rektifizierbarkeit] + Eine stetige Kurve $\gamma\in C^0(I;\R^n)$ heißt rektifizierbar, wenn die Menge aller Längen $S(\mathcal{Z})$ von Polygonen zu Partitionen $\mathcal{Z}$ von $I$ beschränkt ist. In diesem Fall heißt $S(\gamma) \colon \sup\{S(\mathcal{Z})\mid \mathcal{Z} \text{ Partition von } I\}$ die Länge von $\gamma$, in anderen Worten $\forall\epsilon > 0,\; \exists \delta > 0, \;\forall$ Partitionen von $I$ gilt: + \[ + \max_i |t_{i-1} - t_i| < \delta \implies |S(\mathcal{Z}) - S(\gamma)| < \epsilon + \] + \end{definition} + \begin{bsp}[Lipschitz-stetige Kurven] + Sei $\gamma\colon I\to \R$ Lipschitz-stetig mit $|I| < \infty$, dann gilt für alle Zerlegungen $\mathcal{Z}$ von $I$: + \[ + S(\mathcal{Z}) = \sum_{i = 1}^{M}\norm{\gamma(t_i) - \gamma(t_{i-1})} \oldstackrel{\gamma \text{ L-stetig}}{\le} \sum_{i = 1}^{M} L\cdot |t_i - t_{i-1}| = L\cdot |I| + \] + Also ist $\gamma$ rektifizierbar. + \end{bsp} + \begin{satz}[Kurvenlänge stückweiser $C^1$-Kurven] + Sei $\gamma\in C^0([a,b],\R^n)$ eine stetige Kurve mit $[a,b]$ kompakt, $\gamma$ stückweise $C^1$, d.h. $\exists$ Zerlegung $a = s_0 < s_1 <\dots < s_M = b$ mit + \[ + \left.\gamma\right|_{[s_{i-1}, s_i]} \in C^1([s_{i-1},s_i], \R^n),\quad \forall i = 1, \dots, M. + \] + Dann ist $\gamma$ rektifizierbar und hat die Länge \[S(\gamma) = \int_a^b \norm{\gamma'(t)}_2\d t = \sum_{i = 1}^{M}\int_{s_{i-1}}^{s_i} \norm{\gamma'(t)}_2 \d t.\] + Insbesondere hat der Graph einer $C_1$-Funktion $f\in C^1([a,b], \R), \quad \gamma_f(t)\colon (t, f(t))^T$ die Länge \[S(\gamma_f) = \int_a^b \sqrt{1 + f'(t)^2} \d t.\] + \end{satz} + \begin{proof} + Sei $\mathcal{Z} = \{t_0, \dots, t_N\}$ eine Partition, betrachte $\mathcal{Z}^* = \mathcal{Z} \cup \{s_0,\dots, s_M\} = \{x_0,\dots, x_K\}$. Dann gilt + \begin{salign*} + S(\mathcal{Z}) &\leq S(\mathcal{Z}^*)\\ + &= \sum_{i = 1}^{K}\norm{\gamma(x_i) - \gamma(x_{i-1})}\\ + &\stackrel{\text{HDI}}{=} \sum_{i = 1}^{K}\norm{\int_{x_{i-1}}^{x_i} \gamma'(t)\d t}\\ + &\stackrel{\triangle-\text{UGl.}}{\le} \sum_{i = 1}^{K}\int_{x_{i-1}}^{x_i} \norm{\gamma'(t)} \d t\\ + &= \int_a^b\norm{\gamma'(t)}\d t + \end{salign*} + Also ist $\gamma$ rektifizierbar und $S(\gamma) \leq \int_a^b \norm{y'(t)}\d t$. + Z.Z. $\forall \epsilon > 0\; \exists$ Zerlegung mit + \[ + S(\mathcal{Z}) \geq \int_a^b \norm{\gamma'(t)} \d t - \epsilon. + \] + Fixiere dazu ein $\epsilon > 0$. Wähle dann eine Treppenfunktion $\varphi$ auf $[a,b]$ mit \[\norm{\gamma'(t) - \varphi(t)} \le \frac{\epsilon}{2(b-a)} \forall t\in [a,b]\setminus\{s_0,\dots, s_M\}\] + ($\varphi$ existiert, weil $\gamma'(t)$ stückweise stetig ist). Wähle ferner eine (feine) Partition $a = t_0< t_1< \dots < t_N = b$, s.d. $\varphi\big|_{[t_{i-1}, t_i]}$ konstant ist $\forall i = 1,\dots, N$. Dann gilt nämlich + \begin{salign*} + S(Z) &= \sum_{i = 1}^{N}\norm{\int_{t_{i-1}}^{t_i} \gamma'(t) \d t}\\ + &= \sum_{i = 1}^{N}\norm{\int_{t_{i-1}}^{t_i} \varphi(t) + (\gamma'(t) - \varphi(t))\d t}\\ + &\geq \sum_{i = 1}^{N}\left( \norm{\int_{t_{i-1}}^{t_i} \varphi(t)\d t} - \norm{\int_{t_{i-1}}^{t_i} (\gamma'(t) - \varphi(t)) \d t}\right)\\ + &\geq \sum_{i = 1}^{N}\left( \norm{\int_{t_{i-1}}^{t_i} \varphi(t)\d t} - \frac{\epsilon}{2(b-a)} |t_i-t_{i-1}|\right)\\ + &\geq \sum_{i = 1}^{N}\norm{\int_{t_{i-1}}^{t_i} \varphi(t) \d t} - \sum_{i = 1}^{N}\frac{\epsilon}{2(b-a)}|t_i-t_{i-1}|\\ + &= \sum_{i = 1}^{N}\norm{\int_{t_{i-1}}^{t_i} \varphi(t) \d t} - \frac{\epsilon}{2}\\ + &\stackrel{\varphi(t) = \mathrm{const}}{=} \sum_{i = 1}^{N}\int_{t_{i-1}}^{t_i} \norm{\varphi(t)} \d t - \frac{\epsilon}{2}\\ + &= \sum_{i = 1}^{N}\norm{\gamma'(t) + (\varphi(t) - \gamma'(t))} \d t - \frac{\epsilon}{2}\\ + &\stackrel{\text{analog}}{\geq} \sum_{i = 1}^{N}\int_{t_{i-1}}^{t_i} \norm{\gamma'(t)} \d t - \epsilon = \int_a^b\norm{\gamma'(t)} \d t - \epsilon + \end{salign*} + Also existiert für ein beliebiges $\epsilon > 0$ eine Zerlegung $\mathcal{Z}$ mit $S(\mathcal{Z}) \geq \int_a^b\norm{\gamma'(t)}\d t - \epsilon$. Zusammen mit $S(y) \leq \int_a^b\norm{\gamma'(t)}\d t$ folgt $S(\gamma) = \int_a^b \norm{\gamma'(t)} \d t$. + \end{proof} + \begin{bsp} + \begin{enumerate} + \item Kreisbogen: $\gamma(t) = \begin{pmatrix} + r\cos(t)\\ + r\sin(t)\\ + \end{pmatrix},\; \gamma \in C^\infty([0,\varphi], \R^2),\; r > 0,\; \varphi > 0$ fest. Es gilt $S(\gamma) = \int_0^\varphi\norm{\gamma'(t)}_2 \d t = \int_0^\varphi \norm{\begin{pmatrix} + -r\sin(t)\\ r\cos(t) + \end{pmatrix}}_2 \d t = \int_0^\varphi r\d t = r\varphi$. Also ist der Umfang des Einheitskreises genau $\int_0^{2\pi} \underbrace{r}_{=1} \d t = 2\pi$. + \item Zykloide $\gamma\colon [0,2\pi] \to \R^2,\; r(t) = \begin{pmatrix} + t-\sin(t)\\1-\cos(t) + \end{pmatrix}$. + Wir erhalten $\gamma'(t) = \begin{pmatrix} + 1 - \cos(t)\\\sin(t) + \end{pmatrix}$ und daher + $\norm{\gamma'(t)}_2^2 = 1 - 2 \cos(t) + \cos^2(t) + \sin^2(t) = 2 - 2\cos(t) = 4 \sin^2(\frac{t}{2})$. Insgesamt gilt also + \[ + S(\gamma) = \int_0^{2\pi} \underbrace{\left|2\sin\left(\frac{t}{2}\right)\right|}_{\geq 0} \d t \oldstackrel{x = \frac{t}{2}}{=} 4\int_0^\pi \sin(x) \d x = 8\] + \end{enumerate} + \end{bsp} + \section{Parametertransformationen} + \begin{definition}[Parametertransformation] + \begin{enumerate} + \item Sei $\varphi\colon [\alpha, \beta]\! \to\! [a,b]$ eine $C^k$-Abbildung $(k\in \N_0 \cup \infty_+)$ zwischen den Intervallen $[\alpha,\beta]$ und $[a,b]$, + sei außerdem $\varphi$ bijektiv und $\varphi^{-1} \in C^k([a,b],\;[\alpha, \beta])$. + Dann heißt $\varphi$ eine $C^k$-Parametertransformation. + \item Sei weiter $\gamma\colon [a,b]\to\R^n$ eine Kurve. Dann heißt die Kurve $\delta:[\alpha, \beta] \to \R^n,\;\delta \coloneqq \gamma\circ \varphi$ die Umparametrisierung von $\gamma$ (mittels $\varphi$). + \item Die Parametertransformation $\varphi\colon [\alpha,\beta] \to [a,b]$ heißt orientierungstreu (oder orientierungserhaltend), wenn $\varphi$ streng monoton wächst; $\varphi$ heißt orientierungsumkehrend, wenn $\varphi$ streng monoton fällt. + \end{enumerate} + \end{definition} + \begin{bem} + \begin{enumerate} + \item Falls $\varphi$ eine $C^1$-Parametertransformation ist, dann gilt \[\varphi'(t) \neq 0,\;\forall t\in I,\]d.h. $\varphi$ ist ein Diffeomorphismus. Ferner heißt $\varphi$ orientierungstreu, falls $\varphi'(t) > 0$ und orientierungsumkehrend, falls $\varphi'(t) < 0$. + \item Die Bogenlänge $S(\gamma)$ ändert sich nicht beim Umparametrisieren: Seien $\varphi, \varphi^{-1}$ stetig differenzierbar, $\gamma\in C^1([a,b], \R^n)$. Dann gilt für die Bogenlänge + \begin{salign*} + S(\gamma\circ \varphi) &= \int_\alpha^\beta \norm{(\gamma\circ \varphi)'(\tau)} \d \tau\\ + &\stackrel{\text{Kettenregel}}{=} \int_\alpha^\beta\norm{\gamma'(\varphi(\tau)) \cdot \varphi'(\tau)} \d \tau\\ + &= \begin{cases} + \int_\alpha^\beta \norm{\gamma'(\varphi(\tau))}\cdot \varphi'(\tau) \d \tau&\varphi'(\tau) > 0, \tau \in [a,b]\\ + -\int_\alpha^\beta \norm{\gamma'(\varphi(\tau))}\cdot \varphi'(\tau) \d \tau&\varphi'(\tau) < 0, \tau \in [a,b]\\ + \end{cases}\\ + &\stackrel{\substack{t=\varphi(\tau)\\\d t = \varphi'(\tau)\d \tau}}{=} + \begin{cases} + \int_a^b\norm{\gamma'(t)} \d t&\varphi'(\tau) > 0, \tau \in [a,b]\\ + -\int_b^a \norm{\gamma'(t)} \d t &\varphi'(\tau) < 0, \tau \in [a,b]\\ + \end{cases}\\ + &= \int_a^b\norm{\gamma'(t)} \d t\\ + &= S(\gamma) + \end{salign*} + \item Umparametrisierung auf Bogenlänge. Sei $\gamma\colon [a,b] \to \R^n$ eine reguläre $C^1$-Kurve, d.h. $\gamma'(t) \neq 0\forall t\in [a,b]$. Definiere die Abbildung $\sigma\colon [a,b] \to [0, S(\gamma)]$ durch + \[ + \sigma(t) \coloneqq \int_a^t\norm{\gamma'(\tau)} \d \tau \left(= S(\gamma(t))\bigg|_{[a,t]}\right). + \] + Wir können zeigen, dass $\varphi\coloneqq \sigma^{-1}$ eine orientierungstreue $C^1$-Parametertransformation ist und für die (\glqq auf Bogenlänge\grqq) umparametrisierte Kurve $\beta\colon [0,S(\gamma)]\to \R^n$ gilt \[S\left(\beta\big|_{[0,x]}\right) = x,\; \norm{\beta'(x)} = 1,\; \forall x\in [0,S(\gamma)].\] + \begin{proof} + Es gilt $\sigma \in C^1([a,b];\;[0,S(\gamma)])$ mit $\sigma'(t) = \norm{\gamma'(t)} > 0$. Daher ist $\sigma$ streng monoton wachsend und bijektiv. Wegen Satz \ref{umkehrfunktion} folgt + \[ + \underbrace{\sigma^{-1}}_{\eqqcolon \varphi} \in C^1([0,S(y)];\; [a,b]),\; (\sigma^{-1})'(x) = \varphi'(x) = \frac{1}{\norm{\gamma'(\varphi(x))}} > 0. + \] + Also ist $\varphi$ streng monoton wachsend und daher muss $\varphi$ orientierungstreu sein. Für \[\beta\colon [0,S(\gamma)]\to \R^n,\quad \beta \coloneqq \gamma \circ \varphi\] gilt \[\beta'(x) \oldstackrel{\text{Kettenregel}}{=} \gamma'(\varphi(x))\cdot \varphi'(x) = \frac{\gamma'(\varphi(x))}{\norm{\gamma'(\varphi(x))}}.\] Also erhalten wir $\norm{\beta'(x)} = 1\;\forall x\in [0,S(\gamma)]$ und damit $S\left(\beta|_{[0,x]}\right) = \int_0^x \norm{\beta'(s)}\d s = x$. + \end{proof} + \end{enumerate} + \end{bem} +\end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index 0a506e3e33b6126207a813c0d5e73160b468a06a..345a975227e0078b7dcd92e9ea905354d12b22d2 100644 GIT binary patch delta 182590 zcmZUaWmH>jx2_?<-QC??ixewPafhPCi@ODPDH^mmr8q@`dvGsKaV=IT?uWkbxA)oS z>>n9G0FYvQ$7h(y9%R9sX}=5{Eeq9~loA04gTZKwo<1*yQ) zX0Xg4@TluD7hc4TXgK(~SISF&l^(sNAyXOQCr6e$bqrS)(GPA0g^NqECV2i-<{zFv z=vHrOD2;C?YF?rO-?TGGHXUzLp*lOyi@f)H(i{fFHx%>8e+{OI;-z3p%9oj;%7qi$ zgwEYfe@w0AA`c)#dyCGo_Qk|Fu5j{K+cPsZD0dX@4{F&R8a8YC@5hqRsz13y+~porXeGp`PiZl%t`Zqj@hfYTD0&cg!<- 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z%$5~xhg~6WG(K&L2QUKV)VT;Z?~`VLKxD}8>vufa2-#1O>k6V9vEF_Jmz@0Bu;&4? z*Y!}c;)bDg`r-Kc@aI0VANrm*4sjg&=s!-rZ8Su_ zP@APmn`IC^2BZ#yoezU#AgKNyi5@JV5Vn!f`Zb8scTxm)Ww@WUE@m}%1*cGKoUPqG n+-xkI{*hcP9Z=joEZjW2-E7{Yyv1&D@^W#a(9y}M$)o%)b?_Qj diff --git a/analysisII.tex b/analysisII.tex index bf29fa3..ead6073 100644 --- a/analysisII.tex +++ b/analysisII.tex @@ -42,5 +42,6 @@ Rui Yang (\href{mailto:rui.yang@stud.uni-heidelberg.de}{rui.yang@stud.uni-heidel \input{ana17.tex} \input{ana18.tex} \input{ana19.tex} +\input{ana20.tex} \end{document}