From 032f35092a72a787fd4f4a36fde295f900b22e8e Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Tue, 11 Feb 2014 12:23:43 +0100 Subject: [PATCH] Digitalisieren der Vorlesung von 11.02.2014 --- documents/GeoTopo/Arbeitszeit.md | 3 +- documents/GeoTopo/GeoTopo.pdf | Bin 5131072 -> 5142181 bytes documents/GeoTopo/GeoTopo.sublime-project | 2 +- documents/GeoTopo/Kapitel2.tex | 2 + documents/GeoTopo/Kapitel5.tex | 136 ++++++++++++++++++++++ documents/GeoTopo/Vorwort.tex | 6 +- 6 files changed, 144 insertions(+), 5 deletions(-) diff --git a/documents/GeoTopo/Arbeitszeit.md b/documents/GeoTopo/Arbeitszeit.md index 79e3515..c105546 100644 --- a/documents/GeoTopo/Arbeitszeit.md +++ b/documents/GeoTopo/Arbeitszeit.md @@ -75,4 +75,5 @@ in dem Erstellen dieses Skripts steckt: |10.02.2014 | 11:05 - 11:20 | Verbesserungsvorschläge von Marco, Email 1 vom 10.02.2014, umgesetzt. |10.02.2014 | 11:40 - 13:20 | Verbesserungsvorschläge von Jérôme Urhausen, Email 1 vom 10.02.2014, umgesetzt. |11.02.2014 | 05:30 - 06:00 | TikZ'en eines Bildes mit Hilfe von Jérôme Urhausen (Email 1 vom 10.02.2014) -|11.02.2014 | 06:30 - 07:00 | Verbesserungsvorschläge von Jérôme Urhausen, Email 2 vom 10.02.2014, umgesetzt. \ No newline at end of file +|11.02.2014 | 06:30 - 07:00 | Verbesserungsvorschläge von Jérôme Urhausen, Email 2 vom 10.02.2014, umgesetzt. +|11.02.2014 | 09:45 - 12:20 | Digitalisieren der Vorlesung von 11.02.2014 \ No newline at end of file diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 9117ddaf05dc2227d07c9ebc24a7d6378bd55649..b94c46e26ea108b3818309d2959bccc9bdd85eaa 100644 GIT binary patch delta 738880 zcmV(?K-a&(mbn6@?YROHB~V00Eio=N3MJ6g#n=ilHZ_;ATQd}~CTal!HZzx@&oNGu z-~lcIH!-si0t`GDIUq0~Z(?d7JUj|7Ol59obZ8(kIWRUfmjNR+D1UuflO(s1<~x5y z?pw8)G8q8`j@XwSX(i4ft?fu`lh#a{hbFs7)+$gfRd-7}{PpXOD+7RgGd+|v+S<;+ z&Wg;4z~RT|pF4o*^zCUn{qtv2_wO%WfA;woi#w_5Vy!Z#*WaA{deO^ra+AL(vz}hR zJ^k_F_ph{ic=t-{hkuv9Uq60$^&hYQtv>4WFJ^n1r^`C6Gd>4CdH7;F&8Ky`%!@i% 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"file_exclude_patterns": ["*.aux", "*.fdb_latexmk", "*.out", "*.idx", "*.toc"] + "file_exclude_patterns": ["*.aux", "*.fdb_latexmk", "*.out", "*.idx", "*.toc", "*.ilg", "*.thm", "*.ind"] } ], "settings": diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index fa768f2..6397f29 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -81,6 +81,8 @@ Anschaulich ist also ein $n$-dimensionale Mannigfaltigkeit lokal dem $\mdr^n$ ä $(x_1, \dots, x_{n+1}) \mapsto (x_1, \dots, \cancel{x_i}, \dots, x_{n+1})$\footnote{$x_i$ wird rausgenommen}\\ $(x_1, \dots, x_{n}) \mapsto (x_1, \dots, x_{i-1}, \sqrt{1-\sum_{k=1}^n x_k^2}, x_i, \dots, x_n)$, oder $-\sqrt{1-\sum_{k=1}^n x_k^2}$ für $C_i$\\ $S^n = \bigcup_{i=1}^{n+1} (C_i \cup D_i)$ + + Als kompakte Mannigfaltigkeit wird $S^n$ auch \enquote{geschlossene Mannigfaltigkeit}\xindex{Mannigfaltigkeit!geschlossene} genannt. \item $[0,1]$ ist keine Mannigfaltigkeit, denn:\\ Es gibt keine Umgebung von $0$ in $[0,1]$, die homöomorph zu einem offenem Intervall ist. diff --git a/documents/GeoTopo/Kapitel5.tex b/documents/GeoTopo/Kapitel5.tex index fac234c..5557546 100644 --- a/documents/GeoTopo/Kapitel5.tex +++ b/documents/GeoTopo/Kapitel5.tex @@ -1,3 +1,4 @@ +%!TEX root = GeoTopo.tex %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mitschrieb vom 30.01.2014 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -370,3 +371,138 @@ Im folgenden werden diese Begriffe jedoch synonym benutzt. beide Seiten von $T_s S + s$. \end{bemenum} \end{bemerkung} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Mitschrieb vom 11.02.2014 % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Erste und zweite Fundamentalform}%In Vorlesung: §19 +Sei $S \subseteq \mdr^3$ eine reguläre Fläche, $s \in S$, $T_s S$ die Tangentialebene +an $S$ in $s$. + +\begin{bemerkung}%In Vorlesung: Bem.+Def. 19.1 + \begin{bemenum} + \item \label{bem:19.1a} Die Einschränkung des Standardskalarproduktes des $\mdr^3$ auf + $T_s S$ macht $T_s S$ zu einem euklidischen Vektorraum. + \item Sei $F: U \rightarrow V$ eine lokale Parametrisierung von $S$ um + $s$ und $p := F^{-1}(s)$. + + Dann ist $\Set{D_P F(e_1), D_P F(e_2)}$ eine Basis von $T_s S$. + \item Bzgl. der Basis $\Set{D_P F(e_1), D_P F(e_2)}$ hat das + Standardskalarprodukt aus \cref{bem:19.1a} die Darstellungsmatrix + \[ I_S = \begin{pmatrix} + g_{1,1}(s) & g_{1,2}(s)\\ + g_{1,2}(s) & g_{2,2}(s) + \end{pmatrix} = + \begin{pmatrix} + E(s) & F(s) \\ + F(s) & G(s) + \end{pmatrix}\] + mit $\begin{aligned} + g_{i,j} &= g_s(D_P F(e_i), D_P F(e_j))\\ + &= \langle \frac{\partial F}{\partial u_i} (p), \frac{\partial F}{\partial u_j} (p) \rangle \;\;\; i,j \in \Set{1,2} + \end{aligned}$.\\ + Die Matrix $I_S$ heißt \textbf{erste Fundamentalform}\xindex{Fundamentalform!erste} + von $S$ bzgl. der Parametrisierung $F$. + \item $g_{i,j}(s)$ ist eine differenzierbare Funktion von $s$. + \end{bemenum} +\end{bemerkung} + +\begin{bemerkung} + \[\det(I_S) = \| \frac{\partial F}{\partial u_1}(p) \times \frac{\partial F}{\partial u_2}(p)\|^2\] +\end{bemerkung} + +\begin{beweis} + Sei $\frac{\partial F}{\partial u_1}(p) = \begin{pmatrix} + x_1\\ x_2 \\ x_3 + \end{pmatrix}, \;\;\; \frac{\partial F}{\partial u_2}(p) = \begin{pmatrix} + y_1\\ y_2 \\ y3 + \end{pmatrix}$ + + Dann ist $\frac{\partial F}{\partial u_1}(p) \times \frac{\partial F}{\partial u_2}(p) = \begin{pmatrix} + z_1 \\ z_2 \\ z_3 + \end{pmatrix}$ mit + \begin{align*} + z_1 &= x_2 y_3 - x_3 - y_2\\ + z_2 &= x_3 y_1 - x_1 y_3\\ + z_3 &= x_1 y_2 - x_2 y_1 + \end{align*} + $\Rightarrow \|\frac{\partial F}{\partial u_1} (p) \times \frac{\partial F}{\partial u_2} (p)\| = z_1^2 + z_2^2 + z_3^2$\\ + \begin{align*} + \det(I_S) &= g_{1,1} g_{2,2} - g_{1,2}^2\\ + &= \langle \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}\rangle \langle \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}\rangle - \langle \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \begin{pmatrix} y_1 \\ y_2 \\ y_3 \end{pmatrix}\rangle^2\\ + &= (x_1^2 + x_2^2 + x_3^2) (y_1^2 + y_2^2 + y_3^2) - (x_1 y_1 + x_2 y_2 + x_3 y_3)^2 + \end{align*} +\end{beweis} + +\begin{definition}\xindex{Flächenelement}%In Vorlesung: Def.+Bem. 19.3 / Erinnerung + \begin{defenum} + \item Das Differential + \[\mathrm{d} A = \sqrt{\det (I)} \mathrm{d} u_1 \mathrm{d} u_2\] + heißt \textbf{Flächenelement} von $S$ bzgl. der Parametrisierung $F$. + \item \label{def:berechenbares-integral}Für eine Funktion $f: V \rightarrow \mdr$ heißt + \[\int_V f \mathrm{d} A := \int_U f(\underbrace{F(u_1, u_2)}_{=: s}) \sqrt{\det I(s)} \mathrm{d} u_1 \mathrm{d} u_2\] + der \textbf{Wert des Integrals} von $f$ über $V$, falls das Integral rechts + existiert. + \end{defenum} + +\end{definition} + +\begin{bemerkung} + \begin{bemenum} + \item $\int_V f \mathrm{d} A$ ist unabhänig von der gewählten Parametrisierung. + \item Sei $f: S \rightarrow \mdr$ eine Funktion, die im Sinne von + \cref{def:berechenbares-integral} lokal integrierbar ist. + + Dann ist $\int_S f \mathrm{d} A$ wohldefiniert, falls (z.~B.) $S$ + kompakt ist. + + Etwa: $\int_S f \mathrm{d} A = \sum_{i=1}^n \int_{V_i} f \mathrm{d} A - \sum_{i \neq j} \int_{V_i \cap V_j} f \mathrm{d} A + \sum_{i,j,k} \int_{V_i \cap V_j \cap V_k} - \dots$ + \end{bemenum} +\end{bemerkung} + +\begin{beweis} + \begin{enumerate}[label=\alph*)] + \item Mit Transformationsformel + \item Ist dem Leser überlassen + \end{enumerate} +\end{beweis} + +\begin{proposition} + Sei $S \subseteq \mdr^3$ eine reguläre, orientierbare Fläche mit glatten + Normalenfeld $n: S \rightarrow S^2$. + + \begin{propenum} + \item $n$ induziert für jedes $s \in S$ eine lineare Abbildung $d_S n: T_s S \rightarrow T_{n(s)} S^2$ + durch + \[d_s n(x) = \frac{\mathrm{d}}{\mathrm{d} t} n (\underbrace{s \text{\enquote{+}} tx}_{\mathclap{\text{Soll auf Fläche $S$ bleiben}}}) \Bigr |_{t=0}\] + \item $T_{n(s)} S^2 = T_s S$ + \item $d_S n$ ist ein Endomorphismus von $T_s S$ + \item $d_S n$ ist selbstadjungiert bzgl. des Skalarproduktes $I_S$ + \end{propenum} +\end{proposition} + +\begin{beweis}\leavevmode + \begin{enumerate}[label=\alph*)] + \item TODO + \item $T_{n(S)} S^2 = \langle n(s) \rangle^\bot = T_s S$ + \item TODO + \item Zu zeigen: $\forall x,y \in I_s S: \langle x, d_s n (y) \rangle = \langle d_s n(x), y \rangle$ + + Aufgrund der Bilinearität des Skalarproduktes genügt es diese Eigenschaft + für die Basisvektoren zu zeigen. + + Sei $x_i = D_P F(e_i) = \frac{\partial F}{\partial u_i} (p)\;\;\; i = 1,2$ + + \underline{Beh.:} $\langle x_i, d_s n(x_j) \rangle = \langle \frac{\partial^2 F}{\partial u_i \partial u_j} (p), d_s n (x_i) \rangle$ + + $\Rightarrow \langle \frac{\partial^2 F}{\partial u_i \partial u_j} (p), d_s n (x_i) \rangle = \langle x_j, d_s n (x_i) \rangle$ + + \underline{Bew.:} + + \begin{align*} + 0 &= \hphantom{\frac{\mathrm{d}}{\mathrm{d}t} \left (\right.} \langle \frac{\partial F}{\partial u} (p + t e_j), n(p + t e_j) \rangle\\ +\Rightarrow 0 &= \frac{\mathrm{d}}{\mathrm{d}t} \left (\langle \frac{\partial F}{\partial u} (p + t e_j), n(p + t e_j) \rangle \right) \Bigr |_{t=0}\\ + &= \langle \underbrace{\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial F}{\partial u_i} (p + t e_j)}_{\frac{\partial^2 F}{\partial u_j \partial u_i} (p)} \Bigr |_{t=0}, n(s) \rangle + \langle x_i, d_s n \underbrace{D_P F (e_j)}_{x_j}\rangle + \end{align*} + \end{enumerate} +\end{beweis} \ No newline at end of file diff --git a/documents/GeoTopo/Vorwort.tex b/documents/GeoTopo/Vorwort.tex index 9b290ba..3b7734c 100644 --- a/documents/GeoTopo/Vorwort.tex +++ b/documents/GeoTopo/Vorwort.tex @@ -1,8 +1,8 @@ %!TEX root = GeoTopo.tex \chapter*{Vorwort} -Dieses Skript wird/wurde im Wintersemester 2013/2014 +Dieses Skript wurde im Wintersemester 2013/2014 von Martin Thoma geschrieben. Es beinhaltet die Mitschriften aus -der Vorlesung von Prof. Dr. Herrlich sowie die Mitschriften einiger +der Vorlesung von Prof.~Dr.~Herrlich sowie die Mitschriften einiger Übungen und Tutorien. An dieser Stelle möchte ich Herrn~Prof.~Dr.~Herrlich für einige @@ -65,7 +65,7 @@ Häufungspunkten nicht weiter schwer fallen. Diese Vorkenntnisse werden vor allem in \enquote{Analysis I} vermittelt. Außerdem wird vorausgesetzt, dass (affine) Vektorräume, Faktorräume, -lineare Unabhängigkeit und und der projektive Raum $\praum(\mdr)$ aus +lineare Unabhängigkeit, der Spektralsatz und der projektive Raum $\praum(\mdr)$ aus \enquote{Lineare Algebra I} bekannt sind. In \enquote{Lineare Algebra II} wird der Begriff der Orthonormalbasis eingeführt.