From 52ed72ca0c1e0c7b17128c1dea0a6c5a306fdc12 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Tue, 14 Jan 2014 00:04:44 +0100 Subject: [PATCH] Tippfehler korrigiert --- documents/GeoTopo/Arbeitszeit.md | 2 +- documents/GeoTopo/GeoTopo.pdf | Bin 4692569 -> 4692649 bytes documents/GeoTopo/Kapitel1.tex | 6 ++--- documents/GeoTopo/Kapitel2.tex | 42 +++++++++++++++---------------- documents/GeoTopo/Kapitel3.tex | 12 ++++----- documents/GeoTopo/Kapitel4.tex | 12 ++++----- documents/GeoTopo/Readme.md | 5 +++- documents/GeoTopo/shortcuts.sty | 2 ++ 8 files changed, 43 insertions(+), 38 deletions(-) diff --git a/documents/GeoTopo/Arbeitszeit.md b/documents/GeoTopo/Arbeitszeit.md index 6fa8eab..0a2d1bb 100644 --- a/documents/GeoTopo/Arbeitszeit.md +++ b/documents/GeoTopo/Arbeitszeit.md @@ -27,4 +27,4 @@ in dem Erstellen dieses Skripts steckt: |12.01.2014 | 16:00 - 16:15 | `cleveref` benutzt |12.01.2014 | 22:15 - 22:30 | Beweis erstellt, dass Überlagerungen surjektiv sind |12.01.2014 | 23:30 - 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b/documents/GeoTopo/Kapitel1.tex @@ -402,12 +402,12 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{bemerkung} \begin{enumerate}[label=\alph*)] \item Für jeden topologischen Raum ist - $\text{Homöo}(X) := \Set{f: X \rightarrow X | f \text{ ist Homöomorphismus}}$ + $\Homoo(X) := \Set{f: X \rightarrow X | f \text{ ist Homöomorphismus}}$ eine Gruppe.\xindex{Homöomorphismengruppe} \item Jede Isometrie $f:X \rightarrow Y$ zwischen metrischen Räumen ist ein Homöomorphismus. \item $\Iso(X) := \Set{f:X \rightarrow X | f \text{ ist Isometrie}}$ ist - eine Untergruppe von $\text{Homöo}(X)$ für jeden + eine Untergruppe von $\Homoo(X)$ für jeden metrischen Raum $X$. \end{enumerate} \end{bemerkung} @@ -580,7 +580,7 @@ sodass $\pi$ stetig wird. \begin{korollar} Sei $X$ ein topologischer Raum. Dann gilt: \begin{enumerate}[label=\alph*)] - \item $Z(X)$ ist die größte zusammehängede Teilmenge von $X$, + \item $Z(X)$ ist die größte zusammenhängende Teilmenge von $X$, die $x$ enthält. \item $Z(X)$ ist abgeschlossen. \item $X$ ist disjunkte Vereinigung von Zusammenhangskomponenten. diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index bc45a53..fb075e8 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -1,7 +1,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Henriekes Mitschrieb vom 07.11.2013 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\chapter{Mannigfaltigkeiten und Simpizidkomplexe} +\chapter{Mannigfaltigkeiten und Simplizialkomplexe} \section{Topologische Mannigfaltigkeiten} \begin{definition} Sei $X$ ein topologischer Raum und $n \in \mdn$. @@ -120,7 +120,7 @@ $Z$ heißt \textbf{Verklebung} von $X$ und $Y$ längs $U$ und $V$. $Z$ besitzt einen Atlas aus $n$-dimensionalen Karten. - Falls $Z$ hausdoffsch ist, ist $Z$ eine $n$-dimensionale + Falls $Z$ hausdorffsch ist, ist $Z$ eine $n$-dimensionale Mannigfaltigkeit. \end{definition} @@ -352,14 +352,14 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{beispiel} $f: \mdr \rightarrow \mdr, \;\;\; x \mapsto x^3$ ist kein - Diffeomorphismis, aber Homöomorphismus, da mit $g(x) := \sqrt[3]{x}$ + Diffeomorphismus, aber Homöomorphismus, da mit $g(x) := \sqrt[3]{x}$ gilt: $f \circ g = \id_\mdr, \;\;\; g \circ f = \id_\text{\mdr}$ \end{beispiel} \begin{bemerkung} Sei $X$ eine glatte Mannigfaltigkeit. Dann ist - \[\text{Diffeo}(X) := \Set{f:X \rightarrow X | f \text{ ist Diffeomorphismus}}\] - eine Untergruppe von $\text{Homöo}(X)$. + \[\Diffeo(X) := \Set{f:X \rightarrow X | f \text{ ist Diffeomorphismus}}\] + eine Untergruppe von $\Homoo(X)$. \end{bemerkung} \begin{definition} @@ -390,7 +390,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{figure} \centering - \subfloat[Kugelkooridnaten]{ + \subfloat[Kugelkoordinaten]{ \includegraphics[width=0.45\linewidth, keepaspectratio]{figures/spherical-coordinates.pdf} \label{fig:spherical-coordinates} }% @@ -399,7 +399,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \label{fig:solid-of-revolution} }% - \subfloat[Sinus und Cosinus haben keine gemeinsamme Nullstelle]{ + \subfloat[Sinus und Kosinus haben keine gemeinsame Nullstelle]{ \includegraphics[width=0.8\linewidth, keepaspectratio]{figures/sin-cos.pdf} \label{fig:sin-cos} }% @@ -445,7 +445,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \end{korollar} \begin{beweis} - \todo{Hier muss ich nochmals drüberlesen.} + \todo{Hier muss ich nochmals drüber lesen.} \underline{z.Z.:} $F_j^{-1} \circ F_i$ ist Diffeomorphismus \begin{figure}[htp] @@ -526,11 +526,11 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. a_{n1} & \dots & a_{nn} \end{pmatrix} \in \mdr^{(n-1) \times (n-1)}\] - ist diffbar. + ist differenzierbar. $\det A_{ij}$ kann $0$ werden, da: \[\begin{pmatrix}1 & 1\\-1&0\end{pmatrix}\] - \item $\SL_n(\mdr) = \Set{A \in \GL_n(\mdr) | \text{det}(A) = 1} $ \todo{Besser strukturieren} + \item $\SL_n(\mdr) = \Set{A \in \GL_n(\mdr) | \det(A) = 1} $ \todo{Besser strukturieren} $\grad(\det-1)(A) = 0$? @@ -557,23 +557,23 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \item $v_0, \dots, v_k$ sind \textbf{in allgemeiner Lage} $\gdw$ es gibt keinen $(k-1)$-dimensionalen affinen Untervektorraum, der $v_0, \dots, v_k$ enthält \gdw $v_1 - v_0, \dots, v_k - v_0$ sind linear abhängig. - \item $\text{conv}(v_0, \dots, v_k) = \Set{\sum_{i=0}^k \lambda_i v_i | \lambda_i \geq 0, \sum_{i=0}^k \lambda_i = 1} $ + \item $\conv(v_0, \dots, v_k) = \Set{\sum_{i=0}^k \lambda_i v_i | \lambda_i \geq 0, \sum_{i=0}^k \lambda_i = 1} $ \end{enumerate} \end{definition} \begin{definition} \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] - \item Sei $\Delta^k = \text{conv}(e_0, \dots, e_k) \subseteq \mdr^{n+1}$ + \item Sei $\Delta^k = \conv(e_0, \dots, e_k) \subseteq \mdr^{n+1}$ die konvexe Hülle der Standard-Basisvektoren $e_0, \dots, e_k$. Dann heißt $\Delta^k$ \textbf{Standard-Simplex}\xindex{Standard-Simplex} und $k$ die Dimension des Simplex. \item Für Punkte $v_0, \dots, v_k$ im $\mdr^n$ in allgemeiner - Lage heißt $\delta (v_0, \dots, v_k) = \text{conv}(v_0, \dots, v_k)$ + Lage heißt $\delta (v_0, \dots, v_k) = \conv(v_0, \dots, v_k)$ ein \textbf{$k$-Simplex}\xindex{Simplex} in $\mdr^n$. \item Ist $\Delta (v_0, \dots, v_k)$ ein $k$-Simplex und $I = \Set{i_0, \dots, i_r} \subseteq \Set{0, \dots, k}$, - so heißt $s_{i_0, \dots, i_r} := \text{conv}(v_{i_0}, \dots, v_{i_r})$ + so heißt $s_{i_0, \dots, i_r} := \conv(v_{i_0}, \dots, v_{i_r})$ \textbf{Teilsimplex}\xindex{Teilsimplex} oder \textbf{Seite}\xindex{Seite} von $\Delta$. @@ -613,15 +613,15 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \item Eine endliche Menge $K$ von Simplizes im $\mdr^n$ heißt (endlicher) \textbf{Simplizialkomplex}\xindex{Simplizialkomplex}, wenn gilt: - \begin{enumerate}[label=(\roman*)] + \begin{enumerate}[label=(\roman*),ref=\theenumii.\roman*] \item Für $\Delta \in K$ und $S \subseteq \Delta$ Teilsimplex ist $S \in K$ \item Für $\Delta_1, \Delta_2 \in K$ ist - $\Delta_1 \cap \Delta_2$ leer - oder ein Teilsimplex von $\Delta_1$ und von - $\Delta_2$ + $\Delta_1 \cap \Delta_2$ leer oder ein + Teilsimplex von $\Delta_1$ und von + $\Delta_2$ \label{def:simplizialkomplex.ii} \end{enumerate} - \item $|K| := \bigcup_{\Delta \in K} \Delta$ (mit Spurtoplogie) + \item $|K| := \bigcup_{\Delta \in K} \Delta$ (mit Spurtopologie) heißt \textbf{geometrische Realisierung}\xindex{Realisierung!geometrische} von $K$. \item Ist $d = \max \Set{ k | K \text{ enthält } k-\text{Simplex}}$, @@ -655,7 +655,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \label{fig:simplizialkomplex-cube-divided} } - \subfloat[$P$ ist kein Teilsimplex, da Eigenschaft (ii) verletzt ist]{ + \subfloat[$P$ ist kein Teilsimplex, da Eigenschaft \ref{def:simplizialkomplex.ii} verletzt ist]{ \parbox[c][4cm]{5cm}{\centering\input{figures/topology-triangle-no-simplicial-complex.tex}} \label{fig:no-simplizialkomplex-triangles} }% @@ -852,7 +852,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \end{beweis} \begin{korollar}[Der Rand vom Rand ist 0]\label{kor:9.11} - Sei $K$ ein \todo{Warum in Klammern?}{(endlicher)} Simplizialkomplex mit Eckenmenge $V$ + Sei $K$ ein \todo{Warum in Klammern?}{(endlicher)} Simplizialkomplex mit Knotenmenge $V$ und $<$ eine Totalordnung auf $V$. Sei $A_n$ die Menge der $n$-Simplizes in $K$, d.~h. diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index c252ebe..0de8eb5 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -68,13 +68,13 @@ nicht homöotop. \item Sei $X = \mdr^2$ und $a=b=(0,0)$. - Je zwei Wege im $\mdr^2$ mit Anfangs- und Enpunkt $(0,0)$ + Je zwei Wege im $\mdr^2$ mit Anfangs- und Endpunkt $(0,0)$ sind homöotop. \begin{figure} \centering \input{figures/topology-paths-in-r2.tex} - \caption{Zwei Wege im $\mdr^2$ mit Anfangs- und Enpunkt $(0,0)$} + \caption{Zwei Wege im $\mdr^2$ mit Anfangs- und Endpunkt $(0,0)$} \label{fig:paths-from-origin} \end{figure} @@ -98,7 +98,7 @@ \label{fig:torus-three-paths} }% \label{fig:homotop-paths} - \caption{Beispiele für (nicht)-homotopie von Wegen} + \caption{Beispiele für (nicht)-Homotopie von Wegen} \end{figure} \end{beispiel} @@ -551,7 +551,7 @@ Haben wir Häufungspunkt definiert?} \underline{2. Fall}: $p(y_1) \neq p(y_2)$. - Dann seien $U_1$ und $U_2$ disjunkte Umbebungen von $p(y_1)$ + Dann seien $U_1$ und $U_2$ disjunkte Umgebungen von $p(y_1)$ und $p(y_2)$. $\Rightarrow p^{-1}(U_1)$ und $p^{-1}(U_2)$ sind Umgebungen von @@ -805,7 +805,7 @@ $p|V_j: V_j \rightarrow U$ Homöomorphismus. $\xRightarrow{q \text{ offen}} q(W)$ ist offene Umgebung von $p(z) \cdot d(\tilde{p}(z))$. - Sei $U \subseteq q(W)$ offen wie in Defintion~\ref{def:12.1} und + Sei $U \subseteq q(W)$ offen wie in Definition~\ref{def:12.1} und $V \subseteq q^{-1}(U)$ die \todo{Was?}{Komp.} die $\tilde{p}(z)$ enthält. @@ -1051,7 +1051,7 @@ und der Fundamentalgruppe herstellen: \begin{enumerate}[label=\arabic*),ref=\thebeispiel.\arabic*] \item $G = (\mdz, +), X = \mdr, nx = x + n$\label{bsp:gruppenoperation1} \item $G$ operiert auf $X = G$ durch $g \circ h := g \cdot h$ - \item $G$ operatiert auf $X = G$ durch $g \circ h := g \cdot h \cdot g^{-1}$, denn + \item $G$ operiert auf $X = G$ durch $g \circ h := g \cdot h \cdot g^{-1}$, denn \begin{enumerate}[label=\roman*)] \item $1_G \circ h = 1_G \cdot h \cdot 1_G^{-1} = h$ \item \begin{align*} diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index 368c559..174a9a0 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -6,7 +6,7 @@ Axiome\xindex{Axiom} bilden die Grundbausteine jeder mathematischen Theorie. Eine Sammlung aus Axiomen nennt man Axiomensystem\xindex{Axiomensystem}. Da der Begriff des Axiomensystems so grundlegend ist, hat man auch -ein paar sehr grundlegende Vorderungen an ihn: Axiomensysteme sollen +ein paar sehr grundlegende Forderungen an ihn: Axiomensysteme sollen \textbf{widerspruchsfrei} sein, die Axiome sollen möglichst \textbf{unabhängig} sein und \textbf{Vollständigkeit} wäre auch toll. Mit Unabhängigkeit ist gemeint, dass kein Axiom sich aus einem anderem @@ -19,7 +19,7 @@ Vollständigkeit. Ein Axiomensystem gilt als Vollständig, wenn jede Aussage innerhalb des Systems verifizierbar oder falsifizierbar ist. Interessant ist hierbei der Gödelsche Unvollständigkeitssatz, der z.~B. für die Arithmetik beweist, dass nicht alle Aussagen -formal bewiesen oder wiederlegt werden können. +formal bewiesen oder widerlegt werden können. Kehren wir nun jedoch zurück zur Geometrie. Euklid hat in seiner Abhandlung \enquote{Die Elemente} ein Axiomensystem für die Geometrie @@ -65,7 +65,7 @@ aufgestellt. \begin{definition} \begin{enumerate}[label=\alph*)] - \item $P, Q, R$ liegen \textbf{kolinear}\xindex{kolinear}, + \item $P, Q, R$ liegen \textbf{kollinear}\xindex{kollinear}, wenn es $g \in G$ gibt mit $\Set{P, Q, R} \subseteq g$. \item $Q$ \textbf{liegt zwischen}\xindex{liegt zwischen} $P$ und $R$, wenn $d(P, R) = d(P, Q) + d(Q, R)$ @@ -94,7 +94,7 @@ aufgestellt. \begin{enumerate}[label=(\roman*)] \item \enquote{$\subseteq$} folgt direkt aus der Definition von $PR^+$ und $PR^-$\\ \enquote{$\supseteq$}: Sei $Q \in PR \Rightarrow P, Q, R$ - sind kolinear.\\ + sind kollinear.\\ $\stackrel{\ref{axiom:2}}{\Rightarrow} \begin{cases} Q \text{ liegt zwischen } P \text{ und } R \Rightarrow Q \in PR\\ @@ -158,7 +158,7 @@ aufgestellt. \Obda sei $i=1$ und $j=2$, also $\varphi_1(R) = \varphi_2(R)$. \end{behauptung} \begin{behauptung} - Hat eine Isometrie $\varphi$ 3 Fixpunkte, die nicht kolinear sind, + Hat eine Isometrie $\varphi$ 3 Fixpunkte, die nicht kollinear sind, so ist $\varphi = \id_X$. Aus Beh. 1 und Beh. 2 folgt, dass $\varphi_2^{-1} \circ \varphi_1 = \id_X$, @@ -172,7 +172,7 @@ aufgestellt. \end{behauptung} \begin{beweis} Es ist $\varphi(PQ) = \varphi(P) \varphi(Q)$ weil $\varphi$ - wegen \ref{axiom:2} kolinearität erhält. + wegen \ref{axiom:2} Kollinearität erhält. Sei nun $R \in PQ$. Dann ist $d(P, \varphi(R)) \stackrel{P \text{ ist Fixpunkt}}{=} d(\varphi(P), \varphi(R)) = d(P, R)$. Weiter ist $\varphi (PQ^+) = \varphi(P) \varphi(Q)^+ = PQ^+$ diff --git a/documents/GeoTopo/Readme.md b/documents/GeoTopo/Readme.md index 1ab3b84..71fba12 100644 --- a/documents/GeoTopo/Readme.md +++ b/documents/GeoTopo/Readme.md @@ -57,4 +57,7 @@ Was noch kommen soll * ... replace `\usepackage[...]{hyperref}` by `\usepackage{nohyperref}` * In `titlepage.tex`: replace `10cm` by `4cm` * Momentan sind es ca. 60 Seiten in A4. In A5 sind es ca. -4. Version für Sehgeschädigte und Blinde +4. Version für Sehgeschädigte: + * min `12pt`, besser `14pt` + * nicht `article`, `book`, `report` sondern `extarticle` + * Sans serif: Arial, Helvetica (`\usepackage{cmbright}`) diff --git a/documents/GeoTopo/shortcuts.sty b/documents/GeoTopo/shortcuts.sty index 6a40609..c39500f 100644 --- a/documents/GeoTopo/shortcuts.sty +++ b/documents/GeoTopo/shortcuts.sty @@ -80,6 +80,8 @@ \DeclareMathOperator{\Perm}{Perm} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Homoo}{Homöo} +\DeclareMathOperator{\Diffeo}{Diffeo} +\DeclareMathOperator{\conv}{conv} %%%Text %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand\obda{o.~B.~d.~A.\xspace}