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Symbolverzeichnis verbessert

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Martin Thoma 2014-03-19 10:57:57 +01:00
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documents/GeoTopo/GeoTopo.pdf
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documents/GeoTopo/GeoTopo.tex
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@ -45,6 +45,13 @@
\usepackage{tqft} \usepackage{tqft}
\usepackage{xspace} % for new commands; decides weather I want to insert a space after the command \usepackage{xspace} % for new commands; decides weather I want to insert a space after the command
\usepackage[german,nameinlink,noabbrev]{cleveref} % has to be after hyperref, ntheorem, amsthm \usepackage[german,nameinlink,noabbrev]{cleveref} % has to be after hyperref, ntheorem, amsthm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{array,xtab,ragged2e} % for symbol table
\newlength\mylengtha
\newlength\mylengthb
\newcolumntype{P}[1]{>{\RaggedRight}p{#1}}
\tabcolsep=3pt % default: 6pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{acronym} \usepackage{acronym}
\usepackage{cancel} \usepackage{cancel}
\usepackage{shortcuts} \usepackage{shortcuts}

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documents/GeoTopo/Kapitel1.tex
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@ -449,7 +449,7 @@ Die Teilraumtopologie wird auch \textit{Spurtopologie} oder
\begin{bemerkung} \begin{bemerkung}
\begin{bemenum} \begin{bemenum}
\item \xindex{Homöomorphismengruppe}Für jeden topologischen Raum ist \item \xindex{Homöomorphismengruppe}Für jeden topologischen Raum $X$ ist
\[\Homoo(X) := \Set{f: X \rightarrow X | f \text{ ist Homöomorphismus}}\] \[\Homoo(X) := \Set{f: X \rightarrow X | f \text{ ist Homöomorphismus}}\]
eine Gruppe. eine Gruppe.
\item \xindex{Isometrie}Jede Isometrie $f:X \rightarrow Y$ zwischen metrischen \item \xindex{Isometrie}Jede Isometrie $f:X \rightarrow Y$ zwischen metrischen

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documents/GeoTopo/Symbolverzeichnis.tex
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@ -7,81 +7,128 @@
% Mengenoperationen % % Mengenoperationen %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Mengenoperationen} \section*{Mengenoperationen}
$A^C\;\;\;$ Komplement der Menge $A$\\
$\mathcal{P}(M)\;\;\;$ Potenzmenge von $M$\\ Seien $A, B$ und $M$ Mengen.
$\overline{M}\;\;\;$ Abschluss der Menge $M$\\
$\partial M\;\;\;$ Rand der Menge $M$\\ % Set \mylengtha to widest element in first column; adjust
$M^\circ\;\;\;$ Inneres der Menge $M$\\ % \mylengthb so that the width of the table is \columnwidth
$A \times B\;\;\;$ Kreuzprodukt zweier Mengen\\ \settowidth\mylengtha{$A \subsetneq B$}
$A \subseteq B\;\;\;$ Teilmengenbeziehung\\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
$A \subsetneq B\;\;\;$ echte Teilmengenbeziehung\\
$A \setminus B\;\;\;$ $A$ ohne $B$\\ \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$A \cup B\;\;\;$ Vereinigung\\ $A^C $ & Komplement von $A$\\
$A \dcup B\;\;\;$ Disjunkte Vereinigung\\ $\mathcal{P}(M)$ & Potenzmenge von $M$\\
$A \cap B\;\;\;$ Schnitt\\ $\overline{M}$ & Abschluss von $M$\\
$\partial M$ & Rand der Menge $M$\\
$M^\circ$ & Inneres der Menge $M$\\
$A \times B$ & Kreuzprodukt\\
$A \subseteq B$ & Teilmengenbeziehung\\
$A \subsetneq B$ & echte Teilmengenbeziehung\\
$A \setminus B$ & Differenzmenge\\
$A \cup B$ & Vereinigung\\
$A \dcup B$ & Disjunkte Vereinigung\\
$A \cap B$ & Schnitt\\
\end{xtabular}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Geometrie % % Geometrie %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Geometrie} \section*{Geometrie}
$AB\;\;\;$ Gerade durch die Punkte $A$ und $B$\\
$\overline{AB}\;\;\;$ Strecke mit Endpunkten $A$ und $B$\\ \settowidth\mylengtha{$\overline{AB} \cong \overline{CD}$}
$\triangle ABC\;\;\;$ Dreieck mit Eckpunkten $A, B, C$\\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
$\overline{AB} \cong \overline{CD}\;\;\;$ Die Strecken $\overline{AB}$ und $\overline{CD}$ sind isometrisch\\
$|K|\;\;\;$ Geometrische Realisierung des Simplizialkomplexes $K$\\ \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$AB$ & Gerade durch die Punkte $A$ und $B$\\
$\overline{AB}$ & Strecke mit Endpunkten $A$ und $B$\\
$\triangle ABC$ & Dreieck mit Eckpunkten $A, B, C$\\
$\overline{AB} \cong \overline{CD}$& Die Strecken $\overline{AB}$ und $\overline{CD}$ sind isometrisch\\
$|K|$ & Geometrische Realisierung des Simplizialkomplexes~$K$\\
\end{xtabular}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Gruppen % % Gruppen %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Gruppen} \section*{Gruppen}
$\Homoo(X)\;\;\;$ Homöomorphismengruppe\\
$\Iso(X)\;\;\;$ Isometriengruppe\\ Sei $X$ ein topologischer Raum und $K$ ein Körper.
$\GL_n(K)\;\;\;$ Allgemeine lineare Gruppe\footnote{von \textit{\textbf{G}eneral \textbf{L}inear Group}}\\
$\SL_n(K)\;\;\;$ Spezielle lineare Gruppe\\ \settowidth\mylengtha{$\Homoo(X)$}
$\PSL_n(K)\;\;\;$ Projektive lineare Gruppe\\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
$\Perm(X)\;\;\;$ Permutationsgruppe\\
$\Sym(X)\;\;\;$ Symmetrische Gruppe \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$\Homoo(X)$ & Homöomorphis\-men\-gruppe\\
$\Iso(X)$ & Isometrien\-gruppe\\
$\GL_n(K)$ & Allgemeine lineare Gruppe (von \textit{\textbf{G}eneral \textbf{L}inear Group})\\
$\SL_n(K)$ & Spezielle lineare Gruppe\\
$\PSL_n(K)$ & Projektive lineare Gruppe\\
$\Perm(X)$ & Permutations\-gruppe\\
$\Sym(X)$ & Symmetrische Gruppe\\
\end{xtabular}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Wege % % Wege %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Wege} \section*{Wege}
$\gamma: I \rightarrow X\;\;\;$ Ein Weg\\
$[\gamma]\;\;\;$ Homotopieklasse von $\gamma$\\
$\gamma_1 * \gamma_2\;\;\;$ Zusammenhängen von Wegen\\
$\gamma_1 \sim \gamma_2\;\;\;$ Homotopie von Wegen\\
$\overline{\gamma}(x) = \gamma(1-x)\;\;\;$ Inverser Weg\\
$C := \gamma([0,1])\;\;\;$ Bild eines Weges $\gamma$
Sei $\gamma: I \rightarrow X$ ein Weg.
\settowidth\mylengtha{$\gamma_1 \sim \gamma_2$}
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$[\gamma]$ & Homotopieklasse von $\gamma$\\
$\gamma_1 * \gamma_2$ & Zusammenhängen von Wegen\\
$\gamma_1 \sim \gamma_2$ & Homotopie von Wegen\\
$\overline{\gamma}(x)$ & Inverser Weg, also $\overline{\gamma}(x) := \gamma(1-x)$\\
$C$ & Bild eines Weges $\gamma$, also $C := \gamma([0,1])$
\end{xtabular}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Weiteres % % Weiteres %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Weiteres} \section*{Weiteres}
$\fB\;\;\;$ Basis einer Topologie\\
$\calS\;\;\;$ Subbasis einer Topologie\\
$\fB_\delta(x)\;\;\;$ $\delta$-Kugel um $x$\\
$\fT\;\;\;$ Topologie\\
$\atlas\;\;\;$ Atlas\\ \settowidth\mylengtha{$\fB_\delta(x)$}
$\praum\;\;\;$ Projektiver Raum\\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
$\langle \cdot , \cdot \rangle\;\;\;$ Skalarprodukt\\
$X /_\sim\;\;\;$ $X$ modulo $\sim$\\ \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$[x]_\sim\;\;\;$ Äquivalenzklassen von $x$ bzgl. $\sim$\\ $\fB$ & Basis einer Topologie\\
$\| x \|\;\;\;$ Norm von $x$\\ $\fB_\delta(x)$& $\delta$-Kugel um $x$\\
$| x |\;\;\;$ Betrag von $x$\\ $\calS$ & Subbasis einer Topologie\\
$\langle a \rangle\;\;\;$ Erzeugnis von $a$\\ $\fT$ & Topologie\\
\end{xtabular}
\settowidth\mylengtha{$X /_\sim$}
\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$\atlas$ & Atlas\\
$\praum$ & Projektiver Raum\\
$\langle \cdot , \cdot \rangle$ & Skalarprodukt\\
$X /_\sim$ & $X$ modulo $\sim$\\
$[x]_\sim$ & Äquivalenzklassen von $x$ bzgl. $\sim$\\
$\| x \|$ & Norm von $x$\\
$| x |$ & Betrag von $x$\\
$\langle a \rangle$ & Erzeugnis von $a$\\
\end{xtabular}
$S^n\;\;\;$ Sphäre\\ $S^n\;\;\;$ Sphäre\\
$T^n\;\;\;$ Torus\\ $T^n\;\;\;$ Torus\\
$f \circ g\;\;\;$ Verkettung von $f$ und $g$\\ \settowidth\mylengtha{$f^{-1}(M)$}
$\pi_X\;\;\;$ Projektion auf $X$\\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
$f|_U\;\;\;$ $f$ eingeschränkt auf $U$\\
$f^{-1}(M)\;\;\;$ Urbild von $M$\\ \begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$\rang(M)\;\;\;$ Rang von $M$\\ $f \circ g$&Verkettung von $f$ und $g$\\
$\chi(K)\;\;\;$ Euler-Charakteristik von $K$\\ $\pi_X$ &Projektion auf $X$\\
$\Delta^k\;\;\;$ Standard-Simplex\\ $f|_U$ $f$ &eingeschränkt auf $U$\\
$X \# Y\;\;\;$ Verklebung von $X$ und $Y$\\ $f^{-1}(M)$&Urbild von $M$\\
$d_n\;\;\;$ Lineare Abbildung aus \cref{kor:9.11}\\ $\rang(M)$ & Rang von $M$\\
$A \cong B\;\;\;$ $A$ ist isometrisch zu $B$\\ $\chi(K)$ & Euler-Charakteristik von $K$\\
$f_*\;\;\;$ Abbildung zwischen Fundamentalgruppen (vgl. \cpageref{korr:11.5}) $\Delta^k$ & Standard-Simplex\\
$X \# Y$ & Verklebung von $X$ und $Y$\\
$d_n$ & Lineare Abbildung aus \cref{kor:9.11}\\
$A \cong B$& $A$ ist isometrisch zu $B$\\
$f_*$ & Abbildung zwischen Fundamentalgruppen (vgl. \cpageref{korr:11.5})
\end{xtabular}
\onecolumn \onecolumn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -100,21 +147,32 @@ $\mdp = \Set{2, 3, 5, 7, \dots}\;\;\;$ Primzahlen\\
$\mdh = \Set{z \in \mdc | \Im{z} > 0}\;\;\;$ obere Halbebene\\ $\mdh = \Set{z \in \mdc | \Im{z} > 0}\;\;\;$ obere Halbebene\\
$I = [0,1] \subsetneq \mdr\;\;\;$ Einheitsintervall\\ $I = [0,1] \subsetneq \mdr\;\;\;$ Einheitsintervall\\
$f:S^1 \hookrightarrow \mdr^2\;\;\;$ Einbettung der Kreislinie in die Ebene\\ \settowidth\mylengtha{$f:S^1 \hookrightarrow \mdr^2$}
$\pi_1(X,x)\;\;\;$ Fundamentalgruppe im topologischen Raum $X$ um $x \in X$\\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
$\Fix(f)\;\;\;$ Menge der Fixpunkte der Abbildung $f$\\
$\|\cdot\|_2\;\;\;$ 2-Norm; Euklidische Norm\\
$\kappa\;\;\;$ Krümmung\\
$\kappa_{\ts{Nor}}\;\;\;$ Normalenkrümmung\\
$V(f)\;\;\;$ Nullstellenmenge von $f$\footnote{von \textit{\textbf{V}anishing Set}}
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$f:S^1 \hookrightarrow \mdr^2$& Einbettung der Kreislinie in die Ebene\\
$\pi_1(X,x)$ & Fundamentalgruppe im topologischen Raum $X$ um $x \in X$\\
$\Fix(f)$ & Menge der Fixpunkte der Abbildung $f$\\
$\|\cdot\|_2$ & 2-Norm; Euklidische Norm\\
$\kappa$ & Krümmung\\
$\kappa_{\ts{Nor}}$ & Normalenkrümmung\\
$V(f)$ & Nullstellenmenge von $f$\footnotemark
\end{xtabular}
\footnotetext{von \textit{\textbf{V}anishing Set}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Krümmung % % Krümmung %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Krümmung} \section*{Krümmung}
$D_p F: \mdr^2 \rightarrow \mdr^3\;\;\;$ Lineare Abbildung mit Jacobi-Matrix in $p$ (siehe \cpageref{def:Tangentialebene})\\
$T_s S\;\;\;$ Tangentialebene an $S \subseteq \mdr^3$ durch $s \in S$\\ \settowidth\mylengtha{$D_p F: \mdr^2 \rightarrow \mdr^3$}
$d_s n(x)\;\;\;$ Weingarten-Abbildung\\ \setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
$D_p F: \mdr^2 \rightarrow \mdr^3$& Lineare Abbildung mit Jacobi-Matrix in $p$ (siehe \cpageref{def:Tangentialebene})\\
$T_s S$ & Tangentialebene an $S \subseteq \mdr^3$ durch $s \in S$\\
$d_s n(x)$ & Weingarten-Abbildung\\
\end{xtabular}
\index{Faser|see{Urbild}} \index{Faser|see{Urbild}}
\index{kongruent|see{isometrisch}} \index{kongruent|see{isometrisch}}

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documents/GeoTopo/meta/Arbeitszeit.md
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@ -93,4 +93,5 @@ in dem Erstellen dieses Skripts steckt:
|20.02.2014 | 13:00 - 13:45 | 45 | Verbesserungsvorschläge von Jérôme Urhausen, Email 1 vom 20.02.2014, umgesetzt. |20.02.2014 | 13:00 - 13:45 | 45 | Verbesserungsvorschläge von Jérôme Urhausen, Email 1 vom 20.02.2014, umgesetzt.
|20.02.2014 | 19:30 - 20:15 | 45 | Verbesserungsvorschläge von Jérôme Urhausen, Email 2 vom 20.02.2014, umgesetzt. |20.02.2014 | 19:30 - 20:15 | 45 | Verbesserungsvorschläge von Jérôme Urhausen, Email 2 vom 20.02.2014, umgesetzt.
| Zwischenstand | --- | --- | 6081 Minuten => Über 100 Stunden! | Zwischenstand | --- | --- | 6081 Minuten => Über 100 Stunden!
|17.03.2014 | 16:00 - 18:00 | 120 | Textsetzung |17.03.2014 | 16:00 - 18:00 | 120 | Textsetzung
|19.03.2014 | 08:00 - 10:00 | 120 | Verbesserung des Symbolverzeichnisses

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