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Symbolverzeichnis verbessert
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Martin Thoma
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7 changed files with 131 additions and 65 deletions
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@ -45,6 +45,13 @@
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\usepackage{tqft}
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\usepackage{xspace} % for new commands; decides weather I want to insert a space after the command
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\usepackage[german,nameinlink,noabbrev]{cleveref} % has to be after hyperref, ntheorem, amsthm
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{array,xtab,ragged2e} % for symbol table
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\newlength\mylengtha
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\newlength\mylengthb
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\newcolumntype{P}[1]{>{\RaggedRight}p{#1}}
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\tabcolsep=3pt % default: 6pt
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\usepackage{acronym}
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\usepackage{cancel}
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\usepackage{shortcuts}
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@ -449,7 +449,7 @@ Die Teilraumtopologie wird auch \textit{Spurtopologie} oder
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\begin{bemerkung}
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\begin{bemenum}
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\item \xindex{Homöomorphismengruppe}Für jeden topologischen Raum ist
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\item \xindex{Homöomorphismengruppe}Für jeden topologischen Raum $X$ ist
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\[\Homoo(X) := \Set{f: X \rightarrow X | f \text{ ist Homöomorphismus}}\]
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eine Gruppe.
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\item \xindex{Isometrie}Jede Isometrie $f:X \rightarrow Y$ zwischen metrischen
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@ -7,81 +7,128 @@
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% Mengenoperationen %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Mengenoperationen}
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$A^C\;\;\;$ Komplement der Menge $A$\\
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$\mathcal{P}(M)\;\;\;$ Potenzmenge von $M$\\
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$\overline{M}\;\;\;$ Abschluss der Menge $M$\\
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$\partial M\;\;\;$ Rand der Menge $M$\\
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$M^\circ\;\;\;$ Inneres der Menge $M$\\
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$A \times B\;\;\;$ Kreuzprodukt zweier Mengen\\
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$A \subseteq B\;\;\;$ Teilmengenbeziehung\\
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$A \subsetneq B\;\;\;$ echte Teilmengenbeziehung\\
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$A \setminus B\;\;\;$ $A$ ohne $B$\\
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$A \cup B\;\;\;$ Vereinigung\\
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$A \dcup B\;\;\;$ Disjunkte Vereinigung\\
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$A \cap B\;\;\;$ Schnitt\\
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Seien $A, B$ und $M$ Mengen.
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% Set \mylengtha to widest element in first column; adjust
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% \mylengthb so that the width of the table is \columnwidth
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\settowidth\mylengtha{$A \subsetneq B$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$A^C $ & Komplement von $A$\\
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$\mathcal{P}(M)$ & Potenzmenge von $M$\\
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$\overline{M}$ & Abschluss von $M$\\
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$\partial M$ & Rand der Menge $M$\\
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$M^\circ$ & Inneres der Menge $M$\\
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$A \times B$ & Kreuzprodukt\\
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$A \subseteq B$ & Teilmengenbeziehung\\
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$A \subsetneq B$ & echte Teilmengenbeziehung\\
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$A \setminus B$ & Differenzmenge\\
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$A \cup B$ & Vereinigung\\
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$A \dcup B$ & Disjunkte Vereinigung\\
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$A \cap B$ & Schnitt\\
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\end{xtabular}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Geometrie %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Geometrie}
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$AB\;\;\;$ Gerade durch die Punkte $A$ und $B$\\
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$\overline{AB}\;\;\;$ Strecke mit Endpunkten $A$ und $B$\\
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$\triangle ABC\;\;\;$ Dreieck mit Eckpunkten $A, B, C$\\
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$\overline{AB} \cong \overline{CD}\;\;\;$ Die Strecken $\overline{AB}$ und $\overline{CD}$ sind isometrisch\\
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$|K|\;\;\;$ Geometrische Realisierung des Simplizialkomplexes $K$\\
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\settowidth\mylengtha{$\overline{AB} \cong \overline{CD}$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$AB$ & Gerade durch die Punkte $A$ und $B$\\
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$\overline{AB}$ & Strecke mit Endpunkten $A$ und $B$\\
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$\triangle ABC$ & Dreieck mit Eckpunkten $A, B, C$\\
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$\overline{AB} \cong \overline{CD}$& Die Strecken $\overline{AB}$ und $\overline{CD}$ sind isometrisch\\
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$|K|$ & Geometrische Realisierung des Simplizialkomplexes~$K$\\
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\end{xtabular}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Gruppen %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Gruppen}
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$\Homoo(X)\;\;\;$ Homöomorphismengruppe\\
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$\Iso(X)\;\;\;$ Isometriengruppe\\
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$\GL_n(K)\;\;\;$ Allgemeine lineare Gruppe\footnote{von \textit{\textbf{G}eneral \textbf{L}inear Group}}\\
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$\SL_n(K)\;\;\;$ Spezielle lineare Gruppe\\
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$\PSL_n(K)\;\;\;$ Projektive lineare Gruppe\\
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$\Perm(X)\;\;\;$ Permutationsgruppe\\
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$\Sym(X)\;\;\;$ Symmetrische Gruppe
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Sei $X$ ein topologischer Raum und $K$ ein Körper.
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\settowidth\mylengtha{$\Homoo(X)$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$\Homoo(X)$ & Homöomorphis\-men\-gruppe\\
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$\Iso(X)$ & Isometrien\-gruppe\\
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$\GL_n(K)$ & Allgemeine lineare Gruppe (von \textit{\textbf{G}eneral \textbf{L}inear Group})\\
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$\SL_n(K)$ & Spezielle lineare Gruppe\\
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$\PSL_n(K)$ & Projektive lineare Gruppe\\
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$\Perm(X)$ & Permutations\-gruppe\\
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$\Sym(X)$ & Symmetrische Gruppe\\
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\end{xtabular}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Wege %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Wege}
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$\gamma: I \rightarrow X\;\;\;$ Ein Weg\\
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$[\gamma]\;\;\;$ Homotopieklasse von $\gamma$\\
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$\gamma_1 * \gamma_2\;\;\;$ Zusammenhängen von Wegen\\
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$\gamma_1 \sim \gamma_2\;\;\;$ Homotopie von Wegen\\
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$\overline{\gamma}(x) = \gamma(1-x)\;\;\;$ Inverser Weg\\
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$C := \gamma([0,1])\;\;\;$ Bild eines Weges $\gamma$
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Sei $\gamma: I \rightarrow X$ ein Weg.
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\settowidth\mylengtha{$\gamma_1 \sim \gamma_2$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$[\gamma]$ & Homotopieklasse von $\gamma$\\
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$\gamma_1 * \gamma_2$ & Zusammenhängen von Wegen\\
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$\gamma_1 \sim \gamma_2$ & Homotopie von Wegen\\
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$\overline{\gamma}(x)$ & Inverser Weg, also $\overline{\gamma}(x) := \gamma(1-x)$\\
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$C$ & Bild eines Weges $\gamma$, also $C := \gamma([0,1])$
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\end{xtabular}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Weiteres %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Weiteres}
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$\fB\;\;\;$ Basis einer Topologie\\
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$\calS\;\;\;$ Subbasis einer Topologie\\
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$\fB_\delta(x)\;\;\;$ $\delta$-Kugel um $x$\\
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$\fT\;\;\;$ Topologie\\
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$\atlas\;\;\;$ Atlas\\
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$\praum\;\;\;$ Projektiver Raum\\
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$\langle \cdot , \cdot \rangle\;\;\;$ Skalarprodukt\\
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$X /_\sim\;\;\;$ $X$ modulo $\sim$\\
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$[x]_\sim\;\;\;$ Äquivalenzklassen von $x$ bzgl. $\sim$\\
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$\| x \|\;\;\;$ Norm von $x$\\
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$| x |\;\;\;$ Betrag von $x$\\
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$\langle a \rangle\;\;\;$ Erzeugnis von $a$\\
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\settowidth\mylengtha{$\fB_\delta(x)$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$\fB$ & Basis einer Topologie\\
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$\fB_\delta(x)$& $\delta$-Kugel um $x$\\
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$\calS$ & Subbasis einer Topologie\\
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$\fT$ & Topologie\\
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\end{xtabular}
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\settowidth\mylengtha{$X /_\sim$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$\atlas$ & Atlas\\
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$\praum$ & Projektiver Raum\\
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$\langle \cdot , \cdot \rangle$ & Skalarprodukt\\
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$X /_\sim$ & $X$ modulo $\sim$\\
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$[x]_\sim$ & Äquivalenzklassen von $x$ bzgl. $\sim$\\
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$\| x \|$ & Norm von $x$\\
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$| x |$ & Betrag von $x$\\
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$\langle a \rangle$ & Erzeugnis von $a$\\
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\end{xtabular}
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$S^n\;\;\;$ Sphäre\\
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$T^n\;\;\;$ Torus\\
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$f \circ g\;\;\;$ Verkettung von $f$ und $g$\\
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$\pi_X\;\;\;$ Projektion auf $X$\\
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$f|_U\;\;\;$ $f$ eingeschränkt auf $U$\\
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$f^{-1}(M)\;\;\;$ Urbild von $M$\\
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$\rang(M)\;\;\;$ Rang von $M$\\
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$\chi(K)\;\;\;$ Euler-Charakteristik von $K$\\
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$\Delta^k\;\;\;$ Standard-Simplex\\
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$X \# Y\;\;\;$ Verklebung von $X$ und $Y$\\
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$d_n\;\;\;$ Lineare Abbildung aus \cref{kor:9.11}\\
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$A \cong B\;\;\;$ $A$ ist isometrisch zu $B$\\
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$f_*\;\;\;$ Abbildung zwischen Fundamentalgruppen (vgl. \cpageref{korr:11.5})
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\settowidth\mylengtha{$f^{-1}(M)$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$f \circ g$&Verkettung von $f$ und $g$\\
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$\pi_X$ &Projektion auf $X$\\
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$f|_U$ $f$ &eingeschränkt auf $U$\\
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$f^{-1}(M)$&Urbild von $M$\\
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$\rang(M)$ & Rang von $M$\\
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$\chi(K)$ & Euler-Charakteristik von $K$\\
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$\Delta^k$ & Standard-Simplex\\
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$X \# Y$ & Verklebung von $X$ und $Y$\\
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$d_n$ & Lineare Abbildung aus \cref{kor:9.11}\\
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$A \cong B$& $A$ ist isometrisch zu $B$\\
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$f_*$ & Abbildung zwischen Fundamentalgruppen (vgl. \cpageref{korr:11.5})
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\end{xtabular}
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\onecolumn
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -100,21 +147,32 @@ $\mdp = \Set{2, 3, 5, 7, \dots}\;\;\;$ Primzahlen\\
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$\mdh = \Set{z \in \mdc | \Im{z} > 0}\;\;\;$ obere Halbebene\\
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$I = [0,1] \subsetneq \mdr\;\;\;$ Einheitsintervall\\
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$f:S^1 \hookrightarrow \mdr^2\;\;\;$ Einbettung der Kreislinie in die Ebene\\
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$\pi_1(X,x)\;\;\;$ Fundamentalgruppe im topologischen Raum $X$ um $x \in X$\\
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$\Fix(f)\;\;\;$ Menge der Fixpunkte der Abbildung $f$\\
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$\|\cdot\|_2\;\;\;$ 2-Norm; Euklidische Norm\\
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$\kappa\;\;\;$ Krümmung\\
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$\kappa_{\ts{Nor}}\;\;\;$ Normalenkrümmung\\
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$V(f)\;\;\;$ Nullstellenmenge von $f$\footnote{von \textit{\textbf{V}anishing Set}}
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\settowidth\mylengtha{$f:S^1 \hookrightarrow \mdr^2$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$f:S^1 \hookrightarrow \mdr^2$& Einbettung der Kreislinie in die Ebene\\
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$\pi_1(X,x)$ & Fundamentalgruppe im topologischen Raum $X$ um $x \in X$\\
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$\Fix(f)$ & Menge der Fixpunkte der Abbildung $f$\\
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$\|\cdot\|_2$ & 2-Norm; Euklidische Norm\\
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$\kappa$ & Krümmung\\
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$\kappa_{\ts{Nor}}$ & Normalenkrümmung\\
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$V(f)$ & Nullstellenmenge von $f$\footnotemark
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\end{xtabular}
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\footnotetext{von \textit{\textbf{V}anishing Set}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Krümmung %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Krümmung}
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$D_p F: \mdr^2 \rightarrow \mdr^3\;\;\;$ Lineare Abbildung mit Jacobi-Matrix in $p$ (siehe \cpageref{def:Tangentialebene})\\
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$T_s S\;\;\;$ Tangentialebene an $S \subseteq \mdr^3$ durch $s \in S$\\
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$d_s n(x)\;\;\;$ Weingarten-Abbildung\\
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\settowidth\mylengtha{$D_p F: \mdr^2 \rightarrow \mdr^3$}
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\setlength\mylengthb{\dimexpr\columnwidth-\mylengtha-2\tabcolsep\relax}
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\begin{xtabular}{@{} p{\mylengtha} P{\mylengthb} @{}}
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$D_p F: \mdr^2 \rightarrow \mdr^3$& Lineare Abbildung mit Jacobi-Matrix in $p$ (siehe \cpageref{def:Tangentialebene})\\
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$T_s S$ & Tangentialebene an $S \subseteq \mdr^3$ durch $s \in S$\\
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$d_s n(x)$ & Weingarten-Abbildung\\
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\end{xtabular}
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\index{Faser|see{Urbild}}
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\index{kongruent|see{isometrisch}}
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@ -94,3 +94,4 @@ in dem Erstellen dieses Skripts steckt:
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|20.02.2014 | 19:30 - 20:15 | 45 | Verbesserungsvorschläge von Jérôme Urhausen, Email 2 vom 20.02.2014, umgesetzt.
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| Zwischenstand | --- | --- | 6081 Minuten => Über 100 Stunden!
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|17.03.2014 | 16:00 - 18:00 | 120 | Textsetzung
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|19.03.2014 | 08:00 - 10:00 | 120 | Verbesserung des Symbolverzeichnisses
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