From 953f08429e83c957de7f5a960f018136f3723aee Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Tue, 28 Jan 2014 07:48:12 +0100 Subject: [PATCH] =?UTF-8?q?Textsetzung=20(http://tex.stackexchange.com/q/1?= =?UTF-8?q?56058/5645);=20L=C3=B6sung=20von=20=C3=9Cbungsaufgabe=20geTeXt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/GeoTopo/Arbeitszeit.md | 1 + documents/GeoTopo/GeoTopo.pdf | Bin 4945383 -> 4949105 bytes documents/GeoTopo/Kapitel1.tex | 2 +- documents/GeoTopo/Kapitel2.tex | 2 +- documents/GeoTopo/Kapitel3.tex | 2 +- documents/GeoTopo/Kapitel4-UB.tex | 4 ++-- documents/GeoTopo/Kapitel4.tex | 6 ++--- documents/GeoTopo/Loesungen.tex | 37 ++++++++++++++++++++++++++++++ documents/GeoTopo/shortcuts.sty | 13 ++++++++++- 9 files changed, 58 insertions(+), 9 deletions(-) diff --git a/documents/GeoTopo/Arbeitszeit.md b/documents/GeoTopo/Arbeitszeit.md index 7c86a3d..2c918d0 100644 --- a/documents/GeoTopo/Arbeitszeit.md +++ 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zZYLGE0e{~}#lNh|>E_g@GQo!$x<1;-3 zO+R6|!U;43Xrzzo_-V{ApkZJdiAC_!e zfWS6^cK#E(QHJmf^6zo^n|H>iDE%H{Fv>Lif}Z<7VV2h&T|uNMF8na)9{his&EHXo zAEJ7?ZzmNHJMQZU1Az^2Lt3S94_PDm9|pj`viUmatq^#A|> diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 49c118d..fb80187 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -597,7 +597,7 @@ sodass $\pi$ stetig wird. \item Nach \cref{zusammenhangAbschluss} ist $\overline{Z(x)}$ zusammenhängend $\Rightarrow \overline{Z(x)} \subseteq Z(x)$ $\Rightarrow Z(x) = \overline{Z(x)}$ - \item Ist $Z(y) \cap Z(x) \neq \emptyset \xRightarrow{\text{Bem. }\ref{bem:zusammenhangVereinigung}} Z(y) \cup Z(x)$ + \item Ist $Z(y) \cap Z(x) \neq \emptyset \xRightarrow{\crefabbr{bem:zusammenhangVereinigung}} Z(y) \cup Z(x)$ ist zusammenhängend. \\ \begin{align*} \Rightarrow Z(x) \cup Z(y) &\subseteq Z(x) \Rightarrow Z(y) \subseteq Z(x)\\ diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 5eabb16..1147b09 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -191,7 +191,7 @@ \begin{beispiel}\xindex{Neilsche Parabel} \begin{bspenum} \item $F: \mdr^3 \rightarrow \mdr,\;\;\; (x, y, z) \mapsto x^2 + y^2 + z^2 - 1$, - $V(F) = S^2$, $\grad(F) = (2x, 2y, 2z) \xRightarrow{\text{Bem. }\ref{bem:Mannigfaltigkeitskriterium}} S^n$ + $V(F) = S^2$, $\grad(F) = (2x, 2y, 2z) \xRightarrow{\crefabbr{bem:Mannigfaltigkeitskriterium}} S^n$ ist $n$-dimensionale Mannigfaltigkeit in $\mdr^{n+1}$ \item $F: \mdr^2 \rightarrow \mdr, \;\;\; (x,y) \mapsto y^2 - x^3$ \begin{figure}[ht] diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index 5d18620..fa45654 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -986,7 +986,7 @@ und der Fundamentalgruppe herstellen: $\Rightarrow p(\gamma_{g \circ f}) = p(\gamma_g) * \underbrace{(p \circ g)}_{=p} (\gamma_f) = \rho(g) \neq \rho(f)$ \item \underline{$\rho$ ist injektiv}: $\rho(f) = e \Rightarrow p (\gamma_f) \sim \gamma_{x_0}$ $\xRightarrow{\cref{thm:ueberlagerung-weg-satz-12.6}} \gamma_f \sim \gamma_{\tilde{x_0}}$ - $\Rightarrow f(x_0) = \tilde{x_0} \xRightarrow{\text{Bem. }\cref{kor:12.14c}} f = \id_{\tilde{x}}$. + $\Rightarrow f(x_0) = \tilde{x_0} \xRightarrow{\crefabbr{kor:12.14c}} f = \id_{\tilde{x}}$. \item \underline{$\rho$ ist surjektiv}: Sei $[\gamma] \in \pi_1(X, x_0)$, $\tilde{\gamma}$ Lift von $\gamma$ nach $\tilde{x}$ mit Anfangspunkt $\tilde{x_0}$. Der Endpunkt von $\tilde{\gamma}$ diff --git a/documents/GeoTopo/Kapitel4-UB.tex b/documents/GeoTopo/Kapitel4-UB.tex index 9ffbb5e..cec431d 100644 --- a/documents/GeoTopo/Kapitel4-UB.tex +++ b/documents/GeoTopo/Kapitel4-UB.tex @@ -24,10 +24,10 @@ Zeigen Sie: \begin{aufgabeenum} - \item Ist $\triangle ABC$ ein Dreieck, in dem die Seiten + \item \label{ub11:aufg3.a} Ist $\triangle ABC$ ein Dreieck, in dem die Seiten $\overline{AB}$ und $\overline{AC}$ kongruent sind, so sind die Winkel $\angle ABC$ und $\angle BCA$ gleich. - \item Ist $\triangle ABC$ ein beliebiges Dreieck, so liegt + \item \label{ub11:aufg3.b} Ist $\triangle ABC$ ein beliebiges Dreieck, so liegt der längeren Seite der größere Winkel gegenüber und umgekehrt. \item \label{ub11:aufg3.c} Sind $g$ eine Gerade und $P \notin g$ ein Punkt, so gibt diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index 6ec7737..c298357 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -469,7 +469,7 @@ schneiden sich. \caption{Situation aus \cref{def:14.8}} \end{figure} -\begin{bemerkung}\label{kor:14.9}%In Vorlesung: Bemerkung 14.9 +\begin{bemerkung}\label{bem:14.9}%In Vorlesung: Bemerkung 14.9 In einem Dreieck ist jeder Innenwinkel kleiner als jeder nicht anliegende Außenwinkel. \end{bemerkung} @@ -477,7 +477,7 @@ schneiden sich. \begin{figure}[htp] \centering \input{figures/geometry-9.tex} - \caption{Situation aus \cref{kor:14.9}} + \caption{Situation aus \cref{bem:14.9}} \label{fig:bem.14.9} \end{figure} @@ -502,7 +502,7 @@ schneiden sich. \begin{beweis}[von \cref{prop:14.7}] Wäre $\varphi(g)$ nicht parallel zu $g$, so gäbe es einen Schnittpunkt $R$. Dann ist $\angle QPR < \angle RQP^-$ nach - \cref{kor:14.9} und $\angle QPR = \angle RQP^-$, weil + \cref{bem:14.9} und $\angle QPR = \angle RQP^-$, weil $\varphi(\angle RQP') = \angle RPQ$ \end{beweis} diff --git a/documents/GeoTopo/Loesungen.tex b/documents/GeoTopo/Loesungen.tex index 6e2bfbf..2d0efa0 100644 --- a/documents/GeoTopo/Loesungen.tex +++ b/documents/GeoTopo/Loesungen.tex @@ -213,6 +213,43 @@ % %\end{solution} +\begin{solution}[\ref{ub11:aufg3}] + \textbf{Vor.:} Sei $(X, d)$ eine absolute Ebene, $A, B, C \in X$ + und $\triangle ABC$ ein Dreieck. + + \begin{enumerate}[label=(\alph*)] + \item \textbf{Beh.:} $\overline{AB} \cong \overline{AC} \Rightarrow \angle ABC \cong \angle ACB$\\ + \textbf{Bew.:} Sei $\overline{AB} \cong \overline{AC}$.\\ + $\Rightarrow \exists$ Isometrie $\varphi$ mit $\varphi(B) = C$ und + $\varphi(C) = B$ und $\varphi(A) = A$.\\ + $\Rightarrow \varphi(\angle ABC) = \angle ACB$\\ + $\Rightarrow \angle ABC \cong \angle ACB \qed$ + \item \textbf{Beh.:} Der längeren Seite von $\triangle ABC$ liegt der größere Winkel gegenüber und + umgekehrt.\\ + \textbf{Bew.:} Sei $d(A,C) > d(A,B)$. Nach \ref{axiom:3.1} + gibt es $C' \in AC^+$ mit $d(A, C') = d(A,B)$\\ + $\Rightarrow C'$ liegt zwischen $A$ und $C$.\\ + Es gilt $\measuredangle ABC' < \measuredangle ABC$ und + aus \cref{ub11:aufg3.a} folgt: $\measuredangle ABC' = \measuredangle AC' B$.\\ + $\angle BC' A$ ist ein nicht anliegender Außenwinkel zu + $\angle BCA \xRightarrow{\crefabbr{bem:14.9}} \measuredangle BC' A > \measuredangle BCA$\\ + $\Rightarrow \measuredangle BCA < \measuredangle BC' A = \measuredangle ABC' < \measuredangle ABC $ + Sei umgekehrt $\measuredangle ABC > \measuredangle BCA$, + kann wegen 1. Teil von \cref{ub11:aufg3.b} nicht + $d(A,B) > d(A,C)$ gelten.\\ + Wegen \cref{ub11:aufg3.a} kann nicht $d(A,B) = d(A,C)$ + gelten.\\ + $\Rightarrow d(A,B) < d(A, C) \qed$ + \item \textbf{Vor.:} Sei $g$ eine Gerade, $P \in X$ und $P \notin g$\\ + \textbf{Beh.:} $\exists!$ Lot\\ + \textbf{Bew.:} ÜB10 A4(a): Es gibt Geradenspiegelung $\varphi$ + an $g$. $\varphi$ vertauscht die beiden Halbebenen bzgl. + $g$.\\ + $\Rightarrow \varphi(P)P$ schneidet $g$ in $F$. + \todo[inline]{Noch ca. eine halbe Seite} + \end{enumerate} +\end{solution} + \begin{solution}[\ref{ub-tut-24:a1}] Sei $f \parallel h$ und \obda $f \parallel g$. diff --git a/documents/GeoTopo/shortcuts.sty b/documents/GeoTopo/shortcuts.sty index 094858b..0f858dd 100644 --- a/documents/GeoTopo/shortcuts.sty +++ b/documents/GeoTopo/shortcuts.sty @@ -115,7 +115,18 @@ \newlist{bemenum}{enumerate}{1} \setlist[bemenum]{label=\alph*),ref=\textup{\thebemerkung.\alph*}} -\crefalias{bemenumi}{bemerkung} +\crefalias{bemenumi}{bemerkung} +% Commands for local abbreviations +\newcommand\crefabbr[1]{% +\begingroup + \crefname{bemerkung}{\text{Bem.}}{\text{Bem.}}\cref{#1} +\endgroup% +} +\newcommand\Crefabbr[1]{% +\begingroup + \Crefname{bemerkung}{Bem.}{Bem.}\Cref{#1} +\endgroup% +} \newlist{bspenum}{enumerate}{1} \setlist[bspenum]{label=\arabic*),ref=\textup{\thebeispiel.\arabic*}}