From ad0c3c47007e0de29ae684f453774c560bab35d0 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Fri, 21 Feb 2014 11:42:45 +0100 Subject: [PATCH] =?UTF-8?q?Vektorprodukt=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/GeoTopo/Definitionen.tex | 13 +++++++++---- documents/GeoTopo/GeoTopo.pdf | Bin 5226511 -> 5227666 bytes documents/GeoTopo/Kapitel3.tex | 2 +- documents/GeoTopo/Symbolverzeichnis.tex | 5 ++++- .../GeoTopo/definitions/definitionen.pdf | Bin 361595 -> 361595 bytes documents/GeoTopo/meta/Mitwirkende.md | 2 +- .../GeoTopo/other-formats/GeoTopo-A5.pdf | Bin 5279927 -> 5280302 bytes 7 files changed, 15 insertions(+), 7 deletions(-) diff --git a/documents/GeoTopo/Definitionen.tex b/documents/GeoTopo/Definitionen.tex index 4678a1c..c5a76bb 100644 --- a/documents/GeoTopo/Definitionen.tex +++ b/documents/GeoTopo/Definitionen.tex @@ -1,7 +1,7 @@ %!TEX root = GeoTopo.tex -\markboth{Anhang: Definitionen und Sätze}{Anhang: Definitionen und Sätze} -\chapter*{Anhang: Definitionen und Sätze} -\addcontentsline{toc}{chapter}{Anhang: Definitionen und Sätze} +\markboth{Ergänzende Definitionen und Sätze}{Ergänzende Definitionen und Sätze} +\chapter*{Ergänzende Definitionen und Sätze} +\addcontentsline{toc}{chapter}{Ergänzende Definitionen und Sätze} Da dieses Skript in die Geometrie und Topologie einführen soll, sollten soweit wie möglich alle benötigten Begriffe definiert und erklärt werden. Die folgenden @@ -53,4 +53,9 @@ Lineare Algebra entnommen: \begin{satz*}[Binomischer Lehrsatz]\xindex{Lehrsatz!Binomischer}% Sei $x, y \in \mdr$. Dann gilt: \[(x+y)^n = \sum_{k=0}^{n}\binom{n}{k} x^{n-k}y^{k} \;\;\; \forall n \in \mdn_0\] -\end{satz*} \ No newline at end of file +\end{satz*} + +\begin{definition}\xindex{Kreuzprodukt}\index{Vektorprodukt|see{Kreuzprodukt}} + Seien $a, b \in \mdr^3$ Vektoren. + \[ a \times b := \begin{pmatrix}a_1\\b_3\\a_3 \end{pmatrix} \times \begin{pmatrix}a_1\\b_3\\a_3 \end{pmatrix} = \begin{pmatrix}a_2 b_3 - a_3 b_2\\a_3 b_1 - a_1 b_3\\a_1 b_2 - a_2 b_1 \end{pmatrix}\] +\end{definition} \ No newline at end of file diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 5705d0cb2e4b861d7fed7900600fad9150e39f0c..0f61501f7cee573e4c70550cde8ca2a85a436a18 100644 GIT binary patch delta 126188 zcmY(|Q*fZcvH;+CW81c!Y_zd$I~&{hqunGM+qP{dn`C3#w(XO9tIorD_^NuIrg|Qx zr>CR+yTRhZy1{B}V&ho_Xn=EV8I@H@@t)1MaYquEh4Sq*C`VV1DSNm$lz?{`lxKs{?iI)+hBbT<{E{k|LZkx_4D1aI8U zs8IJrc!2GKDB1HqcD-~LYbYtx+q-tn9PH3?j81RP3VMYbJ%x4fJpd=^lW6UnkIb;> 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b/documents/GeoTopo/Kapitel3.tex index 5ba9747..71f8516 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -468,7 +468,7 @@ Wenn $\pi_1(X,x) = \Set{e}$ für ein $x \in X$ gilt, dann wegen Wähle $t_i \in I_i \cap I_{i+1}$, also $\gamma(t_i) \in U \cap V$. Sei $\sigma_i$ Weg in $U \cap V$ von $x_0$ nach $\gamma(t_i) \Rightarrow \gamma$ ist homotop zu - \[\underbrace{\gamma_1 * \overline{\sigma_1}}_{\text{in } U} * \underbrace{\sigma_1 * \gamma_2 * \overline{\sigma_2}}_{\text{in } V} * \dots * \sigma_{n-1} * \gamma_2\] + \[\underbrace{\gamma_1 * \overline{\sigma_1}}_{\text{in } U} * \underbrace{\sigma_1 * \gamma_2 * \overline{\sigma_2}}_{\text{in } V} * \dots * \sigma_{n-1} * \gamma_2 \text{ mit } \gamma_i := \gamma |_{I_i}\] \end{beweis} \begin{beispiel}[Satz von Seifert und van Kampen] diff --git a/documents/GeoTopo/Symbolverzeichnis.tex b/documents/GeoTopo/Symbolverzeichnis.tex index 1a8776b..fff8420 100644 --- a/documents/GeoTopo/Symbolverzeichnis.tex +++ b/documents/GeoTopo/Symbolverzeichnis.tex @@ -19,11 +19,14 @@ $A \setminus B\;\;\;$ $A$ ohne $B$\\ $A \cup B\;\;\;$ Vereinigung\\ $A \dcup B\;\;\;$ Disjunkte Vereinigung\\ $A \cap B\;\;\;$ Schnitt\\ - +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Geometrie % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Geometrie} $AB\;\;\;$ 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 $K$\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Gruppen % diff --git a/documents/GeoTopo/definitions/definitionen.pdf b/documents/GeoTopo/definitions/definitionen.pdf index 7cb075ab14a10c63cec0327fdb49d4772605d9b9..6f4690c8e61a88e39e282db88637cb393527e482 100644 GIT binary patch delta 114 zcmey}A@;jNtf7Umg{g(Pg=Gutk#ZJ8V?*QVC(2pn!L04C%UNyMosG;4EKFU@ogGaK l49(q~-OS9KU0htvoQ+Lf&5Rw5&FmCx2r1c~RnIEO3IM1F9`XPH delta 114 zcmey}A@;jNtf7Umg{g(Pg=Gutk#ZITQ$wTaC(2pn!L04C%UNyMoh@8V%#AFJT`XP9 lOq|?|OdL&}%-l?!EX`eA9SsdkUF{TX2r1c~RnIEO3IMDx9|`~f diff --git a/documents/GeoTopo/meta/Mitwirkende.md b/documents/GeoTopo/meta/Mitwirkende.md index cff9a43..d6ab96d 100644 --- a/documents/GeoTopo/meta/Mitwirkende.md +++ b/documents/GeoTopo/meta/Mitwirkende.md @@ -15,9 +15,9 @@ schicken. * Marco: ein paar Fehlermeldungen * Nilan: Mitschriebe von Tutorien; ein paar Fehlermeldungen * Randecker, Anja: Übungsaufgaben und Lösungen (Übungsleiterin) +* Rocha, Tânia: Mitschriebe von Vorlesungen; ein paar Fehlermeldungen * Sarah: Einige Übungsaufgaben; Hilfestellung als Tutorin beim Verständnis der Beweise / Inhalte * Schickling, Johannes: Mitschriebe der Vorlesung -* Tânia: Mitschriebe von Vorlesungen; ein paar Fehlermeldungen * Thoma, Martin: Erstellen des Grundgerüsts mit Hilfe des Vorlesungsmitschriebs; Beweise; Bilder; Textsetzung; Kontrolle der Korrektheit aller Verbesserungsvorschläge * Urhausen, Jérôme: Beweise; Viele Verbesserungen (Notation und Textsetzung); Bilder diff --git a/documents/GeoTopo/other-formats/GeoTopo-A5.pdf 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