From 83222b58fecfc7dd21d2e22329e65c4aed1ffb91 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 2 Jun 2020 12:10:05 +0200 Subject: [PATCH] use new lecture class for ana1 --- sose2020/ana/lectures | 2 +- ws2019/ana/lectures/analysis.pdf | Bin 637052 -> 686796 bytes ws2019/ana/lectures/analysis.tex | 3 +-- ws2019/ana/lectures/analysis1-2.tex | 8 ++++---- ws2019/ana/lectures/analysis11.tex | 4 ++-- ws2019/ana/lectures/analysis13.tex | 6 +++--- ws2019/ana/lectures/analysis14.tex | 6 +++--- ws2019/ana/lectures/analysis15.tex | 6 +++--- ws2019/ana/lectures/analysis16.tex | 2 +- ws2019/ana/lectures/analysis18.tex | 4 ++-- ws2019/ana/lectures/analysis19.tex | 2 +- ws2019/ana/lectures/analysis20.tex | 4 ++-- ws2019/ana/lectures/analysis21.tex | 4 ++-- ws2019/ana/lectures/analysis22.tex | 2 +- ws2019/ana/lectures/analysis23.tex | 8 ++++---- ws2019/ana/lectures/analysis24.tex | 2 +- ws2019/ana/lectures/analysis25.tex | 4 ++-- ws2019/ana/lectures/analysis26.tex | 2 +- ws2019/ana/lectures/analysis3.tex | 4 ++-- ws2019/ana/lectures/analysis4.tex | 4 ++-- ws2019/ana/lectures/analysis5.tex | 4 ++-- ws2019/ana/lectures/analysis7.tex | 4 ++-- ws2019/ana/lectures/analysis8.tex | 4 ++-- ws2019/ana/lectures/analysis9.tex | 4 ++-- 24 files changed, 46 insertions(+), 47 deletions(-) diff --git a/sose2020/ana/lectures b/sose2020/ana/lectures index ba8188d..1541088 160000 --- a/sose2020/ana/lectures +++ b/sose2020/ana/lectures @@ -1 +1 @@ -Subproject commit ba8188da5cecb3d9ccabf7de434463d71bf68fdb +Subproject commit 1541088353c5e6549eb97723e74d8436ea336278 diff --git a/ws2019/ana/lectures/analysis.pdf b/ws2019/ana/lectures/analysis.pdf index bc0aa2b7e9f98c865bbbe225fe9332563a8e8d0c..4f6bc4e0ab18dc4a98f082cf404d09f6b5bdc9d9 100644 GIT binary patch delta 435867 zcmY(qQ+S|Dw65E+ZL?$Bwr$()IQe7S>e#lObZi?Pn;m1%wb!}W=dP;mYmE2#@Nr8q z^^Y3qr;?ZiJu?G4?9aKSKWnh*%;Ave>DrqRv_NP)FVY|re8}?`Z2Y(qXRGZ(3^5@n zCU_z?vgl93N+EF@?U(1~l-R&Cx{KZ>#^_Ciffjjuy@7$hbqXd))7((^W5wLVaJLbO 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zc9^En&pVk?!Exm)<9QykLMs-rkhsk8!lO0zh^2^xP6|nn3$sEDzRU4wpVh9 z(`|g(u_NO;3W)}n&otn*wF>m#o&DAA9hZ zaKrd8!x%AGHMX~`oR^CyOa%wWqBGHY-ICUiKlAWzZ)sA#44v0O z5@ZG=^v5-T_vD%qCI-sp0j)U|C!|`eZw)BeihF#Vv=ej8Dr&WrgSQB0b*yj}#Ptlc z+5W+uoY>!2=~q;gk}gUd>3$Nzc|NXYFwG88Sj`@$C2e_WY#_uAQHR#qD<3toFO+iM zrcHYiy`QbRteG91S;mQ^&ZW!LB#oGaV!i`7*cfm%9hrzvP?}Iq^^Y iKiAIx@89w8G;{ZaxLaA{3JVAdityvIvxC(YaQ_EA;C|fz diff --git a/ws2019/ana/lectures/analysis.tex b/ws2019/ana/lectures/analysis.tex index 197fcde..5c17a7d 100644 --- a/ws2019/ana/lectures/analysis.tex +++ b/ws2019/ana/lectures/analysis.tex @@ -1,4 +1,4 @@ -\documentclass[titlepage]{../../../lecture} +\documentclass[titlepage]{../../../sose2020/ana/lectures/lecture} \usepackage{standalone} \usepackage{tikz} @@ -15,7 +15,6 @@ \restoregeometry \tableofcontents -\newpage \input{analysis1-2.tex} \input{analysis3.tex} diff --git a/ws2019/ana/lectures/analysis1-2.tex b/ws2019/ana/lectures/analysis1-2.tex index eaf522e..2b6bc4e 100644 --- a/ws2019/ana/lectures/analysis1-2.tex +++ b/ws2019/ana/lectures/analysis1-2.tex @@ -2,9 +2,9 @@ \begin{document} -\section{Grundlagen} +\chapter{Grundlagen} -\subsection{Mengen und Aussagen} +\section{Mengen und Aussagen} \begin{definition} Seien $A$ und $B$ Mengen. @@ -31,7 +31,7 @@ \end{bem} -\subsection{Wahrheitstabellen} +\section{Wahrheitstabellen} \label{sec:wahrheitstafeln} \begin{definition} @@ -129,7 +129,7 @@ \end{bem} -\subsection{Abbildungen} +\section{Abbildungen} \begin{definition}[Abbildungen] Seien $A, B$ Mengen. Eine Abbildung $f$ zwischen $A$ und $B$ $f: A \rightarrow B$ ist eine Vorschrift, die jedem Element $a \in A$ genau ein Element $b \in B$ zugeordnet. \\ diff --git a/ws2019/ana/lectures/analysis11.tex b/ws2019/ana/lectures/analysis11.tex index 378e353..18985f5 100644 --- a/ws2019/ana/lectures/analysis11.tex +++ b/ws2019/ana/lectures/analysis11.tex @@ -2,8 +2,8 @@ \begin{document} -\section{Folgen und Reihen} -\subsection{Folgen} +\chapter{Folgen und Reihen} +\section{Folgen} \begin{definition}[Folgen] Eine Zahlenfolge ist eine Abbildung von $\N$ nach $\R$, d.h. $n \mapsto a_n \in \R$. diff --git a/ws2019/ana/lectures/analysis13.tex b/ws2019/ana/lectures/analysis13.tex index 1067946..4153030 100644 --- a/ws2019/ana/lectures/analysis13.tex +++ b/ws2019/ana/lectures/analysis13.tex @@ -2,7 +2,7 @@ \begin{document} -\subsection{Konvergenz in $\mathbb{C}$} +\section{Konvergenz in $\mathbb{C}$} Eine Folge $(z_n)_{n\in\N}$ komplexer Zahlen $(z_n \in \mathbb{C} \quad \forall n \in \N)$ konvergiert gegen $z \in \mathbb{C}$, falls $\forall \epsilon > 0$ $\exists n_\epsilon \in \N$, s.d. $|z_n - z| < \epsilon$ $\forall n \ge n_\epsilon$ @@ -38,7 +38,7 @@ Aus Definitionen: \end{enumerate} \end{bsp} -\subsection{Unendliche Summe (,,Reihen'')} +\section{Unendliche Summe (,,Reihen'')} \begin{definition} Sei $(a_n)_{n\in\N}$ eine Folge reeller oder komplexer Zahlen. Wir @@ -145,7 +145,7 @@ Aus Definitionen: Alles Falsch (Kostina: ,,Alles Schrot!''), weil $\sum_{k=0}^{\infty} (-1)^{k} $ divergent. \end{bsp} -\subsubsection{Konvergenzkriterien} +\subsection{Konvergenzkriterien} Die Konvergenz einer Reihe ist nichts anderes als die Konvergenz der Folge der Partialsummen. diff --git a/ws2019/ana/lectures/analysis14.tex b/ws2019/ana/lectures/analysis14.tex index 6a720c0..f54beeb 100644 --- a/ws2019/ana/lectures/analysis14.tex +++ b/ws2019/ana/lectures/analysis14.tex @@ -160,7 +160,7 @@ Kostina glaubt, dass das so stimmt, aber offensichtlich ist sie sich nicht siche \end{enumerate} \end{bsp} -\subsection{Umordnen von Reihen} +\section{Umordnen von Reihen} \begin{definition}[Umordnung] Sei $\sum_{n=1}^{\infty} a_n$ eine Reihe und $\tau: \N \to \N$ eine @@ -207,7 +207,7 @@ Kostina glaubt, dass das so stimmt, aber offensichtlich ist sie sich nicht siche gegen $c \in \R$ konvergiert (oder divergiert). \end{bem} -\subsection{Das Cauchy-Produkt von Reihen} +\section{Das Cauchy-Produkt von Reihen} \begin{satz}[Cauchy-Produkt] Seien $\sum_{n=0}^{\infty} a_n$ und $\sum_{n=0}^{\infty} b_n$ @@ -298,7 +298,7 @@ Kostina glaubt, dass das so stimmt, aber offensichtlich ist sie sich nicht siche \end{enumerate} \end{proof} -\subsection{Potenzreihen} +\section{Potenzreihen} \begin{definition} Eine Potenzreihe um den Entwicklungspunkt $z_0 \in \mathbb{C}$ ist diff --git a/ws2019/ana/lectures/analysis15.tex b/ws2019/ana/lectures/analysis15.tex index dad1fa6..59479fb 100644 --- a/ws2019/ana/lectures/analysis15.tex +++ b/ws2019/ana/lectures/analysis15.tex @@ -2,7 +2,7 @@ \begin{document} -\section{Funktionen und Stetigkeit} +\chapter{Funktionen und Stetigkeit} \begin{definition}[Funktion] Es sei $D \subset \R$ eine nichtleere Teilmenge. Eine @@ -19,7 +19,7 @@ .\end{align*} \end{definition} -\subsection{Grenzwerte bei Funktionen} +\section{Grenzwerte bei Funktionen} \begin{definition}[Berührpunkt] Sei $D \subset \R$. Ein Punkt $a \in \R$ heißt Berührpunkt von $D$, falls @@ -199,7 +199,7 @@ \end{itemize} \end{proof} -\subsection{Stetigkeit} +\section{Stetigkeit} \begin{definition}[Stetigkeit] Sei $f: D \to \R$, und $a \in D$. $f$ heißt stetig im Punkt $a$, falls diff --git a/ws2019/ana/lectures/analysis16.tex b/ws2019/ana/lectures/analysis16.tex index 27e19d8..ffd619a 100644 --- a/ws2019/ana/lectures/analysis16.tex +++ b/ws2019/ana/lectures/analysis16.tex @@ -81,7 +81,7 @@ in dieser Umgebung. \end{enumerate} \end{bsp} -\subsection{Weitere Eigenschaften stetiger Funktionen} +\section{Weitere Eigenschaften stetiger Funktionen} \begin{definition}[offene, abgeschlossene, kompakte Mengen] Eine Menge $D \subset \R$ heißt offen, falls $\forall x \in D$ diff --git a/ws2019/ana/lectures/analysis18.tex b/ws2019/ana/lectures/analysis18.tex index 2dbe9f0..aff93e5 100644 --- a/ws2019/ana/lectures/analysis18.tex +++ b/ws2019/ana/lectures/analysis18.tex @@ -29,7 +29,7 @@ trivial. \end{proof} -\subsection{Gleichmäßige Stetigkeit} +\section{Gleichmäßige Stetigkeit} \begin{definition}[gleichmäßige Stetigkeit] Eine Funktion $f\colon D \to \R$, $D \subset \R$ heißt @@ -110,7 +110,7 @@ als gleichmäßige Stetigkeit) \end{bem} -\subsection{Trigonometrische Funktionen} +\section{Trigonometrische Funktionen} \begin{satz} Für $x \in \R$ definiere $\cos(x) := \text{Re}(e^{ix})$ und diff --git a/ws2019/ana/lectures/analysis19.tex b/ws2019/ana/lectures/analysis19.tex index 0852b02..c509551 100644 --- a/ws2019/ana/lectures/analysis19.tex +++ b/ws2019/ana/lectures/analysis19.tex @@ -111,7 +111,7 @@ genauso für $\lim_{x \to 0} \frac{\cos(x) - 1}{x}$. \end{proof} -\subsection{Die Zahl $\pi$} +\section{Die Zahl $\pi$} Ziel: Analytische Definition von $\pi \in \R$. \begin{satz}[und Definition] diff --git a/ws2019/ana/lectures/analysis20.tex b/ws2019/ana/lectures/analysis20.tex index 9bdc254..33aeb7d 100644 --- a/ws2019/ana/lectures/analysis20.tex +++ b/ws2019/ana/lectures/analysis20.tex @@ -2,9 +2,9 @@ \begin{document} -\section{Differentiation} +\chapter{Differentiation} -\subsection{Ableitung} +\section{Ableitung} \begin{definition} Sei $f\colon D \to \R$, $D \subset \R$, eine Funktion. Definiere diff --git a/ws2019/ana/lectures/analysis21.tex b/ws2019/ana/lectures/analysis21.tex index 13834d5..187de90 100644 --- a/ws2019/ana/lectures/analysis21.tex +++ b/ws2019/ana/lectures/analysis21.tex @@ -2,7 +2,7 @@ \begin{document} -\subsection{Mittelwertsatz und Satz von Rolle} +\section{Mittelwertsatz und Satz von Rolle} \begin{definition}[globales / lokales Extremum] Die Funktion $f\colon D \to \R$ hat in @@ -168,7 +168,7 @@ f(b) = f(a)$. \end{proof} -\subsection{Höhere Ableitungen und Satz von Taylor} +\section{Höhere Ableitungen und Satz von Taylor} \begin{definition} Ist $f\colon D \to \R$ differenzierbar und $f'\colon D \to \R$ diff --git a/ws2019/ana/lectures/analysis22.tex b/ws2019/ana/lectures/analysis22.tex index 52873ab..94ab576 100644 --- a/ws2019/ana/lectures/analysis22.tex +++ b/ws2019/ana/lectures/analysis22.tex @@ -170,7 +170,7 @@ trivial. \end{proof} -\subsection{Die Regeln von de l'Hospital} +\section{Die Regeln von de l'Hospital} Ziel: Grenzwerte zu berechnen für $x \to \pm \infty$ oder $f(x) \to \pm \infty$. Grenzwerte vom Typ: diff --git a/ws2019/ana/lectures/analysis23.tex b/ws2019/ana/lectures/analysis23.tex index 1ce0e06..dff49c1 100644 --- a/ws2019/ana/lectures/analysis23.tex +++ b/ws2019/ana/lectures/analysis23.tex @@ -2,19 +2,19 @@ \usepackage{pgf,tikz} \usepackage{pgfplots} -\usetikzlibrary{intersections} +\usetikzlibrary{interchapters} \usetikzlibrary{arrows} \usetikzlibrary{positioning} \tikzset{>=stealth',inner sep=0pt,outer sep=2pt} \begin{document} -\section{Integration} +\chapter{Integration} Es sei $f\colon [a,b] \to \R$ eine Funktion. Ziel: Fläche unter dem Graphen berechnen. -\subsection{Riemannintegral} +\section{Riemannintegral} \begin{definition}[Zerlegungen, Stützpunkte] Eine endliche Zerlegung $Z$ von einem (beschränkten) Intervall $[a,b]$ ist @@ -252,7 +252,7 @@ berechnen. .\] $I = [0,1]$. $\underline{S}_Z(f) = 0 \neq 1 = \overline{S}_Z(f)$. \end{bsp} -\subsection{Eigenschaften des Riemann-Integrals} +\section{Eigenschaften des Riemann-Integrals} \begin{satz}[Additivität] \begin{enumerate} diff --git a/ws2019/ana/lectures/analysis24.tex b/ws2019/ana/lectures/analysis24.tex index c752a36..af57fcf 100644 --- a/ws2019/ana/lectures/analysis24.tex +++ b/ws2019/ana/lectures/analysis24.tex @@ -264,7 +264,7 @@ .\end{align*} \end{bem} -\subsection{Hauptsatz der Differential- und Integralrechnung} +\section{Hauptsatz der Differential- und Integralrechnung} \begin{satz}[Hauptsatz der Differential- und Integralrechnung] Es sei $a < b$, $f \colon [a,b] \to \R $ stetig. diff --git a/ws2019/ana/lectures/analysis25.tex b/ws2019/ana/lectures/analysis25.tex index 74d5b12..2249ed3 100644 --- a/ws2019/ana/lectures/analysis25.tex +++ b/ws2019/ana/lectures/analysis25.tex @@ -33,7 +33,7 @@ .\end{align*} \end{bem} -\subsection{Integrationsformeln} +\section{Integrationsformeln} \begin{lemma}[Partielle Integration] $f, g\colon [a,b] \to \R$ stetig differenzierbar. Dann gilt @@ -107,7 +107,7 @@ .\end{align*} \end{bsp} -\subsection{Uneigentliche Integrale} +\section{Uneigentliche Integrale} \begin{satz}[Uneigentliches R.-Integral Typ 1] Sei $f\colon (a, b] \to \R$ auf $(a, b]$ diff --git a/ws2019/ana/lectures/analysis26.tex b/ws2019/ana/lectures/analysis26.tex index f101683..8fd8569 100644 --- a/ws2019/ana/lectures/analysis26.tex +++ b/ws2019/ana/lectures/analysis26.tex @@ -2,7 +2,7 @@ \begin{document} -\subsection{Konvergenzkriterien für uneigentliche Integrale} +\section{Konvergenzkriterien für uneigentliche Integrale} \begin{satz}[Cauchy-Kriterium] Es sei $-\infty < a < b \le + \infty$ und $f\colon [a,b) \to \R$ lokal diff --git a/ws2019/ana/lectures/analysis3.tex b/ws2019/ana/lectures/analysis3.tex index 9e9921f..d32e5ef 100644 --- a/ws2019/ana/lectures/analysis3.tex +++ b/ws2019/ana/lectures/analysis3.tex @@ -1,7 +1,7 @@ \documentclass{lecture} \begin{document} -\subsection{Vollständige Induktion} +\section{Vollständige Induktion} \begin{bsp} Betrachte die Summe der ersten $n$ natürlichen Zahlen. Es gilt: @@ -87,7 +87,7 @@ $ \forall a, b \in \R$ und $n \in \N$ gilt: .\end{align*} \end{proof} -\subsection{Elemente der Kombinatorik} +\section{Elemente der Kombinatorik} Für $n \in \N$ ist die Fakultät $n!$ rekursiv definiert durch: \[ diff --git a/ws2019/ana/lectures/analysis4.tex b/ws2019/ana/lectures/analysis4.tex index 5a349df..0cab4b9 100644 --- a/ws2019/ana/lectures/analysis4.tex +++ b/ws2019/ana/lectures/analysis4.tex @@ -63,7 +63,7 @@ .\end{align*} \end{proof} -\subsection{Grundlegendes über Zahlenmengen} +\section{Grundlegendes über Zahlenmengen} \[ \N = \{1, 2, 3, \ldots\} \text{ natürliche Zahlen} .\] Auf $\N$ sind die arithmetischen Operationen ,,$+$'' (Addition) und ,,$\cdot$'' @@ -107,7 +107,7 @@ a = \frac{r}{s}, b = \frac{u}{v} \in \Q = \begin{cases} \end{cases} .\] $\Q$ bildet mit der Operation ,,$+$'' und ,,$\cdot$'' einen ,,Körper''. -\subsection{Was ist ein Körper?} +\section{Was ist ein Körper?} Sei $K$ eine Menge mit Operationen ,,$+$'' und ,,$\cdot$''. diff --git a/ws2019/ana/lectures/analysis5.tex b/ws2019/ana/lectures/analysis5.tex index cdff478..0beafc7 100644 --- a/ws2019/ana/lectures/analysis5.tex +++ b/ws2019/ana/lectures/analysis5.tex @@ -101,9 +101,9 @@ Es folgt aus den Eigenschaften: Siehe Lehrbuch \end{proof} -\section{Die Reellen Zahlen} +\chapter{Die Reellen Zahlen} -\subsection{Von den rationalen Zahlen zu den reellen Zahlen} +\section{Von den rationalen Zahlen zu den reellen Zahlen} \begin{lemma}[Irrationalität der Quadratwurzel] Die quadratische Gleichung $x^2 = 2$ besitzt keine rationale diff --git a/ws2019/ana/lectures/analysis7.tex b/ws2019/ana/lectures/analysis7.tex index 33e66f9..0a76904 100644 --- a/ws2019/ana/lectures/analysis7.tex +++ b/ws2019/ana/lectures/analysis7.tex @@ -116,7 +116,7 @@ Fortsetzung Beweis: Für $b=2$ : dijadische Entwicklung \end{bem} -\subsubsection{Zusammenfassung} +\subsection{Zusammenfassung} Beobachtung: Jede reelle Zahl ist ein Grenzwert von einer Folge $(a_n)_{n\in\N}$ rationaler Zahlen. @@ -151,7 +151,7 @@ Konstruktion nach Cantor, 1873 Als nächstes: $\R$ ist ein angeordneter Körper mit ,,$+$ '', ,,$\cdot$'', ,,$>$'' und ist auch ,,vollständig''. -\subsection{Der Körper $\R$} +\section{Der Körper $\R$} Seien $a \in \R, b \in \R$ und $(a_n)_{n\in\N}$, $(b_n)_{n\in\N}$ zugehörige approximierende Folgen rationaler Zahlen. diff --git a/ws2019/ana/lectures/analysis8.tex b/ws2019/ana/lectures/analysis8.tex index 4b6de9e..f174145 100644 --- a/ws2019/ana/lectures/analysis8.tex +++ b/ws2019/ana/lectures/analysis8.tex @@ -75,11 +75,11 @@ .\end{align*} \end{bem} -\subsection{Wichtige Aussage} +\section{Wichtige Aussage} $\R$ ist vollständig, da alle C.F. in $\R$ haben einen Grenzwert in $\R$. -\subsection{Weitere Möglichkeiten, die Vollständigkeit von $\R$ zu charakterisieren} +\section{Weitere Möglichkeiten, die Vollständigkeit von $\R$ zu charakterisieren} \begin{definition}[Maximum, Minimum, obere/untere Schranke, Supremum, Infimum] Sei $M \subset \R, M \neq \emptyset$. diff --git a/ws2019/ana/lectures/analysis9.tex b/ws2019/ana/lectures/analysis9.tex index 551d8c6..faeb56c 100644 --- a/ws2019/ana/lectures/analysis9.tex +++ b/ws2019/ana/lectures/analysis9.tex @@ -134,7 +134,7 @@ Heute: Längstes deutsches Wort! .\] \end{bsp} -\subsection{Mächtigkeit von $\Q$ und $\R$ } +\section{Mächtigkeit von $\Q$ und $\R$ } \begin{definition}[Mächtigkeit] Die Mächtigkeit einer Menge ist die Anzahl ihrer Elemente. @@ -229,7 +229,7 @@ Heute: Längstes deutsches Wort! .\] aber $d_k \neq d_{kk}$ nach Konstruktion. \end{proof} -\subsection{Die Komplexen Zahlen $\C$} +\section{Die Komplexen Zahlen $\C$} \[ \C := \R \times \R = \{ z = (x, y) \mid x, y \in \R\} .\] Addition in $\C$ : $z_1 = (x_1, y_1) \in \C$, $z_2 = (x_2, y_2) \in \C$:

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