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Vorlesung vom 14.11.2013 digitalisiert
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Martin Thoma
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@ -102,5 +102,152 @@ U_i = \Set{(x_0: \dots : x_n) \in \mdp^n(\mdr) | x_i \neq 0} &\rightarrow \mdr^n
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\end{enumerate}
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\end{beispiel}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Mitschrieb vom 14.11.2013 %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{definition}\xindex{Verklebung}
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Seien $X, Y$ $n$-dimensionale Mannigfaltigkeiten, $U \subseteq X$
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und $V \subseteq Y$ offen, $\Phi: U \rightarrow V$ ein Homöomorphismus
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$Z = (X \dcup Y) /_\sim$ mit der von $u \sim \Phi(u) \forall{u \in U}$
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erzeugten Äquivalenzrelation und der von $\sim$ induzierten
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Quotiententopologie.
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$Z$ heißt \textbf{Verklebung} von $X$ und $Y$ längs $U$ und $V$.
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$Z$ besitzt einen Atlas aus $n$-dimensionalen Karten.
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Falls $Z$ hausdoffsch ist, ist $Z$ eine $n$-dimensionale
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Mannigfaltigkeit.
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\end{definition}
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\todo[inline]{Bilder mit Verklebung einfügen}
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\begin{korollar}
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Sind $X, Y$ Mannigfaltigkeiten der Dimension $n$ bzw. $m$, so ist
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$X \times Y$ eine Mannigfaltigkeit der Dimension $n+m$.
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\end{korollar}
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\begin{beweis}
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Produkte von Karten sind Karten. $\qed$
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\end{beweis}
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\begin{beispiel}
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Mannigfaltigkeiten mit Dimension 1:
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\begin{enumerate}[label=\arabic*)]
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\item Offene Intervalle, $\mdr$, $(0,1)$ sind alle homöomorph
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\item $S^1$
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\end{enumerate}
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Mannigfaltigkeiten mit Dimension 2:
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\begin{enumerate}[label=\arabic*)]
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\item $\mdr^2$
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\item $S^2$ (0 Henkel)
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\item $T^2$ (1 Henkel)
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\item oder mehr Henkel, wie z.B. der Zweifachtorus in Abb. \ref{fig:double-torus}
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\end{enumerate}
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\begin{figure}
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\centering
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\includegraphics[width=0.2\linewidth, keepaspectratio]{figures/Double-torus-illustration.png}
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\caption{Zweifachtorus}
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\label{fig:double-torus}
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\end{figure}
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\end{beispiel}
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\begin{korollar}
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Sei $n \in \mdn, F:\mdr^n \rightarrow \mdr$ stetig differenzierbar
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und $X = V(F) := \Set{x \in \mdr^n | F(x) = 0}$ das \enquote{vanishing set}.
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Dann gilt:
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\begin{enumerate}[label=\alph*)]
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\item $X$ ist abgeschlossen in $\mdr^n$
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\item Ist $\text{grad}(F)(X) \neq 0 \forall{x \in X}$, so ist
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$X$ eine Mannigfaltigkeit der Dimension $n-1$. \label{Mannigfaltigkeitskriterium}
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\end{enumerate}
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\end{korollar}
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\begin{beweis}
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\begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*]
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\item Sei $y \in \mdr^n \setminus V(F)$. Weil $F$ stetig ist,
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gibt es $\delta > 0$, sodass $F(\fB_\delta(y)) \subseteq \fB_\varepsilon(F(y))$
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mit $\varepsilon = \frac{1}{2} \|F(y)\|$. Folgt
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$\fB_\delta(y) \cap V(F) = \emptyset \Rightarrow \mdr^n \setminus V(F)$
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ist offen.
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\item Sei $x \in X$ mit $\text{grad}(F)(x) \neq 0$, also
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\obda $\frac{\partial F}{\partial X_1} (x) \neq 0$,
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$x = (x_1, \dots, x_n)$, $x' := (x_2, \dots, x_n) \in \mdr^{n-1}$.
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Der Satz von der impliziten Funktion liefert nun:
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Es gibt Umgebungen $U$ von $x'$ und differenzierbare
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Funktionen $g: U \rightarrow \mdr$, sodass
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$G: U \rightarrow \mdr^n, \; u \mapsto (g(u), u)$
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eine stetige Abbildung auf eine offene Umgebung $V$ von
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$x$ in $X$ ist.
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\end{enumerate}
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$\qed$
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\end{beweis}
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\begin{beispiel}\xindex{Neilsche Parabel}
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\begin{enumerate}[label=\alph*)]
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\item $F: \mdr^3 \rightarrow \mdr,\;\;\; (x, y, z) \mapsto x^2 + y^2 + z^2 - 1$,
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$V(F) = S^2$, $\text{grad}(F) = (2x, 2y, 2z) \xRightarrow{\ref{Mannigfaltigkeitskriterium}} S^n$
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ist $n$-dimensionale Mannigfaltigkeit in $\mdr^{n+1}$
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\item $F: \mdr^2 \rightarrow \mdr, \;\;\; (x,y) \mapsto y^2 - x^3$
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\begin{figure}[ht]
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\centering
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\subfloat[$F(x,y) = y^2 - x^3$]{
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\input{figures/3d-function-semicubical-parabola.tex}
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\label{fig:semicubical-parabola-2d}
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}%
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\subfloat[$y^2 - ax^3 = 0$]{
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\input{figures/2d-semicubical-parabola.tex}
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\label{fig:semicubical-parabola-3d}
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}%
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\label{Neilsche-Parabel}
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\caption{Rechts ist die Neilsche Parabel für verschiedene Parameter $a$.}
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\end{figure}
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Es gilt: $\text{grad}(F) = (-3x^2, 2y)$. Also: $\text{grad}(0,0) = (0,0)$.
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Daher ist Korollar \label{Mannigfaltigkeitskriterium}
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nicht anwendbar, aber $V(F)$ ist trotzdem
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eine 1-dimensionale topologische Mannigfaltigkeit.
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\end{enumerate}
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\end{beispiel}
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\begin{definition}\textbf{Mannigfaltigkeit!mit Rand}
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Sei $X$ ein Hausdorffraum mit abzählbarer Basis der Topologie.
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$X$ heißt $n$-dimensionale \textbf{Mannigfaltigkeit mit Rand},
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wenn es einen Atlas $(U_i, \varphi_i)$ gibt, wobei $U_i \subseteq X_i$
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offen und $\varphi_i$ ein Homöomorphismus auf eine offene
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Teilmenge von
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\[R_{+,0}^n := \Set{(x_1, \dots, x_n) \in \mdr^n | x_m \geq 0}\]
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ist. $R_{+,0}^n$ ist ein \enquote{Halbraum}.
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\end{definition}
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\begin{beispiel}
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\todo[inline]{Viele Bilder: Pair of pants, sphere with a hole, halbraum...}
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\end{beispiel}
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\begin{definition}\xindex{Rand}
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Sei $X$ eine $n$-dimensionale Mannigfaltigkeit mit Rand und
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Atlas $(U_i, \varphi_i)$. Dann heißt
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\[\partial X := \bigcup_{i\in I} \Set{x \in U_i | \varphi_i (x)_n = 0}\]
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\textbf{Rand} von $X$.
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\end{definition}
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$\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
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\begin{definition}\xindex{Kartenwechsel}\xindex{bergangsfunktion}
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Sei $X$ eine $n$-dimensionale Mannigfaltigkeit mit Atlas
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$(U_i, \varphi_i)_{i \in I}$
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\begin{enumerate}[label=\alph*)]
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\item Für $i, j \in I$ mit $U_i, U_j \neq \emptyset$ heißt
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\begin{align*}
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\varphi_{ij} &:= \varphi_j \circ \varphi_i^{-1}\\
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\varphi_i (U_i \cap U_j) &\rightarrow \varphi_j (U_i \cap U_j)
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\end{align*}
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\textbf{Kartenwechsel} oder \textbf{Übergangsfunktion}.
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\end{enumerate}
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\end{definition}
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\todo[inline]{Bilder mit Verklebung einfügen}
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% Die Übungsaufgaben sollen ganz am Ende des Kapitels sein.
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\input{Kapitel2-UB}
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@ -65,6 +65,27 @@
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sort=MengenoperationNSetminus
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}
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\newglossaryentry{cup}
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{
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name={\ensuremath{A \cup B}},
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description={Vereinigung},
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sort=MengenoperationOCup
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}
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\newglossaryentry{dcup}
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{
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name={\ensuremath{A \dcup B}},
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description={Disjunkte Vereinigung},
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sort=MengenoperationOCupD
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}
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\newglossaryentry{cap}
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{
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name={\ensuremath{A \cap B}},
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description={Schnitt},
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sort=MengenoperationOCap
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Zahlenmengen %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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32
documents/GeoTopo/figures/2d-semicubical-parabola.tex
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32
documents/GeoTopo/figures/2d-semicubical-parabola.tex
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@ -0,0 +1,32 @@
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\begin{tikzpicture}
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\begin{axis}[
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legend pos=south east,
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axis x line=middle,
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axis y line=middle,
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grid = major,
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%width=9cm,
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%height=4.5cm,
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grid style={dashed, gray!30},
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xmin= 0, % start the diagram at this x-coordinate
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xmax= 12, % end the diagram at this x-coordinate
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ymin=-10, % start the diagram at this y-coordinate
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ymax= 10, % end the diagram at this y-coordinate
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%axis background/.style={fill=white},
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xlabel=$x$,
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ylabel=$y$,
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%xticklabels={-2,-1.6,...,7},
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tick align=outside,
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%minor tick num=-3,
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enlargelimits=true]
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\addplot[domain=0:12, red, thick,samples=500] {1/3*x^1.5};
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\addplot[domain=0:12, orange, thick,samples=500] {1*x^1.5};
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\addplot[domain=0:12, blue, thick,samples=500] {2*x^1.5};
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\addplot[domain=0:12, red, thick,samples=500] {-1/3*x^1.5};
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\addplot[domain=0:12, orange, thick,samples=500] {-1*x^1.5};
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\addplot[domain=0:12, blue, thick,samples=500] {-2*x^1.5};
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\addlegendentry{$a=\frac{1}{3}$}
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\addlegendentry{$a=1$}
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\addlegendentry{$a=2$}
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\end{axis}
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\end{tikzpicture}
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@ -0,0 +1,31 @@
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\pgfplotsset{
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colormap={whitered}{
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color(0cm)=(white);
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color(1cm)=(orange!75!red)
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}
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}
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\begin{tikzpicture}[scale=0.5]
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\begin{axis}[
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colormap name=whitered,
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width=15cm,
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view={155}{45},
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enlargelimits=false,
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grid=major,
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domain=-5:5,
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y domain=-5:5,
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samples=56, %57 : TeX capacity exceeded, sorry [main memory size=3000000].
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% see also http://tex.stackexchange.com/a/7954/5645
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xlabel=$x$,
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ylabel=$y$,
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zlabel={$z$},
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colorbar,
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colorbar style={
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at={(-0.1,0)},
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anchor=south west,
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height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
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title={$f(x,y)$}
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}
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]
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\addplot3[surf] {y*y-x*x*x};
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\end{axis}
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\end{tikzpicture}
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BIN
documents/GeoTopo/figures/Double-torus-illustration.png
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BIN
documents/GeoTopo/figures/Double-torus-illustration.png
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@ -62,4 +62,7 @@
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\def\GL{\ensuremath{\mathrm{GL}}}
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\newcommand\mapsfrom{\mathrel{\reflectbox{\ensuremath{\mapsto}}}}
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\newcommand\dcup{\mathbin{\dot{\cup}}}
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\newcommand\obda{o.~B.~d.~A.}
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@ -0,0 +1,39 @@
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\documentclass[varwidth=true, border=2pt]{standalone}
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\usepackage{pgfplots}
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\usepackage{tikz}
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\begin{document}
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\begin{tikzpicture}
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\begin{axis}[
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legend pos=south east,
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axis x line=middle,
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axis y line=middle,
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grid = major,
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%width=9cm,
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%height=4.5cm,
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grid style={dashed, gray!30},
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xmin= 0, % start the diagram at this x-coordinate
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xmax= 12, % end the diagram at this x-coordinate
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ymin=-10, % start the diagram at this y-coordinate
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ymax= 10, % end the diagram at this y-coordinate
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%axis background/.style={fill=white},
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xlabel=$x$,
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ylabel=$y$,
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%xticklabels={-2,-1.6,...,7},
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tick align=outside,
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%minor tick num=-3,
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enlargelimits=true]
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\addplot[domain=0:12, red, thick,samples=500] {1/3*x^1.5};
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\addplot[domain=0:12, orange, thick,samples=500] {1*x^1.5};
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\addplot[domain=0:12, blue, thick,samples=500] {2*x^1.5};
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\addplot[domain=0:12, red, thick,samples=500] {-1/3*x^1.5};
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\addplot[domain=0:12, orange, thick,samples=500] {-1*x^1.5};
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\addplot[domain=0:12, blue, thick,samples=500] {-2*x^1.5};
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\addlegendentry{$a=\frac{1}{3}$}
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\addlegendentry{$a=1$}
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\addlegendentry{$a=2$}
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\end{axis}
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\end{tikzpicture}
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\end{document}
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After Width: | Height: | Size: 62 KiB |
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@ -0,0 +1,41 @@
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\documentclass{article}
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\usepackage[pdftex,active,tightpage]{preview}
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\setlength\PreviewBorder{2mm}
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\usepackage{pgfplots}
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\pgfplotsset{compat=1.9}
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\begin{document}
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\begin{preview}
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\pgfplotsset{
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colormap={whitered}{
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color(0cm)=(white);
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color(1cm)=(orange!75!red)
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}
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}
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\begin{tikzpicture}[scale=0.5]
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\begin{axis}[
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colormap name=whitered,
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width=6cm,
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view={155}{45},
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enlargelimits=false,
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grid=major,
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domain=-5:5,
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y domain=-5:5,
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samples=56, %57 : TeX capacity exceeded, sorry [main memory size=3000000].
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% see also http://tex.stackexchange.com/a/7954/5645
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xlabel=$x$,
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ylabel=$y$,
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zlabel={$z$},
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colorbar,
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colorbar style={
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at={(-0.1,0)},
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anchor=south west,
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height=0.25*\pgfkeysvalueof{/pgfplots/parent axis height},
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title={$f(x,y)$}
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}
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]
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\addplot3[surf] {y*y-x*x*x};
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\end{axis}
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\end{tikzpicture}
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\end{preview}
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||||
\end{document}
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35
tikz/3d-function-semicubical-parabola/Makefile
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35
tikz/3d-function-semicubical-parabola/Makefile
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@ -0,0 +1,35 @@
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SOURCE = 3d-function-semicubical-parabola
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DELAY = 80
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DENSITY = 300
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WIDTH = 512
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make:
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pdflatex $(SOURCE).tex -output-format=pdf
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make clean
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clean:
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rm -rf $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot
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gif:
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pdfcrop $(SOURCE).pdf
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convert -verbose -delay $(DELAY) -loop 0 -density $(DENSITY) $(SOURCE)-crop.pdf $(SOURCE).gif
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make clean
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png:
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make
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make svg
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inkscape $(SOURCE).svg -w $(WIDTH) --export-png=$(SOURCE).png
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transparentGif:
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convert $(SOURCE).pdf -transparent white result.gif
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make clean
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svg:
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make
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#inkscape $(SOURCE).pdf --export-plain-svg=$(SOURCE).svg
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pdf2svg $(SOURCE).pdf $(SOURCE).svg
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# Necessary, as pdf2svg does not always create valid svgs:
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inkscape $(SOURCE).svg --export-plain-svg=$(SOURCE).svg
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rsvg-convert -a -w $(WIDTH) -f svg $(SOURCE).svg -o $(SOURCE)2.svg
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inkscape $(SOURCE)2.svg --export-plain-svg=$(SOURCE).svg
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rm $(SOURCE)2.svg
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3
tikz/3d-function-semicubical-parabola/Readme.md
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3
tikz/3d-function-semicubical-parabola/Readme.md
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@ -0,0 +1,3 @@
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Compiled example
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||||
----------------
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