From 12c3f170e2da580b7037c48fa9a84dfe481ea40a Mon Sep 17 00:00:00 2001 From: flavis Date: Fri, 24 Jul 2020 10:48:18 +0200 Subject: [PATCH] add ana22 without graphics --- ana21.tex | 71 ++++++++++++++- ana22.tex | 240 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 963940 -> 985138 bytes analysisII.tex | 1 + 4 files changed, 311 insertions(+), 1 deletion(-) create mode 100644 ana22.tex diff --git a/ana21.tex b/ana21.tex index 9465eec..0421cc8 100644 --- a/ana21.tex +++ b/ana21.tex @@ -181,7 +181,7 @@ \begin{satz}[Erster Hauptsatz über Kurvenintegrale] Sei $D\subset \R^n$ ein Gebiet und $F\in C^0(D,\R^n)$. Dann sind folgend Aussagen äquivalent: - \begin{enumerate} + \begin{enumerate}[(i)] \item F ist konservativ. \item $\int_\gamma F=0$ für alle geschlossenen Integrationswege $\gamma$ in $D$. @@ -189,6 +189,75 @@ Das Kurvenintegral von $F$ in $D$ ist wegunabhängig, d.h. für beliebige Integrationswege $\gamma_1\colon [a,b]\to D,\ \gamma_2\colon [\alpha,\beta]\to D$ mit $\gamma_1(a)=\gamma_2(\alpha)$ und $\gamma_1(b)=\gamma_2(\beta)$ gilt $$\int_{\gamma_1}F=\int_{\gamma_2}F.$$ \end{enumerate} In diesem Fall erhalten wir eine Stammfunktion $\varphi_0\in C^1(D,\R)$ durch $\varphi_0(x)\coloneqq \int_\gamma F$, wobei $\gamma$ ein Integrationsweg von einem gewählten Punkt $x_0\in D$ zu $x\in D$ ist. Die Menge aller Potentiale von $F$ ist gegeben durch $\{\varphi_0+c\,|\,c\in \R\}$. + \label{satz:hauptsatz-1-kurven} \end{satz} +\begin{proof} + (i)$\implies$(ii): Sei $F = \nabla \varphi$, $\varphi \in C^{1}(D, \R)$ und $\gamma\colon [a,b] \to D$ + geschlossener Integrationsweg. Dann folgt + \begin{salign*} + \int_{\gamma} F &\stackrel{\text{Def.}}{=} \int_{a}^{b} (F(\gamma(t)), \gamma'(t)) \d t \\ + &= \int_{a}^{b} (\nabla \varphi(\gamma(t)), \gamma'(t)) \d t \\ + &\stackrel{\gamma \text{ stückweise } C^{1}}{=} + \sum_{i=1}^{M} \int_{s_{i-1}}^{s_i} (\nabla \varphi(\gamma(t)), \gamma'(t)) \d t \\ + &\stackrel{\text{Kettenregel}}{=} \sum_{i=1}^{M} \int_{s_{i-1}}^{s_i} \frac{\d}{\d t} (\varphi \circ \gamma) \d t \\ + &\stackrel{\text{HDI}}{=} \sum_{i=1}^{M} \left( \varphi(\gamma(s_i)) - \varphi(\gamma(s_{i-1})) \right) \\ + &= \varphi(\gamma(b)) - \varphi(\gamma(a)) \\ + &\stackrel{\gamma(b) = \gamma(a)}{=} 0 + .\end{salign*} + + (ii)$\implies$(iii): Nach Umparametrisierung gelte o.E. $[a,b] = [-1, 0] = [\alpha, \beta]$. Seien + $\gamma_1, \gamma_2\colon [-1,0] \to D$ Integrationswege mit gleichem Anfangs und Endpunkt, d.h. + $\gamma_1(-1) = \gamma_2(-1)$ und $\gamma_1(0) = \gamma_2(0)$. Dann betrachte + \begin{align*} + \gamma &\colon [-1, 1] \to D \\ + t& \mapsto \begin{cases} + \gamma_1(t) & t \in [-1, 0] \\ + \gamma_2(-t) & t \in [0, 1] + \end{cases} + .\end{align*} + Dann ist $\gamma$ ein geschlossener Integrationsweg, also folgt + \begin{salign*} + 0 &= \int_{\gamma}^{} F \\ + &= \int_{-1}^{0} (F(\gamma_1(t)), \gamma_1'(t)) \d t + \int_{0}^{1} (F(\gamma_2(-t), - \gamma_2'(-t)) \d t \\ + &\stackrel{s \coloneqq -t}{=} \int_{\gamma_1}^{} F + \int_{0}^{-1} (F(\gamma_2(s)), \gamma_2'(s) \d s \\ + &= \int_{\gamma_1}^{} F - \int_{\gamma_2}^{} F + .\end{salign*} + + (iii) $\implies$ (i): Fixiere $x_0 \in D$ und definiere $\varphi_0\colon D \to \R$ durch + $\varphi_0(x) \coloneqq \int_{\gamma}^{} F$, wobei + $\gamma$ irgendein Integrationsweg von $x_0$ nach $x$ ist. Zu $x \in D$ betrachte + $x + h e_i \in D$ für $|h| \ll 1$. Nach Umparametrisierung gelte o.E. + \begin{align*} + &\gamma_x \colon [-1, 0] \to D \\ + &\gamma_{x + h e_i}\colon [-1, 1] \to D \\ + &\gamma_{x + h e_i} \coloneqq \begin{cases} + \gamma_x(t) & t \in [-1, 0] \\ + x + t h e_i & t \in [0,1] + \end{cases} + .\end{align*} + Dann folgt + \begin{salign*} + \frac{\partial \varphi_0(x)}{\partial x_i} &= \lim_{h \to 0} \frac{\varphi_0(x + h e_i) - \varphi_0(x)}{h} \\ + &= \lim_{h \to 0} \frac{1}{h} \left( \int_{\gamma_{x + h e_i}} F - \int_{\gamma_x}^{} F \right) \\ + &= \lim_{h \to 0} \frac{1}{h} \int_{0}^{1} \left( F(x + t h e_i), \gamma'_{x + h e_i}(t) \right) \d t \\ + &= \lim_{h \to 0} \frac{1}{h} \int_{0}^{1} ( F(\underbrace{x + t h e_i}_{\xrightarrow{h \to 0} F(x)}), h e_i ) \d t \\ + &= \int_{0}^{1} (F(x), e_i) \d t \\ + &= F_i(x) + .\end{salign*} + Damit ist $\varphi_0 \in C^{1}(D, \R)$ und $\nabla \varphi_0 = F$. Das zeigt (i). + + Sei $\gamma$ ein Integrationsweg von $x_0 \in D$ nach $x \in D$. Dann definiere + \[ + \varphi_0(x) \coloneqq \int_{\gamma} F + .\] + Sei weiter $\varphi \in C^{1}(D, \R)$ mit $\nabla \varphi = F$. Dann gilt wegen (i) und (ii): + \[ + \int_{\gamma} F = \varphi(x) - \varphi(x_0) + .\] Damit folgt + \[ + \varphi(x) - \varphi(x_0) = \varphi_0(x) \implies \varphi(x) = \varphi_0(x) + \underbrace{\varphi(x_0)}_{\text{konst}} = \varphi_0(x) + c + .\] +\end{proof} + \end{document} diff --git a/ana22.tex b/ana22.tex new file mode 100644 index 0000000..523b2d4 --- /dev/null +++ b/ana22.tex @@ -0,0 +1,240 @@ +\documentclass{lecture} + +\begin{document} + +\begin{bsp} + \label{bsp:windungsfeld} + \begin{enumerate} + \item Windungsfeld: + \[ + W(x,y) \coloneqq \frac{1}{x^2 + y^2} \begin{pmatrix} -y \\ x \end{pmatrix} + .\] $D \coloneqq \R^2 \setminus \left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix} \right\} $. $W$ ist + nicht konservativ auf $D$ weil mit $\gamma\colon [0, 2\pi] \to \R^2$, + $\gamma(t) \coloneqq \begin{pmatrix} \cos(t) \\ \sin(t) \end{pmatrix} $ ist + \[ + \int_{\gamma} W = 2 \pi \neq 0 + .\] Aber mit $D \coloneqq \left\{ \begin{pmatrix} x \\ y \end{pmatrix} \mid y > 0\right\} $ + ist + \[ + \varphi(x,y) \coloneqq - \arctan\left( \frac{x}{y} \right) + \] ein Potential von $W$ auf $D$, denn + \begin{align*} + \frac{\partial \varphi(x,y)}{\partial x} &= -\frac{1}{1 + \frac{x^2}{y^2}} \frac{1}{y} = - \frac{y}{x^2 + y^2} \\ + \frac{\partial \varphi(x,y)}{\partial y} &= \frac{x}{x^2 + y^2} + .\end{align*} + Die Existenz eines Potentials hängt also auch von $D$ ab. + \item Suche nach einem Potential: + $F\colon \R^2 \to \R^2$, $F(x,y) \coloneqq \begin{pmatrix} y \\ x \end{pmatrix} $. + Falls Potential existiert, dann gilt + \[ + \varphi_0(x,y) \coloneqq \int_{\gamma} F + .\] mit z.B. $\gamma(t) \coloneqq t \begin{pmatrix} x \\ y \end{pmatrix} $, $t \in [0,1]$. + Dann gilt + \begin{salign*} + \int_{\gamma} F &= \int_{0}^{1} \left( F(\gamma(t), \gamma'(t) \right) \d t \\ + &= \int_{0}^{1} \left( \begin{pmatrix} ty \\ tx \end{pmatrix} , \begin{pmatrix} x \\ y \end{pmatrix} \right) \d t \\ + &= \int_{0}^{1} \left( t y x + tx y \right) \d t \\ + &= \int_{0}^{1} 2 t x y \d t \\ + &= xy + .\end{salign*} + Definiere $\varphi = xy$. Dann ist $\frac{\partial \varphi}{\partial x} = y = F_1(x,y)$ + und $\frac{\partial \varphi}{\partial y} x = F_2(x,y)$. Also $\nabla \varphi = F$. + \end{enumerate} +\end{bsp} + +\section{Existenz von Potentialen} + +Sei $D \subseteq \R^{n}$ offen und $F \in C^{1}(D, \R^{n})$ ein konservatives Vektorfeld. Dann existiert +ein $\varphi \in C^2(D, \R)$ mit $\nabla \varphi = F$, d.h. $\frac{\partial \varphi}{\partial x_i} = F_i$ +für $i = 1, \ldots, n$. Da $\varphi$ zweimal stetig differenzierbar, folgt +\begin{align*} +\frac{\partial F_i}{ \partial x_j} = \frac{\partial^2 \varphi}{\partial x_j \partial x_i} += \frac{\partial^2 \varphi}{\partial x_i \partial x_j} = \frac{\partial F_j}{\partial x_i} +.\end{align*} + +Ist also $F \in C^{1}(D, \R^{n})$ konservativ, dann gelten notwendig die \underline{Integrabilitätsbedingungen} +\[ +\frac{\partial F_i}{\partial x_j} - \frac{\partial F_j}{\partial x_i} \equiv 0 \qquad \forall i, j = 1,\ldots,n +.\] +Speziell für $n = 2$: +\[ +\frac{\partial F_2}{\partial x_1} = \frac{\partial F_1}{\partial x_2} +.\] Für $n = 3$: +\[ + \text{rot}(F) \coloneqq \begin{pmatrix} \frac{\partial F_3}{\partial x_2} - \frac{\partial F_2}{\partial x_3} \\ + \frac{\partial F_1}{\partial x_3} - \frac{\partial F_3}{\partial x_1} \\ +\frac{\partial F_2}{\partial x_1} - \frac{\partial F_1}{\partial x_2}\end{pmatrix} = 0 +.\] + +Die Integrabilitätsbedingungen sind nicht hinreichend. + +\begin{bsp}[Windungsfeld] + \[ + W(x,y) \coloneqq \frac{1}{x^2 + y^2} \begin{pmatrix} -y \\ x \end{pmatrix} + .\] Dann gilt + \begin{align*} + \frac{\partial W_y}{\partial x}(x,y) = \frac{\partial}{\partial x} \left( \frac{x}{x^2 + y^2} \right) + = \frac{y^2 - x^2}{(x^2 + y^2)^2} \\ + \frac{\partial W_x}{\partial y} = \frac{\partial}{\partial y} \left( - \frac{y}{x^2 + y^2} \right) + = \frac{y^2 - x^2}{(x^2 + y^2)^2} + .\end{align*} + Also gilt $\frac{\partial W_y}{\partial x} = \frac{\partial W_x}{\partial y}$ auf $D \coloneqq \R^2 \setminus \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix} \right\} $, aber auf $D$ existiert kein Potential + (vgl. \ref{bsp:windungsfeld}). +\end{bsp} + +\begin{definition}[Homotopie] + Sei $D \subseteq \R^{n}$ und $\gamma_0, \gamma_1 \in C\left( [a,b], D \right) $ stetige + Kurven. + \begin{enumerate}[(i)] + \item Es gelte $\gamma_0(a) = A = \gamma_1(a)$ und + $\gamma_0(b) = B = \gamma_1(b)$. $y_0$ und $y_1$ heißen \underline{homotop} in $D$, falls + eine stetige Abbildung $H\colon [a,b] \times [0,1] \to D$ (Homotopie) existiert, s.d. + $H(t, 0) = \gamma_0(t)$ und $H(t,1) = \gamma_1(t)$, $\forall t \in [a,b]$ und + $H(a, s) = A$ und $H(b,s) = B$, $\forall s \in [0,1]$. + + Für $s \in [0,1]$ sind + $\gamma_s^{(t)} \coloneqq H(t,s)$, $t \in [a,b]$ mit $\gamma_s(a) = A$ und $\gamma_s(b) = B$ + stetige Kurven von $A$ nach $B$ in $D$. $H$ heißt stetige Deformation von $\gamma_0$ nach + $\gamma_1$. + \item $\gamma_0$ und $\gamma_1$ seien geschlossen. $\gamma_0$ und $\gamma_1$ heißen + \underline{frei homotop} in $D$ falls eine stetige Abbildung + $H\colon [a,b] \times [0,1] \to D$ mit $H(t,0) = \gamma_0(t)$, $\forall t \in [a,b]$ und + $H(a,s) = H(b,s)$, $\forall x \in [0,1]$ d.h. für $s \in [0,1]$ sind $\gamma_s(t) \coloneqq H(t,s)$ + geschlossene Kurven in $D$. + + $H$ heißt stetige Deformation innerhalb von $D$ der + geschlossenen Kurve $\gamma_0$ nach der geschlossenen Kurve $\gamma_1$. + \item Eine geschlossene Kurve heißt zusammenziehbar in $D$, wenn sie frei homotop zu + einer konstanten Kurve ist, d.h. sie sich in $D$ zu einem Punkt zusammenziehen lässt. + \end{enumerate} +\end{definition} + +\begin{bsp} + \label{bsp:ellipse-und-kreis} + Ellipse: Seien $a, b > 0$. + \[ + \epsilon(t) \coloneqq \begin{pmatrix} a \cos(t)\\ b \sin(t)\end{pmatrix}, \quad t \in [0, 2\pi] + \] ist frei homotop zum Kreis + \[ + K(t) \coloneqq \begin{pmatrix} \cos(t) \\ \sin(t) \end{pmatrix} + \] via der Homotopie + \begin{align*} + &H\colon [0, 2\pi] \times [0,1] \to \R^2 \setminus \left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix} \right\} \\ + &H(t, s) \coloneqq s K(t) + (1 - s) \epsilon(t) + .\end{align*} + Es gilt + \begin{align*} + \Vert H(t,s) \Vert^2 &= (s + a(1-s))^2 \cos^2(t) + (s + b (1-s))^2 \sin^2(t) \\ + &\ge \min(1, a^2, b^2) (\cos^2(t) + \sin^2(t)) \\ + &= \min(1, a^2, b^2) \\ + &> 0 + .\end{align*} + Also $H(t,s) \neq 0$ $\forall t, s$. +\end{bsp} + +\begin{satz}[Zweiter Hauptsatz der Kurvenintegrale: Homotopieinvarianz] + Sei $D \subseteq \R^{n}$ offen, $F \in C^{1}(D, \R^{n})$ erfülle + die Integrabilitätsbedingungen und $\gamma_0, \gamma_1 \colon [a,b] \to D$ Integrationswege. + + Sind $\gamma_0$ und $\gamma_1$ homotop in $D$ mit gemeinsamem Anfangs- und Endpunkt oder + geschlossen und frei homotop in $D$, dann gilt + \[ + \int_{\gamma_1}^{} F = \int_{\gamma_0}^{} F + .\] + \label{satz:hauptsatz-2-kurven} +\end{satz} + +\begin{proof} + ohne Beweis +\end{proof} + +\begin{bsp}[Windungsfeld] + \begin{enumerate} + \item Für Ellipse $\epsilon(t)$ und Kreis $K(t)$ (vgl. Beispiele \ref{bsp:ellipse-und-kreis} und + \ref{bsp:windungsfeld}) gilt + \[ + \int_{K} W = 2 \pi \stackrel{\ref{satz:hauptsatz-2-kurven}}{=} + \int_{\epsilon}^{} W = \int_{0}^{2 \pi} \frac{ab}{a^2 \cos^2 t + b^2 \sin^2 t} \d t + .\] + Also folgt + \[ + \int_{0}^{2\pi} \frac{\d t}{a^2 \cos^2t + b^2 \sin^2 t} = \frac{2 \pi}{ab} + .\] + \item Kurven $\gamma_0, \gamma_1\colon [0, 2\pi] \to \R^2 \setminus \{0\} $: + \[ + \gamma_0 \coloneqq \begin{pmatrix} \cos(t) \\ \sin(t) \end{pmatrix} + \qquad + \gamma_1(t) \coloneqq \begin{pmatrix} \cos(t) \\ - \sin(t) \end{pmatrix} + .\] $\gamma_0$ und $\gamma_1$ sind nicht frei homotop in $\R^2 \setminus \{0\} $, weil + \[ + \int_{\gamma_0} W = 2 \pi \neq - 2\pi = \int_{\gamma_1}^{} W + .\] + \item Kurven $\gamma_0, \gamma_1\colon [0, \pi] \to \R^2 \setminus \{0\}$ + \[ + \gamma_0 \coloneqq \begin{pmatrix} \cos(t) \\ \sin(t) \end{pmatrix} + \qquad + \gamma_1(t) \coloneqq \begin{pmatrix} \cos(t) \\ - \sin(t) \end{pmatrix} + .\] $\gamma_0$, $\gamma_1$ sind nicht homotop in $\R^2 \setminus \{0\} $, weil + \[ + \int_{\gamma_0}^{} W = \int_{0}^{\pi} 1 \d t \neq \int_{0}^{\pi} -1 \d t = \int_{\gamma_1}^{} W \d t + .\] + \end{enumerate} +\end{bsp} + +\begin{definition}[Einfach zusammenhängend] + Sei $D \subseteq \R^{n}$ ein Gebiet. $D$ heißt einfach zusammenhängend, wenn jede geschlossene + Kurve in $D$ frei homotop zu einer konstanten Kurve ist, d.h. jede geschlossene Kurve in $D$ + zusammenziehbar ist. +\end{definition} + +\begin{definition}[Sternförmig] + Ein Gebiet $D \subseteq \R^{n}$ heißt sternförmig, + wenn ein $x_1 \in D$ existiert, s.d. $\forall x \in D$ gilt: + \[ + x_1 + t(x - x_1) \in D \quad \forall t \in [0,1] + .\] D.h. $\forall x \in D$ liegt die Verbindungsstrecke von $x_1$ nach $x$ in $D$. +\end{definition} + +\begin{bem} + Jedes Sterngebiet ist einfach zusammenhängend. +\end{bem} + +\begin{proof} + $H(t,s) \coloneqq x_1 + s(\gamma(t) - x_1) \in D$, $\forall t$, $\forall s \in [0,1]$. +\end{proof} + +\begin{bsp} + \begin{enumerate}[] + \item Jede Kugel $K_r(a)$ ist sternförmig bezüglich $a$, also auch einfach zusammenhängend. + \item Eine gelochte Kreisscheibe $K_1(0) \setminus \{0\} $, $K_1(0) \subseteq \R^2$ + ist kein Sterngebiet und nicht einfach zusammenhängend. + \item Geschlitzte Scheibe $K_1(0) \setminus \{x \in \R^2 \mid x_1 \le 0, x_2= 0\} \subseteq \R^2$ + ist sternförmig, also einfach zusammenhängend. + \item Jede geschlitzte Ebene $\R^2 \setminus S_v$ mit + $S_v \coloneqq \{ t v | t \ge 0, \Vert v \Vert = 1\} $ ist sternförmig mit + Mittelpunkt $(-v)$, also auch einfach zusammenhängend. + \item $R^{n} \setminus \{0\} $ ist kein Sterngebiet, weil $0 \in$ Strecke von $-a$ nach $a$, + aber einfach zusammenhängend für $n \ge 3$. + \end{enumerate} +\end{bsp} + +\begin{satz}[Lemma von Poincaré] + Sei $D \subseteq \R^{n}$ ein einfach zusammenhängendes Gebiet und + $F \in C^{1}(D, \R^{n})$ erfülle die Integrabilitätsbedingunen. Dann + ist $F$ konservativ. +\end{satz} + +\begin{proof} + Sei $\gamma$ geschlossener Integrationsweg in $D$. Da $D$ einfach zusammenhängend, ist + $\gamma$ frei homotop zu einem konstanten Weg $\gamma_C$. Damit folgt + \[ + \int_{\gamma} F + \; \stackrel{\ref{satz:hauptsatz-2-kurven}}{=} \; \int_{\gamma_C} F + \; \stackrel{\text{Def.}}{=} \; + \int_{a}^{b} (F(\gamma_C(t)), \gamma_C'(t)) \d t = 0 + .\] Damit folgt mit \ref{satz:hauptsatz-1-kurven}, dass $F$ konservativ. +\end{proof} + +\begin{proof}[Ende]\end{proof} + +\end{document} diff --git a/analysisII.pdf b/analysisII.pdf index cdc639402a663eb4e380840b64a0c76f099aba0f..f7feca4c39665cdda3534069528d2de3f544c951 100644 GIT binary patch delta 124513 zcmYJ4Wl$VjxULyyaCdiiwfT+t$lPjqOOt4b7OEbW~S@B%ygd(LUj1J*}jQOc61d4>|h#&;Vl z#S|*w36;E!yPx+Wjq#B>Y0)^?%n~MfKKNScW7@xAf9P~biQ)YX)1@IFN$mO+`uWz< zZ)X;fu7LXdUWq?#&o=ofNDVnGfRZzyjbQO{{J`(ZM?r@j*n=b~UpbY}JEzl25@B!% zp9-syrZ(5f5o1AWedUlg;4fNI17IE?B+i@x^G~EaU|hPN1r4hYYBFAlfT>Sx|iNk^{$|>MwC2*WlTp6 zhJ=t^DXU1=Ozgx&W~zkPeK)euAV0Dq3II)#u8z{%T(B4AzA+i;duL-R?xIw`=`l|P z*2I8jXidMzx!h=(aq}e|C~q$5ZOp;g`H;OnFPFy%X?W?TcGw_saeguJg^Fm}Y;SK? 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