From 08b58937cf4d007ce09661f13a0b2a97f7bc567f Mon Sep 17 00:00:00 2001 From: salagne Date: Sat, 18 Jul 2020 13:38:46 +0200 Subject: [PATCH] add ana 21 and minor corrections --- ana20.tex | 5 +- ana21.tex | 194 +++++++++++++++++++++++++++++++++++++++++++++++++ analysisII.pdf | Bin 947614 -> 964533 bytes analysisII.tex | 1 + 4 files changed, 198 insertions(+), 2 deletions(-) create mode 100644 ana21.tex diff --git a/ana20.tex b/ana20.tex index 459afac..e5c9b33 100644 --- a/ana20.tex +++ b/ana20.tex @@ -245,7 +245,8 @@ \begin{subfigure}[b]{0.6\textwidth} \begin{tikzpicture} \begin{axis}[axis equal image, axis lines=middle, xticklabels={0, $\pi$, $2\pi$}, xtick={0,3.14,6.28}, ymin=0,ymax=2.2, xmax=6.5, smooth] - \addplot[domain=0:6.28] ({x-sin(180/3.14 * x)},{1-cos(180/3.14 * x)}); + \addplot[blue, domain=0:6.28] ({x-sin(180/3.14 * x)},{1-cos(180/3.14 * x)}); + \node[blue] at (5.9,1) {$\gamma$}; \draw (3.14,1) circle [radius=1]; \node at (3.14,2) {\textbullet}; \end{axis} @@ -267,7 +268,7 @@ Wir erhalten $\gamma'(t) = \begin{pmatrix} 1 - \cos(t)\\\sin(t) \end{pmatrix}$ und daher - $\norm{\gamma'(t)}_2^2 = 1 - 2 \cos(t) + \cos^2(t) + \sin^2(t) = 2 - 2\cos(t) = 4 \sin^2(\frac{t}{2})$. Insgesamt gilt also + $\norm{\gamma'(t)}_2^2 = 1 - 2 \cos(t) + \cos^2(t) + \sin^2(t) = 2 - 2\cos(t) = 4 \sin^2\!\left(\frac{t}{2}\right)$. Insgesamt gilt also \[ S(\gamma) = \int_0^{2\pi} \left|\smash[b]{\underbrace{2\sin\left(\frac{t}{2}\right)}_{\geq 0}}\right| \d t \oldstackrel{x = \frac{t}{2}}{=} 4\int_0^\pi \sin(x) \d x = 8 \vphantom{\underbrace{2\sin\left(\frac{t}{2}\right)}_{\geq 0}}\] diff --git a/ana21.tex b/ana21.tex new file mode 100644 index 0000000..2519cb4 --- /dev/null +++ b/ana21.tex @@ -0,0 +1,194 @@ +\documentclass{lecture} + +\begin{document} + +\section{Kurvenintegrale} + +\begin{definition}[Integrationsweg] + Eine Kurve $\gamma \in C^0([a,b],\R^n)$ heißt Integrationsweg, falls $\gamma$ stetig und stückweise eine $C^1$-Abbildung ist. +\end{definition} + +\begin{definition}[Skalares Kurvenintegral] + Sei $D \subset \R^n, \ \gamma \colon [a,b] \to D$ ein Integrationsweg und $f\colon D \to \R$ stetig. Dann heißt + \[\int_\gamma f \d s \coloneqq \int_a^b f(\gamma(t)) \cdot \norm{\gamma'(t)} \d t\] + das skalare Kurvenintegral von $f$ längs $\gamma$. Dabei heißt $\d s = \norm{\gamma'(t)} \d t$ das skalare Bogenelement von $\gamma$ und $f$ wird \underline{Skalarfeld} genannt. +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $f \equiv 1$: Das Kurvenintegral $\int_\gamma \d s = \int_a^b \norm{\gamma'(t)} \d t = S(\gamma)$ entspricht der Länge von $\gamma$. + \item Dichtefunktion $\rho(s)$: + \begin{align*} + \rho(\gamma(t)) &: \text{ Dichte verteilt auf } \gamma(t)\\ + \int_\gamma \rho(s) \d s \eqqcolon \mu(\gamma) &: \text{ Gesamtmasse von } \gamma + \end{align*} + \end{enumerate} +\end{bsp} + +\begin{bem} + \begin{enumerate} + \item Das Kurvenintegral ist linear: + \[\int_\gamma (\lambda_1 f_1 + \lambda_2 f_2) \d s = \lambda_1 \int_\gamma f_1\d s + \lambda_2 \int_\gamma f_2 \d s\] + \item Es gilt die Abschätzung + \begin{align*} + \left|\int_\gamma f \d s\right| &= \left|\int_a^b f(\gamma(t)) \norm{\gamma'(t)} \d t\right| \\ + &\le \sup_{s\in [a,b]} \left|f(\gamma(s))\right| \cdot \int_a^b f(\gamma(t)) \norm{\gamma'(t)} \d t \\ + &= \sup_{s\in [a,b]} \left|f(\gamma(s))\right| \cdot S(\gamma). + \end{align*} + \item Das Kurvenintegral ist invariant unter $C^1$-Parametertransformation. Sei $\varphi \colon [\alpha, \beta] \to [a,b]$ eine $C^1$ Parametertransformation. Dann gilt + \begin{salign*} + \int_{\gamma\circ\varphi}f \d s &= \int_\alpha^\beta f(\gamma(\varphi(s))) \norm{\frac{\mathrm{d}}{\mathrm{d}s}\gamma(\varphi(s))} \d s \\ + &\stackrel{\text{Kettenregel}}{=} \int_\alpha^\beta f(\gamma(\varphi(s))) \norm{\gamma'(\varphi(s)) \cdot \frac{\mathrm{d}}{\mathrm{d}s} \varphi(s)} \d s \\ + &= \begin{cases} + \int_\alpha^\beta f(\gamma(\varphi(s))) \norm{\gamma'(\varphi(s))} \left(\frac{\mathrm{d}}{\mathrm{d}s} \varphi(s)\right) \d s, & \frac{\mathrm{d}}{\mathrm{d}s}\varphi(s) > 0 \\ + \int_\alpha^\beta f(\gamma(\varphi(s))) \norm{\gamma'(\varphi(s))} \left(-\frac{\mathrm{d}}{\mathrm{d}s} \varphi(s)\right) \d s, & \frac{\mathrm{d}}{\mathrm{d}s}\varphi(s) < 0 + \end{cases} \\ + &\stackrel{\substack{t=\varphi(s)\\\d t = \varphi'(s)\d s}}{=} + \begin{cases} + \int_a^b f(\gamma(t)) \norm{\gamma'(t)} \d t, & \varphi'(s) > 0 \\ + -\int_b^a f(\gamma(t)) \norm{\gamma'(t)} \d t, & \varphi'(s) < 0 + \end{cases} \\ + &= \int_a^b f(\gamma(t)) \norm{\gamma'(t)} \d t\\ + &= \int_\gamma f \d s + \end{salign*} + \end{enumerate} +\end{bem} + +\begin{definition}[Vektorfeld] + Ein Vektorfeld $F$ auf $D\subset \R^n$ ist eine Abbildung von $D$ nach $\R^n$, d.h. jedem $x\in D$ wird ein Vektor $F(x) \in \R^n$ zugeordnet. +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item Windungsfeld + \[W\colon \R^2\setminus \{0\} \to \R^2, \quad W(x,y) \coloneqq \frac{1}{\norm{(x,y)}_2^2}\begin{pmatrix} -y \\ x \end{pmatrix}\] + \item Gravitationsfeld + \[G\colon \R^3\setminus \{0\} \to \R^3, \quad G(x,y,z) \coloneqq -\frac{1}{\norm{(x,y,z)}_2^2}\begin{pmatrix} x \\ y \\ z\end{pmatrix}\] + \end{enumerate} +\end{bsp} + +\begin{figure}[h] + \centering + \begin{subfigure}[b]{0.4\textwidth} + \begin{tikzpicture}[scale=1] + \begin{axis} + [axis lines=middle, + axis lines=middle, + axis equal image, % Unit vectors for both axes have the same length + xmin=-1, xmax=1.1, % Axis limits + ymin=-1, ymax=1.1, + ticks=none, + xlabel=$x$, + ylabel=$y$ + ] + \foreach \r in {0.4, 0.6, 0.8} + \addplot[domain=0:360*(1-1/(20*\r)), + blue, samples=20*\r, + quiver={u={-y/(x^2+y^2)}, v={x/(x^2+y^2)}, scale arrows=0.1, every arrow/.append style={-latex} }] ({\r*sin(x)},{\r*cos(x)}); % polar coordinates + \end{axis} + \end{tikzpicture} + \subcaption{Beispiel 1: Windungsfeld} + \end{subfigure} + \begin{subfigure}[b]{0.4\textwidth} + \begin{tikzpicture}[scale=1] + \begin{axis} + [axis lines=middle, + axis lines=middle, + axis equal image, % Unit vectors for both axes have the same length + xmin=-1, xmax=1.1, % Axis limits + ymin=-1, ymax=1.1, + ticks=none, + xlabel=$x$, + ylabel=$y$ + ] + \foreach \r in {0.4, 0.6, 0.8} + \addplot[domain=0:360*(1-1/(20*\r)), + red, samples=20*\r, + quiver={u={-x/(x^2+y^2)}, v={-y/(x^2+y^2)}, scale arrows=0.1, every arrow/.append style={-latex} }] ({\r*sin(x+180/(20*\r))},{\r*cos(x+180/(20*\r))}); %polar coordinates + \end{axis} + \end{tikzpicture} + \subcaption{Beispiel 2: Gravitationsfeld bei $z=0$.} + \end{subfigure} +\end{figure} + +\begin{definition}[Vektorielles Kurvenintegral] + Sei $\gamma\colon [a,b] \to D\subset \R^n$ ein Integrationsweg und $F\colon D\to \R^n$ ein stetiges Vektorfeld. Dann ist das (vektorielle) Kurvenintegral von $F$ längs $\gamma$ definiert durch + \[\int_\gamma F = \int_\gamma F\d{\vec s} \coloneqq \int_a^b \underbrace{\left(F(\gamma(t)), \gamma'(t)\right)}_{\text{Skalarprodukt}} \d t = \int_a^b \sum_{i=1}^n F_i(\gamma(t)) \cdot \gamma'_i(t) \d t\] + Alternative Schreibweise: $\int_\gamma F = \int_\gamma F_1\d{x_1} + \dots + F_n\d{x_n} $ +\end{definition} + +\begin{bsp} + Kurvenintegral des Windungsfelds $W\colon \R^2\setminus \{0\} \to \R^2, \ W(x,y) = \frac{1}{x^2+y^2} \begin{pmatrix} -y \\ x \end{pmatrix}$ längs $\gamma(t) \coloneqq \begin{pmatrix} \cos(t) \\ \sin(t) \end{pmatrix}, \ \gamma \colon [0,2\pi] \to \R^2 \setminus \{0\}$. + \begin{align*} + \int_\gamma W &= \int_\gamma -\frac{y}{x^2+y^2}\d x + \frac{x}{x^+y^2}\d y\\ + &= \int_0^{2\pi} \left(-\frac{\sin t}{\cos^2t+\sin^2t}(-\sin t) + \frac{\cos t}{\cos^2t+\sin^2t}\cos t\right) \d t \\ + &= \int_0^{2\pi} \left(\sin^2t+\cos^2t\right) \d t = 2\pi + \end{align*} +\end{bsp} + +\begin{bem} + \begin{enumerate} + \item + Das Kurvenintegral ist linear: $$\int_\gamma (\lambda_1F_1+\lambda_2F_2)=\lambda_1\int_\gamma F_1+\lambda_2\int_\gamma F_2$$ + \item + Standard-Abschätzung: $$\left|\int_\gamma F \right|=\left|\int_a^b\bigl(F(\gamma(t)),\gamma'(t)\bigr)\, \d t \right|\leq \sup_{t\in [a,b]}\norm{F(\gamma(t))}\cdot S(\gamma)$$ + \item + Invarianz unter orientierungstreuen $C^1$-Parametertransformationen. Sei $\varphi \colon [\alpha,\beta]\to [a,b]$ eine $C^1$-Parametertransformation mit $\varphi'(s)>0,\ \forall s\in [\alpha,\beta]$ ($\Longleftrightarrow$ orientierungstreu). Dann gilt \begin{salign*} + \int_{\gamma \circ \varphi}F&=\int_\alpha^\beta \left(F(\gamma(\varphi(s))),\frac{\d}{\d s}\gamma(\varphi(s)) \right) \d s\\ + &=\int_\alpha^\beta \left(F(\gamma(\varphi(s))),\gamma'(\varphi(s))\cdot \frac{\d \varphi}{\d s}(s) \right) \d s\\ + &=\int_\alpha^\beta \left(F(\gamma(\varphi(s))),\gamma'(\varphi(s)) \right)\frac{\d \varphi}{\d s}(s) \d s\\ + &\stackrel{\substack{t=\varphi(s)\\ \d t=\varphi'(s)\d s}}{=}\int_a^b\bigl(F(t),\gamma'(t)\bigr)\d t\\ + &=\int_\gamma F. + \end{salign*} + \end{enumerate} +\end{bem} + +\begin{definition}[Gebiet] + $U\subset \R^n$ heißt Gebiet, falls $U$ offen ist und wegzusammenhängend, d.h. $\forall \,x,y\in U$ existiert $\gamma \in C^0([a,b],U)$ mit $\gamma(a)=x$ und $\gamma(b)=y$. +\end{definition} + +\begin{satz} + Sei $U\subset \R^n$ offen. Dann sind folgende Aussagen äquivalent: \begin{enumerate} + \item + $\forall \,x,y\in U$ existiert ein Integrationsweg $\gamma \colon [a,b]\to U$ mit $\gamma(a)=x$ und $\gamma(b)=y$. + \item + $U$ ist wegzusammenhängend. + \end{enumerate} +\end{satz} + +\begin{proof} + Ohne Beweis. +\end{proof} + +\begin{bsp} + \begin{enumerate} + \item + Sei $U=K_1 \begin{pmatrix} 0 \\ 0\end{pmatrix}\, \cup \, K_1\begin{pmatrix} 3 \\ 0\end{pmatrix}$. $U$ ist kein Gebiet, denn es existiert kein stetiger Weg von $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ nach $\begin{pmatrix} 3 \\ 0 \end{pmatrix}$ in $U$. + \item + $U\subset \R$ Gebiet $\Longleftrightarrow$ $U$ offenes Intervall + \end{enumerate} +\end{bsp} + +\section{Potential} + +\begin{definition}[Geschlossene Kurve] + Eine Kurve $\gamma \in C^0([a,b],\R^n)$ heißt \underline{geschlossen}, falls $\gamma(a)=\gamma(b)$. +\end{definition} + +\begin{definition}[Potential] + Sei $D\subset \R^n$ und $F\in C^0(D,\R^n)$ ein stetiges Vektorfeld. $\varphi \in C^1(D,\R)$ heißt \underline{Potential} oder \underline{Stammfunktion} von $F$ in $D$, falls $\nabla \varphi=F$ gilt. $F$ heißt in diesem Fall \underline{konservativ} auf $D$. +\end{definition} + +\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} + \item F ist konservativ. + \item + $\int_\gamma F=0$ für alle geschlossenen Integrationswege $\gamma$ in $D$. + \item + 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 $\{\varphi_0+c\,|\,c\in \R\}$. +\end{satz} + +\end{document} \ No newline at end of file diff --git a/analysisII.pdf b/analysisII.pdf index 253fb98af78f5396e6c66fa69c8beb64b3509cdc..2642bc7cc89378ea15bdf0aa0a4752ffb0ca5fb3 100644 GIT binary patch delta 106280 zcmZU)bx<5@)a?y}ySux)LvXj??k>UI2X}`sSa65nPVivC-6goY>*btt-+S*@U)6M1 zchxgbb^p`*x7OM_=Y+56g1?pm1|-@Civi99a*~E)0G_3fx~#BbhCh4>16}t?dFrfF zA+1b2Z&5@(~E88+2UwsR(-t(lxVw4h+z4J%}aerB@60=J2;#@ zT`5JAEAjq7!|jBE#}+Xwy)6BFD^zL@V}vCR3uXcoO9a!1QGg& z&}jhmbk(^R!`%_aiS0egIOzaoQwa4~Hck#nK_IWZCP~k1+|Xc2e5LZXyV&Y5jjrA~ zG5m}w@lt4J$Zs%g=h{#bYx=3Jp#m98IMOIwxfH!z6}OS1?3FS7?H+5&5IILmn+{k*0=N z(ts$QnVQoK7?+!E_4^*(u_W6C$cs5YesdeFY08qZEHu7XOD(-P^mN`==d3@f=eNa# z%+8D=3?<4x>z3I|^tF(h#0AF^(Tw&LxtG|9`P&Tb1s2*N`*hxii8$}8bJLp2l$5lZ zXNPz<67abQaC9n4t+5IG*)QCjrUiMB#sc5*)at^@ofMSf6t{nww9D01;zDAr=q4#C zxagmWVa-fS3Dx&f5ak5e4H3+ykqpb6R+$^Z7DFdPrW|0q>EPTZjJ}ps4QO;jSgej6 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zbXr9*J}*6y!A&U*?dzXQNeAbJ(QoE$f}{R;ZZY)#0Bwu#7mz*@zH>v2U(RUlj(oXt WQX1KW&nd~PDyuL{N?y0P!Tdj?qGbgD diff --git a/analysisII.tex b/analysisII.tex index ead6073..3b26f69 100644 --- a/analysisII.tex +++ b/analysisII.tex @@ -43,5 +43,6 @@ Rui Yang (\href{mailto:rui.yang@stud.uni-heidelberg.de}{rui.yang@stud.uni-heidel \input{ana18.tex} \input{ana19.tex} \input{ana20.tex} +\input{ana21.tex} \end{document}