From 81d455fb3af2bf34cd63f87465f9f53f518794a4 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Sun, 12 Apr 2020 16:22:21 +0200 Subject: Retab to spaces, add sections and curvature figure --- an2e_zf.pdf | Bin 67064 -> 80340 bytes an2e_zf.tex | 230 +++++---- fig/plane-curvature.eps | 1213 +++++++++++++++++++++++++++++++++++++++++++++++ fig/plane-curvature.ipe | 356 ++++++++++++++ 4 files changed, 1712 insertions(+), 87 deletions(-) create mode 100644 fig/plane-curvature.eps create mode 100644 fig/plane-curvature.ipe diff --git a/an2e_zf.pdf b/an2e_zf.pdf index cd1b7ac..97801b0 100644 Binary files a/an2e_zf.pdf and b/an2e_zf.pdf differ diff --git a/an2e_zf.tex b/an2e_zf.tex index da6de58..9cb871c 100644 --- a/an2e_zf.tex +++ b/an2e_zf.tex @@ -1,4 +1,4 @@ -\documentclass[a4paper]{article} +\documentclass[a4paper, twocolumn]{article} \usepackage{amssymb} \usepackage{amsmath} @@ -11,7 +11,7 @@ \usepackage{booktabs} \usepackage{rotating} -\usepackage[margin=2cm, bottom=2cm, top=2cm, marginpar=0pt]{geometry} +\usepackage[margin=2cm, marginpar=0pt]{geometry} \usepackage{graphicx} \usepackage{xcolor} @@ -22,7 +22,6 @@ %\usepackage{pgfplots} %\pgfplotsset{compat=1.15} -\usepackage{multicol} \usepackage[colorlinks = true, linkcolor = red!50!black, @@ -40,166 +39,178 @@ \date{Fr\"uhlingsstemester 2020} -\newcommand{\dd}[1]{\ensuremath{~\mathrm{d}#1}} -\newcommand{\deriv}[2]{\ensuremath{\frac{\dd{#1}}{\dd{#2}}}} -\newcommand{\pderiv}[2]{\ensuremath{\frac{\partial#1}{\partial#2}}} +\newcommand{\dd}[2][]{\ensuremath{~\mathrm{d}^{#1} #2}} +\newcommand{\deriv}[3][]{\ensuremath{\frac{\dd[#1]{#2}}{\dd[]{#3^{#1}}}}} +\newcommand{\pderiv}[3][]{\ensuremath{\frac{\partial^{#1} #2}{\partial^{#1} #3}}} \renewcommand{\vec}[1]{\ensuremath{\bm{#1}}} \newcommand{\brpage}[1]{\textcolor{red!70!black}{\small\texttt{S#1}}} \begin{document} -\begin{multicols}{2} \section{Integration \brpage{493,507}} \subsection{Tricks \brpage{495}} Linearit\"at \brpage{495} \[ - \int k(u + v) = k\left(\int u + \int v\right) + \int k(u + v) = k\left(\int u + \int v\right) \] Partialbruchzerlegung \brpage{15,498} \[ - \int \frac{f(x)}{P_n(x)} \dd{x} = \sum_{k=1}^n \int \frac{A_k}{x-r_k}\dd{x} + \int \frac{f(x)}{P_n(x)} \dd{x} = \sum_{k=1}^n \int \frac{A_k}{x-r_k}\dd{x} \] Elementartransformation \brpage{496} \[ - \int f(\lambda x + \ell) \dd{x} = \frac{1}{\lambda} F(\lambda x + \ell) + C + \int f(\lambda x + \ell) \dd{x} = \frac{1}{\lambda} F(\lambda x + \ell) + C \] Partielle Integration \brpage{497} \[ - \int u \dd{v} = uv - \int v \dd{u} + \int u \dd{v} = uv - \int v \dd{u} \] Potenzenregel \brpage{496} \[ - \int u^n \cdot u' = \frac{u^{n+1}}{n+1} + C \qquad n \neq -1 + \int u^n \cdot u' = \frac{u^{n+1}}{n+1} + C \qquad n \neq -1 \] Logaritmusregel \brpage{496} \[ - \int \frac{u'}{u} = \ln|u| + C + \int \frac{u'}{u} = \ln|u| + C \] Allgemeine Substutution \brpage{497}\\ \(x = g(u)\), und \(\dd{x} = g'(u)\dd{u}\) \[ - \int f(x) \dd{x} = \int (f\circ g) ~ g' \dd{u} = \int \frac{f \circ g}{(g^{-1})'\circ g} \dd{u} + \int f(x) \dd{x} = \int (f\circ g) ~ g' \dd{u} = \int \frac{f \circ g}{(g^{-1})'\circ g} \dd{u} \] Universalsubstitution \brpage{504} \begin{align*} - t &= \tan(x/2) & \sin(x) &= \frac{2t}{1+t^2} \\ - \dd{x} &= \frac{2\dd{t}}{1+t^2} & \cos(t) &= \frac{1-t^2}{1+t^2} + t &= \tan(x/2) & \sin(x) &= \frac{2t}{1+t^2} \\ + \dd{x} &= \frac{2\dd{t}}{1+t^2} & \cos(t) &= \frac{1-t^2}{1+t^2} \end{align*} Womit \[ - \int f(\sin(x), \cos(x), \tan(x)) \dd{x} = \int g(t) \dd{t} + \int f(\sin(x), \cos(x), \tan(x)) \dd{x} = \int g(t) \dd{t} \] \subsection{Uneigentliches Integral \brpage{520}} \begin{align*} - \int\limits_a^\infty f \dd{x} &= \lim_{B \to \infty} \int\limits_a^B f \dd{x} \\ - \int\limits_{-\infty}^b f \dd{x} &= \lim_{A \to -\infty} \int\limits_A^b f \dd{x} \\ - \int\limits_{-\infty}^\infty f \dd{x} &= \lim_{\substack{A \to +\infty \\ B \to -\infty}} \int\limits_A^B f \dd{x} + \int\limits_a^\infty f \dd{x} &= \lim_{B \to \infty} \int\limits_a^B f \dd{x} \\ + \int\limits_{-\infty}^b f \dd{x} &= \lim_{A \to -\infty} \int\limits_A^b f \dd{x} \\ + \int\limits_{-\infty}^\infty f \dd{x} &= \lim_{\substack{A \to +\infty \\ B \to -\infty}} \int\limits_A^B f \dd{x} \end{align*} Wenn \(f\) im Punkt \(u \in (a,b)\) nicht definiert ist. \begin{equation} \label{eqn:int-with-pole} - \int\limits_a^b f \dd{x} = - \lim_{\epsilon\to +0} \int\limits_a^{u-\epsilon} f \dd{x} - + \lim_{\delta\to +0} \int\limits_{u+\delta}^b f \dd{x} + \int\limits_a^b f \dd{x} = + \lim_{\epsilon\to +0} \int\limits_a^{u-\epsilon} f \dd{x} + + \lim_{\delta\to +0} \int\limits_{u+\delta}^b f \dd{x} \end{equation} \subsection{Cauchy Hauptwert \brpage{523}} Der C.H. (oder PV f\"ur \emph{Principal Value} auf Englisch) eines uneigentlichen Integrals ist der Wert, wenn in einem Integral wie \eqref{eqn:int-with-pole} beide Grenzwerte mit der gleiche Geschwindigkeit gegen 0 sterben. \[ - \text{C.H.} \int\limits_a^b f \dd{x} = - \lim_{\epsilon\to +0} \left( \int\limits_a^{u-\epsilon} f \dd{x} - + \int\limits_{u+\epsilon}^b f \dd{x} \right) + \text{C.H.} \int\limits_a^b f \dd{x} = + \lim_{\epsilon\to +0} \left( \int\limits_a^{u-\epsilon} f \dd{x} + + \int\limits_{u+\epsilon}^b f \dd{x} \right) \] Zum Beispiel \(x^{-1}\) ist nicht \"uber \(\mathbb{R}\) integrierbar, wegen des Poles bei 0. Aber intuitiv wie die Symmetrie vorschlagt \[ - \text{C.H.} \int\limits^\infty_{-\infty} \frac{1}{x} \dd{x} = 0 + \text{C.H.} \int\limits^\infty_{-\infty} \frac{1}{x} \dd{x} = 0 \] \subsection{Majorant-, Minorantenprinzip und Konvergenzkriterien \brpage{521,473,479,481}} Gilt f\"ur die Funktionen \(0 < f(x) \leq g(x)\) mit \(x \in [a,\infty)\) \[ - \text{konvergiert } \int\limits_a^\infty g \dd{x} - \implies \text{ konvergiert } \int\limits_a^\infty f \dd{x} + \text{konvergiert } \int\limits_a^\infty g \dd{x} + \implies \text{ konvergiert } \int\limits_a^\infty f \dd{x} \] Die selbe gilt umgekehrt f\"ur Divergenz. Wenn \(0 < h(x) \leq f(x)\) \[ - \text{divergiert } \int\limits_a^\infty h \dd{x} - \implies \text{ divergiert } \int\limits_a^\infty f \dd{x} + \text{divergiert } \int\limits_a^\infty h \dd{x} + \implies \text{ divergiert } \int\limits_a^\infty f \dd{x} \] \(g\) und \(h\) hei{\ss}en Majorant und Minorant bzw. \section{Implizite Ableitung \brpage{448}} Alle normale differenziazionsregeln gelten. \[ - \dd{y} = y'\dd{x} + \dd{y} = y'\dd{x} \] %Allgemeiner f\"ur die implizite Funktion \(F(x,y) = 0\) %\[ -% \pderiv{F}{x} + \pderiv{F}{y} y' = 0 +% \pderiv{F}{x} + \pderiv{F}{y} y' = 0 %\] -\end{multicols} -\section{Ebene \brpage{250} und Raumkurven \brpage{263}} -\begin{sidewaystable} -\centering -\renewcommand{\arraystretch}{3} -\begin{tabular}{l *{3}{>{\(\displaystyle}l<{\)}} } -\toprule -\textbf{Ebene Kurven} & \textbf{Explizit } y = f(x) & \textbf{Polar } \vec{r}(\varphi) & \textbf{Parameter } \vec{c}(t) = \left(x(t), y(y)\right) \\ -\midrule -Bogenl\"ange \brpage{251} - & \int\limits_a^b \sqrt{1 + (f')^2} \dd{x} - & \int\limits_\alpha^\beta \sqrt{(r')^2 + r^2} \dd{\varphi} - & \int\limits_{t_0}^{t_1} \sqrt{\dot{x}^2 + \dot{y}^2} \dd{t} = \int\limits_{t_0}^{t_1} |\vec{c}| \dd{t} -\\ -Fl\"ache - & \int\limits_a^b |f(x)| \dd{x} - & \frac{1}{2}\int\limits_\alpha^\beta r(\varphi)^2 \dd{\varphi} - & \frac{1}{2}\int\limits_{t_0}^{t_1} x\dot{y} - \dot{x}y \dd{t} = \frac{1}{2}\int\limits_{t_0}^{t_1}\det(\vec{c},\dot{\vec{c}}) \dd{t} -\\ -\midrule -Rotationsvolumen um \(x\) - & \pi \left|\int\limits_a^b y^2 \dd{x} \right| - & \pi \left|\int\limits_{t_0}^{t_1} y \dot{x} \dd{t} \right| - & \pi \left|\int\limits_\alpha^\beta r^2 \sin^2 \varphi (r'\cos\varphi - r\sin\varphi) \dd{\varphi} \right| -\\ -Rotationsoberfl\"ache um \(x\) - & 2\pi \int\limits_a^b |y| \sqrt{1 + (y')^2} \dd{x} - & 2\pi \int\limits_\alpha^\beta |r\sin(\varphi)| \sqrt{(r')^2 + r^2} \dd{\varphi} - & 2\pi \int\limits_{t_0}^{t_1} |y| \sqrt{\dot{x}^2 + \dot{y}^2} \dd{t} -\\ -% Rotationsvolumen um \(y\) \\ -% Rotationsoberfl\"ache um \(y\) \\ -\midrule -Kr\"ummung \(\kappa\) - & \frac{f''}{\sqrt{1+(f')^2}^3} - & - & \frac{\ddot{y}\dot{x} - \ddot{x}\dot{y}}{\sqrt{\dot{x}^2 + \dot{y}^2}^3} - = \frac{\det(\vec{\dot{c}},\vec{\ddot{c}})}{|\vec{\dot{c}}|^3} -\\ -\bottomrule -\end{tabular} -\end{sidewaystable} +\section{Differentialgeometrie} +\subsection{Ebene \brpage{250} Kurven} -\begin{multicols}{2} -\subsection{Darstellungen} -\begin{figure}[H] +\subsubsection{Darstellungen und Umwanldung} +\begin{figure}[h] \centering \includegraphics[width=.9\linewidth]{fig/plane-curve.eps} -\caption{Die ebene Kurve \(\Lambda(t)\) kann Explizit \(y(x)\) (in diesem Fall nicht), Implizit \(\vec{u}(x,y)\), Polar \(\vec{r}(\varphi)\) oder in Parameterform \((x(t), y(t))\) dargestellt werden.} +\caption{Die ebene Kurve \(\vec{\Lambda}(t)\) kann Explizit \(y(x)\) (in diesem Fall nicht), Implizit \(\vec{u}(x,y) = 0\), Polar \(\vec{r}(\varphi)\) oder in Parameterform \((x(t), y(t))\) dargestellt werden.} +\label{fig:plane-curve} \end{figure} -\subsection{Tangente und Normalvektor} +Sei \(\Lambda: x = \phi(t),\, y = \psi(t), t\in I\) eine glatte Jordankurve. +Beispiel im Abb. \ref{fig:plane-curve}. + +\paragraph{Polar zu Kartesian} +\[ + r = \sqrt{x^2 + y^2} + \qquad + \tan\varphi = y/x +\] +\[ + x = r \cos\varphi + \qquad + y = r \sin\varphi +\] + +\paragraph{Parametrisch zu explizit} +Sei \(\dot{\phi} \neq 0\) oder \(\dot{\psi} \neq 0\). Im Falle \(\dot{\phi} \neq 0\), wechselt \(\dot\phi\) in der Umgebung von \(t\) das Vorzeichen nicht, \(\phi\) ist dort streng monoton. +Daher gilt +\[ + t = \phi^{-1}(x) \quad y = \psi(t) = \psi \circ \phi^{-1}(x) = f(x) +\] +Wenn \(\dot{\psi} \neq 0\) ist dann \(x = \phi \circ \psi^{-1}(y)\) + +\subsubsection{Tangente und Normalvektor \brpage{251,252}} +F\"ur eine ebene Kurve \(\vec{\Lambda}(t)\) \(\tau, t \in I\), der Vektor \(\vec{\dot\Lambda}(\tau)\) ist immer an \(\vec{\Lambda}(\tau)\) tangent. \(\vec{\ddot{\Lambda}}(\tau)\) ist zur Kurve normal. +\begin{align*} + \vec{\dot{\Lambda}} + &= \deriv{y}{x} + = \frac{\dot{y}}{\dot{x}} + = \frac{r'\sin\varphi + r\cos\varphi}{r'\cos\varphi - r\sin\varphi} + \\[.9em] + \vec{\ddot{\Lambda}} + &= \deriv[2]{y}{x} + = \frac{\ddot{y}\dot{x} - \ddot{x}\dot{y}}{\dot{x}^3} +\end{align*} +Man kann auch die Tangentengleichung und die Normalengleichung zur Zeitpunkt \(\tau\) finden +\begin{align*} + T: y - \psi(\tau) &= \frac{\dot{\psi}}{\dot{\phi}}(x - \phi(\tau)) \\ + N: y - \psi(\tau) &= -\frac{\dot{\phi}}{\dot{\psi}}(x - \phi(\tau)) +\end{align*} -\subsection{Kr\"ummung} +\subsubsection{Kr\"ummung und Kr\"ummungsradius \brpage{254}} +Siehe Tab. \ref{tab:plane-curves-big} f\"ur die Rechnungsformeln. \[ - \kappa = \deriv{\phi}{s} = \frac{\ddot{y}}{(1+\dot{y}^2)^{3/2}} + \kappa + = \lim_{\Delta s\to 0} \frac{\Delta \theta}{\Delta s} + = \deriv{\theta}{s} + \qquad + R = 1/\kappa \] -\begin{thebibliography}{1} +\begin{figure}[h] +\centering +\includegraphics[width=.8\linewidth]{fig/plane-curvature} +\caption{Kr\"ummung und Kr\"ummungskreisradien} +\label{fig:plane-curvature} +\end{figure} + +\subsection{Raumkurven \brpage{263}} + +\begin{thebibliography}{3} \bibitem{hsr} \texttt{An2E} Vorlesungen an der Hochschule f\"ur Technik Rapperswil und der dazugeh\"orige Skript, \textit{Dr. Bernhard Zgraggen}, Fr\"uhlingssemester 2020 @@ -209,7 +220,7 @@ Kr\"ummung \(\kappa\) \textit{Bronstein, Semendjajew, Musiol, M\"uhlig}, \texttt{ISBN 978-3-8085-5789-1} \bibitem{mathe2} - Mathematik 2 Lehrbuch für ingenieurwissenschaftliche Studieng\"ange, + Mathematik 2: Lehrbuch für ingenieurwissenschaftliche Studieng\"ange, 2012, 7. Auflage, XII, Springer Berlin, \textit{Albert Fetzer, Heiner Fränkel}, \texttt{ISBN-10 364224114X}, @@ -228,8 +239,53 @@ An2E-ZF is licensed under a Creative Commons Attribution-ShareAlike 4.0 Unported \\\\ You should have received a copy of the license along with this work. If not, see \\\\ -\url{http://creativecommons.org/licenses/by-sa/4.0/} +{\small\url{http://creativecommons.org/licenses/by-sa/4.0/}} } -\end{multicols} -\end{document} \ No newline at end of file + +\onecolumn + +\begin{sidewaystable}[p] +\centering +\caption{Rechnungen bez. ebene Kurven} +\label{tab:plane-curves-big} +\renewcommand{\arraystretch}{3} +\begin{tabular}{l *{3}{>{\(\displaystyle}l<{\)}} } +\toprule +\textbf{Ebene Kurven} & \textbf{Explizit } y = f(x) & \textbf{Polar } \vec{r}(\varphi) & \textbf{Parameter } \vec{c}(t) = \left(x(t), y(y)\right) \\ +\midrule +Bogenl\"ange \brpage{251} + & \int\limits_a^b \sqrt{1 + (f')^2} \dd{x} + & \int\limits_\alpha^\beta \sqrt{(r')^2 + r^2} \dd{\varphi} + & \int\limits_{t_0}^{t_1} \sqrt{\dot{x}^2 + \dot{y}^2} \dd{t} = \int\limits_{t_0}^{t_1} |\vec{c}| \dd{t} +\\ +Fl\"ache + & \int\limits_a^b |f(x)| \dd{x} + & \frac{1}{2}\int\limits_\alpha^\beta r(\varphi)^2 \dd{\varphi} + & \frac{1}{2}\int\limits_{t_0}^{t_1} x\dot{y} - \dot{x}y \dd{t} = \frac{1}{2}\int\limits_{t_0}^{t_1}\det(\vec{c},\dot{\vec{c}}) \dd{t} +\\ +\midrule +Rotationsvolumen um \(x\) + & \pi \left|\int\limits_a^b y^2 \dd{x} \right| + & \pi \left|\int\limits_{t_0}^{t_1} y \dot{x} \dd{t} \right| + & \pi \left|\int\limits_\alpha^\beta r^2 \sin^2 \varphi (r'\cos\varphi - r\sin\varphi) \dd{\varphi} \right| +\\ +Rotationsoberfl\"ache um \(x\) + & 2\pi \int\limits_a^b |y| \sqrt{1 + (y')^2} \dd{x} + & 2\pi \int\limits_\alpha^\beta |r\sin(\varphi)| \sqrt{(r')^2 + r^2} \dd{\varphi} + & 2\pi \int\limits_{t_0}^{t_1} |y| \sqrt{\dot{x}^2 + \dot{y}^2} \dd{t} +\\ +% Rotationsvolumen um \(y\) \\ +% Rotationsoberfl\"ache um \(y\) \\ +\midrule +Kr\"ummung \(\kappa\) + & \frac{f''}{\sqrt{1+(f')^2}^3} + & + & \frac{\ddot{y}\dot{x} - \ddot{x}\dot{y}}{\sqrt{\dot{x}^2 + \dot{y}^2}^3} + = \frac{\det(\vec{\dot{c}},\vec{\ddot{c}})}{|\vec{\dot{c}}|^3} +\\ +\bottomrule +\end{tabular} +\end{sidewaystable} + +\end{document} diff --git a/fig/plane-curvature.eps b/fig/plane-curvature.eps new file mode 100644 index 0000000..64e39da --- /dev/null +++ b/fig/plane-curvature.eps @@ -0,0 +1,1213 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: cairo 1.15.10 (http://cairographics.org) +%%CreationDate: Sun Apr 12 16:21:18 2020 +%%Pages: 1 +%%DocumentData: Clean7Bit +%%LanguageLevel: 2 +%%BoundingBox: 0 1 244 226 +%%EndComments +%%BeginProlog +50 dict begin +/q { gsave } bind def +/Q { grestore } bind def +/cm { 6 array astore concat } bind def +/w { setlinewidth } bind def +/J { setlinecap } bind def +/j { setlinejoin } bind def +/M { setmiterlimit } bind def +/d { setdash } bind def +/m { moveto } bind def +/l { lineto } bind def +/c { curveto } bind def +/h { closepath } bind def +/re { exch dup neg 3 1 roll 5 3 roll moveto 0 rlineto + 0 exch rlineto 0 rlineto closepath } bind def +/S { stroke } bind def +/f { fill } bind def +/f* { eofill } bind def +/n { newpath } bind def +/W { clip } bind def +/W* { eoclip } bind def +/BT { } bind def +/ET { } bind def +/BDC { mark 3 1 roll /BDC pdfmark } bind def +/EMC { mark /EMC pdfmark } bind def +/cairo_store_point { /cairo_point_y exch def /cairo_point_x exch def } def +/Tj { show currentpoint cairo_store_point } bind def +/TJ { + { + dup + type /stringtype eq + { show } { -0.001 mul 0 cairo_font_matrix dtransform rmoveto } ifelse + } forall + currentpoint cairo_store_point +} bind def +/cairo_selectfont { cairo_font_matrix aload pop pop pop 0 0 6 array astore + cairo_font exch selectfont cairo_point_x cairo_point_y moveto } bind def +/Tf { pop /cairo_font exch def /cairo_font_matrix where + { pop cairo_selectfont } if } bind def +/Td { matrix translate cairo_font_matrix matrix concatmatrix dup + /cairo_font_matrix exch def dup 4 get exch 5 get cairo_store_point + /cairo_font where { pop cairo_selectfont } if } bind def +/Tm { 2 copy 8 2 roll 6 array astore /cairo_font_matrix exch def + cairo_store_point /cairo_font where { pop cairo_selectfont } if } bind def +/g { setgray } bind def +/rg { setrgbcolor } bind def +/d1 { setcachedevice } bind def +/cairo_data_source { + CairoDataIndex CairoData length lt + { CairoData CairoDataIndex get /CairoDataIndex CairoDataIndex 1 add def } + { () } ifelse +} def +/cairo_flush_ascii85_file { cairo_ascii85_file status { cairo_ascii85_file flushfile } if } def +/cairo_image { image cairo_flush_ascii85_file } def +/cairo_imagemask { imagemask cairo_flush_ascii85_file } def +%%EndProlog +%%BeginSetup +%%BeginResource: font CMMI12 +%!PS-AdobeFont-1.0: CMMI12 003.002 +%%Title: CMMI12 +%Version: 003.002 +%%CreationDate: Mon Jul 13 16:17:00 2009 +%%Creator: David M. 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