From d73c788a4458700a113ff6c38f189a65af6e7e39 Mon Sep 17 00:00:00 2001 From: Nao Pross Date: Thu, 9 Apr 2020 14:23:18 +0200 Subject: Update table and plane-curve fig --- an2e_zf.pdf | Bin 69262 -> 67064 bytes an2e_zf.tex | 38 ++-- fig/plane-curve.eps | 579 ++++++++-------------------------------------------- fig/plane-curve.ipe | 28 +-- 4 files changed, 116 insertions(+), 529 deletions(-) diff --git a/an2e_zf.pdf b/an2e_zf.pdf index 46bc7a9..cd1b7ac 100644 Binary files a/an2e_zf.pdf and b/an2e_zf.pdf differ diff --git a/an2e_zf.tex b/an2e_zf.tex index a0ee725..da6de58 100644 --- a/an2e_zf.tex +++ b/an2e_zf.tex @@ -145,33 +145,41 @@ Alle normale differenziazionsregeln gelten. \section{Ebene \brpage{250} und Raumkurven \brpage{263}} \begin{sidewaystable} \centering -\renewcommand{\arraystretch}{1.5} +\renewcommand{\arraystretch}{3} \begin{tabular}{l *{3}{>{\(\displaystyle}l<{\)}} } \toprule -\textbf{Ebene Kurven} & \textbf{Explizit} & \textbf{Polar} & \textbf{Parameter} \\ +\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 + (y')^2} \dd{x} + & \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} -\\[1cm] +\\ 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} -\\[1cm] +\\ +\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| -\\[1cm] +\\ 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} -\\[1cm] +\\ % 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} @@ -181,23 +189,19 @@ Rotationsoberfl\"ache um \(x\) \begin{figure}[H] \centering \includegraphics[width=.9\linewidth]{fig/plane-curve.eps} -\caption{Die ebene Kurve \(\Lambda(t)\) kann Explizit (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 \(\Lambda(t)\) 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.} \end{figure} -\subsection{Kr\"ummung} -\[ - \kappa = \deriv{\alpha}{s} = \frac{\ddot{y}}{(1+\dot{y}^2)^{3/2}} -\] +\subsection{Tangente und Normalvektor} +\subsection{Kr\"ummung} \[ - \det(\vec{\dot{c}}, \vec{\ddot{c}})\,|\vec{c}|^{-3} - \quad \stackrel{\text{3D}}{=} \quad - |\vec{\dot{c}}\times\vec{\ddot{c}}|\,|\vec{c}|^{-3} + \kappa = \deriv{\phi}{s} = \frac{\ddot{y}}{(1+\dot{y}^2)^{3/2}} \] \begin{thebibliography}{1} \bibitem{hsr} - \texttt{An2E} Vorlesungen an der Hochschule f\"ur Technik Rapperswil und der dazugehoerige Skript, + \texttt{An2E} Vorlesungen an der Hochschule f\"ur Technik Rapperswil und der dazugeh\"orige Skript, \textit{Dr. Bernhard Zgraggen}, Fr\"uhlingssemester 2020 \bibitem{bronstein} Taschenbuch der Mathematik, @@ -214,7 +218,7 @@ Rotationsoberfl\"ache um \(x\) \end{thebibliography} \section*{Notation} -Rot markierte Zahlen wie zB \brpage{477} sind Hinweise auf die Seiten in der ``Bronstein'': ``Taschenbuch der Mathematik, 10. \"uberarbeitete Auflage''. \texttt{ISBN 978-3-8085-5789-1} +Rot markierte Zahlen wie zB \brpage{477} sind Hinweise auf die Seiten im ``Bronstein'' \cite{bronstein} \section*{License} { \tt diff --git a/fig/plane-curve.eps b/fig/plane-curve.eps index 757a7f5..67d1f0a 100644 --- a/fig/plane-curve.eps +++ b/fig/plane-curve.eps @@ -1,10 +1,10 @@ %!PS-Adobe-3.0 EPSF-3.0 %%Creator: cairo 1.15.10 (http://cairographics.org) -%%CreationDate: Thu Apr 9 11:29:49 2020 +%%CreationDate: Thu Apr 9 13:56:35 2020 %%Pages: 1 %%DocumentData: Clean7Bit %%LanguageLevel: 2 -%%BoundingBox: 0 0 306 225 +%%BoundingBox: 0 1 306 226 %%EndComments %%BeginProlog 50 dict begin @@ -509,213 +509,6 @@ cleartomark % available with a FAQ at: http://scripts.sil.org/OFL. %%EndComments -11 dict begin -/FontType 1 def -/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def -/FontName /f-1-0 def -/FontBBox {-33 -250 945 749 }readonly def -/PaintType 0 def -/FontInfo 9 dict dup begin 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l -\dot{\Lambda}(\tau) -\ddot{\Lambda}(\tau) - - -t = \tau + -- cgit v1.2.1