4 files changed, 116 insertions, 529 deletions

diff --git a/an2e_zf.pdf b/an2e_zf.pdf
index 46bc7a9..cd1b7ac 100644
--- a/an2e_zf.pdf
+++ b/an2e_zf.pdf
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
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-%%CreationDate: Thu Apr  9 11:29:49 2020
+%%CreationDate: Thu Apr  9 13:56:35 2020
 %%Pages: 1
 %%DocumentData: Clean7Bit
 %%LanguageLevel: 2
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+%%BoundingBox: 0 1 306 226
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-%%Creator: David M. Jones
-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
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-%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
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 %%Trailer
diff --git a/fig/plane-curve.ipe b/fig/plane-curve.ipe
index 04defcb..2d2a8d8 100644
--- a/fig/plane-curve.ipe
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