Retab to spaces, add sections and curvature figure

4 files changed, 1712 insertions, 87 deletions

diff --git a/an2e_zf.pdf b/an2e_zf.pdf
index cd1b7ac..97801b0 100644
--- a/an2e_zf.pdf
+++ b/an2e_zf.pdf
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
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+%%EndResource
+%%BeginResource: font CMMI10
+%!PS-AdobeFont-1.0: CMMI10 003.002
+%%Title: CMMI10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+                                                                                                                                                                               
+11 dict begin
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+/FontName /f-1-0 def
+/FontBBox {-32 -250 1048 750 }readonly def
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+/FontInfo 10 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMMI10.) readonly def
+/FullName (CMMI10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle -14.04 def
+/isFixedPitch false def
+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+/ascent 750 def
+end readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 82 /R put
+dup 115 /s put
+dup 120 /x put
+readonly def
+currentdict end
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+%%EndResource
+%%BeginResource: font CMMI10
+%!PS-AdobeFont-1.0: CMMI10 003.002
+%%Title: CMMI10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMMI10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% available with a FAQ at: http://scripts.sil.org/OFL.
+%%EndComments
+                                                                                                                                                                               
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+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMMI10.) readonly def
+/FullName (CMMI10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+/UnderlinePosition -100 def
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+/ascent 750 def
+end readonly def
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+%%EndResource
+%%BeginResource: font CMR10
+%!PS-AdobeFont-1.0: CMR10 003.002
+%%Title: CMR10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% 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-2-0 def
+/FontBBox {-40 -250 1009 750 }readonly def
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR10.) readonly def
+/FullName (CMR10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+end readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 43 /plus put
+readonly def
+currentdict end
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+%%EndResource
+%%BeginResource: font CMR10
+%!PS-AdobeFont-1.0: CMR10 003.002
+%%Title: CMR10
+%Version: 003.002
+%%CreationDate: Mon Jul 13 16:17:00 2009
+%%Creator: David M. Jones
+%Copyright: Copyright (c) 1997, 2009 American Mathematical Society
+%Copyright: (<http://www.ams.org>), with Reserved Font Name CMR10.
+% This Font Software is licensed under the SIL Open Font License, Version 1.1.
+% This license is in the accompanying file OFL.txt, and is also
+% 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-2-1 def
+/FontBBox {-40 -250 1009 750 }readonly def
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+/version (003.002) readonly def
+/Notice (Copyright \050c\051 1997, 2009 American Mathematical Society \050<http://www.ams.org>\051, with Reserved Font Name CMR10.) readonly def
+/FullName (CMR10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+/UnderlinePosition -100 def
+/UnderlineThickness 50 def
+end readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 1 /Delta put
+readonly def
+currentdict end
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