structure; ch1, ch2 beginnings

.gitignore

@@ -301,3 +301,6 @@ TSWLatexianTemp*
 # Uncomment the next line to have this generated file ignored.
 #*Notes.bib
 
+/.idea/
+
+/out/

proseminar.bib

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+@book{heuser,
+    title = {Gew{\"o}hnliche Differentialgleichungen},
+    subtitle = {Einf{\"u}hrung in Lehre und Gebrauch},
+    series = {Mathematische Leitfäden},
+    author = {Heuser, Harro},
+    year = {2009},
+    edition = {6},
+    publisher = {Vieweg+Teubner Verlag},
+    isbn = {978-3-8348-0705-2}
+}
+
+@book{grune,
+    title = {Gew{\"o}hnliche Differentialgleichungen},
+    subtitle = {Eine Einführung aus der Perspektive der dynamischen Systeme},
+    author = {Gr{\"u}ne, Lars and Junge, Oliver},
+    year = {2009},
+    publisher = {Vieweg+Teubner Verlag},
+    series = {Bachelorkurs Mathematik},
+    isbn = {978-3-8348-9261-4}
+}

proseminar.sty

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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%              Required packages and command changes               %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%   required base packages   %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% language and encoding
+\RequirePackage[utf8]{inputenc}
+\RequirePackage[T1]{fontenc}
+\RequirePackage[ngerman]{babel}
+
+% BibLaTeX
+\RequirePackage[hyperref,style=apa]{biblatex}
+
+% math packages
+\RequirePackage{amsmath}   % general math
+\RequirePackage{amsthm}                         % theorems
+\RequirePackage{amssymb}                        % math symbols
+\RequirePackage{mathtools}                      % coloneqq: nice ligatures of := and =:
+\RequirePackage{stmaryrd}                       % lightning symbol
+
+% other packages
+\RequirePackage{enumerate}                      % for alphanumeric enumeration: (i), (ii), ...
+\RequirePackage{csquotes}                       % language dependent correct quote signs
+
+% use sans-serif font Computer Modern Bright
+\RequirePackage{cmbright}                       % sans-serif fonts looking cleaner
+\RequirePackage[lm]{sfmath}                     % bold math symbols in Latin Modern
+
+\DeclareRobustCommand{\sm@ller}{%
+    \dimen@\f@size\p@
+    \ifdim \dimen@ > 12\p@
+        4
+        \dimen@=0.83333\dimen@
+    \else
+        \advance \dimen@ -2\p@
+    \fi
+    \math@fontsfalse
+    \fontsize{\the\dimen@}\z@
+    \selectfont
+}
+\newcommand{\textc}[1]{{\sm@ller\uppercase{#1}}}
+
+
+% load late
+\RequirePackage{hyperref}                  % hyperref package for refs
+\hypersetup{
+    pdftitle={
+        tbd % TODO
+    },
+    pdfsubject={Skript zum Vortrag im Proseminar Algebra},
+    pdfauthor={Niklas Birk},
+    pdfkeywords={mathematics, calculus, analysis, linear algebra, ordinary differential equations, lecture notes, university}
+}
+
+\DeclareTextFontCommand{\emph}{\boldmath\bfseries}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%   new commands and math operatos  %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% common set symbols
+\newcommand{\RR}{\mathbb{R}}
+\newcommand{\CC}{\mathbb{C}}
+
+% common symbols
+\renewcommand{\i}{\mathrm{i}}
+\newcommand{\e}{\mathrm{e}}
+\renewcommand{\d}{\mathrm{d}}
+
+\renewcommand{\Re}{\mathrm{Re}\,}
+\renewcommand{\Im}{\mathrm{Im}\,}
+
+\renewcommand{\vec}[1]{\mathfrak{#1}}
+
+% use the more common variants of greek letters
+\renewcommand{\phi}{\varphi}
+\renewcommand{\epsilon}{\varepsilon}
+
+% symbols for differentiation and integral
+\newcommand{\dx}[1][x]{\ \d #1}                        % differential
+\newcommand{\dfdx}[2]{\frac{\d #1}{\d #2}}             % 1st derivative
+\newcommand{\ddfdx}[3]{\frac{\d^{#3} #1}{\d #2^{#3}}}  % nth derivative
+
+% sets of the form {...; ...}
+\newcommand{\set}[2]{\left\lbrace #1\ \middle|\ #2 \right\rbrace}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%   theorems, definitions, etc.   %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% style for definitions, unnumbered theorems: "Definition ([additional name]) \newline"
+\newtheoremstyle{break*}{}{}{\itshape}{}{\bfseries}{}{\newline}{\thmname{#1}~\thmnote{(#3)}}
+\theoremstyle{break*}
+\newtheorem*{definition}{Definition}
+
+% style for theorems, lemmata, propositions, corollary: "<counter> Theorem ([additional name]) \newline"
+\newtheoremstyle{break}{}{}{\itshape}{}{\bfseries}{}{\newline}{\thmname{#1}~\thmnumber{#2}~\thmnote{(#3)}}
+\theoremstyle{break}
+\newtheorem{theorem}{Satz}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{corollary}[theorem]{Folgerung}
+
+% style for examples: "Beispiel:"
+\newtheoremstyle{nobreak*}{}{}{\normalfont}{}{\bfseries}{}{ }{}
+\theoremstyle{nobreak*}
+\newtheorem*{example*}{Beispiel:}
+
+% proofs with bold name and qed at end
+\renewenvironment{proof}[1][\proofname]{\par\textbf{#1.} }{\nobreak\hfill\ensuremath{\blacksquare}}

proseminar_skript.tex

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+\documentclass[11pt]{scrartcl}
+
+\usepackage{proseminar}
+
+\subject{Proseminar}
+\title{
+    Tbd
+}
+\subtitle{TU Bergakademie Freiberg}
+\author{Niklas Birk}
+\date{16.06.2023 - SS23}
+
+\addbibresource{proseminar.bib}
+\nocite{*}
+
+\makeindex
+
+\begin{document}
+    \maketitle
+    \tableofcontents
+
+    \printbibliography
+
+    \newpage
+
+    \section{Lineare Differentialgleichungssysteme mit konstanten Koeffizienten}\label{sec:01}
+    \input{sections/01_ldgls}
+
+    \section{Existenz und Eindeutigkeit}\label{sec:02}
+    \input{sections/02_existenz_eindeutigkeit}
+
+    \section{Anwendung auf homogene lineare DGLS}\label{sec:03}
+    \input{sections/03_anwendung_auf_jnf}
+
+\end{document}

sections/01_ldgls.tex

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+\begin{definition}
+    Seien $y_1,\dots,y_n: I \subseteq \RR \to \RR$ differenzierbar und $a_{jk} \in \RR$ für $j,k = 1,\dots,n$.
+    Dann heißt
+    \begin{equation}\tag{DGLS}\label{eq:dgls}
+        \begin{aligned}
+            y'_1(x) &= a_{11} y_1(x) + \dots + a_{1n} y_n(x)\\
+            y'_2(x) &= a_{21} y_1(x) + \dots + a_{2n} y_n(x)\\
+            &\vdots\\
+            y'_n(x) &= a_{n1} y_1(x) + \dots + a_{nn} y_n(x)
+        \end{aligned}
+    \end{equation}
+    ein \emph{homogenes lineares Differentialgleichungssystem} (DGLS) (1. Ordnung).\\
+    Das System~\eqref{eq:dgls} lässt sich auch kompakt in der Form
+    \begin{equation*}
+        \vec{y}'(x) = A \vec{y}(x)
+    \end{equation*}
+    schreiben, wobei $\vec{y}(x) = \begin{pmatrix} y_1(x)\\ \vdots\\ y_n(x) \end{pmatrix}, \vec{y}'(x) = \begin{pmatrix} y'_1(x)\\ \vdots\\ y'_n(x) \end{pmatrix}$
+    und $A \in \RR^{n \times n}$
+\end{definition}

sections/02_existenz_eindeutigkeit.tex

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+Eine~\eqref{eq:dgls} zusammen mit einem einem Anfangswertvektor
+\begin{equation*}
+    \vec{y}(x_0) \coloneqq \vec{y}_0 \coloneqq \begin{pmatrix} y_{1_0}\\ \vdots\\ y_{n_0} \end{pmatrix} \in \RR^n
+\end{equation*}
+an der Stelle $x_0 \in \RR$ nennt man ein \emph{C\textc{auchy}-Problem} oder \emph{Anfangswertproblem}.
+
+\begin{theorem}[Existenz- und Eindeutigkeit]
+    Vorgelegt sei ein C\textc{auchy}-Problem
+    \begin{equation}\tag{CP}\label{eq:cp}
+        \vec{y}'(x) = A \vec{y}(x), \qquad \vec{y}_0 \coloneqq \begin{pmatrix} y_{1_0}\\ \vdots\\ y_{n_0} \end{pmatrix}.
+    \end{equation}
+    Dann besitzt~\eqref{eq:cp} eine eindeutig bestimmte Lösung $\vec{y}$ auf $\RR$ mit der Form
+    \begin{equation}\tag{$\ast$}\label{eq:solution}
+        \vec{y}(x) = e^{(x - x_0) A} \vec{y}_0.
+    \end{equation}
+\end{theorem}
+
+\begin{proof}
+    \begin{itemize}
+        \item   \underline{Existenz:}\\
+                Einsetzen in die rechte Seite von~\eqref{eq:solution} in~\eqref{eq:cp} liefert
+                \begin{equation*}
+                    A \vec{y}(x) = A e^{(x - x_0) A} \vec{y}_0.
+                \end{equation*}
+                Zusammen mit~\eqref{eq:} folgt direkt, dass~\eqref{eq:solution} das C\textc{auchy}-Problem löst.
+
+        \item   \underline{Eindeutigkeit:}\\
+                Angenommen $\vec{u}(x)$ sei eine weitere Lösung, d.h.~es gilt $\vec{u}' = A \vec{u},\ \vec{u}(x_0) = \vec{y}_0$.
+                Dann ist
+                \begin{align*}
+                    \dfdx{}{x} \left(  \right)
+                \end{align*}
+    \end{itemize}
+\end{proof}