From dcb2a274a782df9ce1ff7ccea0a2ee39ec06dc36 Mon Sep 17 00:00:00 2001 From: Niklas Birk Date: Thu, 25 May 2023 18:11:02 +0200 Subject: [PATCH] structure; ch1, ch2 beginnings --- .gitignore | 3 + proseminar.bib | 20 +++++ proseminar.sty | 109 +++++++++++++++++++++++++ proseminar_skript.tex | 35 ++++++++ sections/01_ldgls.tex | 19 +++++ sections/02_existenz_eindeutigkeit.tex | 34 ++++++++ sections/03_anwendung_auf_jnf.tex | 0 7 files changed, 220 insertions(+) create mode 100644 proseminar.bib create mode 100644 proseminar.sty create mode 100644 proseminar_skript.tex create mode 100644 sections/01_ldgls.tex create mode 100644 sections/02_existenz_eindeutigkeit.tex create mode 100644 sections/03_anwendung_auf_jnf.tex diff --git a/.gitignore b/.gitignore index 747d42e..b7077ea 100644 --- a/.gitignore +++ b/.gitignore @@ -301,3 +301,6 @@ TSWLatexianTemp* # Uncomment the next line to have this generated file ignored. #*Notes.bib +/.idea/ + +/out/ diff --git a/proseminar.bib b/proseminar.bib new file mode 100644 index 0000000..5b5c138 --- /dev/null +++ b/proseminar.bib @@ -0,0 +1,20 @@ +@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} +} \ No newline at end of file diff --git a/proseminar.sty b/proseminar.sty new file mode 100644 index 0000000..b9ec7d9 --- /dev/null +++ b/proseminar.sty @@ -0,0 +1,109 @@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% 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: " 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}} diff --git a/proseminar_skript.tex b/proseminar_skript.tex new file mode 100644 index 0000000..6b49142 --- /dev/null +++ b/proseminar_skript.tex @@ -0,0 +1,35 @@ +\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} \ No newline at end of file diff --git a/sections/01_ldgls.tex b/sections/01_ldgls.tex new file mode 100644 index 0000000..853226d --- /dev/null +++ b/sections/01_ldgls.tex @@ -0,0 +1,19 @@ +\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} \ No newline at end of file diff --git a/sections/02_existenz_eindeutigkeit.tex b/sections/02_existenz_eindeutigkeit.tex new file mode 100644 index 0000000..24aa5fa --- /dev/null +++ b/sections/02_existenz_eindeutigkeit.tex @@ -0,0 +1,34 @@ +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} \ No newline at end of file diff --git a/sections/03_anwendung_auf_jnf.tex b/sections/03_anwendung_auf_jnf.tex new file mode 100644 index 0000000..e69de29