From 081a50b7b3c848590ea9c15791909cc0b489ebd9 Mon Sep 17 00:00:00 2001 From: Robert Date: Thu, 25 Mar 2021 11:09:32 +0100 Subject: [PATCH] Updated title page --- chapters/real_analysis_2.tex | 2 ++ chapters/sections/lims_and_funcs.tex | 52 +++++++++++++++++++++++++++ chapters/sections/numbers.tex | 2 +- script.pdf | Bin 547450 -> 550651 bytes script.tex | 17 +++++++-- 5 files changed, 70 insertions(+), 3 deletions(-) create mode 100644 chapters/sections/lims_and_funcs.tex diff --git a/chapters/real_analysis_2.tex b/chapters/real_analysis_2.tex index 8556b44..158febb 100644 --- a/chapters/real_analysis_2.tex +++ b/chapters/real_analysis_2.tex @@ -3,4 +3,6 @@ \begin{document} \chapter{Real Analysis: Part II} + + \subfile{sections/lims_and_funcs.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/lims_and_funcs.tex b/chapters/sections/lims_and_funcs.tex new file mode 100644 index 0000000..db4ceee --- /dev/null +++ b/chapters/sections/lims_and_funcs.tex @@ -0,0 +1,52 @@ +\documentclass[../script.tex]{subfiles} +%! TEX root = ../../script.tex + +\begin{document} +\section{Limits and Functions} + +In this chapter we will introduce the notation +\[ + \oball(x) = (x - \epsilon, x + \epsilon) +\] + +\begin{defi} + Let $D \subset \realn$ and $x \in \realn$. $x$ is called a boundary point of $D$ if + \[ + \forall \epsilon > 0: ~~D \cap \oball(x) \ne 0 + \] + The set of all boundary points of $D$ is called closure and is denoted as $\closure{D}$. +\end{defi} + +\begin{eg} + \begin{enumerate}[(i)] + \item $x \in D$ is always a boundary point of $D$, because + \[ + x \in D \cap \oball(x) + \] + + \item Boundary points don't have to be elements of $D$. If $D = (0, 1)$, then $0$ and $1$ are boundary points, because + \[ + \frac{\epsilon}{2} \in (0, 1) \cap \oball(0) = (-\epsilon, \epsilon) ~~\forall \epsilon > 0 + \] + + \item Let $D = \ratn$. Every $x \in \realn$ is a boundary point, because $\forall \epsilon > 0$, $\oball(x)$ contains at least one rational number. I.e. $\closure{\ratn} = \realn$. + \end{enumerate} +\end{eg} + +\begin{rem} + If $x$ is a boundary point, then + \[ + \forall \epsilon > 0 ~\exists y \in D: ~~|x - y| < \epsilon + \] + If $x$ is not a boundary point, then + \[ + \exists \epsilon > 0 ~\forall y \in D: ~~|x - y| \ge \epsilon + \] +\end{rem} + +\begin{thm} + \[ + x \in \realn \text{ is a boundary point of } D \subset \realn \iff \exists \anyseqdef{D} \text{ such that } x_n \rightarrow x + \] +\end{thm} +\end{document} \ No newline at end of file diff --git a/chapters/sections/numbers.tex b/chapters/sections/numbers.tex index a34f6e5..6b8163e 100644 --- a/chapters/sections/numbers.tex +++ b/chapters/sections/numbers.tex @@ -385,7 +385,7 @@ $\intn$ are called integers, and $\ratn$ are called the rational numbers. $\natn The second statement implies that $\ratn$ is a field. \end{rem} -\begin{cor}[Density of the rationals] +\begin{cor}[Density of the rationals]\label{cor:densityrats} $x, y \in \realn, ~x < y$. 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\newcommand{\rseqdef}[1]{\seq{#1} \subset \realn} \newcommand{\cseqdef}[1]{\seq{#1} \subset \cmpln} \newcommand{\rcseqdef}[1]{\seq{#1} \subset \realn \text{ (or } \cmpln \text{)}} +\newcommand{\anyseqdef}[2][x]{\seq{#1} \subset #2} \newcommand{\series}[2][\infty]{\sum_{#2 = 1}^{#1}} \newcommand{\finite}{\text{ finite}} \newcommand{\conj}[1]{\overline{#1}} +\newcommand{\oball}[1][\epsilon]{B_{#1}} +\newcommand{\closure}[1]{\overline{#1}} \newcommand{\conv}[1]{\xrightarrow{\makebox[2em][c]{$\scriptstyle#1$}}} \newcommand{\convinf}{\conv{n \rightarrow \infty}} @@ -111,8 +114,18 @@ \begin{document} \title{Mathematics for Physicists} \author{\href{https://www.github.com/Lauchmelder23/Mathematics}{https://www.github.com/Lauchmelder23/Mathematics}} -\maketitle -\doclicenseThis + +\begin{titlepage} + \begin{center} + \vspace*{5cm} + \Huge{Mathematics for Physicists}\\[1cm] + \Large{https://www.github.com/Lauchmelder23/Mathematics}\\ + \Large{Alma Mater Lipsiensis}\\[0,5cm] + \normalsize{\today} \\[7,5cm] + \end{center} + \doclicenseThis +\end{titlepage} + \tableofcontents \subfile{chapters/FaN.tex}