mathematik-v-zf/Mathematik-V-ZF.tex

\documentclass[8pt,landscape]{article}
\usepackage{multicol}
\usepackage{calc}
\usepackage{bookmark}
\usepackage{ifthen}
\usepackage[a4paper, landscape]{geometry}
\usepackage{hyperref}
% \usepackage{ccicons}
\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{listings}
\usepackage{graphicx}
\usepackage{fontawesome5}
\usepackage{xcolor}
\usepackage{float}
\usepackage{apacite}
\usepackage[
    type={CC},
    modifier={by-sa},
    version={3.0}
]{doclicense}

\graphicspath{{./img/}} 

\definecolor{codegreen}{rgb}{0,0.6,0}
\definecolor{codegray}{rgb}{0.5,0.5,0.5}
\definecolor{codepurple}{rgb}{0.58,0,0.82}
\definecolor{backcolour}{rgb}{0.95,0.95,0.92}

\lstdefinestyle{mystyle}{
    backgroundcolor=\color{backcolour},
    commentstyle=\color{codegreen},
    keywordstyle=\color{magenta},
    numberstyle=\tiny\color{codegray},
    stringstyle=\color{codepurple},
    basicstyle=\ttfamily\footnotesize,
    breakatwhitespace=false,
    breaklines=true,
    captionpos=b,
    keepspaces=true,
    numbers=left,
    numbersep=5pt,
    showspaces=false,
    showstringspaces=false,
    showtabs=false,
    tabsize=2
}

\lstset{style=mystyle}

% To make this come out properly in landscape mode, do one of the following
% 1.
%  pdflatex latexsheet.tex
%
% 2.
%  latex latexsheet.tex
%  dvips -P pdf  -t landscape latexsheet.dvi
%  ps2pdf latexsheet.ps


% If you're reading this, be prepared for confusion.  Making this was
% a learning experience for me, and it shows.  Much of the placement
% was hacked in; if you make it better, let me know...


% 2008-04
% Changed page margin code to use the geometry package. Also added code for
% conditional page margins, depending on paper size. Thanks to Uwe Ziegenhagen
% for the suggestions.

% 2006-08
% Made changes based on suggestions from Gene Cooperman. <gene at ccs.neu.edu>


% To Do:
% \listoffigures \listoftables
% \setcounter{secnumdepth}{0}


% This sets page margins to .5 inch if using letter paper, and to 1cm
% if using A4 paper. (This probably isn't strictly necessary.)
% If using another size paper, use default 1cm margins.
\ifthenelse{\lengthtest { \paperwidth = 11in}}
	{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} }
	{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}}
		{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
		{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
	}

% Turn off header and footer
\pagestyle{empty}


% Redefine section commands to use less space
\makeatletter
\renewcommand{\section}{\@startsection{section}{1}{0mm}%
                                {-1ex plus -.5ex minus -.2ex}%
                                {0.5ex plus .2ex}%x
                                {\normalfont\large\bfseries}}
\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}%
                                {-1explus -.5ex minus -.2ex}%
                                {0.5ex plus .2ex}%
                                {\normalfont\normalsize\bfseries}}
\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}%
                                {-1ex plus -.5ex minus -.2ex}%
                                {1ex plus .2ex}%
                                {\normalfont\small\bfseries}}


\makeatother

% Define BibTeX command
\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}

% Don't print section numbers
% \setcounter{secnumdepth}{0}


\setlength{\parindent}{0pt}
\setlength{\parskip}{0pt plus 0.5ex}

% -----------------------------------------------------------------------

\begin{document}

\raggedright
\footnotesize
\begin{multicols*}{3}


% multicol parameters
% These lengths are set only within the two main columns
%\setlength{\columnseprule}{0.25pt}
\setlength{\premulticols}{1pt}
\setlength{\postmulticols}{1pt}
\setlength{\multicolsep}{1pt}
\setlength{\columnsep}{2pt}

\begin{center}
     \Large{ZF Mathematik V} \\
		 \small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\
		 \small{Jannis Portmann \the\year}
\end{center}

\begin{center}
	\rule{\linewidth}{0.25pt}
\end{center}

\section{Gewöhnliche Differentialgleichungen}
\subsection{1. Ordnung}
$$\frac{dH}{dt} = v_0 - \frac{H(t)}{\tau}$$

Eine Lösung davon
$$H(t) = (H_0 - v_0\tau)^{\frac{-t}{\tau}} + v_0 \tau$$

\subsection{Fliessgleichgewicht}
Für eine Funktion $F$, bei
$$\frac{dF}{dt} = 0$$

\subsection{Satz von Gauss}
$$\iint_A \mathrm{div} \, v \, dA = \oint_C \, v \, dr$$
Flächenintegral der Divergenz von $v$ = Fluss von $v$ durch Rand $C$

\subsection{Satz von Stokes}
$$\iint_A \mathrm{rot} \, v \, dA = \iint_A \zeta \, dA  = \oint_C \, v \, ds$$
Flächenintegral der Rotation von $v$ = Linienintegral von $v$ entlang $C$ (Zirkulation)

\section{Taylor-Reihe}
An der stelle $a$ einer Funtkion $f(x)$
$$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$

\section{Operators}
$$\mathrm{div} \, \vec{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$$

$$\mathrm{rot} \, \vec{u} = \nabla \times \vec{u} = -\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}$$

$$\nabla = \begin{pmatrix}
    \frac{\partial}{\partial x},
    \frac{\partial}{\partial y},
    \frac{\partial}{\partial z}
\end{pmatrix}$$

\scriptsize

\section{Copyleft}

\doclicenseImage \\
Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\
\faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp} \\
\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/wettersysteme-zf} \\
Jannis Portmann, HS20

\section{Referenzen}
\begin{enumerate}
    \item Skript zur Vorlesung
\end{enumerate}

\end{multicols*}

\end{document}