236 lines
7.1 KiB
TeX
236 lines
7.1 KiB
TeX
\documentclass[8pt,landscape]{article}
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\usepackage{multicol}
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\usepackage{calc}
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\usepackage{bookmark}
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\usepackage{ifthen}
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\usepackage[a4paper, landscape]{geometry}
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\usepackage{hyperref}
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\usepackage{ccicons}
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\usepackage{amsmath, amsfonts, amssymb, amsthm}
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\usepackage{listings}
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\usepackage{graphicx}
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\usepackage{fontawesome5}
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\usepackage{xcolor}
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\usepackage{float}
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\usepackage{apacite}
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\usepackage[
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type={CC},
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modifier={by-sa},
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version={3.0}
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]{doclicense}
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\graphicspath{{./img/}}
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\definecolor{codegreen}{rgb}{0,0.6,0}
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\definecolor{codegray}{rgb}{0.5,0.5,0.5}
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\definecolor{codepurple}{rgb}{0.58,0,0.82}
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\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
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\lstdefinestyle{mystyle}{
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backgroundcolor=\color{backcolour},
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commentstyle=\color{codegreen},
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keywordstyle=\color{magenta},
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numberstyle=\tiny\color{codegray},
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stringstyle=\color{codepurple},
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basicstyle=\ttfamily\footnotesize,
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breakatwhitespace=false,
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breaklines=true,
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captionpos=b,
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keepspaces=true,
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numbers=left,
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numbersep=5pt,
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showspaces=false,
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showstringspaces=false,
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showtabs=false,
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tabsize=2
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}
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\lstset{style=mystyle}
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% To make this come out properly in landscape mode, do one of the following
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% 1.
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% pdflatex latexsheet.tex
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%
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% 2.
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% latex latexsheet.tex
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% dvips -P pdf -t landscape latexsheet.dvi
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% ps2pdf latexsheet.ps
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% If you're reading this, be prepared for confusion. Making this was
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% a learning experience for me, and it shows. Much of the placement
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% was hacked in; if you make it better, let me know...
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% 2008-04
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% Changed page margin code to use the geometry package. Also added code for
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% conditional page margins, depending on paper size. Thanks to Uwe Ziegenhagen
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% for the suggestions.
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% 2006-08
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% Made changes based on suggestions from Gene Cooperman. <gene at ccs.neu.edu>
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% To Do:
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% \listoffigures \listoftables
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% \setcounter{secnumdepth}{0}
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% This sets page margins to .5 inch if using letter paper, and to 1cm
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% if using A4 paper. (This probably isn't strictly necessary.)
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% If using another size paper, use default 1cm margins.
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\ifthenelse{\lengthtest { \paperwidth = 11in}}
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{ \geometry{top=.5in,left=.5in,right=.5in,bottom=.5in} }
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{\ifthenelse{ \lengthtest{ \paperwidth = 297mm}}
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{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
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{\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} }
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}
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% Turn off header and footer
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\pagestyle{empty}
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% Redefine section commands to use less space
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\makeatletter
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\renewcommand{\section}{\@startsection{section}{1}{0mm}%
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{-1ex plus -.5ex minus -.2ex}%
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{0.5ex plus .2ex}%x
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{\normalfont\large\bfseries}}
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\renewcommand{\subsection}{\@startsection{subsection}{2}{0mm}%
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{-1explus -.5ex minus -.2ex}%
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{0.5ex plus .2ex}%
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{\normalfont\normalsize\bfseries}}
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\renewcommand{\subsubsection}{\@startsection{subsubsection}{3}{0mm}%
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{-1ex plus -.5ex minus -.2ex}%
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{1ex plus .2ex}%
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{\normalfont\small\bfseries}}
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\makeatother
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% Define BibTeX command
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\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
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T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}
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% Don't print section numbers
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% \setcounter{secnumdepth}{0}
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\setlength{\parindent}{0pt}
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\setlength{\parskip}{0pt plus 0.5ex}
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% -----------------------------------------------------------------------
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\begin{document}
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\raggedright
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\footnotesize
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\begin{multicols*}{3}
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% multicol parameters
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% These lengths are set only within the two main columns
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%\setlength{\columnseprule}{0.25pt}
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\setlength{\premulticols}{1pt}
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\setlength{\postmulticols}{1pt}
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\setlength{\multicolsep}{1pt}
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\setlength{\columnsep}{2pt}
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\begin{center}
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\Large{ZF Mathematik V} \\
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\small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\
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\small{Jannis Portmann \the\year} \\
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{\ccbysa}
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\rule{\linewidth}{0.25pt}
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\end{center}
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\section{Gewöhnliche Differentialgleichungen}
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\subsection{1. Ordnung}
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$$\frac{dH}{dt} = v_0 - \frac{H(t)}{\tau}$$
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Eine Lösung davon
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$$H(t) = (H_0 - v_0\tau)^{\frac{-t}{\tau}} + v_0 \tau$$
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\subsection{Fliessgleichgewicht}
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Für eine Funktion $F$, bei
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$$\frac{dF}{dt} = 0$$
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\section{Vektoranalysis}
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\subsection{Satz von Gauss}
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$$\iint_A \mathrm{div} \, v \, dA = \oint_C \, v \, dr$$
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Flächenintegral der Divergenz von $v$ = Fluss von $v$ durch Rand $C$
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\subsection{Satz von Stokes}
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$$\iint_A \mathrm{rot} \, v \, dA = \iint_A \xi \, dA = \oint_C \, v \, ds$$
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Flächenintegral der Rotation von $v$ = Linienintegral von $v$ entlang $C$ (Zirkulation)
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\vspace{5px}
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\textbf{Bsp} \\
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Für eine Vorticity-Dsik mit $\xi = \xi_0$, $r=2R$ soll $u_\varphi$ bei $r=4R$ berechnet werden. \\
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Der Satz von Stokes lifert:
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$$\xi_0 \cdot (2R)^2 \pi = \int_0^{2\pi}u_\varphi \cdot 4R \cdot d\varphi$$
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nach $u_\varphi$ auflösen: $u_\varphi = \frac{1}{2} \xi_0 R$
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\subsection{Koordinatentransformation}
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Wir verwenden meistens geographische Koordinaten.
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\begin{figure}[H]
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\centering
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\includegraphics[width=.15\textwidth]{1024px-Geographic_coordinates_sphere.png}
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\caption{Geographisches Koorinatensystem}
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\label{fig:geo-coordinates}
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\end{figure}
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\vspace{5px}
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Wir definieren für Kugelkoordinaten einen Würfel mit:
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$$dx = h_1 \, da$$
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$$dy = h_2 \, db$$
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$$dz = h_3 \, dc$$
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wobei jeweils $\vec{e_x} = \vec{e_a}$ etc. \\
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Aus dem obigen folgen mit dem Satz von Gauss:
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$$\mathrm{div} \, v = \frac{1}{h_1 \, h_2} \bigg(\frac{\partial}{\partial a}(u \, h_2) + \frac{\partial}{\partial b}(v \, h_1) \bigg)$$
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Analog mit dem Satz von Stokes:
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$$\xi = \frac{1}{r \, \cos\varphi} \frac{\partial v}{\partial \lambda} - \frac{1}{r}\frac{\partial u}{\partial \varphi} + \frac{\tan \varphi}{r} u$$
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Der letzte Term folgt aus der Produkteregel!
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\section{Taylor-Reihe}
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An der stelle $a$ einer Funtkion $f(x)$
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$$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$
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\section{Operators}
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$$\mathrm{div} \, \vec{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$$
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$$\mathrm{rot} \, \vec{u_{xy}} = \nabla \times \vec{u} = (\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})$$
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$$\mathrm{rot} \, \vec{u_{xyz}} = \nabla \times \vec{u} = (\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}, \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}, \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})$$
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$$\nabla = \begin{pmatrix}
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\frac{\partial}{\partial x},
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\frac{\partial}{\partial y},
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\frac{\partial}{\partial z}
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\end{pmatrix}$$
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\scriptsize
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\section{Copyleft}
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\doclicenseImage \\
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Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\
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\faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp} \\
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\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/mathematik-v-zf} \\
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Jannis Portmann, FS21
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\section{Referenzen}
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\begin{enumerate}
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\item Skript zur Vorlesung
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\end{enumerate}
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\section*{Bildquellen}
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\begin{itemize}
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\item Abb. \ref{fig:geo-coordinates}: E\^(nix) \& ttog, \url{https://de.wikipedia.org/wiki/Geographische_Koordinaten#/media/Datei:Geographic_coordinates_sphere.svg}
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\end{itemize}
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\end{multicols*}
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\end{document}
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