From a69b5a6c69ff7ce416ca8919614f57bf99981722 Mon Sep 17 00:00:00 2001 From: jannisp Date: Thu, 22 Jul 2021 15:56:59 +0200 Subject: [PATCH] Add chapter Koordinatentransformation --- Mathematik-V-ZF.tex | 47 ++++++++++++++----- img/1024px-Geographic_coordinates_sphere.png | Bin 0 -> 128833 bytes 2 files changed, 36 insertions(+), 11 deletions(-) create mode 100644 img/1024px-Geographic_coordinates_sphere.png diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 79a6d07..a4fc898 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -5,7 +5,7 @@ \usepackage{ifthen} \usepackage[a4paper, landscape]{geometry} \usepackage{hyperref} -% \usepackage{ccicons} +\usepackage{ccicons} \usepackage{amsmath, amsfonts, amssymb, amsthm} \usepackage{listings} \usepackage{graphicx} @@ -138,12 +138,10 @@ \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} + \small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\ + \small{Jannis Portmann \the\year} \\ + {\ccbysa} +\rule{\linewidth}{0.25pt} \end{center} \section{Gewöhnliche Differentialgleichungen} @@ -163,16 +161,38 @@ $$\iint_A \mathrm{div} \, v \, dA = \oint_C \, v \, dr$$ Flächenintegral der Divergenz von $v$ = Fluss von $v$ durch Rand $C$ \subsection{Satz von Stokes} -$$\iint_A \mathrm{rot} \, v \, dA = \iint_A \zeta \, dA = \oint_C \, v \, ds$$ +$$\iint_A \mathrm{rot} \, v \, dA = \iint_A \xi \, dA = \oint_C \, v \, ds$$ Flächenintegral der Rotation von $v$ = Linienintegral von $v$ entlang $C$ (Zirkulation) \vspace{5px} \textbf{Bsp} \\ -Für eine Vorticity-Dsik mit $\zeta = \zeta_0$, $r=2R$ soll $u_\varphi$ bei $r=4R$ berechnet werden. \\ +Für eine Vorticity-Dsik mit $\xi = \xi_0$, $r=2R$ soll $u_\varphi$ bei $r=4R$ berechnet werden. \\ Der Satz von Stokes lifert: -$$\zeta_0 \cdot (2R)^2 \pi = \int_0^{2\pi}u_\varphi \cdot 4R \cdot d\varphi$$ -nach $u_\varphi$ auflösen: $u_\varphi = \frac{1}{2} \zeta_0 R$ +$$\xi_0 \cdot (2R)^2 \pi = \int_0^{2\pi}u_\varphi \cdot 4R \cdot d\varphi$$ +nach $u_\varphi$ auflösen: $u_\varphi = \frac{1}{2} \xi_0 R$ + +\subsection{Koordinatentransformation} +Wir verwenden meistens geographische Koordinaten. +\begin{figure}[H] + \centering + \includegraphics[width=.15\textwidth]{1024px-Geographic_coordinates_sphere.png} + \caption{Geographisches Koorinatensystem} + \label{fig:geo-coordinates} + \end{figure} + +\vspace{5px} + +Wir definieren für Kugelkoordinaten einen Würfel mit: +$$dx = h_1 \, da$$ +$$dy = h_2 \, db$$ +$$dz = h_3 \, dc$$ +wobei jeweils $\vec{e_x} = \vec{e_a}$ etc. \\ +Aus dem obigen folgen mit dem Satz von Gauss: +$$\mathrm{div} \, v = \frac{1}{h_1 \, h_2} \bigg(\frac{\partial}{\partial a}(u \, h_2) + \frac{\partial}{\partial b}(v \, h_1) \bigg)$$ +Analog mit dem Satz von Stokes: +$$\xi = \frac{1}{r \, \cos\varphi} \frac{\partial v}{\partial \lambda} - \frac{1}{r}\frac{\partial u}{\partial \varphi} + \frac{\tan \varphi}{r} u$$ +Der letzte Term folgt aus der Produkteregel! \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ @@ -205,6 +225,11 @@ Jannis Portmann, FS21 \item Skript zur Vorlesung \end{enumerate} +\section*{Bildquellen} +\begin{itemize} + \item Abb. \ref{fig:geo-coordinates}: E\^(nix) \& ttog, \url{https://de.wikipedia.org/wiki/Geographische_Koordinaten#/media/Datei:Geographic_coordinates_sphere.svg} +\end{itemize} + \end{multicols*} \end{document} diff --git a/img/1024px-Geographic_coordinates_sphere.png b/img/1024px-Geographic_coordinates_sphere.png new file mode 100644 index 0000000000000000000000000000000000000000..482b4e25065451fe7bc04b54a07ff37444f4b9e6 GIT binary patch literal 128833 zcma&O2{e>{_&)p$LP!Xy>|0cJiL7Z9BAHTliY#T{vscQHC?@-o$r6f0_N{E$lQh{y z_H4<%%zHn5f4}#f_nh~f|9`$`;w$q!pXL5s_jO;_{SO4%3-zr_7qJK#{qJRUb`1Q4#zRY2gJyt&k(NnJLMh-Og76|**mKu> zl7A1~^D(^R$9MLKoT-mnz?IU>i zh**)PZ*8h=s@+1e-=s_+al*e-A&ruW`Ty#rkHcT1QNv=fPp%&b7DMPT@rVbdJk4kv z@{8hc+5tFfx*>Qgvt{h^x*P<4;u==X%#|R0O%i89+8@SGG|1xP)O!StU~t 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