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\begin{document}

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


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\begin{center}
     \Large{ZF Mathematik V} \\
    \small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\
    \small{Jannis Portmann \the\year} \\
    {\ccbysa}
\rule{\linewidth}{0.25pt}
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\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
$$\frac{dF}{dt} = 0$$

\section{Vektoranalysis}
\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}
$$\iint_A \mathrm{rot} \, v \, dA = \iint_A \xi \, dA  = \oint_C \, v \, ds$$
Flächenintegral der Rotation von $v$ = Linienintegral von $v$ entlang $C$ (Zirkulation)

\vspace{5px}

\textbf{Bsp} \\
Für eine Vorticity-Dsik mit $\xi = \xi_0$, $r=2R$ soll $u_\varphi$ bei $r=4R$ berechnet werden. \\
Der Satz von Stokes lifert:
$$\xi_0 \cdot (2R)^2 \pi = \int_0^{2\pi}u_\varphi \cdot 4R \cdot d\varphi$$
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 $f(x)$
$$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$

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

$$\mathrm{rot} \, \vec{u_{xy}} = \nabla \times \vec{u} = (\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})$$
$$\mathrm{rot} \, \vec{u_{xyz}} =  \nabla \times \vec{u} = (\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}, \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}, \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})$$

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

\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/mathematik-v-zf} \\
Jannis Portmann, FS21

\section{Referenzen}
\begin{enumerate}
    \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}

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