From 3c1af1ccb91faa7b5fb2801d2e3942e035abe7c3 Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Thu, 24 Jun 2021 11:16:56 +0200 Subject: [PATCH 01/39] Rename file --- Wettersysteme-ZF.tex => Mathematik-V-ZF.tex | 0 1 file changed, 0 insertions(+), 0 deletions(-) rename Wettersysteme-ZF.tex => Mathematik-V-ZF.tex (100%) diff --git a/Wettersysteme-ZF.tex b/Mathematik-V-ZF.tex similarity index 100% rename from Wettersysteme-ZF.tex rename to Mathematik-V-ZF.tex From 49c4b456da9f423ebd50c5e75f24c3b40df6fc84 Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Thu, 24 Jun 2021 11:56:45 +0200 Subject: [PATCH 02/39] Add taylor series --- Mathematik-V-ZF.tex | 2 ++ 1 file changed, 2 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 469fb04..4195ddd 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -154,6 +154,8 @@ Eine Lösung davon $$H(t) = (H_0 - v_0\tau)^{\frac{-t}{\tau}} + v_0 \tau$$ \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} $$\mathrm{rot} \ u = \nabla \times \vec{u}$$ From 9e21c49fc4da2991b3341967b2d3f2b8ec8441f0 Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Thu, 24 Jun 2021 11:57:16 +0200 Subject: [PATCH 03/39] Change to 3 columns --- Mathematik-V-ZF.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 4195ddd..f6167dd 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -125,7 +125,7 @@ \raggedright \footnotesize -\begin{multicols*}{4} +\begin{multicols*}{3} % multicol parameters From 1eaccc234fa6bfed0dd0ec97306e185e8f4ed958 Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Thu, 24 Jun 2021 11:58:06 +0200 Subject: [PATCH 04/39] Fix typo in taylor series --- Mathematik-V-ZF.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index f6167dd..a8a0cd9 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -155,7 +155,7 @@ $$H(t) = (H_0 - v_0\tau)^{\frac{-t}{\tau}} + v_0 \tau$$ \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ -$$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f''(a)}{3!}(x-a)^3 + ...$$ +$$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ \section{Operators} $$\mathrm{rot} \ u = \nabla \times \vec{u}$$ From 826fd7486d59fb0c9386fceed3b69bb1fc1d2a27 Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Thu, 24 Jun 2021 14:12:16 +0200 Subject: [PATCH 05/39] Add Fliessgleichgewicht --- Mathematik-V-ZF.tex | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index a8a0cd9..2f05bc2 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -153,6 +153,10 @@ $$\frac{dH}{dt} = v_0 - \frac{H(t)}{\tau}$$ 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{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ From 8bf473e564028321bded19843761c76e32a1e5ea Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Thu, 24 Jun 2021 18:05:47 +0200 Subject: [PATCH 06/39] Add Gauss and Stokes --- Mathematik-V-ZF.tex | 8 ++++++++ 1 file changed, 8 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 2f05bc2..bfb0a23 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -157,6 +157,14 @@ $$H(t) = (H_0 - v_0\tau)^{\frac{-t}{\tau}} + v_0 \tau$$ 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 $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ From 3c31741db7dcb8e4a70b1bb604d0226944b04e21 Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Thu, 24 Jun 2021 18:10:26 +0200 Subject: [PATCH 07/39] Add more operators --- Mathematik-V-ZF.tex | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index bfb0a23..9b9091b 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -170,7 +170,9 @@ 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} -$$\mathrm{rot} \ u = \nabla \times \vec{u}$$ +$$\mathrm{div} \, \vec{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}$$ + +$$\mathrm{rot} \, \vec{u} = \nabla \times \vec{u} = -\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}$$ $$\nabla = \begin{pmatrix} \frac{\partial}{\partial x}, From 1e9d3a7786bdb93884b07d15518f474e66ed6f2c Mon Sep 17 00:00:00 2001 From: jannisp Date: Thu, 1 Jul 2021 16:30:50 +0200 Subject: [PATCH 08/39] Change semester to FS21 --- Mathematik-V-ZF.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 9b9091b..1654537 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -188,7 +188,7 @@ $$\nabla = \begin{pmatrix} 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 +Jannis Portmann, FS21 \section{Referenzen} \begin{enumerate} From 91b6105f77b2e05e85639aa4c8ac0320b7311f6e Mon Sep 17 00:00:00 2001 From: jannisp Date: Thu, 1 Jul 2021 16:31:18 +0200 Subject: [PATCH 09/39] Add Stokes example --- Mathematik-V-ZF.tex | 8 ++++++++ 1 file changed, 8 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 1654537..1d7e039 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -165,6 +165,14 @@ Flächenintegral der Divergenz von $v$ = Fluss von $v$ durch Rand $C$ $$\iint_A \mathrm{rot} \, v \, dA = \iint_A \zeta \, dA = \oint_C \, v \, ds$$ Flächenintegral der Rotation von $v$ = Linienintegral von $v$ entlang $C$ (Zirkulation) +\vspace{5px} + +\textbf{Bsp} \\ +Für eine Vorticity-Dsik mit $\zeta = \zeta_0$, $r=2R$ 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: $u_\varphi = \frac{1}{2} \zeta_0 R$ + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ From 47b521446c95e458053999d7c1091fade2819ff7 Mon Sep 17 00:00:00 2001 From: jannisp Date: Fri, 2 Jul 2021 15:12:06 +0200 Subject: [PATCH 10/39] Complete rotation in 2 and 3 dimension --- Mathematik-V-ZF.tex | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 1d7e039..3e4ffc8 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -180,7 +180,8 @@ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x \section{Operators} $$\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}$$ +$$\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}, From 0403a15eea1377b767c7b9d688c919020c20f4b9 Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 5 Jul 2021 12:11:27 +0200 Subject: [PATCH 11/39] Update git URL --- Mathematik-V-ZF.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 3e4ffc8..ac4e832 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -196,7 +196,7 @@ $$\nabla = \begin{pmatrix} \doclicenseImage \\ Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\ \faGlobeEurope \kern 1em \url{https://n.ethz.ch/~jannisp} \\ -\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/wettersysteme-zf} \\ +\faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/mathematik-v-zf} \\ Jannis Portmann, FS21 \section{Referenzen} From efb09471d897dbf9bab6bdeb900830a968f2cf22 Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 5 Jul 2021 15:11:46 +0200 Subject: [PATCH 12/39] Create Vektoranalysis chapter --- Mathematik-V-ZF.tex | 1 + 1 file changed, 1 insertion(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index ac4e832..79a6d07 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -157,6 +157,7 @@ $$H(t) = (H_0 - v_0\tau)^{\frac{-t}{\tau}} + v_0 \tau$$ Für eine Funktion $F$, bei $$\frac{dF}{dt} = 0$$ +\section{Vektoranalysis} \subsection{Satz von Gauss} $$\iint_A \mathrm{div} \, v \, dA = \oint_C \, v \, dr$$ Flächenintegral der Divergenz von $v$ = Fluss von $v$ durch Rand $C$ From a69b5a6c69ff7ce416ca8919614f57bf99981722 Mon Sep 17 00:00:00 2001 From: jannisp Date: Thu, 22 Jul 2021 15:56:59 +0200 Subject: [PATCH 13/39] Add chapter Koordinatentransformation --- Mathematik-V-ZF.tex | 47 ++++++++++++++----- img/1024px-Geographic_coordinates_sphere.png | Bin 0 -> 128833 bytes 2 files changed, 36 insertions(+), 11 deletions(-) create mode 100644 img/1024px-Geographic_coordinates_sphere.png diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 79a6d07..a4fc898 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -5,7 +5,7 @@ \usepackage{ifthen} \usepackage[a4paper, landscape]{geometry} \usepackage{hyperref} -% \usepackage{ccicons} +\usepackage{ccicons} \usepackage{amsmath, amsfonts, amssymb, amsthm} \usepackage{listings} \usepackage{graphicx} @@ -138,12 +138,10 @@ \begin{center} \Large{ZF Mathematik V} \\ - \small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\ - \small{Jannis Portmann \the\year} -\end{center} - -\begin{center} - \rule{\linewidth}{0.25pt} + \small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\ + \small{Jannis Portmann \the\year} \\ + {\ccbysa} +\rule{\linewidth}{0.25pt} \end{center} \section{Gewöhnliche Differentialgleichungen} @@ -163,16 +161,38 @@ $$\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$$ +Der letzte Term folgt aus der Produkteregel! \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ @@ -205,6 +225,11 @@ Jannis Portmann, FS21 \item Skript zur Vorlesung \end{enumerate} +\section*{Bildquellen} +\begin{itemize} + \item Abb. \ref{fig:geo-coordinates}: E\^(nix) \& ttog, \url{https://de.wikipedia.org/wiki/Geographische_Koordinaten#/media/Datei:Geographic_coordinates_sphere.svg} +\end{itemize} + \end{multicols*} \end{document} diff --git a/img/1024px-Geographic_coordinates_sphere.png b/img/1024px-Geographic_coordinates_sphere.png new file mode 100644 index 0000000000000000000000000000000000000000..482b4e25065451fe7bc04b54a07ff37444f4b9e6 GIT binary patch literal 128833 zcma&O2{e>{_&)p$LP!Xy>|0cJiL7Z9BAHTliY#T{vscQHC?@-o$r6f0_N{E$lQh{y z_H4<%%zHn5f4}#f_nh~f|9`$`;w$q!pXL5s_jO;_{SO4%3-zr_7qJK#{qJRUb`1Q4#zRY2gJyt&k(NnJLMh-Og76|**mKu> zl7A1~^D(^R$9MLKoT-mnz?IU>i zh**)PZ*8h=s@+1e-=s_+al*e-A&ruW`Ty#rkHcT1QNv=fPp%&b7DMPT@rVbdJk4kv z@{8hc+5tFfx*>Qgvt{h^x*P<4;u==X%#|R0O%i89+8@SGG|1xP)O!StU~t 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a4fc898..38b5eee 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -194,6 +194,31 @@ Analog mit dem Satz von Stokes: $$\xi = \frac{1}{r \, \cos\varphi} \frac{\partial v}{\partial \lambda} - \frac{1}{r}\frac{\partial u}{\partial \varphi} + \frac{\tan \varphi}{r} u$$ Der letzte Term folgt aus der Produkteregel! +\section{Matrixmethoden} +\subsection{Equilibrium} +Setze $\frac{df_i}{dn_j} = 0$ und löse Gleichungssystem + +\subsection{Jacobimatrix} +$$J = \left( \begin{array}{ccc} + \frac{\partial f_1}{\partial n_1} \ldots \frac{\partial f_1}{\partial n_k} \\ + \vdots \ddots \vdots \\ + \frac{\partial f_k}{\partial n_1} \ldots \frac{\partial f_k}{\partial n_k} +\end{array} \right)$$ + +Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathbb{C}, \lambda= x + iy$ \\ + +\vspace{5px} +\begin{itemize} + \item $x < 0$ für alle $\lambda_i$: stabil \\ + \item $x = 0$ für mindestens ein $\lambda_i$: kann neutral sein \\ + \item $x > 0$ für mindestens ein $\lambda_i$: instabil \\ + \item $y > 0$ für mindestens ein $\lambda_i$: Oszillation um Equilibrium \\ + \item $x$ ist die Konvergenz-/Divergenz-Rate zum/vom Equlibrium \\ + \item $1/y$ ist die Periode der Oszillation +\end{itemize} + + + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ From 42f30ce2a564378e747d4c9704b407bab98fa50a Mon Sep 17 00:00:00 2001 From: jannisp Date: Thu, 22 Jul 2021 16:54:20 +0200 Subject: [PATCH 15/39] Add SIR chapter --- Mathematik-V-ZF.tex | 15 +++++++++++++++ 1 file changed, 15 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 38b5eee..c54dd78 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -218,6 +218,21 @@ Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathb \end{itemize} +\subsection{SIR-Modell} +SIR: Susceptible-Infected-Recovered \\ +\vspace{10px} +$\Lambda$: Geburten- oder Immigrationsrate \\ +$\delta_S, \delta_I, \delta_R$: Sterberaten der jeweiligen (Teil-)populationen +$r$: Erholungsrate von $I$ +$\beta S I$: Mass-action Infektionsrate + +\begin{itemize} + \item Disease-free equilibrium: + $$S_f = \Lambda / \delta_S, I_f=0, R_f=0$$ + \item Endemic equilibrium: + $$S = \frac{\delta_1 + r}{\beta} , I_e=\frac{\Lambda}{\delta_1} - \frac{\delta_S}{\beta}, R_e = \frac{r}{\delta_R}(\frac{\Lambda}{\delta_1 + r} - \frac{\delta_S}{\beta})$$ + \item +\end{itemize} \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ From dcfcdf2b96cf8247f4d48247f3466e94fe2c5cfb Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Fri, 23 Jul 2021 11:36:39 +0200 Subject: [PATCH 16/39] Rename to german --- Mathematik-V-ZF.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index c54dd78..bc00fa1 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -238,7 +238,7 @@ $\beta S I$: Mass-action Infektionsrate An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ -\section{Operators} +\section{Operatoren} $$\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})$$ From cb46456324edc20e7356f50979804c09ac8a8f96 Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Fri, 23 Jul 2021 12:17:52 +0200 Subject: [PATCH 17/39] Add .DS_Store to .gitignore --- .gitignore | 2 ++ 1 file changed, 2 insertions(+) diff --git a/.gitignore b/.gitignore index 0813fba..d6fcb13 100644 --- a/.gitignore +++ b/.gitignore @@ -297,3 +297,5 @@ TSWLatexianTemp* *.glstex # End of https://www.toptal.com/developers/gitignore/api/latex + +.DS_Store \ No newline at end of file From 29c8f0722989773f342e891f4e0f3ebc565b92db Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Fri, 23 Jul 2021 12:18:47 +0200 Subject: [PATCH 18/39] Expand SIR --- Mathematik-V-ZF.tex | 23 +++++++++++++++++++---- img/SIR.png | Bin 0 -> 60406 bytes 2 files changed, 19 insertions(+), 4 deletions(-) create mode 100644 img/SIR.png diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index bc00fa1..893330d 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -219,12 +219,26 @@ Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathb \subsection{SIR-Modell} + SIR: Susceptible-Infected-Recovered \\ -\vspace{10px} + +\begin{figure}[H] + \centering + \includegraphics[width=.3\textwidth]{SIR.png} + \caption{SIR-Modell} + \label{fig:sir} +\end{figure} + +\begin{align*} + \frac{dS}{dt} &= \Lambda - \delta_SS - \beta S I \\ + \frac{dI}{dt} &= \beta S I - \delta_I - rI \\ + \frac{dR}{dt} &= rI - \delta_R \\ +\end{align*} + $\Lambda$: Geburten- oder Immigrationsrate \\ -$\delta_S, \delta_I, \delta_R$: Sterberaten der jeweiligen (Teil-)populationen -$r$: Erholungsrate von $I$ -$\beta S I$: Mass-action Infektionsrate +$\delta_S, \delta_I, \delta_R$: Sterberaten der jeweiligen (Teil-)populationen \\ +$r$: Erholungsrate von $I$ \\ +$\beta S I$: Mass-action Infektionsrate \\ \begin{itemize} \item Disease-free equilibrium: @@ -268,6 +282,7 @@ Jannis Portmann, FS21 \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} + \item Abb. \ref{fig:sir}: Vorlesungsunterlagen \end{itemize} \end{multicols*} diff --git a/img/SIR.png b/img/SIR.png new file mode 100644 index 0000000000000000000000000000000000000000..a54f0d95a768e54e208359b4df14e9916b6a65e0 GIT binary patch literal 60406 zcmag_1ytVZ)&&X!1&X!s;_hA?in}`$cXxMpcemmc_u^jMp|}-yC~n2!PxtQL=NtdM z=iGOUkdWj_9$jnBxmLpEWyRjXVZ(uefxVLu7ghiRgW?4P122Pt2F?(pYwm-AA()v9 z3CT+c2@%RW*qWGI8H0g|hbODPRZ<+r&elwbi<^W1#UXp36sS@W)q_DOK;aO8#6f#%Fr!b-$O>X}q_F 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z+QXf&HAMgTfBZjUf6X{y%f1Jlma^v;X+d+wZLbgR96+x3aw>|8`&G zekBN+{#JwMAOCslBmFU$ZqA5Z{{Q>M$c*99EH33S{`IO`&qTU8RV-}hXti5!{`ZRo z3Bse5oY3Fixc~ST1;Y2!$W^rZ-!E3r0!Sy3U-Cb19s=;6%mu9q^B)s;)|j(#4Sf E2mR3#ssI20 literal 0 HcmV?d00001 From 8e0b941d59e163a53da967a4b37e059e21f74eee Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Sat, 24 Jul 2021 12:31:26 +0200 Subject: [PATCH 19/39] Expand SIR further --- Mathematik-V-ZF.tex | 34 ++++++++++++++++++++++++++++++---- img/SIR-2.png | Bin 0 -> 114865 bytes 2 files changed, 30 insertions(+), 4 deletions(-) create mode 100644 img/SIR-2.png diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 893330d..2a154f1 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -138,7 +138,7 @@ \begin{center} \Large{ZF Mathematik V} \\ - \small{701-0106-00L Mathematik V, bei M. A. Sprenger} \\ + \small{701-0106-00L Mathematik V} \\ \small{Jannis Portmann \the\year} \\ {\ccbysa} \rule{\linewidth}{0.25pt} @@ -222,9 +222,10 @@ Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathb SIR: Susceptible-Infected-Recovered \\ +\subsubsection{Single-Strain SIR} \begin{figure}[H] \centering - \includegraphics[width=.3\textwidth]{SIR.png} + \includegraphics[width=.25\textwidth]{SIR.png} \caption{SIR-Modell} \label{fig:sir} \end{figure} @@ -245,9 +246,34 @@ $\beta S I$: Mass-action Infektionsrate \\ $$S_f = \Lambda / \delta_S, I_f=0, R_f=0$$ \item Endemic equilibrium: $$S = \frac{\delta_1 + r}{\beta} , I_e=\frac{\Lambda}{\delta_1} - \frac{\delta_S}{\beta}, R_e = \frac{r}{\delta_R}(\frac{\Lambda}{\delta_1 + r} - \frac{\delta_S}{\beta})$$ - \item \end{itemize} +Für das Disease-free equilibrium ergeben sich die Eigenwerte aus +$$(-\delta_S - \lambda)(\frac{\beta \Lambda}{\delta_S} - \delta_I - r - \lambda)(- \delta R - \lambda) = 0$$ +also +\begin{itemize} + \item $\lambda_1 = -\delta_S$ + \item $\lambda_2 = -\delta_R$ + \item $\lambda_3 = \frac{\beta \Lambda}{\delta_S} - \delta_I - r$ +\end{itemize} + +\subsubsection*{Reproduktionszahl $R_0$} +$$R_0 = \frac{\beta \Lambda}{\delta_S(\delta_I + r)} = \frac{\beta S_f}{\delta_I + r}$$ +\begin{itemize} + \item $R_0 > 1$: Ausbreitung + \item $R_0 < 1$: Aussterben +\end{itemize} + +\subsubsection{Multi-Strain SIR} +\begin{figure}[H] + \centering + \includegraphics[width=.25\textwidth]{SIR-2.png} + \caption{SIR-Modell mit zwei verschiedenen Erregern} + \label{fig:sir-2} +\end{figure} + +Invasion von Strain (2), wenn $R_0^{(1)} < R_0^{(2)}$ + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ @@ -282,7 +308,7 @@ Jannis Portmann, FS21 \section*{Bildquellen} \begin{itemize} \item Abb. \ref{fig:geo-coordinates}: E\^(nix) \& ttog, \url{https://de.wikipedia.org/wiki/Geographische_Koordinaten#/media/Datei:Geographic_coordinates_sphere.svg} - \item Abb. \ref{fig:sir}: Vorlesungsunterlagen + \item Abb. \ref{fig:sir}, \ref{fig:sir-2}: Vorlesungsunterlagen \end{itemize} \end{multicols*} diff --git a/img/SIR-2.png b/img/SIR-2.png new file mode 100644 index 0000000000000000000000000000000000000000..cee88f1aa428cee5e968bac91dd2334a350ef178 GIT binary patch literal 114865 zcmdSAbzGHO&^8PRN=Sza(jZ+T-6h@K-QC?NDIkrMgmi;+OGtO8q;!ciylbE1IY-~` zd!GNkKR$jN_RZdRtTi)h=9+6}h04i@A|v7=LP0?xi;D>>KtaJtK|#Uf!aoF`B=}9_ zKtVk=GZzw)6BiO9l5?;%F}E^?f)WdjSAC?U*o&L48512n0)rln;enZ>N<~}^1)~6m zM~pxg_Y@-_3}0Kxt|Wv~#YwrdrU9lBr?ZqcpqT+jDb88`iHN*ACdP=fF2^CC>4Def zW=5;==F&5$SJ!nI;`D`Z{ONW4xP$})&oe%BlR3k{5W#RV!bWPPOLc^Whd-2id)37JmQmlvY z_VqBPRLAsiLvpLxft;boRw46Ha@h{`W3*7O4g}KGzxD^Xyy6wuqumVegffc^VE3a~ zKOt7>t zoI#a${R=K@p&TxH*;m4;BbA{EO1NCX1-hY-P~{2SRbF&mbyq04Fgb$vITNDq++MO4 z6YBBr4Q#y?7O;Y~Phw|#q0wb(6xP9x_GL0a<_WeP2Juv|TLnMPhmT{kBx$&1CvT>V 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z+t2?zK%OWZzyJTi|L+&`|7RRlW94Rn)k75u3JRL{D@Ez(6?^V~Z!_{>@DwOC70sXj zz@Z-6cw5oa{_mf|2ToESNEZm-x9!7QzvY9nFWYJPZCFzizMkT&s;)}TX-nV#1H<~- A^Z)<= literal 0 HcmV?d00001 From f7c93c5e599dd1bee733684cb7a8fb7329eff172 Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 26 Jul 2021 11:02:22 +0200 Subject: [PATCH 20/39] Add Oszillation chapter --- Mathematik-V-ZF.tex | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 2a154f1..3fd0ea3 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -274,6 +274,28 @@ $$R_0 = \frac{\beta \Lambda}{\delta_S(\delta_I + r)} = \frac{\beta S_f}{\delta_I Invasion von Strain (2), wenn $R_0^{(1)} < R_0^{(2)}$ +\section{Oszillation} +\subsection{Reibungsfrei} +$$\underbrace{\frac{D^2 \Delta z}{Dt^2}}_\text{Beschleunigung Luftpaket} + \underbrace{N^2 \Delta z}_\text{rücktreibende Kraft} = 0$$ + +wobei $N^2 = \frac{g}{\theta}\frac{\partial \theta}{\partial z}$ die Brunt-Väisälla-Frequenz +\vspace{10px} \\ +Mögliche Lösungen davon +$$\Delta z(t) = A \sin (Nt)$$ +$$\Delta z(t) = B \cos (Nt)$$ +$$\Delta z(t) = C \sin (Nt) + D \cos (Nt)$$ +$$\Delta z(t) = E \sin (Nt + \delta)$$ + +oder in komplexer Schreibweise (Euler-Identität) +$$\Delta z(t) = Ae^{iNt}$$ + +\subsection{Mit Reibung} +$$\frac{D^2 \Delta z}{Dt^2} + N^2 \Delta z + k \frac{D \Delta z}{D t} = 0$$ +Lösung mit Ansatz $\Delta z(t) = A e^{i \omega t}$, führt zu +$$\omega^2 - ik\omega - N^2 = 0$$ +also $\omega_{1,2} = \frac{1}{2}(ik \pm \sqrt{4N^2 - k^2})$ und somit +$$\Delta z(t) = A \exp(-\frac{1}{2}kt)\exp(\frac{1}{2}i\sqrt{4N^2 - k^2}t)$$ + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ From b211b2b092214019b91af0fd7910c305c1f32f6e Mon Sep 17 00:00:00 2001 From: jannisp Date: Tue, 27 Jul 2021 13:43:11 +0200 Subject: [PATCH 21/39] Add Waves chapter --- Mathematik-V-ZF.tex | 45 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 45 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 3fd0ea3..21de5e0 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -296,6 +296,49 @@ $$\omega^2 - ik\omega - N^2 = 0$$ also $\omega_{1,2} = \frac{1}{2}(ik \pm \sqrt{4N^2 - k^2})$ und somit $$\Delta z(t) = A \exp(-\frac{1}{2}kt)\exp(\frac{1}{2}i\sqrt{4N^2 - k^2}t)$$ +\section{Wellengleichung} +\subsection{1D-Welle} +Für die Amplitude $\psi$ +$$\frac{\partial \psi}{\partial t^2} = c^2 \Delta \psi$$ +wobei $\Delta$ der Laplace-Operator ist. Somit im 1D-Fall +$$\frac{\partial \psi}{\partial t^2} = c^2 \frac{\partial^2 \psi}{\partial x^2}$$ +Eine Lösung davon +$$\psi(x,t) = A \sin(kx - \omega t) = A \sin \big[k(x-ct)\big]$$ +wobei $k$ die Wellenzahl und $\omega$ die Kreisfrequenz ist. Es gilt $c = \frac{\omega}{k}$, was gerade der Phasengeschwindigkeit entspricht. \\ +\vspace{5px} +Wir beobachten für $c > 0$ eine Verschiebung des Wellenmusters in positiver x-Richtung. \\ +\subsubsection{Kennzahlen} +Phasengeschwindigkeit $v_p = c = \frac{\omega}{k}$ \\ +Wellenlänge $\lambda = \frac{2\pi}{k}$ \\ +Periode $\tau = \frac{2\pi}{\omega}$ + +\subsubsection{Wellenüberlagerung} +Für die Gruppengeschwindigkeit zweier überlagerter Wellen mit $k_1 \neq k_2$ und $\omega_1 \neq \omega_2$ gilt +$$v_g = \frac{\Delta \omega}{\Delta k}$$ +Dispersion tritt auf, falls $v_g \neq v_p$ (in der Praxis meist der Fall) + +\subsection{2D-Welle} +Für eine Linie konstanter Phase +$$kx + ly - \omega t = \mathrm{const.}$$ +Die Ausbreitung verläuft senkrecht auf diese Linien, was entlang dem Wellenvektor $\vec{h}$ entspricht. \\ + +\subsubsection{Kennzahlen} +Für die Wellenlängen gilt +$$\lambda_x = 2\pi / k, \; \lambda_y = 2\pi / l$$ +bzw. die Wellenlänge entlang der Ausbreitung +$$\lambda = \frac{2\pi}{\sqrt{k^2 + l^2}}$$ +Der Wellenvektor ist +$$\vec{h} = (k,l)$$ +Die Phasengeschwindigkeit +$$v_p = \frac{\omega}{\sqrt{k^2 + l^2}}$$ + +\subsection{Komplexer Wellenansatz} +\begin{itemize} + \item 1D: $A e^{i(kx - \omega t)}$ + \item 2D: $A e^{i(kx + ly - \omega t)}$ + \item 3D: $A e^{i(kx + ly + mz - \omega t)}$ +\end{itemize} + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ @@ -312,6 +355,8 @@ $$\nabla = \begin{pmatrix} \frac{\partial}{\partial z} \end{pmatrix}$$ +$$\Delta \psi = \nabla^2 \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2}$$ + \scriptsize \section{Copyleft} From 7f32f8e37ea66e62a8352da24e73f56060131b6c Mon Sep 17 00:00:00 2001 From: jannisp Date: Thu, 29 Jul 2021 09:43:06 +0200 Subject: [PATCH 22/39] Add Transportgleichung --- Mathematik-V-ZF.tex | 40 +++++++++++++++++++++++++++++++++------- 1 file changed, 33 insertions(+), 7 deletions(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 21de5e0..b3875dd 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -164,7 +164,7 @@ Flächenintegral der Divergenz von $v$ = Fluss von $v$ durch Rand $C$ $$\iint_A \mathrm{rot} \, v \, dA = \iint_A \xi \, dA = \oint_C \, v \, ds$$ Flächenintegral der Rotation von $v$ = Linienintegral von $v$ entlang $C$ (Zirkulation) -\vspace{5px} +\vspace{5pt} \textbf{Bsp} \\ Für eine Vorticity-Dsik mit $\xi = \xi_0$, $r=2R$ soll $u_\varphi$ bei $r=4R$ berechnet werden. \\ @@ -180,8 +180,8 @@ Wir verwenden meistens geographische Koordinaten. \caption{Geographisches Koorinatensystem} \label{fig:geo-coordinates} \end{figure} - -\vspace{5px} + + \vspace{5pt} Wir definieren für Kugelkoordinaten einen Würfel mit: $$dx = h_1 \, da$$ @@ -206,8 +206,7 @@ $$J = \left( \begin{array}{ccc} \end{array} \right)$$ Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathbb{C}, \lambda= x + iy$ \\ - -\vspace{5px} +\vspace{5pt} \begin{itemize} \item $x < 0$ für alle $\lambda_i$: stabil \\ \item $x = 0$ für mindestens ein $\lambda_i$: kann neutral sein \\ @@ -279,7 +278,7 @@ Invasion von Strain (2), wenn $R_0^{(1)} < R_0^{(2)}$ $$\underbrace{\frac{D^2 \Delta z}{Dt^2}}_\text{Beschleunigung Luftpaket} + \underbrace{N^2 \Delta z}_\text{rücktreibende Kraft} = 0$$ wobei $N^2 = \frac{g}{\theta}\frac{\partial \theta}{\partial z}$ die Brunt-Väisälla-Frequenz -\vspace{10px} \\ +\vspace{10pt} \\ Mögliche Lösungen davon $$\Delta z(t) = A \sin (Nt)$$ $$\Delta z(t) = B \cos (Nt)$$ @@ -305,7 +304,7 @@ $$\frac{\partial \psi}{\partial t^2} = c^2 \frac{\partial^2 \psi}{\partial x^2}$ Eine Lösung davon $$\psi(x,t) = A \sin(kx - \omega t) = A \sin \big[k(x-ct)\big]$$ wobei $k$ die Wellenzahl und $\omega$ die Kreisfrequenz ist. Es gilt $c = \frac{\omega}{k}$, was gerade der Phasengeschwindigkeit entspricht. \\ -\vspace{5px} +\vspace{5pt} Wir beobachten für $c > 0$ eine Verschiebung des Wellenmusters in positiver x-Richtung. \\ \subsubsection{Kennzahlen} Phasengeschwindigkeit $v_p = c = \frac{\omega}{k}$ \\ @@ -339,6 +338,33 @@ $$v_p = \frac{\omega}{\sqrt{k^2 + l^2}}$$ \item 3D: $A e^{i(kx + ly + mz - \omega t)}$ \end{itemize} +\section{Transportgleichung} +Sei $\phi(x,y,z,t)$ ein physikalisches Feld mit Punkten $P(x,y,z)$ über die Zeit. + +\subsection{Euler'sche Perspektive} +Beobachtung am Punkt $P(x_0,y_0,z_0)$ wird gemessen: +$$\phi^E(t) = \phi(x_0,y_0,z_0,t)$$ +Die zeitliche Änderung +$$\dot{\phi}^E(t) = \frac{\partial}{\partial t} \phi(x_0,y_0,z_0,t)$$ + +\subsection{Lagrange'sche Perspektive} +Messung wird "mitgetragen", entlang einer Trajektorie $\vec{x}(t)$ +$$\phi^L(t) = \phi(x(t),y(t),z(t),t)$$ +Die zeitliche Änderung (materielle Ableitung) +$$\dot{\phi}^L(t) = \frac{D}{D t} \phi = \frac{d}{dt} \phi \bigg( x(t),y(t),z(t),t \bigg) = \frac{\partial \phi}{\partial t}+\vec{u} \cdot \nabla\phi$$ +Wobei $\vec{u}$ z.B. die Strömungsgeschwindigkeit ist + +\subsection{Beliebiger Pfad} +Als Beispiel bewegen wir uns in einem Flugzeug mit der Geschwindigkeit $\vec{v_F}$, dann gilt für die Veränderung +$$\frac{\partial \phi}{\partial t}\bigg|_F = \frac{D \phi}{D t} + (\vec{v_F} - \vec{u}) \cdot \nabla \phi$$ + +\subsection{Quellen und Senken} +Langrange'sche Perspektive +$$\frac{D \phi}{D t} = s$$ +Analog für Euler +$$\frac{\partial \phi}{\partial t} = s - \vec{u}\cdot\nabla\phi$$ +Man Beachte den Advektionsterm $\vec{u}\cdot\nabla\phi$! + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ From 7859d10953bb28d2ba9fb24b4e52d397986237ec Mon Sep 17 00:00:00 2001 From: jannisp Date: Thu, 29 Jul 2021 09:43:16 +0200 Subject: [PATCH 23/39] Add lecture link --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index 0afbd6a..b100dc3 100644 --- a/README.md +++ b/README.md @@ -1,7 +1,7 @@ # Mathematik V ZF [![CC BY-SA 3.0][cc-by-sa-shield]][cc-by-sa] -Zusammenfassung für die Vorlesung *Mathematik V* bei M. A. Sprenger im FS21. +Zusammenfassung für die Vorlesung [*Mathematik V*](http://www.vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheit.view?lerneinheitId=150657&semkez=2021S&ansicht=LEHRVERANSTALTUNGEN&lang=de) bei M. A. Sprenger und weitere Dozenten im FS21. ## Kompiliertes `.pdf` Findest du hier: https://n.ethz.ch/~jannisp/download/Mathematik-V/ From 48977ffa47efff4c4083e5e10141e837a084844a Mon Sep 17 00:00:00 2001 From: Jannis Portmann Date: Sat, 31 Jul 2021 12:24:02 +0200 Subject: [PATCH 24/39] Start Folgen und Reihen chapter --- Mathematik-V-ZF.tex | 31 +++++++++++++++++++++++++++++++ 1 file changed, 31 insertions(+) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index b3875dd..6df9e89 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -365,6 +365,37 @@ Analog für Euler $$\frac{\partial \phi}{\partial t} = s - \vec{u}\cdot\nabla\phi$$ Man Beachte den Advektionsterm $\vec{u}\cdot\nabla\phi$! +\section{Folgen und Reihen} +Folge: $\{a_n\} = a_1,a_2,...,a_n,a_{n+1},...$ \\ +Reihe: $\{s_n\} = s_1,s_2,...,s_n,s_{n+1},...$, wobei +$$s_n = \sum_{k=1}^{k=n}a_k$$ + +\subsection{Bildungsgesetze} +\textbf{Explizit} +$a_n$ kann direkt berechnet werden, z.B. $a_n = n$ + +\textbf{Rekursiv} +$a_n$ wird als Funktion von $a_{n-1}$ angegeben, z.B. $a_n = a_{n-1} + 1$ + +\subsection{Arithmetische Folge} + +\subsection{Geometrische Folge} +Verhältnis von zwei aufeinander folgenden Gliedern ist konstant +$$\frac{a_{n+1}}{a_n}=q , \, a_n = a_1 q^{n-1}$$ +Daraus folgt die \textbf{geometrische Reihe} +$$s_n = \sum_{k=1}^{k=n}a_1 q^{k-1} = a_1 \frac{1-q^n}{1-q}$$ +für $|q| < 1$ +$$\lim_{n\rightarrow\infty} s_n = a_1 \frac{1}{1-q}$$ + +\subsection{Fraktale Geometrie} +\begin{itemize} + \item $R$ = Anzahl Teillängen + \item $P$ = Löcher/Poren + \item $F$ = Anzahl Felder (1D: $R$, 2D: $R^2$, 3D: $R^3$) - P +\end{itemize} + + + \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ From 7a05d4c181e6cc46ea40fb4a37beed6793676662 Mon Sep 17 00:00:00 2001 From: jannisp Date: Wed, 11 Aug 2021 10:49:12 +0200 Subject: [PATCH 25/39] Add preview link --- README.md | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/README.md b/README.md index b100dc3..f20e303 100644 --- a/README.md +++ b/README.md @@ -4,7 +4,10 @@ Zusammenfassung für die Vorlesung [*Mathematik V*](http://www.vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheit.view?lerneinheitId=150657&semkez=2021S&ansicht=LEHRVERANSTALTUNGEN&lang=de) bei M. A. Sprenger und weitere Dozenten im FS21. ## Kompiliertes `.pdf` -Findest du hier: https://n.ethz.ch/~jannisp/download/Mathematik-V/ +Findest du hier: https://n.ethz.ch/~jannisp/download/Mathematik-V/ + +Vorschau-PDF werden automatisch unter https://server.thisfro.ch/download/latex-previews/ veröffentlicht. +:warning: Achtung: Diese sind meist noch nicht fertig! ## Änderungen Falls du irgendwelche Fehler findest oder Sache ergänzen willst, darfst du die gerne selbst korrigieren/einfügen und einen Pull request öffnen. Ansonsten kontaktiere mich direkt ([jannisp](jannispmailto:jannisp@student.ethz.ch)). From 0d34ad01d0b2a04c1de68c57cf246635b067996a Mon Sep 17 00:00:00 2001 From: jannisp Date: Wed, 11 Aug 2021 10:49:20 +0200 Subject: [PATCH 26/39] Add link to VVZ --- Mathematik-V-ZF.tex | 18 +++++++++++++++--- 1 file changed, 15 insertions(+), 3 deletions(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 6df9e89..05f6e0d 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -138,7 +138,7 @@ \begin{center} \Large{ZF Mathematik V} \\ - \small{701-0106-00L Mathematik V} \\ + \small{\href{http://www.vvz.ethz.ch/Vorlesungsverzeichnis/lerneinheitPre.do?lerneinheitId=150657&semkez=2021S&lang=de}{701-0106-00L}} \\ \small{Jannis Portmann \the\year} \\ {\ccbysa} \rule{\linewidth}{0.25pt} @@ -394,6 +394,18 @@ $$\lim_{n\rightarrow\infty} s_n = a_1 \frac{1}{1-q}$$ \item $F$ = Anzahl Felder (1D: $R$, 2D: $R^2$, 3D: $R^3$) - P \end{itemize} +\subsubsection{Hausdorff-Dimension} +Für Betrachtungen geometrischer Objekte mit Seitenlängen $N(R)$ gilt +$$D = \frac{\log F}{\log R}$$ +Wenn $D$ nicht ganzzahlig $\rightarrow$ Fraktal + +\subsubsection{Präfraktale} +Als Präfraktale werden Fraktale einer bestimmter Ordnung verstanden. Ordnung 5 entspricht 5 Bildungsschritten. Ein ideales Fraktal besteht aus unendlich solcher Schritte. Ordnung 1 entspricht dem \textbf{Generator}. + +\subsection{Anwendung in der Bodenphysik} +\subsubsection{Wassersättigung} +$$\Theta = \frac{\theta(h)}{\theta_s} = \bigg(\frac{h_b}{h}\bigg)^\lambda$$ + \section{Taylor-Reihe} @@ -416,7 +428,7 @@ $$\Delta \psi = \nabla^2 \psi = \frac{\partial^2 \psi}{\partial x^2} + \frac{\pa \scriptsize -\section{Copyleft} +\section*{Copyleft} \doclicenseImage \\ Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\ @@ -424,7 +436,7 @@ Dieses Dokument ist unter (CC BY-SA 3.0) freigegeben \\ \faGit \kern 0.88em \url{https://git.thisfro.ch/thisfro/mathematik-v-zf} \\ Jannis Portmann, FS21 -\section{Referenzen} +\section*{Referenzen} \begin{enumerate} \item Skript zur Vorlesung \end{enumerate} From 5f162223e982ef715587787d20a6238e8059fb9c Mon Sep 17 00:00:00 2001 From: jannisp Date: Wed, 11 Aug 2021 10:54:26 +0200 Subject: [PATCH 27/39] Add Jenkinsfile --- Jenkinsfile | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) create mode 100644 Jenkinsfile diff --git a/Jenkinsfile b/Jenkinsfile new file mode 100644 index 0000000..da57394 --- /dev/null +++ b/Jenkinsfile @@ -0,0 +1,17 @@ +node { + stage('Pull git') { + checkout scm + } + + stage('Build PDF') { + sh 'pdflatex *.tex' + } + + stage('Archive PDF') { + archiveArtifacts artifacts: '*.pdf', followSymlinks: false + } + + stage('Publish PDF') { + sh 'scp -i /root/.ssh/id_rsa *.pdf jannis@192.168.178.45:/var/www/html/download/latex-previews/mathematik-v-zf.pdf' + } +} \ No newline at end of file From c06c8b36dab71b84cc64bb7028ca2e17bd92df61 Mon Sep 17 00:00:00 2001 From: jannisp Date: Wed, 11 Aug 2021 11:10:08 +0200 Subject: [PATCH 28/39] Use correct filenames --- Jenkinsfile | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/Jenkinsfile b/Jenkinsfile index da57394..007a085 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -4,14 +4,14 @@ node { } stage('Build PDF') { - sh 'pdflatex *.tex' + sh 'pdflatex Mathematik-V-ZF.tex' } stage('Archive PDF') { - archiveArtifacts artifacts: '*.pdf', followSymlinks: false + archiveArtifacts artifacts: 'Mathematik-V-ZF.pdf', followSymlinks: false } stage('Publish PDF') { - sh 'scp -i /root/.ssh/id_rsa *.pdf jannis@192.168.178.45:/var/www/html/download/latex-previews/mathematik-v-zf.pdf' + sh 'scp -i /root/.ssh/id_rsa Mathematik-V-ZF.pdf jannis@192.168.178.45:/var/www/html/download/latex-previews/mathematik-v-zf.pdf' } } \ No newline at end of file From 4fc75cee49f97b7ceb5820e99dded1a7b4980abb Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 23 Aug 2021 14:46:22 +0200 Subject: [PATCH 29/39] Update jenkins-ci --- Jenkinsfile | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Jenkinsfile b/Jenkinsfile index 007a085..b59b864 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -4,7 +4,7 @@ node { } stage('Build PDF') { - sh 'pdflatex Mathematik-V-ZF.tex' + sh 'xelatex Mathematik-V-ZF.tex && xelatex Mathematik-V-ZF.tex' } stage('Archive PDF') { @@ -12,6 +12,6 @@ node { } stage('Publish PDF') { - sh 'scp -i /root/.ssh/id_rsa Mathematik-V-ZF.pdf jannis@192.168.178.45:/var/www/html/download/latex-previews/mathematik-v-zf.pdf' + sh 'scp -i /root/.ssh/id_rsa Mathematik-V-ZF.pdf jannis@192.168.178.45:/opt/containers/apache2/html/download/latex-previews/mathematik-v-zf.pdf' } } \ No newline at end of file From c2a492766a416e1d3b8dab0d972f88a92beb70d0 Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 23 Aug 2021 15:00:58 +0200 Subject: [PATCH 30/39] Set correct user --- Jenkinsfile | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Jenkinsfile b/Jenkinsfile index b59b864..9e285a1 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -12,6 +12,6 @@ node { } stage('Publish PDF') { - sh 'scp -i /root/.ssh/id_rsa Mathematik-V-ZF.pdf jannis@192.168.178.45:/opt/containers/apache2/html/download/latex-previews/mathematik-v-zf.pdf' + sh 'scp -i /root/.ssh/id_rsa Mathematik-V-ZF.pdf thisfro@192.168.178.45:/opt/containers/apache2/html/download/latex-previews/mathematik-v-zf.pdf' } -} \ No newline at end of file +} From f2f3a4140c5410ef7db369e92854e3ac5c7473bc Mon Sep 17 00:00:00 2001 From: jannisp Date: Sun, 29 Aug 2021 14:34:37 +0200 Subject: [PATCH 31/39] Use mklatex to build --- Jenkinsfile | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Jenkinsfile b/Jenkinsfile index 9e285a1..4517b56 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -4,7 +4,7 @@ node { } stage('Build PDF') { - sh 'xelatex Mathematik-V-ZF.tex && xelatex Mathematik-V-ZF.tex' + sh 'mklatex -c -xelatex Mathematik-V-ZF.tex' } stage('Archive PDF') { From 248e906daad4b3242ba7da1ac965a847dc01e8e3 Mon Sep 17 00:00:00 2001 From: jannisp Date: Sun, 29 Aug 2021 17:10:27 +0200 Subject: [PATCH 32/39] Add Phasenlinien image and change babel to german --- Mathematik-V-ZF.tex | 13 +++++++++++-- img/phasenlinien.png | Bin 0 -> 128868 bytes 2 files changed, 11 insertions(+), 2 deletions(-) create mode 100644 img/phasenlinien.png diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 05f6e0d..a77feeb 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -1,4 +1,5 @@ \documentclass[8pt,landscape]{article} +\usepackage[ngerman]{babel} \usepackage{multicol} \usepackage{calc} \usepackage{bookmark} @@ -313,13 +314,20 @@ Periode $\tau = \frac{2\pi}{\omega}$ \subsubsection{Wellenüberlagerung} Für die Gruppengeschwindigkeit zweier überlagerter Wellen mit $k_1 \neq k_2$ und $\omega_1 \neq \omega_2$ gilt -$$v_g = \frac{\Delta \omega}{\Delta k}$$ +$$v_g = \frac{d \omega}{d k} \approx \frac{\Delta \omega}{\Delta k}$$ Dispersion tritt auf, falls $v_g \neq v_p$ (in der Praxis meist der Fall) \subsection{2D-Welle} -Für eine Linie konstanter Phase +Für eine Linie konstanter Phase (Phasenlinie) $$kx + ly - \omega t = \mathrm{const.}$$ Die Ausbreitung verläuft senkrecht auf diese Linien, was entlang dem Wellenvektor $\vec{h}$ entspricht. \\ +\begin{figure}[H] + \centering + \includegraphics[width=.25\textwidth]{phasenlinien.png} + \caption{Phasenlinien einer 2D-Welle} + \label{fig:phasenlinien} +\end{figure} + \subsubsection{Kennzahlen} Für die Wellenlängen gilt @@ -445,6 +453,7 @@ Jannis Portmann, FS21 \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} \item Abb. \ref{fig:sir}, \ref{fig:sir-2}: Vorlesungsunterlagen + \item Abb. \ref{fig:phasenlinien}: Jannis Portmann basierend auf Vorlesungsunterlagen, CC BY-SA 3.0 \end{itemize} \end{multicols*} diff --git a/img/phasenlinien.png b/img/phasenlinien.png new file mode 100644 index 0000000000000000000000000000000000000000..e43034e7a223c458635b613c335fcb238279a495 GIT binary patch literal 128868 zcmeFZcU03$v^Go$QUw%2k*YKiDGE}hiF5%4>0L#n_g+F#5Tz+1BGN>p_g({{MnQV7 z3DSG$0YdUlJVo!7^Zoy=b}NlFB5r9aAE!7=K|nxoTvg?UHUR+% zmVkhGj_ertO(4p29DE~m(^kGtP}sq|0RHEOm4WI54Gn^e;Ab)d!f+b`2>wsN*IDod z-kL^8Kn%VU;(wP$^v|m#*fipQekPv7{~_nyr(xh5v&{{yJK#GJ{&(PQTi~Aye|;z1 zBp^KT-LP`{EdhZXf$9zUJKluLiYj@>u+&)pvS3RF=Fi=Zz;R!{E zgTCU^Q#K)ApBl*BKBXxCp8UMZ3CL~Q6Ss`W8U|w@y)i@Jq%pg-s?{rFk~31BaXezX z9d9>NIuk9&5#N&|PBK7f

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zx8~xwGJvSMpY>d}pll1uRVo3LXWC8>wyga_fq(z}-?aZ9*kJz!+`m-&UsCk1$o#+M al?T#y=qyW&dqUa<{4C6@&X$=v-}w)-Vg2m@ literal 0 HcmV?d00001 From aaa42c7619281cf24ca5a8ae2aa883e6d2c53c3f Mon Sep 17 00:00:00 2001 From: jannisp Date: Sun, 29 Aug 2021 17:11:41 +0200 Subject: [PATCH 33/39] typo --- Jenkinsfile | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Jenkinsfile b/Jenkinsfile index 4517b56..43bf94b 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -4,7 +4,7 @@ node { } stage('Build PDF') { - sh 'mklatex -c -xelatex Mathematik-V-ZF.tex' + sh 'latexmk -c -xelatex Mathematik-V-ZF.tex' } stage('Archive PDF') { From 699684a1f1547a989d192cf2d41a8308e5a2fe69 Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 30 Aug 2021 13:34:58 +0200 Subject: [PATCH 34/39] Typo and small addition for ODE --- Mathematik-V-ZF.tex | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index a77feeb..87aec7b 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -150,7 +150,13 @@ $$\frac{dH}{dt} = v_0 - \frac{H(t)}{\tau}$$ Eine Lösung davon -$$H(t) = (H_0 - v_0\tau)^{\frac{-t}{\tau}} + v_0 \tau$$ +$$H(t) = (H_0 - v_0\tau)e^{\frac{-t}{\tau}} + v_0 \tau$$ +kann mit dem Ansatz +$$N(t) = B + A e^{-{t/\tau}}$$ +hergeleitet werden. + +Also gilt für $t \rightarrow \infty$ +$$N(\infty) = v_0 \tau$$ \subsection{Fliessgleichgewicht} Für eine Funktion $F$, bei From c9754a3049ce05c407d531a860159f2847280871 Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 30 Aug 2021 13:40:55 +0200 Subject: [PATCH 35/39] Use jobname --- Jenkinsfile | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/Jenkinsfile b/Jenkinsfile index 43bf94b..4dd844a 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -4,14 +4,14 @@ node { } stage('Build PDF') { - sh 'latexmk -c -xelatex Mathematik-V-ZF.tex' + sh 'latexmk -c -xelatex -jobname=mathematik-v-zf Mathematik-V-ZF.tex' } stage('Archive PDF') { - archiveArtifacts artifacts: 'Mathematik-V-ZF.pdf', followSymlinks: false + archiveArtifacts artifacts: 'mathematik-v-zf.pdf', followSymlinks: false } stage('Publish PDF') { - sh 'scp -i /root/.ssh/id_rsa Mathematik-V-ZF.pdf thisfro@192.168.178.45:/opt/containers/apache2/html/download/latex-previews/mathematik-v-zf.pdf' + sh 'scp -i /root/.ssh/id_rsa mathematik-v-zf.pdf thisfro@192.168.178.45:/opt/containers/apache2/html/download/latex-previews/mathematik-v-zf.pdf' } } From 533d9fbc4d2b8562a566d1a4846fd66e59cf41fb Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 30 Aug 2021 13:44:39 +0200 Subject: [PATCH 36/39] Remove whitespace --- Mathematik-V-ZF.tex | 1 - 1 file changed, 1 deletion(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 87aec7b..8aec21e 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -225,7 +225,6 @@ Eigenwerte $\det(\textbf{J} - \lambda \textbf{I}) = 0$ wobei $\lambda \in \mathb \subsection{SIR-Modell} - SIR: Susceptible-Infected-Recovered \\ \subsubsection{Single-Strain SIR} From 6eac8ab63374aec0ef3c32668f287d8233abb55e Mon Sep 17 00:00:00 2001 From: jannisp Date: Mon, 30 Aug 2021 13:46:35 +0200 Subject: [PATCH 37/39] Add clearing stage --- Jenkinsfile | 4 ++++ 1 file changed, 4 insertions(+) diff --git a/Jenkinsfile b/Jenkinsfile index 4dd844a..6d1d21b 100644 --- a/Jenkinsfile +++ b/Jenkinsfile @@ -1,4 +1,8 @@ node { + stage('Empty workspace') { + sh 'rm -rd *' + } + stage('Pull git') { checkout scm } From 733fb9b2c61c9337083754a5ef13f665d762961a Mon Sep 17 00:00:00 2001 From: jannisp Date: Tue, 31 Aug 2021 12:43:18 +0200 Subject: [PATCH 38/39] Fix typo --- README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/README.md b/README.md index f20e303..f65cb74 100644 --- a/README.md +++ b/README.md @@ -13,7 +13,7 @@ Vorschau-PDF werden automatisch unter https://server.thisfro.ch/download/latex-p Falls du irgendwelche Fehler findest oder Sache ergänzen willst, darfst du die gerne selbst korrigieren/einfügen und einen Pull request öffnen. Ansonsten kontaktiere mich direkt ([jannisp](jannispmailto:jannisp@student.ethz.ch)). ## Copyleft -Ausser den Grafiken unterliegte die Zusammenfassung der [Creative Commons Attribution-ShareAlike 3.0 International License][cc-by-sa]. +Ausser den Grafiken unterliegt die Zusammenfassung der [Creative Commons Attribution-ShareAlike 3.0 International License][cc-by-sa]. [![CC BY-SA 3.0][cc-by-sa-image]][cc-by-sa] From 989f91ba257a0b0e9be28e36a2b93ffefd019982 Mon Sep 17 00:00:00 2001 From: jannisp Date: Tue, 31 Aug 2021 12:43:41 +0200 Subject: [PATCH 39/39] Add example for soil fractals --- Mathematik-V-ZF.tex | 26 +++++++++++++++++++++++--- 1 file changed, 23 insertions(+), 3 deletions(-) diff --git a/Mathematik-V-ZF.tex b/Mathematik-V-ZF.tex index 8aec21e..dffbca9 100644 --- a/Mathematik-V-ZF.tex +++ b/Mathematik-V-ZF.tex @@ -415,12 +415,32 @@ Wenn $D$ nicht ganzzahlig $\rightarrow$ Fraktal \subsubsection{Präfraktale} Als Präfraktale werden Fraktale einer bestimmter Ordnung verstanden. Ordnung 5 entspricht 5 Bildungsschritten. Ein ideales Fraktal besteht aus unendlich solcher Schritte. Ordnung 1 entspricht dem \textbf{Generator}. +\subsection{Beispiel} +Wenden Sie die zugrunde liegende Vorschrift für einen Würfel der Grösse $L=360\mu m$ für unendlich viele Iterationen +an unter der Annahme, dass «die Löcher» Körner sind. In jeder Iteration werden aus einem Würfel der Kantenlänge $l$ 27 kleine Würfel der Kantenlänge $l/3$ erzeugt; 7 werden davon weggenommen (als Loch dargestellt, das hier ein Korn sein soll) und 20 kleine +Würfel bleiben übrig, die dann wieder verkleinert werden. \\ +\vspace{.2cm} +Zeigen sie, dass die in jeder Iteration erzeugten Kornvolumen durch eine geometrische Folge +dargestellt werden. \\ +\vspace{.2cm} +Wir bilden die Folge der in einer Iteration $n$ erzeugten Kornvolumen, $V_n$ , in der Einheit von +Kubikmikrometern um $^3$. In jeder Iteration $n$ wird folgendes Kornvolumen erzeugt: +$$V_n = 20^{n-1} \cdot 7 \cdot \bigg(\frac{360}{3^n}\bigg)^3$$ +Das Verhältnis von zwei aufeinanderfolgenden Folgegliedern ist: +$$\frac{V_{n+1}}{V_n} = \frac{20^{n} \cdot 7 \cdot \big(\frac{360}{3^n}\big)^3}{20^{n-1} \cdot 7 \cdot \big(\frac{360}{3^{n+1}}\big)^3} = \frac{20}{3^3}$$ +dieses Verhältnis ist offensichtlich von der Iterationszahl $n$ unabhängig und konstant, es handelt sich also um eine geometrische Folge. \\ +\vspace{.2cm} +Die kumulierten Kornvolumen bilden dann eine geometrische Reihe $s_n$, mit der Formel +$$s_n = a_1 \frac{1-q^n}{1-q}$$ +wobei $a_1 = V_1$ und $q=\frac{20}{27}$. Wir erhalten +$$s_n = \sum_{i=1}^{i=n} V_i = V_1 \frac{1-(\frac{20}{27})^n}{1-\frac{20}{27}} = ... = 360^3 \bigg(1- \bigg(\frac{20}{27}\bigg)^n\bigg)$$ +Berechnen sie das in unendlich vielen Iterationen erzeugte Kornvolumen $s_\infty$ +$$s_\infty = V_1 \frac{1}{1-\frac{20}{27}} = 360^3 = 46.656\cdot 10^6 \mu m^3$$ + \subsection{Anwendung in der Bodenphysik} \subsubsection{Wassersättigung} $$\Theta = \frac{\theta(h)}{\theta_s} = \bigg(\frac{h_b}{h}\bigg)^\lambda$$ - - \section{Taylor-Reihe} An der stelle $a$ einer Funtkion $f(x)$ $$f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ...$$ @@ -458,7 +478,7 @@ Jannis Portmann, FS21 \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} \item Abb. \ref{fig:sir}, \ref{fig:sir-2}: Vorlesungsunterlagen - \item Abb. \ref{fig:phasenlinien}: Jannis Portmann basierend auf Vorlesungsunterlagen, CC BY-SA 3.0 + \item Abb. \ref{fig:phasenlinien}: Jannis Portmann, basierend auf Vorlesungsunterlagen \end{itemize} \end{multicols*}