From 4313f2c207d5d60171898ccfd4c3b3d0d2fb4a75 Mon Sep 17 00:00:00 2001
From: Roy Seitz <roy.seitz@ost.ch>
Date: Sun, 18 Apr 2021 17:49:56 +0200
Subject: =?UTF-8?q?Pr=C3=A4sentation=20feritg.?=
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 vorlesungen/slides/10/vektorfelder.tex | 82 ++++++++++++++++++++++++++++++++++
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 create mode 100644 vorlesungen/slides/10/vektorfelder.tex

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diff --git a/vorlesungen/slides/10/vektorfelder.tex b/vorlesungen/slides/10/vektorfelder.tex
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+++ b/vorlesungen/slides/10/vektorfelder.tex
@@ -0,0 +1,82 @@
+%
+% iterativ.tex -- Iterative Approximation in \dot x = J x
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+% Erstellt durch Roy Seitz
+%
+% !TeX spellcheck = de_CH
+\bgroup
+\begin{frame}[t]
+  \setlength{\abovedisplayskip}{5pt}
+  \setlength{\belowdisplayskip}{5pt}
+  \frametitle{Als Strömungsfeld}
+  \vspace{-20pt}
+  \begin{columns}[t,onlytextwidth]
+    \begin{column}{0.48\textwidth}
+      \vfil
+      \only<1>{
+        \includegraphics[width=\linewidth,keepaspectratio]
+        {../slides/10/vektorfelder-1.pdf}
+      }
+      \only<2>{
+        \includegraphics[width=\linewidth,keepaspectratio]
+        {../slides/10/vektorfelder-2.pdf}
+      }
+      \only<3>{
+        \includegraphics[width=\linewidth,keepaspectratio]
+        {../slides/10/vektorfelder-3.pdf}
+      }
+      \only<4>{
+        \includegraphics[width=\linewidth,keepaspectratio]
+        {../slides/10/vektorfelder-4.pdf}
+      }
+      \only<5>{
+        \includegraphics[width=\linewidth,keepaspectratio]
+        {../slides/10/vektorfelder-5.pdf}
+      }
+      \only<6->{
+        \includegraphics[width=\linewidth,keepaspectratio]
+        {../slides/10/vektorfelder-6.pdf}
+      }
+      \vfil
+    \end{column}
+    \begin{column}{0.48\textwidth}
+      \begin{block}{Differentialgleichung}
+        \[ 
+        \dot x(t) = J x(t)
+        \quad
+        J = \begin{pmatrix} 0 & -1 \\ 1 & \phantom-0 \end{pmatrix}
+        \quad
+        x_0 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}
+        \]
+      \end{block}
+    
+      \only<2>{
+        Nach einem Schritt der Länge $t$:
+        \[
+        x(t) = x_0 + \dot x t = x_0 + Jx_0t = (1 + Jt)x_0
+        \]
+      }
+
+      \only<3>{
+        Nach zwei Schritten der Länge $t/2$:
+        \[
+        x(t) = \left(1 + \frac{Jt}{2}\right)^2x_0
+        \]
+      }
+
+      \only<4->{
+        Nach n Schritten der Länge $t/n$:
+        \[
+        x(t) = \left(1 + \frac{Jt}{n}\right)^nx_0
+        \]
+      }
+      \only<6->{
+      \[
+      \lim_{n\to\infty}\left(1 + \frac{At}{n}\right)^n = \exp(At)
+      \]
+      }
+    \end{column}
+  \end{columns}
+\end{frame}
+\egroup
-- 
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