From 5f146c5b38ad91022dd0854bc52d1b5086bf9e13 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Andreas=20M=C3=BCller?= Date: Tue, 9 Mar 2021 21:34:40 +0100 Subject: add two slides --- vorlesungen/slides/5/Makefile.inc | 2 + vorlesungen/slides/5/chapter.tex | 2 + vorlesungen/slides/5/exponentialfunktion.tex | 125 +++++++++++++++++++++++++++ vorlesungen/slides/5/logarithmusreihe.tex | 53 ++++++++++++ 4 files changed, 182 insertions(+) create mode 100644 vorlesungen/slides/5/exponentialfunktion.tex create mode 100644 vorlesungen/slides/5/logarithmusreihe.tex (limited to 'vorlesungen/slides/5') diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc index 230df72..03ce407 100644 --- a/vorlesungen/slides/5/Makefile.inc +++ b/vorlesungen/slides/5/Makefile.inc @@ -24,5 +24,7 @@ chapter5 = \ \ ../slides/5/stoneweierstrass.tex \ ../slides/5/potenzreihenmethode.tex \ + ../slides/5/logarithmusreihe.tex \ + ../slides/5/exponentialfunktion.tex \ ../slides/5/chapter.tex diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex index 6544294..31c6d25 100644 --- a/vorlesungen/slides/5/chapter.tex +++ b/vorlesungen/slides/5/chapter.tex @@ -22,3 +22,5 @@ \folie{5/stoneweierstrass.tex} \folie{5/potenzreihenmethode.tex} +\folie{5/logarithmusreihe.tex} +\folie{5/exponentialfunktion.tex} diff --git a/vorlesungen/slides/5/exponentialfunktion.tex b/vorlesungen/slides/5/exponentialfunktion.tex new file mode 100644 index 0000000..698d8a5 --- /dev/null +++ b/vorlesungen/slides/5/exponentialfunktion.tex @@ -0,0 +1,125 @@ +% +% exponentialfunktion.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Exponentialfunktion} +\vspace{-15pt} +\begin{columns}[t,onlytextwidth] +\only<1-6>{% +\begin{column}{0.48\textwidth} +\begin{block}{$x(t) \in\mathbb{R}$} +\vspace{-10pt} +\begin{align*} +\frac{d}{dt}x(t) &= ax(t) &a&\in\mathbb{R} +\\ +x(0) &= c&&\in\mathbb{R} +\intertext{\uncover<2->{Lösung:}} +\uncover<2->{x(t) &= ce^{at}} +\end{align*} +\end{block} +\end{column}} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{$X(t) \in M_n(\mathbb{R})$} +\vspace{-10pt} +\begin{align*} +\frac{d}{dt}X(t) +&= +A +X(t)&A&\in M_n(\mathbb{R}) +\\ +X(0)&=C&&\in M_n(\mathbb{R}) +\intertext{\uncover<4->{gekoppelte Differentialgleichung für +vier Funktionen $x_{ij}(t)$}} +\uncover<5->{\dot{x}_{11} &= \rlap{$a_{11} x_{11}(t) + a_{12} x_{21}(t)$}}\\ +\uncover<5->{\dot{x}_{12} &= \rlap{$a_{11} x_{12}(t) + a_{12} x_{22}(t)$}}\\ +\uncover<5->{\dot{x}_{21} &= \rlap{$a_{21} x_{11}(t) + a_{22} x_{21}(t)$}}\\ +\uncover<5->{\dot{x}_{22} &= \rlap{$a_{21} x_{12}(t) + a_{22} x_{22}(t)$}}\\ +\intertext{\uncover<6->{Lösung:}} +\uncover<6->{X(t) &= \exp(At) C} +\end{align*} +\end{block}} +\end{column} +\only<7-9>{% +\begin{column}{0.48\textwidth} +\begin{block}{Beispiel: Diagonalmatrix} +%$D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$ +\begin{align*} +\frac{d}{dt}X&=DX &&\uncover<8->{\Rightarrow &\dot{x}_{ij}(t) &= \lambda_i x_{ij}(t)} +\\ +X(0)&=C +&&\uncover<8->{\Rightarrow&x_{ij}(t)&=c_{ij}} +\end{align*} +\uncover<9->{% +Lösung: +\[ +x_{ij}(t) =c_{ij}e^{\lambda_i t} +\]} +\end{block} +\end{column}} +\uncover<10->{% +\begin{column}{0.48\textwidth} +\begin{block}{Beispiel: Jordan-Block} +\vspace{-10pt} +\begin{align*} +A&=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix} +\rlap{$\displaystyle,\; +X(t) += +\only<22>{ + e^{\lambda t} + \begin{pmatrix} 1&t/\lambda\\ 0&1 \end{pmatrix} +} +\only<23>{ + \frac{e^{\lambda t}}{\lambda} + \begin{pmatrix} \lambda&t\\ 0&\lambda \end{pmatrix} +} +C +$} +\\ +\uncover<11->{ +\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + \phantom{\lambda}x_{2i}(t),&&x_{1i}(0)&=c_{1i} +} +\\ +\uncover<12->{ +\dot{x}_{2i}(t)&=\phantom{\lambda x_{1i}(t)+\mathstrut}\lambda x_{2i}(t),&&x_{2i}(0)&=c_{2i} +} +\end{align*} +\uncover<13->{% +Lösung:} +\begin{align*} +\uncover<14->{ +x_{2i}(t)&=c_{2i}e^{\lambda t} +} +\\ +\uncover<15->{ +\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + c_{2i}e^{\lambda t} +} +\\ +\only<16-17>{x_{1i\only<16>{,h}}(t)} +\only<18->{\dot{x}_{1i}(t)} +& +\only<16-17>{=c\only<17>{(t)}\lambda e^{\lambda t}} +\only<18>{=\dot{c}(t)\lambda e^{\lambda t} ++ +c(t)\lambda^2 e^{\lambda t}} +\only<19->{=\lambda x_{1i}(t) + \dot{c}(t)\lambda e^{\lambda t}} +\\ +\uncover<20->{\Rightarrow +\dot{c}(t)&= c_{2i}/\lambda +\Rightarrow +c(t) = c_{2i}(0) +tc_{2i}/\lambda +} +\\ +\uncover<21->{ +x_{1i}(t) & =c_{1i}e^{\lambda t} + t(c_{2i}/\lambda)e^{\lambda t} +} +\end{align*} +\end{block} +\end{column}} +\end{columns} +\end{frame} diff --git a/vorlesungen/slides/5/logarithmusreihe.tex b/vorlesungen/slides/5/logarithmusreihe.tex new file mode 100644 index 0000000..85ba0ef --- /dev/null +++ b/vorlesungen/slides/5/logarithmusreihe.tex @@ -0,0 +1,53 @@ +% +% logarithmus.tex +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Logarithmusreihe} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Integralgleichung} +\vspace{-5pt} +\begin{align*} +\log(1+x)&=\int_0^x \frac{1}{1+t}\,dt +\\ +&\uncover<5->{= +\int_0^x +1-t+t^2-t^3+\dots\,dt +} +\\ +\uncover<6->{ +&= +x-\frac{x^2}2+\frac{x^3}{3}-\frac{x^4}{4}+\dots +} +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Geometrische Reihe} +\vspace{-5pt} +\begin{align*} +\frac{1}{1-q}&=1+q+q^2+q^3+\dots +\\ +\uncover<3->{ +\frac{1}{1+q}&=1-q+q^2-q^3+\dots +} +\end{align*} +\uncover<4->{Konvergenzradius $1$} +\end{block}} +\end{column} +\end{columns} +\uncover<7->{% +\begin{block}{Matrix-Logarithmus} +Für $\operatorname{Sp}(A)\subset \{z\in\mathbb{C}\;|\;|z-1|<1\}$ konvergiert +\[ +\log A += +(A-I) - \frac12(A-I)^2 + \frac13(A-I)^3 - \frac14(A-I)^4 + \dots +\] +\end{block}} +\end{frame} -- cgit v1.2.1