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author | Nao Pross <np@0hm.ch> | 2021-04-13 19:48:07 +0200 |
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committer | Nao Pross <np@0hm.ch> | 2021-04-13 19:48:07 +0200 |
commit | d1b602b59a428bea7a59655cd5af34a919e7acf5 (patch) | |
tree | c9ad2469eb5c287d60179e4b57f78373e977a4dc /vorlesungen/slides/7/ableitung.tex | |
parent | Add outline (diff) | |
parent | typos (diff) | |
download | SeminarMatrizen-d1b602b59a428bea7a59655cd5af34a919e7acf5.tar.gz SeminarMatrizen-d1b602b59a428bea7a59655cd5af34a919e7acf5.zip |
Merge branch 'master' of https://github.com/AndreasFMueller/SeminarMatrizen
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-rw-r--r-- | vorlesungen/slides/7/ableitung.tex | 68 |
1 files changed, 68 insertions, 0 deletions
diff --git a/vorlesungen/slides/7/ableitung.tex b/vorlesungen/slides/7/ableitung.tex new file mode 100644 index 0000000..12f9084 --- /dev/null +++ b/vorlesungen/slides/7/ableitung.tex @@ -0,0 +1,68 @@ +% +% ableitung.tex -- Ableitung in der Lie-Gruppe +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Ableitung in der Matrix-Gruppe} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{O}(n)$} +\uncover<2->{% +$s \mapsto A(s)\in\operatorname{O}(n)$ +} +\begin{align*} +\uncover<3->{I +&= +A(s)^tA(s)} +\\ +\uncover<4->{0 += +\frac{d}{ds} I +&= +\frac{d}{ds} (A(s)^t A(s))} +\\ +&\uncover<5->{= +\dot{A}(s)^tA(s) + A(s)^t \dot{A}(s)} +\intertext{\uncover<6->{An der Stelle $s=0$, d.~h.~$A(0)=I$}} +\uncover<7->{0 +&= +\dot{A}(0)^t ++ +\dot{A}(0)} +\\ +\uncover<8->{\Leftrightarrow +\qquad +\dot{A}(0)^t &= -\dot{A}(0)} +\end{align*} +\uncover<9->{% +``Tangentialvektoren'' sind antisymmetrische Matrizen +} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Ableitung in $\operatorname{SL}_2(\mathbb{R})$} +\uncover<2->{% +$s\mapsto A(s)\in\operatorname{SL}_n(\mathbb{R})$ +} +\begin{align*} +\uncover<3->{1 &= \det A(t)} +\\ +\uncover<10->{0 += +\frac{d}{dt}1 +&= +\frac{d}{dt} \det A(t)} +\intertext{\uncover<11->{mit dem Entwicklungssatz kann man nachrechnen:}} +\uncover<12->{0&=\operatorname{Spur}\dot{A}(0)} +\end{align*} +\uncover<13->{``Tangentialvektoren'' sind spurlose Matrizen} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup |