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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-04-11 10:30:05 +0200 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-04-11 10:30:05 +0200 |
commit | 15881729aa3f1293d546a1692a02094ed3f24e2b (patch) | |
tree | 7a051091d6f507ad3ab28dbd227206ad1288f368 /vorlesungen/slides/7/dg.tex | |
parent | phasen (diff) | |
download | SeminarMatrizen-15881729aa3f1293d546a1692a02094ed3f24e2b.tar.gz SeminarMatrizen-15881729aa3f1293d546a1692a02094ed3f24e2b.zip |
phases
Diffstat (limited to 'vorlesungen/slides/7/dg.tex')
-rw-r--r-- | vorlesungen/slides/7/dg.tex | 36 |
1 files changed, 24 insertions, 12 deletions
diff --git a/vorlesungen/slides/7/dg.tex b/vorlesungen/slides/7/dg.tex index 36b1ade..a99cebb 100644 --- a/vorlesungen/slides/7/dg.tex +++ b/vorlesungen/slides/7/dg.tex @@ -15,29 +15,35 @@ Ableitung von $\gamma(t)$ an der Stelle $t$: \begin{align*} \dot{\gamma}(t) -&= +&\uncover<2->{= \frac{d}{d\tau}\gamma(\tau)\bigg|_{\tau=t} +} \\ -&= +&\uncover<3->{= \frac{d}{ds} \gamma(t+s) \bigg|_{s=0} +} \\ -&= +&\uncover<4->{= \frac{d}{ds} \gamma(t)\gamma(s) \bigg|_{s=0} +} \\ -&= +&\uncover<5->{= \gamma(t) \frac{d}{ds} \gamma(s) \bigg|_{s=0} -= +} +\uncover<6->{= \gamma(t) \dot{\gamma}(0) +} \end{align*} \end{block} \vspace{-10pt} +\uncover<7->{% \begin{block}{Differentialgleichung} \vspace{-10pt} \[ @@ -47,33 +53,39 @@ Ableitung von $\gamma(t)$ an der Stelle $t$: \quad A=\dot{\gamma}(0)\in LG \] -\end{block} +\end{block}} \end{column} \begin{column}{0.50\textwidth} +\uncover<8->{% \begin{block}{Lösung} Exponentialfunktion \[ \exp\colon LG\to G : A \mapsto \exp(At) = \sum_{k=0}^\infty \frac{t^k}{k!}A^k \] -\end{block} +\end{block}} \vspace{-5pt} +\uncover<9->{% \begin{block}{Kontrolle: Tangentialvektor berechnen} \vspace{-10pt} \begin{align*} \frac{d}{dt}e^{At} -&= +&\uncover<10->{= \sum_{k=1}^\infty A^k \frac{d}{dt} t^{k}{k!} +} \\ -&= +&\uncover<11->{= \sum_{k=1}^\infty A^{k-1}\frac{t^{k-1}}{(k-1)!} A +} \\ -&= +&\uncover<12->{= \sum_{k=0} A^k\frac{t^k}{k!} A -= +} +\uncover<13->{= e^{At} A +} \end{align*} -\end{block} +\end{block}} \end{column} \end{columns} \end{frame} |