From 680e1e763b8d899b3601b5ab0cf6f1fc2a114e1d Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 3 Jun 2021 18:51:36 +0200
Subject: phases

---
 vorlesungen/slides/2/hilbertraum/laplace.tex | 30 ++++++++++++++++------------
 1 file changed, 17 insertions(+), 13 deletions(-)

(limited to 'vorlesungen/slides/2/hilbertraum/laplace.tex')

diff --git a/vorlesungen/slides/2/hilbertraum/laplace.tex b/vorlesungen/slides/2/hilbertraum/laplace.tex
index 5e0bba9..8f6b196 100644
--- a/vorlesungen/slides/2/hilbertraum/laplace.tex
+++ b/vorlesungen/slides/2/hilbertraum/laplace.tex
@@ -16,46 +16,50 @@ Gegeben: $\Omega\subset\mathbb{R}^n$ ein Gebiet
 \\
 Gesucht: Lösungen von $\Delta u=0$ mit $u_{|\partial\Omega}=0$
 \end{block}
+\uncover<2->{%
 \begin{block}{Funktionen}
 Hilbertraum $H$ der Funktionen $f:\overline{\Omega}\to\mathbb{C}$
 mit $f_{|\partial\Omega}=0$
-\end{block}
+\end{block}}
+\uncover<3->{%
 \begin{block}{Skalarprodukt}
 \[
 \langle f,g\rangle
 =
 \int_{\Omega} \overline{f}(x) g(x)\,d\mu(x)
 \]
-\end{block}
+\end{block}}
+\uncover<4->{%
 \begin{block}{Laplace-Operator}
 \[
 \Delta \psi = \operatorname{div}\operatorname{grad}\psi
 \]
-\end{block}
+\end{block}}
 \end{column}
 \begin{column}{0.52\textwidth}
+\uncover<5->{%
 \begin{block}{Selbstadjungiert}
 \begin{align*}
 \langle f,\Delta g\rangle
-&=
-\int_{\Omega} \overline{f}(x)\operatorname{div}\operatorname{grad}g(x)\,d\mu(x)
+&\uncover<6->{=
+\int_{\Omega} \overline{f}(x)\operatorname{div}\operatorname{grad}g(x)\,d\mu(x)}
 \\
-&=
+&\uncover<7->{=
 \int_{\partial\Omega}
-\underbrace{\overline{f}(x)}_{\displaystyle=0}\operatorname{grad}g(x)\,d\nu(x)
+\underbrace{\overline{f}(x)}_{\displaystyle=0}\operatorname{grad}g(x)\,d\nu(x)}
 \\
-&\qquad
+&\uncover<7->{\qquad
 -
 \int_{\Omega}
 \operatorname{grad}\overline{f}(x)\cdot \operatorname{grad}g(x)
-\,d\mu(x)
+\,d\mu(x)}
 \\
-&=\int_{\Omega}\operatorname{div}\operatorname{grad}\overline{f}(x)g(x)\,d\mu(x)
+&\uncover<8->{=\int_{\Omega}\operatorname{div}\operatorname{grad}\overline{f}(x)g(x)\,d\mu(x)}
 \\
-&=
-\langle \Delta f,g\rangle
+&\uncover<9->{=
+\langle \Delta f,g\rangle}
 \end{align*}
-\end{block}
+\end{block}}
 \end{column}
 \end{columns}
 \end{frame}
-- 
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