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-rw-r--r--vorlesungen/slides/2/hilbertraum/laplace.tex30
1 files changed, 17 insertions, 13 deletions
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}