diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/slides/7/algebraisch.tex | 40 | ||||
| -rw-r--r-- | vorlesungen/slides/7/symmetrien.tex | 64 |
2 files changed, 63 insertions, 41 deletions
diff --git a/vorlesungen/slides/7/algebraisch.tex b/vorlesungen/slides/7/algebraisch.tex index 5b33566..31d209a 100644 --- a/vorlesungen/slides/7/algebraisch.tex +++ b/vorlesungen/slides/7/algebraisch.tex @@ -19,23 +19,25 @@ Längenmessung mit Skalarprodukt \langle \vec{v},\vec{v}\rangle = \vec{v}\cdot \vec{v} -= -\vec{v}^t\vec{v} +\uncover<2->{= +\vec{v}^t\vec{v}} \end{align*} \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<3->{% \begin{block}{Flächeninhalt/Volumen} $n$ Vektoren $V=(\vec{v}_1,\dots,\vec{v}_n)$ \\ Volumen des Parallelepipeds: $\det V$ -\end{block} +\end{block}} \end{column} \end{columns} % \vspace{-7pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<4->{% \begin{block}{Längenerhaltende Transformationen} $A\in\operatorname{GL}_n(\mathbb{R})$ \begin{align*} @@ -44,38 +46,45 @@ $A\in\operatorname{GL}_n(\mathbb{R})$ (A\vec{x}) \cdot (A\vec{y}) -= +\uncover<5->{= (A\vec{x})^t -(A\vec{y}) +(A\vec{y})} \\ +\uncover<6->{ \vec{x}^tI\vec{y} &= -\vec{x}^tA^tA\vec{y} -\Rightarrow I=A^tA +\vec{x}^tA^tA\vec{y}} +\uncover<7->{ +\Rightarrow I=A^tA} \end{align*} -Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$ -\end{block} +\uncover<8->{Begründung: $\vec{e}_i^t B \vec{e}_j = b_{ij}$} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<9->{% \begin{block}{Volumenerhaltende Transformationen} $A\in\operatorname{GL}_n(\mathbb{R})$ \begin{align*} \det(V) &= \det(AV) -= -\det(A)\det(V) +\uncover<10->{= +\det(A)\det(V)} \\ -1&=\det(A) +\uncover<11->{ +1&=\det(A)} \end{align*} +\uncover<10->{ (Produktsatz für Determinante) -\end{block} +} +\end{block}} \end{column} \end{columns} % \vspace{-3pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<12->{% \begin{block}{Orthogonale Matrizen} Längentreue Abbildungen = orthogonale Matrizen: \[ @@ -87,9 +96,10 @@ A \in \operatorname{GL}_n(\mathbb{R}) A^tA=I \} \] -\end{block} +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\uncover<13->{% \begin{block}{``Spezielle'' Matrizen} Volumen-/Orientierungserhaltende Transformationen: \[ @@ -97,7 +107,7 @@ Volumen-/Orientierungserhaltende Transformationen: = \{ A \in \operatorname{GL}_n(\mathbb{R}) \;|\; \det A = 1\} \] -\end{block} +\end{block}} \end{column} \end{columns} diff --git a/vorlesungen/slides/7/symmetrien.tex b/vorlesungen/slides/7/symmetrien.tex index 79f9ef7..35d62d8 100644 --- a/vorlesungen/slides/7/symmetrien.tex +++ b/vorlesungen/slides/7/symmetrien.tex @@ -14,7 +14,7 @@ \begin{column}{0.48\textwidth} \begin{block}{Diskrete Symmetrien} \begin{itemize} -\item +\item<2-> Ebenen-Spiegelung: \[ {\tiny @@ -22,12 +22,13 @@ Ebenen-Spiegelung: } \mapsto {\tiny -\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*}, +\begin{pmatrix*}[r]-x_1\\x_2\\x_3 \end{pmatrix*} } -\; +\uncover<4->{\!,\; \vec{x} \mapsto \vec{x} -2 (\vec{n}\cdot\vec{x}) \vec{n} +} \] \vspace{-10pt} \begin{center} @@ -41,34 +42,39 @@ Ebenen-Spiegelung: \coordinate (C) at ({90+\a}:{\r*cos(90+\a-\b)}); \coordinate (N) at (\a:2); \coordinate (D) at (\a:{\r*cos(\b-\a)}); +\uncover<3->{ \clip (-2.5,-0.45) rectangle (2.5,1.95); -\fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) -- cycle; -\draw[->,color=darkgreen] (O) -- (N); -\node[color=darkgreen] at (N) [above] {$\vec{n}$}; + \fill[color=darkgreen!20] (O) -- ({\a-90}:0.2) arc ({\a-90}:\a:0.2) + -- cycle; + \draw[->,color=darkgreen] (O) -- (N); + \node[color=darkgreen] at (N) [above] {$\vec{n}$}; -\fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) -- cycle; -\fill[color=red] (O) circle[radius=0.06]; -\draw[color=red] ({\a-90}:2) -- ({\a+90}:2); -\fill[color=blue] (C) circle[radius=0.06]; -\draw[color=blue,line width=0.1pt] (A) -- (D); -\node[color=darkgreen] at (D) [below,rotate=\a] {$(\vec{n}\cdot\vec{x})\vec{n}$}; -\draw[color=blue,line width=0.5pt] (A)--(B); + \fill[color=blue!20] (C) -- ($(C)+(\a:0.2)$) arc (\a:{90+\a}:0.2) + -- cycle; + \fill[color=red] (O) circle[radius=0.06]; + \draw[color=red] ({\a-90}:2) -- ({\a+90}:2); + \fill[color=blue] (C) circle[radius=0.06]; + \draw[color=blue,line width=0.1pt] (A) -- (D); + \node[color=darkgreen] at (D) [below,rotate=\a] + {$(\vec{n}\cdot\vec{x})\vec{n}$}; + \draw[color=blue,line width=0.5pt] (A)--(B); -\node[color=blue] at (A) [above right] {$\vec{x}$}; -\node[color=blue] at (B) [above left] {$\vec{x}'$}; + \node[color=blue] at (A) [above right] {$\vec{x}$}; + \node[color=blue] at (B) [above left] {$\vec{x}'$}; -\node[color=red] at (O) [below left] {$O$}; + \node[color=red] at (O) [below left] {$O$}; -\draw[->,color=blue,shorten <= 0.06cm] (O) -- (A); -\draw[->,color=blue,shorten <= 0.06cm] (O) -- (B); + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (A); + \draw[->,color=blue,shorten <= 0.06cm,line width=1.4pt] (O) -- (B); +} \end{tikzpicture} \end{center} \vspace{-5pt} $\vec{n}$ ein Einheitsnormalenvektor auf der Ebene, $|\vec{n}|=1$ -\item +\item<5-> Punkt-Spiegelung: \[ {\tiny @@ -84,13 +90,15 @@ Punkt-Spiegelung: \end{block} \end{column} \begin{column}{0.48\textwidth} +\uncover<6->{% \begin{block}{Kontinuierliche Symmetrien} \begin{itemize} -\item Translation: +\item<7-> Translation: \( \vec{x} \mapsto \vec{x} + \vec{t} \) -\item Drehung: +\item<8-> Drehung: +\vspace{-3pt} \begin{center} \begin{tikzpicture}[>=latex,thick] \def\a{25} @@ -103,14 +111,15 @@ Punkt-Spiegelung: \fill[color=blue!20] (O) -- (90:\r) arc (90:{90+\a}:\r) -- cycle; \node at ({0.5*\a}:1) {$\alpha$}; \node at ({90+0.5*\a}:1) {$\alpha$}; -\draw[->,color=blue] (O) -- (\a:2); -\draw[->,color=darkgreen] (O) -- ({90+\a}:2); +\draw[->,color=blue,line width=1.4pt] (O) -- (\a:2); +\draw[->,color=darkgreen,line width=1.4pt] (O) -- ({90+\a}:2); \end{scope} \draw[->] (-1.1,0) -- (2.3,0) coordinate[label={$x$}]; \draw[->] (0,-0.1) -- (0,2.3) coordinate[label={right:$y$}]; \end{tikzpicture} \end{center} \[ +\uncover<9->{% \begin{pmatrix}x\\y\end{pmatrix} \mapsto \begin{pmatrix} @@ -118,15 +127,18 @@ Punkt-Spiegelung: {\color{blue}\sin\alpha}&{\color{darkgreen}\phantom{-}\cos\alpha} \end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix} +} \] \end{itemize} -\end{block} +\end{block}} \vspace{-10pt} +\uncover<10->{% \begin{block}{Definition} Längen/Winkel bleiben erhalten \\ -$\Rightarrow$ $\exists$ Erhaltungsgrösse -\end{block} +\uncover<11->{% +$\Rightarrow$ $\exists$ Erhaltungsgrösse} +\end{block}} \end{column} \end{columns} \end{frame} |