From c60da4f48d1ad3d97af6fd2aee74eaaf1f76bf3f Mon Sep 17 00:00:00 2001 From: Harry Fuchs Date: Tue, 7 Apr 2020 21:17:33 +0200 Subject: [PATCH] 2019-01-17 --- diffgeoIII/custom.tex | 7 ++ diffgeoIII/edit-this-file.tex | 143 +++++++++++++++++++++++++++++++++- 2 files changed, 146 insertions(+), 4 deletions(-) diff --git a/diffgeoIII/custom.tex b/diffgeoIII/custom.tex index 2caa488..33194ed 100644 --- a/diffgeoIII/custom.tex +++ b/diffgeoIII/custom.tex @@ -12,6 +12,8 @@ \newtheorem{Satz}[Def]{Satz}% \newtheorem{Kor}[Def]{Korollar} \theoremstyle{remark} +\newtheorem{Notation}[Def]{Notation}% +\newtheorem{Idee}[Def]{Idee}% \newtheorem{Frage}[Def]{Frage}% \newtheorem{Bem}[Def]{Bemerkung}% \newtheorem{Ueb}[Def]{Übung}% @@ -54,6 +56,11 @@ \newcommand{ \N }{ \mathbb N } \newcommand{ \K }{ \mathbb K } \newcommand{ \M }{ \mathbb M } +\newcommand{ \bigcupdot }{ \dot \bigcup } +\newcommand{ \cupdot }{ \dot \cup } +\newcommand{ \bw }{{\bigwedge}} \newcommand{\miso}[4]{ \begin{tikzcd} #1 \arrow[r, "#3"', shift left=-0.25ex] \pgfmatrixnextcell #4 \arrow[l, "#2"', shift left=-0.75ex] \end{tikzcd} } \newcommand{\quoteunquote}[1]{ \begin{array}{rcl} && \text{ “ } \\ & #1 & \\ \text{ „ } && \end{array} } + +\newcommand{ \wlw }{\wedge\ldots\wedge} diff --git a/diffgeoIII/edit-this-file.tex b/diffgeoIII/edit-this-file.tex index 73a5f92..50573ac 100644 --- a/diffgeoIII/edit-this-file.tex +++ b/diffgeoIII/edit-this-file.tex @@ -5,8 +5,6 @@ % ref %TODO Nummerierung -TODO 2020-01-16 -TODO 2020-01-17 TODO 2020-01-23 TODO 2020-01-24 TODO 2020-01-31 @@ -3302,6 +3300,8 @@  $v\otimes w$ spannen $(V\otimes W)$ auf $\Rightarrow \bar\psi = \bar\varphi$ +%2020-01-16 + %TODO having enumerations within Def environment \begin{Def} \end{Def} @@ -3499,6 +3499,141 @@  mit $$- \sigma\colon V^*\otimes V^* \to V^* \otimes V^*\quad \text{linear fortgesetzt} - \\ \alpha \otimes \beta \mapsto \beta\otimes \alpha + \sigma\colon V^*\otimes V^* &\to& V^* \otimes V^*\quad \text{linear fortgesetzt} + \\ \alpha \otimes \beta &\mapsto& \beta\otimes \alpha +$$ + +%2020-01-17 + +\begin{Bem} + $\{ \alpha\colon V\times V \to K\mathrel|\alpha\mathrel{\text{symetrisch}} \} =: \operatorname{Sym}_{2,0}(V) \leqslant M_{2,0}(V)$ + $$+ \{ b\in V^*\otimes V^* \mathrel| \sigma (b) = b \} =: (V^*\otimes V^*)^{\sigma} \leqslant V^* \otimes V^* +$$ + wobei + $$+ \sigma : \begin{cases} V^*\otimes V^* &\to V^* \otimes V^* \\ \sum_{i}\alpha_i \otimes \beta_i &\mapsto \sum_i\beta_i\otimes \alpha_i \end{cases} +$$ +\end{Bem} + +* Äußere Algebra + +\begin{Idee} + Koordinatenfreie Determinante +\end{Idee} + +\begin{Def} + Sei $V$ ein $K$-Vektorraum. Die Tensoralgebra ist definiert als: + $$+ T(V) := \bigoplus_{k=0}^\infty V^{\otimes k}, \quad V^{\otimes 0} := K +$$ + mit Multiplikation $\otimes$ + $$+ (v_1\otimes \ldots \otimes v_n) \otimes (w_1\otimes \ldots \otimes w_m) := v_1\otimes \ldots \otimes v_n\otimes w_1\otimes \ldots\otimes w_m\in V^{\otimes (n+m)} +$$ +\end{Def} + +\begin{Def} + $$+ V &=& \operatorname{Lin}(\{ e_1,\ldots, e_n \}) \Rightarrow T(V) \cong K\langle e_1,\ldots, e_n \rangle \quad \text{freie Algebra} + \\ I(V) &:=& \left( v\otimes v \mathrel| v\in V \right) \trianglelefteq T(V), \quad \bigwedge V := T(V) / I(V) \quad \text{äußere Algebra} +$$ +\end{Def} + +\begin{Notation} + $v_1 \wedge \ldots \wedge v_k := [v_1 \otimes \ldots \otimes v_k]_{I(V)}\in \bigwedge V$ +\end{Notation} + +Per Definition: $v \wedge v = 0 \in \bigwedge V, \quad\forall v\in V$. +$$+ \Rightarrow 0 = (v+w)\wedge(v+w) =\underbrace{v\wedge v}_{=0} + v\wedge w ü w\wedge v + \underbrace{w\wedge w}_{=0} = v\wedge w + w\wedge v + \Rightarrow v\wedge w = -w \wedge v +$$ + +$$+ {\bigwedge}^k V &:=& \left[ V^{\otimes k} \right]_{I(V)} \subseteq \bigwedge V + \\ &=& V^{\otimes k}/\left( V^{\otimes k} \cap I(V) \right), \quad V^{\otimes k} \cap I(V) =: I_k(V) + \\ {\bigwedge}^0 V &=& K/(K\cap I(V)) = K/\{ 0 \} = K + \\ {\bigwedge}^1 V &=& V/(V\cap I(V)) = V/\{0\} = V + \\ {\bigwedge}^k &=& \{0\},\quad \mathrel\text{falls} k> n := \dim V +$$ +da $e_1\wedge\ldots\wedge e_n\wedge e_n = 0$ (ein Index immer doppelt) + +\begin{Prop} + $(e_i)_{i=1,\ldots, n} \subsetneq V$ Basis, $\dim\left( {\bigwedge}^kV \right) = \dbinom{n}{k}, \quad (0\leqslant k\leqslant n)$ mit Basis $\{ e_{i_1}\wedge\ldots\wedge e_{i_k} \mathrel| i_1< \ldots < i_k \in \{ 1,\ldots, n\} \}$: + $$+ \dim \left( \bigwedge V \right) = 2^n +$$ +\end{Prop} + +Beweis: + + \operatorname{Lin}\left( \{ e_{i_1} \wedge \ldots\wedge e_{i_k} \mathrel| i_1<\ldots