From d17e504a173c7d2deea2c1a7df2a59a9bba18e7e Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Wed, 18 Dec 2013 12:28:21 +0100 Subject: [PATCH] =?UTF-8?q?Verbesserungsvorschl=C3=A4ge=20von=20Sarah=20(E?= =?UTF-8?q?mail=20vom=2017.12.2013)=20eingearbeitet.=20Vielen=20Dank=20daf?= =?UTF-8?q?=C3=BCr!?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- documents/GeoTopo/GeoTopo.pdf | Bin 5359648 -> 5359562 bytes documents/GeoTopo/Kapitel2.tex | 12 +++++++----- 2 files changed, 7 insertions(+), 5 deletions(-) diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index 671a133d8f667b513a5de63ad6326526e3799541..f242058f73adedd0499e8f763e30e22758dfaad1 100644 GIT binary patch delta 135690 zcmYJ5c_3Ba7JxHlh%#o#JVk@t*_2W=$V`z?=An#9W$3ER6h&qdB2ma#ktjq-L<$L! zB!y5Y-rnc^oc+gsuXXpi%ewoV?|gTkb>GX1Yi4YXZDuT?GKlS@lDegX@`v>1Q*R~a z7nCcetvoZa*BsuEGvi2(zZlL}=EP8x%s`4~h*B)mNd3h~Z&lxF()`XTRcyDrYJYai 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b/documents/GeoTopo/Kapitel2.tex @@ -561,19 +561,21 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{definition} \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] - \item Sei $\Delta^n = \text{conv}(e_0, \dots, e_k) \subseteq \mdr^{n+1}$ \todo{stimmen die indizes?} + \item Sei $\Delta^k = \text{conv}(e_0, \dots, e_k) \subseteq \mdr^{n+1}$ die konvexe Hülle der Standard-Basisvektoren $e_0, \dots, e_k$. - $\Delta^k$ heißt \textbf{Standard-Simplex}\xindex{Standard-Simplex}. + + Dann heißt $\Delta^k$ \textbf{Standard-Simplex}\xindex{Standard-Simplex} + und $k$ die Dimension des Simplex. \item Für Punkte $v_0, \dots, v_k$ im $\mdr^n$ in allgemeiner Lage heißt $\delta (v_0, \dots, v_k) = \text{conv}(v_0, \dots, v_k)$ ein \textbf{$k$-Simplex}\xindex{Simplex} in $\mdr^n$. \item Ist $\Delta (v_0, \dots, v_k)$ ein $k$-Simplex und $I = \Set{i_0, \dots, i_r} \subseteq \Set{0, \dots, k}$, - so heißt $s_{i_0} \dots i_r := \text{conv}(v_{i_0}, \dots, v_{i_r})$ + so heißt $s_{i_0, \dots, i_r} := \text{conv}(v_{i_0}, \dots, v_{i_r})$ \textbf{Teilsimplex}\xindex{Teilsimplex} oder \textbf{Seite}\xindex{Seite} von $\Delta$. - $s_{i_0} \dots i_r$ ist $r$-Simplex. + $s_{i_0, \dots, i_r}$ ist $r$-Simplex. \end{enumerate} \end{definition} @@ -862,7 +864,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. Für $i = 0, \dots, n$ sei $\partial_i \sigma := \Delta(x_0, \dots, \hat{x_i}, \dots, x_n)$ die $i$-te Seite von $\sigma$ und $d_\sigma = d_n \sigma := \sum_{i=0} (-1)^i \partial_i \sigma \in C_{n-1} (K)$ - und $d: C_n(K) \rightarrow C_{n-1}(K)$ die dadurch definierte lineare + und $d_n: C_n(K) \rightarrow C_{n-1}(K)$ die dadurch definierte lineare Abbildung. Dann gilt: $d_{n-1} \circ d_n = 0$