From 6669b959b32b318c9a3cb9692f6bf58449ef220e Mon Sep 17 00:00:00 2001
From: Daniel Bahrdt <d.bahrdt@gmx.de>
Date: Mon, 30 Jun 2008 15:18:27 +0200
Subject: [PATCH] formelsammlung: levi-civita und kronecker-delta

---
 formelsammlung.tex | 31 +++++++++++++++++++++++++++++--
 1 file changed, 29 insertions(+), 2 deletions(-)

diff --git a/formelsammlung.tex b/formelsammlung.tex
index 03b751a..0cdc1c7 100644
--- a/formelsammlung.tex
+++ b/formelsammlung.tex
@@ -1,5 +1,32 @@
 \chapter{Lineare Algebra}
-\section{Identitäten}
+\section{Allgemeines}
+\subsection*{Definitionen}
+\subsubsection*{Levi-Civita-Symbol:}
+	\begin{math}
+	\varepsilon_{12\dots n} = 1 \\
+	\varepsilon_{ij\dots u\dots v\dots} = -\varepsilon_{ij\dots v\dots u\dots}\\
+	\varepsilon_{ij\dots u\dots u\dots} = 0 \\
+	
+	\levicivita{i,j,k} =
+	\begin{cases}
+		+1, & \mbox{falls }(i,j,k,\dots) \mbox{ eine gerade Permutation von } (1,2,3,\dots) \mbox{ ist,} \\
+		-1, & \mbox{falls }(i,j,k,\dots) \mbox{ eine ungerade Permutation von } (1,2,3,\dots) \mbox{ ist,} \\
+		0,  & \mbox{wenn mindestens zwei Indizes gleich sind.}
+	\end{cases}
+	
+	(\vec{a} \times \vec{b})_i = \sum_{j=1}^3 \sum_{k=1}^3 \levicivita{ijk} a_j b_k \\
+	\vec{a} \times \vec{b} = \levicivita{ijk} a_j b_k \vec{e_i} = \levicivita{ijk} a_i b_j \vec{e_k} \\
+	
+	\det A = \levicivita{i_1 i_2 \dots i_n} A_{1i_1} A_{2i_2} \dots A_{ni_n}
+	
+	\end{math}
+\subsubsection*{Kronecker-Delta}
+	$\krondelta{i,j}= \begin{cases} 1 & \mbox{falls } i=j \\ 0 & \mbox{falls } i \neq j \end{cases}$ \\s
+	Die $n\times n$-Einheitsmatrix kann als $(\krondelta{ij})_{i,j\in\{1,\ldots,n\}}$ geschrieben werden.
+
+\section{Matrix-Operationen}
+\subsection*{Inversion}
 \begin{math}
-A^{-1} = \inlinematrix{a & b \\ c & d}^{-1} = \frac{1}{ad - bc} \inlinematrix{d & -b \\ -c & a}
+\hypertarget{fs_mtrx_inv_2d}{A^{-1} = \inlinematrix{a & b \\ c & d}^{-1} = \frac{1}{ad - bc} \inlinematrix{d & -b \\ -c & a}}
 \end{math}
+	
\ No newline at end of file