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formelsammlung: levi-civita und kronecker-delta

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Daniel Bahrdt 2008-06-30 15:18:27 +02:00
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formelsammlung.tex
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\chapter{Lineare Algebra}
\section{Identitäten}
\section{Allgemeines}
\subsection*{Definitionen}
\subsubsection*{Levi-Civita-Symbol:}
\begin{math}
A^{-1} = \inlinematrix{a & b \\ c & d}^{-1} = \frac{1}{ad - bc} \inlinematrix{d & -b \\ -c & a}
\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}
\hypertarget{fs_mtrx_inv_2d}{A^{-1} = \inlinematrix{a & b \\ c & d}^{-1} = \frac{1}{ad - bc} \inlinematrix{d & -b \\ -c & a}}
\end{math}