#+TITLE: The Levi Cevita Symbol
#+SETUPFILE: ../math_options.org
#+LATEX_HEADER: \usepackage{bm}
#+LATEX_HEADER: \usepackage{mathtools}

* Disclaimer
This site as of now just a technology demonstration and its claims
should not be taken as true (even though I myself am pretty confident
they are)
* Levi Cevita symbol
\begin{equation*}
\varepsilon_{ijk} = \left\{\begin{array}{rl}
1,&(i,j,k) \in \{(1,2,3), (2,3,1), (3,1,2)\}\\
-1,&(i,j,k) \in \{(3,2,1), (2,1,3), (1,3,2)\}\\
0,&\text{otherwise}
\right{}
\end{array}
\end{equation*}

\begin{equation*}
\varepsilon_{ijk} = \varepsilon_{jki} = \varepsilon_{kij} = - \varepsilon_{ikj} = - \varepsilon_{jik} = - \varepsilon_{kji} = 1
\end{equation*}

\begin{align*}
\sum_{k=1}^3 \varepsilon_{ijk}\varepsilon_{kmn} = \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm}
\end{align*}