# EpsilonTensor

A fully anti-symmetric tensor with constant components.

A fully anti-symmetric tensor, defined by
\begin{equation}
\epsilon_{m_1... m_k} := \varepsilon_{m_1... m_k}\,\sqrt{|g|}\,,
\end{equation}
where the components of $\varepsilon_{m_1... m_k}$ are 0, $+1$ or $-1$
and $\varepsilon_{01\cdots k}=1$,
independent of the basis, and $g$ denotes the metric
determinant.
This property optionally takes a tensor which indicates the symbol
which should be used as a `KroneckerDelta`

symbol when
writing out the product of two epsilon tensors. Additionally, it takes
a tensor which is the associated metric, from which the signature can
be extracted. See the documentation of `epsilon_to_delta`

for more information on the use of these optional arguments.
When the indices are in different positions it is understood that they
are simply raised with the metric. This in particular implies
\begin{equation}
\epsilon^{m_1... m_k} := g^{m_1 n_1} \cdots g^{m_k n_k}
\epsilon_{n_1... n_k} = \frac{\varepsilon^{m_1... m_k}}{\sqrt{|g|}}\,,
\end{equation}
again with $\varepsilon^{m_1... m_k}$ taking values 0, $+1$ or $-1$
and $\varepsilon^{01\cdots k}=\pm 1$ depending on the signature of the
metric.