EpsilonTensor

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.