My textbook says the following in an appendix on tensor notation:
The tensor $\epsilon_{rst} = \begin{cases} 0 \qquad & \text{unless $r, s,$ and $t$ are distinct} \\ +1 \qquad & \text{if $rst$ is an even permutation of $123$} \\ -1 \qquad & \text{if $rst$ is an odd permutation of $123$} \end{cases}$
The tensor $\epsilon_{ijk}$ is related to determinants: for three contravariant tensors $a^i$, $b^j$, and $c^k$, one verifies that $a^i b^j c^k \epsilon_{ijk}$ is the determinant of the $3 \times 3$ matrix with rows $a^i$, $b^j$, and $c^k$.
I don't understand what this excerpt is saying:
The tensor $\epsilon_{ijk}$ is related to determinants: for three contravariant tensors $a^i$, $b^j$, and $c^k$, one verifies that $a^i b^j c^k \epsilon_{ijk}$ is the determinant of the $3 \times 3$ matrix with rows $a^i$, $b^j$, and $c^k$.
I would greatly appreciate it if people could please take the time to elaborate on this and demonstrate what is being referred to.
If you look at the rule of sarrus for $3\times3$ matrices you find that the determinant is a sum of products of the matrix elements https://en.wikipedia.org/wiki/Rule_of_Sarrus.
The formula $a^ib^jc^k\epsilon_{ijk}$ gives exactly that: It is a sum of products of three matrix elements together with the tensor which is $0,-1$ or $1$.