Specific Question
if $$\epsilon_{ijk} T_{ij} = 0$$
show $$T_{ij} = T_{ji}$$
Reason for asking: The book I have for tensor calculus is very introductory and does not go in depth into the basics of Einstein notation.
What I understand: I understand that I am being asked to prove that the tensor $T_{ij}$ is symmetric but I do not know how to manipulate the equation to do so. I know that ij are the dummy indices and that k is a free index. I also understand that the permutation equals zero if the tensor $T_{ij}$ is symmetric because that would mean that i = j
The general rule with the antisymmetric tensor: if in doubt, multiply by another one and use the $\epsilon \epsilon = \delta\delta-\delta\delta$ identity. More specifically, $$ \epsilon_{ijk} \epsilon_{klm} = \delta_{il} \delta_{jm} - \delta_{im} \delta_{jl} $$ (the way to remember this: cycle the indices on the $\epsilon$s so that the first letter is the same, then the $+$ is on the terms with the indices paired in the same order, the $-$ is on the terms in the opposite order). Applying this operation is generally a good idea: it doesn't lose any information, but can move the antisymmetric tensors to more convenient places.
Applying this to the question at hand, $$ 0 = \epsilon_{klm} \epsilon_{ijk} T_{ij} = (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl}) T_{ij} = T_{lm} - T_{ml} , $$ as required.