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I’m trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I’m reading that have me stumped.

1) $\delta_{i\,j}\delta_{i\,j}$

2) $\delta_{i\,j} \epsilon_{i\,j\,k}$

I have no idea how to approach evaluating these properties. Without knowing i, j, or k, how would I approach? I don’t feel confident using the notation further until I can understand these properties.

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$\delta_{ij}=\begin{cases}1 & \text{if } i=j \\ 0 & \text{otherwise}\end{cases}$

https://en.m.wikipedia.org/wiki/Kronecker_delta

$\epsilon_{ijk}=\begin{cases} sgn(ijk) & \text{as a permutation, if } i,j,k \text{ are different} \\ 0 & \text{otherwise}\end{cases}$

https://en.m.wikipedia.org/wiki/Levi-Civita_symbol

And of course repeated indices (up-down and down-up) are to be summed over (https://en.m.wikipedia.org/wiki/Einstein_notation)

Thus for example your second expression is identically zero

Using the answer given by @boonheT it is clear that $$\delta_{ij}\delta_{ij} = \delta_{ij}$$ and that $$\delta_{ij}\epsilon_{ijk} = 0$$

because when the delta is not zero the epsilon will be zero and vice-versa.

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