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.
$\endgroup$ 12 Answers
$\begingroup$$\delta_{ij}=\begin{cases}1 & \text{if } i=j \\ 0 & \text{otherwise}\end{cases}$
$\epsilon_{ijk}=\begin{cases} sgn(ijk) & \text{as a permutation, if } i,j,k \text{ are different} \\ 0 & \text{otherwise}\end{cases}$
And of course repeated indices (up-down and down-up) are to be summed over ()
Thus for example your second expression is identically zero
$\endgroup$ $\begingroup$Using the answer given by @boonheT it is clear that $$\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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