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Let’s consider this equation for a scalar quantity $f$ as a function of a 3D vector $a$ as: $$ f(\vec a) = S_{ijkk} a_i a_j $$ where $S$ is a tensor of rank 4. Now, I’m not sure what to make of the index $k$ in the expression, as it doesn’t appear on the left-hand side. Is it a typo, meaning there is a $k$ missing somewhere (like $f_k$), or does it mean that it should be summed over $k$ like so: $$f(\vec a) = \sum_i \sum_j \sum_k S_{ijkk} a_i a_j $$ |
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In the Einstein convention, pairs of equal indices to be summed over may appear at the same tensor. For example, the formula ${A_k}^k=tr~A$ is perfectly legitimate. But your formula looks strange, as one usually sums over a lower index and an upper index, whereas you sum over lower indices only, which doesn't make sense in differential geometry unless your metric is flat and Euclidean (and then higher order tensors are very unlikely to occur). |
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You could rewrite your equation as $$ f(\vec a) = S_{ijkl} a_i a_j \delta_{kl} $$ where $\delta_{kl}$ is the Kroneker Delta, if that helps. The last equation you've written is the right idea. I would stress, though, that Einstein notation usually uses one upper and one lower index. This is partially so you can quickly see if your summations and indices are correct. |
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