Classical Mechanics Puzzler

The definition of how to multiply two 2^nd rank tensors to produce another 2^nd rank tensor is given by

Note that the crucial element in such a definition is that a second rank tensor must have two independent indices, i and k in this case . (Note also that summing over the "inner" index j is not the only definition possible, but it is the definition that works properly for orthogonal rotations.)

With this in mind, consider how you might multiply two 3^rd rank tensors to produce another 3^rd rank tensor. Specifically, consider the useful identity

Does this represent such a multiplication? That is, does d_ild_jm - d_imd_jl represent a tensor of third rank. Be sure to explain your argument. (The actual values of the elements e_ijk and d_lm are not important to the argument.)

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