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I (Index Notation) Summing a product of 3 numbers

  1. Sep 29, 2016 #1
    I have just begun reading about Einstein's summation convention and it got me thinking..
    Is it possible to represent ∑aibici with index notation? Since we are only restricted to use an index twice at most I don't think it's possible to construct it using the standard tensors (Levi Cevita and Kronecker Delta). Levi Cevita doesn't work because it's only non-zero when the indices are all different and Kronecker Delta only connects two tensors, leaving the third one behind. It becomes clearer if I think of them in terms of vectors and matrices.
     
    Last edited: Sep 29, 2016
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  3. Sep 29, 2016 #2

    stevendaryl

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    Well, you can introduce a matrix [itex]A_{ijk}[/itex] such that [itex]A_{iii} = 1[/itex] and [itex]A_{ijk} = 0[/itex] if any of the indices differ. Then [itex]\sum_i a_i b_i c_i = A_{ijk} a^i b^j c^k[/itex]. The matrix [itex]A_{ijk}[/itex] might not be a very interesting matrix, mathematically.
     
  4. Sep 29, 2016 #3
    That's exactly what I thought. I suppose this ad hoc matrix is not common because this type of summation rarely appears in Physics
     
  5. Sep 29, 2016 #4

    stevendaryl

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    Well, the most common use of the Einstein summation convention is for manipulations of vectors and tensors. For those uses, one-dimensional matrices such as [itex]a^i[/itex], written with a raised index, corresponds to a vector, and [itex]b_j[/itex], with a lowered index, corresponds to a covector (there is a geometric distinction between vectors and covectors, even though people often ignore the distinction when using Cartesian coordinates for the components). Something with more than one index is a tensor. For example, [itex]g_{ij}[/itex] is the metric tensor, which is used to compute the length of a vector:

    [itex]|\vec{V}| = \sqrt{g_{ij} V^i V^j}[/itex]

    But not every matrix of numbers corresponds to a tensor. The reason why is because tensors transform when you change coordinate systems (for example, changing from Cartesian to polar coordinates). If you define [itex]A_{ijk}[/itex] so that in any coordinate system, [itex]A_{iii} =1[/itex] and [itex]A_{ijk} = 0[/itex] when any two indices are different, then [itex]A[/itex] will not be a tensor.
     
  6. Sep 29, 2016 #5

    Orodruin

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    I would not call ##A_{ijk}## a matrix. Something represented by a matrix generally has two indices (or one in the case of row or column matrices). This would be some sort of multidimensional matrix.

    Seen as a tensor, this object would not be an isotropic tensor, i.e., it would have different components in another frame, unlike the Kronecker delta or the permutation symbol (seen as a Cartesian pseudo-tensor). In the same fashion ##\sum_i a^i b^i c^i## is not an invariant under coordinate transformations.
     
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