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Lecture 14: Einstein Summation Convention . “In any expression containing subscripted variables appearing twice (and only twice) in any term, the subscripted variables are assumed to be summed over.” e.g. Scalar Product will now be written

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lecture 14 einstein summation convention
Lecture 14: Einstein Summation Convention
  • “In any expression containing subscripted variables appearing twice (and only twice) in any term, the subscripted variables are assumed to be summed over.”
  • e.g. Scalar Product
  • will now be written
  • In this lecture we will work in 3D so summation is assumed to be 1 - 3 but can be generalized to N dimensions
  • Note dummy indices do not appear in the ‘answer’. c.f.

Good practice to use Greek letters for these dummy indices

examples
Examples
  • Total Differential

becomes

where

and ‘free index’ runs from 1-3 here (or 1-N in and N-dimensional case)

  • Matrix multiplication

Note the same 2 free indices on each side of the equation

slide3

Trace of a matrix

  • Vector Product

need to define “alternating tensor”

21 cases of zero!

3 cases

3 cases

then (it’s not that hard, honest!)

slide4

There is a “simple” relation between the alternating tensor and the Kronecker delta

  • The proof is simply the evaluation of all 81 cases! (although symmetry arguments can make this easier).
  • If you can get the hang of this, this provides the fastest and most reliable method of proving vector identities
  • Once written down in this form, the order of the terms only matters if they include differential operators (which only act on things to the right- hand-side of them).
example abacab
Example (ABACAB)

rotating indices cyclically doesn’t change sign -th component of

  • Which constitutes a compact proof of the (hopefully familiar) “ABACAB” formula
  • Note we have made much use of the ‘obvious’ identity

ß

further examples
Further Examples
  • curl (grad)

because this term is anti-symmetric (changes sign if  and  are swopped)

but this term symmetric (stays same if  and  are swopped)

so terms ‘cancel in pairs’

  • div (curl)
  • Trace of the unit matrix

as above

final example
Final Example

QED

  • It’s not as hard as it first looks!