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Is this a type-o or intentional?

  • Thread starter tolove
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  • #1
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I've been staring at this confused for a while now, and I've just realized that this might be a type-o. Should I assume that the Levi-Civita symbol is only defined on j,k, and that the i is a type-o, or is there an unwritten rule with this notation that gives the i a meaning?

[itex]\sum[/itex]j[itex]\sum[/itex]k εi,j,kj,k
 

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  • #2
D H
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Nominally that kind of sum would represent ith component of a vector. However, since δjk is zero if j≠k and εijk is zero if j=k, this is a complicated way of writing the zero vector.
 
  • #3
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Minor pedantic point: The word is "typo", which is short for typographical error.
 
  • #4
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Nominally that kind of sum would represent ith component of a vector. However, since δjk is zero if j≠k and εijk is zero if j=k, this is a complicated way of writing the zero vector.
Thank you very much! The i isn't being defined because it doesn't matter what i is in this situation.

And to mark... I didn't know that, thanks
 
  • #5
Ray Vickson
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I've been staring at this confused for a while now, and I've just realized that this might be a type-o. Should I assume that the Levi-Civita symbol is only defined on j,k, and that the i is a type-o, or is there an unwritten rule with this notation that gives the i a meaning?

[itex]\sum[/itex]j[itex]\sum[/itex]k εi,j,kj,k
No typo: the standard ε-symbol, used, eg., in writing vector cross-products in 3 dimensions, is:
[tex] \epsilon_{i j k} = \left\{ \begin{array}{rccl}
1 &\text{ if }& ijk &\text{ is an even permutation of 123}\\
-1&\text{ if }&ijk & \text{ is an odd permutation of 123}\\
0 &&&\text{ otherwise }
\end{array} \right. [/tex]

So, for example, the ith component of ##\vec{C} = \vec{A} \times \vec{B}## is ##C_i = \sum_{j,k} \epsilon_{ijk} A_j B_k##.

BTW: 'type-o' is a category of blood (for blood donations); what you probably mean is 'typo'.
 

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