Kronecker Delta summation (easy)

  • #1
joshmccraney
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Hi PF!

As outlined in my book ##\delta_{ij} \delta_{jk} = \delta_{ik}## but don't we sum over repeated indices (and the ##j## is repeated)? Can someone explain why we do not sum in this situation?

Thanks!
 

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  • #2
blue_leaf77
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Yes it is summed over ##j##. Written explicitly, the sum looks like
$$
\delta_{i1} \delta_{1k} + \delta_{i2} \delta_{2k} + \ldots + \delta_{iN} \delta_{Nk}
$$
You see that if ##i\neq k##, all terms above will vanish. If ##i=k## there will be only one term surviving, therefore that sum can be written compactly as ##\delta_{ik}##.
 
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  • #3
joshmccraney
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Thanks for responding blue_leaf77! Let's assume ##i,j,k## vary from ##1,2,3##. Then we have $$\delta_{11} \delta_{11} + \delta_{22} \delta_{22} + \delta_{33} \delta_{33} = 3 \neq \delta_{ik}$$

Did I miss something?
 
  • #4
PeroK
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Thanks for responding blue_leaf77! Let's assume ##i,j,k## vary from ##1,2,3##. Then we have $$\delta_{11} \delta_{11} + \delta_{22} \delta_{22} + \delta_{33} \delta_{33} = 3 \neq \delta_{ik}$$

Did I miss something?
Why would ##i## and ##k## be summed?

And, if you sum them on the right hand side you would get equality.
 
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  • #5
joshmccraney
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Why would ##i## and ##k## be summed?
That's what blue_leaf wrote.
$$
\delta_{i1} \delta_{1k} + \delta_{i2} \delta_{2k} + \ldots + \delta_{iN} \delta_{Nk}
$$
Since ##i## and ##k## can range from 1,2,3 we would have$$\delta_{11} \delta_{11} + \delta_{21} \delta_{11} + \delta_{31} \delta_{11}+ \delta_{12} \delta_{21}+ \delta_{22} \delta_{21}+ \delta_{32} \delta_{21}+\cdots + \delta_{33} \delta_{33}\\=\delta_{11} \delta_{11}+\delta_{22} \delta_{22}+\delta_{33} \delta_{33}$$ which is why I was summing.
 
  • #6
PeroK
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That's what blue_leaf wrote.


Since ##i## and ##k## can range from 1,2,3 we would have$$\delta_{11} \delta_{11} + \delta_{21} \delta_{11} + \delta_{31} \delta_{11}+ \delta_{12} \delta_{21}+ \delta_{22} \delta_{21}+ \delta_{32} \delta_{21}+\cdots + \delta_{33} \delta_{33}\\=\delta_{11} \delta_{11}+\delta_{22} \delta_{22}+\delta_{33} \delta_{33}$$ which is why I was summing.
blue leaf summed only ##j##, which is a repeated index.
 
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  • #7
joshmccraney
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blue leaf summed only ##j##, which is a repeated index.
Ohhhhh I see now! Shoot, yea I was doing it all wrong! Thank you both!!!
 

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