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Construction of an affine tensor of rank 4

  • #1
132
4

Homework Statement


In En the quantities Bij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components Cijkl and Dijkl for which

kl Cijkl Bkl = Bij + Bji

kl Dijkl Bkl = Bij - Bji

are identities.

Homework Equations



The Attempt at a Solution


Can anyone give me a hint on how to start? I just don't know how to start.
 

Answers and Replies

  • #2
Samy_A
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Homework Statement


In En the quantities Bij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components Cijkl and Dijkl for which

kl Cijkl Bkl = Bij + Bji

kl Dijkl Bkl = Bij - Bji

are identities.

Homework Equations



The Attempt at a Solution


Can anyone give me a hint on how to start? I just don't know how to start.
Maybe the first post of this old thread can help you on the way: https://www.physicsforums.com/threads/isotropic-tensors.106292/
 
  • #3
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4
  • #4
Samy_A
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The link in the first post in that thread is broken. Can you help me by just giving a hint?
There is no link in the first post of that thread.
The tensor mentioned in the first post of that thread was the hint, more specifically ##\delta_{ab}\delta_{cd}##.
 
  • #5
132
4
There is no link in the first post of that thread.
The tensor mentioned in the first post of that thread was the hint, more specifically ##\delta_{ab}\delta_{cd}##.
This is what I did,

Let Cijkl = δik δjl + δil δjk

Multiply both sides by Bkl then take the sum

kl Cijkl Bkl = ∑klik δjl Bkl + δil δjk Bkl)

Then from the dirac delta in the first term k=i and l=j, as for the second term l=i and k=j

Thus ∑kl Cijkl Bkl = Bij + Bji

The same process goes for the second question. Is this correct?
 
  • #6
Samy_A
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This is what I did,

Let Cijkl = δik δjl + δil δjk

Multiply both sides by Bkl then take the sum

kl Cijkl Bkl = ∑klik δjl Bkl + δil δjk Bkl)

Then from the dirac delta in the first term k=i and l=j, as for the second term l=i and k=j

Thus ∑kl Cijkl Bkl = Bij + Bji

The same process goes for the second question. Is this correct?
Yes, that's what I meant.
 
  • #7
132
4
Yes, that's what I meant.
Oh, thanks! But what does that relation mean???
 
  • #8
Samy_A
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Oh, thanks! But what does that relation mean???
I don't know. Contraction of a rank 2 tensor with C gives a symmetric rank 2 tensor. Contraction of a rank 2 tensor with D gives an antisymmetric rank 2 tensor.
Whether is means something, and if so, what it means, I don't know.

Was this an exercise, or is this used somewhere in a theoretical context?
 
  • #9
132
4
I don't know. Contraction of a rank 2 tensor with C gives a symmetric rank 2 tensor. Contraction of a rank 2 tensor with D gives an antisymmetric rank 2 tensor.
Whether is means something, and if so, what it means, I don't know.

Was this an exercise, or is this used somewhere in a theoretical context?
Yes this is an exercise in Tensors, Differential Forms, and Variational Principles by Lovelock and Rund. Problem 2.7. Thank you very much!!!
 

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