# Construction of an affine tensor of rank 4

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1. Jan 25, 2016

### Whitehole

1. The problem statement, all variables and given/known data
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.
2. Relevant equations

3. The attempt at a solution
Can anyone give me a hint on how to start? I just don't know how to start.

2. Jan 25, 2016

### Samy_A

3. Jan 26, 2016

### Whitehole

4. Jan 26, 2016

### Samy_A

The tensor mentioned in the first post of that thread was the hint, more specifically $\delta_{ab}\delta_{cd}$.

5. Jan 26, 2016

### Whitehole

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. Jan 26, 2016

### Samy_A

Yes, that's what I meant.

7. Jan 26, 2016

### Whitehole

Oh, thanks! But what does that relation mean???

8. Jan 26, 2016

### Samy_A

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. Jan 26, 2016

### Whitehole

Yes this is an exercise in Tensors, Differential Forms, and Variational Principles by Lovelock and Rund. Problem 2.7. Thank you very much!!!