The discussion revolves around constructing affine tensors of rank 4 based on given components of an affine tensor of rank 2. Participants are tasked with creating two tensors, Cijkl and Dijkl, that satisfy specific identities involving the components Bij.
Some participants have provided hints and partial constructions, while others express uncertainty about the meaning and application of the relations derived. There is an acknowledgment of the exercise's source, indicating a structured context for the problem.
Participants note the lack of clarity regarding the theoretical implications of the constructed tensors and whether the problem is purely academic or has practical applications.
Maybe the first post of this old thread can help you on the way: https://www.physicsforums.com/threads/isotropic-tensors.106292/Whitehole said: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
∑k ∑l Cijkl Bkl = Bij + Bji
∑k ∑l 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.
The link in the first post in that thread is broken. Can you help me by just giving a hint?Samy_A said:Maybe the first post of this old thread can help you on the way: https://www.physicsforums.com/threads/isotropic-tensors.106292/
There is no link in the first post of that thread.Whitehole said:The link in the first post in that thread is broken. Can you help me by just giving a hint?
This is what I did,Samy_A said: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}##.
Yes, that's what I meant.Whitehole said:This is what I did,
Let Cijkl = δik δjl + δil δjk
Multiply both sides by Bkl then take the sum
∑k∑l Cijkl Bkl = ∑k∑l (δik 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 ∑k∑l Cijkl Bkl = Bij + Bji
The same process goes for the second question. Is this correct?
Oh, thanks! But what does that relation mean?Samy_A said:Yes, that's what I meant.
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.Whitehole said:Oh, thanks! But what does that relation mean?
Yes this is an exercise in Tensors, Differential Forms, and Variational Principles by Lovelock and Rund. Problem 2.7. Thank you very much!Samy_A said: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?