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?
An affine tensor of rank 4 is a mathematical object that describes the linear transformation between two vector spaces, taking into account both the linear and affine components of the transformation.
The purpose of constructing an affine tensor of rank 4 is to understand and model the complex relationships between multiple vector spaces in a more accurate and efficient manner, especially in fields such as physics and engineering.
An affine tensor of rank 4 is different from other types of tensors because it includes both linear and affine components, while other tensors only describe linear transformations.
Affine tensors of rank 4 have various applications in fields such as computer graphics, robotics, and fluid dynamics, where complex transformations between multiple vector spaces need to be accurately represented and calculated.
Constructing an affine tensor of rank 4 can be challenging due to the high dimensionality and complexity involved in representing and calculating the linear and affine components of the transformation accurately. Additionally, the use of appropriate mathematical tools and techniques is crucial in constructing an affine tensor of rank 4.