Help w/ Tensor Calc Homework: Bijk Properties Under Rotations

The idea is that you can make arbitrary coordinate transformations, and you can always find a coordinate transformation that will bring the index to an arbitrary point, so you can always choose it so that the object transforms as a tensor.
  • #1
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Homework Statement


if BijkAjk is a vector for all symmetric tensors Ajk, (but Bijk is not necessarily a tensor),
what are the properties of Bijk under rotations of the basis/coordinate axes?

Homework Equations


The Attempt at a Solution


I am not sure what the question is looking for... though I can say that
BijkAjk=BijkAkj(by symmetry of Ajk)=BikjAjk(by relabeling dummy suffices)
Since Ajk is an arbitrary symmetric matrix, they cancel, giving Bijk=Bikj
So Bijk is symmetric wrt the last 2 suffices...

Thanks in advance!
 
Last edited:
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  • #2
No, B_{ijk} is not symmetric on the last 2 indices, but only the symmetric part survives. That you have showed right.
Now, we can write B_{ijk} = B1_{ijk} + B2_{ijk} where B1 is symmetric on the last 2 indices and B2 is antisymmetric on the last 2 indices. Then,
B_{ijk} A^{jk} = B1_{ijk} A^{jk} (because the antisymmetric part doesn't survive), and your problem said that this product transforms as a vector, so B1_{ijk} transforms on its first index as a vector. Now you can apply the reduction theorem (don't know if it's called this way) to conclude that B1_{ijk} transforms as a tensor under coordinate transformations, since its contraction with any tensor transforms as a vector. About the transformation properties of B2 you can not say anything.
 
  • #3
Thanks, grey_earl.
 
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  • #4
The reduction theorem basically says that if you contract an object with unknown transformation properties with an arbitrary tensor (which means multiplying and summing over equal indices, as you did), and this transforms as a scalar (or if you have one index free, as a vector, etc.), then your original object also transforms as a tensor. I think Schouten's tensor calculus has a proof of this.
 

1. What are Bijk properties?

Bijk properties refer to the properties of the Bijk tensor, which is a third-order tensor that represents the transformation of a vector under rotations. It is commonly used in tensor calculus to describe the change in vectors and tensors under rotations.

2. How are Bijk properties related to rotations?

Bijk properties are specifically related to rotations because they describe the transformation of a vector under rotations. This means that by understanding the Bijk properties, we can better understand how rotations affect vectors and tensors.

3. What is the significance of Bijk properties in tensor calculus?

Bijk properties are significant in tensor calculus because they allow us to characterize the behavior of tensors under rotations. This is important in many fields of science and engineering, such as mechanics, electromagnetics, and fluid dynamics.

4. How are Bijk properties calculated?

Bijk properties can be calculated using the Bijk tensor itself, along with the rotation matrix that describes the rotation being performed. The Bijk tensor is multiplied by the rotation matrix, and the resulting tensor is then compared to the original tensor to determine the Bijk properties.

5. Can Bijk properties be used for other types of transformations besides rotations?

While Bijk properties are primarily used for rotations, they can also be extended to other types of transformations, such as translations and reflections. However, in these cases, the properties may be represented by different tensors, depending on the specific transformation being performed.

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