Transpose of a Tensor Identity

In summary, the identity a\cdotTb = b\cdotTTa is supposed to hold true for all tensors in continuum mechanics. However, when working it out, the same result is not obtained for both sides. The order of computing the dot product does not affect the outcome, as vector algebra is commutative. The original identity is correct and commonly used in math proofs.
  • #1
QuickLoris
12
0
My textbook (regarding continuum mechanics) has the following identity that is supposed to be true for all tensors:

a[itex]\cdot[/itex]Tb = b[itex]\cdot[/itex]TTa

But I don't get the same result for both sides when I work it out.
For each side, I'm doing the dot product last. For example, I compute Tb first and then computer the dot product of a[itex]\cdot[/itex]Tb. Is that right? I tried doing it the other way around also, but it didn't work out that way either.

I'm still pretty new to this subject and teaching it to myself, so I figure I'm multiplying something incorrectly, but I don't understand what.
 
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  • #2
QuickLoris said:
My textbook (regarding continuum mechanics) has the following identity that is supposed to be true for all tensors:

a[itex]\cdot[/itex]Tb = b[itex]\cdot[/itex]TTa

But I don't get the same result for both sides when I work it out.


I don't know what you might be doing wrong, but vector algebra is commutative, as you say.

QuickLoris said:
For each side, I'm doing the dot product last. For example, I compute Tb first and then computer the dot product of a[itex]\cdot[/itex]Tb. Is that right?

The original identity is definitely correct, as it is a common one used as a starting point for other math proofs.
 

1. What is the definition of transpose of a tensor identity?

The transpose of a tensor identity is a mathematical operation that involves changing the rows and columns of a tensor. It is denoted by placing a superscript "T" next to the tensor, and it results in a new tensor with the rows and columns swapped.

2. How is the transpose of a tensor identity calculated?

The transpose of a tensor identity is calculated by reversing the order of the indices in the tensor. For example, if a tensor A has indices (i,j,k), the transpose of A (AT) would have indices (k,j,i).

3. What are the properties of transpose of a tensor identity?

Some of the properties of transpose of a tensor identity include: (1) (AT)T = A, (2) (A+B)T = AT + BT, and (3) (AB)T = BT AT, where A and B are tensors. These properties are similar to the properties of transpose in matrix algebra.

4. What is the significance of transpose of a tensor identity in tensor calculus?

The transpose of a tensor identity is essential in tensor calculus as it allows for the transformation of tensors between different coordinate systems. It also plays a crucial role in simplifying and solving equations involving tensors.

5. Can the transpose of a tensor identity be applied to tensors of any rank?

Yes, the transpose of a tensor identity can be applied to tensors of any rank. It is a fundamental operation in the manipulation of tensors and is applicable to tensors of any order or dimension.

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