About coordinate transformation of tensor.

In summary, the equation (5) is obtained from (4) by rearranging the terms and applying the definitions of the tensors σmn and ωpq.
  • #1
Tensel
7
0
note:The tensors in the new coordinates system are represented by X'.
I have a question about the coordinate transformation of tensor.

σmn=amianjσ'ij; (1)

ωpq=apkaqlω'kl; (2)

In the original coordinates system, we have
σmn = Dmpnqωpq, (3)

Substituting Eqs.(1) and (2) into Eq.(3), one obtains,
amianjσ'ij=Dmpnqapkaqlω'kl (4)

then,
σ'ij=amianjapkaqlDmpnqω'kl (5)

Here is the problem. How was the Eq.(5) obtained from the Eq.(4). I can't really understand, because the items amianjapkaql have changed their position, but their subscript don't change.
Is there any books talking about that?
thanks.
 
Physics news on Phys.org
  • #2


Hello! Thank you for your question. The equation (5) is obtained from (4) by simply rearranging the terms. Let's break it down step by step:

1. Starting with (4), we can see that the term on the left side, amianjσ'ij, is equal to the term on the right side, Dmpnqapkaqlω'kl.

2. Now, we can rearrange the terms on the right side to group the terms with the same subscript together, as follows:

amianjσ'ij=Dmpnq(apkaqlω'kl)

3. We can then use the distributive property to expand the term in parentheses:

amianjσ'ij=Dmpnq(apkω'kl+aqlω'kl)

4. Next, we can use the definition of the tensor ωpq from equation (2) to replace the term apkω'kl with ωpq:

amianjσ'ij=Dmpnq(ωpq+aqlω'kl)

5. Finally, we can use the definition of the tensor σmn from equation (1) to replace the term aqlω'kl with σ'ij:

amianjσ'ij=Dmpnq(ωpq+σ'ij)

6. Therefore, we can see that the left side and the right side of the equation are equal, and we can write it as:

σ'ij=amianjapkaqlDmpnqω'kl

I hope this explanation helps. If you would like to learn more about tensor transformations, I recommend checking out textbooks on tensor analysis or searching for online resources on the topic. Thank you for your interest in this subject!
 

1. What is coordinate transformation of tensor?

Coordinate transformation of tensor is the process of expressing a tensor in terms of different coordinate systems. This allows for the same physical quantity to be described in different ways depending on the chosen coordinate system.

2. Why is coordinate transformation of tensor important in science?

Coordinate transformation of tensor is important in science because it allows for the same physical laws and equations to be applied in different coordinate systems. This enables scientists to study and understand the same phenomenon from different perspectives, leading to a more comprehensive understanding.

3. What is the difference between covariant and contravariant tensors in coordinate transformation?

In coordinate transformation, covariant tensors transform according to the inverse of the transformation matrix, while contravariant tensors transform according to the transformation matrix. This means that the components of covariant tensors change with the change of coordinates, while the components of contravariant tensors remain the same.

4. How does the metric tensor play a role in coordinate transformation of tensors?

The metric tensor, also known as the metric, is a mathematical object that describes the relationship between different coordinate systems. It is used in coordinate transformation to convert between covariant and contravariant tensors, as well as to calculate the length, angle, and inner product of vectors and tensors in different coordinate systems.

5. Can tensors be transformed between any two coordinate systems?

No, tensors can only be transformed between coordinate systems that are related by a smooth, invertible transformation. This means that there must be a one-to-one correspondence between the points in the two coordinate systems and the transformation must be continuous and differentiable. Otherwise, the transformation may result in non-physical or inconsistent results.

Similar threads

  • Differential Geometry
Replies
9
Views
356
Replies
6
Views
2K
  • Differential Geometry
Replies
1
Views
1K
Replies
40
Views
2K
Replies
2
Views
2K
  • Differential Geometry
Replies
11
Views
2K
  • Differential Geometry
Replies
2
Views
841
  • Differential Geometry
Replies
0
Views
597
  • Special and General Relativity
Replies
5
Views
1K
  • Differential Geometry
Replies
1
Views
2K
Back
Top