Is A_{1}B_{i} + A_{2}B_{j} + A_{3}B_{k} a Tensor?

[typhoonss821]

Homework Statement

Let $$A_{i} , B_{j} , C_{k}$$be vectors.
Is $$A_{1} B_{i} + A_{2} B_{j} + A_{3} B_{k}$$ a tensor?

Homework Equations

$$T'_{ijk} = R_{il} R_{jm} R_{kn} T_{ijk}$$

The Attempt at a Solution

If T is a tensor of rank 3, then $$T'_{ijk} = R_{il} R_{jm} R_{kn} T_{ijk}$$ where $$R^\intercal R =I$$. But I am not sure how to use this relation in the problem. > <

I think it is to use the property of vector $$A'_{i} = R_{il} A_{i}$$, but difficult appears when I try this way.

Please help...

 

[nate808]

Hello,

Thank you for your question. To determine if A_{1} B_{i} + A_{2} B_{j} + A_{3} B_{k} is a tensor, we need to understand the properties of tensors and how they behave under transformations.

Firstly, a tensor is a mathematical object that describes a geometric relationship between vectors and scalars. It is characterized by its rank, which is the number of indices it has. In this case, we are dealing with a tensor of rank 2 because we have two indices (i and j).

Now, to determine if A_{1} B_{i} + A_{2} B_{j} + A_{3} B_{k} is a tensor, we need to check if it follows the transformation rule for tensors. As you have mentioned, the transformation rule for a rank 2 tensor is T'_{ij} = R_{il} R_{jm} T_{lm}.

In this case, we have A'_{i} = R_{il} A_{l} and B'_{j} = R_{jm} B_{m}. Therefore, A'_{i} B'_{j} = R_{il} A_{l} R_{jm} B_{m}.

Substituting this into our original expression, we get T'_{ij} = A'_{i} B'_{j} = R_{il} A_{l} R_{jm} B_{m} = R_{il} R_{jm} A_{l} B_{m}.

As you can see, this follows the transformation rule for a rank 2 tensor, therefore A_{1} B_{i} + A_{2} B_{j} + A_{3} B_{k} is a tensor.

I hope this helps to answer your question. Let me know if you have any further questions.