# Difficulty on tensors

1. Aug 2, 2011

### grzz

Is gαβAβ the same as Aβgαβ ?
Thanks for any help.

2. Aug 2, 2011

### HallsofIvy

Staff Emeritus
Yes.

3. Aug 2, 2011

### grzz

But then is
BαβAγ equal to Aγ Bαβ ?

4. Aug 2, 2011

### grzz

I think that the tensors A and B do not commute as the g and A do in the previous example. But I am not sure.
Any help!

5. Aug 3, 2011

### spyphy

Bαβ is not tensor, it is the component of a tensor. The components of a tensor are real or complex numbers. They commute.

6. Aug 3, 2011

### jambaugh

As spyphy says this is just multiplication of numbers (components) so order doesn't matter. The full tensors must be formed by contracting the indices with basis elements. It is there where you see the distinctions in order written:
$\mathbf{B}\otimes\mathbf{A}= B_{\alpha\beta} A^y \mathbf{e}^\alpha\otimes\mathbf{e}^\beta\otimes \mathbf{e}_y =A^y B_{\alpha\beta} \mathbf{e}^\alpha\otimes\mathbf{e}^\beta\otimes \mathbf{e}_y$
but note that:
$\mathbf{B}\otimes\mathbf{A}= B_{\alpha\beta} A^y \mathbf{e}^\alpha\otimes\mathbf{e}^\beta\otimes \mathbf{e}_y \ne B_{\alpha\beta}A^y \mathbf{e}_y\otimes\mathbf{e}^\alpha\otimes\mathbf{e}^\beta = \mathbf{A}\otimes\mathbf{B}$
take your time parsing these and see the distinction.

7. Aug 3, 2011

### grzz

Thanks for the help.
Since $\alpha$ is repeated in g$_{}\beta_{}\alpha$A$^{}\alpha$ then it was clear to me that this is a sum and the g$_{}\beta_{}\alpha$ and the A$^{}\alpha$ are numbers and so commute.

But I thought that A$_{}\beta_{}\alpha$B$^{}\gamma$ represented the product of two tensors. From the little I know I thought that sometimes a tensor is represented by one of its components. That is why I said that the second example may not commute.

8. Aug 3, 2011

### grzz

I am also poor in using latex!

9. Aug 3, 2011

### Fredrik

Staff Emeritus
$A_{\beta\alpha}B^\gamma$ is equal to both the ${}_{\beta\alpha}{}^\gamma$ component of the tensor $A\otimes B$, and the ${}^\gamma{}_{\beta\alpha}$ component of the tensor $B\otimes A$.

Click the quote button if you want to see how I did the LaTeX. Try changing something and use the preview feature to see what it looks like. (To be able to preview, you need to trick the forum software into thinking that you're typing a reply, e.g. by typing at least 4 characters after the quote tags).

Last edited: Aug 3, 2011
10. Aug 4, 2011

### grzz

Thank you because in those last four lines you gave me the best tutorial about LaTeX.

11. Aug 4, 2011