Understanding Tensors: Comparing gαβAβ and Aβgαβ

[grzz]

I am learning about tensors.
Is g_αβA^β the same as A^βg_αβ ?
Thanks for any help.

 

[HallsofIvy]

Yes.

 

[grzz]

But then is
B_αβA^γ equal to A^γ B_αβ ?

 

[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!

 

[spyphy]

But then is
B_αβA^γ equal to A^γ B_αβ ?

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

 

[jambaugh]

But then is
B_αβA^γ equal to A^γ B_αβ ?

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.

 

[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.

 

[grzz]

I am also poor in using latex!

 

[Fredrik]

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.

$A_{\beta\alpha}B^\gamma$ 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).

 

[grzz]

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

 

[Rasalhague]

You can try things out first an online equation editor, such as this one, then copy and paste:

http://www.codecogs.com/latex/eqneditor.php