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Writing tensors in a different way?!

by Physicist
Tags: tensors, writing
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jcsd
#19
Mar14-05, 02:41 PM
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PF Gold
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Yes it's the summation convention your missing physicist, remember that matrices are only (limited) representations of tensors,
Tom Mattson
#20
Mar14-05, 03:21 PM
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Physicist, you're missing a couple of other things, too.

Quote Quote by Physicist
I have a proof to do, starting from the covariant & contravariant field tensors (which are 4 X 4 matrices) & ending with E^2 & B^2.

I couldn't know where did those bold E & B come from? I mean how to transform the calculations from dealing with matrices to the bold symbols?
You need to know that, for any [itex]\mathbb {R} ^3[/itex] vector [itex]\mathbf {A} = A_x \mathbf {i} +A_y \mathbf {j} +A_z \mathbf {k}[/itex], we have:

[tex]\mathbf {A} ^2= \mathbf {A} \cdot \mathbf {A}= A_x^2+A_y^2+A_z^2[/tex]

The other thing you're missing is this issue of matrix multiplication. [itex]F^{\mu \nu }F_{\mu \nu }[/itex] does not mean that you are supposed to multiply the matrix representations of [itex]F[/itex] together. It means that you are supposed to sum over the indices, as I described in my last post. If you were supposed to do matrix multiplication, it would be written as follows:

[tex]
F^{\mu \nu } F_{\nu \lambda }
[/tex]
dextercioby
#21
Mar14-05, 03:24 PM
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Not really,Tom.What u've written is a 4-th rank (2,2) tensor.It doesn't have matrix representation in R^{2}...

Daniel.
jcsd
#22
Mar14-05, 03:31 PM
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No Tom is correcr, but perhaps it's better to treat Matrices as (1,1) tensors, so [itex]F^{\mu}_{\alpha}F^{\alpha}_{\nu} = F^{\mu}_{\nu}[/itex] is the kind of operation that phsyicist is doing.
Physicist
#23
Mar19-05, 04:01 PM
P: 43
Thank you all..

Quote Quote by Tom Mattson
The other thing you're missing is this issue of matrix multiplication. [itex]F^{\mu \nu }F_{\mu \nu }[/itex] does not mean that you are supposed to multiply the matrix representations of [itex]F[/itex] together. It means that you are supposed to sum over the indices, as I described in my last post. If you were supposed to do matrix multiplication, it would be written as follows:

[tex]
F^{\mu \nu } F_{\nu \lambda }
[/tex]
That was the missing point.

Thanks alot


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