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## Writing tensors in a different way?!

Physicist,

What dexter is trying to lead you to is the following:

$$F^{ \mu \nu } F_{ \mu \nu } = \sum _ { \mu = 1} ^ {4} \sum _ { \nu = 1} ^ {4} F^{\mu \nu} F_{\mu \nu}$$

That is the Einstein summation convention. So, you let the indices $\mu$ and $\nu$ each run from 1 to 4 in the double sum, and you should get your answer straightforwardly.
 Recognitions: Gold Member Science Advisor Yes it's the summation convention your missing physicist, remember that matrices are only (limited) representations of tensors,

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Physicist, you're missing a couple of other things, too.

 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 $\mathbb {R} ^3$ vector $\mathbf {A} = A_x \mathbf {i} +A_y \mathbf {j} +A_z \mathbf {k}$, we have:

$$\mathbf {A} ^2= \mathbf {A} \cdot \mathbf {A}= A_x^2+A_y^2+A_z^2$$

The other thing you're missing is this issue of matrix multiplication. $F^{\mu \nu }F_{\mu \nu }$ does not mean that you are supposed to multiply the matrix representations of $F$ 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:

$$F^{\mu \nu } F_{\nu \lambda }$$
 Blog Entries: 9 Recognitions: Homework Help Science Advisor 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.
 Recognitions: Gold Member Science Advisor No Tom is correcr, but perhaps it's better to treat Matrices as (1,1) tensors, so $F^{\mu}_{\alpha}F^{\alpha}_{\nu} = F^{\mu}_{\nu}$ is the kind of operation that phsyicist is doing.

Thank you all..

 Quote by Tom Mattson The other thing you're missing is this issue of matrix multiplication. $F^{\mu \nu }F_{\mu \nu }$ does not mean that you are supposed to multiply the matrix representations of $F$ 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: $$F^{\mu \nu } F_{\nu \lambda }$$
That was the missing point.

Thanks alot

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