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## Homework Statement

Find F

_{μν}, given F

^{μν}=

(0 E

_{x}E

_{y}E

_{z})

(-E

_{x}0 B

_{z}-B

_{y})

(-E

_{y}-B

_{z}0 -B

_{x})

(-E

_{z}B

_{y}-B

_{x}0)

## Homework Equations

g

_{μν}= Diag( -1, 1, 1, 1)

## The Attempt at a Solution

I tried computing F

_{μν}= g

_{μν}F

^{μν}

but this gave

(0 -E

_{x}-E

_{y}-E

_{z})

(-E

_{x}0 B

_{z}-B

_{y})

(-E

_{y}-B

_{z}0 B

_{x})

(-E

_{z}B

_{y}-B

_{x}0)

Which seems wrong to me, since the next question is to find F

_{μν}F

^{νμ}and this is meant to come out as the Lagrangian and from my notes I don't see this working.

I think I'm messing up my index notation, I think I may need an extra metric in there because there are 2 indices downstairs but I don't have the best understanding.

Any help and advice on dealing with pesky index notation would be greatly appreciated, Thanks!

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