Question about the derivation of the energy momentum tensor

[AwesomeTrains]

Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf

Homework Statement

As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. $L=-1/4 F^{μν} F_{μν}$ where $F_{μν}=∂_μA_ν-∂_νA_μ$
I'm stuck at equation (11): $\frac{δL}{δ(∂_μA_λ)}=-F^{μν}$.
Could someone help me understand how they get this result?

Homework Equations

Some tensor product rule maybe?

The Attempt at a Solution

I would get the result if I use the usual product rule. But how do I treat the indices?
Any tips or redirects to where I can read about it is very appreciated :)

Kind regards
Alex

 

[MarcusAgrippa]

Hey I'm trying to follow the derivation given here: http://lampx.tugraz.at/~hadley/ss1/studentpresentations/Bloch08.pdf

Homework Statement

As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. $L=-1/4 F^{μν} F_{μν}$ where $F_{μν}=∂_μA_ν-∂_νA_μ$
I'm stuck at equation (11): $\frac{δL}{δ(∂_μA_λ)}=-F^{μν}$.
Could someone help me understand how they get this result?

Homework Equations

Some tensor product rule maybe?

The Attempt at a Solution

I would get the result if I use the usual product rule. But how do I treat the indices?
Any tips or redirects to where I can read about it is very appreciated :)

Kind regards
Alex

Write the Lagrangian density in terms of the derivatives of the vector potential.
Then use the fact that
$\frac{\partial (\partial_\mu A_\nu)}{\partial (\partial_\sigma A_\tau)} = \delta^\mu_\sigma \delta^\nu_\tau$