I'm going crazy (Electromagnetic Energy-Momentum Tensor)

[LineIntegral]

Homework Statement

The problem is conveniently located here:

http://www.dur.ac.uk/resources/cpt/graduate/lectures/mscps.pdf

Problem no. 31. There's even a solution, here:

http://www.dur.ac.uk/resources/cpt/graduate/lectures/grsolns.pdf

However, I don't understand the solution. How does the antisymmetry of $$F^{\rho\lambda}$$ allow us to rewrite $$\nabla_\rho F_{\sigma\lambda} - \frac{1}{2} \nabla_\sigma F_{\rho\lambda}$$ as $$\frac{1}{2}\left ( \nabla_\rho F_{\sigma\lambda} - \nabla_\lambda F_{\sigma\rho} - \nabla_\sigma F_{\rho\lambda} \right )$$ ? I just don't see it. This implies that $$\nabla_\rho F_{\sigma\lambda} = -\nabla_\lambda F_{\sigma\rho}$$, which doesn't seem right...

Homework Equations

See above.

The Attempt at a Solution

See above.

Any help will be much appreciated!

 

[fzero]

The point is that you have

$$F^{\rho\lambda} \nabla_\rho F_{\sigma\lambda} = \frac{1}{2} (F^{\rho\lambda}-F^{\lambda\rho}) \nabla_\rho F_{\sigma\lambda}.$$

After a relabeling of dummy indices, you find the claimed result.

 

[LineIntegral]

Thanks! Now it finally makes sense... :)