Struggling with MTW Exercises: Any Hints for Exercise 3.18 and 5.1?

[zn52]

hello there,
I have started to tackle the giant book Gravitation by MTW on my own. I would like to see how you people who are having fun in reading this masterpiece faring in solving the exercises. I have read some books on tensor analysis and differential geometry and spacetime geometry as a preparation for MTW. I have also watched the stanford lectures on youtube and downloaded those by Kip Thorne...
Sometimes I spend days to just figure out how to solve one small step of an exercise. It is nice to see that there are no answers, that pushes people to really find out and think by themselves how to find an answer.
I have however not been able to find the answer to some questions. I have reached page 141 without having gone through chapter 4 which I will go back to at chapter 14.

Now my first question is regarding Exercise 3.18 point b) . Please DO NOT answer but provide some hints as to how to manage to get the derivative with respect to a third variable in order to get the last three terms in the equation 3.62 to be proven (the Maxwell's equation).

My second question is regarding exercise 5.1 on page 141. How come the tension and the pressure have the same value ? The field lines here are they the field lines of the electromagnetic force ?

Please note that I'm a graduate student in Physics specialized in Astrophysics (2007).
Thank you very much,
Cheers,
Ziad.

 

[Daverz]

You can replace the dummy indices in the first term in 3.61 in two ways.

 

[zn52]

thank you for your answer.
Yes I have tried to do that but to no avail. The problem is that the derivative in the first term is with respect to one index only (Nu). Would it be possible to have another derivative with respect to another index , i.e. alpha ?
In summary : how to extract from the first term two terms with derivatives with respect to different indices ? I have tried to exploit the symmetry property of the F in order to get the Half but to no avail either...

 

[Daverz]

Nu and alpha are both dummy index in that term, so you can replace them both.

 

[zn52]

I thank you so much for your assistance. after several weeks deep into tensors I have now realized that these are dummy indices. It is not obvious for anyone at the beginning to spot that...Now I can sleep happily at night...
Cheers,