Here is exercise 19.2

On page 233 an expression for the exterior derivative operator is provided in coordinate terms, and involves antisymmetrization brackets. The expression for antisymmetrization brackets containing 3 terms is given explicitly on page 214. Combining these and applying them to dF (using the shorthand notation ) should yield:





The expression for d*F is similar.


The above factor of 1/3 seems to have been left out of Robin's solution.
In the derived equations, all instances of "" should be changed to "" to correct this.


(Note that there are 6 terms in the expression on p.214, but the negative ones are just doubles of the positive ones due to F being antisymmetric, i.e. ...hence the factor of 3 here, rather than 6.)

Indeed the derivation by Robin apparently doesn't manage properly the anti-symmetrisation factor; however the final result is correct: the factor is not .
This because the anti-symmetrisation factor of exterior derivative simplifies with the factor generated by the contraction of F with Levi-Civita tensor in the Hodge star operator.
This is made explicit in my solution to exercise [19.3], in which these factors are worked out in detail; for completeness, I attach also my solution to [19.2], that I didn't post before because too much similar to Robin's one and because reuses the solution to [19.3].

Roberto, when I did this problem, I didn't use the Levi-Civita tensor in the Hodge star operator. Just like Robin, I simply applied the exterior derivative ("d") operator to the *F tensor which is explicitly given (in matrix form) on p.443.

Doing so introduces a factor of 1/3 on the LHS (which Robin left out), which doesn't cancel with anything: I end up with on the RHS.

A quick check confirms that the explicit components of *F given on p.443 agree with the formula (also on that page):




...so I don't know where you get an additional factor of 3 from...?

Yes, OK.
I realised that also in ex. 19.03, on which I relied to solve the 19.02, there is the same problem, and my solution to it contained a factor 3 error.

I reposted a corrected version of my solution to 19.03.

I found the source of the problem.

The "" in the derived maxwell equations is correct, it's the initial expression that's missing a factor of 1/3.

On page 445, the second maxwell equation should read:





Note the fraction on the RHS - it's missing in the text, which is the error.

This is the same expression as in the caption to Fig. 19.1 (last line), though that's expressed in coordinate terms.

Yes, I agree.

Using 4/3 in would lead to correct Maxwell equations both in ex. 19.02 and in ex. 19.03.