|
|
Archived: 07 Aug 2014, 10:06
|
Author |
Message |
robin
Joined: 26 Mar 2010, 04:39 Posts: 109
|
 Exercise [19.02]
Here is exercise 19.2
Attachments:
File comment: Exercise 19.2
Exercise19.2.pdf [27.75 KiB]
Downloaded 169 times
|
21 Jun 2010, 03:37 |
|
 |
deant
Joined: 12 Jul 2010, 07:44 Posts: 154
|
 Re: Exercise [19.02]
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 d F (using the shorthand notation  ) should yield: ![\mathrm{d}F \equiv \partial_{[a}F_{bc]}\\ \\<br />= \frac{1}{3}\left(\partial_{a}F_{bc} + \partial_{b}F_{ca} + \partial_{c}F_{ab}\right)](latexrender/pictures/9291608e3df0b388ded4da1d717eb199.png) 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.)
|
12 Feb 2012, 18:50 |
|
 |
Roberto
Joined: 03 Jun 2010, 15:18 Posts: 136
|
 Re: Exercise [19.02]
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].
|
13 Feb 2012, 15:55 |
|
 |
deant
Joined: 12 Jul 2010, 07:44 Posts: 154
|
 Re: Exercise [19.02]
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...?
|
13 Feb 2012, 19:45 |
|
 |
Roberto
Joined: 03 Jun 2010, 15:18 Posts: 136
|
 Re: Exercise [19.02]
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.
|
17 Feb 2012, 18:30 |
|
 |
deant
Joined: 12 Jul 2010, 07:44 Posts: 154
|
 Re: Exercise [19.02]
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.
|
26 Feb 2012, 10:47 |
|
 |
Roberto
Joined: 03 Jun 2010, 15:18 Posts: 136
|
 Re: Exercise [19.02]
Yes, I agree. Using 4/3 in  would lead to correct Maxwell equations both in ex. 19.02 and in ex. 19.03.
|
26 Feb 2012, 18:38 |
|
 |
|
|