|
|
Post by Admin on Jul 20, 2018 10:30:57 GMT
In section II of chapter 10 on page 461 in the paragraph which begins
"Perhaps using (3), verify that ... the LHS of (3) equals 1."
there is a reference to result (3) which seems to be incorrect and should be a reference to the result immediately under figure 9 on page 461, which Needham calls the Index Theorem.
Vasco/Admin
|
|
|
|
Post by telemeter on Mar 19, 2020 16:23:38 GMT
I think the text is fine. I read the text as intending to verify the Index theorem by independently looking at the LHS and RHS. Looking at the LHS Needham suggests applying (3) to each of C , B1 and B2 separately. Doing this one indeed finds I[C] = 2 I[B1}=0, I[B2}=1 as per the text. Thus you have shown that LHS =1. etc etc.
telemeter
|
|
|
|
Post by Admin on Mar 21, 2020 9:28:29 GMT
telemeter
On page 461 Needham suggests using Poincare's method (3) three times to verify that in figure 9
1) $\mathscr{I}[C]=2$
2) $\mathscr{I}[B_1]=0$
3) $\mathscr{I}[B_2]=1$
The LHS of (3) is equal to 2 in the first case, 0 in the second case and 1 in the third case.
The statement of the Index Theorem under figure 9 on page 461 can then be verified by (on the LHS) substituting 2 for $\mathscr{I}[C]$, 0 for the first term $\mathscr{I}[B_1]$ in the summation on the LHS, and 1 for the second term in the summation
$\displaystyle\sum^{j=g=2}_{j=1}\mathscr{I}[B_j]=\mathscr{I}[B_1]+\mathscr{I}[B_2]=0+1=1$
so that the LHS of the Index Theorem equals 1 not equation (3).
Then the RHS of the Index Theorem is
$\displaystyle\sum^{n=2}_{j=1}\mathscr{I}[s_j]=\mathscr{I}[s_1]+\mathscr{I}[s_2]= \mathscr{I}(\text{dipole})+\mathscr{I}(\text{saddle point})=2+(-1)=1$
and so this verifies the Index Theorem.
So I still think that on page 461 Needham meant to reference the LHS of the Index Theorem not (3).
Vasco
|
|
|
|
Post by telemeter on Mar 23, 2020 12:29:54 GMT
O I see what you mean. Entirely agree. It's the second "(3)" you're referring to. Quite so.
telemeter
|
|
