Editing Jaycall/KillingVectorApproach (section)

==Logarithmic Derivatives of T3 Scale Factors==

Check out my new subsection (under T3 Coordinates) entitled [[User:Tohline/Appendix/Ramblings/T3Integrals#Logarithmic_Derivatives_of_Scale_Factors|Logarithmic Derivatives of Scale Factors]].  First, see if you agree that there is a mistake (typo) in one of your tables of partial derivatives.  Specifically, I think that the correct expression is:
<div align="center">
<math>
\frac{\partial z}{\partial\lambda_2} = - (q^2-1)\frac{\varpi^2 z \ell^2}{\lambda_2} .
</math>
</div>
Second, see if you agree with my derived expressions for <math>\partial\ln h_i/\partial\ln\lambda_j</math>. --[[User:Tohline|Tohline]] 22:28, 31 May 2010 (MDT)

:You're certainly right about the sign error.  I have corrected the expression in the relevant table.  --[[User:Jaycall|Jaycall]] 14:28, 1 June 2010 (MDT)

:I have not yet been able to confirm your expressions for logarithmic derivatives of the scale factors.  I'm not sure what approach you took in deriving them, but since I had already calculated partials of the scale factors (<math>\partial_i h_j</math> and so forth), I derived a little trick to help simplify the work.  Do you agree that

<div align="center">
<math>
\frac{\partial \ln h_i}{\partial \ln \lambda_j} \equiv \frac{\lambda_j}{h_i} \partial_j h_i ?
</math>
</div>

::Yes, this is precisely how I defined the logarithmic derivatives.  And I used your tables of partial derivatives (e.g., <math>\partial\varpi/\partial\lambda_j</math> and <math>\partial z/\partial\lambda_j</math>) to obtain <math>\partial_i h_j</math> and so forth. --[[User:Tohline|Tohline]] 17:30, 1 June 2010 (MDT)

'''<font color="red">Mistake!</font>''' Jay:  I have found a mistake in my original derivation of the logarithmic derivatives of the <math>h_1</math> scale factor.  Perhaps now they will match yours. Check it out and let me know! --[[User:Tohline|Tohline]] 12:41, 4 June 2010 (MDT)

::Joel:  I am finally able to confirm your derivation of the logarithmic derivatives of the scale factors.  I came at them from a different angle with a fresh head, and I am now confident that what you have is correct.  Next I will look into the general case.  --[[User:Jaycall|Jaycall]] 14:29, 10 July 2010 (MDT)