Confusing expression in a paper

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Discussion Overview

The discussion revolves around the interpretation of a mathematical expression found in a paper, specifically focusing on the function defined as f and its components, including the relationship between variables z'', Z', and Zi'. Participants are exploring the implications of differentiating and substituting these variables within the context of the equation presented in the paper.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion over the expression n(z''(Z')) and its interpretation, questioning the relationship between n and z''.
  • Another participant suggests that Z'(z'') comes from a variable substitution detailed in the text, but acknowledges difficulty in rewriting it as z''(Z').
  • A different participant notes that the only occurrence of z'' is in the denominator and proposes to simplify the expression by suppressing n(z'').
  • There is a challenge regarding the complexity of n(z'') and the implications of suppressing it, with one participant asking for clarification on what "suppress" means.
  • One participant suggests differentiating n(z'') with respect to z'' first before substituting, but expresses frustration over the presence of z'' after substitution.
  • Another participant agrees that differentiating implicitly or using the chain rule may complicate the process, emphasizing that the substitution of variables is the main issue.

Areas of Agreement / Disagreement

Participants generally agree that the expression is complex and challenging to interpret, but there is no consensus on the best approach to resolve the confusion or the correct method for substitution and differentiation.

Contextual Notes

Participants note the difficulty in expressing z'' as a function of Z' and the implications of variable substitution on differentiation, indicating potential limitations in the mathematical steps discussed.

natski
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Hi all,

Whilst reading http://www.jhuapl.edu/techdigest/td1703/thomas.pdf I found one of the equations a little difficult to interpret. On page 3, there is a function defined as f. On the denominator there it appears that they have differentiated n(z''(Z')) as a function of z'' and then evaluated it for the case where Z' goes to Zi'.

As far as I can tell, n is only a function of z'' and z'' is not a function of anything else, so the n(z''(Z')) doesn't make much sense to me. I interpretted it as meaning that I write down the expression n(z'') then replace all my z'' s with Z' using an appropriate equation of the form:
Z'=some function of z''.

However, I cannot even write down a clean expression for this! The form of Z', as seen just below eqn 4, cannot be easily rewritten to express z'' as a function of a Z'.

Can anyone help me interpretting this confusing expression?

Thanks!
 
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Reading the text above equation (4), it appears that Z'(z'') originates from a "simple variable substitution" which they then detail, below equation (4).
 
Yes you are right but when I tried to rewrite Z'(z'') as z''(Z') I could not and it seemed to require solving in some kind of non-algebraic way.
 
The only true occurrence of z'' is in the denominator, just supress n(z'') to n.
 
What do you mean suppress? n(z'') is quite a complex function of z''.
 
What I meant was to differentiate implicitly.
 
Umm ok, so I can write:

z''=[n(h) (R+h) Cos(Delta)]/(n(z'') Z') - R

Then are you thinking I should replace all my z''s in n(z'') with this equation? If I do this, I can't differentiate by z'' since now I have got Z' in there too.

I think I should differentiate first n(z'') by z'' first since that's not too hard. I then need to replace z'' with the equation above. However, when I then replace Z' with Zi' I still have z'' floating about which I can't get rid off!
 
Differentiating implicity or using some kind of chain rule or anything will not make life any easier as far as I can tell. The differentation part is the one part that is easily done, it's the replacing z''->Z'->Zi' that's the problem.
 

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