1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Chain rule / Taylor expansion / functional derivative

  1. Jun 19, 2016 #1
    1. The problem statement, all variables and given/known data

    To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0##
    where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)##

    2. Relevant equations

    I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions, but if someone could please give me a clue where to start.

    3. The attempt at a solution
    as above.

    many thanks in advance.
  2. jcsd
  3. Jun 19, 2016 #2

    Charles Link

    User Avatar
    Homework Helper
    Gold Member

    Suggest a Taylor expansion of ## \rho(p',s') ## in just the s variable about s. One comment is since s and s' differ by a differential amount, the the terms of the first inequality differ by only a differential amount. I think the inequality you are trying to prove actually needs a "dz" tacked on to (multiplying) the product of the derivatives. I would be interested in seeing if other readers concur. "dz" does not need to be positive.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Chain rule Taylor Date
Thermal Energy Equation Term - Chain Rule Dec 18, 2017
Homework question using the chain rule -- oil slick spreading on the sea... Sep 20, 2017
Chain rule problem Sep 13, 2017
Chain Rule Question Aug 9, 2017
Chain rule Aug 8, 2017