1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    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!

Double derivatives of F as a function of F and harmonicity

  1. Mar 17, 2013 #1
    1. The problem statement, all variables and given/known data
    Screen_shot_2013_03_17_at_8_12_00_PM.png


    2. Relevant equations
    I can't think of much. u(x,y) is harmonic, so its double derivatives with respect to x and y add up to zero. I'm not 100% sure, but does being harmonic also imply that u satisfies the cauchy riemann equations? That might come in handy in the denominator.


    3. The attempt at a solution
    Where to begin, where to begin. To me, the question sounds very vague. The second derivative of a constant, isn't that just zero? What should I do? I know this is a lousy way to post a question, but I honestly don't even know where to begin. If you have any tips (just to get me started!) that would be great.

    Kind regards
     
    Last edited: Mar 17, 2013
  2. jcsd
  3. Mar 17, 2013 #2

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    I always thought that if u(x,y) is harmonic, then DEL^2 u = 0.
     
  4. Mar 17, 2013 #3
    That was a terrible typo on my part, in my case it means that the second derivative of u with respect to x plus the second derivative of u with respect to y add up to 0, which is the 2D version of what you are saying
     
  5. Mar 18, 2013 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    I really don't understand what you mean by "the equation F(x,y)= constant" can be expressed as u(x,y)= constant with u harmonic" unless you mean that f(x, y) is a harmonic function plus a constant: F(x,y)= u(x,y)+ C.
    Then [itex]\partial^2 F/\partial x^2= \partial^2 u/\partial x^2[/itex] and [itex]\partial^2F/\partial y^2= \partial^2 u/\partial y^2[/itex] and, as you say, the sum of those is 0.
     
  6. Mar 18, 2013 #5
    I'm not sure, I'll email my teacher to ask that, because it doesn't make sense indeed.
     
  7. Mar 18, 2013 #6
    His reply is the following
    consider the set defined by u(x,y)=constant. This is not the only way to describe this set. For example, if u(x,y)=constant also (u(x,y))^2=constant, or take any function F applied to u(x,y) give F(u(x,y))=constant. Now can you choose F such that F(u(x,y)) is harmonic?
     
  8. Mar 19, 2013 #7
    i've been trying to figure this out for a day or so now, but I really don't seem to be making a lot of progress. I tried plugging in F(u(x, y)) with u harmonic, but not a whole lot cancels. Could anyone provide a hint?
     
  9. Mar 20, 2013 #8
    I was thinking, maybe I should use implicit differentiation, as the function is equal to a constant? I don't really see where that gets me, but I am pretty desperate at this point.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Double derivatives of F as a function of F and harmonicity
  1. F primes derivative (Replies: 17)

Loading...