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Partial differentiation & complex analysis

  1. Nov 10, 2008 #1
    1. The problem statement, all variables and given/known data
    Let Δf= d^2f/dx^2+ d^g/dy^2 (laplace equation - Partial Derivatives) Show Δ(f(g(z))= Mod(g'(z))^2 * Δf(w,v) where g(z)=w(x,y)+v(x,y)i


    2. Relevant equations
    we propably need to use cauchy riemman equations: dw/dx = dv/dy and dw/dy = - dv/dx
    and chain rule


    3. The attempt at a solution
    ∆f(g(x,y)) = d^2 f/dw^2*((dw/dx)^2 + (dw/dy)^2) + d^2f/dv^2*((dv/dx)^2 + (dv/dy)^2) + 2*d^2f/dwdv*(dw/dx dv/dx + dw/dy dv/dy) + df/dw*(d^2w/dx^2 + d^2w/dy^2) + df/dv*(d^2v/dx^2 + d^2 v/dy^2).

    If you use the C-R equations this reduces to the identity you stated

    Someone gave me this solution but i cant understand where the term 2*d^2f/dwdv*(dw/dx dv/dx + dw/dy dv/dy) came from?
     
  2. jcsd
  3. Nov 10, 2008 #2
    What's Mod ?

    marlon
     
  4. Nov 11, 2008 #3
    mod is the absolute value
    ie lg'(z)l
     
  5. Nov 12, 2008 #4
    Hi

    The term you cannot understand how to get does not contribute anything to the proof and with the CR equations it quals zero.

    P.s are you a UCL maths student?
     
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