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A Fréchet derivative derivation

  1. Aug 14, 2016 #1
    In Stone & Goldbart's Mathematics for Physics, in section 1.2.1 on the Calculus of Variations, they derive the Fréchet derivative. Part of the derivation is as follows:

    Equation 1: J[y + εη] - J[y] = ∫ { f(x, y + εη, y' + εη') - f(x, y, y') } dx

    Equation 2: J[y + εη] - J[y] = ∫ { εη ∂f/∂y + ε dη/dx ∂f/∂y' + O(ε2) } dx

    To go from equation 1 to 2 a Taylor expansion is (implicitly) used. However, this seems rather complicated and introduces what I consider as the unnecessary complication O(ε2). Instead, I can derive the same result by using the total difference:

    f(x, y + εη, y' + εη') - f(x, y, y') = δf = δx ∂f/∂x + δy ∂f/∂y + δy' ∂f/∂y' = 0 ∂f/∂x + εη ∂f/∂y + ε dη/dx ∂f/∂y' = εη ∂f/∂y + ε dη/dx ∂f/∂y'

    which is the integrand in equation 2, minus the O(ε2) which vanishes in their derivation. Is my method correct? If so, why would one use the more complicated method of a Taylor expansion?
     
    Last edited: Aug 14, 2016
  2. jcsd
  3. Aug 19, 2016 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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