A Fréchet derivative derivation

1. Aug 14, 2016

Boon

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. Aug 19, 2016