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I am trying to make sense of functional derivatives and have a couple of questions bothering me:

  1. Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.

    By chain rule in the continum I intuitively guess that the derivative of the functional F[X] with respect to the functional G[X] is:$$\frac{\delta F}{\delta G} = \int dt \frac{\delta F[X]}{\delta X(t)} \frac{\delta X(t)}{\delta G[X]} = \int dt \frac{\delta F[X]}{\delta X(t)} \left(\frac{\delta G[X]}{\delta X(t)} \right)^{-1} $$ Does this make any sense at all? Is there such thing as a derivative of a functional with respect to another functional given they are both dependent on the same underlying function X(t)?

  2. How would one calculate the functional derivatives $\frac{\delta F[X]}{\delta X(t)}$ and $\frac{\delta G[X]}{\delta X(t)}$? I came across Gateaux derivative and It seems to be similar to a directional derivative. Does it matter what direction is being chosen to evaluate the final functional derivative?

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    $\begingroup$ I am not sure what you are asking. Please rewrite. $\endgroup$
    – William M.
    Commented Dec 11, 2018 at 5:14
  • $\begingroup$ Thanks Will, just edited please let me know if it's still not clear. $\endgroup$
    – ZeroCool
    Commented Dec 11, 2018 at 15:44
  • $\begingroup$ OK. It still makes no sense. However, two points to bring. (1) Frechet derivative is simply the usual derivative on learns from calculus 2 (at least in places like the one I studied where math rigour is imposed over being able to perform calculations) and (2) Gateaux derivatives are directional derivatives and, as always, they depend on the chosen direction. If you want to know the derivatives of a composition $g \circ f,$ then simply apply the chain rule $g'(f(a)) \circ f'(a).$ If you want to know how to calculate $f'$ or $g',$ you need to know the exact form first an apply rules then. $\endgroup$
    – William M.
    Commented Dec 11, 2018 at 18:05
  • $\begingroup$ @ZeroCool you might want to ask on MathOverflow. This site is for more elementary questions (usually pre-functional-derivative stage). $\endgroup$ Commented May 30, 2020 at 22:18

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