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According to Wikipedia, there are two definitions for the differential of a functional (also called the variation). The first is the Fréchet derivative of the functional and the second the Gateaux derivative (a generalization of the directional derivative). Please note the question is related to the differential of a functional and not the functional derivative which comes in the next section in the wikipedia article.

Definition 1: $\delta F_1=F[f+\epsilon \eta]-F[f]$.

Definition 2: $\delta F_2= \displaystyle\lim_{\epsilon \to 0}\frac{F[f+\epsilon \eta]-F[f]} {\epsilon} =\frac{d}{d\epsilon}F[f+\epsilon\eta] \Big|_{\epsilon=0}$.

my questions:

(1) Is it possible to reconcile the two definitions that are smilingly different?

(2) What are the implications of choosing one over the other?

I see a lot of textbooks and authors flip flopping between the two definitions. Understandably, the final result which is usually related to problems in minimization and/or deriving the Euler Lagrange equations is the same no matter which definition is used. But the difference between the two definitions bothers me.

Definition (1) is the natural one if one thinks of extending the definition used in the ordinary calculus of functions (differential of a function). Definition (2) however, is certainly more practical in calculations, for example Taylor expansion of functionals and deriving Euler Lagrange equations.

The same question, however in a different format, has been raised here. The author can't reconcile the Taylor expansion in terms of the two different definitions given above. The answer given is not satisfactory.

In another post, one of the answers suggests that $\delta F_2=\delta F_1 \eta$. But I don's see how this can reconcile the two definitions. This answer by the way goes back to the relation between Fréchet and Gateaux derivatives and is discussed at the end of this wikipedia page.

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  • $\begingroup$ These two definitions are different because they are incorrect. The Frechet derivative does not depend on the choice of $\eta$, whereas the Gateaux derivative does. I suggest you look at their respective Wikipedia pages, as the Functional Derivative page uses less clear notation. Furthermore, Frechet differentiability implies Gateaux differentiability, see here math.stackexchange.com/questions/533172/… $\endgroup$
    – whpowell96
    Commented Oct 16, 2023 at 17:40

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