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In Gelfand Calculus of Variation, chapter 1.7, the variational derivative is defined as:

$$\left.\frac{\partial J}{\partial y}\right|_{x = x_0} = \lim_{\Delta\sigma \rightarrow 0}\frac{J[y+h]-J[y]}{\Delta\sigma},$$ where $h(x)$ is different from zero only in the neighborhood of $x_0$. Also $\Delta\sigma$ is the area lying between the curve $h(x)$ and the x-axis, and $\Delta\sigma$ goes to zero in such a way that both $\max|h(x)|$ and the length of the interval in which $h(x)$ is non-vanishing goes to zero

I wonder if there exist a more precise statement for this definition and if there is any characterization of when such a derivative exist. Also, is there any relation between the variational derivative and the variation (or differential) of a functional defined in Gelfand chapter 1.3.2.

Moreover, what is the purpose of this definition?

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    $\begingroup$ This is the original definition of "Functional derivative" given by Vito Volterra: I have some more comments to add, and I'll do this later. $\endgroup$
    – Daniele Tampieri
    Commented May 29, 2021 at 7:43
  • $\begingroup$ Thanks! Could you name the text by Vito Volterra? $\endgroup$
    – MkSn1999
    Commented May 30, 2021 at 20:08
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    $\begingroup$ The first place to look is chapter I, section II, §1 pp. 22-24 of the standard text Volterra, Vito, Theory of functionals and of integral and integro-differential equations. Dover edition with a preface by Griffith C. Evans, a biography of Vito Volterra and a bibliography of his published works by Sir Edmund Whittaker. Unabridged republ. of the first English transl, New York: Dover Publications, Inc. pp. 39+XVI+226 (1959), MR0100765, ZBL0086.10402. $\endgroup$
    – Daniele Tampieri
    Commented May 30, 2021 at 20:17
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    $\begingroup$ You can also have a look at this Q&A: the unifying concept is the "Functional derivative" one, but since Volterra had in mind the functionals commonly used in the calculus of variation, his definition of functional derivative was a bit involved so, even if you are studying problems derived from, say, modern nonlinear system theory, is better to have a look at the good old sources. $\endgroup$
    – Daniele Tampieri
    Commented May 30, 2021 at 20:22

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