I am studying from
Gelfand, I.M.; Fomin, S.V., Calculus of variations. Transl. from the Russian and edited by Richard A. Silverman., Mineola, NY: Dover Publications. vii, 232 p. (2000). ZBL0964.49001.
In paragraph 12 "Variational Problems with Subsidiary Conditions", I cannot understand the equation (28) in the proof of Theorem 1.
Let $J[y]$ be a functional of the form,
$$J[y]=\int_a^bF(x,y,y')\mathrm d x,$$
where $F$ is twice differentiable with respect of all its arguments.
Now let us give to $y$ an increment $\delta_1y(x)+\delta_2y(x)$, such that $\delta_1y(x)$ does not vanish only in the neighborhood of a chosen point $x_1 \in [a,b]$, and respectively $\delta_2y(x)$ is nonzero only in the neighborhood of $x_2$ which is also in $[a,b]$.
The equation (28) states that, using the variational derivative we can write the corresponding increment $\Delta J$ of the functional $J$ as follow,
$$\Delta J = \left \{ \frac{\delta F}{\delta y} \bigg |_{x=x_1}+\epsilon_1\right \}\Delta \sigma_1+\left \{ \frac{\delta F}{\delta y} \bigg |_{x=x_2}+\epsilon_2\right \}\Delta \sigma_2,$$
where $\Delta \sigma_1 $ and $\Delta \sigma_2$ are the areas between the increments and the x-axis, and $\epsilon_1, \epsilon_2 \to 0$ as $\Delta \sigma_1, \Delta \sigma_2 \to 0$.
Should not the equation (28) be as shown below?
$$\Delta J = \left \{ \frac{\delta J}{\delta y} \bigg |_{x=x_1}+\epsilon_1\right \}\Delta \sigma_1+\left \{ \frac{\delta J}{\delta y} \bigg |_{x=x_2}+\epsilon_2\right \}\Delta \sigma_2.$$
Thanks, Dante.
