Definition:Variational Derivative

Definition
Let $y(x)$ be a real function.

Let $J=J[y]$ be a functional dependent on $y$.

Let $h(x)$ be a real function, which differs from zero only in the neighbourhood of $x_0$.

Consider an increment of functional $\Delta J[y; h]$.

Denote the area between $y(x)+h(x)$ and $y(x)$ (or, equivalently, between $h(x)$ and x-axis) as $\Delta\sigma$.

Let $\Delta\sigma\to 0$ in such a way, that

$\displaystyle \lim_{\Delta\sigma\to 0} \mathrm{max} \left(h(x)\right)=0$

and the length of interval where $h(x)$ differs from 0 would go to 0.

If the ration $\frac{\Delta J[y; h]}{\Delta\sigma}$ converges to a limit as $\Delta\sigma\to 0$, then

$\displaystyle \lim_{\Delta\sigma\to 0}\frac{\Delta J[y; h]}{\Delta\sigma}=\frac{\delta J}{\delta y}\bigg\rvert_{x=x_0}$

where $\frac{\delta J}{\delta y}\big\rvert_{x=x_0}$ is called the variational derivative at the point $x=x_0$ for the function $y=y(x)$.

Also known as
The variational derivative is often seen referred to as the functional derivative.