# Definition:Variational Derivative

## Definition

Let $\map y x$ be a real function.

Let $J=J\sqbrk y$ be a functional dependent on $y$.

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

Consider an increment of functional $\Delta J\sqbrk{y;h}$.

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

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

$\displaystyle\lim_{\Delta\sigma\to 0} \mathrm{max}\paren{\map h x}=0$

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

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

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

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 $y=\map y x$.

## Also known as

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

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 1.7$: The Variational Derivative