# 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

$\ds \lim_{\Delta \sigma \mathop \to 0} \map \max {\map h x} = 0$

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

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

$\ds \lim_{\Delta \sigma \mathop \to 0} \frac {\Delta J \sqbrk {y; h} } {\Delta \sigma} = \intlimits {\frac {\delta J} {\delta y} } {x = x_0} {}$

where $\intlimits {\dfrac {\delta J} {\delta y} } {x = x_0} {}$ 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