Definition:Variational Derivative

Definition
Let $ y \left ( { x } \right ) $ be a real function.

Let $ J = J \left [ { y } \right ] $ be a functional dependent on $ y $.

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

Consider an increment of functional $ \Delta J \left [ { y; h } \right ] $.

Denote the area between $ y \left ( { x } \right ) + h \left ( { x } \right ) $ and $ y \left ( { x } \right ) $ (or, equivalently, between $ h \left ( { x } \right ) $ 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 \left ( { x } \right ) } \right ) = 0 $

and the length of interval where $ h \left ( { x } \right ) $ differs from 0 would go to 0.

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

$ \displaystyle \lim_{ \Delta \sigma \to 0} \frac{ \Delta J \left [ { y; h } \right ] }{ \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 \left ( { x } \right ) $.

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