Editing Functional derivative (section)

{{Short description|Concept in calculus of variation}}
In the [[calculus of variations]], a field of [[mathematical analysis]], the '''functional derivative''' (or '''variational derivative''')<ref name="GiaquintaHildebrandtP18">{{harvp|Giaquinta|Hildebrandt|1996|p=18}}</ref> relates a change in a [[Functional (mathematics)|functional]] (a functional in this sense is a function that acts on functions) to a change in a [[Function (mathematics)|function]] on which the functional depends.

In the calculus of variations, functionals are usually expressed in terms of an [[integral]] of functions, their [[Argument of a function|arguments]], and their [[derivative]]s. In an integrand {{math|''L''}} of a functional, if a function {{math|''f''}} is varied by adding to it another function {{math|''δf''}} that is arbitrarily small, and the resulting integrand is expanded in powers of {{math|''δf''}}, the coefficient of {{math|''δf''}} in the first order term is called the functional derivative.

For example, consider the functional
<math display="block"> J[f] = \int_a^b L( \, x, f(x), f'{(x)} \, ) \, dx \, , </math>
where {{math|''f'' &prime;(''x'') &equiv; ''df''/''dx''}}. If {{math|''f''}} is varied by adding to it a function {{math|''δf''}}, and the resulting integrand {{math|''L''(''x'', ''f'' +''δf'', ''f'' &prime;+''δf'' &prime;)}} is expanded in powers of {{math|''δf''}}, then the change in the value of {{math|''J''}} to first order in {{math|''δf''}} can be expressed as follows:<ref name="GiaquintaHildebrandtP18" /><ref Group = 'Note'>According to {{Harvp|Giaquinta|Hildebrandt|1996|p=18}}, this notation is customary in [[Physics|physical]] literature.</ref>
<math display="block">\begin{align}
\delta J &= \int_a^b \left( \frac{\partial L}{\partial f} \delta f(x) + \frac{\partial L}{\partial f'} \frac{d}{dx} \delta f(x) \right) \, dx \, \\[1ex]
&= \int_a^b \left( \frac{\partial L}{\partial f} - \frac{d}{dx} \frac{\partial L}{\partial f'} \right) \delta f(x) \, dx \, + \, \frac{\partial L}{\partial f'} (b) \delta f(b) \, - \, \frac{\partial L}{\partial f'} (a) \delta f(a)
\end{align} </math>
where the variation in the derivative, {{math|''δf'' &prime;}} was rewritten as the derivative of the variation {{math|(''δf'') &prime;}}, and [[integration by parts]] was used in these derivatives.