# Functional derivative

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Volterra derivative

One of the first concepts of a derivative in an infinite-dimensional space. Let $I ( y)$ be some functional of a continuous function of one variable $y ( x)$; let $x _ {0}$ be some interior point of the segment $[ x _ {1} , x _ {2} ]$; let $y _ {1} ( x) = y _ {0} ( x) + \delta y ( x)$, where the variation $\delta y ( x)$ is different from zero in a small neighbourhood $[ a, b]$ of $x _ {0}$; and let $\sigma = \int _ {a} ^ {b} \delta y ( x) dx$. The limit

$$\lim\limits _ {\begin{array}{c} \sigma \rightarrow 0, \\ a , b \rightarrow x _ {0} \end{array} } \ \frac{I ( y _ {1} ) - I ( y _ {0} ) } \sigma ,$$

assuming that it exists, is called the functional derivative of $I$ and is denoted by $( \delta I ( y _ {0} )/ \delta y) \mid _ {x = x _ {0} }$. For example, for the simplest functional of the classical calculus of variations,

$$I ( y) = \ \int\limits _ { x _ {1} } ^ { {x _ 2 } } F ( x, y, \dot{y} ) dx,$$

the functional derivative has the form

$$\left . \frac{\delta I ( y _ {0} ) }{\delta y } \right | _ {x = x _ {0} } =$$

$$= \ \frac{\partial F ( x _ {0} , y _ {0} ( x _ {0} ), \dot{y} _ {0} ( x _ {0} )) }{\partial y } - { \frac{d}{dx} } \frac{\partial F ( x _ {0} , y ( x _ {0} ), \dot{y} ( x _ {0} )) }{\partial \dot{y} } ,$$

that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of $I ( y)$.

In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the Gâteaux derivative and the Fréchet derivative. But the concept of a functional derivative has been applied with success in numerical methods of the classical calculus of variations (see Variational calculus, numerical methods of).

The existence of the functional derivative of $I$ at $y = y _ {0}$ and $x = x _ {0}$ apparently means that the Fréchet derivative $dI$ of $I$ at $y = y _ {0}$, which is a continuous linear form on the space of admissible infinitesimal variations $z$, is of the form $\int u ( x) \cdot z ( x) dx$ for some continuous function $u$, so that it can be continuously extended to $z =$ the $\delta$- function at $x = x _ {0}$. In the example this happens only if $y _ {0}$ is twice continuously differentiable.