In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative)[1] relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends.
In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, the coefficient of δf in the first order term is called the functional derivative.
For example, consider the functional
where f ′(x) ≡ df/dx. If f is varied by adding to it a function δf, and the resulting integrand L(x, f +δf, f '+δf ′) is expanded in powers of δf, then the change in the value of J to first order in δf can be expressed as follows:[1][Note 1]
where the variation in the derivative, δf ′ was rewritten as the derivative of the variation (δf) ′, and integration by parts was used in these derivatives.
^ ab(Giaquinta & Hildebrandt 1996, p. 18)
Cite error: There are <ref group=Note> tags on this page, but the references will not show without a {{reflist|group=Note}} template (see the help page).
and 25 Related for: Functional derivative information
mathematical analysis, the functionalderivative (or variational derivative) relates a change in a functional (a functional in this sense is a function...
written above equation, it is easy to find the following formula for functionalderivative: δ F [ n e ] δ n = 2 A − 2 B 2 + A e V ( τ 0 ) B + e V ( τ 0 ) ...
spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functionalderivative commonly used in the...
this usage is obsolete, except for functionalderivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of...
The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. The derivative...
real variables. In functional analysis, the functionalderivative defines the derivative with respect to a function of a functional on a space of functions...
z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly...
"flavor" index. This involves functionals over the φ's, functionalderivatives, functional integrals, etc. From a functional point of view this is equivalent...
For organic chemistry, a carbonyl group is a functional group with the formula C=O, composed of a carbon atom double-bonded to an oxygen atom, and it is...
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments...
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative...
\left(\{i,x\}\rightarrow \{i+1,xg(i)\}\right)^{b-a+1}\{a,1\}} The functionalderivative of an iterated function is given by the recursive formula: δ f N...
single point. Accordingly, the necessary condition of extremum (functionalderivative equal zero) appears in a weak formulation (variational form) integrated...
definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation...
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable...
Γ k ( 1 , 1 ) {\displaystyle \Gamma _{k}^{(1,1)}} denotes the functionalderivative of Γ k {\displaystyle \Gamma _{k}} from the left-hand-side and the...
functions, and ε is a scalar. This is recognizable as the Gateaux derivative of the functional. Compute the first variation of J ( y ) = ∫ a b y y ′ d x . {\displaystyle...
In copyright law, a derivative work is an expressive creation that includes major copyrightable elements of a first, previously created original work (the...
three-dimensional boundary. Observe that this expression vanishes implies the functionalderivative vanishes, giving us the Wheeler–DeWitt equation. A similar statement...
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related...
{\displaystyle I[f]} by taking the functionalderivative of the last equality with respect to f {\displaystyle f} and setting the derivative equal to 0. This will...
structure is the structure of an unadorned ion or molecule from which derivatives can be visualized. Parent structures underpin systematic nomenclature...