mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function...

and to define the functional derivative used widely in the calculus of variations. Generally, it extends the idea of the derivative from real-valued functions...

written above equation, it is easy to find the following formula for functional derivative: 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 functional derivative commonly used in the...

of the previous equation is the functional derivative δ J / δ y {\displaystyle \delta J/\delta y} of the functional J {\displaystyle J} . A necessary...

real variables. In functional analysis, the functional derivative defines the derivative with respect to a function of a functional on a space of functions...

with the product rule. Covariant derivative Derivation Exterior derivative Functional derivative Integral Lie derivative Apostol 1967, p. 160; Stewart 2002...

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable...

this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of...

"flavor" index. This involves functionals over the φ's, functional derivatives, functional integrals, etc. From a functional point of view this is equivalent...

as constants. ⁠𝛿 □/𝛿 □⁠ Functional derivative: If f ( y 1 , … , y n ) {\displaystyle f(y_{1},\ldots ,y_{n})} is a functional of several functions, δ f...

Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals; that is, they carry information on how a functional changes...

In 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...

\left(\{i,x\}\rightarrow \{i+1,xg(i)\}\right)^{b-a+1}\{a,1\}} The functional derivative of an iterated function is given by the recursive formula: δ f N...

denote: A change in the value of a variable in calculus. A functional derivative in functional calculus. The (ε, δ)-definition of limits, in mathematics...

Γ k ( 1 , 1 ) {\displaystyle \Gamma _{k}^{(1,1)}} denotes the functional derivative of Γ k {\displaystyle \Gamma _{k}} from the left-hand-side and the...

z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly...

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...

equation cf. Action (physics) Fermat's principle Functional (mathematics) Functional derivative Functional integral Geodesic Isoperimetry Lagrangian Lagrangian...

e. the functional derivative of the vW functional and acknowleding the definition of the Pauli kinetic energy, while the functional derivative of the...

In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative...

definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation...

(OEP) in Kohn-Sham (KS) density functional theory (DFT) is a method to determine the potentials as functional derivatives of the corresponding KS orbital-dependent...

three-dimensional boundary. Observe that this expression vanishes implies the functional derivative vanishes, giving us the Wheeler–DeWitt equation. A similar statement...

exchange-correlation potential, the exchange-correlation kernel – the functional derivative of the exchange-correlation potential with respect to the density...

single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated...

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...

{\displaystyle I[f]} by taking the functional derivative of the last equality with respect to f {\displaystyle f} and setting the derivative equal to 0. This will...

|}_{J=0}.} The derivatives used here are functional derivatives rather than regular derivatives since they are acting on functionals rather than regular...

remains to be established experimentally. Calculus of variations Functional derivative Functional integration Hamiltonian mechanics Lagrangian Lagrangian mechanics...