Functional derivative

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.However, this notion of functional differential is so strong it may not exist,[3] and in those cases a weaker notion, like the Gateaux derivative is preferred.In many practical cases, the functional differential is defined[4] as the directional derivativeNote that this notion of the functional differential can even be defined without a norm.(for example, if there are some boundary conditions imposed) then, and then this is similar in form to the total differential of a functionComparing the last two equations, the functional derivativehas a role similar to that of the partial derivativeis like a continuous version of the summation index[7] One thinks of δF/δρ as the gradient of F at the point ρ, so the value δF/δρ(x) measures how much the functional F will change if the function ρ is changed at the point x.This is a generalization of the Euler–Lagrange equation: indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the principle of least action in Lagrangian mechanics (18th century).that vanishes on the boundary of the region of integration, from a previous section Definition,[Note 4] The third line was obtained by use of a product rule for divergence.The fourth line was obtained using the divergence theorem and the condition thatis also an arbitrary function, applying the fundamental lemma of calculus of variations to the last line, the functional derivative isThis formula is for the case of the functional form given by F[ρ] at the beginning of this section.(See the example Coulomb potential energy functional.)The above equation for the functional derivative can be generalized to the case that includes higher dimensions and higher order derivatives.where the vector r ∈ Rn, and ∇(i) is a tensor whose ni components are partial derivative operators of order i,[Note 5] An analogous application of the definition of the functional derivative yieldsIn the last two equations, the ni components of the tensor[Note 6] The Thomas–Fermi model of 1927 used a kinetic energy functional for a noninteracting uniform electron gas in a first attempt of density-functional theory of electronic structure:Applying the definition of functional derivative,For the classical part of the electron-electron interaction, Thomas and Fermi employed the Coulomb potential energy functionalIn 1935 von Weizsäcker proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it better suit a molecular electron cloud:In physics, it is common to use the Dirac delta functionin place of a generic test function, for yielding the functional derivative at the pointThe definition given in a previous section is based on a relationship that holds for all test functionsEmploying the particular form of the perturbation given by the delta function has the meaning that
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