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.

- Definition
- Functional derivative
- Functional differential
- Properties
- Determining functional derivatives
- Formula
- Examples
- Using the delta function as a test function
- Notes
- Footnotes
- References
- External links

In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integral *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 this section, the functional derivative is defined. Then the functional differential is defined in terms of the functional derivative.

Given a manifold *M* representing (continuous/smooth) functions *ρ* (with certain boundary conditions etc.), and a functional *F* defined as

the **functional derivative** of *F*[*ρ*], denoted *δF/δρ*, is defined through^{ [2] }

where is an arbitrary function. The quantity is called the variation of *ρ*.

In other words,

is a linear functional, so one may apply the Riesz–Markov–Kakutani representation theorem to represent this functional as integration against some measure. Then *δF*/*δρ* is defined to be the Radon–Nikodym derivative of this measure.

One thinks of the function *δF*/*δρ* as the gradient of *F* at the point *ρ* and

as the directional derivative at point *ρ* in the direction of *ϕ*. Then analogous to vector calculus, the inner product with the gradient gives the directional derivative.

The differential (or variation or first variation) of the functional is ^{ [3] }^{ [Note 2] }

Heuristically, is the change in , so we 'formally' have , and then this is similar in form to the total differential of a function ,

where are independent variables. Comparing the last two equations, the functional derivative has a role similar to that of the partial derivative , where the variable of integration is like a continuous version of the summation index .^{ [4] }

Like the derivative of a function, the functional derivative satisfies the following properties, where *F*[*ρ*] and *G*[*ρ*] are functionals:^{ [Note 3] }

- Linearity:
^{ [5] }

where *λ*, *μ* are constants.

- Product rule:
^{ [6] }

- Chain rules:

- If
*F*is a functional and*G*another functional, then^{ [7] } - If
*G*is an ordinary differentiable function (local functional)*g*, then this reduces to^{ [8] }

A formula to determine functional derivatives for a common class of functionals can be written as the integral of a function and its derivatives. 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). The first three examples below are taken from density functional theory (20th century), the fourth from statistical mechanics (19th century).

Given a functional

and a function *ϕ*(** r**) that vanishes on the boundary of the region of integration, from a previous section Definition,

The second line is obtained using the total derivative, where *∂f* /*∂∇**ρ* is a derivative of a scalar with respect to a vector.^{ [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 that *ϕ*=0 on the boundary of the region of integration. Since *ϕ* is also an arbitrary function, applying the fundamental lemma of calculus of variations to the last line, the functional derivative is

where *ρ* = *ρ*(** r**) and

The above equation for the functional derivative can be generalized to the case that includes higher dimensions and higher order derivatives. The functional would be,

where the vector * r*∈ ℝ

^{ [Note 5] }

An analogous application of the definition of the functional derivative yields

In the last two equations, the *n ^{i}* components of the tensor are partial derivatives of

and the tensor scalar product is,

^{ [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:

Since the integrand of *T*_{TF}[*ρ*] does not involve derivatives of *ρ*(* r*), the functional derivative of

For the **electron-nucleus potential**, Thomas and Fermi employed the Coulomb potential energy functional

Applying the definition of functional derivative,

So,

For the classical part of the **electron-electron interaction**, Thomas and Fermi employed the Coulomb potential energy functional

From the definition of the functional derivative,

The first and second terms on the right hand side of the last equation are equal, since * r* and

and the functional derivative of the electron-electron coulomb potential energy functional *J*[*ρ*] is,^{ [10] }

The second functional derivative is

In 1935 von Weizsäcker proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it suit better a molecular electron cloud:

where

Using a previously derived formula for the functional derivative,

and the result is,^{ [11] }

The entropy of a discrete random variable is a functional of the probability mass function.

Thus,

Thus,

Let

Using the delta function as a test function,

Thus,

This is particularly useful in calculating the correlation functions from the partition function in quantum field theory.

A function can be written in the form of an integral like a functional. For example,

Since the integrand does not depend on derivatives of *ρ*, the functional derivative of *ρ*(* r*) is,

The functional derivative of the iterated function is given by:

and

In general:

Putting in N=0 gives:

In physics, it is common to use the Dirac delta function in place of a generic test function , for yielding the functional derivative at the point (this is a point of the whole functional derivative as a partial derivative is a component of the gradient):^{ [12] }

This works in cases when formally can be expanded as a series (or at least up to first order) in . The formula is however not mathematically rigorous, since is usually not even defined.

The definition given in a previous section is based on a relationship that holds for all test functions , so one might think that it should hold also when is chosen to be a specific function such as the delta function. However, the latter is not a valid test function (it is not even a proper function).

In the definition, the functional derivative describes how the functional changes as a result of a small change in the entire function . The particular form of the change in is not specified, but it should stretch over the whole interval on which is defined. Employing the particular form of the perturbation given by the delta function has the meaning that is varied only in the point . Except for this point, there is no variation in .

- ↑ According to Giaquinta & Hildebrandt (1996) , p. 18, this notation is customary in physical literature.
- ↑ Called
*differential*in ( Parr & Yang 1989 , p. 246),*variation*or*first variation*in ( Courant & Hilbert 1953 , p. 186), and*variation*or*differential*in ( Gelfand & Fomin 2000 , p. 11, § 3.2). - ↑ Here the notation is introduced.
- ↑ For a three-dimensional cartesian coordinate system,
- ↑ For example, for the case of three dimensions (
*n*= 3) and second order derivatives (*i*= 2), the tensor ∇^{(2)}has components, - ↑ For example, for the case
*n*= 3 and*i*= 2, the tensor scalar product is,

- 1 2 ( Giaquinta & Hildebrandt 1996 , p. 18)
- ↑ ( Parr & Yang 1989 , p. 246, Eq. A.2).
- ↑ ( Parr & Yang 1989 , p. 246, Eq. A.1).
- ↑ ( Parr & Yang 1989 , p. 246).
- ↑ ( Parr & Yang 1989 , p. 247, Eq. A.3).
- ↑ ( Parr & Yang 1989 , p. 247, Eq. A.4).
- ↑ ( Greiner & Reinhardt 1996 , p. 38, Eq. 6).
- ↑ ( Greiner & Reinhardt 1996 , p. 38, Eq. 7).
- ↑ ( Parr & Yang 1989 , p. 247, Eq. A.6).
- ↑ ( Parr & Yang 1989 , p. 248, Eq. A.11).
- ↑ ( Parr & Yang 1989 , p. 247, Eq. A.9).
- ↑ Greiner & Reinhardt 1996 , p. 37

In mathematics, the **Dirac delta function** is a generalized function or distribution, a function on the space of test functions. It was introduced by physicist Paul Dirac. It is called a function, although it is not a function **R** → **C**.

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In continuum mechanics, the **infinitesimal strain theory** is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body; so that its geometry and the constitutive properties of the material at each point of space can be assumed to be unchanged by the deformation.

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In physics, the **Hamilton–Jacobi equation**, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

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The **Cauchy momentum equation** is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

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In fluid dynamics, **Luke's variational principle** is a Lagrangian variational description of the motion of surface waves on a fluid with a free surface, under the action of gravity. This principle is named after J.C. Luke, who published it in 1967. This variational principle is for incompressible and inviscid potential flows, and is used to derive approximate wave models like the mild-slope equation, or using the averaged Lagrangian approach for wave propagation in inhomogeneous media.

In continuum mechanics, a **compatible** deformation **tensor field** in a body is that *unique* tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. **Compatibility** is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

- Courant, Richard; Hilbert, David (1953). "Chapter IV. The Calculus of Variations".
*Methods of Mathematical Physics*. Vol. I (First English ed.). New York, New York: Interscience Publishers, Inc. pp. 164–274. ISBN 978-0471504474. MR 0065391. Zbl 0001.00501.`|volume=`

has extra text (help). - Frigyik, Béla A.; Srivastava, Santosh; Gupta, Maya R. (January 2008),
*Introduction to Functional Derivatives*(PDF), UWEE Tech Report, UWEETR-2008-0001, Seattle, WA: Department of Electrical Engineering at the University of Washington, p. 7, archived from the original (PDF) on 2017-02-17, retrieved 2013-10-23. - Gelfand, I. M.; Fomin, S. V. (2000) [1963],
*Calculus of variations*, translated and edited by Richard A. Silverman (Revised English ed.), Mineola, N.Y.: Dover Publications, ISBN 978-0486414485, MR 0160139, Zbl 0127.05402 . - Giaquinta, Mariano; Hildebrandt, Stefan (1996),
*Calculus of Variations 1. The Lagrangian Formalism*, Grundlehren der Mathematischen Wissenschaften,**310**(1st ed.), Berlin: Springer-Verlag, ISBN 3-540-50625-X, MR 1368401, Zbl 0853.49001 . - Greiner, Walter; Reinhardt, Joachim (1996), "Section 2.3 – Functional derivatives",
*Field quantization*, With a foreword by D. A. Bromley, Berlin–Heidelberg–New York: Springer-Verlag, pp. 36–38, ISBN 3-540-59179-6, MR 1383589, Zbl 0844.00006 . - Parr, R. G.; Yang, W. (1989). "Appendix A, Functionals".
*Density-Functional Theory of Atoms and Molecules*. New York: Oxford University Press. pp. 246–254. ISBN 978-0195042795.

- "Functional derivative",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

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