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Concept in calculus of variations
mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function
Functional_derivative
Type of derivative in mathematics
considering all partial derivatives simultaneously. In functional analysis, particularly in infinite dimensions, the derivative in this sense is called
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Derivative defined on normed spaces
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
Fréchet_derivative
Generalization of the concept of directional derivative
spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the
Gateaux_derivative
Computational quantum mechanical modelling method to investigate electronic structure
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 )
Density_functional_theory
Instantaneous rate of change (mathematics)
Derivations generalize derivatives to algebraic settings, such as rings. Covariant derivative Derivation Exterior derivative Functional derivative Implicit differentiation
Derivative
Second-order partial differential equation describing motion of mechanical system
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
Euler–Lagrange_equation
Fundamental construction of differential calculus
real variables. In functional analysis, the functional derivative defines the derivative with respect to a function of a functional on a space of functions
Generalizations of the derivative
Generalizations_of_the_derivative
Implementation of the renormalization group
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
Functional renormalization group
Functional_renormalization_group
Fourth letter in the Greek alphabet
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
Delta_(letter)
Theory allowing one to apply mathematical functions to mathematical operators
visible in the functional derivative, which is often called the variational derivative. There are several unrelated uses of the term "functional calculus":
Functional_calculus
Generalisation of the derivative of a function
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable
Weak_derivative
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
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Scientific theory
e. the functional derivative of the vW functional and acknowleding the definition of the Pauli kinetic energy, while the functional derivative of the
Orbital-free density functional theory
Orbital-free_density_functional_theory
Derivative of a function with multiple variables
z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly
Partial_derivative
Functional group (C=O)
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
Carbonyl_group
Area of mathematics
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
Functional_analysis
Initial result in using test functions to find extremum
single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated
Fundamental lemma of the calculus of variations
Fundamental_lemma_of_the_calculus_of_variations
Result of repeatedly applying a mathematical function
\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
Iterated_function
Differential calculus on function spaces
definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation
Calculus_of_variations
Programming paradigm based on applying and composing functions
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative
Functional_programming
Types of mappings in mathematics
Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals; that is, they carry information on how a functional changes
Functional_(mathematics)
Concept in machine learning
{\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
Loss functions for classification
Loss_functions_for_classification
Quantum-mechanical framework for simulating molecules and solids
exchange-correlation potential, the exchange-correlation kernel – the functional derivative of the exchange-correlation potential with respect to the density
Time-dependent density functional theory
Time-dependent_density_functional_theory
Notation used in quantum field theory
"flavor" index. This involves functionals over the φ's, functional derivatives, functional integrals, etc. From a functional point of view this is equivalent
DeWitt_notation
Physical quantity of dimension energy × time
remains to be established experimentally. Calculus of variations Functional derivative Functional integration Hamiltonian mechanics Lagrangian Lagrangian mechanics
Action_(physics)
it is simply the gradient, for continuous variables, it is the functional derivative (a function S → R {\displaystyle S\rightarrow \mathbb {R} } ) the
GENERIC_formalism
Objects that generalize functions
functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions
Distribution (mathematical analysis)
Distribution_(mathematical_analysis)
Concept in copyright law
In copyright law, a derivative work is an expressive creation that includes major copyrightable elements of a first, previously created original work (the
Derivative_work
The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding
Topological_derivative
{\displaystyle \Phi } is a generating functional for irreducible vertex quantities: the first functional derivative with respect to G {\displaystyle G}
Luttinger–Ward_functional
Researcher and lecturer in quantitative finance
with novel functional derivatives with respect to space and time, that models causal relationships that are deployed through time. (“Functional Itô Calculus”
Bruno_Dupire
Instantaneous rate of change of the function
derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. In functional analysis
Directional_derivative
Integration over the space of functions
{R} ^{2}}J(x)K^{-1}(x;y)J(y)\,dx\,dy\right\rbrace .} Now, taking functional derivatives to the definition of W [ J ] {\displaystyle W[J]} and then evaluating
Functional_integration
Formulation of quantum mechanics
{\mathcal {S}}[\varphi ]}{\delta \varphi }}=0} (the left-hand side is a functional derivative; the equation means that the action is stationary under small changes
Path_integral_formulation
Generating function for quantum correlation functions
|}_{J=0}.} The derivatives used here are functional derivatives rather than regular derivatives since they are acting on functionals rather than regular
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Approach to quantum gravity utilizing Wick rotations
three-dimensional boundary. Observe that this expression vanishes implies the functional derivative vanishes, giving us the Wheeler–DeWitt equation. A similar statement
Euclidean_quantum_gravity
Branch of mathematical analysis
Sonin–Letnikov derivative Liouville derivative Caputo derivative Hadamard derivative Marchaud derivative Riesz derivative Miller–Ross derivative Weyl derivative Erdélyi–Kober
Fractional_calculus
Mathematical theory
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
First_variation
Theoretical formalism in condensed matter physics
Indeed, one can see the analogy between the expression relating the functional derivative of V e e S C E [ n ] {\displaystyle V_{\rm {ee}}^{\rm {SCE}}[n]}
Strictly-correlated-electrons density functional theory
Strictly-correlated-electrons_density_functional_theory
Equations for correlation functions in QFT
\varphi }}} is the functional derivative with respect to φ {\displaystyle \varphi } , S {\displaystyle S} is the action functional and T {\displaystyle
Schwinger–Dyson_equation
Function defined on formal languages in computer science
computer science, particularly in formal language theory, the Brzozowski derivative u − 1 S {\displaystyle u^{-1}S} of a set S {\displaystyle S} of strings
Brzozowski_derivative
Quantum-mechanical framework for simulating molecules and solids
(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
Optimized effective potential method
Optimized_effective_potential_method
Physics theorem for symmetries of action
parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations
Noether's_second_theorem
Specialized notation for multivariable calculus
partial derivatives for doing calculations. The Fréchet derivative is the standard way in the setting of functional analysis to take derivatives with respect
Matrix_calculus
In mathematics, the Hadamard derivative is a concept of directional derivative for maps between Banach spaces. It is particularly suited for applications
Hadamard_derivative
Tensor describing energy momentum density in spacetime
stress–energy: The Hilbert stress–energy tensor is defined as the functional derivative T μ ν = − 2 − g δ S m a t t e r δ g μ ν = − 2 − g ∂ ( − g L m a
Stress–energy_tensor
Application of Lagrangian mechanics to field theories
{\displaystyle {\mathcal {S}}} , is a functional of the dependent variables φ i ( s ) {\displaystyle \varphi _{i}(s)} , their derivatives and s itself S [ φ i ] =
Lagrangian_(field_theory)
Effective particle coupling beyond tree level
Γ μ {\displaystyle \Gamma ^{\mu }} can be defined in terms of a functional derivative of the effective action Seff as Γ μ = − 1 e δ 3 S e f f δ ψ ¯ δ
Vertex_function
Group of atoms giving a molecule characteristic properties
a functional group is any substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group
Functional_group
Constraints on possible particle properties
effective action is a functional of the background. The VEV of the stress–energy tensor is then defined as the functional derivative T M N ( x ) ≡ 1 − g
Weinberg–Witten_theorem
Approach to quantum theory
\ldots ,x_{n}\right).} In the functional source formalism, where correlations are derivatives of the generating functional Z [ J ] {\displaystyle Z[J]}
Schwinger's quantum action principle
Schwinger's_quantum_action_principle
Measure of mechanical stress at the atomic scale
The expression of the (local) virial stress can be derived as the functional derivative of the free energy of a molecular system with respect to the deformation
Virial_stress
Expressing a measure as an integral of another
Nikodym derivative". Stack Exchange. April 7, 2018. Brown, Arlen; Pearcy, Carl (1977). Introduction to Operator Theory I: Elements of Functional Analysis
Radon–Nikodym_theorem
Class of chemical compounds
simple hopenes, hopanols and hopanes, but also to extensively functionalized derivatives such as bacteriohopanepolyols (BHPs) and hopanoids covalently
Hopanoids
Description of the dynamics of magnetization in a solid
involving quantum mechanical effects, which is typically defined as the functional derivative of the magnetic free energy with respect to the local magnetization
Landau–Lifshitz–Gilbert equation
Landau–Lifshitz–Gilbert_equation
Concept in differential calculus
second derivative test to functionals. Much like for functions, at a stationary point where the first derivative is zero, the second derivative determines
Second_variation
Statement relating differentiable symmetries to conserved quantities
consisting of functions φ {\displaystyle \varphi } such that all functional derivatives of S {\displaystyle {\mathcal {S}}} at φ {\displaystyle \varphi
Noether's_theorem
Formulation of classical mechanics
Canonical coordinates Fundamental lemma of the calculus of variations Functional derivative Generalized coordinates Hamiltonian mechanics Hamiltonian optics
Lagrangian_mechanics
Programming language
Abstract Machine Language) is a multi-paradigm, general-purpose, high-level, functional programming language which is a dialect of the ML programming language
Caml
Quantum field theory
}^{b}(y)\right)\ } being the generating functional of the free theory. Expanding in g and computing the functional derivatives, we are able to obtain all the n-point
Yang–Mills_theory
Gauge field loop operator
to the functional derivative which acts on functions of functions, functions of loops admit two types of derivatives called the area derivative and the
Wilson_loop
Field theory involving topological effects in physics
} is independent on B {\displaystyle B} and acts similarly to a functional derivative): δ δ B α β S = ∫ M δ δ B α β B ∧ δ B + ∫ M B ∧ δ δ δ B α β B =
Topological quantum field theory
Topological_quantum_field_theory
Operation in differential calculus
In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as: lim h → 0 f ( x + h ) − f ( x − h ) 2
Symmetric_derivative
Nonlinear differential operator used to study conformal mappings
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in
Schwarzian_derivative
Manifold of functions in the calculus of variations
= 0. {\displaystyle \langle J'(u),u\rangle =0.} Here J′ is the functional derivative of J. A. Bahri and P. L. Lions (1988), Morse Index of Some Min-Max
Nehari_manifold
Mathematical formulation of special and general relativity
Fundamental lemma of the calculus of variations Canonical coordinates Functional derivative Generalized coordinates Hamiltonian mechanics Hamiltonian optics
Relativistic Lagrangian mechanics
Relativistic_Lagrangian_mechanics
Approach to linguistics
and functional linguistics nonetheless continue to be used by the Prague linguistic circle and its derivatives, including SILF, Danish functional school
Functional_linguistics
Model in theoretical ecology and statistical mechanics
} is the dynamical susceptibility defined in terms of a functional derivative of the dynamics with respect to a time-dependent perturbation of
Random generalized Lotka–Volterra model
Random_generalized_Lotka–Volterra_model
Chemical compound
organic compound with the formula H2NC6H4NO2. It is a derivative of aniline, carrying a nitro functional group in position 2. It is mainly used as a precursor
2-Nitroaniline
the functional group RnE(=O)xNR2, where x is not zero, E is some element, and each R represents an organic group or hydrogen. It is a derivative of an
Amide_(functional_group)
Mathematical framework
theory of functional connections (TFC) is a mathematical framework for functional interpolation. It provides a method for deriving a functional—a function
Theory of functional connections
Theory_of_functional_connections
Attempt to find a consistent theory of quantum gravity
Here Γ k ( 2 ) {\displaystyle \Gamma _{k}^{(2)}} is the second functional derivative of Γ k {\displaystyle \Gamma _{k}} with respect to the quantum fields
Asymptotic_safety
Rules for computing derivatives of functions
a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are
Differentiation_rules
Chemical compound
the simplest aldehyde with a prochiral methylene such that α-functionalized derivatives (CH3CH(X)CHO) are chiral. If water is available, propionaldehyde
Propionaldehyde
Special mathematical function
prime}}}{\frac {1}{1-\,\scriptstyle (-1)^{\frac {p-1}{2}}\textstyle p^{-s}}}.} The functional equation extends the beta function to the left side of the complex plane
Dirichlet_beta_function
equation cf. Action (physics) Fermat's principle Functional (mathematics) Functional derivative Functional integral Geodesic Isoperimetry Lagrangian Lagrangian
List_of_variational_topics
Notation of differential calculus
standard notation for differentiation. Instead, several notations for the derivative of a function or a dependent variable have been proposed by various mathematicians
Notation_for_differentiation
Chemical structure from which derivatives can be visualized
structure is the structure of an unadorned ion or molecule from which derivatives can be visualized. Parent structures underpin systematic nomenclature
Parent_structure
Type of field appearing in the Lagrangian
statistical field theories stem fundamentally from functional integrations and functional derivatives. Back to the Legendre transforms, The ⟨ ϕ ⟩ {\displaystyle
Source_field
Mathematical function, denoted exp(x) or e^x
function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted e x {\displaystyle e^{x}}
Exponential_function
Approach to library cataloging
Functional Requirements for Bibliographic Records (FRBR; /ˈfɜːrbər/) is a conceptual entity–relationship model developed by the International Federation
Functional Requirements for Bibliographic Records
Functional_Requirements_for_Bibliographic_Records
Quantum version of the classical action
{\displaystyle J(x)} that sources the scalar field. Taking the functional derivative of the Legendre transformation with respect to ϕ ( x ) {\displaystyle
Effective_action
General purpose functional programming language
later ML family (notably Standard ML, Caml, and their derivatives) and influenced subsequent functional language development. ML started development by Robin
ML_(programming_language)
Differential equation with deviating argument
equation that contains a function and some of its derivatives evaluated at different argument values. Functional differential equations find use in mathematical
Functional differential equation
Functional_differential_equation
Design pattern in functional programming to build generic types
programming with monads was largely confined to Haskell and its derivatives, but as functional programming has influenced other paradigms, many languages have
Monad (functional programming)
Monad_(functional_programming)
Relation satisfied by conjugate variables in quantum mechanics
and momentum operators, it can be viewed as a functional, and we may write (using functional derivatives): [ H ^ , Q ^ ] = δ H ^ δ P ^ ⋅ [ P ^ , Q ^ ]
Canonical commutation relation
Canonical_commutation_relation
Measure of difference between two points
Srivastava, Santosh; Gupta, Maya R. (2008). An Introduction to Functional Derivatives (PDF). UWEE Tech Report 2008-0001. University of Washington, Dept
Bregman_divergence
Organic compound with a C=C–OH group
chemistry, enols are a type of functional group or intermediate in organic chemistry. Formally, enols are derivatives of vinyl alcohol, with a C=C−OH
Enol
Determinant in functional analysis
In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order
Functional_determinant
Study of rates of change
as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra. The theory of derivatives is studied more closely
Differential_calculus
Complex-differentiable (mathematical) function
extended to the infinite-dimensional spaces of functional analysis. For instance, the Fréchet or Gateaux derivative can be used to define a notion of a holomorphic
Holomorphic_function
Organic compound containing the functional group R–CH=O
alcohol) is an organic compound containing a functional group with the structure R−CH=O. The functional group itself (without the "R" side chain) can
Aldehyde
Theory of quantum gravity merging quantum mechanics and general relativity
{\displaystyle {\hat {q}}\psi (q)=q\psi (q)} ) and the triads are (functional) derivatives, E i a ~ ^ Ψ ( A ) = − i δ Ψ ( A ) δ A a i . {\displaystyle {\hat
Loop_quantum_gravity
hyperactivity disorder (ADHD) and certain other indications. Several other derivatives including rimiterol, phacetoperane and pipradrol also have more limited
List of methylphenidate analogues
List_of_methylphenidate_analogues
Branch of mathematics
action. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may
Calculus
Generalization of Riemannian manifolds
(t),{\dot {\gamma }}(t)\right)\,dt} in the sense that its functional derivative vanishes among differentiable curves γ: [a, b] → M with fixed endpoints
Finsler_manifold
Chemical compound
it contains both an amine and a thiol functional group. It is often used as the salt of the ammonium derivative [HSCH2CH2NH3]+, including the hydrochloride
Cysteamine
Chemical compound
with the formula H2NC6H4NO2. A yellow solid, it is a derivative of aniline, carrying a nitro functional group in position 3. It is an isomer of 2-nitroaniline
3-Nitroaniline
Technique of representing an aggregate data structure
the structure arbitrarily and update its contents, especially in purely functional programming languages. The zipper was described by Gérard Huet in 1997
Zipper_(data_structure)
FUNCTIONAL DERIVATIVE
FUNCTIONAL DERIVATIVE
Boy/Male
English
The fictional character Jorel father of Superman.
Boy/Male
English
The fictional character Jorel father of Superman.
Boy/Male
American, British, English
Mighty Spearman; The Fictional Character Jorel Father of Superman
Male
Egyptian
, a high Egyptian functionary.
Boy/Male
English
The fictional character Jorel father of Superman.
Male
Egyptian
, a great functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Celtic
, great justiciary, or functionary.
Boy/Male
English
Modern. The fictional character Jorel father of Superman.
Boy/Male
American, Australian, British, Danish, English, Finnish, French, German, Scandinavian
Farmer; The Fictional Character Jorel Father of Superman; Earth Worker
Male
Egyptian
, Functionary of the Interior.
Male
Egyptian
, an Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
Respelling of German Killmann, probably a derivative of Kilian.English
Respelling of German Killmann, probably a derivative of Kilian.English : variant of Gillman.
Boy/Male
French
Fictional swordsman: (ambitious and filled with religious aspirations) from Alexander Dumas's...
Boy/Male
Australian, French
Fictional Swordsman; Ambitious and Filled with Religious Aspirations; From Alexander Dumas's Three Musketeers
Boy/Male
American, British, English
Mighty Spearman; One who Saves; The Fictional Character Jorel Father of Superman
Male
Egyptian
, the son of the functionary Heknofre.
Biblical
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Boy/Male
American, Australian, British, English, French
Mighty Spearman; The Fictional Character Jorel Father of Superman
FUNCTIONAL DERIVATIVE
FUNCTIONAL DERIVATIVE
Boy/Male
Biblical
Reward, recompense.
Boy/Male
Gypsy
Fortune-teller.
Girl/Female
Indian
Shining, Goddess
Girl/Female
Indian, Sanskrit
Flower
Boy/Male
Indian, Punjabi, Sikh
Godly Light
Girl/Female
Tamil
Worship
Surname or Lastname
English (Leicestershire)
English (Leicestershire) : variant of Culver.
Girl/Female
Muslim
Excellent (Name of the wife of caliph Harun al Rashid)
Boy/Male
American, Australian, Biblical, British, Chinese, Christian, Danish, Dutch, English, French, German, Greek, Hebrew, Irish, Jamaican, Lebanese, Portuguese, Swedish, Swiss
Warlike; Loves Horses; Horse Lover; Friend of Horses
Boy/Male
Indian, Tamil
God Murugan; Subburamani
FUNCTIONAL DERIVATIVE
FUNCTIONAL DERIVATIVE
FUNCTIONAL DERIVATIVE
FUNCTIONAL DERIVATIVE
FUNCTIONAL DERIVATIVE
v. i.
To execute or perform a function; to transact one's regular or appointed business.
a.
Fractional.
n.
A derived function; a function obtained from a given function by a certain algebraic process.
n.
Paper fractional currency.
a.
Capable of, or pertaining to, flection or inflection.
a.
Pertaining to the function of an organ or part, or to the functions in general.
a.
Relating to friction; moved by friction; produced by friction; as, frictional electricity.
n.
The office, duties, or functions of a minister, servant, or agent; ecclesiastical, executive, or ambassadorial function or profession.
v. t.
To supply with an organ or organs having a special function or functions.
a.
Of or pertaining to fractions or a fraction; constituting a fraction; as, fractional numbers.
n.
An angle upon which the value of some function depends; -- a term used more especially in connection with elliptic functions.
v. i.
Alt. of Functionate
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
adv.
In a functional manner; as regards normal or appropriate activity.
pl.
of Functionary
a.
Pertaining to, or characterized by, fiction; fictitious; romantic.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
a.
Relatively small; inconsiderable; insignificant; as, a fractional part of the population.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
a.
Pertaining to, or connected with, a function or duty; official.