词条 | Functional (mathematics) |
释义 |
In mathematics, the term functional (as a noun) has at least two meanings.
This article is mainly concerned with the second concept, which arose in the early 18th century as part of the calculus of variations. The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name linear form. Commonly, the space is a space of functions. In this case, the functional is a "function of a function", and some older authors actually define the term "functional" to mean "function of a function". However, the fact that is a space of functions is not mathematically essential, so this older definition is no longer prevalent.{{Citation Needed|date=January 2019}} The term originates in the calculus of variations, where one searches for a function that minimizes a given functional. A particularly important application in physics is searching for a state of a system that minimizes the energy functional. DetailsDualityThe mapping is a function, where {{math|x0}} is an argument of a function {{math|f}}. At the same time, the mapping of a function to the value of the function at a point is a functional; here, {{math|x0}} is a parameter. Provided that {{math|f}} is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals. Definite integralIntegrals such as form a special class of functionals. They map a function into a real number, provided that is real-valued. Examples include
Vector scalar productGiven any vector in a vector space , the scalar product with another vector , denoted or , is a scalar. The set of vectors such that is zero is a vector subspace of , called the null space or kernel of . LocalityIf a functional's value can be computed for small segments of the input curve and then summed to find the total value, the functional is called local. Otherwise it is called non-local. For example: is local while is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass. Equation solving{{Main|Functional equation}}The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation {{math|1=F = G}} between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it is said that an additive function {{math|f}} is one satisfying the functional equation Derivative and integration{{see also|Calculus of variations}}Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals: i.e. they carry information on how a functional changes when the input function changes by a small amount. Richard Feynman used functional integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space. See also
References
1 : Types of functions |
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