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| In [[mathematics]], a '''differential operator''' is an [[Operator (mathematics)|operator]] defined as a function of the [[derivative|differentiation]] operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a [[higher-order function]] in [[computer science]]).
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| This article considers mainly [[linear map|linear]] operators, which are the most common type. However, non-linear differential operators, such as the [[Schwarzian derivative]] also exist.
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| ==Notations==
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| The most common differential operator is the action of taking the [[derivative]] itself. Common notations for taking the first derivative with respect to a variable ''x'' include: | |
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| : <math>{d \over dx}, D,\, D_x,\,</math> and <math>\partial_x.</math>
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| When taking higher, ''n''th order derivatives, the operator may also be written:
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| : <math>{d^n \over dx^n},</math> <math>D^n\,,</math> or <math>D^n_x.\,</math>
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| The derivative of a function ''f'' of an argument ''x'' is sometimes given as either of the following:
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| : <math>[f(x)]'\,\!</math>
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| : <math>f'(x).\,\!</math>
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| The ''D'' notation's use and creation is credited to [[Oliver Heaviside]], who considered differential operators of the form
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| : <math>\sum_{k=0}^n c_k D^k</math>
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| in his study of [[differential equation]]s.
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| One of the most frequently seen differential operators is the [[Laplace operator|Laplacian operator]], defined by
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| :<math>\Delta=\nabla^{2}=\sum_{k=1}^n {\partial^2\over \partial x_k^2}.</math>
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| Another differential operator is the Θ operator, or [[theta operator]], defined by<ref>{{cite web|url=http://mathworld.wolfram.com/ThetaOperator.html|title=Theta Operator|author=E. W. Weisstein|accessdate=2009-06-12}}</ref>
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| :<math>\Theta = z {d \over dz}.</math>
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| This is sometimes also called the '''homogeneity operator''', because its [[eigenfunction]]s are the [[monomial]]s in ''z'':
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| :<math>\Theta (z^k) = k z^k,\quad k=0,1,2,\dots </math>
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| In ''n'' variables the homogeneity operator is given by
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| :<math>\Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}.</math>
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| As in one variable, the [[eigenspace]]s of Θ are the spaces of [[homogeneous polynomial]]s.
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| The result of applying the differential to the left{{Clarify|date=February 2012}} and to the right{{Clarify|date=February 2012}}, and the difference obtained when applying the differential operator to the left and to the right, are denoted by arrows as follows:
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| :<math>f \overleftarrow{\partial_x} g = g \partial_x f</math>
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| :<math>f \overrightarrow{\partial_x} g = f \partial_x g</math>
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| :<math>f \overleftrightarrow{\partial_x} g = f \partial_x g - g \partial_x f.</math>
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| Such a bidirectional-arrow notation is frequently used for describing the [[probability current]] of quantum mechanics.
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| ==Del==
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| {{Main|Del}}
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| The differential operator del, also called nabla operator, is an important [[Euclidean vector|vector]] differential operator. It appears frequently in [[physics]] in places like the differential form of [[Maxwell's Equations]]. In three dimensional [[Cartesian coordinates]], del is defined:
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| :<math>\nabla = \mathbf{\hat{x}} {\partial \over \partial x} + \mathbf{\hat{y}} {\partial \over \partial y} + \mathbf{\hat{z}} {\partial \over \partial z}.</math>
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| Del is used to calculate the [[gradient]], [[curl (mathematics)|curl]], [[divergence]], and [[Laplacian]] of various objects.
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| ==Adjoint of an operator==
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| {{See also|Hermitian adjoint}}
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| Given a linear differential operator T
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| : <math>Tu = \sum_{k=0}^n a_k(x) D^k u</math>
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| the [[Hermitian adjoint|adjoint of this operator]] is defined as the operator <math>T^*</math> such that
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| : <math>\langle Tu,v \rangle = \langle u, T^*v \rangle</math>
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| where the notation <math>\langle\cdot,\cdot\rangle</math> is used for the [[scalar product]] or [[inner product]]. This definition therefore depends on the definition of the scalar product.
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| === Formal adjoint in one variable ===
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| In the functional space of [[square integrable]] functions, the scalar product is defined by
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| : <math>\langle f, g \rangle = \int_a^b f(x) \, \overline{g(x)} \,dx , </math>
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| where the line over ''g(x)'' denotes the complex conjugate of ''g(x)''. If one moreover adds the condition that ''f'' or ''g'' vanishes for <math>x \to a</math> and <math>x \to b</math>, one can also define the adjoint of ''T'' by
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| : <math>T^*u = \sum_{k=0}^n (-1)^k D^k [\overline{a_k(x)}u].\,</math>
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| This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When <math>T^*</math> is defined according to this formula, it is called the '''formal adjoint''' of ''T''.
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| A (formally) '''[[self-adjoint operator|self-adjoint]]''' operator is an operator equal to its own (formal) adjoint.
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| === Several variables ===
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| If Ω is a domain in '''R'''<sup>n</sup>, and ''P'' a differential operator on Ω, then the adjoint of ''P'' is defined in [[Lp space|''L''<sup>2</sup>(Ω)]] by duality in the analogous manner:
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| :<math>\langle f, P^* g\rangle_{L^2(\Omega)} = \langle P f, g\rangle_{L^2(\Omega)}</math>
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| for all smooth ''L''<sup>2</sup> functions ''f'', ''g''. Since smooth functions are dense in ''L''<sup>2</sup>, this defines the adjoint on a dense subset of ''L''<sup>2</sup>: P<sup>*</sup> is a [[densely-defined operator]]. | |
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| === Example ===
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| The [[Sturm–Liouville theory|Sturm–Liouville]] operator is a well-known example of a formal self-adjoint operator. This second-order linear differential operator ''L'' can be written in the form
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| : <math>Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.\;\!</math>
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| This property can be proven using the formal adjoint definition above.
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| : <math>\begin{align}
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| L^*u & {} = (-1)^2 D^2 [(-p)u] + (-1)^1 D [(-p')u] + (-1)^0 (qu) \\
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| & {} = -D^2(pu) + D(p'u)+qu \\
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| & {} = -(pu)''+(p'u)'+qu \\
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| & {} = -p''u-2p'u'-pu''+p''u+p'u'+qu \\
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| & {} = -p'u'-pu''+qu \\
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| & {} = -(pu')'+qu \\
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| & {} = Lu
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| \end{align}</math>
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| This operator is central to [[Sturm–Liouville theory]] where the [[eigenfunctions]] (analogues to [[eigenvectors]]) of this operator are considered.
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| ==Properties of differential operators==
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| Differentiation is [[linearity of differentiation|linear]], i.e.,
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| :<math>D(f+g) = (Df)+(Dg)\,</math>
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| :<math>D(af) = a(Df)\,</math>
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| where ''f'' and ''g'' are functions, and ''a'' is a constant.
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| Any polynomial in ''D'' with function coefficients is also a differential operator. We may also compose differential operators by the rule
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| :<math>(D_1 \circ D_2)(f) = D_1(D_2(f)).\,</math>
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| Some care is then required: firstly any function coefficients in the operator ''D''<sub>2</sub> must be [[differentiable]] as many times as the application of ''D''<sub>1</sub> requires. To get a [[ring (mathematics)|ring]] of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be [[commutative]]: an operator ''gD'' isn't the same in general as ''Dg''. In fact we have for example the relation basic in [[quantum mechanics]]:
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| :<math>Dx - xD = 1.\,</math>
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| The subring of operators that are polynomials in ''D'' with [[constant coefficients]] is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.
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| The differential operators also obey the [[shift theorem]].
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| ==Several variables==
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| The same constructions can be carried out with [[partial derivative]]s, differentiation with respect to different variables giving rise to operators that commute (see [[symmetry of second derivatives]]).
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| ==Coordinate-independent description==
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| In [[differential geometry]] and [[algebraic geometry]] it is often convenient to have a [[coordinate]]-independent description of differential operators between two [[vector bundle]]s. Let ''E'' and ''F'' be two vector bundles over a [[differentiable manifold]] ''M''. An '''R'''-linear mapping of [[vector bundle|sections]] {{nowrap|''P'' : Γ(''E'') → Γ(''F'')}} is said to be a '''''k''th-order linear differential operator''' if it factors through the [[jet bundle]] ''J''<sup>''k''</sup>(''E'').
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| In other words, there exists a linear mapping of vector bundles
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| :<math>i_P: J^k(E) \rightarrow F\,</math>
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| such that
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| :<math>P = i_P\circ j^k</math>
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| where {{nowrap | ''j''<sup>''k''</sup>: Γ(''E'') → Γ(''J''<sup>''k''</sup>(''E''))}} is the prolongation that associates to any section of ''E'' its [[jet (mathematics)|''k''-jet]].
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| This just means that for a given [[vector bundle|sections]] ''s'' of ''E'', the value of ''P''(''s'') at a point ''x'' ∈ ''M'' is fully determined by the ''k''th-order infinitesimal behavior of ''s'' in ''x''. In particular this implies that ''P''(''s'')(''x'') is determined by the [[sheaf (mathematics)|germ]] of ''s'' in ''x'', which is expressed by saying that differential operators are local. A foundational result is the [[Peetre theorem]] showing that the converse is also true: any (linear) local operator is differential.
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| ===Relation to commutative algebra===
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| An equivalent, but purely algebraic description of linear differential operators is as follows: an '''R'''-linear map ''P'' is a ''k''th-order linear differential operator, if for any ''k'' + 1 smooth functions <math>f_0,\ldots,f_k \in C^\infty(M)</math> we have
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| :<math>[f_k,[f_{k-1},[\cdots[f_0,P]\cdots]]=0.</math>
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| Here the bracket <math>[f,P]:\Gamma(E)\rightarrow \Gamma(F)</math> is defined as the commutator
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| :<math>[f,P](s)=P(f\cdot s)-f\cdot P(s).\,</math>
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| This characterization of linear differential operators shows that they are particular mappings between [[module (mathematics)|modules]] over a commutative [[algebra (ring theory)|algebra]], allowing the concept to be seen as a part of [[commutative algebra]].
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| ==Examples==
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| * In applications to the physical sciences, operators such as the [[Laplace operator]] play a major role in setting up and solving [[partial differential equation]]s.
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| * In [[differential topology]] the [[exterior derivative]] and [[Lie derivative]] operators have intrinsic meaning.
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| * In [[abstract algebra]], the concept of a [[derivation (abstract algebra)|derivation]] allows for generalizations of differential operators which do not require the use of calculus. Frequently such generalizations are employed in [[algebraic geometry]] and [[commutative algebra]]. See also [[jet (mathematics)]].
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| * In the development of [[holomorphic function]]s of a [[complex variable]] ''z'' = ''x'' + i ''y'', sometimes a complex function is considered to be a function of two real variables ''x'' and ''y''. Use is made of the [[Wirtinger derivative]]s, which are partial differential operators:
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| ::<math> \frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \quad,\quad \frac{\partial}{\partial\bar{z}}= \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) \ .</math>
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| This approach is also used to study functions of [[several complex variables]] and functions of a [[motor variable]].
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| ==History==
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| The conceptual step of writing a differential operator as something free-standing is attributed to [[Louis François Antoine Arbogast]] in 1800.<ref>James Gasser (editor), ''A Boole Anthology: Recent and classical studies in the logic of George Boole'' (2000), p. 169; [http://books.google.co.uk/books?id=A2Q5Yghl000C&pg=PA169 Google Books].</ref>
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| ==See also==
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| * [[Difference operator]]
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| * [[Delta operator]]
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| * [[Elliptic operator]]
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| * [[Fractional calculus]]
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| * [[Invariant differential operator]]
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| * [[Differential calculus over commutative algebras]]
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| * [[Lagrangian system]]
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| * [[Spectral theory]]
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| * [[Energy operator]]
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| * [[Momentum operator]]
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| * [[DBAR operator]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{springer|title=Differential operator|id=p/d032250}}
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| [[Category:Calculus]]
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| [[Category:Multivariable calculus]]
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| [[Category:Differential operators|*]]
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