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2013-12-15T16:29:07Z

37.201.192.197: /* Limitations and resynthesis */ linked to Instantaneous frequency

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]]).

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.

==Notations==
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:

: <math>{d \over dx}, D,\, D_x,\,</math> and <math>\partial_x.</math>

When taking higher, ''n''th order derivatives, the operator may also be written:

: <math>{d^n \over dx^n},</math> <math>D^n\,,</math> or <math>D^n_x.\,</math>

The derivative of a function ''f'' of an argument ''x'' is sometimes given as either of the following:

: <math>[f(x)]'\,\!</math>
: <math>f'(x).\,\!</math>

The ''D'' notation's use and creation is credited to [[Oliver Heaviside]], who considered differential operators of the form

: <math>\sum_{k=0}^n c_k D^k</math>

in his study of [[differential equation]]s.

One of the most frequently seen differential operators is the [[Laplace operator|Laplacian operator]], defined by

:<math>\Delta=\nabla^{2}=\sum_{k=1}^n {\partial^2\over \partial x_k^2}.</math>

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>

:<math>\Theta = z {d \over dz}.</math>

This is sometimes also called the '''homogeneity operator''', because its [[eigenfunction]]s are the [[monomial]]s in ''z'':

:<math>\Theta (z^k) = k z^k,\quad k=0,1,2,\dots </math>

In ''n'' variables the homogeneity operator is given by

:<math>\Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}.</math>

As in one variable, the [[eigenspace]]s of Θ are the spaces of [[homogeneous polynomial]]s.

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:
:<math>f \overleftarrow{\partial_x} g = g \partial_x f</math>
:<math>f \overrightarrow{\partial_x} g = f \partial_x g</math>
:<math>f \overleftrightarrow{\partial_x} g = f \partial_x g - g \partial_x f.</math>
Such a bidirectional-arrow notation is frequently used for describing the [[probability current]] of quantum mechanics.

==Del==
{{Main|Del}}
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:

:<math>\nabla = \mathbf{\hat{x}} {\partial \over \partial x} + \mathbf{\hat{y}} {\partial \over \partial y} + \mathbf{\hat{z}} {\partial \over \partial z}.</math>

Del is used to calculate the [[gradient]], [[curl (mathematics)|curl]], [[divergence]], and [[Laplacian]] of various objects.

==Adjoint of an operator==
{{See also|Hermitian adjoint}}
Given a linear differential operator T
: <math>Tu = \sum_{k=0}^n a_k(x) D^k u</math>
the [[Hermitian adjoint|adjoint of this operator]] is defined as the operator <math>T^*</math> such that
: <math>\langle Tu,v \rangle = \langle u, T^*v \rangle</math>
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.

=== Formal adjoint in one variable ===

In the functional space of [[square integrable]] functions, the scalar product is defined by

: <math>\langle f, g \rangle = \int_a^b f(x) \, \overline{g(x)} \,dx , </math>

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

: <math>T^*u = \sum_{k=0}^n (-1)^k D^k [\overline{a_k(x)}u].\,</math>

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''.

A (formally) '''[[self-adjoint operator|self-adjoint]]''' operator is an operator equal to its own (formal) adjoint.

=== Several variables ===

If Ω is a domain in '''R'''n, and ''P'' a differential operator on Ω, then the adjoint of ''P'' is defined in [[Lp space|''L''2(Ω)]] by duality in the analogous manner:

:<math>\langle f, P^* g\rangle_{L^2(\Omega)} = \langle P f, g\rangle_{L^2(\Omega)}</math>

for all smooth ''L''2 functions ''f'', ''g''. Since smooth functions are dense in ''L''2, this defines the adjoint on a dense subset of ''L''2: P* is a [[densely-defined operator]].

=== Example ===
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

: <math>Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.\;\!</math>

This property can be proven using the formal adjoint definition above.

: <math>\begin{align}
L^*u & {} = (-1)^2 D^2 [(-p)u] + (-1)^1 D [(-p')u] + (-1)^0 (qu) \\
& {} = -D^2(pu) + D(p'u)+qu \\
& {} = -(pu)''+(p'u)'+qu \\
& {} = -p''u-2p'u'-pu''+p''u+p'u'+qu \\
& {} = -p'u'-pu''+qu \\
& {} = -(pu')'+qu \\
& {} = Lu
\end{align}</math>

This operator is central to [[Sturm–Liouville theory]] where the [[eigenfunctions]] (analogues to [[eigenvectors]]) of this operator are considered.

==Properties of differential operators==

Differentiation is [[linearity of differentiation|linear]], i.e.,

:<math>D(f+g) = (Df)+(Dg)\,</math>

:<math>D(af) = a(Df)\,</math>

where ''f'' and ''g'' are functions, and ''a'' is a constant.

Any polynomial in ''D'' with function coefficients is also a differential operator. We may also compose differential operators by the rule

:<math>(D_1 \circ D_2)(f) = D_1(D_2(f)).\,</math>

Some care is then required: firstly any function coefficients in the operator ''D''2 must be [[differentiable]] as many times as the application of ''D''1 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]]:

:<math>Dx - xD = 1.\,</math>

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.

The differential operators also obey the [[shift theorem]].

==Several variables==

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]]).

==Coordinate-independent description==
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''''k''(''E'').
In other words, there exists a linear mapping of vector bundles

:<math>i_P: J^k(E) \rightarrow F\,</math>

such that

:<math>P = i_P\circ j^k</math>

where {{nowrap | ''j''''k'': Γ(''E'') → Γ(''J''''k''(''E''))}} is the prolongation that associates to any section of ''E'' its [[jet (mathematics)|''k''-jet]].

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.

===Relation to commutative algebra===
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

:<math>[f_k,[f_{k-1},[\cdots[f_0,P]\cdots]]=0.</math>

Here the bracket <math>[f,P]:\Gamma(E)\rightarrow \Gamma(F)</math> is defined as the commutator

:<math>[f,P](s)=P(f\cdot s)-f\cdot P(s).\,</math>

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]].

==Examples==

* 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.

* In [[differential topology]] the [[exterior derivative]] and [[Lie derivative]] operators have intrinsic meaning.

* 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)]].

* 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:
::<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>
This approach is also used to study functions of [[several complex variables]] and functions of a [[motor variable]].

==History==
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>

==See also==
* [[Difference operator]]
* [[Delta operator]]
* [[Elliptic operator]]
* [[Fractional calculus]]
* [[Invariant differential operator]]
* [[Differential calculus over commutative algebras]]
* [[Lagrangian system]]
* [[Spectral theory]]
* [[Energy operator]]
* [[Momentum operator]]
* [[DBAR operator]]

==References==
{{Reflist}}
==External links==
* {{springer|title=Differential operator|id=p/d032250}}

[[Category:Calculus]]
[[Category:Multivariable calculus]]
[[Category:Differential operators|*]]

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