DIFFERENTIAL OPERATOR

In mathematics, a 'differential operator' is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another (in the style of a higher-order function in computer science).
There are certainly reasons not to restrict to ''linear'' operators; for instance the Schwarzian derivative is a well-known non-linear operator. Only the linear case will be addressed here.

Contents

Notations

Adjoint of an operator

Properties of differential operators

Several variables

Coordinate-independent description and relation to commutative algebra

Examples

See also

Notations

The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include:
: $\{d\; over\; dx\}$
: $D,,$ where the variable one is differentiating to is clear, and
: $D\_x,,$ where the variable is declared explicitly.
First derivatives are signified as above, but when taking higher, n-th derivatives, the following alterations are useful:
: $d^n\; over\; dx^n$
: $D^n,$
: $D^n\_x.,$
The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form
: $sum\_\{k=0\}^n\; c\_k\; D^k$
in his study of differential equations.
One of the most frequently seen differential operators is the Laplacian operator, defined by
:$Delta=$
abla^{2}=sum_{k=1}^n {partial^2over partial x_k^2}.
Another differential operator is the Θ operator, defined by
:$Theta\; =\; z\; \{d\; over\; dz\}.$

Adjoint of an operator

Given a linear differential operator
: $Tu\; =\; sum\_\{k=0\}^n\; a\_k(x)\; D^k\; u$
the adjoint of this operator is defined as the operator $T^$
★ such that
: $langle\; u,Tv$
angle = langle T^
★ u, v
angle
where the notation $langlecdot,cdot$
angle is used for the scalar product or inner product. This definition therefore depends on the definition of the scalar product. In the functional space of square integrable functions, the scalar product is defined by
: $langle\; f,\; g$
angle = int_a^b overline{f(x)} , g(x) ,dx.
If one moreover adds the condition that ''f'' and ''g'' vanish for $x\; o\; a$ and $x\; o\; b$, one can also define the adjoint of ''T'' by
: $T^$
★ u = sum_{k=0}^n (-1)^k D^k [a_k(x)u].,
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 $T^$
★ is defined according to this formula, it is called the 'formal adjoint' of ''T''.
A 'self-adjoint' operator is an operator adjoint of itself.
The Sturm-Liouville operator is a well-known example of formal self-adjoint operator. This second order linear differential operators ''L'' can be written in the form
: $Lu\; =\; -(pu\text{'})\text{'}+qu=-(pu\text{'}\text{'}+p\text{'}u\text{'})+qu=-pu\text{'}\text{'}-p\text{'}u\text{'}+qu=(-p)\; D^2\; u\; +(-p\text{'})\; D\; u\; +\; (q)u.;!$
This property can be proven using the formal adjoint definition above.
:
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{matrix}
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 linear, i.e.,
:$D(f+g)\; =\; (Df)+(Dg),$
:$D(af)\; =\; a(Df),$
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
:$(D\_1\; circ\; D\_2)(f)\; =\; D\_1(D\_2(f)).,$
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 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:
:$Dx\; -\; xD\; =\; 1.,$
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.

Several variables

The same constructions can be carried out with partial derivatives, differentiation with respect to different variables giving rise to operators that commute (see symmetry of second derivatives).

Coordinate-independent description and relation to commutative algebra

In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles. Let $E$ and $F$ be two vector bundles over a manifold $M$. A linear mapping of sections $P:\; Gamma(E)$
ightarrow Gamma(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
:$i\_P:\; J^k(E)$
ightarrow F,
such that
:$P\; =\; hat\{i\}\_Pcirc\; j^k$
where $hat\{i\}\_P$ denotes the map induced by $i\_P,$ on sections , and $j^k:Gamma(E)$
ightarrow Gamma(J^k(E)), is the canonical map.
This just means that for a given sections $s$ of $E$, the value of $P(s)$ at a point $xin\; M$ is fully determined by the ''k''-th order infinitesimal behavior of $s$ in $x$. In particular does this imply, that $P(s)(x)$ is determined by the 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 local operator is differential.
An equivalent, but purely algebraic description of linear differential operators is as follows: a linear map $P$ is a ''k''-th order linear differential operator, if for any k+1 smooth functions $f\_0,ldots,f\_k\; in\; C^infty(M)$ we have
:$[f\_k[f\_\{k-1\}[cdots[f\_0,P]cdots]]=0.$
Here the bracket $[f,P]:Gamma(E)$
ightarrow Gamma(F) is defined as the commutator
:$[f,P](s)=P(fcdot\; s)-fcdot\; P(s).,$
This characterization of linear differential operators shows that they are particular mappings between modules over a commutative 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 equations.

★ In differential topology the exterior derivative and Lie derivative operators have intrinsic meaning.

★ In abstract algebra, the concept of a 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).

See also

★ Difference operator

★ Delta operator

★ Elliptic operator

★ Fractional calculus

★ Invariant differential operators