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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 Equation s.

One of the most frequently seen differential operators is the Laplacian Operator , defined by

:\Delta=
abla^{2}=\sum_{k=1}^n {\partial^2\over \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
  • such that

  • u, v angle

    • where the notation \langle, 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

  • (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


  • 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')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u\;\!
This property can be proven using the formal adjoint definition above.
: \begin{matrix}

&=& -(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''1o''D''2)(f) = ''D''1 {Link without Title} .

Some care is then required: firstly any function coefficients in the operator ''D''2 must be :

Dx


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 Derivative s, differentiation with respect to different variables giving rise to operators that commute (see Symmetry Of Second Derivatives ).


COORDINATE-INDEPENDENT DESCRIPTION

In of the Sheaf of Germs of Γ(''E'') at a point ''x'' ∈ ''M'' to the Fibre of ''F'' at ''x'':

:Γ''x''(''E'') → ''F''''x'' .

An operator ''P'' is said to be a ''k''th order differential operator if it factors through the Jet Bundle ''J''k(''E''). In other words, there exists a linear mapping of vector bundles

i


such that ''P'' = ''i''''P'' o ''j''''k'' as in the following composition:

P


A foundational result and characterization is the Peetre Theorem .


EXAMPLES






SEE ALSO