Differential operator Details, Meaning Differential operator Article and Explanation Guide

Differential operator Guide, Meaning , Facts, Information and Description

In mathematics, a differential operator is a linear 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).

Table of contents

1 Notations
2 Operator methods
3 Other forms
4 Properties of differential operators
5 Several variables
6 Coordinate-independent description
7 Examples
8 See also

Notations

The most commonly used differential operator is the action of taking the derivative itself. Common notations for this operator include:

, where the variable one is differentating to is clear, and

, where the variable is declared explicitly.

First derivatives are signified as above, but when taking higher, n-th derivatives, the following alterations are useful:

Operator methods

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

in his study of differential equations.

Other forms

One of the most frequently seen differential operators is the Laplacian operator

Properties of differential operators

Differentiation is linear, ie

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

(D₁oD₂)(f) = D₁ (D₂(f)).

Some care is then required: firstly any function coefficients in the operator D₂ must be differentiable as many times as the application of D₁ 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

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. An operator is a mapping of sections, which maps the stalk of the sheaf of germs of at a point to the fibre of F at x:

An operator P is said to be a k-th order differential operator if it factors through the jet bundle . In other words, there exists a linear mapping of vector bundles

such that as in the following composition:

Examples

In applications to the physical sciences, operators such as the Laplace operator play a major role via the setting up and solution of partial differential equations.

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

In abstract algebra the concept of derivation means that differential operators may still be defined, in the absence of calculus concepts based on geometry.