Differential operator

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 Properties of differential operators 4 Several variables 5 Examples

Notations

Common notations 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, in which he used a method involving the differential operator symbolically to solve differential equations.

The differential operator is not always used to signify differentiation. The act of integration is analogous to "backwards differentiation" (more precisely indefinite integrals are computed as antiderivatives). One can signify the act of integration in terms of the differential operator by using a superscript of -1. For example, the following notations are equivalent:

Properties of differential operators

The differential operator 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 (a point that must be clarified to avoid pathological examples).

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.