Table of contents |
2 Operator methods 3 Properties of differential operators 4 Several variables 5 Examples |
Common notations include:
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:
The differential operator is linear, ie
Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by the rule
(D_{1}oD_{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.
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).
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.