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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).

1 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:

2 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.

Given a linear differential operator

the adjoint of this operator is defined as

Second order linear formally self-adjoint differential operators L can be written in the form

since using the adjoint definition above,

This operator is central to Sturm-Liouville theory where the eigenfunctions (analogues to eigenvectors) of this operator are considered.

This operator L is known as formally self-adjoint, different from the usage to self-adjoint operators in Hilbert spaces, in that if we define the inner product

we desire the evaluation at the endpoints to be zero at infinity, which only occurs under certain conditions.

3 Other forms

One of the most frequently seen differential operators is the Laplacian operatorMultivariate calculus In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. It is denoted by the symbol Δ. Since it can be calculated

4 Properties of differential operators

Differentiation is linearIn mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. It follows from the sum rule in differentiation and the constant factor rule in differentiation. Thus we can say that the act of d, 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 ringIn ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. History See Ring theory Definition and notation A ring is an a 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 mechanicswavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: n 1,2,3,. and angular momentum (increasing across: s p d . Brighter areas correspond to higher probability density for a position measurement. The angular mom:

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.