In mathematics, a differential operator is an 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). For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... For a non-technical overview of the subject, see Calculus. ... In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

There are certainly reasons not to restrict to linear operators; for instance the Schwarzian derivative is a well-known non-linear operator. Only the linear case will be addressed here. In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations. ...

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 differentiating 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:

The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form Oliver Heaviside (May 18, 1850 – February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and...

in his study of differential equations. Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

One of the most frequently seen differential operators is the Laplacian operator, defined by In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...

Another differential operator is the Θ operator, defined by

Adjoint of an operator

Given a linear differential operator

the adjoint of this operator is defined as the operator T ^* such that In mathematics, specifically in functional analysis, each linear operator on a Hilbert space has a corresponding adjoint operator. ...

where the notation 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 In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ...

If one moreover adds the condition that f and g vanish for and , one can also define the adjoint of T by

This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When T ^* 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 In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...

This property can be proven using the formal adjoint definition above.

This operator is central to Sturm-Liouville theory where the eigenfunctions (analogues to eigenvectors) of this operator are considered. In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset... In mathematics, an eigenfunction f of a linear operator A on a function space is an eigenvector of the linear operator; it satisfies for some scalar λ, the corresponding eigenvalue. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

Properties of differential operators

Differentiation is linear, i.e., In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. ...

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

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: In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...

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. In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions. ...

The differential operators also obey the shift theorem. In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (D-operators) and exponential functions. ...

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). In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ... In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a function f(x1, x2, ..., xn) of n variables. ...

Coordinate-independent description and relation to commutative algebra

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 -linear mapping of sections is said to be a k-th order linear differential operator if it factors through the jet bundle . In other words, there exists a linear mapping of vector bundles In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ...

such that

where denotes the map induced by on sections , and is the canonical (or universal) k-th order differential operator.

This just means that for a given sections s of E, the value of P(s) at a point is fully determined by the k-th order infinitesimal behavior of s in x. In particular does this imply, that P(s)(x) is determined by the germ of s in x, which is expressed by saying that differential operators are local. A foundational result is the Peetre theorem showing that the converse is also true: any local operator is differential. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... This article needs to be cleaned up to conform to a higher standard of quality. ...

An equivalent, but purely algebraic description of linear differential operators is as follows: an -linear map P is a k-th order linear differential operator, if for any k+1 smooth functions we have

Here the bracket is defined as the commutator

This characterization of linear differential operators shows that they are particular mappings between modules over a commutative algebra, allowing the concept to be seen as a part of commutative algebra. In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R. In this article, all rings and algebras are assumed to be unital and associative. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ...

Examples

• In applications to the physical sciences, operators such as the Laplace operator play a major role in setting up and solving partial differential equations.

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

• In abstract algebra, the concept of a derivation allows for generalizations of differential operators which do not require the use of calculus. Frequently such generalizations are employed in algebraic geometry and commutative algebra. See also jet (mathematics).

In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator, with many applications. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. ...

See also

• Difference operator
• Delta operator
• Elliptic operator
• Fractional calculus
• Invariant differential operators
• Shift theorem