In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 that accepts a
function and returns another function (in the style of a
higher-order function in
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
).
This article considers mainly
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
differential operators, which are the most common type. However, non-linear differential operators also exist, such as the
Schwarzian derivative.
Definition
An order-
linear differential operator is a map
from a
function space to another function space
that can be written as:
where
is a
multi-index of non-negative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s,
, and for each
,
is a function on some open domain in ''n''-dimensional space. The operator
is interpreted as
Thus for a function
:
A differential operator acting on two functions
is also called a ''bidifferential operator''.
Notations
The most common differential operator is the action of taking the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
.
Common notations for taking the first derivative with respect to a variable ''x'' include:
:
,
,
and
.
When taking higher, ''n''th order derivatives, the operator may be written:
:
,
,
, or
.
The derivative of a function ''f'' of an
argument ''x'' is sometimes given as either of the following:
:
:
The ''D'' notation's use and creation is credited to
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
, who considered differential operators of the form
:
in his study of
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s.
One of the most frequently seen differential operators is the
Laplacian operator, defined by
:
Another differential operator is the Θ operator, or
theta operator, defined by
:
This is sometimes also called the homogeneity operator, because its
eigenfunctions are the
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
s in ''z'':
In ''n'' variables the homogeneity operator is given by
As in one variable, the
eigenspace
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of Θ are the spaces of
homogeneous functions. (
Euler's homogeneous function theorem)
In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself. Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right side of the operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows:
:
:
:
Such a bidirectional-arrow notation is frequently used for describing the
probability current of quantum mechanics.
Del
The differential operator del, also called ''nabla'', is an important
vector differential operator. It appears frequently in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
in places like the differential form of
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
. In three-dimensional
Cartesian coordinates, del is defined as
Del defines the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
, and is used to calculate the
curl,
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
, and
Laplacian of various objects.
Adjoint of an operator
Given a linear differential operator
the
adjoint of this operator is defined as the operator
such that
where the notation
is used for the
scalar product or
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. This definition therefore depends on the definition of the scalar product.
Formal adjoint in one variable
In the functional space of
square-integrable functions on a
real interval , the scalar product is defined by
where the line over ''f''(''x'') denotes the
complex conjugate of ''f''(''x''). If one moreover adds the condition that ''f'' or ''g'' vanishes as
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
is defined according to this formula, it is called the formal adjoint of ''T''.
A (formally)
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a st ...
operator is an operator equal to its own (formal) adjoint.
Several variables
If Ω is a domain in R
''n'', and ''P'' a differential operator on Ω, then the adjoint of ''P'' is defined in
''L''2(Ω) by duality in the analogous manner:
:
for all smooth ''L''
2 functions ''f'', ''g''. Since smooth functions are dense in ''L''
2, this defines the adjoint on a dense subset of ''L''
2: P
* is a
densely defined operator.
Example
The
Sturm–Liouville operator is a well-known example of a formal self-adjoint operator. This second-order linear differential operator ''L'' can be written in the form
:
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.
Properties of differential operators
Differentiation is
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
, i.e.
:
:
where ''f'' and ''g'' are functions, and ''a'' is a constant.
Any
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
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 ''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''. For example we have the relation basic in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
:
:
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 differential operators also obey the
shift theorem.
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, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
:f\left(x_1,\, x_2,\, \ldots,\, x_n\right)
of ''n ...
).
Ring of polynomial differential operators
Ring of univariate polynomial differential operators
If ''R'' is a ring, let
be the
non-commutative polynomial ring over ''R'' in the variables ''D'' and ''X'', and ''I'' the two-sided
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
generated by ''DX'' − ''XD'' − 1. Then the ring of univariate polynomial differential operators over ''R'' is the
quotient ring . This is a
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
. Every element can be written in a unique way as a ''R''-linear combination of monomials of the form
. It supports an analogue of
Euclidean division of polynomials
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common ...
.
Differential modules over