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Differential operators: definitions, orders, linearity, and applications

Overview of differential operators: definitions, order and linearity, ordinary vs partial cases, domains and examples, and their roles in differential equations, physics, and applied mathematics

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: April 10, 2026

In mathematics, and in particular in differential calculus, a differential operator is an operator that takes a function as input and returns another function. Such operators carry out differentiation with respect to one or more variables, possibly combining derivatives of different orders and with coefficients that are themselves functions.

Notation and a simple example

The most basic example is the first derivative. Two common notations are Leibniz's form and Euler's single-letter symbol. In Leibniz's notation the operator is written as

${\tfrac {d}{dx}}$ and in Euler's notation as $D$

For a function f(x), applying the operator gives the derivative f'(x) = (d/dx)f(x) or Df(x). The name Euler's notation is often used when repeated application or polynomial expressions in the derivative are considered.

Types and key features

Order. A differential operator has an order equal to the highest derivative it involves (for example, a second-order operator contains second derivatives).
Linearity. Many differential operators studied in analysis and physics are linear: the derivative operator D satisfies D(af + bg) = aDf + bDg for constants a,b and appropriate functions f,g.
Ordinary vs partial. Ordinary differential operators act on functions of a single variable; partial differential operators involve derivatives with respect to several variables and are used to differentiate multivariable functions.
Domains. Operators are defined on spaces of sufficiently smooth functions (for example C^k or C^∞) and can be extended to act on distributions (generalized functions) in a broader setting.

Typical constructions include polynomials in D with function coefficients (for example a(x)D^2 + b(x)D + c(x)), as well as operators built from partial derivatives like ∂/∂x or the Laplacian. Differential operators are central objects in differential equations, mathematical physics, and many areas of applied mathematics.

First order linear differential operator

Definition

Let $M\subset \mathbb {R} ^{n}$ an open subset. A first order linear differential operator is a mapping

$D\colon C^{1}(M)\to C^{0}(M),$

which through

$u\mapsto \sum _{i=1}^{n}a_{i}(x)\partial _{x_{i}}u$

where $a_{i}$ is a continuous function.

Examples

The most important example of a first order differential operator is the ordinary derivative

$\frac{\mathrm{d}}{\mathrm{d} x}\colon f \mapsto f'.$

The partial derivative

$\frac{\partial}{\partial x_i}\colon f \mapsto \frac{\partial f}{\partial x_i}$

in $x_{i}$ -direction is a partial differential operator of first order.

Other differential operators of this kind are obtained by multiplication with a continuous function. If $a\in C^{0}(M)$ just such a continuous function, then the operator given by

$D = a \frac{\mathrm{d}}{\mathrm{d} x}\colon f \mapsto a f' \quad \text{d. h.} \quad Df(x) = a(x)f'(x)$

operator is again a first-order differential operator.

Three further examples are the operators gradient (grad), divergence (div) and rotation (rot) from vector analysis. They are $\nabla$ denoted by the Nabla symbol , which in the three-dimensional case in Cartesian coordinates has the form

$\nabla = \begin{pmatrix}\frac{\partial}{\partial x_1} \\ \frac{\partial}{\partial x_2} \\ \frac{\partial}{\partial x_3}\end{pmatrix}.$

has.

The Wirtinger derivations

${\frac {\partial }{\partial z}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-\mathrm {i} {\frac {\partial }{\partial y}}\right)$

and

${\frac {\partial }{\partial {\overline {z}}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+\mathrm {i} {\frac {\partial }{\partial y}}\right)$

are two more examples of differential operators. The special thing in these operators is that they can be used to calculate functions $M\subset \mathbb {C} \to \mathbb {C}$ for holomorphism, namely if ∂ $\textstyle \frac{\partial f}{\partial \overline{z}} = 0$ then the function is holomorphic.

Ordinary differential operator

Ordinary differential operators occur in particular in connection with ordinary differential equations.

Definition

Analogous to the definition of the first-order differential operator, an ordinary differential operator of order is a mapping

$D\colon C^{k}(M)\to C^{0}(M),$

which through

$D(f)(x) := \sum_{i=0}^k a_i(x) \left(\frac{\mathrm{d}^i f}{\mathrm{d} x^i}(x)\right)^{\beta_i}$

is given. Here $a_{i}$ for all is again a continuous function. In the case β $\beta_i = 1$ for all this operator is called an ordinary linear differential operator.

Example

The derivative -th order

$\frac{\mathrm{d}^k}{\mathrm{d} x^k}\colon f \mapsto f^{(k)}$

is the simplest case of an ordinary differential operator. It is the special case $\beta_k = 1$ from $a_{i}\equiv 0$ for $i<k,\;a_{k}\equiv 1$ k =

Linear partial differential operator

Definition

Let $M\subset \mathbb {R} ^{n}$ an open subset. A linear partial differential operator of order is a linear operator

$D\colon C^{k}(M)\to C^{0}(M),$

who through

$D(f)(x) := \sum_{|\alpha| \leq k} a_{\alpha}(x) \frac{\partial^\alpha f}{\partial x^\alpha}(x)$

can be represented. Where $a_{\alpha}$ for all multi-indices α $\alpha \in \N^n$ is a continuous function.

Examples

The Laplace operator in Cartesian coordinates is

$\Delta= \nabla^2 = \sum_{k=1}^n \frac{\partial^2}{\partial x_k^2}.$

This is an elementary example of a partial differential operator. Moreover, this is the most important example of an elliptic differential operator. Elliptic differential operators are a special class of partial differential operators.

The operator corresponding to the heat conduction or diffusion equation is

$\Delta - \frac{\partial}{\partial t}.$

This is an example of a parabolic differential operator.

The D'Alembert operator

$\Box\varphi(x,y,z, t)=\frac{1}{c^2}\frac{\partial^2\varphi}{\partial t^2}(x,y,z,t) - \Delta_{(x,y,z)}\varphi(x,y,z,t),$

where corresponds to a velocity, is another important partial differential operator. This is a hyperbolic operator and is used in the wave equation.

Partial differential operator

Definition

A (non-linear) partial differential operator of order is again a mapping

$D\colon C^{k}(M)\to C^{0}(M).$

This is given by

$D(f)(x) := \sum_{i} \sum_{|\alpha| \leq k} a_{\alpha i}(x) \left(\frac{\partial^\alpha f}{\partial x^\alpha}(x)\right)^{i}.$

Here $a_{\alpha i}$ for all α $\alpha \in \N^n$ and continuous functions.

Linear differential operators

In the definitions above, it was already briefly mentioned when an ordinary or a partial differential operator is called linear. For the sake of completeness, the abstract definition of a linear differential operator is now given. This is analogous to the definition of the linear mapping. All the examples given above are linear differential operators, unless otherwise stated.

Definition

Let be an (arbitrary) differential operator. This is called linear if

${D}\,(f+g) = ({D}f) + ({D}g)$

${D}\,(cf) = c\,({D}f)$

holds for all functions $f, g \in C^1(M)$ and all constants

The most prominent example of this is the differential operator

$\frac{\mathrm{d}}{\mathrm{d} x}\colon f \mapsto f',$

which assigns its derivative to a function .

The solution space of a linear differential equation forms a vector space. After Fourier transformation, they can often be traced back to algebraic equations and concepts of linear algebra. Non-linear differential operators are much more difficult to deal with.

Algebra of the differential operators

$\operatorname{Diff}^k(C^k(M))$ denotes the set of all linear differential operators of order $C^{k}(M)$ operating on . The set

$\operatorname{Diff}(C^k(M)) := \bigoplus_{k \geq 0} \operatorname{Diff}^k(C^k(M))$

together with the series connection of linear differential operators is called multiplication

$(\mathrm{D}_1\circ \mathrm{D}_2)(f) = \mathrm{D}_1(\mathrm{D}_2(f))$

to a $\Z_+$ -graded algebra. However, multiplication is generally not commutative. An exception are, for example, differential operators with constant coefficients, where commutativity follows from the interchangeability of the partial derivatives.

One can also formally build power series with the differential operators and exponential functions $\exp (D)$ above them, for example. For calculating with such exponential expressions of linear operators, the Baker-Campbell-Hausdorff formulae apply.

Differential operator on a manifold

Since on manifolds one only has the local coordinate systems in the form of maps and no globally valid coordinate systems available, one must define coordinate-independent differential operators on these. Such differential operators on manifolds are also called geometric differential operators.

Coordinates-invariant definition

Let be a smooth manifold and let $E, F \to M$ vector bundles. A differential operator of order between the intersections of and is a linear mapping

$D \colon \Gamma^\infty(M,E) \to \Gamma^{\infty}(M,F)$

with the following characteristics:

The operator is local, i.e. the following applies

$\operatorname{supp}(Ds) \subseteq \operatorname{supp}(s).$

For $x\in M$ there exist an open environment $U\subseteq M$ of , bundle maps ϕ $\phi \colon E|_{U}\to U\times \mathbb {C} ^{r}$ and ψ $\psi \colon F|_{U}\to U\times \mathbb {C} ^{s}$ and a differential operator ${\tilde {D}}\in \operatorname {Diff} ^{k}(U,\mathbb {C} ^{r},\mathbb {C} ^{s}),$ so that the diagram is
${\begin{array}{ccc}\Gamma _{0}^{\infty }(E\vert _{U})&{\xrightarrow {D}}&\Gamma _{0}^{\infty }(F\vert _{U})\\{\big \downarrow }\phi ^{*}&&{\big \downarrow }\psi ^{*}\\C^{\infty }(U,\mathbb {C} ^{r})&{\xrightarrow {\tilde {D}}}&C^{\infty }(U,\mathbb {C} ^{s})\end{array}}$
. By ϕ $\phi^*$ is $C^{\infty }(U,\mathbb {C} ^{r})$ denoted the pullback of a smooth vector field into the space

Examples

In the following, examples of geometric differential operators are shown.

The set of differential forms forms a smooth vector bundle over a smooth manifold. The Cartan derivative and its adjoint operator are differential operators on this vector bundle.
The Laplace-Beltrami operator and other generalised Laplace operators are differential operators.
The tensor bundle is a vector bundle. For any fixed chosen vector field the mapping $\nabla_X \colon \Gamma^\infty(T^k_lM) \rightarrow \Gamma^\infty(T^k_lM)$ defined by $T \mapsto \nabla_X T$ where $\nabla$ is the covariant derivative, a differential operator.
The Lie derivative is a differential operator on the differential forms.

Symbol of a differential operator

The 2nd order differential operators given in the examples, if one $\partial _{i}$ formally $y_{i}$ replaces the partial derivatives ∂ by variables and considers only the highest - i.e. second - order terms, correspond to a quadratic form in the $y_{i}$ . In the elliptic case all coefficients of the form have the same sign, in the hyperbolic case the sign changes, in the parabolic case the highest order term is missing for one of the $y_{i}$ The corresponding partial differential equations each show very different behaviour. The names come from the analogues to conic section equations.

This can be extended to other cases by the notion of the main symbol of the differential operator. One keeps only terms of the highest order, replaces derivatives by new variables $y_{i}$ and obtains a polynomial in these new variables with which one can characterise the differential operator. For example, it is of the elliptic type if the following holds: the principal symbol is non-zero if at least one $y_{i}$ non-zero. However, there are already "mixed" cases with 2nd order differential operators that cannot be assigned to any of the three classes.

The following definitions capture this again in mathematical precision.

Symbol

Let it be

$P(u)(x) = \sum_{|\alpha|\leq m} b_\alpha(x) \frac{\partial^\alpha}{\partial x^\alpha}u(x)$

a general differential operator of order . The coefficient function $b_{\alpha} \in C^\infty(\R^n)$ can be matrix-valued. The polynomial

$p(x,\xi) = \sum_{|\alpha|\leq m}b_\alpha(x) \left(i\xi\right)^\alpha$

in ξ $\xi \in \R^n$ is called the symbol of . However, since, as already indicated in the introduction, the most important information is to be found in the highest order term, the following definition of the main symbol is usually used.

Main icon

Let again be the differential operator of order defined above. The homogeneous polynomial

$p_m(x,\xi) = \sum_{|\alpha|=m}b_\alpha(x) \left(i\xi\right)^\alpha$

in ξ $\xi \in \R^n$ is called the main symbol of . Often the main symbol is simply called symbol if confusion with the definition given above is excluded.

Examples

The symbol and the main symbol of the Laplace operator Δ $\Delta$ are as follows

$\sum_{i=1}^n -\xi_i^2 = -|\xi|^2.$

Main symbol of a differential operator between vector bundles

Differential operators on manifolds can also be assigned a symbol and a main symbol. Of course, it must be taken into account in the definition that the main symbol and the symbol under map change are defined invariant. Since the map change for symbols is very complicated, one usually restricts oneself to the definition of the main symbol.

Let $D \colon \Gamma^\infty(M,E) \to \Gamma^\infty(M,F)$ be a (coordinate-invariant) differential operator operating between intersections of vector bundles. Let $p\in M$ , ξ $\xi \in T_p^*M$ and $e \in E_p$ . Choose $f \in C^\infty_c(M)$ and $s \in \Gamma^\infty_c(M,E)$ with f(p) = 0 , $\textstyle\mathrm{d}f_p = \xi$ and s(p) = e . Then the expression

$\sigma^k_D(p,\xi) e := \frac{i^k}{k!}D(f^k s)(p)$

independent of the choice of and . The function

$\sigma_D^k(p, \xi) \in \operatorname{Hom}(E_p,F_p)$

is then called the main symbol of .

Pseudo-differential operators

→ Main article: Pseudo-differential operator

The order of a differential operator is always integer and positive. In the theory of pseudo-differential operators this is generalised. Linear differential operators of order with smooth and bounded coefficients can be understood as pseudo-differential operators of the same order. Let $D \colon C^k_c(\R^n) \to C_c(\R^n)$ such a differential operator, then one can $\mathcal{F}^{-1}$ apply to the Fourier transform ${\mathcal {F}}$ and then the inverse Fourier transform That is to say

$(Du)(x)=({\mathcal {F}}^{-1}{\mathcal {F}}Du)(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{\mathrm {i} (x-y)\xi }D(\xi )u(y)\mathrm {d} y\mathrm {d} \xi .$

This is a special case of a pseudo-differential operator

$(Pu)(x)={\frac {1}{(2\pi )^{n}}}\int _{\mathbb {R} ^{n}}\int _{\mathbb {R} ^{n}}e^{\mathrm {i} (x-y)\xi }a(x,y,\xi )u(y)\mathrm {d} y\mathrm {d} \xi .$

This also shows that certain differential operators can be represented as integral operators and thus differential operators and integral operators are not completely opposed.