Differential operator

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 that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear operators, which are the most common type. However, non-linear differential operators, such as the Schwarzian derivative also exist. Assume that there is a map A {displaystyle A} from a function space F 1 {displaystyle {mathcal {F}}_{1}} to another function space F 2 {displaystyle {mathcal {F}}_{2}} and a function f ∈ F 2 {displaystyle fin {mathcal {F}}_{2}} so that f {displaystyle f} is the image of u ∈ F 1 {displaystyle uin {mathcal {F}}_{1}} i.e.,　 f = A ( u )   . {displaystyle f=A(u) .} A differential operator is represented as a linear combination, finitely generated by u {displaystyle u} and its derivatives containing higher degree such as where the set of non-negative integers, α = ( α 1 , α 2 , ⋯ , α n ) {displaystyle alpha =(alpha _{1},alpha _{2},cdots ,alpha _{n})} , is called a multi-index, | α | = α 1 + α 2 + ⋯ + α n {displaystyle |alpha |=alpha _{1}+alpha _{2}+cdots +alpha _{n}} called length, a α ( x ) {displaystyle a_{alpha }(x)} are functions on some open domain in n-dimensional space and D α = D 1 α 1 D 2 α 2 ⋯ D n α n   . {displaystyle D^{alpha }=D_{1}^{alpha _{1}}D_{2}^{alpha _{2}}cdots D_{n}^{alpha _{n}} .} The derivative above is one as functions or, sometimes, distributions or hyperfunctions and D j = − i ∂ ∂ x j {displaystyle D_{j}=-i{frac {partial }{partial x_{j}}}} or sometimes, D j = ∂ ∂ x j {displaystyle D_{j}={frac {partial }{partial x_{j}}}} . The most common differential operator is the action of taking derivative. Common notations for taking the first derivative with respect to a variable x include: When taking higher, nth order derivatives, the operator may also be written: 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, who considered differential operators of the form