structural:scholar_edition, stylistic:tone
Notations

The most common differential operator is the action of taking the derivative. Common notations for taking the first derivative with respect to a variable x include:

When taking higher, n th order derivatives, the operator may 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

in his study of differential equations.

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 monomials in z: Θ (z k) = k z k, k = 0, 1, 2, … {\displaystyle \Theta (z^{k})=kz^{k},\quad k=0,1,2,\dots }

In n variables the homogeneity operator is given by Θ = ∑ k = 1 n x k ∂ ∂ x k. {\displaystyle \Theta =\sum _{k=1}^{n}x_{k}{\frac {\partial }{\partial x_{k}}}.}

As in one variable, the eigenspaces 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.