Differential Operator - Notations

Notations

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

and

When taking higher, nth order derivatives, the operator may also 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, 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:

In n variables the homogeneity operator is given by

As in one variable, the eigenspaces of Θ are the spaces of homogeneous polynomials.

The result of applying the differential to the left and to the right, and the difference obtained when applying the differential operator to the left and to the right, are denoted by arrows as follows:

Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics.