Editing Differential operator (section)

==Notations==
The most common differential operator is the action of taking the [[derivative]]. [[Notation for differentiation|Common notations]] for taking the first derivative with respect to a variable ''x'' include:

: <math>{d \over dx}</math>, <math>D</math>, <math>D_x,</math> and <math>\partial_x</math>.

When taking higher, ''n''th order derivatives, the operator may be written:

: <math>{d^n \over dx^n}</math>, <math>D^n</math>, <math>D^n_x</math>, or <math>\partial_x^n</math>.

The derivative of a function ''f'' of an [[argument of a function|argument]] ''x'' is sometimes given as either of the following:

: <math>[f(x)]'</math>
: <math>f'(x).</math>

The ''D'' notation's use and creation is credited to [[Oliver Heaviside]], who considered differential operators of the form

: <math>\sum_{k=0}^n c_k D^k</math>

in his study of [[differential equation]]s.

One of the most frequently seen differential operators is the [[Laplace operator|Laplacian operator]], defined by

:<math>\Delta = \nabla^2 = \sum_{k=1}^n \frac{\partial^2}{\partial x_k^2}.</math>

Another differential operator is the Θ operator, or [[theta operator]], defined by<ref>{{cite web| url=http://mathworld.wolfram.com/ThetaOperator.html|title=Theta Operator| author=E. W. Weisstein|access-date=2009-06-12}}</ref>

:<math>\Theta = z {d \over dz}.</math>

This is sometimes also called the '''homogeneity operator''', because its [[eigenfunction]]s are the [[monomial]]s in ''z'':
<math display="block">\Theta (z^k) = k z^k,\quad k=0,1,2,\dots </math>

In ''n'' variables the homogeneity operator is given by
<math display="block">\Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}.</math>

As in one variable, the [[eigenspace]]s of Θ are the spaces of [[homogeneous function]]s. ([[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:
:<math>f \overleftarrow{\partial_x} g = g \cdot \partial_x f</math>
:<math>f \overrightarrow{\partial_x} g = f \cdot \partial_x g</math>
:<math>f \overleftrightarrow{\partial_x} g = f \cdot \partial_x g - g \cdot \partial_x f.</math>
Such a bidirectional-arrow notation is frequently used for describing the [[probability current]] of quantum mechanics.