Editing Operator (mathematics) (section)

== Examples ==

=== Analysis (calculus) ===
{{Main|Differential operator|Integral operator}}
From the point of view of [[functional analysis]], [[calculus]] is the study of two linear operators: the [[differential operator]] <math>\frac{\ \mathrm{d}\ }{ \mathrm{d} t }</math>, and the ''[[Volterra operator]]'' <math>\int_0^t</math>. 

=== Fundamental analysis operators on scalar and vector fields ===
{{Main|Vector calculus|Vector field|Scalar field|Gradient|Divergence|Curl (mathematics)|l6=Curl}}

Three operators are key to [[vector calculus]]:
* Grad ([[gradient]]), (with operator symbol [[del|<math>\nabla </math>]]) assigns a vector at every point in a scalar field that points in the direction of greatest rate of change of that field and whose norm measures the absolute value of that greatest rate of change.
* Div ([[divergence]]), (with operator symbol [[del#Divergence|<math>{\nabla \cdot} </math>]]) is a vector operator that measures a vector field's divergence from or convergence towards a given point.
* [[curl (mathematics)|Curl]], (with operator symbol [[del#Curl|<math>\nabla \!\times </math>]]) is a vector operator that measures a vector field's curling (winding around, rotating around) trend about a given point.

As an extension of vector calculus operators to physics, engineering and tensor spaces, grad, div and curl operators also are often associated with [[tensor calculus]] as well as vector calculus.<ref name=Schey-2005>{{cite book |first=H.M. |last=Schey |year=2005 |title=Div, Grad, Curl, and All That |location=New York, NY |publisher=W.W. Norton |isbn=0-393-92516-1}}</ref>

=== Geometry ===
{{Main|General linear group|Isometry}}

In [[geometry]], additional structures on [[vector space]]s are sometimes studied. Operators that map such vector spaces to themselves [[bijective]]ly are very useful in these studies, they naturally form [[group (mathematics)|group]]s by composition.

For example, bijective operators preserving the structure of a vector space are precisely the [[invertible function|invertible]] [[linear operator]]s. They form the [[general linear group]] under composition. However, they ''do not'' form a vector space under operator addition; since, for example, both the identity and −identity are [[invertible]] (bijective), but their sum, 0, is not.

Operators preserving the [[Euclidean metric]] on such a space form the [[isometry group]], and those that fix the origin form a subgroup known as the [[orthogonal group]]. Operators in the orthogonal group that also preserve the orientation of vector tuples form the [[special orthogonal group]], or the group of rotations.

=== Probability theory ===
{{Main|Probability theory}}

Operators are also involved in probability theory, such as [[expected value|expectation]], [[variance]], and [[covariance]], which are used to name both number statistics and the operators which produce them. Indeed, every covariance is basically a [[dot product]]: Every variance is a dot product of a vector with itself, and thus is a [[quadratic norm]]; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the [[Pearson correlation coefficient]]; expected value is basically an integral operator (used to measure weighted shapes in the space).

==== Fourier series and Fourier transform ====
{{Main|Fourier series|Fourier transform}}
The Fourier transform is useful in applied mathematics, particularly physics and signal processing. It is another integral operator; it is useful mainly because it converts a function on one (temporal) domain to a function on another (frequency) domain, in a way effectively [[invertible function|invertible]]. No information is lost, as there is an inverse transform operator. In the simple case of [[periodic function]]s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of [[sine wave]]s and cosine waves:<math display="block">f(t)=\frac{\ a_0\ }{2}+\sum_{n=1}^{\infty}\ a_n\cos(\omega\ n\ t) + b_n\sin(\omega\ n\ t) </math> The tuple {{math|( ''a''{{sub|0}}, ''a''{{sub|1}}, ''b''{{sub|1}}, ''a''{{sub|2}}, ''b''{{sub|2}}, ... )}} is in fact an element of an infinite-dimensional vector space [[Sequence space|{{math|''ℓ''{{i sup|2}} }}]], and thus Fourier series is a linear operator.

When dealing with general function <math>\mathbb{R} \to \mathbb{C}</math>, the transform takes on an [[integral]] form:
:<math display="block">f(t) = {1\over\sqrt{2\pi}}\int_{-\infty}^{+\infty}{g(\omega)\ e^{i\ \omega\ t}\ \mathrm{d}\ \omega} </math>

==== Laplace transform ====
{{Main|Laplace transform}}
The ''Laplace transform'' is another integral operator and is involved in simplifying the process of solving differential equations.

Given {{nobr|{{math|''f'' {{=}} ''f''(''s'')}}}}, it is defined by:<math display="block"> F(s)=\operatorname\mathcal{L}\{f\}(s)=\int_0^\infty e^{-s\ t}\ f(t)\ \mathrm{d}\ t </math>