List of operators

In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators.

In the following ''L'' is an operator

:$L:\mathcal\to\mathcal$

which takes a function $y\in\mathcal$ to another function L in\mathcal. Here, $\mathcal$ and $\mathcal$ are some unspecified function spaces, such as Hardy space, ''L''^p space, Sobolev space, or, more vaguely, the space of holomorphic functions.

{, class="wikitable"
, - style="background:#eaeaea"
! style="text-align: center" , Expression
! style="text-align: center" , Curve
definition
! style="text-align: center" , Variables
! style="text-align: center" , Description
, -
! style="background:#eafaea" colspan=4, Linear transformations
, -
, L y^{(n)} , , , , , , Derivative of ''n''th order
, -
, L \int_a^t y \,dt , , Cartesian, , $y=y(x)$
$x=t$, , Integral, area
, -
, L y\circ f, , , , , , Composition operator
, -
, L \frac{y\circ t+y\circ -t}{2}, , , , , , Even component
, -
, L \frac{y\circ t-y\circ -t}{2}, , , , , , Odd component
, -
, L y\circ (t+1) - y\circ t = \Delta y, , , , , , Difference operator
, -
, L y\circ (t) - y\circ (t-1) = \nabla y, , , , , , Backward difference (Nabla operator)
, -
, L \sum y=\Delta^{-1}y, , , , , , Indefinite sum operator (inverse operator of difference)
, -
, L =-(py')'+qy , , , , , , Sturm–Liouville operator
, -
! style="background:#eafaea" colspan=4, Non-linear transformations
, -
, F y^{ 1 , , , , , , Inverse function
, -
, F t\,y'^{ 1 - y\circ y'^{ 1 , , , , , , Legendre transformation
, -
, F f\circ y, , , , , , Left composition
, -
, F \prod y, , , , , , Indefinite product
, -
, F \frac{y'}{y}, , , , , , Logarithmic derivative
, -
, F {\frac{ty'}{y, , , , , , Elasticity
, -
, F {y \over y'}-{3\over 2}\left({y''\over y'}\right)^2, , , , , , Schwarzian derivative
, -
, F \int_a^t , y', \,dt , , , , , , Total variation
, -
, F \frac{1}{t-a}\int_a^t y\,dt , , , , , , Arithmetic mean
, -
, F \exp \left( \frac{1}{t-a}\int_a^t \ln y\,dt \right) , , , , , , Geometric mean
, -
, F -\frac{y}{y'}, , Cartesian, , $y=y(x)$
$x=t$, , rowspan=3, Subtangent
, -
, F ,y -\frac{yx'}{y'}, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
, -
, F -\frac{r^2}{r'}, , Polar, , $r=r(\phi)$
$\phi=t$
, -
, F \frac{1}{2}\int_a^t r^2 dt, , Polar, , $r=r(\phi)$
$\phi=t$ , , Sector area
, -
, F \int_a^t \sqrt { 1 + y'^2 }\, dt, , Cartesian, , $y=y(x)$
$x=t$, , rowspan=3, Arc length
, -
, F ,y \int_a^t \sqrt { x'^2 + y'^2 }\, dt, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
, -
, F \int_a^t \sqrt { r^2 + r'^2 }\, dt, , Polar, , $r=r(\phi)$
$\phi=t$
, -
, F ,y= \int_a^t\sqrt y''}\, dt , , Cartesian, , $y=y(x)$
$x=t$, , rowspan=3, Affine arc length
, -
, F ,y= \int_a^t\sqrt x'y''-x''y'}\, dt , , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
, -
, F ,y,z\int_a^t\sqrt z(x'y''-y'x'')+z''(xy'-x'y)+z'(x''y-xy'')}, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
$z=z(t)$
, -
, F \frac{y''}{(1+y'^2)^{3/2, , Cartesian, , $y=y(x)$
$x=t$, , rowspan=4, Curvature
, -
, F ,y \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
, -
, F \frac{r^2+2r'^2-rr''}{(r^2+r'^2)^{3/2, , Polar, , $r=r(\phi)$
$\phi=t$
, -
, F ,y,z\frac{\sqrt{(z''y'-z'y'')^2+(x''z'-z''x')^2+(y''x'-x''y')^2{(x'^2+y'^2+z'^2)^{3/2, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
$z=z(t)$
, -
, F \frac{1}{3}\frac{y'}{(y'')^{5/3-\frac{5}{9}\frac{y^2}{(y'')^{8/3, , Cartesian, , $y=y(x)$
$x=t$, , rowspan=2, Affine curvature
, -
, F ,y \frac{x''y-xy''}{(x'y''-x''y')^{5/3-\frac{1}{2}\left frac{1}{(x'y''-x''y')^{2/3\right', , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
, -
, F ,y,z\frac{z(x'y''-y'x'')+z''(xy'-x'y)+z'(x''y-xy'')}{(x'^2+y'^2+z'^2)(x''^2+y''^2+z''^2)}, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$
$z=z(t)$, , Torsion of curves
, -
, X ,y\frac{y'}{yx'-xy'}

Y ,y\frac{x'}{xy'-yx'}, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$, , Dual curve
(tangent coordinates)
, -
, X ,yx+\frac{ay'}{\sqrt {x'^2+y'^2

Y ,yy-\frac{ax'}{\sqrt {x'^2+y'^2, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$, , Parallel curve
, -
, X ,yx+y'\frac{x'^2+y'^2}{x''y'-y''x'}

Y ,yy+x'\frac{x'^2+y'^2}{y''x'-x''y'}, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$, , rowspan=2, Evolute
, -
, F t (r'\circ r^{ 1), , Intrinsic, , $r=r(s)$
$s=t$
, -
, X ,yx-\frac{x'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2

Y ,yy-\frac{y'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 , , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$, , , Involute
, -
, X ,y\frac{(xy'-yx')y'}{x'^2 + y'^2}

Y ,y\frac{(yx'-xy')x'}{x'^2 + y'^2}, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$, , , Pedal curve with pedal point (0;0)
, -
, X ,y\frac{(x'^2-y'^2)y'+2xyx'}{xy'-yx'}

Y ,y\frac{(x'^2-y'^2)x'+2xyy'}{xy'-yx'}, , Parametric
Cartesian, , $x=x(t)$
$y=y(t)$, , , Negative pedal curve with pedal point (0;0)
, -
, X = \int_a^t \cos \left int_a^t \frac{1}{y} \,dt\rightdt

Y = \int_a^t \sin \left int_a^t \frac{1}{y} \,dt\rightdt, , Intrinsic, , $y=r(s)$
$s=t$, , Intrinsic to
Cartesian
transformation
, -
! style="background:#eafaea" colspan=4, Metric functionals
, -
, F \, y\, =\sqrt{\int_E y^2 \, dt}, , , , , , Norm
, -
, F ,y\int_E xy \, dt, , , , , , Inner product
, -
, F ,y\arccos \left frac{\int_E xy \, dt}{\sqrt{\int_E x^2 \, dt}\sqrt{\int_E y^2 \, dt\right/math>, , , , , , Fubini–Study metric
(inner angle)
, -
! style="background:#eafaea" colspan=4, Distribution functionals
, -
, F ,y= x * y = \int_E x(s) y(t - s)\, ds, , , , , , Convolution
, -
, F = \int_E y \ln y \, dt, , , , , , Differential entropy
, -
, F = \int_E yt\,dt, , , , , , Expected value
, -
, F = \int_E \left(t-\int_E yt\,dt\right)^2y\,dt, , , , , , Variance

See also

* List of transforms
* List of Fourier-related transforms
* Transfer operator
* Fredholm operator
* Borel transform
* Glossary of mathematical symbols

Operators
Operators
Operators