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# List of operators

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 Title: List of operators Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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

$L:\mathcal\left\{F\right\}\to\mathcal\left\{G\right\}$

which takes a function $y\in\mathcal\left\{F\right\}$ to another function $L\left[y\right]\in\mathcal\left\{G\right\}$. Here, $\mathcal\left\{F\right\}$ and $\mathcal\left\{G\right\}$ are some unspecified function spaces, such as Hardy space, Lp space, Sobolev space, or, more vaguely, the space of holomorphic functions.

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

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

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

$Y\left[x,y\right]=y+x\text{'}\frac\left\{x\text{'}^2+y\text{'}^2\right\}\left\{y$x'-xy'}
Parametric
Cartesian
$x=x\left(t\right)$
$y=y\left(t\right)$
Evolute
$F\left[r\right]=t \left(r\text{'}\circ r^\right)$ Intrinsic $r=r\left(s\right)$
$s=t$
$X\left[x,y\right]=x-\frac\left\{x\text{'}\int_a^t \sqrt \left\{ x\text{'}^2 + y\text{'}^2 \right\}\, dt\right\}\left\{\sqrt \left\{ x\text{'}^2 + y\text{'}^2 \right\}\right\}$

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

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

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

$Y\left[y\right] = \int_a^t \sin \left\left[\int_a^t \frac\left\{1\right\}\left\{y\right\} \,dt\right\right] dt$
Intrinsic $y=r\left(s\right)$
$s=t$
Intrinsic to
Cartesian
transformation
Metric functionals
$F\left[y\right]=$ y =\sqrt{\int_E y^2 \, dt} Norm
$F\left[x,y\right]=\int_E xy \, dt$ Inner product
$F\left[x,y\right]=\arccos \left\left[\frac\left\{\int_E xy \, dt\right\}\left\{\sqrt\left\{\int_E x^2 \, dt\right\}\sqrt\left\{\int_E y^2 \, dt\right\}\right\}\right\right]$ Fubini-Study metric
(inner angle)
Distribution functionals
$F\left[x,y\right] = x * y = \int_E x\left(s\right) y\left(t - s\right)\, ds$ Convolution
$F\left[y\right] = \int_E y \ln y \, dy$ Differential entropy
$F\left[y\right] = \int_E yt\,dt$ Expected value
$F\left[y\right] = \int_E \left(t-\int_E yt\,dt\right)^2y\,dt$ Variance