# 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

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

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

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

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

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

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

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

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

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