List of operators

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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

See also[edit]