List of operators

From Wikipedia, the free encyclopedia

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, L^p space, Sobolev space, or, more vaguely, the space of holomorphic functions.

Expression	Curve definition	Variables	Description
Linear transformations
$L[y]=y^{(n)} \$			Derivative of n-th 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
$L[y] =-(py')'+qy \,$			Sturm-Liouville operator
$L[y]=\int_0^\infty y(s)\exp{(-ts)}\,ds$			Laplace transform
$L[y]= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty y(s) \exp{(- its)}\,ds$			Fourier transform
$L[y] =\int_0^{\infty} \frac{s^t y(s)}{s}\,ds.$			Mellin transform
$L[y]=2\int_t^\infty \frac{y(s)s\,ds}{\sqrt{s^2-t^2}}.$			Abel transform
$L[y]=-\frac{1}{\pi}\int_t^\infty \frac{y'(s)\,ds}{\sqrt{s^2-t^2}}.$			Inverse Abel transform
$L[y]= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty y(s) \cos ts \sin ts \,ds$			Hartley transform
Non-linear transformations
$F[y]=y^{-1}=\mbox{inv } y \$			Inverse function
$F[y]=t\,\mbox{inv }y' - y\circ \mbox{inv }y'$			Legendre transformation
$F[y]=f\circ y$			Left composition
$F[y]=\frac{y'}{y}$			Logarithmic derivative
$F[y]=\int_a^t \|y'\| \,dt$			Total variation
$F[y]=\frac{1}{t-a}\int_a^t y\,dt$			Mean value
$F[y]=\exp \left( \frac{1}{t-a}\int_a^t \ln y\,dt \right)$			Geometric mean value
$F[y]= -\frac{y}{y'}$	Cartesian	$y = y (x)$ $x = t$	Subtangent
$F[x,y]= -\frac{yx'}{y'}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$
$F[y]= -\frac{y^2}{y'}$	Polar	$y = r (φ)$ $φ = t$
$F[y]=\frac{1}{2}\int_a^t y^2 dt$	Polar	$y = r (φ)$ $φ = t$	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	$x = x (t)$ $y = y (t)$
$F[y]= \int_a^t \sqrt { y^2 + y'^2 }\, dt$	Polar	$y = r (φ)$ $φ = t$
$F[y]=\frac{y''}{(1+y'^2)^{3/2}}$	Cartesian	$y = y (x)$ $x = t$	Curvature
$F[x,y]= \frac{x'y''-y'x''}{(x'^2+y'^2)^{3/2}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$
$F[y]=\frac{y^2+2y'^2-yy''}{(y^2+y'^2)^{3/2}}$	Polar	$y = r (φ)$ $φ = t$
$F[x,y,z]=\frac{\sqrt{(z''y'-z'y'')^2+(x''z'-z''x')^2+(y''x'-x''y')^2}}{(x'^2+y'^2+z'^2)^{3/2}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$ $z = z (t)$
$F[x,y]=\left\| \frac{x''y'''-x'''y''}{(x'y''-x''y')^{5/2}}-\frac{1}{2}\left[\frac{1}{(x'y''-x''y')}\right]''\right\|$	Parametric Cartesian	$x = x (t)$ $y = y (t)$	Affine curvature
$F[x,y,z]=\frac{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}{(x'^2+y'^2+z'^2)(x''^2+y''^2+z''^2)}$	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
$X[y]=t-\frac{1+y'^2}{y''}$ $Y[y]=y+\frac{1+y'^2}{y''}$	Cartesian	$y = y (x)$ $x = t$	Evolute
$X[x,y]=x+y'\frac{x'^2+y'^2}{x''y'-y''x'}$ $Y[x,y]=y+x'\frac{x'^2+y'^2}{y''x'-x''y'}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$
$F[y]=\frac{yy'}{(\mbox{inv }y)'}$	Intrinsic	$y = r (s)$ $s = t$
$X[x,y]=x-\frac{x'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}$ $Y[x,y]=y-\frac{y'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}$	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

Categories: Mathematics-related lists | Functional analysis | Curves

List of operators

From Wikipedia, the free encyclopedia

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