List of mathematic 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

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

which takes a function $y\backslash in\backslash mathcal\{F\}$ to another function $L[y]\backslash in\backslash mathcal\{G\}$. Here, $\backslash mathcal\{F\}$ and $\backslash mathcal\{G\}$ 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
$L[y]=y^\{(n)\}$			Derivative of nth order
$L[y]=\backslash int\_a^t\; y\; \backslash ,dt$	Cartesian	$y=y(x)$ $x=t$	Integral, area
$L[y]=y\backslash circ\; f$			Composition operator
$L[y]=\backslash frac\{y\backslash circ\; t+y\backslash circ\; -t\}\{2\}$			Even component
$L[y]=\backslash frac\{y\backslash circ\; t-y\backslash circ\; -t\}\{2\}$			Odd component
$L[y]=y\backslash circ\; (t+1)\; -\; y\backslash circ\; t\; =\; \backslash Delta\; y$			Difference operator
$L[y]=y\backslash circ\; (t)\; -\; y\backslash circ\; (t-1)\; =\; \backslash nabla\; y$			Backward difference (Nabla operator)
$L[y]=\backslash sum\; y=\backslash Delta^\{-1\}y$			Indefinite sum operator (inverse operator of difference)
$L[y]\; =-(py\text{'})\text{'}+qy$			Sturm–Liouville operator
style="background:#eafaea" colspan=4|Non-linear transformations
$F[y]=y^\{[-1]\}$			Inverse function
$F[y]=t\backslash ,y\text{'}^\{[-1]\}\; -\; y\backslash circ\; y\text{'}^\{[-1]\}$			Legendre transformation
$F[y]=f\backslash circ\; y$			Left composition
$F[y]=\backslash prod\; y$			Indefinite product
$F[y]=\backslash frac\{y\text{'}\}\{y\}$			Logarithmic derivative
$F[y]=\{\backslash frac\{ty\text{'}\}\{y\}\}$			Elasticity
$F[y]=\{y$ \over y'}-{3\over 2}\left({y\over y'}\right)^2			Schwarzian derivative
$F[y]=\backslash int\_a^t\; |y\text{'}|\; \backslash ,dt$			Total variation
$F[y]=\backslash frac\{1\}\{t-a\}\backslash int\_a^t\; y\backslash ,dt$			Arithmetic mean
$F[y]=\backslash exp\; \backslash left(\; \backslash frac\{1\}\{t-a\}\backslash int\_a^t\; \backslash ln\; y\backslash ,dt\; \backslash right)$			Geometric mean
$F[y]=\; -\backslash frac\{y\}\{y\text{'}\}$	Cartesian	$y=y(x)$ $x=t$	rowspan=3|Subtangent
$F[x,y]=\; -\backslash frac\{yx\text{'}\}\{y\text{'}\}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[r]=\; -\backslash frac\{r^2\}\{r\text{'}\}$	Polar	$r=r(\backslash phi)$ $\backslash phi=t$
$F[r]=\backslash frac\{1\}\{2\}\backslash int\_a^t\; r^2\; dt$	Polar	$r=r(\backslash phi)$ $\backslash phi=t$	Sector area
$F[y]=\; \backslash int\_a^t\; \backslash sqrt\; \{\; 1\; +\; y\text{'}^2\; \}\backslash ,\; dt$	Cartesian	$y=y(x)$ $x=t$	rowspan=3|Arc length
$F[x,y]=\; \backslash int\_a^t\; \backslash sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \}\backslash ,\; dt$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[r]=\; \backslash int\_a^t\; \backslash sqrt\; \{\; r^2\; +\; r\text{'}^2\; \}\backslash ,\; dt$	Polar	$r=r(\backslash phi)$ $\backslash phi=t$
$F[y]\; =\; \backslash int\_a^t\backslash sqrt[3]\{y\}\backslash ,\; dt$	Cartesian	$y=y(x)$ $x=t$	rowspan=3|Affine arc length
$F[x,y]\; =\; \backslash int\_a^t\backslash sqrt[3]\{x\text{'}y$-xy'}\, dt	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[x,y,z]=\backslash int\_a^t\backslash sqrt[3]\{z(x\text{'}y$-y'x)+z(xy'-x'y)+z'(xy-xy)}dt	Parametric Cartesian	$x=x(t)$ $y=y(t)$ $z=z(t)$
$F[y]=\backslash frac\{y\}\{(1+y\text{'}^2)^\{3/2\}\}$	Cartesian	$y=y(x)$ $x=t$	rowspan=4|Curvature
$F[x,y]=\; \backslash frac\{x\text{'}y$-y'x}{(x'^2+y'^2)^{3/2}}	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[r]=\backslash frac\{r^2+2r\text{'}^2-rr$}{(r^2+r'^2)^{3/2}}	Polar	$r=r(\backslash phi)$ $\backslash phi=t$
$F[x,y,z]=\backslash frac\{\backslash sqrt\{(zy\text{'}-z\text{'}y$)^2+(xz'-zx')^2+(yx'-xy')^2}}{(x'^2+y'^2+z'^2)^{3/2}}	Parametric Cartesian	$x=x(t)$ $y=y(t)$ $z=z(t)$
$F[y]=\backslash frac\{1\}\{3\}\backslash frac\{y$'}{(y)^{5/3}}-\frac{5}{9}\frac{y^2}{(y)^{8/3}}	Cartesian	$y=y(x)$ $x=t$	rowspan=2|Affine curvature
$F[x,y]=\; \backslash frac\{xy$-xy}{(x'y-xy')^{5/3}}-\frac{1}{2}\left[\frac{1}{(x'y-xy')^{2/3}}\right]	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[x,y,z]=\backslash frac\{z$(x'y-y'x)+z(xy'-x'y)+z'(xy-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
$X[x,y]=\backslash frac\{y\text{'}\}\{yx\text{'}-xy\text{'}\}$ $Y[x,y]=\backslash frac\{x\text{'}\}\{xy\text{'}-yx\text{'}\}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Dual curve (tangent coordinates)
$X[x,y]=x+\backslash frac\{ay\text{'}\}\{\backslash sqrt\; \{x\text{'}^2+y\text{'}^2\}\}$ $Y[x,y]=y-\backslash frac\{ax\text{'}\}\{\backslash sqrt\; \{x\text{'}^2+y\text{'}^2\}\}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Parallel curve
$X[x,y]=x+y\text{'}\backslash frac\{x\text{'}^2+y\text{'}^2\}\{xy\text{'}-y$x'} $Y[x,y]=y+x\text{'}\backslash frac\{x\text{'}^2+y\text{'}^2\}\{y$x'-x''y'}	Parametric Cartesian	$x=x(t)$ $y=y(t)$	rowspan=2|Evolute
$F[r]=t\; (r\text{'}\backslash circ\; r^\{[-1]\})$	Intrinsic	$r=r(s)$ $s=t$
$X[x,y]=x-\backslash frac\{x\text{'}\backslash int\_a^t\; \backslash sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \}\backslash ,\; dt\}\{\backslash sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \}\}$ $Y[x,y]=y-\backslash frac\{y\text{'}\backslash int\_a^t\; \backslash sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \}\backslash ,\; dt\}\{\backslash sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \}\}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	|Involute
$X[x,y]=\backslash frac\{(xy\text{'}-yx\text{'})y\text{'}\}\{x\text{'}^2\; +\; y\text{'}^2\}$ $Y[x,y]=\backslash frac\{(yx\text{'}-xy\text{'})x\text{'}\}\{x\text{'}^2\; +\; y\text{'}^2\}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	|Pedal curve with pedal point (0;0)
$X[x,y]=\backslash frac\{(x\text{'}^2-y\text{'}^2)y\text{'}+2xyx\text{'}\}\{xy\text{'}-yx\text{'}\}$ $Y[x,y]=\backslash frac\{(x\text{'}^2-y\text{'}^2)x\text{'}+2xyy\text{'}\}\{xy\text{'}-yx\text{'}\}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	|Negative pedal curve with pedal point (0;0)
$X[y]\; =\; \backslash int\_a^t\; \backslash cos\; \backslash left[\backslash int\_a^t\; \backslash frac\{1\}\{y\}\; \backslash ,dt\backslash right]\; dt$ $Y[y]\; =\; \backslash int\_a^t\; \backslash sin\; \backslash left[\backslash int\_a^t\; \backslash frac\{1\}\{y\}\; \backslash ,dt\backslash right]\; dt$	Intrinsic	$y=r(s)$ $s=t$	Intrinsic to Cartesian transformation
style="background:#eafaea" colspan=4|Metric functionals
$F[y]=\backslash |y\backslash |=\backslash sqrt\{\backslash int\_E\; y^2\; \backslash ,\; dt\}$			Norm
$F[x,y]=\backslash int\_E\; xy\; \backslash ,\; dt$			Inner product
$F[x,y]=\backslash arccos\; \backslash left[\backslash frac\{\backslash int\_E\; xy\; \backslash ,\; dt\}\{\backslash sqrt\{\backslash int\_E\; x^2\; \backslash ,\; dt\}\backslash sqrt\{\backslash int\_E\; y^2\; \backslash ,\; dt\}\}\backslash right]$			Fubini–Study metric (inner angle)
style="background:#eafaea" colspan=4|Distribution functionals
$F[x,y]\; =\; x\; *\; y\; =\; \backslash int\_E\; x(s)\; y(t\; -\; s)\backslash ,\; ds$			Convolution
$F[y]\; =\; \backslash int\_E\; y\; \backslash ln\; y\; \backslash ,\; dt$			Differential entropy
$F[y]\; =\; \backslash int\_E\; yt\backslash ,dt$			Expected value
$F[y]\; =\; \backslash int\_E\; \backslash left(t-\backslash int\_E\; yt\backslash ,dt\backslash right)^2y\backslash ,dt$			Variance