List of mathematic operators information

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 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	$r=r(\phi )$ $\phi =t$
$F[y]=\int _{a}^{t}{\sqrt[{3}]{y''}}\,dt$	Cartesian	$y=y(x)$ $x=t$	Affine arc length
$F[x,y]=\int _{a}^{t}{\sqrt[{3}]{x'y''-x''y'}}\,dt$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[x,y,z]=\int _{a}^{t}{\sqrt[{3}]{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}}dt$	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	$x=x(t)$ $y=y(t)$	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
$X[x,y]=x+y'{\frac {x'^{2}+y'^{2}}{x''y'-y''x'}}$ $Y[x,y]=y+x'{\frac {x'^{2}+y'^{2}}{y''x'-x''y'}}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Evolute
$F[r]=t(r'\circ r^{[-1]})$	Intrinsic	$r=r(s)$ $s=t$	Evolute
$X[x,y]=x-{\frac {x'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}$ $Y[x,y]=y-{\frac {y'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}$	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
$F[y]=\int _{E}y\ln y\,dt$			Differential entropy
$F[y]=\int _{E}yt\,dt$			Expected value
$F[y]=\int _{E}\left(t-\int _{E}yt\,dt\right)^{2}y\,dt$			Variance

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