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

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

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

See also

• List of transforms
• List of Fourier-related transforms
• Transfer operator
• Fredholm operator
• Borel transform
• Glossary of mathematical symbols