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