List of mathematic operators: Difference between revisions

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In [[mathematics]], an [[operator (mathematics)|operator]] or [[transformation (mathematics)|transform]] is a [[function (mathematics)|function]] from one [[function space|space of functions]] to another. Operators occur commonly in [[engineering]], [[physics]] and mathematics. Many are [[integral operator]]s and [[differential operator]]s.

In the following ''L'' is an operator

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{| class="wikitable"

|- style="background:#eaeaea"

! style="text-align: center" | Expression

! style="text-align: center" | Curve<br>definition

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! style="background:#eafaea" colspan=4|Linear transformations

|-

| <math>L[y]=y^{(n)} \ </math>|| || ||Derivative of ''n-''th order

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| <math>L[y]=\int_a^t y \,dt</math> ||Cartesian||<math>y=y(x)</math><br><math>x=t</math>|| Integral, area

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| <math>L[y]=\sum y=\Delta^{-1}y</math>|| || ||[[Indefinite sum]] operator (inverse operator of difference)

|-

| <math>L[y] =-(py')'+qy \,</math>|| || ||[[Sturm-LiouvilleSturm–Liouville operator]]

|-

! style="background:#eafaea" colspan=4|Non-linear transformations

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| <math>F[y]=y^{[-1]} \ </math> || || ||[[Inverse function]]

|-

| <math>F[y]=t\,y'^{[-1]} - y\circ y'^{[-1]} </math>|| || ||[[Legendre transformation]]

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| <math>F[y]=f\circ y</math>|| || ||Left composition

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| <math>F[y]=\prod y</math>|| || ||[[Indefinite product]]

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| <math>F[y]=\frac{y'}{y}</math>|| || ||[[Logarithmic derivative]]

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| <math>F[y]={\frac{ty'}{y}}</math>|| || ||[[Elasticity of a function|Elasticity]]

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| <math>F[y]={y''' \over y'}-{3\over 2}\left({y''\over y'}\right)^2</math>|| || || [[Schwarzian derivative]]

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| <math>F[y]=\int_a^t |y'| \,dt </math>|| || ||[[Total variation]]

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| <math>F[y]=\frac{1}{t-a}\int_a^t y\,dt </math>|| || ||[[MeanArithmetic valuemean]]

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| <math>F[y]=\exp \left( \frac{1}{t-a}\int_a^t \ln y\,dt \right) </math> || || ||[[Geometric mean value]]

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| <math>F[y]= -\frac{y}{y'}</math>|| Cartesian||<math>y=y(x)</math><br><math>x=t</math>||rowspan=3|[[Subtangent]]

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| <math>F[r]= \int_a^t \sqrt { r^2 + r'^2 }\, dt</math>||Polar||<math>r=r(\phi)</math><br><math>\phi=t</math>

|-

| <math>F[x,y] = \int_a^t\sqrt[3]{y''}\, dt </math> || Cartesian||<math>y=y(x)</math><br><math>x=t</math>||rowspan=3|[[Affine curvature|Affine arc length]]

|-

| <math>F[x,y] = \int_a^t\sqrt[3]{x'y''-x''y'}\, dt </math> || Parametric<br>Cartesian||<math>x=x(t)</math><br><math>y=y(t)</math>

|-

| <math>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<br>Cartesian||<math>x=x(t)</math><br><math>y=y(t)</math><br><math>z=z(t)</math>

|-

| <math>F[y]=\frac{y''}{(1+y'^2)^{3/2}}</math>||Cartesian||<math>y=y(x)</math><br><math>x=t</math>|| rowspan=4|[[Curvature]]

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| <math>F[r]=t (r'\circ r^{[-1]})</math>||Intrinsic||<math>r=r(s)</math><br><math>s=t</math>

|-

|<math>X[x,y]=x-\frac{x'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}</math><br><br><math>Y[x,y]=y-\frac{y'\int_a^t \sqrt { x'^2 + y'^2 }\, dt}{\sqrt { x'^2 + y'^2 }}</math>|| Parametric<br>Cartesian||<math>x=x(t)</math><br><math>y=y(t)</math>||rowspan=1|[[Involute]]

|-

|<math>X[x,y]=\frac{(xy'-yx')y'}{x'^2 + y'^2}</math><br><br><math>Y[x,y]=\frac{(yx'-xy')x'}{x'^2 + y'^2}</math>|| Parametric<br>Cartesian||<math>x=x(t)</math><br><math>y=y(t)</math>||rowspan=1|[[Pedal curve]] with pedal point (0;0)

|-

|<math>X[x,y]=\frac{(x'^2-y'^2)y'+2xyx'}{xy'-yx'}</math><br><br><math>Y[x,y]=\frac{(x'^2-y'^2)x'+2xyy'}{xy'-yx'}</math>|| Parametric<br>Cartesian||<math>x=x(t)</math><br><math>y=y(t)</math>||rowspan=1|[[Negative pedal curve]] with pedal point (0;0)

|-

| <math>X[y] = \int_a^t \cos \left[\int_a^t \frac{1}{y} \,dt\right] dt</math><br><br><math>Y[y] = \int_a^t \sin \left[\int_a^t \frac{1}{y} \,dt\right] dt</math>||Intrinsic||<math>y=r(s)</math><br><math>s=t</math>||Intrinsic to<br>Cartesian<br>transformation

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! style="background:#eafaea" colspan=4|Metric functionals

|-

| <math>F[y]=|\|y|\|=\sqrt{\int_E y^2 \, dt}</math>|| || ||[[norm (mathematics)|Norm]]

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| <math>F[x,y]=\int_E xy \, dt</math>|| || ||[[Inner product]]

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| <math>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-StudyFubini–Study metric]]<br>(inner angle)

|-

! style="background:#eafaea" colspan=4|Distribution functionals

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| <math>F[x,y] = x * y = \int_E x(s) y(t - s)\, ds</math>|| || ||[[Convolution]]

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| <math>F[y] = \int_E y \ln y \, dydt</math>|| || ||[[Differential entropy]]

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| <math>F[y] = \int_E yt\,dt</math>|| || ||[[Expected value]]

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| <math>F[y] = \int_E \left(t-\int_E yt\,dt\right)^2y\,dt</math>|| || ||[[Variance]]

|-

|}

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* [[Transfer operator]]

* [[Fredholm operator]]

* [[Borel summation|Borel transform]]

* [[TableGlossary of mathematical symbols]]

[[Category:Mathematics-related lists|Operators]]

[[Category:Functional analysis|Operators]]

[[Category:Curves|Operators]]

[[ru:Список операторов]]