# List of mathematic operators

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 | $x=x(t)$ $y=y(t)$ | Subtangent |
| $F[r]=-{\frac {r^{2}}{r'}}$ | Polar | $r=r(\phi )$ $\phi =t$ | Subtangent |
| $F[r]={\frac {1}{2}}\int _{a}^{t}r^{2}dt$ | 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 | $x=x(t)$ $y=y(t)$ | Arc length |
| $F[r]=\int _{a}^{t}{\sqrt {r^{2}+r'^{2}}}\,dt$ | Polar | $r=r(\phi )$ $\phi =t$ | Arc length |
| $F[y]=\int _{a}^{t}{\sqrt[{3}]{y''}}\,dt$ | Cartesian | $y=y(x)$ $x=t$ | Affine arc length |
| $F[x,y]=\int _{a}^{t}{\sqrt[{3}]{x'y''-x''y'}}\,dt$ | Parametric Cartesian | $x=x(t)$ $y=y(t)$ | Affine arc length |
| $F[x,y,z]=\int _{a}^{t}{\sqrt[{3}]{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}}dt$ | Parametric Cartesian | $x=x(t)$ $y=y(t)$ $z=z(t)$ | Affine arc length |
| $F[y]={\frac {y''}{(1+y'^{2})^{3/2}}}$ | Cartesian | $y=y(x)$ $x=t$ | Curvature |
| $F[x,y]={\frac {x'y''-y'x''}{(x'^{2}+y'^{2})^{3/2}}}$ | Parametric Cartesian | $x=x(t)$ $y=y(t)$ | Curvature |
| $F[r]={\frac {r^{2}+2r'^{2}-rr''}{(r^{2}+r'^{2})^{3/2}}}$ | Polar | $r=r(\phi )$ $\phi =t$ | Curvature |
| $F[x,y,z]={\frac {\sqrt {(z''y'-z'y'')^{2}+(x''z'-z''x')^{2}+(y''x'-x''y')^{2}}}{(x'^{2}+y'^{2}+z'^{2})^{3/2}}}$ | Parametric Cartesian | $x=x(t)$ $y=y(t)$ $z=z(t)$ | Curvature |
| $F[y]={\frac {1}{3}}{\frac {y''''}{(y'')^{5/3}}}-{\frac {5}{9}}{\frac {y'''^{2}}{(y'')^{8/3}}}$ | 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 |