Coble creep

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
$L[y]=\underset{a}{\overset{t}{\int }}ydt$	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
$L[y]=y\circ (t+1)-y\circ t=\mathrm{\Delta }y$			Difference operator
$L[y]=y\circ (t)-y\circ (t-1)=\mathrm{\nabla }y$			Backward difference (Nabla operator)
$L[y]=\sum y={\mathrm{\Delta }}^{-1}y$			Indefinite sum operator (inverse operator of difference)
$L[y]=-(p{y}^{\prime }{)}^{\prime }+qy$			Sturm–Liouville operator
Non-linear transformations
$F[y]=y^{[-1]}$			Inverse function
$F[y]=ty{\text{'}}^{[-1]}-y\circ y{\text{'}}^{[-1]}$			Legendre transformation
$F[y]=f\circ y$			Left composition
$F[y]=\prod y$			Indefinite product
$F[y]=\frac{y^{\prime }}{y}$			Logarithmic derivative
$F[y]=\frac{t{y}^{\prime }}{y}$			Elasticity
$F[y]=\frac{y^{‴}}{y^{\prime }}-\frac{3}{2}{\left(\frac{y^{″}}{y^{\prime }}\right)}^2$			Schwarzian derivative
$F[y]=\underset{a}{\overset{t}{\int }}\left|y^{\prime }\right|dt$			Total variation
$F[y]=\frac{1}{t-a}\underset{a}{\overset{t}{\int }}ydt$			Arithmetic mean
$F[y]=\exp (\frac{1}{t-a}\underset{a}{\overset{t}{\int }}\ln ydt)$			Geometric mean
$F[y]=-\frac{y}{y^{\prime }}$	Cartesian	$y=y(x)$ $x=t$	Subtangent
$F[x,y]=-\frac{y{x}^{\prime }}{y^{\prime }}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[r]=-\frac{r^2}{r^{\prime }}$	Polar	$r=r(\varphi )$ $\varphi =t$
$F[r]=\frac{1}{2}\underset{a}{\overset{t}{\int }}{r}^{2}dt$	Polar	$r=r(\varphi )$ $\varphi =t$	Sector area
$F[y]=\underset{a}{\overset{t}{\int }}\sqrt{1+y{\text{'}}^2}dt$	Cartesian	$y=y(x)$ $x=t$	Arc length
$F[x,y]=\underset{a}{\overset{t}{\int }}\sqrt{x{\text{'}}^2+y{\text{'}}^2}dt$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[r]=\underset{a}{\overset{t}{\int }}\sqrt{r^2+r{\text{'}}^2}dt$	Polar	$r=r(\varphi )$ $\varphi =t$
$F[x,y]=\underset{a}{\overset{t}{\int }}\sqrt[3]{y^{″}}dt$	Cartesian	$y=y(x)$ $x=t$	Affine arc length
$F[x,y]=\underset{a}{\overset{t}{\int }}\sqrt[3]{x^{\prime }{y}^{″}-x^{″}{y}^{\prime }}dt$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[x,y,z]=\underset{a}{\overset{t}{\int }}\sqrt[3]{z^{‴}(x^{\prime }{y}^{″}-y^{\prime }{x}^{″})+z^{″}(x^{‴}{y}^{\prime }-x^{\prime }{y}^{‴})+z^{\prime }(x^{″}{y}^{‴}-x^{‴}{y}^{″})}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$ $z=z(t)$
$F[y]=\frac{y^{″}}{(1+y{\text{'}}^2{)}^{3/2}}$	Cartesian	$y=y(x)$ $x=t$	Curvature
$F[x,y]=\frac{x^{\prime }{y}^{″}-y^{\prime }{x}^{″}}{(x{\text{'}}^2+y{\text{'}}^2{)}^{3/2}}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[r]=\frac{r^2+2r{\text{'}}^2-r{r}^{″}}{(r^2+r{\text{'}}^2{)}^{3/2}}$	Polar	$r=r(\varphi )$ $\varphi =t$
$F[x,y,z]=\frac{\sqrt{(z^{″}{y}^{\prime }-z^{\prime }{y}^{″}{)}^2+(x^{″}{z}^{\prime }-z^{″}{x}^{\prime }{)}^2+(y^{″}{x}^{\prime }-x^{″}{y}^{\prime }{)}^2}}{(x{\text{'}}^2+y{\text{'}}^2+z{\text{'}}^2{)}^{3/2}}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$ $z=z(t)$
$F[y]=\frac{1}{3}\frac{y^{⁗}}{(y^{″}{)}^{5/3}}-\frac{5}{9}\frac{y^{″}{\text{'}}^2}{(y^{″}{)}^{8/3}}$	Cartesian	$y=y(x)$ $x=t$	Affine curvature
$F[x,y]=\frac{x^{″}{y}^{‴}-x^{‴}{y}^{″}}{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime }{)}^{5/3}}-\frac{1}{2}{\left[{\frac{1}{(x^{\prime }{y}^{″}-x^{″}{y}^{\prime }{)}^{2/3}}}^{″}\right]}^{″}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$
$F[x,y,z]=\frac{z^{‴}(x^{\prime }{y}^{″}-y^{\prime }{x}^{″})+z^{″}(x^{‴}{y}^{\prime }-x^{\prime }{y}^{‴})+z^{\prime }(x^{″}{y}^{‴}-x^{‴}{y}^{″})}{(x{\text{'}}^2+y{\text{'}}^2+z{\text{'}}^2)(x^{\prime }{\text{'}}^2+y^{\prime }{\text{'}}^2+z^{\prime }{\text{'}}^2)}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$ $z=z(t)$	Torsion of curves
$X[x,y]=\frac{y^{\prime }}{y{x}^{\prime }-x{y}^{\prime }}$ $Y[x,y]=\frac{x^{\prime }}{x{y}^{\prime }-y{x}^{\prime }}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Dual curve (tangent coordinates)
$X[x,y]=x+\frac{a{y}^{\prime }}{\sqrt{x{\text{'}}^2+y{\text{'}}^2}}$ $Y[x,y]=y-\frac{a{x}^{\prime }}{\sqrt{x{\text{'}}^2+y{\text{'}}^2}}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Parallel curve
$X[x,y]=x+y^{\prime }\frac{x{\text{'}}^2+y{\text{'}}^2}{x^{″}{y}^{\prime }-y^{″}{x}^{\prime }}$ $Y[x,y]=y+x^{\prime }\frac{x{\text{'}}^2+y{\text{'}}^2}{y^{″}{x}^{\prime }-x^{″}{y}^{\prime }}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Evolute
$F[r]=t(r^{\prime }\circ r^{[-1]})$	Intrinsic	$r=r(s)$ $s=t$
$X[x,y]=x-\frac{x^{\prime }\underset{a}{\overset{t}{\int }}\sqrt{x{\text{'}}^2+y{\text{'}}^2}dt}{\sqrt{x{\text{'}}^2+y{\text{'}}^2}}$ $Y[x,y]=y-\frac{y^{\prime }\underset{a}{\overset{t}{\int }}\sqrt{x{\text{'}}^2+y{\text{'}}^2}dt}{\sqrt{x{\text{'}}^2+y{\text{'}}^2}}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Involute
$X[x,y]=\frac{(x{y}^{\prime }-y{x}^{\prime })y^{\prime }}{x{\text{'}}^2+y{\text{'}}^2}$ $Y[x,y]=\frac{(y{x}^{\prime }-x{y}^{\prime })x^{\prime }}{x{\text{'}}^2+y{\text{'}}^2}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Pedal curve with pedal point (0;0)
$X[x,y]=\frac{(x{\text{'}}^2-y{\text{'}}^2)y^{\prime }+2xy{x}^{\prime }}{x{y}^{\prime }-y{x}^{\prime }}$ $Y[x,y]=\frac{(x{\text{'}}^2-y{\text{'}}^2)x^{\prime }+2xy{y}^{\prime }}{x{y}^{\prime }-y{x}^{\prime }}$	Parametric Cartesian	$x=x(t)$ $y=y(t)$	Negative pedal curve with pedal point (0;0)
$X[y]=\underset{a}{\overset{t}{\int }}\cos \left[\underset{a}{\overset{t}{\int }}\frac{1}{y}dt\right]dt$ $Y[y]=\underset{a}{\overset{t}{\int }}\sin \left[\underset{a}{\overset{t}{\int }}\frac{1}{y}dt\right]dt$	Intrinsic	$y=r(s)$ $s=t$	Intrinsic to Cartesian transformation
Metric functionals
$F[y]=\left|\right|y\left|\right|=\sqrt{\underset{E}{\int }{y}^{2}dt}$			Norm
$F[x,y]=\underset{E}{\int }xydt$			Inner product
$F[x,y]=\arccos \left[\frac{\underset{E}{\int }xydt}{\sqrt{\underset{E}{\int }{x}^{2}dt}\sqrt{\underset{E}{\int }{y}^{2}dt}}\right]$			Fubini-Study metric (inner angle)
Distribution functionals
$F[x,y]=x*y=\underset{E}{\int }x(s)y(t-s)ds$			Convolution
$F[y]=\underset{E}{\int }y\ln ydy$			Differential entropy
$F[y]=\underset{E}{\int }ytdt$			Expected value
$F[y]=\underset{E}{\int }(t-\underset{E}{\int }ytdt{)}^{2}ydt$			Variance

See also

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