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 : ℱ \to 𝒢

which takes a function $y \in ℱ$ to another function $L [y] \in 𝒢$ . Here, $ℱ$ and $𝒢$ 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
$L [y] = \int_{a}^{t} y d t$	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
$L [y] = y \circ (t + 1) - y \circ t = Δ y$			Difference operator
$L [y] = y \circ (t) - y \circ (t - 1) = \nabla y$			Backward difference (Nabla operator)
$L [y] = \sum y = Δ^{- 1} y$			Indefinite sum operator (inverse operator of difference)
$L [y] = - (p y^{'})^{'} + q y$			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
$F [y] = \frac{t y^{'}}{y}$			Elasticity
$F [y] = \frac{y^{‴}}{y^{'}} - \frac{3}{2} {(\frac{y^{″}}{y^{'}})}^{2}$			Schwarzian derivative
$F [y] = \int_{a}^{t} \| y^{'} \| d t$			Total variation
$F [y] = \frac{1}{t - a} \int_{a}^{t} y d t$			Arithmetic mean
$F [y] = \exp (\frac{1}{t - a} \int_{a}^{t} \ln y d t)$			Geometric mean
$F [y] = - \frac{y}{y^{'}}$	Cartesian	$y = y (x)$ $x = t$	Subtangent
$F [x, y] = - \frac{y x^{'}}{y^{'}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$
$F [r] = - \frac{r^{2}}{r^{'}}$	Polar	$r = r (ϕ)$ $ϕ = t$
$F [r] = \frac{1}{2} \int_{a}^{t} r^{2} d t$	Polar	$r = r (ϕ)$ $ϕ = t$	Sector area
$F [y] = \int_{a}^{t} \sqrt{1 + y'^{2}} d t$	Cartesian	$y = y (x)$ $x = t$	Arc length
$F [x, y] = \int_{a}^{t} \sqrt{x'^{2} + y'^{2}} d t$	Parametric Cartesian	$x = x (t)$ $y = y (t)$
$F [r] = \int_{a}^{t} \sqrt{r^{2} + r'^{2}} d t$	Polar	$r = r (ϕ)$ $ϕ = t$
$F [x, y] = \int_{a}^{t} \sqrt[3]{y^{″}} d t$	Cartesian	$y = y (x)$ $x = t$	Affine arc length
$F [x, y] = \int_{a}^{t} \sqrt[3]{x^{'} y^{″} - x^{″} y^{'}} d t$	Parametric Cartesian	$x = x (t)$ $y = y (t)$
$F [x, y, z] = \int_{a}^{t} \sqrt[3]{z^{‴} (x^{'} y^{″} - y^{'} x^{″}) + z^{″} (x^{‴} y^{'} - x^{'} y^{‴}) + z^{'} (x^{″} y^{‴} - x^{‴} y^{″})}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$ $z = z (t)$
$F [y] = \frac{y^{″}}{(1 + y'^{2})^{3 / 2}}$	Cartesian	$y = y (x)$ $x = t$	Curvature
$F [x, y] = \frac{x^{'} y^{″} - y^{'} x^{″}}{(x'^{2} + y'^{2})^{3 / 2}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$
$F [r] = \frac{r^{2} + 2 r'^{2} - r r^{″}}{(r^{2} + r'^{2})^{3 / 2}}$	Polar	$r = r (ϕ)$ $ϕ = t$
$F [x, y, z] = \frac{\sqrt{(z^{″} y^{'} - z^{'} y^{″})^{2} + (x^{″} z^{'} - z^{″} x^{'})^{2} + (y^{″} x^{'} - x^{″} y^{'})^{2}}}{(x'^{2} + y'^{2} + z'^{2})^{3 / 2}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$ $z = z (t)$
$F [y] = \frac{1}{3} \frac{y^{⁗}}{(y^{″})^{5 / 3}} - \frac{5}{9} \frac{y^{″}'^{2}}{(y^{″})^{8 / 3}}$	Cartesian	$y = y (x)$ $x = t$	Affine curvature
$F [x, y] = \frac{x^{″} y^{‴} - x^{‴} y^{″}}{(x^{'} y^{″} - x^{″} y^{'})^{5 / 3}} - \frac{1}{2} {[\frac{1}{(x^{'} y^{″} - x^{″} y^{'})^{2 / 3}}]}^{″}$	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
$X [x, y] = \frac{y^{'}}{y x^{'} - x y^{'}}$ $Y [x, y] = \frac{x^{'}}{x y^{'} - y x^{'}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$	Dual curve (tangent coordinates)
$X [x, y] = x + \frac{a y^{'}}{\sqrt{x'^{2} + y'^{2}}}$ $Y [x, y] = y - \frac{a x^{'}}{\sqrt{x'^{2} + y'^{2}}}$	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
$X [x, y] = x - \frac{x^{'} \int_{a}^{t} \sqrt{x'^{2} + y'^{2}} d t}{\sqrt{x'^{2} + y'^{2}}}$ $Y [x, y] = y - \frac{y^{'} \int_{a}^{t} \sqrt{x'^{2} + y'^{2}} d t}{\sqrt{x'^{2} + y'^{2}}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$	Involute
$X [x, y] = \frac{(x y^{'} - y x^{'}) y^{'}}{x'^{2} + y'^{2}}$ $Y [x, y] = \frac{(y x^{'} - x y^{'}) x^{'}}{x'^{2} + y'^{2}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$	Pedal curve with pedal point (0;0)
$X [x, y] = \frac{(x'^{2} - y'^{2}) y^{'} + 2 x y x^{'}}{x y^{'} - y x^{'}}$ $Y [x, y] = \frac{(x'^{2} - y'^{2}) x^{'} + 2 x y y^{'}}{x y^{'} - y x^{'}}$	Parametric Cartesian	$x = x (t)$ $y = y (t)$	Negative pedal curve with pedal point (0;0)
$X [y] = \int_{a}^{t} \cos [\int_{a}^{t} \frac{1}{y} d t] d t$ $Y [y] = \int_{a}^{t} \sin [\int_{a}^{t} \frac{1}{y} d t] d t$	Intrinsic	$y = r (s)$ $s = t$	Intrinsic to Cartesian transformation
Metric functionals
$F [y] = \| \| y \| \| = \sqrt{\int_{E} y^{2} d t}$			Norm
$F [x, y] = \int_{E} x y d t$			Inner product
$F [x, y] = \arccos [\frac{\int_{E} x y d t}{\sqrt{\int_{E} x^{2} d t} \sqrt{\int_{E} y^{2} d t}}]$			Fubini-Study metric (inner angle)
Distribution functionals
$F [x, y] = x * y = \int_{E} x (s) y (t - s) d s$			Convolution
$F [y] = \int_{E} y \ln y d y$			Differential entropy
$F [y] = \int_{E} y t d t$			Expected value
$F [y] = \int_{E} (t - \int_{E} y t d t)^{2} y d t$			Variance

Coble creep

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