List of mathematic operators explained

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: l{F}\to l{G}

which takes a function

y\in l{F} to another function L[y]\in l{G} . Here, l{F} and l{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 ^{t} _{a} y dt	Cartesian	y = y(x) x = t	Integral, area
L[y] = y\circ f			Composition operator
L[y] = (y\circ t + y\circ - t)/(2)			Even component
L[y] = (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] = ty' ^{[-1]} - y\circ y' ^{[-1]}			Legendre transformation
F[y] = f\circ y			Left composition
F[y] = \prod y			Indefinite product
F[y] = (y')/(y)			Logarithmic derivative
F[y] = { (ty')/(y) }			Elasticity
F[y] = {y''' \over y'} - { 3 \over 2 }\left({y''\over y'}\right) ^{2}			Schwarzian derivative
F[y] = \int ^{t} _{a}	y'	\,dt			Total variation
F[y] = (1)/(t - a) \int ^{t} _{a} ydt			Arithmetic mean
F[y] = \exp \left( (1)/(t - a) \int ^{t} _{a} ln ydt \right)			Geometric mean
F[y] =- (y)/(y')	Cartesian	y = y(x) x = t	Subtangent
F[x,y] =- (yx')/(y')	Parametric Cartesian	x = x(t) y = y(t)
F[r] =- (r ^{2})/(r')	Polar	r = r(\phi) \phi = t
F[r] = (1)/(2) \int ^{t} _{a} r ^{2} dt	Polar	r = r(\phi) \phi = t	Sector area
F[y] = \int ^{t} _{a} \sqrt { 1 + y' ^{2} } dt	Cartesian	y = y(x) x = t	Arc length
F[x,y] = \int ^{t} _{a} \sqrt { x' ^{2} + y' ^{2} } dt	Parametric Cartesian	x = x(t) y = y(t)
F[r] = \int ^{t} _{a} \sqrt { r ^{2} + r' ^{2} } dt	Polar	r = r(\phi) \phi = t
F[y] = \int ^{t\sqrt[ 3 ]{y''}} _{a} dt	Cartesian	y = y(x) x = t	Affine arc length
F[x,y] = \int ^{t\sqrt[ 3 ]{x'y'' - x''y'}} _{a} dt	Parametric Cartesian	x = x(t) y = y(t)
F[x,y,z] = \int ^{t\sqrt[ 3 ]{z'''(x'y'' - y'x'') + z''(x'''y' - x'y''') + z'(x''y''' - x'''y'')}dt} _{a}	Parametric Cartesian	x = x(t) y = y(t) z = z(t)
F[y] = (y'')/(( 1+ y' ^{2)} ^{3/2})	Cartesian	y = y(x) x = t	Curvature
F[x,y] = (x'y'' - y'x'')/((x' ^{2+y'} ^{2)} ^{3/2})	Parametric Cartesian	x = x(t) y = y(t)
F[r] = (r ^{2+2r'} ^{2-rr''})/((r ^{2+r'} ^{2)} ^{3/2})	Polar	r = r(\phi) \phi = t
F[x,y,z] = (\sqrt{(z''y' - z'y'') ^{2+(x''z'-z''x')} ^{2+(y''x'-x''y')} ^{2})/() }	Parametric Cartesian	x = x(t) y = y(t) z = z(t)
F[y] = (1)/(3) (y'''')/(- (y'') ^{5/3}) (5)/(9) (y''' ^{2})/((y'') ^{8/3})	Cartesian	y = y(x) x = t	Affine curvature
F[x,y] = (x''y''' - x'''y'')/(- (x'y'' - x''y') ^{5/3}) (1)/(\left[ 2) (1)/((x'y'' - x''y') ^{2/3}) \right]''	Parametric Cartesian	x = x(t) y = y(t)
F[x,y,z] = (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] = (y')/(yx' - xy') Y[x,y] = (x')/(xy' - yx')	Parametric Cartesian	x = x(t) y = y(t)	Dual curve (tangent coordinates)
X[x,y] = x + (ay')/(\sqrt {x' ^{2+y'} ^{2}) } Y[x,y] = y - (ax')/(\sqrt {x' ^{2+y'} ^{2}) }	Parametric Cartesian	x = x(t) y = y(t)	Parallel curve
X[x,y] = x + y' (x' ^{2+y'} ^{2})/(x''y' - y''x') Y[x,y] = y + x' (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
X[x,y] = x - (x'\int ^{t} _{a} \sqrt { x' ^{2} + y' ^{2})/() dt}{\sqrt { x' ^{2} + y' ^{2} }} Y[x,y] = y - (y'\int ^{t} _{a} \sqrt { x' ^{2} + y' ^{2})/() dt}{\sqrt { x' ^{2} + y' ^{2} }}	Parametric Cartesian	x = x(t) y = y(t)		Involute
X[x,y] = ((xy' - yx')y')/(x' ^{2} + y' ^{2}) Y[x,y] = ((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] = ((x' ^{2-y'} ^{2)y'+2xyx'})/(xy' - yx') Y[x,y] = ((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 ^{t} _{a} \cos \left[\int ^{t} _{a} (1)/(y) dt\right] dt Y[y] = \int ^{t} _{a} \sin \left[\int ^{t} _{a} (1)/(y) dt\right] dt	Intrinsic	y = r(s) s = t	Intrinsic to Cartesian transformation
Metric functionals
F[y] = \	y\	=\sqrt			Norm
F[x,y] = \int _{E} xy dt			Inner product
F[x,y] = \arccos \left[ (\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} ytdt			Expected value
F[y] = \int _{E} \left(t - \int _{E} ytdt\right) ^{2ydt}			Variance

See also

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