View source for List of topics named after Leonhard Euler

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[[File:Leonhard Euler - Jakob Emanuel Handmann (Kunstmuseum Basel).jpg|thumb|200px|Leonhard Euler (1707–1783)]]
In [[mathematics]] and [[physics]], many topics are [[eponym|named in honor]] of Swiss mathematician [[Leonhard Euler]] (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Many of these entities have been given simple yet ambiguous names such as '''Euler's function''', '''Euler's equation''', and '''Euler's formula'''.

Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them ''after'' Euler.<ref>{{cite book |year=2008 |title=Euler's Gem: The polyhedron formula and the birth of topology |first=David S. |last=Richeson|author-link= David Richeson |edition=illustrated |publisher=Princeton University Press |isbn=978-0-691-12677-7 |at=[https://books.google.com/books?id=LB_6VogerHIC&pg=PA86 p. 86]|title-link= Euler's Gem}}</ref><ref>{{cite book|last1=Edwards|first1=Charles Henry|url=https://books.google.com/books?id=PwHQjwEACAAJ|title=Differential equations and boundary value problems|last2=Penney|first2=David E.|last3=Calvis|first3=David|publisher=Pearson Prentice Hall|year=2008|isbn=978-0-13-156107-6|pages=443 (微分方程及边值问题, 2004 edition)}}</ref>

== Conjectures ==
*[[Euler's conjecture (Waring's problem)]]
*[[Euler's sum of powers conjecture]] (disproved by counterexamples)
*[[Mutually orthogonal Latin squares#Euler's conjecture and disproof|Euler's Graeco-Latin square conjecture]]

== Equations ==
Usually, ''Euler's equation'' refers to one of (or a set of) [[differential equation]]s (DEs). It is customary to classify them into [[ordinary differential equation|ODEs]] and [[partial differential equation|PDEs]].

Otherwise, ''Euler's equation'' may refer to a non-differential equation, as in these three cases:
*[[Euler–Lotka equation]], a [[Characteristic polynomial|characteristic equation]] employed in mathematical demography
*[[Euler's pump and turbine equation]]
*[[Binomial transform#Euler transform|Euler transform]] used to accelerate the convergence of an alternating series and is also frequently applied to the [[hypergeometric series]]

=== Ordinary differential equations ===
*[[Euler's equations (rigid body dynamics)|Euler rotation equations]], a set of [[first-order ordinary differential equation|first-order ODEs]] concerning the rotations of a [[rigid body]].
*[[Cauchy–Euler equation|Euler–Cauchy equation]], a linear equidimensional [[differential equation|second-order ODE]] with [[variable coefficient]]s. Its second-order version can emerge from [[Laplace's equation]] in [[polar coordinates]].
*[[Euler–Bernoulli beam equation]], a fourth-order ODE concerning the elasticity of structural beams.
*[[Euler's differential equation]], a first order nonlinear ordinary differential equation

=== Partial differential equations ===
*[[Euler equations (fluid dynamics)|Euler conservation equations]], a set of quasilinear first-order [[hyperbolic equation]]s used in [[fluid dynamics]] for [[inviscid flow]]s. In the (Froude) limit of no external field, they are [[conservation law|conservation equations]].
*[[Euler–Tricomi equation]] – a second-order PDE emerging from Euler conservation equations.
*[[Euler–Poisson–Darboux equation]], a second-order PDE playing important role in solving the [[wave equation]].
*[[Euler–Lagrange equation]], a second-order PDE emerging from minimization problems in [[calculus of variations]].

== Formulas ==
{{columns-list|colwidth=30em|
*[[Euler's formula]], {{math|1=''e''<sup> ''ix''</sup> = cos ''x'' + ''i'' sin ''x''}}
*[[Euler's polyhedral formula]] for planar graphs or polyhedra: {{math|1=''v'' − ''e'' + ''f'' = 2}}, a special case of the [[Euler characteristic]] in topology
*[[Euler's critical load|Euler's formula]] for the critical load of a column: <math>P_\text{cr}=\frac{\pi^2 EI}{(KL)^2}</math>
*[[Euler's continued fraction formula]] connecting a finite sum of products with a finite continued fraction
*[[Euler product formula]] for the [[Riemann zeta function]].
*[[Euler–Maclaurin formula]] ([[Euler's summation formula]]) relating integrals to sums
*[[Euler–Rodrigues formula]] describing the rotation of a vector in three dimensions
*[[Euler's reflection formula]], reflection formula for the [[gamma function]]
*[[Local Euler characteristic formula]]
}}

== Functions ==
*The [[Euler function]], a [[modular form]] that is a prototypical [[q-series]].
*[[Euler's totient function]] (or Euler phi (φ) function) in [[number theory]], counting the number of coprime integers less than an integer.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Totient Function |url=https://mathworld.wolfram.com/TotientFunction.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}</ref>
*[[Euler hypergeometric integral]]
*[[Euler–Riemann zeta function]]

== Identities ==
*[[Euler's identity]] {{math|1=''e''<sup> ''i''&pi;</sup> + 1 = 0}}.
*[[Euler's four-square identity]], which shows that the product of two sums of four squares can itself be expressed as the sum of four squares.
*''Euler's identity'' may also refer to the  [[pentagonal number theorem]].

== Numbers ==
*[[Euler's number]], {{math|1=''e'' = 2.71828&nbsp;.&nbsp;.&nbsp;.&nbsp;}}, the base of the natural logarithm
*[[Euler's idoneal numbers]], a set of 65 or possibly 66 or 67 integers with special properties
*[[Euler numbers]], integers occurring in the coefficients of the Taylor series of 1/cosh&nbsp;''t''
*[[Eulerian number]]s count certain types of permutations.
*[[Euler number (physics)]], the cavitation number in [[fluid dynamics]].
*Euler number (algebraic topology) – now, [[Euler characteristic]], classically the number of vertices minus edges plus faces of a polyhedron.
*Euler number (3-manifold topology) – see [[Seifert fiber space]]
*[[Lucky numbers of Euler]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Lucky Number of Euler |url=https://mathworld.wolfram.com/LuckyNumberofEuler.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}</ref>
*[[Euler's constant]] gamma (''γ''), also known as the Euler–Mascheroni constant
*[[Eulerian integer]]s, more commonly called Eisenstein integers, the algebraic integers of form {{math|''a'' + ''bω''}} where {{mvar|ω}} is a complex cube root of 1.
*[[Euler–Gompertz constant]]

== Theorems ==
*{{annotated link|Euler's homogeneous function theorem}}
*{{annotated link|Euler's infinite tetration theorem}}
*{{annotated link|Euler's rotation theorem}}
*{{annotated link|Euler's theorem (differential geometry)}}
*{{annotated link|Euler's theorem in geometry}}
*{{annotated link|Euler's quadrilateral theorem}}
*[[Euclid–Euler theorem]], characterizing even perfect numbers
*[[Euler's theorem]], on modular exponentiation
*[[Euler's partition theorem]] relating the product and series representations of the Euler function Π(1&nbsp;−&nbsp;''x''<sup>''n''</sup>)
*[[Goldbach–Euler theorem]], stating that sum of 1/(''k''&nbsp;−&nbsp;1), where ''k'' ranges over positive integers of the form ''m''<sup>''n''</sup> for ''m''&nbsp;≥&nbsp;2 and ''n''&nbsp;≥&nbsp;2, equals 1
*{{annotated link|Gram–Euler theorem}}

== Laws ==
{{Main|Euler's laws of motion}}
*[[Euler's laws#Euler's first law|Euler's first law]], the sum of the external forces acting on a rigid body is equal to the rate of change of [[Momentum#Linear momentum of a system of particles|linear momentum]] of the body.
*[[Euler's laws#Euler's second law|Euler's second law]], the sum of the external [[Moment (physics)|moments]] about a point is equal to the rate of change of [[angular momentum]] about that point.

== Other things ==
{{columns-list|colwidth=30em|
*[[2002 Euler]], a minor planet
*[[Euler (crater)]], a lunar impact crater
*[[AMS Euler]] typeface
*[[Euler Mathematical Toolbox]] computer software
*[[Euler Book Prize]], an annual prize for mathematics books
*[[Euler Lecture]], an annual lecture at the [[University of Potsdam]]
*[[Euler Medal]], a prize for research in [[combinatorics]]
*[[Leonhard Euler Gold Medal]], a prize for outstanding results in mathematics and physics
*[[Euler (programming language)]], an Algol derivative
*[[Euler Society]], an American group dedicated to the life and work of Leonhard Euler
*[[Euler Committee of the Swiss Academy of Sciences]], a group dedicated to publishing the Euler's scientific productions
*[[Euler–Fokker genus]], a musical scale
*[[Project Euler]], a collection of programming puzzles
*[[Leonhard Euler Telescope]], a Swiss-operated telescope in Chile
*{{lang|fr|[[:fr:Rue Euler|Rue Euler]]}}, a street in Paris, France<ref>{{cite book |year=1910 |title=Promenades dans toutes les rues de Paris |trans-title=Walks along all of the streets in Paris |edition=VIII<sup>e</sup> arrondissement |first=Félix |last=de Rochegude |publisher=Hachette |page=[https://archive.org/details/promenadesdansto08rochuoft/page/98 98] |url=https://archive.org/details/promenadesdansto08rochuoft }}</ref>
*[[EulerOS]], a CentOS Linux based operating system
*[[French submarine Euler|French submarine ''Euler'']], an early French submarine
*[[Euler square]], a concept in combinatorics
*[[Lagrange, Euler, and Kovalevskaya tops|Euler top]], a special case of a rotating rigid body in classical mechanics
}}

== Topics by field of study ==
Selected topics from above, grouped by subject, and additional topics from the fields of music and physical systems

=== Analysis: derivatives, integrals, and logarithms ===
{{columns-list|colwidth=30em|
*Euler approximation – (see [[Euler's method]])
*The [[Euler integral (disambiguation)|Euler integrals]] of the first and second kind, namely the [[beta function]] and [[gamma function]].
*The [[Euler method]], a method for finding numerical solutions of differential equations
**[[Semi-implicit Euler method]]
**[[Euler–Maruyama method]]
**[[Backward Euler method]]
*[[e (mathematical constant)|Euler's number]] {{math|''e'' ≈ 2.71828}}, the base of the [[natural logarithm]], also known as '''Napier's constant'''.
*The [[Euler substitution]]s for integrals involving a square root.
*[[Euler's summation formula]], a theorem about integrals.
*[[Cauchy–Euler equation]] (or Euler equation), a second-order linear differential equation
*[[Cauchy–Euler operator]]
*[[Euler–Maclaurin formula]] – relation between integrals and sums
*[[Euler–Mascheroni constant]] or Euler's constant {{math|''γ'' ≈ 0.577216}}
*[[Integration using Euler's formula]]
*[[Euler summation]]
*[[Euler–Boole summation]]
}}

=== Geometry and spatial arrangement ===
{{columns-list|colwidth=30em|
*[[Euler angles]] defining a rotation in space
*[[Euler brick]]
*[[Euler's line]] – relation between [[triangle center]]s
*[[Euler operator (digital geometry)|Euler operator]] – set of functions to create [[polygon mesh]]es
*[[Euler filter]]
*[[Euler's rotation theorem]]
*[[Euler spiral]] – a curve whose curvature varies linearly with its arc length
*Euler squares, usually called [[Graeco-Latin square]]s
*[[Euler's theorem in geometry]], relating the [[circumcircle]] and [[incircle]] of a [[triangle]]
*[[Euler's quadrilateral theorem]], an extension of the [[parallelogram law]] to [[convex polygon|convex]] [[quadrilateral]]s
*[[Euler–Rodrigues formula]] concerning [[Euler–Rodrigues formula|Euler–Rodrigues parameters]] and 3D rotation matrices
*[[Cramer–Euler paradox]]
*[[Euler calculus]]
*[[Euler sequence]]
*[[Gram–Euler theorem]]
*[[Euler measure]]
}}

=== Graph theory ===
*[[Euler characteristic]] (formerly called Euler number) in [[algebraic topology]] and [[topological graph theory]], and the corresponding Euler's formula <math display=inline> \chi(S^2)=F-E+V=2</math>
*Eulerian circuit, Euler cycle or [[Eulerian path]] – a path through a graph that takes each edge once
**Eulerian graph has all its vertices spanned by an Eulerian path
*[[Euler class]]
*[[Euler diagram]] – popularly called "Venn diagrams", although some use this term only for a subclass of Euler diagrams.
*[[Euler tour technique]]

=== Music ===
*[[Euler–Fokker genus]]
*[[Euler's tritone]]

=== Number theory ===
*[[Euler's criterion]] – quadratic residues modulo by primes
*[[Euler product]] – [[infinite product]] expansion, indexed by prime numbers of a [[Dirichlet series]]
*[[Euler pseudoprime]]
*[[Euler–Jacobi pseudoprime]]
*[[Euler's totient function]] (or Euler phi (φ) function) in [[number theory]], counting the number of coprime integers less than an integer.
*[[Euler system]]
*[[Euler's factorization method]]

=== Physical systems ===
{{columns-list|colwidth=30em|
*[[Euler's Disk]] – a toy consisting of a circular disk that spins, without slipping, on a surface
*[[Euler's equations (rigid body dynamics)|Euler rotation equations]], in [[rigid body]] dynamics.
*[[Euler equations (fluid dynamics)|Euler conservation equations]] in [[fluid dynamics]].
*[[Euler number (physics)]], the cavitation number in [[fluid dynamics]].
*[[Euler's three-body problem]]
*[[Euler–Bernoulli beam equation]], concerning the elasticity of structural beams.
*[[buckling|Euler formula]] in calculating the buckling load of columns.
*[[Euler–Lagrange equation]]
*[[Euler–Tricomi equation]] – concerns transonic flow
*[[Euler relations]] – Gives relationship between extensive variables in thermodynamics.
*Eulerian observer – An observer "at rest" in spacetime, i.e. with 4-velocity perpendicular to spatial hypersurfaces.<ref>{{cite encyclopedia |last1=Evans |first1=Charles R. |last2=Smarr |first2=Larry L. |last3=Wilson |first3=James R. |date=1986 |title=Numerical Relativistic Gravitational Collapse with Spatial Time Slices |encyclopedia=Astrophysical Radiation Hydrodynamics |volume=188 |pages=491–529 |doi=10.1007/978-94-009-4754-2_15 |isbn=978-94-010-8612-7 |access-date=March 27, 2021 |url=https://rd.springer.com/chapter/10.1007/978-94-009-4754-2_15}}</ref>
*[[Relativistic Euler equations]]
*[[Lagrange, Euler, and Kovalevskaya tops|Euler top]]
*[[Lagrangian and Eulerian specification of the flow field|Eulerian specification of the flow field]]
*[[Nutation|Euler nutation]]
*[[Newton–Euler equations]]
*[[d'Alembert–Euler condition]]
*[[Euler acceleration]] or [[Euler force|force]]
}}

=== Polynomials ===
*[[Euler's homogeneous function theorem]], a theorem about [[homogeneous polynomial]]s.
*[[Euler polynomials]]
*[[Euler spline]] – splines composed of arcs using Euler polynomials<ref>{{cite web |url=http://pages.cs.wisc.edu/~deboor/HAT/fpapers/isobib.pdf |author=Schoenberg |year=1973 |title=bibliography |publisher=University of Wisconsin |url-status=dead |access-date=2007-10-28 |archive-date=2011-05-22 |archive-url=https://web.archive.org/web/20110522075326/http://pages.cs.wisc.edu/~deboor/HAT/fpapers/isobib.pdf }}</ref>

== See also ==
*[[Contributions of Leonhard Euler to mathematics]]

== Notes ==
{{reflist}}

[[Category:Lists of things named after mathematicians|Euler]]
[[Category:Leonhard Euler]]