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| In [[mathematics]] and [[physics]], there are a large number of topics named in honor of [[Leonhard Euler]], many of which include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Unfortunately, many of these entities have been given simple and ambiguous names such as '''Euler's function''', '''Euler's equation''', and '''Euler's formula'''.
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| Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. Physicists and mathematicians sometimes jest that, in an effort to avoid naming everything after Euler, discoveries and theorems are named after the "first person ''after'' Euler to discover it".<ref>{{citation | year=2008 | title=Euler's Gem: The Polyhedron Formula and the Birth of Topology | author1=David S. Richeson | edition=illustrated | publisher=Princeton University Press | isbn=978-0-691-12677-7 | page=86 | url=http://books.google.com/?id=LB_6VogerHIC&pg=PA86&dq=%22person+after+Euler%22}}</ref><ref>{{citation | year=2004 | title=Differential equations and boundary value problems : | author1=C. H. Edwards | author2= David E. Penney | publisher=清华大学出版社 | isbn=978-7-302-09978-9 | page=443 | url=http://books.google.com/?id=51KTl4Fmh2wC&pg=PA443&dq=%22person+after+Euler%22}}</ref>
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| ==Euler's conjectures==
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| *[[Euler's conjecture (Waring's problem)]]
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| *[[Euler's sum of powers conjecture]]
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| (Also see [[Euler's conjecture (disambiguation)|Euler's conjecture]].)
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| ==Euler's equations==
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| *Euler's equation – usually refers to [[Euler's equations (rigid body dynamics)]], [[Euler's formula]], [[Euler's homogeneous function theorem]], or [[Euler's identity]]
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| *[[Euler equations (fluid dynamics)]] in [[fluid dynamics]].
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| *[[Euler's equations (rigid body dynamics)]], concerning the rotations of a [[rigid body]].
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| *[[Euler–Bernoulli beam equation]], concerning the elasticity of structural beams.
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| *[[Cauchy–Euler equation|Euler–Cauchy equation]] (or Euler equation), a second-order linear differential equation
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| *[[Euler–Lagrange equation]] (in regard to minimization problems in [[calculus of variations]])
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| *[[Euler–Lotka equation]] (mathematical demography)
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| *[[Euler–Poisson–Darboux equation]]
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| *[[Euler's pump and turbine equation]]
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| *[[Euler–Tricomi equation]] – concerns transonic flow
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| *[[Binomial transform#Euler transform|Euler transform]] used to accelerate the convergence of an alternating series and is also frequently applied to the [[hypergeometric series]]
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| ==Euler's formulas==
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| *[[Euler's formula]] {{math|1=''e''<sup> ''ix''</sup> = cos ''x'' + ''i'' sin ''x''}} in [[complex analysis]].
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| *[[Planar graph#Euler's formula|Euler's formula]] for planar graphs or polyhedra: {{math|1=''v'' − ''e'' + ''f'' = 2}}
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| *[[Buckling#Buckling in columns|Euler's formula]] for the critical load of a column: <math>P_\text{cr}=\frac{\pi^2 EI}{(KL)^2}</math>
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| *[[Euler's continued fraction formula]]
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| *[[Euler product|Euler product formula]] for the [[Riemann zeta function]].
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| *[[Euler–Maclaurin formula]] ([[Euler's summation formula]]) – relation between integrals and sums
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| *[[Euler–Rodrigues formulas]] concern [[Euler–Rodrigues parameters]] and 3D rotation matrices
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| ==Euler's functions==
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| *The [[Euler function]], a [[modular form]] that is a prototypical [[q-series]].
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| *[[homogeneous function|Euler's homogeneous function theorem]]
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| *[[Euler's totient function]] (or Euler phi (φ) function) in [[number theory]], counting the number of coprime integers less than an integer.
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| *[[Euler hypergeometric integral]]
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| ==Euler's identities==
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| *[[Euler's identity]] {{math|1=''e''<sup> ''i''π</sup> + 1 = 0}}.
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| *[[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.
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| *''Euler's identity'' may also refer to the [[pentagonal number theorem]].
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| ==Euler's numbers==
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| *[[e (mathematical constant)|Euler's number]], {{math|''e'' ≈ 2.71828}}, the base of the [[natural logarithm]], also known as ''Napier's constant''.
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| *[[Euler's idoneal numbers]]
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| *[[Euler number]]s are an [[integer sequence]].
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| *[[Eulerian number]]s are another integer sequence.
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| *[[Euler number (physics)]], the cavitation number in [[fluid dynamics]].
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| *Euler number (topology) – now, [[Euler characteristic]]
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| *[[Lucky numbers of Euler]]
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| *[[Euler–Mascheroni constant]]
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| *[[Eulerian integer]]s are the numbers of form {{math|''a'' + ''bω''}} where {{mvar|ω}} is a complex cube root of 1.
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| ==Euler's theorems==
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| *[[Euler's homogeneous function theorem]], a theorem about [[homogeneous polynomial]]s.
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| *[[Euler's infinite tetration theorem]]
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| *[[Euler's rotation theorem]]
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| *[[Euler's theorem (differential geometry)]] on the existence of the [[principal curvatures]] of a [[surface]] and orthogonality of the associated principal directions.
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| *[[Euler's theorem in geometry]], relating the [[circumcircle]] and [[incircle]] of a [[triangle]].
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| *[[perfect number|Euclid–Euler theorem]]
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| *[[Euler–Fermat theorem]], that {{math|''a''<sup>''φ''(''m'')</sup> ≡ 1 (mod ''m'')}} whenever {{mvar|a}} is [[coprime]] to {{mvar|m}}, and {{mvar|φ}} is [[Euler's totient function]]
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| *Euler's theorem equating the number of [[Partition (number theory)|partitions]] with odd parts and the number of partitions with distinct parts. See [[Glaisher's theorem]].
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| *Euler's adding-up theorem in economics <!-- Is there an article for this? -->
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| ==Euler's laws==
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| {{main|Euler's laws}}
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| *[[Euler's laws#Euler's first law|Euler's first law]], the [[Momentum#Linear momentum of a system of particles|linear momentum]] of a body is equal to the product of the mass of the body and the velocity of its [[center of mass]].
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| *[[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.
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| ==Other things named after Euler==
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| *[[2002 Euler]] (a minor planet)
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| *[[AMS Euler]] typeface
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| *[[Euler (software)]]
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| *[[Euler acceleration]] or [[Euler force|force]]
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| *[[Euler Book Prize]]
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| *[[Euler Medal]], a prize for research in [[combinatorics]]
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| *[[Euler programming language]]
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| *[[Euler Society]], an American group dedicated to the life and work of Leonhard Euler
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| *[[Euler–Fokker genus]]
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| *[[Project Euler]]
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| * [[Leonhard Euler Telescope]]
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| ==Topics by field of study==
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| Selected topics from above, grouped by subject.
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| ===Analysis: derivatives, integrals, and logarithms ===
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| *Euler approximation – (see [[Euler's method]])
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| *[[Euler derivative]] (as opposed to [[Lagrangian derivative]])
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| *The [[Euler integral (disambiguation)|Euler integral]]s of the first and second kind, namely the [[beta function]] and [[gamma function]].
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| *The [[Euler method]], a method for finding numerical solutions of differential equations
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| ** [[Semi-implicit Euler method]]
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| *[[e (mathematical constant)|Euler's number]] {{math|''e'' ≈ 2.71828}}, the base of the [[natural logarithm]], also known as '''Napier's constant'''.
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| *The [[Euler substitution]]s for integrals involving a square root.
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| *[[Euler's summation formula]], a theorem about integrals.
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| *[[Cauchy–Euler equation]] (or Euler equation), a second-order linear differential equation
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| *[[Euler–Maclaurin formula]] – relation between integrals and sums
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| *[[Euler–Mascheroni constant]] or Euler's constant {{math|''γ'' ≈ 0.577216}}
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| ===Geometry and spatial arrangement===
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| *[[Euler angles]] defining a rotation in space.
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| *[[Euler brick]]
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| *[[Euler's line]] – relation between [[triangle center]]s
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| *[[Euler operator]] – set of functions to create [[polygon meshes]]
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| *[[Euler's rotation theorem]]
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| *Euler squares, usually called [[Graeco-Latin square]]s.
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| *[[Euler's theorem in geometry]], relating the [[circumcircle]] and [[incircle]] of a [[triangle]].
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| *[[Euler–Rodrigues formulas]] concern [[Euler–Rodrigues parameters]] and 3D rotation matrices
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| ===Graph theory===
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| *[[Euler characteristic]] (formerly called Euler number) in [[algebraic topology]] and [[topological graph theory]], and the corresponding Euler's formula <math>\scriptstyle \chi(S^2)=F-E+V=2</math>
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| * Eulerian circuit, Euler cycle or [[Eulerian path]] – a path through a graph that takes each edge once
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| ** Eulerian graph has all its vertices spanned by an Eulerian path
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| *[[Euler class]]
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| *[[Euler diagram]] – likely better (but wrongly) known as Venn diagrams, its subclass
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| *[[Euler tour technique]]
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| ===Music===
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| *[[Euler–Fokker genus]]
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| ===Number theory===
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| *[[Euler's criterion]] – quadratic residues modulo by primes
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| *[[Euler product]] – [[infinite product]] expansion, indexed by prime numbers of a [[Dirichlet series]]
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| *[[Euler pseudoprime]]
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| *[[Euler's totient function]] (or Euler phi (φ) function) in [[number theory]], counting the number of coprime integers less than an integer.
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| ===Physical systems===
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| *[[Euler's Disk]] – a toy consisting of a circular disk that spins, without slipping, on a surface
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| *[[Euler equations (fluid dynamics)|Euler equations]] in [[fluid dynamics]].
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| *[[Euler's equations]], concerning the rotations of a [[rigid body]].
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| *[[Euler number (physics)]], the cavitation number in [[fluid dynamics]].
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| *[[Euler's three-body problem]]
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| *[[Euler–Bernoulli beam equation]], concerning the elasticity of structural beams.
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| *[[buckling|Euler formula]] in calculating the buckling load of columns.
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| *[[Euler–Tricomi equation]] – concerns transonic flow
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| ===Polynomials===
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| *[[Euler's homogeneous function theorem]], a theorem about [[homogeneous polynomial]]s.
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| *[[Euler polynomials]]
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| *[[Euler spline]] – composed of classical Euler polynomial arcs<!-- (cred. to [http://pages.cs.wisc.edu/~deboor/HAT/fpapers/isobib.pdf Schoenberg, 1973 – PDF])-->
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| ==See also==
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| *[[Contributions of Leonhard Euler to mathematics]]
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| *[[Euler–Maruyama method]]
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| ==Notes==
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| {{reflist}}
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| [[Category:Lists of things named after mathematicians|Euler]]
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| [[Category:Leonhard Euler]]
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