In mathematics and physics, there is 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.
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".[1][2]
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General "Euler-" mathematical topics
- Euler angles defining a rotation in space.
- Euler approximation – (see Euler method)
- Euler brick
- Euler characteristic in algebraic topology and topological graph theory, and the corresponding Euler's formula
- Euler circle
- Eulerian circuit – (see Eulerian path)
- Euler class
- Euler's constant – (see Euler–Mascheroni constant) (not to be confused with Euler's number)
- Euler cycle – (see Eulerian path)
- Euler's criterion – quadratic residues modulo primes
- Euler derivative (as opposed to Lagrangian derivative)
- Euler diagram – likely more widely (though incorrectly) known as Venn diagram (which has more restrictions)
- Euler's disk – a circular disk that spins, without slipping, on a surface
- Eulerian graph – (see Eulerian path)
- The Euler integrals of the first and second kind, namely the beta function and gamma function.
- Euler's line – relation between triangle centers
- Euler–Mascheroni constant or Euler's constant γ ≈ 0.577216
- Euler's number, e, the base of the natural logarithm.
- Euler operator – set of functions to create polygon meshes
- Euler parameters – (see Euler–Rodrigues parameters)
- Eulerian path, a path through a graph that takes each edge once.
- Euler polynomials
- Euler pseudoprime
- Euler–Rodrigues parameters – concerns Lie groups and quaternions
- Euler's rule – finding amicable numbers
- Euler spline – composed of classical Euler polynomial arcs (cred. to Schoenberg, 1973 – PDF)
- Euler squares, usually called Graeco-Latin squares.
- Euler summation
- Euler system, a collection of cohomology classes.
- Euler's three-body problem
See also: Other things named after Euler
Euler—conjectures
(Also see Euler's conjecture.)
Euler—equations
- Euler's equation – usually refers to Euler's equations (rigid body dynamics), Euler's formula, Euler's homogeneous function theorem, or Euler's identity
- Euler equations (fluid dynamics) in fluid dynamics.
- Euler's equations (rigid body dynamics), concerning the rotations of a rigid body.
- Euler–Bernoulli beam equation, concerning the elasticity of structural beams.
- Euler–Cauchy equation (or Euler equation), a second-order linear differential equation
- Euler–Lagrange equation (in regard to minimization problems)
- Euler–Lotka equation (mathematical demography)
- Euler–Poisson–Darboux equation
- Euler–Tricomi equation – concerns transonic flow
- Euler transform used to accelerate the convergence of an alternating series and is also frequently applied to the hypergeometric series
Euler—formulas
- Euler's formula e ix = cos x + i sin x in complex analysis.
- Euler's formula for planar graphs: v − e + f = 2
- Euler's continued fraction formula
- Euler product formula – for the Riemann zeta function.
- Euler's summation formula, a theorem about integrals.
- Euler–Maclaurin formula – relation between integrals and sums
- Euler–Rodrigues formulas – concerns Euler–Rodrigues parameters and 3D rotation matrices
Euler—functions
- The Euler function, a modular form that is a prototypical q-series.
- Euler's homogeneous function theorem
- Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.
Euler—identities
- Euler's identity e iπ + 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.
Euler—numbers
- Euler's number, e ≈ 2.71828, the base of the natural logarithm, also known as Napier's constant.
- Euler's idoneal numbers
- Euler numbers are an integer sequence.
- Eulerian numbers are another integer sequence.
- Euler number (physics), the cavitation number in fluid dynamics.
- Euler number (topology) – now, Euler characteristic
- Lucky numbers of Euler
Euler—theorems
- Euler's homogeneous function theorem, a theorem about homogeneous polynomials.
- Euler's infinite tetration theorem
- Euler's rotation theorem
- Euler's theorem (differential geometry) on the existence of the principal curvatures of a surface and orthogonality of the associated principal directions.
- Euler's theorem in geometry, relating the circumcircle and incircle of a triangle.
- Euclid-Euler theorem
- Euler–Fermat theorem, that aφ(m) ≡ 1 (mod m) whenever a is coprime to m, and φ is the totient function.
Euler—laws
- Euler's first law, the linear momentum of a body is equal to the product of the mass of the body and the velocity of its center of mass.
- Euler's second law, the sum of the external moments about a point is equal to the rate of change of angular momentum about that point.
Other things named after Euler
- 2002 Euler (an asteroid)
- Euler Medal, a prize for research in combinatorics
- Euler programming language
- Euler (software)
- AMS Euler typeface
- Project Euler
Topics by field of study
Selected topics from above, grouped by subject.
Derivatives and integrals
- Euler approximation – (see Euler's method)
- Euler derivative (as opposed to Lagrangian derivative)
- The 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
- Euler's summation formula, a theorem about integrals.
- Euler–Cauchy equation (or Euler equation), a second-order linear differential equation
- Euler–Maclaurin formula – relation between integrals and sums
Geometry and spatial arrangement
- Euler angles defining a rotation in space.
- Euler brick
- Euler's line – relation between triangle centers
- Euler operator – set of functions to create polygon meshes
- Euler's rotation theorem
- Euler squares, usually called Graeco-Latin squares.
- Euler's theorem in geometry, relating the circumcircle and incircle of a triangle.
- Euler–Rodrigues formulas – concerns Euler–Rodrigues parameters and 3D rotation matrices
Graph theory
- Euler characteristic in algebraic topology and topological graph theory, and the corresponding Euler's formula
- Eulerian circuit – (see Eulerian path)
- Euler class
- Euler cycle – (see Eulerian path)
- Euler diagram – likely better (but wrongly) known as Venn diagram (which has more restrictions)
- Euler's formula for planar graphs: v − e + f = 2
- Eulerian graph – (see Eulerian path)
- Euler number (topology) – now, Euler characteristic
- Eulerian path, a path through a graph that takes each edge once.
Logarithms
- Euler's number, e ≈ 2.71828, the base of the natural logarithm, also known as Napier's constant.
- Euler–Mascheroni constant or Euler's constant γ ≈ 0.577216
Physical systems
- Euler's disk – a circular disk that spins, without slipping, on a surface
- Euler equations in fluid dynamics.
- Euler's equations, concerning the rotations of a rigid body.
- Euler number (physics), the cavitation number in fluid dynamics.
- Euler's three-body problem
- Euler–Bernoulli beam equation, concerning the elasticity of structural beams.
- Euler formula in calculating the buckling load of columns.
- Euler–Tricomi equation – concerns transonic flow
Polynomials
- Euler's homogeneous function theorem, a theorem about homogeneous polynomials.
- Euler polynomials
- Euler spline – composed of classical Euler polynomial arcs (cred. to Schoenberg, 1973 – PDF)
Prime numbers
- Euler's criterion – quadratic residues modulo by primes
- Euler product – infinite product expansion, indexed by prime numbers of a Dirichlet series
- Euler pseudoprime
- Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.
See also
- Contributions of Leonhard Euler to mathematics
- Euler on infinite series
Euler–Maruyama method
Notes
- ^ David S. Richeson (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology (illustrated ed.), Princeton University Press, p. 86, ISBN 9780691126777, http://books.google.com/books?id=LB_6VogerHIC&pg=PA86&dq=%22person+after+Euler%22
- ^ C. H. Edwards; David E. Penney (2004), Differential equations and boundary value problems :, 清华大学出版社, p. 443, ISBN 9787302099789, http://books.google.com/books?id=51KTl4Fmh2wC&pg=PA443&dq=%22person+after+Euler%22
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