List of things named after Leonhard Euler: Difference between revisions

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In mathematics and physics, many topics are 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 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. 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.^[1]^[2]

Conjectures

Equations

Usually, Euler's equation refers to one of (or a set of) differential equations (DEs). It is customary to classify them into ODEs and PDEs.

Otherwise, Euler's equation might refer to a non-differential equation, as in these three cases:

Euler–Lotka equation, a characteristic equation employed in mathematical demography
Euler's pump and turbine equation
Euler transform used to accelerate the convergence of an alternating series and is also frequently applied to the hypergeometric series

Ordinary differential equations

Euler rotation equations, a set of first-order ODEs concerning the rotations of a rigid body.
Euler–Cauchy equation, a linear equidimensional second-order ODE with variable coefficients. Its second-order version can emerge from Laplace equation in polar coordinates.
Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity of structural beams.

Partial differential equations

Euler conservation equations, a set of quasilinear first-order hyperbolic equations used in fluid dynamics for inviscid flows. In the (Froude) limit of no external field, they are 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

Euler's formula, $e ix = cos x + i sin x$
Euler's polyhedral formula for planar graphs or polyhedra: $v - e + f = 2$ , a special case of the Euler characteristic in topology
Euler's formula for the critical load of a column: $P_{\text{cr}}={\frac {\pi ^{2}EI}{(KL)^{2}}}$
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

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.
Euler hypergeometric integral

Identities

Euler's identity $e i π + 1 = 0$ .
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 – 2.71828..., base of natural logarithms
Euler's idoneal numbers, a set of 65 or possibly 66 integers with special properties
Euler numbers – Integers occurring in the coefficients of the Taylor series of 1/cosh t
Eulerian numbers 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
Euler–Mascheroni constant – Relates logarithm and harmonic series
Eulerian integers, more commonly called Eisenstein integers, the algebraic integers of form $a + bω$ where $ω$ is a complex cube root of 1.

Theorems

Euler's homogeneous function theorem – A homogeneous function is a linear combination of its partial derivatives
Euler's infinite tetration theorem – About the limit of iterated exponentiation
Euler's rotation theorem – Movement with a fixed point is rotation
Euler's theorem (differential geometry) – Orthogonality of the directions of the principal curvatures of a surface
Euler's theorem in geometry – On distance between centers of a triangle
Euler's quadrilateral theorem – Relation between the sides of a convex quadrilateral and its diagonals
Euclid–Euler theorem – Characterization of even perfect numbers
Euler's theorem – Theorem on modular exponentiation
Euler's partition theorem – Theorem in number theory

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

2002 Euler (a minor planet)
AMS Euler typeface
Euler (software)
Euler acceleration or force
Euler Book Prize
Euler Medal, a prize for research in combinatorics
Euler programming language
Euler Society, an American group dedicated to the life and work of Leonhard Euler
Euler–Fokker genus
Project Euler
Leonhard Euler Telescope
Rue Euler (a street in Paris, France)^[3]
Euler Park (a public park in Lima, Peru)

Topics by field of study

Selected topics from above, grouped by subject.

Analysis: derivatives, integrals, and logarithms

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
- Semi-implicit Euler method
- Euler–Maruyama method
Euler's number $e \approx 2.71828$ , the base of the natural logarithm, also known as Napier's constant.
The Euler substitutions 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
Euler–Maclaurin formula – relation between integrals and sums
Euler–Mascheroni constant or Euler's constant $γ \approx 0.577216$

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 spiral – a curve whose curvature varies linearly with its arc length
Euler squares, usually called Graeco-Latin squares
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 quadrilaterals
Euler–Rodrigues formula concerning Euler–Rodrigues parameters and 3D rotation matrices

Graph theory

Euler characteristic (formerly called Euler number) in algebraic topology and topological graph theory, and the corresponding Euler's formula $\scriptstyle \chi (S^{2})=F-E+V=2$
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 – incorrectly, but more popularly, known as Venn diagrams, its subclass
Euler tour technique

Music

Euler–Fokker genus

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's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.

Physical systems

Euler's Disk – a toy consisting of a circular disk that spins, without slipping, on a surface
Euler rotation equations, in rigid body 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.
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.^[4]

Polynomials

Euler's homogeneous function theorem, a theorem about homogeneous polynomials.
Euler polynomials
Euler spline – splines composed of arcs using Euler polynomials^[5]

Notes

^ Richeson, David S. (2008). Euler's Gem: The polyhedron formula and the birth of topology (illustrated ed.). Princeton University Press. p. 86. ISBN 978-0-691-12677-7.
^ Edwards, C. H.; Penney, David E. (2004). Differential equations and boundary value problems. 清华大学出版社. p. 443. ISBN 978-7-302-09978-9.
^ de Rochegude, Félix (1910). Promenades dans toutes les rues de Paris [Walks along all of the streets in Paris] (VIII^e arrondissement ed.). Hachette. p. 98.
^ Evans, Charles R.; Smarr, Larry L.; Wilson, James R. (1986). "Numerical Relativistic Gravitational Collapse with Spatial Time Slices". Astrophysical Radiation Hydrodynamics. Vol. 188. pp. 491–529. doi:10.1007/978-94-009-4754-2_15. Retrieved March 27, 2021.
^ Schoenberg (1973). "bibliography" (PDF). University of Wisconsin. Archived from the original (PDF) on 2011-05-22. Retrieved 2007-10-28.

[1] Richeson, David S. (2008). Euler's Gem: The polyhedron formula and the birth of topology (illustrated ed.). Princeton University Press. p. 86. ISBN 978-0-691-12677-7.

[2] Edwards, C. H.; Penney, David E. (2004). Differential equations and boundary value problems. 清华大学出版社. p. 443. ISBN 978-7-302-09978-9.

[3] Rochegude, Félix (1910). Promenades dans toutes les rues de Paris [Walks along all of the streets in Paris] (VIII^e arrondissement ed.). Hachette. p. 98.

[4] Evans, Charles R.; Smarr, Larry L.; Wilson, James R. (1986). "Numerical Relativistic Gravitational Collapse with Spatial Time Slices". Astrophysical Radiation Hydrodynamics. Vol. 188. pp. 491–529. doi:10.1007/978-94-009-4754-2_15. Retrieved March 27, 2021.

[5] Schoenberg (1973). "bibliography" (PDF). University of Wisconsin. Archived from the original (PDF) on 2011-05-22. Retrieved 2007-10-28.

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 *[[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 |access-date=March 27, 2021 |url=https://rd.springer.com/chapter/10.1007/978-94-009-4754-2_15}}</ref>
 }}