Euler equations

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 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.

Conjectures

* Euler's sum of powers conjecture disproved for exponents 4 and 5 during the 20th century; unsolved for higher exponents
* Euler's Graeco-Latin square conjecture proved to be true for and disproved otherwise, during the 20th century

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'' may 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'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 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.
* Euler–Arnold equation, describes the evolution of a velocity field when the Lagrangian flow is a geodesic in a group of smooth transformations.

Formulas

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
* Euler–Riemann zeta function

Identities

* Euler's identity .
* 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, , 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 ''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's constant gamma (''γ''), also known as the Euler–Mascheroni constant
* Eulerian integers, more commonly called Eisenstein integers, the algebraic integers of form where is a complex cube root of 1.
* Euler–Gompertz constant

Theorems

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* 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 − ''x''^''n'')
* Goldbach–Euler theorem, stating that sum of 1/(''k'' − 1), where ''k'' ranges over positive integers of the form ''m''^''n'' for ''m'' ≥ 2 and ''n'' ≥ 2, equals 1
*

Laws

* Euler's first law, the sum of the external forces acting on a rigid body is equal to the rate of change of linear momentum of the body.
* 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

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

Geometry and spatial arrangement

Graph theory

*Euler characteristic (formerly called Euler number) in algebraic topology and topological graph theory, and the corresponding Euler's formula $\chi(S^2)=F-E+V=2$
*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 – 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

Polynomials

*Euler's homogeneous function theorem, a theorem about homogeneous polynomials.
* Euler polynomials
* Euler spline – splines composed of arcs using Euler polynomials

See also

* Contributions of Leonhard Euler to mathematics

Notes

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Euler
Leonhard Euler