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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.
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Leonhard Euler (1707–1783)
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
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 transform used to accelerate the convergence of an alternating series and is also frequently applied to the hypergeometric series
Ordinary differential equations
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.
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.
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:
Euler's continued fraction formula connecting a finite sum of products with a finite continued fraction
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
Functions
Euler's totient function (or Euler phi (φ) function) in number theory, counting the number of coprime integers less than an integer.
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.
Numbers
Euler's number, e = 2.71828..., 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
Euler's constant gamma (γ), also known as the Euler–Mascheroni constant
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 – Relates the product and series representations of the Euler function Π(1-x^n)
Goldbach–Euler theorem – theorem stating that sum of 1/(k−1), where k ranges over positive integers of the form mⁿ for m≥2 and n≥2, equals 1
Laws
Main article: Euler's laws of motionEuler'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 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 Society, an American group dedicated to the life and work of Leonhard Euler
EulerOS, a CentOS Linux based operating system
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[edit]
Euler approximation – (see Euler's method)
The Euler method, a method for finding numerical solutions of differential equationsSemi-implicit Euler method
Euler's number e ≈ 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 ≈ 0.577216
Geometry and spatial arrangement
Euler angles defining a rotation in space
Euler's line – relation between triangle centers
Euler operator – set of functions to create polygon meshes
Euler spiral – a curve whose curvature varies linearly with its arc length
Euler squares, usually called Graeco-Latin squares
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
Eulerian circuit, Euler cycle or Eulerian path – a path through a graph that takes each edge onceEulerian graph has all its vertices spanned by an Eulerian path
Euler diagram - popularly called "Venn diagrams", although some use this term only for a subclass of Euler diagrams.
Music
Number theory
Euler's criterion – quadratic residues modulo by primes
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 number (physics), the cavitation number in fluid dynamics.
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
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 spline – splines composed of arcs using Euler polynomials[5]
See also:
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