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 an excess of things and concepts 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 n = 6 {\displaystyle n=6} 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 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 … {\displaystyle e=2.71828\dots } , 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 = 0.57721 … {\displaystyle \gamma =0.57721\dots } , 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. Euler–Gompertz constant
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, 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 − xn) Goldbach–Euler 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 Gram–Euler theorem
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 χ ( S 2 ) = F − E + V = 2 {\textstyle \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
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