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:
- 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*^{ 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 – 1=e = 2.71828..., base of the natural logarithm
- 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 – γ ≈ 0.5772, the limit of the difference between the harmonic series and the natural logarithm
- 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 the distance between the centers of the circumscribed and inscribed circles of a triangle
- Euler's quadrilateral theorem – Relation between the sides of a convex quadrilateral and its diagonals
- Euclid–Euler theorem – Characterization of the even perfect numbers
- Euler's theorem – Generalization of Fermat's little theorem to non-prime moduli
- Euler's partition theorem – The numbers of partitions with odd parts and with distinct parts are equal

## 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
- Leonhard Euler Gold Medal, a prize for outstanding results in mathematics and physics
- 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
- 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 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
- 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

### 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]}

## See also

## 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*.**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.