200px, Leonhard Euler (1707–1783)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, many topics are
named in honor of Swiss mathematician
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(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.
Conjectures
*
Euler's conjecture (Waring's problem)
In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural numb ...
*
Euler's sum of powers conjecture
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is ...
*
Euler's Graeco-Latin square conjecture
Equations
Usually, ''Euler's equation'' refers to one of (or a set of)
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s (DEs). It is customary to classify them into
ODE
An ode (from grc, ᾠδή, ōdḗ) is a type of lyric poetry. Odes are elaborately structured poems praising or glorifying an event or individual, describing nature intellectually as well as emotionally. A classic ode is structured in three majo ...
s and
PDEs.
Otherwise, ''Euler's equation'' may refer to a non-differential equation, as in these three cases:
*
Euler–Lotka equation In the study of age-structured population growth, probably one of the most important equations is the Euler–Lotka equation. Based on the age demographic of females in the population and female births (since in many cases it is the females that are ...
, a
characteristic equation employed in mathematical demography
*
Euler's pump and turbine equation
*
Euler transform In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to th ...
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
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
.
*
Euler–Cauchy equation, a linear equidimensional
second-order ODE with
variable coefficient
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s. Its second-order version can emerge from
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \nab ...
in
polar coordinates
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
.
*
Euler–Bernoulli beam equation, a fourth-order ODE concerning the elasticity of structural beams.
*
Euler's differential equation In mathematics, Euler's differential equation is a first order nonlinear ordinary differential equation, named after Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geograph ...
, a first order nonlinear ordinary differential equation
Partial differential equations
*
Euler conservation equations, a set of quasilinear first-order
hyperbolic equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...
s used in
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
for
inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, such as ...
s. In the (Froude) limit of no external field, they are
conservation equations.
*
Euler–Tricomi equation In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi.
:
u_+xu_=0. \,
It is elliptic in the ha ...
– a second-order PDE emerging from Euler conservation equations.
*
Euler–Poisson–Darboux equation In mathematics, the Euler–Poisson–Darboux equation is the partial differential equation
: u_+\frac=0.
This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in solving the classical wave ...
, a second-order PDE playing important role in solving the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
.
*
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
, a second-order PDE emerging from minimization problems in
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
.
Formulas
Functions
*The
Euler function
In mathematics, the Euler function is given by
:\phi(q)=\prod_^\infty (1-q^k),\quad , q, A000203
On account of the identity \sum_ d = \sum_ \frac, this may also be written as
:\ln(\phi(q)) = -\sum_^\infty \frac \sum_ d.
Also if a,b\in\mathbb^+ ...
, a
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
that is a prototypical
q-series
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer sym ...
.
*
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
(or Euler phi (φ) function) in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, counting the number of coprime integers less than an integer.
*
Euler hypergeometric integral
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
*
Euler–Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
Identities
*
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
.
*
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
In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that
:\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right ...
.
Numbers
*
Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of a logarithm, base of the natural logarithms. It is the Limit of a sequence, limit ...
, , the base of the natural logarithm
*
Euler's idoneal numbers, a set of 65 or possibly 66 or 67 integers with special properties
*
*
Eulerian number
In combinatorics, the Eulerian number ''A''(''n'', ''m'') is the number of permutations of the numbers 1 to ''n'' in which exactly ''m'' elements are greater than the previous element (permutations with ''m'' "ascents"). They are the coefficients ...
s count certain types of permutations.
*
Euler number (physics)
The Euler number (Eu) is a dimensionless number used in fluid flow calculations. It expresses the relationship between a local pressure drop caused by a restriction and the kinetic energy per volume of the flow, and is used to characterize ener ...
, the cavitation number in
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
.
*Euler number (algebraic topology) – now,
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
, classically the number of vertices minus edges plus faces of a polyhedron.
*Euler number (3-manifold topology) – see
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle ( circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for ...
*
Lucky numbers of Euler Euler's "lucky" numbers are positive integers ''n'' such that for all integers ''k'' with , the polynomial produces a prime number.
When ''k'' is equal to ''n'', the value cannot be prime since is divisible by ''n''. Since the polynomial can be w ...
*
Euler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
gamma (γ), also known as the Euler–Mascheroni constant
*
Eulerian integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
:z = a + b\omega ,
where and are integers and
:\omega = \f ...
s, more commonly called Eisenstein integers, the algebraic integers of form where is a complex cube root of 1.
*
Euler–Gompertz constant In mathematics, the Gompertz constant or Euler–Gompertz constant, denoted by \delta, appears in integral evaluations and as a value of special functions. It is named after Benjamin Gompertz.
It can be defined by the continued fraction
: \delt ...
Theorems
*
*
*
*
*
*
*
*
*
*
*
Laws
*
Euler's first law, the
linear momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and ...
of a body is equal to the product of the mass of the body and the velocity of its
center of mass
In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
.
*
Euler's second law, the sum of the external
moments about a point is equal to the rate of change of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
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
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
(formerly called Euler number) in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and
topological graph theory
In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs.
Embedding a graph in ...
, and the corresponding Euler's formula
*Eulerian circuit, Euler cycle or
Eulerian path
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...
– a path through a graph that takes each edge once
**Eulerian graph has all its vertices spanned by an Eulerian path
*
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of ...
*
Euler diagram
An Euler diagram (, ) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, Ven ...
– incorrectly, but more popularly, known as Venn diagrams, its subclass
*
Euler tour technique
The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented a ...
Music
*
Euler–Fokker genus
In music theory and tuning, an Euler–Fokker genus (plural: genera), named after Leonhard Euler and Adriaan Fokker,Rasch, Rudolph (2000). ''Harry Partch'', p.31-2. Dunn, David, ed. . is a musical scale in just intonation whose pitches can be ex ...
*
Euler's tritone
Number theory
*
Euler's criterion In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,
Let ''p'' be an odd prime and ''a'' be an integer coprime to ''p''. Then
:
a^ \equiv
\begin
\;\;\,1\pmod& \text ...
– quadratic residues modulo by primes
*
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eul ...
–
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product
:
\prod_^ a_n = a_1 a_2 a_3 \cdots
is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a' ...
expansion, indexed by prime numbers of a
Dirichlet series
In mathematics, a Dirichlet series is any series of the form
\sum_^\infty \frac,
where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analyti ...
*
Euler pseudoprime In arithmetic, an odd composite integer ''n'' is called an Euler pseudoprime to base ''a'', if ''a'' and ''n'' are coprime, and
: a^ \equiv \pm 1\pmod
(where ''mod'' refers to the modulo operation).
The motivation for this definition is the f ...
*
Euler–Jacobi pseudoprime In number theory, an odd integer ''n'' is called an Euler–Jacobi probable prime (or, more commonly, an Euler probable prime) to base ''a'', if ''a'' and ''n'' are coprime, and
:a^ \equiv \left(\frac\right)\pmod
where \left(\frac\right) is the J ...
*
Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
(or Euler phi (φ) function) in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, counting the number of coprime integers less than an integer.
*
Euler system In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and ...
*
Euler's factorization method Euler's factorization method is a technique for factoring a number by writing it as a sum of two squares in two different ways. For example the number 1000009 can be written as 1000^2 + 3^2 or as 972^2 + 235^2 and Euler's method gives the factoriza ...
Physical systems
Polynomials
*
Euler's homogeneous function theorem
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''deg ...
, a theorem about
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
s.
*
Euler polynomials
*
Euler spline – splines composed of arcs using Euler polynomials
See also
*
Contributions of Leonhard Euler to mathematics
The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field. His seminal work had a profound impact in numerous areas of mathematics and he is widely cr ...
Notes
{{reflist
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
Leonhard Euler