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Born and educated in Switzerland, he was a mathematical child prodigyProdigies are masters of a specific skill or art, a talent which manifests itself at an early age. One generally accepted definition of a prodigy is a person who, by the age of 10, displays expert proficiency in a field usually only undertaken by adults.. He worked as a professorA professor is a senior teacher and researcher, usually in a college or university. Overview Professors give lectures and seminars in their field of study, such as science or literature. They also do advanced research in their fields and are supposed to d of mathematics in Saint PetersburgSaint Petersburg ( Russian: English transliteration: Sankt-Peterburg), colloquially known as (transliterated "Piter"), formerly known as Leningrad (, 1924- 1991) and Petrograd (, 1914- 1924), is a city located in Northwestern Russia on the Gulf of Finland, later in BerlinBerlin [ bɛrˈliːn ] is the national capital of Germany and its largest city, with 3,387,404 inhabitants (as of September 2004); down from 4. 5 million before World War II. Berlin is located on the rivers Spree and Havel in the northea, and then returned to Saint Petersburg. He holds the world record for most prolific mathematician of all time, his collected work filling 75 volumes. He dominated eighteenth century mathematics and deduced many consequences of the newly invented calculus. He was completely blindBlindness can be defined physiologically as the condition of lacking sight. The definition as it applies to people thus legally classified is, however, more complex. The term "blindness" also applies to partial visual impairment: In North America and most for the last seventeen years of his life, during which time he produced almost half of his total output.
Euler was deeply religious throughout his life. However, the widely told anecdoteAn anecdote is a short tale told about someone who is not present (often because they are already dead) which illustrates one of their character traits, often in a humorous manner and, at any rate, better, it is claimed, than in a long description of thei that Euler challenged Denis Diderot at the court of Catherine the Great with "Sir, (a+b)n/n = x; hence God exists, reply!" is false.
The asteroid 2002 Euler is named in his honour.
Euler, with Daniel Bernoulli, established the law that the torque on a thin elastic beam is proportional to a measure of the elasticity of the material and the moment of inertia of a cross section, about an axis through the center of mass and perpendicular to the plane of the couple.
He also deduced the Euler equations, a set of laws of motion in fluid dynamics, directly from Newton's laws of motion. These equations are formally identical to the Navier-Stokes equations with zero viscosity. They are interesting chiefly because of the existence of
shock waves.He made important contributions to the theory of differential equations. In particular, he is known for creating a series of Euler approximations which are used in computational mechanics . The most famous of these approximations is known as Euler's method .
In number theory he invented the totient function. The totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. For example, φ(8) = 4 since the four numbers 1, 3, 5 and 7 are coprime to 8.
In mathematical analysis, it was Euler who synthesised Leibniz's differential calculus with Newton's method of fluxions.
He established his fame in 1735 by solving the long-standing Basel problem:
where is the Riemann zeta function.
He also showed the usefulness, consistency, and simplicity of defining the exponent of an imaginary number by means of the formula
This is Euler's formula, which establishes the central role of the
exponential function. In essence, all functions studied in elementary analysis are either variations of the exponential function or they are polynomials. What Richard Feynman called "The most remarkable formula in mathematics" (more commonly called Euler's identity) is an easy consequence:In 1735, he defined the Euler-Mascheroni constant useful for differential equations:
He is a co-discoverer of the Euler-Maclaurin formula which is an extremely useful tool for calculation of difficult integrals, sums and series.
Euler wrote Tentamen novae theoriae musicae in 1739 which was an attempt to combine mathematics and music; a biography comments that the work was "for musicians too advanced in its mathematics and for mathematicians too musical".
In economics, he showed that if each factor of production is paid the value of its marginal product, then (under constant returns to scale) the total income and output will be completely exhausted.
In geometry and algebraic topology, there is a relationship called Euler's Formula which relates the number of edges, vertices, and faces of a simply connected polyhedron. Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two. i.e.: F - E + V = 2. The theorem also applies to any planar graph. For nonplanar graphs, there is a generalization: If the graph can be embedded in a manifold M, then F - E + V = χ(M), where χ is the Euler characteristic of the manifold, a constant which is invariant under continuous deformations. The Euler characteristic of a simply-connected manifold such as a sphere or a plane is 2. A generalization of Euler's formula for arbitrary planar graphs exists: F - E + V - C = 1, where C is the number of components in the graph.
In 1736 Euler solved a problem known as the seven bridges of Königsberg, publishing a paper Solutio problematis ad geometriam situs pertinentis which was the earliest application of graph theory or topology.