Leonhard Euler

Leonhard Euler (April 15, 1707–September 18, 1783) was a Swiss mathematician and physicist. He is considered to be one of the greatest mathematicians who ever lived. Leonhard Euler was the first to use the term "function" (defined by Leibniz in 1694) to describe an expression involving various arguments; i.e., y = F(x). He is credited with being one of the first to apply calculus to physics.

Discoveries

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 second moment of area of a cross section, about an axis through the center of mass and perpendicular to the plane of the moment, see Euler-Bernoulli beam equation.

Related Topics:
Torque - Elastic - Beam - Proportional - Second moment of area - Center of mass - Perpendicular - Moment - Euler-Bernoulli beam equation

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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

Related Topics:
Euler equations - Fluid dynamics - Newton's laws of motion - Navier-Stokes equations - Viscosity

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shock waves.

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Euler made important contributions to the theory of differential equations. In particular, he is known for creating a series of approximations which are used in computational mechanics. The most famous of these approximations is known as Euler's method.

Related Topics:
Differential equations - Approximations - Computational mechanics - Euler's method

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In number theory, Euler 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.

Related Topics:
Number theory - Totient function - Positive integer - Coprime

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In mathematical analysis, it was Euler who synthesised Leibniz's differential calculus with Isaac Newton's method of fluxions.

Related Topics:
Leibniz - Differential calculus - Fluxion

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Euler established his fame in 1735 by solving the long-standing Basel problem:

Related Topics:
1735 - Basel problem

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:zeta(2) = sum_{n=1}^infty rac{1}{n^2} = rac{1}{1^2} + rac{1}{2^2} + rac{1}{3^2} + rac{1}{4^2} + cdots = rac{pi^2}{6},

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where zeta(s) is the Riemann zeta function.

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He also showed the usefulness, consistency, and simplicity of defining the exponent of an imaginary number by means of the formula

Related Topics:
Exponent - Imaginary number

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: e^{i heta} = cos heta + isin heta ,.

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This is Euler's formula, which establishes the central role of the

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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:

Related Topics:
Exponential function - Polynomial - Richard Feynman - Euler's identity

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:e^{i pi} +1 = 0 ,.

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Also in 1735, Euler defined the Euler-Mascheroni constant useful for differential equations:

Related Topics:
Euler-Mascheroni constant - Differential equation

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:gamma = lim_{n ightarrow infty } left( 1+ rac{1}{2} + rac{1}{3} + rac{1}{4} + ... + rac{1}{n} - ln(n) ight).

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He is a co-discoverer of the Euler-Maclaurin formula which is an extremely useful tool for calculation of difficult integrals, sums, and series.

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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".

Related Topics:
1739 - Music

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In economics, Euler 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.

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In geometry and algebraic topology, there is a relationship (also 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.

Related Topics:
Geometry - Algebraic topology - Simply connected - Manifold - Euler characteristic

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In 1736 Euler solved, or rather proved insoluble, 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.

Related Topics:
1736 - Seven bridges of Königsberg - Graph theory - Topology

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The solution to the seven bridges problem reduced the land masses to points and the bridges to lines (or edges) connecting those points. Looking at how many lines came into a point gave that point a degree (a point with three lines touching it has a degree of three). An Euler circuit has all its points of even degree. This means it is possible to travel each line exactly once without retracing your steps and end at the same point in which you started. An Euler path has exactly two odd vertices. This means that it is possible to travel each line exactly once without retracing your steps, but you will not end where you began. The seven bridges problem is neither an Euler circuit nor Euler path. Hence, you cannot visit each of the bridges of Königsberg without retracing your steps.

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