He is the physicist, who 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 a set of laws of motion in fluid dynamics from Newton's laws of motion that state:

- The force acting on a small element of a fluid is equal to the rate of change of its momentum.
- The torque acting on a small element of a fluid is equal to the rate of change of its angular momentum.

In mathematics, he established his fame early on by solving a long-standing problem:

- 1/1
^{2}+ 1/2^{2}+ 1/3^{2}+ 1/4^{2}+ ... = π^{2}/ 6

He also showed that for all real numbers *x*,

*e*^{ix}= cos(*x*) +*i*sin(*x*)

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. The most remarkable formula in the world is an easy consequence.

In 1735, he defined the constant gamma? useful for differential equations:

- γ =
**lim**_{n->∞}( 1+ (1/2) + (1/3) + (1/4) ... + (1/*n*) - log(*n*) )

Its value is approximately 0.5772156 and it is still unknown whether it is rational or irrational, let alone algebraic or transcendental.

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

He made important contributions to number theory as well as to the theory of differential equations. His contribution to analysis, for example, came through his synthesis of Leibniz?'s differential calculus with Newton's method of fluxions.

Euler wrote *Tentamen novae theoriae musicae* in 1739 which was an attempt to combine mathematics and music, a comment in a biography of Euler regarding it was 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 is called Euler's Formula which relates the number of edges, vertices, and faces of a convex solid with planar faces and no holes. (i.e. not toroidal). Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two ie:(V + F = 2 + E).

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See also: Mathematician, Physicist, Mathematical constants, Complex number, Euler's conjecture

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