Leonhard Euler

He was born in Bazylea on 15 April 1707

He died in Petersburg on 18 September 1783

Leonhard Euler was a pioneering Swiss mathematician and physicist. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematician function. He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

Mathematical notationHe introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometic functions, the letter e for the base of the natural logarithm (now also known as Euler’s number), the Greek letter Σ for summations and the letter i to denote the imaginary unit.

Graph TheoryIn 1736 Euler solved problem known as Seven Bridges of Konigsberg. Królewiec in Prusy was located by the Pregola River which was situated by two big islands which were connected to each other by 7 bridges. The question was: Is it possible to follow a path that crosses each bridge once and returns to the starting point? Euler discovered that it is not possible.

Euler on the Old Swiss 10 Franc banknote.

1957 Soviet Union stamp commemorating the 250th birthday of Euler.

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as "v − e + f = 2.

The development of infinitesimal calculus was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigor (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):

In 1757 he published an important set of equations for inviscid flow, that are now known as the Euler equations. In differential form, the equations are:

where• ρ is the fluid mass density,• u is the fluid velocity vector, with components u, v, and w,• E = ρ e + ½ ρ ( u2 + v2 + w2 ) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,• p is the pressure,• denotes the tensor product, and• 0 being the zero vector.

Euler is also well known in structural engineering for his formula giving the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness:

where = maximum or critical force (vertical load on column), = modulus of elasticity, = area moment of inertia, = unsupported length of column, = column effective length factor, whose value depends on the conditions of end support of the

column, as follows.For both ends pinned (hinged, free to rotate), = 1.0.For both ends fixed, = 0.50.For one end fixed and the other end pinned, = 0.699....For one end fixed and the other end free to move laterally, = 2.0. is the effective length of the column.

Euler has an extensive bibliography. His best-known books include:

• Mechanica (1736).

• Glossary

imaginary unit - jednostka urojonapolyhedral formula - wielościenna fotmuła

infinitesimal calculus - nieskończony rachunek velocity vector - wektor prędkości

area moment of inertia - geometryczny moment bezwładności

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