Leonhard Euler

1. Leonhard Euler

2. He was born in Bazylea on 15
April 1707
He died in Petersburg on 18
September 1783

3. 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.

4. Mathematical notation
He 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.

5. Graph Theory
In 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.

6. Euler on the Old Swiss 10 Franc banknote.

7. 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.

8. 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):

9. 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.

10. Euler has an extensive bibliography.
His best-known books include:
• Mechanica (1736).

11. • Glossary
imaginary unit - jednostka urojona
polyhedral formula - wielościenna fotmuła
infinitesimal calculus - nieskończony rachunek
velocity vector - wektor prędkości
area moment of inertia - geometryczny moment bezwładności

12. Julia Górska
Nikola Sobieralska
Ic
XX LO Gdańsk