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Leonhard Euler
(Basel, Switzerland, 15 April 1707 - St.
Petersburg, Russia, 18 September 1783)
He was a Swiss mathematician and
physicist. This is the main eighteenth
century mathematician and one of the
largest and most prolific of all time.
He lived in Russia and Germany most of
his life.
At the age of 13 he enrolled at the
University of Basel.
Because of the friendship between the
two families, Euler received private
lessons from Johann Bernoulli, who
quickly discovered the incredible talent
of his new pupil for mathematics.
Contribution to mathematics and
other scientific areas
Mathematical Notation
The Number and Number Theory
Graph Theory And Geometry
Applied Mathematics
Physics and Astronomy
Architecture and Engineering
Mathematical Notation
 Euler introduced several notational conventions through his
numerous and widely circulated textbooks. Most notably,
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 trigonometric 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.
 The use of the Greek letter π to denote the ratio of a circle's
circumference to its diameter was also popularized by Euler,
although it did not originate with him.
 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.
 He also defined the exponential function for
complex numbers, and discovered its relation to
the trigonometric functions. For any real
number φ, Euler's formula states that the complex
exponential function satisfies
Geometric Interpretation of Euler’s
Number Theory
 Euler linked the nature of prime distribution with ideas in
analysis. He proved that the sum of the reciprocals of the
primes diverges. In doing so, he discovered the connection
between the Riemann zeta function and the prime numbers:
the Euler product formula for the Riemann zeta function.
 Euler proved Newton's identities, Fermat's theorem on
sums of two squares, and he made distinct contributions
to Lagrange's four-square theorem.
 He contributed significantly to the theory of perfect
numbers, which had fascinated mathematicians
since Euclid.
Graph theory
 Euler also discovered the formula V − E + F = 2 relating the
number of vertices, edges, and faces of a
convexpolyhedron, and hence of a planar graph.
 In analytic geometry also found that three of the significant
points of a triangle, centroid, orthocenter and circumcentercould obey the same equation, to the same line. In the line
containing the centroid, orthocenter and circumcenter is
called "Euler line" in his honor.
 Euler solved the problem known as THE SEVEN BRIDGES OF
KÖNIGSBERG. The problem is to decide whether it is
possible to follow a path that crosses each bridge exactly
once and returns to the starting point. It is not possible:
there is no Eulerian circuit.
Applied mathematics
 Some of Euler's greatest successes were in solving realworld problems analytically, and in describing numerous
applications of the Bernoulli numbers, Fourier series, Venn
diagrams, Euler numbers, the constants e and π, continued
fractions and integrals.
Physics and astronomy
 Euler development the equation of the elastic curve, which
became the cornerstone of engineering. Aside from successfully
applying his analytic tools to problems in classical mechanics,
Euler also applied to the problems of celestial movements of the
 In the field of mechanics Euler explicitly introduced the concepts
of particle mass and punctual and vector notation to represent
the velocity and acceleration.
 In studied hydrodynamic flow of an ideal fluid incompressible
Euler equations detailing of hydrodynamics.
 Over a hundred years ahead Maxwell predicted the
phenomenon of radiation pressure, fundamental unified theory
of electromagnetism.
 Euler is also credited with using closed curves to
illustrate syllogistic reasoning (1768). These diagrams have
become known as Euler diagrams.
An Euler diagram shows that the set of "four-legged animals" is a
subset of "animals", but all the "mineral" is disjoint (no common
members) with "animals"
Carl Friedrich Gauss
 (April 30, 1777, Brunswick February 23, 1855, Göttingen)
was a mathematician,
astronomer, geodesist, and
German physicist who
contributed significantly to many
fields, including number theory,
mathematical analysis,
differential geometry, statistics,
algebra, geodesy, magnetism
and optics. Considered "the
prince of mathematics" and "the
greatest mathematician since
 Along with Archimedes and
Newton, Gauss is undoubtedly
one of the three geniuses in the
history of mathematics.
The Polygon
 The first contribution of Gauss, with 17 years, mathematics
was the construction of the regular polygon of 17 sides.
Gauss not only managed the construction of the 17-sided
polygon, also found the condition to be met polygons can
be constructed by this method.
 Gauss proved this theorem combining with other geometric
algebraic reasoning. The technique used for the show, has
become one of the most used in math: move a problem
from an initial domain (geometry in this case) to another
(algebra) and solve the latter.
The Disquisitions
 In 1801, when he was 24 years, Gauss published his first major
work "Disquisitiones Arithmeticae". Besides organizing what
already exists on the integers, Gauss contributed ideas. He based
his theory from a congruent number arithmetic used in the proof
of important theorems, perhaps the most famous of all and is the
favorite of Gauss quadratic reciprocity law, which called Gauss
theorem aureus.
 It is also noted the contribution of Gauss's theory of complex
numbers. Also developed a method to decompose the product of
prime numbers in complex numbers.
A New Planet
The discovery of the "new
planet", called Ceres by
Giuseppe Piazzi. It was
necessary to accurately
determine the orbit of Ceres to
put it back to reach the
telescopes, Gauss accept this
challenge and Ceres was
rediscovered a year later, in the
place that had predicted with
detailed calculations. His
technique was to demonstrate
how variations in experimental
source data could be
represented by a bell-shaped
curve (now known as the bell
curve). He also used the least
squares method. Had similar
success in the determination of
the orbit of the asteroid Pallas.
By 1820 Gauss began working in geodesy (determination of
the shape and size of the earth). In 1821 he was
commissioned by the governments of Denmark and
Hannover, Hannover geodetic study. To this end Gauss
invented the heliotrope, an instrument that reflects sunlight
in the specified direction and can reach a distance of 100 km
and enabling alignment of surveying instruments. Working
with data from observations developed a theory on curved
surfaces, the surface characteristics can be known by
measuring the length of curves contained therein.
 From 1831 he began working in the theoretical and experimental
investigation of magnetism. Gauss was able to prove the origin
of the field was in the interior of the earth. Gauss also worked
with the possibilities of the telegraph, his was probably the first
to run in a practical way in seven years ahead of the Morse
 The main feature of the work of Gauss, especially in pure
mathematics is particularly so have reasoned like general.
 After his death it emerged that Gauss had found the double
periodicity of elliptic functions.
Carlos Fernández Martín