Leonhard Euler pronounced Oiler

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Leonhard Euler(pronounced Oiler)
Read Euler, read Euler. He is the master of us
all - Laplace
Euler calculated without apparent effort, as men
breathe, or eagles sustain themselves in the
wind - Arago
Analysis Incarnate
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Biography

• Born Leonhard Euler, in Switzerland
• (April 15, 1707 - September 18, 1783 at age 76)
• His early education was given by his father.
• Entered the University of Basel at 14,
  received Masters in Philosophy at age 17.
• Studied Hebrew Theology, but soon focused on
  mathematics.
• Moved to Russia and found a position at the St.
  Petersburg Academy of Sciences. His efforts
  there helped make Russia a naval power.
• Married Katharina Gsell, a Swiss girl, in 1733.
  He had thirteen children with her, all of whom he
  loved dearly.
• Accepted invitation to move to Prussia, escaping
  political unrest in Russia.
• Frederick the Great, the leader of Prussia, was
  an atheist, and constantly ridiculed Eulers
  faith.
• Euler lost the sight in one eye in 1735, and lost
  the sight in the other in 1766. He had an
  operation to repair them, but both became
  infected. He later said that only his faith in
  God allowed him to bear that torment.
• Produced works almost until the day of his death
  in 1783, working on the black slate of his
  mind. In an astonishing feat, his works became
  more clear after his blindness set in.

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Eulers Worldview

• Raised in a Calvinist home, son of a Protestant
  minister
• Held to the Reformed Worldview all his life
• Held family worship prayer daily in his home
  often preached read Scripture to his children
  every night
• Faced biting criticism from Frederick and
  Voltaire, an atheist and a deist, respectively
• Spent much time writing apologetics to respond to
  these two thinkers

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Eulers Accomplishments

• Wrote a total of 886 works
• His collected works total 74 volumes
• Made first rate discoveries in
• Analysis
• Functions
• Calculus
• Summations
• Combinatorics
• Number Theory
• Higher Algebra
• Convergent series
• Hydromechanics
• Physical Mechanics
• Astronomy
• Topology

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Königsberg Bridge Problem
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Eulers Accomplishments (continued)

• Analyzed
• mechanics
• planetary motion
• ballistics, projectile trajectories
• lunar orbit theory (tides)
• design sailing of ships
• construction architecture
• acoustics, theory of musical harmony
• investment theory
• insurance, annuities, pensions

• Other topics of interest
• chemistry
• medicine
• geography
• cartography
• languages
• philosophy
• apologetics
• religion
• family
• he taught his 13 children and many grandchildren

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Eulers Accomplishments (more)

• Promoted partial solutions to
• Gravitational Problems
• Optic Problems
• Etheric Problems
• Electromagnetic Problems

• This work greatly influenced Riemann and Maxwell
• He wrote textbooks that remained standards for
  hundreds of years
• Wrote research papers at the rate of 800 per year
• The epitome of his mathematical analysis is
  summed up in his formula

eip1 0
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Eulers Textbooks

• Euler wrote three Latin textbooks on the topics
  of Calculus and Pre-Calculus

• The first was the Introductio in Analysin
  Infinitorum (Introduction to the Analysis of the
  Infinite). This is considered by mathematics
  historians to be one of the most influential
  textbooks in history.
• This was Eulers Pre-Calculus textbook, which
  introduced topics that were absolutely required
  for analysis so that the reader almost
  imperceptibly becomes acquainted with the idea of
  the infinite
• He was the first to devise the ingenious teaching
  art of skillfully letting mathematical formulae
  speak for themselves.

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function
e2.71828...
f(x)
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Eulers Introduction(Did you know this?)

• The most important part of this book dealt with
  exponential, logarithmic, and trigonometric
  functions. It was there that Euler first
  introduced important notations such as
• Functional notation f(x)
• The base of natural logarithms e
• The sides of a triangle ABC a, b, c
• The semiperimeter of triangle ABC s
• The summation sign ?
• The imaginary unit ?-1 i

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More Eulerian Textbooks

• Eulers remaining books in the series were
  Institutiones Calculi Differentialis (Methods of
  the Differential Calculus) and Institutiones
  Calculi Integralis (Methods of the Integral
  Calculus)
• Eulers Differential Calculus contains
• Introduction to differential equations
• Discussed various methods for converting
  functions to power series
• Extensive chapters on finding sums of various
  series
• A pair of chapters on finding maxima and minima
• This is especially impressive, because his text
  contains no graphs or charts. All discussion
  given to maxima and minima is done purely
  analytically.
• Eulers Integral Calculus contains
• Integrals of various functions
• Solutions of differential equations
• Integration by infinite series, integration by
  parts, formulas for integration of powers of
  trigonometric functions
• All three books are an exercise in analysis, so
  much so that they contain no applications to
  geometry. The integral is not even used to
  calculate area under a curve.

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Euler functions formulae

• Discovered Eulers identity
• e i x cos(x) i sin(x)
• for any simple closed polyhedron with vertices V,
  edges E, and faces F
• V E F 2
• Euler curvature formula
• ? ?1 cos2? ?2 sin2 ?

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Euler functions formulae

• Number Theory
• Eulers function (or phi-function), ?(n), is
  defined as the number of integers less than n and
  relatively prime to n, i.e. sharing no common
  factor with n. Here are the first 10 values of
  ?(n)
• n 1 2 3 4 5 6 7 8 9
  10
• ?(n) 1 1 2 2 4 2 6 4 6
  4
• ?(10)4 because of all the integers between 1 and
  10 only 1,3,7, 9 share no common factor with
  10. So when n is prime ?(n)n-1 since all
  integers less than n are relatively prime to n.

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Euler functions formulae

• Rigid body motion
• Euler angles
• Hydrodynamics
• the Euler equation
• Dynamics of rigid bodies
• Eulers equation of motion
• Theory of elasticity
• Bernoulli-Euler law
• Trigonometric series
• Euler-Fourier formulas

• Infinite Series
• Eulers constant
• Euler numbers
• Eulers transformations
• DEs Partial Diff Eqs
• Eulers polygonal curves
• Eulers theorem on homogeneous functions
• Calculus of variations
• Euler-Lagrange equation
• Numerical Methods
• Euler-Maclaurin formula

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Euler functions formulae

• Rigid body motion
• Euler angles
• Hydrodynamics
• the Euler equation
• Dynamics of rigid bodies
• Eulers equation of motion
• Theory of elasticity
• Bernoulli-Euler law
• Trigonometric series
• Euler-Fourier formulas

• Infinite Series
• Eulers constant
• Euler numbers
• Eulers transformations
• DEs Partial Diff Eqs
• Eulers polygonal curves
• Eulers theorem on homogeneous functions
• Calculus of variations
• Euler-Lagrange equation
• Numerical Methods
• Euler-Maclaurin formula

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Euler line
The most famous line in the subject of triangle
geometry is named in honor of Leonhard Euler, who
penned more pages of original mathematics than
any other human being.
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Euler line
Gcentroid Ocircumcenter Horthocenter Nnine-poi
nt center LDeLongchamps point
O always lies 1/2 way from H to L


O to H
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Euler stops calculating

• Mathematics was used by Euler as Gods ally.
• He wrote Letters to a German Princess to give
  lessons in mechanics, physical optics, astronomy,
  sound, etc. In it he combined piety and the
  sciences. Their extreme popularity resulted in
  their translation into seven languages.

Euler remained virile and powerful of mind to the
very second of his death despite his total
blindness, which occurred in his seventy seventh
year, on September 18, 1783. That day he had
amused himself by calculating the laws of ascent
of balloons, dined with his family and friends.
Uranus being a recent discovery, Euler outlined
the calculation of its orbit. A little latter he
asked his grandson to be brought in. While
playing he suffered a stroke. Euler ceased
to live and calculate.