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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

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.

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

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

Königsberg Bridge Problem

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

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

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.

function

e2.71828...

f(x)

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

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.

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 ?

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.

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

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

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.

Euler line

Gcentroid Ocircumcenter Horthocenter Nnine-poi

nt center LDeLongchamps point

O always lies 1/2 way from H to L

O to H

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.