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Twenty Other Ideas

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Twenty Other Ideas Countdown of two dozen of Euler s big ideas that don t have his name on them – PowerPoint PPT presentation

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Title: Twenty Other Ideas


1
Twenty Other Ideas
  • Countdown of two dozen of Eulers big ideas that
    dont have his name on them

2
26 - Laplace transform
  • In his 1769 Integral Calculus book, Euler wrote
    the Laplace Transform integral
  • Didnt follow through, like Laplace did
  • Did Laplace really say Read Euler. Read Euler.
    He is the Master of us all!
  • No

3
25 Fourier series
  • 1770s
  • Odd functions only
  • Elliptical orbits
  • Also an early use of subscript-like notation
  • 0, 4, 8, etc.

4
24 - Paddle wheel, Screw propeller
  • Described for 1753 Paris Prize
  • Propulsion of ships without wind
  • 2nd place
  • Actually built about 80 years later

5
23 - Centrifugal pump
  • Invented at the command of Frederick the Great
  • Developed about a hundred years later
  • New patents, often for nautical applications

6
22 Differential equations of fluid dynamics
  • Conservation of mass in a stream line
  • Equation of continuity

7
21 Knights tour
  • and sufficient part of Koenigsburg Bridge
    Problem

8
20 - Statistics of observational data
  • Best fit equations for observation of a comet
  • Used absolute value, not least squares

9
19 Partition numbers
  • Naudes problem
  • How many ways can you write n as a sum?
  • Ramanujan

10
18 Generating functions
  • Invented them to solve the partition problem in
    1741
  • Using the coefficients of a power series to count
    something
  • Relations with recursive calculations

11
17 Zeta function
  • Sum of reciprocals of nth powers
  • Riemann extended it from positive reals to
    complex plane
  • Sum-Product formula -

12
16 Gamma function
  • First letter to Goldbach
  • Generalized n!
  • Suggested fractional derivatives

13
15 FLT n 4
  • First published proof
  • Fermat probably did it
  • Also had a false general proof, never published

14
14 Density of primes
  • Showed
  • diverges

15
13 continued fractions
  • Unless you are a specialist, you dont know
    anything about continued fractions that isnt in
    Eulers first paper.
  • And you probably dont know all of that, either.

16
12 elliptic integrals
  • Summation formula for elliptic integrals
  • Generalizes trigonometric functions
  • Also series for arc length of an ellipse

17
11 - Derangements
  • Permutations that move every element
  • Showed probability approaches 1/e
  • Genoese lottery
  • Command of Frederick II

18
10 integrating factor
  • Reduces order of a differential equation
  • Often attributed to Clairaut
  • Euler was 2 years earlier

19
9 E edges
  • Before Euler, nobody had identified Edges on a
    solid as a mathematical object
  • Descartes came close
  • Counted edges by counting plane angles and
    dividing by 2

20
8 Venn diagrams
  • Venn called them Eulerian Circles
  • Letters to a German Princess
  • Aid to logic
  • See How Euler Did It January, 2004

21
7 Algebra statics Calculus dynamics
  • Calculus is the way to study the world
  • Every problem is an optimization problem

22
6 -
  • Mixed partial derivatives are equal
  • Euler knew of no counterexamples, so he did not
    give continuity conditions

23
5 - Precalculus
  • Introductio in analysin infinitorum
  • All the prerequisites to calculus

24
4 Transit of Venus
  • 1761 and 1769
  • Astronomical unit (distance to sun)
  • Longitude
  • International scientific cooperation
  • Eli Maor, Thomas Pynchon

25
3 - Coauthorship
  • Co-published with Johann Albrecht and with
    Charles on Paris Prize
  • No earlier important work was coauthored
  • Erdos couldnt have functioned without
    coauthorship

26
2 -
  • Modern calculus curriculum
  • First example of chain rule for a transcendental
    function


27
1 - Function
  • Function became a mathematical object
  • Function became an acceptable answer to a problem

28
And thats not all
  • 3-d coordinate systems
  • Best shape for teeth on gears
  • Telescopes and microscopes
  • Logarithms in theory of music
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