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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
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