Fluxions and Fluents. by Jenia Tevelev

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1 Fluxions and Fluents by Jenia Tevelev 1

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$fractions to a wide audience.$

3 Mathematics in the late 16th - early 17th century Simon Stevin ( ) wrote a short pamphlet La Disme, where he introduced decimal fractions to a wide audience. Transition to Hindu-Arabic from Roman numerals was initially resisted, hence the practice of writing amounts on checks 3

Beginnings of Calculus Archimedes (287 BC - 212 BC) The Quadrature of the Parabola : area under the curve using approximations and integration

of logarithms - Laplace said that invention of logarithms by shortening the labors, doubled the life of an astronomer Maybe you don t think of

4 Beginnings of Calculus Archimedes (287 BC BC) The Quadrature of the Parabola : area under the curve using approximations and integration (summation) Kepler ( ) New solid geometry of wine bottles - various optimization problems Napier ( ) Description of the wonderful canon of logarithms - Laplace said that invention of logarithms by shortening the labors, doubled the life of an astronomer Maybe you don t think of logarithms as a device for calculations but have you heard of prosthaphaeresis? How can one multiply numbers quickly using the formula 2cos(x)cos(y)=cos(x+y)+cos(x-y)? 4

Descartes (1596-1650) Discourse on the Method for Rightly Directing One's Reason and

Cartesian coordinates Fermat (1601-1665) Methodus ad disquirendam maximam et minima

f (a)=0 De tangentibus linearum curvarum - construction of tangents to curves Pascal

5 Descartes ( ) Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences - study of analytic geometry using algebra and Cartesian coordinates Fermat ( ) Methodus ad disquirendam maximam et minima -Fermat s principle- in modern day s formulation, if f(x) achieves extremum at x=a then f (a)=0 De tangentibus linearum curvarum - construction of tangents to curves Pascal ( ) Treatise on the sines of the quadrant - area under the sine curve - integral of sin(x) 5

! Barrow (1630-1677) Geometrical Lectures - Fundamental Theorem of Calculus geometric manifestation - relationship between tangents and the area.

6 ! Barrow ( ) Geometrical Lectures - Fundamental Theorem of Calculus geometric manifestation - relationship between tangents and the area. physical interpretation - relationship between velocity and distance. His father decided that he can t become a successful merchant because of his temper and sent him to study theology (!), which in turn get him interested in chronology, then astronomy, and finally geometry (to design telescope lenses). On a journey to the Holy Land, the ship was attacked by pirates. He was the only passenger to join the crew sword in hand to repel the pirates. Barrow took a prestigious position as a Lucasian professor of geometry at Cambridge, where one of his students was Isaac Newton. He later took the same position and Barrow happily became a full-time theologian. 6

7 Gregory ( ) - integral of 1/x is equal to ln(x) Van Heuraut ( ) - Epistola de transmutatione curvarum linearum in rectas - arc length formula It looks like everything that modern students study in Calculus was discovered in the 17th century before Newton. Why is he called an inventor of Calculus? 7

Sir Isaac Newton (1642-1727) was the son of Lincolnshire farmer. In 1660 he entered Trinity College, Cambridge.

8 Sir Isaac Newton ( ) was the son of Lincolnshire farmer. In 1660 he entered Trinity College, Cambridge. His discoveries in mathematics and physics began as early as 1664, and in 1669 he succeeded Barrow as Lucasian professor of mathematics. Later he became warden of the mint and member of parliament, and was knighted by Queen Anne. Newton discovered a method to solve any equation, algebraic or differential (the concept he introduced) using power series, also known as Taylor series (although Taylor was his student). He was guided by an analogy numbers functions decimal fractions power series 8

9 John Maynard Keynes: I believe that the clue to his mind is to be found in his unusual powers of continuous concentrated introspection. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him his secret 9

10 Fluxions and Fluents For Newton, every variable x he considered depended on time t. Nowdays we would use a function notation x=f(t). Newton called the quantity x the fluent. The fluxion is the speed with which x increases. In modern notation, this is f (t), although Newton s notation is still used, especially in physics. 10

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12 Newton applied his new mathematics to solve a staggering number of mathematical and physical problems, many of them contained in Principia (Mathematical Principles of Natural Philosophy). The crowning achievement of Principia was introduction of the Law of Gravity (Law of inverse squares) and the demonstration that it implies Kepler laws. An even harder theorem proved in Principia is that Law of Gravity is the only possible central force field where Laws of Kepler can hold. 12

Robert Hooke (1631-1703): as a secretary of the Royal Society, had to present a new exciting discovery every week, often his own, including Law of

13 Robert Hooke ( ): as a secretary of the Royal Society, had to present a new exciting discovery every week, often his own, including Law of Elasticity, Red Spot of Jupiter, etc. Suggested the Law of Inverse Squares but could not demonstrate mathematically that it implies Kepler Laws. 13

The line joining the sun to a planet sweeps out equal areas in equal times.

14 Kepler s Laws A planet revolves around the sun in an elliptical orbit with the sun in one focus. The line joining the sun to a planet sweeps out equal areas in equal times. The square of the period of revolution of a planet is proportional to the cube of the length of the major axis of its orbit. 14

15 Infamous 4th Law of Kepler 15

16 Let s get to work! 16

17 Group discussion of Newton s The method of fluxions and infinite series (I) Table of contents and Introduction: what s in the book? Comparison of Newton s and modern terminologies. (II) Problems 4 and 5. (III) Problems 6 and 7 (IV) Problem 11 (V) Problems 19 and

18 John Maynard Keynes: I believe that the clue to his mind is to be found in his unusual powers of continuous concentrated introspection. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him his secret Let s look at some examples of power series from Newton s Method of Fluxions and Power Series 18

Historical notes on calculus

Historical notes on calculus Dr. Vladimir Dotsenko Dr. Vladimir Dotsenko Historical notes on calculus 1 / 9 Descartes: Describing geometric figures by algebraic formulas 1637: René Descartes publishes