The Fundamental Theorem of Calculus: History, Intuition, Pedagogy, Proof.

SUNY/Oneonta, October 8, 2010

V. Frederick RickeyWest Point

Isaac Newton 1642 - 1727

• 1702 portrait by Kneller

• The original is in the National Portrait Gallery in London

Newton’s Mathematical Readings

• Barrow Euclid (1655)• Oughtred Clavis (1652)• Descartes 2nd Latin (1659-60)• Schooten Exercitationum (1657)• Viete Opera (1646)• Wallis Arithmetica infinitorum

(1655)• Wallis Tractatus duo (1659)

Took Descartes’s Geometry in hand, tho he had been told it would be very difficult, read some ten pages in it, then stopt, began again, went a little farther than the first time, stopt again, went back again to the beginning, read on til by degrees he made himself master of the whole, to that degree that he understood Descartes’s Geometry better than he had done Euclid.

Descartes’s Geometry, 1637, 1659

Descartes adopted Aristotle’s dictum

The proportion between straight lines and curves is not known and I even believe that it can never be known by man.

van Heuraet on Arc Length, 1659

van Heuraet’s rectification, 1659

Rectification Destroyed

• Aristotle’s dictum and • Descartes’ program

• But the story ends well.

The Fundamental Theorem of Calculus

A Method whereby to square such crooked lines as may be squared.

• van Heuraet swapped arc length for area

• Newton swapped area for a tangent

For Newton

• Mathematical quantities are described by Continuous Motion.

– E.g., Curves are generated by moving points

• In Modern Terms: All variables are functions of time

• Newton said that quantities flow, and so called them fluents.

• How fast they flow – or flex – he called fluxions.

• Par abuse de langu, d/dt ( fluent ) = fluxion

Given an equation involving any number of fluent quantities to find the fluxions and vice versa.

Gottfried Wilhelm von Leibniz

(1646 – 1716)

The Nova methodus of 1684 – the first paper on the differential calculus.

Leibniz proves first FTC in 1690

The Fundamental Theorem of Calculus

Leibniz:

The Sideways Chalk Model

Newton:

The Windshield Wiper Model

The Isochrone Problem

• Find a curve along which a body will descend equal distances in equal times

• Johann Bernoulli reduces it to the Differential Equation √a dx = √y dy.

• Et eorum integralia !• The curve is a semi-cubical parabola, y3 = 9/4 a x2

Johann Bernoulli in 1743

His spirit sees truthHis heart knows justiceHe is an honor to the SwissAnd to all of humanity

• Voltaire

Finding areas under curves

Decompose the region into infinitely many differential areas

1. with parallel lines

2. with lines emanating from a point

3. with tangent lines

4. with normal lines.

We seek the curve where the square of the ordinate BC is the mean proportional between the square of the given length E and the curvilinear figure ABC.

E2 / BC2 = BC2 / Area ABCArea ABC = y4 / a2

By FTC,y dx = 4 y3 dy / a2

Divide by y and integrate

To get a cubical parabola

Johann Bernoulli’s definition of an integral

We have previously shown how to find the differential of a given quantity. Now we show inversely how to find the integral of a differential, i.e., find the quantity from which the differential originates.

E2 / BC2 = BC2 / Area ABCArea ABC = y4 / a2

By FTC,y dx = 4 y3 dy / a2

Divide by y and integrate

To get a cubical parabola

I misread this text.Bernoulli does NOT use FTC but only the notion that an integral is an antiderivative.

• Johann Bernoulli’s best student !

• Leonhard Euler

• 1707-1783

Euler about 1737, age 30• Painting by J. Brucker• 1737 mezzotint by

Sokolov• Black below and above

right eye• Fluid around eye is

infected• “Eye will shrink and

become a raisin”• Ask your

ophthalmologist• Thanks to Florence Fasanelli

Euler’s Calculus Books

• 1748 Introductio in analysin infinitorum399

402

• 1755 Institutiones calculi differentialis676

• 1768 Institutiones calculi integralis462542508

_____2982

• Defines the integral as an antiderivative.

• Gives a careful discussion of approximating a definite integral with a sum of rectangles.

Read Euler, read Euler, he is our teacher in everything.

Laplace as quoted by Libri, 1846

Lisez Euler, lisez Euler, c'est notre maître à tous.

In the 18th century there was no FTC.

This famous work of 1821 began to introduce rigor into the calculus by defining limits, continuity and derivatives and proving theorems about them.

It was never used as a text.

Augustin Cauchy 1789 - 1857

Augustin Cauchy 1823

In his Résumé of 1823, Cauchy

• Gave a careful definition as the limit of a sum of areas of rectangles (evaluated at left endpoints).

• Proved that the integral of a continuous function exists.

• Proved the First FTC in a rigorous way.

• Cauchy’s definition of the integral is a radical break with the past!

• Euler used left sums.• Lacroix and Poisson tried to prove

the sums converge. • Fourier needed to think of the

definite integral as an area.

Rosenstein and Temelli, 2001

At the end of the 19th century, authors had two choices regarding the introduction of the integral:

Either one might define the integral as the limit of a certain sum, or, alternatively,

integration is the inverse operation to differentiation.

If the former introduction is chosen, then one must justify some form of the Fundamental Theorem of Calculus;

if the latter, then the use of the integral in applications becomes the sticking point.

The Name “FTC” in Research Monographs.

• Eduard Goursat uses the term “Fundamental Theorem of Calculus“ in his Cours d'analyse mathematiques (1902).

• Ernest W. Hobson in his Theory of Functions of a Real Variable (1907) has a chapter entitled “The fundamental theorem of the integral calculus.”

• Vallee Poussin in his Cours d'analyse innitesimale (1921) uses the name “relation fondamentale pour le calcul des integrals "

William Anthony Granville

• Text used at West Point 1907-1948 and 1953-1963

Granville 1911

• Granville and Smith define indefinite integration as antidifferentation.

• The definite integral is defined as F(b) – F(a), where F’(x) = f(x).

• Thus there is no FTC in our modern sense.

• They use FTC in the sense of du Bois-Reymond, 1876, 1880.

The Name “FTC” in Textbooks

• Björling (1877) gives the name “Grundsats” to the second fundamental theorem in a Swedish textbook

• G. H. Hardy, A Course of Pure Mathematics (1908), uses the phrase and provides a proof.

• George Thomas uses the phrase in his Calculus (1951).

First use of the name 2nd FTC, 1958

To be continued . . .

after much more research.