1. Chapter 9 Calculus What is Calculus?
Early Results on Areas and Volumes
Maxima, Minima, and Tangents
The “Arithmetica Infinitorum” of Wallis
Newton’s Calculus of Series
The Calculus of Leibniz
Biographical Notes: Wallis, Newton, and Leibniz

2. What is Calculus?
Calculus appeared in 17th century as a system of shortcuts to results obtained by the method of exhaustion
Calculus derives rules for calculations
Problems, solved by calculus include finding areas, volumes (integral calculus), tangents, normals and curvatures (differential calculus) and summing of infinite series
This makes calculus applicable in a wide variety of areas inside and outside mathematics
In traditional approach (method of exhaustions) areas and volumes were computed using subtle geometric arguments
In calculus this was replaced by the set of rules for calculations

3. 17th century calculus
Differentiation and integration of powers of x (including fractional powers) and implicit differentiation of polynomials in x and y
Together with analytic geometry this made possible to find tangents, maxima and minima of all algebraic curves p (x,y) = 0
Newton’s calculus of infinite series (1660s) allowed for differentiation and integration of all functions expressible as power series
Culmination of 17th century calculus: discovery of the Fundamental Theorem of Calculus by Newton and Leibniz (independently)
Features of 17th century calculus:
the concept of limit was not introduced yet
use of “indivisibles” or “infinitesimals”
strong opposition of some well-known philosophers of that time (e.g. Thomas Hobbes)
very often new results were conjectured by analogy with previously discovered formulas and were not rigorously proved

4. 9.2 Early Results on Areas and Volumes
Area ≈ [(1/n)k + (2/n)k + … + (n/n)k](1/n)
→ sum 1k + 2k + … + nk
y = xk
Volume of the solid of revolution:
area of cross-section is π r2
and therefore it is required to compute sum
12k + 22k + 32k +… + n2k
1/n
n/n = 1
2/n
3/n
(n-1)/n

5. First results: Greek mathematicians (method of exhaustion, Archimedes)
Arab mathematician al-Haytham (10th -11th centuries) summed the series 1k + 2k + … + nk for k = 1, 2, 3, 4 and used the result to find the volume of the solid obtained by rotating the parabola about its base
Cavalieri (1635): up to k = 9 and conjectured the formula for positive integers k
Another advance made by Cavalieri was introduction of “indivisibles” which considered areas divided into infinitely thin strips and volumes divided into infinitely thin slices
It was preceded by the work of Kepler on the volumes of solids of revolution (“New Stereometry of wine barrels”, 1615)
Fermat, Descartes and Roberval (1630s) proved the formula for integration of xk (even for fractional values of k)
Torricelly: the solid obtained by rotating y = 1 / x about the x-axis from 1 to has finite volume!
Thomas Hobbes (1672): “to understand this [result] for sense, it is not required that a man should be a geometrician or logician, but that he should be mad”

6. 9.3 Maxima, Minima, and Tangents
The idea of differentiation appeared later than that one of integration
First result: construction of tangent line to spiral r = aθ by Archimedes
No other results until works of Fermat (1629)
“modern” approach:
Fermat’s approach (tangent to y = x2)
E – “small” or “infinitesimal” element which is set equal to zero at the end of all computations
Thus at all steps E ≠ 0 and at the end E = 0
Philosophers of that time did not like such approach

7. Fermat’s method worked well with all polynomials p(x)
Moreover, Fermat extended this approach to curves given by p(x,y) = 0
Completely the latter problem was solved by Sluse (1655) and Hudde (1657)
The formula is equivalent to the use of implicit differentiation

8. 9.4 The “Arithmetica Infinitorum” of Wallis (1655)
An attempt to arithmetize the theory of areas and volumes
Wallis found that ∫01 xpdx = 1/(p+1) for positive integers p (which was already known)
Another achievement: formula for ∫01 xm/ndx
Wallis calculated ∫01 x1/2dx, ∫01 x1/3dx,…, using geometric arguments, and conjectured the general formula for fractional p
Note: observing a pattern for p = 1,2,3, Wallis claimed a formula for all positive p “by induction” and for fractional p “by interpolation” (lack of rigour but a great deal of analogy, intuition and ingenuity) 1
y = x2 1
∫01 x2dx = 1/3
∫01 x1/2dx = 1 - 1/3 = 2/3

9. Wallis’ formula:
Expansion of π as infinite product was known to Viète (before Wallis’ discovery):
Nevertheless Wallis’ formula relates π to the integers through a sequence of rational operations
Moreover, basing on the formula for π Wallis’ found a sequence of fractions he called “hypergeometric”, which as it had been found later occur as coefficients in series expansions of many functions (which led to the class of hypergeometric functions)

10. Other formulas for π related to Wallis’ formula
Continued fraction
(Brouncker):
Series expansion discovered by 15th century Indian mathematicians and rediscovered by Newton, Gregory and Leibniz:
Euler
sub. x = 1

11. 9.5 Newton’s Calculus of Series
Isaac Newton
Most important discoveries in 1665/6
Before he studied the works of Descartes, Viète and Wallis
Contributions to differential calculus (e.g. the chain rule)
Most significant contributions are related to the theory of infinite series
Newton used term-by-term integration and differentiation to find power series representation of many of classical functions, such as tan-1x or log (x+1)
Moreover, Newton developed a method of inverting infinite power series to find inverses of functions (e.g ex from log (x+1))
Unfortunately, Newton’s works were rejected for publication by Royal Society and Cambridge University Press

12. 9.6 The Calculus of Leibniz
The first published paper on calculus was by Gottfried Wilhelm Leibniz (1684)
Leibniz discovered calculus independently
He had better notations than Newton’s
Leibniz was a librarian, a philosopher and a diplomat
“Nova methodus” (1864)
sum, product and quotient rules
notation dy / dx
dy / dx was understood by Leibniz literally as a quotient of infinitesimals dy and dx
dy and dx were viewed as increments of x and y

13. The Fundamental Theorem of Calculus
In “De geometria” (1686) Leibniz introduced the integral sign ∫
∫ f(x) dx meant (for Leibniz) a sum of terms representing infinitesimal areas of height f(x) and width dx
If one applies the difference operator d to such sum it yields the last term f(x) dx
Dividing by dx we obtain the FTC:

14. Leibniz introduced the word “function”
He preferred “closed-form” expressions to infinite series
Evaluation of integral ∫ f(x) dx was for Leibniz the problem of finding a known function whose derivative is f(x)
The search for closed forms led to
the problem of factorization of polynomials and eventually to the Fundamental Theorem of Algebra (integration of rational functions)
the theory of elliptic functions (attempts to integrate 1/√1-x4 )

15. 9.7 Biographical Notes: Wallis, Newton, and Leibniz

16. John Wallis Born: 23 Nov 1616 (Ashford, Kent, England) Died: 28 Oct 1703 (Oxford, England)

17. went to school in Ashford
Wallis’ academic talent was recognized very early
14 years old he was sent to Felsted, Essex to attend the school
He became proficient in Latin, Greek and Hebrew
Mathematics was not considered important in the best schools
Wallis learned rules of arithmetic from his brother
That time mathematics was not consider as a “pure” science in the Western culture
In 1632 he entered Emmanuel College in Cambridge
bachelor of arts degree (topics studied included ethics, metaphysics, geography, astronomy, medicine and anatomy)
Wallis received his Master's Degree in 1640

18. Between 1642 and 1644 he was chaplain at Hedingham, Essex and in London
Wallis became a fellow of Queens College, Cambridge
He relinquished the fellowship when he married in 1645
Wallis was interested in cryptography
Civil War between the Royalists and Parliamentarians began in 1642
Wallis used his skills in cryptography in decoding Royalist messages for the Parliamentarians
Since the appointment to the Savilian Chair in Geometry of Oxford in 1649 by Cromwell Wallis actively worked in mathematics

19. Sir Isaac Newton Born: 4 Jan 1643 (Woolsthorpe, Lincolnshire, England) Died: 31 March 1727 (London, England)

20. A family of farmers
Newton’s father (also Isaac Newton) was a wealthy but completely illiterate man who even could not sign his own name
He died three months before his son was born
Young Newton was abandoned by his mother at the age of three and was left in the care of his grandmother
Newton’s childhood was not happy at all
Newton entered Trinity College (Cambridge) in 1661

21. Newton entered Trinity College (Cambridge) in 1661 to pursue a law degree
Despite the fact that his mother was a wealthy lady he entered as a sizar
He studied philosophy of Aristotle
Newton was impressed by works of Descartes
In his notes “Quaestiones quaedam philosophicae” 1664 (Certain philosophical questions) Newton recorded his thoughts related to mechanics, optics, and the physiology of vision

22. The years 1664 – 66 were the most important in Newton’s mathematical development
By 1664 he became familiar with mathematical works of Descartes, Viète and Wallis and began his own investigations
He received his bachelor's degree in 1665
When the University was closed in the summer of 1665 because of the plague in England, Newton had to return to Lincolnshire
At that time Newton completely devoted himself to mathematics

23. Newton’s fundamental works on calculus “A treatise of the methods of series and fluxions” (1671) (or “De methodis”) and “On analysis by equations unlimited in their number of terms” (1669) (or “De analysis”) were rejected for publication
Nevertheless some people recognized his genius
Isaac Barrow resigned the Lucasian Chair (Cambridge) in 1669 and recommended that Newton be appointed in his place
Newton's first work as Lucasian Prof. was on optics
In particular, using a glass prism Newton discovered the spectrum of white light

24. 1665: Newton discovered inverse square law of gravitation
1687: “Philosophiae naturalis principia mathematica” (Mathematical principles of natural philosophy)
In this work, Newton developed mathematical foundation of the theory of gravitation
This book was published by Royal Society (with the strong support from Edmund Halley)
In 1693 Newton had a nervous breakdown
In 1696 he left Cambridge and accepted a government position in London where he became master of the Mint in 1699
In 1703 he was elected president of the Royal Society and was re-elected each year until his death
Newton was knighted in 1705 by Queen Anne

25. Gottfried Wilhelm von Leibniz Born: 1 July 1646 (Leipzig, Saxony (now Germany) Died: 14 Nov 1716 (Hannover, Hanover (now Germany)

26. An academic family
From the age of six Leibniz was given free access to his father’s library
At the age of seven he entered school in Leipzig
In school he studied Latin
Leibniz had taught himself Latin and Greek by the age of 12
He also studied Aristotle's logic at school
In 1661 Leibniz entered the University of Leipzig
He studied philosophy and mathematics
In 1663 he received a bachelor of law degree for a thesis “De Principio Individui” (“On the Principle of the Individual”)
The beginning of the concept of “monad”
He continued work towards doctorate
Leibniz received a doctorate degree from University of Altdorf (1666)

27. During his visit to the University of Jena (1663) Leibniz learned a little of Euclid
Leibniz idea was to create some “universal logic calculus”
After receiving his degree Leibniz commenced a legal career
From 1672 to 1676 Leibniz developed his ideas related to calculus and obtained the fundamental theorem
Leibniz was interested in summation of infinite series by investigation of the differences between successive terms
He also used term-by term integration to discover series representation of π