Chapter 9 Calculus

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Chapter 9Calculus

• What is Calculus?
• Early Results on Areas and Volumes
• Maxima, Minima, and Tangents
• The Arithmetica Infinitorum of Wallis
• Newtons Calculus of Series
• The Calculus of Leibniz
• Biographical Notes Wallis, Newton, and Leibniz

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9.1 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

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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
• Newtons 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

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9.2 Early Resultson 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 p r2 and therefore it is
required to compute sum 12k 22k 32k n2k
n/n 1
(n-1)/n
1/n
3/n
2/n
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• 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

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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 a? by Archimedes
• No other results until works of Fermat (1629)

modern approach
Fermats 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

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• Fermats 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 equivalentto the use ofimplicit
  differentiation

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9.4 The Arithmetica Infinitorum of Wallis
(1655)

• An attempt to arithmetize the theory of areas and
  volumes
• Wallis found that ?01 xpdx 1/(p1) 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)

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y x2
1
?01 x1/2dx 1 - 1/3 2/3
?01 x2dx 1/3
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• Wallis formula

• Expansion of p as infinite product was known to
  Viète (before Wallis discovery)

• Nevertheless Wallis formula relates p to the
  integers through a sequence of rational
  operations
• Moreover, basing on the formula for p 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)

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Other formulas for p 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
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9.5 Newtons 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 ofinfinite 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 (x1)
• Moreover, Newton developed a method of inverting
  infinite power series to find inverses of
  functions (e.g ex from log (x1))
• Unfortunately, Newtons works were rejected for
  publication by Royal Society and Cambridge
  University Press

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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 Newtons
• 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

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The Fundamental Theorem of Calculus

• In De geometria (1686) Leibniz introducedthe
  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

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• 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/v1-x4 )

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9.7 Biographical NotesWallis, Newton, and
Leibniz
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John WallisBorn 23 Nov 1616 (Ashford, Kent,
England)Died 28 Oct 1703 (Oxford, England)
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• 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

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• 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

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Sir Isaac NewtonBorn 4 Jan 1643 (Woolsthorpe,
Lincolnshire, England)Died 31 March 1727
(London, England)
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• A family of farmers
• Newtons 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
• Newtons childhood was not happy at all
• Newton entered Trinity College (Cambridge) in 1661

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• 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

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• The years 1664 66 were the most important in
  Newtons 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

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• Newtons 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

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• 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

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Gottfried Wilhelm von LeibnizBorn 1 July 1646
(Leipzig, Saxony (now Germany)Died 14 Nov 1716
(Hannover, Hanover (now Germany)
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• An academic family
• From the age of six Leibniz was given free access
  to his fathers 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)

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• 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 p