A very short history of Calculus presentation for MATH 1037 by Alex Karassev - PowerPoint PPT Presentation

PPT – A very short history of Calculus presentation for MATH 1037 by Alex Karassev PowerPoint presentation | free to download - id: 26a95-NWNhO

The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
Title:

A very short history of Calculus presentation for MATH 1037 by Alex Karassev

Description:

Qi Pi can be made arbitrary small. Hence Pi approximate C(R) arbitrarily closely. Elementary ... Pi(R) : Ri (R') = R2:R'2. Suppose that C(R):C(R') R2:R'2 ... – PowerPoint PPT presentation

Number of Views:1311
Avg rating:3.0/5.0
Slides: 35
Provided by: nipis
Category:
Tags:
Transcript and Presenter's Notes

Title: A very short history of Calculus presentation for MATH 1037 by Alex Karassev

1
A very short history of Calculus presentation
for MATH 1037 by Alex Karassev
• Irrational numbers in Greek math
• Theory of Proportion
• The Method of Exhaustion
• The Area of a Parabolic Segment
• 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 Archimedes, Wallis, Newton,
and Leibniz

2
Irrational numbers in Greek math
• Discovery of irrational numbers
• Greeks tried to avoid the use of irrationals
• The infinity was understood as potential for
continuation of a process but not as actual
infinity (static and completed)
• Examples
• 1,2, 3,... but not the set 1,2,3,
• sequence x1, x2, x3, but not the limit x lim
xn
• Paradoxes of Zeno ( 450 BCE) the Dichotomy
• there is no motion because that which is moved
must arrive at the middle before it arrives at
the end
• Approximation of v2 by the sequence of rational
number

3
4.2 Eudoxus Theory of Proportions
• Eudoxus (around 400 350 BCE)
• The theory was designed to deal with (irrational)
lengths using only rational numbers
• Length ? is determined by rational lengths less
than and greater than ?
• Then ?1 ?2 if for any rational r is rational r ?1 )
• Note the theory of proportions can be used to
define irrational numbers Dedekind (1872)
defined v2 as the pair of two sets of positive
rationals Lv2 r r22
(Dedekind cut)

4
The Method of Exhaustion
• was designed to find areas and volumes of
complicated objects (circles, pyramids, spheres)
using
• approximations by simple objects (rectangles,
trianlges, prisms) having known areas (or
volumes)
• the Theory of Proportions

5
Examples
Approximating the pyramid
Approximating the circle
6
Example Area enclosed by a Circle
• Let C(R) denote area of the circle of radius R
• We show that C(R) is proportional to R2
• Inner polygons P1
• Outer polygons Q1 Q2 Q3
• Qi Pi can be made arbitrary small
• Hence Pi approximate C(R) arbitrarily closely
• Elementary geometry shows that Pi is proportional
to R2 . Therefore, for two circles with radii R
and R' we get Pi(R) Ri (R) R2R2
• Suppose that C(R)C(R)
• Then (since Pi approximates C(R)) we can find i
such that Pi (R) Pi (R) contradicts 5)

P2
P1
Q1
Thus Pi(R) Ri (R) R2R2
7
4.4 The area of a Parabolic Segment Archimedes
(287 212 BCE)
• Triangles ?1 , ?2 , ?3 , ?4,
• Note that ?2 ?3 1/4 ?1
• Similarly ?4 ?5 ?6 ?7 1/16 ?1 and so on

Y
Z
S
1
R
4
7
3
2
Q
6
5
O
X
P
Thus A ?1 (11/4 (1/4)2) 4/3 ?1
8
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

9
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

10
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 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
11
• 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
• Cavalieri (1635) up to k 9 and conjectured the
formula for positive integers k
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 infinity 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

12
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

13
• 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 equivalent to the use of implicit
differentiation

14
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)

1
y x2
1
?01 x1/2dx 1 - 1/3 2/3
?01 x2dx 1/3
15
• 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)

16
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
17
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 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 (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

18
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

19
The Fundamental Theorem of Calculus
• In De geometria (1686) Leibniz introduced the
integral sign ?
• Note that ? 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 Fundamental Theorem
of Caculus

20
• 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 )

21
Biographical Notes
• Archimedes
• Wallis
• Newton
• Leibniz

22
Archimedes
• Was born and worked in Syracuse (Greek city in
Sicily) 287 BCE and died in 212 BCE
• Friend of King Hieron II
• Eureka! (discovery of hydrostatic law)
• Invented many mechanisms, some of which were used
for the defence of Syracuse
• Other achievements in mechanics usually
attributed to Archimedes (the law of the lever,
center of mass, equilibrium, hydrostatic
pressure)
• Used the method of exhaustions to show that the
volume of sphere is 2/3 that of the enveloping
cylinder
• According to a legend, his last words were Stay
away from my diagram!, address to a soldier who

23
John Wallis Born 23 Nov 1616 (Ashford, Kent,
England) Died 28 Oct 1703 (Oxford, England)
24
• 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

25
• 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

26
Sir Isaac Newton Born 4 Jan 1643 (Woolsthorpe,
Lincolnshire, England) Died 31 March 1727
(London, England)
27
• 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

28
• 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

29
• 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
• At that time Newton completely devoted himself to
mathematics

30
• 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

31
• 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
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

32
Gottfried Wilhelm von Leibniz Born 1 July 1646
(Leipzig, Saxony (now Germany) Died 14 Nov 1716
(Hannover, Hanover (now Germany)
33
• 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)

34
• 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