صفحه 1:
History of calculus
History of Integration,
History of Differentiation,
and Applications.
صفحه 2:
Contents
۰ 1 Development of calculus
— 1.1 Integral calcul
— 1.2 Differential calculus
— 1.3 Mathematical analysis
— 1.4 Modern calculus
٠ 2 Newton and Leibniz
- 2.1 Newton
— 2.2 Leibniz
+ 3 Integrals
٠ 4 Symbolic methods
+ 5 Calculus of variations
صفحه 3:
Integral calculus
* Calculating volumes and areas, the basic function of
integral calculus, can be traced back 0 6
Moscow papyrus (c. 1820 BC), in which an
Egyptian mathematician successfully calculated the
volume of a pyramidal frustum.
Greek geometers are credited with a significant use of
infinitesimals. Democritus is the first person recorded to
consider seriously the division of objects into an infinite
number of cross-sections, but his inability to rationalize
discrete cross-sections with a cone's smooth slope
prevented him from accepting the idea. At approximately
the same time, Zeno of Elea discredited infinitesimals
further by his articulation of the paradoxes which they
create.
صفحه 4:
۳ Antiphon and later Eudoxus are generally credited with
implementing the method of exhaustion, which made it possible to
compute the area and volume of regions and solids by breaking
them up into an infinite number of recognizable shapes. Archimedes
developed this method further, while also inventing heuristic
methods which resemble modern day concepts somewhat. (See
Archimedes’ Quadrature of the Parabola, The Method, Archimedes
on Spheres & Cylinders.) ۱۲ was not until the time of Newton that
these methods were made obsolete. It should not be thought that
infinitesimals were put on rigorous footing during this time, however.
Only when it was supplemented by a proper geometric proof would
Greek mathematicians accept a proposition as true.
In the third century Liu Hui wrote his Nine Chapters and also Haidao
suanjing (Sea Island Mathematical Manual), which dealt with using
the Pythagorean theorem (already stated in the Nine Chapters),
known in China as the Gougu theorem, to measure the size of
things. He discovered the usage of Cavalieri's principle to find an
accurate formula for the volume of a cylinder, showing a grasp of
elementary concepts associated with the differential and integral
calculus. In the 11th century, the Chinese polymath, Shen Kuo,
developed ‘packing’ equations that dealt with integration.
صفحه 5:
+ Indian mathematicians produced a number of works with
some ideas of calculus. The formula for the sum of the cubes
was first written by Aryabhata circa 500 AD, in order to find
the volume of a cube, which was an important step in the
development of integral calculus.
The next major step in integral calculus came in the 11th
century, when Ibn al-Haytham (known as A/hacen in Europe),
an l|ragi mathematician working in Egypt, devised what is
now known as "“Alhazen's problem", which leads to an
equation of the fourth degree, in his Book of Optics. While
solving this problem, he was the first mathematician to
derive the formula for the sum of the fourth powers, using a
method that is readily generalizable for determining the
general formula for the sum of any integral powers. He
performed an integration in order to find the volume of a
paraboloid, and was able to generalize his result for the
integrals of polynomials up to the fourth degree. He thus
came close to finding a general formula for the integrals of
polynomials, but he was not concerned with any polynomials
higher than the fourth degree.
صفحه 6:
Parallel development
In the 17th century, Pierre de Fermat, among other things, is
credited with an ingenious trick for evaluating the integral of any
power function directly, thus providing a valuable clue to Newton
and Leibniz in their development of the
fundamental theorem of calculus. Fermat also obtained a technique
for finding the centers of gravity of various plane and solid figures,
which influenced further work in quadrature.
At around the same time, there was also a great deal of work being
done by Japanese mathematicians, particularly Kowa Seki. He made
a number of contributions, namely in methods of determining areas
of figures using integrals, extending the method of exhaustion.
While these methods of finding areas were made largely obsolete by
the development of the fundamental theorems by Newton and
Leibniz, they still show that a sophisticated knowledge of
mathematics existed in 17th century Japan.
صفحه 7:
Development of differential
calculus
* The Greek mathematician Archimedes was the first to find the
tangent to a curve, other than a circle, in a method akin to
differential calculus. While studying the spiral, he separated a
point's motion into two components, one radial motion
component and one circular motion component, and then
continued to add the two component motions together thereby
finding the tangent to the curve.
The Indian mathematician-astronomer Aryabhata in 499 used a
notion of infinitesimals and expressed an astronomical problem in
the form of a basic differential equation. Manjula, in the 10th
century, elaborated on this differential equation in a commentary.
This equation eventually led Bhaskara Il in the 12th century to
develop the concept of a derivative representing infinitesimal
change, and he described an early form of "Rolle's theorem".
صفحه 8:
Concept of function
*In the late 12th century, the Persian mathematician,
Sharaf al-Din al-Tusi, introduced the idea of a function. In
his analysis of the equation x? + d = bx? for example, he
begins by changing the equation's form to x*(b — x) = d.
He then states that the question of whether the equation
has a solution depends on whether or not the “function”
on the left side reaches the value d. To determine this, he
finds a maximum value for the function. Sharaf al-Din then
states that if this value is less than d, there are no positive
solutions; if it is equal to d, then there is one solution; and
if it is greater than d, then there are two solutions.
However, his work was never followed up on in either
Europe or the Islamic world.
صفحه 9:
Development of derivative
+ Sharaf al-Din was also the first to discover the derivative of
cubic polynomials. His Treatise on Equations developed
concepts related to differential calculus, such as the
derivative function and the maxima and minima of curves, in
order to solve cubic equations which may not have positive
solutions. For example, in order to solve the equation x + a
= bx, al-Tusi finds the maximum point of the curve .
He usey = bx —x"rivative of the function to find that the
maximum point occurs at 4 at = +n finds the
maximum value for y at by st2(4)2uting دم
back into y = br — 2° . He finds that the equation), _ قن
has a solution i. BE. and al-Tusi thus deduces that the
equation has a positive root if p— «+ 9 , where Dis
the discriminant of the equation. 4
صفحه 10:
Pre-Newton-Leibniz
+ In the 15th century, an early version of the mean value theorem was
first described by Parameshvara (1370-1460) from the
Kerala school of astronomy and mathematics in his commentaries on
Govindasvami and Bhaskara Il.
+ In 17th century Europe, Isaac Barrow, Pierre de Fermat, Blai:
john Wallis and others discussed the idea of a del
particular, in Methodus ad disquirendam maximam et minima and in
De tangentibus linearum curvarum, Fermat developed a method for
determining maxima, minima, and tangents to various curves that
was equivalent to differentiation. Isaac Newton would later write that
his own early ideas about calculus came directly from "Fermat's way
of drawing tangents."
The first proof of Rolle's theorem was given by Michel Rolle in 1691
after the founding of modern calculus. The mean value theorem in its
modern form was stated by Augustin Louis Cauchy (1789-1857) also
after the founding of modern calculus.
صفحه 11:
Mathematical analysis
eek mathematicians such as Eudoxus and ۸ 5 6
informal use of the concepts of limits and convergence when they
used the metho 108 to compute the area and volume of
regions and solids. the 12th century mathematician
Bhaskara Il gave examples of the derivative and
coefficient, along with a statement of what is now known as Rol
theorem.
Mathematical analysis has its roots in work done by
Sangamagrama in the 14th century, along with later mathematician-
astronomers of the Kerala school of astronomy and mathematics,
who described special cases of Taylor series, including the Madhava-
Gregory series of the arctangent, the Madhava-Newton power
of sine and cosine, and the infinite series of n. Yuktibhasa, which
some consider to be the first text on calculus, summarizes these
results. Jslzou21
In the 15th century, a German cardinal named Nicholas of Cusa
argued that rules made for finite quantities lose their validity when
applied to infinite ones, thus putting to rest Zeno's paradoxes.
صفحه 12:
Modern calculus
۰ James Gregory was able to prove a restricted version of the
second fundamental theorem of calculus in the mid-17th
century.
Newton and Leibniz are usually credited with the invention of
modern calculus in the late 17th century. Their most
important contributions were the development of the
fundamental theorem of calculus. Also, Leibniz did a great
deal of work with developing consistent and useful notation
and concepts. Newton was the first to organize the field into
one consistent subject, and also provided some of the first
and most important applications, especially of
integral calculus.
Important contributions were also made by Barrow,
Descartes, de Fermat, Huygens, Wallis and many others.
صفحه 13:
+ Before Newton and Leibniz, the word
“calculus” was a general term used to
refer to any body of mathematics, but in
the following years, "calculus" became a
popular term for a field of mathematics
based upon their insights. The purpose of
this section is to examine Newton and
Leibniz’s investigations into the
developing field of calculus. Specific
importance will be put on the justification
and descriptive terms which they used in
an attempt to understand calculus as
they themselves conceived it.
صفحه 14:
* By the middle of the seventeenth century, European mathematics had
changed its primary repository of knowledge. In comparison to the last
century which maintained Hellenistic mathematics as the starting point for
research, Newton, Leibniz and their contemporaries increasingly looked
towards the works of more modern thinkers. Europe had become home to a
burgeoning mathematical community and with the advent of enhanced
institutional and organizational bases a new level of organization and
academic integration was being achieved. Importantly, however, the
community lacked formalism; instead it consisted of a disordered mass of
various methods, techniques, notations, theories, and paradoxes.
+ Newton came to calculus as part of his investigations in physics and
geometry. He viewed calculus as the scientific description of the
generation of motion and magnitudes. In comparison, Leibniz focused on
the tangent problem and came to believe that calculus was a metaphysical
explanation of change. These differences in approach should neither be
overemphasized nor under appreciated. Importantly, the core of their
insight was the formalization of the inverse properties between the integral
and the differential. This insight had been anticipated by their
predecessors, but they were the first to conceive calculus as a system in
which new rhetoric and descriptive terms were created. Their unique
discoveries lay not only in their imagination, but also in their ability to
synthesize the insights around them into a universal algorithmic process,
thereby forming a new mathematical system.
صفحه 15:
+ Newton completed no definitive publication formalizing his Fluxional
Calculus; rather, many of his mathematical discoveries were
transmitted through correspondence, smaller papers or as
embedded aspects in his other definitive compilations, such as the
Principia and Opticks. Newton would begin his mathematical training
as the chosen heir of Isaac Barrow in Oxford. His incredible aptitude
was recognized early and he quickly learned the current theories. By
1664 Newton had made his first important contribution by
advancing the binomial theorem, which he had extended to include
fractional and negative exponents. Newton succeeded in expanding
the applicability of the binomial theorem by applying the algebra of
finite quantities in an analysis of infinite series. He showed a
willingness to view infinite series not only as approximate devices,
but also as alternative forms of expressing a term.
صفحه 16:
+ Many of Newton's critical insights occurred during the plague years
of 1665-1666 which he later described as, “the prime of my age for
invention and minded mathematics and [natural] philosophy more
than at any time since.” It was during his plague-induced isolation
that the first written conception of Fluxionary Calculus was recorded
in the unpublished De Analysi per Aequationes Numero Terminorum
Infinitas. In this paper, Newton determined the area under a curve by
first calculating a momentary rate of change and then extrapolating
the total area. He began by reasoning about an indefinitely small
triangle whose area is a function of x and y. He then reasoned that
the infinitesimal increase in the abscissa will create a new formula
where xX + o (importantly, o is the letter, not the digit 0). He then
recalculated the area with the aid of the binomial theorem, removed
all quantities containing the letter o and re-formed an algebraic
expression for the area. Significantly, Newton would then “blot out”
the quantities containing o because terms “multiplied by it will be
nothing in respect to the rest”.
صفحه 17:
+ At this point Newton had begun to realize the central property of
inversion. He had created an expression for the area under a curve
by considering a momentary increase at a point. In effect, the
fundamental theorem of calculus was built into his calculations.
While his new formulation offered incredible potential, Newton was
well aware of its logical limitations. He admits that “errors are not to
be disregarded in mathematics, no matter how small” and that what
he had achieved was “shortly explained rather than accurately
demonstrated.”
+ In an effort to give calculus a more rigorous explication and
framework, Newton compiled in 1671 the Methodus Fluxionum et
Serierum Infinitarum. \n this book, Newton's strict empiricism
shaped and defined his Fluxional Calculus. He refused to speculate
on instantaneous motion and infinitesimals; rather he used math as
a methodological tool to explain the physical world. The base of
Newton's revised Calculus became continuity; as such he redefined
his calculations in terms of continual flowing motion. For Newton,
variable magnitudes are not aggregates of infinitesimal elements,
but are generated by the indisputable fact of motion.
صفحه 18:
* Newton attempted to avoid the use of the infinitesimal by forming
calculations based on ratios of changes. In the Methodus Fluxionum he
defined the rate of generated change as a fluxion, which he represented by a
dotted letter, and the quantity generated he defined as a fluent. For
example, if x and y are fluents, then and are their respective fluxions. This
revised calculus of ratios continued to be developed and was maturely
stated in the 1676 text De Quadratura Curvarum where Newton came to
define the present day derivative as the ultimate ratio of change, which he
defined as the ratio between evanescent increments (the ratio of fluxions)
purely at the moment in question. Essentially, the ultimate ratio is the ratio
as the increments vanish into nothingness. Importantly, Newton explained
the existence of the ultimate ratio by appealing to motion;
* “For by the ultimate velocity is meant that, with which the body is moved,
neither before it arrives at its last place, when the motion ceases nor after
but at the very instant when it arrives... the ultimate ratio of evanescent
quantities is to be understood, the ratio of quantities not before they vanish,
not after, but with which they vanish”
* Newton developed his Fluxional Calculus in an attempt to evade the
imprecise use of infinitesimals in his calculations. Foremost a scientist, he
found the imprecise and unverifiable notion of the infinitesimal an unfit base
for calculations.
صفحه 19:
Leibniz
+ While Newton began development of his fluxional calculus in 1665-1666 his
findings did not become widely circulated until later. In the intervening years
Leibniz also strove to create his calculus. In comparison to Newton who came to
math at an early age, Leibniz began his rigorous math studies with a mature
intellect. He was a polymath, and his intellectual interests and achievements
involved metaphysics, law, economics, politics, logic, and mathematics. In order to
understand Leibniz’s reasoning in calculus his background should be kept in mind.
Particularly, his metaphysics which considered the world as an infinite aggregate of
indivisible monads and his plans of creating a precise formal logic whereby, “a
general method in which all truths of the reason would be reduced to a kind of
calculation.” In 1672 Leibniz met the mathematician Huygens who convinced
Leibniz to dedicate significant time to the study of mathematics. By 1673 he had
progressed to reading Pascal's Traité des Sinus du Quarte Cercle and it was during
his largely autodidactic research that Leibniz said a light tuned on. Leibniz, like
Newton, saw the tangent as a ratio but declared it as simply the ratio between
ordinates and abscissas. He continued to argue that the integral was in fact the
sum of the ordinates for infinitesimal intervals in the abscissa, in effect, a sum of
an infinite number of rectangles. From these definitions the inverse relationship
became clear and Leibniz quickly realized the potential to form a whole new
system of mathematics. Where Newton shied away from the use of infinitesimals,
Leibniz made it the cornerstone of his notation and calculus.
صفحه 20:
* In the manuscripts of 25 October - 11 November 1675, Leibniz records his
discoveries and experiments with various forms of notation. He is acutely
aware of the notational terms used and his earlier plans to form a precise
logical symbolism become evident. Eventually, Leibniz denotes the
infinitesimal differences between abscissas and ordinates as dx and dy,
and the summation of infinitely many infinitesimally thin rectangles as a
longs (J ), which became the present integral symbol .
+ Importantly, while Leibniz's notation is used by modern mathematics, his
logical base was different than our current one. Leibniz embraced
infinitesimals and wrote extensively so as, “not to make of the infinitely
small a mystery, as had Pascal.” Towards this end he defined them “not as
a simple and absolute zero, but as a felative zero... that is, as an
evanescent quantity which yet retains the character of that which is
disappearing.” Alternatively, he defines them as, “less then any given
quantity” For Leibniz, the world was an aggregate of infinitesimal points
and the lack of scientific proof for their existence did not trouble him. The
truth of continuity was proven by existence itself. For Leibniz the principle
of continuity and thus the validity of his Calculus was assured. Three
hundred years after Leibniz's work, Abraham Robinson showed that using
infinitesimal quantities in calculus could be given a solid foundation.
صفحه 21:
Rise of calculus
* The rise of Calculus stands out as a unique moment in
mathematics. It is the math of motion and change and
its invention required the creation of a new
mathematical system. Importantly, Newton and Leibniz
did not create the same Calculus and they did not
conceive of modern Calculus. While they were both
involved in the process of creating a mathematical
system to deal with variable quantities their elementary
base was different. For Newton, change was a variable
quantity over time and for Leibniz it was the difference
ranging over a sequence of infinitely close values.
Notably, the descriptive terms each system created to
describe change was different.
صفحه 22:
+ Historically, there was much debate over whether it was Newton or
Leibniz who first “invented" calculus. This argument, the
Leibniz and Newton calculus controversy, involving Leibniz, who was
German, and the Englishman Newton, led to a rift in the European
mathematical community lasting over a century. Leibniz was the first
to publish his investigations; however, it is well established that
Newton had started his work several years prior to Leibniz and had
already developed a theory of tangents by the time Leibniz became
interested in the question. Much of the controversy centers on the
question whether Leibniz had seen certain early manuscripts of
Newton before publishing his own memoirs on the subject. Newton
began his work on calculus no later than 1666, and Leibniz did not
begin his work until 1673. Leibniz visited England in 1673 and again
in 1676, and was shown some of Newton's unpublished writings. He
also corresponded with several English scientists (as well as with
Newton himself), and may have gained access to Newton's
manuscripts through them. It is not known how much this may have
influenced Leibniz. The initial accusations were made by students
and supporters of the two great scientists at the turn of the century,
but after 1711 both of them became personally involved, accusing
each other of plagiarism.
صفحه 23:
Calculus widely accepted
* The priority dispute had an effect of separating English-speaking
mathematicians from those in the continental Europe for many
years and, consequently, slowing down the development of
mathematical analysis. Only in the 1820s, due to the efforts of
the Analytical Society, Leibnizian analytical calculus became
accepted in England. Today, both Newton and Leibniz are given
credit for independently developing the basics of calculus. It is
Leibniz, however, who is credited with giving the new discipline
the name it is known by today: "calculus". Newton's name for it
was "the science of fluents and fluxions".
The work of both Newton and Leibniz is reff ;cted in the notation
used today. Newton introduced the notation “ jor the derivative of
a function f Leibniz introduced the symbol for رن egral
and wrote the derivative of a function y of the variable ay
both of which are still in use. dx’
صفحه 24:
+ Niels Henrik Abel seems to have been the first to consider in a
general way the question as to what differential expressions can be
integrated in a finite form by the aid of ordinary functions, an
investigation extended by Liouville. Cauchy early undertook the
general theory of determining definite integrals, and the subject has
been prominent during the 19th century. Frullani's theorem (1821),
Bierens de Haan's work on the theory (1862) and his elaborate
tables (1867), Dirichlet's lectures (1858) embodied in Meyer's
treatise (1871), and numerous memoirs of Legendre, Poisson, Plana,
Raabe, Sohncke, Schlémilch, Elliott, Leudesdorf, and Kronecker are
among the noteworthy contributions.
Eulerian integrals were first studied by Euler and afterwards
investigated by Legendre, by whom they were classed as Eulerian
integrals of the first and second species, as follows:
1 8
۳ [ete a
0 0
+ although these were not the exact forms of Euler's study.
صفحه 25:
Factorial function
+ If nis an integer, it follows that:
ft e?2""!dr =(n—1)!,
0
* but the integral converges for all positive real n and defines an
analytic continuation of the factorial function to all of the
complex plane except for poles at zero and the negative integers.
To it Legendre assigned the symbol I, and it is now called the
gamma function. Besides being analytic over the positive reals, T
also enjoys the uniquely defining property that log! is convex,
which aesthetically justifies this analytic continuation of the
factorial function over any other analytic continuation. To the
subject Dirichlet has contributed an important theorem (Liouville,
1839), which has been elaborated by Liouville, Catalan, Leslie Ellis,
and others. On the evaluation of F(x) and logl(x) Raabe (1843-44),
Bauer (1859), and Gudermann (1845) have written. Legendre's
great table appeared in 1816.
صفحه 26:
Symbolic methods
+ Symbolic methods may be traced back to Taylor, and the
much debated analogy between successive
differentiation and ordinary exponentials had been
observed by numerous writers before the nineteenth
century. Arbogast (1800) was the first, however, to
separate the symbol of operation from that of quantity
in a differential equation. Francois (1812) and Servois
(1814) seem to have been the first to give correct rules
on the subject. Hargreave (1848) applied these methods
in his memoir on differential equations, and Boole freely
employed them. Grassmann and Hermann Hankel made
great use of the theory, the former in studying equations
, the latter in his theory of complex numbers.
صفحه 27:
Calculus of variations
* The calculus of variations may be said to begin with a problem of
johann Bernoulli's (1696). It immediately occupied the attention of
lakob Bernoulli and the Marquis de |'Hépital, but Euler first elaborated
the subject. His contributions began in 1733, and his Elementa Calculi
Variationum gave to the science its name. Lagrange contributed
extensively to the theory, and Legendre (1786) laid down a method,
not entirely satisfactory, for the discrimination of maxima and minima.
To this discrimination Brunacci (1810), Gauss (1829), Poisson (1831),
Ostrogradsky (1834), and Jacobi (1837) have been among the
contributors. An important general work is that of Sarrus (1842) which
was condensed and improved by Cauchy (1844). Other valuable
treatises and memoirs have been written by Strauch (1849), Jellett
(1850), Hesse (1857), Clebsch (1858), and Carll (1885), but perhaps
the most important work of the century is that of Weierstrass. His
celebrated course on the theory is epoch-making, and it may be
asserted that he was the first to place it on a firm and unquestionable
foundation.
صفحه 28:
Applications
+ The application of the infinitesimal calculus to problems in
physics and astronomy was contemporary with the origin
of the science. All through the eighteenth century these
applications were multiplied, until at its close Laplace and
Lagrange had brought the whole range of the study of
forces into the realm of analysis. To Lagrange (1773) we
owe the introduction of the theory of the potential into
dynamics, although the name “potential function" and the
fundamental memoir of the subject are due to Green
(1827, printed in 1828). The name "potential" is due to
Gauss (1840), and the distinction between potential and
potential function to Clausius. With its development are
connected the names of Dirichlet, Riemann, von Neumann,
Heine, Kronecker, Lipschitz, Christoffel, Kirchhoff, Beltrami,
and many of the leading physicists of the century.
صفحه 29:
Other applications
+ It is impossible in this place to enter into the great variety of other
applications of analysis to physical problems. Among them are the
investigations of Euler on vibrating chords; Sophie Germain on elastic
membranes; Poisson, Lamé, Saint-Venant, and Clebsch on the
elasticity of three-dimensional bodies; Fourier on heat diffusion;
Fresnel on light; Maxwell, Helmholtz, and Hertz on electricity; Hansen,
Hill, and Gyldén on astronomy; Maxwell on spherical harmonics;
Lord Rayleigh on acoustics; and the contributions of Dirichlet, Weber,
Kirchhoff, & Neumann, Lord Kelvin, Clausius, Bjerknes, MacCullagh,
and Fuhrmann to physics in general. The labors of Helmholtz should be
especially mentioned, since he contributed to the theories of
dynamics, electricity, etc., and brought his great analytical powers to
bear on the fundamental axioms of mechanics as well as on those of
pure mathematics.
Furthermore, infinitesimal calculus was introduced into the social
sciences, starting with Neoclassical economics. Today, it is a valuable
tool in mainstream economics.