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A look at how calculus has evolved over hundreds of years and why
calculus pedagogy needs to change Calculus Reordered tells the
remarkable story of how calculus grew over centuries into the
subject we know today. David Bressoud explains why calculus is
credited to seventeenth-century figures Isaac Newton and Gottfried
Leibniz, how it was shaped by Italian philosophers such as Galileo
Galilei, and how its current structure sprang from developments in
the nineteenth century. Bressoud reveals problems with the standard
ordering of its curriculum-limits, differentiation, integration,
and series-and he argues that a pedagogy informed by the historical
evolution of calculus represents a sounder way for students to
learn this fascinating area of mathematics. From calculus's birth
in the Hellenistic Eastern Mediterranean, India, and the Islamic
Middle East, to its contemporary iteration, Calculus Reordered
highlights the ways this essential tool of mathematics came to be.
How our understanding of calculus has evolved over more than three
centuries, how this has shaped the way it is taught in the
classroom, and why calculus pedagogy needs to change Calculus
Reordered takes readers on a remarkable journey through hundreds of
years to tell the story of how calculus evolved into the subject we
know today. David Bressoud explains why calculus is credited to
seventeenth-century figures Isaac Newton and Gottfried Leibniz, and
how its current structure is based on developments that arose in
the nineteenth century. Bressoud argues that a pedagogy informed by
the historical development of calculus represents a sounder way for
students to learn this fascinating area of mathematics. Delving
into calculus's birth in the Hellenistic Eastern
Mediterranean-particularly in Syracuse, Sicily and Alexandria,
Egypt-as well as India and the Islamic Middle East, Bressoud
considers how calculus developed in response to essential questions
emerging from engineering and astronomy. He looks at how Newton and
Leibniz built their work on a flurry of activity that occurred
throughout Europe, and how Italian philosophers such as Galileo
Galilei played a particularly important role. In describing
calculus's evolution, Bressoud reveals problems with the standard
ordering of its curriculum: limits, differentiation, integration,
and series. He contends that the historical order-integration as
accumulation, then differentiation as ratios of change, series as
sequences of partial sums, and finally limits as they arise from
the algebra of inequalities-makes more sense in the classroom
environment. Exploring the motivations behind calculus's discovery,
Calculus Reordered highlights how this essential tool of
mathematics came to be.
"About binomial theorems I'm teeming with a lot of news, With many
cheerful facts about the square on the hypotenuse. " - William S.
Gilbert (The Pirates of Penzance, Act I) The question of
divisibility is arguably the oldest problem in mathematics. Ancient
peoples observed the cycles of nature: the day, the lunar month,
and the year, and assumed that each divided evenly into the next.
Civilizations as separate as the Egyptians of ten thousand years
ago and the Central American Mayans adopted a month of thirty days
and a year of twelve months. Even when the inaccuracy of a 360-day
year became apparent, they preferred to retain it and add five
intercalary days. The number 360 retains its psychological appeal
today because it is divisible by many small integers. The technical
term for such a number reflects this appeal. It is called a
"smooth" number. At the other extreme are those integers with no
smaller divisors other than 1, integers which might be called the
indivisibles. The mystic qualities of numbers such as 7 and 13
derive in no small part from the fact that they are indivisibles.
The ancient Greeks realized that every integer could be written
uniquely as a product of indivisibles larger than 1, what we
appropriately call prime numbers. To know the decomposition of an
integer into a product of primes is to have a complete description
of all of its divisors.
Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book guides us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which mathematics becomes the ultimate tool for modelling physical reality, to the dawn of a radically new and often counter-intuitive age in Albert Einstein's Special Theory of Relativity in which it is the mathematical model which suggests new aspects of that reality. The development of this process is discussed from the modern viewpoint of differential forms. Using this concept, the student learns to compute orbits and rocket trajectories, model flows and force fields, and derive the laws of electricity and magnetism. These exercises and observations of mathematical symmetry enable the student to better understand the interaction of physics and mathematics.
"About binomial theorems I'm teeming with a lot of news, With many
cheerful facts about the square on the hypotenuse. " - William S.
Gilbert (The Pirates of Penzance, Act I) The question of
divisibility is arguably the oldest problem in mathematics. Ancient
peoples observed the cycles of nature: the day, the lunar month,
and the year, and assumed that each divided evenly into the next.
Civilizations as separate as the Egyptians of ten thousand years
ago and the Central American Mayans adopted a month of thirty days
and a year of twelve months. Even when the inaccuracy of a 360-day
year became apparent, they preferred to retain it and add five
intercalary days. The number 360 retains its psychological appeal
today because it is divisible by many small integers. The technical
term for such a number reflects this appeal. It is called a
"smooth" number. At the other extreme are those integers with no
smaller divisors other than 1, integers which might be called the
indivisibles. The mystic qualities of numbers such as 7 and 13
derive in no small part from the fact that they are indivisibles.
The ancient Greeks realized that every integer could be written
uniquely as a product of indivisibles larger than 1, what we
appropriately call prime numbers. To know the decomposition of an
integer into a product of primes is to have a complete description
of all of its divisors.
Meant for advanced undergraduate and graduate students in
mathematics, this lively introduction to measure theory and
Lebesgue integration is rooted in and motivated by the historical
questions that led to its development. The author stresses the
original purpose of the definitions and theorems and highlights
some of the difficulties that were encountered as these ideas were
refined. The story begins with Riemann's definition of the
integral, a definition created so that he could understand how
broadly one could define a function and yet have it be integrable.
The reader then follows the efforts of many mathematicians who
wrestled with the difficulties inherent in the Riemann integral,
leading to the work in the late 19th and early 20th centuries of
Jordan, Borel, and Lebesgue, who finally broke with Riemann's
definition. Ushering in a new way of understanding integration,
they opened the door to fresh and productive approaches to many of
the previously intractable problems of analysis.
This introduction to recent developments in algebraic combinatorics illustrates how research in mathematics actually progresses. The author recounts the dramatic search for and discovery of a proof of a counting formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While it was apparent that the conjecture must be true, the proof was elusive. As a result, researchers became drawn to this problem and made connections to aspects of the invariant theory of Jacobi, Sylvester, Cayley, MacMahon, Schur, and Young; to partitions and plane partitions; to symmetric functions; to hypergeometric and basic hypergeometric series; and, finally, to the six-vertex model of statistical mechanics. This volume is accessible to anyone with a knowledge of linear algebra, and it includes extensive exercises and Mathematica programs to help facilitate personal exploration. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something unique within Proofs and Confirmations.
Meant for advanced undergraduate and graduate students in
mathematics, this lively introduction to measure theory and
Lebesgue integration is rooted in and motivated by the historical
questions that led to its development. The author stresses the
original purpose of the definitions and theorems and highlights
some of the difficulties that were encountered as these ideas were
refined. The story begins with Riemann's definition of the
integral, a definition created so that he could understand how
broadly one could define a function and yet have it be integrable.
The reader then follows the efforts of many mathematicians who
wrestled with the difficulties inherent in the Riemann integral,
leading to the work in the late 19th and early 20th centuries of
Jordan, Borel, and Lebesgue, who finally broke with Riemann's
definition. Ushering in a new way of understanding integration,
they opened the door to fresh and productive approaches to many of
the previously intractable problems of analysis.
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