Calculus reordered: a history of the big ideas
"Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus grew to what we know today. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, and how its current structure is b...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton and Oxford
Princeton University Press
[2019]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | "Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus grew to what we know today. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, 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 presents a sounder way for students to learn this fascinating area of mathematics. Delving into calculus's birth in the Hellenistic Eastern Mediterranean--especially Syracuse in Sicily and Alexandria in 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 instead that the historical order--which follows first 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."--Dust jacket |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | xvi, 224 Seiten Illustrationen, Diagramme 24 cm |
| ISBN: | 9780691218786 9780691181318 |
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| 505 | 8 | 0 | |t Accumulation -- |t Ratios of change -- |t Sequences of partial sums -- |t The algebra of inequalities -- |t Analysis -- |t Reflections on the teaching of calculus -- |t The last word |
| 520 | 3 | |a "Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus grew to what we know today. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, 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 presents a sounder way for students to learn this fascinating area of mathematics. Delving into calculus's birth in the Hellenistic Eastern Mediterranean--especially Syracuse in Sicily and Alexandria in 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 instead that the historical order--which follows first 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."--Dust jacket | |
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Datensatz im Suchindex
| _version_ | 1804180986269794304 |
|---|---|
| adam_text | CONTENTS
PREFACE
XI
CHAPTER
1. ACCUMULATION
I
I.
I.
ARCHIMEDES
AND
THE
VOLUME
OF
THE
SPHERE
1
1.2.
THE
AREA
OF
THE
CIRCLE
AND
THE
ARCHIMEDEAN
PRINCIPLE
7
1.3.
ISLAMIC
CONTRIBUTIONS
11
1.4.
THE
BINOMIAL
THEOREM
17
1.5.
WESTERN
EUROPE
19
1.6.
CAVALIERI
AND
THE
INTEGRAL
FORMULA
21
1.7.
FERMAT S
INTEGRAL
AND
TORRICELLI S
IMPOSSIBLE
SOLID
25
1.8.
VELOCITY
AND
DISTANCE
29
1.9.
ISAAC
BEECKMAN
32
1.10.
GALILEO
GALILEI
AND
THE
PROBLEM
OF
CELESTIAL
MOTION
35
1.11.
SOLVING
THE
PROBLEM
OF
CELESTIAL
MOTION
38
1.12.
KEPLER S
SECOND LAW
42
1.13.
NEWTON S
PRINCIPIA
44
CHAPTER
2. RATIOS
OF
CHANGE
49
2.1.
INTERPOLATION
50
2.2.
NAPIER
AND
THE
NATURAL
LOGARITHM
57
-SIBIIOTHEK
DEUTSCHES
MUSEUM
HLUENCHEY
VIII
CONTENTS
2.3.
THE EMERGENCE
OF
ALGEBRA
64
2.4.
CARTESIAN
GEOMETRY
70
2.5.
PIERRE
DE
FERMAT
75
2.6.
WALLIS S ARITHMETIC
OF
INFINITESIMALS
81
2.7. NEWTON
AND
THE
FUNDAMENTAL
THEOREM
87
2.8. LEIBNIZ
AND
THE
BERNOULLIS
90
2.9.
FUNCTIONS
AND
DIFFERENTIAL EQUATIONS 93
2.10.
THE VIBRATING STRING
99
2.11.
THE
POWER
OF
POTENTIALS
103
2.12.
THE
MATHEMATICS
OF
ELECTRICITY
AND
MAGNETISM
104
CHAPTER
3.
SEQUENCES
OF
PARTIAL SUMS
108
3.1.
SERIES
IN
THE
SEVENTEENTH CENTURY
3.2.
TAYLOR SERIES
3.3.
EULER S INFLUENCE
3.4. D ALEMBERT
AND
THE
PROBLEM
OF
CONVERGENCE
3.5.
LAGRANGE REMAINDER
THEOREM
3.6. FOURIER S
SERIES
110
114
120
125
128
134
CHAPTER
4.
THE
ALGEBRA
OF
INEQUALITIES
141
4.1.
LIMITS
AND
INEQUALITIES
4.2.
CAUCHY
AND
THE
LANGUAGE
OF E AND
6
4.3.
COMPLETENESS
4.4.
CONTINUITY
4.5.
UNIFORM
CONVERGENCE
4.6. INTEGRATION
142
144
149
151
154
157
CONTENTS
CHAPTER
5. ANALYSIS
5.1.
THE
RIEMANN INTEGRAL
5.2.
COUNTEREXAMPLES
TO
THE
FUNDAMENTAL THEOREM
OF
INTEGRAL CALCULUS
5.3. WEIERSTRASS
AND
ELLIPTIC FUNCTIONS
5.4.
SUBSETS
OF
THE
REAL NUMBERS
5.5.
TWENTIETH-CENTURY
POSTSCRIPT
APPENDIX.
REFLECTIONS
ON THE
TEACHING
OF CALCULUS
TEACHING
INTEGRATION
AS
ACCUMULATION
TEACHING
DIFFERENTIATION
AS
RATIOS
OF
CHANGE
TEACHING
SERIES
AS
SEQUENCES
OF
PARTIAL SUMS
TEACHING
LIMITS
AS
THE
ALGEBRA
OF
INEQUALITIES
THE
LAST WORD
NOTES
BIBLIOGRAPHY
INDEX
IMAGE
CREDITS
IX
163
163
166
173
178
183
186
186
189
191
193
196
199
209
215
223
|
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| discipline | Mathematik |
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| indexdate | 2024-07-10T08:44:28Z |
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| spelling | Bressoud, David M. 1950- Verfasser (DE-588)136747221 aut Calculus reordered a history of the big ideas David M. Bressoud Princeton and Oxford Princeton University Press [2019] ©2019 xvi, 224 Seiten Illustrationen, Diagramme 24 cm txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Accumulation -- Ratios of change -- Sequences of partial sums -- The algebra of inequalities -- Analysis -- Reflections on the teaching of calculus -- The last word "Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus grew to what we know today. David Bressoud explains why calculus is credited to Isaac Newton and Gottfried Leibniz in the seventeenth century, 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 presents a sounder way for students to learn this fascinating area of mathematics. Delving into calculus's birth in the Hellenistic Eastern Mediterranean--especially Syracuse in Sicily and Alexandria in 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 instead that the historical order--which follows first 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."--Dust jacket Geschichte gnd rswk-swf Integration Mathematik (DE-588)4072852-3 gnd rswk-swf Infinitesimalanalysis (DE-588)4161657-1 gnd rswk-swf Calculus / Popular works Calculus / History Mathematics / History Calculus Mathematics Cálculo / Obras de divulgación Cálculo / Historia History Popular works Integration Mathematik (DE-588)4072852-3 s Infinitesimalanalysis (DE-588)4161657-1 s Geschichte z DE-604 Digitalisierung Deutsches Museum application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031845393&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bressoud, David M. 1950- Calculus reordered a history of the big ideas Accumulation -- Ratios of change -- Sequences of partial sums -- The algebra of inequalities -- Analysis -- Reflections on the teaching of calculus -- The last word Integration Mathematik (DE-588)4072852-3 gnd Infinitesimalanalysis (DE-588)4161657-1 gnd |
| subject_GND | (DE-588)4072852-3 (DE-588)4161657-1 |
| title | Calculus reordered a history of the big ideas |
| title_alt | Accumulation -- Ratios of change -- Sequences of partial sums -- The algebra of inequalities -- Analysis -- Reflections on the teaching of calculus -- The last word |
| title_auth | Calculus reordered a history of the big ideas |
| title_exact_search | Calculus reordered a history of the big ideas |
| title_full | Calculus reordered a history of the big ideas David M. Bressoud |
| title_fullStr | Calculus reordered a history of the big ideas David M. Bressoud |
| title_full_unstemmed | Calculus reordered a history of the big ideas David M. Bressoud |
| title_short | Calculus reordered |
| title_sort | calculus reordered a history of the big ideas |
| title_sub | a history of the big ideas |
| topic | Integration Mathematik (DE-588)4072852-3 gnd Infinitesimalanalysis (DE-588)4161657-1 gnd |
| topic_facet | Integration Mathematik Infinitesimalanalysis |
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