Mathematics and its history:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York ; Dordrecht ; Heidelberg ; London
Springer
[2010]
|
| Ausgabe: | Third edition |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xxi, 660 Seiten Illustrationen |
| ISBN: | 144196052X 9781441960528 9781461426325 |
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Datensatz im Suchindex
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| adam_text | Contents Preface to the Third Edition vii Preface to the Second Edition ix Preface to the First Edition xi 1 The Theorem of Pythagoras 1.1 Arithmetic and Geometry...................................................... 1.2 Pythagorean Triples............................................................... 1.3 Rational Points on the Circle................................................ 1.4 Right-Angled Triangles......................................................... 1.5 Irrational Numbers............................................................... 1.6 The Definition of Distance................................................... 1.7 Biographical Notes: Pythagoras ......................................... 1 2 4 6 9 11 13 15 2 Greek Geometry 2.1 The Deductive Method......................................................... 2.2 The Regular Polyhedra......................................................... 2.3 Ruler and Compass Constructions...................................... 2.4 Conic Sections ...................................................................... 2.5 Higher-Degree Curves ......................................................... 2.6 Biographical Notes: Euclid................................................... 17 18 20 25 28 31 35 3 Greek Number Theory 3.1 The Role of Number Theory............................................... 3.2 Polygonal, Prime, and Perfect Numbers............................. 3.3 The Euclidean Algorithm...................................................... 3.4 Pell’s
Equation...................................................................... 3.5 The Chord and Tangent Methods......................................... 37 38 38 41 44 48 XV
Contents XVI 3.6 Biographical Notes: Diophantus......................................... 50 4 Infinity in Greek Mathematics 4.1 Fear of Infinity ..................................................................... 4.2 Eudoxus’s Theory of Proportions......................................... 4.3 The Method of Exhaustion................................................... 4.4 The Area of a Parabolic Segment......................................... 4.5 Biographical Notes: Archimedes......................................... 53 54 56 58 63 66 5 Number Theory in Asia 5.1 The Euclidean Algorithm...................................................... 5.2 The Chinese Remainder Theorem ...................................... 5.3 Linear Diophantine Equations ............................................ 5.4 Pell’s Equation in Brahmagupta ......................................... 5.5 Pell’s Equation in Bhâskara II ............................................ 5.6 Rational Triangles.................................................................. 5.7 Biographical Notes: Brahmagupta and Bhâskara................ 69 70 71 74 75 78 81 84 6 Polynomial Equations 87 6.1 Algebra.................................................................................. 88 6.2 Linear Equations and Elimination ...................................... 89 6.3 Quadratic Equations............................................................... 92 6.4 Quadratic Irrationals ............................................................ 95 6.5 The Solution of the Cubic......................................................
97 6.6 Angle Division..................................................................... 99 6.7 Higher-Degree Equations......................................................... 101 6.8 Biographical Notes: Tartaglia, Cardano, and Viète.............103 7 Analytic Geometry 109 7.1 Steps Toward Analytic Geometry.............................................110 7.2 Fermat and Descartes................................................................Ill 7.3 Algebraic Curves...................................................................... 112 7.4 Newton’s Classification of Cubics..........................................115 7.5 Construction of Equations, Bézout’s Theorem.......................118 7.6 The Arithmetization of Geometry ..........................................120 7.7 Biographical Notes: Descartes................................................122
Contents ÄVU g Projective Geometry 127 8.1 Perspective............................................................................... 128 8.2 Anamorphosis............................................................................ 131 8.3 Desargues’s Projective Geometry.............................................132 8.4 The Projective View of Curves................................................136 8.5 The Projective Plane ............................................................... 141 8.6 The Projective Line...................................................................144 8.7 Homogeneous Coordinates...................................................... 147 8.8 Pascal’s Theorem......................................................................150 8.9 Biographical Notes: Desargues and Pascal.............................153 9 Calculus 157 9.1 What Is Calculus?......................................................................158 9.2 Early Results on Areas and Volumes...................................... 159 9.3 Maxima, Minima, and Tangents .............................................162 9.4 The Arilhmetica Irifinito rum of Wallis................................... 164 9.5 Newton’s Calculus of Series ...................................................167 9.6 The Calculus of Leibniz............................................................ 170 9.7 Biographical Notes: Wallis, Newton, and Leibniz................ 172 10 Infinite Series 181 10.1 Early Results............................................................................ 182 10.2 Power Series
............................................................................ 185 10.3 An Interpolation on Interpolation.............................................188 10.4 Summation of Series............................................................... 189 10.5 Fractional Power Series............................................................ 191 10.6 Generating Functions............................................................... 192 10.7 The Zeta Function ...................................................................... 195 10.8 Biographical Notes: Gregory and Euler ................................ 197 11 The Number Theory Revival 203 11.1 Between Diophantus and Fermat............................................ 204 11.2 Fermat’s Little Theorem .........................................................207 11.3 Fermat’s Last Theorem............................................................ 210 11.4 Rational Right-Angled Triangles............................................ 211 11.5 Rational Points on Cubics of Genus 0 ...................................215 11.6 Rational Points on Cubics of Genus 1 ...................................218 11.7 Biographical Notes: Fermat......................................................222
xviii Contents 12 Elliptic Functions 225 12.1 Elliptic and Circular Functions............................................... 226 12.2 Parameterization of Cubic Curves ......................................... 226 12.3 Elliptic Integrals.........................................................................228 12.4 Doubling the Arc of the Lemniscate...................................... 230 12.5 General Addition Theorems ...................................................232 12.6 Elliptic Functions..................................................................... 234 12.7 A Postscript on the Lemniscate................................................236 12.8 Biographical Notes: Abel and Jacobi...................................... 237 13 Mechanics 243 13.1 Mechanics Before Calculus......................................................244 13.2 The Fundamental Theorem of Motion................................... 246 13.3 Kepler’s Laws and the Inverse Square Law............................ 249 13.4 Celestial Mechanics.................................................................. 253 13.5 Mechanical Curves.................................................................. 255 13.6 The Vibrating String ............................................................... 261 13.7 Hydrodynamics.........................................................................265 13.8 Biographical Notes: The Bemoullis ...................................... 267 14 Complex Numbers in Algebra 275 14.1 Impossible Numbers ............................................................... 276 14.2
Quadratic Equations.................................................................. 276 14.3 Cubic Equations.........................................................................277 14.4 Wallis’s Attempt at Geometric Representation...................... 279 14.5 Angle Division.........................................................................281 14.6 The Fundamental Theorem of Algebra...................................285 14.7 The Proofs of d’Alembert and Gauss...................................... 287 14.8 Biographical Notes: d’Alembert............................................ 291 15 Complex Numbers and Curves 295 15.1 Roots and Intersections............................................................ 296 15.2 The Complex Projective Line...................................................298 15.3 Branch Points............................................................................ 301 15.4 Topology of Complex Projective Curves................................304 15.5 Biographical Notes: Riemann...................................................308
Contents xix 16 Complex Numbers and Functions 313 16.1 Complex Functions.................................................................. 314 16.2 Conformal Mapping.................................................................. 318 16.3 Cauchy’s Theorem.................................................................. 319 16.4 Double Periodicity of Elliptic Functions................................322 16.5 Elliptic Curves .........................................................................325 16.6 Uniformization.........................................................................329 16.7 Biographical Notes: Lagrange andCauchy..............................331 17 Differential Geometry 335 17.1 Transcendental Curves............................................................ 336 17.2 Curvature of Plane Curves......................................................340 17.3 Curvature of Surfaces............................................................... 343 17.4 Surfaces of Constant Curvature................................................344 17.5 Geodesics.................................................................................. 346 17.6 The Gauss-Bonnet Theorem...................................................348 17.7 Biographical Notes: Harriot andGauss.................................... 352 18 Non-Euclidean Geometry 359 18.1 The Parallel Axiom.................................................................. 360 18.2 Spherical Geometry.................................................................. 363 18.3 Geometry of Bolyai and
Lobachevsky...................................365 18.4 Beltrami’s Projective Model ...................................................366 18.5 Beltrami’s Conformal Models ................................................369 18.6 The Complex Interpretations...................................................374 18.7 Biographical Notes: Bolyai and Lobachevsky...................... 378 19 Group Theory 383 19.1 The Group Concept.................................................................. 384 19.2 Subgroups and Quotients.........................................................387 19.3 Permutations and Theory of Equations...................................389 19.4 Permutation Groups.................................................................. 393 19.5 Polyhedral Groups .................................................................. 395 19.6 Groups and Geometries............................................................ 398 19.7 Combinatorial Group Theory...................................................401 19.8 Finite Simple Groups............................................................... 404 19.9 Biographical Notes: Galois......................................................409
XX Contents 20 Hypercomplex Numbers 415 20.1 Complex Numbers in Hindsight ......................................... 416 20.2 The Arithmetic of Pairs............................................................417 20.3 Properties of + and x...............................................................419 20.4 Arithmetic of Triples and Quadruples ...................................421 20.5 Quaternions, Geometry, and Physics......................................424 20.6 Octonions..................................................................................428 20.7 Why C, H, and О Are Special.................................................. 430 20.8 Biographical Notes: Hamilton............................................... 433 21 Algebraic Number Theory 439 21.1 Algebraic Numbers.................................................................. 440 21.2 Gaussian Integers..................................................................... 442 21.3 Algebraic Integers..................................................................... 445 21.4 Ideals........................................................................................ 448 21.5 Ideal Factorization .................................................................. 452 21.6 Sums of Squares Revisited......................................................454 21.7 Rings and Fields ..................................................................... 457 21.8 Biographical Notes: Dedekind, Hilbert, and Noether . . . 459 22 Topology 467 22.1 Geometry and Topology
.........................................................468 22.2 Polyhedron Formulas of Descartes and Euler ...................... 469 22.3 The Classification of Surfaces ............................................... 471 22.4 Descartes and Gauss-Bonnet...................................................474 22.5 Euler Characteristic and Curvature......................................... 477 22.6 Surfaces and Planes.................................................................. 479 22.7 The Fundamental Group.........................................................484 22.8 The Poincaré Conjecture.........................................................486 22.9 Biographical Notes: Poincaré...................................................492 23 Simple Groups 495 23.1 Finite Simple Groups and Finite Fields...................................496 23.2 The Mathieu Groups ...............................................................498 23.3 Continuous Groups.................................................................. 501 23.4 Simplicity of SO(3).................................................................. 505 23.5 Simple Lie Groups and Lie Algebras...................................... 509 23.6 Finite Simple Groups Revisited............................................... 513 23.7 The Monster...............................................................................515
Contents XX1 23.8 Biographical Notes: Lie, Killing, and Cartan......................... 518 24 Sets, Logic, and Computation 525 24.1 Sets............................................................................................526 24.2 Ordinals..................................................................................... 528 24.3 Measure..................................................................................... 531 24.4 Axiom of Choice and Large Cardinals................................... 534 24.5 The Diagonal Argument ......................................................... 536 24.6 Computability............................................................................538 24.7 Logic and GodePs Theorem ...................................................541 24.8 Provability and Truth............................................................... 546 24.9 Biographical Notes: Gödel......................................................549 25 Combinatorics 553 25.1 What Is Combinatorics? ......................................................... 554 25.2 The Pigeonhole Principle.........................................................557 25.3 Analysis and Combinatorics ...................................................560 25.4 Graph Theory............................................................................563 25.5 Nonplanar Graphs......................................................................567 25.6 The König Infinity Lemma......................................................571 25.7 Ramsey
Theory.........................................................................575 25.8 Hard Theorems of Combinatorics ......................................... 580 25.9 Biographical Notes: Erdős......................................................584 Bibliography 589 Index 629
|
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| spelling | Stillwell, John 1942- Verfasser (DE-588)128427264 aut Mathematics and its history John Stillwell Third edition New York ; Dordrecht ; Heidelberg ; London Springer [2010] © 2010 xxi, 660 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Geschichte gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Mathematik (DE-588)4037944-9 s Geschichte (DE-588)4020517-4 s DE-604 Geschichte z Erscheint auch als Online-Ausgabe 978-1-4419-6053-5 10.1007/978-1-4419-6053-5 Gefolgt von Concise edition 2020 978-3-030-55192-6 (DE-604)BV047888130 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020420995&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| title | Mathematics and its history |
| title_auth | Mathematics and its history |
| title_exact_search | Mathematics and its history |
| title_full | Mathematics and its history John Stillwell |
| title_fullStr | Mathematics and its history John Stillwell |
| title_full_unstemmed | Mathematics and its history John Stillwell |
| title_short | Mathematics and its history |
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| topic_facet | Mathematik Geschichte |
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