The Britannica Guide to Analysis and Calculus (Math Explained)

296 Pages·2010·5.4 MB·english

by  Erik Gregersen

Published in 2011 by Britannica Educational Publishing
(a trademark of Encyclopædia Britannica, Inc.)
in association with Rosen Educational Services, LLC
29 East 21st Street, New York, NY 10010.
Copyright © 2011 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica,
and the Thistle logo are registered trademarks of Encyclopædia Britannica, Inc. All
rights reserved.
Rosen Educational Services materials copyright © 2011 Rosen Educational Services, LLC.
All rights reserved.
Distributed exclusively by Rosen Educational Services.
For a listing of additional Britannica Educational Publishing titles, call toll free [phone].
First Edition
Britannica Educational Publishing
Michael I. Levy: Executive Editor
J.E. Luebering: Senior Manager
Marilyn L. Barton: Senior Coordinator, Production Control
Steven Bosco: Director, Editorial Technologies
Lisa S. Braucher: Senior Producer and Data Editor
Yvette Charboneau: Senior Copy Editor
Kathy Nakamura: Manager, Media Acquisition
Erik Gregersen: Associate Editor, Astronomy and Space Exploration
Rosen Educational Services
Hope Lourie Killcoyne: Senior Editor and Project Manager
Joanne Randolph: Editor
Nelson Sá: Art Director
Cindy Reiman: Photography Manager
Nicole Russo: Design
Matthew Cauli: Cover Design
Introduction by John Strazzabosco
Library of Congress Cataloging-in-Publication Data
The Britannica guide to analysis and calculus / edited by Erik Gregersen.
p. cm.—(Math explained)
“In association with Britannica Educational Publishing, Rosen Educational Services.”
Includes bibliographical references and index.
ISBN 978-1-61530-220-8 (eBook)
1. Mathematical analysis. 2. Calculus. I. Gregersen, Erik.
QA300.B6955 2011
515—dc22
2010001618
Cover Image Source/Getty Images
On page 12: In this engraving from Isaac Newton’s 18th-century manuscript De methodis
serierum et fluxionum, a hunter adjusts his aim as a group of ancient Greek mathematicians
explain his movements with algebraic formulas. SSPL via Getty Images
On page 20: High school calculus teacher Tom Moriarty writes a problem during a multi-
variable “post-AP” calculus class. Washington Post/Getty Images
On pages 21, 35, 48, 64, 81, 106, 207, 282, 285, 289: Solution of the problem of the
brachistochrone, or curve of quickest descent. The problem was first posed by Galileo,
re-posed by Swiss mathematician Jakob Bernoulli, and solved here by English mathemati-
cian and physicist Isaac Newton. Hulton Archive/Getty Images
C
ontents
Introduction 12
Chapter 1: Measuring
Continuous Change 21
Bridging the Gap Between Arithmetic
and Geometry 22
Discovery of the Calculus and the
Search for Foundations 24 23
Numbers and Functions 25
Number Systems 25
Functions 26 28
The Problem of Continuity 27
Approximations in Geometry 27
Infinite Series 29
The Limit of a Sequence 30
Continuity of Functions 31
Properties of the Real Numbers 32
Chapter 2: Calculus 35
Differentiation 35
Average Rates of Change 36
Instantaneous Rates of Change 36
Formal Definition of the Derivative 38
Graphical Interpretation 40
Higher-Order Derivatives 42
Integration 44
The Fundamental Theorem
of Calculus 44
Antidifferentiation 45
The Riemann Integral 46
Chapter 3: Differential Equations 48
Ordinary Differential Equations 48
Newton and Differential Equations 48 40
Newton’s Laws of Motion 48
Exponential Growth
56
and Decay 50
Dynamical Systems Theory
and Chaos 51
Partial Differential Equations 55
Musical Origins 55
Harmony 55
Normal Modes 55
Partial Derivatives 57
D’Alembert’s Wave Equation 58
Trigonometric Series Solutions 59
Fourier Analysis 62
Chapter 4: Other Areas of Analysis 64 82
Complex Analysis 64
Formal Definition of Complex
Numbers 65
Extension of Analytic Concepts to
Complex Numbers 66
Some Key Ideas of Complex Analysis 68
Measure Theory 70
Functional Analysis 73
Variational Principles and
Global Analysis 76
Constructive Analysis 78
Nonstandard Analysis 79 87
Chapter 5: History of Analysis 81
The Greeks Encounter Continuous
Magnitudes 81
The Pythagoreans and Irrational
Numbers 81
Zeno’s Paradoxes and the Concept
of Motion 83
The Method of Exhaustion 84
Models of Motion in Medieval Europe 85
Analytic Geometry 88
The Fundamental Theorem of Calculus 89
Differentials and Integrals 89
Discovery of the Theorem 91
Calculus Flourishes 94
Elaboration and Generalization 96
Euler and Infinite Series 96
Complex Exponentials 97
Functions 98
Fluid Flow 99
Rebuilding the Foundations 101
93
Arithmetization of Analysis 101
Analysis in Higher Dimensions 103
Chapter 6: Great Figures in
the History of Analysis 106
The Ancient and Medieval Period 106
Archimedes 106
Euclid 112
Eudoxus of Cnidus 115
Ibn al-Haytham 118
Nicholas Oresme 119
Pythagoras 122
Zeno of Elea 123
126
The 17th and 18th Centuries 125
Jean Le Rond d’Alembert 125
Isaac Barrow 129 135
Daniel Bernoulli 131
Jakob Bernoulli 133
Johann Bernoulli 134
Bonaventura Cavalieri 136
Leonhard Euler 137
Pierre de Fermat 140
James Gregory 144
Joseph-Louis Lagrange, comte
de l’Empire 147
Pierre-Simon, marquis de Laplace 150
Gottfried Wilhelm Leibniz 153
Colin Maclaurin 158
Sir Isaac Newton 159
Gilles Personne de Roberval 167
Brook Taylor 168
Evangelista Torricelli 169
John Wallis 170
The 19th and 20th Centuries 173
Stefan Banach 173
Bernhard Bolzano 175
Luitzen Egbertus Jan Brouwer 176
Augustin-Louis, Baron Cauchy 177 160
Richard Dedekind 179
Joseph, Baron Fourier 182
183
Carl Friedrich Gauss 185
David Hilbert 189
Andrey Kolmogorov 191
Henri-Léon Lebesgue 195
Henri Poincaré 196
Bernhard Riemann 200
Stephen Smale 203
Karl Weierstrass 205
Chapter 7: Concepts in Analysis
and Calculus 207
Algebraic Versus Transcendental
Objects 207 215
Argand Diagram 209
Bessel Function 209
Boundary Value 211
Calculus of Variations 212
Chaos Theory 214
Continuity 216
Convergence 217
Curvature 218
Derivative 220
Difference Equation 222
Differential 223
Differential Equation 223
Differentiation 226
Direction Field 227
Dirichlet Problem 228
Elliptic Equation 229
Exact Equation 230
Exponential Function 231
Extremum 233
Fluxion 234
Fourier Transform 234
Function 235 228
Harmonic Analysis 238
Harmonic Function 240
Infinite Series 241
Infinitesimals 243
Infinity 245
Integral 249
Integral Equation 250
Integral Transform 250
Integraph 251
Integration 251
Integrator 252
Isoperimetric Problem 253
232
Kernel 255
Lagrangian Function 255
Laplace’s Equation 256
Laplace Transform 257
Lebesgue Integral 258
Limit 259
Line Integral 260
Mean-Value Theorem 261
Measure 261
Minimum 263
Newton and Infinite Series 263
Ordinary Differential Equation 264
Orthogonal Trajectory 265 247
Parabolic Equation 266