By Erik Gregersen
- Publisher: Rosen Educational Publishing
- Number Of Pages: 304
- Publication Date: 2010-08-15
- ISBN-10 / ASIN: 1615301232
- ISBN-13 / EAN: 9781615301232
The dynamism of the natural world means that it is constantly changing, sometimes rapidly, sometimes gradually. By mathematically interpreting the continuous change that characterizes so many natural processes, analysis and calculus have become indispensable to bridging the divide between mathematics and the sciences. This comprehensive volume examines the key concepts of calculus, providing students with a robust understanding of integration and differentiation. Biographies of important figures will leave readers with an increased appreciation for the sometimes competing theories that informed the early history of the field.
Contents
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
Numbers and Functions 25
Number Systems 25
Functions 26
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
Newton’s Laws of Motion 48
Exponential Growth 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
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
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
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
The 17th and 18th Centuries 125
Jean Le Rond d’Alembert 125
Isaac Barrow 129
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
Richard Dedekind 179
Joseph, Baron Fourier 182
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
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
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
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
Parabolic Equation 266
Partial Differential Equation 267
Planimeter 269
Power Series 269
Quadrature 271
Separation of Variables 271
Singular Solution 272
Singularity 273
Special Function 274
Spiral 276
Stability 278
Sturm-Liouville Problem 279
Taylor Series 280
Variation of Parameters 280
Glossary 282
Bibliography 285
Index 289
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