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TITLE 4 COPYRIGHT 5 CONTENTS 6 INTRODUCTION 13 CHAPTER 1 MEASURING CONTINUOUS CHANGE 22 BRIDGING THE GAP BETWEEN ARITHMETIC AND GEOMETRY 23 DISCOVERY OF THE CALCULUS AND THE SEARCH FOR FOUNDATIONS 25 NUMBERS AND FUNCTIONS 26 NUMBER SYSTEMS 26 FUNCTIONS 27 THE PROBLEM OF CONTINUITY 28 APPROXIMATIONS IN GEOMETRY 28 INFINITE SERIES 30 THE LIMIT OF A SEQUENCE 31 CONTINUITY OF FUNCTIONS 32 PROPERTIES OF THE REAL NUMBERS 33 CHAPTER 2 CALCULUS 36 DIFFERENTIATION 36 AVERAGE RATES OF CHANGE 37 INSTANTANEOUS RATES OF CHANGE 37 FORMAL DEFINITION OF THE DERIVATIVE 39 GRAPHICAL INTERPRETATION 41 HIGHER-ORDER DERIVATIVES 43 INTEGRATION 45 THE FUNDAMENTAL THEOREM OF CALCULUS 45 ANTIDIFFERENTIATION 46 THE RIEMANN INTEGRAL 47 CHAPTER 3 DIFFERENTIAL EQUATIONS 49 ORDINARY DIFFERENTIAL EQUATIONS 49 NEWTON AND DIFFERENTIAL EQUATIONS 49 DYNAMICAL SYSTEMS THEORY AND CHAOS 52 PARTIAL DIFFERENTIAL EQUATIONS 56 MUSICAL ORIGINS 56 PARTIAL DERIVATIVES 58 D’ALEMBERT’S WAVE EQUATION 59 TRIGONOMETRIC SERIES SOLUTIONS 60 FOURIER ANALYSIS 63 CHAPTER 4 OTHER AREAS OF ANALYSIS 65 COMPLEX ANALYSIS 65 FORMAL DEFINITION OF COMPLEX NUMBERS 66 EXTENSION OF ANALYTIC CONCEPTS TO COMPLEX NUMBERS 67 SOME KEY IDEAS OF COMPLEX

Author

ERIK GREGERSEN

Language

EN