MATHEMATICAL BACKGROUND: FOUNDATIONS OF INFINITESIMAL CALCULUS
CONTENTS PART NUMBERS AND FUNCTIONS CHAPTER 1 NUMBERS 11 FIELD AXIOMS 12 ORDER AXIOMS 13 THE COMPLETENESS AXIOM 14 SMALL, MEDIUM AND LARGE NUMBERS CHAPTER 2 FUNCTIONAL IDENTITIES 21 SPECI?C FUNCTIONAL IDENTITIES 22 GENERAL FUNCTIONAL IDENTITIES 23 THE FUNCTION EXTENSION AXIOM 24 ADDITIVE FUNCTIONS 25 THE MOTION OF A PENDULUM PART LIMITS CHAPTER 3 THE THEORY OF LIMITS 31 PLAIN LIMITS 32 FUNCTION LIMITS 33 COMPUTATION OF LIMITS CHAPTER 4 CONTINUOUS FUNCTIONS 41 UNIFORM CONTINUITY 42 THE EXTREME VALUE THEOREM 43 BOLZANO’S INTERMEDIATE VALUE THEOREM 1 VARIABLE DI?ERENTIATION CHAPTER 5 THE THEORY OF DERIVATIVES 51 THE FUNDAMENTAL THEOREM: PART 511 RIGOROUS IN?NITESIMAL JUSTI?CATION 512 RIGOROUS LIMIT JUSTI?CATION 52 DERIVATIVES, EPSILONS AND DELTAS 53 SMOOTHNESS ?CONTINUITY OF FUNCTION AND DERIVATIVE 54 RULES ?SMOOTHNESS 55 THE INCREMENT AND INCREASING 56 INVERSE FUNCTIONS AND DERIVATIVES CHAPTER 6 POINTWISE DERIVATIVES 61 POINTWISE LIMITS 62 POINTWISE DERIVATIVES 63 POINTWISE DERIVATIVES AREN’T ENOUGH FOR INVERSES CHAPTER 7 THE MEAN VALUE THEOREM 71 THE MEAN VALUE THEOREM 72 DARBOUX’S THEOREM 73 CONTINUOUS POINTWISE DERIVATIVES ARE UNIFORM CHAPTER 8 HIGHER ORDER DERIVATIVES 81 TAYLOR’S FORMULA AND BENDING 82 SYMMETRIC DI?ERENCES AND TAYLOR’S FORMULA 83 APPROXIMATION OF SECOND DERIVATIVES 84 THE GENERAL TAYLOR SMALL OH FORMULA 841 THE CONVERSE OF TAYLOR’S THEOREM 85 DIRECT INTERPRETATION OF HIGHER ORDER DERIVATIVES 851 BASIC THEORY OF INTERPOLATION 852 INTERPOLATION WHERE F IS SMOOTH 853 SMOOTHNESS FROM DI?ERENCES INTEGRATION CHAPTER 9 BASIC THEORY OF THE DE?NITE INTEGRAL 91 EXISTENCE OF THE INTEGRAL 92 YOU CAN’T ALWAYS INTEGRATE DISCONTINUOUS FUNCTIONS 93 FUNDAMENTAL THEOREM: PART 94 IMPROPER INTEGRALS 941 COMPARISON OF IMPROPER INTEGRALS 942 A FINITE FUNNEL WITH IN?NITE AREA? MULTIVARIABLE DI?ERENTIATION CHAPTER 10 DERIVATIVES OF MULTIVARIABLE FUNCTIONS DI?ERENTIAL EQUATIONS CHAPTER 11 THEORY OF INITIAL VALUE PROBLEMS 111 EXISTENCE AND UNIQUENESS OF SOLUTIONS 112 LOCAL LINEARIZATION OF DYNAMICAL SYSTEMS 113 ATTRACTION AND REPULSION 114 STABLE LIMIT CYCLES IN?NITE SERIES CHAPTER 12 THE THEORY OF POWER SERIES 121 UNIFORMLY CONVERGENT SERIES 122 ROBINSON’S SEQUENTIAL LEMMA 123 INTEGRATION OF SERIES 124 RADIUS OF CONVERGENCE 125 CALCULUS OF POWER SERIES CHAPTER 13 THE THEORY OF FOURIER SERIES 131 COMPUTATION OF FOURIER SERIES 132 CONVERGENCE FOR PIECEWISE SMOOTH FUNCTIONS