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Contents

Chapter 4. Higher Dimensions 43

§4.1. Bolzano-Weierstrass in Higher Dimensions 43

§4.2. Open and closed sets 48

§4.3. A central theorem of analysis 56

§4.4. The mean value theorem 59

§4.5. Uniform continuity 64

§4.6. The general principle of convergence 66

Chapter 5. Sums and Suchlike V 73

§5.1. Comparison tests W 73

§5.2. Conditional convergence s ? 75

§5.3. Interchanging limits 9 80

§5.4. The exponential function s ? 88

§5.5. The trigonometric functions 9 95

§5.6. The logarithm V 99

§5.7. Powers 9 105

§5.8. The fundamental theorem of algebra 9 109

Chapter 6. Differentiation 117

§6.1. Preliminaries 117

§6.2. The operator norm and the chain rule 123

§6.3. The mean value inequality in higher dimensions 130

Chapter 7. Local Taylor Theorems 135

§7.1. Some one-dimensional Taylor theorems 135

§7.2. Some many-dimensional local Taylor theorems 139

§7.3. Critical points 147

Chapter 8. The Riemann Integral 161

§8.1. Where is the problem ? 161

§8.2. Riemann integration 164

§8.3. Integrals of continuous functions 173

§8.4. First steps in the calculus of variations 9 181

§8.5. Vector-valued integrals 192