Buch, Englisch, 200 Seiten, Format (B × H): 183 mm x 260 mm, Gewicht: 556 g
Buch, Englisch, 200 Seiten, Format (B × H): 183 mm x 260 mm, Gewicht: 556 g
ISBN: 978-0-691-12533-6
Verlag: Princeton University Press
This is the first modern calculus book to be organized axiomatically and to survey the subject's applicability to science and engineering. A challenging exposition of calculus in the European style, it is an excellent text for a first-year university honors course or for a third-year analysis course. The calculus is built carefully from the axioms with all the standard results deduced from these axioms. The concise construction, by design, provides maximal flexibility for the instructor and allows the student to see the overall flow of the development. At the same time, the book reveals the origins of the calculus in celestial mechanics and number theory. The book introduces many topics often left to the appendixes in standard calculus textbooks and develops their connections with physics, engineering, and statistics. The author uses applications of derivatives and integrals to show how calculus is applied in these disciplines. Solutions to all exercises (even those involving proofs) are available to instructors upon request, making this book unique among texts in the field. Focuses on single variable calculus Provides a balance of precision and intuition Offers both routine and demanding exercises
Autoren/Hrsg.
Fachgebiete
Weitere Infos & Material
Preface xi
Acknowledgments xiii
Chapter 1: Functions on Sets
1.1 Sets 1
1.2 Functions 2
1.3 Cardinality 5
Exercises 6
Chapter 2: The Real Numbers
2.1 The Axioms 12
2.2 Implications 14
2.3 Latter-Day Axioms 16
Exercises 16
Chapter 3: Metric Properties
3.1 The Real Line 19
3.2 Distance 20
3.3 Topology 21
3.4 Connectedness 22
3.5 Compactness 23
Exercises 27
Chapter 4: Continuity
4.1 The Definition 30
4.2 Consequences 31
4.3 Combinations of Continuous Functions 33
4.4 Bisection 36
4.5 Subspace Topology 37
Exercises 38
Chapter 5: Limits and Derivatives
5.1 Limits 41
5.2 The Derivative 43
5.3 Mean Value Theorem 46
5.4 Derivatives of Inverse Functions 48
5.5 Derivatives of Trigonometric Functions 50
Exercises 53
Chapter 6: Applications of the Derivative
6.1 Tangents 60
6.2 Newton?s Method 63
6.3 Linear Approximation and Sensitivity 65
6.4 Optimization 66
6.5 Rate of Change 67
6.6 Related Rates 68
6.7 Ordinary Differential Equations 69
6.8 Kepler?s Laws 71
6.9 Universal Gravitation 73
6.10 Concavity 76
6.11 Differentials 79
Exercises 80
Chapter 7: The Riemann Integral
7.1 Darboux Sums 89
7.2 The Fundamental Theorem of Calculus 91
7.3 Continuous Integrands 92
7.4 Properties of Integrals 94
7.5 Variable Limits of Integration 95
7.6 Integrability 96
Exercises 97
Chapter 8: Applications of the Integral
8.1 Work 100
8.2 Area 102
8.3 Average Value 104
8.4 Volumes 105
8.5 Moments 106
8.6 Arclength 109
8.7 Accumulating Processes 110
8.8 Logarithms 110
8.9 Methods of Integration 112
8.10 Improper Integrals 113
8.11 Statistics 115
8.12 Quantum Mechanics 117
8.13 Numerical Integration 118
Exercises 121
Chapter 9: Infinite Series
9.1 Zeno?s Paradoxes 134
9.2 Convergence of Sequences 134
9.3 Convergence of Series 136
9.4 Convergence Tests for Positive Series 138
9.5 Convergence Tests for Signed Series 140
9.6 Manipulating Series 142
9.7 Power Series 145
9.8 Convergence Tests for Power Series 147
9.9 Manipulation of Power Series 149
9.10 Taylor Series 151
Exercises 154
References 163
Index 165
