Calculus for Mathematicians, Computer Scientists, and Physicists꞉ An Introduction to Abstract Mathematics

"The aims of this book are various, but all stem from the author’s wish to present beautiful, interesting, living mathematics, as intuitively and informally as possible, without compromising logical rigor." Table of Contents List of Figures Preface The Language of Mathematics The Nature of Mathematics Sets and Operations Logic Calculus and the ``Real World'' Numbers Natural Numbers Integers Rational Numbers Real Numbers Complex Numbers Functions Basic Definitions Basic Classes of Functions Composition, Iteration, and Inverses Linear Operators Limits and Continuity Order of Vanishing Limits Continuity Sequences and Series Continuity on Intervals Uniform Continuity Extrema of Continuous Functions Continuous Functions and Intermediate Values Applications What is Calculus? Rates of Change Total Change Notation and Infinitesimals Integration Partitions and Sums Basic Examples Abstract Properties of the Integral Integration and Continuity Improper Integrals Differentiation The Derivative Derivatives and Local Behavior Continuity of the Derivative Higher Derivatives The Mean Value Theorem The Mean Value Theorem The Identity Theorem Differentiability of Inverse Functions The Second Derivative and Convexity Indeterminate Limits The Fundamental Theorems Integration and Differentiation Antidifferentiation Sequences of Functions Convergence Series of Functions Power Series Approximating Sequences Log and Exp The Natural Logarithm The Natural Exponential Properties of exp and log The Trigonometric Functions Sine and Cosine Auxiliary Trig Functions Inverse Trig Functions Geometric Definitions Taylor Approximation Numerical Approximation Function Approximation Elementary Functions A Short Course in Complex Analysis Elementary Antidifferentiation Postscript Bibliography Index