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

✍ Scribed by Andrew D. Hwang

• Year: 2008
• Tongue: English
• Leaves: 487
• Category: Library

✦ Synopsis

"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

Table of Contents
List of Figures
Preface
1. The Language of Mathematics
1.1 The Nature of Mathematics
1.2 Sets and Operations
1.3 Logic
1.4 Calculus and the “Real World”
2. Numbers
2.1 Natural Numbers
2.2 Integers
2.3 Rational Numbers
2.4 Real Numbers
2.5 Complex Numbers
3. Functions
3.1 Basic Definitions
3.2 Basic Classes of Functions
3.3 Composition, Iteration, and Inverses
3.4 Linear Operators
4. Limits and Continuity
4.1 Order of Vanishing
4.2 Limits
4.3 Continuity
4.4 Sequences and Series
5. Continuity on Intervals
5.1 Uniform Continuity
5.2 Extrema of Continuous Functions
5.3 Continuous Functions and Intermediate Values
5.4 Applications
6. What is Calculus?
6.1 Rates of Change
6.2 Total Change
6.3 Notation and Infinitesimals
7. Integration
7.1 Partitions and Sums
7.2 Basic Examples
7.3 Abstract Properties of the Integral
7.4 Integration and Continuity
7.5 Improper Integrals
8. Differentiation
8.1 The Derivative
8.2 Derivatives and Local Behavior
8.3 Continuity of the Derivative
8.4 Higher Derivatives
9. The Mean Value Theorem
9.1 The Mean Value Theorem
9.2 The Identity Theorem
9.3 Differentiability of Inverse Functions
9.4 The Second Derivative and Convexity
9.5 Indeterminate Limits
10. The Fundamental Theorems
10.1 Integration and Differentiation
10.2 Antidifferentiation
11. Sequences of Functions
11.1 Convergence
11.2 Series of Functions
11.3 Power Series
11.4 Approximating Sequences
12. Log and Exp
12.1 The Natural Logarithm
12.2 The Natural Exponential
12.3 Properties of exp and log
13. The Trigonometric Functions
13.1 Sine and Cosine
13.2 Auxiliary Trig Functions
13.3 Inverse Trig Functions
13.4 Geometric Definitions
14. Taylor Approximation
14.1 Numerical Approximation
14.2 Function Approximation
15. Elementary Functions
15.1 A Short Course in Complex Analysis
15.2 Elementary Antidifferentiation
Postscript
Bibliography
Index