A First Course in Analysis

✍ Scribed by John B. Conway

• Publisher: Cambridge University Press
• Year: 2017
• Tongue: English
• Leaves: 358
• Series: Cambridge Mathematical Textbooks
• Edition: 1
• Category: Library

✦ Synopsis

This rigorous textbook is intended for a year-long analysis or advanced calculus course for advanced undergraduate or beginning graduate students. Starting with detailed, slow-paced proofs that allow students to acquire facility in reading and writing proofs, it clearly and concisely explains the basics of differentiation and integration of functions of one and several variables, and covers the theorems of Green, Gauss, and Stokes. Minimal prerequisites are assumed, and relevant linear algebra topics are reviewed right before they are needed, making the material accessible to students from diverse backgrounds. Abstract topics are preceded by concrete examples to facilitate understanding, for example, before introducing differential forms, the text examines low-dimensional examples. The meaning and importance of results are thoroughly discussed, and numerous exercises of varying difficulty give students ample opportunity to test and improve their knowledge of this difficult yet vital subject.

✦ Table of Contents

Cover
Half title
Series
Reviews
Title
Copyright
Dedication
Contents
Preface
1 The Real Numbers
1.1 Sets and Functions
1.2 The Real Numbers
1.3 Convergence
1.4 Series
1.5 Countable and Uncountable Sets
1.6 Open Sets and Closed Sets
1.7 Continuous Functions
1.8 Trigonometric Functions
2 Differentiation
2.1 Limits
2.2 The Derivative
2.3 The Sign of the Derivative
2.4 Critical Points
2.5 Some Applications
3 Integration
3.1 The Riemann Integral
3.2 The Fundamental Theorem of Calculus
3.3 The Logarithm and Exponential Functions
3.4 Improper Integrals
3.5 Sets of Measure Zero and Integrability
3.6 The Riemann–Stieltjes Integral
4 Sequences of Functions
4.1 Uniform Convergence
4.2 Power Series
5 Metric and Euclidean Spaces
5.1 Definitions and Examples
5.2 Sequences and Completeness
5.3 Open and Closed Sets
5.4 Continuity
5.5 Compactness
5.6 Connectedness
5.7 The Space of Continuous Functions
6 Differentiation in Higher Dimensions
6.1 Vector-valued Functions
6.2 Differentiable Functions, Part 1
6.3 Orthogonality
6.4 Linear Transformations
6.5 Differentiable Functions, Part 2
6.6 Critical Points
6.7 Tangent Planes
6.8 Inverse Function Theorem
6.9 Implicit Function Theorem
6.10 Lagrange Multipliers
7 Integration in Higher Dimensions
7.1 Integration of Vector-valued Functions
7.2 The Riemann Integral
7.3 Iterated Integration
7.4 Change of Variables
7.5 Differentiation under the Integral Sign
8 Curves and Surfaces
8.1 Curves
8.2 Green's Theorem
8.3 Surfaces
8.4 Integration on Surfaces
8.5 The Theorems of Gauss and Stokes
9 Differential Forms
9.1 Introduction
9.2 Change of Variables for Forms
9.3 Simplexes and Chains
9.4 Oriented Boundaries
9.5 Stokes's Theorem
9.6 Closed and Exact Forms
9.7 Denouement
Bibliography
Index of Terms
Index of Symbols