دانلود کتاب A Friendly Introduction to Analysis

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Cover S Title A Friendly Introduction to Analysis Copyright © 2004, 1999 Pearson Education ISBN 0-13-045796-5 Dedication Contents Preface 1 Purpose and Background 2 Design and Organization 3 Supplements 4 How the 2nd Edition Differs From the 1st Edition 5 Acknowledgments 1 Introduction 1.1 * Algebra of Sets Exercises 1.1 1.2* Relations and Functions Exercises 1.2 1.3* Mathematical Induction Exercises 1.3 1.4* Proof Techniques Exercises 1.4 1.5* Inverse Functions Exercises 1.5 1.6* Finite and Infinite Sets Exercises 1.6 1.7* Ordered Field and a Real Number System Exercises 1.7 1.8* Some Properties of Real Numbers Exercises 1.8 1,9* Review 1.10* Projects Part 1. Fibonacci Numbers Part 2. Lucas Numbers Part 3. Mean of Real Numbers 2 Sequences 2.1 Convergence Exercises 2.1 2.2 Limit Theorems Exercises 2.2 2.3 Infinite Limits Exercises 2.3 2.4 Monotone Sequences Exercises 2.4 2.5 Cauchy Sequences Exercises 2.5 2.6 Subsequences Exercises 2.6 2.7* Review 2.8* Projects Part 1. The Transcendental Number e Part 2. Summable Sequences 3 Limits of Functions 3.1 Limit at Infinity Exercises 3.1 3.2 Limit at a Real Number Exercises 3.2 3.3 Sided Limits Exercises 3.3 3.4* Review 3.5* Projects Part 1. Monotone Functions Part 2. Continued Fractions 4 Continuity 4.1 Continuity of a Function Exercises 4.1 4.2* Discontinuity of a Function Exercises 4.2 4.3 Properties of Continuous Functions Exercises 4.3 4.4 Uniform Continuity Exercises 4.4 4.5* Review 4.6* Projects Part 1. Compact Sets Part 2. Multiplicative, Subadditive, and Additive Functions 5 Differentiation 5.1 Derivative of a Function Exercises 5.1 5.2 Properties of Differentiable Functions Exercises 5.2 5.3 Mean Value Theorems Exercises 5.3 5.4 Higher-Order Derivatives Exercises 5.4 5.5 * L'Hopital's Rule Exercises 5.5 5.6 * Review 5.7 * Projects Part 1. Approximation of Derivatives Part 2. Lipschitz Condition Part 3. Functions of Bounded Variation Part 4. Absolutely Continuous Functions Part 5. Convex Functions 6 Integration 6.1 Riemann Integral Exercises 6.1 6.2 Integrable Functions Exercises 6.2 6.3 Properties of the Riemann Integral Exercises 6.3 6.4 Integration in Relation to Differentiation Exercises 6.4 6.5 Improper Integral Exercises 6.5 6.6 * Special Functions Exercises 6.6 6.7 * Review 6.8 * Projects Part 1. Wallis 's Formula Part 2. Euler's Summation Formula Part 3. Laplace Transforms Part 4. Inverse Laplace Transforms 7 Infinite Series 7.1 Convergence Exercises 7.1 7.2 Tests for Convergence Exercises 7.2 7.3 Ratio and Root Tests Exercises 7.3 7.4 Absolute and Conditional Convergence Exercises 7.4 7.5* Review 7.6 * Projects Part 1. Summation by Parts Part 2. Multiplication of Series Part 3. Infinite Products Part 4. Cantor Set 8 Sequences and Series of Functions 8.1 Pointwise Convergence Exercises 8.1 8.2 Uniform Convergence Exercises 8.2 8.3 Properties of Uniform Convergence Exercises 8.3 8.4 Pointwise and Uniform Convergence of Series Exercises 8.4 8.5 Power Series Exercises 8.5 8.6 Taylor Series Exercises 8.6 8.7 * Review 8.8 * Projects Part 1. Limit Superior Part 2. Irrationality of e Part 3. An Everywhere Continuous but Nowhere Differentiable Function Part 4. Equicontinuity 9 Vector Calculus 9.1* Cartesian Coordinates in R^3 Exercises 9.1 9.2* Vectors in R^3 Exercises 9.2 9.3* Dot Product and Cross Product Exercises 9.3 9.4 Parametric Equations Exercises 9.4 9.5* Lines and Planes in R^3 Exercises 9.5 9.6 Vector-Valued Functions Exercises 9.6 9.7 Arc Length Exercises 9.7 9.8 * Review 9.9 * Projects Part 1. Inner Product Part 2. Polar Coordinates Part 3. Cantor Function 10 Functions of two Variables 10.1 Basic Topology Exercises 10.1 10.2 Limits and Continuity Exercises 10.2 10.3 Partial Derivatives Exercises 10.3 10.4 Differentiation Exercises 10.4 10.5 Directional Derivative Exercises 10.5 10.6 Chain Rule Exercises 10.6 10.7* Review 10.8* Projects Part 1. Operator Method for Solving Differential Equations Part 2. Separable and Homogeneous First-Order Differential Equations 11 Multiple Integration 11.1 Double Integral Exercises 11.1 11.2 Iterated Integrals Exercises 11.2 11.3 Integrals Over General Regions Exercises 11.3 11.4 Line Integrals Exercises 11.4 11.5 Vector Fields and Work Integrals Exercises 11.5 11.6 Gradient Vector Field Exercises 11.6 11.7 Green's Theorem Exercises 11.7 11.8 * Stokes's and Gauss's Theorems Exercises 11.8 11.9* Review 11.10* Projects Part 1. Change of Variables for Double Integrals Part 2. Exact Equations Hints and Solutions to Selected Exercises Greek Alphabet Index of Symbols Index