Advanced Calculus An Introduction to Linear Analysis 1st edition by Leonard Richardson – Ebook PDF Instant Download/Delivery: 1118030672, 9781118030677
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ISBN 10: 1118030672
ISBN 13: 9781118030677
Author: Leonard Richardson
Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers’ transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis. Following an introduction dedicated to writing proofs, the book is divided into three parts: Part One explores foundational one-variable calculus topics from the viewpoint of linear spaces, norms, completeness, and linear functionals. Part Two covers Fourier series and Stieltjes integration, which are advanced one-variable topics. Part Three is dedicated to multivariable advanced calculus, including inverse and implicit function theorems and Jacobian theorems for multiple integrals. Numerous exercises guide readers through the creation of their own proofs, and they also put newly learned methods into practice. In addition, a “Test Yourself” section at the end of each chapter consists of short questions that reinforce the understanding of basic concepts and theorems. The answers to these questions and other selected exercises can be found at the end of the book along with an appendix that outlines key terms and symbols from set theory. Guiding readers from the study of the topology of the real line to the beginning theorems and concepts of graduate analysis, Advanced Calculus is an ideal text for courses in advanced calculus and introductory analysis at the upper-undergraduate and beginning-graduate levels. It also serves as a valuable reference for engineers, scientists, and mathematicians.
Advanced Calculus An Introduction to Linear Analysis 1st Table of contents:
Introduction
PART I: ADVANCED CALCULUS IN ONE VARIABLE �
Chapter 1: Real Numbers and Limits of Sequences
1.1 The Real Number System
1.2 Limits of Sequences & Cauchy Sequences
1.3 The Completeness Axiom and Some Consequences
1.4 Algebraic Combinations of Sequences
1.5 The Bolzano-weierstrass Theorem
1.6 The Nested Intervals Theorem
1.7 The Heine-borel Covering Theorem
1.8 Countability of the Rational Numbers
1.9 Test Yourself
Chapter 2: Continuous Functions
2.1 Limits of Functions
2.2 Continuous Functions
2.3 Some Properties of Continuous Functions
2.4 Extreme Value Theorem and Its Consequences
2.5 The Banach Space C[a, b]
2.6 Test Yourself
Chapter 3: Riemann Integral
3.1 Definition and Basic Properties
3.2 The Darboux Integrability Criterion
3.3 Integrals of Uniform Limits
3.4 The Cauchy-schwarz Inequality
3.5 Test Yourself
Chapter 4: The Derivative
4.1 Derivatives and Differentials
4.2 The Mean Value Theorem
4.3 The Fundamental Theorem of Calculus
4.4 Uniform Convergence and the Derivative
4.5 Cauchy’s Generalized Mean Value Theorem
4.6 Taylor’s Theorem
4.7 Test Yourself
Chapter 5: Infinite Series
5.1 Series of Constants
5.2 Convergence Tests for Positive Term Series
5.3 Absolute Convergence and Products of Series
5.4 The Banach Space L1 and Its Dual Space
5.5 Series of Functions: the Weierstrass M-test
5.6 Power Series
5.7 Real Analytic Functions and C8 Functions
5.8 Weierstrass Approximation Theorem
5.9 Test Yourself
PART II: ADVANCED TOPICS IN ONE VARIABLE
Chapter 6: Fourier Series
6.1 The Vibrating String and Trigonometric Series
6.2 Euler’s Formula and the Fourier Transform
6.3 Bessel’s Inequality and L2
6.4 Uniform Convergence & Riemann Localization
6.5 L2-convergence & the Dual of L2
6.6 Test Yourself
Chapter 7: The Riemann-stieltjes Integral
7.1 Functions of Bounded Variation
7.2 Riemann-stielt Jes Sums and Integrals
7.3 Riemann Stielt Jes Integrability Theorems
7.4 The Riesz Representation Theorem
7.5 Test Yourself
PART III: ADVANCED CALCULUS IN SEVERAL VARIABLES
Chapter 8: Euclidean Space
8.1 Euclidean Space as a Complete Norm Ed Vector Space
8.2 Open Sets and Closed Sets
8.3 Compact Sets
8.4 Connected Sets
8.5 Test Yourself
Chapter 9: Continuous Functions on Euclidean Space
9.1 Limits of Functions
9.2 Continuous Functions
9.3 Continuous Image of a Compact Set
9.4 Continuous Image of a Connected Set
9.5 Test Yourself
Chapter 10: The Derivative in Euclidean Space
10.1 Linear Transformations and Norms
10.2 Differentiable Functions
10.3 The Chain Rule in Euclidean Space
10.3.1 The Mean Value Theorem
10.3.2 Taylor’s Theorem
10.4 Inverse Functions
10.5 Implicit Functions
10.6 Tangent Spaces and Lagrange Multipllers
10.7 Test Yourself
Chapter 11: Riemann Integration in Euclidean Space
11.1 Definition of the Integral
11.2 Lebesgue Null Sets and Jordan Null Sets
11.3 Lebesgue’s Criterion for Riemann Integrability
11.4 Fubini’s Theorem
11.5 Jacobian Theorem for Change of Variables
11.6 Test Yourself
Appendix A: Set Theory
A.1 Terminology and Symbols
A.2 Paradoxes
Problem Solutions
References
Index
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Tags: Leonard Richardson, Advanced Calculus, Linear Analysis