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Description:

Advanced Calculus highlights the connections between calculus and linear algebra and provides a mathematically sophisticated introduction to functional analytic concepts. The book stresses that proofs must be written down, scrutinized step-by-step and rewritten whenever there is doubt. Unlike the competition, this book approaches the rigorous foundations of calculus in a manner that reorients thinking in the directions taken by modern analysis.

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Book details

List price: $154.00 Copyright year: 2008 Publisher: John Wiley & Sons, Incorporated Publication date: 4/25/2008 Binding: Hardcover Pages: 416 Size: 6.00" wide x 9.25" long x 1.00" tall Weight: 1.804 Language: English

AuthorTable of Contents

Leonard F. Richardson, PhD, is Herbert Huey McElveen Professor and Director of Graduate Studies in Mathematics at Louisiana State University, where he is also Assistant Chair of the Department of Mathematics. Dr. Richardson's research interests include harmonic analysis, homogeneous spaces, and representation theory. He is the author of Advanced Calculus: An Introduction to Linear Analysis, also published by Wiley.

Preface

Acknowledgments

Introduction

Advanced Calculus in One Variable

Real Numbers and Limits of Sequences

The Real Number System

Exercises

Limits of Sequences & Cauchy Sequences

Exercises

The Completeness Axiom and Some Consequences

Exercises

Algebraic Combinations of Sequences

Exercises

The Bolzano-Weierstrass Theorem

Exercises

The Nested Intervals Theorem

Exercises

The Heine-Borel Covering Theorem

Exercises

Countability of the Rational Numbers

Exercises

Test Yourself

Exercises

Continuous Functions

Limits of Functions

Exercises

Continuous Functions

Exercises

Some Properties of Continuous Functions

Exercises

Extreme Value Theorem and Its Consequences

Exercises

The Banach Space C[a, b]

Exercises

Test Yourself

Exercises

Riemann Integral

Definition and Basic Properties

Exercises

The Darboux Integrability Criterion

Exercises

Integrals of Uniform Limits

Exercises

The Cauchy-Schwarz Inequality

Exercises

Test Yourself

Exercises

The Derivative

Derivatives and Differentials

Exercises

The Mean Value Theorem

Exercises

The Fundamental Theorem of Calculus

Exercises

Uniform Convergence and the Derivative

Exercises

Cauchy's Generalized Mean Value Theorem

Exercises

Taylor's Theorem

Exercises

Test Yourself

Exercises

Infinite Series

Series of Constants

Exercises

Convergence Tests for Positive Term Series

Exercises

Absolute Convergence and Products of Series

Exercises

The Banach Space l[subscript 1] and Its Dual Space

Exercises

Series of Functions: The Weierstrass M-Test

Exercises

Power Series

Exercises

Real Analytic Functions and C[superscript infinity] Functions

Exercises

Weierstrass Approximation Theorem

Exercises

Test Yourself

Exercises

Advanced Topics in one Variable

Fourier Series

The Vibrating String and Trigonometric Series

Exercises

Euler's Formula and the Fourier Transform

Exercises

Bessel's Inequality and l[subscript 2]

Exercises

Uniform Convergence & Riemann Localization

Exercises

L[superscript 2]-Convergence & the Dual of l[superscript 2]

Exercises

Test Yourself

Exercises

The Riemann-Stieltjes Integral

Functions of Bounded Variation

Exercises

Riemann-Stieltjes Sums and Integrals

Exercises

Riemann-Stieltjes Integrability Theorems

Exercises

The Riesz Representation Theorem

Exercises

Test Yourself

Exercises

Advanced Calculus in Several Variables

Euclidean Space

Euclidean Space as a Complete Normed Vector Space

Exercises

Open Sets and Closed Sets

Exercises

Compact Sets

Exercises

Connected Sets

Exercises

Test Yourself

Exercises

Continuous Functions on Euclidean Space

Limits of Functions

Exercises

Continuous Functions

Exercises

Continuous Image of a Compact Set

Exercises

Continuous Image of a Connected Set

Exercises

Test Yourself

Exercises

The Derivative in Euclidean Space

Linear Transformations and Norms

Exercises

Differentiable Functions

Exercises

The Chain Rule in Euclidean Space

The Mean Value Theorem

Taylor's Theorem

Exercises

Inverse Functions

Exercises

Implicit Functions

Exercises

Tangent Spaces and Lagrange Multipliers

Exercises

Test Yourself

Exercises

Riemann Integration in Euclidean Space

Definition of the Integral

Exercises

Lebesgue Null Sets and Jordan Null Sets

Exercises

Lebesgue's Criterion for Riemann Integrability

Exercises

Fubini's Theorem

Exercises

Jacobian Theorem for Change of Variables

Exercises

Test Yourself

Exercises

Set Theory

Terminology and Symbols

Exercises

Paradoxes

Problem Solutions

References

Index

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