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ISBN13: 978-0821847916

ISBN10: 0821847910 Edition: 2ND 06

Copyright: 2006

Publisher: Brooks/Cole Publishing Co.

Published: 2006

International: No

ISBN10: 0821847910 Edition: 2ND 06

Copyright: 2006

Publisher: Brooks/Cole Publishing Co.

Published: 2006

International: No

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ADVANCED CALCULUS rigorously presents the fundamental concepts of mathematical analysis in the clearest, simplest way, within the context of illuminating examples and stimulating exercises. Emphasizing the unity of the subject, the text shows that mathematical analysis is not a collection of isolated facts and techniques, but rather a coherent body of knowledge. Beyond the intrinsic importance of the actual subject, the author demonstrates that the study of mathematical analysis instills habits of thought that are essential for a proper understanding of many areas of pure and applied mathematics. Students gain a precise understanding of the subject, together with an appreciation of its coherence and significance. The full book is suitable for a year-long course; the first nine chapters are suitable for a one-term course on functions of a single variable. This book is included in the Brooks/Cole Series in Advanced Mathematics.

Benefits:

- NEW! Many changes, some small and some large, have been made in the proofs to emphasize even more strongly the following point of view: Mathematics students must learn to read mathematics, not just listen to it in class, and this book gives them an opportunity to do so. This is a course for students who are learning the art of proof. When the students read a proof, they should feel they understand how mathematicians discovered the proof and so the text avoids 'rabbit out of the hat' proofs. The proofs seem natural to the students and encourage them to believe they could have discovered the proofs.
- NEW! There has been a fundamental conceptual clarification in the development of the material on the structure of the real numbers and the integral.
- NEW! Many new exercises, of varying degrees of difficulty, have been added. A Solutions Manual is available with detailed solutions to selected problems.
- NEW! A number of non-essential concepts have been eliminated and optional material has been placed in a way to make it possible to cover the essential material more easily and fluidly.
- The text's presentation is conversational and mathematically precise.
- In addition to the essential topics, the author includes important topics such as the approximation methods for estimating integrals, the Weierstrass Approximation Theorem, and metric spaces, without disturbing the coherence of the course.
- A wide variety of exercises helps students gain a genuine understanding of the material. The more challenging problems often stimulate the student to carefully reread the relevant sections in order to properly assimilate the material.

**1. TOOLS FOR ANALYSIS.**

The Completeness Axiom and Some of Its Consequences. The Distribution of the Integers and the Rational Numbers. Inequalities and Identities.

**2. CONVERGENT SEQUENCES.**

The Convergence of Sequences. Sequences and Sets. The Monotone Convergence Theorem. The Sequential Compactness Theorem. Covering Properties of Sets.

**3. CONTINUOUS FUNCTIONS.**

Continuity. The Extreme Value Theorem. The Intermediate Value Theorem. Uniform Continuity. The Epsilon-Delta Criterion for Continuity. Images and Inverses; Monotone Functions. Limits.

**4. DIFFERENTIATION.**

The Algebra of Derivatives. Differentiating Inverses and Compositions. The Mean Value Theorem and Its Geometric Consequences. The Cauchy Mean Value Theorem and Its Analytic Consequences. The Notation of Leibnitz.

**5. ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS.**

Solutions of Differential Equations. The Natural Logarithm and the Exponential Functions. The Trigonometric Functions. The Inverse Trigonometric Functions.

**6. INTEGRATION: TWO FUNDAMENTAL THEOREMS.**

Darboux Sums; Upper and Lower Integrals. The Archimedes-Riemann Theorem. Additivity, Monotonicity and Linearity. Continuity and Integrability. The First Fundamental Theorem: Integrating Derivatives. The Second Fundamental Theorem: Differentiating Integrals.

**7. INTEGRATION: FURTHER TOPICS.**

Solutions of Differential Equations. Integration by Parts and by Substitution. The Convergence of Darboux and Riemann Sums. The Approximation of Integrals.

**8. APPROXIMATION BY TAYLOR POLYNOMIALS.**

Taylor Polynomials. The Lagrange Remainder Theorem. The Convergence of Taylor Polynomials. A Power Series for the Logarithm. The Cauchy Integral Remainder Theorem. A Non-Analytic, Infinitely Differentiable Function. The Weierstrass Approximation Theorem.

**9. SEQUENCES AND SERIES OF FUNCTIONS.**

Sequences and Series of Numbers. Pointwise Convergence of Sequences of Functions. Uniform Convergence of Sequences of Functions. The Uniform Limit of Functions. Power Series. A Continuous, Nowhere Differentiable Function.

**10. THE EUCLIDEAN SPACE Rn.**

The Linear Structure of Rn and the Scalar Product. Convergence of Sequences in Rn. Open Sets and Closed Sets in Rn.

**11. CONTINUITY, COMPACTNESS, AND CONNECTEDNESS.**

Continuous Functions and Mappings. Sequential Compactness, Extreme Values and Uniform Continuity. Pathwise Connectedness and the Intermediate Value Theorem. Connectedness and the Intermediate Value Property.

**12. METRIC SPACES.**

Open Sets, Closed Sets, and Sequential Convergence. Completeness and the Contraction Mapping Principle. The Existence Theorem for Nonlinear Differential Equations. Continuous Mappings Between Metric Spaces. Sequentially Compactness and Connectedness.

**13. DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES.**

Limits. Partial Derivatives. The Mean Value Theorem and Directional Derivatives.

**14. LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS.**

First-Order Approximation, Tangent Planes, and Affine Functions. Quadratic Functions, Hessian Matrices, and Second Derivatives. Second-Order Approximations and the Second-Derivative Test.

**15. APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS.**

Linear Mappings and Matrices. The Derivative Matrix and the Differential. The Chain Rule.

**16. IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM.**

Functions of a Single Variable and Maps in the Plane. Stability of Nonlinear Mappings. A Minimization Principle and the General Inverse Function Theorem.

**17. THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS.**

A Scalar Equation in Two Unknowns: Dini's Theorem. The General Implicit Function Theorem. Equations of Surfaces and Curves in R³. Constrained Extrema Problems and Lagrange Multipliers.

**18. INTEGRATING FUNCTIONS OF SEVERAL VARIABLES.**

Integration of Functions on Generalized Rectangles. Continuity and Integrability. Integration over Jordan Domains.

**19. ITERATED INTEGRATION AND CHANGES OF VARIABLES.**

Fubini's Theorem. The Change of Variables Theorem: Statements and Examples. Proof of the Change of Variables Theorem.

**20. LINE AND SURFACE INTEGRALS.**

Arclength and Line Integrals. Surface Area and Surface Integrals. The Integral Formulas of Green and Stokes.

Appendix A: Consequences of the Field and Positivity Axioms.

The Field Axioms and Their Consequences. The Positivity Axioms and Their Consequences.

Appendix B: Linear Algebra.

ISBN10: 0821847910 Edition: 2ND 06

Copyright: 2006

Publisher: Brooks/Cole Publishing Co.

Published: 2006

International: No

ADVANCED CALCULUS rigorously presents the fundamental concepts of mathematical analysis in the clearest, simplest way, within the context of illuminating examples and stimulating exercises. Emphasizing the unity of the subject, the text shows that mathematical analysis is not a collection of isolated facts and techniques, but rather a coherent body of knowledge. Beyond the intrinsic importance of the actual subject, the author demonstrates that the study of mathematical analysis instills habits of thought that are essential for a proper understanding of many areas of pure and applied mathematics. Students gain a precise understanding of the subject, together with an appreciation of its coherence and significance. The full book is suitable for a year-long course; the first nine chapters are suitable for a one-term course on functions of a single variable. This book is included in the Brooks/Cole Series in Advanced Mathematics.

Benefits:

- NEW! Many changes, some small and some large, have been made in the proofs to emphasize even more strongly the following point of view: Mathematics students must learn to read mathematics, not just listen to it in class, and this book gives them an opportunity to do so. This is a course for students who are learning the art of proof. When the students read a proof, they should feel they understand how mathematicians discovered the proof and so the text avoids 'rabbit out of the hat' proofs. The proofs seem natural to the students and encourage them to believe they could have discovered the proofs.
- NEW! There has been a fundamental conceptual clarification in the development of the material on the structure of the real numbers and the integral.
- NEW! Many new exercises, of varying degrees of difficulty, have been added. A Solutions Manual is available with detailed solutions to selected problems.
- NEW! A number of non-essential concepts have been eliminated and optional material has been placed in a way to make it possible to cover the essential material more easily and fluidly.
- The text's presentation is conversational and mathematically precise.
- In addition to the essential topics, the author includes important topics such as the approximation methods for estimating integrals, the Weierstrass Approximation Theorem, and metric spaces, without disturbing the coherence of the course.
- A wide variety of exercises helps students gain a genuine understanding of the material. The more challenging problems often stimulate the student to carefully reread the relevant sections in order to properly assimilate the material.

Table of Contents

**1. TOOLS FOR ANALYSIS.**

The Completeness Axiom and Some of Its Consequences. The Distribution of the Integers and the Rational Numbers. Inequalities and Identities.

**2. CONVERGENT SEQUENCES.**

The Convergence of Sequences. Sequences and Sets. The Monotone Convergence Theorem. The Sequential Compactness Theorem. Covering Properties of Sets.

**3. CONTINUOUS FUNCTIONS.**

Continuity. The Extreme Value Theorem. The Intermediate Value Theorem. Uniform Continuity. The Epsilon-Delta Criterion for Continuity. Images and Inverses; Monotone Functions. Limits.

**4. DIFFERENTIATION.**

The Algebra of Derivatives. Differentiating Inverses and Compositions. The Mean Value Theorem and Its Geometric Consequences. The Cauchy Mean Value Theorem and Its Analytic Consequences. The Notation of Leibnitz.

**5. ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS.**

Solutions of Differential Equations. The Natural Logarithm and the Exponential Functions. The Trigonometric Functions. The Inverse Trigonometric Functions.

**6. INTEGRATION: TWO FUNDAMENTAL THEOREMS.**

Darboux Sums; Upper and Lower Integrals. The Archimedes-Riemann Theorem. Additivity, Monotonicity and Linearity. Continuity and Integrability. The First Fundamental Theorem: Integrating Derivatives. The Second Fundamental Theorem: Differentiating Integrals.

**7. INTEGRATION: FURTHER TOPICS.**

Solutions of Differential Equations. Integration by Parts and by Substitution. The Convergence of Darboux and Riemann Sums. The Approximation of Integrals.

**8. APPROXIMATION BY TAYLOR POLYNOMIALS.**

Taylor Polynomials. The Lagrange Remainder Theorem. The Convergence of Taylor Polynomials. A Power Series for the Logarithm. The Cauchy Integral Remainder Theorem. A Non-Analytic, Infinitely Differentiable Function. The Weierstrass Approximation Theorem.

**9. SEQUENCES AND SERIES OF FUNCTIONS.**

Sequences and Series of Numbers. Pointwise Convergence of Sequences of Functions. Uniform Convergence of Sequences of Functions. The Uniform Limit of Functions. Power Series. A Continuous, Nowhere Differentiable Function.

**10. THE EUCLIDEAN SPACE Rn.**

The Linear Structure of Rn and the Scalar Product. Convergence of Sequences in Rn. Open Sets and Closed Sets in Rn.

**11. CONTINUITY, COMPACTNESS, AND CONNECTEDNESS.**

Continuous Functions and Mappings. Sequential Compactness, Extreme Values and Uniform Continuity. Pathwise Connectedness and the Intermediate Value Theorem. Connectedness and the Intermediate Value Property.

**12. METRIC SPACES.**

Open Sets, Closed Sets, and Sequential Convergence. Completeness and the Contraction Mapping Principle. The Existence Theorem for Nonlinear Differential Equations. Continuous Mappings Between Metric Spaces. Sequentially Compactness and Connectedness.

**13. DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES.**

Limits. Partial Derivatives. The Mean Value Theorem and Directional Derivatives.

**14. LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS.**

First-Order Approximation, Tangent Planes, and Affine Functions. Quadratic Functions, Hessian Matrices, and Second Derivatives. Second-Order Approximations and the Second-Derivative Test.

**15. APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS.**

Linear Mappings and Matrices. The Derivative Matrix and the Differential. The Chain Rule.

**16. IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM.**

Functions of a Single Variable and Maps in the Plane. Stability of Nonlinear Mappings. A Minimization Principle and the General Inverse Function Theorem.

**17. THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS.**

A Scalar Equation in Two Unknowns: Dini's Theorem. The General Implicit Function Theorem. Equations of Surfaces and Curves in R³. Constrained Extrema Problems and Lagrange Multipliers.

**18. INTEGRATING FUNCTIONS OF SEVERAL VARIABLES.**

Integration of Functions on Generalized Rectangles. Continuity and Integrability. Integration over Jordan Domains.

**19. ITERATED INTEGRATION AND CHANGES OF VARIABLES.**

Fubini's Theorem. The Change of Variables Theorem: Statements and Examples. Proof of the Change of Variables Theorem.

**20. LINE AND SURFACE INTEGRALS.**

Arclength and Line Integrals. Surface Area and Surface Integrals. The Integral Formulas of Green and Stokes.

Appendix A: Consequences of the Field and Positivity Axioms.

The Field Axioms and Their Consequences. The Positivity Axioms and Their Consequences.

Appendix B: Linear Algebra.

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