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Hardcover ISBN:  9781470442460 
Product Code:  AMSTEXT/32 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470447625 
Product Code:  AMSTEXT/32.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Hardcover ISBN:  9781470442460 
eBook ISBN:  9781470447625 
Product Code:  AMSTEXT/32.B 
List Price:  $174.00$131.50 
MAA Member Price:  $156.60$118.35 
AMS Member Price:  $139.20$105.20 

Book DetailsPure and Applied Undergraduate TextsVolume: 32; 2018; 360 ppMSC: Primary 00;
This textbook is suitable for a course in advanced calculus that promotes active learning through problem solving. It can be used as a base for a Moore method or inquiry based class, or as a guide in a traditional classroom setting where lectures are organized around the presentation of problems and solutions. This book is appropriate for any student who has taken (or is concurrently taking) an introductory course in calculus. The book includes sixteen appendices that review some indispensable prerequisites on techniques of proof writing with special attention to the notation used the course.
Ancillaries:ReadershipUndergraduate students interested in introduction to proofs.

Table of Contents

Cover

Title page

Contents

Preface

For students: How to use this book

Chapter 1. Intervals

1.1. Distance and neighborhoods

1.2. Interior of a set

Chapter 2. Topology of the real line

2.1. Open subsets of \R

2.2. Closed subsets of \R

Chapter 3. Continuous functions from \R to \R

3.1. Continuity—as a local property

3.2. Continuity—as a global property

3.3. Functions defined on subsets of \R

Chapter 4. Sequences of real numbers

4.1. Convergence of sequences

4.2. Algebraic combinations of sequences

4.3. Sufficient condition for convergence

4.4. Subsequences

Chapter 5. Connectedness and the intermediate value theorem

5.1. Connected subsets of \R

5.2. Continuous images of connected sets

5.3. Homeomorphisms

Chapter 6. Compactness and the extreme value theorem

6.1. Compactness

6.2. Examples of compact subsets of \R

6.3. The extreme value theorem

Chapter 7. Limits of real valued functions

7.1. Definition

7.2. Continuity and limits

Chapter 8. Differentiation of real valued functions

8.1. The families \lobo𝑂 and \lobo𝑜

8.2. Tangency

8.3. Linear approximation

8.4. Differentiability

Chapter 9. Metric spaces

9.1. Definitions

9.2. Examples

9.3. Equivalent metrics

Chapter 10. Interiors, closures, and boundaries

10.1. Definitions and examples

10.2. Interior points

10.3. Accumulation points and closures

Chapter 11. The topology of metric spaces

11.1. Open and closed sets

11.2. The relative topology

Chapter 12. Sequences in metric spaces

12.1. Convergence of sequences

12.2. Sequential characterizations of topological properties

12.3. Products of metric spaces

Chapter 13. Uniform convergence

13.1. The uniform metric on the space of bounded functions

13.2. Pointwise convergence

Chapter 14. More on continuity and limits

14.1. Continuous functions

14.2. Maps into and from products

14.3. Limits

Chapter 15. Compact metric spaces

15.1. Definition and elementary properties

15.2. The extreme value theorem

15.3. Dini’s theorem

Chapter 16. Sequential characterization of compactness

16.1. Sequential compactness

16.2. Conditions equivalent to compactness

16.3. Products of compact spaces

16.4. The Heine–Borel theorem

Chapter 17. Connectedness

17.1. Connected spaces

17.2. Arcwise connected spaces

Chapter 18. Complete spaces

18.1. Cauchy sequences

18.2. Completeness

18.3. Completeness vs. compactness

Chapter 19. A fixed point theorem

19.1. The contractive mapping theorem

19.2. Application to integral equations

Chapter 20. Vector spaces

20.1. Definitions and examples

20.2. Linear combinations

20.3. Convex combinations

Chapter 21. Linearity

21.1. Linear transformations

21.2. The algebra of linear transformations

21.3. Matrices

21.4. Determinants

21.5. Matrix representations of linear transformations

Chapter 22. Norms

22.1. Norms on linear spaces

22.2. Norms induce metrics

22.3. Products

22.4. The space \fml𝐵(𝑆,𝑉)

Chapter 23. Continuity and linearity

23.1. Bounded linear transformations

23.2. The Stone–Weierstrass theorem

23.3. Banach spaces

23.4. Dual spaces and adjoints

Chapter 24. The Cauchy integral

24.1. Uniform continuity

24.2. The integral of step functions

24.3. The Cauchy integral

Chapter 25. Differential calculus

25.1. \lobo𝑂 and \lobo𝑜 functions

25.2. Tangency

25.3. Differentiation

25.4. Differentiation of curves

25.5. Directional derivatives

25.6. Functions mapping into product spaces

Chapter 26. Partial derivatives and iterated integrals

26.1. The mean value theorem(s)

26.2. Partial derivatives

26.3. Iterated integrals

Chapter 27. Computations in \Rⁿ

27.1. Inner products

27.2. The gradient

27.3. The Jacobian matrix

27.4. The chain rule

Chapter 28. Infinite series

28.1. Convergence of series

28.2. Series of positive scalars

28.3. Absolute convergence

28.4. Power series

Chapter 29. The implicit function theorem

29.1. The inverse function theorem

29.2. The implicit function theorem

Chapter 30. Higher order derivatives

30.1. Multilinear functions

30.2. Second order differentials

30.3. Higher order differentials

Appendix A. Quantifiers

Appendix B. Sets

Appendix C. Special subsets of \R

Appendix D. Logical connectives

D.1. Disjunction and conjunction

D.2. Implication

D.3. Restricted quantifiers

D.4. Negation

Appendix E. Writing mathematics

E.1. Proving theorems

E.2. Checklist for writing mathematics

E.3. Fraktur and Greek alphabets

Appendix F. Set operations

F.1. Unions

F.2. Intersections

F.3. Complements

Appendix G. Arithmetic

G.1. The field axioms

G.2. Uniqueness of identities

G.3. Uniqueness of inverses

G.4. Another consequence of uniqueness

Appendix H. Order properties of \R

Appendix I. Natural numbers and mathematical induction

Appendix J. Least upper bounds and greatest lower bounds

J.1. Upper and lower bounds

J.2. Least upper and greatest lower bounds

J.3. The least upper bound axiom for \R

J.4. The Archimedean property

Appendix K. Products, relations, and functions

K.1. Cartesian products

K.2. Relations

K.3. Functions

Appendix L. Properties of functions

L.1. Images and inverse images

L.2. Composition of functions

L.3. The identity function

L.4. Diagrams

L.5. Restrictions and extensions

Appendix M. Functions that have inverses

M.1. Injections, surjections, and bijections

M.2. Inverse functions

Appendix N. Products

Appendix O. Finite and infinite sets

Appendix P. Countable and uncountable sets

Bibliography

Index

Back Cover


Additional Material

Reviews

[This] book is a piece of excellent mathematical writing. The readers—student or teacher— will enjoy precise and clear exposition and careful editing, as well as fine language sparking with a decent dose of humor.
Piotr Sworowski, Mathematical Reviews 
Learning from this book might be a challenging and timeconsuming task, but the reader will be rewarded by a deep understanding of advanced calculus.
Antonín Slavik, Zentralblatt MATH


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This textbook is suitable for a course in advanced calculus that promotes active learning through problem solving. It can be used as a base for a Moore method or inquiry based class, or as a guide in a traditional classroom setting where lectures are organized around the presentation of problems and solutions. This book is appropriate for any student who has taken (or is concurrently taking) an introductory course in calculus. The book includes sixteen appendices that review some indispensable prerequisites on techniques of proof writing with special attention to the notation used the course.
Ancillaries:
Undergraduate students interested in introduction to proofs.

Cover

Title page

Contents

Preface

For students: How to use this book

Chapter 1. Intervals

1.1. Distance and neighborhoods

1.2. Interior of a set

Chapter 2. Topology of the real line

2.1. Open subsets of \R

2.2. Closed subsets of \R

Chapter 3. Continuous functions from \R to \R

3.1. Continuity—as a local property

3.2. Continuity—as a global property

3.3. Functions defined on subsets of \R

Chapter 4. Sequences of real numbers

4.1. Convergence of sequences

4.2. Algebraic combinations of sequences

4.3. Sufficient condition for convergence

4.4. Subsequences

Chapter 5. Connectedness and the intermediate value theorem

5.1. Connected subsets of \R

5.2. Continuous images of connected sets

5.3. Homeomorphisms

Chapter 6. Compactness and the extreme value theorem

6.1. Compactness

6.2. Examples of compact subsets of \R

6.3. The extreme value theorem

Chapter 7. Limits of real valued functions

7.1. Definition

7.2. Continuity and limits

Chapter 8. Differentiation of real valued functions

8.1. The families \lobo𝑂 and \lobo𝑜

8.2. Tangency

8.3. Linear approximation

8.4. Differentiability

Chapter 9. Metric spaces

9.1. Definitions

9.2. Examples

9.3. Equivalent metrics

Chapter 10. Interiors, closures, and boundaries

10.1. Definitions and examples

10.2. Interior points

10.3. Accumulation points and closures

Chapter 11. The topology of metric spaces

11.1. Open and closed sets

11.2. The relative topology

Chapter 12. Sequences in metric spaces

12.1. Convergence of sequences

12.2. Sequential characterizations of topological properties

12.3. Products of metric spaces

Chapter 13. Uniform convergence

13.1. The uniform metric on the space of bounded functions

13.2. Pointwise convergence

Chapter 14. More on continuity and limits

14.1. Continuous functions

14.2. Maps into and from products

14.3. Limits

Chapter 15. Compact metric spaces

15.1. Definition and elementary properties

15.2. The extreme value theorem

15.3. Dini’s theorem

Chapter 16. Sequential characterization of compactness

16.1. Sequential compactness

16.2. Conditions equivalent to compactness

16.3. Products of compact spaces

16.4. The Heine–Borel theorem

Chapter 17. Connectedness

17.1. Connected spaces

17.2. Arcwise connected spaces

Chapter 18. Complete spaces

18.1. Cauchy sequences

18.2. Completeness

18.3. Completeness vs. compactness

Chapter 19. A fixed point theorem

19.1. The contractive mapping theorem

19.2. Application to integral equations

Chapter 20. Vector spaces

20.1. Definitions and examples

20.2. Linear combinations

20.3. Convex combinations

Chapter 21. Linearity

21.1. Linear transformations

21.2. The algebra of linear transformations

21.3. Matrices

21.4. Determinants

21.5. Matrix representations of linear transformations

Chapter 22. Norms

22.1. Norms on linear spaces

22.2. Norms induce metrics

22.3. Products

22.4. The space \fml𝐵(𝑆,𝑉)

Chapter 23. Continuity and linearity

23.1. Bounded linear transformations

23.2. The Stone–Weierstrass theorem

23.3. Banach spaces

23.4. Dual spaces and adjoints

Chapter 24. The Cauchy integral

24.1. Uniform continuity

24.2. The integral of step functions

24.3. The Cauchy integral

Chapter 25. Differential calculus

25.1. \lobo𝑂 and \lobo𝑜 functions

25.2. Tangency

25.3. Differentiation

25.4. Differentiation of curves

25.5. Directional derivatives

25.6. Functions mapping into product spaces

Chapter 26. Partial derivatives and iterated integrals

26.1. The mean value theorem(s)

26.2. Partial derivatives

26.3. Iterated integrals

Chapter 27. Computations in \Rⁿ

27.1. Inner products

27.2. The gradient

27.3. The Jacobian matrix

27.4. The chain rule

Chapter 28. Infinite series

28.1. Convergence of series

28.2. Series of positive scalars

28.3. Absolute convergence

28.4. Power series

Chapter 29. The implicit function theorem

29.1. The inverse function theorem

29.2. The implicit function theorem

Chapter 30. Higher order derivatives

30.1. Multilinear functions

30.2. Second order differentials

30.3. Higher order differentials

Appendix A. Quantifiers

Appendix B. Sets

Appendix C. Special subsets of \R

Appendix D. Logical connectives

D.1. Disjunction and conjunction

D.2. Implication

D.3. Restricted quantifiers

D.4. Negation

Appendix E. Writing mathematics

E.1. Proving theorems

E.2. Checklist for writing mathematics

E.3. Fraktur and Greek alphabets

Appendix F. Set operations

F.1. Unions

F.2. Intersections

F.3. Complements

Appendix G. Arithmetic

G.1. The field axioms

G.2. Uniqueness of identities

G.3. Uniqueness of inverses

G.4. Another consequence of uniqueness

Appendix H. Order properties of \R

Appendix I. Natural numbers and mathematical induction

Appendix J. Least upper bounds and greatest lower bounds

J.1. Upper and lower bounds

J.2. Least upper and greatest lower bounds

J.3. The least upper bound axiom for \R

J.4. The Archimedean property

Appendix K. Products, relations, and functions

K.1. Cartesian products

K.2. Relations

K.3. Functions

Appendix L. Properties of functions

L.1. Images and inverse images

L.2. Composition of functions

L.3. The identity function

L.4. Diagrams

L.5. Restrictions and extensions

Appendix M. Functions that have inverses

M.1. Injections, surjections, and bijections

M.2. Inverse functions

Appendix N. Products

Appendix O. Finite and infinite sets

Appendix P. Countable and uncountable sets

Bibliography

Index

Back Cover

[This] book is a piece of excellent mathematical writing. The readers—student or teacher— will enjoy precise and clear exposition and careful editing, as well as fine language sparking with a decent dose of humor.
Piotr Sworowski, Mathematical Reviews 
Learning from this book might be a challenging and timeconsuming task, but the reader will be rewarded by a deep understanding of advanced calculus.
Antonín Slavik, Zentralblatt MATH