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A Problems Based Course in Advanced Calculus
 
John M. Erdman Portland State University, Portland, OR
A Problems Based Course in Advanced Calculus
Hardcover ISBN:  978-1-4704-4246-0
Product Code:  AMSTEXT/32
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-4762-5
Product Code:  AMSTEXT/32.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-4246-0
eBook: ISBN:  978-1-4704-4762-5
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
A Problems Based Course in Advanced Calculus
Click above image for expanded view
A Problems Based Course in Advanced Calculus
John M. Erdman Portland State University, Portland, OR
Hardcover ISBN:  978-1-4704-4246-0
Product Code:  AMSTEXT/32
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-4762-5
Product Code:  AMSTEXT/32.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-1-4704-4246-0
eBook ISBN:  978-1-4704-4762-5
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 Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 322018; 360 pp
    MSC: 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:

    Readership

    Undergraduate 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
  • 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 time-consuming task, but the reader will be rewarded by a deep understanding of advanced calculus.

      Antonín Slavik, Zentralblatt MATH
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 322018; 360 pp
MSC: 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:

Readership

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 time-consuming task, but the reader will be rewarded by a deep understanding of advanced calculus.

    Antonín Slavik, Zentralblatt MATH
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Instructor's Manual – for instructors who have adopted an AMS textbook for a course and need the instructor's manual
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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