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Introduction to Analysis in Several Variables: Advanced Calculus
 
Michael E. Taylor :  University of North Carolina, Chapel Hill, Chapel Hill, NC
Front Cover for Introduction to Analysis in Several Variables
Available Formats:
Softcover ISBN: 978-1-4704-5669-6
Product Code: AMSTEXT/46
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Electronic ISBN: 978-1-4704-6016-7
Product Code: AMSTEXT/46.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
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Front Cover for Introduction to Analysis in Several Variables
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  • Front Cover for Introduction to Analysis in Several Variables
  • Back Cover for Introduction to Analysis in Several Variables
Introduction to Analysis in Several Variables: Advanced Calculus
Michael E. Taylor :  University of North Carolina, Chapel Hill, Chapel Hill, NC
Available Formats:
Softcover ISBN:  978-1-4704-5669-6
Product Code:  AMSTEXT/46
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Electronic ISBN:  978-1-4704-6016-7
Product Code:  AMSTEXT/46.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Add to cart
Bundle Print and Electronic Formats and Save!
This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.
List Price: $127.50
MAA Member Price: $114.75
AMS Member Price: $102.00
Add bundle to cart
  • Book Details
     
     
    Pure and Applied Undergraduate Texts
    Volume: 46; 2020; 445 pp
    MSC: Primary 26;

    This text was produced for the second part of a two-part sequence on advanced calculus, whose aim is to provide a firm logical foundation for analysis. The first part treats analysis in one variable, and the text at hand treats analysis in several variables.

    After a review of topics from one-variable analysis and linear algebra, the text treats in succession multivariable differential calculus, including systems of differential equations, and multivariable integral calculus. It builds on this to develop calculus on surfaces in Euclidean space and also on manifolds. It introduces differential forms and establishes a general Stokes formula. It describes various applications of Stokes formula, from harmonic functions to degree theory. The text then studies the differential geometry of surfaces, including geodesics and curvature, and makes contact with degree theory, via the Gauss–Bonnet theorem.

    The text also takes up Fourier analysis, and bridges this with results on surfaces, via Fourier analysis on spheres and on compact matrix groups.
    Readership

    Undergraduates interested in analysis in several variables.
  • Table of Contents
     
     
    • Cover
    • Title page
    • Copyright
    • Contents
    • Preface
    • Some basic notation
    • Chapter 1. Background
    • 1.1. One-variable calculus
    • 1.2. Euclidean spaces
    • 1.3. Vector spaces and linear transformations
    • 1.4. Determinants
    • Chapter 2. Multivariable differential calculus
    • 2.1. The derivative
    • 2.2. Inverse function and implicit function theorems
    • 2.3. Systems of differential equations and vector fields
    • Chapter 3. Multivariable integral calculus and calculus on surfaces
    • 3.1. The Riemann integral in 𝑛 variables
    • 3.2. Surfaces and surface integrals
    • 3.3. Partitions of unity
    • 3.4. Sard’s theorem
    • 3.5. Morse functions
    • 3.6. The tangent space to a manifold
    • Chapter 4. Differential forms and the Gauss-Green-Stokes formula
    • 4.1. Differential forms
    • 4.2. Products and exterior derivatives of forms
    • 4.3. The general Stokes formula
    • 4.4. The classical Gauss, Green, and Stokes formulas
    • 4.5. Differential forms and the change of variable formula
    • Chapter 5. Applications of the Gauss-Green-Stokes formula
    • 5.1. Holomorphic functions and harmonic functions
    • 5.2. Differential forms, homotopy, and the Lie derivative
    • 5.3. Differential forms and degree theory
    • Chapter 6. Differential geometry of surfaces
    • 6.1. Geometry of surfaces I: geodesics
    • 6.2. Geometry of surfaces II: curvature
    • 6.3. Geometry of surfaces III: the Gauss-Bonnet theorem
    • 6.4. Smooth matrix groups
    • 6.5. The derivative of the exponential map
    • 6.6. A spectral mapping theorem
    • Chapter 7. Fourier analysis
    • 7.1. Fourier series
    • 7.2. The Fourier transform
    • 7.3. Poisson summation formulas
    • 7.4. Spherical harmonics
    • 7.5. Fourier series on compact matrix groups
    • 7.6. Isoperimetric inequality
    • Appendix A. Complementary material
    • A.1. Metric spaces, convergence, and compactness
    • A.2. Inner product spaces
    • A.3. Eigenvalues and eigenvectors
    • A.4. Complements on power series
    • A.5. The Weierstrass theorem and the Stone-Weierstrass theorem
    • A.6. Further results on harmonic functions
    • A.7. Beyond degree theory—introduction to de Rham theory
    • Bibliography
    • Index
    • Back Cover
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    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
Pure and Applied Undergraduate Texts
Volume: 46; 2020; 445 pp
MSC: Primary 26;

This text was produced for the second part of a two-part sequence on advanced calculus, whose aim is to provide a firm logical foundation for analysis. The first part treats analysis in one variable, and the text at hand treats analysis in several variables.

After a review of topics from one-variable analysis and linear algebra, the text treats in succession multivariable differential calculus, including systems of differential equations, and multivariable integral calculus. It builds on this to develop calculus on surfaces in Euclidean space and also on manifolds. It introduces differential forms and establishes a general Stokes formula. It describes various applications of Stokes formula, from harmonic functions to degree theory. The text then studies the differential geometry of surfaces, including geodesics and curvature, and makes contact with degree theory, via the Gauss–Bonnet theorem.

The text also takes up Fourier analysis, and bridges this with results on surfaces, via Fourier analysis on spheres and on compact matrix groups.
Readership

Undergraduates interested in analysis in several variables.
  • Cover
  • Title page
  • Copyright
  • Contents
  • Preface
  • Some basic notation
  • Chapter 1. Background
  • 1.1. One-variable calculus
  • 1.2. Euclidean spaces
  • 1.3. Vector spaces and linear transformations
  • 1.4. Determinants
  • Chapter 2. Multivariable differential calculus
  • 2.1. The derivative
  • 2.2. Inverse function and implicit function theorems
  • 2.3. Systems of differential equations and vector fields
  • Chapter 3. Multivariable integral calculus and calculus on surfaces
  • 3.1. The Riemann integral in 𝑛 variables
  • 3.2. Surfaces and surface integrals
  • 3.3. Partitions of unity
  • 3.4. Sard’s theorem
  • 3.5. Morse functions
  • 3.6. The tangent space to a manifold
  • Chapter 4. Differential forms and the Gauss-Green-Stokes formula
  • 4.1. Differential forms
  • 4.2. Products and exterior derivatives of forms
  • 4.3. The general Stokes formula
  • 4.4. The classical Gauss, Green, and Stokes formulas
  • 4.5. Differential forms and the change of variable formula
  • Chapter 5. Applications of the Gauss-Green-Stokes formula
  • 5.1. Holomorphic functions and harmonic functions
  • 5.2. Differential forms, homotopy, and the Lie derivative
  • 5.3. Differential forms and degree theory
  • Chapter 6. Differential geometry of surfaces
  • 6.1. Geometry of surfaces I: geodesics
  • 6.2. Geometry of surfaces II: curvature
  • 6.3. Geometry of surfaces III: the Gauss-Bonnet theorem
  • 6.4. Smooth matrix groups
  • 6.5. The derivative of the exponential map
  • 6.6. A spectral mapping theorem
  • Chapter 7. Fourier analysis
  • 7.1. Fourier series
  • 7.2. The Fourier transform
  • 7.3. Poisson summation formulas
  • 7.4. Spherical harmonics
  • 7.5. Fourier series on compact matrix groups
  • 7.6. Isoperimetric inequality
  • Appendix A. Complementary material
  • A.1. Metric spaces, convergence, and compactness
  • A.2. Inner product spaces
  • A.3. Eigenvalues and eigenvectors
  • A.4. Complements on power series
  • A.5. The Weierstrass theorem and the Stone-Weierstrass theorem
  • A.6. Further results on harmonic functions
  • A.7. Beyond degree theory—introduction to de Rham theory
  • Bibliography
  • Index
  • Back Cover
Review Copy – for reviewers who would like to review an AMS book
Desk Copy – for instructors who have adopted an AMS textbook for a course
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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