Softcover ISBN:  9781470456696 
Product Code:  AMSTEXT/46 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Electronic ISBN:  9781470460167 
Product Code:  AMSTEXT/46.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 

Book DetailsPure and Applied Undergraduate TextsVolume: 46; 2020; 445 ppMSC: Primary 26;
This text was produced for the second part of a twopart 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 onevariable 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.ReadershipUndergraduates interested in analysis in several variables.

Table of Contents

Cover

Title page

Copyright

Contents

Preface

Some basic notation

Chapter 1. Background

1.1. Onevariable 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 GaussGreenStokes 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 GaussGreenStokes 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 GaussBonnet 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 StoneWeierstrass theorem

A.6. Further results on harmonic functions

A.7. Beyond degree theory—introduction to de Rham theory

Bibliography

Index

Back Cover


Additional Material

RequestsReview Copy – for reviewers who would like to review an AMS bookDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a coursePermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
This text was produced for the second part of a twopart 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 onevariable 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.
Undergraduates interested in analysis in several variables.

Cover

Title page

Copyright

Contents

Preface

Some basic notation

Chapter 1. Background

1.1. Onevariable 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 GaussGreenStokes 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 GaussGreenStokes 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 GaussBonnet 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 StoneWeierstrass theorem

A.6. Further results on harmonic functions

A.7. Beyond degree theory—introduction to de Rham theory

Bibliography

Index

Back Cover