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Introduction to analysis in several variables : advanced calculus / Michael E. Taylor.

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-variabl...

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Bibliographic Details
Online Access: Full Text (via ProQuest)
Main Author: Taylor, Michael E., 1946- (Author)
Format: eBook
Language:English
Published: Providence, Rhode Island : American Mathematical Society, [2020]
Series:Pure and applied undergraduate texts ; 46.
Subjects:
Calculus.
Functions of several real variables.
Functions of several complex 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 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.

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