Introduction to Analysis in Several Variables: Advanced Calculus

Description

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. 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 theorem 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