Introduction to Analysis in Several Variables: Advanced Calculus (Pure and Applied Undergraduate Texts)

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
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