Mathematical Analysis: Functions of Several Real Variables and Applications

Preface
Contents
1 Sequences and Series of Functions
	1.1 Sequences of Functions: Pointwise and Uniform Convergence
	1.2 First Theorems on Uniform Convergence
	1.3 Theorems on Interchanging Limits and Integrals or Derivatives
	1.4 Uniform Convergence and Monotonicity
	1.5 Series of Functions
	1.6 Power Series
	1.7 Taylor Series
	1.8 Fourier Series
	1.9 The Convergence of Fourier Series
	Appendix to Chap.1
	1.10 The Ascoli-Arzelà Theorem
	1.11 The Weierstrass Approximation Theorem
	1.12 Abel's Theorem on Power Series
2 Metric Spaces and Banach Spaces
	2.1 Introduction
	2.2 Metric Spaces
	2.3 Sequences in a Metric Space: Continuous Functions
	2.4 Vector Spaces: Linear Maps
	2.5 The Vector Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript bold italic n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and Its Dual
	2.6 Normed Vector Spaces
	2.7 The Normed Vector Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript bold italic n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	2.8 Complete Metric Spaces: Banach Spaces
	2.9 Lipschitz Functions: The Contraction Theorem
	2.10 Compact Sets: Continuous Functions on Compact Sets
	2.11 Connected Open Subsets of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript bold italic n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	Appendix to Chap. 2
	2.12 Further Compactness Theorems: Generalised Weierstrass Theorem
3 Functions of Several Variables
	3.1 Round-Up of Topology in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript bold italic n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	3.2 Limits and Continuity
	3.3 Partial Derivatives
	3.4 Higher Derivatives. Schwarz's Theorem
	3.5 Gradient. Differentiability
	3.6 Composite Functions
	3.7 Directional Derivatives
	3.8 Functions with Vanishing Gradient on Connected Sets
	3.9 Homogeneous Functions
	3.10 Functions Defined by Integrals
	3.11 Taylor Formula and Higher-Order Differentials
	3.12 Quadratic Forms. Definite, Semi-definite and Indefinite Matrices
	3.13 Local Maxima and Minima
	3.14 Vector-Valued Functions
	Appendix to Chap.3
	3.15 Convex Functions
	3.16 Complements on Quadratic Forms
	3.17 The Maximum Principle for Harmonic Functions
4 Ordinary Differential Equations
	4.1 Introduction: The Initial Value Problem
	4.2 Cauchy's Local Existence and Uniqueness Theorem
	4.3 First Consequences of Cauchy's Theorem
	4.4 The Global Existence and Uniqueness Theorem: Extension of Solutions
	4.5 Solving First-Order ODEs in Normal Form
	4.6 Solving First-Order ODEs Not in Normal Form
	4.7 Solving Higher-Order Equations
	4.8 Qualitative Study of Solutions
	Appendix to Chap. 4
	4.9 Peano's Theorem
5 Linear Differential Equations
	5.1 General Properties
	5.2 General Integral of Linear ODEs
	5.3 The Method of Variation of Parameters
	5.4 Bernoulli Equations
	5.5 Homogeneous Equations with Constant Coefficients
	5.6 Equations with Constant Coefficients and Special Right-Hand Side
	5.7 Linear Euler Equations
	Appendix to Chap.5
	5.8 Boundary Value Problems
	5.9 Linear Systems
6 Curves and Integrals Along Curves
	6.1 Regular Curves
	6.2 Oriented Curves
	6.3 The Length of a Curve
	6.4 The Integral of a Function Along a Curve
	6.5 The Curvature of a Plane Curve
	6.6 The Cross Product in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R cubed) /StPNE pdfmark [/StBMC pdfmarkR3ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	6.7 Biregular Curves in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R cubed) /StPNE pdfmark [/StBMC pdfmarkR3ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark: Curvature
	Appendix to Chap.6
	6.8 Curves in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R cubed) /StPNE pdfmark [/StBMC pdfmarkR3ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark: Torsion, Frenet Frame
7 Differential One-Forms
	7.1 Vector Fields. Work. Conservative Fields
	7.2 Differential 1-Forms. Line Integrals
	7.3 Exact 1-Forms
	7.4 Exact 1-Forms on the Plane. Simply Connected Open Sets in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R squared) /StPNE pdfmark [/StBMC pdfmarkR2ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	7.5 One-Forms in Space. Irrotational Vector Fields
	Appendix to Chap.7
	7.6 Simply Connected Open Sets in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and Exact 1-Forms
8 Multiple Integrals
	8.1 Double Integrals on Normal Domains
	8.2 Reduction Formulas for Double Integrals
	8.3 Gauss-Green Formulas. The Divergence Theorem. Stokes's Formula
	8.4 Variable Change in Double Integrals
	8.5 Triple Integrals
	8.6 Peano-Jordan Measurable Subsets of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript bold italic n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	8.7 The Riemann Integral in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript bold italic n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	8.8 Properties of Riemann Integrals
	8.9 Summable Functions
	Appendix to Chap.8
	8.10 Jensen's Inequality
	8.11 The Gamma Function. The Measure of the Unit Ball in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript bold italic n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
9 The Lebesgue Integral
	9.1 Introduction
	9.2 Pluri-Intervals. Open Sets. Compact Sets
	9.3 Bounded Measurable Sets
	9.4 Unbounded Measurable Sets
	9.5 Measurable Functions
	9.6 The Lebesgue Integral. Interchanging Limits and Integrals
	9.7 Measure and Integration on Product Spaces
	9.8 Changing Variables in Multiple Integrals
	Appendix to Chap.9
	9.9 Lp Spaces
	9.10 Differentiability of Monotone Functions
	9.11 Functions with Bounded Variation
	9.12 Absolutely Continuous Functions
	9.13 The Indefinite Integral in Lebesgue's Theory
10 Surfaces and Surface Integrals
	10.1 Regular Surfaces
	10.2 Local Coordinates and Change of Parameters
	10.3 The Tangent Plane and the Unit Normal
	10.4 The Area of a Surface
	10.5 Orientable Surfaces: Surfaces with Boundary
	10.6 Surface Integrals
	10.7 Stokes's Formula and the Divergence Theorem
11 Implicit Functions
	11.1 The Implicit Function Theorem for Equations
	11.2 The Implicit Function Theorem for Systems
	11.3 Local and Global Invertibility
	11.4 Constrained Maxima and Minima. Lagrange Multipliers
	Appendix to Chap.11
	11.5 Singular Points of a Plane Curve
12 Manifolds in Rn and k-Forms
	12.1 k-Dimensional Manifolds in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	12.2 The Tangent Space and the Normal Space of a Manifold
	12.3 Measure and Integration on k-Submanifolds in ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R Superscript n) /StPNE pdfmark [/StBMC pdfmarkRnps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
	12.4 The Divergence Theorem
	12.5 Alternating Forms
	12.6 Differential k-Forms
	12.7 Orientable Manifolds. Integration of k-Forms on Manifolds
	12.8 Manifolds with Boundary. Stokes's Formula
	Appendix to Chap.12
	12.9 Exact and Closed Differential Forms
Index