Mathematical Analysis: Functions of Several Real Variables and Applications

✍ Scribed by Nicola Fusco, Paolo Marcellini, Carlo Sbordone

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 678
• Series: UNITEXT, 137
• Edition: 1
• Category: Library

✦ Synopsis

This work is a textbook on Mathematical Analysis written by expert lecturers in the field. This textbook, other than the classical differentiation and integration tools for functions of several real variables, metric spaces, ordinary differential equations, implicit function and so on, also provides opportunities to go deeper into certain topics: among them, the Ascoli-Arzelà theorem, the regularity of convex functions in R^n, L^p spaces and absolutely continuous functions, all topics that are paramount in modern Mathematical Analysis. Other instances include the Weierstrass theorem on polynomial approximation of continuous functions or Peano's existence theorem (typically only existence, without uniqueness) for nonlinear ODEs and systems under general assumptions.

The content is discussed in an elementary way and, at a successive stage, some topics are examined from several, more penetrating, angles. The agile organization of the subject matter helps instructors to effortlessly determine which parts to present during lectures and where to stop. The authors believe that any textbook can contribute to the success of a lecture course only to a point, and the choices made by lecturers are decisive in this respect.

The book is addressed to graduate or undergraduate honors students in Mathematics, Physics, Astronomy, Computer Science, Statistics and Probability, attending Mathematical Analysis courses at the Faculties of Science, Engineering, Economics and Architecture.

✦ Table of Contents

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

✦ Subjects

Numerical Analysis; Modeling; Navier-Stokes; Regular Surface Theory; Riemann Theory