ANALYSIS IN EUCLIDEAN SPACE

Based on notes written during the author's many years of teaching, Analysis in Euclidean Space mainly covers Differentiation and Integration theory in several real variables, but also an array of closely related areas including measure theory, differential geometry, classical theory of curves, geometric measure theory, integral geometry, and others.

With several original results, new approaches and an emphasis on concepts and rigorous proofs, the book is suitable for undergraduate students, particularly in mathematics and physics, who are interested in acquiring a solid footing in analysis and expanding their background. There are many examples and exercises inserted in the text for the student to work through independently.

Analysis in Euclidean Space comprises 21 chapters, each with an introduction summarizing its contents, and an additional chapter containing miscellaneous exercises. Lecturers may use the varied chapters of this book for different undergraduate courses in analysis. The only prerequisites are a basic course in linear algebra and a standard first-year calculus course in differentiation and integration. As the book progresses, the difficulty increases such that some of the later sections may be appropriate for graduate study.

Contents:

• About the Author
• Introduction
• Euclidean Space
• Continuous Functions
• Coordinate Systems, Curves and Surfaces
• Differentiation
• Higher-Order Derivatives
• The Inverse and Implicit Function Theorems
• Regular Sub-Manifolds
• Ordinary Differential Equations
• Linear Partial Differential Equations
• Orthogonal Families of Curves and Surfaces
• Measuring Sets: The Riemann Integral
• The Lebesgue Integral
• Fubini's Theorem and Change of Variables
• Integration on Sub-Manifolds
• Geometric Measure Theory and Integral Geometry
• Line Integrals and Flux
• The Basic Theorems of Vector Analysis
• Conservative and Solenoidal Fields
• Harmonic Functions
• The Divergence and Rotational Equations, Poisson's Equation
• The Dirichlet and Neumann Problems
• Additional Exercises
• Bibliography
• Index

Readership: Can be used as a textbook for undergraduate students studying differentiation theory in several real variables, measure and integration in several real variables, ordinary differential equations, linear partial differential equations, vector analysis, and curves and surfaces. Graduate students may use this book for an introduction to geometric measure theory and integral geometry, as well as advanced topics in vector analysis.

Key Features:

• The inverse and implicit function theorems are combined with Picard's theorem on the existence and uniqueness of solutions for ordinary differential equations to deal with the Cauchy problem for some linear partial differential equations
• Unlike many other textbooks, systems of orthogonal families of curves and surfaces, rigidity of conformal maps in space and Lamé surfaces are carefully studied
• In integration theory, both the Riemann and Lebesgue theories are described and their differences explained
• A multidimensional version of the fundamental theorem of calculus, completely original, is systematically used to prove the basic theorems of vector analysis
• Integration on manifolds is complemented with an introduction to geometric measure theory and integral geometry, another unusual topic in other texts
• An original approach to the theory of conservative and solenoidal fields as it is the Hodge decomposition of vector fields, based on the solution of Dirichlet and Neumann problems, which hardly found in other textbooks
• Additional significant results are proposed as exercises. By studying this text, the student will learn not only the main areas of study, but also understand the connections with other important areas of mathematics