An Introduction to Multivariable Analysis from Vector to Manifold

ISBN-13: 9780817642341 / Angielski / Twarda / 2001 / 295 str.

Piotr Mikusinski;Michael D. Taylor

Opis

Multivariable analysis is an important subject for mathematicians, both pure and applied. Apart from mathematicians, we expect that physicists, mechanical engi neers, electrical engineers, systems engineers, mathematical biologists, mathemati cal economists, and statisticians engaged in multivariate analysis will find this book extremely useful. The material presented in this work is fundamental for studies in differential geometry and for analysis in N dimensions and on manifolds. It is also of interest to anyone working in the areas of general relativity, dynamical systems, fluid mechanics, electromagnetic phenomena, plasma dynamics, control theory, and optimization, to name only several. An earlier work entitled An Introduction to Analysis: from Number to Integral by Jan and Piotr Mikusinski was devoted to analyzing functions of a single variable. As indicated by the title, this present book concentrates on multivariable analysis and is completely self-contained. Our motivation and approach to this useful subject are discussed below. A careful study of analysis is difficult enough for the average student; that of multi variable analysis is an even greater challenge. Somehow the intuitions that served so well in dimension I grow weak, even useless, as one moves into the alien territory of dimension N. Worse yet, the very useful machinery of differential forms on manifolds presents particular difficulties; as one reviewer noted, it seems as though the more precisely one presents this machinery, the harder it is to understand."

Recenzje

"This is a self-contained textbook devoted to multivariable analysis based on nonstandard geometrical methods. The book can be used either as a supplement to a course on single variable analysis or as a semester-long course introducing students to manifolds and differential forms."   -Mathematical Reviews

"The authors strongly motivate the abstract notions by a lot of intuitive examples and pictures. The exercises at the end of each section range from computational to theoretical. The book is highly recommended for undergraduate or graduate courses in multivariable analysis for students in mathematics, physics, engineering, and economics."   -Studia Universitatis Babes-Bolyai, Series Mathematica

"All this [the description on the book's back cover] is absolutely true, but omits any statement attesting to the high quality of the writing and the high level of mathematical scholarship. So, go and order a copy of this attractively produced, and nicely composed, scholarly tome. If you're not teaching this sort of mathematics, this book will inspire you to do so."   -MAA Reviews

Spis treści

1 Vectors and Volumes.- 1.1 Vector Spaces.- 1.2 Some Geometric Machinery for RN.- 1.3 Transformations and Linear Transformations.- 1.4 A Little Matrix Algebra.- 1.5 Oriented Volume and Determinants.- 1.6 Properties of Determinants.- 1.7 Linear Independence, Linear Subspaces, and Bases.- 1.8 Orthogonal Transformations.- 1.9 K-dimensional Volume of Parallelepipeds in RN.- 2 Metric Spaces.- 2.1 Metric Spaces.- 2.2 Open and Closed Sets.- 2.3 Convergence.- 2.4 Continuous Mappings.- 2.5 Compact Sets.- 2.6 Complete Spaces.- 2.7 Normed Spaces.- 3 Differentiation.- 3.1 Rates of Change and Derivatives as Linear Transformations.- 3.2 Some Elementary Properties of Differentiation.- 3.3 Taylor’s Theorem, the Mean Value Theorem, and Related Results.- 3.4 Norm Properties.- 3.5 The Inverse Function Theorem.- 3.6 Some Consequences of the Inverse Function Theorem.- 3.7 Lagrange Multipliers.- 4 The Lebesgue Integral.- 4.1 A Bird’s-Eye View of the Lebesgue Integral.- 4.2 Integrable Functions.- 4.3 Absolutely Integrable Functions.- 4.4 Series of Integrable Functions.- 4.5 Convergence Almost Everywhere.- 4.6 Convergence in Norm.- 4.7 Important Convergence Theorems.- 4.8 Integrals Over a Set.- 4.9 Fubini’s Theorem.- 5 Integrals on Manifolds.- 5.1 Introduction.- 5.2 The Change of Variables Formula.- 5.3 Manifolds.- 5.4 Integrals of Real-valued Functions over Manifolds.- 5.5 Volumes in RN.- 6 K-Vectors and Wedge Products.- 6.1 K-Vectors in RN and the Wedge Product.- 6.2 Properties of A.- 6.3 Wedge Product and a Characterization of Simple K-Vectors.- 6.4 The Dot Product and the Star Operator.- 7 Vector Analysis on Manifolds.- 7.1 Oriented Manifolds and Differential Forms.- 7.2 Induced Orientation, the Differential Operator, and Stokes’ Theorem; What We Can Learn from Simple Cubes.- 7.3 Integrals and Pullbacks.- 7.4 Stokes’Theorem for Chains.- 7.5 Stokes’Theorem for Oriented Manifolds.- 7.6 Applications.- 7.7 Manifolds and Differential Forms: An Intrinsic Point of View.- References.

O autorze

Piotr Mikusinski received his Ph.D. in mathematics from the Institute of Mathematics of the Polish Academy of Sciences. In 1983, he became visiting lecturer at the University of California at Santa Barbara, where he spent two years. He is currently a member of the faculty in the Department of Mathematics at the University of Central Florida in Orlando. His main research interests are the theory of generalized functions and real analysis.