Catalogue


Analysis in vector spaces : a course in advanced calculus /
Mustafa A. Akcoglu, Paul F.A. Bartha, Dzung M. Ha.
imprint
Hoboken, N.J. : Wiley-Interscience, c2009.
description
xii, 465 p. : ill. ; 25 cm.
ISBN
0470148241 (cloth), 9780470148242 (cloth)
format(s)
Book
Holdings
More Details
imprint
Hoboken, N.J. : Wiley-Interscience, c2009.
isbn
0470148241 (cloth)
9780470148242 (cloth)
general note
Includes index.
catalogue key
6783394
A Look Inside
About the Author
Author Affiliation
Mustafa A. Akcoglu, PhD, is Professor Emeritus in the Department of Mathematics at the University of Toronto, Canada. He has authored or coauthored over sixty journal articles on the topics of ergodic theory, functional analysis, and harmonic analysis. Paul F.A. Bartha, PhD, is Associate Professor in the Department of Philosophy at. The University of British Columbia, Canada. He has authored or coauthored journal articles on topics such as probability and symmetry, probabilistic paradoxes, and the general philosophy of science. Dzung Minh Ha, PHD, is Associate Professor in the Department of Mathematics at Ryerson University, Canada. Dr. Ha focuses his research in the areas of ergodic theory operator theory.
Full Text Reviews
Appeared in Choice on 2009-11-01:
This work by Akcoglu (emer., Univ. of Toronto), Bartha (Univ. of British Columbia), and Ha (Ryerson Univ.) combines features of introductory real analysis books (e.g., basics of sets, functions, structure and topology of the real line), with some material on linear algebra and normed vector spaces. After those basics, the book addresses calculus on vector spaces, but at a level well beyond the multivariate calculus appearing in standard "fat" calculus texts such as James Stewart's Calculus (6th ed., 2008) in order to provide the third course in a calculus sequence. The book discusses derivatives of vector-valued functions, and also includes chapters such as "Diffeomorphisms and Manifolds," "Multiple Integrals" (introduced through Jordan sets and volume), "Integration on Manifolds," and "Stokes' Theorem." The authors do not shy away from doing the hard work involved in proving say, the change of variable theorem for integration, the inverse function theorem, and Stokes's theorem--work which is not generally seen in standard calculus books--and thus they are quite correct when they state that students need a firm grip on single-variable calculus and some linear algebra, and a good comfort level with the comprehension and construction of rigorous proofs. Includes many examples and an excellent selection of exercises. Summing Up: Recommended. Upper-division undergraduates and graduate students. D. Robbins Trinity College (CT)
Reviews
Review Quotes
"The authors do not shy away from doing the hard work involved in proving say, the change of variable theorem for integration, the inverse function theorem, and Stokes's theorem--work which is not generally seen in standard calculus books--and thus they are quite correct when they state that students need a firm grip on single-variable calculus and some linear algebra, and a good comfort level with the comprehension and construction of rigorous proofs. Includes many examples and an excellent selection of exercises." ( CHOICE, November 2010)
"The authors do not shy away from doing the hard work involved in proving say, the change of variable theorem for integration, the inverse function theorem, and Stokes's theorem--work which is not generally seen in standard calculus books--and thus they are quite correct when they state that students need a firm grip on single-variable calculus and some linear algebra, and a good comfort level with the comprehension and construction of rigorous proofs. Includes many examples and an excellent selection of exercises." (CHOICE, November 2010)
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Choice, November 2009
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Summaries
Back Cover Copy
A rigorous introduction to calculus in vector spacesThe concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences.The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes:Sets and functionsReal numbersVector functionsNormed vector spacesFirst- and higher-order derivativesDiffeomorphisms and manifoldsMultiple integralsIntegration on manifoldsStokes' theoremBasic point set topologyNumerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants.Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.
Bowker Data Service Summary
A thorough treatment of the portion of analysis that addresses functions of several variables and multiple integrals, this book is self-contained, and the background material in the first few chapters is sufficient to enable well-motivated students to follow the book without undue difficulties.
Main Description
A rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences. The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes: Sets and functions Real numbers Vector functions Normed vector spaces First- and higher-order derivatives Diffeomorphisms and manifolds Multiple integrals Integration on manifolds Stokes theorem Basic point set topology Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants. Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.
Main Description
A rigorous introduction to calculus in vector spaces The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences. The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes: Sets and functions Real numbers Vector functions Normed vector spaces First- and higher-order derivatives Diffeomorphisms and manifolds Multiple integrals Integration on manifolds Stokes' theorem Basic point set topology Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants. Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.
Main Description
A Rigorous Introdution To Calculus In Vector Spaces
Main Description
The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences.The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes:Sets and functionsReal numbersVector functionsNormed vector spacesFirst- and higher-order derivativesDiffeomorphisms and manifoldsMultiple integralsIntegration on manifoldsStokes' theoremBasic point set topologyNumerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants.Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts.
Main Description
This text is intended for a second course in analysis, concentrating on differentiation and integration of functions of several variables. The material is presented as a unified course on vector spaces. The book successfully provides a firm mathematical foundation for further study and established an indispensable background for further studies in mathematics or related disciplines that involve heavy use of mathematical techniques and concepts. The book also provides exposure to a variety of other mathematical topics such as basic point-set topology, measure theory, differential geometry, and the theory of manifolds.
Table of Contents
Prefacep. ix
Background Material
Sets and Functionsp. 3
Sets in Generalp. 3
Sets of Numbersp. 10
Functionsp. 17
Real Numbersp. 31
Review of the Order Relationsp. 32
Completeness of Real Numbersp. 36
Sequences of Real Numbersp. 40
Subsequencesp. 45
Series of Real Numbersp. 50
Intervals and Connected Setsp. 54
Vector Functionsp. 61
Vector Spaces: The Basicsp. 62
Bilinear Functionsp. 82
Multilinear Functionsp. 88
Inner Productsp. 95
Orthogonal Projectionsp. 103
Spectral Theoremp. 109
Differentiation
Normed Vector Spacesp. 123
Preliminariesp. 124
Convergence in Normed Spacesp. 128
Norms of Linear and Multilinear Transformationsp. 135
Continuity in Normed Spacesp. 142
Topology of Normed Spacesp. 156
Derivativesp. 175
Functions of a Real Variablep. 176
Differentiable Functionsp. 190
Existence of Derivativesp. 201
Partial Derivativesp. 205
Rules of Differentiationp. 211
Differentiation of Productsp. 218
Diffeomorphisms and Manifoldsp. 225
The Inverse Function Theoremp. 226
Graphsp. 238
Manifolds in Parametric Representationsp. 243
Manifolds in Implicit Representationsp. 252
Differentiation on Manifoldsp. 260
Higher-Order Derivativesp. 267
Definitionsp. 267
Change of Order in Differentiationp. 270
Sequences of Polynomialsp. 273
Local Extremal Valuesp. 282
Integration
Multiple Integralsp. 287
Jordan Sets and Volumep. 289
Integralsp. 303
Images of Jordan Setsp. 321
Change of Variablesp. 328
Integration on Manifoldsp. 339
Euclidean Volumesp. 340
Integration on Manifoldsp. 345
Oriented Manifoldsp. 353
Integrals of Vector Fieldsp. 361
Integrals of Tensor Fieldsp. 366
Integration on Graphsp. 371
Stokes' Theoremp. 381
Basic Stokes' Theoremp. 382
Flowsp. 386
Flux and Change of Volume in a Flowp. 390
Exterior Derivativesp. 396
Regular and Almost Regular Setsp. 401
Stokes' theorem on Manifoldsp. 412
Appendices
Construction of the real numbersp. 419
Field and Order Axioms in Qp. 420
Equivalence Classes of Cauchy Sequences in Qp. 421
Completeness of Rp. 427
Dimension of a vector spacep. 431
Bases and linearly independent subsetsp. 432
Determinantsp. 435
Permutationsp. 435
Determinants of Square Matricesp. 437
Determinant Functionsp. 439
Determinant of a Linear Transformationp. 443
Determinants on Cartesian Productsp. 444
Determinants in Euclidean Spacesp. 445
Trace of an Operatorp. 448
Partitions of unityp. 451
Partitions of Unityp. 452
Indexp. 455
Table of Contents provided by Ingram. All Rights Reserved.

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