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Algebra and Analysis for Engineers and Scientists - Anthony N. Michel

Algebra and Analysis for Engineers and Scientists

Paperback Published: 4th September 2007
ISBN: 9780817647063
Number Of Pages: 484

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"This is an intermediate level text, with exercises, whose avowed purpose is to provide the science and engineering graduate student with an appropriate modern mathematical (analysis and algebra) background in a succinct, but nontrivial, manner.... T]he book is quite thorough and can serve as a text, for self-study, or as a reference." -Mathematical Reviews

Written for graduate and advanced undergraduate students in engineering and science, this classic book focuses primarily on set theory, algebra, and analysis. Useful as a course textbook, for self-study, or as a reference, the work is intended to:

* provide readers with appropriate mathematical background for graduate study in engineering or science;

* allow students in engineering or science to become familiar with a great deal of pertinent mathematics in a rapid and efficient manner without sacrificing rigor;

* give readers a unified overview of applicable mathematics, enabling them to choose additional, advanced topical courses in mathematics more intelligently.

Whereas these objectives for writing this book were certainly pertinent over twenty years ago when the work was first published, they are even more compelling now. Today's graduate students in engineering or science are expected to be more knowledgeable and sophisticated in mathematics than students in the past. Moreover, today's graduate students in engineering or science are expected to be familiar with a great deal of ancillary material (primarily in the computer science area), acquired in courses that did not even exist a couple of decades ago.

The book is divided into three parts: set theory (Chapter 1), algebra (Chapters 2-4), and analysis (Chapters 5-7). The first two chapters deal with the fundamental concepts of sets, functions, relations and equivalence relations, and algebraic structures. Chapters 3 and 4 cover vector spaces and linear transformations, and finite-dimensional vector spaces and matrices. The last three chapters investigate metric spaces, normed and inner product spaces, and linear operators. Because of its flexible structure, Algebra and Analysis for Engineers and Scientists may be used either in a one- or two-semester course by deleting appropriate sections, taking into account the students' backgrounds and interests.

A generous number of exercises have been integrated into the text, and a section of references and notes is provided at the end of each chapter. Applications of algebra and analysis having a broad appeal are also featured, including topics dealing with ordinary differential equations, integral equations, applications of the contraction mapping principle, minimization of functionals, an example from optimal control, and estimation of random variables.

Supplementary material for students and instructors is available at http: //Michel.Herget.net.

Industry Reviews

"This book is a useful compendium of the mathematics of (mostly) finite-dimensional linear vector spaces (plus two final chapters on infinite-dimensional spaces), which do find increasing application in many branches of engineering and science.... The treatment is thorough; the book will certainly serve as a valuable reference." --American Scientist

"The authors present topics in algebra and analysis for students in engineering and science..... Each chapter is organized to include a brief overview, detailed topical discussions and references for further study. Notes about the references guide the student to collateral reading. Theorems, definitions, and corollaries are illustrated with examples. The student is encouraged to prove some theorems and corollaries as models for proving others in exercises. In most chapters, the authors discuss constructs used to illustrate examples of applications. Discussions are tied together by frequent, well written notes. The tables and index are good. The type faces are nicely chosen. The text should prepare a student well in mathematical matters." --Science Books and Films

"This is an intermediate level text, with exercises, whose avowed purpose is to provide the science and engineering graduate student with an appropriate modern mathematical (analysis and algebra) background in a succinct, but not trivial, manner. After some fundamentals, algebraic structures are introduced followed by linear spaces, matrices, metric spaces, normed and inner product spaces and linear operators.... While one can quarrel with the choice of specific topics and the omission of others, the book is quite thorough and can serve as a text, for self-study, or as a reference." --Mathematical Reviews

"The authors designed a typical work from graduate mathematical lectures: formal definitions, theorems, corollaries, proofs, examples, and exercises. It is to be noted that problems to challenge students' comprehension are interspersed throughout each chapter rather than at the end." --CHOICE

Prefacep. ix
Fundamental Conceptsp. 1
Setsp. 1
Functionsp. 12
Relations and Equivalence Relationsp. 25
Operations on Setsp. 26
Mathematical Systems Considered in This Bookp. 30
References and Notesp. 31
Referencesp. 32
Algebraic Structuresp. 33
Some Basic Structures of Algebrap. 34
Semigroups and Groupsp. 36
Rings and Fieldsp. 46
Modules, Vector Spaces, and Algebrasp. 53
Overviewp. 61
Homomorphismsp. 62
Application to Polynomialsp. 69
References and Notesp. 74
Referencesp. 74
Vector Spaces and Linear Transformationsp. 75
Linear Spacesp. 75
Linear Subspaces and Direct Sumsp. 81
Linear Independence, Bases, and Dimensionp. 85
Linear Transformationsp. 95
Linear Functionalsp. 109
Bilinear Functionalsp. 113
Projectionsp. 119
Notes and Referencesp. 123
Referencesp. 123
Finite-Dimensional Vector Spaces and Matricesp. 124
Coordinate Representation of Vectorsp. 124
Matricesp. 129
Representation of Linear Transformations by Matricesp. 129
Rank of a Matrixp. 134
Properties of Matricesp. 136
Equivalence and Similarityp. 148
Determinants of Matricesp. 155
Eigenvalues and Eigenvectorsp. 163
Some Canonical Forms of Matricesp. 169
Minimal Polynomials, Nilpotent Operators and the Jordan Canonical Formp. 178
Minimal Polynomialsp. 178
Nilpotent Operatorsp. 185
The Jordan Canonical Formp. 190
Bilinear Functionals and Congruencep. 194
Euclidean Vector Spacesp. 202
Euclidean Spaces: Definition and Propertiesp. 202
Orthogonal Basesp. 209
Linear Transformations on Euclidean Vector Spacesp. 216
Orthogonal Transformationsp. 216
Adjoint Transformationsp. 218
Self-Adjoint Transformationsp. 221
Some Examplesp. 227
Further Properties of Orthogonal Transformationsp. 231
Applications to Ordinary Differential Equationsp. 238
Initial-Value Problem: Definitionp. 238
Initial-Value Problem: Linear Systemsp. 244
Notes and Referencesp. 261
Referencesp. 262
Metric Spacesp. 263
Definition of Metric Spacesp. 264
Some Inequalitiesp. 268
Examples of Important Metric Spacesp. 271
Open and Closed Setsp. 275
Complete Metric Spacesp. 286
Compactnessp. 298
Continuous Functionsp. 307
Some Important Results in Applicationsp. 314
Equivalent and Homeomorphic Metric Spaces. Topological Spacesp. 317
Applicationsp. 323
Applications of the Contraction Mapping Principlep. 323
Further Applications to Ordinary Differential Equationsp. 329
References and Notesp. 341
Referencesp. 341
Normed Spaces and Inner Product Spacesp. 343
Normed Linear Spacesp. 344
Linear Subspacesp. 348
Infinite Seriesp. 350
Convex Setsp. 351
Linear Functionalsp. 355
Finite-Dimensional Spacesp. 360
Geometric Aspects of Linear Functionalsp. 363
Extension of Linear Functionalsp. 367
Dual Space and Second Dual Spacep. 370
Weak Convergencep. 372
Inner Product Spacesp. 375
Orthogonal Complementsp. 381
Fourier Seriesp. 387
The Riesz Representation Theoremp. 393
Some Applicationsp. 394
Approximation of Elements in Hilbert Space (Normal Equations)p. 395
Random Variablesp. 397
Estimation of Random Variablesp. 398
Notes and Referencesp. 404
Referencesp. 404
Linear Operatorsp. 406
Bounded Linear Transformationsp. 407
Inversesp. 415
Conjugate and Adjoint Operatorsp. 419
Hermitian Operatorsp. 427
Other Linear Operators: Normal Operators, Projections, Unitary Operators, and Isometric Operatorsp. 431
The Spectrum of an Operatorp. 439
Completely Continuous Operatorsp. 447
The Spectral Theorem for Completely Continuous Normal Operatorsp. 454
Differentiation of Operatorsp. 458
Some Applicationsp. 465
Applications to Integral Equationsp. 465
An Example from Optimal Controlp. 468
Minimization of Functionals: Method of Steepest Descentp. 471
References and Notesp. 473
Referencesp. 473
Indexp. 475
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780817647063
ISBN-10: 0817647066
Audience: General
Format: Paperback
Language: English
Number Of Pages: 484
Published: 4th September 2007
Publisher: BIRKHAUSER BOSTON INC
Country of Publication: US
Dimensions (cm): 23.17 x 15.93  x 2.46
Weight (kg): 0.7

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