Paperback
Published: 4th September 2007
ISBN: 9780817647063
Number Of Pages: 484
"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.
"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
Preface | p. ix |
Fundamental Concepts | p. 1 |
Sets | p. 1 |
Functions | p. 12 |
Relations and Equivalence Relations | p. 25 |
Operations on Sets | p. 26 |
Mathematical Systems Considered in This Book | p. 30 |
References and Notes | p. 31 |
References | p. 32 |
Algebraic Structures | p. 33 |
Some Basic Structures of Algebra | p. 34 |
Semigroups and Groups | p. 36 |
Rings and Fields | p. 46 |
Modules, Vector Spaces, and Algebras | p. 53 |
Overview | p. 61 |
Homomorphisms | p. 62 |
Application to Polynomials | p. 69 |
References and Notes | p. 74 |
References | p. 74 |
Vector Spaces and Linear Transformations | p. 75 |
Linear Spaces | p. 75 |
Linear Subspaces and Direct Sums | p. 81 |
Linear Independence, Bases, and Dimension | p. 85 |
Linear Transformations | p. 95 |
Linear Functionals | p. 109 |
Bilinear Functionals | p. 113 |
Projections | p. 119 |
Notes and References | p. 123 |
References | p. 123 |
Finite-Dimensional Vector Spaces and Matrices | p. 124 |
Coordinate Representation of Vectors | p. 124 |
Matrices | p. 129 |
Representation of Linear Transformations by Matrices | p. 129 |
Rank of a Matrix | p. 134 |
Properties of Matrices | p. 136 |
Equivalence and Similarity | p. 148 |
Determinants of Matrices | p. 155 |
Eigenvalues and Eigenvectors | p. 163 |
Some Canonical Forms of Matrices | p. 169 |
Minimal Polynomials, Nilpotent Operators and the Jordan Canonical Form | p. 178 |
Minimal Polynomials | p. 178 |
Nilpotent Operators | p. 185 |
The Jordan Canonical Form | p. 190 |
Bilinear Functionals and Congruence | p. 194 |
Euclidean Vector Spaces | p. 202 |
Euclidean Spaces: Definition and Properties | p. 202 |
Orthogonal Bases | p. 209 |
Linear Transformations on Euclidean Vector Spaces | p. 216 |
Orthogonal Transformations | p. 216 |
Adjoint Transformations | p. 218 |
Self-Adjoint Transformations | p. 221 |
Some Examples | p. 227 |
Further Properties of Orthogonal Transformations | p. 231 |
Applications to Ordinary Differential Equations | p. 238 |
Initial-Value Problem: Definition | p. 238 |
Initial-Value Problem: Linear Systems | p. 244 |
Notes and References | p. 261 |
References | p. 262 |
Metric Spaces | p. 263 |
Definition of Metric Spaces | p. 264 |
Some Inequalities | p. 268 |
Examples of Important Metric Spaces | p. 271 |
Open and Closed Sets | p. 275 |
Complete Metric Spaces | p. 286 |
Compactness | p. 298 |
Continuous Functions | p. 307 |
Some Important Results in Applications | p. 314 |
Equivalent and Homeomorphic Metric Spaces. Topological Spaces | p. 317 |
Applications | p. 323 |
Applications of the Contraction Mapping Principle | p. 323 |
Further Applications to Ordinary Differential Equations | p. 329 |
References and Notes | p. 341 |
References | p. 341 |
Normed Spaces and Inner Product Spaces | p. 343 |
Normed Linear Spaces | p. 344 |
Linear Subspaces | p. 348 |
Infinite Series | p. 350 |
Convex Sets | p. 351 |
Linear Functionals | p. 355 |
Finite-Dimensional Spaces | p. 360 |
Geometric Aspects of Linear Functionals | p. 363 |
Extension of Linear Functionals | p. 367 |
Dual Space and Second Dual Space | p. 370 |
Weak Convergence | p. 372 |
Inner Product Spaces | p. 375 |
Orthogonal Complements | p. 381 |
Fourier Series | p. 387 |
The Riesz Representation Theorem | p. 393 |
Some Applications | p. 394 |
Approximation of Elements in Hilbert Space (Normal Equations) | p. 395 |
Random Variables | p. 397 |
Estimation of Random Variables | p. 398 |
Notes and References | p. 404 |
References | p. 404 |
Linear Operators | p. 406 |
Bounded Linear Transformations | p. 407 |
Inverses | p. 415 |
Conjugate and Adjoint Operators | p. 419 |
Hermitian Operators | p. 427 |
Other Linear Operators: Normal Operators, Projections, Unitary Operators, and Isometric Operators | p. 431 |
The Spectrum of an Operator | p. 439 |
Completely Continuous Operators | p. 447 |
The Spectral Theorem for Completely Continuous Normal Operators | p. 454 |
Differentiation of Operators | p. 458 |
Some Applications | p. 465 |
Applications to Integral Equations | p. 465 |
An Example from Optimal Control | p. 468 |
Minimization of Functionals: Method of Steepest Descent | p. 471 |
References and Notes | p. 473 |
References | p. 473 |
Index | p. 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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