Elements of Advanced Mathematics, Third Edition PDF Download
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Author: Steven G. Krantz
Publisher: CRC Press
ISBN: 1439898340
Category : Mathematics
Languages : en
Pages : 368
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Book Description
For many years, this classroom-tested, best-selling text has guided mathematics students to more advanced studies in topology, abstract algebra, and real analysis. Elements of Advanced Mathematics, Third Edition retains the content and character of previous editions while making the material more up-to-date and significant. This third edition adds four new chapters on point-set topology, theoretical computer science, the P/NP problem, and zero-knowledge proofs and RSA encryption. The topology chapter builds on the existing real analysis material. The computer science chapters connect basic set theory and logic with current hot topics in the technology sector. Presenting ideas at the cutting edge of modern cryptography and security analysis, the cryptography chapter shows students how mathematics is used in the real world and gives them the impetus for further exploration. This edition also includes more exercises sets in each chapter, expanded treatment of proofs, and new proof techniques. Continuing to bridge computationally oriented mathematics with more theoretically based mathematics, this text provides a path for students to understand the rigor, axiomatics, set theory, and proofs of mathematics. It gives them the background, tools, and skills needed in more advanced courses.
Author: Steven G. Krantz
Publisher: CRC Press
ISBN: 1439898340
Category : Mathematics
Languages : en
Pages : 368
Get Book Here
Book Description
For many years, this classroom-tested, best-selling text has guided mathematics students to more advanced studies in topology, abstract algebra, and real analysis. Elements of Advanced Mathematics, Third Edition retains the content and character of previous editions while making the material more up-to-date and significant. This third edition adds four new chapters on point-set topology, theoretical computer science, the P/NP problem, and zero-knowledge proofs and RSA encryption. The topology chapter builds on the existing real analysis material. The computer science chapters connect basic set theory and logic with current hot topics in the technology sector. Presenting ideas at the cutting edge of modern cryptography and security analysis, the cryptography chapter shows students how mathematics is used in the real world and gives them the impetus for further exploration. This edition also includes more exercises sets in each chapter, expanded treatment of proofs, and new proof techniques. Continuing to bridge computationally oriented mathematics with more theoretically based mathematics, this text provides a path for students to understand the rigor, axiomatics, set theory, and proofs of mathematics. It gives them the background, tools, and skills needed in more advanced courses.
Author: Steven G. Krantz
Publisher: CRC Press
ISBN: 1000593282
Category : Mathematics
Languages : en
Pages : 312
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Book Description
This book has enjoyed considerable use and appreciation during its first four editions. With hundreds of students having learned out of early editions, the author continues to find ways to modernize and maintain a unique presentation. What sets the book apart is the excellent writing style, exposition, and unique and thorough sets of exercises. This edition offers a more instructive preface to assist instructors on developing the course they prefer. The prerequisites are more explicit and provide a roadmap for the course. Sample syllabi are included. As would be expected in a fifth edition, the overall content and structure of the book are sound. This new edition offers a more organized treatment of axiomatics. Throughout the book, there is a more careful and detailed treatment of the axioms of set theory. The rules of inference are more carefully elucidated. Additional new features include: An emphasis on the art of proof. Enhanced number theory chapter presents some easily accessible but still-unsolved problems. These include the Goldbach conjecture, the twin prime conjecture, and so forth. The discussion of equivalence relations is revised to present reflexivity, symmetry, and transitivity before we define equivalence relations. The discussion of the RSA cryptosystem in Chapter 8 is expanded. The author introduces groups much earlier. Coverage of group theory, formerly in Chapter 11, has been moved up; this is an incisive example of an axiomatic theory. Recognizing new ideas, the author has enhanced the overall presentation to create a fifth edition of this classic and widely-used textbook.
Author: Steven G. Krantz
Publisher: CRC Press
ISBN: 9781584883036
Category : Mathematics
Languages : en
Pages : 240
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Book Description
The gap between the rote, calculational learning mode of calculus and ordinary differential equations and the more theoretical learning mode of analysis and abstract algebra grows ever wider and more distinct, and students' need for a well-guided transition grows with it. For more than six years, the bestselling first edition of this classic text has helped them cross the mathematical bridge to more advanced studies in topics such as topology, abstract algebra, and real analysis. Carefully revised, expanded, and brought thoroughly up to date, the Elements of Advanced Mathematics, Second Edition now does the job even better, building the background, tools, and skills students need to meet the challenges of mathematical rigor, axiomatics, and proofs. New in the Second Edition: Expanded explanations of propositional, predicate, and first-order logic, especially valuable in theoretical computer science A chapter that explores the deeper properties of the real numbers, including topological issues and the Cantor set Fuller treatment of proof techniques with expanded discussions on induction, counting arguments, enumeration, and dissection Streamlined treatment of non-Euclidean geometry Discussions on partial orderings, total ordering, and well orderings that fit naturally into the context of relations More thorough treatment of the Axiom of Choice and its equivalents Additional material on Russell's paradox and related ideas Expanded treatment of group theory that helps students grasp the axiomatic method A wealth of added exercises
Author: Richard H. Hammack
Publisher:
ISBN: 9780989472111
Category : Mathematics
Languages : en
Pages : 314
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Book Description
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
Author: John D. Baum
Publisher: Courier Corporation
ISBN: 0486668266
Category : Mathematics
Languages : en
Pages : 164
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Book Description
Topology continues to be a topic of prime importance in contemporary mathematics, but until the publication of this book there were few if any introductions to topology for undergraduates. This book remedied that need by offering a carefully thought-out, graduated approach to point set topology at the undergraduate level. To make the book as accessible as possible, the author approaches topology from a geometric and axiomatic standpoint; geometric, because most students come to the subject with a good deal of geometry behind them, enabling them to use their geometric intuition; axiomatic, because it parallels the student's experience with modern algebra, and keeps the book in harmony with current trends in mathematics. After a discussion of such preliminary topics as the algebra of sets, Euler-Venn diagrams and infinite sets, the author takes up basic definitions and theorems regarding topological spaces (Chapter 1). The second chapter deals with continuous functions (mappings) and homeomorphisms, followed by two chapters on special types of topological spaces (varieties of compactness and varieties of connectedness). Chapter 5 covers metric spaces. Since basic point set topology serves as a foundation not only for functional analysis but also for more advanced work in point set topology and algebraic topology, the author has included topics aimed at students with interests other than analysis. Moreover, Dr. Baum has supplied quite detailed proofs in the beginning to help students approaching this type of axiomatic mathematics for the first time. Similarly, in the first part of the book problems are elementary, but they become progressively more difficult toward the end of the book. References have been supplied to suggest further reading to the interested student.
Author: Harry Lass
Publisher: Courier Corporation
ISBN: 0486471861
Category : Mathematics
Languages : en
Pages : 514
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Book Description
Completely self-contained, this survey explores the important topics in pure and applied mathematics. Each chapter can be read independently of the others, and all subjects are unified by cross-references to the complete work. Numerous worked-out examples appear throughout the text, and review questions and references conclude each section. 1957 edition.
Author: Lynn Harold Loomis
Publisher: World Scientific Publishing Company
ISBN: 9814583952
Category : Mathematics
Languages : en
Pages : 595
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Book Description
An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.
Author: Merle C. Potter
Publisher: Springer
ISBN: 3030170683
Category : Technology & Engineering
Languages : en
Pages : 753
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Book Description
This book is designed to serve as a core text for courses in advanced engineering mathematics required by many engineering departments. The style of presentation is such that the student, with a minimum of assistance, can follow the step-by-step derivations. Liberal use of examples and homework problems aid the student in the study of the topics presented. Ordinary differential equations, including a number of physical applications, are reviewed in Chapter One. The use of series methods are presented in Chapter Two, Subsequent chapters present Laplace transforms, matrix theory and applications, vector analysis, Fourier series and transforms, partial differential equations, numerical methods using finite differences, complex variables, and wavelets. The material is presented so that four or five subjects can be covered in a single course, depending on the topics chosen and the completeness of coverage. Incorporated in this textbook is the use of certain computer software packages. Short tutorials on Maple, demonstrating how problems in engineering mathematics can be solved with a computer algebra system, are included in most sections of the text. Problems have been identified at the end of sections to be solved specifically with Maple, and there are computer laboratory activities, which are more difficult problems designed for Maple. In addition, MATLAB and Excel have been included in the solution of problems in several of the chapters. There is a solutions manual available for those who select the text for their course. This text can be used in two semesters of engineering mathematics. The many helpful features make the text relatively easy to use in the classroom.
Author: John Stillwell
Publisher: Princeton University Press
ISBN: 0691171688
Category : Mathematics
Languages : en
Pages : 440
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Book Description
An exciting look at the world of elementary mathematics Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.
Author: Elsa Abbena
Publisher: CRC Press
ISBN: 1351992201
Category : Mathematics
Languages : en
Pages : 1024
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Book Description
Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.
