Description This book does a very good job conveying what it is like to “do” mathematics as a creative activity. Each of the four chapters presents a topic as a sequence of problems, conjectures, explorations, heuristic approaches, and rigorous justification. Because the number of topics is small there is time for students to explore without being rushed to “learn” the material. Courses that would use this book are not pre-requisite courses and so there is not the pressure to cover the content. Although written for students with minimal background, this book could profitably be used in an upper level course for advanced students or for independent study courses or for senior project/thesis ideas.
Description This textbook, originally published by W. H. Freeman in 1995, is the result of the Five College Calculus Project, a multi-year project funded by the NSF beginning in 1986. The unifying concept of a dynamical system is used throughout to develop calculus in the broader context of scientific questions. “Therefore, differential equations belong at the center of calculus, and technology makes this possible at the introductory level.” The text is aimed at a broad audience as the authors believe “that calculus is one of the great bonds that unifies science. All students should have an opportunity to see how the language and tools of calculus help forge that bond.” (Quotes are from the overview on the book’s home page, which gives an excellent picture of the nature of this unusual book.)
Description Originally published in 1980 by Benjamin/Cummings, the current edition was published by Springer-Verlag in 1985 and is still in print. The publisher makes the PDF versions freely available and allows individuals to print single copies for their own use.
Description Parts I and II deal with two fundamental aspects of combinatorics: enumeration and graph theory. “Enumeration” can mean either counting or listing things. Mathematicians have generally limited their attention to counting, but listing plays an important role in computer science, so we discuss both aspects. After introducing the basic concepts of “graph theory” in Part II, we present a variety of applications of interest in computer science and mathematics. Induction and recursion play a fundamental role in mathematics. The usefulness of recursion in computer science and in its interaction with combinatorics is the subject of Part III. In Part IV we look at “generating functions,” a powerful tool for studying counting problems.
Description This book is written with the author’s conviction that “it is easier to build a course from a base than to extract it from a big book.” The five chapters deal with fundamentals, groups, rings, matrix rings, and linear algebra. Despite the small size there is enough material for a full year. Many of the easy proofs are left as exercises. An instructor using this book should expect to add examples and build on the structure provided.
Description The writing style is informal and conversational. Examples appear before formal definitions. In addition to the usual material for a year long course on groups, rings, and fields, this book treats group actions, modules, and the Galois correspondence. The prerequisite is a good grasp of linear algebra, and overall the level of sophistication is above that of most undergraduate texts. There are appendices with background material in logic, set theory, induction, complex numbers, and linear algebra.
Description The approach in this 165 page book tends to be more sophisticated than other books for the first number theory course, but it motivates much of the material with public key cryptography. It also uses Sage in order to deal with more realistic examples—such as RSA codes based on primes with more than 30 digits. Although published commercially by Springer, the publisher has granted access to a free pdf version that individuals can download, use, and print.
Description This text was originally published by Prentice Hall in 2001. The second (and current) edition published in 2008 is essentially the same with misprints and other errors corrected. For a discussion of this text and the graduate analysis text by the same authors see the review by James Caragal in the UMAP Journal.
Description This text grew out of the lecture notes of a single semester undergraduate course taught at Binghamton University (SUNY) and San Francisco State University, and it has benefited from the comments and suggestions from other instructors who have used the book. From the introduction: 'For many of our students, complex analysis is their first rigorous analysis (if not mathematics) class they take, and these notes reflect this very much. We tried to rely on as few concepts from real analysis as possible. In particular, series and sequences are treated “from scratch.”'
Description Geometry with an Introduction to Cosmic Topology offers an introduction to non-Euclidean geometry through the lens of questions that have ignited the imagination of stargazers since antiquity. What is the shape of the universe? Does the universe have an edge? Is it infinitely big?This text is intended for undergraduate mathematics and physics majors who have completed a multivariable calculus course and are ready for a course that practices the habits of thought needed in advanced courses of the undergraduate mathematics curriculum. The text is also particularly suited to independent study, with essays and other discussions complementing the mathematical content in several sections.
Description This is a highly engaging “textbook” that makes extensive use of the capability for interactive instruction with current software, computers, and the Internet. Although it requires an internet connection, the students need only an up-to-date browser (Firefox recommended) and so there is no need to buy or install any other software. The author has been developing the course material for more than 15 years, and it is both stable and reliable. It resides on a server in the Statistics Department at UC Berkeley and will be there for the foreseeable future.
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