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This is a short text in linear algebra, intended for a one-term course. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues.

Outlines an elementary, one semester course, which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. This book focuses on questions which give analysis its inherent fascination.

The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development. of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. In this book we develop enough abstract algebra to prove that these constructions are impossible. Our approach introduces all the relevant concepts about fields in a way which is more concrete than usual and which avoids the use of quotient structures (and even of the Euclidean algorithm for finding the greatest common divisor of two polynomials). Having the geometrical questions as a specific goal provides motivation for the introduction of the algebraic concepts and we have found that students respond very favourably. We have used this text to teach second-year students at La Trobe University over a period of many years, each time refining the material in the light of student performance.

Offering a unified exposition of calculus and classical real analysis, this textbook presents a meticulous introduction to single¿variable calculus. Throughout, the exposition makes a distinction between the intrinsic geometric definition of a notion and its analytic characterization, establishing firm foundations for topics often encountered earlier without proof. Each chapter contains numerous examples and a large selection of exercises, as well as a ¿Notes and Comments¿ section, which highlights distinctive features of the exposition and provides additional references to relevant literature.This second edition contains substantial revisions and additions, including several simplified proofs, new sections, and new and revised figures and exercises. A new chapter discusses sequences and series of real¿valued functions of a real variable, and their continuous counterpart: improper integrals depending on a parameter. Two new appendices cover a construction of the real numbers using Cauchy sequences, and a self¿contained proof of the Fundamental Theorem of Algebra.In addition to the usual prerequisites for a first course in single¿variable calculus, the reader should possess some mathematical maturity and an ability to understand and appreciate proofs. This textbook can be used for a rigorous undergraduate course in calculus, or as a supplement to a later course in real analysis. The authors¿ A Course in Multivariable Calculus is an ideal companion volume, offering a natural extension of the approach developed here to the multivariable setting.From reviews:[The first edition is] a rigorous, well-presented and original introduction to the core of undergraduate mathematics ¿ first-year calculus. It develops this subject carefully from a foundation of high-school algebra, with interesting improvements and insights rarely found in other books. [¿] This book is a tour de force, and a necessary addition to the library of anyone involved in teaching calculus, or studying it seriously. N.J. Wildberger, Aust. Math. Soc. Gaz.

Throughout, the theory unfolds via over 150 carefully selected problems for students to solve, many of which connect to state-of-the-art research. Inquiry-Based Enumerative Combinatorics is ideal for lower-division undergraduate students majoring in math or computer science, as there are no formal mathematics prerequisites.

It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems.

This textbook provides a unified and concise exploration of undergraduate mathematics by approaching the subject through its history. however, biographical sketches have been omitted. Mathematics and Its History: A Concise Edition is an essential resource for courses or reading programs on the history of mathematics.

This book develops the theory of multivariable analysis, building on the single variable foundations established in the companion volume, Real Analysis: Foundations and Functions of One Variable.

Written to accompany a one- or two-semester course, this text combines rigor and wit to cover a plethora of topics from integers to uncountable sets. It teaches methods such as axiom, theorem, and proof through the mathematics rather than in abstract isolation.

This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces.

The third edition of this concise, popular textbook on elementary differential equations gives instructors an alternative to the many voluminous texts on the market.

Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri's Principle, and, in many cases, the ability to see an integral result from measure theory.

Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book.

The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra.

This book clearly explains the important theorems of single variable calculus. It includes an elementary introduction to complex numbers and complex-valued functions, key applications of calculus, and an introduction to probability and information theory.

This textbook is designed for a one year course covering the fundamentals of partial differential equations, aimed towards advanced undergraduates and beginning graduate students. It features examples and exercises connected to real world problems.

This text plugs a gap in the standard curriculum by linking set theory with analysis. It features a distinctive, detailed treatment of the real numbers system, and combines an introduction to set theory with exposition of the essence of analysis.

Unlike other texts in differential geometry, this book develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. Emphasizes the consequences of a definition and the use of examples and constructions.

For three decades, this classic has been a must-have textbook for transitional courses from calculus to analysis, celebrated for its clear style and simple proofs. This edition adds material on the irrationality of pi, the Baire category theorem and more.

This updated and revised second edition is designed to help students advance from basic calculus to higher-level linear and abstract algebra and number theory. It introduces an array of fundamental structures and shows how to balance intuition and rigor.

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics. It includes exercises and examples at the end of each section.

Based on courses given at Eoetvoes Lorand University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable - systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student's mathematical intuition.