Books in the Springer Undergraduate Mathematics Series series

Written for undergraduate students with a mathematical background, this book is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. It features numerous theoretical and computational examples.

Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject.

This text provides a lively introduction to pure mathematics. It begins with sets, functions and relations, proof by induction and contradiction, complex numbers, vectors and matrices, and provides a brief introduction to group theory.

Vector calculus is the fundamental language of mathematical physics. Each of the eight chapters introduces a new topic, and to facilitate understanding of the material, frequent reference is made to physical applications.

This textbook provides an accessible account of the history of abstract algebra, tracing a range of topics in modern algebra and number theory back to their modest presence in the seventeenth and eighteenth centuries, and exploring the impact of ideas on the development of the subject.Beginning with Gauss's theory of numbers and Galois's ideas, the book progresses to Dedekind and Kronecker, Jordan and Klein, Steinitz, Hilbert, and Emmy Noether. Approaching mathematical topics from a historical perspective, the author explores quadratic forms, quadratic reciprocity, Fermat's Last Theorem, cyclotomy, quintic equations, Galois theory, commutative rings, abstract fields, ideal theory, invariant theory, and group theory. Readers will learn what Galois accomplished, how difficult the proofs of his theorems were, and how important Camille Jordan and Felix Klein were in the eventual acceptance of Galois's approach to the solution of equations. The book also describes the relationship between Kummer's ideal numbers and Dedekind's ideals, and discusses why Dedekind felt his solution to the divisor problem was better than Kummer's. Designed for a course in the history of modern algebra, this book is aimed at undergraduate students with an introductory background in algebra but will also appeal to researchers with a general interest in the topic. With exercises at the end of each chapter and appendices providing material difficult to find elsewhere, this book is self-contained and therefore suitable for self-study.

This undergraduate textbook provides an approachable and thorough introduction to the topic of algebraic number theory, taking the reader from unique factorisation in the integers through to the modern-day number field sieve.

Elementary ideas about groups and rings are then used to study groups of units, quadratic residues and arithmetic functions with applications to enumeration and cryptography. The final part, suitable for third-year students, uses ideas from algebra, analysis, calculus and geometry to study Dirichlet series and sums of squares.

Essential Partial Differential Equations

Motivation comes from everyday experiences of probability, such as that of a dice or cards, the idea of fairness in games of chance, and the random ways in which, say, birthdays are shared or particular events arise.Applications include branching processes, random walks, Markov chains, queues, renewal theory, and Brownian motion.

This book provides an introduction to the mathematical modelling of real world financial markets and the rational pricing of derivatives, which is part of the theory that not only underpins modern financial practice but is a thriving area of mathematical research.

Introduction to Enumeration provides a comprehensive and practical introduction to this subject giving a clear account of fundamental results and a thorough grounding in the use of powerful techniques and tools.Two major themes run in parallel through the book, generating functions and group theory.

Most of the introductory courses on linear algebra develop the basic theory of finite dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix.

Stochastic processes are tools used widely by statisticians and researchers working in the mathematics of finance. This book for self-study provides a detailed treatment of conditional expectation and probability, a topic that in principle belongs to probability theory, but is essential as a tool for stochastic processes.

This book teaches the art of writing mathematics, an essential -and difficult- skill for any mathematics student.

Real Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. This book offers a fresh approach to a core subject and manages to provide a gentle and clear introduction without sacrificing rigour or accuracy.

This is a basic introduction to modern algebra, providing a solid understanding of the axiomatic treatment of groups and then rings, aiming to promote a feeling for the evolutionary and historical development of the subject. It includes problems and fully worked solutions, enabling readers to master the subject rather than simply observing it.

This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. It includes more than 70 exercises, along with complete solutions, that help illustrate and present all concepts.

This textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises).

This book contains a collection of exercises (called "tapas") at undergraduate level, mainly from the fields of real analysis, calculus, matrices, convexity, and optimization. Most of the problems presented here are non-standard and some require broad knowledge of different mathematical subjects in order to be solved.

Methods of Mathematical Modelling

This introduction to Laplace transforms and Fourier series is aimed at second year students in applied mathematics. Mathematics students do not usually meet this material until later in their degree course but applied mathematicians and engineers need an early introduction.

Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises.