This calculus textbook aims to introduce students to the fundamental concepts of derivatives and integrals, balancing accessibility with necessary technical exercises. It is designed for beginners rather than advanced mathematicians, providing a pleasant learning experience while maintaining essential rigor in the subject matter.
Based on the work in algebraic geometry by Norwegian mathematician Niels Henrik Abel (1802–29), this monograph was originally published in 1959 and reprinted later in author Serge Lang's career without revision. The treatment remains a basic advanced text in its field, suitable for advanced undergraduates and graduate students in mathematics. Prerequisites include some background in elementary qualitative algebraic geometry and the elementary theory of algebraic groups. The book focuses exclusively on Abelian varieties rather than the broader field of algebraic groups; therefore, the first chapter presents all the general results on algebraic groups relevant to this treatment. Each chapter begins with a brief introduction and concludes with a historical and bibliographical note. Topics include general theorems on Abelian varieties, the theorem of the square, divisor classes on an Abelian variety, functorial formulas, the Picard variety of an arbitrary variety, the I-adic representations, and algebraic systems of Abelian varieties. The text concludes with a helpful Appendix covering the composition of correspondences.
Focusing on Diophantine approximations, this book explores three key aspects: the formal connections between counting processes and relevant functions, the identification of these functions for classical numbers, and specific asymptotic estimates that are valid in almost all cases. Through significant examples, it aims to deepen understanding of these mathematical concepts and their interrelationships.
Focusing on the infinite dimensional representation theory of semisimple Lie groups, this book specifically examines SL2(R). It highlights the importance of this area in relation to fields like number theory, particularly through Langlands' work. The text aims to simplify the complexities of representation theory, which has evolved rapidly, making it challenging for newcomers. With only basic prerequisites in real analysis and differential equations, it is designed to be accessible to a broad audience, facilitating entry into this advanced subject.
This is a short text in linear algebra, intended for a one-term course. In the first chapter, Lang discusses the relation between the geometry and the algebra underlying the subject, and gives concrete examples of the notions which appear later in the book. 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. The book contains a large number of exercises, some of the routine computational type, while others are conceptual.
Author is well-known and established book author (all Serge Lang books are now published by Springer); Presents a brief introduction to the subject; All manifolds are assumed finite dimensional in order not to frighten some readers; Complete proofs are given; Use of manifolds cuts across disciplines and includes physics, engineering and economics
From the reviews "This is a reprint of the original edition of Lang’s ‘A First Course in Calculus’, which was first published in 1964....The treatment is ‘as rigorous as any mathematician would wish it’....[The exercises] are refreshingly simply stated, without any extraneous verbiage, and at times quite challenging....There are answers to all the exercises set and some supplementary problems on each topic to tax even the most able." --Mathematical Gazette
For many years, Serge Lang has given talks on selected items in mathematics which could be extracted at a level understandable by those who have had calculus. Written in a conversational tone, Lang now presents a collection of those talks as a book covering such topics as: prime numbers, the abc conjecture, approximation theorems of analysis, Bruhat-Tits spaces, and harmonic and symmetric polynomials. Each talk is written in a lively and informal style meant to engage any reader looking for further insight into mathematics.
Focusing on qualitative algebraic geometry, this book serves as an introduction to the Weil-Zariski framework, expanding on lectures from a series of courses initiated by Zariski. It provides a comprehensive overview of Weil's "Foundations" and contextualizes the development of modern algebraic geometry prior to the introduction of sheaves. This reprint preserves the original text, offering readers an authentic experience of the foundational concepts in the field.