1. Mathematical analysis: a modern approach to advanced calculus (1957), by Tom M Apostol.
Mathematical reviews MR0087718.
This is a book on advanced calculus, in that many topics frequently omitted are treated here in detail. On the other hand, the book is easily adaptable to more modest needs. ... There are nearly 500 problems, which include counterexamples, and generalizations of results, as well as straightforward exercises. The presentation is simple and clear. An outstanding feature is the unusually careful motivation provided for each new concept (but with no trace of condescension); likewise, well-chosen comments elucidate the wherefore of proofs. Occasional diagrams, such as to illustrate a suitable smaller neighborhood within a given one, are another welcome feature.
1.2. Review by: R A Good.
Science, New Series 127 (3293) (1958), 292.
The viewpoint of this textbook is excellently summarized by its subtitle "A modern approach to advanced Calculus." The material is serious substantial mathematics; topics traditionally from advanced calculus are supplemented with extra background for function theory. The author states his aim to be a development which is "honest, rigorous, up-to-date, and ... not too pedantic. ... The style is unusually readable. ... This book is difficult for a weak class but should be thoroughly appreciated as a text for the strong student.
1.3. Review by: F M Mears.
Amer. Math. Monthly 65 (6) (1958), 463-464.
This book treats the topics which usually fall under the heading of "Advanced Calculus." It includes rigorous proofs of many of the theorems which are usually considered too difficult for an advanced calculus text, but too elementary for a course on function theory. ... In the author's words, "the book helps to fill the gap between elementary calculus and advanced courses in analysis. Moe important than this, it introduced the reader to some of the abstract thinking that pervades modern mathematics."
The aim has been to provide a development of analysis at the advanced calculus level that is honest, rigorous, up to date and, at the same time, not too pedantic. The second edition differs from the first in many respects. Point set topology is developed in the setting of general metric spaces as well as in Euclidean n-space, and two new chapters have been added on Lebesgue integration. The material on line integrals, vector analysis and surface integrals has been deleted. The order of some chapters has been rearranged, many sections have been completely rewritten, and several new exercises have been added. The development of Lebesgue integration follows the Riesz-Nagy approach which focuses directly on functions and their integrals and does not depend on measure theory. The treatment here is simplified, spread out and somewhat rearranged for presentation at the undergraduate level.
Amer. Math. Monthly 69 (5) (1962), 449-451.
This is Volume I of a two-volume first course in calculus and analytic geometry. It attracts attention not only as the first venture of a young and ambitious publishing firm, but also by its intrinsic qualities, being venturesome, carefully planned and written, rigorous, readable, lively, and the opposite of banal. Designed, according to the preface, to accommodate a variety of backgrounds and interests, its level and tone in fact aim it most naturally at a class of students whose background (or ability) is rather stronger and whose interest is rather more purely mathematical than most colleges are accustomed to, but which we hope accurately anticipates the wave of the future. This book is perhaps most nearly a 1961 counterpart of R Courant's still classic text of 1934. Indeed, it approaches the latter in soundness, clarity and elegance of exposition, and surpasses it in several valuable respects, especially the exercises.
Mathematical reviews MR0214705.
The book is written for those who wish to have some knowledge of calculus and analytic geometry. The presentation differs considerably from what is usually found in standard books of the same nature. Each important new concept is preceded by a historical introduction and, whenever possible, is explained geometrically. Each chapter begins with an introduction and is followed by unsolved examples, some of which are illuminating while others are routine. Also, there are several worked examples to illustrate new concepts.
4.2. Review by: E D Bolker.
Amer. Math. Monthly 77 (1) (1970), 88-89.
This classic book, one of the first in the new wave of calculus texts which broke several years ago, has been improved in its new edition, with "... smaller chapters, ... mean value theorem and routine applications ... introduced ... earlier, ... new illustrative examples, (expanded) applications to physics and engineering, and many new and easier exercises." (Emphasis mine.) It is now easier to use in the classroom and no less stimulating.
4.3. Review (of 1991 reprint) by: John Leamy.
The Mathematics Teacher 84 (3) (1991), 236.
This reprint of a 1967 edition treats integration before differentiation and includes a substantial amount of linear algebra. More than enough material is included for a year's course. The level of theory is higher than in most current texts, and no use is made of colour to highlight results. This text would be a choice for good, highly motivated students.
Amer. Math. Monthly 70 (5) (1963), 587-588.
There is no need to praise again the good writing, shrewd organisation, and lively adult tone of this fine text (see 3.1 above). The second volume if anything surpasses the first in excellence of workmanship, and occasions for detailed criticism are not worth mentioning. The coverage of calculus is extensive, including such "advanced" topics as change of variables in multiple integrals, Green's and Stokes' theorems on line and surface integrals, nth order differential equations and Picard's existence theorem. ... While this book expounds useful technique, and the theory behind it, and while the contact with physics is increased in Volume II, it is still not a course in applied mathematics. It makes no attempt to teach the art of formulating a crude problem into a mathematical problem, and it works out relatively few applications outside mathematics. Without other courses in science where mathematics is systematically used, the student would be left with an excellent idea of the spirit of mathematics, but with only a rather thin idea, based on hearsay rather than experience, of its power.
Sound training in technique is combined with a strong theoretical development. Every effort has been made to convey the spirit of modern mathematics without undue emphasis on formalization.
6.2. Review by: F Brooks.
Mathematical reviews MR0344384.
The second edition provides more exercises than the first, mainly easier ones, some reordering and rewriting of material of the first edition, but mainly it adds a new section on linear algebra.
Mathematical reviews MR0434929.
This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages. ... The presentation is invariably lucid and the book is a real pleasure to read. The author may well be congratulated both for having written such a book and also for his good fortune to have had, during 25 years, such excellent students, who could be taught this sophisticated material - by starting from scratch.
This is the second volume of a 2-volume textbook that evolved from a course offered at the California Institute of Technology. It presupposes a background in number theory comparable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis. Both volumes of this work emphasize classical aspects of a subject which in recent years has undergone a great deal of modern development.
8.2. Review by E Grosswald.
Mathematical reviews MR0422157.
The first reaction of many a reader may well be one of satisfaction (tinged, perhaps, with a whiff of envy) that there still exist groups of students sufficiently gifted and interested in this beautiful subject matter to permit the presentation of a course from which the present volume could evolve. It covers selected topics from the general area specified by the title. Some of them (such as elliptic functions, or Kronecker's approximation theorem) are standard topics; they were included, presumably, mainly in order to achieve the desired (and very remarkable) degree of self-containment of the book. Toward this end, even a proof of Rouché's theorem is included. Needless to say, the presentation of this classical material is as neat and clean, as one has come to expect from the author. On the other hand, the book presents in complete detail some items that are rarely found anywhere, except in original papers. ... The book is beautifully written, discusses some fascinating topics and is a delight to read. In fact, it is surprisingly easy to read, in particular in view of the enormous amount of material presented in less than 200 pages.
The major change is an alternate treatment of the transformation formula for the Dedekind eta function, which appears in a five-page supplement to Chapter 3, inserted at the end of the book (just before the bibliography). Otherwise, the second edition is almost identical to the first. Misprints have been repaired, there are minor changes in the exercises, and the bibliography has been updated.
For many years the author has been urged to develop a text on linear algebra based on material in the second edition of his two-volume Calculus, which presents calculus of functions of one or more variables, integrated with differential equations, infinite series, linear algebra, probability, and numerical analysis. To some extent this was done by others when the two Calculus volumes were translated into Italian and divided into three volumes, the second of which contained the material on linear algebra. The present text is designed to be independent of the Calculus volumes.
This book is a compendium of joint work produced by the authors during the period 1998-2012, most of it published in the American Mathematical Monthly, Math Horizons, Mathematics Magazine, and The Mathematical Gazette. The published papers have been edited, augmented, and rearranged into 15 chapters. Each chapter is preceded by a sample of problem that can be solved by the methods developed in that chapter. Each opening page contains a brief abstract or the chapter's contents. Chapter I, entitled "Mamikon's Sweeping-Tangent Theorem," was the starting point of this collaboration. It describes an innovative and visual approach for solving many standard calculus problems by a geometric method that makes little or no use of formulas. The method was conceived in 1959 by my co-author (who prefers to be called Mamikon), when he was an undergraduate student at Yerevan University in Armenia. When young Mamikon showed his method to Soviet mathematicians they dismissed it out of hand and said "It can't be right. You can't solve calculus problems that easily." Mamikon went on to get a Ph.D. in physics, was appointed a professor or astrophysics at the University of Yerevan, and became an international expert in radiative transfer theory, all the while continuing to develop his powerful geometric methods. Mamikon eventually published a paper outlining them in 1981, but it seems to have escaped notice, probably because it appeared in Russian in an Armenian journal with limited circulation. Mamikon came to California in 1990 to learn more about earthquake preparedness for Armenia. When the Soviet government collapsed he was stranded in the United States without a visa. With the help of a few mathematicians he had met in Sacramento and at UC Davis and who recognized his remarkable talents, Mamikon was granted status as an "alien of extraordinary ability." While working at UC Davis and for the California Department of Education, he further developed his methods into a universal teaching tool using hands-on and computer activities as well as diagrams. He has taught these methods at UC Davis and in Northern California classrooms, ranging from Montessori elementary schools to inner-city public high schools, and he has demonstrated them at teacher conferences. Students and teachers alike have responded enthusiastically, because the methods are vivid and dynamic and don't require the algebraic formalism of trigonometry or calculus. A few years later, Mamikon visited Caltech and convinced me that his methods have the potential to make a significant impact on mathematics education, especially if they are combined with visualization tools of modern technology. Since then, we have jointly published thirty papers, not only on his sweeping-tangent method, but also on a variety of topics in mathematics that are amenable to Mamikon's remarkable geometric insight. I have often described Mamikon as "an artesian well of ideas." It has been a pleasure to work with him in an effort to share many of these ideas with those who enjoy the beauty of mathematics.
11.2. Review by: Brittany Shelton.
Mathematical reviews MR3024916.
New horizons in geometry is a dense compilation of results from a series of papers published over the past 16 years. This book puts a new spin on some well-known topics and introduces us to some previously unknown results. The authors provide a colorful look at many different geometric objects and a visual approach to understanding them. Although the topics in some chapters are related to those in others, each chapter is written independently, so that a reader can read them in any order. ... There are a vast number of color images throughout the entire book. The colors in these images help to distinguish regions and highlight what is being discussed. However, it can still be tricky at times to relate the discussion with the picture. My personal suggestion from experience is to read this book with a paper and pencil. Try to draw the images as they are described and then compare them to the images in the book. Also, take advantage of the online resources that are referenced in several of the chapters. These websites provide animations to help further visualize many of the geometric objects being studied. I would suggest this book to any reader interested in a new way of thinking about geometric objects and their properties.