This is the first modern calculus book to be organized axiomatically and to survey the subject's applicability to science and engineering. A challenging exposition of calculus in the European style, it is an excellent text for a first-year university honors course or for a third-year analysis course. The calculus is built carefully from the axioms with all the standard results deduced from these axioms. The concise construction, by design, provides maximal flexibility for the instructor and allows the student to see the overall flow of the development. At the same time, the book reveals the origins of the calculus in celestial mechanics and number theory. The book introduces many topics often left to the appendixes in standard calculus textbooks and develops their connections with physics, engineering, and statistics. The author uses applications of derivatives and integrals to show how calculus is applied in these disciplines. Solutions to all exercises (even those involving proofs) are available to instructors upon request, making this book unique among texts in the field. Focuses on single variable calculus Provides a balance of precision and intuition Offers both routine and demanding exercises

An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.

This textbook is ideal for a year-long sequence of Honors Calculus at the undergraduate level, or for self-study; it is based on an Honors Calculus course the author has taught at Clark University, where he is an assistant professor of Mathematics. It is intended for all students in the sciences, though the emphasis is on proofs and foundations over computations. Its sincere wish is to awaken in the student a love of mathematical argument, and to inspire her or him to major in mathematics. One novelty of the book is that it includes a substantial amount of ordinary differential equations (ODEs) and multivariable calculus, topics usually not covered in an Honors Calculus course. Applications are also discussed: compound interest and continuous compounding in finance; Newton's method; Hamiltonian systems and Newton's law of gravitation; ODEs inspired from physics and population dynamics; and an introduction to gradient descent and neural networks. There are over 120 exercises. This text is a natural precursor to more advanced undergraduate texts in real analysis, such as Understanding Analysis by Stephen Abbott or Principles of Mathematical Analysis by Walter Rudin.

This book uses elementary versions of modern methods found in sophisticated mathematics to discuss portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level.

This introductory calculus text was developed by the author through his teaching of an honors calculus course at Notre Dame. The book develops calculus, as well as the necessary trigonometry and analytic geometry, from witin the relevant historical context, and yet it is not a textbook in the history of mathematics as such. The notation is modern, and the material is selected to cover the basics of the subject. Special emphasis is placed on pedagogy throughout. Whhile emphasizing the broad applications of the subject, emphasis is placed on the mathematical content of the subject.

Calculus Deconstructed is a thorough and mathematically rigorous exposition of single-variable calculus for readers with some previous exposure to calculus techniques but not to methods of proof. This book is appropriate for a beginning Honors Calculus course assuming high school calculus or a "bridge course" using basic analysis to motivate and illustrate mathematical rigor. It can serve as a combination textbook and reference book for individual self-study. Standard topics and techniques in single-variable calculus are presented in context of a coherent logical structure, building on familiar properties of real numbers and teaching methods of proof by example along the way. Numerous examples reinforce both practical and theoretical understanding, and extensive historical notes explore the arguments of the originators of the subject. No previous experience with mathematical proof is assumed: rhetorical strategies and techniques of proof (reductio ad absurdum, induction, contrapositives, etc.) are introduced by example along the way. Between the text and exercises, proofs are available for all the basic results of calculus for functions of one real variable.