Advanced Calculus

Rating:
Summary: Recommended
Review: The book under review is an easily accessible introduction to Advanced Calculus for those at the upper undergraduate level. As a student, I found it to be a successful mix of rigor and insight, a particularly valuable quality considering the fact that the subject is oftentimes the pons asinorum to abstract reasonings and proofs for most students of mathematics.

The book begins with a relatively "rigorous" refresher on the concepts of sets, functions, and graphs; later on a bunch of so-called field axioms are thrown in, but the author didn't treat these "basic concepts" extensively as interesting subjects in their own right. Some topological concepts are introduced next, then sequence, continuity, and differentiation; in that order. Up to the last item, the semester course was concluded. Notice that this was only page 125 of a 600 page book, hence my inability to comment on subsequent chapters following that elementary discussion of differentiation.

Pictures appropriate to the particular discussion are available; whenever possible the author attempted to provide a more intuitive understanding of a deep mathematical idea by discussing an example from the real world or through a physical interpretation.

The exercises vary in difficulty, but some are particularly hard that a solution manual would be very much welcomed. The hints and answers in the back are too brief to be of much use in demystifying those seemingly mysterious and unmotivated tricks needed in particular solution.

The author encourages the readers to employ any tool of elementary calculus learned earlier, an understandable choice since it was actually his decision not to introduce an axiomatic development of the subject that would compel the readers to deduce a solution to a problem from "scratch".

I would highly recommend the book, even for self-study to the mathematical enthusiasts. Those who desire a not-too-formidable introduction to real analysis may find this classic enjoyable.

Rating:
Summary: A true classic
Review: This book is extremely well written and comprehensive. It is appropriately approaches the subject and addresses the audience (primarily undergraduates) in an elegant manner with a fair amount of challenging problems, though never overestimates the reader's previous knowledge or background. Time and time again, I've found myself digging back in this book finding more depth and content through each pass. This is by far the best Advance Calculus book I've come across for undergraduates.

Rating:
Summary: Great and accessible
Review: This book provides a very readable and insightful account of the material that is usually covered in two semesters of advanced undergraduate courses usually called "Advanced Calculus" or "Real Analysis." Explanations are clear and concepts are well motivated. The problem sets are well-selected, and are do-able after reading the relevant chapters. This book is highly recommended for engineers or scientists wishing to gain a deeper understanding of mathematics, and for math majors preparing for graduate study in real analysis. It is a great book.