Positives The book is highly praised for its thorough and rigorous grounding in real analysis and single-variable calculus, particularly in its first volume, which is suitable for advanced undergraduate students. Reviewers commend its approach of starting with elementary concepts like the real number system, limits, and continuity, but immediately supplying them with precise definitions and proofs. The text is noted for its numerous worked examples, which effectively lead students through complex reasoning and illustrate interesting properties, such as Archimedes’ method for determining pi. Its pedagogical style is described as excellent, lucid, efficient, and unpretentious, preferring a continuous line of exposition while ensuring all points are well-demonstrated and explained. The book also covers topics like Taylor expansions, convergence properties, and Fourier analysis with notable depth and clarity. Furthermore, the almost nine hundred exercises are considered thoughtful and challenging, providing valuable supplementary material. Its broad scope and geometric vision are also highlighted as significant strengths.
Negatives However, the later two volumes, covering multivariable calculus and more advanced topics, are generally considered less strong than the first, with suggestions that better texts might exist for those subjects. The chapter on differential equations, for instance, is described as a collection of specific cases rather than a systematic and comprehensive account, omitting crucial elements like existence theorems or advanced theories. Similarly, the section on complex analysis, while fine, is not comprehensive enough for a full course. Some exercises are noted for being exceptionally difficult, appearing without clear context, or requiring extensive, intricate computations that demand considerable stamina and skill, reflecting an older era of mathematical practice. The book also has a reputation for having some of the hardest problems among its peers, a point that has reportedly led to extreme frustration for at least one reader. Additionally, while applications to physics and geometry are included, they are sometimes too brief to serve as a complete pedagogical resource for those fields.
Conclusion Overall, this book is highly recommended, especially its foundational first volume, for readers seeking a deep, rigorous, and comprehensive introduction to the core concepts of calculus and real analysis. It is best suited for advanced undergraduate mathematics majors and determined students who are prepared to engage with significant intellectual challenges and extensive problem-solving. The book will appeal to those who appreciate a continuous, expository style that prioritizes understanding the underlying rigor and geometric intuition over a more fragmented, formalistic presentation. However, readers looking for a systematic, in-depth treatment of advanced topics like differential equations or complex analysis, or those preferring a less computationally demanding approach to exercises, might find certain aspects less ideal.
