Calculus

In my experience, mathematics textbooks tend to fall into one of two categories: the first being extremely precise but lacking in pedagogy, the other being filled with superficial explanations while lacking in precision. The books of the first category are usually aimed towards students who have already seen the material, and the books of the second category are the kind which are typically required for low-level university courses. This, however, is a superb textbook that fortunately fails to fall into either category: it is an honest, rigorous treatment of calculus, a masterful unity of precise mathematics and lucid explanations.It is the best motivated mathematics textbook I have ever read. In the chapter on trigonometric functions, for example, the author refrains from using the quickest approach, but favours a rigorous approach in which the student can see the motivation behind every step. At the very end of the chapter, Spivak briefly mentions an alternative approach to defining the trigonometric functions, and then proceeds: "While much of this presentation would proceed more rapidly, the definition would be utterly unmotivated; the reasonableness of the definitions would be known to the author, but not to the student, for whom it was intended!". This attitude is a theme throughout the whole book - for the prepared student, a definition or theorem never seems to drop out of the blue, as Spivak goes out of his way to provide a thorough justification for everything.Almost every theorem in the book is proved. For this reason in particular, many novice students find the book very difficult. However, it is possible to overcome this difficulty even if it is the first proof-based book one reads. I can verify because it was the case for me.The exercises make an interesting and quite challenging selection. A lot of them asks the student to prove minor results related to the theory of the chapter. I have personally done more than half the exercises in the book, of which many are among the most rewarding I have ever done.Due to the conversational style of the book, it is ideal for self-study, as it may be able to partly provide the coaching that is usually delegated to an instructor. And due its scope and rigour, it is ideal for serious the sort of people who are more interested in understanding the material than simply passing a course. If you are such a person I cannot recommend this book enough.

I fell in love with this book throughout looks amazing sat on my shelf, and is so much more interesting once open! I found my self struggling at times through my first few years of my undergraduate studies, but this book helped me to discover the areas of maths that I really enjoyed. The open nature of the book, and Spivak's encapsulating way of writing has really sold this book to me, and I beginning-undergraduate students; this book is maybe the first mathematics book that captivated my mind in that way. It opened new doors for me as an undergraduate, and started getting me to think outside the box. It is a fantastic reference book, and you can lose many hours by simply turning to a page that intrigues you!Highly recommended.

I found in this book an invaluable help; it is both clear and rigorous, it does not hide complexity yet it presents it in the most accessible way.The author is clearly both passionate, and well-versed in teaching complex math subjects.A must buy for everybody interested in the foundations of calculus and its applications.

If you are studying for A-level further maths or you are in your first year of a mathematics undergraduate degree, this book might seem a little daunting. That's because it's not really a calculus book. Instead, this book approaches the subject calculus from the underpinning theory of real analysis.By adopting this approach, this book builds on the fundamental foundations of calculus and leaves you well prepared for tackling analysis later on. By introducing theorems with vivid explanations, they easily take form in your mind and then reinforced with challenging and engaging exercises.There is only one reason why this book does not get five stars; the physical print of the book is on paper that is too thin, such that you can see the text on the other side of the pages. For someone with dyslexia, this is very distracting and I have to use a blank coloured sheet of paper under each page to help me read it. This is a shame because the actual content of the book is near faultless.

Awesome book. Essentially covers the same topics as A-Level maths but with much deeper insight and plenty of enlightening questions - many of them can be found on Cambridge problem sheets.A must "read" (work through the problems!) for any student studying/about to study physics or math at university.

My son bought this for himself to study as Uni is off the cards due to this country being dumb and not letting the average person have the right to go to university if they can not afford it. Anyway enough ranting..this book is what I have been told that 99% of people say to read if you are contemplating learning this subject. However a good grasp on algebra is the first stop before attempting this. My son is very bright but even he has had to go back to the beginning of the book and re read. There is an answer book which you could get if you were very keen.

I bought this new from Amazon and they delivered it with damage to corners of the hard cover. Not the usual high standard I expect. Boo hiss.

I don't review much, but this book is worth recommending. It is superb, though not easy reading. Battled my way through it for the second time - and it was even better than the first. Haven't done the problems though.