Calculus

by Michael Spivak

Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis.

Genres: Mathematics, Science, Nonfiction, Textbooks, Calculus, Reference, Academic

647 pages, Hardcover

First published January 1, 1967

About the author

Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. He is the author of the five-volume Comprehensive Introduction to Differential Geometry. He received a Ph.D. from Princeton University under the supervision of John Milnor in 1964.

His book Calculus takes a very rigorous and theoretical approach to introductory calculus. It is used in calculus courses, particularly those with a pure mathematics emphasis, at many universities.

Spivak's book Calculus on Manifolds (often referred to as little Spivak) is also rather infamous as being one of the most difficult undergraduate mathematics textbooks.

Reviews

[Abhi Varma · Rating 5 out of 5]

Many have said it before me, and allow me to say it again: this is the Calculus book. Yes, it really does start you off with (a + b) + c = a + (b + c), and then takes you through a nice variety of topics on a breathtaking journey through epsilon-delta proofs, axiomatic deductions and then inductions, the irrationality of pi, the transcendental nature of e, all sorts of sequences and series, and of course--the main 'body' of calculus that is the practice of integration and differentiation. It holds your hand at the sidewalk, and then casts you into the traffic while crossing the road. It is the very paragon of both coldly beautiful terseness and the warmth that accompanies a clear stream of thought moving rationally from one idea to another. It is conversational, yet sufficiently rigorous; it is formidable in scope yet it travels in steps. But each step is a leap--and don't be surprised if you find yourself poring over a paragraph for hours, before the full meaning sinks in. It is Hemingway wedded with Mathematics--the grand iceberg of thought that is analysis lends weight and a grace of movement to the crystal clear tip of Calculus as it glides through the waters of mathematical thought...


As is visible, I quite love this book. It was my first (real) (no pun intended) introduction to Calculus, and oh was it worth it. I must warn you: this isn't for the pregnant and faint-hearted. Or engineers. But seriously--if you're looking for practical application or pragmatic knowledge then this book isn't exactly for you. If you like--or think you might like--the mathematician's ever-refining pursuit for rigour, for watertight argument, for abstract thought--then you'll love this book. Many say it is second or third year level. I feel, for a serious mathematician, it is first-year level at best, though a reasonably bright and motivated high school student can easily get through the chapters, if not solve the exercises to utmost completion.

Be prepared to use your brain, though. I have to make this disclaimer because sadly, in today's day and age, it is almost too much to ask of someone. It seems to have become almost unfashionable, an unnecessary expenditure of effort. But if you are looking to learn mathematics--in this case, calculus--then get your ass over this book and start engaging the grey matter. It's rewarding.

[Mark Moon · Rating 5 out of 5]

Superb! A pleasure to read, and a pleasure to teach from. I wish I'd learned calculus from this book. Of all the excellent things about it, the best is the selection of exercises. They range from elementary ones useful for practicing the mechanical aspects of the subject to extremely challenging and abstract ones that connect the material in the book to more advanced concepts.

[Jose Moa · Rating 5 out of 5]

For me the best introduction to calculus in one variable, yet absolutely rigurous it is more focussed in explain fundamental concepts in deep that in long full of calculations demostrations;also has some touchs of subtle humour ,some uncommon in a textbook and a lot of remarkable exercises.

Has a lot of explainig clear graphics,give examples of bizarre functions for clearing concepts,give a original introduction to complex variable by convergent complex power series,makes a formal costruction of the real numbers field, and make at this level unusual excursions in more advanced results as the demostration of the irracionality of pi or the demostration of the trancendence of e.

Is a pure mathematics text and has no mention of physical or tecnological use

[Ronald Lett · Rating 5 out of 5]

This is a beautiful exposition of the theory of calculus from the axiomatic approach with full motivation and proof of the major theorems and some interesting facts that are seldom presented at this level. The proofs and exercises are the most elegant that I have seen in any calculus text, as is Spivak's hallmark. All that is necessary is a solid understanding of high school pre-calculus and mathematical curiousity. As other posters have said, if you are going to be a mathematics or physics major, there is no reason to even look at another calculus text. The only ones that come close are the treatments by Courant and Apostol.

[Gal Barkai · Rating 5 out of 5]

This is the book that turned me from an aspiring engineer to an aspiring Mathematician. It is well structured, clear and precises, it made me love the subject because for the first time I saw it presented for the logical wonder it is, and as an assortment of formulas to be used. Not to mention the generous margins for scribbling (a homage to Diophantus and Fermat perhaps) practically beg the reader to try his hand in proving the theorems and exercises stated in it.

I was extremely lucky to spot a used version of this book (in nearly mint condition no less!) sitting under a "for sale" sign at our university's library. Only after picking it up I figured why my Math professors said math is a beautiful and logical subject, I really can't stress how much this book changed my perspective.

[Parsa · Rating 5 out of 5]

I first read this in high school. But it turns out that I didn't really understand it. Now that I'm done with first year university, I realize what a gem this book truly is.
Very well written, well motivated introduction to analysis, with good problems. I also absolutely love the little annotated bibliography near the end, which you can find here :
https://pctex.org/mathpop/wp-content/...

[Dmitri · Rating 3 out of 5]

Not really sure why so many people rate this book so highly. Apart from the gratuitous waste of page real estate there is also some confusing notation as well as an overall feel that is a lot more academic than practical.

[Ilya Zarembsky · Rating 5 out of 5]

Beautiful and merciless

[Simon Mcleish · Rating 5 out of 5]

This review first appeared on my blog here.

Calculus was the very first textbook I read for my university degree. As well as being a fine description of the basics of analysis (mostly real, with a toe in the deep water of complex functions), it is an excelent book to ease the transition from mathematics as taught at school level to the rigours of university mathematics.

Unlike many writers of textbooks in mathematics, Spivak makes a big effort to give more than a dry exposition: theorem - proof - next theorem etc. Considerable attention is paid to motivating the discussion, showing why each result is important (though mainly in the pure mathematics context, applications of calculus being mainly found in the problems at the end of each chapter). Of especial use to the budding mathematician are the points where Spivak discusses potential proof strategies for the theorems, often explaining the pitfalls that student taking a naive approach could fall into. There are even occasional jokes, both in the text and the index.

For students with an interest in how analysis can be used in apparently unrelated parts of mateematics, a number of advanced sections give proofs of such topics as the transcendence of the number e, and a construction of the real numbers from set theoretic principles.

Calculus was not just the first university textbook I read, but one of the best.

[Arden]

this book was the bane of my existence in first year math and yet it also compelled me to (1) not drop the very hard math class and (2) keep taking very hard math classes so i can't fault it that much. although i am in the minority that really prefers calculus on manifolds

[G · Rating 5 out of 5]

Brillante. De todos los libros disponibles sobre cálculo, éste es uno de los mejores. A favor tiene el enfoque matemático que no desestima las aplicaciones ingenieriles, pero prioriza la teoría formal pura. Eso está muy bien porque las matemáticas son teorías de formas puras en las que cada concepto está saturado de teoría. Los mundos matemáticos son densísimos, pero esto no obliga a que los libros de matemáticas sean barrocos. Además, los mundos matemáticos están llenos de vacíos misteriosos que toman la forma de posibles teoremas, de intuiciones de propiedades axiomáticas que todavía no se han demostrado. Si hay convergencias entre estos mundos formales y el mundo humano es más sorprendente que necesario, como lo sugirió Bertrand Russell en su Autobiografía. Este libro posee todos estos felices atributos, incluyendo la dificultad de los ejercicios que propone. Señala propiedades muy difíciles de demostrar, lo cual es parte de la idiosincrasia de los buenos libros de matemáticas. Precisión, intensidad, belleza y saludables fugas hacia lo desconocido.

[Eduardo Kuri · Rating 3 out of 5]

que se vaya alv (no es cierto (si es cierto))

[Ayush Bhat · Rating 5 out of 5]

Love his VERY detailed treatment to the topic of Infinite Series.

[Jesse · Rating 5 out of 5]

This is the best math book I've ever read

[Henne · Rating 4 out of 5]

Like putting an everyday object under a microscope: very interesting, but now there are many more interesting objects I want to put under a microscope.

One day I will return to Spivak's Calculus to complete all the exercises; the suggested reading offers some reassurance that readers who have got as far as completing all the exercises probably know a lot more about advanced mathematics (e.g. functional analysis) than is covered by this book. The reason for this is suggested by the final item of the reading list: where the author suggests that one could learn as much from all of the other items on the reading list by simply reading the “Oeuvres complètes de Niels Henrik Abel”, Abel being the same mathematician who lends his name to the 'Nobel' in mathematics, the Abel Prize.

There are better books on mathematics around, but this is (still) perhaps the best single volume on calculus.

[Deppy. · Rating 5 out of 5]

Ένα πολύ δυνατό βιβλίο το οποίο ‘’χτίζει’’ την μαθηματική σκέψη από την αρχή με σοβαρότητα αλλά και αποδοτικότητα. Θεωρώ πως είναι ένα εύστοχο βιβλίο, το οποίο δεν αναλώνεται σε πολλά και διάφορα.
Ωστόσο,προσωπικά, δεν με βοήθησε ιδιαίτερα το κεφάλαιο «ακολουθίες και σειρές» να κατανοήσω την ύλη που χρειαζόμουν (για αυτό που ζητούσα εγώ εκείνη την στιγμή). Αυτό, το αναφέρω ως προπτυχιακή φοιτήτρια μαθηματικών η οποία προσπάθησε να βρει πηγές ώστε να κατανοήσει το συγκεκριμένο κεφάλαιο καλύτερα και να επιτύχει στις εξετάσεις.
Προφανώς και ο σκοπός του βιβλίου δεν ήταν αυτός, ο σκοπός του βιβλίου είναι η ανάπτυξη της διαίσθησης μας σε σχέση με τα μαθηματικά και η κατανόηση πως η ακρίβεια και η αυστηρότητα δεν αποτελούν εμπόδιο σε αυτή αλλά το φυσικό μέσο με το οποίο διαμορφώνουμε και σκεπτόμαστε τις μαθηματικές ερωτήσεις (όπως αναφέρεται και στον πρόλογο του βιβλίου). Και αυτόν τον σκοπό το έχει πετύχει με εξαιρετικό τρόπο.

[Alex · Rating 5 out of 5]

A solid text and reference on the topic. Were I to want to re-learn calculus, I'd start here.

[Maki · Rating 4 out of 5]

this thing made me want to commit unexistence

[Jose Sanz · Rating 5 out of 5]

Very clear explanations. I did not try to make the exercises, so I don´t know how difficult they are. I suppose they are very, reading other people reviews.

[Calvin · Rating 5 out of 5]

Amazing and classic, my 2nd calculus book and wow this was *hard*. Definitely the most difficult book I've ever read so far. Here are the contents and what I thought of them:

Part 1 - Numbers. A good and easy introduction, showcasing the general proofs-focused theme of the book here.

Part 2 - Limits and Continuity. The most difficult one by far, including topics like supremums and epsilon-delta proofs.

Part 3 - Differentiation and Integration. A mixed bag, naturally the theoretical chapters are more difficult than the more technique-focused ones.

Part 4 - Sequences. Again, a mixed bag. Definitely a close second in difficulty.

Part 5 - Epilogue. Covers fields, which is the first time I've read about them for real, quite nice. Other than that this chapter was short and a nice end to the book.

[Hiatus · Rating 3 out of 5]

A confusingly overrated book. A marginally rigorous book somehow worshipped by those who overemphasize rigor. I find this is bizarre and suspect of social reverberation rather than merit. A treatment partially incompatible with modern courses. I do understand its novelty, and I especially sense its charm from the perspective of a self-learner. Regarding its frequent recommendation to beginning Calculus students, it is far from an ideal book to learn from. It is a good book for interesting problems, however this treatment will not make you any more successful in a Calculus course, perhaps the opposite since you are likely to be in a course that treats Calculus in a different way.

Like many other things you find lauded, it's just okay.

[Eric · Rating 4 out of 5]

Bought this particular textbook because of its connection with the Hampshire College Summer Studies in Mathematics program, if I remember (the index of this, or another, of Spivak's books has an in-joke listing for "Yellow Pigs", not a would-be counterculture insult but rather one of the two mascots of that program, along with the number 17.) As a general thing - it's been a long time since I read this textbook but I have good memories of it.

[Pedro Gasparotti · Rating 4 out of 5]

Este é um livro cuja introdução justifica o restante de suas centenas de páginas com capítulos não tão interessantes quanto o primeiro.

Porém, cada um dos capítulos subsequentes fortalece a justificativa do primeiro.

Um invólucro do conhecimento apriorístico dos números, certamente resistirá ao sofismo da ciência acadêmica e será a fundação de grandes acontecimentos humanos - mesmo que da própria disrupção de axiomas, do produto positivo entre dois inteiros negativos.

[Javier]

Creo que no hay mejor introducción al maravilloso mundo del Análisis Matemático. Las explicaciones son elegantes y detalladas, y los ejercicios fabulosos.
Lo recomendaría a cualquiera que estudie matemáticas. Al principio puede hacerse algo espeso cuando no se está familiarizado con el tipo de razonamiento, pero creo que a la postre merece mucho la pena.

[Dương Ngô · Rating 5 out of 5]

This is one of the best introductions to Real Analysis. "Calculus" of M. Spivak is a pleasure to read and learn from. Besides limits, continuity, derivation, and integration, topics like transcendence of e, irrationality of pi, and construction of the real numbers using Dedekind cuts are beautifully covered.

[Kent Sibilev · Rating 5 out of 5]

It is a very good introduction to Calculus. Considering that this book is a bit old and consequently uses unusual notations, the explanations are extremely good and the author's approach reminds me of the best Leonard Euler's texts.

[Joe Richardson · Rating 5 out of 5]

There's nothing I can adequately say about this. Calculus explained lucidly. I didn't know that had every been done. This book makes you go from struggling in abstraction to realizing that the whole field is intuitive and useful.

[Samuel · Rating 5 out of 5]

Incredibly thorough and well written.

[George Menegakis · Rating 5 out of 5]

By far the best introduction to calculus and real analysis

[Cameron · Rating 4 out of 5]

can't go wrong with Spivak