Calculus Is Calculus by Michael Spivak: A Must-Read for Strong Students?

[Greg Bernhardt]

• Author: Spivak
• Title: Calculus
• Amazon Link: https://www.amazon.com/dp/0914098918/?tag=pfamazon01-20
• Prerequisities: High-school mathematics

Table of Contents:

[LIST]
[*] Preface
[*] Prologue
[LIST]
[*] Basic Properties of Numbers
[*] Numbers of Various Sorts
[/LIST]
[*] Foundations
[LIST]
[*] Functions
[*] Appendix: Ordered Pairs
[*] Graphs
[*] Appendix: Polar Coordinates
[*] Limits
[*] Continuous Functions
[*] Three Hard Theorems
[*] Least Upper Bounds
[*] Appendix: Uniform Continuity
[/LIST]
[*] Derivatives and Integrals
[LIST]
[*] Derivatives
[*] Differentiation
[*] Significance of the Derivative
[*] Appendix: Convexity and Concavity
[*] Inverse Functions
[*] Appendix: Parametric Representation of Curves
[*] Integrals
[*] Appendix: Riemann Sums
[*] Appendix: The Cosmopolitan Integrals
[*] The Fundamental Theorem of Calculus
[*] The Trigonometric Functions
[*] [itex]\pi[/itex] is Irrational
[*] The Logarithm and Exponential Functions
[*] Integration in Elementary Terms
[/LIST]
[*] Infinite Sequences and Infinite Series
[LIST]
[*] Approximation by Polynomial Functions
[*] [itex]e[/itex] is Transcendental
[*] Infinite Sequences
[*] Infinite Series
[*] Uniform Convergence and Power Series
[*] Complex Numbers
[*] Complex Functions
[*] Complex Power Series
[/LIST]
[*] Epilogue
[LIST] 
[*] Fields
[*] Construction of the Real Numbers
[*] Uniqueness of the Real Numbers
[/LIST]
[*] Suggested Reading
[*] Answers (to selected problems)
[*] Glossary of Symbols
[*] Index
[/LIST]

 

[micromass]

This is a wonderful and exciting book. I feel that this is one of the few books that any math major should read. It mainly covers single-variable calculus and it does so very rigorously. Make no mistake about it, the book is rigorous and quite hard. The exercises tend to be very challenging. As such, I would consider the book more an introduction to real analysis than an actual calculus book. If you wish to read this book, I would recommend that you had some experience with calculus already and preferably an experience with proofs too.

 

[Astrum]

This is a wonderful and exciting book. I feel that this is one of the few books that any math major should read. It mainly covers single-variable calculus and it does so very rigorously. Make no mistake about it, the book is rigorous and quite hard. The exercises tend to be very challenging. As such, I would consider the book more an introduction to real analysis than an actual calculus book. If you wish to read this book, I would recommend that you had some experience with calculus already and preferably an experience with proofs too.

I second this.

It's a great book, although if I had used this as my first calculus book... I probably would have been pretty discouraged.

I don't really like how he never used Leibniz notation, but that's not a huge deal.

 

[MarneMath]

I used this for my first Calculus book. Extremely difficult, but it really gives you a feel if you'll enjoy a mathematics major or not. Very friendly and easy to read book. As Micromass said, this book isn't so much an intro to calculus book, but an intro to real analysis. Although, it is at a level less than most real analysis books. I'll say you should use this book if you have taken calculus course and want to review it from a more rigorous view, but don't want to get bogged down with very many new terms and abstract views.

 

[jbunniii]

Probably the best rigorous calculus book for most students. The main alternative would be Apostol volume 1. Spivak's exposition is more conversational and his proofs are somewhat more detailed than Apostol's. Also, Spivak has more interesting exercises, and many of them are quite challenging. Both books are excellent, however.

You would have to be quite a strong student to be able to handle Spivak as your first exposure to calculus, but most people (in the US, at least) will have already taken a computational calculus course in high school anyway, so this isn't really an issue. With that background, you already know what calculus is used for, and how to calculate things, so with Spivak you can focus on why and when those calculations are valid, and how to prove it.

Likewise, most people have a tough time with Rudin's Principles of Mathematical Analysis if it is their first exposure to real analysis. If you have read and worked your way through Spivak, you will have a much easier time with Rudin. You will already understand epsilon-delta proofs and will know many of the theorems, so you will be able to focus on the new material such as topology, and marvel at how clean and efficient Rudin's proofs can be: "wow, Spivak took 20 somewhat grungy lines to prove this, and Rudin did it in only 3 beautiful ones!"

 

[Sankaku]

It's a great book, although if I had used this as my first calculus book... I probably would have been pretty discouraged.

Yes! I think people recommending it to beginners are looking back in time after doing a "normal" calculus course and then finding Spivak. They think how much they like Spivak's insights but forget about the fact that they already understand the mundane parts of the subject.

My mantra in mathematics is: "Preparation trumps everything." Despite how well-written and conversational Spivak is, you would need to be very well prepared to use it as a first text.

 

[mathwonk]

yes. some perspective: as a young college student knowing no calculus, i had all A's in high school math, 800 on math SAT test, had been on the school math team that retired the state contest trophy, and had won several individual state and mid - state math contest titles. i had difficulty gaining admission to, and struggled in, the college course for which then courant, and now spivak, was used as a text.

 

[middleCmusic]

Would this be the appropriate place to ask a question about a specific problem from Spivak? (Not doing it for homework - just self-study)

 

[Jorriss]

Would this be the appropriate place to ask a question about a specific problem from Spivak? (Not doing it for homework - just self-study)

https://www.physicsforums.com/forumdisplay.php?f=156 Here would be more appropriate.

 

[middleCmusic]

https://www.physicsforums.com/forumdisplay.php?f=156 Here would be more appropriate.

Thanks.

 

[synkk]

For anyone who has worked through Spivak Calculus

Hi, I recently picked up "Spivak Calculus, Third edition". I've just finished A levels (which is the equivalent of High School in the US), and should hopefully be going to study mathematics at university in september. I wanted to stay productive throughout the summer so I decided to do two things, one of which is to learn basic programming, and the other is to keep my mathematics skills up to scratch.

Anyway, I have two questions:

1. Should I work through every chapter in the book? Is it worthwhile to do so?
2. Should I work through every problem or just do the harder ones?

To help you judge my ability, here I have attached a difficult paper which some mathematics students sit in the UK.

I understand it's very ambitious to plan to work through this whole book throughout summer, but I figured it'd be fun and would keep my maths skills up.
Thank you very much!

 

[micromass]

It would be very worthwhile to go through the entire book. There are a lot of very nice and useful things in the book. Almost every chapter is directly useful to mathematics (except maybe chapters which prove that e is irrational, but that's beautiful on its own).

You shouldn't do every exercise. Just do the ones you think are challenging or you think have a nice result. If you can do most exercises in a chapter, then you can progress to the next.

If you're going to study math in college, then Spivak will be an awesome preperation for university. Judging from the attached text, I think you should have enough maturity to tackle the text.

 

[HayleySarg]

Interesting. My Calc prof at community college was required to use "Calulus: Single and Multivariable" by Hughes-Hallett.

The first day of class she handed out to each table Spivak's calculus. She noted that we would not be required to buy it, but that it was of superior quality to the book she was supposed to teach with. Consequently, most of our homework problems came from Spivak's calculus.

I found it to be very rigorous but enjoyable. With a strong pre-calc background, I feel most of our class did well with it. However, it was quite obvious who entered the course weak on mathematical knowledge (the non STEM majors mostly). It was dense, and I felt that intimidated them.

I loved it.

 

[mathwonk]

that choice is comparable to, let's see, hot dogs (or worse) as opposed to filet mignon?

 

[preceptor1919]

Do you have a free textbook for spivak's 4th edition?

 

[mathwonk]

I have taught from Spivak to brilliant students several times. It has recently become visible to me that even a great textbook like this does not quite satisfy the need many students have, even outstandingly gifted ones, to motivate the modern precise versions of the concepts such as limits and continuity.

Chapters 5 and 6 just introduce the epsilon and delta limits and continuity without any motivation as to why they are needed. Then chapter 7 states "three hard theorems", whose results seem obvious intuitively, without discussion of why there is a need to prove such things in mathematics.

This evolution took thousands of years, and even though we accept it now, most students still arte at something of a loss as to why we prove these things.


Here is an interesting article by Bolzano, the man who first gave the correct definition of continuity, and proved the intermediate value theorem, although admittedly he used the least upper bound property without realizing he needed to state it as an axiom.


http://www.sciencedirect.com/science/article/pii/0315086080900361

 

[Adgorn]

Quick question, does this book cover both calculus I and II to a fairly comprehensive degree?

 

[mathwonk]

I don't know what you mean by calculus 2. this book covers all one variable calculus. thus it covers infinite series but not 2 variable calculus such as partial derivatives. this is a great book. every strong student will benefit from it.